Calculus Illustrated. Volume 2: Differential Calculus [2, 1 ed.] 9798657083927

Table of contents :
Preface
Limits of sequences
What is calculus about?
Infinite sequences and their long-term trends
The definition of limit
Limits under algebraic operations
Can we add infinities? Subtract? Divide? Multiply?
More properties of limits of sequences
Theorems of Analysis
Compositions
Numbers are limits
The exponential functions
The trigonometric functions
Limits and continuity
Functions
Continuity and discontinuity
Limits of functions: small scale trends
Limits and continuity under algebraic operations
The exponential and trigonometric functions
Limits and continuity under compositions
Continuity of the inverse
Comparison of limits
Global properties of continuous functions
Large-scale behavior and asymptotes
Limits and infinity: computations
Continuity and accuracy
The - definition of limit
The derivative
The Tangent Problem
The difference of a sequence and the difference of a function
The rate of change: the difference quotient
The limit of the difference quotient: the derivative
The derivative is the instantaneous rate of change
The existence of the derivative: differentiability
The derivative as a function
Basic differentiation
Basic differentiation, continued
Free fall
Differentiation
Differentiation over addition and constant multiple: linearity
Change of variables and the derivative
Differentiation over compositions: the Chain Rule
Differentiation over multiplication and division
The rate of change of the rate of change
Repeated differentiation
How to differentiate relations: implicitly
Related rates: radar gun
The derivative of the inverse function
Reversing differentiation
Shooting a cannon
The main theorems of differential calculus
Monotonicity and extreme points
Optimization of functions
What the derivative says about the difference quotient: The Mean Value Theorem
Monotonicity and the sign of the derivative
Concavity and the sign of the second derivative
Derivatives and extrema
Anti-differentiation: the derivative of what function?
Antiderivatives
What we can do with calculus
Magnitudes of functions; L'Hopital's Rule
Linear approximations
The accuracy of the best linear approximation
Solving equations numerically: bisection and Newton's method
Particle in a flow
Differential equations
Motion under forces
Optimization examples
Functions of several variables
Exercises
Exercises: Sets, logic, functions
Exercises: Background
Exercises: Sequences
Exercises: Rates of change
Exercises: Limits and continuity
Exercises: Derivatives
Exercises: Features of graphs
Exercises: Linearization
Exercises: Models
Exercises: Information from the derivatives
Exercises: Computing derivatives
Exercises: Optimization and other applications
Index

Citation preview

✷

❚♦ t❤❡ st✉❞❡♥t✳✳✳

❚♦ t❤❡ st✉❞❡♥t ▼❛t❤❡♠❛t✐❝s ✐s ❛ s❝✐❡♥❝❡✳ ❏✉st ❛s t❤❡ r❡st ♦❢ t❤❡ s❝✐❡♥t✐sts✱ ♠❛t❤❡♠❛t✐❝✐❛♥s ❛r❡ tr②✐♥❣ t♦ ✉♥❞❡rst❛♥❞ ❤♦✇ t❤❡ ❯♥✐✈❡rs❡ ♦♣❡r❛t❡s ❛♥❞ ❞✐s❝♦✈❡r ✐ts ❧❛✇s✳

❲❤❡♥ s✉❝❝❡ss❢✉❧✱ t❤❡② ✇r✐t❡ t❤❡s❡ ❧❛✇s ❛s s❤♦rt st❛t❡♠❡♥ts

❝❛❧❧❡❞ ✏t❤❡♦r❡♠s✑✳ ■♥ ♦r❞❡r t♦ ♣r❡s❡♥t t❤❡s❡ ❧❛✇s ❝♦♥❝❧✉s✐✈❡❧② ❛♥❞ ♣r❡❝✐s❡❧②✱ ❛ ❞✐❝t✐♦♥❛r② ♦❢ t❤❡ ♥❡✇ ❝♦♥❝❡♣ts ✐s ❛❧s♦ ❞❡✈❡❧♦♣❡❞❀ ✐ts ❡♥tr✐❡s ❛r❡ ❝❛❧❧❡❞ ✏❞❡✜♥✐t✐♦♥s✑✳ ❚❤❡s❡ t✇♦ ♠❛❦❡ ✉♣ t❤❡ ♠♦st ✐♠♣♦rt❛♥t ♣❛rt ♦❢ ❛♥② ♠❛t❤❡♠❛t✐❝s ❜♦♦❦✳ ❚❤✐s ✐s ❤♦✇ ❞❡✜♥✐t✐♦♥s✱ t❤❡♦r❡♠s✱ ❛♥❞ s♦♠❡ ♦t❤❡r ✐t❡♠s ❛r❡ ✉s❡❞ ❛s ❜✉✐❧❞✐♥❣ ❜❧♦❝❦s ♦❢ t❤❡ s❝✐❡♥t✐✜❝ t❤❡♦r② ✇❡ ♣r❡s❡♥t ✐♥ t❤✐s t❡①t✳ ❊✈❡r② ♥❡✇ ❝♦♥❝❡♣t ✐s ✐♥tr♦❞✉❝❡❞ ✇✐t❤ ✉t♠♦st s♣❡❝✐✜❝✐t②✳

❉❡✜♥✐t✐♦♥ ✵✳✵✳✶✿ sq✉❛r❡ r♦♦t ❙✉♣♣♦s❡

x✱

a

✐s ❛ ♣♦s✐t✐✈❡ ♥✉♠❜❡r✳ ❚❤❡♥ t❤❡ sq✉❛r❡ r♦♦t ♦❢ x2 = a✳

a

✐s ❛ ♣♦s✐t✐✈❡ ♥✉♠❜❡r

s✉❝❤ t❤❛t

❚❤❡ t❡r♠ ❜❡✐♥❣ ✐♥tr♦❞✉❝❡❞ ✐s ❣✐✈❡♥ ✐♥ ✐t❛❧✐❝s✳ ❚❤❡ ❞❡✜♥✐t✐♦♥s ❛r❡ t❤❡♥ ❝♦♥st❛♥t❧② r❡❢❡rr❡❞ t♦ t❤r♦✉❣❤♦✉t t❤❡ t❡①t✳ ◆❡✇ s②♠❜♦❧✐s♠ ♠❛② ❛❧s♦ ❜❡ ✐♥tr♦❞✉❝❡❞✳

❙q✉❛r❡ r♦♦t √

a

❈♦♥s❡q✉❡♥t❧②✱ t❤❡ ♥♦t❛t✐♦♥ ✐s ❢r❡❡❧② ✉s❡❞ t❤r♦✉❣❤♦✉t t❤❡ t❡①t✳ ❲❡ ♠❛② ❝♦♥s✐❞❡r ❛ s♣❡❝✐✜❝ ✐♥st❛♥❝❡ ♦❢ ❛ ♥❡✇ ❝♦♥❝❡♣t ❡✐t❤❡r ❜❡❢♦r❡ ♦r ❛❢t❡r ✐t ✐s ❡①♣❧✐❝✐t❧② ❞❡✜♥❡❞✳

❊①❛♠♣❧❡ ✵✳✵✳✷✿ ❧❡♥❣t❤ ♦❢ ❞✐❛❣♦♥❛❧ ❲❤❛t ✐s t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❞✐❛❣♦♥❛❧ ♦❢ ❛

1 × 1 sq✉❛r❡❄ ❚❤❡ sq✉❛r❡ ✐s ♠❛❞❡ ♦❢ t✇♦ r✐❣❤t tr✐❛♥❣❧❡s ❛♥❞ t❤❡ a✳ ❚❤❡♥✱ ❜② t❤❡ P②t❤❛❣♦r❡❛♥ ❚❤❡♦r❡♠ ✱ t❤❡ sq✉❛r❡ ♦❢

❞✐❛❣♦♥❛❧ ✐s t❤❡✐r s❤❛r❡❞ ❤②♣♦t❡♥✉s❡✳ ▲❡t✬s ❝❛❧❧ ✐t a ✐s 12 + 12 = 2✳ ❈♦♥s❡q✉❡♥t❧②✱ ✇❡ ❤❛✈❡✿

a2 = 2 . ❲❡ ✐♠♠❡❞✐❛t❡❧② s❡❡ t❤❡ ♥❡❡❞ ❢♦r t❤❡ sq✉❛r❡ r♦♦t✦ ❚❤❡ ❧❡♥❣t❤ ✐s✱ t❤❡r❡❢♦r❡✱

a=

√

2✳

❨♦✉ ❝❛♥ s❦✐♣ s♦♠❡ ♦❢ t❤❡ ❡①❛♠♣❧❡s ✇✐t❤♦✉t ✈✐♦❧❛t✐♥❣ t❤❡ ✢♦✇ ♦❢ ✐❞❡❛s✱ ❛t ②♦✉r ♦✇♥ r✐s❦✳ ❆❧❧ ♥❡✇ ♠❛t❡r✐❛❧ ✐s ❢♦❧❧♦✇❡❞ ❜② ❛ ❢❡✇ ❧✐tt❧❡ t❛s❦s✱ ♦r q✉❡st✐♦♥s✱ ❧✐❦❡ t❤✐s✳

❊①❡r❝✐s❡ ✵✳✵✳✸ ❋✐♥❞ t❤❡ ❤❡✐❣❤t ♦❢ ❛♥ ❡q✉✐❧❛t❡r❛❧ tr✐❛♥❣❧❡ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ s✐❞❡ ♦❢ ✇❤✐❝❤ ✐s

1✳

❚❤❡ ❡①❡r❝✐s❡s ❛r❡ t♦ ❜❡ ❛tt❡♠♣t❡❞ ✭♦r ❛t ❧❡❛st ❝♦♥s✐❞❡r❡❞✮ ✐♠♠❡❞✐❛t❡❧②✳ ▼♦st ♦❢ t❤❡ ✐♥✲t❡①t ❡①❡r❝✐s❡s ❛r❡ ♥♦t ❡❧❛❜♦r❛t❡✳

❚❤❡② ❛r❡♥✬t✱ ❤♦✇❡✈❡r✱ ❡♥t✐r❡❧② r♦✉t✐♥❡ ❛s t❤❡② r❡q✉✐r❡

✉♥❞❡rst❛♥❞✐♥❣ ♦❢✱ ❛t ❧❡❛st✱ t❤❡ ❝♦♥❝❡♣ts t❤❛t ❤❛✈❡ ❥✉st ❜❡❡♥ ✐♥tr♦❞✉❝❡❞✳ ❆❞❞✐t✐♦♥❛❧ ❡①❡r❝✐s❡ s❡ts ❛r❡ ♣❧❛❝❡❞ ✐♥ t❤❡ ❛♣♣❡♥❞✐① ❛s ✇❡❧❧ ❛s ❛t t❤❡ ❜♦♦❦✬s ✇❡❜s✐t❡✿ ❝❛❧❝✉❧✉s✶✷✸✳❝♦♠✳ ❉♦ ♥♦t st❛rt ②♦✉r st✉❞② ✇✐t❤ t❤❡ ❡①❡r❝✐s❡s✦ ❑❡❡♣ ✐♥ ♠✐♥❞ t❤❛t t❤❡ ❡①❡r❝✐s❡s ❛r❡ ♠❡❛♥t t♦ t❡st ✕ ✐♥❞✐r❡❝t❧② ❛♥❞ ✐♠♣❡r❢❡❝t❧② ✕ ❤♦✇ ✇❡❧❧ t❤❡ ❝♦♥❝❡♣ts ❤❛✈❡ ❜❡❡♥ ❧❡❛r♥❡❞✳ ❚❤❡r❡ ❛r❡ s♦♠❡t✐♠❡s ✇♦r❞s ♦❢ ❝❛✉t✐♦♥ ❛❜♦✉t ❝♦♠♠♦♥ ♠✐st❛❦❡s ♠❛❞❡ ❜② t❤❡ st✉❞❡♥ts✳

❚♦ t❤❡ st✉❞❡♥t✳✳✳

✸

❲❛r♥✐♥❣✦ 2 √ (−1) = 1✱ 1✱ 1 = 1✳

■♥ s♣✐t❡ ♦❢ t❤❡ ❢❛❝t t❤❛t ♦♥❡ sq✉❛r❡ r♦♦t ♦❢

t❤❡r❡ ✐s ♦♥❧②

❚❤❡ ♠♦st ✐♠♣♦rt❛♥t ❢❛❝ts ❛❜♦✉t t❤❡ ♥❡✇ ❝♦♥❝❡♣ts ❛r❡ ♣✉t ❢♦r✇❛r❞ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ♠❛♥♥❡r✳

❚❤❡♦r❡♠ ✵✳✵✳✹✿ Pr♦❞✉❝t ♦❢ ❘♦♦ts ❋♦r ❛♥② t✇♦ ♣♦s✐t✐✈❡ ♥✉♠❜❡rs

a

b✱

❛♥❞

√

a·

√

✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐❞❡♥t✐t②✿

b=

√

a·b

❚❤❡ t❤❡♦r❡♠s ❛r❡ ❝♦♥st❛♥t❧② r❡❢❡rr❡❞ t♦ t❤r♦✉❣❤♦✉t t❤❡ t❡①t✳ ❆s ②♦✉ ❝❛♥ s❡❡✱ t❤❡♦r❡♠s ♠❛② ❝♦♥t❛✐♥ ❢♦r♠✉❧❛s❀ ❛ t❤❡♦r❡♠ s✉♣♣❧✐❡s ❧✐♠✐t❛t✐♦♥s ♦♥ t❤❡ ❛♣♣❧✐❝❛❜✐❧✐t② ♦❢ t❤❡ ❢♦r♠✉❧❛ ✐t ❝♦♥t❛✐♥s✳

❋✉rt❤❡r♠♦r❡✱ ❡✈❡r② ❢♦r♠✉❧❛ ✐s ❛ ♣❛rt ♦❢ ❛ t❤❡♦r❡♠✱ ❛♥❞ ✉s✐♥❣ t❤❡ ❢♦r♠❡r ✇✐t❤♦✉t

❦♥♦✇✐♥❣ t❤❡ ❧❛tt❡r ✐s ♣❡r✐❧♦✉s✳ ❚❤❡r❡ ✐s ♥♦ ♥❡❡❞ t♦ ♠❡♠♦r✐③❡ ❞❡✜♥✐t✐♦♥s ♦r t❤❡♦r❡♠s ✭❛♥❞ ❢♦r♠✉❧❛s✮✱ ✐♥✐t✐❛❧❧②✳ ❲✐t❤ ❡♥♦✉❣❤ t✐♠❡ s♣❡♥t ✇✐t❤ t❤❡ ♠❛t❡r✐❛❧✱ t❤❡ ♠❛✐♥ ♦♥❡s ✇✐❧❧ ❡✈❡♥t✉❛❧❧② ❜❡❝♦♠❡ ❢❛♠✐❧✐❛r ❛s t❤❡② ❝♦♥t✐♥✉❡ t♦ r❡❛♣♣❡❛r ✐♥ t❤❡ t❡①t✳ ❲❛t❝❤ ❢♦r ✇♦r❞s ✏✐♠♣♦rt❛♥t✑✱ ✏❝r✉❝✐❛❧✑✱ ❡t❝✳ ❚❤♦s❡ ♥❡✇ ❝♦♥❝❡♣ts t❤❛t ❞♦ ♥♦t r❡❛♣♣❡❛r ✐♥ t❤✐s t❡①t ❛r❡ ❧✐❦❡❧② t♦ ❜❡ s❡❡♥ ✐♥ t❤❡ ♥❡①t ♠❛t❤❡♠❛t✐❝s ❜♦♦❦ t❤❛t ②♦✉ r❡❛❞✳ ❨♦✉ ♥❡❡❞ t♦✱ ❤♦✇❡✈❡r✱ ❜❡ ❛✇❛r❡ ♦❢ ❛❧❧ ♦❢ t❤❡ ❞❡✜♥✐t✐♦♥s ❛♥❞ t❤❡♦r❡♠s ❛♥❞ ❜❡ ❛❜❧❡ t♦ ✜♥❞ t❤❡ r✐❣❤t ♦♥❡ ✇❤❡♥ ♥❡❝❡ss❛r②✳ ❖❢t❡♥✱ ❜✉t ♥♦t ❛❧✇❛②s✱ ❛ t❤❡♦r❡♠ ✐s ❢♦❧❧♦✇❡❞ ❜② ❛ t❤♦r♦✉❣❤ ❛r❣✉♠❡♥t ❛s ❛ ❥✉st✐✜❝❛t✐♦♥✳

Pr♦♦❢✳ ❙✉♣♣♦s❡

A=

√

a

❛♥❞

B=

√

b✳

❚❤❡♥✱ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❞❡✜♥✐t✐♦♥✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿

a = A2

❛♥❞

b = B2 .

❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡✿

❍❡♥❝❡✱

√

a · b = A2 · B 2 = A · A · B · B = (A · B) · (A · B) = (AB)2 . ab = A · B ✱

❛❣❛✐♥ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❞❡✜♥✐t✐♦♥✳

❙♦♠❡ ♣r♦♦❢s ❝❛♥ ❜❡ s❦✐♣♣❡❞ ❛t ✜rst r❡❛❞✐♥❣✳ ■ts ❤✐❣❤❧② ❞❡t❛✐❧❡❞ ❡①♣♦s✐t✐♦♥ ♠❛❦❡s t❤❡ ❜♦♦❦ ❛ ❣♦♦❞ ❝❤♦✐❝❡ ❢♦r s❡❧❢✲st✉❞②✳ ■❢ t❤✐s ✐s ②♦✉r ❝❛s❡✱ t❤❡s❡ ❛r❡ ♠② s✉❣❣❡st✐♦♥s✳ ❲❤✐❧❡ r❡❛❞✐♥❣ t❤❡ ❜♦♦❦✱ tr② t♦ ♠❛❦❡ s✉r❡ t❤❛t ②♦✉ ✉♥❞❡rst❛♥❞ ♥❡✇ ❝♦♥❝❡♣ts ❛♥❞ ✐❞❡❛s✳ ❤♦✇❡✈❡r✱ t❤❛t s♦♠❡ ❛r❡ ♠♦r❡ ✐♠♣♦rt❛♥t t❤❛t ♦t❤❡rs❀ t❤❡② ❛r❡ ♠❛r❦❡❞ ❛❝❝♦r❞✐♥❣❧②✳

❑❡❡♣ ✐♥ ♠✐♥❞✱

❈♦♠❡ ❜❛❝❦ ✭♦r ❥✉♠♣

❢♦r✇❛r❞✮ ❛s ♥❡❡❞❡❞✳ ❈♦♥t❡♠♣❧❛t❡✳ ❋✐♥❞ ♦t❤❡r s♦✉r❝❡s ✐❢ ♥❡❝❡ss❛r②✳ ❨♦✉ s❤♦✉❧❞ ♥♦t t✉r♥ t♦ t❤❡ ❡①❡r❝✐s❡ s❡ts ✉♥t✐❧ ②♦✉ ❤❛✈❡ ❜❡❝♦♠❡ ❝♦♠❢♦rt❛❜❧❡ ✇✐t❤ t❤❡ ♠❛t❡r✐❛❧✳ ❲❤❛t t♦ ❞♦ ❛❜♦✉t ❡①❡r❝✐s❡s ✇❤❡♥ s♦❧✉t✐♦♥s ❛r❡♥✬t ♣r♦✈✐❞❡❞❄ ❋✐rst✱ ✉s❡ t❤❡ ❡①❛♠♣❧❡s✳ ▼❛♥② ♦❢ t❤❡♠ ❝♦♥t❛✐♥ ❛ ♣r♦❜❧❡♠ ✕ ✇✐t❤ ❛ s♦❧✉t✐♦♥✳ ❚r② t♦ s♦❧✈❡ t❤❡ ♣r♦❜❧❡♠ ✕ ❜❡❢♦r❡ ♦r ❛❢t❡r r❡❛❞✐♥❣ t❤❡ s♦❧✉t✐♦♥✳ ❨♦✉ ❝❛♥ ❛❧s♦ ✜♥❞ ❡①❡r❝✐s❡s ♦♥❧✐♥❡ ♦r ♠❛❦❡ ✉♣ ②♦✉r ♦✇♥ ♣r♦❜❧❡♠s ❛♥❞ s♦❧✈❡ t❤❡♠✦ ■ str♦♥❣❧② s✉❣❣❡st t❤❛t ②♦✉r s♦❧✉t✐♦♥ s❤♦✉❧❞ ❜❡ t❤♦r♦✉❣❤❧② ✇r✐tt❡♥✳ ❨♦✉ s❤♦✉❧❞ ✇r✐t❡ ✐♥ ❝♦♠♣❧❡t❡ s❡♥t❡♥❝❡s✱ ✐♥❝❧✉❞✐♥❣ ❛❧❧ t❤❡ ❛❧❣❡❜r❛✳ ❋♦r ❡①❛♠♣❧❡✱ ②♦✉ s❤♦✉❧❞ ❛♣♣r❡❝✐❛t❡ t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡s❡ t✇♦✿

❲r♦♥❣✿

1+1 2

❘✐❣❤t✿

1+1 =2

✹

❚♦ t❤❡ st✉❞❡♥t✳✳✳

❚❤❡ ❧❛tt❡r r❡❛❞s ✏♦♥❡ ❛❞❞❡❞ t♦ ♦♥❡ ✐s t✇♦✑✱ ✇❤✐❧❡ t❤❡ ❢♦r♠❡r ❝❛♥♥♦t ❜❡ r❡❛❞✳ ❨♦✉ s❤♦✉❧❞ ❛❧s♦ ❥✉st✐❢② ❛❧❧ ②♦✉r st❡♣s ❛♥❞ ❝♦♥❝❧✉s✐♦♥s✱ ✐♥❝❧✉❞✐♥❣ ❛❧❧ t❤❡ ❛❧❣❡❜r❛✳ ❋♦r ❡①❛♠♣❧❡✱ ②♦✉ s❤♦✉❧❞ ❛♣♣r❡❝✐❛t❡ t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡s❡ t✇♦✿ ❲r♦♥❣✿

2x = 4 x=2

❘✐❣❤t✿

2x = 4; x = 2.

t❤❡r❡❢♦r❡✱

❚❤❡ st❛♥❞❛r❞s ♦❢ t❤♦r♦✉❣❤♥❡ss ❛r❡ ♣r♦✈✐❞❡❞ ❜② t❤❡ ❡①❛♠♣❧❡s ✐♥ t❤❡ ❜♦♦❦✳ ◆❡①t✱ ②♦✉r s♦❧✉t✐♦♥ s❤♦✉❧❞ ❜❡ t❤♦r♦✉❣❤❧② r❡❛❞✳ ❚❤✐s ✐s t❤❡ t✐♠❡ ❢♦r s❡❧❢✲❝r✐t✐❝✐s♠✿ ▲♦♦❦ ❢♦r ❡rr♦rs ❛♥❞ ✇❡❛❦ s♣♦ts✳ ■t s❤♦✉❧❞ ❜❡ r❡✲r❡❛❞ ❛♥❞ t❤❡♥ r❡✇r✐tt❡♥✳ ❖♥❝❡ ②♦✉ ❛r❡ ❝♦♥✈✐♥❝❡❞ t❤❛t t❤❡ s♦❧✉t✐♦♥ ✐s ❝♦rr❡❝t ❛♥❞ t❤❡ ♣r❡s❡♥t❛t✐♦♥ ✐s s♦❧✐❞✱ ②♦✉ ♠❛② s❤♦✇ ✐t t♦ ❛ ❦♥♦✇❧❡❞❣❡❛❜❧❡ ♣❡rs♦♥ ❢♦r ❛ ♦♥❝❡✲♦✈❡r✳ ◆❡①t✱ ②♦✉ ♠❛② t✉r♥ t♦ ♠♦❞❡❧✐♥❣ ♣r♦❥❡❝ts✳ ❙♣r❡❛❞s❤❡❡ts ✭▼✐❝r♦s♦❢t ❊①❝❡❧ ♦r s✐♠✐❧❛r✮ ❛r❡ ❝❤♦s❡♥ t♦ ❜❡ ✉s❡❞ ❢♦r ❣r❛♣❤✐♥❣ ❛♥❞ ♠♦❞❡❧✐♥❣✳ ❖♥❡ ❝❛♥ ❛❝❤✐❡✈❡ ❛s ❣♦♦❞ r❡s✉❧ts ✇✐t❤ ♣❛❝❦❛❣❡s s♣❡❝✐✜❝❛❧❧② ❞❡s✐❣♥❡❞ ❢♦r t❤❡s❡ ♣✉r♣♦s❡s✱ ❜✉t s♣r❡❛❞s❤❡❡ts ♣r♦✈✐❞❡ ❛ t♦♦❧ ✇✐t❤ ❛ ✇✐❞❡r s❝♦♣❡ ♦❢ ❛♣♣❧✐❝❛t✐♦♥s✳ ♦♣t✐♦♥✳ ●♦♦❞ ❧✉❝❦✦

Pr♦❣r❛♠♠✐♥❣ ✐s ❛♥♦t❤❡r

❚♦ t❤❡ t❡❛❝❤❡r

✺

❚♦ t❤❡ t❡❛❝❤❡r ❚❤❡ ❜✉❧❦ ♦❢ t❤❡ ♠❛t❡r✐❛❧ ✐♥ t❤❡ ❜♦♦❦ ❝♦♠❡s ❢r♦♠ ♠② ❧❡❝t✉r❡ ♥♦t❡s✳ ❚❤❡r❡ ✐s ❧✐tt❧❡ ❡♠♣❤❛s✐s ♦♥ ❝❧♦s❡❞✲❢♦r♠ ❝♦♠♣✉t❛t✐♦♥s ❛♥❞ ❛❧❣❡❜r❛✐❝ ♠❛♥✐♣✉❧❛t✐♦♥s✳ ■ ❞♦ t❤✐♥❦ t❤❛t ❛ ♣❡rs♦♥ ✇❤♦ ❤❛s ♥❡✈❡r ✐♥t❡❣r❛t❡❞ ❜② ❤❛♥❞ ✭♦r ❞✐✛❡r❡♥t✐❛t❡❞✱ ♦r ❛♣♣❧✐❡❞ t❤❡ q✉❛❞r❛t✐❝ ❢♦r♠✉❧❛✱ ❡t❝✳✮ ❝❛♥♥♦t ♣♦ss✐❜❧② ✉♥❞❡rst❛♥❞ ✐♥t❡❣r❛t✐♦♥ ✭♦r ❞✐✛❡r❡♥t✐❛t✐♦♥✱ ♦r q✉❛❞r❛t✐❝ ❢✉♥❝t✐♦♥s✱ ❡t❝✳✮✳ ❍♦✇❡✈❡r✱ ❛ ❧❛r❣❡ ♣r♦♣♦rt✐♦♥ ♦❢ t✐♠❡ ❛♥❞ ❡✛♦rt ❝❛♥ ❛♥❞ s❤♦✉❧❞ ❜❡ ❞✐r❡❝t❡❞ t♦✇❛r❞✿

•

✉♥❞❡rst❛♥❞✐♥❣ ♦❢ t❤❡ ❝♦♥❝❡♣ts ❛♥❞

•

♠♦❞❡❧✐♥❣ ✐♥ r❡❛❧✐st✐❝ s❡tt✐♥❣s✳

❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ t❤✐s ❛♣♣r♦❛❝❤ ✐s t❤❛t ✐t r❡q✉✐r❡s ♠♦r❡ ❛❜str❛❝t✐♦♥ r❛t❤❡r t❤❛♥ ❧❡ss✳ ❱✐s✉❛❧✐③❛t✐♦♥ ✐s t❤❡ ♠❛✐♥ t♦♦❧ ✉s❡❞ t♦ ❞❡❛❧ ✇✐t❤ t❤✐s ❝❤❛❧❧❡♥❣❡✳ ■❧❧✉str❛t✐♦♥s ❛r❡ ♣r♦✈✐❞❡❞ ❢♦r ❡✈❡r② ❝♦♥❝❡♣t✱ ❜✐❣ ♦r s♠❛❧❧✳ ❚❤❡ ♣✐❝t✉r❡s t❤❛t ❝♦♠❡ ♦✉t ❛r❡ s♦♠❡t✐♠❡s ✈❡r② ♣r❡❝✐s❡ ❜✉t s♦♠❡t✐♠❡s s❡r✈❡ ❛s ♠❡r❡ ♠❡t❛♣❤♦rs ❢♦r t❤❡ ❝♦♥❝❡♣ts t❤❡② ✐❧❧✉str❛t❡✳ ❚❤❡ ❤♦♣❡ ✐s t❤❛t t❤❡② ✇✐❧❧ s❡r✈❡ ❛s ✈✐s✉❛❧ ✏❛♥❝❤♦rs✑ ✐♥ ❛❞❞✐t✐♦♥ t♦ t❤❡ ✇♦r❞s ❛♥❞ ❢♦r♠✉❧❛s✳ ■t ✐s ✉♥❧✐❦❡❧② t❤❛t ❛ ♣❡rs♦♥ ✇❤♦ ❤❛s ♥❡✈❡r ♣❧♦tt❡❞ t❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥ ❜② ❤❛♥❞ ❝❛♥ ✉♥❞❡rst❛♥❞ ❣r❛♣❤s ♦r ❢✉♥❝t✐♦♥s✳ ❍♦✇❡✈❡r✱ ✇❤❛t ✐❢ ✇❡ ✇❛♥t t♦ ♣❧♦t ♠♦r❡ t❤❛♥ ❥✉st ❛ ❢❡✇ ♣♦✐♥ts ✐♥ ♦r❞❡r t♦ ✈✐s✉❛❧✐③❡ ❝✉r✈❡s✱ s✉r❢❛❝❡s✱ ✈❡❝t♦r ✜❡❧❞s✱ ❡t❝✳❄

❙♣r❡❛❞s❤❡❡ts ✇❡r❡ ❝❤♦s❡♥ ♦✈❡r ❣r❛♣❤✐❝ ❝❛❧❝✉❧❛t♦rs ❢♦r ✈✐s✉❛❧✐③❛t✐♦♥ ♣✉r♣♦s❡s

❜❡❝❛✉s❡ t❤❡② r❡♣r❡s❡♥t t❤❡ s❤♦rt❡st st❡♣ ❛✇❛② ❢r♦♠ ♣❡♥ ❛♥❞ ♣❛♣❡r✳

■♥❞❡❡❞✱ t❤❡ ❞❛t❛ ✐s ♣❧♦tt❡❞ ✐♥ t❤❡

s✐♠♣❧❡st ♠❛♥♥❡r ♣♦ss✐❜❧❡✿ ♦♥❡ ❝❡❧❧ ✲ ♦♥❡ ♥✉♠❜❡r ✲ ♦♥❡ ♣♦✐♥t ♦♥ t❤❡ ❣r❛♣❤✳ ❋♦r ♠♦r❡ ❛❞✈❛♥❝❡❞ t❛s❦s s✉❝❤ ❛s ♠♦❞❡❧✐♥❣✱ s♣r❡❛❞s❤❡❡ts ✇❡r❡ ❝❤♦s❡♥ ♦✈❡r ♦t❤❡r s♦❢t✇❛r❡ ❛♥❞ ♣r♦❣r❛♠♠✐♥❣ ♦♣t✐♦♥s ❢♦r t❤❡✐r ✇✐❞❡ ❛✈❛✐❧❛❜✐❧✐t② ❛♥❞✱ ❛❜♦✈❡ ❛❧❧✱ t❤❡✐r s✐♠♣❧✐❝✐t②✳ ◆✐♥❡ ♦✉t ♦❢ t❡♥✱ t❤❡ s♣r❡❛❞s❤❡❡t s❤♦✇♥ ✇❛s ✐♥✐t✐❛❧❧② ❝r❡❛t❡❞ ❢r♦♠ s❝r❛t❝❤ ✐♥ ❢r♦♥t ♦❢ t❤❡ st✉❞❡♥ts ✇❤♦ ✇❡r❡ ❧❛t❡r ❛❜❧❡ t♦ ❢♦❧❧♦✇ ♠② ❢♦♦tst❡♣s ❛♥❞ ❝r❡❛t❡ t❤❡✐r ♦✇♥✳ ❆❜♦✉t t❤❡ t❡sts✳ ❚❤❡ ❜♦♦❦ ✐s♥✬t ❞❡s✐❣♥❡❞ t♦ ♣r❡♣❛r❡ t❤❡ st✉❞❡♥t ❢♦r s♦♠❡ ♣r❡❡①✐st✐♥❣ ❡①❛♠❀ ♦♥ t❤❡ ❝♦♥tr❛r②✱ ❛ss✐❣♥♠❡♥ts s❤♦✉❧❞ ❜❡ ❜❛s❡❞ ♦♥ ✇❤❛t ❤❛s ❜❡❡♥ ❧❡❛r♥❡❞✳ ❚❤❡ st✉❞❡♥ts✬ ✉♥❞❡rst❛♥❞✐♥❣ ♦❢ t❤❡ ❝♦♥❝❡♣ts ♥❡❡❞s t♦ ❜❡ t❡st❡❞ ❜✉t✱ ♠♦st ♦❢ t❤❡ t✐♠❡✱ t❤✐s ❝❛♥ ❜❡ ❞♦♥❡ ♦♥❧② ✐♥❞✐r❡❝t❧②✳ ❚❤❡r❡❢♦r❡✱ ❛ ❝❡rt❛✐♥ s❤❛r❡ ♦❢ r♦✉t✐♥❡✱ ♠❡❝❤❛♥✐❝❛❧ ♣r♦❜❧❡♠s ✐s ✐♥❡✈✐t❛❜❧❡✳ ◆♦♥❡t❤❡❧❡ss✱ ♥♦ t♦♣✐❝ ❞❡s❡r✈❡s ♠♦r❡ ❛tt❡♥t✐♦♥ ❥✉st ❜❡❝❛✉s❡ ✐t✬s ❧✐❦❡❧② t♦ ❜❡ ♦♥ t❤❡ t❡st✳ ■❢ ❛t ❛❧❧ ♣♦ss✐❜❧❡✱ ❞♦♥✬t ♠❛❦❡ t❤❡ st✉❞❡♥ts ♠❡♠♦r✐③❡ ❢♦r♠✉❧❛s✳ ■♥ t❤❡ ♦r❞❡r ♦❢ t♦♣✐❝s✱ t❤❡ ♠❛✐♥ ❞✐✛❡r❡♥❝❡ ❢r♦♠ ❛ t②♣✐❝❛❧ ❝❛❧❝✉❧✉s t❡①t❜♦♦❦ ✐s t❤❛t s❡q✉❡♥❝❡s ❝♦♠❡ ❜❡❢♦r❡ ❡✈❡r②t❤✐♥❣ ❡❧s❡✳ ❚❤❡ r❡❛s♦♥s ❛r❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿

•

❙❡q✉❡♥❝❡s ❛r❡ t❤❡ s✐♠♣❧❡st ❦✐♥❞ ♦❢ ❢✉♥❝t✐♦♥s✳

•

▲✐♠✐ts ♦❢ s❡q✉❡♥❝❡s ❛r❡ s✐♠♣❧❡r t❤❛♥ ❧✐♠✐ts ♦❢ ❣❡♥❡r❛❧ ❢✉♥❝t✐♦♥s ✭✐♥❝❧✉❞✐♥❣ t❤❡ ♦♥❡s ❛t ✐♥✜♥✐t②✮✳

•

❚❤❡ s✐❣♠❛ ♥♦t❛t✐♦♥✱ t❤❡ ❘✐❡♠❛♥♥ s✉♠s✱ ❛♥❞ t❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧ ♠❛❦❡ ♠♦r❡ s❡♥s❡ t♦ ❛ st✉❞❡♥t ✇✐t❤

•

❆ q✉✐❝❦ tr❛♥s✐t✐♦♥ ❢r♦♠ s❡q✉❡♥❝❡s t♦ s❡r✐❡s ♦❢t❡♥ ❧❡❛❞s t♦ ❝♦♥❢✉s✐♦♥ ❜❡t✇❡❡♥ t❤❡ t✇♦✳

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❙❡q✉❡♥❝❡s ❛r❡ ♥❡❡❞❡❞ ❢♦r ♠♦❞❡❧✐♥❣✱ ✇❤✐❝❤ s❤♦✉❧❞ st❛rt ❛s ❡❛r❧② ❛s ♣♦ss✐❜❧❡✳

❛ s♦❧✐❞ ❜❛❝❦❣r♦✉♥❞ ✐♥ s❡q✉❡♥❝❡s✳

❋r♦♠ t❤❡ ❞✐s❝r❡t❡ t♦ t❤❡ ❝♦♥t✐♥✉♦✉s

✻

❋r♦♠ t❤❡ ❞✐s❝r❡t❡ t♦ t❤❡ ❝♦♥t✐♥✉♦✉s ■t✬s ♥♦ s❡❝r❡t t❤❛t ❛ ✈❛st ♠❛❥♦r✐t② ♦❢ ❝❛❧❝✉❧✉s st✉❞❡♥ts ✇✐❧❧ ♥❡✈❡r ✉s❡ ✇❤❛t t❤❡② ❤❛✈❡ ❧❡❛r♥❡❞✳ P♦♦r ❝❛r❡❡r ❝❤♦✐❝❡s ❛s✐❞❡✱ ❛ ❢♦r♠❡r ❝❛❧❝✉❧✉s st✉❞❡♥t ✐s ♦❢t❡♥ ✉♥❛❜❧❡ t♦ r❡❝♦❣♥✐③❡ t❤❡ ♠❛t❤❡♠❛t✐❝s t❤❛t ✐s s✉♣♣♦s❡❞ t♦ s✉rr♦✉♥❞ ❤✐♠✳ ❲❤② ❞♦❡s t❤✐s ❤❛♣♣❡♥❄ ❈❛❧❝✉❧✉s ✐s t❤❡ s❝✐❡♥❝❡ ♦❢ ❝❤❛♥❣❡✳ ❋r♦♠ t❤❡ ✈❡r② ❜❡❣✐♥♥✐♥❣✱ ✐ts ♣❡❝✉❧✐❛r ❝❤❛❧❧❡♥❣❡ ❤❛s ❜❡❡♥ t♦ st✉❞② ❛♥❞

❝♦♥t✐♥✉♦✉s ❝❤❛♥❣❡✿ ❝✉r✈❡s ❛♥❞ ♠♦t✐♦♥ ❛❧♦♥❣ ❝✉r✈❡s✳ ❢♦r♠✉❧❛s✳ ❙❦✐❧❧❢✉❧ ♠❛♥✐♣✉❧❛t✐♦♥ ♦❢ t❤♦s❡ ❢♦r♠✉❧❛s ✐s ✇❤❛t

♠❡❛s✉r❡

❚❤❡s❡ ❝✉r✈❡s ❛♥❞ t❤✐s ♠♦t✐♦♥ ❛r❡ r❡♣r❡s❡♥t❡❞

❜②

s♦❧✈❡s ❝❛❧❝✉❧✉s ♣r♦❜❧❡♠s✳ ❋♦r ♦✈❡r ✸✵✵ ②❡❛rs✱

t❤✐s ❛♣♣r♦❛❝❤ ❤❛s ❜❡❡♥ ❡①tr❡♠❡❧② s✉❝❝❡ss❢✉❧ ✐♥ s❝✐❡♥❝❡s ❛♥❞ ❡♥❣✐♥❡❡r✐♥❣✳

❚❤❡ s✉❝❝❡ss❡s ❛r❡ ✇❡❧❧✲❦♥♦✇♥✿

♣r♦❥❡❝t✐❧❡ ♠♦t✐♦♥✱ ♣❧❛♥❡t❛r② ♠♦t✐♦♥✱ ✢♦✇ ♦❢ ❧✐q✉✐❞s✱ ❤❡❛t tr❛♥s❢❡r✱ ✇❛✈❡ ♣r♦♣❛❣❛t✐♦♥✱ ❡t❝✳ ❚❡❛❝❤✐♥❣ ❝❛❧❝✉❧✉s ❢♦❧❧♦✇s t❤✐s ❛♣♣r♦❛❝❤✿ ❆♥ ♦✈❡r✇❤❡❧♠✐♥❣ ♠❛❥♦r✐t② ♦❢ ✇❤❛t t❤❡ st✉❞❡♥t ❞♦❡s ✐s ♠❛♥✐♣✉❧❛t✐♦♥ ♦❢ ❢♦r♠✉❧❛s ♦♥ ❛ ♣✐❡❝❡ ♦❢ ♣❛♣❡r✳ ❇✉t t❤✐s ♠❡❛♥s t❤❛t ❛❧❧ t❤❡ ♣r♦❜❧❡♠s t❤❡ st✉❞❡♥t ❢❛❝❡s ✇❡r❡ ✭♦r ❝♦✉❧❞ ❤❛✈❡ ❜❡❡♥✮ s♦❧✈❡❞ ✐♥ t❤❡ ✶✽t❤ ♦r ✶✾t❤ ❝❡♥t✉r✐❡s✦ ❚❤✐s ✐s♥✬t ❣♦♦❞ ❡♥♦✉❣❤ ❛♥②♠♦r❡✳ ❲❤❛t ❤❛s ❝❤❛♥❣❡❞ s✐♥❝❡ t❤❡♥❄ ❚❤❡ ❝♦♠♣✉t❡rs ❤❛✈❡ ❛♣♣❡❛r❡❞✱ ♦❢ ❝♦✉rs❡✱ ❛♥❞ ❝♦♠♣✉t❡rs ❞♦♥✬t ♠❛♥✐♣✉❧❛t❡ ❢♦r♠✉❧❛s✳

❚❤❡② ❞♦♥✬t ❤❡❧♣ ✇✐t❤ s♦❧✈✐♥❣ ✕ ✐♥ t❤❡ tr❛❞✐t✐♦♥❛❧ s❡♥s❡ ♦❢

t❤❡ ✇♦r❞ ✕ t❤♦s❡ ♣r♦❜❧❡♠s ❢r♦♠ t❤❡ ♣❛st ❝❡♥t✉r✐❡s✳

✐♥❝r❡♠❡♥t❛❧

■♥st❡❛❞ ♦❢

❝♦♥t✐♥✉♦✉s✱

❝♦♠♣✉t❡rs ❡①❝❡❧ ❛t ❤❛♥❞❧✐♥❣

♣r♦❝❡ss❡s✱ ❛♥❞ ✐♥st❡❛❞ ♦❢ ❢♦r♠✉❧❛s t❤❡② ❛r❡ ❣r❡❛t ❛t ♠❛♥❛❣✐♥❣ ❞✐s❝r❡t❡ ✭❞✐❣✐t❛❧✮ ❞❛t❛✳ ❚♦ ✉t✐❧✐③❡

t❤❡s❡ ❛❞✈❛♥t❛❣❡s✱ s❝✐❡♥t✐sts ✏❞✐s❝r❡t✐③❡✑ t❤❡ r❡s✉❧ts ♦❢ ❝❛❧❝✉❧✉s ❛♥❞ ❝r❡❛t❡ ❛❧❣♦r✐t❤♠s t❤❛t ♠❛♥✐♣✉❧❛t❡ t❤❡ ❞✐❣✐t❛❧ ❞❛t❛✳

❚❤❡ s♦❧✉t✐♦♥s ❛r❡ ❛♣♣r♦①✐♠❛t❡ ❜✉t t❤❡ ❛♣♣❧✐❝❛❜✐❧✐t② ✐s ✉♥❧✐♠✐t❡❞✳

❙✐♥❝❡ t❤❡ ✷✵t❤ ❝❡♥t✉r②✱

t❤✐s ❛♣♣r♦❛❝❤ ❤❛s ❜❡❡♥ ❡①tr❡♠❡❧② s✉❝❝❡ss❢✉❧ ✐♥ s❝✐❡♥❝❡s ❛♥❞ ❡♥❣✐♥❡❡r✐♥❣✿ ❛❡r♦❞②♥❛♠✐❝s ✭❛✐r♣❧❛♥❡ ❛♥❞ ❝❛r ❞❡s✐❣♥✮✱ s♦✉♥❞ ❛♥❞ ✐♠❛❣❡ ♣r♦❝❡ss✐♥❣✱ s♣❛❝❡ ❡①♣❧♦r❛t✐♦♥✱ str✉❝t✉r❡ ♦❢ t❤❡ ❛t♦♠ ❛♥❞ t❤❡ ✉♥✐✈❡rs❡✱ ❡t❝✳ ❚❤❡ ❛♣♣r♦❛❝❤ ✐s ❛❧s♦ ❝✐r❝✉✐t♦✉s✿ ❊✈❡r② ❝♦♥❝❡♣t ✐♥ ❝❛❧❝✉❧✉s

st❛rts

✕ ♦❢t❡♥ ✐♠♣❧✐❝✐t❧② ✕ ❛s ❛ ❞✐s❝r❡t❡ ❛♣♣r♦①✐♠❛t✐♦♥

♦❢ ❛ ❝♦♥t✐♥✉♦✉s ♣❤❡♥♦♠❡♥♦♥✦

❈❛❧❝✉❧✉s ✐s t❤❡ s❝✐❡♥❝❡ ♦❢ ❝❤❛♥❣❡✱

❜♦t❤

✐♥❝r❡♠❡♥t❛❧ ❛♥❞ ❝♦♥t✐♥✉♦✉s✳ ❚❤❡ ❢♦r♠❡r ♣❛rt ✕ t❤❡ s♦✲❝❛❧❧❡❞ ❞✐s❝r❡t❡

❝❛❧❝✉❧✉s ✕ ♠❛② ❜❡ s❡❡♥ ❛s t❤❡ st✉❞② ♦❢ ✐♥❝r❡♠❡♥t❛❧ ♣❤❡♥♦♠❡♥❛ ❛♥❞ t❤❡ q✉❛♥t✐t✐❡s

✐♥❞✐✈✐s✐❜❧❡

❜② t❤❡✐r

✈❡r② ♥❛t✉r❡✿ ♣❡♦♣❧❡✱ ❛♥✐♠❛❧s✱ ❛♥❞ ♦t❤❡r ♦r❣❛♥✐s♠s✱ ♠♦♠❡♥ts ♦❢ t✐♠❡✱ ❧♦❝❛t✐♦♥s ♦❢ s♣❛❝❡✱ ♣❛rt✐❝❧❡s✱ s♦♠❡ ❝♦♠♠♦❞✐t✐❡s✱ ❞✐❣✐t❛❧ ✐♠❛❣❡s ❛♥❞ ♦t❤❡r ♠❛♥✲♠❛❞❡ ❞❛t❛✱ ❡t❝✳ ❲✐t❤ t❤❡ ❤❡❧♣ ♦❢ t❤❡ ❝❛❧❝✉❧✉s ♠❛❝❤✐♥❡r② ❝❛❧❧❡❞ ✏❧✐♠✐ts✑✱ ✇❡ ✐♥✈❛r✐❛❜❧② ❝❤♦♦s❡ t♦ tr❛♥s✐t✐♦♥ t♦ t❤❡ ❝♦♥t✐♥✉♦✉s ♣❛rt ♦❢ ❝❛❧❝✉❧✉s✱ ❡s♣❡❝✐❛❧❧② ✇❤❡♥ ✇❡ ❢❛❝❡ ❝♦♥t✐♥✉♦✉s ♣❤❡♥♦♠❡♥❛ ❛♥❞ t❤❡ q✉❛♥t✐t✐❡s

✐♥✜♥✐t❡❧② ❞✐✈✐s✐❜❧❡

❡✐t❤❡r ❜② t❤❡✐r ♥❛t✉r❡ ♦r ❜② ❛ss✉♠♣t✐♦♥✿ t✐♠❡✱

s♣❛❝❡✱ ♠❛ss✱ t❡♠♣❡r❛t✉r❡✱ ♠♦♥❡②✱ s♦♠❡ ❝♦♠♠♦❞✐t✐❡s✱ ❡t❝✳ ❈❛❧❝✉❧✉s ♣r♦❞✉❝❡s ❞❡✜♥✐t✐✈❡ r❡s✉❧ts ❛♥❞ ❛❜s♦❧✉t❡ ❛❝❝✉r❛❝② ✕ ❜✉t ♦♥❧② ❢♦r ♣r♦❜❧❡♠s ❛♠❡♥❛❜❧❡ t♦ ✐ts ♠❡t❤♦❞s✦ ■♥ t❤❡ ❝❧❛ssr♦♦♠✱ t❤❡ ♣r♦❜❧❡♠s ❛r❡ s✐♠♣❧✐✜❡❞ ✉♥t✐❧ t❤❡② ❜❡❝♦♠❡ ♠❛♥❛❣❡❛❜❧❡❀ ♦t❤❡r✇✐s❡✱ ✇❡ ❝✐r❝❧❡ ❜❛❝❦ t♦ t❤❡ ❞✐s❝r❡t❡ ♠❡t❤♦❞s ✐♥ s❡❛r❝❤ ♦❢ ❛♣♣r♦①✐♠❛t✐♦♥s✳ ❲✐t❤✐♥ ❛ t②♣✐❝❛❧ ❝❛❧❝✉❧✉s ❝♦✉rs❡✱ t❤❡ st✉❞❡♥t s✐♠♣❧② ♥❡✈❡r ❣❡ts t♦ ❝♦♠♣❧❡t❡ t❤❡ ✏❝✐r❝❧❡✑✦

▲❛t❡r ♦♥✱ t❤❡

❣r❛❞✉❛t❡ ✐s ❧✐❦❡❧② t♦ t❤✐♥❦ ♦❢ ❝❛❧❝✉❧✉s ♦♥❧② ✇❤❡♥ ❤❡ s❡❡s ❢♦r♠✉❧❛s ❛♥❞ r❛r❡❧② ✇❤❡♥ ❤❡ s❡❡s ♥✉♠❡r✐❝❛❧ ❞❛t❛✳ ■♥ t❤✐s ❜♦♦❦✱ ❡✈❡r② ❝♦♥❝❡♣t ♦❢ ❝❛❧❝✉❧✉s ✐s ✜rst ✐♥tr♦❞✉❝❡❞ ✐♥ ✐ts ❞✐s❝r❡t❡✱ ✏♣r❡✲❧✐♠✐t✑✱ ✐♥❝❛r♥❛t✐♦♥ ✕ ❡❧s❡✇❤❡r❡ t②♣✐❝❛❧❧② ❤✐❞❞❡♥ ✐♥s✐❞❡ ♣r♦♦❢s ✕ ❛♥❞ t❤❡♥ ✉s❡❞ ❢♦r ♠♦❞❡❧✐♥❣ ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s ✇❡❧❧ ❜❡❢♦r❡ ✐ts ❝♦♥t✐♥✉♦✉s ❝♦✉♥t❡r♣❛rt ❡♠❡r❣❡s✳ ❚❤❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❢♦r♠❡r ❛r❡ ❞✐s❝♦✈❡r❡❞ ✜rst ❛♥❞ t❤❡♥ t❤❡ ♠❛t❝❤✐♥❣ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❧❛tt❡r ❛r❡ ❢♦✉♥❞ ❜② ♠❛❦✐♥❣ t❤❡ ✐♥❝r❡♠❡♥t s♠❛❧❧❡r ❛♥❞ s♠❛❧❧❡r✱ ❛t t❤❡ ❞✐s❝r❡t❡ ❝❛❧❝✉❧✉s

∆x→0

−−−−−−−−−−→

❧✐♠✐t ✿

❝♦♥t✐♥✉♦✉s ❝❛❧❝✉❧✉s

❚❤❡ ✈♦❧✉♠❡ ❛♥❞ ❝❤❛♣t❡r r❡❢❡r❡♥❝❡s ❢♦r ❈❛❧❝✉❧✉s ■❧❧✉str❛t❡❞

✼

❚❤❡ ✈♦❧✉♠❡ ❛♥❞ ❝❤❛♣t❡r r❡❢❡r❡♥❝❡s ❢♦r ❈❛❧❝✉❧✉s ■❧❧✉str❛t❡❞ ❚❤✐s ❜♦♦❦ ✐s ❛ ♣❛rt ♦❢ t❤❡ s❡r✐❡s ❈❛❧❝✉❧✉s ■❧❧✉str❛t❡❞✳ ❚❤❡ s❡r✐❡s ❝♦✈❡rs t❤❡ st❛♥❞❛r❞ ♠❛t❡r✐❛❧ ♦❢ t❤❡ ✉♥❞❡r✲ ❣r❛❞✉❛t❡ ❝❛❧❝✉❧✉s ✇✐t❤ ❛ s✉❜st❛♥t✐❛❧ r❡✈✐❡✇ ♦❢ ♣r❡❝❛❧❝✉❧✉s ❛♥❞ ❛ ♣r❡✈✐❡✇ ♦❢ ❡❧❡♠❡♥t❛r② ♦r❞✐♥❛r② ❛♥❞ ♣❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✳ ❇❡❧♦✇ ✐s t❤❡ ❧✐st ♦❢ t❤❡ ❜♦♦❦s ♦❢ t❤❡ s❡r✐❡s✱ t❤❡✐r ❝❤❛♣t❡rs✱ ❛♥❞ t❤❡ ✇❛② t❤❡ ♣r❡s❡♥t ❜♦♦❦ ✭♣❛r❡♥t❤❡t✐❝❛❧❧②✮ r❡❢❡r❡♥❝❡s t❤❡♠✳ 

✶ P❈✲✶ ✶ P❈✲✷ ✶ P❈✲✸ ✶ P❈✲✹ ✶ P❈✲✺



✷ ❉❈✲✶ ✷ ❉❈✲✷ ✷ ❉❈✲✸ ✷ ❉❈✲✹ ✷ ❉❈✲✺ ✷ ❉❈✲✻

❈❛❧❝✉❧✉s ■❧❧✉str❛t❡❞✳ ❱♦❧✉♠❡ ✹✿ ❈❛❧❝✉❧✉s ✐♥ ❍✐❣❤❡r ❉✐♠❡♥s✐♦♥s

❋✉♥❝t✐♦♥s ✐♥ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡s P❛r❛♠❡tr✐❝ ❝✉r✈❡s ❋✉♥❝t✐♦♥s ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s ❚❤❡ ❣r❛❞✐❡♥t ❚❤❡ ✐♥t❡❣r❛❧ ❱❡❝t♦r ✜❡❧❞s 

✺ ❉❊✲✶ ✺ ❉❊✲✷ ✺ ❉❊✲✸ ✺ ❉❊✲✹ ✺ ❉❊✲✺

❈❛❧❝✉❧✉s ■❧❧✉str❛t❡❞✳ ❱♦❧✉♠❡ ✸✿ ■♥t❡❣r❛❧ ❈❛❧❝✉❧✉s

❚❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧ ■♥t❡❣r❛t✐♦♥ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ✐♥t❡❣r❛❧ ❝❛❧❝✉❧✉s ❙❡✈❡r❛❧ ✈❛r✐❛❜❧❡s ❙❡r✐❡s 

✹ ❍❉✲✶ ✹ ❍❉✲✷ ✹ ❍❉✲✸ ✹ ❍❉✲✹ ✹ ❍❉✲✺ ✹ ❍❉✲✻

❈❛❧❝✉❧✉s ■❧❧✉str❛t❡❞✳ ❱♦❧✉♠❡ ✷✿ ❉✐✛❡r❡♥t✐❛❧ ❈❛❧❝✉❧✉s

▲✐♠✐ts ♦❢ s❡q✉❡♥❝❡s ▲✐♠✐ts ❛♥❞ ❝♦♥t✐♥✉✐t② ❚❤❡ ❞❡r✐✈❛t✐✈❡ ❉✐✛❡r❡♥t✐❛t✐♦♥ ❚❤❡ ♠❛✐♥ t❤❡♦r❡♠s ♦❢ ❞✐✛❡r❡♥t✐❛❧ ❝❛❧❝✉❧✉s ❲❤❛t ✇❡ ❝❛♥ ❞♦ ✇✐t❤ ❝❛❧❝✉❧✉s 

✸ ■❈✲✶ ✸ ■❈✲✷ ✸ ■❈✲✸ ✸ ■❈✲✹ ✸ ■❈✲✺

❈❛❧❝✉❧✉s ■❧❧✉str❛t❡❞✳ ❱♦❧✉♠❡ ✶✿ Pr❡❝❛❧❝✉❧✉s

❈❛❧❝✉❧✉s ♦❢ s❡q✉❡♥❝❡s ❙❡ts ❛♥❞ ❢✉♥❝t✐♦♥s ❈♦♠♣♦s✐t✐♦♥s ♦❢ ❢✉♥❝t✐♦♥s ❈❧❛ss❡s ♦❢ ❢✉♥❝t✐♦♥s ❆❧❣❡❜r❛ ❛♥❞ ❣❡♦♠❡tr②

❈❛❧❝✉❧✉s ■❧❧✉str❛t❡❞✳ ❱♦❧✉♠❡ ✺✿ ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s

❖r❞✐♥❛r② ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ❱❡❝t♦r ❛♥❞ ❝♦♠♣❧❡① ✈❛r✐❛❜❧❡s ❙②st❡♠s ♦❢ ❖❉❊s ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ❖❉❊s P❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s

❚❤❡ ✈♦❧✉♠❡s ❝❛♥ ❜❡ r❡❛❞ ✐♥❞❡♣❡♥❞❡♥t❧②✳

❚❤❡ ✈♦❧✉♠❡ ❛♥❞ ❝❤❛♣t❡r r❡❢❡r❡♥❝❡s ❢♦r ❈❛❧❝✉❧✉s ■❧❧✉str❛t❡❞

✽

❆ ♣♦ss✐❜❧❡ s❡q✉❡♥❝❡ ♦❢ ❝❤❛♣t❡rs ✐s ♣r❡s❡♥t❡❞ ❜❡❧♦✇✳ ❆♥ ❛rr♦✇ ❢r♦♠ ❆ t♦ ❇ ♠❡❛♥s t❤❛t ❝❤❛♣t❡r ❇ s❤♦✉❧❞♥✬t ❜❡ r❡❛❞ ❜❡❢♦r❡ ❝❤❛♣t❡r ❆✳

❆❜♦✉t t❤❡ ❛✉t❤♦r

❆❜♦✉t t❤❡ ❛✉t❤♦r P❡t❡r ❙❛✈❡❧✐❡✈ ✐s ❛ ♣r♦❢❡ss♦r ♦❢ ♠❛t❤❡♠❛t✐❝s ❛t ▼❛rs❤❛❧❧ ❯♥✐✈❡rs✐t②✱ ❍✉♥t✲ ✐♥❣t♦♥✱ ❲❡st ❱✐r❣✐♥✐❛✱ ❯❙❆✳ ❆❢t❡r ❛ P❤✳❉✳ ❢r♦♠ t❤❡ ❯♥✐✈❡rs✐t② ♦❢ ■❧❧✐♥♦✐s ❛t ❯r❜❛♥❛✲❈❤❛♠♣❛✐❣♥✱ ❤❡ ❞❡✈♦t❡❞ t❤❡ ♥❡①t ✷✵ ②❡❛rs t♦ t❡❛❝❤✐♥❣ ♠❛t❤❡♠❛t✐❝s✳ P❡t❡r ✐s t❤❡ ❛✉t❤♦r ♦❢ ❛ ❣r❛❞✉❛t❡ t❡①t❜♦♦❦ ❚♦♣♦❧♦❣② ■❧❧✉str❛t❡❞ ♣✉❜❧✐s❤❡❞ ✐♥ ✷✵✶✻✳ ❍❡ ❤❛s ❛❧s♦ ❜❡❡♥ ✐♥✈♦❧✈❡❞ ✐♥ r❡s❡❛r❝❤ ✐♥ ❛❧❣❡❜r❛✐❝ t♦♣♦❧♦❣② ❛♥❞ s❡✈❡r❛❧ ♦t❤❡r ✜❡❧❞s✳ ❍✐s ♥♦♥✲❛❝❛❞❡♠✐❝ ♣r♦❥❡❝ts ❤❛✈❡ ❜❡❡♥✿ ❞✐❣✐t❛❧ ✐♠❛❣❡ ❛♥❛❧②s✐s✱ ❛✉t♦♠❛t❡❞ ✜♥❣❡r♣r✐♥t ✐❞❡♥t✐✜❝❛t✐♦♥✱ ❛♥❞ ✐♠❛❣❡ ♠❛t❝❤✐♥❣ ❢♦r ♠✐s✲ s✐❧❡ ♥❛✈✐❣❛t✐♦♥✴❣✉✐❞❛♥❝❡✳

✾

❈♦♥t❡♥ts Pr❡❢❛❝❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ 

❈❤❛♣t❡r ✶✿ ▲✐♠✐ts ♦❢ s❡q✉❡♥❝❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✶ ❲❤❛t ✐s ❝❛❧❝✉❧✉s ❛❜♦✉t❄ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷ ■♥✜♥✐t❡ s❡q✉❡♥❝❡s ❛♥❞ t❤❡✐r ❧♦♥❣✲t❡r♠ tr❡♥❞s ✳ ✳ ✳ ✳ ✳ ✶✳✸ ❚❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❧✐♠✐t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✹ ▲✐♠✐ts ✉♥❞❡r ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✺ ❈❛♥ ✇❡ ❛❞❞ ✐♥✜♥✐t✐❡s❄ ❙✉❜tr❛❝t❄ ❉✐✈✐❞❡❄ ▼✉❧t✐♣❧②❄ ✳ ✶✳✻ ▼♦r❡ ♣r♦♣❡rt✐❡s ♦❢ ❧✐♠✐ts ♦❢ s❡q✉❡♥❝❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✼ ❚❤❡♦r❡♠s ♦❢ ❆♥❛❧②s✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✽ ❈♦♠♣♦s✐t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✾ ◆✉♠❜❡rs ❛r❡ ❧✐♠✐ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✵ ❚❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✶ ❚❤❡ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳



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✶✷ ✷✶ ✷✽ ✹✺ ✺✺ ✻✷ ✼✶ ✼✾ ✽✺ ✾✵ ✶✵✵

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✶✵✾ ✶✶✺ ✶✷✼ ✶✸✽ ✶✺✶ ✶✺✼ ✶✻✺ ✶✼✶ ✶✼✺ ✶✽✻ ✶✾✹ ✷✵✵ ✷✵✸

❈❤❛♣t❡r ✸✿ ❚❤❡ ❞❡r✐✈❛t✐✈❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶✶ ✸✳✶ ❚❤❡ ❚❛♥❣❡♥t Pr♦❜❧❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✷ ❚❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ ❛ s❡q✉❡♥❝❡ ❛♥❞ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ✸✳✸ ❚❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡✿ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✹ ❚❤❡ ❧✐♠✐t ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✿ t❤❡ ❞❡r✐✈❛t✐✈❡ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✺ ❚❤❡ ❞❡r✐✈❛t✐✈❡ ✐s t❤❡ ✐♥st❛♥t❛♥❡♦✉s r❛t❡ ♦❢ ❝❤❛♥❣❡ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✻ ❚❤❡ ❡①✐st❡♥❝❡ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡✿ ❞✐✛❡r❡♥t✐❛❜✐❧✐t② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✼ ❚❤❡ ❞❡r✐✈❛t✐✈❡ ❛s ❛ ❢✉♥❝t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✽ ❇❛s✐❝ ❞✐✛❡r❡♥t✐❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✾ ❇❛s✐❝ ❞✐✛❡r❡♥t✐❛t✐♦♥✱ ❝♦♥t✐♥✉❡❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✶✵ ❋r❡❡ ❢❛❧❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳



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❈❤❛♣t❡r ✷✿ ▲✐♠✐ts ❛♥❞ ❝♦♥t✐♥✉✐t② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✾ ✷✳✶ ❋✉♥❝t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷ ❈♦♥t✐♥✉✐t② ❛♥❞ ❞✐s❝♦♥t✐♥✉✐t② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✸ ▲✐♠✐ts ♦❢ ❢✉♥❝t✐♦♥s✿ s♠❛❧❧ s❝❛❧❡ tr❡♥❞s ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✹ ▲✐♠✐ts ❛♥❞ ❝♦♥t✐♥✉✐t② ✉♥❞❡r ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s ✷✳✺ ❚❤❡ ❡①♣♦♥❡♥t✐❛❧ ❛♥❞ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s ✳ ✳ ✳ ✷✳✻ ▲✐♠✐ts ❛♥❞ ❝♦♥t✐♥✉✐t② ✉♥❞❡r ❝♦♠♣♦s✐t✐♦♥s ✳ ✳ ✳ ✳ ✷✳✼ ❈♦♥t✐♥✉✐t② ♦❢ t❤❡ ✐♥✈❡rs❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✽ ❈♦♠♣❛r✐s♦♥ ♦❢ ❧✐♠✐ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✾ ●❧♦❜❛❧ ♣r♦♣❡rt✐❡s ♦❢ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✷✳✶✵ ▲❛r❣❡✲s❝❛❧❡ ❜❡❤❛✈✐♦r ❛♥❞ ❛s②♠♣t♦t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✶ ▲✐♠✐ts ❛♥❞ ✐♥✜♥✐t②✿ ❝♦♠♣✉t❛t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✷ ❈♦♥t✐♥✉✐t② ❛♥❞ ❛❝❝✉r❛❝② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✸ ❚❤❡ ε✲δ ❞❡✜♥✐t✐♦♥ ♦❢ ❧✐♠✐t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳



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✷✶✶ ✷✶✾ ✷✷✾ ✷✹✵ ✷✹✽ ✷✺✸ ✷✺✾ ✷✼✹ ✷✽✶ ✷✽✾

❈❤❛♣t❡r ✹✿ ❉✐✛❡r❡♥t✐❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾✼ ✹✳✶ ❉✐✛❡r❡♥t✐❛t✐♦♥ ♦✈❡r ❛❞❞✐t✐♦♥ ❛♥❞ ❝♦♥st❛♥t ♠✉❧t✐♣❧❡✿ ✹✳✷ ❈❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ❛♥❞ t❤❡ ❞❡r✐✈❛t✐✈❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✳✸ ❉✐✛❡r❡♥t✐❛t✐♦♥ ♦✈❡r ❝♦♠♣♦s✐t✐♦♥s✿ t❤❡ ❈❤❛✐♥ ❘✉❧❡ ✳ ✹✳✹ ❉✐✛❡r❡♥t✐❛t✐♦♥ ♦✈❡r ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❛♥❞ ❞✐✈✐s✐♦♥ ✳ ✳ ✳

❧✐♥❡❛r✐t② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

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✷✾✼ ✸✵✹ ✸✶✵ ✸✶✾

❈♦♥t❡♥ts ✹✳✺ ❚❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢ t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ✹✳✻ ❘❡♣❡❛t❡❞ ❞✐✛❡r❡♥t✐❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✳✼ ❍♦✇ t♦ ❞✐✛❡r❡♥t✐❛t❡ r❡❧❛t✐♦♥s✿ ✐♠♣❧✐❝✐t❧② ✹✳✽ ❘❡❧❛t❡❞ r❛t❡s✿ r❛❞❛r ❣✉♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✳✾ ❚❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ✐♥✈❡rs❡ ❢✉♥❝t✐♦♥ ✳ ✳ ✹✳✶✵ ❘❡✈❡rs✐♥❣ ❞✐✛❡r❡♥t✐❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✳✶✶ ❙❤♦♦t✐♥❣ ❛ ❝❛♥♥♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ 

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✸✷✻ ✸✸✶ ✸✸✼ ✸✹✻ ✸✺✵ ✸✺✼ ✸✻✵

❈❤❛♣t❡r ✺✿ ❚❤❡ ♠❛✐♥ t❤❡♦r❡♠s ♦❢ ❞✐✛❡r❡♥t✐❛❧ ❝❛❧❝✉❧✉s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼✵ ✺✳✶ ▼♦♥♦t♦♥✐❝✐t② ❛♥❞ ❡①tr❡♠❡ ♣♦✐♥ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✳✷ ❖♣t✐♠✐③❛t✐♦♥ ♦❢ ❢✉♥❝t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✳✸ ❲❤❛t t❤❡ ❞❡r✐✈❛t✐✈❡ s❛②s ❛❜♦✉t t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✿ ✺✳✹ ▼♦♥♦t♦♥✐❝✐t② ❛♥❞ t❤❡ s✐❣♥ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ ✳ ✳ ✳ ✳ ✳ ✳ ✺✳✺ ❈♦♥❝❛✈✐t② ❛♥❞ t❤❡ s✐❣♥ ♦❢ t❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡ ✳ ✳ ✳ ✳ ✺✳✻ ❉❡r✐✈❛t✐✈❡s ❛♥❞ ❡①tr❡♠❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✳✼ ❆♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥✿ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ✇❤❛t ❢✉♥❝t✐♦♥❄ ✳ ✺✳✽ ❆♥t✐❞❡r✐✈❛t✐✈❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳



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✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❚❤❡ ▼❡❛♥ ❱❛❧✉❡ ❚❤❡♦r❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

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✸✼✵ ✸✽✻ ✸✾✶ ✸✾✽ ✹✵✸ ✹✶✺ ✹✷✷ ✹✷✻

❈❤❛♣t❡r ✻✿ ❲❤❛t ✇❡ ❝❛♥ ❞♦ ✇✐t❤ ❝❛❧❝✉❧✉s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹✶ ✻✳✶ ▼❛❣♥✐t✉❞❡s ♦❢ ❢✉♥❝t✐♦♥s❀ ▲✬❍♦♣✐t❛❧✬s ❘✉❧❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✷ ▲✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✸ ❚❤❡ ❛❝❝✉r❛❝② ♦❢ t❤❡ ❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✹ ❙♦❧✈✐♥❣ ❡q✉❛t✐♦♥s ♥✉♠❡r✐❝❛❧❧②✿ ❜✐s❡❝t✐♦♥ ❛♥❞ ◆❡✇t♦♥✬s ♠❡t❤♦❞ ✳ ✻✳✺ P❛rt✐❝❧❡ ✐♥ ❛ ✢♦✇ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✻ ❉✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✼ ▼♦t✐♦♥ ✉♥❞❡r ❢♦r❝❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✽ ❖♣t✐♠✐③❛t✐♦♥ ❡①❛♠♣❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✾ ❋✉♥❝t✐♦♥s ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

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✹✹✶ ✹✺✶ ✹✻✷ ✹✻✼ ✹✼✶ ✹✼✽ ✹✽✸ ✹✽✾ ✹✾✺

❊①❡r❝✐s❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵✺ ✶ ❊①❡r❝✐s❡s✿ ❙❡ts✱ ❧♦❣✐❝✱ ❢✉♥❝t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ❊①❡r❝✐s❡s✿ ❇❛❝❦❣r♦✉♥❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ❊①❡r❝✐s❡s✿ ❙❡q✉❡♥❝❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ❊①❡r❝✐s❡s✿ ❘❛t❡s ♦❢ ❝❤❛♥❣❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ❊①❡r❝✐s❡s✿ ▲✐♠✐ts ❛♥❞ ❝♦♥t✐♥✉✐t② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ❊①❡r❝✐s❡s✿ ❉❡r✐✈❛t✐✈❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ❊①❡r❝✐s❡s✿ ❋❡❛t✉r❡s ♦❢ ❣r❛♣❤s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ❊①❡r❝✐s❡s✿ ▲✐♥❡❛r✐③❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ❊①❡r❝✐s❡s✿ ▼♦❞❡❧s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ❊①❡r❝✐s❡s✿ ■♥❢♦r♠❛t✐♦♥ ❢r♦♠ t❤❡ ❞❡r✐✈❛t✐✈❡s ✳ ✳ ✶✶ ❊①❡r❝✐s❡s✿ ❈♦♠♣✉t✐♥❣ ❞❡r✐✈❛t✐✈❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ❊①❡r❝✐s❡s✿ ❖♣t✐♠✐③❛t✐♦♥ ❛♥❞ ♦t❤❡r ❛♣♣❧✐❝❛t✐♦♥s

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✺✵✺ ✺✵✾ ✺✶✸ ✺✶✹ ✺✶✼ ✺✷✵ ✺✷✸ ✺✷✾ ✺✸✵ ✺✸✸ ✺✸✺ ✺✸✼

■♥❞❡① ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹✷

❈❤❛♣t❡r ✶✿ ▲✐♠✐ts ♦❢ s❡q✉❡♥❝❡s

❈♦♥t❡♥ts

✶✳✶ ❲❤❛t ✐s ❝❛❧❝✉❧✉s ❛❜♦✉t❄ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷ ■♥✜♥✐t❡ s❡q✉❡♥❝❡s ❛♥❞ t❤❡✐r ❧♦♥❣✲t❡r♠ tr❡♥❞s ✳ ✳ ✳ ✳ ✶✳✸ ❚❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❧✐♠✐t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✹ ▲✐♠✐ts ✉♥❞❡r ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✺ ❈❛♥ ✇❡ ❛❞❞ ✐♥✜♥✐t✐❡s❄ ❙✉❜tr❛❝t❄ ❉✐✈✐❞❡❄ ▼✉❧t✐♣❧②❄ ✶✳✻ ▼♦r❡ ♣r♦♣❡rt✐❡s ♦❢ ❧✐♠✐ts ♦❢ s❡q✉❡♥❝❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✼ ❚❤❡♦r❡♠s ♦❢ ❆♥❛❧②s✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✽ ❈♦♠♣♦s✐t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✾ ◆✉♠❜❡rs ❛r❡ ❧✐♠✐ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✵ ❚❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✶ ❚❤❡ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

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✳ ✶✷ ✳ ✷✶ ✳ ✷✽ ✳ ✹✺ ✳ ✺✺ ✳ ✻✷ ✳ ✼✶ ✳ ✼✾ ✳ ✽✺ ✳ ✾✵ ✳ ✶✵✵

✶✳✶✳ ❲❤❛t ✐s ❝❛❧❝✉❧✉s ❛❜♦✉t❄

❖♥❡ ♦❢ t❤❡ ♠❛✐♥ ❡♥tr② ✇❛②s t♦ ❝❛❧❝✉❧✉s ✐s t❤❡ st✉❞② ♦❢ ♠♦t✐♦♥✳ ❲❡ ♣r❡s❡♥t t❤❡ ✐❞❡❛ ♦❢ ❝❛❧❝✉❧✉s ✐♥ t❤❡s❡ t✇♦ r❡❧❛t❡❞ ♣✐❝t✉r❡s✳ ❋✐rst✱ ✇❡ ❞❡r✐✈❡ t❤❡ s♣❡❡❞ ❢r♦♠ t❤❡ ❞✐st❛♥❝❡ t❤❛t ✇❡ ❤❛✈❡ ❝♦✈❡r❡❞✿

❇❡②♦♥❞ t❤✐s ❝♦♥❝❡✐✈❛❜❧❡ s✐t✉❛t✐♦♥✱ t❤✐s ❢♦r♠✉❧❛ ✐s t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ s♣❡❡❞✳ ❖♥ t❤❡ ✢✐♣ s✐❞❡✱ ✇❡ ❞❡r✐✈❡ t❤❡ ❞✐st❛♥❝❡ ✇❡ ❤❛✈❡ ❝♦✈❡r❡❞ ❢r♦♠ t❤❡ ❦♥♦✇♥ ✈❡❧♦❝✐t②✿

✶✳✶✳ ❲❤❛t ✐s ❝❛❧❝✉❧✉s ❛❜♦✉t❄

✶✸

❚❤❡ t✇♦ ♣r♦❜❧❡♠s ❛r❡ s♦❧✈❡❞✱ r❡s♣❡❝t✐✈❡❧②✱ ✇✐t❤ t❤❡ ❤❡❧♣ ♦❢ t❤❡s❡ t✇♦ ✈❡rs✐♦♥s ♦❢ t❤❡ s❛♠❡ ❡❧❡♠❡♥t❛r② s❝❤♦♦❧ ❢♦r♠✉❧❛✿ s♣❡❡❞ = ❞✐st❛♥❝❡ / t✐♠❡

❛♥❞

❞✐st❛♥❝❡ = s♣❡❡❞ × t✐♠❡

❲❡ s♦❧✈❡ t❤❡ ❡q✉❛t✐♦♥ ❢♦r t❤❡ ❞✐st❛♥❝❡ ♦r ❢♦r t❤❡ s♣❡❡❞ ❞❡♣❡♥❞✐♥❣ ♦♥ ✇❤❛t ✐s ❦♥♦✇♥ ❛♥❞ ✇❤❛t ✐s ✉♥❦♥♦✇♥✳ ❲❤❛t t❛❦❡s t❤✐s ✐❞❡❛ ❜❡②♦♥❞ ❡❧❡♠❡♥t❛r② s❝❤♦♦❧ ✐s t❤❡ ♣♦ss✐❜✐❧✐t② t❤❛t ✈❡❧♦❝✐t② ✈❛r✐❡s ♦✈❡r t✐♠❡✳ ❚❤❡ s✐♠♣❧❡st ❝❛s❡ ✐s ✇❤❡♥ ✐t ✈❛r✐❡s ✐♥❝r❡♠❡♥t❛❧❧②✳ ▲❡t✬s ❜❡ ♠♦r❡ s♣❡❝✐✜❝✳ ❲❡ ✇✐❧❧ ❢❛❝❡ t❤❡ t✇♦ s✐t✉❛t✐♦♥s ❛❜♦✈❡ ❜✉t ✇✐t❤ ♠♦r❡ ❞❛t❛ ❝♦❧❧❡❝t❡❞ ❛♥❞ ♠♦r❡ ✐♥❢♦r♠❛t✐♦♥ ❞❡r✐✈❡❞ ❢r♦♠ ✐t✳ ❋✐rst✱ ✐♠❛❣✐♥❡ t❤❛t ♦✉r s♣❡❡❞♦♠❡t❡r ✐s ❜r♦❦❡♥✳ ❲❤❛t ❞♦ ✇❡ ❞♦ ✐❢ ✇❡ ✇❛♥t t♦ ❡st✐♠❛t❡ ❤♦✇ ❢❛st ✇❡ ❛r❡ ❞r✐✈✐♥❣ ❞✉r✐♥❣ ♦✉r tr✐♣❄ ❲❡ ❧♦♦❦ ❛t t❤❡ ♦❞♦♠❡t❡r s❡✈❡r❛❧ t✐♠❡s ✕ s❛②✱ ❡✈❡r② ❤♦✉r ♦♥ t❤❡ ❤♦✉r ✕ ❞✉r✐♥❣ t❤❡ tr✐♣ ❛♥❞ r❡❝♦r❞ t❤❡ ♠✐❧❡❛❣❡ ♦♥ ❛ ♣✐❡❝❡ ♦❢ ♣❛♣❡r✳ ❚❤❡ ❧✐st ♦❢ ♦✉r ❝♦♥s❡❝✉t✐✈❡ ❧♦❝❛t✐♦♥s ♠✐❣❤t ❧♦♦❦ ❧✐❦❡ t❤✐s✿

• ✐♥✐t✐❛❧ r❡❛❞✐♥❣✿ 10, 000 ♠✐❧❡s

• ❛❢t❡r t❤❡ ✜rst ❤♦✉r✿ 10, 055 ♠✐❧❡s

• ❛❢t❡r t❤❡ s❡❝♦♥❞ ❤♦✉r✿ 10, 095 ♠✐❧❡s • ❛❢t❡r t❤❡ t❤✐r❞ ❤♦✉r✿ 10, 155 ♠✐❧❡s • ❡t❝✳

❲❡ ❝❛♥ ♣❧♦t ✕ ❛s ❛♥ ✐❧❧✉str❛t✐♦♥ ✕ t❤❡ ❧♦❝❛t✐♦♥s ❛❣❛✐♥st t✐♠❡✿

❇✉t ✇❤❛t ❞♦ ✇❡ ❦♥♦✇ ❛❜♦✉t ✇❤❛t t❤❡ s♣❡❡❞ ❤❛s ❜❡❡♥❄ ❲❡ ✇r✐t❡ ❛ q✉✐❝❦ ❢♦r♠✉❧❛✿ s♣❡❡❞ =

❞✐st❛♥❝❡ ❝✉rr❡♥t ❧♦❝❛t✐♦♥ − ❧❛st ❧♦❝❛t✐♦♥ = t✐♠❡ 1

❚❤❡ t✐♠❡ ✐♥t❡r✈❛❧ ✇❛s ❝❤♦s❡♥ t♦ ❜❡ 1 ❤♦✉r✱ s♦ ❛❧❧ ✇❡ ♥❡❡❞ ✐s t♦ ✜♥❞ t❤❡ ❞✐st❛♥❝❡ ❝♦✈❡r❡❞ ❞✉r✐♥❣ ❡❛❝❤ ♦❢ t❤❡s❡ ♦♥❡✲❤♦✉r ♣❡r✐♦❞s✱ ❜② s✉❜tr❛❝t✐♦♥ ✿

• ❞✐st❛♥❝❡ ❝♦✈❡r❡❞ ❞✉r✐♥❣ t❤❡ ✜rst ❤♦✉r✿ 10, 055 − 10, 000 = 55 ♠✐❧❡s❀ s♣❡❡❞ 55 ♠✐❧❡s ❛♥ ❤♦✉r

• ❞✐st❛♥❝❡ ❝♦✈❡r❡❞ ❞✉r✐♥❣ t❤❡ s❡❝♦♥❞ ❤♦✉r✿ 10, 095 − 10, 055 = 40 ♠✐❧❡s❀ s♣❡❡❞ 40 ♠✐❧❡s ❛♥ ❤♦✉r

• ❞✐st❛♥❝❡ ❝♦✈❡r❡❞ ❞✉r✐♥❣ t❤❡ t❤✐r❞ ❤♦✉r✿ 10, 155 − 10, 095 = 60 ♠✐❧❡s❀ s♣❡❡❞ 60 ♠✐❧❡s ❛♥ ❤♦✉r

✶✳✶✳ ❲❤❛t ✐s ❝❛❧❝✉❧✉s ❛❜♦✉t❄

✶✹

• ❡t❝✳

❲❡ s❡❡ ❜❡❧♦✇ ❤♦✇ t❤❡s❡ ♥❡✇ ♥✉♠❜❡rs ❛♣♣❡❛r ❛s t❤❡ ❜❧♦❝❦s t❤❛t ♠❛❦❡ ✉♣ ❡❛❝❤ st❡♣ ♦❢ ♦✉r ❧❛st ♣❧♦t ✭t♦♣✮✿

❲❡ t❤❡♥ ❧♦✇❡r t❤❡s❡ ❜❧♦❝❦s t♦ t❤❡ ❜♦tt♦♠ t♦ ❝r❡❛t❡ ❛ ♥❡✇ ♣❧♦t ✭❜♦tt♦♠✮✳ ❆s ②♦✉ ❝❛♥ s❡❡✱ ✇❡ ✐❧❧✉str❛t❡ t❤❡ ♥❡✇ ❞❛t❛ ✐♥ s✉❝❤ ❛ ✇❛② ❛s t♦ s✉❣❣❡st t❤❛t t❤❡ s♣❡❡❞ r❡♠❛✐♥s ❝♦♥st❛♥t ❞✉r✐♥❣ ❡❛❝❤ ♦❢ t❤❡s❡ ❤♦✉r✲❧♦♥❣ ♣❡r✐♦❞s✳ ❚❤❡ ♣r♦❜❧❡♠ ✐s s♦❧✈❡❞✦ ❲❡ ❤❛✈❡ ❡st❛❜❧✐s❤❡❞ t❤❛t t❤❡ s♣❡❡❞ ❤❛s ❜❡❡♥ ✕ r♦✉❣❤❧② ✕ 55✱ 40✱ ❛♥❞ 60 ♠✐❧❡s ❛♥ ❤♦✉r ❞✉r✐♥❣ t❤♦s❡ t❤r❡❡ t✐♠❡ ✐♥t❡r✈❛❧s✱ r❡s♣❡❝t✐✈❡❧②✳ ◆♦✇ ♦♥ t❤❡ ✢✐♣ s✐❞❡✿ ■♠❛❣✐♥❡ t❤✐s t✐♠❡ t❤❛t ✐t ✐s t❤❡ ♦❞♦♠❡t❡r t❤❛t ✐s ❜r♦❦❡♥✳ ■❢ ✇❡ ✇❛♥t t♦ ❡st✐♠❛t❡ ❤♦✇ ❢❛r ✇❡ ✇✐❧❧ ❤❛✈❡ ❣♦♥❡✱ ✇❡ s❤♦✉❧❞ ❧♦♦❦ ❛t t❤❡ s♣❡❡❞♦♠❡t❡r s❡✈❡r❛❧ t✐♠❡s ✕ s❛②✱ ❡✈❡r② ❤♦✉r ✕ ❞✉r✐♥❣ t❤❡ tr✐♣ ❛♥❞ r❡❝♦r❞ ✐ts r❡❛❞✐♥❣s ♦♥ ❛ ♣✐❡❝❡ ♦❢ ♣❛♣❡r✳ ❚❤❡ r❡s✉❧t ♠❛② ❧♦♦❦ ❧✐❦❡ t❤✐s✿

• ❞✉r✐♥❣ t❤❡ ✜rst ❤♦✉r✿ 35 ♠✐❧❡s ❛♥ ❤♦✉r

• ❞✉r✐♥❣ t❤❡ s❡❝♦♥❞ ❤♦✉r✿ 65 ♠✐❧❡s ❛♥ ❤♦✉r

• ❞✉r✐♥❣ t❤❡ t❤✐r❞ ❤♦✉r✿ 50 ♠✐❧❡s ❛♥ ❤♦✉r • ❡t❝✳

▲❡t✬s ♣❧♦t ♦✉r s♣❡❡❞ ❛❣❛✐♥st t✐♠❡ t♦ ✈✐s✉❛❧✐③❡ ✇❤❛t ❤❛s ❤❛♣♣❡♥❡❞✿

❖♥❝❡ ❛❣❛✐♥✱ ✇❡ ✐❧❧✉str❛t❡ t❤❡ ❞❛t❛ ✐♥ s✉❝❤ ❛ ✇❛② ❛s t♦ s✉❣❣❡st t❤❛t t❤❡ s♣❡❡❞ r❡♠❛✐♥s ❝♦♥st❛♥t ❞✉r✐♥❣ ❡❛❝❤ ♦❢ t❤❡s❡ ❤♦✉r✲❧♦♥❣ ♣❡r✐♦❞s✳ ◆♦✇✱ ✇❤❛t ❞♦❡s t❤✐s t❡❧❧ ✉s ❛❜♦✉t ♦✉r ❧♦❝❛t✐♦♥❄ ❲❡ ✇r✐t❡ ❛ q✉✐❝❦ ❢♦r♠✉❧❛✿ ❞✐st❛♥❝❡ = s♣❡❡❞ × t✐♠❡ = s♣❡❡❞ × 1 ■♥ ❝♦♥tr❛st t♦ t❤❡ ❢♦r♠❡r ♣r♦❜❧❡♠✱ ✇❡ ♥❡❡❞ ❛♥♦t❤❡r ❜✐t ♦❢ ✐♥❢♦r♠❛t✐♦♥✳ ❲❡ ♠✉st ❦♥♦✇ t❤❡ st❛rt✐♥❣ ♣♦✐♥t ♦❢ ♦✉r tr✐♣✱ s❛②✱ t❤❡ 100✲♠✐❧❡ ♠❛r❦✳ ❚❤❡ t✐♠❡ ✐♥t❡r✈❛❧ ✇❛s ❝❤♦s❡♥ t♦ ❜❡ 1 ❤♦✉r s♦ t❤❛t ✇❡ ♥❡❡❞ ♦♥❧② t♦ ❛❞❞✱ ❛♥❞ ❦❡❡♣ ❛❞❞✐♥❣✱ t❤❡ s♣❡❡❞ ❛t ✇❤✐❝❤ ✕ ✇❡ ❛ss✉♠❡ ✕ ✇❡ ❞r♦✈❡ ❞✉r✐♥❣ ❡❛❝❤ ♦❢ t❤❡s❡ ♦♥❡✲❤♦✉r ♣❡r✐♦❞s✿

• t❤❡ ❧♦❝❛t✐♦♥ ❛❢t❡r t❤❡ ✜rst ❤♦✉r✿ 100 + 35 = 135✲♠✐❧❡ ♠❛r❦

• t❤❡ ❧♦❝❛t✐♦♥ ❛❢t❡r t❤❡ t✇♦ ❤♦✉rs✿ 135 + 65 = 200✲♠✐❧❡ ♠❛r❦

• t❤❡ ❧♦❝❛t✐♦♥ ❛❢t❡r t❤❡ t❤r❡❡ ❤♦✉rs✿ 200 + 50 = 250✲♠✐❧❡ ♠❛r❦ • ❡t❝✳

■♥ ♦r❞❡r t♦ ✐❧❧✉str❛t❡ t❤✐s ❛❧❣❡❜r❛✱ ✇❡ ♣❧♦t t❤❡ s♣❡❡❞s ❛s t❤❡s❡ ❜❧♦❝❦s ✭t♦♣✮✿

✶✳✶✳

❲❤❛t ✐s ❝❛❧❝✉❧✉s ❛❜♦✉t❄

✶✺

❚❤❡♥ ✇❡ ✉s❡ t❤❡s❡ ❜❧♦❝❦s t♦ ♠❛❦❡ t❤❡ ❝♦♥s❡❝✉t✐✈❡ st❡♣s ♦❢ t❤❡ st❛✐r❝❛s❡ t♦ s❤♦✇ ❤♦✇ ❤✐❣❤ ✇❡ ❤❛✈❡ t♦ ❝❧✐♠❜ ✭❜♦tt♦♠✮✳ ❚❤❡ ♣r♦❜❧❡♠ ✐s s♦❧✈❡❞✦ ❲❡ ❤❛✈❡ ❡st❛❜❧✐s❤❡❞ t❤❛t ✇❡ ❤❛✈❡ ♣r♦❣r❡ss❡❞ t❤r♦✉❣❤ t❤❡ r♦✉❣❤❧②

250✲♠✐❧❡

135✲✱ 200✲✱

❛♥❞

♠❛r❦s ❞✉r✐♥❣ t❤✐s t✐♠❡✳

❲❡ ♥❡①t ❝♦♥s✐❞❡r ♠♦r❡ ❝♦♠♣❧❡① ❡①❛♠♣❧❡s ♦❢ t❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ ❧♦❝❛t✐♦♥ ❛♥❞ ✈❡❧♦❝✐t②✳ ❋✐rst✱

t♦ ✈❡❧♦❝✐t②✳ ❙✉♣♣♦s❡ t❤❛t t❤✐s t✐♠❡ ✇❡ ❤❛✈❡ ❛

s❡q✉❡♥❝❡

♦❢ ♠♦r❡ t❤❛♥

❢r♦♠ ❧♦❝❛t✐♦♥

30 ❞❛t❛ ♣♦✐♥ts ✭♠♦r❡ ✐s ✐♥❞✐❝❛t❡❞ ❜② ✏✳✳✳✑✮❀

t❤❡② ❛r❡

t❤❡ ❧♦❝❛t✐♦♥s ♦❢ ❛ ♠♦✈✐♥❣ ♦❜❥❡❝t r❡❝♦r❞❡❞ ❡✈❡r② ♠✐♥✉t❡✿ t✐♠❡

♠✐♥✉t❡s

❧♦❝❛t✐♦♥

♠✐❧❡s

0 1 2 3 4 5 6 7 8 9 10 ... 0.00 0.10 0.20 0.30 0.39 0.48 0.56 0.64 0.72 0.78 0.84 ...

❚❤✐s ❞❛t❛ ✐s ❛❧s♦ s❡❡♥ ✐♥ t❤❡ ✜rst t✇♦ ❝♦❧✉♠♥s ♦❢ t❤❡ s♣r❡❛❞s❤❡❡t ✭❧❡❢t✮✿

❊✈❡r② ♣❛✐r ♦❢ ♥✉♠❜❡rs ✐♥ t❤❡ t❛❜❧❡ ✐s t❤❡♥ ♣❧♦tt❡❞ ✭r✐❣❤t✮✳ ❚❤❡ ✏s❝❛tt❡r ♣❧♦t✑ t❤❛t ✐❧❧✉str❛t❡s t❤❡ ❞❛t❛ ❧♦♦❦s ❧✐❦❡ ❛

❝✉r✈❡ ✦

❲❤❛t ❤❛s ❤❛♣♣❡♥❡❞ t♦ t❤❡ ♠♦✈✐♥❣ ♦❜❥❡❝t ❝❛♥ ♥♦✇ ❜❡ r❡❛❞ ❢r♦♠ t❤❡ ❣r❛♣❤✳ ❏✉st ❛s ✐♥ t❤❡ ❧❛st ❡①❛♠♣❧❡✱ ✇❡ ❝♦♥❝❡♥tr❛t❡ ♦♥ t❤❡ ✈❡rt✐❝❛❧ ✐♥❝r❡♠❡♥t ♦❢ t❤❡ st❛✐r❝❛s❡✿

❚❤❡s❡ ❛r❡ t❤❡ r❡s✉❧ts✿ ✶✳ ❚❤❡ ♦❜❥❡❝t ✇❛s ♠♦✈✐♥❣ ✐♥ t❤❡ ♣♦s✐t✐✈❡ ❞✐r❡❝t✐♦♥✳ ✷✳ ■t ✇❛s ♠♦✈✐♥❣ ❢❛✐r❧② ❢❛st ❜✉t t❤❡♥ st❛rt❡❞ t♦ s❧♦✇ ❞♦✇♥✳ ✸✳ ■t st♦♣♣❡❞ ❢♦r ❛ ✈❡r② s❤♦rt ♣❡r✐♦❞✳ ✹✳ ❚❤❡♥ ✐t st❛rt❡❞ t♦ ♠♦✈❡ ✐♥ t❤❡ ♦♣♣♦s✐t❡ ❞✐r❡❝t✐♦♥✳ ✺✳ ❚❤❡♥ ✐t st❛rt❡❞ t♦ s♣❡❡❞ ✉♣ ✐♥ t❤❛t ❞✐r❡❝t✐♦♥✳

✶✳✶✳

❲❤❛t ✐s ❝❛❧❝✉❧✉s ❛❜♦✉t❄

✶✻

❚♦ ✉♥❞❡rst❛♥❞ ❤♦✇ ❢❛st ✇❡ ♠♦✈❡ ♦✈❡r t❤❡s❡ ♦♥❡✲♠✐♥✉t❡ ✐♥t❡r✈❛❧s✱ ✇❡ ❝♦♠♣✉t❡ t❤❡

❞✐✛❡r❡♥❝❡s

♦❢ ❧♦❝❛t✐♦♥s

❢♦r ❡❛❝❤ ♣❛✐r ♦❢ ❝♦♥s❡❝✉t✐✈❡ ❧♦❝❛t✐♦♥s✳ ❋✐rst✱ t❤❡ t❛❜❧❡✳ ❲❡ ✉s❡ t❤❡ ❞❛t❛ ❢r♦♠ t❤❡ r♦✇ ♦❢ ❧♦❝❛t✐♦♥s ❛♥❞ s✉❜tr❛❝t ❡✈❡r② t✇♦ ❝♦♥s❡❝✉t✐✈❡ ❧♦❝❛t✐♦♥s✳ ❚❤✐s ✐s ❤♦✇ t❤❡ ✜rst st❡♣ ✐s ❝❛rr✐❡❞ ♦✉t✿ t✐♠❡

♠✐♥

❧♦❝❛t✐♦♥

♠✐❧❡s

0 0.00 ց

❞✐✛❡r❡♥❝❡ ✈❡❧♦❝✐t②

♠✐❧❡s✴♠✐♥

1 0.10 ↓ 0.10 − 0.00 || 0.10

... ... ... ...

❲❡ ❝♦♠♣✉t❡ t❤✐s ❞✐✛❡r❡♥❝❡ ❢♦r ❡❛❝❤ ♣❛✐r ♦❢ ❝♦♥s❡❝✉t✐✈❡ ❧♦❝❛t✐♦♥s ❛♥❞ t❤❡♥ ♣❧❛❝❡ ✐t ✐♥ ❛ r♦✇ ❢♦r t❤❡ ✈❡❧♦❝✐t✐❡s t❤❛t ✇❡ ❛❞❞❡❞ t♦ t❤❡ ❜♦tt♦♠ ♦❢ ♦✉r t❛❜❧❡✿ t✐♠❡

♠✐♥

❧♦❝❛t✐♦♥

♠✐❧❡s

✈❡❧♦❝✐t②

♠✐❧❡s✴♠✐♥

0 0.00

1 2 3 4 5 6 7 8 9 0.10 0.20 0.30 0.39 0.48 0.56 0.64 0.72 0.78 ց ↓ ց ↓ ց ↓ ց ↓ ց ↓ ց ↓ ց ↓ ց ↓ ց ↓ 0.10 0.10 0.10 0.09 0.09 0.09 0.08 0.07 0.07

... ... ... ...

❲❡ ❤❛✈❡ ❛ ♥❡✇ s❡q✉❡♥❝❡✦ Pr❛❝t✐❝❛❧❧②✱ ✇❡✬❞ r❛t❤❡r ✉s❡ t❤❡ ❝♦♠♣✉t✐♥❣ ❝❛♣❛❜✐❧✐t✐❡s ♦❢ t❤❡ s♣r❡❛❞s❤❡❡t✳ ❊①❛♠♣❧❡ ✶✳✶✳✶✿ s♣r❡❛❞s❤❡❡t ❢♦r♠✉❧❛s

❲❡ ✉s❡ ❢♦r♠✉❧❛s t♦ ♣✉❧❧ ❞❛t❛ ❢r♦♠ ♦t❤❡r ❝❡❧❧s✳ ❚❤❡r❡ ❛r❡ t✇♦ ✇❛②s✳ ❋✐rst✱ t❤❡ ✏❛❜s♦❧✉t❡✧ r❡❢❡r❡♥❝❡✿

❂❘✷❈✸✂✷ ❆♥② ❝❡❧❧ ✇✐t❤ t❤✐s ❢♦r♠✉❧❛ ✇✐❧❧ t❛❦❡ t❤❡ ✈❛❧✉❡ ❝♦♥t❛✐♥❡❞ ✐♥ t❤❡ ❝❡❧❧ ❧♦❝❛t❡❞ ❛t r♦✇

2

❛♥❞ ❝♦❧✉♠♥

3

❛♥❞

sq✉❛r❡ ✐t✿

❙❡❝♦♥❞✱ t❤❡ ✏r❡❧❛t✐✈❡✧ r❡❢❡r❡♥❝❡✿

❂❘❬✷❪❈❬✸❪✂✷ ❆♥② ❝❡❧❧ ✇✐t❤ t❤✐s ❢♦r♠✉❧❛ ✇✐❧❧ t❛❦❡ t❤❡ ✈❛❧✉❡ ❝♦♥t❛✐♥❡❞ ✐♥ t❤❡ ❝❡❧❧ ❧♦❝❛t❡❞

2 r♦✇s ❞♦✇♥ ❛♥❞ 3 ❝♦❧✉♠♥s

r✐❣❤t ❢r♦♠ ✐t ❛♥❞ sq✉❛r❡ ✐t✿

❲❡ ❝♦♠♣✉t❡ t❤❡ ❞✐✛❡r❡♥❝❡s ❜② ♣✉❧❧✐♥❣ ❞❛t❛ ❢r♦♠ t❤❡ ❝♦❧✉♠♥ ♦❢ ❧♦❝❛t✐♦♥s ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✉❧❛✿

❂❘❈❬✲✶❪✲❘❬✲✶❪❈❬✲✶❪

✶✳✶✳ ❲❤❛t ✐s ❝❛❧❝✉❧✉s ❛❜♦✉t❄

✶✼

❍❡r❡✱ t❤❡ t✇♦ ✈❛❧✉❡s ❝♦♠❡ ❢r♦♠ t❤❡ ❧❛st ❝♦❧✉♠♥✱ ❈❬✲✶❪ ✱ s❛♠❡ r♦✇✱ ❘ ✱ ❛♥❞ ❧❛st r♦✇✱ ❘❬✲✶❪ ✳ ❇❡❧♦✇✱ ②♦✉ ❝❛♥ s❡❡ t❤❡ t✇♦ r❡❢❡r❡♥❝❡s ✐♥ t❤❡ ❢♦r♠✉❧❛s ♠❛r❦❡❞ ✇✐t❤ r❡❞ ❛♥❞ ❜❧✉❡ ✭❧❡❢t✮ ❛♥❞ t❤❡ ❞❡♣❡♥❞❡♥❝❡ s❤♦✇♥ ✇✐t❤ t❤❡ ❛rr♦✇s ✭r✐❣❤t✮✿

❲❡ ♣❧❛❝❡ t❤❡ r❡s✉❧t ✐♥ ❛ ♥❡✇ ❝♦❧✉♠♥ ✇❡ ❝r❡❛t❡❞ ❢♦r t❤❡ ✈❡❧♦❝✐t✐❡s✿

❚❤✐s ♥❡✇ ❞❛t❛ ✐s ✐❧❧✉str❛t❡❞ ✇✐t❤ t❤❡ s❡❝♦♥❞ s❝❛tt❡r ♣❧♦t✳ ❚♦ ❡♠♣❤❛s✐③❡ t❤❡ ❢❛❝t t❤❛t t❤❡ ✈❡❧♦❝✐t② ❞❛t❛✱ ✉♥❧✐❦❡ t❤❡ ❧♦❝❛t✐♦♥✱ ✐s r❡❢❡rr✐♥❣ t♦ t✐♠❡ ✐♥t❡r✈❛❧s r❛t❤❡r t❤❛♥ t✐♠❡ ✐♥st❛♥❝❡s✱ ✇❡ ♣❧♦t ✐t ✇✐t❤ ❤♦r✐③♦♥t❛❧ s❡❣♠❡♥ts✳ ■♥ ❢❛❝t✱ t❤❡ ❞❛t❛ t❛❜❧❡ ❝❛♥ ❜❡ r❡❛rr❛♥❣❡❞ ❛s ❢♦❧❧♦✇s t♦ ♠❛❦❡ t❤✐s ♣♦✐♥t ❝❧❡❛r❡r✿ t✐♠❡ ❧♦❝❛t✐♦♥ ✈❡❧♦❝✐t②

0 1 2 3 4 ... 0.00 − 0.10 − 0.20 − 0.30 − .39 − ... · 0.10 · 0.10 · 0.10 · 0.09 · 0.09 ...

❲❤❛t ❤❛s ❤❛♣♣❡♥❡❞ t♦ t❤❡ ♠♦✈✐♥❣ ♦❜❥❡❝t ❝❛♥ ♥♦✇ ❜❡ ❡❛s✐❧② r❡❛❞ ❢r♦♠ t❤❡ s❡❝♦♥❞ ❣r❛♣❤✳ ❚❤❡s❡ ❛r❡ t❤❡ ✜✈❡ st❛❣❡s✿ ✶✳ ❚❤❡ ✈❡❧♦❝✐t② ✇❛s ♣♦s✐t✐✈❡ ✐♥✐t✐❛❧❧② ✭✐t ✇❛s ♠♦✈✐♥❣ ✐♥ t❤❡ ♣♦s✐t✐✈❡ ❞✐r❡❝t✐♦♥✮✳ ✷✳ ❚❤❡ ✈❡❧♦❝✐t② ✇❛s ❢❛✐r❧② ❤✐❣❤ ✭✐t ✇❛s ♠♦✈✐♥❣ ❢❛✐r❧② ❢❛st✮ ❜✉t t❤❡♥ ✐t st❛rt❡❞ t♦ ❞❡❝❧✐♥❡ ✭s❧♦✇ ❞♦✇♥✮✳ ✸✳ ❚❤❡ ✈❡❧♦❝✐t② ✇❛s ③❡r♦ ✭✐t st♦♣♣❡❞✮ ❢♦r ❛ ✈❡r② s❤♦rt ♣❡r✐♦❞✳ ✹✳ ❚❤❡♥ t❤❡ ✈❡❧♦❝✐t② ❜❡❝❛♠❡ ♥❡❣❛t✐✈❡ ✭✐t st❛rt❡❞ t♦ ♠♦✈❡ ✐♥ t❤❡ ♦♣♣♦s✐t❡ ❞✐r❡❝t✐♦♥✮✳ ✺✳ ❆♥❞ t❤❡♥ t❤❡ ✈❡❧♦❝✐t② st❛rt❡❞ t♦ ❜❡❝♦♠❡ ♠♦r❡ ♥❡❣❛t✐✈❡ ✭✐t st❛rt❡❞ t♦ s♣❡❡❞ ✉♣ ✐♥ t❤❛t ❞✐r❡❝t✐♦♥✮✳ ❚❤✉s✱ t❤❡ ❧❛tt❡r s❡t ♦❢ ❞❛t❛ s✉❝❝✐♥❝t❧② r❡❝♦r❞s s♦♠❡ ✐♠♣♦rt❛♥t ❢❛❝ts ❛❜♦✉t t❤❡ ❞②♥❛♠✐❝s ♦❢ t❤❡ ❢♦r♠❡r✳ ◆♦✇✱ ❢r♦♠ ✈❡❧♦❝✐t② t♦ ❧♦❝❛t✐♦♥✳

✶✳✶✳ ❲❤❛t ✐s ❝❛❧❝✉❧✉s ❛❜♦✉t❄

✶✽

❆❣❛✐♥✱ ✇❡ ❝♦♥s✐❞❡r ❛ s❡q✉❡♥❝❡ ♦❢ 30 ❞❛t❛ ♣♦✐♥ts✳ ❚❤❡s❡ ♥✉♠❜❡rs ❛r❡ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ✈❡❧♦❝✐t② ♦❢ ❛♥ ♦❜❥❡❝t r❡❝♦r❞❡❞ ❡✈❡r② ♠✐♥✉t❡✿ t✐♠❡ ✈❡❧♦❝✐t②

♠✐♥✉t❡s ♠✐❧❡s✴❤♦✉r

0 1 2 3 4 5 6 7 8 9 10 ... 0.10 0.20 0.30 0.39 0.48 0.56 0.64 0.72 0.78 0.84 ...

❚❤✐s ❞❛t❛ ✐s ❛❧s♦ s❡❡♥ ✐♥ t❤❡ ✜rst t✇♦ ❝♦❧✉♠♥s ♦❢ t❤❡ s♣r❡❛❞s❤❡❡t ♣❧♦tt❡❞ ♦♥❡ ❜❛r ❛t ❛ t✐♠❡✿

❚❤❡ ❞❛t❛ ✐s ❢✉rt❤❡r♠♦r❡ ✐❧❧✉str❛t❡❞ ❛s ❛ s❝❛tt❡r ♣❧♦t ♦♥ t❤❡ r✐❣❤t✳ ❆❣❛✐♥✱ ✇❡ ❡♠♣❤❛s✐③❡ t❤❡ ❢❛❝t t❤❛t t❤❡ ✈❡❧♦❝✐t② ❞❛t❛ ✐s r❡❢❡rr✐♥❣ t♦ t✐♠❡ ✐♥t❡r✈❛❧s ❜② ♣❧♦tt✐♥❣ ✐ts ✈❛❧✉❡s ✇✐t❤ ❤♦r✐③♦♥t❛❧ ❜❛rs✳ ❚❤❡ ❞❛t❛ ♠❛② ❜❡ ❞❡s❝r✐❜✐♥❣ t❤❡ ❤♦r✐③♦♥t❛❧ s♣❡❡❞ ♦❢ ❛ ❜❛❧❧ r♦❧❧✐♥❣ t❤r♦✉❣❤ ❛ tr♦✉❣❤✿

❚♦ ✜♥❞ ♦✉t ✇❤❡r❡ ✇❡ ❛r❡ ❛t t❤❡ ❡♥❞ ♦❢ ❡❛❝❤ ♦❢ t❤❡s❡ ♦♥❡✲♠✐♥✉t❡ ✐♥t❡r✈❛❧s✱ ✇❡ ❝♦♠♣✉t❡ t❤❡ ❛❞❞✐♥❣ t❤❡ ✈❡❧♦❝✐t✐❡s ♦♥❡ ❛t ❛ t✐♠❡✳ ❚❤✐s ✐s ❤♦✇ t❤❡ ✜rst st❡♣ ✐s ❝❛rr✐❡❞ ♦✉t✱ ✉♥❞❡r t❤❡ ❛ss✉♠♣t✐♦♥ t❤❛t t❤❡ ✐♥✐t✐❛❧ ❧♦❝❛t✐♦♥ ✐s 0✿ t✐♠❡ ♠✐♥ 0 1 ... ✈❡❧♦❝✐t② ♠✐❧❡s 0.10 ... s✉♠ ❧♦❝❛t✐♦♥

♠✐❧❡s✴♠✐♥

↓ 0.00+ 0.10 ... ↑ || 0.00 0.10 ...

❲❡ ♣❧❛❝❡ t❤✐s ❞❛t❛ ✐♥ ❛ ♥❡✇ r♦✇ ❛❞❞❡❞ t♦ t❤❡ ❜♦tt♦♠ ♦❢ ♦✉r t❛❜❧❡✿ t✐♠❡ ✈❡❧♦❝✐t②

♠✐♥ ♠✐❧❡s

❧♦❝❛t✐♦♥

♠✐❧❡s✴♠✐♥

0

1 0.10 ↓ 0.00 → 0.10 →

2 0.20 ↓ 0.30 →

3 0.30 ↓ 0.59 →

4 0.39 ↓ 0.98 →

5 0.48 ↓ 1.46 →

6 0.56 ↓ 2.03 →

7 0.64 ↓ 2.67 →

8 0.72 ↓ 3.39 →

❲❡ ❤❛✈❡ ❛ ♥❡✇ s❡q✉❡♥❝❡✦ Pr❛❝t✐❝❛❧❧②✱ ✇❡ ✉s❡ t❤❡ s♣r❡❛❞s❤❡❡t✳ ❲❡ ❝♦♠♣✉t❡ t❤❡ s✉♠s ❜② ♣✉❧❧✐♥❣ t❤❡ ❞❛t❛ ❢r♦♠ t❤❡ ❝♦❧✉♠♥ ♦❢ ✈❡❧♦❝✐t✐❡s ✉s✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✉❧❛✿ ❂❘❬✲✶❪❈✰❘❈❬✲✶❪

❍❡r❡✱ t❤❡ t✇♦ ✈❛❧✉❡s ❝♦♠❡ ❢r♦♠ t❤❡ s❛♠❡✱ ❈ ✱ ♦r ❧❛st✱ ❈❬✲✶❪ ✱ ❝♦❧✉♠♥ ❛♥❞ t❤❡ s❛♠❡✱ ❘ ✱ ❛♥❞ ❧❛st✱ ❘❬✲✶❪ ✱ r♦✇✱ ❛s ❢♦❧❧♦✇s✿

... ... ... ...

✶✳✶✳ ❲❤❛t ✐s ❝❛❧❝✉❧✉s ❛❜♦✉t❄

✶✾

❲❡ ♣❧❛❝❡ t❤❡ r❡s✉❧t ✐♥ ❛ ♥❡✇ ❝♦❧✉♠♥ ❢♦r ❧♦❝❛t✐♦♥s✿

❚❤❡ ❞❛t❛ ✐s ❛❧s♦ ✐❧❧✉str❛t❡❞ ❛s t❤❡ s❡❝♦♥❞ s❝❛tt❡r ♣❧♦t ♦♥ t❤❡ r✐❣❤t✳ ❲❤❛t ❤❛s ❤❛♣♣❡♥❡❞ t♦ t❤❡ ♠♦✈✐♥❣ ♦❜❥❡❝t ❝❛♥ ♥♦✇ ❜❡ ❡❛s✐❧② r❡❛❞ ❢r♦♠ t❤❡ ✜rst ♦r t❤❡ s❡❝♦♥❞ ♣❧♦t✳ ❚❤❡s❡ ❛r❡ t❤❡ ✜✈❡ st❛❣❡s✿ ✶✳ ❚❤❡ ✈❡❧♦❝✐t② ✐s ♣♦s✐t✐✈❡ ❛♥❞ ❧♦✇✳ ✷✳ ❚❤❡ ✈❡❧♦❝✐t② ✐s ♣♦s✐t✐✈❡ ❛♥❞ ❤✐❣❤✳ ✸✳ ❚❤❡ ✈❡❧♦❝✐t② ✐s t❤❡ ❤✐❣❤❡st✳ ✹✳ ❚❤❡ ✈❡❧♦❝✐t② ✐s ♣♦s✐t✐✈❡ ❛♥❞ ❤✐❣❤✳ ✺✳ ❚❤❡ ✈❡❧♦❝✐t② ✐s ♣♦s✐t✐✈❡ ❛♥❞ ❧♦✇✳ ❲❡✱ ❛❣❛✐♥✱ r❡❛rr❛♥❣❡ t❤❡ ❞❛t❛ t❛❜❧❡ t♦ ♠❛❦❡ t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ t✇♦ t②♣❡s ♦❢ ❞❛t❛ ❝❧❡❛r❡r✿ t✐♠❡ ✈❡❧♦❝✐t② ❧♦❝❛t✐♦♥

0 1 2 3 4 ... · 0.00 · 0.10 · 0.20 · 0.30 · 0.39 ... 0.00 − 0.10 − 0.30 − 0.59 − .98 − ...

❚❤✉s✱ ❛s t❤❡ ❢♦r♠❡r ❞❛t❛ s❡t r❡❝♦r❞s s♦♠❡ ❢❛❝ts ❛❜♦✉t t❤❡ ❞②♥❛♠✐❝s ♦❢ t❤❡ ❧❛tt❡r✱ ✇❡ ❛r❡ ❛❜❧❡ t♦ ✉s❡ t❤✐s ✐♥❢♦r♠❛t✐♦♥ t♦ r❡❝♦✈❡r t❤❡ ❧❛tt❡r✳ ❚❤✐s ✐s ✇❤❛t ✇❡ ❤❛✈❡ ❞✐s❝♦✈❡r❡❞✿

◮

❲❡ ❝❛♥ t❡❧❧ t❤❡ ✈❡❧♦❝✐t② ❢r♦♠ t❤❡ ❧♦❝❛t✐♦♥ ❛♥❞✱ ❝♦♥✈❡rs❡❧②✱ t❤❡ ❧♦❝❛t✐♦♥ ❢r♦♠ t❤❡ ✈❡❧♦❝✐t②✳

■s t❤✐s ✐t t❤♦✉❣❤❄ ▼♦t✐♦♥ ✐s ❛ ❝♦♥t✐♥✉♦✉s ♣❤❡♥♦♠❡♥♦♥✳ ❈❛♥ ✇❡ ✉♥❞❡rst❛♥❞ ✐t ✇✐t❤ t❤❡ ❛❜♦✈❡ ❛♣♣r♦❛❝❤❄ ■❢ ✐t ✐s ❦♥♦✇♥ t❤❛t ♦✉r ❞❛t❛ ✐s ❥✉st ❛ s♥❛♣s❤♦t ♦❢ ❛ ✏❝♦♥t✐♥✉♦✉s✑ ♣r♦❝❡ss✱ ✇❡ ♠❛② ❜❡ ❛❜❧❡ t♦ ❝♦❧❧❡❝t ♠♦r❡ ✐♥❢♦r♠❛t✐♦♥ ✐♥ ♦r❞❡r t♦ ♠❛❦❡ t❤✐s r❡♣r❡s❡♥t❛t✐♦♥ ❜❡tt❡r✳ ❲❡✱ ❢♦r ❡①❛♠♣❧❡✱ ♠❛② ❧♦♦❦ ❛t t❤❡ ♦❞♦♠❡t❡r ❡✈❡r②

✶✳✶✳ ❲❤❛t ✐s ❝❛❧❝✉❧✉s ❛❜♦✉t❄

✷✵

♠✐♥✉t❡✱ ♦r ❡✈❡r② s❡❝♦♥❞✱ ❡t❝✳✱ ✐♥st❡❛❞ ♦❢ ❡✈❡r② ❤♦✉r✳ ❚❤❡ ✐♥✜♥✐t❡ ❞✐✈✐s✐❜✐❧✐t② ♦❢ t❤❡ r❡❛❧ ❧✐♥❡ ✭s❡❡♥ ✐♥ ❱♦❧✉♠❡ ✶✱ ❈❤❛♣t❡r ✶P❈✲✶✮ ❛❧❧♦✇s ✉s t♦ ♣r♦❞✉❝❡ s❡ts ♦❢ ♣♦✐♥ts ♦♥ t❤❡ ♣❧❛♥❡ ✇✐t❤ ❞❡♥s❡r ❛♥❞ ❞❡♥s❡r ♣❛tt❡r♥s✿ ❲❡ ♠❛❦❡ t❤❡ t✐♠❡ ✐♥t❡r✈❛❧s s♠❛❧❧❡r ❛♥❞ s♠❛❧❧❡r ❛♥❞ ✐♥s❡rt ♠♦r❡ ❛♥❞ ♠♦r❡ ✐♥♣✉ts✳ ❲❤❡♥ t❤❡r❡ ❛r❡ ❡♥♦✉❣❤ ♦❢ t❤❡♠✱ t❤❡ ♣♦✐♥ts st❛rt t♦ ❢♦r♠ ❛ ❝✉r✈❡✿

❇♦t❤ ❧♦❝❛t✐♦♥ ❛♥❞ ✈❡❧♦❝✐t② ❛r❡ ❝❤❛♥❣✐♥❣ ❝♦♥t✐♥✉♦✉s❧②✦ ❲❡ ✐♠❛❣✐♥❡ t❤❛t ❛t t❤❡ ❡♥❞ ♦❢ t❤✐s ♣r♦❝❡ss ✇❡ ✇✐❧❧ ❤❛✈❡ ❛♥ ❛❝t✉❛❧ ❝✉r✈❡✳ ❚❤✐s ✐s ♥♦t t❤❡ ❦✐♥❞ ♦❢ ❝✉r✈❡ t❤❛t ✐s ♠❛❞❡ ♦❢ ♠❛r❜❧❡s ♣❧❛❝❡❞ ❝❧♦s❡ t♦❣❡t❤❡r✱ ❜✉t ❛ r♦♣❡✳ ❲❤❛t ❤❛♣♣❡♥s ✏❛t t❤❡ ❡♥❞✑ ✐s st✉❞✐❡❞ ✐♥ ❈❤❛♣t❡r ✷✳ ❚❤✉s✱ t❤✐s ♠❛✐♥ ✐❞❡❛ ♦❢ ❝❛❧❝✉❧✉s ✐s t♦ ❞❡r✐✈❡ ✈❡❧♦❝✐t② ❢r♦♠ ❧♦❝❛t✐♦♥ ✭❢r♦♠ t❤❡ t♦♣ r♦✇ t♦ t❤❡ ❜♦tt♦♠✮ ❛♥❞ ❧♦❝❛t✐♦♥ ❢r♦♠ ✈❡❧♦❝✐t② ✭❢r♦♠ t❤❡ ❜♦tt♦♠ r♦✇ t♦ t❤❡ t♦♣✮✿

■t ✐s ❛ s♣❡❝✐❛❧ ❝❤❛❧❧❡♥❣❡ t♦ s♦❧✈❡ t❤✐s ♣r♦❜❧❡♠ ❢♦r t❤❡ ❝✉r✈❡s✳ ■t ✐s ❛❞❞r❡ss❡❞ ✐♥ ❈❤❛♣t❡r ✸✳ ❲❡✱ ❤♦✇❡✈❡r✱ s❡t t❤❡s❡ ✐❞❡❛s ❛s✐❞❡ ❢♦r ♥♦✇ ❛♥❞ t✉r♥ t♦ s♦♠❡t❤✐♥❣ ♠♦r❡ ✐♠♠❡❞✐❛t❡✿

◮

❲❤❛t ❝❛♥ ✇❡ s❛② ❛❜♦✉t t❤❡ ❧♦♥❣✲t❡r♠ tr❡♥❞s ♦❢ t❤❡ ❧♦❝❛t✐♦♥ ❛♥❞ t❤❡ ✈❡❧♦❝✐t②❄

▲❡t✬s t❛❦❡ ❛♥♦t❤❡r ❧♦♦❦ ❛t t❤❡ ✜rst ❝❤❛rt ❛❜♦✈❡ ❛♥❞ t❤❡ s❡q✉❡♥❝❡s ♣❧♦tt❡❞✳ ■s t❤❡ ❢✉t✉r❡ ♣r❡❞✐❝t❛❜❧❡ ❄ ❚❤❡ ♦❜❥❡❝t ✐s ♠♦✈✐♥❣ ✐♥ t❤❡ ♥❡❣❛t✐✈❡ ❞✐r❡❝t✐♦♥ ✭t♦♣ r♦✇✮✱ ❜✉t ✇❡ ❝❛♥✬t t❡❧❧ ✐❢ t❤✐s ✇✐❧❧ ❝♦♥t✐♥✉❡✳ ❍♦✇❡✈❡r✱ ✐t ✐s s❧♦✇✐♥❣ ❞♦✇♥ ✭❜♦tt♦♠ r♦✇✮✱ ❛♥❞ ✐t ✐s ❝♦♥❝❡✐✈❛❜❧❡ t❤❛t ✐t ✇✐❧❧ st♦♣ ❡✈❡♥t✉❛❧❧②✳ ❚❤❡ ♣❛tt❡r♥ ✐s ❥✉st ❛s ❝❧❡❛r ✐♥ t❤❡ s❡❝♦♥❞ ❝❤❛rt ❛❜♦✈❡ ♦r ✐♥ t❤✐s ❝❤❛rt✿

✶✳✷✳

■♥✜♥✐t❡ s❡q✉❡♥❝❡s ❛♥❞ t❤❡✐r ❧♦♥❣✲t❡r♠ tr❡♥❞s

✷✶

❚❤❡ ♠♦t✐♦♥ s❤♦✇♥ ✐♥ t❤❡ ♥❡①t ♦♥❡ ❤❛s ❞❡✜♥✐t❡❧② st♦♣♣❡❞✿

❚❤✐s ❛♥❞ r❡❧❛t❡❞ ✐ss✉❡s ✭❧✐♠✐t❡❞ t♦ s❡q✉❡♥❝❡s✮ ✇✐❧❧ ❜❡ ❛❞❞r❡ss❡❞ ✐♥ t❤✐s ❝❤❛♣t❡r✳ ❇✉t ✜rst ❛ r❡✈✐❡✇✳

✶✳✷✳ ■♥✜♥✐t❡ s❡q✉❡♥❝❡s ❛♥❞ t❤❡✐r ❧♦♥❣✲t❡r♠ tr❡♥❞s ❲❡ ❢♦r♠❛❧✐③❡ t❤❡ ✇❛② ✇❡ r❡♣r❡s❡♥t s❡q✉❡♥❝❡s ♦❢ ♥✉♠❜❡rs s✉❝❤ ❛s t❤❡ ♦♥❡s ✇❡ s❛✇ ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥✿ t✐♠❡

♠✐♥✉t❡s

❧♦❝❛t✐♦♥

♠✐❧❡s

0 1 2 3 4 5 6 7 8 9 10 ... 0.00 0.10 0.20 0.30 0.39 0.48 0.56 0.64 0.72 0.78 0.84 ...

❲❡ ✜rst ❣✐✈❡ ❛ s❡q✉❡♥❝❡ ❛ ♥❛♠❡✱ s❛②✱

a✱

❛♥❞ t❤❡♥ ❛ss✐❣♥ ❛ s♣❡❝✐✜❝ ✈❛r✐❛t✐♦♥ ♦❢ t❤✐s ♥❛♠❡ t♦ ❡❛❝❤ t❡r♠ ♦❢

t❤❡ s❡q✉❡♥❝❡✿

■♥❞✐❝❡s ♦❢ s❡q✉❡♥❝❡ ✐♥❞❡①✿ t❡r♠✿ ❚❤❡

♥❛♠❡

n 1 2 3 4 5 6 7 ... an a1 a2 a3 a4 a5 a6 a7 ...

♦❢ ❛ s❡q✉❡♥❝❡ ✐s ❛ ❧❡tt❡r✱ ✇❤✐❧❡ t❤❡ s✉❜s❝r✐♣t ❝❛❧❧❡❞ t❤❡

✇✐t❤✐♥ t❤❡ s❡q✉❡♥❝❡✳ ■t r❡❛❞s ✏ a s✉❜

1✑✱ ✏ a

s✉❜

2✑✱

❡t❝✳

❚❤✐s ✐s ✇❤❛t t❤❡ ♥♦t❛t✐♦♥ ♠❡❛♥s✿

■♥❞❡① ♦❢ ❛ t❡r♠

a ↑

♥❛♠❡

■♥❞✐❝❡s s❡r✈❡ ❛s

t❛❣s ✿

✐♥❞❡①

↓

n

✐♥❞❡①

✐♥❞✐❝❛t❡s t❤❡ ♣❧❛❝❡ ♦❢ t❤❡ t❡r♠

✶✳✷✳ ■♥✜♥✐t❡ s❡q✉❡♥❝❡s ❛♥❞ t❤❡✐r ❧♦♥❣✲t❡r♠ tr❡♥❞s

❊①❛♠♣❧❡ ✶✳✷✳✶✿ ❢❛❧❧✐♥❣ ❜❛❧❧

❲❡ ✇❛t❝❤ ❛ ♣✐♥❣✲♣♦♥❣ ❜❛❧❧ ❢❛❧❧✐♥❣ ❞♦✇♥ ❛♥❞ r❡❝♦r❞ ✕ ❛t ❡q✉❛❧ ✐♥t❡r✈❛❧s ✕ ❤♦✇ ❤✐❣❤ ✐t ✐s✳ ❚❤❡ r❡s✉❧t ✐s ❛♥ ❡✈❡r✲❡①♣❛♥❞✐♥❣ str✐♥❣✱ ❛ s❡q✉❡♥❝❡✱ ♦❢ ♥✉♠❜❡rs✳ ■❢ t❤❡ ❢r❛♠❡s ♦❢ t❤❡ ✈✐❞❡♦ ❛r❡ ❝♦♠❜✐♥❡❞ ✐♥t♦ ♦♥❡ ✐♠❛❣❡✱ ✐t ✇✐❧❧ ❧♦♦❦ s♦♠❡t❤✐♥❣ ❧✐❦❡ t❤✐s✿

❲❡ ❤❛✈❡ ❛ ❧✐st ✿

36, 35, 32, 27, 20, 11, 0, ... ❲❡ ❜r✐♥❣ t❤❡♠ ❜❛❝❦ t♦❣❡t❤❡r ✐♥ ♦♥❡ r❡❝t❛♥❣✉❧❛r ♣❧♦t s♦ t❤❛t t❤❡ ❧♦❝❛t✐♦♥ ✈❛r✐❡s ✈❡rt✐❝❛❧❧② ✇❤✐❧❡ t❤❡ t✐♠❡ ♣r♦❣r❡ss❡s ❤♦r✐③♦♥t❛❧❧②✿

❚❤❡ ♣❧♦t ✐s ❝❛❧❧❡❞ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ s❡q✉❡♥❝❡✳ ❆s ❢❛r ❛s t❤❡ ❞❛t❛ ✐s ❝♦♥❝❡r♥❡❞✱ ✇❡ ❤❛✈❡ ❛ ❧✐st ♦❢ ♣❛✐rs✱ t✐♠❡ ❛♥❞ ❧♦❝❛t✐♦♥✱ ❛rr❛♥❣❡❞ ✐♥ ❛ t❛❜❧❡✿ ♠♦♠❡♥t 1 2 3 4 5 6 7 ...

❤❡✐❣❤t 36 35 32 27 20 11 0 ...

♦r

♠♦♠❡♥t✿ 1 2 3 4 5 6 7 ... ❤❡✐❣❤t✿ 36 35 32 27 20 11 0 ...

✷✷

✶✳✷✳

■♥✜♥✐t❡ s❡q✉❡♥❝❡s ❛♥❞ t❤❡✐r ❧♦♥❣✲t❡r♠ tr❡♥❞s

■♥ ♦✉r ❡①❛♠♣❧❡✱ ✇❡ ♥❛♠❡ t❤❡ s❡q✉❡♥❝❡ ♠♦♠❡♥t✿ ❤❡✐❣❤t✿ ❤❡✐❣❤t✿

h

✷✸

❢♦r ✏❤❡✐❣❤t✑✳ ❚❤❡♥ t❤❡ ❛❜♦✈❡

1 h1 || 36

2 h2 || 35

❲❤❡♥ ❛❜❜r❡✈✐❛t❡❞✱ ✐t t❛❦❡s t❤❡ ❢♦r♠ ♦❢ t❤✐s

3 h3 || 32

4 h4 || 27

5 h5 || 20

6 h6 || 11

7 h7 || 0

t❛❜❧❡

t❛❦❡ t❤✐s ❢♦r♠✿

... ... ... ...

❧✐st ✿

h1 = 36, h2 = 35, h3 = 32, h4 = 27, h5 = 20, h6 = 11, h7 = 0, .... ❙♦✱ ✇❡ ✉s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♥♦t❛t✐♦♥✿

a1 = 1, a2 = 1/2, a3 = 1/3, a4 = 1/4, ... ✇❤❡r❡

a

♥❛♠❡

✐s t❤❡

♦❢ t❤❡ s❡q✉❡♥❝❡ ❛♥❞ ❛❞❞✐♥❣ ❛ s✉❜s❝r✐♣t ✐♥❞✐❝❛t❡s ✇❤✐❝❤ t❡r♠ ♦❢ t❤❡ s❡q✉❡♥❝❡ ✇❡ ❛r❡

❢❛❝✐♥❣✳ ❲❡ ✇✐❧❧ st✉❞②

✐♥✜♥✐t❡

s❡q✉❡♥❝❡s✳

❙✉❝❤ ❛ s❡q✉❡♥❝❡ ❝❛♥ ❜❡ ❡①❛♠✐♥❡❞ ✐♥ t❡r♠s ♦❢ ✐ts ❧♦♥❣✲t❡r♠✱ ♦r ❜❡tt❡r ②❡t ✐♥✜♥✐t❡✱ ✶✳ ❚❤❡ s❡q✉❡♥❝❡ ✷✳ ❚❤❡ s❡q✉❡♥❝❡ ✸✳ ❚❤❡ s❡q✉❡♥❝❡ ✹✳ ❚❤❡ s❡q✉❡♥❝❡

1 .9 1 0

1/2 .99 2 1

1/3 .999 3 0

1/4 .9999 4 1

1/5 .99999 5 0

... ... ... ...

t❡♥❞s t♦✇❛r❞ t❡♥❞s t♦✇❛r❞ t❡♥❞s t♦✇❛r❞

tr❡♥❞✳

❚❤❡ ✐❞❡❛ ✐s s✐♠♣❧❡✿

0. 1. + ∞.

❞♦❡s♥✬t t❡♥❞ t♦ ❛♥②t❤✐♥❣✳

❙♦✱ ❛♥ ✐♥✜♥✐t❡ s❡q✉❡♥❝❡ ♦❢ ♥✉♠❜❡rs ✇✐❧❧ s♦♠❡t✐♠❡s ❜❡ ✏❛❝❝✉♠✉❧❛t✐♥❣✑ ❛r♦✉♥❞ ❛ s✐♥❣❧❡ ♥✉♠❜❡r✱ ✐ts ✏❧✐♠✐t✑ ✭★✶ ❛♥❞ ★✷✮✳ ■t✬s ❛s ✐❢ t❤❡ ❣❛♣ ❜❡t✇❡❡♥ t❤❡ ❜♦✉♥❝✐♥❣ ❜❛❧❧ ❛♥❞ t❤❡ ❣r♦✉♥❞ ❜❡❝♦♠❡s ✐♥✈✐s✐❜❧❡✳ ❚❤✐s ✐s♥✬t ❛❧✇❛②s t❤❡ ❝❛s❡✿ ❚❤❡ s❡q✉❡♥❝❡ ♠✐❣❤t ✏r✉♥ ❛✇❛②✑ ✭★✸✮ ♦r ❜♦✉♥❝❡ ❛r♦✉♥❞ ✇✐t❤♦✉t s❧♦✇✐♥❣ ❞♦✇♥ ✭★✹✮✳ ❊✈❡r② ❢✉♥❝t✐♦♥

y = f (x)

✇✐t❤ ❛♥ ❛♣♣r♦♣r✐❛t❡ ❞♦♠❛✐♥ ❝r❡❛t❡s ❛ s❡q✉❡♥❝❡ ✈✐❛ ❛ s✐♠♣❧❡ s✉❜st✐t✉t✐♦♥✿

an = f (n) . ❆ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ♦♥ ❛ r❛② ✐♥ t❤❡ s❡t ♦❢ ✐♥t❡❣❡rs✱ s❡q✉❡♥❝❡✱ ❛♥❞ ✐t ✐s t②♣✐❝❛❧❧② ❣✐✈❡♥ ❜② ✐ts ❢♦r♠✉❧❛✿

an = 1/n,

{p, p + 1, ...}✱

✐s ❝❛❧❧❡❞ ❛♥ ✐♥✜♥✐t❡ s❡q✉❡♥❝❡✱ ♦r s✐♠♣❧②

n = 1, 2, 3, ...

■♥ ❛❞❞✐t✐♦♥ t♦ t❛❜❧❡s ❛♥❞ ❢♦r♠✉❧❛s✱ ❛ s❡q✉❡♥❝❡ ✐s ❝♦♠♠♦♥❧② ❞❡✜♥❡❞ ❜② ❝♦♠♣✉t✐♥❣ ✐ts t❡r♠s ✐♥ ❛

♠❛♥♥❡r✱ ♦♥❡ ❛t ❛ t✐♠❡✳

❝♦♥s❡❝✉t✐✈❡

❉❡✜♥✐t✐♦♥ ✶✳✷✳✷✿ r❡❝✉rs✐✈❡ s❡q✉❡♥❝❡ ❲❡ s❛② t❤❛t ❛ s❡q✉❡♥❝❡ ✐s

r❡❝✉rs✐✈❡

t❡r♠ ❜② ❛ s♣❡❝✐✜❡❞ ❢♦r♠✉❧❛✱ ✐✳❡✳✱

an

✇❤❡♥ ✐ts ♥❡①t t❡r♠ ✐s ❢♦✉♥❞ ❢r♦♠ t❤❡ ❝✉rr❡♥t ❞❡t❡r♠✐♥❡s

an+1 ✳

❊①❛♠♣❧❡ ✶✳✷✳✸✿ ❜❛♥❦ ❛❝❝♦✉♥t ❆ ♣❡rs♦♥ st❛rts t♦ ❞❡♣♦s✐t

$20

❡✈❡r② ♠♦♥t❤ ✐♥ ❤✐s ❜❛♥❦ ❛❝❝♦✉♥t t❤❛t ❛❧r❡❛❞② ❝♦♥t❛✐♥s

✇♦r❞s✱ ✇❡ ❤❛✈❡✿ ♥❡①t ♦r ✇✐t❤ ❛

=

❧❛st

+ 20 ,

r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛ ✿ an+1 = an + 20 ,

♦r ❢♦r t❤❡ s♣r❡❛❞s❤❡❡t✿

❂❘❬✲✶❪❈✰✷✵

$1000✳

■♥ ♦t❤❡r

✶✳✷✳ ■♥✜♥✐t❡ s❡q✉❡♥❝❡s ❛♥❞ t❤❡✐r ❧♦♥❣✲t❡r♠ tr❡♥❞s ❇❡❧♦✇✱ t❤❡ ❝✉rr❡♥t ❛♠♦✉♥t ✐s s❤♦✇♥ ✐♥ ❜❧✉❡✱ ❛♥❞ t❤❡ ♥❡①t ✕ ❝♦♠♣✉t❡❞ ❢r♦♠ t❤❡ ❝✉rr❡♥t ✕ ✐s s❤♦✇♥ ✐♥ r❡❞✿

P❧♦tt✐♥❣ s❡✈❡r❛❧ t❡r♠s ♦❢ t❤❡ s❡q✉❡♥❝❡ ❛t ♦♥❝❡ ❝♦♥✜r♠s t❤❛t t❤❡ s❡q✉❡♥❝❡ ✐s ✐♥❝r❡❛s✐♥❣ ✿

■t ❛❧s♦ ❧♦♦❦s ❧✐❦❡ ❛ str❛✐❣❤t ❧✐♥❡✳ ❆♥♦t❤❡r ♣❡rs♦♥ ❞❡♣♦s✐ts $1000 ✐♥ ❤✐s ❜❛♥❦ ❛❝❝♦✉♥t t❤❛t ♣❛②s 1% ❆P❘ ❝♦♠♣♦✉♥❞❡❞ ❛♥♥✉❛❧❧②✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿ ♥❡①t = ❧❛st · 1.01 ,

♦r ✇✐t❤ ❛ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛ ✿ ♦r ❢♦r t❤❡ s♣r❡❛❞s❤❡❡t✿

bn+1 = bn · 1.01 ,

❂❘❬✲✶❪❈✯✶✳✵✶

❲❡ ♣❧♦t ❛ t❡r♠ ❛♥❞ t❤❡ ♥❡①t ♦♥❡✿

❖♥❧② ❛❢t❡r r❡♣❡❛t✐♥❣ t❤❡ st❡♣ 100 t✐♠❡s ❝❛♥ ♦♥❡ s❡❡ t❤❛t t❤✐s ✐s♥✬t ❥✉st ❛ str❛✐❣❤t ❧✐♥❡✿

✷✹

✶✳✷✳ ■♥✜♥✐t❡ s❡q✉❡♥❝❡s ❛♥❞ t❤❡✐r ❧♦♥❣✲t❡r♠ tr❡♥❞s

✷✺

❲❤❛t ✐❢ ✇❡ ❞❡♣♦s✐t ♠♦♥❡② t♦ ♦✉r ❜❛♥❦ ❛❝❝♦✉♥t ❛♥❞ r❡❝❡✐✈❡ ✐♥t❡r❡st❄ ❚❤❡ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛ ✐s s✐♠♣❧❡❀ ❢♦r ❡①❛♠♣❧❡✿ cn+1 = cn · 1.05 + 2000 .

❍❡r❡✱ t❤❡ ✐♥t❡r❡st ✐s 5% ✇✐t❤ ❛ $2000 ❛♥♥✉❛❧ ❞❡♣♦s✐t✳

❊①❡r❝✐s❡ ✶✳✷✳✹ n

❙❤♦✇ t❤❛t ✐s ❛♥ ✐♥❝r❡❛s✐♥❣ s❡q✉❡♥❝❡✳ ❲❤❛t ❦✐♥❞ ♦❢ s❡q✉❡♥❝❡ ✐s n+1 ✐♥❝r❡❛s✐♥❣ ❛♥❞ ❞❡❝r❡❛s✐♥❣ s❡q✉❡♥❝❡s✳

n+1 ❄ ●✐✈❡ ❡①❛♠♣❧❡s ♦❢ n

❚❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ ❝♦♥❝❡♣ts ✇✐❧❧ ❜❡ r♦✉t✐♥❡❧② ✉s❡❞✳

❉❡✜♥✐t✐♦♥ ✶✳✷✳✺✿ ❛r✐t❤♠❡t✐❝ ♣r♦❣r❡ss✐♦♥ ❆ s❡q✉❡♥❝❡ ❞❡✜♥❡❞ ✭r❡❝✉rs✐✈❡❧②✮ ❜② t❤❡ ❢♦r♠✉❧❛✿ an+1 = an + b

✐s ❝❛❧❧❡❞ ❛♥ ❛r✐t❤♠❡t✐❝ ♣r♦❣r❡ss✐♦♥ ✇✐t❤ b ❛s ✐ts ✐♥❝r❡♠❡♥t✳

❉❡✜♥✐t✐♦♥ ✶✳✷✳✻✿ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥ ❆ s❡q✉❡♥❝❡ ❞❡✜♥❡❞ ✭r❡❝✉rs✐✈❡❧②✮ ❜② t❤❡ ❢♦r♠✉❧❛✿ bn+1 = bn · r

✇✐t❤ r 6= 0✱ ✐s ❝❛❧❧❡❞ ❛ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥ ✇✐t❤ r ❛s ✐ts r❛t✐♦✳ ❲❡ s❛② t❤❛t t❤✐s ✐s✿ • ❛ ❣❡♦♠❡tr✐❝ ❣r♦✇t❤ ✇❤❡♥ r > 1✱ ❛♥❞ • ❛ ❣❡♦♠❡tr✐❝ ❞❡❝❛② ✇❤❡♥ r < 1✳ ❆❧t❡r♥❛t✐✈❡❧②✱ ✐t ✐s ❝❛❧❧❡❞ ❛♥ ❡①♣♦♥❡♥t✐❛❧ ❣r♦✇t❤ ❛♥❞ ❞❡❝❛②✱ r❡s♣❡❝t✐✈❡❧②✳

❚❤❡♦r❡♠ ✶✳✷✳✼✿ ❋♦r♠✉❧❛ ❢♦r ❆r✐t❤♠❡t✐❝ Pr♦❣r❡ss✐♦♥ ❚❤❡

nt❤✲t❡r♠ ❢♦r♠✉❧❛ ❢♦r ❛♥ ❛r✐t❤♠❡t✐❝ ♣r♦❣r❡ss✐♦♥ ✇✐t❤ ✐♥❝r❡♠❡♥t a ✭t❤❛t st❛rts

✇✐t❤ c✮ ✐s

an = c + an ,

n = 0, 1, 2, 3, ...

✶✳✷✳

■♥✜♥✐t❡ s❡q✉❡♥❝❡s ❛♥❞ t❤❡✐r ❧♦♥❣✲t❡r♠ tr❡♥❞s

✷✻

❚❤❡♦r❡♠ ✶✳✷✳✽✿ ❋♦r♠✉❧❛s ❢♦r ●❡♦♠❡tr✐❝ Pr♦❣r❡ss✐♦♥s

❚❤❡

c✮

nt❤✲t❡r♠

❢♦r♠✉❧❛ ❢♦r ❛

❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥ ✇✐t❤ r❛t✐♦ r

✭t❤❛t st❛rts ✇✐t❤

✐s

bn = crn ,

n = 0, 1, 2, 3, ...

❙❡q✉❡♥❝❡s ❛r❡♥✬t ✉s✉❛❧❧② t❤✐s s♣❡❝✐✜❝✳ ❲❡ ❝♦✉❧❞ ✈✐s✉❛❧✐③❡ s❡q✉❡♥❝❡s ❛s t❤❡ ❣r❛♣❤s ♦❢ ❢✉♥❝t✐♦♥s✿

❍♦✇❡✈❡r✱ ✇❡ t❛❦❡ ❛ ❞✐✛❡r❡♥t ❛♣♣r♦❛❝❤ ✐♥ t❤✐s ❝❤❛♣t❡r❀ ✇❡ ✇♦✉❧❞ ❧✐❦❡ t♦ ❛♣♣❧②✱ ❛t ❛ ❧❛t❡r t✐♠❡✱ ✇❤❛t ✇❡ ❤❛✈❡ ❧❡❛r♥❡❞ ❛❜♦✉t s❡q✉❡♥❝❡s t♦ ♦✉r st✉❞② ♦❢ ❢✉♥❝t✐♦♥s✳ ❚❤✐s ✐s ✇❤② ♦✉r ✈✐s✉❛❧✐③❛t✐♦♥s ♦❢ ❣r❛♣❤s ♦❢ s❡q✉❡♥❝❡s ✇✐❧❧ ✉s❡ ❛ ❈❛rt❡s✐❛♥ ❝♦♦r❞✐♥❛t❡ s②st❡♠ ✇✐t❤ t❤❡s❡ ❛①❡s✿

•

t❤❡

❤♦r✐③♦♥t❛❧ ❛①✐s

•

t❤❡

✈❡rt✐❝❛❧ ❛①✐s

✐s t❤❡

✐s t❤❡

n✲❛①✐s✱

❛♥❞

x✲❛①✐s✳

❚❤✐s ❛♣♣r♦❛❝❤ ❛❧❧♦✇s ✉s t♦ ❤❛✈❡ ❛ ♠♦r❡ ❝♦♠♣❛❝t ✇❛② t♦ ✈✐s✉❛❧✐③❡ s❡q✉❡♥❝❡s ✭r✐❣❤t✮ ❛s ♦♥ t❤❡

x✲❛①✐s

s❡q✉❡♥❝❡s ♦❢ ❧♦❝❛t✐♦♥s

✈✐s✐t❡❞ ♦✈❡r ❛♥ ✐♥✜♥✐t❡ ♣❡r✐♦❞ ♦❢ t✐♠❡✳ ❚❤❡ ❧♦♥❣✲t❡r♠ tr❡♥❞ ❜❡❝♦♠❡s ❝❧❡❛r ✇❤❡♥ t❤❡ ♣♦✐♥ts

st♦♣ ✈✐s✐❜❧② ✏♠♦✈✐♥❣✑✳ ❊①❛♠♣❧❡ ✶✳✷✳✾✿ r❡❝✐♣r♦❝❛❧s

❚❤❡ ❣♦✲t♦ ❡①❛♠♣❧❡ ✐s t❤❡ s❡q✉❡♥❝❡ ♦❢ t❤❡ r❡❝✐♣r♦❝❛❧s✿

xn = ■t ❛♣♣❡❛rs t♦

t❡♥❞ t♦ 0✳

1 . n

✶✳✷✳

■♥✜♥✐t❡ s❡q✉❡♥❝❡s ❛♥❞ t❤❡✐r ❧♦♥❣✲t❡r♠ tr❡♥❞s

✷✼

❚❤✐s ❢❛❝t ✐s ❡❛s② t♦ ❝♦♥✜r♠ ♥✉♠❡r✐❝❛❧❧②✿

xn = 1.000, 0.500, 0.333, 0.250, 0.200, 0.167, 0.143, 0.125, 0.111, ... ❊①❛♠♣❧❡ ✶✳✷✳✶✵✿ ♣❧♦tt✐♥❣

❍♦✇❡✈❡r✱ ♥✉♠❡r✐❝❛❧ ❛♥❛❧②s✐s ❛❧♦♥❡ ❝❛♥✬t ❜❡ ✉s❡❞ ❢♦r ❞✐s❝♦✈❡r✐♥❣ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❧✐♠✐t✳ P❧♦tt✐♥❣ t❤❡ −.01 ✜rst 1000 t❡r♠s ♦❢ t❤❡ s❡q✉❡♥❝❡ xn = n ❢❛✐❧s t♦ s✉❣❣❡st t❤❡ tr✉❡ ✈❛❧✉❡ ♦❢ t❤❡ ❧✐♠✐t✿

■♥ ❢❛❝t✱ ✐t ✐s ③❡r♦✳

❊①❛♠♣❧❡ ✶✳✷✳✶✶✿ ❞❡❝✐♠❛❧s

❙❡q✉❡♥❝❡s ❛r❡ ✉❜✐q✉✐t♦✉s✳ ❋♦r ❡①❛♠♣❧❡✱ ❣✐✈❡♥ ❛ r❡❛❧ ♥✉♠❜❡r✱ ✇❡ ❝❛♥ ❡❛s✐❧② ❝♦♥str✉❝t ❛ s❡q✉❡♥❝❡ t❤❛t t❡♥❞s t♦ t❤❛t ♥✉♠❜❡r ✕ ✈✐❛ ✐ts ❞❡❝✐♠❛❧ ❛♣♣r♦①✐♠❛t✐♦♥s✳ ❋♦r ❡①❛♠♣❧❡✱

xn = 0.3, 0.33, 0.333, 0.3333, ...

t❡♥❞s t♦

1/3 .

❊①❛♠♣❧❡ ✶✳✷✳✶✷✿ ❛❧t❡r♥❛t✐♥❣

❲❤❡♥ ❛ s❡q✉❡♥❝❡ ❝❤❛♥❣❡s ✐ts s✐❣♥s ❛❧❧ t❤❡ t✐♠❡✱ ✇❡ ❞♦♥✬t ❡①♣❡❝t ✐t t♦ ❤❛✈❡ ❛ ❧✐♠✐t✳ ❆♥ ❡①❛♠♣❧❡ ✐s ❛♥ ✏❛❧t❡r♥❛t✐♥❣ s❡q✉❡♥❝❡✑✿

an = (−1)n . ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿ ✶✳ ■t✬s ✷✳ ■t✬s

−1 ✇❤❡♥ n ✐s ♦❞❞✳ 1 ✇❤❡♥ n ✐s ❡✈❡♥✳

❚❤❡ s❡q✉❡♥❝❡ ❛❧t❡r♥❛t❡s ❜❡t✇❡❡♥ t❤❡s❡ t✇♦ ♥✉♠❜❡rs✿

✶✳✸✳

❚❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❧✐♠✐t

✷✽

❇✉t ✇❤❛t ✐❢ t❤❡ s✇✐♥❣ ✐s ❞✐♠✐♥✐s❤✐♥❣✱ s✉❝❤ ❛s t❤✐s ✏❛❧t❡r♥❛t✐♥❣ r❡❝✐♣r♦❝❛❧s s❡q✉❡♥❝❡✑✿

1 bn = (−1)n ? n ❚❤❡ ✈❛❧✉❡s ❛♣♣r♦❛❝❤ t❤❡ ✉❧t✐♠❛t❡ ❞❡st✐♥❛t✐♦♥ ✕ ❢r♦♠ ❜♦t❤ s✐❞❡s✿

❊①❡r❝✐s❡ ✶✳✷✳✶✸

❲❤❛t ❝❛♥ ②♦✉ s❛② ❛❜♦✉t t❤❡ ❧✐♠✐t ♦❢ ❛♥ ✐♥t❡❣❡r✲✈❛❧✉❡❞ s❡q✉❡♥❝❡❄

❊①❛♠♣❧❡ ✶✳✷✳✶✹✿ ❩❡♥♦✬s ♣❛r❛❞♦①

❈♦♥s✐❞❡r ❛ s✐♠♣❧❡ s❝❡♥❛r✐♦✿ ❆s ②♦✉ ✇❛❧❦ t♦✇❛r❞ ❛ ✇❛❧❧✱ ②♦✉ ❝❛♥ ♥❡✈❡r r❡❛❝❤ ✐t ❜❡❝❛✉s❡✿ ✶✳ ❖♥❝❡ ②♦✉✬✈❡ ❝♦✈❡r❡❞ ❤❛❧❢ t❤❡ ❞✐st❛♥❝❡✱ t❤❡r❡ ✐s st✐❧❧ ❞✐st❛♥❝❡ ❧❡❢t✳ ✷✳ ❖♥❝❡ ②♦✉✬✈❡ ❝♦✈❡r❡❞ ❤❛❧❢ ♦❢ t❤❛t ❞✐st❛♥❝❡✱ t❤❡r❡ ✐s st✐❧❧ ♠♦r❡ ❧❡❢t✳ ✸✳ ❆♥❞ s♦ ♦♥✳ ❲❡ ❤❛✈❡ ❛ s❡q✉❡♥❝❡✿

❲❡ ♠❛r❦ t❤❡s❡ st❡♣s ❛♥❞ ❞♦ r❡❛❧✐③❡ t❤❛t t❤❡r❡ ❛r❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♦❢ t❤❡♠ t♦ ❜❡ t❛❦❡♥✦ ❍♦✇ ✐s t❤✐s ♣♦ss✐❜❧❡❄ ❆ s❤♦rt ❛♥s✇❡r ✐s✿ ❏✉st ❛s ❛♥ ✐♥t❡r✈❛❧ ♦❢ ✐♥t❡r✈❛❧ ♦❢

t✐♠❡✳

s♣❛❝❡

❝❛♥ ✭❛♥❞ ✐s✮ s♣❧✐t ✐♥t♦ ✐♥✜♥✐t❡❧② ♠❛♥② ♣✐❡❝❡s✱ s♦ ❝❛♥ ❛♥

✭❚❤❡ ✐ss✉❡ ✐s ❛❞❞r❡ss❡❞ ✐♥ ❱♦❧✉♠❡ ✸✱ ❈❤❛♣t❡r ✸■❈✲✺✳✮

✶✳✸✳ ❚❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❧✐♠✐t ❈❛❧❝✉❧✉s✱ ✐♥ ❧❛r❣❡ ♣❛rt✱ ✐s t❤❡ st✉❞② ♦❢ ❤♦✇ t♦ ♣r♦♣❡r❧② ❤❛♥❞❧❡

✐♥✜♥✐t②✳

✶✳✸✳

❚❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❧✐♠✐t

✷✾

❊①❛♠♣❧❡ ✶✳✸✳✶✿ s♦♠❡t❤✐♥❣ ❢r♦♠ ♥♦t❤✐♥❣❄

▲❡t✬s ❡①❛♠✐♥❡ t❤✐s s❡❡♠✐♥❣❧② ❧❡❣✐t✐♠❛t❡ ❝♦♠♣✉t❛t✐♦♥✿

(1) (2) (3) (4) (5) (6)

❄

0 == ❄ == ❄ == ❄ == ❄ == ❄ ==

0 (1 1 1 1 1.

+0 −1) +(1 −1) −1 +1 −1 +(−1 +1) +(−1 +0 +0

❚❤❛t✬s ✐♠♣♦ss✐❜❧❡✦ ❲❤❛t✱

0 = 1❄

+0 +(1 −1) +1 −1 +1) +(−1 +0

+0 +(1 −1) +1 −1 +1) +(−1 +0

+... +... +... +1) ... +...

❲❡ ❛r❡ ❛❞❞✐♥❣ ✵s✳ ❲❡ ✉s❡

❲❡ ✉s❡

0 = 1 − 1. − 1 + 1 = 0.

❲❡ ❛r❡ ❛❞❞✐♥❣ ✵s✳

❍♦✇ ❞✐❞ t❤✐s ❤❛♣♣❡♥❄

❈♦♥s✐❞❡r ❛ ❧✐tt❧❡ st♦r② ❜❡❧♦✇✿

❚❤❡ ♥✉♠❜❡rs r❡❢❡r t♦ t❤❡ ❛♠♦✉♥t ♦❢ s♦✐❧ t❛❦❡♥ ♦✉t✱ ❛♥❞ ♦♥❡ ❝❛♥ s❛② t❤❛t ✇❡ ❣♦t ❖✉r ♠✐st❛❦❡ ✇❛s t♦ ❜❡ t♦♦ ❝❛s✉❛❧ ❛❜♦✉t ❝❛rr②✐♥❣ ♦✉t

✐♥✜♥✐t❡❧② ♠❛♥②

s♦♠❡t❤✐♥❣ ❢r♦♠ ♥♦t❤✐♥❣ ✦

❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s✳

❊①❡r❝✐s❡ ✶✳✸✳✷

❲❤✐❝❤ ♦❢ t❤❡ ✏ =✑ s✐❣♥s ❛❜♦✈❡ ✐s ✐♥❝♦rr❡❝t❄ ❍✐♥t✿ ❚❤✐♥❦ ♦❢ t❤✐s ❝♦♠♣✉t❛t✐♦♥ ❛s ❛ ♣r♦❝❡ss✳

❚❤❡ q✉❡st✐♦♥ ❤❛s ❜❡❡♥✿

◮

❲✐t❤ t❤✐s s❡q✉❡♥❝❡✱ ✇❤❛t ♥✉♠❜❡r ❞♦ ✐ts ✈❛❧✉❡s ❛♣♣r♦❛❝❤❄

❲❡ ❝❛♥ ❛❧s♦ t✉r♥ t❤✐s ❛r♦✉♥❞✿

◮

❲✐t❤ t❤✐s ♥✉♠❜❡r✱ t❤❡ ✈❛❧✉❡s ♦❢ ✇❤❛t s❡q✉❡♥❝❡ ❛♣♣r♦❛❝❤ ✐t❄

✶✳✸✳ ❚❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❧✐♠✐t

✸✵

❙♦✱ t❤❡ ❧✐♠✐t ✐s ❛ ♥✉♠❜❡r ❛♥❞ t❤❡ s❡q✉❡♥❝❡ ❛♣♣r♦①✐♠❛t❡s t❤✐s ♥✉♠❜❡r✿ (1) (2) (3) (4) (5) (6)

❚❤❡ s❡q✉❡♥❝❡ ❚❤❡ s❡q✉❡♥❝❡ ❚❤❡ s❡q✉❡♥❝❡ ❚❤❡ s❡q✉❡♥❝❡ ❚❤❡ s❡q✉❡♥❝❡ ❚❤❡ s❡q✉❡♥❝❡

1 .9 1. 3. 1 0

1/2 .99 1.1 3.1 2 1

1/3 .999 1.01 3.14 3 0

1/4 .9999 1.001 3.141 4 1

1/5 .99999 1.0001 3.1415 5 0

... ... ... ... ... ...

❛♣♣r♦①✐♠❛t❡s 0 . ❛♣♣r♦①✐♠❛t❡s 1 . ❛♣♣r♦①✐♠❛t❡s 1 . ❛♣♣r♦①✐♠❛t❡s π . ❛♣♣r♦❛❝❤❡s ∞ . ❞♦❡s♥✬t ❛♣♣r♦①✐♠❛t❡ ❛♥② ♥✉♠❜❡r.

❙♦✱ ✇❡ ❝❛♥ s✉❜st✐t✉t❡ t❤❡ s❡q✉❡♥❝❡ ❢♦r t❤❡ ♥✉♠❜❡r ✐t ❛♣♣r♦①✐♠❛t❡s ❛♥❞ ❞♦ ✐t ✇✐t❤ ❛♥② ❞❡❣r❡❡ ♦❢ ❛❝❝✉r❛❝② ✦ ❲❡ ✉s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♥♦t❛t✐♦♥ ❢♦r t❤❡ ❧✐♠✐t ♦❢ ❛ s❡q✉❡♥❝❡✿ ▲✐♠✐t ♦❢ s❡q✉❡♥❝❡

an → a

❲❡ ❝❛♥ r❡✇r✐t❡ t❤❡ ❛❜♦✈❡✿ ❧✐st (1) (2) (3) (4) (5) (6)

1 .9 1. 3. 1 0

1/2 .99 1.1 3.1 2 1

1/3 .999 1.01 3.14 3 0

1/4 .9999 1.001 3.141 4 1

1/5 .99999 1.0001 3.1415 5 0

... ... ... ... ... ...

nt❤✲t❡r♠ ❢♦r♠✉❧❛ 1/n → 0 1 − 10−n → 1 1 + 10−n → 1

→0 →1 →1 →π → +∞ n → +∞ → ♥♦t❤✐♥❣

◆♦✇✱ ❧❡t✬s ✜♥❞ t❤❡ ❡①❛❝t ♠❡❛♥✐♥❣ ♦❢ ❧✐♠✐t✳ ❊①❛♠♣❧❡ ✶✳✸✳✸✿ tr❛❥❡❝t♦r②

❚❤❡ ✈❛❧✉❡s ♦❢ ❛ ❝♦♥✈❡r❣❡♥t s❡q✉❡♥❝❡ ❛❝❝✉♠✉❧❛t❡ t♦✇❛r❞ ❛ ♣❛rt✐❝✉❧❛r ♥✉♠❜❡r✱ ❥✉st ❛s ✐♥ t❤❡ ❡①❛♠♣❧❡ ♦❢ ❛ ♠♦✈✐♥❣ ❜❛❧❧✿

❲❡ ❝❛♥ s❡❡ t❤❛t ❛❢t❡r s✉✣❝✐❡♥t❧② ♠❛♥② st❡♣s✱ t❤❡ t❡r♠s ♦❢ t❤❡ s❡q✉❡♥❝❡✱ an ✱ ❜❡❝♦♠❡ ✐♥❞✐st✐♥❣✉✐s❤❛❜❧❡ ❢r♦♠ t❤❡ ❧✐♠✐t✱ a✳ ■t s❡❡♠s t❤❛t✱ s❛②✱ t❤❡ 10t❤ ❞♦t ❤❛s ♠❡r❣❡❞ ✇✐t❤ a✳ ❊①❛♠♣❧❡ ✶✳✸✳✹✿ ❣r❛♣❤

❚❤❡ ❢❛❝t t❤❛t t❤❡ ✈❛❧✉❡s ♦❢ ❛ s❡q✉❡♥❝❡ ❛❝❝✉♠✉❧❛t❡ t♦✇❛r❞ ❛ ♣❛rt✐❝✉❧❛r ♥✉♠❜❡r ♠❡❛♥s t❤❛t✱ ❣❡♦♠❡t✲ r✐❝❛❧❧②✱ ✇❡ ✇✐❧❧ ❛❧✇❛②s s❡❡ ❤♦✇ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ s❡q✉❡♥❝❡ ❡♥❞s ✉♣ ❝❧♦s❡r ❛♥❞ ❝❧♦s❡r t♦ ❛ ♣❛rt✐❝✉❧❛r ❤♦r✐③♦♥t❛❧ ❧✐♥❡ ✭❧❡❢t✮✿

❆t t❤❡ ❡♥❞✱ t❤❡ ❞♦ts s❡❡♠ t♦ ❧❛♥❞ ♦♥ t❤❡ x✲❛①✐s ✭♠✐❞❞❧❡✮✳ ❍♦✇❡✈❡r✱ t❤❡ ❣❛♣ ❜❡❝♦♠❡s ✈✐s✐❜❧❡ ❛s ✇❡ ③♦♦♠ ✐♥ ✭r✐❣❤t✮✳

✶✳✸✳

❚❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❧✐♠✐t

✸✶

❊①❛♠♣❧❡ ✶✳✸✳✺✿ ♥✉♠❡r✐❝❛❧ ❞❛t❛

▲❡t✬s ♥♦✇ ❧♦♦❦ ❛t t❤✐s ✏♣r♦❝❡ss✑ ♥✉♠❡r✐❝❛❧❧②✳ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ s♣❡❝✐✜❝ s❡q✉❡♥❝❡✿

an = 1/n2 . ❲❡ ❝♦♠♣✉t❡ ❛ ❢❡✇ ❞♦③❡♥s ♦❢ ✐ts ✈❛❧✉❡s ❛♥❞ t❤❡♥ ❛s❦✿ ❲❤❛t ❞♦❡s ✐t ♠❡❛♥ t❤❛t ✐t ❛♣♣r♦❛❝❤❡s ❋✐rst✱ ❤♦✇ ❧♦♥❣ ❞♦❡s ✐t t❛❦❡ t♦ ❣❡t ✇✐t❤✐♥

.1

❢r♦♠

a = 0❄

a = 0❄

▲♦♦❦ ✉♣ ✐♥ t❤❡ t❛❜❧❡ ♦❢ ✈❛❧✉❡s✿ ■t t❛❦❡s

4

st❡♣s ✭r❡❞✮✿

❲❡ ❥✉st ❢♦❧❧♦✇ t❤❡ ❧✐st ♦❢ ♥✉♠❜❡rs ✉♥t✐❧ t❤❡② ❜❡❝♦♠❡ ❧❡ss t❤❛♥ t❤❡ t❤r❡s❤♦❧❞

.01

❢r♦♠

a❄

■t t❛❦❡s

11

st❡♣s ✭❣r❡❡♥✮✳

.001

❢r♦♠

a❄

■t t❛❦❡s

32

st❡♣s ✭❜❧✉❡✮✳

❙❡❝♦♥❞✱ ❤♦✇ ❧♦♥❣ ❞♦❡s ✐t t❛❦❡ t♦ ❣❡t ✇✐t❤✐♥ ❚❤✐r❞✱ ❤♦✇ ❧♦♥❣ ❞♦❡s ✐t t❛❦❡ t♦ ❣❡t ✇✐t❤✐♥

❆♥❞ s♦ ♦♥✳ ◆♦ ♠❛tt❡r ❤♦✇ s♠❛❧❧ ❛ ♥✉♠❜❡r ■ ♣✐❝❦✱

❡✈❡♥t✉❛❧❧② an

❆♥♦t❤❡r ✐♥t❡r♣r❡t❛t✐♦♥ ♦❢ t❤✐s ❛♥❛❧②s✐s ✐s ✐♥ t❡r♠s ♦❢ ❛♣♣r♦❛❝❤❡s

a=0

❛❝❝✉r❛❝②✳

❛s ❢♦❧❧♦✇s✿ ✏❚❤❡ s❡q✉❡♥❝❡ ❛♣♣r♦①✐♠❛t❡s

an = 1/n2

0✑✳

❋✐rst✱ ✇❤❛t ✐❢ ✇❡ ♥❡❡❞ t❤❡ ❛❝❝✉r❛❝② t♦ ❜❡

• • •

❙❡❝♦♥❞✱ ✇❤❛t ✐❢ ✇❡ ♥❡❡❞ t❤❡ ❛❝❝✉r❛❝② t♦ ❜❡

❆t ❧❡❛st

❚❤✐r❞✱ ✇❤❛t ✐❢ ✇❡ ♥❡❡❞ t❤❡ ❛❝❝✉r❛❝② t♦ ❜❡

❆t ❧❡❛st

.1❄

✇✐❧❧ ❜❡ t❤❛t ❝❧♦s❡ t♦ ✐ts ❧✐♠✐t✳

❲❡ ✉♥❞❡rst❛♥❞ t❤❡ ✐❞❡❛ t❤❛t

•

4

.1✳

▲♦♦❦ ✉♣ ✐♥ t❤❡ t❛❜❧❡ ♦❢ ✈❛❧✉❡s✿ ❲❡ ♥❡❡❞ t♦ ❝♦♠♣✉t❡

t❡r♠s ♦❢ t❤❡ s❡q✉❡♥❝❡ ♦r ♠♦r❡✳

.01❄ .001❄

11 t❡r♠s✳ 32✳

❆♥❞ s♦ ♦♥✳

◆♦ ♠❛tt❡r ❤♦✇ ♠✉❝❤ ❛❝❝✉r❛❝② ■ ♥❡❡❞✱ t❤❡r❡ ✐s ❛ ✇❛② t♦ ❛❝❝♦♠♠♦❞❛t❡ t❤✐s r❡q✉✐r❡♠❡♥t ❜② ❣❡tt✐♥❣ ❢❛rt❤❡r ❛♥❞ ❢❛rt❤❡r ✐♥t♦ t❤❡ s❡q✉❡♥❝❡

an ✳

❊①❡r❝✐s❡ ✶✳✸✳✻

❍♦✇ ❧♦♥❣ ❞♦❡s ✐t t❛❦❡ t♦ ❣❡t ✇✐t❤✐♥

.0003

❢r♦♠

a = 0❄

❊①❛♠♣❧❡ ✶✳✸✳✼✿ ♥✉♠❡r✐❝❛❧ ❞❛t❛ ❢r♦♠ ❣r❛♣❤

❯♥❢♦rt✉♥❛t❡❧②✱ ♥♦t ❛❧❧ s❡q✉❡♥❝❡s ❛r❡ ❛s s✐♠♣❧❡ ❛s t❤❛t✳ ❚❤❡② ♠❛② ❛♣♣r♦❛❝❤ t❤❡✐r r❡s♣❡❝t✐✈❡ ❧✐♠✐ts ✐♥ ❛ ♥✉♠❜❡r ♦❢ ✇❛②s✱ ❛s ✇❡ ❤❛✈❡ s❡❡♥✳ ❚❤❡② ❞♦♥✬t ❤❛✈❡ t♦ ❜❡ ♠♦♥♦t♦♥❡✿

✶✳✸✳ ❚❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❧✐♠✐t

✸✷

❚❤❡② ♠✐❣❤t ❛♣♣r♦❛❝❤ t❤❡ ❧✐♠✐t ❢r♦♠ ❛❜♦✈❡ ❛♥❞ ❜❡❧♦✇ ❛t t❤❡ s❛♠❡ t✐♠❡✿

❖✉r q✉❡st✐♦♥s ❝❛♥ st✐❧❧ ❜❡ ❛s❦❡❞ ❛♥❞ ❛♥s✇❡r❡❞✿ • ❍♦✇ ❧♦♥❣ ❞♦❡s ✐t t❛❦❡ t♦ ❣❡t ✇✐t❤✐♥ .4 ❢r♦♠ a = 0❄ ▲♦♦❦ ✉♣ ✐♥ t❤❡ ❣r❛♣❤✿ ■t t❛❦❡s 4 st❡♣s t♦ ❣❡t ✇✐t❤✐♥ t❤❡ ❜❛♥❞ ❢r♦♠ −.4 t♦ .4✳ • ❍♦✇ ❧♦♥❣ ❞♦❡s ✐t t❛❦❡ t♦ ❣❡t ✇✐t❤✐♥ .2 ❢r♦♠ a = 0❄ ■t t❛❦❡s 6 st❡♣s t♦ ❣❡t ✇✐t❤✐♥ t❤❡ ❜❛♥❞ ❢r♦♠ −.2 t♦ .2✳ • ❆♥❞ s♦ ♦♥✳ ❚❤✐s ✐s ❤♦✇ ✐t ✐s ❞♦♥❡✿

❊①❡r❝✐s❡ ✶✳✸✳✽

❋♦r t❤❡ ✜rst ❣r❛♣❤ ✐♥ t❤❡ ❡①❛♠♣❧❡✱ ❤♦✇ ❧♦♥❣ ❞♦❡s ✐t t❛❦❡ t♦ ❣❡t ✇✐t❤✐♥ .4✱ .2✱ .1 ❢r♦♠ a = 0❄ ❊①❡r❝✐s❡ ✶✳✸✳✾

❋♦r t❤❡ s❡❝♦♥❞ ❣r❛♣❤ ✐♥ t❤❡ ❡①❛♠♣❧❡✱ ❤♦✇ ❧♦♥❣ ❞♦❡s ✐t t❛❦❡ t♦ ❣❡t ✇✐t❤✐♥ .1 ❢r♦♠ a = 0❄ ❇✉t ✇❤❛t ✐❢ t❤❡ ❛❝❝✉r❛❝② r❡q✉✐r❡♠❡♥ts ❦❡❡♣ ✐♥❝r❡❛s✐♥❣❄ ■t ✇♦✉❧❞ ❜❡ ♥✐❝❡ ✐❢ ✇❡ ❝♦✉❧❞ s♦❧✈❡ t❤❡ ♣r♦❜❧❡♠ ♦♥❝❡ ❛♥❞ ❢♦r ❛❧❧✦ ❇❡❧♦✇ ✇❡ r❡✇r✐t❡ ✇❤❛t ✇❡ ✇❛♥t t♦ s❛② ❛❜♦✉t t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ ❧✐♠✐t ✐♥ ♣r♦❣r❡ss✐✈❡❧② ♠♦r❡ ❛♥❞ ♠♦r❡ ♣r❡❝✐s❡

✶✳✸✳

❚❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❧✐♠✐t

✸✸

t❡r♠s✿

n

y = an

1.

❆s

n → ∞,

2.

❆s

n

✇❡ ❤❛✈❡

❛♣♣r♦❛❝❤❡s

∞,

y

y → a.

❛♣♣r♦❛❝❤❡s

a.

❲❡ ✐♥t❡r♣r❡t ✏❛♣♣r♦❛❝❤✑ ✐♥ t❡r♠s ♦❢ ❞✐st❛♥❝❡✳

3.

❆s

n

✐s ❣❡tt✐♥❣ ❧❛r❣❡r ❛♥❞ ❧❛r❣❡r,

t❤❡ ❞✐st❛♥❝❡ ❢r♦♠

y

t♦

❚❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥

4.

❇② ♠❛❦✐♥❣

n

❧❛r❣❡r ❛♥❞ ❧❛r❣❡r,

✇❡ ♠❛❦❡

❚❤❡ ❛❜s♦❧✉t❡ ✈❛❧✉❡s ❛❜♦✈❡ ❛r❡ t❤❡ ❞✐st❛♥❝❡s ❢r♦♠

an

t♦

a✱

|y − a|

a y

❣♦❡s ❛♥❞

0.

a

✐s

|y − a| .

❛s s♠❛❧❧ ❛s r❡q✉✐r❡❞.

❛s s❤♦✇♥ ❜❡❧♦✇✿

❲❡ ♠❛❦❡ t❤❡ ❞❡✜♥✐t✐♦♥ ❡✈❡♥ ♠♦r❡ ♣r❡❝✐s❡✿

n

y = an

4.

❇② ♠❛❦✐♥❣

n

❧❛r❣❡r ❛♥❞ ❧❛r❣❡r,

5.

❇② ♠❛❦✐♥❣

n

❧❛r❣❡r t❤❛♥ s♦♠❡

N > 0,

✇❡ ♠❛❦❡

|y − a|

❛s s♠❛❧❧ ❛s ♥❡❡❞❡❞.

✇❡ ♠❛❦❡

|y − a|

s♠❛❧❧❡r t❤❛♥ ❛♥② ❣✐✈❡♥

❆❧❣❡❜r❛✐❝❛❧❧②✱ ✇❡ s❡❡ t❤❛t ❢♦r ❡✈❡r② ♠❡❛s✉r❡ ♦❢ ✏❝❧♦s❡♥❡ss✑✱ ❝❛❧❧ ✐t t❤❛t ❝❧♦s❡ t♦ t❤❡ ❧✐♠✐t✳ ■♥ ♦t❤❡r ✇♦r❞s✱

ε

✐s

ε✱

ε > 0.

t❤❡ ❢✉♥❝t✐♦♥✬s ✈❛❧✉❡s ❡✈❡♥t✉❛❧❧② ❜❡❝♦♠❡

t❤❡ ❞❡❣r❡❡ ♦❢ r❡q✉✐r❡❞ ❛❝❝✉r❛❝②✳ ❲❛r♥✐♥❣✦

■t ✐s ε t❤❛t ❝♦♠❡s ✜rst✱ t❤❡♥ N ✳ ❊①❛♠♣❧❡ ✶✳✸✳✶✵✿ ❛❜s♦❧✉t❡ ❛❝❝✉r❛❝② ▲❡t✬s ❝♦♥s✐❞❡r t❤❡ ❧❛st ✐t❡r❛t✐♦♥ ♦❢ ♦✉r ❞❡✜♥✐t✐♦♥ ✐♥ ❧✐❣❤t ♦❢ t❤❡ ❡①❛♠♣❧❡ ❛❜♦✈❡ ❝♦♥❝❡r♥✐♥❣ t❤❡ s❡q✉❡♥❝❡✿

an = 1/n2 .

✶✳✸✳ ❚❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❧✐♠✐t

✸✹

❚❤❡s❡ ❛r❡ t❤❡ N ✬s ✇❡ ❣♦t ❢r♦♠ t❤♦s❡ ε✬s✿ ❇② ♠❛❦✐♥❣ n ❧❛r❣❡r t❤❛♥ s♦♠❡

N > 0, ✇❡ ♠❛❦❡ |an − a| s♠❛❧❧❡r t❤❛♥ ❛♥② ❣✐✈❡♥

ε > 0.

N =4

←−

ε = .1

N = 11

←−

ε = .01

N = 32

←−

ε = .001

◆♦✇✱ ❧❡t✬s ✐♠❛❣✐♥❡ t❤❛t ❛♥② ❞❡❣r❡❡ ♦❢ ❛❝❝✉r❛❝② ε > 0 t❤❛t ♥❡❡❞s t♦ ❜❡ ❛❝❝♦♠♠♦❞❛t❡❞ ♠✐❣❤t ❜❡ s✉♣♣❧✐❡❞ ❛❤❡❛❞ ♦❢ t✐♠❡✳ ▲❡t✬s ✜♥❞ s✉❝❤ ❛♥ n t❤❛t an ✐s ✇✐t❤✐♥ ε ❢r♦♠ a = 0✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ♥❡❡❞ t❤✐s ✐♥❡q✉❛❧✐t② t♦ ❜❡ s❛t✐s✜❡❞✿ 1 1 |an − a| = 2 − 0 = 2 < ε . n n

❲❡ s♦❧✈❡ ✐t ❛♥❞ ❝❤♦♦s❡ N ✿

1 n> √ =N. ε

❚❤✐s ♣r♦✈❡s t❤❛t t❤❡ r❡q✉✐r❡♠❡♥t ❝❛♥ ❜❡ s❛t✐s✜❡❞✳ ❚❤❡♥✱ ❢♦r ❛♥② s✉❝❤ n ✇❡ ❤❛✈❡ |an −a| < ε✱ ❛s r❡q✉✐r❡❞✳ ❚❤❡ ♣r♦❜❧❡♠ ✐s s♦❧✈❡❞ ♦♥❝❡ ❛♥❞ ❢♦r ❛❧❧❀ t❤❡ r❡s✉❧t ❣✐✈❡s ✉s t❤❡ s❛♠❡ ❛♥s✇❡rs ❢♦r t❤❡ t❤r❡❡ ♣❛rt✐❝✉❧❛r ❝❤♦✐❝❡s ♦❢ ε = .1, .01, .001 ❢r♦♠ t❤❡ ❧❛st ❡①❛♠♣❧❡✱ ❛s ✇❡❧❧ ❛s ❢♦r ❛♥② ♦t❤❡r✦ ❋♦r ❡①❛♠♣❧❡✱ ✐❢ ✇❡ ♣✐❝❦ ε = .01✱ ✇❤❛t ✐s N ❄ ❇② t❤❡ ❢♦r♠✉❧❛✱ ✐t ✐s 1 1 1 N=√ = 10 . = = 10−1 .01 (10−2 )1/2

■❢ ✇❡ ♣✐❝❦ ε = .0001✱ ✇❤❛t ✐s N ❄ ❇② t❤❡ ❢♦r♠✉❧❛✱ ✐t ✐s N=√

1 1 1 = = −2 = 102 = 100 . 1/2 10 .0001 (10−4 )

❊①❡r❝✐s❡ ✶✳✸✳✶✶

√

❈❛rr② ♦✉t s✉❝❤ ❛♥ ❛♥❛❧②s✐s ❢♦r an = 1/ n✳ ◆♦✇ t❤❡ ❝♦♥❝❡♣t t❤❛t ♠♦st ♦❢ ❝❛❧❝✉❧✉s ✐s ❜✉✐❧t ✉♣♦♥✿ ❉❡✜♥✐t✐♦♥ ✶✳✸✳✶✷✿ ❧✐♠✐t ♦❢ s❡q✉❡♥❝❡

❲❡ ❝❛❧❧ ♥✉♠❜❡r a t❤❡ ❧✐♠✐t ♦❢ ❛ s❡q✉❡♥❝❡ an ✐❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥ ❤♦❧❞s✿ ◮ ❋♦r ❡❛❝❤ r❡❛❧ ♥✉♠❜❡r ε > 0✱ t❤❡r❡ ❡①✐sts ❛ ♥✉♠❜❡r N s✉❝❤ t❤❛t ❢♦r ❡✈❡r② ♥❛t✉r❛❧ ♥✉♠❜❡r n > N ✱ ✇❡ ❤❛✈❡✿ |an − a| < ε .

❲❡ ❛❧s♦ s❛② t❤❛t t❤❡ ❧✐♠✐t ✐s ✜♥✐t❡✳ ■❢ ❛ s❡q✉❡♥❝❡ ❤❛s ❛ ❧✐♠✐t✱ t❤❡♥ ✇❡ ❝❛❧❧ t❤❡ s❡q✉❡♥❝❡ ❝♦♥✈❡r❣❡♥t ❛♥❞ s❛② t❤❛t ✐t ❝♦♥✈❡r❣❡s ❀ ♦t❤❡r✇✐s❡ ✐t ✐s ❞✐✈❡r❣❡♥t ❛♥❞ ✇❡ s❛② ✐t ❞✐✈❡r❣❡s✳ ❊①❛♠♣❧❡ ✶✳✸✳✶✸✿ ✏❢♦r ❛♥② ✳✳✳ t❤❡r❡ ✐s✑

❚❤❡ ❞❡✜♥✐t✐♦♥ ✐s ❧♦❣✐❝❛❧❧② ❝♦♠♣❧❡①❀ ✐t ✐s ❛ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t✇♦ ❢❛♠✐❧✐❛r ❝♦♥str✉❝ts✳ ❚❤❡ ✜rst ♦♥❡✿ • ❚✇♦ ❢✉♥❝t✐♦♥s ❛r❡ ❡q✉❛❧✱ f = g ✱ ✇❤❡♥ ❋❖❘ ❆◆❨ x ✐♥ t❤❡ ❞♦♠❛✐♥✱ ✇❡ ❤❛✈❡ f (x) = g(x)✳ • ❋❖❘ ❆◆❨ ♥✉♠❜❡r✱ ✐ts sq✉❛r❡ ❝❛♥♥♦t ❜❡ ♥❡❣❛t✐✈❡✳

✶✳✸✳

❚❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❧✐♠✐t • ❋❖❘ ❆◆❨

✸✺

t✇♦ ♦❞❞ ♥✉♠❜❡rs✱ t❤❡✐r s✉♠ ✐s ❡✈❡♥✳

❚❤❡ s❡❝♦♥❞ ♦♥❡✿

• ❚✇♦ ❢✉♥❝t✐♦♥s ❛r❡ ✉♥❡q✉❛❧✱ f 6= g ✱ ✇❤❡♥ ❚❍❊❘❊ ■❙ • ❚❍❊❘❊ ❆❘❊ ♥♦ ❣❛♣s ✐♥ t❤❡ ❣r❛♣❤ ♦❢ y = x2 ✳ • ❚❍❊❘❊ ■❙ ❛ r❡❝t❛♥❣✉❧❛r ❡♥❝❧♦s✉r❡ ✇✐t❤ t❤❡ ❧❛r❣❡st

s✉❝❤ ❛♥

x

t❤❛t

f (x) 6= g(x)✳

♣♦ss✐❜❧❡ ❛r❡❛ ✇❤❡♥

100

②❛r❞s ♦❢ ❢❡♥❝✐♥❣ ✐s

❣✐✈❡♥✳

❚❤✐s ✐s ❤♦✇ t❤❡ t✇♦ ❛r❡ ♦❢t❡♥ ❝♦♠❜✐♥❡❞✿

• ❋❖❘ ❆◆❨ • ❋❖❘ ❆◆❨ • ❚❍❊❘❊ ■❙

♥✉♠❜❡r✱

❚❍❊❘❊ ■❙

❛ ❧❛r❣❡r ♥✉♠❜❡r✳

t✇♦ ♥♦♥✲♣❛r❛❧❧❡❧ ❧✐♥❡s✱

❚❍❊❘❊ ■❙

❛♥ ✐♥t❡rs❡❝t✐♦♥✳

❋❖❘ ❆◆❨

❛ ❧♦❝❛t✐♦♥ ♦♥ ❛ r✉❜❜❡r ❜❛♥❞ t❤❛t ✇✐❧❧ r❡♠❛✐♥ ✐♥ ♣❧❛❝❡

str❡t❝❤❡s ❛♥❞ s❤r✐♥❦s✳

❚❤❡ st❛t❡♠❡♥t ✐♥ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢

❧✐♠✐t

❝♦♠❜✐♥❛t✐♦♥ ♦❢

✐s ❡✈❡♥ ♠♦r❡ ❝♦♠♣❧❡① ✇✐t❤ t❤❡s❡ ♣❤r❛s❡s ❛♣♣❡❛r✐♥❣ t❤r❡❡

t✐♠❡s✿

◮ ❋❖❘ ❆◆❨ r❡❛❧ ♥✉♠❜❡r ε > 0✱ ❚❍❊❘❊ ■❙ ❛ ♥✉♠❜❡r N n > N ✱ ✇❡ ❤❛✈❡ |an − a| < ε .

❋❖❘ ❆◆❨

s✉❝❤ t❤❛t

♥❛t✉r❛❧

♥✉♠❜❡r

❊①❡r❝✐s❡ ✶✳✸✳✶✹ ❘❡st❛t❡ ✏ ❚❍❊❘❊ ✐s ❣✐✈❡♥✑ ❛s

■❙ ❛ r❡❝t❛♥❣✉❧❛r ❡♥❝❧♦s✉r❡ ✇✐t❤ t❤❡ ❧❛r❣❡st ♣♦ss✐❜❧❡ ✏ ❚❍❊❘❊ ■❙ ❛ r❡❝t❛♥❣✉❧❛r ❡♥❝❧♦s✉r❡ s♦ t❤❛t ❋❖❘ ❆◆❨ ✳✳✳✑✳

❛r❡❛ ✇❤❡♥

100

②❛r❞s ♦❢ ❢❡♥❝✐♥❣

❊①❡r❝✐s❡ ✶✳✸✳✶✺ ❙✉❣❣❡st ♠♦r❡ ✐♥st❛♥❝❡s ♦❢ st❛t❡♠❡♥ts ❢♦r t❤❡ ❧❛st ❡①❛♠♣❧❡✳

❚♦ ♣r♦✈❡ t❤❡ st❛t❡♠❡♥t ❢♦r ❛ s♣❡❝✐✜❝ s❡q✉❡♥❝❡✱ ✇❡ ✇✐❧❧ ♥❡❡❞ t♦ ✜♥❞ s✉❝❤ ❛♥

N

❢♦r ❡❛❝❤

ε > 0✳

❊①❛♠♣❧❡ ✶✳✸✳✶✻✿ ❧✐♠✐t ❢r♦♠ ❞❡✜♥✐t✐♦♥ ▲❡t✬s ❛♣♣❧② t❤❡ ❞❡✜♥✐t✐♦♥ t♦

an = 1 + ❙✉♣♣♦s❡ ❛♥

ε>0

(−1)n . n

✐s ❣✐✈❡♥✳ ▲♦♦❦✐♥❣ ❛t t❤❡ ♥✉♠❜❡rs✱ ✇❡ ❞✐s❝♦✈❡r t❤❛t t❤❡② ❛❝❝✉♠✉❧❛t❡ t♦✇❛r❞

t❤✐s t❤❡ ❧✐♠✐t❄ ❲❡ ❛♣♣❧② t❤❡ ❞❡✜♥✐t✐♦♥✳ ▲❡t✬s ✜♥❞ s✉❝❤ ❛♥

t❤❛t

an

✐s ✇✐t❤✐♥

n (−1)n 1 (−1) = < ε. − 1 = |an − a| = 1 + n n n

❲❡ ❝♦♥❝❡♥tr❛t❡ ♦♥ t❤❡ ❧❛st ♣❛rt✿ ❋♦r ❡✈❡r② ✇❡ ❤❛✈❡✿

❈❛♥ ✇❡❄

n

ε✱

✇❡ ♥❡❡❞ t♦ ♣♦✐♥t ♦✉t s✉❝❤ ❛♥

ε

N

❢r♦♠

1✳

■s

a = 1✿

t❤❛t ❢♦r ❡✈❡r②

n>N

1 < ε. n ❖❢ ❝♦✉rs❡✱ t❤❡ ❢r❛❝t✐♦♥ ✇✐❧❧ ❜❡❝♦♠❡ s♠❛❧❧❡r ❛♥❞ s♠❛❧❧❡r ❛s ✇❡ ✐♥❝r❡❛s❡ t❤❡ ❞❡♥♦♠✐♥❛t♦r✳

❆❧❣❡❜r❛✐❝❛❧❧②✱ ✇❡ s♦❧✈❡ t❤✐s ✐♥❡q✉❛❧✐t② ❛s ❢♦❧❧♦✇s✿

n> ❚❤❛t ❣✐✈❡s ✉s t❤❡

N

r❡q✉✐r❡❞ ❜② t❤❡ ❞❡✜♥✐t✐♦♥❀ ✇❡ ❧❡t

N= ❚❤❡♥✱ ❢♦r ❛♥②

n>N

1 . ε

✇❡ ❤❛✈❡

|an − a| < ε✱

1 . ε

❛s r❡q✉✐r❡❞ ❜② t❤❡ ❞❡✜♥✐t✐♦♥✳

❚❤❡ ✇❛② t♦ ✈✐s✉❛❧✐③❡ ❛ tr❡♥❞ ✐♥ t❤✐s ❝♦♥✈❡r❣❡♥t s❡q✉❡♥❝❡ ✐s t♦ ❡♥❝❧♦s❡ t❤❡ ❡♥❞ ♦❢ t❤❡ t❛✐❧ ♦❢ t❤❡ s❡q✉❡♥❝❡ ✐♥ ❛

❜❛♥❞ ✿

✶✳✸✳ ❚❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❧✐♠✐t

✸✻

■t s❤♦✉❧❞ ❜❡✱ ✐♥ ❢❛❝t✱ ❛ ♥❛rr♦✇❡r ❛♥❞ ♥❛rr♦✇❡r ❜❛♥❞❀ ✐ts ✇✐❞t❤ ✐s 2ε✳ ▼❡❛♥✇❤✐❧❡✱ t❤❡ st❛rt✐♥❣ ♣♦✐♥t ♦❢ t❤❡ ❜❛♥❞ ♠♦✈❡s t♦ t❤❡ r✐❣❤t❀ t❤❛t✬s N ✳ ❊①❛♠♣❧❡s ♦❢ ❞✐✈❡r❣❡♥❝❡ ❛r❡ ❜❡❧♦✇✳

❊①❛♠♣❧❡ ✶✳✸✳✶✼✿ ❞✐✈❡r❣❡♥❝❡ t♦ ✐♥✜♥✐t② ❆ s❡q✉❡♥❝❡ ♠❛② t❡♥❞ t♦ ✐♥✜♥✐t②✱ s✉❝❤ ❛s an = n✿

❚❤❡♥ ♥♦ ❜❛♥❞ ✕ ♥♦ ♠❛tt❡r ❤♦✇ ✇✐❞❡ ✕ ✇✐❧❧ ❝♦♥t❛✐♥ t❤❡ s❡q✉❡♥❝❡✬s t❛✐❧✳ ❚❤✐s ❜❡❤❛✈✐♦r✱ ❤♦✇❡✈❡r✱ ❤❛s ❛ ♠❡❛♥✐♥❣❢✉❧ ♣❛tt❡r♥✳ ❇✉t ❤♦✇ ❞♦ ✇❡ ❡①♣❧❛✐♥ t❤❛t t❤❡ s❡q✉❡♥❝❡ ✐s ❛♣♣r♦❛❝❤✐♥❣ ✐♥✜♥✐t②❄ ■t ✇✐❧❧ ❣❡t ♦✈❡r ❛♥② t❤r❡s❤♦❧❞ ♥♦ ♠❛tt❡r ❤♦✇ ❤✐❣❤✿

❉❡✜♥✐t✐♦♥ ✶✳✸✳✶✽✿ ✐♥✜♥✐t❡ ❧✐♠✐t ♦❢ s❡q✉❡♥❝❡ ❲❡ s❛② t❤❛t ❛ s❡q✉❡♥❝❡ an t❡♥❞s t♦ ♣♦s✐t✐✈❡ ✐♥✜♥✐t② ✐❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥ ❤♦❧❞s✿ ◮ ❋♦r ❡❛❝❤ r❡❛❧ ♥✉♠❜❡r R✱ t❤❡r❡ ❡①✐sts ❛ ♥❛t✉r❛❧ ♥✉♠❜❡r N s✉❝❤ t❤❛t ❢♦r ❡✈❡r② ♥❛t✉r❛❧ ♥✉♠❜❡r n > N ✱ ✇❡ ❤❛✈❡✿

an > R . ❲❡ s❛② t❤❛t ❛ s❡q✉❡♥❝❡ an t❡♥❞s t♦ ♥❡❣❛t✐✈❡ ✐♥✜♥✐t② ✐❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥ ❤♦❧❞s✿ ◮ ❋♦r ❡❛❝❤ r❡❛❧ ♥✉♠❜❡r R✱ t❤❡r❡ ❡①✐sts ❛ ♥❛t✉r❛❧ ♥✉♠❜❡r N s✉❝❤ t❤❛t ❢♦r ❡✈❡r② ♥❛t✉r❛❧ ♥✉♠❜❡r n > N ✱ ✇❡ ❤❛✈❡

an < R .

✶✳✸✳ ❚❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❧✐♠✐t

✸✼

■♥ ❡✐t❤❡r ❝❛s❡✱ ✇❡ ❛❧s♦ s❛② t❤❛t t❤❡ ❧✐♠✐t ✐s ✐♥✜♥✐t❡✳ ❊①❛♠♣❧❡ ✶✳✸✳✶✾✿ ✏❢♦r ❛♥② ✳✳✳ t❤❡r❡ ✐s✑

❚❤❡ ❧♦❣✐❝ ♦❢ t❤❡ ❞❡✜♥✐t✐♦♥ ✐s t❤❡ s❛♠❡ ❛s t❤❛t ♦❢ t❤❡ ❧❛st ♦♥❡✿ ◮ ❋❖❘ ❆◆❨ r❡❛❧ ♥✉♠❜❡r R > 0✱ ❚❍❊❘❊ ■❙ ❛ ♥✉♠❜❡r N s✉❝❤ t❤❛t ❋❖❘ ❆◆❨ ♥❛t✉r❛❧ ♥✉♠❜❡r n > N ✱ ✇❡ ❤❛✈❡ an > R .

❚❤❡ ❢♦❧❧♦✇✐♥❣ ♥♦t❛t✐♦♥ ✐s r♦✉t✐♥❡❧② ✉s❡❞✿ ■♥✜♥✐t❡ ❧✐♠✐t ♦❢ s❡q✉❡♥❝❡

an → ±∞ ❊①❛♠♣❧❡ ✶✳✸✳✷✵✿ ❞✐✈❡r❣❡♥❝❡ t♦ ✐♥✜♥✐t②

▲❡t✬s ♣r♦✈❡ t❤❡ ❧❛st ❡①❛♠♣❧❡ an = n✳ ❲❡ ♥❡❡❞ t♦ ♣r♦✈❡ t❤❛t✿ ◮ ❢♦r ❡❛❝❤ r❡❛❧ ♥✉♠❜❡r R✱ t❤❡r❡ ❡①✐sts ❛ ♥❛t✉r❛❧ ♥✉♠❜❡r N s✉❝❤ t❤❛t ❢♦r ❡✈❡r② ♥❛t✉r❛❧ ♥✉♠❜❡r n > N ✱ ✇❡ ❤❛✈❡✿ an = n > R .

❊❛s②✱ ✇❡ ❝❤♦♦s❡ N = R✦ ■♥❞❡❡❞✿ n > N =⇒ an = n > N = R . ❊①❡r❝✐s❡ ✶✳✸✳✷✶

Pr♦✈❡ t❤❛t t❤❡ ❧✐♠✐ts ❛r❡ ✐♥✜♥✐t❡ ❢♦r t❤❡ ❢♦❧❧♦✇✐♥❣ s❡q✉❡♥❝❡s✿ ✭❛✮ −n ❀ ✭❜✮ n2 ❀ ✭❝✮

√

n✳

❊✈❡♥ t❤♦✉❣❤ t❤❡ t✇♦ t②♣❡s ♦❢ ❧♦♥❣✲t❡r♠ ❜❡❤❛✈✐♦r ❛♣♣❡❛r ✈❡r② ❞✐✛❡r❡♥t✱ t❤❡r❡ ✐s ❛ s✐♠✐❧❛r✐t② ✐♥ t❤❡ ❣❡♦♠❡tr②✳ ❈♦♠♣❛r❡✿

• ❖♥ t❤❡ ❧❡❢t✱ t❤❡ s❡q✉❡♥❝❡ ❛♣♣r♦❛❝❤❡s ✭❜✉t ♣♦ss✐❜❧② ♥❡✈❡r r❡❛❝❤❡s✮ ❛ ♣❛rt✐❝✉❧❛r ❧✐♥❡ ♦♥ t❤❡ ♣❧❛♥❡✳ • ❖♥ t❤❡ r✐❣❤t✱ t❤❡ s❡q✉❡♥❝❡ ❛♣♣r♦❛❝❤❡s ✭❜✉t ❞❡✜♥✐t❡❧② ♥❡✈❡r r❡❛❝❤❡s✮ ❛ ❝❡rt❛✐♥ ✐♠❛❣✐♥❛r② ❧✐♥❡✳

❚❤❡r❡ ❛r❡ ❛ t♦t❛❧ ♦❢ t❤r❡❡ ♣♦ss✐❜✐❧✐t✐❡s ❢♦r ❡✈❡r② s❡q✉❡♥❝❡✦

❉❡✜♥✐t✐♦♥ ✶✳✸✳✷✷✿ ❝♦♥✈❡r❣❡♥❝❡

❆♥② s❡q✉❡♥❝❡ s❛t✐s✜❡s ♦♥❡ ♦❢ t❤❡ t❤r❡❡✿ ✶✳ ■t ❤❛s ❛ ✜♥✐t❡ ❧✐♠✐t✱ ❛♥❞ ✇❡ s❛② ✐t ❝♦♥✈❡r❣❡s✳ ✷✳ ■t ❤❛s ❛♥ ✐♥✜♥✐t❡ ❧✐♠✐t✱ ❛♥❞ ✇❡ s❛② ✐t ❞✐✈❡r❣❡s t♦ ✐♥✜♥✐t② ♦r s✐♠♣❧② t❤❛t ✐t

✶✳✸✳ ❚❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❧✐♠✐t

✸✽

❞✐✈❡r❣❡s✳

✸✳ ■t ❤❛s ♥❡✐t❤❡r ❛ ✜♥✐t❡ ♦r ✐♥✜♥✐t❡ ❧✐♠✐t✱ ❛♥❞ ✇❡ s❛② t❤❛t t❤❡ ❧✐♠✐t ❞♦❡s ♥♦t ❡①✐st ♦r s✐♠♣❧② t❤❛t ✐t ❞✐✈❡r❣❡s✳ ❊①❛♠♣❧❡ ✶✳✸✳✷✸✿ ♥♦✲♣❛tt❡r♥ ❞✐✈❡r❣❡♥❝❡

❙♦♠❡ s❡q✉❡♥❝❡s s❡❡♠ t♦ ❤❛✈❡ ♥♦ ♣❛tt❡r♥ ❛t ❛❧❧✱ s✉❝❤ ❛s an = sin n✿

❍❡r❡✱ ♥♦ ❜❛♥❞ ✕ ✉♥❧❡ss ✇✐❞❡ ❡♥♦✉❣❤ ✕ ❝❛♥ ❝♦♥t❛✐♥ t❤❡ s❡q✉❡♥❝❡✬s t❛✐❧✳ ◆♦r ❞♦❡s t❤❡ s❡q✉❡♥❝❡ ❣r♦✇ ✇✐t❤♦✉t ❜♦✉♥❞✳ ❊①❛♠♣❧❡ ✶✳✸✳✷✹✿ ♥♦✲♣❛tt❡r♥ ❝♦♥✈❡r❣❡♥❝❡

■❢✱ ❤♦✇❡✈❡r✱ ✇❡ ❛❧s♦ ❞✐✈✐❞❡ t❤✐s ❡①♣r❡ss✐♦♥ ❜② n✱ r❡s✉❧t✐♥❣ ✐♥

1 sin n , n t❤❡ s✇✐♥❣s st❛rt t♦ ❞✐♠✐♥✐s❤✿

❚❤❡ ❧✐♠✐t ✐s 0✦ ❊①❛♠♣❧❡ ✶✳✸✳✷✺✿ t✇♦ ❧✐♠✐ts❄

❚❤❡ ♥❡①t ❡①❛♠♣❧❡ ✐s

an = 1 + (−1)n + ■t s❡❡♠s t♦ ❛♣♣r♦❛❝❤ t✇♦ ❧✐♠✐ts ❛t t❤❡ s❛♠❡ t✐♠❡✿

1 . n

✶✳✸✳

❚❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❧✐♠✐t

✸✾

■♥❞❡❡❞✱ ♥♦ ♠❛tt❡r ❤♦✇ ♥❛rr♦✇✱ ✇❡ ❝❛♥ ✜♥❞ t✇♦ ❜❛♥❞s t♦ ❝♦♥t❛✐♥ t❤❡ s❡q✉❡♥❝❡✬s t✇♦ t❛✐❧s✳ ❇✉t t❤✐s ♠❡❛♥s t❤❛t ♥♦

s✐♥❣❧❡

❜❛♥❞ ✕ ✐❢ ♥❛rr♦✇ ❡♥♦✉❣❤ ✕ ✇✐❧❧ ❝♦♥t❛✐♥ t❤❡♠✦

❚❤❡ ❜❡❤❛✈✐♦r t✉r♥s ♦✉t t♦ ❜❡

✐rr❡❣✉❧❛r✿

❊①❛♠♣❧❡ ✶✳✸✳✷✻✿ ❛❧t❡r♥❛t✐♥❣ s❡q✉❡♥❝❡

▲❡t✬s ♣✐❝❦ ❛ s✐♠♣❧❡r s❡q✉❡♥❝❡ ❛♥❞ ❞♦ t❤✐s ❛♥❛❧②t✐❝❛❧❧②✳ ▲❡t

an = (−1)n = ■s t❤❡ ❧✐♠✐t

a = 1❄

(

1 −1

■t✬s ♥♦t✦ ■♥❞❡❡❞✱ t❤✐s ❡①♣r❡ss✐♦♥ ✇♦♥✬t ❜❡ ❧❡ss t❤❛♥

a=1

♥❡❡❞ t♦ tr②

a = −1 ♦❢ a✳

✐s ♥♦t t❤❡ ❧✐♠✐t✳ ■s

❡✈❡r②

✐❢

n n

✐s ❡✈❡♥✱ ✐s ♦❞❞✳

■❢ ✐t ✐s✱ t❤❡♥ t❤✐s ✐s ✇❤❛t ♥❡❡❞s t♦ ❜❡ ✏s♠❛❧❧✑✿

|an − a| = |(−1)n − 1| =

✐s✳ ❙♦✱

✐❢

♣♦ss✐❜❧❡ ✈❛❧✉❡

ε

(

0 2

✐❢ ✐❢

n n

✐s ❡✈❡♥✱ ✐s ♦❞❞✳

✐❢ ✇❡ ❝❤♦♦s❡ ✐t t♦ ❜❡✱ s❛②✱

1✱

♥♦ ♠❛tt❡r ✇❤❛t

N

t❤❡ ❧✐♠✐t❄ ❙❛♠❡ st♦r②✳ ■♥ ♦r❞❡r t♦ ♣r♦✈❡ t❤❡ ♥❡❣❛t✐✈❡✱ ✇❡

❊①❡r❝✐s❡ ✶✳✸✳✷✼

❋✐♥✐s❤ t❤❡ ♣r♦♦❢ ✐♥ t❤❡ ❧❛st ❡①❛♠♣❧❡✳

❊①❛♠♣❧❡ ✶✳✸✳✷✽✿ ❞❡❝✐♠❛❧ ❛♣♣r♦①✐♠❛t✐♦♥s

❋♦r ❛ ❣✐✈❡♥ r❡❛❧ ♥✉♠❜❡r✱ ✇❡ ❝❛♥ ❝♦♥str✉❝t ❛ s❡q✉❡♥❝❡ t❤❛t ❛♣♣r♦①✐♠❛t❡s t❤❛t ♥✉♠❜❡r ✕ ✈✐❛ ♦❢ ✐ts ❞❡❝✐♠❛❧ ❛♣♣r♦①✐♠❛t✐♦♥s✳ ❋♦r ❡①❛♠♣❧❡✱ ✇❡ ❤❛✈❡ ❛❧r❡❛❞② s❡❡♥ t❤✐s✿

xn = 0.9, 0.99, 0.999, 0.9999, ...

t❡♥❞s t♦

1.

❋✉rt❤❡r♠♦r❡✱ ✇❡ ❤❛✈❡✿

xn = 0.3, 0.33, 0.333, 0.3333, ...

t❡♥❞s t♦

1/3 .

tr✉♥❝❛t✐♦♥s

✶✳✸✳

❚❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❧✐♠✐t

✹✵

❚❤❡ ✐❞❡❛ ♦❢ ❧✐♠✐t t❤❡♥ ❤❡❧♣s ✉s ✉♥❞❡rst❛♥❞ ✐♥✜♥✐t❡ ❞❡❝✐♠❛❧s✿

• •

❲❤❛t ✐s t❤❡ ♠❡❛♥✐♥❣ ♦❢ ❲❤❛t ✐s t❤❡ ♠❡❛♥✐♥❣ ♦❢

.9999...❄ .3333...❄

■t ✐s t❤❡ ❧✐♠✐t ♦❢ t❤❡ s❡q✉❡♥❝❡ ■t ✐s t❤❡ ❧✐♠✐t ♦❢ t❤❡ s❡q✉❡♥❝❡

0.9, 0.99, 0.999, ...❀ ✐✳❡✳✱ 1 ✳ 0.3, 0.3, 0.333, ...❀ ✐✳❡✳✱ 1/3 ✳

❊①❡r❝✐s❡ ✶✳✸✳✷✾ ❋✐♥❞ t❤❡

nt❤✲t❡r♠

❢♦r♠✉❧❛s ❢♦r t❤❡ t✇♦ s❡q✉❡♥❝❡s ❛❜♦✈❡ ❛♥❞ ❝♦♥✜r♠ t❤❡ ❧✐♠✐ts✳

❊①❛♠♣❧❡ ✶✳✸✳✸✵✿ ❞❡✜♥✐t✐♦♥ ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ❚❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ✐s ❡✈❡♥ ♠♦r❡ ❧♦❣✐❝❛❧❧② ❝♦♠♣❧❡①✿

• • • •

❚❍❊❘❊ ■❙ ❋❖❘ ❆◆❨ ❚❍❊❘❊ ■❙ ❋❖❘ ❆◆❨

❛ ♥✉♠❜❡r

a

s✉❝❤ t❤❛t

ε > 0✱ N s✉❝❤ t❤❛t ♥✉♠❜❡r n > N ✱

r❡❛❧ ♥✉♠❜❡r ❛ ♥✉♠❜❡r ♥❛t✉r❛❧

✇❡ ❤❛✈❡✿

|an − a| < ε . ❚❤❡ ❞❡✜♥✐t✐♦♥ s♣❡❛❦s ♦❢

❛ ♥✉♠❜❡r ✦

■♥ ❡✈❡r② ❡①❛♠♣❧❡✱ ✇❡ ❢♦✉♥❞ ♦♥❡ ♥✉♠❜❡r ❛♥❞ t❤❡♥ ❞❡♠♦♥str❛t❡❞ t❤❛t ✐t

s❛t✐s✜❡s t❤❡ ❞❡✜♥✐t✐♦♥✳ ❇✉t ✇❤❛t ✐❢ t❤❡r❡ ❛r❡ ♦t❤❡rs❄ ❲❡ ♥❡❡❞ t♦ ❥✉st✐❢② ✏t❤❡✑ ✐♥ ✏t❤❡ ❧✐♠✐t✑✳

❚❤❡♦r❡♠ ✶✳✸✳✸✶✿ ❯♥✐q✉❡♥❡ss ♦❢ ▲✐♠✐t ♦❢ ❙❡q✉❡♥❝❡ ❆ s❡q✉❡♥❝❡ ❝❛♥ ❤❛✈❡ ♦♥❧② ♦♥❡ ❧✐♠✐t ✭✜♥✐t❡ ♦r ✐♥✜♥✐t❡✮✳

Pr♦♦❢✳ ❚❤❡ ❣❡♦♠❡tr② ♦❢ t❤❡ ♣r♦♦❢ ✐s ❝❧❡❛r✿ ❲❡ ✇❛♥t t♦ s❡♣❛r❛t❡ t❤❡ t✇♦ ❤♦r✐③♦♥t❛❧ ❧✐♥❡s r❡♣r❡s❡♥t✐♥❣ t✇♦ ♣♦t❡♥t✐❛❧ ❧✐♠✐ts ❜② t✇♦ ♥♦♥✲♦✈❡r❧❛♣♣✐♥❣ ❜❛♥❞s✱ ❛s s❤♦✇♥ ❛❜♦✈❡✳ ❚❤❡♥ t❤❡ t❛✐❧ ♦❢ t❤❡ s❡q✉❡♥❝❡ ✇♦✉❧❞ ❤❛✈❡ t♦ ✜t ♦♥❡ ♦r t❤❡ ♦t❤❡r✱ ❜✉t ♥♦t ❜♦t❤✳

❚❤❡s❡ ❜❛♥❞s ❝♦rr❡s♣♦♥❞ t♦ t✇♦ ✐♥t❡r✈❛❧s ❛r♦✉♥❞ t❤♦s❡

t✇♦ ✏❧✐♠✐ts✑✳ ■♥ ♦r❞❡r ❢♦r t❤❡♠ ♥♦t t♦ ✐♥t❡rs❡❝t✱ t❤❡✐r ✇✐❞t❤ ✭t❤❛t✬s

2ε✦✮

s❤♦✉❧❞ ❜❡ ❧❡ss t❤❛♥ ❤❛❧❢ t❤❡

❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ t✇♦ ♥✉♠❜❡rs✿

❚❤❡ ♣r♦♦❢ ✐s ❜② ❝♦♥tr❛❞✐❝t✐♦♥✳ ❙✉♣♣♦s❡ s✉♣♣♦s❡ ❛❧s♦

a 6= b✳

a

❛♥❞

b

❛r❡ t✇♦ ❧✐♠✐ts✱ ✐✳❡✳✱ ❡✐t❤❡r s❛t✐s✜❡s t❤❡ ❞❡✜♥✐t✐♦♥✱ ❛♥❞

■♥ ❢❛❝t✱ ✇✐t❤♦✉t ❧♦ss ♦❢ ❣❡♥❡r❛❧✐t② ✇❡ ❝❛♥ ❛ss✉♠❡ t❤❛t

ε=

a < b✳

▲❡t

b−a . 2

❚❤❡♥✱ ✇❤❛t ✇❡ ❛r❡ ❣♦✐♥❣ t♦ ✉s❡ ❛t t❤❡ ❡♥❞ ✐s

a + ε = b − ε. ◆♦✇✱ ✇❡ r❡✇r✐t❡ t❤❡ ❞❡✜♥✐t✐♦♥ ❢♦r

◮

❚❤❡r❡ ❡①✐sts ❛ ♥✉♠❜❡r

L

a

❛♥❞

b

s♣❡❝✐✜❝❛❧❧②✿

s✉❝❤ t❤❛t ❢♦r ❡✈❡r② ♥❛t✉r❛❧ ♥✉♠❜❡r

n > L✱

✇❡ ❤❛✈❡

|an − a| < ε . ◆♦✇✱ ✇❡ r❡✇r✐t❡ t❤❡ ❞❡✜♥✐t✐♦♥ ❢♦r

◮

❚❤❡r❡ ❡①✐sts ❛ ♥✉♠❜❡r

M

M

❛s ❧✐♠✐t✿

s✉❝❤ t❤❛t ❢♦r ❡✈❡r② ♥❛t✉r❛❧ ♥✉♠❜❡r

|an − b| < ε .

n > M✱

✇❡ ❤❛✈❡

✶✳✸✳

❚❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❧✐♠✐t

✹✶

■♥ ♦r❞❡r t♦ ❝♦♠❜✐♥❡ t❤❡ t✇♦ st❛t❡♠❡♥ts✱ ✇❡ ♥❡❡❞ t❤❡♠ t♦ ❜❡ s❛t✐s✜❡❞ ❢♦r t❤❡ s❛♠❡ ✈❛❧✉❡s ♦❢

n✳

▲❡t

N = min{L, M } . ❲❡ ❝♦♥❝❧✉❞❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿

•

❋♦r ❡✈❡r② ♥✉♠❜❡r

•

❋♦r ❡✈❡r② ♥✉♠❜❡r

n > N✱

✇❡ ❤❛✈❡

|an − a| < ε . n > N✱

✇❡ ❤❛✈❡

|an − b| < ε .

■♥ ♣❛rt✐❝✉❧❛r✱ ❢♦r ❡✈❡r②

n > N✱

✇❡ ❤❛✈❡✿

an < a + ε = b − ε < an . ❆ ❝♦♥tr❛❞✐❝t✐♦♥✳ ❊①❡r❝✐s❡ ✶✳✸✳✸✷

❋♦❧❧♦✇ t❤❡ ♣r♦♦❢ ❛♥❞ ❞❡♠♦♥str❛t❡ t❤❛t ✐t ✐s ✐♠♣♦ss✐❜❧❡ ❢♦r ❛ s❡q✉❡♥❝❡ t♦ ❤❛✈❡ ❛s ✐ts ❧✐♠✐t✿ ✭❛✮ ❛ r❡❛❧ ♥✉♠❜❡r ❛♥❞

±∞✱

♦r ✭❜✮

−∞

❛♥❞

+∞✳

❚❤❡ t❤❡♦r❡♠ ✐♥❞✐❝❛t❡s t❤❛t t❤❡ ❝♦rr❡s♣♦♥❞❡♥❝❡ ♠❛❦❡s s❡♥s❡✿

❛ ❝♦♥✈❡r❣❡♥t s❡q✉❡♥❝❡

−→

✐ts ❧✐♠✐t ✭❛ r❡❛❧ ♥✉♠❜❡r✮

❈❛♥ ✇❡ r❡✈❡rs❡ t❤✐s ❝♦rr❡s♣♦♥❞❡♥❝❡❄ ◆♦✱ ❜❡❝❛✉s❡ t❤❡r❡ ❛r❡ ♠❛♥② s❡q✉❡♥❝❡s ❝♦♥✈❡r❣✐♥❣ t♦ t❤❡ s❛♠❡ ♥✉♠❜❡r✿

3 + 1/n 3 + 1/n2 3 + 1/2n

3 − 1/n 3 + 2/n ց ↓ ւ ← 3 + (−1)n /n → 3 ր ↑√ տ 3 − 1/ n 3 + 1/ ln n

❍♦✇❡✈❡r✱ ✇❡ ❝❛♥ s❛② t❤❛t ❛ r❡❛❧ ♥✉♠❜❡r ✏✐s✑ ✐ts ❛♣♣r♦①✐♠❛t✐♦♥s✱ ✐✳❡✳✱ ❛❧❧ s❡q✉❡♥❝❡s t❤❛t ❝♦♥✈❡r❣❡ t♦ ✐t✳ ❚❤✉s✱ t❤❡r❡ ❝❛♥ ❜❡ ♥♦ t✇♦ ❧✐♠✐ts ♦❢ ❛ s❡q✉❡♥❝❡ ❛♥❞ ✇❡ ❛r❡ ❥✉st✐✜❡❞ t♦ s♣❡❛❦ ♦❢

t❤❡

❧✐♠✐t✳ ■❢ ✐t ❡①✐sts✱ ✐t✬s

❡✐t❤❡r ❛ ♥✉♠❜❡r ♦r ♦♥❡ ♦❢ t❤❡ t✇♦ ✐♥✜♥✐t✐❡s✳ ❲❡ ✉s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♥♦t❛t✐♦♥ ❢♦r t❤❡ ❧✐♠✐t ♦❢ ❛ s❡q✉❡♥❝❡✿ ▲✐♠✐t

an → a

❛s

n→∞

❛♥❞

lim an = a

n→∞

■t r❡❛❞s ✏t❤❡ ❧✐♠✐t ♦❢ ♦r ✏ an ❝♦♥✈❡r❣❡s t♦

❚❤❡ t❤❡♦r❡♠ ❝❛♥ t❤❡♥ ❜❡ r❡st❛t❡❞ ❛s ❢♦❧❧♦✇s ✭✇❡ ✇✐❧❧ ✉s❡

an

✐s

a✑

a✑✳

=⇒ ❢♦r ❛♥ ✐♠♣❧✐❝❛t✐♦♥✱ ✐✳❡✳✱ ❛♥ ✏✐❢✲t❤❡♥✑

❈♦r♦❧❧❛r② ✶✳✸✳✸✸✿ ❯♥✐q✉❡♥❡ss ♦❢ ▲✐♠✐t ♦❢ ❙❡q✉❡♥❝❡ ■❢ ✇❡ ❤❛✈❡ t✇♦ ❧✐♠✐ts ♦❢ t❤❡ s❛♠❡ s❡q✉❡♥❝❡✱ t❤❡② ❛r❡ ❡q✉❛❧✳

st❛t❡♠❡♥t✮✿

✶✳✸✳

❚❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❧✐♠✐t

✹✷

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿

a = lim xn ❆◆❉ b = lim xn =⇒ a = b n→∞

n→∞

❊①❡r❝✐s❡ ✶✳✸✳✸✹ ❙t❛t❡ ❛s ❛♥ ✐♠♣❧✐❝❛t✐♦♥✿ ✭❛✮ ✏❲❤❡♥ t❤❡r❡ ✐s s♠♦❦❡✱ t❤❡r❡ ✐s ✜r❡✳✑ ✭❜✮ ✏❊✈❡r② sq✉❛r❡ ✐s ❛ r❡❝t❛♥❣❧❡✳✑

❲❡ ✉s❡ ❛ s✐♠✐❧❛r ♥♦t❛t✐♦♥ t♦ ❞❡s❝r✐❜❡ t❤❡ ♣❛rt✐❝✉❧❛r ❦✐♥❞ ♦❢ ❞✐✈❡r❣❡♥t ❜❡❤❛✈✐♦r ✭✐♥✜♥✐t②✮✿

■♥✜♥✐t❡ ❧✐♠✐t an → ±∞

❛s

n→∞

♦r

lim an = ±∞

n→∞

■t r❡❛❞s ✏t❤❡ ❧✐♠✐t ♦❢

an

✐s ✐♥✜✲

♥✐t❡✑ ♦r ✏ an ❞✐✈❡r❣❡s t♦ ✐♥✜♥✲ ✐t②✑✳

❚❤❡ t❤✐r❞ ♣♦ss✐❜✐❧✐t② ✐s t❤❛t t❤❡r❡ ✐s ♥♦ tr❡♥❞✳ ❙✐♥❝❡ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❝♦♥✈❡r❣❡♥❝❡ st❛rts ✇✐t❤ ✏ ❚❍❊❘❊ ♥✉♠❜❡r

a✳✳✳✑✱

■❙

❛

✇❡ ❝❛♥ s❛② t❤❛t ♦t❤❡r✇✐s❡ t❤❡r❡ ✐s ♥♦ ❧✐♠✐t✿

◆♦ ❧✐♠✐t lim an ,

n→∞

♥♦ ❧✐♠✐t

♦r

lim an

n→∞

❉◆❊

an

■t r❡❛❞s ✏t❤❡ ❧✐♠✐t ♦❢

❞♦❡s

♥♦t ❡①✐st✑✳

❊①❛♠♣❧❡ ✶✳✸✳✸✺✿ ♥♦t❛t✐♦♥ ❲❡ ❝❛♥ ♥♦✇ ✇r✐t❡ t❤❡ ❝♦♥❝❧✉s✐♦♥s ✐♥ t❤❡ ❛❜♦✈❡ ❡①❛♠♣❧❡s ✉s✐♥❣ ♦✉r ♥❡✇ ♥♦t❛t✐♦♥✱ ❛s ❢♦❧❧♦✇s✿

1 .9 1. 3. 1 0

1/2 .99 1.1 3.1 2 1

1/3 .999 1.01 3.14 3 0

1/4 .9999 1.001 3.141 4 1

1/5 .99999 1.0001 3.1415 5 0

... ... ... ... ... ...

→ 0. → 1. → 1. → π. → +∞ . ♥♦ ❧✐♠✐t.

❚❤❡ ❧✐♠✐ts ♦❢ s♦♠❡ s♣❡❝✐✜❝ s❡q✉❡♥❝❡s ✭s❡❡♥ ✐♥ ❱♦❧✉♠❡ ✶✱ ❈❤❛♣t❡r ✶P❈✲✶✮ ❝❛♥ ❜❡ ❡❛s✐❧② ❢♦✉♥❞ ❛s s❤♦✇♥ ✐♥ t❤❡ t✇♦ ❢♦r♠✉❧❛s ❜❡❧♦✇✳

❚❤❡♦r❡♠ ✶✳✸✳✸✻✿ ▲✐♠✐t ♦❢ ❈♦♥st❛♥t ❙❡q✉❡♥❝❡ ❋♦r ❛♥② r❡❛❧

c✱

✇❡ ❤❛✈❡✿

lim c = c

n→∞

✶✳✸✳ ❚❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❧✐♠✐t

✹✸

❚❤❡♦r❡♠ ✶✳✸✳✸✼✿ ▲✐♠✐t ♦❢ P♦✇❡r ❙❡q✉❡♥❝❡ ❋♦r ❛♥② ✐♥t❡❣❡r

k✱

✇❡ ❤❛✈❡✿

  0 k lim n = 1 n→∞   +∞

✐❢ ✐❢ ✐❢

k0

Pr♦♦❢✳

❋✐rst✱ t❤❡ ❝❛s❡ ♦❢ k < 0✳ ❙✉♣♣♦s❡ ε > 0 ✐s ❣✐✈❡♥✳ ❲❡ ♥❡❡❞ t♦ ✜♥❞ s✉❝❤ ❛♥ N t❤❛t |nk − 0| = nk < ε ✇❤❡♥❡✈❡r n > N ✳ ❲❡ ❝❛♥ ❡①♣r❡ss s✉❝❤ ❛♥ N ✐♥ t❡r♠s ♦❢ t❤✐s ε❀ ✇❡ ❥✉st ❝❤♦♦s❡✿ N=

√

1/k

ε.

❙❡❝♦♥❞✱ t❤❡ ❝❛s❡ ♦❢ k > 0✳ ❙✉♣♣♦s❡ R > 0 ✐s ❣✐✈❡♥✳ ❲❡ ♥❡❡❞ t♦ ✜♥❞ s✉❝❤ ❛♥ N t❤❛t nk > R ✇❤❡♥❡✈❡r n > N ✳ ❲❡ ❝❛♥ ❡①♣r❡ss s✉❝❤ ❛♥ N ✐♥ t❡r♠s ♦❢ t❤✐s R❀ s✐♠✐❧❛r❧②✱ t♦ t❤❡ ❛❜♦✈❡ ✇❡ ❝❤♦♦s❡✿ N=

√

1/k

R.

❚❤✐s ✐s ✇❤❛t ❛ ❢❡✇ t②♣✐❝❛❧ ❛r✐t❤♠❡t✐❝ ♣r♦❣r❡ss✐♦♥s ❧♦♦❦ ❧✐❦❡✿

❚❤❡ t❤❡♦r❡♠ ❜❡❧♦✇ s✉♠♠❛r✐③❡s t❤❡s❡ ♦❜s❡r✈❛t✐♦♥s✿ ❚❤❡♦r❡♠ ✶✳✸✳✸✽✿ ▲✐♠✐t ♦❢ ❆r✐t❤♠❡t✐❝ Pr♦❣r❡ss✐♦♥ ❋♦r ❛♥② r❡❛❧ ♥✉♠❜❡rs

m, b > 0✱

✇❡ ❤❛✈❡✿

  −∞ lim (b + nm) = b n→∞   +∞ ❊①❡r❝✐s❡ ✶✳✸✳✸✾

Pr♦✈❡ t❤❡ t❤❡♦r❡♠✳ ❚❤✐s ✐s ✇❤❛t ❛ ❢❡✇ t②♣✐❝❛❧ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥s ❧♦♦❦ ❧✐❦❡✿

✐❢ ✐❢ ✐❢

m0

✶✳✸✳

❚❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❧✐♠✐t

✹✹

❚❤❡ t❤❡♦r❡♠ ❜❡❧♦✇ s✉♠♠❛r✐③❡s t❤❡s❡ ♦❜s❡r✈❛t✐♦♥s✿

❚❤❡♦r❡♠ ✶✳✸✳✹✵✿ ▲✐♠✐t ♦❢ ●❡♦♠❡tr✐❝ Pr♦❣r❡ss✐♦♥ ❋♦r ❛♥② r❡❛❧ ♥✉♠❜❡r

r✱

✇❡ ❤❛✈❡✿

 ♥♦ ❧✐♠✐t    0 lim rn = n→∞  1    +∞

✐❢ ✐❢ ✐❢ ✐❢

r ≤ −1 |r| < 1 r=1 r>1

❊①❡r❝✐s❡ ✶✳✸✳✹✶ Pr♦✈❡ t❤❡ t❤❡♦r❡♠✳

❊①❛♠♣❧❡ ✶✳✸✳✹✷✿ ♣♦♣✉❧❛t✐♦♥ ❣r♦✇t❤ ❛♥❞ ❞❡❝❧✐♥❡ ●❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥s ❛r❡ ✉s❡❞ t♦ ♠♦❞❡❧ ♣♦♣✉❧❛t✐♦♥ ❣r♦✇t❤ ❛♥❞ ❞❡❝❧✐♥❡✿

❲❡ ❝♦♥❝❧✉❞❡ ❢r♦♠ t❤❡ t❤❡♦r❡♠ t❤❛t✿

• •

■♥ ❝❛s❡ ♦❢ ❣r♦✇t❤✱ t❤❡ ♣♦♣✉❧❛t✐♦♥ ✇✐❧❧ ❜❡❝♦♠❡ ❛s ❧❛r❣❡ ❛s ✇❡ ❞❡s✐r❡ ✐❢ ✇❡ ✇❛✐t ❧♦♥❣ ❡♥♦✉❣❤✳ ■♥ ❝❛s❡ ♦❢ ❞❡❝❧✐♥❡✱ t❤❡ ♣♦♣✉❧❛t✐♦♥ ✇✐❧❧ ❜❡❝♦♠❡ ❛s s♠❛❧❧ ❛s ✇❡ ❞❡s✐r❡ ✐❢ ✇❡ ✇❛✐t ❧♦♥❣ ❡♥♦✉❣❤✳

❚❤❡ ❢❛❝t t❤❛t ✐♥ t❤❡ ❧❛tt❡r ❝❛s❡ ✇❡ ✇✐❧❧ ❤❛✈❡ ❛ ♣♦♣✉❧❛t✐♦♥ ♦❢ ❧❡ss t❤❛♥ ♦♥❡ ♣❡rs♦♥ ❜✉t ♥❡✈❡r ③❡r♦ ✐♥❞✐❝❛t❡s ❛ ❧✐♠✐t❛t✐♦♥ ♦❢ t❤✐s ♠♦❞❡❧✳

❊①❡r❝✐s❡ ✶✳✸✳✹✸ ❋✐♥❞ t❤❡ ❧✐♠✐t ♦❢ ❡❛❝❤ ♦❢ t❤❡s❡ s❡q✉❡♥❝❡s ♦r s❤♦✇ t❤❛t ✐t ❞♦❡s♥✬t ❡①✐st✿ ✶✳ ✷✳ ✸✳ ✹✳

1, 3, 5, 7, 9, 11, 13, 15, ... .9, .99, .999, .9999, ... 1, −1, 1, −1, ... 1, 1/2, 1/3, 1/4, ...

✶✳✹✳

▲✐♠✐ts ✉♥❞❡r ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s 1, 2, 1, 3,

✺✳ ✻✳ ✼✳ ✽✳

✹✺

1/2, 1/4 , 1/8, ... 3, 5, 7, 11, 13, 17, ... −4, 9, −16, 25, ... 1, 4, 1, 5, 9, ...

❊①❛♠♣❧❡ ✶✳✸✳✹✹✿ s❡r✐❡s

■♥ ❡✐t❤❡r ♦❢ t❤❡ t✇♦ t❛❜❧❡s ❜❡❧♦✇✱ ✇❡ ❤❛✈❡ ❛ s❡q✉❡♥❝❡ ❣✐✈❡♥ ✐♥ t❤❡ ✜rst t✇♦ ❝♦❧✉♠♥s✳

■ts

nt❤✲t❡r♠

❢♦r♠✉❧❛ ✐s ❦♥♦✇♥✳ ❚❤❡ t❤✐r❞ ❝♦❧✉♠♥ s❤♦✇s t❤❡ s❡q✉❡♥❝❡ ♦❢ s✉♠s ✭s❡❡♥ ✐♥ ❱♦❧✉♠❡ ✶✱ ❈❤❛♣t❡r ✶P❈✲✶✮ ♦❢ t❤❡ ✜rst✿

n an sn 1 1 1 2 1 3

1 1 1 1 + 1 2 1 1 1 + + 1 2 3

2

✳ ✳ ✳

✳ ✳ ✳

✳ ✳ ✳

n

1 n

1 1 1 1 + + + ... + 1 2 3 n

1 2 3

❚❤❡

nt❤✲t❡r♠

n an s n 1 1 1 2 1 4

1 1 1 1 + 1 2 1 1 1 + + 1 2 4

✳ ✳ ✳

✳ ✳ ✳

✳ ✳ ✳

n

1 2n

1

3

1 1 1 1 + + + ... + n−1 1 2 4 2

❢♦r♠✉❧❛ ✐s ✉♥❦♥♦✇♥ ❛s ✇❡ ❞♦♥✬t ❦♥♦✇ ❤♦✇ t♦ r❡♣r❡s❡♥t t❤❡s❡ q✉❛♥t✐t✐❡s ✇✐t❤♦✉t ✏✳✳✳✑✳

■♥ ❝♦♥tr❛st t♦ t❤❡ ❧❛st ❡①❛♠♣❧❡✱ ✜♥❞✐♥❣ t❤❡ ❧✐♠✐t ♦❢ s✉❝❤ ❛ s❡q✉❡♥❝❡ ✐s ❛ ❝❤❛❧❧❡♥❣❡ ✭❱♦❧✉♠❡ ✸✱ ❈❤❛♣t❡r ✸■❈✲✺✮✳

❊①❡r❝✐s❡ ✶✳✸✳✹✺

●✐✈❡ ❢♦r♠✉❧❛s ❢♦r t❤❡ ❢♦❧❧♦✇✐♥❣ s❡q✉❡♥❝❡s✿ ✭❛✮ ✭❜✮

bn → +∞

❛s

n→∞

an → 0

❜✉t ✐t✬s ♥♦t ✐♥❝r❡❛s✐♥❣✳

❛s

n→∞

❜✉t ✐t✬s ♥♦t ✐♥❝r❡❛s✐♥❣ ♦r ❞❡❝r❡❛s✐♥❣❀

✶✳✹✳ ▲✐♠✐ts ✉♥❞❡r ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s

■❢ ❡✈❡r② r❡❛❧ ♥✉♠❜❡r

✐s t❤❡ s❡q✉❡♥❝❡ ♦❢ ✐ts ❛♣♣r♦①✐♠❛t✐♦♥s✱ ❞♦❡s t❤❡ ✉s✉❛❧ ❛r✐t❤♠❡t✐❝ ♦♣❡r❛t✐♦♥s ✇✐t❤ ♥✉♠❜❡rs

♠❛t❝❤ t❤♦s❡ ✇✐t❤ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s❡q✉❡♥❝❡s❄ ❨❡s✳ ❲❡ ✇✐❧❧ ❞✐s❝♦✈❡r t❤❡ ❢♦❧❧♦✇✐♥❣✿

◮

▲✐♠✐ts ❜❡❤❛✈❡ ✇❡❧❧ ✇✐t❤ r❡s♣❡❝t t♦ ❛❧❣❡❜r❛✳

❋♦r s✐♠♣❧✐❝✐t②✱ ✇❡ ❛ss✉♠❡ ❜❡❧♦✇ t❤❛t ❛❧❧ t❤❡ s❡q✉❡♥❝❡s ❛r❡ ❞❡✜♥❡❞ ♦♥ t❤❡ s❛♠❡ s❡t ♦❢ ✐♥t❡❣❡rs✳ ❊①❛♠♣❧❡ ✶✳✹✳✶✿ ❛❧❣❡❜r❛ ♦❢ s❡q✉❡♥❝❡s

❲❤❛t ❞♦ ✇❡ ♠❡❛♥ ❜② ❛❞❞✐♥❣✱ ♠✉❧t✐♣❧②✐♥❣✱ ❡t❝✳

t✇♦ s❡q✉❡♥❝❡s❄

❏✉st ❛s ✇✐t❤ ❢✉♥❝t✐♦♥s✱ ✇❡ ❛❞❞✱

♠✉❧t✐♣❧②✱ ❡t❝✳ t❡r♠✲✇✐s❡✿

n an bn an + bn an · bn

1 1 −1 1−1 1·1

❖❢ ❝♦✉rs❡✱ ✇❡ r❡❛❧❧② ♥❡❡❞ ♦♥❧② t❤❡

2 1/2 1 1/2 + 1 1/2 · 1

nt❤

❝♦❧✉♠♥✳

3 1/3 −1 1/3 − 1 1/3 · (−1)

... ... ... ... ...

n 1/n (−1)n 1/n + (−1)n 1/n · (−1)n

... ... ... ... ...

✶✳✹✳

▲✐♠✐ts ✉♥❞❡r ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s

✹✻

❚❤❡ t❤r❡❡ ♠❛✐♥ ❝❧❛ss❡s ♦❢ s❡q✉❡♥❝❡s ✕ ♣♦✇❡r✱ ❛r✐t❤♠❡t✐❝✱ ❛♥❞ ❣❡♦♠❡tr✐❝ ✕ ✐♥tr♦❞✉❝❡❞ ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥ ✇✐❧❧ s❡r✈❡ ❛s ✏s❡❡❞s✑✳ ❲❤❡♥ ❝♦♠❜✐♥❡❞ ✈✐❛ ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s✱ t❤❡② ♣r♦❞✉❝❡ ❛♥ ✐♥✜♥✐t❡ ✈❛r✐❡t② ♦❢ s❡q✉❡♥❝❡s✱ t❤❡ ❧✐♠✐ts ♦❢ ✇❤✐❝❤ ✇❡ s❤♦✉❧❞ ❜❡ ❛❜❧❡ t♦ ✜♥❞✳ ❲❡ ✇✐❧❧ st❛rt ✇✐t❤ ♦✉r st✉❞② ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ s❡q✉❡♥❝❡s ✐♥ ❣❡♥❡r❛❧ ✇✐t❤ ❛ s♣❡❝✐❛❧ ❝❧❛ss ♦❢ s✐♠♣❧❡r s❡q✉❡♥❝❡s✳ ❇❡❧♦✇ ✇❡ ❤❛✈❡ ❛ s❡q✉❡♥❝❡

an

♣❧♦tt❡❞✱ ❛s ✇❡❧❧ ❛s t❤❡ s❡q✉❡♥❝❡ ♦❢ t❤❡ ❞✐st❛♥❝❡s ❢r♦♠

an

t♦

a✱ ✐✳❡✳✱ bn = |an − a|✿

❚❤❡ t❤❡♦r❡♠ ❜❡❧♦✇ s❤♦✇s ✇❤② t❤✐s ✐s ✐♠♣♦rt❛♥t ✭✇❡ ✇✐❧❧ ✉s❡ ✏ ⇐⇒✑ t♦ ✐♥❞✐❝❛t❡ ❛♥ ❡q✉✐✈❛❧❡♥❝❡✱ ✐✳❡✳✱ ❛♥ ✏✐❢✲❛♥❞✲♦♥❧②✲✐❢ ✑ st❛t❡♠❡♥t✮✿

❚❤❡♦r❡♠ ✶✳✹✳✷✿ ❈♦♥✈❡r❣❡♥❝❡ t♦ ❩❡r♦ ❆ s❡q✉❡♥❝❡ ❝♦♥✈❡r❣❡s t♦ ❛ ♥✉♠❜❡r ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ ❧✐♠✐t ♦❢ t❤❡ ❞✐st❛♥❝❡ ❢r♦♠ t❤❡ t❡r♠s ♦❢ t❤✐s s❡q✉❡♥❝❡ t♦ t❤✐s ♥✉♠❜❡r ✐s ③❡r♦✳

■♥ ♦t❤❡r ✇♦r❞s✱ ❢♦r ❛♥② s❡q✉❡♥❝❡

an ✱

✇❡ ❤❛✈❡✿

an → a ⇐⇒ |an − a| → 0 ❯s✐♥❣ t❤❡ ♦t❤❡r ♥♦t❛t✐♦♥ ❢♦r ❧✐♠✐ts✱ ✇❡ ❤❛✈❡✿

lim an = a ⇐⇒ lim |an − a| = 0

n→∞

❙♦✱ ✐❢ t❤❡ ❧✐♠✐t ♦❢

an

✐s

a✱

t❤❡♥ t❤❡ ❧✐♠✐t ♦❢

n→∞

|an − a|

✐s

0✱

❛♥❞ ✈✐❝❡ ✈❡rs❛✳

❚❤❡♥✱ t♦ ✉♥❞❡rst❛♥❞ ❧✐♠✐ts ♦❢ s❡q✉❡♥❝❡s ✐♥ ❣❡♥❡r❛❧✱ ✇❡ ♥❡❡❞ ✜rst t♦ ✉♥❞❡rst❛♥❞ t❤♦s❡ ♦❢ t❤✐s s♠❛❧❧❡r ❝❧❛ss✿

♣♦s✐t✐✈❡ s❡q✉❡♥❝❡s t❤❛t ❝♦♥✈❡r❣❡ t♦ 0✳

❚❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ❜❡❝♦♠❡s s✐♠♣❧❡r✦ ■♥❞❡❡❞✱

✇❤❡♥

◮

❢♦r ❛♥②

ε > 0✱

t❤❡r❡ ✐s ❛♥

N

s✉❝❤ t❤❛t

an < ε

❢♦r ❛❧❧

0 < an → 0

n > N✳

❆s ✇❡ ❦♥♦✇ ✭❱♦❧✉♠❡ ✶✱ ❈❤❛♣t❡r ✶P❈✲✸✮✱ s♦♠❡ s✐♠♣❧❡ ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s ♦♥ t❤❡ ♦✉t♣✉ts ♦❢ ❢✉♥❝t✐♦♥ ♣r♦❞✉❝❡ tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ♣❧❛♥❡ ❛♥❞ ✈✐❝❡ ✈❡rs❛✳ ❇✉t ❤♦✇ ❞♦ t❤❡② ❛✛❡❝t t❤❡ ❧✐♠✐ts❄ Pr❡❞✐❝t❛❜❧②✳ ❲❡ st❛rt ✇✐t❤ ❛❞❞✐t✐♦♥✳ ■❢ t✇♦ ♣❡♦♣❧❡ ❛r❡ ✇❛❧❦✐♥❣ ❛✇❛② ❢r♦♠ ❛ ♣♦❧❡ ✐♥ t❤❡ ♦♣♣♦s✐t❡ ❞✐r❡❝t✐♦♥s✱ ✇❡ ❤❛✈❡ t✇♦ s❡q✉❡♥❝❡s

an

❛♥❞

bn

t❤❛t r❡♣r❡s❡♥t t❤❡✐r ❧♦❝❛t✐♦♥s ✭❧❡❢t✮✳

✭r✐❣❤t✮❄ ❚❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡♠✿

❲❤❛t ✐s t❤❡ s✉♠ ♦❢ t❤❡ t✇♦ s❡q✉❡♥❝❡s

an + b n

✶✳✹✳ ▲✐♠✐ts ✉♥❞❡r ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s

✹✼

❚♦ ❣r❛♣❤✐❝❛❧❧② ❛❞❞ t✇♦ s❡q✉❡♥❝❡s✱ ✇❡ ✢✐♣ t❤❡ s❡❝♦♥❞ ✉♣s✐❞❡ ❞♦✇♥ ❛♥❞ t❤❡♥ ❝♦♥♥❡❝t ❡❛❝❤ ♣❛✐r ♦❢ ❞♦ts ✇✐t❤ ❛ ❜❛r✳ ❚❤❡♥✱ t❤❡ ❧❡♥❣t❤s ♦❢ t❤❡s❡ ❜❛rs ❢♦r♠ t❤❡ ♥❡✇ s❡q✉❡♥❝❡✳ ◆♦✇✱ ✐❢ t❤❡ ♦r✐❣✐♥❛❧ ❜❛rs ❞✐♠✐♥✐s❤✱ t❤❡♥ s♦ ❞♦ t❤❡ ♥❡✇ ♦♥❡s✱ ❛s ❢♦❧❧♦✇s✿ ❚❤❡♦r❡♠ ✶✳✹✳✸✿ ❙✉♠ ❘✉❧❡ ❢♦r ❩❡r♦ ▲✐♠✐t ❙❡q✉❡♥❝❡s ■❢ ❡✐t❤❡r ♦❢ t✇♦ s❡q✉❡♥❝❡s ✇✐t❤ ♥♦♥✲♥❡❣❛t✐✈❡ t❡r♠s ❝♦♥✈❡r❣❡s t♦ ③❡r♦✱ t❤❡♥ s♦ ❞♦❡s t❤❡✐r s✉♠✳

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿

0 ≤ an → 0 ❆◆❉ 0 ≤ bn → 0 =⇒ an + bn → 0 . Pr♦♦❢✳

❙✉♣♣♦s❡ ε > 0 ✐s ❣✐✈❡♥✳ ❋r♦♠ t❤❡ ❞❡✜♥✐t✐♦♥✱ • an → 0 =⇒ t❤❡r❡ ✐s N s✉❝❤ t❤❛t an < ε/2 ❢♦r ❛❧❧ n > N ✱ ❛♥❞ • bn → 0 =⇒ t❤❡r❡ ✐s M s✉❝❤ t❤❛t bn < ε/2 ❢♦r ❛❧❧ n > M ✳ ❚❤❡♥ ❢♦r ❛❧❧ n > max{N, M }✱ ✇❡ ❤❛✈❡✿ an + bn < ε/2 + ε/2 = ε .

❚❤❡r❡❢♦r❡✱ ❜② ❞❡✜♥✐t✐♦♥✱ ✇❡ ❤❛✈❡✿ an + bn → 0✳ ❊①❡r❝✐s❡ ✶✳✹✳✹

Pr♦✈❡ t❤❡ ✈❡rs✐♦♥ ♦❢ t❤❡ ❛❜♦✈❡ t❤❡♦r❡♠ ❢♦r m s❡q✉❡♥❝❡s ✭❛✮ ❢r♦♠ t❤❡ t❤❡♦r❡♠ ❛♥❞ ✭❜✮ ❜② ❣❡♥❡r❛❧✐③✐♥❣ t❤❡ ♣r♦♦❢✳ ❊①❛♠♣❧❡ ✶✳✹✳✺✿ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ ❙✉♠ ❘✉❧❡

1 1 1 1 → 0 =⇒ → 0 ❆◆❉ + 2 → 0. 2 n n n n

❇❡❢♦r❡ ❝♦♥s✐❞❡r✐♥❣ t❤❡ ♣r♦❞✉❝t ♦❢ t✇♦ s❡q✉❡♥❝❡s✱ ❧❡t✬s ❝♦♥s✐❞❡r ✇❤❛t ❤❛♣♣❡♥s ✇❤❡♥ ✇❡ ♠✉❧t✐♣❧② ❛ s❡q✉❡♥❝❡ ❜② ❛ ❝♦♥st❛♥t ♥✉♠❜❡r✳ ❲❡ ❦♥♦✇ t❤❛t s✉❝❤ ♠✉❧t✐♣❧✐❝❛t✐♦♥ s✐♠♣❧② str❡t❝❤❡s t❤❡ ✇❤♦❧❡ ♣❧❛♥❡ ✐♥ t❤❡ ✈❡rt✐❝❛❧ ❞✐r❡❝t✐♦♥ ✭❜♦t❤ ✉♣ ❛♥❞ ❞♦✇♥✱ ❛✇❛② ❢r♦♠ t❤❡ n✲❛①✐s✮✳

✶✳✹✳ ▲✐♠✐ts ✉♥❞❡r ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s

✹✽

❚❤❡ ❣r❛♣❤ ❛❧s♦ str❡t❝❤❡s✳ ❍❡r❡ ✇❡ s❡❡ an ✭r❡❞✮ ❛♥❞ 1.2 · an ✭♦r❛♥❣❡✮✿

❯♥❞❡r s✉❝❤ str❡t❝❤✱ ③❡r♦ r❡♠❛✐♥s ③❡r♦✦ ❚❤❡♦r❡♠ ✶✳✹✳✻✿ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ❢♦r ❩❡r♦ ▲✐♠✐t ❙❡q✉❡♥❝❡s ■❢ ❛ s❡q✉❡♥❝❡ ✇✐t❤ ♥♦♥✲♥❡❣❛t✐✈❡ t❡r♠s ❝♦♥✈❡r❣❡s t♦ ③❡r♦✱ t❤❡♥ s♦ ❞♦❡s ❛♥② ♦❢ ✐ts ♠✉❧t✐♣❧❡s✳

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿

0 ≤ an → 0 =⇒ can → 0

❢♦r ❛♥② r❡❛❧

c > 0.

Pr♦♦❢✳

❙✉♣♣♦s❡ ε > 0 ✐s ❣✐✈❡♥✳ ❋r♦♠ t❤❡ ❞❡✜♥✐t✐♦♥✱ ✇❡ ❤❛✈❡✿ ◮ 0 < an → 0 =⇒ t❤❡r❡ ✐s N s✉❝❤ t❤❛t an < ε/c ❢♦r ❛❧❧ n > N ✳ ❚❤❡♥ ❢♦r ❛❧❧ n > N ✱ ✇❡ ❤❛✈❡✿ c · an < c · ε/c = ε .

❚❤❡r❡❢♦r❡✱ ❜② ❞❡✜♥✐t✐♦♥✱ ✇❡ ❤❛✈❡✿ can → 0✳

❊①❛♠♣❧❡ ✶✳✹✳✼✿ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡

5 1 → 0 =⇒ → 0. n n

❋♦r ♠♦r❡ ❝♦♠♣❧❡① s✐t✉❛t✐♦♥s✱ ✇❡ ♥❡❡❞ t♦ ✉s❡ t❤❡ ❢❛❝t t❤❛t ❝♦♥✈❡r❣❡♥t s❡q✉❡♥❝❡s ❛r❡ ❜♦✉♥❞❡❞ ❛s ❢✉♥❝t✐♦♥s ✭s❡❡♥ ✐♥ ❱♦❧✉♠❡ ✶✱ ❈❤❛♣t❡r ✶P❈✲✷✮✿

✶✳✹✳ ▲✐♠✐ts ✉♥❞❡r ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s

✹✾

❚❤❡♦r❡♠ ✶✳✹✳✽✿ ❇♦✉♥❞❡❞♥❡ss ♦❢ ❈♦♥✈❡r❣❡♥t ❙❡q✉❡♥❝❡s ❆ ❝♦♥✈❡r❣❡♥t s❡q✉❡♥❝❡ ✐s ❜♦✉♥❞❡❞✳

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿

an → a =⇒ |an | < Q

❢♦r s♦♠❡ r❡❛❧

Q.

Pr♦♦❢✳

❚❤❡ ✐❞❡❛ ✐s t❤❛t t❤❡ t❛✐❧ ♦❢ t❤❡ s❡q✉❡♥❝❡ ✜ts ✐♥t♦ ❛ ✭♥❛rr♦✇✮ ❜❛♥❞❀ ♠❡❛♥✇❤✐❧❡✱ t❤❡r❡ ❛r❡ ♦♥❧② ✜♥✐t❡❧② ♠❛♥② t❡r♠s ❧❡❢t✦ ❈❤♦♦s❡ ε = 1✳ ❚❤❡♥ ❜② ❞❡✜♥✐t✐♦♥✱ t❤❡r❡ ✐s s✉❝❤ N t❤❛t ❢♦r ❛❧❧ n > N ✇❡ ❤❛✈❡✿ |an − a| < 1 .

❚❤❡♥✱ ✇❡ ❤❛✈❡✿ |an | = |(an − a) + a| ❲❡ ❛❞❞ ❛♥❞ s✉❜tr❛❝t a. ≤ |an − a| + |a| ❲❡ ✉s❡ t❤❡ ❚r✐❛♥❣❧❡ ■♥❡q✉❛❧✐t② ✭❈❤❛♣t❡r ✶P❈✲✺✮✳ < 1 + |a|. ❲❡ ✉s❡ t❤❡ ✐♥❡q✉❛❧✐t② ❛❜♦✈❡✳

❚♦ ✜♥✐s❤ t❤❡ ♣r♦♦❢✱ ✇❡ ❝❤♦♦s❡✿  Q = max |a1 |, ..., |aN |, 1 + |a| .

❙♦✱ t❤❡ s❡q✉❡♥❝❡ ✜ts ✐♥t♦ ❛ ✭♥♦t ♥❡❝❡ss❛r✐❧② ♥❛rr♦✇✮ ❜❛♥❞✳ ❚❤❡ ♣r♦♦❢ ✐s ✐❧❧✉str❛t❡❞ ❜❡❧♦✇✿

❚❤❡ ❝♦♥✈❡rs❡ ✐s♥✬t tr✉❡✿ ◆♦t ❡✈❡r② ❜♦✉♥❞❡❞ s❡q✉❡♥❝❡ ✐s ❝♦♥✈❡r❣❡♥t✳ ❏✉st tr② an = sin n✳ ❲❡ ✇✐❧❧ s❤♦✇ ❧❛t❡r t❤❛t✱ ✇✐t❤ ❛♥ ❡①tr❛ ❝♦♥❞✐t✐♦♥✱ ❜♦✉♥❞❡❞ s❡q✉❡♥❝❡s ❞♦ ❤❛✈❡ t♦ ❝♦♥✈❡r❣❡✳ ❆ s♠❛❧❧❡r s❡q✉❡♥❝❡ ✇♦✉❧❞ ❜❡ sq✉❡❡③❡❞ ❜❡t✇❡❡♥ t❤❡ ❧❛r❣❡r ♦♥❡ ❛♥❞ ③❡r♦✿ ❈♦r♦❧❧❛r② ✶✳✹✳✾✿ ❙q✉❡❡③❡ ❚❤❡♦r❡♠ ❢♦r ❩❡r♦ ▲✐♠✐t ❙❡q✉❡♥❝❡s ■❢ ❛ s❡q✉❡♥❝❡ ✇✐t❤ ♥♦♥✲♥❡❣❛t✐✈❡ t❡r♠s ❝♦♥✈❡r❣❡s t♦ ③❡r♦✱ t❤❡♥ s♦ ❞♦❡s ❛♥② ♦t❤❡r s♠❛❧❧❡r s❡q✉❡♥❝❡ ✇✐t❤ ♥♦♥✲♥❡❣❛t✐✈❡ t❡r♠s✳

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿

0 < an → 0 ❆◆❉ 0 < bn < an =⇒ bn → 0 . ❊①❡r❝✐s❡ ✶✳✹✳✶✵

Pr♦✈❡ t❤❡ ❝♦r♦❧❧❛r②✳ ❲❡ ❛r❡ ♥♦✇ r❡❛❞② ❢♦r t❤❡ ❣❡♥❡r❛❧ r❡s✉❧ts ♦♥ t❤❡ ❛❧❣❡❜r❛ ♦❢ ❧✐♠✐ts✳

✶✳✹✳ ▲✐♠✐ts ✉♥❞❡r ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s

✺✵

❋✐rst✱ t❤❡ s✉♠♠❛t✐♦♥✿

❚❤❡♦r❡♠ ✶✳✹✳✶✶✿ ❙✉♠ ❘✉❧❡ ❢♦r ▲✐♠✐ts ♦❢ ❙❡q✉❡♥❝❡s ■❢ s❡q✉❡♥❝❡s

a n , bn

❝♦♥✈❡r❣❡✱ t❤❡♥ s♦ ❞♦❡s

an + bn ✳

❋✉rt❤❡r♠♦r❡✱ ✇❡ ❤❛✈❡✿

lim (an + bn ) = lim an + lim bn

n→∞

n→∞

n→∞

Pr♦♦❢✳

❙✉♣♣♦s❡ an → a, bn → b .

❚❤❡♥ ❲❡ ❝♦♠♣✉t❡

|an − a| → 0, |bn − b| → 0 .

|(an + bn ) − (a + b)| = |(an − a) + (bn − b)| ❲❡ r❡❛rr❛♥❣❡ t❤❡ t❡r♠s✳ ≤ |an − a| + |bn − b| ❲❡ ✉s❡ t❤❡ ❚r✐❛♥❣❧❡ ■♥❡q✉❛❧✐t②✳ →0+0 ❲❡ ✉s❡ ❙✉♠ ❘✉❧❡ ❢♦r ❩❡r♦ ▲✐♠✐t ❙❡q✉❡♥❝❡s✳ = 0.

❚❤❡♥✱ ❜② t❤❡ ❧❛st ❝♦r♦❧❧❛r②✱ ✇❡ ❤❛✈❡✿ |(an + bn ) − (a + b)| → 0 .

❚❤❡♥✱ ❜② t❤❡ ✜rst t❤❡♦r❡♠✱ ✇❡ ❤❛✈❡✿ an + bn → a + b . ❲❛r♥✐♥❣✦

❚❤❡r❡ ✐s ♦♥❡ ❝♦♥❞✐t✐♦♥ ✭❝♦♥✈❡r❣❡♥❝❡✮ ❛♥❞ t✇♦ ❝♦♥✲ ❝❧✉s✐♦♥s ✭❝♦♥✈❡r❣❡♥❝❡ ❛♥❞ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❧✐♠✐t ♦❢ t❤❡ s✉♠✮✳

✶✳✹✳

▲✐♠✐ts ✉♥❞❡r ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s

✺✶

❊①❡r❝✐s❡ ✶✳✹✳✶✷

❲❤❛t ✐❢ ♦♥❡ ♦r ❜♦t❤ ♦❢ t❤❡ ❧✐♠✐ts ❛r❡ ✐♥✜♥✐t❡❄

❆s ②♦✉ ❝❛♥ s❡❡✱ s✉❝❤ ❛ ❧✐♠✐t ✐s s✐♠♣❧② ✏s♣❧✐t✑ ✐♥ ❤❛❧❢✳ ❲❤✐❧❡ t❤❡ ❣❡♦♠❡tr✐❝ ✐♥t❡r♣r❡t❛t✐♦♥ ♦❢ ❛❞❞✐t✐♦♥ ✐s ♣✉tt✐♥❣ t✇♦ ❜❛rs t♦❣❡t❤❡r✱ ✇❤❛t ✐s t❤❛t ❢♦r ♠✉❧t✐♣❧✐❝❛t✐♦♥❄ ■❢

a

❛♥❞

b

❛r❡ t❤❡ s✐❞❡s ♦❢ ❛ r❡❝t❛♥❣❧❡✱ t❤❡♥

ab

✐s ✐ts ❛r❡❛✳

❙♦✱ ✇❤❡♥ t✇♦ s❡q✉❡♥❝❡s ❛r❡ ♠✉❧t✐♣❧✐❡❞✱ ✐t ✐s ❛s ✐❢ ✇❡ ✉s❡ ❡❛❝❤ ♣❛✐r ♦❢ t❤❡✐r ✈❛❧✉❡s t♦ ❜✉✐❧❞ ❛ r❡❝t❛♥❣❧❡✿

❚❤❡♥ t❤❡ ❛r❡❛s ♦❢ t❤❡s❡ r❡❝t❛♥❣❧❡s ❢♦r♠ ❛ ♥❡✇ s❡q✉❡♥❝❡ ❛♥❞ t❤❡s❡ ❛r❡❛s ❝♦♥✈❡r❣❡ ✐❢ t❤❡ ✇✐❞t❤s ❛♥❞ t❤❡ ❤❡✐❣❤ts ❝♦♥✈❡r❣❡✿ ❚❤❡♦r❡♠ ✶✳✹✳✶✸✿ Pr♦❞✉❝t ❘✉❧❡ ❢♦r ▲✐♠✐ts ♦❢ ❙❡q✉❡♥❝❡s ■❢ s❡q✉❡♥❝❡s

a n , bn

❝♦♥✈❡r❣❡✱ t❤❡♥ s♦ ❞♦❡s

an · bn ✳

❋✉rt❤❡r♠♦r❡✱ ✇❡ ❤❛✈❡✿

lim (an · bn ) =

n→∞


lim an ·

n→∞

lim bn

n→∞




Pr♦♦❢✳

❙✉♣♣♦s❡

❚❤❡♥✱

❲❡ ❝♦♠♣✉t❡✿

an → a, bn → b. |an − a| → 0, |bn − b| → 0 .

|an · bn − a · b| = |an · bn + (−a · bn + a · bn ) − a · b| = |(an − a) · bn + a · (bn − b)| ≤ |(an − a) · bn | + |a · (bn − b)| = |an − a| · |bn | + |a| · |bn − b| ≤ |an − a| · Q + |a| · |bn − b| → 0 · Q + |a| · 0 = 0. ❚❤❡r❡❢♦r❡✱

an · bn → a · b .

❲❡ ❛❞❞ ❡①tr❛ t❡r♠s ❛♥❞ t❤❡♥ ❢❛❝t♦r✳ ❲❡ ✉s❡ t❤❡ ❚r✐❛♥❣❧❡ ■♥❡q✉❛❧✐t②✳ ❲❡ ✉s❡ t❤❡ ❇♦✉♥❞❡❞♥❡ss ❚❤❡♦r❡♠✳ ❲❡ ✉s❡ ❙❘ ❛♥❞ ❈▼❘✳

✶✳✹✳

▲✐♠✐ts ✉♥❞❡r ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s

✺✷

❚❤❡ ❧✐♠✐t s♣❧✐ts ✐♥ ❤❛❧❢✱ ❛❣❛✐♥✳ ❊①❡r❝✐s❡ ✶✳✹✳✶✹

❲❤❛t ✐❢ ♦♥❡ ♦r ❜♦t❤ ♦❢ t❤❡ ❧✐♠✐ts ❛r❡ ✐♥✜♥✐t❡❄

❚❤❡ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ❢♦❧❧♦✇s ❢r♦♠ t❤❡ t❤❡♦r❡♠✿ ❈♦r♦❧❧❛r② ✶✳✹✳✶✺✿ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ❢♦r ▲✐♠✐ts ♦❢ ❙❡q✉❡♥❝❡s

■❢ s❡q✉❡♥❝❡ an ❝♦♥✈❡r❣❡s✱ t❤❡♥ s♦ ❞♦❡s can ❢♦r ❛♥② r❡❛❧ c✳ ❋✉rt❤❡r♠♦r❡✱ ✇❡ ❤❛✈❡✿
lim c · an = c · lim an n→∞

n→∞

❆s ②♦✉ ❝❛♥ s❡❡✱ ❛ ❝♦♥st❛♥t ✐s s✐♠♣❧② ✏❢❛❝t♦r❡❞✑ ♦✉t ♦❢ ❛ ✭❝♦♥✈❡r❣❡♥t✦✮ ❧✐♠✐t✳ ❊①❡r❝✐s❡ ✶✳✹✳✶✻

❉❡r✐✈❡ t❤❡ ❝♦r♦❧❧❛r② ❢r♦♠ t❤❡ ❧❛st t❤❡♦r❡♠✳

❖♥❡ ❝❛♥ ✉♥❞❡rst❛♥❞ ❞✐✈✐s✐♦♥ ♦❢ s❡q✉❡♥❝❡s ❛s ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✐♥ r❡✈❡rs❡✿ ■❢ t❤❡ ❛r❡❛s ♦❢ t❤❡ r❡❝t❛♥❣❧❡s ❝♦♥✈❡r❣❡ ❛♥❞ s♦ ❞♦ t❤❡✐r ✇✐❞t❤s✱ t❤❡♥ s♦ ❞♦ t❤❡✐r ❤❡✐❣❤ts✳ ❆❧s♦✱ ✇❤❡♥ t✇♦ s❡q✉❡♥❝❡s ❛r❡ ❞✐✈✐❞❡❞✱ ✐t ✐s ❛s ✐❢ ✇❡ ✉s❡ ❡❛❝❤ ♣❛✐r ♦❢ t❤❡✐r ✈❛❧✉❡s t♦ ❜✉✐❧❞ ❛ tr✐❛♥❣❧❡✿

❚❤❡♥ t❤❡ t❛♥❣❡♥ts ♦❢ t❤❡ ❜❛s❡ ❛♥❣❧❡s ♦❢ t❤❡s❡ tr✐❛♥❣❧❡s ❢♦r♠ ❛ ♥❡✇ s❡q✉❡♥❝❡ ❛♥❞ t❤❡② ❝♦♥✈❡r❣❡ ✐❢ t❤❡ ✇✐❞t❤s ❛♥❞ t❤❡ ❤❡✐❣❤ts ❝♦♥✈❡r❣❡✳ ❚❤✐s r❡s✉❧t ✐s ❡s♣❡❝✐❛❧❧② ✐♠♣♦rt❛♥t ✐♥ t❤❡ ❢♦rt❤❝♦♠✐♥❣ ♣❛rts ♦❢ ❝❛❧❝✉❧✉s✿ ❚❤❡♦r❡♠ ✶✳✹✳✶✼✿ ◗✉♦t✐❡♥t ❘✉❧❡ ❢♦r ▲✐♠✐ts ♦❢ ❙❡q✉❡♥❝❡s

■❢ s❡q✉❡♥❝❡s an , bn ❝♦♥✈❡r❣❡✱ t❤❡♥ s♦ ❞♦❡s an /bn ✇❤❡♥❡✈❡r ❞❡✜♥❡❞✱ ❛♥❞

lim

n→∞



an bn



lim an

=

n→∞

lim bn

n→∞

♣r♦✈✐❞❡❞ lim bn 6= 0 . n→∞

Pr♦♦❢✳

an = 1✳ ❙✉♣♣♦s❡ bn → b 6= 0✳ ❋✐rst✱ N s✉❝❤ t❤❛t ❢♦r ❛❧❧ n > N ✇❡ ❤❛✈❡✿

❲❡ ✇✐❧❧ ♦♥❧② ♣r♦✈❡ t❤❡ ❝❛s❡ ♦❢ ♦❢ ❝♦♥✈❡r❣❡♥❝❡✳ ❚❤❡♥ t❤❡r❡ ✐s

|bn − b| < |b|/2 .

❝❤♦♦s❡

ε = |b|/2

✐♥ t❤❡ ❞❡✜♥✐t✐♦♥

✶✳✹✳

▲✐♠✐ts ✉♥❞❡r ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s

✺✸

❚❤❡r❡❢♦r❡✱

|bn | > |b|/2 . ◆❡①t✱

b − bn 1 1 − = bn b bn b |b − bn | = |bn | · |b| |b − bn | < |b/2| · |b| 0 → |b/2| · |b|

❋r♦♠ t❤❡ ❛❜♦✈❡ ✐♥❡q✉❛❧✐t②✳

❲❡ ✉s❡ ❈▼❘✳

= 0. ❚❤❡r❡❢♦r❡✱

❋✐♥❛❧❧②✱ t❤❡ ❣❡♥❡r❛❧ ❝❛s❡ ♦❢ t❤❡

◗✉♦t✐❡♥t ❘✉❧❡

1 1 → . bn b ❢♦❧❧♦✇s ❢r♦♠

Pr♦❞✉❝t ❘✉❧❡ ✿

an 1 1 a = an · →a· = . bn bn b b ❲❛r♥✐♥❣✦

❚❤❡r❡ ❛r❡ t✇♦ ❝♦♥❞✐t✐♦♥s ✭❝♦♥✈❡r❣❡♥❝❡ ❛♥❞ ♥♦♥✲ ③❡r♦ ♦❢ t❤❡ ❧✐♠✐t ♦❢ t❤❡ ❞❡♥♦♠✐♥❛t♦r✮ ❛♥❞ t✇♦ ❝♦♥✲ ❝❧✉s✐♦♥s ✭❝♦♥✈❡r❣❡♥❝❡ ❛♥❞ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❧✐♠✐t ♦❢ t❤❡ r❛t✐♦✮✳

❊①❡r❝✐s❡ ✶✳✹✳✶✽ ❲❤❛t ✐❢ ♦♥❡ ♦r ❜♦t❤ ♦❢ t❤❡ ❧✐♠✐ts ❛r❡ ✐♥✜♥✐t❡❄

❚❤❡ s✉♠♠❛r② r❡s✉❧t ❜❡❧♦✇ s❤♦✇s t❤❛t ✇❤❡♥ ✇❡ r❡♣❧❛❝❡ ❡✈❡r② r❡❛❧ ♥✉♠❜❡r ✇✐t❤ ❛ s❡q✉❡♥❝❡ ❝♦♥✈❡r❣✐♥❣ t♦ ✐t✱ ✐t ✐s st✐❧❧ ♣♦ss✐❜❧❡ t♦ ❞♦ ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s ✇✐t❤ t❤❡♠✿

❚❤❡♦r❡♠ ✶✳✹✳✶✾✿ ❆❧❣❡❜r❛ ♦❢ ▲✐♠✐ts ♦❢ ❙❡q✉❡♥❝❡s

an → a

❙✉♣♣♦s❡

❙❘✿ P❘✿

❛♥❞

bn → b✳

❚❤❡♥

an + b n → a + b an · bn → ab

❈▼❘✿ ◗❘✿

c · an → ca an /bn → a/b

c b 6= 0

❢♦r ❛♥② r❡❛❧ ♣r♦✈✐❞❡❞

❆s ②♦✉ ❝❛♥ s❡❡✱ ✭❝♦♥✈❡r❣❡♥t✦✮ s❡q✉❡♥❝❡s ❛r❡ s✐♠♣❧② r❡♣❧❛❝❡❞ ✇✐t❤ t❤❡✐r ❧✐♠✐ts✳ ❚❤❡s❡ r❡s✉❧ts ❛r❡ ❛❧s♦ ❦♥♦✇♥ ❛s t❤❡

▲✐♠✐ts ❚❤❡♦r❡♠s✳

❊①❛♠♣❧❡ ✶✳✹✳✷✵✿ ✉s✐♥❣ ❧✐♠✐ts r✉❧❡s ▲❡t

an = 7n−2 + ❲❤❛t ✐s ✐ts ❧✐♠✐t ❛s ✐s ❣❡♦♠❡tr✐❝✱ ❛♥❞

n → ∞❄

8 ✐s ❝♦♥st❛♥t✳

2 + 8. 3n

❲❡ r❡❝♦❣♥✐③❡ t❤r❡❡ ✏s❡❡❞✑ s❡q✉❡♥❝❡s✿

n−2

✐s ❛ ♣♦✇❡r s❡q✉❡♥❝❡✱

3n

❚❤❡s❡ s❡q✉❡♥❝❡s ✇✐t❤ ❦♥♦✇♥ ❧✐♠✐ts✱ ❤♦✇❡✈❡r✱ ❛r❡ ❝♦♠❜✐♥❡❞ ❜② ❛❧❣❡❜r❛✳

✶✳✹✳

▲✐♠✐ts ✉♥❞❡r ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s

✺✹

❚❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ❧✐♠✐t ❜❡❧♦✇ ✐s str❛✐❣❤t❢♦r✇❛r❞✱ ❜✉t ❡✈❡r② st❡♣ ❤❛s t♦ ❜❡ ❥✉st✐✜❡❞ ✇✐t❤ t❤❡ r✉❧❡s ♦❢ ❧✐♠✐ts ♣r❡s❡♥t❡❞ ❛❜♦✈❡✳ ❚♦ ✉♥❞❡rst❛♥❞ ✇❤✐❝❤ r✉❧❡ t♦ ❛♣♣❧② ✜rst✱ ♦❜s❡r✈❡ ✇❤❛t t❤❡ ❧❛st ♦♣❡r❛t✐♦♥ ✐s✳ ■t ✐s ❛❞❞✐t✐♦♥✳ ❲❡✱ t❤❡r❡❢♦r❡✱ ✉s❡ t❤❡ ❙✉♠ ❘✉❧❡ ✜rst✱ s✉❜❥❡❝t t♦ ❥✉st✐✜❝❛t✐♦♥✿

lim an = lim (7n−2 +

n→∞

n→∞

2 + 8) 3n

❲❡ ✉s❡ ❙❘✳

= lim (7 · n−2 ) + lim (2 · n→∞

n→∞

1 ) + lim 8 n→∞ 3n

❲❡ ✉s❡ ❈▼❘✳ ❲❡ ✉s❡ t❤❡ ❧✐♠✐ts ❢r♦♠ t❤❡ ❧❛st s❡❝t✐♦♥✳

= 7 · lim n−2 + 2 · lim 3−n + 8 n→∞

n→∞

=7·0+2·0+8 = 8.

❙✉♠ ❘✉❧❡ ✭❛♥❞ t❤❡♥ t❤❡ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ✮ ✇❛s ❥✉st✐✜❡❞✳

❆s ❛❧❧ t❤❡ ❧✐♠✐ts ❡①✐st✱ ♦✉r ✉s❡ ♦❢ t❤❡

❲❛r♥✐♥❣✦ ■t ✐s ❝♦♥s✐❞❡r❡❞ ❛ s❡r✐♦✉s ❡rr♦r ✐❢ ②♦✉ ✉s❡ t❤❡ ❝♦♥✲ ❝❧✉s✐♦♥ ✭t❤❡ ❢♦r♠✉❧❛✮ ♦❢ ♦♥❡ ♦❢ t❤❡s❡ r✉❧❡s ✇✐t❤♦✉t ✈❡r✐❢②✐♥❣ t❤❡ ❝♦♥❞✐t✐♦♥s ✭t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ s❡✲ q✉❡♥❝❡s ✐♥✈♦❧✈❡❞✮✳

❊①❛♠♣❧❡ ✶✳✹✳✷✶✿ ✐♥✜♥✐t❡ ❧✐♠✐ts Pr♦✈❡ t❤❡ ❧✐♠✐t✿

lim (n2 − n) = +∞ .

n→∞

P❧♦tt✐♥❣ t❤❡ ❣r❛♣❤ ❞♦❡s s✉❣❣❡st t❤❛t t❤❡ ❧✐♠✐t ✐s ✐♥✜♥✐t❡✳ ❙✐♥❝❡ t❤❡ ❧❛st ♦♣❡r❛t✐♦♥ ✐s ❛❞❞✐t✐♦♥✱ ✇❡ ❛r❡ s✉♣♣♦s❡❞ t♦ ✉s❡ t❤❡ ❙✉♠ ❘✉❧❡ ✜rst✳ ❍♦✇❡✈❡r✱ ❜♦t❤ ♦❢ t❤❡ t❡r♠s ❣♦ t♦ ∞✱ ✇❤✐❝❤ ♠❛❦❡s t❤❡ ❙✉♠ ❘✉❧❡ ✐♥❛♣♣❧✐❝❛❜❧❡✳ ❆♥ ❛tt❡♠♣t t♦ ❛♣♣❧② t❤❡ ❙✉♠ ❘✉❧❡ ✕ ♦✈❡r t❤✐s ♦❜❥❡❝t✐♦♥ ✕ ✇♦✉❧❞ r❡s✉❧t ✐♥ ❛ ♠❡❛♥✐♥❣❧❡ss ❡①♣r❡ss✐♦♥✿ ???

n2 − n −−−−−→ ∞ − ∞ . ❲❡ ❝❛♥✬t ❝♦♥❝❧✉❞❡ t❤❛t t❤❡ ❧✐♠✐t ❞♦❡s♥✬t ❡①✐st✱ ♦r t❤❛t ✐s ❞♦❡s✦ ❲❡✬✈❡ ❥✉st ❢❛✐❧❡❞ t♦ ✜♥❞ t❤❡ ❛♥s✇❡r✳ ❚❤❡ ✐ss✉❡ ✐s r❡s♦❧✈❡❞ ✐♥ t❤❡ ♥❡①t s❡❝t✐♦♥✳ Pr❡s❡♥t❡❞ ✈❡r❜❛❧❧②✱ t❤❡s❡ r✉❧❡s ❤❛✈❡ t❤❡s❡ ❛❜❜r❡✈✐❛t❡❞ ✈❡rs✐♦♥s✿

• ❚❤❡ ❧✐♠✐t ♦❢ t❤❡

s✉♠ ✐s t❤❡ s✉♠ ♦❢ t❤❡ ❧✐♠✐ts✳

• ❚❤❡ ❧✐♠✐t ♦❢ t❤❡

♣r♦❞✉❝t ✐s t❤❡ ♣r♦❞✉❝t ♦❢ t❤❡ ❧✐♠✐ts✳

• ❚❤❡ ❧✐♠✐t ♦❢ t❤❡

❞✐✛❡r❡♥❝❡ ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ ❧✐♠✐ts✳

• ❚❤❡ ❧✐♠✐t ♦❢ t❤❡

q✉♦t✐❡♥t ✐s t❤❡ q✉♦t✐❡♥t ♦❢ t❤❡ ❧✐♠✐ts ✭❛s ❧♦♥❣ ❛s t❤❛t ♦❢ t❤❡ ❞❡♥♦♠✐♥❛t♦r ✐s♥✬t ③❡r♦✮✳

❲❡ ❝❛♥ ❛❧s♦ r❡♣r❡s❡♥t t❤❡s❡ r✉❧❡s ❛s ❞✐❛❣r❛♠s✳ ❋♦r ❡①❛♠♣❧❡✱ t❤✐s ✐s t❤❡ ❙✉♠ ❘✉❧❡✿

an, bn  + y

lim

−−−−−→ SR lim

a, b  + y

an + bn −−−−−→ lim(an + bn ) = a + b ■♥ t❤❡ ❞✐❛❣r❛♠✱ ✇❡ st❛rt ✇✐t❤ ❛ ♣❛✐r ♦❢ s❡q✉❡♥❝❡s ❛t t❤❡ t♦♣ ❧❡❢t ❛♥❞ t❤❡♥ ✇❡ ♣r♦❝❡❡❞ ✐♥ t✇♦ ✇❛②s✿

✶✳✺✳

❈❛♥ ✇❡ ❛❞❞ ✐♥✜♥✐t✐❡s❄ ❙✉❜tr❛❝t❄ ❉✐✈✐❞❡❄ ▼✉❧t✐♣❧②❄

✺✺

•

r✐❣❤t✿ t❛❦❡ t❤❡ ❧✐♠✐t ♦❢ ❡✐t❤❡r✱ t❤❡♥ ❞♦✇♥✿ ❛❞❞ t❤❡ r❡s✉❧ts❀ ♦r

•

❞♦✇♥✿ ❛❞❞ t❤❡♠✱ t❤❡♥ r✐❣❤t✿ t❛❦❡ t❤❡ ❧✐♠✐t ♦❢ t❤❡ r❡s✉❧t✳

❚❤❡ r❡s✉❧t ✐s t❤❡ s❛♠❡✦ ❋♦r t❤❡

Pr♦❞✉❝t ❘✉❧❡

❛♥❞ t❤❡

◗✉♦t✐❡♥t ❘✉❧❡✱

r❡s♣❡❝t✐✈❡❧②✳

✇❡ ❥✉st r❡♣❧❛❝❡ ✏ +✑ ✇✐t❤ ✏ ·✑ ❛♥❞ ✏ ÷✑

❚❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ ❙✉♠ ❘✉❧❡ ❛♥❞ t❤❡ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ✐s ✇♦rt❤ r❡♠❡♠❜❡r✐♥❣✿

❚❤❡♦r❡♠ ✶✳✹✳✷✷✿ ▲✐♥❡❛r✐t② ❘✉❧❡ ❢♦r ▲✐♠✐ts ♦❢ ❙❡q✉❡♥❝❡s

an → a

❙✉♣♣♦s❡

❛♥❞

m

❛♥❞

b

❛r❡ r❡❛❧ ♥✉♠❜❡rs✳ ❚❤❡♥

man + b → ma + b ❲❤❛t ❛❜♦✉t

✐♥✜♥✐t❡ ❧✐♠✐ts ❄

■❢ ✇❡ r❡♣❧❛❝❡ ✐♥✜♥✐t② ✭♣♦s✐t✐✈❡ ♦r ♥❡❣❛t✐✈❡✮ ✇✐t❤ ❛ s❡q✉❡♥❝❡ t❤❛t ❛♣♣r♦❛❝❤❡s ✐t✱

✇✐❧❧ t❤❡ ❛❧❣❡❜r❛ ♠❛❦❡ s❡♥s❡❄

✶✳✺✳ ❈❛♥ ✇❡ ❛❞❞ ✐♥✜♥✐t✐❡s❄ ❙✉❜tr❛❝t❄ ❉✐✈✐❞❡❄ ▼✉❧t✐♣❧②❄ ❲❡ ❤❛✈❡ ❞❡♠♦♥str❛t❡❞ t❤❛t ✐♥ ♦✉r ❝♦♠♣✉t❛t✐♦♥s ♦❢ ❧✐♠✐ts ✇❡ ❝❛♥ r❡♣❧❛❝❡ ❛♥② s❡q✉❡♥❝❡ ✇✐t❤ ✐ts ❧✐♠✐t ❛♥❞ ❝♦♥t✐♥✉❡ ❞♦✐♥❣ t❤❡ ❛❧❣❡❜r❛✳ ❚❤✐s ❝♦♥❝❧✉s✐♦♥ ❞♦❡s♥✬t ❛♣♣❧② t♦ ❞✐✈❡r❣❡♥t s❡q✉❡♥❝❡s✦

❲❛r♥✐♥❣✦

◆❡✈❡r ❢♦r❣❡t t♦ ❝♦♥✜r♠ t❤❡ ♣r❡❝♦♥❞✐t✐♦♥s ✇❤❡♥ ✉s✲ ✐♥❣ t❤❡s❡ r✉❧❡s✳ ❙❡q✉❡♥❝❡s t❤❛t ❛♣♣r♦❛❝❤ ✐♥✜♥✐t②

❞✐✈❡r❣❡✱ t❡❝❤♥✐❝❛❧❧②✱ ❜✉t t❤❡② ♣r♦✈✐❞❡ ✉s❡❢✉❧ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t t❤❡ ♣❛tt❡r♥

❡①❤✐❜✐t❡❞ ❜② t❤❡ s❡q✉❡♥❝❡s✳ ❙✉❝❤ ❛ s❡q✉❡♥❝❡ ❝❛♥ ❛❧s♦ ❜❡ ✉s❡❞ t♦ ❝r❡❛t❡ ❛ ❝♦♥✈❡r❣❡♥t s❡q✉❡♥❝❡✿

an = m

❛♥❞

bn =

1 . n

❘❡❝❛❧❧ t❤❛t t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ ❣r❛♣❤ ♦❢ ❛ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧ ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ s✐❣♥ ♦❢

a

f (x) = ax2 + bx + c

✭❛ ♣❛r❛❜♦❧❛✮ ✐s

♦♥❧②✿

❚❤❡ ♣✐❝t✉r❡ s✉❣❣❡sts t❤❛t t❤❡ ❧✐♠✐ts ❛r❡ ✐♥✜♥✐t❡✱

+∞

✐♥ t❤❡ ❢♦r♠❡r ❝❛s❡ ❛♥❞

−∞

✐♥ t❤❡ ❧❛tt❡r✳ ❲❡ ❡✈❡♥ p ❦♥♦✇ ♠♦r❡✿ ❚❤❡ ❧❡❛❞✐♥❣ t❡r♠ ✇✐❧❧ ❞❡t❡r♠✐♥❡ t❤❡ ❧❛r❣❡ s❝❛❧❡ s❤❛♣❡ ♦❢ t❤❡ ❣r❛♣❤ ♦❢ ❛♥② ♣♦❧②♥♦♠✐❛❧ ap n + p−1 ap−1 n + ... + a1 n + a0 ✿

✶✳✺✳ ❈❛♥ ✇❡ ❛❞❞ ✐♥✜♥✐t✐❡s❄ ❙✉❜tr❛❝t❄ ❉✐✈✐❞❡❄ ▼✉❧t✐♣❧②❄

✺✻

❚❤❡ ❢♦❧❧♦✇✐♥❣ r❡s✉❧t ♠❛❦❡s t❤❡s❡ ✐❞❡❛s ♣r❡❝✐s❡✿

❚❤❡♦r❡♠ ✶✳✺✳✶✿ ▲✐♠✐ts ♦❢ P♦❧②♥♦♠✐❛❧s ❛t ■♥✜♥✐t② ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ♣♦❧②♥♦♠✐❛❧ ♦❢ ❞❡❣r❡❡ p ✭t❤❡ ❧❡❛❞✐♥❣ ❝♦❡✣❝✐❡♥t ap 6= 0✮✳ ❚❤❡♥ t❤❡ ❧✐♠✐t ♦❢ t❤❡ s❡q✉❡♥❝❡ ❞❡✜♥❡❞ ❜② t❤✐s ❢✉♥❝t✐♦♥ ✐s✿

lim

n→∞



ap np + ap−1 np−1 + ... + a1 n + a0



=

(

+∞ −∞

✐❢ ap > 0 ✐❢ ap < 0

Pr♦♦❢✳ ❚❤❡ ✐❞❡❛ ✐s t♦ ❢❛❝t♦r ♦✉t t❤❡ ❤✐❣❤❡st ♣♦✇❡r ✿ ap np + ap−1 np−1 + ... + a1 n + a0 = np · (ap + ap−1 n−1 + ... + a1 n1−p + a0 n−p ) .

❚❤❡♥ t❤❡ ❧✐♠✐t ♦❢ t❤❡ ✜rst ❢❛❝t♦r ✐s ∞ ❛♥❞ t❤❛t ♦❢ t❤❡ s❡❝♦♥❞ ✐s ap + 0 = ap ✳ ❙♦✱ ❛s ❢❛r ❛s ✐ts ❜❡❤❛✈✐♦r ❛t ∞✱ ❢♦r ❛ ♣♦❧②♥♦♠✐❛❧✱ ♦♥❧② t❤❡ ❧❡❛❞✐♥❣ t❡r♠ ♠❛tt❡rs ✿ lim

n→∞



ap np +ap−1 np−1 + ... + a1 n + a0



= lim ap np = n→∞

(

+∞ −∞

✐❢ ap > 0 ✐❢ ap < 0

❋r♦♠ ♣♦❧②♥♦♠✐❛❧s t♦ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥s✿

❉❡✜♥✐t✐♦♥ ✶✳✺✳✷✿ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥ ❚❤❡ r❛t✐♦ ♦❢ t✇♦ ♣♦❧②♥♦♠✐❛❧s ✐s ❝❛❧❧❡❞ ❛ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥✳

❲❛r♥✐♥❣✦ ❆❧❧ ♣♦❧②♥♦♠✐❛❧s ❛r❡ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥s t♦♦✳

❚❤❡ ♣r♦❜❧❡♠s ✇❡ ❢❛❝❡ ❛r❡ ❞✐✛❡r❡♥t✳

✶✳✺✳ ❈❛♥ ✇❡ ❛❞❞ ✐♥✜♥✐t✐❡s❄ ❙✉❜tr❛❝t❄ ❉✐✈✐❞❡❄ ▼✉❧t✐♣❧②❄ ❊①❛♠♣❧❡ ✶✳✺✳✸✿ r❡❝✐♣r♦❝❛❧ s❡q✉❡♥❝❡

❲❡ ❛r❡ ❛❧r❡❛❞② ❢❛♠✐❧✐❛r ✇✐t❤ s✉❝❤ ❛♥ ✐♠♣♦rt❛♥t s❡q✉❡♥❝❡ ❛s t❤❡ r❡❝✐♣r♦❝❛❧s✿

an =

1 . n

❊✈❡♥ ✇✐t❤ s✉❝❤ ❛ s✐♠♣❧❡ ❢♦r♠✉❧❛✱ ♦♥❝❡ ❞✐✈✐s✐♦♥ ✐s ✐♥tr♦❞✉❝❡❞✱ t❤❡ ❝♦♠♣❧❡①✐t② ✐♥❝r❡❛s❡s ❞r❛♠❛t✐❝❛❧❧②✳ ❲❡ ❝❛♥ s❡❡ ✕ ✐♥ ❝♦♠♣❛r✐s♦♥ t♦ t❤❡ ♣♦❧②♥♦♠✐❛❧ s❡q✉❡♥❝❡s ✕ s♦♠❡ ♥❡✇ ❢❡❛t✉r❡s ✐♥ t❤❡ ❣r❛♣❤ ✭y = 1/x✮✿

▲❡t✬s ❧♦♦❦ ❛t t❤❡ x✲❛①✐s✳ ■❢ t❤❡ ❣r❛♣❤ ❝❛♥✬t ❝r♦ss ✐t✱ ✐t ❤❛s t♦ st❛rt t♦ ❝r❛✇❧✱ ✇✐t❤ ✈✐rt✉❛❧❧② ♥♦ ✉♣ ♦r ❞♦✇♥ ♣r♦❣r❡ss✿

❚❤❡ ♣❤❡♥♦♠❡♥♦♥ ✐s ❛❧s♦ s❡❡♥ ✐♥ t❤❡ ❞❛t❛✿

3 ... 100 ... n 1 2 1/n 1 1/2 1/3 ... 1/100 ... ■❢ ✇❡ ③♦♦♠ ♦✉t✱ t❤❡ ❡♥❞s ♦❢ t❤❡ ❣r❛♣❤ ♠❡r❣❡ ✇✐t❤ t❤❡ ❛①✐s✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ❧✐♠✐t ✐s ③❡r♦✱ ❛s ✇❡ ❦♥♦✇✿ 1 → 0 ❛s n → ∞ . n ❊①❛♠♣❧❡ ✶✳✺✳✹✿ ❝♦♠♣✉t❛t✐♦♥s

❊✈❛❧✉❛t❡ t❤❡ ❧✐♠✐t✿

4n2 − n + 2 . n→∞ 2n2 − 1 P❧♦tt✐♥❣ t❤❡ ❣r❛♣❤ s✉❣❣❡sts t❤❛t t❤❡ ❧✐♠✐t ✐s a = 2✿ lim

✺✼

✶✳✺✳ ❈❛♥ ✇❡ ❛❞❞ ✐♥✜♥✐t✐❡s❄ ❙✉❜tr❛❝t❄ ❉✐✈✐❞❡❄ ▼✉❧t✐♣❧②❄

❙✐♥❝❡ t❤❡ ❧❛st ♦♣❡r❛t✐♦♥ ✐s ❞✐✈✐s✐♦♥✱ s♦ ✇❡ ❛r❡ s✉♣♣♦s❡❞ t♦ ✉s❡ t❤❡ ◗✉♦t✐❡♥t ❘✉❧❡ ✜rst✳ ❍♦✇❡✈❡r✱ ❜♦t❤ t❤❡ ♥✉♠❡r❛t♦r ❛♥❞ t❤❡ ❞❡♥♦♠✐♥❛t♦r ❣♦ t♦ ∞✳ ❚❤❡♥✱ t❤❡ ◗✉♦t✐❡♥t ❘✉❧❡ ✐s ✐♥❛♣♣❧✐❝❛❜❧❡✦ ❆♥ ❛tt❡♠♣t t♦ ❛♣♣❧② t❤❡ ◗✉♦t✐❡♥t ❘✉❧❡ ✕ ❛❣❛✐♥st ♦✉r ❜❡tt❡r ❥✉❞❣❡♠❡♥t ✕ ✇♦✉❧❞ r❡s✉❧t ✐♥ ❛♥ ✐♥❞❡t❡r✲ ♠✐♥❛t❡ ❡①♣r❡ss✐♦♥ ✿ 4n2 − n + 2 ∞ ??? . −−−−−→ 2 2n − 1 ∞

❖♥❝❡ ❛❣❛✐♥✱ ✇❡ ❝❛♥✬t ❝♦♥❝❧✉❞❡ t❤❛t t❤❡ ❧✐♠✐t ❞♦❡s♥✬t ❡①✐st❀ ✇❡✬✈❡ ❥✉st ❢❛✐❧❡❞ t♦ ✜♥❞ t❤❡ ❛♥s✇❡r✳ ❚❤❡ ♣❛t❤ ♦✉t ♦❢ t❤✐s ❝♦♥✉♥❞r✉♠ ❧✐❡s ✐♥ ❛❧❣❡❜r❛✳ ❲❡ ♥❡❡❞ t♦ ❣❡t r✐❞ ♦❢ t❤❡ ✐♥✜♥✐t✐❡s ✐♥ t❤❡ ❢r❛❝t✐♦♥✦ ❍♦✇❄ ❚❤❡ ♠❡t❤♦❞ t❤❛t ♦❢t❡♥ ✇♦r❦s ✐s ❛s ❢♦❧❧♦✇s✿ ◮ ❉✐✈✐❞❡ t❤❡ ♥✉♠❡r❛t♦r ❛♥❞ ❞❡♥♦♠✐♥❛t♦r ❜② ❛♥ ❛♣♣r♦♣r✐❛t❡ ♣♦✇❡r ♦❢ n✳ ❚❤r♦✉❣❤ tr✐❛❧ ❛♥❞ ❡rr♦r✱ ✇❡ ❞❡t❡r♠✐♥❡ t❤❛t ✇❡ s❤♦✉❧❞ ✉s❡ n2 ✿

(4n2 − n + 2)/n2 4n2 − n + 2 = 2n2 − 1 (2n2 − 1)/n2 4 − n1 + n22 = 2 − n12 4−0+0 → 2−0 4 = 2

❉✐✈✐❞❡ ❜② n2 . ◆✉♠❡r❛t♦r ❛♥❞ ❞❡♥♦♠✐♥❛t♦r ❝♦♥s✐st ♦❢ ❢❛♠✐❧✐❛r s❡q✉❡♥❝❡s✳ ❚❤❡✐r ❧✐♠✐ts ❛r❡ ❦♥♦✇♥✳ ❙❘ ❛❧❧♦✇s ✉s t♦ ❡✈❛❧✉❛t❡ t❤❡s❡ t✇♦ ❧✐♠✐ts✳

= 2. ❲❡ ♦♥❧② ✉s❡❞ t❤❡ ◗✉♦t✐❡♥t ❘✉❧❡ ❛t t❤❡ ✈❡r② ❡♥❞✱ ❛❢t❡r ∞/∞ ✭t❤❡ ✐♥❞❡t❡r♠✐♥❛❝②✮ ✇❛s r❡♠♦✈❡❞✳ ■♥ r❡tr♦s♣❡❝t✱ ✇❡ ❛❧s♦ s❡❡ t❤❛t t❤❡ ❧✐♠✐t ✐s t❤❡ r❛t✐♦ ♦❢ t❤❡ ❧❡❛❞✐♥❣ ❝♦❡✣❝✐❡♥ts✳ ❚❤✐s ✐s ♥♦t ❛ ❝♦✐♥❝✐❞❡♥❝❡✳ ❊①❡r❝✐s❡ ✶✳✺✳✺

❚r② t♦ ❞✐✈✐❞❡ ❜② x ❛♥❞ x3 ✐♥st❡❛❞✳ ❚❤❡ ❣❡♥❡r❛❧ ♠❡t❤♦❞ ❢♦r ✜♥❞✐♥❣ s✉❝❤ ❧✐♠✐ts ✐s ❣✐✈❡♥ ❜② t❤❡ t❤❡♦r❡♠ ❜❡❧♦✇✿

✺✽

✶✳✺✳

❈❛♥ ✇❡ ❛❞❞ ✐♥✜♥✐t✐❡s❄ ❙✉❜tr❛❝t❄ ❉✐✈✐❞❡❄ ▼✉❧t✐♣❧②❄

✺✾

❚❤❡♦r❡♠ ✶✳✺✳✻✿ ▲✐♠✐ts ♦❢ ❘❛t✐♦♥❛❧ ❋✉♥❝t✐♦♥s ❛t ■♥✜♥✐t② ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥ f r❡♣r❡s❡♥t❡❞ ❛s ❛ q✉♦t✐❡♥t ♦❢ t✇♦ ♣♦❧②♥♦✲ ♠✐❛❧s ♦❢ ❞❡❣r❡❡s p ❛♥❞ q ✭t❤❡ ❧❡❛❞✐♥❣ ❝♦❡✣❝✐❡♥ts ap 6= 0 ❛♥❞ bq 6= 0✮✳ ❚❤❡♥ t❤❡ ❧✐♠✐t ♦❢ t❤❡ s❡q✉❡♥❝❡ ❞❡✜♥❡❞ ❜② t❤✐s ❢✉♥❝t✐♦♥ ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✿   ±∞ ap np + ap−1 np−1 + ... + a1 n + a0  ap lim = bp n→∞ bq nq + bq−1 nq−1 + ... + b1 n + b0    0

✐❢ p > q ✐❢ p = q ✐❢ p < q

Pr♦♦❢✳ ❚❤❡ ✐❞❡❛ ✐s✱ ❛❣❛✐♥✱ t♦

❞✐✈✐❞❡ ❜② t❤❡ ❤✐❣❤❡st ♣♦✇❡r✳

■❢

p > q✱

✇❡ ❤❛✈❡✿

ap + 0 ap + ap−1 n−1 + ... + a1 n−p+1 + a0 n−p ap np + ap−1 np−1 + ... + a1 n + a0 → = = ±∞ . q q−1 q−p q−p−1 1−p −p bq n + bq−1 n + ... + b1 n + b0 bq n + bq−1 n + ... + b1 n + b0 n 0 ■❢

p = q✱

✇❡ ❤❛✈❡✿

ap np + ap−1 np−1 + ... + a1 n + a0 ap + 0 ap + ap−1 n−1 + ... + a1 n−p+1 + a0 n−p ap → = = . bq nq + bq−1 nq−1 + ... + b1 n + b0 bq + bq−1 n−1 + ... + b1 n1−p + b0 n−p bp + 0 bp ■❢

p < q✱

✇❡ ❤❛✈❡✿

0 ap np + ap−1 np−1 + ... + a1 n + a0 ap np−q + ap−1 np−q−1 + ... + a1 n1−q + a0 n−q → = = 0. q q−1 −1 −q+1 −q bq n + bq−1 n + ... + b1 n + b0 bq + bq−1 n + ... + b1 n + b0 n bq + 0 ❚❤✐s ✐s t❤❡ ❧❡ss♦♥ ✇❡ ❤❛✈❡ r❡✲❧❡❛r♥❡❞✿

◮

❚❤❡ ❧♦♥❣✲t❡r♠ ❜❡❤❛✈✐♦r ♦❢ ♣♦❧②♥♦♠✐❛❧s ✐s ❞❡t❡r♠✐♥❡❞ ❜② t❤❡✐r ❧❡❛❞✐♥❣ t❡r♠s✳

❲❡ ❝♦♥❝❧✉❞❡ t❤❛t ❛ ❧❛r❣❡r ❞❡❣r❡❡ ♣♦❧②♥♦♠✐❛❧ ✇✐❧❧ ❛❧✇❛②s ✏♦✈❡r♣♦✇❡r✑ ♦♥❡ ✇✐t❤ ❛ ❧♦✇❡r ❞❡❣r❡❡✿ ❡✐t❤❡r t❤❡ ♥✉♠❡r❛t♦r t❛❦❡s t❤❡ ❢r❛❝t✐♦♥ t♦ ✐♥✜♥✐t② ♦r t❤❡ ❞❡♥♦♠✐♥❛t♦r t❛❦❡s ✐t t♦ ③❡r♦✳ Pr❛❝t✐❝❛❧❧②✱ ✇❡ ❥✉st ❞r♦♣ t❤❡ ♥♦♥✲❧❡❛❞✐♥❣ t❡r♠s✿

lim

n→∞

ap np +ap−1 np−1 + ... + a1 n + a0 bq nq +bq−1 nq−1 + ... + b1 n + b0

= lim

n→∞

ap n p bq n q

=

ap lim np−q n→∞ bq

❚❤✐s r❡s✉❧ts ✐♥ ❛ ♣♦✇❡r s❡q✉❡♥❝❡✱ ❛♥❞ ✇❡ ❦♥♦✇ ✐ts ❧✐♠✐t ❢r♦♠ t❤❡ t❤❡♦r❡♠ ✐♥ t❤✐s ❝❤❛♣t❡r✳

❊①❛♠♣❧❡ ✶✳✺✳✼✿ r❛t✐♦♥❛❧ s❡q✉❡♥❝❡s ❲✐t❤ t❤❡ t❤❡♦r❡♠ ❛✈❛✐❧❛❜❧❡✱ t❤❡ ❛❧❣❡❜r❛✐❝ tr✐❝❦ ✇❡ ✉s❡❞ ✐♥ t❤❡ ❧❛st ❡①❛♠♣❧❡ ✐s♥✬t ♥❡❝❡ss❛r② ❛♥②♠♦r❡✳ ■♥ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡q✉❡♥❝❡✱ t❤❡ ❞❡❣r❡❡s ❛r❡ ❡q✉❛❧❀ t❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡✿

4n 4n3 − 2n + 2 = = 4. n3 − 1 1

❊①❛♠♣❧❡ ✶✳✺✳✽✿ ♥♦♥✲r❛t✐♦♥❛❧ s❡q✉❡♥❝❡s ❚❤❡ s❡q✉❡♥❝❡ ❜❡❧♦✇ ✐s♥✬t r❛t✐♦♥❛❧✿

1 + (−3)n . 5n

✶✳✺✳

❈❛♥ ✇❡ ❛❞❞ ✐♥✜♥✐t✐❡s❄ ❙✉❜tr❛❝t❄ ❉✐✈✐❞❡❄ ▼✉❧t✐♣❧②❄

✻✵

❚❤❡r❡❢♦r❡✱ t❤❡ t❤❡♦r❡♠ ❞♦❡s♥✬t ❛♣♣❧②✳ ❇✉t t❤❡ ♠❡t❤♦❞ ❞♦❡s✦ ❋✐rst ✇❡ ♥♦t✐❝❡ t❤❛t ◗❘ ❞♦❡s♥✬t ❛♣♣❧②✿

1 + (−3)n 5n (−3)n 1 = n+ 5  5n   n n 1 −3 = + 5 5

yn =

❚❤❡ ♥✉♠❡r❛t♦r ❞✐✈❡r❣❡s ✕ ❉❊❆❉ ❊◆❉✦ ❚❤❡ ✇❛② ♦✉t ✐s ❛❧✇❛②s ❛❧❣❡❜r❛✿ ❲❡ ✇✐❧❧ tr② ❙❘✳ ❲❡ ✉s❡ ♣r♦♣❡rt✐❡s ♦❢ ❡①♣♦♥❡♥ts ❛♥❞ ❞✐s❝♦✈❡r ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥s✳

↓

↓

❚❛❦❡ t❤❡ ❧✐♠✐ts✳

0

0

❇❡❝❛✉s❡ t❤❡ r❛t✐♦s ❛r❡ ✇✐t❤✐♥

→

0

❆❝❝♦r❞✐♥❣ t♦ ❙❘✳

❚❤❡s❡ ❛r❡ t✇♦ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥s ✇✐t❤ t❤❡ r❛t✐♦s✿ ♦✉r ❛♣♣❧✐❝❛t✐♦♥ ♦❢ t❤❡

(−1, 1) .

❙✉♠ ❘✉❧❡

r = 1/5, −3/5✱ t❤❛t s❛t✐s❢② |r| < 0✳

▼❡❛♥✇❤✐❧❡✱

✇❛s ❥✉st✐✜❡❞ ❜② t❤❡ ❢❛❝t t❤❛t t❤❡ t✇♦ ❧✐♠✐ts ❡①✐st✳

❊①❛♠♣❧❡ ✶✳✺✳✾✿ ♥♦♥✲r❛t✐♦♥❛❧ s❡q✉❡♥❝❡s ❚❤❡ s❡q✉❡♥❝❡ ❜❡❧♦✇ ✐s♥✬t r❛t✐♦♥❛❧✿

(−3)n + 1 . 5n + 1

❚❤❡r❡❢♦r❡✱ t❤❡ t❤❡♦r❡♠ ❞♦❡s♥✬t ❛♣♣❧②✳ ❇✉t t❤❡ tr✐❝❦ ✇❡ ✉s❡❞ t♦ ❞❡❛❧ ✇✐t❤ ∞/∞ ❞♦❡s✳ ❲❡ ❞✐✈✐❞❡ ❜♦t❤ n ♥✉♠❡r❛t♦r ❛♥❞ ❞❡♥♦♠✐♥❛t♦r ❜② t❤❡ ✏str♦♥❣❡st✑ s❡q✉❡♥❝❡ ♣r❡s❡♥t✱ 5 ✱ t❤❡♥ s✐♠♣❧✐❢②✿

(−3/5)n + 1/5n 0+0 (−3)n + 1 = → = 0. n n 5 +1 1 + 1/5 1+0 ❊①❡r❝✐s❡ ✶✳✺✳✶✵ ❋✐♥❞ t❤❡ ❧✐♠✐t ♦❢ t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ n ✭❝✮ (−1) /n✳

f (x) = sign(x)

❛♥❞ t❤❡ s❡q✉❡♥❝❡

xn

❣✐✈❡♥ ❜② ✭❛✮

1/n✱

✭❜✮

−1/n✱

❲❤❛t ✐s ✐♥✜♥✐t②❄ ❚❤❡ ♣❧✉s ✭♦r ♠✐♥✉s✮ ✐♥✜♥✐t② ✐s ✐❞❡♥t✐✜❡❞ ✇✐t❤ t❤❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ❛❧❧ s❡q✉❡♥❝❡s ❛♣♣r♦❛❝❤✐♥❣ t❤✐s ✐♥✜♥✐t②✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐❞❡♥t✐t② ✐s r❡❛❞ ✐♥ ❜♦t❤ ❞✐r❡❝t✐♦♥s✿

lim an = +∞ .

n→+∞

◆♦✇✱ ❞♦❡s ✐t ♠❛❦❡ s❡♥s❡ t♦ ❞♦ ❛♥② ❛❧❣❡❜r❛ ✇✐t❤ t❤❡ ✐♥✜♥✐t✐❡s❄ ❚❤✐s s❡❡♠s t♦ ♠❛❦❡ s❡♥s❡✿

∞ + ∞ = ∞. ❇✉t t❤✐s ❞♦❡s♥✬t✿

∞ − ∞ = ??? ❲❡ ✇✐❧❧ ❤❛✈❡ t♦ s❡❡ ✇❤❡♥ t❤❡ ❛❧❣❡❜r❛ ✇✐t❤ t❤♦s❡ ✐♥✜♥✐t❡ ❧✐♠✐ts ♠❛❦❡s s❡♥s❡✳ ❚❤❡

❆❧❣❡❜r❛✐❝ ❘✉❧❡s ♦❢ ▲✐♠✐ts

❛❜♦✈❡ ❤❛✈❡ ❡①❝❡♣t✐♦♥s❀ ✇❡ ❝❛♥ ✐♠❛❣✐♥❡ t❤❛t ♦♥❡ ♦r ❜♦t❤ ♦❢ t❤❡ s❡q✉❡♥❝❡s

❛♣♣r♦❛❝❤ ✐♥✜♥✐t② ♦r t❤❛t t❤❡ ❧✐♠✐t ✐♥ t❤❡ ❞❡♥♦♠✐♥❛t♦r ✐s ❋✐rst✱ ✐❢ t✇♦ s❡q✉❡♥❝❡s ❛♣♣r♦❛❝❤ t❤❡

s❛♠❡

0✳

✐♥✜♥✐t②✱ t❤❡♥ s♦ ❞♦❡s t❤❡✐r s✉♠✳ ❙✐♠✐❧❛r ❢♦r t❤❡ ♣r♦❞✉❝t✿

✶✳✺✳ ❈❛♥ ✇❡ ❛❞❞ ✐♥✜♥✐t✐❡s❄ ❙✉❜tr❛❝t❄ ❉✐✈✐❞❡❄ ▼✉❧t✐♣❧②❄

✻✶

❚❤❡♦r❡♠ ✶✳✺✳✶✶✿ ❆❧❣❡❜r❛ ♦❢ ■♥✜♥✐t❡ ▲✐♠✐ts ♦❢ ❙❡q✉❡♥❝❡s ❙✉♣♣♦s❡

• an → ±∞✱ • bn → ±∞✳

❛♥❞

❚❤❡♥ ✇❡ ❤❛✈❡✿

❙❘✿ P❘✿

an + bn → ±∞ an · bn → +∞

■t ✐s ❥✉st ❛s ✐♠♣♦rt❛♥t ✇❤❛t ✐s ◆❖❚ ❤❡r❡✿ ❉❘❄ an − bn → ??? ◗❘❄ an /bn → ???

❊①❛♠♣❧❡ ✶✳✺✳✶✷✿ ✐♥✜♥✐t❡ ❧✐♠✐ts ❚❤❡ ❙✉♠ ❘✉❧❡ ❛♥❞ t❤❡ Pr♦❞✉❝t ❘✉❧❡✱ ❛s st❛t❡❞ ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥✱ ❛r❡ ✐♥❛♣♣❧✐❝❛❜❧❡ ❜❡❝❛✉s❡ ♥❡✐t❤❡r ♦❢ t❤❡ t✇♦ ❧✐♠✐ts ❡①✐sts✱ ❜✉t t❤❡ ❧❛st t❤❡♦r❡♠ ❛♣♣❧✐❡s✿ n2 + 2n → ∞ ❛♥❞ n2 · 2n → ∞ .

❙❡❝♦♥❞✱ ✐❢ ♦♥❡ ❧✐♠✐t ✐s ✐♥✜♥✐t❡ ❛♥❞ t❤❡ ♦t❤❡r ✐s ♥♦t✱ t❤❡ s✉♠ ♦r ♣r♦❞✉❝t ✐s ✐♥✜♥✐t❡✳ ❋✉rt❤❡r♠♦r❡✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿

❚❤❡♦r❡♠ ✶✳✺✳✶✸✿ ❆❧❣❡❜r❛ ♦❢ ❋✐♥✐t❡ ❛♥❞ ■♥✜♥✐t❡ ▲✐♠✐ts ♦❢ ❙❡q✉❡♥❝❡s ❙✉♣♣♦s❡

• an → a ✱ ❛♥❞ • bn → ±∞ ✳

❚❤❡♥ ✇❡ ❤❛✈❡✿

❙❘✿ P❘✿

an + bn → ±∞ an · bn → ±∞ ♣r♦✈✐❞❡❞ a > 0

❈▼❘✿ ◗❘✿

c · bn → ±∞ ❢♦r ❛♥② r❡❛❧ c > 0 an /bn → 0, bn /an → ±∞ ♣r♦✈✐❞❡❞ a > 0

❊①❛♠♣❧❡ ✶✳✺✳✶✹✿ ✜♥✐t❡ ❛♥❞ ✐♥✜♥✐t❡ ❧✐♠✐ts ❚❤❡ ❙✉♠ ❘✉❧❡ ❛♥❞ t❤❡ Pr♦❞✉❝t ❘✉❧❡✱ ❛s st❛t❡❞ ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥✱ ❛r❡ ✐♥❛♣♣❧✐❝❛❜❧❡ ❜❡❝❛✉s❡ ♦♥❡ ♦❢ t❤❡ t✇♦ ❧✐♠✐ts ❞♦❡s♥✬t ❡①✐st✱ ❜✉t t❤❡ ❧❛st t❤❡♦r❡♠ ❛♣♣❧✐❡s✿ n 1 + 2n → ∞ ❛♥❞ · 2n → ∞ . n n+1

❊①❡r❝✐s❡ ✶✳✺✳✶✺ ❈♦♥s✐❞❡r t❤❡ ❝❛s❡s ♦❢ t❤❡ t❤❡♦r❡♠ ✇❤❡♥ a < 0 ❛♥❞ a = 0✳ ❏✉st✐✜❡❞ ❜② t❤❡s❡ t❤❡♦r❡♠s✱ ✇❡ ♠❛❦❡ t❤✐s ✐♥❢♦r♠❛❧ ❧✐st ❢♦r t❤❡ ❛❧❣❡❜r❛ ♦❢ ✐♥✜♥✐t✐❡s ✿ ♥✉♠❜❡r ♥✉♠❜❡r +∞ −∞

♥✉♠❜❡r

+ + + + /

(+∞) (−∞) (+∞) (−∞) (±∞)

= +∞ = −∞ = +∞ = −∞ =0

✶✳✻✳

▼♦r❡ ♣r♦♣❡rt✐❡s ♦❢ ❧✐♠✐ts ♦❢ s❡q✉❡♥❝❡s

✻✷

❖♥❝❡ ❛❣❛✐♥✱ t❤✐s ✐s♥✬t ♦♥ t❤❡ ❧✐st✿

+∞ − (+∞) = ??? +∞ / (+∞) = ??? ❊①❛♠♣❧❡ ✶✳✺✳✶✻✿ ❚❤❡r❡ ✐s ♥♦

∞−∞

∞−∞

❇❡❤✐♥❞ ❡❛❝❤

∞✱

✭❥✉st ❛s t❤❡r❡ ✐s ♥♦

∞/∞✮✳

❲❤② ♥♦t❄

t❤❡r❡ ♠✉st ❜❡ ❛ s❡q✉❡♥❝❡ ❛♣♣r♦❛❝❤✐♥❣

t❤❡ ♦♥❡ ❤❛♥❞ ✇❡ ❤❛✈❡✿

∞✦

❍♦✇❡✈❡r✱ t❤❡ ♦✉t❝♦♠❡ ✐s ❛♠❜✐❣✉♦✉s✳ ❖♥

an = n → +∞, bn = −n → +∞ =⇒ an − bn = 0 → 0 ; ♦♥ t❤❡ ♦t❤❡r✿

an = n2 → +∞, bn = −n → +∞ =⇒ an − bn = n2 − n → +∞ ,

❜②

▲✐♠✐ts ♦❢ P♦❧②♥♦♠✐❛❧s✳

❚✇♦ s❡❡♠✐♥❣❧② ❧❡❣✐t✐♠❛t❡ ❛♥s✇❡rs ❢♦r t❤❡ s❛♠❡ ❡①♣r❡ss✐♦♥✱

∞ − ∞ ✳✳✳

❲❡ ❤❛✈❡ ❛♥♦t❤❡r ✐♥❞❡t❡r♠✐♥❛t❡ ❡①♣r❡ss✐♦♥✦

❊①❛♠♣❧❡ ✶✳✺✳✶✼✿

1∞

❲❤❛t ❛❜♦✉t t❤✐s ❧✐♠✐t❄

1∞ . ❖❢ ❝♦✉rs❡✱ ✐❢ ✇❡ ❦❡❡♣ ♠✉❧t✐♣❧②✐♥❣

1 ❜② ✐ts❡❧❢✱ ✇❡ ✇✐❧❧ ❛❧✇❛②s ❤❛✈❡ 1✳

❍♦✇❡✈❡r✱ ✇❤❛t ✐❢

1 ✐s ❛❧s♦ t❤❡ ❧✐♠✐t

♦❢ ❛ s❡q✉❡♥❝❡❄ ❉♦ ✇❡ ❤❛✈❡✿

1∞ = lim abnn , n→∞

❢♦r s♦♠❡ ✭♦r ❛♥②❄✮ s❡q✉❡♥❝❡s✿

lim an = 1,

n→∞ ▲❡t✬s tr②

an = 1 +

1 n

lim bn = ∞ ?

n→∞

❛♥❞

bn = n .

❲❡ ✇✐❧❧ s❤♦✇ ❛t t❤❡ ❡♥❞ ♦❢ t❤❡ ❝❤❛♣t❡r t❤❛t t❤❡ ❧✐♠✐t ❡①✐sts✿

 ❙♦✱

1∞

1 1+ n

n

.

❝❛♥ ❜❡ ❛♥♦t❤❡r ✐♥❞❡t❡r♠✐♥❛t❡ ❡①♣r❡ss✐♦♥✦

✶✳✻✳ ▼♦r❡ ♣r♦♣❡rt✐❡s ♦❢ ❧✐♠✐ts ♦❢ s❡q✉❡♥❝❡s

❊✈❡r② s❡q✉❡♥❝❡

✐s

❛ ❢✉♥❝t✐♦♥❀ ✐t ❥✉st ❤❛♣♣❡♥s t♦ ❤❛✈❡ ❛ s♣❡❝✐❛❧ ❦✐♥❞ ♦❢ ❞♦♠❛✐♥✳ ❚❤❡♥✱ ✇❤② ❞♦ t❤❡② ❞❡s❡r✈❡

❛ s♣❡❝✐❛❧ ❛tt❡♥t✐♦♥❄

✶✳✻✳

▼♦r❡ ♣r♦♣❡rt✐❡s ♦❢ ❧✐♠✐ts ♦❢ s❡q✉❡♥❝❡s

✻✸

❆ s❡q✉❡♥❝❡ ❞♦❡s♥✬t ♣r♦❞✉❝❡ ❛ ❢✉♥❝t✐♦♥ ❡❛s✐❧② ✇❤❡♥ ✐t ✐s ❞❡✜♥❡❞

r❡❝✉rs✐✈❡❧②✳

❊①❛♠♣❧❡ ✶✳✻✳✶✿ r❡❝✉rs✐✈❡ ❧✐♠✐ts

❘❡❝✉rs✐✈❡ ❞❡✜♥✐t✐♦♥s ❛r❡ ✈❡r② ❝♦♠♠♦♥❀ ❡✈❡♥ t❤❡ s✐♠♣❧❡st ❜❛♥❦✐♥❣ r❡q✉✐r❡s ♦♥❡ t♦ ✉s❡ t❤❡♠✿ ✶✳ ■❢ ②♦✉ s❛② t❤❛t ②♦✉ ✇✐❧❧ ❝♦♥tr✐❜✉t❡ $2000 ❡✈❡r② ②❡❛r✱ ②♦✉ ❛r❡ st❛t✐♥❣ t❤❛t t❤❡ ♥❡①t ②❡❛r✬s ❜❛❧❛♥❝❡ ✇✐❧❧ ❜❡ $2000 ❤✐❣❤❡r t❤❛♥ t❤❡ ❧❛st✿ an+1 = an + 2000 .

✷✳ ■❢ ②♦✉ s❛② t❤❛t ②♦✉r ❜❛♥❦ ✇✐❧❧ ♣❛② 5% ♣❡r ②❡❛r✱ ②♦✉ ❛r❡ st❛t✐♥❣ t❤❛t t❤❡ ♥❡①t ②❡❛r✬s ❜❛❧❛♥❝❡ ✇✐❧❧ ❜❡ 1.05 t✐♠❡s ❤✐❣❤❡r t❤❛♥ t❤❡ ❧❛st✿ bn+1 = bn · 1.05 .

❞❡r✐✈❡

❖❢ ❝♦✉rs❡✱ ✇❡ t❤❡♥ t❤❡ nt❤✲t❡r♠ ❢♦r♠✉❧❛s ❢♦r t❤❡s❡ s❡q✉❡♥❝❡s✿ ✶✳ r❡♣❡❛t❡❞ ❞❡♣♦s✐ts st❛rt✐♥❣ ❢r♦♠ 0✿ an = 2000 · n ;

✷✳ ❝♦♠♣♦✉♥❞❡❞ ✐♥t❡r❡st st❛rt✐♥❣ ❢r♦♠ $2000✿

bn = 2000 · 1.05n .

❚❤❡♥ ✐t ✐s ❝❧❡❛r ❢r♦♠ t❤❡s❡ ❢♦r♠✉❧❛s t❤❛t t❤❡ ❧✐♠✐ts ❛r❡ ✐♥✜♥✐t❡✳ ❍♦✇❡✈❡r✱ ✇❤❛t ✐❢ ✇❡ ❝❛rr② ♦✉t ❜♦t❤ ♦❢ t❤❡ str❛t❡❣✐❡s❄ ❚❤❡ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛ ✐s st✐❧❧ ✈❡r② ❝❧❡❛r✿ cn+1 = (cn + 2000) · 1.05 .

❇✉t t❤❡r❡ ✐s ♥♦ nt❤✲t❡r♠ ❢♦r♠✉❧❛✦ ❍♦✇ ❞♦ ✇❡ ❡✈❡♥ ♣r♦✈❡ t❤❛t t❤❡ ❧✐♠✐t ✐s ✐♥✜♥✐t❡❄ ■♥❞✐r❡❝t❧②✱ ❜② cn ✇✐t❤ ❡✐t❤❡r an ♦r bn ✳

❝♦♠♣❛r✐♥❣

❚❤✐s s❡❝t✐♦♥ ✐s ❛❜♦✉t

✐♥❞✐r❡❝t ♣r♦♦❢s ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ♦r ❞✐✈❡r❣❡♥❝❡✳

❋✐rst✱ ❝♦♠♣❛r❡ t❤❡s❡ s❡q✉❡♥❝❡s✿

❚❤❡ ♦❜s❡r✈❛t✐♦♥ ✐s ✈❡r② s✐♠♣❧❡ ❜✉t ✉s❡❢✉❧✿ ❖♥❧② t❤❡

t❛✐❧ ♦❢ t❤❡ s❡q✉❡♥❝❡ ♠❛tt❡rs ❢♦r ❝♦♥✈❡r❣❡♥❝❡✦

❚❤❡♦r❡♠ ✶✳✻✳✷✿ ❚r✉♥❝❛t✐♦♥ Pr✐♥❝✐♣❧❡ ❆ s❡q✉❡♥❝❡ ✐s ❝♦♥✈❡r❣❡♥t ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❛❧❧ ♦❢ ✐ts tr✉♥❝❛t✐♦♥s ❛r❡ ❝♦♥✈❡r❣❡♥t✳

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿

❢♦r ❛♥② ✐♥t❡❣❡r

p

an

❛♥❞

n=p,p+1,...

q✳

→ a ⇐⇒ an

n=q,q+1,...

→a

✶✳✻✳

▼♦r❡ ♣r♦♣❡rt✐❡s ♦❢ ❧✐♠✐ts ♦❢ s❡q✉❡♥❝❡s

✻✹

r❡str✐❝t✐♦♥ ♦❢ t❤❡ ❞♦♠❛✐♥ ♦❢ ❛ ❢✉♥❝t✐♦♥ ✭s❡❡♥ ✐♥ ❱♦❧✉♠❡ ✶✱ ❈❤❛♣t❡r ✶P❈✲✹✮✳ ❯s✐♥❣

❚r✉♥❝❛t✐♦♥ ❛♠♦✉♥ts t♦ ❛ t❤❡ ♦t❤❡r ♥♦t❛t✐♦♥ ❢♦r ❧✐♠✐ts✱ ✇❡ ❤❛✈❡✿

lim an = lim an n→∞ n→∞

❊①❡r❝✐s❡ ✶✳✻✳✸

n≥p

= lim an n→∞

n≥q

Pr♦✈❡ t❤❡ t❤❡♦r❡♠✳ ❖♥❡ ❝❛♥ s❡❡ ♦t❤❡r ✐♠♣♦rt❛♥t ✇❛②s t♦ r❡str✐❝t t❤❡ ❞♦♠❛✐♥ ♦❢ ❛ s❡q✉❡♥❝❡✳ ❋♦r ❡①❛♠♣❧❡✱ t❤❡ ❛❧t❡r♥❛t✐♥❣ s❡q✉❡♥❝❡ ❞✐✈❡r❣❡s✱ ❜✉t ✇❤❡♥ r❡str✐❝t❡❞ t♦ t❤❡ ❡✈❡♥ ♦r t❤❡ ♦❞❞ ♥✉♠❜❡rs✱ ✐t ♣r♦❞✉❝❡s t✇♦ s❡q✉❡♥❝❡s✿

❝♦♥✈❡r❣❡♥t

❚❤✐s ❝♦♥✈❡♥✐❡♥t ✐❞❡❛ ✐s s✉♠♠❛r✐③❡❞ ❛s ❢♦❧❧♦✇s✿

❉❡✜♥✐t✐♦♥ ✶✳✻✳✹✿ s✉❜s❡q✉❡♥❝❡ ❆ s❡q✉❡♥❝❡ r❡str✐❝t❡❞ t♦ ❛♥ ✐♥✜♥✐t❡ s✉❜s❡t ♦❢ t❤❡ ✐♥t❡❣❡rs ✐s ❝❛❧❧❡❞ ✐ts

s✉❜s❡q✉❡♥❝❡✳

❋♦r ❡①❛♠♣❧❡✱ t❤❡s❡ ❛r❡ s♦♠❡ s✉❜s❡q✉❡♥❝❡s ♦❢ t❤❡ ❛❧t❡r♥❛t✐♥❣ s❡q✉❡♥❝❡ (−1)n ✿ 2

(−1)2n , (−1)3n , (−1)n .

❚❤❡② ❜❡❤❛✈❡ ❛s ❡①♣❡❝t❡❞✿

❚❤❡♦r❡♠ ✶✳✻✳✺✿ ▲✐♠✐t ♦❢ ❙✉❜s❡q✉❡♥❝❡ ❆ s✉❜s❡q✉❡♥❝❡ ♦❢ ❛♥② ❝♦♥✈❡r❣❡♥t s❡q✉❡♥❝❡ ✐s ❛❧s♦ ❝♦♥✈❡r❣❡♥t✱ ❛♥❞ t❤❡ ❧✐♠✐t ✐s t❤❡ s❛♠❡✳

❚❤❡ ❝♦♥✈❡rs❡ ✐s ✉♥tr✉❡✿ (−1)n ✐s ❞✐✈❡r❣❡♥t✱ ❜✉t (−1)2n = 1 ✐s ❝♦♥✈❡r❣❡♥t✳ ◆♦♥✲str✐❝t ✐♥❡q✉❛❧✐t✐❡s ❜❡t✇❡❡♥ s❡q✉❡♥❝❡s✱ s✉❝❤ ❛s✿ ❛r❡ ♣r❡s❡r✈❡❞ ✉♥❞❡r ❧✐♠✐ts✿

a ← an ≥ bn → b, a ≥ b.

✶✳✻✳

▼♦r❡ ♣r♦♣❡rt✐❡s ♦❢ ❧✐♠✐ts ♦❢ s❡q✉❡♥❝❡s

✻✺

❚❤❡♦r❡♠ ✶✳✻✳✻✿ ❈♦♠♣❛r✐s♦♥ ❚❡st ❢♦r ▲✐♠✐ts ♦❢ ❙❡q✉❡♥❝❡s ■❢

an ≥ bn

❢♦r ❛❧❧

n

❣r❡❛t❡r t❤❛♥ s♦♠❡

N✱

t❤❡♥

lim an ≥ lim bn ,

n→∞

n→∞

♣r♦✈✐❞❡❞ t❤❡ s❡q✉❡♥❝❡s ❝♦♥✈❡r❣❡✳

Pr♦♦❢✳

a

❚❤❡ ♣r♦♦❢ ✐s ❜② ❝♦♥tr❛❞✐❝t✐♦♥✳ ❙✉♣♣♦s❡ ❛❧s♦

❛♥❞

b

❛r❡ t❤❡ ❧✐♠✐ts ♦❢

an

❛♥❞

bn

r❡s♣❡❝t✐✈❡❧② ❛♥❞ s✉♣♣♦s❡

a < b✳

❚❤❡ ❣❡♦♠❡tr② ♦❢ t❤❡ ♣r♦♦❢ ✐s ❝❧❡❛r✿ ❲❡ ✇❛♥t t♦ s❡♣❛r❛t❡ t❤❡ t✇♦ ❤♦r✐③♦♥t❛❧ ❧✐♥❡s r❡♣r❡s❡♥t✐♥❣ t❤❡ t✇♦ ❧✐♠✐ts ❜② t✇♦ ♥♦♥✲♦✈❡r❧❛♣♣✐♥❣ ❜❛♥❞s✱ ❥✉st ❛s ❧❛st t✐♠❡✳ ❚❤❡♥✱ ✐❢ ♥❛rr♦✇ ❡♥♦✉❣❤✱ t❤❡ t❛✐❧s ♦❢ t❤❡ ✏❧❛r❣❡r✑ s❡q✉❡♥❝❡ ✇♦✉❧❞ ❤❛✈❡ t♦ ✜t t❤❡ ✏s♠❛❧❧❡r✑ ❜❛♥❞✳ ❚❤❡s❡ ❜❛♥❞s ❝♦rr❡s♣♦♥❞ t♦ t✇♦ ✐♥t❡r✈❛❧s ❛r♦✉♥❞ t❤♦s❡ t✇♦ ❧✐♠✐ts✳ ■♥ ♦r❞❡r ❢♦r t❤❡♠ t♦ ♥♦t ✐♥t❡rs❡❝t✱ t❤❡✐r ✇✐❞t❤ ✭t❤❛t✬s

2ε✦✮

s❤♦✉❧❞ ❜❡ ❧❡ss t❤❛♥ ❤❛❧❢ t❤❡

❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ t✇♦ ♥✉♠❜❡rs✳

▲❡t

b−a . 2

ε= ❚❤❡♥✱ ✇❤❛t ✇❡ ❛r❡ ❣♦✐♥❣ t♦ ✉s❡ ❛t t❤❡ ❡♥❞ ✐s

a + ε = b − ε. ◆♦✇✱ ✇❡ ✉s❡ t❤❡ ❞❡✜♥✐t✐♦♥ ❢♦r

a

❛♥❞

✶✳ ❚❤❡r❡ ❡①✐sts ❛ ♥❛t✉r❛❧ ♥✉♠❜❡r

b ❛s ❧✐♠✐ts✿ N s✉❝❤ t❤❛t

❢♦r ❡✈❡r② ♥❛t✉r❛❧ ♥✉♠❜❡r

n > L✱

✇❡ ❤❛✈❡

|an − a| < ε . ✷✳ ❚❤❡r❡ ❡①✐sts ❛ ♥❛t✉r❛❧ ♥✉♠❜❡r

K

s✉❝❤ t❤❛t ❢♦r ❡✈❡r② ♥❛t✉r❛❧ ♥✉♠❜❡r

n > M✱

✇❡ ❤❛✈❡

|bn − b| < ε . ■♥ ♦r❞❡r t♦ ❝♦♠❜✐♥❡ t❤❡ t✇♦ st❛t❡♠❡♥ts✱ ✇❡ ♥❡❡❞ t❤❡♠ t♦ ❜❡ s❛t✐s✜❡❞ ❢♦r t❤❡ s❛♠❡ ✈❛❧✉❡s ♦❢

N = max{L, M } . ❚❤❡♥✱ ✶✳ ❋♦r ❡✈❡r② ♥✉♠❜❡r

n > N✱

✇❡ ❤❛✈❡

|an − a| < ε , ✷✳ ❋♦r ❡✈❡r② ♥✉♠❜❡r

n > N✱

♦r

a − ε < an < a + ε .

♦r

b − ε < bn < b + ε .

✇❡ ❤❛✈❡

|bn − b| < ε,

❚❛❦✐♥❣ ♦♥❡ ❢r♦♠ ❡✐t❤❡r ♦❢ t❤❡ t✇♦ ♣❛✐rs ♦❢ ✐♥❡q✉❛❧✐t✐❡s✱ ✇❡ ❤❛✈❡✿

an < a + ε = b − ε < bn . ❆ ❝♦♥tr❛❞✐❝t✐♦♥✳

n✳

▲❡t

✶✳✻✳

▼♦r❡ ♣r♦♣❡rt✐❡s ♦❢ ❧✐♠✐ts ♦❢ s❡q✉❡♥❝❡s

✻✻ ❲❛r♥✐♥❣✦ ❚❤❡ t❤❡♦r❡♠ ❞♦❡s ♥♦t ❝❧❛✐♠ ❝♦♥✈❡r❣❡♥❝❡✳

❊①❡r❝✐s❡ ✶✳✻✳✼

❙❤♦✇ t❤❛t r❡♣❧❛❝✐♥❣ t❤❡ ♥♦♥✲str✐❝t ✐♥❡q✉❛❧✐t②✱ an ≥ bn ✱ ✇✐t❤ ❛ str✐❝t ♦♥❡✱ an > bn ✱ ✇♦♥✬t ♣r♦❞✉❝❡ ❛ str✐❝t ✐♥❡q✉❛❧✐t② ✐♥ t❤❡ ❝♦♥❝❧✉s✐♦♥ ♦❢ t❤❡ t❤❡♦r❡♠✳

❯♥✐q✉❡♥❡ss ❚❤❡♦r❡♠

❚❤❡ s✐t✉❛t✐♦♥ ✐s s✐♠✐❧❛r t♦ t❤❛t ♦❢ t❤❡ ✿ ■❢ t❤❡ ♦♣♣♦s✐t❡ ✐♥❡q✉❛❧✐t② ✇❡r❡ t♦ ❤♦❧❞✱ ✇❡ ❝♦✉❧❞ ✜♥❞ t✇♦ ❜❛♥❞s t♦ ❝♦♥t❛✐♥ t❤❡ t✇♦ s❡q✉❡♥❝❡s✬ t❛✐❧s✳ ❚❤❡♥ t❤❡ ♦r✐❣✐♥❛❧ ✐♥❡q✉❛❧✐t② ✇♦✉❧❞ ❢❛✐❧✿

❚❤✐s ✐s t❤❡ s✉♠♠❛r② ♦❢ t❤❡ t❤❡♦r❡♠✿ an ≥ b n ↓ ↓ a b =⇒ ≥

❋r♦♠ t❤❡ ✐♥❡q✉❛❧✐t② ✐♥ t❤❡ t❤❡♦r❡♠✱ ✇❡ ❝❛♥✬t ❝♦♥❝❧✉❞❡ ❛♥②t❤✐♥❣ ❛❜♦✉t t❤❡ ❡①✐st❡♥❝❡ ♦❢ t❤❡ ❧✐♠✐t✿

▲❡t✬s tr② ❤❛✈✐♥❣ ❛♥ ✐♥❡q✉❛❧✐t② ♦♥

❡✐t❤❡r s✐❞❡ ♦❢ t❤❡ s❡q✉❡♥❝❡✿

❍♦✇❡✈❡r✱ t❤❡②✬❞ ❤❛✈❡ t♦ ✇♦r❦ t♦❣❡t❤❡r ✐♥ ♦r❞❡r t♦ ❝♦♥tr♦❧ t❤❡ s❡q✉❡♥❝❡ t❤❡② ❜♦✉♥❞✦ ■t ✐s ❝❛❧❧❡❞ ❛

sq✉❡❡③❡ ✿

✶✳✻✳

▼♦r❡ ♣r♦♣❡rt✐❡s ♦❢ ❧✐♠✐ts ♦❢ s❡q✉❡♥❝❡s

✻✼

■❢ ✇❡ ❝❛♥ sq✉❡❡③❡ t❤❡ s❡q✉❡♥❝❡ ✉♥❞❡r ✐♥✈❡st✐❣❛t✐♦♥ ❜❡t✇❡❡♥ t✇♦ ❢❛♠✐❧✐❛r s❡q✉❡♥❝❡s✱ ✇❡ ♠✐❣❤t ❜❡ ❛❜❧❡ t♦ s❛② s♦♠❡t❤✐♥❣ ❛❜♦✉t ✐ts ❧✐♠✐t✳ ❙♦♠❡ ❢✉rt❤❡r r❡q✉✐r❡♠❡♥ts ✇✐❧❧ ❜❡ ♥❡❝❡ss❛r②✿

❚❤❡♦r❡♠ ✶✳✻✳✽✿ ❙q✉❡❡③❡ ❚❤❡♦r❡♠ ❢♦r ❙❡q✉❡♥❝❡s ■❢ t❤❡ ✈❛❧✉❡s ♦❢ ❛ s❡q✉❡♥❝❡ ❧✐❡ ❜❡t✇❡❡♥ t❤♦s❡ ♦❢ t✇♦ s❡q✉❡♥❝❡s ✇✐t❤ t❤❡ s❛♠❡ ❧✐♠✐t✱ t❤❡♥ ✐ts ❧✐♠✐t ❛❧s♦ ❡①✐sts ❛♥❞ ✐s ❡q✉❛❧ t♦ t❤❛t ♥✉♠❜❡r✳

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿

an ≤ cn ≤ bn ❢♦r s♦♠❡

N✱

❢♦r ❛❧❧

n>N,

❛♥❞

lim an = lim bn = c ,

n→∞ t❤❡♥ t❤❡ s❡q✉❡♥❝❡

cn

n→∞

❝♦♥✈❡r❣❡s✱ ❛♥❞

lim cn = c .

n→∞

Pr♦♦❢✳ ❚❤❡ ❣❡♦♠❡tr② ♦❢ t❤❡ ♣r♦♦❢ ✐s s❤♦✇♥ ❜❡❧♦✇✿

❙✉♣♣♦s❡

ε>0

✐s ❣✐✈❡♥✳ ❆s ✇❡ ❦♥♦✇✱ ✇❡ ❤❛✈❡ ❢♦r ❛❧❧

c − ε < an < c + ε

❛♥❞

n

❧❛r❣❡r t❤❛♥ s♦♠❡

N✿

c − ε < bn < c + ε .

❚❤❡♥ ✇❡ ❤❛✈❡✿

c − ε < an ≤ cn ≤ bn < c + ε . ❲❛r♥✐♥❣✦ ❚❤❡ t❤❡♦r❡♠ ❞♦❡s ❝❧❛✐♠ ❝♦♥✈❡r❣❡♥❝❡ t❤✐s t✐♠❡✳

✶✳✻✳

▼♦r❡ ♣r♦♣❡rt✐❡s ♦❢ ❧✐♠✐ts ♦❢ s❡q✉❡♥❝❡s

✻✽

❊①❛♠♣❧❡ ✶✳✻✳✾✿ ❛❧t❡r♥❛t✐♥❣ r❡❝✐♣r♦❝❛❧s

❙♦♠❡t✐♠❡s t❤❡ ❝❤♦✐❝❡ ♦❢ t❤❡ sq✉❡❡③❡ ✐s ♦❜✈✐♦✉s✳ ▲❡t✬s ✐♠❛❣✐♥❡ t❤❛t ✇❡ s❡❡ t❤✐s s❡q✉❡♥❝❡ ❢♦r t❤❡ ✜rst t✐♠❡✿ n cn =

(−1) . n

❊①❛♠✐♥✐♥❣ t❤❡ s❡q✉❡♥❝❡ r❡✈❡❛❧s t✇♦ ❜♦✉♥❞✐♥❣ s❡q✉❡♥❝❡s✿

❲❡ ❤❛✈❡✿ −

◆♦✇✱ s✐♥❝❡ ❜♦t❤ an = −

(−1)n 1 1 ≤ ≤ . n n n

1 1 ❛♥❞ bn = ❣♦ t♦ 0✱ ❜② t❤❡ n n

❙q✉❡❡③❡ ❚❤❡♦r❡♠✱ s♦ ❞♦❡s c

n

=

(−1)n ✳ n

❚❤❡ ❡①❛♠♣❧❡ s❤♦✇s t❤❛t t❤❡ ❛❜s♦❧✉t❡ ✈❛❧✉❡ ✐s ❛ ❣♦♦❞ t♦♦❧ ❢♦r ❝♦♥str✉❝t✐♥❣ ❛ sq✉❡❡③❡✳ ❲❡ ❛♣♣❧② t❤✐s ✐❞❡❛ t♦ ❛❧❧ s❡q✉❡♥❝❡s✿ ❈♦r♦❧❧❛r② ✶✳✻✳✶✵✿ ▲✐♠✐t ♦❢ ❆❜s♦❧✉t❡ ❱❛❧✉❡ ❆ s❡q✉❡♥❝❡ ❝♦♥✈❡r❣❡s t♦ ③❡r♦ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐ts ❛❜s♦❧✉t❡ ✈❛❧✉❡ ❞♦❡s✳

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿

an → 0 ⇐⇒ |an | → 0 . ❊①❡r❝✐s❡ ✶✳✻✳✶✶

Pr♦✈❡ t❤❡ ❝♦r♦❧❧❛r②✳ ❊①❡r❝✐s❡ ✶✳✻✳✶✷

❲❤❛t ✐❢ ✐t✬s ♥♦t 0❄ ❊①❛♠♣❧❡ ✶✳✻✳✶✸✿ ❞✐♠✐♥✐s❤✐♥❣ ♦s❝✐❧❧❛t✐♦♥s

▲❡t✬s ✜♥❞ t❤❡ ❧✐♠✐t✱

1 sin n . n→∞ n ❜❡❝❛✉s❡ lim sin n ❞♦❡s ♥♦t ❡①✐st✳ ▲❡t✬s tr② ❛ sq✉❡❡③❡✳ lim

Pr♦❞✉❝t ❘✉❧❡

■t ❝❛♥♥♦t ❜❡ ❝♦♠♣✉t❡❞ ❜② t❤❡ ❚❤✐s ✐s ✇❤❛t ✇❡ ❦♥♦✇ ❢r♦♠ tr✐❣♦♥♦♠❡tr②✿

n→∞

−1 ≤ sin n ≤ 1 .

❍♦✇❡✈❡r✱ t❤✐s sq✉❡❡③❡ ♣r♦✈❡s ♥♦t❤✐♥❣ ❛❜♦✉t t❤❡ ❧✐♠✐t ♦❢ ♦✉r s❡q✉❡♥❝❡✦

✶✳✻✳

▼♦r❡ ♣r♦♣❡rt✐❡s ♦❢ ❧✐♠✐ts ♦❢ s❡q✉❡♥❝❡s

▲❡t✬s tr② ❛♥♦t❤❡r sq✉❡❡③❡✿

1 1 1 − ≤ sin n ≤ . n n n

◆♦✇✱ s✐♥❝❡

lim

n→∞

❢r♦♠ t❤❡

✻✾

❙q✉❡❡③❡ ❚❤❡♦r❡♠✱ ✇❡ ❝♦♥❝❧✉❞❡✿



1 − n



= lim

n→∞

1 = 0, , n

1 sin n = 0 . n→∞ n lim

❊①❡r❝✐s❡ ✶✳✻✳✶✹

❙✉♣♣♦s❡ an ❛♥❞ bn ❛r❡ ❝♦♥✈❡r❣❡♥t✳ Pr♦✈❡ t❤❛t max{an , bn } ❛♥❞ min{an , bn } ❛r❡ ❛❧s♦ ❝♦♥✈❡r❣❡♥t✳ ❍✐♥t✿ ❙t❛rt ✇✐t❤ t❤❡ ❝❛s❡ lim an > lim bn ✳

❚✇♦ P♦❧✐❝❡♠❡♥ ❚❤❡♦r❡♠

❚❤❡ sq✉❡❡③❡ t❤❡♦r❡♠ ✐s ❛❧s♦ ❦♥♦✇♥ ❛s t❤❡ ✿ ■❢ t✇♦ ♣♦❧✐❝❡♠❡♥ ❛r❡ ❡s❝♦rt✐♥❣ ❛ ♣r✐s♦♥❡r ❤❛♥❞❝✉✛❡❞ ❜❡t✇❡❡♥ t❤❡♠✱ ❛♥❞ ❜♦t❤ ♦✣❝❡rs ❣♦ t♦ t❤❡ s❛♠❡✭✦✮ ♣♦❧✐❝❡ st❛t✐♦♥✱ t❤❡♥ ✕ ✐♥ s♣✐t❡ ♦❢ s♦♠❡ ❢r❡❡❞♦♠ t❤❡ ❤❛♥❞❝✉✛s ❛❧❧♦✇ ✕ t❤❡ ♣r✐s♦♥❡r ✇✐❧❧ ❛❧s♦ ❡♥❞ ✉♣ ✐♥ t❤❛t st❛t✐♦♥✿

❙❛♥❞✇✐❝❤ ❚❤❡♦r❡♠

❆♥♦t❤❡r ♥❛♠❡ ✐s t❤❡ ✳ ■t ✐s✱ ♦♥❝❡ ❛❣❛✐♥✱ ❛❜♦✉t ❝♦♥tr♦❧✳ ❆ s❛♥❞✇✐❝❤ ❝❛♥ ❜❡ ❛ ♠❡ss② ❛✛❛✐r✿ ❤❛♠✱ ❝❤❡❡s❡✱ ❧❡tt✉❝❡✱ ❡t❝✳ ❖♥❡ ✇♦♥✬t ✇❛♥t t♦ t♦✉❝❤ t❤❛t ❛♥❞ ✐♥st❡❛❞ t❛❦❡s ❝♦♥tr♦❧ ♦❢ t❤❡ ❝♦♥t❡♥ts ❜② ❦❡❡♣✐♥❣ t❤❡♠ ❜❡t✇❡❡♥ t❤❡ t✇♦ ❜✉♥s✳ ❍❡ t❤❡♥ ❜r✐♥❣s t❤❡ t✇♦ t♦ ❤✐s ♠♦✉t❤ ❛♥❞ t❤❡ r❡st ♦❢ t❤❡ s❛♥❞✇✐❝❤ ❛❧♦♥❣ ✇✐t❤ t❤❡♠✦

✶✳✻✳ ▼♦r❡ ♣r♦♣❡rt✐❡s ♦❢ ❧✐♠✐ts ♦❢ s❡q✉❡♥❝❡s

✼✵

❲❡ s✉♠♠❛r✐③❡ t❤❡ t❤❡♦r❡♠ ❜② ♣♦✐♥t✐♥❣ ♦✉t t❤❡ ✜✈❡ ❤②♣♦t❤❡s❡s ❛♥❞ t❤❡ t✇♦ ❝♦♥❝❧✉s✐♦♥s✿ an ≤(1) ↓(3) a = =⇒

cn

≤(2)

=(5) ↓(6) a =(7) b

=

bn ↓(4) b

❊①❡r❝✐s❡ ✶✳✻✳✶✺

▲✐st t❤❡s❡ st❛t❡♠❡♥ts✳ ❚♦ ♠❛❦❡ ❝♦♥❝❧✉s✐♦♥s ❛❜♦✉t ❞✐✈❡r❣❡♥❝❡ t♦ ✐♥✜♥✐t②✱ ✇❡ ♦♥❧② ♥❡❡❞ t♦ ❝♦♥tr♦❧ ✐t ❢r♦♠ ♦♥❡✱ ❜✉t t❤❡ r✐❣❤t✱ s✐❞❡✿

❚❤❡ s♠❛❧❧❡r s❡q✉❡♥❝❡ ✇✐❧❧ ♣✉s❤ t❤❡ ❧❛r❣❡r t♦ ✐♥✜♥✐t②✱ ❛♥❞ t❤❡ ❧❛r❣❡r s❡q✉❡♥❝❡ ✇✐❧❧ ♣✉s❤ t❤❡ s♠❛❧❧❡r t♦ ♥❡❣❛t✐✈❡ ✐♥✜♥✐t②✳ ❇❡❧♦✇ ✐s ❛♥ ❛♥❛❧♦❣ ♦❢ t❤❡ ❙q✉❡❡③❡ ❚❤❡♦r❡♠ ❢♦r ✐♥✜♥✐t❡ ❧✐♠✐ts✿ ❚❤❡♦r❡♠ ✶✳✻✳✶✻✿ P✉s❤ ❖✉t ❢♦r ▲✐♠✐ts ♦❢ ❙❡q✉❡♥❝❡s

✶✳ ■❢ t❤❡ ✈❛❧✉❡s ♦❢ ❛ s❡q✉❡♥❝❡ ❧✐❡ ❛❜♦✈❡ t❤♦s❡ ♦❢ ❛ s❡q✉❡♥❝❡ t❤❛t ❞✐✈❡r❣❡s t♦ ♣♦s✐t✐✈❡ ✐♥✜♥✐t②✱ t❤❡♥ s♦ ❞♦❡s t❤✐s s❡q✉❡♥❝❡✳ ✷✳ ■❢ t❤❡ ✈❛❧✉❡s ♦❢ ❛ s❡q✉❡♥❝❡ ❧✐❡ ❜❡❧♦✇ t❤♦s❡ ♦❢ ❛ s❡q✉❡♥❝❡ t❤❛t ❞✐✈❡r❣❡s t♦ ♥❡❣❛t✐✈❡ ✐♥✜♥✐t②✱ t❤❡♥ s♦ ❞♦❡s t❤✐s s❡q✉❡♥❝❡✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✐❢ an ≥ bn ❢♦r ❛❧❧ n ❣r❡❛t❡r t❤❛♥ s♦♠❡ N ✱ t❤❡♥ ✇❡ ❤❛✈❡✿ 1. 2.

lim an = −∞ =⇒

n→∞

lim an = +∞ ⇐=

n→∞

❊①❡r❝✐s❡ ✶✳✻✳✶✼

Pr♦✈❡ t❤❡ t❤❡♦r❡♠✳ ❊①❡r❝✐s❡ ✶✳✻✳✶✽

❙✉♣♣♦s❡ ❛ s❡q✉❡♥❝❡ ✐s ❞❡✜♥❡❞ r❡❝✉rs✐✈❡❧② ❜② an+1 = 2an + 1 ✇✐t❤ a0 = 1 .

❉♦❡s t❤❡ s❡q✉❡♥❝❡ ❝♦♥✈❡r❣❡ ♦r ❞✐✈❡r❣❡❄

lim bn = −∞ .

n→∞

lim bn = +∞ .

n→∞

✶✳✼✳

❚❤❡♦r❡♠s ♦❢ ❆♥❛❧②s✐s

✼✶

❚❤✐s ✐s t❤❡ s✉♠♠❛r② ♦❢ t❤❡ t❤❡♦r❡♠✿ an ≤ ↓ +∞

cn

cn

↓ +∞

=⇒

≤

↓ −∞

=⇒

bn ↓ −∞

✶✳✼✳ ❚❤❡♦r❡♠s ♦❢ ❆♥❛❧②s✐s

❚❤❡ t❤❡♦r❡♠s ✐♥ t❤✐s s❡❝t✐♦♥ ✇✐❧❧ ❜❡ ✉s❡❞ t♦ ♣r♦✈❡ ♥❡✇ t❤❡♦r❡♠s✳ ■t ❝❛♥ ❜❡ s❦✐♣♣❡❞ ♦♥ t❤❡ ✜rst r❡❛❞✐♥❣✳ ❲❡ ❛❝❝❡♣t t❤❡ ❢♦❧❧♦✇✐♥❣ ❢✉♥❞❛♠❡♥t❛❧ r❡s✉❧t ✇✐t❤♦✉t ♣r♦♦❢✿ ❚❤❡♦r❡♠ ✶✳✼✳✶✿ ▼♦♥♦t♦♥❡ ❈♦♥✈❡r❣❡♥❝❡ ❚❤❡♦r❡♠ ❊✈❡r② ❜♦✉♥❞❡❞ ❛♥❞ ♠♦♥♦t♦♥✐❝ s❡q✉❡♥❝❡ ✐s ❝♦♥✈❡r❣❡♥t✳

an ✐s an ≤ an+1

■♥ ♦t❤❡r ✇♦r❞s✱ ✐❢ ❛ s❡q✉❡♥❝❡

•

❡✐t❤❡r ✐♥❝r❡❛s✐♥❣✱

•

❜♦✉♥❞❡❞✱

❛♥❞

|an | ≤ Q

❢♦r ❛❧❧

❢♦r s♦♠❡ ♥✉♠❜❡r

n✱

♦r ❞❡❝r❡❛s✐♥❣✱

an ≥ an+1

❢♦r ❛❧❧

n✱

Q✱

t❤❡♥ ✐t ❤❛s ❛ ❧✐♠✐t✳

❙✐♠♣❧② ❦♥♦✇✐♥❣ t❤❡s❡ t✇♦ ❢❛❝ts ♣r♦✈❡s t❤❛t t❤❡r❡ ✐s ❛ ❧✐♠✐t✿

❍♦✇❡✈❡r✱ t❤❡r❡ ✐s ♥♦ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❧✐♠✐t✳ ❚❤❡ r❡s✉❧t ✐s ❛❧s♦ ❦♥♦✇♥ ❛s t❤❡ ❈♦♠♣❧❡t❡♥❡ss

Pr♦♣❡rt② ♦❢ ❘❡❛❧ ◆✉♠❜❡rs✳

❆♥♦t❤❡r r❡s✉❧t ✇✐t❤ r❡❣❛r❞ t♦ t❤❡ ❣❡♦♠❡tr② ♦❢ r❡❛❧ ♥✉♠❜❡rs ✐s ❛❜♦✉t ❛ s❤r✐♥❦✐♥❣ s❡q✉❡♥❝❡ ♦❢ ✐♥t❡r✈❛❧s✿ I = [a, b] ⊃ I1 = [a1 , b1 ] ⊃ I2 = [a2 , b2 ] ⊃ ...

■t ✐s ✐❧❧✉str❛t❡❞ ❜❡❧♦✇✿

✶✳✼✳

❚❤❡♦r❡♠s ♦❢ ❆♥❛❧②s✐s

✼✷

❚❤❡ ❝♦♥❝❧✉s✐♦♥ s✉❣❣❡st❡❞ ❜② t❤❡ ♣✐❝t✉r❡ ✐s ♣r❡s❡♥t❡❞ ❜❡❧♦✇✿ ❚❤❡♦r❡♠ ✶✳✼✳✷✿ ◆❡st❡❞ ■♥t❡r✈❛❧s ❚❤❡♦r❡♠ ✶✳ ❆ s❡q✉❡♥❝❡ ♦❢ ♥❡st❡❞ ❝❧♦s❡❞ ✐♥t❡r✈❛❧s ❤❛s ❛ ♥♦♥✲❡♠♣t② ✐♥t❡rs❡❝t✐♦♥✱ ✐✳❡✳✱ ✐❢ ✇❡ ❤❛✈❡ t✇♦ s❡q✉❡♥❝❡s ♦❢ ♥✉♠❜❡rs

an

❛♥❞

bn

t❤❛t s❛t✐s❢②

a1 ≤ a2 ≤ ... ≤ an ≤ ... ≤ bn ≤ ... ≤ b2 ≤ b1 , t❤❡♥ t❤❡② ❜♦t❤ ❝♦♥✈❡r❣❡✱

an → a, bn → b , ❛♥❞

∞ \

[an , bn ] = [a, b] .

n=1 ✷✳ ■❢✱ ♠♦r❡♦✈❡r✱

bn − an → 0 , t❤❡♥

∞ \

n=1

[an , bn ] = {a} = {b} .

Pr♦♦❢✳

❋♦r ♣❛rt ✭✶✮✱ ♦❜s❡r✈❡ t❤❛t ❛ ♣♦✐♥t x ❜❡❧♦♥❣s t♦ t❤❡ ✐♥t❡rs❡❝t✐♦♥ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐t s❛t✐s✜❡s✿ an ≤ x ≤ bm , ❢♦r ❛❧❧ n, m .

▼❡❛♥✇❤✐❧❡✱ t❤❡ s❡q✉❡♥❝❡s ❝♦♥✈❡r❣❡ ❜② t❤❡ ▼♦♥♦t♦♥❡ ❜② t❤❡ ❈♦♠♣❛r✐s♦♥

❈♦♥✈❡r❣❡♥❝❡ ❚❤❡♦r❡♠✳ ❚❤❡r❡❢♦r❡✱

a≤x≤b

❚❤❡♦r❡♠✳

❋♦r ♣❛rt ✭✷✮✱ ❝♦♥s✐❞❡r✿ ❜② t❤❡ ❙✉♠

0 = lim (bn − an ) = lim bn − lim an = b − a , n→∞

❘✉❧❡✳ ❲❡ t❤❡♥ ❝♦♥❝❧✉❞❡ t❤❛t a = b✳

n→∞

n→∞

✶✳✼✳

❚❤❡♦r❡♠s ♦❢ ❆♥❛❧②s✐s

✼✸

❲❡ ❤❛✈❡ ✐♥❞❡❡❞ ❛ ✏♥❡st❡❞✑ s❡q✉❡♥❝❡ ♦❢ ✐♥t❡r✈❛❧s

I = [a, b] ⊃ I1 = [a1 , b1 ] ⊃ I2 = [a2 , b2 ] ⊃ ... , ✇✐t❤ ❛ s✐♥❣❧❡ ♣♦✐♥t✱ s❛②

A✱

✐♥ ❝♦♠♠♦♥✳

▲❡t✬s r❡❝❛❧❧ s♦♠❡ ❞❡✜♥✐t✐♦♥s ✭s❡❡♥ ✐♥ ❱♦❧✉♠❡ ✶✱ ❈❤❛♣t❡r ✶P❈✲✷✮✳

❉❡✜♥✐t✐♦♥ ✶✳✼✳✸✿ ✉♣♣❡r ❛♥❞ ❧♦✇❡r ❜♦✉♥❞s ♦❢ s❡t ❙✉♣♣♦s❡

•

•

❋♦r

S = [0, 1]✱

❛♥② ♥✉♠❜❡r

❆♥

❆

S

✐s ❛ s❡t ♦❢ r❡❛❧ ♥✉♠❜❡rs✳

✉♣♣❡r ❜♦✉♥❞ ♦❢ S ✐s ❛♥② ♥✉♠❜❡r M t❤❛t s❛t✐s✜❡s✿ x≤M

❢♦r ❛♥②

x≥m

❢♦r ❛♥②

x

✐♥

S.

❧♦✇❡r ❜♦✉♥❞ ♦❢ S ✐s ❛♥② ♥✉♠❜❡r m t❤❛t s❛t✐s✜❡s✿

M ≥1

x

✐♥

S.

✐s ✐ts ✉♣♣❡r ❜♦✉♥❞✳ ❍♦✇❡✈❡r✱ t❤❡s❡ s❡ts ❤❛✈❡ ♥♦ ✉♣♣❡r ❜♦✉♥❞s✿

(−∞, +∞), [0, +∞), {0, 1, 2, 3, ...} . ❲❡ ♥♦✇ t❛❦❡ t❤✐s t♦ t❤❡ ♥❡①t ❧❡✈❡❧✿

❉❡✜♥✐t✐♦♥ ✶✳✼✳✹✿ s❡ts ❜♦✉♥❞❡❞ ❜❡❧♦✇ ❛♥❞ ❛❜♦✈❡ ❆ s❡t t❤❛t ❤❛s ❛♥ ✉♣♣❡r ❜♦✉♥❞ ✐s ❝❛❧❧❡❞

❜♦✉♥❞❡❞ ❛❜♦✈❡✱ ❛♥❞ ❛ s❡t t❤❛t ❤❛s ❛ ❧♦✇❡r

❜♦✉♥❞❡❞ ❜❡❧♦✇✳ ❆ s❡t t❤❛t ❤❛s ❜♦t❤ ✉♣♣❡r ❛♥❞ ❧♦✇❡r ❜♦✉♥❞s ✐s ❝❛❧❧❡❞ ❜♦✉♥❞❡❞ ❀ ♦t❤❡r✇✐s❡✱ ✐t✬s ✉♥❜♦✉♥❞❡❞✳

❜♦✉♥❞ ✐s ❝❛❧❧❡❞

❉❡✜♥✐t✐♦♥ ✶✳✼✳✺✿ s✉♣r❡♠✉♠ ❛♥❞ ✐♥✜♠✉♠ ❙✉♣♣♦s❡

•

S

✐s ❛ s❡t✳

❆♥ ✉♣♣❡r ❜♦✉♥❞ ❢♦r ✇❤✐❝❤ t❤❡r❡ ✐s ♥♦ s♠❛❧❧❡r ✉♣♣❡r ❜♦✉♥❞ ✐s ❝❛❧❧❡❞ t❤❡

❧❡❛st ✉♣♣❡r ❜♦✉♥❞ ❀ ✐t ✐s ❛❧s♦ ❝❛❧❧❡❞ t❤❡ s✉♣r❡♠✉♠✳ ■t ✐s ❞❡♥♦t❡❞ ❛s ❢♦❧❧♦✇s✿ sup S

✶✳✼✳

❚❤❡♦r❡♠s ♦❢ ❆♥❛❧②s✐s

✼✹

•

❆ ❧♦✇❡r ❜♦✉♥❞ ❢♦r ✇❤✐❝❤ t❤❡r❡ ✐s ♥♦ ❧❛r❣❡r ❧♦✇❡r ❜♦✉♥❞ ✐s ❝❛❧❧❡❞ t❤❡

❧♦✇❡r ❜♦✉♥❞ ❀ ✐t ✐s ❛❧s♦ ❝❛❧❧❡❞ t❤❡ ✐♥✜♠✉♠✳

❣r❡❛t❡st

■t ✐s ❞❡♥♦t❡❞ ❛s ❢♦❧❧♦✇s✿

inf S

❇❡❧♦✇ ✇❡ ❥✉st✐❢② t❤❡ ✉s❛❣❡ ♦❢ ✏t❤❡✑✿

❚❤❡♦r❡♠ ✶✳✼✳✻✿ ❯♥✐q✉❡♥❡ss ♦❢ ❙✉♣r❡♠✉♠ ❛♥❞ ■♥✜♠✉♠ • •

❋♦r ❛ ❣✐✈❡♥ s❡t✱ t❤❡r❡ ❝❛♥ ❜❡ ♦♥❧② ♦♥❡ ❧❡❛st ✉♣♣❡r ❜♦✉♥❞✳ ❋♦r ❛ ❣✐✈❡♥ s❡t✱ t❤❡r❡ ❝❛♥ ❜❡ ♦♥❧② ♦♥❡ ❣r❡❛t❡st ❧♦✇❡r ❜♦✉♥❞✳

Pr♦♦❢✳ M = sup S ♠❡❛♥s t❤❛t✿ M ✐s ❛♥ ✉♣♣❡r ❜♦✉♥❞ ♦❢ S ✳ ′ ′ ✷✳ ■❢ M ✐s ❛♥♦t❤❡r ✉♣♣❡r ❜♦✉♥❞ ♦❢ S ✱ t❤❡♥ M ≥ M ✳ ′ ◆♦✇✱ ✐❢ ✇❡ ❤❛✈❡ ❛♥♦t❤❡r M = sup S ✱ t❤❡♥✿ ′ ✶✳ M ✐s ❛♥ ✉♣♣❡r ❜♦✉♥❞ ♦❢ S ✳ ′ ✷✳ ■❢ M ✐s ❛♥♦t❤❡r ✉♣♣❡r ❜♦✉♥❞ ♦❢ S ✱ t❤❡♥ M ≥ M ✳ ′ ❚❤❡r❡❢♦r❡✱ M = M ✳

❚❤✉s✱ ✶✳

❊①❛♠♣❧❡ ✶✳✼✳✼✿ ❧❡❛st ✉♣♣❡r ❜♦✉♥❞ ❋♦r t❤❡ ❢♦❧❧♦✇✐♥❣ s❡ts✱ t❤❡ ❧❡❛st ✉♣♣❡r ❜♦✉♥❞ ✐s

M = 3✿

• S = {1, 2, 3} • S = [1, 3] • S = (1, 3)

′ ❚❤❡ ♣r♦♦❢ ❢♦r t❤❡ ❧❛st ♦♥❡ ✐s ❛s ❢♦❧❧♦✇s✳ ❙✉♣♣♦s❡ M ✐s ❛♥ ✉♣♣❡r ❜♦✉♥❞ ✇✐t❤ ′ 3+M ′ a= ✳ ❇✉t a ❜❡❧♦♥❣s t♦ S ✦ ❚❤❡r❡❢♦r❡✱ M ✐s♥✬t ❛♥ ✉♣♣❡r ❜♦✉♥❞✿

1 < M ′ < 3✳

▲❡t✬s ❝❤♦♦s❡

2

❲❤❛t ✐❢ ✇❡ ❧✐♠✐t

M

′

S

t♦ t❤❡

r❛t✐♦♥❛❧

♥✉♠❜❡rs ♦♥❧② ✐♥

✐s ✐rr❛t✐♦♥❛❧✳ ❚❤❡ ♣r♦♦❢ ❢❛✐❧s✳

(1, 3)❄

❚❤❡♥

a=

3 + M′ 2

✇♦♥✬t ❜❡❧♦♥❣ t♦

S

✇❤❡♥

❲❡ ❦♥♦✇ ♥♦✇ t❤❛t t❤❡r❡ ✐s ❛t ♠♦st ♦♥❡ s✉❝❤ ♥✉♠❜❡r✳ ❲❤❛t ❛❜♦✉t ❛t ❧❡❛st❄

❚❤❡♦r❡♠ ✶✳✼✳✽✿ ❊①✐st❡♥❝❡ ♦❢ ❙✉♣r❡♠✉♠ ❛♥❞ ■♥✜♠✉♠ • •

❆♥② ❜♦✉♥❞❡❞ ❛❜♦✈❡ s❡t ❤❛s ❛ ❧❡❛st ✉♣♣❡r ❜♦✉♥❞✳ ❆♥② ❜♦✉♥❞❡❞ ❜❡❧♦✇ s❡t ❤❛s ❛ ❣r❡❛t❡st ❧♦✇❡r ❜♦✉♥❞✳

Pr♦♦❢✳ ❚❤❡ ✐❞❡❛ ♦❢ t❤❡ ♣r♦♦❢ ✐s t♦ ❝♦♥str✉❝t ♥❡st❡❞ ✐♥t❡r✈❛❧s ✇✐t❤ t❤❡ r✐❣❤t✲❡♥❞ ♣♦✐♥ts ❜❡✐♥❣ ✉♣♣❡r ❜♦✉♥❞s✳ ❲❤❛t s❤♦✉❧❞ ❜❡ t❤❡ ❧❡❢t✲❡♥❞ ♣♦✐♥ts❄

✶✳✼✳

❚❤❡♦r❡♠s ♦❢ ❆♥❛❧②s✐s

❙✉♣♣♦s❡ ❛ s❡t ✶✳ ▲❡t ✷✳ ▲❡t

S

❙✐♥❝❡

U L

S

✼✺

✐s ❣✐✈❡♥✳ ❚❤❡♥✿

S✳ U✳

❜❡ t❤❡ s❡t ♦❢ ❛❧❧ ✉♣♣❡r ❜♦✉♥❞s ♦❢ ❜❡ t❤❡ s❡t ♦❢ ❛❧❧ ❧♦✇❡r ❜♦✉♥❞s ♦❢

✐s ❜♦✉♥❞❡❞✱ ✇❡ ❤❛✈❡✿

U 6= ∅ . ◆♦✇✱ ✐❢ t♦

L✱

S

✐s ❛ s✐♥❣❧❡ ♣♦✐♥t✱ ✇❡ ❛r❡ ❞♦♥❡✳ ■❢ ♥♦t✱ ✇❡ ❤❛✈❡

x, y

✐♥

S

s✉❝❤ t❤❛t

❛♥❞

L 6= ∅ . ❲❡ st❛rt ✇✐t❤✿

◮ a1

✐s ❛♥② ❡❧❡♠❡♥t ♦❢

L✱

❛♥❞

b1

✐s ❛♥② ❡❧❡♠❡♥t ♦❢

U✳

❙✉♣♣♦s❡ ✐♥❞✉❝t✐✈❡❧② t❤❛t ✇❡ ❤❛✈❡ ❝♦♥str✉❝t❡❞ t✇♦ s❡q✉❡♥❝❡s ♦❢ ♥✉♠❜❡rs

ai , bi , s✉❝❤ t❤❛t✿ ✶✳ ✷✳ ✸✳

i = 1, 2, 3..., n ,

ai ✐s ✐♥ L ❛♥❞ bi ✐s ✐♥ U ✳ an ≤ ... ≤ a1 ≤ b1 ≤ ... ≤ bn ✳ 1 bi − ai ≤ i−1 (b1 − a1 )✳ 2

❲❡ ❝♦♥t✐♥✉❡ ✇✐t❤ t❤❡ ✐♥❞✉❝t✐✈❡ st❡♣✿ ▲❡t

1 c = (bn − an ) . 2 ❲❡ ❤❛✈❡ t✇♦ ❝❛s❡s✳

❈❛s❡ ✶✿

c

❜❡❧♦♥❣s t♦

U✳

❚❤❡♥ ❝❤♦♦s❡

an+1 = an

❛♥❞

bn+1 = c .

❛♥❞

bn+1 = c

❚❤❡♥✱

an+1 = an ❈❛s❡ ✷✿

c

❜❡❧♦♥❣s t♦

L✳

✐♥

L

✐♥

❚❤❡♥ ❝❤♦♦s❡

an+1 = c

❛♥❞

bn+1 = bn .

U.

x < y✳

❚❤❡r❡❢♦r❡✱

x ❜❡❧♦♥❣s

✶✳✼✳

❚❤❡♦r❡♠s ♦❢ ❆♥❛❧②s✐s

✼✻

❚❤❡♥✱

an+1 = c ❋✉rt❤❡r♠♦r❡✱

✐♥

L

❛♥❞

bn+1 = bn

✐♥

U.

1 1 1 1 (b − a ) = (b1 − a1 ) . bn+1 − an+1 = (bn − an ) ≤ 1 1 2 2 2n−1 2n

❚❤✉s✱ ❛❧❧ t❤❡ ❝♦♥❞✐t✐♦♥s ❛r❡ s❛t✐s✜❡❞✱ ❛♥❞ ♦✉r s❡q✉❡♥❝❡ ♦❢ ♥❡st❡❞ ✐♥t❡r✈❛❧s ❤❛s ❜❡❡♥ ✐♥❞✉❝t✐✈❡❧② ❜✉✐❧t✳

◆❡st❡❞ ■♥t❡r✈❛❧s ❚❤❡♦r❡♠ ❛♥❞ ❝♦♥❝❧✉❞❡ t❤❛t

❲❡ ❛♣♣❧② t❤❡

an → d ← bn . ❲❤② ✐s

c

❛ ❧❡❛st ✉♣♣❡r ❜♦✉♥❞ ♦❢

❋✐rst✱ s✉♣♣♦s❡

bn → c bn ✐♥ U ✳

❢r♦♠

c

✐s

♥♦t ❛♥ ✉♣♣❡r ❜♦✉♥❞✳ ❚❤❡♥ t❤❡r❡ ✐s x ✐♥ S ✇✐t❤ x > c✳ ■❢ ✇❡ ❝❤♦♦s❡ ε = x − c✱ t❤❡♥

✇❡ ❝♦♥❝❧✉❞❡ t❤❛t

♥♦t

❙❡❝♦♥❞✱ s✉♣♣♦s❡

c

ε = c − y✱

an → c an ✐♥ L✳

✐s

t❤❡♥ ❢r♦♠

❛ss✉♠♣t✐♦♥ t❤❛t

S❄

bn < x

❢♦r ❛❧❧

n>N

❢♦r s♦♠❡

N✳

❚❤✐s ❝♦♥tr❛❞✐❝ts t❤❡ ❛ss✉♠♣t✐♦♥ t❤❛t

❛ ❧❡❛st ✉♣♣❡r ❜♦✉♥❞✳ ❚❤❡♥ t❤❡r❡ ✐s ❛♥ ✉♣♣❡r ❜♦✉♥❞ ✇❡ ❝♦♥❝❧✉❞❡ t❤❛t

an > y

❇❡❧♦✇ ✇❡ ❞❡s❝r✐❜❡ ✇❤❛t ✐t ♠❡❛♥s ❢♦r ❛ s❡t t♦ ❜❡ ❛♥

❢♦r ❛❧❧

n>N

❢♦r s♦♠❡

N✳

y < c✳

■❢ ✇❡ ❝❤♦♦s❡

❚❤✐s ❝♦♥tr❛❞✐❝ts t❤❡

✐♥t❡r✈❛❧ ✿

❚❤❡♦r❡♠ ✶✳✼✳✾✿ ■♥t❡r♠❡❞✐❛t❡ P♦✐♥t ❚❤❡♦r❡♠ ❆ s✉❜s❡t

J

♦❢ t❤❡ r❡❛❧s ✐s ❛♥ ✐♥t❡r✈❛❧ ♦r ❛ s✐♥❣❧❡ ♣♦✐♥t ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐t ❝♦♥t❛✐♥s

❛❧❧ ♦❢ ✐ts ✐♥t❡r♠❡❞✐❛t❡ ♣♦✐♥ts✳

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿

y1 , y2

✐♥

J ❆◆❉ y1 < c < y2 =⇒ c

✐♥

J.

Pr♦♦❢✳

❚❤❡ ✏✐❢ ✑ ♣❛rt ✐s ♦❜✈✐♦✉s✳ ◆♦✇ ❛ss✉♠❡ t❤❛t t❤❡ ❝♦♥❞✐t✐♦♥ ✐s s❛t✐s✜❡❞ ❢♦r s❡t ❜♦✉♥❞❡❞✳ ❚❤❡♥ t❤❡s❡ ❡①✐st ❜② t❤❡

❊①✐st❡♥❝❡ ♦❢ ❙✉♣r❡♠✉♠ ❚❤❡♦r❡♠ ✿

J✳

❙✉♣♣♦s❡ ❛❧s♦ t❤❛t

J

a = inf S, b = sup J . ◆♦t❡ t❤❛t t❤❡s❡ ♠✐❣❤t ♥♦t ❜❡❧♦♥❣ t♦ ✶✳ ✷✳

y1 y2

✐♥ ✐♥

J J

J✳

❍♦✇❡✈❡r✱ ✐❢

a < y1 < c✱ ❛♥❞ c < y2 < b✳ ✇❡ ❤❛✈❡✿ c ✐♥ J ✳ ❚❤❡r❡❢♦r❡✱ J

s✉❝❤ t❤❛t

c

s❛t✐s✜❡s

a ≤ c ≤ b✱

t❤❡♥ t❤❡r❡ ❛r❡ t❤❡s❡ t✇♦✿

s✉❝❤ t❤❛t

❇② t❤❡ ♣r♦♣❡rt②✱ t❤❡♥

❊①❡r❝✐s❡ ✶✳✼✳✶✵

Pr♦✈❡ t❤❡ t❤❡♦r❡♠ ❢♦r t❤❡ ✉♥❜♦✉♥❞❡❞ ❝❛s❡✳

✐s ❛♥ ✐♥t❡r✈❛❧ ✇✐t❤

a, b

❛s ✐ts ❡♥❞✲♣♦✐♥ts✳

✐s

✶✳✼✳

❚❤❡♦r❡♠s ♦❢ ❆♥❛❧②s✐s

✼✼

❊①❛♠♣❧❡ ✶✳✼✳✶✶✿ s✉❜s❡q✉❡♥❝❡s

❚❤❡ ❝♦♥✈❡rs❡ ♦❢ t❤❡ ▼♦♥♦t♦♥❡ ❈♦♥✈❡r❣❡♥❝❡ ❚❤❡♦r❡♠ ❛❜♦✈❡ ✐s✱ ♦❢ ❝♦✉rs❡✱ ❢❛❧s❡✿ ❚❤❡ ❛❧t❡r♥❛t✐♥❣ s❡✲ q✉❡♥❝❡ cn = (−1)n ✐s ❜♦t❤ ❜♦✉♥❞❡❞ ✭✐t ❧✐❡s ✇✐t❤✐♥ [−1, 1]✮ ❛♥❞ ❞✐✈❡r❣❡♥t✳ ❲❡✱ ❤♦✇❡✈❡r✱ ❝❛♥✬t ❤❡❧♣ ♥♦t✐❝✐♥❣ t❤❛t ✐t ✐s ♠❛❞❡ ♦❢ t✇♦ ❝♦♥✈❡r❣❡♥t s❡q✉❡♥❝❡s✿

❚❤❡s❡ t✇♦ ❛r❡

s✉❜s❡q✉❡♥❝❡s ♦❢ cn✳

n ❡✈❡♥ cn = 1 an n ♦❞❞ cn = −1 bn

❲❡ ❝❛♥ ❛❧✇❛②s ❡①tr❛❝t ❛ s✉❜s❡q✉❡♥❝❡ t❤❛t ✐s ❡✐t❤❡r ✐♥❝r❡❛s✐♥❣ ♦r ❞❡❝r❡❛s✐♥❣✿

❚❤❡ r❡❛s♦♥ ✐s t❤❛t t❤❡r❡ ❛r❡ ❛❧✇❛②s ✐♥✜♥✐t❡❧② ♠❛♥② t❡r♠s ❡✐t❤❡r ❜❡❧♦✇ ♦r ❛❜♦✈❡ ❡✈❡r② t❡r♠ ♦❢ t❤❡ s❡q✉❡♥❝❡✳ ❲❡ ❤❛✈❡ r❡❛❝❤❡❞ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❝❧✉s✐♦♥✿ ❚❤❡♦r❡♠ ✶✳✼✳✶✷✿ ❇♦❧③❛♥♦✲❲❡✐❡rstr❛ss ❚❤❡♦r❡♠ ❊✈❡r② ❜♦✉♥❞❡❞ s❡q✉❡♥❝❡ ❤❛s ❛ ❝♦♥✈❡r❣❡♥t s✉❜s❡q✉❡♥❝❡✳

Pr♦♦❢✳

❙✉♣♣♦s❡ xn ✐s s✉❝❤ ❛ s❡q✉❡♥❝❡✳ ❚❤❡♥✱ ✐t ✐s ❝♦♥t❛✐♥❡❞ ✐♥ s♦♠❡ ✐♥t❡r✈❛❧ [a, b]✳ ❚❤❡ ✜rst ♣❛rt ♦❢ t❤❡ ❝♦♥str✉❝t✐♦♥ ✐s t♦ ❝✉t ❝♦♥s❡❝✉t✐✈❡ ✐♥t❡r✈❛❧s ✐♥ ❤❛❧❢ ❛♥❞ ♣✐❝❦ t❤❡ ❤❛❧❢ t❤❛t ❝♦♥t❛✐♥s ✐♥✜♥✐t❡❧② ♠❛♥② ❡❧❡♠❡♥ts ♦❢ t❤❡ s❡t {xn : n = 1, 2, 3...}✳ ❙✐♠✐❧❛r t♦ t❤❡ ♣r❡✈✐♦✉s ♣r♦♦❢s✱ ✇❡ ❛ss✉♠❡ t❤❛t ✇❡ ❤❛✈❡ ❛❧r❡❛❞② ❝♦♥str✉❝t❡❞ s❡q✉❡♥❝❡s✿ ai , bi , i = 1, 2, 3..., n ,

s✉❝❤ t❤❛t✿

✶✳ ❚❤❡ ✐♥t❡r✈❛❧ [ai , bi ] ❝♦♥t❛✐♥s ✐♥✜♥✐t❡❧② ♠❛♥② ❡❧❡♠❡♥ts ♦❢ {xn : n = 1, 2, 3...} . ✷✳ an ≤ ... ≤ a1 ≤ b1 ≤ ... ≤ bn . ✸✳ bi − ai ≤

1

2i−1

(b1 − a1 ) .

✶✳✼✳

❚❤❡♦r❡♠s ♦❢ ❆♥❛❧②s✐s

✼✽

❲❡ ❝♦♥t✐♥✉❡ ✇✐t❤ t❤❡ ✐♥❞✉❝t✐✈❡ st❡♣✿ ▲❡t 1 c = (bn − an ) . 2

❲❡ ❤❛✈❡ t✇♦ ❝❛s❡s✳

❈❛s❡ ✶✿ ■♥t❡r✈❛❧ [an , c] ❝♦♥t❛✐♥s ✐♥✜♥✐t❡❧② ♠❛♥② ❡❧❡♠❡♥ts ♦❢ {xn : n = 1, 2, 3...}✳ ❚❤❡♥ ❝❤♦♦s❡ an+1 = an ❛♥❞ bn+1 = c .

❈❛s❡ ✷✿ ■♥t❡r✈❛❧ [an , c] ❞♦❡s ♥♦t ❝♦♥t❛✐♥ ✐♥✜♥✐t❡❧② ♠❛♥② ❡❧❡♠❡♥ts ♦❢ {xn : n = 1, 2, 3...}✱ t❤❡♥ [c, bn ] ❞♦❡s✳ ❚❤❡♥ ❝❤♦♦s❡ an+1 = c ❛♥❞ bn+1 = bn . ❆s ❜❡❢♦r❡✱

1 1 1 1 bn+1 − an+1 = (bn − an ) ≤ (b1 − a1 ) = n (b1 − a1 ) . n−1 2 22 2

❚❤❡ ✐♥t❡r✈❛❧s ❛r❡ ❝♦♥str✉❝t❡❞ ❛s ❞❡s✐r❡❞❀ t❤❡ ✐♥t❡r✈❛❧s ❛r❡ ③♦♦♠✐♥❣ ✐♥ ♦♥ t❤❡ ♣r♦❣r❡ss✐✈❡❧② ❞❡♥s❡r ❛♥❞ ❞❡♥s❡r ♣❛rts ♦❢ t❤❡ s❡q✉❡♥❝❡✿

◆♦✇ ✇❡ ❛♣♣❧② t❤❡ ◆❡st❡❞

■♥t❡r✈❛❧s ❚❤❡♦r❡♠ t♦ ❝♦♥❝❧✉❞❡ t❤❛t an → d ← bn .

❚❤❡ s❡❝♦♥❞ ♣❛rt ♦❢ t❤❡ ❝♦♥str✉❝t✐♦♥ ✐s t♦ ❝❤♦♦s❡ t❤❡ t❡r♠s ♦❢ t❤❡ s✉❜s❡q✉❡♥❝❡ yk ♦❢ xn ✱ ❛s ❢♦❧❧♦✇s✳ ❲❡ ❥✉st ♣✐❝❦ ❛s yk ❛♥② ❡❧❡♠❡♥t ♦❢ t❤❡ s❡t {xn : n = 1, 2, 3, ...} ✐♥ [ak , bk ] t❤❛t ❝♦♠❡s ❧❛t❡r ✐♥ t❤❡ s❡q✉❡♥❝❡ t❤❛♥ t❤❡ ♦♥❡s ❛❧r❡❛❞② ❛❞❞❡❞✱ ✐✳❡✳✱ y1 , y2 , ..., yk−1 ✳ ❚❤✐s ✐s ❛❧✇❛②s ♣♦ss✐❜❧❡ ❜❡❝❛✉s❡ ✇❡ ❛❧✇❛②s ❤❛✈❡ ✐♥✜♥✐t❡❧② ♠❛♥② ❡❧❡♠❡♥ts ❧❡❢t t♦ ❝❤♦♦s❡ ❢r♦♠✳ ❖♥❝❡ t❤❡ s✉❜s❡q✉❡♥❝❡ yk ✐s ❝♦♥str✉❝t❡❞✱ ✇❡ ❤❛✈❡ yk → d ❜② t❤❡ ❙q✉❡❡③❡ ❚❤❡♦r❡♠✳

✶✳✽✳

❈♦♠♣♦s✐t✐♦♥s

✼✾

✶✳✽✳ ❈♦♠♣♦s✐t✐♦♥s

❆❢t❡r ❛❧❧ ♦❢ t❤❡s❡ ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s✱ ✇❤❛t ❛❜♦✉t t❤❡ ❝♦♠♣♦s✐t✐♦♥s❄ ▲❡t✬s r❡✈✐❡✇✳ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ s❡t ♦❢ ❜♦②s ❛♥❞ ❛ s❡t ♦❢ ❜❛❧❧s ❛♥❞ ✇❡ ❦♥♦✇ t❤❡ ❜♦②s✬ ♣r❡❢❡r❡♥❝❡s ✐♥ ❜❛❧❧s✳ ❢✉♥❝t✐♦♥✳

❝♦❧♦rs

❲❡ ❛❧s♦ ♥♦t❡ t❤❡ ❝♦❧♦rs ♦❢ t❤❡ ❜❛❧❧s✳

❚❤❛t✬s ❛♥♦t❤❡r ❢✉♥❝t✐♦♥✳

❉♦ ✇❡ ❦♥♦✇ t❤❡✐r

❚❤❛t✬s ♦♥❡

♣r❡❢❡r❡♥❝❡s ✐♥

❄ ■t✬s t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ t✇♦ ❢✉♥❝t✐♦♥s✿

■❢ ✇❡ r❡♣r❡s❡♥t t❤❡ t✇♦ ❢✉♥❝t✐♦♥s ❛s

✐♥♣✉t

x

❜❧❛❝❦ ❜♦①❡s

✱ ✇❡ ❝❛♥ ✇✐r❡ t❤❡♠ t♦❣❡t❤❡r✿

❢✉♥❝t✐♦♥

→

F

♦✉t♣✉t

→

y ↓

✐♥♣✉t

y

❢✉♥❝t✐♦♥

→

G

♦✉t♣✉t

→

z

❚❤✉s✱ ✇❡ ✉s❡ t❤❡ ♦✉t♣✉t ♦❢ t❤❡ ❢♦r♠❡r ❛s t❤❡ ✐♥♣✉t ♦❢ t❤❡ ❧❛tt❡r✳ ❚❤❡ ✐❞❡❛ ✐s t❤❛t t❤❡ t✇♦ ✭♦r ♠♦r❡✮ ❢✉♥❝t✐♦♥s ❛r❡ ❛♣♣❧✐❡❞ ❝♦♥s❡❝✉t✐✈❡❧②✿

❉❡✜♥✐t✐♦♥ ✶✳✽✳✶✿ ❝♦♠♣♦s✐t✐♦♥ ♦❢ ❢✉♥❝t✐♦♥s ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ t✇♦ ❢✉♥❝t✐♦♥s ✭✇✐t❤ t❤❡ ❝♦❞♦♠❛✐♥ ♦❢ t❤❡ ❢♦r♠❡r ♠❛t❝❤✐♥❣ t❤❡ ❞♦♠❛✐♥ ♦❢ t❤❡ ❧❛tt❡r✮✿

F :X→Y

❛♥❞

G:Y →Z.

✶✳✽✳

❈♦♠♣♦s✐t✐♦♥s

✽✵ ❚❤❡♥ t❤❡✐r ❝♦♠♣♦s✐t✐♦♥ ✐s t❤❡ ❢✉♥❝t✐♦♥ ✭❢r♦♠ t❤❡ ❞♦♠❛✐♥ ♦❢ t❤❡ ❢♦r♠❡r t♦ t❤❡ ❝♦❞♦♠❛✐♥ ♦❢ t❤❡ ❧❛tt❡r✮ H :X →Z,

✇❤✐❝❤ ✐s ❝♦♠♣✉t❡❞ ❢♦r ❡✈❡r② x ✐♥ X ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦✲st❡♣ ♣r♦❝❡✲ ❞✉r❡✿ x → F (x) = y → G(y) = z . ❈♦♠♣♦s✐t✐♦♥ ♦❢ ❢✉♥❝t✐♦♥s

♥❛♠❡s ♦❢ t❤❡ s❡❝♦♥❞ ❛♥❞ ✜rst ❢✉♥❝t✐♦♥s ↓ ↓

G ◦ F (x) = G F (x) ↑ ↑ ↑ ↑ ♥❛♠❡ ♦❢ t❤❡ ♥❡✇ ❢✉♥❝t✐♦♥ s✉❜st✐t✉t✐♦♥ ❊①❛♠♣❧❡ ✶✳✽✳✷✿ ❝♦♠♣♦s✐t✐♦♥ ♦❢ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s

❚❤✐s s✉❜st✐t✉t✐♦♥ ✐s ❥✉st ❛s ✐♠♠❡❞✐❛t❡ ❢♦r ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s✳
❋♦r ❡①❛♠♣❧❡✱ 3 ◮ y = x2 ✐s s✉❜st✐t✉t❡❞ ✐♥t♦ z = y 3 ✱ r❡s✉❧t✐♥❣ ✐♥ z = x2 ✳ ❚❤❡ ✐❞❡❛ ✐s t❤❡ s❛♠❡ ❛s ❜❡❢♦r❡ ✭❱♦❧✉♠❡ ✶✱ ❈❤❛♣t❡r ✶P❈✲✸✮✿ ■♥s❡rt t❤❡ ✐♥♣✉t ✈❛❧✉❡ ✐♥ ❛❧❧ ♦❢ t❤❡s❡ ❜♦①❡s✳ ❋♦r ❡①❛♠♣❧❡✱ t❤✐s ❢✉♥❝t✐♦♥ ♦♥ t❤❡ ❧❡❢t ✐s ✉♥❞❡rst♦♦❞ ❛♥❞ ❡✈❛❧✉❛t❡❞ ✈✐❛ t❤❡ ❞✐❛❣r❛♠ ♦♥ t❤❡ r✐❣❤t✿

f (y) =

2y 2 − 3y + 7 , y 3 + 2y + 1

f () =

22 − 3 + 7 . 3 + 2 + 1

❚❤✐s ✐s t❤❡ r❡s✉❧t ♦❢ t❤❡ s✉❜st✐t✉t✐♦♥ y = sin x✿

f (sin x) =

2 (sin x) (sin x)

❚❤❡♥✱ ✇❡ ❤❛✈❡

f (sin x) =

2 3

− 3 (sin x) + 7

.

+ 2 (sin x) + 1

2(sin x)2 − 3(sin x) + 7 . (sin x)3 + 2(sin x) + 1

❊①❛♠♣❧❡ ✶✳✽✳✸✿ ❝♦♠♣♦s✐t✐♦♥ ✇✐t❤ s❡q✉❡♥❝❡s

❚❤❡ ✐♥♣✉t ♦❢ ❛ s❡q✉❡♥❝❡ ✕ ✉♥❞❡rst♦♦❞ ❛s ❛ ❢✉♥❝t✐♦♥ ✕ ✐s ❛♥ ✐♥t❡❣❡r✱ ❜✉t t❤❡ ♦✉t♣✉t ❝♦✉❧❞ ❜❡ ❛♥② r❡❛❧ ♥✉♠❜❡r✳ ❚❤❡♥✱ ❝❛♥ ✇❡ ❤❛✈❡ t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ t✇♦ s❡q✉❡♥❝❡s✿ ✐♥t❡❣❡r ✐♥♣✉t, n →

s❡q✉❡♥❝❡

→

r❡❛❧ ♦✉t♣✉t, an ↓? ✐♥t❡❣❡r ✐♥♣✉t, m →

s❡q✉❡♥❝❡

→ r❡❛❧ ♦✉t♣✉t, bm

❚❤❡r❡ ✐s ❛ ♠✐s♠❛t❝❤ ♦❢ t❤❡ ✈❛r✐❛❜❧❡s✦ ❋♦r ❡①❛♠♣❧❡✱ ✇❤❛t ✐s t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡s❡ t✇♦ s❡q✉❡♥❝❡s✿

an = 1/n ❛♥❞ bn = (−1)n ?! ❍♦✇❡✈❡r✱ ✐❢ t❤❡ ✜rst s❡q✉❡♥❝❡s ✐s ✐♥t❡❣❡r✲✈❛❧✉❡❞✱ t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♠❛❦❡s s❡♥s❡✳ ✐♥t❡❣❡r ✐♥♣✉t, n →

❋♦r ❡①❛♠♣❧❡✱ ✐❢ ✇❡ ❤❛✈❡✿

s❡q✉❡♥❝❡

→ ✐♥t❡❣❡r ♦✉t♣✉t, mn ↓ ✐♥t❡❣❡r ✐♥♣✉t, m → an = 2n ❛♥❞ bk = (−1)k ,

s❡q✉❡♥❝❡

→ r❡❛❧ ♦✉t♣✉t, bm

❈♦♠♣♦s✐t✐♦♥s

✶✳✽✳

✽✶

t❤❡♥ t❤❡✐r ❝♦♠♣♦s✐t✐♦♥ ✐s

cn = b2n = (−1)2n . ■t✬s ❛ s✉❜s❡q✉❡♥❝❡ ♦❢

bn ✦

■❢ t❤❡ s❡❝♦♥❞ ♦♥❡ ✐s ❥✉st ❛ ❢✉♥❝t✐♦♥ ♦❢ r❡❛❧ ✈❛r✐❛❜❧❡✱ t❤❡r❡ ✐s ❛ ❝♦♠♣♦s✐t✐♦♥✳ ❋♦r ❡①❛♠♣❧❡✱ t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢

xn = 1/n

❛♥❞

f (x) = x2

✐s

yn = (1/n)2 . ❚❤✐s ✐s t❤❡ ✢♦✇❝❤❛rt ♦❢ s✉❝❤ ❛ ❝♦♠♣♦s✐t✐♦♥✿ ✐♥t❡❣❡r ✐♥♣✉t,

n →

s❡q✉❡♥❝❡

→

r❡❛❧ ♦✉t♣✉t,

↓

r❡❛❧ ✐♥♣✉t,

xn x

→

❢✉♥❝t✐♦♥

→

r❡❛❧ ♦✉t♣✉t,

y

❊①❡r❝✐s❡ ✶✳✽✳✹

❙t❛t❡ ❛s ❛♥ ✐♠♣❧✐❝❛t✐♦♥✿ ✏❊✈❡r② s❡q✉❡♥❝❡ ✐s ❛ ❢✉♥❝t✐♦♥✳✑ ❲❤❛t ❛❜♦✉t t❤❡ ❝♦♥✈❡rs❡❄

❋♦r ❛ ❢✉♥❝t✐♦♥

f

❞❡✜♥❡❞ ♦♥ ❛♥ ✐♥t❡r✈❛❧✱ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡q✉❡♥❝❡ ✐s ❧✐❦❡❧② t♦ ♠❛❦❡ s❡♥s❡✿

yn = f (xn ) ◆♦✇✱ ✇❤❛t ❛❜♦✉t t❤❡ ❧✐♠✐t ♦❢ t❤❡ ♥❡✇ s❡q✉❡♥❝❡❄ ❈❛♥ ✇❡ s❛②✱ s✐♠✐❧❛r t♦ t❤❡ ❢♦✉r r✉❧❡s ♦❢ ❧✐♠✐ts✱ t❤❡ ❧✐♠✐t ♦❢ t❤❡

❝♦♠♣♦s✐t✐♦♥ ✐s t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ ❧✐♠✐ts❄

❚❤❡ q✉❡st✐♦♥ ✐s✿

◮

❲❤❛t ❞♦❡s t❤❡ ❝♦♠♣♦s✐t✐♦♥ ✇✐t❤ ❛ ❢✉♥❝t✐♦♥ ❞♦ t♦ t❤❡ ❧✐♠✐t ♦❢ ❛ ❝♦♥✈❡r❣❡♥t s❡q✉❡♥❝❡❄

▲❡t✬s ❧♦♦❦ ❛t s♦♠❡ ❡①❛♠♣❧❡s✳ ❊①❛♠♣❧❡ ✶✳✽✳✺✿ ❧✐♥❡❛r

❙♦♠❡t✐♠❡s t❤❡ ❛❧❣❡❜r❛ ✐s s✐♠♣❧❡✳ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ s❡q✉❡♥❝❡ ♠✐❛❧✿

x n → a✳

❙✉♣♣♦s❡

f

✐s ❛ ❧✐♥❡❛r ♣♦❧②♥♦✲

f (x) = mx + b . ❲❤❛t ✐s t❤❡ ❧✐♠✐t ♦❢ t❤❡ ❝♦♠♣♦s✐t✐♦♥ s❡q✉❡♥❝❡

yn = f (xn )❄

▲❡t✬s ❝♦♠♣✉t❡✿

lim yn = lim f (xn )

n→∞

n→∞

= lim (mxn + b) n→∞

= m lim xn + b n→∞

❆❝❝♦r❞✐♥❣ t♦ t❤❡ ▲✐♥❡❛r✐t② ❘✉❧❡✳

= ma + b = f (a) . ❚❤✐s ✐s ❥✉st t❤❡ ✈❛❧✉❡ ♦❢ ❡①❛♠♣❧❡✱

f

❛t

a✦

■♥ r❡tr♦s♣❡❝t✱ t❤❡ ❧✐♠✐t ❝❛♥ ❜❡ ❢♦✉♥❞ ❜② ❛ s✐♠♣❧❡ s✉❜st✐t✉t✐♦♥✳ ❋♦r

      n+1 n+1 lim 3 + 7 = 3 · lim + 7 = 3 · 1 + 7 = 10 . n→∞ n→∞ n−1 n−1

❙♦✱ t❤❡ ❧✐♠✐t ♦❢ t❤❡ ❝♦♠♣♦s✐t✐♦♥ ✐s t❤❡

✈❛❧✉❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❛t t❤❡ ❧✐♠✐t✦

✶✳✽✳

❈♦♠♣♦s✐t✐♦♥s

✽✷

❊①❛♠♣❧❡ ✶✳✽✳✻✿ ❝♦♠♣♦s✐t✐♦♥s ▲❡t✬s tr②

• • •

f (x) = x2

❛♥❞ ❛ s❡q✉❡♥❝❡ t❤❛t ❝♦♥✈❡r❣❡s t♦

0✳

❚❤✐s ✐s ✇❤❛t ✇❡ s❡❡ ❜❡❧♦✇✿

x ❞❡♣❡♥❞s ♦♥ n✱ ♠✐❞❞❧❡✿ ❤♦✇ y ❞❡♣❡♥❞s ♦♥ x✱ r✐❣❤t✿ ❤♦✇ y ❞❡♣❡♥❞s ♦♥ n✳

❜♦tt♦♠✿ ❤♦✇

❈❛♥ ✇❡ ♣r♦✈❡ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ t❤❛t ✇❡ s❡❡❄ ❆♥ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ t❤❡ r❡✈❡❛❧s✿

lim xn

n→∞


2

Pr♦❞✉❝t ❘✉❧❡

✐♥ t❤✐s s✐♠♣❧❡ s✐t✉❛t✐♦♥

= lim (xn · xn ) n→∞

= lim xn · lim xn n→∞ n→∞  2 = lim xn , n→∞

♣r♦✈✐❞❡❞ t❤❛t ❧✐♠✐t ❡①✐sts✳

❆ r❡♣❡❛t❡❞ ✉s❡ ♦❢ t❤❡ ❞♦❡s

(xn )p

Pr♦❞✉❝t ❘✉❧❡

❢♦r ❛♥② ♣♦s✐t✐✈❡ ✐♥t❡❣❡r

♣r♦❞✉❝❡s ❛ ♠♦r❡ ❣❡♥❡r❛❧ ❢♦r♠✉❧❛✿ ■❢ ❛ s❡q✉❡♥❝❡

p✱

❙✉♠ ❘✉❧❡

❝♦♥✈❡r❣❡s✱ t❤❡♥ s♦

ip h lim [(xn )p ] = lim xn .

n→∞ ❈♦♠❜✐♥❡❞ ✇✐t❤ t❤❡

xn

❛♥❞

❛♥❞ t❤❡

n→∞

❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡

t❤✐s ♣r♦✈❡s t❤❡ ❢♦❧❧♦✇✐♥❣✿

❚❤❡♦r❡♠ ✶✳✽✳✼✿ ❈♦♠♣♦s✐t✐♦♥ ❘✉❧❡ ❢♦r ▲✐♠✐ts ♦❢ P♦❧②♥♦♠✐❛❧s ■❢ s❡q✉❡♥❝❡

xn

❝♦♥✈❡r❣❡s✱ t❤❡♥ s♦ ❞♦❡s

f (xn ) ❢♦r ❛♥② ♣♦❧②♥♦♠✐❛❧ f ✳

❋✉rt❤❡r♠♦r❡✱

✇❡ ❤❛✈❡✿

lim f (xn ) = f

n→∞

❙✉❝❤ ❛ r❡s✉❧t ✐s ❛❧s♦ ❦♥♦✇♥ ❛s t❤❡ ✐♥♣✉t✳ ■t ✐s ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡

❙✉❜st✐t✉t✐♦♥ ❘✉❧❡ ▲✐♥❡❛r✐t② ❘✉❧❡ ✳



lim xn

n→∞



❜❡❝❛✉s❡ t❤❡ ❧✐♠✐t ✐s s✉❜st✐t✉t❡❞ ✐♥t♦ t❤✐s ❢✉♥❝t✐♦♥ ❛s ✐ts

✶✳✽✳

❈♦♠♣♦s✐t✐♦♥s

✽✸

❊①❛♠♣❧❡ ✶✳✽✳✽✿ ❢r♦♠ ❜♦t❤ s✐❞❡s

❚❤✐s t✐♠❡ ✇❡ ❝❤♦♦s❡ ❛ s❡q✉❡♥❝❡ t❤❛t ❛♣♣r♦❛❝❤❡s 0 ✇❤✐❧❡ ❛❧t❡r♥❛t✐♥❣ ❜❡t✇❡❡♥ t❤❡ s✐❞❡s✿ xn = (−1)n

1 ❛♥❞ f (x) = − sin 5x . n.8

❲❡ s❡❡ t❤❡ s❛♠❡ ♣❛tt❡r♥✦ ❚❤✐s ✐s t❤❡ s✉♠♠❛r② ♦❢ t❤❡ t❤❡♦r❡♠✿ xn → a =⇒ f (xn ) → f (a)

Pr♦✈❡♥ ♦♥❧② ❢♦r ♣♦❧②♥♦♠✐❛❧s✦ ❊①❛♠♣❧❡ ✶✳✽✳✾✿ ❧✐♠✐ts ❜② s✉❜st✐t✉t✐♦♥

■t ✇✐❧❧ ❜❡ ✈❡r② ❡❛s② t♦ ❝♦♠♣✉t❡ s♦♠❡ ❧✐♠✐ts ♥♦✇❀ ❢♦r ❡①❛♠♣❧❡✿ lim

n→∞

lim

n→∞

!    2
1 1 +5 = 3 · 02 − 2 · 0 + 5 = 5 −2· 3· n n ! 2   
3 3 +5 = 3 · 22 − 2 · 2 + 5 = 13 3· 2+ −2· 2+ n n

❲❡ ❛r❡ ✉s✐♥❣ t❤❡ s❛♠❡ ♣♦❧②♥♦♠✐❛❧ ❤❡r❡✿

f (x) = 3x2 − 2x + 5 .

❲❡ s✐♠♣❧② s✉❜st✐t✉t❡ t❤❡ ❛♣♣r♦♣r✐❛t❡ ❧✐♠✐t✱ 0 ❛♥❞ 2 r❡s♣❡❝t✐✈❡❧②✳ ❙♦✱ ✇❡ ❝♦♥❝❧✉❞❡ t❤❛t ❧✐♠✐ts ❜❡❤❛✈❡ ✇❡❧❧ ✇✐t❤ r❡s♣❡❝t t♦ ❝♦♠♣♦s✐t✐♦♥ ✇✐t❤ s♦♠❡ ❢✉♥❝t✐♦♥s✳ ❖♥❝❡ ❛❣❛✐♥✱ ♥❡✇ s❡q✉❡♥❝❡s ❛r❡ ♣r♦❞✉❝❡❞ ✈✐❛ ❝♦♠♣♦s✐t✐♦♥s ✇✐t❤ ❢✉♥❝t✐♦♥s✿ ●✐✈❡♥ ❛ s❡q✉❡♥❝❡ xn ❛♥❞ ❛ ❢✉♥❝t✐♦♥ y = f (x)✱ ❞❡✜♥❡ yn = f (xn ) .

▲❡t✬s ❡①❛♠✐♥❡ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤✐s s❡q✉❡♥❝❡✳ ❊❛❝❤ ♣♦✐♥t xn ♦♥ t❤❡ x✲❛①✐s ♣r♦❞✉❝❡s ❛ ♣♦✐♥t yn ♦♥ t❤❡ y ✲❛①✐s ✈✐❛ f ✿

✶✳✽✳ ❈♦♠♣♦s✐t✐♦♥s

✽✹

❲❡ ❝❛♥ s❡❡ ❤♦✇ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢

xn

t♦

a

❧❡❛❞s t♦ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢

yn

t♦

f (a)✳

■t ✐s ❡❛s②✱ ❤♦✇❡✈❡r✱ t♦ ♣r♦❞✉❝❡ ❛♥ ❡①❛♠♣❧❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ❢♦r ✇❤✐❝❤ t❤✐s ❝❛✉s❛t✐♦♥ ❢❛✐❧s✳ ❇❡❧♦✇✱ ♦♥❝❡ ❛❣❛✐♥✱ ❡❛❝❤ ♣♦✐♥t

xn

♦♥ t❤❡

x✲❛①✐s

♣r♦❞✉❝❡s ❛ ♣♦✐♥t

❍♦✇❡✈❡r✱ ❜❡❝❛✉s❡ ♦❢ t❤❡ ❣❛♣ ✐♥ t❤❡ ❣r❛♣❤✱

yn

yn

♦♥ t❤❡

y ✲❛①✐s

✈✐❛

❞♦❡s♥✬t ❝♦♥✈❡r❣❡ t♦

❞✐s❝♦♥t✐♥✉✐t② ❛❞❞r❡ss❡❞ ✐♥ t❤❡ ♥❡①t ❝❤❛♣t❡r✳

❊①❛♠♣❧❡ ✶✳✽✳✶✵✿ ❞✐✈❡r❣❡♥❝❡ t♦ ✐♥✜♥✐t② ❲❤❛t ✐❢ ✇❡ ❝❤♦♦s❡

xn =

f✿

1 n

❚❤❡♥✱ ♦❜✈✐♦✉s❧②✱ ✇❡ ❤❛✈❡✿

yn =

❛♥❞

f (x) =

1 ? x

1 = n → ∞! 1/n

f (a)✦

❚❤❡ ✐ss✉❡ ✐s ✇✐t❤ ❝♦♥t✐♥✉✐t② ✈s✳

✶✳✾✳

◆✉♠❜❡rs ❛r❡ ❧✐♠✐ts

❚❤❡

✽✺

❈♦♠♣♦s✐t✐♦♥ ❘✉❧❡ ❢❛✐❧s ❢♦r t❤✐s ❢✉♥❝t✐♦♥ t♦♦✦

■♥ ❈❤❛♣t❡r ✷✱ ✇❡ ✇✐❧❧ ✉s❡ t❤✐s ❝♦♥str✉❝t✐♦♥ t♦ st✉❞② t❤❡ ❧✐♠✐ts ♦❢

❢✉♥❝t✐♦♥s r❛t❤❡r t❤❛♥ t❤♦s❡ ♦❢ s❡q✉❡♥❝❡s✳

❆ ❢❡✇ ❡①❛♠♣❧❡s ♦❢ t❤❛t ❛r❡ ♣r❡s❡♥t❡❞ ✐♥ t❤❡ t✇♦ ❢♦❧❧♦✇✐♥❣ s❡❝t✐♦♥s✿ ❲❡ ✇✐❧❧ ❡st❛❜❧✐s❤ s❡✈❡r❛❧ ✐♠♣♦rt❛♥t ❢❛❝ts t❤❛t ✇✐❧❧ ❜❡ ✉s❡❞ t❤r♦✉❣❤♦✉t t❤❡ ❜♦♦❦✳

✶✳✾✳ ◆✉♠❜❡rs ❛r❡ ❧✐♠✐ts

❙♦✱ ❧✐♠✐ts ✭✇❤❡♥ ✜♥✐t❡✮ ❛r❡ ♥✉♠❜❡rs✱ ❛♥❞ ✈✐❝❡ ✈❡rs❛✳ ❍♦✇❡✈❡r✱ ❛s ✇❡ ❥✉st s❛✇✱ ❛ ♥✉♠❜❡r ❝❛♥ ❜❡ t❤❡ ❧✐♠✐t ♦❢ ♠❛♥② s❡q✉❡♥❝❡s✿

.9 .99 .999 .9999 .99999 ... → 1 1. 1.1 1.01 1.001 1.0001 ... → 1 ■♥✜♥✐t❡❧② ♠❛♥②✱ ✐♥ ❢❛❝t✿

a + 1/n

a + 1/(2n) a + 1/(3n) ց ↓ ւ 2 a + 1/n → a ← a + 1/n3 ↑ տ √ ր n a + 1/ n a + 1/2 a + 1/(ln n) ❲❡✱ t❤❡r❡❢♦r❡✱ ❛❝t ✐♥ r❡✈❡rs❡✿

◮

■♥st❡❛❞ ♦❢ ❧♦♦❦✐♥❣ ❢♦r t❤❡ ❧✐♠✐t ♦❢ ❛ s❡q✉❡♥❝❡✱ ✇❡ ✜♥❞ s❡q✉❡♥❝❡s t❤❛t ❝♦♥✈❡r❣❡ t♦ ❛ ♥✉♠❜❡r

✇❡ ❛r❡ ✐♥t❡r❡st❡❞ ✐♥✳ ❲❡ t❤✐♥❦ ♦❢ t❤♦s❡ ❛s ❊①❛♠♣❧❡ ✶✳✾✳✶✿

❛♣♣r♦①✐♠❛t✐♦♥s ♦❢ t❤✐s ♥✉♠❜❡r✳ π

▲❡t✬s ❜✉✐❧❞ t❤❡ ♥❡①t s❡q✉❡♥❝❡ ❢r♦♠ ❣❡♦♠❡tr②✿

◮

❲❡ ❛♣♣r♦①✐♠❛t❡ t❤❡ ❝✐r❝❧❡ ✇✐t❤ r❡❣✉❧❛r ♣♦❧②❣♦♥s✿ ❡q✉❛❧ s✐❞❡s ❛♥❞ ❛♥❣❧❡s✳

❲❡ ♣✉t s✉❝❤ ♣♦❧②❣♦♥s ❛r♦✉♥❞ t❤❡ ❝✐r❝❧❡ s♦ t❤❛t ✐t t♦✉❝❤❡s t❤❡♠ ❢r♦♠ t❤❡ ✐♥s✐❞❡ ✭✏❝✐r❝✉♠s❝r✐❜✐♥❣✑ ♣♦❧②❣♦♥s✮✿

✶✳✾✳

◆✉♠❜❡rs ❛r❡ ❧✐♠✐ts

❋♦r ❡❛❝❤

✽✻

n = 3, 4, 5, ...✱

✇❡ ✜♥❞ t❤❡ ❛r❡❛ ♦❢ t❤❡ ♣♦❧②❣♦♥✳ ▲❡t✬s ❝❛❧❧ ✐t

An ✳

❲❡ ❝❛♥ ❡①❛♠✐♥❡ t❤❡ ❞❛t❛✿

n 3 4 5 6 7 8 9 10 11 12 13 14 15 An 5.196 4.000 3.633 3.464 3.371 3.314 3.276 3.249 3.230 3.215 3.204 3.195 3.188 ■s t❤❡r❡ ❛

❧✐♠✐t ❄

❋✐rst✱ t❤❡ s❡q✉❡♥❝❡ ✐s ❜♦✉♥❞❡❞ ❢r♦♠ ❜❡❧♦✇ ❜②

❚❤❡♥ ✐t ♠✉st ❜❡ ❝♦♥✈❡r❣❡♥t ❜② t❤❡

0✳

❙❡❝♦♥❞✱ ✐t ❛♣♣❡❛rs t♦ ❜❡ ♠♦♥♦t♦♥❡✳

▼♦♥♦t♦♥❡ ❈♦♥✈❡r❣❡♥❝❡ ❚❤❡♦r❡♠✳

♣✉t s✉❝❤ ♣♦❧②❣♦♥s ❛r♦✉♥❞ t❤❡ ❝✐r❝❧❡ s♦ t❤❛t ✐t t♦✉❝❤❡s t❤❡♠ ❢r♦♠ t❤❡ ❚❤❡s❡ ❛r❡❛s

Bn

❛❧s♦ ❝♦♥✈❡r❣❡ t♦

π✳

❲❡ ❝❛❧❧ ✐ts ❧✐♠✐t

♦✉ts✐❞❡

π✦

❲❡ ❝❛♥ ❛❧s♦

✭✏✐♥s❝r✐❜✐♥❣✑ ♣♦❧②❣♦♥s✮✳

❲❡ ♥♦✇ ❤❛✈❡✿

An → π ← Bn . ▲❡t✬s r❡❝❛❧❧ ✭❱♦❧✉♠❡ ✶✱ ❈❤❛♣t❡r ✶P❈✲✶✮ ❤♦✇ r❡❛❧ ♥✉♠❜❡rs ❛r❡ ✐♥tr♦❞✉❝❡❞✳ ❚❤❡ st❛rt✐♥❣ ♣♦✐♥t ✐s t❤❡

♥❛t✉r❛❧ ♥✉♠❜❡rs

✭✉s❡❞ ❢♦r ❝♦✉♥t✐♥❣✮✿

0, 1, 2, 3, ... ❚❤❡ ♥❡①t st❡♣ ✐s t❤❡

✐♥t❡❣❡rs ✿ ..., −3, −2, −1, 0, 1, 2, 3, ...

❚❤❡② ❝❛♥ ❜❡ ✉s❡❞ ❢♦r ❦❡❡♣✐♥❣ r❡❝♦r❞ ♦❢ s♣❛❝❡ ❛♥❞ ❧♦❝❛t✐♦♥s✳ ■♠❛❣✐♥❡ ❢❛❝✐♥❣ ❛ ❢❡♥❝❡ s♦ ❧♦♥❣ t❤❛t ②♦✉ ❝❛♥✬t s❡❡ ✇❤❡r❡ ✐t ❡♥❞s✳ ❲❡ st❡♣

■s t❤❡ ♥✉♠❜❡r ♦❢ ♣❧❛♥❦s

❛✇❛②

✐♥✜♥✐t❡ ❄

❢r♦♠ t❤❡ ❢❡♥❝❡ ♠✉❧t✐♣❧❡ t✐♠❡s ❛♥❞ t❤❡r❡ ✐s st✐❧❧ ♠♦r❡ t♦ s❡❡✿

❲❡ ❛ss✉♠❡ t❤❛t ✐t ✐s✳ ❲❡ ✈✐s✉❛❧✐③❡ t❤❡s❡ ❛s ♠❛r❦✐♥❣s ♦♥ ❛ str❛✐❣❤t ❧✐♥❡✱

❛❝❝♦r❞✐♥❣ t♦ t❤❡ ♦r❞❡r ♦❢ t❤❡ ♣❧❛♥❦s✿

❲❡ ❤❛✈❡ ❛ s❡q✉❡♥❝❡ ✇✐t❤ ❛♥ ✐♥✜♥✐t❡ ❧✐♠✐t✦

✶✳✾✳ ◆✉♠❜❡rs ❛r❡ ❧✐♠✐ts

✽✼

❙♦✱ ✇❡ ③♦♦♠❡❞ ♦✉t t♦ s❡❡ t❤❡ ❢❡♥❝❡✳ ❙✉♣♣♦s❡ ♥♦✇ ✇❡ ③♦♦♠ ✐♥ ♦♥ ❛ s♣♦t ♦♥ t❤❡ ❢❡♥❝❡✳ ❲❤❛t ✐❢ t❤❡r❡ ✐s ❛ s❤♦rt❡r ♣❧❛♥❦ ❜❡t✇❡❡♥ ❡✈❡r② t✇♦ ♣❧❛♥❦s❄ ❲❡ ❧♦♦❦ ❝❧♦s❡r ❛♥❞ ✇❡ s❡❡ ♠♦r❡✿

■❢ ✇❡ ❦❡❡♣ ③♦♦♠✐♥❣ ✐♥✱ t❤❡ r❡s✉❧t ✇✐❧❧ ❧♦♦❦ s✐♠✐❧❛r t♦ ❛ r✉❧❡r ✿

■t✬s ❛s ✐❢ ✇❡ ❛❞❞ ♦♥❡ ♠❛r❦ ❜❡t✇❡❡♥ ❛♥② t✇♦ ❛♥❞ t❤❡♥ ❛❞❞ ❛♥♦t❤❡r ♦♥❡ ❜❡t✇❡❡♥ ❡✐t❤❡r ♦❢ t❤❡ t✇♦ ♣❛✐rs ✇❡ ❤❛✈❡ ❝r❡❛t❡❞✳ ❲❡ ❦❡❡♣ r❡♣❡❛t✐♥❣ t❤✐s st❡♣✳ ■❣♥♦r✐♥❣ t❤❡ ❢❛❝t t❤❛t t❤✐s r✉❧❡r ❣♦❡s ♦♥❧② t♦ 1/16 ♦❢ ❛♥ ✐♥❝❤✱ ❧❡t✬s ✐♠❛❣✐♥❡ t❤❛t t❤❡ ♣r♦❝❡ss ❝♦♥t✐♥✉❡s ✐♥❞❡✜♥✐t❡❧②✿

■s t❤❡ ❞❡♣t❤ ✐♥✜♥✐t❡ ❄ ❲❡ ❛ss✉♠❡ t❤❛t ✐t ✐s✳ ■❢ ✇❡ ♣✐❝❦ ❛ ♣♦✐♥t ❢r♦♠ ❡❛❝❤ r✉♥❣✱ ✇❡ ❤❛✈❡ ❜♦✉♥❞❡❞ s❡q✉❡♥❝❡✳ ❊①❡r❝✐s❡ ✶✳✾✳✷

❲❤❡♥ ❞♦❡s s✉❝❤ ❛ s❡q✉❡♥❝❡ ❝♦♥✈❡r❣❡❄ ■❢ ✇❡ ❛❞❞ ♥✐♥❡ ♠❛r❦s ❛t ❛ t✐♠❡✱ t❤❡ r❡s✉❧t ✐s ❛ ♠❡tr✐❝ r✉❧❡r ✿

✶✳✾✳

◆✉♠❜❡rs ❛r❡ ❧✐♠✐ts

✽✽

❚♦ s❡❡ ✐t ❛♥♦t❤❡r ✇❛②✱ ✇❡ ❛❧❧♦✇ ♠♦r❡ ❛♥❞ ♠♦r❡ ❞❡❝✐♠❛❧s ✐♥ ♦✉r ♥✉♠❜❡rs✿

1.55 : 1/3 : 1: π:

1. .3 1. 3.

1.5 .33 1.0 3.1

1.55 .333 1.00 3.14

1.550 .3333 1.000 3.141

1.5500 .33333 1.0000 3.1415

... ... ... ...

❚❤❡s❡ ❛r❡ ❛❧❧ ❝♦♥✈❡r❣❡♥t s❡q✉❡♥❝❡s✦ ❙♦✱ ✇❡ st❛rt ✇✐t❤ ✐♥t❡❣❡rs ❛s ❧♦❝❛t✐♦♥s ❛♥❞ t❤❡♥ ✕ ❜② ❝✉tt✐♥❣ t❤❡s❡ ✐♥t❡r✈❛❧s ❢✉rt❤❡r ❛♥❞ ❢✉rt❤❡r ✕ ❛❧s♦ ✐♥❝❧✉❞❡ ❢r❛❝t✐♦♥s✱ ✐✳❡✳✱

r❛t✐♦♥❛❧ ♥✉♠❜❡rs✳

❍♦✇❡✈❡r✱ ✇❡ t❤❡♥ r❡❛❧✐③❡ t❤❛t s♦♠❡ ♦❢ t❤❡ ❧♦❝❛t✐♦♥s ❤❛✈❡ ♥♦ ❝♦✉♥t❡r♣❛rts ❛♠♦♥❣ t❤❡s❡ ♥✉♠❜❡rs✳ ❋♦r ❡①❛♠♣❧❡✱ √ 2 ✐s t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❞✐❛❣♦♥❛❧ ♦❢ ❛ 1 × 1 sq✉❛r❡ ✭❛♥❞ ❛ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❡q✉❛t✐♦♥ x2 = 2✮❀ ✐t✬s ♥♦t r❛t✐♦♥❛❧✳ ❲✐t❤♦✉t t❤✐s ♥✉♠❜❡r✱ t❤❡ ❧✐♥❡ ✐s ✐♥❝♦♠♣❧❡t❡✦ ❆s ❛♥ ✐❧❧✉str❛t✐♦♥✱ ❛♥ ✏✐♥❝♦♠♣❧❡t❡✑ r♦♣❡ ✇♦♥✬t ❤❛♥❣✿

❲❤❛t t♦ ❞♦❄ ❲❡ ❛♣♣r♦①✐♠❛t❡ ❡✈❡r② s✉❝❤ ♥✉♠❜❡r ✇✐t❤ ❛ s❡q✉❡♥❝❡ ♦❢ r❛t✐♦♥❛❧ ♥✉♠❜❡rs✳ ❚♦❣❡t❤❡r✱ r❛t✐♦♥❛❧ ❛♥❞ ✐rr❛t✐♦♥❛❧ ♥✉♠❜❡rs ♠❛❦❡ ✉♣ t❤❡ ♦❢ t❤✐s ❧✐♥❡ ❛s

❝♦♠♣❧❡t❡ ❀ t❤❡r❡ ❛r❡ ♥♦ ♠✐ss✐♥❣ ♣♦✐♥ts✳

r❡❛❧ ♥✉♠❜❡rs ❛♥❞ t❤❡ r❡❛❧ ♥✉♠❜❡r ❧✐♥❡✳ ❲❡ t❤✐♥❦ ❈♦♠♣❧❡t❡♥❡ss Pr♦♣❡rt② ♦❢ ❘❡❛❧ ◆✉♠❜❡rs ♣r♦✈❡♥

❚❤❡

❛❜♦✈❡ ❣✉❛r❛♥t❡❡s t❤✐s✳ ❚❤✐s s❡t✉♣ ♣r♦❞✉❝❡s ❛ ❝♦rr❡s♣♦♥❞❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❧♦❝❛t✐♦♥s ♦♥ t❤❡ ❧✐♥❡ ❛♥❞ t❤❡ r❡❛❧ ♥✉♠❜❡rs✿ ❧♦❝❛t✐♦♥ ❲❡ ✇✐❧❧ ❢♦❧❧♦✇ t❤✐s ❝♦rr❡s♣♦♥❞❡♥❝❡ ✐♥

P ←→

♥✉♠❜❡r

❜♦t❤ ❞✐r❡❝t✐♦♥s✱ ❛s ❢♦❧❧♦✇s✿

x

✶✳✾✳

◆✉♠❜❡rs ❛r❡ ❧✐♠✐ts

• •

✽✾

❧♦❝❛t✐♦♥ ♦♥ t❤❡ ❧✐♥❡✳ ❲❡ t❤❡♥ ✜♥❞ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♠❛r❦ ♦♥ t❤❡ ❧✐♥❡✳ ❚❤❛t✬s s♦♠❡ ♥✉♠❜❡r x✳ ❈♦♥✈❡rs❡❧②✱ s✉♣♣♦s❡ x ✐s ❛ ♥✉♠❜❡r✳ ❲❡ t❤✐♥❦ ♦❢ ✐t ❛s ❛ ✏❝♦♦r❞✐♥❛t❡✑ ❛♥❞ ✜♥❞ ✐ts ♠❛r❦ ♦♥ t❤❡ ❧✐♥❡✳ ❚❤❛t✬s t❤❡ ❧♦❝❛t✐♦♥ ♦❢ x✿ s♦♠❡ ♣♦✐♥t P ♦♥ t❤❡ ❧✐♥❡✳ ❋✐rst✱ s✉♣♣♦s❡

P

✐s ❛

t❤❡ ✏❝♦♦r❞✐♥❛t❡✑ ♦❢

P✿

❖♥❝❡ t❤❡ ❝♦♦r❞✐♥❛t❡ s②st❡♠ ✐s ✐♥ ♣❧❛❝❡✱ ✐t ✐s ❛❝❝❡♣t❛❜❧❡ t♦ t❤✐♥❦ ♦❢ ❡✈❡r② ❧♦❝❛t✐♦♥ ❛s ❛ ♥✉♠❜❡r✱ ❛♥❞ ✈✐❝❡ ✈❡rs❛✳ ■♥ ❢❛❝t✱ ✇❡ ♦❢t❡♥ ✇r✐t❡✿

P = x. ❚❤❡ r❡s✉❧t ♠❛② ❜❡ ❞❡s❝r✐❜❡❞ ❛s t❤❡ ✏ 1✲❞✐♠❡♥s✐♦♥❛❧ ❝♦♦r❞✐♥❛t❡ s②st❡♠✑✳ ■t ✐s ❛❧s♦ ❝❛❧❧❡❞ t❤❡ ♦r s✐♠♣❧②

t❤❡ ♥✉♠❜❡r ❧✐♥❡✳

r❡❛❧ ♥✉♠❜❡r ❧✐♥❡

❇✉t ✇❤❛t ❛❜♦✉t t❤❡ ❛❧❣❡❜r❛ ✇❡ r♦✉t✐♥❡❧② ❞♦ ✇✐t❤ t❤❡s❡ ♥✉♠❜❡rs❄ ❖✉r r✉❧❡s ♦❢ ❧✐♠✐ts s❤♦✇ t❤❛t ✇❤❡♥ ✇❡ r❡♣❧❛❝❡ ❡✈❡r② r❡❛❧ ♥✉♠❜❡r ✇✐t❤ ❛ s❡q✉❡♥❝❡ ❝♦♥✈❡r❣✐♥❣ t♦ ✐t✱ ✐t ✐s st✐❧❧ ♣♦ss✐❜❧❡ t♦ ❞♦ ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s ✇✐t❤ t❤❡s❡ r❡♣❧❛❝❡♠❡♥ts✳ ❇❡❧♦✇ ✐s t❤❡ s✉♠♠❛r②✿ ❚❤❡♦r❡♠ ✶✳✾✳✸✿ ❆❧❣❡❜r❛ ♦❢ ▲✐♠✐ts ♦❢ ❙❡q✉❡♥❝❡s

an → a

❙✉♣♣♦s❡

❙❘✿ P❘✿

❛♥❞

bn → b✳

❚❤❡♥

an + b n → a + b an · bn → ab

❈▼❘✿ ◗❘✿

c · an → ca an /bn → a/b

c b 6= 0

❢♦r ❛♥② r❡❛❧ ♣r♦✈✐❞❡❞

❊①❛♠♣❧❡ ✶✳✾✳✹✿ ❛♣♣r♦①✐♠❛t✐♦♥s

❚❤❡s❡ r✉❧❡s s❤♦✇ ✇❤②

❛♣♣r♦①✐♠❛t✐♦♥s ✇♦r❦✳

■♥❞❡❡❞✱ ✇❡ ❝❛♥ t❤✐♥❦ ♦❢ ❛ s❡q✉❡♥❝❡ t❤❛t ❝♦♥✈❡r❣❡s t♦ ❛

♥✉♠❜❡r ❛s ❛ s❡q✉❡♥❝❡ ♦❢ ❜❡tt❡r ❛♥❞ ❜❡tt❡r ❛♣♣r♦①✐♠❛t✐♦♥s✳ ❚❤❡♥ ❝❛rr②✐♥❣ ♦✉t ❛❧❧ t❤❡ ❛❧❣❡❜r❛ ✇✐t❤ t❤❡s❡ s❡q✉❡♥❝❡s ✇✐❧❧ ♣r♦❞✉❝❡ t❤❡ s❛♠❡ r❡s✉❧t ❛s t❤❡ ♦r✐❣✐♥❛❧ ❝♦♠♣✉t❛t✐♦♥ ✐s ♠❡❛♥t t♦ ♣r♦❞✉❝❡✦ ❋♦r ❡①❛♠♣❧❡✱ ❤❡r❡ ✐s s✉❝❤ ❛ s✉❜st✐t✉t✐♦♥✿

(1) 

+ 

(2) 

= 3 

1 5 4 + 2− =3− → 3 n n n   3 2 1− → 3 + 2+ 2 n n 1+

... ❙✐♠✐❧❛r❧②✱ ✐❢ ✇❡ ♥❡❡❞ t♦ ✜♥❞

√

2 + π✱

✇❡ ✉s❡ t✇♦ ❛♣♣r♦①✐♠❛t✐♦♥s ♦❢ t❤❡ t✇♦ ♥✉♠❜❡rs ✭♥♦ ♠❛tt❡r ✇❤✐❝❤

♦♥❡s✮✿

an →

√

2

❛♥❞

bn → π .

❚❤❡♥ t❤❡ ♥❡✇ s❡q✉❡♥❝❡ ❣✐✈❡s ✉s ❛♥ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ s✉♠✿

an + b n →

√

2+π.

❊①❛♠♣❧❡ ✶✳✾✳✺✿ ❞❡❝✐♠❛❧s

❚❤❡s❡ ❧❛✇s ❤❡❧♣ ✉s ❥✉st✐❢② t❤❡ ❢♦❧❧♦✇✐♥❣

tr✐❝❦ ♦❢ ✜♥❞✐♥❣ ❢r❛❝t✐♦♥ r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ ✐♥✜♥✐t❡ ❞❡❝✐♠❛❧s✳

✶✳✶✵✳

❚❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥s

❚❤✐s ✐s ❤♦✇ ✇❡ ❞❡❛❧ ✇✐t❤

✾✵

x = .3333...✿

x = 0.3333... − 10x = 3.3333... −9x = −3.0000...

❙♦❧✈❡ t❤✐s ❡q✉❛t✐♦♥✦

=⇒ x = 1/3

❇❡❤✐♥❞ t❤✐s tr✐❝❦ ✐s ❛ ♣r♦♣❡r ♠❡t❤♦❞✳ ❋✐rst✱ ✇❡ ♥♦t❡ t❤❛t ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ▼♦♥♦t♦♥❡ ❈♦♥✈❡r❣❡♥❝❡ ❚❤❡♦r❡♠✱ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡q✉❡♥❝❡ ❝♦♥✲ ✈❡r❣❡s✿

0 0.3 0.33 0.333 0.3333 ... → x ❲❡ ❝❛❧❧ t❤❡ ❧✐♠✐t ❚❤❡♥ ✇❡ ✉s❡ t❤❡

x✳

❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡

❛♥❞ t❤❡

❉✐✛❡r❡♥❝❡ ❘✉❧❡

t♦ ❝❛rr② ♦✉t t❤❡ ❢♦❧❧♦✇✐♥❣ ❛❧❣❡❜r❛ ♦❢

s❡q✉❡♥❝❡s✿

an : 0 0.3 0.33 0.333 0.3333 ... → x − 10an : 3 3.3 3.33 3.333 3.3333 ... → 10x −9an : −3 −3 −3 −3 −3 ... → −9x

❙♦❧✈❡✿

− 3 = −9x =⇒ x = 1/3

◆♦t❡ t❤❛t ✇❡ ❤❛✈❡ s❤✐❢t❡❞ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ s❡❝♦♥❞ s❡q✉❡♥❝❡✳

❆s ✇❡ ❦♥♦✇✱ ❝♦♠♣✉t✐♥❣ ❛ ♣♦❧②♥♦♠✐❛❧ ♦♥❧② r❡q✉✐r❡s t❤❡ ✜rst t❤r❡❡ ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s✳ ❚❤❛t ✐s ✇❤② ✇❡ ❝❛♥ ❛♣♣r♦①✐♠❛t❡ ♦✉t♣✉ts ♦❢ ❛ ♣♦❧②♥♦♠✐❛❧

f

❢r♦♠ ❛♣♣r♦①✐♠❛t✐♦♥s ♦❢ t❤❡ ✐♥♣✉ts✿

an → a =⇒ f (an ) → f (a) .

❆❧❣❡❜r❛ ♦❢ ▲✐♠✐ts ♦❢ ❙❡q✉❡♥❝❡s ❛♥❞ ✐s ❥✉st ❛♥♦t❤❡r ✈❡rs✐♦♥ ♦❢ ❈♦♠♣♦s✐t✐♦♥ ❘✉❧❡ ❢♦r ▲✐♠✐ts ♦❢ P♦❧②♥♦♠✐❛❧s✳ ❆❝❝♦r❞✐♥❣ t♦ t❤❡ ◗✉♦t✐❡♥t ❘✉❧❡✱ ✇❡ ❤❛✈❡ t❤✐s ♣r♦♣❡rt② ❢♦r r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥s t♦♦ ❛s ❧♦♥❣ ❛s ❚❤✐s ❢♦❧❧♦✇s ❢r♦♠

✇❡ ❛✈♦✐❞ ❞✐✈✐❞✐♥❣ ❜②

0✳

❇✉t ✇❤❛t ❛❜♦✉t ♠♦r❡ ❝♦♠♣❧❡① ❢✉♥❝t✐♦♥s❄

✶✳✶✵✳ ❚❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥s

▲❡t✬s r❡❝❛❧❧ ✜rst ✇❤❛t ✇❡ ❦♥♦✇ ❛❜♦✉t t❤❡ ❛❧❣❡❜r❛ ♦❢ ❡①♣♦♥❡♥ts ✭s❡❡♥ ✐♥ ❱♦❧✉♠❡ ✶✱ ❈❤❛♣t❡r ✶P❈✲✸✮✳ ■t st❛rts ✇✐t❤ t❤✐s s✐♠♣❧❡ ❛❧❣❡❜r❛✿

◮

❘❡♣❡❛t❡❞ ❛❞❞✐t✐♦♥ ✐s

♠✉❧t✐♣❧✐❝❛t✐♦♥ ✿ 2 + 2 + 2 = 2 · 3 ✳

❖♥❡ ❝❛♥ s❛② t❤❛t t❤❛t✬s ❤♦✇ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✇❛s ✏✐♥✈❡♥t❡❞✑ ✕ ❛s r❡♣❡❛t❡❞ ❛❞❞✐t✐♦♥✳ ◆❡①t✿

◮

❘❡♣❡❛t❡❞ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✐s

♣♦✇❡r ✿ 2 · 2 · 2 = 23 ✳

❆♥❞ t❤✐s ✐s t❤❡ ♥♦t❛t✐♦♥ t❤❛t ✇❡ ✉s❡✿

✶✳✶✵✳

❚❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥s

✾✶ ❇❛s❡ ❛♥❞ ❡①♣♦♥❡♥t

❡①♣♦♥❡♥t ↓

b

x

↑

❜❛s❡ ❙♦✱ t❤✐s ♥♦t❛t✐♦♥ ✐s ♥♦t❤✐♥❣ ❜✉t ❛ ❝♦♥✈❡♥t✐♦♥✳ ❋r♦♠ t❤❡ ✐❞❡❛ ♦❢ t❤❡ ❡①♣♦♥❡♥t ❛s ❛ r❡♣❡❛t❡❞ ♠✉❧t✐♣❧✐❝❛t✐♦♥✱ ✇❡ ❞❡r✐✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ r✉❧❡s ✭❢♦r ❛r❜✐tr❛r② a, b > 0✮✿ ❆♥❛❧♦❣②✿ r❡♣❡❛t❡❞ ❛❞❞✐t✐♦♥ ✈s✳ r❡♣❡❛t❡❞ ♠✉❧t✐♣❧✐❝❛t✐♦♥

▼✉❧t✐♣❧✐❝❛t✐♦♥✿

x = 1, 2, ...

❊①♣♦♥❡♥t✐❛t✐♦♥✿

a · ... · a} = ax |a · a · {z

a + ... + a} = a · x |a + a + {z x

x

t✐♠❡s

t✐♠❡s

❘✉❧❡s✿ 1.

a(x + y) = ax + ay

a0 = 1  −x 1 1 x a = = −x a a x+y x y a =a a

2.

(a + b)x = ax + bx

(ab)x = ax bx

3.

a(xy) = (ax)y

x=0

a·0 =0

x = −1, −2, ...

■♥✐t✐❛❧❧②✱ t❤❡ ❡①♣♦♥❡♥t✐❛❧

ax = (−a)(−x)

axy = (ax )y

❢✉♥❝t✐♦♥ ♦❢ ❜❛s❡ a > 0 ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥✿ f (x) = bx

✇✐t❤ t❤❡

❞♦♠❛✐♥ t❤❡ s❡t ♦❢ ❛❧❧ ✐♥t❡❣❡rs✿ Z = {..., −3, −2, −1, 0, 1, 2, 3, ...} .

❍♦✇❡✈❡r✱ t❤❡ ❞♦♠❛✐♥ ♠✐ss❡s s♦♠❡ ♦❢ t❤❡ ♥✉♠❜❡rs t❤❛t ✐♥t❡r❡st ✉s✦ ❊①❛♠♣❧❡ ✶✳✶✵✳✶✿ ❜❛❝t❡r✐❛ ♠✉❧t✐♣❧②✐♥❣

❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ♣♦♣✉❧❛t✐♦♥ ♦❢ ❜❛❝t❡r✐❛ t❤❛t ❞♦✉❜❧❡s ❡✈❡r② ❞❛②✿ pn+1 |{z}

♣♦♣✉❧❛t✐♦♥✿ ❛t t✐♠❡

=2· n+1

❚❤❡ ❣r❛♣❤ ❝♦♥s✐sts ♦❢ ❞✐s❝♦♥♥❡❝t❡❞ ♣♦✐♥ts✿

pn |{z}

❛t t✐♠❡

=⇒ pn = p0 2n . n

✶✳✶✵✳

❚❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥s

✾✷

▲❡t✬s t❤✐♥❦ ♦❢ ✐t ❛s ❛ ❢✉♥❝t✐♦♥✳ ■t ✐s ❣✐✈❡♥ ❜② t❤❡ s❛♠❡ ❢♦r♠✉❧❛✱ ✇✐t❤ x✬s st✐❧❧ ❧✐♠✐t❡❞ t♦ t❤❡ ✐♥t❡❣❡rs✿ p(x) = p0 2x .

◆♦✇✱ ✇❤❛t ✐s t❤❡ ♣♦♣✉❧❛t✐♦♥ ✐♥ t❤❡ ♠✐❞❞❧❡ ♦❢ √ t❤❡ ✜rst ❞❛②❄ ❚♦ ❛♥s✇❡r✱ ✇❡ ❝♦♥s✐❞❡r t❤❡ ❢❛❝t t❤❛t ♠✉❧t✐♣❧②✐♥❣ ❜② 2 ✐s ❡q✉✐✈❛❧❡♥t t♦ ♠✉❧t✐♣❧②✐♥❣ 2 t✇✐❝❡✿ √

2·

√

2 = 2.

❲❡ ❝♦♥❝❧✉❞❡ t❤❛t p(1/2) =

√

2.

❊①❛♠♣❧❡ ✶✳✶✵✳✷✿ ❝♦♠♣♦✉♥❞❡❞ ✐♥t❡r❡st

❈♦♥s✐❞❡r ✇❤❛t ❤❛♣♣❡♥s t♦ ❛ $1000 ❞❡♣♦s✐t ✇✐t❤ 10% ❛♥♥✉❛❧ ✐♥t❡r❡st✱ ❝♦♠♣♦✉♥❞❡❞ ②❡❛r❧②✳ ■t✬s ❛ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥❀ ❛❢t❡r x ②❡❛rs ✇❡ ❤❛✈❡✿ f (x) = 1000 · 1.1x .

✇❤❡r❡ x ✐s ❛ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r✳ ❇✉t ✇❤❛t ✐❢ ■ ✇❛♥t t♦ ✇✐t❤❞r❛✇ ♠② ♠♦♥❡② ✐♥ t❤❡ ♠✐❞❞❧❡ ♦❢ t❤❡ ②❡❛r❄ ■t ✇♦✉❧❞ ❜❡ ❢❛✐r t♦ ❛s❦ t❤❡ ❜❛♥❦ ❢♦r t❤❡ ✐♥t❡r❡st t♦ ❜❡ ❝♦♠♣♦✉♥❞❡❞ ♥♦✇✳ ■t ✇♦✉❧❞ ❛❧s♦ ❜❡ ❢❛✐r ❢♦r t❤❡ ❜❛♥❦ t♦ ❞♦ ✐t ✐♥ s✉❝❤ ❛ ✇❛② t❤❛t t❤❡ ❛♥♥✉❛❧ r❡t✉r♥ r❡♠❛✐♥s t❤❡ s❛♠❡ ❡✈❡♥ ✐❢ ✇❡ ❝♦♠♣♦✉♥❞ t✇✐❝❡✳ ❲❤❛t s❤♦✉❧❞ ❜❡ t❤❡ s❡♠✐✲❛♥♥✉❛❧ ✐♥t❡r❡st r❛t❡❄ ❙✉♣♣♦s❡ t❤❡ ❛♠♦✉♥t ✐s t♦ ❣r♦✇ ❜② ❛ ♣r♦♣♦rt✐♦♥✱ r✳ ❚❤❡♥✱ ✐❢ ❛♣♣❧✐❡❞ ❛❣❛✐♥✱ ✐t ✇✐❧❧ ❣✐✈❡ ♠❡ t❤❡ s❛♠❡ t❡♥ ♣❡r❝❡♥t ❣r♦✇t❤✦ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿ f (.5) = 1000 · r ❛♥❞ r · r = 1.1 .

❚❤❡r❡❢♦r❡✱ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ sq✉❛r❡ r♦♦t✱ ✇❡ ❤❛✈❡ r=

√

1.1 ≈ 1.0488 ,

♦r ❛❜♦✉t 4.9 ♣❡r❝❡♥t✳ ❊①❡r❝✐s❡ ✶✳✶✵✳✸

■❢ ②♦✉r ❜❛♥❦ ♣r♦♠✐s❡s t♦ ♣❛② ②♦✉ 1% ❢♦r t❤❡ ✜rst ②❡❛r ❛♥❞ 2% ❢♦r t❤❡ s❡❝♦♥❞✱ ✇❤❛t s✐♥❣❧❡ ✭❛♥♥✉❛❧✮ ✐♥t❡r❡st ❝❛♥ ✐t ♦✛❡r ✐♥ ♦r❞❡r t♦ ♣❛② ②♦✉ ❛s ♠✉❝❤ ♦✈❡r t❤❡ ♥❡①t t✇♦ ②❡❛rs❄ ❊①♣❧❛✐♥ t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ ❛✈❡r❛❣❡ ✐♥t❡r❡st r❛t❡✳

✶✳✶✵✳

✾✸

❚❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥s

❊①❛♠♣❧❡ ✶✳✶✵✳✹✿ r❛❞✐♦❛❝t✐✈❡ ❞❡❝❛② ❛♥❞ r❛❞✐♦❝❛r❜♦♥ ❞❛t✐♥❣

❚❤❡ r❛❞✐♦❛❝t✐✈❡ ❝❛r❜♦♥ ❧♦s❡s ❤❛❧❢ ♦❢ ✐ts ♠❛ss ♦✈❡r ❛ ❝❡rt❛✐♥ ♣❡r✐♦❞ ♦❢ t✐♠❡ ❝❛❧❧❡❞ t❤❡ ❡❧❡♠❡♥t✳ ■t✬s ❛ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥ ❛❣❛✐♥✿ an+1 = an ·

❤❛❧❢✲❧✐❢❡

♦❢ t❤❡

1 . 2

❯♥❢♦rt✉♥❛t❡❧②✱ n ✐s ♥♦t t❤❡ ♥✉♠❜❡r ♦❢ ②❡❛rs ❜✉t t❤❡ ♥✉♠❜❡r ♦❢ ❤❛❧❢✲❧✐✈❡s✦ ❋♦r ❡①❛♠♣❧❡✱ t❤❡ ♣❡r❝❡♥t❛❣❡ ♦❢ t❤✐s ❡❧❡♠❡♥t✱ 14 ❈✱ ❧❡❢t ✐s ♣❧♦tt❡❞ ❜❡❧♦✇ ❛❣❛✐♥st t✐♠❡✿

❍♦✇❡✈❡r✱ ✇❡ ♦♥❧② ❦♥♦✇ t✇♦ ♣♦✐♥ts ♦♥ t❤❡ ❣r❛♣❤✦ ❙✉♣♣♦s❡ t❤❡ ❤❛❧❢✲❧✐❢❡ ✐s 5730 ②❡❛rs ✭✐✳❡✳✱ t❤❡ t✐♠❡ ✐t t❛❦❡s t♦ ❣♦ ❢r♦♠ 100% t♦ 50%✮✳ ❚❤❡ ♠♦❞❡❧ ♠❡❛s✉r❡s t✐♠❡ ✐♥ ♠✉❧t✐♣❧❡s ♦❢ t❤❡ ❤❛❧❢✲❧✐❢❡✱ 5730 ②❡❛rs✱ ❛♥❞ ❛♥② ♣❡r✐♦❞ s❤♦rt❡r t❤❛♥ t❤❛t ✇✐❧❧ r❡q✉✐r❡ ❛ ♥❡✇ ✐♥s✐❣❤t✳ ❍♦✇ ♠✉❝❤ ✐s ❧❡❢t ❛❢t❡r 5730/2 = 2865 ②❡❛rs❄ ❚❤❡ ❛♥s✇❡r ✐s ❜❡❧♦✇ 75%✿ r 1 ≈ .707 . 2

❲❡ ✜❧❧ t❤❡ ❣❛♣s ✐♥ t❤✐s ♠❛♥♥❡r✿

❋♦r ❛♥② t✇♦ ♥✉♠❜❡rs a ❛♥❞ b✱ ✇❡ ❦♥♦✇ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❡①♣♦♥❡♥t ❢♦r t❤❡ ♥✉♠❜❡r ❤❛❧❢✲✇❛② ❜❡t✇❡❡♥ t❤❡♠✳ ❋♦r ❡①❛♠♣❧❡✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿ x=

√ a+b =⇒ 2x = 2a · 2b . 2

❲❤❛t ✐❢ ✇❡ ❝♦♥t✐♥✉❡ t♦ ♣r♦❞✉❝❡ ♠♦r❡ ❛♥❞ ♠♦r❡ ✈❛❧✉❡s ♦❢ t❤✐s ❢✉♥❝t✐♦♥ ❜② ❞✐✈✐❞✐♥❣ t❤❡ ✐♥t❡r✈❛❧s ✐♥ ❤❛❧❢❄ ❚❤❡ ♥❡✇ ✈❛❧✉❡ ✇✐❧❧ ❛❧✇❛②s ❧✐❡ s❧✐❣❤t❧② ❜❡❧♦✇ t❤❡ ❧✐♥❡ t❤❛t ❝♦♥♥❡❝ts t❤❡ t✇♦ ♣♦✐♥ts ♦♥ t❤❡ ❣r❛♣❤✿

❖♥❡ ❝❛♥ ❛❧s♦ ✐♠❛❣✐♥❡ t❤❛t t❤❡ x✲❛①✐s✱ ❛s t❤❡ ❞♦♠❛✐♥✱ ✐s ❜❡❝♦♠✐♥❣ ♠♦r❡ ❛♥❞ ♠♦r❡ ❞❡♥s❡❧② ❝♦✈❡r❡❞✳ ❚❤❡s❡ ✐♥✐t✐❛❧❧② ❧♦♦s❡ ♣♦✐♥ts st❛rt t♦ ❢♦r♠ ❛ ❝✉r✈❡✿

✶✳✶✵✳

❚❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥s

✾✹

❚❤❡ ❝✉r✈❡ ✇✐❧❧ ❧✐❡ ❜❡❧♦✇ ❛♥② ❝❤♦r❞ t❤❛t ❝♦♥♥❡❝ts t✇♦ ♣♦✐♥ts ♦♥ t❤❡ ❣r❛♣❤✳ ❙✉❝❤ ❛ ❢✉♥❝t✐♦♥ ✐s ❝❛❧❧❡❞ ✏❝♦♥❝❛✈❡ ✉♣✑ ✭❈❤❛♣t❡r ✸✮✳ ▼♦r❡ ❣❡♥❡r❛❧❧②✱ ✐❢

x=

m ✱ n

✇❤❡r❡

m, n

❛r❡ ✐♥t❡❣❡rs ✇✐t❤

n > 0✱

t❤❡♥ ✇❡ s❡t t❤❡

xt❤

♣♦✇❡r

♦❢

b>0

t♦ ❜❡ t❤❡

❢♦❧❧♦✇✐♥❣✿ m

bn =

√ n

bm

❇❡❧♦✇ ✐s t❤❡ s✉♠♠❛r② ♦❢ t❤❡ r✉❧❡s ♦❢ ❡①♣♦♥❡♥ts t❤❛t ✇✐❧❧ r❡❛♣♣❡❛r ♠❛♥② t✐♠❡s✿ ❚❤❡♦r❡♠ ✶✳✶✵✳✺✿ ❘✉❧❡s ♦❢ ❊①♣♦♥❡♥ts

❚❤❡ ✐❞❡♥t✐t✐❡s ❜❡❧♦✇ ❛r❡ s❛t✐s✜❡❞ ❢♦r ❡❛❝❤ r❛t✐♦♥❛❧ a > 0 ❛♥❞ ❡✈❡r② r❡❛❧ x ❛♥❞ y ✿ 1. bx+y = bx by 2. bx cx = (bc)x 3. bxy = (bx )y

❲❤❛t ❞♦❡s t❤❡ ❣r❛♣❤ ♦❢ t❤✐s ♥❡✇ ❢✉♥❝t✐♦♥ ❧♦♦❦ ❧✐❦❡❄ ❲❡ ❤❛✈❡ s❡❡♥ t❤❛t✱ ❛s t❤❡s❡ ❢r❛❝t✐♦♥s ❣❡t ❧❛r❣❡r ❛♥❞ ❧❛r❣❡r ❞❡♥♦♠✐♥❛t♦rs✱ t❤❡ ❞♦♠❛✐♥ ❜❡❝♦♠❡s ❞❡♥s❡r ❛♥❞ ❞❡♥s❡r✱ s♦ ❞♦❡s t❤❡ ❣r❛♣❤ ❛♥❞✱ ❡✈❡♥t✉❛❧❧②✱ ✐t ❜❡❝♦♠❡s ❛ ❝✉r✈❡✦ ❆❢t❡r ❛❧❧✱ t❤❡ ❞♦♠❛✐♥ ❝♦♥t❛✐♥s ❛❧❧ r❛t✐♦♥❛❧ ♥✉♠❜❡rs

Q

❛♥❞ t❤❡ r❛t✐♦♥❛❧ ♥✉♠❜❡rs ❛r❡ s♦ ❞❡♥s❡ t❤❛t

❛♥② ✐♥t❡r✈❛❧✱ ♥♦ ♠❛tt❡r ❤♦✇ s♠❛❧❧✱ ✇✐❧❧ ❝♦♥t❛✐♥ ✐♥✜♥✐t❡❧② ♠❛♥② ♦❢ t❤❡♠ ✭✐♥❞❡❡❞✱ ✐❢

p ❛♥❞ q

❛r❡ r❛t✐♦♥❛❧✱ t❤❡♥

♥♦ ❣❛♣s ✐♥ t❤❡ ❣r❛♣❤✦ ❍♦✇❡✈❡r✱ ✐ts ♣♦✐♥ts r❡♠❛✐♥ ❞✐s❝♦♥♥❡❝t❡❞ ❢r♦♠ ❡❛❝❤ ♦t❤❡r❀ ❛ ❧✐tt❧❡ ❜❧♦✇ ❛♥❞ t❤❡ ❝✉r✈❡ ❢❛❧❧s ❛♣❛rt✿

s♦ ✐s

(p + q)/2✮✳

❚❤❡r❡❢♦r❡✱ t❤❡r❡ ❛r❡

❚❤❡ r❡❛s♦♥ ✐s t❤❛t t❤❡ ♣♦✐♥ts ♦♥ t❤❡ ❣r❛♣❤ ✇✐t❤

✐rr❛t✐♦♥❛❧

x= ❆♥❞ t❤❡ ✐rr❛t✐♦♥❛❧ ♥✉♠❜❡rs ❛r❡ ❛❧s♦ s♦

√

2,

√

x✲❝♦♦r❞✐♥❛t❡s

❛r❡ ♠✐ss✐♥❣✿

3, π .

❞❡♥s❡ t❤❛t ❛♥② ✐♥t❡r✈❛❧✱ ♥♦ ♠❛tt❡r ❤♦✇ s♠❛❧❧✱ ✇✐❧❧ ❝♦♥t❛✐♥ ✐♥✜♥✐t❡❧②

♠❛♥② ♦❢ t❤❡♠✳ ❙♦✱ ❡✈❡♥ t❤♦✉❣❤ t❤❡r❡ ❛r❡ ♥♦ ❣❛♣s ✐♥ t❤❡ ❣r❛♣❤✱ t❤❡r❡ ❛r❡ ✐♥✈✐s✐❜❧❡ ✏❝✉ts✑ ❡✈❡r②✇❤❡r❡✦ ❲❡ ✇✐❧❧ ♠❛❦❡ ❛ st❡♣ t♦✇❛r❞ ❞❡✜♥✐♥❣ t❤❡ ❢✉♥❝t✐♦♥ ❢♦r t❤❡ ✐rr❛t✐♦♥❛❧ ❡①♣♦♥❡♥ts ❜② ✏❛♣♣r♦①✐♠❛t✐♥❣✑ t❤❡♠ ✇✐t❤ r❛t✐♦♥❛❧ ♥✉♠❜❡rs✳

✶✳✶✵✳

❚❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥s

✾✺

❚❤❡ st❛rt✐♥❣ ♣♦✐♥t ✇✐❧❧ ❜❡ t♦ ❝♦♥s✐❞❡r ✇❤❛t ✐s ❤❛♣♣❡♥✐♥❣ ❛r♦✉♥❞ x = 0✳ ❲❡ ❝❛♥ s❡❡ ❛ s❡q✉❡♥❝❡ ♦♥ t❤❡ x✲❛①✐s t❤❛t ❝♦♥✈❡r❣❡s t♦ 0 ❛♥❞ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s❡q✉❡♥❝❡ ♦♥ t❤❡ y ✲❛①✐s ❝♦♥✈❡r❣❡s t♦ 1✿

❚❤❡r❡ ❛r❡ ♥♦ ❥✉♠♣s✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿ ❚❤❡♦r❡♠ ✶✳✶✵✳✻✿ ❊①♣♦♥❡♥t✐❛❧ ❋✉♥❝t✐♦♥ ❆r♦✉♥❞ ❋♦r ❡❛❝❤ r❡❛❧

b > 0✱

0

✇❡ ❤❛✈❡✿

lim bxn = 1

n→∞ ❢♦r ❡✈❡r② s❡q✉❡♥❝❡ ♦❢ ♥✉♠❜❡rs

xn → 0 ✳

Pr♦♦❢✳

▲❡t✬s ❝♦♥s✐❞❡r t❤❡ s✐♠♣❧✐✜❡❞ ✈❡rs✐♦♥ ♦❢ t❤❡ st❛t❡♠❡♥t✿ lim a1/qn = 1 ,

n→∞

✇❤❡r❡ qn ✐s ❛ s❡q✉❡♥❝❡ ♦❢ ✐♥t❡❣❡rs ✇✐t❤ qn → ∞✳ ❚♦ ♣r♦✈❡ t❤❡ ❢♦r♠✉❧❛✱ ✇❡ ♥❡❡❞ t♦ ✜♥❞ ♦✉t ❢♦r ✇❤❛t ✈❛❧✉❡s ♦❢ n t❤❡ ✐♥❡q✉❛❧✐t② ❜❡❧♦✇ ✐s s❛t✐s✜❡❞ ❢♦r ❛♥② ❣✐✈❡♥ ε > 0✿ |

√

qn

b − 1| < ε .

❋♦r b > 1✱ t❤❡ ✐♥❡q✉❛❧✐t② ✐s ❢✉rt❤❡r s✐♠♣❧✐✜❡❞✿ √

qn

b − 1 < ε ⇐⇒

√

qn

b < 1 + ε ⇐⇒ b < (1 + ε)qn .

◆♦✇ ✇❡ ❝❛♥ ❛♥s✇❡r t❤❡ q✉❡st✐♦♥✳ ❲❡ ✉s❡ t❤✐s ❢❛❝t✿ qn → ∞ =⇒ (1 + ε)qn → ∞ .

■t ❢♦❧❧♦✇s t❤❛t t❤❡r❡ ✐s s✉❝❤ ❛♥ N t❤❛t t❤❡ ❧❛st ✐♥❡q✉❛❧✐t② ✐s s❛t✐s✜❡❞ ❢♦r ❛❧❧ n > N ✳ ❚❤❡♥ s♦ ✐s t❤❡ ♦r✐❣✐♥❛❧ ✐♥❡q✉❛❧✐t②✳ ❊①❡r❝✐s❡ ✶✳✶✵✳✼

Pr♦✈✐❞❡ t❤❡ ♠✐ss✐♥❣ ♣❛rts ♦❢ t❤❡ ♣r♦♦❢✳ ❏✉st ❛s ✇❡ ♥❡❡❞ ♦♥❧② ♦♥❡ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧✱ y = x2 ✱ ❛s ❛ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥✦

t❡♠♣❧❛t❡ ❢♦r t❤❡ r❡st✱ ✇❡ r❡❛❧❧② ♦♥❧② ♥❡❡❞ ♦♥❡

✶✳✶✵✳

❚❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥s

❲❤✐❝❤ ♦♥❡ ✐s t❤❡

♦♥❡

✾✻

❄

❊①❛♠♣❧❡ ✶✳✶✵✳✽✿ ❝♦♠♣♦✉♥❞❡❞ ✐♥t❡r❡st ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ♠♦♥❡② ✐♥ t❤❡ ❜❛♥❦ ❛t

$1, 000

10%

❆P❘ ❝♦♠♣♦✉♥❞❡❞ ❛♥♥✉❛❧❧②✳

❚❤❡♥ ❛❢t❡r ❛ ②❡❛r✱ ❣✐✈❡♥

✐♥✐t✐❛❧ ❞❡♣♦s✐t✱ ②♦✉ ❤❛✈❡✿

1000 + 1000 · 0.10 = 1000(1 + 0.1) = 1000 · 1.1 . ❙❛♠❡ ❡✈❡r② ②❡❛r✳ ❆❢t❡r

t

②❡❛rs✱ ✐t✬s

1000 · 1.1t ✳

❲❤❛t ✐❢ ✇❡ ✇❛♥t t♦ ❝♦♠♣♦✉♥❞ ♠♦r❡ ♦❢t❡♥ ✇✐t❤ t❤❡ ❣♦❛❧ ♦❢ r❡❝❡✐✈✐♥❣ t❤❡ s❛♠❡ ✐♥t❡r❡st ❛t t❤❡ ❡♥❞ ♦❢ t❤❡ ②❡❛r❄ ❚❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥ ✐s ❞❡s✐❣♥❡❞ t♦ s♦❧✈❡ t❤✐s ♣r♦❜❧❡♠✳

❙♦✱ t❤❡ ❛❝❝♦✉♥t ✐s ❝♦♠♣♦✉♥❞❡❞ s❡♠✐✲❛♥♥✉❛❧❧②✱ ✇✐t❤ t❤❡ s❛♠❡ ❆P❘ ❛s ❢♦❧❧♦✇s✳

1000 · 0.05✱

t

1 2

②❡❛r✱ ✐t✬s

♦r ❛ t♦t❛❧ ♦❢✿

❆❢t❡r ❛♥♦t❤❡r

❆❢t❡r

❆❢t❡r

1 2

1000 + 1000 · 0.05 = 1000 · 1.05 . ②❡❛r✱ ✐t ✐s

(1000 · 1.05) · 1.05 = 1000 · 1.052 .

②❡❛rs✱

1000 · (1.052 )t = 1000 · 1.052t . ◆♦t❡ t❤❛t ✇❡ ❛r❡ ❣❡tt✐♥❣ ♠♦r❡ ♠♦♥❡② t❤❛♥ ❜❡❢♦r❡✿

1.052 = 1.1025 > 1.1 ✦

◆❡①t✱ ✇❡ tr② t♦ ❝♦♠♣♦✉♥❞ q✉❛rt❡r❧②✱

1000 · 1.0254t . ❆♥❞ s♦ ♦♥✳

5/4 = 1.25%

5% ❝♦♠♣♦✉♥❞❡❞ 1 t✐♠❡❀ 5/2 = 2.5% 5/8 = .635% ❝♦♠♣♦✉♥❞❡❞ 8 t✐♠❡s✿

❇❡❧♦✇ ✇❡ s❡❡ r❡s♣❡❝t✐✈❡❧②✿ ❝♦♠♣♦✉♥❞❡❞

4

t✐♠❡s❀

❲❤❡♥ t❤❡ ✐♥t❡r❡st ✐s ❝♦♠♣♦✉♥❞❡❞

n

1 1000 · 1 + n 1 n

2

t✐♠❡s❀

t✐♠❡s✱ ✇❡ ❤❛✈❡✿



✇❤❡r❡

❝♦♠♣♦✉♥❞❡❞

nt

,

✐s t❤❡ ✐♥t❡r❡st ✐♥ ♦♥❡ ♣❡r✐♦❞✳ ■s t❤❡r❡ ❛ ❜♦✉♥❞ t♦ t❤✐s ❣r♦✇t❤❄

❊①❡r❝✐s❡ ✶✳✶✵✳✾ ❲❤✐❝❤ ♦❢ t❤❡ ❢♦✉r ❛r❡❛s ✐s t❤❡ ❜✐❣❣❡st❄

●❡♥❡r❛❧❧②✱ ❢♦r ❆P❘

✐❢ ❝♦♠♣♦✉♥❞❡❞

n

r

✭❣✐✈❡♥ ❛s ❛ ❞❡❝✐♠❛❧✮ ❛♥❞ ❢♦r t❤❡ ✐♥✐t✐❛❧ ❞❡♣♦s✐t

 r nt , A(t) = A0 1 + n t✐♠❡s ♣❡r ②❡❛r✳

A0 ✱

❛❢t❡r

t ②❡❛rs✱ t❤❡ ❝✉rr❡♥t ❛♠♦✉♥t ✐s

✶✳✶✵✳

❚❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥s

✾✼

❲❤❛t ✐❢ ✇❡ ❝♦♠♣♦✉♥❞❡❞ ♠♦r❡ ❛♥❞ ♠♦r❡ ❢r❡q✉❡♥t❧②❄ ❲✐❧❧ ✇❡ ❜❡ ♣❛✐❞ ✉♥❧✐♠✐t❡❞ ❛♠♦✉♥ts❄ ▲❡t✬s ♣✐❝❦ ❛ s♣❡❝✐✜❝

r=1

✭✐✳❡✳✱

100%

❆P❘✮ ❛♥❞ ♣❧♦t t❤✐s s❡q✉❡♥❝❡✳ ■t ❛♣♣❡❛rs t❤❛t t❤❡ ❣r♦✇t❤ s❧♦✇s ❞♦✇♥✿

❚❤❡ ❛♥s✇❡r t♦ t❤❡ q✉❡st✐♦♥ ✐s ◆♦✱ ❛♥❞ ✇❡ ♣r♦✈❡ t❤✐s ❢❛❝t ❜❡❧♦✇✿ ❚❤❡♦r❡♠ ✶✳✶✵✳✶✵✿ ❡ ❛s ▲✐♠✐t ❚❤❡ ❧✐♠✐t ❜❡❧♦✇ ❡①✐sts✿

lim

n→∞



1 1+ n

n

❲❡ ❤❛✈❡ ❛ s♣❡❝✐❛❧ ♥♦t❛t✐♦♥ ❢♦r t❤✐s ♥✉♠❜❡r✿

e = lim

n→∞

e 

1 1+ n

n

■t ✐s ❦♥♦✇♥ ❛s t❤❡ ✏❊✉❧❡r ♥✉♠✲ ❜❡r✑✳

Pr♦♦❢✳

❋✐rst✱ ✇❡ s❤♦✇ t❤❛t t❤❡ s❡q✉❡♥❝❡

an =



1 1+ n

n

✐s ✐♥❝r❡❛s✐♥❣✳ ❲❡ ❤❛✈❡✿


n+1
1 n+2 n+1 1 + n+1 an+1 n+1
n = n+1
n = an 1 + n1 n n+1  1  n+1  n n+1 n+2 = n+1 n+1 n  2 n+1 n + 2n + 1 − 1 n+1 = 2 n + 2n + 1 n n+1  n+1 1 . = 1− (n + 1)2 n

❋r♦♠ t❤❡

❇✐♥♦♠✐❛❧ ❋♦r♠✉❧❛ ✭s❡❡♥ ✐♥ ❱♦❧✉♠❡ ✶✱ ❈❤❛♣t❡r ✶P❈✲✶✮✱ ✐t ❢♦❧❧♦✇s t❤❛t (1 + b)m > 1 + mb ,

❢♦r ❛♥②

b✳

❲❡ ❥✉st ❝❤♦♦s❡✿

b=

−1 (n + 1)2

❛♥❞

m = n + 1,

✶✳✶✵✳

❚❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥s

✾✽

❛♥❞ t❤❡♥ s✉❜st✐t✉t❡ t❤❡ ✐♥❡q✉❛❧✐t② ✐♥t♦ t❤❡ ✜rst t❡r♠ ♦❢ t❤❡ ❧❛st ❢♦r♠✉❧❛✿ n+1 an+1 = (1 + b)m · an n n+1 > (1 + mb) · n n+1  n+1 1 = 1− 2 (n + 1) n   1 n+1 = 1− n+1 n n n+1 = n+1 n = 1.

■♥ ❛ s✐♠✐❧❛r ❢❛s❤✐♦♥✱ ✇❡ s❤♦✇ t❤❛t t❤❡ s❡q✉❡♥❝❡ bn =



1 1+ n

n+1

✐s ❞❡❝r❡❛s✐♥❣✳ ❙✐♥❝❡ an < bn ✱ ✇❡ ❝♦♥❝❧✉❞❡ t❤❛t ❢♦r♠❡r s❡q✉❡♥❝❡ ✐s ❜♦t❤ ✐♥❝r❡❛s✐♥❣ ❛♥❞ ❜♦✉♥❞❡❞✳ ❚❤❡r❡❢♦r❡✱ ✐t ❝♦♥✈❡r❣❡s ❜② t❤❡ ▼♦♥♦t♦♥❡ ❈♦♥✈❡r❣❡♥❝❡ ❚❤❡♦r❡♠✳ ❚❤❡ ✈❛❧✉❡ ♦❢ t❤✐s s♣❡❝✐❛❧ ♥✉♠❜❡r ✐s ❛♣♣r♦①✐♠❛t❡❧② t❤❡ ❢♦❧❧♦✇✐♥❣✿ e ≈ 2.71828 .

■❢ ②♦✉ ❤❛✈❡ 100% ❆P❘✱ t❤✐s ✐s t❤❡ ♠♦st ②♦✉ ❝❛♥ ❣❡t ❢♦r ②♦✉r ❞♦❧❧❛r ♥♦ ♠❛tt❡r ❤♦✇ ❢r❡q✉❡♥t❧② t❤❡ ✐♥t❡r❡st ✐s ❝♦♠♣♦✉♥❞❡❞✳ ❲❡ ❤❛✈❡ ❞❡♠♦♥str❛t❡❞ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❝❡♣t✿

❉❡✜♥✐t✐♦♥ ✶✳✶✵✳✶✶✿ ♥❛t✉r❛❧ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥ ❚❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥ ex ✇✐t❤ ❜❛s❡ e✱ t❤❡ ❊✉❧❡r ♥✉♠❜❡r✱ ✐s ❝❛❧❧❡❞ t❤❡ ♥❛t✉r❛❧ ❜❛s❡ ❡①♣♦♥❡♥t✳ ■♥ ♦r❞❡r t♦ ✉s❡ t❤✐s ❢✉♥❝t✐♦♥✱ ✇❡ ✇✐❧❧ ♥❡❡❞ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡s✉❧t t❤❛t ✇❡ ✇✐❧❧ ❛❝❝❡♣t ✇✐t❤♦✉t ♣r♦♦❢✿

❚❤❡♦r❡♠ ✶✳✶✵✳✶✷✿ ❊①♣♦♥❡♥t✐❛❧ ❋✉♥❝t✐♦♥ ❛s ▲✐♠✐t ex = lim

n→∞



1+

x n n

❊①❛♠♣❧❡ ✶✳✶✵✳✶✸✿ ❝♦♥t✐♥✉♦✉s❧② ❝♦♠♣♦✉♥❞✐♥❣ ✐♥t❡r❡st ❲❡ ❝♦♥t✐♥✉❡ ✇✐t❤ t❤❡ ❡①❛♠♣❧❡✿ ❲❤❛t ✐❢ t❤❡ ✐♥t❡r❡st ✐s ❝♦♠♣♦✉♥❞❡❞ n t✐♠❡s ❛♥❞ n → ∞❄ ❚❤❡♥ ✇❡ ❤❛✈❡✿  r nt lim A(x) = lim A0 1 + n→∞ n→∞ n  r nt = A0 lim 1 + n→∞ n h  r n it = A0 lim 1 + n→∞ n = A0 (er )t .

❚❤✐s ✐s ❜② ❈▼❘✳ ❚❤✐s ✐s ❜② t❤❡ t❤❡♦r❡♠✳

✶✳✶✵✳

❚❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥s

❚❤✉s✱ ✇✐t❤ ❛♥ ❆P❘ ♦❢

r

✾✾

❛♥❞ ❛♥ ✐♥✐t✐❛❧ ❞❡♣♦s✐t

A0 ✱

❛❢t❡r

t

②❡❛rs ②♦✉ ❤❛✈❡✿

A(t) = A0 ert . ❲❡ s❛② t❤❛t t❤❡ ✐♥t❡r❡st ✐s ❝♦♠♣♦✉♥❞❡❞

❙✉♣♣♦s❡ t❤❡ ❆P❘ ✐s

10%✱

❛♥❞

❝♦♥t✐♥✉♦✉s❧②

A0 = 1000✱ x = 1✳

✳

❚❤❡♥✿

A(1) = 1000 · e1.1 = 1000 · e0.1 ≈ $1, 105 . ❚❤❡ ✐♥t❡r❡st ✐s

$105 > $100✳ 5%✱ ❝♦♠♣♦✉♥❞❡❞ ❝♦♥t✐♥✉♦✉s❧②❄

❍♦✇ ❧♦♥❣ ❞♦❡s ✐t t❛❦❡ t♦ tr✐♣❧❡ ②♦✉r ♠♦♥❡② ✇✐t❤ t❤❡ ❆P❘❂ ❛♥❞ s♦❧✈❡ ❢♦r

t✿

3 = 1 · e0.05t =⇒ ln 3 = 0.05t =⇒ ln 3 t = ≈ 22 0.05

❙❡t

A0 = 1

②❡❛rs✳

❚❤❡r❡ ✐s ❛ ♠♦r❡ s✉❜t❧❡ ♦❜s❡r✈❛t✐♦♥ ❛❜♦✉t t❤❡ ❣r❛♣❤ ♦❢ t❤✐s ❢✉♥❝t✐♦♥✳ ❲❤❛t ♠❛❦❡s

e s♣❡❝✐❛❧ ✐s t❤✐s ♣r♦♣❡rt②✿

■♥ ❢❛❝t✱ ♣❧♦tt✐♥❣ t❤❡ ♣♦✐♥ts

(1/n, e1/n )

❚❤❡ ❣r❛♣❤ ♦❢

y = ex

❛❧♠♦st ♠❡r❣❡s ✇✐t❤ t❤❡ ❧✐♥❡

r❡✈❡❛❧s ❛ str❛✐❣❤t ❧✐♥❡ ✇✐t❤ s❧♦♣❡

❚❤✐s ✐s ❤♦✇ ✇❡ st❛t❡ t❤✐s ❢❛❝t ❛❧❣❡❜r❛✐❝❛❧❧②✿

1✿

y = x + 1 ❛r♦✉♥❞ 0✿

✶✳✶✶✳

❚❤❡ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s

✶✵✵

❚❤❡♦r❡♠ ✶✳✶✵✳✶✹✿ ❋❛♠♦✉s ▲✐♠✐t✿ ❲❡ ❤❛✈❡✿

ex

e xn − 1 =1 n→∞ xn lim

❢♦r ❛♥② s❡q✉❡♥❝❡

❙♦✱

y = ex

xn → 0 ✳

❝✉ts✱ ✐♥ ❛ s❡♥s❡✱ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥s ✐♥t♦ t✇♦ ❤❛❧✈❡s✿ t❤❡ ✏st❡❡♣✑ ♦♥❡s ❛♥❞ t❤❡ ✏s❤❛❧❧♦✇✑

♦♥❡s✿

✶✳✶✶✳ ❚❤❡ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s

▲❡t✬s r❡✈✐❡✇ ✇❤❛t ✇❡ ❦♥♦✇ ❛❜♦✉t t❤❡s❡ ❢✉♥❝t✐♦♥s ✭❱♦❧✉♠❡ ✶✱ ❈❤❛♣t❡r ✶P❈✲✹✮✳ ❖♥❡ ❡♥❝♦✉♥t❡rs ♥✉♠❡r♦✉s ❡①❛♠♣❧❡s ♦❢

♣❡r✐♦❞✐❝ ♣❤❡♥♦♠❡♥❛✳

❚❤❡ s✐♠♣❧❡st ❝❛s❡ ✐s t❤❛t ♦❢ ❛ q✉❛♥t✐t② t❤❛t

❝❤❛♥❣❡s ❜✉t t❤❡♥ ❝♦♠❡s ❜❛❝❦ t♦ ❝❤❛♥❣❡ ❛❣❛✐♥ ✐♥ t❤❡ s❛♠❡ ♠❛♥♥❡r✿

❋✉♥❝t✐♦♥s ✇✐t❤ t❤✐s ❜❡❤❛✈✐♦r ❛r❡ ❝❛❧❧❡❞

♣❡r✐♦❞✐❝✳

❊①❛♠♣❧❡ ✶✳✶✶✳✶✿ ♣❡r✐♦❞✐❝ ❜❡❤❛✈✐♦r

❚❤❡ s✐♠♣❧❡st ♣❡r✐♦❞✐❝ ❜❡❤❛✈✐♦r ✐s ♦s❝✐❧❧❛t✐♦♥ ♦❢ ❛♥ ♦❜❥❡❝t ♦♥ ❛ s♣r✐♥❣ ♦r ❛ str✐♥❣ ♦❢ ❛ ♠✉s✐❝❛❧ ✐♥str✉♠❡♥t✿

❆ ♠♦r❡ ❝♦♠♣❧❡① ❡①❛♠♣❧❡ ✐s t❤❡ tr✐♣ ♦❢ t❤❡ ♠♦♦♥ ❛r♦✉♥❞ t❤❡ s✉♥✳

❚❤❡ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s ✐♥✐t✐❛❧❧② ❝♦♠❡ ❢r♦♠

♣❧❛♥❡ ❣❡♦♠❡tr② ✿

✶✳✶✶✳

❚❤❡ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s

✶✵✶

❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ r✐❣❤t tr✐❛♥❣❧❡ ✇✐t❤ s✐❞❡s a, b, c✱ ✇✐t❤ c ❜❡✐♥❣ t❤❡ ❧♦♥❣❡st ♦♥❡ ❢❛❝✐♥❣ t❤❡ r✐❣❤t ❛♥❣❧❡✳ ■❢ α ✐s t❤❡ ❛♥❣❧❡ ❛❞❥❛❝❡♥t t♦ s✐❞❡ a✱ t❤❡♥ ✇❡ ❞❡✜♥❡ t❤❡ ❝♦s✐♥❡ ❛♥❞ t❤❡ s✐♥❡ ♦❢ t❤✐s ❛♥❣❧❡ ❛s ❢♦❧❧♦✇s✿

a c b sin α = c b sin α tan α = = a cos α cos α =

❚❤❡ ✐♠♣♦rt❛♥❝❡ ♦❢ t❤❡ t❛♥❣❡♥t ✐s s❡❡♥ ✐♥ t❤✐s ❢♦r♠✉❧❛✿

tan α =

r✐s❡ b = = s❧♦♣❡ ♦❢ t❤❡ ❤②♣♦t❡♥✉s❡ c , a r✉♥

✐❢ s✐❞❡ a ❢♦❧❧♦✇s t❤❡ x✲❛①✐s✿

❙❧♦♣❡s ❛r❡ ❢✉rt❤❡r ❞✐s❝✉ss❡❞ ✐♥ ❈❤❛♣t❡r ✸✳ ❲❡ ❛❞♦♣t t❤❡ t♦ π r❛❞✐❛♥s ✿

❝♦♥✈❡♥t✐♦♥

t❤❛t t❤❡ s✐③❡ ♦❢ t❤❡ ❤❛❧❢✲t✉r♥ ❛♥❣❧❡ ✕ ❛♥❞ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❤❛❧❢✲❝✐r❝❧❡ ✕ ✐s ❡q✉❛❧

180 ❞❡❣r❡❡s = π r❛❞✐❛♥s

❲❡✱ t❤❡r❡❢♦r❡✱ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥✈❡rs✐♦♥ ❢♦r♠✉❧❛s ❢♦r q✉❛♥t✐t✐❡s ♠❡❛s✉r❡❞ ✐♥ t❤❡s❡ ✉♥✐ts✿

# ❞❡❣r❡❡s = # ♦❢ r❛❞✐❛♥s ·

180 π

❊①❛♠♣❧❡ ✶✳✶✶✳✷✿ s❤❛❞♦✇

❲❡ ❤❛✈❡ ♦t❤❡r ✐♥t❡r♣r❡t❛t✐♦♥s ♦❢ t❤❡s❡ q✉❛♥t✐t✐❡s✳ ❙✉♣♣♦s❡ ✇❡ ♣❧❛❝❡ ❛ st✐❝❦ ♦❢ ❧❡♥❣t❤ 1 ❛t t❤❡ ❛♥❣❧❡ α ✇✐t❤ t❤❡ ❣r♦✉♥❞✳

✶✳✶✶✳

❚❤❡ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s

✶✵✷

❚❤❡♥✿ • ❲❡ ❝❛♥ t❤✐♥❦ ♦❢ cos α ❛s t❤❡ ❧❡♥❣t❤ ♦❢ ✐ts s❤❛❞♦✇ ✕ ♦♥ t❤❡ ❣r♦✉♥❞ ❛t ♥♦♦♥ ✭❧❡❢t✮✳ • ❲❡ ❝❛♥ t❤✐♥❦ ♦❢ sin α ❛s t❤❡ ❧❡♥❣t❤ ♦❢ ✐ts s❤❛❞♦✇ ✕ ♦♥ t❤❡ ✇❛❧❧ ❛t s✉♥s❡t ✭r✐❣❤t✮✳ ❆❧t❡r♥❛t✐✈❡❧②✱ t❤❡ st✐❝❦ ✐s ✈❡rt✐❝❛❧ ❛♥❞ st✐❧❧ ❛♥❞ ✐t ✐s t❤❡ s✉♥ t❤❛t ✐s ♠♦✈✐♥❣✿

❊①❛♠♣❧❡ ✶✳✶✶✳✸✿ r♦t❛t✐♥❣ r♦❞

❙✉♣♣♦s❡ ❛ r♦❞ ♦❢ ❧❡♥❣t❤ R ✐s r♦t❛t❡❞ ❛r♦✉♥❞ ✐ts ❡♥❞✳ ■❢ ✇❡ ❝❛♥ ❝♦♥tr♦❧ t❤❡ ❛♥❣❧❡✱ θ✱ t❤❡♥ ✇❤❛t ❞♦ ✇❡ ❦♥♦✇ ❛❜♦✉t t❤❡ ♣♦s✐t✐♦♥ ♦❢ ✐ts ♦t❤❡r ❡♥❞ ✐♥ s♣❛❝❡✱ ✐✳❡✳✱ ✐ts x ❛♥❞ y ❝♦♦r❞✐♥❛t❡s❄

❲❡ ✉s❡ t❤❡ ❛❜♦✈❡ ❢♦r♠✉❧❛s✿

x y ❛♥❞ sin θ = . R R ❚❤❡r❡❢♦r❡✱ ♦✉r ❝♦♦r❞✐♥❛t❡s ❛r❡ ❣✐✈❡♥ ❜② t❤❡s❡ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s✿ cos θ =

x = R cos θ ❛♥❞ y = R sin θ . ❚❤❡ ♠♦✈✐♥❣ ❡♥❞ ♦❢ t❤❡ r♦❞✱ ♦❢ ❝♦✉rs❡✱ tr❛❝❡s ♦✉t ❛

❝✐r❝❧❡ ♦❢ r❛❞✐✉s R✿

❚❤✐s ❛♥❛❧②s✐s ✇✐❧❧ ❜❡ ✉s❡❞ ❛s ❛ ❜❛s✐s ❢♦r t❤❡ ♣♦❧❛r ❝♦♦r❞✐♥❛t❡ s②st❡♠ ✭❱♦❧✉♠❡ ✸✱ ❈❤❛♣t❡r ✸■❈✲✹✮✳ ■♥ ❛ tr✐❛♥❣❧❡✱ t❤❡ ✈❛❧✉❡ ♦❢ ❛♥❣❧❡ α ❝❛♥♥♦t ❣♦ ❜❡②♦♥❞ 90 ❞❡❣r❡❡s ❛♥❞ t❤❡ ❢♦r♠✉❧❛s ❛❜♦✈❡ ❣✐✈❡ ✉s 1/4 ♦❢ t❤❡ ✇❤♦❧❡ ❝✐r❝❧❡✳ ❲❡ ❞❡✜♥❡ t❤❡ ❛♥❣❧❡ ❜❡②♦♥❞ 90 ❞❡❣r❡❡s✿

◮ ❚❤❡ ❛♥❣❧❡ ✐s♥✬t ❛♥ ❛♥❣❧❡ ♦❢ ❛ tr✐❛♥❣❧❡ ❜✉t t❤❡ ❛♥❣❧❡ ♦❢

r♦t❛t✐♦♥✳

❲❡ ✇✐❧❧ ✉s❡ t❤❡ ❝✐r❝❧❡ ♦❢ r❛❞✐✉s 1✳ ❇❡❧♦✇✱ t❤❡ ♥❛♠❡ ❢♦r t❤❡ ✐♥❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡ t❤❛t r❡♣r❡s❡♥ts t❤❡ ❛♥❣❧❡ ❛♥❞ r✉♥s ✇✐t❤✐♥ (−∞, +∞) ✐s x✳ ❚❤❡ ♦✉t❧✐♥❡ ♦❢ t❤❡ ❝♦♥str✉❝t✐♦♥ ✐s s❤♦✇♥ ❜❡❧♦✇✿

✶✳✶✶✳

❚❤❡ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s

✶✵✸

❲❡ ❢♦r♠❛❧✐③❡ t❤✐s ❝♦♥str✉❝t✐♦♥ ❜❡❧♦✇✿

❉❡✜♥✐t✐♦♥ ✶✳✶✶✳✹✿ s✐♥❡ ❛♥❞ ❝♦s✐♥❡ ❢✉♥❝t✐♦♥s ❙✉♣♣♦s❡ ❛ r❡❛❧ ♥✉♠❜❡r x ✐s ❣✐✈❡♥✳ ❲❡ ❝♦♥str✉❝t ❛ ❧✐♥❡ s❡❣♠❡♥t ♦❢ ❧❡♥❣t❤ 1 ♦♥ t❤❡ ❈❛rt❡s✐❛♥ ♣❧❛♥❡ ✭✇✐t❤ t❤❡ ❤♦r✐③♦♥t❛❧ ❛①✐s ♥♦t ♠❛r❦❡❞ x✮ st❛rt✐♥❣ ❛t 0 ✇✐t❤ ❛♥❣❧❡ x r❛❞✐❛♥s ❢r♦♠ t❤❡ ❤♦r✐③♦♥t❛❧✱ ❝♦✉♥t❡r❝❧♦❝❦✇✐s❡✳ ❚❤❡♥✿ • ❚❤❡ ❝♦s✐♥❡ ♦❢ x ✐s t❤❡ ❤♦r✐③♦♥t❛❧ ❝♦♦r❞✐♥❛t❡ ♦❢ t❤❡ ❡♥❞ ♦❢ t❤❡ s❡❣♠❡♥t✳ • ❚❤❡ s✐♥❡ ♦❢ x ✐s t❤❡ ✈❡rt✐❝❛❧ ❝♦♦r❞✐♥❛t❡ ♦❢ t❤❡ ❡♥❞ ♦❢ t❤❡ s❡❣♠❡♥t✳ ❚❤❡ ♣♦s✐t✐✈❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ ❛♥❣❧❡✱ ✇❤✐❝❤ ✐s x✱ ✐s

❝♦✉♥t❡r❝❧♦❝❦✇✐s❡ ✭✜rst r♦✇✮✿

x −2π −3π/2 −π −π/2 0 π/2 π 3π/2 2π ... y 0 1 0 −1 0 1 0 −1 0 ... ❚❤❡ ✈❛❧✉❡s ♦❢ t❤❡ s✐♥❡ ❛r❡ ♣r♦✈✐❞❡❞ ✐♥ t❤❡ s❡❝♦♥❞ r♦✇✳ ❲❡ ♣❧♦t t❤❡ ❣r❛♣❤ ♦❢ y = sin x ✕ ♣♦✐♥t ❜② ♣♦✐♥t ✕ ❢♦❧❧♦✇✐♥❣ t❤✐s t❛❜❧❡✿

❚❤❡r❡ ✐s ❛ s✐♠✐❧❛r t❛❜❧❡ ❢♦r t❤❡ ❝♦s✐♥❡✳ ❍♦✇ ❞♦ ✇❡ ✜❧❧ t❤❡ ❣❛♣s ✐♥ t❤❡ ❣r❛♣❤❄ ❲❡ ❞✐✈✐❞❡ t❤❡ ✐♥t❡r✈❛❧s ✐♥ ❤❛❧❢ ❛♥❞ ❛♣♣❧② t❤❡ ❢♦❧❧♦✇✐♥❣ tr✐❣♦♥♦♠❡tr✐❝ ❢♦r♠✉❧❛ ✭s❡❡♥ ✐♥ ❱♦❧✉♠❡ ✶✱ ❈❤❛♣t❡r ✶P❈✲✺✮✿

sin



α+β 2



=±

r

1 − cos α cos β − sin α sin β 2

❋♦r ❡①❛♠♣❧❡✱ t❤✐s ✐s ❤♦✇ ✇❡ ❞❡✜♥❡❞ t❤❡ s✐♥❡ ✭❛♥❞ t❤❡♥ ❝♦s✐♥❡✮ ♦❢ t❤❡ ❛♥❣❧❡ ❤❛❧❢✇❛② ❜❡t✇❡❡♥ 0 ❛♥❞ π/2✿ r r  r  0 + π/2 1 − cos 0 cos π/2 − sin 0 sin π/2 1 − cos 0 cos π/2 − sin 0 sin π/2 1 π = = = . sin = sin 4 2 2 2 2 ❆t t❤❡ ♥❡①t st❛❣❡✱ ✇❡ t❤❡♥ ❝♦♠♣✉t❡ ✇✐t❤ t❤❡ s❛♠❡ ❢♦r♠✉❧❛ t❤❡ s✐♥❡ ✭❛♥❞ t❤❡♥ ❝♦s✐♥❡✮ ♦❢ t❤❡ ❛♥❣❧❡ ❤❛❧❢ ✇❛② ❜❡t✇❡❡♥ 0 ❛♥❞ π/4✿   π 0 + π/4 sin = sin = ... 8 2 ❆♥❞ s♦ ♦♥✳ ❲❡ ❝♦♥t✐♥✉❡ t♦ ❞✐✈✐❞❡ t❤❡ ✐♥t❡r✈❛❧s ✐♥ ❤❛❧❢✱ ♣r♦❞✉❝✐♥❣ ♠♦r❡ ❛♥❞ ♠♦r❡ ♣♦✐♥ts ♦♥ t❤❡ ❣r❛♣❤✿

✶✳✶✶✳

❚❤❡ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s

❚❤❡ st❡♣s ❛❜♦✈❡✱ r❡s♣❡❝t✐✈❡❧②✱ ❛r❡✿

✶✵✹

π/2✱ π/4✱ π/8✱

❛♥❞

π/16✳

❲✐t❤ ♠♦r❡ ✈❛❧✉❡s ❢♦✉♥❞✱ t❤❡s❡ ❛r❡ t❤❡ ❣r❛♣❤s ♦❢ t❤❡ s✐♥❡ ❛♥❞ t❤❡ ❝♦s✐♥❡✿

❊✈❡♥ t❤♦✉❣❤ t❤❡s❡ ❧♦♦❦ ❧✐❦❡ t✇♦ ❝✉r✈❡s✱ ♥♦t ❛❧❧ t❤❡ ❧♦❝❛t✐♦♥s ❤❛✈❡ ❜❡❡♥ ✜❧❧❡❞✦ ■♥❞❡❡❞✱ t❤❡ ♦♥❧② ✈❛❧✉❡s ♦❢

x

2 st❛rt✐♥❣ sin 1❄

✇❡ ❤❛✈❡ t❛❦❡♥ ❝❛r❡ ♦❢ ❝♦♠❡ ❢r♦♠ ❞✐✈✐s✐♦♥ ❜②

❝❧♦s❡ t♦ ❡❛❝❤ ♦t❤❡r t❤❛t ✇❤❛t ✐s ❧❡❢t ✐s ❥✉st ✐♥✈✐s✐❜❧❡ ✏❝✉ts✑✳ ❇✉t ✇❤❛t ✐s

✇✐t❤

π✳

❚❤❡② ❛r❡ s♦

❲❡ ❦♥♦✇ t❤❛t ✇❡ ♥❡❡❞ t♦ ❛♣♣r♦①✐♠❛t❡✱ ❛♥❞ ✇❡ ❛r❡ ♠❛❦✐♥❣ t❤❡ ✜rst st❡♣ ❜❡❧♦✇✳ ❲❡ ❦♥♦✇ t❤❛t

sin 0 = 0 ◆♦✇✱ t❤❡ t✇♦ ❣r❛♣❤s s❡❡♠ t♦ ❝r♦ss t❤❡

❥✉♠♣s✳

y ✲❛①✐s

❛♥❞

❛t t❤❡s❡ t✇♦ ✈❛❧✉❡s ♦❢

❏✉st ❛s ❜❡❢♦r❡✱ ✇❡ ❝♦♥s✐❞❡r s❡q✉❡♥❝❡s ♦♥ t❤❡

❲❤❛t ✐s ❤❛♣♣❡♥✐♥❣ t♦ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ✐s ♠♦r❡✿

y ✬s❄

cos 0 = 1 .

x✲❛①✐s

y✳

▲❡t✬s ♠❛❦❡ s✉r❡ t❤❛t t❤❡r❡ ❛r❡

t❤❛t ❝♦♥✈❡r❣❡ t♦

0✿

❚❤❡ ♣✐❝t✉r❡ s✉❣❣❡sts t❤❛t t❤❡② ❛❧s♦ ❝♦♥✈❡r❣❡ t♦

0✳

♥♦

❆♥❞ t❤❡r❡

✶✳✶✶✳

❚❤❡ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s

✶✵✺

❚❤❡♦r❡♠ ✶✳✶✶✳✺✿

❙✐♥❡ ❛♥❞ ❈♦s✐♥❡ ❆r♦✉♥❞

0

❲❡ ❤❛✈❡✿

lim sin xn = 0

n→∞ ❢♦r ❛♥② s❡q✉❡♥❝❡

lim cos xn = 1

❛♥❞

n→∞

xn → 0 ✳

Pr♦♦❢✳

❚❤❡ ❝♦♥❝❧✉s✐♦♥ ❢♦❧❧♦✇s ❢r♦♠ ❝❡rt❛✐♥ ❢❛❝ts ❢r♦♠ tr✐❣♦♥♦♠❡tr② ✭s❡❡♥ ✐♥ ❱♦❧✉♠❡ ✶✱ ❈❤❛♣t❡r ✶P❈✲✹✮✿

❲❡ ❦♥♦✇ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛❜♦✉t s s❡❣♠❡♥t ♦❢ t❤❡ ❝✐r❝❧❡ ♦❢ r❛❞✐✉s

• •

1

✇✐t❤ ❛♥❣❧❡

x✿

sin x✳ x✳

❚❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ✈❡rt✐❝❛❧ s❡❣♠❡♥t ✐s ❚❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❝✉r✈❡❞ s❡❣♠❡♥t ✐s

❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡✿

0 < sin x < x . ■❢ ✇❡ ❤❛✈❡ ❛ s❡q✉❡♥❝❡

❚❤❡♦r❡♠✳

xn

t❤❛t ❝♦♥✈❡r❣❡s t♦

0✱

t❤❡♥ s♦ ❞♦❡s t❤❡ ♦♥❡ ✐♥ t❤❡ ♠✐❞❞❧❡ ❜② t❤❡

❙q✉❡❡③❡

❲❛r♥✐♥❣✦ ❲❡ ✇✐❧❧ st✐❧❧ ♥❡❡❞ t♦ ❝♦♥✜r♠ t❤❛t ♦✉r ❛♣♣❧✐❝❛t✐♦♥ ♦❢ t❤❡ ✐❞❡❛ ♦❢ ✏❧❡♥❣t❤✑ t♦ ❛ ❝✉r✈❡ ✐s ✈❛❧✐❞ ✭❱♦❧✉♠❡ ✸✱ ❈❤❛♣t❡r ✸■❈✲✸✮✳

❊①❡r❝✐s❡ ✶✳✶✶✳✻

Pr♦✈❡ t❤❡ s❡❝♦♥❞ ❤❛❧❢ ♦❢ t❤❡ t❤❡♦r❡♠✳ ❚❤❡r❡ ❛r❡ ♠♦r❡ s✉❜t❧❡ ♦❜s❡r✈❛t✐♦♥s ❛❜♦✉t t❤❡ ❣r❛♣❤s ♦❢ t❤❡s❡ t✇♦ ❢✉♥❝t✐♦♥s✳ ❋✐rst✱ t❤❡ ❣r❛♣❤ ♦❢

y = sin x

■♥ ❢❛❝t✱ ♣❧♦tt✐♥❣ t❤❡ ♣♦✐♥ts

❛❧♠♦st ♠❡r❣❡s ✇✐t❤ t❤❡ ❧✐♥❡

(1/n, sin 1/n)

y=x

❛r♦✉♥❞

0✿

r❡✈❡❛❧s ❛ str❛✐❣❤t ❧✐♥❡ ✇✐t❤ s❧♦♣❡

1✿

✶✳✶✶✳ ❚❤❡ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s

✶✵✻

❚❤✐s ✐s ❤♦✇ ✇❡ st❛t❡ t❤✐s ❢❛❝t ❛❧❣❡❜r❛✐❝❛❧❧②✿ ❚❤❡♦r❡♠ ✶✳✶✶✳✼✿ ❋❛♠♦✉s ▲✐♠✐t ❢♦r ❙✐♥❡ ❲❡ ❤❛✈❡✿

sin xn =1 n→∞ xn lim

❢♦r ❛♥② s❡q✉❡♥❝❡

xn → 0 ✳

Pr♦♦❢✳

❚❤❡ ❝♦♥❝❧✉s✐♦♥ ❢♦❧❧♦✇s ❢r♦♠ ❝❡rt❛✐♥ tr✐❣♦♥♦♠❡tr② ❢❛❝ts ✭s❡❡♥ ✐♥ ❱♦❧✉♠❡ ✶✱ ❈❤❛♣t❡r ✶P❈✲✹✮✿

❲❡ ❝❛♥ s❡❡ t❤r❡❡ tr✐❛♥❣❧❡s ❛♥❞ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡✐r ❛r❡❛s✿ r❡❞ < ♣✉r♣❧❡ < ♦r❛♥❣❡

❲❡ ✉s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ ❢❛❝ts✿ • ❚❤❡ ❛r❡❛ ♦❢ ❛ tr✐❛♥❣❧❡ ✇✐t❤ ❜❛s❡ 1 ❛♥❞ ❤❡✐❣❤t h ✐s h/2✳ • ❚❤❡ ❛r❡❛ ♦❢ ❛ s❡❝t♦r ♦❢ t❤❡ ❝✐r❝❧❡ ♦❢ r❛❞✐✉s 1 ✇✐t❤ ❛♥❣❧❡ x ✐s x/2✳ ❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡✿ ♦r

1 1 1 sin x < x < tan x , 2 2 2 sin x < x < tan x .

❲❡ ✢✐♣ ✐t ❛♥❞ ♠✉❧t✐♣❧② ❜② sin x✿

sin x > cos x . x ❚❤❡ s❡q✉❡♥❝❡ ♦♥ t❤❡ r✐❣❤t cos xn ❝♦♥✈❡r❣❡s t♦ 1 ✇❤❡♥ xn → 0 ❜② t❤❡ ❧❛st t❤❡♦r❡♠✳ ❚❤❡r❡❢♦r❡✱ s♦ ❞♦❡s 1>

t❤❡ ♦♥❡ ✐♥ t❤❡ ♠✐❞❞❧❡ ❜② t❤❡ ❙q✉❡❡③❡ ❚❤❡♦r❡♠✳

✶✳✶✶✳

❚❤❡ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s

✶✵✼

❲❛r♥✐♥❣✦ ❲❡ ✇✐❧❧ st✐❧❧ ♥❡❡❞ t♦ ❝♦♥✜r♠ t❤❛t ♦✉r ❛♣♣❧✐❝❛t✐♦♥ ♦❢ t❤❡ ✐❞❡❛ ♦❢ ✏❛r❡❛✑ t♦ ❛ ❝✉r✈❡❞ r❡❣✐♦♥ ✐s ✈❛❧✐❞ ✭❱♦❧✉♠❡ ✸✱ ❈❤❛♣t❡r ✸■❈✲✶✮✳

◆❡①t✱ t❤❡ ❣r❛♣❤ ♦❢

y = cos x

■♥❞❡❡❞✱ ♣❧♦tt✐♥❣ t❤❡ ♣♦✐♥ts

❛❧♠♦st ♠❡r❣❡s ✇✐t❤ t❤❡ ❧✐♥❡

(1/n, cos 1/n)

y=1

✇❤❡♥ ❝❧♦s❡ t♦ t❤❡

s❤♦✇s t❤❛t t❤❡ s❧♦♣❡ ❝♦♥✈❡r❣❡s t♦

▲❡t✬s ❝♦♠♣❛r❡ t❤❡ t✇♦ ❛❧❣❡❜r❛✐❝❛❧❧②✿

❈♦r♦❧❧❛r② ✶✳✶✶✳✽✿ ❋❛♠♦✉s ▲✐♠✐t ❢♦r ❈♦s✐♥❡ ❲❡ ❤❛✈❡✿

lim

n→∞

❢♦r ❛♥② s❡q✉❡♥❝❡

xn → 0

cos xn − 1 =0 xn

✳

❊①❡r❝✐s❡ ✶✳✶✶✳✾ Pr♦✈❡ t❤❡ t❤❡♦r❡♠✳

❊①❡r❝✐s❡ ✶✳✶✶✳✶✵ ❲❤❛t ❞♦❡s t❤❡ ❣r❛♣❤ ♦❢ t❤❡ t❛♥❣❡♥t ❧♦♦❦ ❧✐❦❡ ❝❧♦s❡ t♦ t❤❡

❆♥ ❛♥❛❧♦❣♦✉s r❡s✉❧t ✐s tr✉❡ ❢♦r t❤❡ t❛♥❣❡♥t✿

y ✲❛①✐s❄

0✿

y ✲❛①✐s✿

✶✳✶✶✳

❚❤❡ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s

✶✵✽

❈♦r♦❧❧❛r② ✶✳✶✶✳✶✶✿ ❋❛♠♦✉s ▲✐♠✐t ❢♦r ❚❛♥❣❡♥t ❲❡ ❤❛✈❡✿

tan xn =1 n→∞ xn lim

❢♦r ❛♥② s❡q✉❡♥❝❡

xn → 0 ✳

Pr♦♦❢✳

■t ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ❛❜♦✈❡ t❤❡♦r❡♠✱ t❤❡ ❢❛❝t t❤❛t cos xn → 1 ❢♦r ❛♥② s❡q✉❡♥❝❡ xn → 0✱ ❛♥❞ t❤❡ ◗✉♦t✐❡♥t ❘✉❧❡✳ ❚❤✐s ✐s ❛ ❝♦♥✜r♠❛t✐♦♥✿

❈❤❛♣t❡r ✷✿ ▲✐♠✐ts ❛♥❞ ❝♦♥t✐♥✉✐t②

❈♦♥t❡♥ts

✷✳✶ ❋✉♥❝t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷ ❈♦♥t✐♥✉✐t② ❛♥❞ ❞✐s❝♦♥t✐♥✉✐t② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✸ ▲✐♠✐ts ♦❢ ❢✉♥❝t✐♦♥s✿ s♠❛❧❧ s❝❛❧❡ tr❡♥❞s ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✹ ▲✐♠✐ts ❛♥❞ ❝♦♥t✐♥✉✐t② ✉♥❞❡r ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s ✷✳✺ ❚❤❡ ❡①♣♦♥❡♥t✐❛❧ ❛♥❞ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s ✳ ✳ ✷✳✻ ▲✐♠✐ts ❛♥❞ ❝♦♥t✐♥✉✐t② ✉♥❞❡r ❝♦♠♣♦s✐t✐♦♥s ✳ ✳ ✳ ✳ ✷✳✼ ❈♦♥t✐♥✉✐t② ♦❢ t❤❡ ✐♥✈❡rs❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✽ ❈♦♠♣❛r✐s♦♥ ♦❢ ❧✐♠✐ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✾ ●❧♦❜❛❧ ♣r♦♣❡rt✐❡s ♦❢ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s ✳ ✳ ✳ ✳ ✷✳✶✵ ▲❛r❣❡✲s❝❛❧❡ ❜❡❤❛✈✐♦r ❛♥❞ ❛s②♠♣t♦t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✶ ▲✐♠✐ts ❛♥❞ ✐♥✜♥✐t②✿ ❝♦♠♣✉t❛t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✷ ❈♦♥t✐♥✉✐t② ❛♥❞ ❛❝❝✉r❛❝② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✸ ❚❤❡ ε✲δ ❞❡✜♥✐t✐♦♥ ♦❢ ❧✐♠✐t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✵✾ ✶✶✺ ✶✷✼ ✶✸✽ ✶✺✶ ✶✺✼ ✶✻✺ ✶✼✶ ✶✼✺ ✶✽✻ ✶✾✹ ✷✵✵ ✷✵✸

✷✳✶✳ ❋✉♥❝t✐♦♥s

■♥ ❈❤❛♣t❡r ✶✱ ✇❡ ♣r❡s❡♥t❡❞ ❛ ✈❡r② ❜r✐❡❢ ✐♥tr♦❞✉❝t✐♦♥ t♦ ❝❛❧❝✉❧✉s✳ ❲❡ ❝❤♦s❡ t♦ ❡♥t❡r t❤r♦✉❣❤ t❤❡ st✉❞② ♦❢ ♠♦t✐♦♥✳ ❆s ✇❡ ❝♦♠♠♦♥❧② t❤✐♥❦ ♦❢ ♠♦t✐♦♥ ❛s ❛ ❝♦♥t✐♥✉♦✉s ♣r♦❣r❡ss t❤r♦✉❣❤ ♣❤②s✐❝❛❧ s♣❛❝❡✱ ✇❡ t❤✐♥❦ ♦❢ s♣❛❝❡ ❛s ✐♥✜♥✐t❡❧② ❞✐✈✐s✐❜❧❡ ✭✐✳❡✳✱ t❤❡ s❡t ♦❢ r❡❛❧ ♥✉♠❜❡rs✮✳ ■❢ ✇❡ ❛❧s♦ t❤✐♥❦ ♦❢ ♠♦t✐♦♥ ❛s ❛♥ ✐♥❝r❡♠❡♥t❛❧ ♣r♦❣r❡ss t❤r♦✉❣❤ t✐♠❡✱ ✇❡ t❤✐♥❦ ♦❢ t✐♠❡ ❛s ❞✐s❝r❡t❡ ✭✐✳❡✳✱ t❤❡ s❡t ♦❢ ✐♥t❡❣❡rs✮✳ ❚❤❡ ❧❛tt❡r ✐s t❤❡ ✐♥♣✉t ❛♥❞ t❤❡ ❢♦r♠❡r ✐s t❤❡ ♦✉t♣✉t ♦❢ t❤❡s❡ ❢✉♥❝t✐♦♥s t❤❛t ✇❡ ❝❛❧❧ s❡q✉❡♥❝❡s ✦ ❚❤✐s ✐s ❤♦✇ ✇❡ ✈✐s✉❛❧✐③❡ s❡q✉❡♥❝❡s✿

❇✉t ❤♦✇ ❞♦ ✇❡ tr❡❛t ♠♦t✐♦♥ ✇❤❡♥ t❤❡ t✐♠❡ ✈❛r✐❡s ❝♦♥t✐♥✉♦✉s❧②❄ ❚❤✐s r❡q✉✐r❡s ✉s t♦ t❤✐♥❦ ♦❢ t✐♠❡ ❛s t❤❡ r❡❛❧ ♥✉♠❜❡rs✱ t♦♦✳ ❲❡ ♥❡❡❞ ❛ ❞✐✛❡r❡♥t ❝❧❛ss ♦❢ ❢✉♥❝t✐♦♥s ✿ r❡❛❧ ✐♥♣✉t ❛♥❞ r❡❛❧ ♦✉t♣✉t✦

✷✳✶✳

❋✉♥❝t✐♦♥s

✶✶✵

❲❛r♥✐♥❣✦ ■♥ t❤❡ ❧♦♥❣ r✉♥✱ ✇❡ ✇✐❧❧ ❜❡ ❞❡❝✐❞✐♥❣ ♦♥ ♠✉❧t✐♣❧❡ ♦❝✲ ❝❛s✐♦♥s ✇❤❛t ❦✐♥❞s ♦❢ ✐♥♣✉ts ❛♥❞ ♦✉t♣✉ts ✇❡ ♥❡❡❞✳

❘❡❝❛❧❧ ♦✉r ❣❡♥❡r❛❧ ✈✐❡✇ ♦♥ ❢✉♥❝t✐♦♥s✿

❆s ✇❡ ❝❛♥ s❡❡✱ t❤❡r❡ ♠✉st ❜❡ ❛

y

❢♦r ❡❛❝❤

x✱

❜✉t ♥♦t ♥❡❝❡ss❛r✐❧② ✈✐❝❡ ✈❡rs❛✳ ❚❤❡ ✇❤♦❧❡ ❝❛❧❝✉❧✉s ✐s ❜✉✐❧t ♦♥

t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❝❡♣t ❛s ❛ ❢♦✉♥❞❛t✐♦♥✿

❉❡✜♥✐t✐♦♥ ✷✳✶✳✶✿ ❢✉♥❝t✐♦♥ ❆ ❢✉♥❝t✐♦♥ ✐s ❛ r✉❧❡ ♦r ♣r♦❝❡❞✉r❡ f t❤❛t ❛ss✐❣♥s t♦ ❛♥② ❡❧❡♠❡♥t x ✐♥ ❛ s❡t X ✱ ❝❛❧❧❡❞ t❤❡ ✐♥♣✉t s❡t ♦r t❤❡ ❞♦♠❛✐♥ ♦❢ f ✱ ❡①❛❝t❧② ♦♥❡ ❡❧❡♠❡♥t y ✱ ✇❤✐❝❤ ✐s t❤❡♥ ❞❡♥♦t❡❞ ❜②

y = f (x)

♦✉t♣✉t s❡t ♦r t❤❡ ❝♦❞♦♠❛✐♥ ♦❢ ✐♥❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡ ❀ t❤❡ ♦✉t♣✉ts ❛r❡ ❝♦❧❧❡❝t✐✈❡❧② ❝❛❧❧❡❞ t❤❡ ❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡✳ ❲❡ ❛❧s♦ s❛② t❤❛t t❤❡ ✈❛❧✉❡ ♦❢ x ✉♥❞❡r ✐♥ ❛♥♦t❤❡r s❡t

f✳ f

Y✳

❚❤❡ ❧❛tt❡r s❡t ✐s ❝❛❧❧❡❞ t❤❡

❚❤❡ ✐♥♣✉ts ❛r❡ ❝♦❧❧❡❝t✐✈❡❧② ❝❛❧❧❡❞ t❤❡

✐s

y✳

❚❤❡ t❡r♠ ✏✈❛r✐❛❜❧❡✑ ✐s ❡①♣❧❛✐♥❡❞ ❜② t❤❡ ❢❛❝t t❤❛t ✐❢ ✇❡ ❝❛♥ ✇✐❧❧

✈❛r②

✇✐t❤✐♥ s❡t

Y✳

✈❛r②

t❤❡ ✐♥♣✉ts

x

✇✐t❤✐♥ s❡t

X✱

t❤❡ ♦✉t♣✉ts

y

❚❤❡ ✐❞❡❛ ❤❛s ♥♦t❤✐♥❣ t♦ ❞♦ ✇✐t❤ ♥✉♠❜❡rs✳ ❍❡r❡ ✐s ❛♥ ❡①❛♠♣❧❡ ♦❢ ❛ ❢✉♥❝t✐♦♥✿

❚❤❡ ❛rr♦✇s ✐♥❞✐❝❛t❡ ✇❤❛t s♣♦rt ❡❛❝❤ ❜♦② ♣r❡❢❡rs✱ ❛♥❞ ❛s ✇❡ ❣♦ ❞♦✇♥ t❤❡ ❧✐st ♦❢ ❜♦②s ♦♥ t❤❡ ❧❡❢t✱ t❤❡ ❛rr♦✇ ❛✉t♦♠❛t✐❝❛❧❧② tr❛❝❡ t❤❡ ❜❛❧❧s ♦♥ t❤❡ r✐❣❤t✳ ❇❡❧♦✇ ✐s t❤❡ ❝♦♠♠♦♥ ♥♦t❛t✐♦♥✿

❋✉♥❝t✐♦♥ ❢r♦♠ s❡t t♦ s❡t f :X→Y ♦r

f

X −−−−→ Y ■t r❡❛❞s ✏❢✉♥❝t✐♦♥

Y ✑✳

f

❢r♦♠

X

t♦

✷✳✶✳

❋✉♥❝t✐♦♥s

✶✶✶

■♥ t❤✐s ✈♦❧✉♠❡ ✇❡ ✇✐❧❧ ✉s❡ ♣r✐♠❛r✐❧② t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ t②♣❡s ♦❢ ❢✉♥❝t✐♦♥s✿

❉❡✜♥✐t✐♦♥ ✷✳✶✳✷✿ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s ❛♥❞ s❡q✉❡♥❝❡s •

❆

♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥

X ⊂ R✱

✐s ❛ ❢✉♥❝t✐♦♥ ❢r♦♠ ❛ s✉❜s❡t ♦❢ t❤❡ r❡❛❧ ♥✉♠❜❡rs✱

t♦ t❤❡ r❡❛❧ ♥✉♠❜❡rs✿

f : X → R. •

❆

s❡q✉❡♥❝❡

✐s ❛ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥ ✇✐t❤ t❤❡ ❞♦♠❛✐♥

X

❝♦♥s✐st✐♥❣ ♦❢ ❝♦♥✲

s❡❝✉t✐✈❡ ✐♥t❡❣❡rs✳

▲❡t✬s ❝♦♠♣❛r❡✿

•

•

❆ s❡q✉❡♥❝❡

an : X → R✳

✕

❚❤❡ ✐♥♣✉t ✈❛r✐❛❜❧❡ ✐s

✕

❚❤❡ ♦✉t♣✉t ✈❛r✐❛❜❧❡ ✐s

❆ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥

n✱

X ⊂ Z✮✳

❛♥ ✐♥t❡❣❡r ✭

y = an ✱

❛ r❡❛❧ ♥✉♠❜❡r✳

f : X → R✳

✕

❚❤❡ ✐♥♣✉t ✈❛r✐❛❜❧❡ ✐s

✕

❚❤❡ ♦✉t♣✉t ✈❛r✐❛❜❧❡ ✐s

x✱

X ⊂ R✮✳

❛ r❡❛❧ ♥✉♠❜❡r ✭

y = f (x)✱

❛♥♦t❤❡r r❡❛❧ ♥✉♠❜❡r✳

❲❛r♥✐♥❣✦ ❚❤❡ ❧❛tt❡r ❝❧❛ss ✐♥❝❧✉❞❡s t❤❡ ❢♦r♠❡r✳

❲❡ ❝♦♠♣❛r❡ t❤❡ ♥♦t❛t✐♦♥s t♦♦✱ s✐❞❡ ❜② s✐❞❡✿

❋✉♥❝t✐♦♥ ✈s✳ s❡q✉❡♥❝❡ ♥❛♠❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥

↓ f

x ↑




✈s✳

↓ a

n

↑

♥❛♠❡ ♦❢ t❤❡ ✐♥♣✉t ✈❛r✐❛❜❧❡ ✈❛❧✉❡ ♦❢ t❤❡ ✐♥♣✉t ✈❛r✐❛❜❧❡

f

●r❛♣❤s

↓
3 =5 ↑

✈s✳

✈❛❧✉❡ ♦❢ t❤❡ ♦✉t♣✉t ✈❛r✐❛❜❧❡

a

↓ 3

=5 ↑

♣r♦✈✐❞❡ ❛ ✇❛② t♦ ❤❛✈❡ ❛ ❜✐r❞✬s ❡②❡ ✈✐❡✇ ♦❢ t❤❡ ❢✉♥❝t✐♦♥✳

❘❡❝❛❧❧ t❤❛t t❤❡

❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥ ✐s t❤❡ s❡t ♦❢ ♣♦✐♥ts ✐♥ t❤❡ xy✲♣❧❛♥❡ t❤❛t s❛t✐s❢② ty = f (x)✳

❲❡ ❝❛♥ ❡①♣r❡ss

t❤✐s ✐❞❡❛ ✇✐t❤ t❤❡ t❤❡ s❡t✲❜✉✐❧❞✐♥❣ ♥♦t❛t✐♦♥ ✭s❡❡♥ ✐♥ ❱♦❧✉♠❡ ✶✱ ❈❤❛♣t❡r ✶P❈✲✷✮✿

❣r❛♣❤ ♦❢

f = {(x, y) : y = f (x)}

■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ❣r❛♣❤ ✐s t❤❡ s❡t ♦❢ ❛❧❧ ♣♦ss✐❜❧❡ ♣♦✐♥ts ♦♥ t❤❡ ♣❧❛♥❡ ✐♥ t❤❡ ❢♦r♠

(x, f (x))✳

❲❡ ✐♥s❡rt ♠♦r❡ ✐♥♣✉ts ❛s ♥❡❝❡ss❛r②✳ ❲❤❡♥ t❤❡r❡ ❛r❡ ❡♥♦✉❣❤ ♦❢ t❤❡♠✱ t❤❡② st❛rt t♦ ❢♦r♠ ❛ ❝✉r✈❡✦

✷✳✶✳ ❋✉♥❝t✐♦♥s

✶✶✷

❊①❛♠♣❧❡ ✷✳✶✳✸✿ ❣r❛♣❤s ❛s ❝✉r✈❡s

■♥st❡❛❞ ♦❢ ♣r♦❞✉❝✐♥❣ ♠♦r❡ ❞❛t❛✱ ✐t ✐s ❝♦♠♠♦♥ t♦ ✜❧❧ ✐♥ t❤❡ ❣❛♣s ✐♥ ❛ ❣r❛♣❤ ✇✐t❤ ❛ str♦❦❡ ♦❢ ❛ ♣❡♥✿

❚❤❡ ❝♦♠♣✉t❡r ❝❛♥ ❛❧s♦ ♠❛❦❡ ❛ ❣✉❡ss✿

❊①❛♠♣❧❡ ✷✳✶✳✹✿ r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ ❢✉♥❝t✐♦♥s

❚❤❡r❡ ❛r❡ s❡✈❡r❛❧ ✇❛②s t♦ r❡♣r❡s❡♥t ❛♥❞ ✈✐s✉❛❧✐③❡ ❛ ❢✉♥❝t✐♦♥✳ ❚❤❡② ❛r❡ ✐♥t❡r❝♦♥♥❡❝t❡❞✿

■♥ s✉♠♠❛r②✱ ✇❡ s❤♦✉❧❞ ♠♦✈❡ ❢r❡❡❧② ❛♠♦♥❣ t❤❡s❡ tr❡❛t♠❡♥ts ♦❢ ❢✉♥❝t✐♦♥s✿ ❢♦r♠✉❧❛ l

♣r♦❣r❛♠

ց

❧✐st

ր

←→

❣r❛♣❤

❲❤❡♥ ❛ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥ y = f (x) ✐s ❣✐✈❡♥ t♦ ✉s ✇✐t❤♦✉t ❛♥② ♣r✐♦r ❜❛❝❦❣r♦✉♥❞✱ ✇❡ ❝❛♥ st✐❧❧ s✉♣♣❧② ✐t ✇✐t❤ ❛ t❛♥❣✐❜❧❡ r❡♣r❡s❡♥t❛t✐♦♥ ✿ • ❲❡ t❤✐♥❦ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❛s ✐❢ ✐t r❡♣r❡s❡♥ts

♠♦t✐♦♥ ✭x t✐♠❡✱ y ❧♦❝❛t✐♦♥✮✳

❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ♦♥ ❛ ♣✐❡❝❡ ♦❢ ♣❛♣❡r✳ ❲❡ t❤✐♥❦ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❛s ❛ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ t❤❡ r❡❛❧ ❧✐♥❡✳

• ❲❡ ♣❧♦t t❤❡ •

▲❡t✬s r❡✈✐❡✇ t❤❡ ❧❛st ✐t❡♠ ✭s❡❡♥ ✐♥ ❱♦❧✉♠❡ ✶✱ ❈❤❛♣t❡r ✶P❈✲✸✮✿ ◮ ◆✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s ❛r❡ tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ r❡❛❧ ♥✉♠❜❡r ❧✐♥❡✳✳✳ ❛♥❞ ✈✐❝❡ ✈❡rs❛✳

✷✳✶✳

❋✉♥❝t✐♦♥s

✶✶✸

❲❡ r❡♣r❡s❡♥t ❛ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥ ❛s ❛ ❝♦rr❡s♣♦♥❞❡♥❝❡ ❜❡t✇❡❡♥ t❤❡

■♥ ❝♦♥tr❛st t♦ t❤❡

xy ✲♣❧❛♥❡

x✲

❛♥❞ t❤❡

y ✲❛①✐s✿

t❤❛t ❝♦♥t❛✐♥s t❤❡ ❣r❛♣❤✱ t❤❡② ❛r❡ ♣❛r❛❧❧❡❧ r❛t❤❡r t❤❛♥ ♣❡r♣❡♥❞✐❝✉❧❛r t♦ ❡❛❝❤

♦t❤❡r✦ ❋♦r ❡①❛♠♣❧❡✱ t❤❡ ❢✉♥❝t✐♦♥ ❣✐✈❡♥ ❜②

y =x+s s❤✐❢ts t❤❡

x✲❛①✐s

✐♥ t❤❡ ♣♦s✐t✐✈❡ ❞✐r❡❝t✐♦♥ ✇❤❡♥

❛♥❞

s > 0✱

x 7→ x + s

❛♥❞ ✐♥ t❤❡ ♥❡❣❛t✐✈❡ ❞✐r❡❝t✐♦♥ ✇❤❡♥

s < 0✿

■t✬s ❛ r✐❣✐❞ ♠♦t✐♦♥✳ ◆❡①t✱ t❤❡ ❢✉♥❝t✐♦♥ ❣✐✈❡♥ ❜②

y = −x ✐♥t❡r❝❤❛♥❣❡s t✇♦ ♣♦✐♥ts ♦♥ t❤❡

❚❤✐s ✐s ❛❧s♦ ❛ r✐❣✐❞ ♠♦t✐♦♥✳

❛♥❞

x 7→ −x

x✲❛①✐s✿

❲❤❛t t❤❡s❡ tr❛♥s❢♦r♠❛t✐♦♥s ❤❛✈❡ ✐♥ ❝♦♠♠♦♥ ✐s t❤❛t ❡❛❝❤ t❛❦❡s t❤❡

t✇♦ s♣♦ts ❛♥❞ t❤❡♥ ❜r✐♥❣ t❤♦s❡ t✇♦ t♦ t❤❡ ❛ss✐❣♥❡❞ ❧♦❝❛t✐♦♥s ♦♥ t❤❡ ❈♦♥s✐❞❡r ♥♦✇ t❤❡

❢♦❧❞✐♥❣

♦❢ ❛ ♣✐❡❝❡ ♦❢ ✇✐r❡✿

y ✲❛①✐s✳

x✲❛①✐s

❜②

✷✳✶✳

❋✉♥❝t✐♦♥s

✶✶✹

❚❤✐s tr❛♥s❢♦r♠❛t✐♦♥ ✐s ❣✐✈❡♥ ❜② t❤❡ ❛❜s♦❧✉t❡ ✈❛❧✉❡ ❢✉♥❝t✐♦♥✿

y = |x| ❆♥♦t❤❡r t②♣❡ ♦❢ tr❛♥s❢♦r♠❛t✐♦♥ ✐s t❤❡

str❡t❝❤

❛♥❞

♦❢ t❤❡

y =x·k

x

x 7→ |x| . ❛①✐s ❛s ✐❢ ✐t✬s ❛ r✉❜❜❡r str✐♥❣✱ ❣✐✈❡♥ ❜②

❛♥❞

x → x·k.

❲❡ ❣r❛❜ ✐t ❜② t❤❡ ❡♥❞s ❛♥❞ ♣✉❧❧ t❤❡♠ ❛♣❛rt✿

❚❤✐s ✐s ✐♥❞❡❡❞ ❛ ✉♥✐❢♦r♠ str❡t❝❤ ❜❡❝❛✉s❡ t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ ✏str❡t❝❤❡❞ ❜② ❛ ❢❛❝t♦r

k✑

❛s ✏s❤r✉♥❦ ❜② ❛ ❢❛❝t♦r

❛♥②

t✇♦ ♣♦✐♥ts ❞♦✉❜❧❡s✳

❲❡ ✉♥❞❡rst❛♥❞

1/k ✑✳

❆ ❝♦♥st❛♥t ❢✉♥❝t✐♦♥✱ ❣✐✈❡♥ ❜②

f (x) = c ✐s s❡❡♥ ❛s ❛

❝♦❧❧❛♣s❡✳

❛♥❞

x 7→ c ,

■t s❤r✐♥❦s t♦ ❛ s✐♥❣❧❡ ♣♦✐♥t✳ ❚❤❡ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ t❤❡ ❞♦♠❛✐♥ ✐♥t♦ t❤❡ ❝♦❞♦♠❛✐♥ ♣❡r❢♦r♠❡❞ ❜② t❤❡ ❢✉♥❝t✐♦♥ ❝❛♥ ❛❧s♦ ❜❡ s❡❡♥ ✐♥ ✐ts ❣r❛♣❤✿

✷✳✷✳

❈♦♥t✐♥✉✐t② ❛♥❞ ❞✐s❝♦♥t✐♥✉✐t②

✶✶✺

❋✐rst✱ ✇❡ t❛❦❡ t❤❡ x✲❛①✐s ❛s ✐❢ ✐t ✐s ❛ r♦♣❡ ❛♥❞ ❧✐❢t ✐t ✈❡rt✐❝❛❧❧② t♦ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❛♥❞ t❤❡♥ ✇❡ ♣✉s❤ ✐t ❤♦r✐③♦♥t❛❧❧② t♦ t❤❡ y ✲❛①✐s✳ ❊①❡r❝✐s❡ ✷✳✶✳✺

❈❧❛ss✐❢② t❤❡s❡ ❢✉♥❝t✐♦♥s✿ ❢✉♥❝t✐♦♥

♦❞❞ ❡✈❡♥ ♦♥t♦ ♦♥❡✲t♦✲♦♥❡

f (x) = 2x − 1 g(x) = −x + 2 h(x) = 3 ❊①❡r❝✐s❡ ✷✳✶✳✻

❊①♣❧❛✐♥ t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡s❡ t✇♦ ❢✉♥❝t✐♦♥s✿ r

√ x−1 x−1 . ❛♥❞ √ x+1 x+1

✷✳✷✳ ❈♦♥t✐♥✉✐t② ❛♥❞ ❞✐s❝♦♥t✐♥✉✐t②

■❢ ♦♥❡ st❛♥❞s ❛t t❤❡ ❡❞❣❡ ♦❢ ❛ ❝❧✐✛✱ ♦♥❡ s♠❛❧❧ st❡♣ ♠❛② ❧❡❛❞ t♦ ❝❛t❛str♦♣❤✐❝ ❝♦♥s❡q✉❡♥❝❡s✿

❙t❛♥❞✐♥❣ ♦✛ t❤❡ ❡❞❣❡ ✇✐❧❧ ♥♦t✳ ❖♥❡ ❝❛♥ ❝❛♣t✉r❡ t❤✐s ♣❤❡♥♦♠❡♥♦♥ ✇✐t❤ t❤❡ ❣r❛♣❤ ♦❢ t❤✐s ❢✉♥❝t✐♦♥✿ f (x) =

(

1 0

✐❢ x ≤ 0 , ✐❢ x > 0 .

❍❡r❡ x ✐s t❤❡ ♣❡rs♦♥✬s ❤♦r✐③♦♥t❛❧ ♣♦s✐t✐♦♥ ❛♥❞ f (x) ✐s t❤❡ ✈❡rt✐❝❛❧ ♣♦s✐t✐♦♥✿

✷✳✷✳

❈♦♥t✐♥✉✐t② ❛♥❞ ❞✐s❝♦♥t✐♥✉✐t②

✶✶✻

◆♦ ♠❛tt❡r ❤♦✇ s♠❛❧❧ ❛ st❡♣ ✭✐♥ t❤❡ ♣♦s✐t✐✈❡ ❞✐r❡❝t✐♦♥✮ ✐s ✕ 1, 1/2, 1/3, ..., 1/n, ... ✕ ✐t ✐s ❢❛t❛❧✳ ■t ✐s ❝❛❧❧❡❞

❞✐s❝♦♥t✐♥✉✐t② ✳

■♥ t❤✐s ✈♦❧✉♠❡✱ ✇❡ ✇✐❧❧ st✉❞② ❢✉♥❝t✐♦♥s ♦♥ ❛ s♠❛❧❧ s❝❛❧❡✳ ❚❤❛t ✐s ✇❤② t❤❡ ❢✉♥❝t✐♦♥s t❤❛t ♠❛② ♣r♦❞✉❝❡ ❛ ❞r❛♠❛t✐❝❛❧❧② ❞✐✛❡r❡♥t ♦✉t♣✉t ✐♥ r❡s♣♦♥s❡ t♦ ❛ s♠❛❧❧ ❝❤❛♥❣❡ ✐♥ t❤❡ ✐♥♣✉t ✇✐❧❧ ♥♦t ✐♥✐t✐❛❧❧② ❜❡ ❛ s✉❜❥❡❝t ♦❢ ♦✉r st✉❞②✳ ■♥ ❱♦❧✉♠❡ ✸✱ ✇❡ ✇✐❧❧ t✉r♥ t♦ t❤❡ st✉❞② ♦❢ ❢✉♥❝t✐♦♥s ♦♥ ❛ ❧❛r❣❡ s❝❛❧❡ ❛♥❞ s✉❝❤ ❢✉♥❝t✐♦♥s ✇♦♥✬t ❜❡ ❛s ❜✐❣ ❛ ❝♦♥❝❡r♥✳ ❚❤❡ ✐❞❡❛ ❤♦✇ ❛ t✐♥② ❝❤❛♥❣❡ ❝❛♥ ♠❛❦❡ ❛ ❞✐✛❡r❡♥❝❡ ❜❡❝♦♠❡s ❡s♣❡❝✐❛❧❧② ✈✐✈✐❞ ✇❤❡♥ ✇❡ ❧♦♦❦ ❛t ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s ❛s tr❛♥s❢♦r♠❛t✐♦♥s ✿

❚❤❡ ❛rr♦✇s ♥♦t ♦♥❧② t❡❧❧ ✉s ✇❤❛t ❤❛♣♣❡♥s t♦ ❡❛❝❤ ♥✉♠❜❡r ❜✉t t❤❡② ❛❧s♦ s✉❣❣❡st ✇❤❛t ❤❛♣♣❡♥s t♦ t❤❡ ✇❤♦❧❡ x✲❛①✐s✦ ❍❡r❡ ✇❡ s❡❡ ❛ ✈❛r✐❡t② ♦❢ s❤✐❢t✐♥❣✱ str❡t❝❤✐♥❣✱ ❛♥❞ s❤r✐♥❦✐♥❣ ❛t ❞✐✛❡r❡♥t ❧♦❝❛t✐♦♥s✿

❚❤❡r❡ ✐s ♥♦ t❡❛r✐♥❣ t❤♦✉❣❤✦ ❊①❛♠♣❧❡ ✷✳✷✳✶✿ ✐♥t❛❝t

x✲❛①✐s

❚❤❡ ❜❛s✐❝ tr❛♥s❢♦r♠❛t✐♦♥s ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥ ❛r❡✿ s❤✐❢t✐♥❣✱ str❡t❝❤✐♥❣✱ ✢✐♣♣✐♥❣✳ ❚❤❡ ♣♦✐♥ts ❛r❡ ♠♦✈❡❞ ✐♥ ✉♥✐s♦♥✿

❲❡ ❝❛♥ ♠❛❦❡ t❤✐s ✐❞❡❛ ♠♦r❡ ♣r❡❝✐s❡✳ ■♥ ❛ s❤✐❢t✱ ✐❢ x ❝❤❛♥❣❡s ❜② p✱ t❤❡♥ s♦ ❞♦❡s y ✿

✷✳✷✳

❈♦♥t✐♥✉✐t② ❛♥❞ ❞✐s❝♦♥t✐♥✉✐t②

❚❤✐s ✐s ❛ str❡t❝❤ ❜②

■❢

x

❝❤❛♥❣❡s ❜②

p✱

2

❛♥❞ ❛ s❤r✐♥❦ ❜②

t❤❡♥

y

❝❤❛♥❣❡s ❜②

❊①❛♠♣❧❡ ✷✳✷✳✷✿ t❡❛r✐♥❣ ♦❢ ❚❤❡

x✲❛①✐s

✶✶✼

2✿

2p✳

❆♥❞ s♦ ♦♥✳

x✲❛①✐s

♠✐❣❤t ❜❡ t♦r♥ ❜② t❤❡ tr❛♥s❢♦r♠❛t✐♦♥✿

❲❡ ❝❛♥ s❡❡ ❤♦✇ t❤❡ ❝♦❧♦r ♦❢ t❤❡ ♦✉t♣✉t ❝❤❛♥❣❡s ❛❜r✉♣t❧② ❛t ♦♥❡ ❧♦❝❛t✐♦♥✦ ▼❡❛♥✇❤✐❧❡✱ t❤❡r❡ ✐s ❛ ❣❛♣ ✐♥ t❤❡ ❣r❛♣❤ ✕ ❞✐s❝♦♥t✐♥✉✐t②✳ ▲❡t✬s ❧♦♦❦ ❛t t❤❡ ❢♦r♠✉❧❛✿

f (x) =

(

x+8 x

✇❤❡♥ ✇❤❡♥

x > −5 , x ≤ −5 .

❚❤❡ ❢✉♥❝t✐♦♥ ❤❛s ❛ s♣❡❝✐❛❧ ♣♦✐♥t✿ ◆♦ ♠❛tt❡r ❤♦✇ s♠❛❧❧ ❛ ❞❡✈✐❛t✐♦♥ ♦❢ ❛ ❧❛r❣❡ ❞❡✈✐❛t✐♦♥ ♦❢ t❤❡ ♦✉t♣✉t ❢r♦♠

b = f (a)

t♦

y = f (x)✳

x ❢r♦♠ a = −5 ✐s✱ ✐t ✇✐❧❧ ♣r♦❞✉❝❡

❊①❛♠♣❧❡ ✷✳✷✳✸✿ s✐❣♥ ❢✉♥❝t✐♦♥ ❚❤❡ s✐❣♥ ❢✉♥❝t✐♦♥ ❝♦❧❧❛♣s❡s t❤❡

x✲❛①✐s

t♦ t❤r❡❡ ❞✐✛❡r❡♥t ♣♦✐♥ts ♦♥ t❤❡

y ✲❛①✐s✿

❚❤✐s ♠❛② ❜❡ ♦✉r ♠❛✐♥ ♦❜s❡r✈❛t✐♦♥ ❛❜♦✉t t❤✐s ❢✉♥❝t✐♦♥✿ ❊✈❡♥ t❤❡ s♠❛❧❧❡st st❡♣ ❛✇❛② ❢r♦♠ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❢r♦♠

f (0) = 0

t♦

0 ✇✐❧❧ ❝❤❛♥❣❡

y = f (x) = ±1 ✳

❊①❡r❝✐s❡ ✷✳✷✳✹ ❉✐s❝✉ss t❤❡ ❝♦♥t✐♥✉✐t② ♦❢ t❤❡ ✐♥t❡❣❡r ✈❛❧✉❡ ❢✉♥❝t✐♦♥✳

❲❡ ❤❛✈❡ ❝♦♥s✐❞❡r❡❞ s♦♠❡ ✏♣♦♦r❧② ❜❡❤❛✈❡❞✑✱ ❞✐s❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s✳ ❙✉❝❤ ❛ ❢✉♥❝t✐♦♥ ✇♦✉❧❞ ❤❛✈❡ ❛t ❧❡❛st ♦♥❡ s♣❡❝✐❛❧ ♣♦✐♥t✿

◮

◆♦ ♠❛tt❡r ❤♦✇ s♠❛❧❧ ❛ ❞❡✈✐❛t✐♦♥ ♦❢ t❤❡ ✐♥♣✉t

t❤❡ ♦✉t♣✉t

y = f (x)

❢r♦♠

b = f (a)✳

x

❢r♦♠

a

✐s✱ ✐t ✇✐❧❧ ♣r♦❞✉❝❡ ❛ ❧❛r❣❡ ❞❡✈✐❛t✐♦♥ ♦❢

✷✳✷✳

❈♦♥t✐♥✉✐t② ❛♥❞ ❞✐s❝♦♥t✐♥✉✐t②

✶✶✽

❙♦✱ ✇❤❛t ✐s t❤❡ ♦♣♣♦s✐t❡❄ ❚❤✐s ✐s ✇❤❛t ✇❡ ♠❡❛♥ ❜② ❛

❝♦♥t✐♥✉♦✉s ❞❡♣❡♥❞❡♥❝❡

♦❢

y

♦♥

x

✉♥❞❡r ❢✉♥❝t✐♦♥

f✿

◮ ❆ s♠❛❧❧ ❞❡✈✐❛t✐♦♥ ♦❢ t❤❡ ✐♥♣✉t x ❢r♦♠ a ✇✐❧❧ ♣r♦❞✉❝❡ ❛ s♠❛❧❧ ❞❡✈✐❛t✐♦♥ ♦❢ t❤❡ ♦✉t♣✉t y = f (x) ❢r♦♠ b = f (a)✳ ❊①❛♠♣❧❡ ✷✳✷✳✺✿ ❜❛s✐❝ tr❛♥s❢♦r♠❛t✐♦♥s ❚❤❡ ✐❞❡❛ ✐s ✈✐s✉❛❧✐③❡❞ ❜❡❧♦✇✿

❚❤❡ ❝♦❧♦r ♦❢ t❤❡ ♦✉t♣✉t ♥❡✈❡r ❝❤❛♥❣❡s ❛❜r✉♣t❧②✦

❲❡ ❛❧s♦ ♣❧♦t t❤❡ ❣r❛♣❤s ♦❢ t❤❡s❡ ❢✉♥❝t✐♦♥s ❜❡❧♦✇✿

❚❤❡r❡ ❛r❡ ♥♦ ❣❛♣s✦

❈♦♥t✐♥✉♦✉s ❞❡♣❡♥❞❡♥❝✐❡s ❛r❡ ✉❜✐q✉✐t♦✉s ✐♥ ♥❛t✉r❡✿

•

❚❤❡ ❧♦❝❛t✐♦♥ ♦❢ ❛ ♠♦✈✐♥❣ ♦❜❥❡❝t ❝♦♥t✐♥✉♦✉s❧② ❞❡♣❡♥❞s ♦♥ t✐♠❡✳

•

❚❤❡ ♣r❡ss✉r❡ ✐♥ ❛ ❝❧♦s❡❞ ❝♦♥t❛✐♥❡r ❝♦♥t✐♥✉♦✉s❧② ❞❡♣❡♥❞s ♦♥ t❤❡ t❡♠♣❡r❛t✉r❡✳

•

❚❤❡ ❛✐r r❡s✐st❛♥❝❡ ❝♦♥t✐♥✉♦✉s❧② ❞❡♣❡♥❞s ♦♥ t❤❡ ✈❡❧♦❝✐t② ♦❢ t❤❡ ♠♦✈✐♥❣ ♦❜ ❥❡❝t✱ ❡t❝✳

■t ✐s ♦✉r ❣♦❛❧ t♦ ❞❡✈❡❧♦♣ t❤✐s ✐❞❡❛ ✐♥ ❢✉❧❧ ❛♥❞ t♦ ♠❛❦❡ s✉r❡ t❤❛t ♦✉r ♠❛t❤❡♠❛t✐❝❛❧ t♦♦❧s ♠❛t❝❤ t❤❡ ♣❡r❝❡✐✈❡❞ r❡❛❧✐t②✳ ❲❡ t❛❦❡ ❛ ❜❡tt❡r ❧♦♦❦ ❛t t❤❡

❣r❛♣❤s

✳

❲❡ ♠❛❦❡ ✈✐s✐❜❧❡ t❤❡ ✐❞❡❛ t❤❛t ❛ s♠❛❧❧ ❞❡✈✐❛t✐♦♥ ♦❢ ❢r♦♠

b = f (a)✿

x

❢r♦♠

a

✇✐❧❧ ♣r♦❞✉❝❡ ❛ s♠❛❧❧ ❞❡✈✐❛t✐♦♥ ♦❢ ♦❢

y = f (x)

✷✳✷✳

❈♦♥t✐♥✉✐t② ❛♥❞ ❞✐s❝♦♥t✐♥✉✐t②

✶✶✾

❲❡ ❝❛♥ s❡❡ s❡✈❡r❛❧ ✭s♠❛❧❧❡r ❛♥❞ s♠❛❧❧❡r✮ ❞❡✈✐❛t✐♦♥s ♦❢

x ❛♥❞ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❞❡✈✐❛t✐♦♥s ♦❢ y

✭❛❧s♦ s♠❛❧❧❡r

❛♥❞ s♠❛❧❧❡r✮✳ ■t ✐s r❡✈❡❛❧❡❞ t❤❛t ♦✉r ♠❛t❤❡♠❛t✐❝❛❧ t♦♦❧ ✇✐❧❧ ❜❡

❧✐♠✐ts ♦❢ s❡q✉❡♥❝❡s✦

❊①❛♠♣❧❡ ✷✳✷✳✻✿ s✐❣♥ ❢✉♥❝t✐♦♥

❚❤❡ s✐❣♥ ❢✉♥❝t✐♦♥

f (x) = sign(x)

❤❛s ❛ ✈✐s✐❜❧❡

❣❛♣ ❛t x = 0✿

y ♦♥ x ❢❛✐❧s✿ ❙t❛rt✐♥❣ ✇✐t❤ x = 0✱ ❡✈❡♥ t❤❡ t✐♥✐❡st ❞❡✈✐❛t✐♦♥ ♦❢ x✱ s❛②✱ x = .0001✱ ♣r♦❞✉❝❡s ❛ ❥✉♠♣ ✐♥ y ❢r♦♠ sign(0) = 0 t♦ sign(.0001) = 1✳ ❲❡ ✐♥t❡r♣r❡t t❤✐s ♦❜s❡r✈❛t✐♦♥ ✈✐❛ ❧✐♠✐ts✳ ▲❡t✬s ❝♦♥s✐❞❡r t❤❡ ❝♦♠♣♦s✐t✐♦♥s ♦❢ f ✇✐t❤ ❛ ❢❡✇ s❡q✉❡♥❝❡s t❤❛t ❝♦♥✈❡r❣❡ t♦ 0✿ lim xn = 0 . ❚❤✐s ❡①❛♠♣❧❡ s❤♦✇s ❤♦✇ t❤❡ ✐❞❡❛ ♦❢ ❝♦♥t✐♥✉♦✉s ❞❡♣❡♥❞❡♥❝❡ ♦❢

n→∞

❋✐rst✱ tr②

xn = 1/n✳

❲❡ ❤❛✈❡✿

❍♦✇❡✈❡r✱ ✇❤❛t ❛❜♦✉t

  1 lim sign = lim 1 = 1 . n→∞ n→∞ n

xn = −1/n❄

❲❡ ❤❛✈❡✿



1 lim sign − n→∞ n



= lim (−1) = −1 . n→∞

❚❤❡ ❧✐♠✐ts ❞♦♥✬t ♠❛t❝❤✦ ❋✉rt❤❡r♠♦r❡✱ t❤❡ ❢♦❧❧♦✇✐♥❣ ❧✐♠✐t ❞♦❡s♥✬t ❡✈❡♥ ❡①✐st ❛s ✇❡ ❦♥♦✇ ❢r♦♠ ❈❤❛♣t❡r ✶✿

lim sign

n→∞



(−1)n n



= lim (−1)n , n→∞

♥♦ ❧✐♠✐t.

❆❧❧ ♦❢ t❤✐s ♣♦✐♥ts ❛t ❞✐s❝♦♥t✐♥✉✐t②✦

❊①❛♠♣❧❡ ✷✳✷✳✼✿ ❥✉♠♣ ❞✐s❝♦♥t✐♥✉✐t②

❆ ♠♦r❡ ❣❡♥❡r❛❧ ✈✐❡✇ ♦❢ t❤❡ ✏❥✉♠♣ ❞✐s❝♦♥t✐♥✉✐t②✑ ✐s s❤♦✇♥ ❜❡❧♦✇✿

❖♥ t❤❡ r✐❣❤t✱ t❤❡s❡

x✬s

t❤❛t ❛r❡ ✏❝❧♦s❡✑ t♦

a

❞♦♥✬t ❛❧❧ ♣r♦❞✉❝❡

y ✬s

✏❝❧♦s❡✑ t♦

f (a)✳

✷✳✷✳

❈♦♥t✐♥✉✐t② ❛♥❞ ❞✐s❝♦♥t✐♥✉✐t②

✶✷✵

❊①❛♠♣❧❡ ✷✳✷✳✽✿ r♦♣❡

❖♥❡ ♦❢ t❤❡ ✐♥❢♦r♠❛❧ ✇❛②s t♦ s♣❡❛❦ ❛❜♦✉t ❝♦♥t✐♥✉✐t② ✐s t♦ s❛② t❤❛t t❤❡ ❣r❛♣❤ ♦❢ s✉❝❤ ❛ ❢✉♥❝t✐♦♥ ✐s ✏♠❛❞❡ ♦❢ ❛ s✐♥❣❧❡ ♣✐❡❝❡✑✳ ❲❡ ❝❛♥ ❡✈❡♥ t❤✐♥❦ ♦❢ t❤❡ ❣r❛♣❤ ❛s ❛ r♦♣❡✿

❊✈❡♥ t❤♦✉❣❤ ✐t ❧♦♦❦s ✐♥t❛❝t✱ t❤❡r❡ ♠❛② ❜❡ ❛♥ ✐♥✈✐s✐❜❧❡ ❝✉t t❤❛t ❛❧❧♦✇ ✉s t♦ ♣✉❧❧ ✐t ❛♣❛rt✳

❊①❛♠♣❧❡ ✷✳✷✳✾✿ ❝✉ts ✐♥ ❣r❛♣❤s

▲❡t✬s ✐♥✈❡st✐❣❛t❡ t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ ❢✉♥❝t✐♦♥s ❛r♦✉♥❞

f (x) = ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ s❡q✉❡♥❝❡ t♦ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣

y ✬s❄

xn → 1

❛s

x2 − 1 x−1

n→∞

a = 1✿

❛♥❞

g(x) = x + 1 .

✭s✉❝❤ ❛s

1 + 1/n✮✳

❚❤❡s❡ ❛r❡ t❤❡

x✬s✳

❲❤❛t ❤❛♣♣❡♥s

❚❤❡ ❧✐♠✐t ♦❢ t❤❡ ❢♦r♠❡r ✐s ❢♦✉♥❞ ❜② t❤✐s ❝♦♠♣✉t❛t✐♦♥✿

0 x2n − 1 → ? lim f (xn ) = lim n→∞ x→1 xn − 1 0 (xn − 1)(xn + 1) = lim n→∞ xn − 1

❉❊❆❉ ❊◆❉

= lim (xn + 1) n→∞

= lim xn + 1 n→∞

= 2. ❚❤❡ ❧✐♠✐t ♦❢ t❤❡ ❧❛tt❡r ✐s ❢♦✉♥❞ t♦♦✿

lim g(xn ) = lim (xn + 1) = lim xn + 1 = 1 + 1 = 2 .

n→∞

n→∞

n→∞

❚❤❡ t✇♦ ❢✉♥❝t✐♦♥s ❛r❡ ❛❧♠♦st t❤❡ s❛♠❡✱ ❛♥❞ t❤❡ ❞✐✛❡r❡♥❝❡ ✐s s❡❡♥ ✐♥ t❤❡✐r ❣r❛♣❤s ❜❡❧♦✇✳ ❚♦ ❡♠♣❤❛s✐③❡ t❤❡ ❞✐✛❡r❡♥❝❡✱ ✇❡ ✉s❡ ❛ ❧✐tt❧❡ ❝✐r❝❧❡ t♦ ✐♥❞✐❝❛t❡ t❤❡ ♠✐ss✐♥❣ ♣♦✐♥t✿

✷✳✷✳

❈♦♥t✐♥✉✐t② ❛♥❞ ❞✐s❝♦♥t✐♥✉✐t②

✶✷✶

❚❤❡r❡ ✐s ♦♥❧② ♦♥❡ ♣♦✐♥t ♠✐ss✐♥❣ ❢r♦♠ t❤❡ ❢♦r♠❡r ❣r❛♣❤✳ ❍♦✇❡✈❡r✱ ✐❢ ✇❡ t❤✐♥❦ ♦❢ t❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥ ❛s ❛ r♦♣❡✱ ✇❡ r❡❛❧✐③❡ t❤❛t t❤❡ ❢♦r♠❡r ❣r❛♣❤ ❝♦♥s✐sts ♦❢ t✇♦ s❡♣❛r❛t❡ ♣✐❡❝❡s✦ ❊✈❡♥ t❤♦✉❣❤ t❤❡ ❝✉t ✐s ✐♥✈✐s✐❜❧② t❤✐♥✱ ✇❡ ❝❛♥ ♣✉❧❧ t❤❡ ♣✐❡❝❡s ❛♣❛rt✳ ❚❤❡ ❧❛tt❡r ❣r❛♣❤ ✐s ❛ s✐♥❣❧❡ ♣✐❡❝❡❀ ✐t ✐s ✐♥❞❡❡❞ ✏❝♦♥t✐♥✉♦✉s✑✳ ❆ ❧✐❣❤t ❜r❡❡③❡ ✇♦✉❧❞ ❜❧♦✇ ❛♣❛rt t❤❡ ❢♦r♠❡r ❜✉t ♦♥❧② ♠♦✈❡ t❤❡ ❧❛tt❡r✿

❚❤❡ ❞✐s❝♦♥t✐♥✉♦✉s ❣r❛♣❤ ✐♥ t❤❡ ❧❛st ❡①❛♠♣❧❡ ✐s ❡❛s② t♦ ✏r❡♣❛✐r✑ ✭❣❧✉❡✱ s♦❧❞❡r✱ ✇❡❧❞✱ ❡t❝✳✮ ❜② ❛❞❞✐♥❣ ❛ s✐♥❣❧❡ ♣♦✐♥t✳ ❚❤❡ ❥✉♠♣ ❞✐s❝♦♥t✐♥✉✐t② ✐s ❛ ♠♦r❡ ❡①tr❡♠❡✱ ❝❛s❡ ♦❢ ❞✐s❝♦♥t✐♥✉✐t②❀ ✐t ❝❛♥♥♦t ❜❡ ✜①❡❞✳ ❚❤❡ ❡①❛♠♣❧❡s t❡❛❝❤ ✉s ❛ ❧❡ss♦♥✳ ●✐✈❡♥ ❛ ❢✉♥❝t✐♦♥

f

❛♥❞ ❛ ♣♦✐♥t

a✱ ✇❤❡r❡ f

✐s ❞❡✜♥❡❞✱ t❤❡ ❣r❛♣❤ ♦❢

f

❝♦♥s✐sts

♦❢ t❤r❡❡ ♣❛rts✿ ✶✳ t❤❡ ♣❛rt ♦❢ t❤❡ ❣r❛♣❤ ♦❢

f

✇✐t❤

x < a✱

✷✳ t❤❡ ♣❛rt ♦❢ t❤❡ ❣r❛♣❤ ♦❢

f

✇✐t❤

x=a

✸✳ t❤❡ ♣❛rt ♦❢ t❤❡ ❣r❛♣❤ ♦❢

f

✇✐t❤

x > a✳

✭♦♥❡ ♣♦✐♥t✮✱ ❛♥❞

❋♦r t❤✐s ❢✉♥❝t✐♦♥ t♦ ❜❡ ❝♦♥t✐♥✉♦✉s✱ t❤❡s❡ t❤r❡❡ ♣❛rts ✭t❤❡ t✇♦ ♣✐❡❝❡s ♦❢ t❤❡ r♦♣❡ ❛♥❞ ❛ ❞r♦♣ ♦❢ ❣❧✉❡✮ ❤❛✈❡ t♦ ✜t t♦❣❡t❤❡r✿

❍♦✇❄ ❲❡ r❡q✉✐r❡ t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ t②♣❡s ♦❢ ❧✐♠✐ts ❡①✐st ✐♥ t❤❡ ✜rst ♣❧❛❝❡✿

lim f (xn ) = f (a)

n→∞

❢♦r ❡❛❝❤ s❡q✉❡♥❝❡

xn → a

❆♥❞ t❤❡♥ t❤❡② ❤❛✈❡ t♦ ❜❡ ❡q✉❛❧ t♦ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❛t

✇✐t❤

xn < a . xn > a

a✳

❲❡ ❝❛♥ ❣✉❡ss t❤❛t t❤❡ s❡q✉❡♥❝❡s t❤❛t ❛❧t❡r♥❛t❡ ❜❡t✇❡❡♥ t❤❡ s✐❞❡s ✇✐❧❧ ❛❧s♦ ❤❛✈❡ t❤✐s ♣r♦♣❡rt②✳ s✐♠♣❧✐❝✐t②✱ ✇❡ ✐♥❝❧✉❞❡ ❛❧❧ s❡q✉❡♥❝❡s t❤❛t ❝♦♥✈❡r❣❡ t♦

a

✐♥ ❛♥② ♠❛♥♥❡r✿

❙♦✱ ❢♦r

✷✳✷✳

❈♦♥t✐♥✉✐t② ❛♥❞ ❞✐s❝♦♥t✐♥✉✐t②

✶✷✷

❉❡✜♥✐t✐♦♥ ✷✳✷✳✶✵✿ ❝♦♥t✐♥✉✐t② ❆ ❢✉♥❝t✐♦♥

f

✐s ❝❛❧❧❡❞

❝♦♥t✐♥✉♦✉s ❛t ❛ ♣♦✐♥t

x=a

✇❤❡♥

f

✐s ❞❡✜♥❡❞ ❛t

❛♥❞ t❤❡ ❧✐♠✐t ❜❡❧♦✇ ❡①✐sts ❛♥❞ ❡q✉❛❧ t♦ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❛t

x=a

a✿

lim f (xn ) = f (a)

n→∞

❢♦r ❡❛❝❤ s❡q✉❡♥❝❡

❞✐s❝♦♥t✐♥✉♦✉s✳

xn → a✳

■❢ t❤❡ ❧✐♠✐t ❞♦❡s♥✬t ❡①✐st✱ t❤❡ ❢✉♥❝t✐♦♥ ✐s ❝❛❧❧❡❞

❚❤❡ ❢✉♥❝t✐♦♥ ✐s ✏s❛♠♣❧❡❞✑ ❜② ❛ s❡q✉❡♥❝❡ ♦❢ ✐♥♣✉ts t♦ s❡❡ ✇❤❛t ❤❛♣♣❡♥s t♦ t❤❡ ♦✉t♣✉ts✿

❲❛r♥✐♥❣✦ ❈♦♥t✐♥✉✐t② ✐s ❡st❛❜❧✐s❤❡❞ ♦♥❡ ♣♦✐♥t ❛t ❛ t✐♠❡✳

❊①❛♠♣❧❡ ✷✳✷✳✶✶✿ ❧♦❣✐❝ ❚❤❡ ❞❡✜♥✐t✐♦♥ ✐♥❝❧✉❞❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ♣❤r❛s❡✿

◮ ❋❖❘ ❆◆❨

s❡q✉❡♥❝❡ t❤❡ ❝♦♥❞✐t✐♦♥ ✐s s❛t✐s✜❡❞✳

❚❤❡r❡❢♦r❡✱ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❞✐s❝♦♥t✐♥✉✐t② ✇✐❧❧ ✐♥❝❧✉❞❡✿

◮ ❚❍❊❘❊ ■❙

❛ s❡q✉❡♥❝❡ ❢♦r ✇❤✐❝❤ t❤❡ ❝♦♥❞✐t✐♦♥ ✐s

◆❖❚

s❛t✐s✜❡❞✳

■♥ ♦t❤❡r ✇♦r❞s✱ ❛ ✏t❡st✑ s❡q✉❡♥❝❡ ✐s ❝❤♦s❡♥ ❛♥❞ s✉❜❥❡❝t❡❞ t♦ t❤❡ ❝♦♠♣✉t❛t✐♦♥ ✐♥ t❤❡ ❞❡✜♥✐t✐♦♥✳ ■❢ ✐t ❢❛✐❧s✱ t❤❡ ❢✉♥❝t✐♦♥ ✐s ❞✐s❝♦♥t✐♥✉♦✉s✳ ■❢ ✐t ♣❛ss❡❞✱ ✇❡ ✇✐❧❧ ❤❛✈❡ t♦ tr② ❛♥♦t❤❡r✱ ❛♥❞ ♠❛②❜❡ ❛❧❧ ♦❢ t❤❡♠✳

❲❤❡r❡ ❞♦ ❞✐s❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s ❝♦♠❡ ❢r♦♠❄ ❲❡ ❝❛♥ ♦❜s❡r✈❡ t❤❡♠ ✐♥ ♥❛t✉r❡ ❛r♦✉♥❞ ✉s✿

•

❲❡ ✇❛❧❦ ❝♦♥t✐♥✉♦✉s❧② ✕ ✉♥t✐❧ ✇❡ ❢❛❧❧ ♦✛ ❛ ❝❧✐✛✳

•

❲❡ ❝♦♥t✐♥✉♦✉s❧② ✐♥❝r❡❛s❡ t❤❡ ♣r❡ss✉r❡ ✐♥ ❛ ❝❧♦s❡❞ ❝♦♥t❛✐♥❡r ✕ ✉♥t✐❧ ✐t ❡①♣❧♦❞❡s✳

•

❲❡ ❝♦♥t✐♥✉♦✉s❧② ✐♥❝r❡❛s❡ t❤❡ t❡♠♣❡r❛t✉r❡ ♦❢ ❛ ♣✐❡❝❡ ♦❢ ✐❝❡ ✕ ✉♥t✐❧ ✐t ♠❡❧ts✳

❊①❛♠♣❧❡s ♦❢ ❞✐s❝♦♥t✐♥✉✐t② r♦✉t✐♥❡❧② ❝♦♠❡ ❢r♦♠ ❤✉♠❛♥ ❛✛❛✐rs ❜❡❝❛✉s❡ t❤❡ ♦✉t❝♦♠❡s ❛r❡ ♦❢t❡♥ ❞❡s✐❣♥❡❞ t♦ ❜❡

❞✐s❝r❡t❡ ✿

❨❡s ♦r ◆♦✱ ❛❧❧ ♦r ♥♦t❤✐♥❣✱ ❡t❝✳

❊①❛♠♣❧❡ ✷✳✷✳✶✷✿ ❣r❛❞❡s ▲❡t✬s ❝♦♥s✐❞❡r ♣♦ss✐❜❧❡ ♦✉t❝♦♠❡s ✐♥ ❛ ✉♥✐✈❡rs✐t② ❡♥✈✐r♦♥♠❡♥t✳ ❲❡ st❛rt ✇✐t❤ ♣❛ss✴❢❛✐❧✳ ❚❤❡ ♦✉t❝♦♠❡ ❞❡♣❡♥❞s ♦♥ ②♦✉r t♦t❛❧ ✭♦r ❛✈❡r❛❣❡✮ s❝♦r❡✳ ❆ t❤r❡s❤♦❧❞ ✐s t②♣✐❝❛❧❧② ❛ss✐❣♥❡❞✿ ■❢ t❤❡ s❝♦r❡ ✐s ❛❜♦✈❡ ✐t✱ ②♦✉ ♣❛ss❀ ♦t❤❡r✇✐s❡ ②♦✉ ❢❛✐❧ ♥♦ ♠❛tt❡r ❤♦✇ ❝❧♦s❡ ②♦✉ ❛r❡✳ ❚❤❛t✬s ❞✐s❝♦♥t✐♥✉✐t②✦ ❯s✐♥❣ ❧❡tt❡r ❣r❛❞❡s r❡❧✐❡✈❡ t❤✐s s♦♠❡✇❤❛t✱ ❜✉t t❤❡r❡ ❛r❡ st✐❧❧ t❤r❡s❤♦❧❞s✦ ❚❤❡② ✇✐❧❧ r❡♠❛✐♥ ❡✈❡♥ ✐❢ ✇❡

✷✳✷✳

❈♦♥t✐♥✉✐t② ❛♥❞ ❞✐s❝♦♥t✐♥✉✐t②

❝❤♦♦s❡ t♦ ✉s❡

A−✱ B+✱

✶✷✸

❡t❝✳

●❡♥❡r❛❧❧②✱ t❤❡r❡ ❛r❡ ♦♥❧② ❛ ❢❡✇ ♣♦ss✐❜❧❡ ♦✉t♣✉ts ✭t❤❡ ❧❡tt❡r ❣r❛❞❡s✮ ❜✉t t❤❡ ✐♥♣✉t ✭t❤❡ s❝♦r❡✮ ✐s ❛♥ ❛r❜✐tr❛r② r❡❛❧ ♥✉♠❜❡r✳ ❛❜♦✈❡ t❤❡ ❧✐♥❡s t❤❛t

❚♦ tr② t♦ ♣❧♦t t❤❡ ❣r❛♣❤ ♦❢ ❤♦✇ t❤❡ ♦✉t❝♦♠❡s ❞❡♣❡♥❞ ♦♥ t❤❡ s❝♦r❡✱ ✇❡ s❤♦✇

♠✉st

❝♦♥t❛✐♥ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥✳ ❚❤❡r❡ ❝❛♥♥♦t ❜❡ ❛ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥

✇✐t❤ s✉❝❤ ❛ ❣r❛♣❤✳

❊①❛♠♣❧❡ ✷✳✷✳✶✸✿ ✐♥t❡❣❡r ❛♥❞ ❢r❛❝t✐♦♥❛❧ ♣❛rt ❆ ♣❛✐r ♦❢

✉s❡❢✉❧

❞✐s❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s ❛r❡ t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦✳ ❋✐rst ✐s t❤❡

❚❤❡ s❡❝♦♥❞ ✐s t❤❡

❢r❛❝t✐♦♥❛❧ ♣❛rt

✐♥t❡❣❡r ♣❛rt

✿

✿

❊①❛♠♣❧❡ ✷✳✷✳✶✹✿ ✐♥❝♦♠❡ t❛① ❆ s♣❡❝✐❛❧ ❝❛r❡ ❤❛s t♦ ❜❡ t❛❦❡♥ ✐♥ ♦r❞❡r t♦ ❡♥s✉r❡ ❝♦♥t✐♥✉✐t②✳

▲❡t✬s ❝♦♥s✐❞❡r t❤✐s ❡①❛♠♣❧❡ t❤❛t ❞❡❛❧s

✇✐t❤ t❛①❡s ✭❱♦❧✉♠❡ ✶✱ ❈❤❛♣t❡r ✶P❈✲✹✮✳ ❍②♣♦t❤❡t✐❝❛❧❧②✱ s✉♣♣♦s❡ t❤❡ t❛① ❝♦❞❡ s❛②s ❛❜♦✉t t❤❡s❡ t❤r❡❡ ❜r❛❝❦❡ts ♦❢ ✐♥❝♦♠❡✿

❲❡

• • •

$10000✱ t❤❡r❡ ✐s ♥♦ ✐♥❝♦♠❡ t❛①✳ $10000 ❛♥❞ $20000✱ t❤❡ t❛① r❛t❡ ✐s 10%✳ ■❢ ②♦✉r ✐♥❝♦♠❡ ✐s ♦✈❡r $20000✱ t❤❡ t❛① r❛t❡ ✐s 20%✳ ❝❛♥ ❡①♣r❡ss t❤✐s ❛❧❣❡❜r❛✐❝❛❧❧②✳ ❙✉♣♣♦s❡ x ✐s t❤❡ ✐♥❝♦♠❡ ❛♥❞ y = f (x)   ✐❢ x ≤ 10000 , 0 f (x) = .10 ✐❢ 10000 < x ≤ 20000 ,   .20 ✐❢ 20000 < x . ■❢ ②♦✉r ✐♥❝♦♠❡ ✐s ❧❡ss t❤❛♥ ■❢ ②♦✉r ✐♥❝♦♠❡ ✐s ❜❡t✇❡❡♥

✐s t❤❡ t❛① r❛t❡✱ t❤❡♥

✷✳✷✳

❈♦♥t✐♥✉✐t② ❛♥❞ ❞✐s❝♦♥t✐♥✉✐t②

✶✷✹

❚❤❡ ❢✉♥❝t✐♦♥ ✐s✱ ♦❢ ❝♦✉rs❡✱ ❞✐s❝♦♥t✐♥✉♦✉s ✭❧❡❢t✮✿

❲❡ s❡❡ ✇❤② t❤✐s ✐s ❛ ♣r♦❜❧❡♠ ♦♥❝❡ ✇❡ tr② t♦ ❛♣♣❧② t❤✐s ❢♦r♠✉❧❛✳ ■♠❛❣✐♥❡ t❤❛t ②♦✉r ✐♥❝♦♠❡ ❤❛s r✐s❡♥ ❢r♦♠

$10, 000

t♦

$10, 001✳

❍♦✇ ❞♦ ✇❡ ✜① t❤✐s❄

❚❤❡♥ ②♦✉r t❛① ❜✐❧❧ r✐s❡s ❢r♦♠

$0

t♦

$1, 000✦

❲❡ ♥❡❡❞ t♦ ❛ss✉r❡ t❤❛t ❛♥ ✐♥❝r❡❛s❡ ✐♥ t❤❡ ✐♥❝♦♠❡ ✇✐❧❧ ❝❛✉s❡ ♦♥❧② ❛ s♠❛❧❧❡r

✐♥❝r❡❛s❡ ✐♥ t❤❡ t❛① ❜✐❧❧✳ ❚❤✐s r❡q✉✐r❡♠❡♥t ✐♠♣❧✐❡s t❤❡ ♣r✐♥❝✐♣❧❡ ✇❡ ❤❛✈❡ ❞✐s❝✉ss❡❞✿ ❆ ✏s♠❛❧❧✑ ❝❤❛♥❣❡ ✐♥ t❤❡ ✐♥♣✉t ✭t❤❡ ✐♥❝♦♠❡✮ ✇✐❧❧ ❝❛✉s❡ ♦♥❧② ❛ ✏s♠❛❧❧✑ ❝❤❛♥❣❡ ✐♥ t❤❡ ♦✉t♣✉t ✭t❤❡ t❛① ❜✐❧❧✮✳ ❲❡ ♥♦t✐❝❡ t❤❛t t❤✐s r❡q✉✐r❡♠❡♥t ✐s s❛t✐s✜❡❞ ❛s ❧♦♥❣ ❛s t❤❡ ✐♥❝♦♠❡ st❛②s ✇✐t❤✐♥ t❤❡ ❜r❛❝❦❡ts✳

❚❤❡

✐ss✉❡ ❛r✐s❡s ♦♥❧② ❛t t❤❡ tr❛♥s✐t✐♦♥ ♣♦✐♥ts ❛♥❞ ❝❛♥ ❜❡ ❛❞❞r❡ss❡❞ ✐♥ t❤❡ ✇❛② s❤♦✇♥ ♦♥ t❤❡ r✐❣❤t✳ ❚❤❡ ✐♥t❡r♣r❡t❛t✐♦♥ ♦❢ t❤❡ r❡s✉❧t ✐s ♣♦ss✐❜❧❡ ✐❢ ✇❡ ✉♥❞❡rst❛♥❞ t❤❡ ❧❛✇ ❝♦rr❡❝t❧②✳ ❚❤❡ t❛① r❛t❡ ✐s ✏♠❛r❣✐♥❛❧✑❀ ✐✳❡✳✱ ✐t ✐s t❤❡ t❛① r❛t❡ ❛♣♣❧✐❡❞ t♦ t❤❡ ♣❛rt ♦❢ t❤❡ ✐♥❝♦♠❡ t❤❛t ❧✐❡s ✇✐t❤✐♥ t❤❡ ❜r❛❝❦❡t ❛♥❞ t❤❡s❡ t❤r❡❡ ♥✉♠❜❡rs ❛r❡ ♠❡❛♥t t♦ ❜❡ ❛❞❞❡❞ t♦❣❡t❤❡r t♦ ♣r♦❞✉❝❡ ②♦✉r t❛① ❜✐❧❧ ❧✐❛❜✐❧✐t②✳ ❚❤✐s ✐s t❤❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ t❛① ❜✐❧❧ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ ✐♥❝♦♠❡✿

  0 g(x) = .10 · (x − 10000)   .10 · (x − 10000) + .20 · (x − 20000)

✐❢ ✐❢ ✐❢

x ≤ 10000 , 10000 < x ≤ 20000 , 20000 < x .

❚❤❡ ❢✉♥❝t✐♦♥ ✐s ❝♦♥t✐♥✉♦✉s✳ ■♥❞❡❡❞✱ ✇❡ ❤❛✈❡✿

• •

■❢ ■❢

xn → 10000 xn → 10000

❛♥❞ ❛♥❞

x ≤ 10000✱ x > 10000✱

t❤❡♥ t❤❡♥

g(xn ) = 0 → 0 ✳ g(xn ) = .1(xn − 10000) → 0 ✳

❊①❡r❝✐s❡ ✷✳✷✳✶✺

Pr♦✈❡ t❤❛t t❤❡ ❢♦r♠✉❧❛ ❛❜♦✈❡ s❛t✐s✜❡s t❤❡ r❡q✉✐r❡♠❡♥ts✳

Pr♦✈✐❞❡ t❤❡ ♠✐ss✐♥❣ ♣❛rts ♦❢ t❤❡ ♣r♦♦❢ ♦❢

❝♦♥t✐♥✉✐t②✳

❊①❡r❝✐s❡ ✷✳✷✳✶✻

Pr♦✈✐❞❡ ❛ s✐♠✐❧❛r ❛♥❛❧②s✐s ❢♦r t❤❡ ♣♦ss✐❜✐❧✐t② ♦❢ ❛ ✇❡❧❢❛r❡ ♣❛②♠❡♥t ♦❢ ❜❡❧♦✇

$5000✳

❊✈❡♥ ♠♦r❡ ❡①tr❡♠❡ ❡①❛♠♣❧❡s ❛r❡ ❜❡❧♦✇✳ ❊①❛♠♣❧❡ ✷✳✷✳✶✼✿ r❡❝✐♣r♦❝❛❧ ❛t

0

❚❤❡ r❡❝✐♣r♦❝❛❧ ❢✉♥❝t✐♦♥

g(x) = ❤❛s ❛♥

✐♥✜♥✐t❡ ❣❛♣

❛t

x = 0✿

1 x

$1000 t♦ ❛ ♣❡rs♦♥ ✇✐t❤ ❛♥ ✐♥❝♦♠❡

✷✳✷✳ ❈♦♥t✐♥✉✐t② ❛♥❞ ❞✐s❝♦♥t✐♥✉✐t②

✶✷✺

❍❡r❡✱ ✐❢ t❤❡ ❞❡✈✐❛t✐♦♥s ♦❢ x ❢r♦♠ 0 ❞✐✛❡r ✐♥ s✐❣♥✱ t❤❡ ❥✉♠♣ ✐♥ y ♠❛② ❜❡ ✈❡r② ❧❛r❣❡✿ ❢r♦♠

1 1 = −10000 t♦ = 10000 . −.0001 .0001

❆ s✐♠♣❧❡ t❡st s❡q✉❡♥❝❡ ✐s xn = 1/n✳ ❚❤❡ ❝♦♠♣♦s✐t✐♦♥ ✇✐t❤ t❤❡ ❢✉♥❝t✐♦♥ ❣✐✈❡s ✉s t❤❡ ❢♦❧❧♦✇✐♥❣✿ yn = g(xn ) = 1/xn = 1/(1/n) = n → +∞ .

❚❤❡ ❞✐s❝♦♥t✐♥✉✐t② ✐s ♣r♦✈❡♥✳ ❊①❛♠♣❧❡ ✷✳✷✳✶✽✿

sin(1/x)

❛t

0

❚❤❡ s✐♥❡ ♦❢ t❤❡ r❡❝✐♣r♦❝❛❧

  1 g(x) = sin x ♦s❝✐❧❧❛t❡s ✐♥✜♥✐t❡❧② ♠❛♥② t✐♠❡s ❛s x ❛♣♣r♦❛❝❤❡s 0✿

❍❡r❡✱ ❛ ❞❡✈✐❛t✐♦♥ ♦❢ x ❢r♦♠ 0 ♠❛② ✉♥♣r❡❞✐❝t❛❜❧② ♣r♦❞✉❝❡ ❛♥② ♥✉♠❜❡r ❜❡t✇❡❡♥ −1 ❛♥❞ 1✳ ❲❡ ❝❛♥ ❣✉❡ss t❤❛t t❤❡ ❢✉♥❝t✐♦♥ ❝❛♥♥♦t ❜❡ ❝♦♥t✐♥✉♦✉s ❛t 0✦ ❚♦ ❛♣♣r❡❝✐❛t❡ t❤✐s ❝♦♥❝❧✉s✐♦♥ ❢r♦♠ t❤❡ ❣❡♦♠❡tr✐❝ ♣♦✐♥t ♦❢ ✈✐❡✇✱ ✐t ✐s ✐♠♣♦ss✐❜❧❡ t♦ ❛tt❛❝❤ t❤✐s ❣r❛♣❤ t♦ ❛ s✐♥❣❧❡ ♣♦✐♥t ♦♥ t❤❡ y ✲❛①✐s ♥♦ ♠❛tt❡r ✇❤❛t t❤❛t ♣♦✐♥t ♠❛② ❜❡✳ ❆ s✐♠♣❧❡ t❡st s❡q✉❡♥❝❡ ✐s xn = 1/n✳ ❚❤❡ ❝♦♠♣♦s✐t✐♦♥ ✇✐t❤ t❤❡ ❢✉♥❝t✐♦♥ ❣✐✈❡s ✉s t❤❡ ❢♦❧❧♦✇✐♥❣✿ lim yn = lim g(xn ) = lim sin

n→∞

❚❤❡ ❞✐s❝♦♥t✐♥✉✐t② ✐s ♣r♦✈❡♥✳

n→∞

n→∞



1 1/n



= lim sin n ❉◆❊. n→∞

■♥ t❤❡ ♠❡❛♥t✐♠❡✱ t❤❡ ❢✉♥❝t✐♦♥ h(x) = x sin (1/x)

❛❧s♦ ♦s❝✐❧❧❛t❡s ✐♥✜♥✐t❡❧② ♠❛♥② t✐♠❡s ❛s ✇❡ ❛♣♣r♦❛❝❤ 0✳ ❨❡t✱ ❜❡❝❛✉s❡ t❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ t❤❡s❡ ♦s❝✐❧❧❛t✐♦♥s ❞✐♠✐♥✐s❤❡s✱ t❤❡ ❧✐♠✐t ♦❢ t❤❡ ❝♦♠♣♦s✐t✐♦♥ ✇✐t❤ ❛♥② t❡st s❡q✉❡♥❝❡ ❡①✐sts ✭✐t✬s 0✦✮✿

✷✳✷✳

❈♦♥t✐♥✉✐t② ❛♥❞ ❞✐s❝♦♥t✐♥✉✐t②

✶✷✻

❉♦❡s t❤✐s ♠❛❦❡ t❤❡ ❢✉♥❝t✐♦♥ ❝♦♥t✐♥✉♦✉s❄ ◆♦✱ t❤❡r❡ ✐s st✐❧❧ ❛ ♣♦✐♥t ♠✐ss✐♥❣ ❛t 0✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ t✇♦ ♦s❝✐❧❧❛t✐♥❣ ❜r❛♥❝❤❡s ❛r❡ ❞❡t❛❝❤❡❞✳ ▲❡t✬s ❣❧✉❡ t❤❡♠ t♦❣❡t❤❡r✳ ❲❡ s✐♠♣❧② ❞❡✜♥❡ ❛ ♥❡✇ ❢✉♥❝t✐♦♥✿ f (x) =

(

x sin (1/x) 0

✐❢ x 6= 0 , ✐❢ x = 0 .

❊①❛♠♣❧❡ ✷✳✷✳✶✾✿ ♥♦✇❤❡r❡ ❝♦♥t✐♥✉♦✉s

❚❤❡

❉✐r✐❝❤❧❡t ❢✉♥❝t✐♦♥ ✐s ♥♦✇❤❡r❡ ❝♦♥t✐♥✉♦✉s✦ IQ (x) =

(

1 0

■t ✐s ❞❡✜♥❡❞ ❜②✿

✐❢ x ✐s ❛ r❛t✐♦♥❛❧ ♥✉♠❜❡r, ✐❢ x ✐s ❛♥ ✐rr❛t✐♦♥❛❧ ♥✉♠❜❡r.

■♥ ❛♥ ❛tt❡♠♣t t♦ ♣❧♦t ✐ts ❣r❛♣❤✱ ✇❡ ❝❛♥ ♦♥❧② ❞r❛✇ t✇♦ ❤♦r✐③♦♥t❛❧ ❧✐♥❡s ❛♥❞ t❤❡♥ s✐♠♣❧② ♣♦✐♥t ♦✉t s♦♠❡ ♦❢ t❤❡ ♠✐ss✐♥❣ ♣♦✐♥ts ✐♥ ❡✐t❤❡r✿

■❢ ✇❡ ✇❡r❡ t♦ ♣❧♦t ❛❧❧ ♣♦✐♥ts✱ ✇❡✬❞ ❤❛✈❡ ✇❤❛t ❧♦♦❦s ❧✐❦❡ t✇♦ ❝♦♠♣❧❡t❡ str❛✐❣❤t ❧✐♥❡s✳ ❍♦✇❡✈❡r✱ ❛ ❧✐❣❤t ❜r❡❡③❡ ✇♦✉❧❞ ❜❧♦✇ t❤❡♠ ❛♣❛rt✿

✷✳✸✳ ▲✐♠✐ts ♦❢ ❢✉♥❝t✐♦♥s✿ s♠❛❧❧ s❝❛❧❡ tr❡♥❞s

✶✷✼

❊①❡r❝✐s❡ ✷✳✷✳✷✵

❉✐s❝✉ss ❝♦♥t✐♥✉✐t② ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥t❡①t✿

✷✳✸✳ ▲✐♠✐ts ♦❢ ❢✉♥❝t✐♦♥s✿ s♠❛❧❧ s❝❛❧❡ tr❡♥❞s

❚❤❡ s✉❜❥❡❝t ♦❢ ♦✉r st✉❞② ✐s ❝❧❡❛r ♥♦✇✿ ■t ✐s s♠❛❧❧ s❝❛❧❡ ❜❡❤❛✈✐♦r ♦❢ ❢✉♥❝t✐♦♥s✳ ❲❡ ❛s❦✱ ✇❤❛t ❢❡❛t✉r❡s ♦❢ t❤❡ ❣r❛♣❤ ✇♦♥✬t ❞✐s❛♣♣❡❛r ♥♦ ♠❛tt❡r ❤♦✇ ♠✉❝❤ ✇❡ ③♦♦♠ ✐♥❄ ❖♥❡ ♦❢ t❤❡ ♠♦st ❝r✉❝✐❛❧ ♣r♦♣❡rt✐❡s ♦❢ ❛ ❢✉♥❝t✐♦♥ ✐s t❤❡ ✐♥t❡❣r✐t② ♦❢ ✐ts ❣r❛♣❤✿ ■s t❤❡r❡ ❛ ❜r❡❛❦ ♦r ❛ ❝✉t❄ ■♥ ❛❞❞✐t✐♦♥✱ ✐♥ ♦r❞❡r t♦ st✉❞② ♠♦t✐♦♥✱ ✇❡ t②♣✐❝❛❧❧② ❛ss✉♠❡ t❤❛t t♦ ❣❡t ❢r♦♠ ♣♦✐♥t ✈✐s✐t ❡✈❡r② ❧♦❝❛t✐♦♥ ❜❡t✇❡❡♥

A

❛♥❞

A

t♦ ♣♦✐♥t

B✱

✇❡ ❤❛✈❡ t♦

B✿

■❢ t❤❡r❡ ✐s ❛ ❥✉♠♣ ✐♥ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥✱ ✐t ❝❛♥✬t r❡♣r❡s❡♥t ♠♦t✐♦♥✦ ❋✉rt❤❡r♠♦r❡✱ ✐❢ ✐t r❡♣r❡s❡♥ts ❛ tr❛♥s❢♦r♠❛t✐♦♥✱ t❤❡r❡ ✐s t❡❛r✐♥❣✳ ❚❤✉s✱ ✇❡ ✇❛♥t t♦ ✉♥❞❡rst❛♥❞ ✇❤❛t ✐s ❤❛♣♣❡♥✐♥❣ t♦

y = f (x)

✇❤❡♥

x

✐s ✐♥ t❤❡ ✈✐❝✐♥✐t② ♦❢ ❛ ❝❤♦s❡♥ ♣♦✐♥t

x = a✳ ❚❤❡ ♠❛✐♥ t♦♦❧ ✇✐t❤ ✇❤✐❝❤ ✇❡ ❝❤♦♦s❡ t♦ t❡st t❤❡ ❜❡❤❛✈✐♦r ♦❢ ❛ ❢✉♥❝t✐♦♥ ❛r♦✉♥❞ ❛ ♣♦✐♥t ✇✐❧❧ ❜❡✿

◮

❛ s❡q✉❡♥❝❡ ❝♦♥✈❡r❣✐♥❣ t♦ t❤❡ ♣♦✐♥t✱

❲❡ t❤❡♥ ❝♦♥s✐❞❡r t❤❡ ❡✛❡❝t ♦❢

f

xn → a ✱

❜✉t ♥❡✈❡r ❡q✉❛❧ t♦ ✐t✱

an 6= a✳

♦♥ t❤✐s s❡q✉❡♥❝❡✳ ❋✐rst✱ ✇❡ ✏❧✐❢t✑ t❤❡ ♣♦✐♥ts ♦❢ t❤❡ s❡q✉❡♥❝❡ ❢r♦♠ t❤❡

t♦ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥✿

❲❡ t❤❡♥ ❧♦♦❦ ❢♦r ❛ ♣♦ss✐❜❧❡ ❧♦♥❣✲t❡r♠ ♣❛tt❡r♥ ♦❢ ❜❡❤❛✈✐♦r ♦❢ t❤✐s ♥❡✇ s❡q✉❡♥❝❡✿

x✲❛①✐s

✷✳✸✳ ▲✐♠✐ts ♦❢ ❢✉♥❝t✐♦♥s✿ s♠❛❧❧ s❝❛❧❡ tr❡♥❞s

◮

✶✷✽

❉♦ t❤❡s❡ ♣♦✐♥ts ❛❝❝✉♠✉❧❛t❡ t♦ ❛♥♦t❤❡r ♣♦✐♥t ♦♥ t❤❡ ❣r❛♣❤❄

x✲❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ y ✲❝♦♦r❞✐♥❛t❡s❄

❙✐♥❝❡ ✇❡ ❛❧r❡❛❞② ❦♥♦✇ t❤❛t t❤❡ ❜❡❝♦♠❡s✿ ❲❤❛t ✐s ❤❛♣♣❡♥✐♥❣ t♦

t❤❡s❡ ♣♦✐♥ts ❞♦ ❛❝❝✉♠✉❧❛t❡ ✭t♦

x = a✮✱

t❤❡ q✉❡st✐♦♥

❊①❛♠♣❧❡ ✷✳✸✳✶✿ t❤r❡❡ ❢✉♥❝t✐♦♥s ✕ t❤r❡❡ ❜❡❤❛✈✐♦rs ▲❡t✬s st✉❞② t❤❡s❡ t❤r❡❡ ❢✉♥❝t✐♦♥s ❛r♦✉♥❞ t❤❡ ♣♦✐♥t

f (x) = sin x,

g(x) =

a = 0✿ 1 , x

h(x) = sin(1/x) .

❚❤❡ r❡❝✐♣r♦❝❛❧ s❡q✉❡♥❝❡ ✐s ❛♥ ❛♣♣r♦♣r✐❛t❡ ❝❤♦✐❝❡ ❢♦r ❛ t❡st s❡q✉❡♥❝❡✿

xn = ❚❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ ❛ ❢✉♥❝t✐♦♥ ❛♥❞ ❛ s❡q✉❡♥❝❡ ♦❢

1 → 0. n x✬s

❣✐✈❡s ✉s ❛ ♥❡✇ s❡q✉❡♥❝❡✱ ❛ s❡q✉❡♥❝❡ ♦❢

y ✬s✳

❲❡

❤❛✈❡ t❤r❡❡✱ ♦♥❡ ❢♦r ❡❛❝❤ ❢✉♥❝t✐♦♥✿

an = cos(1/n),

1 , bn = 1/n

cn = sin



1 1/n



.

x✲❛①✐s t♦ t❤❡ xn ♦♥ t❤❡ x✲❛①✐s✱ t❤❡♥ (xn , yn ) ♦♥ t❤❡ ❣r❛♣❤✱ ❛♥❞ yn ♦♥ t❤❡ y ✲❛①✐s✳

❚❤✐s ✐s ❤♦✇ t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ t❤❡s❡ s❡q✉❡♥❝❡s ❝❛♥ ❜❡ ✈✐s✉❛❧✐③❡❞✳ ❲❡ ♣r♦❝❡❡❞ ❢r♦♠ t❤❡ ❣r❛♣❤ t♦ t❤❡

❍❡r❡ ✐s

y ✲❛①✐s ❛s ❢♦❧❧♦✇s✿

f (x) = sin x✿

❲❡ s❡❡ t❤❛t t❤❡ s❡q✉❡♥❝❡s

❍❡r❡ ✐s

❲❡ ♣❧♦t

g(x) = 1/x✿

xn

❛♥❞

yn

❧♦♦❦ ✈❡r② s✐♠✐❧❛r✳

✷✳✸✳

✶✷✾

▲✐♠✐ts ♦❢ ❢✉♥❝t✐♦♥s✿ s♠❛❧❧ s❝❛❧❡ tr❡♥❞s

❲❡ s❡❡ t❤❛t xn ❝♦♥✈❡r❣❡s t♦ 0 ❜✉t yn ❣♦❡s t♦ ✐♥✜♥✐t②✳ ❍❡r❡ ✐s h(x) = sin (1/x)✿

❲❡ s❡❡ t❤❛t xn ❝♦♥✈❡r❣❡s t♦ 0 ❜✉t yn ❧♦♦❦s r❛♥❞♦♠❧② s♣r❡❛❞ ❛r♦✉♥❞ t❤❡ ✐♥t❡r✈❛❧ [−1, 1]✳ ❆♥❞ t❤✐s ✐s ✇❤❛t ✇❡ ❞✐s❝♦✈❡r ❛❜♦✉t t❤❡s❡ ♥❡✇ s❡q✉❡♥❝❡s ✇❤❡♥ ✇❡ ❡①♣❧♦r❡ t❤❡♠ ♥✉♠❡r✐❝❛❧❧②✿ an = sin(1/n) → 0,

bn = n → ∞,

cn = sin(n) ♥♦ ❧✐♠✐t.

❚❤✉s✱ t❤❡ ✐♥✐t✐❛❧ ✐❞❡❛ ♦❢ ❤♦✇ t♦ ✜♥❞ ✇❤❛t ✐s ❤❛♣♣❡♥✐♥❣ t♦ y = f (x) ❛s x ✐s ❛♣♣r♦❛❝❤✐♥❣ a ✐s t♦ ♣✐❝❦ ❛ s❡q✉❡♥❝❡ t❤❛t ❛♣♣r♦❛❝❤❡s a✱ ✐✳❡✳✱ xn → a✳ ❚❤❡♥ ✇❡ ❡✈❛❧✉❛t❡ t❤✐s ❧✐♠✐t ♦❢ ❛ ♥❡✇ s❡q✉❡♥❝❡ t❤❛t ❝♦♠❡s ❢r♦♠ s✉❜st✐t✉t✐♦♥✿ lim f (xn ) = ?

n→∞

■❢ ✇❡ t❤✐♥❦ ♦❢ t❤❡ s❡q✉❡♥❝❡ xn ❛s ❛ ❢✉♥❝t✐♦♥✱ t❤❡♥ ✇❡ s❤♦✉❧❞ ✐♥t❡r♣r❡t t❤✐s s✉❜st✐t✉t✐♦♥✱ yn = f (xn ) ,

❛s t❤❡ ❝♦♠♣♦s✐t✐♦♥✱ ❛s ❞✐s❝✉ss❡❞ ✐♥ ❈❤❛♣t❡r ✶✳ ❆ s✐♥❣❧❡ s❡q✉❡♥❝❡ ♠✐❣❤t ♥♦t ❜❡ ❡♥♦✉❣❤ t❤♦✉❣❤✦ ❊①❛♠♣❧❡ ✷✳✸✳✷✿ s✐❣♥ ❢✉♥❝t✐♦♥

❏✉st ♦❜s❡r✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢❛✐❧✉r❡ ♦❢ ❛ s✐♥❣❧❡ s❡q✉❡♥❝❡ t♦ t❡❧❧ ✉s ✇❤❛t ✐s ❣♦✐♥❣ ♦♥ ✇✐t❤ t❤❡ ❢✉♥❝t✐♦♥ sign ❛r♦✉♥❞ 0✳

❲❡ tr② xn = −1/n ❛♥❞ xn = 1/n✿ lim sign(−1/n) = −1, ❜✉t lim sign(1/n) = 1 ,

n→∞

n→∞

❛s ✇❡ ❛♣♣r♦❛❝❤ 0 ❢r♦♠ ♦♥❡ ❞✐r❡❝t✐♦♥ ❛t ❛ t✐♠❡✳ ❚❤❡ ❧✐♠✐ts ❞♦♥✬t ♠❛t❝❤✦ ■♥ ❢❛❝t✱ t❤❡ r❡s✉❧t s✉❣❣❡sts t❤❛t t❤❡ ❧✐♠✐t ♦❢ sign(x) s✐♠♣❧② ❞♦❡s♥✬t ❡①✐st ❛t t❤✐s ♣♦✐♥t✳ ❚❤❡r❡ ✐s ♥♦ tr❡♥❞✦ ❊①❡r❝✐s❡ ✷✳✸✳✸

❲❤❛t t❡st s❡q✉❡♥❝❡ s❤♦✉❧❞ ✇❡ ❝❤♦♦s❡ t♦ ♣r♦✈❡ t❤❡ ❞✐s❝♦♥t✐♥✉✐t② ♦❢ t❤✐s ❢✉♥❝t✐♦♥❄

✷✳✸✳

✶✸✵

▲✐♠✐ts ♦❢ ❢✉♥❝t✐♦♥s✿ s♠❛❧❧ s❝❛❧❡ tr❡♥❞s

❊①❛♠♣❧❡ ✷✳✸✳✹✿ s✐♥❡ ♦❢ r❡❝✐♣r♦❝❛❧ ❆♥♦t❤❡r ❢❛✐❧✉r❡ ✐s ♦❢ ❛ s✐♥❣❧❡ s❡q✉❡♥❝❡ t❡st ✐s ❛s ❢♦❧❧♦✇s✿ lim sin

n→∞

❜✉t lim sin

n→∞



1 1/n



= lim sin n ❞♦❡s♥✬t ❡①✐st✱ n→∞

1 = lim sin(πn) = lim 0 = 0 ❞♦❡s✳ n→∞ πn n→∞

❍❡r❡ ❛r❡ ✜✈❡ t❤❛t ♣❛ss ✭❧❡❢t✮ ❛♥❞ ♦♥❡ t❤❛t ❢❛✐❧s ✭r✐❣❤t✮✿

❲❡ tr② t♦ s✉❜st✐t✉t❡ ❢r♦♠ f ✳

❛❧❧

s❡q✉❡♥❝❡s t❤❛t ❝♦♥✈❡r❣❡ t♦ a ❛♥❞ ❡♥s✉r❡ t❤❡② ❛❧❧ ✐♥❞✉❝❡ t❤❡ s❛♠❡ ❜❡❤❛✈✐♦r

❲❡ ✜♥❛❧❧② s✉♠♠❛r✐③❡ t❤✐s ✐❞❡❛ ❤❡r❡✿

❉❡✜♥✐t✐♦♥ ✷✳✸✳✺✿ ❧✐♠✐t ♦❢ ❢✉♥❝t✐♦♥ ❚❤❡ ❧✐♠✐t ♦❢ ❛ ❢✉♥❝t✐♦♥ f ❛t ❛ ♣♦✐♥t x = a ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ❧✐♠✐t ♦❢ ✐ts ❝♦♠♣♦s✐t✐♦♥ ✇✐t❤ ❛♥② s❡q✉❡♥❝❡ xn ✇✐t❤✐♥ t❤❡ ❞♦♠❛✐♥ ♦❢ f ❡①❝❧✉❞✐♥❣ a t❤❛t ❝♦♥✈❡r❣❡s t♦ a✱ xn → a ✇✐t❤ xn 6= a , ✇❤❡♥ ❛❧❧ t❤❡s❡ ❧✐♠✐ts ❡①✐st✱ ✜♥✐t❡ ♦r ✐♥✜♥✐t❡✱ ❛♥❞ ❛r❡ ❡q✉❛❧ t♦ ❡❛❝❤ ♦t❤❡r✿ yn = f (xn ) → L .

❆s ②♦✉ ❝❛♥ s❡❡✱ ❡✈❡♥ t❤♦✉❣❤ ✇❡ s❡t t❤❡ ✐ss✉❡ ♦❢ ❝♦♥t✐♥✉✐t② ❛s✐❞❡ ❢♦r ❛ ♠♦♠❡♥t✱ t❤❡ ♠❡t❤♦❞ r❡♠❛✐♥s t❤❡ s❛♠❡✿ ❲❡ t❡st t❤❡ ❢✉♥❝t✐♦♥ ✇✐t❤ s❡q✉❡♥❝❡s✳ ■❢ ❛ s❡q✉❡♥❝❡ ♦❢ x✬s ❝♦♥✈❡r❣❡s✱ ❞♦❡s t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s❡q✉❡♥❝❡ ♦❢ y ✬s ❝♦♥✈❡r❣❡ t♦♦❄ ❚❤❛t✬s ❛ t❡st✿

✷✳✸✳ ▲✐♠✐ts ♦❢ ❢✉♥❝t✐♦♥s✿ s♠❛❧❧ s❝❛❧❡ tr❡♥❞s

✶✸✶

❇✉t t❤❡② ❛❧❧ ❤❛✈❡ t♦ ♣❛ss ✐t✳ ❆♥❞ t❤❡ r❡s✉❧ts ❤❛✈❡ t♦ ❜❡ t❤❡ s❛♠❡✦

❊①❛♠♣❧❡ ✷✳✸✳✻✿ ✏❢♦r ❛♥② ✳✳✳ t❤❡r❡ ✐s✑ ❲❡ s❤♦✉❧❞♥✬t ❝♦♥❢✉s❡ t❤❡ t✇♦ ❢❛♠✐❧✐❛r ❝♦♥str✉❝ts ✐♥ t❤❡ ❧❛st ♣❛rt ♦❢ t❤❡ ❞❡✜♥✐t✐♦♥✳ ■t ✐s✿

◮ ❋❖❘ ❆◆❨

s❡q✉❡♥❝❡✳✳✳

❛♥❞ ♥♦t

◮ ❚❍❊❘❊ ■❙

❛ s❡q✉❡♥❝❡✳✳✳

❚❤❡ ❡①❛♠♣❧❡s ❛❜♦✈❡ ✐❧❧✉str❛t❡ t❤❡ ❞✐✛❡r❡♥❝❡✳

❚❤❡ ❞❡✈❡❧♦♣♠❡♥t✱ t❤❡ ♥♦t❛t✐♦♥✱ ❛♥❞ t❤❡ t❡r♠✐♥♦❧♦❣② ♦❢ t❤❡ ❧✐♠✐ts ♦❢ ❢✉♥❝t✐♦♥s ❢♦❧❧♦✇ t❤❛t ❢♦r s❡q✉❡♥❝❡s✳ ❋✉rt❤❡r♠♦r❡✱ t❤❡ ❝♦♠♣❧❡①✐t② ♦❢ t❤❡ ❞❡✜♥✐t✐♦♥s ❢♦r t❤❡ ❢♦r♠❡r ✐s ❞✐♠✐♥✐s❤❡❞ ❜❡❝❛✉s❡ ✇❡ r❡❧② ♦♥ t❤♦s❡ ❢♦r t❤❡ ❧❛tt❡r✳ ❲❡ ♥❡❡❞ t♦ ❥✉st✐❢② ✏t❤❡✑ ✐♥ ✏t❤❡ ❧✐♠✐t✑✳ ■♥❞❡❡❞✱ ❛ ❧✐♠✐t ✐s ❞❡✜♥❡❞ ❛s ❛ ♥✉♠❜❡r t❤❛t s❛t✐s✜❡s ❛ ❝❡rt❛✐♥ ♣r♦♣❡rt② ❛♥❞ t❤❡r❡ ✐s ♥♦ r❡❛s♦♥ t♦ ❛ss✉♠❡ t❤❛t t❤❡r❡ ♠✉st ❜❡ ♦♥❧② ♦♥❡✳

❚❤❡♦r❡♠ ✷✳✸✳✼✿ ❯♥✐q✉❡♥❡ss ♦❢ ▲✐♠✐t ♦❢ ❋✉♥❝t✐♦♥ ■❢ ✇❡ ❤❛✈❡ t✇♦ ❧✐♠✐ts ♦❢ t❤❡ s❛♠❡ ❢✉♥❝t✐♦♥ ❛t t❤❡ s❛♠❡ ♣♦✐♥t✱ t❤❡② ❛r❡ ❡q✉❛❧✳

Pr♦♦❢✳ ■t ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ✉♥✐q✉❡♥❡ss ♦❢ t❤❡ ❧✐♠✐t ♦❢ ❛ s❡q✉❡♥❝❡ ♣r❡s❡♥t❡❞ ✐♥ t❤❡ ❧❛st ❝❤❛♣t❡r✳

❲❡ ✉s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♥♦t❛t✐♦♥ ❢♦r t❤❡ ❧✐♠✐t ♦❢ ❛ ❢✉♥❝t✐♦♥✿

▲✐♠✐t ♦❢ ❢✉♥❝t✐♦♥

f (x) → L

x→a

❛s

❛♥❞

lim f (x) = L .

x→a

■t r❡❛❞s ✏t❤❡ ❧✐♠✐t ♦❢

x

✐s ❛♣♣r♦❛❝❤✐♥❣

a

✐s

f (x) L✑✳

❛s

❲❛r♥✐♥❣✦ ❚❤❡ ✉s❡ ♦❢ ✏ f (x)

→ L

❛s

x → a✑

♠✐❣❤t ♠✐s❧❡❛❞

♦♥❡ ✐♥t♦ t❤✐♥❦✐♥❣ t❤❛t t❤❡ t✇♦ ♣❛rts ❛r❡ ✐❞❡♥t✐❝❛❧✱

f (x) ❝❛♥ ❜❡ ❡q✉❛❧ t♦ L ❜✉t x ❝❛♥♥♦t a✳ ❙♦♠❡ ❧✐❦❡ t♦ ✉s❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿

✇❤✐❧❡ ✐♥ ❢❛❝t ❜❡ ❡q✉❛❧ t♦

f (x) → L

❛s

x → a6= .

❲❡ ✉s❡ ❛ s✐♠✐❧❛r ♥♦t❛t✐♦♥ t♦ ❞❡s❝r✐❜❡ t❤❡ s♣❡❝✐❛❧ ❦✐♥❞ ♦❢ ❞✐✈❡r❣❡♥t ❜❡❤❛✈✐♦r ✭✐♥✜♥✐t②✮✿

■♥✜♥✐t❡ ❧✐♠✐t ♦❢ ❢✉♥❝t✐♦♥

f (x) → ±∞

❛s

x→a

♦r

lim f (x) = ±∞ .

x→a

f (x) ❛s x ✐s ❛♣♣r♦❛❝❤✐♥❣ a ✐s ✐♥✜♥✐t❡✑✳

■t r❡❛❞s ✏t❤❡ ❧✐♠✐t ♦❢

✷✳✸✳ ▲✐♠✐ts ♦❢ ❢✉♥❝t✐♦♥s✿ s♠❛❧❧ s❝❛❧❡ tr❡♥❞s

✶✸✷

❲❡ r❡st❛t❡ t❤❡ t❤❡♦r❡♠✿

L = lim f (x) ❆◆❉ M = lim f (x) =⇒ L = M x→a

x→a

❚❤❡ t❤✐r❞ ♣♦ss✐❜✐❧✐t② ✐s t❤❛t t❤❡r❡ ✐s ♥♦ tr❡♥❞✳ ❙✐♥❝❡ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❝♦♥✈❡r❣❡♥❝❡ st❛rts ✇✐t❤ ✏ ❚❍❊❘❊ ♥✉♠❜❡r

a✳✳✳✑✱

■❙

✇❡ ❝❛♥ s❛② t❤❛t ♦t❤❡r✇✐s❡ t❤❡r❡ ✐s ♥♦ ❧✐♠✐t✿ ❚❤❡ ❧✐♠✐t ❞♦❡s ♥♦t ❡①✐st

lim f (x),

x→a

♥♦ ❧✐♠✐t

♦r

lim f (x)

x→a

❉◆❊✳

■t r❡❛❞s ✏t❤❡ ❧✐♠✐t ♦❢

x

✐s ❛♣♣r♦❛❝❤✐♥❣

a

f (x)

❛s

❞♦❡s ♥♦t

❡①✐st✑✳

❇❡❧♦✇✱ t❤❡ ❧✐♠✐t ❞♦❡s ❡①✐st❀ ✐t✬s ❥✉st ♥♦t ❡q✉❛❧ t♦ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥✿

❚❤❡ ❢✉♥❝t✐♦♥ ✐s ❞✐s❝♦♥t✐♥✉♦✉s✦ ❲❡ ❤❛✈❡ ❛ ♥❡✇ ✇❛② t♦ ❡①♣r❡ss ❝♦♥t✐♥✉✐t②✿ ❚❤❡♦r❡♠ ✷✳✸✳✽✿ ❈♦♥t✐♥✉✐t② ✈✐❛ ▲✐♠✐t

❆ ❢✉♥❝t✐♦♥ f ✐s ❝♦♥t✐♥✉♦✉s ❛t x = a ✐❢ ❛♥❞ ♦♥❧② ✐❢ f ✐s ❞❡✜♥❡❞ ❛t a✱ t❤❡ ❧✐♠✐t ❛t a ❡①✐sts ❛♥❞ ✐s ❡q✉❛❧ t♦ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❛t a✿ lim f (x) = f (a)

x→a

❊①❛♠♣❧❡ ✷✳✸✳✾✿ ❧✐♠✐ts ❢r♦♠ ❝♦♥t✐♥✉✐t②

■❢ t❤❡ ❢✉♥❝t✐♦♥ ✐s ❦♥♦✇♥ t♦ ❜❡ ❝♦♥t✐♥✉♦✉s✱ t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ❧✐♠✐t ✐s tr✐✈✐❛❧✿

lim 3x3 + 2x2 + 1 = 3x3 + 2x2 + 1 x=3 = 3 · 33 + 2 · 32 + 1 x→3 2 2 2 lim ex = ex x=π = eπ x→π lim sin x = sin x x=−1 = sin(−1) x→−1

❊①❛♠♣❧❡ ✷✳✸✳✶✵✿ t❤r❡❡ t❡st s❡q✉❡♥❝❡s

❲❡ ✇✐❧❧ ✉s❡ t❤❡s❡ t❤r❡❡ ✏t❡st s❡q✉❡♥❝❡s✑✿

xn =

1 , n

1 yn = − , n

zn =

(−1)n . n

❛

✷✳✸✳ ▲✐♠✐ts ♦❢ ❢✉♥❝t✐♦♥s✿ s♠❛❧❧ s❝❛❧❡ tr❡♥❞s

✶✸✸

❋✐rst✱ ✇❡ t❛❦❡ f (x) = x2 ❛t a = 0✳ ❚❤❡ ❧✐♠✐ts ❛r❡ t❤❡ s❛♠❡✿ 1 = 0. x→0 n2

lim f (xn ) = lim f (yn ) = lim f (zn ) = lim

x→a

x→a

x→a

❆s ✇❡ ❤❛✈❡ s❡❡♥✱ t❤✐s ❧✐♠✐t ❞♦❡s ♥♦t ❡①✐st✿   1 ✱ ♥♦ ❧✐♠✐t✳ lim sin x→0 x

■♥ ❢❛❝t✱ t❤❡ ✈❛❧✉❡s ♦❢ t❤✐s ❢✉♥❝t✐♦♥ st❛rt t♦ ✜❧❧ t❤❡ ✇❤♦❧❡ ✐♥t❡r✈❛❧ [−1, 1] ❛s ✇❡ ❛♣♣r♦❛❝❤ 0✿

❍♦✇❡✈❡r✱ ✐❢ ✇❡ ♠✉❧t✐♣❧② t❤✐s ❡①♣r❡ss✐♦♥ ❜② x✱ t❤❡ s✇✐♥❣s ✇✐❧❧ st❛rt t♦ ❞✐♠✐♥✐s❤ ♦♥ t❤❡ ✇❛② t♦ 0✿

❲❡ ❤❛✈❡ ❛ ❧✐♠✐t✿

  1 = 0. lim x sin x→0 x

▲❡t✬s ✉s❡ t❤❡ ❛❧t❡r♥❛t✐♥❣ r❡❝✐♣r♦❝❛❧ s❡q✉❡♥❝❡ ❢♦r y = sign(x)✿ 1 xn = (−1)n . n

❚❤❡♥✱ sign(xn ) =

(

1 −1

✐❢ x ✐s ❡✈❡♥, ✐❢ x ✐s ♦❞❞.

❚❤✐s s❡q✉❡♥❝❡ ✐s ❞✐✈❡r❣❡♥t✳ ❚❤❡r❡❢♦r❡✱ t❤❡ r❡q✉✐r❡♠❡♥t ♦❢ t❤❡ ❞❡✜♥✐t✐♦♥ ❢❛✐❧s✱ ❛♥❞ lim sign(x) ❞♦❡s♥✬t x→0 ❡①✐st✳ ❆♥♦t❤❡r ✇❛② t♦ ❝♦♠❡ t♦ t❤✐s ❝♦♥❝❧✉s✐♦♥ ✐s t♦ ❝♦♥❝❡♥tr❛t❡ ♦♥ ♦♥❡ s✐❞❡ ❛t ❛ t✐♠❡✿ 

1 lim sign − n→∞ n   1 lim sign n→∞ n



= lim −1 = −1 , n→∞

= lim 1 n→∞

= 1.

❚❤❡ t✇♦ ❧✐♠✐ts ❛r❡ ❞✐✛❡r❡♥t✱ t❤❡ r❡q✉✐r❡♠❡♥t ❢❛✐❧s✱ ❛♥❞ t❤❛t✬s ✇❤② lim sign(x) ❞♦❡s♥✬t ❡①✐st✳ x→0

❙♦✱ t❤❡ ❜❡❤❛✈✐♦r ♦❢ y = sign(x) ♦♥ t❤❡ ❧❡❢t ❛♥❞ ♦♥ t❤❡ r✐❣❤t✱ ✇❤❡♥ ❝♦♥s✐❞❡r❡❞ s❡♣❛r❛t❡❧②✱ ✐s ✈❡r② r❡❣✉❧❛r✳

✷✳✸✳

✶✸✹

▲✐♠✐ts ♦❢ ❢✉♥❝t✐♦♥s✿ s♠❛❧❧ s❝❛❧❡ tr❡♥❞s

■♥❞❡❡❞✱ ✇❡ ❝❛♥ ❝❤♦♦s❡ ❛♥② s❡q✉❡♥❝❡s ❛♥❞✱ ❛s ❧♦♥❣ ❛s t❤❡② st❛② ♦♥ ♦♥❡ s✐❞❡ ♦❢ 0✱ ✇❡ ❤❛✈❡ t❤❡ s❛♠❡ ❝♦♥❝❧✉s✐♦♥✿ • ■❢ xn → 0 ❛♥❞ xn < 0 ❢♦r ❛❧❧ n✱ t❤❡♥ lim sign(xn ) = −1✳ n→∞

• ■❢ xn → 0 ❛♥❞ xn > 0 ❢♦r ❛❧❧ n✱ t❤❡♥ lim sign(xn ) = 1✳ n→∞

❚♦ t❛❦❡ ❛❞✈❛♥t❛❣❡ ♦❢ t❤✐s ✐♥s✐❣❤t✱ ✇❡ ❝❛♥ ✐♠❛❣✐♥❡ t❤❛t ❡✐t❤❡r t❤❡ ♣❛rt ♦❢ t❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥ t♦ t❤❡ ❧❡❢t ♦❢ x = a ❞✐s❛♣♣❡❛rs ♦r t❤❡ ♣❛rt t♦ t❤❡ r✐❣❤t ❞♦❡s✿

❲❡ ♠❛❦❡ t❤❡ ✐❞❡❛ ♣r❡❝✐s❡ ❜❡❧♦✇✿

❉❡✜♥✐t✐♦♥ ✷✳✸✳✶✶✿ ❧✐♠✐t ❢r♦♠ t❤❡ ❧❡❢t ❛♥❞ r✐❣❤t • ❚❤❡

♦❢ ❛ ❢✉♥❝t✐♦♥ f ❛t ❛ ♣♦✐♥t x = a ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ s❛♠❡ ❧✐♠✐t✱ lim f (xn )✱ ❜✉t ♦♥❧② ❧✐♠✐t❡❞ t♦ t❤❡ s❡q✉❡♥❝❡s xn ✇✐t❤ ❧✐♠✐t ❢r♦♠ t❤❡ ❧❡❢t n→∞

xn → a ❛s n → ∞,

❛♥❞ xn < a ❢♦r ❛❧❧ n ,

✇❤❡♥ ❛❧❧ t❤❡s❡ ❧✐♠✐ts ❡①✐st ❛♥❞ ❛r❡ ❡q✉❛❧ t♦ ❡❛❝❤ ♦t❤❡r✳ ❧✐♠✐t ❢r♦♠ t❤❡ r✐❣❤t ♦❢ ❛ ❢✉♥❝t✐♦♥ f ❛t ❛ ♣♦✐♥t x = a ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ s❛♠❡ ❧✐♠✐t✱ lim f (xn )✱ ❜✉t ♦♥❧② ❧✐♠✐t❡❞ t♦ t❤❡ s❡q✉❡♥❝❡s xn ✇✐t❤

• ❚❤❡

n→∞

xn → a ❛s n → ∞,

❛♥❞ xn > a ❢♦r ❛❧❧ n ,

✇❤❡♥ ❛❧❧ t❤❡s❡ ❧✐♠✐ts ❡①✐st ❛♥❞ ❛r❡ ❡q✉❛❧ t♦ ❡❛❝❤ ♦t❤❡r✳ ❖t❤❡r✇✐s❡✱ ✇❡ s❛② t❤❛t t❤❡ ❧✐♠✐t ❢r♦♠ t❤❡ r✐❣❤t ♦r ❢r♦♠ t❤❡ ❧❡❢t ❈♦❧❧❡❝t✐✈❡❧②✱ t❤❡ t✇♦ ❛r❡ ❝❛❧❧❡❞ ♦♥❡✲s✐❞❡❞ ❧✐♠✐ts✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ♦♥❡✲s✐❞❡❞ ❧✐♠✐ts ❛r❡ t❤❡ ❧✐♠✐ts ♦❢ t❤❡ ❢✉♥❝t✐♦♥ r✐❣❤t ♦❢ t❤❡ ♣♦✐♥t✿ ▲✐♠✐t ❢r♦♠ t❤❡ ❧❡❢t✿ lim− f (x) xa

r❡str✐❝t❡❞

x→a

❞♦❡s ♥♦t ❡①✐st✳

t♦ ❛♥ ✐♥t❡r✈❛❧ t♦ t❤❡ ❧❡❢t ♦r t♦ t❤❡

− − −◦

◦ − −−

❚❤❡ ❢♦❧❧♦✇✐♥❣ ♥♦t❛t✐♦♥ ✐s ✉s❡❞✿

❖♥❡✲s✐❞❡❞ ❧✐♠✐ts ▲❡❢t✿ ❘✐❣❤t✿

lim f (x)

x→a−

lim f (x)

x→a+

• ❚❤❡ ♥♦t❛t✐♦♥ a− s✉❣❣❡sts t❤❛t ✇❡ ♦♥❧② ❝♦♥s✐❞❡r ♥✉♠❜❡rs ♦❢ t❤❡ t②♣❡ a − ε ✇✐t❤ ε > 0✳

• ❚❤❡ ♥♦t❛t✐♦♥ a+ s✉❣❣❡sts t❤❛t ✇❡ ♦♥❧② ❝♦♥s✐❞❡r ♥✉♠❜❡rs ♦❢ t❤❡ t②♣❡ a + ε ✇✐t❤ ε > 0✳

✷✳✸✳ ▲✐♠✐ts ♦❢ ❢✉♥❝t✐♦♥s✿ s♠❛❧❧ s❝❛❧❡ tr❡♥❞s

✶✸✺

❲❛r♥✐♥❣✦ ❙♦♠❡ s♦✉r❝❡s ❛❧s♦ ✉s❡

lim f (x)

xրa

❛♥❞

lim f (x)✳

xցa

❚❤❡s❡ ❛r❡ t❤❡ ✏♦♥❡✲s✐❞❡❞✑ ❧✐♠✐ts✳ ❋♦r t❤❡ ♦r✐❣✐♥❛❧✱ ✏t✇♦✲s✐❞❡❞✑ ❧✐♠✐t✱ t❤❡ q✉❡st✐♦♥ ❜❡❝♦♠❡s✱ ❞♦ t❤❡ t✇♦ ✕ ❧❡❢t ❛♥❞ r✐❣❤t ✕ ❧✐♠✐ts ♠❛t❝❤❄

❚❤❡♦r❡♠ ✷✳✸✳✶✷✿ ▲✐♠✐t ✐♥ ❚❡r♠s ♦❢ ❖♥❡✲❙✐❞❡❞ ▲✐♠✐ts ■❋ ❆◆❉ ❖◆▲❨ ■❋

❚❤❡ ❧✐♠✐t ♦❢ ❛ ❢✉♥❝t✐♦♥ ❡①✐sts

t❤❡ ❧✐♠✐ts ❢r♦♠ t❤❡ ❧❡❢t ❛♥❞

❢r♦♠ t❤❡ r✐❣❤t ♦❢ t❤❡ ♣♦✐♥t ❜♦t❤ ❡①✐st ❛♥❞ ❛r❡ ❡q✉❛❧ t♦ ❡❛❝❤ ♦t❤❡r✳

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿

lim f (x)

x→a

⇐⇒ lim− f (x) = lim+ f (x) .

❡①✐sts

x→a

x→a

■♥ t❤❛t ❝❛s❡✱ ✇❡ ❤❛✈❡✿

lim f (x) = lim+ f (x) = lim− f (x) .

x→a

x→a

x→a

Pr♦♦❢✳ lim f (xn ) ❡①✐sts ❢♦r ❛♥② s❡q✉❡♥❝❡ xn → a ❛♥❞ ✐s t❤❡ s❛♠❡✳ ■♥ n→∞ ♣❛rt✐❝✉❧❛r✱ t❤✐s ✐s tr✉❡ ❢♦r t❤❡ s❡q✉❡♥❝❡s ❧✐♠✐t❡❞ t♦ t❤❡ ♦♥❡s ❛❧❧ ❧❛r❣❡r t❤❛♥ a ❛♥❞ ❛❧❧ s♠❛❧❧❡r t❤❛♥ a✳

❚❤❡ ❡①✐st❡♥❝❡ ♦❢ t❤✐s ❧✐♠✐t ♠❡❛♥s t❤❛t

❚❤❡ ♣r♦♦❢ ♦❢ t❤❡ ❝♦♥✈❡rs❡ ✐s ♦♠✐tt❡❞✳

❊①❛♠♣❧❡ ✷✳✸✳✶✸✿ ♣✐❡❝❡✇✐s❡✲❞❡✜♥❡❞ ❢✉♥❝t✐♦♥s ▲❡t✬s ♣❧♦t t❤❡ ❣r❛♣❤ ♦❢ t❤✐s ❢✉♥❝t✐♦♥✿

lim f (x) ❡①✐sts ❢♦r ❛❧❧ x→a ❢♦r♠✉❧❛ ❢r♦♠ t❤❡ t❤r❡❡✿

❚❤❡ ❧✐♠✐t

  2 − x f (x) = x   (x − 1)2 a

❡①❝❡♣t

lim f (x)

x→1−

x→−1

= lim − x x→−1

=1

✐❢ ✐❢

a = −1, 1✳

lim f (x) = lim − (2 − x) = (2 − (−1)) = 3

x→−1−

✐❢

x < −1 , − 1 < x < 1, x > 1. ❲❤✐❧❡ ❝♦♠♣✉t✐♥❣ t❤❡ ❧✐♠✐ts✱ ✇❡ ♣✐❝❦ t❤❡ r✐❣❤t

lim f (x) = lim + x

x→−1+

lim f (x)

x→1+

❙♦✱ t❤✐s ✐s t❤❡ ❣r❛♣❤ ♠❛❞❡ ♦❢ t❤r❡❡ ❜r❛♥❝❤❡s✱ ♦♥❡ ❢♦r ❡❛❝❤ ❢♦r♠✉❧❛✿

x→−1

= −1

= lim+ (1 − x)2 = (1 − 1)2 = 0 x→1

✷✳✸✳

✶✸✻

▲✐♠✐ts ♦❢ ❢✉♥❝t✐♦♥s✿ s♠❛❧❧ s❝❛❧❡ tr❡♥❞s

❚❤✐s ❡①❛♠♣❧❡ ✇❛s ❛❜♦✉t ✐♥t❡r♣r❡t✐♥❣ t❤❡ ❣r❛♣❤ ✐♥ t❡r♠s ♦❢ ❧✐♠✐ts✳ ◆♦✇✱ t❤❡ ♦t❤❡r ✇❛② ❛r♦✉♥❞✳ ❊①❛♠♣❧❡ ✷✳✸✳✶✹✿ ❢r♦♠ ❧✐♠✐ts t♦ ❣r❛♣❤s

●✐✈❡♥ t❤✐s ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t f ✱ ♣❧♦t ✐ts ❣r❛♣❤✿

lim f (x) = 1

lim f (x) = −1

x→0−

x→0+

x→3−

x→3+

lim f (x) = 0

lim f (x) = 1

f (0)

✉♥❞❡✜♥❡❞

f (3) = 1

❲❡ r❡✇r✐t❡ t❤❡ ♦♥❡✲s✐❞❡❞ ❧✐♠✐ts✿ • ❆s x → 0− ✱ ✇❡ ❤❛✈❡✿ y → 1 ✳ • ❆s x → 0+ ✱ ✇❡ ❤❛✈❡✿ y → −1 ✳ • ❆s x → 3− ✱ ✇❡ ❤❛✈❡✿ y → 0 ✳ • ❆s x → 3+ ✱ ✇❡ ❤❛✈❡✿ y → 1 ✳ ❚❤❡♥ ✇❡ ♣❧♦t t❤❡ r❡s✉❧ts ❜❡❧♦✇ ❝♦♥❝❡♥tr❛t✐♥❣ ♦♥ t❤❡ ❜❡❤❛✈✐♦r ♦❢ f ❝❧♦s❡ t♦ t❤❡s❡ ♣♦✐♥ts✿

■❢ ✇❡ ❝♦♥s✐❞❡r t✇♦ s❡q✉❡♥❝❡s ❝♦♥✈❡r❣✐♥❣ t♦ a ✕ ❢r♦♠ t❤❡ ❧❡❢t ❛♥❞ ❢r♦♠ t❤❡ r✐❣❤t ✕ ✇❡ s❡❡ ❛ s❡q✉❡♥❝❡ ♦❢ ✐♥t❡r✈❛❧s s❤r✐♥❦✐♥❣ t♦✇❛r❞s a✿

❆s ✇❡ ❦♥♦✇✱ t❤❡ ❧✐♠✐t ♦❢ ❛ s❡q✉❡♥❝❡ ✐s ❢✉❧❧② ❞❡t❡r♠✐♥❡❞ ❜② ✐ts t❛✐❧✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ❧✐♠✐t ♦❢ ❛ ❢✉♥❝t✐♦♥ ❛t a ✐s ❢✉❧❧② ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ t❛✐❧ ♦❢ t❤❡ s❡q✉❡♥❝❡ f (xn ) ✇❤❡♥ xn → a✳ ■❢ ✇❡ ❝❛♥ ♦♥❧② s❡❡ t❤❡ ❜❡❤❛✈✐♦r ♦❢ f ✐♥ t❤❡ ✈✐❝✐♥✐t② ✕ ♥♦ ♠❛tt❡r ❤♦✇ s♠❛❧❧ ✕ ♦❢ a✱ ✇❡ st✐❧❧ ❦♥♦✇ t❤❡ ✈❛❧✉❡ ✭❛♥❞ t❤❡ ❡①✐st❡♥❝❡✮ ♦❢ t❤❡ ❧✐♠✐t lim f (x)✿ x→a

✷✳✸✳

✶✸✼

▲✐♠✐ts ♦❢ ❢✉♥❝t✐♦♥s✿ s♠❛❧❧ s❝❛❧❡ tr❡♥❞s

❚❤✐s ✐s ♦✉r ❝♦♥❝❧✉s✐♦♥✿ ❚❤❡♦r❡♠ ✷✳✸✳✶✺✿ ▲✐♠✐ts ❆r❡ ▲♦❝❛❧ ❚❤❡ ❧✐♠✐t ♦❢ ❛ ❢✉♥❝t✐♦♥ ✐s ❞❡t❡r♠✐♥❡❞ ❜② ✐ts ✈❛❧✉❡s ✐♥ t❤❡ ✈✐❝✐♥✐t② ♦❢ t❤❡ ♣♦✐♥t✳

■♥ ♦t❤❡r ✇♦r❞s✱ ✐❢ t✇♦ ❢✉♥❝t✐♦♥s ❝♦♥t❛✐♥s t❤❡ ♣♦✐♥t

ε > 0✱

❛♥❞

g

❛r❡ ❡q✉❛❧ ♦✈❡r ❛♥ ♦♣❡♥ ✐♥t❡r✈❛❧ t❤❛t

a✿

f (x) = g(x) ❢♦r s♦♠❡

f

❢♦r ❛❧❧

t❤❡♥ t❤❡✐r ❧✐♠✐ts ❛t

a

a − ε < x < a + ε, x 6= a , ❝♦✐♥❝✐❞❡ t♦♦✿

lim f (x) = lim g(x) ,

x→a

x→a

✉♥❧❡ss ♥❡✐t❤❡r ❡①✐sts✳

❊①❡r❝✐s❡ ✷✳✸✳✶✻

❘❡st❛t❡ t❤❡ t❤❡♦r❡♠ ✐♥ t❡r♠s ♦❢ r❡str✐❝t✐♦♥s ♦❢ ❢✉♥❝t✐♦♥s✳ ❊①❡r❝✐s❡ ✷✳✸✳✶✼

❙t❛t❡ ❛♥ ❛♥❛❧♦❣ ♦❢ t❤❡ t❤❡♦r❡♠ ❢♦r ♦♥❡✲s✐❞❡❞ ❧✐♠✐ts✳ ❘❡❝❛❧❧ t❤❛t ❢♦r ❛ ❢✉♥❝t✐♦♥ f ❛♥❞ ❛ ♣♦✐♥t a✱ ✇❤❡r❡ f ✐s ❞❡✜♥❡❞✱ t❤❡ ❣r❛♣❤ ♦❢ f ❝♦♥s✐sts ♦❢ t❤r❡❡ ♣❛rts✿ ✶✳ t❤❡ ♣❛rt ♦❢ t❤❡ ❣r❛♣❤ ♦❢ f ✇✐t❤ x < a✱ ✷✳ t❤❡ ♣❛rt ♦❢ t❤❡ ❣r❛♣❤ ♦❢ f ✇✐t❤ x = a ✭♦♥❡ ♣♦✐♥t✮✱ ❛♥❞ ✸✳ t❤❡ ♣❛rt ♦❢ t❤❡ ❣r❛♣❤ ♦❢ f ✇✐t❤ x > a✳ ❋♦r t❤✐s ❢✉♥❝t✐♦♥ t♦ ❜❡ ❝♦♥t✐♥✉♦✉s✱ t❤❡s❡ t❤r❡❡ ♣❛rts ✭t❤❡ t✇♦ ♣✐❡❝❡s ♦❢ t❤❡ r♦♣❡ ❛♥❞ ❛ ❞r♦♣ ♦❢ ❣❧✉❡✮ ❤❛✈❡ t♦ ✜t t♦❣❡t❤❡r✿

✷✳✹✳

▲✐♠✐ts ❛♥❞ ❝♦♥t✐♥✉✐t② ✉♥❞❡r ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s

✶✸✽

❲❡ ♣✉t t❤✐s ✐❞❡❛ ✐♥ t❤❡ ❢♦r♠ ♦❢ ❛ t❤❡♦r❡♠ t❤❛t r❡❧✐❡s ♦♥ t❤❡ ❝♦♥❝❡♣t ♦❢ ♦♥❡✲s✐❞❡❞ ❧✐♠✐t✿

❚❤❡♦r❡♠ ✷✳✸✳✶✽✿ ❈♦♥t✐♥✉✐t② ✈✐❛ ❖♥❡✲❙✐❞❡❞ ❈♦♥t✐♥✉✐t②

❆ ❢✉♥❝t✐♦♥ f ✐s ❝♦♥t✐♥✉♦✉s ❛t x = a ✐❢ ❛♥❞ ♦♥❧② ✐❢ f ✐s ❞❡✜♥❡❞ ❛t a✱ t❤❡ t✇♦ ♦♥❡✲s✐❞❡❞ ❧✐♠✐ts ❡①✐st ❛♥❞ ❜♦t❤ ❡q✉❛❧ t♦ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❛t a✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿ lim f (x) = f (a) = lim+ f (x) .

x→a−

x→a

❊①❡r❝✐s❡ ✷✳✸✳✶✾ ▼❛❦❡ ❛ ❤❛♥❞✲❞r❛✇♥ s❦❡t❝❤ ♦❢ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❛♥❞ ❡✈❛❧✉❛t❡ ✐ts ❝♦♥t✐♥✉✐t②✿

  −3 x2 f (x) =  x

✐❢ ✐❢ ✐❢

x < 0, 0 ≤ x < 1, x > 1.

✷✳✹✳ ▲✐♠✐ts ❛♥❞ ❝♦♥t✐♥✉✐t② ✉♥❞❡r ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s

❚❤❡ ❧✐♠✐t ♣r♦❝❡❞✉r❡ ❢♦r ❛ ❣✐✈❡♥

a ✐s ❛ s♣❡❝✐❛❧ ❦✐♥❞ ♦❢ ❢✉♥❝t✐♦♥✱ ❛ ❢✉♥❝t✐♦♥ ✇❤♦s❡ ✐♥♣✉t ✐s ❛ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥ ✿ ❢✉♥❝t✐♦♥

→

lim

x→a

→

❛ ♥✉♠❜❡r

❊①❡r❝✐s❡ ✷✳✹✳✶ ❲❤❛t ❝❛♥ ②♦✉ s❛② ❛❜♦✉t t❤❡ ❞♦♠❛✐♥ ♦❢ t❤✐s s♣❡❝✐❛❧ ❢✉♥❝t✐♦♥❄

❲❡ ✇✐❧❧ ✉s❡ t❤❡ ❛❧❣❡❜r❛✐❝ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❧✐♠✐ts ♦❢ s❡q✉❡♥❝❡s t♦ ♣r♦✈❡ ✈✐rt✉❛❧❧② ✐❞❡♥t✐❝❛❧ ❢❛❝ts ❛❜♦✉t ❧✐♠✐ts ♦❢ ❢✉♥❝t✐♦♥s✳ ▲❡t✬s r❡✈✐❡✇ t❤❡ ♠❛✐♥ ❛❧❣❡❜r❛✐❝ ♣r♦♣❡rt✐❡s✿

❚❤❡♦r❡♠ ✷✳✹✳✷✿ ❆❧❣❡❜r❛ ♦❢ ▲✐♠✐ts ♦❢ ❙❡q✉❡♥❝❡s

❙✉♣♣♦s❡ an → a ❛♥❞ bn → b✳ ❚❤❡♥ ✇❡ ❤❛✈❡✿ ❙❘✿ P❘✿

an + bn → a + b ❈▼❘✿ ◗❘✿ an · bn → a · b

c · an → c · a an /bn → a/b

❢♦r ❛♥② r❡❛❧ c ♣r♦✈✐❞❡❞ b 6= 0

Pr❡s❡♥t❡❞ ✈❡r❜❛❧❧②✱ t❤❡s❡ r✉❧❡s ❤❛✈❡ t❤❡s❡ ❛❜❜r❡✈✐❛t❡❞ ✈❡rs✐♦♥s✿

• • • •

❙✉♠ ❘✉❧❡ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ Pr♦❞✉❝t ❘✉❧❡ ◗✉♦t✐❡♥t ❘✉❧❡

✿ ❚❤❡ ❧✐♠✐t ♦❢ t❤❡ s✉♠ ✐s t❤❡ s✉♠ ♦❢ t❤❡ ❧✐♠✐ts✳ ✿ ❚❤❡ ❧✐♠✐t ♦❢ ❛ ❝♦♥st❛♥t ♠✉❧t✐♣❧❡ ✐s t❤❡ ❝♦♥st❛♥t ♠✉❧t✐♣❧❡ ♦❢ t❤❡ ❧✐♠✐ts✳

✿ ❚❤❡ ❧✐♠✐t ♦❢ t❤❡ ♣r♦❞✉❝t ✐s t❤❡ ♣r♦❞✉❝t ♦❢ t❤❡ ❧✐♠✐ts✳ ✿

❚❤❡ ❧✐♠✐t ♦❢ t❤❡ q✉♦t✐❡♥t ✐s t❤❡ q✉♦t✐❡♥t ♦❢ t❤❡ ❧✐♠✐ts ❛s ❧♦♥❣ ❛s t❤❡ ❧✐♠✐t ♦❢ t❤❡

❞❡♥♦♠✐♥❛t♦r ✐s♥✬t ③❡r♦✳

✷✳✹✳ ▲✐♠✐ts ❛♥❞ ❝♦♥t✐♥✉✐t② ✉♥❞❡r ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s

✶✸✾

❊❛❝❤ ♣r♦♣❡rt② ✐s ♠❛t❝❤❡❞ ❜② ✐ts ❛♥❛❧♦❣ ❢♦r ❢✉♥❝t✐♦♥s✳ ❋✉rt❤❡r♠♦r❡✱ t❤❡r❡ ❛r❡ ❛♥❛❧♦❣s ❢♦r t❤❡ ♦♥❡✲s✐❞❡❞ ❧✐♠✐ts✳ ■♥ ❢❛❝t✱ ✇❡ ✇✐❧❧ s❡❡ ❛♥❛❧♦❣s ♦❢ s♦♠❡ ♦❢ t❤❡s❡ r✉❧❡s ❢♦r ♥✉♠❡r♦✉s ♥❡✇ ❝♦♥❝❡♣ts ✐♥ t❤❡ ❢♦rt❤❝♦♠✐♥❣ ❝❤❛♣t❡rs✿ ❚❤❡♦r❡♠ ✷✳✹✳✸✿ ❆❧❣❡❜r❛ ♦❢ ▲✐♠✐ts ♦❢ ❋✉♥❝t✐♦♥s

❙✉♣♣♦s❡

❙❘✿ P❘✿

f (x) → F

❛♥❞

g(x) → G

f (x) + g(x) → F + G f (x) · g(x) → F · G

❛s

x→a

✭♦r

x → a−

♦r

x → a+ ✮✳

c · f (x) → c · F f (x)/g(x) → F/G

❈▼❘✿ ◗❘✿

❚❤❡♥

c G 6= 0

❢♦r ❛♥② r❡❛❧ ♣r♦✈✐❞❡❞

▲❡t✬s ❝♦♥s✐❞❡r t❤❡♠ ❛❧❧ st❛rt✐♥❣ ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣✿ ❚❤❡♦r❡♠ ✷✳✹✳✹✿ ❙✉♠ ❘✉❧❡ ♦❢ ▲✐♠✐ts ♦❢ ❋✉♥❝t✐♦♥s

a

■❢ t❤❡ ❧✐♠✐ts ❛t

♦❢ ❢✉♥❝t✐♦♥s

f (x), g(x)

❡①✐st✱ t❤❡♥ s♦ ❞♦❡s t❤❛t ♦❢ t❤❡✐r s✉♠✱

f (x) + g(x)✳ ❋✉rt❤❡r♠♦r❡✱ t❤❡ ❧✐♠✐t ♦❢ t❤❡ s✉♠ ✐s ❡q✉❛❧ t♦ t❤❡ s✉♠ ♦❢ t❤❡ ❧✐♠✐ts✿


lim f (x) + g(x) = lim f (x) + lim g(x) .

x→a

x→a

x→a

Pr♦♦❢✳

❋♦r ❛♥② s❡q✉❡♥❝❡ xn → a✱ ✇❡ ❤❛✈❡ ❜② ❙✉♠ ❘✉❧❡ ❢♦r ❙❡q✉❡♥❝❡s✿ lim (f (x) + g(x)) = lim (f (xn ) + g(xn )) = lim f (xn ) + lim g(xn ) .

x→a

n→∞

n→∞

n→∞

■♥ t❤❡ ❝❛s❡ ♦❢ ✐♥✜♥✐t❡ ❧✐♠✐ts✱ ✇❡ ❢♦❧❧♦✇ t❤❡ r✉❧❡s ♦❢ t❤❡ ❛❧❣❡❜r❛ ♦❢ ✐♥✜♥✐t✐❡s ❛s ✐♥ ❈❤❛♣t❡r ✶✿ ♥✉♠❜❡r ♥✉♠❜❡r +∞ −∞

+ + + +

(+∞) (−∞) (+∞) (−∞)

= +∞ = −∞ = +∞ = −∞

❲❛r♥✐♥❣✦ ■♥❞❡t❡r♠✐♥❛t❡ ❡①♣r❡ss✐♦♥s ❛r❡♥✬t ❤❡r❡✿

∞✱ 0/0✱

❡t❝✳

∞/∞✱ ∞ −

❚❤❡ ♣r♦♦❢s ♦❢ t❤❡ r❡st ♦❢ t❤❡ ♣r♦♣❡rt✐❡s ❛r❡ ✐❞❡♥t✐❝❛❧✳ ❚❤❡ r❡s✉❧ts ❛r❡ ♣r❡s❡♥t❡❞ ❜❡❧♦✇✿ ❚❤❡♦r❡♠ ✷✳✹✳✺✿ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ♦❢ ▲✐♠✐ts ♦❢ ❋✉♥❝t✐♦♥s

■❢ t❤❡ ❧✐♠✐t ❛t

a

♦❢ ❢✉♥❝t✐♦♥

f (x)

❡①✐sts✱ t❤❡♥ s♦ ❞♦❡s t❤❛t ♦❢ ✐ts ♠✉❧t✐♣❧❡✱

cf (x)✳

❋✉rt❤❡r♠♦r❡✱ t❤❡ ❧✐♠✐t ♦❢ t❤❡ ♠✉❧t✐♣❧❡ ✐s ❡q✉❛❧ t♦ t❤❡ ♠✉❧t✐♣❧❡ ♦❢ t❤❡ ❧✐♠✐t✿

lim cf (x) = c · lim f (x) .

x→a

x→a

✷✳✹✳

▲✐♠✐ts ❛♥❞ ❝♦♥t✐♥✉✐t② ✉♥❞❡r ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s

✶✹✵

❚❤❡♦r❡♠ ✷✳✹✳✻✿ Pr♦❞✉❝t ❘✉❧❡ ♦❢ ▲✐♠✐ts ❢♦r ❋✉♥❝t✐♦♥s ■❢ t❤❡ ❧✐♠✐ts ❛t

a ♦❢ ❢✉♥❝t✐♦♥s f (x), g(x) ❡①✐st✱ t❤❡♥ s♦ ❞♦❡s t❤❛t ♦❢ t❤❡✐r ♣r♦❞✉❝t✱

f (x) · g(x)✳ ❋✉rt❤❡r♠♦r❡✱ t❤❡ ❧✐♠✐t ♦❢ t❤❡ ♣r♦❞✉❝t ✐s ❡q✉❛❧ t♦ t❤❡ ♣r♦❞✉❝t ♦❢ t❤❡ ❧✐♠✐ts✿




lim f (x) · g(x) = lim f (x) · lim g(x) .

x→a

x→a

x→a

❚❤❡♦r❡♠ ✷✳✹✳✼✿ ◗✉♦t✐❡♥t ❘✉❧❡ ♦❢ ▲✐♠✐ts ♦❢ ❋✉♥❝t✐♦♥s ■❢ t❤❡ ❧✐♠✐ts ❛t

f (x)/g(x)✱

a

f (x), g(x) lim g(x) 6= 0✳

♦❢ ❢✉♥❝t✐♦♥s

♣r♦✈✐❞❡❞

❡①✐st✱ t❤❡♥ s♦ ❞♦❡s t❤❛t ♦❢ t❤❡✐r r❛t✐♦✱

x→a

❋✉rt❤❡r♠♦r❡✱ t❤❡ ❧✐♠✐t ♦❢ t❤❡ r❛t✐♦ ✐s ❡q✉❛❧ t♦ t❤❡ r❛t✐♦ ♦❢ t❤❡ ❧✐♠✐ts✿

lim

x→a



f (x) g(x)



lim f (x)

=

x→a

lim g(x)

.

x→a

❲❡ ❝❛♥ s❛② t❤❛t t❤❡ ❧✐♠✐t s✐❣♥ ✐s ✏❞✐str✐❜✉t❡❞✑ ♦✈❡r t❤❡s❡ ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s✳ ❚❤❡s❡ r❡s✉❧ts ❛r❡ ❦♥♦✇♥ ❛s t❤❡

▲✐♠✐ts ❘✉❧❡s✳

❚❤❡ ♠❛✐♥ ❜✉✐❧❞✐♥❣ ❜❧♦❝❦s ❛r❡ t❤❡s❡ t✇♦ ❢✉♥❝t✐♦♥s ✕ t❤❡ ❝♦♥st❛♥t ❢✉♥❝t✐♦♥ ❛♥❞ t❤❡ ✐❞❡♥t✐t② ❢✉♥❝t✐♦♥ ✕ ✇✐t❤ s✐♠♣❧❡ ❧✐♠✐ts✿ ❚❤❡♦r❡♠ ✷✳✹✳✽✿ ▲✐♠✐t ♦❢ ❈♦♥st❛♥t ❋✉♥❝t✐♦♥ ❋♦r ❛♥② r❡❛❧

c✱

t❤❡ ❢♦❧❧♦✇✐♥❣ ❧✐♠✐t ❡①✐sts ❛t ❛♥② ♣♦✐♥t

a✿

lim c = c .

x→a

Pr♦♦❢✳

❋♦r ❛♥② s❡q✉❡♥❝❡

xn → a✱

✇❡ ❤❛✈❡✿

lim c = lim c = c .

x→a

▲❡t✬s s❡t

g(x) = c

✐♥

n→∞

Pr♦❞✉❝t ❘✉❧❡ ❛♥❞ ✉s❡ t❤❡ ❧❛st t❤❡♦r❡♠✱ t❤❡♥ lim cf (x) = lim (f (x) · g(x)) = c · (lim g(x)) .

x→a

❚❤❡♥ t❤❡

❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ❢♦❧❧♦✇s✳

x→a

x→a

❊✈❡♥ t❤♦✉❣❤ t❤❡

❘✉❧❡✱ t❤❡ ❢♦r♠❡r ✐s s✐♠♣❧❡r ❛♥❞ ❡❛s✐❡r t♦ ✉s❡✿

❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ✐s ❛❜s♦r❜❡❞ ✐♥t♦ Pr♦❞✉❝t

❚❤❡♦r❡♠ ✷✳✹✳✾✿ ▲✐♠✐t ♦❢ ■❞❡♥t✐t② ❋✉♥❝t✐♦♥ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ❧✐♠✐t ❡①✐sts ❛t ❛♥② ♣♦✐♥t

a✿

lim x = a .

x→a

Pr♦♦❢✳

❋♦r ❛♥② s❡q✉❡♥❝❡

xn → a✱

✇❡ ❤❛✈❡✿

lim x = lim xn = a .

x→a

❆♥② ♣♦❧②♥♦♠✐❛❧ ❝❛♥ ❜❡ ❜✉✐❧t ❢r♦♠

x

n→∞

❛♥❞ ❝♦♥st❛♥ts ❜② ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❛♥❞ ❛❞❞✐t✐♦♥✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ✜rst ✜✈❡

t❤❡♦r❡♠s ❛❧❧♦✇ ✉s t♦ ❝♦♠♣✉t❡ t❤❡ ❧✐♠✐ts ♦❢ ❛❧❧ ♣♦❧②♥♦♠✐❛❧s✳

✷✳✹✳ ▲✐♠✐ts ❛♥❞ ❝♦♥t✐♥✉✐t② ✉♥❞❡r ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s

✶✹✶

❊①❛♠♣❧❡ ✷✳✹✳✶✵✿ ♣♦❧②♥♦♠✐❛❧s ▲❡t

f (x) = x3 + 3x2 − 7x + 8 .

❲❤❛t ✐s ✐ts ❧✐♠✐t ❛s x → 1❄ ❚❤❡ ❝♦♠♣✉t❛t✐♦♥ ✐s str❛✐❣❤t❢♦r✇❛r❞✱ ❜✉t ❡✈❡r② st❡♣ ❤❛s t♦ ❜❡ ❥✉st✐✜❡❞ ✇✐t❤ t❤❡ r✉❧❡s ❛❜♦✈❡✳ ❚♦ ✉♥❞❡rst❛♥❞ ✇❤✐❝❤ r✉❧❡s t♦ ❛♣♣❧② ✜rst✱ ♦❜s❡r✈❡ ✇❤❛t t❤❡ ❧❛st ♦♣❡r❛t✐♦♥ ✐s❀ ✐t✬s ❛❞❞✐t✐♦♥✳ ❲❡ ✉s❡ t❤❡ ❙✉♠ ❘✉❧❡✱ s✉❜❥❡❝t t♦ ❥✉st✐✜❝❛t✐♦♥✿ lim f (x) = lim (x3 + 3x2 − 7x + 8)

x→1

x→1

3

2

= lim x + lim 3x − lim 7x + lim 8 x→1

P❘✱

x→1

x→1

❈▼❘✱

x→1

❈▼❘✱

❈♦♥t✐♥✉❡ ✇✐t❤ ❙❘✳ ❈♦♥t✐♥✉❡ ✇✐t❤

❈❘✳

= lim x · lim x + 3 lim x − 7 lim x + 8 ❈♦♥t✐♥✉❡ ✇✐t❤ ❈❘✳ x→1

2

x→1

2

x→1

x→1

❈♦♥t✐♥✉❡ ✇✐t❤ P❘ ❛♥❞ ❈❘✳

= 1 · lim x2 + 3 lim x2 − 7 · 1 + 8 x→1

x→1

=1·1+3·1−7+8 = 5.

❲✐t❤ t❤✐s ❝♦♠♣❧❡① ❛r❣✉♠❡♥t✱ ✐t ✐s ❡❛s② t♦ ♠✐ss t❤❡ s✐♠♣❧❡ ❢❛❝t t❤❛t t❤❡ ❧✐♠✐t ♦❢ t❤✐s ❢✉♥❝t✐♦♥ ❤❛♣♣❡♥s t♦ ❜❡ ❡q✉❛❧ t♦ ✐ts ✈❛❧✉❡ ✿ lim f (x) = lim (x + 3x − 7x + 8) = x + 3x − 7x + 8 3

x→1

2

3

2

x→1

x=1

= 13 + 3 · 1 2 − 7 · 1 + 8 = 5 .

❲❡ ❦♥♦✇ t❤✐s ✐s tr✉❡ ❜❡❝❛✉s❡ t❤❡ ❢✉♥❝t✐♦♥ ✐s ❝♦♥t✐♥✉♦✉s✦ ❚❤❡ ✐❞❡❛ ✐s ❝♦♥✜r♠❡❞ ❜② t❤❡ ♣❧♦t✿

▲❡t✬s r❡✲st❛t❡ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❝♦♥t✐♥✉✐t②✿

❉❡✜♥✐t✐♦♥ ✷✳✹✳✶✶✿ ❝♦♥t✐♥✉✐t② ❆ ❢✉♥❝t✐♦♥ f ✐s ❝❛❧❧❡❞ ❝♦♥t✐♥✉♦✉s ❛t ♣♦✐♥t a ✐❢ • f (x) ✐s ❞❡✜♥❡❞ ❛t x = a✱ • t❤❡ ❧✐♠✐t ♦❢ f ❡①✐sts ❛t a✱ ❛♥❞ • t❤❡ t✇♦ ❛r❡ ❡q✉❛❧ t♦ ❡❛❝❤ ♦t❤❡r✿ lim f (x) = f (a) .

x→a

❚❤✉s✱ t❤❡ ❧✐♠✐ts ♦❢ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s ❝❛♥ ❜❡ ❢♦✉♥❞ ❜② s✉❜st✐t✉t✐♦♥✳ ❊q✉✐✈❛❧❡♥t❧②✱ ❛ ❢✉♥❝t✐♦♥ f ✐s ❝♦♥t✐♥✉♦✉s ❛t a ✐❢ lim f (xn ) = f (a) ,

n→∞

❢♦r ❛♥② s❡q✉❡♥❝❡ xn → a✿

✷✳✹✳

▲✐♠✐ts ❛♥❞ ❝♦♥t✐♥✉✐t② ✉♥❞❡r ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s

✶✹✷

❆s ✇❡ s❤❛❧❧ s❡❡✱ ❛❧❧ ♣♦❧②♥♦♠✐❛❧s ❛r❡ ❝♦♥t✐♥✉♦✉s✳ ◆♦✇ ❛♥ ❡①❛♠♣❧❡ ♦❢ ❛ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥✿ ❊①❛♠♣❧❡ ✷✳✹✳✶✷✿ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥s

▲❡t✬s ✜♥❞ t❤❡ ❧✐♠✐t ❛t 2 ♦❢

x+1 . x−1 ♦♣❡r❛t✐♦♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥✳ ■t ✐s ❞✐✈✐s✐♦♥✱ s♦ ✇❡ ✉s❡ f (x) =

❆❣❛✐♥✱ ✇❡ ❧♦♦❦ ❛t t❤❡

❧❛st

x+1 x→2 x − 1

✜rst✿

❲❡ ♥♦✇ ❥✉st✐❢② ◗❘ ❜② ♦❜s❡r✈✐♥❣ t❤❛t

lim f (x) = lim

x→2

◗✉♦t✐❡♥t ❘✉❧❡

limx→2 (x + 1) limx→2 (x − 1) 3 = 1 = 3.

lim (x − 1) = 1 6= 0 .

x→2

=

❊①❛♠♣❧❡ ✷✳✹✳✶✸✿ ✐♥❞❡t❡r♠✐♥❛t❡ ❡①♣r❡ss✐♦♥s

▲❡t✬s ✜♥❞ t❤❡ ❧✐♠✐t ❛t 1 ♦❢ t❤❡ ❢✉♥❝t✐♦♥

f (x) =

x2 − 1 . x−1

❙✐♥❝❡ t❤❡ ❧❛st ♦♣❡r❛t✐♦♥ ✐s ❞✐✈✐s✐♦♥✱ ✇❡ ❛r❡ s✉♣♣♦s❡❞ t♦ ✉s❡ t❤❡ ◗✉♦t✐❡♥t ♦❢ t❤❡ ❞❡♥♦♠✐♥❛t♦r ✐s 0✿ lim (x − 1) = 0 .

❘✉❧❡ ✜rst✳

❍♦✇❡✈❡r✱ t❤❡ ❧✐♠✐t

x→1

❚❤❡♥✱ t❤❡

◗✉♦t✐❡♥t ❘✉❧❡

✐s ✐♥❛♣♣❧✐❝❛❜❧❡✳ ❇✉t t❤❡♥ ❛❧❧ ♦t❤❡r r✉❧❡s ♦❢ ❧✐♠✐ts ❛r❡ ❛❧s♦ ✐♥❛♣♣❧✐❝❛❜❧❡✦

❆ ❝❧♦s❡r ❧♦♦❦ r❡✈❡❛❧s t❤❛t t❤✐♥❣s ❛r❡ ❡✈❡♥ ✇♦rs❡❀ ❜♦t❤ t❤❡ ♥✉♠❡r❛t♦r ❛♥❞ t❤❡ ❞❡♥♦♠✐♥❛t♦r ❣♦ t♦ 0 ❛s x ❣♦❡s t♦ 1✳ ❆♥ ❛tt❡♠♣t t♦ ❛♣♣❧② t❤❡ ◗✉♦t✐❡♥t ❘✉❧❡ ✕ ♦✈❡r t❤❡s❡ ♦❜❥❡❝t✐♦♥s ✕ ✇♦✉❧❞ r❡s✉❧t ✐♥ ❛♥ ✐♥❞❡t❡r♠✐♥❛t❡ ❡①♣r❡ss✐♦♥ ✿

x2 − 1 0 ??? ❛s x → 1 −−−−−→ x−1 0

❉❊❆❉ ❊◆❉

❚❤✐s ❞♦❡s♥✬t ♠❡❛♥ t❤❛t t❤❡ ❧✐♠✐t ❞♦❡s♥✬t ❡①✐st❀ ✐t ❞♦❡s♥✬t ♠❡❛♥ ❛♥②t❤✐♥❣✦ ❚❤❡ ♦♥❧② ❝♦♥❝❧✉s✐♦♥ ✇❡ ❝❛♥ ❞r❛✇ ❢r♦♠ t❤✐s ✐s t❤❛t ✇❡ ❤❛✈❡ ❞♦♥❡ s♦♠❡t❤✐♥❣ ✇❡ s❤♦✉❧❞♥✬t ❤❛✈❡✳ ❲❡ ❣♦ ❜❛❝❦ ❛♥❞✱ ✐♥st❡❛❞ ♦❢ ◗❘✱

✷✳✹✳ ▲✐♠✐ts ❛♥❞ ❝♦♥t✐♥✉✐t② ✉♥❞❡r ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s

✶✹✸

❞♦ s♦♠❡ ❛❧❣❡❜r❛✳ ❲❡ ❢❛❝t♦r t❤❡ ♥✉♠❡r❛t♦r ❛♥❞ t❤❡♥ ❝❛♥❝❡❧ t❤❡ ❞❡♥♦♠✐♥❛t♦r ✭t❤❡r❡❜② ❝✐r❝✉♠✈❡♥t✐♥❣ t❤❡ ♥❡❡❞ ❢♦r ◗❘✮✿ x2 − 1 x→1 x − 1 (x − 1)(x + 1) = lim ❢♦r x 6= 1 x→1 x−1

lim f (x) = lim

x→1

= lim (x + 1) x→1

= 2.

❚❤❡ ❝❛♥❝❡❧❧❛t✐♦♥ ✐s ❥✉st✐✜❡❞ ❜② t❤❡ ❢❛❝t t❤❛t x t❡♥❞s t♦ 1 ❜✉t ♥❡✈❡r r❡❛❝❤❡s ✐t✳ ▲❡t✬s ❝♦♥s✐❞❡r ♠♦r❡ ❡①❛♠♣❧❡s ♦❢ ❤♦✇ tr②✐♥❣ t♦ ❛♣♣❧② t❤❡ ❧❛✇s ♦❢ ❧✐♠✐ts ✇✐t❤♦✉t ✈❡r✐❢②✐♥❣ t❤❡✐r ❝♦♥❞✐t✐♦♥s ❝♦✉❧❞ ❧❡❛❞ t♦ ✐♥❞❡t❡r♠✐♥❛t❡ ❡①♣r❡ss✐♦♥s✳ ❊①❛♠♣❧❡ ✷✳✹✳✶✹✿ ✐♥❞❡t❡r♠✐♥❛t❡

0/0

❲❡ ❝❤♦♦s❡ ❛ ❢❡✇ ❛❧❣❡❜r❛✐❝❛❧❧② tr✐✈✐❛❧ s✐t✉❛t✐♦♥s✳ ❋♦r x → 0 ❜❡❧♦✇✱ ♠✐s❛♣♣❧✐❝❛t✐♦♥s ♦❢ t❤❡ ◗✉♦t✐❡♥t ❘✉❧❡ ❧❡❛❞ t♦ t❤❡ s❛♠❡ ♣r♦❜❧❡♠✳ ❲❤❡♥ r❡s♦❧✈❡❞✱ t❤❡ ❛♥s✇❡rs ✈❛r② ✿ ❢✉♥❝t✐♦♥ x2 x x x2 x x x sin(1/x) x

✇r♦♥❣ t✉r♥ ♦✉t❝♦♠❡

r❡❞♦♥❡ ✇✐t❤ ❛❧❣❡❜r❛

0 0 0 ??? −−−−−→ 0 0 ??? −−−−−→ 0 0 ??? −−−−−→ 0

x2 =x →0 x 1 x = →∞ 2 x x x =1 →1 x x sin(1/x) = x sin(1/x) ❉◆❊ x

???

−−−−−→

❉❊❆❉ ❊◆❉

❉❊❆❉ ❊◆❉

❉❊❆❉ ❊◆❉

❉❊❆❉ ❊◆❉

♦✉t❝♦♠❡

❙♦✱ ❛♥② ❛♥s✇❡r ✐s ♣♦ss✐❜❧❡ ✇❤❡♥ ✇❡ ❢❛❝❡ ❛♥ ✐♥❞❡t❡r♠✐♥❛t❡ ❡①♣r❡ss✐♦♥✦ ❊①❛♠♣❧❡ ✷✳✹✳✶✺✿ ✐♥❞❡t❡r♠✐♥❛t❡

∞/∞

◆♦✇✱ t❤❡r❡ ❛r❡ ♦t❤❡r ❦✐♥❞s ♦❢ ✐♥❞❡t❡r♠✐♥❛t❡ ❡①♣r❡ss✐♦♥s✳ ❋♦r x → 0✱ ❛ ♠✐s❛♣♣❧✐❝❛t✐♦♥ ♦❢ t❤❡ ◗✉♦t✐❡♥t ❘✉❧❡ ❧❡❛❞s t♦ t❤❡ ❢♦❧❧♦✇✐♥❣✿ ❢✉♥❝t✐♦♥ ✇r♦♥❣ t✉r♥ ♦✉t❝♦♠❡ 1/x 1/x2 1/x2 1/x

∞ ∞ ∞ ??? −−−−−→ ∞ ???

−−−−−→

❉❊❆❉ ❊◆❉

❉❊❆❉ ❊◆❉

r❡❞♦♥❡ ✇✐t❤ ❛❧❣❡❜r❛ ♦✉t❝♦♠❡ 1/x =x 1/x2 1 1/x2 = 1/x x

→0 →∞

❙♦✱ ❛♥② ❛♥s✇❡r ✐s ♣♦ss✐❜❧❡ ✇❤❡♥ ✇❡ ❢❛❝❡ ❛♥ ✐♥❞❡t❡r♠✐♥❛t❡ ❡①♣r❡ss✐♦♥✦ ❊①❡r❝✐s❡ ✷✳✹✳✶✻

❙❤♦✇ ♠♦r❡ ❡①❛♠♣❧❡s ♦❢ ❞✐✛❡r❡♥t ♦✉t❝♦♠❡s ❢♦r t❤❡ ✐♥❞❡t❡r♠✐♥❛t❡ ❡①♣r❡ss✐♦♥ ❛❜♦✈❡✳

✷✳✹✳ ▲✐♠✐ts ❛♥❞ ❝♦♥t✐♥✉✐t② ✉♥❞❡r ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s

❊①❛♠♣❧❡ ✷✳✹✳✶✼✿ ✐♥❞❡t❡r♠✐♥❛t❡

✶✹✹

∞−∞

❋✐♥❛❧❧②✱ ✇❡ s❡❡ ❤♦✇ ✐♥❞❡t❡r♠✐♥❛t❡ ❡①♣r❡ss✐♦♥s ❛♣♣❡❛r ✉♥❞❡r t❤❡ ❙✉♠ ❘✉❧❡ ✐♥st❡❛❞ ♦❢ t❤❡ ◗✉♦t✐❡♥t ❘✉❧❡✳ ❋♦r

x → 0✱

✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿

❢✉♥❝t✐♦♥

✇r♦♥❣ t✉r♥

♦✉t❝♦♠❡

???

(1/x + 1) − 1/x −−−−−→ ∞ − ∞

❉❊❆❉ ❊◆❉

r❡❞♦♥❡ ✇✐t❤ ❛❧❣❡❜r❛

♦✉t❝♦♠❡

(1/x + 1) − 1/x = 1 → 1

❊①❡r❝✐s❡ ✷✳✹✳✶✽

❙❤♦✇ ♠♦r❡ ❡①❛♠♣❧❡s ♦❢ ❞✐✛❡r❡♥t ♦✉t❝♦♠❡s ❢♦r t❤❡ ✐♥❞❡t❡r♠✐♥❛t❡ ❡①♣r❡ss✐♦♥ ❛❜♦✈❡✳

❊①❛♠♣❧❡ ✷✳✹✳✶✾✿ ❛❧❣❡❜r❛✐❝ tr✐❝❦s

❈♦♠♣✉t❡

√

lim

x→0 ■❢ ✇❡ ♠✐♥❞❧❡ss❧② s✉❜st✐t✉t❡

x = 0✱

✇❡ ❣❡t

0/0✳

x2 + 9 − 3 . x2

❲❤❛t ❞♦❡s ✐t ♠❡❛♥❄ ■t ♠❡❛♥s✿

❉❊❆❉ ❊◆❉ ❙❚❖P✦ ❊r❛s❡ ❡✈❡r②t❤✐♥❣ ❛♥❞ ❞♦ ❛❧❣❡❜r❛✳

❚❤❡ ❣♦❛❧ ✐s t♦ ❝❛♥❝❡❧ t❤❡ ❞❡♥♦♠✐♥❛t♦r✳ ❚❤❡ tr✐❝❦ ✐s t♦ ♠✉❧t✐♣❧② ❜② t❤❡ ❝♦♥❥✉❣❛t❡✱

√

x2 + 9 + 3 ✱

♦❢ t❤❡

♥✉♠❡r❛t♦r ✐♥ ♦r❞❡r t♦ ✏r❛t✐♦♥❛❧✐③❡✑ ✐t✳ ❚❤❡♥ ✇❡ ❤❛✈❡✿

lim

x→0

√

√ √ ( x2 + 9 − 3) · ( x2 + 9 + 3) x2 + 9 − 3 √ = lim x→0 x2 x2 · ( x2 + 9 + 3) (x2 + 9) − 32 √ = lim x→0 x2 ( x2 + 9 + 3) x2 √ = lim x→0 x2 ( x2 + 9 + 3) 1 = lim √ x→0 x2 + 9 + 3 1 =√ 0+9+3 1 = . 6

❆t t❤❡ ❡♥❞✱ t❤❡ ◗✉♦t✐❡♥t ❘✉❧❡ ❛♣♣❧✐❡s ❜❡❝❛✉s❡ t❤❡ ❧✐♠✐t ✐♥ t❤❡ ❞❡♥♦♠✐♥❛t♦r ❡①✐sts ❛♥❞ ✐s ♥♦t

0✳

❊①❛♠♣❧❡ ✷✳✹✳✷✵✿ t✇♦ ❢❛♠♦✉s ❧✐♠✐ts

❲❡ ❛r❡ ✐♥ t❤❡ s❛♠❡ ♣♦s✐t✐♦♥ ✭0/0 ✐♥❞❡t❡r♠✐♥❛❝②✮ ✇✐t❤ t❤❡s❡ t✇♦ ❢❛♠✐❧✐❛r ❧✐♠✐ts✿

sin x =1 x→0 x lim

❛♥❞

1 − cos x = 0. x→0 x lim

❚❤❡② ❛r❡ r❡s♦❧✈❡❞ ❧❛t❡r ✇✐t❤ tr✐❣♦♥♦♠❡tr②✳ ❚❤❡② ✇♦✉❧❞ ♣r♦❞✉❝❡ ✐♥❞❡t❡r♠✐♥❛t❡ ❡①♣r❡ss✐♦♥s ✐❢ ✇❡ tr✐❡❞ t♦ ❛♣♣❧② t❤❡ r✉❧❡s ♦❢ ❧✐♠✐ts ✇✐t❤♦✉t ❝❤❡❝❦✐♥❣ t❤❡✐r ❝♦♥❞✐t✐♦♥s ✜rst✳

✷✳✹✳ ▲✐♠✐ts ❛♥❞ ❝♦♥t✐♥✉✐t② ✉♥❞❡r ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s

✶✹✺ ❲❛r♥✐♥❣✦ ❚❤❡ ❛♥s✇❡r t♦ ❛ ❧✐♠✐t ♣r♦❜❧❡♠ ❝❛♥✬t ❜❡ ✏■t✬s ✐♥❞❡✲ t❡r♠✐♥❛t❡✦✑✳

❊①❛♠♣❧❡ ✷✳✹✳✷✶✿ ❛❧❣❡❜r❛ ✇✐t❤ ❉◆❊

■s ✐t ♣♦ss✐❜❧❡ t❤❛t lim (f (x) + g(x)) ❡①✐sts✱ ❡✈❡♥ t❤♦✉❣❤ lim f (x) ❛♥❞ lim g(x) ❞♦ ♥♦t❄ ■♥ ♦t❤❡r ✇♦r❞s✱ x→a x→a x→a ❝❛♥ t❤❡✐r ❛❞❞✐t✐♦♥✱ f + g ✱ ❝❛♥❝❡❧ t❤❡✐r ✐rr❡❣✉❧❛r ❜❡❤❛✈✐♦r❄ ❖❢ ❝♦✉rs❡❀ ❥✉st ♣✐❝❦ g = −f ✳ ❚❤❡♥ f + g = 0✱ s♦ lim (f + g) = 0✳ ❋♦r s♣❡❝✐✜❝ ❡①❛♠♣❧❡s✱ ✇❡ ❝❛♥ t❛❦❡✿ x→a

• f (x) =

1 ❛♥❞ a = 0✳ ❚❤❡ ❧✐♠✐t ❞♦❡s ♥♦t ❡①✐st✿ x lim

x→0

• ◆❡✐t❤❡r ❞♦❡s t❤✐s✿

1 . x



1 lim − x→0 x

• ❇✉t t❤✐s ♦♥❡ ❞♦❡s✿



.



  1 1 lim + − = lim 0 = 0 . x→0 x x→0 x

❲❡ ❝♦♠❜✐♥❡ ❧✐♠✐ts t❤❛t ❡①✐sts ✇✐t❤ t❤♦s❡ t❤❛t ❞♦♥✬t ❜❡❧♦✇✿ ❚❤❡♦r❡♠ ✷✳✹✳✷✷✿ ❉✐✈❡r❣❡♥❝❡ ✉♥❞❡r ❆❞❞✐t✐♦♥ ■❢

lim f (x)

x→a

❡①✐sts ❜✉t

lim g(x)

x→a

❞♦❡s ♥♦t✱ t❤❡♥

lim (f (x) + g(x))

x→a

❞♦❡s ♥♦t ❡①✐st✳

❲❡ t❛❦❡ ❛ ♥❡✇ ❧♦♦❦ ❛t t❤❡ ❜❡❤❛✈✐♦r ♦❢ x ✐♥ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❧✐♠✐t✳ ■♥st❡❛❞ ♦❢ ❝♦♥❝❡♥tr❛t✐♥❣ ♦♥ ◮ ❤♦✇ x ✐s ❛♣♣r♦❛❝❤✐♥❣ a✱

✇❡ ❝❛♥ ❧♦♦❦ ❛t ◮ ❤♦✇ ❢❛r x ✐s ❢r♦♠ a✳

❲❡ ❝♦♥s✐❞❡r t❤❡ ✐♥❝r❡♠❡♥t✱ ✐✳❡✳✱ t❤❡ ❞✐✛❡r❡♥❝❡ h ❜❡t✇❡❡♥ t❤❡ t✇♦✿

❚❤✉s✱ ✇❡ ❤❛✈❡ ❛♥ ❡q✉✐✈❛❧❡♥❝❡✿ x → a ⇐⇒ h = x − a → 0

❚❤❡♥ ✇❡ r❡✇r✐t❡ t❤❡ ❧✐♠✐t lim f (x) ❜② s✉❜st✐t✉t✐♥❣ h = x − a✱ ❛s ❢♦❧❧♦✇s✿ x→a

❚❤❡♦r❡♠ ✷✳✹✳✷✸✿ ❆❧t❡r♥❛t✐✈❡ ❋♦r♠✉❧❛ ❢♦r ▲✐♠✐t ♦❢ ❋✉♥❝t✐♦♥ ❚❤❡ ❧✐♠✐t ♦❢ ❛ ❢✉♥❝t✐♦♥

f

❛t

a

✐s ❡q✉❛❧ t♦

L

✐❢ ❛♥❞ ♦♥❧② ✐❢

lim f (a + h) = L

h→0

✷✳✹✳

▲✐♠✐ts ❛♥❞ ❝♦♥t✐♥✉✐t② ✉♥❞❡r ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s

✶✹✻

❚❤✐s ✇✐❧❧ ♠❛❦❡ ♦✉r ❝♦♠♣✉t❛t✐♦♥s ❧♦♦❦ ❞✐✛❡r❡♥t✳ ❊①❛♠♣❧❡ ✷✳✹✳✷✹✿ ❡❛s✐❡r ❝❛♥❝❡❧❧❛t✐♦♥

▲❡t✬s ✉s❡ t❤❡ t❤❡♦r❡♠ t♦ t❛❦❡ ❛♥♦t❤❡r ❛♣♣r♦❛❝❤ t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ ❧✐♠✐t✿

(1 + h)2 − 1 x2 − 1 = lim x→1 x − 1 h→0 (1 + h) − 1 1 + 2h + h2 − 1 = lim h→0 1+h−1 2h + h2 = lim h→0 h = lim (2 + h) h→0 = 2 + h h=0 = 2. lim

❆❝❝♦r❞✐♥❣ t♦ t❤❡ t❤❡♦r❡♠✳ ❲❡ ❡①♣❛♥❞✳ ❲❡ s✐♠♣❧✐❢②✳ ❲❡ ❞✐✈✐❞❡✳ ❲❡ s✉❜st✐t✉t❡✳

❚❤✐s t✐♠❡✱ ✇❡ ❞✐❞♥✬t ❤❛✈❡ t♦ ❞♦ ❛♥② ❢❛❝t♦r✐♥❣✦

❆s ✇❡ s❡❡✱ t❤✐s s✉❜st✐t✉t✐♦♥ ♠✐❣❤t ♠❛❦❡ t❤❡ ❛❧❣❡❜r❛ s✐♠♣❧❡r✳ ❊①❡r❝✐s❡ ✷✳✹✳✷✺

❯s❡ t❤❡ t❤❡♦r❡♠ t♦ ❝♦♠♣✉t❡✿

x3 − 1 . x→1 x − 1 lim

❏✉st ❛s ✇✐t❤ s❡q✉❡♥❝❡s✱ ✇❡ ❝❛♥ r❡♣r❡s❡♥t t❤❡

f, g  + y

❙✉♠ ❘✉❧❡ lim

−−−−−→ SR lim

❛s ❛ ❞✐❛❣r❛♠✿

F,G  + y

f + g −−−−−→ lim(f + g) = F + G ■♥ t❤❡ ❞✐❛❣r❛♠✱ ✇❡ st❛rt ✇✐t❤ ❛ ♣❛✐r ♦❢ ❢✉♥❝t✐♦♥s ❛t t❤❡ t♦♣ ❧❡❢t ❛♥❞ t❤❡♥ ✇❡ ❝❛♥ ♣r♦❝❡❡❞ ✐♥ t✇♦ ✇❛②s✿

•

❘✐❣❤t✿ t❛❦❡ t❤❡ ❧✐♠✐t ♦❢ ❡✐t❤❡r❀ t❤❡♥ ❞♦✇♥✿ ❛❞❞ t❤❡ r❡s✉❧ts✳

•

❉♦✇♥✿ ❛❞❞ t❤❡♠❀ t❤❡♥ r✐❣❤t✿ t❛❦❡ t❤❡ ❧✐♠✐t ♦❢ t❤❡ r❡s✉❧t✳

❚❤❡ r❡s✉❧t ✐s t❤❡ s❛♠❡✦ ❋♦r t❤❡ Pr♦❞✉❝t ❘✉❧❡ ❛♥❞ t❤❡ ◗✉♦t✐❡♥t ❘✉❧❡✱ ✇❡ ❥✉st r❡♣❧❛❝❡ ✏ +✑ ✇✐t❤ ✏ ·✑ ❛♥❞ ✏ ÷✑ r❡s♣❡❝t✐✈❡❧②✳

❲✐t❤ t❤❡ ❛♣♣❛r❡♥t ❛❜✉♥❞❛♥❝❡ ♦❢ ❞✐s❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s✱ ♥♦✇ t❤❡ ❣♦♦❞ ♥❡✇s✿

◮

❆ t②♣✐❝❛❧ ❢✉♥❝t✐♦♥ ✇❡ ❡♥❝♦✉♥t❡r ✐s ❝♦♥t✐♥✉♦✉s ❛t ❡✈❡r② ♣♦✐♥t ♦❢ ✐ts ❞♦♠❛✐♥✳

❖❢ ❝♦✉rs❡✱ ✐t ❝❛♥✬t ❜❡ ❝♦♥t✐♥✉♦✉s ♦✉ts✐❞❡ t❤❡ ❞♦♠❛✐♥✳ ❋r♦♠ ❛❧❧ t❤❡ ❢✉♥❝t✐♦♥s ✇❡ ❤❛✈❡ s❡❡♥ s♦ ❢❛r✱ ♦♥❧② ❛ ❢❡✇ ♣✐❡❝❡✇✐s❡✲❞❡✜♥❡❞ ❢✉♥❝t✐♦♥s ❤❛✈❡ ❜❡❡♥ ❡①❝❡♣t✐♦♥s✿ t❤❡ s✐❣♥ ❢✉♥❝t✐♦♥ ❛♥❞ t❤❡ ✐♥t❡❣❡r ✈❛❧✉❡ ❢✉♥❝t✐♦♥✳ ❲❡ ♣r♦✈❡ t❤❡ ❝♦♥t✐♥✉✐t② ♦❢ ❛ ❢✉♥❝t✐♦♥ ❜② s❤♦✇✐♥❣ t❤❡ ❢❛❝t t❤❛t ✐ts ❧✐♠✐t ✐s ❡✈❛❧✉❛t❡❞ ❜② s✉❜st✐t✉t✐♦♥✿

lim F (x) = F (a)

x→a

♦r

F (x) → F (a)

❛s

x→a

❖♥❝❡ ✐t ✐s ♣r♦✈❡♥✱ ✇❡ t✉r♥ t❤✐s ❛r♦✉♥❞ ❛♥❞ ✉s❡ t❤✐s ❢❛❝t ❡✈❡r② t✐♠❡ ✇❡ ♥❡❡❞ t♦ ❝♦♠♣✉t❡ ❛ ❧✐♠✐t✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ t❤❡♦r❡♠ ❤❡❧♣s✿

✷✳✹✳

▲✐♠✐ts ❛♥❞ ❝♦♥t✐♥✉✐t② ✉♥❞❡r ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s

✶✹✼

❚❤❡♦r❡♠ ✷✳✹✳✷✻✿ ❆❧❣❡❜r❛ ❛♥❞ ❈♦♥t✐♥✉✐t② ❙✉♣♣♦s❡

f

❛♥❞

g

❛r❡ ❝♦♥t✐♥✉♦✉s ❛t

❙❘✿ P❘✿

f +g f ·g

x = a✳ c·f f /g

❈▼❘✿ ◗❘✿

❚❤❡♥ s♦ ❛r❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢✉♥❝t✐♦♥s✿

c g(a) 6= 0

❢♦r ❛♥② r❡❛❧ ♣r♦✈✐❞❡❞

Pr♦♦❢✳

❋♦r t❤❡

❙✉♠ ❘✉❧❡✱ ✇❡ tr② t♦ ❡✈❛❧✉❛t❡ t❤❡ ❧✐♠✐t ♦❢ t❤❡ s✉♠ ❛t a ❛s ❢♦❧❧♦✇s✿     lim (f + g)(x) = lim f (x) + g(x)

x→a

❲❡ ✉s❡ ❙❘ ♥❡①t✳

x→a

= lim f (x) + lim g(x) x→a

x→a

❲❡ ✉s❡ ❝♦♥t✐♥✉✐t② ♥❡①t✳

= f (a) + g(a) = (f + g)(a) . ❚❤❡r❡❢♦r❡✱ t❤❡ ❧✐♠✐t ♦❢

f +g

✐s t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥❀ ❤❡♥❝❡✱ ✐t ✐s ❝♦♥t✐♥✉♦✉s ❜② t❤❡ ❞❡✜♥✐t✐♦♥✳ ◆❡①t✱

❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡✱ t❤❡ Pr♦❞✉❝t ❘✉❧❡✱ ❛♥❞ t❤❡ ◗✉♦t✐❡♥t ❘✉❧❡ ❛r❡ ♣r♦✈❡♥ ✇✐t❤ t❤❡ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡✱ t❤❡ Pr♦❞✉❝t ❘✉❧❡✱ ❛♥❞ t❤❡ ◗✉♦t✐❡♥t ❘✉❧❡ ❢♦r ❧✐♠✐ts✱ r❡s♣❡❝t✐✈❡❧②✳ t❤❡

▲❡t✬s r❡✈✐❡✇ t❤❡ ❣❡♦♠❡tr✐❝ ♠❡❛♥✐♥❣ ♦❢ t❤❡s❡ r✉❧❡s✳ ■♥ t❤❡

❙✉♠ ❘✉❧❡✱ g s❡r✈❡s ❛s ❛ ✈❡rt✐❝❛❧ ✏♣✉s❤✑

♦❢ t❤❡ ❣r❛♣❤ ♦❢

t❤❛t ✐❞❡❛✳ ❚❤❡r❡ ❛r❡ ♣✐♥❣✲♣♦♥❣ ❜❛❧❧s ❛rr❛♥❣❡❞ ✐♥ ❛ ❝✉r✈❡✱

f✱

f✳

❚❤❡ ♣✐❝t✉r❡ ❜❡❧♦✇ ✐s ♠❡❛♥t t♦ ✐❧❧✉str❛t❡

♦♥ t❤❡ ❣r♦✉♥❞ ❛♥❞ t❤❡r❡ ✐s ❛❧s♦ ✇✐♥❞✱

g✳

❚❤❡♥✱

t❤❡ ✇✐♥❞✱ ♥♦♥✲✉♥✐❢♦r♠❧② ❜✉t ❝♦♥t✐♥✉♦✉s❧②✱ ❜❧♦✇s t❤❡♠ ❢♦r✇❛r❞✿

❚❤❡ ♣✐♥❣✲♣♦♥❣ ❜❛❧❧s r❡♠❛✐♥ ❛rr❛♥❣❡❞ ✐♥ ❛ ❝✉r✈❡✱

f + g✳

❲❡ ❝❛♥ s❛② t❤❛t ✐❢ t❤❡ ✢♦♦r ❛♥❞ t❤❡ ❝❡✐❧✐♥❣ ♦❢ ❛ t✉♥♥❡❧ r❡♣r❡s❡♥t❡❞ ❜② ❝♦♥t✐♥✉♦✉s❧②✱ t❤❡♥ s♦ ✐s ✐ts ❤❡✐❣❤t✱ ✇❤✐❝❤ ✐s

❖r✱ ✐❢ t❤❡ ✢♦♦r ❛♥❞ t❤❡ ❝❡✐❧✐♥❣✱

f

❛♥❞

−g ✱

g−f

f

❛♥❞

g

r❡s♣❡❝t✐✈❡❧② ❛r❡ ❝❤❛♥❣✐♥❣

✭❧❡❢t✮✿

♦❢ ❛ t✉♥♥❡❧ ❛r❡ ❝❤❛♥❣✐♥❣ ❝♦♥t✐♥✉♦✉s❧②✱ t❤❡♥ s♦ ✐s ✐ts ❤❡✐❣❤t✱

g+f

✭r✐❣❤t✮✳ ❚❤✉s✱ ✐❢ t❤❡r❡ ❛r❡ ♥♦ ❜✉♠♣s ♦♥ t❤❡ ✢♦♦r ❛♥❞ ♥♦ ❜✉♠♣s ♦♥ t❤❡ ❝❡✐❧✐♥❣✱ ②♦✉ ✇♦♥✬t ✭s✉❞❞❡♥❧②✮ ❜✉♠♣ ②♦✉r ❤❡❛❞ ❛s ②♦✉ ✇❛❧❦✳

✷✳✹✳

▲✐♠✐ts ❛♥❞ ❝♦♥t✐♥✉✐t② ✉♥❞❡r ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s

✶✹✽

■♥ t❤❡ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡✱ t❤❡ ♠✉❧t✐♣❧❡ c ✐s t❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ ❛ ✈❡rt✐❝❛❧ str❡t❝❤✴s❤r✐♥❦ ♦❢ t❤❡ r✉❜❜❡r s❤❡❡t t❤❛t ❤❛s t❤❡ ❣r❛♣❤ ♦❢ f ❞r❛✇♥ ♦♥ ✐t✿

■♥ t❤❡ Pr♦❞✉❝t ❘✉❧❡✱ ✇❡ s❛② t❤❛t ✐❢ t❤❡ ✇✐❞t❤ ❛♥❞ t❤❡ ❤❡✐❣❤t✱ f ❛♥❞ g ✱ ♦❢ ❛ r❡❝t❛♥❣❧❡ ❛r❡ ❝❤❛♥❣✐♥❣ ❝♦♥t✐♥✉♦✉s❧②✱ t❤❡♥ s♦ ✐s ✐ts ❛r❡❛✱ f · g ✿

■♥ t❤❡ ◗✉♦t✐❡♥t ❘✉❧❡✱ ✇❡ s❛② t❤❛t ✐❢ t❤❡ ✇✐❞t❤ ❛♥❞ t❤❡ ❤❡✐❣❤t✱ f ❛♥❞ g ✱ ♦❢ ❛ tr✐❛♥❣❧❡ ❛r❡ ❝❤❛♥❣✐♥❣ ❝♦♥t✐♥✉♦✉s❧②✱ t❤❡♥ s♦ ✐s t❤❡ t❛♥❣❡♥t ♦❢ ✐ts ❜❛s❡ ❛♥❣❧❡✱ f /g ✿

❚❤❡ ❧❛st ♦♥❡ st❛♥❞s ♦✉t ❜❡❝❛✉s❡ ♦❢ ✐ts ❡①tr❛ r❡str✐❝t✐♦♥ t❤❛t t❤❡ ❞❡♥♦♠✐♥❛t♦r ✐s♥✬t ③❡r♦ ✕ ❜✉t ♦♥❧② ❛t t❤❡ 1 ✐s ❝♦♥t✐♥✉♦✉s ❛t x = 0❀ t❤❡ ❢❛❝t t❤❛t ✐t ✐s ✉♥❞❡✜♥❡❞ ❛t 1 ✐s ✐rr❡❧❡✈❛♥t✳ ❖♥ ♣♦✐♥t a ✐ts❡❧❢✳ ❋♦r ❡①❛♠♣❧❡✱ x−1 t❤❡ ♦t❤❡r ❤❛♥❞✱ ❞✐✈✐s✐♦♥ ❜② 0 ❝r❡❛t❡s ❛ ❤♦❧❡ ✐♥ t❤❡ ❞♦♠❛✐♥❀ t❤❡ ❢✉♥❝t✐♦♥ ❝❛♥✬t ❜❡ ❝♦♥t✐♥✉♦✉s t❤❡r❡ ❛♥②✇❛②✦ ❆♥ ❛❜❜r❡✈✐❛t❡❞ ✈❡rs✐♦♥ ♦❢ t❤❡ t❤❡♦r❡♠ r❡❛❞s ❛s ❢♦❧❧♦✇s✿ ❈♦r♦❧❧❛r② ✷✳✹✳✷✼✿ ❈♦♥t✐♥✉✐t② ✉♥❞❡r ❆❧❣❡❜r❛

❚❤❡ s✉♠✱ t❤❡ ❞✐✛❡r❡♥❝❡✱ t❤❡ ♣r♦❞✉❝t✱ ❛♥❞ t❤❡ r❛t✐♦ ♦❢ t✇♦ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s ✐s ❝♦♥t✐♥✉♦✉s ✭♦♥ ✐ts ❞♦♠❛✐♥✮✳

✷✳✹✳

▲✐♠✐ts ❛♥❞ ❝♦♥t✐♥✉✐t② ✉♥❞❡r ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s

✶✹✾

❊①❡r❝✐s❡ ✷✳✹✳✷✽

❙❦❡t❝❤ t❤❡ ❣r❛♣❤s ♦❢ t❤❡ ♣♦❧②♥♦♠✐❛❧s ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣❛r❛♠❡t❡rs✿

❞❡❣r❡❡ t❤❡ ❧❡❛❞✐♥❣ ❝♦❡✣❝✐❡♥t

❚♦ ❥✉st✐❢② ♦✉r ❝♦♥❝❧✉s✐♦♥ ❛❜♦✉t t❤❡

nt❤

a b c 1 3 4 −2 2 −1

❝♦♥t✐♥✉✐t② ♦❢ ♣♦❧②♥♦♠✐❛❧s✱ ❧❡t✬s ❝♦♥s✐❞❡r ❛ ❣❡♥❡r❛❧ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛♥

❞❡❣r❡❡ ♣♦❧②♥♦♠✐❛❧✿

a0 + a1 x + a2 x2 + ... + an−1 x + an xn . ❚❤❡♥ ✇❡ ❢♦❧❧♦✇ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡q✉❡♥❝❡ ♦❢ ❝♦♥❝❧✉s✐♦♥s✿ ❚❤❡s❡ ❛r❡ ❝♦♥t✐♥✉♦✉s✿ ❚❤❡s❡ ❛r❡ t♦♦ ❜② P❘✿ ❚❤❡s❡ t♦♦ ❜② ❈▼❘✿ ❚❤✐s ✐s ❝♦♥t✐♥✉♦✉s ❜② ❙❘✿

1 , x , x , 1 , x , x2 , a0 , a1 x , a2 x2 , a0 + a1 x + a2 x2 +

... x , x. ... xn−1 , xn . ... an−1 xn−1 , an xn . ... + an−1 xn−1 + an xn .

❲❡ ❛❧s♦ ❝♦♥❝❧✉❞❡ t❤❛t t❤❡ ❣r❛♣❤ ♦❢ ❛ ♣♦❧②♥♦♠✐❛❧ ❝♦♥s✐sts ♦❢ ❛ s✐♥❣❧❡ ♣✐❡❝❡✳ ❊①❛♠♣❧❡ ✷✳✹✳✷✾✿ ❧✐♠✐ts ❜② s✉❜st✐t✉t✐♦♥

❊✈❛❧✉❛t✐♦♥ ♦❢ t❤❡ ❧✐♠✐t ♦❢ ❛ ♣♦❧②♥♦♠✐❛❧ ✐s ♥♦✇ ❡❧❡♠❡♥t❛r②✿


lim x2 − 17x2 + 7x − 2 = 222 − 17 · 222 + 7 · 22 − 2 .

x→22

❚❤❡ ❢♦❧❧♦✇✐♥❣ r❡s✉❧t ✐s ❛ ✈❡r② ✉s❡❢✉❧ s❤♦rt❝✉t✿ ❚❤❡♦r❡♠ ✷✳✹✳✸✵✿ ❈♦♥t✐♥✉✐t② ♦❢ P♦❧②♥♦♠✐❛❧s ❛♥❞ ❘❛t✐♦♥❛❧ ❋✉♥❝t✐♦♥s

• ❊✈❡r② ♣♦❧②♥♦♠✐❛❧ ✐s ❝♦♥t✐♥✉♦✉s ❛t ❡✈❡r② ♣♦✐♥t✳ • ❊✈❡r② r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥ ✐s ❝♦♥t✐♥✉♦✉s ❛t ❡✈❡r② ♣♦✐♥t ✇❤❡r❡ ✐t ✐s ❞❡✜♥❡❞✳ ❊①❡r❝✐s❡ ✷✳✹✳✸✶

❘❡st❛t❡ ❛s ❛♥ ✐♠♣❧✐❝❛t✐♦♥ ❛♥❞ ❞✐s❝✉ss t❤❡ ❝♦♥✈❡rs❡✿ ✏❊✈❡r② ♣♦❧②♥♦♠✐❛❧ ✐s ❛ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥✑✳

❖♥❝❡ t❤❡ ❝♦♥t✐♥✉✐t② ♦❢ ♣♦❧②♥♦♠✐❛❧s ✐s ❡st❛❜❧✐s❤❡❞✱ t❤❡ ❝♦♥t✐♥✉✐t② ♦❢ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥s ✭❛✇❛② ❢r♦♠ t❤❡ ♣♦✐♥ts ✇❤❡r❡ t❤❡ ❞❡♥♦♠✐♥❛t♦r ✐s ③❡r♦✮ ✐s ♣r♦✈❡♥ ❢r♦♠ t❤❡

◗✉♦t✐❡♥t ❘✉❧❡✳

❲❡ ❛❧s♦ ❝♦♥❝❧✉❞❡ t❤❛t t❤❡ ❣r❛♣❤ ♦❢ ❛

r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥ ❝♦♥s✐sts ♦❢ s❡✈❡r❛❧ ♣✐❡❝❡s ✕ ♦♥❡ ❢♦r ❡❛❝❤ ✐♥t❡r✈❛❧ ♦❢ ✐ts ❞♦♠❛✐♥✳ ❊①❛♠♣❧❡ ✷✳✹✳✸✷✿ r❛t✐♦♥❛❧

❋♦r ❡①❛♠♣❧❡✱ t❤❡ ❢✉♥❝t✐♦♥

f (x) = ❤❛s t❤r❡❡ ❤♦❧❡s✱

x = 1, 2, 3✱

✐♥ ✐ts ❞♦♠❛✐♥✿

(x −

1)2 (x

1 − 2)(x − 3)

1 2 3 − − −− ◦ − − −− ◦ − − −− ◦ − − −− ❚❤❡r❡❢♦r❡✱ ✐ts ❣r❛♣❤ ❤❛s ❢♦✉r ❜r❛♥❝❤❡s✳

❊①❡r❝✐s❡ ✷✳✹✳✸✸

❉✐s❝✉ss t❤❡ ❝♦♥t✐♥✉✐t② ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢✉♥❝t✐♦♥✿

✷✳✹✳ ▲✐♠✐ts ❛♥❞ ❝♦♥t✐♥✉✐t② ✉♥❞❡r ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s

❚❤✐s ✐s ❛ ✢♦✇❝❤❛rt ❢♦r ❧✐♠✐t ❝♦♠♣✉t❛t✐♦♥✿

❚❤❡r❡ ✐s ♦♥❧② ♦♥❡ ✇❛② ♦✉t✦

✶✺✵

✷✳✺✳

❚❤❡ ❡①♣♦♥❡♥t✐❛❧ ❛♥❞ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s

✶✺✶

✷✳✺✳ ❚❤❡ ❡①♣♦♥❡♥t✐❛❧ ❛♥❞ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s

■s t❤❡ ❡①♣♦♥❡♥t✐❛❧

❢✉♥❝t✐♦♥ ❝♦♥t✐♥✉♦✉s❄

▲❡t✬s r❡✈✐❡✇ ❤♦✇ ✇❡ ❝♦♥str✉❝t ✐t✳ ❲❡ ❤❛✈❡ ❞❡✜♥❡❞ ✭✐♥ ❱♦❧✉♠❡ ✶✱ ❈❤❛♣t❡rs ✶P❈✲✶ ❛♥❞ ✶P❈✲✹✮ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥ f (x) = ax , a > 0 ❢♦r ❛❧❧ ✐♥t❡❣❡r ✈❛❧✉❡s ♦❢ x ❛s r❡♣❡❛t❡❞ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❛♥❞ t❤❡♥ ❛❧s♦ ❞❡✜♥❡❞ ❛s r❡♣❡❛t❡❞ ❞✐✈✐s✐♦♥ ✐♥ ❝❛s❡ ♦❢ ❛ ♥❡❣❛t✐✈❡ x ❛♥❞ t❤❡♥ ❛❞❞❡❞ ♠♦r❡ ✈❛❧✉❡s ❜② ❞✐✈✐❞✐♥❣ t❤❡s❡ ✐♥t❡r✈❛❧s ✐♥ ❤❛❧❢ ❛s ♠❛♥② t✐♠❡s ❛s ♥❡❝❡ss❛r②✿

❲❡ ✉s❡❞ t❤❡

❣❡♦♠❡tr✐❝ ♠❡❛♥ t♦ ❞❡✜♥❡ t❤❡ ✈❛❧✉❡ ❛t t❤✐s ♥❡✇ ♣♦✐♥t✿ a

x+y 2

❲❡ ❝❛♥ ❛❧s♦ ❡①t❡♥❞ t❤❡ ❞❡✜♥✐t✐♦♥ t♦ ❛❧❧ r❛t✐♦♥❛❧ p

=

√

ax · ay .

♥✉♠❜❡rs x = pq

aq =

√ q

ap =

❚❤❡ ❣❛♣s ✐♥ t❤❡ ❣r❛♣❤ ❜❡❝♦♠❡ ✐♥✈✐s✐❜❧❡ ❛t ♦♥❝❡✳

❜② ♠❡❛♥s ♦❢ t❤❡ ❢♦r♠✉❧❛✿


p √ q a

◆♦✇✱ ✇❤❛t ❛❜♦✉t y = ex ❢♦r t❤❡ r❡st ♦❢ t❤❡ r❡❛❧ ✈❛❧✉❡s ♦❢ x❄ ❲❡ ✉s❡ ❧✐♠✐ts❀ ❢♦r ❡①❛♠♣❧❡✱ eπ = lim exn , n→∞

❢♦r ❛♥② s❡q✉❡♥❝❡ xn ♦❢ r❛t✐♦♥❛❧ ♥✉♠❜❡rs t❤❛t ❝♦♥✈❡r❣❡s t♦ π ✳ ❖❢ ❝♦✉rs❡✱ s✐♥❝❡ t❤❡r❡ ❛r❡ ✐♥✜♥✐t❡❧② ♠❛♥② s❡q✉❡♥❝❡s ❝♦♥✈❡r❣✐♥❣ t♦ ❛ ♥✉♠❜❡r✱ ✇❡ ♥❡❡❞ t♦ s❤♦✇ t❤❛t t❤❡② ❛❧❧ ♣r♦❞✉❝❡ t❤❡ s❛♠❡ r❡s✉❧t✦ ❲❡ ❛❧r❡❛❞② ❤❛✈❡ ♦♥❡ ❧✐♠✐t ❢r♦♠ ❈❤❛♣t❡r ✶ t❤❛t ❡♥s✉r❡s ❛ s✐♥❣❧❡ ♦✉t❝♦♠❡✿ lim exn = 1

n→∞

❢♦r ❛♥② s❡q✉❡♥❝❡ xn → 0✳ ◆♦✇ t❤❡ r❡st ♦❢ ♥✉♠❜❡rs✿

❉❡✜♥✐t✐♦♥ ✷✳✺✳✶✿ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥ ❢♦r r❡❛❧ ❛r❣✉♠❡♥t ❲❡ ❞❡✜♥❡ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥ ❛t x = a ❛s t❤✐s ❧✐♠✐t✿ ea = lim exn n→∞

❢♦r ❛♥② s❡q✉❡♥❝❡ xn → a ♦❢ r❛t✐♦♥❛❧ ♥✉♠❜❡rs✳ ❲❡ ♥❡❡❞ t♦ ♣r♦✈❡ t❤❛t t❤❡ ❞❡✜♥✐t✐♦♥ ♠❛❦❡s s❡♥s❡✿

✷✳✺✳

❚❤❡ ❡①♣♦♥❡♥t✐❛❧ ❛♥❞ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s

✶✺✷

❚❤❡♦r❡♠ ✷✳✺✳✷✿ ❱❛❧✐❞✐t② ♦❢ ❉❡✜♥✐t✐♦♥ ♦❢ ❊①♣♦♥❡♥t ❚❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥ ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ t❤❡ ❝❤♦✐❝❡ ♦❢ t❤❡ s❡q✉❡♥❝❡✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡ ❢♦r ❛♥② a✿ lim exn = ea ,

n→∞

❋❖❘ ❆◆❨ s❡q✉❡♥❝❡ xn → a✳

Pr♦♦❢✳ ▲❡t hn = xn − a✳ ❚❤❡♥✿ lim exn = lim ea+hn

n→∞

n→∞

= lim ea ehn n→∞ a

= e · lim ehn n→∞

= ea · 1 = ea .

❆❝❝♦r❞✐♥❣ t♦ ❛ r✉❧❡ ♦❢ ❡①♣♦♥❡♥ts✳ ❆❝❝♦r❞✐♥❣ t♦ ❈▼❘✳ ❆❝❝♦r❞✐♥❣ t♦ t❤❡ ❛❜♦✈❡ ❧✐♠✐t✳

▲❡t✬s r❡♣❤r❛s❡ t❤❡ r❡s✉❧t ✐♥ t❡r♠s ♦❢ t❤❡ ❝♦♥t✐♥✉✐t② ♦❢ t❤✐s ❢✉♥❝t✐♦♥✳ ■♥ t❡r♠s ♦❢ ❧✐♠✐ts ♦❢ ❢✉♥❝t✐♦♥s✱ ✇❡ ✜rst r❡st❛t❡ t❤❡ ❧✐♠✐t ❢r♦♠ t❤❡ ❧❛st ❝❤❛♣t❡r✿ lim ex = 1 .

x→0

■♥ ♦t❤❡r ✇♦r❞s✿

lim ex = e0 .

x→0

❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡ ♣r♦✈❡♥ t❤❡ ❝♦♥t✐♥✉✐t② ♦❢ ex ❛t ♦♥❡ s✐♥❣❧❡ ♣♦✐♥t✱ x = 0✳ ❲❡ ♥♦✇ ❞❡r✐✈❡ t❤❡ r❡st ❛s ❢♦❧❧♦✇s✿

❚❤❡♦r❡♠ ✷✳✺✳✸✿ ❈♦♥t✐♥✉✐t② ♦❢ ◆❛t✉r❛❧ ❊①♣♦♥❡♥t✐❛❧ ❋✉♥❝t✐♦♥ ❚❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥ y = ex ✐s ❝♦♥t✐♥✉♦✉s ❛t ❡✈❡r② x✳ ❚❤❡ t❤❡♦r❡♠ ❝♦♥✜r♠s t❤❛t t❤❡ ❣r❛♣❤s ♦❢ y = ex ❞♦❡s ✐♥❞❡❡❞ ❧♦♦❦ ❧✐❦❡ t❤✐s✱ ❡✈❡♥ ✐❢ ✇❡ ③♦♦♠ ✐♥ ♦♥ ❛♥② ♣♦✐♥t✿

❊①❡r❝✐s❡ ✷✳✺✳✹ Pr♦✈❡✿

x

e = lim

n→∞



x n 1+ . n

✷✳✺✳

❚❤❡ ❡①♣♦♥❡♥t✐❛❧ ❛♥❞ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s

✶✺✸

❈♦r♦❧❧❛r② ✷✳✺✳✺✿ ❈♦♥t✐♥✉✐t② ♦❢ ❊①♣♦♥❡♥t✐❛❧ ❋✉♥❝t✐♦♥s

❆♥② ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥ ✭✐✳❡✳✱ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥ ❝♦♥t✐♥✉♦✉s ❛t ❡✈❡r②

y = bx

♦❢ ❛♥② ❜❛s❡ b✮ ✐s

x✳

❊①❡r❝✐s❡ ✷✳✺✳✻

Pr♦✈❡ t❤❡ ❝♦r♦❧❧❛r②✳ ❲❡ ❢♦❧❧♦✇ t❤❡ s❛♠❡ ♣❛t❤ ❢♦r t❤❡ (−∞, ∞)✳

tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s

❛♥❞ ❥✉st✐❢② tr❡❛t✐♥❣ t❤❡s❡ ❢✉♥❝t✐♦♥s ❛s ❞❡✜♥❡❞ ♦♥

❙✉♣♣♦s❡ ❛ r❡❛❧ ♥✉♠❜❡r x ✐s ❣✐✈❡♥✳ ❲❡ ❝♦♥str✉❝t ❛ ❧✐♥❡ s❡❣♠❡♥t ♦❢ ❧❡♥❣t❤ 1 ♦♥ t❤❡ ♣❧❛♥❡✳ ❚❤❡♥✿

• cos x ✐s t❤❡ ❤♦r✐③♦♥t❛❧ ❝♦♦r❞✐♥❛t❡ ♦❢ t❤❡ ❡♥❞ ♦❢ t❤❡ s❡❣♠❡♥t✳ • sin x ✐s t❤❡ ✈❡rt✐❝❛❧ ❝♦♦r❞✐♥❛t❡ ♦❢ t❤❡ ❡♥❞ ♦❢ t❤❡ s❡❣♠❡♥t✳

❲❡ t❤✐♥❦ ♦❢ cos x ❛s t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ s❤❛❞♦✇ ♦❢ t❤❡ st✐❝❦ ♦❢ ❧❡♥❣t❤ 1 ✉♥❞❡r ❛♥❣❧❡ x ✐♥ t❤❡ ❣r♦✉♥❞ ✇❤❡♥ t❤❡ s✉♥ ✐s ❛❜♦✈❡ ✐t ❛♥❞ sin x ❛s t❤❡ ❧❡♥❣t❤ ♦❢ ✐ts s❤❛❞♦✇ ♦♥ t❤❡ ✇❛❧❧ ❛t s✉♥s❡t✳ ■t ✐s t❤❡♥ ♣❧❛✉s✐❜❧❡ t❤❛t ✕ ❛s t❤❡ st✐❝❦ r♦t❛t❡s ✕ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ s❤❛❞♦✇ ❝❤❛♥❣❡s ❝♦♥t✐♥✉♦✉s❧②✳ ❖r t❤❡ st✐❝❦ ✐s st✐❧❧ ❛♥❞ ✐t ✐s t❤❡ s✉♥ t❤❛t ✐s ♠♦✈✐♥❣✳ ❚❤❡♥ t❤❡ s❤❛❞♦✇ ❣✐✈❡s ✉s cos x✱ ✇❤❡r❡ x ✐s ❛ ♠✉❧t✐♣❧❡ ♦❢ t✐♠❡✿

■♥ t❤❡ ❧❛st ❝❤❛♣t❡r✱ ✇❡ ❞❡✜♥❡❞ t❤❡ s✐♥❡ ❛♥❞ ❝♦s✐♥❡ ❛❧❣❡❜r❛✐❝❛❧❧② ❜② ✉s✐♥❣ t❤❡ ❤❛❧❢✲❛♥❣❧❡ ❢♦r♠✉❧❛✳ ❚❤❡ ❢✉♥❝t✐♦♥s✱ t❤❡r❡❢♦r❡✱ ❛r❡ ♦♥❧② ❞❡✜♥❡❞ ♦♥ t❤❡ ❢r❛❝t✐♦♥s ♦❢ π ✿

✷✳✺✳

❚❤❡ ❡①♣♦♥❡♥t✐❛❧ ❛♥❞ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s

■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ❞♦♠❛✐♥ ✐s

✶✺✹

X = {rπ : r r❛t✐♦♥❛❧ }.

❲❤❛t ❛❜♦✉t t❤❡ r❡st❄

❲❤❛t ✐s sin 1❄ ❲❡ ✉s❡ ❧✐♠✐ts❀ ❢♦r ❡①❛♠♣❧❡✱ sin 1 = lim sin π · 1 + 1/n n→∞





.

❖❢ ❝♦✉rs❡✱ s✐♥❝❡ t❤❡r❡ ❛r❡ ✐♥✜♥✐t❡❧② ♠❛♥② s❡q✉❡♥❝❡s ❝♦♥✈❡r❣✐♥❣ t♦ ❛ ♥✉♠❜❡r✱ ✇❡ ♥❡❡❞ t♦ s❤♦✇ t❤❛t t❤❡② ❛❧❧ ♣r♦❞✉❝❡ t❤❡ s❛♠❡ r❡s✉❧t✳ ❲❡ ❛❧r❡❛❞② ❤❛✈❡ t❤❡ t✇♦ ❧✐♠✐ts ❢r♦♠ ❈❤❛♣t❡r ✶ t❤❛t ❡♥s✉r❡ ❛ s✐♥❣❧❡ ♦✉t❝♦♠❡✿ lim sin xn = 0 ❛♥❞

n→∞

lim cos xn = 1

n→∞

❢♦r ❛♥② s❡q✉❡♥❝❡ xn → 0✳ ◆♦✇ t❤❡ r❡st ♦❢ ♥✉♠❜❡rs✿

❉❡✜♥✐t✐♦♥ ✷✳✺✳✼✿ ❙✐♥❡ ❛♥❞ ❈♦s✐♥❡ ❲❡ ❞❡✜♥❡ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ s✐♥❡ ❛♥❞ t❤❡ ❝♦s✐♥❡ ❛t x = a ❛s t❤❡s❡ ❧✐♠✐ts✿ sin a = lim sin xn ❛♥❞ cos a = lim cos xn n→∞

n→∞

❢♦r ❛♥② s❡q✉❡♥❝❡ xn → a ✐♥ X ✳ ❲❡ ♥❡❡❞ t♦ ♣r♦✈❡ t❤❛t t❤❡ ❞❡✜♥✐t✐♦♥ ♠❛❦❡s s❡♥s❡✿

❚❤❡♦r❡♠ ✷✳✺✳✽✿ ❱❛❧✐❞✐t② ♦❢ ❉❡✜♥✐t✐♦♥ ♦❢ ❙✐♥❡ ❛♥❞ ❈♦s✐♥❡ ❚❤❡ ❞❡✜♥✐t✐♦♥ ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ t❤❡ ❝❤♦✐❝❡ ♦❢ t❤❡ s❡q✉❡♥❝❡✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿ lim sin xn = sin a ❛♥❞

n→∞

❢♦r ❛♥② s❡q✉❡♥❝❡ xn → a ❛♥❞ ❛♥② a✳

lim cos xn = cos a ,

n→∞

✷✳✺✳

❚❤❡ ❡①♣♦♥❡♥t✐❛❧ ❛♥❞ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s

✶✺✺

Pr♦♦❢✳

▲❡t hn = xn − a✳ ❚❤❡♥✿ lim sin(xn ) = lim sin(a + hn ) n→∞ = lim sin a · cos h

n→∞

h→0

= lim

n→∞

sin a · cos hn

= sin a · lim cos hn n→∞




= sin a · 1 = sin a .

+ cos a · sin hn +

lim

n→∞



cos a · sin hn

+ cos a · lim sin hn n→∞

+ cos a · 0

❆ tr✐❣ ❢♦r♠✉❧❛ ✭❈❤❛♣t❡r ✶P❈✲✺✮✳


❆❝❝♦r❞✐♥❣ t♦ ❙❘✳ ❆❝❝♦r❞✐♥❣ t♦ ❈▼❘✳ ❋r♦♠ t❤❡ ❛❜♦✈❡ ❧✐♠✐ts✳

❊①❡r❝✐s❡ ✷✳✺✳✾

Pr♦✈❡ t❤❡ t❤❡♦r❡♠ ❢♦r t❤❡ ❝♦s✐♥❡✳ ❊①❡r❝✐s❡ ✷✳✺✳✶✵

❙❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢ ②♦✉r ❡❧❡✈❛t✐♦♥ ❞✉r✐♥❣ ❛ r✐❞❡ ♦♥ t❤❡ ❋❡rr✐s ✇❤❡❡❧✳ ▲❡t✬s r❡♣❤r❛s❡ t❤❡ r❡s✉❧t ✐♥ t❡r♠s ♦❢ t❤❡ ❝♦♥t✐♥✉✐t② ♦❢ t❤❡s❡ t✇♦ ❢✉♥❝t✐♦♥s✳ ■♥ t❡r♠s ♦❢ ❧✐♠✐ts ♦❢ ❢✉♥❝t✐♦♥s✱ ✇❡ ❝❛♥ r❡st❛t❡ t❤❡ t✇♦ ❧✐♠✐ts ❢r♦♠ t❤❡ ❧❛st ❝❤❛♣t❡r✿ lim sin x = 0 ❛♥❞ lim cos x = 1 .

x→0

■♥ ♦t❤❡r ✇♦r❞s✿

x→0

lim sin x = sin 0 ❛♥❞ lim cos x = cos 0 .

x→0

x→0

❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡ ❛❧r❡❛❞② ♣r♦✈❡♥ t❤❡ ❝♦♥t✐♥✉✐t② ♦❢ t❤❡ s✐♥❡ ❛♥❞ t❤❡ ❝♦s✐♥❡ ❛t ♦♥❡ s✐♥❣❧❡ ♣♦✐♥t✱ x = 0✳ ❲❡ ♥♦✇ ❞❡r✐✈❡ t❤❡ r❡st ❛s ❢♦❧❧♦✇s✿ ❚❤❡♦r❡♠ ✷✳✺✳✶✶✿ ❈♦♥t✐♥✉✐t② ♦❢ ❙✐♥❡ ❛♥❞ ❈♦s✐♥❡ ❇♦t❤ t❤❡ ❝♦s✐♥❡✱

y = cos x✱

❛♥❞ t❤❡ s✐♥❡✱

y = sin x✱

❛r❡ ❝♦♥t✐♥✉♦✉s ❛t ❡✈❡r②

x✳

Pr♦♦❢✳

❲❡ r❡♣❡❛t t❤❡ ❛❜♦✈❡ ♣r♦♦❢✿ 

sin a · cos h + cos a · sin h ❆ tr✐❣ ❢♦r♠✉❧❛ ✭❈❤❛♣t❡r ✶P❈✲✺✮✳

= lim sin a · cos h + lim cos a · sin h ❆❝❝♦r❞✐♥❣ t♦ ❙❘✳

lim sin(a + h) = lim

h→0



h→0

h→0

h→0

= sin a · lim cos h h→0

= sin a · 1 = sin a .

+ cos a · lim sin h h→0

+ cos a · 0

❆❝❝♦r❞✐♥❣ t♦ ❈▼❘✳ ❋r♦♠ t❤❡ ❛❜♦✈❡ ❧✐♠✐ts✳

❚❤❡ t❤❡♦r❡♠ ❝♦♥✜r♠s t❤❛t t❤❡ ❣r❛♣❤s ♦❢ y = sin x ❛♥❞ y = cos x ❞♦ ✐♥❞❡❡❞ ❧♦♦❦ ❧✐❦❡ t❤✐s✱ ❡✈❡♥ ✐❢ ✇❡ ③♦♦♠ ✐♥ ♦♥ ❛♥② ♣♦✐♥t✿

✷✳✺✳

❚❤❡ ❡①♣♦♥❡♥t✐❛❧ ❛♥❞ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s

✶✺✻

❇❡❧♦✇✱ ✇❡ r❡✲st❛t❡ t❤❡ ❢❛♠♦✉s ❧✐♠✐ts ❛s ❢❛❝ts ❛❜♦✉t ❝♦♥t✐♥✉✐t②✳ ❊①❡r❝✐s❡ ✷✳✺✳✶✷

❙❤♦✇ t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ❢✉♥❝t✐♦♥s ❛r❡ ❝♦♥t✐♥✉♦✉s ❛t ❡✈❡r② ♣♦✐♥t✿   sin x f (x) = x 1

✐❢ x 6= 0 ✐❢ x = 0

❲❡ ✇♦♥✬t ❣♦ ✐♥t♦ tr✐❣♦♥♦♠❡tr② ❜❡②♦♥❞ t❤❡ t❛♥❣❡♥t✿

  1 − cos x g(x) = x 0

✐❢ x 6= 0 ✐❢ x = 0

❈♦r♦❧❧❛r② ✷✳✺✳✶✸✿ ❈♦♥t✐♥✉✐t② ♦❢ ❚❛♥❣❡♥t

❚❤❡ t❛♥❣❡♥t✱ y = tan x✱ ✐s ❝♦♥t✐♥✉♦✉s ❛t ❡✈❡r② x ✇❤❡r❡ ✐t ✐s ❞❡✜♥❡❞✳ Pr♦♦❢✳

❙✐♥❝❡ tan x =

✇❡ ❛♣♣❧② t❤❡ ◗✉♦t✐❡♥t

sin x , cos x

❘✉❧❡ t♦ ❝♦♥❝❧✉❞❡ t❤❛t ✐t ✐s ❝♦♥t✐♥✉♦✉s ❛t ❡✈❡r② ♣♦✐♥t x ✇✐t❤ cos x 6= 0✳

❊①❛♠♣❧❡ ✷✳✺✳✶✹✿ ❧✐♠✐t ♦❢ t❛♥❣❡♥t

▲❡t✬s ❝♦♥s✐❞❡r ❛ ♣♦✐♥t ✇❤❡r❡ t❤❡ t❛♥❣❡♥t ✐s ✉♥❞❡✜♥❡❞✱ x = π/2✿

❲❡ ❦♥♦✇ t❤❛t ❛s x → π/2✱ ✇❡ ❤❛✈❡✿ sin x → 1 ❛♥❞ cos x → 0 .

❲❡ s❤♦✉❧❞ ❝♦♥❝❧✉❞❡ t❤❛t tan x → ±∞ .

❍♦✇❡✈❡r✱ ✇❤✐❝❤ ✐♥✜♥✐t②❄ ❲❡ t❛❦❡ ✐♥t♦ ❛❝❝♦✉♥t t❤❡ s✐❣♥ ♦❢ cos x ♦♥ t❤❡ t✇♦ s✐❞❡s ♦❢ π/2✿ 0 < cos x → 0 ❛s x → π/2− ❛♥❞ 0 > cos x → 0 ❛s x → π/2+ .

❚❤❡r❡❢♦r❡✱

tan x → +∞ ❛s x → π/2− ❛♥❞ tan x → −∞ ❛s x → π/2+ .

■♥❞❡❡❞✱ t❤❡ ❣r❛♣❤ r❡✈❡❛❧s ❞✐✛❡r❡♥t ❜❡❤❛✈✐♦rs ♦♥ t❤❡ t✇♦ s✐❞❡s ♦❢ π/2✳ ❚❤❡ ♣❛tt❡r♥ r❡♣❡❛ts ✐ts❡❧❢ ❡✈❡r② π ✉♥✐ts ♦♥ t❤❡ x✲❛①✐s✳ ❲❡ ❤❛✈❡ ❛❞❞❡❞ ♠♦r❡ ❢✉♥❝t✐♦♥s t♦ ♦✉r ❝♦❧❧❡❝t✐♦♥ ♦❢ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s✦ ■♥ s✉♠♠❛r②✱ ❛❧❧ ♣♦❧②♥♦♠✐❛❧s✱ ❛❧❧ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥s✱ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥s✱ ❛♥❞ t❤❡ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝✲

✷✳✻✳

▲✐♠✐ts ❛♥❞ ❝♦♥t✐♥✉✐t② ✉♥❞❡r ❝♦♠♣♦s✐t✐♦♥s

✶✺✼

t✐♦♥s ❛r❡ ❝♦♥t✐♥✉♦✉s ✭♦♥ t❤❡✐r ❞♦♠❛✐♥s✮✳ ❚❤❡r❡❢♦r❡✱ t❤❡✐r ❧✐♠✐ts ❛r❡ ❡✈❛❧✉❛t❡❞ ❜② s✉❜st✐t✉t✐♦♥✿

lim ex = e0 , lim sin x = sin 0, ❡t❝✳

x→0

x→0

❈♦♠❜✐♥✐♥❣ t❤❡s❡ ❢✉♥❝t✐♦♥s ✈✐❛ t❤❡ ❢♦✉r ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s ✇✐❧❧ ♣r♦❞✉❝❡ ♠♦r❡ ❝♦♥t✐♥✉♦✉s ✭♦♥ t❤❡✐r ❞♦♠❛✐♥s✮ ❢✉♥❝t✐♦♥s✿ x3 + 3 , ❡t❝✳ x2 + sin x, cos x · ex , sin x + 2x ❍♦✇❡✈❡r✱ ✇❤❛t ❛❜♦✉t t❤❡s❡✿

sin(x2 + 1), ecos x , ❡t❝✳ ❄

✷✳✻✳ ▲✐♠✐ts ❛♥❞ ❝♦♥t✐♥✉✐t② ✉♥❞❡r ❝♦♠♣♦s✐t✐♦♥s

■♥ t❤❡ ❧❛r❣❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s ♦♥ ❢✉♥❝t✐♦♥s✱ ❝♦♠♣♦s✐t✐♦♥s ❛r❡ t❤❡ ♠♦st ✐♠♣♦rt❛♥t✳ ❊①❛♠♣❧❡ ✷✳✻✳✶✿ ❞r✐✈✐♥❣ t❤r♦✉❣❤ t❡rr❛✐♥

❚❤✐s ✐s ❤♦✇ ❝♦♠♣♦s✐t✐♦♥s ♠✐❣❤t ❡♠❡r❣❡✳ ❙✉♣♣♦s❡ ❛ ❝❛r ✐s ❞r✐✈❡♥ t❤r♦✉❣❤ ❛ ♠♦✉♥t❛✐♥ t❡rr❛✐♥ ❛♥❞ ✇❡ ❦♥♦✇ t❤❡ ❢♦❧❧♦✇✐♥❣✿ • ❲❡ ❦♥♦✇ ✇❤❡r❡ ✇❡ ❛r❡ ♦♥ t❤❡ ♠❛♣ ❛t ❡✈❡r② ♠♦♠❡♥t ♦❢ t✐♠❡✳ • ❚❤❡ ♠❛♣ t❡❧❧s ✉s t❤❡ ❛❧t✐t✉❞❡ ❢♦r ❡❛❝❤ ❧♦❝❛t✐♦♥✳ ❚❤❡s❡ t✇♦ ♣✐❡❝❡s ♦❢ ✐♥❢♦r♠❛t✐♦♥ ❛r❡ s❤♦✇♥ ❜❡❧♦✇ ❛s r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ t❤r❡❡ q✉❛♥t✐t✐❡s✿

❲❡ s❡t ✉♣ t✇♦ ❢✉♥❝t✐♦♥s✱ ❢♦r ❧♦❝❛t✐♦♥ ❛♥❞ ❛❧t✐t✉❞❡✱ ❛♥❞ t❤❡✐r ❝♦♠♣♦s✐t✐♦♥ ✐s ✇❤❛t ✇❡ ❛r❡ ✐♥t❡r❡st❡❞ ✐♥✿ • t ✐s t✐♠❡ ✕ ♠❡❛s✉r❡❞ ✐♥ ❤r • x = f (t) ✐s t❤❡ ❧♦❝❛t✐♦♥ ♦❢ t❤❡ ❝❛r ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✐♠❡ ✕ ♠❡❛s✉r❡❞ ✐♥ ♠✐ • y = g(x) ✐s t❤❡ ❡❧❡✈❛t✐♦♥ ♦❢ t❤❡ r♦❛❞ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ ✭❤♦r✐③♦♥t❛❧✮ ❧♦❝❛t✐♦♥ ✕ ♠❡❛s✉r❡❞ ✐♥ ❢t • y = h(t) = g(f (t)) ✐s t❤❡ ❛❧t✐t✉❞❡ ♦❢ t❤❡ r♦❛❞ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✐♠❡ ✕ ♠❡❛s✉r❡❞ ✐♥ ❢t ❚❤❡ ✜rst ❢✉♥❝t✐♦♥ ❞❡s❝r✐❜❡s t❤❡ ♠♦t✐♦♥✱ ❛♥❞ t❤❡ s❡❝♦♥❞ ❢✉♥❝t✐♦♥ ✐s ❧✐t❡r❛❧❧② t❤❡ ♣r♦✜❧❡ ♦❢ t❤❡ r♦❛❞✳ ❖♥❡ ✇♦✉❧❞ ❡①♣❡❝t ❛❧❧ t❤r❡❡ ❢✉♥❝t✐♦♥s ❤❡r❡ t♦ ❜❡ ❝♦♥t✐♥✉♦✉s✿

✷✳✻✳

▲✐♠✐ts ❛♥❞ ❝♦♥t✐♥✉✐t② ✉♥❞❡r ❝♦♠♣♦s✐t✐♦♥s

✶✺✽

■♥ ❣❡♥❡r❛❧✱ t❤✐s ✐s t❤❡ s❡t✉♣✿

❉❡✜♥✐t✐♦♥ ✷✳✻✳✷✿ ❝♦♠♣♦s✐t✐♦♥ ♦❢ ❢✉♥❝t✐♦♥s ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ t✇♦ ❢✉♥❝t✐♦♥s ✭✇✐t❤ t❤❡ ❝♦❞♦♠❛✐♥ ♦❢ t❤❡ ❢♦r♠❡r ♠❛t❝❤✐♥❣ t❤❡ ❞♦♠❛✐♥ ♦❢ t❤❡ ❧❛tt❡r✮✿

F :X→Y ❚❤❡♥ t❤❡✐r

❝♦♠♣♦s✐t✐♦♥

G:Y →Z.

❛♥❞

✐s t❤❡ ❢✉♥❝t✐♦♥ ✭❢r♦♠ t❤❡ ❞♦♠❛✐♥ ♦❢ t❤❡ ❢♦r♠❡r t♦ t❤❡

❝♦❞♦♠❛✐♥ ♦❢ t❤❡ ❧❛tt❡r✮

H :X →Z, ✇❤✐❝❤ ✐s ❝♦♠♣✉t❡❞ ❢♦r ❡✈❡r②

x

✐♥

X

❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦✲st❡♣ ♣r♦❝❡✲

❞✉r❡✿

x → F (x) = y → G(y) = z . ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ♥❡✇ ❢✉♥❝t✐♦♥ ✐s ❣✐✈❡♥ ❜② t❤❡

s✉❜st✐t✉t✐♦♥ ❢♦r♠✉❧❛ ✿

z = H(x) = G(F (x)) . ■t ✐s ❞❡♥♦t❡❞ ❛s ❢♦❧❧♦✇s✿

G◦F

❲❛r♥✐♥❣✦ ❲❡ r❡❛❞ ❝♦♠♣♦s✐t✐♦♥s ❢r♦♠ r✐❣❤t t♦ ❧❡❢t✳

X

❲❡ ❥✉st ❢♦❧❧♦✇ ❢r♦♠

❛❧♦♥❣ t❤❡ ❛rr♦✇s ♦❢

F

t♦

Y

❛♥❞ t❤❡♥ ❛❧♦♥❣ t❤❡ ❛rr♦✇s ♦❢

G

t♦

Z✿

❚❤❡r❡ ♠❛② ❜❡ ♠♦r❡✿

F

X −−−−→ Y

G

H

Q

−−−−→ Z −−−−→ U −−−−→ W −−−−→ ...

■t✬s ❛s ✐❢ ✉♣♦♥ ❛rr✐✈✐♥❣ ❛t ❛ ❧♦❝❛t✐♦♥ ✇❡ ❛r❡ ❣✐✈❡♥ ❞✐r❡❝t✐♦♥s t♦ ❛ ♥❡✇ ❞❡st✐♥❛t✐♦♥✱ ♦♥ ❛♥❞ ♦♥✳ ❚❤✐s ✐s t❤❡ ✏❞❡❝♦♥str✉❝t✐♦♥✑ ♦❢ t❤❡ ♥♦t❛t✐♦♥✿

❈♦♠♣♦s✐t✐♦♥ ♦❢ ❢✉♥❝t✐♦♥s ♥❛♠❡s ♦❢ t❤❡ s❡❝♦♥❞ ❛♥❞ ✜rst ❢✉♥❝t✐♦♥s

G◦F ↑




(x)

♥❛♠❡ ♦❢ t❤❡ ♥❡✇ ❢✉♥❝t✐♦♥

↓ = G

↓
F (x) ↑ ↑ ↑

s✉❜st✐t✉t✐♦♥

✷✳✻✳

▲✐♠✐ts ❛♥❞ ❝♦♥t✐♥✉✐t② ✉♥❞❡r ❝♦♠♣♦s✐t✐♦♥s

■❢ ✇❡ r❡♣r❡s❡♥t t❤❡ t✇♦ ❢✉♥❝t✐♦♥s ❛s ✐♥♣✉t

✶✺✾

❜❧❛❝❦ ❜♦①❡s✱ ✇❡ ❝❛♥ ✇✐r❡ t❤❡♠ t♦❣❡t❤❡r✿

❢✉♥❝t✐♦♥

→

x

F

♦✉t♣✉t

→

y ↓

✐♥♣✉t

❢✉♥❝t✐♦♥

→

y

♦✉t♣✉t

→

G

z

❍❡r❡✱ ✇❡ ✉s❡ t❤❡ ♦✉t♣✉t ♦❢ t❤❡ ❢♦r♠❡r ❛s t❤❡ ✐♥♣✉t ♦❢ t❤❡ ❧❛tt❡r✱ t❤✉s ❝r❡❛t✐♥❣ ❛ ♥❡✇ ❢✉♥❝t✐♦♥✿ ✐♥♣✉t

→ → → ↓

x

♦✉t♣✉t

↓ → → → ❚♦ ♠❛❦❡ ✐t ❝❧❡❛r t❤❛t

Y

z

✐s ♥♦ ❧♦♥❣❡r ❛ ♣❛rt ♦❢ t❤❡ ♣✐❝t✉r❡✱ ✇❡ ❝❛♥ ❛❧s♦ ✈✐s✉❛❧✐③❡ t❤❡ ❝♦♠♣♦s✐t✐♦♥ ❛s ❢♦❧❧♦✇s✿

F

X −−−−→

Y   G y Z

ցH

❚❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ ❞✐❛❣r❛♠ ✐s ❛s ❢♦❧❧♦✇s✿ ❲❤❡t❤❡r ✇❡ ❢♦❧❧♦✇ t❤❡

F ✲t❤❡♥✲G r♦✉t❡ ♦r t❤❡ ❞✐r❡❝t H

r♦✉t❡✱ t❤❡

r❡s✉❧ts ✇✐❧❧ ❜❡ t❤❡ s❛♠❡✳ ■❢ ✇❡ t❤✐♥❦ ♦❢ ❢✉♥❝t✐♦♥s ❛s

❧✐sts ♦❢ ✐♥str✉❝t✐♦♥s✱ ✇❡ ❥✉st ❛tt❛❝❤ t❤❡ ❧✐st ♦❢ t❤❡ ❧❛tt❡r ❛t t❤❡ ❜♦tt♦♠ ♦❢ t❤❡ ❧✐st

♦❢ t❤❡ ❢♦r♠❡r✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ❤❡r❡ ✐s t❤❡ ❧✐st ♦❢

•

❙t❡♣ ✶✿ ❉♦

F✳

•

❙t❡♣ ✷✿ ❉♦

G✳

❚❤❡② ❛r❡ ❡①❡❝✉t❡❞

G ◦ F✿

❝♦♥s❡❝✉t✐✈❡❧② ❀ ②♦✉ ❝❛♥✬t st❛rt ✇✐t❤ t❤❡ s❡❝♦♥❞ ✉♥t✐❧ ②♦✉ ❛r❡ ❞♦♥❡ ✇✐t❤ t❤❡ ✜rst✳

❚❤❡ ❝♦♠♣♦s✐t✐♦♥ ❝❛♥ ❛❧s♦ ❜❡ ✇r✐tt❡♥ ✈✐❛ t❤❡ s✉❜st✐t✉t✐♦♥ ♥♦t❛t✐♦♥✿

❍❡r❡✱

u

(g ◦ f )(x) = g(u)

✐s t❤❡ ✏✐♥t❡r♠❡❞✐❛t❡✑ ✈❛r✐❛❜❧❡✳

u=f (x)

❊①❛♠♣❧❡ ✷✳✻✳✸✿ ❝♦♠♣♦s✐t✐♦♥ ♦❢ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s

❚❤✐s ❢✉♥❝t✐♦♥ ♦♥ t❤❡ ❧❡❢t ✐s ✉♥❞❡rst♦♦❞ ❛♥❞ ❡✈❛❧✉❛t❡❞ ✈✐❛ t❤❡ ❞✐❛❣r❛♠ ♦♥ t❤❡ r✐❣❤t✿

2y 2 − 3y + 7 , y 3 + 2y + 1

f (y) = ❲❡ ❝❛♥ ❞♦ t❤❡ s✉❜st✐t✉t✐♦♥

y=3 f



❜② ✐♥s❡rt✐♥❣

3



= f (y)

3

f () =

22 − 3 + 7 . 3 + 2 + 1

✐♥ ❡❛❝❤ ♦❢ t❤❡s❡ ✇✐♥❞♦✇s✿

= y=3

2 3 3

2 3

−3 3 +7

+2 3 +1

.

✷✳✻✳

▲✐♠✐ts ❛♥❞ ❝♦♥t✐♥✉✐t② ✉♥❞❡r ❝♦♠♣♦s✐t✐♦♥s

▲❡t✬s ✐♥s❡rt

sin x✱

♦r✱ ❜❡tt❡r✱

(sin x)✳

❚❤✐s ✐s t❤❡ r❡s✉❧t ♦❢ t❤❡ s✉❜st✐t✉t✐♦♥

f (sin x) = f (y)

❚❤❡♥✱ ✇❡ ❤❛✈❡

✶✻✵

=

2

2 (sin x) (sin x)

y=sin x

3

y = sin x✿

− 3 (sin x) + 7

.

+ 2 (sin x) + 1

2(sin x)2 − 3(sin x) + 7 . (sin x)3 + 2(sin x) + 1

f (sin x) = (f ◦ sin)(x) = ❊①❡r❝✐s❡ ✷✳✻✳✹

❘❡♣r❡s❡♥t t❤✐s ❢✉♥❝t✐♦♥ ❛s ❛ ❧✐st ♦❢ ✐♥str✉❝t✐♦♥s✿

√ 3

f (x) =

sin x + 2


1/2

.

❊①❡r❝✐s❡ ✷✳✻✳✺

❋✐♥❞ ❛ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❢♦❧❧♦✇✐♥❣ ❢✉♥❝t✐♦♥✿

→

sq✉❛r❡ ✐t

→

t❛❦❡ ✐ts r❡❝✐♣r♦❝❛❧

→

❊①❡r❝✐s❡ ✷✳✻✳✻

❙✉♣♣♦s❡ t❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥

❧✐♠✐ts ❛♥❞ ❝♦♥t✐♥✉✐t②✳ ❆♥ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ Pr♦❞✉❝t ❘✉❧❡

f

✐s ♣r♦✈✐❞❡❞✳ ❍♦✇ ❞♦ ②♦✉ ♣❧♦t t❤❡ ❣r❛♣❤ ♦❢

y = −f (−x − 3) − 1❄

◆♦✇✱

lim

x→a

h

f (x)


2 i

✐♥ ❛ s✐♠♣❧❡ s✐t✉❛t✐♦♥ r❡✈❡❛❧s ❛ ♥❡✇ s❤♦rt❝✉t✿

= lim [f (x) · f (x)] x→a

= lim f (x) · lim f (x) x→a  x→a 2 = lim f (x) . x→a

❆❝❝♦r❞✐♥❣ t♦ P❘✳ ❯♥❞❡r t❤❡ ❛ss✉♠♣t✐♦♥ t❤❛t ❧✐♠✐t ❡①✐sts✳

■t s❡❡♠s t❤❛t ✇❡ ❛r❡ s❛②✐♥❣ t❤❛t ✏t❤❡ ❧✐♠✐t ♦❢ t❤❡ sq✉❛r❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ✐s t❤❡ sq✉❛r❡ ♦❢ t❤❡ ❧✐♠✐t ♦❢ t❤❛t ❢✉♥❝t✐♦♥✑✳ ❏✉st ❧✐❦❡ ✇✐t❤ t❤❡ r❡st ♦❢ t❤❡ ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s✦ ❋✉rt❤❡r♠♦r❡✱ ❛ r❡♣❡❛t❡❞ ✉s❡ ♦❢

Pr♦❞✉❝t ❘✉❧❡

♣r♦❞✉❝❡s ❛ ♠♦r❡ ❣❡♥❡r❛❧ ❢♦r♠✉❧❛✿

 n lim [f (x)n ] = lim f (x) ,

x→a ❢♦r ❛♥② ♥❛t✉r❛❧ ♥✉♠❜❡r

◮

n✳

x→a

❲❡ ❝♦♥❝❧✉❞❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿

❚❤❡ ❧✐♠✐t ♦❢ t❤❡ ♣♦✇❡r ✐s ❡q✉❛❧ t♦ t❤❡ ♣♦✇❡r ♦❢ t❤❡ ❧✐♠✐t✳

❊①❛♠♣❧❡ ✷✳✻✳✼✿ ♣♦✇❡rs

❚❤❡ r✉❧❡ ✐s ✉s❡❢✉❧✿



lim (sin x)

x→0

20





= lim [sin x] x→0

20

= (0)20 = 0 .

▲❡t✬s ❣✐✈❡ t❤✐s ❢♦r♠✉❧❛ ❛ ♥❡✇ ✐♥t❡r♣r❡t❛t✐♦♥❀ ✐t✬s ❛ ❝♦♠♣♦s✐t✐♦♥✿

x →

f

un

→ u →

❲❡ ❤❛✈❡✿

f (x)n = g(f (x))

✇✐t❤

g(u) = un .

❲❤❛t ✐s s♦ s♣❡❝✐❛❧ ❛❜♦✉t t❤✐s ♥❡✇ ❢✉♥❝t✐♦♥❄ ■t ✐s ❝♦♥t✐♥✉♦✉s✦ ■♥ ❜r✐❡❢✱

→ z

✷✳✻✳ ▲✐♠✐ts ❛♥❞ ❝♦♥t✐♥✉✐t② ✉♥❞❡r ❝♦♠♣♦s✐t✐♦♥s

✶✻✶

◮ ❝♦♠♣♦s✐t✐♦♥s ♦❢ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s ❛r❡ ❝♦♥t✐♥✉♦✉s✳ ❊①❛♠♣❧❡ ✷✳✻✳✽✿ ❝♦♥t✐♥✉✐t② ♦❢ ❝♦♠♣♦s✐t✐♦♥s✱ ♠♦t✐♦♥

❚❤❡ ✐❞❡❛ ✐s ✐❧❧✉str❛t❡❞ ❛s ❢♦❧❧♦✇s✳ ■♠❛❣✐♥❡ ✇❡ ❤❛✈❡ t❤r❡❡ ❝✉r✈❡❞ ✇✐r❡s ✇✐t❤ ❛ ❢r❡❡❧② ♠♦✈✐♥❣ ♥✉t ♦♥ ❡❛❝❤✳ ❚❤❡ ♥✉ts ❛r❡ ❝♦♥♥❡❝t❡❞ ❜② t✇♦ r♦❞s✳ ❚❤❡♥✱ ✐❢ t❤❡ ✜rst ♥✉t ✐s ♠♦✈❡❞✱ ✐t ♠♦✈❡s t❤❡ s❡❝♦♥❞✱ ❛♥❞ t❤❡ s❡❝♦♥❞ ♠♦✈❡s t❤❡ t❤✐r❞✿

❊✐t❤❡r ❝♦♥♥❡❝t✐♦♥ ❣✉❛r❛♥t❡❡s ❝♦♥t✐♥✉♦✉s ♠♦t✐♦♥✱ ❛♥❞ s♦ ❞♦❡s t❤❡✐r ❝♦♠❜✐♥❛t✐♦♥✳ ❙✐♠✐❧❛r❧②✱ ❛ ❞r✐✈❡r ❝♦♥tr♦❧s t❤❡ ❛①❧❡ ✇✐t❤ t❤❡ st❡❡r✐♥❣ ✇❤❡❡❧ ✐♥ ❛ ♠✉❧t✐st❛❣❡ ❜✉t ❝♦♥t✐♥✉♦✉s ✇❛②✳ ❊①❛♠♣❧❡ ✷✳✻✳✾✿ ❝♦♥t✐♥✉✐t② ♦❢ ❝♦♠♣♦s✐t✐♦♥s✱ tr❛♥s❢♦r♠❛t✐♦♥s

■♠❛❣✐♥❡ ✇❡ ❝❛rr② ♦✉t t✇♦ tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ ❛ r❡❛❧ ❧✐♥❡ ✐♥ ❛ r♦✇✿

❙✐♠♣❧② ♣✉t✱ ✐❢ ✇❡ ❞✐❞♥✬t t❡❛r t❤❡ r♦♣❡ ❞✉r✐♥❣ t❤❡ ✜rst st❛❣❡✱ ♦r ❞✉r✐♥❣ t❤❡ s❡❝♦♥❞ st❛❣❡✱ t❤❡♥ ✐t✬s ♥♦t t♦r♥✦ ❲❡ ✜rst ❝♦♥s✐❞❡r t❤❡ ❧✐♠✐t ❛t ❛ s✐♥❣❧❡ ♣♦✐♥t✿ ❚❤❡♦r❡♠ ✷✳✻✳✶✵✿ ❈♦♠♣♦s✐t✐♦♥ ❘✉❧❡ ❢♦r ▲✐♠✐ts ■❢ t❤❡ ❧✐♠✐t ❛t

a

♦❢ ❢✉♥❝t✐♦♥

f (x)

❡①✐sts✱ ❛♥❞ ✐s ❡q✉❛❧ t♦

✐ts ❝♦♠♣♦s✐t✐♦♥ ✇✐t❤ ❛♥② ❢✉♥❝t✐♦♥

g

❋✉rt❤❡r♠♦r❡✱ t❤✐s ❧✐♠✐t ✐s ❡q✉❛❧ t♦

❝♦♥t✐♥✉♦✉s ❛t

g(L)✿

lim (g ◦ f )(x) = g(L) .

x→a

Pr♦♦❢✳

❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ s❡q✉❡♥❝❡✱ ❚❤❡♥✱ ✇❡ ❛❧s♦ ❤❛✈❡ ❛♥♦t❤❡r s❡q✉❡♥❝❡✱

xn → a . bn = f (xn ) .

L✳

L✱

t❤❡♥ s♦ ❞♦❡s t❤❛t ♦❢

✷✳✻✳ ▲✐♠✐ts ❛♥❞ ❝♦♥t✐♥✉✐t② ✉♥❞❡r ❝♦♠♣♦s✐t✐♦♥s

✶✻✷

❚❤❡ ❝♦♥❞✐t✐♦♥ f (x) → L ❛s x → a ✐s r❡st❛t❡❞ ❛s ❢♦❧❧♦✇s✿ bn → L ❛s n → ∞ .

❚❤❡r❡❢♦r❡✱ t❤❡ ❝♦♥t✐♥✉✐t② ♦❢ g ✐♠♣❧✐❡s✱ g(bn ) → g(L) ❛s n → ∞ .

■♥ ♦t❤❡r ✇♦r❞s✱

(g ◦ f )(xn ) = g(f (xn )) → g(L) ❛s n → ∞ .

❙✐♥❝❡ s❡q✉❡♥❝❡ xn → a ✇❛s ❝❤♦s❡♥ ❛r❜✐tr❛r✐❧②✱ t❤✐s ❝♦♥❞✐t✐♦♥ ✐s r❡st❛t❡❞ ❛s✱ (g ◦ f )(x) → g(L) ❛s x → a .

❲❡ ❝❛♥ r❡✇r✐t❡ t❤❡ r❡s✉❧t ❛s ❛ s✉❜st✐t✉t✐♦♥ ✿ lim (g ◦ f )(x) = g(L) x→a

L=limx→a f (x)

■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ❝♦♥t✐♥✉♦✉s g ❝❛♥ ❜❡ ♠♦✈❡❞ ♦✉t ♦❢ t❤❡ ❧✐♠✐t t♦ ❜❡ ❝♦♠♣✉t❡❞✿  
lim g f (x) = g lim f (x)

x→a

x→a

■t✬s ❛ s✐♠♣❧✐✜❝❛t✐♦♥ ♠♦✈❡✳ ❊①❛♠♣❧❡ ✷✳✻✳✶✶✿ s✉❜st✐t✉t✐♦♥s

❇❡❝❛✉s❡ t❤❡s❡ t✇♦ ❢✉♥❝t✐♦♥s ❛r❡ ❝♦♥t✐♥✉♦✉s✱ ✇❡ ❝❛♥ s✉❜st✐t✉t❡ t✇✐❝❡✿



lim sin cos x = sin lim cos x = sin cos 0 = sin 1 .

x→0

x→0

❇✉t t❤✐s ✐s t❤❡ s❛♠❡ ❛s ❛ s✐♥❣❧❡ s✉❜st✐t✉t✐♦♥✿



lim sin cos x = lim sin ◦ cos(x) = sin ◦ cos(0) = sin 1 .

x→0

x→0

❚❤❡ ❢♦❧❧♦✇✐♥❣ ✐s ❛ ❝r✉❝✐❛❧ ❝♦♥s❡q✉❡♥❝❡ ♦❢ t❤❡ t❤❡♦r❡♠✿ ❈♦r♦❧❧❛r② ✷✳✻✳✶✷✿ ❈♦♥t✐♥✉✐t② ✉♥❞❡r ❈♦♠♣♦s✐t✐♦♥s

g◦f y = f (a)

f

❚❤❡ ❝♦♠♣♦s✐t✐♦♥

♦❢ ❛ ❢✉♥❝t✐♦♥

❝♦♥t✐♥✉♦✉s ❛t

✐s ❝♦♥t✐♥✉♦✉s ❛t

❝♦♥t✐♥✉♦✉s ❛t

x = a

x = a✳

Pr♦♦❢✳

❲❡ ❛♣♣❧② t❤❡ t❤❡♦r❡♠ ❛♥❞ t❤❡♥ t❤❡ ❝♦♥t✐♥✉✐t② ♦❢ f ✿   lim (g ◦ f )(x) = g lim f (x) = g (f (a)) = (g ◦ f )(a) .

x→a

x→a

❚❤❡ ❣❡♦♠❡tr② ♦❢ t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s ✐s ✐❧❧✉str❛t❡❞ ❜❡❧♦✇✿

❛♥❞ ❛ ❢✉♥❝t✐♦♥

g

✷✳✻✳ ▲✐♠✐ts ❛♥❞ ❝♦♥t✐♥✉✐t② ✉♥❞❡r ❝♦♠♣♦s✐t✐♦♥s

✶✻✸

◆♦✇ ❛❧❣❡❜r❛✳ ❚❤✐s ✐s ❤♦✇ t✇♦ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s ✐♥t❡r❛❝t✳ ❋✐rst✱ ✇❡ ❤❛✈❡ t❤❡ ❝♦♥t✐♥✉✐t② ♦❢ ❡✐t❤❡r ❞❡s❝r✐❜❡❞ s❡♣❛r❛t❡❧②✿ • ✶✳ ❆ s♠❛❧❧ ❞❡✈✐❛t✐♦♥ ♦❢ x ❢r♦♠ a ♣r♦❞✉❝❡s ❛ s♠❛❧❧ ❞❡✈✐❛t✐♦♥ ♦❢ u = f (x) ❢r♦♠ f (a)✳

• ✷✳ ❆ s♠❛❧❧ ❞❡✈✐❛t✐♦♥ ♦❢ u ❢r♦♠ c ♣r♦❞✉❝❡s ❛ s♠❛❧❧ ❞❡✈✐❛t✐♦♥ ♦❢ y = g(u) ❢r♦♠ g(c)✳

■❢ ✇❡ s❡t c = f (a)✱ ✇❡ ❤❛✈❡✿

• ✸✳ ❆ s♠❛❧❧ ❞❡✈✐❛t✐♦♥ ♦❢ u = f (x) ❢r♦♠ c = f (a) ♣r♦❞✉❝❡s ❛ s♠❛❧❧ ❞❡✈✐❛t✐♦♥ ♦❢ y = g(u) = g(f (x)) ❢r♦♠ g(c) = g(f (a))✳

❚❤❛t✬s ❝♦♥t✐♥✉✐t② ♦❢ h = g ◦ f ❛t x = a✦ ❚❤❡ ❞✐❛❣r❛♠ ✐❧❧✉str❛t❡s t❤✐s ♦❜s❡r✈❛t✐♦♥✿ f

−−−−→

x→a

ցg◦f

f (x) → f (a)   g y

g(f (x)) → g(f (a)) ❊①❛♠♣❧❡ ✷✳✻✳✶✸✿

sin

❛♥❞

cos

❋r♦♠ tr✐❣♦♥♦♠❡tr②✱ ✇❡ ❦♥♦✇ t❤❛t cos(x) = sin(π/2 − x) .

❚❤❡r❡❢♦r❡✱ t❤❡ ❝♦♥t✐♥✉✐t② ♦❢ sin ✐♠♣❧✐❡s t❤❡ ❝♦♥t✐♥✉✐t② ♦❢ cos✱ ❛♥❞ ✈✐❝❡ ✈❡rs❛✳ ❊①❛♠♣❧❡ ✷✳✻✳✶✹✿ ❛❧❣❡❜r❛ ♦❢ ❝♦♠♣♦s✐t✐♦♥s

❈♦♥s✐❞❡r t❤❡s❡ t✇♦ ❢✉♥❝t✐♦♥s ❛♥❞ t❤❡✐r ❝♦♠♣♦s✐t✐♦♥✿ y = g(u) = u2 + 2u − 1 , u = f (x) = 2x−3 .

❲❤❛t ✐s t❤❡ ❧✐♠✐t ♦❢ h = g ◦ f ❛t 1❄ ❋✐rst✱ ✇❡ ♥♦t❡ t❤❛t g ✐s ❝♦♥t✐♥✉♦✉s ❛t ❡✈❡r② ♣♦✐♥t ❛s ❛ ♣♦❧②♥♦♠✐❛❧✳ ❚❤❡r❡❢♦r❡✱ ❜② t❤❡ t❤❡♦r❡♠ ✇❡ ❤❛✈❡✿   lim (g ◦ f )(x) = g lim f (x) ,

x→1

x→1

✐❢ t❤❡ ❧✐♠✐t ♦♥ t❤❡ r✐❣❤t ❡①✐sts✳ ■t ❞♦❡s✱ ❜❡❝❛✉s❡ f ✐s ❛ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ❛t x = 1✿ lim f (x) = lim 2x−3 = 2 · 1−3 = 2 .

x→1

x→1

❚❤❡ ❧✐♠✐t ❜❡❝♦♠❡s ❛ ♥✉♠❜❡r✱ u = 2✱ ❛♥❞ t❤✐s ♥✉♠❜❡r ✐s s✉❜st✐t✉t❡❞ ✐♥t♦ g ✿   lim h(x) = g lim f (x) = g(2) = 22 + 2 · 2 − 1 = 7 .

x→1

x→1

✷✳✻✳

▲✐♠✐ts ❛♥❞ ❝♦♥t✐♥✉✐t② ✉♥❞❡r ❝♦♠♣♦s✐t✐♦♥s

✶✻✹

❚❤❡ ❛♥s✇❡r ✐s ✈❡r✐✜❡❞ ❜② ❛ ❞✐r❡❝t ❝♦♠♣✉t❛t✐♦♥ ♦❢ h✿ h(x) = (g ◦ f )(x) = g(f (x)) = u2 + 2u − 1

u=2x−3

= 2x−3


2


+ 2 2x−3 − 1 = 4x−6 + 4x−3 − 1 .

❙✐♥❝❡ t❤❡ ❢✉♥❝t✐♦♥ ✐s ❝♦♥t✐♥✉♦✉s ✭r❛t✐♦♥❛❧ ❛♥❞ ❞❡✜♥❡❞ ❛t 1✮✱ ✇❡ ❤❛✈❡ ❜② s✉❜st✐t✉t✐♦♥✿ lim h(x) = 4x−6 + 4x−3 − 1

x→1

x=1

= 4 · 1−6 + 4 · 1−3 − 1 = 7 .

❊①❛♠♣❧❡ ✷✳✻✳✶✺✿ ✉s✐♥❣ t❤❡ r✉❧❡

❈♦♠♣✉t❡✿ lim

x→0 (x2

❲❡ ♣r♦❝❡❡❞ ❜② ❛ ❣r❛❞✉❛❧ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ r(x) =

❚❤❡ ❧❛st ♦♣❡r❛t✐♦♥ ♦❢ f ✐s ❞✐✈✐s✐♦♥✳ ❚❤❡r❡❢♦r❡✱

1 . + x − 1)3

1 . (x2 + x − 1)3

r(x) = g(f (x)), ✇❤❡r❡ u = f (x) = x2 + x − 1


3

❛♥❞ g(u) =

1 . u

❋✉♥❝t✐♦♥ g ✐s r❛t✐♦♥❛❧❀ ❜✉t ✐s ✐t ❝♦♥t✐♥✉♦✉s❄ ■ts ❞❡♥♦♠✐♥❛t♦r ✐s♥✬t ③❡r♦ ❛t t❤❡ ♣♦✐♥t ✇❡ ❛r❡ ✐♥t❡r❡st❡❞ ✐♥✿



❚❤❡♥ t❤❡

f (0) = x2 + x − 1

3



x=0

= 02 + 0 − 1

3

= −1 6= 0 .

❈♦♠♣♦s✐t✐♦♥ ❘✉❧❡ ❢♦r ▲✐♠✐ts ❛♣♣❧✐❡s ❛♥❞ ♦✉r ❧✐♠✐t ❜❡❝♦♠❡s✿

1 1 ,  = lim r(x) = lim g(f (x)) = g(lim f (x)) = 3 x→0 x→0 x→0 x→0 (x2 + x − 1) limx→0 (x2 + x − 1)3 lim

♣r♦✈✐❞❡❞ t❤❡ ♥❡✇ ❧✐♠✐t ❡①✐sts✳ ◆♦t✐❝❡ t❤❛t t❤❡ ❧✐♠✐t t♦ ❜❡ ❝♦♠♣✉t❡❞ ❤❛s ❜❡❡♥ s✐♠♣❧✐✜❡❞✦ ▲❡t✬s ❝♦♠♣✉t❡ ✐t✳ ❲❡ st❛rt ♦✈❡r ❛♥❞ ❝♦♥t✐♥✉❡ ✇✐t❤ ❛ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ p(x) = x2 + x − 1

❚❤❡ ❧❛st ♦♣❡r❛t✐♦♥ ♦❢ f ✐s t❤❡ ♣♦✇❡r✳ ❚❤❡r❡❢♦r❡✱


3

.

p(x) = g(f (x)), ✇❤❡r❡ u = f (x) = x2 + x − 1 ❛♥❞ g(u) = u3 .

❋✉♥❝t✐♦♥ g ✐s ❛ ♣♦❧②♥♦♠✐❛❧ ❛♥❞✱ t❤❡r❡❢♦r❡✱ ❝♦♥t✐♥✉♦✉s✳ ❚❤❡♥ t❤❡ ❈♦♠♣♦s✐t✐♦♥ ❛♥❞ t❤❡ ❧✐♠✐t ❜❡❝♦♠❡s✿ lim x2 + x − 1

x→0


3

❘✉❧❡ ❢♦r ▲✐♠✐ts ❛♣♣❧✐❡s

h
i3 , = lim p(x) = lim g(f (x)) = g(lim f (x)) = lim x2 + x − 1 x→0

x→0

x→0

x→0

♣r♦✈✐❞❡❞ t❤❡ ♥❡✇ ❧✐♠✐t ❡①✐sts✳ ◆♦t✐❝❡ t❤❛t✱ ❛❣❛✐♥✱ t❤❡ ❧✐♠✐t t♦ ❜❡ ❝♦♠♣✉t❡❞ ❤❛s ❜❡❡♥ s✐♠♣❧✐✜❡❞✦ ▲❡t✬s ❝♦♠♣✉t❡ ✐t✳ ❲❡ r❡❛❧✐③❡ t❤❛t t❤❡ ❢✉♥❝t✐♦♥ x2 + x − 1 ✐s ❛ ♣♦❧②♥♦♠✐❛❧ ❛♥❞✱ t❤❡r❡❢♦r❡✱ ✐ts ❧✐♠✐t ✐s ❝♦♠♣✉t❡❞ ❜② s✉❜st✐t✉t✐♦♥✿ lim x + x − 1 = x + x − 1 2

x→0




2

x=0

❲❤❛t r❡♠❛✐♥s ✐s t♦ ❝♦♠❜✐♥❡ t❤❡ t❤r❡❡ ❢♦r♠✉❧❛s ❛❜♦✈❡✿

= 02 + 0 − 1 = −1 .

1 1  = 3 x→0 (x2 + x − 1) limx→0 (x2 + x − 1)3 1 = [limx→0 (x2 + x − 1)]3 1 = [−1]3 = −1 . lim

✷✳✼✳ ❈♦♥t✐♥✉✐t② ♦❢ t❤❡ ✐♥✈❡rs❡

✶✻✺

❲❤❡♥ t❤❡r❡ ✐s ♥♦ ❝♦♥t✐♥✉✐t② t♦ ❜❡ ✉s❡❞✱ ✇❡ ♠❛② ❤❛✈❡ t♦ ❛♣♣❧② ❛❧❣❡❜r❛ ✭s✉❝❤ ❛s ❢❛❝t♦r✐♥❣✮✱ ♦r tr✐❣♦♥♦♠❡tr②✱ ❡t❝✳ ✐♥ ♦r❞❡r t♦ ✜♥❞ ❛♥♦t❤❡r ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥✳ ❖✉r s❤♦rt ❧✐st ♦❢ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s ❛❧❧♦✇s ✉s t♦ ❝♦♠♣✉t❡ ♠❛♥② ❧✐♠✐ts❀ ✇❡ ❥✉st ❤❛✈❡ t♦ ❥✉st✐❢②✱ ❡✈❡r② t✐♠❡✱ ♦✉r ❝♦♥❝❧✉s✐♦♥ ❜② ♠❛❦✐♥❣ ❛ r❡❢❡r❡♥❝❡ t♦ t❤✐s ❢❛❝t✳ ❊①❛♠♣❧❡ ✷✳✻✳✶✻✿ ❝♦♠♣✉t❛t✐♦♥s

❋✐rst✱







2

1 lim ln 1 + n→∞ n

1 = ln lim 1 + n→∞ n

❜❡❝❛✉s❡ ln x ✐s ❝♦♥t✐♥✉♦✉s ❛t x = 1✳ ❙❡❝♦♥❞✱ lim

n→∞

1 1+ n





=







1 lim 1 + n→∞ n

2

1 

=

= ln 1 = 0 ,

= 12 = 1 ,

❜❡❝❛✉s❡ x2 ✐s ❝♦♥t✐♥✉♦✉s ❛t x = 1✳ ❚❤✐r❞✱ 1 n→∞ 1 + lim

❜❡❝❛✉s❡

1 n

=

limn→∞ 1 +

1 ✐s ❝♦♥t✐♥✉♦✉s ❛t x = 1✳ x

1 n

1 = 1, 1

✷✳✼✳ ❈♦♥t✐♥✉✐t② ♦❢ t❤❡ ✐♥✈❡rs❡

❙♦ ❢❛r✱ t❤❡ ♦♥❧② ❧❛r❣❡ ❝❧❛ss ♦❢ ❢✉♥❝t✐♦♥s ✇❡ ❦♥♦✇ t♦ ❜❡ ❝♦♥t✐♥✉♦✉s ✐s t❤❡ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥s✿ ❆❧❧ ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s✱ ✐♥❝❧✉❞✐♥❣ ❝♦♠♣♦s✐t✐♦♥s✱ ♦♥ r❛t✐♦♥❛❧ √ ❢✉♥❝t✐♦♥s ♣r♦❞✉❝❡ ♠♦r❡ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥s✳ ❚❤❡ ❝♦♥t✐♥✉✐t② ♦❢ s✉❝❤ ❛ s✐♠♣❧❡ ❢✉♥❝t✐♦♥ ❛s t❤❡ sq✉❛r❡ r♦♦t x r❡♠❛✐♥s ✉♥♣r♦✈❡♥✳ ❇✉t t♦ s❡❡ ❤♦✇ ✇❡ s❤♦✉❧❞ ❛♣♣r♦❛❝❤ t❤❡ ♣r♦❜❧❡♠ ✇❡ ❥✉st ♥❡❡❞ t♦ r❡♠❡♠❜❡r ✇❤❡r❡ t❤❡ ❢✉♥❝t✐♦♥ ❝♦♠❡s ❢r♦♠✳ ■t✬s t❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ sq✉❛r✐♥❣ ❢✉♥❝t✐♦♥ y2✦ ▲❡t✬s r❡✈✐❡✇✳ ❊①❛♠♣❧❡ ✷✳✼✳✶✿ r❡❧❛t✐♦♥s

❚❤❡ ✐♥✈❡rs❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ✐s ❜❛s❡❞ ♦♥ t❤❡ s❛♠❡ r❡❧❛t✐♦♥ ❜✉t ✇✐t❤ ❞♦♠❛✐♥ ❛♥❞ ❝♦❞♦♠❛✐♥ ✭♦r ✐♥♣✉t ❛♥❞ ♦✉t♣✉t✮ r❡✈❡rs❡❞✿

❊①❛♠♣❧❡ ✷✳✼✳✷✿ ✐♥✈❡rs❡s

❲❡ ❛❧s♦ ❧✐❦❡ t♦ t❤✐♥❦ ♦❢ t✇♦ ❢✉♥❝t✐♦♥s ❛s ✐♥✈❡rs❡s ✇❤❡♥ ♦♥❡ ✉♥❞♦❡s t❤❡ ❡✛❡❝t ♦❢ t❤❡ ♦t❤❡r✳ ❋♦r ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s✱ t❤❡s❡ ❛r❡ ❛ ❢❡✇ ❡①❛♠♣❧❡s✳ ❆ ❢✉♥❝t✐♦♥ ✐s ✉♥❞♦♥❡ ❜② ✐ts ✐♥✈❡rs❡✱ ❛♥❞ ✈✐❝❡ ✈❡rs❛✿

✷✳✼✳

❈♦♥t✐♥✉✐t② ♦❢ t❤❡ ✐♥✈❡rs❡

♠✉❧t✐♣❧✐❝❛t✐♦♥ ❜②

✶✻✻

❢✉♥❝t✐♦♥

r❡❧❛t✐♦♥

✐♥✈❡rs❡ ❢✉♥❝t✐♦♥

1→3 2→4 2→9

1↔3 2↔4 2↔9

1←3 2←4 2←9

3

sq✉❛r✐♥❣ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥ ❜❛s❡

3

❇✉t ✇❤② ✐s t❤❡ ✐♥✈❡rs❡ ❝♦♥t✐♥✉♦✉s❄

❞✐✈✐s✐♦♥ ❜②

3 x, y ≥ 0✮ 3

sq✉❛r❡ r♦♦t ✭❢♦r ❧♦❣❛r✐t❤♠ ❜❛s❡

❊①❛♠♣❧❡ ✷✳✼✳✸✿ ✐♥✈❡rs❡ ♦❢ tr❛♥s❢♦r♠❛t✐♦♥ ❚❤❡ ❜❛s✐❝ tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ❧✐♥❡ ♣r♦✈✐❞❡ ♠♦r❡ ❡①❛♠♣❧❡s✿

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿

• • •

❚❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ s❤✐❢t

s

✉♥✐ts t♦ t❤❡ r✐❣❤t ✐s t❤❡ s❤✐❢t ♦❢

s

✉♥✐ts t♦ t❤❡ ❧❡❢t✳

❚❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ ✢✐♣ ✐s ❛♥♦t❤❡r ✢✐♣✳ ❚❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ str❡t❝❤ ❜②

k 6= 0

✐s t❤❡ s❤r✐♥❦ ❜②

k

✭✐✳❡✳✱ str❡t❝❤ ❜②

1/k ✮✳

❲❤❡♥ ❡①❡❝✉t❡❞ ❝♦♥s❡❝✉t✐✈❡❧②✱ t❤❡ ❡✛❡❝t ✐s ♥✐❧✳ ❆❧❣❡❜r❛✐❝❛❧❧②✿

s❤✐❢t ✢✐♣ str❡t❝❤

f y =x+s y = −x y =x·k

✈s✳

f −1 x=y−s x = −y x = y/k

❲✐t❤✐♥ t❤✐s ❝❧❛ss✱ ✐t s❡❡♠s t❤❛t ✐ts ✐♥✈❡rs❡ s❤♦✉❧❞ ❜❡ ❝♦♥t✐♥✉♦✉s ✐❢ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ ✐s✳

▲❡t✬s r❡❝❛❧❧ t❤❡ ♣r❡❝✐s❡ ❞❡✜♥✐t✐♦♥✿

❉❡✜♥✐t✐♦♥ ✷✳✼✳✹✿ ✐♥✈❡rs❡ ♦❢ ❢✉♥❝t✐♦♥ ❋✉♥❝t✐♦♥s

F :X →Y

❛♥❞

G:Y →X

❝♦♠❡ ❢r♦♠ t❤❡ s❛♠❡ r❡❧❛t✐♦♥❀ ✐✳❡✳✱ ❢♦r ❛❧❧

❛r❡ ❝❛❧❧❡❞

x

❛♥❞

y✱

✐♥✈❡rs❡

♦❢ ❡❛❝❤ ♦t❤❡r ✐❢ t❤❡②

✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿

F (x) = y ■❋ ❆◆❉ ❖◆▲❨ ■❋ G(y) = x . ❚❤❡ ❝♦♥❞✐t✐♦♥ ✐s ✐❧❧✉str❛t❡❞ ✇✐t❤ t❤✐s ❞✐❛❣r❛♠✿

F

G

x −−−−→ y ⇐⇒ y −−−−→ x ❚❤❡ ♥♦t❛t✐♦♥ ❢♦r

G

✐s ❛s ❢♦❧❧♦✇s✿

■♥✈❡rs❡ ❢✉♥❝t✐♦♥ F −1 ■t r❡❛❞s ✏ F ✐♥✈❡rs❡✑✳

✷✳✼✳

✶✻✼

❈♦♥t✐♥✉✐t② ♦❢ t❤❡ ✐♥✈❡rs❡

❚❤❡♥ ✇❡ ❤❛✈❡ ❛ ❧✐t❡r❛❧ r❡✈❡rs❛❧ ♦❢ ❛rr♦✇s✿ x

F

−−−−→

y

F −1

x ←−−−−−− y ❊①❡r❝✐s❡ ✷✳✼✳✺

❋✐♥❞ t❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ f (x) = 3x2 + 1✳ ❈❤♦♦s❡ ❛♣♣r♦♣r✐❛t❡ ❞♦♠❛✐♥s ❢♦r t❤❡s❡ t✇♦ ❢✉♥❝t✐♦♥s✳ ❊①❡r❝✐s❡ ✷✳✼✳✻

✭❛✮ ❆❧❣❡❜r❛✐❝❛❧❧②✱ s❤♦✇ t❤❛t t❤❡ ❢✉♥❝t✐♦♥ f (x) = x2 ✐s ♥♦t ♦♥❡✲t♦✲♦♥❡✳ ✭❜✮ ●r❛♣❤✐❝❛❧❧②✱ s❤♦✇ t❤❛t t❤❡ ❢✉♥❝t✐♦♥ g(x) = 2x+1 ✐s ♦♥❡✲t♦✲♦♥❡✳ ✭❝✮ ❋✐♥❞ t❤❡ ✐♥✈❡rs❡ ♦❢ g ✳ ◆♦✇ ❝♦♥t✐♥✉✐t②✳ ❊①❛♠♣❧❡ ✷✳✼✳✼✿ t❡❛r✐♥❣

❋✐rst✱ t❤❡ ✐♥✈❡rs❡s ♦❢ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥s ❛❜♦✈❡ ❛r❡ ❝♦♥t✐♥✉♦✉s✳ ❲❤❛t ❛❜♦✉t ♦t❤❡rs❄ ❍♦✇ ❞♦ ✇❡ ❦♥♦✇ t❤❛t ✉♥❞♦✐♥❣ ❛ tr❛♥s❢♦r♠❛t✐♦♥ ❞♦❡s♥✬t t❡❛r t❤❡ r♦♣❡❄ ❲❡❧❧✱ ✇❤❛t ✐s t❤❡ ✐♥✈❡rs❡ ♦❢ t❡❛r✐♥❣❄ ■t✬s ❣❧✉✐♥❣✦

■❢ ✇❡ ♠❛❦❡ s✉r❡ t❤❛t ♦✉r ♦r✐❣✐♥❛❧ tr❛♥s❢♦r♠❛t✐♦♥ ❞♦❡s♥✬t ✐♥❝❧✉❞❡ ❣❧✉✐♥❣ ♦❢ t❤❡ ♣✐❡❝❡s ♦❢ t❤❡ r♦♣❡ t♦❣❡t❤❡r✱ t❤❡ ✐♥✈❡rs❡ ✇✐❧❧ ❜❡ ❝♦♥t✐♥✉♦✉s t♦♦✳ ❇✉t t❤✐s s✐♠♣❧② ♠❡❛♥s t❤❛t t❤❡ ❢✉♥❝t✐♦♥ ❤❛s t♦ ❜❡ ♦♥❡✲t♦✲♦♥❡✳ ❚❤❡ ✐♥✈❡rs❡s ✉♥❞♦ ❡❛❝❤ ♦t❤❡r ✉♥❞❡r ❝♦♠♣♦s✐t✐♦♥ ❀ ❢✉♥❝t✐♦♥s y = f (x) ❛♥❞ x = g(y) ❛r❡ ❝❛❧❧❡❞ ✐♥✈❡rs❡ ♦❢ ❡❛❝❤ ♦t❤❡r ✇❤❡♥ ❢♦r ❛❧❧ x ✐♥ t❤❡ ❞♦♠❛✐♥ ♦❢ f ❛♥❞ ❢♦r ❛❧❧ y ✐♥ t❤❡ ❞♦♠❛✐♥ ♦❢ g ✱ ✇❡ ❤❛✈❡✿ g(f (x)) = x ❆◆❉ f (g(y)) = y

❲❤❛t ✐❢ ♦♥❡ ♦❢ t❤❡s❡ ❢✉♥❝t✐♦♥s✱ s❛② f ✱ ✐s ❝♦♥t✐♥✉♦✉s❄ ❉♦❡s ✐t ♠❛❦❡ t❤❡ ♦t❤❡r✱ g ✱ ❝♦♥t✐♥✉♦✉s t♦♦❄ ❈♦♥s✐❞❡r✐♥❣ t❤❡ ❢❛❝t t❤❛t t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡s ♦❢ t❤❡s❡ ❡q✉❛t✐♦♥s ✭x ❛♥❞ y ✮ ❛r❡ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s✱ ✇❡ ❡①♣❡❝t t❤❡ ❛♥s✇❡r t♦ ❜❡ ❨❡s✳ ❊✈❡r② ❢✉♥❝t✐♦♥ y = f (x) ✇❤✐❝❤ ✐s ♦♥❡✲t♦✲♦♥❡ ✭✐✳❡✳✱ t❤❡r❡ ✐s ♦♥❧② ♦♥❡ x ❢♦r ❡❛❝❤ y ✮ ❤❛s t❤❡ ✐♥✈❡rs❡ x = g(y) ✇❤✐❝❤ ✐s ❛❧s♦ ♦♥❡✲t♦✲♦♥❡ ✭✐✳❡✳✱ t❤❡r❡ ✐s ♦♥❧② ♦♥❡ y ❢♦r ❡❛❝❤ x✮✳ ■❢ ✇❡ t❛❦❡ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢♦r♠❡r ❛♥❞ ✢✐♣ t❤✐s ♣✐❡❝❡ ♦❢ ♣❛♣❡r s♦ t❤❛t t❤❡ x✲❛①✐s ❛♥❞ t❤❡ y ✲❛①✐s ❛r❡ ✐♥t❡r❝❤❛♥❣❡❞✱ ✇❡ ❣❡t t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❧❛tt❡r ✭❛♥❞ ✈✐❝❡ ✈❡rs❛✮✿

✷✳✼✳

❈♦♥t✐♥✉✐t② ♦❢ t❤❡ ✐♥✈❡rs❡

✶✻✽

❚❤❡ s❤❛♣❡s ♦❢ t❤❡ ❣r❛♣❤s ❛r❡ ❡①❛❝t❧② t❤❡ s❛♠❡✦ ■❢ ♦♥❡ ❤❛s ♥♦ ❜r❡❛❦s✱ t❤❡♥ ♥❡✐t❤❡r ❞♦❡s t❤❡ ♦t❤❡r✳ ■t ✐s t❤❡♥ ❝♦♥❝❡✐✈❛❜❧❡ t❤❛t t❤❡ ❝♦♥t✐♥✉✐t② ♦❢ ♦♥❡ ✐♠♣❧✐❡s t❤❡ ❝♦♥t✐♥✉✐t② ♦❢ t❤❡ ♦t❤❡r✳ ❚♦ ❧✐♥❦ t❤❡ ❝♦♥t✐♥✉✐t② ♦❢ t❤❡ t✇♦ ❢✉♥❝t✐♦♥s✱ ✇❡ ❤❛✈❡ t♦ ❜❡ ❝❛r❡❢✉❧ ❛❜♦✉t t❤❡ ❧♦❝❛t✐♦♥ ♦❢ t❤❡ s❡q✉❡♥❝❡✳ ❲❡ ❦♥♦✇ t❤❛t ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❞❡✜♥✐t✐♦♥✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♠♣❧✐❝❛t✐♦♥✿ xn → a =⇒ f (xn ) → f (a)

❚❤❛t✬s t❤❡ ❛ss✉♠♣t✐♦♥✳ ◆♦✇✱ ✐♥ r❡✈❡rs❡✿ f −1 (yn ) → f −1 (b) ⇐= yn → b

❚❤❛t✬s ✇❤❛t ✇❡ ♥❡❡❞ t♦ ♣r♦✈❡✳ ❲❡ ❛❝❝❡♣t t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♠♣♦rt❛♥t ❝♦♥❝❧✉s✐♦♥ ✇✐t❤♦✉t ♣r♦♦❢✿ ❚❤❡♦r❡♠ ✷✳✼✳✽✿ ❈♦♥t✐♥✉✐t② ♦❢ ■♥✈❡rs❡

y = f (x) b = f (a)✳

❚❤❡ ✐♥✈❡rs❡ ♦❢ ❛ ❢✉♥❝t✐♦♥

x = g(y)

❝♦♥t✐♥✉♦✉s ❛t

❝♦♥t✐♥✉♦✉s ❛t

x = a✱

✐❢ ✐t ❡①✐sts✱ ✐s ❛ ❢✉♥❝t✐♦♥

❚❤✉s✱ ❛s t❤❡ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥s ❛r❡ ❝♦♥t✐♥✉♦✉s ✭✇✐t❤✐♥ t❤❡✐r ❞♦♠❛✐♥s✮✱ s♦ ❛r❡ t❤❡✐r ✐♥✈❡rs❡s✳ ■♥ ♣❛rt✐❝✉❧❛r✱ t❤❡ sq✉❛r❡ r♦♦t ❢✉♥❝t✐♦♥ ✐s ❝♦♥t✐♥✉♦✉s ❛♥❞ ✇❡ ❤❛✈❡✿ lim

x→a

p

f (x) =

q

lim f (x) .

x→a

❲❤❛t ❛❜♦✉t t❤❡ r❡st ♦❢ ♣♦✇❡r ❢✉♥❝t✐♦♥s❄ ❘❡❝❛❧❧ t❤❛t s✐♥❝❡ t❤❡ ♦❞❞ ♣♦✇❡rs ❛r❡ ♦♥❡✲t♦✲♦♥❡ ❜✉t t❤❡ ❡✈❡♥ ♣♦✇❡rs ❛r❡♥✬t ♦♥❡✲t♦✲♦♥❡✱ ✇❡ ❤❛❞ t♦ r❡♠♦✈❡ ❤❛❧❢ ♦❢ t❤❡ ❞♦♠❛✐♥s ♦❢ t❤❡ ❧❛tt❡r✿

❚❤❡ ❧♦♦❦ ♦❢ t❤❡ ♥❡✇ ❣r❛♣❤ s✉❣❣❡st t❤❡ ❢♦❧❧♦✇✐♥❣✿ ❚❤❡♦r❡♠ ✷✳✼✳✾✿ ❈♦♥t✐♥✉✐t② ♦❢ ❘♦♦ts

• •

❚❤❡ ♦❞❞ r♦♦ts✱

y=

√ 3

x✳ ❚❤❡ ❡✈❡♥ r♦♦ts✱

y =

x, y = √ 2

√ 5

x, y =

x, y = √ 4

√ 7

x, ...

x, y =

√ 6

❛r❡ ❝♦♥t✐♥✉♦✉s ❛t ❡✈❡r② r❡❛❧

x, ...

❛r❡ ❝♦♥t✐♥✉♦✉s ❛t ❡✈❡r②

x ≥ 0✳

❆♥♦t❤❡r ✐♠♣♦rt❛♥t ♣❛✐r ♦❢ ✐♥✈❡rs❡s ✐s t❤❡ ❡①♣♦♥❡♥t ❛♥❞ t❤❡ ❧♦❣❛r✐t❤♠✳ ❙✐♥❝❡ t❤❡ ❢♦r♠❡r ✐s ❝♦♥t✐♥✉♦✉s✱ t❤❡♥ s♦ ✐s t❤❡ ❧❛tt❡r✳ ❚❤✐s ♠❛❦❡s ♣♦ss✐❜❧❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❞❡✜♥✐t✐♦♥s✿

✷✳✼✳

❈♦♥t✐♥✉✐t② ♦❢ t❤❡ ✐♥✈❡rs❡

✶✻✾

❆s ✇❡ ❦♥♦✇ ✭❛s s❡❡♥ ✐♥ ❱♦❧✉♠❡ ✶✱ ❈❤❛♣t❡r ✶P❈✲✹✮✱ t❤✐s ❢✉♥❝t✐♦♥ ✐s ✐♥❝r❡❛s✐♥❣ ❢♦r ❛❧❧ r❛t✐♦♥❛❧ x✳ ❚❤❡♥✱ ❢r♦♠ t❤❡ ❈♦♠♣❛r✐s♦♥ ❚❤❡♦r❡♠✱ ✇❡ ❝♦♥❝❧✉❞❡ t❤❛t ✐t ✐s ❛❧s♦ ✐♥❝r❡❛s✐♥❣ ♦♥ (−∞, ∞)✳ ❚❤❡r❡❢♦r❡✱ ✐t ✐s ♦♥❡✲t♦✲♦♥❡✳

❉❡✜♥✐t✐♦♥ ✷✳✼✳✶✵✿ ♥❛t✉r❛❧ ❧♦❣❛r✐t❤♠

❚❤❡ ♥❛t✉r❛❧ ❧♦❣❛r✐t❤♠ ❢✉♥❝t✐♦♥✱ ♦r t❤❡ ❧♦❣❛r✐t❤♠ ❢✉♥❝t✐♦♥ ❜❛s❡ e✱ ✐s ❞❡✜♥❡❞ ❛s t❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ ♥❛t✉r❛❧ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥✳

❉❡✜♥✐t✐♦♥ ✷✳✼✳✶✶✿ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥ ❚❤❡

❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥ ❜❛s❡ a > 0 ✐s ❞❡✜♥❡❞ ❜② ✐ts ✈❛❧✉❡s ❢♦r ❡❛❝❤ r❡❛❧ x✿ ax = ex ln a .

❚❤❡ ❣r❛♣❤s ❛r❡ ❥✉st ✢✐♣♣❡❞✿

❲❡ ❛❧r❡❛❞② ❦♥♦✇ t❤❡ ❢♦❧❧♦✇✐♥❣✿

❚❤❡♦r❡♠ ✷✳✼✳✶✷✿ ❈♦♥t✐♥✉✐t② ♦❢ ❊①♣♦♥❡♥t✐❛❧ ❋✉♥❝t✐♦♥ ❚❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥ ♦❢ ❛♥② ❜❛s❡

Pr♦♦❢✳

y = ax

■t ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ❝♦♥t✐♥✉✐t② ♦❢ y = ex ❛♥❞ t❤❡ ❈♦♠♣♦s✐t✐♦♥

✐s ❝♦♥t✐♥✉♦✉s✳

❘✉❧❡ ❢♦r ▲✐♠✐ts✳

❲❡ t❤❡♥ ❞❡r✐✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛❜♦✉t ✐ts ✐♥✈❡rs❡✿

❚❤❡♦r❡♠ ✷✳✼✳✶✸✿ ❈♦♥t✐♥✉✐t② ♦❢ ▲♦❣❛r✐t❤♠ ❚❤❡ ❧♦❣❛r✐t❤♠ ♦❢ ❛♥② ❜❛s❡

y = loga x

✐s ❝♦♥t✐♥✉♦✉s ✭♦♥ ✐ts ❞♦♠❛✐♥

x > 0✮✳

❚❤❡ ❛♥❛❧②s✐s ♦❢ t❤❡ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s ✐s s✐♠✐❧❛r t♦ t❤❛t ❢♦r ♣♦✇❡rs✿ ✇❡ ♥❡❡❞ t♦ r❡❞✉❝❡ t❤❡ ❞♦♠❛✐♥s✱ ❛s ❢♦❧❧♦✇s✿

❉❡✜♥✐t✐♦♥ ✷✳✼✳✶✹✿ ❛r❝s✐♥❡ ❛♥❞ ❛r❝❝♦s✐♥❡ ❚❤❡

❛r❝s✐♥❡

✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ s✐♥❡ ❢✉♥❝t✐♦♥ r❡str✐❝t❡❞ t♦ [−π/2, π/2]✱ ❞❡♥♦t❡❞ ❜② ❡✐t❤❡r ♦❢ t❤❡s❡✿ arcsin y = sin−1 y

❚❤❡ ❛r❝❝♦s✐♥❡ ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ ❝♦s✐♥❡ ❢✉♥❝t✐♦♥ ♦♥ [0, π]✱ ❞❡♥♦t❡❞

✷✳✼✳

❈♦♥t✐♥✉✐t② ♦❢ t❤❡ ✐♥✈❡rs❡

✶✼✵

❜② ❡✐t❤❡r ♦❢ t❤❡s❡✿

arccos y = cos−1 y

❚❤✉s✱ ✇❡ ❤❛✈❡ t✇♦ ♣❛✐rs ♦❢ ✐♥✈❡rs❡ ❢✉♥❝t✐♦♥s✿

y = sin x x = sin−1 y ❞♦♠❛✐♥✿ [−π/2, π/2] ❞♦♠❛✐♥✿ [−1, 1] r❛♥❣❡✿ [−1, 1] r❛♥❣❡✿ [−π/2, π/2] ❚❤❡ ❣r❛♣❤s ❛r❡✱ ♦❢ ❝♦✉rs❡✱ t❤❡ s❛♠❡ ✇✐t❤ ❥✉st

❊①❡r❝✐s❡ ✷✳✼✳✶✺ ❈♦♥❞✉❝t t❤❡ s❛♠❡ ❛♥❛❧②s✐s ❢♦r t❤❡ t❛♥❣❡♥t✳

x

❛♥❞

y

y = cos x x = cos−1 y ❞♦♠❛✐♥✿ [0, π] ❞♦♠❛✐♥✿ [−1, 1] r❛♥❣❡✿ [−1, 1] r❛♥❣❡✿ [0, π]

✐♥t❡r❝❤❛♥❣❡❞✿

✷✳✽✳

❈♦♠♣❛r✐s♦♥ ♦❢ ❧✐♠✐ts

✶✼✶

✷✳✽✳ ❈♦♠♣❛r✐s♦♥ ♦❢ ❧✐♠✐ts

❚♦ s❤♦✇ ❝♦♥t✐♥✉✐t② ♦❢ ❢✉♥❝t✐♦♥s ❜❡②♦♥❞ t❤❡ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥s✱ ✇❡ ✇✐❧❧ ❡♠♣❧♦② s♦♠❡ ✐♥❞✐r❡❝t ❛♥❞ ❞✐r❡❝t ♠❡t❤♦❞s✳ ❲❤❡♥ t✇♦ ❢✉♥❝t✐♦♥ ❛r❡ ❝♦♠♣❛r❛❜❧❡ ✐♥ ✈❛❧✉❡✱ t❤❡ ❧✐♠✐t ♦❢ ♦♥❡ ♠✐❣❤t t❡❧❧ ✉s s♦♠❡t❤✐♥❣ ❛❜♦✉t t❤❛t ❢♦r t❤❡ ♦t❤❡r✿

❚❤❡ ♣✐❝t✉r❡ ✐s ❛♥ ✐❧❧✉str❛t✐♦♥ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢❛❝t✿

❚❤❡♦r❡♠ ✷✳✽✳✶✿ ❈♦♠♣❛r✐s♦♥ ❚❡st ❢♦r ▲✐♠✐ts ♦❢ ❋✉♥❝t✐♦♥s ◆♦♥✲str✐❝t ✐♥❡q✉❛❧✐t✐❡s ❜❡t✇❡❡♥ ❢✉♥❝t✐♦♥s ❛r❡ ♣r❡s❡r✈❡❞ ✉♥❞❡r ❧✐♠✐ts✳

■♥ ♦t❤❡r ✇♦r❞s✱ ✐❢

f (x) ≤ g(x) ❢♦r ❛❧❧

n

❣r❡❛t❡r t❤❛♥ s♦♠❡

N✱

t❤❡♥

lim f (x) ≤ lim g(x),

x→a

x→a

♣r♦✈✐❞❡❞ t❤❡ ❧✐♠✐ts ❡①✐st✳

❊①❡r❝✐s❡ ✷✳✽✳✷ Pr♦✈❡ t❤❡ t❤❡♦r❡♠✳

❲❛r♥✐♥❣✦ ❚❤❡ t❤❡♦r❡♠ ❞♦❡s ♥♦t ❝❧❛✐♠ ❝♦♥✈❡r❣❡♥❝❡✳

❚❤✐s ✐s t❤❡ s✉♠♠❛r② ♦❢ t❤❡ t❤❡♦r❡♠✿

f (x) ≥ g(x) ↓ ↓ L M =⇒ ≥ ❊①❡r❝✐s❡ ✷✳✽✳✸ ❙❤♦✇ t❤❛t r❡♣❧❛❝✐♥❣ t❤❡ ♥♦♥✲str✐❝t ✐♥❡q✉❛❧✐t②

f (x) ≤ g(x) ✇✐t❤ ❛ str✐❝t ♦♥❡ f (x) < g(x) ✇♦♥✬t ♣r♦❞✉❝❡

❛ str✐❝t ✐♥❡q✉❛❧✐t② ✐♥ t❤❡ ❝♦♥❝❧✉s✐♦♥ ♦❢ t❤❡ t❤❡♦r❡♠✳

❋r♦♠ t❤✐s ✭s✐♥❣❧❡✮ ✐♥❡q✉❛❧✐t②✱ ✇❡ ❝❛♥✬t ❝♦♥❝❧✉❞❡ ❛♥②t❤✐♥❣ ❛❜♦✉t t❤❡ ❡①✐st❡♥❝❡ ♦❢ t❤❡ ❧✐♠✐t✿

✷✳✽✳

❈♦♠♣❛r✐s♦♥ ♦❢ ❧✐♠✐ts

✶✼✷

❙♦✱ ❡✈❡♥ ✇❤❡♥ t✇♦ ❢✉♥❝t✐♦♥ ❛r❡ ❝♦♠♣❛r❛❜❧❡ ✐♥ ✈❛❧✉❡✱ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦r ❞✐✈❡r❣❡♥❝❡ ♦❢ ♦♥❡ ✇♦♥✬t t❡❧❧ ✉s ❛♥②t❤✐♥❣ ❛❜♦✉t t❤❛t ♦❢ t❤❡ ♦t❤❡r✳ ❍❛✈✐♥❣

❙✉❝❤ ❛ ❞♦✉❜❧❡ ✐♥❡q✉❛❧✐t② ✐s ❝❛❧❧❡❞ ❛

t✇♦

sq✉❡❡③❡

✐♥❡q✉❛❧✐t✐❡s✱ ♦♥ ❜♦t❤ s✐❞❡s✱ ♠❛② ✇♦r❦ ❜❡tt❡r✿

✱ ❥✉st ❛s ✐♥ t❤❡ ❝❛s❡ ♦❢ s❡q✉❡♥❝❡s✳ ■❢ ✇❡ ❝❛♥ sq✉❡❡③❡ t❤❡ ❢✉♥❝t✐♦♥

✉♥❞❡r ✐♥✈❡st✐❣❛t✐♦♥ ❜❡t✇❡❡♥ t✇♦ ❢❛♠✐❧✐❛r ❢✉♥❝t✐♦♥s✱ ✇❡ ♠✐❣❤t ❜❡ ❛❜❧❡ t♦ s❛② s♦♠❡t❤✐♥❣ ❛❜♦✉t ✐ts ❧✐♠✐t✳ ❙♦♠❡ ❢✉rt❤❡r r❡q✉✐r❡♠❡♥ts ✇✐❧❧ ❜❡ ♥❡❝❡ss❛r②✿

❚❤❡♦r❡♠ ✷✳✽✳✹✿ ❙q✉❡❡③❡ ❚❤❡♦r❡♠ ❢♦r ❋✉♥❝t✐♦♥s ■❢ t❤❡ ✈❛❧✉❡s ♦❢ ❛ ❢✉♥❝t✐♦♥ ❧✐❡ ❜❡t✇❡❡♥ t❤♦s❡ ♦❢ t✇♦ ❢✉♥❝t✐♦♥s ✇✐t❤ ❡q✉❛❧ ❧✐♠✐t✱ t❤❡♥ ✐ts ❧✐♠✐t ❛❧s♦ ❡①✐sts ❛♥❞ ✐s ❡q✉❛❧ t♦ t❤❛t ♥✉♠❜❡r✳

■♥ ♦t❤❡r ✇♦r❞s✱ ✐❢ ✇❡ ❤❛✈❡✿

f (x) ≤ h(x) ≤ g(x) ❢♦r ❛❧❧

x

✇✐t❤✐♥ s♦♠❡ ♦♣❡♥ ✐♥t❡r✈❛❧ t❤❛t ❝♦♥t❛✐♥s

x = a✱

❛♥❞ ✐❢ ✇❡ ❤❛✈❡✿

lim f (x) = lim g(x) = L ,

x→a

x→a

t❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❧✐♠✐t ❡①✐sts ❛♥❞ ✐s ❡q✉❛❧ t♦ t❤❡ s❛♠❡ ♥✉♠❜❡r✿

lim h(x) = L .

x→a

Pr♦♦❢✳ ❋♦r ❛♥② s❡q✉❡♥❝❡

xn → a✱

✇❡ ❤❛✈❡✿

f (xn ) ≤ h(xn ) ≤ g(xn ) . ❲❡ ❛❧s♦ ❤❛✈❡✿

lim f (xn ) = lim g(xn ) = L .

n→∞

❚❤❡r❡❢♦r❡✱ ❜② t❤❡

❙q✉❡❡③❡ ❚❤❡♦r❡♠ ❢♦r ❙❡q✉❡♥❝❡s

n→∞

✇❡ ❤❛✈❡✿

lim h(xn ) = L .

n→∞

✷✳✽✳

❈♦♠♣❛r✐s♦♥ ♦❢ ❧✐♠✐ts

✶✼✸

❲❛r♥✐♥❣✦ ❚❤❡ t❤❡♦r❡♠ ❞♦❡s ❝❧❛✐♠ ❝♦♥✈❡r❣❡♥❝❡✳

❊①❛♠♣❧❡ ✷✳✽✳✺✿ ✜♥❞ ❛ sq✉❡❡③❡ ▲❡t✬s ✜♥❞ t❤❡ ❧✐♠✐t✱ ■t ❝❛♥♥♦t ❜❡ ❝♦♠♣✉t❡❞ ❜② t❤❡ Pr♦❞✉❝t

  1 =? lim x sin x→0 x

❘✉❧❡ ❜❡❝❛✉s❡

  1 lim sin ❞♦❡s ♥♦t ❡①✐st✳ x→0 x

▲❡t✬s tr② t♦ ✜♥❞ ❛ sq✉❡❡③❡ ✐♥st❡❛❞✳ ❚❤✐s ✐s ✇❤❛t ✇❡ ❦♥♦✇ ❢r♦♠ tr✐❣♦♥♦♠❡tr②✿   1 −1 ≤ sin ≤ 1. x

▲❡t✬s ♥♦t❡ t❤❛t t❤✐s sq✉❡❡③❡ ♣r♦✈❡s ♥♦t❤✐♥❣ ❛❜♦✉t t❤❡ ❧✐♠✐t ♦❢ sin(1/x)✿

■t✬s ❥✉st t✇♦ ❜♦✉♥❞s✦ ▲❡t✬s ❝♦♥s✐❞❡r t❤✐s sq✉❡❡③❡ ♣r♦❞✉❝❡❞ ❢r♦♠ t❤❡ ❧❛st ♦♥❡✿   1 ≤ |x| . −|x| ≤ x sin x

■t ❧♦♦❦s ♠♦r❡ ✉s❡❢✉❧✿

◆♦✇✱ s✐♥❝❡ lim (−x) = lim (x) = 0✱ ❜② t❤❡ ❙q✉❡❡③❡ x→0

x→0

❚❤❡♦r❡♠✱ ✇❡ ❤❛✈❡✿

  1 lim x sin = 0. x→0 x

✷✳✽✳ ❈♦♠♣❛r✐s♦♥ ♦❢ ❧✐♠✐ts

✶✼✹

❊①❛♠♣❧❡ ✷✳✽✳✻✿ ❉✐r✐❝❤❧❡t

❲❡ ❝❛♥ ✉s❡ t❤❡ s❛♠❡ ❧♦❣✐❝ t♦ ♣r♦✈❡ t❤❛t t❤❡ ❉✐r✐❝❤❧❡t ❢✉♥❝t✐♦♥ ♠✉❧t✐♣❧✐❡❞ ❜② x ✐s ❝♦♥t✐♥✉♦✉s ❛t ❡①❛❝t❧② ♦♥❡ ♣♦✐♥t✦ ( x ✐❢ x ✐s ❛ r❛t✐♦♥❛❧ ♥✉♠❜❡r, xIQ (x) = 0 ✐❢ x ✐s ❛♥ ✐rr❛t✐♦♥❛❧ ♥✉♠❜❡r. ❆❣❛✐♥✱ t♦ ♣❧♦t ✐ts ❣r❛♣❤✱ ✇❡ ❝❛♥ ♦♥❧② ❞r❛✇ t✇♦ ❧✐♥❡s ❛♥❞ t❤❡♥ ♣♦✐♥t ♦✉t s♦♠❡ ♦❢ t❤❡ ♠✐ss✐♥❣ ♣♦✐♥ts✿

❚❤❡ ❡①❛♠♣❧❡ s❤♦✇s t❤❛t t❤❡ ❛❜s♦❧✉t❡ ✈❛❧✉❡ ✐s ❛ ❣♦♦❞ t♦♦❧ ❢♦r ❝♦♥str✉❝t✐♥❣ ❛ sq✉❡❡③❡✳ ❲❡ ❛♣♣❧② t❤✐s ✐❞❡❛ t♦ ❛❧❧ ❢✉♥❝t✐♦♥s✿ ❈♦r♦❧❧❛r② ✷✳✽✳✼✿ ▲✐♠✐t ♦❢ ❆❜s♦❧✉t❡ ❱❛❧✉❡ ❚❤❡ ❧✐♠✐t ♦❢ ❛ ❢✉♥❝t✐♦♥ ✐s ③❡r♦ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❛t ♦❢ ✐ts ❛❜s♦❧✉t❡ ✈❛❧✉❡ ✐s✳

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿

lim f (x) = 0 ⇐⇒ lim |f (x)| = 0 .

x→a

x→a

❊①❡r❝✐s❡ ✷✳✽✳✽

Pr♦✈❡ t❤❡ ❝♦r♦❧❧❛r②✳ ❊①❛♠♣❧❡ ✷✳✽✳✾✿ ❢❛♠♦✉s ❧✐♠✐t

❚❤❡ ✜rst ♦❢ t❤❡ t✇♦ ❢❛♠♦✉s ❧✐♠✐ts✿ sin x 1 − cos x = 1 ❛♥❞ lim = 0, x→0 x x→0 x lim

❢♦❧❧♦✇s ❜② t❤❡ ❙q✉❡❡③❡ ❚❤❡♦r❡♠ ❢r♦♠ t❤✐s tr✐❣♦♥♦♠❡tr✐❝ ❢❛❝t✿ cos x ≤

sin x ≤ 1. x

❊①❡r❝✐s❡ ✷✳✽✳✶✵

Pr♦✈❡ t❤❡ ♦t❤❡r ♦♥❡ ✐♥ t❤❡ s❛♠❡ ♠❛♥♥❡r✳ ❲❡ ♥♦✇ t✉r♥ t♦ ❞✐✈❡r❣❡♥❝❡✳ ❚♦ ♠❛❦❡ ❝♦♥❝❧✉s✐♦♥s ❛❜♦✉t ❞✐✈❡r❣❡♥❝❡ t♦ ✐♥✜♥✐t②✱ ✇❡ ♦♥❧② ♥❡❡❞ t♦ ❝♦♥tr♦❧ t❤❡ ❢✉♥❝t✐♦♥ ❢r♦♠ ♦♥❡✱ ❜✉t t❤❡ r✐❣❤t✱ s✐❞❡✿

✷✳✾✳

●❧♦❜❛❧ ♣r♦♣❡rt✐❡s ♦❢ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s

✶✼✺

❚❤❡ s♠❛❧❧❡r ❢✉♥❝t✐♦♥ ✇✐❧❧ ♣✉s❤ t❤❡ ❧❛r❣❡r t♦ t❤❡ ♣♦s✐t✐✈❡ ✐♥✜♥✐t② ❛♥❞ t❤❡ ❧❛r❣❡r ❢✉♥❝t✐♦♥ ✇✐❧❧ ♣✉s❤ t❤❡ s♠❛❧❧❡r t♦ t❤❡ ♥❡❣❛t✐✈❡ ✐♥✜♥✐t②✳ ❇❡❧♦✇ ✐s t❤❡ ❛♥❛❧♦❣ ♦❢ t❤❡ ❙q✉❡❡③❡ ❚❤❡♦r❡♠ ❢♦r ✐♥✜♥✐t❡ ❧✐♠✐ts✿ ❚❤❡♦r❡♠ ✷✳✽✳✶✶✿ P✉s❤ ❖✉t ❚❤❡♦r❡♠ ❢♦r ▲✐♠✐ts ♦❢ ❋✉♥❝t✐♦♥s

• ■❢ t❤❡ s♠❛❧❧❡r ❢✉♥❝t✐♦♥ ❛♣♣r♦❛❝❤❡s ♣♦s✐t✐✈❡ ✐♥✜♥✐t②✱ t❤❡♥ s♦ ❞♦❡s t❤❡ ❧❛r❣❡r

♦♥❡✳

• ■❢ t❤❡ ❧❛r❣❡r ❢✉♥❝t✐♦♥ ❛♣♣r♦❛❝❤❡s ♥❡❣❛t✐✈❡ ✐♥✜♥✐t②✱ t❤❡♥ s♦ ❞♦❡s t❤❡ s♠❛❧❧❡r

♦♥❡✳

■♥ ♦t❤❡r ✇♦r❞s✱ ✐❢ f (x) ≤ g(x) ❢♦r ❛❧❧ x ✇✐t❤✐♥ s♦♠❡ ♦♣❡♥ ✐♥t❡r✈❛❧ t❤❛t ❝♦♥t❛✐♥s

a✱ t❤❡♥ ✇❡ ❤❛✈❡✿

lim f (x) = +∞ =⇒ lim g(x) = +∞

x→a

x→a

lim f (x) = −∞ ⇐= lim g(x) = −∞

x→a

x→a

❊①❡r❝✐s❡ ✷✳✽✳✶✷

❙✉❣❣❡st ❛♥ ❡①❛♠♣❧❡ ♦❢ ❤♦✇ t❤✐s t❤❡♦r❡♠ ❛♣♣❧✐❡s✳ ❚❤✐s ✐s t❤❡ s✉♠♠❛r② ♦❢ t❤❡ t❤❡♦r❡♠✿ f (x) ≤ g(x) ↓ +∞ =⇒ ↓ +∞

=⇒

f (x) ≤ g(x) ↓ −∞ ↓ −∞

✷✳✾✳ ●❧♦❜❛❧ ♣r♦♣❡rt✐❡s ♦❢ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s

❚❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❝♦♥t✐♥✉✐t② ✐s ♣✉r❡❧② ❧♦❝❛❧ ❛s ♦♥❧② t❤❡ ❜❡❤❛✈✐♦r ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ✐♥ t❤❡ ✈✐❝✐♥✐t②✱ ♥♦ ♠❛tt❡r ❤♦✇ s♠❛❧❧✱ ♦❢ t❤❡ ♣♦✐♥t ♠❛tt❡rs✳ ◆♦✇✱ ✇❤❛t ✐❢ t❤❡ ❢✉♥❝t✐♦♥ ✐s ❝♦♥t✐♥✉♦✉s ♦♥ ❛ ✇❤♦❧❡ ✐♥t❡r✈❛❧❄ ❲❤❛t ❝❛♥ ✇❡ s❛② ❛❜♦✉t ✐ts ❣❧♦❜❛❧ ❜❡❤❛✈✐♦r❄

✷✳✾✳

●❧♦❜❛❧ ♣r♦♣❡rt✐❡s ♦❢ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s

❲❡ ❦♥♦✇ ✇❤❛t ✇❡

◮

✇❛♥t

✶✼✻

t♦ s❛②✿

❚❤❡ ❣r❛♣❤ ♦❢ ❛ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ ❝♦♥s✐sts ♦❢ ❛ s✐♥❣❧❡ ♣✐❡❝❡✳

❖✉r ✉♥❞❡rst❛♥❞✐♥❣ ♦❢ ❝♦♥t✐♥✉✐t② ♦❢ ❢✉♥❝t✐♦♥s ❤❛s ❜❡❡♥ ❛s t❤❡ ♣r♦♣❡rt② ♦❢ ❤❛✈✐♥❣ ❢❛❝t✱ t❤❡r❡ ❛r❡ ♥♦ ❣❛♣s ✐♥ t❤❡ r❛♥❣❡ ❡✐t❤❡r✳ ❚♦ ❣❡t ❢r♦♠ ♣♦✐♥t

A = f (a)

t♦ ♣♦✐♥t

♥♦ ❣❛♣s ✐♥ t❤❡✐r ❣r❛♣❤s

✳ ■♥

B = f (b)✱

✇❡ ❤❛✈❡ t♦ ✈✐s✐t

❡✈❡r② ♣♦✐♥t ✐♥ ❜❡t✇❡❡♥ ✭♥♦ t❡❧❡♣♦rt❛t✐♦♥✦✮✿

❲❡ ❝❛♥ s❡❡ ✐♥ t❤❡ s❡❝♦♥❞ ♣❛rt ♦❢ t❤❡ ✐❧❧✉str❛t✐♦♥ ❤♦✇ t❤✐s ♣r♦♣❡rt② ♠❛② ❢❛✐❧✳ ❚❤✐s ✐❞❡❛ ✐s ♠♦r❡ ♣r❡❝✐s❡❧② ❡①♣r❡ss❡❞ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣✿

❚❤❡♦r❡♠ ✷✳✾✳✶✿ ■♥t❡r♠❡❞✐❛t❡ ❱❛❧✉❡ ❚❤❡♦r❡♠

❆ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ ♦♥ ❛ ❝❧♦s❡❞ ❜♦✉♥❞❡❞ ✐♥t❡r✈❛❧ ❞♦❡s♥✬t ♠✐ss ❛♥② ♣♦✐♥ts ❜❡t✇❡❡♥ ✐ts ✈❛❧✉❡s ❛t t❤❡ ❡♥❞s ♦❢ t❤❡ ✐♥t❡r✈❛❧✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✐❢ ❛ ❢✉♥❝t✐♦♥ f ✐s ❞❡✜♥❡❞ ❛♥❞ ✐s ❝♦♥t✐♥✉♦✉s ♦♥ ❛♥ ✐♥t❡r✈❛❧ [a, b]✱ t❤❡♥ ❢♦r ❛♥② c ❜❡t✇❡❡♥ f (a) ❛♥❞ f (b)✱ t❤❡r❡ ✐s d ✐♥ [a, b] s✉❝❤ t❤❛t f (d) = c✳ Pr♦♦❢✳ ❚❤❡ ✐❞❡❛ ♦❢ t❤❡ ♣r♦♦❢ ✐s t❤✐s✿ ❲❡ ✇✐❧❧ ♠❛❦❡ t❤❡ ✐♥t❡r✈❛❧s ✐♥ t❤❡

x✲❛①✐s

t❤❛t s❛t✐s❢② t❤❡ ❝♦♥❞✐t✐♦♥ ♦❢ t❤❡

t❤❡♦r❡♠ ♥❛rr♦✇❡r ❛♥❞ ♥❛rr♦✇❡r❀ t❤❡♥ t❤❡ ❝♦♥t✐♥✉✐t② ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ✇✐❧❧ ❡♥s✉r❡ t❤❛t t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ✐♥t❡r✈❛❧s ✐♥ t❤❡

y ✲❛①✐s

✇✐❧❧ ❣❡t ♥❛rr♦✇❡r ❛♥❞ ♥❛rr♦✇❡r t♦♦✿

✷✳✾✳

●❧♦❜❛❧ ♣r♦♣❡rt✐❡s ♦❢ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s

✶✼✼

▼♦r❡ ♣r❡❝✐s❡❧②✱ ✇❡ ✇✐❧❧ ❝♦♥str✉❝t s✉❝❤ ❛ s❡q✉❡♥❝❡ ♦❢

♥❡st❡❞ ✐♥t❡r✈❛❧s

✱

I = [a, b] ⊃ I1 = [a1 , b1 ] ⊃ I2 = [a2 , b2 ] ⊃ ... , t❤❛t t❤❡② ✇✐❧❧ ❤❛✈❡ ♦♥❧② ♦♥❡ ♣♦✐♥t ✐♥ ❝♦♠♠♦♥✳

❈♦♥s✐❞❡r t❤❡ ❤❛❧✈❡s ♦❢

I = [a, b]✿



   a+b a+b a, , ,b . 2 2

❋♦r ❛t ❧❡❛st ♦♥❡ ♦❢ t❤❡♠✱ t❤❡r❡ ✐s ❛ ❝❤❛♥❣❡ ♦❢ ❈❛❧❧ t❤✐s ✐♥t❡r✈❛❧

y = f (x)

❢r♦♠ ❧❡ss t♦ ♠♦r❡ ♦r ❢r♦♠ ♠♦r❡ t♦ ❧❡ss ♦❢

I1 = [a1 , b1 ]✿ f (a1 ) < c, f (b1 ) > c

♦r

f (a1 ) > c, f (b1 ) < c .

◆❡①t✱ ✇❡ ❝♦♥s✐❞❡r t❤❡ ❤❛❧✈❡s ♦❢ t❤✐s ♥❡✇ ✐♥t❡r✈❛❧✿



   a1 + b1 a1 + b1 a1 , , , b1 . 2 2

❖♥❝❡ ❛❣❛✐♥✱ ❢♦r ❛t ❧❡❛st ♦♥❡ ♦❢ t❤❡♠✱ t❤❡ ✈❛❧✉❡s ♦❢

f (a2 ) < c, f (b2 ) > c

f

♦r

❝r♦ss

y = c✳

❈❛❧❧ t❤✐s ✐♥t❡r✈❛❧

f (a2 ) > c, f (b2 ) < c .

❲❡ ❝♦♥t✐♥✉❡✿

◆♦t❡ t❤❛t ✇❤❡♥❡✈❡r

f (an ) = c

♦r

f (bn ) = c✱

✇❡ ❛r❡ ❞♦♥❡✳

❋♦❧❧♦✇✐♥❣ t❤✐s ♣r♦❝❡ss✱ t❤❡ r❡s✉❧t ✐s ❛ s❡q✉❡♥❝❡ ♦❢ ✐♥t❡r✈❛❧s✿

I = [a, b] ⊃ I1 = [a1 , b1 ] ⊃ I2 = [a2 , b2 ] ⊃ ... ❚❤❡② s❛t✐s❢② t❤❡s❡ t✇♦ ♣r♦♣❡rt✐❡s✿

a ≤ a1 ≤ ... ≤ an ≤ ... ≤ bn ≤ ... ≤ b1 ≤ b , ❛♥❞

f (an ) < c, f (bn ) > c ❲❡ ❛❧s♦ ❤❛✈❡✿

♦r

f (an ) > c, f (bn ) < c .

1 1 |bn+1 − an+1 | = |bn − an | = n |b − a| → 0 . 2 2

I2 := [a2 , b2 ]✿

c✳

✷✳✾✳

●❧♦❜❛❧ ♣r♦♣❡rt✐❡s ♦❢ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s

✶✼✽

❇② t❤❡ ◆❡st❡❞ ■♥t❡r✈❛❧s ❚❤❡♦r❡♠ ✭❈❤❛♣t❡r ✶✮✱ t❤❡ s❡q✉❡♥❝❡s ❝♦♥✈❡r❣❡✱ ❛♥❞ ❝♦♥✈❡r❣❡ t♦ t❤❡ s❛♠❡ ✈❛❧✉❡✿

an → d, bn → d . ❋r♦♠ t❤❡ ❝♦♥t✐♥✉✐t② ♦❢ f ✱ ✇❡ t❤❡♥ ❝♦♥❝❧✉❞❡✿

f (an ) → f (d), f (bn ) → f (d) .

❇② t❤❡

❈♦♠♣❛r✐s♦♥ ❚❤❡♦r❡♠✱ ✇❡ ❝♦♥❝❧✉❞❡ ❢r♦♠ t❤❡ ✐♥❡q✉❛❧✐t✐❡s ❛❜♦✈❡✿ f (d) ≤ c, f (d) ≥ c

♦r

f (d) ≥ c, f (d) ≤ c .

❍❡♥❝❡✱ ✇❡ ❤❛✈❡✿

f (d) = c . ❙♦✱ d ✐s t❤❡ ❞❡s✐r❡❞ ♥✉♠❜❡r✳ ❈❤♦♦s✐♥❣ c = 0 ✐♥ t❤❡ t❤❡♦r❡♠ ❣✐✈❡s ✉s t❤❡ ❢♦❧❧♦✇✐♥❣✿ ❈♦r♦❧❧❛r② ✷✳✾✳✷✿

x✲✐♥t❡r❝❡♣ts

♦❢ ❈♦♥t✐♥✉♦✉s ❋✉♥❝t✐♦♥

■❢ ❛ ❝♦♥t✐♥✉♦✉s ♦♥ ❛ ❝❧♦s❡❞ ✐♥t❡r✈❛❧ ❢✉♥❝t✐♦♥ t❛❦❡s ✈❛❧✉❡s ✇✐t❤ ♦♣♣♦s✐t❡ s✐❣♥s ❛t ✐ts ❡♥❞✲♣♦✐♥ts✱ t❤❡♥ ✐t ❤❛s ❛♥

■♥ ♦t❤❡r ✇♦r❞s✱ ✐❢

f

x✲✐♥t❡r❝❡♣t

✐s ❝♦♥t✐♥✉♦✉s ♦♥

✇✐t❤✐♥ t❤✐s ✐♥t❡r✈❛❧✳

[a, b]✱

t❤❡♥ ✇❡ ❤❛✈❡✿

f (a) > 0, f (b) < 0 ❖❘ f (a) < 0, f (b) > 0 =⇒ f (c) = 0

❢♦r s♦♠❡

c

✐♥

[a, b] .

❲❡ ❝❛♥ ♠❛❦❡ t❤✐s ♦❜s❡r✈❛t✐♦♥ ❡✈❡♥ ♠♦r❡ s♣❡❝✐✜❝✱ ❢♦r ♣♦❧②♥♦♠✐❛❧s✳ ❘❡❝❛❧❧ ❤♦✇ t❤❡② ❛r❡ ❝❧❛ss✐✜❡❞✿

■♥❞❡❡❞✱ t❤❡② ❤❛✈❡ ✐♥✜♥✐t❡ ✕ ❜✉t ♣♦ss✐❜❧② ❞✐✛❡r❡♥t ✕ ❧✐♠✐ts ❛t t❤❡ ✐♥✜♥✐t✐❡s✳ ❚❤❡ ♦♥❡s ✐♥ t❤❡ ✜rst r♦✇ ❛r❡✱ t❤❡r❡❢♦r❡✱ ❣✉❛r❛♥t❡❡❞ t♦ t❛❦❡ ❜♦t❤ ♣♦s✐t✐✈❡ ❛♥❞ ♥❡❣❛t✐✈❡ ✈❛❧✉❡s✳ ❲❡ ❝♦♥❝❧✉❞❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿ ❈♦r♦❧❧❛r② ✷✳✾✳✸✿

x✲✐♥t❡r❝❡♣ts

♦❢ P♦❧②♥♦♠✐❛❧

❆♥② ♦❞❞ ❞❡❣r❡❡ ♣♦❧②♥♦♠✐❛❧ ❤❛s ❛♥

❊①❡r❝✐s❡ ✷✳✾✳✹

Pr♦✈✐❞❡ ❞❡t❛✐❧s t♦ t❤❡ ♣r♦♦❢ ♦❢ t❤❡ ❝♦r♦❧❧❛r②✳

x✲✐♥t❡r❝❡♣t✳

✷✳✾✳

●❧♦❜❛❧ ♣r♦♣❡rt✐❡s ♦❢ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s

✶✼✾

❊①❛♠♣❧❡ ✷✳✾✳✺✿ ✐t❡r❛t❡❞ s❡❛r❝❤

■♥t❡r♠❡❞✐❛t❡ ❱❛❧✉❡ ❚❤❡♦r❡♠ ◆❡st❡❞ ■♥t❡r✈❛❧s ❚❤❡♦r❡♠ ✐t❡r❛t❡❞ s❡❛r❝❤ ❢♦r ❛ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❡q✉❛t✐♦♥ f (x) = c

❚❤❡ ♣r♦♦❢ ♦❢ t❤❡ ❛♥

✈✐❛ t❤❡

✐s ♥♦t❤✐♥❣ ♠♦r❡ t❤❛♥

✳ ▲❡t✬s ✉s❡ t❤❡ ♠❡t❤♦❞ t♦ s♦❧✈❡ t❤✐s ❡q✉❛t✐♦♥✿

sin x = 0 . ❲❡ st❛rt ✇✐t❤ t❤❡ ✐♥t❡r✈❛❧

[a1 , b1 ] = [3, 3.5]✳

f (x) = sin x ❞♦❡s ❝❤❛♥❣❡ ✐ts s✐❣♥ f ❛❧s♦ ❝❤❛♥❣❡s ✐ts s✐❣♥✳ ❢♦r♠✉❧❛ ❢♦r an ✐s✿

❚❤❡ ❢✉♥❝t✐♦♥

✐♥t❡r✈❛❧✳ ❲❡ ❞✐✈✐❞❡ t❤❡ ✐♥t❡r✈❛❧ ✐♥ ❤❛❧❢ ❛♥❞ ♣✐❝❦ t❤❡ ❤❛❧❢ ♦✈❡r ✇❤✐❝❤ r❡♣❡❛t t❤✐s ♣r♦❝❡ss s❡✈❡r❛❧ t✐♠❡s✳ ❚❤❡ s♣r❡❛❞s❤❡❡t

♦✈❡r t❤✐s ❚❤❡♥ ✇❡

❂■❋✭❘❬✲✶❪❈❬✸❪✯❘❬✲✶❪❈❬✹❪❁✵✱❘❬✲✶❪❈✱❘❬✲✶❪❈❬✶❪✮ ❛♥❞

bn ✿

✇❤✐❧❡

❂■❋✭❘❬✲✶❪❈❬✶❪✯❘❬✲✶❪❈❬✷❪❁✵✱❘❬✲✶❪❈❬✲✶❪✱❘❬✲✶❪❈✮

dn

✐s t❤❡ ♠✐❞✲♣♦✐♥t ♦❢ t❤❡ ✐♥t❡r✈❛❧✿

❲❡ ❝❛♥ s❡❡ t❤❛t t❤❡ ✈❛❧✉❡s ♦❢ ♣❧♦t ♦❢

(an , f (an ))

an , bn

❝♦♥✈❡r❣❡ t♦

π

❛♥❞ t❤❡ ✈❛❧✉❡s ♦❢

f (an ), f (bn )

t♦

0✳

❚❤❡ s❝❛tt❡r

✐s s❤♦✇♥ ♦♥ t❤❡ r✐❣❤t ❛❧♦♥❣ ✇✐t❤ ✐ts ✈❡rs✐♦♥ ❢♦r ❛ ❧♦❣❛r✐t❤♠✐❝❛❧❧② r❡✲s❝❛❧❡❞

y ✲❛①✐s✳

❚❤❡ t❤❡♦r❡♠ s❛②s t❤❛t t❤❡r❡ ❛r❡ ♥♦ ♠✐ss✐♥❣ ✈❛❧✉❡s ✐♥ t❤❡ ✐♠❛❣❡ ♦❢ ❛♥ ✐♥t❡r✈❛❧✳ ■t✬s ❛❧s♦ ❛♥ ✐♥t❡r✈❛❧✿

▼♦r❡ ❝♦♥❝✐s❡❧②✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿

❈♦r♦❧❧❛r② ✷✳✾✳✻✿ ❘❛♥❣❡ ♦❢ ❈♦♥t✐♥✉♦✉s ❋✉♥❝t✐♦♥ ■❢ t❤❡ ❞♦♠❛✐♥ ♦❢ ❛ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ ✐s ❛♥ ✐♥t❡r✈❛❧✱ t❤❡♥ s♦ ✐s ✐ts r❛♥❣❡✳

Pr♦♦❢✳ ■t ❢♦❧❧♦✇s ❢r♦♠ t❤❡

■♥t❡r♠❡❞✐❛t❡ P♦✐♥t ❚❤❡♦r❡♠

✳

✷✳✾✳

●❧♦❜❛❧ ♣r♦♣❡rt✐❡s ♦❢ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s

✶✽✵

❚❤✐s ✐s ❤♦✇ ❞✐s❝♦♥t✐♥✉✐t② ♠❛② ❝❛✉s❡ t❤❡ r❛♥❣❡ t♦ ❤❛✈❡ ❣❛♣s✿

❚❤❡ s❡❝♦♥❞ ❣r❛♣❤ s❤♦✇s t❤❛t t❤❡ ❝♦♥✈❡rs❡ ♦❢ t❤❡ t❤❡♦r❡♠ ✐s♥✬t tr✉❡✳ ❘❡❝❛❧❧ ✭❢r♦♠ ❱♦❧✉♠❡ ✶✱ ❈❤❛♣t❡r ✶P❈✲✹✮ t❤❛t ❛❧❧ str✐❝t❧② ♠♦♥♦t♦♥✐❝ ❢✉♥❝t✐♦♥s ❛r❡ ♦♥❡✲t♦✲♦♥❡ ❜❡❝❛✉s❡ t❤❡ ❣r❛♣❤ ❝❛♥✬t ❝♦♠❡ ❜❛❝❦ ❛♥❞ ❝r♦ss ❛ ❤♦r✐③♦♥t❛❧ ❧✐♥❡ ❢♦r t❤❡ s❡❝♦♥❞ t✐♠❡✿

❚❤❡ ❝♦♥✈❡rs❡ ✐s tr✉❡ ❜✉t ♦♥❧② ❢♦r ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s✿

❊①❡r❝✐s❡ ✷✳✾✳✼

Pr♦✈❡ t❤❛t ✐❢ ✇❡ ❤❛✈❡ ❛ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ♦♥ ❛♥ ✐♥t❡r✈❛❧ I ✱ t❤❡♥ t❤❡ ❢✉♥❝t✐♦♥ ✐s ♦♥❡✲t♦✲♦♥❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐t ✐s str✐❝t❧② ♠♦♥♦t♦♥✐❝✳ ❊①❛♠♣❧❡ ✷✳✾✳✽✿ s✉♣♣❧② ❛♥❞ ❞❡♠❛♥❞

❍❡r❡ ✐s ❛ ❝♦♥t✐♥✉✐t② ❛r❣✉♠❡♥t ✉s❡❞ ✐♥ ❡❝♦♥♦♠✐❝s✳ ■♥ ❛ t②♣✐❝❛❧ tr❛♥s❛❝t✐♦♥✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿ • ❚❤❡ s✉♣♣❧✐❡r ✐s ✇✐❧❧✐♥❣ t♦ ♣r♦❞✉❝❡ ♠♦r❡ ♦❢ t❤❡ ❝♦♠♠♦❞✐t② ❢♦r ❛ ❤✐❣❤❡r ♣r✐❝❡✳ • ❚❤❡ ❜✉②❡r ✐s ♣r❡♣❛r❡❞ t♦ ❜✉② ♠♦r❡ ❢♦r ❛ ❧♦✇❡r ♣r✐❝❡✳ ❊✐t❤❡r ❢❛❝t ✐♠♣❧✐❡s t❤❛t t❤❡r❡ ✐s ❛ ❢✉♥❝t✐♦♥✿ ❤♦✇ t❤❡ q✉❛♥t✐t② ♦❢ t❤❡ ❝♦♠♠♦❞✐t② ❞❡♣❡♥❞s ♦♥ ♣r✐❝❡✳ ❚❤❡ ❢♦r♠❡r ✐s ✐♥❝r❡❛s✐♥❣ ❛♥❞ t❤❡ ❧❛tt❡r ✐s ❞❡❝r❡❛s✐♥❣✿

■❢ ✇❡ ❛ss✉♠❡ ❛❧s♦ t❤❛t t❤❡ t✇♦ ❢✉♥❝t✐♦♥s ❛r❡ ❝♦♥t✐♥✉♦✉s✱ ✇❡ ❝♦♥❝❧✉❞❡ t❤❛t t❤❡r❡ ♠✉st ❜❡ ❛ ♣r✐❝❡ t❤❛t s❛t✐s✜❡s ❜♦t❤ ♣❛rt✐❡s✳

✷✳✾✳

●❧♦❜❛❧ ♣r♦♣❡rt✐❡s ♦❢ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s

✶✽✶

❊①❡r❝✐s❡ ✷✳✾✳✾ ❲❤❛t ♦t❤❡r ✐♠♣❧✐❝✐t ❛ss✉♠♣t✐♦♥s ❛r❡ ♠❛❞❡ ✐♥ t❤❡ ❡①❛♠♣❧❡❄ P✉t t❤❡ ❡①❛♠♣❧❡ ✐♥ t❤❡ ❢♦r♠ ♦❢ ❛ t❤❡♦r❡♠ ❛♥❞ ♣r♦✈❡ ✐t✳ ❙♦✱ t❤❡ ✐♠❛❣❡ ♦❢ ❛♥ ✐♥t❡r✈❛❧ ✐s ❛♥ ✐♥t❡r✈❛❧✱ ❜✉t ✐s t❤❡ ✐♠❛❣❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ❜♦✉♥❞❡❞❄ ❘❡❝❛❧❧ t❤❛t ❛ ❢✉♥❝t✐♦♥ f ✐s ❝❛❧❧❡❞ ♥✉♠❜❡r Q t❤❛t

❝❧♦s❡❞

✐♥t❡r✈❛❧ ❝❧♦s❡❞❄ ❇✉t ✜rst✱ ✐s t❤❡

❜♦✉♥❞❡❞ ♦♥ ❛♥ ✐♥t❡r✈❛❧ I ✐❢ ✐ts r❛♥❣❡ ✐s ❜♦✉♥❞❡❞✱ ✐✳❡✳✱ t❤❡r❡ ✐s s✉❝❤ ❛ r❡❛❧ |f (x)| ≤ Q

❢♦r ❛❧❧ x ✐♥ I ✳ ❚❤❡ ❧✐♥❦ t♦ t❤❡ ❝✉rr❡♥t s✉❜❥❡❝t ✐s t❤❡ r❡s✉❧t ❜❡❧♦✇✿

❚❤❡♦r❡♠ ✷✳✾✳✶✵✿ ❈♦♥✈❡r❣❡♥t ▼❡❛♥s ❇♦✉♥❞❡❞ ■❢ t❤❡ ❧✐♠✐t ♦❢ ❛ ❢✉♥❝t✐♦♥ ❡①✐sts✱ t❤❡♥ t❤❡ ❢✉♥❝t✐♦♥ ✐s ❜♦✉♥❞❡❞ ♦♥ s♦♠❡ ♦♣❡♥ ✐♥t❡r✈❛❧ t❤❛t ❝♦♥t❛✐♥s t❤✐s ♣♦✐♥t✳

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿

lim f (x)

x→a ❢♦r ❛❧❧

x

✐♥

(a − δ, a + δ)

❡①✐sts

❢♦r s♦♠❡

δ>0

=⇒ |f (x)| ≤ Q ❛♥❞ s♦♠❡

Q✳

❊①❡r❝✐s❡ ✷✳✾✳✶✶ Pr♦✈❡ t❤❡ t❤❡♦r❡♠✳ ❲❡ ❤❛✈❡ ❜❡❡♥ s♣❡❛❦✐♥❣ ✉♥t✐❧ ♥♦✇ ♦❢ ❝♦♥t✐♥✉✐t② ♦♥❧② ♦♥❡ ♣♦✐♥t ❛t ❛ t✐♠❡❀ t❤❡r❡ ✐s ♥♦ ❝✉t ❛t x = a✳ ◆♦✇ s❡ts ✿

❉❡✜♥✐t✐♦♥ ✷✳✾✳✶✷✿ ❢✉♥❝t✐♦♥ ❝♦♥t✐♥✉♦✉s ♦♥ ✐♥t❡r✈❛❧ ❆ ❢✉♥❝t✐♦♥ f ✐s ❝❛❧❧❡❞ ❝♦♥t✐♥✉♦✉s t❤❡ ❞♦♠❛✐♥ ♦❢ f ❛♥❞

♦♥ ❛♥ ✐♥t❡r✈❛❧ I ✐❢ t❤❡ ✐♥t❡r✈❛❧ ✐s ❝♦♥t❛✐♥❡❞ ✐♥

lim f (xn ) = f (a)

n→∞

❢♦r ❛♥② s❡q✉❡♥❝❡ xn ✐♥ I s✉❝❤ t❤❛t xn → a✳

❊①❛♠♣❧❡ ✷✳✾✳✶✸✿ ❝♦♥t✐♥✉✐t② ♦♥ ✐♥t❡r✈❛❧s ■♥ ♣❛rt✐❝✉❧❛r✱ ✇❡ ❤❛✈❡✿ • ❚❤❡ ❢✉♥❝t✐♦♥ 1/x ✐s ❝♦♥t✐♥✉♦✉s ♦♥ (−∞, 0) ❛♥❞ ♦♥ (0, ∞)✳ • ❇✉t ✐t ✐s ♥♦t ❝♦♥t✐♥✉♦✉s ♦♥ (−∞, ∞) ♦r ❡✈❡♥ (−∞, 0) ∪ (0, ∞)✳ ❲❡ ❛❧s♦ ❤❛✈❡✿ √ • ❚❤❡ ❢✉♥❝t✐♦♥ x ✐s ❝♦♥t✐♥✉♦✉s ♦♥ [0, ∞)✳ • ❚❤❡ ❢✉♥❝t✐♦♥s sin−1 x ❛♥❞ cos−1 x ❛r❡ ❝♦♥t✐♥✉♦✉s ♦♥ [−1, 1]✳ ❚❤❡ ❣❧♦❜❛❧ ✈❡rs✐♦♥ ♦❢ t❤❡ ❛❜♦✈❡ t❤❡♦r❡♠ ❣✉❛r❛♥t❡❡s t❤❛t t❤❡ ❢✉♥❝t✐♦♥ ✐s ❜♦✉♥❞❡❞ ♦♥ ❛♥② ❝❧♦s❡❞ ❜♦✉♥❞❡❞ ✐♥t❡r✈❛❧✱ ✐✳❡✳✱ [a, b]✱ ❛s ❢♦❧❧♦✇s✿

✷✳✾✳

✶✽✷

●❧♦❜❛❧ ♣r♦♣❡rt✐❡s ♦❢ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿ ❚❤❡♦r❡♠ ✷✳✾✳✶✹✿ ❈♦♥t✐♥✉♦✉s ▼❡❛♥s ❇♦✉♥❞❡❞ ❆ ❢✉♥❝t✐♦♥ ❝♦♥t✐♥✉♦✉s ♦♥ ❛ ❝❧♦s❡❞ ❜♦✉♥❞❡❞ ✐♥t❡r✈❛❧ ✐s ❜♦✉♥❞❡❞ ♦♥ t❤❡ ✐♥t❡r✈❛❧✳

Pr♦♦❢✳

❙✉♣♣♦s❡✱ t♦ t❤❡ ❝♦♥tr❛r②✱ t❤❛t f ✐s ✉♥❜♦✉♥❞❡❞ ♦♥ ✐♥t❡r✈❛❧ [a, b]✳ ❚❤❡♥ t❤❡r❡ ✐s ❛ s❡q✉❡♥❝❡ xn ✐♥ [a, b] s✉❝❤ t❤❛t f (xn ) → ∞✳ ❚❤❡♥✱ ❜② t❤❡ ❇♦❧③❛♥♦✲❲❡✐❡rstr❛ss ❚❤❡♦r❡♠ ✭❈❤❛♣t❡r ✶✮✱ t❤❡ s❡q✉❡♥❝❡ xn ❤❛s ❛ ❝♦♥✈❡r❣❡♥t s✉❜s❡q✉❡♥❝❡ yk ✿ yk → y .

❚❤✐s ♣♦✐♥t ❜❡❧♦♥❣s t♦ [a, b]✦ ❋r♦♠ t❤❡ ❝♦♥t✐♥✉✐t② ♦❢ t❤❡ ❢✉♥❝t✐♦♥✱ ✐t ❢♦❧❧♦✇s t❤❛t f (yk ) → f (y) .

❚❤✐s ❝♦♥tr❛❞✐❝ts t❤❡ ❢❛❝t t❤❛t yk ✐s ❛ s✉❜s❡q✉❡♥❝❡ ♦❢ ❛ s❡q✉❡♥❝❡ t❤❛t ❞✐✈❡r❣❡s t♦ ∞✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ r❛♥❣❡ ♦r t❤❡ ✐♠❛❣❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ✐s ✇✐t❤✐♥ [−Q, Q]✳ ❊①❡r❝✐s❡ ✷✳✾✳✶✺

❲❤② ❛r❡ ✇❡ ❥✉st✐✜❡❞ t♦ ❝♦♥❝❧✉❞❡ ✐♥ t❤❡ ♣r♦♦❢ t❤❛t t❤❡ ❧✐♠✐t y ♦❢ yk ✐s ✐♥ [a, b]❄ ❊①❛♠♣❧❡ ✷✳✾✳✶✻✿ ♣r❡✲❝♦♥❞✐t✐♦♥s

❲❡ s❤♦✇ ❜❡❧♦✇ t❤❛t t❤❡ t❤❡♦r❡♠ ❢❛✐❧s ✐❢ ♦♥❡ ♦❢ t❤❡ ❝♦♥❞✐t✐♦♥s ✐s ♦♠✐tt❡❞✿ ✶✳ ❚❤❡ ❢✉♥❝t✐♦♥ ✐s ❞✐s❝♦♥t✐♥✉♦✉s ♦♥ ❛ ❝❧♦s❡❞ ❜♦✉♥❞❡❞ ✐♥t❡r✈❛❧✳ ✷✳ ❚❤❡ ❢✉♥❝t✐♦♥ ✐s ❝♦♥t✐♥✉♦✉s ♦♥ ❛ ♥♦t ❝❧♦s❡❞ ❜♦✉♥❞❡❞ ✐♥t❡r✈❛❧✳ ✸✳ ❚❤❡ ❢✉♥❝t✐♦♥ ✐s ❝♦♥t✐♥✉♦✉s ♦♥ ❛ ❝❧♦s❡❞ ✉♥❜♦✉♥❞❡❞ ✐♥t❡r✈❛❧✳

■♥ ❛❧❧ t❤r❡❡ ❝❛s❡s✱ t❤❡ ✉♥❜♦✉♥❞❡❞ ❜❡❤❛✈✐♦r ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ❛s ❛♥ ✐♥✜♥✐t❡ ❧✐♠✐t✿ ✶✳ ✷✳ ✸✳

lim f (x) = +∞

x→c−

lim f (x) = +∞

x→b−

lim f (x) = +∞

x→+∞

❍♦✇❡✈❡r✱ ✇❡ s❤♦✉❧❞♥✬t ❡q✉❛t❡ ✉♥❜♦✉♥❞❡❞♥❡ss ✇✐t❤ ✐♥✜♥✐t❡ ❧✐♠✐ts❀ ❤❡r❡ ✐s t❤❡ ❣r❛♣❤ ♦❢ y = x sin x✿

✷✳✾✳

●❧♦❜❛❧ ♣r♦♣❡rt✐❡s ♦❢ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s

✶✽✸

❊①❛♠♣❧❡ ✷✳✾✳✶✼✿ arctan ❚❤❡ r❛♥❣❡ ♠❛② ❜❡ ❛♥ ♦♣❡♥ ✐♥t❡r✈❛❧✱ ❛s ✐♥ t❤❡ ❝❛s❡ ♦❢ ❛r❝t❛♥❣❡♥t✳ ■ts r❛♥❣❡ ✐s

(−π/2, π/2)✿

❖❢ ❝♦✉rs❡ ✇❡ ❤❛✈❡✿

inf(−π/2, π/2) = −π/2, sup(−π/2, π/2) = π/2 . ❍♦✇❡✈❡r✱ t❤❡s❡ ✈❛❧✉❡s ❛r❡ ♥❡✈❡r r❡❛❝❤❡❞✿

arctan(x) 6= ±π/2 . ■❢ t❤❡ ✐♠❛❣❡✴r❛♥❣❡ ✐s ❛ ❝❧♦s❡❞ ✐♥t❡r✈❛❧✱ t❤❡ ❢✉♥❝t✐♦♥

r❡❛❝❤❡s ✐ts ❡①tr❡♠❡ ✈❛❧✉❡s

✱ ✐✳❡✳✱ t❤❡ ♠❛①✐♠✉♠ ❛♥❞

♠✐♥✐♠✉♠ ♦❢ ✐ts r❛♥❣❡✳

▲❡t✬s r❡❝❛❧❧ t❤❡ r❡❧❡✈❛♥t ❝♦♥❝❡♣ts ✭❢r♦♠ ❱♦❧✉♠❡ ✶✱ ❈❤❛♣t❡r ✶P❈✲✹✮✿

❉❡✜♥✐t✐♦♥ ✷✳✾✳✶✽✿ ❣❧♦❜❛❧ ♠❛①✐♠✉♠ ❛♥❞ ♠✐♥✐♠✉♠ ♣♦✐♥ts ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ❢✉♥❝t✐♦♥

•

❆ ♣♦✐♥t

x=d

y = f (x) ❛♥❞ t❤❡ ✐♥t❡r✈❛❧ [a, b] ✐s ✇✐t❤✐♥ ✐ts ❞♦♠❛✐♥✳ ❣❧♦❜❛❧ ♠❛①✐♠✉♠ ♣♦✐♥t ♦❢ f ♦♥ ✐♥t❡r✈❛❧ [a, b] ✐❢

✐s ❝❛❧❧❡❞ ❛

f (d) ≥ f (x)

❢♦r ❛❧❧

a ≤ x ≤ b.

✷✳✾✳ ●❧♦❜❛❧ ♣r♦♣❡rt✐❡s ♦❢ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s

•

❆ ♣♦✐♥t

x=c

✶✽✹

✐s ❝❛❧❧❡❞ ❛ ❣❧♦❜❛❧ ♠✐♥✐♠✉♠ ♣♦✐♥t ♦❢

f (c) ≤ f (x)

❢♦r ❛❧❧

f

♦♥ ✐♥t❡r✈❛❧

[a, b]

✐❢

a ≤ x ≤ b.

❈♦❧❧❡❝t✐✈❡❧② t❤❡② ❛r❡ ❛❧❧ ❝❛❧❧❡❞ ❣❧♦❜❛❧ ❡①tr❡♠❡ ♣♦✐♥ts✳ ❚❤❡ ✇♦r❞ ✏❣❧♦❜❛❧✑ ✐s ♦❢t❡♥ ♦♠✐tt❡❞✳

❏✉st ❜❡❝❛✉s❡ s♦♠❡t❤✐♥❣ ✐s ❞❡s❝r✐❜❡❞ ❞♦❡s♥✬t ♠❡❛♥ t❤❛t ✐t ❝❛♥ ❜❡ ❢♦✉♥❞✳ ❋♦r ❡①❛♠♣❧❡✱ ♠✐♥✐♠✉♠ ✈❛❧✉❡ ♦♥

[0, ∞)✳

f (x) = 1/x

❤❛s ♥♦

❚❤❡ r❡s✉❧t ❜❡❧♦✇ ❣✉❛r❛♥t❡❡s t❤❡✐r ❡①✐st❡♥❝❡✿

❚❤❡♦r❡♠ ✷✳✾✳✶✾✿ ❊①tr❡♠❡ ❱❛❧✉❡ ❚❤❡♦r❡♠ ❆ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ ♦♥ ❛ ❜♦✉♥❞❡❞ ❝❧♦s❡❞ ✐♥t❡r✈❛❧ ❤❛s ❛ ❣❧♦❜❛❧ ♠❛①✐♠✉♠ ❛♥❞ ❛ ❣❧♦❜❛❧ ♠✐♥✐♠✉♠✳

■♥ ♦t❤❡r ✇♦r❞s✱ ✐❢

f

✐s ❝♦♥t✐♥✉♦✉s ♦♥

[a, b]✱

t❤❡♥ t❤❡r❡ ❛r❡

c, d

✐♥

[a, b]

s✉❝❤ t❤❛t

f (c) ≤ f (x) ≤ f (d) ❢♦r ❛❧❧

x

✐♥

[a, b]✳

Pr♦♦❢✳ ❆❝❝♦r❞✐♥❣ t♦ t❤❡ ❧❛st t✇♦ t❤❡♦r❡♠s✱ t❤❡ ✐♠❛❣❡ ♦❢

[a, b]

✐s ❛ ❜♦✉♥❞❡❞ ✐♥t❡r✈❛❧✳

❊①❡r❝✐s❡ ✷✳✾✳✷✵ Pr♦✈✐❞❡ t❤❡ r❡st ♦❢ t❤❡ ♣r♦♦❢✳ ❍✐♥t✿ t❤❡ ❇♦❧③❛♥♦✲❲❡✐❡rstr❛ss ❚❤❡♦r❡♠ ✭❈❤❛♣t❡r ✶✮✳

❊①❛♠♣❧❡ ✷✳✾✳✷✶✿ ▲❛✛❡r ❝✉r✈❡ ❍❡r❡ ✐s ❝♦♥t✐♥✉✐t② ❛r❣✉♠❡♥t ✉s❡❞ ✐♥ ❡❝♦♥♦♠✐❝s✳ ❚❤❡ ❡①❛❝t ❞❡♣❡♥❞❡♥❝❡ ♦❢ t❤❡ r❡✈❡♥✉❡ ♦♥ t❤❡ ✐♥❝♦♠❡ t❛① r❛t❡ ✐s ✉♥❦♥♦✇♥✳

■t s❡❡♠s ♦❜✈✐♦✉s✱ ❤♦✇❡✈❡r✱ t❤❛t ❜♦t❤

0%

❛♥❞

100%

r❛t❡s ✇✐❧❧ ♣r♦❞✉❝❡ ③❡r♦

r❡✈❡♥✉❡✳

■❢ ✇❡ ❛❧s♦ ❛ss✉♠❡ t❤❛t t❤❡ ❞❡♣❡♥❞❡♥❝❡ ✐s ❝♦♥t✐♥✉♦✉s✱ t❤❡♥ t❤❡r❡ ✐s ❛ ♠❛①✐♠✉♠ ♣♦✐♥t s♦♠❡✇❤❡r❡ ✇✐t❤✐♥ t❤❡ ✐♥t❡r✈❛❧✳ ❲❡ ❝♦♥❝❧✉❞❡ t❤❛t ✐♥❝r❡❛s✐♥❣ t❤❡ r❛t❡ ♠❛② ②✐❡❧❞ ❛ t❛①✲❞❡❝r❡❛s✐♥❣ r❡✈❡♥✉❡✳

❊①❡r❝✐s❡ ✷✳✾✳✷✷ ❲❤❛t ♦t❤❡r ✐♠♣❧✐❝✐t ❛ss✉♠♣t✐♦♥s ❛r❡ ♠❛❞❡ ✐♥ t❤❡ ❡①❛♠♣❧❡❄ P✉t t❤❡ ❡①❛♠♣❧❡ ✐♥ t❤❡ ❢♦r♠ ♦❢ ❛ t❤❡♦r❡♠ ❛♥❞ ♣r♦✈❡ ✐t✳

▼♦r❡ r❡❧❡✈❛♥t ❝♦♥❝❡♣ts ❜❡❧♦✇✿

❉❡✜♥✐t✐♦♥ ✷✳✾✳✷✸✿ ❣❧♦❜❛❧ ♠❛①✐♠✉♠ ❛♥❞ ♠✐♥✐♠✉♠ ✈❛❧✉❡s ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ❢✉♥❝t✐♦♥

•

❆ ♣♦✐♥t

y=M

y = f (x) ❛♥❞ t❤❡ ✐♥t❡r✈❛❧ [a, b] ✐s ✇✐t❤✐♥ ✐ts ❞♦♠❛✐♥✳ f ♦♥ ✐♥t❡r✈❛❧ [a, b] ✐❢

✐s ❝❛❧❧❡❞ t❤❡ ❣❧♦❜❛❧ ♠❛①✐♠✉♠ ✈❛❧✉❡ ♦❢

✷✳✾✳ ●❧♦❜❛❧ ♣r♦♣❡rt✐❡s ♦❢ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s

✶✽✺

✐t ✐s t❤❡ ♠❛①✐♠✉♠ ❡❧❡♠❡♥t ♦❢ t❤❡ r❛♥❣❡ ♦❢

f❀

✐✳❡✳✱

M = max{f (x) : a ≤ x ≤ b} . •

❆ ♣♦✐♥t

y=m

✐s ❝❛❧❧❡❞ t❤❡ ❣❧♦❜❛❧ ♠✐♥✐♠✉♠ ✈❛❧✉❡ ♦❢

✐t ✐s t❤❡ ♠✐♥✐♠✉♠ ❡❧❡♠❡♥t ♦❢ t❤❡ r❛♥❣❡ ♦❢

f❀

f

♦♥ ✐♥t❡r✈❛❧

[a, b]

✐❢

✐✳❡✳✱

m = min{f (x) : a ≤ x ≤ b} . ❈♦❧❧❡❝t✐✈❡❧② t❤❡② ❛r❡ ❛❧❧ ❝❛❧❧❡❞ ❣❧♦❜❛❧ ❡①tr❡♠❡ ✈❛❧✉❡s✳ ❚❤❡ ✇♦r❞ ✏❣❧♦❜❛❧✑ ✐s ♦❢t❡♥ ♦♠✐tt❡❞✳

❚❤✐s ♠❡❛♥s s✐♠♣❧② t❤❛t ✇❡ ❤❛✈❡✿

f

♠❛①✐♠✉♠ ♣♦✐♥t




=

♠❛①✐♠✉♠ ✈❛❧✉❡

❛♥❞

f

♠✐♥✐♠✉♠ ♣♦✐♥t




=

♠✐♥✐♠✉♠ ✈❛❧✉❡

❲❛r♥✐♥❣✦ ❚❤❡ ❛❜s♦❧✉t❡ ♠❛①✐♠✉♠ ❛♥❞ ♠✐♥✐♠✉♠ ✈❛❧✉❡s ❛♥❞ ♣♦✐♥ts ❛r❡ ❛❧s♦ ❝❛❧❧❡❞

✐♠✉♠

❛❜s♦❧✉t❡ ♠❛①✐♠✉♠ ❛♥❞ ♠✐♥✲

✈❛❧✉❡s ❛♥❞ ♣♦✐♥ts✳

❚❤❡♥ t❤❡ ❣❧♦❜❛❧ ♠❛① ✭♦r ♠✐♥✮ ✈❛❧✉❡ ✐s r❡❛❝❤❡❞ ❜② t❤❡ ❢✉♥❝t✐♦♥ ❛t ❛♥② ♦❢ ✐ts ❣❧♦❜❛❧ ♠❛① ✭♦r ♠✐♥✮ ♣♦✐♥ts✳ ❋♦r ❡①❛♠♣❧❡✱

f (x) = sin x

❛tt❛✐♥s ✐ts ♠❛① ✈❛❧✉❡ ♦❢

1

❢♦r ✐♥✜♥✐t❡❧② ♠❛♥② ❝❤♦✐❝❡s ♦❢

x = π/2, 5π/2, ...✳

❲❤② ❛r❡ t❤❡ r❡str✐❝t✐♦♥s ♦❢ t❤❡♦r❡♠ ❡ss❡♥t✐❛❧❄

❊①❛♠♣❧❡ ✷✳✾✳✷✹✿ ❚❤❡ ❢✉♥❝t✐♦♥

1/x

1/x

❞♦❡s♥✬t ❛tt❛✐♥ ✐ts ❧❡❛st ✉♣♣❡r ❜♦✉♥❞ ✈❛❧✉❡ ♦♥

(0, 1]

❛s ✐t ✐s✱ ✐♥ ❢❛❝t✱ ✐♥✜♥✐t❡✿

❚❤❡ t❤❡♦r❡♠ ❞♦❡s♥✬t ❛♣♣❧② ❜❡❝❛✉s❡ t❤❡ ✐♥t❡r✈❛❧ ✐s♥✬t ❝❧♦s❡❞✳ ❙❡❝♦♥❞❧②✱ t❤❡ ❢✉♥❝t✐♦♥ ✐ts ❣r❡❛t❡st ❧♦✇❡r ❜♦✉♥❞ ✈❛❧✉❡✱ ✇❤✐❝❤ ✐s ✐s♥✬t ❜♦✉♥❞❡❞✳

0✱

♦♥

[1, ∞)✳

1/x ❞♦❡s♥✬t ❛tt❛✐♥

❚❤❡ t❤❡♦r❡♠ ❞♦❡s♥✬t ❛♣♣❧② ❜❡❝❛✉s❡ t❤❡ ✐♥t❡r✈❛❧

❊①❛♠♣❧❡ ✷✳✾✳✷✺✿ ❞✐s❝♦♥t✐♥✉♦✉s ❍❡r❡ t❤❡ ♠❛①✐♠✉♠ ✐s♥✬t ❛tt❛✐♥❡❞ ❡✈❡♥ t❤♦✉❣❤ t❤❡ ✐♥t❡r✈❛❧ ✐s ❝❧♦s❡❞ ❛♥❞ ❜♦✉♥❞❡❞✿

✷✳✶✵✳

✶✽✻

▲❛r❣❡✲s❝❛❧❡ ❜❡❤❛✈✐♦r ❛♥❞ ❛s②♠♣t♦t❡s

❚❤❡ t❤❡♦r❡♠ ❞♦❡s♥✬t ❛♣♣❧② ❜❡❝❛✉s❡ f ◆♦t❡ t❤❛t t❤❡ r❡❛s♦♥ ✇❡ ♥❡❡❞ t❤❡ ❛r❡ ❢❛❝✐♥❣ ❤❛s ❛ s♦❧✉t✐♦♥✳

✳

✐s♥✬t ❝♦♥t✐♥✉♦✉s

❊①tr❡♠❡ ❱❛❧✉❡ ❚❤❡♦r❡♠

✐s t♦ ❡♥s✉r❡ t❤❛t t❤❡ ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠ ✇❡

✷✳✶✵✳ ▲❛r❣❡✲s❝❛❧❡ ❜❡❤❛✈✐♦r ❛♥❞ ❛s②♠♣t♦t❡s

●r❛♣❤s ♦❢ ♠♦st ❢✉♥❝t✐♦♥s ❛r❡ ✐♥✜♥✐t❡ ✐♥ s✐③❡ ❛♥❞ ✇♦♥✬t ✜t ♦♥ ❛♥② ♣✐❡❝❡ ♦❢ ♣❛♣❡r✳ ❚❤❡② ❤❛✈❡ t♦ ♣❛♣❡r ❛♥❞ t❤❡② ❞♦ t❤❛t ✐♥ ❛ ♥✉♠❜❡r ♦❢ ❞✐✛❡r❡♥t ✇❛②s✿

❧❡❛✈❡

t❤❡

■♥ ♦r❞❡r t♦ ❞❡s❝r✐❜❡ t❤✐s ❜❡❤❛✈✐♦r ❛❧❣❡❜r❛✐❝❛❧❧②✱ ✇❡ ✇✐❧❧ ❧♦♦❦ ❛t t❤❡ x✲ ❛♥❞ y ✲❝♦♦r❞✐♥❛t❡ ♦❢ t❤❡ ♣♦✐♥ts ♦❢ t❤❡ ❣r❛♣❤ s❡♣❛r❛t❡❧②✳ ❋♦r ❡✐t❤❡r✱ t❤❡r❡ ❛r❡ t✇♦ ♠❛✐♥ ❝❛s❡s✿

❚❤❡r❡ ❝❛♥ ❜❡ ❡✐t❤❡r✿ • ❛ ✜♥✐t❡ ❧✐♠✐t ✭❢r♦♠ ♦♥❡ s✐❞❡✮ ♦r

• ❛♥ ✐♥✜♥✐t❡ ❧✐♠✐t ✭♣♦s✐t✐✈❡ ♦r ♥❡❣❛t✐✈❡✮✳

❚❤❡ ❝♦♦r❞✐♥❛t❡s ❛r❡ ✐♥❞❡♣❡♥❞❡♥t✱ ❛♥❞ t❤❡ t✇♦ ❝❛s❡s ❝❛♥ ❜❡ ❝♦♠❜✐♥❡❞ ✐♥ ❛ ♥✉♠❜❡r ♦❢ ✇❛②s✳

✷✳✶✵✳

✶✽✼

▲❛r❣❡✲s❝❛❧❡ ❜❡❤❛✈✐♦r ❛♥❞ ❛s②♠♣t♦t❡s

❋✐rst✱ ❜♦t❤ ❧✐♠✐ts ❛r❡ ✜♥✐t❡✿

x → a± , y → L± .

❚❤✐s ✐s t❤❡ ❢❛♠✐❧✐❛r ❧✐♠✐t t❤❛t ❤❛s ❜❡❡♥ ❞✐s❝✉ss❡❞ ❢r♦♠ t❤❡ ✈❡r② ❜❡❣✐♥♥✐♥❣ ♦❢ t❤❡ ❝❤❛♣t❡r✿

❙❡❝♦♥❞✱ ♦♥❡ ❧✐♠✐t ✐s ✜♥✐t❡ ❛♥❞ t❤❡ ♦t❤❡r ✐♥✜♥✐t❡✿ x → a± , y → ±∞ ♦r x → ±∞ y → L± .

❚❤❡r❡ ❛r❡ ❡✐❣❤t ♣♦ss✐❜✐❧✐t✐❡s ❤❡r❡✱ ❛❧❧ ✇✐t❤ ❡✐t❤❡r ❤♦r✐③♦♥t❛❧ ♦r ✈❡rt✐❝❛❧ ❛s②♠♣t♦t❡s✿

❚❤✐r❞✱ ❜♦t❤ ❧✐♠✐ts ❛r❡ ✐♥✜♥✐t❡✿ x → ±∞, y → ±∞ .

❚❤❡r❡ ❛r❡ ❢♦✉r ♣♦ss✐❜✐❧✐t✐❡s✱ ❛❧❧ ✇✐t❤ ♥♦ ❛s②♠♣t♦t❡s✿

■♥ ♦r❞❡r t♦ ❞❡t❡r♠✐♥❡ t❤❡ tr❡♥❞ ♦❢ t❤❡ ♣♦✐♥t (x, y) ✐♥ t❤❡ ♣❧❛♥❡✱ ✇❡ ❝❛♥ ✐♠❛❣✐♥❡ ✇❛❧❦✐♥❣ ♦♥ t❤❡ ❝✉r✈❡ ❛♥❞ ❧♦♦❦✐♥❣ ❞♦✇♥ ♦♥ t❤❡ x✲❛①✐s t♦ r❡❝♦r❞ t❤❡ x✲❝♦♦r❞✐♥❛t❡ ❛♥❞ ❧♦♦❦✐♥❣ ❢♦r✇❛r❞ ♦r ❜❛❝❦ t♦ s❡❡ ✇❤❛t ✐s ❤❛♣♣❡♥✐♥❣ t❤❡ y ✲❝♦♦r❞✐♥❛t❡✿

✷✳✶✵✳

▲❛r❣❡✲s❝❛❧❡ ❜❡❤❛✈✐♦r ❛♥❞ ❛s②♠♣t♦t❡s

✶✽✽

❊①❡r❝✐s❡ ✷✳✶✵✳✶

❙✉❣❣❡st ❛ ❢✉♥❝t✐♦♥ ❢♦r ❡❛❝❤ ♦❢ t❤❡s❡ t②♣❡s ♦❢ tr❡♥❞s✳ ❆❧t❡r♥❛t✐✈❡❧②✱ ✇❡ ✐♠❛❣✐♥❡ t❤❛t t❤❡ ❝✉r✈❡s ❛r❡ ❞r❛✇♥ ♦♥ ❛ ♣✐❡❝❡ ♦❢ ♣❛♣❡r✳ ❚❤❡♥✱ ✐❢ ✇❡ ❧♦♦❦ ❛t ✐t ❛t ❛ s❤❛r♣ ❛♥❣❧❡ ❢r♦♠ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ x✲❛①✐s✱ t❤❡ ❝❤❛♥❣❡ ♦❢ x ❜❡❝♦♠❡s ❛❧♠♦st ✐♥✈✐s✐❜❧❡ ❛♥❞ ✇❡ ❝❧❡❛r❧② s❡❡ t❤❡ ❜❡❤❛✈✐♦r ♦❢ t❤❡ y ✲❝♦♦r❞✐♥❛t❡✿

■❢ ✇❡ ❧♦♦❦ ❢r♦♠ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ y ✲❛①✐s✱ ♦♥❧② t❤❡ ❝❤❛♥❣❡ ♦❢ x ✐s s✐❣♥✐✜❝❛♥t✳ ❈♦♥✈❡rs❡❧②✱ ✐❢ ✇❡ ❤❛✈❡ t❤❡ ❧✐♠✐t ❞❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ ❝✉r✈❡✱ ✇❤❛t ❞♦❡s ✐t ❧♦♦❦ ❧✐❦❡❄ ❋♦r ❡①❛♠♣❧❡✱ ✇❡ ♠✐❣❤t ❤❛✈❡✿ • x → +∞ ❛♥❞ y → 0+

• x → 0+ ❛♥❞ y → −∞

■❢ t❤❡ ♣♦✐♥t ❛♣♣r♦❛❝❤❡s ❛ ❧✐♥❡ ✐t ❝❛♥✬t ❝r♦ss ✭y = 0 ❛♥❞ x = 0✮✱ t❤❡ ❝✉r✈❡ st❛rts t♦ ❜❡❝♦♠❡ ♠♦r❡ ❛♥❞ ♠♦r❡ str❛✐❣❤t ❛♥❞ s❧♦✇❧② ❛♣♣r♦❛❝❤❡s t❤❛t ❧✐♥❡✿

✷✳✶✵✳

▲❛r❣❡✲s❝❛❧❡ ❜❡❤❛✈✐♦r ❛♥❞ ❛s②♠♣t♦t❡s

❊✈❡♥t✉❛❧❧② ✐t

❛❧♠♦st

✶✽✾

♠❡r❣❡s ✇✐t❤ t❤✐s ❧✐♥❡✳ ❚❤❡ ❧✐♥❡ ✐s t❤❡♥ ❝❛❧❧❡❞ ❛♥

❛s②♠♣t♦t❡✳

❊①❛♠♣❧❡ ✷✳✶✵✳✷✿ ♥♦ ❛s②♠♣t♦t❡s ❚❤❡ s✐♠♣❧❡st ❝❛s❡✱ ❤♦✇❡✈❡r✱ ✐s t❤❡ ♦♥❡ ✇✐t❤ ♥♦ ❛s②♠♣t♦t❡s✿

y = x, y = x2 , ...

❛t

± ∞, y = ex , y = ln x

+ ∞,

❛t

❛♥❞ ♠❛♥② ♠♦r❡✳

❊①❛♠♣❧❡ ✷✳✶✵✳✸✿ ❡①♣♦♥❡♥t✐❛❧ ❲❡ ♣r❡✈✐♦✉s❧② ❞❡♠♦♥str❛t❡❞ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛❜♦✉t t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥✿

x → −∞, y → 0

x → +∞, y → +∞ .

❛♥❞

❊①❛♠♣❧❡ ✷✳✶✵✳✹✿ ❛r❝t❛♥❣❡♥t ❆s ❛♥ ✐❧❧✉str❛t✐♦♥ ♦❢ t❤❡ ❛s②♠♣t♦t✐❝ ❜❡❤❛✈✐♦r✱ ✇❡ ✇✐❧❧ ❝♦♥s✐❞❡r t❤❡ ✭r❡str✐❝t❡❞✮ t❛♥❣❡♥t

y = tan x

❛♥❞

✐ts ✐♥✈❡rs❡✱ ❛r❝t❛♥❣❡♥t✿

y = tan x ❢♦r

−π/2 < x < π/2✳

x = tan−1 y

❛♥❞

❚❤❡ ❣r❛♣❤s ❛r❡✱ ♦❢ ❝♦✉rs❡✱ t❤❡ s❛♠❡ ✇✐t❤ ❥✉st

x

❛♥❞

y

✐♥t❡r❝❤❛♥❣❡❞✳

▲❡t✬s

❞❡s❝r✐❜❡ t❤❡✐r ❧❛r❣❡✲s❝❛❧❡ ❜❡❤❛✈✐♦rs ❛t ❛ s✐♥❣❧❡ ❛r❡❛ ✭✐♥ ②❡❧❧♦✇✮ ✉s✐♥❣ ❧✐♠✐ts✿

❋✐rst t❤❡ t❛♥❣❡♥t✿

y → +∞ ❈❤❛♥❣✐♥❣

x

t♦ ❜❡ t❤❡ ❞❡♣❡♥❞❡♥t ❛♥❞

y

❛s

x → π/2− .

t♦ ❜❡ t❤❡ ✐♥❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡s✱ ✇❡ s✐♠♣❧② r❡✇r✐t❡ t❤❡ ❛❜♦✈❡

❢♦r t❤❡ ❛r❝t❛♥❣❡♥t✿

y → π/2−

❛s

x → +∞ .

✷✳✶✵✳

▲❛r❣❡✲s❝❛❧❡ ❜❡❤❛✈✐♦r ❛♥❞ ❛s②♠♣t♦t❡s

✶✾✵

❲❛r♥✐♥❣✦ ❉❡s❝r✐❜✐♥❣ ❛s②♠♣t♦t✐❝ ❜❡❤❛✈✐♦r ❛s ✏✢❛tt❡♥✐♥❣✑ ♦❢ t❤❡ ❣r❛♣❤ ✐s ♠✐s❧❡❛❞✐♥❣✱ ❛s t❤❡ ❡①❛♠♣❧❡

y =

√

x

s❤♦✇s✳

❚❤❡ t❤r❡❡ ♣❛tt❡r♥s ♦❢ ❜❡❤❛✈✐♦r ♦❢ ❛ ❢✉♥❝t✐♦♥ ❛s ✐ts ❣r❛♣❤ ❧❡❛✈❡s t❤❡ ✈✐s✐❜❧❡ ♣❛rt ♦❢ t❤❡

xy ✲♣❧❛♥❡

❛r❡ st❛t❡❞

❜❡❧♦✇ ✐♥ ♣r❡❝✐s❡ t❡r♠s✿

❉❡✜♥✐t✐♦♥ ✷✳✶✵✳✺✿ ❢✉♥❝t✐♦♥ ❣♦❡s t♦ ✐♥✜♥✐t② ●✐✈❡♥ ❛ ❢✉♥❝t✐♦♥

f✱

f ❣♦❡s t♦ ✐♥✜♥✐t② ♦♥ t❤❡ r✐❣❤t

✇❡ s❛② t❤❛t

f (xn ) → ±∞

❢♦r ❛♥② s❡q✉❡♥❝❡

✐❢

xn → +∞ .

❲❡ ✉s❡ t❤❡ ♥♦t❛t✐♦♥✿

f (x) → ±∞

❛s

x → +∞

♦r

lim f (x) = ±∞ .

x→+∞ ❯♥❞❡r t❤✐s ❞❡✜♥✐t✐♦♥✱ ✇❡ s❛②

f ❣♦❡s t♦ ✐♥✜♥✐t② ♦♥ t❤❡ ❧❡❢t

✇❤❡♥

x → −∞✳

❉❡✜♥✐t✐♦♥ ✷✳✶✵✳✻✿ ❤♦r✐③♦♥t❛❧ ❛s②♠♣t♦t❡ ❛s ❧✐♠✐t ●✐✈❡♥ ❛ ❢✉♥❝t✐♦♥

❛s②♠♣t♦t❡

♦❢

f

y = f (x)✱

❛ ❧✐♥❡

y =p

❢♦r s♦♠❡ r❡❛❧

p

✐s ❝❛❧❧❡❞ ❛

✐❢

lim f (xn ) = p

n→∞

❢♦r ❛♥② s❡q✉❡♥❝❡

xn → +∞ .

❲❡ ✉s❡ t❤❡ ♥♦t❛t✐♦♥✿

f (x) → p

❛s

x → +∞

♦r✱ ❛❧t❡r♥❛t✐✈❡❧②✱

lim f (x) = p .

x→+∞

❯♥❞❡r t❤✐s ❞❡✜♥✐t✐♦♥✱ ✇❡ ❝❛♥ ❛❧s♦ ❤❛✈❡

x → −∞✳

❉❡✜♥✐t✐♦♥ ✷✳✶✵✳✼✿ ✈❡rt✐❝❛❧ ❛s②♠♣t♦t❡ ❛s ❧✐♠✐t ❆ ❧✐♥❡

x=a

❢♦r s♦♠❡ r❡❛❧

a

✐s ❝❛❧❧❡❞ ❛

lim f (x) = ±∞

n→∞

✈❡rt✐❝❛❧ ❛s②♠♣t♦t❡

❢♦r ❛♥② s❡q✉❡♥❝❡

❲❡ ✉s❡ t❤❡ ♥♦t❛t✐♦♥✿

f (x) → ±∞

❛s

x → a+

♦r✱ ❛❧t❡r♥❛t✐✈❡❧②✱

lim f (x) = ±∞ .

x→a+

❯♥❞❡r t❤✐s ❞❡✜♥✐t✐♦♥✱ ✇❡ ❝❛♥ ❛❧s♦ ❤❛✈❡

❚❤❡ ❢♦✉rt❤ ♣❛tt❡r♥ ✐s ✏♥♦ ♣❛tt❡r♥✑✳

x → a− ✳

♦❢

f

xn → a+ .

✐❢

❤♦r✐③♦♥t❛❧

✷✳✶✵✳

✶✾✶

▲❛r❣❡✲s❝❛❧❡ ❜❡❤❛✈✐♦r ❛♥❞ ❛s②♠♣t♦t❡s

❊①❛♠♣❧❡ ✷✳✶✵✳✽✿ ♥♦t❛t✐♦♥

❲❡ ❝♦❧❧❡❝t t❤✐s ✐♥❢♦r♠❛t✐♦♥ ❢♦r t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥✿ lim ex = 0,

x→−∞

lim ex = +∞ ,

x→+∞

❛♥❞ ❢♦r t❤❡ ❛r❝t❛♥❣❡♥t✿ lim tan−1 = −π/2+ ❛♥❞

x→−∞

lim tan−1 y → π/2− .

x→+∞

❊①❛♠♣❧❡ ✷✳✶✵✳✾✿ s②♠♠❡tr②

❊✈❡♥ t❤♦✉❣❤ t❤❡ ❧❛st t✇♦ ❞❡✜♥✐t✐♦♥s ❧♦♦❦ ✈❡r② ❞✐✛❡r❡♥t✱ t❤❡② ❞❡s❝r✐❜❡ t❤❡ ❝✉r✈❡✳

✐❞❡♥t✐❝❛❧

❜❡❤❛✈✐♦r ♦❢ t❤❡

❲❡ ❝❛♥ s❡❡ t❤✐s s②♠♠❡tr② ✐♥ t❤❡ ❣r❛♣❤ ♦❢ y = 1/x✱ ❛♥❞ ✇❡ ❝❛♥ s❡❡ ✐t ✐♥ t❤❡ ❛❧❣❡❜r❛✿ ❤♦r✐③♦♥t❛❧✿ ✈❡rt✐❝❛❧✿

x y x → +∞ y → 0 x→0 y → +∞

❲❡ ❥✉st ✐♥t❡r❝❤❛♥❣❡ x ❛♥❞ y ✳ ■♥ ♠♦r❡ ❞❡t❛✐❧✿ 1 1 1 1 → 0− ❛s x → −∞, → 0+ ❛s x → +∞, → −∞ ❛s x → 0− , → +∞ ❛s x → 0+ . x x x x

❚❤❡ ❛♥❛❧②s✐s s✉❣❣❡sts t❤❡ ❢♦❧❧♦✇✐♥❣✿ ❚❤❡♦r❡♠ ✷✳✶✵✳✶✵✿ ❆s②♠♣t♦t❡s ♦❢ ■♥✈❡rs❡ ❚❤❡ ✈❡rt✐❝❛❧ ❛s②♠♣t♦t❡s ♦❢ ❛ ❢✉♥❝t✐♦♥ ❛r❡ t❤❡ ❤♦r✐③♦♥t❛❧ ❛s②♠♣t♦t❡s ♦❢ ✐ts ✐♥✈❡rs❡✱ ❛♥❞ ✈✐❝❡ ✈❡rs❛✳

❊①❡r❝✐s❡ ✷✳✶✵✳✶✶

Pr♦✈❡ t❤❡ t❤❡♦r❡♠✳ ❊①❛♠♣❧❡ ✷✳✶✵✳✶✷✿ ◆❡✇t♦♥✬s ▲❛✇ ♦❢ ●r❛✈✐t②

❚❤❡ ❧❛✇ st❛t❡s✿ ◮ ❚❤❡ ❢♦r❝❡ ♦❢ ❣r❛✈✐t② ❜❡t✇❡❡♥ t✇♦ ♦❜❥❡❝ts ✐s ✐♥✈❡rs❡❧② ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ sq✉❛r❡ ♦❢ t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡✐r ❝❡♥t❡rs✳

✷✳✶✵✳

✶✾✷

▲❛r❣❡✲s❝❛❧❡ ❜❡❤❛✈✐♦r ❛♥❞ ❛s②♠♣t♦t❡s

■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ❢♦r❝❡ ✐s ❣✐✈❡♥ ❜② t❤❡ ❢♦r♠✉❧❛✿ F =

C , r2

✇❤❡r❡✿

• C ✐s s♦♠❡ ❝♦♥st❛♥t✱ ❛♥❞ • r ✐s t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❝❡♥t❡rs ♦❢ t❤❡ ♠❛ss ♦❢ t❤❡ t✇♦✳ ❚❤❡♥✱ ✐❢ r > 0 ✐s ✈❛r✐❛❜❧❡✱ ✇❡ ❤❛✈❡✿ lim F (r) = 0 . r→+∞

❚❤✐s ♠❡❛♥s t❤❛t t❤❡ ❢♦r❝❡ ❜❡❝♦♠❡s ♥❡❣❧✐❣✐❜❧❡ ✇❤❡♥ t❤❡ t✇♦ ♦❜❥❡❝ts ❜❡❝♦♠❡ s✉✣❝✐❡♥t❧② ❢❛r ❛✇❛② ❢r♦♠ ❡❛❝❤ ♦t❤❡r✳ ❊①❡r❝✐s❡ ✷✳✶✵✳✶✸

❲❤❛t ✐s ❤❛♣♣❡♥✐♥❣ ✇❤❡♥ t❤❡ t✇♦ ❜♦❞✐❡s ❛r❡ ❣❡tt✐♥❣ ❝❧♦s❡r ❛♥❞ ❝❧♦s❡r t♦ ❡❛❝❤ ♦t❤❡r❄ ❊①❛♠♣❧❡ ✷✳✶✵✳✶✹✿ ◆❡✇t♦♥✬s ▲❛✇ ♦❢ ❈♦♦❧✐♥❣

❲❡ ❞❡r✐✈❡❞ ❢r♦♠ t❤❡ ❧❛✇ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢❛❝t✿ ◮ ❚❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ ❛♥ ♦❜❥❡❝t✬s t❡♠♣❡r❛t✉r❡ ❛♥❞ t❤❡ t❡♠♣❡r❛t✉r❡ ♦❢ t❤❡ ❛t♠♦s♣❤❡r❡ ✐s ❞❡❝❧✐♥✐♥❣ ❡①♣♦♥❡♥t✐❛❧❧②✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿ T − T0 = Cekx , k < 0, C > 0 ,

✇❤❡r❡ T ✐s t❤❡ ❝✉rr❡♥t t❡♠♣❡r❛t✉r❡ ❛♥❞ T0 ✐s t❤❡ ❛♠❜✐❡♥t t❡♠♣❡r❛t✉r❡✳ ❲❡ ❝♦♠♣✉t❡✿ lim T (t) = T0 .

t→+∞

❚❤❡r❡❢♦r❡✱ t❤❡r❡ ✐s ❛ ❤♦r✐③♦♥t❛❧ ❛s②♠♣t♦t❡ y = T0 ✳ ❚❤❡ ❛s②♠♣t♦t❡ ♠❛② ❜❡ ❛♣♣r♦❛❝❤❡❞ ❜② t❤❡ ❣r❛♣❤ ❢r♦♠ ❛❜♦✈❡ ♦r ❢r♦♠ ❜❡❧♦✇✿ • ❆ ✇❛r♠❡r ♦❜❥❡❝t ✐s ❝♦♦❧✐♥❣✿ T − T0 > 0 ❛♥❞ T ց ✳ • ❆ ❝♦♦❧❡r ♦❜❥❡❝t ✐s ✇❛r♠✐♥❣✿ T − T0 < 0 ❛♥❞ T ր ✳

❊①❛♠♣❧❡ ✷✳✶✵✳✶✺✿ ♣❧♦ts ❢r♦♠ ❧✐♠✐ts

❙✉♣♣♦s❡ ✇❡ ❛r❡ ❢❛❝✐♥❣ t❤❡ ♦♣♣♦s✐t❡ ✭✐♥✈❡rs❡✦✮ ♣r♦❜❧❡♠✿ ❙✉♣♣♦s❡ ✇❡ ❦♥♦✇ t❤❡ ❧✐♠✐ts ❛♥❞ ♥♦✇ ✇❡ ♥❡❡❞ t♦ ♣❧♦t t❤❡ ❛s②♠♣t♦t❡s ❛♥❞ ❛ ♣♦ss✐❜❧❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥✳

✷✳✶✵✳

✶✾✸

▲❛r❣❡✲s❝❛❧❡ ❜❡❤❛✈✐♦r ❛♥❞ ❛s②♠♣t♦t❡s

❙✉♣♣♦s❡ t❤✐s ✐s ✇❤❛t ✇❡ ❦♥♦✇ ❛❜♦✉t f ✿ ■t ✐s ❞❡✜♥❡❞ ❢♦r ❛❧❧ x 6= 3✱ ❛♥❞ lim f (x) = 3, lim f (x) = +∞, lim f (x) = −2 .

x→−∞

x→3

x→+∞

▲❡t✬s r❡✇r✐t❡ t❤♦s❡✿ ✶✳ x → −∞, y → 3 ✷✳ x → 3− , y → +∞ ✸✳ x → 3+ , y → +∞ ✹✳ x → +∞, y → −2 ❲❡ ❞r❛✇ r♦✉❣❤ str♦❦❡s t♦ r❡♣r❡s❡♥t t❤❡s❡ ❢❛❝ts ✭❧❡❢t✮✿

❚❤❡ ❛♠❜✐❣✉✐t② ❛❜♦✉t ❤♦✇ t❤❡ ❣r❛♣❤ ❛♣♣r♦❛❝❤❡s t❤❡ ❛s②♠♣t♦t❡s r❡♠❛✐♥s ❛t −∞ ❛♥❞ +∞✳ ❲❡ ❝♦♥♥❡❝t t❤❡ ✐♥✐t✐❛❧ str♦❦❡s ✐♥t♦ ❛ s✐♥❣❧❡ ❣r❛♣❤❀ ✐t ❤❛s t✇♦ ❜r❛♥❝❤❡s✳ ❚✇♦ ♣♦ss✐❜❧❡ ✈❡rs✐♦♥s ♦❢ t❤❡ ❣r❛♣❤ ♦❢ f ❛r❡ s❤♦✇♥ ✭♠✐❞❞❧❡ ❛♥❞ r✐❣❤t✮✳ ❊①❛♠♣❧❡ ✷✳✶✵✳✶✻✿ t❤r❡❡ ♣❛tt❡r♥s

❆ ❤♦r✐③♦♥t❛❧ ❛s②♠♣t♦t❡ ♠❛② ❜❡ ❛♣♣r♦❛❝❤❡❞ ❜② t❤❡ ❣r❛♣❤ ✐s t❤❡s❡ t❤r❡❡ ♠❛✐♥ ✇❛②s✿

❚❤❡② ❛r❡ ❡①❡♠♣❧✐✜❡❞ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ❧✐♠✐ts✿ lim

x→∞



1 1+ x



= 1,

lim

x→+∞



1 2− x



= 2,

lim

x→+∞



1 cos x x

❊①❡r❝✐s❡ ✷✳✶✵✳✶✼

❙✉❣❣❡st ❛ ❢✉♥❝t✐♦♥ t❤❛t ♦s❝✐❧❧❛t❡s ❧✐❦❡ t❤❡ ❧❛st ♦♥❡ ❜✉t ❞♦❡s♥✬t ✢❛tt❡♥ ♦✉t✳ ❊①❡r❝✐s❡ ✷✳✶✵✳✶✽

❯s❡ ❧✐♠✐ts t♦ ❞❡s❝r✐❜❡ t❤❡ ❧❛r❣❡✲s❝❛❧❡ ❜❡❤❛✈✐♦r ♦❢ t❤❡ ❢✉♥❝t✐♦♥ s❤♦✇♥ ❜❡❧♦✇✿



= 0.

✷✳✶✶✳ ▲✐♠✐ts ❛♥❞ ✐♥✜♥✐t②✿ ❝♦♠♣✉t❛t✐♦♥s

✶✾✹

✷✳✶✶✳ ▲✐♠✐ts ❛♥❞ ✐♥✜♥✐t②✿ ❝♦♠♣✉t❛t✐♦♥s ▲❡t✬s s✉♠♠❛r✐③❡ t❤❡ ♣♦ss✐❜❧❡ ♦✉t❝♦♠❡s ♦❢ ♦✉r ❝♦♠♣✉t❛t✐♦♥s ♦❢ ❛ ❧✐♠✐t ✭❛t ❛ ♣♦✐♥t✱ ♦♥❡✲s✐❞❡❞ ♦r t✇♦✲s✐❞❡❞✱ ♦r ❛t ✐♥✜♥✐t②✮✿  L    ±∞ lim =  ♥♦ ❧✐♠✐t    0/0, ∞/∞, ∞ − ∞

→ → → →

■t✬s ❛ ♥✉♠❜❡r✳ ❨♦✉ ❝❛♥ ❞♦ ❛❧❣❡❜r❛ ✇✐t❤ t❤❡ ❧✐♠✐t✳ ❨♦✉ ❝❛♥ ❞♦ s♦♠❡ ❛❧❣❡❜r❛✿ ∞ + ∞ = ∞✱ ❜✉t ♥♦t ❛❧❧✿ ∞ − ∞ = ? ❉♦ ♥♦ ❛❧❣❡❜r❛ ✇✐t❤ t❤❡ ❧✐♠✐t✳ ■t✬s ✐♥❞❡t❡r♠✐♥❛t❡✳ ❙t❛rt ♦✈❡r✦

❊①❛♠♣❧❡ ✷✳✶✶✳✶✿ ❝♦♠♣❧❡t❡ ❛♥❛❧②s✐s

▲❡t✬s ❢✉❧❧② ✐♥✈❡st✐❣❛t❡ t❤✐s ❢✉♥❝t✐♦♥✿  2 x − x f (x) = x2 − 1  1

✐❢ x 6= 1 ,

✐❢ x = 1 .

❚❤❡ ♣♦✐♥t ✇❤❡r❡ t❤❡ t✇♦ ❢♦r♠✉❧❛s ♠✐❣❤t ❝♦♥✢✐❝t ✐s x = 1✳ ❚❤❡ ❢✉♥❝t✐♦♥ ✐s ❞❡✜♥❡❞✱ ❜✉t ❞♦❡s t❤❡ ❧✐♠✐t ❡①✐st❄ ▲❡t✬s ❝♦♠♣✉t❡✿ x2 − x . x→1 x2 − 1

lim f (x) = lim

x→1

P❧✉❣ ✐♥ x = 1 ✭♠✐st❛❦❡✦✮✿

0 11 − 1 = . 12 − 1 0

❉❊❆❉ ❊◆❉

■♥❞❡t❡r♠✐♥❛t❡✦ ❲❡ ❝♦♥❝❧✉❞❡ t❤❛t ✇❡ ♥❡❡❞ t♦ ❞♦ ❛❧❣❡❜r❛ ✇✐t❤ t❤❡ ❢✉♥❝t✐♦♥ ❜❡❢♦r❡ t✉r♥✐♥❣ ❜❛❝❦ t♦ t❤❡ ❧✐♠✐t✳ ❍❡r❡ ✐s t❤❡ ❛❧❣❡❜r❛✿

x2 − x x(x − 1) = lim 2 x→1 x − 1 x→1 (x − 1)(x + 1) x = lim x→1 x + 1 1 1 = = . 1+1 2 lim

x ✐s r❛t✐♦♥❛❧ ❛♥❞✱ x+1 1 t❤❡r❡❢♦r❡✱ ❝♦♥t✐♥✉♦✉s ❛t ❛♥② ♣♦✐♥t ♦❢ ✐ts ❞♦♠❛✐♥✳ ◆♦✇✱ ❛r❡ t❤❡ t✇♦ ♦♥❡✲s✐❞❡❞ ❧✐♠✐ts ❡q✉❛❧❄ ◆♦✿ 1 6= ✳ 2

❚❤❡ ❧❛st st❡♣ ✭♣❧✉❣❣✐♥❣ ✐♥ x = 1✮ ✐s ❥✉st✐✜❡❞ ❜② t❤❡ ❢❛❝t t❤❛t t❤❡ ❢✉♥❝t✐♦♥ ❚❤❛t✬s ✇❤② t❤❡ ❢✉♥❝t✐♦♥ ❤❛s ❛ ❞✐s❝♦♥t✐♥✉✐t② ❛t t❤✐s ♣♦✐♥t ✭❛ r❡♠♦✈❛❜❧❡ ❦✐♥❞✮✳ ❲❡ ❝❛♥ s✐♠♣❧✐❢② t❤❡ ❢♦r♠✉❧❛✿

( x f (x) = x + 1 1

✐❢ x 6= 1 ,

✐❢ x = 1 .

❲❡ ✜♥❞ t❤❡ ❞♦♠❛✐♥ ❜② s♦❧✈✐♥❣✿ x + 1 6= 0 ❛♥❞ x 6= 1✳ ❚❤❡ ❞♦♠❛✐♥ ✐s (−∞, −1) ∪ (−1, ∞) ✳ ❈♦♥s✐❞❡r

x ❛t x = −1✿ x+1

x = ∞. x→−1 x + 1 lim

✷✳✶✶✳ ▲✐♠✐ts ❛♥❞ ✐♥✜♥✐t②✿ ❝♦♠♣✉t❛t✐♦♥s

✶✾✺

❚❤✉s✱ x = −1 ✐s ❛ ✈❡rt✐❝❛❧ ❛s②♠♣t♦t❡✳ ❍♦✇❡✈❡r✱ t❤❡r❡ ✐s st✐❧❧ t❤❡ ✐ss✉❡ ♦❢ ✐♥ ✇❤✐❝❤ ♦❢ t❤❡ ❢♦✉r ❞✐✛❡r❡♥t ✇❛②s t❤❡ ❣r❛♣❤ ❛♣♣r♦❛❝❤❡s t❤❡ ❛s②♠♣t♦t❡✳ ❚♦ t❡❧❧ ✇❤✐❝❤ ♦♥❡✱ ❝♦♠♣✉t❡ t❤❡ ♦♥❡✲s✐❞❡❞ ❧✐♠✐ts ✇❤✐❧❡ ✇❛t❝❤✐♥❣ t❤❡ s✐❣♥s ♦❢ t❤❡ ♥✉♠❡r❛t♦r ❛♥❞ t❤❡ ❞❡♥♦♠✐♥❛t♦r✿   x = x→−1 x + 1  x = lim + x→−1 x + 1 lim −

◆❡①t✱ t❤❡ ❢✉♥❝t✐♦♥✬s ❜❡❤❛✈✐♦r ❛t ∞✿

− =+ = +∞, −  − =− = −∞ . +

x = 1. x→∞ x + 1

lim f (x) = lim

x→∞

❚❤✉s✱ y = 1 ✐s ❛ ❤♦r✐③♦♥t❛❧ ❛s②♠♣t♦t❡✳ ❋✐♥❛❧❧②✱ ✇❡ ❝♦♥✜r♠ t❤❡ ♣❧♦t✿

❏✉st ❛s ❜❡❢♦r❡✱ ❡❛❝❤ ♣r♦♣❡rt② ♦❢ ❧✐♠✐ts ✐s ♠❛t❝❤❡❞ ❜② ✐ts ❛♥❛❧♦❣ ❢♦r ❧✐♠✐ts ❛t ✐♥✜♥✐t②✿

❚❤❡♦r❡♠ ✷✳✶✶✳✷✿ ❆❧❣❡❜r❛ ♦❢ ▲✐♠✐ts ♦❢ ❋✉♥❝t✐♦♥s ❛t ■♥✜♥✐t② ❙✉♣♣♦s❡

f (x) → F

❛♥❞

g(x) → G

f (x) + g(x) → F + G f (x) · g(x) → F G

❙❘✿ P❘✿

❛s

x → +∞

❈▼❘✿ ◗❘✿

✭♦r

−∞✮✳

❚❤❡♥

c · f (x) → cF f (x)/g(x) → F/G

❢♦r ❛♥② r❡❛❧ ✇❤❡♥

c

G 6= 0

❚❤❡♦r❡♠ ✷✳✶✶✳✸✿ ❙✉❜st✐t✉t✐♦♥ ❘✉❧❡ ❢♦r ▲✐♠✐ts ❙✉♣♣♦s❡

f

❛♥❞

g

❛r❡ t✇♦ ❢✉♥❝t✐♦♥s✳ ❚❤❡♥✱ ✐❢

f

✐s ❝♦♥t✐♥✉♦✉s ❛t

t❤❡♥ ✇❡ ❤❛✈❡✿

lim f (g(x)) = f

x→∞



lim g(x)

x→∞



❊①❛♠♣❧❡ ✷✳✶✶✳✹✿ s✉❜st✐t✉t✐♦♥ ❈♦♠♣✉t❡✿ lim

x→−∞



1 e − +3 x x



❙❘ ===

1 + lim 3 x→−∞ x→−∞ x

lim ex − lim

x→−∞

= 0−0+3 = 3.

L = lim g(x)✱ x→∞

✷✳✶✶✳ ▲✐♠✐ts ❛♥❞ ✐♥✜♥✐t②✿ ❝♦♠♣✉t❛t✐♦♥s

✶✾✻

❊①❛♠♣❧❡ ✷✳✶✶✳✺✿ ❤♦r✐③♦♥t❛❧ ❛s②♠♣t♦t❡

❈♦♠♣✉t❡✿

x . x→∞ x + 1 lim

P❧✉❣❣✐♥❣ ✐♥ x = ∞ ✭♠✐st❛❦❡✦✮✱ ❣✐✈❡s ✉s ∞ ∞

❉❊❆❉ ❊◆❉

❋❛❝✐♥❣ ❛♥ ✐♥❞❡t❡r♠✐♥❛t❡ ❡①♣r❡ss✐♦♥✱ ✇❡ ❛r❡ s✉♣♣♦s❡❞ t♦ ❞♦ ❛❧❣❡❜r❛ ✐♥st❡❛❞✳ ❲❤❛t ❛r❡ ✇❡ tr②✐♥❣ t♦ ❛❝❝♦♠♣❧✐s❤❄ ❲❡ ✇❛♥t t♦ ❣❡t r✐❞ ♦❢ t❤❡ ✐♥✜♥✐t✐❡s ✐♥ t❤❡ ♥✉♠❡r❛t♦r ❛♥❞ t❤❡ ❞❡♥♦♠✐♥❛t♦r✳ ❍♦✇ ❛❜♦✉t ✇❡ ❞✐✈✐❞❡ ❜♦t❤ ❜② x❄ x x→∞ x + 1

x x x→∞ (x+1) x

=

lim

lim

1 x→∞ 1 + 1 x limx→∞ 1 ◗❘
=== limx→∞ 1 + x1 1 ❙❘ === = 1. 1+0 =

lim

■t ✇♦r❦❡❞✦ ❚❤❡ ❝♦♥❝❧✉s✐♦♥ ✐s t❤❛t y = 1 ✐s ❛ ❤♦r✐③♦♥t❛❧ ❛s②♠♣t♦t❡✳ ❊①❛♠♣❧❡ ✷✳✶✶✳✻✿ ❛♥♦t❤❡r ❤♦r✐③♦♥t❛❧ ❛s②♠♣t♦t❡

❊✈❛❧✉❛t❡ t❤❡ ❧✐♠✐t ♦❢ t❤✐s ❢✉♥❝t✐♦♥ ❛t ∞✿

x2 . x2 + 1

❙✉❜st✐t✉t✐♥❣ ✭♠✐st❛❦❡✦✮ ❧❡❛❞s t♦

∞ ❉❊❆❉ ❊◆❉ ∞ ✇❤✐❝❤ ✐s ✐♥❞❡t❡r♠✐♥❛t❡✳ ▲❡t✬s ❞✐✈✐❞❡ ❜② x ❛s ❧❛st t✐♠❡✿ x2 = x2 + 1 =

x2 x (x2 +1) x

x x+

1 x

→

∞ ∞

❉❊❆❉ ❊◆❉

■t✬s st✐❧❧ ✐♥❞❡t❡r♠✐♥❛t❡✦ ❲❡ ❤❛✈❡♥✬t ❡❧✐♠✐♥❛t❡❞ t❤❡ ✐♥✜♥✐t✐❡s✳ ▲❡t✬s ❞✐✈✐❞❡ ❜② x ❛❣❛✐♥✱ x x+

1 x

=

x x (x+ x1 ) x

1 1 + x12 1 → 1+0 =

= 1.

❇❡tt❡r ✐❞❡❛✿ ❉✐✈✐❞❡ ❜② x2 ✐♥ t❤❡ ✜rst ♣❧❛❝❡✳ ❍♦✇ ❞♦ ✇❡ ❡✈❛❧✉❛t❡ t❤❡ ❧✐♠✐t ❛t ∞❄ ❚❤❡ s❛♠❡ ✇❛② ✇❡ ❞✐❞ ✐t ❢♦r s❡q✉❡♥❝❡s ✐♥ t❤❡ ❧❛st ❝❤❛♣t❡r✳ ❋♦r ❛ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥✱ t❤❡ ❧❡❛❞✐♥❣ t❡r♠s ♦❢ t❤❡ t✇♦ ♣♦❧②♥♦♠✐❛❧s ❞❡t❡r♠✐♥❡ t❤❡ ❧♦♥❣✲t❡r♠ ❜❡❤❛✈✐♦r ♦❢ t❤❡ ♥✉♠❡r❛t♦r ❛♥❞ ❞❡♥♦♠✐♥❛t♦r ❛♥❞✱ t❤❡r❡❢♦r❡✱ ♦❢ t❤❡ ✇❤♦❧❡ ❢r❛❝t✐♦♥✳

✷✳✶✶✳ ▲✐♠✐ts ❛♥❞ ✐♥✜♥✐t②✿ ❝♦♠♣✉t❛t✐♦♥s

✶✾✼

❊①❛♠♣❧❡ ✷✳✶✶✳✼✿ ❧❡❛❞✐♥❣ t❡r♠s ❛♥❛❧②s✐s

❈♦♥s✐❞❡r✿

❧♦♥❣✲t❡r♠✦

lim

x→∞

♣❛r❛❜♦❧❛✦

❚❤✐s ✐s ✇❤❛t ❞❡t❡r♠✐♥❡s t❤❡ ❧♦♥❣ t❡r♠✿

s❤♦rt✲t❡r♠✳✳✳

}| { z}|{ z ∞ 3x3 − 2x2 + x − 8 → . 2 ∞ − 17x + 5 2x |{z} | {z } ✇❤❡r❡ ✐t ✐s✳✳✳

3 3x2 = x → ∞. 2 2x 2

❚♦ ♠❛❦❡ t❤✐s ✈✐s✐❜❧❡✱ ✇❡ ❞✐✈✐❞❡ ♥✉♠❡r❛t♦r ❛♥❞ ❞❡♥♦♠✐♥❛t♦r ❜② x2 ✳ (3x3 −2x2 +x−8) x2 lim (2x2 −17x+5) x→∞ x2

3x − 2 + x1 − x82 = lim x→∞ + x52 2 − 17 x ∞−2+0−0 = 2−0+0 = ∞.

• ❚❤❡ ❧❡ss♦♥✿ ❚♦ ❣❡t r✐❞ ♦❢ t❤❡ ✐♥❞❡t❡r♠✐♥❛❝②✱ ❣❡t r✐❞ ♦❢ ♦♥❡ ♦❢ t❤❡ ✐♥✜♥✐t✐❡s✳

• ❚❤❡ ♣❧❛♥✿ ❉✐✈✐❞❡ ❜♦t❤ ♣❛rts ♦❢ t❤❡ ❢r❛❝t✐♦♥ ❜② x t♦ t❤❡ ❞❡❣r❡❡ ♦❢ t❤❡ ❞❡♥♦♠✐♥❛t♦r✳

❊①❛♠♣❧❡ ✷✳✶✶✳✽✿ r❛t✐♦♥❛❧

❆♥❛❧②③❡✿ f (x) =

❆t ∞✱ ✇❡ ❤❛✈❡✿

x3 − x . x2 − 6x + 5 x3 −x x2 x2 −6x+5 x2 x − x1 lim x→∞ 1 − 6 + 52 x x

x3 − x lim 2 = lim x→∞ x − 6x + 5 x→∞ =

∞−0 1−0−0 = ∞. =

■t✬s ✐♥✜♥✐t❡✱ ②❡s✱ ❜✉t ✇❤✐❝❤ ✐♥✜♥✐t②❄ ❲❡ ♥❡❡❞ t♦ ✜♥❞ t❤❡ s✐❣♥✳ ❲❡ ❥✉st ✜♥❞ t❤❡ s✐❣♥s ♦❢ t❤❡ ❧❡❛❞✐♥❣ t❡r♠s ♦❢ t❤❡ ♥✉♠❡r❛t♦r ❛♥❞ ❞❡♥♦♠✐♥❛t♦r✳ lim (x3 − x) = +∞

lim (x3 − x) = −∞

x→+∞

x→−∞

x→+∞

x→−∞

lim (x2 − 6x + 5) = +∞

lim (x2 − 6x + 5) = +∞

❈♦♥❝❧✉s✐♦♥✿ ♥♦ ❤♦r✐③♦♥t❛❧ ❛s②♠♣t♦t❡s✳ ◆♦✇ t❤❡ ✈❡rt✐❝❛❧ ❛s②♠♣t♦t❡s✳ ❲❡ ❧♦♦❦ ❛t x3 − x x2 − 6x + 5

❛♥❞ s❡❛r❝❤ ❢♦r 0 ♦❢ t❤❡ ❞❡♥♦♠✐♥❛t♦r✳ ▲❡t✬s ❢❛❝t♦r ❜♦t❤✿

x3 − x = x(x2 − 1) = x(x − 1)(x + 1) x2 − 6x + 5 = (x − 1)(x − 5)

✷✳✶✶✳ ▲✐♠✐ts ❛♥❞ ✐♥✜♥✐t②✿ ❝♦♠♣✉t❛t✐♦♥s

✶✾✽

❲❡ ✉s❡ t❤❡ q✉❛❞r❛t✐❝ ❢♦r♠✉❧❛✿ √ 6 ± 36 · 20 62 − 4 · 5 = x = 2 2 6±4 = = 5, 1 . 2 6±

◆♦✇✱ ✇❡ ❤❛✈❡✿

√

x(x − 1)(x + 1) x(x + 1) = (x − 5)(x − 1) x−5

f (x) =

❢♦r x 6= 1 ✭t❤❛t✬s t❤❡ ❞♦♠❛✐♥✦✮✳ ◆❡①t✿

lim f (x) =

x→1

❜❡❝❛✉s❡

1 1(1 + 1) =− , 1−5 2

x(x + 1) ✐s ❝♦♥t✐♥✉♦✉s ❛t 1✳ ❙♦✱ x = 1 ✐s ♥♦t ❛ ✈❡rt✐❝❛❧ ❛s②♠♣t♦t❡✳ ◆❡①t✿ x−5 lim−

x→5

? x(x + 1) 30 = = ±∞ . x−5 0

❚❤❡r❡❢♦r❡✱ x = 5 ✐s ❛ ✈❡rt✐❝❛❧ ❛s②♠♣t♦t❡✳ ❇✉t ✇❤✐❝❤ ✐♥✜♥✐t②❄ ❲❡ ♥❡❡❞ t♦ ✜♥❞ t❤❡ s✐❣♥✿ +·+ =− −

❲❡ ❝♦♥❝❧✉❞❡✿ lim−

x→5

❙✐♠✐❧❛r❧②✱ ✇❡ ✜♥❞✿

x(x + 1) = −∞ . x−5

lim f (x) = +∞ .

x→5+

◆♦✇ s♦♠❡ ❢❛❝ts ❛❜♦✉t ❧✐♠✐ts ♦❢ s❡q✉❡♥❝❡s ❝❛♥ ❜❡ r❡st❛t❡❞ ❢♦r ❢✉♥❝t✐♦♥s✳ ❋✐rst✱ r❡❝❛❧❧ t❤❡ ❧❡ss♦♥ ❛❜♦✉t t❤❡ ❜❡❤❛✈✐♦r ♦❢ ♣♦❧②♥♦♠✐❛❧s ❛t ∞✿ ◮ ❖♥❧② t❤❡ ❧❡❛❞✐♥❣ t❡r♠ ♠❛tt❡rs✳

❚❤❡ ❝♦♠♣❧❡t❡ st❛t❡♠❡♥t ✐s ❛s ❢♦❧❧♦✇s✿

❚❤❡♦r❡♠ ✷✳✶✶✳✾✿ ▲✐♠✐ts ♦❢ P♦❧②♥♦♠✐❛❧s ❛t ■♥✜♥✐t②

❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ♣♦❧②♥♦♠✐❛❧ ♦❢ ❞❡❣r❡❡ p ✇✐t❤ t❤❡ ❧❡❛❞✐♥❣ ❝♦❡✣❝✐❡♥t ap 6= 0✳ ❚❤❡♥ ✐ts ❧✐♠✐t ✐s ❛s ❢♦❧❧♦✇s✿ lim (ap xp + ap−1 xp−1 + ... + a1 x + a0 ) =

x→±∞

(

±∞ ∓∞

✐❢ ap > 0 ✐❢ ap < 0

❚❤❡♥✱ ✐❢ ✇❡ ③♦♦♠ ♦✉t ♦♥ t❤❡ ❣r❛♣❤ ♦❢ ❛ ♣♦❧②♥♦♠✐❛❧✱ ✇❡ ✇✐❧❧ s❡❡ ❥✉st ❢♦✉r ♣♦ss✐❜❧❡ ♣❛tt❡r♥s

✷✳✶✶✳

▲✐♠✐ts ❛♥❞ ✐♥✜♥✐t②✿ ❝♦♠♣✉t❛t✐♦♥s

✶✾✾

❆ ♠♦r❡ ❣❡♥❡r❛❧ r❡s✉❧t ✐s ❛❜♦✉t r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥s✿

❚❤❡♦r❡♠ ✷✳✶✶✳✶✵✿ ▲✐♠✐ts ♦❢ ❘❛t✐♦♥❛❧ ❋✉♥❝t✐♦♥s ❛t ■♥✜♥✐t②

❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥ r❡♣r❡s❡♥t❡❞ ❛s ❛ q✉♦t✐❡♥t ♦❢ t✇♦ ♣♦❧②♥♦♠✐❛❧s ♦❢ ❞❡❣r❡❡s p ❛♥❞ q✱ ✇✐t❤ t❤❡ ❧❡❛❞✐♥❣ ❝♦❡✣❝✐❡♥ts ap 6= 0, bq 6= 0✳ ❚❤❡♥ ✐ts ❧✐♠✐t ✐s ❛s ❢♦❧❧♦✇s✿   ∞   ap

ap xp + ap−1 xp−1 + ... + a1 x + a0 = lim bp x→+∞ bq xq + bq−1 xq−1 + ... + b1 x + b0    0

✐❢ p > q ✐❢ p = q ✐❢ p < q

❚❤❡ ❧❛st t✇♦ ❝❛s❡s ♣r♦❞✉❝❡ ❤♦r✐③♦♥t❛❧ ❛s②♠♣t♦t❡s✳

❊①❡r❝✐s❡ ✷✳✶✶✳✶✶ ❙t❛t❡ t❤❡ ❛❜♦✈❡ t❤❡♦r❡♠ ❢♦r

x → −∞✳

❚❤❡ ❧♦♥❣✲t❡r♠ ❜❡❤❛✈✐♦r ♦❢ ❛ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥ ✐s ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ ❧❡❛❞✐♥❣ t❡r♠s ♦❢ ✐ts ♥✉♠❡r❛t♦r ❛♥❞ ❞❡♥♦♠✐♥❛t♦r✿

ap xp ap ap xp + ap−1 xp−1 + ... + a1 x + a0 = = lim lim xp−q q q−1 q x→∞ bq x + bq−1 xx + ... + b1 x + b0 x→∞ bp x bp x→∞ lim

❚❤❡ ♣✐❝t✉r❡ ✐s ❝♦♠♣❧✐❝❛t❡❞ ✕ ✐♥ ❝♦♠♣❛r✐s♦♥ t♦ ♣♦❧②♥♦♠✐❛❧s ✕ ❜② t❤❡ ❢❛❝t t❤❛t r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥s ❛❧s♦ ❤❛✈❡ ❤♦r✐③♦♥t❛❧ ❛♥❞ ✈❡rt✐❝❛❧ ❛s②♠♣t♦t❡s✳

❊①❛♠♣❧❡ ✷✳✶✶✳✶✷✿ ♥♦t r❛t✐♦♥❛❧ ❙♦♠❡t✐♠❡s ✇❡ ❢❛❝❡ ✐♥❞❡t❡r♠✐♥❛t❡ ❡①♣r❡ss✐♦♥s ✇✐t❤ ❢✉♥❝t✐♦♥s ♦t❤❡r t❤❛♥ r❛t✐♦♥❛❧✳ ❊✈❛❧✉❛t❡✿

lim

x→∞ ❙✉❜st✐t✉t✐♥❣ ✭♠✐st❛❦❡✦✮ ❣✐✈❡s ✉s✿

√

x2



+1−x .

❉❊❆❉ ❊◆❉

∞−∞

❚❤✐s ✐s ✐♥❞❡t❡r♠✐♥❛t❡✳ ❚❤❡ tr✐❝❦ ✐s t♦ ♠✉❧t✐♣❧② ❛♥❞ ❞✐✈✐❞❡ ❜② t❤❡ ✏❝♦♥❥✉❣❛t❡✑ ♦❢ t❤✐s ❡①♣r❡ss✐♦♥✿

√

x2 + 1 + x .

✷✳✶✷✳

❈♦♥t✐♥✉✐t② ❛♥❞ ❛❝❝✉r❛❝②

✷✵✵

❚❤✐s ✐s ❤♦✇ ✐t ✇♦r❦s✿ lim

√

x→∞


√
x2 + 1 − x x2 + 1 + x (x2 + 1) − x2 √ = lim √ x→∞ x2 + 1 + x x2 + 1 + x 1 = lim √ 2 x→∞ x +1+x 1 = ∞ = 0.

❊①❡r❝✐s❡ ✷✳✶✶✳✶✸

❊✈❛❧✉❛t❡ t❤❡ ❧✐♠✐t✿ lim ln x→∞



 x ✳ x+1

❊①❡r❝✐s❡ ✷✳✶✶✳✶✹

●✐✈❡ ❛♥ ❡①❛♠♣❧❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ❢♦r ❡❛❝❤ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣❛tt❡r♥s ❛s x → ∞✿ ✭❛✮ f (x) → −1✱ ✭❜✮ f (x) → .33✱ ✭❜✮ f (x) → +∞✱ ✭❝✮ f (x) ❞✐✈❡r❣❡s ❜✉t ♥♦t t♦ ✐♥✜♥✐t②✳ ■♥ ❝♦♥❝❧✉s✐♦♥✱ t❤✐s ✐s ✇❤❛t t❤❡ ❣r❛♣❤s ♦❢ s♦♠❡ ❢❛♠✐❧✐❛r ❢✉♥❝t✐♦♥s ❧♦♦❦ ❧✐❦❡ ❛t ❛♥ ❡①tr❡♠❡ ❞✐st❛♥❝❡✿

❊①❡r❝✐s❡ ✷✳✶✶✳✶✺

P❧♦t ✐♥ t❤✐s ♠❛♥♥❡r t❤❡ ❣r❛♣❤s ♦❢ t❤❡ ❢✉♥❝t✐♦♥s ✐♥ t❤✐s s❡❝t✐♦♥✳

✷✳✶✷✳ ❈♦♥t✐♥✉✐t② ❛♥❞ ❛❝❝✉r❛❝②

❚❤❡ ✐❞❡❛ ♦❢ ❝♦♥t✐♥✉✐t② ❝❛♥ ❜❡ ✐♥tr♦❞✉❝❡❞ ❛♥❞ ❥✉st✐✜❡❞ ❜② ❝♦♥s✐❞❡r✐♥❣ t❤❡ ❛❝❝✉r❛❝② ♦❢ ❛ ♠❡❛s✉r❡♠❡♥t✳ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ❝♦❧❧❡❝t✐♦♥ ♦❢ sq✉❛r❡ t✐❧❡s ♦❢ ✈❛r✐♦✉s s✐③❡s ❛♥❞ ✇❡ ♥❡❡❞ t♦ ✜♥❞ t❤❡ ❛r❡❛ A ♦❢ ❡❛❝❤ ♦❢ t❤❡♠ ✐♥ ♦r❞❡r t♦ ❦♥♦✇ ❤♦✇ ♠❛♥② ✇❡ ♥❡❡❞ t♦ ❝♦✈❡r t❤❡ ✇❤♦❧❡ ✢♦♦r✳

✷✳✶✷✳

❈♦♥t✐♥✉✐t② ❛♥❞ ❛❝❝✉r❛❝②

✷✵✶

❚❤❡ ❛♥s✇❡r ✐s✱ ♦❢ ❝♦✉rs❡✱ t♦ ♠❡❛s✉r❡ t❤❡ s✐❞❡✱ x✱ ♦❢ ❡❛❝❤ t✐❧❡ ❛♥❞ t❤❡♥ ❝♦♠♣✉t❡✿

A = x2 . ❋♦r ❡①❛♠♣❧❡✱ ✇❡ ❤❛✈❡✿

x = 10 =⇒ A = 100 . ❇✉t ♠❡❛s✉r❡♠❡♥ts ❛r❡ ♥❡✈❡r ❢✉❧❧② ❛❝❝✉r❛t❡❀ ❢♦r ❡①❛♠♣❧❡✱ ✇❡ ♠✐❣❤t ❤❛✈❡✿ ■♥ ♦t❤❡r ✇♦r❞s✱ x ❧✐❡s ✇✐t❤✐♥ ❛ ❝❡rt❛✐♥ ✐♥t❡r✈❛❧✿

x = 10 ± .3 .

x ✐♥ (10 − .3, 10 + .3) = (9.7, 10.3) .

❆s ❛ r❡s✉❧t✱ t❤❡ ❝♦♠♣✉t❡❞ ✈❛❧✉❡ ♦❢ t❤❡ ❛r❡❛ ♦❢ t❤❡ t✐❧❡ ✕ ✇❤❛t ✇❡ ❝❛r❡ ❛❜♦✉t ✕ ✇✐❧❧ ❛❧s♦ ❤❛✈❡ s♦♠❡ ❡rr♦r✦ ■♥❞❡❡❞✱ t❤❡ ❛r❡❛ ✇♦♥✬t ❜❡ ❥✉st A = 100 ❜✉t

A = (10 ± .3)2 .

■♥ ♦t❤❡r ✇♦r❞s✱ A ❧✐❡s ✇✐t❤✐♥ ❛ ❝❡rt❛✐♥ ✐♥t❡r✈❛❧✿

A ✐s ✐♥ ((10 − .3)2 , (10 + .3)2 ) = (9.72 , 10.32 ) = (94.09, 106.09) . ❚❤✐s ✐s ✐♥❢♦r♠❛t✐✈❡✱ ❜✉t ♦✉r ✐♥✈❡st✐❣❛t✐♦♥ ❣♦❡s ✐♥ t❤❡ ♦♣♣♦s✐t❡ ❞✐r❡❝t✐♦♥✳ ❙✉♣♣♦s❡ ♥❡①t t❤❛t ✇❡ ❝❛♥ ❛❧✇❛②s ✐♠♣r♦✈❡ t❤❡ ❛❝❝✉r❛❝② ♦❢ t❤❡ ♠❡❛s✉r❡♠❡♥t ♦❢ t❤❡ s✐❞❡ ♦❢ t❤❡ t✐❧❡ x ✕ ❛s ♠✉❝❤ ❛s ✇❡ ❧✐❦❡✳ ❚❤❡ q✉❡st✐♦♥ ✐s✱ ❝❛♥ ✇❡ ❛❧s♦ ✐♠♣r♦✈❡ t❤❡ ❛❝❝✉r❛❝② ♦❢ t❤❡ ❝♦♠♣✉t❡❞ ✈❛❧✉❡ ♦❢ A✳ ❈❛♥ ✇❡ ❛❝❤✐❡✈❡ t❤✐s ❛❝❝✉r❛❝② t♦ ❛♥②❜♦❞②✬s s❛t✐s❢❛❝t✐♦♥✱ ❡✈❡♥ ✐❢ t❤✐s st❛♥❞❛r❞ ♦❢ ❛❝❝✉r❛❝② ♠✐❣❤t ❝❤❛♥❣❡❄ ❙✉♣♣♦s❡ ❛❣❛✐♥ t❤❛t x = 10✳ ■❢ t❤❡ ❞❡s✐r❡❞ ❛❝❝✉r❛❝② ♦❢ A ✐s ±3✱ ✇❡

r❡q✉✐r❡ ✿

A ✐s ✐♥ (100 − 3, 100 + 3) = (97, 103) .

❇✉t t❤✐s ✐♥t❡r✈❛❧ ❞♦❡s♥✬t ❝♦♥t❛✐♥ t❤❡ ✐♥t❡r✈❛❧ ✇❡ ❢♦✉♥❞ t♦ ❣✉❛r❛♥t❡❡ t♦ ❝♦♥t❛✐♥ A✿ (94.09, 106.09)✦ ❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡♥✬t ❛❝❤✐❡✈❡❞ t❤❡ ❞❡s✐r❡❞ ❛❝❝✉r❛❝② ♦❢ A ✇✐t❤ t❤❡ ❣✐✈❡♥ ❛❝❝✉r❛❝② ♦❢ ♠❡❛s✉r❡♠❡♥t x✱ ✇❤✐❝❤ ✐s ±.3 ✳ ■❢ t❤❡ ❞❡s✐r❡❞ ❛❝❝✉r❛❝② ♦❢ A ✐s ±7✱ ✇❡ ❤❛✈❡✿ ■t ✐s s❛t✐s✜❡❞✦

A ✐s ✐♥ (94.09, 106.09) ⊂ (100 − 7, 100 + 7) = (93, 107) .

❲❡ ❝❛♥ ❡❛s✐❧② s❤♦✇ t❤❛t t❤❡ ❡rr♦r ±.1 ❢♦r x ♣r♦✈✐❞❡s t❤❡ ♦r✐❣✐♥❛❧✱ s♠❛❧❧❡r t❤r❡s❤♦❧❞ ❢♦r A✿
A ✐s ✐♥ (10 − .1)2 , (10 + .1)2 = (9.92 , 10.12 ) = (98.01, 102.01) ⊂ (100 − 3, 100 + 3) = (97, 103) . ▲❡t✬s r❡♣❤r❛s❡ t❤✐s ♣r♦❜❧❡♠ ✐♥ ♦r❞❡r t♦ s♦❧✈❡ ✐t ❢♦r ❛❧❧ ♣♦ss✐❜❧❡ ✈❛❧✉❡s ♦❢ t❤❡ ❞❡s✐r❡❞ ❛❝❝✉r❛❝② ♦❢ A✳

▲❡t✬s ❛ss✉♠❡ t❤❛t t❤❡ ♠❡❛s✉r❡♠❡♥t ♦❢ t❤❡ s✐❞❡ ✐s a ❛♥❞✱ t❤❡r❡❢♦r❡✱ t❤❡ ❛ss✉♠❡❞ ❛r❡❛ A ✐s a2 ✳ ◆♦✇ s✉♣♣♦s❡ ✇❡ ✇❛♥t t❤❡ ❞❡✈✐❛t✐♦♥ ♦❢ A ❢r♦♠ t❤❡ tr✉❡ ✭✉♥❦♥♦✇♥✮ ❛r❡❛ t♦ ❜❡ s♦♠❡ s♠❛❧❧ ✈❛❧✉❡ ε > 0 ♦r ❜❡tt❡r✳ ❲❤❛t t❤r❡s❤♦❧❞ δ ❢♦r t❤❡ ❞❡✈✐❛t✐♦♥ ♦❢ a ❢r♦♠ x ❞♦ ✇❡ ♥❡❡❞ t♦ ❜❡ ❛❜❧❡ t♦ ❣✉❛r❛♥t❡❡ t❤❛t❄ ❲❡ ✇❛♥t t♦ ❡♥s✉r❡ t❤❛t A ✐s ✇✐t❤✐♥ ε ❢r♦♠ a2 ❜② ♠❛❦✐♥❣ s✉r❡ t❤❛t x ✐s ✇✐t❤✐♥ δ ❢r♦♠ a✿

✷✳✶✷✳

❈♦♥t✐♥✉✐t② ❛♥❞ ❛❝❝✉r❛❝②

❲❡ ❦♥♦✇ t❤✐s✿

✷✵✷


(a − δ)2 , (a + δ)2 .

A ✐s ✐♥

❲❡ ♥❡❡❞ t♦ ❞❡♠♦♥str❛t❡ t❤❛t✿

A ✐s ✐♥ (a2 − ε, a2 + ε) .

❚❤❡ ❧❛tt❡r ✇✐❧❧ ❢♦❧❧♦✇ ❢r♦♠ t❤❡ ❢♦r♠❡r✱ ♣r♦✈✐❞❡❞✿


(a − δ)2 , (a + δ)2 ⊂ (a2 − ε, a2 + ε) .

▼♦r❡ ❣❡♥❡r❛❧❧②✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♥❝❧✉s✐♦♥ ❢♦r ❛♥② ✐♥❝r❡❛s✐♥❣ ❢✉♥❝t✐♦♥ f ✿
f (a − δ), f (a + δ) ⊂ (f (a) − ε, f (a) + ε) .

❚❤✐s ✐♥❝❧✉s✐♦♥ ✐s ✐❧❧✉str❛t❡❞ ❜❡❧♦✇✿

❲❡ r❡♣❤r❛s❡ t❤❡ ❧❛st ✐♥❝❧✉s✐♦♥ ♦❢ ✐♥t❡r✈❛❧s✿

x ✇✐t❤✐♥ δ ❢r♦♠ ❛ =⇒ f (x) ✇✐t❤✐♥ ε ❢r♦♠ ❢✭❛✮ . ❚❤❡ ❞❡✜♥✐t✐♦♥ s✉❣❣❡st❡❞ ❜② t❤❡ ❛❜♦✈❡ ❞✐s❝✉ss✐♦♥ ✐s ❛s ❢♦❧❧♦✇s✿

❉❡✜♥✐t✐♦♥ ✷✳✶✷✳✶✿ ε✲δ ❞❡✜♥✐t✐♦♥ ♦❢ ❝♦♥t✐♥✉✐t② ❆ ❢✉♥❝t✐♦♥ f ✐s ❝❛❧❧❡❞ t❤❛t

❝♦♥t✐♥✉♦✉s

❛t x = a ✐❢ ❢♦r ❛♥② ε > 0 t❤❡r❡ ✐s δ > 0 s✉❝❤

|x − a| < δ =⇒ |f (x) − f (a)| < ε . ■❢ ✇❡ ❝❛♥ ❞❡♠♦♥str❛t❡ t❤❛t t❤✐s ♥❡✇ ❞❡✜♥✐t✐♦♥ ✐s ❡q✉✐✈❛❧❡♥t t♦ t❤❡ ♦♥❡ ♣r❡s❡♥t❡❞ ✐♥ t❤✐s ❝❤❛♣t❡r✱ ✇❡ ❝❛♥ ❛♥s✇❡r ♦✉r q✉❡st✐♦♥ ✐♥ t❤❡ ❛✣r♠❛t✐✈❡✿

◮ ❨❡s✱ ✇❡ ❝❛♥ ❛❧✇❛②s ✐♠♣r♦✈❡ t❤❡ ❛❝❝✉r❛❝② ♦❢ t❤❡ ❝♦♠♣✉t❡❞ ✈❛❧✉❡ ♦❢ A = x2 ✕ t♦ ❛♥②❜♦❞②✬s s❛t✐s❢❛❝t✐♦♥ ✕ ❜② ✐♠♣r♦✈✐♥❣ t❤❡ ❛❝❝✉r❛❝② ♦❢ t❤❡ ♠❡❛s✉r❡♠❡♥t ♦❢ x✳

❊①❡r❝✐s❡ ✷✳✶✷✳✷ Pr♦✈❡ t❤❛t f (x) = x2 ✐s ❝♦♥t✐♥✉♦✉s ❛t x = 0, x = 1, x = a✳

✷✳✶✸✳ ❚❤❡ ε✲δ ❞❡✜♥✐t✐♦♥ ♦❢ ❧✐♠✐t

✷✵✸

❊①❡r❝✐s❡ ✷✳✶✷✳✸

❈❛rr② ♦✉t t❤✐s ❦✐♥❞ ♦❢ ❛♥❛❧②s✐s ❢♦r✿ ❛ t❤❡r♠♦♠❡t❡r ♣✉t ✐♥ ❛ ❝✉♣ ♦❢ ❝♦✛❡❡ t♦ ✜♥❞ ✐ts t❡♠♣❡r❛t✉r❡✳ ❆ss✉♠❡ t❤❛t t❤❡ t❤❡r♠♦♠❡t❡r ❣✐✈❡s ♣❡r❢❡❝t r❡❛❞✐♥❣s✳ ❍✐♥t✿ ■t✬❧❧ t❛❦❡ t✐♠❡ ❢♦r ✐t t♦ ✇❛r♠ ✉♣✳ ❊①❡r❝✐s❡ ✷✳✶✷✳✹

❲❤❛t ✐s t❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ ε ❛♥❞ δ ✇❤❡♥ f ✐s ❧✐♥❡❛r ❄ ❚♦ ❢✉rt❤❡r ✐❧❧✉str❛t❡ t❤❡ ✐❞❡❛✱ ❧❡t✬s ❝♦♥s✐❞❡r ❛ ❞✐✛❡r❡♥t s✐t✉❛t✐♦♥✳ ❙✉♣♣♦s❡ ✇❡ ❞♦♥✬t ❝❛r❡ ❛❜♦✉t t❤❡ ❛r❡❛ ❛♥②♠♦r❡❀ ✇❡ ❥✉st ✇❛♥t t♦ ✜t t❤❡s❡ t✐❧❡s ✐♥t♦ ❛ str✐♣ 10 ✐♥❝❤❡s ✇✐❞❡✳ ❲❡ ✇✐❧❧ ❜❡ t❡st✐♥❣ t❤❡ t✐❧❡s ❛♥❞ ❞❡❝✐❞❡ ❛❤❡❛❞ ♦❢ t✐♠❡ ✇❤❡t❤❡r ✐t ✇✐❧❧ ✜t✿ ■❢ ✐t ✜ts✱ ✐t ✐s ✉s❡❞❀ ♦t❤❡r✇✐s❡✱ ✐t ✐s ❞✐s❝❛r❞❡❞✳ ❚❤❡♥✱ ✇❡ st✐❧❧ ❣❡t ❛ ♠❡❛s✉r❡♠❡♥t a ♦❢ t❤❡ s✐❞❡ ♦❢ t❤❡ t✐❧❡ ❜✉t ♦✉r r❡❛❧ ✐♥t❡r❡st ✐s ✇❤❡t❤❡r a ✐s ❧❡ss ♦r ♠♦r❡ t❤❛♥ 10✳ ❏✉st ❛s ✐♥ t❤❡ ♣r❡✈✐♦✉s ❡①❛♠♣❧❡✱ ✇❡ ❞♦♥✬t ❦♥♦✇ t❤❡ ❛❝t✉❛❧ ❧❡♥❣t❤ x ❡①❛❝t❧②❀ ✐t✬s ❛❧✇❛②s ✇✐t❤✐♥ s♦♠❡ ❧✐♠✐ts✿ 10.0 ± 0.5 ♦r a ± δ ✳ ❍❡r❡ δ ✐s t❤❡ ❛❝❝✉r❛❝② ♦❢ ♠❡❛s✉r❡♠❡♥t ♦❢ a✳ ❚❤❡ ❛❧❣❡❜r❛ ✐s ♠✉❝❤ s✐♠♣❧❡r t❤❛♥ ❜❡❢♦r❡✳ ❋♦r ❡①❛♠♣❧❡✱ ✐❢ t❤❡ ❧❡♥❣t❤ ✐s ♠❡❛s✉r❡❞ ❛s 11✱ ✇❡ ♥❡❡❞ t❤❡ ❛❝❝✉r❛❝② δ = 1 ♦r ❜❡tt❡r t♦ ♠❛❦❡ t❤❡ ❞❡t❡r♠✐♥❛t✐♦♥✳ ■t✬s t❤❡ s❛♠❡ ❢♦r t❤❡ ❧❡♥❣t❤ 9✳ ❇✉t ✇❤❛t ✐❢ t❤❡ ♠❡❛s✉r❡♠❡♥t ✐s ❡①❛❝t❧② a = 10❄ ❊✈❡♥ ✐❢ ✇❡ ❝❛♥ ✐♠♣r♦✈❡ t❤❡ ❛❝❝✉r❛❝②✱ ✐✳❡✳✱ δ ✱ ❛s ❧♦♥❣ ❛s δ > 0✱ ✇❡ ❝❛♥✬t ❦♥♦✇ ✇❤❡t❤❡r x ✐s ❧❛r❣❡r ♦r s♠❛❧❧❡r t❤❛♥ 10✳ ▲❡t✬s ❞❡✜♥❡ ❛ ❢✉♥❝t✐♦♥ t❤❛t ♠❛❦❡s t❤✐s ❞❡❝✐s✐♦♥✿ ( 1 f (x) = 0

✭♣❛ss✮ ✭❢❛✐❧✮

✐❢ x ≤ 10 , ✐❢ x > 10 .

❙✉♣♣♦s❡ ✇❡ ♥❡❡❞ t❤❡ ❛❝❝✉r❛❝② ♦❢ y = f (x) t♦ ❜❡ ε = 0.5✳ ❈❛♥ ✇❡ ❛❝❤✐❡✈❡ t❤✐s ❜② ❞❡❝r❡❛s✐♥❣ δ ❄ ■♥ ♦t❤❡r ✇♦r❞s✱ ❝❛♥ ✇❡ ✜♥❞ δ s✉❝❤ t❤❛t |x − 10| < δ =⇒ |f (x) − 1| < ε ? ❖❢ ❝♦✉rs❡ ♥♦t✿

x > 10 =⇒ |f (x) − 1| = |0 − 1| = 1 . ❚❤✉s✱ t❤❡ ❛♥s✇❡r t♦ ♦✉r q✉❡st✐♦♥ ✐s✿

◮ ◆♦✱ ✇❡ ❝❛♥♥♦t ❛❧✇❛②s ✐♠♣r♦✈❡ t❤❡ ❛❝❝✉r❛❝② ♦❢ t❤❡ ❝♦♠♣✉t❡❞ ✈❛❧✉❡ ♦❢ f (x) ✕ t♦ ❛♥②❜♦❞②✬s s❛t✐s❢❛❝t✐♦♥ ✕ ❜② ✐♠♣r♦✈✐♥❣ t❤❡ ❛❝❝✉r❛❝② ♦❢ t❤❡ ♠❡❛s✉r❡♠❡♥t ♦❢ x✳ ❚❤❡ r❡❛s♦♥ t♦ ❜❡ q✉♦t❡❞ ✐s t❤❛t f ✐s ❞✐s❝♦♥t✐♥✉♦✉s ❛t x = 10✳ ❊①❡r❝✐s❡ ✷✳✶✷✳✺

❈❛rr② ♦✉t t❤✐s ❦✐♥❞ ♦❢ ❛♥❛❧②s✐s ❢♦r✿ t❤❡ t♦t❛❧ t❡st s❝♦r❡ ✈s✳ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❧❡tt❡r ❣r❛❞❡✳ ❲❤❛t ✐❢ ✇❡ ✐♥tr♦❞✉❝❡ ❆✲✱ ❇✰✱ ❡t❝✳❄ ❲❡ ♥❡①t ♣✉rs✉❡ t❤✐s ✐❞❡❛ ♦❢ ❝♦♥t✐♥✉✐t② ♦❢ t❤❡ ❞❡♣❡♥❞❡♥❝❡ ♦❢ y ♦♥ x✿

◮ ❲❡ ❝❛♥ ❡♥s✉r❡ t❤❡ ❞❡s✐r❡❞ ❛❝❝✉r❛❝② ♦❢ y ❜② ✐♥❝r❡❛s✐♥❣ t❤❡ ❛❝❝✉r❛❝② ♦❢ x✳ ❲❡ ❞✐s❝✉ss ❧✐♠✐ts ✜rst✳

✷✳✶✸✳ ❚❤❡ ε✲δ ❞❡✜♥✐t✐♦♥ ♦❢ ❧✐♠✐t ❲❡ ❝❛♥ ❞❡✜♥❡ ❧✐♠✐ts ❛♥❞ ❝♦♥t✐♥✉✐t② ♦❢ ❢✉♥❝t✐♦♥s ✇✐t❤♦✉t ✐♥✈♦❦✐♥❣ ❧✐♠✐ts ♦❢ s❡q✉❡♥❝❡s✳ ▲❡t✬s st❛rt ♦✈❡r✳

✷✳✶✸✳ ❚❤❡ ε✲δ ❞❡✜♥✐t✐♦♥ ♦❢ ❧✐♠✐t

✷✵✹

❊①❛♠♣❧❡ ✷✳✶✸✳✶✿ ✐♠❛❣❡s ♦❢ ✐♥t❡r✈❛❧s ▲❡t✬s ❝♦♠♣❛r❡ ✇❤❛t ❤❛♣♣❡♥s t♦ ♦♣❡♥ ✐♥t❡r✈❛❧s t❤❛t ❝♦♥t❛✐♥ ♦✉r ♣♦✐♥t ♦❢ ✐♥t❡r❡st ✉♥❞❡r ❛ ❢✉♥❝t✐♦♥✳ ❋✐rst✱ s✉♣♣♦s❡ t❤❡ ❢✉♥❝t✐♦♥ ✐s ❝♦♥t✐♥✉♦✉s ✭t♦♣✮✳ ❆s t❤❡ ✐♥t❡r✈❛❧ ✐♥ t❤❡ x✲❛①✐s s❤r✐♥❦s✱ s♦ ❞♦❡s t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ✐♥t❡r✈❛❧ ✐♥ t❤❡ y ✲❛①✐s✿

❙❡❝♦♥❞✱ ❝♦♥s✐❞❡r t❤✐s ❞✐s❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ ✭❜♦tt♦♠✮✳ ❆s t❤❡ ✐♥t❡r✈❛❧ ✐♥ t❤❡ x✲❛①✐s s❤r✐♥❦s✱ ✇❡ ♥♦t✐❝❡ t✇♦ t❤✐♥❣s ❛❜♦✉t t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s❡t ✐♥ t❤❡ y ✲❛①✐s✿ • ■t ❡✈❡♥t✉❛❧❧② ❝❡❛s❡s t♦ ❜❡ ❛♥ ✐♥t❡r✈❛❧✳ ❆ t❡❛r✦ • ■t ❛❧s♦ st♦♣s s❤r✐♥❦✐♥❣✳ ❆ ❣❛♣✦ ❇❡❧♦✇✱ ✇❡ r❡✇r✐t❡ ✇❤❛t ✇❡ ✇❛♥t t♦ s❛② ❛❜♦✉t t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ ❧✐♠✐ts ✐♥ ♣r♦❣r❡ss✐✈❡❧② ♠♦r❡ ❛♥❞ ♠♦r❡ ♣r❡❝✐s❡ t❡r♠s✳ x

❆s x → a, ❆s x ❛♣♣r♦❛❝❤❡s a, ❆s t❤❡ ❞✐st❛♥❝❡ ❢r♦♠ x t♦ a ❛♣♣r♦❛❝❤❡s 0, ❆s |x − a| → 0, ❇② ♠❛❦✐♥❣ |x − a| ❛s s♠❛❧❧❡r ❛♥❞ s♠❛❧❧❡r, ❇② ♠❛❦✐♥❣ |x − a| ❧❡ss t❤❛♥ s♦♠❡ δ > 0,

y = f (x) ✇❡ ❤❛✈❡ y → L. y ❛♣♣r♦❛❝❤❡s L.

t❤❡ ❞✐st❛♥❝❡ ❢r♦♠ y t♦ L ❛♣♣r♦❛❝❤❡s 0. ✇❡ ❤❛✈❡ |y − L| → 0. ✇❡ ♠❛❦❡ |y − L| ❛s s♠❛❧❧ ❛s ♥❡❡❞❡❞. ✇❡ ♠❛❦❡ |y − L| s♠❛❧❧❡r t❤❛♥ ❛♥② ❣✐✈❡♥ ε > 0.

◆♦✇ ✇❡ ❤❛✈❡ ✐t✿

❉❡✜♥✐t✐♦♥ ✷✳✶✸✳✷✿ ε✲δ ❞❡✜♥✐t✐♦♥ ♦❢ ❧✐♠✐t ❚❤❡ ❧✐♠✐t ♦❢ ❛ ❢✉♥❝t✐♦♥ f ❛t x = a ✐s ❛ ♥✉♠❜❡r L✱ ✐❢ ✐t ❡①✐sts✱ s✉❝❤ t❤❛t ❢♦r ❛♥② ε > 0 t❤❡r❡ ✐s s✉❝❤ ❛ δ > 0 t❤❛t 0 < |x − a| < δ =⇒ |f (x) − L| < ε .

❚❤✐s ✐s t❤❡ ❣❡♦♠❡tr✐❝ ♠❡❛♥✐♥❣ ♦❢ t❤❡ ❞❡✜♥✐t✐♦♥✿ ◮ ■❢ x ✐s ✇✐t❤✐♥ δ ❢r♦♠ a✱ t❤❡♥ f (x) ✐s s✉♣♣♦s❡❞ t♦ ❜❡ ✇✐t❤✐♥ ε ❢r♦♠ L✳

■♥ ♦t❤❡r ✇♦r❞s✱ t❤✐s ♣❛rt ♦❢ t❤❡ ❣r❛♣❤ ✜ts ✇✐t❤✐♥ t❤❡ ε✲❜❛♥❞ ❛r♦✉♥❞ t❤❡ ❧✐♥❡ y = L✿

✷✳✶✸✳

❚❤❡ ε✲δ ❞❡✜♥✐t✐♦♥ ♦❢ ❧✐♠✐t

✷✵✺

❚❤✐s ✐s t❤❡ ✏ ε✲δ ❞❡✜♥✐t✐♦♥✑ ♦❢ ❧✐♠✐t ♦❢ ❛ ❢✉♥❝t✐♦♥✳ ■t ♠❛t❝❤❡s t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❧✐♠✐t ♦❢ ❛ s❡q✉❡♥❝❡✱ ✇❤✐❝❤ ❝❛♥ ♥♦✇ ❜❡ ❝❛❧❧❡❞ t❤❡ ✏ ε✲N ❞❡✜♥✐t✐♦♥✑✳

❚❤✐s ✐s ♦✉r ❝♦♥❝❧✉s✐♦♥✿

❚❤❡♦r❡♠ ✷✳✶✸✳✸✿ ❊q✉✐✈❛❧❡♥❝❡ ♦❢ ❉❡✜♥✐t✐♦♥s ♦❢ ▲✐♠✐t ❚❤❡ t✇♦ ❞❡✜♥✐t✐♦♥s ♦❢ t❤❡ ❧✐♠✐t ♦❢ ❛ ❢✉♥❝t✐♦♥ ❛r❡ ❡q✉✐✈❛❧❡♥t✳

Pr♦♦❢✳ [ε✲δ ⇒ ε✲N ] ❙✉♣♣♦s❡ xn → a✳ t❤❡r❡ ✐s ❛ δ > 0 s✉❝❤ t❤❛t

❙✉♣♣♦s❡ ♥♦✇ t❤❛t

ε>0

✐s ❣✐✈❡♥✳ ❆s t❤❡ ❞❡✜♥✐t✐♦♥ ❛❜♦✈❡ ✐s s❛t✐s✜❡❞✱

0 < |x − a| < δ =⇒ |f (x) − L| < ε . ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ✇❡ ❤❛✈❡✿

|xn − a| < δ ❢♦r s♦♠❡

N✳

❚❤❡r❡❢♦r❡✱ ❢♦r ❛❧❧

n > N✱

❢♦r ❛❧❧

n>N

✇❡ ❤❛✈❡✿

|f (xn ) − L| < ε . ❚❤✐s ♠❡❛♥s t❤❛t

f (xn ) → L✳

❊①❡r❝✐s❡ ✷✳✶✸✳✹ Pr♦✈❡ t❤❡ ♦t❤❡r ❤❛❧❢✿

[ε✲N ⇒ ε✲δ]✳

❆s ❛♥ ❡①❡r❝✐s❡✱ ❧❡t✬s ❝♦♠♣✉t❡ t❤❡ ❧✐♠✐t ♦❢ ❛♥ ❛r❜✐tr❛r② ❧✐♥❡❛r ❢✉♥❝t✐♦♥✳ ♣r❡❞✐❝❛❜❧❡✿

❚❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥

δ

❛♥❞

ε

✐s

✷✳✶✸✳ ❚❤❡ ε✲δ ❞❡✜♥✐t✐♦♥ ♦❢ ❧✐♠✐t

✷✵✻

❚❤✐s ✐s ✇❤❛t ✇❡ ❣✉❡ss ❢r♦♠ t❤❡ ♣✐❝t✉r❡✿

❚❤❡♦r❡♠ ✷✳✶✸✳✺✿ ▲✐♠✐t ♦❢ ▲✐♥❡❛r ❋✉♥❝t✐♦♥ ❋♦r ❛ ❧✐♥❡❛r ❢✉♥❝t✐♦♥ f (x) = mx+b ✇✐t❤ m 6= 0✱ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❧✐♠✐t ✐s s❛t✐s✜❡❞ ❢♦r δ = ε/|m| .

Pr♦♦❢✳ |x − a| < δ ⇐⇒ ⇐⇒ ⇐⇒ ⇐⇒ ⇐⇒ ⇐⇒

|x − a| < ε/|m| |m| · |x − a| < ε |m · (x − a)| < ε |mx − ma| < ε |(mx + b) − (ma + b)| < ε |f (x) − f (a)| < ε .

❊①❛♠♣❧❡ ✷✳✶✸✳✻✿ ✉s✐♥❣ t❤❡ ❞❡✜♥✐t✐♦♥ ✕ ❧✐♥❡❛r Pr♦✈❡ t❤❡ ❧✐♠✐t ❜② ✉s✐♥❣ t❤❡ ❞❡✜♥✐t✐♦♥✿ 9 − 4x2 = 6. x→−1.5 3 + 2x lim

▲❡t✬s ♠❛t❝❤ ✇❤❛t ✇❡ ✇❛♥t t♦ ♣r♦✈❡ ✇✐t❤ t❤❡ ❞❡✜♥✐t✐♦♥✿ 9 − 4x2 = |{z} 6 . lim x→−1.5 3 + 2x {z } | L | {z } a

f (x)

❲❡ r❡♣❧❛❝❡ f (x) ✇✐t❤ ❛ s✐♠♣❧❡r ❡①♣r❡ss✐♦♥✱ ❡q✉❛❧ ❢♦r ❛❧❧ x 6= −1.5✿ 9 − 4x2 32 − (2x)2 = 3 + 2x 3 + 2x (3 + 2x)(3 − 2x) = 3 + 2x = 3 − 2x

❢♦r ❛❧❧ x ❢♦r ✇❤✐❝❤ 3 + 2x 6= 0✱ ♦r x 6= 1.5 ✳ ◆♦✇✱ t❤✐s ❡①♣r❡ss✐♦♥ ✐s ❧✐♥❡❛r✦ ❚❤❡♥ t❤❡ ❞❡♣❡♥❞❡♥❝❡ ♦❢ δ ♦♥ ε ✐s s✐♠♣❧❡✱ ❛s ✇❡ ❦♥♦✇✿ δ=

ε . 2

✷✳✶✸✳ ❚❤❡ ε✲δ ❞❡✜♥✐t✐♦♥ ♦❢ ❧✐♠✐t

✷✵✼

❲❡ ♥❡❡❞ t♦ ❞❡♠♦♥str❛t❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿ 0 < |x − (−1.5)|
0 s✉❝❤ t❤❛t✿ f (x)

L a z }| { z}|{ z}|{ 0 < |x − 0 | < δ =⇒ | s✐❣♥(x) − 1 | < ε = 1 . | {z } f (x) ✐s ✇✐t❤✐♥ (0,2)

❲❤❛t x ✇♦✉❧❞ ✈✐♦❧❛t❡ t❤✐s❄

✷✳✶✸✳ ❚❤❡ ε✲δ ❞❡✜♥✐t✐♦♥ ♦❢ ❧✐♠✐t

✷✵✽

❆♥② x < 0 s✐♥❝❡ ✇❡ ❤❛✈❡ t❤❡♥ f (x) = −1✳ ❚❤❡ ❞❡✜♥✐t✐♦♥ ❢❛✐❧s✱ s♦ L = 1 ✐s ♥♦t t❤❡ ❧✐♠✐t✳ ❊①❡r❝✐s❡ ✷✳✶✸✳✶✵

Pr♦✈❡ t❤❛t L = 0 ✐s♥✬t t❤❡ ❧✐♠✐t ❡✐t❤❡r✳ ■♥ ♦r❞❡r t♦ ❧❡❛r♥ ❤♦✇ t♦ ❞✐s♣r♦✈❡ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❛ ❧✐♠✐t✱ ❧❡t✬s st❛t❡ t❤❡ ♥❡❣❛t✐♦♥ ♦❢ t❤❡ ❞❡✜♥✐t✐♦♥✿ ❚❤❡♦r❡♠ ✷✳✶✸✳✶✶✿ ◆♦t ▲✐♠✐t

L ✐s ♥♦t t❤❡ ❧✐♠✐t ♦❢ ❢✉♥❝t✐♦♥ f δ > 0 ❛♥❞ s♦♠❡ x ✇❡ ❤❛✈❡✿

❆ ♥✉♠❜❡r ❢♦r ❛♥②

0 < |x − a| < δ

❛♥❞

❛t

x=a

✐❢ t❤❡r❡ ✐s ❛♥

ε>0

s✉❝❤ t❤❛t

|f (x) − L| ≥ ε .

❚❤❡ ❡①✐st❡♥❝❡ ♦❢ s✉❝❤ ❛♥ ε ✐♥❞✐❝❛t❡s t❤❡ ♣r❡s❡♥❝❡ ♦❢ ❛ ❣❛♣ ✐♥ t❤❡ ❣r❛♣❤✿

❊①❛♠♣❧❡ ✷✳✶✸✳✶✷✿ s✐❣♥ ❢✉♥❝t✐♦♥

▲❡t✬s ♣r♦✈❡ t❤❡ ♥♦♥✲❡①✐st❡♥❝❡ ♦❢ ❧✐♠✐t ❛t a = 0 ❢♦r t❤❡ s✐❣♥ ❢✉♥❝t✐♦♥ f (x) = sign(x)✳ ❙✉♣♣♦s❡ L 6= 0 ✐s ❛♥② ♥✉♠❜❡r✳ ▲❡t✬s ♣✐❝❦ ε = |L|/2✳ ◆♦✇ s✉♣♣♦s❡ δ > 0 ✐s ❛r❜✐tr❛r② ❛♥❞ s✉♣♣♦s❡ ❢♦r s♦♠❡ x 6= 0 ✇❡ ❤❛✈❡✿ 0 < |x| < δ ❛♥❞ |f (x) − L| < ε = |L|/2 .

❚❤❡♥✱ f (x) ✐s ♦❢ t❤❡ s❛♠❡ s✐❣♥ ❛s L✳ ❇✉t ✇❤❛t ❞♦❡s t❤✐s s❛② ❛❜♦✉t −x❄ ❲❡ ❛❧s♦ ❤❛✈❡✿ 0 < | − x| < δ ❛♥❞✱ t❤❡r❡❢♦r❡✱ |f (−x) − L| < ε = |L|/2 .

❚❤❡♥✱ f (−x) ✐s ❛❧s♦ ♦❢ t❤❡ s❛♠❡ s✐❣♥ ❛s L✳ ❚❤✐s ❝♦♥tr❛❞✐❝ts t❤❡ ❢❛❝t t❤❛t f (−x) = −f (x)✳ ▲❡t✬s ♣r♦✈❡ t❤❡ ❈♦♠♣♦s✐t✐♦♥ ❘✉❧❡ ❢♦r t❤✐s ♥❡✇ ❞❡✜♥✐t✐♦♥ ♦❢ ❧✐♠✐t✳

✷✳✶✸✳ ❚❤❡ ε✲δ ❞❡✜♥✐t✐♦♥ ♦❢ ❧✐♠✐t

✷✵✾

❲❡ ❤❛✈❡ t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ ❢✉♥❝t✐♦♥s f ❛♥❞ g ✳ ❚❤❡s❡ ❛r❡ t❤❡ t❤r❡❡ ✐♥st❛♥❝❡s ♦❢ t❤❡ ❞❡✜♥✐t✐♦♥ ✇❡ ❢❛❝❡❀ ✇❡ ♥❡❡❞ t♦ ♣r♦✈❡ ♣❛rt ✸ ❢r♦♠ ♣❛rts ✶ ❛♥❞ ✷✿ ✶✳ y = g(x) → L ❛s x → a✱ ✐✳❡✳✱ ❢♦r ❛♥② 6 ε γ > 0 t❤❡r❡ ✐s s✉❝❤ ❛ δ > 0 t❤❛t

0 < |x − a| < δ =⇒ |g(x) − L| 0 t❤❡r❡ ✐s s✉❝❤ ❛ 6 δ γ > 0 t❤❛t

0 < |y − L| 0 t❤❡r❡ ✐s s✉❝❤ ❛ δ > 0 t❤❛t

0 < |x − a| < δ =⇒ |f (g(x)) − M | < ε . ❆❜♦✈❡ ✇❡ r❡❝♦♥❝✐❧❡ t❤❡ ♣❛r❛♠❡t❡rs✱ ε ❛♥❞ δ ✱ t❤❛t ❛♣♣❡❛r ✐♥ ❡❛❝❤✱ ❜② ♠❛t❝❤✐♥❣ t❤❡ t❤r❡❡ ✈❛r✐❛❜❧❡s✿ ❆ s♠❛❧❧ ❞❡✈✐❛t✐♦♥ ♦❢ x ✭❤♦✇ s♠❛❧❧✿ ✇✐t❤✐♥ δ ✮ ❝❛✉s❡s ❛ s♠❛❧❧ ❞❡✈✐❛t✐♦♥ ♦❢ y ✭❝❛❧❧ ✐t γ ✮✱ ✇❤✐❝❤ ✐♥ t✉r♥ ❝❛✉s❡s ❛ s♠❛❧❧ ❞❡✈✐❛t✐♦♥ ♦❢ z ✭t❤❛t✬s ε✮✳ ❙✉♣♣♦s❡ ♥♦✇ ε > 0 ✐s ❣✐✈❡♥✳ ❚❤❡♥ t❤❡ γ ❢♦✉♥❞✱ ❢r♦♠ t❤✐s ε✱ ✐♥ ♣❛rt ✷ ✐s ❢❡❞ ✐♥t♦ ♣❛rt ✶✱ ♣r♦❞✉❝✐♥❣ ❛ δ ✳ ❈♦♠❜✐♥✐♥❣ t❤❡s❡ t✇♦ st❛t❡♠❡♥ts t♦❣❡t❤❡r✱ ✇❡ ❤❛✈❡✿

0 < |x − a| < δ =⇒ |g(x) − L| < γ . ▲❡t y = g(x) =⇒ |y − L| < γ =⇒ |f (y) − M | < ε . P❛rt ✸ ✐s ♣r♦✈❡♥✦ ❲❡ ✜♥❛❧❧② st❛t❡ t❤❡ r❡s✉❧t✿ ❚❤❡♦r❡♠ ✷✳✶✸✳✶✸✿ ▲✐♠✐t ✉♥❞❡r ❈♦♠♣♦s✐t✐♦♥ ❙✉♣♣♦s❡ ✶✳ ✷✳ ❚❤❡♥

y = g(x) → L ❛s x → a ✱ ❛♥❞ z = f (y) → M ❛s y → L ✳ z = (f ◦ g)(x) → M

❛s

x → a.

❊①❡r❝✐s❡ ✷✳✶✸✳✶✹

Pr♦✈❡ t❤❛t ✐❢ f ❛♥❞ g ❛r❡ ❝♦♥t✐♥✉♦✉s ❛t x = a✱ t❤❡♥ s♦ ❛r❡ max{f, g} ❛♥❞ min{f, g}✳ ❲❡ ❤❛✈❡ r❡q✉✐r❡❞ t❤❛t ✐❢ x ✐s ✇✐t❤✐♥ δ ❢r♦♠ a✱ t❤❡♥ f (x) ✐s ✇✐t❤✐♥ ε ❢r♦♠ L✳ ❲❡ ❛r❡ r❡❢❡rr✐♥❣ ❤❡r❡ t♦ ✐♥t❡r✈❛❧s ✕ ♦♥❡ ❝♦♥t❛✐♥✐♥❣ a ❛♥❞ ❛♥♦t❤❡r L ✕ ❜✉t ♥♦t ❥✉st ❛♥②❀ t❤❡s❡ ✐♥t❡r✈❛❧s ❛r❡ ❝❡♥t❡r❡❞ ❛t t❤❡s❡ ♣♦✐♥ts✳ ❚❤✐s s②♠♠❡tr② r❡q✉✐r❡♠❡♥t ✐s♥✬t ♥❡❝❡ss❛r②✿

❊✈❡♥ ✇❤❡♥ ✇❡ st❛rt ✇✐t❤ ❛ s②♠♠❡tr✐❝ ✐♥t❡r✈❛❧ ❛r♦✉♥❞ a✱ t❤❡ s②♠♠❡tr② ❛r♦✉♥❞ L ♦❝❝✉rs ♦♥❧② ✇❤❡♥ t❤❡ ❢✉♥❝t✐♦♥ ✐s ❧✐♥❡❛r ✿

✷✳✶✸✳ ❚❤❡ ε✲δ ❞❡✜♥✐t✐♦♥ ♦❢ ❧✐♠✐t

✷✶✵

❚❤❡♦r❡♠ ✷✳✶✸✳✶✺✿ ❆❧t❡r♥❛t✐✈❡ ❉❡✜♥✐t✐♦♥ ♦❢ ▲✐♠✐t

❚❤❡ ❧✐♠✐t ♦❢ ❛ ❢✉♥❝t✐♦♥ f ❛t x = a ✐s L ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❢♦r ❛♥② ε1, s✉❝❤ δ2, δ2 > 0 t❤❛t

ε2 > 0 t❤❡r❡ ❛r❡

−δ1 < x − a < δ2 =⇒ −ε1 < f (x) − L < ε2 .

❚❤❡ t❤❡♦r❡♠s s✉♣♣❧② ❛♥ ❛❧t❡r♥❛t✐✈❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❧✐♠✐t✱ ❛♥❞ ♦❢ ❝♦♥t✐♥✉✐t②✳

❊①❡r❝✐s❡ ✷✳✶✸✳✶✻ ✭❛✮ Pr♦✈❡ t❤❡ ❛❜♦✈❡ t❤❡♦r❡♠✳ ✭❜✮ ❯s❡ t❤❡ ❛❜♦✈❡ t❤❡♦r❡♠ t♦ ♣r♦✈❡ t❤❡ ❈♦♠♣♦s✐t✐♦♥ ❚❤❡♦r❡♠ ❛❜♦✈❡✳ ✭❝✮ ❯s❡ t❤❡ ❛❜♦✈❡ t❤❡♦r❡♠ t♦ ♣r♦✈❡ t❤❛t t❤❡ ✐♥✈❡rs❡ ♦❢ ❛ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ ✐s ❝♦♥t✐♥✉♦✉s✳

❈❤❛♣t❡r ✸✿ ❚❤❡ ❞❡r✐✈❛t✐✈❡

❈♦♥t❡♥ts

✸✳✶ ❚❤❡ ❚❛♥❣❡♥t Pr♦❜❧❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✷ ❚❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ ❛ s❡q✉❡♥❝❡ ❛♥❞ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ✳ ✸✳✸ ❚❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡✿ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✹ ❚❤❡ ❧✐♠✐t ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✿ t❤❡ ❞❡r✐✈❛t✐✈❡ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✺ ❚❤❡ ❞❡r✐✈❛t✐✈❡ ✐s t❤❡ ✐♥st❛♥t❛♥❡♦✉s r❛t❡ ♦❢ ❝❤❛♥❣❡ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✻ ❚❤❡ ❡①✐st❡♥❝❡ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡✿ ❞✐✛❡r❡♥t✐❛❜✐❧✐t② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✼ ❚❤❡ ❞❡r✐✈❛t✐✈❡ ❛s ❛ ❢✉♥❝t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✽ ❇❛s✐❝ ❞✐✛❡r❡♥t✐❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✾ ❇❛s✐❝ ❞✐✛❡r❡♥t✐❛t✐♦♥✱ ❝♦♥t✐♥✉❡❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✶✵ ❋r❡❡ ❢❛❧❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✶✶ ✷✶✾ ✷✷✾ ✷✹✵ ✷✹✽ ✷✺✸ ✷✺✾ ✷✼✹ ✷✽✶ ✷✽✾

✸✳✶✳ ❚❤❡ ❚❛♥❣❡♥t Pr♦❜❧❡♠

❚❤❡r❡ ❛r❡ t✇♦ ♠❛✐♥ ✇❛②s t♦ ❡♥t❡r ❝❛❧❝✉❧✉s✿

•

t❤r♦✉❣❤ t❤❡ st✉❞② ♦❢ ♠♦t✐♦♥✱ ❛♥❞

•

t❤r♦✉❣❤ t❤❡ st✉❞② ♦❢ ❝✉r✈❡❞ s❤❛♣❡s ✭✏❣❡♦♠❡tr②✑✮✳

❲❡ st❛rt❡❞ ✇✐t❤ t❤❡ ❢♦r♠❡r ✐♥ ❈❤❛♣t❡r ✶ ❛♥❞ ✇✐❧❧ ❝♦♥t✐♥✉❡ ♦♥ s❤♦rt❧②✳ ■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ ❝♦♥s✐❞❡r t❤❡ ❧❛tt❡r✳ ■♥ ✇❤❛t ❞✐r❡❝t✐♦♥ ✇✐❧❧ ❧✐❣❤t ❜♦✉♥❝❡ ♦✛ ❛ ♠✐rr♦r❄ ❲❡ ❦♥♦✇ t❤❡ ❛♥s✇❡r ✇❤❡♥ t❤❡ ♠✐rr♦r ✐s str❛✐❣❤t ✭❧❡❢t✮✱ ❜✉t ✇❤❛t ❛❜♦✉t ❛ ❝✉r✈❡❞ ♠✐rr♦r ✭♠✐❞❞❧❡✮❄

❚❤❡ r❛② ✐s ❡①tr❡♠❡❧② ♥❛rr♦✇ ❛♥❞ t❤❡r❡ ✐s ❥✉st ❛ s✐♥❣❧❡ ♣♦✐♥t ♦❢ ❝♦♥t❛❝t✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ❧✐❣❤t ❜♦✉♥❝❡s ❛s ✐❢ ♦✛ ❛

str❛✐❣❤t

♠✐rr♦r ❛t t❤❡ ♣♦✐♥t ♦❢ ❝♦♥t❛❝t ✭r✐❣❤t✮✳ ❚♦ s❡❡ t❤❛t✱ ✇❡ ③♦♦♠ ✐♥ ♦♥ t❤❡ ♣♦✐♥t✳

❲❤❛t ❞♦ ✇❡ s❡❡ ❡①❛❝t❧②❄ ❚❤❡ ✜rst ♣♦ss✐❜✐❧✐t② ✐s t❤❛t ✇❡ ♠✐❣❤t s❡❡ t✇♦ ♣♦✐♥ts ❝♦♥♥❡❝t❡❞ ❜② ❛ str❛✐❣❤t ❡❞❣❡✿

✸✳✶✳

❚❤❡ ❚❛♥❣❡♥t Pr♦❜❧❡♠

✷✶✷

❲❡ ❛ss✉♠❡ t❤❛t t❤❡ ❧✐❣❤t ✇✐❧❧ ❤✐t t❤✐s ❡❞❣❡ ❛♥❞ t❤✐s ✐s t❤❡ ♦♥❡ ✇❡ ✉s❡ t♦ ✜♥❞ ✐ts ♣❛t❤ ❛❢t❡r t❤❡ ❝♦♥t❛❝t✳ ❚❤✐s ❧✐♥❡ ❝✉ts ❛❝r♦ss t❤❡ ❝✉r✈❡ ❛♥❞ ✐s ❝❛❧❧❡❞ ❛

s❡❝❛♥t

✭t❤❡ ●r❡❡❦ ❢♦r ✏❝✉t✑✮ ❧✐♥❡✳

❍♦✇ ❞♦ ✇❡ ✜♥❞ t❤❡s❡ ❧✐♥❡s❄ ❙✐♥❝❡ ❛ ♣♦✐♥t ✐s s✉♣♣❧✐❡❞ ❢♦r ❡❛❝❤✱ ❛❧❧ ✇❡ ♥❡❡❞ ✐s t❤❡ ❛♥❣❧❡✳ ■♥ t❤❡ ♣r❡s❡♥❝❡ ♦❢ t❤❡ ❈❛rt❡s✐❛♥ s②st❡♠✱ ✇❡ ✉s❡ t❤❡

s❧♦♣❡s✱ ❛♥❞ t❤❡♥ t❤❡ ♣♦✐♥t✲s❧♦♣❡ ❢♦r♠ ♦❢ t❤❡ ❧✐♥❡✱ t♦ ✜♥❞ t❤♦s❡ ❧✐♥❡s ✭s❡❡♥

✐♥ ❱♦❧✉♠❡ ✶✱ ❈❤❛♣t❡r ✶P❈✲✷✮✳ ▲❡t✬s r❡✈✐❡✇ t❤❡ ♠♦st ❜❛s✐❝ ❝♦♥❝❡♣t✿

❉❡✜♥✐t✐♦♥ ✸✳✶✳✶✿ s❧♦♣❡ A = (x0 , y0 ) t❤❡♥ B = (x1 , y1 )✱ A t♦ B ✐s ❞❡✜♥❡❞ t♦ ❜❡

❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ t✇♦ ♣♦✐♥ts ✐♥ ❛ s♣❡❝✐✜❡❞ ♦r❞❡r✱ ♦♥ t❤❡

xy ✲♣❧❛♥❡✱

s❧♦♣❡

t❤❡♥

=m=

t❤❡ s❧♦♣❡ ♦❢ t❤❡ ❧✐♥❡

r✐s❡ r✉♥

=

❝❤❛♥❣❡ ♦❢ ❝❤❛♥❣❡ ♦❢

y = x

❢r♦♠

❝❤❛♥❣❡ ❢r♦♠ ❝❤❛♥❣❡ ❢r♦♠

❊①❛♠♣❧❡ ✸✳✶✳✷✿ ❝♦♠♣✉t✐♥❣ s❧♦♣❡s ❚❤❡ ❣❡♦♠❡tr✐❝ ♠❡❛♥✐♥❣ ♦❢ t❤❡ ♥✉♠❡r❛t♦r ❛♥❞ ❞❡♥♦♠✐♥❛t♦r ✐s s❡❡♥ ❜❡❧♦✇✿

❍❡r❡ ✇❡ ❤❛✈❡✿

• •

= 8 − 2 = 6✱ ❛♥❞ = 10 − 1 = 9✱ t❤❡r❡❢♦r❡✱ 9 3 • s❧♦♣❡ = = = 1.5 ✳ 6 2 r✉♥

r✐s❡

y0 x0

t♦ t♦

y1 y1 − y0 = x1 x1 − x0

✸✳✶✳

❚❤❡ ❚❛♥❣❡♥t Pr♦❜❧❡♠

✷✶✸

❊①❡r❝✐s❡ ✸✳✶✳✸

❊①♣r❡ss t❤❡ s❧♦♣❡ ✐♥ tr✐❣♦♥♦♠❡tr✐❝ t❡r♠s✳ ❲❤❛t ❞♦❡s t❤❡ s❧♦♣❡ t❡❧❧ ✉s ❛❜♦✉t t❤❡ ❧✐♥❡❄ ❇❡❧♦✇ ✇❡ ❛rr❛♥❣❡ ❛❧❧ ❧✐♥❡❛r ❢✉♥❝t✐♦♥s ❛❝❝♦r❞✐♥❣ t♦ t❤❡✐r s❧♦♣❡s ✭✇✐t❤ t❤❡ s❛♠❡ y ✲✐♥t❡r❝❡♣t✮✿

■t✬s ❛s ✐❢ ✐♥❝r❡❛s✐♥❣ t❤❡ s❧♦♣❡ r♦t❛t❡s t❤❡ ❧✐♥❡ ❝♦✉♥t❡r❝❧♦❝❦✇✐s❡✿

■t ❝❛♥✬t ♠❛❦❡ t❤❡ ❧✐♥❡ ✈❡rt✐❝❛❧ t❤♦✉❣❤✳ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ s♣❡❝✐✜❡❞ ♣♦✐♥t A = (x0 , y0 ) ♦♥ ♦✉r ❧✐♥❡✳ ▲❡t✬s ❝♦♥s✐❞❡r ❛♥ ❛r❜✐tr❛r② ♣♦✐♥t X = (x, y) ♦♥ t❤❡ ❧✐♥❡✿

❚❤❡ r✉♥ ✐s x − x0 ❛♥❞ t❤❡ r✐s❡ ✐s y − y0 ✭❧❡❢t ♦r r✐❣❤t✮✳ ❚❤❡r❡❢♦r❡✱ t❤❡ s❧♦♣❡ ✐s m=

y − y0 . x − x0

■♥ t❤✐s ❢♦r♠✉❧❛✱ x ❝❛♥♥♦t ❜❡ ❡q✉❛❧ t♦ x0 ✳ ■t ❝❛♥♥♦t✱ t❤❡r❡❢♦r❡✱ ❜❡ ✉s❡❞ ❛s ❛ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❧✐♥❡✦ ▲❡t✬s ❝❤❛♥❣❡ ♦✉r ✈✐❡✇ ♦♥ t❤❡ s❧♦♣❡✿ ❢r♦♠ r✐s❡ s❧♦♣❡ = r✉♥ t♦ r✐s❡ = s❧♦♣❡ · r✉♥ .

✸✳✶✳

❚❤❡ ❚❛♥❣❡♥t Pr♦❜❧❡♠

✷✶✹

❲❡ ❤❛✈❡ ❛ ♥❡✇ ✇❛② t♦ r❡♣r❡s❡♥t ❛ ❧✐♥❡✿

❚❤❡♦r❡♠ ✸✳✶✳✹✿ P♦✐♥t✲❙❧♦♣❡ ❋♦r♠ ♦❢ ▲✐♥❡ ❆ ❧✐♥❡ ✇✐t❤ s❧♦♣❡

m ♣❛ss✐♥❣ t❤r♦✉❣❤ ♣♦✐♥t (x0 , y0 ) ✐s ❣✐✈❡♥ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ❧✐♥❡❛r

r❡❧❛t✐♦♥✿

y − y0 = m · (x − x0 ) ❇❡❧♦✇ ✐s t❤❡ ❜r❡❛❦❞♦✇♥ ♦❢ t❤❡ ❢♦r♠✉❧❛✿

P♦✐♥t✲s❧♦♣❡ ❢♦r♠ ♦❢ ❧✐♥❡ r✐s❡ = (y − y0 ) = ♣♦✐♥t

s❧♦♣❡

m

X

↓ y

· r✉♥ · (x − x0 ) ♣♦✐♥t

−

y0 ↑

♣♦✐♥t

= m ·

X

↓ (x

A

−

x0 ) ↑

♣♦✐♥t

A

❊①❛♠♣❧❡ ✸✳✶✳✺✿ ✜♥❞✐♥❣ s❧♦♣❡s ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ t✇♦ ❝♦❧✉♠♥s ♦❢ ♥✉♠❜❡rs ❜② ✐ts ♣♦✐♥ts ♦♥ t❤❡

xy ✲♣❧❛♥❡✳

x, y ✐♥ ❛ s♣r❡❛❞s❤❡❡t✳

❚❤✐s ❞❛t❛ ✐s ✈✐s✉❛❧✐③❡❞ ❛s ❛ ❝✉r✈❡ ❣✐✈❡♥

❚❤❡♥ ✇❡ ❝❛♥ ❝❛rr② ♦✉t ❛ ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ s❧♦♣❡ ❢♦r ❡❛❝❤ ❝♦♥s❡❝✉t✐✈❡

♣❛✐r ♦❢ ♣♦✐♥ts ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✉❧❛✿

❂✭❘❈❬✲✶❪✲❘❬✲✶❪❈❬✲✶❪✮✴✭❘❈❬✲✷❪✲❘❬✲✶❪❈❬✲✷❪✮ ❚❤❡ r❡s✉❧ts ❛r❡ ♣❧❛❝❡❞ ✐♥ t❤❡ t❤✐r❞ ❝♦❧✉♠♥✿

❚❤❡ ❧✐♥❡s ❛r❡ t❤❡♥ ♣❧♦tt❡❞ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ t❤❡♦r❡♠ ❛❜♦✈❡✿

✸✳✶✳ ❚❤❡ ❚❛♥❣❡♥t Pr♦❜❧❡♠

✷✶✺

❊①❛♠♣❧❡ ✸✳✶✳✻✿ ♣❛r❛❜♦❧❛

❚❤❡ ❤✐st♦r② ♦❢ t❤❡ ❛♥❝✐❡♥t ●r❡❡❝❡ t❡❧❧s ❛❜♦✉t t❤❡ ❢❛♠♦✉s ♠❛t❤❡♠❛t✐❝✐❛♥ ❆r❝❤✐♠❡❞❡s ✇❤♦ ❛rr❛♥❣❡❞ t❤❡ s❤✐❡❧❞s ♦❢ t❤❡ s♦❧❞❡rs ❛❧♦♥❣ t❤✐s ❝✉r✈❡ ✐♥ ♦r❞❡r t♦ s❡t ♦♥ ✜r❡ t❤❡ s❤✐♣s ♦❢ t❤❡ ❘♦♠❛♥s t❤❛t ❜❡s✐❡❣❡❞ ❤✐s ❤♦♠❡ ❝✐t② ♦❢ ❙②r❛❝✉s❡✳ ❍❡ ✉s❡❞ t❤❡ ❢❛❝t t❤❛t t❤❡ r❛②s ♦❢ ❧✐❣❤t ❜♦✉♥❝✐♥❣ ❢r♦♠ ❛ ♣❛r❛❜♦❧✐❝ ♠✐rr♦r ✇✐❧❧ ❛❧❧ ♠❡❡t ❛t ♦♥❡ ♣♦✐♥t ❝❛❧❧❡❞ t❤❡ ❢♦❝✉s ♦❢ t❤❡ ♣❛r❛❜♦❧❛✳ ❲❡ ❝❛♥ ❝♦♥✜r♠ t❤❛t ✐❞❡❛ ❜② tr❛❝✐♥❣ t❤❡ ♣❛t❤s ♦❢ t❤❡ ❧✐❣❤t ❜❡❛♠s ❛s t❤❡② ❜♦✉♥❝❡ ♦✛ t❤❡ s❡❝❛♥t ❧✐♥❡s✿

❈♦♥✈❡rs❡❧②✱ ❛ s♦✉r❝❡ ♦❢ ❧✐❣❤t ♣❧❛❝❡❞ ❛t t❤❡ ❢♦❝✉s ♦❢ ❛ ♣❛r❛❜♦❧❛ ✇✐❧❧ ❝r❡❛t❡ ❛ ❜❡❛♠ ♦❢ ♣❛r❛❧❧❡❧ r❛②s ♦❢ ❧✐❣❤t❀ t❤❡ ❢❛❝t ✐s ✉s❡❞ t♦ ❞❡s✐❣♥ ❝❛rs✬ ❤❡❛❞❧✐❣❤ts✿

❆♥♦t❤❡r ❤②♣♦t❤❡t✐❝❛❧ ♣♦ss✐❜✐❧✐t② ✐s t❤❛t ✇❡ ♠✐❣❤t s❡❡ ♥♦ ❝♦r♥❡rs ❡✈❡♥ ❛❢t❡r ③♦♦♠✐♥❣ ✐♥ ♠✉❧t✐♣❧❡ t✐♠❡s✿ ✇❡ ❞❡❛❧ ✇✐t❤ ✐❞❡❛❧✐③❡❞✱ ♠❛t❤❡♠❛t✐❝❛❧ ❝✉r✈❡s✳ ■♥ t❤✐s ❝❛s❡✱ ③♦♦♠✐♥❣ ✐♥ ♣r♦❞✉❝❡s ❛ ✈✐rt✉❛❧❧② str❛✐❣❤t ❧✐♥❡✿

✸✳✶✳

❚❤❡ ❚❛♥❣❡♥t Pr♦❜❧❡♠

✷✶✻

■t ✐s t❤✐s ❧✐♥❡ t❤❛t t❤❡ ❧✐❣❤t ✇✐❧❧ ❤✐t✳ ❚❤❡r❡❢♦r❡✱ ✐ts ♣❛t❤ ❢♦❧❧♦✇✐♥❣ t❤❡ ❝♦♥t❛❝t ❝❛♥ ❜❡ ❞❡r✐✈❡❞ ✐❢ t❤✐s ❧✐♥❡ ✐s ❢♦✉♥❞✳ ❚❤✐s ❧✐♥❡

t♦✉❝❤❡s

t❤❡ ❝✉r✈❡ ❛♥❞ ✐s ❝❛❧❧❡❞ ❛

t❛♥❣❡♥t

✭t❤❡ ●r❡❡❦ ❢♦r ✏t♦✉❝❤✑✮ ❧✐♥❡✳

❍♦✇ ❞♦ ✇❡ ✜♥❞ t❤❡s❡ ✏t♦✉❝❤✐♥❣✑ ❧✐♥❡s❄ ■t ✐s ♠✉❝❤ ♠♦r❡ ❝♦♠♣❧❡①✳ ❊①❛♠♣❧❡ ✸✳✶✳✼✿ ❝✐r❝❧❡

❚❤❡ ♣r♦❜❧❡♠ ♦❢ ✜♥❞✐♥❣ s✉❝❤ ❛ ❧✐♥❡ ❤❛s ❛♥ ❡❛s② s♦❧✉t✐♦♥ ❢♦r s♦♠❡ s♣❡❝✐✜❝ ❝✉r✈❡s✱ s✉❝❤ ❛s ❛ ❝✐r❝❧❡✿

❖♥❝❡ t❤❡② r❡❛❧✐③❡❞ t❤❛t t❤❡ r❛❞✐✉s ❛♥❞ s✉❝❤ ❛ ❧✐♥❡ ❢♦r♠ ❛

90✲❞❡❣r❡❡

❛♥❣❧❡✱ t❤❡ ♣r♦❜❧❡♠ ✇❛s s♦❧✈❡❞ ❜②

t❤❡ ❛♥❝✐❡♥t ●r❡❡❦s ✇✐t❤ ❥✉st r✉❧❡r ❛♥❞ ❝♦♠♣❛ss✳ ■♥❞❡❡❞✱ ♦♥❡ ❥✉st ♣❧♦t ❛♥♦t❤❡r ❝✐r❝❧❡ ✇✐t❤ ❞✐❛♠❡t❡r s❡r✈❡❞ ❜② t❤❡ s❡❣♠❡♥t ❜❡t✇❡❡♥ t❤❡ ♣♦✐♥t ❛♥❞ t❤❡ ❝❡♥t❡r ♦❢ t❤❡ ♦r✐❣✐♥❛❧ ❝✐r❝❧❡✳

❊①❛♠♣❧❡ ✸✳✶✳✽✿ ❤❡❛❞❧✐❣❤ts

❲❤❡r❡ ❞♦ t❤❡ ❧✐❣❤ts ♦❢ ❛ ❝❛r tr❛✈❡❧✐♥❣ ♦♥ ❛ ❝✉r✈② r♦❛❞ ♣♦✐♥t❄

✸✳✶✳ ❚❤❡ ❚❛♥❣❡♥t Pr♦❜❧❡♠

❊①❛♠♣❧❡ ✸✳✶✳✾✿ s❧✐♥❣

■♥ ✇❤❛t ❞✐r❡❝t✐♦♥ ✇♦✉❧❞ ❛ r♦❝❦ r❡❧❡❛s❡❞ ❢r♦♠ ❛ s❧✐♥❣ ❣♦ ✭✈✐❡✇ ❢r♦♠ ❛❜♦✈❡✮❄

❆♥s✇❡r✐♥❣ t❤❡ ❛❜♦✈❡ q✉❡st✐♦♥ ✇✐❧❧ ❛❧s♦ ❛♥s✇❡r t❤❡ q✉❡st✐♦♥s ❜❡❧♦✇✳ ❍♦✇ ✇♦✉❧❞ ❛ ❜✐❧❧✐❛r❞ ❜❛❧❧ ❜♦✉♥❝❡ ♦✛ ❛ ✇❛❧❧❄ ❙❛♠❡ ❛s ❧✐❣❤t✿

❍♦✇ ✇♦✉❧❞ t✇♦ ❜❛❧❧s ❜♦✉♥❝❡ ♦✛ ❡❛❝❤ ♦t❤❡r❄ ❆s ✐❢ t❤❡② ❜♦t❤ ❤✐t ❛ ✇❛❧❧✿

✷✶✼

✸✳✶✳ ❚❤❡ ❚❛♥❣❡♥t Pr♦❜❧❡♠

✷✶✽

◆❡①t✱ ✐♥ ✇❤✐❝❤ ❞✐r❡❝t✐♦♥ ✇✐❧❧ ❛ r❛❞❛r s✐❣♥❛❧ ❜♦✉♥❝❡ ♦✛ t❤❡ s✉r❢❛❝❡ ♦❢ ❛ ♣❧❛♥❡❄ ■❢ ②♦✉ ✇❛♥t t♦ ❦♥♦✇ ❛❤❡❛❞ ♦❢ t✐♠❡✱ ❞❡s✐❣♥ ❛ ♣❧❛♥❡ ✇✐t❤ ♥♦ ❝✉r✈❡s✿

❊①❛♠♣❧❡ ✸✳✶✳✶✵✿ ✉s✐♥❣ ❛ r✉❧❡r

Pr❛❝t✐❝❛❧❧②✱ ❤♦✇ ❞♦ ✇❡ ✜♥❞ t❤❡ t❛♥❣❡♥t ❧✐♥❡ ♦❢ ❛ ❝✉r✈❡ ❞r❛✇♥ ♦♥ ❛ ♣✐❡❝❡ ♦❢ ♣❛♣❡r❄ ❆s ✇❡ ❞♦♥✬t ❤❛✈❡ t❤❡ ❧✉①✉r② ♦❢ ③♦♦♠✐♥❣ ✐♥ ♦♥ ❛ ❞✐❣✐t❛❧ ✐♠❛❣❡✱ ✇❡ ❞r❛✇ str❛✐❣❤t ❧✐♥❡s ❝❧♦s❡r ❛♥❞ ❝❧♦s❡r t♦ t❤❡ ❣r❛♣❤ s♦ t❤❛t t❤❡ ❧❛st ♦♥❡ t♦✉❝❤❡s t❤❡ ❣r❛♣❤ ❛t t❤❛t ♣♦✐♥t✳

❚❤❡ r✉❧❡ ♦❢ t❤✉♠❜ ✐s t❤❛t ♦♥❡ s❤♦✉❧❞ ❡①♣❡❝t ✕ ✇❤❡♥ ③♦♦♠❡❞ ♦✉t ✕ ♦♥❧② ♦♥❡ ♣♦✐♥t ✐♥ ❝♦♠♠♦♥ ❜❡t✇❡❡♥ t❤❡ t❛♥❣❡♥t ❧✐♥❡ ❛♥❞ t❤❡ ♣❛rt ♦❢ t❤❡ ❣r❛♣❤ ♦❢ f ❝❧♦s❡ t♦ t❤❡ ♣♦✐♥t A✳ ❆♥♦t❤❡r ✇❛② t♦ ❦♥♦✇ t❤❛t ②♦✉ ❞✐❞ ✐t r✐❣❤t ✐s t♦ s❡❡ t❤❡ t❛♥❣❡♥t ❧✐♥❡ ❛s ❛♥ ❡❞❣❡ ♦❢ ❛ ♣✐❡❝❡ ♦❢ ♣❛♣❡r❀ t❤❡♥ t❤✐s ♣✐❡❝❡ ❤❛s t♦ ❝♦✈❡r ♥♦♥❡ ♦❢ t❤❡ ✭r❡❧❡✈❛♥t✮ ♣❛rts ♦❢ t❤❡ ❣r❛♣❤✿

❚❤❡r❡ ❛r❡ ❡①❝❡♣t✐♦♥s❀ t❤❡ ❧❛st ✐♠❛❣❡ ❣✐✈❡s ②♦✉ ❛♥ ✐♥✢❡❝t✐♦♥ ♣♦✐♥t ✇✐t❤ t❤❡ t❛♥❣❡♥t ❧✐♥❡ ❝✉tt✐♥❣ t❤❡ ❣r❛♣❤ ✐♥ ❤❛❧❢✳ ❚❤❡ ❛♥❛❧②t✐❝❛❧ ♠❡t❤♦❞ ♦❢ s♦❧✈✐♥❣ t❤✐s ❚❛♥❣❡♥t Pr♦❜❧❡♠ ✐s ♦♥❡ ♦❢ t❤❡ t✇♦ ♠❛✐♥ ♠♦t✐✈❛t✐♦♥s ✭t❤❡ ♦t❤❡r ♦♥❡ ✐s ♠♦t✐♦♥✮ ❢♦r t❤❡ ❞❡✈❡❧♦♣♠❡♥t ✐♥ t❤✐s ❝❤❛♣t❡r✳

✸✳✷✳ ❚❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ ❛ s❡q✉❡♥❝❡ ❛♥❞ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ ❛ ❢✉♥❝t✐♦♥

✷✶✾

▲❡t✬s ❡①❛♠✐♥❡ t❤❡ ❣❡♦♠❡tr② ✜rst✳ ■♥st❡❛❞ ♦❢ ❛ tr✐❛❧✲❛♥❞✲❡rr♦r ♦❢ t❤❡ ❧❛st ❡①❛♠♣❧❡✱ ✇❡ ✉s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛❧❣♦r✐t❤♠✳ ❲❡ ❜✉✐❧❞ ❛ s❡q✉❡♥❝❡ ♦❢ s❡❝❛♥t ❧✐♥❡s✳ ❊❛❝❤ ❧✐♥❡ ♣❛ss❡s t❤r♦✉❣❤ t✇♦ ♣♦✐♥ts✱ t❤❡ ♣♦✐♥t A = (a, f (a)) ✭s❛♠❡ ❢♦r ❛❧❧✮ ❛♥❞ s♦♠❡ ✈❛r✐❛❜❧❡ ♣♦✐♥t B = (b, f (b)) ✇✐t❤ a 6= b✿

❚❤❡s❡ s❡❝❛♥ts ❛r❡ ❡①♣❡❝t❡❞ t♦ ❣❡t ❝❧♦s❡r ❛♥❞ ❝❧♦s❡r t♦ t❤❡ t❛♥❣❡♥t ❛s B ✐s ❣❡tt✐♥❣ ❝❧♦s❡r ❛♥❞ ❝❧♦s❡r t♦ A✳ ❍♦✇❡✈❡r✱ t❤❡ s❡❝❛♥ts ❛r❡ ❛❧s♦ ❜❡❝♦♠✐♥❣ s❤♦rt❡r ❛♥❞ s❤♦rt❡r✱ ❛♥❞ ❧❡ss ❛♥❞ ❧❡ss ✉s❡❢✉❧✳ ■❢ ✇❡ ❡①t❡♥❞ t❤❡s❡ s❡❣♠❡♥ts ✐♥t♦ str❛✐❣❤t ❧✐♥❡s✱ ✇❡ ❝❛♥ ♦❜s❡r✈❡ t❤❡♠ r♦t❛t❡✳ ❚❤❡② ✇✐❧❧ r♦t❛t❡ ❧❡ss ❛♥❞ ❧❡ss ❛s t❤❡② ❛r❡ ❣❡tt✐♥❣ ❝❧♦s❡r ❛♥❞ ❝❧♦s❡r t♦ t❤❡ t❛♥❣❡♥t ❧✐♥❡✳ ❇✉t ❛♥ ❡✈❡♥ ❜❡tt❡r ✐❞❡❛ ✐s ❛❧❣❡❜r❛✐❝✿ ❋♦❧❧♦✇ t❤❡ s❧♦♣❡s ✦ ❙✐♥❝❡ t❤❡② ❛r❡ ♥♦t❤✐♥❣ ❜✉t ♥✉♠❜❡rs✱ ✇❡ ❛r❡ ❞❡❛❧✐♥❣ ✇✐t❤ ❛ s❡q✉❡♥❝❡✳ ❆ s❡q✉❡♥❝❡ ❛♥❞ ✐ts ❧✐♠✐t ✦ ❲❡ ✇✐❧❧✱ ❤♦✇❡✈❡r✱ s❡t ❧✐♠✐ts ❛s✐❞❡ ❢♦r ♥♦✇ ❛♥❞ ❛❞❞r❡ss ✜rst t❤❡ ♦t❤❡r ❝❤❛❧❧❡♥❣❡ ♣♦s❡❞ ❛❜♦✈❡✿ ❤♦✇ t♦ ❞❡❛❧ ✇✐t❤ ❞✐✛❡r❡♥t s❧♦♣❡s ❛t ❞✐✛❡r❡♥t ❧♦❝❛t✐♦♥s✱ ✐✳❡✳✱ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✿

❆♥② ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ s❧♦♣❡ st❛rts ✇✐t❤ ❛ ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ r✐s❡✳

✸✳✷✳ ❚❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ ❛ s❡q✉❡♥❝❡ ❛♥❞ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ ❛ ❢✉♥❝✲ t✐♦♥ ❲❡ ❡♥t❡r ❝❛❧❝✉❧✉s t❤r♦✉❣❤ ❛ st✉❞② ♦❢ ♠♦t✐♦♥✳ ❘❡❝❛❧❧ ❢r♦♠ ❈❤❛♣t❡r ✶ ♦✉r r✉♥♥✐♥❣ ❡①❛♠♣❧❡ ♦❢ ❛ ❜r♦❦❡♥ s♣❡❡❞♦♠❡t❡r ❛♥❞ ❛ ❜r♦❦❡♥ ♦❞♦♠❡t❡r✳ ❲❤❡♥ t❤❡ ✐♥t❡r✈❛❧s ♦❢ t✐♠❡ ❛r❡ ✉♥✐ts✱ ✇❡ ❝❛♥ ❣♦ ❢r♦♠ ❧♦❝❛t✐♦♥s t♦ ✈❡❧♦❝✐t✐❡s ❛♥❞ ❜❛❝❦ ✇✐t❤ t❤❡ t✇♦ s✐♠♣❧❡ ♦♣❡r❛t✐♦♥s ♦❢ ❞✐✛❡r❡♥❝❡ ❛♥❞ s✉♠ ✿

✸✳✷✳ ❚❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ ❛ s❡q✉❡♥❝❡ ❛♥❞ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ ❛ ❢✉♥❝t✐♦♥

✷✷✵

❚❤✐s ✐s t❤❡ s✉♠♠❛r②✿ • ■❢ ❡❛❝❤ t❡r♠ ♦❢ ❛ s❡q✉❡♥❝❡ r❡♣r❡s❡♥ts ❛ ❧♦❝❛t✐♦♥✱ t❤❡ ♣❛✐r✲✇✐s❡ ❞✐✛❡r❡♥❝❡s ✇✐❧❧ ❣✐✈❡ ②♦✉ t❤❡

❆ ❜r♦❦❡♥ s♣❡❡❞♦♠❡t❡r ✐s s✉❜st✐t✉t❡❞ ✇✐t❤ ❛♥ ♦❞♦♠❡t❡r ❛♥❞ ❛ ✇❛t❝❤✳

✈❡❧♦❝✐t✐❡s✳

• ■❢ ❡❛❝❤ t❡r♠ ♦❢ ❛ s❡q✉❡♥❝❡ r❡♣r❡s❡♥ts ❛ ✈❡❧♦❝✐t②✱ t❤❡✐r s✉♠ ✉♣ t♦ t❤❛t ♣♦✐♥t ✇✐❧❧ ❣✐✈❡ ②♦✉ t❤❡

❆ ❜r♦❦❡♥ ♦❞♦♠❡t❡r ✐s s✉❜st✐t✉t❡❞ ✇✐t❤ ❛ s♣❡❡❞♦♠❡t❡r ❛♥❞ ❛ ✇❛t❝❤✳

❧♦❝❛t✐♦♥✳

■♥ t❤❡ ❛❜str❛❝t✱ t❤❡ ♣❛✐r✇✐s❡ ❞✐✛❡r❡♥❝❡s r❡♣r❡s❡♥t t❤❡ ❝❤❛♥❣❡ ✇✐t❤✐♥ t❤❡ s❡q✉❡♥❝❡✱ ❢r♦♠ ❡❛❝❤ ♦❢ ✐ts t❡r♠s t♦ t❤❡ ♥❡①t ✭s❡❡♥ ✐♥ ❱♦❧✉♠❡ ✶✱ ❈❤❛♣t❡r ✶P❈✲✶✮✳

❊①❛♠♣❧❡ ✸✳✷✳✶✿ s❡q✉❡♥❝❡ ❣✐✈❡♥ ❜② ❣r❛♣❤ ■♥ t❤❡ s✐♠♣❧❡st ❝❛s❡✱ ❛ s❡q✉❡♥❝❡ t❛❦❡s ♦♥❧② ✐♥t❡❣❡r ✈❛❧✉❡s✱ t❤❡♥ ♦♥ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ s❡q✉❡♥❝❡✱ ✇❡ ❥✉st ❝♦✉♥t t❤❡ ♥✉♠❜❡r ♦❢ st❡♣s ✇❡ ♠❛❦❡✱ ✉♣ ❛♥❞ ❞♦✇♥✿

❚❤❡s❡ ✐♥❝r❡♠❡♥ts t❤❡♥ ♠❛❦❡ ❛ ♥❡✇ s❡q✉❡♥❝❡ ♣❧♦tt❡❞ ♦♥ t❤❡ r✐❣❤t✳ ❚❤❡ ✐❞❡❛ ❧❡❛❞s t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ ❞❡✜♥✐t✐♦♥✿

❉❡✜♥✐t✐♦♥ ✸✳✷✳✷✿ s❡q✉❡♥❝❡ ♦❢ ❞✐✛❡r❡♥❝❡s ❋♦r ❛ s❡q✉❡♥❝❡ an ✱ ✐ts s❡q✉❡♥❝❡ ♦❢ ❞✐✛❡r❡♥❝❡s✱ ♦r s✐♠♣❧② t❤❡ ❞✐✛❡r❡♥❝❡✱ ✐s ❛ ♥❡✇ s❡q✉❡♥❝❡✱ s❛② dn ✱ ❞❡✜♥❡❞ ❢♦r ❡❛❝❤ n ❜② t❤❡ ❢♦❧❧♦✇✐♥❣✿ dn = an+1 − an .

■t ✐s ❞❡♥♦t❡❞ ❛s ❢♦❧❧♦✇s✿ ∆an = an+1 − an

❚❤♦s❡ ❛r❡ t❤❡ r✐s❡s t❤❛t ✇❡ ✇✐❧❧ s♦♦♥ ❜❡ ✉s✐♥❣ t♦ ❝♦♠♣✉t❡ s♦♠❡ s❧♦♣❡s✳ ❚❤✐s ✐s ❤♦✇ t❤❡ ♥❡✇ s❡q✉❡♥❝❡ ✐s ❜✉✐❧t✿

✸✳✷✳

❚❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ ❛ s❡q✉❡♥❝❡ ❛♥❞ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ ❛ ❢✉♥❝t✐♦♥

✷✷✶

❙❡q✉❡♥❝❡ ♦❢ ❞✐✛❡r❡♥❝❡s ❛ s❡q✉❡♥❝❡✿

a1

a2 ց

✐ts ❞✐✛❡r❡♥❝❡s✿ ❛ ♥❡✇ s❡q✉❡♥❝❡✿ t❤❡ ♥♦t❛t✐♦♥✿

a2 − a1 || d1 || ∆a1

ւ

a3 ց

a3 − a2 || d2 || ∆a2

ւ

ց

a4 ... ւ ... ... ... ... ... ...

a4 − a3 || d3 || ∆a3

❊①❛♠♣❧❡ ✸✳✷✳✸✿ s❡q✉❡♥❝❡ ❣✐✈❡♥ ❜② ❧✐st ❲❤❡♥ ❛ s❡q✉❡♥❝❡ ✐s ❣✐✈❡♥ ❜② ❛ ❧✐st✱ ✇❡ s✉❜tr❛❝t t❤❡ ❧❛st t❡r♠ ❢r♦♠ t❤❡ ❝✉rr❡♥t ♦♥❡ ❛♥❞ ♣✉t t❤❡ r❡s✉❧t ❜❡❧♦✇ ❛s ❢♦❧❧♦✇s✿ ❛ s❡q✉❡♥❝❡✿

2

4 ց

✐ts ❞✐✛❡r❡♥❝❡s✿ ❛ ♥❡✇ s❡q✉❡♥❝❡✿

4−2 || 2

ւ

7 ց

7−4 || 3

ւ

1 ց

1−7 || −6

ւ

... ց ... ... ... ...

❲❡ ❤❛✈❡ ❛ ♥❡✇ ❧✐st✳

❊①❡r❝✐s❡ ✸✳✷✳✹ P❧♦t t❤❡ ❧♦❝❛t✐♦♥ ❛♥❞ t❤❡ ✈❡❧♦❝✐t② ❢♦r t❤❡ ❢♦❧❧♦✇✐♥❣ tr✐♣✿ ✏■ ❞r♦✈❡ ❢❛st✱ t❤❡♥ ❣r❛❞✉❛❧❧② s❧♦✇❡❞ ❞♦✇♥✱ st♦♣♣❡❞ ❢♦r ❛ ✈❡r② s❤♦rt ♠♦♠❡♥t✱ ❣r❛❞✉❛❧❧② ❛❝❝❡❧❡r❛t❡❞✱ ♠❛✐♥t❛✐♥❡❞ s♣❡❡❞✱ ❤✐t ❛ ✇❛❧❧✳✑ ▼❛❦❡ ✉♣ ②♦✉r ♦✇♥ st♦r② ❛♥❞ r❡♣❡❛t t❤❡ t❛s❦✳

❊①❡r❝✐s❡ ✸✳✷✳✺ ❉r❛✇ ❛ ❝✉r✈❡ ♦♥ ❛ ♣✐❡❝❡ ♦❢ ♣❛♣❡r✱ ✐♠❛❣✐♥❡ t❤❛t ✐t r❡♣r❡s❡♥ts ②♦✉r ❧♦❝❛t✐♦♥s✱ ❛♥❞ t❤❡♥ s❦❡t❝❤ ✇❤❛t ②♦✉r ✈❡❧♦❝✐t② ✇♦✉❧❞ ❧♦♦❦ ❧✐❦❡✳ ❘❡♣❡❛t✳

❊①❡r❝✐s❡ ✸✳✷✳✻ ■♠❛❣✐♥❡ t❤❛t t❤❡ ✜rst ❣r❛♣❤ r❡♣r❡s❡♥ts✱ ✐♥st❡❛❞ ♦❢ ❧♦❝❛t✐♦♥s✱ t❤❡ ❜❛❧❛♥❝❡s ♦❢ ❜❛♥❦ ❛❝❝♦✉♥ts✳ ❉❡s❝r✐❜❡ ✇❤❛t ❤❛s ❜❡❡♥ ❤❛♣♣❡♥✐♥❣✳

❊①❛♠♣❧❡ ✸✳✷✳✼✿ ✈❡❧♦❝✐t✐❡s ◆♦✇✱ ✇❤❛t ✐❢ t❤❡ t✐♠❡ ✐♥t❡r✈❛❧s ❛r❡♥✬t ✉♥✐ts❄ ❲❡ st✐❧❧ ❧♦♦❦ ❛t t❤❡ ♦❞♦♠❡t❡r

s❡✈❡r❛❧

t✐♠❡s ❞✉r✐♥❣ t❤❡ tr✐♣ ❛♥❞ r❡❝♦r❞ t❤❡ ♠✐❧❡❛❣❡ ♦♥ ❛ ♣✐❡❝❡ ♦❢ ♣❛♣❡r✳

❇✉t t❤✐s t✐♠❡ ✐t✬s ❡✈❡r② ❤❛❧❢✲❤♦✉r✳ ❚❤✐s ❧✐st ♦❢ ♦✉r ❝♦♥s❡❝✉t✐✈❡

• • • • •

✐♥✐t✐❛❧ r❡❛❞✐♥❣✿

10, 000

♠✐❧❡s

10, 055 ♠✐❧❡s 10, 095 ♠✐❧❡s ❤❛❧❢✲❤♦✉r✿ 10, 155 ♠✐❧❡s

❛❢t❡r t❤❡ ✜rst ❤❛❧❢✲❤♦✉r✿

❛❢t❡r t❤❡ s❡❝♦♥❞ ❤❛❧❢✲❤♦✉r✿ ❛❢t❡r t❤❡ t❤✐r❞ ❡t❝✳

❲❡ ♣❧♦t t❤✐s ❞❛t❛ ✭t♦♣✮✿

❧♦❝❛t✐♦♥s

♠✐❣❤t ❧♦♦❦ ❧✐❦❡ t❤✐s✿

✸✳✷✳ ❚❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ ❛ s❡q✉❡♥❝❡ ❛♥❞ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ ❛ ❢✉♥❝t✐♦♥

✷✷✷

❲❡ ❛❧s♦ ❝♦♠♣✉t❡ t❤❡ ❞✐✛❡r❡♥❝❡s ✭❜♦tt♦♠✮✳ ❲❤❛t ✐s t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡s❄ ❚❤❡② ❛r❡ t❤❡ ❞✐st❛♥❝❡s ❝♦✈❡r❡❞ ❞✉r✐♥❣ ❡❛❝❤ ♦❢ t❤❡s❡ ❤❛❧❢✲❤♦✉r ♣❡r✐♦❞s✱ ❜② s✉❜tr❛❝t✐♦♥ ✿ • ❞✐st❛♥❝❡ ❝♦✈❡r❡❞ ❞✉r✐♥❣ t❤❡ ✜rst ❤❛❧❢✲❤♦✉r✿ 10, 055 − 10, 000 = 55 ♠✐❧❡s • ❞✐st❛♥❝❡ ❝♦✈❡r❡❞ ❞✉r✐♥❣ t❤❡ s❡❝♦♥❞ ❤❛❧❢✲❤♦✉r✿ 10, 095 − 10, 055 = 40 ♠✐❧❡s • ❞✐st❛♥❝❡ ❝♦✈❡r❡❞ ❞✉r✐♥❣ t❤❡ t❤✐r❞ ❤❛❧❢✲❤♦✉r✿ 10, 155 − 10, 095 = 60 ♠✐❧❡s • ❡t❝✳ ❚❤❡② ❛r❡ ❝❛❧❧❡❞ t❤❡ ❞✐s♣❧❛❝❡♠❡♥ts✳ ❊①❛♠♣❧❡ ✸✳✷✳✽✿ t❤r❡❡ r✉♥♥❡rs

❚❤❡ ❣r❛♣❤ ❜❡❧♦✇ s❤♦✇s t❤❡ ♣♦s✐t✐♦♥s ♦❢ t❤r❡❡ r✉♥♥❡rs ✐♥ t❡r♠s ♦❢ t✐♠❡✱ n✳ ❉❡s❝r✐❜❡ ✇❤❛t ❤❛s ❤❛♣♣❡♥❡❞✿

❚❤❡② ❛r❡ ❛❧❧ ❛t t❤❡ st❛rt✐♥❣ ❧✐♥❡ t♦❣❡t❤❡r✱ ❛♥❞ ❛t t❤❡ ❡♥❞✱ t❤❡② ❛r❡ ❛❧❧ ❛t t❤❡ ✜♥✐s❤ ❧✐♥❡✳ ❋✉rt❤❡r♠♦r❡✱ A r❡❛❝❤❡s t❤❡ ✜♥✐s❤ ❧✐♥❡ ✜rst✱ ❢♦❧❧♦✇❡❞ ❜② B ✱ ❛♥❞ t❤❡♥ C ✭✇❤♦ ❛❧s♦ st❛rts ❧❛t❡✮✳ ❚❤✐s ✐s ❤♦✇ ❡❛❝❤ ❞✐❞ ✐t✿ • A st❛rts ❢❛st✱ t❤❡♥ s❧♦✇s ❞♦✇♥✱ ❛♥❞ ❛❧♠♦st st♦♣s ❝❧♦s❡ t♦ t❤❡ ✜♥✐s❤ ❧✐♥❡✳ • B ♠❛✐♥t❛✐♥s t❤❡ s❛♠❡ s♣❡❡❞✳ • C st❛rts ❧❛t❡ ❛♥❞ t❤❡♥ r✉♥s ❢❛st ❛t t❤❡ s❛♠❡ s♣❡❡❞✳ ❲❡ ❝❛♥ s❡❡ t❤❛t A ✐s r✉♥♥✐♥❣ ❢❛st❡r ❜❡❝❛✉s❡ t❤❡ ❞✐st❛♥❝❡ ❢r♦♠ B ✐s ✐♥❝r❡❛s✐♥❣✳ ■t ❜❡❝♦♠❡s s❧♦✇❡r ❧❛t❡r✱ ✇❤✐❝❤ ✐s ✈✐s✐❜❧❡ ❢r♦♠ t❤❡ ❞❡❝r❡❛s✐♥❣ ❞✐st❛♥❝❡✳ ❚❤❡r❡ ✐s ❛♥♦t❤❡r ✇❛②✦ ❲❡ ❝❛♥ ❞✐s❝♦✈❡r t❤✐s ❛♥❞ t❤❡ r❡st ♦❢ t❤❡ ❢❛❝ts ❜② ❡①❛♠✐♥✐♥❣ t❤❡ ❣r❛♣❤s ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡s ♦❢ t❤❡ s❡q✉❡♥❝❡s✿

✸✳✷✳ ❚❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ ❛ s❡q✉❡♥❝❡ ❛♥❞ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ ❛ ❢✉♥❝t✐♦♥

✷✷✸

❋♦r ❡①❛♠♣❧❡✱ ✇❡ ❝❧❡❛r❧② s❡❡ t❤❡ ❝♦♥st❛♥t ✈❡❧♦❝✐t✐❡s ♦❢ t❤❡ t❤r❡❡ ❛t t❤❡ ❡♥❞ ♦❢ t❤❡ r✉♥✳ ■♥ ❣❡♥❡r❛❧✱ ✇❡ ❢❛❝❡ ❢✉♥❝t✐♦♥s ✿

■♥ ♦r❞❡r t♦ ❝♦♠♣✉t❡ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ ❛ ❢✉♥❝t✐♦♥✱ ✇❡ ♥❡❡❞ t♦ s❛♠♣❧❡ ✐t✳ ❚❤✐s ✐s ❤♦✇ ✐t ✐s ❞♦♥❡✳ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛♥ ✐♥t❡r✈❛❧ [a, b]✳ ❲❡ ❝♦♥str✉❝t ✐ts n ❛♥❞ t❤❡♥ ♣❧❛❝❡ n + 1 ♣♦✐♥ts ♦♥ t❤❡ ✐♥t❡r✈❛❧✿

♣❛rt✐t✐♦♥ ❛s ❢♦❧❧♦✇s✳ ❋✐rst✱ ✇❡ ❝❤♦♦s❡ ❛ ♥❛t✉r❛❧ ♥✉♠❜❡r

a = x0 < x1 < x2 < ... < xn−1 < xn = b .

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡ ❛ str✐❝t❧② ✐♥❝r❡❛s✐♥❣ s❡q✉❡♥❝❡ xi ✿

❆s ❛ r❡s✉❧t✱ t❤❡ ✐♥t❡r✈❛❧ ✐s s♣❧✐t ✐♥t♦ n s♠❛❧❧❡r ✐♥t❡r✈❛❧s ♦❢ ♣♦ss✐❜❧② ❞✐✛❡r❡♥t ❧❡♥❣t❤s✿ [x0 , x1 ], [x1 , x2 ], ..., [xn−1 , xn ] .

❲❡ ✇✐❧❧ ✉s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ t❡r♠✐♥♦❧♦❣②✿

❉❡✜♥✐t✐♦♥ ✸✳✷✳✾✿ ♣❛rt✐t✐♦♥ ♦❢ ❛♥ ✐♥t❡r✈❛❧ ❆ ♣❛rt✐t✐♦♥ ♦❢ ❛♥ ✐♥t❡r✈❛❧ [a, b] ✐s ✐ts r❡♣r❡s❡♥t❛t✐♦♥ ❛s t❤❡ ✉♥✐♦♥ ♦❢ ✐♥t❡r✈❛❧s t❤❛t ✐♥t❡rs❡❝t ♦♥❧② ❛t t❤❡✐r ❡♥❞✲♣♦✐♥ts✿ [a, b] = [x0 , x1 ] ∪ [x1 , x2 ] ∪ ... ∪ [xn−1 , xn ] .

✸✳✷✳ ❚❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ ❛ s❡q✉❡♥❝❡ ❛♥❞ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ ❛ ❢✉♥❝t✐♦♥

✷✷✹

❚❤❡s❡ ❡♥❞✲♣♦✐♥ts✱ a = x0 , x1 , x2 , ..., xn−1 , xn = b ,

✇✐❧❧ ❜❡ ❝❛❧❧❡❞ t❤❡ ♥♦❞❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥✳ ❚❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤✐s s❡q✉❡♥❝❡ ❣✐✈❡s ✉s t❤❡ ❧❡♥❣t❤s ♦❢ t❤❡s❡ ✐♥t❡r✈❛❧s✿

❉❡✜♥✐t✐♦♥ ✸✳✷✳✶✵✿ ✐♥❝r❡♠❡♥ts ♦❢ ♣❛rt✐t✐♦♥ ❚❤❡ ✐♥❝r❡♠❡♥ts ♦❢ t❤❡ ♣❛rt✐t✐♦♥ ❛r❡ ❣✐✈❡♥ ❜②✿ ∆xi = xi − xi−1 , i = 1, 2, ..., n .

■t ✐s s✐♠♣❧② ∆x ✇❤❡♥ t❤❡② ❛r❡ ❡q✉❛❧✳ ❚❤♦s❡ ❛r❡ t❤❡ r✉♥s t❤❛t ✇❡ ✇✐❧❧ s♦♦♥ ❜❡ ✉s✐♥❣ t♦ ❝♦♠♣✉t❡ s♦♠❡ s❧♦♣❡s✳

❊①❛♠♣❧❡ ✸✳✷✳✶✶✿ ❞❡s❝r✐♣t✐♦♥ ♦❢ ❛ ❞r✐✈❡ ❙✉♣♣♦s❡ ✇❡ ❞r♦✈❡ ❛r♦✉♥❞ ✇❤✐❧❡ ♣❛②✐♥❣ ❛tt❡♥t✐♦♥ t♦ t❤❡ ❝❧♦❝❦ ❛♥❞ t♦ t❤❡ ♠✐❧❡♣♦sts✳ ❲❡ ♣r♦❞✉❝❡❞ t❤✐s s✐♠♣❧❡ t❛❜❧❡ ✇✐t❤ ✜✈❡ ❝♦❧✉♠♥s✿ t✐♠❡ ✭❤♦✉rs✮✿ ❧♦❝❛t✐♦♥ ✭♠✐❧❡s✮✿

0 2 4 6 8 0 60 160 80 0

❲❤❛t ✇❡r❡ t❤❡ ❞✐s♣❧❛❝❡♠❡♥ts ♦✈❡r t❤❡s❡ ❢♦✉r ♣❡r✐♦❞s ♦❢ t✐♠❡❄ ❲❡ ✉s❡ t❤❡ ❞✐✛❡r❡♥❝❡ ❢♦r♠✉❧❛✿ ❞✐s♣❧❛❝❡♠❡♥t = ❝❤❛♥❣❡ ♦❢ ❧♦❝❛t✐♦♥ ❚❤❡s❡ ❛r❡ t❤❡ ❝♦♠♣✉t❛t✐♦♥s✿ t✐♠❡ ✭❤♦✉rs✮✿

0

2

4

6

8

❧♦❝❛t✐♦♥ ✭♠✐❧❡s✮✿

0

60

160

80

0

60 − 0 =

60

❞✐s♣❧❛❝❡♠❡♥t ✭♠✐❧❡s✮✿ ❞✐s♣❧❛❝❡♠❡♥t ✭♠✐❧❡s✮✿

160 − 60 =

❞✐s♣❧❛❝❡♠❡♥t ✭♠✐❧❡s✮✿ ❞✐s♣❧❛❝❡♠❡♥t ✭♠✐❧❡s✮✿

100 80 − 160 =

−80 0 − 80 = −80

❇❡❝❛✉s❡ t❤❡ t✐♠❡ ✐♥t❡r✈❛❧s ❛r❡ ∆x = 1✱ t❤❡s❡ ❞✐s♣❧❛❝❡♠❡♥ts ❛r❡ ❛❧s♦ t❤❡ ✈❡❧♦❝✐t✐❡s✳ ❚❤❡ ❢♦✉r ❝♦♠♣✉t❡❞ ✈❛❧✉❡s ❛r❡ t❤❡ ❞✐s♣❧❛❝❡♠❡♥ts ♦✈❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♥t❡r✈❛❧s ♦❢ t✐♠❡✿ [0, 2]✱ [2, 4]✱ [4, 6]✱ ❛♥❞ [6, 8]✱ r❡s♣❡❝t✐✈❡❧②✳ ❚❤❡ r❡s✉❧t ✐s t❤✐s t❛❜❧❡✿ t✐♠❡ ✐♥t❡r✈❛❧s ✭❤♦✉rs✮✿ ✈❡❧♦❝✐t② ✭♠✐❧❡s✴❤♦✉r✮✿ ❚❤✐s ✐s t❤❡ s✉♠♠❛r② ♦❢ ✇❤❛t ✇❡ ❤❛✈❡ ❞♦♥❡✿

[0, 2] [2, 4] [4, 6] [6, 8] 60 100 −80 −80

✸✳✷✳ ❚❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ ❛ s❡q✉❡♥❝❡ ❛♥❞ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ ❛ ❢✉♥❝t✐♦♥

✷✷✺

❲❡ ❝❛♥ ❛❧s♦ ❝❤♦♦s❡ t♦ ❛ss✐❣♥ t❤❡ ❢♦✉r ✈❛❧✉❡s t♦✱ s❛②✱ t❤❡ ♠✐❞❞❧❡ ♣♦✐♥ts ♦❢ t❤❡s❡ ✐♥t❡r✈❛❧s✱ ❛s ❢♦❧❧♦✇s✿ t✐♠❡ ✭❤♦✉rs✮✿ ✈❡❧♦❝✐t② ✭♠✐❧❡s✴❤♦✉r✮✿

1 3 5 7 60 100 −80 −80

❙♦✱ ✐♥ ❛❞❞✐t✐♦♥ t♦ t❤❡ ♥♦❞❡s✱ t❤❡ ♣r✐♠❛r② ♥♦❞❡s✱ ✇❡ ♠❛② ❛❧s♦ ❜❡ ❣✐✈❡♥ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ✐♥ ❡❛❝❤ ✐♥t❡r✈❛❧ ♦❢ t❤❡ ♣❛rt✐t✐♦♥✿

❲❡ ❤❛✈❡ ❛♥♦t❤❡r s❡q✉❡♥❝❡✿ c1 ✐♥ [x0 , x1 ], c2 ✐♥ [x1 , x2 ], ..., cn ✐♥ [xn−1 , xn ] .

❚❤❡ ❞❡✜♥✐t✐♦♥ ❜❡❧♦✇ s✉♠♠❛r✐③❡s t❤✐s s❡t✉♣✿

❉❡✜♥✐t✐♦♥ ✸✳✷✳✶✷✿ ❛✉❣♠❡♥t❡❞ ♣❛rt✐t✐♦♥ ♦❢ ✐♥t❡r✈❛❧ ❆♥ ❛✉❣♠❡♥t❡❞ ♣❛rt✐t✐♦♥✱ ♦r s✐♠♣❧② ❛ ♣❛rt✐t✐♦♥✱ ♦❢ ❛♥ ✐♥t❡r✈❛❧ [a, b] ❝♦♥s✐sts ♦❢ t✇♦ s❡q✉❡♥❝❡s✿ ✶✳ ♣r✐♠❛r② ♥♦❞❡s a = x0 , x1 , x2 , ..., xn−1 , xn = b ✷✳ s❡❝♦♥❞❛r② ♥♦❞❡s c1 , c2 , x3 , ..., cn−1 , cn t❤❛t s❛t✐s❢② t❤❡s❡ ✐♥❡q✉❛❧✐t✐❡s✿ x0 ≤ c1 ≤ x1 ≤ c2 ≤ x2 ≤ ... ≤ xn−1 ≤ cn ≤ xn .

✸✳✷✳

❚❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ ❛ s❡q✉❡♥❝❡ ❛♥❞ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ ❛ ❢✉♥❝t✐♦♥

✷✷✻

❲❛r♥✐♥❣✦ ❲❡ ❝❛♥ ❝❤♦♦s❡ s❡❝♦♥❞❛r② ♥♦❞❡s ❢r♦♠ t❤❡ ❧✐st ♦❢ ♣r✐✲ ♠❛r② ♥♦❞❡s ❜❡❝❛✉s❡ t❤❡ ✐♥❡q✉❛❧✐t✐❡s ❛r❡ ♥♦♥✲str✐❝t✳

❆ ♣❛rt✐t✐♦♥ ✐s ❛ ♣r♦♣❡r s❡tt✐♥❣ ❢♦r s❛♠♣❧✐♥❣ ❢✉♥❝t✐♦♥s✳

❊①❛♠♣❧❡ ✸✳✷✳✶✸✿ s❛♠♣❧✐♥❣ ▲❡t✬s ❣❡♥❡r❛❧✐③❡ t❤❡ ❧❛st ❡①❛♠♣❧❡✳ ♠♦r❡ ❣❡♥❡r❛❧ s❡tt✐♥❣✳

❋✐rst✱ ✇❡ ❝♦♥s✐❞❡r t❤❡ tr✐♣ ✇✐t❤ ❛ ❜r♦❦❡♥ s♣❡❡❞♦♠❡t❡r ✐♥ ❛

❲❡ ❤❛✈❡ ❛ t✐♠❡ ✐♥t❡r✈❛❧

[a, b]✳

■♥ ♦r❞❡r t♦ ❡st✐♠❛t❡ ♦✉r s♣❡❡❞✱ ✇❡ ❞❡❝✐❞❡

t♦ ❧♦♦❦ ❛t t❤❡ ♠✐❧❡st♦♥❡s s❡✈❡r❛❧ t✐♠❡s ❞✉r✐♥❣ t❤❡ tr✐♣✳

❚❤❡ ♠♦♠❡♥ts ♦❢ t✐♠❡ ♠❛② ❜❡ r❛♥❞♦♠✿

a = x0 , x1 , x2 , ..., xn−1 , xn = b ✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡ ❛ ♣❛rt✐t✐♦♥ ♦❢ [a, b]✱ ❛♥❞ ✇❡ s❛♠♣❧❡ t❤❡ ❢✉♥❝t✐♦♥ f t❤❛t r❡♣r❡s❡♥ts ♦✉r ❧♦❝❛t✐♦♥ ❛t t❤❡ ♣r✐♠❛r② ♥♦❞❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥✳ ❲❡ ❤❛✈❡ ♦✉r ❛♥s✇❡r ❜✉t ✇❡ ❝❛♥ ❛❧s♦ ❛ss✐❣♥ t❤❡s❡ ♥✉♠❜❡rs t♦ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ❛s ❛ ♠❛tt❡r ♦❢ ❜♦♦❦❦❡❡♣✐♥❣✿ t✐♠❡ ✐♥t❡r✈❛❧s ✭❤♦✉rs✮✿ t✐♠❡ ✭❤♦✉rs✮✿ ✈❡❧♦❝✐t② ✭♠✐❧❡s✴❤♦✉r✮✿

[0, 2] [2, 4] [4, 6] [6, 8] c1 c2 c3 c4 60 100 −80 −80

■♥ t❤❡ ❡①❛♠♣❧❡s✱ ✇❡ ❝❤♦s❡ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ✐♥ ❛ ❝♦♥s✐st❡♥t ✇❛②✳ ❋✐rst ✇❡ ❝❛♥ ❝❤♦♦s❡ ❡q✉❛❧ ✐♥❝r❡♠❡♥ts✿

h = ∆x =

b−a n

❲✐t❤ t❤✐s ❛ss✉♠♣t✐♦♥✱ ✇❡ ♥♦✇ ❝❤♦♦s❡ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s✿

❚❤❡r❡ ❛r❡ t❤r❡❡ ♠❛✐♥ ✏s❝❤❡♠❡s✑ ❢♦r ❝❤♦♦s✐♥❣ s❡❝♦♥❞❛r② ♥♦❞❡s✳ ❖♥❡ ✐s s❡❡♥ ❛❜♦✈❡✿ ❚❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ❛r❡ ♣❧❛❝❡❞ ❛t t❤❡ ❡♥❞ ♦❢ ❡❛❝❤ ✐♥t❡r✈❛❧✳ ■t ✐s ❝❛❧❧❡❞ t❤❡

•

❚❤❡ ♣r✐♠❛r② ♥♦❞❡s ❛r❡

•

❚❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ❛r❡

❚❤✐s ✐s t❤❡

r✐❣❤t✲❡♥❞ s❝❤❡♠❡ ✿

x = a, a + h, a + 2h, ...✳ c = a + h, a + 2h, ...✳

❧❡❢t✲❡♥❞ s❝❤❡♠❡ ✿

•

❚❤❡ ♣r✐♠❛r② ♥♦❞❡s ❛r❡

•

❚❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ❛r❡

x = a, a + h, a + 2h, ...✳ c = a, a + h, ...✳

❆♥♦t❤❡r ❝♦♥✈❡♥✐❡♥t ❝❤♦✐❝❡ ✐s t❤❡

•

❚❤❡ ♣r✐♠❛r② ♥♦❞❡s ❛r❡

•

❚❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ❛r❡

❚❤❡② ❛r❡ ✐❧❧✉str❛t❡❞ ❜❡❧♦✇✿

♠✐❞✲♣♦✐♥t s❝❤❡♠❡ ✿

x = a, a + h, a + 2h, ...✳ c = a + h/2, a + 3h/2, ...✳

✸✳✷✳ ❚❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ ❛ s❡q✉❡♥❝❡ ❛♥❞ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ ❛ ❢✉♥❝t✐♦♥

❚❤❡ ✜♥❛❧ st❡♣ ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✿ ◮ ❲❡ ✉t✐❧✐③❡ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ❛s t❤❡ ✐♥♣✉ts ♦❢ ❛ ♥❡✇ ❢✉♥❝t✐♦♥✳

❚❤❡ ✇❤♦❧❡ ❝♦♥str✉❝t✐♦♥ ✐s ♦✉t❧✐♥❡❞ ❜❡❧♦✇✿

❚❤❡s❡ ❛r❡ t❤❡ st❛❣❡s t❤❛t ✇❡ s❡❡ ❤❡r❡✿ ✶✳ ❛ ❢✉♥❝t✐♦♥✱ ✷✳ ❛ ♣❛rt✐t✐♦♥ ❛♥❞ ✐ts ♣r✐♠❛r② ♥♦❞❡s✱ ✸✳ s❛♠♣❧✐♥❣ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❛t t❤❡ ♣r✐♠❛r② ♥♦❞❡s✱ ✹✳ r❡♠♦✈✐♥❣ t❤❡ ❣r❛♣❤✱ ✺✳ ♣❧♦tt✐♥❣ t❤❡ ❞✐✛❡r❡♥❝❡s✱ ✻✳ ♣❧❛❝✐♥❣ t❤❡s❡ ❞✐✛❡r❡♥❝❡s ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s✳ ❚❤❡ r❡s✉❧t ✐s ❛ ♥❡✇ ❢✉♥❝t✐♦♥✳ ■t ✐s ❞❡✜♥❡❞ ❛❧❣❡❜r❛✐❝❛❧❧② ❛s ❢♦❧❧♦✇s✿

✷✷✼

✸✳✷✳

❚❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ ❛ s❡q✉❡♥❝❡ ❛♥❞ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ ❛ ❢✉♥❝t✐♦♥

✷✷✽

❉❡✜♥✐t✐♦♥ ✸✳✷✳✶✹✿ ❞✐✛❡r❡♥❝❡ ♦❢ ❢✉♥❝t✐♦♥ ❙✉♣♣♦s❡

y = f (x)

❚❤❡♥ t❤❡

✐s ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s

❞✐✛❡r❡♥❝❡

f

♦❢

xk , k = 0, 1, 2, ..., n✱

♦❢ ❛ ♣❛rt✐t✐♦♥✳

✐s ❛ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ♦❢ t❤❡

♣❛rt✐t✐♦♥✱ ❛♥❞ ❞❡♥♦t❡❞✱ ❛s ❢♦❧❧♦✇s✿

∆f (ck ) = f (xk+1 ) − f (xk )

❊①❛♠♣❧❡ ✸✳✷✳✶✺✿ sq✉❛r✐♥❣ ❢✉♥❝t✐♦♥ ❲❡ ❧❡t

• •

f (x) = x2

❛♥❞ ❝❤♦♦s❡ ❛ ♣❛rt✐t✐♦♥ ♦❢

[0, 6]✿

0, 1, 3, 6✳ ❛r❡ 0, 2, 4✳

❚❤❡ ♣r✐♠❛r② ♥♦❞❡s ❛r❡ ❚❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s

❲❡ s❛♠♣❧❡ t❤❡ ❢✉♥❝t✐♦♥✿

f (0) = 0, f (1) = 1, f (3) = 9, f (6) = 36 . ❚❤✐s ✐s t❤❡ s❛♠♣❧❡❞ ❢✉♥❝t✐♦♥✿

✵

✶

✾

✸✻

0

1

3

6

|

❚❤❡♥

|

|

|

∆f (0) = f (1) − f (0) = 12 − 02 = 1 ∆f (2) = f (3) − f (1) = 32 − 12 = 8 ∆f (4) = f (6) − f (3) = 62 − 32 = 27

❚❤✐s ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ s❛♠♣❧❡❞ ❢✉♥❝t✐♦♥✿

✶

✽

✷✼

0

2

4

|

|

|

■❢ ✇❡ ❝❤❛♥❣❡ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s✱ ✇❡✱ ♦❢ ❝♦✉rs❡✱ ❝❤❛♥❣❡ t❤❡ ❞✐✛❡r❡♥❝❡s✳ ❍♦✇❡✈❡r✱ t❤❡ r❡s✉❧ts ❛r❡ t❤❡ s❛♠❡ ♦♥ ❛ ❧❛r❣❡ s❝❛❧❡✱ ❛s ✇❡ s❤❛❧❧ s❡❡ ❧❛t❡r✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ❧❛♥❣✉❛❣❡ ✇✐❧❧ ❜❡ ♦❢t❡♥ ✉s❡❞ t♦ ✐❧❧✉str❛t❡ t❤❡s❡ ♠❛t❤❡♠❛t✐❝❛❧ ✐❞❡❛s✿

❉❡✜♥✐t✐♦♥ ✸✳✷✳✶✻✿ ❞✐s♣❧❛❝❡♠❡♥t ❲❤❡♥ ❛ ❢✉♥❝t✐♦♥ ♦r ❛ s❡q✉❡♥❝❡ ✐s ❝❛❧❧❡❞ ✏❧♦❝❛t✐♦♥✑ ♦r ✏♣♦s✐t✐♦♥✑✱ ✐ts ❞✐✛❡r❡♥❝❡ ✐s ❝❛❧❧❡❞ t❤❡

❞✐s♣❧❛❝❡♠❡♥t✳

❲❤❡♥ ❛ ♣❛rt✐t✐♦♥ ✐s s♣❡❝✐✜❡❞✱ ✇❡ ♠❛② ♦♠✐t t❤❡ s✉❜s❝r✐♣t ❢♦r t❤❡ ♥♦❞❡s✱

x✱ ❛♥❞ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s✱ c✳

❚❤❡♥

✇❡ ❝❛♥ ✉s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ s✐♠♣❧✐✜❡❞ ♥♦t❛t✐♦♥✿

❉✐✛❡r❡♥❝❡ ∆f (c) = f (x + ∆x) − f (x) ❍♦✇ ❞♦ ✇❡ tr❡❛t ♠♦t✐♦♥ ✇❤❡♥ t❤❡ t✐♠❡ ✐♥❝r❡♠❡♥t ✐s♥✬t

1❄

❍♦✇ ❞♦ ✇❡ ✜♥❞ t❤❡

✈❡❧♦❝✐t✐❡s

♦✈❡r t✐♠❡❄ ❚❤❡

❞✐✛❡r❡♥❝❡ ❝♦♥str✉❝t✐♦♥ ♣r❡s❡♥t❡❞ ✐♥ t❤✐s s❡❝t✐♦♥ ♣❛✈❡s t❤❡ ✇❛②✿ ❲❤❡♥❡✈❡r ✇❡ s✉❜tr❛❝t✱ ✇❡ ❛❧s♦ ❞✐✈✐❞❡✳

❊①❡r❝✐s❡ ✸✳✷✳✶✼ ❨♦✉ ❤❛✈❡ r❡❝❡✐✈❡❞ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡♠❛✐❧ ❢r♦♠ ②♦✉r ❜♦ss✿ ✏❚✐♠✱ ▲♦♦❦ ❛t t❤❡ ♥✉♠❜❡rs ✐♥ t❤✐s s♣r❡❛❞s❤❡❡t✳ ❚❤✐s st♦❝❦ s❡❡♠s t♦ ❜❡ ✐♥❝❤✐♥❣ ✉♣✳✳✳ ❛❝t✐♦♥s✳

❉♦❡s ✐t❄

■❢ ❞♦❡s✱ ❤♦✇ ❢❛st❄

❚❤❛♥❦s✳

✕ ❚♦♠✑✳

❉❡s❝r✐❜❡ ②♦✉r

✸✳✸✳

❚❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡✿ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t

✷✷✾

✸✳✸✳ ❚❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡✿ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❲❡ ❤❛✈❡ ❜❡❡♥ ❝♦♠♣✉t✐♥❣ t❤❡

r✐s❡s✳

❚♦ ❣❡t t❤❡ s❧♦♣❡ ❢r♦♠ t❤♦s❡✱ ✇❡ ❥✉st ❞✐✈✐❞❡ ❜② t❤❡

❖♥❝❡ ❛❣❛✐♥ ❧❡t✬s ❝♦♥s✐❞❡r t❤❡ r✉♥♥✐♥❣ ❡①❛♠♣❧❡ ♦❢ ❛

❜r♦❦❡♥ s♣❡❡❞♦♠❡t❡r

❢r♦♠ ❈❤❛♣t❡r ✶✳ ❏✉st ❛s ✐♥ t❤❡ ❧❛st

s❡❝t✐♦♥✱ ✇❡ ✜♥❞ t❤❡ ❞✐s♣❧❛❝❡♠❡♥t ❢♦r ❡❛❝❤ ✐♥t❡r✈❛❧ ♦❢ t✐♠❡✱ ✇❤✐❝❤ ✐s ❤❛❧❢✲❤♦✉r ❧♦♥❣✳ ❞✐✈✐s✐♦♥ ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡s ❜② t❤❡ t✐♠❡ ✐♥❝r❡♠❡♥t

h = .5

r✉♥s✳ ❚❤❡ ♥❡✇ st❡♣ ✐s t❤❡

♣r♦❞✉❝✐♥❣ t❤❡ ✈❡❧♦❝✐t✐❡s✿

❊①❛♠♣❧❡ ✸✳✸✳✶✿ ✈❡❧♦❝✐t②

❲❡ ♥♦✇ ❝♦♥s✐❞❡r ❛ ♠♦r❡ ❡❧❛❜♦r❛t❡ ✈❡rs✐♦♥ ♦❢ t❤❡ ♣r♦❜❧❡♠ ♦❢

✐♥❝r❡♠❡♥t❛❧ ♠♦t✐♦♥✳

❙✉♣♣♦s❡ ✇❡ ❞r♦✈❡ ❛r♦✉♥❞ ✇❤✐❧❡ ♣❛②✐♥❣ ❛tt❡♥t✐♦♥ t♦ t❤❡ ❝❧♦❝❦ ❛♥❞ t♦ t❤❡ ♠✐❧❡♣♦sts✳ ❚❤❡ r❡s✉❧t ✐s t❤✐s s✐♠♣❧❡ t❛❜❧❡ ✇✐t❤

✜✈❡

❝♦❧✉♠♥s✿ t✐♠❡ ✭❤♦✉rs✮✿ ❧♦❝❛t✐♦♥ ✭♠✐❧❡s✮✿

❲❤❛t ✇❛s t❤❡ ✈❡❧♦❝✐t② ♦✈❡r t❤❡s❡

❢♦✉r

0 2 4 6 8 0 60 160 80 0

♣❡r✐♦❞s ♦❢ t✐♠❡❄ ❲❡ ❡st✐♠❛t❡ ✐t ❜② t❤❡ ✏❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✑

❢♦r♠✉❧❛✿ ✈❡❧♦❝✐t②

=

❝❤❛♥❣❡ ♦❢ ❧♦❝❛t✐♦♥ ❝❤❛♥❣❡ ♦❢ t✐♠❡

✸✳✸✳ ❚❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡✿ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t

✷✸✵

❚❤❡s❡ ❛r❡ t❤❡ ❝♦♠♣✉t❛t✐♦♥s✿ t✐♠❡ ✭❤♦✉rs✮✿

0

2

4

6

8

❧♦❝❛t✐♦♥ ✭♠✐❧❡s✮✿

0

60

160

80

0

60 − 0 = 2−0

30

✈❡❧♦❝✐t② ✭♠✐❧❡s✴❤♦✉r✮✿ ✈❡❧♦❝✐t② ✭♠✐❧❡s✴❤♦✉r✮✿ ✈❡❧♦❝✐t② ✭♠✐❧❡s✴❤♦✉r✮✿

160 − 60 = 4−2

✈❡❧♦❝✐t② ✭♠✐❧❡s✴❤♦✉r✮✿

50 80 − 160 = 6−4

❚❤❡ ♥✉♠❡r❛t♦rs ❛♥❞ ❞❡♥♦♠✐♥❛t♦rs ❛r❡ ♥♦t❤✐♥❣ ❜✉t t❤❡ r✐s❡s ❛♥❞ r✉♥s✿

−40

0 − 80 = −40 8−6

❚❤❡ ❢♦✉r ❝♦♠♣✉t❡❞ ✈❛❧✉❡s ❛r❡ t❤❡ ❛✈❡r❛❣❡ ✈❡❧♦❝✐t✐❡s ♦✈❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♥t❡r✈❛❧s ♦❢ t✐♠❡✿ [0, 2]✱ [2, 4]✱ [4, 6]✱ ❛♥❞ [6, 8]✱ r❡s♣❡❝t✐✈❡❧②✳ ❚❤❡ r❡s✉❧t ✐s t❤✐s t❛❜❧❡✿ t✐♠❡ ✐♥t❡r✈❛❧s ✭❤♦✉rs✮✿ ✈❡❧♦❝✐t② ✭♠✐❧❡s✴❤♦✉r✮✿

[0, 2] [2, 4] [4, 6] [6, 8] 30 50 −40 −40

❚❤✐s ✐s t❤❡ s✉♠♠❛r② ♦❢ ✇❤❛t ✇❡ ❤❛✈❡ ❢♦✉♥❞✿

❲❡ ♠❛② ❛❧s♦ ❝❤♦♦s❡ t♦ ❛ss✐❣♥ t❤❡ ❢♦✉r ✈❛❧✉❡s t♦ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ✭t❤❡ ♠✐❞❞❧❡ ♣♦✐♥ts ❛r❡ s❤♦✇♥ ✐♥ t❤❡ ❧❛st ❝❤❛rt✮✱ ❛s ❢♦❧❧♦✇s✿ t✐♠❡ ✭❤♦✉rs✮✿ ✈❡❧♦❝✐t② ✭♠✐❧❡s✴❤♦✉r✮✿

1 3 5 7 30 50 −40 −40

■♥ ❣❡♥❡r❛❧✱ ✇❡ s❡❡ t❤❡ t✐♠❡ ❛♥❞ t❤❡ ❧♦❝❛t✐♦♥ ❛s ❥✉st t✇♦ s❡♣❛r❛t❡ s❡q✉❡♥❝❡s✱ s❛②✱ xn ❛♥❞ yn ✳ ❚❤❡♥ t❤❡ ✈❡❧♦❝✐t② ✐s t❤❡ ✐♥❝r❡♠❡♥t ♦❢ t❤❡ ❧❛tt❡r ♦✈❡r t❤❡ ✐♥❝r❡♠❡♥t ♦❢ t❤❡ ❢♦r♠❡r✳ ❲❡ ♥♦t✐❝❡ t❤❛t t❤♦s❡ t✇♦ ❛r❡ t❤❡ ❞✐✛❡r❡♥❝❡s ♦❢ t❤❡ t✇♦ s❡q✉❡♥❝❡s✿

✸✳✸✳

❚❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡✿ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t

✷✸✶

❉❡✜♥✐t✐♦♥ ✸✳✸✳✷✿ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❢♦r s❡q✉❡♥❝❡s ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t

❚❤❡

♦❢ ❛ s❡q✉❡♥❝❡

yn

✇✐t❤ r❡s♣❡❝t t♦ ❛ s❡q✉❡♥❝❡

t♦ ❜❡ t❤❡ s❡q✉❡♥❝❡ t❤❛t ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢

yn

xn

✐s ❞❡✜♥❡❞

❞✐✈✐❞❡❞ ❜② t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢

xn ✿

yn+1 − yn ∆yn = ∆xn xn+1 − xn ♣r♦✈✐❞❡❞ t❤❡ ❞❡♥♦♠✐♥❛t♦r ✐s ♥♦t ③❡r♦✳

■t ✐s t❤❡ r❡❧❛t✐✈❡ ❝❤❛♥❣❡ ✕ t❤❡

r❛t❡ ♦❢ ❝❤❛♥❣❡ ✕ ♦❢ t❤❡ t✇♦ s❡q✉❡♥❝❡s✳

❆♥❞ ❢♦r ❡❛❝❤ ❝♦♥s❡❝✉t✐✈❡ ♣❛✐r ♦❢ ♣♦✐♥ts✱

✐t ✐s t❤❡ s❧♦♣❡ ♦❢ t❤❡ ❧✐♥❡ ❝♦♥♥❡❝t✐♥❣ t❤❡♠✿

❚❤✐s ✐s ✇❤② ✐t ✐s ❝❛❧❧❡❞ t❤✐s ✇❛②✿

❉✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❞✐✛❡r❡♥❝❡✱ s✉❜tr❛❝t✐♦♥

ւ ∆yn yn+1 − yn = ←− ∆xn xn+1 − xn տ

q✉♦t✐❡♥t✱ ❞✐✈✐s✐♦♥

❞✐✛❡r❡♥❝❡✱ s✉❜tr❛❝t✐♦♥

❊①❛♠♣❧❡ ✸✳✸✳✸✿ ✈❡❧♦❝✐t②✱ ❝♦♥t✐♥✉❡❞ ▲❡t✬s ❝♦♥t✐♥✉❡ ✇✐t❤ t❤❡ ❧❛st ❡①❛♠♣❧❡✳ ❚❤✐s t✐♠❡✱ ✇❡ ❛❧s♦ ♣r♦❝❡❡❞ t♦ ❝♦♠♣✉t❡ t❤❡

❛❝❝❡❧❡r❛t✐♦♥✳

❏✉st ❛s

✇❡ ✉s❡❞ t❤❡ ✏❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✑ ❢♦r♠✉❧❛ t♦ ✜♥❞ t❤❡ ✈❡❧♦❝✐t② ❢r♦♠ t❤❡ ❧♦❝❛t✐♦♥✱ ✇❡ ♥♦✇ ✉s❡ ✐t t♦ ✜♥❞ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ❢r♦♠ t❤❡ ✈❡❧♦❝✐t②✿

❛❝❝❡❧❡r❛t✐♦♥

❲❡ ❛♣♣❧② t❤✐s ❢♦r♠✉❧❛ t♦

t❤r❡❡

=

❝❤❛♥❣❡ ♦❢ ✈❡❧♦❝✐t② ❝❤❛♥❣❡ ♦❢ t✐♠❡

♣❡r✐♦❞s ♦❢ t✐♠❡✳ ❚❤❡s❡ ❛r❡ t❤❡ ❝♦♠♣✉t❛t✐♦♥s✿

t✐♠❡ ✐♥t❡r✈❛❧s ✭❤♦✉rs✮✿ t✐♠❡ ✭❤♦✉rs✮✿ ✈❡❧♦❝✐t② ✭♠✐❧❡s✴❤♦✉r✮✿ ❛❝❝❡❧❡r❛t✐♦♥ ✭♠✐❧❡s✴❤♦✉r✴❤♦✉r✮✿ ❛❝❝❡❧❡r❛t✐♦♥ ✭♠✐❧❡s✴❤♦✉r✴❤♦✉r✮✿ ❛❝❝❡❧❡r❛t✐♦♥ ✭♠✐❧❡s✴❤♦✉r✴❤♦✉r✮✿

[0, 2]

[2, 4]

[4, 6]

[6, 8]

1

3

5

7

30

50

50 − 30 = 3−1

−40

−40

10 −40 − 50 = 5−3

−45

−40 − (−40) = 7−5

0

✸✳✸✳ ❚❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡✿ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t

✷✸✷

❚❤❡ t❤r❡❡ ❝♦♠♣✉t❡❞ ✈❛❧✉❡s ❛r❡ t❤❡ ❛✈❡r❛❣❡ ❛❝❝❡❧❡r❛t✐♦♥s ♦✈❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♥t❡r✈❛❧s ♦❢ t✐♠❡✿ [0, 4]✱ [2, 6]✱ ❛♥❞ [4, 8]✱ r❡s♣❡❝t✐✈❡❧②✳ ❚❤✐s ✐s t❤❡ r❡s✉❧t✿ t✐♠❡ ✐♥t❡r✈❛❧s ✭❤♦✉rs✮✿ ❛❝❝❡❧❡r❛t✐♦♥ ✭♠✐❧❡s✴❤♦✉r✴❤♦✉r✮✿

[0, 4] [2, 6] [4, 8] 10 −45 0

❚❤✐s ✐s t❤❡ s✉♠♠❛r② ♦❢ ✇❤❛t ✇❡ ❤❛✈❡ ❢♦✉♥❞✿

❆❧t❡r♥❛t✐✈❡❧②✱ ✇❡ ♠❛② ❝❤♦♦s❡ t♦ ❛ss✐❣♥ t❤❡ t❤r❡❡ ✈❛❧✉❡s t♦ t❤❡ ♠✐❞❞❧❡ ♣♦✐♥ts ♦❢ t❤❡s❡ ✐♥t❡r✈❛❧s✱ ❛s ❢♦❧❧♦✇s✿ t✐♠❡ ✭❤♦✉rs✮✿ 2 4 6 ❛❝❝❡❧❡r❛t✐♦♥ ✭♠✐❧❡s✴❤♦✉r✴❤♦✉r✮✿ 10 −45 0 ❚❤❡s❡ ❤❛♣♣❡♥ t♦ ❜❡ t❤❡ ♣r✐♠❛r② ♥♦❞❡s ♦❢ ♦✉r ♣❛rt✐t✐♦♥✦ ❊①❛♠♣❧❡ ✸✳✸✳✹✿ ❝♦♠♣✉t✐♥❣ ♠♦t✐♦♥

■♥ ❣❡♥❡r❛❧✱ ✇❤❡♥ t❤❡ ❧♦❝❛t✐♦♥ ✐s ❦♥♦✇♥ ❢♦r ♥✉♠❡r♦✉s ♠♦♠❡♥ts ♦❢ t✐♠❡✱ ✇❡ ❤❛✈❡ t❤❡s❡ s❡q✉❡♥❝❡s✿ • tn ❢♦r t❤❡ t✐♠❡✱ • pn ❢♦r t❤❡ ♣♦s✐t✐♦♥✱ • vn ❢♦r t❤❡ ✈❡❧♦❝✐t②✱ ❛♥❞ • an ❢♦r t❤❡ ❛❝❝❡❧❡r❛t✐♦♥✳ ❚❤❡② ❛r❡ ❝♦♥♥❡❝t❡❞ t♦ ❡❛❝❤ ♦t❤❡r ❜② t❤❡s❡ ❢♦r♠✉❧❛s✿ vn+1 =

pn+1 − pn v − vn ❛♥❞ an+1 = n+1 . tn+1 − tn tn+1 − tn

❇♦t❤ ❛r❡ ♥♦t❤✐♥❣ ❜✉t t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✦

❋♦r ❝♦♠♣✉t❛t✐♦♥s ✇✐t❤ ❛ ❧♦t ♦❢ ❞❛t❛✱ ❛ s♣r❡❛❞s❤❡❡t ✐s ✉s❡❞✳ ❚❤❡ t✇♦ ❢♦r♠✉❧❛s ❤❛✈❡ t❤❡ s❛♠❡ ❢♦r♠✿ ❂✭❘❈❬✲✶❪✲❘❬✲✶❪❈❬✲✶❪✮✴✭❘❈✷✲❘❬✲✶❪❈✷✮

❚❤❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ ✈❡❧♦❝✐t② ✭♦r t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ r❡s♣❡❝t✐✈❡❧②✮ r❡❢❡rs t♦ t❤❡ ❝♦❧✉♠♥ t❤❛t ❝♦♥t❛✐♥s t❤❡ ❧♦❝❛t✐♦♥ ✭♦r ✈❡❧♦❝✐t② r❡s♣❡❝t✐✈❡❧②✮ ✐♥ t❤❡ ♥✉♠❡r❛t♦r ❛♥❞ t♦ t❤❡ ❝♦❧✉♠♥ t❤❛t ❝♦♥t❛✐♥s t❤❡ t✐♠❡ ✐♥ t❤❡ ❞❡♥♦♠✐♥❛t♦r✳ P♦ss✐❜❧❡ ❞❛t❛ ❛♥❞ ❣r❛♣❤s ❢♦r t❤❡s❡ t❤r❡❡ ❛r❡ s❤♦✇♥ ❜❡❧♦✇✿

✸✳✸✳ ❚❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡✿ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t

✷✸✸

❚❤❡ ❣r❛♣❤s ❛r❡ ♠❛❞❡ ♦❢ ❞♦ts ♦♥❧②✦ ❚❤❡ s❡❝♦♥❞ ❝♦❧✉♠♥ ✭✈❡❧♦❝✐t②✮ ❤❛s ♦♥❡ ❢❡✇❡r ❞❛t❛ ♣♦✐♥t ❛♥❞ t❤❡ ♥❡①t ✭❛❝❝❡❧❡r❛t✐♦♥✮ ♦♥❡ ❢❡✇❡r ②❡t✳ ❍♦✇❡✈❡r✱ ✇❤❡♥ ③♦♦♠❡❞ ♦✉t✱ t❤❡ ❣r❛♣❤s ❣✐✈❡ t❤❡ ✐♠♣r❡ss✐♦♥ t❤❛t t❤❡ t❤r❡❡ ❢✉♥❝t✐♦♥s ❤❛✈❡ t❤❡ s❛♠❡ ❞♦♠❛✐♥✳ ❚❤❡ st♦r② ♦❢ ✇❤❛t ❤❛♣♣❡♥❡❞ ❝❛♥ ❜❡ ✐♥❢❡rr❡❞ ❢r♦♠ t❤❡s❡ ❣r❛♣❤s✿ ✶✳ ❚❤❡ ♦❜❥❡❝t ✇❛s ♠♦✈✐♥❣ ❢♦r✇❛r❞✱ t❤❡♥ st♦♣♣❡❞ ❢♦r ❛ ♠♦♠❡♥t✱ t✉r♥❡❞ ❜❛❝❦ ❢♦r ❛ s❤♦rt ✇❤✐❧❡✱ t❤❡♥ st❛rt❡❞ ♠♦✈✐♥❣ ❢♦r✇❛r❞ ❛❣❛✐♥✳ ✷✳ ❚❤❡ ♦❜❥❡❝t ✈❡❧♦❝✐t② ✇❛s ❢♦r✇❛r❞ ❤✐❣❤✱ t❤❡♥ ❧♦✇❡r ❛♥❞ ❧♦✇❡r ✭s❧♦✇❡r ❛♥❞ s❧♦✇❡r✮✱ ✉♥t✐❧ ✐t ✇❛s ③❡r♦ ❛♥❞ t❤❡♥ ❝❤❛♥❣❡❞ ❞✐r❡❝t✐♦♥✱ t❤❡♥ ❢♦r✇❛r❞ ❛❣❛✐♥✱ ❤✐❣❤❡r ❛♥❞ ❤✐❣❤❡r ✭❢❛st❡r ❛♥❞ ❢❛st❡r✮✳ ✸✳ ❚❤❡ ♦❜❥❡❝t ❤❛❞ ♥❡❣❛t✐✈❡ ❛❝❝❡❧❡r❛t✐♦♥ ✭❞❡❝❡❧❡r❛t✐♦♥✮✱ ✇❤✐❝❤ t❤❡♥ ❜❡❝❛♠❡ ♣♦s✐t✐✈❡✱ ❛♥❞ ❝♦♥t✐♥✉❡❞ t♦ ❣r♦✇✳ ❍♦✇❡✈❡r✱ ✐t ✐s ♠✉❝❤ ❡❛s✐❡r t♦ ❞r❛✇ ❝♦♥❝❧✉s✐♦♥s ❛❜♦✉t t❤❡ ✈❡❧♦❝✐t② ❢r♦♠ ✐ts ❣r❛♣❤ t❤❛♥ ❢r♦♠ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ♣♦s✐t✐♦♥✦ ❚❤❡s❡ ❢♦r♠✉❧❛s ✇✐❧❧ ❛❧❧♦✇ ✉s t♦ ❞❡✈❡❧♦♣ r❡❛❧✐st✐❝ ♠♦❞❡❧s ♦❢ ♠♦t✐♦♥✳ ❲❡ ❝❛♥ ❝♦♥t✐♥✉❡ t♦ ✐♥❝r❡❛s❡ t❤❡ ♥✉♠❜❡r ♦❢ ❞❛t❛ ♣♦✐♥ts ❛♥❞✱ ❛s ✇❡ ③♦♦♠ ♦✉t✱ t❤❡ s❝❛tt❡r ♣❧♦ts ✇✐❧❧ ❧♦♦❦ ❧✐❦❡ ❝♦♥t✐♥✉♦✉s ❝✉r✈❡s ✦ ❚❤❡ ❛♥❛❧②s✐s ♣r❡s❡♥t❡❞ ✐♥ t❤✐s s❡❝t✐♦♥ r❡♠❛✐♥s ❢✉❧❧② ❛♣♣❧✐❝❛❜❧❡✳ ■t ❛♠♦✉♥ts t♦ ❧♦♦❦✐♥❣ ❛t ❤♦✇ ❢❛st t❤❡ ✈❡rt✐❝❛❧ ❧♦❝❛t✐♦♥ ✐s ❝❤❛♥❣✐♥❣ r❡❧❛t✐✈❡ t♦ t❤❡ ❝❤❛♥❣❡ ♦❢ t❤❡ ❤♦r✐③♦♥t❛❧ ❧♦❝❛t✐♦♥ ✭❧❡❢t✮✿

❋✉rt❤❡r♠♦r❡✱ ✇❡ r❡♣❧❛❝❡ ✭r✐❣❤t✮ ♦✉r t✐♠❡✲❞❡♣❡♥❞❡♥t q✉❛♥t✐t②✱ ❧♦❝❛t✐♦♥✱ ❢♦r ❛♥♦t❤❡r✱ t❡♠♣❡r❛t✉r❡ ❛s ❥✉st ♦♥❡ ♦❢ t❤❡ ❡①❛♠♣❧❡s ♦❢ t❤❡ ❜r❡❛❞t❤ ♦❢ ❛♣♣❧✐❝❛❜✐❧✐t② ♦❢ t❤❡s❡ ✐❞❡❛s✳ ❚❤✐s ❞✐❝t❛t❡s t❤❡ ♥❡❡❞ ❢♦r ❝♦♥t❡①t✲✐♥❞❡♣❡♥❞❡♥t✱ ♠❛t❤❡♠❛t✐❝❛❧ t❡r♠✐♥♦❧♦❣②✳ ❲❡ ✉s❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿ ◮ ❚❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✐s t❤❡

r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢ t❤❡ ♦✉t♣✉t ♣❡r ❝❤❛♥❣❡ ♦❢ t❤❡ ✐♥♣✉t✳

◆❡①t✱ t❤❡ s❡q✉❡♥❝❡s ♠❛② ❝♦♠❡ ❢r♦♠ s❛♠♣❧❡❞ ❢✉♥❝t✐♦♥s✳ ❆ s❡q✉❡♥❝❡ ♦❢ ✐♥♣✉ts ✇✐❧❧ ♣r♦❞✉❝❡ ❛ s❡q✉❡♥❝❡ ♦❢

✸✳✸✳

❚❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡✿ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t

✷✸✹

♦✉t♣✉ts✿

an = f (xn ) . ❲❡ ❛♣♣r♦❛❝❤❡❞ t❤❡ ♣r♦❜❧❡♠ ❜② ♣❧♦tt✐♥❣ t❤❡ ❧♦❝❛t✐♦♥ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✐♠❡✿

❲❡ t❤❡♥ ❢♦✉♥❞✿ ❛✈❡r❛❣❡ ✈❡❧♦❝✐t②

=

❞✐s♣❧❛❝❡♠❡♥t t✐♠❡

❢♦r ❡❛❝❤ ♦❢ t❤❡ t✐♠❡ ♣❡r✐♦❞s ✭❧❡❢t✮✿

❍♦✇❡✈❡r✱ ✇❤❛t ✐❢ ❜❡❤✐♥❞ t❤✐s ❞❛t❛ ✐s ❛ ❝♦♥t✐♥✉♦✉s❧② ❝❤❛♥❣✐♥❣ ❧♦❝❛t✐♦♥ ✭r✐❣❤t✮❄ ❲✐t❤ s♦ ♠✉❝❤ ✐♥❢♦r♠❛t✐♦♥✱ ❝❛♥ ✇❡ ✜♥❞ t❤❡ ✏❡①❛❝t✑ ✈❡❧♦❝✐t② ❢♦r ❛❧❧ ♦❢ t❤❡s❡ ♠♦♠❡♥ts ♦❢ t✐♠❡❄ ❙✉♣♣♦s❡ ✇❡ ❦♥♦✇ ♦♥❧②

t✇♦

✈❛❧✉❡s ♦❢ ❛ ❢✉♥❝t✐♦♥✿

f (x1 ) = y1 ✇✐t❤

x1 6= x2 ✳

f (x2 ) = y2 ,

❚❤❡♥✱ ✇❤❛t ❝❛♥ ✇❡ s❛② ❛❜♦✉t ✐ts r❛t❡ ♦❢ ❝❤❛♥❣❡❄ ❚❤❡ ❛♥s✇❡r ✐s ❣✐✈❡♥ ❜② t❤❡

t❤r♦✉❣❤ t❤❡s❡ t✇♦ ♣♦✐♥ts ♦♥ t❤❡ ❣r❛♣❤ ♦❢

y = f (x)✿

A = (x1 , y1 ) ❛s ❢♦❧❧♦✇s✿

❛♥❞

❛♥❞

B = (x2 , y2 ) ,

s❧♦♣❡

♦❢ t❤❡ ❧✐♥❡

✸✳✸✳ ❚❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡✿ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t

✷✸✺

❚❤❡ s❧♦♣❡ ✐s✱ ♦❢ ❝♦✉rs❡✱ t❤❡ r✐s❡ ♦✈❡r t❤❡ r✉♥✿ s❧♦♣❡ =

f (x2 ) − f (x1 ) y2 − y1 = . x2 − x1 x2 − x1

❚❤❡ ♥✉♠❡r❛t♦r✱ t❤❡ r✐s❡✱ ✐s t❤❡ ❝❤❛♥❣❡ ♦❢ y ✱ ✇❤✐❝❤ ✐s ❛❧s♦ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ f ✱✿ ∆f = f (x2 ) − f (x1 ) .

❚❤❡ ❞❡♥♦♠✐♥❛t♦r✱ t❤❡ r✉♥✱ ✐s t❤❡ ❝❤❛♥❣❡ ♦❢ x✱ ✇❤✐❝❤ ✇❡ ✇✐❧❧ ❝❛❧❧ t❤❡ ✐♥❝r❡♠❡♥t ♦❢ x✿ ∆x = x2 − x1 .

❚❤❡✐r r❛t✐♦✱ t❤❡ s❧♦♣❡ ♦❢ t❤❡ ❧✐♥❡✱ ✐s t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢ f ✱ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ f ✿ ∆f f (x2 ) − f (x1 ) = ∆x x2 − x1 ❊①❛♠♣❧❡ ✸✳✸✳✺✿ s✐♥❣❧❡ s❡❣♠❡♥t

■♥ t❤❡ ❛❜♦✈❡ ♣✐❝t✉r❡✱ ✇❡ ❤❛✈❡✿ ✶✳ ❚❤❡ ✐♥❝r❡♠❡♥t ♦❢ x ✐s ∆x = x2 − x1 = 8 − 2 = 6 ✳ ✷✳ ❚❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ f ✐s ∆f = f (x2 ) − f (x1 ) = 10 − 1 = 9 ✳ ∆f 9 ✸✳ ❚❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ f ✐s = ✳ ∆x

6

❊①❛♠♣❧❡ ✸✳✸✳✻✿ s♣❡❡❞

❙✉♣♣♦s❡ ✇❡ tr❛✈❡❧ ❜② ❝❛r ❛♥❞ r❡❝♦r❞ ♦✉r ❧♦❝❛t✐♦♥ s❡✈❡r❛❧ t✐♠❡s ✭✜rst r♦✇✮✿ − 100 − − − 250 − − −•− −−− −•− − 0 2

−→

− 100 −−− 250 − − − • − − 75 − − • − − 0 1 2

❲❡ t❤❡♥ ❝♦♠♣✉t❡ t❤❡ ✈❡❧♦❝✐t② ✭✐✳❡✳✱ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✮ ❞✉r✐♥❣ ❡❛❝❤ s❡❣♠❡♥t ♦❢ t✐♠❡ ❛♥❞ ♣❧❛❝❡ t❤❡ ♥✉♠❜❡r ✐♥ ❜❡t✇❡❡♥ ✭s❡❝♦♥❞ r♦✇✮✳ ❇② ❛♥❛❧♦❣②✱ ✐❢ ✇❡ ❦♥♦✇ ♦♥❧② t✇♦ ✈❛❧✉❡s ♦❢ ❛ ❢✉♥❝t✐♦♥ ✭✜rst r♦✇✮ ❛t ❡♥❞s ♦❢ ❛♥ ✐♥t❡r✈❛❧✱ ✇❡ ❝♦♠♣✉t❡ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❛❧♦♥❣ t❤✐s ✐♥t❡r✈❛❧ ✭s❡❝♦♥❞ r♦✇✮✿ −

− − − f (x + ∆x) − ∆f −•− − − −•− ∆x x c x + ∆x f (x)

◆♦✇✱ ✇❤❛t ✐❢ ❛ ❢✉♥❝t✐♦♥ y = f (x) ✐s ❦♥♦✇♥ ❢♦r s❡✈❡r❛❧ ✈❛❧✉❡s ♦❢ x ✇✐t❤✐♥ ❛♥ ✐♥t❡r✈❛❧ [a, b]❄ ❲❡ ❢♦❧❧♦✇ t❤❡ ✐❞❡❛ ♦❢ t❤❡ ❝♦♥str✉❝t✐♦♥ ✐♥ t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤❡ ❝❤❛♣t❡r✳ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❝♦♥str✉❝t❡❞ ❛♥ ❛✉❣♠❡♥t❡❞ ♣❛rt✐t✐♦♥ ❛♥ ✐♥t❡r✈❛❧ [a, b] ❛s ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥✿ a = x0 ≤ c1 ≤ x1 ≤ c2 ≤ x2 ≤ ... ≤ xn−1 ≤ cn ≤ xn = b .

✸✳✸✳

❚❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡✿ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t

✷✸✻

❏✉st ❛s ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥✱ ✇❡ ✉t✐❧✐③❡ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ❛s t❤❡ ✐♥♣✉ts ♦❢ t❤❡ ♥❡✇ ❢✉♥❝t✐♦♥✿

❍♦✇❡✈❡r✱ t❤✐s t✐♠❡ ✇❡ ❛r❡ ♥♦t ✉s✐♥❣ t❤❡ ❞✐✛❡r❡♥❝❡s ♦❢ ✈❛❧✉❡s ✭t❤❡ ❤❡✐❣❤ts ♦❢ t❤❡ st❡♣s✮ ❜✉t t❤❡ ✐♥❝❧✐♥❡s✳ ❲❡ ❞❡✜♥❡ t❤❡ ♠♦st ❢✉♥❞❛♠❡♥t❛❧ ♦♣❡r❛t✐♦♥ ♦❢ ❝❛❧❝✉❧✉s ❜❡❧♦✇✿

s❧♦♣❡s ♦❢ t❤❡

❉❡✜♥✐t✐♦♥ ✸✳✸✳✼✿ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❢♦r ❢✉♥❝t✐♦♥s ❙✉♣♣♦s❡ ❛ ❢✉♥❝t✐♦♥ y = f (x) ✐s ❞❡✜♥❡❞ ❛t t❤❡ ♣r✐♠❛r② ♥♦❞❡s xk , k = 0, 1, 2, ..., n✱ ♦❢ ❛ ♣❛rt✐t✐♦♥✳ ❚❤❡♥ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ f ✐s ❞❡✜♥❡❞ ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ck , k = 1, 2, ..., n ♦❢ t❤❡ ♣❛rt✐t✐♦♥ ❛s t❤✐s ❢r❛❝t✐♦♥✿

f (xk+1 ) − f (xk ) f (xk + ∆xk ) − f (xk ) ∆f (ck ) = = ∆x xk+1 − xk ∆xk ❲❤❡♥ ❛ ♣❛rt✐t✐♦♥ ✐s s♣❡❝✐✜❡❞✱ ✇❡ ♠❛② ♦♠✐t t❤❡ s✉❜s❝r✐♣t ❢♦r t❤❡ ♥♦❞❡s✱ x✱ ❛♥❞ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s✱ c✳ ❚❤❡♥ ✇❡ ❝❛♥ ✉s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ s✐♠♣❧✐✜❡❞ ♥♦t❛t✐♦♥✿

∆f f (x + ∆x) − f (x) (c) = ∆x ∆x ❇❡❧♦✇ ✐s t❤❡ ❜r❡❛❦❞♦✇♥ ♦❢ t❤❡ ♥♦t❛t✐♦♥✿

❉✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❧❞ ❢✉♥❝t✐♦♥ ↓ ∆f ❛ ❢r❛❝t✐♦♥✿ (c) ∆x ր ↑ ♥❡✇ ❢✉♥❝t✐♦♥ ✐♥♣✉t ✈❛r✐❛❜❧❡

❊①❛♠♣❧❡ ✸✳✸✳✽✿ ❣r❛♣❤s ❚❤❡ ❞✐✛❡r❡♥❝❡ ✐s ❛ ♥❡✇ s❡q✉❡♥❝❡✳ ❈❛♥ ✇❡ ♣❧♦t ✐t ❜❛s❡❞ ♦♥ t❤❡ ❣r❛♣❤ ♦❢ f ♦♥❧②❄ ❨❡s✱ ❜✉t ♥♦t ❛t ♦♥❝❡✳ ❲❡ ❞♦ t❤✐s ♣✐❡❝❡ ❜② ♣✐❡❝❡✳ ❲❡ t❛❦❡ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ✭t♦♣ ❧❡❢t✮ ❛♥❞✱ ❛t ♠❛♥② ❧♦❝❛t✐♦♥s✱ ❝✉t ✕ ✉s✐♥❣ t❤❡ ❣r✐❞ ✕ ❛ s❡❣♠❡♥t ♦❢ t❤❡ ❣r❛♣❤ s♦ s❤♦rt t❤❛t ✐t✬s ❛❧♠♦st str❛✐❣❤t ✭t♦♣ r✐❣❤t✮✿

✸✳✸✳ ❚❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡✿ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t

✷✸✼

❚❤❡ ✈❡rt✐❝❛❧ ❧♦❝❛t✐♦♥s ♦❢ t❤❡s❡ ♣✐❡❝❡s ❛r❡ ✐rr❡❧❡✈❛♥t✳ ❲❡ ❧✐♥❡ t❤❡♠ ✉♣ ❜❡❧♦✇ s♦ t❤❛t t❤❡✐r s❧♦♣❡s✱ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ♦❢ t❤❡ ❢✉♥❝t✐♦♥✱ ❛r❡ ❡❛s② t♦ ❡st✐♠❛t❡ ✭❜♦tt♦♠ r✐❣❤t✮✳ ❚❤✐s ❣✐✈❡s ✉s 15 ♥✉♠❜❡rs✳ ❚❤❡s❡ ♣♦✐♥ts ❝❛♥ ❜❡ ❝♦♥♥❡❝t❡❞ t♦ ❢♦r♠ ❛ ❝✉r✈❡ ✭❜♦tt♦♠ ❧❡❢t✮✳ ❚❤✐s ❧♦♦❦s ❧✐❦❡ ❛ ♥❡✇ ❢✉♥❝t✐♦♥✦ ❊①❛♠♣❧❡ ✸✳✸✳✾✿ t❛❜❧❡s

◆♦✇ ✇❡ ♣❧♦t

∆f ❜❛s❡❞ ♦♥ t❤❡ ∆x

✈❛❧✉❡s ♦❢ f ✇✐t❤ t❤❡ ✐♥❝r❡♠❡♥t ♦❢ ∆x = .5✿ x 0 .5 1.0 1.5 2.0 y = f (x) −1 −2 0 1 1

❲❡ ❝♦♠♣✉t❡ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❢r♦♠ t❤✐s ❞❛t❛✿ f (x + ∆x) − f (x) ∆f (x) = . ∆x ∆x

❲❡ ❞♦ t❤❛t ✐♥t❡r✈❛❧ ❜② ✐♥t❡r✈❛❧✿ [0, .5] [.5, 1.0] [1.0, 1.5] [1.5, 2.0] [x, x + ∆x] ∆f (c) = f (x + ∆x) − f (x) = −2 − (−1) 0 − (−2) 1 − 0 1−1 −1 2 1 0 = ∆f (c) = −1/.5 2/.5 1/.5 0/.5 ∆x = −2 4 2 0

❚❤❡ r❡s✉❧ts ❛r❡ ❝♦♥✜r♠❡❞ ❜② ♣❧♦tt✐♥❣ t❤❡s❡ ❞❛t❛ ♣♦✐♥ts✿

❚❤❡s❡ ♥✉♠❜❡rs ❛r❡✱ ♦❢ ❝♦✉rs❡✱ ❥✉st t❤❡ s❧♦♣❡s ♦❢ t❤❡ ❧✐♥❡s t❤❛t ❝♦♥♥❡❝t t❤❡ ❞♦ts✳

✸✳✸✳ ❚❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡✿ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t

✷✸✽

❊①❡r❝✐s❡ ✸✳✸✳✶✵

❚❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥ f ✐s ❣✐✈❡♥ ❜❡❧♦✇✿

❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ♣❛rt✐t✐♦♥ ✇✐t❤✿ [a, b] = [0, 4] ❛♥❞ ∆x = 1 .

❊st✐♠❛t❡ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t

∆f ✳ ❙❤♦✇ ②♦✉r ❝♦♥str✉❝t✐♦♥s ❛♥❞ ❝♦♠♣✉t❛t✐♦♥s✳ ∆x

❊①❛♠♣❧❡ ✸✳✸✳✶✶✿ s♣r❡❛❞s❤❡❡t

❲❡ ♥♦✇ ✉t✐❧✐③❡ ❛ s♣r❡❛❞s❤❡❡t t♦ s♣❡❡❞ ✉♣ t❤✐s ♣r♦❝❡ss✳ ❚❤❡ ❢✉♥❝t✐♦♥ ✐s ❣✐✈❡♥ ❜② ❛ ♣♦ss✐❜❧② ❧❛r❣❡ t❛❜❧❡ ♦❢ ✈❛❧✉❡s✱ ✇✐t❤ t✇♦ ❝♦❧✉♠♥s✿ x ❛♥❞ y = f (x)✳ ❚❤❡♥✱ ❢♦r ❡❛❝❤ ♣❛✐r ♦❢ ❝♦♥s❡❝✉t✐✈❡ ✈❛❧✉❡s✱ ✇❡ ❝♦♠♣✉t❡✿ • t❤❡ ✐♥❝r❡♠❡♥t ♦❢ x✱ • t❤❡ ✐♥❝r❡♠❡♥t ♦❢ y ✱ ❛♥❞ • t❤❡✐r r❛t✐♦✱ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✳ ❚❤❡ ❧❛st ❢♦r♠✉❧❛ ✐s✿ ❂❘❈❬✲✷❪✴❘❈❬✲✸❪

❊①❛♠♣❧❡ ✸✳✸✳✶✷✿ s❡❝❛♥t ❧✐♥❡s

▲❡t✬s ❝♦♥s✐❞❡r f (x) = x2 ✳ ❇❡❧♦✇✱ ✇❡ ✉s❡ s♣r❡❛❞s❤❡❡ts t♦ s❛♠♣❧❡ t❤❡ ❢✉♥❝t✐♦♥ ❛t t❤❡s❡ 5 ♣♦✐♥ts✱ ❞r❛✇ ❧✐♥❡s t❤r♦✉❣❤ ❡✈❡r② t✇♦ ❝♦♥s❡❝✉t✐✈❡ ♦♥❡s✱ ❛♥❞ ❝♦♠♣✉t❡ t❤❡ s❧♦♣❡ ♦❢ ❡❛❝❤✿

✸✳✸✳ ❚❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡✿ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t

✷✸✾

❚❤❡s❡ ❛r❡ ❛ ❝♦✉♣❧❡ ♦❢ ❝♦♠♣✉t❛t✐♦♥s✿ 1−0 • ❚❤❡ s❧♦♣❡ ❢r♦♠ (0, 0) t♦ (1, 1) ✐s = 1✳

1−0 4−1 • ❚❤❡ s❧♦♣❡ ❢r♦♠ (1, 1) t♦ (2, 4) ✐s = 3✳ 2−1 ◆♦✇ f (x) = x3 ✿

❆ ❝♦✉♣❧❡ ♦❢ ❝♦♠♣✉t❛t✐♦♥s✿ 1−0 = 1✳ • ❚❤❡ s❧♦♣❡ ❢r♦♠ (0, 0) t♦ (1, 1) ✐s 1−0 8−1 = 7✳ • ❚❤❡ s❧♦♣❡ ❢r♦♠ (1, 1) t♦ (2, 8) ✐s 2−1

❊①❛♠♣❧❡ ✸✳✸✳✶✸✿ ❞❡♥s❡r s❛♠♣❧✐♥❣

▲❡t✬s ❛♣♣❧② t❤❡ ❢♦r♠✉❧❛ t♦ f (x) = sin x✳ ❉✐✛❡r❡♥t ✈❛❧✉❡s ♦❢ t❤❡ ✐♥❝r❡♠❡♥t ♦❢ x ♣r♦❞✉❝❡ ❞✐✛❡r❡♥t ✈❛❧✉❡s ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✳ ❍♦✇❡✈❡r✱ ❞❡❝r❡❛s✐♥❣ t❤✐s ♥✉♠❜❡r s❤♦✇s ❛ ♣❛tt❡r♥ ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ✿

✸✳✹✳

❚❤❡ ❧✐♠✐t ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✿ t❤❡ ❞❡r✐✈❛t✐✈❡

■♥ t❤❡ ✜rst ❣r❛♣❤ ❜❡❧♦✇✱ ✇❡ s❛♠♣❧❡ t❤✐s ❢✉♥❝t✐♦♥ ❡✈❡r②

✷✹✵

∆x = .1✿

❚❤❡ s❡❝♦♥❞ ❣r❛♣❤ ✐s t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✳ ■t ❧♦♦❦s ❧✐❦❡

y = cos x✱

❡s♣❡❝✐❛❧❧② ✐❢ ✇❡ s❤✐❢t ✐t ❛ ❧✐tt❧❡ t♦

t❤❡ r✐❣❤t✳

❇❛s❡❞ ♦♥ t❤❡s❡ ❞❡✜♥✐t✐♦♥s✱ ✇❡ ❝❛♥ ✉s❡ t❤❡ ❧❛♥❣✉❛❣❡ ♦❢ ♠♦t✐♦♥ ✐♥ ❛ ♠♦r❡ ♣r❡❝✐s❡ ♠❛♥♥❡r✿

❉❡✜♥✐t✐♦♥ ✸✳✸✳✶✹✿ ✈❡❧♦❝✐t② ❲❤❡♥ ❛ ❢✉♥❝t✐♦♥ ♦r ❛ s❡q✉❡♥❝❡ ✐s ❝❛❧❧❡❞ ✏❧♦❝❛t✐♦♥✑ ♦r ✏♣♦s✐t✐♦♥✑✱ ✐ts ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✐s ❝❛❧❧❡❞ t❤❡

✈❡❧♦❝✐t②✳

❉❡✜♥✐t✐♦♥ ✸✳✸✳✶✺✿ ❛❝❝❡❧❡r❛t✐♦♥ ❲❤❡♥ ❛ ❢✉♥❝t✐♦♥ ♦r ❛ s❡q✉❡♥❝❡ ✐s ❝❛❧❧❡❞ ✏✈❡❧♦❝✐t②✑✱ ✐ts ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✐s ❝❛❧❧❡❞ t❤❡

❛❝❝❡❧❡r❛t✐♦♥✳

✸✳✹✳ ❚❤❡ ❧✐♠✐t ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✿ t❤❡ ❞❡r✐✈❛t✐✈❡ ■♥ ❣❡♥❡r❛❧✱ s✉♣♣♦s❡ ❝❛♥ ❢r❡❡❧②

y = f (x)

✐s ❛ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ♦♥ ❛♥ ✐♥t❡r✈❛❧

[a, b]✳

s❛♠♣❧❡✱ ✐✳❡✳✱ t♦ ✜♥❞ ✐ts ✈❛❧✉❡ y = f (x) ❢♦r ❛♥② ❝❤♦✐❝❡ ♦❢ x✳

❚❤❡♥✱ ✇❡ ❤❛✈❡ ❛ ❢✉♥❝t✐♦♥ t❤❛t ✇❡

❚❤❡♥ t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ r❛t❡s ♦❢

❝❤❛♥❣❡ ♣r❡s❡♥t❡❞ ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥ ✐s ❛✈❛✐❧❛❜❧❡ ❢♦r ❡✈❡r② ♣❛✐r ♦❢ ✈❛❧✉❡s ♦❢ ❧✐♥❡ ❢r♦♠

(x0 , f (x0 ))

t♦

(x1 , f (x1 ))

♦♥ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥✿

f (x1 ) − f (x0 ) x1 − x0 ❚❤❡r❡ ❛r❡ ✐♥✜♥✐t❡❧② ♠❛♥② s✉❝❤ ❝♦♠♣✉t❛t✐♦♥s ❢♦r ❛ s✐♥❣❧❡ ❢✉♥❝t✐♦♥✿

x✦

■♥❞❡❡❞✱ ✐t ✐s t❤❡ s❧♦♣❡ ♦❢ t❤❡

✸✳✹✳ ❚❤❡ ❧✐♠✐t ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✿ t❤❡ ❞❡r✐✈❛t✐✈❡

✷✹✶

❚❤❡② ❛r❡♥✬t ✉♥str✉❝t✉r❡❞ t❤♦✉❣❤❀ t❤❡ s❛♠♣❧❡ ♣♦✐♥ts ❛r❡ ❛rr❛♥❣❡❞ ✐♥ ♣❛rt✐t✐♦♥s ♦❢ t❤❡ ✐♥t❡r✈❛❧✿

■♥ ♦r❞❡r t♦ ♠❛❦❡ s❡♥s❡ ♦❢ ❛❧❧ t❤❡ r❛t❡s ♦❢ ❝❤❛♥❣❡ ♦❢ ❛ ❢✉♥❝t✐♦♥✱ ✇❡ ✇✐❧❧ ❜❡ ♠❛❦✐♥❣ t❤❡s❡ ♣❛rt✐t✐♦♥s ❞❡♥s❡r ❛♥❞ ❞❡♥s❡r✳ ❊①❛♠♣❧❡ ✸✳✹✳✶✿ ❞❡♥s❡r s❛♠♣❧✐♥❣

❲❡ ❧❡t t❤❡ s♣r❡❛❞s❤❡❡t ❝♦♠♣✉t❡ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ❢♦r f (x) = x2 ✭❧❡❢t✮ ❛♥❞ ♣❧♦t t❤❡ s❧♦♣❡s ✐♥ ❛ s❡♣❛r❛t❡ ❝❤❛rt ❜❡❧♦✇✿

❲❡ t❤❡♥ r❡♣❡❛t t❤❡ ♣r♦❝❡❞✉r❡ ❢♦r ❛ ❞❡♥s❡r ♣❛tt❡r♥ ♦❢ ♣♦✐♥ts ✭r✐❣❤t✮✳ ◆♦✇ f (x) = x3 ✳ ❲❡ ❝♦♠♣✉t❡ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✉s✐♥❣ ❛ s♣r❡❛❞s❤❡❡t ❢♦r t❤❡ t✇♦ ✈❛❧✉❡s ♦❢ t❤❡ ✐♥❝r❡♠❡♥t h = 1 ❛♥❞ h = .2 ✿

✸✳✹✳ ❚❤❡ ❧✐♠✐t ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✿ t❤❡ ❞❡r✐✈❛t✐✈❡

❚❤✐s ✐s t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t

✷✹✷

∆f ✱ ✐✳❡✳✱ t❤❡ s❛♠♣❧❡❞ s❧♦♣❡s✱ ♦❢ f (x) = 1/x ✭❜♦tt♦♠ r♦✇✮✿ ∆x

❊①❡r❝✐s❡ ✸✳✹✳✷

❉♦ t❤❡ ♥❡✇ ❣r❛♣❤s ❧♦♦❦ ❢❛♠✐❧✐❛r❄ ❊①❛♠♣❧❡ ✸✳✹✳✸✿

sin x

▲❡t✬s t❛❦❡ ❛♥♦t❤❡r ❧♦♦❦ ❛t f (x) = sin x✳ ❆ ❢✉♥❝t✐♦♥ ✐s ❣✐✈❡♥ ❜❡❧♦✇✱ ❛♥❞ s♦ ✐s t❤❡ ✐♥t❡r✈❛❧ [a, b]✳ ❋♦r ❡❛❝❤ n = 2, 3, 4, ...✱ t❤❡ ✐♥❝r❡♠❡♥t ✐s ❢♦✉♥❞✱ ∆x = (b − a)/n✱ ❛♥❞ ✇❡ ❤❛✈❡ n s❡❣♠❡♥ts ✐♥ ♦✉r ♣❛rt✐t✐♦♥ ♦❢ [a, b]✳ ❖♥ ❡❛❝❤ ♦❢ t❤❡ s❡❣♠❡♥ts✱ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✐s ❝♦♠♣✉t❡❞✱ t❤❡ ✈❛❧✉❡ ✐s r❡❝♦r❞❡❞ ❛s t❤❡ ✈❛❧✉❡ ♦❢ ❛ ♥❡✇ ❢✉♥❝t✐♦♥✱ ❛♥❞ t❤❡ r❡s✉❧t ✐s ♣❧♦tt❡❞ ✐♥ t❤❡ ❜♦tt♦♠ r♦✇✿

❲❡ ❢❛❝❡ ❛ s❡q✉❡♥❝❡ ♦❢ ❢✉♥❝t✐♦♥s✳ ❙❤♦✉❧❞ ✇❡ t❤✐♥❦ ♦❢ ✐ts ❧✐♠✐t❄ ❖✉r s❛♠♣❧✐♥❣ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ✐s ❣❡tt✐♥❣ ❞❡♥s❡r ❛♥❞ ❞❡♥s❡r✳ ■♥ t❤❡ ♠❡❛♥t✐♠❡✱ t❤❡ ♣♦✐♥ts t❤❛t ♠❛❦❡ ✉♣ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❛r❡ ❣❡tt✐♥❣ ❝❧♦s❡r ❛♥❞ ❝❧♦s❡r t♦❣❡t❤❡r✳ ❲❤❛t ✐s ❛t t❤❡ ❡♥❞ ♦❢ t❤✐s ♣r♦❝❡ss❄ ❆ ♥❡✇ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ♦♥ t❤❡ ✇❤♦❧❡ ✐♥t❡r✈❛❧✦ ❊✈❡♥ t❤♦✉❣❤ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ❛❜♦✈❡ s❡❡♠ t♦ ♣r♦❞✉❝❡ r❡❝♦❣♥✐③❛❜❧❡ ♥❡✇ ❢✉♥❝t✐♦♥s✱ ✐t ✐s ❛ ❝❤❛❧❧❡♥❣❡ t♦ ❝❛♣t✉r❡ t❤❡ ✇❤♦❧❡ ❢✉♥❝t✐♦♥ ❛t ♦♥❝❡✳ ■♥✐t✐❛❧❧②✱ ♦✉r ❛♣♣r♦❛❝❤ ✇✐❧❧ ❜❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿

✸✳✹✳ ❚❤❡ ❧✐♠✐t ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✿ t❤❡ ❞❡r✐✈❛t✐✈❡ ◮ ❚❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ❛r❡ ❡✈❛❧✉❛t❡❞ ✐♥ t❤❡ ✈✐❝✐♥✐t② ♦❢ ❛

✷✹✸

s✐♥❣❧❡ ♣♦✐♥t✳

❖♥❝❡ ❛❣❛✐♥✱ ✇❡ ❤❛✈❡ ❛ ❢✉♥❝t✐♦♥ t❤❛t ✇❡ ❝❛♥ ❢r❡❡❧② s❛♠♣❧❡✱ ✐✳❡✳✱ t♦ ✜♥❞ ✐ts ✈❛❧✉❡ y = f (x) ❢♦r ❛♥② ❝❤♦✐❝❡ ♦❢ x✱ ❛♥❞ ✇❡ ✇♦✉❧❞ ❧✐❦❡ t♦ ♠❛❦❡ s❡♥s❡ ♦❢ t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢ t❤✐s ❢✉♥❝t✐♦♥ ✐♥ t❤❡ ✈✐❝✐♥✐t② ♦❢ ❛ ♣❛rt✐❝✉❧❛r ✈❛❧✉❡ ♦❢ x✳ ❊①❛♠♣❧❡ ✸✳✹✳✹✿ ❞❡❝r❡❛s❡

∆x

▲❡t✬s ✐♥✈❡st✐❣❛t❡ t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢ t❤✐s ❢✉♥❝t✐♦♥ ✐♥ t❤❡ ✈✐❝✐♥✐t② ♦❢ x = 0✿

❚❤❡s❡ ✈❛❧✉❡s ♦❢ x ❤❛✈❡ ❝♦♠❡ ❢r♦♠ ❛ ♣❛rt✐t✐♦♥ ✇✐t❤ t❤❡ ♦r✐❣✐♥❛❧ ♥♦❞❡ a = x0 = 0 ❛♥❞ t❤❡ ✐♥❝r❡♠❡♥t ∆x = 1✳ ❚❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡✱ ✐✳❡✳✱ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❢♦r t❤✐s ♣❛rt✐t✐♦♥✱ ✈❛r✐❡s✳ ▲❡t✬s ♥♦✇ ❝❤♦♦s❡ ❛ s♠❛❧❧❡r st❡♣ ∆x = .1 ✿

❲❡ ❝❛♥ s❡❡ ❛♥ ❛❧♠♦st str❛✐❣❤t ❧✐♥❡✦ ■♥ ❢❛❝t✱ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✐s ❛❧♠♦st ❝♦♥st❛♥t ✭t❤❡ s❧♦♣❡ ❛♣♣❡❛rs t♦ ❜❡ .1✮✳ ❲❡ ✇❡r❡ ✉s✐♥❣ t✇♦✲♥♦❞❡ ♣❛rt✐t✐♦♥s ❤❡r❡✱ {a, a + ∆x}✳ ❚❤❡ ❞❡r✐✈❛t✐✈❡ ✐s t❤❡ s❧♦♣❡ ♦❢ t❤❡ t❛♥❣❡♥t ❧✐♥❡✳ ❖♥❡ ❝❛♥ ❡st✐♠❛t❡ t❤✐s ♥✉♠❜❡r ❢r♦♠ ❛ ♣✐❝t✉r❡ ❜② ❜❧♦✇✐♥❣ ✉♣ ❛ s♠❛❧❧ ♣✐❡❝❡ s✉rr♦✉♥❞✐♥❣ t❤❡ ♣♦✐♥t ❛♥❞ t❤❡♥ ♣❧❛❝✐♥❣ ✐t ♦♥ ❛ ❣r✐❞✿

❇✉t ✇❤❛t ✐s t❤❡ t❛♥❣❡♥t ❧✐♥❡❄

✸✳✹✳ ❚❤❡ ❧✐♠✐t ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✿ t❤❡ ❞❡r✐✈❛t✐✈❡

✷✹✹

❊①❛♠♣❧❡ ✸✳✹✳✺✿ s❡q✉❡♥❝❡ ♦❢ s❡❝❛♥ts

▲❡t✬s ❝♦♥s✐❞❡r y = x2 ❛t x = 1✳ ❋✐✈❡ ✐♥♣✉ts ❛r❡ ❝❤♦s❡♥ ❝❧♦s❡r ❛♥❞ ❝❧♦s❡r t♦ 1 ❢r♦♠ t❤❡ ❧❡❢t✿ x = −1, 0, .5, .75, .875 .

❚❤❡♥ ✇❡ ✜♥❞ ❝♦rr❡s♣♦♥❞✐♥❣ ♣♦✐♥ts ♦♥ t❤❡ ❣r❛♣❤✿ (x, y) = (−1, 1), (0, 0), (.5, .25), (.75, .5625), (.875, 0.765625) .

❚❤❡♥ ❛ ❧✐♥❡ ✐s ❞r❛✇♥ t❤r♦✉❣❤ (1, 1) ❛♥❞ ❡❛❝❤ ♦❢ t❤❡s❡ ♣♦✐♥ts✳

❚❤❡s❡ ❧✐♥❡s ❛r❡ s✉♣♣♦s❡❞ t♦ ❝♦♥✈❡r❣❡ ♦♥t♦ t❤❡ t❛♥❣❡♥t ❧✐♥❡✳ ❍♦✇❡✈❡r✱ ✐t ✐s ♠✉❝❤ ❡❛s✐❡r t♦ ❝♦♥❝❡♥tr❛t❡ ♦♥ t❤❡ s❧♦♣❡s ♦♥❧②✿ ✐♥st❡❛❞ ♦❢ ❛ s❡q✉❡♥❝❡ ♦❢ ❧✐♥❡s✱ ✇❡ ❤❛✈❡ ❛ s❡q✉❡♥❝❡ ♦❢ ♥✉♠❜❡rs✦ ❋♦r ❡❛❝❤ ♦❢ t❤❡s❡ ✈❛❧✉❡s ♦❢ x✱ ✇❡ ❤❛✈❡ ❛ ❧✐♥❡ ❞r❛✇♥ t❤r♦✉❣❤ (1, 1) ❛♥❞ (x, x2 )✳ ❚❤❡r❡❢♦r❡✱ t❤❡ s❧♦♣❡s ❛r❡✿ x2 − 1 . x−1

❋r♦♠ ♦✉r ❝♦♠♣✉t❛t✐♦♥s ✐♥ t❤❡ ❧❛st ❝❤❛♣t❡r✱ ✇❡ ❦♥♦✇ t❤❛t t❤❡ ❧✐♠✐t ♦❢ t❤✐s ❡①♣r❡ss✐♦♥ ❛s x ✐s ❛♣♣r♦❛❝❤✐♥❣ 1 ✐s 2✳ ❚❤❛t✬s t❤❡ s❧♦♣❡ ♦❢ t❤❡ t❛♥❣❡♥t ❧✐♥❡✦ ❙♦✱ ✐♥st❡❛❞ ♦❢ tr②✐♥❣ t♦ ✉♥❞❡rst❛♥❞ ✇❤❛t t❤❡ t❛♥❣❡♥t ❧✐♥❡ ✐s✱ ✇❡ s✐♠♣❧② ✜♥❞ ✐ts s❧♦♣❡✳ ❲❡ ❞❡✜♥❡ ✐ts s❧♦♣❡ ❛s t❤❡ ❧✐♠✐t ♦❢ s❧♦♣❡s ♦❢ t❤❡ ❝♦r❞s✱ ❝❛❧❧❡❞ t❤❡ s❡❝❛♥t ❧✐♥❡s✱ ✐✳❡✳✱ ❧✐♥❡s ❞❡t❡r♠✐♥❡❞ ❜② t✇♦ ♣♦✐♥ts ♦♥ t❤❡ ❣r❛♣❤✳ ❋♦r ❡❛❝❤ x 6= a✱ t❤❡ s❧♦♣❡ ♦❢ t❤❡ ❧✐♥❡ t❤r♦✉❣❤ (a, f (a)) ❛♥❞ (x, f (x)) ✐s✿ s❧♦♣❡ =

r✐s❡ f (x) − f (a) = . r✉♥ x−a

❚❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ ❛ ❢✉♥❝t✐♦♥ f ❛t a ✐s ❞❡✜♥❡❞ ❢♦r ❡❛❝❤ x 6= a t♦ ❜❡ f (x) − f (a) . x−a

✸✳✹✳ ❚❤❡ ❧✐♠✐t ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✿ t❤❡ ❞❡r✐✈❛t✐✈❡

✷✹✺

❚❤❡♥ t❤❡ s❧♦♣❡ ♦❢ t❤❡ t❛♥❣❡♥t ❧✐♥❡ ✐s t❤❡ ❧✐♠✐t ♦❢ t❤❡s❡ s❧♦♣❡s✳ ❆s ✇❡ ♠♦✈❡ x t♦✇❛r❞ a✱ t❤❡ s❡❝❛♥t ❧✐♥❡s t✉r♥ ❛♥❞ ❛♣♣r♦❛❝❤ t❤❡ t❛♥❣❡♥t ❧✐♥❡✱ ♣r♦✈✐❞❡❞ t❤❡ ❧✐♠✐t ❡①✐sts✳ ❚❤❡ ❛❧❣❡❜r❛✐❝ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤✐s ✐❞❡❛ ✐s ❜❡❧♦✇✿

❉❡✜♥✐t✐♦♥ ✸✳✹✳✻✿ ❞❡r✐✈❛t✐✈❡ ❛t ❛ ♣♦✐♥t ❚❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ f ❛t x = a ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ❧✐♠✐t ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ♦❢ f ❛t x = a ❛s t❤❡ ✐♥❝r❡♠❡♥t ∆x ✐s ❛♣♣r♦❛❝❤✐♥❣ 0✱ ❞❡♥♦t❡❞ ❛s ❢♦❧❧♦✇s✿ df f (x) − f (a) (a) = lim x→a dx x−a

❲❛r♥✐♥❣✦ ❚❤❡ ❞❡r✐✈❛t✐✈❡ ✐s

♥♦t

❛ ❢r❛❝t✐♦♥❀ t❤✐s ✐s ❥✉st ❛ ♥♦t❛✲

t✐♦♥✳

❇❡❧♦✇ ✐s t❤❡ ❜r❡❛❦❞♦✇♥ ♦❢ t❤❡ ♥♦t❛t✐♦♥✿

❉❡r✐✈❛t✐✈❡ ♦❧❞ ❢✉♥❝t✐♦♥ ♥♦t ❛ ❢r❛❝t✐♦♥✿

↓ df (a) dx ր ↑

♥❡✇ ❢✉♥❝t✐♦♥ ✐♥♣✉t ✈❛r✐❛❜❧❡ ❆❧t❡r♥❛t✐✈❡❧② ✇r✐tt❡♥✱ t❤❡ ❢♦r♠✉❧❛ r❡✈❡❛❧s t❤❡ r❡❛s♦♥ ❢♦r t❤❡ ♥♦t❛t✐♦♥✿ ∆f df (a) = lim ∆x→0 ∆x dx

❊①❛♠♣❧❡ ✸✳✹✳✼✿ t❛❜❧❡ ♦❢ ✈❛❧✉❡s ▲❡t✬s ❡st✐♠❛t❡ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ❣✐✈❡♥ ❜② ♥✉♠❜❡rs ♦♥❧②✿ x 5 10 15 y = f (x) 554 344 250

❲❤❛t ✐s

df (10)❄ dx

❲❡ ✉s❡ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✱ ✐✳❡✳✱ t❤❡ s❧♦♣❡ ♦❢ ❛ s❡❝❛♥t ❧✐♥❡✳ ❚❤❡r❡ ❛r❡ s❡✈❡r❛❧ ❝❤♦✐❝❡s✿

❲❡ ❝❛♥ ✉s❡ t❤❡ s❧♦♣❡ ♦❢ ❡✐t❤❡r ♦❢ t❤❡ t✇♦ ❛❞❥❛❝❡♥t s❡❝❛♥t ❧✐♥❡s✿

✸✳✹✳ ❚❤❡ ❧✐♠✐t ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✿ t❤❡ ❞❡r✐✈❛t✐✈❡

✷✹✻

344 − 554 = −42 ✳ 5 250 − 344 = −18.8 ✳ • ❙❧♦♣❡ ♦❢ t❤❡ ✷♥❞ s❡❣♠❡♥t = 5 • ❙❧♦♣❡ ♦❢ t❤❡ ✶st s❡❣♠❡♥t =

❲❡ ❝❛♥ ❛❧s♦ ✉s❡ t❤❡ ❛✈❡r❛❣❡ ♦❢ t❤❡ t✇♦✳

❆✈❡r❛❣❡ s❧♦♣❡ =

−60.8 −42 + (−18.8) = = −30.4 . 2 2

❚❤❡ ❧❛st ♦♣t✐♦♥ ✐s t♦ ✉s❡ t❤❡ t✇♦ s❡❣♠❡♥ts ❛s ❛ s✐♥❣❧❡ ✐♥t❡r✈❛❧✿ ❙❧♦♣❡ =

250 − 554 = −30.4 . 10

■t✬s t❤❡ s❛♠❡ ♥✉♠❜❡r✦ ❊①❡r❝✐s❡ ✸✳✹✳✽

■s t❤✐s ❛ ❝♦✐♥❝✐❞❡♥❝❡❄ ❊①❛♠♣❧❡ ✸✳✹✳✾✿ ❞❡r✐✈❛t✐✈❡ ❢r♦♠ ❣r❛♣❤

❖♥❡ ❝❛♥ ❡st✐♠❛t❡ t❤❡ ❞❡r✐✈❛t✐✈❡ ❢r♦♠ t❤❡ ❣r❛♣❤ ❜② ✉s✐♥❣ ♦✉r ❛❜✐❧✐t② t♦ ♣❧♦t t❛♥❣❡♥t ❧✐♥❡s✳ ▲❡t✬s ✜♥❞ t❤❡ ❞❡r✐✈❛t✐✈❡ ❛t t❤❡ r❡❞ ♣♦✐♥t✿

❚❤❡s❡ ❛r❡ t❤❡ st❡♣s✿ ✶✳ P❧♦t t❤❡ t❛♥❣❡♥t ❧✐♥❡ ✭❣r❡❡♥✮✳ ✷✳ ❯s✐♥❣ t❤❡ ❣r✐❞✱ ❜✉✐❧❞ ❛ r✐❣❤t tr✐❛♥❣❧❡ ✇✐t❤ t❤❡ s❡❣♠❡♥t ♦❢ t❤❡ t❛♥❣❡♥t ❧✐♥❡ ❛s ✐ts ❤②♣♦t❡♥✉s❡✳ ✸✳ ❈♦♠♣✉t❡ t❤❡ s❧♦♣❡✿ r✐s❡ 10.5 s❧♦♣❡ = ≈ = 1.75 . r✉♥ 6 ❚♦ ❜❡tt❡r ❡st✐♠❛t❡ t❤❡ s❧♦♣❡✱ ❞r❛✇ t❤❡ tr✐❛♥❣❧❡ ❛s ❧❛r❣❡ ❛s ♣♦ss✐❜❧❡✳ ❋✐♥❞ t❤❡ ❞❡r✐✈❛t✐✈❡ ❛t t❤❡ ♣♦✐♥t ❣✐✈❡♥ ❜❡❧♦✇✿

❚♦ ✜♥❞ t❤❡ s❧♦♣❡✱ ✇❡ ♥❡❡❞ t❤❡ r✐s❡✳ ❚❤❡ ❤❡✐❣❤t ♦❢ t❤❡ tr✐❛♥❣❧❡ ✐s 6.5✳ ❍♦✇❡✈❡r✱ t❤❡ ♦♥❧② ✇❛② t♦ ❥✉st✐❢②

✸✳✹✳

❚❤❡ ❧✐♠✐t ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✿ t❤❡ ❞❡r✐✈❛t✐✈❡

✉s✐♥❣ t❤❡ ✇♦r❞ ✏r✐s❡✑ ✇❤✐❧❡ ✇❡ ❛r❡ ❣♦✐♥❣

❞♦✇♥

✷✹✼

✐s t♦ ❣✐✈❡ ✐t ❛ ♥❡❣❛t✐✈❡ ✈❛❧✉❡✿

r✐s❡

= −6.5 .

❚❤❡♥ t❤❡ s❧♦♣❡ ❛♥❞ t❤❡ ❞❡r✐✈❛t✐✈❡ ❛r❡ ❛❧s♦ ♥❡❣❛t✐✈❡✿

−6.5 df (3) = ≈ −7.2 . dx 9 ❊①❡r❝✐s❡ ✸✳✹✳✶✵

❋✐♥❞ t❤❡ ❞❡r✐✈❛t✐✈❡ ❛t t❤❡ r❡st ♦❢ t❤❡ ✐♥t❡❣❡r ♣♦✐♥ts✳

❊①❛♠♣❧❡ ✸✳✹✳✶✶✿ ❣r❛♣❤ ❢r♦♠ ❞❡r✐✈❛t✐✈❡s

◆♦✇ ✇❡ ❣♦ ✐♥ t❤❡ ♦♣♣♦s✐t❡ ❞✐r❡❝t✐♦♥✿ P❧♦t t❤❡ ❣r❛♣❤ ❜❛s❡❞ ♦♥ ♥✉♠❡r✐❝❛❧ ❞❛t❛✳ ❙✉♣♣♦s❡ ♦♥❧② t❤❡s❡ ✈❛❧✉❡s ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ ❛r❡ ❦♥♦✇♥✿

df df df df df df (−5) = 2, (−3) = 3, (−1) = 1, (1) = 0, (3) = −2, (5) = −1 . dx dx dx dx dx dx ❊❛❝❤ ✐s t❤❡ s❧♦♣❡ ♦❢ t❤❡ r❡s♣❡❝t✐✈❡ t❛♥❣❡♥t ❧✐♥❡ ❛♥❞✱ t❤❡r❡❢♦r❡✱ t❤❡ r✐s❡ ♦❢ t❤✐s ❧✐♥❡ ✇❤❡♥❡✈❡r t❤❡ r✉♥ ✐s ❡q✉❛❧ t♦

1✳

■♥st❡❛❞✱ ✇❡ ❝❤♦♦s❡ t❤❡ r✉♥ t♦ ❜❡ t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ ♣♦✐♥ts✱ ✐✳❡✳✱

✈❛❧✉❡s ♦❢ t❤❡ r✐s❡ ❛r❡✿

x: r✐s❡✿

2✳

❚❤❡r❡❢♦r❡✱ t❤❡

−5 −3 −1 1 3 5 4 6 2 0 −4 −2

❚❤❡♥ ✇❡ ❤❛✈❡ s✐① tr✐❛♥❣❧❡s ❛♥❞ t❤❡✐r ❤②♣♦t❡♥✉s❡s ❛r❡ ♠❡❛♥t t♦ ♠❛❦❡ ✉♣ r♦✉❣❤ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ s❡❣♠❡♥t ♦❢ t❤❡ ❣r❛♣❤ ♦❢

f

✭❧❡❢t✮✿

■♥ ❧✐❣❤t ♦❢ t❤❡ ❞✐s❝✉ss✐♦♥s ✐♥ t❤❡ ❧❛st ❝❤❛♣t❡r✱ ✇❡ ✇❛♥t

f

t♦ ❜❡

❝♦♥t✐♥✉♦✉s ✦

❚❤❛t ✐s ✇❤② ✇❡ ❛tt❛❝❤

❡❛❝❤ ♣✐❡❝❡ t♦ t❤❡ ❡♥❞ ♦❢ t❤❡ ❧❛st ♦♥❡ ✭r✐❣❤t✮✳

❊①❡r❝✐s❡ ✸✳✹✳✶✷

❚r② t♦ ♣❧♦t t❤❡ ❣r❛♣❤s ♦❢ ♦t❤❡r ❢✉♥❝t✐♦♥s ✇✐t❤ t❤❡s❡ ✈❛❧✉❡s ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡✳

❊①❡r❝✐s❡ ✸✳✹✳✶✸

P❧♦t t❤❡ ❣r❛♣❤ ♦❢

f

❜❛s❡❞ ♦♥ t❤❡s❡ ✈❛❧✉❡s ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡✿

df df df df df df (−5) = −1, (−4) = 1, (−3) = 0, (−2) = 2, (−1) = −2, (0) = 3 . dx dx dx dx dx dx ▼❛❦❡ ✉♣ ②♦✉r ♦✇♥ ♥✉♠❜❡rs ❛♥❞ r❡♣❡❛t✳

◆❡①t✱ ✇❡ ❣♦ ❜❡②♦♥❞ ❛♣♣r♦①✐♠❛t✐♦♥s✳

✸✳✺✳

❚❤❡ ❞❡r✐✈❛t✐✈❡ ✐s t❤❡ ✐♥st❛♥t❛♥❡♦✉s r❛t❡ ♦❢ ❝❤❛♥❣❡

✷✹✽

✸✳✺✳ ❚❤❡ ❞❡r✐✈❛t✐✈❡ ✐s t❤❡ ✐♥st❛♥t❛♥❡♦✉s r❛t❡ ♦❢ ❝❤❛♥❣❡

❚❤❡ ♠❛✐♥ ✐❞❡❛ ♦❢ t❤✐s ❝❤❛♣t❡r ✐s ❛s ❢♦❧❧♦✇s✿ ◮ ❚❤❡ ❞❡r✐✈❛t✐✈❡ ✐s t❤❡ ❧✐♠✐t ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✳

❲❡ ✇✐❧❧ ♥♦✇ ✉s❡ ♦✉r s❦✐❧❧s ✇✐t❤ ❧✐♠✐ts ✭❈❤❛♣t❡r ✷✮ t♦ ❝♦♠♣✉t❡ ❞❡r✐✈❛t✐✈❡s✳ ❊①❛♠♣❧❡ ✸✳✺✳✶✿

x2

❋✐♥❞ t❤❡ ❞❡r✐✈❛t✐✈❡ ❛♥❞ t❤❡ t❛♥❣❡♥t ❧✐♥❡ ♦❢ y = x2 ❛t (1, 1)✿

❯s❡ f (x) = x2 ✱ a = 1 ✐♥ t❤❡ ❞❡✜♥✐t✐♦♥✿ df f (x) − f (a) (a) = lim x→a dx x−a x2 − 12 0 = lim → ? x→1 x − 1 0 (x − 1)(x + 1) = lim x→1 x−1

❉❊❆❉ ❊◆❉

= lim (x + 1) x→1

=1+1

❚❤❛t✬s t❤❡ s❧♦♣❡✦

= 2.

❚❤❡♥ t❤❡ ♣♦✐♥t✲s❧♦♣❡ ❢♦r♠ ♦❢ t❤❡ ❧✐♥❡ ✐s✿ y − 1 = 2(x − 1) .

■♥ ❈❤❛♣t❡r ✷✱ ✇❡ s❛✇ ❛♥ ❛❧t❡r♥❛t✐✈❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❧✐♠✐t✿ ■♥st❡❛❞ ♦❢ ❝♦♥❝❡♥tr❛t✐♥❣ ♦♥ ❤♦✇ x ✐s ❛♣♣r♦❛❝❤✐♥❣ a✱ ✇❡ ❧♦♦❦ ❛t t❤❡ ✭s✐❣♥❡❞✮ ❞✐st❛♥❝❡ h = x − a ❜❡t✇❡❡♥ t❤❡♠✳ ❚❤❡ t✇♦ ❛♣♣r♦❛❝❤❡s ❛r❡ ✐❧❧✉str❛t❡❞ ❜❡❧♦✇✿ |

x

|

❚❤❡♥ ❢♦r ❛♥② ❢✉♥❝t✐♦♥ g ✇❡ ❤❛✈❡✿

•

|

❛

h

• ❛

h

x

|

lim g(x) = lim g(a + h) .

x→a

❲❡ ❛♣♣❧② t❤✐s ✐❞❡❛ t♦ t❤❡ ❧✐♠✐t ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡✳

h→0

▲❡t✬s ❝♦♥s✐❞❡r t❤❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✳ ❍❡r❡✱ ✐♥st❡❛❞ ♦❢ ❝♦♥❝❡♥tr❛t✐♥❣ ♦♥ ❤♦✇ x ✐s ❛♣♣r♦❛❝❤✐♥❣ a✱ ✇❡ ❧♦♦❦ ❛t t❤❡ ✭s✐❣♥❡❞✮ ❞✐st❛♥❝❡ h = x − a ❜❡t✇❡❡♥ t❤❡♠✳ ❚❤❡♥ h → 0✳ ❲❡ s✉❜st✐t✉t❡ h = x − a ✐♥t♦ t❤❡

✸✳✺✳

❚❤❡ ❞❡r✐✈❛t✐✈❡ ✐s t❤❡ ✐♥st❛♥t❛♥❡♦✉s r❛t❡ ♦❢ ❝❤❛♥❣❡

✷✹✾

❞❡✜♥✐t✐♦♥ ✭t♦♣✮ ❛♥❞ ♦❜t❛✐♥ ❛♥ ❛❧t❡r♥❛t✐✈❡ ❢♦r♠✉❧❛ ✭❜♦tt♦♠✮ ❛❢t❡r t❤❡ s✉❜st✐t✉t✐♦♥ x = a + h✿ df f (x) − f (a) (a) = lim x→a dx x−a | {z } h=x−a

f (a + h) − f (a) df (a) = lim h→0 dx h

❚❤✉s✱ ✇❤❛t t❤✐s s✉❜st✐t✉t✐♦♥ ❤❛s ❛❝❝♦♠♣❧✐s❤❡❞ ✐s ❛ ❝❤❛♥❣❡

♦❢ ✈❛r✐❛❜❧❡s ✐♥ t❤❡ ❧✐♠✐t✳

❚❤❡ ❣❡♦♠❡tr② ♦❢ t❤❡ ❧✐♠✐t r❡♠❛✐♥s t❤❡ s❛♠❡✿

❍❡r❡✱ t❤❡ s❡❝❛♥t s❡❣♠❡♥ts ❛r❡ ❣❡tt✐♥❣ s❤♦rt❡r ❛♥❞ s❤♦rt❡r s♦ t❤❛t ✐t ✐s ❤❛r❞ t♦ t❡❧❧ ✇❤❛t ✐s ❣♦✐♥❣ ♦♥✳ ❲❡ ❡①t❡♥❞ t❤❡♠ ✐♥t♦ ❧✐♥❡s✳ ❆s ②♦✉ ❝❛♥ s❡❡✱ t❤❡s❡ ❧✐♥❡s ❦❡❡♣ t✉r♥✐♥❣✿ ✷♥❞ ✐s ❝❧♦s❡r t♦ t❤❡ t❛♥❣❡♥t t❤❛♥ ✶st✱ ❡t❝✳ ❚❤❡ ❝♦♠♣✉t❛t✐♦♥ ✐s ❡♥t✐r❡❧② ❛❜♦✉t ❛♥❣❧❡s ✦ ❲❛r♥✐♥❣✦

❚❤❡ ❞❡r✐✈❛t✐✈❡ ✐s

♥♦t

t❤❡ ❧✐♠✐t ♦❢

f ✱ lim f (x)✱ x→a

❛ ❧✐♠✐t ♦❢ ❛ ♥❡✇ ❢✉♥❝t✐♦♥ ♠❛❞❡ ✭❞❡r✐✈❡❞✮ ❢r♦♠

❊①❛♠♣❧❡ ✸✳✺✳✷✿

x2

❚❤✐s s✉❜st✐t✉t✐♦♥ ♠❛❦❡s ❝♦♠♣✉t❛t✐♦♥s ❡❛s✐❡r s♦♠❡t✐♠❡s✳ ▲❡t✬s ❝♦♥s✐❞❡r t❤❡ ❧❛st ❡①❛♠♣❧❡ ❛❣❛✐♥✿ f (x) = x2 .

❈♦♠♣✉t❡ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❛♥❞ ✐ts ❧✐♠✐t✿ f (1 + h) − f (1) df (1) = lim h→0 dx h 2 0 (1 + h) − 12 → ? = lim h→0 h 0 12 + 2h + h − 1 = lim h→0 h 2h + h2 = lim h→0 h = lim (2 + h) h→0

= 2 + 0 = 2.

❉❊❆❉ ❊◆❉

❚❤❡ ❝❛♥❝❡❧❧❛t✐♦♥✳

❜✉t

f✳

✸✳✺✳

❚❤❡ ❞❡r✐✈❛t✐✈❡ ✐s t❤❡ ✐♥st❛♥t❛♥❡♦✉s r❛t❡ ♦❢ ❝❤❛♥❣❡

✷✺✵

❊①❡r❝✐s❡ ✸✳✺✳✸

❋✐♥❞

df (−1) dx

❛♥❞

df (0) dx

❢♦r

f (x) = x2 ✳

❊①❡r❝✐s❡ ✸✳✺✳✹

❋✐♥❞

df (1) dx

❢♦r

f (x) = x2 + 1✳

❊①❡r❝✐s❡ ✸✳✺✳✺

❋✐♥❞

df (1) dx

❢♦r

f (x) = x3 ✳

❊①❛♠♣❧❡ ✸✳✺✳✻✿

❋✐♥❞

df (1) dx

❢♦r

1/x

f (x) =

1 ✳ x

df f (1 + h) − f (1) = lim h→0 dx h 1 1 − 0 → ? ❉❊❆❉ = lim 1+h 1 h→0 h 0 1 − 1+h 1+h 1+h = lim h→0   h 1 − (1 − h) 1 · = lim h→0 1+h h   −h 1 = lim · h→0 1+h h   1 ❚❤❡ ❝❛♥❝❡❧❧❛t✐♦♥✳ = lim − h→0 1+h 1 =− = −1 . 1+0

❊◆❉

❆t t❤❡ ❡♥❞✱ r❡❝♦❣♥✐③✐♥❣ t❤❛t t❤✐s ✐s ❛ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥ ❝♦♥t✐♥✉♦✉s ♦♥ ✐ts ❞♦♠❛✐♥✱ ✇❡ s✐♠♣❧② ♣❧✉❣ ✐♥ t❤❡ ✈❛❧✉❡✳ ❊①❡r❝✐s❡ ✸✳✺✳✼

❋✐♥❞

df (1) dx

❢♦r

f (x) = 1/x2 ✳

◆♦t❡ t❤❛t ❡✈❡r② t✐♠❡ ✇❡ ❝♦♠♣✉t❡ t❤❡ ❞❡r✐✈❛t✐✈❡ ❛s ❛ ✭s♣❡❝✐❛❧ ❦✐♥❞ ♦❢✮ ❧✐♠✐t✱ ✇❡ ♠✐❣❤t ❣♦ t❤r♦✉❣❤ t❤❡ s❛♠❡ st❛❣❡s✿

✸✳✺✳

✷✺✶

❚❤❡ ❞❡r✐✈❛t✐✈❡ ✐s t❤❡ ✐♥st❛♥t❛♥❡♦✉s r❛t❡ ♦❢ ❝❤❛♥❣❡

◮ ◗✉♦t✐❡♥t ❘✉❧❡ ✭❡rr♦r✦✮ ✳✳✳ ❧❡❛❞✐♥❣ t♦ ✳✳✳ ❛♥ ✐♥❞❡t❡r♠✐♥❛t❡ ❡①♣r❡ss✐♦♥ ✳✳✳ ❧❡❛❞✐♥❣ t♦ ✳✳✳ ❛❧❣❡❜r❛

✳✳✳ ❧❡❛❞✐♥❣ t♦ ✳✳✳ t❤❡ ❝❛♥❝❡❧❧❛t✐♦♥ ✳✳✳ ❧❡❛❞✐♥❣ t♦ ✳✳✳ s✉❜st✐t✉t✐♦♥

❲❡ s❤♦✉❧❞ ❥✉st ✐❣♥♦r❡ ✇❤❛t ✇❡ ❦♥♦✇ t♦ ❜❡ ❛ ❞❡❛❞✲❡♥❞ ❛♥❞ ❣♦ str❛✐❣❤t t♦ ❛❧❣❡❜r❛✦ ❙♦♠❡t✐♠❡s ❛❧❣❡❜r❛ ✐s ♥♦t ❡♥♦✉❣❤ t❤♦✉❣❤✳ ❊①❛♠♣❧❡ ✸✳✺✳✽✿ tr✐❣♦♥♦♠❡tr②

❋✐rst f (x) = sin x✳ ❚❤❡♥ ∆f ∆x

sin h − sin 0 h sin h ◆♦ ❝❛♥❝❡❧❧❛t✐♦♥✳ = h →1 ❛s h → 0 . =

x=0

❚❤❡ ❧❛st st❡♣ ✐s s✐♠♣❧② t❤❡ ❢♦❧❧♦✇✐♥❣ ❢❛♠♦✉s ❧✐♠✐t ❢r♦♠ ❈❤❛♣t❡r ✷✿ sin x = 1. x→0 x lim

❚❤❡ r❡s✉❧t ✐s ❝♦♥✜r♠❡❞ ✇✐t❤ ❛ s♣r❡❛❞s❤❡❡t✿

■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡

❣r❛♣❤ ♦❢

y = sin x

❝r♦ss❡s t❤❡

y ✲❛①✐s

❛t

45

❞❡❣r❡❡s✳

❊①❡r❝✐s❡ ✸✳✺✳✾

❋✐♥❞

df (π) ❢♦r f (x) = sin x✳ dx

❊①❡r❝✐s❡ ✸✳✺✳✶✵

❋✐♥❞

df (π) ❢♦r f (x) = sin x + x✳ dx

❊①❛♠♣❧❡ ✸✳✺✳✶✶✿ tr✐❣♦♥♦♠❡tr②

❙❡❝♦♥❞✱ f (x) = cos x✳ ❚❤❡♥ ∆f ∆x

cos h − cos 0 h cos h − 1 = ◆♦ ❝❛♥❝❡❧❧❛t✐♦♥✳ h →0 ❛s h → 0 .

= x=0

✸✳✺✳

✷✺✷

❚❤❡ ❞❡r✐✈❛t✐✈❡ ✐s t❤❡ ✐♥st❛♥t❛♥❡♦✉s r❛t❡ ♦❢ ❝❤❛♥❣❡

❚❤❡ ❧❛st st❡♣ ✐s s✐♠♣❧② t❤❡ ♦t❤❡r ❢❛♠♦✉s ❧✐♠✐t ❢r♦♠ ❈❤❛♣t❡r ✷✿ lim

x→0

1 − cos x = 0. x

❚❤❡ r❡s✉❧t ✐s ❝♦♥✜r♠❡❞ ✇✐t❤ ❛ s♣r❡❛❞s❤❡❡t✿

■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡

❣r❛♣❤ ♦❢

y = cos x

❝r♦ss❡s t❤❡

y ✲❛①✐s

❤♦r✐③♦♥t❛❧❧②✳

❊①❡r❝✐s❡ ✸✳✺✳✶✷

❋✐♥❞

df (π) ❢♦r f (x) = cos x✳ dx

❊①❡r❝✐s❡ ✸✳✺✳✶✸

❋✐♥❞

df (0) ❢♦r f (x) = sin x + cos x✳ dx

❊①❛♠♣❧❡ ✸✳✺✳✶✹✿ ❡①♣♦♥❡♥t✐❛❧

▲❡t f (x) = ex ✳ ❚❤❡♥

∆f ∆x

x=0

eh − e0 = h h e −1 = ◆♦ ❝❛♥❝❡❧❧❛t✐♦♥✳ h →1 ❛s h → 0 .

❚❤❡ ❧❛st st❡♣ ✐s s✐♠♣❧② t❤❡ ❢♦❧❧♦✇✐♥❣ ❢❛♠♦✉s ❧✐♠✐t ❢r♦♠ ❈❤❛♣t❡r ✷✿ ex − 1 = 1. x→0 x lim

❚❤❡ r❡s✉❧t ✐s ❝♦♥✜r♠❡❞ ✇✐t❤ ❛ s♣r❡❛❞s❤❡❡t✿

✸✳✻✳ ❚❤❡ ❡①✐st❡♥❝❡ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡✿ ❞✐✛❡r❡♥t✐❛❜✐❧✐t②

✷✺✸

■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ❣r❛♣❤ ♦❢ y = ex ❝r♦ss❡s t❤❡ y ✲❛①✐s ❛t 45 ❞❡❣r❡❡s✳ ■t ❢♦❧❧♦✇s t❤❛t t❤❡ ♥❛t✉r❛❧ ❜❛s❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥ ❝✉ts ❛❧❧ t❤❡ r❡st ✐♥ t✇♦ ❤❛❧✈❡s✿ st❡❡♣ ❛♥❞ s❤❛❧❧♦✇✳ ❊①❡r❝✐s❡ ✸✳✺✳✶✺

❋✐♥❞

df (1) ❢♦r f (x) = ex ✳ dx

❊①❛♠♣❧❡ ✸✳✺✳✶✻✿

x3

❈♦♠♣✉t❡ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ f (x) = x3 ❛t x = a✳ ▲❡t✬s t❛❦❡ ❛❧❧ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ❛t ♦♥❝❡✿ x3 − a3 = x2 + xa + a2 → 3a2 ❛s x → a . x−a

■♥ ♣❛rt✐❝✉❧❛r✱ ✇❡ ❝♦♥❝❧✉❞❡ t❤❛t t❤❡ ❣r❛♣❤ ❝r♦ss❡s t❤❡ x✲❛①✐s ❜② t♦✉❝❤✐♥❣ ✐t✿

▲❡t✬s ❝❤❡❝❦✿ • ■❢ ✇❡ tr② a = −4✱ t❤❡ ❞❡r✐✈❛t✐✈❡ ✐s 48✳ • ■❢ ✇❡ tr② a = 0✱ t❤❡ ❞❡r✐✈❛t✐✈❡ ✐s 0✳ • ■❢ ✇❡ tr② a = 4✱ t❤❡ ❞❡r✐✈❛t✐✈❡ ✐s 48✳ ❚❤❡ r❡s✉❧ts ♠❛t❝❤ t❤❡ ♣✐❝t✉r❡✳

✸✳✻✳ ❚❤❡ ❡①✐st❡♥❝❡ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡✿ ❞✐✛❡r❡♥t✐❛❜✐❧✐t② ❲❡ ❤❛✈❡ ❝♦♠♣✉t❡❞ s✐♠♣❧✐✜❡❞ ❢♦r♠✉❧❛s ❢♦r

✸✳✻✳ ❚❤❡ ❡①✐st❡♥❝❡ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡✿ ❞✐✛❡r❡♥t✐❛❜✐❧✐t②

✷✺✹

✶✳ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts✱ ❛♥❞ t❤❡♥ ✷✳ t❤❡ ❞❡r✐✈❛t✐✈❡s✳ ❚❤❡ ❢♦r♠❡r ♣r❡❝❡❞❡s t❤❡ ❧❛tt❡r ❛♥❞ t❛❦❡s ❝❛r❡ ♦❢ ❛❧❧ t❤❡ ❛❧❣❡❜r❛✳ ❚❤❡ ❧❛tt❡r r❡q✉✐r❡s ❛♣♣❧✐❝❛t✐♦♥s ♦❢ t❤❡ ♠❡t❤♦❞s ♦❢ ❝♦♠♣✉t✐♥❣ ❧✐♠✐ts ♣r❡s❡♥t❡❞ ✐♥ ❈❤❛♣t❡r ✶✳ ❚❤❡ ♣r♦❝❡ss ❛♥❞ t❤❡ ♠❡t❤♦❞s ♦❢ ✜♥❞✐♥❣ t❤❡ ❡①♣❧✐❝✐t ❢♦r♠✉❧❛s ❢♦r ❢✉♥❝t✐♦♥s ❣✐✈❡♥ ❜② ❡①♣❧✐❝✐t ❢♦r♠✉❧❛s ✐s ❝❛❧❧❡❞ ❞✐✛❡r❡♥t✐❛t✐♦♥✳

❲❛r♥✐♥❣✦ ❚❤❡ ✇♦r❞ ✏❞✐✛❡r❡♥t✐❛t❡✑ ❤❛s ♥♦t❤✐♥❣ t♦ ❞♦ ✇✐t❤ ✏❞✐st✐♥❣✉✐s❤✑ ♦r ✏t❡❧❧ ❛♣❛rt✑✳

❙♦♠❡ ❧✐♠✐ts ❞♦♥✬t ❡①✐st✳ ❚❤❡♥✱ ❛s ❛ ❧✐♠✐t✱ t❤❡ ❞❡r✐✈❛t✐✈❡✱ df f (x) − f (a) (a) = lim , x→a dx x−a

♠✐❣❤t ♥♦t ❡①✐st ❡✐t❤❡r✳

❊①❛♠♣❧❡ ✸✳✻✳✶✿ ✉♥❞❡✜♥❡❞ ❚♦ ❜❡❣✐♥ ✇✐t❤✱ ✐❢ ❛ ❢✉♥❝t✐♦♥ ✐s ✉♥❞❡✜♥❡❞ ❛t a✱ t❤❡ ♥✉♠❡r❛t♦r ✐s ✉♥❞❡✜♥❡❞ ❛♥❞ ♥❡✐t❤❡r ✐s t❤❡ ❞❡r✐✈❛t✐✈❡ df ❛t a✳ ❋♦r ❡①❛♠♣❧❡✱ (0) ❞♦❡s♥✬t ❡①✐st ❢♦r f (x) = 1/x✳ dx

❊①❛♠♣❧❡ ✸✳✻✳✷✿ sign(x) ▲❡t✬s ❝♦♥s✐❞❡r ❛ ❞✐s❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥✱ y = f (x) = sign(x) .

❲❡ s❡❡ ❜❡❧♦✇ t❤❛t ❛s x ✐s ❛♣♣r♦❛❝❤✐♥❣ 0✱ t❤❡ s❡❝❛♥t ❧✐♥❡s ❜❡❝♦♠❡ ♠♦r❡ ❛♥❞ ♠♦r❡ st❡❡♣ ❛♥❞ ❛♣♣r♦❛❝❤ ❛ ✈❡rt✐❝❛❧ ♦♥❡✿

❚❤✐s ♠❡❛♥s t❤❛t s❧♦♣❡s → +∞✳ ❙♦✱ t❤❡✐r ❧✐♠✐t

df (a) ✐s ✐♥✜♥✐t❡✳ dx

❚❤✐s ✐s ❤♦✇ t❤✐s ❢❛❝t ❡✛❡❝ts t❤❡ ❛❧❣❡❜r❛✿ sign(x) − sign(0) df (0) = lim x→0 dx x−0 sign(x) = lim x→0 x 1 = lim x→0 |x| = ∞.

❚❤✐s ❧✐♠✐t ❞♦❡s ♥♦t ❡①✐st✳ ❚❤❡ ❞❡r✐✈❛t✐✈❡ ❞♦❡s♥✬t ❡①✐st✳

✸✳✻✳ ❚❤❡ ❡①✐st❡♥❝❡ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡✿ ❞✐✛❡r❡♥t✐❛❜✐❧✐t②

✷✺✺

❊✈❡♥ ✐❢ ✇❡ s✉♣♣♦s❡ t❤❛t t❤❡ ❢✉♥❝t✐♦♥ ✐s ❞❡✜♥❡❞ ❛t x = a ❛♥❞ t❤❡ s❡❝❛♥t ❧✐♥❡s ❛r❡ ❞❡✜♥❡❞✱ ✐t ✐s ♣♦ss✐❜❧❡ t❤❛t ❛s h → 0 t❤❡② ❞♦ ♥♦t t❡♥❞ t♦✇❛r❞ ❛♥② ♣❛rt✐❝✉❧❛r ❧✐♥❡✳

■♥ ♦✉r ♠♦t✐♦♥ ♠❡t❛♣❤♦r✱ t❤❡ ❢✉♥❝t✐♦♥ r❡♣r❡s❡♥ts t❤❡ ❧♦❝❛t✐♦♥ ❛♥❞ t❤❡ ❞❡r✐✈❛t✐✈❡ ✐s t❤❡ ✈❡❧♦❝✐t②✳ ■❢ t❤❡ ❧♦❝❛t✐♦♥ ❝❤❛♥❣❡s ✐♥st❛♥t❧② ✭❛s ✐♥ t❤❡ ❧❛st ❡①❛♠♣❧❡✮✱ t❤❡ ✈❡❧♦❝✐t② ♠✉st ❜❡ ✐♥✜♥✐t❡✿ ✈❡❧♦❝✐t② =

❞✐st❛♥❝❡ 6= 0 . t✐♠❡ = 0

❲❤❡♥ t❤✐s ❧✐♠✐t ❞♦❡s ❡①✐st✱ ✇❤❛t ❞♦❡s ✐t t❡❧❧ ✉s❄ ■t t❡❧❧s ✉s t❤❛t t❤❡r❡ ✐s ❛ t❛♥❣❡♥t ❧✐♥❡ ❛t t❤❛t ❧♦❝❛t✐♦♥✳ ❲❤❛t ❞♦❡s t❤❛t t❡❧❧ ✉s❄ ■t t❡❧❧s ✉s t❤❛t t❤❡r❡ ✐s ♥♦ ❜r❡❛❦ ✐♥ t❤❡ ❣r❛♣❤✦

❚❤❡♦r❡♠ ✸✳✻✳✸✿ ❉❡r✐✈❛t✐✈❡ ◆❡❡❞s ❈♦♥t✐♥✉✐t② ■❢ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢

f

❡①✐sts ❛t

a✱

t❤❡♥

f

✐s ❝♦♥t✐♥✉♦✉s ❛t

a✳

Pr♦♦❢✳ ▲❡t✬s ❝♦♥s✐❞❡r t❤❡ r✐s❡✱ ❢♦r x 6= a✱ ❛♥❞ r❡✇r✐t❡ ✐t ✇✐t❤ t❤❡ ❤❡❧♣ ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❛s ❢♦❧❧♦✇s✿ f (x) − f (a) =

f (x) − f (a) · (x − a) . x−a

■t✬s t❤❡ s❛♠❡ ❢✉♥❝t✐♦♥✳ ▲❡t✬s t❛❦❡ t❤❡ ❧✐♠✐t ♦❢ t❤✐s ❢✉♥❝t✐♦♥ ❛s x → a✿ f (x) − f (a) · (x − a) = f (x) − f (a) | {z } | {z } x−a | {z } ↓ ↓ ↓ f ′ (a) · 0 =⇒ 0

❙♦✱ t❤❡ ❧✐♠✐t ♦❢ t❤❡ ✜rst ❢❛❝t♦r ❡①✐sts ❛♥❞ ✐s ❡q✉❛❧ t♦ f ′ (a)✱ ❛♥❞ t❤❡ ❧✐♠✐t ♦❢ t❤❡ s❡❝♦♥❞ ❢❛❝t♦r ❡①✐sts ❛♥❞ ✐s ❡q✉❛❧ t♦ 0✳ ❚❤❡r❡❢♦r❡✱ t❤❡ Pr♦❞✉❝t ❘✉❧❡ ❛♣♣❧✐❡s✱ ❛♥❞ ✇❡ ❝♦♥❝❧✉❞❡ t❤❛t t❤❡ ❧✐♠✐t ♦❢ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ✐s ❛❧s♦ ③❡r♦✿ f (x) − f (a) =

❚❤❡ ❝♦♥❝❧✉s✐♦♥✱

f (x) − f (a) · (x − a) → f ′ (a) · 0 = 0 . x−a lim (f (x) − f (a)) = 0 ,

x→a

✐s r❡✇r✐tt❡♥ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❙✉♠ ❘✉❧❡ ❛s ❢♦❧❧♦✇s✿

lim f (x) − lim f (a) = 0 .

x→a

x→a

◆♦✇✱ s✐♥❝❡ t❤❡ ❧✐♠✐t ♦❢ ❛ ❝♦♥st❛♥t ❢✉♥❝t✐♦♥ ✐s t❤❡ ❝♦♥st❛♥t✱ lim f (a) = f (a)✱ ✇❡ ❤❛✈❡✿ x→a

lim f (x) = f (a) .

x→a

❚❤❡ ✐❞❡♥t✐t② ♠❡❛♥s t❤❛t f ✐s ❝♦♥t✐♥✉♦✉s ❛t a✳ ■♥❞❡❡❞✱ ♦♥ ♠❛♥② ♦❝❝❛s✐♦♥s✱ ✇❡ ③♦♦♠❡❞ ✐♥ ♦♥ t❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥ t♦ ❞✐s❝♦✈❡r t❤❛t ✐t ❧♦♦❦s ❧✐❦❡ ❛ str❛✐❣❤t ❧✐♥❡✳ ❚❤❡ ✐❞❡❛ s✉❣❣❡sts t❤❛t t❤♦s❡ ❢✉♥❝t✐♦♥s ✇❡r❡ ❝♦♥t✐♥✉♦✉s✦ ❲❡ ✇✐❧❧ ❛♣♣❧② t❤✐s t❡r♠ t♦ t❤❡s❡ ❢✉♥❝t✐♦♥s✿

❉❡✜♥✐t✐♦♥ ✸✳✻✳✹✿ ❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥ ❲❤❡♥ t❤❡ ❧✐♠✐t ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❛t a✱ f (x) − f (a) , x→a x−a lim

❡①✐sts✱ ✇❡ s❛② t❤❛t f ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t a✳

✸✳✻✳ ❚❤❡ ❡①✐st❡♥❝❡ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡✿ ❞✐✛❡r❡♥t✐❛❜✐❧✐t②

✷✺✻

❚❤✉s✱ ✇❡ s❛②✿ ◮ ❊✈❡r② ❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥ ✐s ❛❧s♦ ❝♦♥t✐♥✉♦✉s✳ ❊①❡r❝✐s❡ ✸✳✻✳✺

Pr❡s❡♥t t❤❡ ❧❛st st❛t❡♠❡♥t ❛s ❛♥ ✐♠♣❧✐❝❛t✐♦♥✳ ❙t❛t❡ t❤❡ ❝♦♥✈❡rs❡ t♦♦✳ ❚❤❡ ❝♦♥✈❡rs❡ ✐s♥✬t tr✉❡✳ ❊①❛♠♣❧❡ ✸✳✻✳✻✿ ❛❜s♦❧✉t❡ ✈❛❧✉❡ ❢✉♥❝t✐♦♥

■❢ ✇❡ ③♦♦♠ ✐♥ ♦♥ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❛❜s♦❧✉t❡ ✈❛❧✉❡ ❢✉♥❝t✐♦♥ ❛r♦✉♥❞ (0, 0)✱ ✐t ✇♦♥✬t ❜❡❝♦♠❡ ❛ str❛✐❣❤t ❧✐♥❡✿

❚❤❡ ❝♦r♥❡r ♦❢ t❤❡ ❱ ✇♦♥✬t ❞✐s❛♣♣❡❛r ❡✈❡♥ ❛❢t❡r ♠✉❧t✐♣❧❡ tr✐❡s✳ ▲❡t✬s ❝♦♥✜r♠ t❤✐s ❛❧❣❡❜r❛✐❝❛❧❧② ❜② tr②✐♥❣ t♦ ❝♦♠♣✉t❡ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ f (x) = |x| ❛t x = 0✳ ❲❡ ❤❛✈❡✿ |0 + h| − |0| df (0) = lim h→0 dx h |h| . = lim h→0 h

❲❡✬✈❡ s❡❡♥ t❤✐s ❧✐♠✐t ❜❡❢♦r❡✦ ❈♦♥s✐❞❡r t❤❡s❡ ♦♥❡✲s✐❞❡❞ ❧✐♠✐ts✿ −h |h| = lim− = −1 h→0 h→0 h h h |h| lim+ = lim+ =1 h→0 h h→0 h lim−

❚❤❡② ❛r❡ ♥♦t ❡q✉❛❧✱ s♦ t❤❡ ❧✐♠✐t ❞♦❡s ♥♦t ❡①✐st✳ ❲❡ s❡❡ t❤✐s ❢❛❝t ❜❡❧♦✇✿

❚❤❡ s❡❝❛♥t ❧✐♥❡s s✐♠♣❧② ❢♦❧❧♦✇ t❤❡ ❧✐♥❡ ♦❢ t❤❡ ❣r❛♣❤ ✐ts❡❧❢✳ ❲❡ ❝❛♥ ✈✐s✉❛❧✐③❡ t❤❡s❡ t✇♦ ♠❛✐♥ ❝❧❛ss❡s ♦❢ ❢✉♥❝t✐♦♥s✿

✸✳✻✳

❚❤❡ ❡①✐st❡♥❝❡ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡✿ ❞✐✛❡r❡♥t✐❛❜✐❧✐t②

✷✺✼

❊①❡r❝✐s❡ ✸✳✻✳✼

P❧❛❝❡ ✇✐t❤✐♥ t❤❡ ❞✐❛❣r❛♠✿ |x|, sign x, x3 ✳ ❈♦♠♣❛r❡✿

• ❈♦♥t✐♥✉♦✉s✿ t❤❡r❡ ✐s ♥♦ ❜r❡❛❦ ♦r ❣❛♣✳

• ❉✐✛❡r❡♥t✐❛❜❧❡✿ t❤❡r❡ ✐s ♥♦ ❝♦r♥❡r ♦r ❝✉s♣✳

❆s ✇❡ ❥✉st s❛✇✱ ✇❤❡♥ t❤❡r❡ ❛r❡ t✇♦ ❝❛♥❞✐❞❛t❡s t♦ ❜❡ ❛ t❛♥❣❡♥t✱ t❤❡r❡ ✐s ♥♦ t❛♥❣❡♥t✦ ▲❡t✬s tr② t♦ ✉s❡ t❤❡ ❣❡♦♠❡tr✐❝ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ ✕ ✈✐❛ s❡❝❛♥t ❧✐♥❡s✳ ■♥ ❢❛❝t✱ ✇❡ ✇✐❧❧ ✉s❡ s❡❝❛♥t r❛②s✳ ❆s ✇❡ ❛♣♣r♦❛❝❤ a s❡♣❛r❛t❡❧② ❢r♦♠ t❤❡ ❧❡❢t ❛♥❞ r✐❣❤t✱ t❤❡② t✉r♥✱ ❛♥❞ t❤❡ ❡♥❞ r❡s✉❧t ✐s ♦♥❡ ♦❢ t❤❡s❡ t✇♦✿

• ❚✇♦ r❛②s✳ ❚❤❡♥ f ✐s ♥♦t ❞✐✛❡r❡♥t✐❛❜❧❡✿

• ❖♥❡ ❧✐♥❡✳ ❚❤❡♥ f ✐s ❞✐✛❡r❡♥t✐❛❜❧❡✿

❆❧❣❡❜r❛✐❝❛❧❧②✱ ✇❡ ✈❡r✐❢② t❤❛t t❤❡ t✇♦ ♦♥❡✲s✐❞❡❞ ❧✐♠✐ts ❛r❡ ❡q✉❛❧ t♦ ❡❛❝❤ ♦t❤❡r✿

lim = lim+

h→0− ❊①❡r❝✐s❡ ✸✳✻✳✽

❲❤❛t ✐❢ t❤❡ ❡♥❞ r❡s✉❧t ✐s

♦♥❡ r❛②❄

h→0

✸✳✻✳ ❚❤❡ ❡①✐st❡♥❝❡ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡✿ ❞✐✛❡r❡♥t✐❛❜✐❧✐t②

✷✺✽

❊①❛♠♣❧❡ ✸✳✻✳✾✿ s♠♦♦t❤ ❣r❛♣❤s

❚❤❡r❡ ❛r❡ ❢✉♥❝t✐♦♥s t❤❡ ❣r❛♣❤ ♦❢ ✇❤✐❝❤ ❧♦♦❦ s♠♦♦t❤ ✭♥♦ ❝✉s♣s✮ ❛♥❞ ②❡t t❤❡② ❛r❡ ♥♦t ❞✐✛❡r❡♥t✐❛❜❧❡✿

■♥❞❡❡❞✱ ✐❢ ✇❡ ③♦♦♠ ✐♥ ♦♥ t❤❡ ♦r✐❣✐♥✱ t❤❡ ❧✐♥❡ ✇✐❧❧ ❛♣♣❡❛r ✈❡rt✐❝❛❧✦ ❈♦♥s✐❞❡r ❛ s♣❡❝✐✜❝ ❢✉♥❝t✐♦♥✿ f (x) =

√ 3

x

❛t x = 0 ✭❧❡❢t✮✿

❚❤❡ ❧✐♠✐t ✐s ✐♥✜♥✐t❡ ❡✈❡♥ ✇✐t❤♦✉t ❛ ❝♦♠♣✉t❛t✐♦♥✿ df (0) = ∞ . dx

❍♦✇ ❞♦ ✇❡ ❦♥♦✇❄ ❚❤✐s ❢✉♥❝t✐♦♥ ✐s t❤❡ ✐♥✈❡rs❡ ♦❢ x = y 3 ✭r✐❣❤t✮ ❛♥❞ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❧❛tt❡r ❛t 0 √ 3 ✐s 0❀ t❤❡r❡❢♦r❡✱ ✐ts t❛♥❣❡♥t ❧✐♥❡ ✐s ❤♦r✐③♦♥t❛❧ ❛t 0✳ ❚❤❡♥✱ t❤❡ t❛♥❣❡♥t ❧✐♥❡ ♦❢ y = x ❛t 0 ✐s ✈❡rt✐❝❛❧✦ ❊①❛♠♣❧❡ ✸✳✻✳✶✵✿ q✉❛❞r❛t✐❝

❈♦♠♣✉t❡

df (2) ❢r♦♠ t❤❡ ❞❡✜♥✐t✐♦♥ ❢♦r dx f (x) = −x2 − x .

❉❡✜♥✐t✐♦♥✿

f (2 + h) − f (2) df (2) = lim . h→0 dx h

❚♦ ❝♦♠♣✉t❡ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✱ ✇❡ ♥❡❡❞ t♦ s✉❜st✐t✉t❡ t✇✐❝❡✳ ■♥ f (x) = −x2 − x✱ ✇❡ r❡♣❧❛❝❡ x ✇✐t❤ 2 + h ❛♥❞ t❤❡♥ ✇❡ r❡♣❧❛❝❡ x ✇✐t❤ 2✿ f (2 + h) = −(2 + h)2 − (2 + h), f (2) = −22 − 2 .

✸✳✼✳

❚❤❡ ❞❡r✐✈❛t✐✈❡ ❛s ❛ ❢✉♥❝t✐♦♥

✷✺✾

◆♦✇✱ ✇❡ s✉❜st✐t✉t❡ ✐♥t♦ t❤❡ ❞❡✜♥✐t✐♦♥✿ df [−(2 − h)2 − (2 − h)] − [−22 − 2] (2) = lim h→0 dx h −4 − 4h − h2 − 2 − h + 4 + 2 = lim h→0 h −5h − h2 = lim h→0 h = lim (−5 − h) h→0

= −5 − 0 = 5.

❊①❛♠♣❧❡ ✸✳✻✳✶✶✿ ✐♥❝♦♠❡ t❛①

❘❡❝❛❧❧ t❤❡ ❤②♣♦t❤❡t✐❝❛❧ t❛① ❝♦❞❡ ❢r♦♠ ❈❤❛♣t❡r ✶✳ ■❢ x ✐s t❤❡ ✐♥❝♦♠❡✱ t❤❡♥ t❤❡ ♠❛r❣✐♥❛❧ t❛① ❝♦♠♣✉t❡❞ ❜② t❤❡ ❢♦r♠✉❧❛✿   ✐❢ x ≤ 10000 , 0 f (x) = .10 ✐❢ 10000 < x ≤ 20000 ,   .20 ✐❢ 20000 < x .

r❛t❡ ✐s

❚❤❡ t❛① ❜✐❧❧ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ ✐♥❝♦♠❡ ✐s ❛s ❢♦❧❧♦✇s✿

▲❡t✬s ♣❧♦t ❜♦t❤✿

  ✐❢ x ≤ 10000 , 0 g(x) = .10 · (x − 10000) ✐❢ 10000 < x ≤ 20000 ,   .10 · (x − 10000) + .20 · (x − 20000) ✐❢ 20000 < x .

▲♦♦❦✐♥❣ ❛t t❤❡ ❣r❛♣❤s✱ ✇❡ s❡❡ t❤❛t t❤❡ s❧♦♣❡s ♦❢ t❤❡ ❧❛tt❡r ❛r❡ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ❢♦r♠❡r✦ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ❢♦r♠❡r ✐s t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❧❛tt❡r✿ dg (x) = f (x) , dx

❡①❝❡♣t ❢♦r t❤❡ ✈❛❧✉❡s ♦❢ x ✇❤❡r❡ g ✐s♥✬t ❞✐✛❡r❡♥t✐❛❜❧❡✿ x = 10, 000 ❛♥❞ x = 20, 000✳ ❚❤❡ ♣♦✐♥ts ✇❤❡r❡ t❤❡ ❢✉♥❝t✐♦♥ ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ❢♦r♠ t❤❡ ❞♦♠❛✐♥

♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡✳

✸✳✼✳ ❚❤❡ ❞❡r✐✈❛t✐✈❡ ❛s ❛ ❢✉♥❝t✐♦♥

❚❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✐s ❛ ❢✉♥❝t✐♦♥✱ ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥✳ ❲❤❛t ❛❜♦✉t t❤❡ ❞❡r✐✈❛t✐✈❡❄ ❲❤❛t ✐❢ ✇❡ ❝♦♥str✉❝t t❤❡ s❡❝❛♥t ❛♥❞ t❤❡ t❛♥❣❡♥t ❧✐♥❡s t♦ t❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥ ❛t

❛❧❧ ♣♦✐♥ts ❛t t❤❡ s❛♠❡

✸✳✼✳

❚❤❡ ❞❡r✐✈❛t✐✈❡ ❛s ❛ ❢✉♥❝t✐♦♥

✷✻✵

t✐♠❡❄ ■t ❧♦♦❦s ❝♦♠♣❧✐❝❛t❡❞✿

❚❤❡ s♦❧✉t✐♦♥ ✐s ❛❧❣❡❜r❛✐❝✿ ❲❡ ✜♥❞✱ ♦r ❛tt❡♠♣t t♦ ✜♥❞✱ t❤❡ ❞❡r✐✈❛t✐✈❡ ✕ ❛s t❤❡ ❧✐♠✐t ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t

a ♦❢ t❤❡ ❞♦♠❛✐♥ ♦❢ f ✳ ❚❤❡s❡ a✬s ❛r❡ t❤❡ ✐♥♣✉ts ♦❢ t❤❡ ♥❡✇ ❢✉♥❝t✐♦♥✱ ❛♥❞ t❤❡ ❢♦✉♥❞ ✈❛❧✉❡s ♦❢ df (a) ❛r❡ t❤❡ ♦✉t♣✉ts✳ ❚❤❡ r❡s✉❧t ✐s t❤❡ ❞❡r✐✈❛t✐✈❡ ❢✉♥❝t✐♦♥ ✦ ■ts ❞♦♠❛✐♥ ❝♦♥s✐sts ♦❢ ❛❧❧ ♣♦✐♥ts dx

✕ ❛t ❡✈❡r② ♣♦✐♥t t❤❡ ❞❡r✐✈❛t✐✈❡ ✇❤❡r❡

f

✐s ❞✐✛❡r❡♥t✐❛❜❧❡✳

❚❤✐s ✐s ❛♥ ❛❧t❡r♥❛t✐✈❡✱ ♠♦r❡ ❝♦♠♣❛❝t✱ ♥♦t❛t✐♦♥ ❢♦r t❤❡ ❞❡r✐✈❛t✐✈❡✿

❉❡r✐✈❛t✐✈❡

f ′ (x) =

df (x) dx

■t r❡❛❞s ✏ f ♣r✐♠❡✑✳

❙✐♥❝❡

x

✐s t❤❡ s❛♠❡ ✐♥❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡ ❢♦r ❜♦t❤✱ ✐t ♠❛❦❡s s❡♥s❡ t♦ ✉s❡ t❤❡ t❤❛t ♥❛♠❡ ❢♦r t❤❡ ✐♥♣✉t ♦❢ t❤❡

❞❡r✐✈❛t✐✈❡✿

y = f (x) −→ z = f ′ (x) . ❚❤✉s✱ ❛ ♥❡✇ ❢✉♥❝t✐♦♥ ❤❛s ❜❡❡♥

❞❡r✐✈❡❞

❢r♦♠ t❤❡ ♦❧❞✱ ❛♥❞ s♦ ❛r❡ t❤❡ ♥❛♠❡s✿ t❤❡ ♥❛♠❡ ♦❢ t❤❡ ♥❡✇ ❢✉♥❝t✐♦♥

♠❛❦❡s ❛ r❡❢❡r❡♥❝❡ t♦ t❤❡ ❢✉♥❝t✐♦♥ ✐t ❝❛♠❡ ❢r♦♠✳ ▲❡t✬s ❞❡❝♦♥str✉❝t t❤❡ ♥♦t❛t✐♦♥✿

▲❛❣r❛♥❣❡ ♥♦t❛t✐♦♥ ❛♥❞ ▲❡✐❜♥✐③ ♥♦t❛t✐♦♥ ❢♦r ❞❡r✐✈❛t✐✈❡

♥❛♠❡ ♦❢ ❢✉♥❝t✐♦♥

♥❛♠❡ ♦❢ ✐♥♣✉t

♥❛♠❡ ♦❢ ❢✉♥❝t✐♦♥

♥❛♠❡ ♦❢ ✐♥♣✉t

f

x

f

x

f′

x

df dx

x

df (x) dx df f′ = dx

f ′ (x) =

❚❤❡ ❝❤♦✐❝❡ ❜❡t✇❡❡♥ t❤❡ t✇♦ ✐s ❛ ♠❛tt❡r ♦❢ ❝♦♥✈❡♥✐❡♥❝❡✳

❲❛r♥✐♥❣✦ ❚❤❡ ♥❛♠❡ ♦❢ t❤❡

♦✉t♣✉t

♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ ❞♦❡s♥✬t

♠❛t❝❤ t❤❛t ♦❢ t❤❡ ❢✉♥❝t✐♦♥✳

■♥ ♦r❞❡r t♦ ❜✉✐❧❞ ❛ t❤❡♦r② ❛r♦✉♥❞ t❤❡ ♥❡✇ ❝♦♥❝❡♣t✱ ✇❡ st❛rt ✇✐t❤ t❤❡ s✐♠♣❧❡st s✐t✉❛t✐♦♥s✳ ❈♦♥s✐❞❡r t❤✐s

◮

♦❜✈✐♦✉s st❛t❡♠❡♥t ❛❜♦✉t ♠♦t✐♦♥✿

✏■❢ ■ ❛♠ st❛♥❞✐♥❣ st✐❧❧✱ ♠② s♣❡❡❞ ✐s ③❡r♦✳✑

✸✳✼✳

❚❤❡ ❞❡r✐✈❛t✐✈❡ ❛s ❛ ❢✉♥❝t✐♦♥

✷✻✶

■❢ ❛ ❢✉♥❝t✐♦♥ y = f (x) r❡♣r❡s❡♥ts t❤❡ ♣♦s✐t✐♦♥✱ ✇❡ ❝❛♥ r❡st❛t❡ t❤✐s ♠❛t❤❡♠❛t✐❝❛❧❧②✳ ❲❡ ❢♦❧❧♦✇ ✇❤❛t ✇❡ ❦♥♦✇ ❛❜♦✉t t❤❡ ❞✐✛❡r❡♥❝❡s ♦❢ s❡q✉❡♥❝❡s ✭s❡❡♥ ✐♥ ❱♦❧✉♠❡ ✶✱ ❈❤❛♣t❡r ✶P❈✲✶✮ ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣✿

❚❤❡♦r❡♠ ✸✳✼✳✶✿ ❉✐✛❡r❡♥❝❡ ♦❢ ❈♦♥st❛♥t ❋✉♥❝t✐♦♥ ■❢ ❛ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ ✐♥t❡r✈❛❧ [a, b] ✐s ❝♦♥st❛♥t ♦✈❡r t❤❡ ♥♦❞❡s ♦❢ [a, b]✱ t❤❡♥ t❤❡ ❢✉♥❝t✐♦♥ ❤❛s ❛ ③❡r♦ ❞✐✛❡r❡♥❝❡ ❢♦r ❛❧❧ s❡❝♦♥❞❛r② ♥♦❞❡s ✐♥ t❤❡ ♣❛rt✐t✐♦♥✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿ f (x) = ❝♦♥st❛♥t =⇒ ∆f (c) = 0 .

❲❡ ❥✉st ❞✐✈✐❞❡ ❜② ∆x t♦ ♣r♦✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿

❚❤❡♦r❡♠ ✸✳✼✳✷✿ ❉✐✛❡r❡♥❝❡ ◗✉♦t✐❡♥t ♦❢ ❈♦♥st❛♥t ❋✉♥❝t✐♦♥ ■❢ ❛ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ ✐♥t❡r✈❛❧ [a, b] ✐s ❝♦♥st❛♥t ♦✈❡r t❤❡ ♥♦❞❡s ♦❢ [a, b]✱ t❤❡♥ t❤❡ ❢✉♥❝t✐♦♥ ❤❛s ❛ ③❡r♦ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❢♦r ❛❧❧ s❡❝♦♥❞❛r② ♥♦❞❡s ✐♥ t❤❡ ♣❛rt✐t✐♦♥✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿ f (x) = ❝♦♥st❛♥t =⇒

∆f (c) = 0 . ∆x

❚❛❦✐♥❣ t❤❡ ❧✐♠✐t ∆x → 0 ♣r♦✈❡s t❤❡ ❢♦❧❧♦✇✐♥❣✿

❚❤❡♦r❡♠ ✸✳✼✳✸✿ ❉❡r✐✈❛t✐✈❡ ♦❢ ❈♦♥st❛♥t ❋✉♥❝t✐♦♥ ■❢ ❛ ❢✉♥❝t✐♦♥ ✐s ❝♦♥st❛♥t ♦♥ ❛♥ ♦♣❡♥ ✐♥t❡r✈❛❧ I ✱ t❤❡♥ ✐ts ❞❡r✐✈❛t✐✈❡ ✐s ③❡r♦ ❢♦r ❛❧❧ x ✐♥ I ✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿ f (x) = ❝♦♥st❛♥t =⇒

■♥ s✉♠♠❛r②✿ f (x) = ❝♦♥st❛♥t =⇒ ∆f (c) = 0 =⇒

df (x) = 0 . dx

df ∆f (c) = 0 =⇒ (x) = 0 . ∆x dx

❊①❡r❝✐s❡ ✸✳✼✳✹ Pr♦✈✐❞❡ ❞❡t❛✐❧s ♦❢ t❤❡ ♣r♦♦❢✳ ❚❤❡ ❝♦♥✈❡rs❡ ✐s ❛❧s♦ tr✉❡ ❜✉t ✐ts ♣r♦♦❢ ✐s ♣♦st♣♦♥❡❞ ✉♥t✐❧ ❈❤❛♣t❡r ✺✳

❊①❛♠♣❧❡ ✸✳✼✳✺✿ ♣✐❡❝❡✇✐s❡ ❝♦♥st❛♥t ❢✉♥❝t✐♦♥s ■s ✐t ♣♦ss✐❜❧❡ t♦ ❤❛✈❡ ❛ ❢✉♥❝t✐♦♥ ✇✐t❤ ③❡r♦ ❞❡r✐✈❛t✐✈❡ ❜✉t ♥♦t ❝♦♥st❛♥t❄ ❩♦♦♠✐♥❣ ✐♥ r❡✈❡❛❧s ❣❛♣s✿

✸✳✼✳

❚❤❡ ❞❡r✐✈❛t✐✈❡ ❛s ❛ ❢✉♥❝t✐♦♥

✷✻✷

❊①❛♠♣❧❡ ✸✳✼✳✻✿ ❈❛♥t♦r✬s st❛✐r❝❛s❡

❚♦ ❢❛❝❡ t❤❡ ❝❤❛❧❧❡♥❣❡✱ ❧❡t✬s ❜✉✐❧❞ ❛ ❢✉♥❝t✐♦♥ t❤❛t ❣❡ts ❝❧♦s❡ t♦ t❤✐s ✐♠♣♦ss✐❜✐❧✐t②✳ ❲❡ ❝✉t t❤❡ r❡❝t❛♥❣❧❡ ✐♥ ❤❛❧❢ ✈❡rt✐❝❛❧❧② ❛♥❞ ✐♥ t❤r❡❡ ❤♦r✐③♦♥t❛❧❧②✱ t❤❡♥ ♣❧❛❝❡ ❛ ❤♦r✐③♦♥t❛❧ s❡❣♠❡♥t ✐♥ t❤❡ ♠✐❞❞❧❡✿

❚❤❡ ✇❡ ❞♦ t❤❡ s❛♠❡ ✇✐t❤ t❤❡ t✇♦ r❡❝t❛♥❣❧❡s ❧♦❝❛t❡❞ ❞✐❛❣♦♥❛❧❧② ❢r♦♠ t❤❡ s❡❣♠❡♥t✱ ❛♥❞ s♦ ♦♥✳ ❲❤❡♥ ✇❡ r❡♠♦✈❡ t❤❡ s❝❛✛♦❧❞✐♥❣✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❣r❛♣❤✿

❚❤❡ ❢✉♥❝t✐♦♥ ✐s ❦♥♦✇♥ t♦ ❜❡ ❝♦♥t✐♥✉♦✉s ❛♥❞ ✐♥❝r❡❛s✐♥❣ ❡✈❡♥ t❤♦✉❣❤ ♠❛❞❡ ❡♥t✐r❡❧② ♦❢ ❤♦r✐③♦♥t❛❧ ♣✐❡❝❡s✳ ❊①❛♠♣❧❡ ✸✳✼✳✼✿ ❣❡♥❡r❛❧ ❧✐♥❡❛r ❢✉♥❝t✐♦♥

▲❡t✬s ❝♦♠♣✉t❡ t❤❡ ❞❡r✐✈❛t✐✈❡ ❛t x = a ♦❢ ❛ ❧✐♥❡❛r ❢✉♥❝t✐♦♥ f (x) = mx + b .

❋✐rst✱ t❤❡ ❣❡♦♠❡tr②✿

❊✈❡r② s❡❝❛♥t ❧✐♥❡ ❝♦♥♥❡❝ts ♦✉r ♣♦✐♥t ♦❢ ✐♥t❡r❡st✱ (a, f (a))✱ t♦ ❛♥♦t❤❡r ♣♦✐♥t ♦♥ t❤❡ ❣r❛♣❤✱ (a + h, f (a + h))✱ ✇❤❡r❡ h = ∆x ✐s t❤❡ ✐♥❝r❡♠❡♥t ♦❢ t❤❡ ✐♥❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡✳ ❚❤❡♥✱ t❤❡ s❡❝❛♥t ❧✐♥❡ ❧✐❡s ❡♥t✐r❡❧② ✇✐t❤✐♥ t❤❡ str❛✐❣❤t ❧✐♥❡ t❤❛t ✐s t❤❡ ❣r❛♣❤ ♦❢ f ✳ ■t t❤❡♥ ❤❛s t❤❡ s❛♠❡ s❧♦♣❡✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ❞❡r✐✈❛t✐✈❡ ♠✉st ❜❡ m✦

✸✳✼✳

❚❤❡ ❞❡r✐✈❛t✐✈❡ ❛s ❛ ❢✉♥❝t✐♦♥

✷✻✸

◆♦✇✱ ❛❧❣❡❜r❛✿ ❚❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢ ❧♦❝❛t✐♦♥✳ ■♥❞❡❡❞✿

f (a + h) − f (a) ∆f (a) = ∆x h [m(a + h) + b] − [ma + b] = h mh = h =m df (a) dx ❚❤❡ r❡s✉❧t ✐s ❛

❙✉❜st✐t✉t❡

x=a+h

✐♥t♦ t❤❡ ❢♦r♠✉❧❛ ♦❢

f.

❙✐♠♣❧✐❢②✳ ❚❤❡♥ ❞✐✈✐❞❡✳

=⇒

∆f (a) = m . h→0 ∆x

= lim

♥✉♠❜❡r✳

❍♦✇❡✈❡r✱ s✐♥❝❡ ✐s ✐t ✐♥❞❡♣❡♥❞❡♥t ♦❢ t❤❡ ❝❤♦s❡♥

a✱

✇❡ tr❡❛t ✐t ❛s ❛

❢✉♥❝t✐♦♥✱ ❛

❝♦♥st❛♥t ❢✉♥❝t✐♦♥✿

❚❤❡ ❡①❛♠♣❧❡ ♣r♦✈❡s t❤❡ ✜rst ❤❛❧❢ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ st❛t❡♠❡♥t✿

◮

❚❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ❧✐♥❡❛r ❢✉♥❝t✐♦♥ ✐s ❝♦♥st❛♥t ❛♥❞✱ ❝♦♥✈❡rs❡❧②✱ ❛ ❝♦♥st❛♥t ❢✉♥❝t✐♦♥ ❝❛♥ ♦♥❧②

❜❡ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ❧✐♥❡❛r ❢✉♥❝t✐♦♥✳ ❚❤❡ s❡❝♦♥❞ ❤❛❧❢ ✇✐❧❧ r❡♠❛✐♥ ❛ ❝♦♥❥❡❝t✉r❡ ❢♦r ♥♦✇✳ ❚♦ ❝♦♥t✐♥✉❡ ✇✐t❤ ♦✉r t❤❡♦r②✱ s✉♣♣♦s❡ t❤✐s t✐♠❡ t❤❛t t❤❡r❡ ❛r❡

t✇♦ r✉♥♥❡rs❀ ✇❡ ❤❛✈❡ ❛ s❧✐❣❤t❧② ❧❡ss ♦❜✈✐♦✉s

❢❛❝t ❛❜♦✉t ♠♦t✐♦♥✿

◮

✏■❢ t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t✇♦ r✉♥♥❡rs ✐s♥✬t ❝❤❛♥❣✐♥❣✱ t❤❡♥ t❤❡② r✉♥ ✇✐t❤ t❤❡ s❛♠❡ s♣❡❡❞✳✑

■t✬s ❛s ✐❢ t❤❡② ❛r❡ ❤♦❧❞✐♥❣ t❤❡ t✇♦ ❡♥❞s ♦❢ ❛ ♣♦❧❡ ✇✐t❤♦✉t ♣✉❧❧✐♥❣ ♦r ♣✉s❤✐♥❣✿

■t ✐s ❡✈❡♥ ♣♦ss✐❜❧❡ t❤❛t t❤❡② s♣❡❡❞ ✉♣ ❛♥❞ s❧♦✇ ❞♦✇♥ ❛❧❧ t❤❡ t✐♠❡✳ ❚❤❡ ✈❡❧♦❝✐t② ✐s t❤❡ s❛♠❡ ❜❡❝❛✉s❡ t❤❡② ♠♦✈❡ ❧✐❦❡ ❛ s✐♥❣❧❡ ❜♦❞②✳ ❖♥❝❡ ❛❣❛✐♥✱ ❢♦r ❢✉♥❝t✐♦♥s

y = f (x)

❛♥❞

y = g(x)

r❡♣r❡s❡♥t✐♥❣ t❤❡✐r ♣♦s✐t✐♦♥✱ ✇❡ ❝❛♥ r❡st❛t❡ t❤✐s ✐❞❡❛

♠❛t❤❡♠❛t✐❝❛❧❧② ✐♥ ♦r❞❡r t♦ ❝♦♥✜r♠ t❤❛t ♦✉r t❤❡♦r② ♠❛❦❡s s❡♥s❡✳ ❊①❛♠♣❧❡ ✸✳✼✳✽✿ s❤✐❢t ♦❢ s❡q✉❡♥❝❡

❚❤✐s ✐s ✇❤❛t t❤❡ ❝♦♥str✉❝t✐♦♥ ❧♦♦❦s ❧✐❦❡ ❢♦r s❡q✉❡♥❝❡s✳ ❲❡ s❤✐❢t t❤❡ s❡q✉❡♥❝❡ t♦ ♣r♦❞✉❝❡ ❛ ♥❡✇ s❡q✉❡♥❝❡

bn

✭t♦♣✮✿

an

❜❡❧♦✇ ❜②

1

✉♥✐t ✉♣

✸✳✼✳

❚❤❡ ❞❡r✐✈❛t✐✈❡ ❛s ❛ ❢✉♥❝t✐♦♥

✷✻✹

❇❡❝❛✉s❡ t❤❡ ✉♣s ❛♥❞ ❞♦✇♥s r❡♠❛✐♥ t❤❡ s❛♠❡✱ t❤❡ s❡q✉❡♥❝❡s ♦❢ ❞✐✛❡r❡♥❝❡s ♦❢ t❤❡s❡ t✇♦ s❡q✉❡♥❝❡s ❛r❡ ✐❞❡♥t✐❝❛❧ ✭❜♦tt♦♠✮✳ ❲❡ ❢♦❧❧♦✇ ✇❤❛t ✇❡ ❦♥♦✇ ❛❜♦✉t t❤❡ ❞✐✛❡r❡♥❝❡s ♦❢ s❡q✉❡♥❝❡s ✭s❡❡♥ ✐♥ ❱♦❧✉♠❡ ✶✱ ❈❤❛♣t❡r ✶P❈✲✷✮✿ ◮ ■❢ t✇♦ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ ✐♥t❡r✈❛❧ [a, b] ❞✐✛❡r ❜② ❛ ❝♦♥st❛♥t✱ t❤❡♥

t❤❡② ❤❛✈❡ t❤❡ s❛♠❡ ❞✐✛❡r❡♥❝❡s✳

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿ f (x) − g(x) = K =⇒ ∆f (c) = ∆g(c) .

❚❤❡ ❢✉♥❝t✐♦♥s ❜❡❧♦✇ ❛r❡ ❝♦❧♦r❡❞ ❛❝❝♦r❞✐♥❣❧②✿

■♥ ♦t❤❡r ✇♦r❞s✱ ✐❢ t❤❡ ❤❡✐❣❤t ♦❢ ❛ t✉♥♥❡❧ ✐s ❝♦♥st❛♥t✱ t❤❡ ✢♦♦r ❛♥❞ t❤❡ ❝❡✐❧✐♥❣ ♠✉st ❜❡ ✐❞❡♥t✐❝❛❧ ✐♥ s❤❛♣❡✳ ❇❡❧♦✇ ✐s t❤❡ ❛❧❣❡❜r❛✐❝ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤✐s ✐❞❡❛✿

❚❤❡♦r❡♠ ✸✳✼✳✾✿ ❉✐✛❡r❡♥❝❡ ♦❢ ❋✉♥❝t✐♦♥s ❚❤❛t ❉✐✛❡r ❜② ❈♦♥st❛♥t ■❢ t✇♦ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ ✐♥t❡r✈❛❧ [a, b] ❞✐✛❡r ❜② ❛ ❝♦♥st❛♥t✱ t❤❡♥ t❤❡② ❤❛✈❡ t❤❡ s❛♠❡ ❞✐✛❡r❡♥❝❡s✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿ f (x) − g(x) = ❝♦♥st❛♥t =⇒ ∆f (c) = ∆g(c) .

❲❡ ❥✉st ❞✐✈✐❞❡ ❜② ∆x t♦ ♣r♦✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿

✸✳✼✳

❚❤❡ ❞❡r✐✈❛t✐✈❡ ❛s ❛ ❢✉♥❝t✐♦♥

✷✻✺

❚❤❡♦r❡♠ ✸✳✼✳✶✵✿ ❉✐✛❡r❡♥❝❡ ◗✉♦t✐❡♥t ♦❢ ❋✉♥❝t✐♦♥s ❚❤❛t ❉✐✛❡r ❜② ❈♦♥✲ st❛♥t ■❢ t✇♦ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ ✐♥t❡r✈❛❧ [a, b] ❞✐✛❡r ❜② ❛ ❝♦♥st❛♥t✱ t❤❡♥ t❤❡② ❤❛✈❡ t❤❡ s❛♠❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿ f (x) − g(x) = ❝♦♥st❛♥t =⇒

∆g ∆f (c) = (c) . ∆x ∆x

❚❛❦✐♥❣ t❤❡ ❧✐♠✐t ∆x → 0 ♣r♦✈❡s t❤❡ ❢♦❧❧♦✇✐♥❣✿

❚❤❡♦r❡♠ ✸✳✼✳✶✶✿ ❉❡r✐✈❛t✐✈❡ ♦❢ ❋✉♥❝t✐♦♥s ❚❤❛t ❉✐✛❡r ❜② ❈♦♥st❛♥t ■❢ t✇♦ ❞✐✛❡r❡♥t✐❛❜❧❡ ♦♥ ♦♣❡♥ ✐♥t❡r✈❛❧ I ❢✉♥❝t✐♦♥s ❞✐✛❡r ❜② ❛ ❝♦♥st❛♥t✱ t❤❡♥ t❤❡✐r ❞❡r✐✈❛t✐✈❡s ❛r❡ ❡q✉❛❧✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿ f (x) − g(x) = ❝♦♥st❛♥t =⇒

df dg (x) = (x) . dx dx

■♥ s✉♠♠❛r②✿ f (x) − g(x) = ❝♦♥st❛♥t =⇒ ∆f (c) = ∆g(c) =⇒

∆f ∆g df dg (c) = (c) =⇒ (x) = (x) . ∆x ∆x dx dx

❍❡r❡✱ ✇❡ ♠♦✈❡ ❢r♦♠ ❞✐s♣❧❛❝❡♠❡♥ts t♦ ❛✈❡r❛❣❡ ✈❡❧♦❝✐t✐❡s t♦ ✐♥st❛♥t❛♥❡♦✉s ✈❡❧♦❝✐t✐❡s✳

❊①❡r❝✐s❡ ✸✳✼✳✶✷ Pr♦✈✐❞❡ ❞❡t❛✐❧s ♦❢ t❤❡ ♣r♦♦❢✳ ❍♦✇❡✈❡r✱ ✐❢ ✇❡ ❛r❡ ✇✐❧❧✐♥❣ t♦ ✉s❡ ❉❡r✐✈❛t✐✈❡

♦❢ ❈♦♥st❛♥t ❋✉♥❝t✐♦♥✱ t❤✐s ♣r♦♦❢ ♠✐❣❤t ❜❡ ♠✉❝❤ s❤♦rt❡r✿

f (x) − g(x) = ❝♦♥st❛♥t =⇒

d(f − g) df dg (x) = 0 =⇒ (x) = (x) . dx dx dx

❚❤❡ ❧❛st ✐♠♣❧✐❝❛t✐♦♥ ✐s t♦ ❜❡ ♣r♦✈❡❞ s❤♦rt❧②✳ ❚❤❡ ❝♦♥❝❧✉s✐♦♥ ❝❛♥ ❜❡ ❝♦♥✜r♠❡❞ ❜② ❧♦♦❦✐♥❣ ❛t t❤❡ ❣r❛♣❤s✿

■❢ t❤❡② ❞✐✛❡r ❜② ❛ ❝♦♥st❛♥t✱ ♦♥❡ ✐s ❥✉st ❛ ✈❡rt✐❝❛❧❧② s❤✐❢t❡❞ ✈❡rs✐♦♥ ♦❢ t❤❡ ♦t❤❡r✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ❣r❛♣❤s ❤❛✈❡ ❡①❛❝t❧② t❤❡ s❛♠❡ s❤❛♣❡✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ❣r❛♣❤s ❤❛✈❡ ❡①❛❝t❧② t❤❡ s❛♠❡ s❧♦♣❡s ❛t t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❧♦❝❛t✐♦♥s✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ❞❡r✐✈❛t✐✈❡s ❛r❡ ❡①❛❝t❧② t❤❡ s❛♠❡✳

❊①❛♠♣❧❡ ✸✳✼✳✶✸✿ t❤r❡❡ r✉♥♥❡rs ❚❤❡ ❣r❛♣❤ s❤♦✇s t❤❡ ♣♦s✐t✐♦♥s ♦❢ t❤r❡❡ r✉♥♥❡rs ❛s ❢✉♥❝t✐♦♥s ♦❢ t✐♠❡✳ ❉❡s❝r✐❜❡ ✇❤❛t ❤❛s ❤❛♣♣❡♥❡❞✳

✸✳✼✳

❚❤❡ ❞❡r✐✈❛t✐✈❡ ❛s ❛ ❢✉♥❝t✐♦♥

✷✻✻

❍❡r❡✬s ✇❤❛t ❤❛♣♣❡♥❡❞✿ • ❘✉♥♥❡r A st❛rts ❢❛st ❛♥❞ t❤❡♥ s❧♦✇s ❞♦✇♥✱ ❜✉t r❡❛❝❤❡s t❤❡ ✜♥✐s❤ ❧✐♥❡ ✜rst✳ • ❘✉♥♥❡r B ♠❛✐♥t❛✐♥s t❤❡ s❛♠❡ s♣❡❡❞✳ • ❘✉♥♥❡r C st❛rts ❧❛t❡ ❛♥❞ t❤❡♥ r✉♥s ❢❛st ❛♥❞ ❛rr✐✈❡s ❛t t❤❡ s❛♠❡ t✐♠❡ ❛s B ✳ ❲❡ ❡st✐♠❛t❡ t❤❡ s❧♦♣❡s ♦❢ t❤❡ ❣r❛♣❤ ❛♥❞ ❞✐s❝♦✈❡r s❡✈❡r❛❧ ✈❛❧✉❡s ♦❢ t❤❡ ✈❡❧♦❝✐t✐❡s✳ ❚❤❡ t❤r❡❡ ❣r❛♣❤s ❛r❡ s❦❡t❝❤❡❞ ❤❡r❡✿

❊①❡r❝✐s❡ ✸✳✼✳✶✹

❋✐♥❞ t❤❡ ❡①❛❝t ✈❛❧✉❡s ♦❢ t❤❡ ❝♦♥st❛♥t ✈❡❧♦❝✐t✐❡s✳ ❊①❡r❝✐s❡ ✸✳✼✳✶✺

❉❡s❝r✐❜❡ ✇❤❛t ❤❛s ❤❛♣♣❡♥❡❞ ❤❡r❡✿

✸✳✼✳

❚❤❡ ❞❡r✐✈❛t✐✈❡ ❛s ❛ ❢✉♥❝t✐♦♥

✷✻✼

❊①❛♠♣❧❡ ✸✳✼✳✶✻✿ ❣❡♥❡r❛❧ q✉❛❞r❛t✐❝ ❢✉♥❝t✐♦♥

❲❤❛t ✐❢ ✇❡ ♣✐❝❦ ❛ q✉❛❞r❛t✐❝ ❢✉♥❝t✐♦♥ t❤✐s t✐♠❡❄ ▲❡t✬s ✜♥❞ f ′ = ∆f f (x + h) − f (x) (x) = ∆x h (a[x + h]2 + b[x + h] + c) − (ax2 + bx + c) = h ax2 + 2axh + ah2 + bx + bh + c − ax2 − bx − c = h 2axh + ah2 + bh = h = 2ax + ah + b . df (x) dx

df ❢♦r f (x) = ax2 + bx + c✳ dx

❚❤❡ t❡r♠s ✇✐t❤♦✉t h ❝❛♥❝❡❧✳ ❚❤❛t✬s ✇❤② ✇❡ ❝❛♥ ❞✐✈✐❞❡ ❜② h!

=⇒

= lim (2ax + ah + b) h→0 = (2ax + ah + b)

h=0

= 2ax + b .

❆s ✇❡ ❝❛♥ s❡❡✱ t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❛♥❞ t❤❡ ❞❡r✐✈❛t✐✈❡ ✐s ❛ s♠❛❧❧ ✈❡rt✐❝❛❧ s❤✐❢t✳ ❲❡ ❝❛♥ ♥♦✇ ♠❛t❝❤ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♣❛rts ♦❢ t❤❡ ❣r❛♣❤s ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❛♥❞ ✐ts ❞❡r✐✈❛t✐✈❡ ✭t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✐s✱ ❤♦✇❡✈❡r✱ ✇❤❛t ✇❡ ♣❧♦t ❜❡❧♦✇✮✿

❲❡ s❡❡ ❤♦✇ t❤❡ ❛r❡❛s ♦❢ t❤❡ ❣r❛♣❤ ♦❢ f ✇✐t❤ ♣♦s✐t✐✈❡✴♥❡❣❛t✐✈❡ df ❣r❛♣❤ ♦❢ f ′ = ✇✐t❤ ♣♦s✐t✐✈❡✴♥❡❣❛t✐✈❡ ✈❛❧✉❡s✳

s❧♦♣❡s ❝♦rr❡s♣♦♥❞ t♦ t❤❡ ❛r❡❛ ♦❢ t❤❡

dx

❚❤❡ ❡①❛♠♣❧❡ ♣r♦✈❡s t❤❡ ✜rst ❤❛❧❢ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ st❛t❡♠❡♥t✿ ◮ ❚❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ q✉❛❞r❛t✐❝ ❢✉♥❝t✐♦♥ ✐s ❧✐♥❡❛r ❛♥❞✱ ❝♦♥✈❡rs❡❧②✱ ❛ ❧✐♥❡❛r ❢✉♥❝t✐♦♥ ❝❛♥ ♦♥❧② ❜❡

t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ q✉❛❞r❛t✐❝ ❢✉♥❝t✐♦♥✳

❚❤❡ s❡❝♦♥❞ ❤❛❧❢ ✇✐❧❧ r❡♠❛✐♥ ❛ ❝♦♥❥❡❝t✉r❡ ❢♦r ♥♦✇✳ ❊①❡r❝✐s❡ ✸✳✼✳✶✼

Pr♦✈❡ t❤❛t t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛♥ ♦❞❞ ❢✉♥❝t✐♦♥ ✐s ❡✈❡♥✱ ❛♥❞ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛♥ ❡✈❡♥ ❢✉♥❝t✐♦♥ ✐s ♦❞❞✳

✸✳✼✳

❚❤❡ ❞❡r✐✈❛t✐✈❡ ❛s ❛ ❢✉♥❝t✐♦♥

✷✻✽

❊①❛♠♣❧❡ ✸✳✼✳✶✽✿ r♦❧❧✐♥❣ ❜❛❧❧

❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ r♦❧❧✐♥❣ ❜❛❧❧ t❤❛t ❤✐ts ❛ ✇❛❧❧ ❛♥❞ t❤❡♥ ❣♦❡s ❜❛❝❦✳

❚❤✐s t✐♠❡✱ t❤❡ ♠♦t✐♦♥ ✐s ❞❡s❝r✐❜❡❞ ✐♥ t❡r♠s ♦❢ ✐ts ✈❡❧♦❝✐t② r❛t❤❡r t❤❛♥ ✐ts ❧♦❝❛t✐♦♥✦ ❲❡ ✇❛♥t t♦ ✜♥❞ t❤❡ ❧♦❝❛t✐♦♥ ✕ ♦❢ t❤❡ ❝❡♥t❡r ♦❢ t❤❡ ❜❛❧❧ ✕ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✐♠❡✳ ❋✐rst✱ ✇❡ ❛❞❞ t❤❡ x✲❛①✐s t♦ t❤❡ ♣✐❝t✉r❡✳ ❲❡ ♠❛❦❡ t❤❡ ♣♦s✐t✐✈❡ ❞✐r❡❝t✐♦♥ t♦ ❜❡ t♦ t❤❡ r✐❣❤t ♦❢ t❤❡ ✇❛❧❧✳ ❚❤❡♥ ✇❡ ♦❜s❡r✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛❜♦✉t t❤❡ ✈❡❧♦❝✐t②✿ ✶✳ ❚❤❡ ✈❡❧♦❝✐t② ♦❢ t❤❡ ❜❛❧❧ ✐s ✐♥✐t✐❛❧❧② ♥❡❣❛t✐✈❡ ❛♥❞ ❝♦♥st❛♥t✳ ✷✳ ❆s t❤❡ ❜❛❧❧ t♦✉❝❤❡s t❤❡ ✇❛❧❧ ❛♥❞ st❛rts t♦ ❝♦♥tr❛❝t✱ t❤❡ s♣❡❡❞ ❞❡❝❧✐♥❡s✳ ❚❤✐s ♠❡❛♥s t❤❛t t❤❡ ✈❡❧♦❝✐t② ✐♥❝r❡❛s❡s✦ ✸✳ ❚❤❡ ✈❡❧♦❝✐t② ✐♥❝r❡❛s❡s ✉♥t✐❧ ✐t r❡❛❝❤❡s 0✳ ✹✳ ❚❤❡ ✈❡❧♦❝✐t② t❤❡♥ ❝♦♥t✐♥✉❡s t♦ ✐♥❝r❡❛s❡ ❛s t❤❡ ❜❛❧❧ st❛rts t♦ ❡①♣❛♥❞✳ ✺✳ ❋✐♥❛❧❧②✱ ❛s t❤❡ ❜❛❧❧ ❧❡❛✈❡s t❤❡ ✇❛❧❧✱ t❤❡ ✈❡❧♦❝✐t② t❤❛t ❤❛s ❜❡❡♥ r❡❛❝❤❡❞ ❜❡❝♦♠❡s ❝♦♥st❛♥t ❛♥❞ ♣♦s✐t✐✈❡✳ ❚❤❡s❡ ❛r❡ ♣❧❛✉s✐❜❧❡ ✈❛❧✉❡s✿

❲❡ ❛ss✉♠❡✱ ♥❛t✉r❛❧❧②✱ t❤❛t t❤❡ ✈❡❧♦❝✐t② ✐s ❝❤❛♥❣✐♥❣ ❝♦♥t✐♥✉♦✉s❧②✳ ❚❤❡♥ ✇❡ ❝♦♥♥❡❝t t❤❡s❡ ✈❛❧✉❡s ✐♥t♦ ❛ ❝✉r✈❡ ❢♦❧❧♦✇✐♥❣ t❤❡ ❞❡s❝r✐♣t✐♦♥ ❛❜♦✈❡✿

◆❡①t✱ ❜❛s❡❞ ♦♥ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ✈❡❧♦❝✐t②✱ ✇❡ ♣❧♦t t❤❡ ❧♦❝❛t✐♦♥✳ ❍❡r❡ ✇❡ ✉s❡ t❤❡ ❢❛❝ts ❡st❛❜❧✐s❤❡❞ ❛❜♦✈❡✿ • ❚❤❡ ♣♦s✐t✐♦♥ ✐s ❛ ❧✐♥❡❛r ❧♦❝❛t✐♦♥ ❢✉♥❝t✐♦♥ ✇❤❡♥ t❤❡ ✈❡❧♦❝✐t② ✐s ❝♦♥st❛♥t✳ • ❚❤❡ ♣♦s✐t✐♦♥ ✐s ❛ q✉❛❞r❛t✐❝ ❧♦❝❛t✐♦♥ ❢✉♥❝t✐♦♥ ✇❤❡♥ t❤❡ ✈❡❧♦❝✐t② ✐s ❧✐♥❡❛r✳ ❋✉rt❤❡r♠♦r❡✱ ✇❡ ❛ss✉♠❡✱ ✈❡r② ♥❛t✉r❛❧❧②✱ t❤❛t t❤❡ ❧♦❝❛t✐♦♥ ✐s ❝❤❛♥❣✐♥❣ ❝♦♥t✐♥✉♦✉s❧②✳ ❲❡ t❤❡♥ ❝♦♥♥❡❝t t❤❡ ❧✐♥❡❛r ❛♥❞ q✉❛❞r❛t✐❝ ♣✐❡❝❡s ✐♥t♦ ❛ s✐♥❣❧❡ ❝✉r✈❡✿

✸✳✼✳

❚❤❡ ❞❡r✐✈❛t✐✈❡ ❛s ❛ ❢✉♥❝t✐♦♥

✷✻✾

❊①❡r❝✐s❡ ✸✳✼✳✶✾

❉♦❡s t❤❡ ❣r❛♣❤ ♦❢ ❧♦❝❛t✐♦♥ t♦✉❝❤ t❤❡ t✲❛①✐s❄ ❉♦❡s t❤❡ ❣r❛♣❤ ♦❢ ❧♦❝❛t✐♦♥ ❤❛✈❡ ❝✉s♣s ❛t t❤❡ ♠♦♠❡♥ts ✇❤❡♥ t❤❡ ❜❛❧❧ r❡❛❝❤❡s ❛♥❞ ❧❡❛✈❡s t❤❡ ✇❛❧❧❄ ❊①❡r❝✐s❡ ✸✳✼✳✷✵

❲❤❛t ✐❢ t❤✐s ✐s ❛ ❜✐❧❧✐❛r❞ ❜❛❧❧ ❛♥❞ t❤❡ ❝♦❧❧✐s✐♦♥ ✐s ♣❡r❢❡❝t❧② r✐❣✐❞❄ P❧♦t t❤❡ r❡s✉❧t✐♥❣ ❢✉♥❝t✐♦♥s ❛♥❞ ❞✐s❝✉ss t❤❡✐r ❝♦♥t✐♥✉✐t② ❛♥❞ ❞✐✛❡r❡♥t✐❛❜✐❧✐t②✳ ❊①❛♠♣❧❡ ✸✳✼✳✷✶✿ ❧✐♥❡❛r ❞❡♥s✐t②

❙✉♣♣♦s❡ ✇❡ ❛r❡ ❣✐✈❡♥ ❛ ✉♥✐❢♦r♠ ♠❡t❛❧ r♦❞✿

❲❡ ❞❡✜♥❡ ❢♦r s✉❝❤ ❛ r♦❞✿

♠❛ss . ❧❡♥❣t❤ ❲❤❛t ✐❢ t❤❡ r♦❞ ✐s♥✬t ✉♥✐❢♦r♠✱ ❧✐❦❡ ❛♥ ❛❧❧♦②❄ ❋♦r ❡①❛♠♣❧❡✱ ✇❡ ♠✐❣❤t ❤❛✈❡ t✇♦ ♣✐❡❝❡s ♦❢ ❞✐✛❡r❡♥t ♠❡t❛❧s ♣❛rt✐❛❧❧② ♠❡❧t❡❞ t♦❣❡t❤❡r✿ ❛✈❡r❛❣❡ ❧✐♥❡❛r ❞❡♥s✐t② =

❙✉♣♣♦s❡ t❤❡ ❞❡♥s✐t✐❡s ❛r❡ 1 ❧❜✴✐♥ ❛♥❞ 2 ❧❜s✴✐♥ ❛t t❤❡ ❡♥❞s r❡s♣❡❝t✐✈❡❧②✳ ❚❤❡♥ t❤❡ ❞❡♥s✐t② ♦❢ t❤✐s ❛❧❧♦② ✇✐❧❧ ❣r❛❞✉❛❧❧② ❝❤❛♥❣❡ ❢r♦♠ 1 t♦ 2✿

✸✳✼✳

❚❤❡ ❞❡r✐✈❛t✐✈❡ ❛s ❛ ❢✉♥❝t✐♦♥

✷✼✵

❚❤✐s ✐s t❤❡ ♠❛ss ❢✉♥❝t✐♦♥ ♦❢ x✱ t❤❡ ❧♦❝❛t✐♦♥✿ y = m(x) ✐s t❤❡ ♠❛ss ♦❢ t❤❡ r♦❞ ❢r♦♠ 0 t♦ x✳ ❈♦♥s✐❞❡r ❤♦✇ m ✐s ❝❤❛♥❣✐♥❣✿

❲❡ ❛r❡ ✐♥ t❤❡ s❛♠❡ ♣♦s✐t✐♦♥ ✇❡ ✇❡r❡ ✇❤❡♥ ✇❡ ♠♦✈❡❞ ❢r♦♠ ❝♦♥st❛♥t t♦ ✈❛r✐❛❜❧❡ ✈❡❧♦❝✐t✐❡s✦ ❚❛❦❡ ❛ s♠❛❧❧ ♣✐❡❝❡ ♦❢ t❤❡ r♦❞ ❛t ❧♦❝❛t✐♦♥ x✱ ∆x ❧♦♥❣✱ ❛♥❞ ❧❡t✬s ❝❛❧❧ ✐ts ♠❛ss ∆m✳ ❚❤❡♥✱ ❢♦r t❤✐s ♣✐❡❝❡✱ ✇❡ ❤❛✈❡✿ ∆m m(x + ∆x) − m(x) ♠❛ss = = . ❛✈❡r❛❣❡ ❞❡♥s✐t② = ❧❡♥❣t❤ ∆x ∆x ❚❤❡♥ ✇❡ ❞❡✜♥❡ t❤❡ ❧✐♥❡❛r ❞❡♥s✐t② ❛s t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ♠❛ss ❢✉♥❝t✐♦♥✿

∆m = m′ (x) . ∆x→0 ∆x

❧✐♥❡❛r ❞❡♥s✐t② ❛t x = lim ❊①❛♠♣❧❡ ✸✳✼✳✷✷✿ s❦❡t❝❤ t❤❡ ❞❡r✐✈❛t✐✈❡

❲❡ st❛rt ✇✐t❤ t❤❡ ❣r❛♣❤ ♦❢ f ✐♥ r❡❞ ✭✜rst r♦✇✮ ❛♥❞ ❧♦♦❦ ❢♦r ✐ts ♣♦s✐t✐✈❡ ❛♥❞ ♥❡❣❛t✐✈❡ s❧♦♣❡s ✭s❡❝♦♥❞ r♦✇✮✿

❖❢ ❝♦✉rs❡✱ ✇❡ ❛r❡ ❧♦♦❦✐♥❣ ❛t t❤❡ ✐♥❝r❡❛s✐♥❣ ❛♥❞ ❞❡❝r❡❛s✐♥❣ ❜❡❤❛✈✐♦r✳ ❲❡ ✇r✐t❡ + ❢♦r ✐♥❝r❡❛s✐♥❣ ❛♥❞ − ❢♦r ❞❡❝r❡❛s✐♥❣ ✭t❤✐r❞ r♦✇✮✳ ❚❤✐s ✐♥❢♦r♠❛t✐♦♥ ✐s s✉✣❝✐❡♥t t♦ ♣❧♦t ❛ r♦✉❣❤ ❣r❛♣❤ ♦❢ f ′ ✭❧❛st r♦✇✮✳ ❆s ❛ s❤♦rt❝✉t✱ ✇❡ ❝❛♥ ✜rst ❧♦♦❦ ❛t t❤❡ ♣♦✐♥ts ✇✐t❤ ❛ ❤♦r✐③♦♥t❛❧ t❛♥❣❡♥t✱ ✐✳❡✳✱ ❢♦r t❤♦s❡ x✬s ✇❤❡r❡ f ′ (x) = 0✳ ❚❤❡s❡ ✇✐❧❧ t②♣✐❝❛❧❧② s❡♣❛r❛t❡ ✐♥t❡r✈❛❧s ✇❤❡r❡ f ′ > 0 ❢r♦♠ f ′ < 0✳ ❲❡ ❛❧s♦ ♥♦t✐❝❡ t❤❡ ✢❛tt❡♥✐♥❣ ♦❢ t❤❡ ❣r❛♣❤ ♦♥ t❤❡ ❢❛r r✐❣❤t❀ t❤❡ r❡s✉❧t ✐s ❛❧s♦ ❛ ✢❛tt❡♥✐♥❣ ♦❢ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ ✕ t♦✇❛r❞ 0✳

✸✳✼✳

❚❤❡ ❞❡r✐✈❛t✐✈❡ ❛s ❛ ❢✉♥❝t✐♦♥

✷✼✶ ❲❛r♥✐♥❣✦

❊✈❡♥ t❤♦✉❣❤ f ′ ✐s ❞❡r✐✈❡❞ ❢r♦♠ f ✱ ❛s ②♦✉ ❝❛♥ s❡❡✱ t❤❡r❡ ✐s ♥♦ ✇❛② t♦ ❝♦♥❥✉r❡ t❤❡ ✇❤♦❧❡ ❣r❛♣❤ ♦❢ f ′ ❢r♦♠ t❤❡ ❧♦♦❦s ♦❢ t❤❡ ❣r❛♣❤ ♦❢ f ✳ ❲❡ s❤♦✉❧❞♥✬t ❡①♣❡❝t t❤❡ ❣r❛♣❤ ♦❢ f ′ t♦ ❡♠❡r❣❡ ❢r♦♠ s❤✐❢t✐♥❣ t❤❡ ❣r❛♣❤ ♦❢ f ✱ str❡t❝❤✐♥❣ ♦r s❤r✐♥❦✐♥❣✱ ♦r ✢✐♣♣✐♥❣✱ ❡t❝✳ ❊①❛♠♣❧❡ ✸✳✼✳✷✸✿ q✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s

❲❡ ❝❛♥ ❞♦ t❤✐s ♠♦r❡ str❛t❡❣✐❝❛❧❧②✱ ✇✐t❤ ❢❡✇❡r ♣♦✐♥ts t♦ t❡st✳ ▲❡t✬s ❛♣♣❧② t❤✐s ♠❡t❤♦❞ t♦ t❤❡ ❣r❛♣❤ ♦❢ y = sin x✳ ■t ✐s r❡♣❡t✐t✐✈❡✿

❚❤❡ r❡s✉❧t✐♥❣ ❣r❛♣❤ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ ❧♦♦❦s ❧✐❦❡ t❤❛t ♦❢ y = cos x✦ ❲❡ ✇✐❧❧ s❤♦✇ ❜❡❧♦✇ t❤❛t t❤✐s ✐s♥✬t ❛ ❝♦✐♥❝✐❞❡♥❝❡✳ ▲❡t✬s s✉♠♠❛r✐③❡✳ ❆s ∆x ✐s ❛♣♣r♦❛❝❤✐♥❣ ③❡r♦✱ t❤❡r❡ ❛r❡ ♠♦r❡ ❛♥❞ ♠♦r❡ ♥♦❞❡s ✐♥ t❤❡ ♣❛rt✐t✐♦♥ ❛♥❞ ♠♦r❡ ❛♥❞ ♠♦r❡ ♣♦✐♥ts ❢♦✉♥❞ ♦♥ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥✳ ❆s ❛ r❡s✉❧t✱ t❤❡r❡ ❛r❡ ♠♦r❡ ❛♥❞ ♠♦r❡ ✈❛❧✉❡s ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❢♦✉♥❞✳ ❊✈❡♥t✉❛❧❧②✱ t❤❡ ❢♦r♠❡r ♣♦✐♥ts ❢♦r♠ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❛♥❞ t❤❡ ❧❛tt❡r ❢♦r♠ t❤❡ ❣r❛♣❤ ♦❢ ✐ts ❞❡r✐✈❛t✐✈❡✿

◆❡①t✱ ✇❡ ❝♦♥s✐❞❡r t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ t❤❛t r❡♣r❡s❡♥ts ❛ tr❛♥s❢♦r♠❛t✐♦♥ ✿

▲❡t✬s st❛rt ✇✐t❤ t❤❡ ❜❛s✐❝ tr❛♥s❢♦r♠❛t✐♦♥s✳ ❋✐rst ❛ s❤✐❢t y = x + k✿

✸✳✼✳

❚❤❡ ❞❡r✐✈❛t✐✈❡ ❛s ❛ ❢✉♥❝t✐♦♥

✷✼✷

❚❤❡ ❞❡r✐✈❛t✐✈❡ ✐s 1✳ ◆❡①t✱ ❛ ✢✐♣ y = −x✿

❚❤❡ ❞❡r✐✈❛t✐✈❡ ✐s −1✳

❲❡ ❝♦♥❝❧✉❞❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿ ◮ ❋♦r r✐❣✐❞ ♠♦t✐♦♥ ✭♥♦ ❝❤❛♥❣❡s ♦❢ ❞✐st❛♥❝❡s✮✱ t❤❡ ❞❡r✐✈❛t✐✈❡ ✐s 1 ♦r −1✳

◆❡①t✱ ❛ str❡t❝❤ y = kx (k > 0)✿

❚❤❡ ❞❡r✐✈❛t✐✈❡ ✐s k ✳ ❚❤❡ str❡t❝❤ y = 2x ✐s ✉♥✐❢♦r♠ ❛s t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ ❛♥② t✇♦ ♣♦✐♥ts ❞♦✉❜❧❡s✳ ■t ✐s ❛❧s♦ t❤❡ ❞❡r✐✈❛t✐✈❡✦ ■♥ ❣❡♥❡r❛❧✱ t❤✐s r❛t✐♦ k ✐s t❤❡ r❛t✐♦ ♦❢ t❤❡ ❧❡♥❣t❤s ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s❡❣♠❡♥ts ✐♥ t❤❡ r❛♥❣❡ ❛♥❞ t❤❡ ❞♦♠❛✐♥✳ ❋✉rt❤❡r♠♦r❡✱ ❛ s❤r✐♥❦ ✐s ❛ str❡t❝❤ ✇✐t❤ k < 1✳ ❚❤❡ ❝♦❧❧❛♣s❡✱ f (x) = c✱ ♠❛❦❡s ❛❧❧ ❞✐st❛♥❝❡s ❡q✉❛❧ t♦ 0✿

❚❤❡r❡❢♦r❡✱ ✐ts ❞❡r✐✈❛t✐✈❡ ✐s 0 t♦♦✳ ❲❡ ❝♦♥❝❧✉❞❡✿ ◮ ❚❤❡ ❞❡r✐✈❛t✐✈❡ ❞❡s❝r✐❜❡s ❤♦✇ t❤❡ r❡❧❛t✐✈❡

❧♦❝❛t✐♦♥s ♦❢ ♣♦✐♥ts ❝❤❛♥❣❡ ✉♥❞❡r t❤❡ tr❛♥s❢♦r♠❛t✐♦♥✳

❚❤✐s ✐s t❤❡ s✉♠♠❛r② ♦❢ t❤❡ ❢✉♥❝t✐♦♥s ✇❡ ❤❛✈❡ ❝♦♥s✐❞❡r❡❞✿

✸✳✼✳

❚❤❡ ❞❡r✐✈❛t✐✈❡ ❛s ❛ ❢✉♥❝t✐♦♥

✷✼✸

❚❤❡✐r ❞❡r✐✈❛t✐✈❡s ❛r❡ r❡s♣❡❝t✐✈❡❧②✿ 1, 1, −1, 2✱ ❛♥❞ 1/2✳ ❚❤❡② ❛r❡ ❛❧s♦ t❤❡ str❡t❝❤✐♥❣ ❢❛❝t♦rs✦ ❲❡ ❛❧s♦ ♣❧♦t t❤❡ ❣r❛♣❤s ♦❢ t❤❡s❡ ❢✉♥❝t✐♦♥s ❜❡❧♦✇ t♦ s❡❡ t❤❡s❡ ❢❛❝t♦rs ❛s t❤❡ s❧♦♣❡s✿

❲❡ ❦♥♦✇ ✭❢r♦♠ ❱♦❧✉♠❡ ✶✱ ❈❤❛♣t❡r ✶P❈✲✸✮ t❤❛t ✉♥❞❡r ❛ ❧✐♥❡❛r ♣♦❧②♥♦♠✐❛❧ f (x) = mx + b✱ t❤❡ ❞✐st❛♥❝❡s ✐♥❝r❡❛s❡ ❜② ❛ ❢❛❝t♦r ♦❢ |m| ✇❤❡♥ |m| > 1✱ ♦r ❞❡❝r❡❛s❡❞ ❜② ❛ ❢❛❝t♦r ♦❢ |m| ✇❤❡♥ |m| < 1✳ ❚❤✐s str❡t❝❤✴s❤r✐♥❦ ❢❛❝t♦r ✐s t❤❡ s❛♠❡ ❡✈❡r②✇❤❡r❡✳ ❇✉t m ✐s t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ f ✦ ❊①❛♠♣❧❡ ✸✳✼✳✷✹✿ tr❛♥s❢♦r♠❛t✐♦♥s

▲❡t✬s ❝♦♥s✐❞❡r t❤✐s ❢✉♥❝t✐♦♥ ❣✐✈❡♥ ❜② ✐ts ✈❛❧✉❡s✿ 6 7 8 9 10 x 0 1 2 3 4 5 y 0 2 5 7 8 8.5 8.5 8 7 4 2

❲❡ ❛ss✉♠❡ t❤❛t t❤❡ ❢✉♥❝t✐♦♥ ❝♦♥t✐♥✉❡s ❜❡t✇❡❡♥ t❤❡s❡ ✈❛❧✉❡s ✐♥ ❛ ❧✐♥❡❛r ❢❛s❤✐♦♥✳ ❋♦r ❡①❛♠♣❧❡✱ t❤❡ ✐♥t❡r✈❛❧ [0, 1] ✐s ♠❛♣♣❡❞ t♦ [0, 2] ❧✐♥❡❛r❧② ✭y = 2x✮✱ t❤❡ ✐♥t❡r✈❛❧ [1, 2] t♦ [2, 5]✱ ❡t❝✳ ❚❤❡ ❛❝t✉❛❧ ❢♦r♠✉❧❛s ❞♦♥✬t ♠❛tt❡r✳ ❚❤❡♥✱ t❤❡ 1✲✉♥✐t s❡❣♠❡♥ts ♦♥ t❤❡ x✲❛①✐s ❛r❡ str❡t❝❤❡❞ ❛♥❞ s❤r✉♥❦ ❛t ❞✐✛❡r❡♥t r❛t❡s✱ ❛♥❞ t❤❡ ♦♥❡s ❜❡②♦♥❞ x = 6 ❛r❡ ❛❧s♦ ✢✐♣♣❡❞ ♦✈❡r ✭❧❡❢t✮✿

❚❤❡ ❞❡r✐✈❛t✐✈❡s ❛r❡ r❡s♣❡❝t✐✈❡❧②✿ 2, 3, 2, 1, 1/2, 0, −1/2, −1, −3 .

❙♦✱ t❤❡ ❛❜s♦❧✉t❡ ✈❛❧✉❡s ♦❢ t❤❡s❡ ♥✉♠❜❡rs ❛r❡ t❤❡ str❡t❝❤ ❝♦❡✣❝✐❡♥ts ♦❢ t❤❡s❡ s❡❣♠❡♥ts ♦❢ t❤❡ ❞♦♠❛✐♥✳ ❲❡ ✉s❡ t❤❡s❡ ❢❛❝t♦rs ❛s s❧♦♣❡s ❢♦r t❤❡ ♣❛t❝❤❡s t❤❛t ♠❛❦❡ ✉♣ t❤❡ ❣r❛♣❤ ✭r✐❣❤t✮✳

✸✳✽✳

❇❛s✐❝ ❞✐✛❡r❡♥t✐❛t✐♦♥

✷✼✹

❊①❡r❝✐s❡ ✸✳✼✳✷✺ ❊st✐♠❛t❡ t❤❡ ❞❡r✐✈❛t✐✈❡ ❢r♦♠ t❤❡ ❢♦✉rt❤ ♣✐❝t✉r❡ ❛♥❞ t❤❡♥ ❝♦♥✜r♠ t❤❡ r❡s✉❧t ❜② ❧♦♦❦✐♥❣ ❛t t❤❡ s❧♦♣❡s ✐♥ t❤❡ s❡❝♦♥❞✿

■♥ s✉♠♠❛r②✱ ❛♥ ❛❜str❛❝t ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥

y = f (x) ❤❛s ❜❡❡♥ ❣✐✈❡♥ t❤r❡❡ t❛♥❣✐❜❧❡

r❡♣r❡s❡♥t❛t✐♦♥s ❛♥❞ ♥♦✇

✇❡ ❛❧s♦ ❤❛✈❡ t❤r❡❡ t❛♥❣✐❜❧❡ ✐♥t❡r♣r❡t❛t✐♦♥s ♦❢ ✐ts ❞❡r✐✈❛t✐✈❡✿

1. 2. 3.

❋✉♥❝t✐♦♥ ✐s s❡❡♥ ❛s

■ts ❞❡r✐✈❛t✐✈❡ ✐s

❧♦❝❛t✐♦♥

✈❡❧♦❝✐t②

❣r❛♣❤

s❧♦♣❡ ♦❢ t❤❡ t❛♥❣❡♥t ❧✐♥❡

tr❛♥s❢♦r♠❛t✐♦♥

str❡t❝❤✴s❤r✐♥❦ r❛t❡

✸✳✽✳ ❇❛s✐❝ ❞✐✛❡r❡♥t✐❛t✐♦♥ ▲❡t✬s r❡✈✐❡✇ ♦✉r ❝✉rr❡♥t✱ ❛♥❞ t❤❡♥ ✐♥tr♦❞✉❝❡ ❛ ♥❡✇✱ ♥♦t❛t✐♦♥✳ ■♥✐t✐❛❧❧②✱ ✇❡ ❞❡❛❧ ✇✐t❤ ♦♥❡ ♣♦✐♥t ❛t ❛ t✐♠❡✱

x = a✳

❋✐rst✱ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✿

∆f f (a + ∆x) − f (a) f (x) − f (a) (a) = = ∆x ∆x x−a ❆♥❞ t❤✐s ✐s t❤❡ ❞❡r✐✈❛t✐✈❡✿

df ∆f (a) = lim (a) ∆x→0 ∆x dx ❲❡ ❝❛♥ ♦♠✐t t❤❡ ♠❡♥t✐♦♥ ♦❢ t❤❡ ✐♥♣✉t ✈❛r✐❛❜❧❡ ❛♥❞ ❞❡❛❧ ✇✐t❤ ♦♥❧② t❤❡ ♥❛♠❡s ♦❢ t❤❡s❡ ❢✉♥❝t✐♦♥s✿

∆f ∆x

❛♥❞

df dx

◆♦✇✱ ✐t ✐s ❛❧s♦ ♣♦ss✐❜❧❡ t♦ ❧❡❛✈❡ t❤❡ ♦r✐❣✐♥❛❧ ❢✉♥❝t✐♦♥ ✉♥♥❛♠❡❞✱ ❛s ✐♥✿

y = x2 . ■♥ t❤❛t ❝❛s❡✱ ✇❡ ♠❛② ✉s❡ t❤❡ ♥❛♠❡ ♦❢ t❤❡ ♦✉t♣✉t ✈❛r✐❛❜❧❡ ♦❢ t❤✐s ❢✉♥❝t✐♦♥ ❢♦r t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❛♥❞ t❤❡ ❞❡r✐✈❛t✐✈❡✿

❉✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❛♥❞ ❞❡r✐✈❛t✐✈❡ ∆y ∆x ■♥ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✱ ✇❡ ✉s❡ t❤❡ ●r❡❡❦ ❧❡tt❡r

❛♥❞

∆✱

dy dx

✇❤✐❝❤ st❛♥❞s ❢♦r ✏❞✐✛❡r❡♥❝❡✑✱ ❛s ❢♦❧❧♦✇s✿

✸✳✽✳

❇❛s✐❝ ❞✐✛❡r❡♥t✐❛t✐♦♥

✷✼✺

• ∆x

✐s t❤❡ ❝❤❛♥❣❡ ♦❢

x

✭t❤❡ r✉♥✮✱ ❛♥❞

• ∆y

✐s t❤❡ ❝❤❛♥❣❡ ♦❢

y

✭t❤❡ r✐s❡✮✳

❚❤❡ ❢♦r♠❡r ♣r♦❞✉❝❡s t❤❡ ❧❛tt❡r ✇✐t❤ t❤❡ ✉s❡ ♦❢ t❤❡ ❣r❛♣❤ ♦❢

y = f (x)✳

❋♦r ❡✈❡r② ❝❤♦✐❝❡ ♦❢

∆x✱ ✇❡ ♠❛❦❡ t❤❛t

st❡♣ t♦ t❤❡ ❧❡❢t ♦r t♦ t❤❡ r✐❣❤t ❛♥❞ ✜♥❞ ❛♥♦t❤❡r ♣♦✐♥t ♦♥ t❤❡ ❣r❛♣❤✿

◆❡①t✱ ✐♥ t❤❡ ❞❡r✐✈❛t✐✈❡✱ ✇❡ ✉s❡ t❤❡ ❧❡tt❡r

d

t❤❛t✱ t♦♦✱ st❛♥❞s ❢♦r ✏❞✐✛❡r❡♥❝❡✑✳ ❚❤❡ ❞❡r✐✈❛t✐✈❡ ✐s t❤❡ ❧✐♠✐t ♦❢

❛ ❢r❛❝t✐♦♥✱ ❜✉t t❤✐s ❞♦❡s♥✬t ♠❡❛♥ t❤❛t ✕ ✐♥ s♣✐t❡ ♦❢ t❤❡ ♥♦t❛t✐♦♥ ✕ ✐t ✐s ❛ ❢r❛❝t✐♦♥ t♦♦✦ ❍♦✇❡✈❡r✱ ♦♥❝❡ t❤❡ ❞❡r✐✈❛t✐✈❡ ✐s ❦♥♦✇♥✱ ❛t

x = a✱

✐t ❣✐✈❡s ✉s t❤❡ s❧♦♣❡ ♦❢ t❤❡ t❛♥❣❡♥t ❧✐♥❡✳ ❏✉st ❛s ❛❜♦✈❡✱ ✇❡ ❝♦♥❝❡♥tr❛t❡ ♦♥ ❛

s✐♥❣❧❡ ❧✐♥❡ ✕ t❤❡ t❛♥❣❡♥t ❧✐♥❡ ✕ ❛♥❞ ✉s❡ t❤✐s ♥♦t❛t✐♦♥✿

• dx

❢♦r t❤❡ r✉♥✱ ❛♥❞

• dy

❢♦r t❤❡ r✐s❡✳

❋♦r ❡✈❡r② ❝❤♦✐❝❡ ♦❢

dx✱

✇❡ ♠❛❦❡ t❤❛t st❡♣ t♦ t❤❡ ❧❡❢t ♦r t♦ t❤❡ r✐❣❤t ❛♥❞ ✜♥❞ ❛♥♦t❤❡r ♣♦✐♥t ♦♥ t❤❡ t❛♥❣❡♥t

❧✐♥❡✿

❙♦✱ t❤❡ ❞❡r✐✈❛t✐✈❡ ✐s ❛ ❢r❛❝t✐♦♥✱ t♦♦✿

■❢ ✐♥ t❤❡ ❛❜♦✈❡ ♣✐❝t✉r❡✱ t❤❡ s❧♦♣❡ ✐s

dy = dx 2✱

❝❤❛♥❣❡ ♦❢ ❝❤❛♥❣❡ ♦❢

y . x

✇❡ ❤❛✈❡ ❛ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡s❡ t✇♦ ✈❛r✐❛❜❧❡s✿

dy = 2 dx . ❲❡ tr❡❛t t❤❡s❡ t✇♦ q✉❛♥t✐t✐❡s✱

dx

❛♥❞

dy ✱

❛s ❛

♥❡✇ s❡t ♦❢ ✈❛r✐❛❜❧❡s✳

❚❤❡② ❞❡♣❡♥❞ ♦♥ ❡❛❝❤ ♦t❤❡r✱ ❛s ❢♦❧❧♦✇s✿

❉❡✜♥✐t✐♦♥ ✸✳✽✳✶✿ ❞✐✛❡r❡♥t✐❛❧s y = f (x) ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t x = a ❛♥❞ ✐ts ❞❡r✐✈❛t✐✈❡ ✐s f ′ (a)✳ ❚❤❡♥ t❤❡ ❞✐✛❡r❡♥t✐❛❧✱ dx✱ ♦❢ x ❛♥❞ t❤❡ ❞✐✛❡r❡♥t✐❛❧✱ dy ✱ ♦❢ y ❛r❡ t✇♦ r❡❛❧ ✈❛r✐❛❜❧❡s

❙✉♣♣♦s❡ ❛ ❢✉♥❝t✐♦♥

r❡❧❛t❡❞ t♦ ❡❛❝❤ ♦t❤❡r ❜② t❤❡ ❡q✉❛t✐♦♥✿

dy = f ′ (a) · dx

✸✳✽✳ ❇❛s✐❝ ❞✐✛❡r❡♥t✐❛t✐♦♥

✷✼✻

❙✉❝❤ ❛♥ ❡①♣r❡ss✐♦♥ ✐s ❝❛❧❧❡❞ ❛ ❞✐✛❡r❡♥t✐❛❧ ❢♦r♠ ✭t♦ ❜❡ ❝♦♥s✐❞❡r❡❞ ✐♥ ❱♦❧✉♠❡ ✸✱ ❈❤❛♣t❡rs ✸■❈✲✷ ❛♥❞ ✸■❈✲✹✮✳

❆t ❡✈❡r② ❧♦❝❛t✐♦♥ a✱ t❤❡ ❞❡♣❡♥❞❡♥❝❡ ♦❢ t❤❡ t✇♦ ✈❛r✐❛❜❧❡s ✐s ✈❡r② s✐♠♣❧❡✿ ❧✐♥❡❛r✦ ❆♥♦t❤❡r ❛♣♣r♦❛❝❤ t♦ ♥♦t❛t✐♦♥ ✐s t♦ ♣r❡s❡♥t ❞✐✛❡r❡♥t✐❛t✐♦♥ ❛s ❛ ❢✉♥❝t✐♦♥✱ ❛ ❢✉♥❝t✐♦♥ ♦❢ ❢✉♥❝t✐♦♥s ✿

❉✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❛♥❞ ❞❡r✐✈❛t✐✈❡ ∆ d (f ) ❛♥❞ (f ) ∆x dx

❍❡r❡ ❜♦t❤ t❤❡ ✐♥♣✉t ❛♥❞ t❤❡ ♦✉t♣✉t ❛r❡ ❢✉♥❝t✐♦♥s✦ ❆❧s♦ ❝♦♥✈❡♥✐❡♥t s♦♠❡t✐♠❡s ✐s t❤❡ s✉❜st✐t✉t✐♦♥ ♥♦t❛t✐♦♥ ❢♦r ❡✈❛❧✉❛t✐♥❣ ❛ ❢✉♥❝t✐♦♥ ❛t ❛ ♣❛rt✐❝✉❧❛r ♣♦✐♥t✿ ∆f ∆x

❊①❛♠♣❧❡ ✸✳✽✳✷✿ ❝♦♠♣✉t❛t✐♦♥s

x=a

∆y , ∆x

x=a

df , dx

x=a

dy , dx

. x=a

❲❡ ♠❛②✱ ✐♥ ❢❛❝t✱ ♦♠✐t t❤❡ ♥❛♠❡s ❛❧t♦❣❡t❤❡r ❛♥❞ ♣r❡s❡♥t ♦♥❧② t❤❡ ❢♦r♠✉❧❛ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ t❤❡♦r❡♠s ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥✳ ❚❤✐s ✐s ✇❤❛t ✇❡ ✇r✐t❡ ❢♦r t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts✿ ∆ (3x + 5) = 3 ∆x ∆ (−x2 + 7) = −2x − 2h ∆x

= 3

∆ =⇒ (3x + 5) ∆x

x=0 ∆ 2 =⇒ (−x + 7) ∆x

x=1, h=.1

❚❤✐s ✐s ✇❤❛t ✇❡ ✇r✐t❡ ❢♦r t❤❡ ❞❡r✐✈❛t✐✈❡s✿ d (−x2 + 7) = −2x dx d (3x2 + 2x + 1) = 6x + 2 dx

❆♥❞ ♥♦✇ ✐♥ t❤❡ ▲❛❣r❛♥❣❡ ♥♦t❛t✐♦♥✿ (−x2 + 7)′ = −2x (3x2 + 2x + 1)′ = 6x + 2

=⇒ (−x2 + 7)′

= −2x x=−1

x=−1

x=1, h=.1

x=1

= 6x + 2

= −2x

x=1

=⇒ (3x2 + 2x + 1)′

x=0

= −2x − 2h

d 2 =⇒ (−x + 7) dx

x=1 d 2 =⇒ (3x + 2x + 1) dx

=3

x=1

= 6x + 2

= −2

x=−1

x=−1

= −2.2

= −4

= −2 = −4

✸✳✽✳ ❇❛s✐❝ ❞✐✛❡r❡♥t✐❛t✐♦♥

✷✼✼ ❲❛r♥✐♥❣✦ ❲❡ ❝❛♥✬t s✉❜st✐t✉t❡

x=1

✐♥t♦

(−x2 + 7)′

✉♥t✐❧ ❛♥

❡①♣❧✐❝✐t r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ❧❛tt❡r ✐s ❢♦✉♥❞✳

❊①❡r❝✐s❡ ✸✳✽✳✸

❈♦♠♣✉t❡✿ ∆ (−x + 2) ❛t x = 5 ∆x d (−x + 2) ❛t x = 5 dx d x (e ) ❛t x = −1 dx (sin x)′

❛t x = π/2

❊①❡r❝✐s❡ ✸✳✽✳✹

Pr♦✈❡ ♣❧♦tt✐♥❣ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ ❛ q✉❛❞r❛t✐❝ ❢✉♥❝t✐♦♥ ✇✐❧❧ ♣r♦❞✉❝❡ ❛ str❛✐❣❤t ❧✐♥❡✳ ❊①❛♠♣❧❡ ✸✳✽✳✺✿ s♣r❡❛❞s❤❡❡t

❈♦♠♣✉t✐♥❣ ♠✐❣❤t ❣✐✈❡ ✉s ❡✈✐❞❡♥❝❡ t♦ ♠❛❦❡ ❛♥ ❡❞✉❝❛t❡❞ ❣✉❡ss ❛❜♦✉t t❤❡ ❞❡r✐✈❛t✐✈❡s ♦❢ ♣❛rt✐❝✉❧❛r ❢✉♥❝t✐♦♥s✳ ❈♦♥s✐❞❡r ❛ ♣❛r❛❜♦❧❛✱ s❛②✱ y = f (x) = −(x − 1.5)2 + 3✳ ❘❡❝❛❧❧ ❤♦✇ ✐♥ s❡❛r❝❤ ♦❢ ❛ ♣❛tt❡r♥✱ ✇❡ ♠❛❦❡ t❤❡ ♥♦❞❡s ♦❢ ♦✉r ♣❛rt✐t✐♦♥ ❞❡♥s❡r ❛♥❞ ❞❡♥s❡r✿

❆s ✇❡ ♣r♦❞✉❝❡ ♠♦r❡ ❛♥❞ ♠♦r❡ ♣♦✐♥ts ♦♥ t❤✐s ✐♥t❡r✈❛❧✱ ✇❡ r❡❛❧✐③❡ t❤❛t t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✐s ❛ str❛✐❣❤t ❧✐♥❡✦ ❲❡ ✇✐❧❧ s❡❡ ♠♦r❡ ♣❛tt❡r♥s ✐❢ ✇❡ ❝♦❧❧❡❝t t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ♦❢ t❤❡ ❢✉♥❝t✐♦♥s ✇❡ ❤❛✈❡ s❡❡♥ ♣r❡✈✐♦✉s❧②✳ ❚❤❡s❡ ❛r❡ t❤♦s❡ ♦❢ t❤❡ ♣♦✇❡r ❢✉♥❝t✐♦♥s✿

✸✳✽✳ ❇❛s✐❝ ❞✐✛❡r❡♥t✐❛t✐♦♥

✷✼✽

■t ❛♣♣❡❛rs t❤❛t t❤❡ ♣♦✇❡r ❣♦❡s ❞♦✇♥ ❜② ♦♥❡✦ ❊①❡r❝✐s❡ ✸✳✽✳✻

❙❦❡t❝❤ t❤❡ ♥❡①t ♣❛✐r✳ ❊①❛♠♣❧❡ ✸✳✽✳✼✿

x2

❲❡ ❤❛✈❡ ♠❛❞❡ ♣r♦❣r❡ss ✇✐t❤ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ f (x) = x2 .

❋r♦♠ ❛ s✐♥❣❧❡ s♣❡❝✐✜❝ ♣♦✐♥t✱ t♦ ❛ s✐♥❣❧❡ ✉♥s♣❡❝✐✜❡❞ ♣♦✐♥t✱ t♦ ❛❧❧ ♣♦✐♥ts ❛t ♦♥❝❡✱ ❛ ♥❡✇ ❢✉♥❝t✐♦♥✦ ❍♦✇❡✈❡r✱ t❤❡ ❛❧❣❡❜r❛✐❝ st❡♣s ❛r❡ t❤❡ s❛♠❡✿ a=1

r❡♣❧❛❝❡ 1 ✇✐t❤ a

r❡♣❧❛❝❡ a ✇✐t❤ x

∆f (1) = ∆x f (1 + h) − f (1) = h (1 + h)2 − 12 = h 2 1 + 2h + h − 1 = h 2h + h2 = h

∆f (a) = ∆x f (a + h) − f (a) = h (a + h)2 − a2 = h 2 a + 2ah + h − a2 = h 2ah + h2 = h

∆f (x) = ∆x f (x + h) − f (x) = h (x + h)2 − x2 = h 2 x + 2xh + h − x2 = h 2xh + h2 = h

❚❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✳✳✳

=2+h

= 2a + h

= 2x + h

❚❤❡ ♥✉♠❡r❛t♦r ✐s ❞✐✈✐❞❡❞ ❜② h.

→2+0

→ 2a + 0

→ 2x + 0

❚❤❡ ❧✐♠✐t ✐s ❡✈❛❧✉❛t❡❞ ❜②✳✳✳

=2

= 2a

= 2x

s✉❜st✐t✉t✐♦♥ ❜❡❝❛✉s❡ t❤❡ ❢✉♥❝t✐♦♥✳✳✳

✐s ✇r✐tt❡♥ ❢r♦♠ t❤❡ ❞❡✜♥✐t✐♦♥✳ ❚❤❡ ❢✉♥❝t✐♦♥ ✐s s♣❡❝✐✜❡❞✳ ❚❤❡ ♥✉♠❡r❛t♦r ✐s ❡①♣❛♥❞❡❞✳ ❚❤❡ t❡r♠s ✇✐t❤♦✉t h ❛r❡ ❝❛♥❝❡❧❡❞✳

✐s ❝♦♥t✐♥✉♦✉s ✇✐t❤ r❡s♣❡❝t t♦ h. ❖✉r ♣r♦❣r❡ss ✐s ✐❧❧✉str❛t❡❞ ✐♥ t❡r♠s ♦❢ t❤❡ t❛♥❣❡♥t ❧✐♥❡s✿

✸✳✽✳ ❇❛s✐❝ ❞✐✛❡r❡♥t✐❛t✐♦♥

✷✼✾

■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ ✇✐❧❧ ✜♥❞ t❤❡ ❞❡r✐✈❛t✐✈❡s ♦❢ s♦♠❡ ✐♠♣♦rt❛♥t ❢✉♥❝t✐♦♥s✳ ❚❤❡ ❢✉♥❝t✐♦♥s ❛r❡ ✈❡r② ❞✐✛❡r❡♥t✱ ❜✉t t❤❡ ❝♦♠♣✉t❛t✐♦♥s ✇✐❧❧ ❤❛✈❡ ❛ ❧♦t ✐♥ ❝♦♠♠♦♥✳ ❲❡ ♥❡❡❞ t♦ ✜♥❞ t❤❡ ❧✐♠✐t ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✱ f (x + h) − f (x) ∆f = ❛s h → 0 . ∆x h

❲❤❡♥❡✈❡r f ✐s ❝♦♥t✐♥✉♦✉s ❛t x✱ t❤❡ ❧✐♠✐t ♦❢ t❤❡ ♥✉♠❡r❛t♦r ✐s 0✳ ❆♥❞ s♦ ✐s t❤❡ ❧✐♠✐t ♦❢ t❤❡ ❞❡♥♦♠✐♥❛t♦r✦ 0 ❚❤✉s✱ ✇❡ ✇✐❧❧ ❢❛❝❡✱ ❡✈❡r② t✐♠❡✱ t❤❡ s❛♠❡ ♣r♦❜❧❡♠✿ ❛♥ ✐♥❞❡t❡r♠✐♥❛t❡ ❡①♣r❡ss✐♦♥ ♦❢ t❤❡ t②♣❡ ✳ ❊✈❡r② t✐♠❡✱ 0 ✐t ✐s t♦ ❜❡ r❡s♦❧✈❡❞✱ ❛♥❞ ♥♦t ❜② t❤❡ r✉❧❡s ♦❢ ❧✐♠✐ts ❜✉t ❜② ❛❧❣❡❜r❛✦ ❚❤✐s ✐s t❤❡ ♠♦st ❝❤❛❧❧❡♥❣✐♥❣ st❡♣✿ ◮ ❲❡ ✇✐❧❧ ♥❡❡❞ t♦

❢❛❝t♦r t❤❡ ♥✉♠❡r❛t♦r ✐♥ ♦r❞❡r t♦ ❝❛♥❝❡❧ h ❢r♦♠ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✳

❊①❛♠♣❧❡ ✸✳✽✳✽✿ ❡①♣❛♥❞✐♥❣ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ x3 ❚❤❡ ♣♦✇❡r ❢✉♥❝t✐♦♥s ✜rst✳ ❲❡ ❛❧r❡❛❞② ❦♥♦✇ t❤❡ ❞❡r✐✈❛t✐✈❡ ❢♦r t❤❡ ❧✐♥❡❛r ❛♥❞ t❤❡ q✉❛❞r❛t✐❝ ♣♦✇❡rs✿ (x1 )′ = 1, (x2 )′ = 2x1 .

◆♦✇ ❝✉❜✐❝✿

f (x) = x3 .

❲❡ ❢♦❧❧♦✇ t❤❡ ❛♣♣r♦❛❝❤ ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥✱ ❦❡❡♣✐♥❣ ✐♥ ♠✐♥❞ t❤❛t x ✐s ✜①❡❞ ❛s ❢❛r ❛s t❤❡ ❧✐♠✐t ✐s ❝♦♥❝❡r♥❡❞✿ ∆f (x) ∆x

∆f (x) ∆x

f (x + h) − f (x) h (x + h)3 − x3 = h x3 + 3x2 h + 3xh2 + h3 − x2 = h 3x2 h + 3xh2 + h3 = h

❚❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✐s ✇r✐tt❡♥ ❢r♦♠ t❤❡ ❞❡✜♥✐t✐♦♥✳

= 3x2 + 3xh + h2

❚❤❡ ♥✉♠❡r❛t♦r ✐s ❞✐✈✐❞❡❞ ❜② h.

= 3x2 + 3xh + h2

❚❤✐s ✐s t❤❡ s✐♠♣❧✐✜❡❞ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✳

=

❛s h → 0, → 3x2 + 3x · 0 + 02

❚❤❡ ❢✉♥❝t✐♦♥ ✐s s♣❡❝✐✜❡❞✳ ❚❤❡ ♥✉♠❡r❛t♦r ✐s ❡①♣❛♥❞❡❞✳ ❚❤❡ t❡r♠s ✇✐t❤♦✉t h ❛r❡ ❝❛♥❝❡❧❧❡❞✳

❚❤❡ ❧✐♠✐t ✐s t❤❡♥ ❡✈❛❧✉❛t❡❞ ❜② s✉❜st✐t✉t✐♦♥ h = 0... ❜❡❝❛✉s❡ t❤❡ ❡①♣r❡ss✐♦♥ ✐s ❝♦♥t✐♥✉♦✉s ✇✐t❤ r❡s♣❡❝t t♦ h.

= 3x2

❲❡ ♥♦t✐❝❡ s♦♠❡t❤✐♥❣✿ ❆❧❧ t❤❡ t❡r♠s ❞✐s❛♣♣❡❛r ❡①❝❡♣t ❢♦r t❤♦s❡ ✇✐t❤ h ✐♥ t❤❡ ✜rst ♣♦✇❡r✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ✐s ❞❡r✐✈❡❞ ❢r♦♠ t❤❡ ❇✐♥♦♠✐❛❧ ❋♦r♠✉❧❛ ✭s❡❡♥ ✐♥ ❱♦❧✉♠❡ ✶✱ ❈❤❛♣t❡r ✶P❈✲✶✮✿

❚❤❡♦r❡♠ ✸✳✽✳✾✿ ❍✐❣❤❡st ❚❡r♠s ♦❢ ❇✐♥♦♠✐❛❧ ❊①♣❛♥s✐♦♥

❚❤❡ nt❤ ♣♦✇❡r✱ (x + h)n✱ ❤❛s n + 1 t❡r♠s✱ ❛♥❞ t❤❡ ♦♥❡ ✇✐t❤ h ❤❛s ❝♦❡✣❝✐❡♥t ❡q✉❛❧ t♦ n✿ (x + h)n = xn + nxn−1 h + t❡r♠s ✇✐t❤ h2 , h3 , ... ❲❡ ✇✐❧❧ ✉s❡ t❤✐s ❢❛❝t ❧❛t❡r✳ ❆♥ ❛❧t❡r♥❛t✐✈❡ ❛♣♣r♦❛❝❤ t♦ ❢❛❝t♦r✐♥❣ r❡❧✐❡s ♦♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢❛❝t♦r✐♥❣ ❢♦r♠✉❧❛ ✿ an − bn = (a − b)(an−1 + an−2 b + ... + abn−2 + bn−1 )

✸✳✽✳ ❇❛s✐❝ ❞✐✛❡r❡♥t✐❛t✐♦♥

✷✽✵

❊①❛♠♣❧❡ ✸✳✽✳✶✵✿ ❢❛❝t♦r✐♥❣ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ x3 ❲❡ ✉s❡ ✐t ❛s ❢♦❧❧♦✇s ✭n = 3✮✿ (x + h)3 − x3 h→0 h (x + h − x)((x + h)2 + (x + h)x + x2 ) = lim h→0 h   = lim (x + h)2 + (x + h)x + x2

(x3 )′ = lim

h→0

= (x + 0)2 + (x + 0)x + x2 = 3x2 .

❲❡ ❤❛✈❡ ❛❧s♦ ❞✐s❝♦✈❡r❡❞ ❛ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✿
∆ x3 = (x + h)2 + (x + h)x + x2 . ∆x

❲❡ ♣❧♦t ❛❧❧ t❤r❡❡ ❜❡❧♦✇✿

❆ ♣❛tt❡r♥ st❛rts t♦ ❡♠❡r❣❡✿ n

(xn )′

1

(x1 )′

= 1x0

1

x

0

2

(x2 )′

= 2x1

2

x

1

3

(x3 )′

= 3x2

3

x

2

...

...

...

...

...

?

1000 (x1000 )′ = 1000x999 1000 x ...

...

...

...

...

n

(xn )′

= nxn−1

?

n

x

999

n−1

■t s❡❡♠s t❤❛t t❤❡ ♣♦✇❡r ♣r❡s❡♥t ✐♥ t❤❡ ❞❡r✐✈❛t✐✈❡ ✐s ♦♥❡ ❧♦✇❡r t❤❛♥ t❤❛t ✐♥ t❤❡ ♦r✐❣✐♥❛❧✱ ✇❤✐❧❡ t❤❡ ♦r✐❣✐♥❛❧ ♣♦✇❡r ❛♣♣❡❛rs ❛s ❛ ♠✉❧t✐♣❧❡✳ ▲❡t✬s ❣♦ ♦✈❡r t❤❡ ❝♦♠♣✉t❛t✐♦♥✱ st❛rt✐♥❣ ✇✐t❤ t❤❡ ❞✐✛❡r❡♥❝❡✿

✸✳✾✳ ❇❛s✐❝ ❞✐✛❡r❡♥t✐❛t✐♦♥✱ ❝♦♥t✐♥✉❡❞

✷✽✶

❚❤❡♦r❡♠ ✸✳✽✳✶✶✿ ■♥t❡❣❡r P♦✇❡r ❋♦r♠✉❧❛ ❢♦r ❉✐✛❡r❡♥❝❡ ▲❡t n ❜❡ ❛ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r✳ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ❧❡❢t✲❡♥❞ ♣❛rt✐t✐♦♥❀ ✐✳❡✳✱ t❤❡ ♥♦❞❡s ❛r❡ x = a, a + h ❛♥❞ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡ ✐s c = a✳ ❚❤❡♥ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ y = xn ✐s ❣✐✈❡♥ ❛t c ❜② t❤❡ ❢♦r♠✉❧❛✿ n



∆(x ) = h (c + h)

n−1

+ (c + h)

n−2 1

1 n−2

c + ... + (c + h) c

+c

n−1



.

Pr♦♦❢✳ ❚❤❡ ♣r♦♦❢ ♦♥❧② ♥❡❡❞s ♦♥❡ ♦❢ t❤❡ t✇♦ ❢♦r♠✉❧❛s ❛❜♦✈❡✳ ◆♦✇ ✇❡ ❥✉st ❞✐✈✐❞❡ ❜② h = ∆x✿

❚❤❡♦r❡♠ ✸✳✽✳✶✷✿ ■♥t❡❣❡r P♦✇❡r ❋♦r♠✉❧❛ ❢♦r ❉✐✛❡r❡♥❝❡ ◗✉♦t✐❡♥t ▲❡t n ❜❡ ❛ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r✳ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ❧❡❢t✲❡♥❞ ♣❛rt✐t✐♦♥❀ ✐✳❡✳✱ t❤❡ ♥♦❞❡s ❛r❡ x = a, a + h ❛♥❞ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡ ✐s c = a✳ ❚❤❡♥ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ y = xn ✐s ❣✐✈❡♥ ❛t c ❜② t❤❡ ❢♦r♠✉❧❛✿ ∆ n (x ) = (c + h)n−1 + (c + h)n−2 c1 + ... + (c + h)1 cn−2 + cn−1 . ∆x

◆♦✇ ✇❡ ❥✉st ❧❡t h = ∆x → 0✳ ❚❤❡♥ ❡❛❝❤ t❡r♠ ❛❜♦✈❡ ❜❡❝♦♠❡s cn−1 ✳ ❚❤❡r❡ ❛r❡ n ♦❢ t❤❡♠✿

❚❤❡♦r❡♠ ✸✳✽✳✶✸✿ ■♥t❡❣❡r P♦✇❡r ❋♦r♠✉❧❛ ❢♦r ❉❡r✐✈❛t✐✈❡

▲❡t n ❜❡ ❛ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r✳ ❚❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ y = xn ✐s ❣✐✈❡♥ ❜② d n (x ) = nxn−1 dx

❚❤❡ ❧✐♠✐t ❞r❛♠❛t✐❝❛❧❧② s✐♠♣❧✐✜❡s t❤❡ ❢♦r♠✉❧❛✦

✸✳✾✳ ❇❛s✐❝ ❞✐✛❡r❡♥t✐❛t✐♦♥✱ ❝♦♥t✐♥✉❡❞ ❚❤❡r❡ ❛r❡ ♠♦r❡ ❛♣♣❛r❡♥t ♣❛tt❡r♥s t♦ ❜❡ ❝♦♥✜r♠❡❞✿

❚❤❡s❡ ❛r❡ ♦✉r s✐♠♣❧❡st ❣✉❡ss❡s✳ ❋♦r t❤❡ ✜rst ♦♥❡✱ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t s❡❡♠s t♦ ❜❡ t❤❡

✸✳✾✳ ❇❛s✐❝ ❞✐✛❡r❡♥t✐❛t✐♦♥✱ ❝♦♥t✐♥✉❡❞

✷✽✷

♥❡❣❛t✐✈❡ ❛♥❞ ✉♣s✐❞❡✲❞♦✇♥ ✈❡rs✐♦♥ ♦❢ t❤❛t ♦❢ f ❀ t❤❛t✬s ✇❤② ♦✉r ✜rst ❣✉❡ss ✐s −1/x✳ ❍♦✇❡✈❡r✱ t❤❡ s❡❝♦♥❞ ❣r❛♣❤ ✐s ❛♣♣r♦❛❝❤✐♥❣ t❤❡ x✲❛①✐s ❢❛st❡r✦ ▼❛②❜❡ ✐t✬s −1/x2 ❄ ❨❡s✳ ❊①❡r❝✐s❡ ✸✳✾✳✶

❚r② t♦ ❛♥s✇❡r t❤❡s❡ q✉❡st✐♦♥s✳ ❚❤❡ ■♥t❡❣❡r P♦✇❡r ❋♦r♠✉❧❛ ✐s ♥♦✇ t♦ ❜❡ t❡st❡❞ ❢♦r ♦t❤❡r ✈❛❧✉❡s ♦❢ n✳ ❊①❛♠♣❧❡ ✸✳✾✳✷✿ r❡❝✐♣r♦❝❛❧

▲❡t✬s ♥♦✇ tr② ♥❡❣❛t✐✈❡ ♣♦✇❡rs✳ ❈♦♠♣✉t❡✿  ′ 1 = lim h→0 x = lim

1 x+h

− h

1 x

❲❡ ❞♦ ❝♦♠♠♦♥ ❞❡♥♦♠✐♥❛t♦rs ❤❡r❡✳

x−(x+h) (x+h)x

h −h 1 = lim h→0 (x + h)x h −1 = lim h→0 (x + h)x 1 =− 2 x h→0

❈❛♥❝❡❧ h. ❚❤❡ ❞❡♥♦♠✐♥❛t♦r ✐s ❝♦♥t✐♥✉♦✉s ❢♦r h < |x|. ❚❤❛t✬s ✇❤② ✇❡ ❝❛♥ ❥✉st s✉❜st✐t✉t❡ h = 0.

❍♦✇ ❞♦❡s t❤✐s ✜t ✐♥t♦ ♦✉r ❢♦r♠✉❧❛❄ ❲❡ r❡✇r✐t❡✿ 1 1 = x−1 , 2 = x−2 , ... x x

❙♦✱

(x−1 )′ = −x−2 .

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡ n = −1✱ ❛♥❞ t❤❡ ❢♦r♠✉❧❛ st✐❧❧ ✇♦r❦s✳ ❲❡ ❤❛✈❡ ❛❧s♦ ❞✐s❝♦✈❡r❡❞✿

❲❡ ♣❧♦t ❛❧❧ t❤r❡❡ ❜❡❧♦✇✿


∆ −1 x−1 = . ∆x (x + h)x

✸✳✾✳

❇❛s✐❝ ❞✐✛❡r❡♥t✐❛t✐♦♥✱ ❝♦♥t✐♥✉❡❞

✷✽✸

❊①❛♠♣❧❡ ✸✳✾✳✸✿ sq✉❛r❡ r♦♦t

◆❡①t✱ ❢r❛❝t✐♦♥❛❧ ♣♦✇❡rs✳ ❈♦♠♣✉t❡✿

√

√ x+h− x h→0 √ h √ √ √ x+h− x x+h+ x ·√ = lim √ h→0 h x+h+ x (x + h) − x = lim √ √ h→0 h( x + h + x) 1 = lim √ √ h→0 x + h + x 1 =√ √ x + h + x

√
′ x = lim

■s t❤✐s ✐♥❞❡t❡r♠✐♥❛t❡❄ ❚❤✐s ✐s t❤❡ ✏r❛t✐♦♥❛❧✐③❛t✐♦♥ tr✐❝❦✑✦

❇❡❝❛✉s❡ t❤❡ ❢✉♥❝t✐♦♥ ✐s ❝♦♥t✐♥✉♦✉s ❢♦r

❲❡ s✉❜st✐t✉t❡

h < |x|.

h = 0.

h=0

1 = √ . 2 x

❍♦✇ ❞♦❡s t❤✐s ✜t ✐♥t♦ ♦✉r ❢♦r♠✉❧❛❄ ❲❡ r❡✇r✐t❡✿

√ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡

n = 1/2✱

1 x = x1/2 , √ = x−1/2 , ... x

❛♥❞ t❤❡ ❢♦r♠✉❧❛ r❡♠❛✐♥s ✈❛❧✐❞✳ ❲❡ ❤❛✈❡ ❛❧s♦ ❞✐s❝♦✈❡r❡❞✿


1 ∆ x1/2 = √ √ . ∆x x+h+ x ❲❡ ♣❧♦t ❛❧❧ t❤r❡❡ ❜❡❧♦✇✿

❲❡ ❛❧s♦ ♥♦t✐❝❡ t❤❛t✿

f ′ (x) → +∞ ■t ❢♦❧❧♦✇s t❤❛t t❤❡ ❣r❛♣❤ ♦❢

f

❜❡❝♦♠❡s ✈❡rt✐❝❛❧ ❛t

❛s

x → 0+ .

x = 0✳

❚❤❡ ❢✉♥❝t✐♦♥ ✐s ♥♦t ❞✐✛❡r❡♥t✐❛❜❧❡ ❤❡r❡✳

❚❤❡ ❣❡♥❡r❛❧ ❝❛s❡✱ t♦ ❜❡ ♣r♦✈❡♥ ❧❛t❡r✱ ✐s ❛s ❢♦❧❧♦✇s✳ ❋♦r ❛♥② r❡❛❧ ♥✉♠❜❡r

(xr )′ = rxr−1

❲❡ ✇✐❧❧ ♥❡①t ❝♦♠♣✉t❡ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡

❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥✳

r 6= 0✱

✇❡ ❤❛✈❡✿

✸✳✾✳ ❇❛s✐❝ ❞✐✛❡r❡♥t✐❛t✐♦♥✱ ❝♦♥t✐♥✉❡❞

✷✽✹

❊①❛♠♣❧❡ ✸✳✾✳✹✿ ❞❡r✐✈❛t✐✈❡ ♦❢ ex ❛t 0 ❇❡❢♦r❡ ❛❞❞r❡ss✐♥❣ t❤❡ ❣❡♥❡r❛❧ ❝❛s❡✱ ❧❡t✬s ❝♦♥s✐❞❡r ✐ts ❞❡r✐✈❛t✐✈❡ ❛t x = 0✳ ❲❡ ❦♥♦✇ t❤❛t t❤❡ ❣r❛♣❤ ♦❢ y = ex ❝r♦ss❡s t❤❡ y ✲❛①✐s ❛t 45 ❞❡❣r❡❡s✿

❚❤✐s ♠❡❛♥s t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ❢❛♠♦✉s ❧✐♠✐t✿ ex − 1 = 1, x→0 x lim

❤❛s ♥♦✇ ❛ ♥❡✇ ✐♥t❡r♣r❡t❛t✐♦♥✿

x ′ (e )

= 1. x=0

❲❡ ♥♦✇ ✉s❡ t❤✐s r❡s✉❧t t♦ ✜♥❞ t❤❡ ❞❡r✐✈❛t✐✈❡ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥ ❜❛s❡ e✳ ❘❡♠❛r❦❛❜❧②✱ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥ ✐s ✐ts❡❧❢✿

❚❤❡♦r❡♠ ✸✳✾✳✺✿ ❉✐✛❡r❡♥❝❡ ♦❢ ❊①♣♦♥❡♥t✐❛❧ ❋✉♥❝t✐♦♥ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ❧❡❢t✲❡♥❞ ♣❛rt✐t✐♦♥❀ ✐✳❡✳✱ t❤❡ ♥♦❞❡s ❛r❡ x = a, a + h ❛♥❞ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡ ✐s c = a✳ ❚❤❡♥ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ ex ✐s ❣✐✈❡♥ ❛t c ❜②✿ ∆(ex ) = (eh − 1) · ec .

Pr♦♦❢✳ ▲❡t f (x) = ex ✳ ❲❡ ❝♦♠♣✉t❡ ❛t c✿ ∆f = ea+h − ea = ea eh − ea

❆❝❝♦r❞✐♥❣ t♦ ❛ ❢♦r♠✉❧❛✳ ❚❤❡♥ ✇❡ ❢❛❝t♦r✳

= ea (eh − 1) .

❍❡r❡ a ✐s ♦✉r s❡❝♦♥❞❛r② ♥♦❞❡✦

❚❤❡♦r❡♠ ✸✳✾✳✻✿ ❉✐✛❡r❡♥❝❡ ◗✉♦t✐❡♥t ♦❢ ❊①♣♦♥❡♥t✐❛❧ ❋✉♥❝t✐♦♥ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ❧❡❢t✲❡♥❞ ♣❛rt✐t✐♦♥❀ ✐✳❡✳✱ t❤❡ ♥♦❞❡s ❛r❡ x = a, a + h ❛♥❞ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡ ✐s c = a✳ ❚❤❡♥ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ ex ✐s ❣✐✈❡♥ ❛t c ❜②✿ eh − 1 c ∆ x (e ) = ·e . ∆x h

✸✳✾✳ ❇❛s✐❝ ❞✐✛❡r❡♥t✐❛t✐♦♥✱ ❝♦♥t✐♥✉❡❞

✷✽✺

Pr♦♦❢✳

▲❡t f (x) = ex ✳ ❲❡ ❝♦♠♣✉t❡ ❛t c✿ ea (eh − 1) ∆f = ∆x h h e −1 . = ea · h ❚❤❡♦r❡♠ ✸✳✾✳✼✿ ❉❡r✐✈❛t✐✈❡ ♦❢ ❊①♣♦♥❡♥t✐❛❧ ❋✉♥❝t✐♦♥ ❚❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢

ex

✐s ❣✐✈❡♥ ❜②✿

d x (e ) = ex dx Pr♦♦❢✳

▲❡t f (x) = ex ✳ ❲❡ ❤❛✈❡ ❛t c✿ ∆f eh − 1 = ea · ∆x h → ea · 1

❛s h → 0

= ea .

❚❤❡ ❧❛st st❡♣ ✐s ❥✉st✐✜❡❞ ❜② t❤❡ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ❛♥❞ t❤❡ ❢❛♠♦✉s ❧✐♠✐t ❛❜♦✈❡✳ ▲❡t✬s ❝♦♠♣❛r❡ t❤❡ ❢♦r♠✉❧❛s ❢♦r t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ ex ❛♥❞ ✐ts ❞❡r✐✈❛t✐✈❡✿ ∆ x eh − 1 ✭❆✮ (e ) = · ex h ∆x d x (e ) = ex ✭❇✮ dx

❲❡ ❦♥♦✇ t❤❛t k=

eh − 1 > 1. h

❙♦✱ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❞✐✛❡rs ❢r♦♠ t❤❛t ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦♥❧② ❜② ❛ ✈❡rt✐❝❛❧ str❡t❝❤ ❜② ❛ ❢❛❝t♦r k✦ ❆s h ✐s ❛♣♣r♦❛❝❤✐♥❣ 0✱ t❤❡ str❡t❝❤ ❞✐♠✐♥✐s❤❡s ❛♥❞ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢♦r♠❡r ✐s ❛♣♣r♦❛❝❤✐♥❣ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❧❛tt❡r✿

✸✳✾✳ ❇❛s✐❝ ❞✐✛❡r❡♥t✐❛t✐♦♥✱ ❝♦♥t✐♥✉❡❞

✷✽✻

❚❤❡ ❢♦r♠✉❧❛ ✐♥ t❤❡ ▲❛❣r❛♥❣❡ ♥♦t❛t✐♦♥ ✐s ❛s ❢♦❧❧♦✇s✿ (ex )′ = ex

❊①❛♠♣❧❡ ✸✳✾✳✽✿

bx

❚❤❡ ❛❜♦✈❡ ❝♦♠♣✉t❛t✐♦♥ ❝❛♥ ❜❡ ❡❛s✐❧② ❛♣♣❧✐❡❞ t♦ t❤❡ ❣❡♥❡r❛❧ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥✱ ❜❛s❡ b✳ ❈♦♥s✐❞❡r✱ f (x) = bx , b > 0✿ f (x + h) − f (x) h→0 h bx+h − bx = lim h→0 h bx bh − bx = lim h→0 h bx (bh − 1) = lim h→0 h bh − 1 x = b lim h→0 h b0+h − b0 x = b lim h→0 h x x ′ = b · (b ) .

(bx )′ = f ′ (x) = lim

x=0

0 → ? ❲❡ ♥❡❡❞ ❛❧❣❡❜r❛✦ 0

❲❡ ✉s❡ ❛ r✉❧❡ ♦❢ ❡①♣♦♥❡♥ts✳ ❚❤❡♥ ❢❛❝t♦r ❛♥❞ ❛♣♣❧② ❈▼❘✳ ❉♦❡s t❤✐s ❧✐♠✐t ❡①✐st❄ ■t ✐s ❛ ❢❛♠✐❧✐❛r ♦♥❡✦ ◆♦ s✐♠♣❧✐✜❝❛t✐♦♥ ❤❡r❡✦

❚❤✐s ❧✐♠✐t ✐s t❤❡ s❧♦♣❡ ♦❢ t❤❡ ❝✉r✈❡ ❛t t❤❡ y ✲✐♥t❡r❝❡♣t✱ ✐✳❡✳✱ f ′ (0)✱ ✐❢ ✐t ❡①✐sts✳

■♥❞❡❡❞✱ t❤❡ ♥❛t✉r❛❧ ❜❛s❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥ ✐s ❛ s♣❡❝✐❛❧ ♦♥❡✦ ❲❡ ✇♦✉❧❞ ❧✐❦❡ t♦ ✜♥❞ t❤❡ ❞❡r✐✈❛t✐✈❡s ♦❢ s✐♥❡ ❛♥❞ ❝♦s✐♥❡ ♥❡①t✳ ❊①❛♠♣❧❡ ✸✳✾✳✾✿ ❞❡r✐✈❛t✐✈❡ ♦❢

sin

❛t

0

❇❡❢♦r❡ ❛❞❞r❡ss✐♥❣ t❤❡ ❣❡♥❡r❛❧ ❝❛s❡✱ ❧❡t✬s r❡❝❛❧❧ t❤❡✐r ❞❡r✐✈❛t✐✈❡s ❛t x = 0✳ ❲❡ ❦♥♦✇ t❤❛t t❤❡ ❣r❛♣❤ ♦❢ y = sin x ❝r♦ss❡s t❤❡ y ✲❛①✐s ❛t 45 ❞❡❣r❡❡s✿

✸✳✾✳ ❇❛s✐❝ ❞✐✛❡r❡♥t✐❛t✐♦♥✱ ❝♦♥t✐♥✉❡❞

✷✽✼

❚❤✐s ♠❡❛♥s t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ❢❛♠♦✉s tr✐❣♦♥♦♠❡tr✐❝ ❧✐♠✐t✱ lim

x→0

❤❛s ♥♦✇ ❛ ♥❡✇ ✐♥t❡r♣r❡t❛t✐♦♥✿

sin x = 1, x

(sin x)′

= 1. x=0

❲❡ ❛❧s♦ ❦♥♦✇ t❤❛t t❤❡ ❣r❛♣❤ ♦❢ y = cos x ❝r♦ss❡s t❤❡ y ✲❛①✐s ❤♦r✐③♦♥t❛❧❧②✿

❚❤✐s ♠❡❛♥s t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ❢❛♠♦✉s tr✐❣♦♥♦♠❡tr✐❝ ❧✐♠✐t✱ lim

x→0

❤❛s ♥♦✇ ❛ ♥❡✇ ✐♥t❡r♣r❡t❛t✐♦♥✿

1 − cos x = 0, x

(cos x)′

= 0. x=0

❲❡ ♥♦✇ ✉s❡ t❤❡s❡ r❡s✉❧ts t♦ ✜♥❞ t❤❡ ❞❡r✐✈❛t✐✈❡ ❢✉♥❝t✐♦♥s ♦❢ s✐♥❡ ❛♥❞ ❝♦s✐♥❡✳ ❘❡♠❛r❦❛❜❧②✱ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ♦♥❡ ✐s t❤❡ ♦t❤❡r✱ ✉♣ t♦ ❛ s✐❣♥✳

❚❤❡♦r❡♠ ✸✳✾✳✶✵✿ ❉✐✛❡r❡♥❝❡ ♦❢ ❙✐♥❡ ❛♥❞ ❈♦s✐♥❡ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ♠✐❞✲♣♦✐♥t ♣❛rt✐t✐♦♥❀ ✐✳❡✳✱ t❤❡ ♥♦❞❡s ❛r❡ x = a, a + h ❛♥❞ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡ ✐s c = a + h/2✳ ❚❤❡♥ t❤❡ ❞✐✛❡r❡♥❝❡s ♦❢ y = sin x ❛♥❞ y = cos x ❛r❡ ❣✐✈❡♥ ❛t c ❜② t❤❡ ❢♦❧❧♦✇✐♥❣✿ ∆(sin x) = 2 sin(h/2)· cos c ❛♥❞ ∆(cos x) = −2 sin(h/2)· sin c .

Pr♦♦❢✳ ❋✐rst f (x) = sin x✳ ❲❡ ✉s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✉❧❛ ✭s❡❡♥ ✐♥ ❱♦❧✉♠❡ ✶✱ ❈❤❛♣t❡r ✶P❈✲✹✮✿ sin u − sin v = 2 sin

u−v u+v cos . 2 2

❚❤✐s ❢♦r♠✉❧❛ ✐s t❤❡ r❡❛s♦♥ ✇❤② ✇❡ ❝❤♦♦s❡ t❤✐s ♣❛rt✐❝✉❧❛r ♣❛rt✐t✐♦♥✳ ❲❡ ❝♦♠♣✉t❡ ❛t c✿ ∆f = sin(a + h) − sin(a) = 2 sin(h/2) cos(a + h/2) .

❍❡r❡ a + h/2 ✐s ❡①❛❝t❧② t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡✳

❚❤❡ ♣r♦♦❢ ♦❢ t❤❡ s❡❝♦♥❞ ✐❞❡♥t✐t② ✉s❡s ❛♥♦t❤❡r tr✐❣ ❢♦r♠✉❧❛✿ cos u − cos v = −2 sin

u−v u+v sin . 2 2

✸✳✾✳ ❇❛s✐❝ ❞✐✛❡r❡♥t✐❛t✐♦♥✱ ❝♦♥t✐♥✉❡❞

✷✽✽

❚❤❡♦r❡♠ ✸✳✾✳✶✶✿ ❉✐✛❡r❡♥❝❡ ◗✉♦t✐❡♥t ♦❢ ❙✐♥❡ ❛♥❞ ❈♦s✐♥❡ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ♠✐❞✲♣♦✐♥t ♣❛rt✐t✐♦♥❀ ✐✳❡✳✱ t❤❡ ♥♦❞❡s ❛r❡ x = a, a + h ❛♥❞ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡ ✐s c = a + h/2✳ ❚❤❡♥ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ♦❢ y = sin x ❛♥❞ y = cos x ❛r❡ ❣✐✈❡♥ ❛t c ❜② t❤❡ ❢♦❧❧♦✇✐♥❣✿ sin(h/2) ∆ sin(h/2) ∆ (sin x) = · cos c ❛♥❞ (cos x) = − · sin c . ∆x h/2 ∆x h/2

Pr♦♦❢✳ ❋✐rst f (x) = sin x✳ ❲❡ ❝♦♠♣✉t❡ ❛t c ❢r♦♠ t❤❡ ❧❛st t❤❡♦r❡♠✿ ∆f 2 sin(h/2) cos(a + h/2) = ∆x h sin(h/2) · cos(a + h/2) . = h/2

❚❤❡♦r❡♠ ✸✳✾✳✶✷✿ ❉❡r✐✈❛t✐✈❡ ♦❢ ❙✐♥❡ ❛♥❞ ❈♦s✐♥❡ ❚❤❡ ❞❡r✐✈❛t✐✈❡s ♦❢ y = sin x ❛♥❞ y = cos x ❛r❡ ❣✐✈❡♥ ❜②✿ d d (sin x) = cos x ❛♥❞ (cos x) = − sin x dx dx

Pr♦♦❢✳ ❋✐rst f (x) = sin x✳ ❲❡ ❤❛✈❡ ❛t c✿ sin(h/2) ∆f = · cos(a + h/2) ∆x h/2 →1 · cos a = cos a .

❛s h → 0

❚❤❡ ❧❛st st❡♣ ✐s t❤❡ ✜rst ❢❛♠♦✉s ❧✐♠✐t ❛❜♦✈❡ ❝♦♠❜✐♥❡❞ ✇✐t❤ t❤❡ ❝♦♥t✐♥✉✐t② ♦❢ cos x✳ ❚❤❡ ♣r♦♦❢ ♦❢ t❤❡ s❡❝♦♥❞ ✐❞❡♥t✐t② ✉s❡s t❤❡ ♦t❤❡r ❢❛♠♦✉s ❧✐♠✐t ❛❜♦✈❡✳

❊①❡r❝✐s❡ ✸✳✾✳✶✸ Pr♦✈✐❞❡ ❛ ♣r♦♦❢ ♦❢ t❤❡ ♠✐ss✐♥❣ ♣❛rt✳ ▲❡t✬s ❝♦♠♣❛r❡ t❤❡ ❢♦r♠✉❧❛s ❢♦r t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ sin x ✇✐t❤ ✐ts ❞❡r✐✈❛t✐✈❡✿ sin(h/2) ∆ (sin x) = · cos(x + h/2) . h/2 ∆x d (sin x) = cos x . ✭❇✮ dx

✭❆✮

❲❡ ❦♥♦✇ t❤❛t 0
0

✐s ❛ ♣♦❧②♥♦♠✐❛❧ ♦❢ ❞❡❣r❡❡

6= 0✮✿

n − 1✱

f (x) = an xn +an−1 xn−1 +... +a2 x2 +a1 x +a0 f ′ (x) = nan xn−1 +(n − 1)an−1 xn−2 +... +2a2 x +a1 ❊①❡r❝✐s❡ ✹✳✶✳✶✷

Pr♦✈❡ t❤❡ t❤❡♦r❡♠✳

✹✳✷✳ ❈❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ❛♥❞ t❤❡ ❞❡r✐✈❛t✐✈❡

❊①❛♠♣❧❡ ✹✳✷✳✶✿ tr❛♥s❢♦r♠❛t✐♦♥s

❈♦♥s✐❞❡r✿ • ❚❤❡ ✜rst tr❛♥s❢♦r♠❛t✐♦♥ ✐s ❛ str❡t❝❤ ❜② ❛ ❢❛❝t♦r ♦❢ 2✱ ✐✳❡✳✱ t❤❡ ❞❡r✐✈❛t✐✈❡ ✐s 2✳ • ❚❤❡ s❡❝♦♥❞ tr❛♥s❢♦r♠❛t✐♦♥ ✐s ❛ str❡t❝❤ ❜② ❛ ❢❛❝t♦r ♦❢ 3✱ ✐✳❡✳✱ t❤❡ ❞❡r✐✈❛t✐✈❡ ✐s 3✳ • ❚❤❡♥ t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ t✇♦ tr❛♥s❢♦r♠❛t✐♦♥s ✐s ❛ str❡t❝❤ ❜② ❛ ❢❛❝t♦r ♦❢ 3 · 2 = 6✱ ✐✳❡✳✱ t❤❡ ❞❡r✐✈❛t✐✈❡ ✐s 6✳ ■t ❤❛♣♣❡♥s ❛s ❢♦❧❧♦✇s✿

❲❡ ♠✉❧t✐♣❧② t❤❡ ❞❡r✐✈❛t✐✈❡s✳ ❇✉t str❡t❝❤❡s ❛r❡ ❥✉st ❧✐♥❡❛r ❢✉♥❝t✐♦♥s ❛♥❞ t❤❡ str❡t❝❤ ❢❛❝t♦rs ❛r❡ t❤❡✐r s❧♦♣❡s✳ ❇✉t t❤❡ s❧♦♣❡s ❛r❡ t❤❡ ❞❡r✐✈❛t✐✈❡s ♦❢ t❤❡s❡ ❢✉♥❝t✐♦♥s✿ tr❛♥s❢♦r♠❛t✐♦♥ ❞❡r✐✈❛t✐✈❡ ✜rst tr❛♥s❢♦r♠❛t✐♦♥ str❡t❝❤ ❜② 2 ✷ s❡❝♦♥❞ tr❛♥s❢♦r♠❛t✐♦♥ str❡t❝❤ ❜② 3 ✸ t❤❡✐r ❝♦♠♣♦s✐t✐♦♥ str❡t❝❤ ❜② 2 · 3 = 6 2 · 3 = 6

✹✳✷✳

❈❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ❛♥❞ t❤❡ ❞❡r✐✈❛t✐✈❡

✸✵✺

❊①❡r❝✐s❡ ✹✳✷✳✷ ❲❤❛t ✐❢ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥s ❛r❡ s❤✐❢ts❄ ❲❤❛t ✐❢ t❤❡② ❛r❡ ❝♦♠❜✐♥❛t✐♦♥s ♦❢ str❡t❝❤❡s ❛♥❞ s❤✐❢ts❄

❊①❛♠♣❧❡ ✹✳✷✳✸✿ ❧✐♥❡❛r ❢✉♥❝t✐♦♥s ▲❡t✬s ❞♦ t❤❡ ❛❧❣❡❜r❛✳ ❲❡ s❡❡ t❤❡✐r ❞❡r✐✈❛t✐✈❡s ❛♥❞✱ ✇❤✐❝❤ ✐s t❤❡ s❛♠❡ t❤✐♥❣ ❢♦r ❧✐♥❡❛r ❢✉♥❝t✐♦♥s✱ t❤❡✐r ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts✿

❢✉♥❝t✐♦♥s✿

x

❞❡r✐✈❛t✐✈❡s✿

= qt ◦

y y

=⇒

dx ∆x = ∆t dt

=q ×

∆y dy = dx ∆x ∆y dy = = m(qt) = mqt =⇒ ∆t dt

= mx

=m

=⇒

=m·q =

∆x ∆y dx dy · = · ∆t ∆x dt dx

■♥ ❡✐t❤❡r ❝❛s❡✱ ✇❡ s❡❡ ❤♦✇ t❤❡ ✐♥t❡r♠❡❞✐❛t❡ ✈❛r✐❛❜❧❡✱ ✇❤❡t❤❡r ✐t ✐s t❤❡ ❞✐✛❡r❡♥❝❡

dx✱

∆x

♦r t❤❡ ❞✐✛❡r❡♥t✐❛❧

✐s ❝❛♥❝❡❧❡❞ ♦r ✏❝❛♥❝❡❧❡❞✑✿

∆y ∆x ∆y · = ∆t ∆x ∆t dy dx dy · = dt dx dt ❈♦♥❝❧✉s✐♦♥✿ ❲❤✐❝❤❡✈❡r r♦✉t❡ ②♦✉ t❛❦❡ t❤r♦✉❣❤ t❤❡ ❞✐❛❣r❛♠ ✕ ✜rst ❝♦♠♣♦s✐t✐♦♥ t❤❡♥ ❞✐✛❡r❡♥t✐❛t✐♦♥ ♦r t❤❡ ♦♣♣♦s✐t❡ ✕ t❤❡ r❡s✉❧t ✐s t❤❡ s❛♠❡✦

▲❡t✬s r❡❝❛❧❧ t❤❛t ✇❡ ❝❛♥ ✐♥t❡r♣r❡t ❡✈❡r② ❝♦♠♣♦s✐t✐♦♥ ❛s ❛ ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s✳ ❚❤❡ ✈❛r✐❛❜❧❡s ♦❢ t❤❡ ❢✉♥❝t✐♦♥s ✇❡ ❛r❡ ❝♦♥s✐❞❡r✐♥❣ ❛r❡ q✉❛♥t✐t✐❡s ✇❡ ♠❡❡t ✐♥ ❡✈❡r②❞❛② ❧✐❢❡✳ ❋r❡q✉❡♥t❧②✱ t❤❡r❡ ❛r❡ ♠✉❧t✐♣❧❡ ✇❛②s t♦ ♠❡❛s✉r❡ t❤❡s❡ q✉❛♥t✐t✐❡s✿

•

❧❡♥❣t❤ ❛♥❞ ❞✐st❛♥❝❡✿ ✐♥❝❤❡s✱ ♠✐❧❡s✱ ♠❡t❡rs✱ ❦✐❧♦♠❡t❡rs✱ ✳✳✳✱ ❧✐❣❤t ②❡❛rs

•

❛r❡❛✿ sq✉❛r❡ ✐♥❝❤❡s✱ sq✉❛r❡ ♠✐❧❡s✱ ✳✳✳✱ ❛❝r❡s

•

✈♦❧✉♠❡✿ ❝✉❜✐❝ ✐♥❝❤❡s✱ ❝✉❜✐❝ ♠✐❧❡s✱ ✳✳✳✱ ❧✐t❡rs✱ ❣❛❧❧♦♥s

•

t✐♠❡✿ ♠✐♥✉t❡s✱ s❡❝♦♥❞s✱ ❤♦✉rs✱ ✳✳✳✱ ②❡❛rs

•

✇❡✐❣❤t✿ ♣♦✉♥❞s✱ ❣r❛♠s✱ ❦✐❧♦❣r❛♠s✱ ❦❛r❛ts

•

t❡♠♣❡r❛t✉r❡✿ ❞❡❣r❡❡s ♦❢ ❈❡❧s✐✉s✱ ♦❢ ❋❛❤r❡♥❤❡✐t

•

♠♦♥❡②✿ ❞♦❧❧❛rs✱ ❡✉r♦s✱ ♣♦✉♥❞s✱ ②❡♥

•

❡t❝✳

❆❧♠♦st ❛❧❧ ❝♦♥✈❡rs✐♦♥ ❢♦r♠✉❧❛s ❛r❡ ❥✉st ♠✉❧t✐♣❧✐❝❛t✐♦♥s✱ s✉❝❤ ❛s t❤✐s ♦♥❡✿

#

♦❢ ♠❡t❡rs

=#

· 1000 .

♦❢ ❦✐❧♦♠❡t❡rs

❲❛r♥✐♥❣✦ ❲❡ ❞♦♥✬t ❝♦♥✈❡rt ✏♣♦✉♥❞s t♦ ❦✐❧♦s✑✱ ✇❡ ❝♦♥✈❡rt t❤❡

♥✉♠❜❡r ♦❢ ▲❡t✬s t✉r♥ t♦ ♠♦t✐♦♥✿

♣♦✉♥❞s t♦ t❤❡

♥✉♠❜❡r ♦❢

❦✐❧♦s✳

✹✳✷✳

❈❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ❛♥❞ t❤❡ ❞❡r✐✈❛t✐✈❡

• ■❢ t❤❡ ❞✐st❛♥❝❡ ✐s ♠❡❛s✉r❡❞ ✐♥

♠✐♥✉t❡✳

✸✵✻

✐♥❝❤❡s ❛♥❞ t✐♠❡ ✐♥ ♠✐♥✉t❡s✱ t❤❡ ✈❡❧♦❝✐t② ✐s ♠❡❛s✉r❡❞ ✐♥ ✐♥❝❤❡s ♣❡r

❢❡❡t✱ t❤❡ ✈❡❧♦❝✐t② ✐s ♥♦✇ ♠❡❛s✉r❡❞ ✐♥ ❢❡❡t ♣❡r ♠✐♥✉t❡✳ ❇✉t ✐❢ t❤❡ t✐♠❡ ✐s ♠❡❛s✉r❡❞ ✐♥ s❡❝♦♥❞s✱ t❤❡ ✈❡❧♦❝✐t② ✐s ♠❡❛s✉r❡❞ ✐♥ ✐♥❝❤❡s ♣❡r s❡❝♦♥❞✳

• ◆♦✇✱ ✐❢ t❤❡ ❞✐st❛♥❝❡ ✐s ♠❡❛s✉r❡❞ ✐♥ •

❲❡ ❛r❡ ❞❡❛❧✐♥❣ ✇✐t❤ t❤❡ s❛♠❡ ❢✉♥❝t✐♦♥s ❥✉st ♠❡❛s✉r❡❞ ✐♥ ❞✐✛❡r❡♥t ✉♥✐ts✳ ❍♦✇ ❞♦ ✇❡ tr❛♥s✐t✐♦♥ ❜❡t✇❡❡♥ t❤❡ t❤r❡❡❄ ❊①❛♠♣❧❡ ✹✳✷✳✹✿ t✐♠❡ ✉♥✐ts

❙✉♣♣♦s❡ t❤❛t ✇❡ ❤❛✈❡ ❛ r❡❧❛t✐♦♥ ✭❛ ❢✉♥❝t✐♦♥ f ✮ ❜❡t✇❡❡♥ • x✱ t✐♠❡ ✐♥ ♠✐♥✉t❡s✱ ❛♥❞ • y ✱ ❧♦❝❛t✐♦♥ ✐♥ ✐♥❝❤❡s✳ ❲❤❛t ✐❢ ✇❡ ♥❡❡❞ t♦ s✇✐t❝❤ t♦ t✱ t✐♠❡ ✐♥ s❡❝♦♥❞s❄ ❚❤❡ ❛❧❣❡❜r❛ ✐s s✐♠♣❧❡✿ x=

1 · t. 60

❚❤❡ ❝♦♠♣❧❡t❡ ❞❡♣❡♥❞❡♥❝❡ ✐s ❛s ❢♦❧❧♦✇s✿ ·

1

f

−−→ x −−−−→ y t −−−60

◆♦✇✱ t♦ s❡❡ t❤❡ ♥❡✇ ❣r❛♣❤s✱ ✇❡ ❝♦♠❜✐♥❡ t❤❡ ❣r❛♣❤ ♦❢ f ✇✐t❤ ❛ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ t❤❡ x✲❛①✐s✱ ❛s ❢♦❧❧♦✇s✿

❲❡ ❦♥♦✇ t❤❛t ✇❡ ✇✐❧❧ ❜❡ ❝♦✈❡r✐♥❣ 60 t✐♠❡s ❧❡ss ❢♦r ❡✈❡r② s❡❝♦♥❞ t❤❛♥ ❢♦r ❛ ♠✐♥✉t❡✳ ❚❤❛t✬s t❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ✈❡❧♦❝✐t✐❡s✦ ❲❡ ✇r✐t❡ ✐t ❛s ❢♦❧❧♦✇s✿ dy 1 dy = · . dt 60 dx ❊①❛♠♣❧❡ ✹✳✷✳✺✿ ❞✐st❛♥❝❡ ✉♥✐ts

❙✉♣♣♦s❡ ❛❣❛✐♥ ✇❡ ❤❛✈❡ ❛ r❡❧❛t✐♦♥ ✭❛ ❢✉♥❝t✐♦♥ f ✮ ❜❡t✇❡❡♥ • x✱ t✐♠❡ ✐♥ ♠✐♥✉t❡s✱ ❛♥❞ • y ✱ ❧♦❝❛t✐♦♥ ✐♥ ✐♥❝❤❡s✳ ❲❤❛t ✐❢ ✇❡ ♥❡❡❞ t♦ s✇✐t❝❤ t♦ z ✱ ❧♦❝❛t✐♦♥ ✐♥ ❢❡❡t❄ ❚❤❡ ❛❧❣❡❜r❛ ✐s s✐♠♣❧❡✿ z=

1 ·y. 12

❚❤❡ ❝♦♠♣❧❡t❡ ❞❡♣❡♥❞❡♥❝❡ ✐s ❛s ❢♦❧❧♦✇s✿ f

·

1

x −−−−→ y −−−12 −−→ z

◆♦✇✱ t♦ s❡❡ t❤❡ ♥❡✇ ❣r❛♣❤s✱ ✇❡ ❝♦♠❜✐♥❡ t❤❡ ❣r❛♣❤ ♦❢ f ✇✐t❤ ❛ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ t❤❡ y ✲❛①✐s✱ ❛s ❢♦❧❧♦✇s✿

✹✳✷✳

❈❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ❛♥❞ t❤❡ ❞❡r✐✈❛t✐✈❡

❲❡ ❦♥♦✇ t❤❛t ✇❡ ✇✐❧❧ ❜❡ ❝♦✈❡r✐♥❣

12

✸✵✼

✐♥❝❤❡s ❢♦r ❡✈❡r② ❢♦♦t✳ ❚❤❛t✬s t❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ✈❡❧♦❝✐t✐❡s✦

❲❡ ✇r✐t❡ ✐t ❛s ❢♦❧❧♦✇s✿

1 dy dz = · . dx 12 dx ❲❛r♥✐♥❣✦ ❊✈❡♥ t❤♦✉❣❤ ✇❡ s❡❡ ✈❡rt✐❝❛❧ ❛♥❞ ❤♦r✐③♦♥t❛❧ str❡t❝❤✲ ✐♥❣✴s❤r✐♥❦✐♥❣ ♦❢ t❤❡ ♦r✐❣✐♥❛❧ ❣r❛♣❤✱ ✐t✬s ❡♥t✐r❡❧② ✉♣ t♦ ✉s t♦ ♠❛r❦ t❤❡ ✉♥✐ts ♦♥ t❤❡ ♥❡✇ ❛①❡s t♦ ♠❛t❝❤ t❤❡ ♦❧❞✳ ❚❤❡ ❣r❛♣❤ ✇✐❧❧ r❡♠❛✐♥ t❤❡ s❛♠❡✦

❊①❡r❝✐s❡ ✹✳✷✳✻ ❲❤❛t ✐s t❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ s❡❝♦♥❞s ❛♥❞ ❢❡❡t❄

❍♦✇ ❞♦ s✉❝❤ s✐♠♣❧❡ s✉❜st✐t✉t✐♦♥s ❛✛❡❝t ❝❛❧❝✉❧✉s ❛s ✇❡ ❦♥♦✇ ✐t❄ ■❢

y = f (x) ✐s ❛ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t✇♦ q✉❛♥t✐t✐❡s ❝❛❧❧ t❤❡♠

t

❛♥❞

z

x

❛♥❞

y✱

t❤❡♥ ❡✐t❤❡r ♦♥❡ ♠❛② ❜❡ r❡♣❧❛❝❡❞ ✇✐t❤ ❛ ♥❡✇ ✈❛r✐❛❜❧❡✳ ▲❡t✬s

r❡s♣❡❝t✐✈❡❧② ❛♥❞ s✉♣♣♦s❡ t❤❡s❡ r❡♣❧❛❝❡♠❡♥ts ❛r❡ ❣✐✈❡♥ ❜② s♦♠❡ ❢✉♥❝t✐♦♥s✿

•

❈❛s❡ ✶✿

x = g(t)

•

❈❛s❡ ✷✿

z = h(y)

❚❤❡s❡ s✉❜st✐t✉t✐♦♥s ❝r❡❛t❡ ♥❡✇ r❡❧❛t✐♦♥s✿

•

❈❛s❡ ✶✿

y = k(t) = f (g(t))

•

❈❛s❡ ✷✿

z = k(x) = h(f (x))

❚❤❡ ❣r❛♣❤ ✇✐❧❧ tr❛♥s❢♦r♠ ✐♥t♦ t❤❛t ♦❢ t❤❡ ❞✐✛❡r❡♥t✳ ■t✬s ✏❜❡❢♦r❡

f✑

✈s✳ ✏❛❢t❡r

1. t → 2.

f ✑✿

g

❝♦♠♣♦s✐t✐♦♥

→ x →

x →

f f

♦❢

f

✇✐t❤ t❤✐s ❢✉♥❝t✐♦♥✳

→ y

→ y →

h

❚❤❡

→ z

♦r❞❡r✱

❤♦✇❡✈❡r✱ ✐s

✹✳✷✳

✸✵✽

❈❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ❛♥❞ t❤❡ ❞❡r✐✈❛t✐✈❡

❚❤✐s ✐s ✇❤② t❤❡ ❛♥s✇❡rs ✇✐❧❧ ❜❡ ❞✐✛❡r❡♥t✳ ❚❤✐s ✐s ❢♦r ❈❛s❡ ✶✿ ❚❤❡♦r❡♠ ✹✳✷✳✼✿ ▲✐♥❡❛r ❈❤❛✐♥ ❘✉❧❡ ■

■❢ y = f (x) ✐s ❛ ❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥ ❛♥❞ m ✐s ❛ r❡❛❧ ♥✉♠❜❡r✱ t❤❡♥ ✇❡ ❤❛✈❡✿ d f (mt) = mf ′ (mt) . dt

❚❤✉s✱ t❤❡ ❣r❛♣❤s ♦❢ t❤❡ ♥❡✇ q✉❛♥t✐t✐❡s ❞❡s❝r✐❜✐♥❣ ♠♦t✐♦♥ ❛r❡ s✐♠♣❧② r❡✲s❝❛❧❡❞ ✈❡rs✐♦♥s ♦❢ t❤❡ ❣r❛♣❤s ♦❢ t❤❡ ♦❧❞ ♦♥❡s✳ ❚❤✐s ✐s ❢♦r ❈❛s❡ ✷✿ ❚❤❡♦r❡♠ ✹✳✷✳✽✿ ▲✐♥❡❛r ❈❤❛✐♥ ❘✉❧❡ ■■

■❢ y = f (x) ✐s ❛ ❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥ ❛♥❞ m ✐s ❛ r❡❛❧ ♥✉♠❜❡r✱ t❤❡♥ ✇❡ ❤❛✈❡✿ d mf (x) = mf ′ (x) . dx

❚❤✉s✱ t❤❡ q✉❛♥t✐t✐❡s ❞❡s❝r✐❜✐♥❣ ♠♦t✐♦♥ ❛r❡ s✐♠♣❧② r❡♣❧❛❝❡❞ ✇✐t❤ t❤❡✐r ✈❡rt✐❝❛❧❧② str❡t❝❤❡❞ ✈❡rs✐♦♥s ♦❢ t❤❡ ♦❧❞ ♦♥❡s✳

✳ ❚❤❡ ♥❡✇ ❣r❛♣❤s ❛r❡ t❤❡

♠✉❧t✐♣❧❡s

❲❛r♥✐♥❣✦

■❢ ✇❡ ❛r❡ t♦ ❝❤❛♥❣❡ ♦✉r ✉♥✐t t♦ ❛ ❧♦❣❛r✐t❤♠✐❝ s❝❛❧❡ ✭❢♦r ❡①❛♠♣❧❡✱ x = 10t ✮✱ t❤❡♥ t❤❡ ❡✛❡❝t ♦♥ t❤❡ ❞❡r✐✈❛t✐✈❡ ✇✐❧❧ ♥♦t ❜❡ ♣r♦♣♦rt✐♦♥❛❧✳

✹✳✷✳

❈❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ❛♥❞ t❤❡ ❞❡r✐✈❛t✐✈❡

✸✵✾

❊①❛♠♣❧❡ ✹✳✷✳✾✿ ❢♦✉r ✈❛r✐❛❜❧❡s

❘❡❝❛❧❧ t❤❡ ❡①❛♠♣❧❡ ✇❤❡♥ ✇❡ ❤❛✈❡ ❛ ❢✉♥❝t✐♦♥ f t❤❛t r❡❝♦r❞s t❤❡ t❡♠♣❡r❛t✉r❡ ✕ ✐♥ ❋❛❤r❡♥❤❡✐t ✕ ❛s ❛ ❢✉♥❝t✐♦♥ f ♦❢ t✐♠❡ ✕ ✐♥ ♠✐♥✉t❡s ✕ r❡♣❧❛❝❡❞ ✇✐t❤ ❛♥♦t❤❡r t♦ r❡❝♦r❞ t❤❡ t❡♠♣❡r❛t✉r❡ ✐♥ ❈❡❧s✐✉s ❛s ❛ ❢✉♥❝t✐♦♥ g ♦❢ t✐♠❡ ✐♥ s❡❝♦♥❞s✿ • s t✐♠❡ ✐♥ s❡❝♦♥❞s • m t✐♠❡ ✐♥ ♠✐♥✉t❡s • F t❡♠♣❡r❛t✉r❡ ✐♥ ❋❛❤r❡♥❤❡✐t • C t❡♠♣❡r❛t✉r❡ ✐♥ ❈❡❧s✐✉s ❚❤❡ ❝♦♥✈❡rs✐♦♥ ❢♦r♠✉❧❛s ❛r❡✿ m = s/60 , ❛♥❞

C = (F − 32)/1.8 .

❚❤❡s❡ ❛r❡ t❤❡ r❡❧❛t✐♦♥s ❜❡t✇❡❡♥ t❤❡ ❢♦✉r q✉❛♥t✐t✐❡s✿

g:

s/60

f

(F −32)/1.8

s −−−−−−→ m −−−−→ F −−−−−−−−−−→ C .

❆♥❞ t❤✐s ✐s t❤❡ ♥❡✇ ❢✉♥❝t✐♦♥✿ ❚❤❡♥✱ ❜② t❤❡

F = k(s) = (f (s/60) − 32)/1.8 .

▲✐♥❡❛r ❈❤❛✐♥ ❘✉❧❡✱ ✇❡ ❤❛✈❡✿

dF dC dm 1 1 dF = = · f ′ (m) · . ds dC dm ds 1.8 60 ❊①❡r❝✐s❡ ✹✳✷✳✶✵

Pr♦✈✐❞❡ ❛ s✐♠✐❧❛r ❛♥❛❧②s✐s ❢♦r t❤❡ s✐③❡s ♦❢ s❤♦❡s ❛♥❞ ❝❧♦t❤✐♥❣✳ ❊①❛♠♣❧❡ ✹✳✷✳✶✶✿ s✐♥❡ ❛♥❞ ❝♦s✐♥❡ ✐♥ ❞❡❣r❡❡s

❚❤❡ ❝♦♥✈❡rs✐♦♥ ♦❢ t❤❡ ♥✉♠❜❡r ♦❢ ❞❡❣r❡❡s y t♦ t❤❡ ♥✉♠❜❡r ♦❢ r❛❞✐❛♥s x ✐s✿

x= ❚❤❡♥✱

▲❡t✬s ❞❡♥♦t❡ ✭❥✉st ♦♥❝❡✮

π dx = . dy 180

s✐♥❡ ❛♥❞ ❝♦s✐♥❡ ❢♦r ❞❡❣r❡❡s ❜② sind y ❛♥❞ cosd y r❡s♣❡❝t✐✈❡❧②✿ sind y = sin

❚❤❡♥✱

π y. 180

 π   π  y ❛♥❞ cosd y = cos y . 180 180

 π  d d sind y = sin y dy dy 180  π π  = cos y 180 180 π = cosd y . 180 ❲ ❤❛✈❡ ❞✐s❝♦✈❡r❡❞ t❤❛t t❤❡s❡ ❢✉♥❝t✐♦♥ ❞♦♥✬t ❝♦♥♥❡❝t t♦ ❡❛❝❤ ♦t❤❡r ❛s s✐♠♣❧② ❛s t❤❡ ♦r✐❣✐♥❛❧ ♦♥❡s✿ (sin x)′ = cos x✦ ❙✐♠✐❧❛r❧②✱ ❝❤♦♦s✐♥❣ ❛ ❜❛s❡ ♥♦t ❡q✉❛❧ t♦ e ✇✐❧❧ r✉✐♥ t❤❡ ✐♥t✐♠❛t❡ ❝♦♥♥❡❝t✐♦♥ ♦❢ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥ t♦ ✐ts ❞❡r✐✈❛t✐✈❡✿ (ex )′ = ex ✳

✹✳✸✳

❉✐✛❡r❡♥t✐❛t✐♦♥ ♦✈❡r ❝♦♠♣♦s✐t✐♦♥s✿ t❤❡ ❈❤❛✐♥ ❘✉❧❡

✸✶✵

❚❤✐s ✐s t❤❡ s✉♠♠❛r② ♦❢ t❤❡ ❈❤❛✐♥ ❘✉❧❡✿ d

′ f (g(x)) −−−dx−−→ f ′ (g(x))g x (x)   ❈❘ u=g(x)  y

s✉❜st✐t✉t✐♦♥

d

f (u)

−−−du −−→

f ′ (u)

❚❤❡ ♠❡t❤♦❞ ❛❧❧♦✇s ✉s t♦ ❣❡t ❢r♦♠ ❧❡❢t t♦ r✐❣❤t ❛t t❤❡ t♦♣ ✭❞✐✛❡r❡♥t✐❛t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦

x✮ ❜② t❛❦✐♥❣ ❛ u✱ t❤❡ ❈❤❛✐♥

❞❡t♦✉r✳ ❲❡ ❢♦❧❧♦✇ t❤❡ ♣❛t❤ ❛r♦✉♥❞ t❤❡ sq✉❛r❡✿ s✉❜st✐t✉t✐♦♥✱ ❞✐✛❡r❡♥t✐❛t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ ❘✉❧❡ ❢♦r♠✉❧❛ ✇✐t❤ ❜❛❝❦✲s✉❜st✐t✉t✐♦♥✳ ❚❤❡ ❞✐r❡❝t ♣❛t❤ ✐s✱ ♦❢ ❝♦✉rs❡✱ t❤❡ ❧✐♠✐t✳

✹✳✸✳ ❉✐✛❡r❡♥t✐❛t✐♦♥ ♦✈❡r ❝♦♠♣♦s✐t✐♦♥s✿ t❤❡ ❈❤❛✐♥ ❘✉❧❡ ❍♦✇ ❞♦❡s ♦♥❡ ❡①♣r❡ss t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ t✇♦ ❢✉♥❝t✐♦♥s ✐♥ t❡r♠s ♦❢ t❤❡✐r ❞❡r✐✈❛t✐✈❡s❄ ❚❤❡ ❛♥s✇❡r s✉❣❣❡st❡❞ ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥ ✐s✿

◮

❲❤❡♥ ②♦✉

❝♦♠♣♦s❡

❢✉♥❝t✐♦♥s✱ ②♦✉

♠✉❧t✐♣❧②

t❤❡✐r ❞❡r✐✈❛t✐✈❡s✳

❊①❛♠♣❧❡ ✹✳✸✳✶✿ ♠♦t✐♦♥

Pr♦❜❧❡♠✿

❙✉♣♣♦s❡ ❛ ❝❛r ✐s ❞r✐✈❡♥ t❤r♦✉❣❤ ❛ ♠♦✉♥t❛✐♥ t❡rr❛✐♥✳ ■ts ❧♦❝❛t✐♦♥ ❛♥❞ ✐ts s♣❡❡❞✱ ❛s s❡❡♥ ♦♥ ❛

♠❛♣✱ ❛r❡ ❦♥♦✇♥✳ ❚❤❡

❣r❛❞❡

♦❢ t❤❡ r♦❛❞ ✐s ❛❧s♦ ❦♥♦✇♥✳ ❍♦✇ ❢❛st ✐s t❤❡ ❝❛r ❝❧✐♠❜✐♥❣❄

❲❡ s❡t ✉♣ t❤r❡❡ ✈❛r✐❛❜❧❡s✿ t❤❡ t✐♠❡

t✱

t❤❡ ✭❤♦r✐③♦♥t❛❧✮ ❧♦❝❛t✐♦♥

t✇♦ ❦♥♦✇♥ ❢✉♥❝t✐♦♥s✿

f

❚❤❡ ❣r❛♣❤ ♦❢ t❤❡ s❡❝♦♥❞ ❢✉♥❝t✐♦♥

g

x✱

❛♥❞ t❤❡ ❛❧t✐t✉❞❡

y✳

❚❤❡r❡ ❛r❡ ❛❧s♦

g

t −−−−→ x −−−−→ y ✐s ❧✐t❡r❛❧❧② t❤❡ ♣r♦✜❧❡ ♦❢ t❤❡ r♦❛❞✳ ❚❤❡✐r ❝♦♠♣♦s✐t✐♦♥ ✐s ✇❤❛t ✇❡

❛r❡ ✐♥t❡r❡st❡❞ ✐♥✦ ❲❡ ❛❧r❡❛❞② ❦♥♦✇ ✭❢r♦♠ ❈❤❛♣t❡r ✷✮ t❤❛t ✐❢ t❤❡ ❧♦❝❛t✐♦♥✱ t❤❡ ❛❧t✐t✉❞❡✱ t✐♠❡ ❛s ✇❡❧❧✱

x = f (t)✱

❞❡♣❡♥❞s ❝♦♥t✐♥✉♦✉s❧② ♦♥ t✐♠❡ ❛♥❞

z = g(x)✱ ❝♦♥t✐♥✉♦✉s❧② ❞❡♣❡♥❞s ♦♥ ❧♦❝❛t✐♦♥✱ t❤❡♥ t❤❡ ❛❧t✐t✉❞❡ ❞❡♣❡♥❞s ❝♦♥t✐♥✉♦✉s❧② ♦♥ z = (g ◦ f )(t)✳ ❲❡ s❤❛❧❧ ❛❧s♦ s❡❡ ❧❛t❡r t❤❛t t❤❡ ❞✐✛❡r❡♥t✐❛❜✐❧✐t② ♦❢ ❜♦t❤ ❢✉♥❝t✐♦♥s ✐♠♣❧✐❡s

t❤❡ ❞✐✛❡r❡♥t✐❛❜✐❧✐t② ♦❢ t❤❡ ❝♦♠♣♦s✐t✐♦♥✳

❍♦✇❡✈❡r✱ ❧❡t✬s ✜rst ❞✐s♣♦s❡ ♦❢ t❤❡ ✏◆❛✐✈❡ ❈♦♠♣♦s✐t✐♦♥ ❘✉❧❡✑✿

❄❄❄

(f ◦ g)′ === f ′ ◦ g ′ ❲❡ ❝❛rr② ♦✉t ❛ ✏✉♥✐t ❛♥❛❧②s✐s✑ t♦ s❤♦✇ t❤❛t s✉❝❤ ❛ ❢♦r♠✉❧❛ s✐♠♣❧②

• t ✐s t✐♠❡ ♠❡❛s✉r❡❞ ✐♥ ❤r✳ • x = f (t) ✐s t❤❡ ❧♦❝❛t✐♦♥ ♦❢ • y = g(x) ✐s t❤❡ ❛❧t✐t✉❞❡ ♦❢

❝❛♥♥♦t

❜❡ tr✉❡✳ ❙✉♣♣♦s❡

t❤❡ ❝❛r ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✐♠❡ ✕ ♠❡❛s✉r❡❞ ✐♥ ♠✐✳ t❤❡ r♦❛❞ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ ✭❤♦r✐③♦♥t❛❧✮ ❧♦❝❛t✐♦♥ ✕ ♠❡❛s✉r❡❞ ✐♥ ❢t✳

✹✳✸✳

❉✐✛❡r❡♥t✐❛t✐♦♥ ♦✈❡r ❝♦♠♣♦s✐t✐♦♥s✿ t❤❡ ❈❤❛✐♥ ❘✉❧❡

✸✶✶

• y = h(t) = g(f (t)) ✐s t❤❡ ❛❧t✐t✉❞❡ ♦❢ t❤❡ r♦❛❞ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✐♠❡ ✕ ♠❡❛s✉r❡❞ ✐♥ ❢t✳ ❚❤❡♥✱ ♠✐ ✳ • f ′ (t) ✐s t❤❡ ✭❤♦r✐③♦♥t❛❧✮ ✈❡❧♦❝✐t② ♦❢ t❤❡ ❝❛r ♦♥ t❤❡ r♦❛❞ ✕ ♠❡❛s✉r❡❞ ✐♥ ❤r ❢t • g ′ (x) ✐s t❤❡ r❛t❡ ♦❢ ✐♥❝❧✐♥❡ ✭s❧♦♣❡✮ ♦❢ t❤❡ r♦❛❞ ✕ ♠❡❛s✉r❡❞ ✐♥ ✱ ✇✐t❤ t❤❡ ✐♥♣✉t st✐❧❧ ♠❡❛s✉r❡❞ ✐♥ ♠✐ ♠✐✳ ■t ❞♦❡s♥✬t ❡✈❡♥ ♠❛tt❡r ♥♦✇ ✇❤❛t h′ ✐s ♠❡❛s✉r❡❞ ✐♥❀ ❥✉st tr② t♦ ❝♦♠♣♦s❡ t❤❡s❡ t✇♦ ❢✉♥❝t✐♦♥s✳✳✳ ■t ✐s ✐♠♣♦ss✐❜❧❡ ❜❡❝❛✉s❡ t❤❡ ✉♥✐ts ♦❢ t❤❡ ♦✉t♣✉t ♦❢ t❤❡ ❢♦r♠❡r ❛♥❞ t❤❡ ✐♥♣✉t ♦❢ t❤❡ ❧❛tt❡r ❞♦♥✬t ♠❛t❝❤✦ ❍♦✇❡✈❡r✱ t❤✐s

✐s ♣♦ss✐❜❧❡✿

❢t ♠✐ ❢t · = ❀ ❝♦♠♣❛r❡❞ t♦ ❤r ♠✐ ❤r ❢t ′ • h (t) ✐s t❤❡ ❛❧t✐t✉❞❡ ♦❢ t❤❡ r♦❛❞ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✐♠❡ ✕ ♠❡❛s✉r❡❞ ✐♥ ✳ ❤r ❚❤✐s ✐s ✇❤② ✐t ♠❛❦❡s s❡♥s❡✿ ✶✳ ❍♦✇ ❢❛st ②♦✉ ❛r❡ ❝❧✐♠❜✐♥❣ ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ ②♦✉r ❤♦r✐③♦♥t❛❧ s♣❡❡❞✳ ✷✳ ❍♦✇ ❢❛st ②♦✉ ❛r❡ ❝❧✐♠❜✐♥❣ ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ s❧♦♣❡ ♦❢ t❤❡ r♦❛❞✳

• f (t) · g (x) ✐s t❤❡✐r ♣r♦❞✉❝t ✕ ♠❡❛s✉r❡❞ ✐♥ ′

′

❚❤✐s ✐s t❤❡ ♣r♦❜❧❡♠ ✇❡ ❢❛❝❡✳ ❲❡ ❛r❡ ❣✐✈❡♥ t✇♦ ❢✉♥❝t✐♦♥s f ❛♥❞ g ❛♥❞ t❤❡✐r ❝♦♠♣♦s✐t✐♦♥✿

■❢ t❤❡ t✇♦ ❞❡r✐✈❛t✐✈❡s ❛r❡ ❛❧s♦ ❦♥♦✇♥ ✭♣♦ss✐❜❧② ❛t ❛ s✐♥❣❧❡ ❧♦❝❛t✐♦♥ ❡❛❝❤✮✱ ✜♥❞ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❝♦♠♣♦✲ s✐t✐♦♥✿

✹✳✸✳ ❉✐✛❡r❡♥t✐❛t✐♦♥ ♦✈❡r ❝♦♠♣♦s✐t✐♦♥s✿ t❤❡ ❈❤❛✐♥ ❘✉❧❡

✸✶✷

❆❝❝♦r❞✐♥❣ t♦ ♦✉r ❛♥❛❧②s✐s✱ t❤❡ s❧♦♣❡ ♦❢ t❤❡ ❝♦♠♣♦s✐t✐♦♥ ✐s s✉♣♣♦s❡❞ t♦ ❜❡ t❤❡ ♣r♦❞✉❝t ♦❢ t❤❡ s❧♦♣❡s ♦❢ t❤❡ t✇♦ ❢✉♥❝t✐♦♥s✦ ❖♥❡ ✇♦✉❧❞♥✬t ❣✉❡ss t❤✐s ❢r♦♠ t❤❡ ♣✐❝t✉r❡✳✳✳ ❲❡ ❞♦ ❦♥♦✇ ❢r♦♠ t❤❡ ❧❛st s❡❝t✐♦♥ t❤❛t t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ t✇♦ ❧✐♥❡❛r ❢✉♥❝t✐♦♥s ✐s t❤❡ ♣r♦❞✉❝t ♦❢ t❤❡ t✇♦ ❞❡r✐✈❛t✐✈❡s ✭s❧♦♣❡s✮✳ ❍♦✇❡✈❡r✱ ❛s ❢❛r ❛s s✐♥❣❧❡ ❧♦❝❛t✐♦♥ ✐s ❝♦♥❝❡r♥❡❞✱ ❛❧❧ ❢✉♥❝t✐♦♥s ❛r❡ ❧✐♥❡❛r ✭❛♣♣r♦①✐♠❛t❡❧②✮✿

❲❡ ❤❛✈❡✱ t❤❡r❡❢♦r❡✱ str♦♥❣ ❡✈✐❞❡♥❝❡ ✐♥ s✉♣♣♦rt ♦❢ ♦✉r ❝♦♥❥❡❝t✉r❡✳ ❚❤❡ ❞❡♣❡♥❞❡♥❝❡ ♦❢ t❤❡ ✈❛r✐❛❜❧❡s ✐s ❛s ❢♦❧❧♦✇s✿ f

g

t −−−−→ x −−−−→ y

❘❡❝❛❧❧ t❤❛t ✇❡ ❤❛✈❡ ♥♦t✐❝❡❞ t❤✐s ♣❛tt❡r♥ ♦❢ ❝❛♥❝❡❧❧❛t✐♦♥✿ ∆y ∆y ∆x = · ∆t ∆x ∆t dy dx dy = · dt dx dt

❯♥❢♦rt✉♥❛t❡❧②✱ ❞❡r✐✈❛t✐✈❡s ❛r❡♥✬t ❢r❛❝t✐♦♥s t♦ ❜❡ ❝❛♥❝❡❧❡❞✦ ❇✉t ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ❛r❡✱ ❛♥❞ ✇❡ ❝❛♥ ❝❛♥❝❡❧ ✭✇❤❡♥ ∆x 6= 0✮✳ ❲❡ st❛rt ✇✐t❤ t❤❡♠✳ ❚❤❡ ♠❛✐♥ ❞✐✛❡r❡♥❝❡ ❢r♦♠ t❤❡ r✉❧❡s ❢♦r ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ✇❡ ❝♦♥s✐❞❡r❡❞ ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥ ✐s t❤❛t✿ • ❚❤❡r❡ ❛r❡

t✇♦ ♣❛rt✐t✐♦♥s✿ ❢♦r t ❛♥❞ ❢♦r x✳

• ❆ ❢✉♥❝t✐♦♥ f ♠✉st ♠❛♣ t❤❡ ♣❛rt✐t✐♦♥ ❢♦r t t♦ t❤❡ ♣❛rt✐t✐♦♥ ♦❢ x✳

❇❡❧♦✇✱ k = ∆t ❛♥❞ h = ∆x✿

✹✳✸✳ ❉✐✛❡r❡♥t✐❛t✐♦♥ ♦✈❡r ❝♦♠♣♦s✐t✐♦♥s✿ t❤❡ ❈❤❛✐♥ ❘✉❧❡

✸✶✸

❲❡ ❝❛♥ s❡❡ t❤❛t✱ ✇❤❡t❤❡r y ❞❡♣❡♥❞s ♦♥ x ♦r t✱ t❤❡ ❞✐✛❡r❡♥❝❡s ❛r❡ t❤❡ s❛♠❡✳ ▲❡t✬s ❝❛r❡❢✉❧❧② st❛t❡ t❤✐s tr✐✈✐❛❧ ❢❛❝t✿

❚❤❡♦r❡♠ ✹✳✸✳✷✿ ❈❤❛✐♥ ❘✉❧❡ ❢♦r ❉✐✛❡r❡♥❝❡s ❚❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ t✇♦ ❢✉♥❝t✐♦♥s ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ s❡❝♦♥❞ ❢✉♥❝t✐♦♥✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ❢♦r ❛♥② ❢✉♥❝t✐♦♥ x = f (t) ❞❡✜♥❡❞ ❛t t✇♦ ❛❞❥❛❝❡♥t ♥♦❞❡s t ❛♥❞ t + ∆t ♦❢ ❛ ♣❛rt✐t✐♦♥✱ ❛♥❞ ❛♥② ❢✉♥❝t✐♦♥ y = g(x) ❞❡✜♥❡❞ ❛t t❤❡ t✇♦ ❛❞❥❛❝❡♥t ♥♦❞❡s x = f (t) ❛♥❞ x + ∆x = f (t + ∆t) ♦❢ ❛ ♣❛rt✐t✐♦♥✱ ✇❡ ❤❛✈❡ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ✭❞❡✜♥❡❞ ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s c ❛♥❞ q = f (c) ✇✐t❤✐♥ t❤❡s❡ ❡❞❣❡s ♦❢ t❤❡ t✇♦ ♣❛rt✐t✐♦♥s r❡s♣❡❝t✐✈❡❧②✮ s❛t✐s❢②✿

∆(g ◦ f )(c) = ∆g(q)

❚♦ ❣❡t t♦ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts✱ ✇❡ ❥✉st ♠✉❧t✐♣❧② ♦♥ t❤❡ ❧❡❢t ❛♥❞ ♦♥ t❤❡ r✐❣❤t r❡s♣❡❝t✐✈❡❧② ❜② 1 ∆x 1 = · , ∆t ∆x ∆t

♣r♦❞✉❝✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣✿

❚❤❡♦r❡♠ ✹✳✸✳✸✿ ❈❤❛✐♥ ❘✉❧❡ ❢♦r ❉✐✛❡r❡♥❝❡ ◗✉♦t✐❡♥ts ❚❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ t✇♦ ❢✉♥❝t✐♦♥s ✐s ❡q✉❛❧ t♦ t❤❡ ♣r♦❞✉❝t ♦❢ t❤❡ t✇♦ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ❢♦r ❛♥② ❢✉♥❝t✐♦♥ x = f (t) ❞❡✜♥❡❞ ❛t t✇♦ ❛❞❥❛❝❡♥t ♥♦❞❡s t ❛♥❞ t + ∆t ♦❢ ❛ ♣❛rt✐t✐♦♥✱ ❛♥❞ ❛♥② ❢✉♥❝t✐♦♥ y = g(x) ❞❡✜♥❡❞ ❛t t❤❡ t✇♦ ❛❞❥❛❝❡♥t ♥♦❞❡s x = f (t) ❛♥❞ x + ∆x = f (t + ∆t) ♦❢ ❛ ♣❛rt✐t✐♦♥✱ ✇❡ ❤❛✈❡ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ✭❞❡✜♥❡❞ ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s c ❛♥❞ q = f (c) ✇✐t❤✐♥ t❤❡s❡ ❡❞❣❡s ♦❢ t❤❡ t✇♦ ♣❛rt✐t✐♦♥s r❡s♣❡❝t✐✈❡❧②✮ s❛t✐s❢②✱ ♣r♦✈✐❞❡❞ ∆x 6= 0✿ ∆(g ◦ f ) ∆g ∆f (c) = (q) · (c) ∆t ∆x ∆t

✹✳✸✳ ❉✐✛❡r❡♥t✐❛t✐♦♥ ♦✈❡r ❝♦♠♣♦s✐t✐♦♥s✿ t❤❡ ❈❤❛✐♥ ❘✉❧❡

✸✶✹

Pr♦♦❢✳

❚❤❡ ❢♦r♠✉❧❛ ❢♦r ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ✐s ❞❡❞✉❝❡❞ ❛s ❢♦❧❧♦✇s✿ ∆(g ◦ f ) (g ◦ f )(t + ∆t) − (g ◦ f )(t) (c) = ∆t ∆t g(f (t + ∆t)) − g(f (t)) f (t + ∆t) − f (t) = f (t + ∆t) − f (t) ∆t g(x + ∆x) − g(x) f (t + ∆t) − f (t) = ∆x ∆t ∆g ∆f = (q) · (c) . ∆x ∆t

❚❤❡ ❧✐♠✐t ❣✐✈❡s ✉s t❤❡ ❞❡r✐✈❛t✐✈❡✿ ❚❤❡♦r❡♠ ✹✳✸✳✹✿ ❈❤❛✐♥ ❘✉❧❡ ❢♦r ❉❡r✐✈❛t✐✈❡s

❚❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ ❛ ❢✉♥❝t✐♦♥ ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t ❛ ♣♦✐♥t ❛♥❞ ❛ ❢✉♥❝t✐♦♥ ❞✐✛❡r❡♥✲ t✐❛❜❧❡ ❛t t❤❡ ✈❛❧✉❡ ♦❢ t❤❛t ♣♦✐♥t ✉♥❞❡r t❤❡ ✜rst ❢✉♥❝t✐♦♥ ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t t❤❛t ♣♦✐♥t✱ ❛♥❞ ✐ts ❞❡r✐✈❛t✐✈❡ ✐s ❡q✉❛❧ t♦ t❤❡ ♣r♦❞✉❝t ♦❢ t❤❡ t✇♦ ❞❡r✐✈❛t✐✈❡s✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✐❢ x = f (t) ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t t = c ❛♥❞ y = g(x) ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t x = q = f (c)✱ t❤❡♥ ✇❡ ❤❛✈❡✿ d(g ◦ f ) dg df (c) = (q) · (c) dt dx dt Pr♦♦❢✳

◆♦✇ ✇❡ ❛r❡ t♦ t❛❦❡ t❤❡ ❧✐♠✐t ♦❢ t❤❡ ❢♦r♠✉❧❛✱ ✇✐t❤ c = t✱ ❛s ∆t → 0 .

❙✐♥❝❡ x = x(t) ✐s ❝♦♥t✐♥✉♦✉s✱ ✇❡ ❝♦♥❝❧✉❞❡ t❤❛t ✇❡ ❛❧s♦ ❤❛✈❡✿ ∆x → 0✳ ❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡✿ ∆g = ∆t ↓

dg dt

∆g (f (t)) · ∆x ↓

=

dg (f (t)) dx

∆f (t) ∆t ↓

·

df (t) dt

❚❤❡ ♣r♦❜❧❡♠ ✇✐t❤ t❤❡ ♣r♦♦❢ ✐s✿ ❲❡ ❛ss✉♠❡❞ t❤❛t ∆x 6= 0✦ ❲❤❛t ✐❢ x = f (t) ✐s ❝♦♥st❛♥t ✐♥ t❤❡ ✈✐❝✐♥✐t② ♦❢ t❄ ❆ ❝♦♠♣❧❡t❡ ♣r♦♦❢ ✇✐❧❧ ❜❡ ♣r♦✈✐❞❡❞ ❧❛t❡r✳ ❊①❡r❝✐s❡ ✹✳✸✳✺

❋✐♥❞ ❛♥♦t❤❡r✱ ♥♦♥✲❝♦♥st❛♥t✱ ❡①❛♠♣❧❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ x = f (t) s✉❝❤ t❤❛t ∆f ♠❛② ❜❡ ③❡r♦ ❡✈❡♥ ❢♦r s♠❛❧❧ ✈❛❧✉❡s ♦❢ ∆t✳ ❲❛r♥✐♥❣✦ ❚❤❡ ✐♥♣✉t ✈❛r✐❛❜❧❡s ✐♥ t❤❡ ❢♦r♠✉❧❛ ❞♦♥✬t ♠❛t❝❤✳

✹✳✸✳ ❉✐✛❡r❡♥t✐❛t✐♦♥ ♦✈❡r ❝♦♠♣♦s✐t✐♦♥s✿ t❤❡ ❈❤❛✐♥ ❘✉❧❡

✸✶✺

❚❤❡ ❢♦r♠✉❧❛ ✐♥ t❤❡ ▲❛❣r❛♥❣❡ ♥♦t❛t✐♦♥ ✐s ❛s ❢♦❧❧♦✇s✿ (g ◦ f )′ (t) = g ′ (f (t)) · f ′ (t) ❊①❛♠♣❧❡ ✹✳✸✳✻✿ ❧✐♥❡❛r ❛♥❞ q✉❛❞r❛t✐❝

❋✐♥❞ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢✿

y = (1 + 2x)2 .

❚❤❡ ❢✉♥❝t✐♦♥ ✐s ❝♦♠♣✉t❡❞ ✐♥ t✇♦ ❝♦♥s❡❝✉t✐✈❡ st❡♣s ✭t❤❛t✬s ❤♦✇ ✇❡ ❦♥♦✇ t❤✐s ✐s ❛ ❝♦♠♣♦s✐t✐♦♥✦✮✿ • ❙t❡♣ ✶✿ ❋r♦♠ x ✇❡ ❝♦♠♣✉t❡ 1 + 2x✳ • ❙t❡♣ ✷✿ ❲❡ sq✉❛r❡ t❤❡ ♦✉t❝♦♠❡ ♦❢ t❤❡ ✜rst st❡♣✳ ❲❡ t❤❡♥ ✐♥tr♦❞✉❝❡ ❛♥ ❛❞❞✐t✐♦♥❛❧✱ ❞✐s♣♦s❛❜❧❡✱ ✈❛r✐❛❜❧❡ ✐♥ ♦r❞❡r t♦ st♦r❡ t❤❡ ♦✉t❝♦♠❡ ♦❢ st❡♣ ✶✿ u = 1 + 2x .

❚❤❡♥ st❡♣ ✷ ❜❡❝♦♠❡s✿

y = u2 .

❚❤✐s ✐s t❤❡ ❞❡♣❡♥❞❡♥❝❡ ♦❢ t❤❡ ✈❛r✐❛❜❧❡s✿ x→u→y

◆♦✇ t❤❡ ❞❡r✐✈❛t✐✈❡s✿ u = 1 + 2x y = u2

❈❤❛✐♥ ❘✉❧❡

du =2 dx dy =⇒ = 2u du dy du dy = · = 2u · 2 = 4u =⇒ dx du dx =⇒

❉♦♥❡❄ ◆♦✳ ❚❤❡ ❛♥s✇❡r ♠✉st ❜❡ ✐♥ t❡r♠s ♦❢ x✦ ▲❛st st❡♣✿ ❙✉❜st✐t✉t❡ u = 1 + 2x✳ ❚❤❡♥ t❤❡ ❛♥s✇❡r ✐s 4(1 + 2x)✳ ❚♦ ✈❡r✐❢②✱ ❡①♣❛♥❞✱ 1 + 4x + 4x2 ✱ t❤❡♥ ✉s❡ P❋ ✐♥ ❝♦♠❜✐♥❛t✐♦♥ ✇✐t❤ ❙❘ ❛♥❞ ❈▼❘✳ ❊①❡r❝✐s❡ ✹✳✸✳✼

❱❡r✐❢② t❤❡ r❡s✉❧t✿ ❊①♣❛♥❞✱ 1 + 4x + 4x2 ✱ t❤❡♥ ✉s❡ P❋ ✐♥ ❝♦♠❜✐♥❛t✐♦♥ ✇✐t❤ ❙❘ ❛♥❞ ❈▼❘✳ ❊①❡r❝✐s❡ ✹✳✸✳✽

❋✐♥❞ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ y = (2 − x)3 . ❊①❛♠♣❧❡ ✹✳✸✳✾✿ ❧✐♥❡❛r ❛♥❞ r♦♦t

◆♦✇ ❛ ✈❡r② s✐♠♣❧❡ ❡①❛♠♣❧❡ t❤❛t ❞♦❡s♥✬t ❛❧❧♦✇ ✉s t♦ ❝✐r❝✉♠✈❡♥t t❤❡ ❈❤❛✐♥ ❘✉❧❡✳ ▲❡t y=

√

3x + 1 .

✹✳✸✳

❉✐✛❡r❡♥t✐❛t✐♦♥ ♦✈❡r ❝♦♠♣♦s✐t✐♦♥s✿ t❤❡ ❈❤❛✐♥ ❘✉❧❡

✸✶✻

❚❤✐s ✐s t❤❡ ❛❜❜r❡✈✐❛t❡❞ ❝♦♠♣✉t❛t✐♦♥ ✭❞❡❝♦♠♣♦s✐t✐♦♥✱ t❤❡ ❞❡r✐✈❛t✐✈❡s✱ ❈❘✱ ❜❛❝❦✲s✉❜st✐t✉t✐♦♥✮✿

x → u = 3x + 1 → y =

√

u

= 3x + 1} |x → u{z du =3 dx

√ u → y= u | {z } 1 dy = √ du 2 u | {z } dy du dy 1 1 = · =3· √ =3· √ dx dx du 2 u 2 3x + 1

❊①❛♠♣❧❡ ✹✳✸✳✶✵✿ t❤r❡❡ ❢✉♥❝t✐♦♥s

❋✐♥❞ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢✿

z=e

❚❤r❡❡

√

3x+1

❢✉♥❝t✐♦♥s t❤✐s t✐♠❡✿

x → 3x + 1 = u →

√

u = y → ey = z

❋♦rt✉♥❛t❡❧②✱ ✇❡ ❛❧r❡❛❞② ❦♥♦✇ ❛ ❧♦t ❢r♦♠ t❤❡ ❧❛st ❡①❛♠♣❧❡✳ ❡①tr❛ st❡♣✿

x → u = 3x + 1 → y =

√

❲❡ ❥✉st

❛♣♣❡♥❞

t❤❛t s♦❧✉t✐♦♥ ✇✐t❤ ♦♥❡

u → z = ey

x = 3x + 1} | → u{z du =3 dx √ u → y= u | {z } 1 dy = √ du 2 u | {z } du dy 1 dy = · =3· √ dx dx du 2 u y → z = ey {z } | dz y =e dy | } {z  √ dz dz 1 du dy 1 e 3x+1 · = · = 3 · √ · ey = 3 √ dx dx du dy 2 u 2 3x + 1 ❲❡ ❤❛✈❡ ❛♣♣❧✐❡❞ t❤❡ ❈❤❛✐♥ ❘✉❧❡ t✇✐❝❡✦

❚❤❡

❧❡ss♦♥

✇❡ ❤❛✈❡ ❧❡❛r♥❡❞ ✐s✿ t❤r❡❡ ❢✉♥❝t✐♦♥s ✕ t❤r❡❡ ❞❡r✐✈❛t✐✈❡s ✕ ♠✉❧t✐♣❧② t❤❡♠✿

x →u →y →z dz du dy dz = · · dx dx du dy ❚❤❡s❡ ✏❢r❛❝t✐♦♥s✑ ❛♣♣❡❛r t♦ ❝❛♥❝❡❧ ❛❣❛✐♥✿

dz du dy dz 6 du 6 dy dz · · · = . · = dx du dy dx 6 du 6 dy dx

✹✳✸✳

❉✐✛❡r❡♥t✐❛t✐♦♥ ♦✈❡r ❝♦♠♣♦s✐t✐♦♥s✿ t❤❡ ❈❤❛✐♥ ❘✉❧❡

❚❤✐s ✐s t❤❡

●❡♥❡r❛❧✐③❡❞ ❈❤❛✐♥ ❘✉❧❡

✸✶✼

❛❜♦✉t t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❝♦♠♣♦s✐t✐♦♥ ✭❛ ✏❝❤❛✐♥✑✦✮ ♦❢

n

❢✉♥❝t✐♦♥s✳

❚❤❡ s❤♦rt ✈❡rs✐♦♥ ♦❢ t❤❡ ❈❤❛✐♥ ❘✉❧❡ s❛②s✿

◮

❚❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❝♦♠♣♦s✐t✐♦♥ ✐s t❤❡ ♣r♦❞✉❝t ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡s✱ ❛s ❢✉♥❝t✐♦♥s✳

❊①❛♠♣❧❡ ✹✳✸✳✶✶✿ ❝♦♠♣♦s✐t✐♦♥ ♦❢ ❞❡r✐✈❛t✐✈❡s

❍♦✇❡✈❡r✱ ✐❢ ✇❡ ✜① t❤❡ ❧♦❝❛t✐♦♥

x = a✱

✇❡ ❝❛♥ ♠❛❦❡ s❡♥s❡ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❝♦♠♣♦s✐t✐♦♥ ❛s t❤❡

❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡s✱ ❛❢t❡r ❛❧❧✳

■♥❞❡❡❞✱ s✉♣♣♦s❡ ❛t ♣♦✐♥t

a

✇❡ ❤❛✈❡ t❤❡ ❞❡r✐✈❛t✐✈❡

dy = m. dx ▲❡t✬s✱ ❛❣❛✐♥✱ t❤✐♥❦ ♦❢ t❤❡

❞✐✛❡r❡♥t✐❛❧s dx ❛♥❞ dy

❛s t✇♦ ♥❡✇ ✈❛r✐❛❜❧❡s ✕ r❡❧❛t❡❞ t♦ ❡❛❝❤ ♦t❤❡r ❜② t❤❡

❛❜♦✈❡ ❡q✉❛t✐♦♥✿

❚❤❡♥ ✇❡ t❤✐♥❦ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡✱

m✱

♥♦t ❛s ❛ ♥✉♠❜❡r ❜✉t ❛s ❛

❧✐♥❡❛r ❢✉♥❝t✐♦♥ ✿

dy = m · dx . ■❢ ♥♦✇ t❤❡r❡ ✐s ❛♥♦t❤❡r ✈❛r✐❛❜❧❡

t✱

✇❡ t❤✐♥❦ ♦❢

q

❛s ❛ ❧✐♥❡❛r ❢✉♥❝t✐♦♥✿

dx = q · dt . ❚❤❡♥✱ ✇❡ ❤❛✈❡ t♦ s✉❜st✐t✉t❡

q✿

x = x(t) = qt =⇒ dx = q · dt ◦ ◦ ◦ y = y(x) = mx =⇒ dy = m · dx y = y(x(t)) = m(qt) ⇐⇒ dy = m · (q · dt) ❲❡ ❤❛✈❡ t❤❡ ❝♦♠♣♦s✐t✐♦♥✦

❲❡ ❝❛♥ ✉s❡ t❤❡ ❈❤❛✐♥ ❘✉❧❡ t♦ ✜♥❞ ❢♦r♠✉❧❛s ❢♦r ♦t❤❡r ✐♠♣♦rt❛♥t ❢✉♥❝t✐♦♥s✿ ❚❤❡♦r❡♠ ✹✳✸✳✶✷✿ ❉❡r✐✈❛t✐✈❡ ♦❢ ●❡♥❡r❛❧ ❊①♣♦♥❡♥t✐❛❧ ❋✉♥❝t✐♦♥ ❋♦r ❛♥②

b > 0✱

✇❡ ❤❛✈❡✿

(bx )′ = bx ln b

Pr♦♦❢✳

❲❡ r❡♣r❡s❡♥t t❤✐s ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥ ✐♥ t❡r♠s ♦❢ t❤❡ ♥❛t✉r❛❧ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥✿ x

bx = eln b = ex ln b .

✹✳✸✳ ❉✐✛❡r❡♥t✐❛t✐♦♥ ♦✈❡r ❝♦♠♣♦s✐t✐♦♥s✿ t❤❡ ❈❤❛✐♥ ❘✉❧❡

✸✶✽

❚❤❡♥✱ ❜② t❤❡ ❈❤❛✐♥ ❘✉❧❡ ✇❡ ❤❛✈❡✿ (bx )′ = ex ln b


′

❈❘ === ex ln b · (x ln b)′ = bx · ln b .

❊①❡r❝✐s❡ ✹✳✸✳✶✸

❯s❡ t❤❡ ✐❞❡❛ ❢r♦♠ t❤❡ ♣r♦♦❢ ❛❜♦✈❡ t♦ ✜♥❞ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ xx ✳ ❊①❛♠♣❧❡ ✹✳✸✳✶✹✿ tr✐❣ ❛♥❞ q✉❛❞r❛t✐❝

❉✐✛❡r❡♥t✐❛t❡✿

y = sin(x2 ) .

❲❡ ❞❡❝♦♠♣♦s❡ ✜rst ❜② ✐♥tr♦❞✉❝✐♥❣ ❛♥ ❡①tr❛ ✈❛r✐❛❜❧❡✿ x → x 2 = u → u2 = y .

❚❤❡♥✱ ✇❡ ✜♥❞ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ t✇♦ ❢✉♥❝t✐♦♥s ❛♥❞ t❤❡♥ ♠✉❧t✐♣❧② t❤❡♠✿ du = (x2 )′ = 2x dx dy = (sin u)′ = cos u du du dy · = 2x · cos u dx du

❇② t❤❡ ❈❤❛✐♥ ❘✉❧❡✿

dy du dy = · = 2x · cos u = 2x cos x2 . dx dx du

❊①❡r❝✐s❡ ✹✳✸✳✶✺

❋✐♥❞ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ y = (sin x)3 ✳ ▼❛❦❡ ✉♣ ②♦✉r ♦✇♥ ❝♦♠♣♦s✐t✐♦♥s ❛♥❞ r❡♣❡❛t✳ ▲❡t✬s r❡♣r❡s❡♥t t❤❡ ❙✉♠ ❘✉❧❡✱ t❤❡ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡✱ ❛♥❞ t❤❡ ❈❤❛✐♥ ❘✉❧❡ ❛s ❞✐❛❣r❛♠s✿ d

f, g  + y

−−−dx−−→ SR d

′ ′ f ,g  + y

f + g −−−dx−−→ (f + g)′ = f ′ + g ′

d

−−−dx−−→ f  ×c CM R y d

cf

′ f  ×c y

−−−dx−−→ (cf )′ = cf ′

d

f,g  ◦ y

−−−dx−−→ CR d

′ ′ f ,g  × y

f ◦ g −−−dx−−→ (f ◦ g)′ = f ′ g ′

■♥ ❛❧❧ ♦❢ t❤❡s❡ ❞✐❛❣r❛♠s✱ ✇❡ st❛rt ✇✐t❤ ❛ ♣❛✐r ♦❢ ❢✉♥❝t✐♦♥s ❛t t❤❡ t♦♣ ❧❡❢t ❛♥❞ t❤❡♥ ✇❡ ♣r♦❝❡❡❞ ✐♥ t✇♦ ✇❛②s✿ • ❘✐❣❤t✿ ❉✐✛❡r❡♥t✐❛t❡ t❤❡♠❀ t❤❡♥ ❞♦✇♥✿ ❞♦ ❛❧❣❡❜r❛ ✇✐t❤ t❤❡ r❡s✉❧ts✳

• ❉♦✇♥✿ ❉♦ t❤❡ ❛❧❣❡❜r❛ ✇✐t❤ t❤❡♠❀ t❤❡♥ r✐❣❤t✿ ❞✐✛❡r❡♥t✐❛t❡ t❤❡ r❡s✉❧t✳

❚❤❡ r❡s✉❧t ✐s t❤❡ s❛♠❡✦

❍♦✇❡✈❡r✱ ✇❡ s❤♦✉❧❞♥✬t ♥❛✐✈❡❧② ❛ss✉♠❡ t❤❛t t❤❡ s❛♠❡ ✇✐❧❧ ❤❛♣♣❡♥ t♦ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❛♥❞ ❞✐✈✐s✐♦♥✳

✹✳✹✳

❉✐✛❡r❡♥t✐❛t✐♦♥ ♦✈❡r ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❛♥❞ ❞✐✈✐s✐♦♥

✸✶✾

✹✳✹✳ ❉✐✛❡r❡♥t✐❛t✐♦♥ ♦✈❡r ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❛♥❞ ❞✐✈✐s✐♦♥ ❲❤❛t ❤❛♣♣❡♥s t♦ t❤❡ ♦✉t♣✉t ❢✉♥❝t✐♦♥ ♦❢ ❞✐✛❡r❡♥t✐❛t✐♦♥ ❛s ✇❡ ♣❡r❢♦r♠ s✉❝❤ ❛♥ ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥ ❛s

t✐♣❧✐❝❛t✐♦♥

♠✉❧✲

✇✐t❤ t❤❡ ✐♥♣✉t ❢✉♥❝t✐♦♥s❄

▲❡t✬s ❝♦♥s✐❞❡r ♠✉❧t✐♣❧✐❝❛t✐♦♥✳ ❊①❛♠♣❧❡ ✹✳✹✳✶✿ t❛r♣

❈♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦❜❧❡♠✿ ■❢ t✇♦ ❣r♦✉♣s ♦❢ r✉♥♥❡rs ❛r❡ ✉♥❢♦❧❞✐♥❣ ❛ t❛r♣ ✭♦r ✉♥❢✉r❧✐♥❣ ❛ ✢❛❣✮ ✇❤✐❧❡ r✉♥♥✐♥❣ ❡❛st ❛♥❞ ♥♦rt❤✱ r❡s♣❡❝t✐✈❡❧②✱ ✇❤❛t ✐s ❤❛♣♣❡♥✐♥❣ t♦ t❤❡ ❛r❡❛ ♦❢ t❤✐s r❡❝t❛♥❣❧❡❄ ❚❤❡② ♠❛② ❜❡ r✉♥♥✐♥❣ ❛t ❞✐✛❡r❡♥t s♣❡❡❞s✿

▲❡t✬s ✜rst ♠❛❦❡ s✉r❡ ✇❡ ❛✈♦✐❞ t❤❡ s♦✲❝❛❧❧❡❞ ✏◆❛✐✈❡ Pr♦❞✉❝t ❘✉❧❡✑✿

❄❄❄

(f · g)′ === f ′ · g ′ . ❚❤❡ ❢♦r♠✉❧❛ ✐s ❡①tr❛♣♦❧❛t❡❞ ❢r♦♠ t❤❡ ❙✉♠ ❘✉❧❡ ❜✉t ✐t s✐♠♣❧② ✐♥ t❤❡ t❡r♠s ♦❢ ♠♦t✐♦♥ ❛♥❞ t❛❦❡ ❛ ❣♦♦❞ ❧♦♦❦ ❛t t❤❡

✉♥✐ts✳

❝❛♥♥♦t

❜❡ tr✉❡✳ ▲❡t✬s r❡❝❛st t❤❡ ♣r♦❜❧❡♠

❙✉♣♣♦s❡

• x ✐s t✐♠❡ ♠❡❛s✉r❡❞ ✐♥ s❡❝✳ • y = f (x) ✐s t❤❡ ❧♦❝❛t✐♦♥ ♦❢ t❤❡ ✜rst ♣❡rs♦♥ ✕ ♠❡❛s✉r❡❞ ✐♥ ❢t✳ • y = g(x) ✐s t❤❡ ❧♦❝❛t✐♦♥ ♦❢ t❤❡ s❡❝♦♥❞ ♣❡rs♦♥ ✕ ♠❡❛s✉r❡❞ ✐♥ ❢t✳

❚❤❡♥

❢t

• f ′ (x)

✐s t❤❡ ✈❡❧♦❝✐t② ♦❢ t❤❡ ✜rst ♣❡rs♦♥ ✕ ♠❡❛s✉r❡❞ ✐♥

• g ′ (x)

✐s t❤❡ ✈❡❧♦❝✐t② ♦❢ t❤❡ s❡❝♦♥❞ ♣❡rs♦♥ ✕ ♠❡❛s✉r❡❞ ✐♥

s❡❝

✳ ❢t

s❡❝

✳

❙✉♣♣♦s❡ t❤❡② ❛r❡ r✉♥♥✐♥❣ ✐♥ t✇♦ ♣❡r♣❡♥❞✐❝✉❧❛r ❞✐r❡❝t✐♦♥s ✭❡❛st ❛♥❞ ♥♦rt❤✮✳ ❲❡ ❞r❛✇ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❝❧✉s✐♦♥s✿

2 ✐s t❤❡ ❛r❡❛ ♦❢ t❤❡ r❡❝t❛♥❣❧❡ ❡♥❝❧♦s❡❞ ❜② t❤❡ t✇♦ ♣❡rs♦♥s ✕ ♠❡❛s✉r❡❞ ✐♥ ❢t ✳

• y = f (x) · g(x) ❚❤❡r❡❢♦r❡✱

• y = (f (x) · g(x)) • f (x)′ · g(x)′

′

✐s t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢ t❤❡ ❛r❡❛ ✕ ♠❡❛s✉r❡❞ ✐♥

✐s ❛♥ ✉♥❦♥♦✇♥ q✉❛♥t✐t② ✕ ♠❡❛s✉r❡❞ ✐♥

❢t s❡❝

·

❢t s❡❝

=

❢t

2

s❡❝ 2 ❢t ✦ s❡❝2

✳ ▼❡❛♥✇❤✐❧❡✱

❲❡ ❞♦ ♥♦t✐❝❡ ♥♦✇ t❤❛t t❤❡ ♣r♦❞✉❝t ♦❢ t❤❡ ❧♦❝❛t✐♦♥ ❛♥❞ ✈❡❧♦❝✐t② ❣✐✈❡s t❤❡ r✐❣❤t ✉♥✐ts✿ ′ ′ ✶✳ ♠✉❧t✐♣❧✐❡❞✿ f f, g g ✱ ❛♥❞ ′ ′ ✷✳ ❝r♦ss✲♠✉❧t✐♣❧✐❡❞✿ f g, g f ✳ ❲❡ ❝♦♥❝❧✉❞❡ t❤❛t ❛ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ s♦♠❡ ♦❢ t❤❡s❡ ♠✉st ♠❛❦❡ ✉♣ t❤❡ ❛♥s✇❡r✳

❚❤❡ ♣r♦❞✉❝t ♦❢ t✇♦ ❢✉♥❝t✐♦♥s ✐s ✐♥t❡r♣r❡t❡❞ ❛s t❤❡ ❜❡❧♦✇✿

❛r❡❛s ♦❢ t❤❡ r❡❝t❛♥❣❧❡s ❢♦r♠❡❞ ❜② t❤❡ ❢✉♥❝t✐♦♥s✱ ✐❧❧✉str❛t❡❞

✹✳✹✳ ❉✐✛❡r❡♥t✐❛t✐♦♥ ♦✈❡r ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❛♥❞ ❞✐✈✐s✐♦♥

✸✷✵

❆s t❤❡ ✇✐❞t❤ ❛♥❞ t❤❡ ❞❡♣t❤ ❛r❡ ✐♥❝r❡❛s✐♥❣✱ s♦ ✐s t❤❡ ❛r❡❛ ♦❢ t❤❡ r❡❝t❛♥❣❧❡✳ ❲❡ ❝❛♥ s❡❡ t❤❛t t❤❡ ✐♥❝r❡❛s❡ ♦❢ t❤❡ ❛r❡❛ ❝❛♥♥♦t ❜❡ ❡①♣r❡ss❡❞ ❡♥t✐r❡❧② ✐♥ t❡r♠s ♦❢ t❤❡ ✐♥❝r❡❛s❡s ♦❢ t❤❡ ✇✐❞t❤ ❛♥❞ ❞❡♣t❤✦ ❚❤✐s ✐♥❝r❡❛s❡ ✐s s♣❧✐t ✐♥t♦ t✇♦ ♣❛rts ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ t✇♦ t❡r♠s✿ ◮ ❲❤❡♥ ✇❡

♠✉❧t✐♣❧② ❢✉♥❝t✐♦♥s✱ ✇❡ ❝r♦ss✲♠✉❧t✐♣❧② t❤❡ ❢✉♥❝t✐♦♥s ❛♥❞ t❤❡✐r ❞❡r✐✈❛t✐✈❡s✳

❲❡ st❛rt ✇✐t❤ t❤❡ ❞✐✛❡r❡♥❝❡s✿

❚❤❡♦r❡♠ ✹✳✹✳✷✿ Pr♦❞✉❝t ❘✉❧❡ ❢♦r ❉✐✛❡r❡♥❝❡s ❚❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ ♣r♦❞✉❝t ♦❢ t✇♦ ❢✉♥❝t✐♦♥s ✐s ❢♦✉♥❞ ❛s t❤❡ s✉♠ ♦❢ t❤❡ ♣r♦❞✉❝t ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❛♥❞ t❤❡ ♦t❤❡r ❢✉♥❝t✐♦♥✬s ❞✐✛❡r❡♥❝❡✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ❢♦r ❛♥② t✇♦ ❢✉♥❝t✐♦♥s f, g ❞❡✜♥❡❞ ❛t t❤❡ ❛❞❥❛❝❡♥t ♥♦❞❡s x ❛♥❞ x + ∆x ♦❢ ❛ ♣❛rt✐t✐♦♥✱ t❤❡ ❞✐✛❡r❡♥❝❡s ✭❞❡✜♥❡❞ ❛t t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s❡❝♦♥❞❛r② ♥♦❞❡ c✮ s❛t✐s❢②✿ ∆(f · g)(c) = f (x + ∆x) · ∆g(c) + ∆f (c) · g(x)

Pr♦♦❢✳ ❚❤❡ tr✐❝❦ ✐s t♦ ✐♥s❡rt ❡①tr❛ t❡r♠s✿ ∆(f · g)(c) = (f · g)(x + ∆x) − (f · g)(x) = f (x + ∆x) · g(x + ∆x) − f (x) · g(x) = f (x + ∆x) · g(x + ∆x)−f (x + ∆x) · g(x) + f (x + ∆x) · g(x) − f (x) · g(x) = f (x + ∆x) · (g(x + ∆x) − g(x)) + (f (x + ∆x) − f (x)) · g(x) = f (x + ∆x) · ∆g(c) + ∆f (c) · g(x) .

◆♦✇✱ ❥✉st ❞✐✈✐❞❡ ❜② ∆x✿

❚❤❡♦r❡♠ ✹✳✹✳✸✿ Pr♦❞✉❝t ❘✉❧❡ ❢♦r ❉✐✛❡r❡♥❝❡ ◗✉♦t✐❡♥ts ❚❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ t❤❡ ♣r♦❞✉❝t ♦❢ t✇♦ ❢✉♥❝t✐♦♥s ✐s ❢♦✉♥❞ ❛s t❤❡ s✉♠ ♦❢ t❤❡ ♣r♦❞✉❝t ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❛♥❞ t❤❡ ♦t❤❡r ❢✉♥❝t✐♦♥✬s ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ❢♦r ❛♥② t✇♦ ❢✉♥❝t✐♦♥s f, g ❞❡✜♥❡❞ ❛t t❤❡ ❛❞❥❛❝❡♥t ♥♦❞❡s x ❛♥❞ x + ∆x ♦❢ ❛ ♣❛rt✐t✐♦♥✱ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ✭❞❡✜♥❡❞ ❛t t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s❡❝♦♥❞❛r② ♥♦❞❡ c✮ s❛t✐s❢②✿ ∆(f · g) ∆g ∆f (c) = f (x + ∆x) · (c) + (c) · g(x) ∆x ∆x ∆x

❲❡ ❛❧r❡❛❞② ❦♥♦✇ t❤❛t ✐❢ t❤❡ ✇✐❞t❤ ❛♥❞ t❤❡ ❤❡✐❣❤t ✭f ❛♥❞ g ✮ ♦❢ ❛ r❡❝t❛♥❣❧❡ ❛r❡ ❝❤❛♥❣✐♥❣ ❝♦♥t✐♥✉♦✉s❧②✱ t❤❡♥ s♦ ✐s ✐ts ❛r❡❛ ✭f · g ✮✿

✹✳✹✳ ❉✐✛❡r❡♥t✐❛t✐♦♥ ♦✈❡r ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❛♥❞ ❞✐✈✐s✐♦♥

✸✷✶

❲❡ s❤❛❧❧ ❛❧s♦ s❡❡ t❤❛t t❤❡ ❞✐✛❡r❡♥t✐❛❜✐❧✐t② ♦❢ ❜♦t❤ ❞✐♠❡♥s✐♦♥s ✐♠♣❧✐❡s t❤❡ ❞✐✛❡r❡♥t✐❛❜✐❧✐t② ♦❢ t❤❡ ❛r❡❛✱ ❛s ❢♦❧❧♦✇s✿ ❚❤❡♦r❡♠ ✹✳✹✳✹✿ Pr♦❞✉❝t ❘✉❧❡ ❢♦r ❉❡r✐✈❛t✐✈❡s

❚❤❡ ♣r♦❞✉❝t ♦❢ t✇♦ ❢✉♥❝t✐♦♥s ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t ❛ ♣♦✐♥t ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t t❤❛t ♣♦✐♥t ❛♥❞ ✐ts ❞❡r✐✈❛t✐✈❡ ✐s ❢♦✉♥❞ ❛s t❤❡ s✉♠ ♦❢ t❤❡ ♣r♦❞✉❝t ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❛♥❞ t❤❡ ♦t❤❡r ❢✉♥❝t✐♦♥✬s ❞❡r✐✈❛t✐✈❡✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ❢♦r ❛♥② t✇♦ ❢✉♥❝t✐♦♥s f, g ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t x✱ ✇❡ ❤❛✈❡✿ dg df d(f · g) (x) = f (x) · (x) + (x) · g(x) dx dx dx Pr♦♦❢✳

❚❤❡ ❧✐♠✐t ✇✐t❤ c = x✿ ∆g ∆f ∆(f · g)(x) = f (x + ∆x) · (c) + (c) · g(x) ∆x ∆x ∆x ↓ ↓ ↓ dg df f (x) · (x) + (x) · g(x) dx dx

❛s ∆x → 0 .

❚❤❡ ✜rst ❧✐♠✐t ✐s ❥✉st✐✜❡❞ ❜② t❤❡ ❢❛❝t t❤❛t f ✱ ❛s ❛ ❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥✱ ✐s ❝♦♥t✐♥✉♦✉s✳ ❚❤❡ ❢♦r♠✉❧❛ ✐♥ t❤❡ ▲❛❣r❛♥❣❡ ♥♦t❛t✐♦♥ ✐s ❛s ❢♦❧❧♦✇s✿ (f · g)′ (x) = f (x) · g ′ (x) + f ′ (x) · g(x)

■♥ s✉♠♠❛r②✱ t❤✐s ✐s ❤♦✇ t❤❡ ❝r♦ss✲♠✉❧t✐♣❧✐❝❛t✐♦♥ ✇♦r❦s✿ ✜rst s❡❝♦♥❞

❢✉♥❝t✐♦♥s ❞❡r✐✈❛t✐✈❡s f g

f′ g′

❊①❛♠♣❧❡ ✹✳✹✳✺✿ r♦✉t✐♥❡

▲❡t

y = xex .

−→ (f g)′ = f g ′ + f ′ g

✹✳✹✳

❉✐✛❡r❡♥t✐❛t✐♦♥ ♦✈❡r ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❛♥❞ ❞✐✈✐s✐♦♥

✸✷✷

❚❤❡♥✱ ✇❡ ❤❛✈❡✿

du = (x)′ = 1, dx dv =⇒ = (ex )′ = ex . dx

u =x

=⇒

v = ex ❆♣♣❧②

Pr♦❞✉❝t ❘✉❧❡

✈✐❛ ✏❝r♦ss✲♠✉❧t✐♣❧✐❝❛t✐♦♥✑✱ t❤❡ ✐❞❡❛ ♦❢ ✇❤✐❝❤ ❝♦♠❡s ❢r♦♠ t❤❡ ♣✐❝t✉r❡ ❛❜♦✈❡✿

dy = x · ex + 1 · ex = ex (x + 1) . dx ◆❡①t✱ t❤❡ ❞❡r✐✈❛t✐✈❡s ✉♥❞❡r

❞✐✈✐s✐♦♥✳

❊①❛♠♣❧❡ ✹✳✹✳✻✿ ✉♥✐t ❛♥❛❧②s✐s ▲❡t✬s ✜rst ♠❛❦❡ s✉r❡ ✇❡ ❛✈♦✐❞ t❤❡ s♦✲❝❛❧❧❡❞ ✏◆❛✐✈❡ ◗✉♦t✐❡♥t ❘✉❧❡✑✿

❄❄❄

(f /g)′ === f ′ /g ′ . ❲❡ ❝❛♥ r❡♣❡❛t t❤❡ ✏✉♥✐t ❛♥❛❧②s✐s✑ t♦ s❤♦✇ t❤❛t s✉❝❤ ❛ ❢♦r♠✉❧❛ s✐♠♣❧②

❝❛♥♥♦t

❜❡ tr✉❡✳ ❚❤❡ r✉♥♥❡rs

st✐❧❧ ❛r❡ r✉♥♥✐♥❣ ✐♥ t✇♦ ♣❡r♣❡♥❞✐❝✉❧❛r ❞✐r❡❝t✐♦♥s✱ ❛♥❞ ✇❡ ❤❛✈❡✿

• y = f (x)/g(x)

✐s ✉♥✐t❧❡ss✱ ❛♥❞ t❤❡♥

• y = (f (x)/g(x))′ ′

• f (x) /g(x)

′

✐s ♠❡❛s✉r❡❞ ✐♥

1

s❡❝

✱ ✇❤✐❧❡

✐s ✉♥✐t❧❡ss✦

❚❤❡ ❣❡♥❡r❛❧ ❝❛s❡ ♦❢ ❞✐✈✐s✐♦♥ ✐s ♠♦r❡ ❝♦♠♣❧❡①✿

❚❤❡♦r❡♠ ✹✳✹✳✼✿ ◗✉♦t✐❡♥t ❘✉❧❡ ❢♦r ❉✐✛❡r❡♥❝❡s ❋♦r ❛♥② t✇♦ ❢✉♥❝t✐♦♥s f, g ❞❡✜♥❡❞ ❛t t❤❡ ❛❞❥❛❝❡♥t ♥♦❞❡s x ❛♥❞ x + ∆x ♦❢ ❛ ♣❛rt✐t✐♦♥✱ t❤❡ ❞✐✛❡r❡♥❝❡s ✭❞❡✜♥❡❞ ❛t t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s❡❝♦♥❞❛r② ♥♦❞❡ c✮ s❛t✐s❢②✿ ∆(f /g)(c) =

f (x + ∆x) · ∆g(c) − ∆f (c) · g(x) g(x)g(x + ∆x)

Pr♦♦❢✳ ❲❡ st❛rt ✇✐t❤ t❤❡ ❝❛s❡

f = 1✳

❚❤❡♥ ✇❡ ❤❛✈❡✿

1 1 − g(x + ∆x) g(x) g(x) − g(x + ∆x) . = g(x + ∆x)g(x)

∆(1/g)(x) =

◆♦✇ t❤❡ ❣❡♥❡r❛❧ ❢♦r♠✉❧❛ ❢♦❧❧♦✇s ❢r♦♠ t❤❡

Pr♦❞✉❝t ❘✉❧❡✳

❚❤❡♦r❡♠ ✹✳✹✳✽✿ ◗✉♦t✐❡♥t ❘✉❧❡ ❢♦r ❉✐✛❡r❡♥❝❡ ◗✉♦t✐❡♥ts ❋♦r ❛♥② t✇♦ ❢✉♥❝t✐♦♥s f, g ❞❡✜♥❡❞ ❛t t❤❡ ❛❞❥❛❝❡♥t ♥♦❞❡s x ❛♥❞ x + ∆x ♦❢ ❛ ♣❛rt✐t✐♦♥✱ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ✭❞❡✜♥❡❞ ❛t t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s❡❝♦♥❞❛r② ♥♦❞❡ c✮ s❛t✐s❢②✿ ∆g (c) − ∆f (c) · g(x) f (x + ∆x) · ∆x ∆(f /g) ∆x (c) = ∆x g(x) · g(x + ∆x)

✹✳✹✳ ❉✐✛❡r❡♥t✐❛t✐♦♥ ♦✈❡r ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❛♥❞ ❞✐✈✐s✐♦♥

✸✷✸

♣r♦✈✐❞❡❞ g(x), g(x + ∆x) 6= 0✳ Pr♦♦❢✳

❲❡ st❛rt ✇✐t❤ t❤❡ ❝❛s❡ f = 1✳ ❚❤❡♥ ✇❡ ❤❛✈❡✿ g(x) − g(x + ∆x) ∆(1/g)(x) = ∆x ∆xg(x + ∆x)g(x) g(x + ∆x) − g(x) 1 · =− ∆x g(x + ∆x) · g(x) 1 ∆g =− (c) · ∆x g(x + ∆x) · g(x)

✇✐t❤ c = x .

◆♦✇ t❤❡ ❣❡♥❡r❛❧ ❢♦r♠✉❧❛ ❢♦❧❧♦✇s ❢r♦♠ t❤❡ Pr♦❞✉❝t ❘✉❧❡✳

❲❡ ❛❧r❡❛❞② ❦♥♦✇ t❤❛t ✐❢ t❤❡ ✇✐❞t❤ ❛♥❞ t❤❡ ❤❡✐❣❤t ✭f ❛♥❞ g ✮ ♦❢ ❛ tr✐❛♥❣❧❡ ❛r❡ ❝❤❛♥❣✐♥❣ ❝♦♥t✐♥✉♦✉s❧②✱ t❤❡♥ s♦ ✐s t❤❡ t❛♥❣❡♥t ♦❢ ✐ts ❜❛s❡ ❛♥❣❧❡ ✭f /g ✮✿

❲❡ s❤❛❧❧ ❛❧s♦ s❡❡ t❤❛t t❤❡ ❞✐✛❡r❡♥t✐❛❜✐❧✐t② ♦❢ ❡✐t❤❡r ❞✐♠❡♥s✐♦♥ ✐♠♣❧✐❡s t❤❡ ❞✐✛❡r❡♥t✐❛❜✐❧✐t② ♦❢ t❤❡ t❛♥❣❡♥t✱ ❛s ❢♦❧❧♦✇s✿ ❚❤❡♦r❡♠ ✹✳✹✳✾✿ ◗✉♦t✐❡♥t ❘✉❧❡ ❢♦r ❉❡r✐✈❛t✐✈❡s

❋♦r ❛♥② t✇♦ ❢✉♥❝t✐♦♥s f, g ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t x✱ ✇❡ ❤❛✈❡✿ f (x) · d(f /g) (x) = dx

df − dx (x) · g(x) 2 g(x)

dg (x) dx

♣r♦✈✐❞❡❞ g(x) 6= 0✳ Pr♦♦❢✳

❲❡ st❛rt ✇✐t❤ t❤❡ ❝❛s❡ f = 1✳ ❚❤❡♥ ✇❡ ❤❛✈❡✿ ∆(1/g)(x) ∆g 1 =− (c) · ∆x ∆x g(x + ∆x) · g(x) dg 1 → − (x) · dx g(x) · g(x)

✇✐t❤ c = x ❛s ∆x → 0 .

❚❤❡ ❧✐♠✐t ♦❢ t❤❡ s❡❝♦♥❞ ❢r❛❝t✐♦♥ ✐s ❥✉st✐✜❡❞ ❜② t❤❡ ❢❛❝t t❤❛t g ✱ ❛s ❛ ❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥✱ ✐s ❝♦♥t✐♥✉♦✉s✳ ❆❧t❡r♥❛t✐✈❡❧②✱ ✇❡ r❡♣r❡s❡♥t t❤❡ r❡❝✐♣r♦❝❛❧ ♦❢ g ❛s ❛ ❝♦♠♣♦s✐t✐♦♥✿ z=

1 dz 1 dy dz 1 ′ 1 g (x) , =⇒ z = , y = g(x) =⇒ = − 2, = g ′ (x) =⇒ =− g(x) y dy y dx dx g(x)2

❜② t❤❡ ❈❤❛✐♥ ❘✉❧❡✳ ◆♦✇ t❤❡ ❣❡♥❡r❛❧ ❢♦r♠✉❧❛ ❢♦❧❧♦✇s ❢r♦♠ t❤❡ Pr♦❞✉❝t ❘✉❧❡✳

✹✳✹✳ ❉✐✛❡r❡♥t✐❛t✐♦♥ ♦✈❡r ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❛♥❞ ❞✐✈✐s✐♦♥

✸✷✹

❚❤❡ ❢♦r♠✉❧❛ ✐♥ t❤❡ ▲❛❣r❛♥❣❡ ♥♦t❛t✐♦♥ ✐s ❛s ❢♦❧❧♦✇s✿ 

f (x) g(x)

′

=

f ′ (x) · g(x) − f (x) · g ′ (x) g(x)2

❚❤❡ ❢♦r♠✉❧❛ ✐s s✐♠✐❧❛r t♦ t❤❡ Pr♦❞✉❝t ❘✉❧❡ ✐♥ t❤❡ s❡♥s❡ t❤❛t ✐t ❛❧s♦ ✐♥✈♦❧✈❡s ❝r♦ss✲♠✉❧t✐♣❧✐❝❛t✐♦♥ ✿ ✜rst s❡❝♦♥❞

❢✉♥❝t✐♦♥s ❞❡r✐✈❛t✐✈❡s ′

f g

f g′

 ′ f f g′ − f ′g −→ = g g2

❊①❛♠♣❧❡ ✹✳✹✳✶✵✿ ❞❡r✐✈❛t✐✈❡ ♦❢ t❛♥❣❡♥t

(tan x)

′

′ sin x = cos x ◗❘ (sin x)′ cos x − sin x(cos x)′ === (cos x)2 cos x cos x − sin x(− sin x) = cos2 x 2 cos x + sin2 x = ❯s❡ t❤❡ P②t❤❛❣♦r❡❛♥ ❚❤❡♦r❡♠✳ cos2 x 1 = ❚❤✐s ✐s sec2 x . cos2 x 

■♥ t❤❡ ▲❡✐❜♥✐③ ♥♦t❛t✐♦♥✱ t❤✐s ✐s t❤❡ ❢♦r♠ ♦❢ t❤❡ Pr♦❞✉❝t ❘✉❧❡ ✿ d du dv (uv) = ·v+ ·u dx dx dx

❛♥❞ t❤❡ ◗✉♦t✐❡♥t ❘✉❧❡✿ dv du ·v− ·u d u dx = dx dx v v2 ❊①❛♠♣❧❡ ✹✳✹✳✶✶✿ P♦✇❡r ❋♦r♠✉❧❛ ♣r♦♦❢

■♥ t❤❡ ❧❛st ❝❤❛♣t❡r✱ ✇❡ ♣r♦✈❡❞ t❤❡ P♦✇❡r ❋♦r♠✉❧❛ ❢♦r ♣♦s✐t✐✈❡ ✐♥t❡❣❡rs✿ (xm )′ = mxm−1 .

▲❡t✬s ♣r♦✈❡ t❤❡ ♥❡❣❛t✐✈❡ ✐♥t❡❣❡rs✳ ❲❡ tr❡❛t t❤❡ ❢✉♥❝t✐♦♥ ❛s ❛ ❢r❛❝t✐♦♥✿ x−m =

1 . xm

❲❡ ❞✐✛❡r❡♥t✐❛t❡ t❤❡ ♥✉♠❡r❛t♦r ❛♥❞ t❤❡ ❞❡♥♦♠✐♥❛t♦r✿ u=1 =⇒ u′ = 0 v = xm =⇒ v ′ = mxm−1 ❆❝❝♦r❞✐♥❣ t♦ t❤❡ Pr♦❞✉❝t ❋♦r♠✉❧❛✳

✹✳✹✳ ❉✐✛❡r❡♥t✐❛t✐♦♥ ♦✈❡r ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❛♥❞ ❞✐✈✐s✐♦♥

✸✷✺

❲❡ ❛♣♣❧② t❤❡ ◗✉♦t✐❡♥t ❘✉❧❡ ♥♦✇✿ (x

′ 1 = xm 0 · xm − 1 · mxm−1 = (xm )2 m−1 mx =− m 2 (x ) xm−1 = −m 2m x = −mx−m−1 . 

−m ′

)

■❢ ✇❡ r❡♣❧❛❝❡ −m ✇✐t❤ n✱ ✇❡ ❤❛✈❡ t❤❡ ❢❛♠✐❧✐❛r ❢♦r♠✉❧❛✿ (xn )′ = nxn−1 . ❊①❛♠♣❧❡ ✹✳✹✳✶✷✿ r❛t✐♦

❋✐♥❞

 √

x 2 x +1

❈♦♥s✐❞❡r✿

◆♦ ♥❡❡❞ t♦ s✐♠♣❧✐❢②✳

.

du 1 = √ dx 2 x dv = 2x v = x2 + 1 =⇒ dx u =

❚❤❡♥✱

√

′

x

=⇒

√ 1 √ (x2 + 1) − x · 2x   d u 2 x = . dx v (x2 + 1)2

❊①❛♠♣❧❡ ✹✳✹✳✶✸✿ ❢r♦♠ ❧✐♠✐t t♦ ❞❡r✐✈❛t✐✈❡

❚❤✐s ✐s ❛ ❞✐✛❡r❡♥t ❦✐♥❞ ♦❢ ❡①❛♠♣❧❡✳ ❊✈❛❧✉❛t❡✿ 2x − 32 lim . x→5 x − 5

■t✬s ❥✉st ❛ ❧✐♠✐t✳ ❇✉t ✇❡ r❡❝♦❣♥✐③❡ t❤❛t t❤✐s ✐s t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ s♦♠❡ ❢✉♥❝t✐♦♥✳ ❲❡ ❝♦♠♣❛r❡ t❤❡ ❡①♣r❡ss✐♦♥ t♦ t❤❡ ❢♦r♠✉❧❛ ✐♥ t❤❡ ❞❡✜♥✐t✐♦♥✿ f (x) − f (a) , x→a x−a

f ′ (a) = lim

❛♥❞ ♠❛t❝❤✳ ❙♦✱ ✇❡ ❤❛✈❡ ❤❡r❡✿

a = 5, f (x) = 2x , f (5) = 25 = 32 .

❚❤❡r❡❢♦r❡✱ ♦✉r ❧✐♠✐t ✐s ❡q✉❛❧ t♦ f ′ (5) ❢♦r f (x) = 2x ✳ ❈♦♠♣✉t❡✿ f ′ (x) = (2x )′ = 2x ln 2 ,

s♦

2x − 32 = f ′ (5) = 25 ln 2 = 32 ln 2 . x→5 x − 5 lim

✹✳✺✳

❚❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢ t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡

✸✷✻

✹✳✺✳ ❚❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢ t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡

❲❡ s❛✇ ✐♥ ❈❤❛♣t❡r ✸ ❤♦✇ t♦ ❞❡r✐✈❡ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ❢r♦♠ t❤❡ ❧♦❝❛t✐♦♥✱ ❛s t✇♦ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts✿

❉◗

❉◗

❧♦❝❛t✐♦♥ −−−−−→ ✈❡❧♦❝✐t② −−−−−→ ❛❝❝❡❧❡r❛t✐♦♥

❋✉rt❤❡r♠♦r❡✱ ✐❢ ❛ ❢✉♥❝t✐♦♥ ✐s ❦♥♦✇♥ ❛t t❤❡ ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥✱ ✐ts ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✐s ❛❧s♦ ❛ ❢✉♥❝t✐♦♥ ✕ ❦♥♦✇♥ ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s✳ ❈❛♥ ✇❡ tr❡❛t t❤❡ ❧❛tt❡r ❛s ❛ ❢✉♥❝t✐♦♥ t♦♦❄ ❊①❛♠♣❧❡ ✹✳✺✳✶✿ ❛❝❝❡❧❡r❛t✐♦♥

❙✉♣♣♦s❡ t❤❡ t✐♠❡ ✐s ✐♥❝r❡❛s✐♥❣ ❜② 1 s♦ t❤❛t ✇❡ ♦♥❧② ♥❡❡❞ t♦ ❧♦♦❦ ❛t t❤❡ ❞✐✛❡r❡♥❝❡s✳ ❲❡ ♣r♦❣r❡ss ❢r♦♠ t❤❡ ❧♦❝❛t✐♦♥s ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥ ♦❢ t❤❡ t✐♠❡ ❧✐♥❡ t♦ t❤❡ ✈❡❧♦❝✐t✐❡s ❞❡✜♥❡❞ ♦♥ t❤❡ ❡❞❣❡s t♦ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥s ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s ❛❣❛✐♥✿ ❧♦❝❛t✐♦♥✿ − 2 −−− 5 −−− 10 − ✈❡❧♦❝✐t②✿ − − • − 5 − 3 = 3 −•− 10 − 5 = 5 − • − − ❛❝❝❡❧❡r❛t✐♦♥✿ − −?− −−− 5−3=2 −−− −?− − t✐♠❡✿ 1.0 1.5 2.0 2.5 3.0 ●❡♥❡r❛❧❧②✱ ✐❢ ✇❡ ❦♥♦✇ ♦♥❧② t❤r❡❡ ✈❛❧✉❡s ♦❢ ❛ ❢✉♥❝t✐♦♥ ✭✜rst ❧✐♥❡✮✱ ✇❡ ❝❛♥ ❝♦♠♣✉t❡ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ❛❧♦♥❣ t❤❡ t✇♦ ✐♥t❡r✈❛❧s ✭s❡❝♦♥❞ ❧✐♥❡✮✱ ❛♥❞ t❤❡♥ ♣❧❛❝❡ t❤❡ r❡s✉❧ts ❛t t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❡❞❣❡✿ ❢✉♥❝t✐♦♥✿ − f (x1 ) − − − ∆f ❉◗✿ − − • − ∆x2 ❉◗ ♦❢ ❉◗✿ −

−?−

✈❛r✐❛❜❧❡✿

x1

−−−

f (x2 ) −•−

∆f ∆x3

c2

−

∆f ∆x2

c3 − c2 x2

− − − f (x3 ) − ∆f −•− − ∆x3 −−− c3

−?− x3

−

❚♦ ✜♥❞ t❤❡ ❝❤❛♥❣❡ ♦❢ t❤✐s ♥❡✇ ❢✉♥❝t✐♦♥✱ ✇❡ ❝❛rr② ♦✉t t❤❡ s❛♠❡ ♦♣❡r❛t✐♦♥ ❛♥❞ ♣❧❛❝❡ t❤❡ r❡s✉❧t ✐♥ t❤❡ ♠✐❞❞❧❡ ✭t❤✐r❞ ❧✐♥❡✮✳ ▲❡t✬s r❡✈✐❡✇ t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✳ ❋✐rst✱ ✇❡ ❤❛✈❡ ❛♥ ❛✉❣♠❡♥t❡❞ ♣❛rt✐t✐♦♥ ♦❢ ❛♥ ✐♥t❡r✈❛❧ [a, b]✳ ❲❡ ♣❛rt✐t✐♦♥ ✐t ✐♥t♦ n ✐♥t❡r✈❛❧s ✇✐t❤ t❤❡ ❤❡❧♣ ♦❢ t❤❡ ♥♦❞❡s ✭t❤❡ ❡♥❞✲♣♦✐♥ts ♦❢ t❤❡ ✐♥t❡r✈❛❧s✮✿

a = x0 , x1 , x2 , ..., xn−1 , xn = b . ❲❡ ❛❧s♦ ♣r♦✈✐❞❡ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s✿

c1 ✐♥ [x0 , x1 ], c2 ✐♥ [x1 , x2 ], ..., cn ✐♥ [xn−1 , xn ] . ❆ ❢✉♥❝t✐♦♥ ✐s ❞❡✜♥❡❞ ♦♥ t❤❡ ❢♦r♠❡r✱ ❛♥❞ ✐ts ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦♥ t❤❡ ❧❛tt❡r✿

✹✳✺✳ ❚❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢ t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡

✸✷✼

❙✉♣♣♦s❡ ❛ ❢✉♥❝t✐♦♥ y = f (x) ✐s ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s xk , k = 0, 1, 2, ..., n✳ ❚❤❡♥ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ f ✐s ❞❡✜♥❡❞ ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣✿ ∆f f (xk+1 ) − f (xk ) (ck ) = ∆x xk+1 − xk

❢♦r ❡❛❝❤ k = 1, 2, ..., n ✳ ❚❤❡ ♥❡✇ ❢✉♥❝t✐♦♥ r❡♣r❡s❡♥ts t❤❡ s❧♦♣❡s ♦❢ t❤❡ s❡❝❛♥t ❧✐♥❡s ♦✈❡r t❤❡ ♥♦❞❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥✿

■t ✐s ♥♦✇ ❡s♣❡❝✐❛❧❧② ✐♠♣♦rt❛♥t t❤❛t ✇❡ ❤❛✈❡ ✉t✐❧✐③❡❞ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ❛s t❤❡ ✐♥♣✉ts ♦❢ t❤❡ ♥❡✇ ❢✉♥❝t✐♦♥✳ ■♥❞❡❡❞✱ ✇❡ ❝❛♥ ♥♦✇ ❝❛rr② ♦✉t ❛ s✐♠✐❧❛r ❝♦♥str✉❝t✐♦♥ ✇✐t❤ t❤✐s ♥❡✇ ❢✉♥❝t✐♦♥ ❛♥❞ ✜♥❞ t❤❡ t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢ t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡✳ ❲❡ ❛r❡ ✐♥ t❤❡ s❛♠❡ ♣♦s✐t✐♦♥ ❛s ✇❤❡♥ ✇❡ st❛rt❡❞✿ ❲❡ ❤❛✈❡ ♥♦✇ ❛ ♥❡✇ ❛✉❣♠❡♥t❡❞ ♣❛rt✐t✐♦♥ ✿

❚❤❡ ❝♦♠♣❧❡t❡ ❝♦♥str✉❝t✐♦♥ ✐s ❛s ❢♦❧❧♦✇s✳ ❚❤❡ ✐♥t❡r✈❛❧ ✐s [p, q], ✇✐t❤ p = c0 ❛♥❞ q = cn .

❲❡ ♣❛rt✐t✐♦♥ ✐t ✐♥t♦ n − 1 ✐♥t❡r✈❛❧s ✇✐t❤ t❤❡ ❤❡❧♣ ♦❢ t❤❡ ♥♦❞❡s t❤❛t ✉s❡❞ t♦ ❜❡ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ✐♥ t❤❡ ♦r✐❣✐♥❛❧ ♣❛rt✐t✐♦♥✿ p = c1 , c2 , c3 , ..., cn−1 , cn = b .

❚❤❡♥ t❤❡ ✐♥❝r❡♠❡♥ts ❛r❡✿ ∆ck = ck+1 − ck .

◆♦✇✱ ✇❤❛t ❛r❡ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s❄ ❚❤❡ ♣r✐♠❛r② ♥♦❞❡s ♦❢ t❤❡ ❧❛st ♣❛rt✐t✐♦♥ ♦❢ ❝♦✉rs❡✦ ■♥❞❡❡❞✱ ✇❡ ❤❛✈❡✿ x1 ✐♥ [c1 , c2 ], x2 ✐♥ [c2 , c3 ], ..., xn−1 ✐♥ [cn−1 , cn ] .

✹✳✺✳ ❚❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢ t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡

✸✷✽

❲❡ ❛♣♣❧② t❤❡ s❛♠❡ ❝♦♥str✉❝t✐♦♥ t♦ t❤❡ ❢✉♥❝t✐♦♥ g=

∆f . ∆x

❚❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ g ✐s ❞❡✜♥❡❞ ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ♦❢ t❤❡ ♥❡✇ ♣❛rt✐t✐♦♥ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣✿ g(ck+1 ) − g(ck ) ∆g (xk ) = ∆x ck+1 − ck

❢♦r ❡❛❝❤ k = 1, 2, ..., n − 1 ✳ ■t ✐s ✈✐s✉❛❧✐③❡❞ ❛s ❢♦❧❧♦✇s✿

❲❡ ♣✉t t❤❡ t✇♦ ❢♦r♠✉❧❛s t♦❣❡t❤❡r✿

❉❡✜♥✐t✐♦♥ ✹✳✺✳✷✿ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❚❤❡ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ ❛ ❢✉♥❝t✐♦♥ f ❞❡✜♥❡❞ ❛t t❤❡ ♣r✐♠❛r② ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t❀ ✐✳❡✳✱ ✐t ✐s ❞❡✜♥❡❞ ❛t t❤❡ ♣r✐♠❛r② ♥♦❞❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥ ✭❛♥❞ ❞❡♥♦t❡❞✮ ❛s ❢♦❧❧♦✇s✿ ∆2 f (xk ) = ∆x2

− ∆f (c ) ∆x k ck+1 − ck

∆f (c ) ∆x k+1

❢♦r ❡❛❝❤ k = 1, 2, ..., n − 1 ✳ ❚❤❡r❡ ❛r❡✿ • n + 1 ✈❛❧✉❡s ♦❢ f ✭❛t t❤❡ ♣r✐♠❛r② ♥♦❞❡s✮✱

∆f ✭❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s✮✱ ❛♥❞ ∆x ∆2 f ✭❛t t❤❡ ♣r✐♠❛r② ♥♦❞❡s ❡①❝❡♣t a ❛♥❞ b✮✳ • n − 1 ✈❛❧✉❡s ♦❢ ∆x2

• n ✈❛❧✉❡s ♦❢

❲❡ ✇✐❧❧ ♦❢t❡♥ ♦♠✐t t❤❡ s✉❜s❝r✐♣ts ❢♦r t❤❡ s✐♠♣❧✐✜❡❞ ♥♦t❛t✐♦♥✿

❙❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ∆2 f (x) = ∆x2

∆f (c ∆x

+ ∆c) − ∆c

∆f (c) ∆x

❊①❛♠♣❧❡ ✹✳✺✳✸✿ ❝✉r✈❛t✉r❡ ❆s ✇❡ ❦♥♦✇✱ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ ❛ ❧✐♥❡❛r ❢✉♥❝t✐♦♥ ✐s ❝♦♥st❛♥t✳ ❚❤❡ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✐s✱ t❤❡r❡❢♦r❡✱ ③❡r♦✳ ❲❡ ❝♦♥❝❧✉❞❡ t❤❛t ❛ ♥♦♥✲③❡r♦ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✐♥❞✐❝❛t❡s ❛ ♥♦♥✲❧✐♥❡❛r ❣r❛♣❤✿

✹✳✺✳

❚❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢ t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡

✸✷✾

❆❜♦✈❡✱ t❤❡ s❧♦♣❡s r❡♠❛✐♥ t❤❡ s❛♠❡✱ 2✱ ❛t ✜rst❀ t❤❡r❡❢♦r❡✱ t❤❡ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✐s ③❡r♦✿ ∆2 f = 0. ∆x2

❚❤❡♥✱ t❤❡ s❧♦♣❡ ❝❤❛♥❣❡s t♦ 1✱ ❛♥❞ t❤✐s ❝❤❛♥❣❡ ✐s t❤❡ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✭❛ss✉♠✐♥❣ ∆x = 1✮✿ ∆2 f = −1 . ∆x2

❆s ❛♥♦t❤❡r ✇❛② t♦ s❡❡ t❤✐s ✐❞❡❛✱ ✐♠❛❣✐♥❡ ②♦✉rs❡❧❢ ❞r✐✈✐♥❣ ❛❧♦♥❣ ❛ str❛✐❣❤t ♣❛rt ♦❢ t❤❡ r♦❛❞ ❛♥❞ s❡❡✐♥❣ ❛ ♣❛rt✐❝✉❧❛r tr❡❡ ❛❤❡❛❞ ✭♥♦ ❝✉r✈❛t✉r❡✮✱ t❤❡♥✱ ❛s ②♦✉ st❛rt t♦ t✉r♥✱ t❤❡ tr❡❡s st❛rt t♦ ♣❛ss ②♦✉r ✜❡❧❞ ♦❢ ✈✐s✐♦♥ ❢r♦♠ r✐❣❤t t♦ ❧❡❢t ✭❝✉r✈❛t✉r❡✮✿

❋✉rt❤❡r♠♦r❡✱ ❤✐❣❤❡r ✈❛❧✉❡s ♦❢ t❤❡ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♠❡❛♥ ❤✐❣❤❡r y = f (x)✳

❝✉r✈❛t✉r❡ ♦❢ t❤❡ ❣r❛♣❤ ♦❢

❊①❛♠♣❧❡ ✹✳✺✳✹✿ s✐♥❡ ❛♥❞ ❝♦s✐♥❡

▲❡t✬s ❞✐✛❡r❡♥t✐❛t❡ sin x ❢♦r t❤❡ s❡❝♦♥❞ t✐♠❡✳ ■♥ ❈❤❛♣t❡r ✸✱ ✇❡ ❢♦✉♥❞ ✐ts ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦✈❡r ❛ ♠✐❞✲♣♦✐♥t ♣❛rt✐t✐♦♥ ✇✐t❤ ❛ s✐♥❣❧❡ ✐♥t❡r✈❛❧✳ ❚❤✐s t✐♠❡ ✇❡ ✇✐❧❧ ♥❡❡❞ ❛t ❧❡❛st t✇♦ ✐♥t❡r✈❛❧s✿ • t❤r❡❡ ♥♦❞❡s x✿ a − h✱ a✱ ❛♥❞ a + h✱ ❛♥❞ • t✇♦ s❡❝♦♥❞❛r② ♥♦❞❡s c✿ a − h/2 ❛♥❞ a + h/2✳ ❲❡ ✉s❡ t❤❡ t✇♦ ❢♦r♠✉❧❛s ❢♦r t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ♦❢ sin x ❛♥❞ cos x ❢r♦♠ ❈❤❛♣t❡r ✸✳ ❲❡ ✇r✐t❡ t❤❡ ❢♦r♠❡r ❢♦r t❤❡ t✇♦ s❡❝♦♥❞❛r② ♥♦❞❡s✱ ❜✉t ✇❡ r❡✇r✐t❡ t❤❡ ❧❛tt❡r ❢♦r t❤❡ ♣❛rt✐t✐♦♥ ✇✐t❤ t✇♦ ♥♦❞❡s a − h/2, a + h/2 ❛♥❞ ❛ s✐♥❣❧❡ s❡❝♦♥❞❛r② ♥♦❞❡ x = a✿ sin(h/2) ∆ sin(h/2) ∆ (sin x) = · cos c , (cos x) = − · sin a . ∆x h/2 ∆x h/2

✹✳✺✳

❚❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢ t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡

❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡ ❛t

✸✸✵

a✿

  ∆ ∆ ∆2 (sin x) = (sin x) (a) ∆x2 ∆x ∆x   ∆ sin(h/2) = · cos c ∆x h/2 sin(h/2) ∆ cos (a) = h/2 ∆x   sin(h/2) sin(h/2) = − · sin a h/2 h/2 2  sin(h/2) · sin a . =− h/2

❆❝❝♦r❞✐♥❣ t♦ t❤❡ ✜rst ❢♦r♠✉❧❛✳

❆❝❝♦r❞✐♥❣ t♦ ❈▼❘✳

❆❝❝♦r❞✐♥❣ t♦ t❤❡ s❡❝♦♥❞ ❢♦r♠✉❧❛✳

■t✬s ❥✉st t❤❡ ♦r✐❣✐♥❛❧ ❢✉♥❝t✐♦♥ ✉♣s✐❞❡ ❞♦✇♥ ❛♥❞ str❡t❝❤❡❞✦ ❙✐♠✐❧❛r❧②✱ ✇❡ ✜♥❞✿

∆ sin(h/2) ∆2 (cos x) = − (cos x) = − · sin c =⇒ ∆x h/2 ∆x2



sin(h/2) h/2

2

· cos a .

❊①❛♠♣❧❡ ✹✳✺✳✺✿ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥ ❋♦r t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥✱ ✇❡ ♥❡❡❞ ❛ ❧❡❢t✲❡♥❞ ♣❛rt✐t✐♦♥ ✇✐t❤ t✇♦ ✐♥t❡r✈❛❧s✿

• •

x✿ a − h✱ a✱ ❛♥❞ a + h✱ ❛♥❞ t✇♦ s❡❝♦♥❞❛r② ♥♦❞❡s c✿ a − h ❛♥❞ a✳ ❚❤❡♥✱ ✇❡ ✜♥❞ ❛t a✿ t❤r❡❡ ♥♦❞❡s

eh − 1 −h/2 c ∆2 x ∆ x (e ) = (e ) = e · e =⇒ ∆x h ∆x2



eh − 1 −h/2 e h

2

· ea .

■t✬s ❥✉st t❤❡ ♦r✐❣✐♥❛❧ ❢✉♥❝t✐♦♥ str❡t❝❤❡❞ ✈❡rt✐❝❛❧❧②✦

❚❤✐s ❝♦♥str✉❝t✐♦♥ ✇✐❧❧ ❜❡ r❡♣❡❛t❡❞❧② ✉s❡❞ ❢♦r ❛♣♣r♦①✐♠❛t✐♦♥s ❛♥❞ s✐♠✉❧❛t✐♦♥s✳

■t ✇✐❧❧ ❜❡ ❢♦❧❧♦✇❡❞✱ ✇❤❡♥

♥❡❝❡ss❛r②✱ ❜② t❛❦✐♥❣ ✐ts ❧✐♠✐t✿

❉❡✜♥✐t✐♦♥ ✹✳✺✳✻✿ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡ ❚❤❡

s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ f ✐s ❞❡✜♥❡❞ t♦ ❜❡ ❧✐♠✐t ♦❢ t❤❡ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡

q✉♦t✐❡♥t✿

∆2 f d2 f (x) = lim (xk ) = lim ∆x→0 ∆x2 ∆x→0 dx2

− ∆f (c ) ∆x k ck+1 − ck

∆f (c ) ∆x k+1

❲❡ ❛❝❝❡♣t t❤❡ ❢♦❧❧♦✇✐♥❣ ✇✐t❤♦✉t ♣r♦♦❢✿

❚❤❡♦r❡♠ ✹✳✺✳✼✿ ❙❡❝♦♥❞ ❉❡r✐✈❛t✐✈❡ ❚❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡ ✐s t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡✿

d2 f = (f ′ )′ dx2

❆s ✇❡ r❡♣❡❛t❡❞❧② ❞✐✛❡r❡♥t✐❛t❡ t❤❡ s❛♠❡ ❢✉♥❝t✐♦♥

f✱

✇❡ ❥✉st ❛❞❞ ♠♦r❡ ❛♥❞ ♠♦r❡ ✏♣r✐♠❡s✑✿

f ′′ , f ′′′ , f ′′′′ , ...

✹✳✻✳ ❘❡♣❡❛t❡❞ ❞✐✛❡r❡♥t✐❛t✐♦♥

✸✸✶

✹✳✻✳ ❘❡♣❡❛t❡❞ ❞✐✛❡r❡♥t✐❛t✐♦♥ ❊①❛♠♣❧❡ ✹✳✻✳✶✿

sin

▲❡t✬s ❝♦♥t✐♥✉❡ t♦ ❞✐✛❡r❡♥t✐❛t❡ t❤❡ s✐♥❡✿ (sin x)′ (cos x)′ (− sin x)′ (− cos x)′

= cos x = − sin x =⇒ (sin x)′′ = − sin x = − cos x =⇒ (sin x)′′′ = − cos x = sin x =⇒ (sin x)′′′′ = sin x

❆♥❞ ✇❡ ❛r❡ ❜❛❝❦ ✇❤❡r❡ ✇❡ st❛rt❡❞✱ ✐✳❡✳✱ t❤❡ ❞✐✛❡r❡♥t✐❛t✐♦♥ ♣r♦❝❡ss ❢♦r t❤✐s ♣❛rt✐❝✉❧❛r ❢✉♥❝t✐♦♥ ✐s ❝②❝❧✐❝✦ ❲❡ ✉s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ t❡r♠✐♥♦❧♦❣② ❛♥❞ ♥♦t❛t✐♦♥✿ ❈♦♥s❡❝✉t✐✈❡ ❞❡r✐✈❛t✐✈❡s

❢✉♥❝t✐♦♥

f

f (0)

✜rst ❞❡r✐✈❛t✐✈❡

f′

f (1)

s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡

f ′′ = (f ′ )′

f (2) = f (1)

t❤✐r❞ ❞❡r✐✈❛t✐✈❡

f ′′′ = (f ′′ )′ f (3) = f (2)

...

...

nt❤ ❞❡r✐✈❛t✐✈❡

f

...

...

(n)

= f

df dx   d2 f d df = dx2 dx  dx  3 df d d2 f = dx3 dx dx2


′
′

...


(n−1) ′

d dn f = n dx dx



dn−1 f dxn−1



...

❚❤✉s✱ ❛ ❣✐✈❡♥ ❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥ ♠❛② ♣r♦❞✉❝❡ ❛ s❡q✉❡♥❝❡ ♦❢ ❢✉♥❝t✐♦♥s ✿ d dx

f →

♦r

→ f′ → d

d

d dx d

→ f ′′ → ... → f (n) → ... d

d

f −−−dx−−→ f ′ −−−dx−−→ f ′′ −−−dx−−→ ... −−−dx−−→ f (n) −−−dx−−→ ...

❲❡ ❥✉st ♥❡❡❞ t❤❡ ♦✉t❝♦♠❡ ♦❢ ❡❛❝❤ st❡♣ t♦ ❜❡ ❞✐✛❡r❡♥t✐❛❜❧❡ ❛s ✇❡❧❧✦ ◆♦t❡ t❤❛t✱ ❢♦r ❛ ✜①❡❞ x✱ t❤❡ s❡q✉❡♥❝❡ ♦❢ ♥✉♠❜❡rs✿

f (x), f ′ (x), f ′′ (x), ..., f (n) (x), ...

✐s ❥✉st t❤❛t✱ ❛ s❡q✉❡♥❝❡✱ ❛ ❝♦♥❝❡♣t ❢❛♠✐❧✐❛r ❢r♦♠ ❈❤❛♣t❡r ✶✳ ❍♦✇❡✈❡r✱ ❛s t❤❡ ❡①❛♠♣❧❡ ♦❢ sin x s❤♦✇s✱ t❤✐s s❡q✉❡♥❝❡ ❞♦❡s♥✬t ❤❛✈❡ t♦ ❝♦♥✈❡r❣❡✿ (sin x)(n)

x=0

, n = 0, 1, 2, 3, ... ❛r❡ ❡q✉❛❧ t♦ 0, −1, 0, 1, 0, ...

✭❲❡ ✇✐❧❧ s❡❡ ✐♥ ❱♦❧✉♠❡ ✸✱ ❈❤❛♣t❡r ✸■❈✲✺✱ t❤❛t s♦♠❡ s♣❡❝✐❛❧ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡s ❞♦ ♣r♦❞✉❝❡ ❛ s❡q✉❡♥❝❡ ❝♦♥✈❡r❣❡♥t t♦ t❤❡ ❢✉♥❝t✐♦♥✳✮ ▲❡t✬s tr② t♦ ❝♦♠♣✉t❡ ❛s ♠❛♥② ❝♦♥s❡❝✉t✐✈❡ ❞❡r✐✈❛t✐✈❡s ❛s ♣♦ss✐❜❧❡✱ ♦r ❡✈❡♥ ❛❧❧ ♦❢ t❤❡♠✱ ❢♦r t❤❡ ❢✉♥❝t✐♦♥s ❜❡❧♦✇✳

✹✳✻✳ ❘❡♣❡❛t❡❞ ❞✐✛❡r❡♥t✐❛t✐♦♥

✸✸✷

❊①❛♠♣❧❡ ✹✳✻✳✷✿ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r ♣♦✇❡rs

❚❤❡ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r ♣♦✇❡rs ❛r❡ ❤❛♥❞❧❡❞ ✇✐t❤ t❤❡ Pr♦❞✉❝t ❋♦r♠✉❧❛✿ (xn )′ = nxn−1 .

❚❤❡ ♣♦✇❡r ❞❡❝r❡❛s❡s ❜② 1 ❡✈❡r② t✐♠❡✳ ❚❤❡r❡❢♦r❡✱ (xn )(n+1) = 0 .

❚❤❡♥✱ ✐t st❛②s 0✿

(xn )(n+1) = (xn )(n+2) = ... = 0 .

❚❤❡ ♣♦✇❡rs ✐♥ t❤❡ s❡q✉❡♥❝❡ ♦❢ ❝♦♥s❡❝✉t✐✈❡ ❞❡r✐✈❛t✐✈❡s ❞❡❝r❡❛s❡ t♦ 0 ❛♥❞ t❤❡♥ r❡♠❛✐♥ ❝♦♥st❛♥t✳ ❊①❛♠♣❧❡ ✹✳✻✳✸✿ ❡①♣♦♥❡♥t

❚❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥ ♥❡①t✳ ❙✐♥❝❡ ✇❡ ❤❛✈❡✿

(ex )′ = ex , (ex )(n) = ex .

❚❤❡ ❢✉♥❝t✐♦♥ r❡♠❛✐♥s t❤❡ s❛♠❡✦ ❚❤❡ s❡q✉❡♥❝❡ ♦❢ ❝♦♥s❡❝✉t✐✈❡ ❞❡r✐✈❛t✐✈❡s ✐s ❝♦♥st❛♥t✳ ❊①❛♠♣❧❡ ✹✳✻✳✹✿ s✐♥❡ ❛♥❞ ❝♦s✐♥❡

❚❤❡ tr✐❣ ❢✉♥❝t✐♦♥s✳ ❙❛♠❡ ❢♦r ❜♦t❤ s✐♥❡ ❛♥❞ ❝♦s✐♥❡✿ (sin x)(4n) = sin x (cos x)(4n) = cos x

❚❤❡ s❡q✉❡♥❝❡ ♦❢ ❝♦♥s❡❝✉t✐✈❡ ❞❡r✐✈❛t✐✈❡s ✐s ❝②❝❧✐❝ ❢♦r ❜♦t❤ ❢✉♥❝t✐♦♥s✳ ❊①❛♠♣❧❡ ✹✳✻✳✺✿ ♥❡❣❛t✐✈❡ ✐♥t❡❣❡r ♣♦✇❡rs

❚❤❡ ♥❡❣❛t✐✈❡ ✐♥t❡❣❡r ♣♦✇❡rs ❛r❡ s✉❜❥❡❝t❡❞ t♦ t❤❡ P♦✇❡r ❋♦r♠✉❧❛ ❛❣❛✐♥✿ (x−1 )′ = −1x−2 (−x−2 )′ = 2x−3 ...

❚❤❡ ♣♦✇❡r ❣♦❡s ❞♦✇♥ ❜② 1 ❡✈❡r② t✐♠❡ ❛♥❞✱ ❛s ❛ r❡s✉❧t✱ t❡♥❞s t♦ −∞✳ ❚❤❡ s❡q✉❡♥❝❡ ♦❢ ❝♦♥s❡❝✉t✐✈❡ ❞❡r✐✈❛t✐✈❡s ❞♦❡s♥✬t st♦♣✳ ❊①❡r❝✐s❡ ✹✳✻✳✻

❙❤♦✇ t❤❛t t❤❡ s❛♠❡ ❤❛♣♣❡♥s ✇✐t❤ ❛❧❧ ♥♦♥✲✐♥t❡❣❡r ♣♦✇❡rs✳ ❉✐✛❡r❡♥t✐❛t✐♦♥ ♣r♦❞✉❝❡s ❛ ❞✐✛❡r❡♥t ❞②♥❛♠✐❝s ❢♦r ❞✐✛❡r❡♥t ❢✉♥❝t✐♦♥s✿

✹✳✻✳ ❘❡♣❡❛t❡❞ ❞✐✛❡r❡♥t✐❛t✐♦♥

✸✸✸

❘❡♣❡❛t❡❞ ❉✐✛❡r❡♥t✐❛t✐♦♥ ♦❢ ❇❛s✐❝ ❋✉♥❝t✐♦♥s d

x dx d

xn 1 n

−−−dx−−→

sin xx  d  dx

−−−dx−−→

e−x d x dx ex

dx ←−− −→

d dx

−−−−−→

nxn−1 1 − 2 n

d

d

d

... −−−dx−−→ ❝♦♥st❛♥t −−−dx−−→ d 2 dx − − − − −→ ... ❞✐✈❡r❣❡♥t n3

−−−dx−−→ d dx

−−−−−→

0

d

d

cos  x  d y dx

− cos x ←−−dx−−− − sin x d

−e−x

♣❡r✐♦❞✐❝ ❛❧t❡r♥❛t✐♥❣

❝♦♥st❛♥t

❲❛r♥✐♥❣✦ ❙t❛rt✐♥❣ ✐♥ ❱♦❧✉♠❡ ✹✱ ❈❤❛♣t❡r ✹❍❉✲✷✱ ✇❡ ✇✐❧❧ s❡❡ t❤❛t t❤❡ ❢✉♥❝t✐♦♥ ❛♥❞ ✐ts ❞❡r✐✈❛t✐✈❡ ❛r❡ t✇♦ ❛♥✐♠❛❧s ♦❢ ✈❡r② ❞✐✛❡r❡♥t ❜r❡❡❞s✳ ❆s ❛ r❡s✉❧t✱ t❤❡ ❞②♥❛♠✲ ✐❝s ❞✐s❝✉ss❡❞ ❛❜♦✈❡ ✇✐❧❧ ❞✐s❛♣♣❡❛r ✐♥ ❤✐❣❤❡r ❞✐♠❡♥✲ s✐♦♥s✳

❚❤❡ r❡♣❡❛t❡❞ ❞✐✛❡r❡♥t✐❛t✐♦♥ ♣r♦❝❡ss ♠❛② ❢❛✐❧ t♦ ❝♦♥t✐♥✉❡ ✇❤❡♥ t❤❡ nt❤ ❞❡r✐✈❛t✐✈❡ ✐s ♥♦t ❞✐✛❡r❡♥t✐❛❜❧❡✱ ✐✳❡✳✱ ✇❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❧✐♠✐t ❞♦❡s ♥♦t ❡①✐st✿ f (n−1) (a + h) − f (n−1) (a) . h→0 h

f (n) (a) = lim

❇❡❧♦✇ ✐s t❤❡ s✉♠♠❛r② ♦❢ t❤✐s ✐❞❡❛✿

❉❡✜♥✐t✐♦♥ ✹✳✻✳✼✿ n t✐♠❡s ❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥ ❆ ❢✉♥❝t✐♦♥ f ✐s ❝❛❧❧❡❞ t✇✐❝❡✱ t❤r✐❝❡✱ ✳✳✳✱ n t✐♠❡s ❞✐✛❡r❡♥t✐❛❜❧❡ ✇❤❡♥ f ′ , f ′′ , f ′′′ , ..., f (n) ❡①✐sts r❡s♣❡❝t✐✈❡❧②✳ ❲❤❡♥ t❤❡s❡ ❞❡r✐✈❛t✐✈❡s ❛r❡ ❛❧s♦ ❝♦♥t✐♥✲ ✉♦✉s✱ ✇❡ ❝❛❧❧ t❤❡ ❢✉♥❝t✐♦♥ n t✐♠❡s ❝♦♥t✐♥✉♦✉s❧② ❞✐✛❡r❡♥t✐❛❜❧❡✳

❊①❛♠♣❧❡ ✹✳✻✳✽✿ ♥♦t t✇✐❝❡ ❞✐✛❡r❡♥t✐❛❜❧❡ ❚❤✐s ❢✉♥❝t✐♦♥ ✐s ❞✐✛❡r❡♥t✐❛❜❧❡✿ f (x) =

(

−x2 x2

✐❢ x < 0 , ✐❢ x ≥ 0 .

■ts ❣r❛♣❤ ❧♦♦❦s s♠♦♦t❤ ❛♥❞ t❤❡r❡ ✐s ♥♦ ❞♦✉❜t ✐♥ ✇❤✐❝❤ ❞✐r❡❝t✐♦♥ ❛ ❜❡❛♠ ♦❢ ❧✐❣❤t ✇♦✉❧❞ ❜♦✉♥❝❡ ♦✛ s✉❝❤ ❛ s✉r❢❛❝❡✿

✹✳✻✳ ❘❡♣❡❛t❡❞ ❞✐✛❡r❡♥t✐❛t✐♦♥

✸✸✹

❇✉t t❤❡ ❣r❛♣❤ ♦❢ ✐ts ❞❡r✐✈❛t✐✈❡ ❞♦❡s♥✬t ❧♦♦❦ s♠♦♦t❤✦ ▲❡t✬s ❝♦♠♣✉t❡ t❤❡ ❞❡r✐✈❛t✐✈❡s✳ ■t ✐s ❡❛s② ❢♦r x 6= 0 ❜❡❝❛✉s❡ t❤❡r❡ ✐s ♦♥❧② ♦♥❡ ❢♦r♠✉❧❛✿ f (x) =

(

−2x 2x

✐❢ x < 0 , ✐❢ x > 0 .

❋♦r t❤❡ ❝❛s❡ ♦❢ x = 0✱ ✇❡ ❝♦♥s✐❞❡r t❤❡ t✇♦ ♦♥❡✲s✐❞❡❞ ❧✐♠✐ts✿ lim−

h→0

f (h) −h2 f (0 + h) − f (0) = lim− = lim = lim (−h) = 0 h→0 h h→0 h→0 h h

lim+

h→0

f (0 + h) − f (0) h2 f (h) = lim+ = lim = lim h = 0 h→0 h h→0 h→0 h h

❚❤❡② ♠❛t❝❤✦ ❚❤❡r❡❢♦r❡✱

f ′ (0) = 0 .

❲❡ ❤❛✈❡ ❞✐s❝♦✈❡r❡❞ t❤❛t f ′ (x) = 2|x|✳ ■t✬s ♥♦t ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t 0✦ ❚❤❡ ♦r✐❣✐♥❛❧ ❢✉♥❝t✐♦♥ ✐s♥✬t t✇✐❝❡ ❞✐✛❡r❡♥t✐❛❜❧❡✳ ❊①❛♠♣❧❡ ✹✳✻✳✾✿ ❤✐❡r❛r❝❤②

▼♦r❡ ❡①❛♠♣❧❡s ♦❢ t❤✐s ❦✐♥❞✿

1 ✐s ❞✐s❝♦♥t✐♥✉♦✉s ❛t x = 0✳ x 1 ✷✳ g(x) = x sin ✐s ❝♦♥t✐♥✉♦✉s ❛t x = 0 ❜✉t ♥♦t ❞✐✛❡r❡♥t✐❛❜❧❡✳ x 1 ✸✳ h(x) = x2 sin ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t x = 0 ❜✉t ♥♦t t✇✐❝❡ ❞✐✛❡r❡♥t✐❛❜❧❡✳ x ❲❡ ❛r❡ ❛ss✉♠✐♥❣ ❛❧s♦ t❤❛t t❤❡ ✈❛❧✉❡ ❛t 0 ✐s 0 ❢♦r ❡❛❝❤ ♦❢ t❤❡♠✳ ❲❡ ♣❧♦t t❤❡♠ ❜❡❧♦✇✿

✶✳ f (x) = sin

✹✳✻✳ ❘❡♣❡❛t❡❞ ❞✐✛❡r❡♥t✐❛t✐♦♥

✸✸✺

❚❤✐s ✐s t❤❡ r❡❛s♦♥✐♥❣✿ ✶✳ ❚❤❡ t✇♦ ❧✐♥❡s ❞♦♥✬t ❢♦r♠ ❛ sq✉❡❡③❡✳ ✷✳ ❚❤❡ t✇♦ ❧✐♥❡s ❞♦ ❢♦r♠ ❛ sq✉❡❡③❡ ❛♥❞ ✐t ❣✉❛r❛♥t❡❡s ❝♦♥t✐♥✉✐t② ❛t x = 0✳ ❇✉t t❤❡ t✇♦ ❧✐♥❡s ❛r❡ ❛❧s♦ t✇♦ s❡❝❛♥t ❧✐♥❡s ✇✐t❤ ❞✐✛❡r❡♥t s❧♦♣❡s❀ t❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡ ♥♦ ❞✐✛❡r❡♥t✐❛❜✐❧✐t② ❛t x = 0✳ ✸✳ ❚❤❡ t✇♦ ❝✉r✈❡s ❢♦r♠ ❛ sq✉❡❡③❡ ❛♥❞ ❣✉❛r❛♥t❡❡ ❝♦♥t✐♥✉✐t② ❛t x = 0✳ ❇✉t t❤❡② ❛❧s♦ ❤❛✈❡ t❤❡ s❛♠❡ s❧♦♣❡✱ 0✱ ❛t x = 0✦ ❚❤✐s ❣✉❛r❛♥t❡❡s t❤❛t t❤❡ s❧♦♣❡s ♦❢ t❤❡ s❡❝❛♥t ❧✐♥❡s ✇✐❧❧ ❝♦♥✈❡r❣❡ t♦ 0❀ t❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡ ❞✐✛❡r❡♥t✐❛❜✐❧✐t② ❛t x = 0✳ ■t ♠✐❣❤t ♥♦t ❜❡ t✇✐❝❡ ❞✐✛❡r❡♥t✐❛❜❧❡ t❤♦✉❣❤✳ ❊①❡r❝✐s❡ ✹✳✻✳✶✵

Pr♦✈❡ t❤❡ ❛❜♦✈❡ st❛t❡♠❡♥ts✳ ❇❡❧♦✇ ✇❡ ✈✐s✉❛❧✐③❡ t❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡s❡ ❝❧❛ss❡s ♦❢ ❢✉♥❝t✐♦♥s✿

❲❤❛t ✐s t❤❡ ❣❡♦♠❡tr✐❝ ♠❡❛♥✐♥❣ ♦❢ t❤❡s❡ ❤✐❣❤❡r ❞❡r✐✈❛t✐✈❡s ❢♦r ❛ ❣✐✈❡♥ ❢✉♥❝t✐♦♥❄ ▲❡t✬s ❝♦♥s✐❞❡r t❤❡ ✜rst ❞❡r✐✈❛t✐✈❡✳ ■t r❡♣r❡s❡♥ts t❤❡ s❧♦♣❡s ♦❢ t❤❡ ❢✉♥❝t✐♦♥✳ ❚❤❡♥ t❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡ r❡♣r❡s❡♥ts t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢ t❤❡s❡ s❧♦♣❡s✳ ❈♦♥s✐❞❡r t❤❡ ✇❤❡❡❧ ♦❢ s❧♦♣❡s✿

✹✳✻✳

❘❡♣❡❛t❡❞ ❞✐✛❡r❡♥t✐❛t✐♦♥

✸✸✻

❲❡ ❝❛♥ ✐♠❛❣✐♥❡ t❤❡ t❛♥❣❡♥t ❧✐♥❡s r♦t❛t❡✳ ❚❤✐s ✐s ❤♦✇ ❝❤❛♥❣✐♥❣ s❧♦♣❡s ❛r❡ s❡❡♥ ❛s r♦t❛t✐♥❣ t❛♥❣❡♥ts ✐♥ t❤❡ r✐❣❤t ❞✐r❡❝t✐♦♥✿

❙♣❡❝✐✜❝❛❧❧②✱ ✇❡ s❡❡✿ ✶✳ ❉❡❝r❡❛s✐♥❣ s❧♦♣❡s ✷✳ ■♥❝r❡❛s✐♥❣ s❧♦♣❡s

⇐⇒

t❛♥❣❡♥ts r♦t❛t❡ ❝❧♦❝❦✇✐s❡✳

⇐⇒

t❛♥❣❡♥ts r♦t❛t❡ ❝♦✉♥t❡r✲❝❧♦❝❦✇✐s❡✳

❚❤✐s ♠❛t❝❤❡s ♦✉r ❝♦♥✈❡♥t✐♦♥ ❢r♦♠ tr✐❣♦♥♦♠❡tr② t❤❛t ❝♦✉♥t❡r✲❝❧♦❝❦✇✐s❡ ✐s t❤❡ ♣♦s✐t✐✈❡ ❞✐r❡❝t✐♦♥ ❢♦r r♦t❛t✐♦♥s✳ ❊✈❡♥ t❤♦✉❣❤ ✇❡ t②♣✐❝❛❧❧② ❤❛✈❡ ❢✉♥❝t✐♦♥s ✇✐t❤ t❤❡

✈✐s✐❜❧❡

nt❤

❞❡r✐✈❛t✐✈❡ ❢♦r ❡❛❝❤ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r

n✱

♦♥❧② t❤❡ ✜rst

t✇♦ r❡✈❡❛❧ s♦♠❡t❤✐♥❣ ❛❜♦✉t t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ♦r✐❣✐♥❛❧ ❢✉♥❝t✐♦♥✳ ❇❡❧♦✇✱ ✇❡ ✏❞❡r✐✈❡✑ t❤❡ ❣r❛♣❤ ♦❢ ′′ t❤❡ ❞❡r✐✈❛t✐✈❡ f ❢♦r t❤❛t ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❜② ❧♦♦❦✐♥❣ ❛t t❤❡ s❧♦♣❡s ♦❢ f ❀ t❤❡♥ ✇❡ ✏❞❡r✐✈❡✑ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ′′ ′ ′ ′ ❞❡r✐✈❛t✐✈❡ f ♦❢ f ❢♦r t❤❛t ♦❢ f ❜② ❧♦♦❦✐♥❣ ❛t ✐ts s❧♦♣❡s ♦❢ f ✿

❲❤❛t ✐s ♥❡✇❄ ❚❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥

• •

❖❧❞✿ ❲❡ ❝♦♠♣❛r❡ t❤❡ ′ t❤❡ ✜rst ❞❡r✐✈❛t✐✈❡ f ✳

s❤❛♣❡s

◆❡✇✿ ❲❡ ❝♦♠♣❛r❡ t❤❡

❛♥❞

f ′′ ✳

❆❜♦✈❡ ✇❡ ❝♦♠♣❛r❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣❛✐rs✿

♦❢ t❤❡ ♣❛t❝❤❡s ♦❢ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥

s❤❛♣❡s

♦❢ t❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡

f

f

♦❢ t❤❡ ♣❛t❝❤❡s ♦❢ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥

f ′′ ✳

❚❤❡r❡ ❛r❡ t❤r❡❡ ♠❛✐♥ ❧❡✈❡❧s ♦❢ ❛♥❛❧②s✐s ♦❢ ❛ ❢✉♥❝t✐♦♥✿

t♦ t❤❡ s✐❣♥ ♦❢ t❤❡

f

✈❛❧✉❡s

t♦ t❤❡ s✐❣♥ ♦❢ t❤❡

♦❢

✈❛❧✉❡s

✹✳✼✳ ❍♦✇ t♦ ❞✐✛❡r❡♥t✐❛t❡ r❡❧❛t✐♦♥s✿ ✐♠♣❧✐❝✐t❧②

✸✸✼

•

❆♥❛❧②s✐s ❛t ❧❡✈❡❧ 0✿ t❤❡ ✈❛❧✉❡s ♦❢ f ✳ ❲❡ ❛s❦✱ ❤♦✇ ❧❛r❣❡❄ ❚❤❡ ✜♥❞✐♥❣s ❛r❡ ❛❜♦✉t t❤❡ ✈❛❧✉❡s✱ x✲ ❛♥❞

•

❆♥❛❧②s✐s ❛t ❧❡✈❡❧ 1✿ t❤❡ s❧♦♣❡s ♦❢ f ✳ ❲❡ ❛s❦✱ ✉♣ ♦r ❞♦✇♥❄ ❚❤❡ ✜♥❞✐♥❣s ❛r❡ ❛❜♦✉t t❤❡ ❛♥❣❧❡s✱

•

❆♥❛❧②s✐s ❛t ❧❡✈❡❧ 2✿ t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢ t❤❡ s❧♦♣❡s ♦❢ f ✳ ❲❡ ❛s❦✱ ❝♦♥❝❛✈❡ ✉♣ ♦r ❞♦✇♥❄ ❚❤❡ ✜♥❞✐♥❣s ❛r❡ ❛❜♦✉t t❤❡ ❝❤❛♥❣❡ ♦❢ st❡❡♣♥❡ss✱ ❝♦♥❝❛✈✐t②✱ t❡❧❧✐♥❣ ❛ ♠❛①✐♠✉♠ ❢r♦♠ ❛ ♠✐♥✐♠✉♠✱ ❡t❝✳

y ✲✐♥t❡r❝❡♣ts✱ ❛s②♠♣t♦t❡s ❛♥❞ ♦t❤❡r ❧❛r❣❡✲s❝❛❧❡ ❜❡❤❛✈✐♦r✱ ♣❡r✐♦❞✐❝✐t②✱ ❡t❝✳

✐♥❝r❡❛s✐♥❣✴❞❡❝r❡❛s✐♥❣ ❜❡❤❛✈✐♦r✱ ❝r✐t✐❝❛❧ ♣♦✐♥ts✱ ❡t❝✳

❲❡ ❝❛♥ ❣♦ ♦♥ ❛♥❞ ❝♦♥t✐♥✉❡ t♦ ❞✐s❝♦✈❡r ♠♦r❡ ❛♥❞ ♠♦r❡ s✉❜t❧❡ ❜✉t ❧❡ss ❛♥❞ ❧❡ss s✐❣♥✐✜❝❛♥t ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❢✉♥❝t✐♦♥✳ ❊①❛♠♣❧❡ ✹✳✻✳✶✶✿ ♠♦t✐♦♥

❚❤✐s t❤r❡❡✲❧❡✈❡❧ ❛♥❛❧②s✐s ❛❧s♦ ❛♣♣❧✐❡s t♦ ♦✉r st✉❞② ♦❢ ♠♦t✐♦♥✱ ❛s ❢♦❧❧♦✇s✳ ❚❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ✈❡❧♦❝✐t② ❛♥❞✱ t❤❡r❡❢♦r❡✱ t❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ♣♦s✐t✐♦♥✱ ✐s ❝❛❧❧❡❞ t❤❡ ❛❝❝❡❧✲ ❡r❛t✐♦♥✳ ❚❤❡ ❝♦♥❝❡♣t ❛❧❧♦✇s ♦♥❡ t♦ ❛❞❞ ❛♥♦t❤❡r ❧❡✈❡❧ ♦❢ ❛♥❛❧②s✐s ♦❢ ♠♦t✐♦♥✿ • ❆♥❛❧②s✐s ❛t ❧❡✈❡❧ 0✿ t❤❡ ❧♦❝❛t✐♦♥✱ ✇❤❡r❡❄ • ❆♥❛❧②s✐s ❛t ❧❡✈❡❧ 1✿ t❤❡ ✈❡❧♦❝✐t②✱ ❤♦✇ ❢❛st❄ ❢♦r✇❛r❞ ♦r ❜❛❝❦❄ • ❆♥❛❧②s✐s ❛t ❧❡✈❡❧ 2✿ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥✱ ❤♦✇ ❧❛r❣❡ ✐s t❤❡ ❢♦r❝❡❄ ❙✉♣♣♦s❡ t ✐s t✐♠❡ ❛♥❞ y ✐s t❤❡ ✈❡rt✐❝❛❧ ❞✐♠❡♥s✐♦♥✱ t❤❡ ❤❡✐❣❤t✳ ◆♦✇ ✇❡ t✉r♥ t♦ t❤❡ s♣❡❝✐✜❝ ❝❛s❡ ♦❢ ❢r❡❡ ❢❛❧❧✳ ❚❤❡s❡ ❛r❡ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s✿ • y0 ✐s t❤❡ ✐♥✐t✐❛❧ ❤❡✐❣❤t✱ y0 = y ✳ t=0

• vy ✐s t❤❡ ✐♥✐t✐❛❧ ✈❡rt✐❝❛❧ ❝♦♠♣♦♥❡♥t ♦❢ ✈❡❧♦❝✐t②✱

❚❤❡♥✱ ✇❡ ❤❛✈❡✿

dy ✳ dt t=0

dy d2 y 1 = −g . = vy −gt =⇒ y = y0 + vy t − gt2 =⇒ 2 dt dt2

◆♦✇✱ ❢r♦♠ t❤❡ ♣♦✐♥t ♦❢ t❤❡ ♣❤②s✐❝s ♦❢ t❤❡ s✐t✉❛t✐♦♥✱ t❤❡ ❞❡r✐✈❛t✐♦♥ s❤♦✉❧❞ ❣♦ ✐♥ t❤❡ ♦♣♣♦s✐t❡ ❞✐r❡❝t✐♦♥✿ • ❲❤❡♥ t❤❡r❡ ✐s ♥♦ ❢♦r❝❡✱ t❤❡ ✈❡❧♦❝✐t② ✐s ❝♦♥st❛♥t✳ • ❲❤❡♥ t❤❡ ❢♦r❝❡ ✐s ❝♦♥st❛♥t✱ t❤❡ ✈❡❧♦❝✐t② ✐s ❧✐♥❡❛r ♦♥ t✐♠❡✱ ❡t❝✳ ❍♦✇❡✈❡r✱ ❛t t❤✐s ♣♦✐♥t ✇❡ ❛r❡ st✐❧❧ ✉♥❛❜❧❡ t♦ ❛♥s✇❡r t❤❡s❡ q✉❡st✐♦♥s✿ • ❍♦✇ ❞♦ ✇❡ ❦♥♦✇ t❤❛t ♦♥❧② t❤❡ ❞❡r✐✈❛t✐✈❡s ♦❢ ❝♦♥st❛♥t ❢✉♥❝t✐♦♥s ❛♥❞ ♥♦♥❡ ♦t❤❡rs ❛r❡ ③❡r♦❄ • ❍♦✇ ❞♦ ✇❡ ❦♥♦✇ t❤❛t ♦♥❧② t❤❡ ❞❡r✐✈❛t✐✈❡s ♦❢ ❧✐♥❡❛r ❢✉♥❝t✐♦♥s ❛♥❞ ♥♦♥❡ ♦t❤❡rs ❛r❡ ❝♦♥st❛♥t❄ • ❍♦✇ ❞♦ ✇❡ ❦♥♦✇ t❤❛t ♦♥❧② t❤❡ ❞❡r✐✈❛t✐✈❡s ♦❢ q✉❛❞r❛t✐❝ ❢✉♥❝t✐♦♥s ❛♥❞ ♥♦♥❡ ♦t❤❡rs ❛r❡ ❧✐♥❡❛r❄ ❚❤✐s r❡✈❡rs❡❞ ♣r♦❝❡ss ✐s ❝❛❧❧❡❞ ❛♥t✐❞✐✛❡r❡♥t✐❛t✐♦♥✳ ❙♦ ❢❛r✱ ✇❡ ❝❛♥♥♦t ❥✉st✐❢② ❡✈❡♥ t❤✐s s✐♠♣❧❡st ❝♦♥❝❧✉s✐♦♥✿ f ′ = 0 =⇒ f = c, ❢♦r s♦♠❡ r❡❛❧ ♥✉♠❜❡r c .

❲❡ ✇✐❧❧ st✉❞② t❤❡s❡ ❛♥❞ r❡❧❛t❡❞ q✉❡st✐♦♥s ✐♥ ❈❤❛♣t❡r ✺✳

✹✳✼✳ ❍♦✇ t♦ ❞✐✛❡r❡♥t✐❛t❡ r❡❧❛t✐♦♥s✿ ✐♠♣❧✐❝✐t❧② ■t s❡❡♠s t❤❛t ✇❡ ❝❛♥ ✜♥❞ t❤❡ t❛♥❣❡♥t ❧✐♥❡ t♦ ❛♥② ❝✉r✈❡✦ ❨❡s✱ ✇✐t❤ ❛ ❝❛✈❡❛t✿ ❚❤✐s ❝✉r✈❡ ✐s t❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥✳ ❍♦✇ ❞♦ ✇❡ ✜♥❞ t❤❡ t❛♥❣❡♥t t♦ ❛ ❝✐r❝❧❡❄ ■t ✐s ❣✐✈❡♥ ❜② ❛ r❡❧❛t✐♦♥✿ x2 + y 2 = 1 .

✹✳✼✳

❍♦✇ t♦ ❞✐✛❡r❡♥t✐❛t❡ r❡❧❛t✐♦♥s✿ ✐♠♣❧✐❝✐t❧②

❘❡❝❛❧❧ t❤❛t ❛

❢✉♥❝t✐♦♥

✸✸✽

✐s ❛ s♣❡❝✐❛❧ ❦✐♥❞ ♦❢ r❡❧❛t✐♦♥✳ ❚❤❡s❡ t✇♦ ❛r❡ t❤❡ t✇♦ ♠❛✐♥ ❝♦♥str✉❝ts ❝❛❧❝✉❧✉s r❡❧✐❡s

♦♥✿

•

❆ r❡❧❛t✐♦♥✿ ✏❲❤❛t s♣♦rts ❤❛✈❡ ❜❡❡♥ ♣❧❛②❡❞ ❜② ✇❤❛t ❜♦②s t♦❞❛②❄✑

•

❆ ❢✉♥❝t✐♦♥✿ ✏❲❤✐❝❤ s♣♦rt ❞♦❡s t❤❡ ❜♦②

♣r❡❢❡r

t♦ ♣❧❛②❄✑

❚❤❡ ❞✐✛❡r❡♥❝❡ ✐s t❤❛t ✐♥ t❤❡ ❧❛tt❡r ❝❛s❡ ❡✈❡r②♦♥❡ ❤❛s ❛ ♣r❡❢❡r❡♥❝❡ ❛♥❞ ❡①❛❝t❧② ♦♥❡✿

r❡❧❛t✐♦♥s ❄

❲❡ ❝❛♥ ❞✐✛❡r❡♥t✐❛t❡ ❢✉♥❝t✐♦♥s❀ ❝❛♥ ✇❡ ❞✐✛❡r❡♥t✐❛t❡

❘❡❝❛❧❧ ✭❢r♦♠ ❱♦❧✉♠❡ ✶✱ ❈❤❛♣t❡r ✶P❈✲✷✮ t❤❛t r❡❧❛t✐♦♥s ❛r❡ r❡♣r❡s❡♥t❡❞ ❜② ❡q✉❛t✐♦♥s✳ ❯s✉❛❧❧②✱ ❡q✉❛t✐♦♥s ❢♦r

x2 |{z}

x ✐s ❛ ♥✉♠❜❡r

x

❛r❡ t♦ ❜❡ s♦❧✈❡❞✿

− |{z} 1 = 0.

◆♦✇ ✜♥❞ ❛ ♣❛rt✐❝✉❧❛r ♥✉♠❜❡r

x ✐s ❛ ♥✉♠❜❡r

+

t❤❛t s❛t✐s✜❡s t❤❡ ❡q✉❛t✐♦♥✳

❛ ♥✉♠❜❡r

❚❤❡ ❡q✉❛t✐♦♥s ✇❡ ❛r❡ ✐♥t❡r❡st❡❞ ✐♥ ❛r❡ ❡q✉❛t✐♦♥s ♦❢

x2 |{z}

x

y2 |{z}

= 0.

x✬s

❛♥❞

y ✬s✱

s✉❝❤ ❛s✿

◆♦✇ ✜♥❞ ❛ ♣❛rt✐❝✉❧❛r ❢✉♥❝t✐♦♥

y = y(x)

y ✐s ❛ ❢✉♥❝t✐♦♥

■♥ ♦t❤❡r ✇♦r❞s✱ ❛❢t❡r t❤❡ s✉❜st✐t✉t✐♦♥✱ t❤❡ ❡q✉❛t✐♦♥ s❤♦✉❧❞ ❜❡ tr✉❡ ❢♦r ❛♥❞ ♦♥ t❤❡ r✐❣❤t ❛r❡ ✐❞❡♥t✐❝❛❧ ✭x

+ x = 2x✱

✐♠♣❧✐❝✐t❧② ❞❡✜♥❡s t❤✐s ❢✉♥❝t✐♦♥✳ ❆s ❡①♣❧✐❝✐t ❜② s♦❧✈✐♥❣ t❤❡ ❡q✉❛t✐♦♥ ❢♦r y ✿ y=

√

❛❧❧ x✱

✐✳❡✳✱ t❤❡ ❢✉♥❝t✐♦♥s ♦♥ t❤❡ ❧❡❢t

❡t❝✳✮✳

❚❤❡ ❡q✉❛t✐♦♥

y = y(x)

t❤❛t s❛t✐s✜❡s t❤❡ ❡q✉❛t✐♦♥✳

1 − x2

✇❡ ❤❛✈❡ ❞♦♥❡ ✐♥ t❤❡ ♣❛st✱ ✇❡ ❝♦✉❧❞ ♠❛❦❡ t❤❡ ❢✉♥❝t✐♦♥

♦r

√ y = − 1 − x2 .

❇✉t ✇❡ ♦♥❧② ✇❛♥t t❤❡ t❛♥❣❡♥t✱ ✐✳❡✳✱ t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢ t❤✐s ❢✉♥❝t✐♦♥✦ ❲❡ ✇✐❧❧ ❧❡❛✈❡

y

✉♥s♣❡❝✐✜❡❞✳

❲❡ ✇✐❧❧ r❡❧② ♦♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢❛❝t✿

◮ ■❢ t✇♦ ❢✉♥❝t✐♦♥s ❛r❡ ✐❞❡♥t✐❝❛❧✱ ❢♦r ❛❧❧ ♥♦❞❡s x ♦❢ ❛ ♣❛rt✐t✐♦♥✱ t❤❡♥ s♦ ❛r❡ t❤❡✐r ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ❢♦r ❛❧❧ s❡❝♦♥❞❛r② ♥♦❞❡s c✿ f (x) = g(x)

❢♦r ❛❧❧

x =⇒

∆g ∆f (c) = (c) ∆x ∆x

❢♦r ❛❧❧

c

❊①❛♠♣❧❡ ✹✳✼✳✶✿ ❝✐r❝❧❡

❲❡ st❛rt ✇✐t❤ ✜♥❞✐♥❣ t❤❡

s❡❝❛♥t ❧✐♥❡ √

t❤r♦✉❣❤ t❤❡ t✇♦ ♣♦✐♥ts ♦♥ t❤❡ ❝✐r❝❧❡ ♦❢ r❛❞✐✉s

√ ! 2 2 , 2 2

❛♥❞ ❛♥② ♦t❤❡r ♣♦✐♥t

(x, y) .

1

❝❡♥t❡r❡❞ ❛t

0✿

✹✳✼✳

❍♦✇ t♦ ❞✐✛❡r❡♥t✐❛t❡ r❡❧❛t✐♦♥s✿ ✐♠♣❧✐❝✐t❧②

✸✸✾

❚②♣✐❝❛❧❧②✱ ❛ ❝✉r✈❡ ❤❛s ❜❡❡♥ t❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥ y = x2 ✱ y = sin x✱ ❡t❝✳✱ ❣✐✈❡♥ ❡①♣❧✐❝✐t❧②✳ ❚❤✐s t✐♠❡ t❤❡ ❡q✉❛t✐♦♥ ✐s✿ x2 + y 2 = 1 . ❚♦ ✜♥❞ t❤❡ s❧♦♣❡ ♦❢ t❤❡ s❡❝❛♥t ❧✐♥❡✱ ✇❡ ♥❡❡❞ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ t❤❡ ❢✉♥❝t✐♦♥✳ ❇✉t t❤❡r❡ ✐s ♥♦ ❢✉♥❝t✐♦♥✦ ◆♦t ❡①♣❧✐❝✐t ❛♥②✇❛②✳ ❚❤❡ ✐❞❡❛ ✐s t♦ ❝♦♥s✐❞❡r t❤❡ ❛❜♦✈❡ ❡q✉❛t✐♦♥ ❛s ❛ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ t✇♦ ✈❛r✐❛❜❧❡s✳ ■♥ ❢❛❝t✱ ✇❡ t❤✐♥❦ ♦❢ y = y(x) ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ x✱ ✐✳❡✳✿ x2 + y(x)2 = 1 . ❲❡ ✇✐❧❧ ❛❧s♦ ❛ss✉♠❡✿

√

2 ❛r❡ ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ t❤❡ x✲❛①✐s✳ √2 2 • ❚❤❡ t✇♦ y ✲✈❛❧✉❡s y0 ❛♥❞ y1 = ❛r❡ ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ t❤❡ y ✲❛①✐s✳ 2

• ❚❤❡ t✇♦ x✲✈❛❧✉❡s x0 ❛♥❞ x1 =

❲❡ ❛♣♣❧② t❤❡

❈❤❛✐♥ ❘✉❧❡ t♦ ❜♦t❤ s✐❞❡s ♦❢ t❤❡ ❡q✉❛t✐♦♥✿
∆ x2 + y 2 ∆x ∆ 2 ∆ 2 x + y ∆x ∆x ∆y (x0 + x1 ) + (y0 + y1 ) ∆x ∆y ∆x

=

∆ (1) ∆x

=⇒

=0

=⇒

=0

=⇒

=−

x0 + x1 y0 + y1

❢♦r y0 + y1 6= 0 .

❲❡ ❤❛✈❡ ❢♦✉♥❞ ❛ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❜✉t ✐t ✐s st✐❧❧ ✐♠♣❧✐❝✐t ✕ ❜❡❝❛✉s❡ ✇❡ ❞♦♥✬t ❤❛✈❡ ❛ ❢♦r♠✉❧❛ ❢♦r y = y(x)✳ ❋♦rt✉♥❛t❡❧②✱ ✇❡ ❞♦♥✬t ♥❡❡❞ t❤❡ ✇❤♦❧❡ ❢✉♥❝t✐♦♥✱ ❥✉st t❤♦s❡ t✇♦ ♣♦✐♥ts ♦♥ ✐ts ❣r❛♣❤✳ ❲❡ s✉❜st✐t✉t❡ t❤❡s❡ ✐♥t♦ t❤❡ ❢♦r♠✉❧❛ ❛❜♦✈❡ t♦ ✜♥❞✿

y0 + ∆y =− ∆x x0 +

√

2 2 √ 2 2

.

■♥ ♣❛rt✐❝✉❧❛r✱ ❢♦r t❤❡ ♣♦✐♥t (x0 , y0 ) = (0, 1)✱ t❤❡ s❧♦♣❡ ✐s √ 2 √ . m=− 1+ 2 ❚❤❡♥✱ ❢r♦♠ t❤❡

♣♦✐♥t✲s❧♦♣❡ ❢♦r♠✉❧❛ ✇❡ ♦❜t❛✐♥ t❤❡ ❛♥s✇❡r✿ √

√ 2 2 √ y− =− 2 1+ 2

√ ! 2 x− . 2

❲❡ ❝❛♥ ❛✉t♦♠❛t❡ t❤✐s ❢♦r♠✉❧❛ ❛♥❞ ✜♥❞ ♠♦r❡ s❡❝❛♥t ❧✐♥❡s ✇✐t❤ ❛ s♣r❡❛❞s❤❡❡t✿

✹✳✼✳ ❍♦✇ t♦ ❞✐✛❡r❡♥t✐❛t❡ r❡❧❛t✐♦♥s✿ ✐♠♣❧✐❝✐t❧②

✸✹✵

❲❤❛t ❛❜♦✉t t❤❡ ❞❡r✐✈❛t✐✈❡ ❄ ❲❡ ✇✐❧❧ r❡❧② ♦♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢❛❝t✿ ◮ ■❢ t❤❡ ✈❛❧✉❡s ♦❢ t✇♦ ❢✉♥❝t✐♦♥s ❛r❡ ❡q✉❛❧ ❢♦r ❛❧❧ x✱ t❤❡♥ s♦ ❛r❡ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡✐r ❞❡r✐✈❛t✐✈❡s✿ f (x) = g(x) ❢♦r ❛❧❧ x =⇒ f ′ (x) = g ′ (x) ❢♦r ❛❧❧ x

❲❡ ❝❛♥ ♣✉t ✐t s✐♠♣❧② ❛s t❤✐s✿ ◮ ■❢ t✇♦ ❢✉♥❝t✐♦♥s ❛r❡ ✐❞❡♥t✐❝❛❧✱ t❤❡♥ s♦ ❛r❡ t❤❡✐r ❞❡r✐✈❛t✐✈❡s❀ ✐✳❡✳✱ f = g =⇒ f ′ = g ′

❉✐✛❡r❡♥t✐❛t✐♥❣ ❛♥ ❡q✉❛t✐♦♥ ♦❢ ❢✉♥❝t✐♦♥s ❛♥❞ ✜♥❞✐♥❣ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ❜② t❤✐s ❡q✉❛t✐♦♥ ✐s ❝❛❧❧❡❞ ✐♠♣❧✐❝✐t ❞✐✛❡r❡♥t✐❛t✐♦♥✳ ▲❡t✬s ❝♦♥s✐❞❡r t✇♦ ❡①❛♠♣❧❡s ♦❢ ❤♦✇ t❤✐s ✐❞❡❛ ♠❛② ❤❡❧♣ ✉s ✇✐t❤ ✜♥❞✐♥❣ t❛♥❣❡♥ts t♦ ✐♠♣❧✐❝✐t ❝✉r✈❡s✳ ❊①❛♠♣❧❡ ✹✳✼✳✷✿ ❝✐r❝❧❡

❋✐♥❞ t❤❡ t❛♥❣❡♥t ❧✐♥❡ ❢♦r t❤❡ ❝✐r❝❧❡ ♦❢ r❛❞✐✉s 1 ❝❡♥t❡r❡❞ ❛t 0 ❛t t❤❡ ♣♦✐♥t

√ ! 2 2 , ✳ 2 2

√

❚②♣✐❝❛❧❧②✱ ❛ ❝✉r✈❡ ❤❛s ❜❡❡♥ t❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥ y = x2 ✱ y = sin x✱ ❡t❝✳✱ ❣✐✈❡♥ ❡①♣❧✐❝✐t❧②✳ ❚❤✐s t✐♠❡ t❤❡ ❡q✉❛t✐♦♥ ✐s✿ x2 + y 2 = 1 .

❚♦ ✜♥❞ t❤❡ s❧♦♣❡ ♦❢ t❤❡ t❛♥❣❡♥t ❧✐♥❡✱ ✇❡ ♥❡❡❞ t❤❡ ❞❡r✐✈❛t✐✈❡✱ ❜✉t t❤❡r❡ ✐s ♥♦ ❢✉♥❝t✐♦♥ t♦ ❞✐✛❡r❡♥t✐❛t❡✦ ❖✉r ❛♣♣r♦❛❝❤ ✐s t♦ ❞✐✛❡r❡♥t✐❛t❡ t❤❡ ❡q✉❛t✐♦♥ ❛❜♦✈❡ ❛s ❛ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ t✇♦ ✈❛r✐❛❜❧❡s✳ ❆s ✇❡

✹✳✼✳

❍♦✇ t♦ ❞✐✛❡r❡♥t✐❛t❡ r❡❧❛t✐♦♥s✿ ✐♠♣❧✐❝✐t❧②

✸✹✶

❞✐✛❡r❡♥t✐❛t❡✱ ✇❡ t❤✐♥❦ ♦❢ y = y(x) ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ x✳ ❚❤❡r❡❢♦r❡✱ y 2 = y(x)2 ✐s t❤❡ ❝♦♠♣♦s✐t✐♦♥✿ y2

y

✈❛r✐❛❜❧❡s✿ x −−−−→ y − −−−−→ z dy 2y ❞❡r✐✈❛t✐✈❡s✿ dx ❲❡ ❞✐✛❡r❡♥t✐❛t❡ t❤❡ ❡q✉❛t✐♦♥ ✭t❤❡ ✐❞❡♥t✐t② ♦❢ ❢✉♥❝t✐♦♥s✮✿

x2 + y 2 = 1 . ❚❤✐s ✐s t❤❡ r❡s✉❧t✱ ✈✐❛ t❤❡ ❈❤❛✐♥ ❘✉❧❡✿


d x2 + y 2 dx d d 2 x + y2 dx dx dy 2x + 2y dx dy dx

=

d (1) =⇒ dx

=0

=⇒

=0

=⇒

=−

x y

❢♦r y 6= 0 .

❲❡ ❤❛✈❡ ❢♦✉♥❞ ❛ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❞❡r✐✈❛t✐✈❡✱ ❜✉t ✐t ✐s st✐❧❧ ✐♠♣❧✐❝✐t ✕ ❜❡❝❛✉s❡ ✇❡ ❞♦♥✬t ❤❛✈❡ ❛ ❢♦r♠✉❧❛ ❢♦r y = y(x)✳ ❋♦rt✉♥❛t❡❧②✱ ✇❡ ❞♦♥✬t ♥❡❡❞ t❤❡ ✇❤♦❧❡ ❢✉♥❝t✐♦♥✱ ❥✉st ❛ s✐♥❣❧❡ ♣♦✐♥t ♦♥ ✐ts ❣r❛♣❤✿ √ √ 2 2 , y= . x= 2 2 ❲❡ s✉❜st✐t✉t❡ t❤❡s❡ ✐♥t♦ t❤❡ ❢♦r♠✉❧❛ ❛❜♦✈❡ t♦ ✜♥❞✿ x dy =− = −1 . √ √ dx √2 y √2 2 2 x=

2

, y=

x=

2

2

, y=

2

❋✐♥❛❧❧②✱ ❢r♦♠ t❤❡ ♣♦✐♥t✲s❧♦♣❡ ❢♦r♠✉❧❛ ✇❡ ♦❜t❛✐♥ t❤❡ ❛♥s✇❡r✿ √ √ ! 2 2 y− = −1 x − . 2 2 √ ❲❡ ❝♦✉❧❞ ✉s❡ t❤❡ ❡①♣❧✐❝✐t ❢♦r♠✉❧❛ y = 1 − x2 ✇✐t❤ t❤❡ s❛♠❡ r❡s✉❧t✿

√

dy ❈❘ −2x x === √ =− , 2 dx 1 − x2 2 1−x

2 ✳ ❍♦✇❡✈❡r✱ t❤❡ ✐♠♣❧✐❝✐t ❛♣♣r♦❛❝❤ r❡q✉✐r❡❞ ❛ s✐♥❣❧❡ ❞✐✛❡r❡♥t✐❛t✐♦♥ 2 ✭(y 2 )′ = 2y ✮ ✐♥ ❝♦♠♣❛r✐s♦♥✳ ❆❧s♦✱ t❤❡ ❢♦r♠✉❧❛ ✐s ♦♥❧② ❡①♣❧✐❝✐t ❢♦r t❤❡ ✉♣♣❡r ❤❛❧❢√♦❢ t❤❡ ❝✐r❝❧❡✳ ❋♦r ❛ ♣♦✐♥t ❜❡❧♦✇ t❤❡ x✲❛①✐s✱ ✇❡✬❞ ♥❡❡❞ t♦ st❛rt ♦✈❡r ❛♥❞ ✉s❡ t❤❡ ♦t❤❡r ❢♦r♠✉❧❛✱ y = − 1 − x2 ✳

❛❢t❡r ✇❡ s✉❜st✐t✉t❡ x =

dy ◆♦ ♠❛tt❡r ✇❤❛t t❤❡ ❛♣♣r♦❛❝❤ ✐s✱ t❤❡ ❞❡r✐✈❛t✐✈❡ ✐s ✉♥❞❡✜♥❡❞ ❛t x = ±1 ❜❡❝❛✉s❡ t❤❡ ❞❡♥♦♠✐♥❛t♦r dx ✐s 0✳ ❍♦✇ ❞♦ ✇❡ ✜♥❞ t❤❡ t❛♥❣❡♥t❄ ❋r♦♠ t❤❡ ✐♠♣❧✐❝✐t ❢♦r♠✉❧❛✱ ✇❡ ❝❛♥ ♣r♦❝❡❡❞ ✐♥ t✇♦ ❞✐r❡❝t✐♦♥s✿ ր y ❞❡♣❡♥❞s ♦♥ x ց x ❞❡♣❡♥❞s ♦♥ y

x2 + y 2 = 1

❚❤❡♥✱ ✇❡ ❝❛♥ tr② ✐♠♣❧✐❝✐t ❞✐✛❡r❡♥t✐❛t✐♦♥ ♦❢ t❤❡ s❛♠❡ ❡q✉❛t✐♦♥ ✕ ❜✉t ✇✐t❤ r❡s♣❡❝t t♦ y t❤✐s t✐♠❡✳ ❚❤❡ ❝♦♠♣✉t❛t✐♦♥ ✐s ✈❡r② s✐♠✐❧❛r✱ ❛♥❞ t❤❡ r❡s✉❧t ✐s✿

y dx =− . dy x ❚❤❡ ❢♦r♠✉❧❛ ✐s ❞❡✜♥❡❞ ❢♦r y = 0✱ ❛t t❤❡ ♣♦✐♥ts (−1, 0), (1, 0)✳ ❚❤❡♥✱ t❤❡ t❛♥❣❡♥t ❧✐♥❡ ✐s x − 1 = 0(y − 0)✱ ♦r x = 1✳

dx = 0 ❛t t❤❡s❡ ♣♦✐♥ts✳ ❚❤❡r❡❢♦r❡✱ dy

✹✳✼✳

❍♦✇ t♦ ❞✐✛❡r❡♥t✐❛t❡ r❡❧❛t✐♦♥s✿ ✐♠♣❧✐❝✐t❧②

✸✹✷

❊①❛♠♣❧❡ ✹✳✼✳✸✿ ❋♦❧✐✉♠ ♦❢ ❉❡s❝❛rt❡s

❚❤✐s ❝✉r✈❡ ✐s ❣✐✈❡♥ ❜② t❤❡ ❡q✉❛t✐♦♥

x3 + y 3 = 6xy

♣❧♦tt❡❞ ❜❡❧♦✇✿

▲❡t✬s ✜♥❞ s♦♠❡ t❛♥❣❡♥ts✳ ❲❡ ❞✐✛❡r❡♥t✐❛t❡ t❤❡ ❡q✉❛t✐♦♥ ❛s ❜❡❢♦r❡✿


d d x3 + y 3 = (6xy) . dx dx

❲❤✐❧❡ ✉s✐♥❣ t❤❡

❈❤❛✐♥ ❘✉❧❡✱ ✇❡ ♥♦t✐❝❡ t❤❛t ❡✈❡r② t✐♠❡ ✇❡ s❡❡ y ✱ t❤❡ ❢❛❝t♦r

dy ❛❧s♦ ❛♣♣❡❛rs✿ dx

d 3 d d (x ) + (y 3 ) = 6 (xy) =⇒ dx dx dx   dy dy 3x2 + 3y 2 · . =6 y+x dx dx ❙♦❧✈❡ ❢♦r

dy ✿ dx dy dy = 6y + 6x =⇒ dx dx dy (3y 2 − 6x) = 6y − 3x2 =⇒ dx dy 6y − 3x2 = 2 . ❆s ❧♦♥❣ ❛s t❤✐s ✐s♥✬t (0, 0). dx 3y − 6x 3x2 + 3y 2

❚❤❡ ❡♥❞ r❡s✉❧t ✐s✿ ■❢ ✇❡ ❦♥♦✇ t❤❡ ❧♦❝❛t✐♦♥ (x, y)✱ ✇❡ ❦♥♦✇ t❤❡ s❧♦♣❡ ♦❢ t❤❡ t❛♥❣❡♥t ❛t t❤❛t ♣♦✐♥t✳ ❋♦r dy ❡①❛♠♣❧❡✱ ❛t t❤❡ t✐♣ ♦❢ t❤❡ ❝✉r✈❡ ✇❡ ❤❛✈❡ x = y ✳ ❚❤❡r❡❢♦r❡✱ t❤❡ s❧♦♣❡ ✐s = −1✳ dx ❊①❡r❝✐s❡ ✹✳✼✳✹

❋✐♥❞ t❤❡ t❛♥❣❡♥ts ❛t t❤❡ ♦r✐❣✐♥✳ ■♥ ❜♦t❤ ❡①❛♠♣❧❡s✱ ✇❡ ❝❤♦s❡ ❝❛❧❝✉❧✉s ♦✈❡r ❛❧❣❡❜r❛✱ ❛♥❞ t❤❡ ❝❤♦✐❝❡ ❧❡❞ ✕ ✈✐❛ t❤❡ ❈❤❛✐♥ ❘✉❧❡ ✕ t♦ s✐♠♣❧❡r ❝♦♠♣✉t❛t✐♦♥s✳ ◆♦t❡ t❤❛t ✐♥ ❡✐t❤❡r ❡①❛♠♣❧❡✱ ✇❡ ❝❛♥ ❝✉t t❤❡ ❝✉r✈❡ ✐♥t♦ ♣✐❡❝❡s ❡❛❝❤ ♦❢ ✇❤✐❝❤ ✐s t❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥✿

✹✳✼✳ ❍♦✇ t♦ ❞✐✛❡r❡♥t✐❛t❡ r❡❧❛t✐♦♥s✿ ✐♠♣❧✐❝✐t❧②

✸✹✸

❊①❡r❝✐s❡ ✹✳✼✳✺

❋✐♥❞ ♦t❤❡r ✇❛②s t♦ ❞♦ ✐t✳ ■♥ s✉♠♠❛r②✱ ✐❢ t❤❡r❡ ✐s ❛ ❝✉r✈❡ ❣✐✈❡♥ ❜② ❛ r❡❧❛t✐♦♥✱ ✇❡ ❝❛♥ ❞❡❝❧❛r❡ ♦♥❡ ♦❢ t❤❡ ✈❛r✐❛❜❧❡ ✐♥❞❡♣❡♥❞❡♥t ❛♥❞ t❤❡ ♦t❤❡r ❞❡♣❡♥❞❡♥t ❛♥❞ ✜♥❞ ❛♥ ✐♠♣❧✐❝✐t r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❧❛tt❡r ♦✈❡r t❤❡ ❢♦r♠❡r✱ ✇❤✐❝❤ ✐s t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s❧♦♣❡✳ ◆♦✇✱ ✐♠♣❧✐❝✐t ❞✐✛❡r❡♥t✐❛t✐♦♥ ❛❧s♦ ❤❡❧♣s ✇✐t❤ s✐t✉❛t✐♦♥s ✇❤❡♥ t✇♦ ♦r ♠♦r❡ q✉❛♥t✐t✐❡s ❞❡♣❡♥❞ ♦♥ ❡❛❝❤ ♦t❤❡r ✈✐❛ ❛ r❡❧❛t✐♦♥ ❛s ✇❡❧❧ ♦♥ ❛ ❝♦♠♠♦♥ ✐♥❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡✱ ✉s✉❛❧❧② t❤❡ t✐♠❡✳ ■❢ ✇❡ ❞✐✛❡r❡♥t✐❛t❡ t❤✐s r❡❧❛t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ t✐♠❡✱ ✇❡ ❣❡t ❛ r❡❧❛t✐♦♥ ❛♠♦♥❣ t❤❡s❡ q✉❛♥t✐t✐❡s ❛♥❞ t❤❡✐r ❞❡r✐✈❛t✐✈❡s✳ ❚❤❡ r❡s✉❧t ✐s ❝❛❧❧❡❞ t❤❡ r❡❧❛t❡❞ r❛t❡s✳ ❊①❛♠♣❧❡ ✹✳✼✳✻✿ ❛✐r ❜❛❧❧♦♦♥

❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛♥ ❛✐r ❜❛❧❧♦♦♥✱ s♣❤❡r✐❝❛❧ ✐♥3 s❤❛♣❡✳ ❆✐r ✐s ♣✉♠♣❡❞ ✐♥ ✐t ❛♥❞ ❛t ❛ ❝❡rt❛✐♥ ♠♦♠❡♥t ♦❢ t✐♠❡ ✐t ✇❛s r❡❝♦r❞❡❞ t♦ ❜❡ ❛t t❤❡ r❛t❡ ♦❢ 5✐♥ /s❡❝ ✳ ❍♦✇ ❢❛st ❞♦❡s t❤❡ r❛❞✐✉s ❣r♦✇❄ ❙t❡♣ ✶ ✐♥ ✇♦r❞ ♣r♦❜❧❡♠s ✐s t♦ ✐♥tr♦❞✉❝❡ ✈❛r✐❛❜❧❡s✳ ▲❡t • t ❜❡ t✐♠❡✱ • V ❜❡ t❤❡ ✈♦❧✉♠❡✱ ❛♥❞ • r ❜❡ t❤❡ r❛❞✐✉s✳ ❲❡ ❝❛♥ ❛❧s♦ ♥❛♠❡ t❤❛t s♣❡❝✐✜❝ ♠♦♠❡♥t ♦❢ t✐♠❡✱ s❛②✱ t0 ✳ ◆❡①t✱ V ❞❡♣❡♥❞s ♦♥ t✳ ❊✈❡♥ t❤♦✉❣❤ t❤✐s ❞❡♣❡♥❞❡♥❝❡ ✇✐❧❧ r❡♠❛✐♥ ✉♥❦♥♦✇♥✱ ✇❡ ❞♦ ❦♥♦✇ ✐ts ❞❡r✐✈❛t✐✈❡ ❛t t❤❛t ♠♦♠❡♥t t0 ✳ ❚❤❡♥✱ dV = 5, dt t=t0

❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❝♦♥❞✐t✐♦♥✳ ❋✉rt❤❡r♠♦r❡✱ t❤✐s ✐s ❛ s♣❤❡r❡✱ s♦ t❤❡ ❦♥♦✇♥ ❢♦r♠✉❧❛ ❢♦r ✐ts ✈♦❧✉♠❡ ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✿ 4 V = πr3 . 3

❍❡r❡ ✇❡ s❡❡ t❤❛t V ❛❧s♦ ❞❡♣❡♥❞s ♦♥ r❀ ❛❧t♦❣❡t❤❡r✱ t❤✐s ✐s t❤❡ ❞❡♣❡♥❞❡♥❝✐❡s ✇❡ ❢❛❝❡✿ t → r ց ↓ V

❲❡ ❝♦✉❧❞ r❡✈❡rs❡ t❤❡ ❧❛st ❛rr♦✇ ❜② ✜♥❞✐♥❣ t❤❡ ✐♥✈❡rs❡✿ r=

r 3

3 V, 4π

❛♥❞ ❞❡❛❧ ✇✐t❤ t❤✐s ❝♦♠♣❧❡① ❛❧❣❡❜r❛✳ ■♥st❡❛❞✱ ✇❡ ❞✐✛❡r❡♥t✐❛t❡ t❤❡ ❡q✉❛t✐♦♥ ✐ts❡❧❢✳ ❲✐t❤ r❡s♣❡❝t t♦ t✱ ♦❢ ❝♦✉rs❡✦ ❚❤✉s✱ ✐❢ t✇♦ ✈❛r✐❛❜❧❡s ❛r❡ r❡❧❛t❡❞ ✭✈✐❛ ❛♥ ❡q✉❛t✐♦♥✮✱ t❤❡♥ s♦ ❛r❡ t❤❡✐r ❞❡r✐✈❛t✐✈❡s✱ ✐✳❡✳✱ t❤❡

✹✳✼✳

❍♦✇ t♦ ❞✐✛❡r❡♥t✐❛t❡ r❡❧❛t✐♦♥s✿ ✐♠♣❧✐❝✐t❧②

✸✹✹

r❛t❡s ♦❢ ❝❤❛♥❣❡✳ ❍❡♥❝❡✱ ✏r❡❧❛t❡❞ r❛t❡s✑✳ ❑❡❡♣✐♥❣ ✐♥ ♠✐♥❞ t❤❛t ❜♦t❤

V

❛♥❞

r

❛r❡ ❢✉♥❝t✐♦♥s ♦❢ t✐♠❡✱ ✇❡ ❞✐✛❡r❡♥t✐❛t❡ t❤❡ r❡❧❛t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦

t✿

4 V = πr3 . 3

❚❤❡ ❧❡❢t✲❤❛♥❞ s✐❞❡ ✐s ✈❡r② s✐♠♣❧❡✿

❜✉t ✐♥ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡✱

r(t)3

d dV V = , dt dt

✐s ❛ ❝♦♠♣♦s✐t✐♦♥✦ ❲❡ ❞✐✛❡r❡♥t✐❛t❡ ❜② t❤❡ ❈❤❛✐♥ ❘✉❧❡✿

d dt ❚❤✉s✱ ✇❡ ❤❛✈❡✿

❘❡❝❛❧❧ t❤❛t t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢

V

✐s



4 3 πr 3

r

4 dr = π · 3r2 . 3 dt

dr dV = 4π · 3r2 . dt dt 5 ✭❛t t = t0 ✮✱ s♦ ✇❡ ❤❛✈❡✿

5 = 4πr2 ❚❤❡r❡❢♦r❡✱ t❤❡ r❛t❡s ♦❢ ❣r♦✇t❤ ♦❢



dr 5 dr . =⇒ = dt dt 4πr2

❢♦r t❤❡ ❝❛s❡s

r = 1, r = 2, r = 3

❛r❡✿

dr 5 = dt 4π dr 5 r=2: = dt 16π 5 dr = r=3: dt 36π r=1:

❙♦✱ t❤❡ ❡✛❡❝t ♦♥ t❤❡ r❛❞✐✉s ♦❢ ♣✉♠♣✐♥❣ ✐♥ t❤❡ ❛✐r ✐s s♠❛❧❧❡r ❢♦r ❧❛r❣❡r r❛❞✐✐✳

❊①❡r❝✐s❡ ✹✳✼✳✼

❍♦✇ ❢❛st ✐s t❤❡ s✉r❢❛❝❡ ❛r❡❛ ❣r♦✇✐♥❣❄

❊①❛♠♣❧❡ ✹✳✼✳✽✿ s❧✐❞✐♥❣ ❧❛❞❞❡r

❙✉♣♣♦s❡ ❛

10✲❢♦♦t

❧❛❞❞❡r st❛♥❞s ❛❣❛✐♥st t❤❡ ✇❛❧❧ ❛♥❞ ✐ts ❜♦tt♦♠ ✐s s❧✐❞✐♥❣ ❛✇❛② ❢r♦♠ t❤❡ ✇❛❧❧✳ ❍♦✇

❢❛st ✐s t❤❡ t♦♣ ♠♦✈✐♥❣ ✇❤❡♥ ✐t ✐s

6

❢t ❢r♦♠ t❤❡ ✢♦♦r ❛♥❞ t❤❡ ❜♦tt♦♠ ✐s s❧✐❞✐♥❣ ❛t

■♥tr♦❞✉❝❡ ✈❛r✐❛❜❧❡s✿

• x t❤❡ ❞✐st❛♥❝❡ ♦❢ t❤❡ ❜♦tt♦♠ ❢r♦♠ t❤❡ ✇❛❧❧✱ • y t❤❡ ❞✐st❛♥❝❡ ♦❢ t❤❡ t♦♣ ❢r♦♠ t❤❡ ✢♦♦r✱ ❜♦t❤ • t t❤❡ t✐♠❡✳

❢✉♥❝t✐♦♥s ♦❢

2

❢t✴s❡❝❄

✹✳✼✳ ❍♦✇ t♦ ❞✐✛❡r❡♥t✐❛t❡ r❡❧❛t✐♦♥s✿ ✐♠♣❧✐❝✐t❧②

✸✹✺

▲❡t✬s ❞❡♥♦t❡ t❤❡ ♠♦♠❡♥t ♦❢ t✐♠❡ t0 ✳ ❲❡ tr❛♥s❧❛t❡ t❤❡ ✐♥❢♦r♠❛t✐♦♥✱ ❛s ✇❡❧❧ ❛s t❤❡ q✉❡st✐♦♥✱ ❛❜♦✉t t❤❡ ✈❛r✐❛❜❧❡s ✐♥t♦ ❡q✉❛t✐♦♥s✿ q✉❛♥t✐t✐❡s✿

❢✉♥❝t✐♦♥s✿

❛❧✇❛②s x2 + y 2 = 102 (x(t))2 + (y(t))2 = 102 ❢♦r ❛❧❧ t ♥♦✇ ♥♦✇ ♥♦✇ ♥♦✇

x(t0 ) =?

x =? dx =2 dt y=6 dy =? dt

x′ (t0 ) = 2 y(t0 ) = 6 y ′ (t0 ) =?

❚❤❛t✬s ❛ ♣✉r❡❧② ♠❛t❤❡♠❛t✐❝❛❧ ♣r♦❜❧❡♠ t♦ ❜❡ s♦❧✈❡❞✳ ❲❡ ❞✐✛❡r❡♥t✐❛t❡ t❤❡ ❡q✉❛t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ✐♥❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡✱ t✿
d 2 d x + y2 = (100) ❲❡ ❛♣♣❧② ❈❘ t✇✐❝❡✳ dt dt dx dy dy 2x + 2y =0 ❙♦❧✈❡ ❢♦r . dt dt dt x dx dy =− dt y dt

❚❤✐s ✐s ❛ r❡❧❛t✐♦♥ ♦❢ ❢♦✉r ❢✉♥❝t✐♦♥s✦ ❆t t❤❡ s♣❡❝✐✜❝ ♠♦♠❡♥t t = t0 ✱ ✇❡ ✜♥❞ x = 8 ❢r♦♠ x2 + y 2 = 100✳ ❋r♦♠ t❤✐s ❛♥❞ t❤❡ ❢❛❝t t❤❛t y = 6✱ ✇❡ ❝♦♥❝❧✉❞❡✿ ❊①❡r❝✐s❡ ✹✳✼✳✾

dy 8 8 =− 2=− . dt t=t0 6 3

❙♦❧✈❡ t❤❡ ♣r♦❜❧❡♠ ❢♦r t❤❡ ♠♦♠❡♥t ✇❤❡♥ t❤❡ ❧❛❞❞❡r ✐s ✉♣r✐❣❤t✳ ❊①❡r❝✐s❡ ✹✳✼✳✶✵

❙♦❧✈❡ t❤❡ ♣r♦❜❧❡♠ ❢♦r t❤❡ ♠♦♠❡♥t ✇❤❡♥ t❤❡ ❧❛❞❞❡r ❤✐ts t❤❡ ✢♦♦r✳ ❊①❡r❝✐s❡ ✹✳✼✳✶✶

P❧♦t y = y(t) ❜❛s❡❞ ♦♥ t❤✐s ✐♥❢♦r♠❛t✐♦♥✳ ❊①❡r❝✐s❡ ✹✳✼✳✶✷

❙❡t ✉♣ ❛♥❞ s♦❧✈❡ ❛ s✐♠✐❧❛r ♣r♦❜❧❡♠ ❢♦r t❤❡ ❝r❛♥❦s❤❛❢t✿

✹✳✽✳

✸✹✻

❘❡❧❛t❡❞ r❛t❡s✿ r❛❞❛r ❣✉♥

✹✳✽✳ ❘❡❧❛t❡❞ r❛t❡s✿ r❛❞❛r ❣✉♥

Pr♦❜❧❡♠✿ ❙✉♣♣♦s❡ ②♦✉ ❛r❡ ❞r✐✈✐♥❣ ❛t ❛ s♣❡❡❞ ♦❢ 80 ♠♣❤ ✇❤❡♥ ②♦✉ s❡❡ ❛ ♣♦❧✐❝❡ ❝❛r ♣♦s✐t✐♦♥❡❞ 40 ❢❡❡t ♦✛ t❤❡ r♦❛❞✳ ❲❤❛t ✐s t❤❡ r❛❞❛r ❣✉♥✬s r❡❛❞✐♥❣❄

❋✐rst✱ ❤♦✇ ❞♦❡s t❤❡ r❛❞❛r ❣✉♥ ✇♦r❦❄ ■♥ ❢❛❝t✱ ❤♦✇ ❞♦❡s ❛ r❛❞❛r ✇♦r❦❄ ❆ s✐❣♥❛❧ ✐s s❡♥t✱ ✐t ❜♦✉♥❝❡s ♦✛ ❛♥ ♦❜❥❡❝t✱ ❛♥❞✱ ✇❤❡♥ ✐t ❝♦♠❡s ❜❛❝❦✱ t❤❡ t✐♠❡ ❧❛♣s❡ ✐s r❡❝♦r❞❡❞✳ ❚❤❡♥✱ t❤❡ ❞✐st❛♥❝❡ t♦ t❤❡ ♦❜❥❡❝t ✐s ❝♦♠♣✉t❡❞ ❛s ❢♦❧❧♦✇s✿ S = s✐❣♥❛❧✬s s♣❡❡❞ · t✐♠❡ ♣❛ss❡❞ |

❆ r❛❞❛r ❣✉♥ ❞♦❡s t❤✐s t✇✐❝❡✳

{z

❦♥♦✇♥

} |

{z

♠❡❛s✉r❡❞

}

❆ s✐❣♥❛❧ ✐s s❡♥t✱ ✐t ❝♦♠❡s ❜❛❝❦✱ t❤❡ t✐♠❡ ✐s ♠❡❛s✉r❡❞ ❛♥❞ t❤❡ ❞✐st❛♥❝❡ ❝♦♠♣✉t❡❞✳ ❚❤❡♥ t❤❡ s❡❝♦♥❞ t✐♠❡✿ • S1 = s♣❡❡❞ · t✐♠❡✱ ❛t t✐♠❡ t = t1 ✱

• S2 = s♣❡❡❞ · t✐♠❡✱ ❛t t✐♠❡ t = t2 ✳

❚❤❡♥✱ t❤❡ r❡❛❞✐♥❣ ✐s ❝♦♠♣✉t❡❞ ❛s t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ s✐❣♥❛❧ s♦✉r❝❡ ✭t❤❡ ♣♦❧✐❝❡ ❝❛r✮ ❛♥❞ t❤❡ t❛r❣❡t ✭②♦✉r ❝❛r✮✿ ❡st✐♠❛t❡❞ s♣❡❡❞ =

❝❤❛♥❣❡ ♦❢ ❞✐st❛♥❝❡ . t✐♠❡ ❜❡t✇❡❡♥ s✐❣♥❛❧s

◆♦ r❛❞❛r ❣✉♥ ❝❛♥ ❞♦ ❜❡tt❡r t❤❛♥ t❤❛t✦

■❢ t❤❡ s✐❣♥❛❧ ✐s ❝♦♥t✐♥✉❡❞ t♦ ❜❡ ❡♠✐tt❡❞✱ t❤❡ ❞✐st❛♥❝❡ ✐s ❛ ❢✉♥❝t✐♦♥ S = S(t) ❦♥♦✇♥ ❛t ❛ s❡q✉❡♥❝❡ ♦❢ ♠♦♠❡♥ts ♦❢ t✐♠❡ t0 , t1 , ...✳ ❚❤❡♥ t❤❡ r❛❞❛r ❝♦♠♣✉t❡s t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ S ✿ ❘❛❞❛r r❡❛❞✐♥❣ = ✇❤❡r❡

∆S , ∆t

✹✳✽✳

✸✹✼

❘❡❧❛t❡❞ r❛t❡s✿ r❛❞❛r ❣✉♥

• ∆S = Sn+1 − Sn ✐s t❤❡ ❝❤❛♥❣❡ ♦❢ t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ t✇♦ ❝❛rs✱ • ∆t = h = tn+1 − tn ✐s t❤❡ t✐♠❡ ❜❡t✇❡❡♥ s✐❣♥❛❧s✳

◆♦✇ t❤❡ q✉❡st✐♦♥✱ ✐s t❤❡ r❡❛❞✐♥❣ ♦❢ t❤❡ r❛❞❛r ❣✉♥ 80 ♠✴❤❄ ❚♦ ❣❡t ❛♥ ✐❞❡❛ ♦❢ ✇❤❛t ❝❛♥ ❤❛♣♣❡♥✱ ❝♦♥s✐❞❡r t❤✐s ❡①tr❡♠❡ ❡①❛♠♣❧❡✿ ❲❤❛t ✐❢ ②♦✉ ❛r❡ ❥✉st ♣❛ss✐♥❣ ✐♥ ❢r♦♥t ♦❢ t❤❡ ♣♦❧✐❝❡ ❝❛r✱ ❧✐❦❡ t❤✐s❄

■t ✐s ❝♦♥❝❡✐✈❛❜❧❡ t❤❛t ❛t t✐♠❡ t1 ②♦✉r ❝❛r ✐s t❤❡ s❛♠❡ ❞✐st❛♥❝❡ ❢r♦♠ t❤❡ ✐♥t❡rs❡❝t✐♦♥ ❛s ✐t ✐s ♣❛st t❤❡ ✐♥t❡rs❡❝t✐♦♥ ❛t t✐♠❡ t2 ✳ ❚❤❡♥ ∆S = 0 =⇒

❚❤❡ r❡❛❞✐♥❣ ✐s ♦✛ ❜② ❛ ❧♦t✦

∆S = 0. ∆t

❲❡ ❝❛♥ s❡❡ t❤❡ s♦✉r❝❡ ♦❢ t❤❡ ❡rr♦r ❜❡❧♦✇✿

❚❤❡ r❛❞❛r ❣✉♥ t❡❧❧s ♦♥❧② ❤♦✇ ❢❛st t❤❡ ♦t❤❡r ❝❛r ✐s ❣♦✐♥❣ t❤r♦✉❣❤ t❤❡s❡ ❝♦♥❝❡♥tr✐❝ ❝✐r❝❧❡s✱ ♥♦t ✐ts ❛❝t✉❛❧ s♣❡❡❞✳ ❚❤❡ r❡❛s♦♥ ❢♦r t❤❡ ❡rr♦r ✐s t❤❛t t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ r❡❞ ❝❛r ✐s ❛s❦❡✇✳ ▼❡❛♥✇❤✐❧❡ t❤❡ r❡❛❞✐♥❣ ❢♦r t❤❡ ❣r❡❡♥ ❝❛r ✇♦✉❧❞ ❜❡ ❛❝❝✉r❛t❡ ❜❡❝❛✉s❡ ✐ts ❞✐r❡❝t✐♦♥ ✐s str❛✐❣❤t ❛t t❤❡ r❛❞❛r✳ ▲❡t✬s st❛rt ♦✈❡r✳ ❲❡ ❤❛✈❡ ❛ r✐❣❤t tr✐❛♥❣❧❡ t❤❛t ❝❤❛♥❣❡s ✐ts s❤❛♣❡ ✇✐t❤ t✐♠❡✿

❚❤❡s❡ ❛r❡ t❤❡ q✉❛♥t✐t✐❡s✿

✹✳✽✳

✸✹✽

❘❡❧❛t❡❞ r❛t❡s✿ r❛❞❛r ❣✉♥

• S ✱ t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ ♣♦❧✐❝❡ ❝❛r t♦ ②♦✉rs

• P ✱ t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ ②♦✉r ❝❛r t♦ t❤❡ ✐♥t❡rs❡❝t✐♦♥

• t✱ t❤❡ t✐♠❡✱ t❤❡ ✐♥❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡

• D = 40✱ ❞✐st❛♥❝❡ ❢r♦♠ t❤❡ ♣♦❧✐❝❡ ❝❛r t♦ t❤❡ r♦❛❞

❙✐♥❝❡ 80 ♠❧✴❤ ✐s ②♦✉r s♣❡❡❞✱ ✇❡ ❤❛✈❡ ❞♦❡s ♠❡❛s✉r❡ ✐♥ r❡❛❧✐t② ✐s

dS ✦ dt

dP = 80✳ ❚❤❛t✬s ✇❤❛t t❤❡ r❛❞❛r ❣✉♥ ✐s ♠❡❛♥t t♦ ❞❡t❡❝t✳ ❇✉t ✇❤❛t ✐t dt

❚❤✐s ✐s ✇❤❛t ✇❡ ♥❡❡❞ t♦ ✜♥❞ ♦✉t✿ ◮ ❍♦✇ ❣♦♦❞ ❛♥ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ r❡❛❧ ✈❡❧♦❝✐t②

dP dS ✐s t❤❡ ♣❡r❝❡✐✈❡❞ ✈❡❧♦❝✐t② ❄ dt dt

❖✉r s♣r❡❛❞s❤❡❡t ❝♦♥t❛✐♥s ❛ ❝♦❧✉♠♥ ♦❢ ❧♦❝❛t✐♦♥s P ♦❢ ②♦✉r ❝❛r ✭❞✐st❛♥❝❡s t♦ t❤❡ ✐♥t❡rs❡❝t✐♦♥✮ ❢♦✉♥❞ ❜② ♠❡❛♥s ♦❢ t❤❡ ❢❛♠✐❧✐❛r ❢♦r♠✉❧❛ Pn+1 = Pn + 80h ,

✇❤❡r❡ h ✐s t❤❡ ✐♥❝r❡♠❡♥t ♦❢ t✐♠❡✳ ❚❤❡ ♥❡①t ❝♦❧✉♠♥ ✐s ❢♦r t❤❡ ❞✐st❛♥❝❡ S t♦ t❤❡ ♣♦❧✐❝❡ ❝❛r ✭♣❧♦tt❡❞ ✜rst✮✱ ❢♦✉♥❞ ✈✐❛ t❤❡ P②t❤❛❣♦r❡❛♥ ❚❤❡♦r❡♠ ✿ P 2 + D2 = S 2 .

❚❤❡ t❤✐r❞ ❝♦❧✉♠♥ ✐s t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ S ✭♣❧♦tt❡❞ s❡❝♦♥❞✮✿

❆s ✇❡ ❝❛♥ s❡❡✱ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ✐s t❤❡ ❜❡st ✇❤❡♥ t❤❡ r❡❞ ❝❛r ✐s ❛✇❛② ❢r♦♠ t❤❡ ✐♥t❡rs❡❝t✐♦♥✳ ❇✉t✱ ✇✐t❤✐♥ 75 ❢❡❡t ❢r♦♠ t❤❡ ✐♥t❡rs❡❝t✐♦♥✱ t❤❡ r❡❛❞✐♥❣ ✇✐❧❧ ❜❡ ❧❡ss t❤❛♥ 70 ♠♣❤✦ ❋r♦♠ t❤✐s ❛♣♣r♦①✐♠❛t✐♦♥✱ ✇❡ ♠♦✈❡ t♦ ❛ ♠♦r❡ ♣r❡❝✐s❡ ❛♥❛❧②s✐s✳ ❲❡ ❤❛✈❡ ❛ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ t✇♦ ✈❛r✐❛❜❧❡s ✈✐❛ t❤❡ P②t❤❛❣♦r❡❛♥ ❚❤❡♦r❡♠✿ P 2 + D2 = S 2 ← ❚❤❡s❡ ❛r❡♥✬t ♥✉♠❜❡rs✱ ❜✉t ✈❛r✐❛❜❧❡s✱ ✐✳❡✳✱ ❢✉♥❝t✐♦♥s✳

✹✳✽✳

✸✹✾

❘❡❧❛t❡❞ r❛t❡s✿ r❛❞❛r ❣✉♥

❚❤✐s ❡q✉❛t✐♦♥ ❝♦♥♥❡❝ts P ❛♥❞ S ✱ ❜✉t ♥♦t

dP dS ❛♥❞ ②❡t✳ ❲❡ ❞✐✛❡r❡♥t✐❛t❡ t❤❡ ❡q✉❛t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ t✿ dt dt



d d = P 2 + D2 S 2 =⇒ dt dt dS dD dP = 2S · + 2D =⇒ 2P · dt dt dt |{z} =0

P·

dP dt

dS =⇒ dt P dP = . S dt

=S·

dS dt

❚❤✉s✱ ✇❡ ✜♥❛❧❧② ❤❛✈❡ ❛ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡s❡ ❢✉♥❝t✐♦♥s✳ ❲❡ ♣❧♦t t❤✐s ❢✉♥❝t✐♦♥ ❜❡❧♦✇✱ t♦ ❝♦♥✜r♠ ♦✉r ❡❛r❧✐❡r ❝♦♥❝❧✉s✐♦♥s✿

❆♥♦t❤❡r ✇❛② t♦ ❛♣♣r♦❛❝❤ t❤❡ ♣r♦❜❧❡♠ ✐s t❤❡ P❖❱ ♦❢ t❤❡ ❞r✐✈❡r✳ ▲❡t α ❜❡ t❤❡ ❛♥❣❧❡ ❜❡t✇❡❡♥ t❤❡ r♦❛❞ ❛❤❡❛❞ ♦❢ ②♦✉ ❛♥❞ t❤❡ ❞✐r❡❝t✐♦♥ t♦ t❤❡ ♣♦❧✐❝❡ ❝❛r✿

❚❤❡♥✿ cos α =

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿

P . S

❘❛❞❛r r❡❛❞✐♥❣ = ❛❝t✉❛❧ s♣❡❡❞ · cos α .

❚❤✐s ✐s ❤♦✇ ❞♦❡s α ❝❤❛♥❣❡ ❛s ②♦✉ ❞r✐✈❡✿

❚❤❡s❡ ❛r❡ t❤❡ ❝♦♥❝❧✉s✐♦♥s✿ • ❊❛r❧②✱ α ✐s ❝❧♦s❡ t♦ 0✱ s♦ cos α ❝❧♦s❡ t♦ 1✱ ❛♥❞✱ t❤❡r❡❢♦r❡✱

dS ✐s ❝❧♦s❡ t♦ 80✳ dt

✹✳✾✳

❚❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ✐♥✈❡rs❡ ❢✉♥❝t✐♦♥

•

❚❤❡♥✱ ❛s

•

■♥ t❤❡ ♠✐❞❞❧❡✱ ✇❡ ❤❛✈❡

•

❆s

•

▲❛t❡r✱

α

α

✐♥❝r❡❛s❡s✱

♣❛ss❡s

α

π ✱ cos α 2

❛♣♣r♦❛❝❤❡s

cos α α=

❞❡❝r❡❛s❡s t♦✇❛r❞

0✱

❛♥❞ s♦ ❞♦❡s

dS ✳ dt

dS π ✱ cos α = 0✱ = 0✳ 2 dt

❞❡❝r❡❛s❡s t♦ ♥❡❣❛t✐✈❡ ✈❛❧✉❡s✱ ❛♥❞ s♦ ❞♦❡s

π✱

❛♥❞

cos α

❈♦♥❝❧✉s✐♦♥✿ ❚❤❡ r❛❞❛r ❣✉♥ ❛❧✇❛②s

❯♥❧❡ss✱ t❤❡ ♣♦❧✐❝❡ ❝❛r ✐s

✸✺✵

❛♣♣r♦❛❝❤❡s

1✱

❛♥❞✱ t❤❡r❡❢♦r❡✱

✉♥❞❡r ❡st✐♠❛t❡s ②♦✉r s♣❡❡❞✿

dS ✳ dt dS ❛♣♣r♦❛❝❤❡s 80✳ dt

dS < 80 . dt

♦♥ t❤❡ r♦❛❞✦

■♥ t❤❛t ❝❛s❡✱ ✇❤❛t ❝❛♥ ②♦✉ ❞♦ t♦ ✏✐♠♣r♦✈❡✑ t❤❡ r❡❛❞✐♥❣❄ ❲❤❛t ❞♦ ②♦✉ ✇❛♥t

α

t♦ ❜❡ ✕ ❛s ❧❛r❣❡ ❛s ♣♦ss✐❜❧❡✦

❊①❡r❝✐s❡ ✹✳✽✳✶

❈♦♥s✐❞❡r t❤❡ P❖❱ ♦❢ t❤❡ ♣♦❧✐❝❡♠❛♥ ❛♥❞ t❤❡ ♦t❤❡r ❛♥❣❧❡

β

♦❢ t❤❡ tr✐❛♥❣❧❡✳

❊①❡r❝✐s❡ ✹✳✽✳✷

❙❤♦✇ t❤❛t✱ ✇✐t❤ ♣r❡♣❛r❛t✐♦♥✱ ✐t ✐s ♣♦ss✐❜❧❡ ❢♦r t❤❡ ♣♦❧✐❝❡♠❛♥ t♦ ✜♥❞ t❤❡ ❡①❛❝t ✈❛❧✉❡ ♦❢ t❤❡ ♦t❤❡r ❝❛r✬s s♣❡❡❞✳

✹✳✾✳ ❚❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ✐♥✈❡rs❡ ❢✉♥❝t✐♦♥

❚❤❡ ❧✐st ♦❢ ❢✉♥❝t✐♦♥s ✇❡ ❝❛♥♥♦t ❞✐✛❡r❡♥t✐❛t❡ ②❡t r❡♠❛✐♥s ❧♦♥❣✿

√

x, ln x, arcsin x, ...

❲❤❛t ❞♦ t❤❡s❡ ❤❛✈❡ ✐♥ ❝♦♠♠♦♥❄ ❚❤❡② ❛r❡ t❤❡ ✐♥✈❡rs❡s ♦❢ ❢✉♥❝t✐♦♥s ✇✐t❤ ❦♥♦✇♥ ❞❡r✐✈❛t✐✈❡s✦

✹✳✾✳

❚❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ✐♥✈❡rs❡ ❢✉♥❝t✐♦♥

✸✺✶

▲❡t✬s r❡❝❛❧❧ ✭❢r♦♠ ❱♦❧✉♠❡ ✶✱ ❈❤❛♣t❡r ✶P❈✲✸✮ t❤❛t ❢♦r ❛ ❣✐✈❡♥ ♦♥❡✲t♦✲♦♥❡ ❛♥❞ ♦♥t♦ ❢✉♥❝t✐♦♥ −1 ✐s t❤❡ ❢✉♥❝t✐♦♥✱ x = f (y)✱ t❤❛t s❛t✐s✜❡s

✐♥✈❡rs❡

y = f (x)✱

f −1 (y) = x ⇐⇒ f (x) = y . ❚❤❡ ✐❞❡❛ ✐s t❤❛t ❛ ❢✉♥❝t✐♦♥ ❛♥❞ ✐ts ✐♥✈❡rs❡ r❡♣r❡s❡♥t t❤❡

• x

❛♥❞

y

❛r❡ r❡❧❛t❡❞ ✇❤❡♥

y = F (x)✱

• x

❛♥❞

y

❛r❡ r❡❧❛t❡❞ ✇❤❡♥

x = F −1 (y)✳

s❛♠❡ r❡❧❛t✐♦♥ ✿

♦r

❋♦r ❡①❛♠♣❧❡✱ t❤❡s❡ ❛r❡ ♣❛✐rs ♦❢ ❢✉♥❝t✐♦♥s ✐♥✈❡rs❡ t♦ ❡❛❝❤ ♦t❤❡r✿

y =x+2

✈s✳

y = 3x

✈s✳

2

y=x y = ex ❈❛♥ ✇❡ ❡①♣r❡ss

✈s✳ ✈s✳

x=y−2 1 x= y 3 √ x= y x = ln y

❢♦r ❢♦r

x, y ≥ 0 y>0

t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ✐♥✈❡rs❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ✐♥ t❡r♠s ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❄

❲❡ ✇✐❧❧ ✉t✐❧✐③❡ t❤❡ ❢♦❧❧♦✇✐♥❣✱ ❛❧❣❡❜r❛✐❝✱ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ✐♥✈❡rs❡ ✭s❡❡♥ ✐♥ ❱♦❧✉♠❡ ✶✱ ❈❤❛♣t❡r ✶P❈✲✸✮✿

f −1 (f (x))
= x f f −1 (y) = y

❢♦r ❛❧❧ ❢♦r ❛❧❧

x y

❍❡r❡ ✐s ❛ ✢♦✇❝❤❛rt r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤✐s ✐❞❡❛✿

x →

f

→ y →

f −1

→ x (s❛♠❡ x)

y →

f −1

→ x →

f

→ y (s❛♠❡ y)

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❤❡♥ ❝♦♠❜✐♥❡❞ t❤❡② ❦✐❧❧ ❡❛❝❤ ♦t❤❡r✿

▲❡t✬s r❡❝❛❧❧ t❤❛t t❤❡ ✐♥✈❡rs❡ ✏✉♥❞♦❡s✑ t❤❡ ❡✛❡❝t ♦❢ t❤❡ ❢✉♥❝t✐♦♥✿

❲❡ ❝❛♥ s❡❡ ❤♦✇ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢

x

❜❡❝♦♠❡s t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢

y✱

✉♥❞❡r

f

❛♥❞

f −1 ✿

✐ts

✹✳✾✳

❚❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ✐♥✈❡rs❡ ❢✉♥❝t✐♦♥

✸✺✷

❚❤❡r❡ ✐s ♥♦ ♦t❤❡r r❡❧❛t✐♦♥✦ ❊①❛♠♣❧❡ ✹✳✾✳✶✿ tr❛♥s❢♦r♠❛t✐♦♥s

❚❤❡ ✐❞❡❛ t❤❛t t❤❡ ✐♥✈❡rs❡ r❡✈❡rs❡s t❤❡ ❡✛❡❝t ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❛♣♣❧✐❡s ❡s♣❡❝✐❛❧❧② ✇❡❧❧ t♦ tr❛♥s❢♦r♠❛t✐♦♥s ✿

◆♦✇✱ ✇❤❛t ✐s t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✭❛♥❞ t❤❡ ❞❡r✐✈❛t✐✈❡✮ ♦❢ ❛ tr❛♥s❢♦r♠❛t✐♦♥❄ ■t ✐s ✐ts str❡t❝❤ r❛t✐♦✳ ◆♦✇✱ t❤✐s ✐s ❝♦♠♠♦♥ s❡♥s❡✿ ◮ ■❢ f str❡t❝❤❡s t❤❡ x✲❛①✐s ❛t t❤❡ r❛t❡ ♦❢ k ✭❛t x = a✮✱ t❤❡♥ f −1 s❤r✐♥❦s t❤❡ y ✲❛①✐s ❛t t❤❡ r❛t❡ ♦❢ k ✭❛t b = f (a)✮✳ ❙♦✱ ✇❡ ❤❛✈❡ ❛ ♠❛t❝❤✿ 1 ❢♦r f −1 . k ❢♦r f ❛♥❞ k

■t✬s t❤❡ r❡❝✐♣r♦❝❛❧ ✦ ❊①❛♠♣❧❡ ✹✳✾✳✷✿ ❧✐♥❡❛r ❢✉♥❝t✐♦♥s

❲❡ ❝❛♥ ❛❧s♦ ❣✉❡ss t❤✐s r❡❧❛t✐♦♥ ❢r♦♠ t❤❡ ❢♦❧❧♦✇✐♥❣ s✐♠♣❧❡ ♣✐❝t✉r❡ t❤❛t ❛♣♣❧✐❡s t♦ ❧✐♥❡❛r ❢✉♥❝t✐♦♥s✿

❆s t❤❡ xy ✲♣❧❛♥❡ ✐s ✢✐♣♣❡❞ ❛❜♦✉t t❤❡ ❞✐❛❣♦♥❛❧✱ t❤✐s ✐s ✇❤❛t ❤❛♣♣❡♥s✿ s❧♦♣❡ ♦❢ f =

1 1 ❝❤❛♥❣❡ ♦❢ y A . = = = ❝❤❛♥❣❡ ♦❢ x B B/A s❧♦♣❡ ♦❢ f −1

✹✳✾✳

❚❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ✐♥✈❡rs❡ ❢✉♥❝t✐♦♥

❲❡ ❤❛✈❡ ❝♦♥❥❡❝t✉r❡❞ ❛ ❢♦r♠✉❧❛✿

✸✺✸ 1 ∆f −1 = ∆f . ∆y ∆x

❊✈❡♥ t❤♦✉❣❤ t❤❡ ❞❡r✐✈❛t✐✈❡s ❛r❡♥✬t ❢r❛❝t✐♦♥s✱ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts✱ ✐✳❡✳✱ t❤❡ s❧♦♣❡s ♦❢ t❤❡ s❡❝❛♥t ❧✐♥❡s✱ ❛r❡✿

❚❤✐s ❛♥❛❧②s✐s ♣r♦✈❡s t❤❡ ❢♦❧❧♦✇✐♥❣✿

❚❤❡♦r❡♠ ✹✳✾✳✸✿ ❉✐✛❡r❡♥❝❡ ◗✉♦t✐❡♥t ♦❢ ■♥✈❡rs❡ ❚❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ t❤❡ ✐♥✈❡rs❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ✐s ❢♦✉♥❞ ❛s t❤❡ r❡❝✐♣r♦❝❛❧ ♦❢ t❤❡ ✐ts ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ❢♦r ❛♥② ❢✉♥❝t✐♦♥ f ❞❡✜♥❡❞ ❛t t✇♦ ❛❞❥❛❝❡♥t ♥♦❞❡s x ❛♥❞ x + ∆x ♦❢ ❛ ♣❛rt✐t✐♦♥ ✇✐t❤ f (x) 6= f (x + ∆x) s♦ t❤❛t ✐ts ✐♥✈❡rs❡ ❢✉♥❝t✐♦♥ f −1 ✐s ❞❡✜♥❡❞ ❛t t❤❡ t✇♦ ❛❞❥❛❝❡♥t ♥♦❞❡s f (x) ❛♥❞ f (x + ∆x) ♦❢ ❛ ♣❛rt✐t✐♦♥✱ ✇❡ ❤❛✈❡ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ✭❞❡✜♥❡❞ ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s c ❛♥❞ q ✇✐t❤✐♥ t❤❡s❡ ❡❞❣❡s ♦❢ t❤❡ t✇♦ ♣❛rt✐t✐♦♥s r❡s♣❡❝t✐✈❡❧②✮ s❛t✐s❢②✿ ∆f −1 (q) = ∆y

1 ∆f (c) ∆x

❋♦r t❤❡ ❞❡r✐✈❛t✐✈❡s✱ ✇❡ ❥✉st t❛❦❡ t❤❡ ❧✐♠✐t ∆x → 0✿

❚❤❡♦r❡♠ ✹✳✾✳✹✿ ❉❡r✐✈❛t✐✈❡ ♦❢ ■♥✈❡rs❡ ❚❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ✐♥✈❡rs❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ✐s ❢♦✉♥❞ ❛s t❤❡ r❡❝✐♣r♦❝❛❧ ♦❢ t❤❡ ✐ts ❞❡r✐✈❛t✐✈❡✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ❢♦r ❛♥② ♦♥❡✲t♦✲♦♥❡ ❢✉♥❝t✐♦♥ y = f (x) ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t x = a✱ ✐ts

✹✳✾✳

❚❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ✐♥✈❡rs❡ ❢✉♥❝t✐♦♥

✸✺✹

✐♥✈❡rs❡ x = f −1 (y) ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t b = f (a)✱ ❛♥❞ ✇❡ ❤❛✈❡✿ df −1 (b) = dy

1 df (a) dx

❲❛r♥✐♥❣✦ ❚❤❡ ✈❛r✐❛❜❧❡s ✐♥ t❤❡ ❢♦r♠✉❧❛ ❞♦♥✬t ♠❛t❝❤✳

Pr♦♦❢✳ ❚❤❡ ❢♦r♠✉❧❛ ❢♦❧❧♦✇s ❢r♦♠ t❤❡s❡ ❧✐♠✐ts✿

∆x ∆y · =1 ∆y ∆x ↓ ↓ dy dx · =1 dy dx

❛s

∆x → 0, ∆y → 0 .

❚❤❡ ❢❛❝t t❤❛t

∆x → 0 =⇒ ∆y → 0 ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ❝♦♥t✐♥✉✐t② ♦❢

f✳ (a, b)✱ dx ❛♥❞ dy

❚♦ ❜❡ ♣r❡❝✐s❡✱ ✇❡ ♥❡❡❞ t♦ ❝♦♥❝❡♥tr❛t❡ ♦♥ ❛ s✐♥❣❧❡ ♣♦✐♥t ✐ts t❛♥❣❡♥t ❧✐♥❡✳ ❚❤❡♥ t❤❡ ❞❡r✐✈❛t✐✈❡s

dy dx

x=a

❛r❡ ✐♥❞❡❡❞ ❢r❛❝t✐♦♥s ❛♥❞ t❤❡ r❡❝✐♣r♦❝❛❧s ♦❢ ❡❛❝❤ ♦t❤❡r✿

✇❤❡r❡

y=b

b = f (a)✱

♦♥ t❤❡ ❣r❛♣❤ ♦❢

y = f (x)

❛♥❞

✹✳✾✳

❚❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ✐♥✈❡rs❡ ❢✉♥❝t✐♦♥

❊①❛♠♣❧❡ ✹✳✾✳✺✿

✸✺✺

ln x

❙✉♣♣♦s❡ ✇❡ ✇❛♥t t♦ ✜♥❞ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❧♦❣❛r✐t❤♠✳ ❲❡✬❧❧ ✉s❡ ♦♥❧② ✐ts ❞❡✜♥✐t✐♦♥ ✈✐❛ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥✱ ❛s ❢♦❧❧♦✇s✳ ❲❡ ❞✐✛❡r❡♥t✐❛t❡ t❤✐s ❡q✉❛t✐♦♥✱ ✇❤✐❝❤ ❛♠♦✉♥ts t♦ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❧♦❣❛r✐t❤♠✱ ♦❢ ❢✉♥❝t✐♦♥s✿ eln x = x .

❚❤❡ ✢♦✇ ❝❤❛rt ❜❡❧♦✇ s❤♦✇s t❤❡ ❞❡♣❡♥❞❡♥❝✐❡s✿ x → u = ln x → y = eu

❇② t❤❡ ❈❤❛✐♥ ❘✉❧❡✱ ✇❡ ❤❛✈❡✿ (eln x )′ = (ln x)′ · eu = (ln x)′ eln x ◆♦✇ ❜❛❝❦✲s✉❜st✐t✉t✐♦♥✳ = (ln x)′ x ❈❛♥❝❡❧❧❛t✐♦♥✳ ′ = (x) = 1 .

❚❤❡r❡❢♦r❡✱ (ln x)′ =

1 , x

(ln x)′ =

1 x

✇❤❡♥❡✈❡r x > 0✳ ❲❡ ❤❛✈❡ ❛ ✉s❡❢✉❧ ❢♦r♠✉❧❛✿

❙✐♠✐❧❛r❧②✱ ✇❡ ❝❛♥ ✜♥❞ t❤❡ ❞❡r✐✈❛t✐✈❡s ♦❢ sin−1 x✱ cos−1 x✱ ❡t❝✳ ❊①❛♠♣❧❡ ✹✳✾✳✻✿ ❛♥♦t❤❡r ♣r♦♦❢

❋♦r ❛♥ ❛❧t❡r♥❛t✐✈❡ ♣r♦♦❢ ♦❢ t❤❡ t❤❡♦r❡♠✱ ✇❡ ❞✐✛❡r❡♥t✐❛t❡ t❤❡ ❡q✉❛t✐♦♥✿ f −1 (f (x)) = x .

❚❤❡♥✱ ❜② t❤❡ ❈❤❛✐♥

❘✉❧❡✱ ✇❡ ❤❛✈❡✿ df −1 df ∆f −1 ∆f = 1 ❛♥❞ = 1. ∆y ∆x dy dx

❚❤❡ ♦t❤❡r ❡q✉❛t✐♦♥ ♣r♦❞✉❝❡s t❤❡ s❛♠❡ r❡s✉❧t✦ ❚♦ ♠❛❦❡ t❤❡ ❢♦r♠✉❧❛ ✐♥ t❤❡ t❤❡♦r❡♠ ♠♦r❡ ✉s❡❢✉❧✱ ✇❡ ❛❞❞ ❜❛❝❦✲s✉❜st✐t✉t✐♦♥✿ ❈♦r♦❧❧❛r② ✹✳✾✳✼✿ ❉❡r✐✈❛t✐✈❡ ♦❢ ■♥✈❡rs❡ ❲✐t❤ ❇❛❝❦✲❙✉❜st✐t✉t✐♦♥

❋♦r ❛♥② ♦♥❡✲t♦✲♦♥❡ ❢✉♥❝t✐♦♥ y = f (x) ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t x = a✱ ✐ts ✐♥✈❡rs❡ x = f −1 (y) ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t b = f (a)✱ ❛♥❞ ✇❡ ❤❛✈❡✿ df −1 (b) = dy

❚❤❡ ❢♦r♠✉❧❛s r❡✇r✐tt❡♥ ✐♥ t❤❡ ▲❛❣r❛♥❣❡ ♥♦t❛t✐♦♥ ❛r❡ ❛s ❢♦❧❧♦✇s✿
′ f −1 (f (a)) =

1 . f ′ (a)

1 df (f −1 (b)) dx

✹✳✾✳

❚❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ✐♥✈❡rs❡ ❢✉♥❝t✐♦♥

✸✺✻

❆ ❜❡tt❡r ✈❡rs✐♦♥ ✐s ❜❡❧♦✇✿
′ f −1 (b) = ❊①❛♠♣❧❡ ✹✳✾✳✽✿

1 f ′ (f −1 (b))

arcsin x

❋✐♥❞ (sin−1 y)′ ✳ ❚❤❡r❡ ✐s ♥♦ ❢♦r♠✉❧❛ ❢♦r t❤✐s ❢✉♥❝t✐♦♥✱ ❜✉t ✐ts ♠❡❛♥✐♥❣ ✐s ✭❢♦r −π/2 ≤ x ≤ π/2✮ ❛s ❢♦❧❧♦✇s✿ y = sin x, ♦r x = sin−1 y . ❙✐♥❝❡ (sin x)′ = cos x✱ ✇❡ ❝♦♥❝❧✉❞❡✿ (sin−1 y)′ =

1 1 = . cos x cos(sin−1 y)

❚❤❛t ❝♦✉❧❞ s❡r✈❡ ❛s t❤❡ ❛♥s✇❡r✱ ❜✉t ✐t✬s t♦♦ ❝✉♠❜❡rs♦♠❡ ❛♥❞ s❤♦✉❧❞ ❜❡ s✐♠♣❧✐✜❡❞✳ ❲❡ ♥❡❡❞ t♦ ❡①♣r❡ss cos x ✐♥ t❡r♠s ♦❢ sin x✱ ✇❤✐❝❤ ✐s y ✿

❋r♦♠ t❤❡ P②t❤❛❣♦r❡❛♥ ✇❡ ❝♦♥❝❧✉❞❡✿

❚❤❡r❡❢♦r❡✱

❚❤❡♦r❡♠✱

sin2 x + cos2 x = 1 , p cos x = p1 − sin2 x = 1 − y2 .

1 . (sin−1 y)′ = p 1 − y2

❲❡ ❝❛♥ ❛♣♣❧② t❤❡ t❤❡♦r❡♠ t♦ ♦t❤❡r tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s✳ ❚❤❡s❡ ❛r❡ t❤❡ r❡s✉❧ts✿ (sin−1 x)′ (cos−1 x)′ (tan−1 x)′

1 1 − x2 1 = −√ 1 − x2 1 = 1 + x2 =√

❊①❡r❝✐s❡ ✹✳✾✳✾

Pr♦✈❡ t❤❡ ❢♦r♠✉❧❛s✳ ❊①❡r❝✐s❡ ✹✳✾✳✶✵

❙✐♥❝❡ (sin−1 x)′ = −(cos−1 x)′ ✱ ❞♦❡s ✐t ♠❡❛♥ t❤❛t sin−1 x = − cos−1 x❄ ❊✈❡♥ t❤♦✉❣❤ ✐t ✐s ✐♠♣♦ss✐❜❧❡ t♦ s❡❡ ✐t ✐♥ t❤❡s❡ ❢♦r♠✉❧❛s✱ ✇❡ r❡♣❡❛t✿

✹✳✶✵✳

❘❡✈❡rs✐♥❣ ❞✐✛❡r❡♥t✐❛t✐♦♥ ◮

✸✺✼

❚❤❡ ❞❡r✐✈❛t✐✈❡s ♦❢ ✐♥✈❡rs❡s ❛r❡ t❤❡ r❡❝✐♣r♦❝❛❧s ♦❢ ❡❛❝❤ ♦t❤❡r✳

❲❡ ❝❛♥ r❡✇r✐t❡ t❤❡ ■♥✈❡rs❡ ❘✉❧❡ ✐♥ t❤❡ ▲❡✐❜♥✐③ ♥♦t❛t✐♦♥ ✐♥ t❡r♠s ♦❢ t❤❡ ✈❛r✐❛❜❧❡s ♦♥❧②✿

1 dx = dy . dy dx ❆♥ ❡✈❡♥ ❜❡tt❡r ✈❡rs✐♦♥ ✐s ❜❡❧♦✇✿

dx dy · =1 dy dx ❚❤❡

P♦✇❡r ❋♦r♠✉❧❛

✐s ❛ r❡s✉❧t ❢r♦♠ ❧❛st ❝❤❛♣t❡r ✇✐t❤ ❛♥ ✐♥❝♦♠♣❧❡t❡ ✭✐♥t❡❣❡rs ♦♥❧②✮ ♣r♦♦❢✳ ❲❡ st❛rt ✇✐t❤

r❡❝✐♣r♦❝❛❧ ♣♦✇❡rs ❜❡❝❛✉s❡ t❤❡② ❛r❡ r♦♦ts✱ ✐✳❡✳✱ t❤❡ ✐♥✈❡rs❡s ♦❢ t❤❡ ♣♦✇❡rs✿ ❚❤❡♦r❡♠ ✹✳✾✳✶✶✿ ❉❡r✐✈❛t✐✈❡ ♦❢ ❘❡❝✐♣r♦❝❛❧ P♦✇❡rs ❋♦r ❛♥② ♣♦s✐t✐✈❡ ✐♥t❡❣❡r

n✱

✇❡ ❤❛✈❡✿

1

1 1 dy n = y n −1 dy n

Pr♦♦❢✳

❚❤❡ ✐♥✈❡rs❡ ♦❢

1

x = yn

✐s

y = xn ✳

1

dy n dy

=

1 dxn dx

❚❤❡r❡❢♦r❡✱

=

1 = nxn−1

1 

n−1 = 1

n yn

1 ny

n−1 n

=

1 1 −1 yn . n

❲❡ ✉s❡ t❤❡ ❛❜♦✈❡ ❢♦r♠✉❧❛ t♦ ♣r♦✈❡ t❤❡ ❣❡♥❡r❛❧ ♦♥❡✿ ❚❤❡♦r❡♠ ✹✳✾✳✶✷✿ ❉❡r✐✈❛t✐✈❡ ♦❢ ❘❛t✐♦♥❛❧ P♦✇❡rs ❋♦r ❛♥② ♣♦s✐t✐✈❡ ✐♥t❡❣❡rs

n

❛♥❞

m✱

✇❡ ❤❛✈❡✿

m

m m dy n = y n −1 dy n

❊①❡r❝✐s❡ ✹✳✾✳✶✸

Pr♦✈❡ t❤❡ t❤❡♦r❡♠✳ ❲❛r♥✐♥❣✦ ❚❤❡ ❢❛❝t t❤❛t

✹✳✶✵✳ ❘❡✈❡rs✐♥❣ ❞✐✛❡r❡♥t✐❛t✐♦♥ ❲❡ ❤❛✈❡ ❡♥❝♦✉♥t❡r❡❞ t❤❡ ❢♦❧❧♦✇✐♥❣ q✉❡st✐♦♥ s❡✈❡r❛❧ t✐♠❡s ❜② ♥♦✇✿

(xπ )′ = πxπ−1

✇✐❧❧ r❡♠❛✐♥ ✉♥♣r♦✈❡♥✳

✹✳✶✵✳

❘❡✈❡rs✐♥❣ ❞✐✛❡r❡♥t✐❛t✐♦♥ ◮

✸✺✽

❲❤❡♥ ✇❡ ❦♥♦✇ t❤❡ ✈❡❧♦❝✐t② ❛t ❡✈❡r② ♠♦♠❡♥t ♦❢ t✐♠❡✱ ❤♦✇ ❞♦ ✇❡ ✜♥❞ t❤❡ ❧♦❝❛t✐♦♥❄

❚❤❡ q✉❡st✐♦♥ ❛♣♣❧✐❡s ❡q✉❛❧❧② t♦ t❤❡ ✈❡❧♦❝✐t② ❛❝q✉✐r❡❞ ❢r♦♠ t❤❡ ❧♦❝❛t✐♦♥ ❛s ✐ts ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦r ❛s ✐ts ❞❡r✐✈❛t✐✈❡✳ ❚♦ ❛♥s✇❡r t❤❡ q✉❡st✐♦♥✱ ✇❡ ♥❡❡❞ t♦ ✏r❡✈❡rs❡✲❡♥❣✐♥❡❡r✑ t❤❡✐r ❡✛❡❝t ♦♥ ❛ ❢✉♥❝t✐♦♥✳ ❆♥♦t❤❡r s✐♠♣❧❡ ❡①❛♠♣❧❡ t❤❛t ✇❡ ❤❛✈❡ s❡❡♥ s❡✈❡r❛❧ t✐♠❡s ✐s t❤❛t ♦❢ ❛ ❜r♦❦❡♥ s♣❡❡❞♦♠❡t❡r ❛♥❞ ✐ts ✏✐♥✈❡rs❡✑✱ t❤❡ ♣r♦❜❧❡♠ ♦❢ ❛ ❜r♦❦❡♥ ♦❞♦♠❡t❡r✿

■t ✐s s♦❧✈❡❞ ❜② s✐♠♣❧❡ ❛❞❞✐t✐♦♥✳ ❚❤❡ ♣r♦❜❧❡♠ ✐s s✐♠♣❧❡r✿ ■❢ ✇❡ ❦♥♦✇ t❤❡ ❞✐s♣❧❛❝❡♠❡♥ts ❞✉r✐♥❣ ❡❛❝❤ ♦❢ t❤❡ t✐♠❡ ♣❡r✐♦❞s✱ ❝❛♥ ✇❡ ✜♥❞ ♦✉r ❧♦❝❛t✐♦♥❄ ❏✉st ❛❞❞ t❤❡♠ t♦❣❡t❤❡r t♦ ✜♥❞ t❤❡ t♦t❛❧ ❞✐s♣❧❛❝❡♠❡♥t✦ ❚❤✐s ✐s ❛❜♦✉t t❤❡ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ❢✉♥❝t✐♦♥ ❢✉♥❝t✐♦♥

y = f (x)

y = g(x)

❞❡✜♥❡❞ ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s✱

❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s✱

x✱

♦❢ t❤❡ ♣❛rt✐t✐♦♥ s♦ t❤❛t

g

c✱

❞✐✛❡r❡♥❝❡✳

♦❢ ❛ ♣❛rt✐t✐♦♥✳ ❍♦✇ ❞♦ ✇❡ ✜♥❞ ❛

✐s ✐ts ❞✐✛❡r❡♥❝❡✿

∆f (c) = g(c)? ■♥ ♦t❤❡r ✇♦r❞s✱ t❤✐s ✐s ✇❤❛t ✇❡ ❢❛❝❡✿ ❙♦❧✈❡✿

∆f = g

❙♦❧✈✐♥❣ t❤✐s ❡q✉❛t✐♦♥ ✐s♥✬t ❤❛r❞✳ ❙✉♣♣♦s❡ t❤✐s ❢✉♥❝t✐♦♥

g

✐s ❦♥♦✇♥ ❜✉t

f

✐s♥✬t✱ ❡①❝❡♣t ❢♦r ♦♥❡ ✭✐♥✐t✐❛❧✮ ✈❛❧✉❡✿

y0 = f (a)✳

❚❤❡♥ t❤❡ ❛❜♦✈❡

❡q✉❛t✐♦♥ ❜❡❝♦♠❡s✿

∆f (c1 ) = f (x0 + ∆x1 ) − f (x0 ) = g(c1 ) ,

❛♥❞ ✇❡ ❝❛♥ s♦❧✈❡ ✐t✿

f (x1 ) = f (x0 + ∆x1 ) = f (x0 ) + g(c1 ) . ❲❡ ❝♦♥t✐♥✉❡ ✐♥ t❤✐s ♠❛♥♥❡r ❛♥❞ ✜♥❞ t❤❡ r❡st ♦❢ t❤❡ ✈❛❧✉❡s ♦❢

f✿

f (xk+1 ) = f (xk + ∆xk ) = f (xk ) + g(ck ) . ❚❤✐s ❢♦r♠✉❧❛ ✐s

r❡❝✉rs✐✈❡ ✿

❲❡ ♥❡❡❞ t♦ ❦♥♦✇ t❤❡ ❧❛st ✈❛❧✉❡ ♦❢

❡①♣❧✐❝✐t ❢♦r♠✉❧❛✱ t❤❡ s♦❧✉t✐♦♥ ✐s ✈❡r② s✐♠♣❧❡✦ ◆♦✇✱ t❤❡

❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✳

f

✐♥ ♦r❞❡r t♦ ✜♥❞ t❤❡ ♥❡①t✳ ❚❤♦✉❣❤ ♥♦t ❛♥

✹✳✶✵✳ ❘❡✈❡rs✐♥❣ ❞✐✛❡r❡♥t✐❛t✐♦♥

✸✺✾

❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ❢✉♥❝t✐♦♥ y = v(x) ❞❡✜♥❡❞ ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s✱ c✱ ♦❢ ❛ ♣❛rt✐t✐♦♥✳ ❍♦✇ ❞♦ ✇❡ ✜♥❞ ❛ ❢✉♥❝t✐♦♥ y = p(x) ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s✱ x✱ ♦❢ t❤❡ ♣❛rt✐t✐♦♥ s♦ t❤❛t v ✐s ✐ts ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✿ ∆p (c) = v(c)? ∆x

■♥ ♦t❤❡r ✇♦r❞s✱ t❤✐s ✐s ✇❤❛t ✇❡ ❢❛❝❡✿ ❙♦❧✈❡✿

∆p =v ∆x

❲❡ ❥✉st ❢♦❧❧♦✇ ❡①❛❝t❧② t❤❡ ♣r♦❝❡ss ❛❜♦✈❡✳ ❙✉♣♣♦s❡ t❤✐s ❢✉♥❝t✐♦♥ v ✐s ❦♥♦✇♥ ❜✉t p ✐s♥✬t✱ ❡①❝❡♣t ❢♦r ♦♥❡ ✭✐♥✐t✐❛❧✮ ✈❛❧✉❡✿ y0 = p(a)✳ ❚❤❡♥ t❤❡ ❛❜♦✈❡ ❡q✉❛t✐♦♥ ❜❡❝♦♠❡s✿ ❛♥❞ ✇❡ ❝❛♥ s♦❧✈❡ ✐t✿

p(x0 + ∆x1 ) − p(x0 ) ∆p (c1 ) = = v(c1 ) , ∆x ∆x1

p(x1 ) = p(x0 + ∆x1 ) = p(x0 ) + v(c1 )∆x1 .

❲❡ ❝♦♥t✐♥✉❡ ✐♥ t❤✐s ♠❛♥♥❡r ❛♥❞ ✜♥❞ t❤❡ r❡st ♦❢ t❤❡ ✈❛❧✉❡s ♦❢ f ✿ p(xk+1 ) = p(xk + ∆xk ) = p(xk ) + v(ck )∆xk .

❚❤✐s ❢♦r♠✉❧❛ ✐s ❛❧s♦ r❡❝✉rs✐✈❡✱ ❜✉t✱ ✇✐t❤✐♥ t❤✐s ❧✐♠✐t❛t✐♦♥✱ t❤❡ ♣r♦❜❧❡♠ ♦❢ r❡✈❡rs✐♥❣ t❤❡ ❡✛❡❝t ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✐s s♦❧✈❡❞✦ ❏✉st ❛s ✐♥ t❤❡ ❧❛st ❝❤❛♣t❡r✱ ✇❡ ✐♥✐t✐❛❧❧② ❢❛❝❡ t❤❡ ♣r♦❜❧❡♠✿ ❲❡ ♥♦✇ ❣♦ ✐♥ r❡✈❡rs❡✿

♣♦s✐t✐♦♥ → ✈❡❧♦❝✐t② → ❛❝❝❡❧❡r❛t✐♦♥ ♣♦s✐t✐♦♥ ← ✈❡❧♦❝✐t② ← ❛❝❝❡❧❡r❛t✐♦♥

❚❤❡s❡ t✇♦ ♣r♦❜❧❡♠s ❛r❡ s✐♠✐❧❛r t♦ t❤❡ ♦♥❡ ♦❢ ✜♥❞✐♥❣ t❤❡ ✐♥✈❡rs❡ ♦❢ ❛ ❢✉♥❝t✐♦♥✳ ❚❤✐s ✐s ❤♦✇ ✐♥✈❡rs❡ ❢✉♥❝t✐♦♥s ❛♣♣❡❛r ✐♥ ❛❧❣❡❜r❛❀ t❤❡② ❝♦♠❡ ❢r♦♠ s♦❧✈✐♥❣ ❡q✉❛t✐♦♥s✱ ❢♦r x✿ x2 = 4 =⇒ x = 2 x 2 = 8 =⇒ x = 3 sin x = 0 =⇒ x = 0

√

✈✐❛ · ✈✐❛ log2 (·) ✈✐❛ sin−1 (·)

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❤❛t ❞♦ ✇❡ ❞♦ ✐❢ ✇❡ ❦♥♦✇ t❤❡ ♦✉t♣✉t ♦❢ ❛ ❢✉♥❝t✐♦♥ ❛♥❞ ✇❛♥t t♦ ❦♥♦✇ t❤❡ ✐♥♣✉t❄ ■♥✐t✐❛❧❧②✱ ✇❡ ❝❛♥ ♦♥❧② r❡❝♦❧❧❡❝t ❛ ♣❛st ❡①♣❡r✐❡♥❝❡ ✇✐t❤ ❛ ❢✉♥❝t✐♦♥✦ ❋♦r ❛ r❡♣❡❛t❡❞ ✉s❡✱ ✇❡ ❞❡✈❡❧♦♣ t❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥✳ ❙✐♠✐❧❛r❧②✱ ✇❤❛t ❞♦ ✇❡ ❞♦ ✐❢ ✇❡ ❦♥♦✇ t❤❡ r❡s✉❧t ♦❢ ❞✐✛❡r❡♥t✐❛t✐♦♥ ❛♥❞ ✇❛♥t t♦ ❦♥♦✇ ✇❤❡r❡ ✐t ❝❛♠❡ ❢r♦♠❄ ❲❡ ♥❡❡❞ t♦ s♦❧✈❡ t❤❡s❡✱ ❢♦r ❡①❛♠♣❧❡✿ f ′ = 2x =⇒ f = x2 f ′ = cos x =⇒ f = sin x f ′ = ex =⇒ f = ex

■♥ ♦t❤❡r ✇♦r❞s✱ t❤✐s ✐s ✇❤❛t ✇❡ ❢❛❝❡✿ ❙♦❧✈❡✿

dp =v dx

❚❤✐s ❡q✉❛t✐♦♥ ✐s♥✬t ❛s ❡❛s② ❛s t❤❡ ❧❛st✳ ■♥✐t✐❛❧❧②✱ ✇❡ ❝❛♥ ♦♥❧② r❡❝♦❧❧❡❝t ❛ ♣❛st ❡①♣❡r✐❡♥❝❡ ✇✐t❤ ❞✐✛❡r❡♥t✐❛t✐♦♥✦ ❋♦r ❛ r❡♣❡❛t❡❞ ✉s❡✱ ✇❡ ✇✐❧❧ ♥❡❡❞ t♦ ❞❡✈❡❧♦♣ t❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ✭❈❤❛♣t❡r ✺✮✳ ❇❡❝❛✉s❡ t❤❡ ❞❡r✐✈❛t✐✈❡ ✐s ♥♦t ❛ ❢r❛❝t✐♦♥ ❜✉t ❛ ❧✐♠✐t ♦❢ ❛ ❢r❛❝t✐♦♥✱ t❤❡r❡ ✐s ♥♦ s✐♥❣❧❡✱ ❡✈❡♥ r❡❝✉rs✐✈❡✱ ❢♦r♠✉❧❛✳ ❚❤✐s ♣r♦❝❡ss ✐s ❝❛❧❧❡❞ ❛♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥✳

✹✳✶✶✳

❙❤♦♦t✐♥❣ ❛ ❝❛♥♥♦♥

✸✻✵

❊①❛♠♣❧❡ ✹✳✶✵✳✶✿ ❢r❡❡ ❢❛❧❧

❚❤❡ ✐♠♣♦rt❛♥❝❡ ♦❢ t❤✐s ✏✐♥✈❡rs❡✑ ♣r♦❜❧❡♠ st❡♠s ❢r♦♠ t❤❡ ♥❡❡❞ t♦ ✜♥❞ ❧♦❝❛t✐♦♥ ❢r♦♠ ✈❡❧♦❝✐t② ♦r ✈❡❧♦❝✐t② ❢r♦♠ ❛❝❝❡❧❡r❛t✐♦♥✳ ❋♦r ❡①❛♠♣❧❡✱ t❤✐s ✐s ✇❤❛t ✇❡ ❞❡r✐✈❡ ❢r♦♠ ♦✉r ❡①♣❡r✐❡♥❝❡ ✇✐t❤ ❞✐✛❡r❡♥t✐❛t✐♦♥✳ ❚♦ ✉♥❞❡rst❛♥❞ ❢r❡❡ ❢❛❧❧✱ ✇❡ ✇✐❧❧ ♥❡❡❞ t♦ ♣r♦✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ st❛t❡♠❡♥ts✿ ✶✳ ❋♦r t❤❡ ❤♦r✐③♦♥t❛❧ ❝♦♠♣♦♥❡♥t✿ ❚❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ✐s ③❡r♦✳

=⇒

❚❤❡ ✈❡❧♦❝✐t② ✐s ❝♦♥st❛♥t✳

=⇒

❚❤❡

❧♦❝❛t✐♦♥ ✐s ❛ ❧✐♥❡❛r ❢✉♥❝t✐♦♥✳ ✷✳ ❋♦r t❤❡ ✈❡rt✐❝❛❧ ❝♦♠♣♦♥❡♥t✿ ❚❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ✐s ❝♦♥st❛♥t✳

=⇒

=⇒

❚❤❡ ✈❡❧♦❝✐t② ✐s ❛ ❧✐♥❡❛r ❢✉♥❝t✐♦♥✳

❚❤❡ ❧♦❝❛t✐♦♥ ✐s ❛ q✉❛❞r❛t✐❝ ❢✉♥❝t✐♦♥✳

❲❡ ✐❧❧✉str❛t❡ t❤❡ ✐❞❡❛ ♦❢ ❛♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥ ✇✐t❤ ❛ ❞✐❛❣r❛♠✿

x2 → 2x →

d dx  −1 d dx

❲❡✬✈❡ ❢♦✉♥❞ ♦♥❡ s♦❧✉t✐♦♥✳ ❆r❡ t❤❡r❡ ❛♥② ♦t❤❡rs❄ ❨❡s✱

→ 2x → x2

(x2 + 1)′ = 2x✳

❆♥❞ ♠♦r❡✿

x2 + 1 ր 2x → ց ❆s ❛ ❢✉♥❝t✐♦♥ ✕ ❛ ❢✉♥❝t✐♦♥ ♦❢ ❢✉♥❝t✐♦♥s ✕

d dx

x2 x2 − 1

✐s♥✬t ♦♥❡✲t♦✲♦♥❡✦

❊①❡r❝✐s❡ ✹✳✶✵✳✷

❲❡ ❝❛♥ ♠❛❦❡ ❛♥② ❢✉♥❝t✐♦♥ ♦♥❡✲t♦✲♦♥❡ ❜② r❡str✐❝t✐♥❣ ✐ts ❞♦♠❛✐♥✳ ❍♦✇ ✇♦✉❧❞ t❤❛t ✇♦r❦ ❢♦r ❞✐✛❡r❡♥t✐❛✲ t✐♦♥❄

❲❡ ❛♣♣❧② t❤❡s❡ ✐❞❡❛s t♦ ❛ s♣❡❝✐✜❝ ♣r♦❜❧❡♠ ❛❜♦✉t ♠♦t✐♦♥✳

✹✳✶✶✳ ❙❤♦♦t✐♥❣ ❛ ❝❛♥♥♦♥

❆ s♦❝❝❡r ❜❛❧❧ r♦❧❧✐♥❣ ♦♥ ❛ ❤♦r✐③♦♥t❛❧ ♣❧❛♥❡ ✇✐❧❧ ❤❛✈❡ ❛ ❝♦♥st❛♥t ✈❡❧♦❝✐t②✿

✹✳✶✶✳ ❙❤♦♦t✐♥❣ ❛ ❝❛♥♥♦♥

✸✻✶

❆ ♣✐♥❣✲♣♦♥❣ ❜❛❧❧ t❤r♦✇♥ ✉♣ ✐♥ t❤❡ ❛✐r ❣♦❡s ✉♣✱ s❧♦✇s ❞♦✇♥ ✉♥t✐❧ ✐t st♦♣s ❢♦r ❛♥ ✐♥st❛♥t✱ ❛♥❞ t❤❡♥ ❛❝❝❡❧❡r❛t❡s t♦✇❛r❞ t❤❡ s✉r❢❛❝❡✿

❲❤❛t ✐❢ ✇❡ ❞♦ ❜♦t❤✿ ❲❡ r♦❧❧ ❛ s♦❝❝❡r ❜❛❧❧ ❤♦r✐③♦♥t❛❧❧② ❛♥❞ t❤r♦✇ ❛ ♣✐♥❣✲♣♦♥❣ ❜❛❧❧ ✈❡rt✐❝❛❧❧②❄ ▲❡t✬s tr② t♦ ❢♦❧❧♦✇ ❜♦t❤ ❜❛❧❧s ❛t t❤❡ s❛♠❡ t✐♠❡✿

❲❡✬❞ ❤❛✈❡ t♦ ✢② t❤r♦✉❣❤ t❤❡ ❛✐r ❛s ✐❢ t❤r♦✇♥ ❛t ❛♥ ❛♥❣❧❡✦ ❖✉r ✉♥❞❡rst❛♥❞✐♥❣ ✐s t❤❛t ❛ t❤r♦✇♥ ❜❛❧❧ ♠♦✈❡s ✐♥ ❜♦t❤ ✈❡rt✐❝❛❧ ❛♥❞ ❤♦r✐③♦♥t❛❧ ❞✐r❡❝t✐♦♥s✱ s✐♠✉❧t❛♥❡♦✉s❧② ❛♥❞ ✐♥❞❡♣❡♥❞❡♥t❧②✿

❚❤❡ ❞②♥❛♠✐❝s ✐s ✈❡r② ❞✐✛❡r❡♥t✿ ✶✳ ■♥ t❤❡ ❤♦r✐③♦♥t❛❧ ❞✐r❡❝t✐♦♥✱ ❛s t❤❡r❡ ✐s ♥♦ ❢♦r❝❡ ❝❤❛♥❣✐♥❣ t❤❡ ✈❡❧♦❝✐t②✱ t❤❡ ❧❛tt❡r r❡♠❛✐♥s ❝♦♥st❛♥t✳ ✷✳ ▼❡❛♥✇❤✐❧❡✱ t❤❡ ✈❡rt✐❝❛❧ ✈❡❧♦❝✐t② ✐s ❝♦♥st❛♥t❧② ❝❤❛♥❣❡❞ ❜② t❤❡ ❢♦r❝❡ ♦❢ ❣r❛✈✐t②✳ ▲❡t✬s ♥♦✇ ✉s❡ t❤❡s❡ ❞❡s❝r✐♣t✐♦♥s t♦ r❡♣r❡s❡♥t t❤❡ ♠♦t✐♦♥ ♠❛t❤❡♠❛t✐❝❛❧❧②✳ ❘❡❝❛❧❧ ❤♦✇ ❡❛r❧✐❡r ✐♥ t❤✐s ❝❤❛♣t❡r ✇❡ ✉s❡❞ t❤❡s❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts t♦ ✜♥❞ ✈❡❧♦❝✐t② ❛♥❞ t❤❡♥ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ❢r♦♠ t❤❡ ❧♦❝❛t✐♦♥✿ ∆p pn+1 − pn ∆v vn+1 − vn vn = = ❛♥❞ an = = , ∆t

h

∆t

h

✹✳✶✶✳

❙❤♦♦t✐♥❣ ❛ ❝❛♥♥♦♥ h

✇❤❡r❡

✸✻✷

✐s t❤❡ ✐♥❝r❡♠❡♥t ♦❢ t✐♠❡✳

❚❤❡s❡ ❢♦r♠✉❧❛s ❤❛✈❡ ❛❧s♦ ❜❡❡♥ s♦❧✈❡❞ ❢♦r

pn+1

❛♥❞

vn+1

r❡s♣❡❝t✐✈❡❧② ✐♥ ♦r❞❡r t♦ ❜❡ ❛❜❧❡ t♦ ♠♦❞❡❧ ❧♦❝❛t✐♦♥

❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✐♠❡✿

vn+1 = vn + han ❚❤❡s❡ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛s ❛r❡ ❝❛❧❧❡❞ t❤❡

❛♥❞

pn+1 = pn + hvn .

❘✐❡♠❛♥♥ s✉♠s

✳

❚❤✐s ✐s ✇❤❛t t❤❡ r❡s✉❧ts ♠✐❣❤t ❧♦♦❦ ❧✐❦❡✿

❚❤✐s t✐♠❡ ✇❡ ❤❛✈❡

t✇♦

s✉❝❤ s❡q✉❡♥❝❡s✱ ♦♥❡ ❢♦r ❤♦r✐③♦♥t❛❧ ❛♥❞ ♦♥❡ ❢♦r ✈❡rt✐❝❛❧✳

❲❡ ❝♦♥str✉❝t t❤❡ ❈❛rt❡s✐❛♥ ❝♦♦r❞✐♥❛t❡ s②st❡♠ ✐♥ t❤❡ ♠♦st ❝♦♥✈❡♥✐❡♥t ✇❛②✿

•

❚❤❡

x✲❛①✐s

✐s ❤♦r✐③♦♥t❛❧✳

•

❚❤❡

y ✲❛①✐s

✐s ✈❡rt✐❝❛❧✳

❍♦✇❡✈❡r✱ ✇❡ ❛❜❛♥❞♦♥ t❤❡ ❢❛♠✐❧✐❛r

• t

y = f (x)

s❡t✉♣✦ ❲❡ ❤❛✈❡

✐s t✐♠❡✳

• x

✐s t❤❡ ❤♦r✐③♦♥t❛❧ ❞✐♠❡♥s✐♦♥✱ t❤❡ ❞❡♣t❤✳

• y

✐s t❤❡ ✈❡rt✐❝❛❧ ❞✐♠❡♥s✐♦♥✱ t❤❡ ❤❡✐❣❤t✳

❊✐t❤❡r ♦❢ t❤❡ t✇♦ ❚❤❡✐r

t❤r❡❡ ✈❛r✐❛❜❧❡s

❣r❛♣❤s

s♣❛t✐❛❧

✈❛r✐❛❜❧❡s ❞❡♣❡♥❞s ♦♥ t❤❡

❛r❡ ♣❧♦tt❡❞ ❜❡❧♦✇ ✭❧❡❢t✮✿

t❡♠♣♦r❛❧

✈❛r✐❛❜❧❡✳

♥♦✇✿

✹✳✶✶✳

✸✻✸

❙❤♦♦t✐♥❣ ❛ ❝❛♥♥♦♥

▼❡❛♥✇❤✐❧❡✱ t❤❡ ✭r✐❣❤t✮✳

♣❛t❤

♦❢ t❤❡ ❜❛❧❧ ✇✐❧❧ ❛♣♣❡❛r t♦ ❛♥ ♦❜s❡r✈❡r ❛s ❛ ❝✉r✈❡ ✐♥ t❤❡✱ ✈❡rt✐❝❛❧❧② ❛❧✐❣♥❡❞✱ xy ✲♣❧❛♥❡

❍✐st♦r✐❝❛❧❧②✱ ♦♥❡ ♦❢ t❤❡ ✈❡r② ✜rst ❛♣♣❧✐❝❛t✐♦♥s ♦❢ ❝❛❧❝✉❧✉s ✇❛s ✐♥ ❜❛❧❧✐st✐❝s✳ ❇❡❢♦r❡ ❝❛❧❝✉❧✉s✱ ♦♥❡ ❤❛❞ t♦ r❡s♦rt t♦ tr✐❛❧ ❛♥❞ ❡rr♦r ❛♥❞ ✇❛t❝❤✐♥❣ ✇❤❡r❡ t❤❡ ❝❛♥♥♦♥❜❛❧❧s ✇❡r❡ ❧❛♥❞✐♥❣✳ ❆ ✇❡❧❧✲❞❡s✐❣♥❡❞ t❡st ♠❛② ♣r♦✈✐❞❡ ♦♥❡ ✇✐t❤ ❛ t❛❜❧❡ ✭✐✳❡✳✱ ❛ ❢✉♥❝t✐♦♥✮ t❤❛t ❣✐✈❡s t❤❡ s❤♦t ❧❡♥❣t❤ ❢♦r ❡❛❝❤ ❛♥❣❧❡ ♦❢ t❤❡ ❜❛rr❡❧✳ ❍♦✇❡✈❡r✱ s✉❝❤ ❛ r❡❢❡r❡♥❝❡ t❛❜❧❡ ♠❛② ♣r♦✈❡ ✉s❡❧❡ss ✇❤❡♥ ♦♥❡ ✐s t♦ s❤♦♦t ❢r♦♠ ❛♥ ❡❧❡✈❛t❡❞ ♣♦s✐t✐♦♥✱ ♦r ❛t ❛♥ ❡❧❡✈❛t❡❞ t❛r❣❡t✱ ♦r ♦✈❡r ❛♥ ♦❜st❛❝❧❡✳ ❋r♦♠ ❛ 200✲❢♦♦t ❡❧❡✈❛t✐♦♥✱ ❛ ❝❛♥♥♦♥ ✐s ✜r❡❞ ❤♦r✐③♦♥t❛❧❧② ❛t 200 ❢❡❡t ♣❡r s❡❝♦♥❞✳ ❍♦✇ ❢❛r ✇✐❧❧ t❤❡ ❝❛♥♥♦♥❜❛❧❧ ❣♦❄

◮

P❘❖❇▲❊▼✿

❲❡ ✇✐❧❧ ✜♥❞ t❤❡ ✇❤♦❧❡ ♣❛t❤✦ ▲❡t h = ∆t ❜❡ t❤❡ ✐♥❝r❡♠❡♥t ♦❢ t✐♠❡✳ ❲❡ ❤❛✈❡ t❤❡s❡ s✐① s❡q✉❡♥❝❡s ✇✐t❤ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ❝♦♠♣✉t❡❞ ❢♦✉r t✐♠❡s✿ ❤♦r✐③♦♥t❛❧ ✈❡rt✐❝❛❧ ♣♦s✐t✐♦♥ xn yn xn+1 − xn yn+1 − yn ✈❡❧♦❝✐t② vn = un = ❉◗ ❛❝❝❡❧❡r❛t✐♦♥

h vn+1 − vn an = h

h un+1 − un bn = h

❉◗

◆♦✇✱ ❢r♦♠ t❤❡ ♣♦✐♥t ♦❢ ♠♦❞❡❧✐♥❣✱ t❤❡ ❞❡r✐✈❛t✐♦♥ s❤♦✉❧❞ ❣♦ ✐♥ t❤❡ ♦♣♣♦s✐t❡ ❞✐r❡❝t✐♦♥✳ ❲❡ ❣♦ ✐♥ r❡✈❡rs❡✿ t❤❡ ✈❡❧♦❝✐t② ❛♥❞ t❤❡♥ t❤❡ ❧♦❝❛t✐♦♥ ❢r♦♠ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥✳ ❲❤❡♥ ✇❡ s♦❧✈❡ t❤❡ ❛❜♦✈❡ ❡q✉❛t✐♦♥s✱ ✇❡ ❡♥❞ ✉♣ ✇✐t❤ t❤❡s❡ ❢♦✉r r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛s ✭t❤❡ ❘✐❡♠❛♥♥ s✉♠s✮ ❢♦r ♦✉r s✐① s❡q✉❡♥❝❡s✿ ❛❝❝❡❧❡r❛t✐♦♥ ✈❡❧♦❝✐t② ♣♦s✐t✐♦♥

❤♦r✐③♦♥t❛❧

✈❡rt✐❝❛❧

an bn vn+1 = vn + han un+1 = un + hbn ❘❙ xn+1 = xn + hvn yn+1 = yn + hun ❘❙

◆♦✇ ✐♥ t❤❡ s♣❡❝✐✜❝ ❝❛s❡ ♦❢ ❢r❡❡ ❢❛❧❧✱ t❤❡r❡ ✐s ❥✉st ♦♥❡ ❢♦r❝❡✱ t❤❡ ❣r❛✈✐t②✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ❤♦r✐③♦♥t❛❧ ❛❝❝❡❧❡r❛t✐♦♥ ✐s ③❡r♦ ❛♥❞ t❤❡ ✈❡rt✐❝❛❧ ❛❝❝❡❧❡r❛t✐♦♥ ✐s ❝♦♥st❛♥t ✭❢❡❡t ♣❡r s❡❝♦♥❞ sq✉❛r❡❞✮✿ a = 0, b = −32 .

◆❡①t✱ ✇❡ ❛❝q✉✐r❡ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s✿ ✐♥✐t✐❛❧ ❧♦❝❛t✐♦♥✿ ✐♥✐t✐❛❧ ✈❡❧♦❝✐t②✿ ❚❤❡s❡ ❢♦✉r ♥✉♠❜❡rs s❡r✈❡ ❛s t❤❡ ✐♥✐t✐❛❧ t✐♠❡ t0

t1 = t0 + h

t❡r♠s

❛❝❝❡❧❡r❛t✐♦♥ ✈❡❧♦❝✐t② ♣♦s✐t✐♦♥ ❛❝❝❡❧❡r❛t✐♦♥ ✈❡❧♦❝✐t② ♣♦s✐t✐♦♥

t2 = t1 + h ...

x x0 = 0 v0 = 200

y y0 = 200 u0 = 0

♦❢ ♦✉r ❢♦✉r s❡q✉❡♥❝❡s✿ ❤♦r✐③♦♥t❛❧ a0 = 0 v1 = 200 + .1 · 0 x1 = 0 + .1 · 200 a1 = 0 v2 = v1 + ha1 x2 = x1 + hv1 ...

✈❡rt✐❝❛❧ b0 = −32 u1 = 0 + .1 · (−32) y1 = 200 + .1 · 0 b1 = −32 u2 = u1 + hb1 y3 = y1 n + hu1

✹✳✶✶✳

❙❤♦♦t✐♥❣ ❛ ❝❛♥♥♦♥

✸✻✹

❲❡ ✉s❡ t❤❡ ❢♦r♠✉❧❛s t♦ ❡✈❛❧✉❛t❡ t❤❡ ❧♦❝❛t✐♦♥ ❡✈❡r②

h = .1 s❡❝♦♥❞✳

❲❡ t❛❦❡ t❤❡ s♣r❡❛❞s❤❡❡t ♣r❡s❡♥t❡❞ ❛❜♦✈❡✱

❝♦♣② t❤❡ ❝♦❧✉♠♥s ❢♦r ❛❝❝❡❧❡r❛t✐♦♥✱ ✈❡❧♦❝✐t②✱ ❛♥❞ ♣♦s✐t✐♦♥✱ ❛♥❞ ♣❛st❡ ♥❡①t✿

❖❢ ❝♦✉rs❡✱ ❢♦r t❤❡ ❤♦r✐③♦♥t❛❧ ✈❛❧✉❡s✱ ✇❡ r❡♣❧❛❝❡ ❛❝❝❡❧❡r❛t✐♦♥ ✇✐t❤

a = 0✳

❊①❛♠♣❧❡ ✹✳✶✶✳✶✿ ❤♦✇ ❢❛r

❚♦ ✜♥❞ ✇❤❡♥ ❛♥❞ ✇❤❡r❡ t❤❡ ❜❛❧❧ ❤✐ts t❤❡ ❣r♦✉♥❞✱ ✇❡ s❝r♦❧❧ ❞♦✇♥ t♦ ✜♥❞ t❤❡ r♦✇ ✇✐t❤

t = 3.5✳

■t ❤❛♣♣❡♥s ❛❜♦✉t

❚❤❡♥✱ t❤❡ ✈❛❧✉❡ ♦❢

x

❛t t❤❡ t✐♠❡ ✐s ❛❜♦✉t

x = 700✳

y

❝❧♦s❡ t♦

0✳

❍♦✇ ❢❛st ✐s ✐t ❣♦✐♥❣ ❛t

t❤❡ t✐♠❡❄ ❲❡ ✉s❡ t❤❡ ✈❡❧♦❝✐t② ✈❛❧✉❡s ❢r♦♠ t❤❡ s❛♠❡ r♦✇✿

√ ◆♦✇ ✇❡ ❝♦♠❜✐♥❡ t❤❡

x✲

❛♥❞

2002 + 1082 ≈ 227.3

y ✲❝♦❧✉♠♥s

❢❡❡t ♣❡r s❡❝♦♥❞.

t♦ ♣❧♦t t❤❡ ♣❛t❤✿

❲✐t❤ t❤❡ s♣r❡❛❞s❤❡❡t✱ ✇❡ ❝❛♥ ❛s❦ ❛♥❞ ❛♥s✇❡r ❛ ✈❛r✐❡t② ♦❢ q✉❡st✐♦♥s ❛❜♦✉t s✉❝❤ ♠♦t✐♦♥✳ ✐♥tr♦❞✉❝❡ t❤❡

❛♥❣❧❡

❚❤❡ ✈❡❧♦❝✐t② ♦❢ t❤❡ ❝❛♥♥♦♥❜❛❧❧

❇✉t ✜rst✱ ❧❡t✬s

♦❢ t❤❡ ❜❛rr❡❧ ♦❢ t❤❡ ❝❛♥♥♦♥ ✐♥t♦ t❤❡ ♠♦❞❡❧✳

200 ❢❡❡t ♣❡r s❡❝♦♥❞ ✇❡ ❤❛✈❡ ❜❡❡♥ ✉s✐♥❣ ✐s t❤❡ ✏♠✉③③❧❡ ✈❡❧♦❝✐t②✑✱ ✐✳❡✳✱ t❤❡ s♣❡❡❞✱ s✱ ✇✐t❤ ✇❤✐❝❤ ❧❡❛✈❡s t❤❡ ♠✉③③❧❡ ✕ ♥♦ ♠❛tt❡r ✇❤❛t t❤❡ ❛♥❣❧❡✱ α✱ ✐s✳ ❚❤❛t✬s ✇❤❡r❡ t❤❡ ✐♥✐t✐❛❧ ❤♦r✐③♦♥t❛❧ ❛♥❞

t❤❡ ✈❡rt✐❝❛❧ ✈❡❧♦❝✐t✐❡s ❝♦♠❡ ❢r♦♠✿

✹✳✶✶✳

❙❤♦♦t✐♥❣ ❛ ❝❛♥♥♦♥

✸✻✺

❚❤❡ ❢♦r♠✉❧❛s ❝♦♠❡ ❢r♦♠ tr✐❣♦♥♦♠❡tr②✿

v0 = s cos α

u0 = s sin α .

❛♥❞

❲❡ ✉s❡ t❤❡♠ ❜❡❧♦✇ t♦ ♣r♦✈✐❞❡ t❤❡ ✐♥✐t✐❛❧ ✈❛❧✉❡s ♦❢ t❤❡ t✇♦ ✈❡❧♦❝✐t✐❡s✿

❲❡ ❝❛♥ ❢r❡❡❧② ❡♥t❡r t❤❡ ❞❛t❛ ❢♦r t❤❡ ❢♦❧❧♦✇✐♥❣ ✭❤✐❣❤❧✐❣❤t❡❞ ✐♥ ❣r❡❡♥✮✿

•

t❤❡ ✐♥✐t✐❛❧ s♣❡❡❞

•

t❤❡ ✐♥✐t✐❛❧ ❛♥❣❧❡

•

t❤❡ ✐♥✐t✐❛❧ ❧♦❝❛t✐♦♥

•

❛❧❧ ❛❝❝❡❧❡r❛t✐♦♥s

❚❤❡ r❡st ✐s ❝♦♠♣✉t❡❞ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ s❛♠❡ ❢♦r♠✉❧❛s ❛s ❜❡❢♦r❡✳

❊①❛♠♣❧❡ ✹✳✶✶✳✷✿ ❧♦♥❣❡st s❤♦t ■s ✐t r❡❛❧❧② tr✉❡ t❤❛t

45

❞❡❣r❡❡s ✐s t❤❡ ❜❡st ❛♥❣❧❡ t♦ s❤♦♦t ❢♦r ❛ ❧♦♥❣❡r ❞✐st❛♥❝❡❄ ❚♦ t❡st t❤❡ ✐❞❡❛✱ ✇❡

tr② t♦ s❤♦♦t ✇✐t❤ ❛♥ ❛♥❣❧❡ ❥✉st ❛❜♦✈❡ ❛♥❞ ❥✉st ❜❡❧♦✇✿

■t ❛♣♣❡❛rs t❤❛t t❤❡ ♦♥❡ ✐♥ t❤❡ ♠✐❞❞❧❡ ✐s t❤❡ ❜❡st✱ ❜✉t ✇❡ ❝❛♥✬t ♣r♦✈❡ t❤✐s ✇✐t❤ ❥✉st t❤❡ ♥✉♠❡r✐❝❛❧ ♠❡t❤♦❞s✳

◆♦✇✱ ✇❤❛t ✐❢ ✇❡ tr② t♦ s❤♦♦t ❢r♦♠ ❛ ❤✐❧❧ ❛❣❛✐♥✱ s❛②✱

500

❢❡❡t ❤✐❣❤❄

✹✳✶✶✳ ❙❤♦♦t✐♥❣ ❛ ❝❛♥♥♦♥

✸✻✻

■t✬s ♥♦t t❤❡ ❜❡st ❛♥②♠♦r❡✦ ❊①❡r❝✐s❡ ✹✳✶✶✳✸

❙❤♦✇ t❤❛t t❤❡ ❜❡st s❤♦t ✇✐❧❧ ❜❡❝♦♠❡ ♠♦r❡ ❛♥❞ ♠♦r❡ ✢❛t ❛s t❤❡ ❡❧❡✈❛t✐♦♥ ❣r♦✇s✳ ❊①❛♠♣❧❡ ✹✳✶✶✳✹✿ ✈❛r✐❛❜❧❡ ❣r❛✈✐t②

❲❤❛t ❤❛♣♣❡♥s ✐❢ t❤❡ ❣r❛✈✐t② s✉❞❞❡♥❧② ❞✐s❛♣♣❡❛rs❄ ■♥ t❤❡ ❝♦❧✉♠♥ ❢♦r t❤❡ ✈❡rt✐❝❛❧ ❛❝❝❡❧❡r❛t✐♦♥✱ ✇❡ ❥✉st r❡♣❧❛❝❡ −32 ✇✐t❤ 0 ❛❢t❡r ❛ ❢❡✇ r♦✇s✿

❚❤❡ ❝❛♥♥♦♥❜❛❧❧ ✢✐❡s ♦✛ ♦♥ ❛ t❛♥❣❡♥t✳ ❊①❛♠♣❧❡ ✹✳✶✶✳✺✿ ✈❛r✐❛❜❧❡ ❣r❛✈✐t②

❲❤❛t ❤❛♣♣❡♥s ✐❢ t❤❡ ❣r❛✈✐t② st❛rts t♦ ✐♥❝r❡❛s❡❄ ❲❡ tr② t♦ ✐♥❝r❡❛s❡ t❤❡ ❞♦✇♥ ❛❝❝❡❧❡r❛t✐♦♥ 1 ❢♦♦t ♣❡r s❡❝♦♥❞ sq✉❛r❡❞ ♣❡r s❡❝♦♥❞✿

❚❤❡ tr❛❥❡❝t♦r② ❧♦♦❦s st❡❡♣❡r ❛♥❞ st❡❡♣❡r✱ ❜✉t ✐s t❤❡r❡ ❛ ✈❡rt✐❝❛❧ ❛s②♠♣t♦t❡ ❄ ❲❡ ❝❛♥✬t ❛♥s✇❡r ✇✐t❤ ❥✉st t❤❡ ♥✉♠❡r✐❝❛❧ ♠❡t❤♦❞s✳

✹✳✶✶✳ ❙❤♦♦t✐♥❣ ❛ ❝❛♥♥♦♥

✸✻✼

❊①❛♠♣❧❡ ✹✳✶✶✳✻✿ ❤♦r✐③♦♥t❛❧ ❣r❛✈✐t②

❲❤❛t ❤❛♣♣❡♥s ✐❢ t❤❡ ❣r❛✈✐t② ✐s ❤♦r✐③♦♥t❛❧ ✐♥st❡❛❞❄ ❚❤❡ ♠♦t✐♦♥ ✇✐❧❧ ❜❡ ❛❧♦♥❣ ❛ ♣❛r❛❜♦❧❛ t❤❛t ❧✐❡s ♦♥ ✐ts s✐❞❡✱ ♦❢ ❝♦✉rs❡✳ ❇✉t ✇❤❛t ✐❢ t❤❡r❡ ✐s ❜♦t❤ ✈❡rt✐❝❛❧ ❛♥❞ ❤♦r✐③♦♥t❛❧ ❣r❛✈✐t②❄ ▲❡t✬s ♠♦❞✐❢② t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ❝♦❧✉♠♥s ❛❝❝♦r❞✐♥❣❧② ❜② r❡♣❧❛❝✐♥❣ 0✬s ✇✐t❤ −32 ✐♥ t❤❡ ❤♦r✐③♦♥t❛❧ ❛❝❝❡❧❡r❛t✐♦♥ ❝♦❧✉♠♥✿

■s t❤✐s ❛ ♣❛r❛❜♦❧❛❄ ❊①❡r❝✐s❡ ✹✳✶✶✳✼

❊①♣❧❛✐♥ t❤❡ r❡s✉❧ts ✐♥ t❤❡ ❧❛st ❡①❛♠♣❧❡✳ ■♥ s♣✐t❡ ♦❢ t❤❡s❡ ♥✉♠❡r♦✉s ❡①❛♠♣❧❡s✱ ✇❡ ❝❛♥ ♦♥❧② ❞♦ ♦♥❡ ❛t ❛ t✐♠❡✦ ❚❤❡ ❝♦♥❝❧✉s✐♦♥s ✇❡ ❞r❛✇ ❛r❡ ❛❧s♦ s♣❡❝✐✜❝ t♦ t❤❡s❡ s✐t✉❛t✐♦♥s ✭❛❝❝❡❧❡r❛t✐♦♥s✱ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s✱ ❡t❝✳✮✳ ❚❤✐s ✐s ✇❤② ✇❡ ♥♦✇ ❝♦♥s✐❞❡r t❤❡ ❝♦♥t✐♥✉♦✉s ❝❛s❡✱ ✐✳❡✳✱ ✇❡ t❛❦❡ t❤❡ ❧✐♠✐t ♦❢ ❡✈❡r②t❤✐♥❣ ❛❜♦✈❡✿ h = ∆t → 0

❚❤✐s t✐♠❡✱ ✐♥st❡❛❞ ♦❢ s✐① s❡q✉❡♥❝❡s✱ ✇❡ ❤❛✈❡ t❤❡s❡ s✐① ❢✉♥❝t✐♦♥s ♦❢ t✐♠❡✿ x✱ t❤❡ ❞❡♣t❤✱ t❤❡ ❤♦r✐③♦♥t❛❧ ❧♦❝❛t✐♦♥ y ✱ t❤❡ ❤❡✐❣❤t✱ t❤❡ ✈❡rt✐❝❛❧ ❧♦❝❛t✐♦♥

v = x′ ✱ t❤❡ ❤♦r✐③♦♥t❛❧ ✈❡❧♦❝✐t② u = y ′ ✱ t❤❡ ✈❡rt✐❝❛❧ ✈❡❧♦❝✐t②

a = v ′ ✱ t❤❡ ❤♦r✐③♦♥t❛❧ ❛❝❝❡❧❡r❛t✐♦♥ b = u′ ✱ t❤❡ ✈❡rt✐❝❛❧ ❛❝❝❡❧❡r❛t✐♦♥

❚❤❡r❡ ✐s ♥♦ t✐♠❡ ✐♥❝r❡♠❡♥t ❛s ❛ ♣❛r❛♠❡t❡r ❛♥②♠♦r❡✦ ◆♦✇ t❤❡ s♣❡❝✐✜❝ ❝❛s❡ ♦❢ ❢r❡❡ ❢❛❧❧✿ a = 0, b = −g .

❲❡ ❤❛✈❡ ❧❡❛r♥❡❞ ✐♥ t❤✐s ❝❤❛♣t❡r t❤❛t✿ ✶✳ ❚❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧ ✐s ❧✐♥❡❛r✳ ✷✳ ❚❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ❧✐♥❡❛r ♣♦❧②♥♦♠✐❛❧ ✐s ❝♦♥st❛♥t✳ ❲❡ ✇✐❧❧ s❤♦✇ ✐♥ ❈❤❛♣t❡r ✺ t❤❛t✱ ❝♦♥✈❡rs❡❧②✱ ✇❡ ❤❛✈❡✿ ✶✳ ❚❤❡ ♦♥❧② ❢✉♥❝t✐♦♥ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ✇❤✐❝❤ ✐s ❧✐♥❡❛r ✐s ❛ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧✳ ✷✳ ❚❤❡ ♦♥❧② ❢✉♥❝t✐♦♥ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ✇❤✐❝❤ ✐s ❝♦♥st❛♥t ✐s ❛ ❧✐♥❡❛r ♣♦❧②♥♦♠✐❛❧✳ ❋r♦♠ t❤❡ ❧❛tt❡r✱ ✇❡ ❝♦♥❝❧✉❞❡ ❛❜♦✉t ❢r❡❡ ❢❛❧❧ ✿ ✶✳ ❚❤❡ ❤♦r✐③♦♥t❛❧ ♣♦s✐t✐♦♥ x = x(t) ✐s ❧✐♥❡❛r✳ ✷✳ ❚❤❡ ✈❡rt✐❝❛❧ ♣♦s✐t✐♦♥ y = y(t) ✐s q✉❛❞r❛t✐❝✳ ❲❤❛t ♠❛❦❡s t❤❡s❡ t✇♦ ❢✉♥❝t✐♦♥s s♣❡❝✐✜❝ ❛r❡ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ✿

✹✳✶✶✳ ❙❤♦♦t✐♥❣ ❛ ❝❛♥♥♦♥ x0 ✱ t❤❡ ✐♥✐t✐❛❧ ❞❡♣t❤✱ x0 = x(0) y0 ✱ t❤❡ ✐♥✐t✐❛❧ ❤❡✐❣❤t✱ y0 = y(0)

❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡✿

✸✻✽ dx dt t=0 dy u0 ✱ t❤❡ ✐♥✐t✐❛❧ ✈❡rt✐❝❛❧ ❝♦♠♣♦♥❡♥t ♦❢ ✈❡❧♦❝✐t②✱ u(0) = dt t=0 v0 ✱ t❤❡ ✐♥✐t✐❛❧ ❤♦r✐③♦♥t❛❧ ❝♦♠♣♦♥❡♥t ♦❢ ✈❡❧♦❝✐t②✱ v(0) =

x = x0 +v0 t

1 y = y0 +u0 t − gt2 2

❚❤❡s❡ t✇♦ ❡q✉❛t✐♦♥s ❛❧❧♦✇ ✉s t♦ s♦❧✈❡ ❛ ✈❛r✐❡t② ♦❢ ♣r♦❜❧❡♠s ❛❜♦✉t ♠♦t✐♦♥✳ ❲❡ ❝❛rr② t❤✐s ♦✉t ❢♦r x ❛♥❞ y s❡♣❛r❛t❡❧② ❛♥❞ t❤❡ r❡s✉❧ts ❛r❡ s❤♦✇♥ ✐♥ t❤❡ s♣r❡❛❞s❤❡❡t✿

❊①❛♠♣❧❡ ✹✳✶✶✳✽✿ ❤♦✇ ❢❛r

▲❡t✬s r❡✈✐s✐t t❤❡ ♣r♦❜❧❡♠ ❛❜♦✉t ❛ s♣❡❝✐✜❝ s❤♦t ✇❡ s♦❧✈❡❞ ♥✉♠❡r✐❝❛❧❧②✳ ❖✉r ❡q✉❛t✐♦♥s ❜❡❝♦♠❡✿ x = 200t y = 200 −16t2

◆♦✇✱ ❛♥❛❧②t✐❝❛❧❧②✱ t❤❡ ❤❡✐❣❤t ❛t t❤❡ ❡♥❞ ✐s y0 ✱ s♦ t♦ ✜♥❞ ✇❤❡♥ ✐t ❤❛♣♣❡♥❡❞✱ ✇❡ s❡t y = 0✱ ♦r 200 − 16t2 = 0 ,

❛♥❞ s♦❧✈❡ ❢♦r t✳ ❚❤❡♥✱ t❤❡ t✐♠❡ ♦❢ ❧❛♥❞✐♥❣ ✐s✿ t1 =

r

√ 5 2 200 = . 16 2

❚♦ ✜♥❞ ✇❤❡r❡ ✐t ❤❛♣♣❡♥❡❞✱ ✇❡ s✉❜st✐t✉t❡ t❤✐s ✈❛❧✉❡ ♦❢ t ✐♥t♦ x❀ t❤❡ ❧♦❝❛t✐♦♥ ✐s✿ √ 5 2 x1 = 200t1 = 200 ≈ 707 . 2

❚❤❡ r❡s✉❧t ♠❛t❝❤❡s ♦✉r ❡st✐♠❛t❡✦ ❲❡ ✇✐❧❧ ♣r♦✈❡ ✐♥ ❈❤❛♣t❡r ✻ t❤❛t t❤❡ ❧♦♥❣❡st ❞✐st❛♥❝❡ ✐s ❛❝❤✐❡✈❡❞ ✇❤❡♥ s❤♦t ❛t 45 ❞❡❣r❡❡s✳ ❊①❛♠♣❧❡ ✹✳✶✶✳✾✿ ❛❝❝✉r❛❝②

❋r♦♠ t❤❡ ♣r❛❝t✐❝❛❧ ♣♦✐♥t ♦❢ ✈✐❡✇✱ ♥♦ s❤♦t ✐s ♣❡r❢❡❝t❧② ❛❝❝✉r❛t❡✳ ❊✈❡♥ ✇❤❡♥ t❤❡ ♠❛t❤❡♠❛t✐❝s ✐s s❡❡♥ ❛s ♣❡r❢❡❝t✱ ♦✉r ❧✐♠✐t❡❞ ❦♥♦✇❧❡❞❣❡ ♦❢ t❤❡ ♠❛♥② ♣❛r❛♠❡t❡rs t❤❛t ❛✛❡❝t t❤❡ ❛❝❝✉r❛❝② ♦❢ ♦✉r s❤♦t ✇✐❧❧ ♠❛❦❡ ✉s ✉♥❞❡r✲ ♦r ♦✈❡r❡st✐♠❛t❡ t❤❡ ✜♥❛❧ ❧♦❝❛t✐♦♥ ♦❢ t❤❡ ❝❛♥♥♦♥❜❛❧❧✳

✹✳✶✶✳

✸✻✾

❙❤♦♦t✐♥❣ ❛ ❝❛♥♥♦♥

▲❡t✬s t❛❦❡ ❥✉st ♦♥❡✿ t❤❡ ❛❝❝✉r❛❝② ♦❢ ♦✉r ♠❡❛s✉r❡♠❡♥t ♦❢ t❤❡ ❤❡✐❣❤t ♦❢ t❤❡ ❤✐❧❧✳ ■❢ t❤❡ ❤❡✐❣❤t✱ H ✱ ✈❛r✐❡s✱ t❤❡♥ s♦ ✇✐❧❧ t❤❡ ♣❧❛❝❡♠❡♥t✱ D✱ ♦❢ t❤❡ s❤♦t✳ ❚❤❡ ❣♦♦❞ ♥❡✇s ✐s t❤❛t t❤❡ s♠❛❧❧❡r ❡rr♦r ✐♥ t❤❡ ❢♦r♠❡r ✇✐❧❧ ❧❡❛❞ t♦ ❛ s♠❛❧❧❡r ❡rr♦r ✐♥ t❤❡ ❧❛tt❡r✦

■♥ ❢❛❝t✱ ✇❡ ❝❛♥ ❛❝❤✐❡✈❡ r❡q✉✐r❡❞ ❞❡❣r❡❡ ♦❢ ❛❝❝✉r❛❝②✱ ε✱ ♦❢ ♦✉r s❤♦t ✐❢ ✇❡ ❝❛♥ ❡♥s✉r❡ ❛ s✉✣❝✐❡♥t ❛❝❝✉r❛❝②✱ δ✱ ♦❢ t❤❡ ♠❡❛s✉r❡♠❡♥t ♦❢ t❤❡ ❤❡✐❣❤t ♦❢ t❤❡ ❤✐❧❧✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ❞❡♣❡♥❞❡♥❝❡ ♦❢ D ♦♥ H ✐s ✿ ❚❤❡ ❞✐st❛♥❝❡ ♦❢ t❤❡ s❤♦t ❞❡♣❡♥❞s ❝♦♥t✐♥✉♦✉s❧② ♦♥ t❤❡ ✐♥✐t✐❛❧ ❡❧❡✈❛t✐♦♥✳ ❛♥②

❝♦♥t✐♥✉♦✉s

❊①❡r❝✐s❡ ✹✳✶✶✳✶✵

Pr♦✈❡ t❤✐s st❛t❡♠❡♥t ❛❧❣❡❜r❛✐❝❛❧❧②✳ ❊①❡r❝✐s❡ ✹✳✶✶✳✶✶

❲❤❛t ✐❢ ✐♥st❡❛❞ ♦❢ t❛r❣❡t s❤♦♦t✐♥❣ t❤✐s ✇❛s ❛ ❣❛♠❡ ♦❢ t❡♥♥✐s❄ ❊①❡r❝✐s❡ ✹✳✶✶✳✶✷

Pr♦✈❡ t❤❛t t❤❡ ❞✐st❛♥❝❡ ♦❢ t❤❡ s❤♦t ❞❡♣❡♥❞s ❝♦♥t✐♥✉♦✉s❧② ♦♥ t❤❡ ✐♥✐t✐❛❧

✳

❛♥❣❧❡

■t s♦ ❤❛♣♣❡♥s t❤❛t t❤❡ ♣❛t❤ ✐s t❤❡ ❣r❛♣❤ ♦❢ y ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ x✦ ■♥ t❤❛t ❝❛s❡✱ t❤✐s ❞❡♣❡♥❞❡♥❝❡ ❝❛♥ ❛❧s♦ ❜❡ ❢♦✉♥❞ ❜② s✉❜st✐t✉t✐♦♥ ♦❢ t ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ x ✐♥t♦ y(t)✳ ■s t❤❡ ♣❛t❤ ❛ ♣❛r❛❜♦❧❛❄ ■t ❧♦♦❦s ❧✐❦❡ ♦♥❡ ✇❤❡♥ ♣❧♦tt❡❞ ♣♦✐♥t ❜② ♣♦✐♥t ✭❛s ❛❜♦✈❡✮✱ ❜✉t ❧❡t✬s ❞♦ t❤❡ ❛❧❣❡❜r❛ ♥♦✇✳ ❋r♦♠ t❤❡ ❡q✉❛t✐♦♥ ❢♦r x✱ x = x 0 + v0 t ,

✇❡ ❝♦♥❝❧✉❞❡ t❤❛t

t = (x − x0 )/v0

✇❤❡♥❡✈❡r v0 6= 0✳ ❚❤❡♥✱ ✇❡ s✉❜st✐t✉t❡ ✐♥t♦ t❤❡ ❡q✉❛t✐♦♥ ❢♦r y✿ y = y0 +

u0 g (x − x0 ) − 2 (x − x0 )2 v0 2v0

❚❤✐s ✐s ❛ q✉❛❞r❛t✐❝ ❢✉♥❝t✐♦♥✦ ❲❡ ❤❛✈❡ ♣r♦✈❡♥ ✐t✿ ◮

❚❤❡ tr❛❥❡❝t♦r② ♦❢ t❤❡ ❝❛♥♥♦♥❜❛❧❧ ✐s ❛ ♣❛r❛❜♦❧❛✳

❚❤❛t✬s ✇❤② t❤❡ ♣❛t❤ tr❛❝❡❞ ✐♥ t❤❡ s❦② ✐s ❝✉r✈❡❞ t❤✐s ✇❛②✳ ❊①❡r❝✐s❡ ✹✳✶✶✳✶✸

❲❤❛t ✐s t❤❡ ♣❛t❤ ✇❤❡♥ v0 = 0❄ ❊①❡r❝✐s❡ ✹✳✶✶✳✶✹

❙❤♦✇ t❤❛t t❤❡ ❝♦❡✣❝✐❡♥t uv ✐s t❤❡ t❛♥❣❡♥t ♦❢ t❤❡ ❛♥❣❧❡ ♦❢ t❤❡ s❤♦t✳

❈❤❛♣t❡r ✺✿ ❚❤❡ ♠❛✐♥ t❤❡♦r❡♠s ♦❢ ❞✐✛❡r❡♥✲ t✐❛❧ ❝❛❧❝✉❧✉s

❈♦♥t❡♥ts

✺✳✶ ▼♦♥♦t♦♥✐❝✐t② ❛♥❞ ❡①tr❡♠❡ ♣♦✐♥ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✳✷ ❖♣t✐♠✐③❛t✐♦♥ ♦❢ ❢✉♥❝t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✳✸ ❲❤❛t t❤❡ ❞❡r✐✈❛t✐✈❡ s❛②s ❛❜♦✉t t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✿ ❚❤❡ ▼❡❛♥ ❱❛❧✉❡ ❚❤❡♦r❡♠ ✺✳✹ ▼♦♥♦t♦♥✐❝✐t② ❛♥❞ t❤❡ s✐❣♥ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✳✺ ❈♦♥❝❛✈✐t② ❛♥❞ t❤❡ s✐❣♥ ♦❢ t❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✳✻ ❉❡r✐✈❛t✐✈❡s ❛♥❞ ❡①tr❡♠❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✳✼ ❆♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥✿ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ✇❤❛t ❢✉♥❝t✐♦♥❄ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✳✽ ❆♥t✐❞❡r✐✈❛t✐✈❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✼✵ ✸✽✻ ✸✾✶ ✸✾✽ ✹✵✸ ✹✶✺ ✹✷✷ ✹✷✻

✺✳✶✳ ▼♦♥♦t♦♥✐❝✐t② ❛♥❞ ❡①tr❡♠❡ ♣♦✐♥ts

❚❤✐s ✐s ❤♦✇ ✇❡ ❤❛✈❡ ❜❡❡♥ ❛❜❧❡ t♦ s♦❧✈❡ ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠s s♦ ❢❛r✳ ❊①❛♠♣❧❡ ✺✳✶✳✶✿ ❢❛r♠❡r

▲❡t✬s r❡✈✐❡✇ t❤❡ ❢♦❧❧♦✇✐♥❣✱ ♣♦ss✐❜❧② ❢❛♠✐❧✐❛r✱ ♣r♦❜❧❡♠ ✭s❡❡♥ ✐♥ ❱♦❧✉♠❡ ✶✱ ❈❤❛♣t❡r ✶P❈✲✷✮✿ ❆ ❢❛r♠❡r ✇✐t❤ 100 ②❛r❞s ♦❢ ❢❡♥❝✐♥❣ ♠❛t❡r✐❛❧ ✇❛♥ts t♦ ❜✉✐❧❞ ❛s ❧❛r❣❡ ❛ r❡❝t❛♥❣✉❧❛r ❡♥❝❧♦s✉r❡ ❛s ♣♦ss✐❜❧❡ ❢♦r ❤✐s ❝❛tt❧❡✿

❲❡ ❞❡✜♥❡ t❤❡ ✈❛r✐❛❜❧❡s✿ t❤❡ ✇✐❞t❤ W ✭t❤❡ ❞❡♣t❤ D = 50 − W ✮ ❛♥❞ t❤❡ ❛r❡❛ A✳ ❚❤❡ t❤❡ ❛r❡❛ ✐s ❛ ❢✉♥❝t✐♦♥ ♦❢ W ✿ A = W (50 − W ) .

❲❡ ❝❤♦♦s❡ ❛ ♠❡❛♥✐♥❣❢✉❧ ❞♦♠❛✐♥ ❢♦r t❤❡ ❢✉♥❝t✐♦♥✿ [0, 50]✳

❚❤❡♥ t❤❡ ♣r♦❜❧❡♠ ❜❡❝♦♠❡s✿ ◮ ❲❤❛t ✐s t❤❡ ❧❛r❣❡st ♦✉t♣✉t ♦❢ t❤❡ ❢✉♥❝t✐♦♥ A(W ) = −W 2 + 50W ✇✐t❤ 0 ≤ W ≤ 50❄

✺✳✶✳ ▼♦♥♦t♦♥✐❝✐t② ❛♥❞ ❡①tr❡♠❡ ♣♦✐♥ts

✸✼✶

■♥ ✐ts s✐♠♣❧❡st ✐♥t❡r♣r❡t❛t✐♦♥✱ t❤❡ ❛♥s✇❡r t♦ t❤❡ q✉❡st✐♦♥ ✐s ❢♦✉♥❞ ❜② ❡①❛♠✐♥✐♥❣ t❤❡ ❞❛t❛ ♣r♦❞✉❝❡❞ ❜② t❤✐s ❢♦r♠✉❧❛ ❛♥❞ ❝❤♦♦s✐♥❣ t❤❡ ❧❛r❣❡st ✈❛❧✉❡ ❢r♦♠ t❤❡ A✲❝♦❧✉♠♥ ✭t❤❡ ♦✉t♣✉ts✮✿

❚❤❡ ♠❛①✐♠❛❧ ✈❛❧✉❡ ♦❢ t❤❡ ❛r❡❛ ❛♣♣❡❛rs t♦ ❜❡ 25 · 25 = 625✳ ❚♦ ❝♦♥✜r♠✱ ✇❡ ❡✈❛❧✉❛t❡ t❤❡ ♠♦♥♦t♦♥✐❝✐t② ♦❢ t❤❡ ❞❡♣❡♥❞❡♥❝❡ ♦❢ A ♦♥ W ✳ ❲❡ ❡①❛♠✐♥❡ t❤❡ ❞❛t❛ ♣r♦❞✉❝❡❞ ❜② t❤✐s ❢♦r♠✉❧❛ ❛♥❞ ♥♦t✐❝❡ ❛ ♣❛tt❡r♥ ♦❢ ❣r♦✇t❤ ❛♥❞ ❞❡❝❧✐♥❡ ♦❢ t❤❡ ✈❛❧✉❡s ✐♥ t❤❡ A✲❝♦❧✉♠♥✳ ❖♥❡ ❝❛♥ ❛❧s♦ ✇❛t❝❤ t❤❡ s✐❣♥ ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡s ♦❢ A❀ t❤❡② ❛r❡ ❡❛s✐❧② ❝♦♠♣✉t❡❞✳ ❚❤❡♥✱ t❤❡ ✈❛❧✉❡s ❛r❡ ✐♥❝r❡❛s✐♥❣ ✇❤❡♥ t❤❡ ❞✐✛❡r❡♥❝❡ ✐s ♣♦s✐t✐✈❡✦ ❚❤❡s❡ ❛r❡ t❤❡ ♣♦✐♥ts ✇❤❡r❡ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ♦❢ ♦✉r ❢✉♥❝t✐♦♥ ❛r❡ ♣♦s✐t✐✈❡ t♦♦✳ ❲❡ ♦❜s❡r✈❡ t❤❛t ♦♥ t❤❡ ✐♥t❡r✈❛❧ [0, 25]✱ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✐s ♣♦s✐t✐✈❡ ❛♥❞ ❡❧s❡✇❤❡r❡ ✐s ♥❡❣❛t✐✈❡✳ ◆♦✇✱ ❤❡r❡ ✐s ❛ s❧✐❣❤t❧② ❞✐✛❡r❡♥t ❛♣♣r♦❛❝❤✳ ❊①❛♠✐♥✐♥❣ t❤❡ ❣r❛♣❤ r❡✈❡❛❧s t❤❛t t❤❡ ♠❛①✐♠✉♠ ✈❛❧✉❡ ❧✐❡s s♦♠❡✇❤❡r❡ ✐♥ t❤❡ ❛r❡❛ ✇❤❡r❡ t❤❡ ❣r❛♣❤ ✐s t❤❡ ✢❛tt❡st✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤✐s ✐s ✇❤❡r❡ t❤❡ s❧♦♣❡s ♦❢ t❤❡ s❡❝❛♥t ❧✐♥❡s ❛r❡ ❝❧♦s❡st t♦ ③❡r♦✳ ❇✉t t❤❡s❡ s❧♦♣❡s ❛r❡ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ♦❢ ♦✉r ❢✉♥❝t✐♦♥✳ ▲❡t✬s ✜♥❞ t❤❡♠✳ ❋r♦♠ t❤❡ P♦✇❡r ❋♦r♠✉❧❛✱ ✇❡ ❤❛✈❡✿ ∆A = −2W − h + 50 . ∆W

❲❡ ❝♦♥❝❧✉❞❡ t❤❛t ♦♥ ✐♥t❡r✈❛❧ [25, 25 + h]✱ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✐s −h✳ ❚❤✐s ✐s ♣♦t❡♥t✐❛❧❧② t❤❡ ♥✉♠❜❡r ❝❧♦s❡st t♦ 0 ✭♣r♦✈✐❞❡❞ h ✐s s♠❛❧❧ ❡♥♦✉❣❤✮✳ ❲❡ ❤❛✈❡ ❛rr✐✈❡❞ ❛t t❤❡ s❛♠❡ r❡s✉❧t ❛s ❛❜♦✈❡✦ ❇❡❝❛✉s❡ ♦❢ t❤❡ ❣❛♣s ✐♥ t❤❡ ❞❛t❛ ❛♥❞ t❤❡ ❣r❛♣❤✱ ✇❡ ❝❛♥✬t ❜❡ ❝♦♠♣❧❡t❡❧② s✉r❡ ✇❡✬✈❡ ❢♦✉♥❞ t❤❡ ❜❡st ❛♥s✇❡r ♦r t❤❛t ✇❡✬✈❡ ❢✉❧❧② ❝❧❛ss✐✜❡❞ ❛❧❧ ♣♦✐♥ts✿

❉❡♣❡♥❞✐♥❣ ♦♥ ❤♦✇ ❡①♣❡♥s✐✈❡ ❡✈❡r② ❢♦♦t ♦❢ ❢❡♥❝✐♥❣ ✐s✱ ✇❡ ♠❛② ❝❤♦♦s❡ t♦ ❝♦♥s✐❞❡r A ❛s ❛ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ♦♥ t❤❡ ✇❤♦❧❡ [0, 50]✳ ▲❡t✬s r❡✈✐❡✇ t❤❡ t♦♦❧s ❛t ♦✉r ❞✐s♣♦s❛❧✳ ❊①❛♠✐♥✐♥❣ t❤❡ ❣r❛♣❤ r❡✈❡❛❧s t❤❛t t❤❡ ♠❛①✐♠✉♠ ✈❛❧✉❡ ❧✐❡s s♦♠❡✇❤❡r❡ ✐♥ t❤❡ ❛r❡❛ ✇❤❡r❡ t❤❡ ❣r❛♣❤ ✐s ❝❧♦s❡st t♦ t❤❡ ❤♦r✐③♦♥t❛❧✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤✐s ✐s ✇❤❡r❡ t❤❡ s❧♦♣❡ ♦❢ t❤❡ t❛♥❣❡♥t ❧✐♥❡ ✐s ③❡r♦✳ ❇✉t t❤✐s ♥✉♠❜❡r ✐s t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ♦✉r ❢✉♥❝t✐♦♥✳ ❋r♦♠ t❤❡ P♦✇❡r

✺✳✶✳

▼♦♥♦t♦♥✐❝✐t② ❛♥❞ ❡①tr❡♠❡ ♣♦✐♥ts

✸✼✷

❋♦r♠✉❧❛✱ ✇❡ ❤❛✈❡✿ ❲❡ s❡t ✐t ❡q✉❛❧ t♦

dA = −2W + 50 . dW 0

❛♥❞ s♦❧✈❡✿

−2W + 50 = 0 . ❲❡ ❝♦♥❝❧✉❞❡ t❤❛t ❛t

W = 25✱

t❤❡ ❞❡r✐✈❛t✐✈❡ ✐s

0✳

❆♥❞ t❤❡r❡ ❛r❡ ♥♦ ♦t❤❡r ♣♦✐♥ts ❧✐❦❡ t❤❛t✳

❋✉rt❤❡r♠♦r❡✱ ❡①❛♠✐♥✐♥❣ t❤❡ ❣r❛♣❤ r❡✈❡❛❧s t❤❛t t❤❡ ❢✉♥❝t✐♦♥ ✐s ✐♥❝r❡❛s✐♥❣ ✇❤❡r❡ t❤❡ s❧♦♣❡s ♦❢ t❤❡ t❛♥❣❡♥t ❧✐♥❡s ❛r❡ ♣♦s✐t✐✈❡✱ ❛♥❞ ❞❡❝r❡❛s✐♥❣ ✇❤❡r❡ t❤❡② ❛r❡ ♥❡❣❛t✐✈❡✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ♠♦♥♦t♦♥✐❝✐t② ✐s ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ s✐❣♥ ♦❢ t❤❡ s❧♦♣❡ ♦❢ t❤❡ t❛♥❣❡♥t ❧✐♥❡✳ ❲❡ ❝♦♥❝❧✉❞❡ t❤❛t ♦♥ t❤❡ ✐♥t❡r✈❛❧ ❞❡r✐✈❛t✐✈❡ ✐s str✐❝t❧② ♣♦s✐t✐✈❡ ❛♥❞ ♦♥ ❛♥ ✐♥❡q✉❛❧✐t②✿

dA >0 dW

❍♦✇ ❝❛♥ ✇❡ ✜♥❞ ♦✉t ❛❜♦✉t

❛♥②

(25, 50)

✇❤❡♥

(0, 25)✱ t❤❡

✐t ✐s str✐❝t❧② ♥❡❣❛t✐✈❡✳ ❚❤✐s ❛♥❛❧②s✐s ❛♠♦✉♥ts t♦ s♦❧✈✐♥❣

W < 25

❛♥❞

dA 25 .

❣✐✈❡♥ ❢✉♥❝t✐♦♥ ✇❤❡t❤❡r ❛♥❞ ✇❤❡r❡ ✐t ❤❛s ♠♦♥♦t♦♥✐❝✐t② ✐♥t❡r✈❛❧s ❛♥❞ ✐ts

♠❛①✴♠✐♥ ♣♦✐♥ts❄ ❚❤❡ ❛♥s✇❡r ✇❡ ❤❛✈❡ ❞✐s❝♦✈❡r❡❞ ✐s

✇✐t❤ t❤❡ ❞❡r✐✈❛t✐✈❡

❜✉t ✇❡ ✇✐❧❧ r❡❛❝❤ t❤✐s ❣♦❛❧ ✐♥ s❡✈❡r❛❧

st❛❣❡s✳ ❋✐rst✱ s♦♠❡ ❜❛❝❦❣r♦✉♥❞✳ ❲❡ ✉♥❞❡rst❛♥❞ ✐♥❝r❡❛s✐♥❣ ❢✉♥❝t✐♦♥s ❛s ♦♥❡s ✇✐t❤ ❣r❛♣❤s r✐s✐♥❣✱ ❛♥❞ ❞❡❝r❡❛s✐♥❣ ❢✉♥❝t✐♦♥s ❛s ♦♥❡s ✇✐t❤ ❣r❛♣❤s ❢❛❧❧✐♥❣✳

❲❡ ❛❧s♦ ✈✐s✉❛❧✐③❡ ♠♦♥♦t♦♥✐❝✐t② ♦❢ ❢✉♥❝t✐♦♥s ✐♥ t❡r♠s ♦❢ ♣❛rts ♦❢ t❤❡ ❣r❛♣❤ ❛❜♦✈❡ ♦r ❜❡❧♦✇ ♦t❤❡r

♣❛rts✳ ❍♦✇❡✈❡r✱ t❤❡ ♣r❡❝✐s❡ ❞❡✜♥✐t✐♦♥ ♠✉st r❡❧② ♦♥ ❝♦♠♣❛r✐♥❣ t❤❡ ✈❛❧✉❡s

t✇♦ ♣♦✐♥ts ❛t ❛ t✐♠❡✳

❘❡❝❛❧❧ t❤❡ ❞❡✜♥✐t✐♦♥ ✭s❡❡♥ ✐♥ ❱♦❧✉♠❡ ✶✱ ❈❤❛♣t❡r ✶P❈✲✹✮✿

❉❡✜♥✐t✐♦♥ ✺✳✶✳✷✿ ✐♥❝r❡❛s✐♥❣ ❛♥❞ ❞❡❝r❡❛s✐♥❣ ❢✉♥❝t✐♦♥s I ♦❢ ✐ts ❞♦♠❛✐♥✳ ❚❤❡ ❢✉♥❝t✐♦♥ y = f (x) ✐s ❝❛❧❧❡❞ ✐♥❝r❡❛s✐♥❣ ♦♥ I ✐❢✱ ❢♦r ❛❧❧ a, b ✐♥ I ✱ ✇❡ ❤❛✈❡✿

❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ❢✉♥❝t✐♦♥ ✶✳

y = f (x)

✐❢ ✷✳ ❚❤❡ ❢✉♥❝t✐♦♥ ✐s ❝❛❧❧❡❞

t❤❡♥

❞❡❝r❡❛s✐♥❣ ♦♥ I ✐❢

❚❤❡ ❢✉♥❝t✐♦♥ ✐s ❛❧s♦ ❝❛❧❧❡❞

a ≤ b,

❛♥❞ ❛ s✉❜s❡t

a ≤ b,

t❤❡♥

str✐❝t❧② ✐♥❝r❡❛s✐♥❣

f (a) ≤ f (b) . ✐❢✱ ❢♦r ❛❧❧

a, b

✐♥

I✱

✇❡ ❤❛✈❡✿

f (a) ≥ f (b) . ❛♥❞

str✐❝t❧② ❞❡❝r❡❛s✐♥❣

r❡s♣❡❝t✐✈❡❧②

✐❢ t❤❡s❡ ♣❛✐rs ♦❢ ✈❛❧✉❡s ❝❛♥♥♦t ❜❡ ❡q✉❛❧❀ ✐✳❡✳✱ ✇❡ r❡♣❧❛❝❡ t❤❡ ♥♦♥✲str✐❝t ✐♥❡q✉❛❧✐t② s✐❣♥s ✏ ≤✑ ❛♥❞ ✏ ≥✑ ✇✐t❤ str✐❝t ✏ ✑✳

❝❛❧❧❡❞

❲❤❡♥ t❤❡ s❡t

I

♠♦♥♦t♦♥❡

❛♥❞

str✐❝t❧② ♠♦♥♦t♦♥❡✳

❈♦❧❧❡❝t✐✈❡❧②✱ t❤❡s❡ ❢✉♥❝t✐♦♥s ❛r❡

✐s ❛♥ ✐♥t❡r✈❛❧✱ t❤❡ ♣r♦❜❧❡♠ r❡♣r❡s❡♥ts ❛ s✐❣♥✐✜❝❛♥t ❝❤❛❧❧❡♥❣❡ ✭❛s s❡❡♥ ✐♥ ❱♦❧✉♠❡ ✶✱ ❈❤❛♣t❡r

✶P❈✲✹✮ ❜❡❝❛✉s❡ ❞r❛✇✐♥❣ s✉❝❤ ❝♦♥❝❧✉s✐♦♥s ♠❡❛♥s ❝♦♠♣❛r✐♥❣

✐♥✜♥✐t❡❧② ♠❛♥②

♣❛✐rs ♦❢ ♣♦✐♥ts✿

✺✳✶✳

▼♦♥♦t♦♥✐❝✐t② ❛♥❞ ❡①tr❡♠❡ ♣♦✐♥ts

✸✼✸

❊①❛♠♣❧❡ ✺✳✶✳✸✿ tr❛♥s❢♦r♠❛t✐♦♥s ❲❤❛t ✐s t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ ❞❡✜♥✐t✐♦♥ ✐❢ ✇❡ ❧♦♦❦ ❛t ❢✉♥❝t✐♦♥s ❛s ❛♥❞ y ✲❛①✐s s✐❞❡ ❜② s✐❞❡✿

tr❛♥s❢♦r♠❛t✐♦♥s ❄ ▲❡t✬s ♣❧❛❝❡ t❤❡ x✲

❆♥ ✐♥❝r❡❛s✐♥❣ ❢✉♥❝t✐♦♥ ❞♦❡s♥✬t ✢✐♣ ❛♥② ♣❛rts ♦❢ t❤❡ ✐♥t❡r✈❛❧ ❛♥❞ ❛ ❞❡❝r❡❛s✐♥❣ ✢✐♣s ❛❧❧✦ ◆❡①t✱ ♠❛①✐♠❛ ❛♥❞ ♠✐♥✐♠❛✿

❊✈❡♥ t❤♦✉❣❤ ✇❡ ✉♥❞❡rst❛♥❞ t❤❡ ♠❛①✐♠❛ ❛♥❞ ♠✐♥✐♠❛ ♦❢ ❢✉♥❝t✐♦♥s ❛s t❤♦s❡ ❧♦❝❛t✐♦♥s ♦♥ t❤❡ ❣r❛♣❤s ❛❜♦✈❡ ♦r ❜❡❧♦✇ ❛❧❧ ♦t❤❡rs✱ t❤❡ ♣r❡❝✐s❡ ❞❡✜♥✐t✐♦♥ ♠✉st r❡❧②✱ ♦♥❝❡ ❛❣❛✐♥✱ ♦♥ ❝♦♠♣❛r✐♥❣ t❤✐s ❧♦❝❛t✐♦♥ t♦ ✿

❛ t✐♠❡

♦♥❡ ♣♦✐♥t ❛t

❉❡✜♥✐t✐♦♥ ✺✳✶✳✹✿ ❣❧♦❜❛❧ ♠❛①✐♠❛ ❛♥❞ ♠✐♥✐♠❛ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ❢✉♥❝t✐♦♥ y = f (x) ❛♥❞ ❛ s✉❜s❡t I ♦❢ ✐ts ❞♦♠❛✐♥✳ ✶✳ ◆✉♠❜❡r x = c ✐s ❝❛❧❧❡❞ ❛ ♦❢ f ♦♥ I ✐❢ f (c) ✐s t❤❡ ♠❛①✐♠✉♠ ✈❛❧✉❡ ♦❢ t❤❡ r❛♥❣❡ ♦❢ f ♦♥ I ✱ ✐✳❡✳✱

❣❧♦❜❛❧ ♠❛①✐♠✉♠ ♣♦✐♥t

f (c) ≥ f (x) ❢♦r ❛❧❧ x ✐♥ I ,

❣❧♦❜❛❧ ♠❛①✐♠✉♠ ✈❛❧✉❡ ♦❢ f ♦♥ I ✳ ❣❧♦❜❛❧ ♠✐♥✐♠✉♠ ♣♦✐♥t ♦❢ f ♦♥ I ✐❢ f (c) ✐s t❤❡

t❤❡♥ y = f (c) ✐s ❝❛❧❧❡❞ t❤❡ ✷✳ ◆✉♠❜❡r x = c ✐s ❝❛❧❧❡❞ ❛ ♠✐♥✐♠✉♠ ✈❛❧✉❡ ♦❢ t❤❡ r❛♥❣❡ ♦❢ f ♦♥ I ✱ ✐✳❡✳✱

f (c) ≤ f (x) ❢♦r ❛❧❧ x ✐♥ I ,

✺✳✶✳

▼♦♥♦t♦♥✐❝✐t② ❛♥❞ ❡①tr❡♠❡ ♣♦✐♥ts t❤❡♥

✸✼✹

❣❧♦❜❛❧ ♠✐♥✐♠✉♠ ✈❛❧✉❡ ❣❧♦❜❛❧ ❡①tr❡♠❡ ♣♦✐♥ts ❛♥❞ ✈❛❧✉❡s

y = f (c)

❲❡ ❝❛❧❧ t❤❡s❡

✐s ❝❛❧❧❡❞ t❤❡

♦❢

f

I✳

♦♥

✱ ♦r ❡①tr❡♠❛✳

❆s ②♦✉ ❝❛♥ s❡❡✱ t❤❡r❡ ❝❛♥ ❜❡ ♠❛♥② ♠❛①

♣♦✐♥ts

✭t❤♦s❡ ❛r❡

x✬s✮

❜✉t ♦♥❧② ♦♥❡ ♠❛①

✈❛❧✉❡

✭t❤✐s ✐s ❛

y ✮✳

■♥❞❡❡❞✱ t❤❡r❡ ❝❛♥ ❜❡ ♦♥❧② ♦♥❡ ✏❤✐❣❤❡st ❡❧❡✈❛t✐♦♥✑ ✐♥ ❛♥ ❛r❡❛ ❜✉t t❤❡r❡ ❝❛♥ ❜❡ ♠❛♥② ✏❤✐❣❤❡st ❧♦❝❛t✐♦♥s✑✳ ❊①❛♠♣❧❡ ✺✳✶✳✺✿ ♠❛♥② ♠❛①✐♠✉♠ ♣♦✐♥ts

❖✈❡r t❤❡ ✐♥t❡r✈❛❧

y = 1✱

I = (−∞, +∞)✱

❜✉t ♠❛♥② ❣❧♦❜❛❧ ♠❛①✐♠✉♠

t❤❡ ❢✉♥❝t✐♦♥

♣♦✐♥ts

f (x) = sin x

❤❛s ♦♥❧② ♦♥❡ ❣❧♦❜❛❧ ♠❛①✐♠✉♠

✱

✈❛❧✉❡

✱

x = π/2 + 2πk, k = 0, ±1, ±2, ...

❙✐♠✐❧❛r❧②✱ t❤❡ ❢✉♥❝t✐♦♥ ❤❛s ♦♥❧② ♦♥❡ ❣❧♦❜❛❧ ♠✐♥✐♠✉♠ ✈❛❧✉❡✱

y = −1✱ ❜✉t ♠❛♥② ❣❧♦❜❛❧ ♠✐♥✐♠✉♠ ♣♦✐♥ts✿

x = −π/2 + 2πk, k = 0, ±1, ±2, ... ❚❤❡ ❢✉♥❝t✐♦♥ ❝❤❛♥❣❡s ✐ts ♠♦♥♦t♦♥✐❝✐t② ❛t ✐ts ❡①tr❡♠❡ ♣♦✐♥ts✳

❲❡ ✇✐❧❧ ❧✐♠✐t ♦✉r ❛tt❡♥t✐♦♥ t♦

❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s

✐♥ ♦r❞❡r t♦ ❛✈♦✐❞ t❤❡ s✐t✉❛t✐♦♥ ✇❤❡♥ t❤❡s❡ ✈❛❧✉❡s ✕ t❤❡

s✉♣r❡♠✉♠ ❛♥❞ t❤❡ ✐♥✜♠✉♠ ✕ ❛r❡ ♥❡✈❡r r❡❛❝❤❡❞ ✭❈❤❛♣t❡r ✶✮✿

❚❤❡ st❛rt✐♥❣ ♣♦✐♥t ♦❢ ♦✉r ❛♥❛❧②s✐s ✇✐❧❧ ❜❡✱ ❛s ❜❡❢♦r❡✱ t❤❡ ❞✐✛❡r❡♥❝❡s

∆f

♦❢

f✳

❚❤❡✐r

s✐❣♥s

❞❡t❡r♠✐♥❡ ✇❤❡t❤❡r

✺✳✶✳

▼♦♥♦t♦♥✐❝✐t② ❛♥❞ ❡①tr❡♠❡ ♣♦✐♥ts

✸✼✺

t❤❡ ❢✉♥❝t✐♦♥ ❣♦❡s ✉♣ ♦r ❞♦✇♥ ❢r♦♠ ♥♦❞❡ t♦ ♥♦❞❡ ♦❢ t❤❡ ♣❛rt✐t✐♦♥✿

❚❤❡ ❞✐✛❡r❡♥❝❡ ∆f ❛♥❞ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t

∆f ❝♦♠❡ ❤❛♥❞ ✐♥ ❤❛♥❞✿ ∆x

f (xk ) ≤ f (xk+1 ) ⇐⇒ ∆f (ck ) ≥ 0 ⇐⇒

∆f (ck ) ≥ 0 ∆x

❚❤❡ ❞✐s❝r❡t❡ ❝❛s❡ ✐s s♦❧✈❡❞✦ ❲❡ ✇r✐t❡ t❤❡ s✉♠♠❛r② ❜❡❧♦✇✿

❚❤❡♦r❡♠ ✺✳✶✳✻✿ ▼♦♥♦t♦♥✐❝✐t② ❢r♦♠ ❙✐❣♥ ♦❢ ❉✐✛❡r❡♥❝❡ ❙✉♣♣♦s❡ ❛ ❢✉♥❝t✐♦♥ f ✐s ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ ❛♥ ✐♥t❡r✈❛❧✳ ❚❤❡♥✿ • ❚❤❡ ❢✉♥❝t✐♦♥ f ✐s ✭str✐❝t❧②✮ ✐♥❝r❡❛s✐♥❣ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐ts ❞✐✛❡r❡♥❝❡ ∆f ✐s ♥♦♥✲♥❡❣❛t✐✈❡ ✭♣♦s✐t✐✈❡✮✳ • ❚❤❡ ❢✉♥❝t✐♦♥ f ✐s ✭str✐❝t❧②✮ ❞❡❝r❡❛s✐♥❣ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐ts ❞✐✛❡r❡♥❝❡ ∆f ✐s ♥♦♥✲♣♦s✐t✐✈❡ ✭♥❡❣❛t✐✈❡✮✳ ◆♦✇ ✇❡ ❥✉st ❞✐✈✐❞❡ ❜② ∆x > 0✿

❚❤❡♦r❡♠ ✺✳✶✳✼✿ ▼♦♥♦t♦♥✐❝✐t② ❢r♦♠ ❙✐❣♥ ♦❢ ❉✐✛❡r❡♥❝❡ ◗✉♦t✐❡♥t ❙✉♣♣♦s❡ ❛ ❢✉♥❝t✐♦♥ f ✐s ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ ❛♥ ✐♥t❡r✈❛❧✳ ❚❤❡♥✿ • ❚❤❡ ❢✉♥❝t✐♦♥ f ✐s ✭str✐❝t❧②✮ ✐♥❝r❡❛s✐♥❣ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐ts ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ∆f ✐s ♥♦♥✲♥❡❣❛t✐✈❡ ✭♣♦s✐t✐✈❡✮✳ ∆x • ❚❤❡ ❢✉♥❝t✐♦♥ f ✐s ✭str✐❝t❧②✮ ❞❡❝r❡❛s✐♥❣ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐ts ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ∆f ✐s ♥♦♥✲♣♦s✐t✐✈❡ ✭♥❡❣❛t✐✈❡✮✳ ∆x

❚❤❡ ❝♦♥❝❧✉s✐♦♥s ❛r❡ ❝♦♥✜r♠❡❞ ❜② t❤❡s❡ ❡①❛♠♣❧❡s✿

✺✳✶✳

▼♦♥♦t♦♥✐❝✐t② ❛♥❞ ❡①tr❡♠❡ ♣♦✐♥ts

✸✼✻

❍♦✇ ❞♦❡s t❤✐s ❤❡❧♣ ✇✐t❤ t❤❡ st✉❞② ♦❢ ♠♦♥♦t♦♥✐❝✐t② ♦❢ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ♦♥ ✐♥t❡r✈❛❧s✱ ✐✳❡✳✱ t❤❡ ✏❝♦♥t✐♥✉♦✉s✑ ❝❛s❡❄ ❚❛❦✐♥❣ t❤❡ ❧✐♠✐t ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✕ ✇✐t❤ ❛ ♣❛rt✐❝✉❧❛r s✐❣♥ ✕ ✇✐❧❧ t❡❧❧ ✉s ❛❜♦✉t t❤❡

❞❡r✐✈❛t✐✈❡

✳

s✐❣♥ ♦❢ t❤❡

❇✉t ✜rst✱ ❧❡t✬s r❡✈✐❡✇ t❤❛t ✇❡ ❛❧r❡❛❞② ❦♥♦✇✳ ❲❤❛t ❞♦ ✇❡ ❦♥♦✇ ❛❜♦✉t ❛ ❣❡♥❡r❛❧ ❧✐♥❡❛r ❢✉♥❝t✐♦♥❄ ■ts ❞❡r✐✈❛t✐✈❡ ✐s ✐ts s❧♦♣❡✿

f (x) = mx + b f ′ (x) = m ❙♦✱ ✇❡ ❞❡r✐✈❡ ✐ts ♠♦♥♦t♦♥✐❝✐t② ❢r♦♠ t❤❡ s✐❣♥ ♦❢ ✐ts ❞❡r✐✈❛t✐✈❡ ❜② ♠❛♥✐♣✉❧❛t✐♥❣ t❤❡s❡ ✐♥❡q✉❛❧✐t✐❡s ✭❛s s❡❡♥ ✐♥ ❱♦❧✉♠❡ ✶✱ ❈❤❛♣t❡r ✶P❈✲✺✮✿

m < 0 =⇒ mu + b > mv + b m = 0 =⇒ mu + b = mv + b m > 0 =⇒ mu + b < mv + b

✐❢ ✐❢ ✐❢

u < v =⇒ f u < v =⇒ f u < v =⇒ f

✐s str✐❝t❧② ❞❡❝r❡❛s✐♥❣ ✐s ❝♦♥st❛♥t ✐s str✐❝t❧② ✐♥❝r❡❛s✐♥❣

❲❡ ❤❛✈❡ ❞❡♠♦♥str❛t❡❞ t❤❛t ♦♥❧② t❤❡ s❧♦♣❡s ♠❛tt❡r✿

❚❤✐s ✐s t❤❡ s✉♠♠❛r②✿ ❚❤❡♦r❡♠ ✺✳✶✳✽✿ ▼♦♥♦t♦♥✐❝✐t② ♦❢ ▲✐♥❡❛r ❋✉♥❝t✐♦♥s ❆ ❧✐♥❡❛r ♣♦❧②♥♦♠✐❛❧ ✐s✿ ✶✳ str✐❝t❧② ✐♥❝r❡❛s✐♥❣ ✇❤❡♥ ✐ts ❞❡r✐✈❛t✐✈❡ ✐s ♣♦s✐t✐✈❡✱ ❛♥❞ ✷✳ str✐❝t❧② ❞❡❝r❡❛s✐♥❣ ✇❤❡♥ ✐ts ❞❡r✐✈❛t✐✈❡ ✐s ♥❡❣❛t✐✈❡✳

❚❤✐♥❣s ❜❡❝♦♠❡ ♠♦r❡ ❝♦♠♣❧❡① ✐❢ ✇❡ ♥❡❡❞ t♦ ❛♥❛❧②③❡ ❛ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧✿

f (x) = ax2 + bx + c, a 6= 0 . ❘❡❝❛❧❧ t❤❛t t❤❡ ❣r❛♣❤s ♦❢ ❛❧❧ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧s ❛r❡ ❛r❡ ❛❧❧ ❝♦♠❡ ❢r♦♠ tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ ✈❡rt❡① ♦❢ t❤❡ ♣❛r❛❜♦❧❛✿

t❤❡

♣❛r❛❜♦❧❛

♣❛r❛❜♦❧❛s

y = x2 ✳

v=−

b . 2a

✭❱♦❧✉♠❡ ✶✱ ❈❤❛♣t❡r ✶P❈✲✹✮✳ ■♥ ❢❛❝t✱ t❤❡②

❲❤❛t ♠❛tt❡rs ❡s♣❡❝✐❛❧❧②✱ ✐s t❤❡ ❧♦❝❛t✐♦♥ ♦❢ t❤❡

✺✳✶✳

▼♦♥♦t♦♥✐❝✐t② ❛♥❞ ❡①tr❡♠❡ ♣♦✐♥ts

✸✼✼

❚❤❡ r❡❛s♦♥ ✐s t❤❛t t❤❡ ❧✐♥❡ x = v ✐s t❤❡ ❛①✐s ♦❢ s②♠♠❡tr② ♦❢ t❤✐s ♣❛r❛❜♦❧❛✳ ❚❤❡♥✱ ✇❡ ❝♦♥❝❧✉❞❡✿ • ■❢ a > 0✱ t❤❡♥ f ✐s str✐❝t❧② ❞❡❝r❡❛s✐♥❣ ♦♥ (−∞, v) ❛♥❞ str✐❝t❧② ✐♥❝r❡❛s✐♥❣ ♦♥ (v, +∞)✳

• ■❢ a < 0✱ t❤❡♥ f ✐s str✐❝t❧② ✐♥❝r❡❛s✐♥❣ ♦♥ (−∞, v) ❛♥❞ str✐❝t❧② ❞❡❝r❡❛s✐♥❣ ♦♥ (v, +∞)✳

❇✉t✱ ✇❡ ✇✐❧❧ ❛❧s♦ ❞✐s❝♦✈❡r✱ t❤❡ ✈❡rt❡① ✐s ✇❤❡r❡ t❤❡ ❞❡r✐✈❛t✐✈❡

❝❤❛♥❣❡s ✐ts s✐❣♥ ✦ ■♥❞❡❡❞✱ ✐ts ❞❡r✐✈❛t✐✈❡✱

f ′ (x) = 2ax + b ,

✐s ❛ ❧✐♥❡❛r ❢✉♥❝t✐♦♥✱ ❛♥❞ ✇❡ ❡❛s✐❧② ❞❡r✐✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿ • ■❢ a > 0✱ t❤❡♥ f ′ < 0 ♦♥ (−∞, v) ❛♥❞ f ′ > 0 ♦♥ (v, +∞)✳

• ■❢ a < 0✱ t❤❡♥ f ′ > 0 ♦♥ (−∞, v) ❛♥❞ f ′ < 0 ♦♥ (v, +∞)✳

❚❤✐s ✐s t❤❡ ❧✐♥❦✿

❇❡❧♦✇ ✐s t❤❡ ❝♦♥❝❧✉s✐♦♥✿ ❚❤❡♦r❡♠ ✺✳✶✳✾✿ ▼♦♥♦t♦♥✐❝✐t② ♦❢ ◗✉❛❞r❛t✐❝ ❋✉♥❝t✐♦♥s ❆ q✉❛❞r❛t✐❝ ❢✉♥❝t✐♦♥ ✐s✿ ✶✳ str✐❝t❧② ✐♥❝r❡❛s✐♥❣ ✇❤❡♥ ✐ts ❞❡r✐✈❛t✐✈❡ ✐s ♣♦s✐t✐✈❡✱ ❛♥❞ ✷✳ str✐❝t❧② ❞❡❝r❡❛s✐♥❣ ✇❤❡♥ ✐ts ❞❡r✐✈❛t✐✈❡ ✐s ♥❡❣❛t✐✈❡✳

❚❤❡ t✇♦ t❤❡♦r❡♠s ♠❛t❝❤✦ ❊①❛♠♣❧❡ ✺✳✶✳✶✵✿ r❡❝t❛♥❣✉❧❛r ❡♥❝❧♦s✉r❡

❋♦r t❤❡ ✜rst ❡①❛♠♣❧❡ ♦❢ t❤✐s s❡❝t✐♦♥✱ t❤❡ ✈❡rt❡① ♦❢ t❤❡ ♣❛r❛❜♦❧❛ ✐s ❛t v=

❞②♥❛♠✐❝s

0 + 50 = 25 . 2

❲❡ t❤❡♥ ❞❡r✐✈❡ t❤❡ ♦❢ t❤✐s s✐t✉❛t✐♦♥✿ • ❆s ✇❡ ✐♥❝r❡❛s❡ t❤❡ ✇✐❞t❤ ❢r♦♠ 0✱ t❤❡ ❛r❡❛ ❛❧s♦ ✐♥❝r❡❛s❡s✳ • ❆s t❤❡ ✇✐❞t❤ r❡❛❝❤❡s 25✱ t❤❡ ❛r❡❛ r❡❛❝❤❡s ✐ts ♠❛①✐♠✉♠ ✈❛❧✉❡ ♦❢ 625✳ • ❆s ✇❡ ✐♥❝r❡❛s❡ t❤❡ ✇✐❞t❤ ♣❛st 25✱ t❤❡ ❛r❡❛ st❛rts t♦ ❞❡❝r❡❛s❡✳ ❲✐t❤ ♠♦r❡ ❛♥❞ ♠♦r❡ ❝♦♠♣❧❡① ❢✉♥❝t✐♦♥s✱ t❤❡ ❛♥❛❧②s✐s ❜❡❝♦♠❡s ♠♦r❡ ❛♥❞ ♠♦r❡ ❝❤❛❧❧❡♥❣✐♥❣✳

❢r♦♠ ❧♦❝❛❧ t♦ ❣❧♦❜❛❧

❲❡ ✇✐❧❧ t❛❦❡ ❛♥ ✐♥❞✐r❡❝t ❛♣♣r♦❛❝❤✿ ✳ ❊✈❡♥ t❤♦✉❣❤ ♦✉r ✐♥t❡r❡st ✐s t❤❡ ♠♦♥♦t♦♥✐❝✐t② ♦❢ ❢✉♥❝t✐♦♥s ♦♥ ✇❤♦❧❡ ✐♥t❡r✈❛❧s ❛♥❞ t❤❡ ❣❧♦❜❛❧ ❡①tr❡♠❛✱ ✇❡ ✇✐❧❧ ✜rst s♦❧✈❡ t❤❡ ❧♦❝❛❧ ♣r♦❜❧❡♠s✿ ✶✳ ■s t❤❡ ❢✉♥❝t✐♦♥ str✐❝t❧② ♠♦♥♦t♦♥❡ ✐♥ t❤❡

✈✐❝✐♥✐t② ♦❢ ❛ ♣♦✐♥t❄

✷✳ ■s t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❛t ❛ ♣❛rt✐❝✉❧❛r ♣♦✐♥t t❤❡ ❧❛r❣❡st ♦r t❤❡ s♠❛❧❧❡st ✐♥ t❤❡

✈✐❝✐♥✐t② ♦❢ t❤❡ ♣♦✐♥t❄

✺✳✶✳

▼♦♥♦t♦♥✐❝✐t② ❛♥❞ ❡①tr❡♠❡ ♣♦✐♥ts

✸✼✽

■♥ ♦✉r s❡❛r❝❤ ❢♦r t❤❡ ❡①tr❡♠❛✱ t❤❡ ❢♦r♠❡r ❛r❡ t❤❡ ❧♦s❡rs ❛♥❞ t❤❡ ❧❛tt❡r ❛r❡ t❤❡ ♣♦t❡♥t✐❛❧ ✇✐♥♥❡rs ♦❢ t❤✐s ❝♦♥t❡st✿

❲❡ ✇✐❧❧ ♥♦✇ ❧❡❛r♥ ❤♦✇ t♦ ✉s❡ t❤❡

❞❡r✐✈❛t✐✈❡ t♦ ✜♥❞ t❤❡ ✐♥t❡r✈❛❧s ♦❢ ♠♦♥♦t♦♥✐❝✐t② ❛♥❞ t❤❡ ❡①tr❡♠❡ ♣♦✐♥ts✳

❚❤❡ r❡❛s♦♥ ❢♦r ♦✉r ❛♣♣r♦❛❝❤ ✐s t❤❛t t❤❡ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t t❤❡ ❢✉♥❝t✐♦♥✬s ❜❡❤❛✈✐♦r t❤❛t t❤❡ ❞❡r✐✈❛t✐✈❡ ❡♥❝♦❞❡s ✐s ✳ ■♥❞❡❡❞✱ ♥♦ ♠❛tt❡r ❤♦✇ s♠❛❧❧ ❛ ♣✐❡❝❡ ♦❢ t❤❡ ❣r❛♣❤ ❛r♦✉♥❞ t❤❡ ♣♦✐♥t (a, f (a)) ②♦✉ ❦❡❡♣✱ t❤❡ ❞❡r✐✈❛t✐✈❡ f ′ (a) ❛t t❤❛t ♣♦✐♥t ✇✐❧❧ r❡♠❛✐♥ t❤❡ s❛♠❡✳

❧♦❝❛❧

❚❤❡ t✇♦ ❞❡✜♥✐t✐♦♥s ❜❡❧♦✇ ❛r❡ t❤❡ st❡♣♣✐♥❣ st♦♥❡s t♦✇❛r❞ ❛ r❡❛❧✐st✐❝ ♠❡t❤♦❞ ❢♦r ✜♥❞✐♥❣ ✇❤❛t ✐s ❞❡✜♥❡❞ ❛❜♦✈❡✳ ❚❤❡ ❣❧♦❜❛❧ ❛♥❛❧②s✐s ♦❢ ♠♦♥♦t♦♥✐❝✐t② ✐s s✉♣♣♦s❡❞ t♦ s✉♣♣❧② ✉s ✇✐t❤ t❤❡ ✐♥t❡r✈❛❧s ✇❤❡r❡ t❤❡ ❢✉♥❝t✐♦♥ ✐s ♠♦♥♦t♦♥❡✿

❲✐t❤ t❤❡ ❧♦❝❛❧ ❛♣♣r♦❛❝❤✱ ✐♥ ❝♦♥tr❛st✱ ♦♥❧② t❤❡ ❜❡❤❛✈✐♦r ✐♥ t❤❡ ✈✐❝✐♥✐t② ♦❢ t❤❡ ♣♦✐♥t ♠❛tt❡rs✿

✺✳✶✳

▼♦♥♦t♦♥✐❝✐t② ❛♥❞ ❡①tr❡♠❡ ♣♦✐♥ts

✸✼✾

▲❡t✬s ♠❛❦❡ ✐t ♣r❡❝✐s❡✿

❉❡✜♥✐t✐♦♥ ✺✳✶✳✶✶✿ ❧♦❝❛❧❧② ✐♥❝r❡❛s✐♥❣ ❛♥❞ ❞❡❝r❡❛s✐♥❣

❧♦❝❛❧❧② ✐♥❝r❡❛s✐♥❣

✶✳ ❆ ❢✉♥❝t✐♦♥ f ✐s ❛t x = c ✐❢ ❢♦r ❛❧❧ x ✐♥ s♦♠❡ ♦♣❡♥ ✐♥t❡r✈❛❧ I ✇✐t❤✐♥ t❤❡ ❞♦♠❛✐♥ ♦❢ f ❝♦♥t❛✐♥✐♥❣ c✱ ✇❡ ❤❛✈❡✿ • f (x) ≤ f (c) ❢♦r ❛❧❧ x < c✱ ❛♥❞ • f (x) ≥ f (c) ❢♦r ❛❧❧ x > c✳ ✷✳ ❆ ❢✉♥❝t✐♦♥ f ✐s ❛t x = c ✐❢ ❢♦r ❛❧❧ x ✐♥ s♦♠❡ ♦♣❡♥ ✐♥t❡r✈❛❧ I ✇✐t❤✐♥ t❤❡ ❞♦♠❛✐♥ ♦❢ f ❝♦♥t❛✐♥✐♥❣ c✱ ✇❡ ❤❛✈❡✿ • f (x) ≥ f (c) ❢♦r ❛❧❧ x < c✱ ❛♥❞ • f (x) ≤ f (c) ❢♦r ❛❧❧ x > c✳ ❲❡ ❛❧s♦ s❛② t❤❛t f ✐s ❛t t❤❡s❡ ♣♦✐♥ts✳

❧♦❝❛❧❧② ❞❡❝r❡❛s✐♥❣

❧♦❝❛❧❧② ♠♦♥♦t♦♥❡

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿ x
x

t❤❡r❡❢♦r❡

s❧♦♣❡s ≤ 0 t❤❡r❡❢♦r❡

t❤❡r❡❢♦r❡

t❤❡r❡❢♦r❡

f ′ (c) ≥ 0 ❆◆❉ f ′ (c) ≤ 0 0 ≤ f ′ (c) ≤ 0 0 = f ′ (c) = 0

❋♦r ❝♦♠♣❧❡t❡♥❡ss✱ ✇❡ ❞❡♠♦♥str❛t❡ ❛❧❣❡❜r❛✐❝❛❧❧② t❤❛t t❤❡ s❧♦♣❡s ♦❢ t❤❡s❡ s❡❝❛♥ts ❤❛✈❡ t❤❡s❡ s✐❣♥s✳ ❋✐rst✱ t❛❦❡ ❛♥② ✭s❡❝❛♥t✮ ❧✐♥❡ t❤r♦✉❣❤ (x, f (x)) ❛♥❞ (c, f (c)) ✇✐t❤ x < c✳ ❚❤❡♥ f (x) ≤ f (c) ✇❤❡♥ x ✐s ❝❧♦s❡ ❡♥♦✉❣❤ t♦ c✳ ❲❤②❄ ❇❡❝❛✉s❡ c ✐s ❛ ❧♦❝❛❧ ♠❛① ✭r❡✈✐❡✇ t❤❡ ❞❡✜♥✐t✐♦♥✮✳ ❚❤❡♥ ✇❡ ❤❛✈❡✿ ≥0

z }| { f (c) − f (x) ≥ 0. s❧♦♣❡ = c−x | {z } >0

❙❡❝♦♥❞✱ t❛❦❡ ❛♥② ✭s❡❝❛♥t✮ ❧✐♥❡ t❤r♦✉❣❤ (x, f (x)) ❛♥❞ (c, f (c)) ✇✐t❤ x > c✳ ❚❤❡♥ f (x) ≤ f (c) ✇❤❡♥ x ✐s ❝❧♦s❡ ❡♥♦✉❣❤ t♦ c✱ ❜❡❝❛✉s❡ c ✐s ❛ ❧♦❝❛❧ ♠❛①✳ ❚❤❡♥ ✇❡ ❤❛✈❡✿ ≥0

z }| { f (c) − f (x) s❧♦♣❡ = ≤ 0. c−x | {z } 0

♦r

f ′ (c) < 0 .

✺✳✶✳

✸✽✺

▼♦♥♦t♦♥✐❝✐t② ❛♥❞ ❡①tr❡♠❡ ♣♦✐♥ts

❲❛r♥✐♥❣✦ ❆♥ ❛❧t❡r♥❛t✐✈❡ t❡r♠✐♥♦❧♦❣② t♦ ✉s❡ ✐s✿

•

✏❛❜s♦❧✉t❡✑ ❡①tr❡♠❡ ♣♦✐♥ts ✐♥st❡❛❞ ♦❢ ✏❣❧♦❜❛❧✑✱

•

✏r❡❧❛t✐✈❡✑ ❡①tr❡♠❡ ♣♦✐♥ts ✐♥st❡❛❞ ♦❢ ✏❧♦❝❛❧✑✳

❛♥❞

❊①❛♠♣❧❡ ✺✳✶✳✷✻✿ ♠❛①✐♠✐③❡ ♦♣❡♥ ❜♦①

▲❡t✬s ❝♦♥s✐❞❡r ❛ ❞✐✛❡r❡♥t ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠✿ ❚❤❡ ❝♦r♥❡rs ❛r❡ t♦ ❜❡ ❝✉t ❢r♦♠ ❛ 10 × 10 ♣✐❡❝❡ ♦❢ ❝❛r❞❜♦❛r❞ t♦ ❝r❡❛t❡ ❛♥ ♦♣❡♥ ❜♦① ♦❢ t❤❡ ❧❛r❣❡st ♣♦ss✐❜❧❡ ✈♦❧✉♠❡✳

▲❡t✬s ❞❡♥♦t❡ t❤❡ s✐❞❡ ♦❢ t❤❡ ❧✐tt❧❡ sq✉❛r❡ ❜② x✳ ❚❤❡♥ x ❜❡❝♦♠❡s t❤❡ ❤❡✐❣❤t ♦❢ t❤❡ ❜♦① ✇✐t❤ t❤❡ ✇✐❞t❤ ❡q✉❛❧ t♦ 10 − 2x✳ ❚❤❡♥ t❤❡ ✈♦❧✉♠❡ ♦❢ t❤❡ ❜♦① ✐s V = x(10 − 2x)2 = 4x3 − 40x2 + 100x .

❚❤❡ ❢✉♥❝t✐♦♥ ✐s ❝✉❜✐❝✦ ❲❡ ♥❡❡❞ t♦ ✜♥❞ t❤❡ ❧❛r❣❡st ♣♦ss✐❜❧❡ ✈❛❧✉❡ ♦❢ t❤✐s ❢✉♥❝t✐♦♥ ✭❢♦r 0 ≤ x ≤ 5✮✱ ❜✉t✱ ✉♥❢♦rt✉♥❛t❡❧②✱ t❤❡ ❝♦♠♣❧❡t❡♥❡ss ♦❢ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t t❤❡ q✉❛❞r❛t✐❝ ❢✉♥❝t✐♦♥s ✐s♥✬t ♠❛t❝❤❡❞ ❜② ✇❤❛t ✇❡ ❦♥♦✇ ❛❜♦✉t t❤❡s❡✳ ■❢ ✇❡ ♣❧♦t ✐ts ❣r❛♣❤✱ ✇❡ s❡❡ t❤❡ ❤✐❣❤❡st ♣♦✐♥t ✇✐t❤✐♥ t❤✐s ✐♥t❡r✈❛❧✱ ❜✉t✱ ✇✐t❤♦✉t s②♠♠❡tr② t♦ r❡❧② ♦♥✱ ✇❡ ❝❛♥✬t ❦♥♦✇ ✐ts ❡①❛❝t ✈❛❧✉❡✿

❲❡ ❦♥♦✇✱ ❤♦✇❡✈❡r✱ ❢r♦♠ ❋❡r♠❛t✬s ✐✳❡✳✱ t❤✐s x s❛t✐s✜❡s t❤❡ ❡q✉❛t✐♦♥✿

❚❤❡♦r❡♠

t❤❛t ✇❡ ❝❛♥ ✜♥❞ t❤✐s ♣♦✐♥t ❛s ♦♥❡ ✇✐t❤ ❛ ③❡r♦ ❞❡r✐✈❛t✐✈❡❀

V ′ = (4x3 − 40x2 + 100x)′ = 12x2 − 80x + 100 = 4(3x2 − 20x + 25) = 0 .

❚❤❡♥✱ ❢r♦♠ t❤❡ ◗✉❛❞r❛t✐❝

✱ ✇❡ ❤❛✈❡✿

❋♦r♠✉❧❛

x=

20 ±

√

202 − 4 · 3 · 25 20 ± 10 5 = = 5, . 2·3 6 3

❚❤❡r❡ ❛r❡ ♥♦ ♦t❤❡r ❝❛♥❞✐❞❛t❡s ❢♦r t❤✐s ♠❛① ♣♦✐♥t✦ ❚❤❡r❡❢♦r❡✱ t❤❡ ❧❛tt❡r ♦♥❡ ✐s t❤❡ ❛♥s✇❡r✳ ❲❡ ❝❛♥ ❛❧s♦ ❝♦♥✜r♠ t❤❛t ❛❧❧ ♣♦✐♥ts ❜❡t✇❡❡♥ t❤❡ t✇♦ ❛r❡ ✐♥❝r❡❛s✐♥❣ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ▲♦❝❛❧ ▼♦♥♦t♦♥✐❝✐t② ❚❤❡♦r❡♠

✺✳✷✳

✸✽✻

❖♣t✐♠✐③❛t✐♦♥ ♦❢ ❢✉♥❝t✐♦♥s

❜② s♦❧✈✐♥❣ t❤✐s ✐♥❡q✉❛❧✐t②✿

V ′ = 4(3x2 − 20x + 25) > 0 . 5 3

■♥❞❡❡❞✱ ✇❡ ❤❛✈❡ t❤❡ t✇♦ r♦♦ts✱ 5 ❛♥❞ ✱ ❛♥❞ ❜❡t✇❡❡♥ t❤❡♠ V ′ ✇✐❧❧ r❡♠❛✐♥ ♥❡❣❛t✐✈❡✳ ❚❤✐s ✐s ❤♦✇ t❤❡ ♠♦♥♦t♦♥✐❝✐t② ♣r♦❜❧❡♠ ✐s s♦❧✈❡❞ s♦ ❢❛r✿ f (xk ) ≤ f (xk+1 ) ⇐⇒ ∆f (ck ) ≥ 0 ⇐⇒

∆f df (ck ) ≥ 0 =⇒ ≥0 ∆x dx

❚❤❡ ❧❛st ❛rr♦✇ ❣♦❡s ✐♥ ♦♥❡ ❞✐r❡❝t✐♦♥✦

✺✳✷✳ ❖♣t✐♠✐③❛t✐♦♥ ♦❢ ❢✉♥❝t✐♦♥s

❆❝❝♦r❞✐♥❣ t♦ ❋❡r♠❛t✬s ❚❤❡♦r❡♠✱ t❤❡ ♣♦✐♥ts ✇✐t❤ ❛ ③❡r♦ ❞❡r✐✈❛t✐✈❡ ✐♥❝❧✉❞❡ ❛❧❧ ❧♦❝❛❧ ❡①tr❡♠❛ ❛s ✇❡❧❧ ❛s s♦♠❡ ♦t❤❡r ♣♦✐♥ts✳ ❲❡ ❛❞❞ ❛❧❧ ♦❢ t❤♦s❡ t♦ ♦✉r ❧✐st✿ ◮ ❚❤❡ ♣♦✐♥ts ✇✐t❤ ❛ ③❡r♦ ❞❡r✐✈❛t✐✈❡ ❛r❡

❝❛♥❞✐❞❛t❡s

❢♦r ❣❧♦❜❛❧ ❡①tr❡♠❡ ♣♦✐♥ts✳

❋♦r ❡①❛♠♣❧❡✱ ❜❡❧♦✇ ✇❡ ❤❛✈❡ s✐① ❝❛♥❞✐❞❛t❡s✿

❚❤❡ ❜❧✉❡ ♣♦✐♥ts ❤❛✈❡ ❜❡❡♥ ❛❧r❡❛❞② ❡❧✐♠✐♥❛t❡❞ ❢r♦♠ t❤❡ ❝♦♥t❡st✳ ❚❤❡ s✐① ❝♦rr❡s♣♦♥❞✐♥❣ ✈❛❧✉❡s ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❛r❡ st✐❧❧ t♦ ❜❡ ❝♦♠♣❛r❡❞ t♦ ✜♥❞ t❤❡ ✜♥❛❧ ✇✐♥♥❡r ♦r ✇✐♥♥❡rs✳ ❊①❛♠♣❧❡ ✺✳✷✳✶✿ ♠❛①✐♠✐③❡ ❝❧♦s❡❞ ❜♦①

▲❡t✬s ♠♦❞✐❢② ♦✉r ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠✿ ❚❤❡ ❝♦r♥❡rs ❛r❡ t♦ ❜❡ ❝✉t ❢r♦♠ ❛ 10 × 20 ♣✐❡❝❡ ♦❢ ❝❛r❞❜♦❛r❞ t♦ ❝r❡❛t❡ ❛ ❝❧♦s❡❞ ❜♦① ♦❢ t❤❡ ❧❛r❣❡st ♣♦ss✐❜❧❡ ✈♦❧✉♠❡✳

▲❡t y ❜❡ t❤❡ ✇✐❞t❤ ♦❢ t❤❡ ❜♦①❀ t❤❡♥ 2x + 2y = 20✱ ♦r y = 10 − x✳ ❚❤❡ ✈♦❧✉♠❡ ♦❢ t❤❡ ❜♦① ✐s t❤❡♥✿ V = x(10 − 2x)(10 − x) = 2x3 − 40x2 + 100x ,

✇❤❡r❡ x ✐s t❤❡ s✐❞❡ ♦❢ t❤❡ ❧✐tt❧❡ sq✉❛r❡ ✉♥❞❡r t❤❡ r❡str✐❝t✐♦♥ 0 ≤ x ≤ 5✳ P❧♦tt✐♥❣ t❤❡ ❣r❛♣❤ ✭✇✐t❤ ❛ s♣r❡❛❞s❤❡❡t✮ s✉❣❣❡sts t❤❛t t❤❡r❡ ✐s ✐♥❞❡❡❞ ❛ ❧♦❝❛❧ ♠❛①✐♠✉♠✿

✺✳✷✳

✸✽✼

❖♣t✐♠✐③❛t✐♦♥ ♦❢ ❢✉♥❝t✐♦♥s

❲❤❛t ✐s ✐ts ❡①❛❝t ✈❛❧✉❡❄ ❆❝❝♦r❞✐♥❣ t♦ ❋❡r♠❛t✬s ❚❤❡♦r❡♠✱ s✐♥❝❡ t❤❡ ❢✉♥❝t✐♦♥ ✐s ❞✐✛❡r❡♥t✐❛❜❧❡✱ t❤❡ ♣♦✐♥t✱ c✱ ❤❛s t♦ s❛t✐s❢② f ′ (c) = 0✳ ❋✐♥❞ t❤❡ ❞❡r✐✈❛t✐✈❡✿ V ′ (x) = (2x3 − 30x2 + 100x)′ = 6x2 − 60x + 100 .

❚❤❡♥ s♦❧✈❡ t❤✐s ❡q✉❛t✐♦♥✿

V ′ (x) = 6x2 − 60x + 100 = 0 ,

♦r ❇② t❤❡ ◗✉❛❞r❛t✐❝

✱ ✇❡ ❤❛✈❡✿

3x2 − 30x + 50 = 0 .

❋♦r♠✉❧❛

c=

30 ±

√

√ 302 − 4 · 3 · 50 30 ± 10 3 = . 2·3 6

❚❤❡ s♠❛❧❧❡r ❛♥s✇❡r✱ c ≈ 2.1✱ ✐s t❤❡ ♠❛①✐♠✉♠✳ ❊①❛♠♣❧❡ ✺✳✷✳✷✿ ❡①tr❡♠❛ ♦❢

sin

❈♦♥s✐❞❡r f (x) = sin x✳ ❲❡ ❢♦❧❧♦✇ t❤❡ s❛♠❡ ♣❧❛♥✳ ❋✐rst✱ t❤❡ ❞❡r✐✈❛t✐✈❡ ✐s ❢♦✉♥❞ ❛♥❞ s❡t ❡q✉❛❧ t♦ 0✿ d (sin x) = cos x = 0 . dx

❚❤❡ ❡q✉❛t✐♦♥ ♣r♦❞✉❝❡s t❤❡ s❛♠❡ ❧✐st ♦❢ ❝❛♥❞✐❞❛t❡s✿ x = kπ + π/2, k = 0, ±1, ±2, ...

❏✉st ❢r♦♠ t❤✐s ❢❛❝t ❛❧♦♥❡✱ ✇❡ ❝❛♥✬t t❡❧❧ ✇❤✐❝❤ ♦♥❡s ❛r❡ ♠❛①✐♠❛ ❛♥❞ ✇❤✐❝❤ ♦♥❡s ❛r❡ ♠✐♥✐♠❛✳ ❍♦✇❡✈❡r✱ ✇❡ ❦♥♦✇ t❤❡ ❢♦❧❧♦✇✐♥❣✿ ( −1 ✐❢ k ✐s ♦❞❞, sin (kπ + π/2) = 1 ✐❢ k ✐s ❡✈❡♥. ❚❤❡r❡❢♦r❡✱ t❤❡ ❢♦r♠❡r ❛r❡ t❤❡ ♠✐♥✐♠❛ ❛♥❞ t❤❡ ❧❛tt❡r ❛r❡ t❤❡ ♠❛①✐♠❛✳ ❚❤✐s ❝♦♥❝❧✉s✐♦♥ ❝♦♥✜r♠s ✇❤❛t ✇❡ ❦♥♦✇ ❛❜♦✉t t❤✐s ❢✉♥❝t✐♦♥✿

❈❛♥ t❤❡r❡ ❜❡ ♦t❤❡r ❝❛♥❞✐❞❛t❡s ❢♦r ❣❧♦❜❛❧ ❡①tr❡♠❛❄

✺✳✷✳

❖♣t✐♠✐③❛t✐♦♥ ♦❢ ❢✉♥❝t✐♦♥s

✸✽✽

❲❤❡♥ t❤❡ ♣r♦❜❧❡♠ ❝❛❧❧s ❢♦r ❧✐♠✐t✐♥❣ ♦✉r ❛tt❡♥t✐♦♥ t♦ ❛ ❝❧♦s❡❞ ✐♥t❡r✈❛❧✱ ✐ts ❡♥❞✲♣♦✐♥ts ❝❛♥♥♦t ❜❡ ❧♦❝❛❧ ❡①tr❡♠❛ ❜❡❝❛✉s❡ t❤❡ ❢✉♥❝t✐♦♥ ✐s ♦♥❧② ❞❡✜♥❡❞ ♦♥ ♦♥❡ s✐❞❡ ♦❢ s✉❝❤ ❛ ♣♦✐♥t✳ ❲❡ s✐♠♣❧② ❛❞❞ t❤❡ t✇♦ t♦ ♦✉r ❧✐st✿ ◮ ❚❤❡ ❡♥❞✲♣♦✐♥ts ♦❢ t❤❡ ✐♥t❡r✈❛❧ ❛r❡

❝❛♥❞✐❞❛t❡s ❢♦r ❣❧♦❜❛❧ ❡①tr❡♠❡ ♣♦✐♥ts✳

❋♦r ❡①❛♠♣❧❡✱ ❜❡❧♦✇ ✐❢ ✇❡ ✐❣♥♦r❡❞ t❤❡ r✐❣❤t✲♠♦st ♣♦✐♥t✱ ✇❡✬❞ ❣✐✈❡ ✉♣ t❤❡ ❣❧♦❜❛❧ ♠✐♥✐♠✉♠✿

❆❝❝♦r❞✐♥❣ t♦ t❤❡ ❊①tr❡♠❡ ❱❛❧✉❡ ❚❤❡♦r❡♠✱ ❛ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ ♦♥ ❛ ❜♦✉♥❞❡❞ ❝❧♦s❡❞ ✐♥t❡r✈❛❧ ❤❛s ❛ ❣❧♦❜❛❧ ♠❛①✐♠✉♠ ❛♥❞ ❛ ❣❧♦❜❛❧ ♠✐♥✐♠✉♠✿

❲❡ ❝♦♥❝❧✉❞❡✿ ◮ ❚❤❡ ❧✐st ♦❢ ❛❧❧ ♣♦✐♥ts ♦❢ ③❡r♦ ❞❡r✐✈❛t✐✈❡ ✇✐t❤ t❤❡ ❡♥❞✲♣♦✐♥ts ❛❞❞❡❞ ✇✐❧❧ ❛❧✇❛②s ❝♦♥t❛✐♥ ❛❧❧

❡①tr❡♠❡ ♣♦✐♥ts✳

❚❤✐s ✐s t❤❡ ❥✉st✐✜❝❛t✐♦♥ ♦❢ ♦✉r str❛t❡❣②✳ ❙tr✐♣♣❡❞ ♦❢ ❛❧❧ t❤❡ ✐♥❝✐❞❡♥t❛❧ ❞❡t❛✐❧s ♦❢ ❛ ✇♦r❞ ♣r♦❜❧❡♠✱ t❤✐s ✐s ✇❤❛t ❛ s♦❧✉t✐♦♥ ♦❢ ❛♥ ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠ ✇✐❧❧ ❧♦♦❦ ❧✐❦❡✳ ❊①❛♠♣❧❡ ✺✳✷✳✸✿ ♦♣t✐♠✐③❡ ♦♥ ❝❧♦s❡❞ ✐♥t❡r✈❛❧

❋✐♥❞ ❣❧♦❜❛❧ ❡①tr❡♠❡ ♣♦✐♥ts ♦❢

f (x) = x3 − 3x ♦♥ [−2, 3] .

❙t❡♣ ✶✿ ❋✐♥❞ t❤❡ ♣♦✐♥ts ✇✐t❤ ③❡r♦ ❞❡r✐✈❛t✐✈❡✳ ❈♦♠♣✉t❡✿

f ′ (x) = 3x2 − 3 .

❙❡t ✐t ❡q✉❛❧ t♦ 0 ❛♥❞ s♦❧✈❡ ❢♦r x✳

3x2 − 3 = 0 =⇒ x2 = 1 =⇒ x = ±1 .

❙t❡♣ ✷✿ ❈♦♠♣❛r❡ t❤❡ ✈❛❧✉❡s ♦❢ f ❛t t❤❡s❡ ♣♦✐♥ts ❛♥❞ ❛t t❤❡ ❡♥❞✲♣♦✐♥ts ♦❢ t❤❡ ✐♥t❡r✈❛❧❀ ✜♥❞ t❤❡ s♠❛❧❧❡st

✺✳✷✳ ❖♣t✐♠✐③❛t✐♦♥ ♦❢ ❢✉♥❝t✐♦♥s

✸✽✾

❛♥❞ t❤❡ ❧❛r❣❡st✿ ❈❛♥❞✐❞❛t❡s ′

f (x) = 0 f ′ (x) = 0 a b

x 1 −1 −2 3

f (x) f (1) f (1) f (−2) f (3)

= x3 − 3x = x(x2 − 3) = 1(12 − 3) = −1((−1)2 − 3) = −2((−2)2 − 3) = 3(32 − 3)

= = = =

✈❛❧✉❡s ❝❧❛ss✐✜❝❛t✐♦♥ −2 ❣❧♦❜❛❧ ♠✐♥ 2 ♣♦ss✐❜❧② ❧♦❝❛❧ ❜✉t ♥♦t ❣❧♦❜❛❧ ♠❛① −2 ❣❧♦❜❛❧ ♠✐♥ 18 ❣❧♦❜❛❧ ♠❛①

❚❤❡s❡ ♣♦✐♥ts ❛♥❞ ❤♦✇ t❤❡② ❛r❡ ❝❧❛ss✐✜❡❞ ❛r❡ ✈✐s✐❜❧❡ ♦♥ t❤❡ ❣r❛♣❤✿

❆❝❝♦r❞✐♥❣ t♦ ❋❡r♠❛t✬s ❚❤❡♦r❡♠✱ t❤❡r❡ ❛r❡ ♥♦ ♦t❤❡r ❝❛♥❞✐❞❛t❡s✳ ❚❤❡ t❤❡♦r❡♠✱ ❤♦✇❡✈❡r✱ ❧❡❛✈❡s ❛♥ ♦♣t✐♦♥ ♦❢ ❛ ♥♦♥✲❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥✳

❊①❛♠♣❧❡ ✺✳✷✳✹✿ ♥♦♥✲❞✐✛❡r❡♥t✐❛❜❧❡ ❚❤❡ ❛❜s♦❧✉t❡ ✈❛❧✉❡ ❢✉♥❝t✐♦♥ f (x) = |x| ❤❛s ✐ts ❣❧♦❜❛❧ ♠✐♥✐♠✉♠ ❛t x = 0❀ ❤♦✇❡✈❡r✱ ✐t✬s ♥♦t ❞✐✛❡r❡♥t✐❛❜❧❡✿

■♥ ❢❛❝t✱ ✇❡ ❤❛✈❡✿ ′

f (x) =

  −1  

✉♥❞❡✜♥❡❞ 1

✐❢ x < 0, ✐❢ x = 0, ✐❢ x > 0.

❚❤❡r❡❢♦r❡✱ t❤❡ ❣❧♦❜❛❧ ♠✐♥✐♠✉♠ ♦♥ ❛♥② ✐♥t❡r✈❛❧ [a, b] ✇✐t❤ a < 0 < b ✐s ❛t x = 0✱ ❛♥❞ t❤❡ ❣❧♦❜❛❧ ♠❛①✐♠✉♠ ❛t a ♦r b✳

❊①❡r❝✐s❡ ✺✳✷✳✺ ❈♦♥s✐❞❡r t❤❡ ❞❡r✐✈❛t✐✈❡ ❛♥❞ t❤❡ ❡①tr❡♠❡ ♣♦✐♥ts ♦❢ t❤✐s ❢✉♥❝t✐♦♥✿

✺✳✷✳

❖♣t✐♠✐③❛t✐♦♥ ♦❢ ❢✉♥❝t✐♦♥s

✸✾✵

❲❡ ♠❛② ❤❛✈❡ t♦ ❛❞❞ t❤♦s❡ t♦ ♦✉r ❧✐st t♦♦✿

◮

❚❤❡ ♣♦✐♥ts ✇✐t❤ ✉♥❞❡✜♥❡❞ ❞❡r✐✈❛t✐✈❡ ❛r❡

❝❛♥❞✐❞❛t❡s ❢♦r ❣❧♦❜❛❧ ❡①tr❡♠❡ ♣♦✐♥ts✳

❚❤❡ ♣♦✐♥ts ✇✐t❤ ❡✐t❤❡r ✉♥❞❡✜♥❡❞ ♦r ③❡r♦ ❞❡r✐✈❛t✐✈❡ ❛r❡ ♦❢t❡♥ ❝❛❧❧❡❞ ✏❝r✐t✐❝❛❧ ♣♦✐♥ts✑✳ ❚❤✉s✱ ♦✉r ❧✐st ♦❢ ✈❛❧✉❡s ♦❢

x

t♦ ❜❡ ❝❤❡❝❦❡❞ ✐s ❝♦♠♣r✐s❡❞ ♦❢ ❥✉st t✇♦ t②♣❡s✿

t❤❡ ❝r✐t✐❝❛❧ ♣♦✐♥ts ❛♥❞ t❤❡

❡♥❞✲♣♦✐♥ts✳ ❚❤❡♥✱ ❢r♦♠ t❤✐s ❧✐st✱ ✇❡ ✜♥❞✿

•

❚❤❡

x✬s

✇✐t❤ t❤❡ ❧❛r❣❡st ✈❛❧✉❡ ♦❢

•

❚❤❡

x✬s

✇✐t❤ t❤❡ s♠❛❧❧❡st ✈❛❧✉❡ ♦❢

y

❛r❡ t❤❡ ❣❧♦❜❛❧ ♠❛①✐♠❛✳

y

❛r❡ t❤❡ ❣❧♦❜❛❧ ♠✐♥✐♠❛✳

❊①❛♠♣❧❡ ✺✳✷✳✻✿ ❝✉❜✐❝ ♣♦❧②♥♦♠✐❛❧

❆♥❛❧②③❡ t❤❡ ❢✉♥❝t✐♦♥

f (x) = 2x3 − 3x2 − 12x + 1 ♦♥

[−2, 3]✳

❈♦♠♣✉t❡ t❤❡ ❞❡r✐✈❛t✐✈❡✿

f ′ (x) = 6x2 − 6x − 12 . ❙❡t ✐t ❡q✉❛❧ t♦ ③❡r♦ ❛♥❞ s♦❧✈❡ t❤❡ ❡q✉❛t✐♦♥✿

6x2 − 6x − 12 x2 − x − 2 ◗❋✿

= 0 =⇒ = 0 =⇒ p 1 ± 1 − (−2)4 x = 2 1±3 = = 2, 1 . 2

❈♦♠♣❛r❡ t❤❡ ✈❛❧✉❡s✿ ❈❛♥❞✐❞❛t❡s ′

f (x) = 0 f ′ (x) = 0 a b ❆♥s✇❡r✿

• •

x 2 −1 −2 3

f (x) = 6x2 − 6x − 12 y 3 2 2 · 2 − 3 · 2 − 12 · 2 + 1 = 16 − 12 − 24 + 1 = −19 3 2 2 · (−1) − 3 · (−1) − 12 · (−1) + 1 = −2 − 3 + 12 + 1 = 8 2 · (−2)3 − 3 · (−2)2 − 12 · (−2) + 1 = −16 − 12 + 24 + 1 = −3 2 · 33 − 3 · 32 − 12 · 3 + 1 = 54 − 27 − 36 + 1 = −8

✈❛❧✉❡ ✐s y = 8 ❛tt❛✐♥❡❞ ❛t x = −1✱ ❛ ❣❧♦❜❛❧ ♠❛① ♣♦✐♥t✳ ❚❤❡ ❣❧♦❜❛❧ ♠✐♥ ✈❛❧✉❡ ✐s y = −19 ❛tt❛✐♥❡❞ ❛t x = 2✱ ❛ ❣❧♦❜❛❧ ♠✐♥ ♣♦✐♥t✳

❚❤❡ ❣❧♦❜❛❧ ♠❛①

✺✳✸✳

❲❤❛t t❤❡ ❞❡r✐✈❛t✐✈❡ s❛②s ❛❜♦✉t t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✿ ❚❤❡ ▼❡❛♥ ❱❛❧✉❡ ❚❤❡♦r❡♠

✸✾✶

❲❛r♥✐♥❣✦ ❙✉❜st✐t✉t❡ t❤❡ ✈❛❧✉❡s ✐♥t♦ t❤❡ ♦r✐❣✐♥❛❧ ❢✉♥❝t✐♦♥✱ ♥♦t t❤❡ ❞❡r✐✈❛t✐✈❡✳

❚❤✐s ✐s ❛ s✉♠♠❛r② ♦❢ ♦✉r ♠❡t❤♦❞✿ ❚❤❡♦r❡♠ ✺✳✷✳✼✿ ●❧♦❜❛❧ ❊①tr❡♠❛ ✈✐❛ ❉❡r✐✈❛t✐✈❡

❙✉♣♣♦s❡ f ✐s ❝♦♥t✐♥✉♦✉s ♦♥ [a, b] ❛♥❞ c ✐♥ [a, b] ✐s ❛ ❣❧♦❜❛❧ ❡①tr❡♠❡ ♣♦✐♥t✳ ❚❤❡♥✱ ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ♠✉st ❜❡ s❛t✐s✜❡❞✿ ✶✳ f ′ (c) = 0✱ ♦r ✷✳ f ′ (c) ✐s ✉♥❞❡✜♥❡❞✱ ♦r ✸✳ c = a ♦r c = b✳ ❊①❡r❝✐s❡ ✺✳✷✳✽

❘❡❞♦ t❤❡ t✇♦ ❡①❛♠♣❧❡s ❛❜♦✉t ♠❛①✐♠✐③✐♥❣ t❤❡ ✈♦❧✉♠❡ ♦❢ t❤❡ ❜♦① ❜② ❢♦❧❧♦✇✐♥❣ t❤✐s ♠❡t❤♦❞✳

✺✳✸✳ ❲❤❛t t❤❡ ❞❡r✐✈❛t✐✈❡ s❛②s ❛❜♦✉t t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✿ ❚❤❡ ▼❡❛♥ ❱❛❧✉❡ ❚❤❡♦r❡♠ ❚❤❡ ❋❡r♠❛t✬s ❚❤❡♦r❡♠ ✐s ❛♥ ❡①❛♠♣❧❡ ♦❢ ❛ t❤❡♦r❡♠ t❤❡ ❝♦♥✈❡rs❡ ♦❢ ✇❤✐❝❤ ✐s♥✬t tr✉❡✳ ❚❤✐s ✐s ✇❤❛t ✐t ❞♦❡s ❛♥❞ ❞♦❡s ♥♦t s❛② ❛❜♦✉t ❛ ❢✉♥❝t✐♦♥ ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t x = c✿

x = c ✐s ■♥ ❝♦♥tr❛st✱ t❤❡ ❝♦♥✈❡rs❡ ♦❢ t❤❡

❛ ❧♦❝❛❧ =⇒ ❡①tr❡♠❡ ♣♦✐♥t 6⇐=

f ′ (c) = 0

▲♦❝❛❧ ▼♦♥♦t♦♥✐❝✐t② ❚❤❡♦r❡♠ ✐s tr✉❡ ❜✉t ❤❛s♥✬t ❜❡❡♥ ♣r♦✈❡♥ ②❡t✿

x = c ✐s

❛ ❧♦❝❛❧ =⇒ ♠♦♥♦t♦♥❡ ♣♦✐♥t ⇐=

f ′ (c) ≥ 0

❘❡❝❛❧❧ ❢r♦♠ t❤❡ ❧❛st ❝❤❛♣t❡r✱ t❤❛t t❤❡r❡ ❛r❡ ♦t❤❡r st❛t❡♠❡♥ts ✐♥ t❤❡ ❝♦♥✈❡rs❡s ♦❢ ✇❤✐❝❤ ✇❡ ❛r❡ ✐♥t❡r❡st❡❞✿ ✶✳ ❚❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ❝♦♥st❛♥t ❢✉♥❝t✐♦♥ ✐s ③❡r♦✱ ❜✉t ❛r❡ t❤❡ ❝♦♥st❛♥ts t❤❡ ♦♥❧② ❢✉♥❝t✐♦♥s ✇✐t❤ t❤✐s ♣r♦♣❡rt②❄ ✷✳ ❚❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ❧✐♥❡❛r ♣♦❧②♥♦♠✐❛❧ ✐s ❝♦♥st❛♥t✱ ❜✉t ❛r❡ t❤❡ ❧✐♥❡❛r ♣♦❧②♥♦♠✐❛❧s t❤❡ ♦♥❧② ❢✉♥❝t✐♦♥s ✇✐t❤ t❤✐s ♣r♦♣❡rt②❄ ✸✳ ❚❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧ ✐s ❧✐♥❡❛r✱ ❜✉t ❛r❡ t❤❡ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧s t❤❡ ♦♥❧② ❢✉♥❝t✐♦♥s ✇✐t❤ t❤✐s ♣r♦♣❡rt②❄ ❊①❡r❝✐s❡ ✺✳✸✳✶

❚❤❡ ✜rst q✉❡st✐♦♥ ❛s❦s ✐❢ ✇❡ ❝❛♥ ❜✉✐❧❞ ❛ ❝✉r✈❡ ✕ ♦t❤❡r t❤❛♥ ❛ ❤♦r✐③♦♥t❛❧ ❧✐♥❡ ✕ ✇✐t❤ ❛❧❧ t❛♥❣❡♥t ❧✐♥❡s ❤♦r✐③♦♥t❛❧✳ ❚r② ✐t✦ ❊①❛♠♣❧❡ ✺✳✸✳✷✿ ❢r❡❡ ❢❛❧❧

❆♥s✇❡r✐♥❣ ❨❡s t♦ t❤❡ s❡❝♦♥❞ ❛♥❞ t❤✐r❞ q✉❡st✐♦♥s s♦❧✈❡s t❤❡ ♣r♦❜❧❡♠ ♦❢ ❢r❡❡ ❢❛❧❧ ❛s ♣r❡s❡♥t❡❞ ✐♥ ❈❤❛♣t❡r

✺✳✸✳ ❲❤❛t t❤❡ ❞❡r✐✈❛t✐✈❡ s❛②s ❛❜♦✉t t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✿ ❚❤❡ ▼❡❛♥ ❱❛❧✉❡ ❚❤❡♦r❡♠

✸✾✷

✹✿ ❝♦♥st❛♥t ❢♦r❝❡ =⇒ ❝♦♥st❛♥t ❛❝❝❡❧❡r❛t✐♦♥ =⇒ ❧✐♥❡❛r ✈❡❧♦❝✐t② =⇒ q✉❛❞r❛t✐❝ ♣♦s✐t✐♦♥ ❢✉♥❝t✐♦♥ ❲❡ ❤❛✈❡ ❥✉st✐✜❡❞✱ t❤❡r❡❢♦r❡✱ ♦✉r ❢♦r♠✉❧❛✿ 1 y(t) = y0 + v0 t − gt2 . 2

❋✉rt❤❡r♠♦r❡✱ t❤❡ tr❛❥❡❝t♦r② ♦❢ ❛ ❜❛❧❧ t❤r♦✇♥ ❢♦r✇❛r❞ ✐s ❛ ♣❛r❛❜♦❧❛✦ ❚❤❡ ♠❛✐♥ t❤❡♦r❡♠ ♦❢ t❤✐s s❡❝t✐♦♥ ✇✐❧❧ ❤❡❧♣ ✇✐t❤ t❤❡s❡ ❛♥❞ ♦t❤❡r q✉❡st✐♦♥s✳ ❊①❛♠♣❧❡ ✺✳✸✳✸✿ ❞r✐✈✐♥❣

▲❡t✬s ✐♥t❡r♣r❡t t❤❡ ❝♦♥❞✐t✐♦♥s ♦❢ ❋❡r♠❛t✬s ❚❤❡♦r❡♠ ✐♥ t❡r♠s ♦❢ ♠♦t✐♦♥✿ • x ✐s t✐♠❡✳ • f (x) ✐s t❤❡ ❧♦❝❛t✐♦♥ ❛t t✐♠❡ x✳ • f ′ (x) ✐s t❤❡ ✈❡❧♦❝✐t② ❛t t✐♠❡ x✳ ❙♦✱ ✇❡ ❛ss✉♠❡ t❤❛t t❤❡ ✈❡❧♦❝✐t② ❛❧✇❛②s ♠❛❦❡s s❡♥s❡❀ ✐✳❡✳✱ t❤❡r❡ ❛r❡ ♥♦ s✉❞❞❡♥ ❝❤❛♥❣❡s ♦❢ ❞✐r❡❝t✐♦♥✱ ❜✉♠♣s✱ ♦r ❝r✉s❤❡s ✭❛♥❞ ♥♦ t❡❧❡♣♦rt❛t✐♦♥✦✮✳ ◆♦✇ ✇❡ ✐♥t❡r♣r❡t t❤❡ ❝♦♥❝❧✉s✐♦♥ ♦❢ ❋❡r♠❛t✬s ❚❤❡♦r❡♠ ✐♥ t❡r♠s ♦❢ ♠♦t✐♦♥✿ ◮ (⇒) ❲❤❡♥❡✈❡r ✇❡ ❛r❡ t❤❡ ❢❛rt❤❡st ❢r♦♠ ❤♦♠❡ ♦r ❛♥② ❧♦❝❛t✐♦♥✱ ✇❡ st♦♣✱ ❛t ❧❡❛st ❢♦r ❛♥ ✐♥st❛♥t✳ ❇✉t ♥♦t ❝♦♥✈❡rs❡❧②✿ ◮ (6⇐) ❊✈❡♥ ✐❢ ✇❡ st♦♣✱ ✇❡ ❝❛♥ r❡s✉♠❡ ❞r✐✈✐♥❣ ✐♥ t❤❡ s❛♠❡ ❞✐r❡❝t✐♦♥✳ ■♠❛❣✐♥❡ t❤❛t ②♦✉✿ • ❧❡❢t ❤♦♠❡ ❛t 1 ♣♠✱ • ❞✐❞ s♦♠❡ ❞r✐✈✐♥❣✱ ❛♥❞ t❤❡♥ • ❝❛♠❡ ❤♦♠❡ ❛t 2 ♣♠✳ ◗✉❡st✐♦♥✿ ❲❤❛t ❝❛♥ s♦♠❡♦♥❡ ✇❤♦ st❛②❡❞ ❤♦♠❡ ✐♥❢❡r ❛❜♦✉t ②♦✉r s♣❡❡❞ ❞✉r✐♥❣ t❤✐s t✐♠❡❄ ❋♦r s✐♠♣❧✐❝✐t②✱ ✇❡ ❛ss✉♠❡ t❤❛t ②♦✉ ❞r♦✈❡ ♦♥ ❛ str❛✐❣❤t r♦❛❞✳ ❚❤❡♥ t❤✐s ✐s ✇❤❛t t❤❛t ♣❡rs♦♥ ❦♥♦✇s ✭❧❡❢t✮✿

❚❤❡ ♣♦ss✐❜✐❧✐t✐❡s ❛r❡ ♥✉♠❡r♦✉s ✭r✐❣❤t✮✳ ❨♦✉ ♠❛② ❤❛✈❡ ❞r✐✈❡♥ s❧♦✇❧②✱ t❤❡♥ ❢❛st✱ ❜✉t ♦♥❡ t❤✐♥❣ ✐s ❝❡rt❛✐♥✿ ②♦✉ ❝❛♠❡ ❜❛❝❦ ❤♦♠❡✳ ❆♥❞ t♦ ❝♦♠❡ ❜❛❝❦✱ ②♦✉ ❤❛❞ t♦ t✉r♥ ❛r♦✉♥❞✳ ❆♥❞ t♦ t✉r♥ ❛r♦✉♥❞✱ ②♦✉ ❤❛❞ t♦ st♦♣✳ ❙♦✱ s♣❡❡❞ ✇❛s ③❡r♦ ❛t ❧❡❛st ♦♥❝❡✦ ❚❤❡r❡ ✐s ♥♦ ❦♥♦✇❧❡❞❣❡ ✇❤❡♥ t❤✐s ❤❛♣♣❡♥❡❞ t❤♦✉❣❤✳ ▲❡t✬s ♠❛❦❡ t❤❡ ♦❜s❡r✈❛t✐♦♥ ♠❛❞❡ ✐♥ t❤❡ ❧❛st ❡①❛♠♣❧❡ ♣✉r❡❧② ♠❛t❤❡♠❛t✐❝❛❧ ❛♥❞ t✉r♥ ✐t ✐♥t♦ ❛ t❤❡♦r❡♠✿ ❚❤❡♦r❡♠ ✺✳✸✳✹✿ ❘♦❧❧❡✬s ❚❤❡♦r❡♠

❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ❢✉♥❝t✐♦♥✿ ✶✳ f ✐s ❝♦♥t✐♥✉♦✉s ♦♥ [a, b]✳ ✷✳ f ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ♦♥ (a, b)✳ ❚❤❡♥✱ ✐❢ f (a) = f (b) ,

✺✳✸✳ ❲❤❛t t❤❡ ❞❡r✐✈❛t✐✈❡ s❛②s ❛❜♦✉t t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✿ ❚❤❡ ▼❡❛♥ ❱❛❧✉❡ ❚❤❡♦r❡♠

✸✾✸

✇❡ ❤❛✈❡✿

f ′ (c) = 0

❋❖❘ ❙❖▼❊ c

✐♥

(a, b) ✳

❲❡ ❛❣❛✐♥ ❛ss✉♠❡ t❤❛t x ✐s t✐♠❡✱ ❧✐♠✐t❡❞ t♦ ✐♥t❡r✈❛❧ [a, b]✳ ■♥ t❤❡ t❤❡♦r❡♠✱ ★✶ ♠❡❛♥s t❤❛t ②♦✉ ❞♦♥✬t ❧❡❛♣ ❛♥❞ ★✷ ♠❡❛♥s t❤❛t ②♦✉ ❞♦♥✬t ❝r❛s❤✳ ❲❡ ❛❧s♦ ❛ss✉♠❡ t❤❛t ②♦✉✬✈❡ ❝♦♠❡ ❜❛❝❦✳ ❲❡ ❛❧r❡❛❞② ❦♥♦✇ t❤❛t t❤✐s s♣❡❝✐❛❧ ♣♦✐♥t ✐♥ t✐♠❡ ✇❛s ✇❤❡♥ ②♦✉ ✇❡r❡ ❢❛rt❤❡st ❛✇❛② ❢r♦♠ ❤♦♠❡ ✭✐♥ ❡✐t❤❡r ❞✐r❡❝t✐♦♥✮❀ ②♦✉ ✇❡r❡♥✬t ♠♦✈✐♥❣ t❤❡♥✳ ❚❤❡ ✐❞❡❛ ♦❢ t❤❡ ♣r♦♦❢ ✐s ❜❡❧♦✇✿

Pr♦♦❢✳

❙✉♣♣♦s❡ f ❤❛s ♦♥ [a, b]✿ • ❛ ❣❧♦❜❛❧ ♠❛①✐♠✉♠ ❛t x = c ❛♥❞ • ❛ ❣❧♦❜❛❧ ♠✐♥✐♠✉♠ ❛t x = d✳ ❚❤❡s❡ ❝♦♥❝❧✉s✐♦♥s ❛r❡ ❥✉st✐✜❡❞ ❜② t❤❡ ❊①tr❡♠❡ ❱❛❧✉❡ ❚❤❡♦r❡♠ ✭t❤❛t✬s ✇❤② ✇❡ ♥❡❡❞ t♦ ❛ss✉♠❡ ❝♦♥t✐♥✉✐t②✦✮✳ ❈❛s❡ ✶✿ ❊✐t❤❡r c ♦r d ✐s ♥♦t ❛♥ ❡♥❞✲♣♦✐♥t✱ a ♦r b✳ ❲❡ ♥♦✇ ✉s❡ t❤❡ ❋❡r♠❛t✬s ❚❤❡♦r❡♠ ✿ ❊✈❡r② ❣❧♦❜❛❧ ❡①tr❡♠❡ ♣♦✐♥t ❤❛s 0 ❞❡r✐✈❛t✐✈❡ ✇❤❡♥ ✐t✬s ♥♦t ❛♥ ❡♥❞✲♣♦✐♥t✳ ■t ❢♦❧❧♦✇s t❤❛t f ′ (c) = 0 ♦r f ′ (d) = 0✱ ❛♥❞ ✇❡ ❛r❡ ❞♦♥❡✳ ❈❛s❡ ✷✿ ❇♦t❤ c ❛♥❞ d ❛r❡ ❡♥❞✲♣♦✐♥ts✱ a ♦r b✳ ❚❤❡♥ f (c) = f (a) f (d) = f (a)

♦r ♦r

f (c) = f (b) f (d) = f (b)



❇✉t t❤❡s❡ ❛r❡ ❡q✉❛❧✦

❚❤❡r❡❢♦r❡✱ f (c) = f (d) .

❇✉t t❤✐s ♠❡❛♥s t❤❛t max f = min !

❚❤❡r❡❢♦r❡ f ♠✉st ❜❡ ❝♦♥st❛♥t ♦♥ [a, b]✳ ❚❤❡r❡❢♦r❡✱ f ′ (x) = 0 ❢♦r ❛❧❧ x ✐♥ (a, b)✳ ❙♦✱ ❛♠♦♥❣ ❛❧❧ ❤♦r✐③♦♥t❛❧ ❧✐♥❡s✱ t❤❡r❡ ✐s ❛t ❧❡❛st ♦♥❡ t❤❛t t♦✉❝❤❡s t❤❡ ❣r❛♣❤✿

✺✳✸✳ ❲❤❛t t❤❡ ❞❡r✐✈❛t✐✈❡ s❛②s ❛❜♦✉t t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✿ ❚❤❡ ▼❡❛♥ ❱❛❧✉❡ ❚❤❡♦r❡♠

✸✾✹

❚❤❡ t❤❡♦r❡♠ s❛②s t❤❛t ✐❢ ②♦✉ ♣❛ss❡❞ t❤❡ s❛♠❡ ❧♦❝❛t✐♦♥ t✇✐❝❡✱ ②♦✉ ♠✉st ❤❛✈❡ st♦♣♣❡❞ ❛t s♦♠❡ ♠♦♠❡♥t ❞✉r✐♥❣ t❤✐s t✐♠❡✳ ❚❤❡ ❢♦r♠❡r ❝♦♥❞✐t✐♦♥✱ f (a) = f (b) ,

✐s r❡✇r✐tt❡♥ ❛s ❢♦❧❧♦✇s✿ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ❞✐✛❡r❡♥❝❡ ✐s ③❡r♦

f (b) − f (a) = 0 . ∆f = 0 .

■t ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ f ♦✈❡r t❤❡ ♣❛rt✐t✐♦♥ ♦❢ [a, b] ✇✐t❤ n = 1, x0 = a, x1 = b✳ ❍❡r❡ ❛r❡ ♦t❤❡r ✇❛②s t♦ ✐♥t❡r♣r❡t t❤✐s ❢♦r♠✉❧❛✿ ✶✳ ❚❤❡ ❝❤❛♥❣❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ♦✈❡r t❤❡ ✐♥t❡r✈❛❧ ✐s ③❡r♦✳ ✷✳ ❚❤❡ r✐s❡ ♦✈❡r t❤❡ ✐♥t❡r✈❛❧ ✐s ③❡r♦✳ ✸✳ ❚❤❡ ❞✐s♣❧❛❝❡♠❡♥t ♦✈❡r t❤❡ t✐♠❡ ✐♥t❡r✈❛❧ ✐s ③❡r♦✳ ❚❤❡ ❝♦♥❞✐t✐♦♥ ❝❛♥✱ ❢✉rt❤❡r♠♦r❡✱ ❜❡ r❡✇r✐tt❡♥ ❛s ❢♦❧❧♦✇s✿ f (b) − f (a) = 0. b−a

■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✱ t❤❡ s❧♦♣❡✱ ✐s ③❡r♦ t♦♦✿ ∆f = 0. ∆x

❍❡r❡ ❛r❡ ♦t❤❡r ✇❛②s t♦ ✐♥t❡r♣r❡t t❤✐s ❢♦r♠✉❧❛✿ ✶✳ ❚❤❡ ❛✈❡r❛❣❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦✈❡r t❤❡ ✐♥t❡r✈❛❧ ✐s ③❡r♦✳ ✷✳ ❚❤❡ s❧♦♣❡ ♦❢ t❤❡ s❡❝❛♥t ❧✐♥❡ ✐s ③❡r♦✳ ✸✳ ❚❤❡ ❛✈❡r❛❣❡ ✈❡❧♦❝✐t② ♦✈❡r t❤❡ t✐♠❡ ✐♥t❡r✈❛❧ ✐s ③❡r♦✳ ❙♦✱ t❤❡ r❡str✐❝t✐♦♥s ♦❢ t❤❡ t❤❡♦r❡♠ ❛r❡ ✈❡r② ❧✐♠✐t✐♥❣✿ ■t ♠✉st ❜❡ ❛ ③❡r♦ ✈❡❧♦❝✐t②✦ ❈❛♥ ✐t ❜❡ 100 ♠✴❤❄ ▲❡t✬s ✐♥✈❡st✐❣❛t❡✳ ❘♦❧❧❡✬s ❚❤❡♦r❡♠ s❛②s✿ ✶✳ ❚❤❡ ❛✈❡r❛❣❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ✐s ③❡r♦ =⇒ t❤❡ ✐♥st❛♥t❛♥❡♦✉s r❛t❡ ♦❢ ❝❤❛♥❣❡ ✐s ③❡r♦ ❛t s♦♠❡ ♣♦✐♥t✳ ✷✳ ❚❤❡ s❧♦♣❡ ♦❢ t❤❡ s❡❝❛♥t ❧✐♥❡ ✐s ③❡r♦ =⇒ t❤❡ s❧♦♣❡ ♦❢ t❤❡ t❛♥❣❡♥t ❧✐♥❡ ✐s ③❡r♦ ❛t s♦♠❡ ♣♦✐♥t✳ ✸✳ ❚❤❡ ❛✈❡r❛❣❡ ✈❡❧♦❝✐t② ✐s ③❡r♦ =⇒ t❤❡ ✐♥st❛♥t❛♥❡♦✉s ✈❡❧♦❝✐t② ✐s ③❡r♦ ❛t s♦♠❡ ♣♦✐♥t✳ ❚❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ t❤❡ ❤♦r✐③♦♥t❛❧ s❡❝❛♥t ❛♥❞ t❛♥❣❡♥t ❧✐♥❡s ✐s ♦✉t❧✐♥❡❞ ❜❡❧♦✇✿

✺✳✸✳

❲❤❛t t❤❡ ❞❡r✐✈❛t✐✈❡ s❛②s ❛❜♦✉t t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✿ ❚❤❡ ▼❡❛♥ ❱❛❧✉❡ ❚❤❡♦r❡♠

❲❤❛t ❛❜♦✉t ♥♦♥✲❤♦r✐③♦♥t❛❧ ❧✐♥❡s t❤❛t ❛❧s♦ ❲❤❛t ❤❛♣♣❡♥s ✐❢ ✇❡

✸✾✺

t♦✉❝❤ t❤❡ ❣r❛♣❤❄

s❦❡✇ t❤❡ ❣r❛♣❤ ❛❜♦✈❡❄ ■t✬s s✐♠♣❧❡ ✐♠❛❣❡ ❡❞✐t✐♥❣✿

❚❤❡ ♣✐❝t✉r❡ s✉❣❣❡sts ✇❤❛t ❤❛♣♣❡♥s t♦ t❤❡ ❡♥t✐t✐❡s ♣❛rt✐❝✐♣❛t✐♥❣ ✐♥ ❘♦❧❧❡✬s t❤❡♦r❡♠✿

• ❚❤❡ t❛♥❣❡♥ts ♦❢ t❤♦s❡ s♣❡❝✐❛❧ ♣♦✐♥ts t❤❛t ✉s❡❞ t♦ ❜❡ ❤♦r✐③♦♥t❛❧ ❤❛✈❡ ❜❡❝♦♠❡ ✐♥❝❧✐♥❡❞✳

• ❚❤❡ s❡❝❛♥t ❧✐♥❡ t❤❛t ❝♦♥♥❡❝ts t❤❡ ❡♥❞✲♣♦✐♥ts t❤❛t ✉s❡❞ t♦ ❜❡ ❤♦r✐③♦♥t❛❧ ❤❛s ❜❡❝♦♠❡ ✐♥❝❧✐♥❡❞✳

❇✉t t❤❡s❡ ❧✐♥❡s ❤❛✈❡ r❡♠❛✐♥❡❞

♣❛r❛❧❧❡❧ ✦

❙♦✱ ✇❡ ♥❡❡❞ t♦ ❧✐♥❦ t❤❡s❡ t✇♦✿

• t❤❡ s❧♦♣❡s ♦❢ t❤❡ t❛♥❣❡♥t ❧✐♥❡s ✭t❤❛t✬s t❤❡ ✈❡❧♦❝✐t②✮ ❛♥❞

• t❤❡ s❧♦♣❡ ♦❢ t❤❡ ❧✐♥❡ ❜❡t✇❡❡♥ (a, f (a)) ❛♥❞ (b, f (b)) ✭t❤❛t✬s t❤❡ ❛✈❡r❛❣❡ ✈❡❧♦❝✐t②✮✳ ❊①❛♠♣❧❡ ✺✳✸✳✺✿ s♣❡❡❞✐♥❣

❚♦ ✐❧❧✉str❛t❡ t❤❡ ✐❞❡❛✱ ❧❡t✬s r❡✈✐s✐t t❤❡ ✐ss✉❡ ♦❢

s♣❡❡❞✐♥❣✳

❆ r❛❞❛r ❣✉♥ ❝❛t❝❤❡s t❤❡ ✐♥st❛♥t❛♥❡♦✉s s♣❡❡❞ ♦❢ ②♦✉r ❝❛r✳ ❇② ❧❛✇✱ ✐t s❤♦✉❧❞♥✬t ❜❡ ❛❜♦✈❡ 70 ♠✴❤✳ ❲❤❡♥ t❤❡ r❛❞❛r ❣✉♥ s❤♦✇s ❛♥②t❤✐♥❣ ❛❜♦✈❡✱ ②♦✉✬r❡ ❝❛✉❣❤t✳ ◆♦✇✱ s✉♣♣♦s❡ t❤❡r❡ ✐s ♥♦ r❛❞❛r ❣✉♥✳ ■♠❛❣✐♥❡ ✐♥st❡❛❞ t❤❛t ❛ ♣♦❧✐❝❡♠❛♥ ♦❜s❡r✈❡❞ ②♦✉ ❞r✐✈✐♥❣ ❜② ✭❛t ❛ ❧❡❣❛❧ s♣❡❡❞✮ ❛♥❞ t❤❡♥✱ ❛❢t❡r 1 ❤♦✉r✱ ❛♥♦t❤❡r ♣♦❧✐❝❡♠❛♥ ♦❜s❡r✈❡❞ ②♦✉ ❞r✐✈✐♥❣ ❜② ✕ ❜✉t 100 ♠✐❧❡s ❛✇❛②✦ ❚❤❡♥ t❤❡ ❛✈❡r❛❣❡ s♣❡❡❞ ♦❢ ②♦✉r ❝❛r ✇❛s 100 ♠✴❤✳ ❉✐❞ ②♦✉ ✈✐♦❧❛t❡ t❤❡ ❧❛✇❄ ❈♦♥s✐❞❡r✐♥❣ t❤❛t ♥♦✲♦♥❡ ❝❛♥ t❡st✐❢② t♦ ❤❛✈❡ s❡❡♥ ②♦✉ ❞r✐✈❡ ❛❜♦✈❡ t❤❡ s♣❡❡❞ ❧✐♠✐t✱ ❝❛♥ t❤❡ t✇♦ ♣♦❧✐❝❡♠❡♥ ❝♦♠♣❛r❡ ♥♦t❡s ❛♥❞ ♣r♦✈❡ t❤❛t ②♦✉ ❞✐❞❄ ❚❤❡ ❛♥❛❧②s✐s ✇❡ ♣✉rs✉❡ ❤❡r❡ ❛❧❧♦✇s t❤❡♠ t♦ ✐♥❢❡r t❤❛t✱ ②❡s✱ ②♦✉r ✐♥st❛♥t❛♥❡♦✉s ✈❡❧♦❝✐t② ✇❛s 100 ♠✴❤ ❛t s♦♠❡ ♣♦✐♥t✳ ❚♦ ♠❛❦❡ t❤❡✐r ❝❛s❡ ❛❣❛✐♥st ②♦✉ r♦❝❦✲s♦❧✐❞✱ t❤❡②✬❞ q✉♦t❡ t❤❡ t❤❡♦r❡♠ ❜❡❧♦✇✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ✐s ❛ ❝❡♥tr❛❧ r❡s✉❧t ♦❢ ❝❛❧❝✉❧✉s✿

✺✳✸✳

❲❤❛t t❤❡ ❞❡r✐✈❛t✐✈❡ s❛②s ❛❜♦✉t t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✿ ❚❤❡ ▼❡❛♥ ❱❛❧✉❡ ❚❤❡♦r❡♠

✸✾✻

❚❤❡♦r❡♠ ✺✳✸✳✻✿ ▼❡❛♥ ❱❛❧✉❡ ❚❤❡♦r❡♠

❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ❢✉♥❝t✐♦♥✿ ✶✳ f ✐s ❝♦♥t✐♥✉♦✉s ♦♥ [a, b]✳ ✷✳ f ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ♦♥ (a, b)✳ ❚❤❡♥ f (b) − f (a) = f ′ (c) b−a

❋❖❘ ❙❖▼❊ c ✐♥ (a, b) ✳ ❲❤❛t ❤❛♣♣❡♥s ✐❢ ✇❡ ❤❛✈❡

f (a) = f (b)

❤❡r❡❄ ❚❤❡♥ t❤❡ ❧❡❢t✲❤❛♥❞ s✐❞❡ ✐s

❝♦♥❝❧✉s✐♦♥ ♦❢ ❘♦❧❧❡✬s ❚❤❡♦r❡♠✳ ❚❤✐s ♠❡❛♥s t❤❛t

0✱

❤❡♥❝❡

0 = f ′ (c)✳

❲❡ ❤❛✈❡ t❤❡

▼❱❚ ✐s ♠♦r❡ ❣❡♥❡r❛❧ t❤❛♥ ❘❚✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ❧❛tt❡r

✐s ❛♥ ✐♥st❛♥❝❡✱ ❛ ♥❛rr♦✇ ❝❛s❡ ♦❢ ▼❱❚✳ ❚❤❡ ♣r♦♦❢ ♦❢ ▼❱❚✱ ❤♦✇❡✈❡r✱ ✇✐❧❧ r❡❧② ♦♥ ❘❚✳ ❚❤❡ ✐❞❡❛ ✐s t♦ ✏s❦❡✇✑ t❤❡ ❣r❛♣❤ ♦❢ ▼❱❚ ❜❛❝❦ t♦ ❘❚✳ ❚❤✐s ✐s t❤❡ ♦✉t❧✐♥❡ ♦❢ t❤❡ ♣r♦♦❢✿

Pr♦♦❢✳

▲❡t✬s r❡♥❛♠❡

f

✐♥

❘♦❧❧❡✬s ❚❤❡♦r❡♠

❛s

h

h ❝♦♥t✐♥✉♦✉s ♦♥ [a, b]✳ ✷✳ h ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ♦♥ [a, b]✳ ✸✳ h(a) = h(b)✳ ❙✉♣♣♦s❡ y = L(x) ✐s t❤❡ ❧✐♥❡❛r ❢✉♥❝t✐♦♥

t♦ ✉s❡ ✐t ❧❛t❡r✳ ❚❤❡♥ ✐ts ❝♦♥❞✐t✐♦♥s t❛❦❡ t❤✐s ❢♦r♠✿

✶✳

r❡♣r❡s❡♥t❡❞ ❜② t❤❡ ❧✐♥❡ ❜❡t✇❡❡♥

(a, f (a))

❛♥❞

(b, f (b))✳

❚❤❡♥✱

✐ts ❞❡r✐✈❛t✐✈❡ ✐s s✐♠♣❧② t❤❡ s❧♦♣❡ ♦❢ t❤❡ ❧✐♥❡✿

L′ (x) =

◆♦✇ ❜❛❝❦ t♦

f✳

f (b) − f (a) . b−a

❚❤✐s ✐s t❤❡ ❦❡② st❡♣❀ ❧❡t

h(x) = f (x) − L(x) . ▲❡t✬s ✈❡r✐❢② t❤❡ ❝♦♥❞✐t✐♦♥s ❛❜♦✈❡✳ ❋✐rst✱

h

✐s ❝♦♥t✐♥✉♦✉s ♦♥

[a, b]

❛s t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ t✇♦ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s ✭❙❘✮✳ ❈♦♥❞✐t✐♦♥ ★✶

❛❜♦✈❡ ✐s s❛t✐s✜❡❞✦ ◆❡①t✱

h

✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ♦♥

(a, b)

❛s t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ t✇♦ ❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥s ✭❙❘✮✳ ❈♦♥❞✐t✐♦♥

✺✳✸✳ ❲❤❛t t❤❡ ❞❡r✐✈❛t✐✈❡ s❛②s ❛❜♦✉t t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✿ ❚❤❡ ▼❡❛♥ ❱❛❧✉❡ ❚❤❡♦r❡♠

✸✾✼

★✷ ❛❜♦✈❡ ✐s s❛t✐s✜❡❞✦ ❚❤❡ ❞❡r✐✈❛t✐✈❡ ✐s s✐♠♣❧❡✿ h′ (x) = f ′ (x) −

f (b) − f (a) . b−a

❲❡ ❛❧s♦ ❤❛✈❡✿ f (a) = L(a), f (b) = L(b) =⇒ h(a) = 0, h(b) = 0 =⇒ h(a) = h(b) .

❈♦♥❞✐t✐♦♥ ★✸ ❛❜♦✈❡ ✐s s❛t✐s✜❡❞✦ ❚❤✉s✱ h s❛t✐s✜❡s t❤❡ ❝♦♥❞✐t✐♦♥s ♦❢ ❘❚✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ❝♦♥❝❧✉s✐♦♥ ✐s s❛t✐s✜❡❞ t♦♦✿ h′ (c) = 0

❢♦r s♦♠❡ c ✐♥ (a, b)✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿ f ′ (c) −

f (b) − f (a) = 0. b−a

●❡♦♠❡tr✐❝❛❧❧②✱ c ✐s ❢♦✉♥❞ ❜② s❤✐❢t✐♥❣ t❤❡ s❡❝❛♥t ❧✐♥❡ ✉♥t✐❧ ✐t t♦✉❝❤❡s t❤❡ ❣r❛♣❤✿

❊①❡r❝✐s❡ ✺✳✸✳✼

P❧♦t ②♦✉r ♦✇♥ ❣r❛♣❤ ❛♥❞ ✜♥❞ t❤❡s❡ s♣❡❝✐❛❧ t❛♥❣❡♥ts✳ ❲❡ ✇✐❧❧ ✉s❡ t❤❡ ▼❡❛♥ ❱❛❧✉❡ ❚❤❡♦r❡♠ t♦ ❞❡r✐✈❡ ❢❛❝ts ❛❜♦✉t t❤❡ ✈❛❧✉❡s ♦❢ ❛ ❢✉♥❝t✐♦♥ ✇❡ ❦♥♦✇ ♥♦t❤✐♥❣ ❛❜♦✉t ❢r♦♠ ❛ ♣r✐♦r✐ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t ✐ts ❞❡r✐✈❛t✐✈❡✿

❊①❛♠♣❧❡ ✺✳✸✳✽✿ ❝♦♥st❛♥t ❢✉♥❝t✐♦♥

▲❡t✬s tr② ♦♥❡ ♦❢ t❤❡ ❝♦♥✈❡rs❡s ♠❡♥t✐♦♥❡❞ ✐♥ ❡❛r❧✐❡r ✐♥ t❤✐s s❡❝t✐♦♥✿ ❚❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ❝♦♥st❛♥t ❢✉♥❝t✐♦♥ ✐s ③❡r♦✱ ❜✉t ❛r❡ t❤❡ ❝♦♥st❛♥ts t❤❡ ♦♥❧② ❢✉♥❝t✐♦♥s ✇✐t❤ t❤✐s ♣r♦♣❡rt②❄ ❨❡s✳ ❙✉♣♣♦s❡ f ′ = 0 ♦♥ [A, B]✳ ❚❤❡♥ ❢♦r ❛♥② a < b ✐♥ t❤❡ ✐♥t❡r✈❛❧✱ ✇❡ ❤❛✈❡ ❢r♦♠ t❤❡ ▼❡❛♥ ❱❛❧✉❡ ❚❤❡♦r❡♠✿

❚❤❡♥✱

f (b) − f (a) = f ′ (c) = 0 . b−a f (b) = f (a) .

✺✳✹✳ ▼♦♥♦t♦♥✐❝✐t② ❛♥❞ t❤❡ s✐❣♥ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡

✸✾✽

❚❤❡ ❢✉♥❝t✐♦♥ ✐s ❝♦♥st❛♥t✦

✺✳✹✳ ▼♦♥♦t♦♥✐❝✐t② ❛♥❞ t❤❡ s✐❣♥ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡

❲❤❛t ❞♦❡s ❛ ♣r✐♦r✐ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ t❡❧❧ ✉s ❛❜♦✉t ✐ts ♠♦♥♦t♦♥✐❝✐t②❄ ❘❡❝❛❧❧ t❤❛t t❤❡ ▲♦❝❛❧ ▼♦♥♦t♦♥✐❝✐t② ❚❤❡♦r❡♠ st❛t❡s t❤❛t✱ ❢♦r ❛ ❢✉♥❝t✐♦♥ y = f (x) ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t x = c✱ ✐❢ x = c ✐s ❛♥ ✐♥❝r❡❛s✐♥❣ ♣♦✐♥t ♦❢ y = f (x)✱ t❤❡♥ f ′ (c) ≥ 0✳ ■♥st❡❛❞ ♦❢ ♣r♦✈✐♥❣ t❤❡ ❝♦♥✈❡rs❡ ♦❢ t❤✐s ❧♦❝❛❧ r❡s✉❧t✱ ✇❡ ✇✐❧❧ ✉s❡ t❤❡ ▼❡❛♥ ❱❛❧✉❡ ❚❤❡♦r❡♠ t♦ ♣r♦✈❡ ❛ ❣❧♦❜❛❧ r❡s✉❧t✦ ❲❡ ✐♥❝❧✉❞❡ t❤❡ ❞✐s❝r❡t❡ ❝❛s❡ ❢♦r ❝♦♠♣❧❡t❡♥❡ss✿

❚❤❡♦r❡♠ ✺✳✹✳✶✿ ▼♦♥♦t♦♥✐❝✐t② ❢r♦♠ ❙✐❣♥ ♦❢ ❉✐✛❡r❡♥❝❡ ◗✉♦t✐❡♥t ❙✉♣♣♦s❡ f ✐s ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦♥ ❛ ❝❧♦s❡❞ ✐♥t❡r✈❛❧ I ✳ ❚❤❡♥✿ ✶✳ ❚❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ f ✐s ♥♦♥✲♥❡❣❛t✐✈❡ ♦♥ I ✐❢ ❛♥❞ ♦♥❧② ✐❢ f ✐s ✐♥❝r❡❛s✲ ✐♥❣ ♦♥ I ✳ ✷✳ ❚❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ f ✐s ♥♦♥✲♣♦s✐t✐✈❡ ♦♥ I ✐❢ ❛♥❞ ♦♥❧② ✐❢ f ✐s ❞❡❝r❡❛s✐♥❣ ♦♥ I ✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿

∆f ≥ 0 ⇐⇒ f ր ∆x ∆f ✷✳ ≤ 0 ⇐⇒ f ց ∆x

✶✳

❚❤❡ t❤❡♦r❡♠ ❢♦r t❤❡ ❞❡r✐✈❛t✐✈❡s ❜❡❧♦✇ s❡❡♠s ❛❧♠♦st ✐❞❡♥t✐❝❛❧ ❜✉t ✐t ❞♦❡s♥✬t ❥✉st ❢♦❧❧♦✇ ✇✐t❤ ∆x → 0 ❛s ♠❛♥② t✐♠❡s ❜❡❢♦r❡✳ ❲❡✬❧❧ ♥❡❡❞ t♦ ✉s❡ t❤❡ ▼❡❛♥ ❱❛❧✉❡ ❚❤❡♦r❡♠ t♦ ♣r♦✈❡ ✇❤❛t ✇❡ ✇❛♥t✿

❚❤❡♦r❡♠ ✺✳✹✳✷✿ ▼♦♥♦t♦♥✐❝✐t② ❢r♦♠ ❙✐❣♥ ♦❢ ❉❡r✐✈❛t✐✈❡ ❙✉♣♣♦s❡ f ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ♦♥ ❛♥ ♦♣❡♥ ✐♥t❡r✈❛❧ I ✳ ❚❤❡♥✿ ✶✳ ❚❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ f ✐s ♥♦♥✲♥❡❣❛t✐✈❡ ♦♥ I ✐❢ ❛♥❞ ♦♥❧② ✐❢ f ✐s ✐♥❝r❡❛s✐♥❣ ♦♥ I ✳ ✷✳ ❚❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ f ✐s ♥♦♥✲♣♦s✐t✐✈❡ ♦♥ I ✐❢ ❛♥❞ ♦♥❧② ✐❢ f ✐s ❞❡❝r❡❛s✐♥❣ ♦♥ I ✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿ df ≥ 0 ⇐⇒ f ր dx df ✷✳ ≤ 0 ⇐⇒ f ց dx

✶✳

Pr♦♦❢✳ (⇐) ■❢ f ✐s ✐♥❝r❡❛s✐♥❣ ♦♥ I ✱ ❡✈❡r② ♣♦✐♥t c ✐♥ I ✐s ❛♥ ✐♥❝r❡❛s✐♥❣ ♣♦✐♥t ♦❢ f ✳ ❚❤❡r❡❢♦r❡✱ ❜② t❤❡ ▼♦♥♦t♦♥✐❝✐t② ❚❤❡♦r❡♠✱ ✇❡ ❤❛✈❡ f ′ (c) ≥ 0✳ (⇒) ❙✉♣♣♦s❡ a, b ❛r❡ ✐♥ I ❛♥❞ a < b✳ ❲❡ ♥❡❡❞ t♦ s❤♦✇ t❤❛t f (a) ≤ f (b)✳ ❇② t❤❡ ▼❡❛♥

✇❡ ❤❛✈❡✿

▲♦❝❛❧

❱❛❧✉❡ ❚❤❡♦r❡♠✱

f (b) − f (a) = f ′ (c) b−a ❢♦r s♦♠❡ c ✐♥ (a, b)✳ ◆♦ ♠❛tt❡r ✇❤❛t c ✐s✱ ❜② ❛ss✉♠♣t✐♦♥ ♦❢ t❤❡ t❤❡♦r❡♠✱ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ✐s

✺✳✹✳

▼♦♥♦t♦♥✐❝✐t② ❛♥❞ t❤❡ s✐❣♥ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡

✸✾✾

♥♦♥✲♥❡❣❛t✐✈❡✳ ❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡

f (b) − f (a) ≥ 0. b−a ◆♦✇ ♦❜s❡r✈❡ t❤❛t ✐♥ t❤✐s ❢r❛❝t✐♦♥✱ t❤❡ ❞❡♥♦♠✐♥❛t♦r ✐s b − a > 0✳ ❛❧s♦ ♣♦s✐t✐✈❡✿ f (b) − f (a) ≥ 0✳ ❍❡♥❝❡✱ f (b) ≥ f (a)✳

❚❤❡r❡❢♦r❡✱ t❤❡ ♥✉♠❡r❛t♦r ♠✉st ❜❡

❊①❛♠♣❧❡ ✺✳✹✳✸✿ t❤r❡❡ ❢❛♠✐❧✐❛r ❢✉♥❝t✐♦♥s

❲❡ ✉s❡ t❤❡ ❞❡r✐✈❛t✐✈❡s t♦ ❛♥❛❧②③❡ t❤❡s❡ ❢✉♥❝t✐♦♥s✿

(1) f (x) = 3x2 + 1 f ′ (x) < 0 fց

♦♥

✐❢

=⇒

x < −1/2

(−∞, −1/2)

1 x ′ g (x) < 0

gց

♦♥

✐❢

x 0 hր

♦♥

❛♥❞

f ′ (x) > 0

❛♥❞

fր

=⇒

(2) g(x) =

❛♥❞ ❛♥❞

=⇒ ❢♦r ❛❧❧

x

f ′ (x) = 6x + 3

♦♥

✐❢

x > −1/2 =⇒

(−1/2, ∞)

1 x2 g ′ (x) < 0 ✐❢ x > 0 g ′ (x) = −

gց

♦♥

=⇒

=⇒ =⇒

(0, ∞)

h′ (x) = ex

=⇒

=⇒

(−∞, ∞)

❲✐t❤ ❥✉st t❤✐s ❞❛t❛✱ ✇❡ ❝❛♥ s❦❡t❝❤ ✈❡r② r♦✉❣❤ ❣r❛♣❤s ♦❢ t❤❡s❡ t❤r❡❡ ❢✉♥❝t✐♦♥s✿

❊✈❡♥ t❤♦✉❣❤ ✇❡ ❤❛✈❡ s❤♦✇♥ t❤❛t t❤❡ ❢✉♥❝t✐♦♥s ❤❛✈❡ ♥♦ ♦t❤❡r ❡①tr❡♠❛✱ t❤❡ ❝✉r✈❡s ❝♦✉❧❞ ❤❛✈❡ t❤❡s❡ ✏✇✐❣❣❧❡s✑✳ ❚❤✐s ✐ss✉❡ ✇✐❧❧ ❜❡ ❛❞❞r❡ss❡❞ s❤♦rt❧②✳

❚♦ ✐❞❡♥t✐❢② ♠♦♥♦t♦♥✐❝✐t②✱ t❤❡r❡❢♦r❡✱ ✇✐❧❧ r❡q✉✐r❡ ❛ ❞❡r✐✈❛t✐✈❡✱ ❛s ❢♦❧❧♦✇s✿

s✐❣♥ ❛♥❛❧②s✐s

✭s❡❡♥ ✐♥ ❱♦❧✉♠❡ ✶✱ ❈❤❛♣t❡r ✶P❈✲✹✮ ♦❢ t❤❡

✺✳✹✳

✹✵✵

▼♦♥♦t♦♥✐❝✐t② ❛♥❞ t❤❡ s✐❣♥ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡

❚❤❡ ♠❛✐♥ t♦♦❧ ✐s t❤❡ ■♥t❡r♠❡❞✐❛t❡

❱❛❧✉❡ ❚❤❡♦r❡♠

✭❈❤❛♣t❡r ✷✮✿

◮ ■❢ ❛ ❢✉♥❝t✐♦♥ ✐s ❝♦♥t✐♥✉♦✉s✱ ✐ts s✐❣♥ ❝❛♥ ♦♥❧② ❝❤❛♥❣❡ ✇❤❡r❡ ✐t ✐s ③❡r♦✳ ❊①❛♠♣❧❡ ✺✳✹✳✹✿ q✉❛❞r❛t✐❝

❘❡❝❛❧❧ t❤✐s ❡①❛♠♣❧❡✿

f (x) = x3 − 3x .

❖♥ ✇❤❛t ✐♥t❡r✈❛❧s ✐s t❤✐s ❢✉♥❝t✐♦♥ ✐♥❝r❡❛s✐♥❣ ❛♥❞ ❞❡❝r❡❛s✐♥❣❄ ❋✐rst✱

f ′ (x) = 3x2 − 3 .

■♥ ♦r❞❡r t♦ ✉s❡ t❤❡ ▼♦♥♦t♦♥✐❝✐t② ❚❤❡♦r❡♠✱ ✇❡ ♥❡❡❞ t♦ ✜♥❞ t❤♦s❡ x✬s t❤❛t ♣r♦❞✉❝❡ f ′ (x) > 0 ♦r f ′ (x) < 0✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ♥❡❡❞ t♦ s♦❧✈❡ t❤♦s❡ ✐♥❡q✉❛❧✐t✐❡s✳ ❲❡ st❛rt ✇✐t❤ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❡q✉❛t✐♦♥ f ′ (x) = 0✱ ❞♦♥❡ ♣r❡✈✐♦✉s❧②✿ 3x2 − 3 x2 − 1 x2 x

= 0 =⇒ = 0 =⇒ = 1 =⇒ = ±1 .

❚❤❡s❡ t✇♦ ♣♦✐♥ts ❝✉t t❤r❡❡ ✐♥t❡r✈❛❧s ❢r♦♠ t❤❡ r❡❛❧ ❧✐♥❡✿ • (−∞, −1), • (−1, 1), • (1, ∞).

❲❡ ♥❡❡❞ t♦ ❦♥♦✇ t❤❡

s✐❣♥ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡

♦♥ ❡❛❝❤✳

❙✐♥❝❡ f ′ ✐s ❝♦♥t✐♥✉♦✉s✱ t❤❡ s✐❣♥ ♦❢ f ′ ❝❛♥ ♦♥❧② ❝❤❛♥❣❡ ❛t −1 ♦r 1✳ ❚❤❡r❡❢♦r❡✱ ✇❡ ❥✉st ♥❡❡❞ t♦ s❛♠♣❧❡ ♦♥❡ ♣♦✐♥t ✇✐t❤✐♥ ❡❛❝❤ ✐♥t❡r✈❛❧ t♦ ❞❡t❡r♠✐♥❡ t❤❡ s✐❣♥ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦♥ t❤❡ ✇❤♦❧❡ ✐♥t❡r✈❛❧✿ • P✐❝❦ x = −2✱ t❤❡♥ f ′ (−2) = 3 · (−2)2 − 3 = 9 > 0 ✳ ❚❤❡r❡❢♦r❡✱ f ′ (x) > 0 ❢♦r x ✐♥ (−∞, −1)✳ ❚❤❡r❡❢♦r❡✱ f ր ♦♥ (−∞, −1)✳ • P✐❝❦ x = 0✱ t❤❡♥ f ′ (0) = −3 < 0✳ ❚❤❡r❡❢♦r❡✱ f ′ (x) < 0 ❢♦r x ✐♥ (−1, 1)✳ ❚❤❡r❡❢♦r❡✱ f ց ♦♥ (−1, 1)✳ • P✐❝❦ x = 2✱ t❤❡♥ f ′ (2) = 3 · 22 − 3 = 9 > 0✳ ❚❤❡r❡❢♦r❡✱ f ′ (x) > 0 ❢♦r x ✐♥ (1, ∞)✳ ❚❤❡r❡❢♦r❡✱ f ր ♦♥ (1, ∞)✳ ▲❡t✬s ♣✉t t❤✐s ❞❛t❛ ✐♥ ❛ t❛❜❧❡✿ x : (−∞, −1) (−1, 1) (1, ∞) f: ր ց ր

❚❤✐s ✐s t❤❡ ❛♥s✇❡r✦

✺✳✹✳

▼♦♥♦t♦♥✐❝✐t② ❛♥❞ t❤❡ s✐❣♥ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡

✹✵✶

❋✉rt❤❡r♠♦r❡✱ t❤❡s❡ ❛rr♦✇s ❛r❡ ❝❧♦s❡ t♦ ❢♦r♠✐♥❣ ❛ ❝✉r✈❡✱ ❡s♣❡❝✐❛❧❧② ❛❢t❡r t❤✐s ♠♦❞✐✜❝❛t✐♦♥✿

f: ր x:

· −1 max

ց

· 1 min

ր

❲❡ ❤❛✈❡ ❛❧s♦ ✕ ❛✉t♦♠❛t✐❝❛❧❧② ✕ ❝❧❛ss✐✜❡❞ t❤❡ ❡①tr❡♠❡ ♣♦✐♥ts✦ ❲❡ ❝♦♥✜r♠ t❤❡ r❡s✉❧t ❜② ♣❧♦tt✐♥❣✿

❊①❛♠♣❧❡ ✺✳✹✳✺✿ s✐♥❡

❈♦♥s✐❞❡r f (x) = sin x ❛❣❛✐♥✳ ❲❡ ❛❧r❡❛❞② ❦♥♦✇ t❤❛t t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❛♥❞ t❤❡ ❞❡r✐✈❛t✐✈❡ ❛r❡ ③❡r♦✱ cos x = 0✱ ❛t t❤❡s❡ ❧♦❝❛t✐♦♥s✿ π x = + kπ, k = 0, ±1, ±2, ... 2 ❚❤❡s❡ ❛r❡ t❤❡ ♣♦✐♥ts ✇❤❡r❡ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❛♥❞ t❤❡ ❞❡r✐✈❛t✐✈❡ ❝♦✉❧❞✱ ♣♦t❡♥t✐❛❧❧②✱ ❝❤❛♥❣❡ t❤❡✐r s✐❣♥✦ ❙✐♥❝❡ t❤❡ t✇♦ ❞✐✛❡r ❜② ❛ ♣♦s✐t✐✈❡ ♠✉❧t✐♣❧❡✿

∆ sin(h/2) sin(h/2) d (cos x) = − · sin x = · (cos x) , ∆x h/2 h/2 dx t❤❡② ❝❤❛♥❣❡ s✐❣♥s t♦❣❡t❤❡r✳ ❋r♦♠ ✇❤❛t ✇❡ ❦♥♦✇ ❡✈❡♥ ♠♦r❡ ❛❜♦✉t ❝♦s✐♥❡✱ t❤❡ s✐❣♥ ❞♦❡s ❝❤❛♥❣❡ ❡✈❡r② t✐♠❡✿ · · · y = sin x : ր ց ր ց ր · · · x : −π/2 π/2 3π/2 5π/2 7π/2 9π/2 y = cos x : 0 + 0 − 0 + 0 − 0 + 0 − ❚❤✐s ❝♦♥❝❧✉s✐♦♥ ❝♦♥✜r♠s ✇❤❛t ✇❡ ❦♥♦✇ ❛❜♦✉t t❤✐s ❢✉♥❝t✐♦♥✿

✺✳✹✳

▼♦♥♦t♦♥✐❝✐t② ❛♥❞ t❤❡ s✐❣♥ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡

✹✵✷

❊①❛♠♣❧❡ ✺✳✹✳✻✿ ❡①♣♦♥❡♥t ▲❡t✬s ❝♦♥s✐❞❡r t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥ f (x) = ax ❢♦r ❛❧❧ a > 0✳ ❋✐rst t❤❡ ❞❡r✐✈❛t✐✈❡s✿ f ′ (x) = (ax )′ = ax ln a .

❚❤❡r❡❢♦r❡✱ ❜② t❤❡

▼♦♥♦t♦♥✐❝✐t② ❚❤❡♦r❡♠✱ ❢♦r ❛❧❧ x ✇❡ ❤❛✈❡✿ a>1 =⇒ f ′ (x) > 0 0 < a < 1 =⇒ f ′ (x) < 0

■t ❢♦❧❧♦✇s t❤❛t t❤❡ ❢✉♥❝t✐♦♥ ✐s ❡✐t❤❡r ❛❧❧ ✐♥❝r❡❛s✐♥❣ ♦r ❛❧❧ ❞❡❝r❡❛s✐♥❣✿

❚❤❡♥✱ ✇❤❡♥ a > 1✱ ✇❡ ❤❛✈❡ t❤❡ ❣❧♦❜❛❧ ♠✐♥✐♠✉♠ ♦♥ ✐♥t❡r✈❛❧ [a, b] ❛t x = a ❛♥❞ t❤❡ ❣❧♦❜❛❧ ♠❛①✐♠✉♠ ❛t b✳

❊①❛♠♣❧❡ ✺✳✹✳✼✿ ❝✉❜❡ ▲❡t✬s ❝♦♥s✐❞❡r f (x) = x3 ✳

❚❤❡ ❞❡r✐✈❛t✐✈❡ ✐s f ′ (x) = 3x2 ✳ ❚❤❡r❡❢♦r❡✱ ❜② t❤❡

▼♦♥♦t♦♥✐❝✐t② ❚❤❡♦r❡♠✱ ✇❡ ❤❛✈❡✿

♦♥ (−∞, 0) f ′ > 0 =⇒ f ր ❛t 0 f′ = 0 ♦♥ (0, +∞) f ′ > 0 =⇒ f ր ❚❤❡r❡ ✐s ♠♦r❡ ❤❡r❡✿ ❲❡ ❤❛✈❡ ❢♦✉♥❞ ❛♥ ❡①❛♠♣❧❡ ♦❢ ❛ str✐❝t❧② ✐♥❝r❡❛s✐♥❣ ❢✉♥❝t✐♦♥ ✇✐t❤ ❛ ③❡r♦ ❞❡r✐✈❛t✐✈❡ ❛t ♦♥❡ ♣♦✐♥t✦ ❚❤❡s❡ ❛r❡ ❛ ♠♦❞✐✜❡❞ ✈❡rs✐♦♥ ♦❢ t❤❡ ❛❜♦✈❡ t❤❡♦r❡♠s✿

❚❤❡♦r❡♠ ✺✳✹✳✽✿ ❙tr✐❝t ▼♦♥♦t♦♥✐❝✐t② ✈✐❛ ❉✐✛❡r❡♥❝❡ ◗✉♦t✐❡♥t ❙✉♣♣♦s❡ f ✐s ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦♥ ❛ ❝❧♦s❡❞ ✐♥t❡r✈❛❧ I ✳ ❚❤❡♥✱ ✇❡ ❤❛✈❡✿ ✶✳ ❚❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ f ✐s ♣♦s✐t✐✈❡ ♦♥ I ✐❢ ❛♥❞ ♦♥❧② ✐❢ f ✐s str✐❝t❧②

✺✳✺✳

❈♦♥❝❛✈✐t② ❛♥❞ t❤❡ s✐❣♥ ♦❢ t❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡

✹✵✸

✐♥❝r❡❛s✐♥❣ ♦♥ I ✳ ✷✳ ❚❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ f ✐s ♥❡❣❛t✐✈❡ ♦♥ I ✐❢ ❛♥❞ ♦♥❧② ✐❢ f ✐s str✐❝t❧② ❞❡❝r❡❛s✐♥❣ ♦♥ I ✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿

∆f > 0 ⇐⇒ f ր str✐❝t❧② ∆x ∆f ✷✳ < 0 ⇐⇒ f ց str✐❝t❧② ∆x

✶✳

❚❤❡ ❝♦♥t✐♥✉♦✉s ❝❛s❡ ✐s ♠♦r❡ s✉❜t❧❡✿

❚❤❡♦r❡♠ ✺✳✹✳✾✿ ❙tr✐❝t ▼♦♥♦t♦♥✐❝✐t② ✈✐❛ ❉❡r✐✈❛t✐✈❡

❙✉♣♣♦s❡ f ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ♦♥ ❛♥ ♦♣❡♥ ✐♥t❡r✈❛❧ I ✳ ❚❤❡♥✱ ✇❡ ❤❛✈❡✿ ✶✳ ■❢ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ f ✐s ♣♦s✐t✐✈❡ ♦♥ I ✱ t❤❡♥ f ✐s str✐❝t❧② ✐♥❝r❡❛s✐♥❣ ♦♥ I ✳ ✷✳ ■❢ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ f ✐s ♥❡❣❛t✐✈❡ ♦♥ I ✱ t❤❡♥ f ✐s str✐❝t❧② ❞❡❝r❡❛s✐♥❣ ♦♥ I ✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿

df > 0 =⇒ f ր str✐❝t❧② dx df ✷✳ < 0 =⇒ f ց str✐❝t❧② dx

✶✳

Pr♦♦❢✳ ❏✉st r❡♣❧❛❝❡ ❡❛❝❤ ✏ ≥✑ ✐♥ t❤❡ ♣r♦♦❢ ♦❢ t❤❡

▼♦♥♦t♦♥✐❝✐t② ❚❤❡♦r❡♠ ✇✐t❤ ✏ >✑✳

❲❡ ❞❡♠♦♥str❛t❡❞ ✐♥ t❤❡ ❧❛st ❡①❛♠♣❧❡ t❤❛t t❤❡ ❝♦♥✈❡rs❡ ❢❛✐❧s✳

❚❤❡ r❡❛s♦♥ ❢♦r s✉❝❤ ❛ ❞✐✛❡r❡♥❝❡ ❢r♦♠ t❤❡

♥♦♥✲str✐❝t ❝❛s❡ ✐s✱ r♦✉❣❤❧②✱ t❤❛t t❤❡ ❧✐♠✐t ♦❢ ♣♦s✐t✐✈❡ ♥✉♠❜❡rs ♠✐❣❤t ❜❡ ③❡r♦✳

❊①❡r❝✐s❡ ✺✳✹✳✶✵ ❙❤♦✇ t❤❛t ✐❢ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ✐s ③❡r♦✱ ✐t ✐s ❛ ❝♦♥st❛♥t ❢✉♥❝t✐♦♥✱ ✐♥ t✇♦ ✇❛②s✿ ✭❛✮ ❜② ♠♦❞✐❢②✐♥❣ t❤❡ ♣r♦♦❢ ♦❢ t❤❡ ▼♦♥♦t♦♥✐❝✐t② ❚❤❡♦r❡♠✱ ✭❜✮ ❜② ❛♣♣❧②✐♥❣ t❤❡ ▼♦♥♦t♦♥✐❝✐t② ❚❤❡♦r❡♠✳

❆ ❜♦♥✉s r❡s✉❧t ✐s ❜❡❧♦✇✿

❈♦r♦❧❧❛r② ✺✳✹✳✶✶✿ ❖♥❡✲t♦✲♦♥❡ ❢r♦♠ ❉❡r✐✈❛t✐✈❡

❙✉♣♣♦s❡ f ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ♦♥ ❛♥ ♦♣❡♥ ✐♥t❡r✈❛❧ I ✳ ❚❤❡♥✱ ✐❢ f ′ > 0 ♦♥ I ♦r f ′ < 0 ♦♥ I ✱ t❤❡♥ f ✐s ♦♥❡✲t♦✲♦♥❡ ♦♥ I ✳ ❊①❡r❝✐s❡ ✺✳✹✳✶✷ Pr♦✈❡ t❤❡ ❝♦r♦❧❧❛r②✳ ❲❤❛t ❛❜♦✉t t❤❡ ❝♦♥✈❡rs❡❄

✺✳✺✳ ❈♦♥❝❛✈✐t② ❛♥❞ t❤❡ s✐❣♥ ♦❢ t❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡

❘❡❝❛❧❧ ❤♦✇ ❡❛s② ✐t ✐s t♦ s❡❡ t❤❡ ✐❞❡❛ ❜❡❤✐♥❞ ♠♦♥♦t♦♥✐❝✐t② ❜② ③♦♦♠✐♥❣ ✐♥ ♦♥ t❤❡ ❣r❛♣❤ ✇❤❡♥ ✐t ✐s ♠❛❞❡ ♦❢ ♦✈❡r❧❛♣♣✐♥❣ ❞♦ts✿

✺✳✺✳

❈♦♥❝❛✈✐t② ❛♥❞ t❤❡ s✐❣♥ ♦❢ t❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡

✹✵✹

❲❡ ✇✐❧❧ ❞♦ t❤❡ s❛♠❡ ❢♦r ❛ s✉❜t❧❡r ✐❞❡❛✳ ❚❤❡ ♣✐❝t✉r❡ ❜❡❧♦✇ ✐♥❢♦r♠❛❧❧② r❡✈❡❛❧s t❤❡ ♠❡❛♥✐♥❣ ♦❢ ✉♣✇❛r❞ ❛♥❞ ❞♦✇♥✇❛r❞

❝♦♥❝❛✈✐t② ✿

❲❤❡♥ ✇❡ ❣✉❛r❛♥t❡❡ t❤❡ ♦♥❡ ♦r t❤❡ ♦t❤❡r✱ ✇❡ ❡❧✐♠✐♥❛t❡ t❤❡ ♣♦ss✐❜✐❧✐t② ♦❢ ✏✇✐❣❣❧②✑ ❝✉r✈❡s✦ ❲❡ ✇✐❧❧ ❛❧s♦ ❜❡ ❛❜❧❡ t♦ t❡❧❧ ♠❛①✐♠❛ ❢r♦♠ ♠✐♥✐♠❛✳ ▲❡t✬s ③♦♦♠ ✐♥✳ ❚❤❡r❡ ❛r❡

t❤r❡❡ ♣♦✐♥ts t❤✐s t✐♠❡✿

❚❤❡ ♣❛tt❡r♥ ✐s ❝❧❡❛r✿

◮

■t ✐s ❝♦♥❝❛✈❡ ✉♣ ✇❤❡♥ t❤❡ ♠✐❞❞❧❡ ♣♦✐♥t ❧✐❡s ❜❡❧♦✇ t❤❡ ❧✐♥❡ ❝♦♥♥❡❝t✐♥❣ t❤❡ ♦t❤❡r t✇♦✱ ❛♥❞

❝♦♥❝❛✈❡ ❞♦✇♥ ✇❤❡♥ ✐t ✐s ❛❜♦✈❡✳ ▲❡t✬s ✐♥✈❡st✐❣❛t❡ t❤❡ ❛❧❣❡❜r❛✳ ❚❤❡ t✇♦ ♣♦✐♥ts ♦♥ t❤❡

y ✲❛①✐s

❛r❡

f (c − h)

❛♥❞

f (c + h)✿

✺✳✺✳ ❈♦♥❝❛✈✐t② ❛♥❞ t❤❡ s✐❣♥ ♦❢ t❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡

✹✵✺

❚❤❡ y ✲✈❛❧✉❡ ♦❢ t❤❡ ♠✐❞✲♣♦✐♥t ✐s t❤❡✐r ❛✈❡r❛❣❡✿ f (c − h) + f (c + h) . 2

❚❤❡♥✱ t❤❡ ❝♦♥❝❛✈❡ ✉♣ ❛♥❞ ❝♦♥❝❛✈❡ ❞♦✇♥ ❝♦♥❞✐t✐♦♥s t❛❦❡ t❤❡ ❢♦r♠ ♦❢ t❤❡s❡ ✐♥❡q✉❛❧✐t✐❡s r❡s♣❡❝t✐✈❡❧②✿ f (c) ≤

f (c − h) + f (c + h) 2

❚❤❡② ❛r❡ ✐❧❧✉str❛t❡❞ ❜❡❧♦✇✿

❛♥❞

f (c) ≥

f (c − h) + f (c + h) . 2

❲❤❛t ❞♦❡s t❤✐s ❡①♣r❡ss✐♦♥ ❤❛✈❡ t♦ ❞♦ ✇✐t❤ t❤❡ ❞✐✛❡r❡♥❝❡s ♦r t❤❡ ❞❡r✐✈❛t✐✈❡s❄ ▲❡t✬s r❡✲❛rr❛♥❣❡ t❤❡ t❡r♠s ✐♥ t❤❡ ✜rst ✐♥❡q✉❛❧✐t②✿ f (c + h) − 2f (c) + f (c − h) ≥ 0 .

❆ ❜✐t ♠♦r❡ ❛♥❞ ✇❡ s❡❡ t✇♦ ❞✐✛❡r❡♥❝❡s ✿ 

   f (c + h) − f (c) − f (c) − f (c − h) ≥ 0 .

■♥ ❢❛❝t✱ t❤✐s ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡s ✭♦r t❤❡ ❝❤❛♥❣❡ ♦❢ t❤❡ ❝❤❛♥❣❡ ❡t❝✳✮✦ ❲❤❡♥ t❤❡ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ ✐s s♠❛❧❧❡r t❤❛♥ t❤❡ ✜rst✱ ✇❡ ❤❛✈❡ ✐t ❝♦♥❝❛✈❡ ❞♦✇♥❀ ♦t❤❡r✇✐s❡✱ ✉♣✿

✺✳✺✳

❈♦♥❝❛✈✐t② ❛♥❞ t❤❡ s✐❣♥ ♦❢ t❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡

✹✵✻

■❢ ✇❡ ❛ss✉♠❡ t❤❡ ♠✐❞✲♣♦✐♥t s❡❝♦♥❞❛r② ♥♦❞❡s✱ t❤❡ ❝♦♥❝❛✈✐t② ❝♦♥❞✐t✐♦♥ ❛❜♦✈❡ ❜❡❝♦♠❡s✿

∆f (c + h/2) − ∆f (c − h/2) ≥ 0 . h > 0✱

■t ♦♥❧② t❛❦❡s ❞✐✈✐s✐♦♥ ❜②

t✇✐❝❡✱ t♦ ❛rr✐✈❡ ❛t

t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t

❛t

x = c✿

∆f (c + h/2) ∆f (c − h/2) − h h ≥ 0, h ❚❤✐s ✐s ❥✉st t❤❡

s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t

✭❈❤❛♣t❡r ✹✮✿

∆2 f (c) ≥ 0 ∆x2 ❙♦✱ t❤❡r❡ ✐s ❛ ♠❛t❝❤✿

s✐❣♥

•

❲❡ ❦♥♦✇ t❤❛t t❤❡

•

❲❡ ❤❛✈❡ ❞✐s❝♦✈❡r❡❞ t❤❛t t❤❡

♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦r t❤❡ ❞❡r✐✈❛t✐✈❡ t❡❧❧s ✉s t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥

✐♥❝r❡❛s✐♥❣ ❛♥❞ ❞❡❝r❡❛s✐♥❣ ❜❡❤❛✈✐♦r✳

s✐❣♥

♦❢ t❤❡ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦r t❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡ t❡❧❧s ✉s t❤❡

❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ✉♣✇❛r❞ ❛♥❞ ❞♦✇♥✇❛r❞ ❝♦♥❝❛✈✐t②✳

❲❡ ♥❡❡❞ ❛ ♠♦r❡ ❣❡♥❡r❛❧ ♣♦✐♥t ♦❢ ✈✐❡✇✿ ❲❤❛t ✐❢ t❤❡ ♣♦✐♥ts ❛r❡♥✬t ❡q✉❛❧❧② s♣❛❝❡❞❄ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ t❤r❡❡ ❝♦♥s❡❝✉t✐✈❡ ♣♦✐♥ts

❲❡ st✐❧❧ ♥❡❡❞ t♦ ❡①♣r❡ss

C

✐♥ t❡r♠s ♦❢

A

A, B, C

❛♥❞

♦♥ t❤❡

x✲❛①✐s✿

B✿ C = αA + βB ,

❢♦r s♦♠❡ ♣❛✐r ♦❢ ♥✉♠❜❡rs ❚❤❡ ♥✉♠❜❡rs

α

❛♥❞

β

α, β ≥ 0

✇✐t❤

α + β = 1✳

■♥ ♦t❤❡r ✇♦r❞s✱

C

✐s ❛

✇❡✐❣❤t❡❞ ❛✈❡r❛❣❡

♦❢

A

❛♥❞

B✳

❛r❡ t❤❡ ✏✇❡✐❣❤ts✑✿

❚♦ ❧♦♦❦ ❛t t❤✐s ❞✐✛❡r❡♥t❧②✱ t❤❡s❡ t✇♦ ♥✉♠❜❡rs

α

❛♥❞

β

❣✐✈❡ ✉s t❤❡ r❡❧❛t✐✈❡ ♣♦s✐t✐♦♥ ♦❢

C

✇✐t❤✐♥ t❤❡ ✐♥t❡r✈❛❧

[A, B]✳ ◆♦✇ t❤❡

y ✬s✳

❈♦♥s✐❞❡r t❤❡ ♣♦✐♥t✿

X = αf (A) + βf (B) . ❖♥❝❡ ❛❣❛✐♥✱ t❤✐s ✐s ❛ ♣♦s✐t✐♦♥ ♦❢

X

✇❡✐❣❤t❡❞ ❛✈❡r❛❣❡

f (A) ❛♥❞ f (B)✳ ❚❤❡ ❜② f (A) ❛♥❞ f (B)✿

♦❢

✇✐t❤✐♥ t❤❡ ✐♥t❡r✈❛❧ ❢♦r♠❡❞

♥✉♠❜❡rs

α

❛♥❞

β

❛❧s♦ ❣✐✈❡ ✉s t❤❡ r❡❧❛t✐✈❡

✺✳✺✳

❈♦♥❝❛✈✐t② ❛♥❞ t❤❡ s✐❣♥ ♦❢ t❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡

✹✵✼

❊①❡r❝✐s❡ ✺✳✺✳✶ ❙❤♦✇ t❤❛t ❢♦r ❛ ❧✐♥❡❛r ❢✉♥❝t✐♦♥✱ t❤❡ ✇❡✐❣❤ts r❡♠❛✐♥ t❤❡ s❛♠❡✿ f (αA + βB) = αf (A) + βf (B) .

❲❤❛t ❝❛♥ ②♦✉ s❛② ❛❜♦✉t t❤❡ ❝♦♥✈❡rs❡❄ ❚❤❡ ❝♦♥❝❛✈✐t② ✐s ❞❡t❡r♠✐♥❡❞ ❜② ✇❤❡t❤❡r f (C) ✐s ❜❡❧♦✇ ♦r ❛❜♦✈❡ X ✱ ❛s ❢♦❧❧♦✇s✿

❉❡✜♥✐t✐♦♥ ✺✳✺✳✷✿ ❝♦♥❝❛✈❡ ♦♥ ♣❛rt✐t✐♦♥ ❙✉♣♣♦s❡ t❤r❡❡ ❝♦♥s❡❝✉t✐✈❡ ♥♦❞❡s A, B, C ♦❢ ❛ ♣❛rt✐t✐♦♥ s❛t✐s❢② C = αA + βB ,

✇✐t❤ α ≥ 0 ❛♥❞ β ≥ 0 ❛♥❞ α + β = 1✳ ❋♦r ❛ ❢✉♥❝t✐♦♥ f ❞❡✜♥❡❞ ♦♥ t❤❡s❡ ♥♦❞❡s✱ ✇❡ ❞❡✜♥❡✿ ✶✳ ❚❤❡ ❢✉♥❝t✐♦♥ f ✐s ❝❛❧❧❡❞ ❝♦♥❝❛✈❡ ✉♣ ❛t B ✇❤❡♥ f (C) ≤ αf (A) + βf (B) .

✷✳ ❚❤❡ ❢✉♥❝t✐♦♥ f ✐s ❝❛❧❧❡❞ ❝♦♥❝❛✈❡

❞♦✇♥ ❛t B ✇❤❡♥

f (C) ≥ αf (A) + βf (B) .

❲❤❡♥ α = β = 1/2✱ t❤❡ ❞❡✜♥✐t✐♦♥ ♣r♦❞✉❝❡s t❤❡ ✐♥❡q✉❛❧✐t✐❡s ✐♥ t❤❡ ❛♥❛❧②s✐s ✇❡ ♣r❡s❡♥t❡❞ ✐♥ t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤❡ s❡❝t✐♦♥✳ ❲❡ ✇✐❧❧ ♦❝❝❛s✐♦♥❛❧❧② ✉s❡ t❤❡ ♥♦t❛t✐♦♥✿

❈♦♥❝❛✈✐t② • ⌣ ❝♦♥❝❛✈❡ ✉♣ • ⌢ ❝♦♥❝❛✈❡ ❞♦✇♥

❆❜♦✈❡ ✇❡ ♣r♦✈❡❞ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r t❤❡ ❝❛s❡ ♦❢ ❡q✉❛❧❧② ❞✐str✐❜✉t❡❞ ♥♦❞❡s✿

❚❤❡♦r❡♠ ✺✳✺✳✸✿ ❈♦♥❝❛✈✐t② ♦♥ P❛rt✐t✐♦♥ ❙✉♣♣♦s❡ ❛ ❢✉♥❝t✐♦♥ f ✐s ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ ✐♥t❡r✈❛❧ [a, b]✳ ❚❤❡♥ ✇❡ ❤❛✈❡✿ ✶✳ ❚❤❡ ❢✉♥❝t✐♦♥ f ✐s ❝♦♥❝❛✈❡ ✉♣ ♦♥ [a, b] ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✐s ♥♦♥✲♥❡❣❛t✐✈❡✳ ✷✳ ❚❤❡ ❢✉♥❝t✐♦♥ f ✐s ❝♦♥❝❛✈❡ ❞♦✇♥ ♦♥ [a, b] ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✐s ♥♦♥✲♣♦s✐t✐✈❡✳

✺✳✺✳

❈♦♥❝❛✈✐t② ❛♥❞ t❤❡ s✐❣♥ ♦❢ t❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡

✹✵✽

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿

∆2 f ≥0 ∆x2 ∆2 f 2. f ⌢ ⇐⇒ ≤0 ∆x2

1. f ⌣ ⇐⇒

Pr♦♦❢✳

❙✉♣♣♦s❡ t❤r❡❡ ❝♦♥s❡❝✉t✐✈❡ ♥♦❞❡s xi−1 , xi , xi+1 ♦❢ ❛ ♣❛rt✐t✐♦♥ s❛t✐s❢② xi = αxi−1 + βxi+1 ,

❛♥❞ f (xi ) ≤ αf (xi−1 ) + βf (xi+1 ) ,

❢♦r s♦♠❡ ♣❛✐r α ≥ 0 ❛♥❞ β ≥ 0 ✇✐t❤ α + β = 1✳

▲❡t✬s ✜rst ✜♥❞ α ❛♥❞ β ✳ ❈♦♥s✐❞❡r t❤❡ ✜rst ❡q✉❛t✐♦♥ ♦❢ t❤❡ ❞❡✜♥✐t✐♦♥ r❡✇r✐tt❡♥✿ xi = α(xi − ∆xi−1 ) + β(xi + ∆xi ) .

❈❛♥❝❡❧❧❛t✐♦♥ ♣r♦❞✉❝❡s t❤❡ ❢♦❧❧♦✇✐♥❣✿ 0 = α(−∆xi−1 ) + β(∆xi ) .

❚❤❡r❡❢♦r❡✱ α=

∆xi−1 ∆xi , β= . ∆xi−1 + ∆xi ∆xi−1 + ∆xi

❚❤❡ ❝♦♥❝❛✈✐t② ✐s ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ s✐❣♥ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡①♣r❡ss✐♦♥✿ αf (xi−1 ) + βf (xi+1 ) − f (xi ) ≥ 0 .

❲❤❛t ✐s ✐ts ♠❡❛♥✐♥❣❄ ❲❡ s✉❜st✐t✉t❡ α ❛♥❞ β ❛♥❞ ♦✉r ❡①♣r❡ss✐♦♥ ❜❡❝♦♠❡s✿ ∆xi ∆xi−1 f (xi−1 ) + f (xi+1 ) − f (xi ) ≥ 0 . ∆xi−1 + ∆xi ∆xi−1 + ∆xi

▲❡t✬s r❡❛rr❛♥❣❡ t❤❡ t❡r♠s✿ ∆xi f (xi−1 ) + ∆xi−1 f (xi+1 ) − (∆xi−1 + ∆xi )f (xi ) ≥ 0 ,

❛♥❞ ❢❛❝t♦r✿



∆xi−1 f (xi+1 ) − f (xi ) − ∆xk f (xi ) − f (xi−1 ) ≥ 0 .

■♥ ♣❛r❡♥t❤❡s❡s ✇❡ s❡❡ t❤❡ t✇♦ ❞✐✛❡r❡♥❝❡s ♦❢ f ❡✈❛❧✉❛t❡❞ ❛t t❤❡ t✇♦ ❛❞❥❛❝❡♥t ✐♥t❡r✈❛❧s [xi−1 , xi ], [xi , xi+1 ]✳ ❚♦ s❡❡ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts✱ ❧❡t✬s ❞✐✈✐❞❡ t❤✐s ❜② ∆xi−1 ∆xi ✿ ∆f ∆f f (xi+1 ) − f (xi ) f (xi ) − f (xi−1 ) − ≥ 0 ♦r (ci ) − (ci−1 ) ≥ 0 . ∆xi ∆xi−1 ∆x ∆x

■t ♦♥❧② t❛❦❡s ❞✐✈✐s✐♦♥ ❜② ∆ci t♦ ❛rr✐✈❡ ❛t t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✿ ∆2 f (xi ) ≥ 0 . ∆x2

✺✳✺✳

❈♦♥❝❛✈✐t② ❛♥❞ t❤❡ s✐❣♥ ♦❢ t❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡

✹✵✾

❊①❛♠♣❧❡ ✺✳✺✳✹✿ s♣r❡❛❞s❤❡❡t ❚❤❡ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ✇✐t❤ ❛ s♣r❡❛❞s❤❡❡t✳ ❘❡❝❛❧❧ ✭❈❤❛♣t❡r ✹✮ t❤❛t ✐t ❛❧s♦ r❡♣r❡s❡♥ts t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ✇❤❡♥ t❤❡ ♦r✐❣✐♥❛❧ ❢✉♥❝t✐♦♥ r❡♣r❡s❡♥ts t❤❡ ❧♦❝❛t✐♦♥✿

❚❤❡ tr❛♥s✐t✐♦♥s ❢r♦♠

◆♦✇✱ t♦ t❤❡

f

t♦

∆f ∆x

❛♥❞ ❢r♦♠

∆f ∆x

t♦

∆2 f ∆x2

❛r❡ ✐♠♣❧❡♠❡♥t❡❞ ✇✐t❤ t❤❡ s❛♠❡ ❢♦r♠✉❧❛✳

❝♦♥t✐♥✉♦✉s ❝❛s❡ ✿

❆ ❢✉♥❝t✐♦♥ ✐s ❝♦♥❝❛✈❡ ✇❤❡♥ ❛♥② ♦❢ ✐ts s❛♠♣❧✐♥❣s ✐s✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ s❡❣♠❡♥ts ♦❢ t❤❡ s❡❝❛♥t ❧✐♥❡s ✭✏❝♦r❞s✑✮ ❧✐❡ ❡✐t❤❡r ❛❜♦✈❡ ♦r ❜❡❧♦✇ t❤❡ ❣r❛♣❤✿

❆❧❧ ♦❢ t❤❡ ❝♦r❞s✱ ♥♦t ❥✉st t❤❡ ♦♥❡s t❤❛t ❝♦♥♥❡❝t t❤❡ ❡♥❞s✦ ❲❡ ❤❛✈❡ t♦ ❝❛r❡❢✉❧❧② ♣❤r❛s❡ t❤❡s❡ r❡q✉✐r❡♠❡♥ts✿

❉❡✜♥✐t✐♦♥ ✺✳✺✳✺✿ ❝♦♥❝❛✈✐t② ♦♥ ✐♥t❡r✈❛❧ ✶✳ ■❢ ❢♦r ❡❛❝❤

α + β = 1✱

x

✐♥

I✱

❡❛❝❤

k, h > 0✱

❛♥❞ ❛♥② ♣❛✐r ♦❢ ♣♦s✐t✐✈❡

α

❛♥❞

β

✇✐t❤

✇❡ ❤❛✈❡✿

f (x) ≤ αf (x − k) + βf (x + h) , ✇✐t❤✐♥ t❤❡ ✐♥t❡r✈❛❧

I✱

t❤❡♥ t❤❡ ❢✉♥❝t✐♦♥

f

I✳

✐s ❝❛❧❧❡❞

❝♦♥❝❛✈❡ ✉♣ ♦♥ ✐♥t❡r✈❛❧

✷✳ ❲❤❡♥ t❤❡ ♦♣♣♦s✐t❡ ✐♥❡q✉❛❧✐t② ✐s s❛t✐s✜❡❞✿

f (x) ≥ αf (x − k) + βf (x + h) , t❤❡ ❢✉♥❝t✐♦♥

f

✐s ❝❛❧❧❡❞

❝♦♥❝❛✈❡ ❞♦✇♥ ♦♥ ✐♥t❡r✈❛❧ I ✳

✺✳✺✳

❈♦♥❝❛✈✐t② ❛♥❞ t❤❡ s✐❣♥ ♦❢ t❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡ ❚❤❡ ❢✉♥❝t✐♦♥ ✐s

✹✶✵

str✐❝t❧② ❝♦♥❝❛✈❡ ✭✉♣ ♦r ❞♦✇♥✮ ✐❢ t❤❡ ✐♥❡q✉❛❧✐t② ✐s str✐❝t✳

❊①❡r❝✐s❡ ✺✳✺✳✻

■s ❛ ❧✐♥❡❛r ❢✉♥❝t✐♦♥ ❝♦♥❝❛✈❡ ✉♣ ♦r ❞♦✇♥❄ ❚❤❡ ❞❡r✐✈❛t✐✈❡ ♠❛❦❡s ❝♦♠♣❧❡① t❤✐♥❣s ❧♦♦❦ s✐♠♣❧❡✿ ❚❤❡♦r❡♠ ✺✳✺✳✼✿ ❈♦♥❝❛✈✐t② ❚❤❡♦r❡♠

❙✉♣♣♦s❡ ❛ ❢✉♥❝t✐♦♥ f ✐s t✇✐❝❡ ❞✐✛❡r❡♥t✐❛❜❧❡ ♦♥ ❛♥ ♦♣❡♥ ✐♥t❡r✈❛❧ I ✳ ❚❤❡♥✱ ✇❡ ❤❛✈❡✿ ✶✳ ■❢ f ′′ ≥ 0 ♦♥ I ✱ t❤❡♥ f ✐s ❝♦♥❝❛✈❡ ✉♣ ♦♥ I ✳ ✷✳ ■❢ f ′′ ≤ 0 ♦♥ I ✱ t❤❡♥ f ❝♦♥❝❛✈❡ ❞♦✇♥ ♦♥ I ✳ Pr♦♦❢✳

■❢

f ′′ ≥ 0

I✱

♦♥

t❤❡♥ ❜② t❤❡

▼♦♥♦t♦♥✐❝✐t② ❚❤❡♦r❡♠✱ f ′ ✐s ✐♥❝r❡❛s✐♥❣✳

❚❤❡r❡❢♦r❡✱

f ′ (s) ≤ f ′ (t)

❢♦r ❛❧❧

s 0

=⇒ f ⌣

g ′′ (x) < 0

❢♦r

1 =⇒ x 1 = − 2 ❢♦r x 6= 0 =⇒ x 1 = 3 ❢♦r x 6= 0 2x x < 0 =⇒ g ⌢ ♦♥ (−∞, 0)

g ′′ (x) > 0

❢♦r

x > 0 =⇒ g ⌣

(2) g(x)

=

g ′ (x) g ′′ (x)

(3) h(x)

♦♥

(0, ∞)

= ex =⇒

h′ (x)

= ex =⇒

h′′ (x)

= ex

h′′ (x) > 0

=⇒ h ⌣

❯s✐♥❣ t❤✐s ❞❛t❛ ❝♦♠❜✐♥❡❞ ✇✐t❤ t❤❡ ✐♥❝r❡❛s✐♥❣✴❞❡❝r❡❛s✐♥❣ ❜❡❤❛✈✐♦r ❡st❛❜❧✐s❤❡❞ ❡❛r❧✐❡r✱ ✇❡ ❝❛♥ s❦❡t❝❤ ❜② ❤❛♥❞ t❤❡ ❣r❛♣❤s ♦❢ t❤❡s❡ t❤r❡❡ ❢✉♥❝t✐♦♥s✿

❊①❡r❝✐s❡ ✺✳✺✳✶✵ P♦✐♥t ♦✉t ✇❤❡r❡ t❤❡ ❣r❛♣❤s ❛r❡ ✐♥❛❞❡q✉❛t❡✳

◆♦✇✱ t❤❡ ♣♦✐♥ts ✇❤❡r❡ t❤❡ ❜❡❤❛✈✐♦r ❝❤❛♥❣❡s ❛r❡ ❡s♣❡❝✐❛❧❧② ✐♠♣♦rt❛♥t✿

✺✳✺✳ ❈♦♥❝❛✈✐t② ❛♥❞ t❤❡ s✐❣♥ ♦❢ t❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡

✹✶✷

❉❡✜♥✐t✐♦♥ ✺✳✺✳✶✶✿ ✐♥✢❡❝t✐♦♥ ♣♦✐♥t ❚❤❡ ♣♦✐♥ts ✇❤❡r❡ t❤❡ ❢✉♥❝t✐♦♥ ❝❤❛♥❣❡s ✐ts ❝♦♥❝❛✈✐t② ❛r❡ ❝❛❧❧❡❞ ✐♥✢❡❝t✐♦♥ ♣♦✐♥ts ❀ ✐✳❡✳✱ t❤❡s❡ ❛r❡ s✉❝❤ ♣♦✐♥ts c t❤❛t t❤❡ ❢✉♥❝t✐♦♥✬s ❝♦♥❝❛✈✐t② ♦♥ s♦♠❡ ♦♣❡♥ ✐♥t❡r✈❛❧ (a, c) ✐s ♦♣♣♦s✐t❡ ♦❢ t❤❡ ❝♦♥❝❛✈✐t② ♦♥ s♦♠❡ ♦♣❡♥ ✐♥t❡r✈❛❧ (c, b)✳

❲❛r♥✐♥❣✦ ❚❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ✜rst ❞❡r✐✈❛t✐✈❡ ❛t ✐♥✢❡❝t✐♦♥ ♣♦✐♥ts ❝❛♥ ❜❡ ❛r❜✐tr❛r②❀ ✐t ✐s 0 ❢♦r f (x) = x3 ❛t 0 ❛♥❞ ♥♦♥✲③❡r♦ ❢♦r t❤❡ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s ❜❡❧♦✇✳

❊①❛♠♣❧❡ ✺✳✺✳✶✷✿ s✐♥❡ ❛♥❞ ❝♦s✐♥❡ ■❢ ✇❡ ✉s❡ t❤❡ ❚r✐❣ ❋♦r♠✉❧❛s✿

t✇✐❝❡✱ ✇❡ ❤❛✈❡✿

(sin x)′ = cos x, (cos x)′ = − sin x. (sin x)′′ = − sin x, (cos x)′′ = − cos x.

❆s ✇❡ ❦♥♦✇✱ t❤❡ t✇♦ ❢✉♥❝t✐♦♥s ❛❧t❡r♥❛t❡ ✕ ♣❡r✐♦❞✐❝❛❧❧② ❡✈❡r② π ✕ ❜❡t✇❡❡♥ ♣♦s✐t✐✈❡ ❛♥❞ ♥❡❣❛t✐✈❡ ✈❛❧✉❡s✿

+ 0 − + 0 − 0 + 0− ❚❤❡r❡❢♦r❡✱ t❤❡ t✇♦ ❢✉♥❝t✐♦♥s ❛❧t❡r♥❛t❡ ✕ ♣❡r✐♦❞✐❝❛❧❧② ❡✈❡r② π ✕ ❜❡t✇❡❡♥ ❝♦♥❝❛✈❡ ✉♣ ❛♥❞ ❝♦♥❝❛✈❡ ❞♦✇♥ ❜❡❤❛✈✐♦r✿ ⌢ ⌢ ⌢ ր ց ր ց ր ց ⌣ ⌣ ⌣ ❚❤❡s❡ ❛r❡ t❤❡ ✐♥✢❡❝t✐♦♥ ♣♦✐♥ts ♦❢ t❤❡ ❢✉♥❝t✐♦♥✳ ❚❤❡r❡ ❛r❡ ♣❛r❛❧❧❡❧s ✇✐t❤ ♦✉r ❛❝t✐✈✐t✐❡s ❡❛r❧✐❡r ✐♥ t❤✐s ❝❤❛♣t❡r✿

• ❲❤❡♥ ✉s✐♥❣ t❤❡ ▼♦♥♦t♦♥✐❝✐t② ❚❤❡♦r❡♠✱ ✇❡ ❝♦♠♣❛r❡ t❤❡ s❤❛♣❡s ♦❢ t❤❡ ♣❛t❝❤❡s ♦❢ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ f ✱ ց ♦r ր✱ t♦ t❤❡ s✐❣♥s ♦❢ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ✜rst ❞❡r✐✈❛t✐✈❡ f ′ ✱ + ♦r −✳

• ▼❡❛♥✇❤✐❧❡✱ ✇❤❡♥ ✉s✐♥❣ t❤❡ ❈♦♥❝❛✈✐t② ❚❤❡♦r❡♠✱ ✇❡ ❝♦♠♣❛r❡ t❤❡ s❤❛♣❡s ♦❢ t❤❡ ♣❛t❝❤❡s ♦❢ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ f ✱ ⌣ ♦r ⌢✱ t♦ t❤❡ s✐❣♥s ♦❢ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡ f ′′ ✱ + ♦r −✳

❚❤❡s❡ ♦✉t❝♦♠❡s ❛r❡ s❤♦✇♥ ❜❡❧♦✇✿

✺✳✺✳

❈♦♥❝❛✈✐t② ❛♥❞ t❤❡ s✐❣♥ ♦❢ t❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡

✹✶✸

❖♥❡ ❝❛♥ s❡❡ ❤♦✇ ✇❡ ✉s❡ ❛ ❤✐❣❤❡r ❧❡✈❡❧ ♦❢ ❛♥❛❧②s✐s ✐♥ ❝♦♠♣❛r✐s♦♥ t♦ t❤❡ ✜rst ❞❡r✐✈❛t✐✈❡✿

❚❤❡ ✇✐❣❣❧❡s ❛r❡ ❣♦♥❡✦ ❲❛r♥✐♥❣✦

❆s t❤❡ s❡❝♦♥❞ ❞✐❛❣r❛♠ ✐♥❞✐❝❛t❡✱ ♠♦♥♦t♦♥✐❝✐t② ❛♥❞ ❝♦♥❝❛✈✐t② ❛r❡ t✇♦ ✐♥❞❡♣❡♥❞❡♥t ❝❤❛r❛❝t❡r✐st✐❝s ♦❢ ❛ ❢✉♥❝t✐♦♥❀ ♦♥❡ s❤♦✉❧❞ ♥❡✈❡r tr② t♦ ✜❣✉r❡ ♦✉t ♦♥❡ ❢r♦♠ t❤❡ ♦t❤❡r✳

❚❤❡ ❝♦♥❝❧✉s✐♦♥ ♦❢ t❤❡ ❈♦♥❝❛✈✐t② ❚❤❡♦r❡♠ ✐s s❡❡♥ ❢r♦♠ t❤❡ ❢♦❧❧♦✇✐♥❣✿ ❆s t❤❡ ✜rst ❞❡r✐✈❛t✐✈❡ r❡♣r❡s❡♥ts t❤❡ s❧♦♣❡s ♦❢ t❤❡ ❢✉♥❝t✐♦♥✱ t❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡ r❡♣r❡s❡♥ts t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢ t❤❡s❡ s❧♦♣❡s✳ ❆s ②♦✉ ❝❛♥ s❡❡✱ t❤❡ s❧♦♣❡s ✐♥❝r❡❛s❡ ✇❤❡♥ t❤❡ ❧✐♥❡s ❛r❡ r♦t❛t❡❞ ✐♥ t❤❡ ❝♦✉♥t❡r✲❝❧♦❝❦✇✐s❡ ❞✐r❡❝t✐♦♥✿

❚❤❡r❡❢♦r❡✱ t❤❡ t❛♥❣❡♥t ❧✐♥❡s r♦t❛t❡ ❛s ❢♦❧❧♦✇s✿ • ❉❡❝r❡❛s✐♥❣ s❧♦♣❡s =⇒ t❛♥❣❡♥t ❧✐♥❡s r♦t❛t❡ ❝❧♦❝❦✇✐s❡✳

• ■♥❝r❡❛s✐♥❣ s❧♦♣❡s =⇒ t❛♥❣❡♥t ❧✐♥❡s r♦t❛t❡ ❝♦✉♥t❡r✲❝❧♦❝❦✇✐s❡✳

✺✳✺✳

❈♦♥❝❛✈✐t② ❛♥❞ t❤❡ s✐❣♥ ♦❢ t❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡

✹✶✹

❊①❛♠♣❧❡ ✺✳✺✳✶✸✿ ❝❧✐♠❜✐♥❣

❲❡ ❝❛♥ ❛♣♣r❡❝✐❛t❡ t❤✐s ✐❞❡❛ ✐❢ ✇❡ ✐♠❛❣✐♥❡ ❤♦✇ ✇❡ ❞r✐✈❡ ♦♥ t❤✐s r♦❛❞✿

❊✈❡♥ t❤♦✉❣❤ ✇❡ ❛r❡ ❝❧✐♠❜✐♥❣ ✐♥ ❡✐t❤❡r ❝❛s❡✱ ✇❤❛t ❤❛♣♣❡♥s t♦ t❤❡ ❜❡❛♠s ✐s ❞✐✛❡r❡♥t✿ • ❲❤❡♥ t❤❡ r♦❛❞ ✐s ❝♦♥❝❛✈❡ ❞♦✇♥✱ t❤❡ ❤❡❛❞❧✐❣❤ts ♣♦✐♥t ✉♣ ❛❜♦✈❡ t❤❡ r♦❛❞✳ ❚❤❡ ❞r✐✈❡r ♠✐❣❤t ❛♥t✐❝✐♣❛t❡ ❧❡ss ♥❡❡❞ ❢♦r ❣❛s✳ • ❲❤❡♥ t❤❡ r♦❛❞ ✐s ❝♦♥❝❛✈❡ ✉♣✱ t❤❡ ❤❡❛❞❧✐❣❤ts ♣♦✐♥t ❞♦✇♥ ✐♥t♦ t❤❡ r♦❛❞✳ ❚❤❡ ❞r✐✈❡r ♠✐❣❤t ❛♥t✐❝✐♣❛t❡ ♠♦r❡ ♥❡❡❞ ❢♦r ❣❛s✳ ❆❧❧ ❣r❛♣❤s ❛r❡ ♠❛❞❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡✐❣❤t ♣✐❡❝❡s✱ ❝❧❛ss✐✜❡❞ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ s✐❣♥ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ ❛♥❞ t❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡✿

❊①❛♠♣❧❡ ✺✳✺✳✶✹✿ ♣✐❡❝✐♥❣ t♦❣❡t❤❡r

❇❡❧♦✇✱ ✇❡ st❛rt ✇✐t❤ t❤❡ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t t❤❡ ♠♦♥♦t♦♥✐❝✐t② ❛♥❞ ❝♦♥❝❛✈✐t② ♦❢ ❛ ❢✉♥❝t✐♦♥ ♦♥ s❡✈❡r❛❧ ✐♥t❡r✈❛❧s✱ t❤❡♥ ♣✐❝❦ ❛♣♣r♦♣r✐❛t❡ ♣✐❡❝❡s ❢r♦♠ t❤❡ ❛❜♦✈❡ t❛❜❧❡✱ ❛♥❞ t❤❡♥ ❣❧✉❡ t❤❡♠ t♦❣❡t❤❡r t♦ ❢♦r♠ ❛ ❝♦♥t✐♥✉♦✉s ❝✉r✈❡✿

✺✳✻✳ ❉❡r✐✈❛t✐✈❡s ❛♥❞ ❡①tr❡♠❛

✹✶✺

❍❡r❡ ✐s ②❡t ❛♥♦t❤❡r ✇❛② t♦ ✐♥t❡r♣r❡t t❤❡ t❡r♠✐♥♦❧♦❣② ❛♥❞ t❤❡ ♥♦t❛t✐♦♥✿ ✏❢❡❡❧✐♥❣ ✉♣✑

✏❢❡❡❧✐♥❣ ❞♦✇♥✑

⌣ ¨

⌢ ¨

✺✳✻✳ ❉❡r✐✈❛t✐✈❡s ❛♥❞ ❡①tr❡♠❛

❘❡❝❛❧❧ ✇❤❛t t❤❡ ❋❡r♠❛t✬s ❚❤❡♦r❡♠ ❞♦❡s ❛♥❞ ❞♦❡s ♥♦t s❛② ❛❜♦✉t ❛ ❢✉♥❝t✐♦♥ ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t

x=c

✐s

❛ ❧♦❝❛❧ ♠❛①✴♠✐♥

=⇒ 6⇐=

x = c✿

f ′ (c) = 0

❲❤❡♥ ❝❛♥ ✇❡ r❡✈❡rs❡ t❤❡ ❛rr♦✇❄ ❲❤❡♥

f ′ (c) = 0✱

t❤❡r❡ ❛r❡ t❤❡s❡ t❤r❡❡ ♣♦ss✐❜✐❧✐t✐❡s ❢♦r

max, min,

c✿ ♥❡✐t❤❡r.

❆t t❤❡✐r s✐♠♣❧❡st✱ t❤❡② ❧♦♦❦ ❧✐❦❡ t❤✐s✿

■t ✐s ♣♦ss✐❜❧❡ t♦ t❡❧❧ ♦♥❡ ❢r♦♠ ❛♥♦t❤❡r ❜② ❧♦♦❦✐♥❣ ❛t t❤❡ ❞❡r✐✈❛t✐✈❡ ✐♥ t❤❡ ✈✐❝✐♥✐t② ♦❢ t❤❡ ♣♦✐♥t✳

✺✳✻✳

❉❡r✐✈❛t✐✈❡s ❛♥❞ ❡①tr❡♠❛

✹✶✻

❊①❛♠♣❧❡ ✺✳✻✳✶✿ ❝✉❜✐❝

❈♦♥s✐❞❡r ❛❣❛✐♥✿

f (x) = x3 − 3x .

❲❡ ❦♥♦✇ t❤❛t f ′ (x) = 3x2 − 3 = 0 ❢♦r x = ±1 ❛♥❞ ♥♦♥❡ ♦t❤❡rs✳ ❚❤❡ ❞❡r✐✈❛t✐✈❡ ✐s 0✱ s♦ t❤❡s❡ ❡①tr❡♠❡ ♣♦✐♥ts✳

♠❛② ❜❡

❍♦✇ ❞♦ ✇❡ ✜♥❞ ♦✉t❄ ❲❡ ❧♦♦❦ ❛t t❤❡ s✐❣♥s ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ ✿ • f ′ > 0 ❛♥❞ f ր t♦ t❤❡ ❧❡❢t ♦❢ −1 ✳ • f ′ < 0 ❛♥❞ f ց t♦ t❤❡ r✐❣❤t ♦❢ −1✳ ❚❤✐s ❝❛♥ ♦♥❧② ❤❛♣♣❡♥ ✇❤❡♥ x = −1 ✐s ❛ ❧♦❝❛❧ ♠❛①✳ ❚❤❡ ♦♣♣♦s✐t❡ ❢♦r x = 1✳ ❲❡ ❝♦♥✜r♠ ✇✐t❤ ❛ ♣❧♦t✿

❲❡ ❤❛✈❡ ❧❡❛r♥❡❞ t❤❛t ✇❡ ❝❛♥ ❝❧❛ss✐❢② ❝r✐t✐❝❛❧ ♣♦✐♥ts ❜② ❧♦♦❦✐♥❣ ❛t t❤❡ ❝❤❛♥❣❡ րց max

♦❢ ♠♦♥♦t♦♥✐❝✐t② ✿

ցր min

❋✉rt❤❡r♠♦r❡✱ t❤❡ ❢✉♥❝t✐♦♥✬s ♠♦♥♦t♦♥✐❝✐t② ✐s ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ s✐❣♥ ♦❢ ✐ts ✜rst ❞❡r✐✈❛t✐✈❡✿

❲❡ s❡❡ t❤❡

▼♦♥♦t♦♥✐❝✐t② ❚❤❡♦r❡♠ ♦♥ t❤❡ ❧❡❢t ❛♥❞ ♦✉r ❝♦♥❝❧✉s✐♦♥ ♦♥ t❤❡ r✐❣❤t✳

❚❤❡ s✉♠♠❛r② r❡s✉❧t ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✿ ❚❤❡♦r❡♠ ✺✳✻✳✷✿ ❋✐rst ❉❡r✐✈❛t✐✈❡ ❚❡st

■❢ t❤❡ ❞❡r✐✈❛t✐✈❡ ❝❤❛♥❣❡s ✐ts s✐❣♥ ❛t ❛ ♣♦✐♥t✱ t❤❡ ♣♦✐♥t ✐s ❛♥ ❡①tr❡♠✉♠✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✳ ❙✉♣♣♦s❡ f ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ♦♥ ❛♥ ♦♣❡♥ ✐♥t❡r✈❛❧ I t❤❛t ❝♦♥t❛✐♥s ♣♦✐♥t x = c✳ ❚❤❡♥✱ ✇❡ ❤❛✈❡✿ ✶✳ ■❢ f ′ (x) ≥ 0 ❢♦r ❛❧❧ x < c ❆◆❉ f ′ (x) ≤ 0 ❢♦r ❛❧❧ x > c ✇✐t❤✐♥ I ✱ t❤❡♥ c ✐s ❛ ❧♦❝❛❧ ♠❛①✐♠✉♠ ♣♦✐♥t✳ ✷✳ ■❢ f ′ (x) ≤ 0 ❢♦r ❛❧❧ x < c ❆◆❉ f ′ (x) ≥ 0 ❢♦r ❛❧❧ x > c ✇✐t❤✐♥ I ✱ t❤❡♥ c ✐s ❛ ❧♦❝❛❧ ♠✐♥✐♠✉♠ ♣♦✐♥t✳

✺✳✻✳

❉❡r✐✈❛t✐✈❡s ❛♥❞ ❡①tr❡♠❛

✹✶✼

✸✳ ■❢

f ′ (x) ≤ 0

❢♦r ❛❧❧

x

✐♥

I ❖❘ f ′ (x) ≥ 0

❢♦r ❛❧❧

x

✐♥

I✱

t❤❡♥

c

✐s ♥❡✐t❤❡r ❛

❧♦❝❛❧ ♠❛①✐♠✉♠ ♥♦r ♠✐♥✐♠✉♠ ♣♦✐♥t✳

Pr♦♦❢✳

❙✉♣♣♦s❡ I = (a, b)✳ ■❢ f ′ (x) ≥ 0 ❢♦r ❛❧❧ a < x < c✱ t❤❡♥ ❜② t❤❡ ▼♦♥♦t♦♥✐❝✐t② ❚❤❡♦r❡♠✱ f (x) ≤ f (c) ❢♦r ❛❧❧ a < x < c✳ ■❢ f ′ (x) ≤ 0 ❢♦r ❛❧❧ c < x < b✱ t❤❡♥ ❜② t❤❡ ▼♦♥♦t♦♥✐❝✐t② ❚❤❡♦r❡♠✱ f (x) ≤ f (c) ❢♦r ❛❧❧ c < x < b✳ ❚❤✉s✱ f (x) ≤ f (c) ❢♦r ❛❧❧ a < x < b, x 6= c✳ ❚❤❡♥✱ c ✐s ❛ ❧♦❝❛❧ ♠❛①✐♠✉♠✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿ • ■❢ f ′ (x) ❝❤❛♥❣❡s ✐ts s✐❣♥ ❛t x = c ❢r♦♠ + t♦ −✱ t❤❡♥ c ✐s ❛ ❧♦❝❛❧ ♠❛①✐♠✉♠ ♣♦✐♥t✳ • ■❢ f ′ (x) ❝❤❛♥❣❡s ✐ts s✐❣♥ ❛t x = c ❢r♦♠ − t♦ +✱ t❤❡♥ c ✐s ❛ ❧♦❝❛❧ ♠✐♥✐♠✉♠ ♣♦✐♥t✳

❚♦ ❝❧❛ss✐❢② ❝r✐t✐❝❛❧ ♣♦✐♥ts✱ ❛❣❛✐♥✱ ✇✐❧❧ r❡q✉✐r❡ ❛ s✐❣♥ ❛♥❛❧②s✐s ✭s❡❡♥ ✐♥ ❱♦❧✉♠❡ ✶✱ ❈❤❛♣t❡r ✶P❈✲✹✮ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡✳ ■t ✇❛s ❞♦♥❡ ✐♥ t❤❡ ❧❛st ❡①❛♠♣❧❡ ❢♦r ❛ ♣♦❧②♥♦♠✐❛❧✳ ■♥ ❝❛s❡ ♦❢ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥✱ ✇❡ ❤❛✈❡ t♦ t❛❦❡ ✐♥t♦ ❛❝❝♦✉♥t t❤❡ ♣♦ss✐❜✐❧✐t② t❤❛t ❛ ❝❤❛♥❣❡ ♦❢ s✐❣♥ ❤❛♣♣❡♥s ❛❝r♦ss ❛ ✈❡rt✐❝❛❧ ❛s②♠♣t♦t❡✿

✳ ❊①❛♠♣❧❡ ✺✳✻✳✸✿ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥

▲❡t✬s ❛♥❛❧②③❡ t❤✐s ❢✉♥❝t✐♦♥✿ f (x) = √

❋✐rst✱ t❤❡ ❞♦♠❛✐♥ ✐s ❛❧❧ x ❡①❝❡♣t ± 3✳

x3 . x2 − 3

◆❡①t✱ ✇❡ ❞✐✛❡r❡♥t✐❛t❡✿ ′

f (x) = = = =



′ x3 x2 − 3 3x2 (x2 − 3) − x3 2x ❚❤✐s ✇✐❧❧ ❤❛✈❡ t♦ ❜❡ s✉❜❥❡❝t❡❞ t♦ s✐❣♥ ❛♥❛❧②s✐s✳ (x2 − 3)2 x4 − 9x2 ❲❡ ♥❡❡❞ t♦ ❢❛❝t♦r ✐t✦ (x2 − 3)2 x2 (x − 3)(x + 3) √ √ . (x − 3)2 (x + 3)2

◆❡①t✱ t❤❡ ❝r✐t✐❝❛❧ ♣♦✐♥ts ❛r❡ t❤❡ ♦♥❡s ✇❤❡r❡ ♦♥❡ ♦❢ t❤❡s❡ t✇♦ t❤✐♥❣s ❤❛♣♣❡♥s✿ • ❚❤❡ ❞❡r✐✈❛t✐✈❡ ✐s ③❡r♦✱ ✐✳❡✳✱ t❤❡ ♥✉♠❡r❛t♦r ✐s ③❡r♦✳ ❲❡ r❡❛❞ t❤❡s❡ ❢r♦♠ ✐ts ❢❛❝t♦rs ✭❛s s❡❡♥ ✐♥ ❱♦❧✉♠❡ ✶✱ ❈❤❛♣t❡r ✶P❈✲✹✮✿ x = 0, 3, −3 .

• ❚❤❡ ❞❡r✐✈❛t✐✈❡ ✐s ✉♥❞❡✜♥❡❞✱ ✐✳❡✳✱ t❤❡ ❞❡♥♦♠✐♥❛t♦r ✐s ③❡r♦✳ ❲❡ r❡❛❞ t❤❡s❡ ❢r♦♠ ✐ts ❢❛❝t♦rs✿ √ √ x = − 3, 3 .

✺✳✻✳

❉❡r✐✈❛t✐✈❡s ❛♥❞ ❡①tr❡♠❛

✹✶✽

❆❝❝♦r❞✐♥❣ t♦ t❤❡ ■♥t❡r♠❡❞✐❛t❡ ❞❡r✐✈❛t✐✈❡ ❝❤❛♥❣❡ ✐ts s✐❣♥✳

❱❛❧✉❡ ❚❤❡♦r❡♠✱

❛t t❤❡s❡ ♣♦✐♥ts ❛♥❞ ❛t t❤❡s❡ ♣♦✐♥ts ♦♥❧② ♠❛② t❤❡

❲❡ ♥♦✇ ❧✐st ❛❧❧ t❤❡ ❢❛❝t♦rs✳ ❚❤❡② ❛r❡ s✐♠♣❧❡ ❡♥♦✉❣❤ ❢♦r ✉s t♦ ❞❡t❡r♠✐♥❡ ✇❤❡t❤❡r ❛♥❞ ✇❤❡r❡ t❤❡② ❝❤❛♥❣❡ t❤❡✐r s✐❣♥s✿ ❢❛❝t♦rs

x2 x−3 x + 3√ (x − √3)2 (x + 3)2

s✐❣♥s

❞♦♠❛✐♥

x= f′ f

+ − − + + ··· + ր

+ + + + − − − − 0 + + + + + + + + + 0 + • ··· ◦ ··· √ −3 − 3 0 ·

−

ց

✳✳ ✳ ✳✳ ✳ ✳✳ ✳ ✳✳ ✳

−

ց

0 + + + − − − − + + + + + + 0 + + + + + • · · · √◦ · · · 0 3 0 ·

−

ց

✳✳ ✳ ✳✳ ✳ ✳✳ ✳ ✳✳ ✳

− ց

+ + 0 + + + + + + + • ··· → x 3 0

·

+ ր

❚❤❡♥ ✇❡ ❣♦ ✈❡rt✐❝❛❧❧② ❛♥❞ ❞❡t❡r♠✐♥❡ t❤❡ s✐❣♥ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ ✉s✐♥❣✿ + · − = −✱ ❡t❝✳ ❚❤❡ ✐♥❝r❡❛s✐♥❣ ❛♥❞ ❞❡❝r❡❛s✐♥❣ ❜❡❤❛✈✐♦r ♦❢ f ✐s t❤❡♥ ❞❡r✐✈❡❞✳ ❚❤❡ ❡①tr❡♠❛ ❛r❡ ✈✐s✉❛❧❧② ❝❧❛ss✐✜❡❞✳ ❲❡ ❛❧s♦ ❞❡t❡❝t t❤❡ t✇♦ ✈❡rt✐❝❛❧ ❛s②♠♣t♦t❡s✳ ❚❤❡ r❡s✉❧t✐♥❣ ❞✐❛❣r❛♠ ❝❛♥ s❡r✈❡ ❛s ❛ ❣✉✐❞❡ ❢♦r ❛ r♦✉❣❤ s❦❡t❝❤ ♦❢ t❤❡ ❣r❛♣❤✿

❚❤✐s ❞❛t❛ ❝♦❧❧❡❝t❡❞ ✐♥ t❤❡ t❛❜❧❡ ✐s s✉✣❝✐❡♥t ❢♦r ✉s t♦ ♣❧♦t ❜② ❤❛♥❞ ❛ ❜❡tt❡r ❣r❛♣❤✿

❊✈❡♥ t❤❡♥✱ ❡✈❡r② ♣❛rt ♦❢ t❤❡ ❣r❛♣❤ ♠❛② ❧✐❡ ❤✐❣❤❡r ♦r ❧♦✇❡r t❤❛♥ t❤❡s❡✳ ❋✉rt❤❡r♠♦r❡✱ ✇❡ ❝❛♥✬t ❣✉❛r❛♥t❡❡ t❤✐s ❝♦♥❝❛✈✐t② ✇✐t❤♦✉t t❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡✦ ❊①❛♠♣❧❡ ✺✳✻✳✹✿ ❝♦♥✈❡rs❡

❚❤❡ ❝♦♥✈❡rs❡ ♦❢ t❤❡ t❤❡♦r❡♠ ✐s ❛s ❢♦❧❧♦✇s✿ ? ❆ ♣♦✐♥t c ✐s max ♦r min ♦❢ f =⇒ f ′ ❝❤❛♥❣❡s ✐ts s✐❣♥ ❛t x = c✳ ■t ❢❛✐❧s ❛s t❤❡ ❢♦❧❧♦✇✐♥❣ ❡①❛♠♣❧❡ s❤♦✇s✿

✺✳✻✳

❉❡r✐✈❛t✐✈❡s ❛♥❞ ❡①tr❡♠❛

✹✶✾

❊①❡r❝✐s❡ ✺✳✻✳✺

❉❡✈✐s❡ ❛ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❛❜♦✈❡ ❢✉♥❝t✐♦♥ ❛♥❞ s❤♦✇ t❤❛t t❤❡ ❝♦♥✈❡rs❡ ♦❢ t❤❡ t❤❡♦r❡♠ ❢❛✐❧s✳ ❲❤❛t ❞♦ ②♦✉ ♥♦t✐❝❡ ❛❜♦✉t f ′ ❄ ❲❡ ❝❛♥ ❝❧❛ss✐❢② t❤❡ ❝r✐t✐❝❛❧ ♣♦✐♥ts ✇✐t❤ t❤❡ ❤❡❧♣ ♦❢ ❥✉st t❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡ ❛t t❤❛t ♣♦✐♥t✿

❚❤❡♦r❡♠ ✺✳✻✳✻✿ ❙❡❝♦♥❞ ❉❡r✐✈❛t✐✈❡ ❚❡st

❙✉♣♣♦s❡ f ✐s t✇✐❝❡ ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t x = c ❛♥❞ s✉♣♣♦s❡ f ′ (c) = 0✳ ❚❤❡♥✱ ✇❡ ❤❛✈❡✿ • ■❢ f ′′ (c) ≤ 0✱ t❤❡♥ c ✐s ❛ ❧♦❝❛❧ ♠❛① ♣♦✐♥t✳ • ■❢ f ′′ (c) ≥ 0✱ t❤❡♥ c ✐s ❛ ❧♦❝❛❧ ♠✐♥ ♣♦✐♥t✳ ❲❛r♥✐♥❣✦ ■❢

f ′′ (c) = 0✱

t❤❡♥ t❤❡ t❡st ❢❛✐❧s✳

❊①❡r❝✐s❡ ✺✳✻✳✼

❈♦♥s✐❞❡r t❤❡s❡ ❢✉♥❝t✐♦♥s✿ • f (x) = x4 • f (x) = x3

❊①❛♠♣❧❡ ✺✳✻✳✽✿ ❝✉❜✐❝✱ ❝♦♥t✐♥✉❡❞

▲❡t✬s ❝♦♥s✐❞❡r ❛❣❛✐♥✿

f (x) = x3 − 3x .

❲❡ ❝♦✉❧❞ ❤❛✈❡ s❛✈❡❞ s♦♠❡ t✐♠❡ ✇✐t❤ t❤❡ ❝r✐t✐❝❛❧ ♣♦✐♥ts ✇❡ ❢♦✉♥❞✱ x = ±1✱ ✐❢ ✇❡ ❤❛❞ ❢♦r❣♦♥❡ t❤❡ s✐❣♥ ❛♥❛❧②s✐s ♦❢ t❤❡ ✜rst ❞❡r✐✈❛t✐✈❡ f ′ ✳ ■♥st❡❛❞✱ ✇❡ t❛❦❡ t❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡✿ f ′ (x) = 3x2 − 3 =⇒ f ′′ (x) = 6x .

❍❡r❡ ✐s t❤❡ s✐❣♥ ❛♥❛❧②s✐s ♦❢ t❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡✿ • f ′′ (−1) = −6 < 0 =⇒ x = −1 ✐s ❛ ❧♦❝❛❧ ♠❛①✳ • f ′′ (1) = 6 > 0 =⇒ x = 1 ✐s ❛ ❧♦❝❛❧ ♠✐♥✳

✺✳✻✳

❉❡r✐✈❛t✐✈❡s ❛♥❞ ❡①tr❡♠❛

✹✷✵

❊①❡r❝✐s❡ ✺✳✻✳✾

❈❛rr② ♦✉t ❛ s✐♠✐❧❛r ❛♥❛❧②s✐s ❢♦r t❤❡ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥ ✐♥ t❤❡ ❧❛st ❡①❛♠♣❧❡✳ ❊①❛♠♣❧❡ ✺✳✻✳✶✵✿ ❛♥♦t❤❡r r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥

▲❡t f (x) =

❍❡r❡ ✇❡ ✇✐❧❧ s✐♠♣❧② ❝❧❛ss✐❢② t❤❡ ❝r✐t✐❝❛❧ ♣♦✐♥ts✳

x2 . x2 − 1

❲❡ ❝♦♠♣✉t❡ t❤❡ ✜rst ❞❡r✐✈❛t✐✈❡✿ 2x(x2 − 1) − x2 · 2x (x2 − 1)2 3 2x − 2x − 2x3 = (x2 − 1)2 2x . =− 2 (x − 1)2

f ′ (x) =

❲❡ ❤❛❞ t♦ s✐♠♣❧✐❢② ❛♥❞ ❢❛❝t♦r t❤✐s ❡①♣r❡ss✐♦♥ ✐♥ ♦r❞❡r t♦ ❢❛❝✐❧✐t❛t❡ t❤❡ ♥❡①t ❞✐✛❡r❡♥t✐❛t✐♦♥✳ ❲❡ ❝❛♥ ❛❧s♦ ❡❛s✐❧② ♦❜s❡r✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿ f ′ (x) = 0 ⇐⇒ x = 0 .

❚❤✐s ✐s t❤❡ ♦♥❧② ❝r✐t✐❝❛❧ ♣♦✐♥t✳ ❲❡ ❝♦♠♣✉t❡ t❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡✿

f ′′ (x) = −

2(x2 − 1)2 − 2x · 2(x2 − 1)2x . (x2 − 1)4

◆♦ ♥❡❡❞ t♦ s✐♠♣❧✐❢②✦ ❲❡ ❥✉st ♥❡❡❞ ✐ts s✐❣♥ ❛t t❤❡ ❝r✐t✐❝❛❧ ♣♦✐♥t✿

2(02 − 1)2 − 2 · 0 · 2(02 − 1)2 · 0 (02 − 1)4 = −2 < 0 .

f ′′ (0) = −

❚❤✐s ✐s ❛ ♠❛①✐♠✉♠✦ ❊①❛♠♣❧❡ ✺✳✻✳✶✶✿ ❛♥♦t❤❡r r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥✱ ❝♦♥t✐♥✉❡❞

❲❡ ❝♦♥t✐♥✉❡ ✇✐t❤ f (x) =

❲❡ ♥♦✇ ❧♦♦❦ ❢♦r ❛ ♠♦r❡ ❝♦♠♣❧❡t❡ ♣✐❝t✉r❡✳

x2 . x2 − 1

❲❡ st❛rt ❛t t❤❡ ❜♦tt♦♠✿ • ❉♦♠❛✐♥ ✐s ❛❧❧ r❡❛❧s ❡①❝❡♣t x = ±1 ✳ • ❱❡rt✐❝❛❧ ❛s②♠♣t♦t❡s✿ x = 1, x = −1 ✳ • ❍♦r✐③♦♥t❛❧ ❛s②♠♣t♦t❡✿ y = 1 ✳

◆♦✇ t❤❡ ❞❡r✐✈❛t✐✈❡s✳ ❚❤✐s t✐♠❡ ✇❡ ❤❛✈❡ t♦ s✐♠♣❧✐❢② f ′′ ✐♥ ♦r❞❡r t♦ ❢❛❝t♦r✿ 2(x2 − 1)2 − 2x · 2(x2 − 1)2x (x2 − 1)4 2 2(x − 1)(x2 − 1 − 4x2 ) =− (x2 − 1)4 2(x − 1)(x + 1)(3x2 + 1) = . (x2 − 1)4

f ′′ (x) = −

✺✳✻✳

❉❡r✐✈❛t✐✈❡s ❛♥❞ ❡①tr❡♠❛

✹✷✶

❲❡ ♥♦✇ ♥❡❡❞ t♦ ✜♥❞ t❤❡ s✐❣♥s ♦❢ t❤❡ ❢❛❝t♦rs ❛♥❞ t❤❡♥ t❤❡ s✐❣♥s ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡s✳ ❲❡ t❛❦❡ ♥♦t❡ ♦❢ t❤❡ ❞♦♠❛✐♥ ❛♥❞ ♦♥❧② ❧✐st t❤❡ ❢❛❝t♦rs t❤❛t ♠❛② ❝❤❛♥❣❡ s✐❣♥ ✭❱♦❧✉♠❡ ✶✱ ❈❤❛♣t❡r ✶P❈✲✹✮✿ ❢❛❝t♦rs

s✐❣♥s

−x

+

f′

+

f x−1 x+1

ր − −

f ′′ f

❞♦♠❛✐♥ x= f

+

⌣ ··· ր

✳✳ ✳ ✳✳ ✳ ✳✳ ✳ − 0

✳✳ ✳ ✳✳ ✳

+

0

+

0

− −

✳✳ ✳ ✳✳ ✳ ✳✳ ✳

− −

ր − +

− +

ց ց ❆❝❝♦r❞✐♥❣ t♦ t❤❡ ▼♦♥♦t♦♥✐❝✐t② ❚❤❡♦r❡♠✳ − 0 + + + +

−

−

−

✳✳ ✳ ✳✳ ✳

+

⌢ ⌢ ⌢ ⌣ ◦ ··· • ··· ◦ ··· −1 0 1

✳✳ ✳ ✳✳ ✳ ✳✳ ✳

·

ր ⌢ ց

✳✳ ✳ ✳✳ ✳ ✳✳ ✳

❆❝❝♦r❞✐♥❣ t♦ t❤❡ ❈♦♥❝❛✈✐t② ❚❤❡♦r❡♠✳ →x

ց

❚❤❡ ❧❛st ❞✐❛❣r❛♠ ❝❛♥ s❡r✈❡ ❛s ❛ ❣✉✐❞❡ ❢♦r ❛ r♦✉❣❤ s❦❡t❝❤ ♦❢ t❤❡ ❣r❛♣❤✿

❚❤✐s ❞❛t❛ ✐s s✉✣❝✐❡♥t ❢♦r ✉s t♦ ♣❧♦t ❜② ❤❛♥❞ ❛ ❜❡tt❡r ❣r❛♣❤✿

❊①❡r❝✐s❡ ✺✳✻✳✶✷

❙♦❧✈❡ ❛ r❡✈❡rs❡❞ ♣r♦❜❧❡♠✿ ❙❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥ f t❤❛t s❛t✐s✜❡s ❛❧❧ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣✿ • f ′ > 0 ♦♥ (−∞, 0) ❛♥❞ (1, 2)❀ • f ′ < 0 ♦♥ (0, 1) ❛♥❞ (2, ∞)❀ • f ′′ > 0 ♦♥ (−∞, −1) ❛♥❞ (2, ∞)❀ • f ′′ < 0 ♦♥ (−1, 2)✳

✺✳✼✳

❆♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥✿ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ✇❤❛t ❢✉♥❝t✐♦♥❄

✹✷✷

✺✳✼✳ ❆♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥✿ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ✇❤❛t ❢✉♥❝t✐♦♥❄ ❇❡❢♦r❡ t❤❡

▼❡❛♥ ❱❛❧✉❡ ❚❤❡♦r❡♠✱

✇❡ ❤❛✈❡ ♦♥❧② ❜❡❡♥ ❛❜❧❡ t♦ ✜♥❞ ❢❛❝ts ❛❜♦✉t t❤❡ ❞❡r✐✈❛t✐✈❡ ❢r♦♠ t❤❡ ❢❛❝ts

❛❜♦✉t t❤❡ ❢✉♥❝t✐♦♥✳ ❚❤✐s ✐s ❛ s❤♦rt ❧✐st ♦❢ ❢❛♠✐❧✐❛r ❢❛❝ts ✭❈❤❛♣t❡r ✸✮✿ ✐♥❢♦ ❛❜♦✉t

f

0 f

✐s ❝♦♥st❛♥t

1 f

✐s ❧✐♥❡❛r

2 f

✐s q✉❛❞r❛t✐❝

✐♥❢♦ ❛❜♦✉t f ′ ✐s ③❡r♦✳

=⇒ ? ⇐= =⇒ f ′ ? ⇐= =⇒ f ′ ? ⇐=

f′

✐s ❝♦♥st❛♥t✳

0

✐s ❧✐♥❡❛r✳

1

❚❤❡ ❛rr♦✇s r❡♣r❡s❡♥t ❞✐✛❡r❡♥t✐❛t✐♦♥✱ t❤❡ r❡✈❡rs❡ ❛rr♦✇s r❡♣r❡s❡♥t ✇❤❛t ✇❡ ✇✐❧❧ ❝❛❧❧

❛♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥✳

❇✉t ❛r❡ t❤❡s❡ ❛rr♦✇s r❡✈❡rs✐❜❧❡❄ ■❢ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ✐s ③❡r♦✱ ❞♦❡s ✐t ♠❡❛♥ t❤❛t t❤❡ ❢✉♥❝t✐♦♥ ✐s ❝♦♥st❛♥t❄ ❨❡s✱ ✇❡ ♣r♦✈❡❞ ✐t ❡❛r❧✐❡r ✐♥ t❤✐s ❝❤❛♣t❡r✳ ❚❤✐s t✐♠❡✱ ✇❡ ❤❛✈❡ ❛ t♦♦❧ t♦ ♣r♦✈❡ t❤❡s❡ ❢❛❝ts✱ t❤❡ ♦♥

[a, b]

❛♥❞ ❞✐✛❡r❡♥t✐❛❜❧❡ ♦♥

(a, b)✱

▼❡❛♥ ❱❛❧✉❡ ❚❤❡♦r❡♠ ✿

■❢

f

✐s ❝♦♥t✐♥✉♦✉s

t❤❡♥

f (b) − f (a) = f ′ (c) , b−a ❢♦r s♦♠❡

c

(a, b).

✐♥

❚❤❡ t❤❡♦r❡♠ ✇✐❧❧ ❤❡❧♣ ✉s ✇✐t❤ t❤❡ ❢❛❝ts ❛❜♦✉t t❤❡ ❢✉♥❝t✐♦♥ ❞❡r✐✈❡❞ ❢r♦♠ t❤❡ ❢❛❝ts ❛❜♦✉t ✐ts ❞❡r✐✈❛t✐✈❡✳ ❲❡ st❛rt ♦✈❡r✳ ❈♦♥s✐❞❡r t❤✐s

◮

♦❜✈✐♦✉s

st❛t❡♠❡♥t ❛❜♦✉t ♠♦t✐♦♥✿

✏■❢ ♠② s♣❡❡❞ ✐s ③❡r♦✱ ■ ❛♠ st❛♥❞✐♥❣ st✐❧❧ ✭❛♥❞ ✈✐❝❡ ✈❡rs❛✮✳✑

■❢ ❛ ❢✉♥❝t✐♦♥

y = f (x)

r❡♣r❡s❡♥ts t❤❡ ♣♦s✐t✐♦♥✱ ✇❡ ❝❛♥ r❡st❛t❡ t❤✐s ♠❛t❤❡♠❛t✐❝❛❧❧②✳

Pr♦✈✐♥❣ t❤❡ ♠❛t❤❡♠❛t✐❝❛❧ ✈❡rs✐♦♥ ♦❢ t❤✐s st❛t❡♠❡♥t ✇✐❧❧ ❝♦♥✜r♠ t❤❛t ♦✉r t❤❡♦r② ♠❛t❝❤❡s t❤❡ r❡❛❧✐t② ❛♥❞ t❤❡ ❝♦♠♠♦♥ s❡♥s❡✳ ❲❡ ❢♦❧❧♦✇ t❤❡ s❛♠❡ t❤r❡❡ st❡♣s st❛rt✐♥❣ ✇✐t❤ t❤❡ ❞✐✛❡r❡♥❝❡s✿

❚❤❡♦r❡♠ ✺✳✼✳✶✿ ❈♦♥st❛♥❝② ✈s✳ ❩❡r♦ ❉✐✛❡r❡♥❝❡ ❆ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ ✐♥t❡r✈❛❧ [a, b] ❤❛s ❛ ③❡r♦ ❞✐✛❡r❡♥❝❡ ❢♦r ❛❧❧ s❡❝♦♥❞❛r② ♥♦❞❡s ✐♥ t❤❡ ♣❛rt✐t✐♦♥ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤✐s ❢✉♥❝t✐♦♥ ✐s ❝♦♥st❛♥t ♦✈❡r t❤❡ ♥♦❞❡s ♦❢ [a, b]✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡ ❢♦r ❛❧❧ s❡❝♦♥❞❛r② ♥♦❞❡s✿ ∆f = 0 ⇐⇒ f = ❝♦♥st❛♥t.

Pr♦♦❢✳ ∆f (ci ) = 0 =⇒ f (xi ) − f (xi−1 ) = 0 =⇒ f (xi ) = f (xi−1 ) .

✺✳✼✳ ❆♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥✿ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ✇❤❛t ❢✉♥❝t✐♦♥❄

✹✷✸

❚❤❡♦r❡♠ ✺✳✼✳✷✿ ❈♦♥st❛♥❝② ✈s✳ ❩❡r♦ ❉✐✛❡r❡♥❝❡ ◗✉♦t✐❡♥t ❆ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ ✐♥t❡r✈❛❧ [a, b] ❤❛s ❛ ③❡r♦ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❢♦r ❛❧❧ s❡❝♦♥❞❛r② ♥♦❞❡s ✐♥ t❤❡ ♣❛rt✐t✐♦♥ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤✐s ❢✉♥❝t✐♦♥ ✐s ❝♦♥st❛♥t ♦✈❡r t❤❡ ♥♦❞❡s ♦❢ [a, b]✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡ ❢♦r ❛❧❧ s❡❝♦♥❞❛r② ♥♦❞❡s✿ ∆f = 0 ⇐⇒ f = ❝♦♥st❛♥t . ∆x

Pr♦♦❢✳ ∆f (ci ) = 0 =⇒ ∆f (ci ) = 0. ∆x

❚❤❡ ♣r♦♦❢s ❤❛✈❡ ❜❡❡♥ s♦ ❡❛s② ❜❡❝❛✉s❡ ✇❡ ♦♥❧② ❤❛❞ t♦ ❝♦♥s✐❞❡r ❝♦♥s❡❝✉t✐✈❡ ♣♦✐♥ts ♦❢ t❤❡ ♣❛rt✐t✐♦♥✳ ❚❤❡r❡ ✐s ♥♦ s✉❝❤ t❤✐♥❣ ✇❤❡♥ t❤❡ ❢✉♥❝t✐♦♥ ✐s ❞❡✜♥❡❞ ♦♥ t❤❡ ✇❤♦❧❡ ✐♥t❡r✈❛❧✿

❚❤❡♦r❡♠ ✺✳✼✳✸✿ ❈♦♥st❛♥❝② ✈s✳ ❩❡r♦ ❉❡r✐✈❛t✐✈❡ ❆ ❞✐✛❡r❡♥t✐❛❜❧❡ ♦♥ ♦♣❡♥ ✐♥t❡r✈❛❧ I ❢✉♥❝t✐♦♥ ❤❛s ❛ ③❡r♦ ❞❡r✐✈❛t✐✈❡ ❢♦r ❛❧❧ x ✐♥ I ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤✐s ❢✉♥❝t✐♦♥ ✐s ❝♦♥st❛♥t ♦♥ I ✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡ ♦♥ I ✿ f ′ = 0 ⇐⇒ f = ❝♦♥st❛♥t.

Pr♦♦❢✳ ❚♦ ♣r♦✈❡ t❤❛t f ✐s ❝♦♥st❛♥t✱ ✐t s✉✣❝❡s t♦ s❤♦✇ t❤❛t f (a) = f (b) ,

❢♦r ❛❧❧ a, b ✐♥ I ✳ ❆ss✉♠❡ a < b ❛♥❞ ✉s❡ t❤❡ ▼❡❛♥ ❱❛❧✉❡ ❚❤❡♦r❡♠ ✇✐t❤ ✐♥t❡r✈❛❧ (a, b) ✐♥s✐❞❡ t❤❡ ✐♥t❡r✈❛❧ I✿ f (b) − f (a) = f ′ (c) , b−a

❢♦r s♦♠❡ c ✐♥ (a, b)✳ ❚❤✐s ✐s 0 ❜② ❛ss✉♠♣t✐♦♥✳ ❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡✿

❢♦r ❛❧❧ ♣❛✐rs a, b✳ ❍❡♥❝❡

f (b) − f (a) = 0, b−a f (b) − f (a) = 0 ,

✺✳✼✳

❆♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥✿ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ✇❤❛t ❢✉♥❝t✐♦♥❄

✹✷✹

♦r

f (a) = f (b) . ❚❤❡ ❝♦♥✈❡rs❡ ✇❛s ♣r♦✈❡♥ ✐♥ ❈❤❛♣t❡r ✹✳

◆♦t❡ t❤❛t t❤❡ ♣r♦♦❢ ♦❢ t❤❡ ▼♦♥♦t♦♥✐❝✐t② ❚❤❡♦r❡♠s ✐s ✐❞❡♥t✐❝❛❧ t♦ t❤❡ ♦♥❡ ❛❜♦✈❡ ✇✐t❤ ❡❛❝❤ ✏ =✑ r❡♣❧❛❝❡❞ ✇✐t❤ ✏ >✑✳

❊①❡r❝✐s❡ ✺✳✼✳✹ ❲❤❛t ✐❢

f′ = 0

♦♥ t❤❡ ✉♥✐♦♥ ♦❢ t✇♦ ♦♣❡♥ ✐♥t❡r✈❛❧s❄

❙✉♣♣♦s❡ ♥♦✇ t❤❛t t❤❡r❡ ❛r❡

t✇♦

r✉♥♥❡rs r✉♥♥✐♥❣ ✇✐t❤ t❤❡ s❛♠❡ s♣❡❡❞❀ ✇❤❛t ❝❛♥ ✇❡ s❛② ❛❜♦✉t t❤❡✐r ♠✉t✉❛❧

❧♦❝❛t✐♦♥s❄ ❚❤❡② ❛r❡ ♥♦t✱ ♦❢ ❝♦✉rs❡✱ st❛♥❞✐♥❣ st✐❧❧✱ ❜✉t t❤❡②

❛r❡ st✐❧❧ r❡❧❛t✐✈❡ t♦ ❡❛❝❤ ♦t❤❡r✦

❲❡ ❤❛✈❡ ❛ s❧✐❣❤t❧②

❧❡ss ♦❜✈✐♦✉s ❢❛❝t ❛❜♦✉t ♠♦t✐♦♥✿

◮

✏■❢ t✇♦ r✉♥♥❡rs r✉♥ ✇✐t❤ t❤❡ s❛♠❡ s♣❡❡❞✱ t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡♠ ✐s♥✬t ❝❤❛♥❣✐♥❣ ✭❛♥❞ ✈✐❝❡

✈❡rs❛✮✑✳ ■t✬s ❛s ✐❢ t❤❡② ❛r❡ ❤♦❧❞✐♥❣ t❤❡ t✇♦ ❡♥❞s ♦❢ ❛ ♣♦❧❡ ✇✐t❤♦✉t ♣✉❧❧✐♥❣ ♦r ♣✉s❤✐♥❣✳

❚❤❡ ❢❛❝t r❡♠❛✐♥s ✈❛❧✐❞ ❡✈❡♥ ✐❢ t❤❡② s♣❡❡❞ ✉♣ ❛♥❞ s❧♦✇ ❞♦✇♥ ❛❧❧ t❤❡ t✐♠❡✳ ❚❤❡② ♠♦✈❡ ❛s ✐❢ ❛ s✐♥❣❧❡ ❜♦❞②✿

❖♥❝❡ ❛❣❛✐♥✱ ❢♦r ❢✉♥❝t✐♦♥s

y = F (x)

❛♥❞

y = G(x)

r❡♣r❡s❡♥t✐♥❣ t❤❡✐r ♣♦s✐t✐♦♥s✱ ✇❡ ❝❛♥ r❡st❛t❡ t❤✐s ✐❞❡❛

♠❛t❤❡♠❛t✐❝❛❧❧② ✐♥ ♦r❞❡r t♦ ❝♦♥✜r♠ t❤❛t ♦✉r t❤❡♦r② ♠❛❦❡s s❡♥s❡✳ ❲❡ ❤❛✈❡ t♦ ❢♦❧❧♦✇ t❤❡ s❛♠❡ t❤r❡❡ st❡♣s st❛rt✐♥❣ ✇✐t❤ t❤❡ ❞✐✛❡r❡♥❝❡s✿

❚❤❡♦r❡♠ ✺✳✼✳✺✿ ❆♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥ ❢♦r ❉✐✛❡r❡♥❝❡s ❚✇♦ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ ✐♥t❡r✈❛❧ [a, b] ❤❛✈❡ t❤❡ s❛♠❡ ❞✐✛❡r❡♥❝❡s ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡②❞✐✛❡r ❜② ❛ ❝♦♥st❛♥t✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿ ∆F (c) = ∆G(c) ⇐⇒ F (x) − G(x) = ❝♦♥st❛♥t.

❚❤❡♦r❡♠ ✺✳✼✳✻✿ ❆♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥ ❢♦r ❉✐✛❡r❡♥t ◗✉♦t✐❡♥ts ❚✇♦ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ ✐♥t❡r✈❛❧ [a, b] ❤❛✈❡ t❤❡ s❛♠❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡② ❞✐✛❡r ❜② ❛ ❝♦♥st❛♥t✳

✺✳✼✳ ❆♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥✿ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ✇❤❛t ❢✉♥❝t✐♦♥❄

✹✷✺

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿ ∆G ∆F (c) = (c) ⇐⇒ F (x) − G(x) = ❝♦♥st❛♥t. ∆x ∆x

❖♥❝❡ ❛❣❛✐♥✱ t❤✐s ❤❛s ❜❡❡♥ ❡❛s② ❜❡❝❛✉s❡ ✇❡ ♦♥❧② ❤❛❞ t♦ ❝♦♥s✐❞❡r ❝♦♥s❡❝✉t✐✈❡ ♣♦✐♥ts ♦❢ t❤❡ ♣❛rt✐t✐♦♥✳ ❋♦r ❛ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ♦♥ t❤❡ ✇❤♦❧❡ ✐♥t❡r✈❛❧✱ ✇❡✬❞ ♥❡❡❞ t❤❡ ▼❡❛♥ ❱❛❧✉❡ ❚❤❡♦r❡♠✳ ❍❡r❡ ✐♥st❡❛❞✱ t❤❡ t❤❡♦r❡♠ ❛❜♦✉t t❤❡ ❝♦♥st❛♥❝② ♦❢ ❛ ❢✉♥❝t✐♦♥ ✇✐t❤ ❛ ③❡r♦ ❞❡r✐✈❛t✐✈❡ ✐s ✉s❡❞✿

❚❤❡♦r❡♠ ✺✳✼✳✼✿ ❆♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥ ❢♦r ❉❡r✐✈❛t✐✈❡s ❚✇♦ ❞✐✛❡r❡♥t✐❛❜❧❡ ♦♥ ♦♣❡♥ ✐♥t❡r✈❛❧ I ❢✉♥❝t✐♦♥s ❤❛✈❡ t❤❡ s❛♠❡ ❞❡r✐✈❛t✐✈❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡② ❞✐✛❡r ❜② ❛ ❝♦♥st❛♥t✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿ F ′ (x) = G′ (x) ⇐⇒ F (x) − G(x) = ❝♦♥st❛♥t.

Pr♦♦❢✳ ❉❡✜♥❡ ❚❤❡♥✱ ❜② t❤❡ ❙✉♠ ❘✉❧❡✱ ✇❡ ❤❛✈❡✿

h(x) = F (x) − G(x) .

h′ (x) = (F (x) − G(x))′ = F ′ (x) − G′ (x) = 0 ,

❢♦r ❛❧❧ x✳ ❚❤❡♥ h ✐s ❝♦♥st❛♥t✱ ❜② t❤❡ ❈♦♥st❛♥❝② ✈s✳ ❩❡r♦ ❉❡r✐✈❛t✐✈❡ ❚❤❡♦r❡♠✳ ❚❤❡ ❝♦♥✈❡rs❡ ✇❛s ♣r♦✈❡♥ ✐♥ ❈❤❛♣t❡r ✹✳ ■♥ ❛❞❞✐t✐♦♥ t♦ t❤❡ ♠♦t✐♦♥ ✐♥t❡r♣r❡t❛t✐♦♥✱ t❤❡r❡ ✐s ❛❧s♦ ♦♥❡ ✐♥ t❡r♠s ♦❢ ❣❡♦♠❡tr②✳ ❚❤❡ t❤❡♦r❡♠ s❛②s✿ ◮ ■❢ t❤❡ ❣r❛♣❤s ♦❢ y = F (x) ❛♥❞ y = G(x) ❤❛✈❡ ♣❛r❛❧❧❡❧ t❛♥❣❡♥t ❧✐♥❡s ❢♦r ❡✈❡r② ✈❛❧✉❡ ♦❢ x✱ t❤❡♥ t❤❡ ❣r❛♣❤ ♦❢ F ✐s ❛ ✈❡rt✐❝❛❧ s❤✐❢t ♦❢ t❤❡ ❣r❛♣❤ ♦❢ G ✭❛♥❞ ✈✐❝❡ ✈❡rs❛✮✳

❲❡ ❝❛♥ ✉♥❞❡rst❛♥❞ t❤✐s ✐❞❡❛ ✐❢ ✇❡ ✐♠❛❣✐♥❡ ❛ t✉♥♥❡❧ ❛♥❞ ❛ ♣❡rs♦♥ ✇❤♦s❡ ❤❡❛❞ ✐s t♦✉❝❤✐♥❣ t❤❡ ❝❡✐❧✐♥❣✳ ■❢ t❤❡ ❝❡✐❧✐♥❣ ✐s s❧♦♣❡❞ ❞♦✇♥✱ s❤♦✉❧❞ ❤❡ ❜❡ ❝♦♥❝❡r♥❡❞ ❛❜♦✉t ❤✐tt✐♥❣ ❤✐s ❤❡❛❞❄ ◆♦t ✐❢ t❤❡ ✢♦♦r ✐s s❧♦♣❡❞ ❞♦✇♥ ❛s ♠✉❝❤✦ ■♥ ♦t❤❡r ✇♦r❞s✱ ✐❢ t❤❡ s❧♦♣❡ ♦❢ t❤❡ t✉♥♥❡❧✬s t♦♣ ✐s ❡q✉❛❧ t♦ t❤❡ s❧♦♣❡ ♦❢ t❤❡ ❜♦tt♦♠ ❛t ❡✈❡r② ❧♦❝❛t✐♦♥✱ t❤❡♥ t❤❡ ❤❡✐❣❤t ♦❢ t❤❡ t✉♥♥❡❧ r❡♠❛✐♥s t❤❡ s❛♠❡ t❤r♦✉❣❤♦✉t ✐ts ❧❡♥❣t❤✳ ❚❤❡r❡ ❛r❡ ✐♥✜♥✐t❡❧② ♠❛♥② ❢✉♥❝t✐♦♥s ✇✐t❤ t❤❡ s❛♠❡ ❞❡r✐✈❛t✐✈❡ F ′ = f ✿

✺✳✽✳

❆♥t✐❞❡r✐✈❛t✐✈❡s

✹✷✻

❙♦✱ ❡✈❡♥ ✐❢ ✇❡ ❝❛♥ r❡❝♦✈❡r t❤❡ ❢✉♥❝t✐♦♥ ❞❡r✐✈❛t✐✈❡✱ s✉❝❤ ❛s

G = F +C

F

F ′ ✱ t❤❡r❡ ❛r❡ ♠❛♥② C ✳ ❆r❡ t❤❡r❡ ♦t❤❡rs❄

❢r♦♠ ✐ts ❞❡r✐✈❛t✐✈❡

❢♦r ❛♥② ❝♦♥st❛♥t r❡❛❧ ♥✉♠❜❡r

♦t❤❡rs ✇✐t❤ t❤❡ s❛♠❡ ◆♦t ❛❝❝♦r❞✐♥❣ t♦ t❤❡

t❤❡♦r❡♠✳

❲❛r♥✐♥❣✦ ■t✬s ♦♥❧② tr✉❡ ✇❤❡♥ t❤❡ ❞♦♠❛✐♥ ✐s ❛♥ ✐♥t❡r✈❛❧✳

❇❛s❡❞ ♦♥ t❤❡ t❤❡♦r❡♠✱ ✇❡ ❝❛♥ ♥♦✇ ✉♣❞❛t❡ t❤✐s ❧✐st ♦❢ s✐♠♣❧❡ ❜✉t ✐♠♣♦rt❛♥t ❢❛❝ts✿

✐♥❢♦ ❛❜♦✉t

0 f 1 f 2 f ... ❲❡ ♥♦t✐❝❡ ❛ ♣❛tt❡r♥✿

f

f ⇐⇒ ⇐⇒ ⇐⇒

✐s ❝♦♥st❛♥t✳ ✐s ❧✐♥❡❛r✳ ✐s q✉❛❞r❛t✐❝✳

✐s ❛ ♣♦❧②♥♦♠✐❛❧ ♦❢ ❞❡❣r❡❡

n

❲❡ ✉s❡ t❤❡ ❧❛st t✇♦ ❢❛❝ts t♦ ❥✉st✐❢② ♦✉r ❛♥❛❧②s✐s ♦❢

✐♥❢♦ f ′ ✐s f ′ ✐s f ′ ✐s

❛❜♦✉t

f′

③❡r♦✳ ❝♦♥st❛♥t✳ ❧✐♥❡❛r✳

✐❢ ❛♥❞ ♦♥❧② ✐❢

f′

0 1 ...

✐s ❛ ♣♦❧②♥♦♠✐❛❧ ♦❢ ❞❡❣r❡❡

❢r❡❡ ❢❛❧❧ ✿

n − 1✳

❋✉♥❝t✐♦♥s ♦❢ t✐♠❡ ❚❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ✐s ❝♦♥st❛♥t✳ ❚❤❡ ✈❡❧♦❝✐t② ✐s ❧✐♥❡❛r✳

=⇒ =⇒

❚❤❡ ❧♦❝❛t✐♦♥ ✐s q✉❛❞r❛t✐❝✳

❊①❡r❝✐s❡ ✺✳✼✳✽ ❉❡r✐✈❡ t❤❛t t❤❡ tr❛❥❡❝t♦r② ♦❢ ❛ ❜❛❧❧ t❤r♦✇♥ ✉♥❞❡r ❛♥ ❛♥❣❧❡ ✐s ❛ ♣❛r❛❜♦❧❛✳

✺✳✽✳ ❆♥t✐❞❡r✐✈❛t✐✈❡s

❊✈❡r② ❣♦♦❞ ♣r♦❜❧❡♠ ❤❛s ❛ ✏r❡✈❡rs❡❞✑ ❝♦✉♥t❡r♣❛rt✱ ✉s✉❛❧❧② ❤❛r❞❡r✳ ❚❤✐s ♦♥❡ ✐s ❢♦r ❞✐✛❡r❡♥t✐❛t✐♦♥✿

❉❡✜♥✐t✐♦♥ ✺✳✽✳✶✿ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦♥ ♣❛rt✐t✐♦♥ ❙✉♣♣♦s❡ ❛ ❢✉♥❝t✐♦♥ ✐♥t❡r✈❛❧

I✳

f

✐s ❞❡✜♥❡❞ ♦♥ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ ❛ ❝❧♦s❡❞

❚❤❡♥ ❛ ❢✉♥❝t✐♦♥

F

❞❡✜♥❡❞ ♦♥ t❤❡ ♥♦❞❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥ t❤❛t s❛t✐s✜❡s

✺✳✽✳

❆♥t✐❞❡r✐✈❛t✐✈❡s

✹✷✼

t❤❡ ❡q✉❛t✐♦♥✿

∆F (c) = f (c) ∆x c✱

❢♦r ❛❧❧ s❡❝♦♥❞❛r② ♥♦❞❡s

✐s ❝❛❧❧❡❞ ❛♥

❛♥t✐❞❡r✐✈❛t✐✈❡

♦❢

f✳

■♥ t❤❡ ♣r❡✈✐♦✉s ❝❤❛♣t❡rs✱ ✇❡ ❢♦✉♥❞ ❛ r❡❝✉rs✐✈❡ s♦❧✉t✐♦♥ ♦❢ t❤✐s ❡q✉❛t✐♦♥ ❜② s♦❧✈✐♥❣ t❤❡ ❡q✉❛t✐♦♥ ❢♦r t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❛s ✐♥ t❤❡ ❝❛s❡ ♦❢ ♣♦s✐t✐♦♥ ❢r♦♠ ✈❡❧♦❝✐t②✿

pn+1 − pn =⇒ pn+1 = pn + vn ∆t . ∆t

vn = ■t✬s ♣✉r❡ ❛❧❣❡❜r❛✦

❚❤❡ ❝♦♥t✐♥✉♦✉s ❝❛s❡ ✐s ❜② ❢❛r ♠♦r❡ ❝♦♠♣❧❡①✿

❉❡✜♥✐t✐♦♥ ✺✳✽✳✷✿ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦♥ ✐♥t❡r✈❛❧ ❙✉♣♣♦s❡ ❛ ❢✉♥❝t✐♦♥ ❢✉♥❝t✐♦♥

F

f

❞❡✜♥❡❞ ♦♥

✐s ❞❡✜♥❡❞ ♦♥ ❛♥ ♦♣❡♥ ✐♥t❡r✈❛❧

I

I✳

❚❤❡♥ ❛ ❞✐✛❡r❡♥t✐❛❜❧❡

t❤❛t s❛t✐s✜❡s t❤❡ ❡q✉❛t✐♦♥✿

dF (x) = f (x) dx ❢♦r ❛❧❧

x✱

✐s ❝❛❧❧❡❞ ❛♥

❛♥t✐❞❡r✐✈❛t✐✈❡

♦❢

f✳

❲❡ ✉s❡ ✏❛♥✑ ❜❡❝❛✉s❡ t❤❡r❡ ❛r❡ ♠❛♥② ❛♥t✐❞❡r✐✈❛t✐✈❡s ❢♦r ❡❛❝❤ ❢✉♥❝t✐♦♥✳ ❆s ✇❡ ❦♥♦✇ ❢r♦♠ t❤❡

❚❤❡♦r❡♠✱

✐❢

F

✐s ❛♥ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢

tr✉❡ ♦♥❧② ❢♦r ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ♦♥

f✱

t❤❡♥ s♦ ✐s

✐♥t❡r✈❛❧s✳

❲❡ ❝❛♥ t❤✐♥❦ ♦❢ t❤❡ ❞❡✜♥✐t✐♦♥ ❛s ❛♥

●✐✈❡♥

F + C✱

✇❤❡r❡

C

❆♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥

✐s ❛♥② ❝♦♥st❛♥t✱ ✇❤✐❧❡ t❤❡ ❝♦♥✈❡rs❡ ✐s

❡q✉❛t✐♦♥✱ ❛♥ ❡q✉❛t✐♦♥ ❢♦r ❢✉♥❝t✐♦♥s✿ f,

s♦❧✈❡ ❢♦r

F

●✐✈❡♥

∆F =f ∆x

f,

s♦❧✈❡ ❢♦r

F

dF =f dx

❊①❛♠♣❧❡ ✺✳✽✳✸✿ x2 ❋♦r ❡①❛♠♣❧❡✱ t❤❡ t❛s❦ ♠❛② ❜❡ t♦ ✜♥❞ ❛ ❢✉♥❝t✐♦♥

F

t❤❛t s❛t✐s✜❡s t❤❡ ❡q✉❛t✐♦♥✿

F ′ (x) = x2 . ❆s ✐t ✐s ♦❢t❡♥ t❤❡ ❝❛s❡ ✇✐t❤ ❡q✉❛t✐♦♥s✱ t❤❡r❡ ✐s ♥♦ ❢♦r♠✉❧❛ t❤❛t ❞✐r❡❝t❧② ❣✐✈❡s ❛ s♦❧✉t✐♦♥✳ ❚❤❡r❡ ✐s ♥♦

♣r✐♦r ❡①♣❡r✐❡♥❝❡s ✇✐t❤ ❞✐✛❡r❡♥t✐❛t✐♦♥

❧✐♠✐t ❡✐t❤❡r✳

❚❤❡ ✐♥✐t✐❛❧ ✐❞❡❛ ✐s t❤❡♥ t♦ tr② t♦ r❡❝❛❧❧ ✳ ❊✈❡♥ ✐❢ 2 ②♦✉✬✈❡ ♥❡✈❡r ✜♥✐s❤❡❞ ❞✐✛❡r❡♥t✐❛t✐♥❣ ❛ ❢✉♥❝t✐♦♥ ✇✐t❤ x ❛s t❤❡ ❛♥s✇❡r✱ ②♦✉ ♠✐❣❤t ❤❛✈❡ ❞♦♥❡ s♦♠❡t❤✐♥❣ ❝❧♦s❡✿

(x3 )′ = 3x2 . ❆❧♠♦st✦ ❚♦ ❣❡t r✐❞ ♦❢

3✱

tr②

t♦ ❞✐✈✐❞❡✿ ♠❛②❜❡

x3 /3❄

▲❡t✬s t❡st✿

(x3 /3)′ = (x3 )/3 = 3x2 /3 = x2 . ❆s ✐t ✐s ♦❢t❡♥ t❤❡ ❝❛s❡ ✇✐t❤ ❡q✉❛t✐♦♥s✱ tr✐❛❧✲❛♥❞✲❡rr♦r ♠✐❣❤t ❥✉st ✇♦r❦✦

✺✳✽✳

❆♥t✐❞❡r✐✈❛t✐✈❡s

❲❡ ❝❛♥ r❡st❛t❡ t❤❡

✹✷✽

❆♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥ ❚❤❡♦r❡♠

❛s ❢♦❧❧♦✇s✿

❈♦r♦❧❧❛r② ✺✳✽✳✹✿ ❙❡t ♦❢ ❆♥t✐❞❡r✐✈❛t✐✈❡s

❙✉♣♣♦s❡ F ✐s ❛♥② ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ f ❞❡✜♥❡❞ ♦♥ ❛♥ ♦♣❡♥ ✐♥t❡r✈❛❧✳ ❚❤❡♥ t❤❡ s❡t ♦❢ ❛❧❧ ♦❢ ✐ts ❛♥t✐❞❡r✐✈❛t✐✈❡s ✐s {F + C : C r❡❛❧ } . ❆s ✐t ✐s ♦❢t❡♥ t❤❡ ❝❛s❡ ✇✐t❤ ❡q✉❛t✐♦♥s✱ t❤❡r❡ s❡❡♠s t♦ ❜❡ ♠❛♥② ✭✐♥✜♥✐t❡❧② ♠❛♥②✮ s♦❧✉t✐♦♥s✳ ✐♠♣♦rt❛♥t ❝♦♥❝❧✉s✐♦♥ ✐s t❤❛t

✐t s✉✣❝❡s t♦ ✜♥❞ ❥✉st ♦♥❡ ❛♥t✐✲❞❡r✐✈❛t✐✈❡✦

❇✉t ❛ ✈❡r②

❲❛r♥✐♥❣✦

❚❤❡ ❢♦r♠✉❧❛ F + C ✇♦r❦s ♦♥❧② ✇❤❡♥ t❤❡ ❢✉♥❝t✐♦♥ ✐s ❞❡✜♥❡❞ ♦♥ ❛♥ ✐♥t❡r✈❛❧✳ ❚❤✐s ✐s ✇❤❛t t❤✐s s❡t ♦❢ ❢✉♥❝t✐♦♥s ✇♦✉❧❞ ❧♦♦❦ ❧✐❦❡✿

■t ✐s t❤❡ s♦❧✉t✐♦♥ s❡t ♦❢ t❤❡ ❡q✉❛t✐♦♥ ❛♥❞ ✐t ♠❛② ❜❡ ❝❛❧❧❡❞

❊①❡r❝✐s❡ ✺✳✽✳✺✿ ❋✐♥❞ ❛❧❧

F

t❤❡

❛♥t✐❞❡r✐✈❛t✐✈❡✳

x3

t❤❛t s❛t✐s❢② t❤❡ ❡q✉❛t✐♦♥✿

F ′ (x) = x3 . ❊①❡r❝✐s❡ ✺✳✽✳✻ ❙✉♣♣♦s❡ ❛ ❢✉♥❝t✐♦♥

f

✐s ❞❡✜♥❡❞ ♦♥ ❛♥ ♦♣❡♥ ✐♥t❡r✈❛❧

✶✳ ❚❤❡ ❣r❛♣❤s ♦❢ t✇♦ ❞✐✛❡r❡♥t ❛♥t✐❞❡r✐✈❛t✐✈❡s ♦❢ ✷✳ ❋♦r ❡✈❡r② ♣♦✐♥t

(x, y)

✇✐t❤

x

✇✐t❤✐♥

I✱

I ✳ Pr♦✈❡ t❤❛t t❤❡ f ♥❡✈❡r ✐♥t❡rs❡❝t✳

t❤❡r❡ ✐s ❛♥ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢

t❤r♦✉❣❤ ✐t✳

❚❤❡ ♣r♦❜❧❡♠ t❤❡♥ ❜❡❝♦♠❡s t❤❡ ♦♥❡ ♦❢ ✜♥❞✐♥❣ ❛ s✐♥❣❧❡ ❢✉♥❝t✐♦♥

•

❢r♦♠ ✐ts ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t

•

❢r♦♠ ✐ts ❞❡r✐✈❛t✐✈❡

∆F ✱ ∆x

❢♦❧❧♦✇✐♥❣✿

F✱

❡✐t❤❡r

♦r

dF ✳ dx

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ r❡❝♦♥str✉❝t t❤❡ ❢✉♥❝t✐♦♥ ❢r♦♠ ❛ ✏✜❡❧❞ ♦❢ s❧♦♣❡s✑✿

f

t❤❡ ❣r❛♣❤ ♦❢ ✇❤✐❝❤ ♣❛ss❡s

✺✳✽✳

❆♥t✐❞❡r✐✈❛t✐✈❡s

✹✷✾

❖♥❡ ❝❛♥ ✐♠❛❣✐♥❡ ❛ ✢♦✇✐♥❣ ❧✐q✉✐❞ ✇✐t❤ t❤❡ ❞✐r❡❝t✐♦♥ ❦♥♦✇♥ ❛t ❡✈❡r② ❧♦❝❛t✐♦♥✳ ❍♦✇ ❞♦ ✇❡ ✜♥❞ t❤❡ ♣❛t❤ ♦❢ ❛ ♣❛rt✐❝✉❧❛r ♣❛rt✐❝❧❡❄ ❚❤❡ ♣r♦❝❡ss ♦❢ r❡❝♦♥str✉❝t✐♥❣ ❛ ❢✉♥❝t✐♦♥✱ F ✱ ❢r♦♠ ✐ts ❞❡r✐✈❛t✐✈❡✱ f ✱ ✐s ❝❛❧❧❡❞ ❛♥t✐✲ ❞✐✛❡r❡♥t✐❛t✐♦♥✳ ❚❤❡ ❛♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥ ♣r♦❜❧❡♠ ❤❛s ❜❡❡♥ s♦❧✈❡❞ ♦♥ s❡✈❡r❛❧ ♦❝❝❛s✐♦♥s ❢♦r t❤❡ ❢♦r♠❡r✱ ❞✐s❝r❡t❡ ❝❛s❡ ✕ ✈❡❧♦❝✐t② ❢r♦♠ ❛❝❝❡❧❡r❛t✐♦♥ ❛♥❞ ❧♦❝❛t✐♦♥ ❢r♦♠ ✈❡❧♦❝✐t② ✕ ✈✐❛ t❤❡s❡ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛s✿

F (xn+1 ) = F (xn ) + f (cn )∆xn . ❋♦r ❡❛❝❤ ❧♦❝❛t✐♦♥✱ ✇❡ ❧♦♦❦ ✉♣ t❤❡ ✈❡❧♦❝✐t②✱ ✜♥❞ t❤❡ ♥❡①t ❧♦❝❛t✐♦♥✱ ❛♥❞ r❡♣❡❛t✿

■❢ t❤❡ ♥♦❞❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥ ❛r❡ ❝❧♦s❡ ❡♥♦✉❣❤ t♦ ❡❛❝❤ ♦t❤❡r✱ t❤❡s❡ ♣♦✐♥ts ❢♦r♠ ❝✉r✈❡s✿

❋♦r t❤❡ ❧❛tt❡r✱ ❝♦♥t✐♥✉♦✉s ❝❛s❡✱ t❤✐s ✐s ❛ ❝❤❛❧❧❡♥❣✐♥❣ ♣r♦❜❧❡♠✿ ❍♦✇ ❞♦❡s ♦♥❡ ♣❧♦t ❛ ❝✉r✈❡ t❤❛t ❢♦❧❧♦✇s t❤❡s❡ ✕ ✐♥✜♥✐t❡❧② ♠❛♥② ✕ t❛♥❣❡♥ts❄ ❚♦ ❜❡❣✐♥ ✇✐t❤✱ ✇❡ ❥✉st tr② t♦ r❡✈❡rs❡ ❞✐✛❡r❡♥t✐❛t✐♦♥✳ ❲❡ ✇✐❧❧ tr② t♦ ❝♦♥str✉❝t ❛ t❤❡♦r② ♦❢ ❛♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥ t❤❛t ♠❛t❝❤❡s ✕ t♦ t❤❡ ❞❡❣r❡❡ ♣♦ss✐❜❧❡ ✕ t❤❛t ♦❢ ❞✐✛❡r❡♥t✐❛t✐♦♥✳

✺✳✽✳

❆♥t✐❞❡r✐✈❛t✐✈❡s

❍❡r❡ ✐s ❛ s❤♦rt ❧✐st

✹✸✵

♦❢ ❞❡r✐✈❛t✐✈❡s ♦❢ ❢✉♥❝t✐♦♥s ✭❢♦r ❛❧❧ x ❢♦r ✇❤✐❝❤ t❤❡ ❢✉♥❝t✐♦♥ ✐s ❞✐✛❡r❡♥t✐❛❜❧❡✮✿ ❢✉♥❝t✐♦♥ −→ ❞❡r✐✈❛t✐✈❡ xr

ex

rxr−1 1 x ex

sin x

cos x

cos x

− sin x

ln x

❛♥t✐❞❡r✐✈❛t✐✈❡ ←− ❢✉♥❝t✐♦♥ ❚♦ ✜♥❞ ❛♥t✐❞❡r✐✈❛t✐✈❡s✱ r❡✈❡rs❡ t❤❡ ♦r❞❡r✿ ◮ ❘❡❛❞ ❡❛❝❤ ❧✐♥❡ ❢r♦♠ r✐❣❤t t♦ ❧❡❢t✦ ❊①❛♠♣❧❡ ✺✳✽✳✼✿

sin

❛♥❞

cos

❲❤❛t ✐s ❛♥ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢ cos x❄ ❲❡ ♥❡❡❞ t♦ s♦❧✈❡ ❢♦r F ✿ F ′ (x) = cos x .

❏✉st ✜♥❞ cos x ✐♥ t❤❡ r✐❣❤t ❝♦❧✉♠♥✳ ❚❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❢✉♥❝t✐♦♥ ♦♥ t❤❡ ❧❡❢t ✐s sin x✳ ❚❤❛t✬s t❤❡ ❛♥s✇❡r✿ F (x) = sin x✦ ❲❤❛t ✐s ❛♥ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢ sin x❄ ❙♦❧✈❡✿ F ′ (x) = sin x .

❏✉st ✜♥❞ sin x ✐♥ t❤❡ r✐❣❤t ❝♦❧✉♠♥✳ ■t✬s ♥♦t t❤❡r❡✳✳✳ ❜✉t − sin x ✐s✦ ❚❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❢✉♥❝t✐♦♥ ♦♥ t❤❡ ❧❡❢t ✐s cos x✳ ❚❤❡♥ ✭❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡✮ t❤❡ s♦❧✉t✐♦♥ ♠✉st ❜❡ − sin x✳ ■t ✐s t❤❛t s✐♠♣❧❡✦ ❲❡ ❥✉st ♠❛② ♥❡❡❞ s♦♠❡ t✇❡❛❦✐♥❣ t♦ ♠❛❦❡ t❤❡ ❢♦r♠✉❧❛s ❛❜♦✉t t♦ ❡♠❡r❣❡ ❛s ❡❛s② t♦ ❛♣♣❧② ❛s t❤❡ ♦r✐❣✐♥❛❧ ♦♥❡s✳ ❋♦r ❡①❛♠♣❧❡✱ ❧❡t✬s ✜♥❞ ❛♥t✐❞❡r✐✈❛t✐✈❡s ♦❢ xn ✳ ❯s❡ t❤❡ P♦✇❡r ❋♦r♠✉❧❛ ❢♦r ❞✐✛❡r❡♥t✐❛t✐♦♥ ✭t❤❡ ✜rst r♦✇✮✱ ❞✐✈✐❞❡ ❜② r✱ ❛♥❞ ❛♣♣❧② t❤❡ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ✿ r ′

(x ) = rx

r−1

1 =⇒ (xr )′ = xr−1 =⇒ r

❲❡ t❤❡♥ s✐♠♣❧✐❢② t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ❜② s❡tt✐♥❣ r − 1 = s✿ 

❊①❛♠♣❧❡ ✺✳✽✳✽✿

1 s+1 x s+1

′



1 r x r

′

= xs .

√ 3

❙♦❧✈❡ ❢♦r F ✿

F ′ (x) =

√ 3

x = x1/3 .

❲❡ ❝❤♦♦s❡ s = 1/3 ✐♥ t❤❡ ❢♦r♠✉❧❛✳ ❚❤❡♥✱ ✇❡ ❤❛✈❡✿ F (x) =

1 3 x1/3+1 = x4/3 + C . 1/3 + 1 4

= xr−1 .

✺✳✽✳

❆♥t✐❞❡r✐✈❛t✐✈❡s

✹✸✶

P♦✇❡r ❋♦r♠✉❧❛

❲❡ ♠❛❦❡ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ t❤❡ ❧❡❢t✲❤❛♥❞ s✐❞❡ ❛♥❞ ✇❡ ❤❛✈❡ t❤❡

❢♦r ❛♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥✳ ❲❡

❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿ ❆♥ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢ ❇✉t ✇❤❛t ✐❢

s = −1❄

xs

✐s

1 s+1 x ✱ s+1

♣r♦✈✐❞❡❞

s 6= −1✳

❚❤❡♥ ✇❡ r❡❛❞ t❤❡ ❛♥s✇❡r ❢r♦♠ t❤❡ ♥❡①t ❧✐♥❡ ✐♥ t❤❡ t❛❜❧❡✿ ❆♥ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢

x−1

✐s

ln |x|✳

❙♦✱ t❤❛t t❤❡ r✉❧❡ ✏t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ♣♦✇❡r ❢✉♥❝t✐♦♥ ✐s ❛ ♣♦✇❡r ❢✉♥❝t✐♦♥ ♦❢ ❞❡❣r❡❡ t❤❡

0✲♣♦✇❡r✱

1 ❧♦✇❡r✑

❤❛s ❛♥ ❡①❝❡♣t✐♦♥✱

❛♥❞ t❤❡ r✉❧❡ ✏t❤❡ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ♣♦✇❡r ❢✉♥❝t✐♦♥ ✐s ❛ ♣♦✇❡r ❢✉♥❝t✐♦♥ ♦❢ ❞❡❣r❡❡

1

❤✐❣❤❡r✑ ❤❛s

❛♥ ❡①❝❡♣t✐♦♥ t♦♦✳ ❚❛❦✐♥❣ t❤❡ r❡st ♦❢ t❤❡s❡ r♦✇s✱ ✇❡ ❤❛✈❡ ❛

❧✐st ♦❢ ❛♥t✐❞❡r✐✈❛t✐✈❡s

❢✉♥❝t✐♦♥

−→

xs

❛♥t✐❞❡r✐✈❛t✐✈❡

1 s+1 x , s+1

1 x

ln |x|

ex

ex

sin x

− cos x

cos x

sin x

❞❡r✐✈❛t✐✈❡

←−

♦❢ ❢✉♥❝t✐♦♥s✱ ♦♥ ♦♣❡♥ ✐♥t❡r✈❛❧s✿

s 6= −1

❢✉♥❝t✐♦♥

❊①❛♠♣❧❡ ✺✳✽✳✾✿ ❞♦♠❛✐♥s

❊❛❝❤ ❢♦r♠✉❧❛ ✐s ♦♥❧② ✈❛❧✐❞ ♦♥ ❛♥ ♦♣❡♥ ✐♥t❡r✈❛❧ ♦♥ ✇❤✐❝❤ t❤❡ ❛♥t✐❞❡r✐✈❛t✐✈❡ ✐s ❞❡✜♥❡❞✳ ❋♦r ❡①❛♠♣❧❡✱ ✇❡ ✐♥t❡r♣r❡t t❤❡ s❡❝♦♥❞ r♦✇ ❛s ❢♦❧❧♦✇s✿

• ln(x)

✐s ❛♥ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢

• ln(−x)

1 x

✐s ❛♥ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢

◆❡①t✱ ✇❡ ✇✐❧❧ ♥❡❡❞ t❤❡ ❋✐rst✱ ❝♦♥s✐❞❡r t❤❡

♦♥ t❤❡ ✐♥t❡r✈❛❧

1 x

(0, +∞)✱

♦♥ t❤❡ ✐♥t❡r✈❛❧

❛♥❞

(−∞, 0)✳

r✉❧❡s ♦❢ ❛♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥✳

❙✉♠ ❘✉❧❡ ❢♦r ❉❡r✐✈❛t✐✈❡s

✭❈❤❛♣t❡r ✹✮✿ ❚❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ s✉♠ ✐s t❤❡ s✉♠ ♦❢ t❤❡

❞❡r✐✈❛t✐✈❡s❀ ✐✳❡✳✱

(f + g)′ = f ′ + g ′ . ▲❡t✬s r❡❛❞ t❤❛t ❢♦r♠✉❧❛ ❢r♦♠ r✐❣❤t t♦ ❧❡❢t✿ ❚❤❡♦r❡♠ ✺✳✽✳✶✵✿ ❙✉♠ ❘✉❧❡ ❢♦r ❆♥t✐❞❡r✐✈❛t✐✈❡s ❆♥ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ s✉♠ ✐s t❤❡ s✉♠ ♦❢ t❤❡ ❛♥t✐❞❡r✐✈❛t✐✈❡s✳

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿ ✐❢

F G

✐s ❛♥ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢ ✐s ❛♥ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢

f g✱

t❤❡♥

F +G

✐s ❛♥ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢

f + g✳

❛♥❞

✺✳✽✳

❆♥t✐❞❡r✐✈❛t✐✈❡s

✹✸✷

Pr♦♦❢✳

❲❡ ❛♣♣❧② t❤❡

❙✉♠ ❘✉❧❡ ❋♦r ❉❡r✐✈❛t✐✈❡s t♦ ❝♦♥✜r♠✿ (F (x) + G(x))′ = F ′ (x) + G′ (x) = f (x) + g(x) .

❊①❡r❝✐s❡ ✺✳✽✳✶✶

❲❤❛t ❛❜♦✉t t❤❡ ❝♦♥✈❡rs❡❄ ❊①❛♠♣❧❡ ✺✳✽✳✶✷✿ s✉♠s

❙♦❧✈❡ ❢♦r F ✿

F ′ (x) = x2 + sin x .

❚❤❡ ❡q✉❛t✐♦♥ ✐s s♦❧✈❡❞ ❜② s♦❧✈✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ ❡q✉❛t✐♦♥s✿ ❢♦r G : G′ (x) = x2 ❛♥❞ ❢♦r H : H ′ (x) = sin x . ❚❤❡ s♦❧✉t✐♦♥s ❛r❡ ❢♦✉♥❞ ✐♥ t❤❡ t❛❜❧❡✿ 1 G(x) = x3 ❛♥❞ H(x) = − cos x . 3

❆❝❝♦r❞✐♥❣ t♦ t❤❡ t❤❡♦r❡♠✱ ✇❡ ❤❛✈❡ ❛♥ ❛♥s✇❡r✿ 1 F (x) = x3 − cos x . 3 ❊①❡r❝✐s❡ ✺✳✽✳✶✸

❯s✐♥❣ t❤✐s r✉❧❡✱ ✜♥❞ ❛♥ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢ ln x2 ✳ ❈♦♠♣❛r❡✿ ❚❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ s✉♠ ✐s t❤❡ s✉♠ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡s✳ ❚❤❡ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ s✉♠ ✐s t❤❡ s✉♠ ♦❢ t❤❡ ❛♥t✐❞❡r✐✈❛t✐✈❡s✳ ❲❛r♥✐♥❣✦ ❚❤❡ ♠✐s♠❛t❝❤ ❜❡t✇❡❡♥ ✏ ❛♥ ❛♥t✐❞❡r✐✈❛t✐✈❡✑ ❛♥❞ ✏ t❤❡ ❛♥t✐❞❡r✐✈❛t✐✈❡✑ ✐s♥✬t ❛❝❝✐❞❡♥t❛❧✳

❙✐♠✐❧❛r❧②✱ ❝♦♥s✐❞❡r t❤❡ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ t❤❡ ♠✉❧t✐♣❧❡ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡❀ ✐✳❡✳✱

❘✉❧❡ ❢♦r ❉❡r✐✈❛t✐✈❡s ✭❈❤❛♣t❡r ✹✮✿

❚❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ♠✉❧t✐♣❧❡ ✐s

(cf )′ = cf ′ .

▲❡t✬s r❡❛❞ t❤❛t ❢♦r♠✉❧❛ ❢r♦♠ r✐❣❤t t♦ ❧❡❢t✿ ❚❤❡♦r❡♠ ✺✳✽✳✶✹✿ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ❢♦r ❆♥t✐❞❡r✐✈❛t✐✈❡s ❆♥ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ♠✉❧t✐♣❧❡ ✐s t❤❡ ♠✉❧t✐♣❧❡ ♦❢ t❤❡ ❛♥t✐❞❡r✐✈❛t✐✈❡✳

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿ ✐❢

F ✐s ❛♥❞ ❛♥t✐❞❡r✐✈❛t✐✈❡ c ✐s ❛ ❝♦♥st❛♥t✱

♦❢

f

t❤❡♥

cF

✐s ❛♥ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢

cf ✳

❛♥❞

✺✳✽✳ ❆♥t✐❞❡r✐✈❛t✐✈❡s

✹✸✸

Pr♦♦❢✳

❲❡ ❛♣♣❧② t❤❡ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ❋♦r ❉❡r✐✈❛t✐✈❡s ✿ (cF (x))′ = cF ′ (x) = cf (x) . ❊①❡r❝✐s❡ ✺✳✽✳✶✺

❲❤❛t ❛❜♦✉t t❤❡ ❝♦♥✈❡rs❡❄ ❊①❛♠♣❧❡ ✺✳✽✳✶✻✿ ❝♦♥st❛♥t ♠✉❧t✐♣❧❡s

❙♦❧✈❡ ❢♦r F ✿

F ′ (x) = 3 sin x .

❲❡ s♦❧✈❡ t❤❡ ❡q✉❛t✐♦♥ ❜② s♦❧✈✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉❛t✐♦♥✿ ❢♦r G : G′ (x) = sin x . ❚❤❡ s♦❧✉t✐♦♥ ✐s ❢♦✉♥❞ ✐♥ t❤❡ t❛❜❧❡✿ G(x) = − cos x .

❆❝❝♦r❞✐♥❣ t♦ t❤❡ t❤❡♦r❡♠✱ ✇❡ ❤❛✈❡ ❛♥ ❛♥s✇❡r✿

F (x) = 3(− cos x) . ❊①❡r❝✐s❡ ✺✳✽✳✶✼

❯s✐♥❣ t❤✐s r✉❧❡✱ ✜♥❞ ❛♥ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢ ex+3 ✳ ❈♦♠♣❛r❡✿ ❚❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ❝♦♥st❛♥t ♠✉❧t✐♣❧❡ ✐s t❤❡ ❝♦♥st❛♥t ♠✉❧t✐♣❧❡ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡✳ ❚❤❡ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ❝♦♥st❛♥t ♠✉❧t✐♣❧❡ ✐s t❤❡ ❝♦♥st❛♥t ♠✉❧t✐♣❧❡ ♦❢ t❤❡ ❛♥t✐❞❡r✐✈❛t✐✈❡✳ ❲❡ ❝❛♥ ❤❛♥❞❧❡ ❝♦♠♣♦s✐t✐♦♥s ❥✉st ❛s ❡❛s✐❧② ❜✉t ♦♥❧② ✇❤❡♥ ♦♥❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥s ✐s ❧✐♥❡❛r✳ ❲❡ r❡❛❞ t❤❡ ▲✐♥❡❛r ❈❤❛✐♥ ❘✉❧❡ ✭❈❤❛♣t❡r ✹✮ ❢r♦♠ r✐❣❤t t♦ ❧❡❢t✿ d f (mx + b) = mf ′ (mx + b) , dx

♣r♦❞✉❝✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣✿ ❚❤❡♦r❡♠ ✺✳✽✳✶✽✿ ▲✐♥❡❛r ❈♦♠♣♦s✐t✐♦♥ ❘✉❧❡ ❢♦r ❆♥t✐❞❡r✐✈❛t✐✈❡s ■❢

t❤❡♥

F ✐s ❛♥ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢ f m 6= 0, b ❛r❡ ❝♦♥st❛♥ts✱ 1 F (mx + b) m

✐s ❛♥ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢

❛♥❞

f (mx + b)✳

Pr♦♦❢✳

❲❡ ❛♣♣❧② t❤❡ ▲✐♥❡❛r ❈❤❛✐♥ ❘✉❧❡ ❛♥❞ t❤❡ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ❢♦r ❉✐✛❡r❡♥t✐❛t✐♦♥ ✿ 

1 F (mx + b) a

′

=

1 1 (F (mx + b))′ = mF ′ (mx + b) = F ′ (mx + b) = f (mx + b) . m m

✺✳✽✳

❆♥t✐❞❡r✐✈❛t✐✈❡s

✹✸✹

❊①❡r❝✐s❡ ✺✳✽✳✶✾ ❯s✐♥❣ t❤✐s r✉❧❡✱ ✜♥❞ ❛♥ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢

e2x+1 ✳

❊①❡r❝✐s❡ ✺✳✽✳✷✵ ❲❤❛t ❛❜♦✉t t❤❡ ♦t❤❡r ▲✐♥❡❛r ❈❤❛✐♥ ❘✉❧❡ ✭❈❤❛♣t❡r ✹✮❄

❆s ✇❡ ❦♥♦✇ ❢r♦♠ t❤❡

❆♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥ ❚❤❡♦r❡♠✱ ❡✈❡r② ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ❝♦♠❡s ✇✐t❤ ✐♥✜♥✐t❡❧②

♠❛♥② ♦t❤❡rs✿

F → F +C ♦♥ ❡✈❡r② ♦♣❡♥ ✐♥t❡r✈❛❧✳ ❚♦❣❡t❤❡r t❤❡② ❢♦r♠

❊①❛♠♣❧❡ ✺✳✽✳✷✶✿ ❢r❡❡ ❢❛❧❧

❲❡ ❝❛♥ ♠❛❦❡ ♦✉r ❛♥❛❧②s✐s ♦❢

❢r❡❡ ❢❛❧❧

t❤❡

❢♦r ❡✈❡r② r❡❛❧

C,

❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥✳

♠♦r❡ s♣❡❝✐✜❝✿

❋✉♥❝t✐♦♥s ♦❢ t✐♠❡

❚❤❡ ❝♦♥st❛♥ts

C

❚❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ✐s ❝♦♥st❛♥t✿

a = −g ✳

=⇒

❚❤❡ ✈❡❧♦❝✐t② ✐s ❧✐♥❡❛r✿

v = −gt + C ✳

=⇒

❚❤❡ ❧♦❝❛t✐♦♥ ✐s q✉❛❞r❛t✐❝✿

p = −gt2 /2 + Ct + K ✳

❛♥❞

K

❝♦✈❡r ❛❧❧ ♣♦ss✐❜❧❡ tr✐♣s ♦❢ t❤❡ ❜❛❧❧✳

❇❡❧♦✇ ✐s t❤❡ st❛♥❞❛r❞ ♥♦t❛t✐♦♥ ❢♦r

t❤❡

❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ❢✉♥❝t✐♦♥

f✿

❆♥t✐❞❡r✐✈❛t✐✈❡ ❍❡r❡✱

Z

Z

f dx

✐s r❡❢❡rr❡❞ t♦ ❛s t❤❡

✏✐♥t❡❣r❛❧ s✐❣♥✑✳

■t ❧♦♦❦s ❧✐❦❡ ❛ str❡t❝❤❡❞ ❧❡tt❡r ✏❙✑✱ ✇❤✐❝❤ st❛♥❞s ❢♦r ✏s✉♠♠❛t✐♦♥✑✿

❚❤❡ s✉♠♠❛t✐♦♥ ❜❡✐♥❣ r❡❢❡rr❡❞ t♦ ❤❡r❡ ✐s t❤❡ ♦♥❡ ✐♥ t❤❡ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛ ❢♦r ❞✐s❝r❡t❡ ❛♥t✐✲❞❡r✐✈❛t✐✈❡✿

F (xn+1 ) = F (xn )

✰ f (c )∆x n

n

.

❚❤❡ ♥♦t❛t✐♦♥ ✐❞❡❛ ❢♦r ❛♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥ ♠❛t❝❤❡s t❤❛t ❢♦r ❞✐✛❡r❡♥t✐❛t✐♦♥ ✇❤❡♥ ✇r✐tt❡♥ ✐♥ ❛ ❝❡rt❛✐♥ ✇❛②✿


d 3x2 + cos x = 6x − sin x . dx

❲❡ s❡❡ t❤❡ ✏♣❛r❡♥t❤❡s❡s✑ ♦❢ t❤✐s ❢✉♥❝t✐♦♥ ♦❢ ❢✉♥❝t✐♦♥s✳ ❚❤✐s ✐s ❤♦✇ t❤❡ ♥❡✇ ♥♦t❛t✐♦♥ ✐s ❞❡❝♦♥str✉❝t❡❞✿

✺✳✽✳ ❆♥t✐❞❡r✐✈❛t✐✈❡s

✹✸✺

❆♥t✐❞❡r✐✈❛t✐✈❡ ❛ s♣❡❝✐✜❝ ❢✉♥❝t✐♦♥

Z

❛ s♣❡❝✐✜❝ ❢✉♥❝t✐♦♥

↓

3x2

+ cos x

↑

❧❡❢t ❛♥❞




↓

x3 + sin x + C

dx = ↑

↑

r✐❣❤t ✏♣❛r❡♥t❤❡s❡s✑

✐♥❞✐❝❛t❡s t❤❛t t❤❡r❡ ❛r❡ ♦t❤❡rs

❚❤✐s ✐s ❤♦✇ ✇❡ r❡✇r✐t❡ t❤❡ ❛❜♦✈❡ ❧✐st✿

Z

xs dx =

Z

Z

Z

1 s+1 x + C, s+1

❢♦r

s 6= −1

1 dx = ln x + C Z x ex dx = ex + C sin x dx = − cos x + C cos x dx = sin x + C

❲❡ r❡st❛t❡ t❤❡ r✉❧❡s t♦♦✳ ❙✉♠ ❘✉❧❡✿

Z

(f + g) dx =

Z

f dx +

Z

g dx

❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡✿

Z

(cf ) dx = c

Z

f dx

▲✐♥❡❛r ❈♦♠♣♦s✐t✐♦♥ ❘✉❧❡✿

Z

1 f (mx + b) dx = m

Z

f (t) dt

t=mx+b

❲✐t❤ t❤❡s❡ r✉❧❡s✱ ✇❤❡♥ ❛♣♣❧✐❝❛❜❧❡✱ ❛♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥ ✐s ✈❡r② s✐♠✐❧❛r t♦ ❞✐✛❡r❡♥t✐❛t✐♦♥✳

❊①❛♠♣❧❡ ✺✳✽✳✷✷✿ r✉❧❡s ♦❢ ❛♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥ ❋✐♥❞ ❛♥ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢

3x2 + 5ex + cos x . ❖♥❡ ❝❛♥ ✐♠❛❣✐♥❡ ✇❤❛t ❤❡✬❞ ❞♦ t♦ ❞✐✛❡r❡♥t✐❛t❡ ❛♥❞ t❤❡♥ ❢♦❧❧♦✇ t❤❡ s❛♠❡ st❡♣s ❜✉t ✇✐t❤ t❤❡ ❛♥t✐✲ ❞✐✛❡r❡♥t✐❛t✐♦♥ ❢♦r♠✉❧❛s ❛♥❞ r✉❧❡s ✉s❡❞ ✐♥st❡❛❞✳ ❉✐✛❡r❡♥t✐❛t✐♦♥✿

(3x2 + 5ex + cos x)′ = (3x2 )′ + 5(ex )′ + (3 sin x)′ = 3(x2 )′ + 5(ex )′ + 3(sin x)′ = 3 · 6x + 5ex + 3 cos x .

❙❘ ❈▼❘ ❚❛❜❧❡

✺✳✽✳

❆♥t✐❞❡r✐✈❛t✐✈❡s

✹✸✻

❆♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥✿

Z

2

x

′

(3x + 5e + cos x) dx =

Z

Z

2

Z

x

(3x ) dx + 5(e ) dx + (3 sin x)dx Z Z Z 2 x = 3 (x ) dx + 5 (e ) dx + 3 (sin x)dx = 3 · x3 /3 + 5ex + 3(− cos x) + C .

❙❘ ❈▼❘ ❚❛❜❧❡

❏✉st ❛s ✇❤❡♥ s♦❧✈✐♥❣ ❡q✉❛t✐♦♥s✱ ✇❡ ❝❛♥ ❡❛s✐❧② ❝♦♥✜r♠ t❤❛t ♦✉r ❝♦♠♣✉t❛t✐♦♥s ✇❡r❡ ❝♦rr❡❝t✱ ❜② s✉❜st✐✲ t✉t✐♦♥✳ ■♥ t❤✐s ❝❛s❡✱ ✇❡

❞✐✛❡r❡♥t✐❛t❡ t❤❡ ❛♥t✐❞❡r✐✈❛t✐✈❡ ✿

(x3 + 5ex + 3 sin x)′ = (x3 )′ + 5(ex )′ + (3 sin x)′ = 3x2 + 5ex + 3 cos x . ❚❤✐s ✐s t❤❡ ♦r✐❣✐♥❛❧ ❢✉♥❝t✐♦♥✦ ❚❤❡ ❛♥s✇❡r ❝❤❡❝❦s ♦✉t✳

❊①❡r❝✐s❡ ✺✳✽✳✷✸

❋✐♥❞ t❤❡ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢

5e3x+2 − ee ✳

❇❡❧♦✇✱ ✇❡ ❤❛✈❡ t❤❡s❡ t✇♦ ❞✐❛❣r❛♠s t♦ ✐❧❧✉str❛t❡ t❤❡ ✐♥t❡r❛❝t✐♦♥ ♦❢ ❛♥t✐❞❡r✐✈❛t✐✈❡s ✇✐t❤ ❛❧❣❡❜r❛✿

f, g  + y

R

←−−−− R

′

f, g  + y ′

R

′

f + g ←−−−− f + g

f ←−−−−  ·c y

′

R

←−−−−

cf

f′   ·c y

cf ′

❚❤❡ ❛rr♦✇s ♦❢ ❞✐✛❡r❡♥t✐❛t✐♦♥ ❛r❡ r❡✈❡rs❡❞✦ ❲❡ st❛rt ✇✐t❤ ❛ ♣❛✐r ♦❢ ❢✉♥❝t✐♦♥s ❛t t♦♣ r✐❣❤t✱ t❤❡♥ ✇❡ ♣r♦❝❡❡❞ ✐♥ t✇♦ ✇❛②s✿

•

❧❡❢t✿ ❛♥t✐✲❞✐✛❡r❡♥t✐❛t❡ t❤❡♠✱ t❤❡♥ ❞♦✇♥✿ ❛❞❞ t❤❡ r❡s✉❧ts❀ ♦r

•

❞♦✇♥✿ ❛❞❞ t❤❡♠✱ t❤❡♥ ❧❡❢t✿ ❛♥t✐✲❞✐✛❡r❡♥t✐❛t❡ t❤❡ r❡s✉❧ts✳

❚❤❡ r❡s✉❧t ✐s t❤❡ s❛♠❡✦ ❙♦ ❢❛r✱ t❤✐s ✐s ✈❡r② s✐♠✐❧❛r t♦ ❞✐✛❡r❡♥t✐❛t✐♦♥✳ ❚❤❡ str❛t❡❣② ✐s t❤❡ s❛♠❡✿ ❞✐✈✐❞❡ ❛♥❞ ❝♦♥❝✉r✳ ❙♣❧✐t ❛❞❞✐t✐♦♥ ✇✐t❤ t❤❡ ❙✉♠ ❘✉❧❡✱ t❤❡♥ ❢❛❝t♦r ♦✉t t❤❡ ❝♦❡✣❝✐❡♥ts ✇✐t❤ t❤❡ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡✱ t❤❡♥ ❛♣♣❧② t❤❡ t❛❜❧❡ r❡s✉❧ts t♦ t❤❡s❡ ♣✐❡❝❡s✳ ❯♥❢♦rt✉♥❛t❡❧②✱ t❤✐s ✐s ✇❤❡r❡ t❤❡ s✐♠✐❧❛r✐t✐❡s st♦♣✳ ❚❤❡r❡ ✐s ♥♦ ❛♥❛❧♦❣ ♦❢ t❤❡ Pr♦❞✉❝t ❘✉❧❡ ❢♦r t❤❡ ❉❡r✐✈❛t✐✈❡s✿ ❚❤❡

❞❡r✐✈❛t✐✈❡

♦❢ t❤❡ ♣r♦❞✉❝t ♦❢ t✇♦ ❢✉♥❝t✐♦♥s

❝❛♥

❜❡ ❡①♣r❡ss❡❞ ✐♥ t❡r♠s ♦❢ t❤❡✐r

❞❡r✐✈❛t✐✈❡s

❛♥t✐❞❡r✐✈❛t✐✈❡

♦❢ t❤❡ ♣r♦❞✉❝t ♦❢ t✇♦ ❢✉♥❝t✐♦♥s

❝❛♥♥♦t

❛♥❞ t❤❡ ❢✉♥❝t✐♦♥s t❤❡♠s❡❧✈❡s✳

❚❤❡

❜❡ ❡①♣r❡ss❡❞ ✐♥ t❡r♠s ♦❢ t❤❡✐r

❛♥t✐❞❡r✐✈❛t✐✈❡s

❛♥❞ t❤❡ ❢✉♥❝t✐♦♥s t❤❡♠s❡❧✈❡s✳

❙✐♠✐❧❛r❧② t❤❡r❡ ✐s ♥♦ ◗✉♦t✐❡♥t ❘✉❧❡✱ ♥♦r t❤❡ ❈❤❛✐♥ ❘✉❧❡✱ ❢♦r ❛♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥✳ ❚❤✐s ❞✐✛❡r❡♥❝❡ ❤❛s ♣r♦❢♦✉♥❞ ❝♦♥s❡q✉❡♥❝❡s✳ ❲❡ ❝❛♥ st❛rt ✇✐t❤ ❥✉st t❤❡s❡ ❢✉♥❝t✐♦♥s✿

xs , sin x, ex . ❚❤❡♥ ✕ ❜② ❛♣♣❧②✐♥❣ t❤❡ ❢♦✉r ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s✱ ❝♦♠♣♦s✐t✐♦♥✱ ❛♥❞ ✐♥✈❡rt✐♥❣ ✕ ✇❡ ❝❛♥ ❝♦♥str✉❝t ❛ ❣r❡❛t ✈❛r✐❡t② ♦❢ ❢✉♥❝t✐♦♥s✳ ▲❡t✬s ❝❛❧❧ t❤❡♠ ✏❡❧❡♠❡♥t❛r② ❢✉♥❝t✐♦♥s✑✳ ❇❡❝❛✉s❡ ♦❢ t❤❡ ✇❛② t❤❡② ❛r❡ ❝♦♥str✉❝t❡❞✱

❛❧❧

♦❢

t❤❡♠ ❝❛♥ ❜❡ ❡❛s✐❧② ❞✐✛❡r❡♥t✐❛t❡❞ ✇✐t❤ t❤❡ r✉❧❡s ♦❢ ❞✐✛❡r❡♥t✐❛t✐♦♥ t❤✉s ♣r♦❞✉❝✐♥❣ ♦t❤❡r ❡❧❡♠❡♥t❛r② ❢✉♥❝t✐♦♥s ✭❧❡❢t✮✿

✺✳✽✳

❆♥t✐❞❡r✐✈❛t✐✈❡s

✹✸✼

❍♦✇❡✈❡r✱ ❝♦♥tr❛r② t♦ ✇❤❛t t❤❡ ❛❜♦✈❡ ❧✐st ♠✐❣❤t s✉❣❣❡st✱ ❛♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥ ✇✐❧❧ ♦❢t❡♥ t❛❦❡ ✉s ♦✉ts✐❞❡ ♦❢ t❤❡ r❡❛❧♠ ♦❢ ❡❧❡♠❡♥t❛r② ❢✉♥❝t✐♦♥s ✭r✐❣❤t✮✳ ❋♦r ❡①❛♠♣❧❡✱ ❛ ♥❡✇ ❢✉♥❝t✐♦♥✱ ❝❛❧❧❡❞ t❤❡

●❛✉ss ❡rr♦r ❢✉♥❝t✐♦♥✱ ♠✉st

❜❡ ❝r❡❛t❡❞ ❢♦r t❤✐s ✐♠♣♦rt❛♥t ❛♥t✐❞❡r✐✈❛t✐✈❡✿

2 erf(x) = √ π ❚❤❡ r❡s✉❧t ✇✐❧❧ ❜❡ t❤❡ s❛♠❡ ✐❢ ✇❡

❡①❝❧✉❞❡

2

e−x dx .

❢r♦♠ t❤❡ ✏❡❧❡♠❡♥t❛r② ❢✉♥❝t✐♦♥s✑ ❡✐t❤❡r t❤❡ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s

♦r t❤❡ ❡①♣♦♥❡♥t✳ ❚❤❡ r❡s✉❧t ✇✐❧❧ ❜❡ t❤❡ s❛♠❡ ✐❢ ✇❡ ❲✐t❤ t❤❡ ❤❡❧♣ ♦❢ t❤❡

Z

✐♥❝❧✉❞❡

❆♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥ ❚❤❡♦r❡♠✱

♠♦r❡ ❢✉♥❝t✐♦♥s t♦ t❤❡ ❧✐st✳

✇❡ ❝❛♥ ❝❧❛✐♠ t❤❛t ✇❡ ❤❛✈❡ ❢♦✉♥❞

❛❧❧

t❤❡s❡ ❢✉♥❝t✐♦♥s ♦♥ ♦✉r ❧✐st✱ ♦✈❡r ♦♣❡♥ ✐♥t❡r✈❛❧s ✇✐t❤✐♥ t❤❡ ❞♦♠❛✐♥s ♦❢ ❛♥t✐❞❡r✐✈❛t✐✈❡s✱ ✐✳❡✳✱

❛♥t✐❞❡r✐✈❛t✐✈❡s ♦❢

t❤❡

❛♥t✐❞❡r✐✈❛t✐✈❡

♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❢✉♥❝t✐♦♥✿

❉❡✜♥✐t✐♦♥ ✺✳✽✳✷✹✿ ❣❡♥❡r❛❧ ❛♥t✐❞❡r✐✈❛t✐✈❡ ❋♦r ❛ ❣✐✈❡♥ ❢✉♥❝t✐♦♥

f✱

t❤❡ ❣❡♥❡r❛❧ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢ f ♦✈❡r ♦♣❡♥ ✐♥t❡r✈❛❧ I

❞❡✜♥❡❞ ❜②✿

✇❤❡r❡

F

Z

f dx = F (x) + C ,

✐s ❛♥② ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢

f

♦♥

I✱

✐✳❡✳✱

F′ = f✱

♦❢ ❛❧❧ s✉❝❤ ❢✉♥❝t✐♦♥s ♦✈❡r ❛❧❧ ♣♦ss✐❜❧❡ r❡❛❧ ♥✉♠❜❡rs ❝❛❧❧❡❞

t❤❡ ✐♥❞❡✜♥✐t❡ ✐♥t❡❣r❛❧

♦❢

✐s

✉♥❞❡rst♦♦❞ ❛s ❛ ❝♦❧❧❡❝t✐♦♥

C✳

❚❤✐s ❝♦❧❧❡❝t✐♦♥ ✐s ❛❧s♦

f✳

❚❤❡s❡ ❞✐❛❣r❛♠s ✐❧❧✉str❛t❡ ❤♦✇ ❞✐✛❡r❡♥t✐❛t✐♦♥ ❛♥❞ ❛♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥ ✉♥❞♦ ❡❛❝❤ ♦t❤❡r✿

f

→

Z

F → ■♥ t❤❡ s❡❝♦♥❞ r♦✇✱ ✇❡ s❡❡ t❤❛t

d dx

 dx d  dx

d  dx

→ F → → f

→

Z

→

f

→ F +C

 dx

✐s♥✬t ✐♥✈❡rt✐❜❧❡ ✭❱♦❧✉♠❡ ✶✱ ❈❤❛♣t❡r ✶P❈✲✸✮✳

❊①❛♠♣❧❡ ✺✳✽✳✷✺✿ ❤♦✇ ♠❛♥② ❛♥t✐❞❡r✐✈❛t✐✈❡s❄ ❚❤❡r❡ ❛r❡ ✐♥✜♥✐t❡❧② ♠❛♥② ❛♥t✐❞❡r✐✈❛t✐✈❡s ❜✉t t❤❡r❡ ✐s ♠♦r❡ t♦ ✐t✳ ▲❡t✬s t❛❦❡ ❛ ♠♦r❡ ❝❛r❡❢✉❧ ❧♦♦❦ ❛t ♦♥❡ ❧✐♥❡ ♦♥ t❤❡ ❧✐st✿

Z

1 ❄❄❄ dx === ln |x| + C, x

x 6= 0 .

❚❤✐s ❢♦r♠✉❧❛ ✐s ✐♥t❡♥❞❡❞ t♦ ♠❡❛♥ t❤❡ ❢♦❧❧♦✇✐♥❣✿ ✶✳ ❲❡ ❤❛✈❡ ❝❛♣t✉r❡❞ ✐♥✜♥✐t❡❧② ♠❛♥② ✕ ♦♥❡ ❢♦r ❡❛❝❤ r❡❛❧ ♥✉♠❜❡r

❛❧❧ ♦❢ t❤❡♠✳ ❆♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥ ❚❤❡♦r❡♠

✷✳ ❲❡ ❤❛✈❡ ❝❛♣t✉r❡❞ ❍♦✇❡✈❡r✱ t❤❡

❛♣♣❧✐❡s ♦♥❧② t♦

C

✕ ❛♥t✐❞❡r✐✈❛t✐✈❡s✳

♦♥❡ ✐♥t❡r✈❛❧ ❛t ❛ t✐♠❡✳

▼❡❛♥✇❤✐❧❡✱ t❤❡

✺✳✽✳

❆♥t✐❞❡r✐✈❛t✐✈❡s

✹✸✽

❞♦♠❛✐♥ ♦❢ 1/x ❝♦♥s✐sts ♦❢ t✇♦ r❛②s (−∞, 0) ❛♥❞ (0, +∞)✳ ❆s ❛ r❡s✉❧t✱ ✇❡ s♦❧✈❡ t❤✐s ♣r♦❜❧❡♠ s❡♣❛r❛t❡❧② ♦♥ ❡✐t❤❡r ♦❢ t❤❡ t✇♦ ✐♥t❡r✈❛❧s✳ ❚❤❡♥ t❤❡ ❛♥t✐❞❡r✐✈❛t✐✈❡s ♦❢ 1/x ❛r❡✿ • ln(−x) + C ♦♥ (−∞, 0)✱ ❛♥❞ • ln(x) + C ♦♥ (0, +∞)✳ ❇✉t ✐❢ ♥♦✇ ✇❡ ✇❡r❡ t♦ ❝♦♠❜✐♥❡ ❡❛❝❤ ♦❢ t❤❡s❡ ♣❛✐rs ♦❢ ❢✉♥❝t✐♦♥s ✐♥t♦ ♦♥❡✱ F ✱ ❞❡✜♥❡❞ ♦♥ (−∞, 0)∪(0, +∞)✱ ✇❡ ✇♦✉❧❞ r❡❛❧✐③❡ t❤❛t✱ ❡✈❡r② t✐♠❡✱ t❤❡ t✇♦ ❝♦♥st❛♥ts ♠✐❣❤t ❜❡ ❞✐✛❡r❡♥t✿ ❆❢t❡r ❛❧❧✱ t❤❡② ❤❛✈❡ ♥♦t❤✐♥❣ t♦ ❞♦ ✇✐t❤ ❡❛❝❤ ♦t❤❡r✦ ❲❡ ✐❧❧✉str❛t❡ t❤❡ ✇r♦♥❣ ✭✐♥❝♦♠♣❧❡t❡✮ ❛♥s✇❡r ♦♥ t❤❡ ❧❡❢t✱ ❛♥❞ t❤❡ r✐❣❤t ♦♥❡ ♦♥ t❤❡ r✐❣❤t✿

❚❤❡ ✐♠❛❣❡ ♦♥ t❤❡ ❧❡❢t✱ ❛s ✇❡❧❧ ❛s t❤❡ ❢♦r♠✉❧❛ ✇❡ st❛rt❡❞ ✇✐t❤✱ ♠✐❣❤t s✉❣❣❡st t❤❛t ❛❧❧ ♦❢ t❤❡ ❢✉♥❝t✐♦♥✬s ❛♥t✐❞❡r✐✈❛t✐✈❡s ❛r❡ ❡✈❡♥ ❢✉♥❝t✐♦♥s✳ ❚❤❡ ✐♠❛❣❡ ♦♥ t❤❡ r✐❣❤t s❤♦✇s ❛ s✐♥❣❧❡ ❛♥t✐❞❡r✐✈❛t✐✈❡ ✭✐♥ r❡❞✮ ❜✉t 1 ✐ts t✇♦ ❜r❛♥❝❤❡s ❞♦♥✬t ❤❛✈❡ t♦ ♠❛t❝❤✦ ❆❧❣❡❜r❛✐❝❛❧❧②✱ t❤❡ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢ ✕ ♦♥ t❤❡ ✇❤♦❧❡ ❞♦♠❛✐♥ ✕ x ✐s ❣✐✈❡♥ ❜② t❤✐s ♣✐❡❝❡✇✐s❡✲❞❡✜♥❡❞ ❢✉♥❝t✐♦♥✿ ( ln(−x) + C ❢♦r x ✐♥ (−∞, 0), F (x) = ln(x) + K ❢♦r x ✐♥ (0, +∞). ■t ❤❛s

t✇♦ ♣❛r❛♠❡t❡rs ✐♥st❡❛❞ ♦❢ t❤❡ ✉s✉❛❧ ♦♥❡✳

❊①❡r❝✐s❡ ✺✳✽✳✷✻ ❱❡r✐❢② t❤❛t t❤✐s ✐s ❛♥ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢ 1/x✳

❊①❡r❝✐s❡ ✺✳✽✳✷✼ ■♥ ❛ s✐♠✐❧❛r ❢❛s❤✐♦♥✱ ❡①❛♠✐♥❡ t❤❡ P♦✇❡r ❋♦r♠✉❧❛ ❛❜♦✈❡ ❢♦r s < −1✳

❊①❛♠♣❧❡ ✺✳✽✳✷✽✿ ✜♥❞ ❣r❛♣❤s ❚❤❡ ❛♥t✐❞❡r✐✈❛t✐✈❡s ♦♥ ♦✉r ❧✐st ✇❡r❡ ❞✐s❝♦✈❡r❡❞ ❜② r❡❛❞✐♥❣ t❤❡ r❡s✉❧ts ♦❢ ❞✐✛❡r❡♥t✐❛t✐♦♥ ❜❛❝❦✇❛r❞s✳ ❲❡ ❝❛♥ ❞♦ t❤❡ s❛♠❡ ❢♦r ❣r❛♣❤s✳ ❇❡❧♦✇✱ t❤❡ ❞❡r✐✈❛t✐✈❡✬s ❣r❛♣❤ ✭❣r❡❡♥✮ ✇❛s ❢♦✉♥❞ ❢r♦♠ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ✭r❡❞✮ ❜② ❧♦♦❦✐♥❣ ❛t t❤❡ ♠♦♥♦t♦♥✐❝ ❜❡❤❛✈✐♦r ♦❢ f ✿ ❡✐t❤❡r f ′ > 0 ♦r f ′ < 0✱ ❛♥❞ ❧♦❝❛❧ ❡①tr❡♠❡ ♣♦✐♥ts✿ f ′ = 0✳

✺✳✽✳

❆♥t✐❞❡r✐✈❛t✐✈❡s

■♥ s✉♠♠❛r②✱ ✇❡ ❧♦♦❦ ❛t ց, ր ♦❢ f t♦ ✜♥❞ +, − ♦❢ f ′ ✳ ■♥ r❡✈❡rs❡✱ ✇❡ ❧♦♦❦ ❛t +, − ♦❢ f ′ t♦ ✜♥❞ ց, ր ♦❢ f ✳ ❍❡r❡ ✐s ❛♥ ❡①❛♠♣❧❡ ❤♦✇ t❤❡ ❣r❛♣❤ ♦❢ f ✐s ❢♦✉♥❞ ❢r♦♠ t❤❡ ❣r❛♣❤ ♦❢ ✐ts ❞❡r✐✈❛t✐✈❡✿

❚♦ s❤♦✇ t❤❡ ❛♥s✇❡r✱ ✇❡ ❤❛✈❡ t♦ s❤♦✇ ❛ ♠✉❧t✐t✉❞❡ ♦❢ ❛♥t✐❞❡r✐✈❛t✐✈❡s✿

❊①❡r❝✐s❡ ✺✳✽✳✷✾

❋✐♥❞ t❤❡ ✐♥✢❡❝t✐♦♥ ♣♦✐♥ts✳ ❚❤✐s st✉❞② ❝♦♥t✐♥✉❡s ✐♥ ❱♦❧✉♠❡ ✸✱ ❈❤❛♣t❡r ✸■❈✲✶✳

✹✸✾

✺✳✽✳

❆♥t✐❞❡r✐✈❛t✐✈❡s

✹✹✵

❲❡ ❤❛✈❡ ❜❡❡♥ ❢♦❧❧♦✇✐♥❣ ✭❛♥❞ ❝♦♥t✐♥✉❡✮ t♦ ❢♦❧❧♦✇ ♦♥❡ ♦❢ t❤❡ ♠♦st ❝r✉❝✐❛❧ ✐❞❡❛s ♦❢ t❤❡ ❜♦♦❦✿

lim

∆x→0



❞✐s❝r❡t❡ ❝❛❧❝✉❧✉s



=

❝❛❧❝✉❧✉s

❈❤❛♣t❡r ✻✿ ❲❤❛t ✇❡ ❝❛♥ ❞♦ ✇✐t❤ ❝❛❧❝✉❧✉s

❈♦♥t❡♥ts

✻✳✶ ▼❛❣♥✐t✉❞❡s ♦❢ ❢✉♥❝t✐♦♥s❀ ▲✬❍♦♣✐t❛❧✬s ❘✉❧❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✷ ▲✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✸ ❚❤❡ ❛❝❝✉r❛❝② ♦❢ t❤❡ ❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✹ ❙♦❧✈✐♥❣ ❡q✉❛t✐♦♥s ♥✉♠❡r✐❝❛❧❧②✿ ❜✐s❡❝t✐♦♥ ❛♥❞ ◆❡✇t♦♥✬s ♠❡t❤♦❞ ✻✳✺ P❛rt✐❝❧❡ ✐♥ ❛ ✢♦✇ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✻ ❉✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✼ ▼♦t✐♦♥ ✉♥❞❡r ❢♦r❝❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✽ ❖♣t✐♠✐③❛t✐♦♥ ❡①❛♠♣❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✾ ❋✉♥❝t✐♦♥s ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✹✶ ✹✺✶ ✹✻✷ ✹✻✼ ✹✼✶ ✹✼✽ ✹✽✸ ✹✽✾ ✹✾✺

✻✳✶✳ ▼❛❣♥✐t✉❞❡s ♦❢ ❢✉♥❝t✐♦♥s❀ ▲✬❍♦♣✐t❛❧✬s ❘✉❧❡

❍♦✇ ❞♦❡s ♦♥❡ ❝♦♠♣❛r❡ t❤❡ ♠❛❣♥✐t✉❞❡ ♦r t❤❡ ✏str❡♥❣t❤✑ ♦❢ t✇♦ ❢✉♥❝t✐♦♥s ♦♥ ❛ ❧❛r❣❡ s❝❛❧❡❄ ❋♦r ❡①❛♠♣❧❡✱ ❧❡t✬s t❛❦❡ ❛ ❧♦♦❦ ❛t t❤❡

♣♦✇❡r ❢✉♥❝t✐♦♥s✱ s❛② x ✈s✳ x2 ✱ ♦r x2

✈s✳

x3 ✿

❚❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❧❛tt❡r ♦✈❡rt❛❦❡s t❤❛t ♦❢ t❤❡ ❢♦r♠❡r ❛♥❞ t❤❡♥ ❜❡❝♦♠❡s ✐♥❝♦♠♣❛r❛❜❧❡ ❤✐❣❤❡r✳

❚❤❡♥ t❤❡

❛♥s✇❡r ✐s ♦❜✈✐♦✉s✿

◮

❚❤❡ ❤✐❣❤❡r t❤❡ ♣♦✇❡r✱ t❤❡ str♦♥❣❡r t❤❡ ❢✉♥❝t✐♦♥✳

❆ s✐♠✐❧❛r r✉❧❡ ❛♣♣❧✐❡s t♦

◮

♣♦❧②♥♦♠✐❛❧s ✿

❚❤❡ ❤✐❣❤❡r t❤❡ ❞❡❣r❡❡✱ t❤❡ str♦♥❣❡r t❤❡ ♣♦❧②♥♦♠✐❛❧✳

❇✉t ✇❤❛t ❞♦❡s ✏str♦♥❣❡r✑ ♠❡❛♥❄ ❚♦ ❝♦♠♣❛r❡ t✇♦ ❢✉♥❝t✐♦♥s✱ t❤❡ ✜rst t❤✐♥❣ t♦ ❧♦♦❦ ❛t ✐s t❤❡

❞✐✛❡r❡♥❝❡

st❛♥❞❛r❞✱

x2

❛♥❞

x2 + x

♦❢ t❤❡ t✇♦ ❢✉♥❝t✐♦♥s✳ ❍♦✇❡✈❡r✱ ❜② t❤✐s

✻✳✶✳ ▼❛❣♥✐t✉❞❡s ♦❢ ❢✉♥❝t✐♦♥s❀ ▲✬❍♦♣✐t❛❧✬s ❘✉❧❡

✹✹✷

❛r❡ ✐♥✜♥✐t❡❧② ❢❛r ❛♣❛rt✦ ❆t ❧❡❛st✱ t❤❛t✬s ✇❤❛t t❤❡ ❧✐♠✐t ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ ✐s ❛t ✐♥✜♥✐t②✳ ■♥st❡❛❞✱ ✇❡ ❧♦♦❦ ❛t ✐s t❤❡ r❛t✐♦ ♦❢ t❤❡ t✇♦ ❢✉♥❝t✐♦♥s✳ ❈♦♥s✐❞❡r ❛❣❛✐♥✿

x2

❛♥❞

x2 + x .

❇② t❤✐s st❛♥❞❛r❞✱ t❤❡② ❛r❡ t❤❡ s❛♠❡✦ ❆t ❧❡❛st✱ t❤❛t✬s ✇❤❛t t❤❡ ❧✐♠✐t ♦❢ t❤✐s r❛t✐♦ ✐s ❛t ✐♥✜♥✐t②✿

x2 →1 x2 + x

❛s

x → ∞.

❊①❛♠♣❧❡ ✻✳✶✳✶✿ ❤♦r✐③♦♥t❛❧ ❛s②♠♣t♦t❡s

❲❡ ❡♥❝♦✉♥t❡r❡❞ t❤✐s ♣r♦❜❧❡♠ ✐♥❞✐r❡❝t❧② ✇❤❡♥ ❝♦♠♣✉t✐♥❣ ❧✐♠✐ts ❛t ✐♥✜♥✐t② ♦❢ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥s ✭❈❤❛♣t❡r ✷✮✳ ❘❡❝❛❧❧ t❤❡ ♠❡t❤♦❞ ✇❡ ✉s❡❞✿

❲❡ ❞✐✈✐❞❡ ❜♦t❤ ♥✉♠❡r❛t♦r ❛♥❞ ❞❡♥♦♠✐♥❛t♦r ❜② t❤❡ ❤✐❣❤❡st ❛✈❛✐❧❛❜❧❡

♣♦✇❡r✳ ❋♦r ❡①❛♠♣❧❡✿

3x3 + x − 7 /x3 3x3 + x − 7 = 2x3 + 100 2x3 + 100 /x3 3 + x−2 − 7x−3 = 2 + 100x−3 3 → 2

❛s

x → ∞.

❲❡ ❝♦♥❝❧✉❞❡ t❤❛t t❤❡s❡ t✇♦ ❢✉♥❝t✐♦♥s✱

f (x) = 3x3 + x − 7

❛♥❞

g(x) = 2x3 + 100 ,

❤❛✈❡ t❤❡ ✏s❛♠❡ ♣♦✇❡r✑ ❛t ✐♥✜♥✐t②✳ ❲❡ ❝❛♥ s❡❡ ❤♦✇ t❤❡✐r ❣r❛♣❤s st❛② t♦❣❡t❤❡r ❜❡❧♦✇✿

❊✈❡♥ t❤♦✉❣❤ t❤❡ ❞✐✛❡r❡♥❝❡ ❣♦❡s t♦ ✐♥✜♥✐t②✱ t❤❡ ♣r♦♣♦rt✐♦♥ r❡♠❛✐♥s ✈✐s✐❜❧②

3

❲❡ r❡❝♦❣♥✐③❡ ❛ ✈❡rt✐❝❛❧ str❡t❝❤✦ ❚❤❡ t✇♦ ❢✉♥❝t✐♦♥s ❜❡❤❛✈❡ r♦✉❣❤❧② t❤❡ s❛♠❡✳

t♦

2✿

✻✳✶✳ ▼❛❣♥✐t✉❞❡s ♦❢ ❢✉♥❝t✐♦♥s❀ ▲✬❍♦♣✐t❛❧✬s ❘✉❧❡

✹✹✸

❍♦✇❡✈❡r✱ ✐❢ ✇❡ r❡♣❧❛❝❡ t❤❡ ♣♦✇❡r ♦❢ t❤❡ ♥✉♠❡r❛t♦r ✇✐t❤

4✱

✇❡ s❡❡ t❤✐s✿

3x4 + x − 7 f (x) = g(x) 2x3 + 100 3x4 + x − 7 /x4 = 2x3 + 100 /x4 3 + x−3 − 7x−4 = 2x−1 + 100x−4 →∞ ❲❡ ❝♦♥❝❧✉❞❡ t❤❛t

3x4 + x − 7

x → ∞.

❛s

✐s ✏str♦♥❣❡r✑ ❛t ✐♥✜♥✐t② t❤❛♥

2x3 + 100✳

❲❡ ❝❛♥ s❡❡ ❤♦✇ q✉✐❝❦❧② t❤❡

❢♦r♠❡r r✉♥s ❛✇❛② ❜❡❧♦✇✿

❚❤✐s ✐s♥✬t ❛ str❡t❝❤✦ ❋✉rt❤❡r ❡♥❧❛r❣✐♥❣ t❤❡ ❞♦♠❛✐♥ ✇✐❧❧ ❝❛✉s❡ t❤❡ ♣❧♦t t♦ s❤r✐♥❦ ❤♦r✐③♦♥t❛❧❧② ❛♥❞ t❤❡ s♠❛❧❧❡r ❢✉♥❝t✐♦♥ ✇✐❧❧ ❞✐s❛♣♣❡❛r ✐♥t♦ t❤❡

❚❤✉s t♦ ❝♦♠♣❛r❡ t✇♦ ❢✉♥❝t✐♦♥s

f

❛♥❞

x✲❛①✐s✳

g ✱ ❛t ✐♥✜♥✐t② ♦r ❛t ❛ ♣♦✐♥t✱ ✇❡ ❢♦r♠ ❛ ❢r❛❝t✐♦♥ ❢r♦♠ t❤❡♠ ❛♥❞ ❡✈❛❧✉❛t❡

t❤❡ ❧✐♠✐t ♦❢ t❤❡ r❛t✐♦✿

lim

f (x) =? g(x)

◆♦ ♠❛tt❡r ✇❤❛t✬s ✐♥ t❤❡ ❜♦①✱ t❤❡ ❞❡✜♥✐t✐♦♥ ✐s t❤❡ s❛♠❡✿

❉❡✜♥✐t✐♦♥ ✻✳✶✳✷✿ ♦r❞❡rs ♦❢ ♠❛❣♥✐t✉❞❡ ♦❢ ❢✉♥❝t✐♦♥s ■❢ t❤❡ ❧✐♠✐t ❜❡❧♦✇✱ ✇✐t❤

x → ±∞

♦r

x → a± ,

✐s ✐♥✜♥✐t❡ ♦r ✐ts r❡❝✐♣r♦❝❛❧ ✐s ③❡r♦✱

✇❡ s❛② t❤❛t ✇❡ s❛② t❤❛t

f f

f (x) =∞ lim g(x)

✐s ♦❢ ❤✐❣❤❡r ♦r❞❡r t❤❛♥ ❛♥❞

g

♦r

g✳

g(x) = 0, lim f (x)

❲❤❡♥ t❤✐s ❧✐♠✐t ✐s ❛ ♥♦♥✲③❡r♦ ♥✉♠❜❡r✱

❛r❡ ♦❢ t❤❡ s❛♠❡ ♦r❞❡r✳

❲❡ ✉s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♥♦t❛t✐♦♥ ❢♦r t❤❡ ❢♦r♠❡r ❝❛s❡✿

❍✐❣❤❡r ♦r❞❡r f >> g g = o(f ) ❚❤❡ ❧❛tt❡r r❡❛❞s ✏❧✐tt❧❡ ♦✑✳

❲❤❡♥ ♠❛❣♥✐✜❡❞✱ t❤❡✐r ❣r❛♣❤s ❧♦♦❦ ❧✐❦❡ t❤♦s❡ ✐♥ t❤❡ ❧❛st ❡①❛♠♣❧❡✿

✻✳✶✳ ▼❛❣♥✐t✉❞❡s ♦❢ ❢✉♥❝t✐♦♥s❀ ▲✬❍♦♣✐t❛❧✬s ❘✉❧❡

✹✹✹

❲❡ ✉s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♥♦t❛t✐♦♥ ❢♦r t❤❡ ❧❛tt❡r ❝❛s❡✿

❊q✉❛❧ ♦r❞❡r f ∼g

❲❤❡♥ ♠❛❣♥✐✜❡❞✱ ❡✐t❤❡r ❣r❛♣❤ ❧♦♦❦s ❧✐❦❡ ❛ str❡t❝❤❡❞ ✈❡rs✐♦♥ ♦❢ t❤❡ ♦t❤❡r✿

❖♥❡ ♦❢ t❤❡ ♠♦st ✐♠♣♦rt❛♥t ✉s❡s ♦❢ t❤❡ ♥♦t❛t✐♦♥ ✐s ✐♥ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ ✿ ∆y . ∆x→0 ∆x

f ′ (a) = lim

❯s✐♥❣ t❤❡ ❙✉♠ ❘✉❧❡✱ ✐t ❝❛♥ ❜❡ r❡✇r✐tt❡♥ ❛s ❢♦❧❧♦✇s✿ lim

∆x→0

♦r



 ∆y ′ − f (a) = 0 , ∆x

∆y − f ′ (a)∆x = 0. ∆x→0 ∆x lim

■♥ ♦t❤❡r ✇♦r❞s✱

∆y − f ′ (a)∆x > xn >> ... >> x2 >> x >> ❇❡❧♦✇✱ ✇❡ s❡❡ ♣♦✇❡rs ❜❡❧♦✇

1

♦♥ t❤❡ r✐❣❤t ❛♥❞ ❛❜♦✈❡

1

√

x >>

√ 3

+∞✿

x >> ... >> 1

♦♥ t❤❡ ❧❡❢t✿

❚❤❡ r✐❣❤t ✐s t♦ ❜❡ ❛♣♣❡♥❞❡❞ ❛t t❤❡ ❜♦tt♦♠ ♦❢ t❤❡ ❧❡❢t✳ ❆♠♦♥❣ ♣♦❧②♥♦♠✐❛❧s✱ t❤❡ ❞❡❣r❡❡ ♣❧❛②s t❤❡ r♦❧❡ ♦❢ t❤❡ ♦r❞❡r✿

❚❤❡♦r❡♠ ✻✳✶✳✹✿ ▲✐tt❧❡ ♦ ❢♦r P♦❧②♥♦♠✐❛❧s

❆ ♣♦❧②♥♦♠✐❛❧ ❤❛s ❛ ❤✐❣❤❡r ♦r❞❡r t❤❛♥ ❛♥♦t❤❡r ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐ts ❞❡❣r❡❡ ✐s ❤✐❣❤❡r✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ❢♦r ❛♥② t✇♦ ♣♦❧②♥♦♠✐❛❧s P ❛♥❞ Q✱ ✇❡ ❤❛✈❡✿ P = o(Q) ⇐⇒ deg P < deg Q . ❆♥❞✱ ❢✉rt❤❡r♠♦r❡✱

P ∼ Q ⇐⇒ deg P = deg Q . ❊①❡r❝✐s❡ ✻✳✶✳✺ Pr♦✈❡ t❤❡ t❤❡♦r❡♠✳

✻✳✶✳ ▼❛❣♥✐t✉❞❡s ♦❢ ❢✉♥❝t✐♦♥s❀ ▲✬❍♦♣✐t❛❧✬s ❘✉❧❡

✹✹✻

❲❡ ❤❛✈❡✱ t❤❡r❡❢♦r❡✱ ❡①♣❛♥❞❡❞ ♦✉r ❤✐❡r❛r❝❤②✿

... >> nt❤

❞❡❣r❡❡ ♣♦❧②♥♦♠✐❛❧

>> ... >> 2♥❞

❞❡❣r❡❡ ♣♦❧②♥♦♠✐❛❧

>> 1st

❞❡❣r❡❡ ♣♦❧②♥♦♠✐❛❧

❚❤❡♥✱ ✇❤❛t ✐s t❤❡ r❡❧❛t✐♦♥ t♦ t❤❡ ❧✐st ♦❢ t❤❡ ♣♦✇❡rs t❤❛t ✇❡ st❛rt❡❞ ✇✐t❤❄ ❚❤❡ ♠❡❛♥✐♥❣ ♦❢ ❡❛❝❤ ✐t❡♠ ✐♥ t❤❡ n s❡q✉❡♥❝❡ ✐s t❤❡ s✐♠♣❧❡st r❡♣r❡s❡♥t❛t✐✈❡✱ x ✱ ♦❢ ❛ ✇❤♦❧❡ ❝❧❛ss ♦❢ ❢✉♥❝t✐♦♥s✿ t❤❡ ♣♦❧②♥♦♠✐❛❧s ♦❢ nt❤ ❞❡❣r❡❡✳ ❊①❡r❝✐s❡ ✻✳✶✳✻

❈❛♥ ②♦✉ ❢✉rt❤❡r ❡♥❧❛r❣❡ t❤❡ ❝❧❛ss❡s ♦❢ ❢✉♥❝t✐♦♥s ✐♥ t❤✐s s❡q✉❡♥❝❡❄ ❍✐♥t✿ ❛❞❞

√

x✳

❊①❡r❝✐s❡ ✻✳✶✳✼

❙✉❣❣❡st ❛ ✏▲✐tt❧❡ ♦ ❚❤❡♦r❡♠ ❋♦r ❘❛t✐♦♥❛❧ ❋✉♥❝t✐♦♥s✑✳ Pr♦✈❡ ✐t✳

ex

❊①❛♠♣❧❡ ✻✳✶✳✽✿ ♦r❞❡r ❛t

❲❤❡r❡ ❞♦❡s

ex

✜t ✐♥ t❤✐s ❤✐❡r❛r❝❤②❄ ❇❡❧♦✇✱ ✇❡ ❝♦♠♣❛r❡

ex

x100 ✿

❛♥❞

❚❤❡ ❢♦r♠❡r s❡❡♠s str♦♥❣❡r ❜✉t✱ ✉♥❢♦rt✉♥❛t❡❧②✱ t❤❡ ♠❡t❤♦❞s ♦❢ ❞✐✈✐❞✐♥❣ ❜② t❤❡ ❤✐❣❤❡r ♣♦✇❡r ❞♦❡s♥✬t ❛♣♣❧② ❤❡r❡✳ ❲❡ ✇✐❧❧ ❜❡ ❧♦♦❦✐♥❣ ❢♦r ❛♥♦t❤❡r ♠❡t❤♦❞✳

◆❡①t✱ ✇❡ ❝♦♥s✐❞❡r ❤♦✇ t♦ ❝♦♠♣❛r❡ t❤❡ ♠❛❣♥✐t✉❞❡s ♦❢ t✇♦ ❢✉♥❝t✐♦♥s ✐♥ t❤❡ ✈✐❝✐♥✐t② ♦❢ ❛ ♣♦✐♥t✳ ❚❤❡ ✐❞❡❛ ♦❢ ✉s✐♥❣ ❧✐♠✐ts t♦ ❞❡t❡r♠✐♥❡ t❤✐s r❡❧❛t✐✈❡ ♦r❞❡r ❛♣♣❧✐❡s✳

0

❊①❛♠♣❧❡ ✻✳✶✳✾✿ ♦r❞❡r ❛t

❈♦♥s✐❞❡r t❤❡ ❢✉♥❝t✐♦♥s t❤❛t ❢♦r♠ t❤✐s ❢r❛❝t✐♦♥ ✐♥ t❤❡ ✈✐❝✐♥✐t② ♦❢

7 3 + x−2 − 7x−3 →− −3 2 + 10x 10

❛s

x → 0.

❲❡ ❝♦♥❝❧✉❞❡ t❤❛t t❤❡s❡ t✇♦ ❢✉♥❝t✐♦♥s ❤❛✈❡ t❤❡ ✏s❛♠❡ ♦r❞❡r✑ ❛t ❤✐❣❤❡r ♦r❞❡r ❛t

0

0✿

0✳

❍♦✇❡✈❡r✱ ❜❡❧♦✇ t❤❡ ♥✉♠❡r❛t♦r ✐s ♦❢

t❤❛♥ t❤❡ ❞❡♥♦♠✐♥❛t♦r✿

3 + x−2 − 7x−4 →∞ 2 + 10x−3 ❲❡ ❛r❡ t❤✉s ❛❜❧❡ t♦ ❥✉st✐❢② t❤❡ ❢♦❧❧♦✇✐♥❣ ❤✐❡r❛r❝❤② ❛t

... >>

1/x

x → 0.

0✿

1 1 1 1 1 >> ... >> 2 >> >> √ >> √ >> ... >> 1 3 n x x x x x

❊①❛♠♣❧❡ ✻✳✶✳✶✵✿ r❡❝✐♣r♦❝❛❧s

❇❡❧♦✇✱ ✇❡ ❝♦♠♣❛r❡

❛s

❛♥❞

1/x2 ✿

✻✳✶✳ ▼❛❣♥✐t✉❞❡s ♦❢ ❢✉♥❝t✐♦♥s❀ ▲✬❍♦♣✐t❛❧✬s ❘✉❧❡

✹✹✼

❍♦✇❡✈❡r✱ t❤❡ ♠❡t❤♦❞ ❢❛✐❧s ❢♦r ♦t❤❡r t②♣❡s ♦❢ ❢✉♥❝t✐♦♥s❀ ❢♦r ❡①❛♠♣❧❡✱ ✇❤❡r❡ ❞♦❡s ln x ✜t ✐♥ t❤✐s ❤✐❡r❛r❝❤②❄ ❲❡ ✇✐❧❧ ❜❡ ❧♦♦❦✐♥❣ ❢♦r ❛♥♦t❤❡r ♠❡t❤♦❞✳ ▲❡t✬s ♦❜s❡r✈❡ t❤❛t t❤❡ ♦r❞❡r ♦❢ ❛ ❢✉♥❝t✐♦♥ ✭❛t ✐♥✜♥✐t②✮ ✐s ❞❡t❡r♠✐♥❡❞ ❜② ✐ts ❝❤❛♥❣❡✱ ✐✳❡✳✱ ✐ts ❞✐✛❡r❡♥❝❡✳ ❙♦✱ ❤❡r❡ ✐s ❛♥ ✐❞❡❛✿ ◮ ❚♦ ❝♦♠♣❛r❡ t✇♦ ❢✉♥❝t✐♦♥s✱ ✇❡ ❝♦♠♣❛r❡ t❤❡✐r ❞✐✛❡r❡♥❝❡s ✐♥st❡❛❞✳

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❝♦♠♣✉t❡✿

∆f (x) . x→∞ ∆g lim

❲❤❛t ❛❜♦✉t t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ❄ ❚❤❡ ❞❡♥♦♠✐♥❛t♦rs ❝❛♥❝❡❧✿ ∆f ∆x ∆g ∆x

=

∆f . ∆g

❙♦✱ t❤✐s ✐❞❡❛ ✐s ❥✉st ❛s ✈❛❧✐❞✿ ◮ ❚♦ ❝♦♠♣❛r❡ t✇♦ ❢✉♥❝t✐♦♥s✱ ✇❡ ❝♦♠♣❛r❡ t❤❡✐r ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ✐♥st❡❛❞✳

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❝♦♠♣✉t❡✿

∆f (x) lim ∆x ∆g x→∞ (x) ∆x

.

❋✐♥❛❧❧②✱ t❤❡ ♦r❞❡r ♦❢ ❛ ❢✉♥❝t✐♦♥ ✐s ❞❡t❡r♠✐♥❡❞ ❜② ✐ts r❛t❡ ♦❢ ❣r♦✇t❤✱ ✐✳❡✳✱ ✐ts ❞❡r✐✈❛t✐✈❡✳ ❲❡ t❛❦❡ t❤❡ ✐❞❡❛ t♦ t❤❡ ♥❡①t ❧❡✈❡❧✿ ◮ ❚♦ ❝♦♠♣❛r❡ t✇♦ ❢✉♥❝t✐♦♥s✱ ✇❡ ❝♦♠♣❛r❡ t❤❡✐r ❞❡r✐✈❛t✐✈❡s ✐♥st❡❛❞✳

❆❢t❡r ❛❧❧✱ t❤❡ ❞❡r✐✈❛t✐✈❡ ✐s ❛ ❧✐♠✐t✱ ❜② ❞❡✜♥✐t✐♦♥✳ ❲❡ ❤❛✈❡ ❝♦♠♣✉t❡❞ s♦ ♠❛♥② ❞❡r✐✈❛t✐✈❡s ❜② ♥♦✇ t❤❛t ✇❡ ❝❛♥ ✉s❡ t❤❡ r❡s✉❧ts t♦ ❡✈❛❧✉❛t❡ s♦♠❡ ♦t❤❡r ❧✐♠✐ts✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❝♦♠♣✉t❡✿ df (x) dx lim dg x→∞ (x) dx

.

❲❤❡t❤❡r x ✐s ❛♣♣r♦❛❝❤✐♥❣ ✐♥✜♥✐t❡ ♦r ❛ ♥✉♠❜❡r✱ t❤❡ ❛♣♣r♦❛❝❤ ✐s t❤❡ s❛♠❡✿ ❚❤❡♦r❡♠ ✻✳✶✳✶✶✿ ▲✬❍♦♣✐t❛❧✬s ❘✉❧❡

❙✉♣♣♦s❡ ❢✉♥❝t✐♦♥s f ❛♥❞ g ❛r❡ ❝♦♥t✐♥✉♦✉s❧② ❞✐✛❡r❡♥t✐❛❜❧❡✳ ❚❤❡♥✱ ❢♦r ❡✐t❤❡r ♦❢ t❤❡ t✇♦ t②♣❡s ♦❢ ❧✐♠✐ts✿ x → ±∞ ❛♥❞ x → a± ,

✻✳✶✳

✹✹✽

▼❛❣♥✐t✉❞❡s ♦❢ ❢✉♥❝t✐♦♥s❀ ▲✬❍♦♣✐t❛❧✬s ❘✉❧❡

✇❡ ❤❛✈❡✿ lim

f (x) f ′ (x) = lim ′ g(x) g (x)

✇❤❡♥❡✈❡r t❤❡ ❧❛tt❡r ❧✐♠✐t ❡①✐sts ✭❛s ❛ ♥✉♠❜❡r ♦r ✐♥✜♥✐t②✮✱ ❛♥❞ ♣r♦✈✐❞❡❞ t❤❛t t❤❡ ❧❡❢t✲❤❛♥❞ s✐❞❡ ✐s ❛♥ ✐♥❞❡t❡r♠✐♥❛t❡ ❡①♣r❡ss✐♦♥✿ lim f (x) = lim g(x) = 0 ,

♦r lim f (x) = lim g(x) = ∞ . ❲❛r♥✐♥❣✦ ▲✬❍♦♣✐t❛❧✬s ❘✉❧❡ ✐s

♥♦t

t❤❡ ◗✉♦t✐❡♥t ❘✉❧❡ ✭♦❢ ❧✐♠✐ts

♦r ♦❢ ❞❡r✐✈❛t✐✈❡s✮✳ ❚❤❡ ❢♦r♠❡r ✐s✱ ✐♥ ❢❛❝t✱ ❛ ✇❛② t♦ r❡s♦❧✈❡ ✐♥❞❡t❡r♠✐♥❛❝② t❤❛t ♣r❡❝❧✉❞❡s ❛♣♣❧✐❝❛t✐♦♥ ♦❢ t❤❡ ❧❛tt❡r✳

Pr♦♦❢✳

❲❡ ❝❛♥ ❥✉st✐❢② t❤✐s ✐❞❡❛ ❢♦r t❤❡ ❢♦❧❧♦✇✐♥❣ s✐♠♣❧✐✜❡❞ ❝❛s❡✳ ▲❡t c ❜❡ t❤❡ ♣♦✐♥t ✇❡ ❛r❡ ✐♥t❡r❡st❡❞ ✐♥ ❛♥❞ s✉♣♣♦s❡ t❤❛t ✇❡ ❤❛✈❡ t✇♦ ❢✉♥❝t✐♦♥s f ❛♥❞ g ✇✐t❤✿ f (c) = g(c) = 0 .

❲❡ t❤❡♥ ❝❛♥ ✐♥s❡rt t❤❡s❡ ❡①♣r❡ss✐♦♥s ✐♥t♦ t❤❡✐r r❛t✐♦✿ f (x) − 0 f (x) − f (c) f (x) = = = g(x) g(x) − 0 g(x) − g(c)

f (x)−f (c) x−c g(x)−g(c) x−c

.

❲❡ ❛❧s♦ ❞✐✈✐❞❡❞ ❜♦t❤ ❜② g(x) − g(c)✱ ❢♦r x 6= c✳ ❲❡ ❝♦♥❝❧✉❞❡ ❢r♦♠ t❤❡ ◗✉♦t✐❡♥t

❘✉❧❡

♦❢ ❧✐♠✐ts✿

f ′ (c) f (x) → ′ , g(x) g (c)

♣r♦✈✐❞❡❞ t❤❛t t❤❡ ❞❡r✐✈❛t✐✈❡s ❡①✐st ❛♥❞ g ′ (c) 6= 0✳ ❊①❛♠♣❧❡ ✻✳✶✳✶✷✿ ❢r❛❝t✐♦♥

❈♦♠♣✉t❡

x2 − 3 . x→∞ 2x2 − x + 1 lim

■t✬s ✐♥❞❡t❡r♠✐♥❛t❡✱ ∞/∞✳ ❚❤❡ ❦♥♦✇♥ ♠❡t❤♦❞ ✐s t♦ ❞✐✈✐❞❡ ❜② t❤❡ ❤✐❣❤❡st ♣♦✇❡r✳ ■♥st❡❛❞✱ ✇❡ ❛♣♣❧② ▲✬❍♦♣✐t❛❧✬s ❘✉❧❡ ✿ = lim

x→∞

2x . 4x − 1

❇✉t ✐t✬s st✐❧❧ ✐♥❞❡t❡r♠✐♥❛t❡✦ ❚❤❛t✬s ❛ ❣♦♦❞ ♥❡✇s✿ ❲❡ ❝❛♥ ❛♣♣❧② ▲✬❍♦♣✐t❛❧✬s ❘✉❧❡ ❛❣❛✐♥✦ ❚❤✐s ✐s t❤❡ r❡s✉❧t✿ = lim

x→∞

1 2 = . 4 2

✻✳✶✳

✹✹✾

▼❛❣♥✐t✉❞❡s ♦❢ ❢✉♥❝t✐♦♥s❀ ▲✬❍♦♣✐t❛❧✬s ❘✉❧❡

❲❛r♥✐♥❣✦ ❇❡❢♦r❡ ②♦✉ ❛♣♣❧② ▲✬❍♦♣✐t❛❧✬s ❘✉❧❡✱ ✈❡r✐❢② t❤❛t t❤✐s ✐s ✐♥❞❡❡❞ ❛♥ ✐♥❞❡t❡r♠✐♥❛t❡ ❡①♣r❡ss✐♦♥✳

❊①❛♠♣❧❡ ✻✳✶✳✶✸✿ s✐♠♣❧✐✜❝❛t✐♦♥ ❈♦♠♣✉t❡ lim

x→1

ln x x−1

❘❡s✉❧t✐♥❣ ✐♥ = lim

1 x

∞ ? ❆♣♣❧② ▲❘✳ ∞

1 1 = lim x→1 x = 1. x→1

❊①❛♠♣❧❡ ✻✳✶✳✶✹✿ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥ ❈♦♠♣✉t❡

ex x→∞ x2

∞ ? ❆♣♣❧② ▲❘✳ ∞ ∞ ? ❆♣♣❧② ▲❘✳ ❘❡s✉❧t✐♥❣ ✐♥ ∞

❘❡s✉❧t✐♥❣ ✐♥

lim

ex x→∞ 2x ex = lim x→∞ 2 = ∞. = lim

❚❤❡r❡❢♦r❡✱

ex >> x2 .

■♥ ❢❛❝t✱ ♥♦ ♠❛tt❡r ❤♦✇ ❤✐❣❤ t❤❡ ❞❡❣r❡❡ ✐s✱ ❛ ♣♦❧②♥♦♠✐❛❧ ❝♦♠❡s t♦ ③❡r♦ ❛❢t❡r ❛ s✉✣❝✐❡♥t ♥✉♠❜❡r ♦❢ ❞✐✛❡r❡♥✲ t✐❛t✐♦♥s✳ ▼❡❛♥✇❤✐❧❡✱ ♥♦t❤✐♥❣ ❤❛♣♣❡♥s t♦ t❤❡ ❡①♣♦♥❡♥t✿ ex = ∞. x→∞ xn lim

❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡ ♦✉r ❤✐❡r❛r❝❤② ❛♣♣❡♥❞❡❞✿ ex >> ... >> xn >> ... >> x2 >> x >>

√

x >>

√ 3

x >> ... >> 1

❊①❛♠♣❧❡ ✻✳✶✳✶✺✿ ❧❛r❣❡r t❤❛♥ ❡①♣♦♥❡♥t✐❛❧ ■s t❤❡r❡ ❛r❡ ❛ ❧❛r❣❡r ♦♥❡ t❤❛♥ ex ❄ ❨❡s✱ e2x ✳ ▲❛r❣❡r t❤❛♥ t❤❛t❄ ❨❡s✱ ex ✳ ❆♥❞ s♦ ♦♥✳ 2

❊①❡r❝✐s❡ ✻✳✶✳✶✻ ❆❞❞ ♠♦r❡ t♦ t❤❡ ❧✐st✳ ❖♥ t❤❡ r✐❣❤t t♦♦✳

❊①❡r❝✐s❡ ✻✳✶✳✶✼ ■s t❤❡r❡ ❛ ❧❛r❣❡st ❢✉♥❝t✐♦♥❄

❊①❡r❝✐s❡ ✻✳✶✳✶✽ ■s ✐t ♣♦ss✐❜❧❡ t♦ ✐♥s❡rt ❢✉♥❝t✐♦♥s ✐♥ t❤✐s ❧✐st❄ x2 >> f >> x ❛♥❞ x >> g >>

√

x

✻✳✶✳ ▼❛❣♥✐t✉❞❡s ♦❢ ❢✉♥❝t✐♦♥s❀ ▲✬❍♦♣✐t❛❧✬s ❘✉❧❡

✹✺✵

❲❤❛t ❛❜♦✉t ♦t❤❡r ✐♥❞❡t❡r♠✐♥❛t❡ ❡①♣r❡ss✐♦♥s❄

❇❡❧♦✇ ❛r❡ t❤❡ ♣♦ss✐❜❧❡ t②♣❡s ♦❢ ❧✐♠✐ts t❤❛t ❝♦rr❡s♣♦♥❞ t♦

❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s✿

•

♣r♦❞✉❝ts✿

•

❞✐✛❡r❡♥❝❡s✿

•

♣♦✇❡rs✿

0·∞ ∞−∞ 00 , ∞0 , 1∞ ❲❛r♥✐♥❣✦ ❆♥ ✐♥❞❡t❡r♠✐♥❛t❡ ❡①♣r❡ss✐♦♥ ♦♥ ✐ts ♦✇♥ ✐s ♠❡❛♥✐♥❣✲ ❧❡ss✳ ❲❤❡♥ ✐t ❝♦♠❡s ❢r♦♠ ❛ ❧✐♠✐t t♦ ❜❡ ❡✈❛❧✉❛t❡❞✱ ✐ts ♠❡❛♥✐♥❣ ✐s ❉❊❆❉ ❊◆❉✳

❚❤❡ ✐❞❡❛ ✐s t♦ ❝♦♥✈❡rt t❤❡♠ t♦ ❢r❛❝t✐♦♥s✳

❊①❛♠♣❧❡ ✻✳✶✳✶✾✿ ♣r♦❞✉❝ts ❊✈❛❧✉❛t❡✿

lim x ln x =?

x→0+

❍♦✇ ❞♦ ✇❡ ❛♣♣❧② ▲✬❍♦♣✐t❛❧✬s ❘✉❧❡ ❤❡r❡❄ ❲❡ ❝♦♥✈❡rt t♦ ❛ ❢r❛❝t✐♦♥ ❜② ❞✐✈✐❞✐♥❣ ❜② t❤❡ r❡❝✐♣r♦❝❛❧✿

= lim+ x→0

= lim+ x→0

ln x

❘❡s✉❧t✐♥❣ ✐♥

1 x 1 x

∞ ? ∞

❆♣♣❧② ▲❘✳

❙✐♠♣❧✐❢②✦

− x12

= lim+ −x = 0 . x→0

❊①❛♠♣❧❡ ✻✳✶✳✷✵✿ ❡①♣♦♥❡♥ts ❊✈❛❧✉❛t❡

1

lim x x =?

x→∞

❍♦✇ ❞♦ ✇❡ ❝♦♥✈❡rt t♦ ❛ ❢r❛❝t✐♦♥❄ ❯s❡ t❤❡ ❧♦❣❛r✐t❤♠✦ 1

ln x x =

1 ln x . x

❚❤❡♥✱ ✇❡ ✉s❡ t❤❡ ❢❛❝t t❤❛t t❤❡ ❧♦❣❛r✐t❤♠ ✐s ❝♦♥t✐♥✉♦✉s ✭t❤❡ ✏♠❛❣✐❝ ✇♦r❞s✑✮✿

    1 1 = lim ln x x ln lim x x x→∞  x→∞  1 ln x = lim x→∞ x ln x = lim x→∞ x 1 ▲❘

=== lim

x→∞

❚❤❡r❡❢♦r❡✱

1

x

1

❚❤✐s ✐s ❛ ❢r❛❝t✐♦♥✦ ❘❡s✉❧t✐♥❣ ✐♥

= 0.

lim x x = e0 = 1 .

x→∞

∞ . ∞

✻✳✷✳

✹✺✶

▲✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥s

❊①❡r❝✐s❡ ✻✳✶✳✷✶

❲❤❛t ♣r♦♣❡rt② ♦❢ ❧✐♠✐ts ❞✐❞ ✇❡ ✉s❡ ❛t t❤❡ ❡♥❞❄ ❊①❡r❝✐s❡ ✻✳✶✳✷✷

Pr♦✈❡ x >> ln x >>

√

x ❛t ∞✳

❊①❡r❝✐s❡ ✻✳✶✳✷✸

❈♦♠♣❛r❡ ln x ❛♥❞

1 ❛t 0✳ xn

❊①❛♠♣❧❡ ✻✳✶✳✷✹✿ ♠✐s❛♣♣❧✐❝❛t✐♦♥

❚❤❡ ❝♦♥❞✐t✐♦♥ ♦❢ t❤❡♦r❡♠ t❤❛t r❡q✉✐r❡s t❤❡ r❛t✐♦ t♦ ❜❡ ❛♥ ✐♥❞❡t❡r♠✐♥❛t❡ ❡①♣r❡ss✐♦♥ ♠✉st ❜❡ ✈❡r✐✜❡❞✳ ❲❤❛t ❝❛♥ ❤❛♣♣❡♥ ♦t❤❡r✇✐s❡ ✐s ✐❧❧✉str❛t❡❞ ❜❡❧♦✇✳ ■❢ ✇❡ ❛♣♣❧② ▲✬❍♦♣✐t❛❧✬s ❘✉❧❡✱ ✇❡ ❣❡t t❤❡ ❢♦❧❧♦✇✐♥❣✿
′ 1 + x1 1 + x1 ▲❘❄ − x12 1 lim x = +∞ . = = = = lim = lim = lim
′ 1 2 x→+∞ 1 + 2 x→+∞ x→+∞ 1 + 1 x→+∞ − 3 2 2 x x x ❇✉t t❤❡ ♦r✐❣✐♥❛❧ ❧✐♠✐t ❞♦❡s♥✬t ♣r♦❞✉❝❡ ❛♥ ✐♥❞❡t❡r♠✐♥❛t❡ ❡①♣r❡ss✐♦♥✦ ❚❤✐s ♠❛❦❡s t❤❡ ◗✉♦t✐❡♥t ❧✐♠✐ts ❛♣♣❧✐❝❛❜❧❡✿
1 + x1 ◗❘ limx→+∞ 1 + x1 1
= = 1. lim === 1 1 x→+∞ 1 + 2 1 limx→+∞ 1 + x2 x

❘✉❧❡

❢♦r

❆ ♠✐s♠❛t❝❤✦ ▲✬❍♦♣✐t❛❧✬s ❘✉❧❡ ✐s ✐♥❛♣♣❧✐❝❛❜❧❡ ❜❡❝❛✉s❡ t❤❡ ❧✐♠✐ts ♦❢ t❤❡ ♥✉♠❡r❛t♦r ❛♥❞ t❤❡ ❞❡♥♦♠✐♥❛t♦r ♦❢ t❤❡ ♦r✐❣✐♥❛❧ ❢r❛❝t✐♦♥ ❡①✐st ✦ ❍❡r❡ ✐s ❛♥♦t❤❡r ❡①❛♠♣❧❡✿

2x x2 ▲❘❄ ==== lim = 2. x→1 1 x→1 x lim

❊①❡r❝✐s❡ ✻✳✶✳✷✺

❉❡s❝r✐❜❡ t❤✐s ❝❧❛ss ♦❢ ❢✉♥❝t✐♦♥s✿ o(1)✳ ■♥ s✉♠♠❛r②✱ ✐♥ ❝♦♠♣✉t✐♥❣ t❤❡ ❧✐♠✐t ♦❢ ❛ ❢r❛❝t✐♦♥✱ ✐t ✐s ♦♥❧② ♣♦ss✐❜❧❡ t♦ ❢♦❧❧♦✇

lim

f g

◗✉♦t✐❡♥t ❘✉❧❡

→

lim f lim g

▲✬❍♦♣✐t❛❧✬s ❘✉❧❡

→

lim f ′ lim g ′

ր ց

♦♥❡

♦❢ t❤❡s❡ t✇♦ r♦✉t❡s✿

✻✳✷✳ ▲✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥s

■t ✐s ❛ r❡❛s♦♥❛❜❧❡ str❛t❡❣② t♦ ❛♥s✇❡r ❛ q✉❡st✐♦♥ t❤❛t ②♦✉ ❞♦♥✬t ❦♥♦✇ ❤♦✇ t♦ ❛♥s✇❡r ❜② ❛♥s✇❡r✐♥❣ ✐♥st❡❛❞ ❛♥♦t❤❡r ♦♥❡ ✕ ❝❧♦s❡ t♦ t❤❡ ♦r✐❣✐♥❛❧ ✕ ✇✐t❤ ❛ ❦♥♦✇♥ ❛♥s✇❡r✳ ❋♦r ❡①❛♠♣❧❡✿ √ • ❲❤❛t ✐s t❤❡ sq✉❛r❡ r♦♦t ♦❢ 4.1❄ ■ ❦♥♦✇ t❤❛t 4 = 2✱ s♦ ■✬❧❧ s❛② t❤❛t ✐t✬s ❛❜♦✉t 2✳

✻✳✷✳ ▲✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥s

✹✺✷

• ❲❤❛t ✐s t❤❡ sq✉❛r❡ r♦♦t ♦❢ 4.3❄ ■t✬s ❛❜♦✉t 2✳

• ❲❤❛t ✐s t❤❡ sq✉❛r❡ r♦♦t ♦❢ 3.9❄ ■t✬s ❛❜♦✉t 2✳

❆♥❞ s♦ ♦♥✳

❚❤❡s❡ ❛r❡ ❛❧❧ r❡❛s♦♥❛❜❧❡ ❡st✐♠❛t❡s✱ ❜✉t t❤❡② ❛r❡ ❛❧❧ t❤❡ s❛♠❡✿ x 3.8 3.9 4.0 4.1 4.2 y 2 2 2 2 2

❖✉r ✐♥t❡r♣r❡t❛t✐♦♥ ♦❢ t❤✐s ♦❜s❡r✈❛t✐♦♥ ✐s t❤❛t ✇❡ s♣❡❛❦ ♦❢ ❛♣♣r♦①✐♠❛t✐♥❣ t❤❡ ❢✉♥❝t✐♦♥ f (x) = ❝♦♥st❛♥t ❢✉♥❝t✐♦♥ C(x) = 2✿

√

x ❜② ❛

■t ✐s ❛ ✈❛❧✐❞ ❜✉t ❝r✉❞❡ ❛♣♣r♦①✐♠❛t✐♦♥✦ ❆♥ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ ❛ ❢✉♥❝t✐♦♥ ✐s ❥✉st ❛♥♦t❤❡r ❢✉♥❝t✐♦♥✳ ❚♦ ♠❛❦❡ s❡♥s❡ ♦❢ ✐t✱ ❧❡t✬s ❝♦♥s✐❞❡r t❤❡ ❡rr♦r ♦❢ ❛♥ ❛♣♣r♦①✐♠❛t✐♦♥ ❛s t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ t✇♦ ❢✉♥❝t✐♦♥s✳ ❋♦r t❤❡ ❝❛s❡ ♦❢ ❛ ❝♦♥st❛♥t ❛♣♣r♦①✐♠❛t✐♦♥✱ t❤❡ ❡rr♦rs ❛r❡ ✈✐s✐❜❧❡ ❜❡❧♦✇ ❛s t❤❡ s❡❣♠❡♥ts ✐♥ t❤❡ ②❡❧❧♦✇ ❣❛♣ ❜❡t✇❡❡♥ t❤❡ t✇♦ ❣r❛♣❤s✿

❙♦✱ t❤❡ ❡rr♦r ♦❢ ❛♥ ❛♣♣r♦①✐♠❛t✐♦♥ ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✿ E(x) = |f (x) − C| .

❇✉t ✇❤❛t C ❞♦ ✇❡ ❝❤♦♦s❡❄ ❊✈❡♥ t❤♦✉❣❤ t❤❡ ❝❤♦✐❝❡ ✐s ♦❜✈✐♦✉s✱ ❧❡t✬s r✉♥ t❤r♦✉❣❤ t❤❡ ❛r❣✉♠❡♥t ❛♥②✇❛②✳ ❚❤✐s ✐s t❤❡ r❡q✉✐r❡♠❡♥t ✇❡ ♣✉t ❢♦r✇❛r❞✿ ◮ ❚❤❡ ❡rr♦r ❞✐♠✐♥✐s❤❡s ❛s x ✐s ❣❡tt✐♥❣ ❝❧♦s❡r t♦ a✳

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ✇♦✉❧❞ ❧✐❦❡ t❤❡ ❢♦❧❧♦✇✐♥❣ t♦ ❜❡ s❛t✐s✜❡❞✿ ❚❤❡r❡❢♦r❡✱ ✇❡ r❡q✉✐r❡✿

E(x) = |f (x) − C| → 0 ❛s x → a . f (x) → C ❛s x → a .

❲❡ ❞✐s❝♦✈❡r t❤❛t ✐t s✉✣❝❡s t♦ ❛s❦ f t♦ ❜❡ ❝♦♥t✐♥✉♦✉s ❛t a ❛♥❞ t♦ ❝❤♦♦s❡ C = f (a)✦ ❲❡ ❛r❡ ❞♦♥❡✳ ▲❡t✬s t❛❦❡ t❤✐s t♦ t❤❡ ♥❡①t ❧❡✈❡❧✿

✻✳✷✳ ▲✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥s

◮

✹✺✸

❈❛♥ ✇❡ ❞♦ ❜❡tt❡r t❤❛♥ t❤❡ ❤♦r✐③♦♥t❛❧ ❧✐♥❡❄

❉❡✜♥✐t❡❧②✿

❲❡ ❛❧r❡❛❞② ❦♥♦✇ t❤❛t t❤❡ t❛♥❣❡♥t ❧✐♥❡ ✏❛♣♣r♦①✐♠❛t❡s✑ t❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥❀ ✐♥ ❢❛❝t✱ ✐t ✇❛s ❞❡s✐❣♥❡❞ t♦ ❞♦ t❤❛t ❡①❛❝t❧②✦ ❚❤❡ ✐❞❡❛ ✇❛s t❤❛t ✇❤❡♥ ②♦✉ ③♦♦♠ ✐♥ ♦♥ t❤❡ ♣♦✐♥t✱ t❤❡ t❛♥❣❡♥t ❧✐♥❡ ✇✐❧❧ ♠❡r❣❡ ✇✐t❤ t❤❡ ❣r❛♣❤✿

❍♦✇❡✈❡r✱ t❤❡r❡ ❛r❡ ♠❛♥② str❛✐❣❤t ❧✐♥❡s t❤❛t ❝❛♥ ❜❡ ✉s❡❞ t♦ ❛♣♣r♦①✐♠❛t❡✱ ✐♥❝❧✉❞✐♥❣ t❤❡ ❤♦r✐③♦♥t❛❧ ❧✐♥❡✿

❲❤❛t ✐s s♦ s♣❡❝✐❛❧ ❛❜♦✉t t❤❡ ❢✉♥❝t✐♦♥ r❡♣r❡s❡♥t❡❞ ❜② t❤❡ t❛♥❣❡♥t ❧✐♥❡❄ ❲❡ ❛r❡ ❛❜♦✉t t♦ st❛rt ✉s✐♥❣ ♠♦r❡ ♣r❡❝✐s❡ ❧❛♥❣✉❛❣❡✳ ❘❡❝❛❧❧ t❤❡ ♣♦✐♥t✲s❧♦♣❡ ❢♦r♠ ✭s❡❡♥ ✐♥ ❱♦❧✉♠❡ ✶✱ ❈❤❛♣t❡r

✻✳✷✳

▲✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥s

✶P❈✲✷✮ ♦❢ t❤❡ ❧✐♥❡ t❤r♦✉❣❤

✹✺✹

(x0 , y0 )

✇✐t❤ s❧♦♣❡

m

❛s ❛ r❡❧❛t✐♦♥✿

y − y0 = m(x − x0 ) . ❙♣❡❝✐✜❝❛❧❧②✱ ✇❡ r❡q✉✐r❡ t❤❡ ❝❛♥❞✐❞❛t❡s t♦ ❛t ❧❡❛st ♣❛ss t❤♦✉❣❤ t❤❡ ♣♦✐♥t ♦❢ ✐♥t❡r❡st✿

x0 = a, y0 = f (a) . ❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡✿

y − f (a) = m(x − a) . ❆t t❤✐s ♣♦✐♥t✱ ✇❡ ♠❛❦❡ ❛♥ ✐♠♣♦rt❛♥t st❡♣ ❛♥❞ st♦♣ ❧♦♦❦✐♥❣ ❛t t❤✐s ❛s ❛♥ ❡q✉❛t✐♦♥ ♦❢ ❛ ❧✐♥❡ ❜✉t ❛s ❛ ❢♦r♠✉❧❛ ❢♦r ❛ ♥❡✇

❢✉♥❝t✐♦♥ ✿

L(x) = f (a) + m(x − a) x ✐s 1✱ ✇❤✐❧❡ f ❛t x = a✳

■t ✐s ❛ ❧✐♥❡❛r ❢✉♥❝t✐♦♥✿ ❚❤❡ ♣♦✇❡r ♦❢

t❤❡ r❡st ♦❢ t❤❡ ♣❛r❛♠❡t❡rs ❛r❡ ❝♦♥st❛♥t✳ ❚❤❛t✬s ✇❤② ✐t ✐s

❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ ◆♦✇✱ ❛❧❣❡❜r❛✳ ❖♥❝❡ ❛❣❛✐♥✱ ❧❡t✬s ❧♦♦❦ ❛t t❤❡ ❡rr♦r✱ ✐✳❡✳✱ t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ t✇♦ ❢✉♥❝t✐♦♥s✿

❝❛❧❧❡❞ ❛

E(x) = |f (x) − L(x)| . ❆❧❧ ♦❢ t❤❡ t❤r❡❡ ❢✉♥❝t✐♦♥s ❛r❡

❝♦♥t✐♥✉♦✉s

❛t

a✦

❚❤❡r❡❢♦r❡✱ t❤❡ ❡rr♦r ✇✐❧❧ ❞✐♠✐♥✐s❤ ❛s

E(x) = |f (x) − L(x)| → 0 ■♥ ♦t❤❡r ✇♦r❞s✱ ❡✈❡r② ❧✐♥❡❛r ❢✉♥❝t✐♦♥

y = L(x)

❛s

x

✐s ❣❡ts ❝❧♦s❡r t♦

a✿

x → a.

❞❡✜♥❡❞ ❛s ❛❜♦✈❡ ✇✐❧❧ ♣❛ss t❤❡ t❡st t❤❛t ✇❡ ✉s❡❞ t♦ ❝❤♦♦s❡

t❤❡ ❜❡st ❝♦♥st❛♥t ❛♣♣r♦①✐♠❛t✐♦♥✿

❚❤❛t✬s ♥♦t ❣♦♦❞ ❡♥♦✉❣❤ ❛♥②♠♦r❡✦ ▲❡t✬s ❧♦♦❦ ❛t t❤❡s❡ ❧✐♥❡❛r ❢✉♥❝t✐♦♥s✱

❛♥❣❧❡

y = L(x)✱

❛s ❛ ❣r♦✉♣✳ ❲❤❡♥ ②♦✉ ③♦♦♠ ✐♥ ♦♥ t❤❡ ♣♦✐♥t✱ ✇❡ r❡❛❧✐③❡ t❤❡

❜❡t✇❡❡♥ t❤❡ ❧✐♥❡s ✐s ♣r❡s❡r✈❡❞ ✉♥❞❡r t❤❡ ♠❛❣♥✐✜❝❛t✐♦♥✿

❍♦✇ ❞♦ ✇❡ ❝❤♦♦s❡ t❤❡ ✏❜❡st✑ ❛♣♣r♦①✐♠❛t✐♦♥❄

❚❤❡ ♦♥❧② ❞✐✛❡r❡♥❝❡ ❛♠♦♥❣ t❤❡♠ ✐s ❤♦✇ ❢❛st t❤❡ ❡rr♦r ✐s

❞✐♠✐♥✐s❤✐♥❣✳ ❚❤✐s ✐s t❤❡ ♣❧❛♥✿

◮

❲❡ ✇✐❧❧ ❝❤♦♦s❡ t❤❡ ❧✐♥❡❛r ❢✉♥❝t✐♦♥ ✇✐t❤ t❤❡ ❡rr♦r ❝♦♥✈❡r❣✐♥❣ t♦

❙♣❡❝✐✜❝❛❧❧②✱ ✇❡

❝♦♠♣❛r❡

0

t❤❡ ❢❛st❡st ♣♦ss✐❜❧❡✳

t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ❡rr♦r ✇✐t❤ t❤❛t ♦❢ t❤❡ ❞✐st❛♥❝❡ ❢✉♥❝t✐♦♥✿

E(x) = |f (x) − L(x)| → 0

✈s✳

|∆x| = |x − a| → 0 .

✻✳✷✳

▲✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥s

✹✺✺

❍♦✇ ❞♦ ✇❡ ❝♦♠♣❛r❡ t❤❡ ✏s♣❡❡❞ ♦❢ ❝♦♥✈❡r❣❡♥❝❡✑❄ ❲❡ ❦♥♦✇ t❤❡ ❛♥s✇❡r ❢r♦♠ ♦✉r st✉❞② ♦❢ t❤❡ ♠❛❣♥✐t✉❞❡s ♦❢ ❢✉♥❝t✐♦♥ ❡❛r❧✐❡r ✐♥ t❤✐s ❝❤❛♣t❡r✿ ▲♦♦❦ ❛t t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡✐r r❛t✐♦✦ ❚❤❡♥✱

E(x)

f (x) − L(x) →0 x−a

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿

|∆x|

❝♦♥✈❡r❣❡s ❢❛st❡r t❤❛♥ ❛s

✐❢ t❤❡ ❧✐♠✐t ♦❢ t❤✐s r❛t✐♦ ✐s ③❡r♦❀ ✐✳❡✳✱

x → a.

f (x) − L(x) = o(x − a) .

❙✉r♣r✐s✐♥❣❧②✱ t❤✐s ❣❡♥❡r❛❧ ❝♦♥❞✐t✐♦♥ ✐s s✉✣❝✐❡♥t t♦ ♠❛❦❡ t❤❡ s❧♦♣❡

m

♦❢

L

s♣❡❝✐✜❝✦

❚❤❡♦r❡♠ ✻✳✷✳✶✿ ❇❡st ▲✐♥❡❛r ❆♣♣r♦①✐♠❛t✐♦♥

❙✉♣♣♦s❡ f ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t x = a ❛♥❞ L(x) = f (a) + m(x − a)

✐s ❛♥② ♦❢ ✐ts ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥s✳ ❚❤❡♥✱ f (x) − L(x) = 0 ⇐⇒ m = f ′ (a) . x→a x−a lim

Pr♦♦❢✳ ❆s

x → a✱

✇❡ ❤❛✈❡✿

f (x) − (f (a) + m(x − a)) f (x) − f (a) = −m x−a x−a → f ′ (a) −m . ❲❛r♥✐♥❣✦ ❚❤❡

❜❡st

❧✐♥❡❛r

❛♣♣r♦①✐♠❛t✐♦♥

✐s

♦❢t❡♥

s✐♠♣❧②

❝❛❧❧❡❞ t❤❡ ✏❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥✑✳

❊①❛♠♣❧❡ ✻✳✷✳✷✿ r♦♦t ▲❡t✬s ❛♣♣r♦①✐♠❛t❡

√

4.1✳

√

x ❜② ❤❛♥❞✳ ■♥ ❢❛❝t✱ t❤❡ ♦♥❧② ❲❡ ❝❛♥✬t ❝♦♠♣✉t❡ √ ♠❡❛♥✐♥❣ ♦❢ x 2 x = 4.1✳ ■♥ t❤❛t s❡♥s❡✱ t❤❡ ❢✉♥❝t✐♦♥ f (x) = x ✐s ✿

✉♥❦♥♦✇♥

❏✉st ❛ ❢❡✇ ❡①❝❡♣t✐♦♥s ❛r❡ ❚❤❡♥✱ ✇❡ ❝❛♥ ✉s❡ t❤❡

f (4) =

√

4 = 2✱ f (9) =

❝♦♥st❛♥t ❛♣♣r♦①✐♠❛t✐♦♥

√

9 = 3✱

√

4.1 ✐s t❤❛t ✐t ✐s s✉❝❤ ❛ ♥✉♠❜❡r t❤❛t

❡t❝✳ ❲❡ ✇✐❧❧ ✉s❡ t❤❡s❡ ♣♦✐♥ts ❛s ✏❛♥❝❤♦rs✑✳

❛♥❞ ❞❡❝❧❛r❡✿

√

=

4.1 ≈ 2 .

❲❡ ❛r❡ ❞♦♥❡ ✐❢ t❤❡ q✉❛❧✐t② ♦❢ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ✐s s✉✣❝✐❡♥t✳

✻✳✷✳

▲✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥s

❏✉st ❛s t❤❡ ❢✉♥❝t✐♦♥

✹✺✻

f (x) =

√

x

✐s ✉♥❦♥♦✇♥✱ s♦ ✐s ✐ts ❞❡r✐✈❛t✐✈❡✿

1 f ′ (x) = √ . 2 x ❇✉t✱ ❥✉st ❛s

f

✐s ❦♥♦✇♥ ❛t

x = 4✱

s♦ ✐s

f ′✿ 1 1 f ′ (4) = √ = . 4 2 4

❲❤❛t ❝❛♥ ✇❡ ❞♦ ✇✐t❤ t❤✐s ✐♥❢♦r♠❛t✐♦♥❄ ❚❤❡ ❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢

f

✐s

❦♥♦✇♥ ✐♥ t❤❡ s❡♥s❡ t❤❛t✱ ❛s ❛ ❧✐♥❡❛r ❢✉♥❝t✐♦♥✱ ✐t ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞

❜② ❤❛♥❞✿

L(x) = f (a) +f ′ (a)(x − a)

❚❤✐s ❢✉♥❝t✐♦♥ ✐s ❛

r❡♣❧❛❝❡♠❡♥t

❢♦r

=2 √ f (x) = x ✐♥

❋✐♥❛❧❧②✱ ✇❡ ❝♦♠♣✉t❡ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥

1 + (x − 4) . 4

t❤❡ ✈✐❝✐♥✐t② ♦❢ t❤❡ ✏❛♥❝❤♦r ♣♦✐♥t✑

x = 4✳

❜② ❤❛♥❞ ✿

1 L(4.1) = 2 + (4.1 − 4) 4 1 = 2 + · .1 4 = 2 + 0.025 = 2.025 . ❖✉r ❝♦♥❝❧✉s✐♦♥ ✐s t❤❛t ❢♦r t❤❡

❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ √

✇❡ ❤❛✈❡✿

4.1 ≈ 2.025 .

❊①❡r❝✐s❡ ✻✳✷✳✸ ❋✐♥❞ t❤❡ ❜❡st ❝♦♥st❛♥t ❛♥❞ ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥s ♦❢

f (x) =

√

3.99✳

■♥ s✉♠♠❛r②✱ t❤❡ ❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ✐s ❣✐✈❡♥ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✉❧❛✿

L(x) = f (a) + f ′ (a)(x − a) ❲❛r♥✐♥❣✦ ❉♦♥✬t ♣❧✉❣ ✐♥ t❤❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❞❡r✐✈❛t✐✈❡ ✐♥t♦ t❤❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥✳

❊①❡r❝✐s❡ ✻✳✷✳✹ ❋✐♥❞ t❤❡ ❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢

f (x) = x1/3

❛t

a = 1✳

❯s❡ ✐t t♦ ❡st✐♠❛t❡

1.11/3 ✳

✻✳✷✳

▲✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥s

✹✺✼

❊①❛♠♣❧❡ ✻✳✷✳✺✿ s✐♥❡ ❛♥❞ ❝♦s✐♥❡ ❋✐♥❞✐♥❣ t❤❡ ❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ❛♠♦✉♥ts t♦ ✜♥❞✐♥❣ t❤❡ t❛♥❣❡♥t ❧✐♥❡ ❛t t❤❡ ♣♦✐♥t✳

❙♦♠❡ ❛r❡

❛❧r❡❛❞② ❦♥♦✇♥✿

▲❡t✬s r❡❞♦ t❤❡♠✿

❢✉♥❝t✐♦♥s✿ ❡✈❛❧✉❛t❡❞ ❛t

0:

x=0

sin x

❞❡r✐✈❛t✐✈❡s✿ ❡✈❛❧✉❛t❡❞ ❛t

y = sin x sin x

0:


′


′ sin x x=0

t❛♥❣❡♥ts✿

y − 0 = 1 · (x − 0)

❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥s✿

L(x) = x

y = cos x = sin 0 = 0 cos x x=0
′ = cos x cos x
′ = cos 0 = 1 cos x x=0

= cos 0 = 1 = − sin x = − sin 0 = 0

y − 1 = 0 · (x − 0) l(x) = 1

❚❤❡♥✱ ✇❡ ❤❛✈❡

sin .2 ≈ .2, sin −.01 ≈ −.01,

❡t❝✳

❛♥❞

cos .2 ≈ 1, cos −.01 ≈ 1,

❡t❝✳

❊①❡r❝✐s❡ ✻✳✷✳✻ ❲❤❛t ❛r❡ t❤❡ ❜❡st ❝♦♥st❛♥t ❛♣♣r♦①✐♠❛t✐♦♥s ♦❢ t❤❡ s✐♥❡ ❛♥❞ t❤❡ ❝♦s✐♥❡❄

❊①❡r❝✐s❡ ✻✳✷✳✼ ❯s❡ t❤❡ ❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢

f (x) =

❊①❛♠♣❧❡ ✻✳✷✳✽✿ ♣✐❡❝❡✇✐s❡✲❞❡✜♥❡❞ ■❢ ✇❡ ✇❡r❡ t♦ ❞❡s✐❣♥ ❛ ❝❛❧❝✉❧❛t♦r✱ ✇❡✬❞

√

sin x

t♦ ❡st✐♠❛t❡

p sin π/2 ✳

♥❡❡❞ ❡♥♦✉❣❤ ✏❛♥❝❤♦r✑ ♣♦✐♥ts ✐♥ ♦r❞❡r t♦ ♣r♦❞✉❝❡ ❛♥ ❛♣♣r♦①✲

✐♠❛t✐♦♥ ♦❢ t❤❡ ✇❤♦❧❡✱ ✉♥❦♥♦✇♥✱ ❢✉♥❝t✐♦♥✳

❋♦r

f (x) =

√

x✱

✇❡ ✉s❡ ✐t ♦r ✇❡ ❧♦♦❦ ❢♦r ❛♥♦t❤❡r ♥✉♠❜❡r ✇✐t❤ ❛ ❦♥♦✇♥ sq✉❛r❡ r♦♦t✳ ❋♦r ❡①❛♠♣❧❡✱

√

10 ≈

√

9 = 3✱

4✱ 1 = 1✱

❛s ❧♦♥❣ ❛s t❤❡ ♥✉♠❜❡r ✐s ✏❝❧♦s❡✑ t♦

❛♥❞ s♦ ♦♥✳

❋♦r t❤❡ ❝♦♥st❛♥t ❛♣♣r♦①✐♠❛t✐♦♥✱ ✇❡ s✇✐t❝❤ ❤❛❧❢✲✇❛② ❜❡t✇❡❡♥ t❤❡ ♣♦✐♥ts✿

√

.99 ≈

√

✻✳✷✳ ▲✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥s

✹✺✽

❋♦r t❤❡ ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥✱ ✇❡ s✇✐t❝❤ ✇❤❡r❡✈❡r t❤❡ t❛♥❣❡♥ts ✐♥t❡rs❡❝t✿

❊①❡r❝✐s❡ ✻✳✷✳✾ ❲❤❛t ♠❛❦❡s ❛♣♣r♦①✐♠❛t✐♥❣ t❤❡ ❢✉♥❝t✐♦♥ ❛r♦✉♥❞ 0 s❡❡♠ t♦ ❜❡ ♣♦♦r❄ ❘❡♣❧❛❝✐♥❣ ❛ ❢✉♥❝t✐♦♥ ✇✐t❤ ✐ts ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ✐s ❝❛❧❧❡❞ ❧✐♥❡❛r✐③❛t✐♦♥✳ ▲✐♥❡❛r✐③❛t✐♦♥s ♠❛❦❡ ❛ ❧♦t ♦❢ t❤✐♥❣s ♠✉❝❤ s✐♠♣❧❡r✳

❊①❛♠♣❧❡ ✻✳✷✳✶✵✿ ❞✐✛❡r❡♥t✐❛t✐♦♥ ❛♥❞ ✐♥t❡❣r❛t✐♦♥ ❚❤❡ s✐♥❡ ❛♥❞ ❝♦s✐♥❡ ❛r❡ r❡❧❛t❡❞ t♦ ❡❛❝❤ ♦t❤❡r ✈✐❛ ❞✐✛❡r❡♥t✐❛t✐♦♥ ❛♥❞ ✐♥t❡❣r❛t✐♦♥✳ ❆r❡ t❤❡✐r ❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥s r❡❧❛t❡❞ ✐♥ t❤❛t ♠❛♥♥❡r❄ ❨❡s✱ ❜✉t ♥♦t ❡①❛❝t❧②✳

✻✳✷✳ ▲✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥s ❉✐✛❡r❡♥t✐❛t✐♦♥✿

✹✺✾

d → cos x dx d ❛♣♣r♦①✐♠❛t✐♦♥s✿ x → → 1 dx d → ❝♦♥st❛♥t ♣♦❧②♥♦♠✐❛❧s✿ ❧✐♥❡❛r → dx

❢✉♥❝t✐♦♥s✿

sin x

→

■♥t❡❣r❛t✐♦♥✿ ❢✉♥❝t✐♦♥s✿

sin x

❛♣♣r♦①✐♠❛t✐♦♥s✿ x ♣♦❧②♥♦♠✐❛❧s✿

→ →

❧✐♥❡❛r →

Z

Z

Z

→ − cos x + C →

1 2 x +C 2

→ q✉❛❞r❛t✐❝

❚❤❡ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢ sin x ✐s t❤❡ q✉❛❞r❛t✐❝ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ − cos x✿

▲✐♥❡❛r✐③❛t✐♦♥s ♠❛② ❤❡❧♣ ❡①♣❧❛✐♥✐♥❣ t❤❡ ✐❞❡❛s ♦❢ ❝♦♥t✐♥✉✐t② ❛♥❞ ❞✐✛❡r❡♥t✐❛❜✐❧✐t②✳ ❊①❛♠♣❧❡ ✻✳✷✳✶✶✿ ❝♦♥t✐♥✉✐t②

❆❧❧ ❧✐♥❡❛r ❢✉♥❝t✐♦♥s✱ L(x) = mx + b, m 6= 0✱ ❛r❡✱ ♦❢ ❝♦✉rs❡✱ ❝♦♥t✐♥✉♦✉s✳ ❋✉rt❤❡r♠♦r❡✱ t❤❡ r❡❧❛t✐♦♥✱ ✐♥ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❝♦♥t✐♥✉✐t② ✭❈❤❛♣t❡r ✷✮✱ ❜❡t✇❡❡♥ ε ❛♥❞ δ ✐s ✈❡r② s✐♠♣❧❡✿ ❚♦ ❡♥s✉r❡ |x − a| < δ =⇒ |L(x) − L(a)| < ε ,

✇❡ s✐♠♣❧② ❝❤♦♦s❡✿ δ=

1 ε. m

✻✳✷✳ ▲✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥s

✹✻✵

❲❡ ♥♦✇ ✐♥t❡r♣r❡t t❤❡s❡ t✇♦✱ ❛s ❜❡❢♦r❡✿ • δ = ∆x ✐s t❤❡ ❛❝❝✉r❛❝② ♦❢ t❤❡ ♠❡❛s✉r❡♠❡♥t ♦❢ x✳ • ε = ∆y ✐s t❤❡ ❛❝❝✉r❛❝② ♦❢ t❤❡ ✐♥❞✐r❡❝t ❡✈❛❧✉❛t✐♦♥ ♦❢ y = L(x)✳ ❚❤❡♥✱ ♦❢ ❝♦✉rs❡✱ ✇❡ ❤❛✈❡✿ ∆y = m∆y ♦r ∆y = L′ (a)∆y . ❲❡ ❝❛♥ ❤❛✈❡ ❛ s✐♠✐❧❛r✱ ❜✉t t❤✐s t✐♠❡ ❛♣♣r♦①✐♠❛t❡✱ ❛♥❛❧②s✐s ❢♦r ❛♥② ❢✉♥❝t✐♦♥f ✿ ◮ ❲❡ ❝❛♥ ♠❛❦❡ ∆y = f (a + ∆x) − f (a) ❛s s♠❛❧❧ ❛s ✇❡ ❧✐❦❡ ❜② ❝❤♦♦s✐♥❣ ∆x s♠❛❧❧ ❡♥♦✉❣❤✳

❍♦✇ ❞♦ ✇❡ ✜♥❞ t❤✐s ∆x❄ ❲❡ ❧✐♥❡❛r✐③❡ ✿

❚❤❡ ❢✉♥❝t✐♦♥ ✐s r❡♣❧❛❝❡❞ ✇✐t❤ ✐ts ❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ❛t x = a ❛♥❞ t❤❡ ❛♥❛❧②s✐s ✐♥ t❤❡ ❧❛st ❡①❛♠♣❧❡ ✐s ❛♣♣❧✐❡❞ t♦ t❤✐s ❢✉♥❝t✐♦♥✿ f ′ (a) =

dy dx

x=a

≈

∆y . ∆x

❚❤❡♥✱ t❤❡ ❝❤❛♥❣❡ ♦❢ t❤❡ ♦✉t♣✉t ✈❛r✐❛❜❧❡ ✐s ❛♣♣r♦①✐♠❛t❡❧② ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ❝❤❛♥❣❡ ♦❢ t❤❡ ✐♥♣✉t ✈❛r✐❛❜❧❡✿ ∆y ≈ f ′ (a) ∆x

▼♦r❡ ♣r❡❝✐s❡❧②✱ t❤❡s❡ t✇♦ ❢✉♥❝t✐♦♥s ❛r❡ ♦❢ t❤❡ s❛♠❡ ♦r❞❡r ❛s ❡①♣❧❛✐♥❡❞ ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥✿ ∆y ∼ f ′ (a) ∆x .

❍♦✇ ✇❡❧❧ t❤✐s ❛♣♣r♦①✐♠❛t✐♦♥ ✇♦r❦s ✐s ❞✐s❝✉ss❡❞ ✐♥ t❤❡ ♥❡①t s❡❝t✐♦♥✳ ❚❤✐s ✐❞❡❛ ❝❛♥ ❛❧s♦ ❜❡ ❡①♣r❡ss❡❞ ✈✐❛ t❤❡ ❞✐✛❡r❡♥t✐❛❧s ✿ dy = f ′ (a) dx

❚❤❡ ❡q✉❛t✐♦♥ ✐s✱ ✐♥ ❢❛❝t✱ t❤❡✐r ❞❡✜♥✐t✐♦♥✳ ❊①❛♠♣❧❡ ✻✳✷✳✶✷✿ ❡rr♦r ❡st✐♠❛t✐♦♥

▲❡t✬s r❡✈✐s✐t t❤❡ ❡①❛♠♣❧❡ ♦❢ ❡✈❛❧✉❛t✐♥❣ t❤❡ ❛r❡❛ A = f (x) = x2 ♦❢ sq✉❛r❡ t✐❧❡s ✇❤❡♥ t❤❡✐r ❞✐♠❡♥s✐♦♥s ❛r❡ ❝❧♦s❡ t♦ 10 × 10✳ ❚❤❡ ❛r❡❛ ✐s s✉♣♣♦s❡❞ t♦ ❜❡ 100✱ ❜✉t ❤♦✇ ❝❧♦s❡ t♦ t❤❡ tr✉t❤ ✐s t❤✐s ♥✉♠❜❡r❄ ❚❤❡ q✉❡st✐♦♥ ♠❛② ❜❡ ✐♠♣♦rt❛♥t ❢♦r ♣❛ss✐♥❣ ❛ q✉❛❧✐t② ✐♥s♣❡❝t✐♦♥✳ ❲❡ ❧♦♦❦ ❛t ❤♦✇ t❤❡ ❡rr♦r ♦❢ t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ❛r❡❛ ❞❡♣❡♥❞s ♦♥ t❤❡ ❡rr♦r ♦❢ t❤❡ ♠❡❛s✉r❡♠❡♥t ♦❢ t❤❡ s✐❞❡ x✿

✻✳✷✳

▲✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥s

✹✻✶

❙✉♣♣♦s❡ t❤❡ ❞❡s✐r❡❞ ❛❝❝✉r❛❝② ♦❢ ✇❡ ❞✐s❝♦✈❡r❡❞ t❤❛t

∆x = .2

A

✐s

∆A = 5✱

✇❤❛t s❤♦✉❧❞ ❜❡ t❤❡ ❛❝❝✉r❛❝②

∆x

♦❢

x❄

❇② ❜r✉t❡ ❢♦r❝❡✱

✐s ❛♣♣r♦♣r✐❛t❡✿

A = (10 ± .2)2 = 102 ± 2 · 10 · .2 + .22 ♦r 100.04 ± 4 . ❚❤✐s t✐♠❡✱ ✐♥st❡❛❞✱ ✇❡ ❣❡t ❛ q✉✐❝❦ ✏❜❛❧❧♣❛r❦✑ ✜❣✉r❡ ❜② ✉s✐♥❣ t❤❡ ❧✐♥❡❛r✐③❛t✐♦♥ ♦❢ ❲❡ ✜♥❞ t❤❡ ❞❡r✐✈❛t✐✈❡✱

2x✱

❛t

f (x) = x2

❛t

x = 10✳

x = 10✿ f ′ (10) = 2 · 10 = 20 .

❚❤❡♥ ✇❡ ✉s❡ t❤❡ ❢♦r♠✉❧❛ ❢♦r ❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥✿

L(x) = 10 + 20(x − 10) . ❚❤❡♥

|∆A| = |A(10) − A(x)| ≈ |L(10) − L(x)| = |20(x − 10)| = 20|∆x| . ❲❡ ❝❛♥ s❡❡ t❤❡ r❡❧❛t✐♦♥ ❜❡❧♦✇✿

❚❤❡r❡❢♦r❡✱

|∆x| ≈ 5/20 = .25 . ❙♦✱ ✐♥ ♦r❞❡r t♦ ❛❝❤✐❡✈❡ t❤❡ ❛❝❝✉r❛❝② ♦❢ ♦♥❡ ✇✐❧❧ ♥❡❡❞ t❤❡

1/4✲✐♥❝❤

4 sq✉❛r❡ ✐♥❝❤❡s ♦❢ t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ❛r❡❛ ♦❢ ❛ 10 × 10 t✐❧❡✱

❛❝❝✉r❛❝② ♦❢ t❤❡ ♠❡❛s✉r❡♠❡♥t ♦❢ t❤❡ s✐❞❡ ♦❢ t❤❡ t✐❧❡✳

❊①❡r❝✐s❡ ✻✳✷✳✶✸

❲❤❛t ✐s t❤❡ ❛❝❝✉r❛❝② ♦❢ t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ❛r❡❛ ✐❢ t❤❡ ❛❝❝✉r❛❝② ❢♦r t❤❡ s✐❞❡ ✐s

.03❄

✻✳✸✳

❚❤❡ ❛❝❝✉r❛❝② ♦❢ t❤❡ ❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥

✹✻✷

❊①❡r❝✐s❡ ✻✳✷✳✶✹

❲❤❛t ✐s t❤❡ ❛❝❝✉r❛❝② ♦❢ t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ❝✐r❝✉♠❢❡r❡♥❝❡ ✭t❤❡ ❧❡♥❣t❤✮ ♦❢ ❛ ❝✐r❝❧❡ ✐❢ ✐ts r❛❞✐✉s ✐s ❢♦✉♥❞ t♦ ❜❡

20 ± 1

✐♥❝❤❡s❄ ❲❤❛t ❛❜♦✉t t❤❡ ❛r❡❛❄

❊①❡r❝✐s❡ ✻✳✷✳✶✺

❲❤❛t ✐s t❤❡ ❛❝❝✉r❛❝② ♦❢ t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ s✉r❢❛❝❡ ❛r❡❛ ♦❢ t❤❡ ❊❛rt❤ ✐❢ t❤❡ r❛❞✐✉s ✐s ❢♦✉♥❞ t♦ ❜❡

6, 360 ± 30

❦✐❧♦♠❡t❡rs❄ ❲❤❛t ❛❜♦✉t t❤❡ ✈♦❧✉♠❡❄

❆s ❛ s✉♠♠❛r②✱ ❜❡❧♦✇ ✇❡ ✐❧❧✉str❛t❡ ❤♦✇ ✇❡ ❛tt❡♠♣t t♦ ❛♣♣r♦①✐♠❛t❡ ❛ ❢✉♥❝t✐♦♥ ❛r♦✉♥❞ t❤❡ ♣♦✐♥t

(1, 1)

✇✐t❤

❝♦♥st❛♥t ❢✉♥❝t✐♦♥s ✜rst✿ ❢r♦♠ t❤♦s❡ ✇❡ ❝❤♦♦s❡ t❤❡ ❤♦r✐③♦♥t❛❧ ❧✐♥❡ t❤r♦✉❣❤ t❤❡ ♣♦✐♥t✳ ❚❤✐s ❧✐♥❡ t❤❡♥ ❜❡❝♦♠❡s ♦♥❡ ♦❢ ♠❛♥② ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥s ♦❢ t❤❡ ❝✉r✈❡ t❤❛t ♣❛ss t❤r♦✉❣❤ t❤❡ ♣♦✐♥t✿ ❢r♦♠ t❤♦s❡ ✇❡ ❝❤♦♦s❡ t❤❡ t❛♥❣❡♥t ❧✐♥❡✳ ❚❤❡ t✇♦ st❡♣s ❛r❡ s❤♦✇♥ ❜❡❧♦✇✿

■♥ ❱♦❧✉♠❡ ✸✱ ❈❤❛♣t❡r ✸■❈✲✺✱ ✇❡ ✇✐❧❧ s❡❡ t❤❛t t❤❡s❡ ❛r❡ ❥✉st t❤❡ t✇♦ ✜rst st❡♣s ✐♥ t❤❡ ✐♥✜♥✐t❡ s❡q✉❡♥❝❡ ♦❢ ❛♣♣r♦①✐♠❛t✐♦♥s✳

✻✳✸✳ ❚❤❡ ❛❝❝✉r❛❝② ♦❢ t❤❡ ❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥

❆♥ ✏❛♣♣r♦①✐♠❛t✐♦♥✑ ✐s ♠❡❛♥✐♥❣❧❡ss ✐❢ ✐t ❝♦♠❡s ❛s ❛ s✐♥❣❧❡ ♥✉♠❜❡r✳

3.14 t♦ π ❄ ❊✈❡♥ ❣✐✈❡ π ❛ ❞❡✜♥✐t✐✈❡

❲❡ ♥❡❡❞ t♦ ❦♥♦✇ ♠♦r❡ ✐♥ ♦r❞❡r t♦ ♠❛❦❡ t❤✐s ♥✉♠❜❡r ✉s❡❢✉❧✳ ❋♦r ❡①❛♠♣❧❡✱ ❤♦✇ ❝❧♦s❡ ✐s ♠♦r❡ ✐♠♣♦rt❛♥t ✐s t❤❡ q✉❡st✐♦♥✿ ❤♦✇ ❝❧♦s❡ ✐s

π

t♦

3.14❄

r❛♥❣❡ ♦❢ ♣♦ss✐❜❧❡ ✈❛❧✉❡s✿

❲❤❛t✬s ✐♠♣♦rt❛♥t ✐s t❤❛t

π

❝❛♥♥♦t ❜❡ ❛♥②✇❤❡r❡ ❡❧s❡✦

❚❤❡ ❛♥s✇❡r t♦ t❤❡ q✉❡st✐♦♥ ✇✐❧❧

✻✳✸✳

❚❤❡ ❛❝❝✉r❛❝② ♦❢ t❤❡ ❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥

❖r✱ ❢r♦♠ t❤❡ ❧❛st s❡❝t✐♦♥✱ ❤♦✇ ❝❧♦s❡ ✐s

√

4.1 t♦ 2.025❄

❚❤❡ ❛♥s✇❡r ✇✐❧❧ t❡❧❧ ✉s ✇❤❡r❡ t❤❡ tr✉t❤ ❧✐❡s✱ ✇✐t❤ ❋✐rst✱ t❤❡

✹✻✸

❝♦♥st❛♥t ❛♣♣r♦①✐♠❛t✐♦♥✳

❛♥ ❛❜s♦❧✉t❡ ❝❡rt❛✐♥t② ✐❢ ♥♦t ❛❜s♦❧✉t❡ ❛❝❝✉r❛❝②✳

❊✈❡♥ t❤♦✉❣❤ t❤❡ ❢✉♥❝t✐♦♥ y = f (x) ❝♦✐♥❝✐❞❡s ✇✐t❤ y = C = f (a) ❛t x = a✱ ✐t ♠❛② r✉♥ ❛✇❛② ✈❡r② ❢❛st ❛♥❞ ✈❡r② ❢❛r ❛❢t❡r✇❛r❞s✳ ❍♦✇ ❢❛r❄ ❚❤❡ ♦♥❧② ❧✐♠✐t ✐s t❤❡ r❛t❡ ♦❢ ❣r♦✇t❤ ♦❢ f ✱ ✐✳❡✳✱ ✐ts ❞❡r✐✈❛t✐✈❡✿

❛ ♣r✐♦r✐ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t |f |✳ ❲❡ ✇✐❧❧ ❤❛✈❡ ❛ r❛♥❣❡ ♦❢ ♣♦ss✐❜❧❡ ✈❛❧✉❡s ❢♦r f (x)✱ ❢♦r ❛❧❧ x✦ ❚❤❡ r❡s✉❧t ✐s ❛ ❢✉♥♥❡❧ t❤❛t ❝♦♥t❛✐♥s t❤❡ ✭✉♥❦♥♦✇♥✮ ❙♦✱ ✇❡ ❝❛♥ ♣r❡❞✐❝t t❤❡ ❜❡❤❛✈✐♦r ♦❢ f ✐❢ ✇❡ ❤❛✈❡

′

❣r❛♣❤ ♦❢ y = f (x)✿

■♥❞❡❡❞✱ ✇❡ ❝❛♥ ❝♦♥❝❧✉❞❡ ❢r♦♠ t❤❡

▼❡❛♥ ❱❛❧✉❡ ❚❤❡♦r❡♠ t❤❛t

|f (x) − C(x)| = |f (x) − f (a)| = |f ′ (t) · (x − a)| = |f ′ (t)| · |x − a| ≤ K|x − a| ,

✐❢ ✇❡ ♦♥❧② ❦♥♦✇ t❤❛t |f ′ (t)| ≤ K ❢♦r ❛❧❧ t ❜❡t✇❡❡♥ a ❛♥❞ x✳ ❊①❛♠♣❧❡ ✻✳✸✳✶✿ ❛♣♣r♦①✐♠❛t❡ sq✉❛r❡ r♦♦t

√

√

❍♦✇ ❝❧♦s❡ ✐s 4.1 t♦ 2❄ ❲❡ ♥❡❡❞ t♦ ✜♥❞ ❛ ❜♦✉♥❞ ❢♦r t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ f (x) = x✳ ❍❡r❡ ✇❡ t❛❦❡ ✐♥t♦ ❛❝❝♦✉♥t t❤❛t f ′ ✐s ❞❡❝r❡❛s✐♥❣ ❛♥❞✱ t❤❡r❡❢♦r❡✱ ✐ts ❧❛r❣❡st ✈❛❧✉❡ ✐s ✐♥ t❤❡ ❜❡❣✐♥♥✐♥❣✿ 1 1 f ′ (t) = √ ≤ √ ≤ .25 . 2 t 2 4 √ ❚❤❛t ❝♦✉❧❞ ❜❡ K ✳ ❚❤❡♥ ✇❡ ❤❛✈❡ ❜♦✉♥❞s ❢♦r t❤❡ ✉♥❦♥♦✇♥ 4.1✿ √ | 4.1 − 2| ≤ .25 · |4.1 − 4| = .025 .

■♥ ♦t❤❡r ✇♦r❞s✱ ◆❡①t✱ t❤❡

√

4.1 ❧✐❡s ✇✐t❤✐♥ [2 − .025, 2 + .025] = [1.975, 2.025] .

❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥✳

❚❤❡ t✇♦ ❣r❛♣❤s ❜❡❧♦✇ ❤❛✈❡ t❤❡ s❛♠❡ t❛♥❣❡♥t ❧✐♥❡ ❜✉t t❤❡ ✜rst ♦♥❡ s❡❡♠s ❜❡tt❡r ❛♣♣r♦①✐♠❛t❡❞✿

✻✳✸✳ ❚❤❡ ❛❝❝✉r❛❝② ♦❢ t❤❡ ❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥

✹✻✹

❲❤❛t ❝❛✉s❡s t❤❡ ❞✐✛❡r❡♥❝❡❄ ❚❤❡ ❧❛tt❡r ✐s ♠♦r❡ ❝✉r✈❡❞ ✭❝♦♥❝❛✈❡✮✦ ❚❤❡ q✉❛♥t✐t② t❤❛t ♠❛❦❡s t❤❡ s❧♦♣❡s✱ ✐✳❡✳✱ t❤❡ ❞❡r✐✈❛t✐✈❡s✱ ❝❤❛♥❣❡ ✐s t❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡✳ ❲❡ ❤❛✈❡ ❛♥ ❛♥❛❧♦❣②✿ • ❲❡ ❦♥♦✇ t❤❡ ❛❝❝✉r❛❝② ♦❢ t❤❡ ❜❡st ❝♦♥st❛♥t ❛♣♣r♦①✐♠❛t✐♦♥ ✐❢ ✇❡ ❤❛✈❡ ❛ ❜♦✉♥❞ ♦♥ t❤❡ ♠❛❣♥✐t✉❞❡ ♦❢

t❤❡ ✜rst ❞❡r✐✈❛t✐✈❡✳

• ❲❡ ❦♥♦✇ t❤❡ ❛❝❝✉r❛❝② ♦❢ ♦❢ t❤❡ ❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ✐❢ ✇❡ ❤❛✈❡ ❛ ❜♦✉♥❞ ♦♥ t❤❡ ♠❛❣♥✐t✉❞❡ ♦❢

t❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡✳

❚❤❡ ❧❛tt❡r t❛❦❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✿ ❚❤❡♦r❡♠ ✻✳✸✳✷✿ ❊rr♦r ❇♦✉♥❞ ❢♦r ❇❡st ▲✐♥❡❛r ❆♣♣r♦①✐♠❛t✐♦♥

❙✉♣♣♦s❡ f ✐s t✇✐❝❡ ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t x = a ❛♥❞ L(x) = f (a) + f ′ (a)(x − a) ✐s ✐ts ❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ❛t a✳ ❚❤❡♥ ✇❡ ❤❛✈❡✿ 1 E(x) = |f (x) − L(x)| ≤ K(x − a)2 2

✇❤❡r❡ K ✐s ❛ ❜♦✉♥❞ ♦❢ t❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡ ♦♥ t❤❡ ✐♥t❡r✈❛❧ ❢r♦♠ a t♦ x✱ ✐✳❡✳✱ |f ′′ (t)| ≤ K ❢♦r ❛❧❧ t ✐♥ t❤✐s ✐♥t❡r✈❛❧.

❚❤❡ t❤❡♦r❡♠ ❝❧❛✐♠s t❤❛t E = o (x − a)2




❖♥❝❡ K ✐s ✜①❡❞✱ ✇❡ ❤❛✈❡ ❛ r❛♥❣❡ ♦❢ ♣♦ss✐❜❧❡ ✈❛❧✉❡s ♦❢ t❤❡ ❢✉♥❝t✐♦♥ y = f (x)✦ ❚❤❡ r❡s✉❧t ✐s✱ ❛❣❛✐♥✱ ❛ ❢✉♥♥❡❧ t❤❛t ❝♦♥t❛✐♥s t❤❡ ✭✉♥❦♥♦✇♥✮ ❣r❛♣❤ ♦❢ y = f (x)✿

✻✳✸✳

❚❤❡ ❛❝❝✉r❛❝② ♦❢ t❤❡ ❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥

✹✻✺

❚❤✐s t✐♠❡✱ t❤❡ t✇♦ ❡❞❣❡s ♦❢ t❤❡ ❢✉♥♥❡❧ ❛r❡ ♣❛r❛❜♦❧❛s✿ 1 y = L(x) ± K(x − a)2 . 2

❚❤❡r❡❢♦r❡✱ t❤❡ ✈❛❧✉❡ ♦❢ K ♠❛❦❡s t❤❡ ❢✉♥♥❡❧ ♣r♦♣♦rt✐♦♥❛❧❧② ✇✐❞❡r ♦r ♥❛rr♦✇❡r✳ ❚❤❡ ♣r❛❝t✐❝❛❧ ♠❡❛♥✐♥❣ ♦❢ t❤❡ t❤❡♦r❡♠ ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✿ ❋♦r ❡❛❝❤ x✱ t❤✐s ❡rr♦r ❜♦✉♥❞✱

✐s t❤❡ ❛❝❝✉r❛❝②

1 ε(x) = K(x − a)2 , 2

♦❢ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ✐♥ t❤❡ s❡♥s❡ t❤❛t t❤❡ ✐♥t❡r✈❛❧ ❧♦❝❛t❡❞ ♦♥ t❤❡ y✲❛①✐s 

L(x) − ε(x), L(x) − ε(x)

❝♦♥t❛✐♥s t❤❡ tr✉❡ ♥✉♠❜❡r✱ f (x)✳ ❲❡ s❡❡ t❤✐s ❜❡❧♦✇✿



❚❤✐s ✐♥t❡r✈❛❧ ✐s ❛ ✈❡rt✐❝❛❧ ❝r♦ss✲s❡❝t✐♦♥ ♦❢ t❤❡ ❢✉♥♥❡❧✳ ❊①❛♠♣❧❡ ✻✳✸✳✸✿ r♦♦t

❲❡ ❝♦♥t✐♥✉❡ ✇✐t❤ t❤❡ ❧❛st ❡①❛♠♣❧❡✿

√

4.1 ≈ L(4.1) = 2.025 .

❖♥❝❡ ✐s ✉♥s❛t✐s❢❛❝t♦r② ❜❡❝❛✉s❡ ✐t ❞♦❡s♥✬t r❡❛❧❧② t❡❧❧ ✉s ❛♥②t❤✐♥❣ ❛❜♦✉t t❤❡ tr✉❡ ✈❛❧✉❡ √ ❛❣❛✐♥✱ t❤❡ ❛♥s✇❡r √ ♦❢ 4.1✳ ■♥ ❢❛❝t✱ 4.1 6= 2.025✦

▲❡t✬s ❛♣♣❧② t❤❡ t❤❡♦r❡♠✳ ❋✐rst✱ ✇❡ ✜♥❞ t❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡✿ √

1 x =⇒ f ′ (x) = √ =⇒ 2 x  ′  ′ 1 1 −1/2 ′′ √ f (x) = x = ❆❝❝♦r❞✐♥❣ t♦ P♦✇❡r ❋♦r♠✉❧❛✳ 2 2 x  1 1 = − x−1/2−1 2 2 1 −3/2 =− x . 4

f (x)

=

✻✳✸✳

❚❤❡ ❛❝❝✉r❛❝② ♦❢ t❤❡ ❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥

✹✻✻

❙♦✱ ✇❡ ♥❡❡❞ ❛ ❜♦✉♥❞ ❢♦r t❤✐s ❢✉♥❝t✐♦♥✿ |f ′′ (x)| =

1 −3/2 x 4

♦✈❡r t❤❡ ✐♥t❡r✈❛❧ [4, 4.1]✳ ❚❤✐s ✐s ❛ s✐♠♣❧❡✱ ❞❡❝r❡❛s✐♥❣ ❢✉♥❝t✐♦♥✿

❚❤❡r❡❢♦r❡✱ |f ′′ (x)| ≤ |f ′′ (4)| =

1 −3/2 1 4 = = 0.03125 . 4 32

❚❤✐s ✐s ♦✉r ❜❡st ❝❤♦✐❝❡ ❢♦r K ✦ ❆❝❝♦r❞✐♥❣ t♦ t❤❡ t❤❡♦r❡♠✱ ♦✉r ❝♦♥❝❧✉s✐♦♥ ✐s t❤❛t t❤❡ ❡rr♦r ♦❢ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❝❛♥♥♦t ❜❡ ❧❛r❣❡r t❤❛♥ t❤❡ ❢♦❧❧♦✇✐♥❣✿ 1 E(x) = |f (x) − L(x)| ≤ 0.03125 · (x − 4)2 . 2

❙♣❡❝✐✜❝❛❧❧②✱

1 √ E(4.1) = 4.1 − L(4.1) ≤ 0.03125 · (4.1 − 4)2 , 2 √ 4.1 − 2.025 ≤ 0.00015625 .

♦r

❚❤❡r❡❢♦r❡✱ √ ✇❡ ❝♦♥❝❧✉❞❡✿ ◮ 4.1 ✐s ✇✐t❤✐♥ ε = 0.00015625 ❢r♦♠ 2.025 ✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿ 2.025 − 0.00015625 ≤

♦r 2.02484375 ≤

√ √

4.1 ≤ 2.025 + 0.00015625 ,

4.1 ≤ 2.02515625 .

❲❡ ❤❛✈❡ ❢♦✉♥❞ ❛♥ ✐♥t❡r✈❛❧ t❤❛t ✐s ❣✉❛r❛♥t❡❡❞ t♦ ❝♦♥t❛✐♥ t❤❡ ♥✉♠❜❡r ✇❡ ❛r❡ ❧♦♦❦✐♥❣ ❢♦r✦ ❊①❛♠♣❧❡ ✻✳✸✳✹✿ s✐♥

❆♣♣r♦①✐♠❛t❡ sin .01✳ ◆♦t❡ t❤❛t t❤❡ ❝♦♥st❛♥t ❛♣♣r♦①✐♠❛t✐♦♥ ✐s sin .01 ≈ sin 0 = 0✳ ❙♦✱ ✇❡ ❤❛✈❡ a = 0✳ ◆♦✇ ✇❡ ❝♦♠♣✉t❡✿ f (x) = sin x f ′ (x) = cos x

❚❤✉s✱

=⇒ L(x) = 0 + cos x

f ′′ (x) = − sin x =⇒ |f ′′ (x)| = sin x

❛♥❞✱ ❢✉rt❤❡r♠♦r❡✱ t❤❡ ❛❝❝✉r❛❝② ✐s ❛t ✇♦rst

x=0

(x − 0) =⇒ L(x) = x

=⇒ |f ′′ (x)| ≤ 1 = K

sin .01 ≈ .01 ,

1 ε = K(x − a)2 = .5 · 1 · .012 = .00005 . 2

✻✳✹✳

❙♦❧✈✐♥❣ ❡q✉❛t✐♦♥s ♥✉♠❡r✐❝❛❧❧②✿ ❜✐s❡❝t✐♦♥ ❛♥❞ ◆❡✇t♦♥✬s ♠❡t❤♦❞

✹✻✼

❚❤❡r❡❢♦r❡✱ .01 − .00005 = .00995 ≤ sin .01 ≤ .01005 = .01 + .00005 .

◆♦t❡ t❤❛t t❤❡ ❝❤♦✐❝❡ ♦❢ K ✐♥ t❤❡ ❧❛st ❡①❛♠♣❧❡ ✇❛s ❜❡st ♣♦ss✐❜❧❡✳ ❚❤✐s ✇❛s✱ t❤❡r❡❢♦r❡✱ ❛ ✇♦rst✲❝❛s❡ s❝❡♥❛r✐♦✳ ■♥ ❝♦♥tr❛st✱ ❤❡r❡ K = 1 ✐s♥✬t t❤❡ ❜❡st ♣♦ss✐❜❧❡ ❝❤♦✐❝❡✳ ❇② ❧✐♠✐t✐♥❣ ♦✉r ❛tt❡♥t✐♦♥ t♦ t❤❡ r❡❧❡✈❛♥t ♣❛rt ♦❢ t❤❡ ❣r❛♣❤ ♦❢ |f ′′ | ✇❡ ❞✐s❝♦✈❡r ❛ ❜❡tt❡r ✈❛❧✉❡ ❢♦r t❤❡ ❜♦✉♥❞ K = .01 ✳ ❚❤✐s ✐♠♣r♦✈❡s t❤❡ ❛❝❝✉r❛❝② ❜② ❛ ❢❛❝t♦r ♦❢ .01 ✳ ❑♥♦✇✐♥❣ t❤❡ ❝♦♥❝❛✈✐t② ♦❢ t❤❡ ❢✉♥❝t✐♦♥ t❤❛t ✇❡ ❛♣♣r♦①✐♠❛t❡ ❝✉ts t❤❡ ❢✉♥♥❡❧ ✐♥ ❤❛❧❢✳

❯♥❞❡r t❤❡ r❡str✐❝t✐♦♥s ♦❢ t❤❡ t❤❡♦r❡♠✱ ✇❡ ❝♦♥❝❧✉❞❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿ • ❲❤❡♥ f ✐s ❝♦♥❝❛✈❡ ✉♣ ✭✐✳❡✳✱ f ′′ > 0✮✱ ✇❡ ❤❛✈❡ ❢♦r ❡❛❝❤ x ✐♥ [a, b]✿ 1 L(x) ≤ f (x) ≤ L(x) + K(x − a)2 . 2 • ❲❤❡♥ f ✐s ❝♦♥❝❛✈❡ ❞♦✇♥ ✭✐✳❡✳✱ f ′′ < 0✮✱ ✇❡ ❤❛✈❡ ❢♦r ❡❛❝❤ x ✐♥ [a, b]✿ 1 L(x) − K(x − a)2 ≤ f (x) ≤ L(x) . 2

✻✳✹✳ ❙♦❧✈✐♥❣ ❡q✉❛t✐♦♥s ♥✉♠❡r✐❝❛❧❧②✿ ❜✐s❡❝t✐♦♥ ❛♥❞ ◆❡✇t♦♥✬s ♠❡t❤♦❞

❲❤❛t ❞♦❡s ✐t ♠❡❛♥ t♦ s♦❧✈❡

❛♥ ❡q✉❛t✐♦♥

❄ ▲❡t✬s r❡✈✐❡✇ ✭❱♦❧✉♠❡ ✶✱ ❈❤❛♣t❡r ✶P❈✲✺✮✳

❲❡ st❛rt ✇✐t❤✿ ◮ ●✐✈❡♥ ❛ ❢✉♥❝t✐♦♥ y = f (x)✱ ✜♥❞ s✉❝❤ ❛ d t❤❛t f (d) = 0✳

●❡♦♠❡tr✐❝❛❧❧②✱ ✇❡ s♣❡❛❦ ♦❢ t❤❡ x✲✐♥t❡r❝❡♣ts ♦❢ t❤❡ ❢✉♥❝t✐♦♥✿

✻✳✹✳ ❙♦❧✈✐♥❣ ❡q✉❛t✐♦♥s ♥✉♠❡r✐❝❛❧❧②✿ ❜✐s❡❝t✐♦♥ ❛♥❞ ◆❡✇t♦♥✬s ♠❡t❤♦❞

✹✻✽

❆❧❣❡❜r❛✐❝ ♠❡t❤♦❞s ❛r❡ ♣r❡❝✐s❡ ❜✉t ♦♥❧② ❛♣♣❧② t♦ ❛ ✈❡r② ♥❛rr♦✇ ❝❧❛ss ♦❢ ❢✉♥❝t✐♦♥s ✭s✉❝❤ ❛s ♣♦❧②♥♦♠✐❛❧s ♦❢ ❞❡❣r❡❡ ❜❡❧♦✇

5✮✳

❲❤❛t ❛❜♦✉t ❛♣♣r♦①✐♠❛t❡ s♦❧✉t✐♦♥s ❄ ❙♦❧✈✐♥❣ t❤❡ ❡q✉❛t✐♦♥

y = f (x)

✏♥✉♠❡r✐❝❛❧❧②✑ ♠❡❛♥s

t❤❡ ❢♦❧❧♦✇✐♥❣✿

◮ ❚❤❡♥

dn

❋✐♥❞ ❛ s❡q✉❡♥❝❡ ♦❢ ♥✉♠❜❡rs

dn

s✉❝❤ t❤❛t

dn → d

f (d) = 0✳

❛♥❞

d✳

❛r❡ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥s ♦❢ t❤❡ ✉♥❦♥♦✇♥

▲❡t✬s r❡❝❛❧❧ t❤❡ ♣r♦♦❢ ♦❢ t❤❡ ■♥t❡r♠❡❞✐❛t❡ ❱❛❧✉❡ ❚❤❡♦r❡♠ ✭❈❤❛♣t❡r ✶✮✳ ✐t❡r❛t❡❞ s❡❛r❝❤ ❢♦r ❛ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❡q✉❛t✐♦♥

f (x) = 0✳

❲❡ ✐♥t❡r♣r❡t❡❞ t❤✐s ♣r♦♦❢ ❛s ❛♥

❲❡ ❝♦♥str✉❝t❡❞ ❛ s❡q✉❡♥❝❡ ♦❢ ♥❡st❡❞ ✐♥t❡r✈❛❧s ❜②

❝✉tt✐♥❣ t❤❡♠ ✐♥ ❤❛❧❢ ❛❣❛✐♥ ❛♥❞ ❛❣❛✐♥✳ ■t ✐s ❝❛❧❧❡❞ ❜✐s❡❝t✐♦♥✳ ❆ ❢✉♥❝t✐♦♥

f

✐s ❞❡✜♥❡❞ ❛♥❞ ✐s ❝♦♥t✐♥✉♦✉s ♦♥ ❛♥ ✐♥t❡r✈❛❧

[a, b]

✇✐t❤

f (a) < 0, f (b) > 0 . ❲❡ ✇❛♥t t♦ ❛♣♣r♦①✐♠❛t❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿

d ❲❡ st❛rt ✇✐t❤ t❤❡ ❤❛❧✈❡s ♦❢

✐♥

[a, b]

s✉❝❤ t❤❛tf (d)

= 0.

[a, b]✿ 

1 a, (a + b) 2

❖✈❡r ♦♥❡ ♦❢ t❤❡♠ ✭♦r ❜♦t❤✮✱ t❤❡ ❢✉♥❝t✐♦♥

f (a) · f



f



❛♥❞



 1 (a + b), b . 2

❝❤❛♥❣❡s ✐ts s✐❣♥✳ ❚❤✐s ❢❛❝t ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ❛s ❢♦❧❧♦✇s✿

1 (a + b) 2



< 0 ❖❘ f

❲❤✐❝❤❡✈❡r ✐t ✐s✱ ✇❡ r❡♥❛♠❡ t❤❡ ❡♥❞s ♦❢ t❤✐s ✐♥t❡r✈❛❧

a1



❛♥❞

 1 (a + b) · f (b) < 0 . 2

b1 ✿

◆❡①t✱ ✇❡ ❝♦♥s✐❞❡r t❤❡ ❤❛❧✈❡s ♦❢ t❤✐s ♥❡✇ ✐♥t❡r✈❛❧ ❛♥❞ s♦ ♦♥✳ ❲❡ ❝♦♥t✐♥✉❡ ✇✐t❤ t❤✐s ♣r♦❝❡ss ❛♥❞ t❤❡ r❡s✉❧t ✐s t✇♦ s❡q✉❡♥❝❡s ♦❢ ♥✉♠❜❡rs✿

an

❛♥❞

bn .

✻✳✹✳ ❙♦❧✈✐♥❣ ❡q✉❛t✐♦♥s ♥✉♠❡r✐❝❛❧❧②✿ ❜✐s❡❝t✐♦♥ ❛♥❞ ◆❡✇t♦♥✬s ♠❡t❤♦❞

✹✻✾

❊①❡r❝✐s❡ ✻✳✹✳✶

❲r✐t❡ ❛ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛ ❢♦r t❤❡s❡ ✐♥t❡r✈❛❧s✳

❚❤❡s❡ t✇♦ s❡q✉❡♥❝❡s ♦❢ ♥✉♠❜❡rs ❢♦r♠ ❛ ✏♥❡st❡❞✑ s❡q✉❡♥❝❡ ♦❢ ✐♥t❡r✈❛❧s✿

[a, b] ⊃ [a1 , b1 ] ⊃ [a2 , b2 ] ⊃ ... ❲❤❛t ✐s s♣❡❝✐❛❧ ❛❜♦✉t t❤❡s❡ ✐♥t❡r✈❛❧s ✐s t❤❛t ♦♥ ❡❛❝❤ ♦❢ t❤❡♠

f

❝❤❛♥❣❡s ✐ts s✐❣♥✿

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿

f (an ) < 0, f (bn ) > 0

♦r

f (an ) > 0, f (bn ) < 0 .

❲❡ ❝♦♥❝❧✉❞❡ t❤❛t t❤❡ s❡q✉❡♥❝❡s ❝♦♥✈❡r❣❡ t♦ t❤❡ s❛♠❡ ✈❛❧✉❡✱

an → d, bn → d . ❋✉rt❤❡r♠♦r❡✱ ❢r♦♠ t❤❡ ❝♦♥t✐♥✉✐t② ♦❢

f✱

✇❡ ❝♦♥❝❧✉❞❡ t❤❛t t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❛❧s♦ ❝♦♥✈❡r❣❡✿

f (an ) → f (d), f (bn ) → f (d) . ❚❤❡r❡❢♦r❡✱

d

✐s ❛♥ ✭✉♥❦♥♦✇♥✮ s♦❧✉t✐♦♥✿

f (d) = 0 . ❚❤❡ ♣r♦❜❧❡♠ ✐s s♦❧✈❡❞✦ ❊①❛♠♣❧❡ ✻✳✹✳✷✿

sin x = 0

▲❡t✬s r❡✈✐❡✇ ❤♦✇ t❤❡ ❜✐s❡❝t✐♦♥ ♠❡t❤♦❞ s♦❧✈❡s ❛ s♣❡❝✐✜❝ ❡q✉❛t✐♦♥✿

sin x = 0 . ❲❡ st❛rt❡❞ ✇✐t❤ t❤❡ ✐♥t❡r✈❛❧

[a1 , b1 ] = [3, 3.5]

❛♥❞ ✉s❡❞ t❤❡ ❢♦❧❧♦✇✐♥❣ s♣r❡❛❞s❤❡❡t ❢♦r♠✉❧❛ ❢♦r

❂■❋✭❘❬✲✶❪❈❬✸❪✯❘❬✲✶❪❈❬✹❪❁✵✱❘❬✲✶❪❈✱❘❬✲✶❪❈❬✶❪✮ ❛♥❞

bn ✿

❚❤❡ r❡s✉❧ts ❛r❡ ❜❡❧♦✇✿

❂■❋✭❘❬✲✶❪❈❬✶❪✯❘❬✲✶❪❈❬✷❪❁✵✱❘❬✲✶❪❈❬✲✶❪✱❘❬✲✶❪❈✮

an ✿

✻✳✹✳

✹✼✵

❙♦❧✈✐♥❣ ❡q✉❛t✐♦♥s ♥✉♠❡r✐❝❛❧❧②✿ ❜✐s❡❝t✐♦♥ ❛♥❞ ◆❡✇t♦♥✬s ♠❡t❤♦❞

❚❤❡ ✈❛❧✉❡s ♦❢ an , bn ❛r❡ ✈✐s✐❜❧② ❛♣♣r♦❛❝❤✐♥❣ π ✭❛♥❞ s♦ ❞♦❡s dn ✱ t❤❡ ♠✐❞✲♣♦✐♥t ♦❢ t❤❡ ✐♥t❡r✈❛❧✮ ✇❤✐❧❡ t❤❡ ✈❛❧✉❡s ♦❢ f (an ), f (bn ) ❛♣♣r♦❛❝❤ 0✳ ❊①❡r❝✐s❡ ✻✳✹✳✸

❙♦❧✈❡ t❤❡ ❡q✉❛t✐♦♥ sin x = .2✳ ❚❤❡r❡ ❛r❡ ♦t❤❡r ♠❡t❤♦❞s ❢♦r s♦❧✈✐♥❣ f (x) = 0✳ ❖♥❡ ✐s t♦ ✉s❡ t❤❡ ❧✐♥❡❛r✐③❛t✐♦♥ ♦❢ f ❛s ❛ s✉❜st✐t✉t❡✱ ♦♥❡ s✉❝❤ ❛♣♣r♦①✐♠❛t✐♦♥ ❛t ❛ t✐♠❡✳ ❙✉♣♣♦s❡ ❛ ❢✉♥❝t✐♦♥ f ✐s ❣✐✈❡♥ ❛s ✇❡❧❧ ❛s t❤❡ ✐♥✐t✐❛❧ ❡st✐♠❛t❡ x0 ♦❢ ❛ s♦❧✉t✐♦♥ d ♦❢ t❤❡ ❡q✉❛t✐♦♥ f (x) = 0✳ ❲❡ r❡♣❧❛❝❡ f ✐♥ t❤✐s ❡q✉❛t✐♦♥ ✇✐t❤ ✐ts ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ L ❛t t❤✐s ✐♥✐t✐❛❧ ♣♦✐♥t✿ L(x) = f (x0 ) + f ′ (x0 )(x − x0 ) .

❚❤❡♥ ✇❡ s♦❧✈❡ t❤❡ ❡q✉❛t✐♦♥ L(x) = 0 ❢♦r x✿ L(x) = f (x0 ) + f ′ (x0 )(x − x0 ) = 0 .

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ✜♥❞ t❤❡ ✐♥t❡rs❡❝t✐♦♥ ♦❢ t❤❡ t❛♥❣❡♥t ❧✐♥❡ ✇✐t❤ t❤❡ x✲❛①✐s✿

❚❤❡ ❡q✉❛t✐♦♥✱ ✇❤✐❝❤ ✐s ❧✐♥❡❛r✱ ✐s ❡❛s② t♦ s♦❧✈❡✳ ❚❤❡ ♣♦✐♥t ♦❢ ✐♥t❡rs❡❝t✐♦♥ ✐s x1 = x0 −

f (x0 ) . f ′ (x0 )

❚❤❡♥ ✇❡ r❡♣❡❛t t❤❡ ♣r♦❝❡ss ❢♦r x1 ✳ ❆♥❞ s♦ ♦♥✳✳✳ ❚❤✐s ✐s ❝❛❧❧❡❞ ◆❡✇t♦♥✬s

✳ ■t ✐s ❛ s❡q✉❡♥❝❡ ♦❢ ♥✉♠❜❡rs ❣✐✈❡♥ r❡❝✉rs✐✈❡❧②✿

♠❡t❤♦❞

xn+1 = xn −

f (xn ) f ′ (xn )

❲❛r♥✐♥❣✦

❚❤❡ ♠❡t❤♦❞ ❢❛✐❧s ✇❤❡♥ ✐t r❡❛❝❤❡s ❛ ♣♦✐♥t ✇❤❡r❡ t❤❡ ❞❡r✐✈❛t✐✈❡ ✐s ❡q✉❛❧ t♦ ✭♦r ❡✈❡♥ ❝❧♦s❡ t♦✮

❊①❛♠♣❧❡ ✻✳✹✳✹✿ ◆❡✇t♦♥✬s ♠❡t❤♦❞

▲❡t✬s ✉s❡ ◆❡✇t♦♥✬s ♠❡t❤♦❞ t♦ s♦❧✈❡ t❤❡ s❛♠❡ ❡q✉❛t✐♦♥ ❛s ❛❜♦✈❡✿ sin x = 0 .

❚❤❡ s♣r❡❛❞s❤❡❡t ❢♦r♠✉❧❛ ❢♦r xn ✐s ❛s ❢♦❧❧♦✇s✿ ❂❘❬✲✶❪❈✲❙■◆✭❘❬✲✶❪❈✮✴❈❖❙✭❘❬✲✶❪❈✮

❲❡ st❛rt ✇✐t❤ x0 = 3✳ ❚❤❡ s❡q✉❡♥❝❡ ❝♦♥✈❡r❣❡s t♦ π ✈❡r② q✉✐❝❦❧② ✭❧❡❢t✮✿

0✳

✻✳✺✳ P❛rt✐❝❧❡ ✐♥ ❛ ✢♦✇

✹✼✶

❍♦✇❡✈❡r✱ ❛s ♦✉r ❝❤♦✐❝❡ ♦❢ x0 ❝❤❛♥❣❡s ❛♥❞ ❣❡ts ❝❧♦s❡r t♦ π/2✱ t❤❡ ✈❛❧✉❡ ♦❢ x1 ❜❡❝♦♠❡s ❧❛r❣❡r ❛♥❞ ❧❛r❣❡r✳ ■♥ ❢❛❝t✱ ✐t ♠✐❣❤t ❜❡ s♦ ❧❛r❣❡ t❤❛t t❤❡ s❡q✉❡♥❝❡ ✇♦♥✬t ✉❧t✐♠❛t❡❧② ❝♦♥✈❡r❣❡ t♦ π ❜✉t t♦ 2π, 100π ✱ ❡t❝✳ ■❢ ✇❡ ❝❤♦♦s❡ x0 = π/2 ❡①❛❝t❧②✱ t❤❡ t❛♥❣❡♥t ❧✐♥❡ ✐s ❤♦r✐③♦♥t❛❧ ❛♥❞ t❤❡ ❛❧❣♦r✐t❤♠ ❜r❡❛❦s ❞♦✇♥✦ ❊①❡r❝✐s❡ ✻✳✹✳✺

❙♦❧✈❡ t❤❡ ❡q✉❛t✐♦♥ sin x = .2✳

✻✳✺✳ P❛rt✐❝❧❡ ✐♥ ❛ ✢♦✇ Pr❡✈✐♦✉s❧②✱ ✇❡ ❝♦♥s✐❞❡r❡❞ ❛♥ ❡①❛♠♣❧❡ ♦❢ ❤♦✇ ❢✉♥❝t✐♦♥s ❛♣♣❡❛r ❛s ❞✐r❡❝t r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ t❤❡ ❞②♥❛♠✐❝s ❣✐✈❡♥ ❜② ❛♥ ✐♥❞✐r❡❝t ❞❡s❝r✐♣t✐♦♥✳ ❋♦r ❡①❛♠♣❧❡✱ ❛♥ ♦❜❥❡❝t✬s ✈❡❧♦❝✐t② ✐s ❞❡r✐✈❡❞ ❢r♦♠ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥✱ ❛♥❞ t❤❡♥ t❤❡ ❧♦❝❛t✐♦♥ ❢r♦♠ t❤❡ ✈❡❧♦❝✐t②✳ ❚❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ✭♦r t❤❡ ❢♦r❝❡✮ ✐s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✐♠❡✱ ❛♥❞ s♦ ✐s t❤❡ ✈❡❧♦❝✐t②✳ ❆ ❞✐✛❡r❡♥t✱ ❛♥❞ ❥✉st ❛s ✐♠♣♦rt❛♥t✱ ❡①❛♠♣❧❡ ♦❢ s✉❝❤ ❡♠❡r❣❡♥❝❡ ✐s ✢♦✇s ♦❢ ❧✐q✉✐❞s✳ ❚❤❡ ❞✐✛❡r❡♥❝❡ ✐s t❤❛t ❤❡r❡ t❤❡ ✈❡❧♦❝✐t② ✐s ❛ ❢✉♥❝t✐♦♥ ♦❢ ❧♦❝❛t✐♦♥✳ ❚❤❡ t❛s❦ ✐s t♦ ❞❡t❡r♠✐♥❡ t❤❡ ❧♦❝❛t✐♦♥ ♦❢ ❛ ♣❛rt✐❝❧❡ ✭❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✐♠❡✮ ✐♥ ❛ ✢♦✇ ♦❢ ❧✐q✉✐❞✿

❲❡ st❛rt ✇✐t❤ ❛ ❞✐s❝r❡t❡ ♠♦❞❡❧ ❛♥❞ ❧✐♠✐t ♦✉rs❡❧✈❡s t♦ t❤❡ 1✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡✿ ♣✐♣❡s ❛♥❞ ❝❛♥❛❧s✳ ❙✉♣♣♦s❡ t❤❡r❡ ✐s ❛ ♣✐♣❡ ✇✐t❤ t❤❡ ✈❡❧♦❝✐t② ♦❢ t❤❡ str❡❛♠ ♠❡❛s✉r❡❞ s♦♠❡❤♦✇ ❛t ❡❛❝❤ ❧♦❝❛t✐♦♥✳ ❙✐♠✐❧❛r❧②✱ t❤✐s ✐s ❛ ❝❛♥❛❧ ✇✐t❤ t❤❡ ✇❛t❡r t❤❛t ❤❛s t❤❡ ❡①❛❝t s❛♠❡ ✈❡❧♦❝✐t② ✕ ♣❛r❛❧❧❡❧ t♦ t❤❡ ❝❛♥❛❧ ✕ ❛t ❛❧❧ ❧♦❝❛t✐♦♥s ❛❝r♦ss ✐t✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ✈❡❧♦❝✐t② ♦♥❧② ✈❛r✐❡s ❛❧♦♥❣ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ♣✐♣❡ ♦r t❤❡ ❝❛♥❛❧✿

✻✳✺✳ P❛rt✐❝❧❡ ✐♥ ❛ ✢♦✇

✹✼✷

❚❤❛t✬s ✇❤❛t ♠❛❦❡s t❤❡ ♣r♦❜❧❡♠ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧✳ ❖✉r ❣♦❛❧ ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✿

◮ ❚r❛❝❡ ❛ s✐♥❣❧❡ ♣❛rt✐❝❧❡ ♦❢ t❤✐s str❡❛♠✳ ❲❡ ✇✐❧❧ s✐♠♣❧② ❛♣♣❧② t❤❡ ❢❛♠✐❧✐❛r ❢♦r♠✉❧❛ t♦ ❝♦♠♣✉t❡ t❤❡ ❧♦❝❛t✐♦♥ ❢r♦♠ t❤❡ ✈❡❧♦❝✐t②✿ ❞✐s♣❧❛❝❡♠❡♥t = ✈❡❧♦❝✐t② · t✐♠❡ ✐♥❝r❡♠❡♥t . ❆ ✜①❡❞ t✐♠❡ ✐♥❝r❡♠❡♥t ∆t ✐s s✉♣♣❧✐❡❞ ❛❤❡❛❞ ♦❢ t✐♠❡ ❡✈❡♥ t❤♦✉❣❤ ✐♥ ❣❡♥❡r❛❧ ✐t ❝❛♥ ❛❧s♦ ❜❡ ✈❛r✐❛❜❧❡✳ ❲❡ st❛rt ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ q✉❛♥t✐t✐❡s ♣r♦✈✐❞❡❞ ❜② t❤❡ ♠♦❞❡❧ ✇❡ ❛r❡ t♦ ✐♠♣❧❡♠❡♥t✿

• t❤❡ ✐♥✐t✐❛❧ t✐♠❡ t0 ✱ ❛♥❞ • t❤❡ ✐♥✐t✐❛❧ ❧♦❝❛t✐♦♥ p0 ✳

❚❤❡② ❛r❡ ♣❧❛❝❡❞ ✐♥ t❤❡ ✜rst r♦✇ ♦❢ t❤❡ t❛❜❧❡❀ ❢♦r ❡①❛♠♣❧❡✿ ✐t❡r❛t✐♦♥ n t✐♠❡ tn ✈❡❧♦❝✐t② vn ❧♦❝❛t✐♦♥ pn ✐♥✐t✐❛❧✿ 0 3.5 −− 22 ❚❤✐s ✐s t❤❡ st❛rt✐♥❣ ♣♦✐♥t✳ ❲❡ ✇♦✉❧❞ ❧✐❦❡ t♦ ❦♥♦✇ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡s❡ q✉❛♥t✐t✐❡s ❛t ❡✈❡r② ♠♦♠❡♥t ♦❢ t✐♠❡✱ ✐♥ t❤❡s❡ ✐♥❝r❡♠❡♥ts✳ ❆s ✇❡ ♣r♦❣r❡ss ✐♥ t✐♠❡ ❛♥❞ s♣❛❝❡✱ ♥❡✇ ♥✉♠❜❡rs ❛r❡ ♣❧❛❝❡❞ ✐♥ t❤❡ ♥❡①t r♦✇ ♦❢ ♦✉r s♣r❡❛❞s❤❡❡t✳ ❚❤✐s ✐s ❤♦✇ t❤❡ s❡❝♦♥❞ r♦✇✱ n = 1, t1 = t0 + ∆t✱ ✐s ✜♥✐s❤❡❞✳ ❚❤❡ ❝✉rr❡♥t ✈❡❧♦❝✐t② v1 ✐s ✐♥ t❤❡ ✜rst ❝❡❧❧ ♦❢ t❤❡ r♦✇ ❛♥❞ t❤❡ ✐♥✐t✐❛❧ ❧♦❝❛t✐♦♥ p0 ✐s ✐♥ t❤❡ ❧❛st ❝❡❧❧ ♦❢ t❤❡ ❧❛st r♦✇✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛ ✐s ♣❧❛❝❡❞ ✐♥ t❤❡ s❡❝♦♥❞ ❝❡❧❧ ♦❢ t❤❡ ♥❡✇ r♦✇ ♦❢ ♦✉r s♣r❡❛❞s❤❡❡t✿

◮ ♥❡①t ❧♦❝❛t✐♦♥ = ✐♥✐t✐❛❧ ❧♦❝❛t✐♦♥ + ❝✉rr❡♥t ✈❡❧♦❝✐t② · t✐♠❡ ✐♥❝r❡♠❡♥t✳

❲❡ ❝♦♥t✐♥✉❡ ✇✐t❤ t❤❡ r❡st ✐♥ t❤❡ s❛♠❡ ♠❛♥♥❡r✳

❇❡❧♦✇ ✐s t❤❡ r❡❝✉rs✐✈❡ ❛❧❣♦r✐t❤♠ ❢♦r ❝♦♠♣✉t✐♥❣ r♦✇ n + 1 ❢r♦♠ r♦✇ n✿ ❋✐rst ❝♦❧✉♠♥ ✐s t❤❡ t✐♠❡ tn ✿ ◆❡①t ♠♦♠❡♥t ♦❢ t✐♠❡ = ❧❛st ♠♦♠❡♥t ♦❢ t✐♠❡ + t✐♠❡ ✐♥❝r❡♠❡♥t✳ tn+1 = tn + ∆t ❙❡❝♦♥❞ ❝♦❧✉♠♥ ✐s t❤❡ ✈❡❧♦❝✐t② vn ✿ ◆❡①t ✈❡❧♦❝✐t② ❡①♣❧✐❝✐t❧② ❞❡♣❡♥❞s ♦♥ t❤❡ ✈❛❧✉❡s ✐♥ t❤❡ ♣r❡✈✐♦✉s r♦✇✳ vn+1 = ❛♥② ❝♦♠❜✐♥❛t✐♦♥ ♦❢ tn , vn , pn ❚❤✐r❞ ❝♦❧✉♠♥ ✐s t❤❡ ❧♦❝❛t✐♦♥ pn ✿ ◆❡①t ❧♦❝❛t✐♦♥ = ❧❛st ❧♦❝❛t✐♦♥ + ❝✉rr❡♥t ✈❡❧♦❝✐t② · t✐♠❡ ✐♥❝r❡♠❡♥t✳ pn+1 = pn + vn+1 · ∆t

❚❤✐s ❞❡♣❡♥❞❡♥❝❡ ✐s s❤♦✇♥ ❜❡❧♦✇ ❢♦r ∆t = .1✿

✐t❡r❛t✐♦♥ n t✐♠❡ tn ✈❡❧♦❝✐t② vn ✐♥✐t✐❛❧✿ 0 t0 = 3.5 −− ↓ 1 t1 = 3.5 + .1 → v1 = 33 ↓ 2 t2 =? → v2 =? ↓

ւ → ւ → ւ

❧♦❝❛t✐♦♥ pn p0 = 22 ↓ p1 = 22 + 33 · .1 ↓ p2 =? ↓

❆❧s♦✱ ✐♥ ❛ ✢♦✇✱ t❤❡ ❝✉rr❡♥t ✈❡❧♦❝✐t② ♦❢ ❛ ♣❛rt✐❝❧❡ ❞❡♣❡♥❞s ✕ s♦♠❡❤♦✇ ✕ ♦♥ t❤❡ ❝✉rr❡♥t t✐♠❡ ♦r ✐ts ✭❧❛st✮ ❧♦❝❛t✐♦♥✱ ❛s ✐♥❞✐❝❛t❡❞ ❜② t❤❡ ❛rr♦✇s✳ ❚❤✐s ❞❡♣❡♥❞❡♥❝❡ ♠❛② ❜❡ ❛♥ ❡①♣❧✐❝✐t ❢♦r♠✉❧❛ ♦r ✐t ♠❛② ❝♦♠❡ ❢r♦♠ t❤❡ ✐♥str✉♠❡♥ts✬ r❡❛❞✐♥❣s✳

✻✳✺✳

P❛rt✐❝❧❡ ✐♥ ❛ ✢♦✇

✹✼✸

❆s ✇❡ ♣r♦❣r❡ss ✐♥ t✐♠❡ ❛♥❞ s♣❛❝❡✱ ❛ ♥✉♠❜❡r ✐s s✉♣♣❧✐❡❞ ❛♥❞ ❛r❡ ♣❧❛❝❡❞ ✐♥ ❡❛❝❤ ♦❢ t❤❡ ❝♦❧✉♠♥s ♦❢ ♦✉r s♣r❡❛❞s❤❡❡t ♦♥❡ r♦✇ ❛t ❛ t✐♠❡✿

tn , vn , pn , n = 1, 2, 3, ... ❲❡ t❤❡♥ ♣❧♦t t❤❡ t✐♠❡ ❛♥❞ ❧♦❝❛t✐♦♥ ♦♥ t❤❡ ♣❧❛♥❡ ✇✐t❤ ❛①❡s

t

✭❤♦r✐③♦♥t❛❧✮ ❛♥❞

p

✭✈❡rt✐❝❛❧✮✳ ❚❤❡ r❡s✉❧t ✐s ❛

s❡q✉❡♥❝❡ ♦❢ ♣♦✐♥ts ❞❡✈❡❧♦♣✐♥❣ ♦♥❡ r♦✇ ❛t ❛ t✐♠❡ t❤❛t ♠✐❣❤t ❧♦♦❦ ❧✐❦❡ t❤✐s✿

❚❤❡ r❡s✉❧t ✐s ❛ ❣r♦✇✐♥❣ t❛❜❧❡ ♦❢ ✈❛❧✉❡s✿

✐t❡r❛t✐♦♥ ✐♥✐t✐❛❧✿

0 1 ... 1000 ...

❚❤❡ r❡s✉❧t ♠❛② ❜❡ s❡❡♥ ❛s t❤r❡❡ s❡q✉❡♥❝❡s

n

t✐♠❡

tn

✈❡❧♦❝✐t②

−− 33 ... 4 ...

3.5 3.6 ... 103.5 ... tn , vn , pn

vn

❧♦❝❛t✐♦♥

pn

22 25.3 ... 336 ...

♦r ❛s t❤❡ t❛❜❧❡ ♦❢ ✈❛❧✉❡s ♦❢ t✇♦

❢✉♥❝t✐♦♥s

♦❢

t✳

❚❤❡

❣r❛♣❤ ♦❢ t❤❡ ♣♦s✐t✐♦♥ ♠✐❣❤t ❧♦♦❦ ❧✐❦❡ t❤✐s✿

▲❡t✬s t❡st ❛ ❢❡✇ ✐♥st❛♥❝❡s ♦❢ t❤✐s ❛❧❣♦r✐t❤♠ ✇✐t❤ ❛ s♣r❡❛❞s❤❡❡t✳ ❲❡ ✉s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✉❧❛ ❢♦r t❤❡ ♣♦s✐t✐♦♥✿

❂❘❬✲✶❪❈✰❘❈❬✲✶❪✯✭❘❈❬✲✷❪✲❘❬✲✶❪❈❬✲✷❪✮ ❊①❛♠♣❧❡ ✻✳✺✳✶✿ ❝♦♥st❛♥t ✈❡❧♦❝✐t② ❙✉♣♣♦s❡ t❤❛t t❤❡ ✈❡❧♦❝✐t② ♦❢ t❤❡ str❡❛♠ ✐♥ t❤❡ ♣✐♣❡ ✐s ❝♦♥st❛♥t✿

vn+1 = .5 . ❚❤❡ s✐♠✉❧❛t✐♦♥ s❤♦✇s t❤❛t t❤❡ ♣❛rt✐❝❧❡ ♣r♦❣r❡ss❡s ✐♥ ❛ ✉♥✐❢♦r♠ ❢❛s❤✐♦♥✿

✻✳✺✳ P❛rt✐❝❧❡ ✐♥ ❛ ✢♦✇

✹✼✹

❊①❡r❝✐s❡ ✻✳✺✳✷

❋✐♥❞ t❤❡ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ ♣♦s✐t✐♦♥✳ ❊①❛♠♣❧❡ ✻✳✺✳✸✿ ❝✉r✈❡❞ ♣✐♣❡

❙✉♣♣♦s❡ t❤❛t t❤❡r❡ ✐s ❛ s♦✉r❝❡ ♦❢ ✇❛t❡r ✐♥ t❤❡ ♠✐❞❞❧❡ ♦❢ t❤❡ ♣✐♣❡ ❜✉t ✐t ✐s ❛❧s♦ ❜❡♥t ❞♦✇♥ s♦ t❤❛t t❤❡ s♣❡❡❞ ✐s ❤✐❣❤❡r ❛✇❛② ❢r♦♠ t❤✐s ❧♦❝❛t✐♦♥✿

❙♣❡❝✐✜❝❛❧❧②✱ ✇❡ ❛ss✉♠❡ t❤❛t t❤❡ ✈❡❧♦❝✐t② ♦❢ t❤❡ str❡❛♠ ✐s ❞✐r❡❝t❧② ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ❞✐st❛♥❝❡ ❢r♦♠ t❤❡ s♦✉r❝❡✿ vn+1 = .5 · pn .

❲❡ ❝❛♥ s❡❡ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ✐♥ t❤❡ s✐♠✉❧❛t✐♦♥✿

❊①❡r❝✐s❡ ✻✳✺✳✹

❋✐♥❞ t❤❡ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ ♣♦s✐t✐♦♥✳ ❊①❡r❝✐s❡ ✻✳✺✳✺

▼♦❞✐❢② t❤❡ ❢♦r♠✉❧❛s ❢♦r t❤❡ ❝❛s❡ ♦❢ ❛ ✈❛r✐❛❜❧❡ t✐♠❡ ✐♥❝r❡♠❡♥t✿ ∆tn+1 = tn+1 − tn ✳ ◆❡①t✱ ❛ ❝♦♥t✐♥✉♦✉s ✢♦✇✳ ❲❤❛t ✐s t❤❡ ❞✐✛❡r❡♥❝❡❄ ❚❤❡ t✐♠❡ t ✈❛r✐❡s ♦✈❡r ❛ ✇❤♦❧❡ ✐♥t❡r✈❛❧ [t0 , ∞) ❛♥❞ t❤❡ ♣❛rt✐❝❧❡ ✐s ♣r♦❣r❡ss✐♥❣

✻✳✺✳ P❛rt✐❝❧❡ ✐♥ ❛ ✢♦✇

✹✼✺

❝♦♥t✐♥✉♦✉s❧② t❤r♦✉❣❤ ❛♥ ✐♥t❡r✈❛❧ ♦❢ s♣❛❝❡ p = y(t)✳ ❚❤✐s ✐s t❤❡ ❛♥s✇❡r❀ ❤♦✇❡✈❡r✱ ✇❤❛t ✐s t❤❡ q✉❡st✐♦♥❄ ❲❡ ♣♦s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦❜❧❡♠✿ ◮ ❙✉♣♣♦s❡ t❤❡ ✈❡❧♦❝✐t② ✐s ❣✐✈❡♥ ❜② ❛♥ ❡①♣❧✐❝✐t ❢♦r♠✉❧❛ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ❧♦❝❛t✐♦♥ z = f (y) ❞❡✜♥❡❞ ♦♥ ❛♥ ✐♥t❡r✈❛❧ J ♦❢ s♣❛❝❡✳ ■s t❤❡r❡ ❛♥ ❡①♣❧✐❝✐t ❢♦r♠✉❧❛ ❢♦r t❤❡ ❧♦❝❛t✐♦♥ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✐♠❡ y = y(t) ❞❡✜♥❡❞ ♦♥ ❛♥ ✐♥t❡r✈❛❧ I ♦❢ t✐♠❡❄

❲❡ st❛rt ❢r♦♠ t❤❡ ❞✐s❝r❡t❡ ❝❛s❡✳ ❋✐rst✱ ✇❡ r❡❝❛st ♦✉r r❡❝✉rs✐✈❡ r❡❧❛t✐♦♥✱ pn+1 = pn + vn+1 · ∆t ,

❛s ✐❢ t❤✐s s❡q✉❡♥❝❡ ♦❢ ❧♦❝❛t✐♦♥s ❝♦♠❡s ❢r♦♠ s❛♠♣❧✐♥❣ ♦✉r ❢✉♥❝t✐♦♥ y ✿ pn = y(tn ) .

❚❤❡♥✱ ✇❡ ❛❧s♦ ❤❛✈❡✿ vn = f (pn ) .

❚❤✐s ❤♦❧❞s ❢♦r ❡✈❡r② ∆t > 0✳ ❍❡r❡ f ✐s ❦♥♦✇♥ ✇❤✐❧❡ y ✐s ✉♥❦♥♦✇♥✳ ❙❡❝♦♥❞✱ ✇✐t❤ t = tn ✱ t❤❡ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛ ❜❡❝♦♠❡s✿ y(t + ∆t) = y(t) + f (y(t + ∆t)) · ∆t .

❚❤❡r❡❢♦r❡✱

y(t + ∆t) − y(t) = f (y(t + ∆t)) . ∆t

❚❛❦✐♥❣ t❤❡ ❧✐♠✐t ♦✈❡r ∆t → 0 ❣✐✈❡s ✉s t❤❡ ❢♦❧❧♦✇✐♥❣✿

y ′ (t) = f (y(t)) ,

♣r♦✈✐❞❡❞ t❤❡ ❧✐♠✐ts ❡①✐st ♦❢ ❝♦✉rs❡✳ ❚❤❡ ❛❜♦✈❡ ❡q✉❛t✐♦♥ ✐s ❝❛❧❧❡❞ ❛ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥✳ ■ts ❛❜❜r❡✈✐❛t❡❞ ✈❡rs✐♦♥ ✐s ❜❡❧♦✇✿ y ′ = f (y)

❚❤❡ ❛tt✐t✉❞❡ ✐s ✐❞❡♥t✐❝❛❧ t♦ t❤❛t ❢♦r ❛♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥✳ ❲❡ ❛❣❛✐♥ t❤✐♥❦ ♦❢ ❛♥ ❡q✉❛t✐♦♥✱ ❛♥ ❡q✉❛t✐♦♥ ❢♦r ❢✉♥❝t✐♦♥s✿ ❙♦❧✈❡ ❢♦r y

❙♦❧✈❡ ❢♦r y

dy = x2 dx

dy = y2 dx

❚❤❡ ❧❛tt❡r ✐s ♦❜✈✐♦✉s❧② ♠♦r❡ ❝♦♠♣❧❡①✦ ❙✉❝❤ ❡q✉❛t✐♦♥s ❛r❡ ❝♦♥s✐❞❡r❡❞ ❜❡❧♦✇ ❛♥❞ t❤r♦✉❣❤♦✉t t❤❡ r❡st ♦❢ ❝❛❧❝✉❧✉s✳ ❚❤❡ t②♣✐❝❛❧ r❡q✉✐r❡♠❡♥ts ❢♦r t❤❡ ❢✉♥❝t✐♦♥s ✐♥✈♦❧✈❡❞ ❛r❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿ • y = y(t) ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ♦♥ t❤❡ ✐♥t❡r✈❛❧ I ✳

• z = f (y) ✐s ❝♦♥t✐♥✉♦✉s ♦♥ t❤❡ ✐♥t❡r✈❛❧ J ✳

✻✳✺✳ P❛rt✐❝❧❡ ✐♥ ❛ ✢♦✇

✹✼✻

❊①❛♠♣❧❡ ✻✳✺✳✻✿ ❝✉r✈❡❞ ♣✐♣❡✱ ❝♦♥t✐♥✉❡❞

■❢ t❤❡ ♣✐♣❡✬s s❧♦♣❡ ✈❛r✐❡s ❢r♦♠ ❧♦❝❛t✐♦♥ t♦ ❧♦❝❛t✐♦♥✱ t❤❡ ✈❡❧♦❝✐t② ♦❢ t❤❡ ✢♦✇ ✇✐❧❧ ❞❡♣❡♥❞ ♦♥ t❤❡ ❧♦❝❛t✐♦♥ t♦♦✳ ❋♦r ❡①❛♠♣❧❡✱ ✐❢ t❤❡ ♣✐♣❡ ✐s ❝✉r✈❡❞ ❞♦✇♥✱ t❤❡ ✈❡❧♦❝✐t② ♦❢ t❤❡ ✢♦✇ ✇✐❧❧ ❜❡ ❤✐❣❤❡r t❤❡ ❢❛rt❤❡r ❛✇❛② t❤❡ ♣♦✐♥t ✐s ❢r♦♠ t❤❡ ♦r✐❣✐♥✳ ▲❡t✬s ❛❣❛✐♥ ❝♦♥s✐❞❡r t❤❡ ❝❛s❡ ✇❤❡♥ t❤❡ ✈❡❧♦❝✐t② ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ❧♦❝❛t✐♦♥✿ vn+1 = .5 · pn .

❋♦r t❤❡ ❝♦♥t✐♥✉♦✉s ❝❛s❡✱ ✇❡ ❤❛✈❡ ❛ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥✿

y ′ = .5 · y . ❲❡ ❛❧r❡❛❞② ❦♥♦✇ ✐ts s♦❧✉t✐♦♥✿

y(t) = Ce.5t ,

❢♦r ❛♥② C ✳ ▲❡t✬s ❝♦♥s✐❞❡r t❤❡ ❞✐s❝r❡t❡ ❝❛s❡ ♦❢ ✢♦✇s ♦♥ t❤❡ ♣❧❛♥❡ ♥♦✇✳ ❖♥❡ ❝❛♥ ✐♠❛❣✐♥❡ t❤❡ ✢♦✇ ♦❢ r❛✐♥✇❛t❡r ❞♦✇♥ ❛ ♠♦✉♥t❛✐♥ t❡rr❛✐♥✿

❖♥❡ ❝❛♥ ❛❧s♦ ✐♠❛❣✐♥❡ t❤❛t t❤✐s ✐s ❛ ❜❛❧❧ r♦❧❧✐♥❣ ❞♦✇♥ t❤❡ s❧♦♣❡ ❢♦❧❧♦✇✐♥❣ t❤❡s❡ ❞✐r❡❝t✐♦♥s✿ ❈♦♠♣✉t❛t✐♦♥❛❧❧②✱ ✐♥st❡❛❞ ♦❢ t✇♦ ✭✈❡❧♦❝✐t② ✕ ❧♦❝❛t✐♦♥✮✱ ♦✉r t❛❜❧❡ ✇✐❧❧ ❤❛✈❡ ❢♦✉r ♠❛✐♥ ❝♦❧✉♠♥s✿ ✐t❡r❛t✐♦♥ t✐♠❡ ✈❡❧♦❝✐t② ✈❡❧♦❝✐t② ❧♦❝❛t✐♦♥ ❧♦❝❛t✐♦♥ ... t v x u y ... n 0 3.5 −− 22 −− 3 ... 1 3.6 33 25.3 4 3.5 ... ... ... ... ... ... ... ... ❚♦ s❦❡t❝❤ t❤❡ ♣❛t❤ ♦❢ ❛ ♣❛rt✐❝❧❡ ✐♥ t❤❡ str❡❛♠✱ ✇❡ ♦♥❧② s❤♦✇ t❤❡ t✇♦ s♣❛t✐❛❧ ❝♦♦r❞✐♥❛t❡s ❛♥❞ ❤✐❞❡ t❤❡ t✐♠❡✿

✻✳✺✳ P❛rt✐❝❧❡ ✐♥ ❛ ✢♦✇

✹✼✼

❊①❛♠♣❧❡ ✻✳✺✳✼✿ ❝✉r✈❡❞ s✉r❢❛❝❡

❏✉st ❛s ✇✐t❤ t❤❡ ♣✐♣❡✱ ✐❢ t❤❡ t❡rr❛✐♥✬s s❧♦♣❡ ✈❛r✐❡s ❢r♦♠ ❧♦❝❛t✐♦♥ t♦ ❧♦❝❛t✐♦♥✱ t❤❡ ✈❡❧♦❝✐t② ♦❢ t❤❡ ✢♦✇ ✇✐❧❧ ❞❡♣❡♥❞ ♦♥ t❤❡ ❧♦❝❛t✐♦♥ t♦♦✳ ❋♦r ❡①❛♠♣❧❡✱ ✐❢ t❤❡ t❡rr❛✐♥ ✐s ❝✉r✈❡❞ ❞♦✇♥✱ t❤❡ ✈❡❧♦❝✐t② ♦❢ t❤❡ ✢♦✇ ✇✐❧❧ ❜❡ ❤✐❣❤❡r t❤❡ ❢❛rt❤❡r ❛✇❛② t❤❡ ♣♦✐♥t ✐s ❢r♦♠ t❤❡ t♦♣ ♦❢ t❤❡ ♠♦✉♥t❛✐♥✿

▲❡t t❤❡ ✈❡❧♦❝✐t② ❜❡ ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ❧♦❝❛t✐♦♥ ❢♦r ❜♦t❤ ❤♦r✐③♦♥t❛❧ ❛♥❞ ✈❡rt✐❝❛❧✳✿ vn+1 = .2 · xn ❛♥❞ un+1 = .2 · yn .

❚❤✐s ✐s t❤❡ r❡s✉❧t ♦❢ t❤❡ s✐♠✉❧❛t✐♦♥ ❢♦r t✇♦ ♣❛rt✐❝❧❡s✿

❚❤❡ ♣❛rt✐❝❧❡s ❛r❡ r✉♥♥✐♥❣ ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ❛✇❛② ❢r♦♠ t❤❡ ❝❡♥t❡r✱ ❢❛st❡r ❛♥❞ ❢❛st❡r✳ ❊①❛♠♣❧❡ ✻✳✺✳✽✿ ✇❤✐r❧

❋♦r ♠♦r❡ ❝♦♠♣❧❡① ♣❛tt❡r♥s✱ t❤❡ ✈❡rt✐❝❛❧ ❛♥❞ ❤♦r✐③♦♥t❛❧ ✇✐❧❧ ❜❡ ✐♥t❡r❞❡♣❡♥❞❡♥t✳ ❋♦r ❡①❛♠♣❧❡✱ t❤❡ ❤♦r✐③♦♥t❛❧ ✈❡❧♦❝✐t② ♠❛② ❜❡ ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ✈❡rt✐❝❛❧ ❧♦❝❛t✐♦♥ ❛♥❞ t❤❡ ✈❡rt✐❝❛❧ ✈❡❧♦❝✐t② ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ♥❡❣❛t✐✈❡ ♦❢ t❤❡ ❤♦r✐③♦♥t❛❧ ❧♦❝❛t✐♦♥✿ vn+1 = .2 · yn ❛♥❞ un+1 = −.2 · xn .

❚❤✐s ✐s t❤❡ r❡s✉❧t ♦❢ t❤❡ s✐♠✉❧❛t✐♦♥✿

❊①❡r❝✐s❡ ✻✳✺✳✾

❲❤❛t ❛r❡ t❤❡ ❝♦♥t✐♥✉♦✉s ♠♦❞❡❧s ❢♦r t❤❡ ✢♦✇s ❞❡s❝r✐❜❡❞ ✐♥ t❤❡ ❡①❛♠♣❧❡s❄ ❙♦❧✈❡ t❤❡♠✳

✻✳✻✳ ❉✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s

✹✼✽

❊①❡r❝✐s❡ ✻✳✺✳✶✵ ❙✉❣❣❡st ♠♦❞❡❧s ♦❢ ✢♦✇ ❢♦r ♦t❤❡r ❧♦❝❛t✐♦♥s ♦♥ t❤✐s s✉r❢❛❝❡✳

✻✳✻✳ ❉✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ ✇✐❧❧ r❡✈✐❡✇ ❛♥❞ s✉♠♠❛r✐③❡ ♦✉r ❡♥❝♦✉♥t❡rs ✇✐t❤ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✳ ❆ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ✐s ❛♥ ❡q✉❛t✐♦♥ t❤❛t r❡❧❛t❡s t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ t♦ t❤❡ ❢✉♥❝t✐♦♥✬s ✈❛❧✉❡s✳

❊①❛♠♣❧❡ ✻✳✻✳✶✿ ③❡r♦ ❞❡r✐✈❛t✐✈❡ ❚❤❡ s✐♠♣❧❡st ❡①❛♠♣❧❡ ♦❢ ❛ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ✐s✿ f ′ (x) = 0 ❢♦r ❛❧❧ x .

❲❡ ❛❧r❡❛❞② ❦♥♦✇ t❤❡ s♦❧✉t✐♦♥✿ • ❈♦♥st❛♥t ❢✉♥❝t✐♦♥s ❛r❡ s♦❧✉t✐♦♥s✳ • ❈♦♥✈❡rs❡❧②✱ ♦♥❧② ❝♦♥st❛♥t ❢✉♥❝t✐♦♥s ❛r❡ s♦❧✉t✐♦♥s✳

❊①❛♠♣❧❡ ✻✳✻✳✷✿ ❛♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥ ❲❡ ❝❛♥ r❡♣❧❛❝❡ 0 ✇✐t❤ ❛♥② ❢✉♥❝t✐♦♥✿ f ′ (x) = g(x) ❢♦r ❛❧❧ x .

❲❡ ❛❧r❡❛❞② ❦♥♦✇ t❤❡ s♦❧✉t✐♦♥✿ • ❆ s♦❧✉t✐♦♥ f ❤❛s t♦ ❜❡ ❛♥ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢ g ✳ • ❈♦♥✈❡rs❡❧②✱ ♦♥❧② ❛♥t✐❞❡r✐✈❛t✐✈❡s ♦❢ g ❛r❡ s♦❧✉t✐♦♥s✳

■♥ s✉♠♠❛r②✱ ❛ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ✐s ❛♥ ❡q✉❛❧✐t② ♦❢ ❢✉♥❝t✐♦♥s t❤❛t ♠❛② ✐♥❝❧✉❞❡ ❞❡r✐✈❛t✐✈❡s✳ ❚♦ s♦❧✈❡ t❤❡ ❡q✉❛t✐♦♥ ✐s t♦ ✜♥❞ ❛❧❧ ♣♦ss✐❜❧❡ ❢✉♥❝t✐♦♥s t❤❛t s❛t✐s❢② ✐t✳ ▲❡t✬s t❛❦❡ ❛ ❧♦♦❦ ❛t t❤❡ ♣r♦❜❧❡♠ ✇✐t❤ ❛ ❢r❡s❤ ❡②❡✳ ❋♦r t❤❡ ✜rst ❦✐♥❞ ♦❢ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥✱ ✇❡ ✇✐❧❧ ❜❡ ❧♦♦❦✐♥❣ ❢♦r ❢✉♥❝t✐♦♥s y = y(x) t❤❛t s❛t✐s❢② t❤❡ ❡q✉❛t✐♦♥✱ y ′ (x) = g(x) ❢♦r ❛❧❧ x .

❲❤❛t ✇❡ ❦♥♦✇ ❢r♦♠ t❤❡ ❡q✉❛t✐♦♥ ✐s t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ y ′ (x) ♦❢ y ❛t ❡✈❡r② ♣♦✐♥t x✱ ❜✉t ✇❡ ❞♦♥✬t ❦♥♦✇ t❤❡ ✈❛❧✉❡ y(x) ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ✐ts❡❧❢✳ ❚❤❡♥✱ ❢♦r ❡✈❡r② x✱ ✇❡ ❦♥♦✇ t❤❡ s❧♦♣❡ ♦❢ t❤❡ t❛♥❣❡♥t ❧✐♥❡ ❛t (x, y(x))✳ ❆s y(x) ✐s ✉♥❦♥♦✇♥✱ ✐♥ ♦r❞❡r t♦ ✈✐s✉❛❧✐③❡ t❤❡ ❞❛t❛✱ ✇❡ ♣❧♦t t❤❡ s❛♠❡ s❧♦♣❡ ❢♦r ❡✈❡r② ♣♦✐♥t (x, y) ♦♥ ❛ ❣✐✈❡♥ ✈❡rt✐❝❛❧ ❧✐♥❡✿

✻✳✻✳ ❉✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s

✹✼✾

❚❤✉s✱ ❢♦r ❡❛❝❤ x = c✱ ✇❡ ✐♥❞✐❝❛t❡ t❤❡ ❛♥❣❧❡ α✱ ✇✐t❤ g(c) = tan α✱ ♦❢ t❤❡ ✐♥t❡rs❡❝t✐♦♥ ♦❢ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ✉♥❦♥♦✇♥ ❢✉♥❝t✐♦♥ y = y(x) ❛♥❞ t❤❡ ✈❡rt✐❝❛❧ ❧✐♥❡ x = c✳ ■t ✐s ❛s ✐❢ ✇❡ ❛r❡ t♦ ❝r❡❛t❡ ❛ ❢❛❜r✐❝ ❢r♦♠ t✇♦ t②♣❡s ♦❢ t❤r❡❛❞s✳ ❚❤❡ ✈❡rt✐❝❛❧ ♦♥❡s ❤❛✈❡ ❛❧r❡❛❞② ❜❡❡♥ ♣❧❛❝❡❞✱ ❛♥❞ t❤❡ ✇❛② ❛t ✇❤✐❝❤ t❤❡ t❤r❡❛❞s ♦❢ t❤❡ s❡❝♦♥❞ t②♣❡ ❛r❡ t♦ ❜❡ ✇❡❛✈❡❞ ✐♥ ❤❛s ❜❡❡♥ ✐♥❞✐❝❛t❡❞✳ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ✐s t♦ ❞❡✈✐s❡ ❛ ❢✉♥❝t✐♦♥ t❤❛t ✇♦✉❧❞ ❝r♦ss t❤❡s❡ ❧✐♥❡s ❛t t❤❡s❡ ❡①❛❝t ❛♥❣❧❡s✳ ■s ✐t ❛❧✇❛②s ♣♦ss✐❜❧❡ t♦ ❤❛✈❡ t❤❡s❡ ❢✉♥❝t✐♦♥s❄ ❨❡s✱ ❛t ❧❡❛st ✇❤❡♥ g ✐s ❝♦♥t✐♥✉♦✉s✳ ❍♦✇ ❞♦ ✇❡ ✜♥❞ t❤❡♠❄ ❆♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥✳ ❊①❛♠♣❧❡ ✻✳✻✳✸✿ ♣♦s✐t✐♦♥ ❢r♦♠ ✈❡❧♦❝✐t②

❆ ❢❛♠✐❧✐❛r ✐♥t❡r♣r❡t❛t✐♦♥ ✐s t❤❛t ♦❢ ❛♥ ♦❜❥❡❝t t❤❡ ✈❡❧♦❝✐t② v(x) = y ′ (x) ♦❢ ✇❤✐❝❤ ✐s ❦♥♦✇♥ ❛t ❛♥② ♠♦♠❡♥t ♦❢ t✐♠❡ x ❛♥❞ ✐ts ❧♦❝❛t✐♦♥ y = y(x) ✐s t♦ ❜❡ ❢♦✉♥❞✳

❚❤❡s❡ s♦❧✉t✐♦♥s ✜❧❧ t❤❡ ♣❧❛♥❡ ✇✐t❤♦✉t ✐♥t❡rs❡❝t✐♦♥s✳ ❚❤❡② ❛r❡ ❥✉st t❤❡ ✈❡rt✐❝❛❧❧② s❤✐❢t❡❞ ✈❡rs✐♦♥s ♦❢ ♦♥❡ ♦❢ t❤❡♠✳ ❆♥♦t❤❡r ❦✐♥❞ ♦❢ ❛ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ✐s ♠♦r❡ ❝❤❛❧❧❡♥❣✐♥❣✿ ◮ ❚❤❡ ❞❡r✐✈❛t✐✈❡ ❞❡♣❡♥❞s ♦♥ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ❢✉♥❝t✐♦♥✳

❲❡ ❛r❡ ❧♦♦❦✐♥❣ ❢♦r ❢✉♥❝t✐♦♥s y = y(x) t❤❛t s❛t✐s❢② t❤❡ ❡q✉❛t✐♦♥✱ y ′ (x) = h(y(x)) ❢♦r ❛❧❧ x .

❲❤❛t ✇❡ ❦♥♦✇ ❢r♦♠ t❤❡ ❡q✉❛t✐♦♥ ✐s t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ y ′ (x) ♦❢ y ❛t ❡✈❡r② ♣♦✐♥t (x, y) ❡✈❡♥ t❤♦✉❣❤ ✇❡ ❞♦♥✬t ❦♥♦✇ t❤❡ ✈❛❧✉❡ y(x) ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ✐ts❡❧❢✳ ❚❤❡♥✱ ❢♦r ❡✈❡r② y ✱ ✇❡ ❦♥♦✇ t❤❡ s❧♦♣❡ ♦❢ t❤❡ t❛♥❣❡♥t ❧✐♥❡ ❛t (x, y)✳ ❆s y(x) ✐s ✉♥❦♥♦✇♥✱ ✐♥ ♦r❞❡r t♦ ✈✐s✉❛❧✐③❡ t❤❡ ❞❛t❛✱ ✇❡ ♣❧♦t t❤❡ s❛♠❡ s❧♦♣❡ ❢♦r ❡✈❡r② ♣♦✐♥t (x, y) ♦♥ ❛ ❣✐✈❡♥ ❤♦r✐③♦♥t❛❧ ❧✐♥❡✿

✻✳✻✳ ❉✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s

✹✽✵

❚❤✉s✱ ❢♦r ❡❛❝❤ y = d✱ ✇❡ ✐♥❞✐❝❛t❡ t❤❡ ❛♥❣❧❡ α✱ ✇✐t❤ h(d) = tan α✱ ♦❢ t❤❡ ✐♥t❡rs❡❝t✐♦♥ ♦❢ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ✉♥❦♥♦✇♥ ❢✉♥❝t✐♦♥ y = y(x) ❛♥❞ t❤❡ ✈❡rt✐❝❛❧ ❧✐♥❡ x = c ✇✐t❤ y(c) = d✳ ■t ✐s ❛s ✐❢ ✇❡ ❛r❡ t♦ ❝r❡❛t❡ ❛ ❢❛❜r✐❝ ❢r♦♠ t✇♦ t②♣❡s ♦❢ t❤r❡❛❞s✳ ❚❤❡ ❤♦r✐③♦♥t❛❧ ♦♥❡s ❤❛✈❡ ❛❧r❡❛❞② ❜❡❡♥ ♣❧❛❝❡❞ ❛♥❞ t❤❡ ✇❛② ❛t ✇❤✐❝❤ t❤❡ t❤r❡❛❞s ♦❢ t❤❡ s❡❝♦♥❞ t②♣❡ ❛r❡ t♦ ❜❡ ✇❡❛✈❡❞ ✐♥ ❤❛s ❜❡❡♥ ✐♥❞✐❝❛t❡❞✳ ■s ✐t ❛❧✇❛②s ♣♦ss✐❜❧❡ t♦ ❤❛✈❡ t❤❡s❡ ❢✉♥❝t✐♦♥s❄ ❨❡s✱ ❛t ❧❡❛st ✇❤❡♥ h ✐s ❞✐✛❡r❡♥t✐❛❜❧❡✳ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ✐s t♦ ❞❡✈✐s❡ ❛ ❢✉♥❝t✐♦♥ t❤❛t ✇♦✉❧❞ ❝r♦ss t❤❡s❡ ❧✐♥❡s ❛t t❤❡s❡ ❡①❛❝t ❛♥❣❧❡s✳ ❍♦✇ ❞♦ ✇❡ ✜♥❞ t❤❡♠❄ ■t ✐s ❤❛r❞ ♦r ✐♠♣♦ss✐❜❧❡ ✭❱♦❧✉♠❡ ✺✱ ❈❤❛♣t❡r ✺❉❊✲✶✮✳

❊①❛♠♣❧❡ ✻✳✻✳✹✿ ✢♦✇ ❆♥ ✐♥t❡r♣r❡t❛t✐♦♥ ✐s ❛ str❡❛♠ ♦❢ ❧✐q✉✐❞ ✇✐t❤ ✐ts ✈❡❧♦❝✐t② ❦♥♦✇♥ ❛t ❡✈❡r② ❧♦❝❛t✐♦♥ y ❛♥❞ ✇❡ ♥❡❡❞ t♦ tr❛❝❡ t❤❡ ♣❛t❤ ♦❢ ❛ ♣❛rt✐❝❧❡ ✐♥✐t✐❛❧❧② ❧♦❝❛t❡❞ ❛t ❛ s♣❡❝✐✜❝ ♣❧❛❝❡ y0 ✳

❊①❛♠♣❧❡ ✻✳✻✳✺✿ ♣♦♣✉❧❛t✐♦♥ ❣r♦✇t❤ ❚❤✐s ✐s ✇❤❛t ❤❛♣♣❡♥s ✇❤❡♥ t❤❡ ✈❡❧♦❝✐t② ✐s ❡q✉❛❧ t♦ t❤❡ ❧♦❝❛t✐♦♥✿ y′ = y .

❚❤❡ ♣♦♣✉❧❛t✐♦♥ ❞♦✉❜❧❡s ❡✈❡r② ✉♥✐t ♦❢ t✐♠❡✦ ❚♦ s♦❧✈❡✱ ✇❤❛t ✐s t❤❡ ❢✉♥❝t✐♦♥ ❡q✉❛❧ t♦ ✐ts ❞❡r✐✈❛t✐✈❡❄ ■t✬s t❤❡ ❡①♣♦♥❡♥t y = ex ✱ ♦❢ ❝♦✉rs❡✳ ❍♦✇❡✈❡r✱ ❛❧❧ ♦❢ ✐ts ♠✉❧t✐♣❧❡s y = Cex ❛r❡ ❛❧s♦ s♦❧✉t✐♦♥s✿

✻✳✻✳

❉✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s

❖♥❝❡ ❛❣❛✐♥✱ t❤❡s❡ s♦❧✉t✐♦♥s

✹✽✶

✜❧❧

t❤❡ ♣❧❛♥❡ ✇✐t❤♦✉t ✐♥t❡rs❡❝t✐♥❣✳

❊①❡r❝✐s❡ ✻✳✻✳✻ ❲❤❛t ✐s t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ t❤❡ ♣❧❛♥❡ t❤❛t ❝r❡❛t❡s ❛❧❧ t❤❡s❡ ❝✉r✈❡s ❢r♦♠ ♦♥❡❄

❊①❡r❝✐s❡ ✻✳✻✳✼ ❈❛♥ t❤❡ ✈❡❧♦❝✐t② ❜❡ r❡❛❧❧② ✏❡q✉❛❧✑ t♦ t❤❡ ❧♦❝❛t✐♦♥❄

❊①❡r❝✐s❡ ✻✳✻✳✽ ❲❤❛t ❤❛♣♣❡♥s ✇❤❡♥ t❤❡ ✈❡❧♦❝✐t② ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ❧♦❝❛t✐♦♥❀ ✐✳❡✳✱

y ′ = ky? ❈♦♠♣❛r❡ ❛♥❞ ❝♦♥tr❛st✿

y ′ (x) = g(x)

y ′ (x) = h(y(x))

t❤❡ s❧♦♣❡s ❛r❡ t❤❡ s❛♠❡ ❛❧♦♥❣ ❛♥②

✈❡rt✐❝❛❧ ❧✐♥❡

❤♦r✐③♦♥t❛❧ ❧✐♥❡

t❤❡ ✈❡❧♦❝✐t② ✐s ❦♥♦✇♥ ❛t ❛♥②

t✐♠❡

❧♦❝❛t✐♦♥

❉❊✿

❖❢ ❝♦✉rs❡✱ ❛❧s♦ ❝♦♠♠♦♥ ❛r❡ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s t❤❛t ❤❛✈❡ ♥❡✐t❤❡r ♦❢ t❤❡s❡ ♣❛tt❡r♥s✿

❊✈❡♥ t❤♦✉❣❤ ✇❡ ❞♦♥✬t ❝♦♥s✐❞❡r t❤❡ ♠❡t❤♦❞s ♦❢ s♦❧✈✐♥❣ t❤❡s❡ ❡q✉❛t✐♦♥s ❤❡r❡ ✭s❡❡ ❱♦❧✉♠❡ ✺✱ ❈❤❛♣t❡r ✺❉❊✲✶✮✱ ♦♥❝❡ ✇❡ ❤❛✈❡ ❛ ❝❛♥❞✐❞❛t❡ ❢✉♥❝t✐♦♥✱ ✐t ✐s ❡❛s② t♦ t❡st ✐t✳

❊①❛♠♣❧❡ ✻✳✻✳✾✿ ❛♥t✐❞✐✛❡r❡♥t✐❛t✐♦♥ ❚❤✐s ❤❛s ❜❡❡♥ ♦✉r ❛♣♣r♦❛❝❤ t♦ ❛♥t✐❞✐✛❡r❡♥t✐❛t✐♦♥ s♦ ❢❛r✳ ❙♦❧✈❡✿ 2

y ′ (x) = xex . ▲❡t✬s tr②✿

1 2 y = ex . 2

❉✐✛❡r❡♥t✐❛t✐♦♥ ✭✇✐t❤ t❤❡ ❈❤❛✐♥ ❘✉❧❡✮ ❝♦♥✜r♠s t❤❛t t❤✐s ✐s✱ ✐♥❞❡❡❞✱ ❛ s♦❧✉t✐♦♥✿

′

y (x) =



1 x2 e 2

′

1 2 2 = 2xex = xex . 2

❊①❛♠♣❧❡ ✻✳✻✳✶✵✿ ♠✐❣r❛t✐♦♥ ❈♦♥s✐❞❡r✿

y ′ = my + b, m 6= 0 .

✻✳✻✳ ❉✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s

✹✽✷

■t ❝❛♥ r❡♣r❡s❡♥t ♣♦♣✉❧❛t✐♦♥ ❣r♦✇t❤✴❞❡❝❛② ❛❝❝♦♠♣❛♥✐❡❞ ❜② ♠✐❣r❛t✐♦♥ ❛t ❛ ❝♦♥st❛♥t r❛t❡✳ ❚r②✿ y(x) = −

b + Cemt , m

✇❤❡r❡ C ✐s ❛♥② r❡❛❧ ♥✉♠❜❡r✳ ❲❡ s✉❜st✐t✉t❡ t❤✐s ❡①♣r❡ss✐♦♥ ✐♥t♦ t❤❡ ❧❡❢t✲❤❛♥❞ s✐❞❡ ♦❢ t❤❡ ❡q✉❛t✐♦♥✿ 

b − + Cemt m

′

= Cemt


′

= Cmemt .

❲❡ ♥♦✇ s✉❜st✐t✉t❡ ✐t ✐♥t♦ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ♦❢ t❤❡ ❡q✉❛t✐♦♥✿ 

b m − + Cemt m



+ b = −b + mCemt + b = Cmemt .

❈♦♥✜r♠❡❞✦ ❊①❡r❝✐s❡ ✻✳✻✳✶✶

❈♦♥✜r♠ t❤❛t t❤❡ ❢✉♥❝t✐♦♥ y = r + Ce−kt ✐s ❛ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ y ′ = k · (r − y) . ❋✉rt❤❡r♠♦r❡✱ ♦♥❝❡ ✇❡ ❤❛✈❡ ❛ ❢❛♠✐❧② ♦❢ ❝✉r✈❡s✱ ✐t ✐s ❡❛s② t♦ ✜♥❞ t❤❡ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ✐t ❝❛♠❡ ❢r♦♠✳ ❊①❛♠♣❧❡ ✻✳✻✳✶✷✿ ❝✐r❝❧❡s

❲❤❛t ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ❞♦❡s t❤❡ ❢❛♠✐❧② ♦❢ ❛❧❧ ❝♦♥❝❡♥tr✐❝ ❝✐r❝❧❡s ❛r♦✉♥❞ 0 s❛t✐s❢②❄

❚❤✐s ❢❛♠✐❧② ✐s ❣✐✈❡♥ ❜②✿

x2 + y 2 = r2 , r❡❛❧ r .

❲❡ s✐♠♣❧② ❞✐✛❡r❡♥t✐❛t❡ t❤❡ ❡q✉❛t✐♦♥ ✐♠♣❧✐❝✐t❧② ✭❈❤❛♣t❡r ✹✮✿ x2 + y 2 = r2 =⇒ 2x + 2yy ′ = 0 .

❚❤❛t✬s t❤❡ ❡q✉❛t✐♦♥✳ ❊①❛♠♣❧❡ ✻✳✻✳✶✸✿ ❤②♣❡r❜♦❧❛s

❲❤❛t ❛❜♦✉t t❤✐s ❢❛♠✐❧② ♦❢ ❤②♣❡r❜♦❧❛s❄

✻✳✼✳

▼♦t✐♦♥ ✉♥❞❡r ❢♦r❝❡s

✹✽✸

❚❤✐s ❢❛♠✐❧② ✐s ❣✐✈❡♥ ❜②✿

xy = C,

r❡❛❧

C.

❆❣❛✐♥✱ ✇❡ ❞✐✛❡r❡♥t✐❛t❡ t❤❡ ❡q✉❛t✐♦♥ ✐♠♣❧✐❝✐t❧②✳ ❲❡ ❤❛✈❡✿

y + xy ′ = 0 .

❊①❡r❝✐s❡ ✻✳✻✳✶✹ ❲❤❛t ✐s t❤❡ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ❢♦r t❤❡s❡ ♣❛r❛❜♦❧❛s

y = x2 +C ❄

❲❤❛t ❛❜♦✉t

y = Cx2 ❄

❖r

x+C = y 2 ❄

✻✳✼✳ ▼♦t✐♦♥ ✉♥❞❡r ❢♦r❝❡s

■♥ t❤✐s s❡❝t✐♦♥ ✇❡ ✇✐❧❧ s✉♠♠❛r✐③❡ t❤❛t ✇❡ ❤❛✈❡ ❧❡❛r♥❡❞ ❛❜♦✉t ♠♦❞❡❧✐♥❣ ♠♦t✐♦♥✳ ❚❤❡ ❧✐st ♦❢ t❤❡ ♠❛✐♥ q✉❛♥t✐t✐❡s ♦❢ t❤❡ ◆❡✇t♦♥✐❛♥ ♣❤②s✐❝s✱ ✐♥ ❞✐♠❡♥s✐♦♥

1✱

✐s ♣r❡s❡♥t❡❞ ❜❡❧♦✇✿

❉❡✜♥✐t✐♦♥ ✻✳✼✳✶✿ q✉❛♥t✐t✐❡s ♦❢ ◆❡✇t♦♥✐❛♥ ♣❤②s✐❝s ❙✉♣♣♦s❡ ❛ ❢✉♥❝t✐♦♥

r

✐s ❞❡✜♥❡❞ ❛t t❤❡ ♣r✐♠❛r② ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ ❛ ❝❧♦s❡❞

✐♥t❡r✈❛❧ ♦r ♦♥ ❛♥ ♦♣❡♥ ✐♥t❡r✈❛❧✳ ■❢ t❤❡ ✐♥❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡ t❤❡ ❢✉♥❝t✐♦♥

r

t ✐s ❝❛❧❧❡❞ ✏t✐♠❡✑

❛♥❞

✐s ❝❛❧❧❡❞ ✏❧♦❝❛t✐♦♥✧ ♦r ✏♣♦s✐t✐♦♥✑✱ t❤❡♥ t❤❡ r❡❧❛t❡❞ q✉❛♥t✐t✐❡s ❛r❡

❞❡✜♥❡❞ ❛s ❢✉♥❝t✐♦♥s ♦❢ t✐♠❡ ❛♥❞ ❞❡♥♦t❡❞ ❛s ❢♦❧❧♦✇s✿

q✉❛♥t✐t②

✐♥❝r❡♠❡♥t❛❧

♥♦❞❡s

❧♦❝❛t✐♦♥✿

r

♣r✐♠❛r②

r

❞✐s♣❧❛❝❡♠❡♥t✿

D = ∆r

s❡❝♦♥❞❛r②

D = dr

✈❡❧♦❝✐t②✿

v=

∆r ∆t

s❡❝♦♥❞❛r②

v=

♠♦♠❡♥t✉♠✿

p = mv

s❡❝♦♥❞❛r②

p = mv

✐♠♣✉❧s❡✿

J = ∆p

♣r✐♠❛r②

J = dp

♣r✐♠❛r②

d2 r a= 2 dt

2

❛❝❝❡❧❡r❛t✐♦♥✿

∆r a= ∆t2

❝♦♥t✐♥✉♦✉s

dr dt

♣❤②s✐❝s ♥♦t❛t✐♦♥

= r˙

= r¨

❋♦r ❡①❛♠♣❧❡✱ t❤❡ ✉♣✇❛r❞ ❝♦♥❝❛✈✐t② ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❜❡❧♦✇ ✐s ♦❜✈✐♦✉s✱ ✇❤✐❝❤ ✐♥❞✐❝❛t❡s ❛ ♣♦s✐t✐✈❡ ❛❝❝❡❧❡r❛t✐♦♥✿

✻✳✼✳

▼♦t✐♦♥ ✉♥❞❡r ❢♦r❝❡s

✹✽✹

▲❡t✬s st❛t❡ s♦♠❡ ❡❧❡♠❡♥t❛r② ❢❛❝ts ❛❜♦✉t t❤❡ ◆❡✇t♦♥✐❛♥ ♣❤②s✐❝s✳

◆❡✇t♦♥✬s ❋✐rst ▲❛✇ ■❢ t❤❡ ♥❡t ❢♦r❝❡ ✐s ③❡r♦✱ t❤❡♥ t❤❡ ✈❡❧♦❝✐t②

v

♦❢ t❤❡ ♦❜❥❡❝t ✐s ❝♦♥st❛♥t✿

F = 0 =⇒ v =

❝♦♥st❛♥t

❚❤❡ ❧❛✇ ❝❛♥ ❜❡ r❡st❛t❡❞ ✇✐t❤♦✉t ✐♥✈♦❦✐♥❣ t❤❡ ❣❡♦♠❡tr② ♦❢ t✐♠❡✳ ■❢ t❤❡ ♥❡t ❢♦r❝❡ ✐s ③❡r♦✱ t❤❡♥ t❤❡ ❞✐s♣❧❛❝❡♠❡♥t

∆r

♦❢ t❤❡ ♦❜❥❡❝t ✐s ❝♦♥st❛♥t✿

F = 0 =⇒ ∆r =

❝♦♥st❛♥t

.

❚❤❡ ❧❛✇ s❤♦✇s t❤❛t t❤❡ ♦♥❧② ♣♦ss✐❜❧❡ t②♣❡ ♦❢ ♠♦t✐♦♥ ✐♥ s✉❝❤ ❛ ❢♦r❝❡✲❧❡ss ❡♥✈✐r♦♥♠❡♥t ✐s ✉♥✐❢♦r♠❀ ✐✳❡✳✱ ✐t ✐s ❛ r❡♣❡❛t❡❞ ❛❞❞✐t✐♦♥✿

r(t + ∆t) = r(t) + c . ◆❡✇t♦♥✬s ❙❡❝♦♥❞ ▲❛✇ ❚❤❡ ♥❡t ❢♦r❝❡ ♦♥ ❛♥ ♦❜❥❡❝t ✐s ❡q✉❛❧ t♦ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ✐ts ♠♦♠❡♥t✉♠

F =

∆p ∆t

♦r

p✿

F = p˙

◆❡✇t♦♥✬s ❚❤✐r❞ ▲❛✇ ■❢ ♦♥❡ ♦❜❥❡❝t ❡①❡rts ❛ ❢♦r❝❡

F1

♦♥ ❛♥♦t❤❡r ♦❜ ❥❡❝t✱ t❤❡ ❧❛tt❡r s✐♠✉❧t❛♥❡♦✉s❧② ❡①❡rts ❛ ❢♦r❝❡

❢♦r♠❡r✱ ❛♥❞ t❤❡ t✇♦ ❢♦r❝❡s ❛r❡ ❡①❛❝t❧② ♦♣♣♦s✐t❡✿

F1 = −F2 ❚❤❡♦r❡♠ ✻✳✼✳✷✿ ▲❛✇ ♦❢ ❈♦♥s❡r✈❛t✐♦♥ ♦❢ ▼♦♠❡♥t✉♠

❙✉♣♣♦s❡ ❛ s②st❡♠ ♦❢ ♦❜❥❡❝ts ✐s ✏❝❧♦s❡❞✑✿

• •

❚❤❡r❡ ✐s ♥♦ ❡①❝❤❛♥❣❡ ♦❢ ♠❛tt❡r ✇✐t❤ ✐ts s✉rr♦✉♥❞✐♥❣s✳ ❚❤❡r❡ ❛r❡ ♥♦ ❡①t❡r♥❛❧ ❢♦r❝❡s✳

❚❤❡♥ t❤❡ t♦t❛❧ ♠♦♠❡♥t✉♠ ✐s ❝♦♥st❛♥t✿

p=

■♥ ♦t❤❡r ✇♦r❞s✱

J = dp = 0 .

❝♦♥st❛♥t

F2

♦♥ t❤❡

✻✳✼✳

▼♦t✐♦♥ ✉♥❞❡r ❢♦r❝❡s

✹✽✺

Pr♦♦❢✳

❈♦♥s✐❞❡r t✇♦ ♦❜❥❡❝ts ✐♥t❡r❛❝t✐♥❣ ✇✐t❤ ❡❛❝❤ ♦t❤❡r✳ ❇② t❤❡ ❚❤✐r❞ ▲❛✇✱ t❤❡ ❢♦r❝❡s ❜❡t✇❡❡♥ t❤❡♠ ❛r❡ ❡①❛❝t❧② ♦♣♣♦s✐t❡✿ F1 = −F2 .

❉✉❡ t♦ t❤❡ ❙❡❝♦♥❞ ▲❛✇✱ ✇❡ ❝♦♥❝❧✉❞❡ t❤❛t

p′1 = −p′2 .

❋r♦♠ t❤❡ ❙✉♠ ❘✉❧❡✱ ✇❡ ❤❛✈❡✿

(p1 + p2 )′ = 0 .

❊①❡r❝✐s❡ ✻✳✼✳✸

❙t❛t❡ t❤❡ ❡q✉❛t✐♦♥ ♦❢ ♠♦t✐♦♥ ❢♦r ❛ ✈❛r✐❛❜❧❡✲♠❛ss s②st❡♠ ✭s✉❝❤ ❛s ❛ r♦❝❦❡t✮✳ ❍✐♥t✿ ❆♣♣❧② t❤❡ s❡❝♦♥❞ ❧❛✇ t♦ t❤❡ ❡♥t✐r❡✱ ❝♦♥st❛♥t✲♠❛ss s②st❡♠✳ ❊①❡r❝✐s❡ ✻✳✼✳✹

❈r❡❛t❡ ❛ s♣r❡❛❞s❤❡❡t t❤❛t ❝♦♠♣✉t❡s ❛❧❧ ♦❢ t❤❡s❡ q✉❛♥t✐t✐❡s✳ ❙✉♣♣♦s❡ ✇❡ ❦♥♦✇ t❤❡ ❢♦r❝❡s ❛✛❡❝t✐♥❣ ❛ ♠♦✈✐♥❣ ♦❜❥❡❝t✳ ❍♦✇ ❝❛♥ ✇❡ ♣r❡❞✐❝t ✐ts ❞②♥❛♠✐❝s❄ ❆ss✉♠✐♥❣ ❛ ✜①❡❞ ♠❛ss✱ t❤❡ t♦t❛❧ ❢♦r❝❡ ❣✐✈❡s ✉s ♦✉r ❛❝❝❡❧❡r❛t✐♦♥✳ ❚❤❡♥✱ ✇❡ ❛♣♣❧② t❤❡ s❛♠❡ ❢♦r♠✉❧❛ ✭✐✳❡✳✱ ❛♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥✮ t♦ ❝♦♠♣✉t❡✿ • t❤❡ ✈❡❧♦❝✐t② ❢r♦♠ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥✱ ❛♥❞ t❤❡♥ • t❤❡ ❧♦❝❛t✐♦♥ ❢r♦♠ t❤❡ ✈❡❧♦❝✐t②✳

❲❡ ❛❧r❡❛❞② ❦♥♦✇ ❤♦✇ t♦ ✜♥❞ t❤❡ ❧♦❝❛t✐♦♥ ❢r♦♠ t❤❡ ✈❡❧♦❝✐t② ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❛♥❛❧②s✐s ♦❢ ♠♦t✐♦♥ ♦❢ ❛ ✢✉✐❞ ✢♦✇ ❡❛r❧✐❡r ✐♥ t❤✐s ❝❤❛♣t❡r✳ ❚❤✐s ✇✐❧❧ ❜❡ ❛ ❢♦❧❧♦✇✲✉♣✳ ❇❡❧♦✇ ✇❡ ❡①❛♠✐♥❡ t❤❡ ❞✐s❝r❡t❡

♠♦❞❡❧ ♦❢ ♠♦t✐♦♥✳

❆ ✜①❡❞ t✐♠❡ ✐♥❝r❡♠❡♥t ∆t ✐s s✉♣♣❧✐❡❞ ❛❤❡❛❞ ♦❢ t✐♠❡ ❡✈❡♥ t❤♦✉❣❤ ✐t ❝❛♥ ❛❧s♦ ❜❡ ✈❛r✐❛❜❧❡✳ ❲❡ st❛rt ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ t❤r❡❡ q✉❛♥t✐t✐❡s t❤❛t ❝♦♠❡ ❢r♦♠ t❤❡ s❡t✉♣ ♦❢ t❤❡ ♠♦t✐♦♥✿ • t❤❡ ✐♥✐t✐❛❧ t✐♠❡ t0 ✱

• t❤❡ ✐♥✐t✐❛❧ ✈❡❧♦❝✐t② v0 ✱ ❛♥❞

• t❤❡ ✐♥✐t✐❛❧ ❧♦❝❛t✐♦♥ p0 ✳

❚❤❡② ❛r❡ ♣❧❛❝❡❞ ✐♥ t❤❡ ❢♦✉r ❝♦♥s❡❝✉t✐✈❡ ❝❡❧❧s ♦❢ t❤❡ ✜rst r♦✇ ♦❢ t❤❡ t❛❜❧❡✿ ✐♥✐t✐❛❧✿

✐t❡r❛t✐♦♥ n t✐♠❡ tn ❛❝❝❡❧❡r❛t✐♦♥ an ✈❡❧♦❝✐t② vn ❧♦❝❛t✐♦♥ pn 0

3.5

−−

33

22

❆♥♦t❤❡r q✉❛♥t✐t② t❤❛t ❝♦♠❡s ❢r♦♠ t❤❡ s❡t✉♣ ✐s ◮ t❤❡ ❝✉rr❡♥t ❛❝❝❡❧❡r❛t✐♦♥ a1 ✳

■t ✇✐❧❧ ❜❡ ♣❧❛❝❡❞ ✐♥ t❤❡ ♥❡①t r♦✇✳ ❚❤✐s ✐s t❤❡ st❛rt✐♥❣ ♣♦✐♥t✳ ❲❡ ✇♦✉❧❞ ❧✐❦❡ t♦ ❦♥♦✇ t❤❡ ✈❛❧✉❡s ♦❢ ❛❧❧ ♦❢ t❤❡s❡ q✉❛♥t✐t✐❡s ❛t ❡✈❡r② ♠♦♠❡♥t ♦❢ t✐♠❡✱ ✐♥ t❤❡s❡ ✐♥❝r❡♠❡♥ts✳ ❆s ✇❡ ♣r♦❣r❡ss ✐♥ t✐♠❡ ❛♥❞ s♣❛❝❡✱ ♥❡✇ ♥✉♠❜❡rs ❛r❡ ♣❧❛❝❡❞ ✐♥ t❤❡ ♥❡①t r♦✇ ♦❢ ♦✉r t❛❜❧❡✳ ❚❤✐s ✐s ❤♦✇ t❤❡ s❡❝♦♥❞ r♦✇✱ n = 1, t1 = t0 + ∆t✱ ✐s ❝♦♠♣❧❡t❡❞✳ ❚❤❡ ❝✉rr❡♥t ❛❝❝❡❧❡r❛t✐♦♥ a0 ❣✐✈❡♥ ✐♥ t❤❡ ✜rst ❝❡❧❧ ♦❢ t❤❡ s❡❝♦♥❞ r♦✇✳ ❚❤❡ ❝✉rr❡♥t ✈❡❧♦❝✐t② v1 ✐s ❢♦✉♥❞ ❛♥❞ ♣❧❛❝❡❞ ✐♥ t❤❡ s❡❝♦♥❞ ❝❡❧❧ ♦❢ t❤❡ s❡❝♦♥❞ r♦✇ ♦❢ ♦✉r t❛❜❧❡✿

✻✳✼✳

▼♦t✐♦♥ ✉♥❞❡r ❢♦r❝❡s

✹✽✻

◮ ❝✉rr❡♥t ✈❡❧♦❝✐t② = ✐♥✐t✐❛❧ ✈❡❧♦❝✐t② + ❝✉rr❡♥t ❛❝❝❡❧❡r❛t✐♦♥ · t✐♠❡ ✐♥❝r❡♠❡♥t✳

❚❤❡ s❡❝♦♥❞ q✉❛♥t✐t② ✇❡ ✉s❡ ✐s t❤❡ ✐♥✐t✐❛❧ ❧♦❝❛t✐♦♥ p0 ✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ✐s ♣❧❛❝❡❞ ✐♥ t❤❡ t❤✐r❞ ❝❡❧❧ ♦❢ t❤❡ s❡❝♦♥❞ r♦✇✿ ◮ ❝✉rr❡♥t ❧♦❝❛t✐♦♥ = ✐♥✐t✐❛❧ ❧♦❝❛t✐♦♥ + ❝✉rr❡♥t ✈❡❧♦❝✐t② · t✐♠❡ ✐♥❝r❡♠❡♥t✳

❚❤✐s ❞❡♣❡♥❞❡♥❝❡ ✐s s❤♦✇♥ ❜❡❧♦✇✿ ✐♥✐t✐❛❧✿

✐t❡r❛t✐♦♥ n t✐♠❡ tn ❛❝❝❡❧❡r❛t✐♦♥ an

❝✉rr❡♥t✿ ❚❤❡s❡ ❛r❡

0

3.6

−−

1

t1

66

✈❡❧♦❝✐t② vn

❧♦❝❛t✐♦♥ pn

33 ↓ v1

22 ↓ p1

→

→

r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛s✱ ❥✉st ❛s ❜❡❢♦r❡✳

❲❡ ❝♦♥t✐♥✉❡ ✇✐t❤ t❤❡ r❡st ✐♥ t❤❡ s❛♠❡ ♠❛♥♥❡r✳ ❆s ✇❡ ♣r♦❣r❡ss ✐♥ t✐♠❡ ❛♥❞ s♣❛❝❡✱ ♥✉♠❜❡rs ❛r❡ s✉♣♣❧✐❡❞ ❛♥❞ ♣❧❛❝❡❞ ✐♥ ❡❛❝❤ ♦❢ t❤❡ ❢♦✉r ❝♦❧✉♠♥s ♦❢ ♦✉r s♣r❡❛❞s❤❡❡t ♦♥❡ r♦✇ ❛t ❛ t✐♠❡✿ tn , an , vn , pn , n = 1, 2, 3, ...

❚❤❡ ✜rst q✉❛♥t✐t② ✐♥ ❡❛❝❤ r♦✇ ✇❡ ❝♦♠♣✉t❡ ✐s t❤❡ t✐♠❡✿ tn+1 = tn + ∆t .

❚❤❡ ♥❡①t ✐s t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ an+1 ✇❤✐❝❤ ♠❛② ❜❡ ❝♦♥st❛♥t ✭s✉❝❤ ❛s ✐♥ t❤❡ ❝❛s❡ ♦❢ ❛ ❢r❡❡✲❢❛❧❧✐♥❣ ♦❜❥❡❝t✮ ♦r ♠❛② ❡①♣❧✐❝✐t❧② ❞❡♣❡♥❞ ♦♥ t❤❡ ✈❛❧✉❡s ✐♥ t❤❡ ♣r❡✈✐♦✉s r♦✇✳

❞❛t❛

❲❤❡r❡ ❞♦❡s t❤❡ ❝✉rr❡♥t ❛❝❝❡❧❡r❛t✐♦♥ ❝♦♠❡ ❢r♦♠❄ ■t ♠❛② ❝♦♠❡ ❛s ♣✉r❡ ✿ ❚❤❡ ❝♦❧✉♠♥ ✐s ✜❧❧❡❞ ✇✐t❤ ♥✉♠❜❡rs ❛❤❡❛❞ ♦❢ t✐♠❡ ♦r ✐t ✐s ❜❡✐♥❣ ✜❧❧❡❞ ❛s ✇❡ ♣r♦❣r❡ss ✐♥ t✐♠❡ ❛♥❞ s♣❛❝❡✳ ❚❤❡r❡ ♠❛② ❛❧s♦ ❜❡ ❛♥ ❡①♣❧✐❝✐t✱ ❢✉♥❝t✐♦♥❛❧ ❞❡♣❡♥❞❡♥❝❡ ♦❢ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ✭♦r t❤❡ ❢♦r❝❡✮ ♦♥ t❤❡ r❡st ♦❢ t❤❡ q✉❛♥t✐t✐❡s✳ ❚❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ♠❛② ❞❡♣❡♥❞ ♦♥ t❤❡ ❢♦❧❧♦✇✐♥❣✿ ✶✳ t❤❡ ❝✉rr❡♥t t✐♠❡✱ ❡✳❣✳✱ an+1 = sin tn+1 s✉❝❤ ❛s ✇❤❡♥ ✇❡ s♣❡❡❞ ✉♣ t❤❡ ❝❛r✱ ♦r ✷✳ t❤❡ ❧❛st ❧♦❝❛t✐♦♥✱ ❡✳❣✳✱ an+1 = 1/p2n s✉❝❤ ❛s ✇❤❡♥ t❤❡ ❣r❛✈✐t② ❞❡♣❡♥❞s ♦♥ t❤❡ ❞✐st❛♥❝❡ t♦ t❤❡ ♣❧❛♥❡t✱ ♦r ✸✳ t❤❡ ❧❛st ✈❡❧♦❝✐t②✱ ❡✳❣✳✱ an+1 = −vn s✉❝❤ ❛s ✇❤❡♥ t❤❡ ❛✐r r❡s✐st❛♥❝❡ ✇♦r❦s ✐♥ t❤❡ ♦♣♣♦s✐t❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ ✈❡❧♦❝✐t②✱ ✹✳ ♦r ❛❧❧ t❤r❡❡✳ ❊①❡r❝✐s❡ ✻✳✼✳✺

❉r❛✇ ❛rr♦✇s ✐♥ t❤❡ ❛❜♦✈❡ t❛❜❧❡ t♦ ✐❧❧✉str❛t❡ t❤❡s❡ ❞❡♣❡♥❞❡♥❝✐❡s✳ ❙✐♠♣❧❡ ❡①❛♠♣❧❡s ♦❢ ❝❛s❡ ✶ ❛❜♦✈❡ ❛r❡ ❛❞❞r❡ss❡❞ ❜❡❧♦✇✳ ▼♦r❡ ❡①❛♠♣❧❡s ♦❢ ❝❛s❡ ✶ ❛r❡ ❞✐s❝✉ss❡❞ ✐♥ ❱♦❧✉♠❡ ✸✱ ❈❤❛♣t❡r ✸■❈✲✶✱ ❛♥❞ ✐♥ t❤❡ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ s❡tt✐♥❣ ✐♥ ❱♦❧✉♠❡ ✹✱ ❈❤❛♣t❡r ✹❍❉✲✷✳ ❈❛s❡ ✷ ❛♥❞ ❝❛s❡ ✸ ❛r❡ ❝♦♥s✐❞❡r❡❞ ❢✉rt❤❡r ✐♥ ❱♦❧✉♠❡ ✺✱ ❈❤❛♣t❡r ✺❉❊✲✶✳ ❚❤❡ nt❤ ✐t❡r❛t✐♦♥ ♦❢ t❤❡ ✈❡❧♦❝✐t② vn ✐s ❝♦♠♣✉t❡❞✿ • ❈✉rr❡♥t ✈❡❧♦❝✐t② = ❧❛st ✈❡❧♦❝✐t② + ❝✉rr❡♥t ❛❝❝❡❧❡r❛t✐♦♥ · t✐♠❡ ✐♥❝r❡♠❡♥t✱ • vn+1 = vn + an · ∆t✳

❚❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ✈❡❧♦❝✐t② ❛r❡ ♣❧❛❝❡❞ ✐♥ t❤❡ s❡❝♦♥❞ ❝♦❧✉♠♥ ♦❢ ♦✉r t❛❜❧❡✳ ❚❤❡ nt❤ ✐t❡r❛t✐♦♥ ♦❢ t❤❡ ❧♦❝❛t✐♦♥ pn ✐s ❝♦♠♣✉t❡❞✿ • ❈✉rr❡♥t ❧♦❝❛t✐♦♥ = ❧❛st ❧♦❝❛t✐♦♥ + ❝✉rr❡♥t ✈❡❧♦❝✐t② · t✐♠❡ ✐♥❝r❡♠❡♥t✱

• pn+1 = pn + vn · ∆t✳

✻✳✼✳

▼♦t✐♦♥ ✉♥❞❡r ❢♦r❝❡s

✹✽✼

❚❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ❧♦❝❛t✐♦♥ ❛r❡ ♣❧❛❝❡❞ ✐♥ t❤❡ t❤✐r❞ ❝♦❧✉♠♥ ♦❢ ♦✉r t❛❜❧❡✳ ❚❤❡ r❡s✉❧t ✐s ❛ ❣r♦✇✐♥❣ t❛❜❧❡ ♦❢ ✈❛❧✉❡s✿ ✐t❡r❛t✐♦♥ n t✐♠❡ tn ✐♥✐t✐❛❧✿ 0 3.5 1 3.6 ... ... 1000 103.5 ... ...

❛❝❝❡❧❡r❛t✐♦♥ an ✈❡❧♦❝✐t② vn ❧♦❝❛t✐♦♥ pn −− 33 22 66 38.5 25.3 ... ... ... 666 4 336 ... ... ...

❚❤❡ r❡s✉❧t ♠❛② ❜❡ s❡❡♥ ❛s ❢♦✉r s❡q✉❡♥❝❡s tn , an , vn , pn ♦r ❛s t❤❡ t❛❜❧❡ ♦❢ ✈❛❧✉❡s ♦❢ t❤r❡❡

❢✉♥❝t✐♦♥s ♦❢ t✳

❊①❡r❝✐s❡ ✻✳✼✳✻

■♠♣❧❡♠❡♥t ❛ ✈❛r✐❛❜❧❡ t✐♠❡ ✐♥❝r❡♠❡♥t✿ ∆tn+1 = tn+1 − tn ✳ ❊①❛♠♣❧❡ ✻✳✼✳✼✿ r♦❧❧✐♥❣ ❜❛❧❧

❆ r♦❧❧✐♥❣ ❜❛❧❧ ✐s ✉♥❛✛❡❝t❡❞ ❜② ❤♦r✐③♦♥t❛❧ ❢♦r❝❡s✳ ❚❤❡r❡❢♦r❡✱ an = 0 ❢♦r ❛❧❧ n✳ ❚❤❡ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛s ❢♦r t❤❡ ❤♦r✐③♦♥t❛❧ ♠♦t✐♦♥ s✐♠♣❧✐❢② ❛s ❢♦❧❧♦✇s✿ • ❚❤❡ ✈❡❧♦❝✐t② vn+1 = vn + an · ∆t = vn = v0 ✐s ❝♦♥st❛♥t✳ • ❚❤❡ ♣♦s✐t✐♦♥ pn+1 = pn + vn · ∆t = pn + v0 · ∆t ❣r♦✇s ❛t ❡q✉❛❧ ✐♥❝r❡♠❡♥ts✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ♣♦s✐t✐♦♥ ❞❡♣❡♥❞s ❧✐♥❡❛r❧② ♦♥ t❤❡ t✐♠❡✳ ❊①❛♠♣❧❡ ✻✳✼✳✽✿ ❢❛❧❧✐♥❣ ❜❛❧❧

❆ ❢❛❧❧✐♥❣ ❜❛❧❧ ✐s ✉♥❛✛❡❝t❡❞ ❜② ❤♦r✐③♦♥t❛❧ ❢♦r❝❡s ❛♥❞ t❤❡ ✈❡rt✐❝❛❧ ❢♦r❝❡ ✐s ❝♦♥st❛♥t✿ an = a ❢♦r ❛❧❧ n✳ ❚❤❡ ✜rst ♦❢ t❤❡ t✇♦ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛s ❢♦r t❤❡ ✈❡rt✐❝❛❧ ♠♦t✐♦♥ s✐♠♣❧✐✜❡s ❛s ❢♦❧❧♦✇s✿ • ❚❤❡ ✈❡❧♦❝✐t② vn+1 = vn + an · ∆t = vn + a · ∆t ❣r♦✇s ❛t ❡q✉❛❧ ✐♥❝r❡♠❡♥ts✳ • ❚❤❡ ♣♦s✐t✐♦♥ pn+1 = pn + vn · ∆t ❣r♦✇s ❛t ❧✐♥❡❛r❧② ✐♥❝r❡❛s✐♥❣ ✐♥❝r❡♠❡♥ts✳ ■t ❢♦❧❧♦✇s t❤❛t t❤❡ ♣♦s✐t✐♦♥ ❞❡♣❡♥❞s q✉❛❞r❛t✐❝❛❧❧② ♦♥ t❤❡ t✐♠❡✳ ❲❡ ♥♦✇ t✉r♥ t♦ ♠♦t✐♦♥ ✐♥

❞✐♠❡♥s✐♦♥ 2 s✉❝❤ ❛s ✐♥ t❤❡ ❝❛s❡ ♦❢ ❛♥❣❧❡❞ ✢✐❣❤t✳

❖♥❡ ❝♦♥s✐❞❡rs t✇♦ ❢✉♥❝t✐♦♥s ❢♦r ❡❛❝❤ ♦❢ t❤❡ ❛❜♦✈❡ q✉❛♥t✐t✐❡s✿ ✈❡rt✐❝❛❧ ❛♥❞ ❤♦r✐③♦♥t❛❧ ❝♦♠♣♦♥❡♥ts✳ ❋✉rt❤❡r✲ ♠♦r❡✱ ✐♥st❡❛❞ ♦❢ t❤r❡❡ ✭❛❝❝❡❧❡r❛t✐♦♥ ✕ ✈❡❧♦❝✐t② ✕ ❧♦❝❛t✐♦♥✮✱ t❤❡r❡ ✇✐❧❧ ❜❡ s✐① ♠❛✐♥ ❝♦❧✉♠♥s ✿ t✐♠❡

❤♦r✐③✳ ❤♦r✐③✳ ❤♦r✐③✳ ✈❡rt✳ ✈❡rt✳ ✈❡rt✳ ... ❛❝❝❡❧❡r❛t✐♦♥ ✈❡❧♦❝✐t② ♣♦s✐t✐♦♥ ❛❝❝❡❧❡r❛t✐♦♥ ✈❡❧♦❝✐t② ♣♦s✐t✐♦♥ ... t an vn xn bn un yn ... n 0 3.5 −− 33 22 −10 5 3 ... 3.6 66 38.5 25.3 −15 4 3.5 ... 1 ... ... ... ... ... ... ... ... ... 666 4 336 14 66 4 ... 1000 103.5 ... ... ... ... ... ... ... ... ... ❲❡ ✇♦✉❧❞ ❤❛✈❡ ♥✐♥❡ ❝♦❧✉♠♥s ✇❤❡♥ t❤❡ ♠♦❞❡❧ ✐s t❤r❡❡✲❞✐♠❡♥s✐♦♥❛❧✱ s✉❝❤ ❛s ✐♥ t❤❡ ❝❛s❡ ♦❢ ❛♥ ♦❜❥❡❝t t❤r♦✇♥ ✇✐t❤ ❛ s✐❞❡ ✇✐♥❞✳ ❊①❛♠♣❧❡ ✻✳✼✳✾✿ ❝❛♥♥♦♥

❆ ❢❛❧❧✐♥❣ ❜❛❧❧ ✐s ✉♥❛✛❡❝t❡❞ ❜② ❤♦r✐③♦♥t❛❧ ❢♦r❝❡s ❛♥❞ t❤❡ ✈❡rt✐❝❛❧ ❢♦r❝❡ ✐s ❝♦♥st❛♥t✿

x : an+1 = 0;

y : bn+1 = −g .

◆♦✇ r❡❝❛❧❧ t❤❡ s❡t✉♣ ❝♦♥s✐❞❡r❡❞ ♣r❡✈✐♦✉s❧②✿ ❋r♦♠ ❛ 200 ❢❡❡t ❡❧❡✈❛t✐♦♥✱ ❛ ❝❛♥♥♦♥ ✐s ✜r❡❞ ❤♦r✐③♦♥t❛❧❧② ❛t 200 ❢❡❡t ♣❡r s❡❝♦♥❞✳

✻✳✼✳

▼♦t✐♦♥ ✉♥❞❡r ❢♦r❝❡s

✹✽✽

❚❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ❛r❡ ❢♦r

• •

x

❛♥❞

y

❚❤❡ ✐♥✐t✐❛❧ ❧♦❝❛t✐♦♥ ✐s ❣✐✈❡♥ ❜②✿ ❚❤❡ ✐♥✐t✐❛❧ ✈❡❧♦❝✐t② ✐s ❣✐✈❡♥ ❜②✿

r❡s♣❡❝t✐✈❡❧②✿

x0 = 0 ❛♥❞ y0 = 200✳ v0 = 200 ❛♥❞ yu0 = 0✳

❚❤❡♥ ✇❡ ❤❛✈❡ t✇♦ ♣❛✐rs ♦❢ r❡❝✉rs✐✈❡ ❡q✉❛t✐♦♥s ✐♥❞❡♣❡♥❞❡♥t ♦❢ ❡❛❝❤ ♦t❤❡r✿

x: y:

vn+1 = v0 , un+1 = un −g∆t

xn+1 = xn +vn ∆t yn+1 = yn +un ∆t

■♠♣❧❡♠❡♥t❡❞ ✇✐t❤ ❛ s♣r❡❛❞s❤❡❡t✱ t❤❡ ❢♦r♠✉❧❛s ♣r♦❞✉❝❡ t❤❡ s❛♠❡ r❡s✉❧ts ❛s t❤❡ ❡①♣❧✐❝✐t ❢♦r♠✉❧❛s ❞✐❞ ❜❡❢♦r❡✿

❊①❛♠♣❧❡ ✻✳✼✳✶✵✿ ✏s♦❧❛r s②st❡♠✑ ❚❤❡

❣r❛✈✐t② ✐s ❝♦♥st❛♥t

✭✐♥❞❡♣❡♥❞❡♥t ♦❢ ❧♦❝❛t✐♦♥✮ ✐♥ t❤❡ ❢r❡❡✲❢❛❧❧ ♠♦❞❡❧✿

■❢ ✇❡ ♥♦✇ ❧♦♦❦ ❛t t❤❡ s♦❧❛r s②st❡♠✱ ✇✐❧❧ t❤❡ ❣r❛✈✐t② ❜❡ ❝♦♥st❛♥t t♦♦❄ ❝❡rt❛✐♥❧② s❡❡♠s ♣♦ss✐❜❧❡✿

❚❤❡ ❞✐r❡❝t✐♦♥ ✇✐❧❧✱ ♦❢ ❝♦✉rs❡✱ ♠❛tt❡r✦

❊①❡r❝✐s❡ ✻✳✼✳✶✶ ■♠♣❧❡♠❡♥t t❤❡ ♠♦❞❡❧ ❛♥❞ ❡①❛♠✐♥❡ t❤❡ tr❛ ❥❡❝t♦r✐❡s✳

▲♦♦❦✐♥❣ ❢r♦♠ t❤❡ ❊❛rt❤✱ ✐t

✻✳✽✳

❖♣t✐♠✐③❛t✐♦♥ ❡①❛♠♣❧❡s

✹✽✾

❊①❡r❝✐s❡ ✻✳✼✳✶✷

❙✉❣❣❡st ❛ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❞❡♣❡♥❞❡♥❝❡ ♦❢ t❤❡ ❢♦r❝❡ ♦♥ t❤❡ ❧♦❝❛t✐♦♥ t❤❛t ✇♦✉❧❞ ♣r♦❞✉❝❡ s✉❝❤ ❛ s♣✐r❛❧ tr❛❥❡❝t♦r②✿

✻✳✽✳ ❖♣t✐♠✐③❛t✐♦♥ ❡①❛♠♣❧❡s

❊①❛♠♣❧❡ ✻✳✽✳✶✿ ❧♦♥❣❡st s❤♦t

■♥ ❈❤❛♣t❡r ✸✱ ✇❡ ❝♦♥✜r♠❡❞ ✕ ♥✉♠❡r✐❝❛❧❧② ✕ t❤❡ ❝♦♠♠♦♥ ❦♥♦✇❧❡❞❣❡ ❢❛❝t t❤❛t 45 ❞❡❣r❡❡s ✐s t❤❡ ❜❡st ❛♥❣❧❡ t♦ s❤♦♦t ❢♦r ❧♦♥❣❡r ❞✐st❛♥❝❡✿

▲❡t✬s ❛♣♣❧② t❤❡ ♠❡t❤♦❞s ♦❢ ❞✐✛❡r❡♥t✐❛❜❧❡ ❝❛❧❝✉❧✉s ✭❈❤❛♣t❡r ✹✮ t♦

♣r♦✈❡ t❤✐s ❢❛❝t✳

❘❡❝❛❧❧ t❤❡ ❞②♥❛♠✐❝s ♦❢ t❤✐s ♠♦t✐♦♥✳ ❚❤❡ ❤♦r✐③♦♥t❛❧ ❝♦♦r❞✐♥❛t❡ ❛♥❞ t❤❡ ✈❡rt✐❝❛❧ ❝♦♦r❞✐♥❛t❡ ❛r❡ ❣✐✈❡♥ ❛s t❤❡s❡ ❢✉♥❝t✐♦♥s ♦❢ t✐♠❡ t✿ ( x = x0 +v0 t 1 y = y0 +u0 t − gt2 2 ❚❤❡② ❛r❡ s✉❜❥❡❝t t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s✿ • x0 ✐s t❤❡ ✐♥✐t✐❛❧ ❞❡♣t❤✱ • v0 ✐s t❤❡ ✐♥✐t✐❛❧ ❤♦r✐③♦♥t❛❧ ❝♦♠♣♦♥❡♥t ♦❢ ✈❡❧♦❝✐t②✱ • y0 ✐s t❤❡ ✐♥✐t✐❛❧ ❤❡✐❣❤t✱ ❛♥❞ • u0 ✐s t❤❡ ✐♥✐t✐❛❧ ✈❡rt✐❝❛❧ ❝♦♠♣♦♥❡♥t ♦❢ ✈❡❧♦❝✐t②✳ ❲❡ s❤♦♦t ❢r♦♠ t❤❡ ❣r♦✉♥❞ ❛t ③❡r♦ ❡❧❡✈❛t✐♦♥✿

x0 = y0 = 0 . ❲❡ ❛❧s♦ ❛ss✉♠❡ t❤❛t t❤❡ ✐♥✐t✐❛❧ s♣❡❡❞ ✐s 1 ❛♥❞ t❤❛t ✇❡ s❤♦♦t ✉♥❞❡r ❛♥❣❧❡ α✳ ❚❤❡♥✱ ✇❡ ❤❛✈❡ t❤❡ s♣❡❝✐✜❝

✻✳✽✳

❖♣t✐♠✐③❛t✐♦♥ ❡①❛♠♣❧❡s

✹✾✵

❞②♥❛♠✐❝s ♦❢ t❤✐s ✢✐❣❤t r❡♣r❡s❡♥t❡❞ ❜② t✇♦ ❢✉♥❝t✐♦♥s ♦❢ t✐♠❡✿ 

❲❡ s✉❜st✐t✉t❡✿

(

v0 = cos α u0 = sin α

x = cos α · t

1 y = sin α · t − gt2 2

❲❡ ✇✐❧❧ ♥♦✇ ❡①♣r❡ss t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ s❤♦t ❛s t❤❡ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ❛♥❣❧❡ α✳ ◆♦✇✱ t❤❡ ❣r♦✉♥❞ ✐s r❡❛❝❤❡❞ ❛t t❤❡ ♠♦♠❡♥t t ✇❤❡♥ y = 0✳ ❲❡✱ t❤❡♥✱ ♥❡❡❞ t♦ s♦❧✈❡ t❤✐s s②st❡♠ ♦❢ ❡q✉❛t✐♦♥s✿ ( x = cos α · t

1 0 = sin α · t − gt2 2

❚❤❡ s❡❝♦♥❞ ✐s ❛ q✉❛❞r❛t✐❝ ❡q✉❛t✐♦♥ ♦❢ t✳ ❲❡ ❞✐s❝❛r❞ t❤❡ st❛rt✐♥❣ ♠♦♠❡♥t✱ t = 0✱ ❛♥❞ ❞✐✈✐❞❡ ❜② t 6= 0✿ 1 sin α − gt = 0 . 2

❲❡ s♦❧✈❡ ❢♦r t✱ t=

2 sin α , g

❛♥❞ s✉❜st✐t✉t❡ ✐♥t♦ t❤❡ ❡q✉❛t✐♦♥ ❢♦r x✿ x = cos α · t = cos α ·

2 sin α . g

❚❤✐s ✐s t❤❡ ✜♥❛❧ r❡s✉❧t✱ t❤❡ ❞❡♣t❤ ♦❢ t❤❡ s❤♦t ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ❛♥❣❧❡ α✿ D(α) =

2 sin α cos α g

❲❡ ♥❡❡❞ t♦ ✜♥❞ t❤❡ ✈❛❧✉❡ ♦❢ α t❤❛t ♠❛①✐♠✐③❡s D✳ ❙✐♥❝❡ D(0) = D(π/2) = 0✱ t❤❡ ❛♥s✇❡r ❧✐❡s ✇✐t❤✐♥ t❤✐s ✐♥t❡r✈❛❧✳ ❲❡ ❞✐✛❡r❡♥t✐❛t❡✿ D′ (α) =


2 2 cos α cos α + sin α(− sin α) = (cos2 α − sin2 α) . g g

❲❡ s❡t ✐t ❡q✉❛❧ t♦ 0 ❛♥❞ ❝♦♥❝❧✉❞❡✿

D′ (α) = 0 =⇒ cos2 α = sin2 α =⇒ cos α = sin α =⇒ α = ❊①❡r❝✐s❡ ✻✳✽✳✷

❯s❡ ❛ tr✐❣♦♥♦♠❡tr✐❝ ❢♦r♠✉❧❛ t♦ ✜♥✐s❤ t❤❡ s♦❧✉t✐♦♥ ✇✐t❤♦✉t ❞✐✛❡r❡♥t✐❛t✐♦♥✳ ❊①❡r❝✐s❡ ✻✳✽✳✸

❲❤❛t ✐❢ ✇❡ s❤♦♦t ❢r♦♠ ❛ ❤✐❧❧❄

π . 4

✻✳✽✳

❖♣t✐♠✐③❛t✐♦♥ ❡①❛♠♣❧❡s

✹✾✶

❊①❛♠♣❧❡ ✻✳✽✳✹✿ ❧❛②✐♥❣ ❛ ♣✐♣❡

❙✉♣♣♦s❡ ✇❡ ❛r❡ t♦ ❧❛② ❛ ♣✐♣❡ ❢r♦♠ ♣♦✐♥t A t♦ ♣♦✐♥t B ❧♦❝❛t❡❞ ♦♥❡ ❦✐❧♦♠❡t❡r ❡❛st ❛♥❞ ♦♥❡ ❦✐❧♦♠❡t❡r ♥♦rt❤ ♦❢ A✳ ❚❤❡ ❝♦st ♦❢ ❧❛②✐♥❣ ❛ ♣✐♣❡ ✐s ♥♦r♠❛❧❧② $50 ♣❡r ♠❡t❡r✱ ❡①❝❡♣t ❢♦r ❛ r♦❝❦② str✐♣ ♦❢ ❧❛♥❞ 200 ♠❡t❡rs ✇✐❞❡ t❤❛t ❣♦❡s ❡❛st✲✇❡st❀ ❤❡r❡ t❤❡ ♣r✐❝❡ ✐s $100 ♣❡r ♠❡t❡r✳ ❲❤❛t ✐s t❤❡ ❧♦✇❡st ❝♦st t♦ ❧❛② t❤✐s ♣✐♣❡❄

❋✐rst ✇❡ ❞✐s❝♦✈❡r t❤❛t ✐t ❞♦❡s♥✬t ♠❛tt❡r ✇❤❡r❡ t❤❡ ♣❛t❝❤ ✐s ❧♦❝❛t❡❞ ❛♥❞ ❛ss✉♠❡ t❤❛t ✇❡ ❤❛✈❡ t♦ ❝r♦ss ✐t ✜rst✳ ❲❡ ♣r♦❝❡❡❞ ❢r♦♠ A t♦ ♣♦✐♥t P ♦♥ t❤❡ ♦t❤❡r s✐❞❡ ♦❢ t❤❡ ♣❛t❝❤ ❛♥❞ t❤❡♥ t♦ B ✳ ❚❤❡♥ t❤❡ ❝♦st C ✐s C = |AP | · 100 + |P B| · 50 .

❲❡ ♥❡①t ❞❡♥♦t❡ ❜② x t❤❡ ❞✐st❛♥❝❡ ❢r♦♠ P t♦ t❤❡ ♣♦✐♥t ❞✐r❡❝t❧② ❛❝r♦ss ❢r♦♠ A✳ ❚❤❡♥ |AP | =

❚❤❡r❡❢♦r❡✱ ✇❡ ❛r❡ t♦

√

2002 + x2 ❛♥❞ |P B| =

p (1000 − x)2 + 8002 .

♠✐♥✐♠✐③❡ t❤❡ ❝♦st ❢✉♥❝t✐♦♥✿ C(x) =

√

2002 + x2 · 100 +

p (1000 − x)2 + 8002 · 50 .

◆♦✇✱ ✇❡ ❝♦♥✈❡rt t❤✐s ❢♦r♠✉❧❛ ✐♥t♦ ❛ s♣r❡❛❞s❤❡❡t ❢♦r♠✉❧❛✿

❂❙◗❘❚✭✷✵✵✂✷✰❘❈❬✲✶❪✂✷✮✯✶✵✵✰❙◗❘❚✭✭✶✵✵✵✲❘❈❬✲✶❪✮✂✷✰✽✵✵✂✷✮✯✺✵

P❧♦tt✐♥❣ t❤❡ ❝✉r✈❡✱ ❛♥❞ t❤❡♥ ③♦♦♠✐♥❣ ✐♥✱ s✉❣❣❡sts t❤❛t t❤❡ ♦♣t✐♠❛❧ ❝❤♦✐❝❡ ❢♦r x ✐s ❛❜♦✉t 81.5 ♠❡t❡rs ✇✐t❤ t❤❡ ❝♦st ❛❜♦✉t $82, 498.50✿

✻✳✽✳

❖♣t✐♠✐③❛t✐♦♥ ❡①❛♠♣❧❡s

✹✾✷

▲❡t✬s ❝♦♥✜r♠ t❤❡ r❡s✉❧t ✇✐t❤ ❝❛❧❝✉❧✉s✳ ❉✐✛❡r❡♥t✐❛t❡✿ C ′ (x) =

√

2002 + x2 · 100 +

′ p (1000 − x)2 + 8002 · 50

−2(1000 − x) 2x · 100 + p = √ · 50 . 2 2 2 200 + x 2 (1000 − x)2 + 8002

❚❤❡ ❡q✉❛t✐♦♥ C ′ (x) = 0 ♣r♦✈❡s ✐ts❡❧❢ t♦♦ ❝♦♠♣❧❡① t♦ ❜❡ s♦❧✈❡❞ ❡①❛❝t❧②✳

■♥st❡❛❞✱ ✇❡ ❧♦♦❦ ❜❛❝❦ ❛t t❤❡ ♣✐❝t✉r❡ t♦ r❡❝♦❣♥✐③❡ t❤❡ t❡r♠s ✐♥ t❤✐s ❡①♣r❡ss✐♦♥ ❛s ❢r❛❝t✐♦♥s ♦❢ s✐❞❡s ♦❢ t❤❡s❡ t✇♦ tr✐❛♥❣❧❡s✿ ❚❤❡♥✱ C ′ = 0 ✐❢ ❛♥❞ ♦♥❧② ✐❢

[ \ C ′ (x) = cos AP C · 100 − cos BP D · 50 . [ 50 cos AP C . = 100 \ cos BP D

■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ♦♣t✐♠❛❧ ♣r✐❝❡ ✐s r❡❛❝❤❡❞ ✇❤❡♥ t❤❡ r❛t✐♦ ♦❢ t❤❡ ❝♦s✐♥❡s ♦❢ t❤❡ t✇♦ ❛♥❣❧❡s ❛t P ❛r❡ ❡q✉❛❧ t♦ t❤❡ r❛t✐♦ ♦❢ t❤❡ t✇♦ ♣r✐❝❡s✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ♠❛❦✐♥❣ t❤❡ ♣r✐❝❡ ♦❢ ❧❛②✐♥❣ t❤❡ ♣✐♣❡ ❛❝r♦ss t❤❡ ♣❛t❝❤ ♠♦r❡ ❡①♣❡♥s✐✈❡ ✇✐❧❧ ♠❛❦❡ t❤✐s ♣❛rt ♦❢ t❤❡ ♣❛t❤ t♦ ❝r♦ss ♠♦r❡ ❞✐r❡❝t❧②✳ ❙✐♥❝❡ ❝♦s✐♥❡ ✐s ❛ ❞❡❝r❡❛s✐♥❣ \ [ ❢✉♥❝t✐♦♥ ♦♥ (0, π/2)✱ ✇❡ ❝♦♥❝❧✉❞❡ t❤❛t BP D ✐s ❡①♣r❡ss❡❞ ✉♥✐q✉❡❧② ✐♥ t❡r♠s ♦❢ AP C✳

❊①❡r❝✐s❡ ✻✳✽✳✺

❙❤♦✇ t❤❛t✱ ✐♥❞❡❡❞✱ t❤❡ ❧♦❝❛t✐♦♥ ♦❢ t❤❡ str✐♣ ❞♦❡s♥✬t ♠❛tt❡r✳ ❍✐♥t✿ ●❡♦♠❡tr②✳ ❊①❛♠♣❧❡ ✻✳✽✳✻✿ r❡❢r❛❝t✐♦♥

❙✉♣♣♦s❡ ❧✐❣❤t ✐s ♣❛ss✐♥❣ ❢r♦♠ ♦♥❡ ♠❡❞✐✉♠ t♦ ❛♥♦t❤❡r✳ ■t ✐s ❦♥♦✇♥ t❤❛t t❤❡ s♣❡❡❞ ♦❢ ❧✐❣❤t t❤r♦✉❣❤ t❤❡ ✜rst ✐s v1 ✱ ❛♥❞ t❤r♦✉❣❤ t❤❡ s❡❝♦♥❞ v2 ✳ ❲❡ r❡❧② ♦♥ t❤❡ ♣r✐♥❝✐♣❧❡ t❤❛t ❧✐❣❤t ❢♦❧❧♦✇s t❤❡ ♣❛t❤ ♦❢ ❢❛st❡st s♣❡❡❞ ❛♥❞ ✜♥❞ t❤❡ ❛♥❣❧❡ ♦❢ r❡❢r❛❝t✐♦♥✳ ❆ s✐♠✐❧❛r ❜✉t s✐♠♣❧❡r s❡t✲✉♣✿

✻✳✽✳

❖♣t✐♠✐③❛t✐♦♥ ❡①❛♠♣❧❡s

✹✾✸

▲❡t x ❜❡ t❤❡ ♣❛r❛♠❡t❡r✱ t❤❡♥ t❤❡ t✐♠❡ ✐t t❛❦❡s t♦ ❣❡t ❢r♦♠ A t♦ B ✐s✿ |AP | |AP | + v2 p √v1 2 1 + (1 − x)2 1+x + . = v1 v2

T (x) =

❲❡ ❞✐✛❡r❡♥t✐❛t❡✱ ❥✉st ❛s ✐♥ t❤❡ ❧❛st ❡①❛♠♣❧❡✿ 1 x −(1 − x) 1 √ + p v1 1 + x2 v2 1 + (1 − x)2 1 1 [ \ = cos AP C − cos BP D. v1 v2

T ′ (x) =

❆♥❞ T ′ = 0 ✐❢ ❛♥❞ ♦♥❧② ✐❢

[ v1 cos AP C = . v2 \ cos BP D

❚❤❡r❡❢♦r❡✱ ❧✐❣❤t ❢♦❧❧♦✇s t❤❡ ♣❛t❤ ✇✐t❤ t❤❡ r❛t✐♦ ♦❢ t❤❡ ❝♦s✐♥❡s ♦❢ t❤❡ t✇♦ ❛♥❣❧❡s ❛t P ❡q✉❛❧ t♦ t❤❡ r❛t✐♦ ♦❢ t❤❡ t✇♦ ♣r♦♣❛❣❛t✐♦♥ s♣❡❡❞s✳ ❊①❡r❝✐s❡ ✻✳✽✳✼

❋✐♥❞ t❤❡ ❞✐st❛♥❝❡ ❢r♦♠ t❤❡ ♣♦✐♥t (1, 1) t♦ t❤❡ ♣❛r❛❜♦❧❛ y = −x2 ❜② t✇♦ ♠❡t❤♦❞s✿ ✭❛✮ ✜♥❞ t❤❡ ♠✐♥✐♠❛❧ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ ♣♦✐♥t ❛♥❞ t❤❡ ❝✉r✈❡✱ ❛♥❞ ✭❜✮ ✜♥❞ t❤❡ ❧✐♥❡ ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ ❝✉r✈❡ ❛♥❞ ✐ts ❧❡♥❣t❤✳ ❊①❛♠♣❧❡ ✻✳✽✳✽✿ ♥✉♠❡r✐❝❛❧ ♦♣t✐♠✐③❛t✐♦♥

❚❤❡ ❞✐✛❡r❡♥t✐❛t✐♦♥ ♠✐❣❤t ♣r♦❞✉❝❡ ❛ ❢✉♥❝t✐♦♥ s♦ ❝♦♠♣❧❡① t❤❛t s♦❧✈✐♥❣ t❤❡ ❡q✉❛t✐♦♥ f ′ = 0 ❛♥❛❧②t✐❝❛❧❧② ✐s ✐♠♣♦ss✐❜❧❡✳ ■♥ t❤❛t ❝❛s❡✱ ✇❡ ❝❛♥ ❛♣♣❧② ♦♥❡ ♦❢ t❤❡ ✐t❡r❛t✐✈❡ ♣r♦❝❡ss❡s ♦❢ s♦❧✈✐♥❣ ❡q✉❛t✐♦♥ ❞✐s❝✉ss❡❞ ❡❛r❧✐❡r ✐♥ t❤✐s ❝❤❛♣t❡r✳ ❆❧t❡r♥❛t✐✈❡❧②✱ ✇❡ ❞❡s✐❣♥ ❛ ♣r♦❝❡ss t❤❛t ❢♦❧❧♦✇s t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ ❛s ✐t ❛❧✇❛②s ♣♦✐♥ts ❛t t❤❡ ♥❡❛r❡st ♠❛①✐♠✉♠✿

✻✳✽✳ ❖♣t✐♠✐③❛t✐♦♥ ❡①❛♠♣❧❡s

✹✾✹

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ♠♦✈❡ ❛❧♦♥❣ t❤❡ x✲❛①✐s ❛♥❞ t❤❡♥✿ ′ • ❲❡ st❡♣ r✐❣❤t ✇❤❡♥ f > 0✳ • ❲❡ st❡♣ ❧❡❢t ✇❤❡♥ f ′ < 0✳

❲❡ r❡✈❡rs❡ t❤❡ ❞✐r❡❝t✐♦♥ ✇❤❡♥ ✇❡ ❧♦♦❦ ❢♦r ❛ ♠✐♥✐♠✉♠✳ ❲❡ ♠❛❦❡ ❛ st❡♣ ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ❞❡r✐✈❛t✐✈❡ s♦ t❤❛t ✐ts ♠❛❣♥✐t✉❞❡ ❛❧s♦ ♠❛tt❡rs❀ ✇❡ ♠♦✈❡ ❢❛st❡r ✇❤❡♥ ✐t✬s ❤✐❣❤❡r✳

❆s ②♦✉ ❝❛♥ s❡❡✱ t❤❡ ♠♦t✐♦♥ ✇✐❧❧ s❧♦✇ ❞♦✇♥ ❛s ✇❡ ❣❡ts ❝❧♦s❡r t♦ t❤❡ ❞❡st✐♥❛t✐♦♥✱ ✇❤✐❝❤ ✐s ❝♦♥✈❡♥✐❡♥t✳ ❙♦✱ ✇❡ ❜✉✐❧❞ ❛ s❡q✉❡♥❝❡ ♦❢ ❛♣♣r♦①✐♠❛t✐♦♥s r❡❝✉rs✐✈❡❧② ✿ xn+1 = xn + h · f ′ (xn ) ,

✇❤❡r❡ h ✐s t❤❡ ❝♦❡✣❝✐❡♥t ♦❢ ♣r♦♣♦rt✐♦♥❛❧✐t②✳ ❋♦r ❡①❛♠♣❧❡✱ ❢♦r f (x) = sin x✱ ✇❡ ❤❛✈❡✿ ❚❤❡ s♣r❡❛❞s❤❡❡t ❢♦r♠✉❧❛ ✐s✿

xn+1 = xn + h · cos(xn ) .

❂❘❬✲✶❪❈✰❘✷❈✯❈❖❙✭❘❬✲✶❪❈✮

✻✳✾✳

❋✉♥❝t✐♦♥s ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s

✹✾✺

❊s♣❡❝✐❛❧❧② ❢♦r ❢✉♥❝t✐♦♥s ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s ✭❱♦❧✉♠❡ ✹✱ ❈❤❛♣t❡r ✹❍❉✲✸✮✱ t❤❡ ♠❡t❤♦❞ ✐s ❝❛❧❧❡❞ t❤❡ ❣r❛❞✐❡♥t ❞❡s❝❡♥t ✭✇❤✐❝❤ ✐♥❝❧✉❞❡s t❤❡ ❛s❝❡♥t✮✳

✻✳✾✳ ❋✉♥❝t✐♦♥s ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s

❊①❛♠♣❧❡ ✻✳✾✳✶✿ ❝❛tt❧❡ ❡♥❝❧♦s✉r❡

❘❡❝❛❧❧ t❤❡ ♣r♦❜❧❡♠ ✭❈❤❛♣t❡r ✷✮✿ ❆ ❢❛r♠❡r ✇✐t❤ 100 ②❛r❞s ♦❢ ❢❡♥❝✐♥❣ ♠❛t❡r✐❛❧ ✇❛♥ts t♦ ❜✉✐❧❞ ❛s ❧❛r❣❡ ❛ r❡❝t❛♥❣✉❧❛r ❡♥❝❧♦s✉r❡ ❛s ♣♦ss✐❜❧❡ ❢♦r ❤✐s ❝❛tt❧❡✳ ❚❤✐s ✐s ❤♦✇ t❤❡ ❛♥❛❧②s✐s st❛rts✳ ❚❤❡ s❝♦♣❡ ♦❢ ♣♦ss✐❜✐❧✐t✐❡s ✐s ✐♥✜♥✐t❡✿

❲❡ ❛r❡ s✉♣♣♦s❡❞ t♦ ✜♥❞ t❤❡ ❞✐♠❡♥s✐♦♥s ♦❢ ❛ r❡❝t❛♥❣❧❡ ✕ t❤❡ ✇✐❞t❤ ❛♥❞ t❤❡ ❞❡♣t❤ ✕ ✇✐t❤ t❤❡ ❧❛r❣❡st ❛r❡❛✳

❲❡ st❛rt ❜② r❛♥❞♦♠❧② ❝❤♦♦s✐♥❣ ♣♦ss✐❜❧❡ ♠❡❛s✉r❡♠❡♥ts ♦❢ t❤❡ ❡♥❝❧♦s✉r❡ ❛♥❞ ❝♦♠♣✉t❡ ✐ts ❛r❡❛ ✇✐t❤ t❤❡ ❢♦r♠✉❧❛✿ ❆r❡❛ = ✇✐❞t❤ · ❞❡♣t❤ .

❲❡ st❛rt ✇✐t❤ ❛ r❡❝t❛♥❣❧❡ ✇✐t❤ ✇✐❞t❤ 20 ❛♥❞ ✐♥❝r❡❛s❡ t❤❡ ❞❡♣t❤✿ • 20 ❜② 20 ❣✐✈❡s ✉s ❛♥ ❛r❡❛ ♦❢ 400 sq✉❛r❡ ②❛r❞s✳ • 20 ❜② 30 ❣✐✈❡s ✉s ❛♥ ❛r❡❛ ♦❢ 600 sq✉❛r❡ ②❛r❞s✳ • 20 ❜② 40 ❣✐✈❡s ✉s ❛♥ ❛r❡❛ ♦❢ 800 sq✉❛r❡ ②❛r❞s✱ ❡t❝✳ ❖❢ ❝♦✉rs❡✱ t❤❡ ❛r❡❛ ✐s ❣❡tt✐♥❣ ❜✐❣❣❡r ❛♥❞ ❜✐❣❣❡r❀ ❤♦✇❡✈❡r✱ 30 ❜② 30 ❣✐✈❡s ✉s 900✦ ❚❤❡ ♣❛tt❡r♥ ✐s ✉♥❝❧❡❛r✳ ❲❡ ✇✐❧❧ ♥❡❡❞ t♦ ❝♦❧❧❡❝t ♠♦r❡ ❞❛t❛✳ ▲❡t✬s s♣❡❡❞ ✉♣ t❤✐s ♣r♦❝❡ss ✇✐t❤ ❛ s♣r❡❛❞s❤❡❡t✳ ❲❡ ✐♥tr♦❞✉❝❡ t❤❡ ✈❛r✐❛❜❧❡s✿ • w ✐s ❢♦r t❤❡ ✇✐❞t❤✱ ❛♥❞ • d ✐s ❢♦r t❤❡ ❞❡♣t❤✱ t❤❡♥

✻✳✾✳ ❋✉♥❝t✐♦♥s ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s

✹✾✻

• a ✐s ❢♦r t❤❡ ❛r❡❛✿

a = w · d.

❲❡ ✜rst ♥❡❡❞ t♦ ❝♦♠♣✐❧❡ ❛❧❧ ♣♦ss✐❜❧❡ ❝♦♠❜✐♥❛t✐♦♥s ♦❢ t❤❡ ✇✐❞t❤ ✭❝♦❧✉♠♥ W ✮ ❛♥❞ t❤❡ ❞❡♣t❤ ✭❝♦❧✉♠♥ D✮✳ ❲❡ ❝❤♦♦s❡ t♦ ❣♦ ❡✈❡r② 10 ②❛r❞s✳ ❚❤❡♥ t❤❡ t✇♦ q✉❛♥t✐t✐❡s✱ ✐♥❞❡♣❡♥❞❡♥t❧②✱ r✉♥ t❤r♦✉❣❤ t❤❡s❡ 11 ♥✉♠❜❡rs✿ ✇✐❞t❤ = 0, 10, 20, ..., 100 ❛♥❞ ❞❡♣t❤ = 0, 10, 20, ..., 100 . ❚♦❣❡t❤❡r✱ t❤❡r❡ ❛r❡ 11 · 11 = 121 ♣♦ss✐❜❧❡ ❝♦♠❜✐♥❛t✐♦♥s✳ ❚❤❡ ✜rst ❝❤❛❧❧❡♥❣❡ ✐s t♦ ❧✐st ❛❧❧ ♣♦ss✐❜❧❡ ♣❛✐rs ♦❢ ✇✐❞t❤ ❛♥❞ ❞❡♣t❤s ✐♥ ❛ s♣r❡❛❞s❤❡❡t✳ ❚❤❡ s✐♠♣❧❡st ❛♣♣r♦❛❝❤ ✐s t♦ ✜① ♦♥❡ ✈❛❧✉❡ ♦❢ w✱ st❛rt✐♥❣ ✇✐t❤ 0✱ ❛♥❞ t❤❡♥ st❛rt ✈❛r②✐♥❣ t❤❡ ✈❛❧✉❡ ♦❢ d ✉♥t✐❧ ✇❡ r❡❛❝❤ 10✱ t❤❡♥ ✇❡ s❡t w ❡q✉❛❧ t♦ 10 ❛♥❞ s♦ ♦♥✳ ❖♥❝❡ ✇❡ ❤❛✈❡ t❤❡♠ ❛❧❧✱ ✇❡ ❛❧s♦ ❤❛✈❡ ❛❧❧ t❤❡ ❛r❡❛s t♦♦❀ ✇❡ ❥✉st ❝♦♠♣✉t❡ t❤❡ ❛r❡❛✱ ❝♦❧✉♠♥ A✱ ✇✐t❤ t❤❡ ❢♦r♠✉❧❛✿ ❂❘❈❬✲✷❪✯❘❈❬✲✶❪

❚❤✐s ✐s t❤❡ r❡s✉❧t ✭❧❡❢t✮✿

❚❤✐s ❧✐st ❛rr❛♥❣❡♠❡♥t ♦❢ t❤❡ ❞❛t❛ ✐s ✐♥❝♦♥✈❡♥✐❡♥t✳ ❚♦ ✐♥✈❡st✐❣❛t❡✱ ❧❡t✬s ♣❧♦t t❤❡s❡ ♣❛✐rs ✭r✐❣❤t✮✳ ❚♦✲ ❣❡t❤❡r✱ t❤❡② ❢♦r♠ ❛♥ 11 × 11 sq✉❛r❡ ♦❢ ♣♦ss✐❜❧❡ ❝♦♠❜✐♥❛t✐♦♥s✱ ✇✐t❤ ✐ts ✇✐❞t❤ ❛♥❞ ❞❡♣t❤ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ✇✐❞t❤ ❛♥❞ ❞❡♣t❤ ♦❢ t❤❡ ❡♥❝❧♦s✉r❡✦ ■t ❛♣♣❡❛rs t❤❛t ✐t ✐s ❜❡tt❡r t♦ ❛rr❛♥❣❡ t❤❡s❡ ♣❛✐rs ✐♥ ❛ t❛❜❧❡ t❤❛♥ ✐♥ ❛ ❧✐st✳ ❲❡ ❝❤♦♦s❡ t♦ ❝♦♥s✐❞❡r t❤❡ ❞✐♠❡♥s✐♦♥s ❡✈❡r② 10 ②❛r❞s ✈✐❛ t❤❡s❡ t✇♦ s❡ts ♥❛♠❡❞ ❛❢t❡r t❤❡s❡ t✇♦ q✉❛♥✲ t✐t✐❡s✿ W = {0, 10, ..., 100} ❛♥❞ D = {0, 10, ..., 100} .

❚❤❡ t❛❜❧❡✱ ❛♥❞ t❤❡ s♣r❡❛❞s❤❡❡t✱ t❛❦❡s t❤❡ ❢♦r♠✿

W \D 0 10 ... 100 0 1

✳✳ ✳

100

❲❡ ♣✉t t❤❡ ❞❛t❛ ✐♥ ❛ s♣r❡❛❞s❤❡❡t t♦ ❝♦♠♣✉t❡ t❤❡ ❛r❡❛ ♦❢ t❤❡ ❡♥❝❧♦s✉r❡ ✇✐t❤ t❤❡s❡ ❞✐♠❡♥s✐♦♥s ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❢♦r♠✉❧❛✿ ❂❘❈✷✯❘✷❈

r❡❢❡rr✐♥❣ t♦ t❤❡ s❛♠❡ r♦✇ ❛♥❞ s❡❝♦♥❞ ❝♦❧✉♠♥✱ ❛♥❞ t❤❡ s❡❝♦♥❞ r♦✇ ❛♥❞ t❤❡ s❛♠❡ ❝♦❧✉♠♥✱ r❡s♣❡❝t✐✈❡❧②✳ ❆s ❛ r❡s✉❧t✱ t❤❡ t❛❜❧❡ ✐s ✜❧❧❡❞ ✇✐t❤ t❤❡s❡ ✈❛❧✉❡s ✭s❤♦✇♥ ❢♦r t❤❡ 11 × 11 t❛❜❧❡✮✿

✻✳✾✳

❋✉♥❝t✐♦♥s ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s

✹✾✼

❚❤✐s ✐s t❤❡ ❝♦♠♣❧❡t❡ t❛❜❧❡ ✭101 × 101✮✿

◆♦✇ ✇❡ ❝♦❧♦r ✕ ❛✉t♦♠❛t✐❝❛❧❧② ✕ t❤❡ ❝❡❧❧s ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❛r❡❛ ✐t ❝♦rr❡s♣♦♥❞s t♦✿

❚❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❛r❡❛ ❣r♦✇s ❛s ✇❡ ✐♥❝r❡❛s❡ t❤❡ ✇✐❞t❤ ♦r t❤❡ ❞❡♣t❤✱ ❜✉t t❤❡ ❢❛st❡st ❣r♦✇t❤ s❡❡♠s t♦ ❜❡ ✐♥ s♦♠❡ ❞✐❛❣♦♥❛❧ ❞✐r❡❝t✐♦♥✳ ❆♥② ❢♦r♠✉❧❛ ✇✐t❤ t✇♦ ✐♥❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡s ❛♥❞ ♦♥❡ ❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡ ❝❛♥ ❜❡ st✉❞✐❡❞ ✐♥ t❤✐s ♠❛♥♥❡r✿

a = wd ♦r z = x + y , ❙✉❝❤ ❛♥ ❡①♣r❡ss✐♦♥ ✐s ❝❛❧❧❡❞ ❛

❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✳

❚❤❡ ♥♦t❛t✐♦♥ ✐s ❛s ❢♦❧❧♦✇s✿

g(w, d) = wd ♦r f (x, y) = x + y . ❊①❛♠♣❧❡ ✻✳✾✳✷✿ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s

▲❡t

f (x, y) = x + y . ❲❡ ✐❧❧✉str❛t❡ t❤✐s ♥❡✇ ❢✉♥❝t✐♦♥ ❜❡❧♦✇✳ ❋✐rst✱ ❜② ❝❤❛♥❣✐♥❣ ✕ ✐♥❞❡♣❡♥❞❡♥t❧② ✖ t❤❡ t✇♦ ✈❛r✐❛❜❧❡s✱ ✇❡ ❝r❡❛t❡ ❛ t❛❜❧❡ ♦❢ ♥✉♠❜❡rs ✭❧❡❢t✮✳ ❲❡ ❝❛♥ ❢✉rt❤❡r♠♦r❡ ❝♦❧♦r t❤✐s ❛rr❛② ♦❢ ❝❡❧❧s ✭♠✐❞❞❧❡✮ s♦ t❤❛t t❤❡ ❝♦❧♦r ♦❢ t❤❡ (x, y)✲❝❡❧❧ ✐s ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ ✈❛❧✉❡ ♦❢ z ✿

✻✳✾✳

❋✉♥❝t✐♦♥s ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s

❚❤❡ ✈❛❧✉❡ ♦❢

z

✹✾✽

❝❛♥ ❛❧s♦ ❜❡ ✈✐s✉❛❧✐③❡❞ ❛s t❤❡ ❡❧❡✈❛t✐♦♥ ♦❢ ❛ ♣♦✐♥t ❛t t❤❛t ❧♦❝❛t✐♦♥ ✭r✐❣❤t✮✳

❙♦✱ t❤❡ ♠❛✐♥ ♠❡t❛♣❤♦r ❢♦r ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s ✇✐❧❧ ❜❡

t❡rr❛✐♥ ✿

❊❛❝❤ ❧✐♥❡ ✐♥❞✐❝❛t❡s ❛ ❝♦♥st❛♥t ❡❧❡✈❛t✐♦♥✳ ❊①❛♠♣❧❡ ✻✳✾✳✸✿ ❞✐st❛♥❝❡

❚❤❡ ❞✐st❛♥❝❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❈❛rt❡s✐❛♥ ♣❧❛♥❡ ✭s❡❡♥ ✐♥ ❱♦❧✉♠❡ ✶✱ ❈❤❛♣t❡r ✶P❈✲✺✮ ❝r❡❛t❡s ❛ ❢✉♥❝t✐♦♥ ♦❢

(x, y) t♦ t❤❡ p z = x2 + y 2 .

t✇♦ ✈❛r✐❛❜❧❡s✳ ❚❤✐s ✐s t❤❡ ❞✐st❛♥❝❡ ❢r♦♠ ❛ ♣♦✐♥t

❙❧✐❣❤t❧② s✐♠♣❧❡r ✐s t❤❡ sq✉❛r❡ ♦❢ t❤❡ ❞✐st❛♥❝❡ ❢r♦♠ ❛ ♣♦✐♥t

♦r✐❣✐♥✿✿

(x, y)

t♦ t❤❡ ♦r✐❣✐♥✿

z = x2 + y 2 . ❲❡ ❝r❡❛t❡ ❛ t❛❜❧❡ ♦❢ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ❡①♣r❡ss✐♦♥ ♦♥ t❤❡ ❧❡❢t ✐♥ ❛ s♣r❡❛❞s❤❡❡t ✇✐t❤ t❤❡ ❢♦r♠✉❧❛✿

❂❘❈✶✂ ✷✰❘✶❈✂ ✷ ❲❡ t❤❡♥ ❝♦❧♦r t❤❡ ❝❡❧❧s✿

❚❤❡ ♥❡❣❛t✐✈❡ ✈❛❧✉❡s ♦❢

z

❛r❡ ✐♥ ❜❧✉❡ ❛♥❞ t❤❡ ♣♦s✐t✐✈❡ ❛r❡ ✐♥ r❡❞✳ ❚❤❡ ❝✐r❝✉❧❛r ♣❛tt❡r♥ ✐s ❝❧❡❛r✳

❚❤❡ ♣❛tt❡r♥ s❡❡♠s t♦ ❜❡ ♠❛❞❡ ❢r♦♠ ❝♦♥❝❡♥tr✐❝ ❝✐r❝❧❡s ✇✐t❤ t❤❡ r❛❞✐✉s t❤❛t ✈❛r✐❡s ✇✐t❤

❋♦r ❡❛❝❤

z✱

✇❡ ❤❛✈❡ ❛

r❡❧❛t✐♦♥

❲❡ ❛❧s♦ r❡♣r❡s❡♥t ❛ ❢✉♥❝t✐♦♥

p

❜❡t✇❡❡♥

x

❛♥❞

z✿

y✳

❞✐❛❣r❛♠♠❛t✐❝❛❧❧② ❛s ❛

❜❧❛❝❦ ❜♦①

t❤❛t ♣r♦❝❡ss❡s t❤❡ ✐♥♣✉ts ❛♥❞ ♣r♦❞✉❝❡s t❤❡

✻✳✾✳ ❋✉♥❝t✐♦♥s ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s

✹✾✾

♦✉t♣✉t✿ ✐♥♣✉ts

❢✉♥❝t✐♦♥

♦✉t♣✉t

x ց

→

p

ր

y

■♥st❡❛❞✱ ✇❡ ✇♦✉❧❞ ❧✐❦❡ t♦ s❡❡ ❛ s✐♥❣❧❡ ✐♥♣✉t ✈❛r✐❛❜❧❡✱

(x, y)✱

z

❞❡❝♦♠♣♦s❡❞ ✐♥t♦ t✇♦

x

❛♥❞

y

t♦ ❜❡ ♣r♦❝❡ss❡❞

❜② t❤❡ ❢✉♥❝t✐♦♥ ❛t t❤❡ s❛♠❡ t✐♠❡ ✿

(x, y) →

→ z

p

❚❤❡ ❞✐✛❡r❡♥❝❡ ❢r♦♠ ❛❧❧ t❤❡ ❢✉♥❝t✐♦♥s ✇❡ ❤❛✈❡ s❡❡♥ ✉♥t✐❧ ♥♦✇ ✐s t❤❡ ♥❛t✉r❡ ♦❢ t❤❡ ✐♥♣✉t✳ ❙♦✱ ❡✈❡♥ t❤♦✉❣❤ ✇❡ s♣❡❛❦ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✱ t❤❡ ✐❞❡❛ ♦❢ ❢✉♥❝t✐♦♥ r❡♠❛✐♥s t❤❡ s❛♠❡✿

◮ ❚❤❡r❡ ✐s ❛ s❡t ✭❞♦♠❛✐♥✮ ❛♥❞ ❛♥♦t❤❡r s❡t ✭❝♦❞♦♠❛✐♥✮✱

❛♥❞ t❤❡ ❢✉♥❝t✐♦♥ ❛ss✐❣♥s t♦ ❡❛❝❤ ❡❧❡♠❡♥t

♦❢ t❤❡ ❢♦r♠❡r ❛♥ ❡❧❡♠❡♥t ♦❢ t❤❡ ❧❛tt❡r✳ ❚❤❡ ✐❞❡❛ ✐s r❡✢❡❝t❡❞ ✐♥ t❤❡ ♥♦t❛t✐♦♥ ✇❡ ✉s❡✿

F :X→Z ♦r

F

X −−−−→ Z ❆ ❝♦♠♠♦♥ ✇❛② t♦ ✈✐s✉❛❧✐③❡ t❤❡ ❝♦♥❝❡♣t ♦❢ ❢✉♥❝t✐♦♥ ✕ ❡s♣❡❝✐❛❧❧② ✇❤❡♥ t❤❡ s❡ts ❝❛♥♥♦t ❜❡ r❡♣r❡s❡♥t❡❞ ❜② ♠❡r❡ ❧✐sts ✕ ✐s t♦ ❞r❛✇ s❤❛♣❡❧❡ss ❜❧♦❜s ❝♦♥♥❡❝t❡❞ ❜② ❛rr♦✇s✿

■♥ ❝♦♥tr❛st t♦ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥✱ ❤♦✇❡✈❡r✱ t❤❡ ❞♦♠❛✐♥ ✐s ❛ s✉❜s❡t ♦❢ t❤❡ ❋♦r ❡①❛♠♣❧❡✱ ✇❡ ❤❛✈❡ ❢♦r

(x, y)✲♣❧❛♥❡✳

f (x, y) = x + y ✿

(0, 0) → 0, (0, 1) → 1, (1, 0) → 1, (1, 1) → 2, (1, 2) → 3, (2, 1) → 3 ... ❊❛❝❤ ❛rr♦✇ ❝❧❡❛r❧② ✐❞❡♥t✐✜❡s t❤❡ ✐♥♣✉t ✕ ❛♥ ❡❧❡♠❡♥t ♦❢ ✕ ❛♥ ❡❧❡♠❡♥t ♦❢

Z

X

✕ ♦❢ t❤✐s ♣r♦❝❡❞✉r❡ ❜② ✐ts ❜❡❣✐♥♥✐♥❣✱ ❛♥❞ t❤❡ ♦✉t♣✉t

✕ ❜② ✐ts ❡♥❞✳

❚❤✐s ✐s t❤❡ ♥♦t❛t✐♦♥ ❢♦r t❤❡ ♦✉t♣✉t ♦❢ ❛ ❢✉♥❝t✐♦♥

F

✇❤❡♥ t❤❡ ✐♥♣✉t ✐s

■♥♣✉t ❛♥❞ ♦✉t♣✉t ♦❢ ❢✉♥❝t✐♦♥

F (x, y) = z ♦r

F : (x, y) → z ■t r❡❛❞s ✏ F ♦❢

(x, y)

✐s

z ✑✳

❲❡ st✐❧❧ ❤❛✈❡✿

F(

✐♥♣✉t

)=

✐♥♣✉t

→

♦✉t♣✉t

❛♥❞

F :

♦✉t♣✉t

.

x✿

✻✳✾✳

❋✉♥❝t✐♦♥s ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s

t❤r❡❡ ✈❛r✐❛❜❧❡s r❡❧❛t❡❞ t♦ ❡❛❝❤ ♦t❤❡r✱ ❜✉t t❤✐s r❡❧❛t✐♦♥ ✐s ✉♥❡q✉❛❧✿ ❞❡♣❡♥❞❡♥t ♦♥ t❤❡ ✐♥♣✉t✳ ❚❤❛t ✐s ✇❤② ✇❡ s❛② t❤❛t t❤❡ ✐♥♣✉ts ❛r❡ t❤❡ ✐♥❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡s ✇❤✐❧❡ t❤❡ ♦✉t♣✉t ✐s t❤❡ ❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡✳ ❊①❛♠♣❧❡ ✻✳✾✳✹✿ ✢♦✇❝❤❛rts r❡♣r❡s❡♥t ❢✉♥❝t✐♦♥s

❋✉♥❝t✐♦♥s ❛r❡

❡①♣❧✐❝✐t r❡❧❛t✐♦♥s✳

✺✵✵

❚❤❡r❡ ❛r❡

❚❤❡ t✇♦ ✐♥♣✉t ✈❛r✐❛❜❧❡s ❝♦♠❡ ✜rst ❛♥❞✱ t❤❡r❡❢♦r❡✱ t❤❡ ♦✉t♣✉t ✐s

❋♦r ❡①❛♠♣❧❡✱ ❢♦r ❛ ❣✐✈❡♥ ✐♥♣✉t

(x, y)✱

✇❡ ✜rst s♣❧✐t ✐t✿

x

❛♥❞

y

❛r❡ t❤❡ t✇♦

♥✉♠❡r✐❝❛❧

✐♥♣✉ts✳ ❚❤❡♥ ✇❡

❞♦ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥s❡❝✉t✐✈❡❧②✿

• • •

❛❞❞

x

y✱ ❜② 2✱

❛♥❞

♠✉❧t✐♣❧②

❛♥❞ t❤❡♥

sq✉❛r❡✳

❙✉❝❤ ❛ ♣r♦❝❡❞✉r❡ ❝❛♥ ❜❡ ❝♦♥✈❡♥✐❡♥t❧② ✈✐s✉❛❧✐③❡❞ ✇✐t❤ ❛ ✏✢♦✇❝❤❛rt✑✿

(x, y) →

→ u →

x+y

u·2

→ z →

z2

→ v

❋✉♥❝t✐♦♥s ♦❢ t✇♦ ✈❛r✐❛❜❧❡s ❝♦♠❡ ❢r♦♠ ♠❛♥② s♦✉r❝❡s ❛♥❞ ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ✐♥ ❞✐✛❡r❡♥t ❢♦r♠s✿

❆♥

•

❛ ❧✐st ♦❢ ✐♥str✉❝t✐♦♥s ✭❛♥ ❛❧❣♦r✐t❤♠✮

•

❛♥ ❛❧❣❡❜r❛✐❝ ❢♦r♠✉❧❛

•

❛ ❧✐st ♦❢ ♣❛✐rs ♦❢ ✐♥♣✉ts ❛♥❞ ♦✉t♣✉ts

•

❛ ❣r❛♣❤

•

❛ tr❛♥s❢♦r♠❛t✐♦♥

❛❧❣❡❜r❛✐❝ r❡♣r❡s❡♥t❛t✐♦♥

❣✐✈❡ ✐t ❛ ♥❛♠❡✱ s❛②

f✱

✐s ❡①❡♠♣❧✐✜❡❞ ❜②

z = x2 y ✳

■♥ ♦r❞❡r t♦ ♣r♦♣❡r❧② ✐♥tr♦❞✉❝❡ t❤✐s ❛s ❛ ❢✉♥❝t✐♦♥✱ ✇❡

❛♥❞ ✇r✐t❡✿

f (x, y) = x2 y . ▲❡t✬s ❡①❛♠✐♥❡ t❤✐s ♥♦t❛t✐♦♥✿

❋✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s

♥❛♠❡✿

z ↑

=

❞❡♣❡♥❞❡♥t

f ↑

(

x, y ↑ ↑

❢✉♥❝t✐♦♥

✈❛r✐❛❜❧❡

) =

x2 y ↑↑

✐♥❞❡♣❡♥❞❡♥t

✐♥❞❡♣❡♥❞❡♥t

✈❛r✐❛❜❧❡s

✈❛r✐❛❜❧❡s

❊①❛♠♣❧❡ ✻✳✾✳✺✿ ♣❧✉❣ ✐♥ ✈❛❧✉❡s ■♥s❡rt ♦♥❡ ✐♥♣✉t ✈❛❧✉❡ ✐♥ ❛❧❧ ♦❢ t❤❡s❡ ❜♦①❡s ❛♥❞ t❤❡ ♦t❤❡r ✐♥ t❤♦s❡ ❝✐r❝❧❡s✳ ❋♦r ❡①❛♠♣❧❡✱ t❤✐s ❢✉♥❝t✐♦♥✿

f (x) =

2x2 y − 3y + 7 , y 3 + 2x + 1

❝❛♥ ❜❡ ✉♥❞❡rst♦♦❞ ❛♥❞ ❡✈❛❧✉❛t❡❞ ✈✐❛ t❤✐s ❞✐❛❣r❛♠✿

f () = ❚❤✐s ✐s ❤♦✇

f (3, 0)

22 −3 +7 .

3 + 2 + 1

✐s ❡✈❛❧✉❛t❡❞✿

f



2

✵ − 3 ✵ +7 23 ✵ 3 , = . 3 ✵ +2 3 +1



■♥ s✉♠♠❛r②✱

◮ ✏ x✑

❛♥❞ ✏ y ✑ ✐♥ ❛ ❢♦r♠✉❧❛ s❡r✈❡ ❛s ❛

♣❧❛❝❡❤♦❧❞❡rs

❢♦r✿ ♥✉♠❜❡rs✱ ✈❛r✐❛❜❧❡s✱ ❛♥❞ ✇❤♦❧❡ ❢✉♥❝t✐♦♥s✳

✻✳✾✳

❋✉♥❝t✐♦♥s ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s

✺✵✶

❍♦✇ ❞♦ ✇❡ st✉❞② ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s❄ ❲❡ ✉s❡ ✇❤❛t ✇❡ ❦♥♦✇ ❛❜♦✉t ❢✉♥❝t✐♦♥s ♦❢ s✐♥❣❧❡ ✈❛r✐❛❜❧❡✳ ❆❜♦✈❡ ✇❡ ❧♦♦❦❡❞ ❛t t❤❡ ❝✉r✈❡s ♦❢ ❝♦♥st❛♥t ❡❧❡✈❛t✐♦♥ ♦❢ t❤❡ s✉r❢❛❝❡s✳

❆♥ ❛❧t❡r♥❛t✐✈❡ ✐❞❡❛ ✐s ❛ s✉r✈❡②✐♥❣

♠❡t❤♦❞✿

◮

■♥ ♦r❞❡r t♦ st✉❞② ❛ t❡rr❛✐♥✱ ✇❡ ❝♦♥❝❡♥tr❛t❡ ♦♥ t❤❡ t✇♦ ♠❛✐♥ ❞✐r❡❝t✐♦♥s✳

■♠❛❣✐♥❡ t❤❛t ✇❡ ❞r✐✈❡ s♦✉t❤✲♥♦rt❤ ❛♥❞ ❡❛st✲✇❡st s❡♣❛r❛t❡❧② ✇❛t❝❤✐♥❣ ❤♦✇ t❤❡ ❡❧❡✈❛t✐♦♥ ❝❤❛♥❣❡s✿

❲❡ ❝❛♥ ❡✈❡♥ ✐♠❛❣✐♥❡ t❤❛t ✇❡ ❞r✐✈❡ ❛r♦✉♥❞ ❛ ❝✐t② ♦♥ ❛ ❤✐❧❧ ❛♥❞ t❤❡s❡ tr✐♣s ❢♦❧❧♦✇ t❤❡ str❡❡t ❣r✐❞✿

❊❛❝❤ ♦❢ t❤❡s❡ tr✐♣s ❝r❡❛t❡s ❛ ❢✉♥❝t✐♦♥ ♦❢ s✐♥❣❧❡ ✈❛r✐❛❜❧❡✱ ❚♦ ✈✐s✉❛❧✐③❡✱ ❝♦♥s✐❞❡r t❤❡ ♣❧♦t ♦❢

F (x, y) = sin(xy)

x

♦r

y✳

♦♥ t❤❡ ❧❡❢t✿

❲❡ ♣❧♦t t❤❡ s✉r❢❛❝❡ ❛s ❛ ✏✇✐r❡✲❢r❛♠❡✑ ♦♥ t❤❡ r✐❣❤t✳ ❊❛❝❤ ✇✐r❡ ✐s ❛ s❡♣❛r❛t❡ tr✐♣✳ ❚❤❡ ❣r❛♣❤s ♦❢ t❤❡s❡ ❢✉♥❝t✐♦♥s ❛r❡ t❤❡ s❧✐❝❡s ❝✉t ❜② t❤❡ ✈❡rt✐❝❛❧ ♣❧❛♥❡s ❛❧✐❣♥❡❞ ✇✐t❤ t❤❡ ❛①❡s ❢r♦♠ t❤❡ s✉r❢❛❝❡ t❤❛t ✐s t❤❡ ❣r❛♣❤ ♦❢

F✿

✻✳✾✳ ❋✉♥❝t✐♦♥s ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s

✺✵✷

❆s ❛ ✜rst st❡♣✱ ✇❡ ❝❛♥ st✉❞② t❤❡ ♠♦♥♦t♦♥✐❝✐t② ♦❢ t❤❡s❡ ❢✉♥❝t✐♦♥s ✇✐t❤ t❤❡ ❤❡❧♣ ♦❢ t❤❡✐r ❞❡r✐✈❛t✐✈❡s✳ ❊①❛♠♣❧❡ ✻✳✾✳✻✿ s❛❞❞❧❡

▲❡t✬s ♣❧♦t t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥✿ z = xy .

❚❤✐s ✐s ✇❤❛t t❤❡ ❣r❛♣❤s ♦❢ t❤❡s❡ r❡❧❛t✐♦♥s ❧♦♦❦ ❧✐❦❡ ♣❧♦tt❡❞ ❢♦r ✈❛r✐♦✉s z ✬s❀ t❤❡② ❛r❡ ❝✉r✈❡s ❝❛❧❧❡❞ ❤②♣❡r❜♦❧❛s ✿

■♥st❡❛❞✱ ✇❡ ✜① ♦♥❡ ✐♥❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡ ❛t ❛ t✐♠❡✳ ❲❡ ✜① y ✜rst✿

♣❧❛♥❡ y y y y y

❡q✉❛t✐♦♥ ❝✉r✈❡ ❧✐♥❡ ✇✐t❤ ❧✐♥❡ ✇✐t❤ ❧✐♥❡ ✇✐t❤ ❧✐♥❡ ✇✐t❤ ❧✐♥❡ ✇✐t❤

=2 z =x·2 =1 z =x·1 =0 z =x·0=0 = −1 z = x · (−1) = −2 z = x · (−2)

❚❤❡ ✈✐❡✇ s❤♦✇♥ ❜❡❧♦✇ ✐s ❢r♦♠ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ y ✲❛①✐s✿

s❧♦♣❡ 2 s❧♦♣❡ 1 s❧♦♣❡ 0 s❧♦♣❡ 1 s❧♦♣❡ − 2

✻✳✾✳ ❋✉♥❝t✐♦♥s ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s

✺✵✸

❚❤❡ ❞❛t❛ ❢♦r ❡❛❝❤ ❧✐♥❡ ❝♦♠❡s ❢r♦♠ t❤❡ x✲❝♦❧✉♠♥ ♦❢ t❤❡ s♣r❡❛❞s❤❡❡t ❛♥❞ ♦♥❡ ♦❢ t❤❡ z ✲❝♦❧✉♠♥s✳ ❚❤❡s❡ ❧✐♥❡s ❣✐✈❡ t❤❡ ❧✐♥❡s ♦❢ ❡❧❡✈❛t✐♦♥ ♦❢ t❤✐s t❡rr❛✐♥ ✐♥ ❛ ♣❛rt✐❝✉❧❛r✱ s❛②✱ ❡❛st✲✇❡st ❞✐r❡❝t✐♦♥✳ ❚❤✐s ✐s ❡q✉✐✈❛❧❡♥t t♦ ❝✉tt✐♥❣ t❤❡ ❣r❛♣❤ ❜② ❛ ✈❡rt✐❝❛❧ ♣❧❛♥❡ ♣❛r❛❧❧❡❧ t♦ t❤❡ xz ✲♣❧❛♥❡✳ ❲❡ ✜① x s❡❝♦♥❞✿

♣❧❛♥❡

❡q✉❛t✐♦♥ ❝✉r✈❡ x=2 z = 2 · y ❧✐♥❡ ✇✐t❤ x=1 z = 1 · y ❧✐♥❡ ✇✐t❤ x=0 z = 0 · y = 0 ❧✐♥❡ ✇✐t❤ x = −1 z = (−1) · y ❧✐♥❡ ✇✐t❤ x = −2 z = (−2) · y ❧✐♥❡ ✇✐t❤

s❧♦♣❡ 2 s❧♦♣❡ 1 s❧♦♣❡ 0 s❧♦♣❡ 1 s❧♦♣❡ − 2

❚❤✐s ✐s ❡q✉✐✈❛❧❡♥t t♦ ❝✉tt✐♥❣ t❤❡ ❣r❛♣❤ ❜② ❛ ✈❡rt✐❝❛❧ ♣❧❛♥❡ ♣❛r❛❧❧❡❧ t♦ t❤❡ yz ✲♣❧❛♥❡✳ ❚❤❡ ✈✐❡✇ s❤♦✇♥ ❜❡❧♦✇ ✐s ❢r♦♠ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ x✲❛①✐s✿

❚❤❡ ❞❛t❛ ❢♦r ❡❛❝❤ ❧✐♥❡ ❝♦♠❡s ❢r♦♠ t❤❡ y ✲r♦✇ ♦❢ t❤❡ s♣r❡❛❞s❤❡❡t ❛♥❞ ♦♥❡ ♦❢ t❤❡ z ✲r♦✇s✳ ❚❤❡s❡ ❧✐♥❡s ❣✐✈❡ t❤❡ ❧✐♥❡s ♦❢ ❡❧❡✈❛t✐♦♥ ♦❢ t❤✐s t❡rr❛✐♥ ✐♥ ❛ ♣❛rt✐❝✉❧❛r✱ s❛②✱ ♥♦rt❤✲s♦✉t❤ ❞✐r❡❝t✐♦♥✳ ❊①❛♠♣❧❡ ✻✳✾✳✼✿ ❛♥♦t❤❡r s✉r❢❛❝❡

▲❡t✬s tr② ❛ s✐♠✐❧❛r ❛♥❛❧②s✐s ❢♦r t❤❡ ❢✉♥❝t✐♦♥ ❜❡❧♦✇✿ z = y − x2 .

❇❡❧♦✇ ✐s t❤❡ ❛rr❛② ♦❢ ✭❝♦❧♦r❡❞✮ ♦✉t♣✉ts ❛♥❞ t❤❡ ❣r❛♣❤✿

■❢ y ✐s ✜①❡❞ ❛t y = b✱ ✇❡ ❤❛✈❡ ❛ ❢✉♥❝t✐♦♥ ♦❢ x✿ b − x2 .

■ts ❞❡r✐✈❛t✐✈❡ ✇✐t❤ r❡s♣❡❝t t♦ x ✐s −2x✳ ❚❤❡r❡❢♦r❡✱ ✐❢ ✇❡ ♠♦✈❡ ❛❧♦♥❣ t❤❡ x✲❛①✐s✱ ✇❡ ❣♦ ✉♣ ❛♥❞ t❤❡♥ ❞♦✇♥ ♥♦ ♠❛tt❡r ✇❤❡r❡ t❤✐s ❧✐♥❡ ✐s✳ ■t✬s ❛ ♣❛r❛❜♦❧❛✦ ■❢ x ✐s ✜①❡❞ ❛t x = a✱ ✇❡ ❤❛✈❡ ❛ ❢✉♥❝t✐♦♥ ♦❢ y ✿ y − a2 .

■ts ❞❡r✐✈❛t✐✈❡ ✇✐t❤ r❡s♣❡❝t t♦ y ✐s 1✳ ❚❤❡r❡❢♦r❡✱ ✐❢ ✇❡ ♠♦✈❡ ❛❧♦♥❣ t❤❡ y ✲❛①✐s✱ ✇❡ ❣♦ ✉♣ ♥♦ ♠❛tt❡r ✇❤❡r❡ t❤✐s ❧✐♥❡ ✐s✳ ■t✬s ❛ str❛✐❣❤t ❧✐♥❡✦

✻✳✾✳

❋✉♥❝t✐♦♥s ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s

❊①❡r❝✐s❡ ✻✳✾✳✽ ✶✳ ❙❦❡t❝❤ ❛ str❛✐❣❤t ❧✐♥❡ ✇✐t❤ ♣❛r❛❜♦❧❛s ❤❛♥❣✐♥❣ ❢r♦♠ ✐t✳ ✷✳ ❙❦❡t❝❤ ❛ ♣❛r❛❜♦❧❛ ✇✐t❤ str❛✐❣❤t ❧✐♥❡s ❧❡❛♥❡❞ ❛❣❛✐♥st ✐t✳

❊①❡r❝✐s❡ ✻✳✾✳✾ Pr♦✈✐❞❡ ❛ s✐♠✐❧❛r ❛♥❛❧②s✐s ❢♦r

f (x, y) = x2 + y 2 ✳

❚❤✐s st✉❞② ✇✐❧❧ ❝♦♥t✐♥✉❡ ✐♥ ❱♦❧✉♠❡ ✸✱ ❈❤❛♣t❡r ✸■❈✲✹✱ ❛♥❞ t❤❡♥ ✐♥ ❱♦❧✉♠❡ ✹✱ ❈❤❛♣t❡r ✹❍❉✲✸✳

✺✵✹

❊①❡r❝✐s❡s

❈♦♥t❡♥ts ✶ ❊①❡r❝✐s❡s✿ ❙❡ts✱ ❧♦❣✐❝✱ ❢✉♥❝t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺✵✺

✷ ❊①❡r❝✐s❡s✿ ❇❛❝❦❣r♦✉♥❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺✵✾

✸ ❊①❡r❝✐s❡s✿ ❙❡q✉❡♥❝❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺✶✸

✹ ❊①❡r❝✐s❡s✿ ❘❛t❡s ♦❢ ❝❤❛♥❣❡

✺✶✹

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺✶✼

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺✷✵

✺ ❊①❡r❝✐s❡s✿ ▲✐♠✐ts ❛♥❞ ❝♦♥t✐♥✉✐t② ✻ ❊①❡r❝✐s❡s✿ ❉❡r✐✈❛t✐✈❡s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺✷✸

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺✷✾

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺✸✵

✼ ❊①❡r❝✐s❡s✿ ❋❡❛t✉r❡s ♦❢ ❣r❛♣❤s ✽ ❊①❡r❝✐s❡s✿ ▲✐♥❡❛r✐③❛t✐♦♥ ✾ ❊①❡r❝✐s❡s✿ ▼♦❞❡❧s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺✸✸

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺✸✺

✶✵ ❊①❡r❝✐s❡s✿ ■♥❢♦r♠❛t✐♦♥ ❢r♦♠ t❤❡ ❞❡r✐✈❛t✐✈❡s ✶✶ ❊①❡r❝✐s❡s✿ ❈♦♠♣✉t✐♥❣ ❞❡r✐✈❛t✐✈❡s

✶✷ ❊①❡r❝✐s❡s✿ ❖♣t✐♠✐③❛t✐♦♥ ❛♥❞ ♦t❤❡r ❛♣♣❧✐❝❛t✐♦♥s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺✸✼

✶✳ ❊①❡r❝✐s❡s✿ ❙❡ts✱ ❧♦❣✐❝✱ ❢✉♥❝t✐♦♥s

❊①❡r❝✐s❡ ✶✳✶

❊①❡r❝✐s❡ ✶✳✸

❘❡♣r❡s❡♥t t❤❡ ❢♦❧❧♦✇✐♥❣ s❡t ✐♥ t❤❡ s❡t✲❜✉✐❧❞✐♥❣ ♥♦✲

❲❤❛t ❛r❡ t❤❡ ♠❛①✱ ♠✐♥✱ ❛♥❞ ❛♥② ❜♦✉♥❞s ♦❢ t❤❡ s❡t

t❛t✐♦♥✿

♦❢ ✐♥t❡❣❡rs❄ ❲❤❛t ❛❜♦✉t

R❄

X = [0, 1] ∪ [2, 3] = ? ❊①❡r❝✐s❡ ✶✳✹ ■s t❤❡ ❝♦♥✈❡rs❡ ♦❢ ❛ tr✉❡ st❛t❡♠❡♥t tr✉❡❄

❊①❡r❝✐s❡ ✶✳✷ ❊①❡r❝✐s❡ ✶✳✺

❙✐♠♣❧✐❢②✿

■s t❤❡ ❝♦♥✈❡rs❡ ♦❢ t❤❡ ❝♦♥✈❡rs❡ ♦❢ ❛ tr✉❡ st❛t❡♠❡♥t

{x > 0 : x

✐s ❛ ♥❡❣❛t✐✈❡ ✐♥t❡❣❡r

}.

tr✉❡❄

✶✳ ❊①❡r❝✐s❡s✿ ❙❡ts✱ ❧♦❣✐❝✱ ❢✉♥❝t✐♦♥s

✺✵✻

❊①❡r❝✐s❡ ✶✳✻

❊①❡r❝✐s❡ ✶✳✶✺

❙t❛t❡ t❤❡ ❝♦♥✈❡rs❡ ♦❢ t❤✐s st❛t❡♠❡♥t✿ ✏❚❤❡ ❝♦♥✈❡rs❡

●✐✈❡ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❛ ❝✐r❝❧❡✳

♦❢ t❤❡ ❝♦♥✈❡rs❡ ♦❢ ❛ tr✉❡ st❛t❡♠❡♥t ✐s tr✉❡✑✳ ❊①❡r❝✐s❡ ✶✳✶✻ ❊①❡r❝✐s❡ ✶✳✼

❙✉♣♣♦s❡ t❤❡ ❝♦st ✐s

❘❡♣r❡s❡♥t t❤❡s❡ s❡ts ❛s ✐♥t❡rs❡❝t✐♦♥s ❛♥❞ ✉♥✐♦♥s✿

♠✐❧❡s✳ ■♥t❡r♣r❡t t❤❡ ❢♦❧❧♦✇✐♥❣ st♦r✐❡s ✐♥ t❡r♠s ♦❢

✶✳

(0, 5)

✷✳

{3}

✸✳

∅

✹✳

{x : x > 0 ❖❘ x

✺✳

{x : x

f (x)

❞♦❧❧❛rs ❢♦r ❛ t❛①✐ tr✐♣ ♦❢

✶✳ ▼♦♥❞❛②✱ ■ t♦♦❦ ❛ t❛①✐ t♦ t❤❡ st❛t✐♦♥

5

x f✳

♠✐❧❡s

❛✇❛②✳ ✷✳ ❚✉❡s❞❛②✱ ■ t♦♦❦ ❛ t❛①✐ t♦ t❤❡ st❛t✐♦♥ ❜✉t t❤❡♥ r❡❛❧✐③❡❞ t❤❛t ■ ❧❡❢t s♦♠❡t❤✐♥❣ ❛t ❤♦♠❡ ❛♥❞ ✐s ❛♥ ✐♥t❡❣❡r}

✐s ❞✐✈✐s✐❜❧❡ ❜②

❤❛❞ t♦ ❝♦♠❡ ❜❛❝❦✳

6}

✸✳ ❲❡❞♥❡s❞❛②✱ ■ t♦♦❦ ❛ t❛①✐ t♦ t❤❡ st❛t✐♦♥ ❛♥❞ ■ ❣❛✈❡ ♠② ❞r✐✈❡r ❛ ✜✈❡ ❞♦❧❧❛r t✐♣✳

❊①❡r❝✐s❡ ✶✳✽

❚r✉❡ ♦r ❢❛❧s❡✿

✹✳ ❚❤✉rs❞❛②✱ ■ t♦♦❦ ❛ t❛①✐ t♦ t❤❡ st❛t✐♦♥ ❜✉t t❤❡

0 = 1 =⇒ 0 = 1❄

❞r✐✈❡r ❣♦t ❧♦st ❛♥❞ ❞r♦✈❡ ✜✈❡ ❡①tr❛ ♠✐❧❡s✳ ✺✳ ❋r✐❞❛②✱ ■ ❤❛✈❡ ❜❡❡♥ t❛❦✐♥❣ ❛ t❛①✐ t♦ t❤❡ st❛✲

❊①❡r❝✐s❡ ✶✳✾

t✐♦♥ ❛❧❧ ✇❡❡❦ ♦♥ ❝r❡❞✐t❀ ■ ♣❛② ✇❤❛t ■ ♦✇❡

Pr♦✈❡ ♦r ❞✐s♣r♦✈❡✿

t♦❞❛②✳ ❲❤❛t ✐❢ t❤❡r❡ ✐s ❛♥ ❡①tr❛ ❝❤❛r❣❡ ♣❡r r✐❞❡ ♦❢

max{max A, max B} = max(A ∪ B) . ❊①❡r❝✐s❡ ✶✳✶✵

❝♦♥❝❧✉s✐♦♥s ②♦✉ ❛rr✐✈❡ t♦

A✱

A❄

❞♦❧✲

❊①❡r❝✐s❡ ✶✳✶✼

✭❛✮ ■❢✱ st❛rt✐♥❣ ✇✐t❤ ❛ st❛t❡♠❡♥t ❝❧✉❞❡ ❛❜♦✉t

m

❧❛rs❄

0 = 1✱

A✱

❛❢t❡r ❛ s❡r✐❡s ♦❢

✇❤❛t ❝❛♥ ②♦✉ ❝♦♥✲

✭❜✮ ■❢✱ st❛rt✐♥❣ ✇✐t❤ ❛ st❛t❡♠❡♥t

❛❢t❡r ❛ s❡r✐❡s ♦❢ ❝♦♥❝❧✉s✐♦♥s ②♦✉ ❛rr✐✈❡ t♦

✇❤❛t ❝❛♥ ②♦✉ ❝♦♥❝❧✉❞❡ ❛❜♦✉t

f : A → B

❛♥❞

g : C → D

❜❡ t✇♦ ♣♦ss✐✲

❜❧❡ ❢✉♥❝t✐♦♥s✳ ❋♦r ❡❛❝❤ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢✉♥❝t✐♦♥s✱ st❛t❡ ✇❤❡t❤❡r ♦r ♥♦t ②♦✉ ❝❛♥ ❝♦♠♣✉t❡

0 = 0✱

f ◦ g✿

• D⊂B

A❄

• C⊂A

❊①❡r❝✐s❡ ✶✳✶✶

❲❡ ❦♥♦✇ t❤❛t ✏■❢ ✐t r❛✐♥s✱ t❤❡ r♦❛❞ ❣❡ts ✇❡t✑✳ ❉♦❡s ✐t ♠❡❛♥ t❤❛t ✐❢ t❤❡ r♦❛❞ ✐s ✇❡t✱ ✐t ❤❛s r❛✐♥❡❞❄

• B⊂D • B=C ❊①❡r❝✐s❡ ✶✳✶✽

❊①❡r❝✐s❡ ✶✳✶✷

❆ ❣❛r❛❣❡ ❧✐❣❤t ✐s ❝♦♥tr♦❧❧❡❞ ❜② ❛ s✇✐t❝❤ ❛♥❞✱ ❛❧s♦✱ ✐t ♠❛② ❛✉t♦♠❛t✐❝❛❧❧② t✉r♥ ♦♥ ✇❤❡♥ ✐t s❡♥s❡s ♠♦t✐♦♥ ❞✉r✐♥❣ ♥✐❣❤tt✐♠❡✳ ■❢ t❤❡ ❧✐❣❤t ✐s ❖❋❋✱ ✇❤❛t ❞♦ ②♦✉ ❝♦♥❝❧✉❞❡❄

❊①❡r❝✐s❡ ✶✳✶✸

❋✉♥❝t✐♦♥

y = f (x)

✐s ❣✐✈❡♥ ❜❡❧♦✇ ❜② ❛ ❧✐st ♦❢ s♦♠❡

♦❢ ✐ts ✈❛❧✉❡s✳ ▼❛❦❡ s✉r❡ t❤❡ ❢✉♥❝t✐♦♥ ✐s ♦♥t♦✳

x −1 0 1 2 3 4 5 y = f (x) −1 4 5 2 ❊①❡r❝✐s❡ ✶✳✶✾

■❢ ❛♥ ❛❞✈❡rt✐s❡♠❡♥t ❝❧❛✐♠s t❤❛t ✏❆❧❧ ♦✉r s❡❝♦♥❞✲ ❤❛♥❞ ❝❛rs ❝♦♠❡ ✇✐t❤ ✇♦r❦✐♥❣ ❆❈✑✱ ✇❤❛t ✐s t❤❡ ❡❛s✲ ✐❡st ✇❛② t♦ ❞✐s♣r♦✈❡ t❤❡ s❡♥t❡♥❝❡❄

❋✉♥❝t✐♦♥

y = f (x)

✐s ❣✐✈❡♥ ❜❡❧♦✇ ❜② ❛ ❧✐st ♦❢ s♦♠❡

♦❢ ✐ts ✈❛❧✉❡s✳ ❆❞❞ ♠✐ss✐♥❣ ✈❛❧✉❡s ✐♥ s✉❝❤ ❛ ✇❛② t❤❛t t❤❡ ❢✉♥❝t✐♦♥ ✐s ♦♥❡✲t♦✲♦♥❡✳

❊①❡r❝✐s❡ ✶✳✶✹

❚❡❛❝❤❡rs ♦❢t❡♥ s❛② t♦ t❤❡ st✉❞❡♥t✬s ♣❛r❡♥ts✿ ✏■❢ ②♦✉r st✉❞❡♥t ✇♦r❦s ❤❛r❞❡r✱ ❤❡✬❧❧ ✐♠♣r♦✈❡✑✳

▲❡t

❲❤❡♥ ❤❡

✇♦♥✬t ✐♠♣r♦✈❡ ❛♥❞ t❤❡ ♣❛r❡♥ts ❝♦♠❡ ❜❛❝❦ t♦ t❤❡

x −1 0 1 2 3 4 5 y = f (x) −1 0 5 0 ❊①❡r❝✐s❡ ✶✳✷✵

h(x) = sin2 x + sin3 x ❛s g ◦ f ♦❢ t✇♦ ❢✉♥❝t✐♦♥s y = f (x)

t❡❛❝❤❡r✱ ❤❡ ✇✐❧❧ ❛♥s✇❡r✿ ✏❍❡ ❞✐❞♥✬t ✐♠♣r♦✈❡✱ t❤❛t

❘❡♣r❡s❡♥t t❤❡ ❢✉♥❝t✐♦♥

♠❡❛♥s ❤❡ ❞✐❞♥✬t ✇♦r❦ ❤❛r❞❡r✑✳ ❆♥❛❧②③❡✳

t❤❡ ❝♦♠♣♦s✐t✐♦♥

✶✳ ❊①❡r❝✐s❡s✿ ❙❡ts✱ ❧♦❣✐❝✱ ❢✉♥❝t✐♦♥s

❛♥❞

✺✵✼

❊①❡r❝✐s❡ ✶✳✷✽

z = g(y)✳

❙✉♣♣♦s❡ ❛ ❢✉♥❝t✐♦♥ t❤❡ ❧♦❣❛r✐t❤♠ ❜❛s❡ ❊①❡r❝✐s❡ ✶✳✷✶

❋✉♥❝t✐♦♥

f ♣❡r❢♦r♠s t❤❡ ♦♣❡r❛t✐♦♥✿ ✏t❛❦❡ 2 ♦❢ ✑✱ ❛♥❞ ❢✉♥❝t✐♦♥ g ♣❡r❢♦r♠s✿

✏t❛❦❡ t❤❡ sq✉❛r❡ r♦♦t ♦❢ ✑✳

y = f (x)

✐s ❣✐✈❡♥ ❜❡❧♦✇ ❜② ❛ ❧✐st ✐ts ✈❛❧✲

t❤❡ ✐♥✈❡rs❡s ♦❢

f

❛♥❞

✉❡s✳ ❋✐♥❞ ✐ts ✐♥✈❡rs❡ ❛♥❞ r❡♣r❡s❡♥t ✐t ❜② ❛ s✐♠✐❧❛r

t❤❡s❡ ❢♦✉r ❢✉♥❝t✐♦♥s✳

t❛❜❧❡✳

❝♦❞♦♠❛✐♥s✳

x 0 1 2 3 4 y = f (x) 0 1 2 4 3

✭❛✮ ❱❡r❜❛❧❧② ❞❡s❝r✐❜❡

✭❜✮ ❋✐♥❞ t❤❡ ❢♦r♠✉❧❛s ❢♦r

✭❝✮ ●✐✈❡ t❤❡♠ ❞♦♠❛✐♥s ❛♥❞

❊①❡r❝✐s❡ ✶✳✷✾

✶✳ ❘❡♣r❡s❡♥t t❤❡ ❢✉♥❝t✐♦♥

❊①❡r❝✐s❡ ✶✳✷✷

t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ t✇♦

❋✐♥❞ t❤❡ ❢♦r♠✉❧❛s ♦❢ t❤❡ ✐♥✈❡rs❡s ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣

f (x) = (x + 1)3 ❀

❢✉♥❝t✐♦♥s✿ ✭❛✮

g✳

✭❜✮

g(x) = ln(x3 )✳

p x2 − 1 ❛s ❢✉♥❝t✐♦♥s f ❛♥❞ g ✳ h(x) =

y = g(x) = 2x − 1✳

✷✳ Pr♦✈✐❞❡ ❛ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❝♦♠♣♦s✐t✐♦♥

f (g(x))

♦❢

2

f (u) = u + u

❛♥❞

❊①❡r❝✐s❡ ✶✳✷✸

❆r❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢✉♥❝t✐♦♥s ✐♥✈❡rt✐❜❧❡❄ ✭❛✮

y = f (n)

✐s t❤❡ ♥✉♠❜❡r ♦❢ st✉❞❡♥ts ✐♥ ②♦✉r ❝❧❛ss ✇❤♦s❡ ❜✐rt❤✲

y = f (t) ✐s t❤❡ t♦t❛❧ ❛❝❝✉♠✉❧❛t❡❞ r❛✐♥❢❛❧❧ ✐♥ ✐♥❝❤❡s t ♦♥ ❛ ❣✐✈❡♥

❞❛② ✐s ♦♥ t❤❡

nt❤

❞❛② ♦❢ t❤❡ ②❡❛r✳ ✭❜✮

❞❛② ✐♥ ❛ ♣❛rt✐❝✉❧❛r ❧♦❝❛t✐♦♥✳

♦❢

f (u) = sin u

g(x) =

❛♥❞

√

♦❢

f, g ✱

✜♥❞ t❤❡ t❛❜❧❡ ♦❢

f ◦ g✿

f (u) = u2 − 3u + 2

❛♥❞

y = f (g(x))

g(x) = x✳

❊①❡r❝✐s❡ ✶✳✸✷

y = g(x) 0 4 3 0 1

y 0 1 2 3 4

z = f (y) 4 4 0 1 2

❲❤❛t ✐s t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ ❢✉♥❝✲ t✐♦♥

f (x) = 3x2 + 1❄

❍✐♥t✿

❈❤♦♦s❡ ❛♣♣r♦♣r✐❛t❡

❞♦♠❛✐♥s✳

❊①❡r❝✐s❡ ✶✳✸✸

✶✳ ❘❡♣r❡s❡♥t t❤❡ ❢✉♥❝t✐♦♥

❲❤❛t ✐❢ t❤❡ ❧❛st r♦✇s ✇❡r❡ ♠✐ss✐♥❣❄

h(x) =

√

❝♦♠♣♦s✐t✐♦♥ ♦❢ t✇♦ ❢✉♥❝t✐♦♥s✳ ❊①❡r❝✐s❡ ✶✳✷✺

✷✳ ❘❡♣r❡s❡♥t t❤❡ ❢✉♥❝t✐♦♥

♦❢ t✇♦ ❢✉♥❝t✐♦♥s✿

✸✳ ❘❡♣r❡s❡♥t t❤❡ ❢✉♥❝t✐♦♥

p h(x) = 2x3 + x .

p(t) = sin

p t2 − 1

❛s

❊①❡r❝✐s❡ ✶✳✸✹

h(x) = 2 sin3 x + sin x + 5

❛s t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ t✇♦ ❢✉♥❝t✐♦♥s ♦♥❡ ♦❢ ✇❤✐❝❤ ✐s tr✐❣♦♥♦♠❡tr✐❝✳

❊①❡r❝✐s❡ ✶✳✷✼

f ◦ g ❢♦r t❤❡ ❢✉♥❝t✐♦♥s ❣✐✈❡♥ ❜② f (u) = u + u ❛♥❞ g(x) = 3❄ ✭❛✮ ❲❤❛t ✐s t❤❡ ❝♦♠♣♦s✐t✐♦♥ f ◦ g ❢♦r t❤❡ ❢✉♥❝t✐♦♥s ❣✐✈❡♥ ❜② √ f (u) = 2 ❛♥❞ g(x) = x❄ ✭❛✮ ❲❤❛t ✐s t❤❡ ❝♦♠♣♦s✐t✐♦♥

2

❊①❡r❝✐s❡ ✶✳✸✺

h(x) = e ❢✉♥❝t✐♦♥s f ❛♥❞ g ✳

✭❛✮ ❘❡♣r❡s❡♥t t❤❡ ❢✉♥❝t✐♦♥ ❝♦♠♣♦s✐t✐♦♥ ♦❢ t✇♦

p t2 − 1 ❛s t❤❡

t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ ❢♦✉r ❢✉♥❝t✐♦♥s✳

❊①❡r❝✐s❡ ✶✳✷✻

❘❡♣r❡s❡♥t t❤❡ ❢✉♥❝t✐♦♥

k(t) =

x − 1 ❛s t❤❡

❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤r❡❡ ❢✉♥❝t✐♦♥s✳

❘❡♣r❡s❡♥t t❤❡ ❢✉♥❝t✐♦♥ ❜❡❧♦✇ ❛s t❤❡ ❝♦♠♣♦s✐t✐♦♥

f ◦g

y = f (g(x))

x✳

Pr♦✈✐❞❡ ❛ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❝♦♠♣♦s✐t✐♦♥

●✐✈❡♥ t❤❡ t❛❜❧❡s ♦❢ ✈❛❧✉❡s ♦❢

x 0 1 2 3 4

Pr♦✈✐❞❡ ❛ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❝♦♠♣♦s✐t✐♦♥

❊①❡r❝✐s❡ ✶✳✸✶

❊①❡r❝✐s❡ ✶✳✷✹

✈❛❧✉❡s ♦❢

❊①❡r❝✐s❡ ✶✳✸✵

x3 −1

✱ ❛s t❤❡

✭❜✮ Pr♦✈✐❞❡

❢♦r♠✉❧❛s ❢♦r t❤❡ t✇♦ ♣♦ss✐❜❧❡ ❝♦♠♣♦s✐t✐♦♥s ♦❢ t❤❡ t✇♦ ❢✉♥❝t✐♦♥s✿ ✏t❛❦❡ t❤❡ ❧♦❣❛r✐t❤♠ ❜❛s❡ ✏t❛❦❡ t❤❡ sq✉❛r❡ r♦♦t ♦❢ ✑✳

2

♦❢ ✑ ❛♥❞

❋✉♥❝t✐♦♥

y = f (x)

✐s ❣✐✈❡♥ ❜❡❧♦✇ ❜② ❛ ❧✐st ✐ts ✈❛❧✲

✉❡s✳ ❋✐♥❞ ✐ts ✐♥✈❡rs❡ ❛♥❞ r❡♣r❡s❡♥t ✐t ❜② ❛ s✐♠✐❧❛r t❛❜❧❡✳

x 0 1 2 3 4 y = f (x) 1 2 0 4 3

✶✳ ❊①❡r❝✐s❡s✿ ❙❡ts✱ ❧♦❣✐❝✱ ❢✉♥❝t✐♦♥s

❊①❡r❝✐s❡ ✶✳✸✻

●✐✈❡ ❡①❛♠♣❧❡s ♦❢ ❢✉♥❝t✐♦♥s t❤❛t ❛r❡ t❤❡✐r ♦✇♥ ✐♥✲ ✈❡rs❡s✳

✺✵✽

✐♥✈❡rs❡❄

x 0 1 2 3 4 y = f (x) 7 5 3 4 6

❊①❡r❝✐s❡ ✶✳✸✼

P❧♦t t❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ s❤♦✇♥ ❜❡❧♦✇✱ ✐❢ ♣♦ss✐❜❧❡✿

❊①❡r❝✐s❡ ✶✳✹✸

❋✉♥❝t✐♦♥s y = f (x) ❛♥❞ u = g(y) ❛r❡ ❣✐✈❡♥ ❜❡❧♦✇ ❜② t❛❜❧❡s ♦❢ s♦♠❡ ♦❢ t❤❡✐r ✈❛❧✉❡s✳ Pr❡s❡♥t t❤❡ ❝♦♠✲ ♣♦s✐t✐♦♥ u = h(x) ♦❢ t❤❡s❡ ❢✉♥❝t✐♦♥s ❜② ❛ s✐♠✐❧❛r t❛❜❧❡✿ x 0 1 2 3 4 y = f (x) 1 1 2 0 2

y 0 1 2 3 4 u = g(y) 3 1 2 1 0 ❊①❡r❝✐s❡ ✶✳✹✹ ❊①❡r❝✐s❡ ✶✳✸✽

P❧♦t t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ✐♥✈❡rs❡ ♦❢ t❤✐s ❢✉♥❝t✐♦♥✱ ✐❢ ♣♦ss✐❜❧❡✿

❋✉♥❝t✐♦♥ y = f (x) ✐s ❣✐✈❡♥ ❜❡❧♦✇ ❜② ❛ ❧✐st ♦❢ s♦♠❡ ♦❢ ✐ts ✈❛❧✉❡s✳ ❆❞❞ ♠✐ss✐♥❣ ✈❛❧✉❡s ✐♥ s✉❝❤ ❛ ✇❛② t❤❛t t❤❡ ❢✉♥❝t✐♦♥ ✐s ♦♥❡✲t♦ ♦♥❡✳

x −1 0 1 2 3 4 5 y = f (x) −1 4 5 2 ❊①❡r❝✐s❡ ✶✳✹✺

1 ❛♥❞ x−1 t❤❡ ❣r❛♣❤ ♦❢ ✐ts ✐♥✈❡rs❡✳ ■❞❡♥t✐❢② ✐ts ✐♠♣♦rt❛♥t ❢❡❛✲ t✉r❡s✳

P❧♦t t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ f (x) =

❊①❡r❝✐s❡ ✶✳✹✻

❊①❡r❝✐s❡ ✶✳✸✾

❘❡♣r❡s❡♥t t❤✐s ❢✉♥❝t✐♦♥✿ h(x) = tan(2x) ❛s t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ t✇♦ ❢✉♥❝t✐♦♥s ♦❢ ✈❛r✐❛❜❧❡s x ❛♥❞ y ✳ ❊①❡r❝✐s❡ ✶✳✹✵

x3 + 1 , ❛s t❤❡ x3 − 1 ❝♦♠♣♦s✐t✐♦♥ ♦❢ t✇♦ ❢✉♥❝t✐♦♥s ♦❢ ✈❛r✐❛❜❧❡s x ❛♥❞ y ✳ ❘❡♣r❡s❡♥t t❤✐s ❢✉♥❝t✐♦♥✿ h(x) =

❊①❡r❝✐s❡ ✶✳✹✶

❋✉♥❝t✐♦♥ y = f (x) ✐s ❣✐✈❡♥ ❜❡❧♦✇ ❜② ❛ ❧✐st ♦❢ ✐ts ✈❛❧✉❡s✳ ■s t❤❡ ❢✉♥❝t✐♦♥ ♦♥❡✲t♦ ♦♥❡❄ ❲❤❛t ❛❜♦✉t ✐ts ✐♥✈❡rs❡❄

x 0 1 2 3 4 y = f (x) 0 1 2 1 2 ❊①❡r❝✐s❡ ✶✳✹✷

❋✉♥❝t✐♦♥ y = f (x) ✐s ❣✐✈❡♥ ❜❡❧♦✇ ❜② ❛ ❧✐st ♦❢ ✐ts ✈❛❧✉❡s✳ ■s t❤❡ ❢✉♥❝t✐♦♥ ♦♥❡✲t♦ ♦♥❡❄ ❲❤❛t ❛❜♦✉t ✐ts

✭❛✮ ❆❧❣❡❜r❛✐❝❛❧❧②✱ s❤♦✇ t❤❛t t❤❡ ❢✉♥❝t✐♦♥ f (x) = x2 ✐s ♥♦t ♦♥❡✲t♦✲♦♥❡✳ ✭❜✮ ●r❛♣❤✐❝❛❧❧②✱ s❤♦✇ t❤❛t t❤❡ ❢✉♥❝t✐♦♥ g(x) = 2x+1 ✐s ♦♥❡✲t♦✲♦♥❡✳ ✭❝✮ ❋✐♥❞ t❤❡ ✐♥✈❡rs❡ ♦❢ g ✳ ❊①❡r❝✐s❡ ✶✳✹✼

❉❡s❝r✐❜❡ ✕ ❜♦t❤ ❣❡♦♠❡tr✐❝❛❧❧② ❛♥❞ ❛❧❣❡❜r❛✐❝❛❧❧② ✕ t✇♦ ❞✐✛❡r❡♥t tr❛♥s❢♦r♠❛t✐♦♥s t❤❛t ♠❛❦❡ ❛ 1 × 1 sq✉❛r❡ ✐♥t♦ ❛ 2 × 3 r❡❝t❛♥❣❧❡✳

✷✳ ❊①❡r❝✐s❡s✿ ❇❛❝❦❣r♦✉♥❞

✺✵✾

✷✳ ❊①❡r❝✐s❡s✿ ❇❛❝❦❣r♦✉♥❞

❊①❡r❝✐s❡ ✷✳✶

❊①❡r❝✐s❡ ✷✳✶✵

❋✐♥❞ t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ❧✐♥❡ ♣❛ss✐♥❣ t❤r♦✉❣❤ t❤❡ ♣♦✐♥ts (−1, 1) ❛♥❞ (−1, 5)✳

❚❤❡ t❛①✐ ❝❤❛r❣❡s $1.75 ❢♦r t❤❡ ✜rst q✉❛rt❡r ♦❢ ❛ ♠✐❧❡ ❛♥❞ $0.35 ❢♦r ❡❛❝❤ ❛❞❞✐t✐♦♥❛❧ ✜❢t❤ ♦❢ ❛ ♠✐❧❡✳ ❋✐♥❞ ❛ ❧✐♥❡❛r ❢✉♥❝t✐♦♥ ✇❤✐❝❤ ♠♦❞❡❧s t❤❡ t❛①✐ ❢❛r❡ f ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ♥✉♠❜❡r ♦❢ ♠✐❧❡s ❞r✐✈❡♥✱ x✳

❊①❡r❝✐s❡ ✷✳✷

❋♦r t❤❡ ♣♦✐♥ts P = (0, 1)✱ Q = (1, 2)✱ ❛♥❞ R = (−1, 2)✱ ❞❡t❡r♠✐♥❡ t❤❡ ♣♦✐♥ts t❤❛t ❛r❡ s②♠♠❡tr✐❝ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❛①✐s ❛♥❞ t❤❡ ♦r✐❣✐♥✳ ❊①❡r❝✐s❡ ✷✳✸

❈♦♥s✐❞❡r tr✐❛♥❣❧❡ ABC ✐♥ t❤❡ ♣❧❛♥❡ ✇❤❡r❡ A = (3, 2)✱ B = (3, −3)✱ C = (−2, −2)✳ ❋✐♥❞ t❤❡ ❧❡♥❣t❤s ♦❢ t❤❡ s✐❞❡s ♦❢ t❤❡ tr✐❛♥❣❧❡✳

❊①❡r❝✐s❡ ✷✳✶✶

❋✐♥❞ t❤❡ ✈❛❧✉❡ ♦❢ k s♦ t❤❛t t❤❡ ❧✐♥❡ ❝♦♥t❛✐♥✐♥❣ t❤❡ ♣♦✐♥ts (−6, 0) ❛♥❞ (k, −5) ✐s ♣❛r❛❧❧❡❧ t♦ t❤❡ ❧✐♥❡ ❝♦♥t❛✐♥✐♥❣ t❤❡ ♣♦✐♥ts (4, 3) ❛♥❞ (1, 7)✳

❊①❡r❝✐s❡ ✷✳✶✷

■s t❤✐s ❛ ♣❛r❛❜♦❧❛❄ ❊①❡r❝✐s❡ ✷✳✹

❋✐♥❞ ❛❧❧ x s✉❝❤ t❤❛t t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ ♣♦✐♥ts (3, −8) ❛♥❞ (x, −6) ✐s 5✳ ❊①❡r❝✐s❡ ✷✳✺

❋✐♥❞ t❤❡ ♣❡r✐♠❡t❡r ♦❢ t❤❡ tr✐❛♥❣❧❡ ✇✐t❤ t❤❡ ✈❡rt✐❝❡s ❛t (3, −1)✱ (3, 6)✱ ❛♥❞ (−6, −5)✳ ❊①❡r❝✐s❡ ✷✳✻

❋✐♥❞ t❤❡ ♣♦✐♥t ♦♥ t❤❡ x✲❛①✐s t❤❛t ✐s ❡q✉✐❞✐st❛♥t ❢r♦♠ t❤❡ ♣♦✐♥ts (−1, 5) ❛♥❞ (6, 4)✳

❊①❡r❝✐s❡ ✷✳✶✸

❋✐♥❞ t❤❡ ❡q✉❛t✐♦♥ s❛t✐s✜❡❞ ❜② ❛❧❧ ♣♦✐♥ts t❤❛t ❧✐❡ 2 ✉♥✐ts ❛✇❛② ❢r♦♠ t❤❡ ♣♦✐♥t (−1, −2) ❛♥❞ ❜② ♥♦ ♦t❤❡r ♣♦✐♥ts✳

❊①❡r❝✐s❡ ✷✳✶✹ ❊①❡r❝✐s❡ ✷✳✼

❋✐♥❞ t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ ♣♦✐♥ts ♦❢ ✐♥t❡rs❡❝✲ t✐♦♥ ♦❢ t❤❡ ❝✐r❝❧❡ (x − 1)2 + (y − 2)2 = 6 ✇✐t❤ t❤❡ ❛①❡s✳ ❊①❡r❝✐s❡ ✷✳✽

❋♦r t❤❡ ♣♦❧②♥♦♠✐❛❧s ❣r❛♣❤❡❞ ❜❡❧♦✇✱ ✜♥❞ t❤❡ ❢♦❧✲ ❧♦✇✐♥❣✿ s♠❛❧❧❡st ♣♦ss✐❜❧❡ ❞❡❣r❡❡ s✐❣♥ ♦❢ t❤❡ ❧❡❛❞✐♥❣ ❝♦❡✣❝✐❡♥t ❞❡❣r❡❡ ✐s ♦❞❞✴❡✈❡♥

1 2 3

❙♦❧✈❡ t❤❡ s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s✿ (

x − y = −1, 2x + y = 0.

❊①❡r❝✐s❡ ✷✳✾

❆ ♠♦✈✐❡ t❤❡❛t❡r ❝❤❛r❣❡s $10 ❢♦r ❛❞✉❧ts ❛♥❞ $6 ❢♦r ❝❤✐❧❞r❡♥✳ ❖♥ ❛ ♣❛rt✐❝✉❧❛r ❞❛② ✇❤❡♥ 320 ♣❡♦♣❧❡ ♣❛✐❞ ❛♥ ❛❞♠✐ss✐♦♥✱ t❤❡ t♦t❛❧ r❡❝❡✐♣ts ✇❡r❡ $3120✳ ❍♦✇ ♠❛♥② ✇❡r❡ ❛❞✉❧ts ❛♥❞ ❤♦✇ ♠❛♥② ✇❡r❡ ❝❤✐❧❞r❡♥❄

❊①❡r❝✐s❡ ✷✳✶✺

❋✐♥❞ ❛ ♣♦ss✐❜❧❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❢✉♥❝t✐♦♥ ♣❧♦tt❡❞ ❜❡✲ ❧♦✇✿

✷✳ ❊①❡r❝✐s❡s✿ ❇❛❝❦❣r♦✉♥❞

✺✶✵

❊①❡r❝✐s❡ ✷✳✷✶

❋✐♥❞ t❤❡ ✐♠♣❧✐❡❞ ❞♦♠❛✐♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥✿

x−1 ln(x2 + 1) sin x . x+1

❊①❡r❝✐s❡ ✷✳✷✷

❋✐♥❞ t❤❡ ✐♠♣❧✐❡❞ ❞♦♠❛✐♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥✿

(x − 1)(x2 + 1)2x .

❊①❡r❝✐s❡ ✷✳✶✻

❆ ❢❛❝t♦r② ✐s t♦ ❜❡ ❜✉✐❧t ♦♥ ❛ ❧♦t ♠❡❛s✉r✐♥❣ ❜②

320

❢t✳

240

❢t

❆ ❜✉✐❧❞✐♥❣ ❝♦❞❡ r❡q✉✐r❡s t❤❛t ❛ ❧❛✇♥

❊①❡r❝✐s❡ ✷✳✷✸

♦❢ ✉♥✐❢♦r♠ ✇✐❞t❤ ❛♥❞ ❡q✉❛❧ ✐♥ ❛r❡❛ t♦ t❤❡ ❢❛❝t♦r②

❋✐♥✐s❤ t❤❡ s❡♥t❡♥❝❡✿ ✏■❢ ❛ ❢✉♥❝t✐♦♥ ❢❛✐❧s t❤❡ ❤♦r✐③♦♥✲

♠✉st s✉rr♦✉♥❞ t❤❡ ❢❛❝t♦r②✳ ❲❤❛t ♠✉st t❤❡ ✇✐❞t❤

t❛❧ ❧✐♥❡ t❡st✱ t❤❡♥✳✳✳✑

♦❢ t❤❡ ❧❛✇♥ ❜❡❄ ❊①❡r❝✐s❡ ✷✳✷✹

❊①❡r❝✐s❡ ✷✳✶✼

❆ ❢❛❝t♦r② ♦❝❝✉♣✐❡s ❛ ❧♦t ♠❡❛s✉r✐♥❣ ❢t✳

240

❢t ❜②

320

❊①♣❧❛✐♥ t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡s❡ t✇♦ ❢✉♥❝t✐♦♥s✿

r

❆ ❜✉✐❧❞✐♥❣ ❝♦❞❡ r❡q✉✐r❡s t❤❛t ❛ ❧❛✇♥ ♦❢ ✉♥✐✲

❢♦r♠ ✇✐❞t❤ ❛♥❞ ❡q✉❛❧ ✐♥ ❛r❡❛ t♦ t❤❡ ❢❛❝t♦r② ♠✉st s✉rr♦✉♥❞ t❤❡ ❢❛❝t♦r②✳ ❲❤❛t ♠✉st t❤❡ ✇✐❞t❤ ♦❢ t❤❡

x−1 x+1

❛♥❞

√ x−1 √ . x+1

❧❛✇♥ ❜❡❄ ❊①❡r❝✐s❡ ✷✳✷✺ ❊①❡r❝✐s❡ ✷✳✶✽

(x2 + 1)(x + 1)(x − 1) = 0✳ 2 ✐♥❡q✉❛❧✐t② (x + 1)(x + 1)(x − 1) > 0✳

✭❛✮ ❙♦❧✈❡ t❤❡ ❡q✉❛t✐♦♥ ✭❜✮ ❙♦❧✈❡ t❤❡

❊①❡r❝✐s❡ ✷✳✶✾

y = f (x) ✐s ❣✐✈❡♥ ❜❡❧♦✇✳ ✭❛✮ ❋✐♥❞ s✉❝❤ ❛ y t❤❛t t❤❡ ♣♦✐♥t (2, y) ❜❡❧♦♥❣s t♦ t❤❡ ❣r❛♣❤✳ ✭❜✮ ❋✐♥❞ s✉❝❤ ❛♥ x t❤❛t t❤❡ ♣♦✐♥t (x, 3) ❜❡❧♦♥❣s t♦ t❤❡ ❣r❛♣❤✳ ✭❝✮ ❋✐♥❞ s✉❝❤ ❛♥ x t❤❛t t❤❡ ♣♦✐♥t (x, x) ❜❡❧♦♥❣s t♦ t❤❡ ❣r❛♣❤✳ ❙❤♦✇ ②♦✉r ❞r❛✇✲

❈❧❛ss✐❢② t❤❡s❡ ❢✉♥❝t✐♦♥s✿ ♦❞❞

❢✉♥❝t✐♦♥

❡✈❡♥

♦♥t♦

♦♥❡✲t♦✲♦♥❡

f (x) = 2x − 1 g(x) = −x + 2 h(x) = 3

❚❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥

✐♥❣✳

❊①❡r❝✐s❡ ✷✳✷✻

❚❤❡ ❣r❛♣❤ ♦❢ y = f (x) y = −f (x + 5) − 6✳

✐s ♣❧♦tt❡❞ ❜❡❧♦✇✳

❙❦❡t❝❤

❊①❡r❝✐s❡ ✷✳✷✼

■s t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ t✇♦ ❢✉♥❝t✐♦♥s t❤❛t ❛r❡ ♦❞❞✲ ✴❡✈❡♥ ♦❞❞✴❡✈❡♥❄ ❊①❡r❝✐s❡ ✷✳✷✵

▼❛❦❡ ❛ ✢♦✇❝❤❛rt ❛♥❞ t❤❡♥ ♣r♦✈✐❞❡ ❛ ❢♦r♠✉❧❛ ❢♦r

y = f (x) t❤❛t r❡♣r❡s❡♥ts ❛ ♣❛r❦✐♥❣ ❢❡❡ ❢♦r ❛ st❛② ♦❢ x ❤♦✉rs✳ ■t ✐s ❝♦♠♣✉t❡❞ ❛s ❢♦❧❧♦✇s✿ ❢r❡❡ ❢♦r t❤❡ ✜rst ❤♦✉r ❛♥❞ $1 ♣❡r ❤♦✉r ❜❡②♦♥❞✳ t❤❡ ❢✉♥❝t✐♦♥

❊①❡r❝✐s❡ ✷✳✷✽

❋✐♥❞ t❤❡ ❢♦r♠✉❧❛s ♦❢ t❤❡ ✐♥✈❡rs❡s ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢✉♥❝t✐♦♥s✿ ✭❛✮

f (x) = (x + 1)3 ❀

✭❜✮

g(x) = ln(x3 )✳

✷✳ ❊①❡r❝✐s❡s✿ ❇❛❝❦❣r♦✉♥❞

✺✶✶

❊①❡r❝✐s❡ ✷✳✷✾

❊①❡r❝✐s❡ ✷✳✸✸

❖♥❡ ♦❢ t❤❡ ❣r❛♣❤s ❜❡❧♦✇ ✐s t❤❛t ♦❢

y = arctan x✳

❲❤❛t ❛r❡ t❤❡ ♦t❤❡rs❄

❚❤❡ ❣r❛♣❤s ❜❡❧♦✇ ❛r❡ ♣❛r❛❜♦❧❛s✳

❖♥❡ ✐s

y = x2 ✳

❲❤❛t ✐s t❤❡ ♦t❤❡r❄

❊①❡r❝✐s❡ ✷✳✸✹ ❊①❡r❝✐s❡ ✷✳✸✵ ❚❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❧♦✇✳

❙❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢

❚❤❡

y = f (x) ✐s y = 2f (3x)

❣r❛♣❤

❜❡❧♦✇

❣✐✈❡♥ ❜❡✲

f (x) = A sin x + B

❛♥❞ t❤❡♥

♥✉♠❜❡rs✳

✐s

t❤❡

❣r❛♣❤

A

❢♦r s♦♠❡

♦❢

❛♥❞

t❤❡

B✳

❢✉♥❝t✐♦♥

❋✐♥❞ t❤❡s❡

y = f (−x) − 1✳

❊①❡r❝✐s❡ ✷✳✸✺

❊①❡r❝✐s❡ ✷✳✸✶ ❚❤❡ ❣r❛♣❤ ❞r❛✇♥ ✇✐t❤ ❛ s♦❧✐❞ ❧✐♥❡ ✐s

y = x3 ✳

f ✐s ❣✐✈❡♥ ❜❡❧♦✇✳ ❙❦❡t❝❤ y = 2f (x + 2) + 2✳ ❊①♣❧❛✐♥ ❤♦✇ ②♦✉

❚❤❡ ❣r❛♣❤ ♦❢ ❢✉♥❝t✐♦♥

t❤❡

❣r❛♣❤ ♦❢

❣❡t

✐t✳ ❲❤❛t

❛r❡ t❤❡ ♦t❤❡r t✇♦❄

❊①❡r❝✐s❡ ✷✳✸✻ ❇② tr❛♥s❢♦r♠✐♥❣ t❤❡ ❣r❛♣❤ ♦❢ ♦❢ t❤❡ ❢✉♥❝t✐♦♥

❊①❡r❝✐s❡ ✷✳✸✷ ❚❤❡ ❣r❛♣❤ ♦❢ ♦♥❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥s ❜❡❧♦✇ ✐s ❲❤❛t ✐s t❤❡ ♦t❤❡r❄

y = ex ✱ ♣❧♦t t❤❡ ❣r❛♣❤

f (x) = 2ex−3 ✳

■❞❡♥t✐❢② t❤❡ ❞♦♠❛✐♥✱

t❤❡ r❛♥❣❡✱ ❛♥❞ t❤❡ ❛s②♠♣t♦t❡s✳

y = ex ✳ ❊①❡r❝✐s❡ ✷✳✸✼ ❍❛❧❢ ♦❢ t❤❡ ❣r❛♣❤ ♦❢ ❛♥ ❡✈❡♥ ❢✉♥❝t✐♦♥ ✐s s❤♦✇♥ ❜❡✲ ❧♦✇❀ ♣r♦✈✐❞❡ t❤❡ ♦t❤❡r ❤❛❧❢✿

✷✳ ❊①❡r❝✐s❡s✿ ❇❛❝❦❣r♦✉♥❞

✺✶✷

❊①❡r❝✐s❡ ✷✳✸✽

❊①❡r❝✐s❡ ✷✳✹✹

❍❛❧❢ ♦❢ t❤❡ ❣r❛♣❤ ♦❢ ❛♥ ♦❞❞ ❢✉♥❝t✐♦♥ ✐s s❤♦✇♥ ❛❜♦✈❡❀

●✐✈❡

♣r♦✈✐❞❡ t❤❡ ♦t❤❡r ❤❛❧❢✳

❛r❡♥✬t ♣♦❧②♥♦♠✐❛❧s✳

❊①❡r❝✐s❡ ✷✳✸✾

❊①❡r❝✐s❡ ✷✳✹✺

y = f (x) ✐s ♦❢ y = 2f (x)

❚❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥

❣✐✈❡♥ ❜❡✲

❧♦✇✳

❛♥❞ t❤❡♥

❙❦❡t❝❤ t❤❡ ❣r❛♣❤

y = 2f (x) − 1✳

❡①❛♠♣❧❡s

♦❢

♦❞❞

❛♥❞

❡✈❡♥

❢✉♥❝t✐♦♥s

t❤❛t

■s t❤❡ ✐♥✈❡rs❡ ♦❢ ❛♥ ♦❞❞✴❡✈❡♥ ❢✉♥❝t✐♦♥ ♦❞❞✴❡✈❡♥❄

❊①❡r❝✐s❡ ✷✳✹✻

y = sin x✱ f (x) = 2 sin(x − 3)✳

❇② tr❛♥s❢♦r♠✐♥❣ t❤❡ ❣r❛♣❤ ♦❢

♣❧♦t t❤❡

❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥

■❞❡♥t✐❢②

✐ts r❛♥❣❡✳

❊①❡r❝✐s❡ ✷✳✹✼ ●✐✈❡ ❡①❛♠♣❧❡s ♦❢ ❛♥ ❡✈❡♥ ❢✉♥❝t✐♦♥✱ ❛♥ ♦❞❞ ❢✉♥❝t✐♦♥✱ ❛♥❞ ❛ ❢✉♥❝t✐♦♥ t❤❛t✬s ♥❡✐t❤❡r✳ Pr♦✈✐❞❡ ❢♦r♠✉❧❛s✳

❊①❡r❝✐s❡ ✷✳✹✵ ❚❤❡ ❣r❛♣❤ ❛❜♦✈❡ ✐s ❛ ♣❛r❛❜♦❧❛✳ ❋✐♥❞ ✐ts ❢♦r♠✉❧❛✳

❊①❡r❝✐s❡ ✷✳✹✶ ❚❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❧♦✇✳

❙❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢

1 y = f (x − 1)✳ 2

y = f (x) ✐s 1 y = f (x) 2

❣✐✈❡♥ ❜❡✲ ❛♥❞ t❤❡♥

❊①❡r❝✐s❡ ✷✳✹✷ ❲❤❛t ✐s t❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡s❡ t✇♦ ❢✉♥❝t✐♦♥s❄

❊①❡r❝✐s❡ ✷✳✹✸ P❧♦t t❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥ t❤❛t ✐s ❜♦t❤ ♦❞❞ ❛♥❞ ❡✈❡♥✳

✸✳ ❊①❡r❝✐s❡s✿ ❙❡q✉❡♥❝❡s

✺✶✸

✸✳ ❊①❡r❝✐s❡s✿ ❙❡q✉❡♥❝❡s

❊①❡r❝✐s❡ ✸✳✶

❊①❡r❝✐s❡ ✸✳✾

Pr❡s❡♥t t❤❡ ✜rst 5 t❡r♠s ♦❢ t❤❡ s❡q✉❡♥❝❡✿

■♥ t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ ❡❛❝❤ ②❡❛r✱ ❛ ♣❡rs♦♥ ♣✉ts $5000 ✐♥ ❛ ❜❛♥❦ t❤❛t ♣❛②s 3% ❝♦♠♣♦✉♥❞❡❞ ❛♥♥✉❛❧❧②✳ ❍♦✇ ♠✉❝❤ ❞♦❡s ❤❡ ❤❛✈❡ ❛❢t❡r 15 ②❡❛rs❄

a1 = 1,

an+1 = −(an + 1) .

❊①❡r❝✐s❡ ✸✳✶✵

❊①❡r❝✐s❡ ✸✳✷

❘❡♣r❡s❡♥t ✐♥ s✐❣♠❛ ♥♦t❛t✐♦♥✿ −1 − 2 − 3 − 4 − 5 − ... − 10 . ❊①❡r❝✐s❡ ✸✳✸

❋✐♥❞ t❤❡ ❢♦❧❧♦✇✐♥❣ s✉♠✿ −1 − 2 − 3 − 4 − 5 − ... − 10 .

❆♥ ♦❜❥❡❝t ❢❛❧❧✐♥❣ ❢r♦♠ r❡st ✐♥ ❛ ✈❛❝✉✉♠ ❢❛❧❧s ❛♣✲ ♣r♦①✐♠❛t❡❧② 16 ❢❡❡t t❤❡ ✜rst s❡❝♦♥❞✱ 48 ❢❡❡t t❤❡ s❡❝♦♥❞ s❡❝♦♥❞✱ 80 ❢❡❡t t❤❡ t❤✐r❞ s❡❝♦♥❞✱ 112 ❢❡❡t t❤❡ ❢♦✉rt❤ s❡❝♦♥❞✱ ❛♥❞ s♦ ♦♥✳ ❍♦✇ ❢❛r ✇✐❧❧ ✐t ❢❛❧❧ ✐♥ 11 s❡❝♦♥❞s❄ ❊①❡r❝✐s❡ ✸✳✶✶

❊✈❛❧✉❛t❡ t❤❡ ❧✐♠✐t ✐❢ ✐t ❡①✐sts✿ (−1)n . n→∞ n lim

❊①❡r❝✐s❡ ✸✳✹

❋✐♥❞ t❤❡ s❡q✉❡♥❝❡ ♦❢ s✉♠s ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡✲ q✉❡♥❝❡✿ −1, 2, −4, 8, −5, .. . ❊①❡r❝✐s❡ ✸✳✺

❈♦♠♣✉t❡

4 X

n2 ✳

n=1

❊①❡r❝✐s❡ ✸✳✶✷

❊✈❛❧✉❛t❡ t❤❡ ❧✐♠✐t✿ lim ln n→∞



 n ✳ n+1

❊①❡r❝✐s❡ ✸✳✶✸

●✐✈❡ ❛♥ ❡①❛♠♣❧❡ ♦❢ ❛ s❡q✉❡♥❝❡ ❢♦r ❡❛❝❤ ♦❢ t❤❡ ❢♦❧✲ ❧♦✇✐♥❣✿ ✭❛✮ an → 0 ❛s n → ∞✱ ✭❜✮ an → 1 ❛s n → ∞✱ ✭❝✮ an → +∞ ❛s n → ∞✱ ✭❞✮ an ❞✐✈❡r❣❡s ❜✉t ♥♦t t♦ ✐♥✜♥✐t②✳

❊①❡r❝✐s❡ ✸✳✻

n ✐s ❛♥ ✐♥❝r❡❛s✐♥❣ s❡q✉❡♥❝❡✳ ❲❤❛t n+1 n+1 ❦✐♥❞ ♦❢ s❡q✉❡♥❝❡ ✐s ❄ ●✐✈❡ ❡①❛♠♣❧❡s ♦❢ ✐♥✲ n

❊①❡r❝✐s❡ ✸✳✶✹

❊①❡r❝✐s❡ ✸✳✼

❊①❡r❝✐s❡ ✸✳✶✺

❋✐♥❞ t❤❡ ♥❡①t ✐t❡♠ ✐♥ ❡❛❝❤ ❧✐st✿

❊①♣❧❛✐♥ ✇❤② t❤❡ ❧✐♠✐t lim sin n ❞♦❡s ♥♦t ❡①✐st✳

❙❤♦✇ t❤❛t

❝r❡❛s✐♥❣ ❛♥❞ ❞❡❝r❡❛s✐♥❣ s❡q✉❡♥❝❡s✳

✶✳ 7, 14, 28, 56, 112, ... ✷✳ 15, 27, 39, 51, 63, ... ✸✳ 197, 181, 165, 149, 133, ... ❊①❡r❝✐s❡ ✸✳✽

❆ ♣✐❧❡ ♦❢ ❧♦❣s ❤❛s 50 ❧♦❣s ✐♥ t❤❡ ❜♦tt♦♠ ❧❛②❡r✱ 49 ❧♦❣s ✐♥ t❤❡ ♥❡①t ❧❛②❡r✱ 48 ❧♦❣s ✐♥ t❤❡ ♥❡①t ❧❛②❡r✱ ❛♥❞ s♦ ♦♥✱ ✉♥t✐❧ t❤❡ t♦♣ ❧❛②❡r ❤❛s 1 ❧♦❣✳ ❍♦✇ ♠❛♥② ❧♦❣s ❛r❡ ✐♥ t❤❡ ♣✐❧❡❄

❲r✐t❡ ❛ ❢♦r♠✉❧❛ ❢♦r t❤❡ nt❤ t❡r♠ ♦❢ t❤❡ s❡q✉❡♥❝❡✿ 7 15 31 1 3 , − , ... − , , − , 2 4 8 16 32

n→∞

❊①❡r❝✐s❡ ✸✳✶✻

✭❛✮ ❙t❛t❡ t❤❡ ❙q✉❡❡③❡ ❚❤❡♦r❡♠✳ ✭❜✮ ●✐✈❡ ❛♥ ❡①❛♠✲ ♣❧❡ ♦❢ ✐ts ❛♣♣❧✐❝❛t✐♦♥✳ ❊①❡r❝✐s❡ ✸✳✶✼

Pr❡s❡♥t t❤❡ ✜rst 5 t❡r♠s ♦❢ t❤❡ s❡q✉❡♥❝❡ ❛♥❞ t❡❧❧ ✐❢ ✐t ✐s ❝♦♥✈❡r❣❡♥t✿ a1 = 1,

an+1 = (an − 1)2 .

✹✳ ❊①❡r❝✐s❡s✿ ❘❛t❡s ♦❢ ❝❤❛♥❣❡

✺✶✹

✹✳ ❊①❡r❝✐s❡s✿ ❘❛t❡s ♦❢ ❝❤❛♥❣❡

❊①❡r❝✐s❡ ✹✳✶

❚❤r❡❡ str❛✐❣❤t ❧✐♥❡s ❛r❡ s❤♦✇♥ ❜❡❧♦✇✳ ❲❤❛t ✐s s♦ s♣❡❝✐❛❧ ❛❜♦✉t t❤❡♠❄ ❋✐♥❞ t❤❡✐r s❧♦♣❡s✳

❣✐✈❡♥ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ❞❛t❛✿ x y = f (x) −1 0 0 2 1 3 −1 2 3 −2 4 0

❊①❡r❝✐s❡ ✹✳✻

❋✐♥❞ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ❢♦r t❤❡ ❢✉♥❝t✐♦♥ ❣✐✈❡♥ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ❞❛t❛✿ ❊①❡r❝✐s❡ ✹✳✷

y = f (x) x −1 2 2 1 3 3 5 3 7 −2 5 9

❚❤r❡❡ str❛✐❣❤t ❧✐♥❡s ❛r❡ s❤♦✇♥ ❜❡❧♦✇✳ ❋✐♥❞ t❤❡✐r ❡q✉❛t✐♦♥s✳

❊①❡r❝✐s❡ ✹✳✼

P❧♦t t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❛✈❡r❛❣❡ ✈❡❧♦❝✐t② ❢♦r t❤❡ ❢♦❧✲ ❧♦✇✐♥❣ ♣♦s✐t✐♦♥ ❢✉♥❝t✐♦♥✿ ❊①❡r❝✐s❡ ✹✳✸

✭❛✮ ❙✉♣♣♦s❡ ❞✉r✐♥❣ t❤❡ ✜rst 2 s❡❝♦♥❞s ♦❢ ✐ts ✢✐❣❤t ❛♥ ♦❜❥❡❝t ♣r♦❣r❡ss❡❞ ❢r♦♠ ♣♦✐♥t (0, 0) t♦ (1, 0) t♦ (2, 0)✳ ❲❤❛t ✇❛s ✐ts ❛✈❡r❛❣❡ ✈❡❧♦❝✐t② ❛♥❞ ❛✈❡r❛❣❡ ❛❝❝❡❧❡r❛t✐♦♥❄ ✭❜✮ ❲❤❛t ✐❢ t❤❡ ❧❛st ♣♦✐♥t ✐s (1, 1) ✐♥st❡❛❞❄ ❊①❡r❝✐s❡ ✹✳✹

❙✉♣♣♦s❡ t ✐s t✐♠❡ ❛♥❞ x ✐s t❤❡ ♣r✐❝❡ ♦❢ ❜r❡❛❞✳ ❲❤❛t ❝❛♥ ②♦✉ s❛② ❛❜♦✉t ✐ts ❞②♥❛♠✐❝s❄ ❇❡ ❛s s♣❡❝✐✜❝ ❛s ♣♦ss✐❜❧❡✳

❊①❡r❝✐s❡ ✹✳✽

❲❤❛t ❛r❡ t❤❡ s❡❝❛♥t ❧✐♥❡s ♦❢ t❤❡ ❛❜s♦❧✉t❡ ✈❛❧✉❡ ❢✉♥❝t✐♦♥ f (x) = |x|❄ ❊①❡r❝✐s❡ ✹✳✾

✭❛✮ ❙t❛t❡ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❛✈❡r❛❣❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ❛t ♣♦✐♥t a✳ ✭❜✮ ❋✐♥❞ ✐t ❢♦r t❤❡ ❢✉♥❝t✐♦♥ f (x) = x2 + 3 ❛t a = 1 ❛♥❞ h = .5✳ ❊①❡r❝✐s❡ ✹✳✶✵ ❊①❡r❝✐s❡ ✹✳✺

❋✐♥❞ t❤❡ ❛✈❡r❛❣❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ❢♦r t❤❡ ❢✉♥❝t✐♦♥

❚❤❡ ♣♦s✐t✐♦♥ ❛♥❞ t❤❡ ✈❡❧♦❝✐t② ❛r❡ ♣❧♦tt❡❞ ❜❡❧♦✇✳ P❧♦t t❤❡ ❛❝❝❡❧❡r❛t✐♦♥✳

✹✳ ❊①❡r❝✐s❡s✿ ❘❛t❡s ♦❢ ❝❤❛♥❣❡

✺✶✺

t✐♠❡ x ✭✐♥ s❡❝♦♥❞s✮✳ ✭❛✮ ❲❤❛t ❞♦❡s t❤❡ s❧♦♣❡ ♦❢ t❤❡ ❝✉r✈❡ r❡♣r❡s❡♥t❄ ✭❜✮ ❉❡s❝r✐❜❡ ✇❤❛t ❤❛♣♣❡♥❡❞ t♦ t❤❡ ♣❧❛♥❡✳

❊①❡r❝✐s❡ ✹✳✶✶

❚❤❡ ♣✐❝t✉r❡❞ ❣r❛♣❤ r❡♣r❡s❡♥ts t❤❡ ♥✉♠❜❡r ♦❢ ♠♦sq✉✐t♦❡s ✐♥ ❛ ❝❡rt❛✐♥ ❛r❡❛ ♦✈❡r t❤❡ ♣❡r✐♦❞ ♦❢ 150 ❞❛②s✳ ❲❤❛t ❤❛♣♣❡♥❡❞ t♦ ✭❛✮ t❤❡ ♠♦sq✉✐t♦ ♣♦♣✉❧❛✲ t✐♦♥ ❛♥❞ ✭❜✮ ✐ts r❛t❡ ♦❢ ❣r♦✇t❤❄ ❊①♣❧❛✐♥✳

❊①❡r❝✐s❡ ✹✳✶✺

❋r♦♠ t❤❡ ❞❡✜♥✐t✐♦♥✱ ❝♦♠♣✉t❡ t❤❡ ❛✈❡r❛❣❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ❢♦r t❤❡ ❢✉♥❝t✐♦♥ f (x) = x2 + 1 ❛t a = 2 ✇✐t❤ h = .2 ❛♥❞ h = .1✳ ❊①♣❧❛✐♥ t❤❡ ❞✐✛❡r❡♥❝❡✳ ❊①❡r❝✐s❡ ✹✳✶✻

✭❛✮ ❈♦♠♣✉t❡ t❤❡ ❛✈❡r❛❣❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ❢♦r t❤❡ ❢✉♥❝t✐♦♥ f (x) = 3x2 − x ❛t a = 1 ❛♥❞ h = .5✳ ✭❜✮ ❋✐♥❞ t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ s❡❝❛♥t t♦ t❤❡ ❣r❛♣❤ ♦❢ y = f (x) ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤✐s ❛✈❡r❛❣❡ r❛t❡ ♦❢ ❝❤❛♥❣❡✳ ❊①❡r❝✐s❡ ✹✳✶✷

❊❛❝❤ ♦❢ t❤❡s❡ str❛✐❣❤t ❧✐♥❡s ❛r❡ ❞r♦✇♥ t❤r♦✉❣❤ t✇♦ ♣♦✐♥t ♦❢ t❤❡ ❣r❛♣❤✳ ❲❤❛t ❞♦ t❤❡② t❡❧❧ ✉s ❛❜♦✉t t❤❡ ❢✉♥❝t✐♦♥❄

❊①❡r❝✐s❡ ✹✳✶✼

❚❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥ f (x) ✐s ❣✐✈❡♥ ❜❡❧♦✇✳ ❊st✐✲ ∆f ❢♦r ♠❛t❡ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ∆x x = 0, 4, ❛♥❞ 6 ❛♥❞ ∆x = .5✳

❊①❡r❝✐s❡ ✹✳✶✽ ❊①❡r❝✐s❡ ✹✳✶✸

✭❛✮ ❙t❛t❡ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❛✈❡r❛❣❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢ ❢✉♥❝t✐♦♥ f ❛t ♣♦✐♥t x = a✳ ✭❜✮ ❙❦❡t❝❤ ❛♥ ✐❧❧✉s✲ tr❛t✐♦♥ ♦❢ t❤✐s ❞❡✜♥✐t✐♦♥ ❢♦r f (x) = x2 ✳ ❊①❡r❝✐s❡ ✹✳✶✹

❚❤❡ ♣✐❝t✉r❡❞ ❣r❛♣❤ r❡♣r❡s❡♥ts t❤❡ ❛❧t✐t✉❞❡ ✭✐♥ t❤♦✉s❛♥❞s ♦❢ ❢❡❡t✮ ♦❢ ❛ ♣❧❛♥❡ ❛❜♦✈❡ t❤❡ ❣r♦✉♥❞ ❛t

❚❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥ f (x) ✐s ❣✐✈❡♥ ❜❡❧♦✇✳ ❊s✲ t✐♠❛t❡ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❢♦r x = 2, 4, 9 ❛♥❞ ∆x = 1✳

✹✳ ❊①❡r❝✐s❡s✿ ❘❛t❡s ♦❢ ❝❤❛♥❣❡

✺✶✻

❊①❡r❝✐s❡ ✹✳✶✾

❚❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥

f

✐s ❣✐✈❡♥ ❜❡❧♦✇✳ ❊st✐♠❛t❡

t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❛♥❞

∆x = 2, 1, .5✳

∆f ∆x

❢♦r

x=1

❊①❡r❝✐s❡ ✹✳✷✵

❊①♣❧❛✐♥ t❤✐s ♣✐❝t✉r❡✿

❊①❡r❝✐s❡ ✹✳✷✶

❚❤❡ s❡❝❛♥t ❧✐♥❡ ♦❢ t❤❡ s✐❣♥ ❢✉♥❝t✐♦♥ ❛r❡ s❤♦✇♥ ❜❡✲ ❧♦✇✳ ❲❤❛t ❞♦ t❤❡② t❡❧❧ ②♦✉ ❛❜♦✉t t❤❡ ❞✐✛❡r❡♥t✐❛✲ ❜✐❧✐t② ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❛t

x = 0❄

❊①❡r❝✐s❡ ✹✳✷✷

❨♦✉ ❤❛✈❡ r❡❝❡✐✈❡❞ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡♠❛✐❧ ❢r♦♠ ②♦✉r ❜♦ss✿ ✏❚✐♠✱ ▲♦♦❦ ❛t t❤❡ ♥✉♠❜❡rs ✐♥ t❤✐s s♣r❡❛❞✲ s❤❡❡t✳ ❚❤✐s st♦❝❦ s❡❡♠s t♦ ❜❡ ✐♥❝❤✐♥❣ ✉♣✳✳✳ ❉♦❡s ✐t❄ ■❢ ✐t ❞♦❡s✱ ❤♦✇ ❢❛st❄ ❚❤❛♥❦s✳ ✕ ❚♦♠✑✳ ❉❡s❝r✐❜❡ ②♦✉r ❛❝t✐♦♥s✳

✺✳ ❊①❡r❝✐s❡s✿ ▲✐♠✐ts ❛♥❞ ❝♦♥t✐♥✉✐t②

✺✶✼

✺✳ ❊①❡r❝✐s❡s✿ ▲✐♠✐ts ❛♥❞ ❝♦♥t✐♥✉✐t②

❊①❡r❝✐s❡ ✺✳✶

❊①❡r❝✐s❡ ✺✳✽

P❧♦t t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ y = f (x)✱ ✇❤❡r❡ x ✐s t❤❡ ✐♥❝♦♠❡ ✭✐♥ t❤♦✉s❛♥❞s ♦❢ ❞♦❧❧❛rs✮ ❛♥❞ f (x) ✐s t❤❡ t❛① ❜✐❧❧ ✭✐♥ t❤♦✉s❛♥❞s ♦❢ ❞♦❧❧❛rs✮ ❢♦r t❤❡ ✐♥❝♦♠❡ ♦❢ x✱ ✇❤✐❝❤ ✐s ❝♦♠♣✉t❡❞ ❛s ❢♦❧❧♦✇s✿ ♥♦ t❛① ♦♥ t❤❡ ✜rst $10, 000✱ t❤❡♥ 5% ❢♦r t❤❡ ♥❡①t $10, 000✱ ❛♥❞ 10% ❢♦r t❤❡ r❡st ♦❢ t❤❡ ✐♥❝♦♠❡✳ ■♥✈❡st✐❣❛t❡ ✐ts ❧✐♠✐ts ❛♥❞ ❝♦♥t✐♥✉✐t②✳

✭❛✮ ❙t❛t❡ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❛♥ ✐♥✜♥✐t❡ ❧✐♠✐t✳ ✭❜✮ ❯s❡ t❤❡ ❞❡✜♥✐t✐♦♥ t♦ ♣r♦✈❡ t❤❛t lim x3 = +∞✳ x→+∞

❊①❡r❝✐s❡ ✺✳✾

❋✐♥❞ t❤❡ ❤♦r✐③♦♥t❛❧ ❛s②♠♣t♦t❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥✿ f (x) = 3 −

❊①❡r❝✐s❡ ✺✳✷

P❧♦t t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ y = f (x)✱ ✇❤❡r❡ x ✐s t✐♠❡ ✐♥ ❤♦✉rs ❛♥❞ y = f (x) ✐s t❤❡ ♣❛r❦✐♥❣ ❢❡❡ ♦✈❡r x ❤♦✉rs✱ ✇❤✐❝❤ ✐s ❝♦♠♣✉t❡❞ ❛s ❢♦❧❧♦✇s✿ ❢r❡❡ ❢♦r t❤❡ ✜rst ❤♦✉r✱ t❤❡♥ $1 ♣❡r ❡✈❡r② ❢✉❧❧ ❤♦✉r ❢♦r t❤❡ ♥❡①t 3 ❤♦✉rs✱ ❛♥❞ ❛ ✢❛t ❢❡❡ ♦❢ $5 ❢♦r ❛♥②t❤✐♥❣ ❧♦♥❣❡r✳ ■♥✈❡st✐❣❛t❡ ✐ts ❧✐♠✐ts ❛♥❞ ❝♦♥t✐♥✉✐t②✳

1 . x

❊①❡r❝✐s❡ ✺✳✶✵

❇② ❝♦♠♣✉t✐♥❣ ♥❡❝❡ss❛r② ❧✐♠✐ts✱ ✜♥❞ t❤❡ ✈❡rt✐❝❛❧ ❛s②♠♣t♦t❡s ♦❢ t❤❡ ❢✉♥❝t✐♦♥✿ f (x) =

x . (x − 1)(x + 2)2

❊①❡r❝✐s❡ ✺✳✸

❆ ❝♦♥tr❛❝t♦r ♣✉r❝❤❛s❡s ❣r❛✈❡❧ ♦♥❡ ❝✉❜✐❝ ②❛r❞ ❛t ❛ t✐♠❡✳ ❆ ❣r❛✈❡❧ ❞r✐✈❡✇❛② x ②❛r❞s ❧♦♥❣ ❛♥❞ 4 ②❛r❞s ✇✐❞❡ ✐s t♦ ❜❡ ♣♦✉r❡❞ t♦ ❛ ❞❡♣t❤ ♦❢ 1.5 ❢♦♦t✳ ❋✐♥❞ ❛ ❢♦r♠✉❧❛ ❢♦r f (x)✱ t❤❡ ♥✉♠❜❡r ♦❢ ❝✉❜✐❝ ②❛r❞s ♦❢ ❣r❛✈❡❧ t❤❡ ❝♦♥tr❛❝t♦r ❜✉②s✱ ❛ss✉♠✐♥❣ t❤❛t ❤❡ ❜✉②s 10 ♠♦r❡ ❝✉❜✐❝ ②❛r❞s ♦❢ ❣r❛✈❡❧ t❤❛♥ ❛r❡ ♥❡❡❞❡❞✳ ■♥✲ ✈❡st✐❣❛t❡ ✐ts ❧✐♠✐ts ❛♥❞ ❝♦♥t✐♥✉✐t②✳ ❊①❡r❝✐s❡ ✺✳✹

❊①♣❧❛✐♥ ✇❤② t❤❡ ❧✐♠✐t lim sin x→0

1 ❞♦❡s ♥♦t ❡①✐st✳ x

❊①❡r❝✐s❡ ✺✳✶✶

●✐✈❡ ❛♥ ❡①❛♠♣❧❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ✇✐t❤ t✇♦ ✈❡rt✐❝❛❧ ❛s②♠♣t♦t❡s✿ x = 0 ❛♥❞ x = 2✳ ❊①❡r❝✐s❡ ✺✳✶✷

●✐✈❡ ❛♥ ❡①❛♠♣❧❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ✇✐t❤ ❛ ❤♦r✐③♦♥t❛❧ ❛s②♠♣t♦t❡✿ y = −1✱ ❛♥❞ ❛ ✈❡rt✐❝❛❧ ❛s②♠♣t♦t❡✿ x = 2✳ ❊①❡r❝✐s❡ ✺✳✶✸

■❞❡♥t✐❢② ❛❧❧ ✐♠♣♦rt❛♥t ❢❡❛t✉r❡s ♦❢ t❤✐s ❣r❛♣❤✿ ❊①❡r❝✐s❡ ✺✳✺

❙❦❡t❝❤ t❤❡ ❣r❛♣❤s ♦❢ t❤r❡❡ ❢✉♥❝t✐♦♥s ✇✐t❤ t❤❡ t❤r❡❡ ❞✐✛❡r❡♥t t②♣❡s ♦❢ ❞✐s❝♦♥t✐♥✉✐t✐❡s✳ ❉❡s❝r✐❜❡ t❤❡s❡ ❞✐s❝♦♥t✐♥✉✐t✐❡s ✇✐t❤ ❧✐♠✐ts✳ ❊①❡r❝✐s❡ ✺✳✻

✭❛✮ ❙t❛t❡ t❤❡ ε✲δ ❞❡✜♥✐t✐♦♥ ♦❢ ❧✐♠✐t✳ ✭❜✮ ❯s❡ t❤❡ ❞❡✜♥✐t✐♦♥ t♦ ♣r♦✈❡ t❤❛t lim x2 = 0✳ x→0

❊①❡r❝✐s❡ ✺✳✼

✭❛✮ ❙t❛t❡ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❧✐♠✐t✳ ✭❜✮ ❯s❡ t❤❡ ❞❡✜♥✐✲ t✐♦♥ t♦ ♣r♦✈❡ t❤❛t lim x3 6= 3✳ x→0

❊①❡r❝✐s❡ ✺✳✶✹

❊①♣r❡ss t❤❡ ❛s②♠♣t♦t❡s ♦❢ t❤✐s ❢✉♥❝t✐♦♥ ❛s ❧✐♠✐ts ❛♥❞ ✐❞❡♥t✐❢② ♦t❤❡r ♦❢ ✐ts ✐♠♣♦rt❛♥t ❢❡❛t✉r❡s✿

✺✳ ❊①❡r❝✐s❡s✿ ▲✐♠✐ts ❛♥❞ ❝♦♥t✐♥✉✐t②

✺✶✽

❊①❡r❝✐s❡ ✺✳✷✵

f ✐s ❝♦♥t✐♥✉♦✉s [a, ∞)✑❄

❚r✉❡ ♦r ❢❛❧s❡✿ ✏■❢ ✐s ❜♦✉♥❞❡❞ ♦♥

♦♥

[a, ∞)✱

t❤❡♥

f

❊①❡r❝✐s❡ ✺✳✷✶

❚r✉❡ ♦r ❢❛❧s❡✿

✏❊✈❡r② ❢✉♥❝t✐♦♥ ✐s ❜♦✉♥❞❡❞ ♦♥ ❛

❝❧♦s❡❞ ❜♦✉♥❞❡❞ ✐♥t❡r✈❛❧✑❄

❊①❡r❝✐s❡ ✺✳✷✷

❚r✉❡ ♦r ❢❛❧s❡✿ ❛

❝❧♦s❡❞✴♦♣❡♥

✏■❢ ❛ ❢✉♥❝t✐♦♥ ✐s ❝♦♥t✐♥✉♦✉s ♦♥ ✐♥t❡r✈❛❧✱

❝❧♦s❡❞✴♦♣❡♥ ✐♥t❡r✈❛❧✑❄

❊①❡r❝✐s❡ ✺✳✶✺

❊①♣r❡ss t❤❡ ❛s②♠♣t♦t❡s ♦❢ t❤✐s ❢✉♥❝t✐♦♥ ❛s ❧✐♠✐ts

t❤❡♥

✐ts

❞♦♠❛✐♥

✐s

❛

❨♦✉ ❤❛✈❡ ❢♦✉r ♦♣t✐♦♥s t♦

❝♦♥s✐❞❡r✳

❛♥❞ ✐❞❡♥t✐❢② ♦t❤❡r ♦❢ ✐ts ✐♠♣♦rt❛♥t ❢❡❛t✉r❡s✿ ❊①❡r❝✐s❡ ✺✳✷✸

❋✐♥❞ t❤❡ ❤♦r✐③♦♥t❛❧ ❛s②♠♣t♦t❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥✿

f (x) =

x . (x − 1)(x + 2)2

❊①❡r❝✐s❡ ✺✳✷✹

lim

❲❤❛t ❝❛♥ ②♦✉ s❛② ❛❜♦✉t

❊①❡r❝✐s❡ ✺✳✶✻

x→0

√

x❄

❊①❡r❝✐s❡ ✺✳✷✺

❲❤❛t ✐s s♦ s♣❡❝✐❛❧ ❛❜♦✉t t❤❡ ❢✉♥❝t✐♦♥ s❤♦✇♥ ❜❡❧♦✇❄ ❲❤❛t ✐s ✐ts ❢♦r♠✉❧❛❄

✭❛✮ ●✐✈❡ ❛♥ ❡①❛♠♣❧❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ✇✐t❤ t✇♦ ❞✐✛❡r❡♥t ❤♦r✐③♦♥t❛❧ ❛s②♠♣t♦t❡s✳

✭❜✮ ❲❤② ❝❛♥✬t ❛ r❛t✐♦♥❛❧

❢✉♥❝t✐♦♥ ❤❛✈❡ ♠♦r❡ t❤❛♥ ♦♥❡ ❤♦r✐③♦♥t❛❧ ❛s②♠♣t♦t❡❄

❊①❡r❝✐s❡ ✺✳✷✻

❈♦♠♣✉t❡ t❤❡ ♦♥❡✲s✐❞❡❞ ❧✐♠✐ts ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❜❡❧♦✇ ❛t

x = −1

x = 3✿   −x + 1 f (x) = x2 + 1  x e ❛♥❞

✐❢ ✐❢ ✐❢

x < −1 −1≤x3

❊①❡r❝✐s❡ ✺✳✷✼ ❊①❡r❝✐s❡ ✺✳✶✼

❇② ❝♦♠♣✉t✐♥❣ ❛ ❝❡rt❛✐♥ ❧✐♠✐t✱ ✜♥❞ t❤❡ ❤♦r✐③♦♥t❛❧

f ✐s (a, b)✑❄

❚r✉❡ ♦r ❢❛❧s❡✿ ✏■❢ ✐s ❜♦✉♥❞❡❞ ♦♥

❝♦♥t✐♥✉♦✉s ♦♥

(a, b)✱

t❤❡♥

f

❛s②♠♣t♦t❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥✿

f (x) = ❊①❡r❝✐s❡ ✺✳✶✽

❚r✉❡ ♦r ❢❛❧s❡✿ ✏■❢ ❜♦✉♥❞❡❞ ♦♥

f

✐s ❝♦♥t✐♥✉♦✉s ♦♥

[a, b]✱

t❤❡♥

f

✐s

[a, b]✑❄

3x3 − 1 . x(5x2 − 7)

❊①❡r❝✐s❡ ✺✳✷✽

❚❤❡ ❜❛s❡ s❛❧❛r② ♦❢ ❛ s❛❧❡s♠❛♥ ✇♦r❦✐♥❣ ♦♥ ❝♦♠♠✐s✲

$20, 000✳ ❋♦r ❡❛❝❤ $10, 000 ♦❢ s❛❧❡s ❜❡②♦♥❞ $50, 000✱ ❤❡ ✐s ♣❛✐❞ ❛ $1, 000 ❝♦♠♠✐ss✐♦♥✳ ▲❡t f (x)

s✐♦♥ ✐s ❊①❡r❝✐s❡ ✺✳✶✾

f ✐s [a, b)✑❄

❚r✉❡ ♦r ❢❛❧s❡✿ ✏■❢ ✐s ❜♦✉♥❞❡❞ ♦♥

r❡♣r❡s❡♥t ❤✐s s❛❧❛r② ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ❧❡✈❡❧ ♦❢ ❤✐s ❝♦♥t✐♥✉♦✉s ♦♥

[a, b)✱

t❤❡♥

f

s❛❧❡s

x✳

✭❛✮ ❙❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥✳ ✭❜✮

❉✐s❝✉ss t❤❡ ❝♦♥t✐♥✉✐t② ♦❢

f✳

✺✳ ❊①❡r❝✐s❡s✿ ▲✐♠✐ts ❛♥❞ ❝♦♥t✐♥✉✐t②

✺✶✾

❊①❡r❝✐s❡ ✺✳✷✾

f a = 3✿

❢✉♥❝t✐♦♥

❚❤❡ ❜❛s❡ s❛❧❛r② ♦❢ ❛ s❛❧❡s♠❛♥ ✇♦r❦✐♥❣ ♦♥ ❝♦♠✲

❛♥❞

$20, 000✳ ❋♦r ❡❛❝❤ $10, 000 ♦❢ s❛❧❡s ❜❡✲ ②♦♥❞ $50, 000✱ ❤❡ ✐s ♣❛✐❞ ❛ $1, 000 ❝♦♠♠✐ss✐♦♥✳ ▲❡t y = f (x) r❡♣r❡s❡♥t ❤✐s s❛❧❛r② ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ❧❡✈❡❧ ♦❢ ❤✐s s❛❧❡s x✳ ✭❛✮ ❙❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ♠✐ss✐♦♥ ✐s

  x+1 f (x) = x2 + 1  10

❢✉♥❝t✐♦♥✳ ✭❜✮ ❉❡s❝r✐❜❡ t❤❡ ❝♦♥t✐♥✉✐t② ❛♥❞ t❤❡ ❞✐❢✲

f✳

❢❡r❡♥t✐❛❜✐❧✐t② ♦❢ ✏❋♦r ❡❛❝❤

$1

$1

✭❝✮ ❲❤❛t ✐❢ ✇❡ ❤❛✈❡ ✐♥st❡❛❞

$50, 000✱

♦❢ s❛❧❡s ❜❡②♦♥❞

❤❡ ✐s ♣❛✐❞ ❛

❝♦♠♠✐ss✐♦♥✑❄

❞❡✜♥❡❞ ❜❡❧♦✇ ✐s ❝♦♥t✐♥✉♦✉s ❛t

✐❢ ✐❢ ✐❢

a = 0

x 0✳

t✐♦♥✱ ❛♥❞ ❛ ❢✉♥❝t✐♦♥ t❤❛t✬s ♥❡✐t❤❡r✳ Pr♦✈✐❞❡ ❢♦r♠✉✲ ❧❛s✳

❊①❡r❝✐s❡ ✼✳✶✵

❊①❡r❝✐s❡ ✼✳✸

✭❛✮ ❋✐♥❞ ✐ts ❞♦♠❛✐♥✳

❚❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥

❚❡st ✇❤❡t❤❡r t❤❡ ❢♦❧❧♦✇✐♥❣ t❤r❡❡ ❢✉♥❝t✐♦♥s ❛r❡ ❡✈❡♥✱ ♦❞❞✱ ♦r ♥❡t❤❡r✿ ✭❛✮

3

f (x) = x + 1❀

✭❜✮ t❤❡ ❢✉♥❝t✐♦♥

t❤❡ ❣r❛♣❤ ♦❢ ✇❤✐❝❤ ✐s ❛ ♣❛r❛❜♦❧❛ s❤✐❢t❡❞ ♦♥❡ ✉♥✐t

y = f (x)

✐s ❣✐✈❡♥ ❜❡❧♦✇✳

✭❜✮ ❉❡t❡r♠✐♥❡ ✐♥t❡r✈❛❧s ♦♥

✇❤✐❝❤ t❤❡ ❢✉♥❝t✐♦♥ ✐s ❞❡❝r❡❛s✐♥❣ ♦r ✐♥❝r❡❛s✐♥❣✳ ✭❝✮ Pr♦✈✐❞❡

x✲❝♦♦r❞✐♥❛t❡s

♦❢ ✐ts r❡❧❛t✐✈❡ ♠❛①✐♠❛ ❛♥❞

♠✐♥✐♠❛✳ ✭❞✮ ❋✐♥❞ ✐ts ❛s②♠♣t♦t❡s✳

✉♣❀ ✭❝✮ t❤❡ ❢✉♥❝t✐♦♥ ✇✐t❤ t❤✐s ❣r❛♣❤✿

❊①❡r❝✐s❡ ✼✳✶✶

■❢ ❛ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥ ❤❛s

❊①❡r❝✐s❡ ✼✳✹

10

✈❡rt✐❝❛❧ ❛s②♠♣t♦t❡s✱

❤♦✇ ♠❛♥② ❜r❛♥❝❤❡s ❞♦❡s ✐ts ❣r❛♣❤ ❤❛✈❡❄

❋✐♥❞ ❤♦r✐③♦♥t❛❧ ❛s②♠♣t♦t❡s ♦❢ t❤❡s❡ ❢✉♥❝t✐♦♥s✿ ❊①❡r❝✐s❡ ✼✳✶✷

❉❡t❡r♠✐♥❡ ✇❤✐❝❤ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ st❛t❡♠❡♥ts ❛r❡ tr✉❡ ❛♥❞ ✇❤✐❝❤ ❛r❡ ❢❛❧s❡✳ ✶✳ ❚❤❡ ❢✉♥❝t✐♦♥

❊①❡r❝✐s❡ ✼✳✺

■s

sin x/2

♣❡r✐♦❞✳

❛t ❧❡❛st ♦♥❡ ✐♥♣✉t ✇❤✐❝❤ ♣r♦❞✉❝❡s ❛ s♠❛❧❧❡st ❛ ♣❡r✐♦❞✐❝ ❢✉♥❝t✐♦♥❄

■❢ ✐t ✐s✱ ✜♥❞ ✐ts

♦✉t♣✉t ✈❛❧✉❡✳

❨♦✉ ❤❛✈❡ t♦ ❥✉st✐❢② ②♦✉r ❝♦♥❝❧✉s✐♦♥ ❛❧❣❡✲

❜r❛✐❝❛❧❧②✳

sin x ♦♥ t❤❡ ❞♦♠❛✐♥ (−π, π) ❤❛s

✷✳ ❚❤❡ ❢✉♥❝t✐♦♥

f (x) = x3

✇✐t❤ ❞♦♠❛✐♥

(−3, 3)

✼✳ ❊①❡r❝✐s❡s✿ ❋❡❛t✉r❡s ♦❢ ❣r❛♣❤s

❤❛s ❛t ❧❡❛st ♦♥❡ ✐♥♣✉t ✇❤✐❝❤ ♣r♦❞✉❝❡s ❛ ❧❛r❣❡st ♦✉t♣✉t ✈❛❧✉❡✳ ✸✳ ❚❤❡ ❢✉♥❝t✐♦♥ f (x) = x3 ✇✐t❤ ❞♦♠❛✐♥ [−3, 3] ❤❛s ❛t ❧❡❛st ♦♥❡ ✐♥♣✉t ✇❤✐❝❤ ♣r♦❞✉❝❡s ❛ ❧❛r❣❡st ♦✉t♣✉t ✈❛❧✉❡✳ ✹✳ ❚❤❡ ❢✉♥❝t✐♦♥ f (x) = x3 ✇✐t❤ ❞♦♠❛✐♥ [−3, 3] ❤❛s ❛t ❧❡❛st ♦♥❡ ✐♥♣✉t ✇❤✐❝❤ ♣r♦❞✉❝❡s ❛ s♠❛❧❧❡st ♦✉t♣✉t ✈❛❧✉❡✳

✺✷✹

❊①❡r❝✐s❡ ✼✳✶✼

❙❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢ ❢✉♥❝t✐♦♥ y = f (x) s❛t✐s❢②✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s✿ lim− f = 1✱ lim+ f = 3✱ f ✐s x→2

x→2

✐♥❝r❡❛s✐♥❣ ♦♥ (−1, 0)✱ lim f = −1✱ lim f = ∞✳ x→−∞

x→+∞

❊①❡r❝✐s❡ ✼✳✶✽

❚❤❡ ❣r❛♣❤ ♦❢ ❢✉♥❝t✐♦♥ f ✐s ❣✐✈❡♥ ❜❡❧♦✇✳ ▲✐st ❛t ❧❡❛st ✜✈❡ ♦❢ ✐ts ♠❛✐♥ ❢❡❛t✉r❡s✳

✺✳ ❚❤❡ ❢✉♥❝t✐♦♥ sin x ♦♥ t❤❡ ❞♦♠❛✐♥ [−π, π] ❤❛s ❛t ❧❡❛st ♦♥❡ ✐♥♣✉t ✇❤✐❝❤ ♣r♦❞✉❝❡s ❛ s♠❛❧❧❡st ♦✉t♣✉t ✈❛❧✉❡✳

❊①❡r❝✐s❡ ✼✳✶✸

●✐✈❡ ❛♥ ❡①❛♠♣❧❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ t❤❛t ✐s ❜♦t❤ ♦❞❞ ❛♥❞ ❡✈❡♥ ❜✉t ♥♦t ♣❡r✐♦❞✐❝✳ ❊①❡r❝✐s❡ ✼✳✶✾ ❊①❡r❝✐s❡ ✼✳✶✹

❚❤❡ ❣r❛♣❤ ♦❢ f ✐s ❣✐✈❡♥ ❜❡❧♦✇✳ ❋✐♥❞ ❛❧❧ t❤❡ ❛s②♠♣✲ t♦t❡s ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❛♥❞ ❞❡s❝r✐❜❡ t❤❡♠ ❛s ❧✐♠✐ts✳

❆ s❦❡t❝❤ ♦❢ t❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥ f ✐s ❣✐✈❡♥ ❜❡❧♦✇✳ Pr♦✈✐❞❡ t❤❡ ✐♠♣♦rt❛♥t ❧✐♠✐ts ♦❢ f t❤❛t ❞❡s❝r✐❜❡ ✐ts ❜❡❤❛✈✐♦r✳

❊①❡r❝✐s❡ ✼✳✶✺

❚❤❡ ❣r❛♣❤ ♦❢ ❢✉♥❝t✐♦♥ f ✐s ❣✐✈❡♥ ❜❡❧♦✇✳ ❙❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ f ′ ✐♥ t❤❡ s♣❛❝❡ ✉♥❞❡r t❤❡ ❣r❛♣❤ ♦❢ f ✳ ■❞❡♥t✐❢② ❛❧❧ ✐♠♣♦rt❛♥t ♣♦✐♥ts ❛♥❞ ❢❡❛t✉r❡s ♦♥ t❤❡ ❣r❛♣❤✳

❊①❡r❝✐s❡ ✼✳✷✵

❆ s❦❡t❝❤ ♦❢ t❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥ f ✐s ❣✐✈❡♥ ❜❡❧♦✇✳ ▲✐st ❛t ❧❡❛st ✜✈❡ ♦❢ ✐ts ♠❛✐♥ ❢❡❛t✉r❡s✳

❊①❡r❝✐s❡ ✼✳✶✻

❙❦❡t❝❤ t❤❡ ❣r❛♣❤s ♦❢ ❢✉♥❝t✐♦♥s ✇✐t❤ t❤❡s❡ ❢❡❛t✉r❡s✿ ✭❛✮ f ❤❛s ❛ ❧♦❝❛❧ ♠❛①✐♠✉♠ ❛t x = 2❀ ✭❜✮ g ❤❛s ❛ ✈❡rt✐❝❛❧ ❛s②♠♣t♦t❡ x = 1❀ ✭❝✮ h ❤❛s ❛ ❤♦r✐③♦♥t❛❧ ❛s②♠♣t♦t❡ y = −1✳

❊①❡r❝✐s❡ ✼✳✷✶

❚❤❡ ❣r❛♣❤ ♦❢ ❢✉♥❝t✐♦♥ f ✐s ❣✐✈❡♥ ❜❡❧♦✇✳ ❙❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ f ′ ✐♥ t❤❡ s♣❛❝❡ ✉♥❞❡r t❤❡

✼✳ ❊①❡r❝✐s❡s✿ ❋❡❛t✉r❡s ♦❢ ❣r❛♣❤s

✺✷✺

❣r❛♣❤ ♦❢ f ✳ ■❞❡♥t✐❢② ❛❧❧ ✐♠♣♦rt❛♥t ♣♦✐♥ts ♦♥ t❤❡ ❣r❛♣❤✳

❊①❡r❝✐s❡ ✼✳✷✺ ❊①❡r❝✐s❡ ✼✳✷✷

❚❤❡ ❣r❛♣❤ ♦❢ ❢✉♥❝t✐♦♥ f ✐s ❣✐✈❡♥ ❜❡❧♦✇✳ ❙❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ f ′ ✐♥ t❤❡ s♣❛❝❡ ✉♥❞❡r t❤❡ ❣r❛♣❤ ♦❢ f ✳ ■❞❡♥t✐❢② ❛❧❧ ✐♠♣♦rt❛♥t ♣♦✐♥ts ♦♥ t❤❡ ❣r❛♣❤✳

❚❤❡ ❣r❛♣❤ ♦❢ ❢✉♥❝t✐♦♥ f ✐s ❣✐✈❡♥ ❜❡❧♦✇✳ ❙❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ f ′ ✐♥ t❤❡ s♣❛❝❡ ✉♥❞❡r t❤❡ ❣r❛♣❤ ♦❢ f ✳ ■❞❡♥t✐❢② ❛❧❧ ✐♠♣♦rt❛♥t ♣♦✐♥ts ♦♥ t❤❡ ❣r❛♣❤✳

❊①❡r❝✐s❡ ✼✳✷✻ ❊①❡r❝✐s❡ ✼✳✷✸

❚❤❡ ❣r❛♣❤ ♦❢ ❢✉♥❝t✐♦♥ f ✐s ❣✐✈❡♥ ❜❡❧♦✇✳ ❙❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ f ′ ✐♥ t❤❡ s♣❛❝❡ ✉♥❞❡r t❤❡ ❣r❛♣❤ ♦❢ f ✳ ■❞❡♥t✐❢② ❛❧❧ ✐♠♣♦rt❛♥t ♣♦✐♥ts ♦♥ t❤❡ ❣r❛♣❤✳

❚❤❡ ❣r❛♣❤ ♦❢ ❢✉♥❝t✐♦♥ f ✐s ❣✐✈❡♥ ❜❡❧♦✇✳ ❙❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ f ′ ✐♥ t❤❡ s♣❛❝❡ ✉♥❞❡r t❤❡ ❣r❛♣❤ ♦❢ f ✳ ■❞❡♥t✐❢② ❛❧❧ ✐♠♣♦rt❛♥t ♣♦✐♥ts ♦♥ t❤❡ ❣r❛♣❤✳

❊①❡r❝✐s❡ ✼✳✷✼ ❊①❡r❝✐s❡ ✼✳✷✹

❚❤❡ ❣r❛♣❤ ♦❢ ❢✉♥❝t✐♦♥ f ✐s ❣✐✈❡♥ ❜❡❧♦✇✳ ❙❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ f ′ ✐♥ t❤❡ s♣❛❝❡ ✉♥❞❡r t❤❡ ❣r❛♣❤ ♦❢ f ✳ ■❞❡♥t✐❢② ❛❧❧ ✐♠♣♦rt❛♥t ♣♦✐♥ts ♦♥ t❤❡ ❣r❛♣❤✳

✭❛✮ ❙❤♦✇ t❤❛t ♦♥❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥s ❜❡❧♦✇ ✐s t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ♦t❤❡r✳ ✭❜✮ ❊①♣❧❛✐♥ ❤♦✇ t❤❡ ❤♦r✐✲ ③♦♥t❛❧ ❛s②♠♣t♦t❡ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ ❛✛❡❝ts t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥✳ ✭❝✮ ❲❤❛t ❛❜♦✉t t❤❡ ❤♦r✐③♦♥t❛❧ ❛s②♠♣t♦t❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ✈s✳ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡❄

✼✳ ❊①❡r❝✐s❡s✿ ❋❡❛t✉r❡s ♦❢ ❣r❛♣❤s

✺✷✻

❊①❡r❝✐s❡ ✼✳✷✽

❉❡s❝r✐❜❡ t❤❡ ❜❡❤❛✈✐♦r ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ♣❧♦tt❡❞ ❜❡✲ ❧♦✇✿

❊①❡r❝✐s❡ ✼✳✸✶

f ✐s ❣✐✈❡♥ ❜❡❧♦✇✳ ❋♦r ✇❤❛t ✈❛❧✉❡s ♦❢ x ❛r❡ f (x), f ′ (x), f ′′ (x) ♣♦s✐t✐✈❡✱ ♥❡❣❛t✐✈❡ ♦r ③❡r♦❄ ❋✐❧❧ ✐♥ t❤❡ ❜❧❛♥❦s ✇✐t❤ +✱ −✱ ♦r 0✿ ❚❤❡ ❣r❛♣❤ ♦❢

x f (x) f ′ (x) f ′′ (x) −1.5 0 1 2 2.5 ❊①❡r❝✐s❡ ✼✳✷✾

❉❡s❝r✐❜❡ t❤❡ ❝♦♥❝❛✈✐t② ♦❢ t❤✐s ❢✉♥❝t✐♦♥✿

❊①❡r❝✐s❡ ✼✳✸✵

❚❤❡ ❣r❛♣❤ ♦❢

f

✐s ❣✐✈❡♥ ❜❡❧♦✇✳

❈♦♠♣❧❡t❡❧② ❞❡✲

s❝r✐❜❡ t❤❡ ❜❡❤❛✈✐♦r ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❜② ✉s✐♥❣ s✉❝❤ ✇♦r❞s ❛s ✏✐♥❝r❡❛s✐♥❣✴❞❡❝r❡❛s✐♥❣✑✱

✏❝♦♥❝❛✈❡ ✉♣✴✲

❞♦✇♥✑✱ ✏♠❛①✴♠✐♥✑✱ ✏❛s②♠♣t♦t❡s✑✱ ❡t❝✳

✼✳ ❊①❡r❝✐s❡s✿ ❋❡❛t✉r❡s ♦❢ ❣r❛♣❤s

❊①❡r❝✐s❡ ✼✳✸✷

❚❤❡ ❣r❛♣❤ ♦❢ f ✐s ❣✐✈❡♥ ❜❡❧♦✇✳ ❋♦r ✇❤❛t ✈❛❧✉❡s ♦❢ x ❛r❡ f (x), f ′ (x), f ′′ (x) ♣♦s✐t✐✈❡✱ ♥❡❣❛t✐✈❡ ♦r ③❡r♦❄ ❋✐❧❧ ✐♥ t❤❡ ❜❧❛♥❦s ✇✐t❤ +✱ −✱ ♦r 0✿

✺✷✼

❛❧❧ ✐♠♣♦rt❛♥t ♣♦✐♥ts ♦♥ t❤❡ ❣r❛♣❤✳

x f (x) f ′ (x) f ′′ (x) −1.5 0 1 2 2.5

❊①❡r❝✐s❡ ✼✳✸✼

❚❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ f ′ ♦❢ ❢✉♥❝t✐♦♥ f ✐s ❣✐✈❡♥ ❜❡❧♦✇✳ ❙❦❡t❝❤ ❛ ♣♦ss✐❜❧❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ f ✐ts❡❧❢ ✐♥ t❤❡ s♣❛❝❡ ✉♥❞❡r t❤❡ ❣r❛♣❤ ♦❢ f ′ ✳ ■❞❡♥t✐❢② ❛❧❧ ✐♠♣♦rt❛♥t ♣♦✐♥ts ♦♥ t❤❡ ❣r❛♣❤✳

❊①❡r❝✐s❡ ✼✳✸✸

❙❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢ f ✇✐t❤ t❤❡s❡ ✈❛❧✉❡s ♦❢ f (x), f ′ (x), f ′′ (x)✿ x f (x) f ′ (x) f ′′ (x) −1 + + + 0 − 0 − 1 + − 0 − 2 3 − + ❊①❡r❝✐s❡ ✼✳✸✹

❊①❡r❝✐s❡ ✼✳✸✽

❚❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ f ′ ♦❢ ❢✉♥❝t✐♦♥ f ✐s ❣✐✈❡♥ ❜❡❧♦✇✳ ❙❦❡t❝❤ ❛ ♣♦ss✐❜❧❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ f ✐ts❡❧❢ ✐♥ t❤❡ s♣❛❝❡ ✉♥❞❡r t❤❡ ❣r❛♣❤ ♦❢ f ′ ✉♥❞❡r t❤❡ ❛ss✉♠♣t✐♦♥ t❤❛t f (0) = 0✳

❙❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥ f t❤❛t ✐s ❝♦♥t✐♥✉♦✉s ♦♥ [1, 5] ❛♥❞ ❤❛s ❣❧♦❜❛❧ ♠✐♥✐♠✉♠ ❛t 1✱ ❣❧♦❜❛❧ ♠❛①✐✲ ♠✉♠ ❛t 5✱ ❧♦❝❛❧ ♠❛①✐♠✉♠ ❛t 2✱ ❛♥❞ ❧♦❝❛❧ ♠✐♥✐♠✉♠ ❛t 4✳ ❊①❡r❝✐s❡ ✼✳✸✺

❙❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥ f ✇✐t❤ t❤❡ ❢♦❧❧♦✇✲ ✐♥❣ ❢❡❛t✉r❡s✿ ✭❛✮ ✐t ❤❛s ❛ r❡♠♦✈❛❜❧❡ ❞✐s❝♦♥t✐♥✉✐t② ❛t x = −1❀ ✭❜✮ ✐t ❤❛s ❛ ✈❡rt✐❝❛❧ ❛s②♠♣t♦t❡ x = 1❀ ✭❝✮ ✐t ✐s ❝♦♥t✐♥✉♦✉s ❜✉t ♥♦t ❞✐✛❡r❡♥t✐❛❜❧❡ x = 3❀ ✭❞✮ ✐t ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ❡✈❡r②✇❤❡r❡ ❡❧s❡❀ ✭❡✮ ✐t ❤❛s ♥♦ ❤♦r✐③♦♥t❛❧ ❛s②♠♣t♦t❡s✳ ❊①❡r❝✐s❡ ✼✳✸✻

❚❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ f ′ ♦❢ ❢✉♥❝t✐♦♥ f ✐s ❣✐✈❡♥ ❜❡❧♦✇✳ ❙❦❡t❝❤ ❛ ♣♦ss✐❜❧❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ f ✐ts❡❧❢ ✐♥ t❤❡ s♣❛❝❡ ✉♥❞❡r t❤❡ ❣r❛♣❤ ♦❢ f ′ ✳ ■❞❡♥t✐❢②

❊①❡r❝✐s❡ ✼✳✸✾

❙✉♣♣♦s❡ ②♦✉ ❛r❡ t♦✇✐♥❣ ❛ tr❛✐❧❡r✲❤♦♠❡✳ ❉✉r✐♥❣ t❤❡ ✜rst ❢❡✇ ♠✐♥✉t❡s✱ ❡✈❡r② t✐♠❡ ②♦✉ ❧♦♦❦ ❛t t❤❡ r❡❛r ✈✐❡✇ ♠✐rr♦r ②♦✉ ❝❛♥ s❡❡ ♦♥❧② t❤❡ ❧♦✇❡r ♣❛rt ♦❢ t❤❡ ❤♦♠❡✳ ▲❛t❡r✱ ❡✈❡r② t✐♠❡ ②♦✉ ❧♦♦❦ ②♦✉ ❝❛♥ s❡❡ ♦♥❧② t❤❡ t♦♣ ♣❛rt✳ ❉✐s❝✉ss t❤❡ ♣r♦✜❧❡ ♦❢ t❤❡ r♦❛❞✳

✼✳ ❊①❡r❝✐s❡s✿ ❋❡❛t✉r❡s ♦❢ ❣r❛♣❤s

✺✷✽

❊①❡r❝✐s❡ ✼✳✹✵

❲❤✐❧❡ ❞r✐✈✐♥❣✱ ②♦✉ ♥♦t✐❝❡ t❤❛t ❢♦r ❛ ❢❡✇ s❡❝♦♥❞s ②♦✉r ❤❡❛❞❧✐❣❤ts ♣♦✐♥t ❛t t❤❡ ❝❛r ♦♥ t❤❡ ♦♣♣♦s✐t❡ ❧❛♥❡✳ ❲❤❛t ❝❛♥ ②♦✉ s❛② ❛❜♦✉t t❤❡ r♦❛❞❄

❊①❡r❝✐s❡ ✼✳✹✶

f ✐s ❣✐✈❡♥ ′ ❞❡r✐✈❛t✐✈❡ f ♦❢ f ✿

❚❤❡ ❣r❛♣❤ ♦❢ ❢✉♥❝t✐♦♥ ❣r❛♣❤ ♦❢ t❤❡

❜❡❧♦✇✳ ❙❦❡t❝❤ t❤❡

❊①❡r❝✐s❡ ✼✳✹✷

❙❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢ ❛ ❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥ ❤❛s ❛ ❣❧♦❜❛❧ ♠❛①✐♠✉♠ ❛t

2✱

❛ ❧♦❝❛❧ ♠✐♥✐♠✉♠ ❛t

5✱

−1✱

f

t❤❛t

❛ ❧♦❝❛❧ ♠❛①✐♠✉♠ ❛t

❛♥ ✐♥✢❡❝t✐♦♥ ♣♦✐♥t ❛t

❛♥❞ ❛ ❤♦r✐③♦♥t❛❧ ❛s②♠♣t♦t❡

4✱

y = 1✳

❊①❡r❝✐s❡ ✼✳✹✸

✭❛✮ ❆♥❛❧②③❡ t❤❡ ✜rst ❛♥❞ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡s ♦❢ t❤❡

f (x) = x4 −2x2 ✳ ❣r❛♣❤ ♦❢ f ✳

❢✉♥❝t✐♦♥ t❤❡

✭❜✮ ❯s❡ ♣❛rt ✭❛✮ t♦ s❦❡t❝❤

❊①❡r❝✐s❡ ✼✳✹✹

h′ (x) = 0

❢♦r ❛❧❧ x (a, b)✱ t❤❡♥✳✳✳✑✳ ✭❜✮ ❋✐♥✐s❤ t❤❡ st❛t❡♠❡♥t ✏■❢ f ′ (x) = g ′ (x) ❢♦r ❛❧❧ x ✐♥ (a, b)✱ t❤❡♥✳✳✳✑✳ ✭❝✮ ❯s❡

✭❛✮ ❋✐♥✐s❤ t❤❡ st❛t❡♠❡♥t ✏■❢ ✐♥

♣❛rt ✭❛✮ t♦ ♣r♦✈❡ ♣❛rt ✭❜✮✳

❊①❡r❝✐s❡ ✼✳✹✺

❋✐♥❞ t❤❡ ✈❡rt✐❝❛❧ ❛♥❞ ❤♦r✐③♦♥t❛❧ ❛s②♠♣t♦t❡s ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❛♥❞ ❞❡s❝r✐❜❡ t❤❡♠ ❛s ❧✐♠✐ts✿

f (x) =

2x2 . x2 − 1

✽✳ ❊①❡r❝✐s❡s✿ ▲✐♥❡❛r✐③❛t✐♦♥

✺✷✾

✽✳ ❊①❡r❝✐s❡s✿ ▲✐♥❡❛r✐③❛t✐♦♥

❊①❡r❝✐s❡ ✽✳✶

❊①❡r❝✐s❡ ✽✳✶✶

●✐✈❡ ❛ ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ♣❧♦tt❡❞

●✐✈❡ ❛♥ ❡①❛♠♣❧❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ t❤❡ ❜❡st ❧✐♥❡❛r ❛♣✲

❜❡❧♦✇ ❛t

x=5

❛♥❞ ❛t

x = 2✿

♣r♦①✐♠❛t✐♦♥ ♦❢ ✇❤✐❝❤ ❝♦✐♥❝✐❞❡ ✇✐t❤ t❤❡ ❝♦♥st❛♥t ❛♣♣r♦①✐♠❛t✐♦♥✳

❊①❡r❝✐s❡ ✽✳✷

❯s❡ ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ ♠❛t❡

f (x) = sin x

t♦ ❡st✐✲

sin .02.

❊①❡r❝✐s❡ ✽✳✸

❋✐♥❞ t❤❡ ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢

a = 1✳

❯s❡ ✐t t♦ ❡st✐♠❛t❡

f (x) = ln x

❛t

ln .99✳

❊①❡r❝✐s❡ ✽✳✹

a = 1.

❯s❡ ✐t t♦ ❡st✐♠❛t❡

√

x

❛t

f (x) = sin 3x sin −.02.

❛t

❋✐♥❞ t❤❡ ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢

√

f (x) =

1.1.

❊①❡r❝✐s❡ ✽✳✺

❋✐♥❞ t❤❡ ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢

a = 0.

❯s❡ ✐t t♦ ❡st✐♠❛t❡

❊①❡r❝✐s❡ ✽✳✻

f (x) = √ 1.03.

❋✐♥❞ t❤❡ ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ ❛t

a = 0.

❯s❡ ✐t t♦ ❡st✐♠❛t❡

√

1 + 3x

❊①❡r❝✐s❡ ✽✳✼

❋✐♥❞ t❤❡ ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ t♦ ❡st✐♠❛t❡

√ 3

26.9.

❊①❡r❝✐s❡ ✽✳✽

❋✐♥❞ t❤❡ ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢

a = 1✳

❯s❡ ✐t t♦ ❡st✐♠❛t❡

1.1

1/3

f (x) = x1/3

✳

❊①❡r❝✐s❡ ✽✳✾

❯s❡ ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ t♦ ❡st✐♠❛t❡

sin π/2✳

❊①❡r❝✐s❡ ✽✳✶✵

❯s❡ ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ t♦ ❡st✐♠❛t❡

sin π/4✳

❛t

✾✳ ❊①❡r❝✐s❡s✿ ▼♦❞❡❧s

✺✸✵

✾✳ ❊①❡r❝✐s❡s✿ ▼♦❞❡❧s

❊①❡r❝✐s❡ ✾✳✶

❊①❡r❝✐s❡ ✾✳✽

❚❤❡ ♣♦♣✉❧❛t✐♦♥ ♦❢ ❛ ❝✐t② ❤❛s ❞♦✉❜❧❡❞ ✐♥ 10 ②❡❛rs✳ ❆ss✉♠✐♥❣ ❡①♣♦♥❡♥t✐❛❧ ❣r♦✇t❤✱ ❤♦✇ ❧♦♥❣ ❞♦❡s ✐t t❛❦❡ t♦ tr✐♣❧❡❄

❙❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢ ②♦✉r ❡❧❡✈❛t✐♦♥ ❞✉r✐♥❣ ❛ tr✐♣ ♦♥ ❛ ❋❡rr✐s ✇❤❡❡❧✳

❊①❡r❝✐s❡ ✾✳✷

❚❤❡ ♣♦♣✉❧❛t✐♦♥ ♦❢ ❛ ❝✐t② ❤❛s ❞♦✉❜❧❡❞ ✐♥ 10 ②❡❛rs✳ ❆ss✉♠✐♥❣ ❡①♣♦♥❡♥t✐❛❧ ❣r♦✇t❤✱ ❤♦✇ ♠✉❝❤ ❞♦❡s ✐t ❣r♦✇ ❡✈❡r② ②❡❛r❄ ❊①❡r❝✐s❡ ✾✳✸

Pr♦✈✐❞❡ ❛ ❢♦r♠✉❧❛ ❢♦r ♠♦❞❡❧✐♥❣ r❛❞✐♦❛❝t✐✈❡ ❞❡❝❛②✳ ❲❤❛t ✐s t❤❡ ❤❛❧❢✲❧✐❢❡ ♦❢ ❛♥ ❡❧❡♠❡♥t❄

❊①❡r❝✐s❡ ✾✳✾

❆ ❝✉♣ ♦❢ ❤♦t ❝❤♦❝♦❧❛t❡ ❤❛s t❡♠♣❡r❛t✉r❡ 80 ❞❡❣r❡❡s ✐♥ ❛ r♦♦♠ ❦❡♣t ❛t 20 ❞❡❣r❡❡s✳ ❆❢t❡r ❛♥ ❤♦✉r t❤❡ ❝❤♦❝♦❧❛t❡ ❝♦♦❧s t♦ 60 ❞❡❣r❡❡s✳ ✭✶✮ ❆ss✉♠✐♥❣ ◆❡✇✲ t♦♥✬s ▲❛✇ ♦❢ ❈♦♦❧✐♥❣✱ ✇❤❛t ✐s t❤❡ t❡♠♣❡r❛t✉r❡ ♦❢ t❤❡ ❝❤♦❝♦❧❛t❡ ❛❢t❡r ❛♥♦t❤❡r ❤♦✉r✳ ✭✷✮ Pr♦✈✐❞❡ t❤❡ ❢♦r♠✉❧❛ ❢♦r ◆❡✇t♦♥✬s ▲❛✇ ♦❢ ❈♦♦❧✐♥❣ ❛♥❞ ❡①♣❧❛✐♥✳

❊①❡r❝✐s❡ ✾✳✹

❚❤❡ ♣♦♣✉❧❛t✐♦♥ ♦❢ ❛ ❝✐t② ❞❡❝❧✐♥❡s ❜② 10% ❡✈❡r② ②❡❛r✳ ❍♦✇ ❧♦♥❣ ✇✐❧❧ ✐t t❛❦❡ t♦ ❞r♦♣ t♦ 50% ♦❢ t❤❡ ❝✉rr❡♥t ♣♦♣✉❧❛t✐♦♥❄ ❊①❡r❝✐s❡ ✾✳✺

❚❤❡ ❢✉♥❝t✐♦♥ y = f (x) s❤♦✇♥ ❜❡❧♦✇ r❡♣r❡s❡♥ts t❤❡ ❧♦❝❛t✐♦♥ ✭✐♥ ♠✐❧❡s✮ ♦❢ ❛ ❤✐❦❡r ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✐♠❡ ✭✐♥ ❤♦✉rs✮✳ ❙❦❡t❝❤ t❤❡ ❤✐❦❡r✬s ✈❡❧♦❝✐t② ❛s t❤❡ ❞✐✛❡r✲ ❡♥❝❡ q✉♦t✐❡♥t✳

❊①❡r❝✐s❡ ✾✳✶✵

❚❤❡ ✈❡❧♦❝✐t② ♦❢ t❤❡ ♦❜❥❡❝t ❛t t✐♠❡ t ✐s ❣✐✈❡♥ ❜② v(t) = 1 + 3t2 . ■❢ ❛t t✐♠❡ t = 1 t❤❡ ♦❜❥❡❝t ✐s ❛t ♣♦s✐t✐♦♥ x = 4, ✇❤❡r❡ ✐s ✐t ❛t t✐♠❡ t = 0❄

❊①❡r❝✐s❡ ✾✳✶✶

❚❤❡ ❣r❛♣❤s ♦❢ t❤❡ ✈❡❧♦❝✐t② ❛♥❞ t❤❡ ♣♦s✐t✐♦♥ ♦❢ ❛ ♠♦✈✐♥❣ ♦❜❥❡❝t ❛r❡ s❤♦✇♥ ❜❡❧♦✇✳ ❉❡s❝r✐❜❡ ✇❤❛t ✐s ❤❛♣♣❡♥✐♥❣✳

❊①❡r❝✐s❡ ✾✳✻

❆ ❝✐t② ❧♦s❡s 3% ♦❢ ✐ts ♣♦♣✉❧❛t✐♦♥ ❡✈❡r② ②❡❛r✳ ❍♦✇ ❧♦♥❣ ✇✐❧❧ ✐t t❛❦❡ t♦ ❧♦s❡ 20%❄ ❊①❡r❝✐s❡ ✾✳✼

❆ ❝❛r st❛rt ♠♦✈✐♥❣ ❡❛st ❢r♦♠ t♦✇♥ ❆ ❛t ❛ ❝♦♥st❛♥t s♣❡❡❞ ♦❢ 60 ♠✐❧❡s ❛♥ ❤♦✉r✳ ❚♦✇♥ ❇ ✐s ❧♦❝❛t❡❞ 10 ♠✐❧❡s s♦✉t❤ ♦❢ ❆✳ ❘❡♣r❡s❡♥t t❤❡ ❞✐st❛♥❝❡ ❢r♦♠ t♦✇♥ ❇ t♦ t❤❡ ❝❛r ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✐♠❡✳

❊①❡r❝✐s❡ ✾✳✶✷

❚❤❡ ❣r❛♣❤s ♦❢ t❤❡ ✈❡❧♦❝✐t② ❛♥❞ t❤❡ ♣♦s✐t✐♦♥ ♦❢ ❛ ♠♦✈✐♥❣ ♦❜❥❡❝t ❛r❡ s❤♦✇♥ ❜❡❧♦✇✳ ❉❡s❝r✐❜❡ ✇❤❛t ✐s ❤❛♣♣❡♥✐♥❣✳

✾✳ ❊①❡r❝✐s❡s✿ ▼♦❞❡❧s

✺✸✶

♦❢ f ✳ x 0 1 2 3

✈❡❧♦❝✐t②

❛❝❝❡❧❡r❛t✐♦♥

0 + 0 −

+ − − −

❊①❡r❝✐s❡ ✾✳✶✾

❊①❡r❝✐s❡ ✾✳✶✸

❙✉♣♣♦s❡ t❤❡ ❛❧t✐t✉❞❡✱ ✐♥ ♠❡t❡rs✱ ♦❢ ❛♥ ♦❜❥❡❝t ✐s ❣✐✈❡♥ ❜② t❤❡ ❢✉♥❝t✐♦♥ t2 + t✱ ✇❤❡r❡ t ✐s t✐♠❡✱ ✐♥ s❡❝♦♥❞s✳ ❲❤❛t ✐s t❤❡ ✈❡❧♦❝✐t② ✇❤❡♥ t❤❡ ❛❧t✐t✉❞❡ ✐s 12 ♠❡t❡rs❄ ❊①❡r❝✐s❡ ✾✳✶✹

❚❤❡ ✈❡❧♦❝✐t② ♦❢ t❤❡ ♦❜❥❡❝t ❛t t✐♠❡ t ✐s ❣✐✈❡♥ ❜② v(t) = 1 + et . ■❢ ❛t t✐♠❡ t = 0 t❤❡ ♦❜❥❡❝t ✐s ❛t x = 2, ✇❤❡r❡ ✐s ✐t ❛t t✐♠❡ t = 1❄

❚❤❡ ♣♦♣✉❧❛t✐♦♥ ♦❢ ❜❡❡t❧❡s ✐♥ ❛ ❝❡rt❛✐♥ ❧♦❝❛t✐♦♥ ✐s ♣r♦❥❡❝t❡❞ t♦ ❣r♦✇ ❛t t❤❡ r❛t❡ 10, 000 + 2, 000x2 ♣❡r ♠♦♥t❤✱ ✇❤❡r❡ x ✐s t❤❡ ♥✉♠❜❡r ♦❢ ♠♦♥t❤s ♣❛ss❡❞ s✐♥❝❡ t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤✐s ②❡❛r✳ ❲❤❛t ✐s t❤❡ ♣r♦✲ ❥❡❝t❡❞ ♣♦♣✉❧❛t✐♦♥ ❛t t❤❡ ❡♥❞ ♦❢ ❉❡❝❡♠❜❡r ♥❡①t ②❡❛r ✐❢ t❤❡ ❝✉rr❡♥t ♣♦♣✉❧❛t✐♦♥ ✐s 1, 000, 000❄ ❊①❡r❝✐s❡ ✾✳✷✵

❚❤❡ ❤❡✐❣❤t ♦❢ t❤❡ ❜❛❧❧ ✭✐♥ ❢❡❡t✮ t s❡❝♦♥❞s ❛❢t❡r ✐t ✐s t❤r♦✇♥ ✐s ❣✐✈❡♥ ❜② f (t) = −16t2 + 8t + 6✳ ❊①♣❧❛✐♥ t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ ♥✉♠❜❡rs −16, 8, 6✳ ❊①❡r❝✐s❡ ✾✳✷✶

❚❤❡ ❣r❛♣❤ ♦❢ t❤❡ ✈❡❧♦❝✐t② ♦❢ ❛ ❝❛r ✐s ❣✐✈❡♥ ❜❡❧♦✇✳ P❧♦t t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ r❡♣r❡s❡♥t✐♥❣ t❤❡ ❞✐s✲ t❛♥❝❡ ♦❢ t❤❡ ❝❛r ❢r♦♠ t❤❡ st❛rt✐♥❣ ♣♦✐♥t✳

❊①❡r❝✐s❡ ✾✳✶✺

❚❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ♦❢ ❛♥ ♦❜❥❡❝t ❛t t✐♠❡ t ✐s ❣✐✈❡♥ ❜② a(t) = 3t. ■❢ ❛t t✐♠❡ t = 1 t❤❡ ✈❡❧♦❝✐t② ♦❢ ♦❜❥❡❝t ✐s ❛t v(1) = −1, ✇❤❛t ✐s ✐t ❛t t✐♠❡ t = 0❄ ❊①❡r❝✐s❡ ✾✳✶✻

❙✉♣♣♦s❡ s(t) r❡♣r❡s❡♥ts t❤❡ ♣♦s✐t✐♦♥ ♦❢ ❛ ♣❛rt✐❝❧❡ ❛t t✐♠❡ t ❛♥❞ v(t) ✐ts ✈❡❧♦❝✐t②✳ ■❢ v(t) = sin t − cos t ❛♥❞ t❤❡ ✐♥✐t✐❛❧ ♣♦s✐t✐♦♥ ✐s s(0) = 0, ✜♥❞ t❤❡ ♣♦s✐t✐♦♥ s(1). ❊①❡r❝✐s❡ ✾✳✷✷ ❊①❡r❝✐s❡ ✾✳✶✼

❙✉♣♣♦s❡ t❤❡ s♣❡❡❞ ♦❢ ❛ ❝❛r ✇❛s ❣r♦✇✐♥❣ ❝♦♥t✐♥✉✲ ♦✉s❧② ❢♦❧❧♦✇✐♥❣ t❤❡ r✉❧❡ 55 + 5t ♣❡r ❤♦✉r✱ ✇❤❡r❡ t ✐s t❤❡ ♥✉♠❜❡r ♦❢ ❤♦✉rs ♣❛ss❡❞ s✐♥❝❡ ✐t ✇❛s 250 ♠✐❧❡s ❛✇❛② ❢r♦♠ ❛ ❝✐t②✳ ❍♦✇ ❢❛r ✐s ✐t ❢r♦♠ t❤❡ ❝✐t② ❛❢t❡r 3 ❤♦✉rs ♦❢ ❞r✐✈✐♥❣ t♦✇❛r❞s ✐t❄

❊①❡r❝✐s❡ ✾✳✶✽

▲❡t x r❡♣r❡s❡♥t t❤❡ t✐♠❡ ♣❛ss❡❞ s✐♥❝❡ t❤❡ ❝❛r ❧❡❢t t❤❡ ❝✐t②✳ ❚❤❡ t❛❜❧❡ ❜❡❧♦✇ t❡❧❧s ❢♦r ✇❤❛t ✈❛❧✉❡s ♦❢ x t❤❡ ✈❡❧♦❝✐t② ❛♥❞ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ♦❢ t❤❡ ❝❛r ❛r❡ ♣♦s✐t✐✈❡✱ ♥❡❣❛t✐✈❡✱ ♦r ③❡r♦✳ ▲❡t f (x) r❡♣r❡s❡♥t t❤❡ ❞✐st❛♥❝❡ ♦❢ t❤❡ ❝❛r ❢r♦♠ t❤❡ ❝✐t②✳ ❙❦❡t❝❤ t❤❡ ❣r❛♣❤

❙✉♣♣♦s❡ t❤❡ s♣❡❡❞ ♦❢ ❛ ❝❛r ✇❛s ❝❤❛♥❣✐♥❣ ❝♦♥t✐♥✉✲ ♦✉s❧② ❢♦❧❧♦✇✐♥❣ t❤❡ r✉❧❡ 60 − t2 ♣❡r ❤♦✉r✱ ✇❤❡r❡ t ✐s t❤❡ ♥✉♠❜❡r ♦❢ ❤♦✉rs ♣❛ss❡❞ s✐♥❝❡ ♥♦♦♥✳ ❋✐♥❞ t❤❡ ❛✈❡r❛❣❡ s♣❡❡❞ ♦❢ t❤❡ ❝❛r ❜❡t✇❡❡♥ 1 ♣♠ ❛♥❞ 3 ♣♠✳ ❊①❡r❝✐s❡ ✾✳✷✸

❙✉♣♣♦s❡ t❤❡ ❛❧t✐t✉❞❡✱ ✐♥ ♠❡t❡rs✱ ♦❢ ❛♥ ♦❜❥❡❝t ✐s ❣✐✈❡♥ ❜② t❤❡ ❢✉♥❝t✐♦♥ y = t2 + t, t ≥ 0,

✇❤❡r❡ t ✐s t✐♠❡✱ ✐♥ s❡❝✳ ❲❤❛t ✐s t❤❡ ✈❡❧♦❝✐t② ✇❤❡♥ t❤❡ ❛❧t✐t✉❞❡ ✐s 12 ♠❡t❡rs❄

✾✳ ❊①❡r❝✐s❡s✿ ▼♦❞❡❧s

✺✸✷

❊①❡r❝✐s❡ ✾✳✷✹

❊①❡r❝✐s❡ ✾✳✷✺

10% ❡✈❡r② ②❡❛r✳ ❍♦✇ ❧♦♥❣ ✇✐❧❧ ✐t t❛❦❡ t♦ ❞r♦♣ t♦ 50% ♦❢ t❤❡ ❝✉rr❡♥t

❚❤❡ ❛r❡❛ ♦❢ ❛ ❝✐r❝❧❡ ✐s ✐♥❝r❡❛s✐♥❣ ❛t ❛ r❛t❡ ♦❢

♣♦♣✉❧❛t✐♦♥❄

♦❢ t❤❡ ❝✐r❝❧❡ ✐♥❝r❡❛s✐♥❣ ✇❤❡♥ t❤❡ ❛r❡❛ ✐s

❚❤❡ ♣♦♣✉❧❛t✐♦♥ ♦❢ ❛ ❝✐t② ❞❡❝❧✐♥❡s ❜②

5 sq✉❛r❡

❝❡♥t✐♠❡t❡rs ♣❡r s❡❝♦♥❞✳ ❆t ✇❤❛t r❛t❡ ✐s t❤❡ r❛❞✐✉s

2

❝♠❄

❊①❡r❝✐s❡ ✾✳✷✻ ❋✐♥❞ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ♦❢ ❛ ❢r❡❡ ❢❛❧❧✐♥❣ ♦❜ ❥❡❝t ❢r♦♠ t❤✐s ❞❛t❛✿

✶✵✳ ❊①❡r❝✐s❡s✿ ■♥❢♦r♠❛t✐♦♥ ❢r♦♠ t❤❡ ❞❡r✐✈❛t✐✈❡s

✺✸✸

✶✵✳ ❊①❡r❝✐s❡s✿ ■♥❢♦r♠❛t✐♦♥ ❢r♦♠ t❤❡ ❞❡r✐✈❛t✐✈❡s

❊①❡r❝✐s❡ ✶✵✳✶

❊①❡r❝✐s❡ ✶✵✳✾

✭❛✮ ❆♥❛❧②③❡ t❤❡ ❢✉♥❝t✐♦♥ ❞❡r✐✈❛t✐✈❡s✳

f

❣✐✈❡♥ ❜❡❧♦✇ ❛♥❞ ✐ts

❙❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥

✭❜✮ ❯s❡ ♣❛rt ✭❛✮ t♦ s❦❡t❝❤ t❤❡ ❣r❛♣❤

❏✉st✐❢② t❤❡ ❣r❛♣❤ ❜② st✉❞②✐♥❣ t❤❡

f✳ f (x) =

x2 + 7x + 3 . x2

√

xe−x ✳ ❞❡r✐✈❛t✐✈❡s ♦❢ f ✳ f (x) =

❊①❡r❝✐s❡ ✶✵✳✶✵

✭✶✮ ❙t❛t❡ ❘♦❧❧❡✬s ❚❤❡♦r❡♠ ❛♥❞ ✐❧❧✉str❛t❡ ✐t ✇✐t❤ ❛ s❦❡t❝❤✳ ✭❜✮ ◗✉♦t❡ ❛♥❞ st❛t❡ t❤❡ t❤❡♦r❡♠✭s✮ ♥❡❝❡s✲ ❊①❡r❝✐s❡ ✶✵✳✷

s❛r② t♦ ♣r♦✈❡ ✐t✳

f (x) =

❋✐♥❞ ❛❧❧ ❝r✐t✐❝❛❧ ♣♦✐♥ts ♦❢ t❤❡ ❢✉♥❝t✐♦♥

✭❝✮ ❲❤❛t t❤❡♦r❡♠ ❢♦❧❧♦✇s ❢r♦♠

✐t❄

3

2x − 6x + 7✳

❊①❡r❝✐s❡ ✶✵✳✶✶

❈♦♠♣✉t❡ t❤❡ ✜rst ❛♥❞ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡s ♦❢ t❤❡

❊①❡r❝✐s❡ ✶✵✳✸

❙✉♣♣♦s❡ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ❢✉♥❝t✐♦♥

ln x + ln x2 ✳

f

✐s

❢✉♥❝t✐♦♥

′

f (x) =

f (x) = x3 − 3x

❛♥❞ ✉s❡ t❤❡♠ t♦ s❦❡t❝❤

✐ts ❣r❛♣❤✳

✭❛✮ ❖♥ ✇❤❛t ✐♥t❡r✈❛❧s✱ ✐❢ ❛♥②✱ ✐s ❢ ✐♥✲

❝r❡❛s✐♥❣❄ ✭❜✮ ❖♥ ✇❤✐❝❤ ✐♥t❡r✈❛❧s✱ ✐❢ ❛♥②✱ ✐s

f

❝♦♥✲

❝❛✈❡ ❞♦✇♥❄ ❍✐♥t✿ s✐♠♣❧✐❢② ✜rst✳

❊①❡r❝✐s❡ ✶✵✳✶✷

✭❛✮ ❋✐♥✐s❤ t❤❡ st❛t❡♠❡♥t ✏■❢

(a, b)✱

h′ (x) = 0

x

✐♥

t❤❡♥✳✳✳✑✳ ✭❜✮ ❯s❡ t❤❡ t❤❡♦r❡♠ ✐♥ ♣❛rt ✭❛✮ t♦

❊①❡r❝✐s❡ ✶✵✳✹

♣r♦✈❡ t❤❛t ✐❢ t✇♦ ❢✉♥❝t✐♦♥s

❋✐♥❞ ❛❧❧ ❧♦❝❛❧ ♠❛①✐♠❛ ❛♥❞ ♠✐♥✐♠❛ ♦❢ t❤❡ ❢✉♥❝t✐♦♥

t✐✈❡s✱ t❤❡♥ t❤❡② ❞✐✛❡r ❜② ❛ ❝♦♥st❛♥t✳

f (x) = x3 − 3x − 1✳

❢♦r ❛❧❧

f, g

❤❛✈❡ ❡q✉❛❧ ❞❡r✐✈❛✲

❊①❡r❝✐s❡ ✶✵✳✶✸

❙❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❣✐✈❡♥ ❜❡❧♦✇✳ Pr♦✲

❊①❡r❝✐s❡ ✶✵✳✺

✈✐❞❡ ❥✉st✐✜❝❛t✐♦♥ ❢♦r ❡❛❝❤ ❢❡❛t✉r❡ ♦❢ t❤❡ ❣r❛♣❤✿ ✭❛✮ ❆♥❛❧②③❡ t❤❡ ✜rst ❛♥❞ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡s ♦❢ t❤❡

f (x) = x4 −2x2 ✳ ❣r❛♣❤ ♦❢ f ✳

❢✉♥❝t✐♦♥ ✐ts

✭❜✮ ❯s❡ ♣❛rt ✭❛✮ t♦ s❦❡t❝❤

f (x) =

x2 + 7x + 3 . x

❊①❡r❝✐s❡ ✶✵✳✶✹

❊①❡r❝✐s❡ ✶✵✳✻

❙✉♣♣♦s❡ t❤❡ ❢✉♥❝t✐♦♥s t❤❛t ❢♦❧❧♦✇ ❛r❡ ❞✐✛❡r❡♥t✐❛❜❧❡✳ ✭❛✮ ❋✐♥✐s❤ t❤❡ st❛t❡♠❡♥t ✏■❢

h′ (x) = 0

❢♦r ❛❧❧

(a, b)✱ t❤❡♥✳✳✳✑✳ ✭❜✮ ❋✐♥✐s❤ t❤❡ st❛t❡♠❡♥t f (x) = g ′ (x) ❢♦r ❛❧❧ x ✐♥ (a, b)✱ t❤❡♥✳✳✳✑✳

✐♥

x ✏■❢

✭❛✮ ❙t❛t❡ t❤❡ ▼❡❛♥ ❱❛❧✉❡ ❚❤❡♦r❡♠✳ ✭❜✮ ❱❡r✐❢② t❤❛t t❤❡ ❢✉♥❝t✐♦♥

f (x) =

x x+2

s❛t✐s✜❡s t❤❡ ❤②♣♦t❤❡s❡s

♦❢ t❤❡ t❤❡♦r❡♠ ♦♥ t❤❡ ✐♥t❡r✈❛❧

[1, 4]✳

′

❊①❡r❝✐s❡ ✶✵✳✶✺ ❊①❡r❝✐s❡ ✶✵✳✼

❙❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥

✭✶✮ ❙t❛t❡ t❤❡ ▼❡❛♥ ❱❛❧✉❡ ❚❤❡♦r❡♠ ❛♥❞ ✐❧❧✉str❛t❡

f (x) = x4 − x2 ✳

Pr♦✈✐❞❡ ❥✉st✐✜❝❛t✐♦♥ ❢♦r ❡❛❝❤ ❢❡❛t✉r❡ ♦❢ t❤❡ ❣r❛♣❤✳

✐t ✇✐t❤ ❛ s❦❡t❝❤✳ ✭❜✮ ❯s❡ t❤❡ t❤❡♦r❡♠ t♦ ♣r♦✈❡ t❤❛t ✐❢ t✇♦ ❢✉♥❝t✐♦♥s ❤❛✈❡ ❡q✉❛❧ ❞❡r✐✈❛t✐✈❡s✱ t❤❡♥ t❤❡② ❞✐✛❡r ❜② ❛ ❝♦♥st❛♥t✳

❊①❡r❝✐s❡ ✶✵✳✶✻

❋✐♥❞ ❣❧♦❜❛❧ ♠❛①✐♠❛ ❛♥❞ ♠✐♥✐♠❛ ♦❢ t❤❡ ❢✉♥❝t✐♦♥✱

f (x) = x3 − 3x

♦♥ t❤❡ ✐♥t❡r✈❛❧

[−2, 10]✳

❊①❡r❝✐s❡ ✶✵✳✽

❯s❡ ◆❡✇t♦♥✬s ♠❡t❤♦❞ ❢♦r

x1 = −1

t♦ ✜♥❞

②♦✉✬✈❡ ❢♦✉♥❞❄

x3 ✳

f (x) = x5 + 2

✇✐t❤

❲❤❛t ✐s t❤❡ ♠❡❛♥✐♥❣ ♦❢ ✇❤❛t

❊①❡r❝✐s❡ ✶✵✳✶✼

✭❛✮ ❙t❛t❡ t❤❡ ▼❡❛♥ ❱❛❧✉❡ ❚❤❡♦r❡♠✳ ❡①❛♠♣❧❡ ♦❢ ✐ts ❛♣♣❧✐❝❛t✐♦♥✳

✭❜✮ ●✐✈❡ ❛♥

✶✵✳ ❊①❡r❝✐s❡s✿ ■♥❢♦r♠❛t✐♦♥ ❢r♦♠ t❤❡ ❞❡r✐✈❛t✐✈❡s

❊①❡r❝✐s❡ ✶✵✳✶✽

❋✐♥❞ t❤❡ ❧♦❝❛❧ ♠❛①✐♠✉♠ ❛♥❞ ♠✐♥✐♠✉♠ ♣♦✐♥ts ♦❢ t❤❡ ❢✉♥❝t✐♦♥ f (x) = x3 − 3x✳ ❊①❡r❝✐s❡ ✶✵✳✶✾

■❢ ❡✈❡r② ♣♦✐♥t ♦♥ t❤❡ ❣r❛♣❤ ♦❢ y = f (x) ✐s ❛ ❝r✐t✐❝❛❧ ♣♦✐♥t✱ ✇❤❛t ❞♦❡s t❤❡ ❣r❛♣❤ ❧♦♦❦ ❧✐❦❡❄ ❊①❡r❝✐s❡ ✶✵✳✷✵

❚❤❡ ❣r❛♣❤ ♦❢ ❢✉♥❝t✐♦♥ f ✐s ❣✐✈❡♥ ❜❡❧♦✇✳ ✭❛✮ ❆t ✇❤❛t ♣♦✐♥ts ✐s f ❝♦♥t✐♥✉♦✉s❄ ✭❜✮ ❆t ✇❤❛t ♣♦✐♥ts ❞♦❡s t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ f ❡①✐st❄

❊①❡r❝✐s❡ ✶✵✳✷✶

✭❛✮ ❈♦♠♣✉t❡ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ f (x) = 3x2 − x ❛t a = 1 ❢r♦♠ t❤❡ ❞❡✜♥✐t✐♦♥ ✭✐✳❡✳✱ ❛s ❛ ❧✐♠✐t✮✳ ✭❜✮ ❋✐♥❞ t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ❧✐♥❡ t❛♥❣❡♥t t♦ t❤❡ ❣r❛♣❤ ♦❢ y = f (x) ❛t t❤❡ ♣♦✐♥t ❝♦rr❡s♣♦♥❞✐♥❣ t♦ a = 1✳ ❊①❡r❝✐s❡ ✶✵✳✷✷

■♥❞✐❝❛t❡ ✇❤✐❝❤ t❤❡ ❢♦❧❧♦✇✐♥❣ st❛t❡♠❡♥ts ❜❡❧♦✇ ✐s tr✉❡ ♦r ❢❛❧s❡ ✭♥♦ ♣r♦♦❢ ♥❡❝❡ss❛r②✮✿ ✶✳ ■❢ t❤❡ ❢✉♥❝t✐♦♥ f ✐s ✐♥❝r❡❛s✐♥❣✱ t❤❡♥ s♦ ✐s f −1 . ✷✳ ❚❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥ ❤❛s ❛♥ ❛s②♠♣t♦t❡✳ ✸✳ ■❢ f ′ (c) = 0✱ t❤❡♥ c ✐s ❛ ❧♦❝❛❧ ♠❛①✐♠✉♠ ♦r ❛ ❧♦❝❛❧ ♠✐♥✐♠✉♠ ♦❢ f ✳ ✹✳ ■❢ ❛ ❢✉♥❝t✐♦♥ ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ t❤❡♥ ✐t ✐s ❝♦♥✲ t✐♥✉♦✉s✳ ✺✳ ■❢ t✇♦ ❢✉♥❝t✐♦♥s ❛r❡ ❡q✉❛❧✱ t❤❡✐r ❞❡r✐✈❛t✐✈❡s ❛r❡ ❛❧s♦ ❡q✉❛❧✳ ✻✳ ■❢ t✇♦ ❢✉♥❝t✐♦♥s ❛r❡ ❡q✉❛❧✱ t❤❡✐r ❛♥t✐✲ ❞❡r✐✈❛t✐✈❡s ❛r❡ ❛❧s♦ ❡q✉❛❧✳

✺✸✹

✶✶✳ ❊①❡r❝✐s❡s✿ ❈♦♠♣✉t✐♥❣ ❞❡r✐✈❛t✐✈❡s

✺✸✺

✶✶✳ ❊①❡r❝✐s❡s✿ ❈♦♠♣✉t✐♥❣ ❞❡r✐✈❛t✐✈❡s

❊①❡r❝✐s❡ ✶✶✳✶

❊①❡r❝✐s❡ ✶✶✳✶✶

❈❛❧❝✉❧❛t❡ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ f (x) =

❊✈❛❧✉❛t❡

x2 . x2 − 1

dy ❢♦r y = sin e2x . dx

❊①❡r❝✐s❡ ✶✶✳✶✷

❊✈❛❧✉❛t❡ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ♦❢ f (x) = xesin x .

❊①❡r❝✐s❡ ✶✶✳✷

p

❘❡♣r❡s❡♥t t❤✐s ❢✉♥❝t✐♦♥ h(x) = x2 − 1 ❛s t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ t✇♦ ❢✉♥❝t✐♦♥s✳ ❋✐♥❞ ✐ts ❞❡r✐✈❛t✐✈❡✳ ❊①❡r❝✐s❡ ✶✶✳✸

❈❛❧❝✉❧❛t❡ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ f (x) = xπ + π x + x + π ✐♥❞✐❝❛t✐♥❣ t❤❡ r✉❧❡s ②♦✉ ✉s❡✳ ❊①❡r❝✐s❡ ✶✶✳✹

❈❛❧❝✉❧❛t❡ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ f (x) = logπ x + logx π ✐♥❞✐❝❛t✐♥❣ t❤❡ r✉❧❡s ②♦✉ ✉s❡✳

❊①❡r❝✐s❡ ✶✶✳✶✸

❙✉♣♣♦s❡ f ′ (1) = 2✱ g ′ (2) = 3✱ ❛♥❞ h′ (1) = 6✱ ✇❤❡r❡ h = g ◦ f ✳ ❲❤❛t ✐s f (1)❄ ❊①❡r❝✐s❡ ✶✶✳✶✹

■s ✐t ♣♦ss✐❜❧❡ t❤❛t ❜♦t❤ F (x) ❛♥❞ F (2x) ❛r❡ ❜♦t❤ ❛♥t✐❞❡r✐✈❛t✐✈❡s ♦❢ s♦♠❡ ❢✉♥❝t✐♦♥ f ❄ ❊①❡r❝✐s❡ ✶✶✳✶✺

■s sin x + 3x ❛♥ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢ cos x2 ❄

❊①❡r❝✐s❡ ✶✶✳✺

❙✉♣♣♦s❡ f (1) = 3 ❛♥❞ f ′ (1) = 2✳ ❯s❡ t❤✐s ✐♥❢♦r♠❛✲ t✐♦♥ t♦ ✜❧❧ ✐♥ t❤❡ ❜❧❛♥❦s✿
′ f −1 ( ) =

.

■s ✐t ♣♦ss✐❜❧❡ t❤❛t ❜♦t❤ F (x) ❛♥❞ F (2x) ❛r❡ ❜♦t❤ ❛♥t✐❞❡r✐✈❛t✐✈❡s ♦❢ s♦♠❡ ♥♦♥✲③❡r♦ ❢✉♥❝t✐♦♥ f ❄ ❊①❡r❝✐s❡ ✶✶✳✶✼

❊①❡r❝✐s❡ ✶✶✳✻

❊✈❛❧✉❛t❡ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ f (x) = x2 ex ✳

❉✐✛❡r❡♥t✐❛t❡ t❤✐s✿

❊①❡r❝✐s❡ ✶✶✳✶✽

g(t) = t cos t sin t .

❋✐♥❞ t❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡ ♦❢ h(x) = x2 + x + 1. ❲❤❛t ❞♦❡s ✐t t❡❧❧ ②♦✉ ❛❜♦✉t t❤❡ s❤❛♣❡ ♦❢ t❤❡ ❣r❛♣❤ ♦❢ f ?

❊①❡r❝✐s❡ ✶✶✳✼

❉✐✛❡r❡♥t✐❛t❡✿

❊①❡r❝✐s❡ ✶✶✳✶✻

ln(sin x) . x

❊①❡r❝✐s❡ ✶✶✳✶✾

❋✐♥❞ t❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡ ♦❢ h(x) = 2xπ ✳

❊①❡r❝✐s❡ ✶✶✳✽

❈♦♠♣✉t❡ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ f (x) = ex

2 +3x

✳

❊①❡r❝✐s❡ ✶✶✳✷✵

❈♦♠♣✉t❡ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ f (x) = ln(3x + 2)✳ ❊①❡r❝✐s❡ ✶✶✳✾

❊✈❛❧✉❛t❡


d sin x · ex+1 ✳ dx

❊①❡r❝✐s❡ ✶✶✳✷✶

❋✐♥❞ t❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡ ♦❢ h(x) = xex ✳

❊①❡r❝✐s❡ ✶✶✳✶✵


d cos t + et ✳ ❊✈❛❧✉❛t❡ dx

❛❜❧❡s✳

❊①❡r❝✐s❡ ✶✶✳✷✷

❍✐♥t✿ ✇❛t❝❤ t❤❡ ✈❛r✐✲

❋✐♥❞ t❤❡ ❞❡r✐✈❛t✐✈❡s ♦❢ t❤❡ ❢✉♥❝t✐♦♥s✿ ✭❛✮ 3xe + eπ ✱ ✭❜✮ 7 ln x + (1/x) − ln 2.

✶✶✳ ❊①❡r❝✐s❡s✿ ❈♦♠♣✉t✐♥❣ ❞❡r✐✈❛t✐✈❡s

❊①❡r❝✐s❡ ✶✶✳✷✸

❉✐✛❡r❡♥t✐❛t❡ g(t) =

√

x cos x .

❊①❡r❝✐s❡ ✶✶✳✷✹

❊✈❛❧✉❛t❡

dy ❢♦r dx y=

√

ex .

❊①❡r❝✐s❡ ✶✶✳✷✺

❋✐♥❞ t❤❡ s❧♦♣❡s ♦❢ t❤❡ t❛♥❣❡♥t ❧✐♥❡s t♦ t❤❡ ❡❧❧✐♣s❡ x2 + 2y 2 = 1 ❛t t❤❡ ♣♦✐♥ts ✇❤❡r❡ ✐t ❝r♦ss❡s t❤❡ ❞✐❛❣♦♥❛❧ ❧✐♥❡ y = x✳ ❊①❡r❝✐s❡ ✶✶✳✷✻

❯s❡ ✐♠♣❧✐❝✐t ❞✐✛❡r❡♥t✐❛t✐♦♥ t♦ ✜♥❞ ❛♥ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ❧✐♥❡ t❛♥❣❡♥t t♦ t❤❡ ❝✉r✈❡ x1/2 + xy = 2 ♣❛ss✐♥❣ t❤r♦✉❣❤ t❤❡ ♣♦✐♥t (1, 1)✳ ❊①❡r❝✐s❡ ✶✶✳✷✼

❊✈❛❧✉❛t❡

dy ❢♦r y = sin cos(−x)✳ dx

❊①❡r❝✐s❡ ✶✶✳✷✽

❋✐♥❞ ❛♥ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ❧✐♥❡ t❛♥❣❡♥t t♦ t❤❡ ❝✉r✈❡ xy = 1 ♣❛ss✐♥❣ t❤r♦✉❣❤ t❤❡ ♣♦✐♥t (1, 1). ❊①❡r❝✐s❡ ✶✶✳✷✾

❙✉♣♣♦s❡ x sin y + y 2 = x✳ ❋✐♥❞

dy ✳ dx

✺✸✻

✶✷✳ ❊①❡r❝✐s❡s✿ ❖♣t✐♠✐③❛t✐♦♥ ❛♥❞ ♦t❤❡r ❛♣♣❧✐❝❛t✐♦♥s

✺✸✼

✶✷✳ ❊①❡r❝✐s❡s✿ ❖♣t✐♠✐③❛t✐♦♥ ❛♥❞ ♦t❤❡r ❛♣♣❧✐❝❛t✐♦♥s

❊①❡r❝✐s❡ ✶✷✳✶

10

❊①❡r❝✐s❡ ✶✷✳✶✵

❢t✴s❡❝✱ ❤♦✇ ❢❛st ✐s t❤❡ t♦♣ ♦❢ t❤❡ ❧❛❞✲

▲❡t A = f (r) ❜❡ t❤❡ ❛r❡❛ ♦❢ ❛ ❝✐r❝❧❡ ✇✐t❤ r❛❞✐✉s r ❛♥❞ r = h(t) ❜❡ t❤❡ r❛❞✐✉s ♦❢ t❤❡ ❝✐r❝❧❡ ❛t t✐♠❡ t✳ ❲❤✐❝❤ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ st❛t❡♠❡♥ts ❝♦rr❡❝t❧② ♣r♦✲

❞❡r s❧✐❞✐♥❣ ❞♦✇♥ t❤❡ ✇❛❧❧ ✇❤❡♥ t❤❡ ❜♦tt♦♠ ♦❢ t❤❡

✈✐❞❡s ❛ ♣r❛❝t✐❝❛❧ ✐♥t❡r♣r❡t❛t✐♦♥ ♦❢ t❤❡ ❝♦♠♣♦s✐t✐♦♥

❆ ❧❛❞❞❡r

❢t ❧♦♥❣ r❡sts ❛❣❛✐♥st ❛ ✈❡rt✐❝❛❧ ✇❛❧❧✳ ■❢

t❤❡ ❜♦tt♦♠ ♦❢ t❤❡ ❧❛❞❞❡r s❧✐❞❡s ❛✇❛② ❢r♦♠ t❤❡ ✇❛❧❧ ❛t t❤❡ r❛t❡ ❧❛❞❞❡r ✐s

6

1

f (h(t))❄

❢t ❢r♦♠ t❤❡ ✇❛❧❧❄

✶✳ ❚❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ r❛❞✐✉s ❛t t✐♠❡

❊①❡r❝✐s❡ ✶✷✳✷

❚❤❡ ❛r❡❛ ♦❢ ❛ ❝✐r❝❧❡ ✐s ✐♥❝r❡❛s✐♥❣ ❛t ❛ r❛t❡ ♦❢

2

❝♠ ✴s❡❝✳

5

❆t ✇❤❛t r❛t❡ ✐s t❤❡ r❛❞✐✉s ♦❢ t❤❡ ❝✐r❝❧❡

✐♥❝r❡❛s✐♥❣ ✇❤❡♥ t❤❡ ❛r❡❛ ✐s

2

❝♠❄

t❤❡ ❧✐♥❡ t❛♥❣❡♥t t♦ t❤❡ ❝✉r✈❡ t❤r♦✉❣❤ t❤❡ ♣♦✐♥t

x2 + y 2 = x

t✳

♣❛ss✐♥❣

r❛❞✐✉s

h(t)✳

✺✳ ❚❤❡ t✐♠❡

t

✇❤❡♥ t❤❡ ❛r❡❛ ✇✐❧❧ ❜❡

A = f (r)✳

✻✳ ❚❤❡ t✐♠❡

t

✇❤❡♥ t❤❡ r❛❞✐✉s ✇✐❧❧ ❜❡

(0, 0)✳

❊①❡r❝✐s❡ ✶✷✳✹

r = h(t)✳

Pr♦✈✐❞❡ ❢♦r♠✉❧❛s ❢♦r ✐t❡♠s ✶✲✻✳

❚✇♦ ❝❛rs st❛rt ❢r♦♠ t❤❡ s❛♠❡ ♣♦✐♥t✳ ♠✐✴❤✳

❛t t✐♠❡

✹✳ ❚❤❡ ❛r❡❛ ♦❢ t❤❡ ❝✐r❝❧❡ ✇❤✐❝❤ ❛t t✐♠❡ t ❤❛s

❯s❡ ✐♠♣❧✐❝✐t ❞✐✛❡r❡♥t✐❛t✐♦♥ t♦ ✜♥❞ ❛♥ ❡q✉❛t✐♦♥ ♦❢

60

t✳

✸✳ ❚❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ r❛❞✐✉s ♦❢ ❛ ❝✐r❝❧❡ ✇✐t❤ ❛r❡❛

A = f (r)

❊①❡r❝✐s❡ ✶✷✳✸

♥♦rt❤ ❛t

✷✳ ❚❤❡ ❛r❡❛ ♦❢ t❤❡ ❝✐r❝❧❡ ❛t t✐♠❡

t✳

❖♥❡ tr❛✈❡❧s

♠✐✴❤ ❛♥❞ t❤❡ ♦t❤❡r tr❛✈❡❧s ❡❛st ❛t

25

❍♦✇ ❢❛st ✐s t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡♠ ✐♥✲

❝r❡❛s✐♥❣ t✇♦ ❤♦✉rs ❧❛t❡r❄

❊①❡r❝✐s❡ ✶✷✳✶✶

❚❤❡ ❛r❡❛ ♦❢ ❛ r❡❝t❛♥❣❧❡ ✐s

100

sq✳ ❢❡❡t✳ ✭❛✮ ❊①♣r❡ss

t❤❡ ♣❡r✐♠❡t❡r ♦❢ t❤❡ r❡❝t❛♥❣❧❡ ✐♥ t❡r♠s ♦❢ ✐ts ✇✐❞t❤✳ ✭❜✮ ❋✐♥❞ t❤❡ ♠✐♥✐♠❛❧ ♣♦ss✐❜❧❡ ♣❡r✐♠❡t❡r✳ ✭❝✮ ❋✐♥❞

❊①❡r❝✐s❡ ✶✷✳✺

❚❤❡ ✈♦❧✉♠❡ ♦❢ ❛ ❝✉❜❡ ✐s ✐♥❝r❡❛s✐♥❣ ❛t ❛ r❛t❡ ♦❢

24

t❤❡ ♠❛①✐♠❛❧ ♣♦ss✐❜❧❡ ♣❡r✐♠❡t❡r✳

3

❝♠ ✴♠✐♥✳ ❍♦✇ ❢❛st ✐s t❤❡ ❡❞❣❡ ♦❢ t❤❡ ❝✉❜❡ ✐♥❝r❡❛s✲ ✐♥❣ ✇❤❡♥ t❤❡ ✈♦❧✉♠❡ ✐s

8

3

❝♠

?

❊①❡r❝✐s❡ ✶✷✳✶✷

❘❡st❛t❡ ✭❜✉t ❞♦ ♥♦t s♦❧✈❡✮ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦❜❧❡♠ ❛❧✲

❊①❡r❝✐s❡ ✶✷✳✻

❙✉♣♣♦s❡

2 3

xy + x y = 1✳

❣❡❜r❛✐❝❛❧❧②✿ ✏❲❤❛t ❛r❡ t❤❡ ❞✐♠❡♥s✐♦♥s ♦❢ t❤❡ r❡❝t✲

dy ✳ dx

❋✐♥❞

❛♥❣❧❡ ✇✐t❤ t❤❡ s♠❛❧❧❡st ♣♦ss✐❜❧❡ ♣❡r✐♠❡t❡r ❛♥❞ ❛r❡❛ ✜①❡❞ ❛t

100❄✑

❊①❡r❝✐s❡ ✶✷✳✼

❋✐♥❞ ❛♥ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ❧✐♥❡ t❛♥❣❡♥t t♦ t❤❡ ❝✉r✈❡

x sin y = x

❛t t❤❡ ♣♦✐♥t

(1, π/2)✳

❊①❡r❝✐s❡ ✶✷✳✶✸

❋✐♥❞ t❤❡ ♣♦✐♥t ♦♥ t❤❡ ♣❛r❛❜♦❧❛ ❝❧♦s❡st t♦ t❤❡ ♣♦✐♥t

y 2 = 2x

t❤❛t ✐s

(1, 4)✳

❊①❡r❝✐s❡ ✶✷✳✽

❯s❡ ✐♠♣❧✐❝✐t ❞✐✛❡r❡♥t✐❛t✐♦♥ t♦ ✜♥❞ ❛♥ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ❧✐♥❡ t❛♥❣❡♥t t♦ t❤❡ ❝✉r✈❡ t❤r♦✉❣❤ t❤❡ ♣♦✐♥t

3x + 2y = 7

♣❛ss✐♥❣

(1, 2)✳

❊①❡r❝✐s❡ ✶✷✳✶✹

❋✐♥❞ t❤❡ ♣♦✐♥t ♦♥ t❤❡ ❝✐r❝❧❡ t❤❛t ✐s ❝❧♦s❡st t♦ t❤❡ ♦r✐❣✐♥✳

(x − 1)2 + (y − 2)2 = 3

❊①❡r❝✐s❡ ✶✷✳✾

❚❤❡ ♣❡r✐♠❡t❡r ♦❢ ❛ r❡❝t❛♥❣❧❡ ✐s

10 ❢❡❡t✳

✭❛✮ ❊①♣r❡ss

t❤❡ ❛r❡❛ ♦❢ t❤❡ r❡❝t❛♥❣❧❡ ✐♥ t❡r♠s ♦❢ ✐ts ✇✐❞t❤✳ ✭❜✮

❊①❡r❝✐s❡ ✶✷✳✶✺

❋✐♥❞ t❤❡ ♠✐♥✐♠❛❧ ♣♦ss✐❜❧❡ ❛r❡❛✳ ✭❝✮ ❋✐♥❞ t❤❡ ♠❛①✲

❋✐♥❞ t❤❡ ❞✐♠❡♥s✐♦♥s ♦❢ t❤❡ r❡❝t❛♥❣❧❡ ♦❢ ❧❛r❣❡st ❛r❡❛

✐♠❛❧ ♣♦ss✐❜❧❡ ❛r❡❛✳

t❤❛t ❝❛♥ ❜❡ ✐♥s❝r✐❜❡❞ ✐♥ ❛ ❝✐r❝❧❡ ♦❢ r❛❞✐✉s

1✳

✶✷✳ ❊①❡r❝✐s❡s✿ ❖♣t✐♠✐③❛t✐♦♥ ❛♥❞ ♦t❤❡r ❛♣♣❧✐❝❛t✐♦♥s

❊①❡r❝✐s❡ ✶✷✳✶✻

✺✸✽

❊①❡r❝✐s❡ ✶✷✳✷✻

❋✐♥❞ t❤❡ ♣♦✐♥t ♦♥ t❤❡ ♣❛r❛❜♦❧❛

y 2 = 2x

t❤❛t ✐s

✈❛❧✉❡s ♦❢ t❤❡ ❢✉♥❝t✐♦♥

(2, 2)✳

❝❧♦s❡st t♦ t❤❡ ♣♦✐♥t

❋✐♥❞ t❤❡ ❣❧♦❜❛❧ ♠❛①✐♠✉♠ ❛♥❞ ♠✐♥✐♠✉♠ ♣♦✐♥ts ❛♥❞ ✐♥t❡r✈❛❧

❊①❡r❝✐s❡ ✶✷✳✶✼

❆ ♣✐❡❝❡ ♦❢ ✇✐r❡

10

[−2, 5].

f (x) = 2x3 − 6x + 5

♦♥ t❤❡

❊①❡r❝✐s❡ ✶✷✳✷✼

♠ ❧♦♥❣ ✐s ❝✉t ✐♥t♦

2

♣✐❡❝❡s✳ ❖♥❡

♣✐❡❝❡ ✐s ❜❡♥t ✐♥t♦ ❛ sq✉❛r❡ ❛♥❞ t❤❡ ♦t❤❡r ✐s ❜❡♥t ✐♥t♦ ❛♥ ❡q✉✐❧❛t❡r❛❧ tr✐❛♥❣❧❡✳ ❍♦✇ s❤♦✉❧❞ t❤❡ ✇✐r❡ ❜❡ ❝✉t s♦ t❤❛t t❤❡ t♦t❛❧ ❛r❡❛ ❡♥❝❧♦s❡❞ ✐s ✭❛✮ ❛ ♠❛①✐♠✉♠❄ ✭❜✮ ❛ ♠✐♥✐♠✉♠❄

❆ ❢❛r♠❡r ✇✐t❤

750

❢t ♦❢ ❢❡♥❝✐♥❣ ✇❛♥ts t♦ ❡♥❝❧♦s❡ ❛

r❡❝t❛♥❣✉❧❛r ❛r❡❛ ❛♥❞ t❤❡♥ ❞✐✈✐❞❡ ✐t ✐♥t♦ ❢♦✉r ♣❡♥s ✇✐t❤ ❢❡♥❝✐♥❣ ♣❛r❛❧❧❡❧ t♦ ♦♥❡ s✐❞❡ ♦❢ t❤❡ r❡❝t❛♥❣❧❡✳ ❲❤❛t ✐s t❤❡ ❧❛r❣❡st ♣♦ss✐❜❧❡ t♦t❛❧ ❛r❡❛ ♦❢ t❤❡ ❢♦✉r ♣❡♥s❄

❊①❡r❝✐s❡ ✶✷✳✶✽

❊①❡r❝✐s❡ ✶✷✳✷✽

❋✐♥❞ t❤❡ ♣♦✐♥t ♦♥ t❤❡ ❧✐♥❡ t♦ t❤❡ ♦r✐❣✐♥✳

y = 1 − 2x t❤❛t ✐s ❝❧♦s❡st

❋✐♥❞ t✇♦ ♣♦s✐t✐✈❡ ✐♥t❡❣❡rs s✉❝❤ t❤❛t t❤❡ s✉♠ ♦❢ t❤❡ ✜rst ♥✉♠❜❡r ❛♥❞ ❢♦✉r t✐♠❡s t❤❡ s❡❝♦♥❞ ♥✉♠❜❡r ✐s

1000

❛♥❞ t❤❡ ♣r♦❞✉❝t ❛s ❧❛r❣❡ ❛s ♣♦ss✐❜❧❡✳

❊①❡r❝✐s❡ ✶✷✳✶✾

❋✐♥❞ t❤❡ ♣♦✐♥t ♦♥ t❤❡ ❧✐♥❡ t♦ t❤❡ ♦r✐❣✐♥✳

y = −2x

t❤❛t ✐s ❝❧♦s❡st

❊①❡r❝✐s❡ ✶✷✳✷✾

❋✐♥❞ t✇♦ ♣♦s✐t✐✈❡ ♥✉♠❜❡rs

100 ❊①❡r❝✐s❡ ✶✷✳✷✵

❆ ❢❛r♠❡r ❤❛s

x, y

✇❤♦s❡ ♣r♦❞✉❝t ✐s

❛♥❞ ✇❤♦s❡ s✉♠ ✐s ❛ ♠✐♥✐♠✉♠✳

❊①❡r❝✐s❡ ✶✷✳✸✵

100 ②❛r❞s ♦❢ ❢❡♥❝✐♥❣✳

❲❤❛t ❛r❡ t❤❡ ❞✐✲

♠❡♥s✐♦♥s ♦❢ ❛♥ ❡♥❝❧♦s✉r❡ t❤❛t ❤❛s t❤❡ ❧❛r❣❡st ❛r❡❛❄

❙❡t ✉♣ ❜✉t ❞♦ ♥♦t s♦❧✈❡ t❤❡ ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠ ❢♦r t❤❡ ❢♦❧❧♦✇✐♥❣ s✐t✉❛t✐♦♥✿ ✏■❢

1200 ❝♠2

♦❢ ♠❛t❡r✐❛❧

✐s ❛✈❛✐❧❛❜❧❡ t♦ ♠❛❦❡ ❛ ❜♦① ✇✐t❤ ❛ sq✉❛r❡ ❜❛s❡ ❛♥❞ ❛♥ ♦♣❡♥ t♦♣✱ ✜♥❞ t❤❡ ❧❛r❣❡st ♣♦ss✐❜❧❡ ✈♦❧✉♠❡ ♦❢ t❤❡

❊①❡r❝✐s❡ ✶✷✳✷✶

■t ✐s ❦♥♦✇♥ t❤❛t ❛ ❢❛r♠❡r ✇✐t❤ s❤♦✉❧❞ ❜✉✐❧❞ ❛

25✲❜②✲25

100 ②❛r❞s ♦❢ ❢❡♥❝✐♥❣

❜♦①✳✑

②❛r❞ ❡♥❝❧♦s✉r❡ ✐♥ ♦r❞❡r t♦

❤❛✈❡ t❤❡ ❧❛r❣❡st ❛r❡❛✳ ❲❤❛t ✐❢ ❤❡ ❤❛s

200

②❛r❞s❄

❊①❡r❝✐s❡ ✶✷✳✸✶

❙❡t ✉♣ ❜✉t ❞♦ ♥♦t s♦❧✈❡ t❤❡ ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠ ❢♦r t❤❡ ❢♦❧❧♦✇✐♥❣ s✐t✉❛t✐♦♥✿ ✏❆ ♣♦st❡r ✐s t♦ ❤❛✈❡ ❛♥

❊①❡r❝✐s❡ ✶✷✳✷✷

❆ ❢❛r♠❡r ❤❛s

100

❛r❡❛ ♦❢ ②❛r❞s ♦❢ ❢❡♥❝✐♥❣✳ ❲❤❛t ❛r❡ t❤❡

❞✐♠❡♥s✐♦♥s ♦❢ ❛♥ ❡♥❝❧♦s✉r❡ t❤❛t ❤❛s t❤❡ ❧❛r❣❡st

180

✐♥

2

✇✐t❤

❛♥❞ t❤❡ s✐❞❡s ❛♥❞ ❛

1✲✐♥❝❤

♠❛r❣✐♥s ❛t t❤❡ ❜♦tt♦♠

2✲✐♥❝❤ ♠❛r❣✐♥ ❛t t❤❡ t♦♣✳

❲❤❛t

❞✐♠❡♥s✐♦♥s ✇✐❧❧ ❣✐✈❡ t❤❡ ❧❛r❣❡st ♣r✐♥t❡❞ ❛r❡❛❄✑

♣❡r✐♠❡t❡r❄ ❊①❡r❝✐s❡ ✶✷✳✸✷ ❊①❡r❝✐s❡ ✶✷✳✷✸

❋✐♥❞ t✇♦ ♥✉♠❜❡rs

❋✐♥❞ ❛ ♣♦s✐t✐✈❡ ♥✉♠❜❡r s✉❝❤ t❤❛t t❤❡ s✉♠ ♦❢ t❤❡

x, y

✇❤♦s❡ s✉♠ ✐s

2

❛♥❞ ✇❤♦s❡

♥✉♠❜❡r ❛♥❞ ✐ts r❡❝✐♣r♦❝❛❧ ✐s ❛s s♠❛❧❧ ❛s ♣♦ss✐❜❧❡✳

♣r♦❞✉❝t ✐s ❛ ♠❛①✐♠✉♠✳ ❊①❡r❝✐s❡ ✶✷✳✸✸

❙❡t ✉♣ ❜✉t ❞♦ ♥♦t s♦❧✈❡ t❤❡ ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠

❊①❡r❝✐s❡ ✶✷✳✷✹

❙❡t ✉♣ ❜✉t ❞♦ ♥♦t s♦❧✈❡ t❤❡ ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠ ❢♦r t❤❡ ❢♦❧❧♦✇✐♥❣ s✐t✉❛t✐♦♥✿ ✏❆♠♦♥❣ ❛❧❧ r❡❝t❛♥❣❧❡s ✐♥s❝r✐❜❡❞ ✐♥ ❛ ❝✐r❝❧❡ ♦❢ r❛❞✐✉s ✶✱ ✜♥❞ t❤❡ ♦♥❡ ✇✐t❤

❢♦r t❤❡ ❢♦❧❧♦✇✐♥❣ s✐t✉❛t✐♦♥✿ ✏❆♠♦♥❣ ❛❧❧ r✐❣❤t tr✐❛♥✲ ❣❧❡s ✇✐t❤ ❛r❡❛

10✱

✜♥❞ t❤❡ ♦♥❡ ✇✐t❤ t❤❡ s♠❛❧❧❡st

♣❡r✐♠❡t❡r✑✳

t❤❡ ❧❛r❣❡st ❛r❡❛✑✳ ❊①❡r❝✐s❡ ✶✷✳✸✹

■❢ ❛♥ ♦♣❡♥ ❜♦① ✐s t♦ ❜❡ ♠❛❞❡ ❢r♦♠ ❛ t✐♥ s❤❡❡t

❊①❡r❝✐s❡ ✶✷✳✷✺

8

✐♥✳

sq✉❛r❡ ❜② ❝✉tt✐♥❣ ♦✉t ✐❞❡♥t✐❝❛❧ sq✉❛r❡s ❢r♦♠ ❡❛❝❤

❙❡t ✉♣ ❛♥❞ s♦❧✈❡ t❤❡ ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠ ❢♦r t❤❡

❝♦r♥❡r ❛♥❞ ❜❡♥❞✐♥❣ ✉♣ t❤❡ r❡s✉❧t✐♥❣ ✢❛♣s✱ ❞❡t❡r✲

❢♦❧❧♦✇✐♥❣ s✐t✉❛t✐♦♥✿

♠✐♥❡ t❤❡ ❞✐♠❡♥s✐♦♥s ♦❢ t❤❡ ❧❛r❣❡st ❜♦① t❤❛t ❝❛♥ ❜❡

y=π

✏❋✐♥❞ t❤❡ ♣♦✐♥t ♦♥ t❤❡ ❧✐♥❡

t❤❛t ✐s ❝❧♦s❡st t♦ t❤❡ ♦r✐❣✐♥✳✑

♠❛❞❡✳

✶✷✳ ❊①❡r❝✐s❡s✿ ❖♣t✐♠✐③❛t✐♦♥ ❛♥❞ ♦t❤❡r ❛♣♣❧✐❝❛t✐♦♥s

❊①❡r❝✐s❡ ✶✷✳✸✺

❆ ❢❛r♠❡r ✇❛♥ts t♦ ❢❡♥❝❡ ❛♥ ❛r❡❛ ♦❢ 1.5 ♠✐❧❧✐♦♥ sq✉❛r❡ ❢❡❡t ✐♥ ❛ r❡❝t❛♥❣✉❧❛r ✜❡❧❞ ❛♥❞ t❤❡♥ ❞✐✈✐❞❡ ✐t ✐♥ ❤❛❧❢ ✇✐t❤ ❛ ❢❡♥❝❡ ♣❛r❛❧❧❡❧ t♦ ♦♥❡ ♦❢ t❤❡ s✐❞❡s ♦❢ t❤❡ r❡❝t❛♥❣❧❡✳ ❍♦✇ ❝❛♥ ❤❡ ❞♦ t❤✐s s♦ ❛s t♦ ♠✐♥✐♠✐③❡ t❤❡ ❝♦st ♦❢ t❤❡ ❢❡♥❝❡❄

✺✸✾

◆❡①t✳✳✳

■♥❞❡①

❛❝❝❡❧❡r❛t✐♦♥✱ ✷✹✵✱ ✹✽✸ ❆❧❣❡❜r❛ ❛♥❞ ❈♦♥t✐♥✉✐t②✱ ✶✹✼ ❆❧❣❡❜r❛ ♦❢ ❋✐♥✐t❡ ❛♥❞ ■♥✜♥✐t❡ ▲✐♠✐ts ♦❢ ❙❡q✉❡♥❝❡s✱ ✻✶ ❆❧❣❡❜r❛ ♦❢ ■♥✜♥✐t❡ ▲✐♠✐ts ♦❢ ❙❡q✉❡♥❝❡s✱ ✻✶ ❛❧❣❡❜r❛ ♦❢ ✐♥✜♥✐t✐❡s✱ ✻✶✱ ✶✸✾ ❆❧❣❡❜r❛ ♦❢ ▲✐♠✐ts ♦❢ ❋✉♥❝t✐♦♥s✱ ✶✸✾ ❆❧❣❡❜r❛ ♦❢ ▲✐♠✐ts ♦❢ ❋✉♥❝t✐♦♥s ❛t ■♥✜♥✐t②✱ ✶✾✺ ❆❧❣❡❜r❛ ♦❢ ▲✐♠✐ts ♦❢ ❙❡q✉❡♥❝❡s✱ ✺✸✱ ✽✾✱ ✶✸✽ ❆❧t❡r♥❛t✐✈❡ ❉❡✜♥✐t✐♦♥ ♦❢ ▲✐♠✐t✱ ✷✶✵ ❆❧t❡r♥❛t✐✈❡ ❋♦r♠✉❧❛ ❢♦r ▲✐♠✐t ♦❢ ❋✉♥❝t✐♦♥✱ ✶✹✺ ❆♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥ ❢♦r ❉❡r✐✈❛t✐✈❡s✱ ✹✷✺ ❆♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥ ❢♦r ❉✐✛❡r❡♥❝❡s✱ ✹✷✹ ❆♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥ ❢♦r ❉✐✛❡r❡♥t ◗✉♦t✐❡♥ts✱ ✹✷✹ ❛♥t✐❞❡r✐✈❛t✐✈❡✱ ✹✸✹✱ ✹✸✺ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦♥ ✐♥t❡r✈❛❧✱ ✹✷✼ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦♥ ♣❛rt✐t✐♦♥✱ ✹✷✻ ❛r❝❝♦s✐♥❡✱ ✶✻✾ ❛r❝s✐♥❡✱ ✶✻✾ ❛r✐t❤♠❡t✐❝ ♣r♦❣r❡ss✐♦♥✱ ✷✺✱ ✹✸ ❆r✐t❤♠❡t✐❝ Pr♦❣r❡ss✐♦♥✿ ❋♦r♠✉❧❛✱ ✷✺ ❆s②♠♣t♦t❡s ♦❢ ■♥✈❡rs❡✱ ✶✾✶ ❛✉❣♠❡♥t❡❞ ♣❛rt✐t✐♦♥ ♦❢ ✐♥t❡r✈❛❧✱ ✷✷✺ ❜❛s❡ ♦❢ ❡①♣♦♥❡♥t✱ ✾✶ ❇❡st ▲✐♥❡❛r ❆♣♣r♦①✐♠❛t✐♦♥✱ ✹✺✺ ❇♦❧③❛♥♦✲❲❡✐❡rstr❛ss ❚❤❡♦r❡♠✱ ✼✼ ❜♦✉♥❞❡❞ ❜❡❧♦✇ ❛♥❞ ❛❜♦✈❡ s❡ts✱ ✼✸ ❜♦✉♥❞❡❞ ❢✉♥❝t✐♦♥✱ ✶✽✶ ❜♦✉♥❞❡❞ s❡q✉❡♥❝❡✱ ✼✼ ❇♦✉♥❞❡❞♥❡ss ♦❢ ❈♦♥✈❡r❣❡♥t ❙❡q✉❡♥❝❡s✱ ✹✾ ❈❤❛✐♥ ❘✉❧❡ ❢♦r ❉❡r✐✈❛t✐✈❡s✱ ✸✶✹ ❈❤❛✐♥ ❘✉❧❡ ❢♦r ❉✐✛❡r❡♥❝❡ ◗✉♦t✐❡♥ts✱ ✸✶✸ ❈❤❛✐♥ ❘✉❧❡ ❢♦r ❉✐✛❡r❡♥❝❡s✱ ✸✶✸ ❝♦❞♦♠❛✐♥ ♦❢ ❢✉♥❝t✐♦♥✱ ✶✶✵ ❈♦♠♣❛r✐s♦♥ ❚❡st ❢♦r ▲✐♠✐ts ♦❢ ❋✉♥❝t✐♦♥s✱ ✶✼✶ ❈♦♠♣❛r✐s♦♥ ❚❡st ❢♦r ▲✐♠✐ts ♦❢ ❙❡q✉❡♥❝❡s✱ ✻✺ ❝♦♠♣♦s✐t✐♦♥✱ ✼✾✱ ✽✵✱ ✶✺✽✱ ✶✻✶✱ ✶✻✷✱ ✷✵✾✱ ✸✵✽✱ ✸✶✸✱ ✸✶✹ ❈♦♠♣♦s✐t✐♦♥ ❘✉❧❡ ❢♦r ▲✐♠✐ts✱ ✶✻✶ ❈♦♠♣♦s✐t✐♦♥ ❘✉❧❡ ❢♦r ▲✐♠✐ts ♦❢ P♦❧②♥♦♠✐❛❧s✱ ✽✷ ❝♦♥❝❛✈❡ ❢✉♥❝t✐♦♥ ♦♥ ✐♥t❡r✈❛❧✱ ✹✵✾ ❝♦♥❝❛✈❡ ❢✉♥❝t✐♦♥ ♦♥ ♣❛rt✐t✐♦♥✱ ✹✵✼ ❈♦♥❝❛✈✐t② ♦♥ P❛rt✐t✐♦♥✱ ✹✵✼ ❈♦♥❝❛✈✐t② ❚❤❡♦r❡♠✱ ✹✶✵ ❈♦♥st❛♥❝② ✈s✳ ❩❡r♦ ❉❡r✐✈❛t✐✈❡✱ ✹✷✸ ❈♦♥st❛♥❝② ✈s✳ ❩❡r♦ ❉✐✛❡r❡♥❝❡✱ ✹✷✷ ❈♦♥st❛♥❝② ✈s✳ ❩❡r♦ ❉✐✛❡r❡♥❝❡ ◗✉♦t✐❡♥t✱ ✹✷✸ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ❢♦r ❆♥t✐❞❡r✐✈❛t✐✈❡s✱ ✹✸✷ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ❢♦r ❉❡r✐✈❛t✐✈❡s✱ ✸✵✷ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ❢♦r ❉✐✛❡r❡♥❝❡ ◗✉♦t✐❡♥ts✱ ✸✵✶ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ❢♦r ❉✐✛❡r❡♥❝❡s✱ ✸✵✶ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ❢♦r ▲✐♠✐ts ♦❢ ❙❡q✉❡♥❝❡s✱ ✺✷ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ❢♦r ❩❡r♦ ▲✐♠✐t ❙❡q✉❡♥❝❡s✱ ✹✽

❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ♦❢ ▲✐♠✐ts ♦❢ ❋✉♥❝t✐♦♥s✱ ✶✸✾ ❝♦♥st❛♥t s❡q✉❡♥❝❡✱ ✹✷ ❝♦♥t✐♥✉✐t②✱ ✶✷✷✱ ✶✹✶✱ ✶✺✷✱ ✶✺✸✱ ✷✵✵ ❈♦♥t✐♥✉✐t② ♦❢ ❊①♣♦♥❡♥t✐❛❧ ❋✉♥❝t✐♦♥✱ ✶✻✾ ❈♦♥t✐♥✉✐t② ♦❢ ❊①♣♦♥❡♥t✐❛❧ ❋✉♥❝t✐♦♥s✱ ✶✺✸ ❈♦♥t✐♥✉✐t② ♦❢ ■♥✈❡rs❡✱ ✶✻✽ ❈♦♥t✐♥✉✐t② ♦❢ ▲♦❣❛r✐t❤♠✱ ✶✻✾ ❈♦♥t✐♥✉✐t② ♦❢ ◆❛t✉r❛❧ ❊①♣♦♥❡♥t✐❛❧ ❋✉♥❝t✐♦♥✱ ✶✺✷ ❈♦♥t✐♥✉✐t② ♦❢ P♦❧②♥♦♠✐❛❧s ❛♥❞ ❘❛t✐♦♥❛❧ ❋✉♥❝t✐♦♥s✱ ✶✹✾ ❈♦♥t✐♥✉✐t② ♦❢ ❘♦♦ts✱ ✶✻✽ ❈♦♥t✐♥✉✐t② ♦❢ ❙✐♥❡ ❛♥❞ ❈♦s✐♥❡✱ ✶✺✺ ❈♦♥t✐♥✉✐t② ♦❢ ❚❛♥❣❡♥t✱ ✶✺✻ ❈♦♥t✐♥✉✐t② ✉♥❞❡r ❆❧❣❡❜r❛✱ ✶✹✽ ❈♦♥t✐♥✉✐t② ✉♥❞❡r ❈♦♠♣♦s✐t✐♦♥s✱ ✶✻✷ ❈♦♥t✐♥✉✐t② ✈✐❛ ▲✐♠✐t✱ ✶✸✷ ❈♦♥t✐♥✉✐t② ✈✐❛ ❖♥❡✲❙✐❞❡❞ ❈♦♥t✐♥✉✐t②✱ ✶✸✽ ❝♦♥t✐♥✉✐t②✱ ❡♣s✐❧♦♥✲❞❡❧t❛✱ ✷✵✷ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥✱ ✶✷✷✱ ✶✸✷✱ ✶✸✽✱ ✶✹✶✱ ✶✹✼✕✶✹✾✱ ✶✺✺✱ ✶✺✻✱ ✶✻✽✱ ✶✼✻✱ ✶✼✽✱ ✶✼✾✱ ✶✽✹✱ ✷✺✺✱ ✸✾✷✱ ✸✾✻ ❈♦♥t✐♥✉♦✉s ▼❡❛♥s ❇♦✉♥❞❡❞✱ ✶✽✷ ❝♦♥t✐♥✉♦✉s ♦♥ ✐♥t❡r✈❛❧✱ ✶✽✶ ❝♦♥✈❡r❣❡♥❝❡✱ ✸✼ ❈♦♥✈❡r❣❡♥❝❡ t♦ ❩❡r♦✱ ✹✻ ❈♦♥✈❡r❣❡♥t ▼❡❛♥s ❇♦✉♥❞❡❞✱ ✶✽✶ ❝♦♥✈❡r❣❡♥t s❡q✉❡♥❝❡✱ ✸✹✱ ✸✼ ❝♦s✐♥❡✱ ✶✵✸✱ ✶✺✹✱ ✶✺✺ ❞❡❝r❡❛s✐♥❣ ❢✉♥❝t✐♦♥✱ ✸✼✷✱ ✸✼✻✱ ✸✼✼ ❞❡❝r❡❛s✐♥❣ ♣♦✐♥t✱ ✸✼✾ ❞❡✜♥✐t✐♦♥ ♦❢ ❧✐♠✐t✱ ✷✵✹ ❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡✱ ✶✶✵ ❞❡r✐✈❛t✐✈❡✱ ✷✻✵✱ ✷✻✶✱ ✷✻✺✱ ✷✾✾✱ ✸✵✷✱ ✸✵✽✱ ✸✶✹✱ ✸✾✷✱ ✸✾✻ ❞❡r✐✈❛t✐✈❡ ❛t ❛ ♣♦✐♥t✱ ✷✹✺ ❉❡r✐✈❛t✐✈❡ ◆❡❡❞s ❈♦♥t✐♥✉✐t②✱ ✷✺✺ ❉❡r✐✈❛t✐✈❡ ♦❢ ❈♦♥st❛♥t ❋✉♥❝t✐♦♥✱ ✷✻✶ ❉❡r✐✈❛t✐✈❡ ♦❢ ❊①♣♦♥❡♥t✐❛❧ ❋✉♥❝t✐♦♥✱ ✷✽✺ ❉❡r✐✈❛t✐✈❡ ♦❢ ❋✉♥❝t✐♦♥s ❚❤❛t ❉✐✛❡r ❜② ❈♦♥st❛♥t✱ ✷✻✺ ❉❡r✐✈❛t✐✈❡ ♦❢ ●❡♥❡r❛❧ ❊①♣♦♥❡♥t✐❛❧ ❋✉♥❝t✐♦♥✱ ✸✶✼ ❉❡r✐✈❛t✐✈❡ ♦❢ ■♥✈❡rs❡✱ ✸✺✸✱ ✸✺✺ ❉❡r✐✈❛t✐✈❡ ♦❢ P♦❧②♥♦♠✐❛❧✱ ✸✵✹ ❉❡r✐✈❛t✐✈❡ ♦❢ ❘❛t✐♦♥❛❧ P♦✇❡rs✱ ✸✺✼ ❉❡r✐✈❛t✐✈❡ ♦❢ ❘❡❝✐♣r♦❝❛❧ P♦✇❡rs✱ ✸✺✼ ❉❡r✐✈❛t✐✈❡ ♦❢ ❙✐♥❡ ❛♥❞ ❈♦s✐♥❡✱ ✷✽✽ ❉❡r✐✈❛t✐✈❡ ✈✐❛ ▲✐tt❧❡ ♦✱ ✹✹✺ ❉✐✛❡r❡♥❝❡ ♦❢ ❈♦♥st❛♥t ❋✉♥❝t✐♦♥✱ ✷✻✶ ❉✐✛❡r❡♥❝❡ ♦❢ ❊①♣♦♥❡♥t✐❛❧ ❋✉♥❝t✐♦♥✱ ✷✽✹ ❞✐✛❡r❡♥❝❡ ♦❢ ❢✉♥❝t✐♦♥✱ ✷✷✽✱ ✷✻✶✱ ✷✻✹✱ ✷✾✾✱ ✸✶✸✱ ✸✷✵ ❉✐✛❡r❡♥❝❡ ♦❢ ❋✉♥❝t✐♦♥s ❚❤❛t ❉✐✛❡r ❜② ❈♦♥st❛♥t✱ ✷✻✹ ❞✐✛❡r❡♥❝❡ ♦❢ s❡q✉❡♥❝❡✱ ✷✷✵✱ ✷✷✶✱ ✸✵✶✱ ✸✷✷ ❉✐✛❡r❡♥❝❡ ♦❢ ❙✐♥❡ ❛♥❞ ❈♦s✐♥❡✱ ✷✽✼ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✱ ✷✸✶✱ ✷✸✻✱ ✷✹✷✱ ✷✻✺✱ ✷✾✾✱ ✸✵✶✱ ✸✶✸ ❉✐✛❡r❡♥❝❡ ◗✉♦t✐❡♥t ♦❢ ❈♦♥st❛♥t ❋✉♥❝t✐♦♥✱ ✷✻✶

✺✹✸ ❉✐✛❡r❡♥❝❡ ◗✉♦t✐❡♥t ♦❢ ❊①♣♦♥❡♥t✐❛❧ ❋✉♥❝t✐♦♥✱ ✷✽✹ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ ❢✉♥❝t✐♦♥✱ ✷✻✶ ❉✐✛❡r❡♥❝❡ ◗✉♦t✐❡♥t ♦❢ ❋✉♥❝t✐♦♥s ❚❤❛t ❉✐✛❡r ❜② ❈♦♥st❛♥t✱ ✷✻✺ ❉✐✛❡r❡♥❝❡ ◗✉♦t✐❡♥t ♦❢ ■♥✈❡rs❡✱ ✸✺✸ ❉✐✛❡r❡♥❝❡ ◗✉♦t✐❡♥t ♦❢ ❙✐♥❡ ❛♥❞ ❈♦s✐♥❡✱ ✷✽✽ ❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥✱ ✷✺✺✱ ✸✾✷✱ ✸✾✻ ❞✐✛❡r❡♥t✐❛❧✱ ✷✼✺ ❞✐s❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥✱ ✶✶✻✱ ✶✶✾✱ ✶✷✷✱ ✷✵✸ ❞✐s♣❧❛❝❡♠❡♥t✱ ✷✷✽✱ ✹✽✸ ❞✐✈❡r❣❡♥❝❡ t♦ ✐♥✜♥✐t②✱ ✸✻ ❉✐✈❡r❣❡♥❝❡ ✉♥❞❡r ❆❞❞✐t✐♦♥✱ ✶✹✺ ❞✐✈❡r❣❡♥t s❡q✉❡♥❝❡✱ ✸✹✱ ✸✼ ❉◆❊✱ ✹✷✱ ✶✸✷ ❞♦♠❛✐♥ ♦❢ ❢✉♥❝t✐♦♥✱ ✶✶✵ ❡✱ ✾✼✱ ✾✽ ❡ ❛s ▲✐♠✐t✱ ✾✼ ❊q✉✐✈❛❧❡♥❝❡ ♦❢ ❉❡✜♥✐t✐♦♥s ♦❢ ▲✐♠✐t✱ ✷✵✺ ❊rr♦r ❇♦✉♥❞ ❢♦r ❇❡st ▲✐♥❡❛r ❆♣♣r♦①✐♠❛t✐♦♥✱ ✹✻✹ ❊①✐st❡♥❝❡ ♦❢ ❙✉♣r❡♠✉♠ ❛♥❞ ■♥✜♠✉♠✱ ✼✹ ❡①♣♦♥❡♥t✱ ✾✶ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥✱ ✾✹✱ ✾✺✱ ✾✽✱ ✶✺✶✕✶✺✸✱ ✶✻✾✱ ✸✶✼ ❊①♣♦♥❡♥t✐❛❧ ❋✉♥❝t✐♦♥ ❆r♦✉♥❞ ✵✱ ✾✺ ❊①♣♦♥❡♥t✐❛❧ ❋✉♥❝t✐♦♥ ❛s ▲✐♠✐t✱ ✾✽ ❡①tr❡♠❛✱ ✸✼✸ ❊①tr❡♠❡ ❱❛❧✉❡ ❚❤❡♦r❡♠✱ ✶✽✹ ❋❛♠♦✉s ▲✐♠✐t ❢♦r ❈♦s✐♥❡✱ ✶✵✼ ❋❛♠♦✉s ▲✐♠✐t ❢♦r ❊①♣♦♥❡♥t✐❛❧✱ ✶✵✵ ❋❛♠♦✉s ▲✐♠✐t ❢♦r ❙✐♥❡✱ ✶✵✻ ❋❛♠♦✉s ▲✐♠✐t ❢♦r ❚❛♥❣❡♥t✱ ✶✵✽ ❋❛♠♦✉s ▲✐♠✐ts✱ ✶✹✹ ❋❡r♠❛t✬s ❚❤❡♦r❡♠✱ ✸✽✷ ❋✐rst ❉❡r✐✈❛t✐✈❡ ❚❡st✱ ✹✶✻ ❋r♦♠ ▲♦❝❛❧ t♦ ●❧♦❜❛❧ ❊①tr❡♠❛✱ ✸✽✶ ❋r♦♠ ▲♦❝❛❧ t♦ ●❧♦❜❛❧ ▼♦♥♦t♦♥✐❝✐t②✱ ✸✽✵ ❢✉♥❝t✐♦♥✱ ✶✶✵✱ ✶✶✶ ❢✉♥❝t✐♦♥ ❣♦❡s t♦ ✐♥✜♥✐t②✱ ✶✾✵ ❣❡♥❡r❛❧ ❛♥t✐❞❡r✐✈❛t✐✈❡✱ ✹✸✼ ❣❡♦♠❡tr✐❝ ❣r♦✇t❤ ❛♥❞ ❞❡❝❛②✱ ✷✺ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥✱ ✷✺✱ ✷✻✱ ✹✹ ●❡♦♠❡tr✐❝ Pr♦❣r❡ss✐♦♥✿ ❋♦r♠✉❧❛✱ ✷✻ ●❧♦❜❛❧ ❊①tr❡♠❛ ✈✐❛ ❉❡r✐✈❛t✐✈❡✱ ✸✾✶ ❣❧♦❜❛❧ ❡①tr❡♠❡ ♣♦✐♥t✱ ✶✽✸ ❣❧♦❜❛❧ ❡①tr❡♠❡ ✈❛❧✉❡✱ ✶✽✹ ❣❧♦❜❛❧ ♠❛①✐♠✉♠ ♣♦✐♥t✱ ✶✽✸✱ ✸✼✸ ❣❧♦❜❛❧ ♠❛①✐♠✉♠ ✈❛❧✉❡✱ ✶✽✹✱ ✸✼✸ ❣❧♦❜❛❧ ♠✐♥✐♠✉♠ ♣♦✐♥t✱ ✶✽✸✱ ✸✼✸ ❣❧♦❜❛❧ ♠✐♥✐♠✉♠ ✈❛❧✉❡✱ ✶✽✹✱ ✸✼✸ ❣r❛♣❤ ♦❢ ❢✉♥❝t✐♦♥✱ ✶✶✶ ❣r❡❛t❡st ❧♦✇❡r ❜♦✉♥❞✱ ✼✸ ❍✐❣❤❡st ❚❡r♠s ♦❢ ❇✐♥♦♠✐❛❧ ❊①♣❛♥s✐♦♥✱ ✷✼✾ ❤♦r✐③♦♥t❛❧ ❛s②♠♣t♦t❡✱ ✶✾✵ ✐♠♣✉❧s❡✱ ✹✽✸ ✐♥❝r❡❛s✐♥❣ ❢✉♥❝t✐♦♥✱ ✸✼✷✱ ✸✼✻✱ ✸✼✼ ✐♥❝r❡❛s✐♥❣ ♣♦✐♥t✱ ✸✼✾

✐♥❝r❡♠❡♥ts ♦❢ ♣❛rt✐t✐♦♥✱ ✷✷✹ ✐♥❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡✱ ✶✶✵ ✐♥❞✐❝❡s ♦❢ t❡r♠s ♦❢ s❡q✉❡♥❝❡✱ ✷✶ ✐♥✜♠✉♠✱ ✼✸ ✐♥✜♥✐t❡ ❧✐♠✐t ♦❢ s❡q✉❡♥❝❡✱ ✸✻✱ ✸✼✱ ✹✷ ✐♥✢❡❝t✐♦♥ ♣♦✐♥t✱ ✹✶✷ ■♥t❡❣❡r P♦✇❡r ❋♦r♠✉❧❛ ❢♦r ❉❡r✐✈❛t✐✈❡✱ ✷✽✶ ■♥t❡❣❡r P♦✇❡r ❋♦r♠✉❧❛ ❢♦r ❉✐✛❡r❡♥❝❡✱ ✷✽✶ ■♥t❡❣❡r P♦✇❡r ❋♦r♠✉❧❛ ❢♦r ❉✐✛❡r❡♥❝❡ ◗✉♦t✐❡♥t✱ ✷✽✶ ■♥t❡r♠❡❞✐❛t❡ P♦✐♥t ❚❤❡♦r❡♠✱ ✼✻ ■♥t❡r♠❡❞✐❛t❡ ❱❛❧✉❡ ❚❤❡♦r❡♠✱ ✶✼✻✱ ✹✻✽ ✐♥t❡r✈❛❧✱ ✼✻ ✐♥✈❡rs❡ ❢✉♥❝t✐♦♥✱ ✶✻✻✱ ✶✻✽✱ ✶✾✶✱ ✸✺✸✱ ✸✺✺ ▲✬❍♦♣✐t❛❧✬s ❘✉❧❡✱ ✹✹✼ ▲❛❣r❛♥❣❡ ♥♦t❛t✐♦♥✱ ✷✻✵ ▲❛✇ ♦❢ ❈♦♥s❡r✈❛t✐♦♥ ♦❢ ▼♦♠❡♥t✉♠✱ ✹✽✹ ❧❡❛st ✉♣♣❡r ❜♦✉♥❞✱ ✼✸✱ ✼✹ ▲❡✐❜♥✐③ ♥♦t❛t✐♦♥✱ ✷✻✵ ❧✐♠✐t ❢r♦♠ t❤❡ ❧❡❢t✱ ✶✸✹ ❧✐♠✐t ❢r♦♠ t❤❡ r✐❣❤t✱ ✶✸✹ ▲✐♠✐t ✐♥ ❚❡r♠s ♦❢ ❖♥❡✲❙✐❞❡❞ ▲✐♠✐ts✱ ✶✸✺ ▲✐♠✐t ♦❢ ❆❜s♦❧✉t❡ ❱❛❧✉❡✱ ✻✽✱ ✶✼✹ ▲✐♠✐t ♦❢ ❆r✐t❤♠❡t✐❝ Pr♦❣r❡ss✐♦♥✱ ✹✸ ▲✐♠✐t ♦❢ ❈♦♥st❛♥t ❋✉♥❝t✐♦♥✱ ✶✹✵ ▲✐♠✐t ♦❢ ❈♦♥st❛♥t ❙❡q✉❡♥❝❡✱ ✹✷ ❧✐♠✐t ♦❢ ❢✉♥❝t✐♦♥✱ ✶✸✶✱ ✶✸✷✱ ✶✸✽✕✶✹✵✱ ✶✹✺✱ ✶✼✷✱ ✶✽✶✱ ✶✾✹✱ ✷✵✽✱ ✷✶✵✱ ✷✹✺ ❧✐♠✐t ♦❢ ❢✉♥❝t✐♦♥ ❛t ♣♦✐♥t✱ ✶✸✵ ▲✐♠✐t ♦❢ ●❡♦♠❡tr✐❝ Pr♦❣r❡ss✐♦♥✱ ✹✹ ▲✐♠✐t ♦❢ ■❞❡♥t✐t② ❋✉♥❝t✐♦♥✱ ✶✹✵ ▲✐♠✐t ♦❢ ▲✐♥❡❛r ❋✉♥❝t✐♦♥✱ ✷✵✻ ▲✐♠✐t ♦❢ P♦✇❡r ❙❡q✉❡♥❝❡✱ ✹✸ ❧✐♠✐t ♦❢ s❡q✉❡♥❝❡✱ ✸✵✱ ✸✹✱ ✹✶✱ ✶✸✽ ▲✐♠✐t ♦❢ ❙✉❜s❡q✉❡♥❝❡✱ ✻✹ ▲✐♠✐t ✉♥❞❡r ❈♦♠♣♦s✐t✐♦♥✱ ✷✵✾ ▲✐♠✐ts ❆r❡ ▲♦❝❛❧✱ ✶✸✼ ▲✐♠✐ts ♦❢ P♦❧②♥♦♠✐❛❧s ❛t ■♥✜♥✐t②✱ ✺✻✱ ✶✾✽ ▲✐♠✐ts ♦❢ ❘❛t✐♦♥❛❧ ❋✉♥❝t✐♦♥s ❛t ■♥✜♥✐t②✱ ✺✾✱ ✶✾✾ ▲✐♥❡❛r ❈❤❛✐♥ ❘✉❧❡ ■✱ ✸✵✽ ▲✐♥❡❛r ❈❤❛✐♥ ❘✉❧❡ ■■✱ ✸✵✽ ▲✐♥❡❛r ❈♦♠♣♦s✐t✐♦♥ ❘✉❧❡ ❢♦r ❆♥t✐❞❡r✐✈❛t✐✈❡s✱ ✹✸✸ ❧✐♥❡❛r ❢✉♥❝t✐♦♥✱ ✷✶✹✱ ✸✼✻✱ ✸✼✼ ▲✐♥❡❛r✐t② ♦❢ ❉✐✛❡r❡♥t✐❛t✐♦♥✱ ✸✵✸ ▲✐♥❡❛r✐t② ❘✉❧❡ ❢♦r ▲✐♠✐ts ♦❢ ❙❡q✉❡♥❝❡s✱ ✺✺ ▲✐tt❧❡ ♦ ❢♦r P♦❧②♥♦♠✐❛❧s✱ ✹✹✺ ▲♦❝❛❧ ❊①tr❡♠❛✱ ✸✽✷ ❧♦❝❛❧ ❡①tr❡♠❛✱ ✸✽✵ ❧♦❝❛❧ ♠❛①✐♠✉♠ ♣♦✐♥t✱ ✸✽✵ ❧♦❝❛❧ ♠✐♥✐♠✉♠ ♣♦✐♥t✱ ✸✽✵ ▲♦❝❛❧ ▼♦♥♦t♦♥✐❝✐t②✱ ✸✽✶ ▲♦❝❛❧ ▼♦♥♦t♦♥✐❝✐t② ❛♥❞ ❊①tr❡♠❛✱ ✸✽✵ ❧♦❝❛❧❧② ❞❡❝r❡❛s✐♥❣✱ ✸✼✾ ❧♦❝❛❧❧② ✐♥❝r❡❛s✐♥❣✱ ✸✼✾ ❧♦❝❛t✐♦♥✱ ✹✽✸ ❧♦❣❛r✐t❤♠✱ ✶✻✾ ❧♦✇❡r ❜♦✉♥❞✱ ✼✸ ▼❡❛♥ ❱❛❧✉❡ ❚❤❡♦r❡♠✱ ✸✾✻

✺✹✹ ♠♦♠❡♥t✉♠✱ ✹✽✸ ▼♦♥♦t♦♥❡ ❈♦♥✈❡r❣❡♥❝❡ ❚❤❡♦r❡♠✱ ✼✶ ♠♦♥♦t♦♥❡ ❢✉♥❝t✐♦♥✱ ✸✼✷✱ ✸✼✻✱ ✸✼✼ ♠♦♥♦t♦♥❡ s❡q✉❡♥❝❡✱ ✼✶ ▼♦♥♦t♦♥✐❝✐t② ❢r♦♠ ❙✐❣♥ ♦❢ ❉❡r✐✈❛t✐✈❡✱ ✸✾✽ ▼♦♥♦t♦♥✐❝✐t② ❢r♦♠ ❙✐❣♥ ♦❢ ❉✐✛❡r❡♥❝❡✱ ✸✼✺ ▼♦♥♦t♦♥✐❝✐t② ❢r♦♠ ❙✐❣♥ ♦❢ ❉✐✛❡r❡♥❝❡ ◗✉♦t✐❡♥t✱ ✸✼✺✱ ✸✾✽ ▼♦♥♦t♦♥✐❝✐t② ♦❢ ▲✐♥❡❛r ❋✉♥❝t✐♦♥s✱ ✸✼✻ ▼♦♥♦t♦♥✐❝✐t② ♦❢ ◗✉❛❞r❛t✐❝ ❋✉♥❝t✐♦♥s✱ ✸✼✼ ♥ t✐♠❡s ❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥✱ ✸✸✸ ♥❛t✉r❛❧ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥✱ ✾✽ ♥❛t✉r❛❧ ❧♦❣❛r✐t❤♠ ❢✉♥❝t✐♦♥✱ ✶✻✾ ♥❡st❡❞ ✐♥t❡r✈❛❧s✱ ✼✶✱ ✼✸✱ ✶✼✼✱ ✹✻✽ ◆❡st❡❞ ■♥t❡r✈❛❧s ❚❤❡♦r❡♠✱ ✼✷ ◆❡✇t♦♥✬s ❋✐rst ▲❛✇✱ ✹✽✹ ◆❡✇t♦♥✬s ❙❡❝♦♥❞ ▲❛✇✱ ✹✽✹ ◆❡✇t♦♥✬s ❚❤✐r❞ ▲❛✇✱ ✹✽✹ ♥♦ ❧✐♠✐t✱ ✹✷ ♥♦❞❡ ♦❢ ♣❛rt✐t✐♦♥✱ ✷✷✸ ◆♦t ▲✐♠✐t✱ ✷✵✽ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥✱ ✶✶✶ ♦♥❡✲s✐❞❡❞ ❧✐♠✐ts✱ ✶✸✹ ❖♥❡✲t♦✲♦♥❡ ❢r♦♠ ❉❡r✐✈❛t✐✈❡✱ ✹✵✸ ♦r❞❡rs ♦❢ ♠❛❣♥✐t✉❞❡ ♦❢ ❢✉♥❝t✐♦♥s✱ ✹✹✸ ♣❛rt✐t✐♦♥ ♦❢ ✐♥t❡r✈❛❧✱ ✷✷✸✱ ✷✷✺✱ ✷✷✽ P♦✐♥t✲❙❧♦♣❡ ❋♦r♠ ♦❢ ▲✐♥❡✱ ✷✶✹ ♣♦❧②♥♦♠✐❛❧✱ ✺✻✱ ✶✼✽ ♣♦✇❡r s❡q✉❡♥❝❡✱ ✹✸ ♣r✐♠❛r② ♥♦❞❡s✱ ✷✷✺ Pr♦❞✉❝t ❘✉❧❡ ❢♦r ❉❡r✐✈❛t✐✈❡s✱ ✸✷✶ Pr♦❞✉❝t ❘✉❧❡ ❢♦r ❉✐✛❡r❡♥❝❡ ◗✉♦t✐❡♥ts✱ ✸✷✵ Pr♦❞✉❝t ❘✉❧❡ ❢♦r ❉✐✛❡r❡♥❝❡s✱ ✸✷✵ Pr♦❞✉❝t ❘✉❧❡ ❢♦r ▲✐♠✐ts ♦❢ ❋✉♥❝t✐♦♥s✱ ✶✹✵ Pr♦❞✉❝t ❘✉❧❡ ❢♦r ▲✐♠✐ts ♦❢ ❙❡q✉❡♥❝❡s✱ ✺✶ P✉s❤ ❖✉t ❚❤❡♦r❡♠ ❢♦r ▲✐♠✐ts ♦❢ ❋✉♥❝t✐♦♥s✱ ✶✼✺ P✉s❤ ❖✉t ❚❤❡♦r❡♠ ❢♦r ▲✐♠✐ts ♦❢ ❙❡q✉❡♥❝❡s✱ ✼✵ ◗✉♦t✐❡♥t ◗✉♦t✐❡♥t ◗✉♦t✐❡♥t ◗✉♦t✐❡♥t ◗✉♦t✐❡♥t

❘✉❧❡ ❘✉❧❡ ❘✉❧❡ ❘✉❧❡ ❘✉❧❡

❢♦r ❢♦r ❢♦r ❢♦r ❢♦r

❉❡r✐✈❛t✐✈❡s✱ ✸✷✸ ❉✐✛❡r❡♥❝❡ ◗✉♦t✐❡♥ts✱ ✸✷✷ ❉✐✛❡r❡♥❝❡s✱ ✸✷✷ ▲✐♠✐ts ♦❢ ❋✉♥❝t✐♦♥s✱ ✶✹✵ ▲✐♠✐ts ♦❢ ❙❡q✉❡♥❝❡s✱ ✺✷

❘❛♥❣❡ ♦❢ ❈♦♥t✐♥✉♦✉s ❋✉♥❝t✐♦♥✱ ✶✼✾ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥✱ ✺✻✱ ✺✾ r❡❝✉rs✐✈❡✱ ✷✸✱ ✷✺ r❡♣❡❛t❡❞ ❛❞❞✐t✐♦♥✱ ✾✶ r❡♣❡❛t❡❞ ♠✉❧t✐♣❧✐❝❛t✐♦♥✱ ✾✶

❘♦❧❧❡✬s ❚❤❡♦r❡♠✱ ✸✾✷ ❘✉❧❡s ♦❢ ❊①♣♦♥❡♥ts✱ ✾✹ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡✱ ✸✸✵ ❙❡❝♦♥❞ ❉❡r✐✈❛t✐✈❡ ❚❡st✱ ✹✶✾ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✱ ✸✷✽ s❡❝♦♥❞❛r② ♥♦❞❡s✱ ✷✷✺ s❡q✉❡♥❝❡✱ ✷✶✱ ✶✶✶ s❡q✉❡♥❝❡ ♦❢ ❞✐✛❡r❡♥❝❡s✱ ✷✷✵✱ ✸✷✷ ❙❡t ♦❢ ❆♥t✐❞❡r✐✈❛t✐✈❡s✱ ✹✷✽ s✐♥❡✱ ✶✵✸✱ ✶✺✹✕✶✺✻ ❙✐♥❡ ❛♥❞ ❈♦s✐♥❡ ❆r♦✉♥❞ ✵✱ ✶✵✺ s❧♦♣❡✱ ✷✶✷✱ ✷✶✹✱ ✸✼✻✱ ✸✼✼ s♣r❡❛❞s❤❡❡t✱ ✹✵✾ ❙q✉❡❡③❡ ❚❤❡♦r❡♠ ❢♦r ❋✉♥❝t✐♦♥s✱ ✶✼✷ ❙q✉❡❡③❡ ❚❤❡♦r❡♠ ❢♦r ❙❡q✉❡♥❝❡s✱ ✻✼ ❙q✉❡❡③❡ ❚❤❡♦r❡♠ ❢♦r ❩❡r♦ ▲✐♠✐t ❙❡q✉❡♥❝❡s✱ ✹✾ ❙tr✐❝t ▼♦♥♦t♦♥✐❝✐t② ✈✐❛ ❉❡r✐✈❛t✐✈❡✱ ✹✵✸ ❙tr✐❝t ▼♦♥♦t♦♥✐❝✐t② ✈✐❛ ❉✐✛❡r❡♥❝❡ ◗✉♦t✐❡♥t✱ ✹✵✷ str✐❝t❧② ❞❡❝r❡❛s✐♥❣ ❢✉♥❝t✐♦♥✱ ✸✼✷ str✐❝t❧② ✐♥❝r❡❛s✐♥❣ ❢✉♥❝t✐♦♥✱ ✸✼✷ s✉❜s❡q✉❡♥❝❡✱ ✻✹✱ ✼✼ s✉❜st✐t✉t✐♦♥✱ ✽✵✱ ✶✺✽ s✉❜st✐t✉t✐♦♥ ♥♦t❛t✐♦♥✱ ✻✹✱ ✶✺✾ ❙✉❜st✐t✉t✐♦♥ ❘✉❧❡ ❢♦r ▲✐♠✐ts✱ ✶✾✺ ❙✉♠ ❘✉❧❡ ❢♦r ❆♥t✐❞❡r✐✈❛t✐✈❡s✱ ✹✸✶ ❙✉♠ ❘✉❧❡ ❢♦r ❉❡r✐✈❛t✐✈❡s✱ ✷✾✾ ❙✉♠ ❘✉❧❡ ❢♦r ❉✐✛❡r❡♥❝❡ ◗✉♦t✐❡♥ts✱ ✷✾✾ ❙✉♠ ❘✉❧❡ ❢♦r ❉✐✛❡r❡♥❝❡s✱ ✷✾✾ ❙✉♠ ❘✉❧❡ ❢♦r ▲✐♠✐ts ♦❢ ❙❡q✉❡♥❝❡s✱ ✺✵ ❙✉♠ ❘✉❧❡ ❢♦r ❩❡r♦ ▲✐♠✐t ❙❡q✉❡♥❝❡s✱ ✹✼ ❙✉♠ ❘✉❧❡ ♦❢ ▲✐♠✐ts ♦❢ ❋✉♥❝t✐♦♥s✱ ✶✸✾ s✉♣r❡♠✉♠✱ ✼✸✱ ✼✹ t❛♥❣❡♥t✱ ✶✺✻ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s✱ ✶✵✸ ❚r✉♥❝❛t✐♦♥ Pr✐♥❝✐♣❧❡ ❢♦r ❙❡q✉❡♥❝❡s✱ ✻✸ ❯♥✐q✉❡♥❡ss ♦❢ ▲✐♠✐t ♦❢ ❋✉♥❝t✐♦♥✱ ✶✸✶ ❯♥✐q✉❡♥❡ss ♦❢ ▲✐♠✐t ♦❢ ❙❡q✉❡♥❝❡✱ ✹✵✱ ✹✶ ❯♥✐q✉❡♥❡ss ♦❢ ❙✉♣r❡♠✉♠ ❛♥❞ ■♥✜♠✉♠✱ ✼✹ ✉♣♣❡r ❜♦✉♥❞✱ ✼✸ ❱❛❧✐❞✐t② ♦❢ ❉❡✜♥✐t✐♦♥ ♦❢ ❙✐♥❡ ❛♥❞ ❈♦s✐♥❡✱ ✶✺✹ ✈❛r✐❛❜❧❡s✱ ✺✵✵ ✈❡❧♦❝✐t②✱ ✷✹✵✱ ✹✽✸ ✈❡rt✐❝❛❧ ❛s②♠♣t♦t❡✱ ✶✾✵ ①✲✐♥t❡r❝❡♣t✱ ✶✼✽ ①✲✐♥t❡r❝❡♣ts ♦❢ ❈♦♥t✐♥✉♦✉s ❋✉♥❝t✐♦♥✱ ✶✼✽ ①✲✐♥t❡r❝❡♣ts ♦❢ P♦❧②♥♦♠✐❛❧✱ ✶✼✽