Calculus Illustrated. Volume 1: Precalculus [1, 1 ed.] 1694326977, 9781694326973

Table of contents :
Preface
Calculus of sequences
What is calculus about?
The real number line
Sequences
Repeated addition and repeated multiplication
How to find nth-term formulas for sequences
The algebra of exponents
The Binomial Formula
The sequence of differences: velocity
The sequence of the sums: displacement
Sums of differences and differences of sums: motion
The algebra of sums and differences
Sets and functions
Sets and relations
Functions
Sequences are numerical functions
How numerical functions emerge: optimization
Set building
The xy-plane: where graphs live...
Linear relations
Relations vs. functions
A function as a black box
Give the function a domain...
The graph of a function
Linear functions
Algebra creates functions
The image: the range of values of a function
Compositions of functions
Operations on sets
Piecewise-defined functions
Numerical functions are transformations of the line
Functions with regularities: one-to-one and onto
Compositions of functions
The inverse of a function
Units conversions and changes of variables
Transforming the axes transforms the plane
Changing a variable transforms the graph of a function
The graph of a quadratic polynomial is a parabola
The main classes of functions
The simplest functions
Monotonicity and the extreme values of functions
Functions with symmetries
Quadratic polynomials
The polynomial functions
The rational functions
The root functions
The exponential functions
The logarithmic functions
The trigonometric functions
Algebra and geometry
The arithmetic operations on functions
The algebra of compositions
Solving equations
The algebra of logarithms
The Cartesian system for the Euclidean plane
The Euclidean plane: distances
Trigonometry and the wave function
The Euclidean plane: angles
From geometry to calculus
Solving inequalities
Exercises
Exercises: Algebra
Exercises: Sequences
Exercises: Sets and logic
Exercises: Coordinate system
Exercises: Linear algebra
Exercises: Polynomials
Exercises: Relations and functions
Exercises: Graphs
Exercises: Compositions
Exercises: Transformations
Exercises: Basic models
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❞✿

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✉♥❞❡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✇♦✳

•

❙❡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

❆❜♦✉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 ✶✿ ❈❛❧❝✉❧✉s ♦❢ s❡q✉❡♥❝❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵

✶✳✶ ❲❤❛t ✐s ❝❛❧❝✉❧✉s ❛❜♦✉t❄ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷ ❚❤❡ r❡❛❧ ♥✉♠❜❡r ❧✐♥❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✸ ❙❡q✉❡♥❝❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✹ ❘❡♣❡❛t❡❞ ❛❞❞✐t✐♦♥ ❛♥❞ r❡♣❡❛t❡❞ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✳ ✳ ✳ ✳ ✶✳✺ ❍♦✇ t♦ ✜♥❞ nt❤✲t❡r♠ ❢♦r♠✉❧❛s ❢♦r s❡q✉❡♥❝❡s ✳ ✳ ✳ ✳ ✳ ✶✳✻ ❚❤❡ ❛❧❣❡❜r❛ ♦❢ ❡①♣♦♥❡♥ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✼ ❚❤❡ ❇✐♥♦♠✐❛❧ ❋♦r♠✉❧❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✽ ❚❤❡ s❡q✉❡♥❝❡ ♦❢ ❞✐✛❡r❡♥❝❡s✿ ✈❡❧♦❝✐t② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✾ ❚❤❡ s❡q✉❡♥❝❡ ♦❢ t❤❡ s✉♠s✿ ❞✐s♣❧❛❝❡♠❡♥t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✵ ❙✉♠s ♦❢ ❞✐✛❡r❡♥❝❡s ❛♥❞ ❞✐✛❡r❡♥❝❡s ♦❢ s✉♠s✿ ♠♦t✐♦♥ ✶✳✶✶ ❚❤❡ ❛❧❣❡❜r❛ ♦❢ s✉♠s ❛♥❞ ❞✐✛❡r❡♥❝❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

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✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✵ ✷✵ ✷✹ ✸✻ ✹✷ ✺✵ ✺✻ ✻✸ ✼✷ ✽✺ ✾✶

 ❈❤❛♣t❡r ✷✿ ❙❡ts ❛♥❞ ❢✉♥❝t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✽

✷✳✶ ❙❡ts ❛♥❞ r❡❧❛t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷ ❋✉♥❝t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✸ ❙❡q✉❡♥❝❡s ❛r❡ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✹ ❍♦✇ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s ❡♠❡r❣❡✿ ♦♣t✐♠✐③❛t✐♦♥ ✳ ✷✳✺ ❙❡t ❜✉✐❧❞✐♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✻ ❚❤❡ xy ✲♣❧❛♥❡✿ ✇❤❡r❡ ❣r❛♣❤s ❧✐✈❡✳✳✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✼ ▲✐♥❡❛r r❡❧❛t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✽ ❘❡❧❛t✐♦♥s ✈s✳ ❢✉♥❝t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✾ ❆ ❢✉♥❝t✐♦♥ ❛s ❛ ❜❧❛❝❦ ❜♦① ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✵ ●✐✈❡ t❤❡ ❢✉♥❝t✐♦♥ ❛ ❞♦♠❛✐♥✳✳✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✶ ❚❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✷ ▲✐♥❡❛r ❢✉♥❝t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✸ ❆❧❣❡❜r❛ ❝r❡❛t❡s ❢✉♥❝t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✹ ❚❤❡ ✐♠❛❣❡✿ t❤❡ r❛♥❣❡ ♦❢ ✈❛❧✉❡s ♦❢ ❛ ❢✉♥❝t✐♦♥ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✶ ❖♣❡r❛t✐♦♥s ♦♥ s❡ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✷ P✐❡❝❡✇✐s❡✲❞❡✜♥❡❞ ❢✉♥❝t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✸ ◆✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s ❛r❡ tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ❧✐♥❡ ✳ ✳ ✸✳✹ ❋✉♥❝t✐♦♥s ✇✐t❤ r❡❣✉❧❛r✐t✐❡s✿ ♦♥❡✲t♦✲♦♥❡ ❛♥❞ ♦♥t♦ ✳ ✳ ✳ ✳ ✸✳✺ ❈♦♠♣♦s✐t✐♦♥s ♦❢ ❢✉♥❝t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✻ ❚❤❡ ✐♥✈❡rs❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✼ ❯♥✐ts ❝♦♥✈❡rs✐♦♥s ❛♥❞ ❝❤❛♥❣❡s ♦❢ ✈❛r✐❛❜❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✽ ❚r❛♥s❢♦r♠✐♥❣ t❤❡ ❛①❡s tr❛♥s❢♦r♠s t❤❡ ♣❧❛♥❡ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✾ ❈❤❛♥❣✐♥❣ ❛ ✈❛r✐❛❜❧❡ tr❛♥s❢♦r♠s t❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥ ✸✳✶✵ ❚❤❡ ❣r❛♣❤ ♦❢ ❛ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧ ✐s ❛ ♣❛r❛❜♦❧❛ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

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✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

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✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

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

 ❈❤❛♣t❡r ✸✿ ❈♦♠♣♦s✐t✐♦♥s ♦❢ ❢✉♥❝t✐♦♥s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

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

 ❈❤❛♣t❡r ✹✿ ❚❤❡ ♠❛✐♥ ❝❧❛ss❡s ♦❢ ❢✉♥❝t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾✶

✹✳✶ ❚❤❡ s✐♠♣❧❡st ❢✉♥❝t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾✶ ✹✳✷ ▼♦♥♦t♦♥✐❝✐t② ❛♥❞ t❤❡ ❡①tr❡♠❡ ✈❛❧✉❡s ♦❢ ❢✉♥❝t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵✶ ✹✳✸ ❋✉♥❝t✐♦♥s ✇✐t❤ s②♠♠❡tr✐❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶✵

❈♦♥t❡♥ts

✾

✹✳✹ ◗✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✳✺ ❚❤❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s

✸✷✽

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✹✶

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✺✸

✹✳✻ ❚❤❡ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥s ✹✳✼ ❚❤❡ r♦♦t ❢✉♥❝t✐♦♥s

✸✷✶

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✳✽ ❚❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✳✾ ❚❤❡ ❧♦❣❛r✐t❤♠✐❝ ❢✉♥❝t✐♦♥s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✳✶✵ ❚❤❡ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

 ❈❤❛♣t❡r ✺✿ ❆❧❣❡❜r❛ ❛♥❞ ❣❡♦♠❡tr②

✸✻✵ ✸✼✸ ✸✼✽

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✾✹

✺✳✶ ❚❤❡ ❛r✐t❤♠❡t✐❝ ♦♣❡r❛t✐♦♥s ♦♥ ❢✉♥❝t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✾✹

✺✳✷ ❚❤❡ ❛❧❣❡❜r❛ ♦❢ ❝♦♠♣♦s✐t✐♦♥s

✹✵✷

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺✳✸ ❙♦❧✈✐♥❣ ❡q✉❛t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✵✾

✺✳✹ ❚❤❡ ❛❧❣❡❜r❛ ♦❢ ❧♦❣❛r✐t❤♠s

✹✶✽

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺✳✺ ❚❤❡ ❈❛rt❡s✐❛♥ s②st❡♠ ❢♦r t❤❡ ❊✉❝❧✐❞❡❛♥ ♣❧❛♥❡ ✺✳✻ ❚❤❡ ❊✉❝❧✐❞❡❛♥ ♣❧❛♥❡✿ ❞✐st❛♥❝❡s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✷✾

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✸✼

✺✳✼ ❚r✐❣♦♥♦♠❡tr② ❛♥❞ t❤❡ ✇❛✈❡ ❢✉♥❝t✐♦♥

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✺✶

✺✳✽ ❚❤❡ ❊✉❝❧✐❞❡❛♥ ♣❧❛♥❡✿ ❛♥❣❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✻✺

✺✳✾ ❋r♦♠ ❣❡♦♠❡tr② t♦ ❝❛❧❝✉❧✉s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✼✺

✺✳✶✵ ❙♦❧✈✐♥❣ ✐♥❡q✉❛❧✐t✐❡s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✽✵

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✽✼

❊①❡r❝✐s❡s

✶ ❊①❡r❝✐s❡s✿ ❆❧❣❡❜r❛

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷ ❊①❡r❝✐s❡s✿ ❙❡q✉❡♥❝❡s

✹✽✼

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✾✵

✸ ❊①❡r❝✐s❡s✿ ❙❡ts ❛♥❞ ❧♦❣✐❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✾✶

✹ ❊①❡r❝✐s❡s✿ ❈♦♦r❞✐♥❛t❡ s②st❡♠

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✾✷

✺ ❊①❡r❝✐s❡s✿ ▲✐♥❡❛r ❛❧❣❡❜r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✾✸

✻ ❊①❡r❝✐s❡s✿ P♦❧②♥♦♠✐❛❧s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✾✺

✼ ❊①❡r❝✐s❡s✿ ❘❡❧❛t✐♦♥s ❛♥❞ ❢✉♥❝t✐♦♥s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✾✼

✽ ❊①❡r❝✐s❡s✿ ●r❛♣❤s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺✵✵

✾ ❊①❡r❝✐s❡s✿ ❈♦♠♣♦s✐t✐♦♥s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✵ ❊①❡r❝✐s❡s✿ ❚r❛♥s❢♦r♠❛t✐♦♥s

✺✵✷

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺✵✺

✶✶ ❊①❡r❝✐s❡s✿ ❇❛s✐❝ ♠♦❞❡❧s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺✵✼

■♥❞❡①

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺✶✵

❈❤❛♣t❡r ✶✿ ❈❛❧❝✉❧✉s ♦❢ s❡q✉❡♥❝❡s

❈♦♥t❡♥ts

✶✳✶ ❲❤❛t ✐s ❝❛❧❝✉❧✉s ❛❜♦✉t❄ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷ ❚❤❡ r❡❛❧ ♥✉♠❜❡r ❧✐♥❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✸ ❙❡q✉❡♥❝❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✹ ❘❡♣❡❛t❡❞ ❛❞❞✐t✐♦♥ ❛♥❞ r❡♣❡❛t❡❞ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✳ ✳ ✳ ✶✳✺ ❍♦✇ t♦ ✜♥❞ nt❤✲t❡r♠ ❢♦r♠✉❧❛s ❢♦r s❡q✉❡♥❝❡s ✳ ✳ ✳ ✳ ✶✳✻ ❚❤❡ ❛❧❣❡❜r❛ ♦❢ ❡①♣♦♥❡♥ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✼ ❚❤❡ ❇✐♥♦♠✐❛❧ ❋♦r♠✉❧❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✽ ❚❤❡ s❡q✉❡♥❝❡ ♦❢ ❞✐✛❡r❡♥❝❡s✿ ✈❡❧♦❝✐t② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✾ ❚❤❡ s❡q✉❡♥❝❡ ♦❢ t❤❡ s✉♠s✿ ❞✐s♣❧❛❝❡♠❡♥t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✵ ❙✉♠s ♦❢ ❞✐✛❡r❡♥❝❡s ❛♥❞ ❞✐✛❡r❡♥❝❡s ♦❢ s✉♠s✿ ♠♦t✐♦♥ ✶✳✶✶ ❚❤❡ ❛❧❣❡❜r❛ ♦❢ s✉♠s ❛♥❞ ❞✐✛❡r❡♥❝❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✳✶✳ ❲❤❛t ✐s ❝❛❧❝✉❧✉s ❛❜♦✉t❄

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

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✵ ✷✵ ✷✹ ✸✻ ✹✷ ✺✵ ✺✻ ✻✸ ✼✷ ✽✺ ✾✶

✶✳✶✳ ❲❤❛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✐♠❡

❚❤❡ ❡q✉❛t✐♦♥ ✐s s♦❧✈❡❞ ❢♦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✳ ▲❡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✐✈✐♥❣❄ ❲❡ ❧♦♦❦ ❛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 ❜❡❡♥❄ ◆♦t❤✐♥❣ ✇✐t❤♦✉t ❛❧❣❡❜r❛✦ ❋♦rt✉♥❛t❡❧②✱ t❤❡ ❛❧❣❡❜r❛ ✐s s✐♠♣❧❡✿ s♣❡❡❞ =

❞✐st❛♥❝❡ t✐♠❡

❚❤❡ 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

• ❞✐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❝✳

✶✳✶✳ ❲❤❛t ✐s ❝❛❧❝✉❧✉s ❛❜♦✉t❄

✶✷

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

❲❡ t❤❡♥ ♣❧♦t t❤❡s❡ ♥❡✇ ♥✉♠❜❡rs ❛❣❛✐♥st 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✐♦♥❄ ◆♦t❤✐♥❣✱ ✇✐t❤♦✉t ❛❧❣❡❜r❛✦ ❋♦rt✉♥❛t❡❧②✱ ✇❡ ❝❛♥ ❥✉st ✉s❡ t❤❡ s❛♠❡ ❢♦r♠✉❧❛ ❛s ❜❡❢♦r❡✿ ❞✐st❛♥❝❡ = s♣❡❡❞ × t✐♠❡ ■♥ ❝♦♥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❛✱ ✇❡ ✉s❡ t❤❡ s♣❡❡❞s ❛s t❤❡ ❤❡✐❣❤ts ♦❢ t❤❡ ❝♦♥s❡❝✉t✐✈❡ st❡♣s ♦❢ t❤❡ st❛✐r❝❛s❡✿

✶✳✶✳

❲❤❛t ✐s ❝❛❧❝✉❧✉s ❛❜♦✉t❄

✶✸

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

250✲♠✐❧❡

135✲✱ 200✲✱

❛♥❞

♠❛r❦s ❞✉r✐♥❣ t❤✐s t✐♠❡✳

❖✉r ❛❜✐❧✐t② t♦ ✉s❡ ♥❡❣❛t✐✈❡ ♥✉♠❜❡rs ❛❧❧♦✇s ✉s t♦ tr❡❛t t❤❡ ❞❛t❛ t❤❡ ❡①❛❝t s❛♠❡ ✇❛② ❡✈❡♥ ✇❤❡♥ ✇❡ ❝❤❛♥❣❡ ❞✐r❡❝t✐♦♥s✳

❆s t❤❡ ✇♦r❞s ✏s♣❡❡❞✑ ❛♥❞ ✏❞✐st❛♥❝❡✑ ✐♠♣❧② t❤❛t t❤❡s❡ q✉❛♥t✐t✐❡s ❛r❡ ♣♦s✐t✐✈❡✱ ✇❡ s♣❡❛❦ ♦❢

✏✈❡❧♦❝✐t②✑ ❛♥❞ ✏❞✐s♣❧❛❝❡♠❡♥t✑ ✐♥st❡❛❞ ❛s✿

✈❡❧♦❝✐t②

=

❞✐s♣❧❛❝❡♠❡♥t

❛♥❞

t✐♠❡

❞✐s♣❧❛❝❡♠❡♥t

=

✈❡❧♦❝✐t②

·

t✐♠❡

❈❤❛♥❣✐♥❣ ❣❡❛rs✱ ♦♥❡ ♦❢ t❤❡ ❡❛s✐❡r ❝♦♥❝❧✉s✐♦♥s ✇❡ ❞❡r✐✈❡ ❢r♦♠ t❤❡s❡ ❢♦r♠✉❧❛s ✐s t❤❡ ❢♦❧❧♦✇✐♥❣ s✐♠♣❧❡ st❛t❡♠❡♥t✿

◮

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

■♥❞❡❡❞✱ ✐❢ t❤❡ t✐♠❡ ✐♥❝r❡♠❡♥t ✐s ♣♦s✐t✐✈❡ ❛♥❞ t❤❡ ✈❡❧♦❝✐t② ✐s ♣♦s✐t✐✈❡ t♦♦✱ t❤❡♥ s♦ ✐s t❤❡✐r ♣r♦❞✉❝t✱ t❤❡ ❞✐s♣❧❛❝❡♠❡♥t✱ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ s❡❝♦♥❞ ❢♦r♠✉❧❛✳ ▲❡t✬s ❡①♣❧♦r❡ t❤❡ ❧♦❣✐❝ ♦❢ t❤✐s st❛t❡♠❡♥t✳ ■t ❝❛♥ ❜❡ r❡❝❛st ❛s ❛♥

◮ ■❋

t❤❡ ✈❡❧♦❝✐t② ✐s ♣♦s✐t✐✈❡✱

❚❍❊◆

✐♠♣❧✐❝❛t✐♦♥✱

✐✳❡✳✱ ❛♥ ✏✐❢✲t❤❡♥✑ st❛t❡♠❡♥t✿

t❤❡ ♠♦t✐♦♥ ✐s ✐♥ t❤❡ ♣♦s✐t✐✈❡ ❞✐r❡❝t✐♦♥✳

❊①❡r❝✐s❡ ✶✳✶✳✶

❘❡st❛t❡ ❛s ❛♥ ✐♠♣❧✐❝❛t✐♦♥ ❡❛❝❤ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ st❛t❡♠❡♥ts✿ ✭❛✮ ❡✈❡r② sq✉❛r❡ ✐s ❛ r❡❝t❛♥❣❧❡❀ ✭❜✮ ♣❛r❛❧❧❡❧ 2 2 2 ❧✐♥❡s ❞♦♥✬t ✐♥t❡rs❡❝t❀ ✭❝✮ (a + b) = a + 2ab + b ✳

❲❡ ✇✐❧❧ ✉s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥✈❡♥✐❡♥t ❛❜❜r❡✈✐❛t✐♦♥ t❤r♦✉❣❤♦✉t t❤❡ t❡①t✿ ■♠♣❧✐❝❛t✐♦♥

=⇒ ■t r❡❛❞s ✏t❤❡♥✑✱ ✏t❤❡r❡❢♦r❡✑✱ ♦r ✏✐♠♣❧✐❡s t❤❛t✑✳

❚❤❡♥✱ t❤❡ ❛❜♦✈❡ st❛t❡♠❡♥t t❛❦❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ❛❜❜r❡✈✐❛t❡❞ ❢♦r♠✿

◮

❚❤❡ ✈❡❧♦❝✐t② ✐s ♣♦s✐t✐✈❡

=⇒

t❤❡ ♠♦t✐♦♥ ✐s ✐♥ t❤❡ ♣♦s✐t✐✈❡ ❞✐r❡❝t✐♦♥✳

◆♦✇✱ ✇❡ ❝❛♥ tr② t♦ ✏✢✐♣✑ t❤❡ ✐♠♣❧✐❝❛t✐♦♥ ♦❢ t❤✐s st❛t❡♠❡♥t✱ ✇✐t❤♦✉t ❛ss✉♠✐♥❣ t❤❛t t❤❡ r❡s✉❧t ✇✐❧❧ ❜❡ tr✉❡✿

◮

❚❤❡ ✈❡❧♦❝✐t② ✐s ♣♦s✐t✐✈❡

⇐=

t❤❡ ♠♦t✐♦♥ ✐s ✐♥ t❤❡ ♣♦s✐t✐✈❡ ❞✐r❡❝t✐♦♥✳

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

◮

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

❚❤❡ ❧❛tt❡r ✐s ❝❛❧❧❡❞ t❤❡

◮ ■❋

❝♦♥✈❡rs❡

=⇒

t❤❡ ✈❡❧♦❝✐t② ✐s ♣♦s✐t✐✈❡✳

♦❢ t❤❡ ♦r✐❣✐♥❛❧ st❛t❡♠❡♥t✳ ■t✬s ❛❧s♦ ❛♥ ✐♠♣❧✐❝❛t✐♦♥ st❛t❡❞ ❛s✿

t❤❡ ♠♦t✐♦♥ ✐s ✐♥ t❤❡ ♣♦s✐t✐✈❡ ❞✐r❡❝t✐♦♥✱

❚❍❊◆

t❤❡ ✈❡❧♦❝✐t② ✐s ♣♦s✐t✐✈❡✳

✶✳✶✳

❲❤❛t ✐s ❝❛❧❝✉❧✉s ❛❜♦✉t❄

✶✹

❚❤❡ ❝♦♥✈❡rs❡ ✐s tr✉❡ ❛s ✇❡❧❧✦ ❚❤✐s ✐s t❤❡ ❛❜❜r❡✈✐❛t✐♦♥ ✇❡ ✉s❡❞✿

■♠♣❧✐❝❛t✐♦♥

⇐= ■t

r❡❛❞s

✏✇❤❡♥❡✈❡r✑✱

✏♣r♦✲

✈✐❞❡❞✑✱ ♦r ✏♦♥❧② ✐❢ ✑✳

❊①❡r❝✐s❡ ✶✳✶✳✷ ❙t❛t❡ t❤❡ ❝♦♥✈❡rs❡ ♦❢ ❡❛❝❤ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ st❛t❡♠❡♥ts✿ ✭❛✮ ❡✈❡r② sq✉❛r❡ ✐s ❛ r❡❝t❛♥❣❧❡❀ ✭❜✮ ♣❛r❛❧❧❡❧ ❧✐♥❡s ❞♦♥✬t ✐♥t❡rs❡❝t❀ ✭❝✮

2x = 2y

✇❤❡♥

x = y✳ ❲❛r♥✐♥❣✦ ❚❤❡ ❝♦♥✈❡rs❡ ♦❢ ❛ tr✉❡ st❛t❡♠❡♥t ❞♦❡s ♥♦t ❤❛✈❡ t♦ ❜❡ tr✉❡❀ ❡①❛♠♣❧❡✿

x = 1 =⇒ x2 = 1✳

❊①❡r❝✐s❡ ✶✳✶✳✸ ❙✉❣❣❡st ②♦✉r ♦✇♥ ❡①❛♠♣❧❡ ♦❢ ❛ tr✉❡ st❛t❡♠❡♥t t❤❡ ❝♦♥✈❡rs❡ ♦❢ ✇❤✐❝❤ ✐s ❢❛❧s❡✳

■♥ ♦✉r ❝❛s❡✱ t❤❡ ✐♠♣❧✐❝❛t✐♦♥s ❣♦ ❜♦t❤ ✇❛②s✦ ❈♦♠❜✐♥❡❞✱ t❤❡ st❛t❡♠❡♥t ❛♥❞ ✐ts ❝♦♥✈❡rs❡ ❢♦r♠ ❛♥

◮

■❋ ❆◆❉ ❖◆▲❨ ■❋

❚❤❡ ✈❡❧♦❝✐t② ✐s ♣♦s✐t✐✈❡

❡q✉✐✈❛❧❡♥❝❡ ✿

t❤❡ ♠♦t✐♦♥ ✐s ✐♥ t❤❡ ♣♦s✐t✐✈❡ ❞✐r❡❝t✐♦♥✳

❲❡ ✇✐❧❧ ✉s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥✈❡♥✐❡♥t ❛❜❜r❡✈✐❛t✐♦♥✿

❊q✉✐✈❛❧❡♥❝❡

⇐⇒ ■t r❡❛❞s ✏✐❢ ❛♥❞ ♦♥❧② ✐❢ ✑ ♦r ✏✐s ❡q✉✐✈❛❧❡♥t t♦✑✳

❚❤❡♥ ♦✉r st❛t❡♠❡♥t ✐s ✇r✐tt❡♥ ❛s ❢♦❧❧♦✇s✿

◮

❚❤❡ ✈❡❧♦❝✐t② ✐s ♣♦s✐t✐✈❡

⇐⇒

t❤❡ ♠♦t✐♦♥ ✐s ✐♥ t❤❡ ♣♦s✐t✐✈❡ ❞✐r❡❝t✐♦♥✳

❚❤❡ t✇♦ ♣❛rts ♦❢ ❛♥ ❡q✉✐✈❛❧❡♥❝❡ ❛r❡ ✐♥t❡r❝❤❛♥❣❡❛❜❧❡✦ ❚❤r♦✉❣❤♦✉t t❤❡ t❡①t✱ ✇❡ ✇✐❧❧ ❛♣♣❧② t❤✐s ❛♥❛❧②s✐s t♦ ❛❧❧ st❛t❡♠❡♥ts ✇❡ ♠❛❦❡ ✐♥ ♦r❞❡r t♦ ❡♥s✉r❡ t❤❛t ✇❡ ❦♥♦✇ ❡①❛❝t❧② ✇❤❛t ✇❡ ❛r❡ s❛②✐♥❣ ❛♥❞ ✇❤❛t ✇❡ ❛r❡

♥♦t

s❛②✐♥❣✳

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

♠✐♥

❧♦❝❛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✮✿

t❤❡② ❛r❡

✶✳✶✳

❲❤❛t ✐s ❝❛❧❝✉❧✉s ❛❜♦✉t❄

✶✺

❚❤❡ ❞❛t❛ ✐s ❢✉rt❤❡r♠♦r❡ ✐❧❧✉str❛t❡❞ ❛s ❛ ✏s❝❛tt❡r ♣❧♦t✑ ✭r✐❣❤t✮✳ ■t st❛rts t♦ ❧♦♦❦ ❧✐❦❡ ❛

❝✉r✈❡ ✦

❲❛r♥✐♥❣✦ ❚❤❡ ♣❧♦t ✐s ♥♦t❤✐♥❣ ❜✉t ❛ ✈✐s✉❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ❞❛t❛✳

❲❤❛t ❤❛s ❤❛♣♣❡♥❡❞ t♦ t❤❡ ♠♦✈✐♥❣ ♦❜❥❡❝t ❝❛♥ ♥♦✇ ❜❡ r❡❛❞ ❢r♦♠ t❤❡ ❣r❛♣❤✿

•

■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✐♦♥✳

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

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

❊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♦✇ sq✉❛r❡ ✐t✿

2

❛♥❞ ❝♦❧✉♠♥

3

❛♥❞

✶✳✶✳

❲❤❛t ✐s ❝❛❧❝✉❧✉s ❛❜♦✉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✿

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

❂❘❈❬✲✶❪✲❘❬✲✶❪❈❬✲✶❪ ❍❡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✿

✶✳✶✳ ❲❤❛t ✐s ❝❛❧❝✉❧✉s ❛❜♦✉t❄

✶✼

❚❤✐s ♥❡✇ ❞❛t❛ ✐s ✐❧❧✉str❛t❡❞ ✇✐t❤ t❤❡ s❡❝♦♥❞ s❝❛tt❡r ♣❧♦t✳ ❲❤❛t ❤❛s ❤❛♣♣❡♥❡❞ t♦ t❤❡ ♠♦✈✐♥❣ ♦❜❥❡❝t ❝❛♥ ♥♦✇ ❜❡ ❡❛s✐❧② r❡❛❞ ❢r♦♠ t❤❡ s❡❝♦♥❞ ❣r❛♣❤✿ • ❚❤❡ ✈❡❧♦❝✐t② ✇❛s ♣♦s✐t✐✈❡ ✐♥✐t✐❛❧❧② ✭✐t ✇❛s ♠♦✈✐♥❣ ✐♥ t❤❡ ♣♦s✐t✐✈❡ ❞✐r❡❝t✐♦♥✮❀

• t❤❡ ✈❡❧♦❝✐t② ✇❛s ❢❛✐r❧② ❤✐❣❤ ✭✐t ✇❛s ♠♦✈✐♥❣ ❢❛✐r❧② ❢❛st✮ ❜✉t t❤❡♥ ✐t st❛rt❡❞ t♦ ❞❡❝❧✐♥❡ ✭s❧♦✇ ❞♦✇♥✮❀ • t❤❡ ✈❡❧♦❝✐t② ✇❛s ③❡r♦ ✭✐t st♦♣♣❡❞✮ ❢♦r ❛ ✈❡r② s❤♦rt ♣❡r✐♦❞❀

• t❤❡♥ 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♦♠❡ ❢❛❝ts ❛❜♦✉t t❤❡ q✉❛❧✐t❛t✐✈❡ ❛♥❞ q✉❛♥t✐t❛t✐✈❡ ❜❡❤❛✈✐♦r ♦❢ t❤❡ ❢♦r♠❡r✳ ❊①❡r❝✐s❡ ✶✳✶✳✺

❚❤❡ ♣❧♦t ♦❢ t❤❡ ❧♦❝❛t✐♦♥ ✐s s❤♦✇♥ ❜❡❧♦✇✿

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

♠✐♥ ♠✐❧❡s✴♠✐♥

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✿

✶✳✶✳ ❲❤❛t ✐s ❝❛❧❝✉❧✉s ❛❜♦✉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 ✇❤❡r❡ ✇❡ ❛r❡ ❛t t❤❡ ❡♥❞ ♦❢ ❡❛❝❤ ♦❢ t❤❡s❡ ♦♥❡✲♠✐♥✉t❡ ✐♥t❡r✈❛❧s✱ ✇❡ ❝♦♠♣✉t❡ t❤❡ s✉♠ ♦❢ ✈❡❧♦❝✐t② ✭❞✐s♣❧❛❝❡♠❡♥t✮ ❢♦r ❡❛❝❤ ✐♥t❡r✈❛❧ ❜② ♣✉❧❧✐♥❣ t❤❡ ❞❛t❛ ❢r♦♠ t❤❡ r♦✇ ♦❢ ✈❡❧♦❝✐t✐❡s ✇✐t❤ t❤❡ ♣r❡✈✐♦✉s r❡s✉❧t✳ ❚❤✐s ✐s ❤♦✇ t❤❡ ✜rst ♦♥❡ ✐s ❝♦♠♣✉t❡❞✱ ✉♥❞❡r t❤❡ ❛ss✉♠♣t✐♦♥ t❤❛t t❤❡ ✐♥✐t✐❛❧ ❧♦❝❛t✐♦♥ ✐s 0✿ t✐♠❡ ✈❡❧♦❝✐t②

♠✐♥ ♠✐❧❡s✴♠✐♥

s✉♠ ❧♦❝❛t✐♦♥

♠✐❧❡s

0

1 0.10 ↓ 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 →

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

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

✶✳✶✳ ❲❤❛t ✐s ❝❛❧❝✉❧✉s ❛❜♦✉t❄

✶✾

❚❤❡ ❞❛t❛ ✐s ❛❧s♦ ✐❧❧✉str❛t❡❞ ❛s t❤❡ s❡❝♦♥❞ s❝❛tt❡r ♣❧♦t ♦♥ t❤❡ r✐❣❤t✳ ❚❤✉s✱ ❛s t❤❡ ❢♦r♠❡r ❞❛t❛ s❡t r❡❝♦r❞s s♦♠❡ ❢❛❝ts ❛❜♦✉t t❤❡ q✉❛♥t✐t❛t✐✈❡ ❜❡❤❛✈✐♦r ♦❢ t❤❡ ❧❛tt❡r✱ ✇❡ ❛r❡ ❛❜❧❡ t♦ ❝♦♠❜✐♥❡ t❤✐s ✐♥❢♦r♠❛t✐♦♥ t♦ r❡❝♦✈❡r t❤❡ ❧❛tt❡r✳ ❊①❡r❝✐s❡ ✶✳✶✳✻

❚❤❡ ♣❧♦t ♦❢ t❤❡ ✈❡❧♦❝✐t② ✐s s❤♦✇♥ ❜❡❧♦✇✿

❉❡s❝r✐❜❡ ❤♦✇ t❤❡ ✈❡❧♦❝✐t② ❛♥❞ t❤❡ ❧♦❝❛t✐♦♥ ❤❛✈❡ ❜❡❡♥ ❝❤❛♥❣✐♥❣✳ ❆❝t ✐t ♦✉t ❜② st❡♣♣✐♥❣ ❧❡❢t ❛♥❞ r✐❣❤t✳ ❙❦❡t❝❤ t❤❡ ♣❧♦t ♦❢ t❤❡ ❧♦❝❛t✐♦♥✳

❚❤✐s ✐s ✇❤❛t ✇❡ ❤❛✈❡ ❞✐s❝♦✈❡r❡❞✿

◮

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

▲❡t✬s ♠❛❦❡ s✉r❡ ✇❡ ❦♥♦✇ ✇❤❛t ✇❡ ❛r❡ ❛♥❞ ❛r❡ ♥♦t s❛②✐♥❣✳ ❲❡ ❛r❡ s❛②✐♥❣✿

◮ ■❋

✇❡ ❦♥♦✇ t❤❡ ❧♦❝❛t✐♦♥✱

❚❍❊◆

✇❡ ❦♥♦✇ t❤❡ ✈❡❧♦❝✐t②✳

❲❡ ❛r❡ ♥♦t s❛②✐♥❣ t❤❡ ❝♦♥✈❡rs❡✳ ❚❤✐s ✇♦✉❧❞ ❜❡ ❢❛❧s❡✿ ❲❡ ❦♥♦✇ t❤❡ ✈❡❧♦❝✐t② ❚❤❡ ❝♦rr❡❝t✱ ✏♣❛rt✐❛❧✑✱ ❝♦♥✈❡rs❡ ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✿

◮ ■❋

✇❡ ❦♥♦✇ t❤❡ ✈❡❧♦❝✐t②

❆◆❉

✇❡ ❦♥♦✇ t❤❡ ✐♥✐t✐❛❧ ❧♦❝❛t✐♦♥✱

❚❍❊◆

6=⇒

✇❡ ❦♥♦✇ t❤❡ ❧♦❝❛t✐♦♥✳

✇❡ ❦♥♦✇ t❤❡ ❧♦❝❛t✐♦♥✳

❊①❡r❝✐s❡ ✶✳✶✳✼

✭❛✮ ❨♦✉r ❧♦❝❛t✐♦♥ ✐s r❡❝♦r❞❡❞ ❡✈❡r② ❤❛❧❢✲❤♦✉r✱ s❤♦✇♥ ❜❡❧♦✇✳ ❊st✐♠❛t❡ ②♦✉r ✈❡❧♦❝✐t② ❛s ✐t ❝❤❛♥❣❡s ✇✐t❤ t✐♠❡✳ ✭❜✮ ❨♦✉r ✈❡❧♦❝✐t② ✐s r❡❝♦r❞❡❞ ❡✈❡r② ❤❛❧❢✲❤♦✉r✱ s❤♦✇♥ ❜❡❧♦✇✳ ❊st✐♠❛t❡ ②♦✉r ❧♦❝❛t✐♦♥ ❛s ✐t ❝❤❛♥❣❡s

✶✳✷✳

❚❤❡ r❡❛❧ ♥✉♠❜❡r ❧✐♥❡

✷✵

✇✐t❤ t✐♠❡✳ t✐♠❡

0 .5 1 1.5 2

❧♦❝❛t✐♦♥

t✐♠❡

20 30 20 20 50

✈❡❧♦❝✐t②

0 .5 1 1.5 2

5 3 10 10 −10

❲❡ ❝❛♥ ❝♦♥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❤❡ ❧♦❝❛t✐♦♥✱ ❢♦r ❛♥♦t❤❡r✱ t❤❡ t❡♠♣❡r❛t✉r❡ ❛s ❛ s✐♥❣❧❡ ❡①❛♠♣❧❡ ♦❢ t❤❡ ❜r❡❛❞t❤ ♦❢ ❛♣♣❧✐❝❛❜✐❧✐t② ♦❢ t❤❡s❡ ✐❞❡❛s✳

✶✳✷✳ ❚❤❡ r❡❛❧ ♥✉♠❜❡r ❧✐♥❡

❚❤❡ st❛rt✐♥❣ ♣♦✐♥t ♦❢ st✉❞②✐♥❣ ♥✉♠❜❡rs ✐s t❤❡

♥❛t✉r❛❧ ♥✉♠❜❡rs

✿

0, 1, 2, 3, ... ❚❤❡② ❛r❡ ✐♥✐t✐❛❧❧② ✉s❡❞ ❢♦r ❝♦✉♥t✐♥❣✳ ❚❤❡ ♥❡①t st❡♣ ✐s t❤❡

✐♥t❡❣❡rs

✿

..., −3, −2, −1, 0, 1, 2, 3, ... ❚❤❡② ❝❛♥ ❜❡ ✉s❡❞ ❢♦r st✉❞②✐♥❣ t❤❡ s♣❛❝❡ ❛♥❞ ❧♦❝❛t✐♦♥s✱ ❛s ❢♦❧❧♦✇s✳ ■♠❛❣✐♥❡ ❢❛❝✐♥❣ ❛ ❢❡♥❝❡ s♦ ❧♦♥❣ t❤❛t ②♦✉ ❝❛♥✬t s❡❡ ✇❤❡r❡ ✐t ❡♥❞s✳ ❲❡ st❡♣ ❛♥❞ t❤❡r❡ ✐s st✐❧❧ ♠♦r❡ t♦ s❡❡✿

❛✇❛②

❢r♦♠ t❤❡ ❢❡♥❝❡ ♠✉❧t✐♣❧❡ t✐♠❡s

✶✳✷✳

❚❤❡ r❡❛❧ ♥✉♠❜❡r ❧✐♥❡

■s t❤❡ ♥✉♠❜❡r ♦❢ ♣❧❛♥❦s

✷✶

✐♥✜♥✐t❡ ❄

■t ♠❛② ❜❡✳ ❋♦r

❝♦♥✈❡♥✐❡♥❝❡✱

✇❡ ✇✐❧❧ ❥✉st ❛ss✉♠❡ t❤❛t ✇❡ ❝❛♥ ❣♦ ♦♥ ✇✐t❤

t❤✐s ❢♦r ❛s ❧♦♥❣ ❛s ♥❡❝❡ss❛r②✳ ❲❡ ✈✐s✉❛❧✐③❡ t❤❡s❡ ❛s ♠❛r❦✐♥❣s ♦♥ ❛ str❛✐❣❤t ❧✐♥❡✱ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ♦r❞❡r ♦❢ t❤❡ ♣❧❛♥❦s✿

❚❤❡ ❛ss✉♠♣t✐♦♥ ✐s t❤❛t t❤❡ ❧✐♥❡ ❛♥❞ t❤❡ ♠❛r❦✐♥❣s ❝♦♥t✐♥✉❡ ✇✐t❤♦✉t st♦♣♣✐♥❣ ✐♥ ❜♦t❤ ❞✐r❡❝t✐♦♥s✱ ✇❤✐❝❤ ✐s ❝♦♠♠♦♥❧② r❡♣r❡s❡♥t❡❞ ❜② ✏ ...✑✳ ❚❤❡ s❛♠❡ ✐❞❡❛ ❛♣♣❧✐❡s t♦ t❤❡

♠✐❧❡st♦♥❡s

♦♥ t❤❡ r♦❛❞❀ t❤❡② ❛r❡ ❛❧s♦ ♦r❞❡r❡❞

❛♥❞ ♠✐❣❤t ❝♦♥t✐♥✉❡ ✐♥❞❡✜♥✐t❡❧②✳ ❙♦✱ ✇❡ ③♦♦♠❡❞

♦✉t

t♦ s❡❡ t❤❡ ❢❡♥❝❡✳ ❙✉♣♣♦s❡ ♥♦✇ ✇❡ ③♦♦♠

✐♥

♦♥ ❛ ❧♦❝❛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♦ ❛

■t✬s ❛s ✐❢ ✇❡ ❛❞❞

♦♥❡ ♠❛r❦

r✉❧❡r ✿

❜❡t✇❡❡♥ ❛♥② t✇♦ ❛♥❞ t❤❡♥ ❛❞❞ ❛♥♦t❤❡r ♦♥❡ ❜❡t✇❡❡♥ ❡✐t❤❡r ♦❢ t❤❡ t✇♦ ♣❛✐rs ✇❡

❤❛✈❡ ❝r❡❛t❡❞✳ ❲❡ ❦❡❡♣ r❡♣❡❛t✐♥❣ t❤✐s st❡♣✳ ❊✈❡♥ t❤♦✉❣❤ t❤✐s r✉❧❡r ❣♦❡s ♦♥❧② t♦ ✐♠❛❣✐♥❡ t❤❛t t❤❡ ♣r♦❝❡ss ❝♦♥t✐♥✉❡s ✐♥❞❡✜♥✐t❡❧②✿

1/16

♦❢ ❛♥ ✐♥❝❤✱ ✇❡ ❝❛♥

✶✳✷✳ ❚❤❡ r❡❛❧ ♥✉♠❜❡r ❧✐♥❡

✷✷

■s t❤❡ ❞❡♣t❤ ✐♥✜♥✐t❡ ❄ ■t ♠❛② ❜❡✳ ❋♦r ❝♦♥✈❡♥✐❡♥❝❡✱ ✇❡ ✇✐❧❧ ❥✉st ❛ss✉♠❡ t❤❛t ✇❡ ❝❛♥ ❣♦ ♦♥ ✇✐t❤ t❤✐s ❢♦r ❛s ❧♦♥❣ ❛s ♥❡❝❡ss❛r②✳ ■❢ ✇❡ ❛❞❞ ♥✐♥❡ ♠❛r❦s ❛t ❛ t✐♠❡✱ t❤❡ r❡s✉❧t ✐s ❛ ♠❡tr✐❝ r✉❧❡r ✿

❍❡r❡✱ ✇❡ ❣♦ ❢r♦♠ ♠❡t❡rs t♦ ❞❡❝✐♠❡t❡rs✱ t♦ ❝❡♥t✐♠❡t❡rs✱ t♦ ♠✐❧❧✐♠❡t❡rs✱ ❡t❝✳ ❚♦ s❡❡ ✐t ❛♥♦t❤❡r ✇❛②✱ ✇❡ ❛❧❧♦✇ ♠♦r❡ ❛♥❞ ♠♦r❡ ❞❡❝✐♠❛❧s ✐♥ ♦✉r ♥✉♠❜❡rs✿ 1.55 : 1. 1.5 1.55 1.550 1.5500 ... 1/3 : .3 .33 .333 .3333 .33333 ... π : 3. 3.1 3.14 3.141 3.1415 ...

■♥ ♦r❞❡r t♦ ✈✐s✉❛❧✐③❡ ❛❧❧ ♥✉♠❜❡rs✱ ✇❡ ✜rst ❛rr❛♥❣❡ t❤❡ ✐♥t❡❣❡rs ✐♥ ❛ ❧✐♥❡ ❛♥❞ t❤❡♥ t❤❡ ❧✐♥❡ ♦❢ ♥✉♠❜❡rs ✐s ❜✉✐❧t✳ ■t t❛❦❡s s❡✈❡r❛❧ st❡♣s✳ ❙t❡♣ ✶✿ ❉r❛✇ ❛ ❧✐♥❡✱ ❝❛❧❧❡❞ ❛♥ ❛①✐s ✭❤♦r✐③♦♥t❛❧ ✇❤❡♥ ❝♦♥✈❡♥✐❡♥t✮✿

❙t❡♣ ✷✿ ❈❤♦♦s❡ ♦♥❡ ♦❢ t❤❡ t✇♦ ❡♥❞s ♦❢ t❤❡ ❧✐♥❡ ❛s t❤❡ ♣♦s✐t✐✈❡ ❞✐r❡❝t✐♦♥ ✭t❤❡ ♦♥❡ ♦♥ t❤❡ r✐❣❤t ✇❤❡♥ ❝♦♥✈❡♥✐❡♥t✮✱ t❤❡♥ t❤❡ ♦t❤❡r ✐s t❤❡ ♥❡❣❛t✐✈❡ ✿

❙t❡♣ ✸✿ ❙❡t ❛ ♣♦✐♥t O ✭❛ ❧❡tt❡r✱ ♥♦t ❛ ♥✉♠❜❡r✮ ❛s t❤❡ ♦r✐❣✐♥ ✿

❙t❡♣ ✹✿ ❈❤♦♦s❡ ❛ s❡❣♠❡♥t ♦❢ t❤❡ ❧✐♥❡ ❛s t❤❡ ✉♥✐t ♦❢ ❧❡♥❣t❤✿

✶✳✷✳

❚❤❡ r❡❛❧ ♥✉♠❜❡r ❧✐♥❡

❙t❡♣ ✺✿

✷✸

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

❞✐r❡❝t✐♦♥✱ ❛♥❞ ♥❡❣❛t✐✈❡ ✐♥ t❤❡ ♥❡❣❛t✐✈❡ ❞✐r❡❝t✐♦♥✮ ❛♥❞ ❛❞❞ ♠❛r❦s✱ ❝❛❧❧❡❞ t❤❡

O

✭♣♦s✐t✐✈❡ ✐♥ t❤❡ ♣♦s✐t✐✈❡

❝♦♦r❞✐♥❛t❡s ✿

❙t❡♣ ✻✿ ❉✐✈✐❞❡ t❤❡ s❡❣♠❡♥ts ❢✉rt❤❡r ✐♥t♦ ❢r❛❝t✐♦♥s ♦❢ t❤❡ ✉♥✐t✱ ❡t❝✳✿

❚❤❡ ❡♥❞ r❡s✉❧t ❞❡♣❡♥❞s ♦♥ ✇❤❛t t❤❡ ❜✉✐❧❞✐♥❣ ❜❧♦❝❦ ✐s✳ ■t ♠❛② ❝♦♥t❛✐♥ ❣❛♣s ❛♥❞ ❧♦♦❦ ❧✐❦❡ ❛ r✉❧❡r ✭♦r ❛ ❝♦♠❜✮ ❛s ❞✐s❝✉ss❡❞ ❛❜♦✈❡✳ ■t ♠❛② ❛❧s♦ ❜❡ s♦❧✐❞ ❛♥❞ ❧♦♦❦ ❧✐❦❡ ❛ t✐❧❡ ♦r ❛ ❞♦♠✐♥♦ ♣✐❡❝❡✿

❙♦✱ ✇❡ 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

✐rr❛t✐♦♥❛❧ ♥✉♠❜❡rs ❝♦♠❡ ✐♥t♦ ♣❧❛②✳ ❚♦❣❡t❤❡r✱ t❤❡② ♠❛❦❡ ✉♣ t❤❡ r❡❛❧ ♥✉♠❜❡rs ❛♥❞ t❤❡ r❡❛❧ ♥✉♠❜❡r ❧✐♥❡✳ ❲❡ t❤✐♥❦ ♦❢ t❤✐s ❧✐♥❡ ❛s ❝♦♠♣❧❡t❡ ❀ t❤❡r❡ ❛r❡ ♥♦ ♠✐ss✐♥❣ ♣♦✐♥ts✳ ❆s ❛♥ ✐❧❧✉str❛t✐♦♥✱ ❛♥ r❛t✐♦♥❛❧✳ ❚❤❛t✬s ❤♦✇ t❤❡

✏✐♥❝♦♠♣❧❡t❡✑ r♦♣❡ ✇♦♥✬t ❤❛♥❣✿

❲❡ ✉s❡ t❤✐s s❡t✉♣ t♦ ♣r♦❞✉❝❡ ❛ ❝♦rr❡s♣♦♥❞❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❧♦❝❛t✐♦♥s ♦♥ t❤❡ ❧✐♥❡ ❛♥❞ t❤❡ r❡❛❧ ♥✉♠❜❡rs✿

❧♦❝❛t✐♦♥

❲❡ ✇✐❧❧ ❢♦❧❧♦✇ t❤✐s ❝♦rr❡s♣♦♥❞❡♥❝❡ ✐♥

P ←→

♥✉♠❜❡r

❜♦t❤ ❞✐r❡❝t✐♦♥s✱ ❛s ❢♦❧❧♦✇s✿

x

✶✳✸✳

❙❡q✉❡♥❝❡s

✷✹

✶✳ ❋✐rst✱ s✉♣♣♦s❡

P

✐s ❛

t❤❡ ✏❝♦♦r❞✐♥❛t❡✑ ♦❢

P✿

✷✳ ❈♦♥✈❡rs❡❧②✱ s✉♣♣♦s❡ ❚❤❛t✬s t❤❡

❧♦❝❛t✐♦♥

❧♦❝❛t✐♦♥ ♦♥ t❤❡ ❧✐♥❡✳ s♦♠❡ ♥✉♠❜❡r x✳

x

♦❢

✐s ❛

x✿

♥✉♠❜❡r✳

s♦♠❡ ♣♦✐♥t

❲❡ t❤❡♥ ✜♥❞ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♠❛r❦ ♦♥ t❤❡ ❧✐♥❡✳ ❚❤❛t✬s

❲❡ t❤✐♥❦ ♦❢ ✐t ❛s ❛ ✏❝♦♦r❞✐♥❛t❡✑ ❛♥❞ ✜♥❞ ✐ts ♠❛r❦ ♦♥ t❤❡ ❧✐♥❡✳

P

♦♥ t❤❡ ❧✐♥❡✳

❖♥❝❡ t❤✐s s②st❡♠ ♦❢ ❝♦♦r❞✐♥❛t❡s ✐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❡❛t❡❞ ❛ ✈✐s✉❛❧ ♠♦❞❡❧ ♦❢ t❤❡ r❡❛❧ ♥✉♠❜❡rs✳

r❡❛❧ ♥✉♠❜❡r ❧✐♥❡

❉❡♣❡♥❞✐♥❣ ♦♥ t❤❡ r❡❛❧ ♥✉♠❜❡r ♦r ❛ ❝♦❧❧❡❝t✐♦♥ ♦❢

♥✉♠❜❡rs t❤❛t ✇❡ ❛r❡ tr②✐♥❣ t♦ ✈✐s✉❛❧✐③❡✱ ✇❡ ❝❤♦♦s❡ ✇❤❛t ♣❛rt ♦❢ t❤❡ r❡❛❧ ❧✐♥❡ t♦ ❡①❤✐❜✐t❀ ❢♦r ❡①❛♠♣❧❡✱ t❤❡ ③❡r♦ ♠❛② ♦r ♠❛② ♥♦t ❜❡ ✐♥ t❤❡ ♣✐❝t✉r❡✳ ❲❡ ❛❧s♦ ❤❛✈❡ t♦ ❝❤♦♦s❡ ❛♥ ❛♣♣r♦♣r✐❛t❡ ❧❡♥❣t❤ ♦❢ t❤❡ ✉♥✐t s❡❣♠❡♥t ✐♥ ♦r❞❡r ❢♦r t❤❡ ♥✉♠❜❡rs t♦ ✜t ✐♥✳ ■♥ ❛❞❞✐t✐♦♥ t♦ t❤❡ r✉❧❡r✱ ❛♥♦t❤❡r ✇❛② t♦ ✈✐s✉❛❧✐③❡ ♥✉♠❜❡rs ✐s ✇✐t❤ ❧❡✈❡❧s ♦❢ ❣r❛② ❛r❡ ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡ ♥✉♠❜❡rs ❜❡t✇❡❡♥

0

❛♥❞

255✳

❝♦❧♦rs✳

■♥ ❢❛❝t✱ ✐♥ ❞✐❣✐t❛❧ ✐♠❛❣✐♥❣ t❤❡

❆ s❤♦rt❡r s❝❛❧❡ ✕

1, 2, ..., 20

✕ ✐s ✉s❡❞ ✐♥

t❤❡ ✐❧❧✉str❛t✐♦♥ ❜❡❧♦✇ ✭t♦♣✮✿

■t ✐s ❛❧s♦ ♦❢t❡♥ ❝♦♥✈❡♥✐❡♥t t♦ ❛ss♦❝✐❛t❡ ❜❧✉❡ ✇✐t❤ ♥❡❣❛t✐✈❡ ❛♥❞ r❡❞ ✇✐t❤ ♣♦s✐t✐✈❡ ♥✉♠❜❡rs ✭❜♦tt♦♠✮✳ ❊①❡r❝✐s❡ ✶✳✷✳✶

❚❤✐♥❦ ♦❢ ♦t❤❡r ❡①❛♠♣❧❡s ✇❤❡♥ ♥✉♠❜❡rs ❛r❡ ✈✐s✉❛❧✐③❡❞ ✇✐t❤ ❝♦❧♦rs✳

✶✳✸✳ ❙❡q✉❡♥❝❡s

❚❤❡ ❧✐sts ♦❢ ♥✉♠❜❡rs ❝♦♥s✐❞❡r❡❞ ✐♥ t❤❡ ✜rst s❡❝t✐♦♥ ❛r❡

s❡q✉❡♥❝❡s ✿

t❤❡ ❧♦❝❛t✐♦♥s ❛♥❞ t❤❡ ✈❡❧♦❝✐t✐❡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✿

✶✳✸✳ ❙❡q✉❡♥❝❡s

✷✺

❲❡ ✐❣♥♦r❡ t❤❡ t✐♠❡ ❢♦r ♥♦✇ ❛♥❞ ❝♦♥❝❡♥tr❛t❡ ♦♥ t❤❡ ❧♦❝❛t✐♦♥s ♦♥❧②✳ ❲❡ ❤❛✈❡ t❤❡ ✜rst ❢❡✇ ✐♥ ❛ ❧✐st ✿

36, 35, 32, 27, 20, 11, 0 . ❚❤✐s ❞❛t❛ ❝❛♥ ❜❡ ✈✐s✉❛❧✐③❡❞ ❜② ♣❧❛❝✐♥❣ t❤❡ ❜❛❧❧ ❛t ❡✈❡r② ❝♦♦r❞✐♥❛t❡ ❧♦❝❛t✐♦♥ ♦♥ t❤❡ r❡❛❧ ❧✐♥❡✱ ♦r✐❡♥t❡❞ ✈❡rt✐❝❛❧❧② ♦r ❤♦r✐③♦♥t❛❧❧②✿

❚❤♦✉❣❤ ♥♦t ✉♥❝♦♠♠♦♥✱ t❤✐s ♠❡t❤♦❞ ♦❢ ✈✐s✉❛❧✐③❛t✐♦♥ ♦❢ ♠♦t✐♦♥✱ ♦r ♦❢ s❡q✉❡♥❝❡s ✐♥ ❣❡♥❡r❛❧✱ ❤❛s ✐ts ❞r❛✇❜❛❝❦s✿ ❖✈❡r❧❛♣♣✐♥❣ ♠❛② ❜❡ ✐♥❡✈✐t❛❜❧❡ ❛♥❞ t❤❡ ♦r❞❡r ♦❢ ❡✈❡♥ts ✐s ❧♦st ✭✉♥❧❡ss ✇❡ ❛❞❞ ❧❛❜❡❧s✮✳ ❆ ♠♦r❡ ♣♦♣✉❧❛r ❛♣♣r♦❛❝❤ ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✳ ❚❤❡ ✐❞❡❛ ✐s t♦ s❡♣❛r❛t❡ t✐♠❡ ❛♥❞ s♣❛❝❡✱ t♦ ❣✐✈❡ ❛ s❡♣❛r❛t❡ r❡❛❧ ❧✐♥❡✱ ❛♥ ❛①✐s✱ t♦ ❡❛❝❤ ♠♦♠❡♥t ♦❢ t✐♠❡✱ ❛♥❞ t❤❡♥ ❜r✐♥❣ t❤❡♠ ❜❛❝❦ t♦❣❡t❤❡r ✐♥ ♦♥❡ r❡❝t❛♥❣✉❧❛r ♣❧♦t✿

❚❤❡ ❧♦❝❛t✐♦♥ ✈❛r✐❡s ✕ ❛s ✐t ❞♦❡s ✕ ✈❡rt✐❝❛❧❧② ✇❤✐❧❡ t❤❡ t✐♠❡ ♣r♦❣r❡ss❡s ❤♦r✐③♦♥t❛❧❧②✳ ❚❤❡ r❡s✉❧t ✐s s✐♠✐❧❛r t♦ t❤❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ t❤❡ ❢r❛♠❡s ♦❢ t❤❡ ✈✐❞❡♦ ❛s s❡❡♥ ❛❜♦✈❡✳ ❚❤❡ ♣❧♦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

✶✳✸✳

❙❡q✉❡♥❝❡s

✷✻

❚❤❡ t❛❜❧❡ ✐s ❥✉st ❛s ❡✛❡❝t✐✈❡ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ❞❛t❛ ✐❢ ✇❡ ✢✐♣ ✐t❀ ✐t✬s ♠♦r❡ ❝♦♠♣❛❝t✿ ♠♦♠❡♥t✿ ❤❡✐❣❤t✿

1 2 3 4 5 6 7 36 35 32 27 20 11 0 ❲❛r♥✐♥❣✦ ■t ✐s ❡♥t✐r❡❧② ❛ ♠❛tt❡r ♦❢ ❝♦♥✈❡♥✐❡♥❝❡ t♦ r❡♣r❡s❡♥t ♦✉r ❞❛t❛ ❛s ❛ t✇♦✲❝♦❧✉♠♥ t❛❜❧❡ ✭❡s♣❡❝✐❛❧❧② ✐♥ ❛ s♣r❡❛❞s❤❡❡t✮ ♦r ❛ t✇♦✲r♦✇ t❛❜❧❡✳

■♥ ❡✐t❤❡r ❝❛s❡✱

✐t✬s ❛ ❧✐st ♦❢ ♣❛✐rs ♦❢ ♥✉♠❜❡rs✳

❙♦✱ t❤❡ ♠♦st ❝♦♠♠♦♥ ✇❛② t♦ ✈✐s✉❛❧✐③❡ ❛ s❡q✉❡♥❝❡ ♦❢

♥✉♠❜❡rs

✐s ❛s ❛ s❡q✉❡♥❝❡ ♦❢

♣♦✐♥ts

♦♥ ❛ s❡q✉❡♥❝❡ ♦❢

✈❡rt✐❝❛❧ ❛①❡s✿

■t ✐s ❛❧s♦ ❝♦♠♠♦♥ t♦ r❡♣r❡s❡♥t t❤❡ s❛♠❡ ♥✉♠❜❡rs ❛s ✈❡rt✐❝❛❧

❜❛rs ✿

❲❛r♥✐♥❣✦ ❚❤❡ ❣r❛♣❤ ✐s ❥✉st ❛ ✈✐s✉❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ❞❛t❛✳

❚♦ r❡♣r❡s❡♥t ❛ s❡q✉❡♥❝❡ ❛❧❣❡❜r❛✐❝❛❧❧②✱ ✇❡ ✜rst ❣✐✈❡ ✐t ❛ ♥❛♠❡✱ 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✑✱

✐♥❞❡①

✐♥❞✐❝❛t❡s t❤❡ ♣❧❛❝❡ ♦❢ t❤❡ t❡r♠

❡t❝✳

❚❤❡ ❧❡tt❡r ✏♥✑ ✐s ♦❢t❡♥ t❤❡ ♣r❡❢❡rr❡❞ ❝❤♦✐❝❡ ❢♦r t❤❡ ✐♥❞❡① ❜❡❝❛✉s❡ ✐t ♠✐❣❤t st❛♥❞ ❢♦r ✏♥❛t✉r❛❧ ♥✉♠❜❡rs✑✿

1, 2, 3, 4, ...✳

❆s ❜❡❢♦r❡✱ ✏✳✳✳✑ ✐♥❞✐❝❛t❡s ❛ ❝♦♥t✐♥✉✐♥❣ ♣❛tt❡r♥✿ ❚❤❡ ✐♥❞✐❝❡s ❝♦♥t✐♥✉❡ t♦ ❣r♦✇ ✐♥❝r❡♠❡♥t❛❧❧②✳

✶✳✸✳

❙❡q✉❡♥❝❡s

✷✼

❊①❛♠♣❧❡ ✶✳✸✳✷✿ ❢❛❧❧✐♥❣ ❜❛❧❧

❋♦r t❤❡ ❧❛st ❡①❛♠♣❧❡✱ ❧❡t✬s ♥❛♠❡ t❤❡ s❡q✉❡♥❝❡ ♠♦♠❡♥t✿ ❤❡✐❣❤t✿ ❤❡✐❣❤t✿

1 h1 || 36

h

❢♦r ✏❤❡✐❣❤t✑✳ ❚❤❡♥ t❤❡ ❛❜♦✈❡

2 h2 || 35

3 h3 || 32

4 h4 || 27

5 h5 || 20

6 h6 || 11

7 h7 || 0

t❛❜❧❡

t❛❦❡ t❤✐s ❢♦r♠✿

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

❚❤✐s ✐s t❤❡ s❛♠❡ t❛❜❧❡ ❛❧✐❣♥❡❞ ✈❡rt✐❝❛❧❧②✿ ♠♦♠❡♥t

1 2 3 4 5 6 7 ..

❤❡✐❣❤t

❤❡✐❣❤t

h1 h2 h3 h4 h5 h6 h7 ..

36 35 32 27 20 11 0 ..

= = = = = = =

❊✐t❤❡r t❛❜❧❡ ✐s ❛ ❧✐st ♦❢ ✐❞❡♥t✐t✐❡s t❤❛t ❝❛♥ ❜❡ ✇r✐tt❡♥ ✐♥ ❛♥② ♦r❞❡r✿

h3 = 32 h5 = 20 h7 = 0 h1 = 36 h2 = 35

h4 = 27

h4 = 27 h6 = 11

▲❡t✬s ❞❡❝♦♥str✉❝t t❤❡ ♥♦t❛t✐♦♥✿ ■♥❞❡① ♦❢ ❛ t❡r♠

a ↑

✐♥❞❡①

↓

n

♥❛♠❡

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ s♣❡❝✐❢② ❛ s❡q✉❡♥❝❡ ✜rst ❛♥❞ t❤❡♥ s♣❡❝✐❢② t❤❡ ❧♦❝❛t✐♦♥ ♦❢ t❤❡ t❡r♠ ✇✐t❤✐♥ t❤❡ s❡q✉❡♥❝❡✳ ■♥❞✐❝❡s s❡r✈❡ ❛s

t❛❣s ✿

❆ s❡q✉❡♥❝❡ ❝❛♥ ❝♦♠❡ ❢r♦♠ ❛ ❧✐st ♦r ❛ t❛❜❧❡ ✉♥❧❡ss ✐t✬s ✐♥✜♥✐t❡✳ ■♥✜♥✐t❡ s❡q✉❡♥❝❡s ♦❢t❡♥ ❝♦♠❡ ❢r♦♠ ❊①❛♠♣❧❡ ✶✳✸✳✸✿ s❡q✉❡♥❝❡ ♦❢ r❡❝✐♣r♦❝❛❧s

❚❤❡ ❢♦r♠✉❧❛✿

an = 1/n , ❣✐✈❡s r✐s❡ t♦ t❤❡ s❡q✉❡♥❝❡✱

a1 = 1, a2 = 1/2, a3 = 1/3, a4 = 1/4, ...

❢♦r♠✉❧❛s✳

✶✳✸✳ ❙❡q✉❡♥❝❡s

✷✽

■♥❞❡❡❞✱ r❡♣❧❛❝✐♥❣

n

✐♥ t❤❡ ❢♦r♠✉❧❛ ✇✐t❤

♦♥❡✱ ❛s ❢♦❧❧♦✇s✳ ❲❡ ❡♥t❡r

n

1✱

2✱ 3✱ ❡t❝✳ ♣r♦❞✉❝❡s t❤❡ ♥✉♠❜❡rs ♦♥ t❤❡ ❧✐st ♦♥❡ ❜② an ❛♣♣❡❛rs ❛t t❤❡ ❡♥❞ ♦❢ t❤❡ ❝♦♠♣✉t❛t✐♦♥✳ ■♥ ♦t❤❡r ❜❧❛♥❦ ❜♦① ✭✇❤❡r❡ n ✉s❡❞ t♦ ❜❡✮ ✐♥ t❤❡ ❢♦r♠✉❧❛✿

t❤❡♥

✐♥t♦ t❤❡ ❢♦r♠✉❧❛✱ ❛♥❞

✇♦r❞s✱ ✇❡ ♣❧❛❝❡ t❤❡ ❝✉rr❡♥t ✈❛❧✉❡ ♦❢

n

✐♥s✐❞❡ ❛

a ✐♥s❡rt

1 .  ↑

=

↑ n

✐♥s❡rt

n

■t ✐s ❝❛❧❧❡❞ ✏s✉❜st✐t✉t✐♦♥✑✳ ❲❡ ❞♦ t❤✐s s❡✈❡♥ t✐♠❡s ❜❡❧♦✇✿

n

1

2

3

4

5

6

7

...

an a1 a2 a3 a4 a5 a6 a7 ... || 1 n

|| 1 1

|| 1 2

|| 1 3

|| 1 4

|| 1 5

|| 1 6

|| 1 7

... ...

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

❂✶✴❘❈❬✲✶❪ ❲❡ ❛❧s♦ ♣❧♦t t❤❡s❡ ✈❛❧✉❡s✿

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

an = 1/n, n = 1, 2, 3, ...

❊①❡r❝✐s❡ ✶✳✸✳✹ ❲r✐t❡ ❛ ❢❡✇ t❡r♠s ♦❢ t❤❡ s❡q✉❡♥❝❡s ❣✐✈❡♥ ❜② t❤❡ ❢♦r♠✉❧❛s✿ ✶✳ ✷✳

an = 3n − 1 1 bn = 1 + n

❲❡ ✇✐❧❧ s❛② t❤❛t t❤✐s ✐s t❤❡

nt❤✲t❡r♠

❢♦r♠✉❧❛ ♦❢ t❤❡ s❡q✉❡♥❝❡✳

❇❡❧♦✇ ✐s t❤❡ s✐♠♣❧❡st ❦✐♥❞✳

❉❡✜♥✐t✐♦♥ ✶✳✸✳✺✿ ❝♦♥st❛♥t s❡q✉❡♥❝❡ ❆ ❝♦♥st❛♥t s❡q✉❡♥❝❡ ❤❛s ❛❧❧ ✐ts t❡r♠s ❡q✉❛❧ t♦ ❡❛❝❤ ♦t❤❡r✳

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡

a1 = a2 = ... = c ❢♦r s♦♠❡ ♥✉♠❜❡r

c✳

✶✳✸✳

❙❡q✉❡♥❝❡s

❚❤✉s✱

❡✈❡r②

✷✾

❢♦r♠✉❧❛ ✐s ❝❛♣❛❜❧❡ ♦❢ ❝r❡❛t✐♥❣ ❛♥ ✐♥✜♥✐t❡ s❡q✉❡♥❝❡

an ✳

❋♦r ❡①❛♠♣❧❡✱ ✇❡ ❝❛♥ t❛❦❡ t❤❡s❡✿

• an = n

• bn = n 2

• cn = n3 •

❡t❝✳

❚❤❡② ♠❛❦❡ ✉♣ ❛ ✇❤♦❧❡ ❝❧❛ss ♦❢ s❡q✉❡♥❝❡s✳

❉❡✜♥✐t✐♦♥ ✶✳✸✳✻✿ ♣♦✇❡r s❡q✉❡♥❝❡ ❋♦r ❡✈❡r② ♣♦s✐t✐✈❡ ✐♥t❡❣❡r

p✱

❛

♣♦✇❡r s❡q✉❡♥❝❡✱

♦r ❛

p✲s❡q✉❡♥❝❡✱

✐s ❣✐✈❡♥ ❜② t❤❡

❢♦r♠✉❧❛✿

an = np ,

n = 1, 2, 3, ...

❚❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ s❡q✉❡♥❝❡s ❜❡❝♦♠❡s ❝❧❡❛r ✐❢ ✇❡ ③♦♦♠ ♦✉t ❢r♦♠ t❤❡✐r ❣r❛♣❤s ✭p

■♥❞❡❡❞✱ t❤❡ ❧❛r❣❡r t❤❡ ♣♦✇❡r

p✱

= 1, 2, ..., 7✮✿

t❤❡ ❢❛st❡r t❤❡ s❡q✉❡♥❝❡ ❣r♦✇s✳

❚❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ❝♦♥s❡❝✉t✐✈❡ t❡r♠s ♦❢ ❡❛❝❤ s❡q✉❡♥❝❡ ✐s ❛❧s♦ ❝❧❡❛r✿ ■t ❣r♦✇s✦ ❲❡ ✉s❡ t❤❡s❡ ✇♦r❞s✿

•

✏❣r♦✇t❤✑ ♦r ✏✐♥❝r❡❛s❡✑ ✇❤❡♥ ✇❡ s❡❡ t❤❡ ❣r❛♣❤ t❤❛t

r✐s❡s

•

✏❞❡❝❧✐♥❡✑ ♦r ✏❞❡❝r❡❛s❡✑ ✇❤❡♥ ✇❡ s❡❡ t❤❡ ❣r❛♣❤ t❤❛t

❞r♦♣s

❧❡❢t t♦ r✐❣❤t✱ ❛♥❞ ❧❡❢t t♦ r✐❣❤t✱

❛s ❢♦❧❧♦✇s✿

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

❡✈❡r② ♣❛✐r ♦❢ ❝♦♥s❡❝✉t✐✈❡ t❡r♠s

❡①❛♠♣❧❡✱ t❤❡ s❡q✉❡♥❝❡ ♦❢ t❤❡ ❢❛❧❧✐♥❣ ❜❛❧❧✱

36, 35, 32, 27, 20, 11, 0 ,

♦❢ t❤❡ s❡q✉❡♥❝❡✳

❋♦r

✶✳✸✳

❙❡q✉❡♥❝❡s

✸✵

✐s ❞❡❝r❡❛s✐♥❣ ❜❡❝❛✉s❡

36 > 35 > 32 > 27 > 20 > 11 > 0 . ❋♦r t❤❡ ❣❡♥❡r❛❧ ❝❛s❡✱ ✇❡ ✇r✐t❡✿

a1 > a2 > a3 > a4 > a5 > a6 . ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ❝✉rr❡♥t t❡r♠✱

an ✱

✐s ❧❛r❣❡r t❤❛♥ t❤❡ ♥❡①t✱

≥

❧❛st

an+1 ✿

♥❡①t

❚❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤✐s ✐♥❡q✉❛❧✐t② ✐s ❝❧❡❛r ✇❤❡♥ ✇❡ ③♦♦♠ ✐♥ ♦♥ t❤❡ ❣r❛♣❤ ✭r✐❣❤t✮✿

❖♥ ❧❡❢t✱ t❤❡ s❡q✉❡♥❝❡ ✐s ✐♥❝r❡❛s✐♥❣✿ ♥❡①t

≥

❧❛st

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

❉❡✜♥✐t✐♦♥ ✶✳✸✳✼✿ ✐♥❝r❡❛s✐♥❣ s❡q✉❡♥❝❡ ❛♥❞ ❞❡❝r❡❛s✐♥❣ s❡q✉❡♥❝❡ •

an

❆ s❡q✉❡♥❝❡

✐s ❝❛❧❧❡❞

✐♥❝r❡❛s✐♥❣

✐❢✱ ❢♦r ❛❧❧

n✱

✇❡ ❤❛✈❡

an ≤ an+1 . •

❆ s❡q✉❡♥❝❡ ✐s ❝❛❧❧❡❞

❞❡❝r❡❛s✐♥❣

✐❢✱ ❢♦r ❛❧❧

n✱

✇❡ ❤❛✈❡

an ≥ an+1 . ❈♦❧❧❡❝t✐✈❡❧②✱ t❤❡② ❛r❡ ❝❛❧❧❡❞

♠♦♥♦t♦♥❡✳

❲❛r♥✐♥❣✦ ❇♦t❤ ✐♥❝r❡❛s✐♥❣ ❛♥❞ ❞❡❝r❡❛s✐♥❣ s❡q✉❡♥❝❡s ♠❛② ❤❛✈❡ s❡❣♠❡♥ts ✇✐t❤ ♥♦ ❝❤❛♥❣❡❀ ❢✉rt❤❡r♠♦r❡✱ ❛ ❝♦♥st❛♥t s❡q✉❡♥❝❡ ✐s ❜♦t❤ ✐♥❝r❡❛s✐♥❣ ❛♥❞ ❞❡❝r❡❛s✐♥❣✳

❊①❛♠♣❧❡ ✶✳✸✳✽✿ ♣r♦✈✐♥❣ ♠♦♥♦t♦♥✐❝✐t② ❲❤❡♥ t❤❡ s❡q✉❡♥❝❡ ✐s ❣✐✈❡♥ ❜② ✐ts ❢♦r♠✉❧❛✱ ✇❡ ✉s❡ ✐t ❞✐r❡❝t❧②✳ ❚❤❡ s❡q✉❡♥❝❡ ✐♥❝r❡❛s✐♥❣ ❛s ❢♦❧❧♦✇s✳ ❲❡ ♥❡❡❞ t♦ s❤♦✇ t❤❛t ❢♦r ❛❧❧

n

an = n2

✐s ♣r♦✈❡♥ t♦ ❜❡

✇❡ ❤❛✈❡✿

n2 < (n + 1)2 . ❲❡ s✐♠♣❧② ❡①♣❛♥❞ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡✿

(n + 1)2 = n2 + 2n + 1 . ❆s

n

✐s ♣♦s✐t✐✈❡✱ t❤❡ ❧❛st ♣❛rt✱

❙✐♠✐❧❛r❧②✱ ✇❡ s❤♦✇ t❤❛t

1 n

2n + 1✱

✐s ♣♦s✐t✐✈❡ t♦♦✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ❡①♣r❡ss✐♦♥ ✐s ❧❛r❣❡r t❤❛♥

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

n < n + 1 =⇒

1 1 > . n n+1

❚❤❡ s❡q✉❡♥❝❡

1, −1, 1, −1, 1, −1, 1, −1, 1, −1, ... ✐s ♥❡✐t❤❡r ✐♥❝r❡❛s✐♥❣ ♥♦r ❞❡❝r❡❛s✐♥❣✱ ✐✳❡✳✱ ✐t✬s ♥♦t ♠♦♥♦t♦♥❡✳

n2 ✳

✶✳✸✳ ❙❡q✉❡♥❝❡s

✸✶

❊①❡r❝✐s❡ ✶✳✸✳✾

❙❤♦✇ t❤❛t

1 ✐s ❞❡❝r❡❛s✐♥❣✳ n2

❊①❡r❝✐s❡ ✶✳✸✳✶✵

❙❤♦✇ t❤❛t ❛❧❧ ♣♦✇❡r s❡q✉❡♥❝❡s ❛r❡ ✐♥❝r❡❛s✐♥❣✳ ❆ ♠❛❥♦r r❡❛s♦♥ ✇❤② ✇❡ st✉❞② s❡q✉❡♥❝❡s ✐s t❤❛t✱ ✐♥ ❛❞❞✐t✐♦♥ t♦ t❛❜❧❡s ❛♥❞ ❢♦r♠✉❧❛s✱ ❛ s❡q✉❡♥❝❡ ❝❛♥ ❜❡ ❞❡✜♥❡❞ ❜② ❝♦♠♣✉t✐♥❣ ✐ts t❡r♠s ✐♥ ❛ ❝♦♥s❡❝✉t✐✈❡ ♠❛♥♥❡r✱ ♦♥❡ ❛t ❛ t✐♠❡✳ ❊①❛♠♣❧❡ ✶✳✸✳✶✶✿ r❡❣✉❧❛r ❞❡♣♦s✐ts

❆ ♣❡rs♦♥ st❛rts t♦ ❞❡♣♦s✐t $20 ❡✈❡r② ♠♦♥t❤ ✐♥ ❤✐s ❜❛♥❦ ❛❝❝♦✉♥t t❤❛t ❛❧r❡❛❞② ❝♦♥t❛✐♥s $1000✳ ❚❤❡♥✱ ❛❢t❡r t❤❡ ✜rst ♠♦♥t❤ t❤❡ ❛❝❝♦✉♥t ❝♦♥t❛✐♥s✿

$1000 + $20 = $1020 , ❛❢t❡r t❤❡ s❡❝♦♥❞✿

$1020 + $20 = $1040 , ❛♥❞ s♦ ♦♥✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿

♥❡①t = ❧❛st + 20 .

▲❡t✬s ♠❛❦❡ t❤✐s ❛❧❣❡❜r❛✐❝✳ ❙✉♣♣♦s❡ an ✐s t❤❡ ❛♠♦✉♥t ✐♥ t❤❡ ❜❛♥❦ ❛❝❝♦✉♥t ❛❢t❡r n ♠♦♥t❤s✱ t❤❡♥ ✇❡ ❤❛✈❡ ❛ ❢♦r♠✉❧❛ ❢♦r t❤✐s s❡q✉❡♥❝❡✿ an+1 = an + 20 . ❍♦✇ ♠✉❝❤ ✇✐❧❧ ❤❡ ❤❛✈❡ ❛❢t❡r 50 ②❡❛rs❄ ❲❡✬❞ ❤❛✈❡ t♦ ❝❛rr② ♦✉t 50 · 12 = 600 ❛❞❞✐t✐♦♥s✳ ❋♦r t❤❡ s♣r❡❛❞s❤❡❡t✱ t❤❡ ❢♦r♠✉❧❛ r❡❢❡rs t♦ t❤❡ ❧❛st r♦✇ ❛♥❞ ❛❞❞s 20✱ ❛s ❢♦❧❧♦✇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 ❧✐♥❡✳

✶✳✸✳ ❙❡q✉❡♥❝❡s

✸✷

❉❡✜♥✐t✐♦♥ ✶✳✸✳✶✷✿ r❡❝✉rs✐✈❡ s❡q✉❡♥❝❡ ❲❡ s❛② t❤❛t ❛ s❡q✉❡♥❝❡ ✐s r❡❝✉rs✐✈❡ ✇❤❡♥ ✐ts ♥❡①t t❡r♠ ✐s ❢♦✉♥❞ ❢r♦♠ t❤❡ ❝✉rr❡♥t t❡r♠ ❜② ❛ s♣❡❝✐✜❡❞ ❢♦r♠✉❧❛✱ ✐✳❡✳✱ an ❞❡t❡r♠✐♥❡s an+1 ✳ ❚❤✐s ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ ❝♦♠♣✉t✐♥❣ ❛ s❡q✉❡♥❝❡ ❞✐r❡❝t❧②✱ s✉❝❤ ❛s an = n2 ✱ ❛♥❞ r❡❝✉rs✐✈❡❧②✱ s✉❝❤ ❛s an+1 = an + 20✿ n an 1 → a1

n 1

2 → a2

2

3 → a3

3

..

..

..

an a1 ↓ a2 ↓ a3 ↓ ..

❚❤❡ ❢♦❧❧♦✇✐♥❣ ✇✐❧❧ ❜❡ 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✳

❊①❡r❝✐s❡ ✶✳✸✳✶✹ ■❢ t❤❡ ✐♥❝r❡♠❡♥t ✐s ③❡r♦✱ t❤❡ s❡q✉❡♥❝❡ ✐s✳✳✳

❊①❛♠♣❧❡ ✶✳✸✳✶✺✿ ❝♦♠♣♦✉♥❞❡❞ ✐♥t❡r❡st ❆♥ ❛r✐t❤♠❡t✐❝ ♣r♦❣r❡ss✐♦♥ ❞❡s❝r✐❜❡s ❛ r❡♣❡t✐t✐✈❡ ♣r♦❝❡ss✳ ❆❧s♦ r❡♣❡t✐t✐✈❡ ✐s t❤❡ ❢♦❧❧♦✇✐♥❣ t②♣✐❝❛❧ s✐t✉✲ ❛t✐♦♥✳ ❆ ♣❡rs♦♥ ❞❡♣♦s✐ts $1000 ✐♥ ❤✐s ❜❛♥❦ ❛❝❝♦✉♥t t❤❛t ♣❛②s 1% ❆P❘ ❝♦♠♣♦✉♥❞❡❞ ❛♥♥✉❛❧❧②✳ ❚❤❡♥✱ ❛❢t❡r t❤❡ ✜rst ②❡❛r✱ t❤❡ ✐♥t❡r❡st ✐s $1000 · .01 = $10 ,

❛♥❞ t❤❡ t♦t❛❧ ❛♠♦✉♥t ❜❡❝♦♠❡s $1010✳ ❆❢t❡r t❤❡ s❡❝♦♥❞ ②❡❛r✱ t❤❡ ✐♥t❡r❡st ✐s $1010 · .01 = $10.10 ,

❛♥❞ s♦ ♦♥✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ t♦t❛❧ ❛♠♦✉♥t ✐s ♠✉❧t✐♣❧✐❡❞ ❜② .01 ❛t t❤❡ ❡♥❞ ♦❢ ❡❛❝❤ ②❡❛r ❛♥❞ t❤❡♥ ❛❞❞❡❞ t♦ t❤❡ t♦t❛❧✳ ❆♥ ❡✈❡♥ s✐♠♣❧❡r ✇❛② t♦ ❛❧❣❡❜r❛✐❝❛❧❧② ❞❡s❝r✐❜❡ t❤✐s ✐s t♦ s❛② t❤❛t t❤❡ t♦t❛❧ ❛♠♦✉♥t ✐s ♠✉❧t✐♣❧✐❡❞ ❜② 1.01 ❛t t❤❡ ❡♥❞ ♦❢ ❡❛❝❤ ②❡❛r✱ ❛s ❢♦❧❧♦✇s✳ ❆❢t❡r t❤❡ ✜rst ②❡❛r✱ t❤❡ t♦t❛❧ ✐s ❡q✉❛❧ t♦ $1000 · 1.01 = $1010 .

❆❢t❡r t❤❡ s❡❝♦♥❞ ②❡❛r✱ t❤❡ t♦t❛❧ ✐s ❡q✉❛❧ t♦ $1010 · 1.01 = $1020.1 ,

❛♥❞ s♦ ♦♥✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿

♥❡①t = ❧❛st · 1.01 .

▲❡t✬s ♠❛❦❡ t❤✐s ❛❧❣❡❜r❛✐❝✳ ❙✉♣♣♦s❡ an ✐s t❤❡ ❛♠♦✉♥t ✐♥ t❤❡ ❜❛♥❦ ❛❝❝♦✉♥t ❛❢t❡r n ②❡❛rs✳ ❚❤❡♥ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛ ✿ an+1 = an · 1.01 .

✶✳✸✳ ❙❡q✉❡♥❝❡s

✸✸

❍♦✇ ♠✉❝❤ ✇✐❧❧ ❤❡ ❤❛✈❡ ❛❢t❡r 50 ②❡❛rs❄ ❲❡✬❞ ❤❛✈❡ t♦ ❝❛rr② ♦✉t 50 ♠✉❧t✐♣❧✐❝❛t✐♦♥s✳ ❋♦r t❤❡ s♣r❡❛❞s❤❡❡t✱ t❤❡ ❢♦r♠✉❧❛ r❡❢❡rs t♦ t❤❡ ❧❛st r♦✇ ✭ ❘❬✲✶❪ ✮ ❛♥❞ ♠✉❧t✐♣❧✐❡s ❜② 1.01✱ ❛s ❢♦❧❧♦✇s✿ ❂❘❬✲✶❪❈✯✶✳✵✶

❲❡ ♣❧♦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 ❧✐♥❡✿

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

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

❊①❛♠♣❧❡ ✶✳✸✳✶✼✿ ♣♦♣✉❧❛t✐♦♥ ❧♦ss ■❢ t❤❡ ♣♦♣✉❧❛t✐♦♥ ♦❢ ❛ ❝✐t② ❞❡❝❧✐♥❡s ❜② 3% ❡✈❡r② ②❡❛r✱ ✐t ✐s ❧❡❢t ✇✐t❤ 97% ♦❢ ✐ts ♣♦♣✉❧❛t✐♦♥ ❛t t❤❡ ❡♥❞

✶✳✸✳

✸✹

❙❡q✉❡♥❝❡s

♦❢ ❡❛❝❤ ②❡❛r✳ ❚❤❡ r❡s✉❧t ✐s ❢♦✉♥❞ ❜② ♠✉❧t✐♣❧②✐♥❣ ❜② .97✱ ❡✈❡r② t✐♠❡✳ ❲❡ ❤❛✈❡✱ t❤❡r❡❢♦r❡✿ ❛❢t❡r ✸ ②❡❛rs

}| { z ((1, 000, 000 · 0.97) ·0.97) · 0.97 . {z } | ❛❢t❡r ✶ ②❡❛r | {z } ❛❢t❡r ✷ ②❡❛rs

❆♥❞ s♦ ♦♥✳ ❲❤❛t ✇✐❧❧ ❜❡ t❤❡ ♣♦♣✉❧❛t✐♦♥ ❛❢t❡r 50 ②❡❛rs❄ ❲❡✬❞ ❤❛✈❡ t♦ ❝❛rr② ♦✉t 50 ♠✉❧t✐♣❧✐❝❛t✐♦♥s✳ ❚❤❡ ❧♦♥❣✲t❡r♠ tr❡♥❞ ✐s ❝❧❡❛r ❢r♦♠ t❤❡ ❣r❛♣❤✿

❚❤✐s ✐s ❛ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥ ✇✐t❤ r❛t✐♦ r = .97✱ ✐✳❡✳✱ ❛ ❣❡♦♠❡tr✐❝ ❞❡❝❛②✳ ❚❤❡ s❡q✉❡♥❝❡ ✐s ❞❡❝r❡❛s✐♥❣ ❛♥❞ ❡✈❡♥t✉❛❧❧② t❤❡r❡ ✐s ❛❧♠♦st ♥♦❜♦❞② ❧❡❢t✳

❊①❛♠♣❧❡ ✶✳✸✳✶✽✿ ❞❡♣♦s✐ts ❛♥❞ ✐♥t❡r❡st✱ t♦❣❡t❤❡r ❲❤❛t ✐❢ ✇❡ ❞❡♣♦s✐t ♠♦♥❡② t♦ ♦✉r ❜❛♥❦ ❛❝❝♦✉♥t ❛♥❞ r❡❝❡✐✈❡ ✐♥t❡r❡st❄ ❚❤❡ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛ ✐s s✐♠♣❧❡✱ ❢♦r ❡①❛♠♣❧❡✿ an+1 = an · 1.05 + 2000 .

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

❊①❡r❝✐s❡ ✶✳✸✳✶✾ ❲❤❛t ❞♦❡s ❛ 2% ✐♥✢❛t✐♦♥ ❞♦ t♦ ❛ ❞♦❧❧❛r ❤✐❞❞❡♥ ✐♥ t❤❡ ♠❛ttr❡ss❄ ❆♥② ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥✱ ♦r s❡✈❡r❛❧ ♦♣❡r❛t✐♦♥s t♦❣❡t❤❡r✱ ❝❛♥ ♣r♦❞✉❝❡ ❛ r❡❝✉rs✐✈❡ s❡q✉❡♥❝❡✿ a0 → a0 → a0 →

❛❞❞ 2 ❞✐✈✐❞❡ ❜② 3 sq✉❛r❡ ✐t

→ a1 → → a1 → → a1 →

❛❞❞ 2 ❞✐✈✐❞❡ ❜② 3 sq✉❛r❡ ✐t

→ ... → → ... → → ... →

❛❞❞ 2 ❞✐✈✐❞❡ ❜② 3 sq✉❛r❡ ✐t

→ an → ... → an → ... → an → ...

❊①❡r❝✐s❡ ✶✳✸✳✷✵ ❍♦✇ ❞♦ t❤❡s❡ r❡❝✉rs✐✈❡ s❡q✉❡♥❝❡s ❞❡♣❡♥❞ ♦♥ t❤❡ ✈❛❧✉❡ ♦❢ a0 ❄ ❲❤❡♥ ❛ s❡q✉❡♥❝❡ ✐s ❞❡✜♥❡❞ r❡❝✉rs✐✈❡❧②✱ ✇❡✬❞ ♥❡❡❞ t♦ ❝❛rr② ♦✉t t❤✐s ❞❡✜♥✐t✐♦♥ n t✐♠❡s ✐♥ ♦r❞❡r t♦ ✜♥❞ t❤❡ nt❤ t❡r♠✳ ❚❤✐s ✐s ✐♥ ❝♦♥tr❛st t♦ s❡q✉❡♥❝❡s ❞❡✜♥❡❞ ❞✐r❡❝t❧② ✈✐❛ ✐ts nt❤✲t❡r♠ ❢♦r♠✉❧❛✱ s✉❝❤ ❛s an = n2 ✱ t❤❛t r❡q✉✐r❡s ❛ s✐♥❣❧❡ ❝♦♠♣✉t❛t✐♦♥ t♦ ✜♥❞ ❛♥② t❡r♠✳ ❇❡❧♦✇✱ ✇❡ ❦❡❡♣ ♠✉❧t✐♣❧②✐♥❣✱ ❥✉st ❛s ✐♥ t❤❡ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥✱ ❜✉t t❤✐s t✐♠❡ t❤❡ ♠✉❧t✐♣❧❡ ❣r♦✇s✳

❉❡✜♥✐t✐♦♥ ✶✳✸✳✷✶✿ ❢❛❝t♦r✐❛❧ ❚❤❡

❢❛❝t♦r✐❛❧

✐s t❤❡ s❡q✉❡♥❝❡ ❞❡✜♥❡❞ ✭r❡❝✉rs✐✈❡❧②✮ ❛s ❢♦❧❧♦✇s✿ a0 = 1,

an = an−1 · n, n ≥ 1 ;

✶✳✸✳

❙❡q✉❡♥❝❡s

✸✺

✐✳❡✳✱

an = 1 · 2 · ... · (n − 1) · n . ■t ✐s ❞❡♥♦t❡❞ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣✿

n! ■t r❡❛❞s ✏ n✲❢❛❝t♦r✐❛❧✑✳

❋♦r ❡①❛♠♣❧❡✱ t❤✐s ✐s ❤♦✇ ❛ ❢❡✇ ✐♥✐t✐❛❧ t❡r♠s ❛r❡ ❝♦♠♣✉t❡❞✳ ❲❡ ♣r♦❣r❡ss ❢r♦♠ ❧❡❢t t♦ r✐❣❤t ✐♥ t❤❡ ❜♦tt♦♠ r♦✇ ❜② ❢♦❧❧♦✇✐♥❣ t❤❡ ❛rr♦✇ ♥❡①t ❛rr♦✇

↓

♣♦✐♥ts✿

ր

t♦ ✜♥❞ ❛♥❞ ♠✉❧t✐♣❧② ❜② t❤❡ ❝✉rr❡♥t ✈❛❧✉❡ ♦❢

n✱

t❤❡♥ ♣❧❛❝✐♥❣ t❤❡ r❡s✉❧t ✇❤❡r❡ t❤❡

n 0

1 2 3 4 ... ր ↓ ր ↓ ր ↓ ր ↓ ... 1 2 6 24 ... an = n! 1

❚❤❡ ❢❛❝t♦r✐❛❧ ❡①❤✐❜✐ts ❛ ✈❡r② ❢❛st ❣r♦✇t❤✱ ❡✈❡♥t✉❛❧❧②✿

❨♦✉ ❝❛♥ s❡❡ ❤♦✇ t❤❡ ❢❛❝t♦r✐❛❧ ✭❜❧✉❡✮ st❛②s ❜❡❤✐♥❞ t❤❡ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥ ✇✐t❤ r❛t✐♦

r = 10 ✭r❡❞✮ ❜✉t t❤❡♥

❧❡❛♣s ❛❤❡❛❞✳ ❚❤❡ ❢❛❝t♦r✐❛❧ ❛♣♣❡❛rs ❢r❡q✉❡♥t❧② ✐♥ ❝❛❧❝✉❧✉s ❛♥❞ ❡❧s❡✇❤❡r❡✳ ■t s✉✣❝❡s t♦ ♣♦✐♥t ♦✉t ❢♦r ♥♦✇ t❤❛t ✐t ❝♦✉♥ts t❤❡ ♥✉♠❜❡r ♦❢ ✇❛②s ♦♥❡ ❝❛♥

♣❡r♠✉t❡

♦❜❥❡❝ts✳

❊①❛♠♣❧❡ ✶✳✸✳✷✷✿ ♣❡r♠✉t❛t✐♦♥s

❙✉♣♣♦s❡ ✇❡ ❤❛✈❡

5

❣✉❡sts t♦ ❜❡ ♣❧❛❝❡❞ ❛r♦✉♥❞ t❤❡ t❛❜❧❡✳ ❚❤❡r❡ ❛r❡ ❣✉❡sts✿

5

s❡❛ts✿

s❡❛ts✿

A, B, C, D, E −→ 1, 2, 3, 4, 5 ■♥ ❤♦✇ ♠❛♥② ✇❛②s ❝❛♥ ✇❡ ❞♦ t❤✐s❄ ❲❡ st❛rt ✇✐t❤ t❤❡ ✜rst s❡❛t✳ ❚❤❡r❡ ❛r❡

5

❣✉❡sts t♦ ❝❤♦♦s❡ ❢r♦♠✳ ❖♥❝❡ ❛ ❝❤♦✐❝❡ ✐s ♠❛❞❡✱ ✇❡ ❤❛✈❡

4

❣✉❡sts ❧❡❢t✳

5

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

5 · 4 = 20

4

❣✉❡sts t♦ ❜❡ ♣❧❛❝❡❞ ❛t t❤❡ s❡❝♦♥❞ s❡❛t✳ ❚❤❡

❝❤♦✐❝❡s✳ ❖♥❝❡ ❛ ❝❤♦✐❝❡ ✐s ♠❛❞❡✱ ✇❡ ❤❛✈❡

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

❝❤♦✐❝❡s ✇❡ ❤❛❞✱ ✇❡ ❤❛✈❡

20 · 3 = 60

20

❝❤♦✐❝❡s ✇❡ ❤❛❞✱ ✇❡ ❤❛✈❡

3

3

❣✉❡sts ❧❡❢t✳

❣✉❡sts t♦ ❜❡ ♣❧❛❝❡❞ ❛t t❤❡ t❤✐r❞ s❡❛t✳ ❚❤❡

❝❤♦✐❝❡s✳ ❖♥❝❡ ❛ ❝❤♦✐❝❡ ✐s ♠❛❞❡✱ ✇❡ ❤❛✈❡

2

❣✉❡sts ❧❡❢t✳

✶✳✹✳

❘❡♣❡❛t❡❞ ❛❞❞✐t✐♦♥ ❛♥❞ r❡♣❡❛t❡❞ ♠✉❧t✐♣❧✐❝❛t✐♦♥ 60 ❝❤♦✐❝❡s 60 · 2 = 120 ❝❤♦✐❝❡s✳

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

✸✻

✇❡ ❤❛❞✱ ✇❡ ❤❛✈❡

2

❣✉❡sts t♦ ❜❡ ♣❧❛❝❡❞ ❛t t❤❡ ❢♦✉rt❤ s❡❛t✳ ❚❤❡

❚❤❡r❡ ✐s ♦♥❧② ♦♥❡ ❣✉❡st ❧❡❢t ❛♥❞ ✐t ✐s ♣❧❛❝❡❞ ❛t t❤❡ ✜❢t❤ s❡❛t✳ ❙♦✱ t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ❝❤♦✐❝❡s ✐s

5 · 4 · 3 · 2 · 1 = 120 . ■♥ ❣❡♥❡r❛❧ ✇❡ ❤❛✈❡ t♦ ♣❧❛❝❡

•

❚❤❡ ✜rst ♦❜❥❡❝t ❤❛s

•

❚❤❡ s❡❝♦♥❞

•

❚❤❡ t❤✐r❞

•

✳✳✳

•

❚❤❡ ❧❛st ❤❛s

n−1

n−2 1

n

n

♦❜❥❡❝ts ✐♥t♦

n

s❧♦ts✱ ♦♥❡ ❜② ♦♥❡✿

♦♣t✐♦♥s✳

♦♣t✐♦♥s✳

♦♣t✐♦♥s✳

♦♣t✐♦♥ t❤❡ ❧❡❢t✳

❙✐♥❝❡ t❤❡ ❝❤♦✐❝❡s ❛r❡ ✐♥❞❡♣❡♥❞❡♥t ♦❢ ❡❛❝❤ ♦t❤❡r✱ t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ s✉❝❤ ♣❧❛❝❡♠❡♥ts ✐s

n · (n − 1) · ... · 2 · 1 = n! ❚❤❡♦r❡♠ ✶✳✸✳✷✸✿ ◆✉♠❜❡r ♦❢ P❡r♠✉t❛t✐♦♥s ❚❤❡ ♥✉♠❜❡r ♦❢ ❛❧❧ ♣♦ss✐❜❧❡

♣❡r♠✉t❛t✐♦♥s ♦❢ n ♦❜❥❡❝ts ✐s ❡q✉❛❧ t♦ n✲❢❛❝t♦r✐❛❧✳

❊①❡r❝✐s❡ ✶✳✸✳✷✹ ■♥ ❤♦✇ ♠❛♥② ✇❛②s ❝❛♥ ②♦✉ ❛rr❛♥❣❡

9

♣❧❛②❡rs ❛t t❤❡

9

♣♦s✐t✐♦♥s ♦♥ t❤❡ ❜❛s❡❜❛❧❧ ✜❡❧❞❄

❲❛r♥✐♥❣✦

❚❤❡ ❡①♣r❡ss✐♦♥ n! ✐s ❥✉st ❛♥ ❛❜❜r❡✈✐❛t✐♦♥ ♦❢ t❤❡ r❡❝✉rs✐✈❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❢❛❝t♦r✐❛❧✱ ♥♦t ✐ts nt❤✲t❡r♠ ❢♦r♠✉❧❛✳

✶✳✹✳ ❘❡♣❡❛t❡❞ ❛❞❞✐t✐♦♥ ❛♥❞ r❡♣❡❛t❡❞ ♠✉❧t✐♣❧✐❝❛t✐♦♥

■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ ✇✐❧❧ t❛❦❡ ❛ ❜❡tt❡r ❧♦♦❦ ❛t t❤❡ ❛r✐t❤♠❡t✐❝ ❛♥❞ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥s✱ s✐❞❡ ❜② s✐❞❡✳ ❘❡♠❡♠❜❡r 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✿

♣♦✇❡r ✿ 2 · 2 · 2 = 23✳ ■♥ ❡✐t❤❡r ❝❛s❡✱ ✇❡ ❝❤♦♦s❡ t♦ ✉s❡ ❛♥ ❛❜❜r❡✈✐❛t❡❞ ♥♦t❛t✐♦♥ ❢♦r r❡♣❡t✐t✐✈❡ ♦♣❡r❛t✐♦♥s✳ ◮

❘❡♣❡❛t❡❞ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✐s

❲❛r♥✐♥❣✦

❙♦♠❡ ❝❛❧❝✉❧❛t♦rs ❛♥❞ s♦♠❡ ❝♦♠♣✉t❡r ♣r♦❣r❛♠s ✇✐❧❧ ❣✐✈❡ ②♦✉✿ ✲✶✂ ✷❂✶ .

✶✳✹✳

❘❡♣❡❛t❡❞ ❛❞❞✐t✐♦♥ ❛♥❞ r❡♣❡❛t❡❞ ♠✉❧t✐♣❧✐❝❛t✐♦♥

✸✼

❇② ❛❞♦♣t✐♥❣ t❤✐s ♥♦t❛t✐♦♥✱ ✇❡ ❝r❡❛t❡ ✕ ✐♥ ❛❞❞✐t✐♦♥ t♦ t❤❡ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛s ✕ t❤❡

nt❤✲t❡r♠

❢♦r♠✉❧❛s ❢♦r

t❤❡s❡ s❡q✉❡♥❝❡s✱ ❛s ❢♦❧❧♦✇s✳ ❚❤❡♦r❡♠ ✶✳✹✳✶✿ ❋♦r♠✉❧❛s ❢♦r ❆r✐t❤♠❡t✐❝ ❛♥❞ ●❡♦♠❡tr✐❝ Pr♦❣r❡ss✐♦♥s

nt❤✲t❡r♠ ❢♦r♠✉❧❛ ❢♦r ❛♥ ❛r✐t❤♠❡t✐❝ ♣r♦❣r❡ss✐♦♥ ✇✐t❤ ✐♥❝r❡♠❡♥t a ✭t❤❛t st❛rts ✇✐t❤ a✮ ✐s an = a · n , n = 1, 2, 3, ...

✶✳ ❚❤❡

nt❤✲t❡r♠ ❢♦r♠✉❧❛ ❢♦r ❛ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥ ✇✐t❤ r❛t✐♦ a ✭t❤❛t st❛rts ✇✐t❤ a✮ ✐s an = an , n = 1, 2, 3, ...

✷✳ ❚❤❡

❙♦✱ ✇❡ ❢❛❝❡ t✇♦ ❛♥❛❧♦❣♦✉s

❝♦♥✈❡♥t✐♦♥s ♦❢ ❛❧❣❡❜r❛✿

•

❆ r❡❛❧ ♥✉♠❜❡r

a

t❤❛t ❛♣♣❡❛rs

n

t✐♠❡s t♦ ❜❡ ❛❞❞❡❞ ✐s r❡♣❧❛❝❡❞ ✇✐t❤

a · n✳

•

❆ r❡❛❧ ♥✉♠❜❡r

a

t❤❛t ❛♣♣❡❛rs

n

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

an ✳

▲❡t✬s r❡❝❛❧❧ t❤❡ ♥♦t❛t✐♦♥ ❛♥❞ t❤❡ t❡r♠✐♥♦❧♦❣② ❢♦r t❤❡ ❧❛tt❡r✿ ❇❛s❡ ❛♥❞ ❡①♣♦♥❡♥t

❡①♣♦♥❡♥t

a

↓

n

↑

❜❛s❡

❲❡ ✇✐❧❧ ♣✉rs✉❡ t❤✐s ❝♦♥✈❡♥✐❡♥t ❛♥❛❧♦❣② ❢✉rt❤❡r ❛♥❞ r❡✲❞✐s❝♦✈❡r s♦♠❡ ❢❛♠✐❧✐❛r ❛❧❣❡❜r❛✐❝ ♣r♦♣❡rt✐❡s✳ ❆❞❞✐t✐♦♥ ♦r ♠✉❧t✐♣❧✐❝❛t✐♦♥✱ ✇❡ ✇✐❧❧ ❛s❦ t❤❡ s❛♠❡ q✉❡st✐♦♥✿

◮

❍♦✇ ♠❛♥② t✐♠❡s ❞♦❡s

a

❛♣♣❡❛r❄

❚❤❡ ✜rst s❡t ♦❢ ♣r♦♣❡rt✐❡s ✐s ❛❜♦✉t t❤❡ ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s ❝❛rr✐❡❞ ♦✉t ✇✐t❤ t❤❡ ✈❛❧✉❡s ♦❢ t✇♦ s❡q✉❡♥❝❡s✳ ❚❤❡ r❡♣❡t✐t✐♦♥s ❛r❡ ❝❛rr✐❡❞ ♦✉t ✐♥ ♣❛r❛❧❧❡❧ ❛♥❞ t❤❡♥ ❝♦♠❜✐♥❡❞ t♦❣❡t❤❡r✳ ❙♦✱ ✇✐t❤ ❛ t♦t❛❧ ♦❢

n+m

a ❛♣♣❡❛rs n t✐♠❡s✱ t❤❡♥ m t✐♠❡s✱

t✐♠❡s✿ ❆♥❛❧♦❣②✿ r❡♣❡❛t❡❞ ❛❞❞✐t✐♦♥ ✈s✳ r❡♣❡❛t❡❞ ♠✉❧t✐♣❧✐❝❛t✐♦♥

n = 1, 2, 3, ... ❈♦♥✈❡♥t✐♦♥✿

r❡♣❡❛t❡❞ ❛❞❞✐t✐♦♥

= ♠✉❧t✐♣❧✐❝❛t✐♦♥

a + ... + a} = a · n |a + a + {z n t✐♠❡s

❘❡♣❡❛t❡❞

n

t✐♠❡s✱

a + ... + a} = a · n |a + a + {z n t✐♠❡s

t❤❡♥

m

t✐♠❡s ♠♦r❡✳

+ |a + a + {z a + ... + a} m t✐♠❡s

❈♦✉♥t✿

=a·m

= |a + a + {z a + ... + a} n+m t✐♠❡s

Pr♦♣❡rt② ✶✿

a · (n + m) = a · n + a · m

r❡♣❡❛t❡❞ ♠✉❧t✐♣❧✐❝❛t✐♦♥

= ♣♦✇❡r

a · ... · a} = an |a · a · {z n t✐♠❡s

a · ... · a} = an |a · a · {z n t✐♠❡s

· |a · a · {z a · ... · a}

= am

m t✐♠❡s

= |a · a · {z a · ... · a} n+m t✐♠❡s n+m

a

= an · am

❚❤✐s ♣r♦♣❡rt② ❢♦r ❛❞❞✐t✐♦♥

a · (n + m) = a · n + a · m

✐s ❝❛❧❧❡❞ t❤❡

❉✐str✐❜✉t✐✈❡ Pr♦♣❡rt②✳ ■t ✏❞✐str✐❜✉t❡s✑ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦✈❡r ❛❞❞✐t✐♦♥ ❛♥❞ t❤❡r❡❜② ✉♥❞♦❡s t❤❡ ❡✛❡❝t

♦❢ ❢❛❝t♦r✐♥❣✳ ❚❤❡ ♦t❤❡r ❢♦r♠✉❧❛ ✐s ✐ts ❛♥❛❧♦❣✉❡✿

✶✳✹✳

❘❡♣❡❛t❡❞ ❛❞❞✐t✐♦♥ ❛♥❞ r❡♣❡❛t❡❞ ♠✉❧t✐♣❧✐❝❛t✐♦♥

✸✽

❚❤❡♦r❡♠ ✶✳✹✳✷✿ ❆❞❞✐t✐♦♥✲▼✉❧t✐♣❧✐❝❛t✐♦♥ ❘✉❧❡ ♦❢ ❊①♣♦♥❡♥ts ❋♦r ❡✈❡r②

a>0

❛♥❞ ❛♥② ♥❛t✉r❛❧ ♥✉♠❜❡rs

n

❛♥❞

m✱

✇❡ ❤❛✈❡✿

an+m = an · am ❲❡ ❝❛♥ ❛❧s♦ s❡❡ ❤♦✇ t❤✐s ❢♦r♠✉❧❛ t✉r♥s ❛❞❞✐t✐♦♥ ✐♥t♦ ♠✉❧t✐♣❧✐❝❛t✐♦♥✳ ❊✈❡r② ♣r♦♣❡rt②✴r✉❧❡ t❤❛t ✇❡ ❞✐s❝♦✈❡r ✇✐❧❧ ♦❢t❡♥ ♦♣❡r❛t❡ ❛s ❛

s❤♦rt❝✉t✳

❊①❛♠♣❧❡ ✶✳✹✳✸✿ ❢♦r♠✉❧❛s ❛r❡ s❤♦rt❝✉ts ❲❡ ✉s❡ t❤❡ ❢♦r♠✉❧❛ t♦

❡①♣❛♥❞ t❤❡ ❡①♣r❡ss✐♦♥ ♦♥ t❤❡ ❧❡❢t✿ 23+2 = 23 · 22 .

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

❝♦♥tr❛❝t ❡①♣r❡ss✐♦♥s✿

23 · 22 = 23+2 = 25 . ❲❛r♥✐♥❣✦ ❲❡ ❞♦♥✬t ✏❞✐str✐❜✉t❡✑ ❡①♣♦♥❡♥t✐❛t✐♦♥ ♦✈❡r ❛❞❞✐t✐♦♥✿

an+m 6= an + am ✳ ❚❤❡ s❡❝♦♥❞ s❡t ♦❢ ♣r♦♣❡rt✐❡s ✐s ❛❧s♦ ❛❜♦✉t t❤❡ ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s ❝❛rr✐❡❞ ♦✉t ✕ ✐♥ ♣❛r❛❧❧❡❧ ✕ ✇✐t❤ t❤❡ ✈❛❧✉❡s ♦❢ t✇♦ s❡q✉❡♥❝❡s✳ ❚❤✐s t✐♠❡ t❤❡② ❤❛✈❡ ❞✐✛❡r❡♥t ❜❛s❡s✳ ❙♦✱ ❛♣♣❡❛rs

n

a ❛♣♣❡❛rs n t✐♠❡s✱ ❛♥❞ b ❛♣♣❡❛rs n t✐♠❡s✱ s♦ a&b

t✐♠❡s t♦♦✿

❆♥❛❧♦❣②✿ r❡♣❡❛t❡❞ ❛❞❞✐t✐♦♥ ✈s✳ r❡♣❡❛t❡❞ ♠✉❧t✐♣❧✐❝❛t✐♦♥

n = 1, 2, 3, ...

r❡♣❡❛t❡❞ ❛❞❞✐t✐♦♥

= ♠✉❧t✐♣❧✐❝❛t✐♦♥

r❡♣❡❛t❡❞ ♠✉❧t✐♣❧✐❝❛t✐♦♥

a · ... · a} = an |a · a · {z

a + ... + a} = a · n |a + a + {z

❈♦♥✈❡♥t✐♦♥✿

n t✐♠❡s

n t✐♠❡s

❘❡♣❡❛t❡❞

n

a · ... · a} = an |a · a · {z

a + ... + a} = a · n |a + a + {z

t✐♠❡s✳

n t✐♠❡s

n t✐♠❡s

❘❡♣❡❛t❡❞

n

· b| · b · {z b · ... · }b = bn

+ |b + b + {z b + ... + }b = b · n

t✐♠❡s✳

n t✐♠❡s

n t✐♠❡s

❈♦✉♥t✿

= (a · b) · ... · (a · b) | {z }

= (a + b) + ... + (a + b) | {z } n t✐♠❡s

Pr♦♣❡rt② ✷✿

= ♣♦✇❡r

n t✐♠❡s

(a · b)n = an · bn

(a + b) · n = a · n + b · n

❚❤✐s ♣r♦♣❡rt② ❢♦r ❛❞❞✐t✐♦♥

(a + b) · n = a · n + b · n

✐s✱ ♦♥❝❡ ❛❣❛✐♥✱ t❤❡

❉✐str✐❜✉t✐✈❡ Pr♦♣❡rt②✳

■t ✏❞✐str✐❜✉t❡s✑ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦✈❡r ❛❞❞✐t✐♦♥✳

♣r♦♣❡rt② ❢♦r ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✏❞✐str✐❜✉t❡s✑ ❡①♣♦♥❡♥t✐❛t✐♦♥ ♦✈❡r ♠✉❧t✐♣❧✐❝❛t✐♦♥✿

❚❤❡♦r❡♠ ✶✳✹✳✹✿ ❉✐str✐❜✉t✐✈❡ ❘✉❧❡ ♦❢ ❊①♣♦♥❡♥ts ❋♦r ❡✈❡r②

a, b > 0

❛♥❞ ❡✈❡r② ♥❛t✉r❛❧

n✱

✇❡ ❤❛✈❡✿

(a · b)n = an · bn

❚❤❡ ❛♥❛❧♦❣♦✉s

✶✳✹✳

❘❡♣❡❛t❡❞ ❛❞❞✐t✐♦♥ ❛♥❞ r❡♣❡❛t❡❞ ♠✉❧t✐♣❧✐❝❛t✐♦♥

✸✾

❊①❛♠♣❧❡ ✶✳✹✳✺✿ ❢♦r♠✉❧❛s ❛r❡ s❤♦rt❝✉ts ❲❡ ✉s❡ t❤❡ ❢♦r♠✉❧❛ t♦

❡①♣❛♥❞ t❤❡ ❡①♣r❡ss✐♦♥ ♦♥ t❤❡ ❧❡❢t✿ (2 · 3)5 = 25 · 35 .

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

❝♦♥tr❛❝t ❡①♣r❡ss✐♦♥s✿

25 · 35 = (2 · 3)5 = 65 . ❲❛r♥✐♥❣✦ ❲❡ ❛❧s♦ ❝❛♥✬t ✏❞✐str✐❜✉t❡✑ ❡①♣♦♥❡♥t✐❛t✐♦♥ ♦✈❡r ❛❞✲ ❞✐t✐♦♥ t❤✐s ✇❛②✿

(a + b)n 6= an + bn ✳

■♥ ❝♦♥tr❛st t♦ t❤❡ ♣r♦♣❡rt✐❡s ❛❜♦✈❡✱ t❤❡ ♥❡①t s❡t ✐s ❛❜♦✉t ❝♦♥s❡❝✉t✐✈❡ ❝♦♠♣✉t❛t✐♦♥s ✭✏❝♦♠♣♦s✐t✐♦♥s✑✮✳ ❲❡ r❡♣❡❛t t❤❡ r❡♣❡❛t❡❞✳ ❙♦✱ ✐♥ ❡❛❝❤ r♦✇ t♦t❛❧ ♦❢

n·m

a

❛♣♣❡❛rs

n

t✐♠❡s✱ ❛♥❞ t❤❡r❡ ❛r❡

m

r♦✇s ✐♥ t❤❡ t❛❜❧❡✱ s♦

a

❛♣♣❡❛rs ❛

t✐♠❡s✿

❆♥❛❧♦❣②✿ r❡♣❡❛t❡❞ ❛❞❞✐t✐♦♥ ✈s✳ r❡♣❡❛t❡❞ ♠✉❧t✐♣❧✐❝❛t✐♦♥

n = 1, 2, 3, ...

r❡♣❡❛t❡❞ ❛❞❞✐t✐♦♥

= ♠✉❧t✐♣❧✐❝❛t✐♦♥

r❡♣❡❛t❡❞ ♠✉❧t✐♣❧✐❝❛t✐♦♥

a + ... + a} = a · n |a + a + {z

a · ... · a} = an |a · a · {z

+ |a + a + {z a + ... + a} = a · n

·a a · ... · a} = an | · a · {z

❈♦♥✈❡♥t✐♦♥✿

n t✐♠❡s

n t✐♠❡s

❘❡♣❡❛t❡❞

n

t✐♠❡s✱

a a · ... · a} = an | · a · {z

a + ... + a} = a · n |a + a + {z

1.

n t✐♠❡s

n t✐♠❡s

m

t✐♠❡s✳

2. ✳ ✳ ✳

m.

n t✐♠❡s

✳ ✳ ✳

n t✐♠❡s

✳ ✳ ✳

+ |a + a + {z a + ... + a} = a · n

|

Pr♦♣❡rt② ✸✿

{z

nm t✐♠❡s

}

✳ ✳ ✳

✳ ✳ ✳

·a a · ... · a} = an | · a · {z n t✐♠❡s

n t✐♠❡s

❈♦✉♥t✿

= ♣♦✇❡r

|{z}

|

m t✐♠❡s

a · (n · m) = (a · n) · m

{z

nm t✐♠❡s

}

|{z}

m t✐♠❡s n m

a(n·m) = (a )

❚❤✐s ♣r♦♣❡rt② ❢♦r ❛❞❞✐t✐♦♥

✐s ❝❛❧❧❡❞ t❤❡

❆ss♦❝✐❛t✐✈✐t② Pr♦♣❡rt②

a · (n · m) = (a · n) · m ♦❢ ♠✉❧t✐♣❧✐❝❛t✐♦♥✳

■t ♠❡❛♥s t❤❛t ♠✉❧t✐♣❧✐❝❛t✐♦♥s ❝❛♥ ❜❡ r❡✲❣r♦✉♣❡❞

❛r❜✐tr❛r✐❧②✿ t❤❡ ♠✐❞❞❧❡ ♥✉♠❜❡r ❝❛♥ ❜❡ ✏❛ss♦❝✐❛t❡❞✑ ✇✐t❤ t❤❡ ❧❛st ♦♥❡ ♦r t❤❡ ♥❡①t ♦♥❡✳ ❚❤❡ ♦t❤❡r ❢♦r♠✉❧❛ ✐s ✐ts ❛♥❛❧♦❣✉❡✿

❚❤❡♦r❡♠ ✶✳✹✳✻✿ ▼✉❧t✐♣❧✐❝❛t✐♦♥✲❊①♣♦♥❡♥t✐❛t✐♦♥ ❘✉❧❡ ♦❢ ❊①♣♦♥❡♥ts ❋♦r ❡✈❡r② r❡❛❧

a>0

❛♥❞ ❡✈❡r② ♥❛t✉r❛❧ ♥✉♠❜❡rs

an·m = an

n

❛♥❞


m

❲❡ ❝❛♥ ❛❧s♦ s❡❡ ❤♦✇ t❤✐s ❢♦r♠✉❧❛ t✉r♥s ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✐♥t♦ ✏❡①♣♦♥❡♥t✐❛t✐♦♥✑✳

m✱

✇❡ ❤❛✈❡✿

✶✳✹✳

❘❡♣❡❛t❡❞ ❛❞❞✐t✐♦♥ ❛♥❞ r❡♣❡❛t❡❞ ♠✉❧t✐♣❧✐❝❛t✐♦♥

✹✵

❊①❛♠♣❧❡ ✶✳✹✳✼✿ ❢♦r♠✉❧❛s ❛r❡ s❤♦rt❝✉ts ❲❡ ✉s❡ t❤❡ ❢♦r♠✉❧❛ t♦

❡①♣❛♥❞

t❤❡ ❡①♣r❡ss✐♦♥ ♦♥ t❤❡ ❧❡❢t✿

23·4 = (23 )4 . ❖r ❣♦✐♥❣ ❜❛❝❦✇❛r❞✱ ✇❡ ✉s❡ t❤❡ ❢♦r♠✉❧❛ t♦

❝♦♥tr❛❝t ❡①♣r❡ss✐♦♥s✿

(23 )4 = 23·4 = 212 . ❙♦ ❢❛r✱ ✇❡ ❛r❡ ❢❛❝✐♥❣ ♥♦t❤✐♥❣ ❜✉t ❛

❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥ ✇✐t❤ r❛t✐♦ a ❛♥❞ n = 1, 2, 3, ...✿

❲❤❛t ❛❜♦✉t n = 0❄ ❚❤❡r❡ s❡❡♠s t♦ ❜❡ ❛ ♣❧❛❝❡ ❢♦r ✐t ♥❡❛r t❤❡ ♦r✐❣✐♥✳ ❇✉t ✇❤❛t ✇♦✉❧❞ ❜❡ t❤❡ ♦✉t❝♦♠❡ ✕ ❛♥❞ t❤❡ ♠❡❛♥✐♥❣ ✕ ♦❢ r❡♣❡❛t✐♥❣ ❛♥ ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥ ③❡r♦ t✐♠❡s ❄ ❲❡ ❦♥♦✇ ✇❤❛t ✐t ✐s ❢♦r ❛❞❞✐t✐♦♥❀ ✇❡ ❝❤♦♦s❡✿

a·0=0

■♥ ♦t❤❡r ✇♦r❞s✱ a ❛❞❞❡❞ t♦ ✐ts❡❧❢ 0 t✐♠❡s ✐s 0✳ ❚❤❛t✬s ❥✉st ❛♥♦t❤❡r ❝♦♥✈❡♥t✐♦♥ ✦ ❲❡ ❛❞♦♣t ♦♥❡ ❢♦r ♠✉❧t✐♣❧✐❝❛t✐♦♥ t♦♦❀ ✇❡ ❝❤♦♦s❡ ❛s ❢♦❧❧♦✇s ♦♥❝❡ ❛♥❞ ❢♦r ❛❧❧✳

❉❡✜♥✐t✐♦♥ ✶✳✹✳✽✿ ❡①♣♦♥❡♥t ❡q✉❛❧ t♦ ③❡r♦ ❚❤❡ 0t❤ ♣♦✇❡r ♦❢ ❛♥② ♥✉♠❜❡r ✐s 1✿

a0 = 1 ❇✉t ✇❤②❄ ❲❤② ♥♦t 0❄ ❲❤② ♥♦t ❛♥② ♦t❤❡r ♥✉♠❜❡r❄ ❇❡❝❛✉s❡ ✇❡ ✇❛♥t t❤❡ t❤r❡❡ ♣r♦♣❡rt✐❡s st✐❧❧ t♦ ❜❡ ✈❛❧✐❞✦ ❚❤✐s ✇❛② ✇❡ ❝❛♥ ❝♦♥t✐♥✉❡ t♦ ✉s❡ t❤❡♠ ❛s ✐❢ ♥♦t❤✐♥❣ ❤❛s ❝❤❛♥❣❡❞✳

❚❤❡♦r❡♠ ✶✳✹✳✾✿ ❩❡r♦ ❊①♣♦♥❡♥t ❘✉❧❡ ❚❤❡ r✉❧❡s ♦❢ ❡①♣♦♥❡♥ts ❤♦❧❞ ✇❤❡♥

a0 = 1

❜✉t ❢❛✐❧ ❢♦r ❛♥② ♦t❤❡r ❝❤♦✐❝❡ ♦❢

Pr♦♦❢✳ ▲❡t✬s ❝❤❡❝❦✳ ❲❡ ♣❧✉❣ ✐♥ n = 0 ♦r m = 0 ❛♥❞ ✉s❡ ♦✉r ❝♦♥✈❡♥t✐♦♥✿ Pr♦♣❡rt② ✶✿ an+m = an · am n = 0

Pr♦♣❡rt② ✷✿ an bn Pr♦♣❡rt② ✸✿ a

nm

= (ab)n

n m

= (a )

n=0 n=0

=⇒ a0+m = a0 · am ⇐⇒ am = 1 · am ❚❘❯❊ =⇒ a0 b0 0m

=⇒ a

m = 0 =⇒ an0

❊①❡r❝✐s❡ ✶✳✹✳✶✵ Pr♦✈✐❞❡ t❤❡ r❡st ♦❢ t❤❡ ♣r♦♦❢ ♦❢ t❤❡ t❤❡♦r❡♠✳

= (ab)0

0 m

= (a )

= (an )0

⇐⇒ 1 · 1 = 1 0

⇐⇒ a = 1

⇐⇒ a0 = 1

m

❚❘❯❊

❚❘❯❊ ❚❘❯❊

a0 ✳

✶✳✹✳

❘❡♣❡❛t❡❞ ❛❞❞✐t✐♦♥ ❛♥❞ r❡♣❡❛t❡❞ ♠✉❧t✐♣❧✐❝❛t✐♦♥

✹✶

❊①❡r❝✐s❡ ✶✳✹✳✶✶

❙t❛t❡ t❤❡ t❤❡♦r❡♠ ❛s ❛♥ ❡q✉✐✈❛❧❡♥❝❡✳

❊①❡r❝✐s❡ ✶✳✹✳✶✷

❙✐♠♣❧✐❢②✿ ✭❛✮ ❘❡♣❡❛t✳

50 · 53 ✱

✭❜✮

(4 · 3)2 ✱

✭❝✮

(33 )3 ✱

✭❞✮

13+3 ✱

✭❡✮

51 · 31 ✱

✭❢ ✮

22·2 ✳

▼❛❦❡ ✉♣ ❛ ❢❡✇ ♦❢ ②♦✉r ♦✇♥✳

❚❤❡ t❤r❡❡ ♣r♦♣❡rt✐❡s ❛r❡ st✐❧❧ s❛t✐s✜❡❞✱ ❛♥❞ ✇❡ ✇✐❧❧ ❝♦♥t✐♥✉❡ t♦ ✉s❡ t❤❡♠❀ ❝❤♦♦s✐♥❣ ❛♥②t❤✐♥❣ ❜✉t

a0 = 1

✇♦✉❧❞ ❤❛✈❡ r✉✐♥❡❞ t❤❡♠✳ ❋r♦♠ ♥♦✇ ♦♥✱ t❤❡ ❢♦r♠✉❧❛ ❢♦r

❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥ an = an

❝❛♥ st❛rt ✇✐t❤ ❛♥ ✐♥❞❡① ✈❛❧✉❡

n=0

❛♥❞ ❛ ③❡r♦t❤ t❡r♠

a0 ✿

❊①❡r❝✐s❡ ✶✳✹✳✶✸

❊①♣❧❛✐♥ t❤❡ ♠❡❛♥✐♥❣ ♦❢

a0

✐♥ t❤❡ ❝♦♠♣♦✉♥❞❡❞ ✐♥t❡r❡st s❡q✉❡♥❝❡✳

❇❡❧♦✇ ✐s t❤❡ s✉♠♠❛r② ♦❢ t❤❡ ❛❧❣❡❜r❛ ♦❢ ❡①♣♦♥❡♥ts ✭✇❤❡r❡

a

❛♥❞

b

❛♥② ❛r❡ r❡❛❧ ♥✉♠❜❡rs✮✿

❆♥❛❧♦❣②✿ r❡♣❡❛t❡❞ ❛❞❞✐t✐♦♥ ✈s✳ r❡♣❡❛t❡❞ ♠✉❧t✐♣❧✐❝❛t✐♦♥

▼✉❧t✐♣❧✐❝❛t✐♦♥✿

n = 1, 2, 3, ...

❊①♣♦♥❡♥t✐❛t✐♦♥✿

a + ... + a} = a · n |a + a + {z n t✐♠❡s

n=0 ❘✉❧❡s✿

1.

a·0 =0

a · (n + m) = a · n + a · m

a · ... · a} = an |a · a · {z

(a + b) · n = a · n + b · n

2.

a · (n · m) = (a · n) · m

3.

n t✐♠❡s

a0 = 1

an+m = an · am

(a · b)n = an · bn an·m = (an )m

∧ ◆♦t❡ t❤❛t t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡s ♦❢ t❤❡ r✉❧❡s ❛r❡ ♠❛t❝❤❡❞✿ ❏✉st r❡♣❧❛❝❡ ✏ +✑ ✇✐t❤ ✏ ·✑ ❛♥❞ ✏ ·✑ ✇✐t❤ ✏ ✑ ✐♥ t❤❡

✜rst ❝♦❧✉♠♥ ❛♥❞ ②♦✉ ❣❡t t❤♦s❡ ✐♥ t❤❡ s❡❝♦♥❞✳ ❋♦r ❡①❛♠♣❧❡✱ t❤✐s ✐s ❤♦✇ r✉❧❡ ✶ ✇♦r❦s✿

❚❤❡ t✇♦ r✉❧❡s ❛r❡ tr✉❧②

♣❛r❛❧❧❡❧✳

a · (n + m) = a · n + a · m a ∧ (n + m) = a ∧ n · a ∧ m ❍♦✇❡✈❡r✱ t❤✐s ✐s ♥♦t t❤❡ ❝❛s❡ ✕ ✇❛r♥✐♥❣✦ ✕ ❢♦r t❤❡ ❧❡❢t✲❤❛♥❞ s✐❞❡s ♦❢ t❤❡

r✉❧❡s✳ ❚❤❡ r❡❛s♦♥ ✐s t❤❛t s♦♠❡ ♦❢ t❤❡ ❛❧❣❡❜r❛ ✐♥ t❤❡ ❧❡❢t✲❤❛♥❞ s✐❞❡s ❤❛s ❝♦♠❡ ❢r♦♠

❝♦✉♥t✐♥❣ t❤❡ r❡♣❡t✐t✐♦♥s✱

✐❞❡♥t✐❝❛❧❧② ❢♦r ❜♦t❤ ❝♦❧✉♠♥s✳ ❙♦✱ ✇❡ ✉s❡ t❤❡ ❢♦r♠✉❧❛s t♦ ✏s✐♠♣❧✐❢②✑ ❡①♣r❡ss✐♦♥s✳ ❉❡♣❡♥❞✐♥❣ ♦♥ t❤❡ ❝✐r❝✉♠st❛♥❝❡s✱ ✐t ♠✐❣❤t ♠❡❛♥ t♦ ❡①♣❛♥❞ ♦r ✐t ♠✐❣❤t ♠❡❛♥ t♦ ❝♦♥tr❛❝t✳ ❲❡ ✇✐❧❧ ❢✉rt❤❡r ❝♦♥t✐♥✉❡ t♦ ❡①♣❛♥❞ t❤❡ ✐❞❡❛ ♦❢ ❡①♣♦♥❡♥t ✐♥ ❈❤❛♣t❡r ✹✳ ❆ r❡❧❛t❡❞

❝♦♥✈❡♥t✐♦♥ ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✿

✶✳✺✳

❍♦✇ t♦ ✜♥❞ nt❤✲t❡r♠ ❢♦r♠✉❧❛s ❢♦r s❡q✉❡♥❝❡s

✹✷

❉❡✜♥✐t✐♦♥ ✶✳✹✳✶✹✿ ❢❛❝t♦r✐❛❧ ♦❢ ③❡r♦ ❚❤❡

❢❛❝t♦r✐❛❧ ♦❢ ③❡r♦

✐s

1✿ 0! = 1

❊①❡r❝✐s❡ ✶✳✹✳✶✺ ❉❡✈✐s❡ ♥♦t❛t✐♦♥ ❢♦r r❡♣❡❛t❡❞ ❡①♣♦♥❡♥t✐❛t✐♦♥✳

✶✳✺✳ ❍♦✇ t♦ ✜♥❞ nt❤✲t❡r♠ ❢♦r♠✉❧❛s ❢♦r s❡q✉❡♥❝❡s ▲❡t✬s r❡✈✐❡✇ ❤♦✇ t❤❡ ❛❧❣❡❜r❛ ❞✐s❝✉ss❡❞ ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥ ❛❧❧♦✇s ✉s t♦ ♣r♦❞✉❝❡ ❛ ❝♦✉♣❧❡ ♦❢ ❡①♣❧✐❝✐t ❢♦r♠✉❧❛s✳ ❋✐rst✱ ❢♦r ❛♥ ❛r✐t❤♠❡t✐❝ ♣r♦❣r❡ss✐♦♥✱ ✇❡ ❤❛✈❡ ❛ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛✿

an+1 = an + b , ✇❤✐❝❤✱ ✇r✐tt❡♥ ❡①♣❧✐❝✐t❧②✱ t❛❦❡s t❤✐s ❢♦r♠✿

an = a0 + |b + b + {z b + ... + }b . n t✐♠❡s

❋✐♥❛❧❧②✱ ✇❡ ❤❛✈❡ ❛♥♦t❤❡r r❡♣r❡s❡♥t❛t✐♦♥ ♥♦✇✱ ✇✐t❤♦✉t ✏✳✳✳✑✦

❚❤❡♦r❡♠ ✶✳✺✳✶✿ ❋♦r♠✉❧❛ ❢♦r ❆r✐t❤♠❡t✐❝ Pr♦❣r❡ss✐♦♥ nt❤✲t❡r♠ ❢♦r♠✉❧❛ ❢♦r ❛♥ ❛r✐t❤♠❡t✐❝ ♣r♦❣r❡ss✐♦♥ ✇✐t❤ ✐♥❝r❡♠❡♥t b ❛♥❞ ✐♥✐t✐❛❧ t❡r♠ a0 ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✿

❚❤❡

an = a0 + b · n,

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

❙❡❝♦♥❞✱ ❢♦r ❛ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥✱ ✇❡ ❤❛✈❡ ❛ ❢♦r♠✉❧❛✿

an+1 = an · r , ✇❤✐❝❤✱ ✇r✐tt❡♥ ❡①♣❧✐❝✐t❧②✱ t❛❦❡s t❤✐s ❢♦r♠✿

an = a0 · r| · r · {z r · ... · r} . n t✐♠❡s

❋✐♥❛❧❧②✱ ✇❡ ❤❛✈❡ ❛♥♦t❤❡r r❡♣r❡s❡♥t❛t✐♦♥ ♥♦✇✱ ✇✐t❤♦✉t ✏✳✳✳✑✦

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

a0

nt❤✲t❡r♠

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

r

❛♥❞ ✐♥✐t✐❛❧ t❡r♠

✐s t❤❡ ❢♦❧❧♦✇✐♥❣✿

an = a0 · rn , ❚❤❡s❡ t✇♦ s❡q✉❡♥❝❡s✱ ❣✐✈❡♥ ♦r✐❣✐♥❛❧❧② ❜② t❤❡✐r

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

r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛s✱

❛r❡ ♥♦✇ ♣r❡s❡♥t❡❞ ❜② t❤❡✐r

nt❤✲t❡r♠

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

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

✶✳✺✳

❍♦✇ t♦ ✜♥❞ nt❤✲t❡r♠ ❢♦r♠✉❧❛s ❢♦r s❡q✉❡♥❝❡s

✹✸

❊①❛♠♣❧❡ ✶✳✺✳✸✿ ✜♥❞✐♥❣ t❤❡ nt❤ t❡r♠ ❲❤❛t ✐s t❤❡ ❢♦r♠✉❧❛ ❢♦r t❤✐s s❡q✉❡♥❝❡✿

1, 1/2, 1/4, 1/8, ...? ❋✐rst✱ ✇❡ ♥♦t✐❝❡ t❤❛t t❤❡s❡ ❛r❡ ❛❧❧ ❢r❛❝t✐♦♥s ❛♥❞ t❤❡✐r ♥✉♠❡r❛t♦rs ❛r❡ ❥✉st

1✬s✿

1 1 1 1 , , , , ... 1 2 4 8 ❙❡❝♦♥❞✱ t❤❡ ❞❡♥♦♠✐♥❛t♦r ✐s ♠✉❧t✐♣❧✐❡❞ ❜② ❜②

1 ✳ 2

2 ❡✈❡r② t✐♠❡✳

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

❚❤❡ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛ ✐s t❤❡♥✿

an+1 = an · ■t ✐s ❛ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥✦ ■ts r❛t✐♦ ✐s t❤❡♦r❡♠✱ ✇❡ ❤❛✈❡

r = 1/2

1 . 2

❛♥❞ ✐ts ✐♥✐t✐❛❧ t❡r♠ ✐s

a0 = 1✳

❆❝❝♦r❞✐♥❣ t♦ t❤❡ ❧❛st

 n 1 1 = n. an = 1 · 2 2

■t ✐s ♣♦ss✐❜❧❡ t♦ s❡❡ t❤❡ ♣❛tt❡r♥ ❢r♦♠ t❤❡ ❜❡❣✐♥♥✐♥❣✿ ❚❤❡ ❞❡♥♦♠✐♥❛t♦rs ❛r❡ t❤❡ ♣♦✇❡rs ♦❢

1 1 1 a0 = 1, a1 = , a2 = 2 , a3 = 3 , ... 2 2 2 ❲✐t❤ t❤✐s ❢♦r♠✉❧❛✱ ✇❡ ❝❛♥ ♣❧♦t ♠♦r❡ t❡r♠s✿

❊①❡r❝✐s❡ ✶✳✺✳✹ ❲❤❛t ✐s t❤❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ s❡q✉❡♥❝❡ ✐❢ ✇❡ r❡q✉✐r❡ ✐t t♦ st❛rt ✇✐t❤

❊①❛♠♣❧❡ ✶✳✺✳✺✿ ❛❧t❡r♥❛t✐♥❣ s❡q✉❡♥❝❡ ❲❤❛t ✐s t❤❡ ❢♦r♠✉❧❛ ❢♦r t❤✐s s❡q✉❡♥❝❡✿

1, −1, 1, −1, ...? ❚❤❡ ♣❧♦t ✐s s✐♠♣❧❡✿

a1 = 3

✐♥st❡❛❞❄

2✱

✐✳❡✳✱

✶✳✺✳

❍♦✇ t♦ ✜♥❞ nt❤✲t❡r♠ ❢♦r♠✉❧❛s ❢♦r s❡q✉❡♥❝❡s

❋✐rst✱ ✇❡ ♥♦t✐❝❡ t❤❡s❡ ♥✉♠❜❡rs ❛r❡ ❥✉st ❛❧❧

✹✹

1✬s

❛♥❞ ♦♥❧② t❤❡ s✐❣♥

❛❧t❡r♥❛t❡s✳

❲❡ ✇r✐t❡ ✐t ✐♥ ❛ ♠♦r❡

❝♦♥✈❡♥✐❡♥t ❢♦r♠✿

a0 = 1, a1 = −1, a2 = 1, a3 = −1, ... ❚❤❡ ♣❛tt❡r♥ ✐s ❝❧❡❛r ❛♥❞ t❤❡ ❢♦r♠✉❧❛ ❝❛♥ ❜❡ ✇r✐tt❡♥ ❢♦r t❤❡ t✇♦ ❝❛s❡s s❡♣❛r❛t❡❧②✿

an = ❚❤✐s q✉❛❧✐✜❡s ❛s ✐ts

nt❤✲t❡r♠



−1 1

✐❢ ✐❢

n n

✐s ❡✈❡♥, ✐s ♦❞❞.

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

an = (−1)n+1 . ❲❡ ✇❡r❡ ❛❜❧❡ t♦ ❣❡t r✐❞ ♦❢ ✏✳✳✳✑✦

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

nt❤✲t❡r♠

❢♦r♠✉❧❛ ❢♦r t❤❡

❛❧t❡r♥❛t✐♥❣ s❡q✉❡♥❝❡✱

a0 = 1, a1 = −1, a2 = 1, a3 = −1, ..., ✐s ❣✐✈❡♥ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣✿

an = (−1)n ,

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

❊①❡r❝✐s❡ ✶✳✺✳✼

❲❤❛t ✐s t❤❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ s❡q✉❡♥❝❡ ✐❢ ✐t st❛rts ✇✐t❤

a1 = 1

✐♥st❡❛❞❄

❊①❡r❝✐s❡ ✶✳✺✳✽

❚❤❡ ❛❧t❡r♥❛t✐♥❣ s❡q✉❡♥❝❡ ❛❜♦✈❡ ✐s ❛ ❴❴❴❴❴❴ s❡q✉❡♥❝❡✳

❊①❡r❝✐s❡ ✶✳✺✳✾

P♦✐♥t ♦✉t ❛ ♣❛tt❡r♥ ✐♥ ❡❛❝❤ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡q✉❡♥❝❡s ❛♥❞ s✉❣❣❡st ❛ ❢♦r♠✉❧❛ ❢♦r ✐ts ♣♦ss✐❜❧❡✿ ✶✳ ✷✳ ✸✳ ✹✳ ✺✳ ✻✳

1, 3, 5, 7, 9, 11, 13, 15, ... .9, .99, .999, .9999, ... 1/2, −1/4, 1/8, −1/16, ... 1, 1/2, 1/3, 1/4, ... 1, 1/2, 1/4 , 1/8, ... 2, 3, 5, 7, 11, 13, 17, ...

nt❤ t❡r♠ ✇❤❡♥❡✈❡r

✶✳✺✳

❍♦✇ t♦ ✜♥❞ nt❤✲t❡r♠ ❢♦r♠✉❧❛s ❢♦r s❡q✉❡♥❝❡s ✼✳ ✽✳

✹✺

1, −4, 9, −16, 25, ... 3, 1, 4, 1, 5, 1, 9, ...

❊①❛♠♣❧❡ ✶✳✺✳✶✵✿ r❡❣✉❧❛r ❞❡♣♦s✐ts✱ r❡s✉❧ts ❝♦♠♣✉t❡❞ ❆ ♣❡rs♦♥ st❛rts t♦ ❞❡♣♦s✐t

$20

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

$1000✿

an+1 = an + 20 . ❲❡ ❤❛✈❡ ♥♦✇ ❛ ❢♦r♠✉❧❛ ❢♦r t❤❡

nt❤

t❡r♠✿

an = 1000 + 20 · n , ❛ss✉♠✐♥❣ t❤❛t

a0 = 1000✿

❊①❡r❝✐s❡ ✶✳✺✳✶✶ ❍♦✇ ❧♦♥❣ ✇✐❧❧ ✐t t❛❦❡ t♦ ❞♦✉❜❧❡ ②♦✉r ♠♦♥❡②❄

❊①❛♠♣❧❡ ✶✳✺✳✶✷✿ ❝♦♠♣♦✉♥❞❡❞ ✐♥t❡r❡st✱ r❡s✉❧ts ❝♦♠♣✉t❡❞ ❆ ♣❡rs♦♥ ❞❡♣♦s✐ts

$1000

✐♥t♦ ❤✐s ❜❛♥❦ ❛❝❝♦✉♥t t❤❛t ♣❛②s

1%

✐♥t❡r❡st ❡✈❡r② ②❡❛r✿

an+1 = an · 1.01 . ❲❡ ❤❛✈❡ ♥♦✇ ❛ ❢♦r♠✉❧❛✿

an+1 = 1000 · 1.01n . ❚❤✐s ✐s t❤❡ ❣r❛♣❤✿

a0 ❞♦❧❧❛rs t♦ ❤✐s ❜❛♥❦ ❛❝❝♦✉♥t✳ ❙✉♣♣♦s❡ t❤❡ ❛❝❝♦✉♥t ♣❛②s R✱ t❤❡ ❞❡❝✐♠❛❧ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ❆P❘ ❝♦♠♣♦✉♥❞❡❞ ❛♥♥✉❛❧❧②✱ ✐✳❡✳✱ .10 ❢♦r 10 ♣❡r❝❡♥t ❡t❝✳ ❚❤❡ t♦t❛❧ ❛♠♦✉♥t ✐s t❤❡♥ ♠✉❧t✐♣❧✐❡❞ ❜② 1 + R ❛t t❤❡ ❡♥❞ ♦❢ ❡❛❝❤ ②❡❛r✳ ◆♦✇✱ ✐❢ an st❛♥❞s ❢♦r t❤❡ ❛♠♦✉♥t ✐♥ t❤❡ ❜❛♥❦ ❛❝❝♦✉♥t ❛❢t❡r n ②❡❛rs✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛✿

❍❡r❡ ✐s t❤❡ ❣❡♥❡r❛❧ s❡t✉♣✳ ❆ ♣❡rs♦♥ ❞❡♣♦s✐ts

an+1 = an · (1 + R) .

✶✳✺✳ ❍♦✇ t♦ ✜♥❞ nt❤✲t❡r♠ ❢♦r♠✉❧❛s ❢♦r s❡q✉❡♥❝❡s

✹✻

❆ ✈❡r❜❛❧ ❞❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ ♠♦❞❡❧ ✕ t❤❡ ❣r♦✇t❤ ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ s✐③❡ ♦❢ ❝✉rr❡♥t ❛♠♦✉♥t ✕ r❡✈❡❛❧s t❤❛t t❤✐s ✐s ❛ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥✳ ■ts nt❤✲t❡r♠ ❢♦r♠✉❧❛ ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✿

an = a0 · (1 + R)n , n = 1, 2, 3, ... ❊①❡r❝✐s❡ ✶✳✺✳✶✸

❍♦✇ ❧♦♥❣ ✇✐❧❧ ✐t t❛❦❡ t♦ ❞♦✉❜❧❡ ②♦✉r ♠♦♥❡②❄ ❊①❛♠♣❧❡ ✶✳✺✳✶✹✿ ❜❛❝t❡r✐❛ ♠✉❧t✐♣❧②✐♥❣

❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ♣♦♣✉❧❛t✐♦♥ ♦❢ ❜❛❝t❡r✐❛ t❤❛t ❞♦✉❜❧❡s ❡✈❡r② ❞❛②✳ ❋♦r ❡①❛♠♣❧❡✱ ✇❡ ❝❛♥ ✐♠❛❣✐♥❡ t❤❛t ❡✈❡r② ♦♥❡ ♦❢ t❤❡♠ ❞✐✈✐❞❡s ✐♥ ❤❛❧❢ ❡✈❡r② ❞❛②✳ ▲❡t pn ❜❡ t❤❡ ♥✉♠❜❡r ♦❢ ❜❛❝t❡r✐❛ ❛❢t❡r n ❞❛②s✿

pn+1 |{z}

♣♦♣✉❧❛t✐♦♥✿ ❛t t✐♠❡

=2 n+1

pn |{z}

❛t t✐♠❡

n

❚♦ ❦♥♦✇ pn ❢♦r ❛❧❧ n✬s✱ ✇❡ ♥❡❡❞ t♦ ❦♥♦✇ t❤❡ ✐♥✐t✐❛❧ ♣♦♣✉❧❛t✐♦♥ p0 ✳ ❚❤✐s ✐s t❤❡ ❣r❛♣❤✿

❲❡ ❤❛✈❡ ❛ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥✳ ■ts nt❤✲t❡r♠ ❢♦r♠✉❧❛ ✐s✿

pn = p0 2 n . ❊①❛♠♣❧❡ ✶✳✺✳✶✺✿ r❛❞✐♦❛❝t✐✈❡ ❞❡❝❛② ❛♥❞ r❛❞✐♦❝❛r❜♦♥ ❞❛t✐♥❣

■t ✐s ❦♥♦✇♥ t❤❛t ♦♥❝❡ ❛ tr❡❡ ✐s ❝✉t✱ t❤❡ r❛❞✐♦❛❝t✐✈❡ ❝❛r❜♦♥ ✐t ❝♦♥t❛✐♥s st❛rts t♦ ❞❡❝❛②✳ ■t ❧♦s❡s ❤❛❧❢ ♦❢ ✐ts ♠❛ss ♦✈❡r ❛ ♣❡r✐♦❞ ♦❢ t✐♠❡ ♦❢ ❛ ❝❡rt❛✐♥ ❧❡♥❣t❤ ❝❛❧❧❡❞ t❤❡ ❤❛❧❢✲❧✐❢❡ ♦❢ t❤❡ ❡❧❡♠❡♥t✳ ❚❤❡ ❧♦ss✱ t❤❡r❡❢♦r❡✱ ❢♦❧❧♦✇s t❤❡ ❢❛♠✐❧✐❛r ❡①♣♦♥❡♥t✐❛❧ ❞❡❝❛② ♠♦❞❡❧✿

an+1 = an ·

1 . 2

❆ ✈❡r❜❛❧ ❞❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ ♠♦❞❡❧ ✕ t❤❡ ❞❡❝❛② ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ❝✉rr❡♥t ❛♠♦✉♥t ✕ r❡✈❡❛❧s t❤❛t t❤✐s ✐s ❛ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥✳ ❚❤❡ ❞✐✣❝✉❧t② ✐s t❤❛t ❤❡r❡ n ✐s ♥♦t t❤❡ ♥✉♠❜❡r ♦❢ ②❡❛rs ❜✉t t❤❡ ♥✉♠❜❡r ♦❢ ❤❛❧❢✲❧✐✈❡s✦ ❋♦r ❡①❛♠♣❧❡✱ t❤❡ ♣❡r❝❡♥t❛❣❡ ♦❢ t❤✐s ❡❧❡♠❡♥t✱ 14 ❈✱ ❧❡❢t ✐s ♣❧♦tt❡❞ ❜❡❧♦✇ ✉♥❞❡r t❤❡ ❛ss✉♠♣t✐♦♥ t❤❛t ✐ts ❤❛❧❢✲❧✐❢❡ ✐s 5730 ②❡❛rs ✭✐✳❡✳✱ t❤❡ t✐♠❡ ✐t t❛❦❡s t♦ ❣♦ ❢r♦♠ 100% t♦ 50%✮✿

✶✳✺✳ ❍♦✇ t♦ ✜♥❞ nt❤✲t❡r♠ ❢♦r♠✉❧❛s ❢♦r s❡q✉❡♥❝❡s

✹✼

❚❤❡ ✐❞❡❛ ✇❡ ✇✐❧❧ ❞❡✈❡❧♦♣ ✐s ❛s ❢♦❧❧♦✇s✿ • ❋✐♥❞ ②♦✉ t❤❡ ❡❧❡♠❡♥t✬s ❤❛❧❢✲❧✐❢❡✳ • ▼❡❛s✉r❡ t❤❡ ♣❡r❝❡♥t❛❣❡ ♦❢ t❤❡ ❡❧❡♠❡♥t ♣r❡s❡♥t ✈s✳ t❤❡ ❛♠♦✉♥t ♥♦r♠❛❧❧② ♣r❡s❡♥t✳ • ❈❛❧❝✉❧❛t❡ t❤❡ t✐♠❡ ✇❤❡♥ tr❡❡ ✇❛s ❝✉t✳ ❙✉♣♣♦s❡ ❛ ♣❛r❝❤♠❡♥t ❤❛s 74% ♦❢ 14 ❈ ❧❡❢t✳ ❍♦✇ ♦❧❞ ✐s ✐t❄ ▲❡t✬s t❛❦❡ ❛ ❝❧♦s❡r ❧♦♦❦ ❛t t❤❡ ❣r❛♣❤✿

❯♥❢♦rt✉♥❛t❡❧②✱ t❤❡ ♠♦❞❡❧ ♠❡❛s✉r❡s t✐♠❡ ✐♥ ♠✉❧t✐♣❧❡s ♦❢ t❤❡ ❤❛❧❢✲❧✐❢❡✱ 5730 ②❡❛rs✳ ❆♥② ♣❡r✐♦❞ s❤♦rt❡r t❤❛♥ t❤❛t ✐s ♦✉t ♦❢ r❡❛❝❤ ❢♦r ♥♦✇✳ ❲❡ ❝❛♥ tr② t♦ ❡st✐♠❛t❡ t❤❡ ❛♥s✇❡r ❜② ❛ss✉♠✐♥❣ t❤❛t t❤✐s ✐s ❛♥ ❛r✐t❤♠❡t✐❝ ♣r♦❣r❡ss✐♦♥✱ ②❡❛r❧②✿

❚❤❡ ♣❡r✐♦❞✱ t❤❡r❡❢♦r❡✱ ✐s ❡st✐♠❛t❡❞ t♦ ❜❡ ❝❧♦s❡ t♦✿ 1 1 ✲❧✐❢❡ = 4 2



1 ✲❧✐❢❡ 2



= 2865 .

■♥ ❈❤❛♣t❡r ✹✱ ✇❡ ✇✐❧❧ ❧❡❛r♥ ❤♦✇ t♦ ❞❡❛❧ ✇✐t❤ ❢r❛❝t✐♦♥❛❧ t✐♠❡ ♣❡r✐♦❞s ❛♥❞ s♦❧✈❡ t❤❡ ♣r♦❜❧❡♠ ❡①❛❝t❧②✳ ❊①❛♠♣❧❡ ✶✳✺✳✶✻✿ ◆❡✇t♦♥✬s ▲❛✇ ♦❢ ❈♦♦❧✐♥❣

❆ ❝♦♦❧❡r ♦❜❥❡❝t ✐♥ ❛ ✇❛r♠❡r ❡♥✈✐r♦♥♠❡♥t ❤❡❛ts ✉♣✱ ❛♥❞ ❛ ✇❛r♠❡r ♦❜❥❡❝ts ❝♦♦❧s ❞♦✇♥ ✐♥ ❛ ❝♦❧❞❡r ❡♥✈✐r♦♥♠❡♥t✳ ❍♦✇ ❢❛st t❤❛t ❤❛♣♣❡♥s ❞❡♣❡♥❞s ♦♥ t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ♦❜❥❡❝t✬s t❡♠♣❡r❛t✉r❡ Tn ✭❛t t✐♠❡ n✮ ❛♥❞ t❤❡ r♦♦♠ t❡♠♣❡r❛t✉r❡ R✳ ❚❤✐s ♥✉♠❜❡r ❞❡t❡r♠✐♥❡s t❤❡ ♥❡✇ t❡♠♣❡r❛t✉r❡ Tn+1 ✳ ❚❤❡ ❧❛✇ st❛t❡s t❤❛t

✶✳✺✳ ❍♦✇ t♦ ✜♥❞ nt❤✲t❡r♠ ❢♦r♠✉❧❛s ❢♦r s❡q✉❡♥❝❡s

✹✽

t❤❡ r❛t❡ ♦❢ ❝♦♦❧✐♥❣ ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤✐s ❞✐✛❡r❡♥❝❡✿ Tn+1 − R = (Tn − R) · k ,

❢♦r s♦♠❡ k < 1✳ ❲❡✱ t❤❡r❡❢♦r❡✱ ❤❛✈❡ ❛ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛✿ Tn+1 = R + (Tn − R) · k .

❯♥❧❡ss R = 0✱ t❤✐s ✐s ♥♦t ❛ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥ ✭❜✉t Tn − R ✐s✮✳ ❚❤❡r❡ ❛r❡ t❤r❡❡ ❝❛s❡s ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ ✐♥✐t✐❛❧ t❡♠♣❡r❛t✉r❡✿ • T0 > R✿ ❝♦♦❧✐♥❣✱ • T0 = R✿ ✉♥❝❤❛♥❣✐♥❣✱ • T0 < R✿ ✇❛r♠✐♥❣✳ ❲❡ ♣❧♦t ❜❡❧♦✇ s❡✈❡r❛❧ s❡q✉❡♥❝❡s ✇✐t❤ ✈❛r✐♦✉s ✐♥✐t✐❛❧ t❡♠♣❡r❛t✉r❡s✿

❊①❡r❝✐s❡ ✶✳✺✳✶✼

❋✐♥❞ t❤❡ nt❤✲t❡r♠ ❢♦r♠✉❧❛ ❢♦r t❤❡ s❡q✉❡♥❝❡ ✐♥ t❤❡ ❛❜♦✈❡ ❡①❛♠♣❧❡✳ ❈♦❧❧❡❝t✐✈❡❧②✱ t❤❡s❡ ❡①❛♠♣❧❡s ❛r❡ ❝❛❧❧❡❞ ❡①♣♦♥❡♥t✐❛❧ ♠♦❞❡❧s ❀ t❤❡② ❛r❡ ❝❤❛r❛❝t❡r✐③❡❞ ❜② t❤❡ ❞②♥❛♠✐❝s t❤❛t ❝♦♠❡s ❢r♦♠ r❡♣❡❛t❡❞ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❜② t❤❡ s❛♠❡ ♥✉♠❜❡r✳ ❚❤❡r❡ ❛r❡ ♠❛♥② ♦t❤❡r t②♣❡s ♦❢ ♠♦❞❡❧s✳ ❊①❛♠♣❧❡ ✶✳✺✳✶✽✿ ♥✉♠❜❡r ♦❢ ♣❧❛②s ✐♥ r♦✉♥❞ r♦❜✐♥

■❢ ✇❡ ❤❛✈❡ n t❡❛♠s t♦ ♣❧❛② ❡❛❝❤ ♦t❤❡r ❡①❛❝t❧② ♦♥❝❡✱ ❤♦✇ ♠❛♥② ❣❛♠❡s ❞♦ ✇❡ ❤❛✈❡ t♦ ♣❧❛♥ ❢♦r❄ ❆ t❛❜❧❡ ❝♦♠♠♦♥❧② ✉s❡❞ ❢♦r s✉❝❤ ❛ t♦✉r♥❛♠❡♥t ✐s ❜❡❧♦✇✿

❚❤❡ t❛❜❧❡ r❡✈❡❛❧s t❤❡ ❢♦❧❧♦✇✐♥❣✿ ❚❤❡ ✜rst t❡❛♠ ✐s t♦ ♣❧❛② n − 1 ❣❛♠❡s✳ ▲✐❦❡✇✐s❡✱ t❤❡ s❡❝♦♥❞ ❛❧s♦ ✐s t♦ ♣❧❛② n − 1 ❣❛♠❡s ❜✉t ♦♥❡ ❧❡ss ✐s ❛❝t✉❛❧❧② ❝♦✉♥t❡❞ ❛s ✐t ✐s ❛❧r❡❛❞② ♦♥ t❤❡ ✜rst ❧✐st✳ ❚❤❡ t❤✐r❞ ✐s t♦ ♣❧❛② n − 1 ❣❛♠❡s ❜✉t t✇♦ ❧❡ss ✐s ❛❝t✉❛❧❧② ❝♦✉♥t❡❞ ❛s t❤❡② ❛r❡ ❛❧r❡❛❞② ♦♥ t❤❡ ✜rst ❛♥❞ s❡❝♦♥❞ ❧✐sts✳ ❆♥❞ s♦ ♦♥✳ ❚❤❡ t♦t❛❧ ✐s (n − 1) + (n − 2) + ... + 2 + 1 .

❲❡ ❝❛♥ tr❡❛t t❤❡s❡ n − 1 ♥✉♠❜❡rs ❛s ❛ r❡❝✉rs✐✈❡ s❡q✉❡♥❝❡✿

a1 = 1, an+1 = an + n .

❍♦✇ ❞♦ ✇❡ ✜♥❞ ❛♥ ❡①♣❧✐❝✐t✱ ❞✐r❡❝t ❢♦r♠✉❧❛ ❢♦r t❤❡ nt❤ t❡r♠ ♦❢ t❤✐s s❡q✉❡♥❝❡❄

✶✳✺✳

❍♦✇ t♦ ✜♥❞ nt❤✲t❡r♠ ❢♦r♠✉❧❛s ❢♦r s❡q✉❡♥❝❡s

✹✾

❚❤❡ t❛❜❧❡ s✉❣❣❡sts t❤❡ ❛♥s✇❡r✳ ❚❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ❝❡❧❧s ✐♥ ❛♥ 2 ♦♥ t❤❡ ❞✐❛❣♦♥❛❧✱ ✐t✬s n − n✳ ❋✐♥❛❧❧②✱ ✇❡ t❛❦❡ ♦♥❧② ❤❛❧❢ ♦❢ t❤♦s❡✿

n × n t❛❜❧❡ (n2 − n)/2✳

✐s

n2 ✳

❲✐t❤♦✉t t❤❡ ♦♥❡s

n ❝♦♥s❡❝✉t✐✈❡ ✐♥t❡❣❡rs st❛rt✐♥❣ ❢r♦♠ 1 ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✿

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

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

n(n + 1) . 2

❲❡ ✇❡r❡ ❛❜❧❡ t♦ ❣❡t r✐❞ ♦❢ ✏✳✳✳✑✦ ✭❚❤❡ ❢♦r♠✉❧❛ ✇✐❧❧ ✜♥❞ ❛♥♦t❤❡r ✉s❡ ✐♥ ❈❤❛♣t❡r ✸■❈✲✶✳✮

❊①❡r❝✐s❡ ✶✳✺✳✶✾

❙❤♦✇ t❤❛t t❤❡ s✉♠ ♦❢ ❝♦♥s❡❝✉t✐✈❡

♦❞❞

✐♥t❡❣❡rs s❛t✐s✜❡s t❤❡ ❢♦❧❧♦✇✐♥❣✿

1 + 3 + 5 + 7 + ... + (2n − 1) = n2 . ❈❛♥ ✇❡ ❛❧✇❛②s ✜♥❞ ❛♥ ❡①♣❧✐❝✐t ❢♦r♠✉❧❛❄ s❡q✉❡♥❝❡ ✇✐t❤♦✉t

nt❤✲t❡r♠

◆♦✳

❲❡ ❤❛✈❡ ❛❧r❡❛❞② s❡❡♥ ❛♥ ❡①❛♠♣❧❡ ♦❢ ❛ r❡❝✉rs✐✈❡❧② ❞❡✜♥❡❞

❢♦r♠✉❧❛✱ t❤❡ ❢❛❝t♦r✐❛❧✱

n! ✳

❊①❛♠♣❧❡ ✶✳✺✳✷✵✿ ❞❡♣♦s✐ts ❛♥❞ ✐♥t❡r❡st✱ t♦❣❡t❤❡r

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

an+1 = an · 1.05 + 200 .

❇✉t ✐s t❤❡r❡ ❛

❞✐r❡❝t nt❤✲t❡r♠ ❢♦r♠✉❧❛❄ an =



■t✬s t♦♦ ❝✉♠❜❡rs♦♠❡ t♦ ❜❡ ♦❢ ❛♥② ✉s❡✿




... (a0 · 1.05 + 200) · 1.05 + 200 ... r❡♣❡❛t ♥ t✐♠❡s



· 1.05 + 200 .

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

❊①❛♠♣❧❡ ✶✳✺✳✷✶✿ r❡str✐❝t❡❞ ❣r♦✇t❤

❙✉♣♣♦s❡ ♦✉r ❜❛❝t❡r✐❛ ❧✐✈❡ ✐♥ ❛

❥❛r✳

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

✶✳ ❚❤❡ r❛t❡ ♦❢ r❡♣r♦❞✉❝t✐♦♥ ✇✐❧❧ st✐❧❧ ❜❡ ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ❝✉rr❡♥t ♣♦♣✉❧❛t✐♦♥ ✕ ❜✉t ♦♥❧② ✇❤❡♥ t❤❡ ♣♦♣✉❧❛t✐♦♥ s✐③❡ ✐s ✏s♠❛❧❧✑ ❛♥❞ t❤❡ ❡✛❡❝ts ♦❢ t❤❡ r❡str✐❝t❡❞ r♦♦♠ ❛r❡ ♥❡❣❧✐❣✐❜❧❡❀ ❛♥❞ ✷✳ ❖♥❝❡ ✐t ❣❡ts ✏❧❛r❣❡✑✱ st❛r✈❛t✐♦♥ st❛rts ✕ t❤❡ ❣r♦✇t❤ r❛t❡ ✇✐❧❧ ❞❡❝r❡❛s❡ ❛t ❛ r❛t❡ ♣r♦♣♦rt✐♦♥❛❧ t♦ ❤♦✇ ❝❧♦s❡ t❤❡ ♣♦♣✉❧❛t✐♦♥ ✐s t♦ t❤❡ t❤❡♦r❡t✐❝❛❧

❝❛rr②✐♥❣ ❝❛♣❛❝✐t②✳

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

pn+1 = 2pn

1000✳

❚❤❡♥ t❤❡ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛

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

pn+1 = 2pn ·

1000 − pn . 1000

❚❤❡ ♥❡✇ ❢❛❝t♦r ✐♥❞✐❝❛t❡s ❤♦✇ ❝❧♦s❡ ✇❡ ❛r❡✱ ♣r♦♣♦rt✐♦♥❛❧❧②✱ t♦ t❤✐s ❧✐♠✐t✳ ▲❡t✬s t❛❦❡ ❛ s♣❡❝✐✜❝ ❡①❛♠♣❧❡✳ ■❢ t❤❡ ❝✉rr❡♥t ♣♦♣✉❧❛t✐♦♥ ✐s

pn+1 = 2 · 800 · ■t ❣♦❡s

❞♦✇♥

✉♣

t❤❡ ♥❡①t ✐s

1000 − 800 = 320 . 1000

❜❡❝❛✉s❡ ✐t ✐s t♦♦ ❝❧♦s❡ t♦ t❤❡ ❝❛♣❛❝✐t②✳ ❇✉t t❤❡ ♥❡①t ♦♥❡✱

pn+2 = 2 · 320 · ✐s

pn = 800✱

1000 − 320 = 435.2 , 1000

❛❣❛✐♥✦ ■t ♠✐❣❤t ❝♦♥t✐♥✉❡ t♦ ❣♦ ✉♣ ❛♥❞ ❞♦✇♥✳

✶✳✻✳

❚❤❡ ❛❧❣❡❜r❛ ♦❢ ❡①♣♦♥❡♥ts

✺✵

■♥ ❣❡♥❡r❛❧✱ ✇❡ ❞❡✜♥❡ ❛ s❡q✉❡♥❝❡ t♦ r❡♣r❡s❡♥t t❤❡ ♣r♦♣♦rt✐♦♥ ♦❢ t❤❡ ❧❛r❣❡st ♣♦ss✐❜❧❡ ♣♦♣✉❧❛t✐♦♥ ❛s ❢♦❧❧♦✇s✿ an+1 = ran (1 − an ) ,

✇❤❡r❡ r > 0 ✐s ❛ ♣❛r❛♠❡t❡r✳ ❙♦✱ t❤❡ ✜rst ❢❛❝t♦r tr✐❡s t♦✱ s❛②✱ ❞♦✉❜❧❡ t❤❡ ♣♦♣✉❧❛t✐♦♥✱ ✇❤✐❧❡ t❤❡ s❡❝♦♥❞ ❤♦❧❞s ✐t ❜❛❝❦✳ ❋♦r t❤❡ s♣r❡❛❞s❤❡❡t✱ t❤❡ ❢♦r♠✉❧❛ r❡❢❡rs t♦ t❤❡ ♣r❡✈✐♦✉s st❛t❡✱ ❘❬✲✶❪❈ ✿

❂❘✷❈✷✯❘❬✲✶❪❈✯✭✶✲❘❬✲✶❪❈✮ ✇❤✐❧❡ ❘✷❈✷ ✐s t❤❡ ❝❡❧❧ t❤❛t ❝♦♥t❛✐♥s t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ♣❛r❛♠❡t❡r r✳ ❲❡ ♣r❡s❡♥t ❛ ❝♦♠♣✉t❛t✐♦♥ ✇✐t❤ ❛ s♣❡❝✐✜❝ ♣❛r❛♠❡t❡r r = 3.9 ✭❛♥❞ a1 = .5✮✿

❙✉❝❤ ❛ s❡q✉❡♥❝❡ ✐s ❝❛❧❧❡❞ ❛ ♣❛r❛♠❡t❡r r✿

❧♦❣✐st✐❝ s❡q✉❡♥❝❡✳

■ts ❞②♥❛♠✐❝s ❞r❛♠❛t✐❝❛❧❧② ❞❡♣❡♥❞s ♦♥ t❤❡ ❝❤♦✐❝❡ ♦❢ t❤❡

❆❜♦✈❡✱ t❤❡ s❡q✉❡♥❝❡ st❛rts ✇✐t❤ 1/2 ❛♥❞ t❤❡♥ s❤♦✇s s❡✈❡r❛❧ ❞✐✛❡r❡♥t t②♣❡s ♦❢ ❞②♥❛♠✐❝s✿ ❞❡❝❛②✱ ❡q✉✐✲ ❧✐❜r✐✉♠✱ ♦s❝✐❧❧❛t✐♥❣ ❝♦♥✈❡r❣❡♥❝❡ t♦ ❡q✉✐❧✐❜r✐✉♠✱ ❛♥❞ ❝❤❛♦s✳ ❚❤❡r❡ ✐s ♥♦ nt❤✲t❡r♠ ❢♦r♠✉❧❛✳ ❚❤❡ ❝❤❛♦t✐❝ ♦♥❡ ❧♦♦❦s s✐♠✐❧❛r t♦ ❛ ❝♦♠♣✉t❡r✲❣❡♥❡r❛t❡❞ r❛♥❞♦♠ s❡q✉❡♥❝❡ ❜❡❧♦✇✿

❊①❡r❝✐s❡ ✶✳✺✳✷✷

❲❤❛t ✐❢ ♥❡✇ ❜❛❝t❡r✐❛ ❛r❡ ✐♥tr♦❞✉❝❡❞ t♦ t❤❡ ❥❛r ❛t ❛ ❝♦♥st❛♥t r❛t❡❄

✶✳✻✳ ❚❤❡ ❛❧❣❡❜r❛ ♦❢ ❡①♣♦♥❡♥ts

▲❡t✬s r❡✈✐❡✇ ♦✉r ❛♥❛❧②s✐s ❢r♦♠ ❡❛r❧✐❡r ✇❤❡r❡ t❤❡ ❡①♣♦♥❡♥ts ✕ ❜② ❛♥❛❧♦❣② ✇✐t❤ ♣r♦❞✉❝ts ✕ ❝♦♠❡ ❢r♦♠✿

✶✳✻✳

❚❤❡ ❛❧❣❡❜r❛ ♦❢ ❡①♣♦♥❡♥ts

✺✶

❆♥❛❧♦❣②✿ r❡♣❡❛t❡❞ ❛❞❞✐t✐♦♥ ✈s✳ r❡♣❡❛t❡❞ ♠✉❧t✐♣❧✐❝❛t✐♦♥

▼✉❧t✐♣❧✐❝❛t✐♦♥✿

n = 1, 2, 3, ...

❊①♣♦♥❡♥t✐❛t✐♦♥✿

a + ... + a} = a · n |a + a + {z

n t✐♠❡s

n t✐♠❡s

n=0

❘✉❧❡s✿ 1.

a · ... · a} = an |a · a · {z a0 = 1

a·0 =0

a(n + m) = a · n + a · m

an+m = an · am

(ab)n = an bn

(a + b)n = a · n + b · n

2.

anm = (an )m

a · (nm) = (a · n) · m

3. ❊①❡r❝✐s❡ ✶✳✻✳✶

❘❡♣r❡s❡♥t ❛s ❛ ♣♦✇❡r ♦❢ 3✿ ✭❛✮ 33 · 9 ✱ ✭❜✮ (9 · 27)5 ✳ ❚❤✐s ♥❡✇ ❛❧❣❡❜r❛ ❤❡❧♣s ✉s t♦ ✉♥❞❡rst❛♥❞ ❜❡st✿ ❇✉t ✇❡ ❝❛♥ ✐♠♣r♦✈❡ ✐t ❡✈❡♥ ❢✉rt❤❡r✿

❢❛❝t♦r✐♥❣

♦❢ ✐♥t❡❣❡rs✳ ❋♦r ❡①❛♠♣❧❡✱ t❤❡ ❧❛st ❢❛❝t♦r✐③❛t✐♦♥ ❜❡❧♦✇ ✐s

20 = 4 · 5 = 2 · 2 · 5 = 22 51 . 20 = 22 30 51 .

▲❡t✬s r❡✈✐❡✇ ✇❤❛t ✇❡ ❦♥♦✇ ❛❜♦✉t ❢❛❝t♦r✐♥❣✿ ❚❤❡♦r❡♠ ✶✳✻✳✷✿ ❘✉❧❡s ♦❢ ❋❛❝t♦r✐♥❣ ■♥t❡❣❡rs

•

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

> 1✿

2, 3, 5, 7, 11, 13, ... •

❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❆r✐t❤♠❡t✐❝✿

•

❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❆r✐t❤♠❡t✐❝✿ ❚❤✐s r❡♣r❡s❡♥t❛t✐♦♥ ✐s ✉♥✐q✉❡✱

❊✈❡r② ✐♥t❡❣❡r ❝❛♥ ❜❡ r❡♣r❡✲

s❡♥t❡❞ ❛s t❤❡ ♣r♦❞✉❝t ♦❢ ♣r✐♠❡s✳

✉♣ t♦ t❤❡ ♦r❞❡r ♦❢ t❤❡ ❢❛❝t♦rs✿

q = 2m · 3n · 5k · ... •

❚❤❡ ♠✉❧t✐♣❧✐❝✐t② ♦❢ ❛ ♣r✐♠❡ ❢❛❝t♦r ✐s ❤♦✇ ♠❛♥② t✐♠❡s ✐t ❛♣♣❡❛rs ✐♥ t❤❡ ❢❛❝t♦r✐③❛t✐♦♥✳

❚❤❡♥✱ ✐♥ t❤❡ ♣r✐♠❡ ❢❛❝t♦r✐③❛t✐♦♥

20 = 22 30 51 ,

t❤❡ ♠✉❧t✐♣❧✐❝✐t② ♦❢ 2 ✐s 2✱ ♦❢ 5 ✐s 1✱ ❛♥❞ t❤❡ r❡st ❛r❡ 0s✳ ❆♥❞ t❤❡r❡ ❛r❡ ♥♦ ♦t❤❡r ❢❛❝t♦r✐③❛t✐♦♥s✦ ❊①❛♠♣❧❡ ✶✳✻✳✸✿ ♣r✐♠❡ ❢❛❝t♦r✐③❛t✐♦♥

▲❡t✬s ✜♥❞ t❤❡ ♣r✐♠❡ ❢❛❝t♦r✐③❛t✐♦♥ ♦❢ 1176✳ ❖✉r str❛t❡❣② ✐s t♦ ❞✐✈✐❞❡ ❜② ❧❛r❣❡r ❛♥❞ ❧❛r❣❡r ♣r✐♠❡s✱ s✇✐t❝❤✐♥❣ t♦ t❤❡ ♥❡①t ♦♥❡ ✇❤❡♥ ✇❡ ❝❛♥✬t✳ ❲❡ st❛rt ✇✐t❤ 2 ❛♥❞ t❤❡♥ ❣♦ ✉♣✳ ❲❡ tr② t♦ ❞✐✈✐❞❡ ❜② 2 ❛♥❞ ❞✐s❝♦✈❡r t❤❛t ✐t ✐s ♣♦ss✐❜❧❡✿ 1176/2 = 588 .

❲❡ st❛rt t❤❡ ❧✐st ♦❢ ❢❛❝t♦rs ✇✐t❤ t❤✐s 2✳ ❲❡ ✇✐❧❧ ❝♦♥t✐♥✉❡ t♦ tr② t♦ ❞✐✈✐❞❡ ❜② 2✳ ■t ✐s ♣♦ss✐❜❧❡✿ 588/2 = 294 ,

✶✳✻✳

❚❤❡ ❛❧❣❡❜r❛ ♦❢ ❡①♣♦♥❡♥ts

✺✷

❛♥❞ ✇❡ ❛❞❞ t❤✐s 2 t♦ ♦✉r ❧✐st ♦❢ ❢❛❝t♦rs✿ 2, 2✳ ❲❡ ❝♦♥t✐♥✉❡ t♦ ❞✐✈✐❞❡ ❜② 2✿

294/2 = 147 . ■t ✇♦r❦s ❛♥❞ ✇❡ ❤❛✈❡✿ 2, 2, 2✳ ❇✉t t❤❡ r❡s✉❧t ✐s ♦❞❞✦ ❲❡✱ t❤❡r❡❢♦r❡✱ ♠♦✈❡ ♦♥ t♦ t❤❡ ♥❡①t ♣r✐♠❡✱ 3✿

147/3 = 49 . ■t ✇♦r❦s ❛♥❞ ♦✉r ❧✐st ❜❡❝♦♠❡s✿ 2, 2, 2, 3✳ ❚❤❡ r❡s✉❧t ✐s ♥♦t ❞✐✈✐s✐❜❧❡ ❜② 3 ❛♥❞ ✇❡✱ t❤❡r❡❢♦r❡✱ ♠♦✈❡ ♦♥ t♦ t❤❡ ♥❡①t ♣r✐♠❡✱ 5✳ ❲❡ ❢❛✐❧ t♦ ❞✐✈✐❞❡ ❜② 5✳ ❚❤❡ ♥❡①t ♣r✐♠❡ ✇♦r❦s✿

49/7 = 7 . ❚❤✐s ❧❛st ♥✉♠❜❡r ✐s ❛❧s♦ ♣r✐♠❡ ❛♥❞ ✇❡ ❛r❡ ❞♦♥❡ ✇✐t❤ ♦✉r ❧✐st✿

2, 2, 2, 3, 7, 7 . ❚❤❡ ♣r✐♠❡ ❢❛❝t♦r✐③❛t✐♦♥ ✐s ❛s ❢♦❧❧♦✇s✿

1176 = 2 · 2 · 2 · 3 · 7 · 7 = 23 31 50 72 . ❍❡r❡✱ 3, 1, 0, 2 ❛r❡ t❤❡ r❡s♣❡❝t✐✈❡ ♠✉❧t✐♣❧✐❝✐t✐❡s✳ ❊①❡r❝✐s❡ ✶✳✻✳✹

❋❛❝t♦r✐③❡ ❡❛❝❤ ✐♥t♦ t❤❡ ♣r♦❞✉❝t ♦❢ ♣♦✇❡rs ♦❢ ♣r✐♠❡s✿ ✭❛✮ 500 ✱ ✭❜✮ 660 ✱ ✭❝✮ 1024 ✳ ❲❡ ❛r❡ ❢❛♠✐❧✐❛r ✇✐t❤

❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥s✳

❚❤❡② st❛rt ✇✐t❤ ✐♥❞❡① n = 0✿

❍♦✇❡✈❡r✱ t❤❡r❡ ❛r❡ ♦t❤❡r ✈❛❧✉❡s ♦❢ n t❤❛t ♠✐❣❤t ❛❧s♦ ✐♥t❡r❡st ✉s✦ ❊①❛♠♣❧❡ ✶✳✻✳✺✿ ❜❛❝t❡r✐❛ ♠✉❧t✐♣❧②✐♥❣✱ t❤❡ ♣❛st

❙✉♣♣♦s❡ ♦✉r ❜❛❝t❡r✐❛ ❞♦✉❜❧❡ ✐♥ ♥✉♠❜❡r ❡✈❡r② ❞❛② ❛♥❞ t❤❡ ❝✉rr❡♥t ♣♦♣✉❧❛t✐♦♥ ✐s 1000✳ ❚❤❡♥✱ t♦♠♦rr♦✇ ✐t ✇✐❧❧ ❜❡ 1, 000 · 2✱ ❛♥❞ s♦ ♦♥✳ ❆❢t❡r n ❞❛②s✱ ✐t✬s

1, 000 · 2n .

❇✉t ❤♦✇ ♠❛♥② ✇❡r❡ t❤❡r❡ ②❡st❡r❞❛② ❄ ❍❛❧❢ ♦❢ ✐t✱ 500✦ ❉♦❡s t❤✐s ♥✉♠❜❡r ✜t ✐♥t♦ ♦✉r ❣❡♦♠❡tr✐❝ ♣r♦❣r❡s✲ s✐♦♥ an ❄ ❈❛♥ ✇❡ ❝❤♦♦s❡ n = −1❄ ▲❡t✬s tr②✿ ❄

a−1 = 500 = 1, 000/2 == 1, 000 · 2−1 . ■t s❡❡♠s t❤❛t ✐❢ ✇❡ ❣♦ ✐♥t♦ t❤❡ ❢✉t✉r❡✱ ✇❡ ♠✉❧t✐♣❧②✱ ❛♥❞ ✐❢ ✇❡ ❣♦ t♦ t❤❡ ♣❛st✱ ✇❡✬❧❧ ♥❡❡❞ t♦

❞✐✈✐❞❡✳

❏✉st ❛s ❡❛r❧✐❡r ✐♥ t❤✐s ❝❤❛♣t❡r ✇❡ ❢❛❝❡ ♥❡✇ ❝✐r❝✉♠st❛♥❝❡s ❛♥❞✱ ♦♥❝❡ ❛❣❛✐♥✱ ✇❡ ❛s❦ ♦✉rs❡❧✈❡s✿ ❈❛♥ ✇❡ ♣r♦❝❡❡❞ ✇✐t❤♦✉t ❝❤❛♥❣✐♥❣ t❤❡ r✉❧❡s ❄ ❈❛♥ t❤❡ ❡①♣♦♥❡♥ts ❜❡

♥❡❣❛t✐✈❡ ❄

❲❡ st❛rt ✇✐t❤ n = −1✳ ❈❛♥ −1 ❜❡ t❤❡ ❡①♣♦♥❡♥t✱ s✉❝❤ ❛s 2−1 ❄ ❆♥❞ ✇❤❛t ✇♦✉❧❞ ❜❡ t❤❡ ♦✉t❝♦♠❡ ✕ ❛♥❞ t❤❡ ♠❡❛♥✐♥❣ ✕ ♦❢ r❡♣❡❛t✐♥❣ ❛♥ ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥ ♠✐♥✉s 1 t✐♠❡s ❄✦

✶✳✻✳

❚❤❡ ❛❧❣❡❜r❛ ♦❢ ❡①♣♦♥❡♥ts

❲❡ ✇✐❧❧ ♥❡❡❞ ❛♥♦t❤❡r

✺✸

❝♦♥✈❡♥t✐♦♥

✦ ❲❡ ❝❤♦♦s❡✱ ❢♦r

a 6= 0✱

a−1 =

t❤❡ ❢♦❧❧♦✇✐♥❣

♥♦t❛t✐♦♥

✿

1 . a

❚❤❡ ❝❤♦✐❝❡ ✐s ❞✐❝t❛t❡❞ ❜② ♦✉r ❞❡s✐r❡ ❢♦r t❤❡ t❤r❡❡ ♣r♦♣❡rt✐❡s t♦ ❜❡ s❛t✐s✜❡❞✦

❚❤❡♦r❡♠ ✶✳✻✳✻✿ ❊①♣♦♥❡♥t ❊q✉❛❧ t♦ −1 ❚❤❡ r✉❧❡s ♦❢ ❡①♣♦♥❡♥ts ❤♦❧❞ ❢♦r

a−1 = 1/a

❜✉t ❢❛✐❧ ❢♦r ❛♥② ♦t❤❡r ❝❤♦✐❝❡ ♦❢

a−1 ✳

Pr♦♦❢✳ ▲❡t✬s ❝❤❡❝❦✳ ❲❡ ♣❧✉❣ ✐♥ ♣r❡s❡♥t❡❞ ❛❜♦✈❡✿

n = ±1 ❛♥❞ m = ±1✱ ✉s❡ ♦✉r ❝♦♥✈❡♥t✐♦♥✱ ❛♥❞ t❤❡♥ ❛♣♣❧② ♦♥❡ ♦❢ t❤❡ ♣r♦♣❡rt✐❡s

✶✿

an+m = an · am n = −1, m = 1 =⇒ a−1+1

✷✿

an bn

= (ab)n

n = −1

✸✿

anm

= (an )m

n = −1, m = 1 =⇒ a(−1)1

=⇒ a−1 b−1 = (ab)−1

n = 1, m = −1 =⇒ a1(−1)

1 ·a a 1 1 1 ⇐⇒ · = a b ab 1 ⇐⇒ a−1 = a ⇐⇒ a−1 = a−1

= a−1 · a1 ⇐⇒ a0 =

= (a−1 )1 = (a1 )−1

❚❘❯❊ ❚❘❯❊ ❚❘❯❊ ❚❘❯❊

❊①❡r❝✐s❡ ✶✳✻✳✼ Pr♦✈✐❞❡ t❤❡ r❡st ♦❢ t❤❡ ♣r♦♦❢✳

❝♦♥✈❡♥t✐♦♥✿ ❊①❡r❝✐s❡ ✶✳✻✳✽

❙♦✱ t❤✐s ✐s ♦✉r

▼✉❧t✐♣❧②✐♥❣ ❜② a ♠❡❛♥s ❞✐✈✐❞✐♥❣ ❜② a −1

❘❡♣r❡s❡♥t ❛s ❡①♣♦♥❡♥ts✿ ✭❛✮

.2 ✱

.25 ✱

✭❜✮

✭❝✮

✳

.0001 ✳

◆♦✇✱ t❤❡ r❡st ♦❢ t❤❡ ♥❡❣❛t✐✈❡ ♥✉♠❜❡rs✳ ❋✐rst✱ ❧❡t✬s t❛❦❡ ❛♥♦t❤❡r ❧♦♦❦ ❛t ♦✉r ❛♥❛❧♦❣②✳ ❛❞❞✐t✐♦♥✱ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❜② ❛

♥❡❣❛t✐✈❡

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

✐♥t❡❣❡r ♠❡❛♥s r❡♣❡❛t❡❞

s✉❜tr❛❝t✐♦♥

✭t❤❡ ✏✐♥✈❡rs❡✑ ♦❢ ❛❞❞✐t✐♦♥✮✿

a(−n) = −an = 0 − a − a − a − ... − a . ❙✐♠✐❧❛r❧②✱ ✇❤✐❧❡ ❡①♣♦♥❡♥t✐❛t✐♦♥ ❜② ❛ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r ♠❡❛♥s r❡♣❡❛t❡❞ ♠✉❧t✐♣❧✐❝❛t✐♦♥✱ ❡①♣♦♥❡♥t✐❛t✐♦♥ ❜② ❛

♥❡❣❛t✐✈❡

✐♥t❡❣❡r ♠❡❛♥s r❡♣❡❛t❡❞

❞✐✈✐s✐♦♥

✭t❤❡ ✏✐♥✈❡rs❡✑ ♦❢ ♠✉❧t✐♣❧✐❝❛t✐♦♥✮✿

a−n = 1 ÷ a ÷ a ÷ a ÷ ... ÷ a . ❖✉r ❝♦♥✈❡♥t✐♦♥ ❢♦r ❛♥②

n = 1, 2, 3, ...

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

❉❡✜♥✐t✐♦♥ ✶✳✻✳✾✿ ♥❡❣❛t✐✈❡ ❡①♣♦♥❡♥t ❚❤❡ ♣♦✇❡r ♦❢ t❤❡ ♥❡❣❛t✐✈❡ ♦❢ ❛ ♥✉♠❜❡r ✐s t❤❡ ❞✐✈✐s✐♦♥ ❜② t❤❡ ♣♦✇❡r ♦❢ t❤❛t ♥✉♠❜❡r✿

a−n =

1 an

✶✳✻✳

❚❤❡ ❛❧❣❡❜r❛ ♦❢ ❡①♣♦♥❡♥ts

✺✹

❚♦ ❝♦♥✜r♠✱ ✇❡ ♦❜s❡r✈❡ t❤✐s✿

−1 −1 a−1 · ... · a−1} = |a · a · {z n t✐♠❡s

||

1 ÷a ÷ a ÷ a ÷ ... ÷ a {z } |

=

n t✐♠❡s

a−1


n

|| n 1 a

❚❤❡♦r❡♠ ✶✳✻✳✶✵✿ ◆❡❣❛t✐✈❡ ❊①♣♦♥❡♥t ❘✉❧❡ ❚❤❡ r✉❧❡s ♦❢ ❡①♣♦♥❡♥ts ❤♦❧❞ ❢♦r ❝❤♦✐❝❡ ♦❢

−n

a

a−n = 1/an ,

✇❤❡r❡

n > 0✱

❜✉t ❢❛✐❧ ❢♦r ❛♥② ♦t❤❡r

✳

❊①❡r❝✐s❡ ✶✳✻✳✶✶

Pr♦✈✐❞❡ t❤❡ ♣r♦♦❢ ♦❢ t❤❡ t❤❡♦r❡♠✳ ❊①❛♠♣❧❡ ✶✳✻✳✶✷✿ ❢♦r♠✉❧❛s ❛r❡ s❤♦rt❝✉ts

❖♥❝❡ ❛❣❛✐♥✱ ✇❡ s❡❡ t❤❡s❡ ♣r♦♣❡rt✐❡s ❛s s❤♦rt❝✉ts ✿ 25 = 25 · 2−2 = 25−2 = 23 . 22 ❊①❡r❝✐s❡ ✶✳✻✳✶✸

❘❡♣r❡s❡♥t ❛s ❛ ♣♦✇❡r ♦❢ 4✿ ✭❛✮ 43 · .25 ✱ ✭❜✮ .25/42 ✳ ❘❡♣r❡s❡♥t ❛s ❛ ♣♦✇❡r ♦❢ 2✳ ❍❡r❡ ✐s t❤❡ s✉♠♠❛r② ♦❢ ✇❤❛t ✇❡ ❤❛✈❡ ❡st❛❜❧✐s❤❡❞✿ ❆♥❛❧♦❣②✿ r❡♣❡❛t❡❞ ❛❞❞✐t✐♦♥ ✈s✳ r❡♣❡❛t❡❞ ♠✉❧t✐♣❧✐❝❛t✐♦♥

▼✉❧t✐♣❧✐❝❛t✐♦♥✿

❊①♣♦♥❡♥t✐❛t✐♦♥✿

❈♦♥✈❡♥t✐♦♥s✿ n = 1, 2, ...

a + ... + a} = a · n |a + a + {z n t✐♠❡s

n=0 n = −1, −2, ...

a·0 =0

0 −a − a − a − ... − a = a · (−n) | {z } n t✐♠❡s

= (−a) + (−a) + ... + (−a) = (−a) · n {z } | n t✐♠❡s

❘✉❧❡s✿ 1. 2. 3.

= −(a a + ... + a}) = −(a · n) | + a + {z n t✐♠❡s

a · (n + m) = a · n + a · m (a + b) · n = a · n + b · n a · (n · m) = (a · n) · m

a a · ... · a} = an | · a · {z n t✐♠❡s

a0 = 1

1 ÷a ÷ a ÷ a ÷ ... ÷ a = a−n | {z } n t✐♠❡s  n 1 1 1 1 = · · ... · = a |a a {z a} n t✐♠❡s

= 1 ÷ (a a · ... · a}) = | · a · {z n t✐♠❡s

1 an

an+m = an · am

(a · b)n = an bn

an·m = (an )m

❙♦✱ t❤❡ t❤r❡❡ ♣r♦♣❡rt✐❡s ❛r❡ st✐❧❧ s❛t✐s✜❡❞✱ ❛♥❞ ✇❡ ✇✐❧❧ ❝♦♥t✐♥✉❡ t♦ ✉s❡ t❤❡♠ ✇✐t❤ ♥♦ r❡❣❛r❞ ❢♦r ❛ ♣♦ss✐❜❧❡ ♥❡❣❛t✐✈✐t② ♦❢ t❤❡ ❡①♣♦♥❡♥t✳ ❚❤❡r❡ ✐s ♥♦ ♥❡❡❞ t♦ ❝♦✉♥t ❛♥②♠♦r❡✳ ❚❤❡s❡ ❛r❡ t❤❡ ♣♦ss✐❜❧❡ ✈❛❧✉❡s ♦❢ n ❢r♦♠ ♥♦✇ ♦♥✿ ..., −3, −2, −1, 0, 1, 2, 3, ...

✶✳✻✳

❚❤❡ ❛❧❣❡❜r❛ ♦❢ ❡①♣♦♥❡♥ts

✺✺

■♥ t❤❡ ❣r❛♣❤✱ ✇❡ ♥♦t✐❝❡ t❤❛t ✇❡ ✭❞❡❝r❡❛s❡✮✿

♠✉❧t✐♣❧②

❜② a ❛s ✇❡ ♠♦✈❡ r✐❣❤t ✭✐♥❝r❡❛s❡✮ ❛♥❞

❞✐✈✐❞❡ ❜② a ❛s ✇❡ ♠♦✈❡ ❧❡❢t

❍♦✇❡✈❡r✱ ✐❢ ✇❡ st❛rt ❛t ❛♥② ♣♦✐♥t ❛♥❞ ♣r♦❝❡❡❞ t♦ t❤❡ r✐❣❤t✱ ✇❡ ♦♥❧② ♠✉❧t✐♣❧②✳ ❚❤✐s ✐s ♦✉r ❝♦♥❝❧✉s✐♦♥✳ ❚❤❡♦r❡♠ ✶✳✻✳✶✹✿ ▼♦♥♦t♦♥✐❝✐t② ♦❢ ●❡♦♠❡tr✐❝ Pr♦❣r❡ss✐♦♥ ❆ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥

• •

rn

✐s

r > 1✱ ❛♥❞ ✐❢ 0 < r < 1✳

✐♥❝r❡❛s✐♥❣ ✐❢ ❞❡❝r❡❛s✐♥❣

❚❤✐s ✐s ✇❤❛t t❤❡ ❣r❛♣❤s ❧♦♦❦ ❧✐❦❡✿

❊①❡r❝✐s❡ ✶✳✻✳✶✺

Pr♦✈❡ t❤❡ t❤❡♦r❡♠✳ ❊①❡r❝✐s❡ ✶✳✻✳✶✻

■❢ ♦✉r ❜❛❝t❡r✐❛ ❞♦✉❜❧❡ ✐♥ ♥✉♠❜❡r ❡✈❡r② ❞❛② ❛♥❞ t❤❡ ❝✉rr❡♥t ♣♦♣✉❧❛t✐♦♥ ✐s 1024✱ ❤♦✇ ♠❛♥② ✇❡r❡ t❤❡r❡ t✇♦ ❞❛②s ❛❣♦❄ ❚❤❡ ✐♥t❡❣❡rs ❛s t❤❡ ♣♦ss✐❜❧❡ ♠♦♠❡♥ts ♦❢ t✐♠❡ st✐❧❧ ♠✐ss s♦♠❡ ♦❢ t❤❡ ♥✉♠❜❡rs t❤❛t ♠✐❣❤t ✐♥t❡r❡st ✉s ✐♥ t❤❡s❡ ♠♦❞❡❧s✦ ❋♦r ❡①❛♠♣❧❡✱ s✉♣♣♦s❡ ♦✉r ❜❛❝t❡r✐❛ ❞♦✉❜❧❡ ✐♥ ♥✉♠❜❡r ❡✈❡r② ❞❛② st❛rt✐♥❣ ✇✐t❤ 1✳ ❲❤❛t ❤❛♣♣❡♥s ❛❢t❡r 10.5 ❞❛②s❄ ❲❡✬❧❧ ♥❡❡❞ t♦ ✜❣✉r❡ ♦✉t t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤✐s✿ n = 10.5,

❲❡ ❛❞❞r❡ss t❤❡ ❢r❛❝t✐♦♥❛❧ ❡①♣♦♥❡♥ts ✐♥ ❈❤❛♣t❡r ✹✳

210.5 = ?

✶✳✼✳

❚❤❡ ❇✐♥♦♠✐❛❧ ❋♦r♠✉❧❛

✺✻

✶✳✼✳ ❚❤❡ ❇✐♥♦♠✐❛❧ ❋♦r♠✉❧❛

❲❡ ❦♥♦✇ t❤❛t ✇❡ ❛r❡♥✬t s✉♣♣♦s❡❞ t♦ ❞✐str✐❜✉t❡

♣♦✇❡rs ♦✈❡r ❛❞❞✐t✐♦♥ ✭❛♥ ✉♥❢♦r❣✐✈❛❜❧❡ ♠✐st❛❦❡✦✮✿

(a + b)2 6= a2 + b2 . ■♥st❡❛❞✱ ✇❡ ✜♥❞ ✇❤❛t ✐t ✐s ❡q✉❛❧ t♦ ❜② ❞✐str✐❜✉t✐♥❣

♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦✈❡r ❛❞❞✐t✐♦♥ ✿

(a + b)2 = (a + b) · (a + b) = (a + b) · a + (a + b) · b = (a · a + b · a) + (a · b + b · b) = a2 + 2ab + b2 . ❲❛r♥✐♥❣✦ ❲❡ r❡♣❡❛t t❤❡ ❡q✉❛❧ s✐❣♥ ✏ =✑ ❡✈❡r② t✐♠❡ ❜❡❝❛✉s❡ ✐t st❛♥❞s ❢♦r t❤❡ ✈❡r❜ ✏✐s✑ ❛♥❞ ✇❡ ✇❛♥t ♦✉r s❡♥t❡♥❝❡s t♦ ❜❡ ❣r❛♠♠❛t✐❝❛❧❧② ❝♦rr❡❝t✳

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

(a + b)2

❚❤❡ ✏❡♥❞✑ t❡r♠s ❝♦rr❡s♣♦♥❞ t♦ t❤❡ t✇♦ s♠❛❧❧❡r sq✉❛r❡s✱ ❝♦rr❡s♣♦♥❞ t♦ t❤❡ r❡❝t❛♥❣❧❡s✱

a×b

❛♥❞

b × a✳

❛s t❤❡

a×a

❛♥❞

❛r❡❛ ♦❢ ❛ sq✉❛r❡ ✇✐t❤ s✐❞❡ a + b✿

b × b✱

❛♥❞ t❤❡ ✏✐♥t❡r♠❡❞✐❛t❡✑ t❡r♠s

❚❤❡ r❡s✉❧t ✐s ❛ ❤❛♥❞② ❢♦r♠✉❧❛✿

(a + b)2 = a2 + 2ab + b2 ■t ✐s ✈❡r② ✉s❡❢✉❧✱ ❡s♣❡❝✐❛❧❧② ✇❤❡♥ ✇❡ ♥❡❡❞ t♦

❢❛❝t♦r

❛♥ ❡①♣r❡ss✐♦♥❀ ✇❡ s✐♠♣❧② r❡❛❞ t❤❡ ❢♦r♠✉❧❛ ❢r♦♠ r✐❣❤t t♦

❧❡❢t✿

a2 + 2ab + b2 = (a + b)2 . ❊①❛♠♣❧❡ ✶✳✼✳✶✿ ❛ ❝♦♠♣❧❡t❡ sq✉❛r❡ ❚♦ ❢❛❝t♦r

4x2 + 4x + 1✱

✇❡ ♠❛t❝❤ ✐t ✇✐t❤ t❤❡ ❧❡❢t✲❤❛♥❞ s✐❞❡ ♦❢ t❤❡ ❢♦r♠✉❧❛ ❛❜♦✈❡ ✭✇❡ ❛ss✉♠❡ t❤❛t t❤❡

♥✉♠❜❡rs ❛r❡ ♣♦s✐t✐✈❡✮ ❛s ❢♦❧❧♦✇s✿

a2 + 2ab + b2 = (a + b)2 2 4x + 4x + 1 = ? 2 2 2 =⇒ a = 4x ... b =1 =⇒ a = 2x ... b=1 =⇒ 4x2 + 4x + 1 = (2x + 1)2

❲❡ ♠❛t❝❤ ✈❡rt✐❝❛❧❧②✳ ❚❤❡ ❡q✉❛t✐♦♥s t❤❛t ❝♦♠❡ ❢r♦♠ t❤❡ ♠❛t❝❤✐♥❣✳ ❚❤❡ ❡q✉❛t✐♦♥s ❛r❡ s♦❧✈❡❞✳ ❚❤❡ ♠✐❞❞❧❡ t❡r♠ ❛❧s♦ ❝❤❡❝❦s ♦✉t✳

✶✳✼✳

❚❤❡ ❇✐♥♦♠✐❛❧ ❋♦r♠✉❧❛

✺✼

❲❡ ❤❛✈❡ ❛ ❝♦♠♣❧❡t❡ sq✉❛r❡✿

4x2 + 4x + 1 = (2x + 1)2 . ■t ✐s ♠♦r❡ ❝♦♠♣❛❝t t❤❛♥ t❤❡ ♦r✐❣✐♥❛❧ ❛♥❞ ✇❡ ❝❛♥ ♥♦t✐❝❡ ❝❡rt❛✐♥ ❢❛❝ts ❛❜♦✉t ✐t t❤❛t ✉s❡❞ t♦ ❜❡ ✐♥✈✐s✐❜❧❡✳ ❋♦r ❡①❛♠♣❧❡✱ t❤❡ ❡①♣r❡ss✐♦♥ ❝❛♥✬t ❜❡ ♥❡❣❛t✐✈❡✦

❲❛r♥✐♥❣✦ ❲❡ ❞♦ ♥♦t ♣✉t t❤❡ ❡q✉❛❧ s✐❣♥ ✏ =✑ ❜❡t✇❡❡♥ ❡q✉❛✲ t✐♦♥s ❜❡❝❛✉s❡ ❡❛❝❤ ✐s ❛ s❡♣❛r❛t❡ s❡♥t❡♥❝❡✱ ❛♥❞ t❤❡ q✉❛♥t✐t✐❡s ❛r❡♥✬t ❡q✉❛❧✳

❊①❡r❝✐s❡ ✶✳✼✳✷ ❋❛❝t♦r

9x2 + 12xy + 4y 2 ✳

❲❤❛t ❛❜♦✉t t❤❡

❤✐❣❤❡r ♣♦✇❡rs ❄ ❚♦ ❣❡t t❤❡ ❝✉❜✐❝ ♣♦✇❡r ❛♥❞ ❛❜♦✈❡✱ ✇❡ s✐♠♣❧② ❝♦♥t✐♥✉❡ ♠✉❧t✐♣❧②✐♥❣ ❜② (a+b) ❉✐str✐❜✉t✐✈❡ Pr♦♣❡rt② ✿

❛♥❞ ❛♣♣❧②✐♥❣ t❤❡

(a + b)3 = (a + b)2 · (a + b) = (a2 + 2ab + b2 ) · (a + b) = (a2 + 2ab + b2 ) · a + (a2 + 2ab + b2 ) · b = (a2 · a + 2ab · a + b2 · a) + (a2 · b + 2ab · b + b2 · b) = (a3 + 2a2 b + ab2 ) + (a2 b + 2ab2 + b3 ) = a3 + 3a2 b + 3ab2 + b3 . ❚❤❡ r❡s✉❧t ✐s ❛♥♦t❤❡r ❤❛♥❞② ❢♦r♠✉❧❛✿

(a + b)3 = a3 + 3a2 b + 3ab2 + b3

❊①❛♠♣❧❡ ✶✳✼✳✸✿ ❛ ❝♦♠♣❧❡t❡ ❝✉❜❡ ❚♦ ❢❛❝t♦r

x3 + 3x2 + 3x + 1✱

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

a3 + 3a2 b x3 + 3x2 =⇒ a3 = x3 .. =⇒ a = x .. 3 =⇒ x + 3x2

+ 3ab2 + b3 = (a + b)3 + 3x + 1 = ? 3 .. b =1 .. b=1 + 3x + 1 = (x + 1)3

❚❤❡ ✏✐♥t❡r♠❡❞✐❛t❡✑ t❡r♠s ❛❧s♦ ❝❤❡❝❦s ♦✉t✳

❊①❡r❝✐s❡ ✶✳✼✳✹ ■♥t❡r♣r❡t

(a+b)3 ❛s t❤❡ ✈♦❧✉♠❡ ♦❢ ❛ ❝✉❜❡ ✇✐t❤ s✐❞❡ a+b ❛♥❞ ✐❧❧✉str❛t❡ t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ ✏✐♥t❡r♠❡❞✐❛t❡✑

t❡r♠s ✐♥ t❤❡ ❛❜♦✈❡ ❢♦r♠✉❧❛✳

❚❤❡ t✇♦✲t❡r♠ ❡①♣r❡ss✐♦♥

a+b

✐s ❝❛❧❧❡❞ ❛

❜✐♥♦♠✐❛❧✳

■♥ t❤❡ ❢♦r♠✉❧❛ ✭m

=2

❛♥❞

m=3

❛❜♦✈❡✮✱

(a + b)m = am + ... + bm , ✇❡ ❝❛❧❧ t❤❡ ❧❡❢t✲❤❛♥❞ s✐❞❡ ❛

❜✐♥♦♠✐❛❧ ♣♦✇❡r

❛♥❞ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ t❤❡

❜✐♥♦♠✐❛❧ ❡①♣❛♥s✐♦♥✳

❲❤❡r❡ t❤❡s❡ t❡r♠s ❝♦♠❡ ❢r♦♠❄ ❲❡ ♣✐❝❦ ♦♥❡ t❡r♠ ❢r♦♠ ❡❛❝❤ ❜✐♥♦♠✐❛❧ ❛♥❞ t❤❡r❡ ❛r❡ ❛ t♦t❛❧

(a + b) · (a + b) · ... · (a + b) =

s✉♠ ♦❢ t❡r♠s ♦❢ t❤❡ t②♣❡

an · b k .

m

♦❢ t❤❡♠✿

✶✳✼✳

❚❤❡ ❇✐♥♦♠✐❛❧ ❋♦r♠✉❧❛

✺✽

❆s ✇❡ ❛rr❛♥❣❡ t❤❡ t❡r♠s ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❞❡❝r❡❛s✐♥❣ ♣♦✇❡rs ♦❢ a✱ ❛ ♣❛tt❡r♥ st❛rts t♦ ❡♠❡r❣❡✦ ❆s t❤❡ ♣♦✇❡r ♦❢ a ✐s ❞❡❝r❡❛s✐♥❣✱ t❤❛t ♦❢ b ✐s ✐♥❝r❡❛s✐♥❣✳ ▼♦r❡♦✈❡r✱ t❤❡ a b✱ ✐✳❡✳✱ t❤❡ s✉♠ ♦❢ t❤❡ t✇♦ ♣♦✇❡rs✱ ✐s m = 3✱ ❥✉st ❛s ✐t ✇❛s m = 2 ✐♥ t❤❡ ❧❛st ❢♦r♠✉❧❛✳ ▲❡t✬s ♠❛❦❡ ❛ t❛❜❧❡ ❢♦r t❤❡ ❜✐♥♦♠✐❛❧s t❤❛t st❛rts ✇✐t❤ m = 1✿

❝♦♠❜✐♥❡❞ ♣♦✇❡r ♦❢ ❛♥❞

m=1

(a + b)1 =

♣♦✇❡rs✿

1

m=2

(a + b)

♣♦✇❡rs✿

2

m=3

2

+

a0 b 1

= 1+0 =

0+1

+

2a1 b1

+

a0 b2

= 2+0 =

1+1

=

0+2

+

3a2 b1

+

3a1 b2

+

a0 b3

= 3+0 =

2+1

=

1+2

=

0+3

=

(a + b)3 =

♣♦✇❡rs✿

3

m=4

(a + b)

♣♦✇❡rs✿

4

4

a1 b0 2 0

ab

a3 b0

+

?a b

+

?a b

+

? a1 b3 +

= 4+0 =

3+1

=

2+2

=

1+3

=

4 0

ab

3 1

2 2

a0 b4

= 0+4

❲❡ ❛❧s♦ ✐♥❝❧✉❞❡❞ ✕ ❛s ❛ ❣✉❡ss ✕ t❤❡ ❝❛s❡ m = 4✳ ❲❡ s❡❡ t❤❛t m + 1 ✐s t❤❡ ♥✉♠❜❡r ♦❢ ✇❛②s ✇❡ ❝❛♥ r❡♣r❡s❡♥t m ❛s ❛ s✉♠ ♦❢ t✇♦ ♥♦♥✲♥❡❣❛t✐✈❡ ♥✉♠❜❡rs✳ ❇❡❧♦✇ ✇❡ ♣✉t t❤❡ r❡s✉❧ts ✐♥ ❛ s♣r❡❛❞s❤❡❡t ✇✐t❤ t❤❡ ❢♦r♠✉❧❛✿

❂❘❈✷✰❘✷❈ ❚❤❡ ❡①♣❛♥s✐♦♥s ❛❜♦✈❡ ❢♦❧❧♦✇ t❤❡ ❞✐❛❣♦♥❛❧ ❛s ♠❛r❦❡❞✿

■♥ ❡❛❝❤✱ t❤❡ ❝♦♠❜✐♥❡❞ ♣♦✇❡r ♦❢ a ❛♥❞ b r❡♠❛✐♥s t❤❡ s❛♠❡ ❛♥❞ ✐t ❣r♦✇s ❜② 1 ❛s ✇❡ ♠♦✈❡ t♦ t❤❡ ♥❡①t ❞✐❛❣♦♥❛❧✳ ❚❤❡ ❜❡❤❛✈✐♦r ♦❢ t❤❡ ♣♦✇❡rs ✐s ❝❧❡❛r❀ ✐t✬s t❤❡ ✜rst✳

❝♦❡✣❝✐❡♥ts t❤❛t ♣r❡s❡♥t ❛ ❝❤❛❧❧❡♥❣❡✳ ▲❡t✬s ❞❡s❝r✐❜❡ t❤❡ ❢♦r♠❡r

❚❤❡♦r❡♠ ✶✳✼✳✺✿ ◆✉♠❜❡r ♦❢ ❚❡r♠s ✐♥ ❇✐♥♦♠✐❛❧ ❊①♣❛♥s✐♦♥

mt❤ ♣♦✇❡r (a+b)m ♦❢ t❤❡ ❜✐♥♦♠✐❛❧ (a+b) ❝♦♥s✐sts ♦❢ m+1 an bk ✱ ❡❛❝❤ ♦❢ ✇❤✐❝❤ ❤❛s t❤❡ s✉♠ ♦❢ ♣♦✇❡rs ♦❢ a ❛♥❞ b ❡q✉❛❧ t♦

❚❤❡ ❡①♣❛♥s✐♦♥ ♦❢ t❤❡ t❡r♠s ♦❢ t❤❡ ❢♦r♠

m✳ Pr♦♦❢✳

■❢ ❛❧❧ ♦❢ t❤❡ t❡r♠s ♦❢ t❤❡ ❡①♣❛♥s✐♦♥ ♦❢ (a + b)m ❤❛✈❡ t❤❡ s✉♠ ♦❢ ♣♦✇❡rs ❡q✉❛❧ t♦ m✱ t❤❡♥ ✇❤❛t ❤❛♣♣❡♥s ✐♥ t❤❡ ♥❡①t✿ (a + b)m+1 = (a + b)m · (a + b) ?

❉✐str✐❜✉t✐✈❡ Pr♦♣❡rt②

❆❝❝♦r❞✐♥❣ t♦ t❤❡ ✱ ❡❛❝❤ ♦❢ t❤❡ t❡r♠s ♦❢ t❤❡ ❡①♣❛♥s✐♦♥ ♦❢ (a + b)m ✐s ♠✉❧t✐♣❧✐❡❞ ❜② a ♦r ❜② b✳ ❚❤❡r❡❢♦r❡✱ t❤❡ t♦t❛❧ ♣♦✇❡r ❣♦❡s ✉♣ ❜② 1✦ ❊①❡r❝✐s❡ ✶✳✼✳✻

Pr♦✈✐❞❡ t❤❡ ♠✐ss✐♥❣ ♣❛rts ♦❢ t❤❡ ♣r♦♦❢✳ ❖♥❝❡ ❛❣❛✐♥✱ ✇❡ ❤❛✈❡✿

(a + b)m = s✉♠ ♦❢ t❡r♠s ♦❢ t❤❡ t②♣❡ an · bk .

✶✳✼✳ ❚❤❡ ❇✐♥♦♠✐❛❧ ❋♦r♠✉❧❛

✺✾

❲❡ ✇❛♥t t♦ ❦♥♦✇ ❤♦✇ ♠❛♥② t✐♠❡s ❡❛❝❤ t❡r♠ ❛♣♣❡❛r ✐♥ t❤❡ ❡①♣❛♥s✐♦♥✳ ■♥ t❤❡ s✉♠♠❛r② ❜❡❧♦✇✱ ✇❡ s❤♦✇ t❤❡ ♣♦✇❡rs ❜✉t t❤❡ ❝♦❡✣❝✐❡♥ts ♦❢ t❤❡ t❡r♠s r❡♠❛✐♥ ✉♥❦♥♦✇♥✿

❲❤❛t ❛r❡ t❤❡ ❝♦❡✣❝✐❡♥ts❄

(a + b)m = am b0 + ? am−1 b1 + ? am−2 b2 +...+ ? a1 bm−1 + a0 bm m = m + 0 = (m − 1) + 1 = (m − 2) + 2 = ... = 1 + (m − 1) = 0 + m

❆s ②♦✉ ❝❛♥ s❡❡✱ t❤❡ ❧❛st r♦✇ ❥✉st s❤♦✇s ❛❧❧ ♣♦ss✐❜❧❡ ✇❛②s t♦ r❡♣r❡s❡♥t m ❛s t❤❡ s✉♠ ♦❢ t✇♦ ♥♦♥✲♥❡❣❛t✐✈❡ ✐♥t❡❣❡rs✳ ▲❡t✬s t❛❦❡ ❛ ❧♦♦❦ ❛t t❤❡ ❝♦❡✣❝✐❡♥ts✿ m=1

❝♦❡✣❝✐❡♥ts✿

m=2

❝♦❡✣❝✐❡♥ts✿ m=3

❝♦❡✣❝✐❡♥ts✿ m=4

(a + b)1 = a1 +

b1

1

1

(a + b)2 = a2 +

2a1 b1

1

b2

+

2

(a + b)3 = a3 +

1

3a2 b1

1

3a1 b2

+

3

(a + b)4 = a4 +

b3

+

3

? a3 b 1 +

1

? a2 b2 +

? a1 b3 + b4

❝♦❡✣❝✐❡♥ts✿ 1 4 ? ❲❡✬✈❡ ♠❛❞❡ ❛ ❣✉❡ss ❢♦r t❤❡ ❧❛st r♦✇✱ ❜✉t t❤❡ ♠✐❞❞❧❡ t❡r♠ r❡♠❛✐♥s ✉♥❦♥♦✇♥✳ ❆♥ ❛❝t✉❛❧ ❝♦♠♣✉t❛t✐♦♥ r❡✈❡❛❧s t❤❛t ✐t✬s 6✳ ❚❤❡ t❛❜❧❡ ♦❢ ❝♦❡✣❝✐❡♥ts ❜❡❝♦♠❡s✿ m=1 m=2 m=3 m=4

1 1 1 1

1 2 3 4

1 3 6

1 4

4

1

1

❆♥ ❡①❛♠✐♥❛t✐♦♥ ♦❢ t✇♦ ❝♦♥s❡❝✉t✐✈❡ r♦✇s ❛❧❧♦✇s ✉s t♦ ♦❜s❡r✈❡ ❤♦✇ t❤✐♥❣s ❞❡✈❡❧♦♣ ✐♥ t❤✐s t❛❜❧❡✿ m=3 1 + 3 || 4 m=4 1

3 + 3 || 4 6

3 + 1 || 6 4

❲❡ ❝❛♥ ❣✉❡ss t❤❛t ❡✈❡r② t✇♦ ❛❞❥❛❝❡♥t ❝♦❡✣❝✐❡♥ts ✐♥ ❡❛❝❤ r♦✇ ❛r❡ ❛❞❞❡❞ ❛♥❞ ♣❧❛❝❡❞ ✐♥ t❤❡ ♥❡①t r♦✇✳ ❚❤❡ t❛❜❧❡ ✐s ❛ s❡q✉❡♥❝❡ ♦❢ s❡q✉❡♥❝❡s✱ ❛♥❞ t❤❡ t❡r♠s ✐♥ t❤❡ ♥❡①t r♦✇ ❛r❡ ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ ♦♥❡s ✐♥ t❤❡ ❧❛st✳ ■t ✐s ♠♦r❡ ❝♦♥✈❡♥✐❡♥t✱ ❤♦✇❡✈❡r✱ t♦ ❛rr❛♥❣❡ t❤❡ t❡r♠s ✐♥ ❛♥ ✐s♦s❝❡❧❡s ✐♥st❡❛❞ ♦❢ ❛ r✐❣❤t tr✐❛♥❣❧❡ ❛s ❛❜♦✈❡✳ ❚❤✐s ✐s ❤♦✇ ✇❡ ❜✉✐❧❞ t❤❡ 4t❤ r♦✇ ❢r♦♠ t❤❡ 3r❞✿ ❦♥♦✇♥✿ m = 3 1 3 3 1 ւ ց 1+3 ւ ց 3+3 ւ ց 3+1 ւ ց ✉♥❦♥♦✇♥✿ m = 4 1 4 6 4 1 ❊❛❝❤ ❡♥tr② ✐s t❤❡ s✉♠ ♦❢ t❤❡ t✇♦ ❡♥tr✐❡s ❛❜♦✈❡ ✐t✳ ❚❤❡ r❡s✉❧t ✐s ❝❛❧❧❡❞ t❤❡ P❛s❝❛❧ ❚r✐❛♥❣❧❡ ✿ 1 m=0 m=1 1 1 1 2 1 m=2 m=3 1 3 3 1 m=4 1 4 6 4 1 m=5 1 5 10 10 5 1 ... . . . . . .

.

✶✳✼✳

❚❤❡ ❇✐♥♦♠✐❛❧ ❋♦r♠✉❧❛

✻✵

❊①❡r❝✐s❡ ✶✳✼✳✼

❊①♣❧❛✐♥ t❤❡ ❡♥tr② ❛t t❤❡ t♦♣ ♦❢ t❤❡ tr✐❛♥❣❧❡✳ ❚❤❡ ❝♦♠♣✉t❛t✐♦♥ ♠❛② ❝♦♥t✐♥✉❡ ✐♥❞❡✜♥✐t❡❧② ❢♦❧❧♦✇✐♥❣ t❤✐s ♣r♦❝❡❞✉r❡✿ X

Y ց

ւ

X +Y

❲❡ s✉♠♠❛r✐③❡ t❤❡ r❡s✉❧t ❜❡❧♦✇✳ ❚❤❡♦r❡♠ ✶✳✼✳✽✿ ❇✐♥♦♠✐❛❧ ❚❤❡♦r❡♠ ■❢ t✇♦ ❝♦♥s❡❝✉t✐✈❡ t❡r♠s ♦❢ t❤❡ ❡①♣❛♥s✐♦♥ ♦❢

X · an bm−n t❤❡♥ t❤❡ ❡①♣❛♥s✐♦♥ ♦❢

(a + b)m+1

❛♥❞

(a + b)m

❛r❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿

Y · an−1 bm−n+1 ,

❝♦♥t❛✐♥s t❤❡ ❢♦❧❧♦✇✐♥❣ t❡r♠✿

(X + Y ) · an bm−n+1 . Pr♦♦❢✳

(Xan bm−n + Y an−1 bm−n+1 )(a + b) = Xan bm−n a + Y an−1 bm−n+1 a + Xan bm−n b + Y an−1 bm−n+1 b = Xan+1 bm−n + Y an bm−n+1 + Xan bm−n+1 + Y an−1 bm−n+2 = Xan+1 bm−n + (X + Y )an bm−n+1 + Y an−1 bm−n+2 . ❊①❡r❝✐s❡ ✶✳✼✳✾

Pr♦✈✐❞❡ t❤❡ ♠✐ss✐♥❣ ♣❛rts ♦❢ t❤❡ ♣r♦♦❢✳ ❊①❛♠♣❧❡ ✶✳✼✳✶✵✿ P❛s❝❛❧ ❚r✐❛♥❣❧❡✱ ❝♦♠♣✉t❡❞

❚❤✐s ✐s ❤♦✇ t❤❡ P❛s❝❛❧ ❚r✐❛♥❣❧❡ ✐s ✉s❡❞✳ ❲❡ t❛❦❡ ✐ts 5t❤ r♦✇ ❛♥❞ ♣r♦❞✉❝❡ ❛ ♥❡✇ ❜✐♥♦♠✐❛❧ ❡①♣❛♥s✐♦♥✿ m=5 (a + b)5 =

1 5 10 10 5 1 5 4 3 2 2 3 4 1 · a + 5 · a b + 10 · a b + 10 · a b + 5 · ab + 1 · b5

❖♥❡ ❝❛♥ ❡❛s✐❧② ❝♦♠♣✉t❡✱ r❡❝✉rs✐✈❡❧②✱ ❛ ❧❛r❣❡ ♣❛rt ♦❢ t❤❡ P❛s❝❛❧ tr✐❛♥❣❧❡ ✇✐t❤ ❛ s♣r❡❛❞s❤❡❡t✿

❲❡ ✉s❡ t❤❡ ❢♦r♠✉❧❛✿

❂❘❬✲✶❪❈❬✲✶❪✰❘❬✲✶❪❈❬✶❪

❙♦✱ ❢♦r ❡❛❝❤ m✱ t❤❡ ❝♦❡✣❝✐❡♥ts ♦❢ ❛♥ ❡①♣❛♥s✐♦♥ ❢♦r♠ ❛ s❡q✉❡♥❝❡ ♦❢ m + 1 ✐♥t❡❣❡rs✳ ❚❤❡② ❛r❡ s✐♠♣❧❡ ✐♥ t❤❡ ✈❡r② ❜❡❣✐♥♥✐♥❣ ❛♥❞ t❤❡ ✈❡r② ❡♥❞✿ n 0 1 ... m − 1 m 1 m ... m 1

✶✳✼✳

❚❤❡ ❇✐♥♦♠✐❛❧ ❋♦r♠✉❧❛

✻✶

▲❡t✬s ❡①❛♠✐♥❡ t❤❡ r❡st✳

❉❡✜♥✐t✐♦♥ ✶✳✼✳✶✶✿ ❜✐♥♦♠✐❛❧ ❝♦❡✣❝✐❡♥t ❚❤❡

nt❤

t❡r♠ ♦❢ t❤✐s s❡q✉❡♥❝❡ ✐s ❝❛❧❧❡❞ t❤❡

❜②

❜✐♥♦♠✐❛❧ ❝♦❡✣❝✐❡♥t

❛♥❞ ✐s ❞❡♥♦t❡❞

  m n ■t r❡❛❞s ✏ m ❝❤♦♦s❡

n✑✳

❚❤❡ ✜rst ♥✉♠❜❡r ❣✐✈❡s t❤❡ ✈❡rt✐❝❛❧ ❧♦❝❛t✐♦♥ ❛♥❞ t❤❡ s❡❝♦♥❞ ❤♦r✐③♦♥t❛❧ ❧♦❝❛t✐♦♥ ✇✐t❤✐♥ t❤❡ P❛s❝❛❧ tr✐❛♥❣❧❡✳ ❆ ❝❡rt❛✐♥ s②♠♠❡tr② ♦❢ t❤❡ P❛s❝❛❧ tr✐❛♥❣❧❡ ✐s s❡❡♥ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✉❧❛✳

❊①❡r❝✐s❡ ✶✳✼✳✶✷ ❙❤♦✇ t❤❛t

    m m = . n m−n

❊①❡r❝✐s❡ ✶✳✼✳✶✸ ❙❤♦✇ t❤❛t

  m = m. 2

■♥ t❤❡ ♥❡✇ ♥♦t❛t✐♦♥✱ t❤❡ P❛s❝❛❧ ❚r✐❛♥❣❧❡ t❛❦❡s t❤❡ ❢♦r♠✿

  0 0     1 1 0 1       2 2 2 0 1 2         3 3 3 3 0 1 2 3 . . . . . ❚❤❡r❡❢♦r❡✱ t❤❡ ✜rst ♥✉♠❜❡r ✐♥❞✐❝❛t❡s t❤❡ r♦✇✱ ❛♥❞ t❤❡ s❡❝♦♥❞ ♥✉♠❜❡r t❤❡ ♣♦s✐t✐♦♥ ✇✐t❤✐♥ t❤❡ r♦✇✳ ■♥ t❤❡ ♥❡✇ ♥♦t❛t✐♦♥✱ t❤❡

❇✐♥♦♠✐❛❧ ❚❤❡♦r❡♠

✐s r❡st❛t❡❞ ❛s ❢♦❧❧♦✇s✳

❈♦r♦❧❧❛r② ✶✳✼✳✶✹✿ ❙✉♠ ♦❢ ❇✐♥♦♠✐❛❧ ❈♦❡✣❝✐❡♥ts ❋♦r ❡✈❡r② ♣♦s✐t✐✈❡ ✐♥t❡❣❡rs

m

❛♥❞

n ≤ m✱

✇❡ ❤❛✈❡✿

      m m m+1 + = n n+1 n+1 ❚❤❛t✬s ❛

r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛

❡①♣r❡ss✐♦♥ ✐♥ t❡r♠s ♦❢ t❤❡

✦ ❲❤❛t ✐s t❤❡

❢❛❝t♦r✐❛❧

n

t❤✲t❡r♠ ❢♦r♠✉❧❛

❢♦r t❤✐s s❡q✉❡♥❝❡✱ ♦r r❛t❤❡r s❡q✉❡♥❝❡s❄ ■t ❝❛♥ ❜❡

✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ✐s ❛♥ ✐♠♣♦rt❛♥t r❡s✉❧t✿

✶✳✼✳

❚❤❡ ❇✐♥♦♠✐❛❧ ❋♦r♠✉❧❛

✻✷

❚❤❡♦r❡♠ ✶✳✼✳✶✺✿ ❇✐♥♦♠✐❛❧ ❈♦❡✣❝✐❡♥t ❋♦r ❡✈❡r② ♣♦s✐t✐✈❡ ✐♥t❡❣❡rs

m

❛♥❞

n ≤ m✱

✇❡ ❤❛✈❡✿

  m m! = n!(m − n)! n

Pr♦♦❢✳

❲❡ ❝♦♥✜r♠ t❤❡ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛✿     m m m! m! + = + n n+1 n!(m − n)! (n + 1)!(m − n − 1)! m! m! = + n!(m − n − 1)!(m − n) n!(m − n − 1)!(n + 1) m!(n + 1) + m!(m − n) = n!(m − n − 1)!(m − n)(n
+ 1) m! (n + 1) + (m − n) = (n + 1)!(m − (n
+ 1))! m! m + 1 = (n + 1)!(m   − (n + 1))! m+1 = . n+1

❊①❡r❝✐s❡ ✶✳✼✳✶✻

❲❤② ❞♦❡s t❤✐s ❝♦♠♣✉t❛t✐♦♥ ♣r♦✈❡ t❤❡ ❢♦r♠✉❧❛❄

❲❛r♥✐♥❣✦

❊✈❡♥ t❤♦✉❣❤ ❛ ❢r❛❝t✐♦♥✱ ❛ ❜✐♥♦♠✐❛❧ ❝♦❡✣❝✐❡♥t ✐s ❛♥ ✐♥t❡❣❡r✳

❊①❛♠♣❧❡ ✶✳✼✳✶✼✿ ❜✐♥♦♠✐❛❧ ❝♦❡✣❝✐❡♥ts✱ ❝♦♠♣✉t❡❞ ❚♦ t❡st t❤❡ ❢♦r♠✉❧❛✱ ✇❡ s✉❜st✐t✉t❡ m = 5 ❛♥❞ n = 3✿   5 5! 5! 2·3·4·5 = = = = 2 · 5 = 10 . 3 3!(5 − 3)! 3!2! 2·3·2

❚❤❡ r❡s✉❧t ♠❛t❝❤❡s t❤❡ ♦♥❡ ✐♥ t❤❡ P❛s❝❛❧ ❚r✐❛♥❣❧❡✳ ❚❤❡ ❜✐♥♦♠✐❛❧ ❝♦❡✣❝✐❡♥ts ❤❛✈❡ ❛♥♦t❤❡r ✐♥t❡r♣r❡t❛t✐♦♥✳ ❙✐♥❝❡ t❤❡ ♣♦✇❡r ✐s ❥✉st ❛ r❡♣❡❛t❡❞ ♠✉❧t✐♣❧✐❝❛t✐♦♥✱ (a + b)m = (a + b) · (a + b) · ... · (a + b) , {z } | m t✐♠❡s

t❤❡r❡ ✇✐❧❧ ❜❡ ♦♥❡ t❡r♠ ✐♥ t❤❡ ❜✐♥♦♠✐❛❧ ❡①♣❛♥s✐♦♥ ❢♦r ❡❛❝❤ ❝❤♦✐❝❡ ♦❢ ❡✐t❤❡r a ♦r b ❢r♦♠ ❡❛❝❤ ♦❢ t❤❡ ❢❛❝t♦rs✿ an bm−n = a · ... · a} · b| · b {z · ... · }b . | · a {z n t✐♠❡s

❲❡ ❞r❛✇ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❝❧✉s✐♦♥✿

m−n t✐♠❡s

✶✳✽✳ ❚❤❡ s❡q✉❡♥❝❡ ♦❢ ❞✐✛❡r❡♥❝❡s✿ ✈❡❧♦❝✐t②

✻✸

❚❤❡♦r❡♠ ✶✳✼✳✶✽✿ ❈❤♦♦s❡ ♥ ❋r♦♠ ♠

❚❤❡ ❜✐♥♦♠✐❛❧ ❝♦❡✣❝✐❡♥t ❢r♦♠ m ♦❜❥❡❝ts✳

  m n

✐s t❤❡ ♥✉♠❜❡r ♦❢ ✇❛②s ✇❡ ❝❛♥ ❝❤♦♦s❡ n ♦❜❥❡❝ts

❊①❛♠♣❧❡ ✶✳✼✳✶✾✿ ♥✉♠❜❡r ♦❢ ✇❛②s t♦ ❝❤♦♦s❡ ❛ t❡❛♠

■♥ ❤♦✇ ♠❛♥② ✇❛②s ❝❛♥ ♦♥❡ ❢♦r♠ ❛ t❡❛♠ ♦❢ 5 ❢r♦♠ 20 ♣❧❛②❡rs❄ ❲❡ ❝♦♠♣✉t❡✿   20! 20 = 5 5!15! 15! · 16 · 17 · 18 · 19 · 20 = 5!15! 16 · 17 · 18 · 19 · 20 = 2·3·4·5 16 · 17 · 18 · 19 = 2·3 = 16 · 17 · 3 · 19 = 15, 504 . ❊①❡r❝✐s❡ ✶✳✼✳✷✵

❍♦✇ ♠❛♥② ❞✐✛❡r❡♥t ❤❛♥❞s ♦❢ 5 ❛r❡ t❤❡r❡ ✐♥ ❛ ❞❡❝❦ ♦❢ 52 ❝❛r❞s❄

✶✳✽✳ ❚❤❡ s❡q✉❡♥❝❡ ♦❢ ❞✐✛❡r❡♥❝❡s✿ ✈❡❧♦❝✐t② ◆✉♠❜❡rs ❛r❡ s✉❜❥❡❝t t♦ ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s✿ ❛❞❞✐t✐♦♥✱ s✉❜tr❛❝t✐♦♥✱ ♠✉❧t✐♣❧✐❝❛t✐♦♥✱ ❛♥❞ ❞✐✈✐s✐♦♥✳ ❙✐♥❝❡ t❤❡ t❡r♠s ♦❢ s❡q✉❡♥❝❡s ❛r❡ ♥✉♠❜❡rs✱ ❛ ♣❛✐r ♦❢ s❡q✉❡♥❝❡s ❝❛♥ ❜❡ ❛❞❞❡❞ ✭s✉❜tr❛❝t❡❞✱ ❡t❝✳✮ t♦ ♣r♦❞✉❝❡ ❛ ♥❡✇ ♦♥❡✳ ■♥ ❛❞❞✐t✐♦♥✱ t❤❡r❡ ❛r❡ t✇♦ ♦♣❡r❛t✐♦♥s t❤❛t ❛♣♣❧② t♦ ❛ s✐♥❣❧❡ s❡q✉❡♥❝❡ ❛♥❞ ♣r♦❞✉❝❡ ❛ ♥❡✇ s❡q✉❡♥❝❡ t❤❛t t❡❧❧s ✉s ❛ ❧♦t ❛❜♦✉t t❤❡ ♦r✐❣✐♥❛❧ s❡q✉❡♥❝❡✳ ❚❤❡s❡ ♦♣❡r❛t✐♦♥s ❛r❡✿ s✉❜tr❛❝t✐♥❣ t❤❡ ❝♦♥s❡❝✉t✐✈❡ t❡r♠s ♦❢ t❤❡ s❡q✉❡♥❝❡ ❛♥❞ ❛❞❞✐♥❣ ✐ts t❡r♠s r❡♣❡❛t❡❞❧②✳ ❲❡ s❛✇ t❤❡♠ ✐♥ ❛❝t✐♦♥ ✐♥ t❤❡ ✜rst s❡❝t✐♦♥✿

❚❤✐s ✐s t❤❡ s✉♠♠❛r②✿ • ■❢ ❡❛❝❤ t❡r♠ ♦❢ ❛ s❡q✉❡♥❝❡ r❡♣r❡s❡♥ts ❛ ❧♦❝❛t✐♦♥✱ t❤❡ ♣❛✐r✲✇✐s❡ ❞✐✛❡r❡♥❝❡s ✇✐❧❧ ❣✐✈❡ ②♦✉ t❤❡

❛♥❞

✈❡❧♦❝✐t✐❡s✱

• ■❢ ❡❛❝❤ t❡r♠ ♦❢ ❛ s❡q✉❡♥❝❡ r❡♣r❡s❡♥ts t❤❡ ✈❡❧♦❝✐t②✱ t❤❡✐r s✉♠ ✉♣ t♦ t❤❛t ♣♦✐♥t ✇✐❧❧ ❣✐✈❡ ②♦✉ t❤❡ ❧♦❝❛t✐♦♥

✶✳✽✳ ❚❤❡ s❡q✉❡♥❝❡ ♦❢ ❞✐✛❡r❡♥❝❡s✿ ✈❡❧♦❝✐t②

✻✹

✭♦r ❞✐s♣❧❛❝❡♠❡♥t✮✳ ■♥ t❤✐s s❡❝t✐♦♥ ✇❡ st❛rt ♦♥ t❤❡ ♣❛t❤ ♦❢ ❞❡✈❡❧♦♣♠❡♥t ♦❢ ❛♥ ✐❞❡❛ t❤❛t ❝✉❧♠✐♥❛t❡s ✇✐t❤ t❤❡ ✜rst ❢✉♥❞❛♠❡♥t❛❧ ❝♦♥❝❡♣t ♦❢ ❝❛❧❝✉❧✉s✱ t❤❡ ❞❡r✐✈❛t✐✈❡ ✭❈❤❛♣t❡r ✷❉❈✲✸✮✳ ❚❤❡ ♣❛✐r✇✐s❡ ❞✐✛❡r❡♥❝❡s r❡♣r❡s❡♥t t❤❡ ❝❤❛♥❣❡ ✇✐t❤✐♥ t❤❡ s❡q✉❡♥❝❡✱ ❢r♦♠ ❡❛❝❤ ♦❢ ✐ts t❡r♠s t♦ t❤❡ ♥❡①t✳

❊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 ✐♥ t❤❡ ❜♦tt♦♠ r♦✇ ❛s ❢♦❧❧♦✇s✿ ❛ s❡q✉❡♥❝❡✿ 2 ✐ts ❞✐✛❡r❡♥❝❡s✿ ❛ ♥❡✇ s❡q✉❡♥❝❡✿

4 ց

4−2 || 2

ւ

7 ց

7−4 || 3

ւ

1 ց

1−7 || −6

ւ

... ց ... ... ... ...

❲❡ ❤❛✈❡ ❛ ♥❡✇ ❧✐st✳

❊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✳

❉❡✜♥✐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❡❞ ❜② ∆an = an+1 − an

❲❛r♥✐♥❣✦ ❚❤❡ s②♠❜♦❧ ∆ ❛♣♣❧✐❡s t♦ t❤❡ ✇❤♦❧❡ s❡q✉❡♥❝❡ an ✱ ❛♥❞ ∆a s❤♦✉❧❞ ❜❡ s❡❡♥ ❛s t❤❡ ♥❛♠❡ ♦❢ t❤❡ ♥❡✇ s❡q✉❡♥❝❡❀ t❤❡ ♥♦t❛t✐♦♥ ❢♦r t❤❡ ❞✐✛❡r❡♥❝❡ ✐s ❛♥ ❛❜✲ ❜r❡✈✐❛t✐♦♥ ❢♦r (∆a)n ✳

❚❤✐s ✐s ✇❤❛t t❤❡ ❞❡✜♥✐t✐♦♥ s❛②s✿

✶✳✽✳

❚❤❡ s❡q✉❡♥❝❡ ♦❢ ❞✐✛❡r❡♥❝❡s✿ ✈❡❧♦❝✐t②

✻✺

❙❡q✉❡♥❝❡ ♦❢ ❞✐✛❡r❡♥❝❡s ❛ s❡q✉❡♥❝❡✿ ✐ts ❞✐✛❡r❡♥❝❡s✿ ❛ ♥❡✇ s❡q✉❡♥❝❡✿ t❤❡ ♥♦t❛t✐♦♥✿

a1

a2 ց

a2 − a1 || d1 || ∆a1

ւ

a3 ց

a3 − a2 || d2 || ∆a2

ւ

ց

a4 − a3 || d3 || ∆a3

a4 ... ւ ... ... ... ... ... ...

❲❛r♥✐♥❣✦ n = q ✱ t❤❡♥ t❤❡ n = q + 1 ❜✉t ✐t ❝♦✉❧❞ ✇✐t❤ n = q ♦r ❛♥② ♦t❤❡r

■❢ t❤❡ ♦r✐❣✐♥❛❧ s❡q✉❡♥❝❡ st❛rts ✇✐t❤ ♥❡✇ s❡q✉❡♥❝❡ st❛rts ✇✐t❤ ❛❧s♦ ❜❡ ❛rr❛♥❣❡❞ t♦ st❛rt ✐♥❞❡①✳

◆♦✇✱ ✇❤❛t ❛❜♦✉t s❡q✉❡♥❝❡s ❣✐✈❡♥ ❜②

❢♦r♠✉❧❛s ❄

❛r✐t❤♠❡t✐❝ ♣r♦❣r❡ss✐♦♥

❚❤❡ ✜rst ♦♥❡ ✐s t❤❡

▲❡t✬s ❝♦♥s✐❞❡r ❛ ❝♦✉♣❧❡ ♦❢ s♣❡❝✐✜❝ s❡q✉❡♥❝❡s✳

❛♥❞ ✐t ✐s ✈❡r② s✐♠♣❧❡✳

❚❤❡♦r❡♠ ✶✳✽✳✹✿ ❉✐✛❡r❡♥❝❡ ♦❢ ❆r✐t❤♠❡t✐❝ Pr♦❣r❡ss✐♦♥ ❚❤❡ s❡q✉❡♥❝❡ ♦❢ ❞✐✛❡r❡♥❝❡s ♦❢ ❛♥ ❛r✐t❤♠❡t✐❝ ♣r♦❣r❡ss✐♦♥ ✇✐t❤ ✐♥❝r❡♠❡♥t m ✐s ❛ ❝♦♥st❛♥t s❡q✉❡♥❝❡ ✇✐t❤ t❤❡ ✈❛❧✉❡ ❡q✉❛❧ t♦ m✳

Pr♦♦❢✳ ❲❡ s✐♠♣❧② ❝♦♠♣✉t❡ ❢r♦♠ t❤❡ ❞❡✜♥✐t✐♦♥✿

dn = ∆(a0 + mn) = an+1 − an = (a0 + b(m + 1)) − (a0 + mn) = m . ❚❤❡ t❤❡♦r❡♠ ❝❛♥ ❜❡ r❡❝❛st ❛s ❛♥ ✐♠♣❧✐❝❛t✐♦♥✱ ❛♥ ✏✐❢✲t❤❡♥✑ st❛t❡♠❡♥t✿

◮ ■❋ an

✐s ❛♥ ❛r✐t❤♠❡t✐❝ ♣r♦❣r❡ss✐♦♥ ✇✐t❤ ✐♥❝r❡♠❡♥t

❝♦♥st❛♥t s❡q✉❡♥❝❡ ✇✐t❤ t❤❡ ✈❛❧✉❡ ❡q✉❛❧ t♦

m✱ ❚❍❊◆

✐ts s❡q✉❡♥❝❡ ♦❢ ❞✐✛❡r❡♥❝❡s ✐s ❛

m✳

❲❡ ❝❛♥ ❛❧s♦ ✉s❡ ♦✉r ❝♦♥✈❡♥✐❡♥t ❛❜❜r❡✈✐❛t✐♦♥✿

◮ an

✐s ❛♥ ❛r✐t❤♠❡t✐❝ ♣r♦❣r❡ss✐♦♥ ✇✐t❤ ✐♥❝r❡♠❡♥t

❝♦♥st❛♥t s❡q✉❡♥❝❡ ✇✐t❤ t❤❡ ✈❛❧✉❡ ❡q✉❛❧ t♦

m =⇒

✐ts s❡q✉❡♥❝❡ ♦❢ ❞✐✛❡r❡♥❝❡s ♦❢

an

✐s ❛

m✳

❚❤✐s ✐s ✇❤❛t t❤❡ ❣r❛♣❤s ♦❢ t❤✐s ♣❛✐r ♦❢ s❡q✉❡♥❝❡s ♠❛② ❧♦♦❦ ❧✐❦❡✱ ③♦♦♠❡❞ ♦✉t✱ ❢♦r t❤❡ ❢♦❧❧♦✇✐♥❣ t❤r❡❡ ❝❤♦✐❝❡s ♦❢ t❤❡ ✐♥❝r❡♠❡♥t

m✿

▲❡t✬s t❛❦❡ ❛ ❧♦♦❦ ❛t ❛

❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥ an = arn

✇✐t❤

a>0

❛♥❞

r > 0✳

✶✳✽✳ ❚❤❡ s❡q✉❡♥❝❡ ♦❢ ❞✐✛❡r❡♥❝❡s✿ ✈❡❧♦❝✐t②

✻✻

❊①❛♠♣❧❡ ✶✳✽✳✺✿ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥ an = 3n ❚❤✐s ✐s ❛ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥ ✇✐t❤ r❛t✐♦ r = 3✳ ▲❡t✬s ❝♦♠♣✉t❡ t❤❡ ❞✐✛❡r❡♥❝❡✿ ∆an = an+1 − an = 3n+1 − 3n = 3n · 3 − 3 n = 3n (3 − 1) = 3n · 2 .

■t✬s ❛ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥ ✇✐t❤ r = 3✱ ❛❣❛✐♥✦ ■s t❤❡r❡ ❛ ♣❛tt❡r♥❄ ▲❡t✬s ♣❧♦t t❤❡ ❣r❛♣❤ ♦❢ ❛ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥ an = arn ✇✐t❤ a > 0 ❛♥❞ r > 0✳ ❚❤❡r❡ ❛r❡ t✇♦ ❝❛s❡s✱ ❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ ❝❤♦✐❝❡ ♦❢ r❛t✐♦ r ✭❣r♦✇t❤ ♦r ❞❡❝❛②✮✿

❲❤❛t ❞♦ t❤❡ s❡q✉❡♥❝❡ ♦❢ ❞✐✛❡r❡♥❝❡s ✭s❡❝♦♥❞ r♦✇✮ ❧♦♦❦ ❧✐❦❡❄ ❲❡ ♥♦t✐❝❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿ • ■t ✐s ♣♦s✐t✐✈❡ ❛♥❞ ✐♥❝r❡❛s✐♥❣✱ ✇✐t❤ s♣❡❡❞✐♥❣ ✉♣ ✇❤❡♥ t❤❡ r❛t✐♦ r ✐s ❧❛r❣❡r t❤❛♥ 1✳ • ■t ✐s ♥❡❣❛t✐✈❡ ❛♥❞ ✐♥❝r❡❛s✐♥❣✱ ✇✐t❤ s❧♦✇✐♥❣ ❞♦✇♥ ✇❤❡♥ 0 < r < 1✳

■t ❛❧s♦ r❡s❡♠❜❧❡s t❤❡ ♦r✐❣✐♥❛❧ s❡q✉❡♥❝❡✦

❚❤❡♦r❡♠ ✶✳✽✳✻✿ ❉✐✛❡r❡♥❝❡ ♦❢ ●❡♦♠❡tr✐❝ Pr♦❣r❡ss✐♦♥ ❚❤❡ s❡q✉❡♥❝❡ ♦❢ ❞✐✛❡r❡♥❝❡s ♦❢ ❛ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥ ✐s ❛ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥ ✇✐t❤ t❤❡ s❛♠❡ r❛t✐♦✳

Pr♦♦❢✳ ■❢ ✇❡ ❤❛✈❡ ❛ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥ ✇✐t❤ r❛t✐♦ r ❛♥❞ ✐♥✐t✐❛❧ t❡r♠ a✱ ✐ts ❢♦r♠✉❧❛ ✐s an = arn ✳ ❚❤❡r❡❢♦r❡✱ dn = ∆(arn ) = an+1 − an = arn+1 − arn = a(r − 1) · rn .

❇✉t t❤❛t✬s t❤❡ ❢♦r♠✉❧❛ ♦❢ ❛ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥ ✇✐t❤ r❛t✐♦ r ❛♥❞ ✐♥✐t✐❛❧ t❡r♠ a(r − 1)✳ ❚❤❡ t❤❡♦r❡♠ ❝❛♥ ❜❡ r❡st❛t❡❞ ❛s ❛♥ ✐♠♣❧✐❝❛t✐♦♥✿ ◮ ■❋ an ✐s ❛ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥ ✇✐t❤ r❛t✐♦ r✱ ❚❍❊◆ ✐ts s❡q✉❡♥❝❡ ♦❢ ❞✐✛❡r❡♥❝❡s ✐s ❛ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥ ✇✐t❤ r❛t✐♦ r✳

❆❧s♦✿ ◮ an ✐s ❛ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥ ✇✐t❤ r❛t✐♦ r =⇒ ✐ts s❡q✉❡♥❝❡ ♦❢ ❞✐✛❡r❡♥❝❡s ✐s ❛ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥ ✇✐t❤ r❛t✐♦ r✳

✶✳✽✳ ❚❤❡ s❡q✉❡♥❝❡ ♦❢ ❞✐✛❡r❡♥❝❡s✿ ✈❡❧♦❝✐t②

✻✼

❊①❛♠♣❧❡ ✶✳✽✳✼✿ ❛❧t❡r♥❛t✐♥❣ s❡q✉❡♥❝❡ ❚❤❡ s❡q✉❡♥❝❡ ♦❢ ❞✐✛❡r❡♥❝❡s ♦❢ t❤❡ ❛❧t❡r♥❛t✐♥❣ s❡q✉❡♥❝❡ an = (−1)n ✐s ❝♦♠♣✉t❡❞ ❜❡❧♦✇✿ n

∆ ((−1) ) = (−1)

n+1

n

− (−1) =



(−1) − 1, n ✐s ❡✈❡♥ = 1 − (−1), n ✐s ♦❞❞



−2, n ✐s ❡✈❡♥ = 2(−1)n+1 . 2, n ✐s ♦❞❞

❊①❡r❝✐s❡ ✶✳✽✳✽ ❲❤❛t ✐s t❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ s❡q✉❡♥❝❡ ❛❜♦✈❡ ❛♥❞ ✐ts ❞✐✛❡r❡♥❝❡❄

❊①❛♠♣❧❡ ✶✳✽✳✾✿ ❞✐✛❡r❡♥❝❡s ❛r❡ ✈❡❧♦❝✐t✐❡s ❲❡ ❝❛♥ ✉s❡ ❝♦♠♣✉t❡rs t♦ s♣❡❡❞ ✉♣ t❤❡s❡ ❝♦♠♣✉t❛t✐♦♥s✳ ❋♦r ❡①❛♠♣❧❡✱ ♦♥❡ ♠❛② ❤❛✈❡ ❜❡❡♥ r❡❝♦r❞✐♥❣ ♦♥❡✬s ❧♦❝❛t✐♦♥s ❛♥❞ ♥♦✇ ♥❡❡❞s t♦ ✜♥❞ t❤❡ ✈❡❧♦❝✐t✐❡s✳ ❍❡r❡ ✐s ❛ s♣r❡❛❞s❤❡❡t ❢♦r♠✉❧❛ ❢♦r t❤❡ s❡q✉❡♥❝❡ ♦❢ ❞✐✛❡r❡♥❝❡s ✭✈❡❧♦❝✐t✐❡s✮✿ ❂❘❈❬✲✶❪✲❘❬✲✶❪❈❬✲✶❪

❲❤❡t❤❡r t❤❡ s❡q✉❡♥❝❡ ❝♦♠❡s ❢r♦♠ ❛ ❢♦r♠✉❧❛ ♦r ✐t✬s ❥✉st ❛ ❧✐st ♦❢ ♥✉♠❜❡rs✱ t❤❡ ❢♦r♠✉❧❛ ❛♣♣❧✐❡s ❡q✉❛❧❧②✿

❆s ❛ r❡s✉❧t✱ ❛ ❝✉r✈❡ ❤❛s ♣r♦❞✉❝❡❞ ❛ ♥❡✇ ❝✉r✈❡✿

❲❤✐❧❡ t❤❡ ✜rst ❣r❛♣❤ t❡❧❧s ✉s t❤❛t ✇❡ ❛r❡ ♠♦✈✐♥❣ ❢♦r✇❛r❞ ❛♥❞ t❤❡♥ ❜❛❝❦✇❛r❞✱ ✐t ✐s ❡❛s✐❡r t♦ ❞❡r✐✈❡ ❜❡tt❡r ❞❡s❝r✐♣t✐♦♥ ❢r♦♠ t❤❡ s❡❝♦♥❞✿ s♣❡❡❞ ✉♣ ❢♦r✇❛r❞✱ t❤❡♥ s❧♦✇ ❞♦✇♥✱ t❤❡♥ s♣❡❡❞ ✉♣ ❜❛❝❦✇❛r❞✳

❊①❡r❝✐s❡ ✶✳✽✳✶✵ ❉❡s❝r✐❜❡ ✇❤❛t ❤❛s ❤❛♣♣❡♥❡❞ r❡❢❡rr✐♥❣ t♦✱ s❡♣❛r❛t❡❧②✱ t❤❡ ✜rst ❣r❛♣❤ ❛♥❞ t❤❡ s❡❝♦♥❞ ❣r❛♣❤✳

❊①❡r❝✐s❡ ✶✳✽✳✶✶ ■♠❛❣✐♥❡✱ ✐♥st❡❛❞✱ t❤❛t t❤❡ ✜rst ❝♦❧✉♠♥ ♦❢ t❤❡ s♣r❡❛❞s❤❡❡t ❛❜♦✈❡ ✐s ✇❤❡r❡ ②♦✉ ❤❛✈❡ ❜❡❡♥ r❡❝♦r❞✐♥❣ t❤❡ ♠♦♥t❤❧② ❜❛❧❛♥❝❡ ♦❢ ②♦✉r ❜❛♥❦ ❛❝❝♦✉♥t✳ ❲❤❛t ❞♦❡s t❤❡ s❡❝♦♥❞ ❝♦❧✉♠♥ r❡♣r❡s❡♥t❄ ❉❡s❝r✐❜❡ ✇❤❛t ❤❛s ❜❡❡♥ ❤❛♣♣❡♥✐♥❣ ✇✐t❤ ②♦✉r ✜♥❛♥❝❡s r❡❢❡rr✐♥❣ t♦✱ s❡♣❛r❛t❡❧②✱ t❤❡ ✜rst ❣r❛♣❤ ❛♥❞ t❤❡ s❡❝♦♥❞ ❣r❛♣❤✳ ❚❤✐s ✐s t❤❡ t✐♠❡ ❢♦r s♦♠❡ t❤❡♦r②✳ ❈♦♥s✐❞❡r t❤✐s ♦❜✈✐♦✉s st❛t❡♠❡♥t ❛❜♦✉t ♠♦t✐♦♥✿ ◮ ■❋ ■ ❛♠ st❛♥❞✐♥❣ st✐❧❧✱ ❚❍❊◆ ♠② ✈❡❧♦❝✐t② ✐s ③❡r♦✳

✶✳✽✳

❚❤❡ s❡q✉❡♥❝❡ ♦❢ ❞✐✛❡r❡♥❝❡s✿ ✈❡❧♦❝✐t②

✻✽

❲❡ ❝❛♥ ❛❧s♦ s❛②✿

◮ ■❋

♠② ✈❡❧♦❝✐t② ✐s ③❡r♦✱

❚❍❊◆

■ ❛♠ st❛♥❞✐♥❣ st✐❧❧✳

❲❡ s❡❡ t❤❡ ✐♠♣❧✐❝❛t✐♦♥s ❣♦✐♥❣ ❜♦t❤ ✇❛②s❀ t❤❡ ❧❛tt❡r ✐s t❤❡

❝♦♥✈❡rs❡

♦❢ t❤❡ ♦r✐❣✐♥❛❧ st❛t❡♠❡♥t ✭❛♥❞ ✈✐❝❡

✈❡rs❛✦✮✳ ▲❡t✬s ✉s❡ s②♠❜♦❧s t♦ r❡st❛t❡ t❤❡s❡ st❛t❡♠❡♥ts ♠♦r❡ ❝♦♠♣❛❝t❧②✿

=⇒ ⇐=

■ ❛♠ st❛♥❞✐♥❣ st✐❧❧ ■ ❛♠ st❛♥❞✐♥❣ st✐❧❧

♠② ✈❡❧♦❝✐t② ✐s ③❡r♦✳ ♠② ✈❡❧♦❝✐t② ✐s ③❡r♦✳

❚❤❡ ❛❜❜r❡✈✐❛t✐♦♥ ♦❢ t❤❡ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ t✇♦ ✐s ❛♥ ❡q✉✐✈❛❧❡♥❝❡✱ ❛♥ ✏✐❢✲❛♥❞✲♦♥❧②✲✐❢ ✑ st❛t❡♠❡♥t✿

⇐⇒

■ ❛♠ st❛♥❞✐♥❣ st✐❧❧

♠② ✈❡❧♦❝✐t② ✐s ③❡r♦✳

■t✬s ❥✉st ❛♥♦t❤❡r ✇❛② ♦❢ s❛②✐♥❣ t❤❡ s❛♠❡ t❤✐♥❣✳ ■❢ ✇❡ s❡t t❤❡ ♠♦t✐♦♥ ♣♦✐♥t ♦❢ ✈✐❡✇ ❛s✐❞❡✱ ❤❡r❡ ✐s t❤❡ ❣❡♥❡r❛❧ st❛t❡♠❡♥t✳

❚❤❡♦r❡♠ ✶✳✽✳✶✷✿ ❉✐✛❡r❡♥❝❡ ♦❢ ❈♦♥st❛♥t ❙❡q✉❡♥❝❡ ❆ s❡q✉❡♥❝❡ ✐s ❝♦♥st❛♥t ■❋ ❆◆❉ ❖◆▲❨ ■❋ ✐ts s❡q✉❡♥❝❡ ♦❢ ❞✐✛❡r❡♥❝❡s ✐s ③❡r♦✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿ an ✐s ❝♦♥st❛♥t ⇐⇒ ∆an = 0 .

Pr♦♦❢✳ ❉✐r❡❝t✿

an = c

❢♦r ❛❧❧

n =⇒ an+1 − an = c − c = 0 =⇒ ∆an = 0 .

❈♦♥✈❡rs❡✿

an+1 = an = c

❢♦r ❛❧❧

n ⇐= an+1 − an = 0 ⇐= ∆an = 0 .

❊①❡r❝✐s❡ ✶✳✽✳✶✸ Pr♦✈❡ t❤❛t t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ ❛♥ ❛r✐t❤♠❡t✐❝ ♣r♦❣r❡ss✐♦♥ ✐s ❝♦♥st❛♥t ❛♥❞✱ ❝♦♥✈❡rs❡❧②✱ t❤❛t ✐❢ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ ❛ s❡q✉❡♥❝❡ ✐s ❛ ❝♦♥st❛♥t s❡q✉❡♥❝❡✱ t❤❡♥ t❤❡ s❡q✉❡♥❝❡ ✐s ❛♥ ❛r✐t❤♠❡t✐❝ ♣r♦❣r❡ss✐♦♥✳

❈♦♥s✐❞❡r ❛♥♦t❤❡r ♦❜✈✐♦✉s st❛t❡♠❡♥t ❛❜♦✉t ♠♦t✐♦♥✿

◮ ■❋

■ ❛♠ ♠♦✈✐♥❣ ❢♦r✇❛r❞✱

❚❍❊◆

♠② ✈❡❧♦❝✐t② ✐s ♣♦s✐t✐✈❡✳

❆♥❞✱ ❝♦♥✈❡rs❡❧②✿

◮ ■❋

♠② ✈❡❧♦❝✐t② ✐s ♣♦s✐t✐✈❡✱

❚❍❊◆

■ ❛♠ ♠♦✈✐♥❣ ❢♦r✇❛r❞✳

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡ t❤✐s ♣❛✐r ♦❢ st❛t❡♠❡♥ts✿

❖r✐❣✐♥❛❧✿

■ ❛♠ ♠♦✈✐♥❣ ❢♦r✇❛r❞

❈♦♥✈❡rs❡✿

■ ❛♠ ♠♦✈✐♥❣ ❢♦r✇❛r❞

❊①❡r❝✐s❡ ✶✳✽✳✶✹

=⇒ ⇐=

♠② ✈❡❧♦❝✐t② ✐s ♣♦s✐t✐✈❡✳ ♠② ✈❡❧♦❝✐t② ✐s ♣♦s✐t✐✈❡✳

❲❤❛t ✐s t❤❡ ❝♦♥✈❡rs❡ ♦❢ t❤❡ ❝♦♥✈❡rs❡❄

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

❚❤❡♦r❡♠ ✶✳✽✳✶✺✿ ▼♦♥♦t♦♥✐❝✐t② ❚❤❡♦r❡♠ ❢♦r ❙❡q✉❡♥❝❡s ❆ s❡q✉❡♥❝❡ ✐s ✐♥❝r❡❛s✐♥❣✴❞❡❝r❡❛s✐♥❣ ■❋ ❆◆❉ ❖◆▲❨ ■❋ t❤❡ s❡q✉❡♥❝❡ ♦❢ ❞✐✛❡r✲ ❡♥❝❡s ✐s ♣♦s✐t✐✈❡✴♥❡❣❛t✐✈❡ ♦r ③❡r♦✱ r❡s♣❡❝t✐✈❡❧②✳

✶✳✽✳ ❚❤❡ s❡q✉❡♥❝❡ ♦❢ ❞✐✛❡r❡♥❝❡s✿ ✈❡❧♦❝✐t②

✻✾

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿ an an an

✐s ✐♥❝r❡❛s✐♥❣ ✐s ❞❡❝r❡❛s✐♥❣ ✐s ❝♦♥st❛♥t

⇐⇒ ∆an ≥ 0 . ⇐⇒ ∆an ≤ 0 . ⇐⇒ ∆an = 0 .

Pr♦♦❢✳ an+1 ≥ an ❢♦r ❛❧❧ n ⇐⇒ an+1 − an ≥ 0 ⇐⇒ ∆an ≥ 0 .

■t✬s ❥✉st ❛♥♦t❤❡r ✇❛② ♦❢ s❛②✐♥❣ t❤❡ s❛♠❡ t❤✐♥❣✳ ❙✉♣♣♦s❡ ♥♦✇ t❤❛t t❤❡r❡ ❛r❡ t✇♦ r✉♥♥❡rs❀ t❤❡♥ ✇❡ ❤❛✈❡ ❛ ❧❡ss ♦❜✈✐♦✉s ❢❛❝t ❛❜♦✉t ♠♦t✐♦♥✿ ◮ ■❋ t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t✇♦ r✉♥♥❡rs ✐s♥✬t ❝❤❛♥❣✐♥❣✱ ❚❍❊◆ t❤❡② ❛r❡ r✉♥♥✐♥❣ ✇✐t❤ t❤❡ s❛♠❡ ✈❡❧♦❝✐t②✳

❆♥❞ ✈✐❝❡ ✈❡rs❛✿ ◮ ■❋ t✇♦ r✉♥♥❡rs ❛r❡ r✉♥♥✐♥❣ ✇✐t❤ t❤❡ s❛♠❡ ✈❡❧♦❝✐t②✱ ❚❍❊◆ t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡♠ ✐s♥✬t

❝❤❛♥❣✐♥❣✳

■t✬s ❛s ✐❢ t❤❡② ❛r❡ ❤♦❧❞✐♥❣ t❤❡ t✇♦ ❡♥❞s ♦❢ ❛ ♣♦❧❡✿

❚❤❡ ❝♦♥❝❧✉s✐♦♥ ❤♦❧❞s ❡✈❡♥ ✐❢ t❤❡② s♣❡❡❞ ✉♣ ❛♥❞ s❧♦✇ ❞♦✇♥ ❛❧❧ t❤❡ t✐♠❡✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿ ◮ ❚❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t✇♦ r✉♥♥❡rs ✐s♥✬t ❝❤❛♥❣✐♥❣ ■❋ ❆◆❉ ❖◆▲❨ ■❋ t❤❡② ❛r❡ r✉♥♥✐♥❣ ✇✐t❤

t❤❡ s❛♠❡ ✈❡❧♦❝✐t②✳

❖♥❝❡ ❛❣❛✐♥✱ ❢♦r s❡q✉❡♥❝❡s an ❛♥❞ bn r❡♣r❡s❡♥t✐♥❣ t❤❡✐r r❡s♣❡❝t✐✈❡ ♣♦s✐t✐♦♥s ❛t t✐♠❡ n✱ ✇❡ ❝❛♥ r❡st❛t❡ t❤✐s ✐❞❡❛ ♠❛t❤❡♠❛t✐❝❛❧❧② ✐♥ ♦r❞❡r t♦ ❝♦♥✜r♠ t❤❛t ♦✉r t❤❡♦r② ♠❛❦❡s s❡♥s❡✳

❈♦r♦❧❧❛r② ✶✳✽✳✶✻✿ ❉✐✛❡r❡♥❝❡ ✉♥❞❡r ❙✉❜tr❛❝t✐♦♥ ❚✇♦ s❡q✉❡♥❝❡s ❞✐✛❡r ❜② ❛ ❝♦♥st❛♥t ■❋ ❆◆❉ ❖◆▲❨ ■❋ ✐❢ t❤❡✐r s❡q✉❡♥❝❡s ♦❢ ❞✐✛❡r❡♥❝❡s ❛r❡ ❡q✉❛❧✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿ an − bn ✐s ❝♦♥st❛♥t ⇐⇒ ∆an = ∆bn .

Pr♦♦❢✳ ❚❤❡ ❝♦r♦❧❧❛r② ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ❉✐✛❡r❡♥❝❡ ♦❢ ❈♦♥st❛♥t ❙❡q✉❡♥❝❡ ❚❤❡♦r❡♠ ❛❜♦✈❡✳

❊①❛♠♣❧❡ ✶✳✽✳✶✼✿ s❤✐❢t ♦❢ s❡q✉❡♥❝❡ ❲❡ s❤✐❢t t❤❡ s❡q✉❡♥❝❡ an ❜❡❧♦✇ ❜② 1 ✉♥✐t ✉♣ t♦ ♣r♦❞✉❝❡ ❛ ♥❡✇ s❡q✉❡♥❝❡ bn ✭t♦♣✮✿

✶✳✽✳ ❚❤❡ s❡q✉❡♥❝❡ ♦❢ ❞✐✛❡r❡♥❝❡s✿ ✈❡❧♦❝✐t②

✼✵

❇❡❝❛✉s❡ t❤❡ ✉♣s ❛♥❞ ❞♦✇♥s r❡♠❛✐♥ t❤❡ s❛♠❡✱ t❤❡ s❡q✉❡♥❝❡s ♦❢ ❞✐✛❡r❡♥❝❡s ♦❢ t❤❡s❡ t✇♦ s❡q✉❡♥❝❡s ❛r❡ ✐❞❡♥t✐❝❛❧ ✭❜♦tt♦♠✮✳ ❊①❡r❝✐s❡ ✶✳✽✳✶✽

❲❤❛t ✐❢ t❤❡ t✇♦ r✉♥♥❡rs ❤♦❧❞✐♥❣ t❤❡ ♣♦❧❡ ❛❧s♦ st❛rt t♦ ♠♦✈❡ t❤❡✐r ❤❛♥❞s ❜❛❝❦ ❛♥❞ ❢♦rt❤❄ ❲❡ ❝❛♥ ✉s❡ t❤❡ ❧❛tt❡r t❤❡♦r❡♠ t♦ ✇❛t❝❤ ❛❢t❡r t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ t✇♦ r✉♥♥❡rs✳ ❆ ♠❛t❝❤✐♥❣ st❛t❡♠❡♥t ❛❜♦✉t ♠♦t✐♦♥ ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✿ ◮ ■❋ t❤❡ ❞✐st❛♥❝❡ ❢r♦♠ ♦♥❡ ♦❢ t❤❡ t✇♦ r✉♥♥❡rs t♦ t❤❡ ♦t❤❡r ✐s ✐♥❝r❡❛s✐♥❣✱ ❚❍❊◆ t❤❡ ❢♦r♠❡r✬s

✈❡❧♦❝✐t② ✐s ❤✐❣❤❡r✳

❈♦♥✈❡rs❡❧②✿ ◮ ■❋ t❤❡ ✈❡❧♦❝✐t② ♦❢ ♦♥❡ r✉♥♥❡r ✐s ❤✐❣❤❡r t❤❛♥ t❤❡ ♦t❤❡r✱ ❚❍❊◆ t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡♠ ✐s

✐♥❝r❡❛s✐♥❣✳

❊①❡r❝✐s❡ ✶✳✽✳✶✾

❈♦♠❜✐♥❡ t❤❡ t✇♦ st❛t❡♠❡♥ts ✐♥t♦ ♦♥❡✳ ❲❡ ❝❛♥ r❡st❛t❡ t❤✐s ♠❛t❤❡♠❛t✐❝❛❧❧②✳ ❈♦r♦❧❧❛r② ✶✳✽✳✷✵✿ ▼♦♥♦t♦♥✐❝✐t② ❛♥❞ ❙✉❜tr❛❝t✐♦♥

❚❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t✇♦ s❡q✉❡♥❝❡s ✐s ✐♥❝r❡❛s✐♥❣ ■❋ ❆◆❉ ❖◆▲❨ ■❋ t❤❡ ❢♦r♠❡r✬s ❞✐✛❡r❡♥❝❡ ✐s ❜✐❣❣❡r t❤❛♥ t❤❡ ❧❛tt❡r✬s✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿ an − bn an − bn

✐s ✐♥❝r❡❛s✐♥❣ ✐s ❞❡❝r❡❛s✐♥❣

⇐⇒ ∆an ≥ ∆bn , ⇐⇒ ∆an ≤ ∆bn .

Pr♦♦❢✳

❚❤❡ ❝♦r♦❧❧❛r② ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ▼♦♥♦t♦♥✐❝✐t② ❚❤❡♦r❡♠ ❢♦r ❙❡q✉❡♥❝❡s ❛❜♦✈❡✳ ❊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 ❤❛♣♣❡♥❡❞✿

✶✳✽✳

❚❤❡ s❡q✉❡♥❝❡ ♦❢ ❞✐✛❡r❡♥❝❡s✿ ✈❡❧♦❝✐t②

✼✶

❚❤❡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 • B • C

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

❞✐✛❡r❡♥❝❡s

♦❢ t❤❡ s❡q✉❡♥❝❡s✿

❊①❡r❝✐s❡ ✶✳✽✳✷✷ ❙✉♣♣♦s❡ ❛ s❡q✉❡♥❝❡ ✐s ❣✐✈❡♥ ❜② t❤❡ ❣r❛♣❤ ❢♦r ✈❡❧♦❝✐t② ❛❜♦✈❡✳ ❙❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤✐s s❡q✉❡♥❝❡✳ ❲❤❛t ✐s ✐ts ♠❡❛♥✐♥❣❄

❊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 ♦❢ t❤r❡❡ ❜❛♥❦ ❛❝❝♦✉♥ts✳ ❉❡s❝r✐❜❡ ✇❤❛t ❤❛s ❜❡❡♥ ❤❛♣♣❡♥✐♥❣✳

❊①❛♠♣❧❡ ✶✳✽✳✷✻✿ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ s❡q✉❡♥❝❡ ❍♦✇ ❞♦ ✇❡ tr❡❛t ♠♦t✐♦♥ ✇❤❡♥ t❤❡ t✐♠❡ ✐♥❝r❡♠❡♥t ✐s♥✬t

1❄

❲❤❛t ✐s t❤❡

✈❡❧♦❝✐t②

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

t❤❡♥❄

xn

❛♥❞

yn ✳

❚❤❡♥ t❤❡

✶✳✾✳

❚❤❡ s❡q✉❡♥❝❡ ♦❢ t❤❡ s✉♠s✿ ❞✐s♣❧❛❝❡♠❡♥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❡♥❝❡ q✉♦t✐❡♥t ♦❢ t✇♦ s❡q✉❡♥❝❡s ♦❢ xn ❛♥❞ yn ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ s❡q✉❡♥❝❡ t❤❛t ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ yn ❞✐✈✐❞❡❞ ❜② t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ xn ✿

∆yn yn+1 − yn = , ∆xn xn+1 − xn ♣r♦✈✐❞❡❞ t❤❡ ❞❡♥♦♠✐♥❛t♦r ✐s ♥♦t ③❡r♦✳ ■t ✐s t❤❡ r❡❧❛t✐✈❡ ❝❤❛♥❣❡ ✕ t❤❡ s❡q✉❡♥❝❡s ✭❢♦r ❡❛❝❤ ❝♦♥s❡❝✉t✐✈❡ ♣❛✐r ♦❢ ♣♦✐♥ts✱ ✐t ✐s t❤❡ s❧♦♣❡✮✿

r❛t❡ ♦❢ ❝❤❛♥❣❡ ✕ ♦❢ t❤❡ t✇♦

❚❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ t❤❡ s❡q✉❡♥❝❡ ♦❢ t❤❡ ❧♦❝❛t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ s❡q✉❡♥❝❡ ♦❢ t✐♠❡ ✐s t❤❡ ✈❡❧♦❝✐t②✳ ✭❲❡ ✇✐❧❧ ❝♦♥t✐♥✉❡ t❤✐s st✉❞② ✐♥ ❈❤❛♣t❡r ✷✳✮

❊①❛♠♣❧❡ ✶✳✽✳✷✼✿ ❞✐✛❡r❡♥❝❡ ♦❢ r❛♥❞♦♠ s❡q✉❡♥❝❡ ■❢ ✇❡ t❛❦❡ ❛ s❡q✉❡♥❝❡ ♦❢ r❛♥❞♦♠ ♥✉♠❜❡rs✱ ✇✐t❤ ✐ts ✈❛❧✉❡s s♣r❡❛❞ ❜❡t✇❡❡♥ −1 ❛♥❞ 1✱ ✐ts ❞✐✛❡r❡♥❝❡ ✐s ❛❧s♦ r❛♥❞♦♠ ❜✉t t❤❡ ✈❛❧✉❡s ❛r❡ s♣r❡❛❞ ❜❡t✇❡❡♥ −2 ❛♥❞ 2✿

❊①❡r❝✐s❡ ✶✳✽✳✷✽ P♦✐♥t ♦✉t ❛♥❞ ❡①♣❧❛✐♥ t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ t✇♦ ❣r❛♣❤s✳

✶✳✾✳ ❚❤❡ s❡q✉❡♥❝❡ ♦❢ t❤❡ s✉♠s✿ ❞✐s♣❧❛❝❡♠❡♥t

■♥ t❤❡ ✜rst s❡❝t✐♦♥✱ ✇❡ s❛✇ ❤♦✇ t❤❡ s❡q✉❡♥❝❡s ♦❢ ❧♦❝❛t✐♦♥s ❛♥❞ ✈❡❧♦❝✐t✐❡s ✐♥t❡r❛❝t✳ ❲❡ t♦♦❦ ❛ ❝❧♦s❡r ❧♦♦❦ ❛t t❤❡ tr❛♥s✐t✐♦♥ ❢r♦♠ t❤❡ ❢♦r♠❡r t♦ t❤❡ ❧❛tt❡r ❛♥❞ ♥♦✇ ✐♥ r❡✈❡rs❡✿

✶✳✾✳ ❚❤❡ s❡q✉❡♥❝❡ ♦❢ t❤❡ s✉♠s✿ ❞✐s♣❧❛❝❡♠❡♥t

✼✸

■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ st❛rt ♦♥ t❤❡ ♣❛t❤ ♦❢ ❞❡✈❡❧♦♣♠❡♥t ♦❢ ❛♥ ✐❞❡❛ t❤❛t ❝✉❧♠✐♥❛t❡s ✇✐t❤ t❤❡ s❡❝♦♥❞ ❢✉♥❞❛♠❡♥t❛❧ ❝♦♥❝❡♣t ♦❢ ❝❛❧❝✉❧✉s✱ t❤❡ ✐♥t❡❣r❛❧ ✭❈❤❛♣t❡r ✸■❈✲✶✮✳ ❚❤❡ s✉♠ r❡♣r❡s❡♥ts t❤❡ t♦t❛❧✐t② ♦❢ t❤❡ ✏❜❡❣✐♥♥✐♥❣✑ ♦❢ ❛ s❡q✉❡♥❝❡✱ ❢♦✉♥❞ ❜② ❛❞❞✐♥❣ ❡❛❝❤ ♦❢ ✐ts t❡r♠s t♦ t❤❡ ♥❡①t✱ ✉♣ t♦ t❤❛t ♣♦✐♥t✳

❊①❛♠♣❧❡ ✶✳✾✳✶✿ s❡q✉❡♥❝❡s ❣✐✈❡♥ ❜② ❧✐sts ❲❡ ❥✉st ❛❞❞ t❤❡ ❝✉rr❡♥t t❡r♠ t♦ ✇❤❛t ✇❡ ❤❛✈❡ ❛❝❝✉♠✉❧❛t❡❞ s♦ ❢❛r✿ s❡q✉❡♥❝❡✿ s✉♠s✿

♥❡✇ s❡q✉❡♥❝❡✿

2 4 7 1 −1 ↓ ↓ ↓ ↓ ↓ 2 2 + 4 = 6 6 + 7 = 13 13 + 1 = 14 14 + (−1) = 13 ↓ ↓ ↓ ↓ ↓ 2 6 13 14 13

... ...

... ...

❲❡ ❤❛✈❡ ❛ ♥❡✇ ❧✐st✦

❊①❛♠♣❧❡ ✶✳✾✳✷✿ s❡q✉❡♥❝❡s ❣✐✈❡♥ ❜② ❣r❛♣❤s ❲❡ tr❡❛t t❤❡ ❣r❛♣❤ ♦❢ ❛ s❡q✉❡♥❝❡ ❛s ✐❢ ♠❛❞❡ ♦❢ ❜❛rs ❛♥❞ t❤❡♥ ❥✉st st❛❝❦ ✉♣ t❤❡s❡ ❜❛rs ♦♥ t♦♣ ♦❢ ❡❛❝❤ ♦t❤❡r ♦♥❡ ❜② ♦♥❡✿

❚❤❡s❡ st❛❝❦❡❞ ❜❛rs ✕ ♦r r❛t❤❡r t❤❡ ♣r♦❝❡ss ♦❢ st❛❝❦✐♥❣ ✕ ♠❛❦❡ ❛ ♥❡✇ s❡q✉❡♥❝❡✳

❯♥❧✐❦❡ t❤❡ ❞✐✛❡r❡♥❝❡✱ t❤❡ s✉♠ ♠✉st ❜❡ ❞❡✜♥❡❞ ✭❛♥❞ ❝♦♠♣✉t❡❞✮ ✐♥ ❛ r❡❝✉rs✐✈❡ ♠❛♥♥❡r✳

❉❡✜♥✐t✐♦♥ ✶✳✾✳✸✿ s❡q✉❡♥❝❡ ♦❢ s✉♠s ❋♦r ❛ s❡q✉❡♥❝❡

an ✱ ✐ts s❡q✉❡♥❝❡ ♦❢ s✉♠s✱ ♦r s✐♠♣❧② t❤❡ s✉♠✱ ✐s ❛ ♥❡✇ s❡q✉❡♥❝❡ sn n ≥ m ✇✐t❤✐♥ t❤❡ ❞♦♠❛✐♥ ♦❢ an ❜② t❤❡ ❢♦❧❧♦✇✐♥❣

❞❡✜♥❡❞ ❛♥❞ ❞❡♥♦t❡❞ ❢♦r ❡❛❝❤ ✭r❡❝✉rs✐✈❡✮ ❢♦r♠✉❧❛✿

sm = 0,

sn+1 = sn + an+1

✶✳✾✳ ❚❤❡ s❡q✉❡♥❝❡ ♦❢ t❤❡ s✉♠s✿ ❞✐s♣❧❛❝❡♠❡♥t

✼✹

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿

sn = am + am+1 + ... + an

❊①❛♠♣❧❡ ✶✳✾✳✹✿ ❛❧t❡r♥❛t✐♥❣ s❡q✉❡♥❝❡

an

▲❡t✬s ❞♦ s♦♠❡ ❛❧❣❡❜r❛✳ ❍❡r❡

✐s t❤❡ ♦r✐❣✐♥❛❧ s❡q✉❡♥❝❡ ❛♥❞

n 1 2 3

an 1 −1 1

✳ ✳ ✳

✳ ✳ ✳

sn

✐s t❤❡ ♥❡✇ ♦♥❡✿

sn 1 1−1 1−1+1

= = = =

sn 1 0 1

✳ ✳ ✳

✳ ✳ ✳

n (−1)n 1 − 1 + 1 − ... + (−1)n = 1

♦r

0

❚❤❡ r❡s✉❧t✐♥❣ s❡q✉❡♥❝❡ ✐s ❛❧s♦ ✏❛❧t❡r♥❛t✐♥❣✑✦

❆ ❝♦♠♠♦♥❧② ✉s❡❞ ♥♦t❛t✐♦♥ ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✿

❙✐❣♠❛ ♥♦t❛t✐♦♥ ❢♦r s✉♠♠❛t✐♦♥

sn = am + am+1 + ... + an =

n X

ak

k=m

▲❡t✬s t❛❦❡ ❛ ❝❧♦s❡r ❧♦♦❦ ❛t t❤❡ ♥❡✇ ♥♦t❛t✐♦♥✳ ❚❤❡ ✜rst ❝❤♦✐❝❡ ♦❢ ❤♦✇ t♦ r❡♣r❡s❡♥t t❤❡ s✉♠ ♦❢ ❛ s❡❣♠❡♥t ✕ ❢r♦♠

m

t♦

n

✕ ♦❢ ❛ s❡q✉❡♥❝❡

an

✐s t❤✐s✿

am |{z}

st❡♣ 1

+am+1 +... | {z } st❡♣ 2

+a +... |{z}k

st❡♣ k

+a . |{z}n

st❡♣ n−m

❚❤✐s ♥♦t❛t✐♦♥ r❡✢❡❝ts t❤❡ r❡❝✉rs✐✈❡ ♥❛t✉r❡ ♦❢ t❤❡ ♣r♦❝❡ss ❜✉t ✐t ❝❛♥ ❛❧s♦ ❜❡ r❡♣❡t✐t✐✈❡ ❛♥❞ ❝✉♠❜❡rs♦♠❡✳ ❚❤❡ s❡❝♦♥❞ ❝❤♦✐❝❡ ✐s ♠♦r❡ ❝♦♠♣❛❝t✿

n X

ak .

k=m ❍❡r❡ t❤❡ ●r❡❡❦ ❧❡tt❡r

Σ

st❛♥❞s ❢♦r t❤❡ ❧❡tt❡r ❙ ♠❡❛♥✐♥❣ ✏s✉♠✑✳

❙✐❣♠❛ ♥♦t❛t✐♦♥ ❜❡❣✐♥♥✐♥❣

3 X k=0


k 2 + k = 20

−→

❛♥❞ ❡♥❞ ✈❛❧✉❡s ❢♦r

↓ 3 X

k2 + k

k=0

↑

k




= 20 ↑

❛ s♣❡❝✐✜❝ s❡q✉❡♥❝❡

❛ s♣❡❝✐✜❝ ♥✉♠❜❡r

❲❛r♥✐♥❣✦ ■t ✇♦✉❧❞ ♠❛❦❡ s❡♥s❡ t♦ ❤❛✈❡ ✏ k s✐❣♠❛✿

k=3 X k=0


k2 + k .

= 0✑

❛❜♦✈❡ t❤❡

✶✳✾✳

✼✺

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

❊①❛♠♣❧❡ ✶✳✾✳✺✿ ❡①♣❛♥❞✐♥❣ ❢r♦♠ s✐❣♠❛ ♥♦t❛t✐♦♥

❚❤❡ ❝♦♠♣✉t❛t✐♦♥ ❛❜♦✈❡ ✐s ❡①♣❛♥❞❡❞ ❤❡r❡✿ k k2 + k 0 02 + 0 = 0 3 X

+

2


1 1 +1 =2 k2 + k = 2 22 + 2 = 6

k=0

+ +

3 32 + 3 = 12 = 20

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

❍♦✇ ✇✐❧❧ t❤❡ s✉♠ ❝❤❛♥❣❡ ✐❢ ✇❡ r❡♣❧❛❝❡ k = 0 ✇✐t❤ k = 1✱ ♦r k = −1❄ ❲❤❛t ✐❢ ✇❡ r❡♣❧❛❝❡ 3 ❛t t❤❡ t♦♣ ✇✐t❤ 4❄ ❊①❛♠♣❧❡ ✶✳✾✳✼✿ ❝♦♥tr❛❝t✐♥❣ s✉♠♠❛t✐♦♥

❚❤✐s ✐s ❤♦✇ ✇❡ ❝♦♥tr❛❝t t❤❡ s✉♠♠❛t✐♦♥✿ 2

2

2

2

1 + 2 + 3 + ... + 17 =

n X

k2 .

k=1

❚❤✐s ✐s ♦♥❧② ♣♦ss✐❜❧❡ ✐❢ ✇❡ ✜♥❞ t❤❡ nt❤✲t❡r♠ ❢♦r♠✉❧❛ ❢♦r t❤❡ s❡q✉❡♥❝❡❀ ✐♥ t❤✐s ❝❛s❡✱ ak = k 2 ✳ ❆♥❞ t❤✐s ✐s ❤♦✇ ✇❡ ❡①♣❛♥❞ ❜❛❝❦ ❢r♦♠ t❤✐s ❝♦♠♣❛❝t ♥♦t❛t✐♦♥✱ ❜② ♣❧✉❣❣✐♥❣ t❤❡ ✈❛❧✉❡s ♦❢ k = 1, 2, ..., 17 ✐♥t♦ t❤❡ ❢♦r♠✉❧❛✿ 17 X k=1

❙✐♠✐❧❛r❧②✱ ✇❡ ❤❛✈❡✿

k 2 = |{z} 12 + |{z} 22 + |{z} 32 +... + |{z} 172 . k=1

k=2

k=3

k=17

10

1+

X 1 1 1 1 1 + 2 + 3 + ... + 10 = . k 2 2 2 2 2 k=0

❊①❡r❝✐s❡ ✶✳✾✳✽

❈♦♥✜r♠ t❤❛t ✇❡ ❝❛♥ st❛rt ❛t ❛♥② ♦t❤❡r ✐♥✐t✐❛❧ ✐♥❞❡① ✐❢ ✇❡ ❥✉st ♠♦❞✐❢② t❤❡ ❢♦r♠✉❧❛✿ ?

?

X 1 X 1 1 1 1 1 1 + + 2 + 3 + ... + 10 = = = ... 2 2 2 2 2k−1 2k−2 k=? k=? ❊①❡r❝✐s❡ ✶✳✾✳✾

❈♦♥tr❛❝t t❤✐s s✉♠♠❛t✐♦♥✿ 1+

1 1 1 + + =? 3 9 27

❊①❡r❝✐s❡ ✶✳✾✳✶✵

❊①♣❛♥❞ t❤✐s s✉♠♠❛t✐♦♥✿

4 X k=0

(k/2) = ?

✶✳✾✳ ❚❤❡ s❡q✉❡♥❝❡ ♦❢ t❤❡ s✉♠s✿ ❞✐s♣❧❛❝❡♠❡♥t

✼✻

❊①❡r❝✐s❡ ✶✳✾✳✶✶

❘❡✇r✐t❡ ✉s✐♥❣ t❤❡ s✐❣♠❛ ♥♦t❛t✐♦♥✿ ✶✳ ✷✳ ✸✳ ✹✳ ✺✳ ✻✳ ✼✳

1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 .9 + .99 + .999 + .9999 1/2 − 1/4 + 1/8 − 1/16 1 + 1/2 + 1/3 + 1/4 + ... + 1/n 1 + 1/2 + 1/4 + 1/8 2 + 3 + 5 + 7 + 11 + 13 + 17 1 − 4 + 9 − 16 + 25

❊①❛♠♣❧❡ ✶✳✾✳✶✷✿ ❜✐♥♦♠✐❛❧s

■♥ t❤✐s ♥♦t❛t✐♦♥✱ t❤❡ ❇✐♥♦♠✐❛❧ ❚❤❡♦r❡♠ r❡❛❞s✿

m

(a + b) =

m   X m n=0

n

am−n bn .

❋♦r ❡①❛♠♣❧❡✿

4

(a + b) =

4   X 4 n=0

❚❤❡♥ ✇❡ ❤❛✈❡✿

n

a4−n bn .

          4 4−0 0 4 4−1 1 4 4−2 2 4 4−3 3 4 4−4 4 = a b + a b + a b + a b + a b 0 1 2 3 4 = 1 · a4 b0 + 4 · a3 b1 + 6 · a2 b2 + 4 · a1 b3 + 1 · a0 b4 = a4 + 4a3 b + 6a2 b2 + 4ab3 + b4 . ❚❤❡ ♥♦t❛t✐♦♥ ❛♣♣❧✐❡s t♦ ❛❧❧ s❡q✉❡♥❝❡s✱ ❜♦t❤ ✜♥✐t❡ ❛♥❞ ✐♥✜♥✐t❡✳ ❋♦r ✐♥✜♥✐t❡ s❡q✉❡♥❝❡s✱ r❡❝♦❣♥✐③❡❞ ❜② ✏✳✳✳✑ ❛t t❤❡ ❡♥❞✱ t❤❡ s✉♠ s❡q✉❡♥❝❡ ✐s ❛❧s♦ ❝❛❧❧❡❞ ✏♣❛rt✐❛❧ s✉♠s✑ ❛s ✇❡❧❧ ❛s ✏s❡r✐❡s✑ ✭t♦ ❜❡ ❞✐s❝✉ss❡❞ ✐♥ ❈❤❛♣t❡r ✸■❈✲✺✮✳ ❚❤✐s ✐s t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ s❡q✉❡♥❝❡ ♦❢ s✉♠s ✇r✐tt❡♥ ✇✐t❤ t❤❡ s✐❣♠❛ ♥♦t❛t✐♦♥✿ ❙❡q✉❡♥❝❡ ♦❢ s✉♠s

❛ s❡q✉❡♥❝❡✿ ✐ts s✉♠s✿

t❤❡ s❡q✉❡♥❝❡ ♦❢ s✉♠s✿

t❤❡ s✐❣♠❛ ♥♦t❛t✐♦♥✿

a1 ↓ a1 a1

a2 ↓ + a2 =

↓ s1 || 1 X ak k=1

a3 ↓ s2 s2

↓ s2 || 2 X ak k=1

+ a3 =

a4 ↓ s3 s3 ↓ s3 || 3 X ak k=1

+ a4 =

... ...

s4 s4 ↓ s4 || 4 X ak

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

k=1

❇❡❧♦✇ ✐s t❤❡ s✐♠♣❧❡st r❡s✉❧t ❛❜♦✉t t❤❡ s✉♠s✳ ■t ✐s st✐❧❧ ❝♦♥s✐❞❡r❛❜❧② ♠♦r❡ ❝❤❛❧❧❡♥❣✐♥❣ t❤❛♥ ♠♦st r❡s✉❧ts ❛❜♦✉t t❤❡ ❞✐✛❡r❡♥❝❡s t❤❛t ✇❡ s❛✇ ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥✳

✶✳✾✳

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

✼✼

❚❤❡♦r❡♠ ✶✳✾✳✶✸✿ ❙✉♠ ♦❢ ❆r✐t❤♠❡t✐❝ Pr♦❣r❡ss✐♦♥

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

m

❛♥❞ ❛

0

✐♥✐t✐❛❧ t❡r♠ ✐s

❛ ✭q✉❛❞r❛t✐❝✮ s❡q✉❡♥❝❡ ❣✐✈❡♥ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣✿ n X

(mk) =

k=1

n(n + 1) m. 2

❊①❡r❝✐s❡ ✶✳✾✳✶✹ Pr♦✈❡ t❤❡ t❤❡♦r❡♠✳ ❍✐♥t✿ ❯s❡ t❤❡ ❡①❛♠♣❧❡ ❛❜♦✉t r♦✉♥❞ r♦❜✐♥s ♣r❡s❡♥t❡❞ ❡❛r❧✐❡r ✐♥ t❤✐s ❝❤❛♣t❡r✳

❊①❛♠♣❧❡ ✶✳✾✳✶✺✿ ❢r♦♠ s❡q✉❡♥❝❡s t♦ s❡r✐❡s ❲❤❛t ❞♦❡s t❤❡ s✉♠

1 1 1 + + + ... 2 4 8

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

❖♥ t❤❡ ♦♥❡ ❤❛♥❞✱ ❛❞❞✐♥❣ t❤❡ ❛r❡❛s ♦❢ t❤❡s❡ sq✉❛r❡s ✇✐❧❧ ♥❡✈❡r ❣♦ ♦✈❡r

1 ✭t❤❡ ❛r❡❛ ♦❢ t❤❡ ❜✐❣ sq✉❛r❡✮✱ ❛♥❞✱

♦♥ t❤❡ ♦t❤❡r✱ t❤❡s❡ sq✉❛r❡s s❡❡♠ t♦ ❡①❤❛✉st t❤✐s sq✉❛r❡ ❡♥t✐r❡❧②✳ ❙♦✱ ❡✈❡♥ t❤❡

✐♥✜♥✐t❡

s✉♠ s♦♠❡t✐♠❡s

♠❛❦❡s s❡♥s❡ ✭t♦ ❜❡ ❝♦♥s✐❞❡r❡❞ ✐♥ ❈❤❛♣t❡r ✸■❈✲✺✮✳ ❖❢ ❝♦✉rs❡✱ t❤✐s ✐s ❛ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥✳

▲❡t✬s t❛❦❡ ❛ ❧♦♦❦ ❛t ❛

❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥

❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ ❝❤♦✐❝❡ ♦❢ r❛t✐♦

r

✇✐t❤

an = arn

✭❣r♦✇t❤ ♦r ❞❡❝❛②✮✿

✇✐t❤

a>0

❛♥❞

r > 0✳

❚❤❡r❡ ❛r❡ t✇♦ ❝❛s❡s✱

✶✳✾✳ ❚❤❡ s❡q✉❡♥❝❡ ♦❢ t❤❡ s✉♠s✿ ❞✐s♣❧❛❝❡♠❡♥t

✼✽

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

•

■t ✐s ✐♥❝r❡❛s✐♥❣ ✇✐t❤ s♣❡❡❞✐♥❣ ✉♣ ✇❤❡♥ ✐ts r❛t✐♦

•

■t ✐s ✐♥❝r❡❛s✐♥❣ ✇✐t❤ s❧♦✇✐♥❣ ❞♦✇♥ ✇❤❡♥

r

✐s ❧❛r❣❡r t❤❛♥

1✳

0 < r < 1✳

■t ❛❧s♦ r❡s❡♠❜❧❡s t❤❡ ♦r✐❣✐♥❛❧ s❡q✉❡♥❝❡✦ ❚❤❡♦r❡♠ ✶✳✾✳✶✻✿ ❙✉♠ ♦❢ ●❡♦♠❡tr✐❝ Pr♦❣r❡ss✐♦♥ ❚❤❡ s❡q✉❡♥❝❡ ♦❢ s✉♠s ♦❢ ❛ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥ ✇✐t❤ r❛t✐♦ ♣r♦❣r❡ss✐♦♥ ✇✐t❤ t❤❡ s❛♠❡ r❛t✐♦ ❛♥❞ ❛ ❝♦♥st❛♥t s❡q✉❡♥❝❡✳

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿

n X

r 6= 1

✐s ❛ ❣❡♦♠❡tr✐❝

ark = Arn + C ,

k=1

❢♦r s♦♠❡ r❡❛❧ ♥✉♠❜❡rs

A

❛♥❞

C✳

Pr♦♦❢✳

❇❡❧♦✇✱ ✇❡ ✉s❡ ❛ ❝❧❡✈❡r tr✐❝❦ t♦ ❣❡t r✐❞ ♦❢ ✏✳✳✳✑✳ ❲❡ ✇r✐t❡ t❤❡

nt❤

s✉♠

sn ✱

sn = ar0 +ar1 +ar2 +... +arn−1 +arn ❛♥❞ t❤❡♥ ♠✉❧t✐♣❧② ✐t ❜②

r✿


rsn = r ar0 +ar1 +ar2 +... +arn−1 +arn = ar1 + ar2 + ar3 +... + arn + arn+1 ◆♦✇ ✇❡ s✉❜tr❛❝t t❤❡s❡ t✇♦✿

= ar0

sn rsn

=

ar1

+ar1

+ar2

+... +arn−1

+arn

+ ar2

+ ar3

+... + arn

+ arn+1

sn − rsn = ar0 − ar1 +ar1 − ar2 +ar2 − ar3 +... +arn−1 − arn +arn − arn+1 = ar0

−arn+1

❲❡ ❝❛♥❝❡❧ t❤❡ t❡r♠s t❤❛t ❛♣♣❡❛r t✇✐❝❡ ✐♥ t❤❡ ❧❛st r♦✇ ❛♥❞ ✏✳✳✳✑ ✐s ❣♦♥❡✦ ❚❤❡r❡❢♦r❡✱

sn (1 − r) = a − arn+1 . ❚❤✉s✱ ✇❡ ❤❛✈❡ ❛♥ ❡①♣❧✐❝✐t ❢♦r♠✉❧❛ ❢♦r t❤❡

sn =

nt❤

t❡r♠ ♦❢ t❤❡ s✉♠✿

a a a (1 − rn+1 ) = − · rn+1 + . 1−r 1−r 1−r

✶✳✾✳

✼✾

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

❚❤❡ ❢♦r♠❡r t❡r♠ ✐s t❤❡ ❣❡♦♠❡tr✐❝ ♣❛rt✱ ❛♥❞ t❤❡ ❧❛tt❡r ✐s t❤❡ ❝♦♥st❛♥t✿ A=−

ar a , C= . 1−r 1−r

❊①❡r❝✐s❡ ✶✳✾✳✶✼

❋✐♥❞ t❤❡ ❡①♣❧✐❝✐t ❢♦r♠✉❧❛ ❢♦r t❤❡ s❡q✉❡♥❝❡ ♦❢ s✉♠s ♦❢ t❤❡ ❛❧t❡r♥❛t✐♥❣ s❡q✉❡♥❝❡ an = (−1)n ✳ ❊①❡r❝✐s❡ ✶✳✾✳✶✽

❯s❡ t❤❡ tr✐❝❦ t♦ ♣r♦✈❡ t❤❡ t❤❡♦r❡♠ ❛❜♦✉t ❛r✐t❤♠❡t✐❝ ♣r♦❣r❡ss✐♦♥s✳ ❲❛r♥✐♥❣✦ ❖✉r ❛❜✐❧✐t② t♦ ♣r♦❞✉❝❡ ❛♥ ❡①♣❧✐❝✐t ❢♦r♠✉❧❛ ❢♦r t❤❡

nt❤

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

t✐♦♥✱ ♥♦t ❛ r✉❧❡✳

❊①❛♠♣❧❡ ✶✳✾✳✶✾✿ s✉♠s ❛r❡ ❞✐s♣❧❛❝❡♠❡♥ts

❲❡ ❝❛♥ ✉s❡ ❝♦♠♣✉t❡rs t♦ s♣❡❡❞ ✉♣ t❤❡s❡ ❝♦♠♣✉t❛t✐♦♥s✳ ❋♦r ❡①❛♠♣❧❡✱ ♦♥❡ ♠❛② ❤❛✈❡ ❜❡❡♥ r❡❝♦r❞✐♥❣ ♦♥❡✬s ✈❡❧♦❝✐t✐❡s ❛♥❞ ♥♦✇ ❧♦♦❦✐♥❣ ❢♦r t❤❡ ❧♦❝❛t✐♦♥✳ ❚❤✐s ✐s ❛ ❢♦r♠✉❧❛ ❢♦r ❛ s♣r❡❛❞s❤❡❡t ✭t❤❡ ❧♦❝❛t✐♦♥s✮✿ ❂❘❬✲✶❪❈✰❘❈❬✲✶❪

❲❤❡t❤❡r t❤❡ s❡q✉❡♥❝❡ ❝♦♠❡s ❢r♦♠ ❛ ❢♦r♠✉❧❛ ♦r ✐t✬s ❥✉st ❛ ❧✐st ♦❢ ♥✉♠❜❡rs✱ t❤❡ ❢♦r♠✉❧❛ ❛♣♣❧✐❡s✿

❆s ❛ r❡s✉❧t✱ ❛ ❝✉r✈❡ ❤❛s ♣r♦❞✉❝❡❞ ❛ ♥❡✇ ❝✉r✈❡✿

❊①❡r❝✐s❡ ✶✳✾✳✷✵

❉❡s❝r✐❜❡ ✇❤❛t ❤❛s ❤❛♣♣❡♥❡❞ r❡❢❡rr✐♥❣ t♦✱ s❡♣❛r❛t❡❧②✱ t❤❡ ✜rst ❣r❛♣❤ ❛♥❞ t❤❡ s❡❝♦♥❞ ❣r❛♣❤✳ ❊①❡r❝✐s❡ ✶✳✾✳✷✶

■♠❛❣✐♥❡ t❤❛t t❤❡ ✜rst ❝♦❧✉♠♥ ♦❢ t❤❡ s♣r❡❛❞s❤❡❡t ✐s ✇❤❡r❡ ②♦✉ ❤❛✈❡ ❜❡❡♥ r❡❝♦r❞✐♥❣ ②♦✉r ♠♦♥t❤❧② ❞❡♣♦s✐t✴✇✐t❤❞r❛✇❛❧s ❛t ②♦✉r ❜❛♥❦ ❛❝❝♦✉♥t✳ ❲❤❛t ❞♦❡s t❤❡ s❡❝♦♥❞ ❝♦❧✉♠♥ r❡♣r❡s❡♥t❄ ❉❡s❝r✐❜❡ ✇❤❛t

✶✳✾✳

✽✵

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

❤❛s ❜❡❡♥ ❤❛♣♣❡♥✐♥❣ r❡❢❡rr✐♥❣ t♦✱ s❡♣❛r❛t❡❧②✱ t❤❡ ✜rst ❣r❛♣❤ ❛♥❞ t❤❡ s❡❝♦♥❞ ❣r❛♣❤✳ ❚❤✐s ✐s t❤❡ t✐♠❡ ❢♦r s♦♠❡ t❤❡♦r②✳ ❘❡❝❛❧❧ ❢r♦♠ t❤❡ ❧❛st s❡❝t✐♦♥ t❤✐s ♣❛✐r ♦❢ ♦❜✈✐♦✉s st❛t❡♠❡♥ts ❛❜♦✉t ♠♦t✐♦♥✿ ◮ ■ ❛♠ st❛♥❞✐♥❣ st✐❧❧ ■❋ ❆◆❉ ❖◆▲❨ ■❋ ♠② ✈❡❧♦❝✐t② ✐s ③❡r♦✳

■❢ t❤❡ ✈❡❧♦❝✐t② ✐s r❡♣r❡s❡♥t❡❞ ❜② ❛ s❡q✉❡♥❝❡✱ ✐ts s✉♠ ✐s t❤❡ ❧♦❝❛t✐♦♥✳ ❲❡ ❝❛♥ t❤❡♥ r❡st❛t❡ t❤❡ ❛❜♦✈❡ ♠❛t❤❡✲ ♠❛t✐❝❛❧❧②✳ ❚❤❡♦r❡♠ ✶✳✾✳✷✷✿ ❈♦♥st❛♥t ❙❡q✉❡♥❝❡ ❛s ❙✉♠ ❚❤❡ s❡q✉❡♥❝❡ ♦❢ s✉♠s ♦❢ ❛ s❡q✉❡♥❝❡ ✐s ❝♦♥st❛♥t ❤❛s ♦♥❧② ③❡r♦ ✈❛❧✉❡s st❛rt✐♥❣ ❢r♦♠ s♦♠❡ ✐♥❞❡①

■❋ ❆◆❉ ❖◆▲❨ ■❋ N✳

t❤❡ s❡q✉❡♥❝❡

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿

n X

ak

✐s ❝♦♥st❛♥t

k=m

⇐⇒ an = 0

❢♦r ❛❧❧

n≥N.

Pr♦♦❢✳

n X

k=m

ak = c ❢♦r ❛❧❧ n ⇐⇒ an+1 =

n+1 X

k=m

ak −

n X

k=m

ak = c − c = 0 ⇐⇒ an+1 = 0 .

❍❡r❡ ✐s ❛♥♦t❤❡r ❡q✉✐✈❛❧❡♥❝❡ st❛t❡♠❡♥ts ❛❜♦✉t ♠♦t✐♦♥✿ ◮ ■ ❛♠ ♠♦✈✐♥❣ ❢♦r✇❛r❞ ■❋ ❆◆❉ ❖◆▲❨ ■❋ ♠② ✈❡❧♦❝✐t② ✐s ♣♦s✐t✐✈❡✳

❲❡ ❝❛♥ r❡st❛t❡ t❤✐s ♠❛t❤❡♠❛t✐❝❛❧❧② ✉s✐♥❣ t❤❡ s✉♠s✳ ❚❤❡♦r❡♠ ✶✳✾✳✷✸✿ ▼♦♥♦t♦♥✐❝✐t② ♦❢ ❙✉♠ ❚❤❡ s❡q✉❡♥❝❡ ♦❢ s✉♠s ♦❢ ❛ s❡q✉❡♥❝❡ ✐s ✐♥❝r❡❛s✐♥❣

■❋ ❆◆❉ ❖◆▲❨ ■❋

♦❢ t❤❡ s❡q✉❡♥❝❡ ❛r❡ ♥♦♥✲♥❡❣❛t✐✈❡✳

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿

n X

k=m n X

ak

✐s ✐♥❝r❡❛s✐♥❣

⇐⇒ an ≥ 0 .

ak

✐s ❞❡❝r❡❛s✐♥❣

⇐⇒ an ≤ 0 .

k=m

Pr♦♦❢✳

n+1 X

k=m

ak ≥

n X

k=m

ak ❢♦r ❛❧❧ n ⇐⇒ an+1 =

n+1 X

k=m

ak −

◆♦✇ s✉♣♣♦s❡ ❥✉st ❧✐❦❡ ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥✱ t❤❛t t❤❡r❡ ❛r❡ t✇♦ r✉♥♥❡rs✿

n X

k=m

ak ≥ 0 .

t❤❡ t❡r♠s

✶✳✾✳

✽✶

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

❚❤❡♥✱ ✇❡ ❤❛✈❡✿ ◮ t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t✇♦ r✉♥♥❡rs ✐s♥✬t ❝❤❛♥❣✐♥❣ ■❋ ❆◆❉ ❖◆▲❨ ■❋ t❤❡② ❛r❡ r✉♥♥✐♥❣ ✇✐t❤

t❤❡ s❛♠❡ ✈❡❧♦❝✐t②✳

❲❡ r❡st❛t❡ t❤✐s ♠❛t❤❡♠❛t✐❝❛❧❧②✳ ❈♦r♦❧❧❛r② ✶✳✾✳✷✹✿ ❙✉❜tr❛❝t✐♥❣ ❙✉♠s ♦❢ ❙❡q✉❡♥❝❡s

❚❤❡ s❡q✉❡♥❝❡s ♦❢ s✉♠s ♦❢ t✇♦ s❡q✉❡♥❝❡s ❞✐✛❡r ❜② ❛ ❝♦♥st❛♥t ■❋ ❆◆❉ ❖◆▲❨ ■❋ t❤❡ s❡q✉❡♥❝❡s ❛r❡ ❡q✉❛❧ st❛rt✐♥❣ ✇✐t❤ s♦♠❡ t❡r♠ N ✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿ n X

k=m

ak −

n X

k=m

bk ✐s ❝♦♥st❛♥t ⇐⇒ an = bn ❢♦r ❛❧❧ n ≥ N .

Pr♦♦❢✳

❚❤❡ ❝♦r♦❧❧❛r② ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ❈♦♥st❛♥t

❙❡q✉❡♥❝❡ ❛s ❙✉♠

❛❜♦✈❡✳

❊①❛♠♣❧❡ ✶✳✾✳✷✺✿ s❤✐❢t ♦❢ s❡q✉❡♥❝❡

❲❡ ❤❛✈❡ ❜❡❧♦✇ t✇♦ ❞✐✛❡r❡♥t s❡q✉❡♥❝❡s an ❛♥❞ bn t❤❛t ❜❡❝♦♠❡ ✐❞❡♥t✐❝❛❧ ❛❢t❡r 3 t❡r♠s✿

❚❤❡ r❡s✉❧t ✐s t❤❛t t❤❡ s✉♠ ♦❢ t❤❡ ❧❛tt❡r s❡q✉❡♥❝❡ ✐s ❥✉st ❛ ✈❡rt✐❝❛❧ s❤✐❢t ♦❢ t❤❡ s✉♠ ♦❢ t❤❡ ❢♦r♠❡r✿

❚♦ st❛t❡ t❤✐s ❛❧❣❡❜r❛✐❝❛❧❧②✱ ✇❡ ❤❛✈❡ ❢♦r ❡❛❝❤ n✿ n X

k=m

ak =

n X

bk + C ,

k=m

✇❤❡r❡ C ✐s s♦♠❡ ♥✉♠❜❡r✳ ❚❤❡ ♦✉t❝♦♠❡ ✐s t❤❡ s❛♠❡ ✇❤❡♥ t❤❡ t✇♦ s❡q✉❡♥❝❡s an ❛♥❞ bn ❛r❡ ✐❞❡♥t✐❝❛❧ ❜✉t t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡✐r s❡q✉❡♥❝❡s ♦❢ s✉♠s st❛rts ❛t ❞✐✛❡r❡♥t ♣♦✐♥ts✿

✶✳✾✳ ❚❤❡ s❡q✉❡♥❝❡ ♦❢ t❤❡ s✉♠s✿ ❞✐s♣❧❛❝❡♠❡♥t

✽✷

❊①❡r❝✐s❡ ✶✳✾✳✷✻ ❲❤❛t ✐s t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ ♥✉♠❜❡r

C❄

❲❡ ❝❛♥ ✉s❡ t❤❡ t❤❡♦r❡♠s t♦ ✇❛t❝❤ ❢♦r t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ t✇♦ r✉♥♥❡rs✿

◮

t❤❡ ❞✐st❛♥❝❡ ❢r♦♠ ♦♥❡ ♦❢ t❤❡ t✇♦ r✉♥♥❡rs t♦ t❤❡ ♦t❤❡r ✐s ✐♥❝r❡❛s✐♥❣

■❋ ❆◆❉ ❖◆▲❨ ■❋

t❤❡

❢♦r♠❡r✬s ✈❡❧♦❝✐t② ✐s ❤✐❣❤❡r✳ ❲❡ ❝❛♥ r❡st❛t❡ t❤✐s ♠❛t❤❡♠❛t✐❝❛❧❧② ✉s✐♥❣ t❤❡ s✉♠s✳

❈♦r♦❧❧❛r② ✶✳✾✳✷✼✿ ❙✉❜tr❛❝t✐♥❣ ❙✉♠s✿ ▼♦♥♦t♦♥✐❝✐t②

❚❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ s❡q✉❡♥❝❡s ♦❢ s✉♠s ♦❢ t✇♦ s❡q✉❡♥❝❡s ✐s ✐♥❝r❡❛s✐♥❣ ■❋ ❆◆❉ ❖◆▲❨ ■❋ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ t❡r♠s ♦❢ t❤❡ ❢♦r♠❡r ❛r❡ ❧❛r❣❡r t❤❛♥ ♦r ❡q✉❛❧ t♦ t❤♦s❡ ♦❢ t❤❡ ❧❛tt❡r✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿ n X

k=m n X

k=m

ak − ak −

n X

k=m n X

bk

✐s ✐♥❝r❡❛s✐♥❣

⇐⇒ an ≥ bn .

bk

✐s ❞❡❝r❡❛s✐♥❣

⇐⇒ an ≤ bn .

k=m

Pr♦♦❢✳ ❚❤❡ ❝♦r♦❧❧❛r② ❢♦❧❧♦✇s ❢r♦♠ ▼♦♥♦t♦♥✐❝✐t② ♦❢ ❙✉♠ ❛❜♦✈❡✳

❍❡r❡ ✐s ❛♥♦t❤❡r ✇❛② t♦ ❧♦♦❦ ❛t t❤❡ st❛t❡♠❡♥t t❤❡ ❢❛st❡r ❝♦✈❡rs t❤❡ ❧♦♥❣❡r ❞✐st❛♥❝❡✳ t❤❡ ✈❛❧✉❡s ♦❢ t✇♦ s✉♠s✳ ❈♦♥s✐❞❡r t❤✐s s✐♠♣❧❡ ❛❧❣❡❜r❛✿

a ≤ b A ≤ B a+A ≤ b+B ❚❤❡ r✉❧❡ ❛♣♣❧✐❡s ❡✈❡♥ ✐❢ ✇❡ ❤❛✈❡ ♠♦r❡ t❤❛♥ ❥✉st t✇♦ t❡r♠s✿

am ≤ b m am+1 ≤ bm+1 ✳ ✳ ✳

✳ ✳ ✳

✳ ✳ ✳

aq ≤ bq am + ... + aq ≤ bm + ... + bq ❚❤❡ s✉♠♠❛t✐♦♥ ✐s ✐❧❧✉str❛t❡❞ ❜❡❧♦✇✿

■t ✐s ❛❜♦✉t ❝♦♠♣❛r✐♥❣

✶✳✾✳

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

❊①❛♠♣❧❡ ✶✳✾✳✷✽✿ t❤r❡❡ r✉♥♥❡rs✱ ❝♦♥t✐♥✉❡❞

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

■t✬s ❡❛s② t♦ ❞❡s❝r✐❜❡ ❤♦✇ t❤❡② ❛r❡ ♠♦✈✐♥❣✿ • A st❛rts ❢❛st ❛♥❞ t❤❡ s❧♦✇s ❞♦✇♥✳ • B ♠❛✐♥t❛✐♥s t❤❡ s❛♠❡ s♣❡❡❞✳ • C st❛rts ❧❛t❡ ❛♥❞ t❤❡♥ r✉♥s ❢❛st✳ ❇✉t ✇❤❡r❡ ❛r❡ t❤❡②✱ ❛t ❡✈❡r② ♠♦♠❡♥t❄ ❚❤❡r❡ ❛r❡ s❡✈❡r❛❧ ♣♦ss✐❜❧❡ ❛♥s✇❡rs✿

❲❤✐❝❤ ♦♥❡ ✐s t❤❡ r✐❣❤t ♦♥❡ ❞❡♣❡♥❞s ♦♥ t❤❡ st❛rt✐♥❣ ♣♦✐♥t✳ ❖❢ ❝♦✉rs❡✱ ❛ s✐♠♣❧❡ ❡①❛♠✐♥❛t✐♦♥ ♦❢ t❤❡ ✜rst ❣r❛♣❤ ❞♦❡s♥✬t ♣r♦✈❡ t❤❛t t❤❡ t❤r❡❡ r✉♥♥❡rs ✇✐❧❧ ❛rr✐✈❡ ❛t t❤❡ ✜♥✐s❤ ❧✐♥❡ ❛t t❤❡ s❛♠❡ t✐♠❡✳ ❋✉rt❤❡r♠♦r❡✱ ✐❢ t❤❡ r❡q✉✐r❡♠❡♥t t❤❛t t❤❡② ❛❧❧ st❛rt ❛t t❤❡ s❛♠❡ ❧♦❝❛t✐♦♥ ✐s ❧✐❢t❡❞✱ t❤❡ r❡s✉❧t ✇✐❧❧ ❜❡ ❞✐✛❡r❡♥t✱ ❢♦r ❡①❛♠♣❧❡✿

❊①❡r❝✐s❡ ✶✳✾✳✷✾

❙✉❣❣❡st ♦t❤❡r ❣r❛♣❤s t❤❛t ♠❛t❝❤ t❤❡ ❞❡s❝r✐♣t✐♦♥ ❛❜♦✈❡✳

✽✸

✶✳✾✳ ❚❤❡ s❡q✉❡♥❝❡ ♦❢ t❤❡ s✉♠s✿ ❞✐s♣❧❛❝❡♠❡♥t

✽✹

❊①❡r❝✐s❡ ✶✳✾✳✸✵

P❧♦t t❤❡ ❧♦❝❛t✐♦♥ ❛♥❞ t❤❡ ✈❡❧♦❝✐t② ❢♦r t❤❡ ❢♦❧❧♦✇✐♥❣ tr✐♣✿ ✏■ ❞r♦✈❡ s❧♦✇❧②✱ ❣r❛❞✉❛❧❧② s♣❡❡❞ ✉♣✱ st♦♣♣❡❞ ❢♦r ❛ ✈❡r② s❤♦rt ♠♦♠❡♥t✱ ❛♥❞ st❛rt❡❞ ❜✉t ✐♥ t❤❡ ♦♣♣♦s✐t❡ ❞✐r❡❝t✐♦♥✱ q✉✐❝❦❧② ❛❝❝❡❧❡r❛t❡❞✱ ❛♥❞ ❢r♦♠ t❤❛t ♣♦✐♥t ♠❛✐♥t❛✐♥❡❞ t❤❡ s♣❡❡❞✳✑ ▼❛❦❡ ✉♣ ②♦✉r ♦✇♥ st♦r② ❛♥❞ r❡♣❡❛t t❤❡ t❛s❦✳

❊①❡r❝✐s❡ ✶✳✾✳✸✶

❉r❛✇ ❛ ❝✉r✈❡ ♦♥ ❛ ♣✐❡❝❡ ♦❢ ♣❛♣❡r✱ ✐♠❛❣✐♥❡ t❤❛t ✐t r❡♣r❡s❡♥ts ②♦✉r ✈❡❧♦❝✐t②✱ ❛♥❞ t❤❡♥ s❦❡t❝❤ ✇❤❛t ②♦✉r ❧♦❝❛t✐♦♥s ✇♦✉❧❞ ❧♦♦❦ ❧✐❦❡✳ ❘❡♣❡❛t✳

❍❡r❡ ✐s ❛♥♦t❤❡r tr✐✈✐❛❧ st❛t❡♠❡♥t ❛❜♦✉t ♠♦t✐♦♥ ✿ ❞✐st❛♥❝❡ ❝♦✈❡r❡❞ ❞✉r✐♥❣ t❤❡ ✶st ❤♦✉r

+ =

❞✐st❛♥❝❡ ❝♦✈❡r❡❞ ❞✉r✐♥❣ t❤❡ ✷♥❞ ❤♦✉r ❞✐st❛♥❝❡ ❞✉r✐♥❣ t❤❡ t✇♦ ❤♦✉rs

❚❤❡ st❛t❡♠❡♥t ✐s ❛❜♦✉t t❤❡ ❢❛❝t t❤❛t ✇❤❡♥ ❛❞❞✐♥❣✱ ✇❡ ❝❛♥ ❝❤❛♥❣❡ t❤❡ ♦r❞❡r ♦❢ t❡r♠s ❢r❡❡❧②❀ t❤✐s ✐s ❝❛❧❧❡❞ t❤❡ ❆ss♦❝✐❛t✐✈✐t② Pr♦♣❡rt② ♦❢ ❛❞❞✐t✐♦♥✳ ❆t ✐ts s✐♠♣❧❡st✱ ✐t ❛❧❧♦✇s ✉s t♦ r❡♠♦✈❡ t❤❡ ♣❛r❡♥t❤❡s❡s✿

(am + am+1 + ... + aq−1 + aq ) + (aq+1 + aq+2 + ... + ar−1 + ar ) = am + am+1 + ... + aq−1 + aq + aq+1 + aq+2 + ... + ar−1 + ar = am + am+1 + ... +ar−1 + ar . ❚❤❡ t❤r❡❡ s✉♠s ❛r❡ s❤♦✇♥ ❜❡❧♦✇✿

❆♥ ❛❜❜r❡✈✐❛t❡❞ ✈❡rs✐♦♥ ♦❢ t❤✐s ✐❞❡♥t✐t② ✐s ❛s ❢♦❧❧♦✇s✳ ❚❤❡♦r❡♠ ✶✳✾✳✸✷✿ ❆❞❞✐t✐✈✐t② ❢♦r ❙✉♠s ❚❤❡ s✉♠ ♦❢ t❤❡ s✉♠s ♦❢ t✇♦ ❝♦♥s❡❝✉t✐✈❡ s❡❣♠❡♥ts ♦❢ ❛ s❡q✉❡♥❝❡ ✐s t❤❡ s✉♠ ♦❢ t❤❡ ❝♦♠❜✐♥❡❞ s❡❣♠❡♥t✳

■♥ ♦t❤❡r ✇♦r❞s✱ ❢♦r ❛♥② s❡q✉❡♥❝❡

q X

k=m

an

ak +

❛♥❞ ❢♦r ❛♥②

r X

k=q+1

ak =

m, q, r ✇✐t❤ m ≤ q ≤ r✱ ✇❡ ❤❛✈❡✿

r X

ak

k=m

❊①❛♠♣❧❡ ✶✳✾✳✸✸✿ ❘✐❡♠❛♥♥ s✉♠s ♦❢ s❡q✉❡♥❝❡s

❍♦✇ ❞♦ ✇❡ ❞❡❛❧ ✇✐t❤ ♠♦t✐♦♥ ✇❤❡♥ t❤❡ t✐♠❡ ♠♦♠❡♥ts ❛r❡♥✬t ✐♥t❡❣❡rs❄ ❲❤❛t ✐s t❤❡ ❞✐s♣❧❛❝❡♠❡♥t t❤❡♥❄ ❙✉♣♣♦s❡

xn

✐s t❤❡ s❡q✉❡♥❝❡ ♦❢ ❧♦❝❛t✐♦♥s ❛♥❞

vn

t❤❡ s❡q✉❡♥❝❡ ♦❢ ✈❡❧♦❝✐t✐❡s✿

✶✳✶✵✳

❙✉♠s ♦❢ ❞✐✛❡r❡♥❝❡s ❛♥❞ ❞✐✛❡r❡♥❝❡s ♦❢ s✉♠s✿ ♠♦t✐♦♥

❚❤❡♥ t❤❡✐r

❘✐❡♠❛♥♥ s✉♠

❛♥❞ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢

✽✺

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

xn ✿

n X

vn

vn ∆xn .

k=1

❲❡ ❦♥♦✇ ❢r♦♠ t❤❡ ❧❛st s❡❝t✐♦♥ t❤❛t ✐❢

yn

✐s t❤❡ ♣♦s✐t✐♦♥✱ t❤❡♥ t❤❡ ✈❡❧♦❝✐t② ✐s

vn = ∆yn /∆xn ✳

❚❤❡r❡✲

❢♦r❡✱ t❤❡ ❘✐❡♠❛♥♥ s✉♠ ♦❢ t❤❡ s❡q✉❡♥❝❡ ♦❢ t❤❡ ✈❡❧♦❝✐t② ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ s❡q✉❡♥❝❡ ♦❢ t✐♠❡ ✐s t❤❡ ❞✐s♣❧❛❝❡♠❡♥t✳ ❖❢ ❝♦✉rs❡✱ t❤✐s ✐s ❛❧s♦ t❤❡ t♦t❛❧ ❛r❡❛ ♦❢ t❤❡ r❡❝t❛♥❣❧❡s✳

❊①❛♠♣❧❡ ✶✳✾✳✸✹✿ s✉♠ ♦❢ r❛♥❞♦♠ s❡q✉❡♥❝❡ ❲❡ t❛❦❡ ❛ ❝♦♠♣✉t❡r✲❣❡♥❡r❛t❡❞ r❛♥❞♦♠ s❡q✉❡♥❝❡ ✭t❤❡ ✈❛❧✉❡s s♣r❡❛❞ ❜❡t✇❡❡♥

−1

❛♥❞

1✮✿

❚❤❡♥ ✐ts s✉♠ ✐s ❛❧s♦ r❛♥❞♦♠ ❜✉t ♠✐❣❤t ❡①❤✐❜✐t ❛♣♣❛r❡♥t ♣❡r✐♦❞s ♦❢ ❣r♦✇t❤ ❛♥❞ ❞❡❝❧✐♥❡ ✭✏str❡❛❦s✑✮✳ ❚❤❡ ♥✉♠❜❡rs ✐♥ t❤❡ ♦r✐❣✐♥❛❧ s❡q✉❡♥❝❡ ❝❛♥ r❡♣r❡s❡♥t t❤❡ ♦✉t❝♦♠❡ ♦❢ ♣❧❛②✐♥❣ ❛ ❤❛♥❞ ♦❢ ❝❛r❞s ✇✐t❤ t❤❡ ♣♦ss✐❜✐❧✐t② ♦❢ ✇✐♥♥✐♥❣ ♦r ❧♦s✐♥❣ ❛♥ ❛♠♦✉♥t ✇✐t❤✐♥

$1✳

❚❤❡♥ t❤❡ s❡q✉❡♥❝❡ ♦❢ s✉♠s r❡♣r❡s❡♥ts t❤❡ ❛♠♦✉♥t

t❤❡ ♣❡rs♦♥ ❤❛s ❛t ❡✈❡r② ♠♦♠❡♥t ♦❢ t✐♠❡✳

✶✳✶✵✳ ❙✉♠s ♦❢ ❞✐✛❡r❡♥❝❡s ❛♥❞ ❞✐✛❡r❡♥❝❡s ♦❢ s✉♠s✿ ♠♦t✐♦♥ ■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ st❛rt ♦♥ t❤❡ ♣❛t❤ ♦❢ ❞❡✈❡❧♦♣♠❡♥t ♦❢ ❛♥ ✐❞❡❛ t❤❛t ❝✉❧♠✐♥❛t❡s ✇✐t❤ t❤❡ ❝♦r♥❡rst♦♥❡ r❡s✉❧t ♦❢ ❝❛❧❝✉❧✉s✱

t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s

❲❡ ❦♥♦✇ t❤❛t

✭❈❤❛♣t❡r ✸■❈✲✶✮✳

❛❞❞✐t✐♦♥ ❛♥❞ s✉❜tr❛❝t✐♦♥ ✉♥❞♦ ❡❛❝❤ ♦t❤❡r ❀ ✐t ♠❛❦❡s s❡♥s❡ t❤❡♥ t❤❛t t❤❡ ♦♣❡r❛t✐♦♥s ♦❢ ♠❛❦✐♥❣

t❤❡ s❡q✉❡♥❝❡ ♦❢ ❞✐✛❡r❡♥❝❡s ❛♥❞ ♠❛❦✐♥❣ t❤❡ s❡q✉❡♥❝❡ ♦❢ s✉♠s ✇✐❧❧ ❝❛♥❝❡❧ ❡❛❝❤ ♦t❤❡r t♦♦✦

❊①❛♠♣❧❡ ✶✳✶✵✳✶✿ ❜r♦❦❡♥ ♦❞♦♠❡t❡r ✕ ❜r♦❦❡♥ s♣❡❡❞♦♠❡t❡r ❲❡ ❦♥♦✇ ❤♦✇ t♦ ❣❡t t❤❡ ✈❡❧♦❝✐t② ❢r♦♠ t❤❡ ❧♦❝❛t✐♦♥✱ ❛♥❞ t❤❡ ❧♦❝❛t✐♦♥ ❢r♦♠ t❤❡ ✈❡❧♦❝✐t②✳ ❲❡ ❡①♣❡❝t t❤❛t ❡①❡❝✉t✐♥❣ t❤❡s❡ t✇♦ ♦♣❡r❛t✐♦♥s ❝♦♥s❡❝✉t✐✈❡❧② s❤♦✉❧❞ ❜r✐♥❣ ✉s ❜❛❝❦ ✇❤❡r❡ ✇❡ st❛rt❡❞✳ ▲❡t✬s t❛❦❡ ❛♥♦t❤❡r ❧♦♦❦ ❛t t❤❡ ❡①❛♠♣❧❡ ♦❢

t✇♦

❝♦♠♣✉t❛t✐♦♥s ❛❜♦✉t ♠♦t✐♦♥ ✕ ❛ ❜r♦❦❡♥ ♦❞♦♠❡t❡r ❛♥❞

✶✳✶✵✳

❙✉♠s ♦❢ ❞✐✛❡r❡♥❝❡s ❛♥❞ ❞✐✛❡r❡♥❝❡s ♦❢ s✉♠s✿ ♠♦t✐♦♥

❛ ❜r♦❦❡♥ s♣❡❡❞♦♠❡t❡r ✕ ♣r❡s❡♥t❡❞ ✐♥ t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤✐s ❝❤❛♣t❡r✳ ❚❤❡ t❡r♠✐♥♦❧♦❣② ❤❛s ♥♦✇ ❜❡❡♥ ❞❡✈❡❧♦♣❡❞✿ ❊✈❡r② t✐♠❡ ✇❡ s♣❡❛❦ ♦❢ ❛ s❡q✉❡♥❝❡✱ ✇❡ ❛❧s♦ s♣❡❛❦ ♦❢ t❤❡ s❡q✉❡♥❝❡ ♦❢ ✐ts ❞✐✛❡r❡♥❝❡s ❛♥❞ t❤❡ s❡q✉❡♥❝❡ ♦❢ ✐ts s✉♠s✳ ■♥ t❤❡ ✜rst ❞✐❛❣r❛♠✱ ♦♥❡ ✜rst t❛❦❡s t❤❡ ✈❡❧♦❝✐t② ❞❛t❛ ❛♥❞ ❛❝q✉✐r❡s t❤❡ ❞✐s♣❧❛❝❡♠❡♥ts ✈✐❛ t❤❡ s✉♠s✱ t❤❡♥ s♦♠❡♦♥❡ ❡❧s❡ t❛❦❡s t❤✐s ❞✐s♣❧❛❝❡♠❡♥t ❞❛t❛ ❛♥❞ ❛❝q✉✐r❡s t❤❡ ✈❡❧♦❝✐t✐❡s ❜② ✉s✐♥❣ t❤❡ ❞✐✛❡r❡♥❝❡s✿

❲❡ ❛r❡ ❜❛❝❦ t♦ t❤❡ ♦r✐❣✐♥❛❧ s❡q✉❡♥❝❡✳ ■♥ t❤❡ s❡❝♦♥❞ ❞✐❛❣r❛♠✱ ♦♥❡ ✜rst t❛❦❡s t❤❡ ❧♦❝❛t✐♦♥ ❞❛t❛ ❛♥❞ ❛❝q✉✐r❡s t❤❡ ✈❡❧♦❝✐t✐❡s ✈✐❛ t❤❡ ❞✐✛❡r❡♥❝❡s✱ t❤❡♥ s♦♠❡♦♥❡ ❡❧s❡ t❛❦❡s t❤✐s ✈❡❧♦❝✐t② ❞❛t❛ ❛♥❞ ❛❝q✉✐r❡s t❤❡ ❧♦❝❛t✐♦♥s ❜② ✉s✐♥❣ t❤❡ s✉♠s✿

❲❡ ❛r❡ ❜❛❝❦ t♦ t❤❡ ♦r✐❣✐♥❛❧ s❡q✉❡♥❝❡ ✭♣r♦✈✐❞❡❞ ✇❡ st❛rt ❛t t❤❡ s❛♠❡ ✐♥✐t✐❛❧ ✈❛❧✉❡✮✳ ❊①❛♠♣❧❡ ✶✳✶✵✳✷✿ s❡q✉❡♥❝❡s ❣✐✈❡♥ ❜② ❧✐sts

❇❡❧♦✇ ✇❡ ❤❛✈❡ ❛ s❡q✉❡♥❝❡ ❣✐✈❡♥ ❜② ❛ ❧✐st✳ ❲❡ ❝♦♠♣✉t❡ ✐ts s❡q✉❡♥❝❡ s✉♠s ❛♥❞ t❤❡♥ ❝♦♠♣✉t❡ t❤❡

✽✻

✶✳✶✵✳

❙✉♠s ♦❢ ❞✐✛❡r❡♥❝❡s ❛♥❞ ❞✐✛❡r❡♥❝❡s ♦❢ s✉♠s✿ ♠♦t✐♦♥

✽✼

s❡q✉❡♥❝❡ ♦❢ ❞✐✛❡r❡♥❝❡s ♦❢ t❤❡ r❡s✉❧t✿ ❛ s❡q✉❡♥❝❡✿ ✐ts s✉♠s✿

3 ↓ 3 3 +

1 ↓ 1

= 4 4 +

↓ t❤❡ s❡q✉❡♥❝❡ ♦❢ s✉♠s✿ 3 t❤❡ ❞✐✛❡r❡♥❝❡s✿

−2 ↓

0 ↓ 0

= 4 ... 4 + (−2)

↓ 4 ց

❛ ♥❡✇ s❡q✉❡♥❝❡✿

4−3 || 1

ւ

↓ 4 ց

4−4 || 0

ւ

ց

2−4 || −2

... ...

= 2 2 ↓ 2 ւ

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

❲❡ ❛r❡ ❜❛❝❦ t♦ t❤❡ ♦r✐❣✐♥❛❧ s❡q✉❡♥❝❡✦ ❊①❡r❝✐s❡ ✶✳✶✵✳✸

❲❤❛t ❤❛♣♣❡♥❡❞ t♦ t❤❡ ✈❡r② ✜rst t❡r♠❄ ❊①❡r❝✐s❡ ✶✳✶✵✳✹

❙t❛rt ✇✐t❤ t❤❡ s❡q✉❡♥❝❡ ✐♥ t❤❡ ❧❛st ❡①❛♠♣❧❡ ❛♥❞ ✉s❡ t❤❡ ❞✐❛❣r❛♠s t♦ s❤♦✇ t❤❛t t❤❡ s✉♠s ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡s ❣✐✈❡ ✉s t❤❡ ♦r✐❣✐♥❛❧ s❡q✉❡♥❝❡✳ ❊①❛♠♣❧❡ ✶✳✶✵✳✺✿ s❡q✉❡♥❝❡s ❣✐✈❡♥ ❜② ❣r❛♣❤s

❏✉st ❝♦♠♣❛r✐♥❣ t❤❡ ✐❧❧✉str❛t✐♦♥s ❛❜♦✈❡ ❞❡♠♦♥str❛t❡s t❤❛t t❤❡ t✇♦ ♦♣❡r❛t✐♦♥s ✕ t❤❡ ❞✐✛❡r❡♥❝❡ ❛♥❞ t❤❡ s✉♠ ✕ ✉♥❞♦ t❤❡ ❡✛❡❝t ♦❢ ❡❛❝❤ ♦t❤❡r✳ ❚❤❡ t✇♦ ♦♣❡r❛t✐♦♥s ❛r❡ s❤♦✇♥ t♦❣❡t❤❡r ❜❡❧♦✇✿

❆s ②♦✉ ❝❛♥ s❡❡ ✐♥ t❤❡ ♣✐❝t✉r❡✱ t❤❡ s✉♠ ✭❧❡❢t t♦ r✐❣❤t✮ st❛❝❦s ✉♣ t❤❡ t❡r♠s ♦❢ t❤❡ s❡q✉❡♥❝❡ ♦♥ t♦♣ ♦❢ ❡❛❝❤ ♦t❤❡r✱ ✇❤✐❧❡ t❤❡ ❞✐✛❡r❡♥❝❡ ✭r✐❣❤t t♦ ❧❡❢t✮ t❛❦❡s t❤❡s❡ ❛♣❛rt✳ ▲❡t✬s t❛❦❡ ❝❛r❡ ♦❢ t❤❡

❛❧❣❡❜r❛✳

❚❤❡s❡ ❛r❡ t❤❡ t✇♦ ❢❛❝ts ✇❡ ✇✐❧❧ ❜❡ ✉s✐♥❣✿ ✶✳ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ s❡q✉❡♥❝❡✱ an ✳ ❲❡ ❝♦♠♣✉t❡ ✐ts

❞✐✛❡r❡♥❝❡✱ ❛ ♥❡✇ s❡q✉❡♥❝❡✿

bn+1 = an+1 − an . ✷✳ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ s❡q✉❡♥❝❡✱ ck ✳ ❲❡ ❝♦♠♣✉t❡ ✐ts

dn =

s✉♠✱ ❛ ♥❡✇ s❡q✉❡♥❝❡✿ n X

ck ,

k=1

♦r✱ r❡❝✉rs✐✈❡❧②✿

dn+1 = dn + cn+1 .

✶✳✶✵✳

❙✉♠s ♦❢ ❞✐✛❡r❡♥❝❡s ❛♥❞ ❞✐✛❡r❡♥❝❡s ♦❢ s✉♠s✿ ♠♦t✐♦♥

✽✽

❲❡ ✉s❡ t❤✐s s❡t✉♣ t♦ ❛♥s✇❡r t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ q✉❡st✐♦♥s✳ ❚❤❡ ✜rst q✉❡st✐♦♥ ✇❡ ✇♦✉❧❞ ❧✐❦❡ t♦ ❛♥s✇❡r ✐s✿

◮

❲❤❛t ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ s✉♠ ❄

❲❡ st❛rt ✇✐t❤ cn ✳ ❚❤❡♥✱ ✇❡ ❤❛✈❡ ❢r♦♠ ✭✷✮ ❛♥❞ ✭✶✮✱ r❡s♣❡❝t✐✈❡❧②✿

dn+1 = dn + cn+1 ❛♥❞ bn+1 = dn+1 − dn . ❲❡ s✉❜st✐t✉t❡ t❤❡ ✜rst ❢♦r♠✉❧❛ ✐♥t♦ t❤❡ s❡❝♦♥❞ ✭❛♥❞ t❤❡♥ ❝❛♥❝❡❧✮✿

bn+1 = dn+1 − dn = (dn + cn+1 ) − dn = cn+1 . ❆s ✇❡ ❝❛♥ s❡❡✱ t❤❡ ❛♥s✇❡r ✐s✿

◮

❚❤❡ ♦r✐❣✐♥❛❧ s❡q✉❡♥❝❡✳

❚❤❡ s❡❝♦♥❞ q✉❡st✐♦♥ ✇❡ ✇♦✉❧❞ ❧✐❦❡ t♦ ❛♥s✇❡r ✐s✿

◮

❲❤❛t ✐s t❤❡ s✉♠ ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ ❄

❲❡ st❛rt ✇✐t❤ an ✳ ❚❤❡♥✱ ✇❡ ❤❛✈❡ ❢r♦♠ ✭✶✮ ❛♥❞ ✭✷✮✱ r❡s♣❡❝t✐✈❡❧②✿

bk = ak − ak−1 ❛♥❞ dn =

n X

bk .

k=1

❲❡ s✉❜st✐t✉t❡ t❤❡ ✜rst ❢♦r♠✉❧❛ ✐♥t♦ t❤❡ s❡❝♦♥❞ ✭❛♥❞ t❤❡♥ ❝❛♥❝❡❧✮✿

dn =

n X

bk =

k=1

n X k=1

(ak − ak−1 ) = (a2 − a1 ) + (a3 − a2 ) + (a4 − a3 ) + ... + (an − an−1 ) = −a1 + an .

❆s ✇❡ ❝❛♥ s❡❡✱ t❤❡ ❛♥s✇❡r ✐s✿

◮

❚❤❡ ♦r✐❣✐♥❛❧ s❡q✉❡♥❝❡ ♣❧✉s ❛ ♥✉♠❜❡r✳

❲❡ s✉♠♠❛r✐③❡ t❤❡s❡ r❡s✉❧ts ✐♥ t❤❡ ❢♦r♠ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ ❢❛r✲r❡❛❝❤✐♥❣ t❤❡♦r❡♠s✳ ❚❤❡♦r❡♠ ✶✳✶✵✳✻✿ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s ♦❢ ❙❡q✉❡♥❝❡s ■

❚❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ s✉♠ ♦❢ ❛ s❡q✉❡♥❝❡ ✐s t❤❛t s❡q✉❡♥❝❡❀ ✐✳❡✳✱ ❢♦r ❛❧❧ n✱ ✇❡ ❤❛✈❡✿ ∆

X n k=1

❚❤❡ t✇♦ ♦♣❡r❛t✐♦♥s

ak



= an

❝❛♥❝❡❧ ❡❛❝❤ ♦t❤❡r✦ ❚❤❡♦r❡♠ ✶✳✶✵✳✼✿ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s ♦❢ ❙❡q✉❡♥❝❡s ■■

❚❤❡ s✉♠ ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ ❛ s❡q✉❡♥❝❡ ✐s t❤❛t s❡q✉❡♥❝❡ ♣❧✉s ❛ ❝♦♥st❛♥t ♥✉♠❜❡r❀ ✐✳❡✳✱ ❢♦r ❛❧❧ n✱ ✇❡ ❤❛✈❡✿ n X k=1

❚❤❡ t✇♦ ♦♣❡r❛t✐♦♥s ✕ ❛❧♠♦st ✕ ❝❛♥❝❡❧ ❡❛❝❤ ♦t❤❡r✱ ❛❣❛✐♥✦


∆bk = bn + C

✶✳✶✵✳ ❙✉♠s ♦❢ ❞✐✛❡r❡♥❝❡s ❛♥❞ ❞✐✛❡r❡♥❝❡s ♦❢ s✉♠s✿ ♠♦t✐♦♥ ❊①❛♠♣❧❡ ✶✳✶✵✳✽✿ ❢✉♥❞❛♠❡♥t❛❧ t❤❡♦r❡♠s✱ ❝♦♠♣✉t❡❞

❋♦r ❧❛r❣❡r s❡ts ♦❢ ❞❛t❛✱ ✇❡ ✉s❡ ❛ s♣r❡❛❞s❤❡❡t✳ ❘❡❝❛❧❧ t❤❡ ❢♦r♠✉❧❛s✿ • ❋r♦♠ ❛ s❡q✉❡♥❝❡ t♦ ✐ts s✉♠✿ ❂❘❬✲✶❪❈✰❘❈❬✲✶❪

• ❋r♦♠ ❛ s❡q✉❡♥❝❡ t♦ ✐ts ❞✐✛❡r❡♥❝❡✿

❂❘❈❬✲✶❪✲❘❬✲✶❪❈❬✲✶❪

❲❤❛t ✐❢ ✇❡ ❝♦♠❜✐♥❡ t❤❡ t✇♦ ❝♦♥s❡❝✉t✐✈❡❧②❄ ❋r♦♠ ❛ s❡q✉❡♥❝❡ t♦ ✐ts ❞✐✛❡r❡♥❝❡ t♦ t❤❡ s✉♠ ♦❢ t❤❡ ❧❛tt❡r✿

■t✬s t❤❡ s❛♠❡ ❝✉r✈❡✦ ◆♦✇ ✐♥ t❤❡ ♦♣♣♦s✐t❡ ♦r❞❡r✱ ❢r♦♠ ❛ s❡q✉❡♥❝❡ t♦ ✐ts s✉♠ t♦ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ ❧❛tt❡r✿

■t✬s t❤❡ s❛♠❡ ❝✉r✈❡✦ ❊①❡r❝✐s❡ ✶✳✶✵✳✾

❲❤❛t ✇♦✉❧❞ t❤❡ r❡s✉❧t✐♥❣ ❝✉r✈❡ ❧♦♦❦ ❧✐❦❡ ✐❢ ✇❡ st❛rt❡❞ ❛t ❛♥♦t❤❡r ♣♦✐♥t❄ ❊①❛♠♣❧❡ ✶✳✶✵✳✶✵✿ ❢❛❧❧✐♥❣ ❜❛❧❧✱ ❛❝❝❡❧❡r❛t✐♦♥

■♥ ❛♥ ❡①❛♠♣❧❡ ❢r♦♠ ❡❛r❧✐❡r ✐♥ t❤✐s ❝❤❛♣t❡r✱ ✇❡ ✈✐❡✇❡❞ t❤❡ ❡①♣❡r✐♠❡♥t❛❧ ❞❛t❛ ♦❢ t❤❡ ❤❡✐❣❤ts ♦❢ ❛ ♣✐♥❣✲ ♣♦♥❣ ❜❛❧❧ ❢❛❧❧✐♥❣ ❞♦✇♥✿

✽✾

✶✳✶✵✳

❙✉♠s ♦❢ ❞✐✛❡r❡♥❝❡s ❛♥❞ ❞✐✛❡r❡♥❝❡s ♦❢ s✉♠s✿ ♠♦t✐♦♥

❏✉st ❛s ❜❡❢♦r❡✱ ✇❡ ✉s❡ ❛ s♣r❡❛❞s❤❡❡t t♦ ♣❧♦t t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ pn ✱ ✐✳❡✳✱ t❤❡ ✈❡❧♦❝✐t②✱ vn ✭♣✉r♣❧❡✮✿

❧♦❝❛t✐♦♥ s❡q✉❡♥❝❡✱ pn ✭❣r❡❡♥✮✳ ❲❡ t❤❡♥ ❝♦♠♣✉t❡ t❤❡

■t ❧♦♦❦s ❧✐❦❡ ❛ str❛✐❣❤t ❧✐♥❡✳ ❇✉t t❤✐s t✐♠❡✱ ✇❡ t❛❦❡ ♦♥❡ ♠♦r❡ st❡♣✿ ❲❡ ❝♦♠♣✉t❡ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ ✈❡❧♦❝✐t② s❡q✉❡♥❝❡✳ ■t ✐s t❤❡ ❛❝❝❡❧❡r❛t✐♦♥✱ an ✭❜❧✉❡✮✳ ■t ❛♣♣❡❛rs ❝♦♥st❛♥t✦ ❚❤❡r❡ ♠✐❣❤t ❜❡ ❛ ❧❛✇ ♦❢ ♥❛t✉r❡ ❤❡r❡✳ ❊①❛♠♣❧❡ ✶✳✶✵✳✶✶✿ s❤♦♦t✐♥❣ ❛ ❝❛♥♥♦♥

▲❡t✬s ❛❝❝❡♣t t❤❡ ♣r❡♠✐s❡ ♣✉t ❢♦r✇❛r❞ ✐♥ t❤❡ ❧❛st ❡①❛♠♣❧❡✱ t❤❛t t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ♦❢ ❢r❡❡ ❢❛❧❧ ✐s ❝♦♥st❛♥t✳ ❚❤❡♥ ✇❡ ❝❛♥ tr② t♦ ♣r❡❞✐❝t t❤❡ ❜❡❤❛✈✐♦r ♦❢ ❛♥ ♦❜❥❡❝t s❤♦t ✐♥ t❤❡ ❛✐r ✕ ❢r♦♠ ❛♥② ✐♥✐t✐❛❧ ❤❡✐❣❤t ❛♥❞ ✇✐t❤ ❛♥② ✐♥✐t✐❛❧ ✈❡❧♦❝✐t②✳ ❚❤❡ ❞✐r❡❝t✐♦♥ ♦❢ ♦✉r ❝♦♠♣✉t❛t✐♦♥ ✐s ♦♣♣♦s✐t❡ t♦ t❤❛t ♦❢ t❤❡ ❧❛st ❡①❛♠♣❧❡✿ ❲❡ ❛ss✉♠❡ t❤❛t ✇❡ ❦♥♦✇ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥✱ t❤❡♥ ❞❡r✐✈❡ t❤❡ ✈❡❧♦❝✐t②✱ ❛♥❞ t❤❡♥ ❞❡r✐✈❡ t❤❡ ❧♦❝❛t✐♦♥ ✭❛❧t✐t✉❞❡✮ ♦❢ t❤❡ ♦❜❥❡❝t ✐♥ t✐♠❡✳ ❲❤✐❧❡ ✇❡ ✉s❡❞ ❞✐✛❡r❡♥❝❡s ✐♥ t❤❡ ❧❛st ❡①❛♠♣❧❡✱ ✇❡ ✉s❡ s✉♠s ♥♦✇✿

❆❜♦✈❡ ✇❡ s❤♦✇ ❛ ♣r♦❥❡❝t✐❧❡ ❧❛✉♥❝❤❡❞ ❢r♦♠ ❛ 100✲♠❡t❡r t❛❧❧ ❜✉✐❧❞✐♥❣ ✈❡rt✐❝❛❧❧② ✉♣ ✐♥ t❤❡ ❛✐r ✇✐t❤ ❛ s♣❡❡❞ ♦❢ 100 ♠❡t❡rs ♣❡r s❡❝♦♥❞ ✭t❤❡ ❣r❛✈✐t② ❝❛✉s❡s ❛❝❝❡❧❡r❛t✐♦♥ ♦❢ −9.8 ♠❡t❡rs ♣❡r s❡❝♦♥❞ sq✉❛r❡❞✮✳

✾✵

✶✳✶✶✳ ❚❤❡ ❛❧❣❡❜r❛ ♦❢ s✉♠s ❛♥❞ ❞✐✛❡r❡♥❝❡s

✾✶

❲❡ ❝❛♥ s❡❡ t❤❛t ✐t ✇✐❧❧ r❡❛❝❤ ✐ts ❤✐❣❤❡st ♣♦✐♥t ✐♥ ❛❜♦✉t 20 s❡❝♦♥❞s ❛♥❞ ✇✐❧❧ ❤✐t t❤❡ ❣r♦✉♥❞ ✐♥ ❛❜♦✉t

40 s❡❝♦♥❞s✳

❊①❡r❝✐s❡ ✶✳✶✵✳✶✷

❍♦✇ ❤✐❣❤ ❞♦❡s t❤❡ ♣r♦❥❡❝t✐❧❡ ❣♦ ✐♥ t❤❡ ❛❜♦✈❡ ❡①❛♠♣❧❡❄ ❊①❡r❝✐s❡ ✶✳✶✵✳✶✸

❯s✐♥❣ t❤❡ ❛❜♦✈❡ ❡①❛♠♣❧❡✱ ❤♦✇ ❧♦♥❣ ✇✐❧❧ ✐t t❛❦❡ ❢♦r t❤❡ ♣r♦❥❡❝t✐❧❡ t♦ r❡❛❝❤ t❤❡ ❣r♦✉♥❞ ✐❢ ✜r❡❞ ❞♦✇♥ ❄ ❊①❡r❝✐s❡ ✶✳✶✵✳✶✹

❯s❡ t❤❡ ❛❜♦✈❡ ♠♦❞❡❧ t♦ ❞❡t❡r♠✐♥❡ ❤♦✇ ❧♦♥❣ ✐t ✇✐❧❧ t❛❦❡ ❢♦r ❛♥ ♦❜❥❡❝t t♦ r❡❛❝❤ t❤❡ ❣r♦✉♥❞ ✐❢ ✐t ✐s ❞r♦♣♣❡❞✳ ▼❛❦❡ ✉♣ ②♦✉r ♦✇♥ q✉❡st✐♦♥s ❛❜♦✉t t❤❡ s✐t✉❛t✐♦♥ ❛♥❞ ❛♥s✇❡r t❤❡♠✳ ❘❡♣❡❛t✳ ❊①❡r❝✐s❡ ✶✳✶✵✳✶✺

❙✉♣♣♦s❡ t❤❡ t✐♠❡ ♠♦♠❡♥ts ❛r❡ ❣✐✈❡♥ ❜② ❛♥♦t❤❡r s❡q✉❡♥❝❡ ✭❛♥ ❛r✐t❤♠❡t✐❝ ♣r♦❣r❡ss✐♦♥✮✳ ❈♦♠♣✉t❡ t❤❡ ✈❡❧♦❝✐t② ❛♥❞ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ❢r♦♠ t❤❡ t❛❜❧❡ ❜❡❧♦✇✿ t✐♠❡ ❤❡✐❣❤t n 1 2 3 4 5 6 7

tn .00 .05 .10 .15 .20 .25 .30

an 36 35 32 25 20 11 0

❚❤✐s st✉❞② ♦❢ ♠♦t✐♦♥ ❝♦♥t✐♥✉❡s t❤r♦✉❣❤♦✉t t❤❡ ❜♦♦❦✳

✶✳✶✶✳ ❚❤❡ ❛❧❣❡❜r❛ ♦❢ s✉♠s ❛♥❞ ❞✐✛❡r❡♥❝❡s ❲❤❛t ❤❛♣♣❡♥s t♦ t❤❡✐r ❞✐✛❡r❡♥❝❡s ❛♥❞ t❤❡✐r s✉♠s ❛❢t❡r ❛♥ ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥ ✐s ❝❛rr✐❡❞ ♦✉t ✇✐t❤ ❛ ♣❛✐r ♦❢ s❡q✉❡♥❝❡s❄ ❚❤❡r❡ ❛r❡ ❛ ❢❡✇ s❤♦rt❝✉t ♣r♦♣❡rt✐❡s✳ ❍❡r❡ ✐s ❛♥ ❡❧❡♠❡♥t❛r② st❛t❡♠❡♥t ❛❜♦✉t ♠♦t✐♦♥ ✿ ◮ ■❋ t✇♦ r✉♥♥❡rs ❛r❡ r✉♥♥✐♥❣ ❛✇❛② ❢r♦♠ ❛ ♣♦st✱ ❚❍❊◆ t❤❡✐r r❡❧❛t✐✈❡ ✈❡❧♦❝✐t② ✐s t❤❡ s✉♠ ♦❢

t❤❡✐r r❡s♣❡❝t✐✈❡ ✈❡❧♦❝✐t✐❡s✳

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

❚❤❡ ✐❞❡❛ ✇❤② ✇❡ ❛❞❞ t❤❡✐r ❞✐✛❡r❡♥❝❡s ✇❤❡♥ ✇❡ ❛❞❞ s❡q✉❡♥❝❡s ✐s ✐❧❧✉str❛t❡❞ ❜❡❧♦✇✿

✶✳✶✶✳ ❚❤❡ ❛❧❣❡❜r❛ ♦❢ s✉♠s ❛♥❞ ❞✐✛❡r❡♥❝❡s

✾✷

❍❡r❡✱ t❤❡ ❜❛rs t❤❛t r❡♣r❡s❡♥t t❤❡ ❝❤❛♥❣❡ ♦❢ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ s❡q✉❡♥❝❡ ❛r❡ st❛❝❦❡❞ ♦♥ t♦♣ ♦❢ ❡❛❝❤ ♦t❤❡r✳ ❚❤❡ ❤❡✐❣❤ts ❛r❡ t❤❡♥ ❛❞❞❡❞ t♦ ❡❛❝❤ ♦t❤❡r✱ ❛♥❞ s♦ ❛r❡ t❤❡ ❤❡✐❣❤t ❞✐✛❡r❡♥❝❡s✳ ❚❤❡ ❛❧❣❡❜r❛ ❜❡❤✐♥❞ t❤✐s ❣❡♦♠❡tr② ✐s ✈❡r② s✐♠♣❧❡✿ (A + B) − (a + b) = (A − a) + (B − b) .

■t✬s t❤❡ ❆ss♦❝✐❛t✐✈❡ ❘✉❧❡ ♦❢ ❛❞❞✐t✐♦♥✳

❚❤❡ ✐❞❡❛ ❛❜♦✈❡ ✐s ❡q✉❛❧❧② ❛♣♣❧✐❝❛❜❧❡ t♦ r✉♥♥❡rs ✇❤♦ ❝❤❛♥❣❡ ❤♦✇ ❢❛st t❤❡② r✉♥❀ ✇❡ s♣❡❛❦ ♦❢ s❡q✉❡♥❝❡s✿

❚❤❡♦r❡♠ ✶✳✶✶✳✶✿ ❙✉♠ ❘✉❧❡ ❢♦r ❉✐✛❡r❡♥❝❡s ❚❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ s✉♠ ♦❢ t✇♦ s❡q✉❡♥❝❡s ✐s t❤❡ s✉♠ ♦❢ t❤❡✐r ❞✐✛❡r❡♥❝❡s✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ❢♦r ❛♥② t✇♦ s❡q✉❡♥❝❡s an , bn ✱ t❤❡✐r s❡q✉❡♥❝❡s ♦❢ ❞✐✛❡r❡♥❝❡s s❛t✐s❢②✿ ∆(an + bn ) = ∆an + ∆bn

Pr♦♦❢✳ ∆(an + bn ) = (an+1 + bn+1 ) − (an + bn ) = (an+1 − an ) + (bn+1 − bn ) = ∆an + ∆bn .

❊①❛♠♣❧❡ ✶✳✶✶✳✷✿ ❞✐✛❡r❡♥❝❡ ♦❢ s✉♠ ❈♦♥s✐❞❡r t❤❡ s✉♠ ♦❢ ❛♥ ❛r✐t❤♠❡t✐❝ ♣r♦❣r❡ss✐♦♥ ❛♥❞ ❛ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥✿ ∆(a + mn + arn ) = ∆(a + mn) + ∆(arn ) = m + arn (r − 1) .

◆♦✇✱ ❞✐✛❡r❡♥❝❡s ❛♥❞ s✉♠s ❛r❡ ♠❛t❝❤❡❞ ✉♣ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠s ♦❢ ❈❛❧❝✉❧✉s ♦❢ ❙❡q✉❡♥❝❡s ♣r❡s❡♥t❡❞ ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥✦ ■t s❤♦✉❧❞ ❜❡ ♥♦ s✉r♣r✐s❡ t❤❡♥ t❤❛t t❤❡r❡ ✐s ❛ ♠❛t❝❤✐♥❣ t❤❡♦r❡♠ ❛❜♦✉t s✉♠s✳ ❲❤❡♥ t✇♦ s❡q✉❡♥❝❡s ❛r❡ ❛❞❞❡❞ t♦ ❡❛❝❤ ♦t❤❡r✱ ✇❤❛t ❤❛♣♣❡♥s t♦ t❤❡✐r s✉♠s❄ ❚❤✐s s✐♠♣❧❡ ❛❧❣❡❜r❛✱ t❤❡ ❆ss♦❝✐❛t✐✈❡ Pr♦♣❡rt② ❝♦♠❜✐♥❡❞ ✇✐t❤ t❤❡ ❈♦♠♠✉t❛t✐✈❡ Pr♦♣❡rt②✱ t❡❧❧s t❤❡ ✇❤♦❧❡ st♦r②✿ r❡✲❛rr❛♥❣❡ t❤❡ s✉♠ ♦❢ ❢♦✉r ♥✉♠❜❡rs✿

= (a + b), a + b + + + = (A + B) A + B = (a + A) + (b + B) = a + A + b + B

❚❤❡ r✉❧❡ ❛♣♣❧✐❡s ❡✈❡♥ ✐❢ ✇❡ ❤❛✈❡ ♠♦r❡ t❤❛♥ ❥✉st t✇♦ t❡r♠s❀ ✐t✬s ❛❜♦✉t r❡✲❛rr❛♥❣✐♥❣ t❡r♠s ♦❢ s❡q✉❡♥❝❡s✿ r❡✲❛rr❛♥❣❡ t❤❡ s✉♠ ♦❢ t✇♦ s❡q✉❡♥❝❡s✿

ap + bp ap+1 + bp+1

✳✳ ✳✳ ✳✳ ✳ ✳ ✳

= (ap + bp )+ = (ap+1 + bp+1 )+

aq + bq = = (ap + ... + aq ) + (bp + ... + bq ) =

✳✳ ✳

(aq + bq ) (ap + bp )+

... +(aq + bq )

✶✳✶✶✳

❚❤❡ ❛❧❣❡❜r❛ ♦❢ s✉♠s ❛♥❞ ❞✐✛❡r❡♥❝❡s

✾✸

❚❤❡ s✉♠♠❛t✐♦♥ ✐s ✐❧❧✉str❛t❡❞ ❜❡❧♦✇✿

❆♥ ❛❜❜r❡✈✐❛t❡❞ ✈❡rs✐♦♥ ♦❢ t❤✐s ❢♦r♠✉❧❛ ✐s ❛s ❢♦❧❧♦✇s✳ ❚❤❡♦r❡♠ ✶✳✶✶✳✸✿ ❙✉♠ ❘✉❧❡ ❢♦r ❙✉♠s ❚❤❡ s✉♠ ♦❢ t❤❡ s✉♠s ♦❢ t✇♦ s❡q✉❡♥❝❡s ✐s t❤❡ s✉♠ ♦❢ t❤❡ s❡q✉❡♥❝❡ ♦❢ t❤❡ s✉♠s✳

■♥ ♦t❤❡r ✇♦r❞s✱ ✐❢

an

❛♥❞

bn

❛r❡ s❡q✉❡♥❝❡s✱ t❤❡♥✱ ❢♦r ❛♥②

s❡q✉❡♥❝❡s ♦❢ s✉♠s s❛t✐s❢②✿

q X n=p

an +

q X n=p

bn =

q X

p, q

✇✐t❤

p ≤ q✱

t❤❡✐r

(an + bn )

n=p

❊①❡r❝✐s❡ ✶✳✶✶✳✹

❉❡r✐✈❡ t❤✐s t❤❡♦r❡♠ ❢r♦♠ t❤❡ ❧❛st ♦♥❡✱ r❡✈❡rs❡✳ ❍❡r❡ ✐s ❛♥♦t❤❡r s✐♠♣❧❡ st❛t❡♠❡♥t ❛❜♦✉t

♠♦t✐♦♥ ✿

◮ ■❋ t❤❡ ❞✐st❛♥❝❡ ✐s r❡✲s❝❛❧❡❞✱ s✉❝❤ ❛s ❢r♦♠ ♠✐❧❡s t♦ ❦✐❧♦♠❡t❡rs✱ ❚❍❊◆ s♦ ✐s t❤❡ ✈❡❧♦❝✐t② ✕ ❛t t❤❡ s❛♠❡ ♣r♦♣♦rt✐♦♥✳ ❚❤❡ ✐❞❡❛ ✇❤② ❛ ♣r♦♣♦rt✐♦♥❛❧ ❝❤❛♥❣❡ ❝❛✉s❡s t❤❡ s❛♠❡ ♣r♦♣♦rt✐♦♥❛❧ ❝❤❛♥❣❡ ✐♥ t❤❡ ❞✐✛❡r❡♥❝❡s ✐s ✐❧❧✉str❛t❡❞ ❜❡❧♦✇ ✭tr✐♣❧✐♥❣✮✿

✶✳✶✶✳ ❚❤❡ ❛❧❣❡❜r❛ ♦❢ s✉♠s ❛♥❞ ❞✐✛❡r❡♥❝❡s

✾✹

❍❡r❡✱ ✐❢ t❤❡ ❤❡✐❣❤ts tr✐♣❧❡✱ t❤❡♥ s♦ ❞♦ t❤❡ ❤❡✐❣❤t ❞✐✛❡r❡♥❝❡s✳ ❚❤❡ ❛❧❣❡❜r❛ ❜❡❤✐♥❞ t❤✐s ❣❡♦♠❡tr② ✐s ✈❡r② s✐♠♣❧❡✿ kA − ka = k(A − a) .

■t✬s t❤❡ ❉✐str✐❜✉t✐✈❡ ❘✉❧❡✳ ❚❤✐s ✐s ❤♦✇ ✐t ❛♣♣❧✐❡s t♦ s❡q✉❡♥❝❡s✳

❚❤❡♦r❡♠ ✶✳✶✶✳✺✿ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ❢♦r ❉✐✛❡r❡♥❝❡s ❚❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ ❛ ♠✉❧t✐♣❧❡ ♦❢ ❛ s❡q✉❡♥❝❡ ✐s t❤❡ ♠✉❧t✐♣❧❡ ♦❢ t❤❡ s❡q✉❡♥❝❡✬s ❞✐✛❡r❡♥❝❡✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ❢♦r ❛♥② s❡q✉❡♥❝❡ an ✱ t❤❡ s❡q✉❡♥❝❡ ♦❢ ❞✐✛❡r❡♥❝❡s s❛t✐s✜❡s✿ ∆(kan ) = k∆an

Pr♦♦❢✳ ∆(kan ) = kan+1 k − kan = kan+1 k − kan = k∆an .

❚❤❡ t❤❡♦r❡♠ ❝❛♥ ❛❧s♦ ❜❡ ✐♥t❡r♣r❡t❡❞ ❛s ❢♦❧❧♦✇s✿ ■❢ t❤❡ ❞✐st❛♥❝❡s ❛r❡ ♣r♦♣♦rt✐♦♥❛❧❧② ✐♥❝r❡❛s❡❞✱ t❤❡♥ s♦ ❛r❡ t❤❡ ✈❡❧♦❝✐t✐❡s ♥❡❡❞❡❞ t♦ ❝♦✈❡r t❤❡♠✱ ✐♥ t❤❡ s❛♠❡ ♣❡r✐♦❞ ♦❢ t✐♠❡✳ ■s t❤❡r❡ ❛ ♠❛t❝❤✐♥❣ st❛t❡♠❡♥t ❛❜♦✉t s✉♠s❄ ❨❡s✱ ❜✉t ❧❡t✬s ✜rst ❧♦♦❦ ❛t ♠♦t✐♦♥ ❛❣❛✐♥✿ ■❢ ②♦✉r ✈❡❧♦❝✐t② ✐s tr✐♣❧❡❞✱ t❤❡♥ s♦ ✐s t❤❡ ❞✐st❛♥❝❡ ②♦✉ ❤❛✈❡ ❝♦✈❡r❡❞✳ ❲❤❡♥ ❛ s❡q✉❡♥❝❡ ✐s ♠✉❧t✐♣❧✐❡❞ ❜② ❛ ❝♦♥st❛♥t✱ ✇❤❛t ❤❛♣♣❡♥s t♦ ✐ts s✉♠s❄ ❚❤✐s s✐♠♣❧❡ ❛❧❣❡❜r❛✱ t❤❡ ❉✐str✐❜✉t✐✈❡ Pr♦♣❡rt②✱ t❡❧❧s t❤❡ ✇❤♦❧❡ st♦r②✿ k · ( a + b) = ka + kb

t❛❦❡ ♦✉t ❛ ❝♦♠♠♦♥ ❢❛❝t♦r✿

❚❤❡ r✉❧❡ ❛♣♣❧✐❡s ❡✈❡♥ ✐❢ ✇❡ ❤❛✈❡ ♠♦r❡ t❤❛♥ ❥✉st t✇♦ t❡r♠s❀ ✐t✬s ❛❜♦✉t ❢❛❝t♦r✐♥❣✿ k k

t❛❦❡ ♦✉t ❛ ❝♦♠♠♦♥ ❢❛❝t♦r✿

· ·

ap ap+1

✳✳ ✳✳ ✳✳ ✳ ✳ ✳

= k · ap + = k · ap+1 +

✳✳ ✳

= k · aq k · aq = k · ap + ... +k · aq = k · (ap + ... + aq )

✶✳✶✶✳ ❚❤❡ ❛❧❣❡❜r❛ ♦❢ s✉♠s ❛♥❞ ❞✐✛❡r❡♥❝❡s

✾✺

❚❤✐s s✉♠♠❛t✐♦♥ ✐s ✐❧❧✉str❛t❡❞ ❜❡❧♦✇✿

❆♥ ❛❜❜r❡✈✐❛t❡❞ ✈❡rs✐♦♥ ♦❢ t❤✐s ❢♦r♠✉❧❛ ✐s ❛s ❢♦❧❧♦✇s✳ ❚❤❡♦r❡♠ ✶✳✶✶✳✻✿ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ❢♦r ❙✉♠s

❚❤❡ s✉♠ ♦❢ ❛ ♠✉❧t✐♣❧❡ ♦❢ ❛ s❡q✉❡♥❝❡ ✐s t❤❡ ♠✉❧t✐♣❧❡ ♦❢ ✐ts s✉♠✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✐❢ an ✐s ❛ s❡q✉❡♥❝❡✱ t❤❡♥ ❢♦r ❛♥② p, q ✇✐t❤ p ≤ q ❛♥❞ ❛♥② r❡❛❧ k ✱ ✐ts s❡q✉❡♥❝❡ ♦❢ s✉♠s s❛t✐s✜❡s✿ q X

(kan ) = k

n=p

q X

an

n=p

❊①❡r❝✐s❡ ✶✳✶✶✳✼

❉❡r✐✈❡ t❤✐s t❤❡♦r❡♠ ❢r♦♠ t❤❡ ❧❛st ♦♥❡✱ r❡✈❡rs❡✳ ◆♦✇ ✇❡ ❣♦ ❜❡②♦♥❞ ❛❞❞✐t✐♦♥ ❛♥❞ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❜② ❛ ❝♦♥st❛♥t✳ ▲❡t✬s ✐♠❛❣✐♥❡ t❤✐s✿ ◮ ■❢ 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❤❡ ♣r♦❞✉❝t ♦❢ t✇♦ s❡q✉❡♥❝❡s ✐s ✐♥t❡r♣r❡t❡❞ ❛s t❤❡ ❛r❡❛s ♦❢ t❤❡ r❡❝t❛♥❣❧❡s ❢♦r♠❡❞ ❜② t❤❡ s❡q✉❡♥❝❡s✱ ✐❧❧✉str❛t❡❞ ❜❡❧♦✇✿

✶✳✶✶✳ ❚❤❡ ❛❧❣❡❜r❛ ♦❢ s✉♠s ❛♥❞ ❞✐✛❡r❡♥❝❡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❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ♦❢ t❤❡ ❢♦r♠✉❧❛ ❜❡❧♦✇✳

❚❤❡♦r❡♠ ✶✳✶✶✳✽✿ Pr♦❞✉❝t ❘✉❧❡ ❢♦r ❉✐✛❡r❡♥❝❡s ❚❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ ♣r♦❞✉❝t ♦❢ t✇♦ s❡q✉❡♥❝❡s ✐s ❢♦✉♥❞ ❛s ❛ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡s❡ s❡q✉❡♥❝❡s ❛♥❞ ❡✐t❤❡r ♦❢ t❤❡ t✇♦ ❞✐✛❡r❡♥❝❡s✳ ❙♣❡❝✐✜❝❛❧❧②✱ ❢♦r ❛♥② t✇♦ s❡q✉❡♥❝❡s an , bn ✱ t❤❡ s❡q✉❡♥❝❡ ♦❢ ❞✐✛❡r❡♥❝❡s ♦❢ t❤❡ ♣r♦❞✲ ✉❝t s❛t✐s✜❡s✿ ∆(an · bn ) = an+1 · ∆bn + ∆an · bn

Pr♦♦❢✳ ∆(an · bn ) = an+1 · bn+1 − an · bn = an+1 · bn+1 − an+1 · bn + an+1 · bn − an · bn = an+1 · (bn+1 ) − bn ) + (an+1 − an ) · bn = an+1 · ∆b + ∆an · bn .

■♥s❡rt t❡r♠s✳ ❋❛❝t♦r✳

❊①❛♠♣❧❡ ✶✳✶✶✳✾✿ ❞✐✛❡r❡♥❝❡ ♦❢ ♣r♦❞✉❝t ❈♦♥s✐❞❡r t❤❡ ♣r♦❞✉❝t ♦❢ ❛♥ ❛r✐t❤♠❡t✐❝ ♣r♦❣r❡ss✐♦♥ ❛♥❞ ❛ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥s✿ ∆(mn · arn ) = ∆(mn) · arn + mn∆(arn ) = marn + mnarn (r − 1) .

❊①❛♠♣❧❡ ✶✳✶✶✳✶✵✿ ❞✐✛❡r❡♥❝❡ ♦❢ sq✉❛r❡ ❚❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ sq✉❛r❡ s❡q✉❡♥❝❡ an = n2 ✐s ❝♦♠♣✉t❡❞ ✇✐t❤ t❤❡ Pr♦❞✉❝t ❘✉❧❡ ✿ ∆(n2 ) = ∆(n · n) = (n + 1) · ∆(n) + ∆(n) · (n) = (n + 1) + (n) = 2n + 1 .

■t✬s ❛♥ ❛r✐t❤♠❡t✐❝ ♣r♦❣r❡ss✐♦♥✱ ❝♦♥✜r♠❡❞ ❜② t❤❡ s❡❝♦♥❞ ❣r❛♣❤ ❜❡❧♦✇✿

❊①❡r❝✐s❡ ✶✳✶✶✳✶✶ ❋✐♥❞ ❛ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ ♣♦✇❡r s❡q✉❡♥❝❡✱ an = nm ✱ ✉s✐♥❣ t❤❡ ❇✐♥♦♠✐❛❧ ❚❤❡♦r❡♠✳

❊①❛♠♣❧❡ ✶✳✶✶✳✶✷✿ ❞✐✛❡r❡♥❝❡ ♦❢ s❡q✉❡♥❝❡ ♦❢ r❡❝✐♣r♦❝❛❧s ▲❡t✬s ✜♥❞ t❤❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ r❡❝✐♣r♦❝❛❧s✿   1 1 n − (n + 1) 1 1 = − = =− . ∆ n n+1 n n(n + 1) n(n + 1)

✶✳✶✶✳

❚❤❡ ❛❧❣❡❜r❛ ♦❢ s✉♠s ❛♥❞ ❞✐✛❡r❡♥❝❡s

✾✼

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

❚❤❡♦r❡♠ ✶✳✶✶✳✶✸✿ ◗✉♦t✐❡♥t ❘✉❧❡ ❢♦r ❉✐✛❡r❡♥❝❡s ❚❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ q✉♦t✐❡♥t ♦❢ t✇♦ s❡q✉❡♥❝❡s ✐s ❢♦✉♥❞ ❛s ❛ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡s❡ s❡q✉❡♥❝❡s ❛♥❞ ❡✐t❤❡r ♦❢ t❤❡ t✇♦ ❞✐✛❡r❡♥❝❡s✳ ❙♣❡❝✐✜❝❛❧❧②✱ ❢♦r ❛♥② t✇♦ s❡q✉❡♥❝❡s an , bn ✱ t❤❡ s❡q✉❡♥❝❡ ♦❢ ❞✐✛❡r❡♥❝❡s ♦❢ t❤❡ q✉♦✲ t✐❡♥t s❛t✐s✜❡s✿ ∆



an bn



=

an+1 · ∆bn + ∆an · bn bn bn+1

❊①❡r❝✐s❡ ✶✳✶✶✳✶✹ Pr♦✈❡ t❤❡ t❤❡♦r❡♠✳

❚❤❡r❡ ❛r❡ ♥♦ ♠❛t❝❤✐♥❣ st❛t❡♠❡♥ts ❢♦r ❍❡r❡ ♣r❡❝❛❧❝✉❧✉s ❛s ❛

s✉♠s

♣r❡✈✐❡✇ ♦❢ ❝❛❧❝✉❧✉s

❛s s✐♠♣❧❡ ❛s t❤❡s❡ t✇♦✳

❡♥❞s✱ ❛♥❞ ♣r❡❝❛❧❝✉❧✉s ❛s ❛

♣r❡r❡q✉✐s✐t❡ ❢♦r ❝❛❧❝✉❧✉s

st❛rts✳

❈❤❛♣t❡r ✷✿ ❙❡ts ❛♥❞ ❢✉♥❝t✐♦♥s

❈♦♥t❡♥ts ✷✳✶ ❙❡ts ❛♥❞ r❡❧❛t✐♦♥s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✳✷ ❋✉♥❝t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✸ ❙❡q✉❡♥❝❡s ❛r❡ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✳✹ ❍♦✇ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s ❡♠❡r❣❡✿ ♦♣t✐♠✐③❛t✐♦♥

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✳✺ ❙❡t ❜✉✐❧❞✐♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✻ ❚❤❡

xy ✲♣❧❛♥❡✿

✇❤❡r❡ ❣r❛♣❤s ❧✐✈❡✳✳✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✳✼ ▲✐♥❡❛r r❡❧❛t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✽ ❘❡❧❛t✐♦♥s ✈s✳ ❢✉♥❝t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✾ ❆ ❢✉♥❝t✐♦♥ ❛s ❛ ❜❧❛❝❦ ❜♦① ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✵ ●✐✈❡ t❤❡ ❢✉♥❝t✐♦♥ ❛ ❞♦♠❛✐♥✳✳✳ ✷✳✶✶ ❚❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✳✶✷ ▲✐♥❡❛r ❢✉♥❝t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✸ ❆❧❣❡❜r❛ ❝r❡❛t❡s ❢✉♥❝t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✹ ❚❤❡ ✐♠❛❣❡✿ t❤❡ r❛♥❣❡ ♦❢ ✈❛❧✉❡s ♦❢ ❛ ❢✉♥❝t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

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

✷✳✶✳ ❙❡ts ❛♥❞ r❡❧❛t✐♦♥s

■♥ ♠❛t❤❡♠❛t✐❝s✱ ✇❡ r❡❢❡r t♦ ❛♥② ❧♦♦s❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ♦❜❥❡❝ts ♦r ❡♥t✐t✐❡s ✕ ♦❢ ❛♥② ♥❛t✉r❡ ✕ ❛s ❛ s❡t✳ ❋♦r ❡①❛♠♣❧❡✱ ✐s t❤✐s ❛ ❝✐r❝❧❡ ♦❢ ♠❛r❜❧❡s t❤❛t ✇❡ s❡❡ ✐♥ ❛ ❜❛❣❄ ◆♦✱ t❤❡ ♠❛r❜❧❡s ✐t ✐s ♠❛❞❡ ♦❢ ❛r❡♥✬t ❝♦♥♥❡❝t❡❞ t♦ ❡❛❝❤ ♦t❤❡r ♦r t♦ ❛♥② ❧♦❝❛t✐♦♥✳ ❖♥❡ s❤❛❦❡ ❛♥❞ t❤❡ ❝✐r❝❧❡ ✐s ❣♦♥❡✿

❊①❛♠♣❧❡ ✷✳✶✳✶✿ s❡ts ❛s ❧✐sts

❙❡ts ❣✐✈❡♥ ❡①♣❧✐❝✐t❧② ✕ ❛s ❧✐sts ✕ ❛r❡ t❤❡ s✐♠♣❧❡st ♦♥❡s✿ • ❆ r♦st❡r ♦❢ st✉❞❡♥ts✿ ❆❞❛♠s✱ ❆❞❦✐♥s✱ ❆rr♦✇s✱ ✳✳✳ • ❆ ❧✐st ♦❢ ♥✉♠❜❡rs✿ 1, 2, 3, 4, ... • ❆ ❧✐st ♦❢ ♣❧❛♥❡ts✿ ▼❡r❝✉r②✱ ❱❡♥✉s✱ ❊❛rt❤✱ ▼❛rs✱ ✳✳✳ ❚❤❡ ♦r❞❡r ❛t ✇❤✐❝❤ t❤❡② ❛♣♣❡❛r ♦♥ t❤❡ ❧✐st ✐s ♥♦t ❛ ♣❛rt ♦❢ t❤❡ ✐♥❢♦r♠❛t✐♦♥ ✇❡ ❝❛r❡ ❛❜♦✉t ✇❤❡♥ ✇❡ s♣❡❛❦ ♦❢ s❡ts✳ ❍❡r❡ ✐s ❛ ❜❛❣ ♦❢ ♥✉♠❜❡rs✿

✷✳✶✳

❙❡ts ❛♥❞ r❡❧❛t✐♦♥s

✾✾

❊①❛♠♣❧❡ ✷✳✶✳✷✿ ✏s❡ts✑ ❚❤❡ ✐❞❡❛ ♦❢ s❡t ❝♦♥tr❛sts ✇✐t❤ s✉❝❤ ❡①♣r❡ss✐♦♥s ❛s ✏❛ s❡t ♦❢ s✐❧✈❡r✇❛r❡✑ ✇❤❡♥ t❤❡ ✇♦r❞ ✏s❡t✑ s✉❣❣❡sts ❛ ❝❡rt❛✐♥ str✉❝t✉r❡✿ s♣❡❝✐✜❝ t②♣❡s ♦❢ ❦♥✐✈❡s ❛♥❞ ❢♦r❦s ✇✐t❤ ❛ s♣❡❝✐✜❝ ♣❧❛❝❡ ✐♥ t❤❡ ❜♦①✳ ■t ✐s t❤❡ s❛♠❡ s❡t✱ ♠❛t❤❡♠❛t✐❝❛❧❧②✱ ✇❤❡t❤❡r t❤❡ ✐t❡♠s ❛r❡ ❛rr❛♥❣❡❞ ✐♥ ❛ ❜♦① ♦r ♣✐❧❡❞ ✉♣ ♦♥ t❤❡ ❝♦✉♥t❡r✳ ❆ s❡t ♦❢ ❡♥❝②❝❧♦♣❡❞✐❛ ❝♦♥s✐sts ♦❢ ❜♦♦❦s t❤❛t ❝❛♥ ❜❡ ❛rr❛♥❣❡❞ ❛❧♣❤❛❜❡t✐❝❛❧❧② ♦r ❝❤r♦♥♦❧♦❣✐❝❛❧❧② ♦r r❛♥❞♦♠❧②✳

❲❛r♥✐♥❣✦

❊✈❡♥ t❤♦✉❣❤ ✇❡ tr② t♦ ♣r♦✈✐❞❡ ❛ ♣r❡❝✐s❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❡✈❡r② ♥❡✇ ❝♦♥❝❡♣t✱ t❤❡ ✐❞❡❛ ♦❢ s❡t ✐s s♦ ❣❡♥❡r❛❧ t❤❛t ✇❡ ✇✐❧❧ ❤❛✈❡ t♦ r❡❧② ♦♥ ❡①❛♠♣❧❡s✳ ❲❤❛t ❝r❡❛t❡s ❛ s❡t ✐s ♦✉r ❦♥♦✇❧❡❞❣❡ ♦r ❛❜✐❧✐t② t♦ ❞❡t❡r♠✐♥❡ ✇❤❡t❤❡r ❛♥ ♦❜❥❡❝t

❜❡❧♦♥❣s ♦r ❞♦❡s ♥♦t ❜❡❧♦♥❣

✐t✳ ❆ ❧✐st ✐s ♦♥❡ s✉❝❤ ♣♦ss✐❜✐❧✐t②✳ ❆♥♦t❤❡r ✐s ❛ ❝♦♥❞✐t✐♦♥ t♦ ❜❡ ✈❡r✐✜❡❞✳

❊①❛♠♣❧❡ ✷✳✶✳✸✿ s❡ts ✈✐❛ ❝♦♥❞✐t✐♦♥s ❆ r♦st❡r ♦❢ ❛ ❝❧❛ss ♣r♦❞✉❝❡s ❛ s❡t ♦❢ t❤❡ st✉❞❡♥ts ✐♥ t❤✐s ❝❧❛ss✳

❢❡♠❛❧❡

■t✬s ❛ ❧✐st✦

❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ t❤❡

st✉❞❡♥ts ✐♥ t❤❡ ❝❧❛ss ❛❧s♦ ❢♦r♠ ❛ s❡t ❡✈❡♥ ✐❢ t❤❡r❡ ✐s ♥♦ s✉❝❤ ❧✐st❀ ✇❡ ❝❛♥ ❥✉st ❣♦ ❞♦✇♥ t❤❡ r♦st❡r

❛♥❞ ❞❡t❡r♠✐♥❡ ✐❢ ❛ st✉❞❡♥t ❜❡❧♦♥❣s t♦ t❤✐s ♥❡✇ s❡t✳ ❙✐♠✐❧❛r❧②✱ t❤❡ st✉❞❡♥ts ✇✐t❤ ❛♥ ❆ ♦♥ t❤❡ ❧❛st t❡st ❛❧s♦ ✕ ✐♠♣❧✐❝✐t❧② ✕ ❢♦r♠ ❛ s❡t✳

❊①❛♠♣❧❡ ✷✳✶✳✹✿ s❡ts ✐♥ ♠❛t❤ ❆ ❧♦t ♦❢ s❡ts ❡①❛♠✐♥❡❞ ❡❛r❧② ✐♥ t❤✐s ❜♦♦❦ ✇✐❧❧ ❜❡ s❡ts ♦❢

♥✉♠❜❡rs ❀

t❤❡♥ ✇❡ ❦♥♦✇ t❤❛t

♥✉♠❜❡r ❞✐✈✐s✐❜❧❡ ❜②

2❄

2

❜❡❧♦♥❣s t♦ ✐t ❜✉t

3

♥✉♠❜❡rs✳

❋♦r ❡①❛♠♣❧❡✱ t❛❦❡ t❤❡ s❡t ♦❢

❡✈❡♥

❞♦❡s ♥♦t✳ ❲❡ s✐♠♣❧② ❝❤❡❝❦ t❤❡ ❝♦♥❞✐t✐♦♥✿ ■s t❤❡

❆♥♦t❤❡r ❡①❛♠♣❧❡ ❢r♦♠ ❢❛♠✐❧✐❛r ♣❛rts ♦❢ ♠❛t❤❡♠❛t✐❝s ✐s s❡ts ♦❢

♣♦✐♥ts

♦♥ t❤❡

♣❧❛♥❡✿ str❛✐❣❤t ❧✐♥❡s✱ tr✐❛♥❣❧❡s✱ ❝✐r❝❧❡s ❛♥❞ ♦t❤❡r ❝✉r✈❡s✱ ❡t❝✳✿

❲❡ ❝❛♥ ❛❧✇❛②s t❡❧❧ ✇❤❡t❤❡r ❛ ♣♦✐♥t ❜❡❧♦♥❣s t♦ t❤❡ s❡t✦

❊①❛♠♣❧❡ ✷✳✶✳✺✿ ♥♦♥✲s❡ts ■❢ t❤❡ ❝♦♥❞✐t✐♦♥ ✐s ✈❛❣✉❡✱ ✇❡ ❞♦♥✬t ❤❛✈❡ ❛ s❡t✿ ✏✐♥t❡r❡st✐♥❣ ♥♦✈❡❧s✑✱ ✏❜❛❞ ♣❛✐♥t✐♥❣s✑✱ ❡t❝✳ ❝♦♥❞✐t✐♦♥ ✐s ♥♦♥s❡♥s✐❝❛❧✱ ✇❡ ❞♦♥✬t ❤❛✈❡ ❛ s❡t ❡✐t❤❡r✿ ✏❢❛st tr❡❡s✑✱ ✏❜❧✉❡ ♥✉♠❜❡rs✑✱ ❡t❝✳

❲❤❡♥ t❤❡

t♦

✷✳✶✳

❙❡ts ❛♥❞ r❡❧❛t✐♦♥s

✶✵✵

❊①❡r❝✐s❡ ✷✳✶✳✻ ●✐✈❡ ②♦✉r ♦✇♥ ❡①❛♠♣❧❡s ♦❢ ✭❛✮ s❡ts ❛s ❧✐sts✱ ✭❜✮ s❡ts ❞❡✜♥❡❞ ✈✐❛ ❝♦♥❞✐t✐♦♥s✱ ❛♥❞ ✭❝✮ ♥♦♥✲s❡ts✳

■♥ t❤❡ r❡st ♦❢ t❤✐s ❝❤❛♣t❡r ✇❡ ✇✐❧❧ ✉s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡①❛♠♣❧❡✳ ❚❤❡s❡

✜✈❡ ❜♦②s

❢♦r♠ ❛ s❡t✿

❖♥ t❤❡ ♦♥❡ ❤❛♥❞✱ t❤❡② ❛r❡ ✐♥❞✐✈✐❞✉❛❧s ❛♥❞ ❝❛♥ ❛❧✇❛②s ❜❡ t♦❧❞ ❢r♦♠ ❡❛❝❤ ♦t❤❡r✳ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ t❤❡② ❛r❡ ✉♥r❡❧❛t❡❞ t♦ ❡❛❝❤ ♦t❤❡r✿ ❲❡ ❝❛♥ ❧✐st t❤❡♠ ✐♥ ❛♥② ♦r❞❡r✱ ✇❡ ❝❛♥ ❛rr❛♥❣❡ t❤❡♠ ✐♥ ❛ ❝✐r❝❧❡✱ ❛ sq✉❛r❡✱ ♦r ❛t r❛♥❞♦♠❀ ✇❡ ❝❛♥ ❝❤❛♥❣❡ t❤❡ ❞✐st❛♥❝❡s ❜❡t✇❡❡♥ t❤❡♠✱ ❛♥❞ s♦ ♦♥✳ ■t✬s t❤❡ s❛♠❡ s❡t✦ ❚❤❡ ♠❡♠❜❡rs ♦❢ ❛ s❡t ❛r❡ ❝❛❧❧❡❞ ✐ts

❡❧❡♠❡♥ts✳

❖✉r s❡t ✐s ♥♦t❤✐♥❣ ❜✉t ❛

•

❚♦♠

•

❑❡♥

•

❙✐❞

•

◆❡❞

•

❇❡♥

❧✐st ✿

❖r✿ ✏❚♦♠✱ ❑❡♥✱ ❙✐❞✱ ◆❡❞✱ ❇❡♥✑✱ ✐♥ ❛♥② ♦r❞❡r✳

❲❛r♥✐♥❣✦

❡❧❡♠❡♥ts ♦❢ ❛ s❡t t❡r♠s ♦❢ ❛ s❡q✉❡♥❝❡

❆s t❤❡r❡ ✐s ♥♦ ♦r❞❡r✱ t❤❡

❛r❡♥✬t

t♦ ❜❡ ❝♦♥❢✉s❡❞ ✇✐t❤ t❤❡

❛s t❤❡

❧❛tt❡r ❛r❡ ♦r❞❡r❡❞✳

❚❤❡r❡ ✐s ❛ s♣❡❝✐✜❝ ♠❛t❤❡♠❛t✐❝❛❧ ♥♦t❛t✐♦♥ ❢♦r ✜♥✐t❡ s❡ts❀ ✇❡ ♣✉t t❤❡ ❧✐st ✐♥

❜r❛❝❡s ✿

▲✐st ♥♦t❛t✐♦♥ ❢♦r s❡ts

{A, B, C, D} ■t r❡❛❞s ✏t❤❡ s❡t ✇✐t❤ ❡❧❡♠❡♥ts

A, B, C, D✑✳ ❆❧❧ ♦❢ t❤❡s❡ ❛r❡ ❡q✉❛❧❧② ✈❛❧✐❞ r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ ♦✉r s❡t✿

{ ❚♦♠ , ❑❡♥ , ❙✐❞ , ◆❡❞ , ❇❡♥ } = { ◆❡❞ , ❑❡♥ , ❚♦♠ , ❇❡♥ , ❙✐❞ } = { ❇❡♥ , ❑❡♥ , ❙✐❞ , ❚♦♠ , ◆❡❞ } = ... ❊①❡r❝✐s❡ ✷✳✶✳✼ ❍♦✇ ♠❛♥② s✉❝❤ r❡♣r❡s❡♥t❛t✐♦♥s ❛r❡ t❤❡r❡❄ ❡❧❡♠❡♥ts❄

❍✐♥t✿

■♥ ❤♦✇ ♠❛♥② ✇❛②s ❝❛♥ ②♦✉ ♣❡r♠✉t❡ t❤❡s❡ ✜✈❡

✷✳✶✳

❙❡ts ❛♥❞ r❡❧❛t✐♦♥s

✶✵✶

❏✉st ❛s t❤❡ ❜♦②s ❤❛✈❡ ♥❛♠❡s✱ t❤❡ s❡t ❛❧s♦ ♥❡❡❞s ♦♥❡✳ ❲❡ ❝❛♥ ❝❛❧❧ t❤✐s s❡t ✏❚❡❛♠✑✱ ♦r ✏❇♦②s✑✱ ❡t❝✳ ❚♦ ❦❡❡♣ t❤✐♥❣s ❝♦♠♣❛❝t✱ ❧❡t✬s ❣✐✈❡ ✐t ❛ s❤♦rt ♥❛♠❡✱ s❛②

X={

❚♦♠

,

X✿ ❑❡♥

,

❙✐❞

❲❡ s❛② t❤❡♥ t❤❛t ❚♦♠ ✭❑❡♥✱ ❡t❝✳✮ ✐s ❛♥ ❡❧❡♠❡♥t ♦❢ s❡t

•

❚♦♠

❜❡❧♦♥❣s

• X ❝♦♥t❛✐♥s

t♦

X✱

X✱

,

◆❡❞

,

❇❡♥

}.

❛s ✇❡❧❧ ❛s✿

♦r

❚♦♠✳

❏✉st ❛s ✇❡ ✇❛♥t t♦ ❜❡ ❝❧❡❛r ✇❤❡♥ t✇♦ ♥✉♠❜❡rs ❛r❡ ❡q✉❛❧✱ ✇❡ ✇❛♥t t❤❡ s❛♠❡ ❝❧❛r✐t② ❢♦r s❡ts✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ✇✐❧❧ ❜❡ ❛ss✉♠❡❞ t♦ ❜❡ ❦♥♦✇♥✳

❉❡✜♥✐t✐♦♥ ✷✳✶✳✽✿ ❡q✉❛❧ s❡ts ❚✇♦ s❡ts

• •

X

❛♥❞

Y

❛r❡ s❛✐❞ t♦ ❜❡

❡✈❡r② ❡❧❡♠❡♥t ♦❢ ❡✈❡r② ❡❧❡♠❡♥t ♦❢

X Y

❡q✉❛❧

t♦ ❡❛❝❤ ♦t❤❡r ✐❢✿

✐s ❛❧s♦ ❛♥ ❡❧❡♠❡♥t ♦❢ ✐s ❛❧s♦ ❛♥ ❡❧❡♠❡♥t ♦❢

Y✱ X✳

❛♥❞ ❝♦♥✈❡rs❡❧②✱

❘❡♣❡t✐t✐♦♥s ❛r❡♥✬t ❛❧❧♦✇❡❞✦ ❖r✱ ❛t ❧❡❛st✱ t❤❡② ❛r❡ t♦ ❜❡ ❡❧✐♠✐♥❛t❡❞✿

{

❚♦♠

,

❑❡♥

,

❙✐❞

,

◆❡❞

,

❇❡♥

,

❇❡♥

r❡♠♦✈❡ r❡♣❡t✐t✐♦♥s✦

} −−−−−−−−−−−−−−−→ {

❚♦♠

,

❑❡♥

,

❙✐❞

,

◆❡❞

,

❇❡♥

}

■t✬s t❤❡ s❛♠❡ s❡t✦ ❲❡ ❝❛♥ ❢♦r♠ ♦t❤❡r s❡ts ❢r♦♠ t❤❡ s❛♠❡ ❡❧❡♠❡♥ts✳ ❲❡ ❝❛♥ ❝♦♠❜✐♥❡ t❤♦s❡ ✜✈❡ ❡❧❡♠❡♥ts ✐♥t♦ ❛♥② s❡t ✇✐t❤ ❛♥② ♥✉♠❜❡r ♦❢ ❡❧❡♠❡♥ts ❛s ❧♦♥❣ ❛s t❤❡r❡ ✐s ♥♦ r❡♣❡t✐t✐♦♥❀ ❢♦r ❡①❛♠♣❧❡✱ ✇❡ ❝❛♥ ❝r❡❛t❡ t❤❡s❡ ♥❡✇ s❡ts✿

T = { ❚♦♠ }, K = { ❑❡♥ }, S = { ❙✐❞ }, N = { A = { ❚♦♠ , ❑❡♥ }, B = { ❙✐❞ , ◆❡❞ }, ... Q = { ❚♦♠ , ❑❡♥ , ❙✐❞ }, ...

◆❡❞

},

...

❚❤❡ ❢♦❧❧♦✇✐♥❣ ✇✐❧❧ ❜❡ r♦✉t✐♥❡❧② ✉s❡❞✳

❉❡✜♥✐t✐♦♥ ✷✳✶✳✾✿ s✉❜s❡t ❆ s❡t

A

✐s ❝❛❧❧❡❞ ❛

s✉❜s❡t

♦❢ ❛ s❡t

X

✐❢ ❡✈❡r② ❡❧❡♠❡♥t ♦❢

A

✐s ❛❧s♦ ❛♥ ❡❧❡♠❡♥t ♦❢

X✳ ❚❤✐s ✐s ❤♦✇ ✇❡ ♠❛r❦ s✉❜s❡ts ✇❤❡♥ t❤❡ s❡t ✐s s❤♦✇♥✿

❊①❡r❝✐s❡ ✷✳✶✳✶✵ ❍♦✇ ♠❛♥② s✉❜s❡ts ♦❢

3

❡❧❡♠❡♥ts ❞♦❡s t❤❡ s❡t ❤❛✈❡❄ ❍✐♥t✿ ■♥ ❤♦✇ ♠❛♥② ✇❛②s ❝❛♥ ②♦✉ ❝❤♦♦s❡ t❤r❡❡

❡❧❡♠❡♥ts ♦✉t ♦❢ ✜✈❡❄

❲❡ ✇✐❧❧ ✉s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♥♦t❛t✐♦♥ t♦ ❝♦♥✈❡② t❤❛t ✐❞❡❛✿

❙✉❜s❡t A⊂X

✷✳✶✳

❙❡ts ❛♥❞ r❡❧❛t✐♦♥s

✶✵✷

❚❤❡ ♥♦t❛t✐♦♥ r❡s❡♠❜❧❡s t❤❡ ♦♥❡ ❢♦r ♥✉♠❜❡rs✿

1 < 2, 3 < 5✱

❡t❝✳ ■♥❞❡❡❞✱ ❛ s✉❜s❡t ✐s✱ ✐♥ ❛ s❡♥s❡✱ ✏s♠❛❧❧❡r✑

t❤❛♥ t❤❡ s❡t t❤❛t ❝♦♥t❛✐♥s ✐t✳

❲❛r♥✐♥❣✦

❆ s✉❜s❡t ❞♦❡s♥✬t ❤❛✈❡ t♦ ❜❡ ❧✐t❡r❛❧❧② s♠❛❧❧❡r ❜❡✲ ❝❛✉s❡ ❛ s❡t ✐s ❛ s✉❜s❡t ♦❢ ✐ts❡❧❢✳ ❋✉rt❤❡r♠♦r❡✱ ❛♥ ✐♥✜♥✐t❡ s❡t ♠✐❣❤t ❤❛✈❡ ❛ s✉❜s❡t ❥✉st ❛s ✐♥✜♥✐t❡✳✳✳ ❊①❛♠♣❧❡ ✷✳✶✳✶✶✿ ♣❧❛♥❡ s❤❛♣❡s ❲❡ s❡❡ s✉❜s❡ts ♦❢ ❣❡♦♠❡tr✐❝ ✜❣✉r❡s ✐♥ t❤❡ ♣❧❛♥❡✿

■♥ ❈❤❛♣t❡r ✶✱ ✇❡ st❛rt❡❞ ♦✉r st✉❞② ♦❢ ♥✉♠❜❡rs ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ s❡ts✳ ❲❡ ❝❤♦s❡ t♦ s♣❡❛❦ ♦❢ ❧♦❝❛t✐♦♥s s♣❛❝❡❞ ♦✈❡r ❛♥ ✐♥✜♥✐t❡ str❛✐❣❤t ❧✐♥❡ ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡

✐♥t❡❣❡rs✱ ❞❡♥♦t❡❞ ❜②

Z = {..., −3, −2, −1, 0, 1, 2, 3, ...} , ❛♥❞ ✇❡ s♣♦❦❡ ♦❢ t✐♠❡ ♠♦♠❡♥ts s♣❛❝❡❞ ♦✈❡r t❤❡

♥❛t✉r❛❧ ♥✉♠❜❡rs ✿

N = {0, 1, 2, 3, ...} .

■♥ t❤❡ ♠❡❛♥t✐♠❡✱ t❤❡ s❡t ♦❢ r❡❛❧ ♥✉♠❜❡rs ✐s ❞❡♥♦t❡❞ ❜②

R✳

■t ✐s ✈✐s✉❛❧✐③❡❞ ❛s t❤❡

x✲❛①✐s✿

❲❡ ❝❛♥ ❡①♣r❡ss t❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡s❡ s❡ts ✉s✐♥❣ t❤❡ ♥❡✇ ♥♦t❛t✐♦♥✿

N ⊂ Z ⊂ R. ❊①❡r❝✐s❡ ✷✳✶✳✶✷ ■s ❛ s❡q✉❡♥❝❡ ❛ s❡t❄

❚♦ ❝♦♥t✐♥✉❡ ✇✐t❤ ♦✉r ❡①❛♠♣❧❡✱ s✉♣♣♦s❡ t❤❡r❡ ✐s

❛♥♦t❤❡r✱ ✉♥r❡❧❛t❡❞✱ s❡t✱ s❛② Y ✱ t❤❡ s❡t ♦❢ t❤❡s❡ ❢♦✉r ❜❛❧❧s✿

✷✳✶✳ ❙❡ts ❛♥❞ r❡❧❛t✐♦♥s

✶✵✸

❏✉st ❛s X ✱ s❡t Y ❤❛s ♥♦ str✉❝t✉r❡✳ ❏✉st ❛s X ✱ ✐t✬s ❥✉st ❛ ❧✐st✿ Y

= { ❜❛s❦❡t❜❛❧❧ , t❡♥♥✐s , ❜❛s❡❜❛❧❧ , ❢♦♦t❜❛❧❧ } = { ❢♦♦t❜❛❧❧ , ❜❛s❡❜❛❧❧ , t❡♥♥✐s , ❜❛s❦❡t❜❛❧❧ } = ...

❲❡ ❝❛♥ r❡♠♦✈❡ ❜❛❧❧s ❢r♦♠ t❤❡ s❡t✱ ❝r❡❛t✐♥❣ s✉❜s❡ts ♦❢ Y ✳ ◆♦✇✱ ❧❡t✬s ♣✉t t❤❡ t✇♦ s❡ts✱ X ❛♥❞ Y ✱ ♥❡①t t♦ ❡❛❝❤ ♦t❤❡r ❛♥❞ ❛s❦ ♦✉rs❡❧✈❡s✿ ❆r❡ t❤❡s❡ t✇♦ s❡ts r❡❧❛t❡❞ t♦ ❡❛❝❤ ♦t❤❡r s♦♠❡❤♦✇❄

❨❡s✱ ❜♦②s ❧✐❦❡ t♦ ♣❧❛② s♣♦rts✦ ▲❡t✬s ♠❛❦❡ t❤✐s ✐❞❡❛ s♣❡❝✐✜❝✳ ❊❛❝❤ ❜♦② ♠❛② ❜❡ ✐♥t❡r❡st❡❞ ✐♥ ❛ ♣❛rt✐❝✉❧❛r s♣♦rt ♦r ❤❡ ♠❛② ♥♦t✳ ❋♦r ❡①❛♠♣❧❡✱ s✉♣♣♦s❡ t❤✐s ✐s ✇❤❛t ✇❡ ❦♥♦✇✿ • ❚♦♠ ❧✐❦❡s ❜❛s❦❡t❜❛❧❧✳

• ❇❡♥ ❧✐❦❡s ❜❛s❦❡t❜❛❧❧ ❛♥❞ t❡♥♥✐s✳ • ❑❡♥ ❧✐❦❡s ❜❛s❡❜❛❧❧ ❛♥❞ ❢♦♦t❜❛❧❧✳ • ❇❡♥ ❧✐❦❡s ❢♦♦t❜❛❧❧✳

❆♥❞ t❤❛t✬s ❛❧❧ ❡❛❝❤ ❧✐❦❡s✳ ❆s ❛ r❡s✉❧t✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿ ◮ ❆♥ ❡❧❡♠❡♥t ♦❢ s❡t X ✐s r❡❧❛t❡❞ t♦ ❛♥ ❡❧❡♠❡♥t ♦❢ s❡t Y ✳

■♥ ♦r❞❡r t♦ ✈✐s✉❛❧✐③❡ t❤❡s❡ r❡❧❛t✐♦♥s✱ ❧❡t✬s ❝♦♥♥❡❝t ❡❛❝❤ ❜♦② ✇✐t❤ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❜❛❧❧ ❜② ❛ ❧✐♥❡ s❡❣♠❡♥t ✇✐t❤ ❛rr♦✇s ❛t t❤❡ ❡♥❞s✱ ✇❤✐❧❡ t❤❡ t✇♦ s❡ts ♠❛② ❜❡ ♣❧❛❝❡❞ ❛r❜✐tr❛r✐❧② ❛❣❛✐♥st ❡❛❝❤ ♦t❤❡r✿

❚❤✐s ✈✐s✉❛❧✐③❛t✐♦♥ ❤❡❧♣s ✉s ❞✐s❝♦✈❡r t❤❛t ◆❡❞ ❞♦❡s♥✬t ❧✐❦❡ s♣♦rts ❛t ❛❧❧✳ ❆s ②♦✉ ❝❛♥ s❡❡✱ t❤✐s ✐s ❛ t✇♦✲s✐❞❡❞ ❝♦rr❡s♣♦♥❞❡♥❝❡✿ ◆❡✐t❤❡r ♦❢ t❤❡ t✇♦ ❡❧❡♠❡♥ts ❛t t❤❡ ❡♥❞s ♦❢ t❤❡ ❧✐♥❡ ❝♦♠❡s ✜rst ♦r s❡❝♦♥❞✳ ❚❤❡ s❛♠❡ ❛♣♣❧✐❡s t♦ t❤❡ s❡ts✿ ◆❡✐t❤❡r ♦❢ t❤❡ t✇♦ s❡ts ❝♦♠❡s ✜rst ♦r s❡❝♦♥❞✳ ■♥ ❢❛❝t✱ ✇❡ ❝❛♥ ❞❡r✐✈❡ t❤❡s❡ ♥❡✇ ❢❛❝ts ❛❜♦✉t t❤❡ ♣r❡❢❡r❡♥❝❡s ❡✐t❤❡r ❢r♦♠ t❤❡ ♦r✐❣✐♥❛❧ ❧✐st ♦r ❢r♦♠ t❤❡ ✐♠❛❣❡ ♦♥ t❤❡ r✐❣❤t✿ • ❇❛s❦❡t❜❛❧❧ ✐s ❧✐❦❡❞ ❜② ❚♦♠ ❛♥❞ ❇❡♥✳ • ❚❡♥♥✐s ✐s ❧✐❦❡❞ ❜② ❇❡♥✳

• ❇❛s❡❜❛❧❧ ✐s ❧✐❦❡❞ ❜② ❑❡♥✳

• ❋♦♦t❜❛❧❧ ✐s ❧✐❦❡❞ ❜② ❑❡♥ ❛♥❞ ❙✐❞✳

✷✳✶✳ ❙❡ts ❛♥❞ r❡❧❛t✐♦♥s

✶✵✹

❲❡ ❤❛✈❡✱ t❤❡r❡❢♦r❡✱ ❛ ❧✐st ♦❢ ♣❛✐rs ✿ • ❚♦♠ ✫ ❜❛s❦❡t❜❛❧❧ • ❇❡♥ ✫ ❜❛s❦❡t❜❛❧❧ • ❇❡♥ ✫ t❡♥♥✐s

• ❑❡♥ ✫ ❜❛s❡❜❛❧❧ • ❑❡♥ ✫ ❢♦♦t❜❛❧❧ • ❇❡♥ ✫ ❢♦♦t❜❛❧❧

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

❉❡✜♥✐t✐♦♥ ✷✳✶✳✶✸✿ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ s❡ts ❆♥② s❡t ♦❢ ♣❛✐rs (x, y)✱ ✇✐t❤ x t❛❦❡♥ ❢r♦♠ ❛ s❡t X ❛♥❞ y ❢r♦♠ ❛ s❡t Y ✱ ✐s ❝❛❧❧❡❞ ❛ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ s❡ts X ❛♥❞ Y ✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤✐s ✐s ❛ s❡t ♦❢ ❛rr♦✇s✳ ❚❤❡r❡ ♠❛② ❜❡ ♠❛♥② ❞✐✛❡r❡♥t r❡❧❛t✐♦♥s ❜❡t✇❡❡♥ ❛ ♣❛✐r s❡ts❀ ❧❡t✬s ❝❛❧❧ t❤✐s ♦♥❡ R✿

❲❛r♥✐♥❣✦ ❲❡ ❞♦♥✬t r❡q✉✐r❡ ❡✈❡r② ❡❧❡♠❡♥t t♦ ❤❛✈❡ ❛ ❝♦rr❡✲ s♣♦♥❞✐♥❣ ❡❧❡♠❡♥t ✐♥ t❤❡ ♦t❤❡r s❡t✳

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

ց r❡❧❛t✐♦♥✿

❉♦❡s t❤❡ ❜♦② ❧✐❦❡ t❤❡ ❜❛❧❧❄

❜❛❧❧

ր

❚❘❯❊

→

ր

❋❆▲❙❊

ց

❘❡❧❛t❡❞✦ ×

◆♦t r❡❧❛t❡❞✦

❲❤❡♥ t❤❡ s❡ts ❛r❡ ❧✐sts✱ r❡❧❛t✐♦♥s ❛r❡ t❛❜❧❡s✳ ▲❡t✬s ♠❛❦❡ ❛ t❛❜❧❡ ❢♦r R✦ ❲❡ ♣✉t t❤❡ ❜♦②s ✐♥ t❤❡ ❧❡❢t♠♦st ❝♦❧✉♠♥ ❛♥❞ t❤❡ ❜❛❧❧s ✐♥ t❤❡ t♦♣ r♦✇✳ ❚❤❡r❡ ❛r❡ 20 ❝❡❧❧s✿

■❢ t❤❡ ❜♦② ❧✐❦❡s t❤❡ s♣♦rt✱ ✇❡ ♣✉t ❛ ♠❛r❦ ✐♥ t❤❡ ❜♦②✬s r♦✇ ❛♥❞ t❤❡ ❜❛❧❧✬s ❝♦❧✉♠♥ ✭❧❡❢t✮✿

✷✳✶✳ ❙❡ts ❛♥❞ r❡❧❛t✐♦♥s

✶✵✺

❖r✱ ✇❡ ❝❛♥ ♣✉t t❤❡ ❜♦②s ✐♥ t❤❡ ✜rst r♦✇ ❛♥❞ t❤❡ ❜❛❧❧s ✐♥ t❤❡ ✜rst ❝♦❧✉♠♥ ✭r✐❣❤t✮✦ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ✢✐♣ t❤❡ t❛❜❧❡ ❛❜♦✉t ✐ts ❞✐❛❣♦♥❛❧✳ ❚❤❡s❡ ❛r❡ t✇♦ ✈✐s✉❛❧✐③❛t✐♦♥s ♦❢ t❤❡ s❛♠❡ r❡❧❛t✐♦♥✳ ❚❤✐s ✐s ✇❤❛t s✉❝❤ ❛ ✈✐s✉❛❧✐③❛t✐♦♥ ❧♦♦❦s ❧✐❦❡ ✇❤❡♥ ✇❡ ✉s❡ ❛ s♣r❡❛❞s❤❡❡t ✐♥st❡❛❞✿

❊①❡r❝✐s❡ ✷✳✶✳✶✹

❇❛s❡❞ ♦♥ t❤❡ r❡❧❛t✐♦♥

R

♣r❡s❡♥t❡❞ ❛❜♦✈❡✱ ❝r❡❛t❡ ❛ ♥❡✇ r❡❧❛t✐♦♥ ❝❛❧❧❡❞✱ s❛②✱

S✱

t❤❛t r❡❧❛t❡s t❤❡ ❜♦②s

❛♥❞ t❤❡ s♣♦rts t❤❡② ❞♦♥✬t ❧✐❦❡✳ ●✐✈❡ ❛♥ ❛rr♦✇ r❡♣r❡s❡♥t❛t✐♦♥s ❛♥❞ ❛ t❛❜❧❡ r❡♣r❡s❡♥t❛t✐♦♥s ♦❢

S✳

❊①❡r❝✐s❡ ✷✳✶✳✶✺

❆r❡ t❤❡r❡ ❛♥② r❡❧❛t✐♦♥s ♦♥ t❤❡ s✉❜s❡ts ♦❢ t❤❡ t✇♦ s❡ts❄

❆♥② ❝♦❧❧❡❝t✐♦♥ ✭❛ s❡t✮ ♦❢ ♠❛r❦s ✐♥ s✉❝❤ ❛ t❛❜❧❡ ❝r❡❛t❡s ❛ r❡❧❛t✐♦♥✱ ❛♥❞ ❝♦♥✈❡rs❡❧②✱ ❛ r❡❧❛t✐♦♥ ✐s ♥♦t❤✐♥❣ ❜✉t

❛ ❝♦❧❧❡❝t✐♦♥ ♦❢ ♠❛r❦s ✐♥ t❤✐s t❛❜❧❡✳ ❚❤r♦✉❣❤♦✉t t❤❡ ❡❛r❧② ♣❛rt ♦❢ t❤✐s ❜♦♦❦✱ ✇❡ ✇✐❧❧ ❝♦♥❝❡♥tr❛t❡ ♦♥ s❡ts t❤❛t ❝♦♥s✐st ♦❢ ♥✉♠❜❡rs✳ ❊✈❡♥ t❤♦✉❣❤ t❤❡ s❡t ♦❢ ♥✉♠❜❡rs ❞♦❡s ❤❛✈❡ ❛ str✉❝t✉r❡ ✭❈❤❛♣t❡r ✶✮✱ t❤❡ ✐❞❡❛s ♣r❡s❡♥t❡❞ ❛❜♦✈❡ st✐❧❧ ❛♣♣❧②✳ ❚♦ ✐❧❧✉str❛t❡ t❤❡s❡ ✐❞❡❛s✱ ❤♦✇ ❛❜♦✉t ✇❡ s✐♠♣❧② r❡♥❛♠❡ t❤❡ ❜♦②s ❛s ♥✉♠❜❡rs✱ ❜❛❧❧s ❛s ♥✉♠❜❡rs t♦♦✱

1 − 4✳

1 − 5❄

❆♥❞ ✇❡ r❡♥❛♠❡ t❤❡

❚❤❡♥ t❤❡ ❢♦r♠❡r ❜❡❧♦♥❣ t♦ t❤❡ s❡t ♦❢ r❡❛❧ ♥✉♠❜❡rs✱ ✇❤✐❧❡ t❤❡ ❧❛tt❡r ❜❡❧♦♥❣ t♦

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

❚❤❡s❡ ❛①❡s ❛r❡ ❛❧s♦ ❧❛❜❡❧❡❞ t♦ ❛✈♦✐❞ ❝♦♥❢✉s✐♦♥ ❜❡t✇❡❡♥ t❤❡ t✇♦ ✈❡r② ❞✐✛❡r❡♥t s❡ts✳

❋✉rt❤❡r♠♦r❡✱ ♦♥ t❤❡

r✐❣❤t✱ t❤❡ t❛❜❧❡ ✐s r♦t❛t❡❞ ✭90 ❞❡❣r❡❡s ❝♦✉♥t❡r❝❧♦❝❦✇✐s❡✮✳ ❚❤✐s t❛❜❧❡ ✐s t❤❡♥ ❝❛❧❧❡❞ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ r❡❧❛t✐♦♥✳ ❚❤❡ ♣❧❛❝❡♠❡♥ts ♦❢ t❤❡ t✇♦ s❡ts ✇✐t❤✐♥ t❤❡ t❛❜❧❡s ❝❛♥ st✐❧❧ ❜❡ ✐♥t❡r❝❤❛♥❣❡❞✳ ❊①❡r❝✐s❡ ✷✳✶✳✶✻

❲❤❡♥ t❤❡ r♦✇s ❛♥❞ t❤❡ ❝♦❧✉♠♥s ❛r❡ ✐♥t❡r❝❤❛♥❣❡❞✱ ✐s t❤❡r❡ ❛♥②t❤✐♥❣ t❤❛t ✐s ♣r❡s❡r✈❡❞❄

❊①❡r❝✐s❡ ✷✳✶✳✶✼

❋✐♥✐s❤ t❤❡ s❡♥t❡♥❝❡✿ ✏❚❤✐s r❡♥❛♠✐♥❣ ♦❢ t❤❡ ❜♦②s ✐s ❛ ❴❴❴✳✑

❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ t❤❡ ❡❧❡♠❡♥ts ♦❢ t❤❡ s❡ts r❡♥❛♠❡❞ ❛s ♥✉♠❜❡rs ✭❧❡❢t✮✱ t❤❡♥ ✇❡ ❝❛♣t✉r❡ t❤❡ r❡❧❛t✐♦♥ ❛s ❛ ❧✐st ♦❢ ♣❛✐rs ♦❢ ❡❧❡♠❡♥ts ♦❢

X

❛♥❞

Y

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

✷✳✷✳

❋✉♥❝t✐♦♥s

✶✵✻

❜② t❤❡ s♣r❡❛❞s❤❡❡ts✿

■t ✐s ❝❛❧❧❡❞ ❛ ✏s❝❛tt❡r ❝❤❛rt✑✳ ❊①❛♠♣❧❡ ✷✳✶✳✶✽✿ ♥❡t✇♦r❦s ❛s r❡❧❛t✐♦♥s

❚❤❡ ♣❧♦t ❜❡❧♦✇ r❡♣r❡s❡♥ts ❛ ♣♦ss✐❜❧❡ ♥❡t✇♦r❦ ♦❢

❢r✐❡♥❞s❤✐♣

❛♠♦♥❣ t❤❡ ❜♦②s✿

❲❡ ❝❛♥ st✐❧❧ r❡♣r❡s❡♥t t❤✐s ❛s ❛ r❡❧❛t✐♦♥❀ ✇❡ ❥✉st ❝❤♦♦s❡ t❤❡ s❡ts

X

❛♥❞

Y

t♦ ❜❡ t❤❡ s❛♠❡✳ ❚❤❡ ♥❛t✉r❡

♦❢ t❤✐s ❦✐♥❞ ♦❢ r❡❧❛t✐♦♥ ✐s♥✬t ❥✉st t✇♦✲s✐❞❡❞❀ ✐t✬s ✏s②♠♠❡tr✐❝✑✿ ■❢ ❚♦♠ ✐s ❛ ❢r✐❡♥❞ ♦❢ ❇❡♥✱ t❤❡♥ ❇❡♥ ✐s ❛❧s♦ ❛ ❢r✐❡♥❞ ♦❢ ❚♦♠✱ ❛♥❞ ✈✐❝❡ ✈❡rs❛✳ ❚❤❛t ✐s ✇❤② ❡❛❝❤ ❛rr♦✇ ✐♥ t❤❡ ❞✐❛❣r❛♠ ♦♥ t❤❡ ❧❡❢t ✐s r❡♣r❡s❡♥t❡❞ ❜②

t✇♦

♠❛r❦s ✐♥ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ r❡❧❛t✐♦♥ ♦♥ t❤❡ r✐❣❤t✳

❊①❡r❝✐s❡ ✷✳✶✳✶✾

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

X❄

✷✳✷✳ ❋✉♥❝t✐♦♥s

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

•

✏❲❤❛t s♣♦rts ❤❛s t❤❡ ❜♦② ♣❧❛②❡❞ t♦❞❛②❄✑ t♦✿

•

✏❲❤✐❝❤ s♣♦rt ❞♦❡s t❤❡ ❜♦②

♣r❡❢❡r

t♦ ♣❧❛②❄✑

❚❤❡ ✐❞❡❛ ✐s t❤❛t ❡✈❡r②♦♥❡✱ ❡✈❡♥ ◆❡❞✱ ❤❛s ❛ ♣r❡❢❡r❡♥❝❡ ❛♥❞ ❡①❛❝t❧② ♦♥❡✳ ❚❤✐s ✐s t❤❡ tr❛♥s✐t✐♦♥✿

✷✳✷✳

❋✉♥❝t✐♦♥s

✶✵✼

❚♦ ♠❛❦❡ s❡♥s❡ ♦❢ t❤✐s ♥❡✇ s❡t✉♣✱ ✇❡ ❤❛❞ t♦ ❡r❛s❡ ♦♥❡ ♦❢ t❤❡ t✇♦ ❛rr♦✇s t❤❛t st❛rt ❛t ❇❡♥ ❛♥❞ ♦♥❡ ♦❢ t❤❡ t✇♦ ❛rr♦✇s t❤❛t st❛rt ❛t ❑❡♥ ❛♥❞ ✇❡ ❤❛❞ t♦ ❛❞❞ ❛♥ ❛rr♦✇ ❢♦r ◆❡❞✳ ■♥ ❛ r❡❧❛t✐♦♥✱ t❤❡ t✇♦ s❡ts ✐♥✈♦❧✈❡❞ ♣❧❛② ❡q✉❛❧ r♦❧❡s✳ ■♥st❡❛❞✱ ✇❡ ♥♦✇ t❛❦❡ t❤❡ ♣♦✐♥t ♦❢ ✈✐❡✇ ♦❢ t❤❡ ❜♦②s✳ ❲❡ ✇✐❧❧ ❡①♣❧♦r❡ ❛ ♥❡✇ r❡❧❛t✐♦♥✿

•

❚♦♠ ♣r❡❢❡rs ❜❛s❦❡t❜❛❧❧✳

•

❇❡♥ ♣r❡❢❡rs ❜❛s❦❡t❜❛❧❧✳

•

◆❡❞ ♣r❡❢❡rs t❡♥♥✐s✳

•

❑❡♥ ♣r❡❢❡rs ❢♦♦t❜❛❧❧✳

•

❙✐❞ ♣r❡❢❡rs ❢♦♦t❜❛❧❧✳

❲❡ ♠♦✈❡ ❢r♦♠ ♦✉r t✇♦✲❡♥❞❡❞ ❛rr♦✇s ✭♦r ❧✐♥❡ s❡❣♠❡♥ts✮ t♦ r❡❣✉❧❛r ❛rr♦✇s✿

❲❡ ♠❛✐♥t❛✐♥ t❤❡ r✉❧❡✿ t❤❡r❡ ✐s ❡①❛❝t❧② ♦♥❡ ❣♦♥❡✿

X

❝♦♠❡s ✜rst✱

Y

y

❢♦r ❡❛❝❤

x✳

❆s ❛ r❡s✉❧t✱ t❤❡ ❡q✉❛❧✐t② ❜❡t✇❡❡♥ t❤❡ t✇♦ s❡ts ✐s

s❡❝♦♥❞✳

❚❤✐s ✐s ❛ s♣❡❝✐❛❧ ❦✐♥❞ ♦❢ r❡❧❛t✐♦♥ ❝❛❧❧❡❞ ❛

❢✉♥❝t✐♦♥ ❀ ❧❡t✬s ❝❛❧❧ t❤✐s ♦♥❡ F ✳

❲❤❛t ♠❛❦❡s ✐t s♣❡❝✐❛❧ ✐s t❤❛t t❤❡r❡

✐s ❡①❛❝t❧② ♦♥❡ ❜❛❧❧ ❢♦r ❡❛❝❤ ❜♦②✳ ❇❡❧♦✇ ✐s t❤❡ ❝♦♠♠♦♥ ♥♦t❛t✐♦♥✿

❋✉♥❝t✐♦♥ ❢r♦♠ s❡t t♦ s❡t F :X→Y ♦r

F

X −−−−→ Y ■t r❡❛❞s ✏❢✉♥❝t✐♦♥ t♦

❊❛❝❤ ❡❧❡♠❡♥t ♦❢

X

F

❢r♦♠

X

Y ✑✳

❤❛s ♦♥❧② ♦♥❡ ❛rr♦✇ ♦r✐❣✐♥❛t✐♥❣ ❢r♦♠ ✐t✳ ❚❤❡♥✱ t❤❡ t❛❜❧❡ ♦❢ t❤✐s ❦✐♥❞ ♦❢ r❡❧❛t✐♦♥ ♠✉st

❤❛✈❡ ❡①❛❝t❧② ♦♥❡ ♠❛r❦ ✐♥ ❡❛❝❤ r♦✇✿

✷✳✷✳

❋✉♥❝t✐♦♥s

❖✉r ❢✉♥❝t✐♦♥

✶✵✽

✐s ❛ ♣r♦❝❡❞✉r❡ t❤❛t ❛♥s✇❡rs t❤❡ q✉❡st✐♦♥✿ ❲❤✐❝❤ ❜❛❧❧ ❞♦❡s t❤✐s ❜♦② ♣r❡❢❡r t♦ ♣❧❛② ✇✐t❤❄ ■♥ ❛❧❧ t❤❡s❡ q✉❡st✐♦♥s✦ ❈♦♥✈❡rs❡❧②✱ ❛ ❢✉♥❝t✐♦♥ ✐s ♥♦t❤✐♥❣ ❜✉t t❤❡s❡ ❛♥s✇❡rs✳ ❊❛❝❤ ❛rr♦✇ ❝❧❡❛r❧② ✐♥♣✉t ✕ ❛♥ ❡❧❡♠❡♥t ♦❢ X ✕ ♦❢ t❤✐s ♣r♦❝❡❞✉r❡ ❜② ✐ts ❜❡❣✐♥♥✐♥❣ ❛♥❞ t❤❡ ♦✉t♣✉t ✕ ❛♥ ❡❧❡♠❡♥t ♦❢

F

❢❛❝t✱ ✐t ❛♥s✇❡rs ✐❞❡♥t✐✜❡s t❤❡

Y

✕ ❜② ✐ts ❡♥❞✳

❊❛❝❤ ❛rr♦✇ ✐♥ t❤❡ ❞✐❛❣r❛♠ ♦❢

F

❝♦rr❡s♣♦♥❞s t♦ ❛ r♦✇ ♦❢ t❤❡ t❛❜❧❡ ✭❛♥❞ ✈✐❝❡ ✈❡rs❛✮✳

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

❚❤❡ ❢✉♥❝t✐♦♥

F

❛❧❣❡❜r❛✐❝

❚❤❡ ✐♥❢♦r♠❛t✐♦♥

♠❛♥♥❡r✱ ❛s ❢♦❧❧♦✇s✿

✐s t❤❡♥ ❛ q✉❡st✐♦♥✲❛♥s✇❡r✐♥❣ ♠❛❝❤✐♥❡✿ ■❢ ②♦✉ ✐♥♣✉t t❤❡ ♥❛♠❡ ♦❢ t❤❡ ❜♦②✱ ✐t ✇✐❧❧ ♣r♦❞✉❝❡ t❤❡

♥❛♠❡ ♦❢ t❤❡ ❜❛❧❧ ❤❡ ♣r❡❢❡rs ❛s t❤❡ ♦✉t♣✉t✳

F

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

✇❤❡♥ t❤❡ ✐♥♣✉t ✐s

x✿

■♥♣✉t ❛♥❞ ♦✉t♣✉t ♦❢ ❢✉♥❝t✐♦♥

F (x) = y ♦r

F : x 7→ y ■t r❡❛❞s✿ ✏ F ♦❢

x

✐s

y ✑✳

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡

F(

✐♥♣✉t

)=

♦✉t♣✉t

❛♥❞

F :

✐♥♣✉t

7→

♦✉t♣✉t✳

❏✉st ❧✐❦❡ ❛♥② r❡❧❛t✐♦♥✱ ❛ ❢✉♥❝t✐♦♥ ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ✐♥ ❢✉❧❧ ❜② ♣r♦✈✐❞✐♥❣ ❛ ❧✐st ♦❢ ♣❛✐rs✱ ✐t✬s t❤❡

x

❛♥❞

y✳

❚❤✐s t✐♠❡✱

❧✐st ♦❢ ❛❧❧ ✐♥♣✉ts ❛♥❞ t❤❡✐r ♦✉t♣✉ts ✿

F( F( F( F( F(

) ) ❇❡♥ ) ❑❡♥ ) ❙✐❞ ) ❚♦♠

◆❡❞

= = = = =

❜❛s❦❡t❜❛❧❧ t❡♥♥✐s ❜❛s❦❡t❜❛❧❧ ❢♦♦t❜❛❧❧ ❢♦♦t❜❛❧❧

❚❤✐s ♥♦t❛t✐♦♥ ✇✐❧❧ ❜❡✱ ❜② ❢❛r✱ t❤❡ ♠♦st ❝♦♠♠♦♥ ✇❛② ♦❢ r❡♣r❡s❡♥t✐♥❣ ❢✉♥❝t✐♦♥s✳ ❚❤r♦✉❣❤♦✉t t❤❡ ❡❛r❧② ♣❛rt ♦❢ t❤✐s ❜♦♦❦✱ ✇❡ ✇✐❧❧ ❝♦♥❝❡♥tr❛t❡ ♦♥ ❢✉♥❝t✐♦♥s t❤❡ ✐♥♣✉ts ❛♥❞ t❤❡ ♦✉t♣✉ts ♦❢ ✇❤✐❝❤ ❛r❡

♥✉♠❜❡rs✳

❚♦ ✐❧❧✉str❛t❡ t❤✐s ✐❞❡❛✱ ❧❡t✬s ❛❣❛✐♥

1 − 4✳

r❡♥❛♠❡

t❤❡ ❜♦②s ❛s ♥✉♠❜❡rs✱

1 − 5✱

❚❤❡ t❛❜❧❡ ♦❢ ♦✉r r❡❧❛t✐♦♥ t❛❦❡s t❤✐s ❢♦r♠ ✭s❡❡♥ ♦♥ t❤❡ ❧❡❢t✮✿

❛♥❞ r❡♥❛♠❡ t❤❡ ❜❛❧❧s ❛s ♥✉♠❜❡rs✱

✷✳✷✳

❋✉♥❝t✐♦♥s

❲❤❛t ♠❛❦❡s t❤❡ t❛❜❧❡ ♦❢ ❛ ✈❛❧✉❡s ♦❢

F

✶✵✾

❢✉♥❝t✐♦♥

s♣❡❝✐❛❧ ✐s t❤❛t ✐t ♠✉st ❤❛✈❡ ❡①❛❝t❧② ♦♥❡ ♠❛r❦ ✐♥ ❡❛❝❤

❝♦❧✉♠♥✳

❚❤❡

❤❛✈❡ ❛❧s♦ ❜❡❡♥ r❡✇r✐tt❡♥ ✭❝❡♥t❡r✮✳ ❲❡ t❤❡♥ r♦t❛t❡ t❤❡ t❛❜❧❡ ❝♦✉♥t❡r❝❧♦❝❦✇✐s❡ ✭r✐❣❤t✮ ❜❡❝❛✉s❡ ✐t

✐s tr❛❞✐t✐♦♥❛❧ t♦ ❤❛✈❡ t❤❡ ✐♥♣✉ts ❛❧♦♥❣ ❛ ❤♦r✐③♦♥t❛❧ ❧✐♥❡ ✭❧❡❢t t♦ r✐❣❤t✮ ❛♥❞ t❤❡ ♦✉t♣✉ts ❛❧♦♥❣ ❛ ✈❡rt✐❝❛❧ ❧✐♥❡ ✭❜♦tt♦♠ t♦ t♦♣✮✳ ❚❤❡ ❧❛tt❡r t❛❜❧❡ ✐s ❝❛❧❧❡❞ t❤❡

❣r❛♣❤

♦❢ t❤❡ ❢✉♥❝t✐♦♥✳ ❚❤❡ ❛rr♦✇s ❛r❡ st✐❧❧ t❤❡r❡✿

❊①❡r❝✐s❡ ✷✳✷✳✶

❋✐♥✐s❤ t❤❡ s❡♥t❡♥❝❡✿ ✏❚❤✐s r❡♥❛♠✐♥❣ ♦❢ t❤❡ ❜♦②s ✭❛♥❞ t❤❡ ❜❛❧❧s✮ ✐s ❛❧s♦ ❛ ❴❴❴✳✑

❲❡ ❝❛♥ ♣✉t t❤❡ ❞❛t❛✱ ❛s ❜❡❢♦r❡✱ ✐♥ ❛

s♣r❡❛❞s❤❡❡t

❛♥❞ t❤❡♥ ♣❧♦t ✐t ❛✉t♦♠❛t✐❝❛❧❧②✿

❚❤❡r❡ ✐s ♦♥❧② ♦♥❡ ❝r♦ss ✐♥ ❡✈❡r② r♦✇✦ ❊①❛♠♣❧❡ ✷✳✷✳✷✿ r❡❧❛t✐♦♥s ❛♥❞ ❢✉♥❝t✐♦♥ ✐♥ s♣r❡❛❞s❤❡❡ts

❍❡r❡ ✐s ❛♥ ❡①❛♠♣❧❡ ♦❢ ❤♦✇ ❝♦♠♠♦♥ s♣r❡❛❞s❤❡❡ts ❛r❡ ❞✐s❝♦✈❡r❡❞ t♦ ❝♦♥t❛✐♥ r❡❧❛t✐♦♥s ❛♥❞ ❢✉♥❝t✐♦♥s✳ ❇❡❧♦✇✱ ✇❡ ❤❛✈❡ ❛ ❧✐st ♦❢ ❢❛❝✉❧t② ♠❡♠❜❡rs ✐♥ t❤❡ ✜rst ❝♦❧✉♠♥ ❛♥❞ ❛ ❧✐st ♦❢ ❢❛❝✉❧t② ❝♦♠♠✐tt❡❡s ✐♥ t❤❡ ✜rst r♦✇✳ ❆ ❝r♦ss ♠❛r❦ ✐♥❞✐❝❛t❡s ♦♥ ✇❤✐❝❤ ❝♦♠♠✐tt❡❡ t❤✐s ❢❛❝✉❧t② ♠❡♠❜❡r s✐ts✱ ✇❤✐❧❡ ✏❈✑ st❛♥❞s ❢♦r ✏❝❤❛✐r✑✿

✷✳✷✳

❋✉♥❝t✐♦♥s

✶✶✵

❚❤❡ t❛❜❧❡ ❣✐✈❡s ❛ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡s❡ s❡ts✿ X = { ❢❛❝✉❧t② } ❛♥❞ Y = { ❝♦♠♠✐tt❡❡s }✳ ❍♦✇❡✈❡r✱ t❤✐s ✐s ♥♦t ❛ ❢✉♥❝t✐♦♥✳ ❇✉t t❤❡r❡ ✐s ❛ ❢✉♥❝t✐♦♥ F : Y → X ✐♥❞✐❝❛t✐♥❣ t❤❡ ❝❤❛✐r ♦❢ t❤❡ ❝♦♠♠✐tt❡❡✳ ❊①❡r❝✐s❡ ✷✳✷✳✸

❚❤✐♥❦ ♦❢ ♦t❤❡r ❢✉♥❝t✐♦♥s ♣r❡s❡♥t ✐♥ t❤❡ s♣r❡❛❞s❤❡❡t✳ ❊①❡r❝✐s❡ ✷✳✷✳✹

❙✉❣❣❡st ❢✉♥❝t✐♦♥s ✐♥ t❤❡ s✐t✉❛t✐♦♥ ✇❤❡♥ ❛♥ ❡♠♣❧♦②❡r ♠❛✐♥t❛✐♥s ❛ ❧✐st ♦❢ ❡♠♣❧♦②❡❡s✱ ✇✐t❤ ❡❛❝❤ ♣❡rs♦♥ ✐❞❡♥t✐✜❡❞ ❛s ❛ ♠❡♠❜❡r ♦❢ ♦♥❡ ♦❢ t❤❡ ♣r♦❥❡❝ts✳ ❊①❡r❝✐s❡ ✷✳✷✳✺

❲❤❛t ❢✉♥❝t✐♦♥s ❞♦ ②♦✉ s❡❡ ❜❡❧♦✇❄

❆ ❝♦♠♠♦♥ ✇❛② t♦ ✈✐s✉❛❧✐③❡ t❤❡ ❝♦♥❝❡♣t ♦❢ s❡t ✕ ❡s♣❡❝✐❛❧❧② ✇❤❡♥ t❤❡ s❡ts ❝❛♥♥♦t ❜❡ r❡♣r❡s❡♥t❡❞ ❜② ♠❡r❡ ❧✐sts ✕ ✐s t♦ ❞r❛✇ ❛ s❤❛♣❡❧❡ss ❜❧♦❜ ✐♥ ♦r❞❡r t♦ s✉❣❣❡st t❤❡ ❛❜s❡♥❝❡ ♦❢ ❛♥② ✐♥t❡r♥❛❧ str✉❝t✉r❡ ♦r r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ❡❧❡♠❡♥ts✿

❆ ❝♦♠♠♦♥ ✇❛② t♦ ✈✐s✉❛❧✐③❡ t❤❡ ✐❞❡❛ ♦❢ ❛ ❢✉♥❝t✐♦♥ ❜❡t✇❡❡♥ s✉❝❤ s❡ts ✐s t♦ ❞r❛✇ ❛ ❢❡✇ ❛rr♦✇s✳

✷✳✷✳ ❋✉♥❝t✐♦♥s

✶✶✶

❚❤❡ ♥❡✇ ❝♦♥❝❡♣t ✐s ❝❡♥tr❛❧ t♦ ♦✉r st✉❞②✳

❉❡✜♥✐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❤❡r s❡t Y ✳ ❚❤❡ ❧❛tt❡r s❡t ✐s ❝❛❧❧❡❞ t❤❡ ♦✉t♣✉t s❡t ♦r t❤❡ ❝♦❞♦♠❛✐♥ ♦❢ f ✳ ❚❤❡ ✐♥♣✉ts ❛r❡ ❝♦❧❧❡❝t✐✈❡❧② ❝❛❧❧❡❞ t❤❡ ✐♥❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡ ❀ t❤❡ ♦✉t♣✉ts ❛r❡ ❝♦❧❧❡❝t✐✈❡❧② ❝❛❧❧❡❞ t❤❡ ❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡✳ ❲❡ ❛❧s♦ s❛② t❤❛t t❤❡ ✈❛❧✉❡ ♦❢ x ✉♥❞❡r f ✐s y ✳ ❚❤✐s ❞❡✜♥✐t✐♦♥ ❢❛✐❧s ❢♦r ❛ r❡❧❛t✐♦♥ t❤❛t ❤❛s t♦♦ ❢❡✇ ♦r t♦♦ ♠❛♥② ❛rr♦✇s ❢♦r ❛ ❣✐✈❡♥ x✳ ❇❡❧♦✇✱ ✇❡ ✐❧❧✉str❛t❡ ❤♦✇ t❤❡ r❡q✉✐r❡♠❡♥t ♠❛② ❜❡ ✈✐♦❧❛t❡❞✱ ✐♥ t❤❡ ❞♦♠❛✐♥ ✭❧❡❢t✮✿

❚❤❡s❡ ❛r❡ ♥♦t ❢✉♥❝t✐♦♥s✳ ▼❡❛♥✇❤✐❧❡✱ ✇❡ ❛❧s♦ s❡❡ ✇❤❛t s❤♦✉❧❞♥✬t ❜❡ r❡❣❛r❞❡❞ ❛s ✈✐♦❧❛t✐♦♥s✱ ✐♥ t❤❡ ❝♦❞♦♠❛✐♥ ✭r✐❣❤t✮✳ ❇❡❧♦✇ ✐s t❤❡ s✉♠♠❛r② ♦❢ t❤❡ ♠❛✐♥ ❝♦♥str✉❝t✐♦♥ ✐♥ t❤✐s s❡❝t✐♦♥✳

❚❤❡♦r❡♠ ✷✳✷✳✼✿ ❲❤❡♥ ❘❡❧❛t✐♦♥ ■s ❋✉♥❝t✐♦♥ Y ❛r❡ s❡ts ❛♥❞ R ✐s ❛ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ X ❛♥❞ Y ✳ ❚❤❡♥✿ • ❘❡❧❛t✐♦♥ R r❡♣r❡s❡♥ts s♦♠❡ ❢✉♥❝t✐♦♥ F ❢r♦♠ X t♦ Y ✱ F : X → Y, ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❢♦r ❡❛❝❤ x ✐♥ X t❤❡r❡ ✐s ❡①❛❝t❧② ♦♥❡ y ✐♥ Y s✉❝❤ t❤❛t x ❛♥❞ y ❛r❡ r❡❧❛t❡❞ ❜② R✳ • ❘❡❧❛t✐♦♥ R r❡♣r❡s❡♥ts s♦♠❡ ❢✉♥❝t✐♦♥ G ❢r♦♠ Y t♦ X ✱ G : Y → X, ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❢♦r ❡❛❝❤ y ✐♥ Y t❤❡r❡ ✐s ❡①❛❝t❧② ♦♥❡ x ✐♥ X s✉❝❤ t❤❛t x ❛♥❞ y ❛r❡ r❡❧❛t❡❞ ❜② R✳

❙✉♣♣♦s❡

X

❛♥❞

❊①❡r❝✐s❡ ✷✳✷✳✽ ❙♣❧✐t ❡✐t❤❡r ♣❛rt ✐♥t♦ ❛ st❛t❡♠❡♥t ❛♥❞ ✐ts ❝♦♥✈❡rs❡✳ ❲❤❡♥ ♦✉r s❡ts ❛r❡ s❡ts ♦❢ ♥✉♠❜❡rs✱ t❤❡ r❡❧❛t✐♦♥s ❛r❡ ♦❢t❡♥ ❣✐✈❡♥ ❜② ❢♦r♠✉❧❛s✳ ■♥ t❤❛t ❝❛s❡✱ t❤❡ ❛❜♦✈❡ ✐ss✉❡ ✐s r❡s♦❧✈❡❞ ✇✐t❤ ❛❧❣❡❜r❛✳

✷✳✸✳

❙❡q✉❡♥❝❡s ❛r❡ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s

✶✶✷

❊①❡r❝✐s❡ ✷✳✷✳✾ ❲❤❛t ❢✉♥❝t✐♦♥ ❝❛♥ ②♦✉ t❤✐♥❦ ♦❢ ❢r♦♠ t❤❡ s❡t

X

♦❢ t❤❡ ❜♦②s t♦ t❤❡ s❡ts ♦❢✿ ❧❡tt❡rs✱ ♥✉♠❜❡rs✱ ❝♦❧♦rs✱

❣❡♦❣r❛♣❤✐❝ ❧♦❝❛t✐♦♥s❄ ❚❤✐♥❦ ♦❢ ♦t❤❡rs✳

✷✳✸✳ ❙❡q✉❡♥❝❡s ❛r❡ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s

■♥ ❈❤❛♣t❡r ✶✱ ✇❡ ✈✐s✉❛❧✐③❡❞ ❛ s❡q✉❡♥❝❡ ♦❢ ♣♦s✐t✐♦♥s ♦❢ ❛ ❢❛❧❧✐♥❣ ❜❛❧❧ ❜② ✏s❡♣❛r❛t✐♥❣ s♣❛❝❡ ❛♥❞ t✐♠❡✑✳ ❲❡ ❣❛✈❡ t❤❡ ❢♦r♠❡r ❛ r❡❛❧ ♥✉♠❜❡r ❧✐♥❡ ❛♥❞ t❤❡ ❧❛tt❡r ❛ ❧✐♥❡ ♦❢ ✐♥t❡❣❡rs✿

❇✉t t❤❡ ❧❛tt❡r ✐s ❛❧s♦ ❛ s✉❜s❡t ♦❢ t❤❡ r❡❛❧ ♥✉♠❜❡rs✿

{1, 2, 3, 4, 5, 6, 7} ⊂ R . ❲❡ ❤❛✈❡✱ t❤❡r❡❢♦r❡✱ ❛ ❢✉♥❝t✐♦♥✳ ■♥ ❢❛❝t✱ t❤❡ s❡q✉❡♥❝❡ ✇❛s r❡♣r❡s❡♥t❡❞ ❛s ❛ ❧✐st ♦❢ ♣❛✐rs ♦❢ ✐♥♣✉ts ❛♥❞ ♦✉t♣✉ts✱ ❥✉st ❛s ❛♥② ❢✉♥❝t✐♦♥✿ t✐♠❡

n 1 2 3 4 5 6 7 X

❧♦❝❛t✐♦♥

an 36 35 32 27 20 11 0 Y

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

❉❡✜♥✐t✐♦♥ ✷✳✸✳✶✿ s❡q✉❡♥❝❡ ❛s ❢✉♥❝t✐♦♥ ❆

s❡q✉❡♥❝❡

✐s ❛ ❢✉♥❝t✐♦♥ ❢r♦♠ ❛ s✉❜s❡t ♦❢ t❤❡ ✐♥t❡❣❡rs✱

♥✉♠❜❡rs✿

A ⊂ Z✱

t♦ t❤❡ r❡❛❧

a : A → R. ❚❤❡ ❢♦❧❧♦✇✐♥❣ ✇✐❧❧ ❜❡ ♦✉r ♠❛✐♥ ✐♥t❡r❡st✳

❉❡✜♥✐t✐♦♥ ✷✳✸✳✷✿ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥ ❆

♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥

✐s ❛ ❢✉♥❝t✐♦♥ ❢r♦♠ ❛ s✉❜s❡t ♦❢ t❤❡ r❡❛❧ ♥✉♠❜❡rs✱

t♦ t❤❡ r❡❛❧ ♥✉♠❜❡rs✿

f : X → R.

X ⊂ R✱

✷✳✸✳ ❙❡q✉❡♥❝❡s ❛r❡ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s

✶✶✸

❙✐♥❝❡ ❡✈❡r② s✉❜s❡t ♦❢ t❤❡ ✐♥t❡❣❡rs ✐s ❛❧s♦ ❛ s✉❜s❡t ♦❢ t❤❡ r❡❛❧s✱

A ⊂ Z ⊂ R, ❡✈❡r② s❡q✉❡♥❝❡ ✐s ❛❧s♦ ❛ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥✳ ❍♦✇❡✈❡r✱ ♥♦t ❡✈❡r② ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥ ✐s ❛ s❡q✉❡♥❝❡✦ ❙✐♠♣❧② ♣✉t✱ a1/2 ❞♦❡s♥✬t ♠❛❦❡ s❡♥s❡✳ ▲❡t✬s ❝♦♠♣❛r❡✿

• ❆ s❡q✉❡♥❝❡✿ ❚❤❡ ✐♥♣✉t ✈❛r✐❛❜❧❡ ✐s n✱ ❛♥ ✐♥t❡❣❡r❀ t❤❡ ♦✉t♣✉t ✈❛r✐❛❜❧❡ ✐s y = an ✱ ❛ r❡❛❧ ♥✉♠❜❡r✳

• ❆ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥✿ ❚❤❡ ✐♥♣✉t ✈❛r✐❛❜❧❡ ✐s x✱ ❛ r❡❛❧ ♥✉♠❜❡r❀ t❤❡ ♦✉t♣✉t ✈❛r✐❛❜❧❡ ✐s y = f (x)✱ ❛♥♦t❤❡r r❡❛❧ ♥✉♠❜❡r✳ ❲❡ ❝♦♠♣❛r❡ t❤❡ ♥♦t❛t✐♦♥ t♦♦✱ s✐❞❡ ❜② s✐❞❡✿ ❋✉♥❝t✐♦♥ ✈s✳ s❡q✉❡♥❝❡

♥❛♠❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥

↓ f

x ↑




✈s✳

↓ a

n

↑

♥❛♠❡ ♦❢ t❤❡ ✐♥♣✉t ✈❛r✐❛❜❧❡ ✈❛❧✉❡ ♦❢ t❤❡ ✐♥♣✉t ✈❛r✐❛❜❧❡

f

↓
3 =5 ↑

✈s✳

a

✈❛❧✉❡ ♦❢ t❤❡ ♦✉t♣✉t ✈❛r✐❛❜❧❡

↓ 3

=5 ↑

❇✉t ✐s t❤✐s tr❛♥s✐t✐♦♥ ✕ t♦ ♠♦r❡ ❛ ❝♦♠♣❧❡① str✉❝t✉r❡ ✕ ✇♦rt❤ ✐t❄ ■❢ X ✐s t❤❡ s❡t ♦❢ t✐♠❡ ♠♦♠❡♥ts ❛♥❞ Y ✐s t❤❡ s❡t ♦❢ ❧♦❝❛t✐♦♥s ♦♥ t❤❡ r♦❛❞✱ ✇❡ ❝❛♥ s❡❡ ❛ ✇❛② t♦ st✉❞② ♠♦t✐♦♥ ✦ ■♥❞❡❡❞✱ ❛ ❢✉♥❝t✐♦♥ F : X → Y ❛♥s✇❡rs t❤❡ q✉❡st✐♦♥✿

◮ ❆t t❤✐s ♠♦♠❡♥t ♦❢ t✐♠❡✱ x✱ ✇❤❛t ✐s t❤❡ ❧♦❝❛t✐♦♥✱ y ✱ ✇❤❡r❡ ✇❡ ❛r❡❄

❚❤✐s ✐s ❛ ❢✉♥❝t✐♦♥ ❜❡❝❛✉s❡ ✇❡ ❝❛♥✬t ❜❡ ❛t t✇♦ ❧♦❝❛t✐♦♥s ❛t t❤❡ s❛♠❡ t✐♠❡✦ ❊①❛♠♣❧❡ ✷✳✸✳✸✿ ❢✉♥❝t✐♦♥s ❛r✐s❡ ❢r♦♠ ♠♦t✐♦♥

❍❡r❡ ✐s ❛ ✈❡r② s✐♠♣❧❡ ❡①❛♠♣❧❡✿ ❙✉♣♣♦s❡ ✇❡ ♠♦✈❡ t♦ t❤❡ ♥❡①t ♠✐❧❡st♦♥❡ ❡✈❡r② ♠✐♥✉t❡ ❢♦r 2 ♠✐♥✉t❡s✱ st❛rt✐♥❣ ❛t t❤❡ 0 ❧♦❝❛t✐♦♥ ♦♥ t❤❡ r❡❛❧ ❧✐♥❡✳ ❚♦ ♠❛❦❡ t❤✐s ♠♦r❡ ♣r❡❝✐s❡✱ ✇❡ ♠❛② ❛s❦✿ ◮ ❆t t✐♠❡ x✱ ✇❤✐❝❤ ♠✐❧❡st♦♥❡ y = F (x) ❞✐❞ ✇❡ s❡❡ ❧❛st❄ ❚❤❡♥ t❤❡ ❧✐st ♦❢ ✈❛❧✉❡s ♦❢ F ❛♣♣❡❛rs✿ t✐♠❡✱ X ✜rst ♠♦♠❡♥t s❡❝♦♥❞ ♠♦♠❡♥t t❤✐r❞ ♠♦♠❡♥t

❧♦❝❛t✐♦♥s✱ Y ✜rst ♠✐❧❡st♦♥❡ s❡❝♦♥❞ ♠✐❧❡st♦♥❡ t❤✐r❞ ♠✐❧❡st♦♥❡

❲✐t❤ ❛ s❡q✉❡♥❝❡✱ ✇❡ ✇♦✉❧❞ ❝❤♦♦s❡ t❤❡ ❞♦♠❛✐♥ t♦ ❜❡

X = {0, 1, 2} . ❛♥❞ t❤❡ ❝♦❞♦♠❛✐♥ t♦ ❜❡

Y = R.

✷✳✸✳

✶✶✹

❙❡q✉❡♥❝❡s ❛r❡ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s

❚❤❡♥ ♦✉r ❧✐st ❛♥❞ t❤❡ t❛❜❧❡ ❛r❡ ❛s ❢♦❧❧♦✇s✿ t✐♠❡✱ X 1 2 3 ❋✐♥❛❧❧②✱ t❤✐s ✐s t❤❡

❣r❛♣❤

❧♦❝❛t✐♦♥s✱ Y 1 2 3

❛♥❞

t✐♠❡ ❧♦❝❛t✐♦♥ 1 2 3

♦❢ F ✿

1 2 3 × × ×

❉r✐✈✐♥❣ ❛t ❛ ❝♦♥st❛♥t s♣❡❡❞✱ ✇❡ ♣r♦❣r❡ss 2 ♠✐❧❡s ❡✈❡r② ♠✐♥✉t❡✳ ❲❡ ♣r♦❞✉❝❡ ♠♦r❡ ❞❛t❛ ✇✐t❤ ❛ s♣r❡❛❞✲ s❤❡❡t✿

❚❤❡ ❣r❛♣❤ ♠✐❣❤t ❣✐✈❡ ❛♥ ✐♠♣r❡ss✐♦♥ t❤❛t ✇❡ s❦✐♣ ♠✐❧❡st♦♥❡s✦ ❚❤❡♥✱ ✐♥ ♦r❞❡r t♦ ❝❛♣t✉r❡ ♦✉r ♠♦t✐♦♥ ♠♦r❡ t❤♦r♦✉❣❤❧②✱ ✇❡ s✐♠♣❧② st❛rt ❛s❦✐♥❣ t❤❡ s❛♠❡ q✉❡st✐♦♥✿ ❆t t✐♠❡ x✱ ✇❤✐❝❤ ♠✐❧❡st♦♥❡ y = F (x) ❞✐❞ ✇❡ s❡❡ ❧❛st❄ ❜✉t ❡✈❡r② 30 s❡❝♦♥❞s✳ ❲❡ ✐♥tr♦❞✉❝❡ ❤❛❧❢✲♠✐♥✉t❡ ♠❛r❦s t♦ ♦✉r s❡t ♦❢ ✐♥♣✉ts✿

■s t❤✐s st✐❧❧ ❛ s❡q✉❡♥❝❡❄ ❨❡s✱ ✐❢ n ♠❡❛s✉r❡s ❤❛❧❢✲♠✐♥✉t❡s✳ ❲✐t❤ ❛ ❢✉♥❝t✐♦♥✱ ✇❡ s✐♠♣❧② ❦❡❡♣ t❤❡ s❡t ♦❢ ♦✉t♣✉ts Y = R ❛♥❞ ❝❤❛♥❣❡ t❤❡ s❡t ♦❢ ✐♥♣✉ts X ❢r♦♠ ✭❡✈❡r② ♠✐❧❡✮✿ t♦ ✭❡✈❡r② 1/2✲♠✐❧❡✮✿

X = {0, 1, 2, 3, 4, 5, ..., 9} X = {0, .5, 1, 1.5, 2, 2.5, 3, ..., 8.5, 9, 9.5, 10} .

❚❤❡ ♣r♦❜❧❡♠ ✐s s♦❧✈❡❞ ✕ ✉♥t✐❧ ✇❡ ❝❤♦♦s❡ t♦ ❞r✐✈❡ ❡✈❡♥ ❢❛st❡r✳ ❉r✐✈✐♥❣ 4 ♠✐❧❡s ♣❡r ♠✐♥✉t❡ ✇✐❧❧ r❡q✉✐r❡ t❤❡ ♦✉t♣✉ts t♦ ❜❡ ✭❡✈❡r② 1/4✲♠✐❧❡✮✿ ❆♥❞ s♦ ♦♥✿

X = {0, .25, .5, .75, 1, 1.25, 1.5, ... , 9.75, 10} .

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

✷✳✸✳ ❙❡q✉❡♥❝❡s ❛r❡ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s

✶✶✺

❧✐♥❡✳ ■t st❛rts t♦ ❧♦♦❦ ❧✐❦❡ ❛ r✉❧❡r ✿

❲❡ ❝♦♠♠♦♥❧② t❤✐♥❦ ♦❢ ♠♦t✐♦♥ ❛s ❛ ❝♦♥t✐♥✉♦✉s ♣r♦❣r❡ss t❤r♦✉❣❤ ♣❤②s✐❝❛❧ s♣❛❝❡✳ ❚❤✐s r❡q✉✐r❡s ✉s t♦ t❤✐♥❦ ♦❢ ❜♦t❤ s♣❛❝❡ ❛♥❞ t✐♠❡ ❛s ✐♥✜♥✐t❡❧② ❞✐✈✐s✐❜❧❡✳ ❇✉t ❤♦✇ ❞♦ ✇❡ ✈✐s✉❛❧✐③❡ s✉❝❤ ❢✉♥❝t✐♦♥s❄ ❲❡ st✐❧❧ r❡♣r❡s❡♥t t❤❡♠ ❛s s❡q✉❡♥❝❡s ♦❢ ♣❛✐rs ♦❢ ♥✉♠❜❡rs ✕ ❛♥❞ t❤❡♥ ♣❧♦t t❤❡✐r ❣r❛♣❤s ✕ ❜✉t ✇✐t❤ ❛ ❝❧❡❛r ✉♥❞❡rst❛♥❞✐♥❣ t❤❛t s♦♠❡ ♦❢ t❤❡ ✐♥♣✉ts ❛r❡ ♠✐ss✐♥❣✳

❲❡ ✐♥s❡rt ♠♦r❡ ✐♥♣✉ts ❛s ♥❡❝❡ss❛r②✳ ❲❤❡♥ t❤❡r❡ ❛r❡ ❡♥♦✉❣❤ ♦❢ t❤❡♠✱ t❤❡② st❛rt t♦ ❢♦r♠ ❛ ❝✉r✈❡✦ ❊①❡r❝✐s❡ ✷✳✸✳✹

❆ ❝❛r st❛rts ♠♦✈✐♥❣ ✇❡st ❢r♦♠ t♦✇♥ ❆ ❛t ❛ ❝♦♥st❛♥t s♣❡❡❞ ♦❢ 40 ♠✐❧❡s ❛♥ ❤♦✉r✳ ❚♦✇♥ ❇ ✐s ❧♦❝❛t❡❞ 50 ♠✐❧❡s ❡❛st ♦❢ ❆✳ ❘❡♣r❡s❡♥t t❤❡ ❞✐st❛♥❝❡ ❢r♦♠ t♦✇♥ ❇ t♦ t❤❡ ❝❛r ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✐♠❡✳ ❊①❡r❝✐s❡ ✷✳✸✳✺

❆ ❝❛r st❛rts ♠♦✈✐♥❣ ✇❡st ❢r♦♠ t♦✇♥ ❆ ❛t ❛ ❝♦♥st❛♥t s♣❡❡❞ ♦❢ 40 ♠✐❧❡s ❛♥ ❤♦✉r✳ ❆t t❤❡ s❛♠❡ t✐♠❡✱ ❛♥♦t❤❡r ❝❛r st❛rts ♠♦✈✐♥❣ ✇❡st ❢r♦♠ t♦✇♥ ❆ ❛t ❛ ❝♦♥st❛♥t s♣❡❡❞ ♦❢ 50 ♠✐❧❡s ❛♥ ❤♦✉r✳ ❘❡♣r❡s❡♥t t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❝❛rs ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✐♠❡✳ ❲❤❛t ✐❢ t❤❡ s❡❝♦♥❞ ❝❛r ✐s ♠♦✈✐♥❣ ❡❛st t♦♦❄ ❲❤❛t ✐❢ ✐t st❛rts 1 ❤♦✉r ❧❛t❡❄ ❊①❛♠♣❧❡ ✷✳✸✳✻✿ ❣r❛♣❤s ❛s ❝✉r✈❡s

■♥st❡❛❞ ♦❢ ♣r♦❞✉❝✐♥❣ ♠♦r❡ ❞❛t❛✱ ✐t ✐s ❝♦♠♠♦♥ t♦ ✜❧❧ ✐♥ t❤❡ ❣❛♣s ✐♥ ❛ ❣r❛♣❤ ✇✐t❤ ❛ str♦❦❡ ♦❢ ❛ ♣❡♥✿

✳✳✳ ♦r ✇✐t❤ ❛ ❝❧✐❝❦✿

✷✳✸✳

❙❡q✉❡♥❝❡s ❛r❡ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s

✶✶✻

❚❤❡ ❝♦♠♣✉t❡r ❝❛♥ ❛❧s♦ ♠❛❦❡ ❛ ❣✉❡ss ❛♥❞ t❤❡ r❡s✉❧t ✐s✱ ❛❣❛✐♥✱ ❛ ❝✉r✈❡✦ ❍❡r❡ ✐s ❛♥♦t❤❡r ❡①❛♠♣❧❡✿

❊①❡r❝✐s❡ ✷✳✸✳✼

❘❡♣r❡s❡♥t ❛ r♦✉♥❞ tr✐♣✳

❇② ❝❤♦♦s✐♥❣ ❛♥ ❛♣♣r♦♣r✐❛t❡ s❡t

Y

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

t❡♠♣❡r❛t✉r❡✱ ♣r❡ss✉r❡✱ ♣♦♣✉❧❛t✐♦♥✱ ♠♦♥❡②✱ ❡t❝✳ ❇♦t❤ ❢✉♥❝t✐♦♥s ❛♥❞ s❡q✉❡♥❝❡s ❝❛♥ ❜❡ ♣❛rt✐❛❧❧② ♦r ❢✉❧❧② r❡♣r❡s❡♥t❡❞ ❜② ❧✐sts ♦❢ ✈❛❧✉❡s✳ ■♥ ❛❞❞✐t✐♦♥✱ t❤❡② ❝❛♥ ❛❧s♦ ❜❡ ❞❡✜♥❡❞ ❜②

❢♦r♠✉❧❛s✳

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

an = n2

❛♥❞

f (x) = x2 .

❇♦t❤ ❛r❡ ❞❡s❝r✐❜❡❞ ❜② t❤❡ ❝♦♠♠❛♥❞✿ ✏❙q✉❛r❡ t❤❡ ✐♥♣✉t✦✑ ❚❤❡✐r t❛❜❧❡s ♦❢ ✈❛❧✉❡s ❛r❡ ✐❞❡♥t✐❝❛❧✱ ✐♥✐t✐❛❧❧②✿

n y = n2 0 0 1 1 2 4 9 3

x y = x2 0 0 1 1 2 4 9 3

✳ ✳ ✳

✳ ✳ ✳

✳ ✳ ✳

✳ ✳ ✳

❯♥❧✐❦❡ t❤❡ t❛❜❧❡ ♦❢ t❤❡ s❡q✉❡♥❝❡✱ t❤❡ t❛❜❧❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ♠✐ss❡s ♠♦r❡ r♦✇s✱ ♥♦t ❥✉st ❛t t❤❡ ❡♥❞ ❜✉t ❛❧s♦ t❤❡s❡✿ ❢♦r

x = .5, x =

√

2,

❡t❝✳ ❖♥❡ ❝❛♥ ❛❧s♦ s❡❡ t❤❡ ❞✐✛❡r❡♥❝❡ ✐❢ ✇❡ ♣❧♦t t❤❡ t✇♦ ❣r❛♣❤s t♦❣❡t❤❡r✿

❇❡t✇❡❡♥ ❛♥② t✇♦ ✐♥♣✉t ♦❢ t❤❡ s❡q✉❡♥❝❡ ✭r❡❞✮✱ t❤❡ ❢✉♥❝t✐♦♥ ♠✐❣❤t ❤❛✈❡ ❛ ✇❤♦❧❡ ✐♥t❡r✈❛❧ ♦❢ ❡①tr❛ ✐♥♣✉ts ✭❜❧✉❡✮✳ ❚❤✉s✱ ❡✈❡r② ❢✉♥❝t✐♦♥ ❡①❛♠♣❧❡✑ ✐s ♣r♦✈✐❞❡❞

y = f (x) ❝r❡❛t❡s n ❜② an = (−1) ✳

❛ s❡q✉❡♥❝❡

an = f (n)✱

❜✉t ♥♦t ♥❡❝❡ss❛r✐❧② ✈✐❝❡ ✈❡rs❛✳ ❆ ✏❝♦✉♥t❡r✲

❊①❡r❝✐s❡ ✷✳✸✳✽

●✐✈❡ ❡①❛♠♣❧❡s ♦❢ ♦t❤❡r s❡q✉❡♥❝❡s t❤❛t ❞♦♥✬t ♣r♦❞✉❝❡ ❢✉♥❝t✐♦♥s ✐♥ t❤✐s ♠❛♥♥❡r✳

✷✳✹✳ ❍♦✇ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s ❡♠❡r❣❡✿ ♦♣t✐♠✐③❛t✐♦♥

✶✶✼

✷✳✹✳ ❍♦✇ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s ❡♠❡r❣❡✿ ♦♣t✐♠✐③❛t✐♦♥

◮ P❘❖❇▲❊▼✿ ❆ ❢❛r♠❡r ✇✐t❤ 100 ②❛r❞s ♦❢ ❢❡♥❝✐♥❣ ♠❛t❡r✐❛❧ ✇❛♥ts t♦ ❜✉✐❧❞ ❛s ❧❛r❣❡ ❛ r❡❝t❛♥❣✉❧❛r ❡♥❝❧♦s✉r❡ ❛s ♣♦ss✐❜❧❡ ❢♦r ❤✐s ❝❛tt❧❡✳

❚❤❡ s❝♦♣❡ ♦❢ ♣♦ss✐❜✐❧✐t✐❡s ✐s ✐♥✜♥✐t❡✿

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

■t ♠❛❦❡s s❡♥s❡ ✐❢ ❜② ✏t❤❡ ❧❛r❣❡st ❡♥❝❧♦s✉r❡✑ ✇❡ ✇✐❧❧ ♠❡❛♥ t❤❡ ♦♥❡ ✇✐t❤ t❤❡ ❧❛r❣❡st ❛r❡❛✳ ❲❡ ✇✐❧❧ ❣♦ t❤r♦✉❣❤ ♠✉❧t✐♣❧❡ st❛❣❡s✱ ✐♥✐t✐❛❧❧② r❡❧②✐♥❣ ♦♥❧② ♦♥ ♦✉r ❝♦♠♠♦♥ s❡♥s❡ ✭❛♥❞ s♦♠❡ ♠✐❞❞❧❡ s❝❤♦♦❧ ♠❛t❤✮✳

✭✶✮ ❚r✐❛❧ ❛♥❞ ❡rr♦r✳

❲❡ st❛rt ❜② r❛♥❞♦♠❧② ❝❤♦♦s✐♥❣ ♣♦ss✐❜❧❡ ♠❡❛s✉r❡♠❡♥ts ♦❢ t❤❡ ❡♥❝❧♦s✉r❡ ❛♥❞ ❝♦♠♣✉t❡

✐ts ❛r❡❛ ✇✐t❤ t❤❡ ❢♦r♠✉❧❛✿ ❆r❡❛ ❲❡ st❛rt ✇✐t❤ ❛ r❡❝t❛♥❣❧❡ ✇✐t❤ ✇✐❞t❤

20

=

✇✐❞t❤

·

❛♥❞ ✐♥❝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✳✳✳

❖❢ ❝♦✉rs❡✱ t❤❡ ❛r❡❛ ✐s ❣❡tt✐♥❣ ❜✐❣❣❡r ❛♥❞ ❜✐❣❣❡r❀ ❤♦✇❡✈❡r✱ ❲❡ ✇✐❧❧ ♥❡❡❞ t♦ ❝♦❧❧❡❝t ♠♦r❡ ❞❛t❛✳

❞❡♣t❤✳

30

❜②

30

❣✐✈❡s ✉s

900✦

❚❤❡ ♣❛tt❡r♥ ✐s ✉♥❝❧❡❛r✳

▲❡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❤❡♥

• a

✐s ❢♦r t❤❡ ❛r❡❛✿

a = w · d.

✭✷✮ ❈♦❧❧❡❝t✐♥❣ ❞❛t❛ ✐♥ ❛ s♣r❡❛❞s❤❡❡t✳ ❝♦❧✉♠♥

W✱

❛♥❞ t❤❡ ❞❡♣t❤✱ ❝♦❧✉♠♥

❞❡♥t❧②✱ r✉♥ t❤r♦✉❣❤ t❤❡s❡

11

✇✐❞t❤ ❚♦❣❡t❤❡r✱ t❤❡r❡ ❛r❡

D✳

❲❡ ✜rst ♥❡❡❞ t♦ ❝♦♠♣✐❧❡ ❛❧❧ ♣♦ss✐❜❧❡ ❝♦♠❜✐♥❛t✐♦♥s ♦❢ t❤❡ ✇✐❞t❤✱

❲❡ ❝❤♦♦s❡ t♦ ❣♦ ❡✈❡r②

10

②❛r❞s✳ ❚❤❡♥ t❤❡ t✇♦ q✉❛♥t✐t✐❡s✱ ✐♥❞❡♣❡♥✲

♥✉♠❜❡rs✿

= 0, 10, 20, ..., 100

11 · 11 = 121

❛♥❞ ❞❡♣t❤

= 0, 10, 20, ..., 100 .

♣♦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❤❡♥

✷✳✹✳

✶✶✽

❍♦✇ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s ❡♠❡r❣❡✿ ♦♣t✐♠✐③❛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✮✿

❯♥❢♦rt✉♥❛t❡❧②✱ ✇❡ ❝❛♥✬t ❥✉st ❧♦♦❦ t❤r♦✉❣❤ t❤✐s ❝♦❧✉♠♥ ❛♥❞ ✜♥❞ t❤❡ ❧❛r❣❡st ♥✉♠❜❡r✦ ❚❤❡ r❡❛s♦♥ ✐s t❤❛t ✇❡ ♥❡❡❞ t♦ t❡st ✇❤❡t❤❡r ❛ ❣✐✈❡♥ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ✇✐❞t❤ ❛♥❞ ❞❡♣t❤ ✉s❡s ❡①❛❝t❧② 100 ②❛r❞s ♦❢ t❤❡ ❢❡♥❝✐♥❣ ♠❛t❡r✐❛❧✳ ■s t❤❡r❡ ❛ ❜❡tt❡r ✇❛②❄ ❚♦ ✐♥✈❡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✳ ✭✸✮ ❊st❛❜❧✐s❤✐♥❣ s❡ts ❢♦r t❤❡ ✈❛r✐❛❜❧❡s✳

t✇♦ s❡ts ♥❛♠❡❞ ❛❢t❡r t❤❡s❡ t✇♦ q✉❛♥t✐t✐❡s✿

❲❡ ❝❤♦♦s❡ t♦ ❝♦♥s✐❞❡r t❤❡ ❞✐♠❡♥s✐♦♥s ❡✈❡r② 10 ②❛r❞s ✈✐❛ t❤❡s❡

W = {0, 10, ..., 100}

❛♥❞

❚❤❡ t❛❜❧❡✱ ❛♥❞ t❤❡ s♣r❡❛❞s❤❡❡t✱ t❛❦❡s t❤❡ ❢♦r♠✿

D = {0, 10, ..., 100} .

W \D 0 10 ... 100 0 ? 1

✳✳ ✳

100

?

❲❤❛t ❞♦ ✇❡ ✜❧❧ ✐t ✇✐t❤❄ ❲❡ ❝❛♥✱ ❛♥❞ ✇✐❧❧✱ ✜❧❧ ✐t ✇✐t❤ t❤❡ ❛r❡❛s ❜✉t t❤❡r❡ ✐s s♦♠❡t❤✐♥❣ ♠♦r❡ ✉r❣❡♥t✿ t❤❡ ♣❡r✐♠❡t❡r ♦❢ t❤❡ ❡♥❝❧♦s✉r❡✳ ■♥❞❡❡❞✱ s✉❝❤ ❡♥❝❧♦s✉r❡s ❛s 0 × 0 ❛♥❞ 100 × 100 s✐♠♣❧② ❞♦♥✬t ♠❛❦❡ s❡♥s❡✦ ✭✹✮ ❊st❛❜❧✐s❤✐♥❣ ❛ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ✈❛r✐❛❜❧❡s✳

✇❤♦❧❡ 100 ②❛r❞s ♦❢ ❢❡♥❝✐♥❣✳ ❚❤✐s ♠❛❦❡s ✉♣ ✐ts ♣❡r✐♠❡t❡r✿

❋♦r s✐♠♣❧✐❝✐t②✱ ✇❡ ❛ss✉♠❡ t❤❛t ✇❡ ❛r❡ t♦ ✉s❡ t❤❡

❚❤❡r❡❢♦r❡✱ ❛ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ s❡ts W ❛♥❞ D ✐s ❞❡✜♥❡❞ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣✿ ◮ ❚✇♦ ♥✉♠❜❡rs w ❛♥❞ d ❛r❡ r❡❧❛t❡❞ ✇❤❡♥ t❤❡② ❢♦r♠ ❛♥ r❡❝t❛♥❣❧❡ t❤❡ ♣❡r✐♠❡t❡r p ♦❢ ✇❤✐❝❤ ✐s 100✿ p = 2(w + d) = 100 .

❲❡ ✜❧❧ t❤❡ t❛❜❧❡ ✇✐t❤ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ✈❛❧✉❡s ♦❢ t❤❡ ♣❡r✐♠❡t❡r✿ ❂✷✯✭❘❈✷✰❘✷❈✮

❚❤✐s ✐s t❤❡ r❡s✉❧t✿

✷✳✹✳

✶✶✾

❍♦✇ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s ❡♠❡r❣❡✿ ♦♣t✐♠✐③❛t✐♦♥

◆❡①t✱ ✇❡ ♠❛r❦ t❤❡ ❛❝❝❡♣t❛❜❧❡ ✈❛❧✉❡s✱ ❡q✉❛❧ t♦ 100✱ ♦❢ t❤❡ ♣❡r✐♠❡t❡r ✐♥ ②❡❧❧♦✇❀ t❤❡s❡ ❛r❡ t❤❡ ♦♥❧② ♦♥❡s ❛❧❧♦✇❡❞✳ ✭✺✮ ❱✐s✉❛❧✐③✐♥❣ t❤❡ r❡❧❛t✐♦♥✳ ❋♦r ♠♦r❡ ❛❝❝✉r❛t❡ r❡s✉❧ts✱ ✇❡ ♥❡❡❞ ♠♦r❡ ❞❛t❛❀ ✇❡ ❣♦ ❡✈❡r② s✐♥❣❧❡ ②❛r❞✿

W = {0, 1, 2, ..., 100}

❛♥❞

D = {0, 1, 2, ..., 100} .

❚❤❡♥ t❤❡ ♠❛♥✉❛❧ ❞❛t❛ ❛♥❛❧②s✐s ❛❜♦✈❡ ✐s♥✬t ♣♦ss✐❜❧❡ ❛♥②♠♦r❡✿ ❲❡ ❤❛✈❡ ❛ 101 × 101 t❛❜❧❡ ✇✐t❤ 10, 201 ♣❛✐rs✳ ❲❡ ✜❧❧ t❤❡ t❛❜❧❡ ✇✐t❤ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ♣❡r✐♠❡t❡r p ✉s✐♥❣ t❤❡ s❛♠❡ s♣r❡❛❞s❤❡❡t✳ ❲❡ t❤❡♥ ❤✐❣❤❧✐❣❤t ✕ ❛✉t♦♠❛t✐❝❛❧❧② ✕ t❤❡ ❝❡❧❧s ✇❤❡r❡ t❤❡ ✈❛❧✉❡ ✐s ❡①❛❝t❧② 100✿

❚❤❡s❡ ❛r❡ t❤❡ ♦♥❧② ♦♥❡s ❛❧❧♦✇❡❞ ❛♥❞ t❤❡② s❡❡♠ t♦ ❢♦r♠ ❛ str❛✐❣❤t ❧✐♥❡✦ ✭✻✮ ❈♦♠♣✉t✐♥❣ t❤❡ q✉❛♥t✐t② t♦ ❜❡ ♠❛①✐♠✐③❡❞✳ ❲❡ ♥❡①t ✉s❡ 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❛♠❡ ❝♦❧✉♠♥s✱ r❡s♣❡❝t✐✈❡❧②✳ ❆s ❛ r❡s✉❧t✱ t❤❡ t❛❜❧❡ ✐s ✜❧❧❡❞ ✇✐t❤ t❤❡s❡ ✈❛❧✉❡s ✭s❤♦✇♥ ❢♦r t❤❡ 11 × 11 t❛❜❧❡✮✿

■❢ ✇❡ ❜r✐♥❣ t❤❡ ②❡❧❧♦✇ ❧✐♥❡ ♦❢ t❤❡ ❛❧❧♦✇❛❜❧❡ ♣❡r✐♠❡t❡rs t♦ t❤✐s t❛❜❧❡✱ ✇❡ ♥♦t✐❝❡ t❤❛t t❤❡ ❧❛r❣❡st ❛❧❧♦✇❛❜❧❡ ❛r❡❛s s❡❡♠ t♦ ❜❡ ❜❡t✇❡❡♥ 20 × 30 ❛♥❞ 30 × 20✳ ◆♦✇ ✇❡ ❝♦♥s✐❞❡r t❤❡ ❜✐❣❣❡r t❛❜❧❡ ✭101 × 101✮ ❛♥❞ ❝♦❧♦r ✕ ❛✉t♦♠❛t✐❝❛❧❧② ✕ t❤❡ ❝❡❧❧s ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❛r❡❛ ✐t ❝♦♥t❛✐♥s✿

✷✳✹✳

✶✷✵

❍♦✇ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s ❡♠❡r❣❡✿ ♦♣t✐♠✐③❛t✐♦♥

❚❤❡ ✈❛❧✉❡s ❣r♦✇ ✕ ❢r♦♠ r❡❞ t♦ ❣r❡❡♥ ✕ ✐♥ t❤❡ ❞✐❛❣♦♥❛❧ ❞✐r❡❝t✐♦♥✦ ✭✼✮ ❊st✐♠❛t✐♥❣ t❤❡ ♠❛①✐♠✉♠✳ ▼❛t❝❤✐♥❣ t❤❡ ♣✐❝t✉r❡s ♦❢ t❤❡ ♣❡r✐♠❡t❡rs ❛♥❞ t❤❡ ❛r❡❛s✱ ✇❡ ❞✐s❝♦✈❡r t❤❛t

t❤❡ ❧❛r❣❡st ❛r❡❛ ♠✉st ❜❡ s♦♠❡✇❤❡r❡ ✭❤❛❧❢✇❛②❄✮ ❜❡t✇❡❡♥ t❤❡ t✇♦ ❡①tr❡♠❡s✱ 0 × 50 ❛♥❞ 50 × 0✱ ❛t t❤❡ t✇♦ ❡♥❞s ♦❢ t❤❡ ②❡❧❧♦✇ ❧✐♥❡✳ ▼❛②❜❡ ❡✈❡♥ s♦♠❡✇❤❡r❡ ✭❤❛❧❢✇❛②❄✮ ❜❡t✇❡❡♥ 20 ❛♥❞ 30✳ ❈♦✉❧❞ ✐t ❜❡ 25 × 25❄

■t✬s ❛ ❢❛✐r ❣✉❡ss ❜✉t t❤❡r❡ ♠✉st ❜❡ ❛ ❜❡tt❡r ✇❛②✦ ❚❤❡ ♣r♦❜❧❡♠ ✐s t❤❛t s❡❧❡❝t✐♥❣ t❤❡ ❛❧❧♦✇❛❜❧❡ ❞❛t❛ ❢r♦♠ t❤❡ ✇❤♦❧❡ t❛❜❧❡ ♦❢ ♣❛✐rs ♦❢ w ❛♥❞ d ✐s t♦♦ ❝✉♠❜❡rs♦♠❡✳ ■t ✇♦✉❧❞ ❤❡❧♣ ✐❢ ✇❡ ❤❛❞ ❛ ❞✐r❡❝t ✢♦✇ ♦❢ ❞❛t❛ ❢r♦♠ w ✭♦r d✮ t♦ a✱ ❛s ❛ ❢✉♥❝t✐♦♥✳ ❚♦ t❤❛t ❡♥❞✱ ✇❡ r❡♣r❡s❡♥t t❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ W ❛♥❞ D ❛s ❛ ❢✉♥❝t✐♦♥✳ ✭✽✮ ❊st❛❜❧✐s❤✐♥❣ ❛ ❢✉♥❝t✐♦♥ ❢r♦♠ t❤❡ r❡❧❛t✐♦♥✳ ❲❡ ❡①♣r❡ss d ✐♥ t❡r♠s ♦❢ w ✳ ❲❡ t❛❦❡ ♦✉r r❡❧❛t✐♦♥

2(w + d) = 100 ❛♥❞ s♦❧✈❡ ✐t ❢♦r d✿

d = 50 − w .

❚❤❡♥ t❤❡r❡ ✐s ❡①❛❝t❧② ♦♥❡ d ❢♦r ❡❛❝❤ w❀ ✐t✬s ❛ ❢✉♥❝t✐♦♥✦ ❍❡r❡ ✐s ✐ts ❧✐st ♦❢ ✈❛❧✉❡s ✭10 ②❛r❞s ❛t ❛ t✐♠❡✮✿

w 0 10 20 30 40 50

d 50 40 30 20 10 0

■❢ ✇❡ ❝❤♦♦s❡ t♦ ❣♦ 1 ②❛r❞ ❛t t✐♠❡✱ ✇❡ ❤❛✈❡ 51 r♦✇s ❛♥❞ ✇❡ ❤❛✈❡ t♦ ♣✉t t❤♦s❡ ✐♥ ❛ ♥❡✇ s♣r❡❛❞s❤❡❡t✳ ❚❤❡ ✜rst ❝♦❧✉♠♥ ✐s ❢♦r t❤❡ ✇✐❞t❤ w r✉♥♥✐♥❣ t❤r♦✉❣❤✿ 0, 1, 2, ..., 50✳ ❚❤❡ s❡❝♦♥❞ ✐s ❢♦r t❤❡ ❞❡♣t❤ d✱ ❡✈❛❧✉❛t❡❞ ❜②

❂✺✵✲❘❈❬✲✶❪ r❡❢❡rr✐♥❣ t♦ t❤❡ ♣r❡✈✐♦✉s ❝♦❧✉♠♥✳ ✭✾✮ ❱✐s✉❛❧✐③✐♥❣ t❤❡ ❢✉♥❝t✐♦♥✳ ❚❤❡s❡ t✇♦ ❝♦❧✉♠♥s ❛r❡ s❡❡♥ ♦♥ t❤❡ ❧❡❢t ❜❡❧♦✇ ❛♥❞ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❞❡♣t❤

❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ✇✐❞t❤ ✐s ❛ str❛✐❣❤t ❧✐♥❡ ✭♠✐❞❞❧❡✮✿

❚❤❡ ❛r❡❛ a ✐s ❛❧s♦ ❝♦♠♣✉t❡❞ ✐♥ t❤❡ ♥❡①t ❝♦❧✉♠♥ ❛s t❤❡ ♣r♦❞✉❝t ♦❢ w ❛♥❞ d✿

❂❘❈❬✲✷❪✯❘❈❬✲✶❪

✷✳✹✳ ❍♦✇ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s ❡♠❡r❣❡✿ ♦♣t✐♠✐③❛t✐♦♥

✶✷✶

❛♥❞ t❤❡♥ ♣❧♦tt❡❞ ❛❣❛✐♥st t❤❡ ✇✐❞t❤ ✭r✐❣❤t✮✳ ❙✐♥❝❡ t❤❡r❡ ✐s ❡①❛❝t❧② ♦♥❡

a

❢♦r ❡❛❝❤

w✱

t❤✐s ✐s ❛❧s♦ ❛ ❢✉♥❝t✐♦♥✦

■ts ❣r❛♣❤ ✐s ❛ ❝✉r✈❡✳

✭✶✵✮ ❊st✐♠❛t✐♥❣ t❤❡ ♠❛①✐♠✉♠✳ ❝♦rr❡s♣♦♥❞✐♥❣ ❛r❡❛

a = 25 · 25 = 625

▲♦♦❦✐♥❣ ❛t t❤❡ ❧❛st ♣❧♦t✱

w = 25

s❡❡♠s t♦ ❜❡ ❛ ❝❧❡❛r ❝❤♦✐❝❡ ✇✐t❤ t❤❡

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

❊①❡r❝✐s❡ ✷✳✹✳✶ ❲❤❛t ❤❛♣♣❡♥s ✐❢✱ ✐♥st❡❛❞✱ ✇❡ ❡①♣r❡ss

w

✐♥ t❡r♠s ♦❢

d❄

❚❤❡ ♣r♦❜❧❡♠ s❡❡♠s t♦ ❜❡ s♦❧✈❡❞✱ ❜✉t ✉♥❢♦rt✉♥❛t❡❧②✱ t❤❡ ♣❧♦t ❤❛s ❣❛♣s ✦ ❲❤❛t ✐❢ ✇❡ ❤❛✈❡ ♦✈❡r❧♦♦❦❡❞ ❛ ✇✐❞t❤ t❤❛t ✐t ❣✐✈❡s ✉s t❤❡ ❛r❡❛ ❜✐❣❣❡r t❤❛♥

625❄

▲❡t✬s ❝♦♥s✐❞❡r t❤❡ ❢✉♥❝t✐♦♥ t❤❛t ✇❡ ❤❛✈❡ ❜✉✐❧t ✐♥ t❤✐s s♣r❡❛❞s❤❡❡t✿

a

❞❡♣❡♥❞s ♦♥

w

♦♥❧②✳

❲❤❛t ✐s t❤✐s

❢✉♥❝t✐♦♥❄ ❲✐t❤ ♠♦r❡ ♠✐❞❞❧❡ s❝❤♦♦❧ ❛❧❣❡❜r❛✱ ✇❡ ♠❛❦❡ t❤✐s ❢✉♥❝t✐♦♥ ❡①♣❧✐❝✐t✿

a = wd = w(50 − w) . ❲❡ ❝❛♥ ♥♦✇ ❡❛s✐❧② ♣❧♦t

100

♦r

100, 000

♣♦✐♥ts ❛t ❛s s♠❛❧❧ ✐♥❝r❡♠❡♥ts ❛s ✇❡ ❧✐❦❡✿

❲✐t❤ ♥♦ ✈✐s✐❜❧❡ ❣❛♣s✱ t❤❡ ❛♥s✇❡r r❡♠❛✐♥s t❤❡ s❛♠❡✿

w = 25, a = 625 . ❆r❡ t❤❡r❡ ✐♥✈✐s✐❜❧❡ ❣❛♣s❄ ❍♦✇ ❝❛♥ ✇❡ ❜❡ s✉r❡❄ ❚❤❡ ❣r❛♣❤ ♦❢ ❛ q✉❛❞r❛t✐❝ ❢✉♥❝t✐♦♥✱ s✉❝❤ ❛s

a = w(50 − w) = −w2 + 50w , ✐s ❛ ♣❛r❛❜♦❧❛ ✭❈❤❛♣t❡r ✹✮✳ ■ts ✏t✐♣✑✱ ❝❛❧❧❡❞ t❤❡ ✈❡rt❡①✱ ❧✐❡s ❤❛❧❢✲✇❛② ❜❡t✇❡❡♥ t❤❡s❡ t✇♦ ♣♦✐♥ts ✭0 ❛♥❞

x=

50✮✿

0 + 50 = 25 . 2

▲❡t✬s r❡✈✐❡✇ ♦✉r s♦❧✉t✐♦♥✳ ❲❡ ♥❛♠❡❞ t❤❡ q✉❛♥t✐t✐❡s t❤❛t ❛♣♣❡❛r ✐♥ t❤❡ ✐♥✐t✐❛❧ ♣r♦❜❧❡♠ ❛♥❞ t❤❡♥ tr❛♥s❧❛t❡❞ ✐ts s❡♥t❡♥❝❡s ✐♥t♦ ❛❧❣❡❜r❛✳ ❚❤❡ r❡s✉❧t ✇❛s t❤❡ ❢♦❧❧♦✇✐♥❣ ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠ ✿

◮

w ❛♥❞ d w + d = 50✳

❋✐♥❞ s✉❝❤ ✈❛❧✉❡s ♦❢

t♦ t❤❡ r❡❧❛t✐♦♥

❚❤❡♥ ✉s✐♥❣ t❤❡ ❢✉♥❝t✐♦♥

◮

t❤❛t

0 ≤ w ≤ 50, 0 ≤ d ≤ 50

❛♥❞

a = wd

✐s t❤❡ ❧❛r❣❡st✱ s✉❜❥❡❝t

d = 50−w ❞❡r✐✈❡❞ ❢r♦♠ t❤❡ r❡❧❛t✐♦♥ t♦ ❡❧✐♠✐♥❛t❡ d ❢r♦♠ t❤❡ ♣r♦❜❧❡♠ ❜② s✉❜st✐t✉t✐♦♥✿

❋✐♥❞ s✉❝❤ ❛ ✈❛❧✉❡ ♦❢

w

t❤❛t

0 ≤ w ≤ 50

❛♥❞

a = w(50 − w)

✐s t❤❡ ❧❛r❣❡st✳

❊①❡r❝✐s❡ ✷✳✹✳✷ ❙♦❧✈❡ ❛ ♠♦❞✐✜❡❞ ♣r♦❜❧❡♠✿ ❆ r✐✈❡r ✐s ❛❞❥❛❝❡♥t t♦ t❤❡ ❡♥❝❧♦s✉r❡✱ ✇❤✐❝❤ ✇✐❧❧ ❤❛✈❡✱ ❝♦♥s❡q✉❡♥t❧②✱ t❤r❡❡ s✐❞❡s✳

✷✳✹✳ ❍♦✇ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s ❡♠❡r❣❡✿ ♦♣t✐♠✐③❛t✐♦♥

✶✷✷

❊①❡r❝✐s❡ ✷✳✹✳✸

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

❙❡♠✐❝✐r❝❧❡s ❛r❡

❛tt❛❝❤❡❞ t♦ t❤❡ r❡❝t❛♥❣❧❡s✿

❲❡✬✈❡ s♦❧✈❡❞ t❤❡ ♣r♦❜❧❡♠✱ ❜✉t ♦✉r ❦♥♦✇❧❡❞❣❡ ✐s t♦♦ ❧✐♠✐t❡❞ ✇❤❡♥ ❢✉♥❝t✐♦♥s ❛r❡ ♠♦r❡ ❝♦♠♣❧❡① t❤❛♥ ❥✉st ❛ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧✳ ❈❛❧❝✉❧✉s ✭❈❤❛♣t❡r ✷❉❈✲✸✮ ✇✐❧❧ ❤❡❧♣✳ ❊①❛♠♣❧❡ ✷✳✹✳✹✿ ♦♣t✐♠✐③❛t✐♦♥

❋✐♥❞ t✇♦ ♥✉♠❜❡rs ✇❤♦s❡ ❞✐✛❡r❡♥❝❡ ✐s

100

❛♥❞ t❤❡ ♣r♦❞✉❝t ✐s ❛ ♠✐♥✐♠✉♠✳

❙t❡♣ ✶✳ ❉❡❝♦♥str✉❝t✿ ✶✳ t✇♦ ♥✉♠❜❡rs✱ ✇❤♦s❡ ✷✳ ❞✐✛❡r❡♥❝❡ ✐s

100✱

❛♥❞

✸✳ t❤❡ ♣r♦❞✉❝t ✐s ❛ ♠✐♥✐♠✉♠✳ ❚r❛♥s❧❛t❡✿

x ✐s t❤❡ ✜rst ♥✉♠❜❡r✱ y x − y = 100; ♣r♦❞✉❝t✿ p = xy ✱ ♠✐♥✐♠✐③❡ p✳

✶✳ ✐♥tr♦❞✉❝❡ t❤❡ ✏✈❛r✐❛❜❧❡s✑✿

✐s t❤❡ s❡❝♦♥❞ ♥✉♠❜❡r❀

✷✳ ❝♦♥str❛✐♥t✿ ✸✳

p

✐s t❤❡✐r

❚❤✐s ✐s ❛ ♠❛t❤ ♣r♦❜❧❡♠ ♥♦✇✳ ❙t❡♣ ✷✳

❊❧✐♠✐♥❛t❡ t❤❡ ❡①tr❛ ✈❛r✐❛❜❧❡s t♦ ❝r❡❛t❡ ❛ ❢✉♥❝t✐♦♥ ♦❢ s✐♥❣❧❡ ✈❛r✐❛❜❧❡ t♦ ❜❡ ♠❛①✐♠✐③❡❞ ♦r

♠✐♥✐♠✐③❡❞✳ ❚❤❡ ❝♦♥str❛✐♥t✱ ❛♥ ❡q✉❛t✐♦♥ ❝♦♥♥❡❝t✐♥❣ t❤❡ ✈❛r✐❛❜❧❡s✱ ✐s

x − y = 100 . ❙♦❧✈❡ t❤❡ ❡q✉❛t✐♦♥ ❢♦r

y✿ y = x − 100 ,

❛♥❞ ❡❧✐♠✐♥❛t❡

y

❢r♦♠

p

❜② s✉❜st✐t✉t✐♦♥✿

p = xy = x(x − 100) . ❙t❡♣ ✸✳ ❖♣t✐♠✐③❡ t❤✐s ❢✉♥❝t✐♦♥✿

p(x) = x(x − 100) = x2 − 100x . ❚❤❡ t✇♦ ❡♥❞✲♣♦✐♥ts ♦❢ t❤❡ ✐♥t❡r✈❛❧ ❛r❡

0

❛♥❞

100❀

t❤❡r❡❢♦r❡✱ t❤❡ ✈❡rt❡① ♦❢ t❤✐s ♣❛r❛❜♦❧❛ ❝♦rr❡s♣♦♥❞s

t♦✿

x = 50 . ❙t❡♣ ✹✳ Pr♦✈✐❞❡ t❤❡ ❛♥s✇❡r ✉s✐♥❣ t❤❡ ♦r✐❣✐♥❛❧ ❧❛♥❣✉❛❣❡ ♦❢ t❤❡ ♣r♦❜❧❡♠✿ ❙✉❜st✐t✉t❡

x ✐♥t♦ y ✱ ❛s ❢♦❧❧♦✇s✿

y = x − 100 = 50 − 100 = −50 . ❆♥s✇❡r✿ ❚❤❡ t✇♦ ♥✉♠❜❡rs ❛r❡

50

❛♥❞

−50✳

❚❤✐s ✐s t❤❡ s✉♠♠❛r② ♦❢ t❤❡ ❛♥❛❧②s✐s ♦❢ t❤❡ ❢✉♥❝t✐♦♥ t❤❛t r❡♣r❡s❡♥ts t❤❡ ♣r❡❢❡r❡♥❝❡s ♦❢ t❤❡ ❜♦②s ❢♦r ❞✐✛❡r❡♥t ❣❛♠❡s ♣r❡s❡♥t❡❞ ❡❛r❧✐❡r ✐♥ t❤❡ ❝❤❛♣t❡r✿

✷✳✹✳

❍♦✇ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s ❡♠❡r❣❡✿ ♦♣t✐♠✐③❛t✐♦♥

✶✷✸

❚❤❡ t✇♦ ❣r❛♣❤s r❡♣r❡s❡♥t t❤❡ s❛♠❡ ❢✉♥❝t✐♦♥✦ ❚❤❡② ♦♥❧② ❧♦♦❦ ❞✐✛❡r❡♥t ❜❡❝❛✉s❡ ✇❡ ❤❛✈❡ r❡❛rr❛♥❣❡❞ t❤❡ ❡❧❡♠❡♥ts ♦❢ t❤❡ ❞♦♠❛✐♥✱ X ✱ ❛♥❞ t❤❡ ❝♦❞♦♠❛✐♥✱ Y ✳ ❙✉❝❤ ❛ ♠♦✈❡ ✐s ♥♦ ❧♦♥❣❡r ♣♦ss✐❜❧❡ ✇❤❡♥ ✇❡ ❞❡❛❧ ✇✐t❤ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s ❜❡❝❛✉s❡ ♥✉♠❜❡rs ❤❛✈❡ ❛♥ ✐♥❤❡r❡♥t str✉❝t✉r❡✱ ❛♥ ♦r❞❡r✳ ❲❡ ✉s❡ t❤❡ t❛❜❧❡s ❛♥❞ t❤❡ ❣r❛♣❤s ♦❢ ❢✉♥❝t✐♦♥s t♦ ❞✐s❝♦✈❡r ♣❛tt❡r♥s ✐♥ t❤❡ ❞❛t❛✳ ❍♦✇❡✈❡r✱ t❤✐s ✐s ♦♥❧② ♣♦ss✐❜❧❡ ✇❤❡♥ t❤❡ s❡ts t❤❡♠s❡❧✈❡s ❤❛✈❡ str✉❝t✉r❡✳ ❋♦r ❡①❛♠♣❧❡✱ ❛ ❞❡❝❦ ♦❢ ❝❛r❞s r❡♠❛✐♥s t❤❡ s❛♠❡ ❞❡❝❦ ❛❢t❡r ✐t✬s ❜❡❡♥ s❤✉✤❡❞ ❜✉t t❤❡r❡ ✐s ❛❧s♦ ❛ ❤✐❡r❛r❝❤✐❝❛❧ r❡❧❛t✐♦♥ ✇✐t❤✐♥ t❤❡ ❞❡❝❦ t❤❛t ♠❛❦❡s ❛❧❧ t❤❡ ❞✐✛❡r❡♥❝❡ t♦ t❤❡ ♣❧❛②❡rs✳ ❚❤❡ s✐♠♣❧❡st ❡①❛♠♣❧❡ ♦❢ ❛ s❡t ✇✐t❤ ❛ str✉❝t✉r❡ ✐s ❛ s❡t ♦❢ ❧♦❝❛t✐♦♥s ♦♥ ❛ str❛✐❣❤t r♦❛❞✳ ❲❡ ❝❤♦♦s❡ ♠✐❧❡st♦♥❡s t♦ ❜❡ s✉❝❤ ❛s ❛ s❡t✳ ■t ✐s t❤❡✐r ♦r❞❡r t❤❛t ♠❛❦❡s ✐t ✐♠♣♦ss✐❜❧❡ t♦ r❡s❤✉✤❡ t❤❡♠ ✇✐t❤♦✉t ❧♦s✐♥❣ ✐♠♣♦rt❛♥t ✐♥❢♦r♠❛t✐♦♥✳ ❲❡ ✇✐❧❧ ✉s❡ t❤❛t t♦ ♦✉r ❛❞✈❛♥t❛❣❡✳ ❲❡ ✈✐s✉❛❧✐③❡ t❤❡ s❡t ♦❢ ♠✐❧❡st♦♥❡s ❛s ♠❛r❦✐♥❣s ♦♥ ❛ str❛✐❣❤t ❧✐♥❡✱ ❛❝❝♦r❞✐♥❣ t♦ t❤❡✐r ♦r❞❡r ✭... < 1 < 2 < 3 < ...✮✿

❚❤❡ s❛♠❡ r❡♣r❡s❡♥t❛t✐♦♥ ✐s ❛❧s♦ ✉s❡❞ ❢♦r t✐♠❡✳ ❊✈❡r② ♠❛r❦✐♥❣ ♦♥ ❛ ❧✐♥❡ ✭❛♥♦t❤❡r ❧✐♥❡✦✮ ✐♥❞✐❝❛t❡s ❛ ♠♦♠❡♥t ♦❢ t✐♠❡ ✇❤❡♥ s♦♠❡ r❡♣❡❛t❛❜❧❡ ❡✈❡♥t✱ s✉❝❤ ❛s ❛ ❜❡❧❧ r✐♥❣✐♥❣ ♦r ❛ ❝❧♦❝❦✬s ❤❛♥❞ ♣❛ss✐♥❣ ❛ ♣❛rt✐❝✉❧❛r ♣♦s✐t✐♦♥✱ ♦❝❝✉rs✳ ❊①❛♠♣❧❡ ✷✳✹✳✺✿ ❤✐❞❞❡♥ ♣❛tt❡r♥s

❚❤❡ t❛❜❧❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ♦♥ t❤❡ ❧❡❢t ❤❛s ♥♦ ❛♣♣❛r❡♥t ♣❛tt❡r♥ ✕ ✉♥t✐❧ ✇❡ r❡✲❛rr❛♥❣❡ t❤❡ r♦✇s ❛❝❝♦r❞✐♥❣ t♦ t❤✐s ♦r❞❡r✿

❙✐♠✐❧❛r❧②✱ ❛ s❡❡♠✐♥❣❧② r❛♥❞♦♠ ❧✐st ♦❢ ♣❛✐rs ♦❢ ♥✉♠❜❡rs✱ x ❛♥❞ y = F (x)✱ ♣r♦❞✉❝❡s ❛ str❛✐❣❤t ❧✐♥❡ ✇❤❡♥ ♣❧♦tt❡❞ ❛❣❛✐♥st ♣r♦♣❡r❧② ♦r❞❡r❡❞ ♥✉♠❜❡rs✿

✷✳✺✳ ❙❡t ❜✉✐❧❞✐♥❣

✶✷✹

✷✳✺✳ ❙❡t ❜✉✐❧❞✐♥❣

❙♦ ❢❛r✱ ❛❧❧ ♥✉♠❡r✐❝❛❧ s❡ts ✐♥ t❤✐s ❝❤❛♣t❡r ❤❛✈❡ ❜❡❡♥ s✉❜s❡ts ♦❢ t❤❡ s❡t ♦❢ r❡❛❧ ♥✉♠❜❡rs R✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ♥✉♠❡r✐❝❛❧ s❡ts ❡♠❡r❣❡ ❛s ❞♦♠❛✐♥s ❛♥❞ ❝♦❞♦♠❛✐♥s ♦❢ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s✳ ❚❤❡② ♠❛② ❛❧s♦ ❝♦♠❡ ❢r♦♠ s♦❧✈✐♥❣ ❡q✉❛t✐♦♥s✳ ❊①❛♠♣❧❡ ✷✳✺✳✶✿ s❡ts ❢r♦♠ ❡q✉❛t✐♦♥s

❯♥❧❡ss ❡♥t✐r❡❧② ♥♦♥s❡♥s✐❝❛❧✱ ❡✈❡r② st❛t❡♠❡♥t ✐♥ ♠❛t❤❡♠❛t✐❝s ✐s tr✉❡ ♦r ❢❛❧s❡✿ 1 + 1 = 2 ❚❘❯❊ 1 + 1 = 3 ❋❆▲❙❊

❲❤❛t ❛❜♦✉t t❤✐s✿

x + 1 = 2 ❚❘❯❊ ❖❘ ❋❆▲❙❊❄

■t ❞❡♣❡♥❞s ♦♥ x✱ ♦❢ ❝♦✉rs❡✳ ❲❡ ❝❛♥✱ t❤❡r❡❢♦r❡✱ ✉s❡ ❡q✉❛t✐♦♥s t♦ ❢♦r♠ s❡ts✳ ❈♦♥s✐❞❡r t❤❡s❡✿ • ❲❡ ❢❛❝❡ t❤❡ ❡q✉❛t✐♦♥ x + 2 = 5✳ ❆❢t❡r s♦♠❡ ✇♦r❦✱ ✇❡ ✜♥❞✿ x = 3✳ • ❲❡ ❢❛❝❡ t❤❡ ❡q✉❛t✐♦♥ 3x = 15✳ ❆❢t❡r s♦♠❡ ✇♦r❦✱ ✇❡ ✜♥❞✿ x = 5✳ ■s t❤❡r❡ ♠♦r❡❄ • ❲❡ ❢❛❝❡ t❤❡ ❡q✉❛t✐♦♥ x2 − 3x + 2 = 0✳ ❆❢t❡r s♦♠❡ ✇♦r❦✱ ✇❡ ✜♥❞✿ x = 1✳ ■s t❤❛t ✐t❄ • ❲❡ ❢❛❝❡ t❤❡ ❡q✉❛t✐♦♥ x2 + 1 = 0✳ ❆❢t❡r ❛❧❧ t❤❡ ✇♦r❦✱ ✇❡ ❝❛♥✬t ✜♥❞ ❛♥② x✳ ❙❤♦✉❧❞ ✇❡ ❦❡❡♣ tr②✐♥❣❄ ❍❡r❡✱ x ✐s ❛ ❧❛❜❡❧ t❤❛t st❛♥❞s ❢♦r ❛♥ ✉♥s♣❡❝✐✜❡❞ ♥✉♠❜❡r t❤❛t ✐s ♠❡❛♥t t♦ s❛t✐s❢② t❤✐s ❝♦♥❞✐t✐♦♥✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ s❡❡❦ s✉❝❤ ♥✉♠❜❡rs t❤❛t✱ ✇❤❡♥ t❤❡② r❡♣❧❛❝❡ x ✐♥ t❤❡ ❡q✉❛t✐♦♥✱ ✇❡ s❡❡ ❛ tr✉❡ st❛t❡♠❡♥t✳ ■t ❝♦✉❧❞ s✐♠♣❧② ❜❡ tr✐❛❧ ❛♥❞ ❡rr♦r✳ ▲❡t✬s t❛❦❡ t❤❡ ✜rst ❡q✉❛t✐♦♥✿ x + 2 = 5.

❲❡ r❡♣❧❛❝❡ x ✇✐t❤ ❛ ♥✉♠❜❡r ❛❜♦✈❡ ♦r ✇r✐t❡ ❛ ♥✉♠❜❡r ✐♥ t❤❡ ❜❧❛♥❦ sq✉❛r❡ ❜❡❧♦✇✿  + 1 = 5.

❋♦r ❡①❛♠♣❧❡✿ • ■s x = 1 ❛ s♦❧✉t✐♦♥❄ P❧✉❣ ✐t ✐♥ t❤❡ ❡q✉❛t✐♦♥✿ x + 2 = 5 ❜❡❝♦♠❡s (1) + 2 = 5✳ ❋❆▲❙❊✳ ❚❤✐s ✐s ♥♦t ❛ s♦❧✉t✐♦♥✳

✷✳✺✳ ❙❡t ❜✉✐❧❞✐♥❣

•

■s

x=2

✶✷✺

x+2=5

❛ s♦❧✉t✐♦♥❄ P❧✉❣ ✐t ✐♥ t❤❡ ❡q✉❛t✐♦♥✿

❜❡❝♦♠❡s

(2) + 2 = 5✳ ❋❆▲❙❊✳

❚❤✐s ✐s ♥♦t

❛ s♦❧✉t✐♦♥✳

(3) + 2 = 5✳ ❚❘❯❊✳

•

■s

•

❙❤♦✉❧❞ ✇❡ st♦♣ ♥♦✇❄ ❲❤② ✇♦✉❧❞ ✇❡❄ ❋♦r ❛❧❧ ✇❡ ❦♥♦✇✱ t❤❡r❡ ♠❛② ❜❡ ♠♦r❡ s♦❧✉t✐♦♥s✳

x=3

❛ s♦❧✉t✐♦♥❄ P❧✉❣ ✐t ✐♥ t❤❡ ❡q✉❛t✐♦♥✿

x+2 = 5

❜❡❝♦♠❡s

❚❤✐s ✐s ❛

s♦❧✉t✐♦♥✳

❲❡ ♥❡✈❡r s❛② t❤❛t ✇❡ ❤❛✈❡ ❢♦✉♥❞ ✏t❤❡✑ s♦❧✉t✐♦♥ ✉♥❧❡ss ✇❡ ❦♥♦✇ ❢♦r s✉r❡ t❤❛t t❤❡r❡ ✐s ♦♥❧② ♦♥❡✳

❊①❡r❝✐s❡ ✷✳✺✳✷

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

❊①❡r❝✐s❡ ✷✳✺✳✸

❙♦❧✈❡ t❤❡s❡ ❡q✉❛t✐♦♥s✿

(a) x2 + 2x + 1 = 0,

(b)

x = 0, x

(c)

❇✉t ✇❤❛t ❞♦❡s ✐t ♠❡❛♥ t♦ s♦❧✈❡ ❛♥ ❡q✉❛t✐♦♥❄ ❲❡ ❤❛✈❡ tr✐❡❞ t♦ ✜♥❞

x = 1. x

x t❤❛t s❛t✐s✜❡s t❤❡ ❡q✉❛t✐♦♥✳✳✳

❇✉t ✇❤❛t

❛r❡ ✇❡ s✉♣♣♦s❡❞ t♦ ❤❛✈❡ ❛t t❤❡ ❡♥❞ ♦❢ ♦✉r ✇♦r❦❄ ▲❡t✬s ❣♦ ❜❛❝❦ t♦ ♦✉r r✉♥♥✐♥❣ ❡①❛♠♣❧❡ ♦❢ ❜♦②s ❛♥❞ ❜❛❧❧s✿

■t t❡❧❧s ✉s ✇❤❛t ❣❛♠❡ ❡❛❝❤ ❜♦② ♣r❡❢❡rs✳ ❲❤❛t ❛❜♦✉t t❤❡ ♦t❤❡r ✇❛② ❛r♦✉♥❞❄ ❲❤✐❝❤ ❜♦②s ♣r❡❢❡r ❛ ♣❛rt✐❝✉❧❛r ❣❛♠❡❄

•

❲❤✐❝❤ ❜♦②s ♣r❡❢❡r ❜❛s❦❡t❜❛❧❧❄

•

❲❤✐❝❤ ❜♦②s ♣r❡❢❡r t❡♥♥✐s❄ ✏❥✉st ◆❡❞✑✳

•

❲❤✐❝❤ ❜♦②s ♣r❡❢❡r ❜❛s❡❜❛❧❧❄ ✏◆♦ ♦♥❡✑✳

•

❲❤✐❝❤ ❜♦②s ♣r❡❢❡r ❢♦♦t❜❛❧❧❄ ✏❑❡♥ ❛♥❞ ❙✐❞ ♦♥❧②✑✳

❚❤❡ ❛♥s✇❡r ✐s♥✬t ✏❚♦♠✑✱ ❛♥❞ ✐t ✐s♥✬t ✏❇❡♥✑❀ ✐t✬s ✏❚♦♠ ❛♥❞ ❇❡♥ ❛♥❞

♥♦❜♦❞② ❡❧s❡✑✳

❇✉t ❡❛❝❤ q✉❡st✐♦♥ ✕ ♦♥❡ ❢♦r ❡❛❝❤ ❡❧❡♠❡♥t ♦❢ t❤❡ ❝♦❞♦♠❛✐♥

Y

✕ ✐s ❛❧s♦ ❛♥ ❡q✉❛t✐♦♥ ✿

•

❋✐♥❞

•

❋✐♥❞

x

✇✐t❤

F (x) =

t❡♥♥✐s✳ ◆❡❞ ✐s t❤❡ s♦❧✉t✐♦♥✳

•

❋✐♥❞

x

✇✐t❤

F (x) =

❜❛s❡❜❛❧❧✳ ◆♦ s♦❧✉t✐♦♥s✳

•

❋✐♥❞

x

✇✐t❤

F (x) =

❢♦♦t❜❛❧❧✳ ❑❡♥ ❛♥❞ ❙✐❞ ❛r❡ t❤❡ s♦❧✉t✐♦♥s✳

x

✇✐t❤

F (x) =

❜❛s❦❡t❜❛❧❧✳ ❚♦♠ ✐s ❛ s♦❧✉t✐♦♥✱ ❛♥❞ ❇❡♥ ✐s ❛ s♦❧✉t✐♦♥✳ ❈♦♠❜✐♥❡❞✱ ❚♦♠ ❛♥❞ ❇❡♥

❛r❡ t❤❡ s♦❧✉t✐♦♥s✳

❚❤✐s ✐s ❤♦✇ ✇❡ ✉♥❞❡rst❛♥❞ t❤✐s ✐❞❡❛✿

◮

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

t❤❡ ❡q✉❛t✐♦♥✱ ❣✐✈❡s ✉s ❛ tr✉❡ st❛t❡♠❡♥t✳

x

✐s ❛♥ ❡❧❡♠❡♥t t❤❛t✱ ✇❤❡♥ ♣✉t ✐♥ t❤❡ ♣❧❛❝❡ ♦❢

x

✐♥

✷✳✺✳

❙❡t ❜✉✐❧❞✐♥❣

✶✷✻

❍♦✇❡✈❡r✱ ✇❡ ♠✉st ♣r❡s❡♥t

❛❧❧ x✬s t❤❛t s❛t✐s❢② t❤❡ ❡q✉❛t✐♦♥✳

■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ❛♥s✇❡r ✐s ❛

•

❚❤❡ s♦❧✉t✐♦♥ s❡t ♦❢ t❤❡ ❡q✉❛t✐♦♥

F (x) =

❜❛s❦❡t❜❛❧❧ ✐s

•

❚❤❡ s♦❧✉t✐♦♥ s❡t ♦❢ t❤❡ ❡q✉❛t✐♦♥

F (x) =

t❡♥♥✐s ✐s

•

❚❤❡ s♦❧✉t✐♦♥ s❡t ♦❢ t❤❡ ❡q✉❛t✐♦♥

F (x) =

❜❛s❡❜❛❧❧ ❤❛s ♥♦♥ ❡❧❡♠❡♥ts✳

•

❚❤❡ s♦❧✉t✐♦♥ s❡t ♦❢ t❤❡ ❡q✉❛t✐♦♥

F (x) =

❢♦♦t❜❛❧❧ ✐s

X✳

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

{

{

◆❡❞

{

❚♦♠✱ ❇❡♥

s❡t ✿

}✳

}✳

❑❡♥✱ ❙✐❞

}✳

❚❤✐s ✐s t❤❡ t❡r♠✐♥♦❧♦❣② ✇❡ ✇✐❧❧ r♦✉t✐♥❡❧② ✉s❡✳

❉❡✜♥✐t✐♦♥ ✷✳✺✳✹✿ ❡q✉❛t✐♦♥ ❛♥❞ ✐ts s♦❧✉t✐♦♥ b ✐s ♦♥❡ ♦❢ t❤❡ ❡❧❡♠❡♥ts ♦❢ Y ✳ ❚❤❡ s♦❧✉t✐♦♥ s❡t ♦❢ t❤❡ ❡q✉❛t✐♦♥ f (x) = b ✐s t❤❡ s❡t ♦❢ ❛❧❧ x ✐♥ X t❤❛t ♠❛❦❡ t❤❡ ❡q✉❛t✐♦♥ tr✉❡✳

❙✉♣♣♦s❡ ❚♦

f :X→Y

✐s ❛ ❢✉♥❝t✐♦♥ ❛♥❞

s♦❧✈❡ ❛♥ ❡q✉❛t✐♦♥

■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ s♦❧✉t✐♦♥ s❡t ✐s

t❤❡

♠❡❛♥s t♦ ✜♥❞ ✐ts s♦❧✉t✐♦♥ s❡t✳

s♦❧✉t✐♦♥✦

◆❡①t✱ t❤❡ s❡ts ❛❜♦✈❡ ❝❛♥ ❜❡ ♣r❡s❡♥t❡❞ ✐♥ t❤✐s s♣✐r✐t✿

❙❡t✲❜✉✐❧❞✐♥❣ ♥♦t❛t✐♦♥ n ❚❤❡ ❡①♣r❡ss✐♦♥ st❛♥❞s ❢♦r t❤❡ s❡t ♦❢

x:

❝♦♥❞✐t✐♦♥ ❢♦r

x

o

❛❧❧ x t❤❛t s❛t✐s❢② t❤❡ ❝♦♥❞✐t✐♦♥✳

❲❤❛t ❦✐♥❞ ♦❢ ❝♦♥❞✐t✐♦♥❄ ❆♥ ❡q✉❛t✐♦♥✱

❛s ❛❜♦✈❡✳ ❆♥② ❝♦♥❞✐t✐♦♥ ❛s ❧♦♥❣ ❛s ✐t ✐s s♣❡❝✐✜❝ ❡♥♦✉❣❤ ❢♦r ✉s t♦ ✉♥❛♠❜✐❣✉♦✉s❧② ❛♥s✇❡r t❤❡ q✉❡st✐♦♥ ✏❞♦❡s

x

s❛t✐s❢② ✐t❄✑✳ ❋♦r ❡①❛♠♣❧❡✿

{ ❚❤❡ s❡t ❢r♦♠ ✇❤✐❝❤ ✇❡ ♣✐❝❦

x✬s

st✉❞❡♥t✿ ✷✵ ②❡❛rs ♦❧❞

}.

♦♥❡ ❛t ❛ t✐♠❡ ✐s ♣r❡s❡♥t❡❞ ♦r ❛ss✉♠❡❞ t♦ ❜❡ ❦♥♦✇♥✳

❲❛r♥✐♥❣✦ ▼❛♥② s♦✉r❝❡s ❛❧s♦ ✉s❡✿

{x|

❝♦♥❞✐t✐♦♥ ❢♦r

x }.

❋♦r ❡①❛♠♣❧❡✱ t❤❡ ❡q✉❛t✐♦♥s ❛❜♦✈❡ ❛r❡ s❡❡♥ ❛s ❝♦♥❞✐t✐♦♥s✳ ❇❡❧♦✇ ✇❡ ❧✐st t❤❡✐r s♦❧✉t✐♦♥ s❡ts ✭❧❡❢t✮✿

{x, {x, {x, {x,

❜♦② ❜♦② ❜♦② ❜♦②

❚❤❡s❡ ❞❡s❝r✐♣t✐♦♥ ❝❛♥ s♦♠❡t✐♠❡s ❜❡ ♦❢

X

: : : :

F (x) = F (x) = F (x) = F (x) =

s✐♠♣❧✐✜❡❞

} = { ❚♦♠✱ ❇❡♥ } t❡♥♥✐s } = { ◆❡❞ } ❜❛s❡❜❛❧❧ } ={ } ❢♦♦t❜❛❧❧ } = { ❑❡♥✱ ❙✐❞ } ❜❛s❦❡t❜❛❧❧

✭r✐❣❤t✮✳ ❖♥❡ ❝❛♥ ✐♠❛❣✐♥❡ t❤❛t ✇❡ s✐♠♣❧② ✇❡♥t ♦✈❡r t❤❡ ❧✐st

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

✇✐❧❧ ❜❡ r♦✉t✐♥❡❧② ❛♣♣❧✐❡❞ t♦ t❤✐s s❡t✳

❊♠♣t② s❡t ∅

❊①❡r❝✐s❡ ✷✳✺✳✺ ❙❤♦✇ t❤❛t t❤❡ ❡♠♣t② s❡t ✐s ❛ s✉❜s❡t ♦❢ ❛♥② s❡t✳

✷✳✺✳

❙❡t ❜✉✐❧❞✐♥❣

✶✷✼

❊①❡r❝✐s❡ ✷✳✺✳✻

❙✐♠♣❧✐❢② t❤❡ ❢♦❧❧♦✇✐♥❣ s❡ts✿

{x, {y, {y,

❜♦② ❜❛❧❧ ❜❛❧❧

: : :

❤✐s s❤✐rt ✐s r❡❞

}

✐s ♣r❡❢❡rr❡❞ ❜② t✇♦ ❜♦②s ✐s r♦✉♥❞

}

}

❊①❛♠♣❧❡ ✷✳✺✳✼✿ ✐♥❝❧✉s✐♦♥ ✈s✳ ✐♠♣❧✐❝❛t✐♦♥

❍❡r❡ ✐s ❛♥ ✐♥t❡r♣r❡t❛t✐♦♥ ♦❢ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ s✉❜s❡t✳ ❚❤❡ ❞❡✜♥✐t✐♦♥ s❛②s t❤❛t ✐s s❛t✐s✜❡❞✿

◮ ■❋ x

❜❡❧♦♥❣s t♦

A✱ ❚❍❊◆ x

❜❡❧♦♥❣s t♦

A⊂B

✇❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣

B❀

♦r

◮x

❜❡❧♦♥❣s t♦

A =⇒ x

❜❡❧♦♥❣s t♦

B✳

❚❤❡ ❝♦♥✈❡rs❡ ♦❢ t❤✐s ✐♠♣❧✐❝❛t✐♦♥ ✐s ❢❛❧s❡ ✇❤❡♥

◮x

s❛t✐s✜❡s t❤❡ ❝♦♥❞✐t✐♦♥ ❢♦r

A =⇒ x

A 6= B ✳

❋✉rt❤❡r♠♦r❡✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿

s❛t✐s✜❡s t❤❡ ❝♦♥❞✐t✐♦♥ ❢♦r

B✳

■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ❧❛tt❡r ❝♦♥❞✐t✐♦♥ ✐s ❧❡ss r❡str✐❝t✐✈❡✳

❊①❛♠♣❧❡ ✷✳✺✳✽✿ s♦❧✉t✐♦♥ s❡ts ♦❢ ❡q✉❛t✐♦♥s

▲❡t✬s t❛❦❡ ❛♥♦t❤❡r ❧♦♦❦ ❛t t❤❡ ❡q✉❛t✐♦♥s ❛❜♦✈❡✱ ❛ss✉♠✐♥❣ t❤❛t t❤❡ ✏❛♠❜✐❡♥t✑ s❡t ✐s t❤❡ s❡t ♦❢ r❡❛❧ ♥✉♠❜❡rs✿ ❡q✉❛t✐♦♥✿

❛♥s✇❡r❄

s♦❧✉t✐♦♥ s❡t✿

x+2=5 x=3 3x = 15 x=5 2 x − 3x + 2 = 0 x = 1 ❛♥❞✳✳✳ x2 + 2x + 1 = 0 ♥♦ x?

{3} {5} {1, 2} { }

❚❤✐s ✐s ❤♦✇ ✇❡ ✈✐s✉❛❧✐③❡ t❤❡s❡ ❢♦✉r s❡ts✿

❇❡❧♦✇ ✇❡ ✉s❡ t❤❡ s❡t✲❜✉✐❧❞✐♥❣ ♥♦t❛t✐♦♥ ❛❣❛✐♥ ♦♥ t❤❡ ❧❡❢t✱ ❛♥❞ t❤❡♥ ♦♥ t❤❡ r✐❣❤t✱ ✇❡ s❡❡ ❛♥♦t❤❡r✱ s✐♠♣❧❡r✱ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ s❡t✿

{x : {x : {x : {x :

x + 2 = 5} 3x = 15} x2 − 3x + 2 = 0} x2 + 1 = 0}

= {3} = {5} = {1, 2} ={ }=∅

❚❤❡ s✐♠♣❧❡st ✇❛② t♦ r❡♣r❡s❡♥t ❛ s❡t ✐s✱ ♦❢ ❝♦✉rs❡✱ ❛ ❧✐st✳

❊①❡r❝✐s❡ ✷✳✺✳✾

❙♦❧✈❡ t❤❡s❡ ❡q✉❛t✐♦♥s✿

x = x,

1 = 1,

1 = 0.

❊①❛♠♣❧❡ ✷✳✺✳✶✵✿ s❡ts ❢r♦♠ ✐♥❡q✉❛❧✐t✐❡s

❯♥❧❡ss ❡♥t✐r❡❧② ♥♦♥s❡♥s✐❝❛❧✱ ❡✈❡r② st❛t❡♠❡♥t ✐♥ ♠❛t❤❡♠❛t✐❝s ✐s tr✉❡ ♦r ❢❛❧s❡✿

1 ≤ 2 ❚❘❯❊ 1 ≤ 0 ❋❆▲❙❊ ❲❤❛t ❛❜♦✉t t❤✐s✿

1 ≤ x ❚❘❯❊ ❖❘ ❋❆▲❙❊❄

✷✳✺✳ ❙❡t ❜✉✐❧❞✐♥❣

✶✷✽

■t ❞❡♣❡♥❞s ♦♥

x✱

♦❢ ❝♦✉rs❡✳ ❲❡ ❝❛♥✱ t❤❡r❡❢♦r❡✱ ✉s❡ ✐♥❡q✉❛❧✐t✐❡s t♦ ❢♦r♠ s❡ts✿

{x, {x, {x, {x,

: 3≤ x r❡❛❧ : 1 ≤ x < 2 r❡❛❧ : x≤0 r❡❛❧ : 1 ≥ x ≥ 2 r❡❛❧

} } } }

❚❤❡s❡ ❛r❡ ❛❧s♦ s✉❜s❡ts ♦❢ t❤❡ r❡❛❧ ♥✉♠❜❡r ❧✐♥❡✿

❇❡❝❛✉s❡ ✐t✬s ✐♥✜♥✐t❡✱ ❛s ✇❡ ③♦♦♠ ✐♥ ♦♥ t❤❡ r❡❛❧ ❧✐♥❡✱ ✇❡ s❡❡ ❥✉st ❛s ♠❛♥② ♥✉♠❜❡rs ❛s ❜❡❢♦r❡✳ ❚❤✐s ✐s t❤❡ r❡❛s♦♥ ✇❤② t❤❡r❡ ✐s ♥♦ s✉❝❤ t❤✐♥❣ ❛s t❤❡ ❧✐st ♦❢ ❛❧❧ r❡❛❧ ♥✉♠❜❡rs✦ ❙✐♥❝❡ ✇❡ ❝❛♥✬t t❡st t❤❡♠ ♦♥❡ ❜② ♦♥❡✱ ✈✐s✉❛❧✐③❛t✐♦♥ ❜❡❝♦♠❡s ❡s♣❡❝✐❛❧❧② ✐♠♣♦rt❛♥t✳

❊①❡r❝✐s❡ ✷✳✺✳✶✶ ❙♦❧✈❡ t❤❡s❡ ✐♥❡q✉❛❧✐t✐❡s✿

x ≤ x,

1 ≤ 1,

x < x,

1 < 0.

❲❤❡♥ ✐♥❡q✉❛❧✐t✐❡s ❛r❡ ✐♥✈♦❧✈❡❞✱ t❤❡ s❡t✲❜✉✐❧❞✐♥❣ ♥♦t❛t✐♦♥ ✐s ✉s❡❞ ❛❧♦♥❣ ✇✐t❤ ❛ ♠♦r❡ ❝♦♠♣❛❝t ♥♦t❛t✐♦♥✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ✇✐❧❧ ❜❡ ✉♥✐✈❡rs❛❧❧② ✉s❡❞✳

❉❡✜♥✐t✐♦♥ ✷✳✺✳✶✷✿ ✐♥t❡r✈❛❧ ❆ ♥♦♥✲❡♠♣t② s✉❜s❡t ♦❢ t❤❡ r❡❛❧ ❧✐♥❡ ❞❡✜♥❡❞ ❜② ❛♥ ✐♥❡q✉❛❧✐t② ♦r t✇♦ ✐♥❡q✉❛❧✐t✐❡s ✐s ❝❛❧❧❡❞ ❛♥ ✐♥t❡r✈❛❧✳

❲❡ st❛rt ✇✐t❤ s❡ts ♦❢ r❡❛❧ ♥✉♠❜❡rs ❧♦❝❛t❡❞ ❜❡t✇❡❡♥ t✇♦ s♣❡❝✐✜❡❞ ♥✉♠❜❡rs

a < b✿

■♥t❡r✈❛❧ ♥♦t❛t✐♦♥✱ s❡❣♠❡♥ts −−• −−• −−◦ −−◦

• −− ◦ −− • −− ◦ −−

{x : {x : {x : {x :

a≤x≤b a≤x 1✱

♣♦✐♥t

x

❜❡❝♦♠❡s

x·k✳

x/k ✳

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

k

t♦ ❜❡

❧❡ss t❤❛♥

1✳

✐♥t❡r♣r❡t t❤❡ ❢♦r♠❡r st❛t❡♠❡♥t t♦ ✐♥❝❧✉❞❡ t❤❡ ❧❛tt❡r ✐❢ ✇❡ ✉♥❞❡rst❛♥❞ ✏str❡t❝❤❡❞ ❜② ❛ ❢❛❝t♦r ❜② ❛ ❢❛❝t♦r

1/k ✑✳

❚❤✐s ✐s ❤♦✇ ✇❡ ❝❛♥ ❞❡s❝r✐❜❡ ✐t✿

str❡t❝❤ ❜② k

x −−−−−−−−−−→ x · k ❖❢ ❝♦✉rs❡✱

k=1

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

y = f (x) = x · k . ❊①❡r❝✐s❡ ✸✳✸✳✹ ❲❤❛t ✐s t❤❡ ♠❡❛♥✐♥❣ ♦❢

|m|

✐♥ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ ❣✐✈❡♥ ❜②

f (x) = mx + b❄

❊①❡r❝✐s❡ ✸✳✸✳✺ ❲❤❛t ✐s t❤❡ ♠❡❛♥✐♥❣ ♦❢

b

✐♥ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ ❣✐✈❡♥ ❜②

f (x) = mx + b❄

❚❤❡♥ ✇❡ ❝❛♥

k✑

❛s ✏s❤r✉♥❦

✸✳✸✳

◆✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s ❛r❡ tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ❧✐♥❡

✷✷✶

❊①❛♠♣❧❡ ✸✳✸✳✻✿ ❝♦❧♦r✐♥❣ ♥✉♠❜❡rs

❲❡ ✇✐❧❧ ♦❢t❡♥ ❝♦❧♦r ♥✉♠❜❡rs ❛❝❝♦r❞✐♥❣ t♦ t❤❡✐r ✈❛❧✉❡s✳ ❚❤✐s ✐s t❤❡ s✉♠♠❛r② ♦❢ t❤❡ ❢✉♥❝t✐♦♥s ✇❡ ❤❛✈❡ ❝♦♥s✐❞❡r❡❞✿

❖♥❡ ❝❛♥ ❝♦♠♣❛r❡ t❤❡ ❝♦❧♦rs ♦❢ X t♦ t❤♦s❡ ♦❢ X ❛s ✐t ❧❛♥❞s ♦♥ y ❛♥❞ r❡❛❝❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❝❧✉s✐♦♥s✿ • ❚❤❡ ❝♦❧♦rs ❛r❡ ❝❧❡❛r❧② s❤✐❢t❡❞ ✐♥ t❤❡ ✜rst t✇♦ ✐♠❛❣❡s✳ • ■♥ t❤❡ t❤✐r❞✱ t❤❡ ❜❧✉❡ ❛♥❞ t❤❡ r❡❞ ❛r❡ ✐♥t❡r❝❤❛♥❣❡❞✳ • ■♥ t❤❡ ❢♦✉rt❤✱ t❤❡ ❝♦❧♦rs ❞♦♥✬t ❝❤❛♥❣❡ ♦♥ Y ❛s ❢❛st ❛s ♦♥ X ✱ ✇❤✐❧❡ ✐♥ t❤❡ ✜❢t❤✱ t❤❡② ❛r❡ ❝❤❛♥❣✐♥❣ ❢❛st❡r✳ ❲❡ ❛❧s♦ ♣❧♦t t❤❡ ❣r❛♣❤s ♦❢ t❤❡s❡ ❢✉♥❝t✐♦♥s ❜❡❧♦✇✿

❲❡ ❝❛♥✱ t❤❡r❡❢♦r❡✱ ❝❧❛ss✐❢② t❤❡ tr❛♥s❢♦r♠❛t✐♦♥s ❜❛s❡❞ ♦♥ t❤❡ s❧♦♣❡s ♦❢ t❤❡✐r ❣r❛♣❤s✿ • s❤✐❢t ✭r✐❣✐❞ ♠♦t✐♦♥✮✱ s❧♦♣❡ 1❀ • ✢✐♣ ✭r✐❣✐❞ ♠♦t✐♦♥✮✱ s❧♦♣❡ −1❀ • str❡t❝❤ ❜② 2✱ s❧♦♣❡ 2❀ • s❤r✐♥❦ ❜② 2✱ s❧♦♣❡ 1/2✳ ❆ ✈❡r② s✐♠♣❧❡✱ ❜✉t ✐♠♣♦rt❛♥t✱ ❝❧❛ss ♦❢ ❢✉♥❝t✐♦♥ ✐s t❤❡ ❝♦♥st❛♥t ❢✉♥❝t✐♦♥s✱ f (x) = c✳ ❲❤❛t ❞♦❡s t❤✐s ❢✉♥❝t✐♦♥ ❞♦ t♦ t❤❡ x✲❛①✐s❄ ❚❤❡r❡ ✐s ♦♥❧② ♦♥❡ ♦✉t♣✉t✿

❚❤❡ r❡❛❧ ❧✐♥❡ ✐s s❤r✉♥❦ t♦ ❛ s✐♥❣❧❡ ♣♦✐♥t❀ ✇❡ ❝❛♥ ❝❛❧❧ t❤✐s tr❛♥s❢♦r♠❛t✐♦♥ ❝♦❧❧❛♣s❡✳ ❝♦❧❧❛♣s❡

x −−−−−−−−→ c

❖❢ ❝♦✉rs❡✱ ♦♥❡ ♠❛② s❡❡ ✐t ❛s ❛♥ ❡①tr❡♠❡ ❝❛s❡ ♦❢ ❛ str❡t❝❤✴s❤r✐♥❦✱ ✇✐t❤ ❛ ❢❛❝t♦r k = 0✳ ❚❤❡ s❧♦♣❡ ♦❢ t❤❡ ❣r❛♣❤ ✐s✱ ♦❢ ❝♦✉rs❡✱ 0✳ ■♥ ❣❡♥❡r❛❧✱ ✐t ✐s t②♣✐❝❛❧ t♦ ❤❛✈❡ ❞✐✛❡r❡♥t str❡t❝❤✐♥❣ ❢❛❝t♦rs ❛t ❞✐✛❡r❡♥t ❧♦❝❛t✐♦♥s✳ ❊①❛♠♣❧❡ ✸✳✸✳✼✿ tr❛♥s❢♦r♠❛t✐♦♥ ❢r♦♠ ❧✐st ♦❢ ✈❛❧✉❡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

✸✳✸✳

◆✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s ❛r❡ tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ❧✐♥❡

✷✷✷

❲❡ ❛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❝✳ ❚❤❡ 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✮✿

❲❡ ❛❧s♦ ♣❧♦t t❤❡ ❣r❛♣❤ ♦❢ t❤✐s ❢✉♥❝t✐♦♥ ♦♥ t❤❡ r✐❣❤t❀ t❤❡ str❡t❝❤✐♥❣ ❢❛❝t♦rs ❜❡❝♦♠❡ t❤❡ s❧♦♣❡s✦ ❙♦✱ ❛t ✐ts s✐♠♣❧❡st✱ ❛ ♥♦♥✲❧✐♥❡❛r ❢✉♥❝t✐♦♥ ✐s ❛ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ❧✐♥❡❛r ♣❛t❝❤❡s✳

❊①❡r❝✐s❡ ✸✳✸✳✽ ❈♦♠♣✉t❡ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❢♦r t❤✐s ❢✉♥❝t✐♦♥✱ ✐✳❡✳✱ t❤❡ s❧♦♣❡s ♦❢ t❤❡s❡ ♣❛t❝❤❡s✳ ❲❤❛t ✐s t❤❡ r❡❧❛t✐♦♥ ♦❢ ❡❛❝❤ s❧♦♣❡ t♦ ✇❤❛t ❤❛♣♣❡♥s t♦ t❤❡ s❡❣♠❡♥t❄

❊①❡r❝✐s❡ ✸✳✸✳✾ ❘❡♣r❡s❡♥t t❤❡ ❢♦❧❧♦✇✐♥❣ ❢✉♥❝t✐♦♥ ❣✐✈❡♥ ❜② ✐ts ✈❛❧✉❡s ❛s ❛ tr❛♥s❢♦r♠❛t✐♦♥✿

x 0 1 2 3 4 5 6 7 8 9 10 y 6 5 4 3 3 3 4 5 6 6 6 ▼❛❦❡ ✉♣ ②♦✉r ♦✇♥ ❢✉♥❝t✐♦♥s ❛♥❞ t❤❡♥ r❡♣r❡s❡♥t t❤❡♠ ❛s tr❛♥s❢♦r♠❛t✐♦♥s✳ ❘❡♣❡❛t✳ ❋✉rt❤❡r♠♦r❡✱ ✐t ✐s ❛❧s♦ ♣♦ss✐❜❧❡✱ ❛♥❞ ❧✐❦❡❧②✱ ❢♦r ❛ str❡t❝❤ ❢❛❝t♦r ✭♦r r❛t❡✮ ♦❢ ❛ ❢✉♥❝t✐♦♥ t♦ ✈❛r② t❤r♦✉❣❤♦✉t t❤❡ ❞♦♠❛✐♥ ♦❢ ❛ ❢✉♥❝t✐♦♥✳ ✭❚❤✐s ✐ss✉❡ ✐s ❛❞❞r❡ss❡❞ ✐♥ ❈❤❛♣t❡r ✷❉❈✲✸✳✮

❝♦♥t✐♥✉♦✉s❧②

❊①❛♠♣❧❡ ✸✳✸✳✶✵✿ ✜①❡❞ ♣♦✐♥t ❚❤✐s ✐s ❛ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ✕ ❧♦❝❛t✐♦♥ ❞❡♣❡♥❞❡♥t ✕ s❤✐❢t✐♥❣✱ str❡t❝❤✐♥❣✱ ❛♥❞ s❤r✐♥❦✐♥❣✿

❆♠♦♥❣ ❛❧❧ t❤✐s ❝♦♠♣❧❡①✐t②✱ t❤❡ ♣♦✐♥t ♠❛r❦❡❞ ✇✐t❤ st❛r ✐s ✜①❡❞✳

❊①❡r❝✐s❡ ✸✳✸✳✶✶ ❙❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ t❤❛t r❡♣r❡s❡♥ts t❤❡ ❛❜♦✈❡ tr❛♥s❢♦r♠❛t✐♦♥✳ ■♥ s✉♠♠❛r②✱ ❛♥ ❛❜str❛❝t ❢✉♥❝t✐♦♥ y = f (x) ❤❛s ❜❡❡♥ ❣✐✈❡♥ t❤r❡❡ ❤❛✈❡ t❤r❡❡ t❛♥❣✐❜❧❡ ✐♥t❡r♣r❡t❛t✐♦♥s ♦❢ ✐ts r❛t❡ ♦❢ ❝❤❛♥❣❡✿

1. 2. 3.

❋✉♥❝t✐♦♥ ✐s s❡❡♥ ❛s ❧♦❝❛t✐♦♥ ❣r❛♣❤ tr❛♥s❢♦r♠❛t✐♦♥

t❛♥❣✐❜❧❡ r❡♣r❡s❡♥t❛t✐♦♥s ❛♥❞ ♥♦✇ ✇❡ ❛❧s♦

■ts r❛t❡ ♦❢ ❝❤❛♥❣❡ ✐s ✈❡❧♦❝✐t② s❧♦♣❡ ♦❢ t❤❡ t❛♥❣❡♥t ❧✐♥❡ str❡t❝❤✴s❤r✐♥❦ r❛t❡

✸✳✸✳

◆✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s ❛r❡ tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ❧✐♥❡

✷✷✸

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

❙t✐❧❧✱ t❤❡ ♠♦st ❝♦♠♠♦♥ ✇❛② t♦ ✈✐s✉❛❧✐③❡ ❛ ❢✉♥❝t✐♦♥ ✇✐❧❧ r❡♠❛✐♥s ♣❧♦tt✐♥❣ ✐ts

❣r❛♣❤✳ ❚❤❡ ♣✐❝t✉r❡ ❜❡❧♦✇

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

❋✐rst✱ ✇❡ t❛❦❡ t❤❡

x✲❛①✐s

❛s ✐❢ ✐t ✐s ❛ ✭❝♦❧♦r❡❞✮ r♦♣❡ ❛♥❞ ❧✐❢t ✐t ✈❡rt✐❝❛❧❧② t♦ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥✱

❛♥❞ s❡❝♦♥❞✱ ✇❡ ♣✉s❤ ✐t ❤♦r✐③♦♥t❛❧❧② t♦ t❤❡

y ✲❛①✐s

✭♣r❛❝t✐❝❛❧❧②✱ ✇❡ ❥✉st s❤r✐♥❦ t❤❡ ✐♠❛❣❡✮✳ ❆t t❤❡ ❡♥❞✱

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

❚❤❡r❡ ✐s ❛❧s♦ ❛ ✢✐♣ ✐♥ t❤❡ ♠✐❞❞❧❡ ✇✐t❤ ❢♦❧❞s✳

❊①❡r❝✐s❡ ✸✳✸✳✶✸

❉r❛✇ ②♦✉r ♦✇♥ ❣r❛♣❤ ❛♥❞ t❤❡♥ ✐♥t❡r♣r❡t t❤❡ ❢✉♥❝t✐♦♥ ❛s ❛ tr❛♥s❢♦r♠❛t✐♦♥✳ ❘❡♣❡❛t✳

❊①❛♠♣❧❡ ✸✳✸✳✶✹✿ t❡❛r✐♥❣ ❛♥❞ ❞✐s❝♦♥t✐♥✉✐t②

■♥ ❣❡♥❡r❛❧✱ ❛ tr❛♥s❢♦r♠❛t✐♦♥ ❝❛♥ t♦ ❡❛❝❤ ♦t❤❡r ♦♥ t❤❡

x✲❛①✐s✱

t❡❛r t❤✐s r♦♣❡✳ ❲❡ ❝❛♥ s❡❡ ❤♦✇ t✇♦ ♥✉♠❜❡rs✱ 4 ❛♥❞ 5✱ t❤❛t ❛r❡ ❝❧♦s❡

❛r❡ ♥♦✇ ❢❛r ❛♣❛rt ♦♥ t❤❡

y ✲❛①✐s

✭r✐❣❤t✮✿

❲❡ ❝❛♥ ❛❧s♦ s❡❡ ❛ ❣❛♣ ✐♥ t❤❡ ❣r❛♣❤ ✭r✐❣❤t✮✳ ✭❚❤❡ ✐ss✉❡ ♦❢ ✏❝♦♥t✐♥✉✐t②✑ ✐s ❛❞❞r❡ss❡❞ ✐♥ ❈❤❛♣t❡r ✷❉❈✲✷✳✮

❊①❡r❝✐s❡ ✸✳✸✳✶✺

❉❡s❝r✐❜❡ t❤❡ ❢✉♥❝t✐♦♥ t❤❡ ❣r❛♣❤ ♦❢ ✇❤✐❝❤ ✐s ❣✐✈❡♥ ❜❡❧♦✇ ❛s ❛ tr❛♥s❢♦r♠❛t✐♦♥✿

✸✳✹✳ ❋✉♥❝t✐♦♥s ✇✐t❤ r❡❣✉❧❛r✐t✐❡s✿ ♦♥❡✲t♦✲♦♥❡ ❛♥❞ ♦♥t♦

✷✷✹

❊①❡r❝✐s❡ ✸✳✸✳✶✻

Pr♦✈✐❞❡ t❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥ ✭❛ tr❛♥s❢♦r♠❛t✐♦♥✮ t❤❛t t❡❛rs t❤✐s r♦♣❡ ❛♥❞ t❤❡♥✿ ✭❛✮ ♣✉❧❧s t❤❡ t✇♦ ❤❛❧✈❡s ❛♣❛rt✱ ♦r ✭❜✮ ♦✈❡r❧❛♣s t❤❡✐r ❡♥❞s✳

❊①❛♠♣❧❡ ✸✳✸✳✶✼✿ s✐❣♥ ❢✉♥❝t✐♦♥

❚❤❡ s✐❣♥ ❢✉♥❝t✐♦♥ ❝♦❧❧❛♣s❡s t❤❡

x✲❛①✐s

t♦ t❤r❡❡ ❞✐✛❡r❡♥t ♣♦✐♥ts ♦♥ t❤❡

y ✲❛①✐s✿

❊①❡r❝✐s❡ ✸✳✸✳✶✽

❲❤❛t ❞♦❡s t❤❡ ✐♥t❡❣❡r ✈❛❧✉❡ ❢✉♥❝t✐♦♥

y = [x]

❞♦ t♦ t❤❡

x✲❛①✐s❄

✸✳✹✳ ❋✉♥❝t✐♦♥s ✇✐t❤ r❡❣✉❧❛r✐t✐❡s✿ ♦♥❡✲t♦✲♦♥❡ ❛♥❞ ♦♥t♦

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

■t ❞♦❡s✳ ❇✉t ✇❤❛t ❛❜♦✉t t❤❡ ❝♦❧✉♠♥s ❄ ■❢ ✇❡ ❢✉rt❤❡r st✉❞② t❤✐s ❢✉♥❝t✐♦♥✱ ✇❡ ♠✐❣❤t ♥♦t✐❝❡ t✇♦ ❞✐✛❡r❡♥t ❜✉t r❡❧❛t❡❞ ✏✐rr❡❣✉❧❛r✐t✐❡s✑✳ ❋✐rst✱ ♥♦ ♦♥❡ s❡❡♠s t♦ ❧✐❦❡ ❜❛s❡❜❛❧❧✦ ❚❤❡r❡ ✐s ♥♦ ❛rr♦✇ ❡♥❞✐♥❣ ❛t t❤❡ ❜❛s❡❜❛❧❧✱ ❛♥❞ ✐ts ❝♦❧✉♠♥ ❤❛s ♥♦ ♠❛r❦s✳ ▲❡t✬s ♠♦❞✐❢② t❤❡ ❢✉♥❝t✐♦♥ s❧✐❣❤t❧②✿ ❑❡♥ ❝❤❛♥❣❡s ❤✐s ♣r❡❢❡r❡♥❝❡ ❢r♦♠ ❢♦♦t❜❛❧❧ t♦ ❜❛s❡❜❛❧❧✳ ❚❤❡♥✱ t❤❡r❡ ✐s ❛♥ ❛rr♦✇ ❢♦r ❡❛❝❤ ❜❛❧❧✱ ❛♥❞ ❛❧❧ ❝♦❧✉♠♥s ✐♥ t❤❡ t❛❜❧❡ ❤❛✈❡ ♠❛r❦s✿

✸✳✹✳ ❋✉♥❝t✐♦♥s ✇✐t❤ r❡❣✉❧❛r✐t✐❡s✿ ♦♥❡✲t♦✲♦♥❡ ❛♥❞ ♦♥t♦

✷✷✺

■♥ t❤❡ ❣r❛♣❤✱ t❤❡r❡ ✐s ❛ ❞♦t ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ❡❛❝❤ ❤♦r✐③♦♥t❛❧ ❧✐♥❡✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❝❡♣t ✐s ❝r✉❝✐❛❧✳

❉❡✜♥✐t✐♦♥ ✸✳✹✳✶✿ ♦♥t♦ ❢✉♥❝t✐♦♥ ❆ ❢✉♥❝t✐♦♥

F :X→Y

✐s ❝❛❧❧❡❞ ♦♥t♦ ✇❤❡♥ t❤❡r❡ ✐s ❛♥

x ❢♦r ❡❛❝❤ y ✇✐t❤ F (x) = y ✳

■♥ ♦t❤❡r ✇♦r❞s✱ ♥♦ ♣♦t❡♥t✐❛❧ ♦✉t♣✉t ✐s ✏✇❛st❡❞✑✳ ❇❡❧♦✇✱ t❤❡ r❡❛s♦♥ ❢♦r t❤✐s t❡r♠✐♥♦❧♦❣② ✐s ❡①♣❧❛✐♥❡❞✿

❲❡ st❛rt ✇✐t❤ ❛♥ ❡❧❡♠❡♥t ♦❢

X

❛♥❞ ❜r✐♥❣ ✐t ✕ ❛❧♦♥❣ t❤❡ ❛rr♦✇ ✕ t♦ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❡❧❡♠❡♥t ♦❢

❢✉♥❝t✐♦♥ ✐s ♦♥t♦ ✐❢ ❛❧❧ ❡❧❡♠❡♥ts ♦❢

Y

Y✳

❚❤❡

❛r❡ ❝♦✈❡r❡❞✳

❙❡❝♦♥❞✱ ❜♦t❤ ❚♦♠ ❛♥❞ ❇❡♥ ♣r❡❢❡r ❜❛s❦❡t❜❛❧❧✦ ❚❤❡ t✇♦ ❛rr♦✇s ❝♦♥✈❡r❣❡ ♦♥ t❤❡ ❜❛s❦❡t❜❛❧❧✱ ❛♥❞ ✇❡ ❝❛♥ ❛❧s♦ s❡❡ t❤❛t ✐ts ❝♦❧✉♠♥ ❤❛s t✇♦ ♠❛r❦s✳ ❲❡ ♥♦t❡ t❤❡ s❛♠❡ ❛❜♦✉t t❤❡ ❢♦♦t❜❛❧❧✳ ❚❤❡ ❛❜♦✈❡ ❢✉♥❝t✐♦♥ ✐s ♠♦❞✐✜❡❞✿ ❚♦♠ ❛♥❞ ❇❡♥ ❤❛✈❡ ❧❡❢t✳ ❚❤❡♥✱ ♥♦ t✇♦ ❛rr♦✇s ❝♦♥✈❡r❣❡ ♦♥ ♦♥❡ ❜❛❧❧✱ ❛♥❞ ♥♦ ❝♦❧✉♠♥ ❤❛s ♠♦r❡ t❤❛♥ ♦♥❡ ♠❛r❦✿

■♥ t❤❡ ❣r❛♣❤✱ t❤❡r❡ ❝❛♥ ♦♥❧② ❜❡ ♦♥❡ ❞♦t✱ ♦r ♥♦♥❡✱ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ❡❛❝❤ ❤♦r✐③♦♥t❛❧ ❧✐♥❡✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❝❡♣t ✐s ❝r✉❝✐❛❧✳

✸✳✹✳

✷✷✻

❋✉♥❝t✐♦♥s ✇✐t❤ r❡❣✉❧❛r✐t✐❡s✿ ♦♥❡✲t♦✲♦♥❡ ❛♥❞ ♦♥t♦

❉❡✜♥✐t✐♦♥ ✸✳✹✳✷✿ ♦♥❡✲t♦✲♦♥❡ ❢✉♥❝t✐♦♥ ❆ ❢✉♥❝t✐♦♥ F : X → Y ✐s ❝❛❧❧❡❞ y ✇✐t❤ F (x) = y ✳

♦♥❡✲t♦✲♦♥❡

✇❤❡♥ t❤❡r❡ ✐s ❛t ♠♦st ♦♥❡ x ❢♦r ❡❛❝❤

❇❡❧♦✇✱ t❤❡ r❡❛s♦♥ ❢♦r t❤✐s t❡r♠✐♥♦❧♦❣② ✐s ❡①♣❧❛✐♥❡❞✿

❲❡ st❛rt ✇✐t❤ ❛♥ ❡❧❡♠❡♥t ♦❢ X ❛♥❞ ❜r✐♥❣ ✐t ✕ ❛❧♦♥❣ t❤❡ ❛rr♦✇ ✕ t♦ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❡❧❡♠❡♥t ♦❢ Y ✳ ❚❤❡ ❢✉♥❝t✐♦♥ ✐s ♦♥❡✲t♦✲♦♥❡ ✐❢ ❡✈❡r② ❡❧❡♠❡♥t ♦❢ Y ✐s ❝♦✈❡r❡❞ ♦♥❧② ♦♥❝❡✱ ✐❢ ❛t ❛❧❧✳ ■♥ s✉♠♠❛r②✱ t❤❡ t✇♦ ❝♦♥❝❡♣ts ❛r❡ ♥♦t ❛❜♦✉t ❤♦✇ ♠❛♥② ❛rr♦✇s ♦r✐❣✐♥❛t❡ ❢r♦♠ ❡❛❝❤ x ✕ ✐t✬s ❛❧✇❛②s ♦♥❡ ✕ ❜✉t ❛❜♦✉t ❤♦✇ ♠❛♥② ❛rr♦✇s ❛rr✐✈❡ ❛t ❡❛❝❤ y ✳ ❚❤❡ ❧♦❣✐❝ ♦❢ t❤❡ t✇♦ ❞❡✜♥✐t✐♦♥s ✐s q✉✐t❡ ❞✐✛❡r❡♥t✿

• ❖♥t♦✿

❋❖❘ ❊❆❈❍ x ❚❍❊❘❊ ■❙ y s✉❝❤ t❤❛t y = F (x)✳

• ❖♥❡✲t♦✲♦♥❡✿

■❋ F (x1 ) = F (x2 ) ❚❍❊◆ x1 = x2 ✳

◆♦✇ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s✳ ❚❤❡s❡ ❢✉♥❝t✐♦♥s ❛r❡ r❡♣r❡s❡♥t❡❞ ❜② t❤❡✐r tr❛♥s❢♦r♠❛t✐♦♥s ❛♥❞ ❜② t❤❡✐r ❣r❛♣❤s✳ ❲❡ ✇✐❧❧ ❢♦❧❧♦✇ t❤❡s❡ t✇♦ ❛♣♣r♦❛❝❤❡s t♦ ❞✐s❝♦✈❡r ✇❤❡t❤❡r ❛ ❢✉♥❝t✐♦♥ s❛t✐s✜❡s ♦♥❡ ♦❢ t❤❡ t✇♦ ❞❡✜♥✐t✐♦♥s✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ✇✐❧❧ ❜❡ ♦✉r ❛ss✉♠♣t✐♦♥✿

◮ ❚❤❡ ❝♦❞♦♠❛✐♥ ✐s t❤❡ s❡t ♦❢ ❛❧❧ r❡❛❧s R✳

❊①❛♠♣❧❡ ✸✳✹✳✸✿ ❜❛s✐❝ tr❛♥s❢♦r♠❛t✐♦♥s ▲❡t✬s ❝♦♥s✐❞❡r t❤❡ tr❛♥s❢♦r♠❛t✐♦♥s ❢r♦♠ t❤❡ ❧❛st s❡❝t✐♦♥✳ ❚❤❡✐r ❞❡s❝r✐♣t✐♦♥s ✕ ❛♥❞ ✐❧❧✉str❛t✐♦♥s ✕ t❡❧❧ t❤❡ ✇❤♦❧❡ st♦r②✳ ❲❡ ❥✉st ♥❡❡❞ t♦ ❧♦♦❦ ❛t ❤♦✇ t❤❡ ❛rr♦✇s ❛rr✐✈❡ ❛t t❤❡ y ✲❛①✐s✳ ❚❤❡ ❧❡❢t ❛♥❞ r✐❣❤t s❤✐❢ts✿

❚❤❡ ✢✐♣✿

❚❤❡ str❡t❝❤ ❛♥❞ t❤❡ s❤r✐♥❦✿

❚❤❡② ❛r❡ ❛❧❧ ❜♦t❤ ♦♥❡✲t♦✲♦♥❡ ❛♥❞ ♦♥t♦✦

✸✳✹✳ ❋✉♥❝t✐♦♥s ✇✐t❤ r❡❣✉❧❛r✐t✐❡s✿ ♦♥❡✲t♦✲♦♥❡ ❛♥❞ ♦♥t♦

✷✷✼

❲❤❛t ✇♦✉❧❞ ♠❛❦❡ ❛ tr❛♥s❢♦r♠❛t✐♦♥ t♦ ❜❡ ♥♦t ♦♥❡✲t♦✲♦♥❡❄ ❆♥② ❢♦❧❞✐♥❣ ♣r❡s❡♥ts ❛ ✈✐s✐❜❧❡ ♣r♦❜❧❡♠✿

❆♥❞ s♦ ❞♦❡s ❛♥② ❝♦❧❧❛♣s✐♥❣✿

❙♦✱ ✐❢ ✇❡ ✐♠❛❣✐♥❡ t❤❛t t❤❡

x✲❛①✐s✱ X ✱

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

y ✲❛①✐s✱ Y ✱

❝❛♥ ✉♥❞❡rst❛♥❞ t❤✐s ❢✉♥❝t✐♦♥ ❛s ❛ tr❛♥s❢♦r♠❛t✐♦♥✳ ❚❤✐s ❝❛♥ ❤❛♣♣❡♥ ✐♥ t❤❡s❡ ❢♦✉r ❜❛s✐❝ ✇❛②s✿

❚❤❡ q✉❡st✐♦♥s ✇❡ ❛s❦ ❛r❡ t❤❡s❡ t✇♦✿

•

❖♥t♦✿ ❉♦❡s

•

❖♥❡✲t♦✲♦♥❡✿ ❉♦❡s

X

❝♦✈❡r t❤❡ ✇❤♦❧❡

X

Y❄

❝♦✈❡r ❛♥② ❧♦❝❛t✐♦♥ ♦♥

Y

♥♦ ♠♦r❡ t❤❛♥ ♦♥❝❡❄

◆♦✇✱ t❤❡ ❣r❛♣❤s✳ ❚❤❡ ✐❞❡❛ ✐s t♦ ✏s❛♠♣❧❡✑ t❤❡ ❢✉♥❝t✐♦♥ ❛♥❞ s❡❡ ❤♦✇ t❤❡ ❛rr♦✇s ❜❡❤❛✈❡✳ ❊①❛♠♣❧❡ ✸✳✹✳✹✿ ♦♥❡✲t♦✲♦♥❡✱ ♦♥t♦ ❢r♦♠ ❣r❛♣❤

▲❡t✬s st❛rt ✇✐t❤ t❤✐s ❝❛s❡✿ ❚❤❡ ❞♦♠❛✐♥ ✐s t❤❡ r❡❛❧s ♥♦♥✲♥❡❣❛t✐✈❡ r❡❛❧s✱

Y = [0, ∞)✳

■❢ t❤❡ ❣r❛♣❤ ♦❢ ❛

X = R = (−∞, +∞)✱ ✇❤✐❧❡ t❤❡ ❝♦❞♦♠❛✐♥ ✐s t❤❡ ❢✉♥❝t✐♦♥ f : X → Y ✐s ❣✐✈❡♥✱ ❝❛♥ ✇❡ ❞❡t❡r♠✐♥❡

✇❤❡t❤❡r t❤✐s ❢✉♥❝t✐♦♥ ✐s ♦♥❡✲t♦✲♦♥❡ ♦r ♦♥t♦ ❜② ❥✉st ❡①❛♠✐♥✐♥❣ t❤❡ ❣r❛♣❤❄ ❲❡ ❛rr❛♥❣❡ t❤❡ ❢♦✉r ♦♣t✐♦♥s ✐♥ ❛

2×2

t❛❜❧❡✿

✇❡

✸✳✹✳ ❋✉♥❝t✐♦♥s ✇✐t❤ r❡❣✉❧❛r✐t✐❡s✿ ♦♥❡✲t♦✲♦♥❡ ❛♥❞ ♦♥t♦

✷✷✽

❲❡ ♥♦t✐❝❡ t❤❛t✿ ✶✳ ❚❤❡ ❛rr♦✇s ✐♥ t❤❡ ✜rst r♦✇ s❡❡♠ t♦ ❝♦✈❡r t❤❡ ✇❤♦❧❡

Y✳

✷✳ ❚❤❡ ❛rr♦✇s ✐♥ t❤❡ ✜rst ❝♦❧✉♠♥ s❡❡♠ ♥♦t t♦ ❝♦✈❡r ❛♥② ❧♦❝❛t✐♦♥ ♦♥

Y

t✇✐❝❡✳

❲❡ ❝♦♥❝❧✉❞❡ t❤❛t✿ ✶✳ ❚❤❡ ❢✉♥❝t✐♦♥s ✐♥ t❤❡ ✜rst r♦✇ ❛r❡ ♦♥t♦✳ ✷✳ ❚❤❡ ❢✉♥❝t✐♦♥s ✐♥ t❤❡ ✜rst ❝♦❧✉♠♥ ❛r❡ ♦♥❡✲t♦✲♦♥❡✳ ❲❡ ❞r❛✇ t❤❡s❡ ❝♦♥❝❧✉s✐♦♥s ✇✐t❤ t❤❡ ✉♥❞❡rst❛♥❞✐♥❣ t❤❛t t❤❡② ❛r❡ ❜❛s❡❞ ♦♥ t❤❡ ♣❛rt✐❛❧ ✐♥❢♦r♠❛t✐♦♥ ♣r♦✈✐❞❡❞ ❜② t❤❡ ❣r❛♣❤✳ ❚❤❡ ❝❛s❡ ❢♦r ♥♦t ♦♥t♦ ❛♥❞ ♥♦t ♦♥❡✲t♦✲♦♥❡ ✐s ❜❡tt❡r❀ ✇❡ ❝❛♥ ❞❡✜♥✐t❡❧② s❡❡ ❛♥❞ ♠❛r❦ ✇✐t❤ ❝✐r❝❧❡s t❤❡ ❢♦❧❧♦✇✐♥❣✿

• •

❚❤❡ ❛rr♦✇s ✐♥ t❤❡ s❡❝♦♥❞ r♦✇ ❛r❡ ♠✐ss✐♥❣ ❛ s♣❡❝✐✜❝ ✈❛❧✉❡ ✐♥

Y✳

❚❤❡ ❛rr♦✇s ✐♥ t❤❡ s❡❝♦♥❞ ❝♦❧✉♠♥ ❛r❡ ❤✐tt✐♥❣ ❛♥ ✐❞❡♥t✐❝❛❧ s♣❡❝✐✜❝ ✈❛❧✉❡ ✐♥

Y✳

❊①❡r❝✐s❡ ✸✳✹✳✺

❍♦✇ ❛r❡ t❤❡ ❝♦♥❝❧✉s✐♦♥s ✐♥ t❤❡ ❛❜♦✈❡ ❡①❛♠♣❧❡ ❛✛❡❝t❡❞ ❜② ❞✐✛❡r❡♥t ❝❤♦✐❝❡s ♦❢ ✭❛✮ ❝♦❞♦♠❛✐♥s✱ ✭❜✮ ❞♦♠❛✐♥s❄ ❊①❛♠♣❧❡ ✸✳✹✳✻✿ ♣♦✇❡r ❢✉♥❝t✐♦♥s✱ ♦♥t♦

❙✉♣♣♦s❡ t❤✐s t✐♠❡ t❤❡ ❞♦♠❛✐♥ ❛♥❞ t❤❡ ❝♦❞♦♠❛✐♥ ❛r❡ t❤❡ r❡❛❧s✱ X = f (x) = x2 ✐s ♥♦t ♦♥t♦ ❜✉t g(x) = x3 ✐s✳ ■t ✐s ❡❛s② t♦ ♣r♦✈❡ t❤❡ ❢♦r♠❡r✿

x2 6= −1 ,

Y = R✳

❋❖❘ ❊❆❈❍ x.

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

❊①❡r❝✐s❡ ✸✳✹✳✼

❍♦✇ ❝❛♥ ✇❡ ✏♠❛❦❡✑ t❤❡s❡ ❢✉♥❝t✐♦♥s ♦♥t♦✿ ✭❛✮

x2 ✱

✭❜✮

|x| ❄

❚❤❡♥ t❤❡ ❢✉♥❝t✐♦♥

✸✳✹✳ ❋✉♥❝t✐♦♥s ✇✐t❤ r❡❣✉❧❛r✐t✐❡s✿ ♦♥❡✲t♦✲♦♥❡ ❛♥❞ ♦♥t♦

❲❤❛t ♠❛❦❡s ❛ ❞✐✛❡r❡♥❝❡❄

✷✷✾

❚❤❡ ❣r❛♣❤ ❞♦❡s♥✬t ♣r♦❣r❡ss ❜❡❧♦✇ ❛ ❝❡rt❛✐♥ ❧✐♥❡✦

❚❤❡ ❢♦❧❧♦✇✐♥❣ ✐s ❛ ✉s❡❢✉❧

♦❜s❡r✈❛t✐♦♥✿

◮ ❆ ❢✉♥❝t✐♦♥ ✐s ♦♥t♦ ■❋ ❆◆❉ ❖◆▲❨ ■❋

✐ts ❣r❛♣❤ ❛♥❞ ❛♥② ❤♦r✐③♦♥t❛❧ ❧✐♥❡ ❤❛✈❡ ❛t ❧❡❛st ♦♥❡ ♣♦✐♥t

✐♥ ❝♦♠♠♦♥✳ ❊①❛♠♣❧❡ ✸✳✹✳✽✿ ♣♦✇❡r ❢✉♥❝t✐♦♥s✱ ♦♥❡✲t♦✲♦♥❡

❚❤❡ ❢✉♥❝t✐♦♥

f (x) = x2

✐s ♥♦t ♦♥❡✲t♦✲♦♥❡ ❜✉t

g(x) = x3

✐s✳ ■t ✐s ❡❛s② t♦ ♣r♦✈❡ t❤❡ ❢♦r♠❡r✿

12 = (−1)2 = 1 . ❚♦ ♣r♦✈❡ t❤❡ ❧❛tt❡r ✇✐❧❧ r❡q✉✐r❡ s♦♠❡ ❛❧❣❡❜r❛ ❡✈❡♥ t❤♦✉❣❤ t❤❡ ✐❞❡❛ ✐s ❝❧❡❛r✿

❊①❡r❝✐s❡ ✸✳✹✳✾

❍♦✇ ❝❛♥ ✇❡ ✏♠❛❦❡✑ t❤❡s❡ ❢✉♥❝t✐♦♥s ♦♥❡✲t♦✲♦♥❡✿ ✭❛✮

x2 ✱

✭❜✮

|x| ❄

❲❤❛t ♠❛❦❡s ❛ ❞✐✛❡r❡♥❝❡❄ ❚✇♦ ♣♦✐♥ts ♦♥ t❤❡ ❣r❛♣❤ ❤❛✈❡ t❤❡ s❛♠❡ ❤❡✐❣❤t ❛❜♦✈❡ t❤❡

x✲❛①✐s✦

❚❤❡ ❢♦❧❧♦✇✐♥❣

✐s ❛ ✉s❡❢✉❧ ♦❜s❡r✈❛t✐♦♥✿

◮

❆ ❢✉♥❝t✐♦♥ ✐s ♦♥❡✲t♦✲♦♥❡

■❋ ❆◆❉ ❖◆▲❨ ■❋

t❤❡ ✐♥t❡rs❡❝t✐♦♥ ♦❢ ✐ts ❣r❛♣❤ ❛♥❞ ❛♥② ❤♦r✐③♦♥t❛❧

❧✐♥❡ ❝♦♥t❛✐♥s ❛t ♠♦st ♦♥❡ ♣♦✐♥t✳ ◆♦t✐❝❡ t❤❡ ❝♦♥♥❡❝t✐♦♥ ✇✐t❤ t❤❡ ❱❡rt✐❝❛❧ ▲✐♥❡ ❚❡st ❢♦r ❘❡❧❛t✐♦♥s ✭✏✐s t❤✐s ❛ ❢✉♥❝t✐♦♥❄✑✮ ❢r♦♠ ❈❤❛♣t❡r ✷✿

❊①❛♠♣❧❡ ✸✳✹✳✶✵✿ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧s

◗✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧s✱

F (x) = ax2 + bx + c, a 6= 0 , ❛r❡ ♥❡✐t❤❡r ♦♥❡✲t♦✲♦♥❡ ♥♦r ♦♥t♦✿

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

✸✳✹✳

✷✸✵

❋✉♥❝t✐♦♥s ✇✐t❤ r❡❣✉❧❛r✐t✐❡s✿ ♦♥❡✲t♦✲♦♥❡ ❛♥❞ ♦♥t♦

❊①❡r❝✐s❡ ✸✳✹✳✶✶

■❞❡♥t✐❢② ❛♥❞ ❝❧❛ss✐❢② t❤❡ ❢✉♥❝t✐♦♥s ❜❡❧♦✇✿

❊①❡r❝✐s❡ ✸✳✹✳✶✷

❆ ❢✉♥❝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❡ ✸✳✹✳✶✸

❲❤❛t ❝♦❞♦♠❛✐♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❣✐✈❡♥ ❛❜♦✈❡ s❤♦✉❧❞ ✇❡ ❛ss✉♠❡ t♦ ❛ss✉r❡ t❤❛t ✐t ✐s ♦♥t♦❄ ❲❡ s✉♠♠❛r✐③❡ t❤❡ r❡s✉❧ts ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♠♣♦rt❛♥t t❤❡♦r❡♠✳ ❚❤❡♦r❡♠ ✸✳✹✳✶✹✿ ❍♦r✐③♦♥t❛❧ ▲✐♥❡ ❚❡st ✶✳ ❆ ❢✉♥❝t✐♦♥ ✐s

❧❡❛st

✷✳ ❆ ❢✉♥❝t✐♦♥ ✐s ❤❛✈❡

♦♥t♦

✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐ts ❣r❛♣❤ ❛♥❞ ❛♥② ❤♦r✐③♦♥t❛❧ ❧✐♥❡ ❤❛✈❡

❛t

♦♥❡ ♣♦✐♥t ✐♥ ❝♦♠♠♦♥✳

❛t ♠♦st

♦♥❡✲t♦✲♦♥❡

✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐ts ❣r❛♣❤ ❛♥❞ ❛♥② ❤♦r✐③♦♥t❛❧ ❧✐♥❡

♦♥❡ ♣♦✐♥t ✐♥ ❝♦♠♠♦♥✳

❊①❡r❝✐s❡ ✸✳✹✳✶✺

❇r❡❛❦ ❡✐t❤❡r ♣❛rt ♦❢ t❤❡ t❤❡♦r❡♠ ✐♥t♦ ❛ st❛t❡♠❡♥t ❛♥❞ ✐ts ❝♦♥✈❡rs❡✳ ■♥ ❧✐❣❤t ♦❢ t❤❡ t❤❡♦r❡♠✱ ✇❡ r❡❞♦ t❤❡ t❛❜❧❡ ♣r❡s❡♥t❡❞ ❡❛r❧✐❡r ❜② ❝♦✉♥t✐♥❣ t❤❡ ♥✉♠❜❡r ♦❢ ✐♥t❡rs❡❝t✐♦♥s ♦❢ ❤♦r✐③♦♥t❛❧ ❧✐♥❡s ✇✐t❤ t❤❡ ❣r❛♣❤✿

■♥ ❡❛❝❤ ❝❡❧❧ ♦❢ t❤❡ t❛❜❧❡✱ ✇❡ s❡❡ ❛♥ ❡①❛♠♣❧❡ ♦❢ ❤♦✇ ✕ ✐♥ t❤❡ s✐♠♣❧❡st ♣♦ss✐❜❧❡ ✇❛② ✕ t❤❡s❡ t✇♦ ❝♦♥❞✐t✐♦♥s ❝❛♥ ❜❡ ✈✐♦❧❛t❡❞✱ ❢♦r ❛ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥✳

✸✳✹✳

✷✸✶

❋✉♥❝t✐♦♥s ✇✐t❤ r❡❣✉❧❛r✐t✐❡s✿ ♦♥❡✲t♦✲♦♥❡ ❛♥❞ ♦♥t♦

❊①❡r❝✐s❡ ✸✳✹✳✶✻

■❧❧✉str❛t❡ ❡❛❝❤ ♦❢ t❤❡ ❢♦✉r ♣♦ss✐❜✐❧✐t✐❡s ❛❜♦✈❡ ✇✐t❤ ❛ ❝❤♦✐❝❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ❣✐✈❡♥ ❜② ❛ ❧✐st ♦❢ ✈❛❧✉❡s✳ ▲✐♥❡❛r ❢✉♥❝t✐♦♥s ❛r❡ ❡❛s② t♦ ❝❤❛r❛❝t❡r✐③❡ t❤✐s ✇❛②✿

■♥❞❡❡❞✱ ❛s ✇❡ ❦♥♦✇ ❢r♦♠ ❣❡♦♠❡tr②✱ t✇♦ str❛✐❣❤t ❧✐♥❡s ❤❛✈❡ ❡①❛❝t❧② ♦♥❡ ✐♥t❡rs❡❝t✐♦♥ ✉♥❧❡ss t❤❡② ❛r❡ ♣❛r❛❧❧❡❧❀ t❤❡r❡❢♦r❡ t❤❡ ❍♦r✐③♦♥t❛❧ ▲✐♥❡ ❚❡st ✐s ♣❛ss❡❞ ❜② ❛❧❧ ❧✐♥❡❛r ❢✉♥❝t✐♦♥s ❡①❝❡♣t t❤❡ ♦♥❡s ✇✐t❤ ③❡r♦ s❧♦♣❡✳ ❚❤♦s❡ ❛r❡ ❝♦♥st❛♥t ❢✉♥❝t✐♦♥s✳ ❍♦✇❡✈❡r✱ ✇❡ ♥❡❡❞ t♦ ♣r♦✈❡ t❤❡s❡ ❢❛❝ts ❛❧❣❡❜r❛✐❝❛❧❧②✳ ❊①❛♠♣❧❡ ✸✳✹✳✶✼✿ ❧✐♥❡❛r ♣♦❧②♥♦♠✐❛❧s

❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ❧✐♥❡❛r ❢✉♥❝t✐♦♥ ✇✐t❤ s❧♦♣❡ 3 ❛♥❞ y ✲✐♥t❡r❝❡♣t 2✿ F (x) = 3x + 2 .

■s ✐t ♦♥❡✲t♦✲♦♥❡❄ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ t✇♦ ✐♥♣✉ts x1 ❛♥❞ x2 ✳ ❈❛♥ t❤❡✐r ♦✉t♣✉ts ❜❡ ❡q✉❛❧ ✉♥❞❡r F ❄ ▲❡t✬s tr②✿ ❙✉♣♣♦s❡ F (x1 ) = F (x2 )✳ ❲❡ s✉❜st✐t✉t❡ ❛♥❞ ♦❜t❛✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣✿ 3x1 + 2 = 3x2 + 2 .

❈❛♥❝❡❧✐♥❣ 2 ♣r♦❞✉❝❡s t❤❡ ❢♦❧❧♦✇✐♥❣✿ 3x1 = 3x2 .

❋✐♥❛❧❧②✱ ✇❡ ❞✐✈✐❞❡ ❜② 3✿ x1 = x2 .

◆♦✱ t❤❡ ♦✉t♣✉ts ❛r❡ ❡q✉❛❧ ♦♥❧② ✇❤❡♥ t❤❡ ✐♥♣✉ts ❛r❡✦ ■s ✐t ♦♥t♦❄ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛♥ ✐♥♣✉t y ✳ ■s t❤❡r❡ ❛♥ x t❛❦❡♥ t♦ y ✉♥❞❡r F ❄ ❲❡ ❥✉st ♥❡❡❞ t♦ s♦❧✈❡ t❤❡ ❡q✉❛t✐♦♥ F (x) = y ❢♦r x✱ ❢♦r ❡❛❝❤ y ✳ ❲❡ ❤❛✈❡ ❛♥ ❡q✉❛t✐♦♥✿ 3x + 2 = y ,

✇✐t❤ ❛♥ ✉♥s♣❡❝✐✜❡❞ y ✳ ◆♦ ♠❛tt❡r ✇❤❛t y ✐s t❤♦✉❣❤✱ ✇❡ s✉❜tr❛❝t 2 ❛♥❞ t❤❡♥ ❞✐✈✐❞❡ ❜② 3✱ ♣r♦❞✉❝✐♥❣✿ x=

y−2 . 3

❨❡s✱ t❤❡r❡ ✐s s✉❝❤ ❛♥ x✱ ❢♦r ❡❛❝❤ y ✦ ❚❤❡♦r❡♠ ✸✳✹✳✶✽✿ ▲✐♥❡❛r ❋✉♥❝t✐♦♥s✱ ❖♥❡✲t♦✲♦♥❡ ❖♥t♦ ❆ ❧✐♥❡❛r ❢✉♥❝t✐♦♥ ✇✐t❤ s❧♦♣❡

m

❛♥❞

y ✲✐♥t❡r❝❡♣t b✱

F (x) = mx + b , ✐s ❜♦t❤ ♦♥❡✲t♦✲♦♥❡ ❛♥❞ ♦♥t♦ ❛s ❧♦♥❣ ❛s

m 6= 0✳

✸✳✹✳

✷✸✷

❋✉♥❝t✐♦♥s ✇✐t❤ r❡❣✉❧❛r✐t✐❡s✿ ♦♥❡✲t♦✲♦♥❡ ❛♥❞ ♦♥t♦

Pr♦♦❢✳ ❖♥❡✲t♦✲♦♥❡✳

❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ t✇♦ ✐♥♣✉ts x1 ❛♥❞ x2 ✳ ❚❤❡♥✿ F (x1 ) = F (x2 ) =⇒ mx1 + b = mx2 + b =⇒ mx1 = mx2 =⇒ x1 = x2 .

■t✬s t❤❡ s❛♠❡ ✐♥♣✉t✳ ❖♥t♦✳

❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛♥ ♦✉t♣✉t y ✳ ❲❡ s♦❧✈❡ t❤❡ ❡q✉❛t✐♦♥ F (x) = y ❢♦r x✿ mx + b = y =⇒ x =

y−b . m

❚❤❡r❡ ✐s s✉❝❤ ❛♥ x✳ ❊①❡r❝✐s❡ ✸✳✹✳✶✾

❇r❡❛❦ t❤❡ t❤❡♦r❡♠ ✐♥t♦ ❛ st❛t❡♠❡♥t ❛♥❞ ✐ts ❝♦♥✈❡rs❡✳ ❚❤❡ ♣✐❝t✉r❡ ❝♦♥✜r♠s t❤❡ ❝♦♥❝❧✉s✐♦♥✿

❊①❛♠♣❧❡ ✸✳✹✳✷✵✿ r❡❝✐♣r♦❝❛❧

▲❡t✬s ♣r♦✈❡ ❛❧❣❡❜r❛✐❝❛❧❧② t❤❛t f (x) = 1/x ✐s ♦♥❡✲t♦✲♦♥❡ ❜✉t ♥♦t ♦♥t♦✳ ❚❤❡r❡ ✐s ❛♥ ❡q✉❛t✐♦♥ ❢♦r ❡❛❝❤ y ✇✐t❤ r❡s♣❡❝t t♦ x✿ y = 1/x .

❲❡ s♦❧✈❡ ❡❛❝❤ ❢♦r x✿ y = 1/x =⇒ x = 1/y .

❆ s✐♥❣❧❡ s♦❧✉t✐♦♥ ❢♦r ❡❛❝❤ y ✱ ❤❡♥❝❡ t❤❡ ❢✉♥❝t✐♦♥ ✐s ♦♥❡✲t♦✲♦♥❡✳ ❇✉t t❤❡r❡ ✐s ♥♦ s♦❧✉t✐♦♥ ✇❤❡♥ y = 0✱ ❤❡♥❝❡ t❤❡ ❢✉♥❝t✐♦♥ ✐s♥✬t ♦♥t♦✳ ❖❢ ❝♦✉rs❡✱ t❤❡ ❛♥s✇❡r ❞❡♣❡♥❞s ♦♥ ②♦✉r ❝❤♦✐❝❡ ♦❢ t❤❡ ❝♦❞♦♠❛✐♥✿ Y = {y : y 6= 0} ♠❛❦❡s f : R → Y ♦♥t♦✳ ❊①❡r❝✐s❡ ✸✳✹✳✷✶

Pr♦✈❡ ❛❧❣❡❜r❛✐❝❛❧❧② t❤❛t f (x) = 1/x2 ✐s ♥❡✐t❤❡r ♦♥❡✲t♦✲♦♥❡ ♥♦r ♥♦t ♦♥t♦✳ ❙❤♦✇ ❤♦✇ t❤❡ ❛♥s✇❡r ❝❤❛♥❣❡s ✇✐t❤ ②♦✉r ❝❤♦✐❝❡ ♦❢ t❤❡ ❝♦❞♦♠❛✐♥✳ ❊①❡r❝✐s❡ ✸✳✹✳✷✷

❈❧❛ss✐❢② t❤❡ ❢✉♥❝t✐♦♥ f (x) = x3 − x ❛❝❝♦r❞✐♥❣ t♦ t❤❡s❡ t✇♦ ❞❡✜♥✐t✐♦♥s✳ Pr♦✈❡ ❛❧❣❡❜r❛✐❝❛❧❧②✳ ❚❤❡♦r❡♠ ✸✳✹✳✷✸✿ ❖♥❡✲t♦✲♦♥❡ ❛♥❞ ❖♥t♦ ✈s✳ ■♠❛❣❡

F : X → Y ❝♦❞♦♠❛✐♥✱ Y ✳ ❆ ❢✉♥❝t✐♦♥ F : X → Y ✐s

✶✳ ❆ ❢✉♥❝t✐♦♥

✷✳

❡❧❡♠❡♥t ♦❢ t❤❡ ❝♦❞♦♠❛✐♥ ❡♠♣t②✳

✐s ♦♥t♦ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐ts ✐♠❛❣❡ ✐s t❤❡ ✇❤♦❧❡

♦♥❡✲t♦✲♦♥❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ ♣r❡✐♠❛❣❡ ♦❢ ❡✈❡r②

Y

✐s ❛ s✐♥❣❧❡ ❡❧❡♠❡♥t ♦❢ t❤❡ ❞♦♠❛✐♥✱

X✱

♦r ✐t✬s

✸✳✹✳ ❋✉♥❝t✐♦♥s ✇✐t❤ r❡❣✉❧❛r✐t✐❡s✿ ♦♥❡✲t♦✲♦♥❡ ❛♥❞ ♦♥t♦

✷✸✸

❊①❡r❝✐s❡ ✸✳✹✳✷✹

❇r❡❛❦ ❡✐t❤❡r ♣❛rt ♦❢ t❤❡ t❤❡♦r❡♠ ✐♥t♦ ❛ st❛t❡♠❡♥t ❛♥❞ ✐ts ❝♦♥✈❡rs❡✳

❊①❡r❝✐s❡ ✸✳✹✳✷✺

Pr♦✈❡ t❤❡ t❤❡♦r❡♠✳

❚♦ s✉♠♠❛r✐③❡✱ t❤❡ r❡str✐❝t✐♦♥s ✐♥ t❤❡s❡ t✇♦ ❞❡✜♥✐t✐♦♥s ❝❛♥ ❜❡ ✈✐♦❧❛t❡❞ ✇❤❡♥ t❤❡r❡ ❛r❡ t♦♦ ❢❡✇ ♦r t♦♦ ♠❛♥② ❛rr♦✇s ❛rr✐✈✐♥❣ t♦ ❛ ❣✐✈❡♥

y✳

❚❤❡s❡ ✈✐♦❧❛t✐♦♥s ❛r❡ s❡❡♥ ✐♥ t❤❡ ❝♦❞♦♠❛✐♥✳ ❚❤✐s ♦♥❡ ✐s ♥♦t ♦♥t♦ ✿

❚❤❛t ♦♥❡ ✐s ♥♦t ♦♥❡✲t♦✲♦♥❡ ✿

❊①❛♠♣❧❡ ✸✳✹✳✷✻✿ ❜♦t❤ ♦♥❡✲t♦✲♦♥❡ ❛♥❞ ♦♥t♦

❲❤❛t ❢✉♥❝t✐♦♥s ❛r❡ t❤❡ ♠♦st ✏r❡❣✉❧❛r✑❄ ❚❤❡ ♦♥❡s t❤❛t ❛r❡ ❜♦t❤ ♦♥❡✲t♦✲♦♥❡ ❛♥❞ ♦♥t♦✱ s♦♠❡t✐♠❡s ❝❛❧❧❡❞

❜✐❥❡❝t✐♦♥s ✿

❚❤❡ ❢✉♥❝t✐♦♥ ♠❛② ❧♦♦❦ ♣❧❛✐♥ ❜✉t ✐t ❤❛s ❛♥ ✐♠♣♦rt❛♥t ♣r♦♣❡rt②✿ t❤❡ ❜♦②s ❛♥❞ t❤❡ ❜❛❧❧s ❝❛♥ ❜❡ ✉s❡❞ ❛s s✉❜st✐t✉t❡s ♦❢ ❡❛❝❤ ♦t❤❡r✦ ❋♦r ❡①❛♠♣❧❡✱ ②♦✉ ❞♦♥✬t ❤❛✈❡ t♦ r❡♠❡♠❜❡r t❤❡ ♥❛♠❡ ♦❢ ❡✈❡r② ❜♦② ❜✉t ❥✉st s❛② ✏t❤❡ ♦♥❡ t❤❛t ♣❧❛②s ❜❛s❦❡t❜❛❧❧✑ t♦ ✐❞❡♥t✐❢② ❚♦♠ ✇✐t❤♦✉t ❛ ❝❤❛♥❝❡ ♦❢ ❝♦♥❢✉s✐♦♥✳

❊①❡r❝✐s❡ ✸✳✹✳✷✼

❲❤❛t ❛r❡ t❤❡ s♠❛❧❧❡st s❡t

X

❛♥❞ t❤❡ s♠❛❧❧❡st s❡t

Y

❢♦r ✇❤✐❝❤ ❛ ❢✉♥❝t✐♦♥

♦♥❡✲t♦✲♦♥❡✱ ✭❜✮ ♥♦t ♦♥t♦❄

F :X→Y

❝❛♥ ❜❡✿ ✭❛✮ ♥♦t

❊①❡r❝✐s❡ ✸✳✹✳✷✽

❙❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥

f

❣✐✈❡♥ ❜② ✐ts ❧✐st ♦❢ ✈❛❧✉❡s ❜❡❧♦✇✳ ■s ✐t ♦♥❡✲t♦✲♦♥❡❄

x 1 2 3 4 5 y = f (x) 1 2 0 3 1 ▼❛❦❡ ✉♣ ②♦✉r ♦✇♥ ❢✉♥❝t✐♦♥s ❜② ♣r♦✈✐❞✐♥❣ t❤❡✐r ❧✐sts ♦❢ ✈❛❧✉❡s ❛♥❞ t❡st t❤❡ t✇♦ ❞❡✜♥✐t✐♦♥s✳ ❘❡♣❡❛t✳

❊①❛♠♣❧❡ ✸✳✹✳✷✾✿ ♣♦✇❡r ❢✉♥❝t✐♦♥s✱ s②♠♠❡tr✐❡s

❚❤❡ ♣♦✇❡r ❢✉♥❝t✐♦♥s✱

y = xn , n = ..., −3, −2, −1, 0, 1, 2, 3, ..., ❝❛♥ ♥♦✇ ❜❡ ❝❧❛ss✐✜❡❞✿

✸✳✺✳

❈♦♠♣♦s✐t✐♦♥s ♦❢ ❢✉♥❝t✐♦♥s

✷✸✹

❊①❡r❝✐s❡ ✸✳✹✳✸✵

❲❤✐❝❤ ♦♥❡s ❛r❡ ♦♥t♦❄ ❊①❡r❝✐s❡ ✸✳✹✳✸✶

Pr♦✈❡ t❤❛t ❛ ♦♥❡✲t♦✲♦♥❡ ❢✉♥❝t✐♦♥ ❝❛♥✬t ❤❛✈❡ ♠✐rr♦r s②♠♠❡tr②✳

❊①❡r❝✐s❡ ✸✳✹✳✸✷

❲❤❛t ❦✐♥❞ ♦❢ ❢✉♥❝t✐♦♥ ✐s ✏♠❛♥②✲t♦✲♦♥❡✑❄ ❲❤❛t ❛❜♦✉t ✏♦♥❡✲t♦✲♠❛♥②✑❄

■♥ s✉♠♠❛r②✱ ✇❡ ♣r❡s❡♥t t❤❡ t✇♦ s✐♠♣❧❡st ✇❛②s t❤❡ t✇♦ ❝♦♥❞✐t✐♦♥s ❝❛♥ ❜❡

◆♦t ♦♥t♦✿

• −→ • •

◆♦t ✶✲✶✿

✈✐♦❧❛t❡❞ ✿

• −→ • ր •

✸✳✺✳ ❈♦♠♣♦s✐t✐♦♥s ♦❢ ❢✉♥❝t✐♦♥s

❇❛❝❦ t♦ ♦✉r ❜♦②s✲❛♥❞✲❜❛❧❧s ❡①❛♠♣❧❡✱ ❧❡t✬s ♥♦t❡ t❤❡

❝♦❧♦rs ♦❢ t❤❡ ❜❛❧❧s✳

❚❤✐s ❤❛s ♥♦t❤✐♥❣ t♦ ❞♦ ✇✐t❤ t❤❡ ❜♦②s✱

❛♥❞ ✐t ❝r❡❛t❡s ❛ ♥❡✇ ❢✉♥❝t✐♦♥✿

■t ✐s ❛ ❢✉♥❝t✐♦♥

G:Y →Z

❢r♦♠ t❤❡ s❡t ♦❢ ❛❧❧ ❜❛❧❧s t♦ t❤❡ ♥❡✇ s❡t

Z

♦❢ t❤❡ ♠❛✐♥ ❝♦❧♦rs✳

❊①❡r❝✐s❡ ✸✳✺✳✶

■s t❤❡ ♥❡✇ ❢✉♥❝t✐♦♥ ♦♥❡✲t♦✲♦♥❡ ♦r ♦♥t♦❄

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

♣r❡❢❡r❡♥❝❡s ✐♥ ❝♦❧♦rs ❄

■♥ ❛ s❡♥s❡✳ ❲❡ ❝❛♥ ❥✉st

✸✳✺✳

❈♦♠♣♦s✐t✐♦♥s ♦❢ ❢✉♥❝t✐♦♥s

✷✸✺

■❢ ✇❡ st❛rt ✇✐t❤ ❛ ❜♦② ♦♥ t❤❡ ❧❡❢t✱ ✇❡ ❝❛♥ ❝♦♥t✐♥✉❡ ✇✐t❤ t❤❡ ❛rr♦✇s ❛❧❧ t❤❡ ✇❛② t♦ t❤❡ r✐❣❤t✳ ❚❤✐s ✇❛②✱ ✇❡ ✇✐❧❧ ❦♥♦✇ t❤❡ ❝♦❧♦r ♦❢ t❤❡ ❜❛❧❧ t❤❡ ❜♦② ✐s ♣❧❛②✐♥❣ ✇✐t❤✿

❚❤✐s ✐s ❛ ♥❡✇ ❢✉♥❝t✐♦♥✱ s❛②

H : X → Z✱

❢r♦♠ t❤❡ s❡t ♦❢ ❜♦②s

X

t♦ t❤❡ s❡t

Z

♦❢ t❤❡ ♠❛✐♥ ❝♦❧♦rs✿

❊①❡r❝✐s❡ ✸✳✺✳✷ ■s t❤❡ ♥❡✇ ❢✉♥❝t✐♦♥ ♦♥❡✲t♦✲♦♥❡ ♦r ♦♥t♦❄ ■♥ ❣❡♥❡r❛❧✱ t❤✐s ✐s t❤❡ s❡t✉♣✿ F

X −−−−→ Y

G

−−−−→ Z

❚❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❝❡♣t ✐s ✈❡r② ✐♠♣♦rt❛♥t✳

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

❚❤❡♥ t❤❡✐r

❝♦♠♣♦s✐t✐♦♥

F :X→Y

❛♥❞

G:Y →Z.

✐s t❤❡ ❢✉♥❝t✐♦♥ ✭❢r♦♠ t❤❡ ❞♦♠❛✐♥ ♦❢ t❤❡ ❢♦r♠❡r t♦ t❤❡

❝♦❞♦♠❛✐♥ ♦❢ t❤❡ ❧❛tt❡r✮

✇❤✐❝❤ ✐s ❝♦♠♣✉t❡❞ ❢♦r ❡✈❡r②

x

✐♥

H :X →Z, X

❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦✲st❡♣ ♣r♦❝❡✲

❞✉r❡✿ ■t ✐s ❞❡♥♦t❡❞ ❜②

x 7→ F (x) = y 7→ G(y) = z G◦F

✸✳✺✳

❈♦♠♣♦s✐t✐♦♥s ♦❢ ❢✉♥❝t✐♦♥s

❲❡ ❥✉st ❢♦❧❧♦✇ ❢r♦♠

X

✷✸✻

F

❛❧♦♥❣ t❤❡ ❛rr♦✇s ♦❢

t♦

Y

❛♥❞ t❤❡♥ ❛❧♦♥❣ t❤❡ ❛rr♦✇s ♦❢

G

t♦

Z✿

❲❛r♥✐♥❣✦ ❚❤❡ r❡q✉✐r❡♠❡♥t t❤❛t ✏t❤❡ ❝♦❞♦♠❛✐♥ ♦❢ t❤❡ ❢♦r♠❡r ✐s t❤❡ ❞♦♠❛✐♥ ♦❢ t❤❡ ❧❛tt❡r✑ ❝♦✉❧❞ ❜❡ r❡♣❧❛❝❡❞ ✇✐t❤ ✏t❤❡ ❞♦♠❛✐♥

■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ♥❡✇ ❢✉♥❝t✐♦♥ ✐s ❡✈❛❧✉❛t❡❞ ❜② t❤❡

❝♦♥t❛✐♥s

t❤❡ ❝♦❞♦♠❛✐♥✑✳

s✉❜st✐t✉t✐♦♥ ❢♦r♠✉❧❛ ✿

z = H(x) = G(F (x)) ❚❤✐s ✐s t❤❡ ✏❞❡❝♦♥str✉❝t✐♦♥✑ ♦❢ t❤❡ ♥♦t❛t✐♦♥✿

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

G◦F ↑




↓ ↓

(x)

= G

F (x)

↑ ↑ ↑

♥❛♠❡ ♦❢ t❤❡ ♥❡✇ ❢✉♥❝t✐♦♥

s✉❜st✐t✉t✐♦♥

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



F ✑✳

❚❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦♥ r✐❣❤t r❡❛❞s ✏ G ♦❢

F

x✑✳

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

x

❜❧❛❝❦ ❜♦①❡s✱ ✇❡ ❝❛♥ ✇✐r❡ t❤❡♠ t♦❣❡t❤❡r✿

❢✉♥❝t✐♦♥

→

❢r♦♠ ✐♥s✐❞❡ ♦✉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✳ ❚♦ ♠❛❦❡ ✐t ❝❧❡❛r t❤❛t

Y

✐s ♥♦ ❧♦♥❣❡r ❛ ♣❛rt ♦❢ t❤❡ ♣✐❝t✉r❡✱ ✇❡ ❝❛♥ ❛❧s♦ ✈✐s✉❛❧✐③❡ t❤❡ ❝♦♠♣♦s✐t✐♦♥ ❛s ❢♦❧❧♦✇s✿

F

X −−−−→ ցH

Y   G y Z

❚❤❡ ♠❡❛♥✐♥❣ ♦❢ 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 ♦❢

G ◦ F✿

✸✳✺✳

❈♦♠♣♦s✐t✐♦♥s ♦❢ ❢✉♥❝t✐♦♥s

•

❙t❡♣ ✶✿ ❉♦

F✳

•

❙t❡♣ ✷✿ ❉♦

G✳

✷✸✼

❝♦♥s❡❝✉t✐✈❡❧② ❀ ②♦✉ ❝❛♥✬t st❛rt ✇✐t❤ t❤❡ s❡❝♦♥❞ ✉♥t✐❧ ②♦✉ ❛r❡ ❞♦♥❡ ✇✐t❤ t❤❡ ✜rst✳ ▲❡t✬s t❡st t❤✐s ✐❞❡❛ ♦♥ ♦✉r ❡①❛♠♣❧❡✳ ❲❡ t❛❦❡ t❤❡ t✇♦ ❧✐sts ♦❢ ✈❛❧✉❡s ❛♥❞ t❤❡♥ ❝r♦ss✲r❡❢❡r❡♥❝❡ t❤❡♠ ✭❢r♦♠ ❧❡❢t

❚❤❡② ❛r❡ ❡①❡❝✉t❡❞

t♦ r✐❣❤t✮✿

F( F( F( F( F(

) ) ❇❡♥ ) ❑❡♥ ) ❙✐❞ ) ❚♦♠

◆❡❞

= = = = =

❜❛s❦❡t❜❛❧❧

G( G( G( G(

t❡♥♥✐s

,

❜❛s❦❡t❜❛❧❧ ❢♦♦t❜❛❧❧ ❢♦♦t❜❛❧❧

) = t❡♥♥✐s ) = ❢♦♦t❜❛❧❧ ) = ❜❛s❡❜❛❧❧ ) = ❜❛s❦❡t❜❛❧❧

♦r❛♥❣❡ ②❡❧❧♦✇

−→

❜r♦✇♥ ✇❤✐t❡

H( H( H( H( H(

) ) ❇❡♥ ) ❑❡♥ ) ❙✐❞ ) ❚♦♠

◆❡❞

= = = = =

❄ ❄ ❄ ❄ ❄

❲❡ ✐❣♥♦r❡ ❛♥② ❛❧✐❣♥♠❡♥t ❜❡t✇❡❡♥ t❤❡ t✇♦ ❧✐sts✳ ❲❡ t❛❦❡ t❤❡ ✜rst ❡♥tr② ✐♥ t❤❡ s❡❝♦♥❞ ❧✐st✱

G(

)=

❜❛s❦❡t❜❛❧❧

♦r❛♥❣❡✱

❛♥❞ r❡♣❧❛❝❡ ✏❜❛s❦❡t❜❛❧❧✑✱ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ✜rst ❡♥tr② ♦❢ t❤❡ ✜rst ❧✐st✱ ✇✐t❤

G



F(

❚♦♠



)

=

F(

❚♦♠

)✳

❚❤✐s ✐s t❤❡ r❡s✉❧t✿

♦r❛♥❣❡✳

❚❤❡r❡❢♦r❡✱

H(

❚♦♠

)=

♦r❛♥❣❡✳

❚❤✐s ✐s t❤❡ ✜rst ❡♥tr② ✐♥ t❤❡ ♥❡✇ ❧✐st✳ ❊①❡r❝✐s❡ ✸✳✺✳✹

❋✐♥✐s❤ t❤❡ ❧✐st✳

❖♥❝❡ ❛❣❛✐♥✱ ❛❧❣❡❜r❛✐❝❛❧❧②✱ ❝♦♠♣♦s✐t✐♦♥ ✐s ♥♦t❤✐♥❣ ❜✉t

s✉❜st✐t✉t✐♦♥ ✦

❯♥❢♦rt✉♥❛t❡❧②✱ t❤❡ ❞♦♠❛✐♥s ♦❢ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s ❛r❡ ♦❢t❡♥ ✐♥✜♥✐t❡✱ ❛♥❞ ✐t ✐s ✐♠♣♦ss✐❜❧❡ t♦ ❥✉st ❢♦❧❧♦✇ t❤❡ ❛rr♦✇s✳ ❲❡ ❞❡❛❧ ✇✐t❤ ❢♦r♠✉❧❛s✳ ❋♦rt✉♥❛t❡❧②✱ s✉❜st✐t✉t✐♦♥ ✐s ❥✉st ❛s s✐♠♣❧❡ ❢♦r ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s✳ ❊①❛♠♣❧❡ ✸✳✺✳✺✿ ❝♦♠♣♦s✐t✐♦♥ ♦❢ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s

❚❤✐s t✐♠❡✱ ✇❡ s✉❜st✐t✉t❡ ♦♥❡ ❢♦r♠✉❧❛ ✐♥t♦ ❛♥♦t❤❡r✳ ❋♦r ❡①❛♠♣❧❡✱ ❝♦♥s✐❞❡r t❤❡s❡ t✇♦ ❢✉♥❝t✐♦♥s✿

x2 =y

X −−−−−−→ Y ❚❤❡♥✱

y = x2

✐s s✉❜st✐t✉t❡❞ ✐♥t♦

z = y3

y 3 =z

−−−−−−→ Z

r❡s✉❧t✐♥❣ ✐♥✿

z = x2


3

.

■t ✐s t❤❡ s❛♠❡ ❢✉♥❝t✐♦♥✱ ❛♥❞ ✐t ✐s ❝♦♠♣✉t❡❞ ❜② t❤❡ s❛♠❡

t✇♦ st❡♣s✦

❆ s✐♠♣❧✐✜❝❛t✐♦♥ ♠✐❣❤t ♠❛❦❡ t❤❡

❡①tr❛ ✇♦r❦ ✇♦rt❤✇❤✐❧❡✿

z = x6 . ❊①❛♠♣❧❡ ✸✳✺✳✻✿ s✉❜st✐t✉t✐♦♥ ♦❢ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s

❚❤❡ ✐❞❡❛ ♦❢ ❤♦✇ t❤❡ s✉❜st✐t✉t✐♦♥ ✐s ❡①❡❝✉t❡❞ ✐s t❤❡ s❛♠❡ ❛s ✐♥ ❈❤❛♣t❡r ✷✿ ■♥s❡rt t❤❡ ✐♥♣✉t ✈❛❧✉❡ ✐♥ ❛❧❧ ♦❢ t❤❡s❡ ❜♦①❡s✳ ❙✉♣♣♦s❡ 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✐t✐♦♥s ♦❢ ❢✉♥❝t✐♦♥s

✷✸✽

Pr❡✈✐♦✉s❧②✱ ✇❡ ❞✐❞ t❤❡ s✉❜st✐t✉t✐♦♥ y = 3 ❜② ✐♥s❡rt✐♥❣ 3 ✐♥ ❡❛❝❤ ♦❢ t❤❡s❡ ✇✐♥❞♦✇s✿ f



3



=

2

2 3 3

3

−3 3 +7

+2 3 +1

.

❚❤✐s t✐♠❡✱ ❧❡t✬s ✐♥s❡rt sin x✱ ♦r✱ ❜❡tt❡r✱ (sin x)✳ ❚❤✐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

◆♦t❡ t❤❛t ✐❢ ②♦✉ ❞♦♥✬t ❦♥♦✇ ✇❤❛t t❤❡ s✐♥❡ ❢✉♥❝t✐♦♥ ✭❈❤❛♣t❡r ✹✮ ❞♦❡s✱ ✐t ♠❛❦❡s ♥♦ ❞✐✛❡r❡♥❝❡✦ ❚❤❡ ♦♥❧② t❤✐♥❣ t❤❛t ♠❛tt❡rs ✐s t❤❛t ✇❡ ❦♥♦✇ t❤❛t t❤✐s ✐s ❛ ❢✉♥❝t✐♦♥✳ ❊①❛♠♣❧❡ ✸✳✺✳✼✿ ❝♦♠♣♦s✐t✐♦♥ ❢r♦♠ t❛❜❧❡s

◆❡①t✱ ✇❤❛t ❛❜♦✉t r❡♣r❡s❡♥t✐♥❣ ❢✉♥❝t✐♦♥s ❜② t❤❡✐r t❛❜❧❡s❄ ❚❤❡s❡ ❛r❡ t❤❡ t❛❜❧❡s ♦❢ F ❛♥❞ G ❛❜♦✈❡✿

❍♦✇ ❞♦ ✇❡ ❝♦♠❜✐♥❡ t❤❡♠ t♦ ✜♥❞ H ❄ ❚❤❡ ❛❧✐❣♥♠❡♥t ✇❡ ♠✐❣❤t s❡❡ ✐s ♠❡❛♥✐♥❣❧❡ss✳ ❲❡ ♥❡❡❞ t♦ ❛❧✐❣♥ ✇❤❛t t❤❡ t✇♦ ❤❛✈❡ ✐♥ ❝♦♠♠♦♥✱ Y ✳ ❚❤❡♥ ✇❡ st❛rt ✇✐t❤ ❛♥ x ✐♥ X ✱ ✉s❡ F t♦ ✜♥❞ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ✈❛❧✉❡ ♦❢ y ✐♥ Y ✱ t❤❡♥ ✉s❡ G t♦ ✜♥❞ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ✈❛❧✉❡ ♦❢ z ✭♦♥❡ s✉❝❤ st❡♣ ✐s ✐❧❧✉str❛t❡❞ ❜❡❧♦✇✮✿

❊①❛♠♣❧❡ ✸✳✺✳✽✿ ❝♦♠♣♦s✐t✐♦♥ ❢r♦♠ ❣r❛♣❤s

❲❤❛t ✐❢ ✇❡ ❤❛✈❡ ♦♥❧② t❤❡✐r ❣r❛♣❤s❄ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ t❤❡ t✇♦ ❣r❛♣❤s ♦❢ u = f (x) ❛♥❞ y = g(u) ,

s✐❞❡ ❜② s✐❞❡ ❛♥❞ ✇❡ ♥❡❡❞ t♦ ✜♥❞ t❤❡ ❝♦♠♣♦s✐t✐♦♥✱ g ◦ f ✳ ▲❡t✬s t❛❦❡ ❛ s✐♥❣❧❡ ✈❛❧✉❡✿ d = g(f (a)) .

❚❤❡♥✱ ✇❡ ✉s❡ t❤❡ ✜rst ❣r❛♣❤ t♦ ✜♥❞ c = f (a) ♦♥ t❤❡ ✈❡rt✐❝❛❧ ❛①✐s✱ tr❛✈❡❧ ❛❧❧ t❤❡ ✇❛② t♦ t❤❡ ❤♦r✐③♦♥t❛❧ ❛①✐s ♦❢ t❤❡ s❡❝♦♥❞✱ ✜♥❞ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ c ♦♥ ✐t✱ ❛♥❞ ✜♥❛❧❧② ✜♥❞ d = g(c) ♦♥ t❤❡ ✈❡rt✐❝❛❧ ❛①✐s✳

✸✳✺✳

❈♦♠♣♦s✐t✐♦♥s ♦❢ ❢✉♥❝t✐♦♥s

✷✸✾

●r❛♣❤s ❞♦♥✬t ❞♦ ❛ ✈❡r② ❣♦♦❞ ❥♦❜ ♦❢ ✈✐s✉❛❧✐③✐♥❣ ❝♦♠♣♦s✐t✐♦♥s✳✳✳ ❊①❛♠♣❧❡ ✸✳✺✳✾✿ ❝♦♠♣♦s✐t✐♦♥ ♦❢ tr❛♥s❢♦r♠❛t✐♦♥s

❲❤❛t ✐❢ ✇❡✱ ✐♥st❡❛❞✱ t❤✐♥❦ ♦❢ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s ❛s tr❛♥s❢♦r♠❛t✐♦♥s❄ ❋♦r ❡①❛♠♣❧❡✿ • ❚❤❡ ✜rst tr❛♥s❢♦r♠❛t✐♦♥ ✐s ❛ str❡t❝❤ ❜② ❛ ❢❛❝t♦r ♦❢ 2✳ • ❚❤❡ s❡❝♦♥❞ tr❛♥s❢♦r♠❛t✐♦♥ ✐s ❛ str❡t❝❤ ❜② ❛ ❢❛❝t♦r ♦❢ 3✳ ❲❡ ✈✐s✉❛❧✐③❡ t❤❡ ✜rst ❜② ❞♦✉❜❧✐♥❣ ❡✈❡r② sq✉❛r❡ ❛♥❞ t❤❡ s❡❝♦♥❞ ❜② tr✐♣❧✐♥❣✿

❆s ❛ r❡s✉❧t t❤❡r❡ ❛r❡ s✐① sq✉❛r❡s ✐♥ t❤❡ ❧❛st r♦✇ ❢♦r ❡❛❝❤ sq✉❛r❡ ✐♥ t❤❡ ✜rst✳ ■♥❞❡❡❞✱ t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ t✇♦ tr❛♥s❢♦r♠❛t✐♦♥s ✐s ❛ str❡t❝❤ ❜② ❛ ❢❛❝t♦r ♦❢ 3 · 2 = 6✳ ❙♦✱ ✇❡ ❥✉st ❝❛rr② ♦✉t t✇♦ tr❛♥s❢♦r♠❛t✐♦♥s ✐♥ ❛ r♦✇✳ ❈♦♥s✐❞❡r t❤✐s ❛❧s♦✿ • ❚❤❡ ✜rst tr❛♥s❢♦r♠❛t✐♦♥ ✐s ❛ s❤✐❢t ✭r✐❣❤t✮ ❜② ❛ 3✳ • ❚❤❡ s❡❝♦♥❞ tr❛♥s❢♦r♠❛t✐♦♥ ✐s ❛ ✢✐♣✳ • ❚❤❡ t❤✐r❞ tr❛♥s❢♦r♠❛t✐♦♥ ✐s ❛ s❤✐❢t ✭r✐❣❤t✮ ❜② ❛ 5✳ ❚❤❡♥ ✇❤❛t t❤❡✐r ❝♦♠♣♦s✐t✐♦♥ ❞♦❡s ✐s s❤♦✇♥ ❜❡❧♦✇✿

❊①❡r❝✐s❡ ✸✳✺✳✶✵

■❧❧✉str❛t❡✱ ❛s ❛❜♦✈❡✱ t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢✿ ❛ s❤✐❢t ❧❡❢t ❜② 5✱ ❛ str❡t❝❤ ❜② 2✱ ❛♥❞ ❛ ✢✐♣✳ ❊①❡r❝✐s❡ ✸✳✺✳✶✶

■❧❧✉str❛t❡ t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ t✇♦ tr❛♥s❢♦r♠❛t✐♦♥s s❤♦✇♥ ❜❡❧♦✇✿

■❢ ✇❡ t❤✐♥❦ ♦❢ ❢✉♥❝t✐♦♥s ❛s ❧✐sts ♦❢ ✐♥str✉❝t✐♦♥s ✭✇✐t❤ ♥♦ ❢♦r❦s✮✱ t❤❡♥ ❡❛❝❤ ♦❢ t❤❡♠ ✐s ❛❧r❡❛❞② ❛ ❝♦♠♣♦s✐t✐♦♥✦ ❚❤❡ st❡♣s ♦♥ t❤❡ ❧✐st ❛r❡ t❤❡ ❢✉♥❝t✐♦♥s t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ ✇❤✐❝❤ ❝r❡❛t❡s t❤❡ ❢✉♥❝t✐♦♥✳ ❋♦r ❡①❛♠♣❧❡✱ ◮ F ✿ ❆❞❞ 3✱ ♠✉❧t✐♣❧② ❜② −2✱ s✉❜tr❛❝t 1✳

❚❤✐s ✐s ❝❛❧❧❡❞ ❛ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ F ✳ ■❢✱ ❢✉rt❤❡r♠♦r❡✱ t❤❡r❡ ✐s ❛♥♦t❤❡r ❢✉♥❝t✐♦♥✱ s❛②✱ ◮ G✿ ❙✉❜tr❛❝t 2✱ ❛♣♣❧② sin✳

✸✳✺✳

❈♦♠♣♦s✐t✐♦♥s ♦❢ ❢✉♥❝t✐♦♥s

✷✹✵

◆♦✇✱ ✇❡ ❥✉st ❛❞❞ t❤❡ ❧❛tt❡r ❧✐st t♦ t❤❡ ❜♦tt♦♠ ♦❢ t❤❡ ❢♦r♠❡r✿

◮ G ◦ F ✿ ❆❞❞ 3✱ ♠✉❧t✐♣❧② ❜② −2✱ s✉❜tr❛❝t 1✱ s✉❜tr❛❝t 2✱ ❛♣♣❧② sin✳

❖❢ ❝♦✉rs❡✱ ✇❡ ❝❛♥ ❤❛✈❡ ❝♦♠♣♦s✐t✐♦♥s ♦❢ ♠❛♥② ❢✉♥❝t✐♦♥s ✐♥ ❛ r♦✇ ❛s ❧♦♥❣ t❤❡ ♦✉t♣✉t ♦❢ ❡❛❝❤ ❢✉♥❝t✐♦♥ ♠❛t❝❤❡s t❤❡ ✐♥♣✉t ♦❢ t❤❡ ♥❡①t✿

■t✬s ❛s ✐❢ t❤❡ ✜rst ❢✉♥❝t✐♦♥ ❣✐✈❡s ✉s ❞✐r❡❝t✐♦♥ t♦ ❛ ❞❡st✐♥❛t✐♦♥✱ ❛♥❞ ❛t t❤❛t ❞❡st✐♥❛t✐♦♥✱ ✇❡ r❡❝❡✐✈❡ t❤❡ ❞✐r❡❝t✐♦♥s t♦ ♦✉r ♥❡①t ❞❡st✐♥❛t✐♦♥ ✇❤❡r❡ ✇❡ ❣❡t ❢✉rt❤❡r ❞✐r❡❝t✐♦♥s✱ ❛♥❞ s♦ ♦♥✳✳✳ ❧✐❦❡ ❛ tr❡❛s✉r❡ ❤✉♥t✳ ❲❡ r❡♣r❡s❡♥t ❢✉♥❝t✐♦♥s ❛s

❜❧❛❝❦ ❜♦①❡s t❤❛t ♣r♦❝❡ss t❤❡ ✐♥♣✉t ❛♥❞ ♣r♦❞✉❝❡ t❤❡ ♦✉t♣✉t✿

✐♥♣✉t ❢✉♥❝t✐♦♥ ♦✉t♣✉t x → f → y ✐♥♣✉t ❢✉♥❝t✐♦♥ ♦✉t♣✉t y → g → z ✐♥♣✉t ❢✉♥❝t✐♦♥ ♦✉t♣✉t z → h → u ❇❡❝❛✉s❡ ♦❢ t❤❡ ♠❛t❝❤✱ ✇❡ ❝❛♥ ❝❛rr② ♦✈❡r t❤❡ ♦✉t♣✉t t♦ t❤❡ ♥❡①t ❧✐♥❡ ✕ ❛s t❤❡ ✐♥♣✉t ♦❢ t❤❡ ♥❡①t ❢✉♥❝t✐♦♥✳ ❚❤✐s

❝❤❛✐♥ ♦❢ ❡✈❡♥ts ❝❛♥ ❜❡ ❛s ❧♦♥❣ ❛s ✇❡ ❧✐❦❡✿ x1 →

f1

→ x2 →

f2

→ x3 →

f3

→ x4 →

f4

→ x5 → ...

❊①❛♠♣❧❡ ✸✳✺✳✶✷✿ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ ✢♦✇❝❤❛rt ❙✉❝❤ ❛ ❞❡❝♦♠♣♦s✐t✐♦♥ ✇✐❧❧ ❛❧❧♦✇ ✉s t♦ st✉❞② t❤❡ ❢✉♥❝t✐♦♥ ♦♥❡ ♣✐❡❝❡ ❛t ❛ t✐♠❡✿

❊①❡r❝✐s❡ ✸✳✺✳✶✸ ❘❡♣r❡s❡♥t t❤❡ ❢♦❧❧♦✇✐♥❣ ❢✉♥❝t✐♦♥ ❜② ❛ s✐♥❣❧❡ ❢♦r♠✉❧❛✿

x →

♠✉❧t✐♣❧② ❜② 2

→ y →

❛❞❞ 5

→ z →

❞✐✈✐❞❡ ❜② 3

→ u

❊①❡r❝✐s❡ ✸✳✺✳✶✹ ❘❡♣r❡s❡♥t t❤❡ ❢✉♥❝t✐♦♥ h(x) = (x − 1)2 + (x − 1)3 ❛s t❤❡ ❝♦♠♣♦s✐t✐♦♥ g ◦ f ♦❢ t✇♦ ❢✉♥❝t✐♦♥s y = f (x) ❛♥❞ z = g(y)✳

✸✳✺✳

❈♦♠♣♦s✐t✐♦♥s ♦❢ ❢✉♥❝t✐♦♥s

✷✹✶

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

❚❤✐s ✐s ❤♦✇ t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ s❡✈❡r❛❧ ❢✉♥❝t✐♦♥s r❡♣r❡s❡♥t❡❞ ❜② ❢♦r♠✉❧❛s ✐s ❝♦♠♣✉t❡❞ ✇✐t❤ ❛ s♣r❡❛❞✲ s❤❡❡t✳ ❲❡ st❛rt ✇✐t❤ ❥✉st ❛ ❧✐st ♦❢ ♥✉♠❜❡rs ✐♥ t❤❡ ✜rst ❝♦❧✉♠♥✳ ❚❤❡♥ ✇❡ ♣r♦❞✉❝❡ t❤❡ ✈❛❧✉❡s ✐♥ t❤❡ ♥❡①t ❝♦❧✉♠♥ ♦♥❡ r♦✇ ❛t ❛ t✐♠❡✳ ❍♦✇❄ ❲❡ ✐♥♣✉t ❛ ❢♦r♠✉❧❛ ✐♥ t❤❡ ♥❡①t ❝♦❧✉♠♥ ✇✐t❤ ❛ r❡❢❡r❡♥❝❡ t♦ t❤❡ ❧❛st ♦♥❡✳ ❋♦r ❡①❛♠♣❧❡✱ ✇❡ ❤❛✈❡ ✐♥ t❤❡ s❡❝♦♥❞ ❛♥❞ t❤✐r❞ ❝♦❧✉♠♥s✱ r❡s♣❡❝t✐✈❡❧②✿ ❂❘❈❬✲✶❪✯✷

❂❘❈❬✲✶❪✰✺

r❡❢❡rr✐♥❣ t♦ t❤❡ ❝❡❧❧ t♦ ✐ts ❧❡❢t✳ ❊❛❝❤ ♦❢ t❤❡ t✇♦ ❝♦♥s❡❝✉t✐✈❡ ❝♦❧✉♠♥s ✐s ❛ ❧✐st ♦❢ ✈❛❧✉❡s ♦❢ ❛ ❢✉♥❝t✐♦♥ ✭❧❡❢t✮✿

■❢ ✇❡ ❤✐❞❡ t❤❡ ♠✐❞❞❧❡ ❝♦❧✉♠♥ ✭r✐❣❤t✮✱ ✇❡ ❤❛✈❡ t❤❡ ❧✐st ♦❢ ✈❛❧✉❡s ♦❢ t❤❡ ❝♦♠♣♦s✐t✐♦♥✳ ❲❡ ❝❛♥ ❤❛✈❡ ❛s ♠❛♥② ✐♥t❡r♠❡❞✐❛t❡ ❝♦❧✉♠♥s ❛s ✇❡ ❧✐❦❡✳ ❊①❛♠♣❧❡ ✸✳✺✳✶✻✿ ❢✉♥❝t✐♦♥s ❣✐✈❡♥ ❜② ❧✐sts✱ ❝r♦ss✲r❡❢❡r❡♥❝✐♥❣

❲❤❛t ✐❢ t❤❡ ❢✉♥❝t✐♦♥s ❛r❡ ❣✐✈❡♥ ❜② ♥♦t❤✐♥❣ ❜✉t t❤❡✐r ❧✐sts ♦❢ ✈❛❧✉❡s❄ ❚❤❡♥ ✇❡ ♥❡❡❞ t♦ ✜♥❞ ❛ ♠❛t❝❤ ❢♦r t❤❡ ♦✉t♣✉t ♦❢ t❤❡ ✜rst ❢✉♥❝t✐♦♥ ❛♠♦♥❣ t❤❡ ✐♥♣✉ts ♦❢ t❤❡ s❡❝♦♥❞✳ ●✐✈❡♥ t❤❡ t❛❜❧❡s ♦❢ ✈❛❧✉❡s ♦❢ f, g ✱ ✜♥❞ t❤❡ t❛❜❧❡ ♦❢ ✈❛❧✉❡s ♦❢ g ◦ f ✿ x 0 1 2 3 4

y = f (x) 1 0 2 4 2

y 0 1 2 3 4

❢♦❧❧♦✇❡❞ ❜②

z = g(y) 0 3 5 1 2

✐s

x 0 1 2 3 4

z = g(f (x)) ? ? ? ? ?

.

❲❡ ♥❡❡❞ t♦ ✜❧❧ t❤❡ s❡❝♦♥❞ ❝♦❧✉♠♥ ♦❢ t❤❡ ❧❛st t❛❜❧❡✳ ❋✐rst✱ ✇❡ ♥❡❡❞ t♦ ♠❛t❝❤ t❤❡ ♦✉t♣✉ts ♦❢ f ✇✐t❤ t❤❡ ✐♥♣✉ts ♦❢ g ✱ ❛s ❢♦❧❧♦✇s✿

❆❧t❡r♥❛t✐✈❡❧②✱ ✇❡ r❡✲❛rr❛♥❣❡ t❤❡ r♦✇s ♦❢ g ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ✈❛❧✉❡s ♦❢ y ❛♥❞ t❤❡♥ r❡♠♦✈❡ t❤❡ y ✲❝♦❧✉♠♥s✿ x 0 1 2 3 4

y 1 0 2 4 2

y 1 0 2 4 2

z 3 0 5 2 5

−→

x 0 1 2 3 4

z 3 0 5 2 5

✸✳✺✳

❈♦♠♣♦s✐t✐♦♥s ♦❢ ❢✉♥❝t✐♦♥s

✷✹✷

❊①❡r❝✐s❡ ✸✳✺✳✶✼ ●✐✈❡♥ t❤❡ t❛❜❧❡s ♦❢ ✈❛❧✉❡s ♦❢

f, g

❜❡❧♦✇✱ ✜♥❞ t❤❡ t❛❜❧❡ ♦❢ ✈❛❧✉❡s ♦❢

x 0 1 2 3 4

y = f (x) 0 2 3 0 1

y 0 1 2 3

g ◦ f✿

z = g(y) 4 4 0 1

❊①❛♠♣❧❡ ✸✳✺✳✶✽✿ ❝♦♠♣♦s✐t✐♦♥ ✇✐t❤ s♣r❡❛❞s❤❡❡t✱ ❝r♦ss✲r❡❢❡r❡♥❝✐♥❣ ❋♦r t❤✐s t❛s❦✱ ✇❡ ❤❛✈❡ t♦ ✉s❡ t❤❡ s❡❛r❝❤ ❢❡❛t✉r❡ ♦❢ t❤❡ s♣r❡❛❞s❤❡❡t✿

❋♦r ❡①❛♠♣❧❡✱ t❤❡ s❡❛r❝❤ ♠❛② ❜❡ ❡①❡❝✉t❡❞ ✇✐t❤ ❛ ✏❧♦♦❦✲✉♣✑ ❢✉♥❝t✐♦♥✿

❂❱▲❖❖❑❯P✭❘❈❬✲✻❪✱❘✸❈❬✲✹❪✿❘✶✽❈❬✲✸❪✱✷✮

❊①❛♠♣❧❡ ✸✳✺✳✶✾✿ t✇♦ ❢✉♥❝t✐♦♥s✱ t✇♦ ❝♦♠♣♦s✐t✐♦♥s ❈♦♥s✐❞❡r t❤❡s❡ ❢✉♥❝t✐♦♥s✿

f (x) = x2 , ❇♦t❤

f ◦g

❛♥❞

g◦f

g(y) = y + 1 .

♠❛❦❡ s❡♥s❡✳ ❲❡ ❥✉st ♥❡❡❞ t♦ ✐♥❞✐❝❛t❡ ✇❤✐❝❤ ✐♥♣✉t✱

t❤❡ ♦t❤❡r ❢✉♥❝t✐♦♥✳ ❚❤❡ ❢♦r♠❡r✿

f (x) = x2 ,

x = g(y) = y + 1 .

❲❡ ♥♦✇ ❦♥♦✇ ✇❤❛t t♦ s✉❜st✐t✉t❡✿

(f ◦ g)(y) = f (g(y)) = (y + 1)2 . ❚❤❡ ❧❛tt❡r✿

y = f (x) = x2 ,

g(y) = y + 1 .

❲❡ ♥♦✇ ❦♥♦✇ ✇❤❛t t♦ s✉❜st✐t✉t❡✿

(g ◦ f )(y) = g(f (x)) = (x2 ) + 1 .

❊①❛♠♣❧❡ ✸✳✺✳✷✵✿ ✉♥s♣❡❝✐✜❡❞ s✉❜st✐t✉t✐♦♥ ❨♦✉ ♠❛② ❡♥❝♦✉♥t❡r ♣r♦❜❧❡♠s st❛t❡❞ ❛s ❢♦❧❧♦✇s✿

x

♦r

y✱

✇✐❧❧ ❛s t❤❡ ♦✉t♣✉t ♦❢

✸✳✺✳

❈♦♠♣♦s✐t✐♦♥s ♦❢ ❢✉♥❝t✐♦♥s

✷✹✸

◮ ❋✐♥❞ t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥s✿ f (x) =

❙✐♥❝❡ ❜♦t❤ ❢✉♥❝t✐♦♥s ❤❛✈❡ t❤❡

s❛♠❡

x , 1+x

g(x) = sin x .

✐♥♣✉t ✈❛r✐❛❜❧❡✱ ✐t ✐s ✉♥❝❧❡❛r ❤♦✇ t♦ s✉❜st✐t✉t❡✦ ■s ✐t f t❤❡♥ g ✱ ♦r

g t❤❡♥ f ❄ ▲❡t✬s s♦rt t❤✐s ♦✉t ❛♥❞ ❞♦ ❜♦t❤✳

❚❤❡ ✢♦✇❝❤❛rt ♦❢ t❤❡ ❝♦♠♣♦s✐t✐♦♥ f ◦ g ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✿ x →

g

→ y →

f

→ z

❚❤✐s ♠❡❛♥s t❤❛t ✐♥ ♦r❞❡r t♦ ♠❛❦❡ t❤❡ ♦✉t♣✉t ♦❢ t❤❡ ✜rst t♦ ♠❛t❝❤ t❤❡ ✐♥♣✉t ♦❢ t❤❡ s❡❝♦♥❞✱ ✇❡ ❤❛✈❡ t♦ r❡♥❛♠❡ t❤❡ ✈❛r✐❛❜❧❡s ✿ x →

sin x

→ y →

y 1+y

→ z

❚❤✐s ✐s t❤❡ ♥❡✇ ✈❡rs✐♦♥ ♦❢ t❤❡ ♣r♦❜❧❡♠✿ ◮ ❋✐♥❞ t❤❡ ❝♦♠♣♦s✐t✐♦♥✱ f ◦ g ✱ ♦❢ t❤❡ ❢✉♥❝t✐♦♥s✿ f (y) =

◆♦✇ ✇❡ s✉❜st✐t✉t❡

y , 1+y

y = g(x) = sin x .

y = sin x ✐♥t♦ f (y) =

y , 1+y

❛s ❢♦❧❧♦✇s✿ (f ◦ g)(x) = f (y) = f (g(x)) y sin x = = . 1+y 1 + sin x

◆♦✇✱ t❤❡ r❡✈❡rs❡✳ ❚❤❡ ✢♦✇❝❤❛rt ♦❢ t❤❡ ❝♦♠♣♦s✐t✐♦♥ g ◦ f ✿ x →

f

→ y →

g

→ y →

sin y

→ z

❖♥❝❡ ❛❣❛✐♥✱ ✇❡ ❤❛✈❡ t♦ r❡♥❛♠❡ t❤❡ ✈❛r✐❛❜❧❡s✿ x →

x 1+x

◆♦✇ ✇❡ s✉❜st✐t✉t❡ y=

❛s ❢♦❧❧♦✇s✿

→ z

x ✐♥t♦ g(y) = sin y , 1+x

(g ◦ f )(x) = g(y) = g(f (x))  x . = sin y = sin 1+x

❊①❡r❝✐s❡ ✸✳✺✳✷✶

❘❡♣r❡s❡♥t t❤❡ ❢✉♥❝t✐♦♥ h(x) = 2(sin x)3 + 4 sin x + 5 ❛s t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ t✇♦ ❢✉♥❝t✐♦♥s ♦♥❡ ♦❢ ✇❤✐❝❤ ✐s tr✐❣♦♥♦♠❡tr✐❝✳ ❍✐♥t✿ ■t ❞♦❡s♥✬t ♠❛tt❡r ✇❤❛t sin ❞♦❡s✳ ❊①❡r❝✐s❡ ✸✳✺✳✷✷

√

✭❛✮ ❘❡♣r❡s❡♥t t❤❡√❢✉♥❝t✐♦♥ h(x) = x − 1 ❛s t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ t✇♦ ❢✉♥❝t✐♦♥s✳ ✭❜✮ ❘❡♣r❡s❡♥t t❤❡ 2 ❢✉♥❝t✐♦♥ √ k(t) = t − 1 ❛s t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤r❡❡ ❢✉♥❝t✐♦♥s✳ ✭❝✮ ❘❡♣r❡s❡♥t t❤❡ ❢✉♥❝t✐♦♥ p(t) = sin t2 − 1 ❛s t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ ❢♦✉r ❢✉♥❝t✐♦♥s✳

✸✳✻✳ ❚❤❡ ✐♥✈❡rs❡ ♦❢ ❛ ❢✉♥❝t✐♦♥

✷✹✹

❊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❡ ✸✳✺✳✷✹

❘❡♣r❡s❡♥t t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡s❡ t✇♦ ❢✉♥❝t✐♦♥s✿ f (x) = 1/x ❛♥❞ g(y) =

h ♦❢ ✈❛r✐❛❜❧❡ x✳ ❉♦♥✬t s✐♠♣❧✐❢②✳

y2

y ✱ ❛s ❛ s✐♥❣❧❡ ❢✉♥❝t✐♦♥ −3

❊①❡r❝✐s❡ ✸✳✺✳✷✺

❋✉♥❝t✐♦♥ y = f (x) ✐s ❣✐✈❡♥ ❜❡❧♦✇ ❜② ❛ ❧✐st ♦❢ ✐ts ✈❛❧✉❡s✳ ■s t❤❡ ❢✉♥❝t✐♦♥ ♦♥❡✲t♦✲♦♥❡❄

x 0 1 2 3 4 y = f (x) 0 1 2 1 2

✸✳✻✳ ❚❤❡ ✐♥✈❡rs❡ ♦❢ ❛ ❢✉♥❝t✐♦♥

❖✉r r✉♥♥✐♥❣ ❡①❛♠♣❧❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ❛♥s✇❡rs t❤❡ q✉❡st✐♦♥✿ ❲❤✐❝❤ ❜❛❧❧ ✐s t❤✐s ❜♦② ♣❧❛②✐♥❣ ✇✐t❤❄

▲❡t✬s t✉r♥ t❤✐s ❛r♦✉♥❞✿ ❲❤✐❝❤ ❜♦② ✐s ♣❧❛②✐♥❣ ✇✐t❤ t❤✐s ❜❛❧❧❄ ❚❤❡ s♦❧✉t✐♦♥ ✇♦✉❧❞ s❡❡♠ t♦ ❜❡ ❛ s✐♠♣❧❡ r❡✈❡rs❛❧ ♦❢ t❤❡ ❛rr♦✇s✿

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

• ❚❤❡r❡ ❛r❡ t✇♦ ❜♦②s ♣❧❛②✐♥❣ ✇✐t❤ t❤❡ ❜❛s❦❡t❜❛❧❧ ✕ t✇♦ ❛♥s✇❡rs✳

✸✳✻✳ ❚❤❡ ✐♥✈❡rs❡ ♦❢ ❛ ❢✉♥❝t✐♦♥

•

✷✹✺

❚❤❡r❡ ✐s ♥♦ ❜♦② ♣❧❛②✐♥❣ ✇✐t❤ t❤❡ ❜❛s❡❜❛❧❧ ✕ ♥♦ ❛♥s✇❡r✳

❚❤✐s ♠❡❛♥s t❤❛t t❤❡r❡ ✐s ♥♦ ❢✉♥❝t✐♦♥ t❤✐s t✐♠❡✦ ◗✉❡st✐♦♥✿ ❯♥❞❡r ✇❤❛t ❝✐r❝✉♠st❛♥❝❡s ✇♦✉❧❞ s✉❝❤ ❛ r❡✈❡rs❛❧ ♦❢ ❛rr♦✇s ♠❛❦❡ s❡♥s❡❄ ❆❧❧ ❢✉♥❝t✐♦♥s ❢r♦♠ ❡✐t❤❡r ❢r♦♠

X

t♦

X

Y

t♦

Y

❛r❡ ❛❧s♦ r❡❧❛t✐♦♥s ❜❡t✇❡❡♥

♦r ❢r♦♠

Y

t♦

X✳

X

❛♥❞

Y✳

❍♦✇❡✈❡r✱ ♥♦t ❡✈❡r② r❡❧❛t✐♦♥ ✐s ❛ ❢✉♥❝t✐♦♥ ✕

❚❤❡ r❡❛s♦♥s ❛r❡ t❤❡ s❛♠❡✿ t♦♦ ♠❛♥② ♦r t♦♦ ❢❡✇ ❛rr♦✇s st❛rt✐♥❣ ❛t ❛♥

❡❧❡♠❡♥t ♦❢ t❤❡ ❞♦♠❛✐♥ s❡t✳ ❆♥ ❡s♣❡❝✐❛❧❧② ✐♠♣♦rt❛♥t q✉❡st✐♦♥ ✐s✿ ❈❛♥ ✇❡ r❡✈❡rs❡ t❤❡ ❛rr♦✇s ♦❢ ❛ ❢✉♥❝t✐♦♥ s♦ t❤❛t t❤❛t t❤❡ s❛♠❡ r❡❧❛t✐♦♥ ✐s ♥♦✇ s❡❡♥ ❛s ❛ ♥❡✇✱ ✏✐♥✈❡rs❡✑✱ ❢✉♥❝t✐♦♥❄

■❢ t❤❡ ❛♥s✇❡r ✐s ◆♦✱ ❝❛♥ ✇❡ s❡❡ t❤❡ r❡❛s♦♥ ❜② ❧♦♦❦✐♥❣ ❛t t❤❡

♦r✐❣✐♥❛❧ ❢✉♥❝t✐♦♥❄

•

❋✐rst✱ s♦♠❡

•

❙❡❝♦♥❞✱ s♦♠❡

y ✬s

✐♥

Y

❤❛✈❡ ♥♦ ❝♦rr❡s♣♦♥❞✐♥❣

y ✬s

✐♥

Y

x✬s

✐♥

X✳

■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ❢✉♥❝t✐♦♥ ✐s♥✬t ♦♥t♦✦

❤❛✈❡ t✇♦ ♦r ♠♦r❡ ❝♦rr❡s♣♦♥❞✐♥❣

x✬s

✐♥

X✳

■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ❢✉♥❝t✐♦♥ ✐s♥✬t

♦♥❡✲t♦✲♦♥❡✦

❙♦✱ t❤❡ ♦r✐❣✐♥❛❧ ❢✉♥❝t✐♦♥ ❧❛❝❦s ❡✐t❤❡r ♦❢ t❤❡ t✇♦ t②♣❡s ♦❢ r❡❣✉❧❛r✐t② ❢♦r t❤✐s t♦ ❜❡ ♣♦ss✐❜❧❡✳ ❲❤❛t ❦✐♥❞ ♦❢ ❢✉♥❝t✐♦♥ ✇♦✉❧❞ ♠❛❦❡ t❤✐s ♣♦ss✐❜❧❡❄ ❆ ❢✉♥❝t✐♦♥ t❤❛t ✐s ❜♦t❤ ♦♥❡✲t♦✲♦♥❡ ❛♥❞ ♦♥t♦ ✿

❲❡ ❤❛✈❡ ❛❞❞❡❞ ❛♥ ❡①tr❛ ❜❛❧❧ ✭s♦❝❝❡r✮ ❛♥❞ ❤❛✈❡ r❡✲❞r♦✇♥ t❤❡ ❛rr♦✇s❀ t❤❡r❡ ✐s ❡①❛❝t❧② ♦♥❡ ❛rr♦✇ ❢♦r ❡❛❝❤ ❜❛❧❧✳ ❚❤✐s ✐s ❛ ✈❡r② s✐♠♣❧❡✱ ❛❧♠♦st ✉♥✐♥t❡r❡st✐♥❣✱ ❦✐♥❞ ♦❢ ❢✉♥❝t✐♦♥✿ ❡❛❝❤ ❜♦② ❤♦❧❞s ❛ s✐♥❣❧❡ ❜❛❧❧✱ ❛♥❞ ❡✈❡r② ❜❛❧❧ ✐s ❤❡❧❞ ❜② ❛ s✐♥❣❧❡ ❜♦②✳ ❆s ❛ r❡s✉❧t✱ t❤❡ ♦♥❧② ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ t✇♦ s❡ts ✐s ✐♥ t❤❡ ♥❛♠❡s✿ ❜♦②s

❜❛❧❧s

❚♦♠

❜❛s❦❡t❜❛❧❧

❇❡♥

t❡♥♥✐s

◆❡❞

❜❛s❡❜❛❧❧

❑❡♥

❢♦♦t❜❛❧❧

❙✐❞

s♦❝❝❡r

■♥❞❡❡❞✱ ✐❢ ②♦✉ ❞♦♥✬t r❡♠❡♠❜❡r ❚♦♠✬s ♥❛♠❡✱ ②♦✉ ❥✉st s❛② ✏t❤❡ ❜♦② ✇❤♦ ♣❧❛②s ✇✐t❤ t❤❡ ❜❛s❦❡t❜❛❧❧✑✳ ❖r✱ ✐❢ ②♦✉ ❞♦♥✬t r❡♠❡♠❜❡r ✇❤❛t t❤❛t r❡❞ ❜❛❧❧ ✐s ❢♦r✱ ②♦✉ ❥✉st s❛② ✏t❤❡ ❣❛♠❡ ❙✐❞ ♣❧❛②s✑✳ ❚❤❡r❡ ✐s ♥♦ ❛♠❜✐❣✉✐t② ✐♥ t❤✐s s✉❜st✐t✉t✐♦♥✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ✐s ✈❡r② ✐♠♣♦rt❛♥t✳

❉❡✜♥✐t✐♦♥ ✸✳✻✳✶✿ ✐♥✈❡rs❡ ♦❢ ❢✉♥❝t✐♦♥ ❙✉♣♣♦s❡

❢✉♥❝t✐♦♥

F : X → Y ✐s ❛ ❢✉♥❝t✐♦♥✳ ❆ ❢✉♥❝t✐♦♥ G : Y → X ✐s ❝❛❧❧❡❞ ♦❢ F ✇❤❡♥✱ ❢♦r ❛❧❧ x ❛♥❞ y ✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♠❛t❝❤✿ F (x) = y

✐❢ ❛♥❞ ♦♥❧② ✐❢

G(y) = x

❛♥ ✐♥✈❡rs❡

✸✳✻✳

❚❤❡ ✐♥✈❡rs❡ ♦❢ ❛ ❢✉♥❝t✐♦♥

✷✹✻

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿

F

G

x −−−−→ y ⇐⇒ y −−−−→ x ❲❡ ✇✐❧❧ r❡❧② ♦♥ t❤✐s ✐♠♣♦rt❛♥t ❢❛❝t✳

❚❤❡♦r❡♠ ✸✳✻✳✷✿ ❖♥❡✲t♦✲♦♥❡ ❖♥t♦ ✈s✳ ■♥✈❡rs❡ ❆ ❢✉♥❝t✐♦♥ ✐s ❜♦t❤ ♦♥❡✲t♦✲♦♥❡ ❛♥❞ ♦♥t♦ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐t ❤❛s ❛♥ ✐♥✈❡rs❡✳

❊①❡r❝✐s❡ ✸✳✻✳✸ Pr♦✈❡ t❤❡ t❤❡♦r❡♠✳

❯♥❞❡r t❤✐s ❝♦♥❞✐t✐♦♥✱ t❤❡ ❛rr♦✇s ❝❛♥ ❜❡ s❛❢❡❧② r❡✈❡rs❡❞✿

❆s t❤❡ ❢✉♥❝t✐♦♥ ✐s ❞❡✜♥❡❞ ✐♥❞✐r❡❝t❧②✱ ✇❡ ♥❡❡❞ ❛♥ ❛ss✉r❛♥❝❡ t❤❛t t❤❡r❡ ✐s ♦♥❧② ♦♥❡✳

❚❤❡♦r❡♠ ✸✳✻✳✹✿ ❯♥✐q✉❡♥❡ss ♦❢ ■♥✈❡rs❡ ❚❤❡r❡ ❝❛♥ ❜❡ ♦♥❧② ♦♥❡ ✐♥✈❡rs❡ ❢♦r ❛ ❢✉♥❝t✐♦♥✳

❊①❡r❝✐s❡ ✸✳✻✳✺ Pr♦✈❡ t❤❡ t❤❡♦r❡♠✳

❚❤✐s ❥✉st✐✜❡s ✉s✐♥❣ ✏t❤❡ ✐♥✈❡rs❡✑ ❢r♦♠ ♥♦✇ ♦♥✳ ❚❤❡ ✐♥✈❡rs❡ ♦❢ ❛ ❢✉♥❝t✐♦♥

F

✐s ❞❡♥♦t❡❞ ❛s ❢♦❧❧♦✇s✿

■♥✈❡rs❡ ❢✉♥❝t✐♦♥

F −1 ■t r❡❛❞s ✏ F ✐♥✈❡rs❡✑✳

❍❡r❡ ✏ F ✑ ✐s t❤❡

♥❛♠❡ ♦❢ t❤❡ ♦❧❞ ❢✉♥❝t✐♦♥ ❛♥❞ ✏ F −1✑ ✐s t❤❡ ♥❛♠❡ ♦❢ t❤❡ ♥❡✇ ❢✉♥❝t✐♦♥ ✇✐t❤ ❛ r❡❢❡r❡♥❝❡ t♦ t❤❡

♦♥❡ ✐t ❝❛♠❡ ❢r♦♠✳

❲❛r♥✐♥❣✦

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

2−1 =

1 ✳ 2

✭❲❛r♥✐♥❣

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

❆♥ ✐❞❡❛ t♦ ❤♦❧❞ ♦♥ t♦ ✐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)✳

♦r

2

2✳✮

s❛♠❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ s❡ts X ❛♥❞ Y ✿

✸✳✻✳

❚❤❡ ✐♥✈❡rs❡ ♦❢ ❛ ❢✉♥❝t✐♦♥

✷✹✼

❚❤❡r❡ ✐s ♥♦ ♣r❡❢❡rr❡❞ ❛❧✐❣♥♠❡♥t✿

❚❤❡ ❝❤♦✐❝❡ ❜❡t✇❡❡♥

F

❛♥❞

F −1 ✐s t❤❡ ❝❤♦✐❝❡ ♦❢ t❤❡ ✏r♦❧❡s✑

❢♦r

X

❛♥❞

Y ✱ ✐♥♣✉t ♦r ♦✉t♣✉t✱ ❞♦♠❛✐♥ ♦r ❝♦❞♦♠❛✐♥✳

❊①❡r❝✐s❡ ✸✳✻✳✻

❊①♣❧❛✐♥ t❤❡ ♣✐❝t✉r❡ ❜❡❧♦✇✿

❈♦♠♣♦s✐t✐♦♥s ♣r♦✈✐❞❡ ❛ ❞✐✛❡r❡♥t ♣♦✐♥t ♦❢ ✈✐❡✇ ♦♥ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ✐♥✈❡rs❡✳ ◆♦t✐❝❡ t❤❛t t❤❡ ❞♦♠❛✐♥ ♦❢ t❤❡ ♥❡✇ ❢✉♥❝t✐♦♥ ✕ t❤❡ ✐♥✈❡rs❡ ✕ ✇♦✉❧❞ ❤❛✈❡ t♦ ❜❡ t❤❡ ❝♦❞♦♠❛✐♥ ♦❢ t❤❡ ♦r✐❣✐♥❛❧✦ ❚❤❡✐r ❝♦♠♣♦s✐t✐♦♥ t❤❡♥ ♠❛❦❡s s❡♥s❡✿

❲❤❛t ✐s ✉♥✉s✉❛❧ ❛❜♦✉t t❤✐s ❝♦♠♣♦s✐t✐♦♥ ✐s t❤❛t ✇❡ ❤❛✈❡ ♠❛❞❡ ✕ t❤r♦✉❣❤ t❤❡ t✇♦ ❢✉♥❝t✐♦♥s ✕ ❛ ❢✉❧❧ ❝✐r❝❧❡ ❢r♦♠ ❜♦②s t♦ ❜❛❧❧s ❛♥❞ ❜❛❝❦ t♦ ❜♦②s✳

❋✉rt❤❡r♠♦r❡✱ ✇✐t❤ ❡✈❡r② t✇♦ ❝♦♥s❡❝✉t✐✈❡ ❛rr♦✇s✱ ✇❡ ❛rr✐✈❡ t♦ ♦✉r

st❛rt✐♥❣ ♣♦✐♥t✱ ❛ ❜♦②✳ ❋✉rt❤❡r♠♦r❡✱ t❤❡r❡ ✐s t❤❡ ❛♥♦t❤❡r ✇❛② ✭t❤❡ ♦♣♣♦s✐t❡✮ t♦ ❜✉✐❧❞ ❛ ❝♦♠♣♦s✐t✐♦♥ ❢r♦♠ t❤❡s❡ t✇♦✿

✸✳✻✳

❚❤❡ ✐♥✈❡rs❡ ♦❢ ❛ ❢✉♥❝t✐♦♥

✷✹✽

❲❡ ❤❛✈❡ ♠❛❞❡ ✕ t❤r♦✉❣❤ t❤❡ t✇♦ ❢✉♥❝t✐♦♥s ✕ ❛ ❢✉❧❧ ❝✐r❝❧❡ ❢r♦♠ ❜❛❧❧s t♦ ❜♦②s ❛♥❞ ❜❛❝❦ t♦ ❜❛❧❧s✳ ❊✈❡r② t✐♠❡✱ ✇❡ ❛rr✐✈❡ t♦ ♦✉r st❛rt✐♥❣ ♣♦✐♥t✱ ❛ ❜❛❧❧✳ ❚❤❡s❡ ❛r❡ t❤❡ t✇♦ ❝♦♠♣♦s✐t✐♦♥s ♥❡①t t♦ ❡❛❝❤ ♦t❤❡r✿

❍❡r❡ ✐s ❛ ✢♦✇❝❤❛rt r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤✐s ✐❞❡❛✿

x → y →

F G

→ y →

→ x ✳✳✳ s❛♠❡ x!

G

→ x →

→ y ✳✳✳ s❛♠❡ y!

F

❲❡ ❢❡❡❞ t❤❡ ♦✉t♣✉t ♦❢ F ✐♥t♦ G✱ ❛♥❞ ✈✐❝❡ ✈❡rs❛✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ✐s ❝r✉❝✐❛❧✳ ❚❤❡♦r❡♠ ✸✳✻✳✼✿ ■♥✈❡rs❡ ✈✐❛ ❈♦♠♣♦s✐t✐♦♥s

F : X → Y ✐s ❛ ❢✉♥❝t✐♦♥ t❤❛t ✐s ❜♦t❤ ♦♥❡✲t♦✲♦♥❡ G : Y → X ✐s t❤❡ ✐♥✈❡rs❡ ♦❢ F ✐❢ ❛♥❞ ♦♥❧② ✐❢ • G(F (x)) = x ❋❖❘ ❊❆❈❍ x✱ ❆◆❉ • F (G(y)) = y ❋❖❘ ❊❆❈❍ y ✳

❙✉♣♣♦s❡ ❢✉♥❝t✐♦♥

❊①❡r❝✐s❡ ✸✳✻✳✽

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


F −1 ◦ F (x) = x
F ◦ F −1 (y) = y ▼❛❦✐♥❣ ❝✐r❝❧❡s ✐♥ t❤❡ ❞✐❛❣r❛♠ ❜❡❧♦✇ ✇♦♥✬t ❝❤❛♥❣❡ t❤❡ ✈❛❧✉❡ ♦❢ x ♦r y ✿

x ↑

F

−−−−→ F −1

y ↓

x ←−−−−−− y

❛♥❞ ♦♥t♦✳

❚❤❡♥ ❛

✸✳✻✳

❚❤❡ ✐♥✈❡rs❡ ♦❢ ❛ ❢✉♥❝t✐♦♥

✷✹✾

❉❡✜♥✐t✐♦♥ ✸✳✻✳✾✿ ✐♥✈❡rt✐❜❧❡ ❢✉♥❝t✐♦♥ ❆ ❢✉♥❝t✐♦♥ t❤❛t ✐s ❜♦t❤ ♦♥❡✲t♦✲♦♥❡ ❛♥❞ ♦♥t♦ ✐s ❛❧s♦ ❝❛❧❧❡❞ ✐♥✈❡rt✐❜❧❡ ✭♦r ❛ ❜✐❥❡❝✲ t✐♦♥✮✳ ❙✉❝❤ ❛ ❢✉♥❝t✐♦♥ ✐s ✈❡r② ❡❛s② t♦ ✐❧❧✉str❛t❡✿

❙✉❝❤ ❛ ❢✉♥❝t✐♦♥ ❝r❡❛t❡s ❛ ❝♦rr❡s♣♦♥❞❡♥❝❡ ❜❡t✇❡❡♥ t✇♦ s❡ts t❤❛t ♠❛❦❡s ✐t ❧♦♦❦ ❧✐❦❡ t❤✐s ✐s t❤❡ s❛♠❡ s❡t✳

❊①❡r❝✐s❡ ✸✳✻✳✶✵ ❙✉♣♣♦s❡ A, B, C ❛r❡ t❤❡ s❡ts ♦❢ t❤❡ ♦♥❡✲t♦✲♦♥❡✱ t❤❡ ♦♥t♦✱ ❛♥❞ t❤❡ ✐♥✈❡rt✐❜❧❡ ❢✉♥❝t✐♦♥s✱ r❡s♣❡❝t✐✈❡❧②✳ ❲❤❛t ✐s t❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡s❡ s❡ts❄ ◆♦✇✱ t❤❡ ♥✉♠❡r✐❝❛❧

❢✉♥❝t✐♦♥s✳✳✳

❊①❛♠♣❧❡ ✸✳✻✳✶✶✿ ✐♥✈❡rs❡ ♦❢ tr❛♥s❢♦r♠❛t✐♦♥ ❲❤❛t ✐s t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ ✐♥✈❡rs❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ✇❤❡♥ s❡❡♥ ❛s ❛ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ t❤❡ ❧✐♥❡❄ ■t ✐s ❛ tr❛♥s❢♦r♠❛t✐♦♥ t❤❛t ✇♦✉❧❞ r❡✈❡rs❡ t❤❡ ❡✛❡❝t ♦❢ t❤❡ ♦r✐❣✐♥❛❧✿

❏✉st ❜② ❡①❛♠✐♥✐♥❣ t❤❡s❡ s✐♠♣❧❡ tr❛♥s❢♦r♠❛t✐♦♥s✱ ✇❡ ❞✐s❝♦✈❡r t❤❡ ❢♦❧❧♦✇✐♥❣✿ ✶✳ ❚❤❡ ✐♥✈❡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✮✳ ■♥ ❢❛❝t✱ ✇❡ r❡❛❧✐③❡ t❤❛t t❤❡② ♣❛✐r ✉♣ ✿ ✶✳ ❚❤❡ s❤✐❢t s ✉♥✐ts t♦ t❤❡ r✐❣❤t ❛♥❞ t❤❡ s❤✐❢t ♦❢ s ✉♥✐ts t♦ t❤❡ ❧❡❢t ❛r❡ t❤❡ ✐♥✈❡rs❡s ♦❢ ❡❛❝❤ ♦t❤❡r✳ ✷✳ ❚❤❡ ✢✐♣ ✐s t❤❡ ✐♥✈❡rs❡ ✇✐t❤ ✐ts❡❧❢✳ ✸✳ ❚❤❡ str❡t❝❤ ❜② k 6= 0 ❛♥❞ t❤❡ s❤r✐♥❦ ❜② k ❛r❡ t❤❡ ✐♥✈❡rs❡s ♦❢ ❡❛❝❤ ♦t❤❡r✳ ❲❤❡♥ ❡①❡❝✉t❡❞ ❝♦♥s❡❝✉t✐✈❡❧② ✭✐♥ ❡✐t❤❡r ♦r❞❡r✮✱ t❤❡ ❡✛❡❝t ✐s ♥✐❧✳ ❆❧❣❡❜r❛✐❝❛❧❧②✱ ✇❡ ❤❛✈❡ t❤❡s❡ ♣❛✐rs ♦❢ ❢✉♥❝t✐♦♥s✿ f ✶✳ s❤✐❢t y y ✷✳ ✢✐♣ ✸✳ str❡t❝❤ y

= g −1 ✈s✳ g = f −1 =x+s x=y−s = −x x = −y =x·k x = y/k

❲❤❛t ❛❜♦✉t t❤❡ ❢♦❧❞❄ ■t ❝❛♥✬t ❜❡ ✉♥❞♦♥❡ s✐♥❝❡ ❛♥② t✇♦ ♣♦✐♥ts t❤❛t ❛r❡ ❜r♦✉❣❤t t♦❣❡t❤❡r ❜❡❝♦♠❡

✸✳✻✳

❚❤❡ ✐♥✈❡rs❡ ♦❢ ❛ ❢✉♥❝t✐♦♥

✷✺✵

✐♥❞✐st✐♥❣✉✐s❤❛❜❧❡✳ ❆♥② ❢✉rt❤❡r tr❛♥s❢♦r♠❛t✐♦♥s ✇✐❧❧ ♣r♦❞✉❝❡ t❤❡ s❛♠❡ ♦✉t♣✉t✿

x=a ցF րF

x=b

G?

y −−−−−→ x

❚❤✐s ❢✉♥❝t✐♦♥ ✐s♥✬t ♦♥❡✲t♦✲♦♥❡✦ ❆♥❞ ♥❡✐t❤❡r ✐s t❤❡ ❝♦❧❧❛♣s❡✳

❊①❛♠♣❧❡ ✸✳✻✳✶✷✿ ✐♥✈❡rs❡ ♦❢ ❧✐st

❍♦✇ ❞♦ ✇❡ ✜♥❞ t❤❡ ✐♥✈❡rs❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ❣✐✈❡♥ ❜② ✐ts

x 0 1 f= 2 3 4 ...

y = f (x) 1 0 2 1 3 ...

❧✐st ♦❢ ✈❛❧✉❡s ✿

=

x 0 1 2 3 4 ...

→ → → → → → ...

y 1 0 2 1 3 ...

❚❤❡ t❛❜❧❡ ✐s ✉♥❞❡rst♦♦❞ ❛s ✐❢ t❤❡r❡ ❛r❡ ❛rr♦✇s ❣♦✐♥❣ ❤♦r✐③♦♥t❛❧❧② ❧❡❢t t♦ r✐❣❤t✳ ❚❤❛t ✐s ✇❤② ✏r❡✈❡rs✐♥❣ t❤❡ ❛rr♦✇s✑ ♠❡❛♥s ✐♥t❡r❝❤❛♥❣✐♥❣ t❤❡ ❝♦❧✉♠♥s✿

f −1

x 0 1 = 2 3 4 ...

→ ← ← ← ← ← ...

y 1 0 2 1 3 ...

=

y 1 0 2 1 3 ...

→ → → → → → ...

x 0 1 2 3 4 ...

=

y 0 1 1 2 3 ...

→ → → → → → ...

x 1 0 3 2 4 ...

y 0 1 1 2 3 ...

=

x = f −1 (y) 1 0 . 3 2 4 ...

■t ♠❛②✱ ♦r ♠❛② ♥♦t✱ ❜❡❝♦♠❡ ❝❧❡❛r t❤❛t t❤❡ ♥❡✇ ❢✉♥❝t✐♦♥ ✐s♥✬t ❛ ❢✉♥❝t✐♦♥✦ ❚♦ ♠❛❦❡ s✉r❡✱ ✐t✬s ❛ ❣♦♦❞ ✐❞❡❛✱ −1 ❛t t❤❡ ❡♥❞✱ t♦ ❛rr❛♥❣❡ t❤❡ ✐♥♣✉ts ✐♥ t❤❡ ✐♥❝r❡❛s✐♥❣ ♦r❞❡r✳ ❚❤❡♥ ✇❡ ❝❧❡❛r❧② s❡❡ t❤❡ ❝♦♥✢✐❝t✿ f (1) = 0 −1 ❛♥❞ f (1) = 3✳ ❚❤❡ ♦r✐❣✐♥❛❧ ❢✉♥❝t✐♦♥✱ f ✱ ✇❛s♥✬t ♦♥❡✲t♦✲♦♥❡✦

❚❤❡ ❣❡♥❡r❛❧ r✉❧❡ ❢♦r

◮

✜♥❞✐♥❣ t❤❡ ✐♥✈❡rs❡

❚❤❡ ✐♥✈❡rs❡ ♦❢

y = f (x)

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

✐s ❢♦✉♥❞ ❜② s♦❧✈✐♥❣ t❤✐s ❡q✉❛t✐♦♥ ❢♦r

x❀

✐✳❡✳✱

❚❤✐s ♠❡t❤♦❞ r❡s✉❧ts ✐♥ ❛ s✉❝❝❡ss ♦♥❧② ✇❤❡♥ t❤❡r❡ ✐s ❡①❛❝t❧② ♦♥❡ s♦❧✉t✐♦♥✱

x✱

❊①❛♠♣❧❡ ✸✳✻✳✶✸✿ ✐♥✈❡rs❡ ♦❢ ❧✐♥❡❛r ♣♦❧②♥♦♠✐❛❧

❚♦ ✜♥❞ t❤❡ ✐♥✈❡rs❡ ♦❢ ❛ ❧✐♥❡❛r ♣♦❧②♥♦♠✐❛❧

f (x) = 3x − 7 , s❡t ❛♥❞ s♦❧✈❡ t❤❡ ❡q✉❛t✐♦♥ ✭r❡❧❛t✐♦♥✮✱ ❛s ❢♦❧❧♦✇s✿

y = 3x − 7 =⇒ y + 7 = 3x =⇒ ❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡✿

f −1 (y) =

y+7 . 3

y+7 = x. 3

x = f −1 (y)✳ ❢♦r ❡❛❝❤

y✳

✸✳✻✳

❚❤❡ ✐♥✈❡rs❡ ♦❢ ❛ ❢✉♥❝t✐♦♥

✷✺✶

■❢ ✐t ✐s ♥♦t ❦♥♦✇♥ ❛❤❡❛❞ ♦❢ t✐♠❡ ✇❤❡t❤❡r t❤❡ ❢✉♥❝t✐♦♥ ✐s ♦♥❡✲t♦✲♦♥❡✱ t❤✐s ❢❛❝t ✐s ❡st❛❜❧✐s❤❡❞✱ ❛✉t♦♠❛t✐✲ ❝❛❧❧②✱ ❛s ❛ ♣❛rt ♦❢ ✜♥❞✐♥❣ t❤❡ ✐♥✈❡rs❡✳ ❋♦r ❡①❛♠♣❧❡✱ t♦ ✜♥❞ t❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ q✉❛❞r❛t✐❝ ❢✉♥❝t✐♦♥ f (x) = x2 ,

✇❡ s❡t ❛♥❞ s♦❧✈❡✿

√ y = x2 =⇒ ± y = x .

❚❤❡ ± s✐❣♥ ✐♥❞✐❝❛t❡s t❤❛t t❤❡r❡ ❛r❡ t✇♦ s♦❧✉t✐♦♥s ✭x > 0✮✳ ❚❤❡ ♦r✐❣✐♥❛❧ ❢✉♥❝t✐♦♥ ✇❛s♥✬t ✐♥✈❡rt✐❜❧❡✦ ❆ ❧✐♥❡❛r ❢✉♥❝t✐♦♥✱ f (x) = mx + b ,

✐s ♦♥❡✲t♦✲♦♥❡ ❛♥❞ ♦♥t♦ ✇❤❡♥❡✈❡r m 6= 0✳ ❆❧❣❡❜r❛✐❝❛❧❧②✱ ✇❡ ❥✉st s♦❧✈❡ t❤❡ ❡q✉❛t✐♦♥ y = mx + b ❢♦r x✳ ❚❤❡ ❛❧❣❡❜r❛✐❝ r❡s✉❧t ✐s ❜❡❧♦✇✳ ❚❤❡♦r❡♠ ✸✳✻✳✶✹✿ ■♥✈❡rs❡ ♦❢ ▲✐♥❡❛r P♦❧②♥♦♠✐❛❧ ❚❤❡ ✐♥✈❡rs❡ ♦❢ ❛ ❧✐♥❡❛r ♣♦❧②♥♦♠✐❛❧

f (x) = mx + b, m 6= 0 , ✐s ❛❧s♦ ❛ ❧✐♥❡❛r ♣♦❧②♥♦♠✐❛❧✱ ❛♥❞ ✐ts s❧♦♣❡ ✐s t❤❡ r❡❝✐♣r♦❝❛❧ ♦❢ t❤❛t ♦❢ t❤❡ ♦r✐❣✐♥❛❧✿

f −1 (y) =

1 b y− . m m

❲❡ ❦♥♦✇ t❤❛t t❤❡ s❡t ♦❢ ✐♥✈❡rt✐❜❧❡ ❢✉♥❝t✐♦♥s ✐s s♣❧✐t ✐♥t♦ ♣❛✐rs ♦❢ ✐♥✈❡rs❡s✳ ❲❡ ❝❛♥ ❜❡ ♠♦r❡ s♣❡❝✐✜❝ ✇✐t❤ t❤❡ s❡t ♦❢ ❛❧❧ ❧✐♥❡❛r ♣♦❧②♥♦♠✐❛❧s✳ ❚❤❡ ♣❛✐rs ❤❛✈❡ r❡❝✐♣r♦❝❛❧ s❧♦♣❡s✱ ❢♦r ❡①❛♠♣❧❡✿ • 2 ❛♥❞ 1/2

• −2 ❛♥❞ −1/2 • 1 ❛♥❞ 1

• −1 ❛♥❞ −1

• ❡t❝✳

❲❡ ❝❛♥ s❡❡ t❤❡s❡ ♣❛✐rs ♦❢ ❛ st❡❡♣❡r ❧✐♥❡ ❛♥❞ ❛ s❤❛❧❧♦✇❡r ❧✐♥❡✿

❊①❡r❝✐s❡ ✸✳✻✳✶✺

❲❤❛t ❞♦ ♥❡❡❞ t♦ ❞♦ t♦ t❤✐s s❤❡❡t ♦❢ ♣❛♣❡r ✐♥ ♦r❞❡r t♦ ♠❛❦❡ t❤❡ ❢♦r♠❡r ❧❛♥❞ ♦♥ t❤❡ ❧❛tt❡r❄ ◆❡①t✱ ❧❡t✬s tr② t♦ ✐♠❛❣✐♥❡ ❤♦✇ s♦♠❡ ♥❡✇

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

✸✳✻✳

❚❤❡ ✐♥✈❡rs❡ ♦❢ ❛ ❢✉♥❝t✐♦♥

❙♦♠❡ ❡♠❡r❣❡❞ ❛s t❤❡

✷✺✷

❛❜❜r❡✈✐❛t✐♦♥s

2 = 2 · 3✱ ❧❡❛❞s t♦ ❛ ♥❡✇ ♦♣❡r❛t✐♦♥✿

t♦ ❛ ♥❡✇ ♦♣❡r❛t✐♦♥✿

❡①♣♦♥❡♥t✳

2+2+ 2 · 2 · 2 · 2 = 24 ✱ ❧❡❛❞s

❢♦r r❡♣❡❛t❡❞ ❢❛♠✐❧✐❛r ♦♣❡r❛t✐♦♥s❀ ❢♦r ❡①❛♠♣❧❡✱ r❡♣❡❛t❡❞ ❛❞❞✐t✐♦♥✱

♠✉❧t✐♣❧✐❝❛t✐♦♥✳

▼❡❛♥✇❤✐❧❡✱ r❡♣❡❛t❡❞ ♠✉❧t✐♣❧✐❝❛t✐♦♥✱

❇✉t ✇❤❛t ❛❜♦✉t s✉❜tr❛❝t✐♦♥ ❛♥❞ ❞✐✈✐s✐♦♥❄

❊①❛♠♣❧❡ ✸✳✻✳✶✻✿ s✉❜tr❛❝t✐♦♥ ❛s ✐♥✈❡rs❡

❙✉♣♣♦s❡ ■ ❦♥♦✇ ❤♦✇ t♦ ❛❞❞✳ ❆♥s✇❡r✿

$7✳

Pr♦❜❧❡♠✿

❲✐t❤

$5

✐♥ ♠② ♣♦❝❦❡t✱ ❤♦✇ ♠✉❝❤ ❞♦ ■ ❛❞❞ t♦ ❤❛✈❡

$12❄

❍♦✇ ❞♦ ■ ❦♥♦✇❄ ❙♦❧✈❡ t❤❡ ❡q✉❛t✐♦♥✿

5 + x = 12 . ❚❤✐s ❡q✉❛t✐♦♥ ❧❡❛❞s t♦ ❛ ♥❡✇ ♦♣❡r❛t✐♦♥✱

s✉❜tr❛❝t✐♦♥ ✿ x = 12 − 5✳

❖❢ ❝♦✉rs❡✱ t❤❡r❡ ✐s ❛❧s♦ ❛ ♥❡✇

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

5

✐s

5✳

❊①❛♠♣❧❡ ✸✳✻✳✶✼✿ ❞✐✈✐s✐♦♥ ❛s ✐♥✈❡rs❡

❙✉♣♣♦s❡ ♥♦✇ ■ ❦♥♦✇ ❤♦✇ t♦ ♠✉❧t✐♣❧②✳ Pr♦❜❧❡♠✿ ■❢ ■ ✇❛♥t t♦ ♠❛❦❡ ❛ t❛❜❧❡

2✲❜②✲4✬s

❞♦ ■ ♥❡❡❞❄ ❆♥s✇❡r✿

5✳

20

✐♥❝❤❡s ✇✐❞❡✱ ❤♦✇ ♠❛♥②

❍♦✇ ❞♦ ■ ❦♥♦✇❄ ❙♦❧✈❡ t❤❡ ❡q✉❛t✐♦♥✿

4x = 20 . ❚❤✐s ❡q✉❛t✐♦♥ ❧❡❛❞s t♦ ❛ ♥❡✇ ♦♣❡r❛t✐♦♥✱ t✐♦♥ ✭❜②

❞✐✈✐s✐♦♥ ✿ x =

4✮✳

20 ✳ 4

❉✐✈✐s✐♦♥ ✭❜②

4✮

✐s t❤❡ ✐♥✈❡rs❡ ♦❢ ♠✉❧t✐♣❧✐❝❛✲

❊①❛♠♣❧❡ ✸✳✻✳✶✽✿ sq✉❛r❡ r♦♦t ❛s ✐♥✈❡rs❡

Pr♦❜❧❡♠✿ ■❢ ■ ✇❛♥t t♦ ♠❛❦❡ ❛ sq✉❛r❡ t❛❜❧❡ ✇✐t❤ ❛♥ ❛r❡❛ ✷✺ sq✉❛r❡ ❢❡❡t✱ ✇❤❛t s❤♦✉❧❞ ❜❡ t❤❡ ✇✐❞t❤ ♦❢ t❤❡ t❛❜❧❡❄ ❙♦❧✈❡ t❤❡ ❡q✉❛t✐♦♥✿

x · x = 25 =⇒ x2 = 25 =⇒ x = ❚❤✉s✱ ✇❡ ❤❛✈❡ ❛ ♥❡✇ ♦♣❡r❛t✐♦♥✿

sq✉❛r❡ r♦♦t✳

√

25 .

■t ✐s t❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ sq✉❛r✐♥❣ ❢✉♥❝t✐♦♥✳

❊①❛♠♣❧❡ ✸✳✻✳✶✾✿ ❝✉❜✐❝ r♦♦t ❛s ✐♥✈❡rs❡

Pr♦❜❧❡♠✿ ❲❤❛t ✐s t❤❡ s✐❞❡ ♦❢ ❛ ❜♦① ✐❢ ✐ts ✈♦❧✉♠❡ ✐s ❦♥♦✇♥ t♦ ❜❡

x3 = 8 =⇒ x =

√ 3

8

❝✉❜✐❝ ❢❡❡t❄ ❙♦❧✈❡ t❤❡ ❡q✉❛t✐♦♥✿

8.

❚❤❡ ❝✉❜✐❝ r♦♦t ✐s t❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ ❝✉❜✐❝ ♣♦✇❡r✳

❚❤✉s✱ s♦❧✈✐♥❣ ❡q✉❛t✐♦♥s r❡q✉✐r❡s ✉s t♦

✉♥❞♦

s♦♠❡ ❢✉♥❝t✐♦♥ ♣r❡s❡♥t ✐♥ t❤❡ ❡q✉❛t✐♦♥✿

1. x+2 = 5 =⇒ (x+2)−2 = 5−2 =⇒ x = 3 2. 3.

x·3 = 6 =⇒ (x·3)/3 = 6/3 √ √ x2 = 4 x2 = 4 =⇒

=⇒ x = 2 =⇒ x = 2 (x, y ≥ 0)

❲❡ ❤❛✈❡ ✏❝❛♥❝❡❧❧❛t✐♦♥✑ ♦♥ t❤❡ ❧❡❢t ❛♥❞ s✐♠♣❧✐✜❝❛t✐♦♥ ♦♥ t❤❡ r✐❣❤t✳ ❲❡ ❛r❡ ❞❡❛❧✐♥❣ ✇✐t❤ ❢✉♥❝t✐♦♥s✦ ❆♥❞ s♦♠❡ ❢✉♥❝t✐♦♥s ✶✳ ❚❤❡ ❛❞❞✐t✐♦♥ ♦❢

2

✉♥❞♦ t❤❡ ❡✛❡❝t ♦❢ ♦t❤❡rs ✿

✐s ✉♥❞♦♥❡ ❜② t❤❡ s✉❜tr❛❝t✐♦♥ ♦❢

✷✳ ❚❤❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❜②

3

2✱

❛♥❞ ✈✐❝❡ ✈❡rs❛✳

✐s ✉♥❞♦♥❡ ❜② t❤❡ ❞✐✈✐s✐♦♥ ❜②

✸✳ ❚❤❡ s❡❝♦♥❞ ♣♦✇❡r ✐s ✉♥❞♦♥❡ ❜② t❤❡ sq✉❛r❡ r♦♦t ✭❢♦r ❊❛❝❤ ♦❢ t❤❡s❡ ✉♥❞♦❡s t❤❡ ❡✛❡❝t ♦❢ ✐ts ❝♦✉♥t❡r♣❛rt

3✱

❛♥❞ ✈✐❝❡ ✈❡rs❛✳

x ≥ 0✮✱

❛♥❞ ✈✐❝❡ ✈❡rs❛✳

✉♥❞❡r s✉❜st✐t✉t✐♦♥ ✿

✸✳✻✳

❚❤❡ ✐♥✈❡rs❡ ♦❢ ❛ ❢✉♥❝t✐♦♥

✷✺✸

✶✳ ❙✉❜st✐t✉t✐♥❣ y = x + 2 ✐♥t♦ x = y − 2 ❣✐✈❡s ✉s x = x✳

1 ✷✳ ❙✉❜st✐t✉t✐♥❣ y = 3x ✐♥t♦ x = y ❣✐✈❡s ✉s x = x✳ 3 √ 2 ✸✳ ❙✉❜st✐t✉t✐♥❣ y = x ✐♥t♦ x = y ❣✐✈❡s ✉s x = x✱ ❢♦r x, y ≥ 0✳

❆♥❞ ✈✐❝❡ ✈❡rs❛✿

✶✳ ❙✉❜st✐t✉t✐♥❣ x = y − 2 ✐♥t♦ y = x + 2 ❣✐✈❡s ✉s y = y ✳

1 ✷✳ ❙✉❜st✐t✉t✐♥❣ x = y ✐♥t♦ y = 3x ❣✐✈❡s ✉s y = y ✳ 3 √ ✸✳ ❙✉❜st✐t✉t✐♥❣ x = y ✐♥t♦ y = x2 ❣✐✈❡s ✉s y = y ✱ ❢♦r x, y ≥ 0✳ ❲❛r♥✐♥❣✦ ❇♦t❤ ❝❛♥❝❡❧❧❛t✐♦♥s ♠❛tt❡r✳

❆s ✇❡ ❦♥♦✇✱ ✐t ✐s ♠♦r❡ ♣r❡❝✐s❡ t♦ s❛② t❤❛t t❤❡② ✉♥❞♦ ❡❛❝❤ ♦t❤❡r ✉♥❞❡r ❝♦♠♣♦s✐t✐♦♥ ✿ ❚✇♦ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s y = f (x) ❛♥❞ x = g(y) ❛r❡ ✐♥✈❡rs❡ ♦❢ ❡❛❝❤ ♦t❤❡r ✇❤❡♥ ❢♦r ❡✈❡r② x ✐♥ t❤❡ ❞♦♠❛✐♥ ♦❢ f ❛♥❞ ❢♦r ❡✈❡r② y ✐♥ t❤❡ ❞♦♠❛✐♥ ♦❢ g ✱ ✇❡ ❤❛✈❡✿ g(f (x)) = x ❛♥❞ f (g(y)) = y . ❚❤✐s ✐s ❛♥ ❛❧t❡r♥❛t✐✈❡ ✇❛② ♦❢ ✇r✐t✐♥❣ t❤❡s❡✿ ■♥✈❡rs❡s ✐♥ s✉❜st✐t✉t✐♦♥ ♥♦t❛t✐♦♥

g(y)

=x

f (x)

❛♥❞

y=f (x)

❚❤✉s✱ ✇❡ ❤❛✈❡ t❤r❡❡ ♣❛✐rs ♦❢ ✐♥✈❡rs❡ ❢✉♥❝t✐♦♥s✿

f (x) = x + 2

✈s✳

f (x) = 3x

✈s✳

2

✈s✳

f (x) = x

=y x=g(y)

f −1 (y) = y − 2 1 f −1 (y) = y 3 √ f −1 (y) = y ❢♦r x, y ≥ 0

◆❡①t✱ ✐t ✐s r❡❛s♦♥❛❜❧❡ t♦ ❛s❦✿ ❲❤❛t ✐s t❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ✐♥✈❡rs❡❄

❣r❛♣❤

♦❢ ❛ ❢✉♥❝t✐♦♥ ❛♥❞ t❤❡ ❣r❛♣❤ ♦❢ ✐ts

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❤❛t ❞♦ ✇❡ ♥❡❡❞ t♦ ❞♦ ✇✐t❤ t❤❡ ❣r❛♣❤ ♦❢ f t♦ ❣❡t t❤❡ ❣r❛♣❤ f −1 ❄ ❚❤❡ ❛♥s✇❡r ✐s✿ ❍❛r❞❧② ❛♥②t❤✐♥❣✳ ❆❢t❡r ❛❧❧✱ ❛ ❢✉♥❝t✐♦♥ ❛♥❞ ✐ts ✐♥✈❡rs❡ r❡♣r❡s❡♥t t❤❡ s❛♠❡ r❡❧❛t✐♦♥✳ ❚❤❡ ❣r❛♣❤ ♦❢ f ✐❧❧✉str❛t❡s ❤♦✇ y ❞❡♣❡♥❞s ♦♥ x ✕ ❛s ✇❡❧❧ ❛s ❤♦✇ x ❞❡♣❡♥❞s ♦♥ y ✳ ❆♥❞ t❤❡ ❧❛tt❡r ✐s ✇❤❛t ❞❡t❡r♠✐♥❡s f −1 ✦ ❙♦✱ t❤❡r❡ ✐s ♥♦ ♥❡❡❞ ❢♦r ❛ ♥❡✇ ❣r❛♣❤❀ t❤❡ ❣r❛♣❤ ♦❢ f −1 ✐s t❤❡ ❣r❛♣❤ ♦❢ f ✳ ❚❤❡ ♦♥❧② ✐ss✉❡ ✐s t❤❛t t❤❡ x✲ ❛♥❞ t❤❡ y ✲❛①✐s ♣♦✐♥t ✐♥ t❤❡ ✇r♦♥❣ ❞✐r❡❝t✐♦♥s✳ ■t✬s ❛♥ ❡❛s② ✜①✳ ❊①❛♠♣❧❡ ✸✳✻✳✷✵✿ ♣♦✐♥ts ♦♥ t❤❡ ❣r❛♣❤ ♦❢ ✐♥✈❡rs❡

❙✉♣♣♦s❡ ✇❡ ❛r❡ tr❛♥s✐t✐♦♥✐♥❣ ❢r♦♠ f t♦ ✐ts ✐♥✈❡rs❡ f −1 ✿

x 2 f= 3 8 ...

y = f (x) 5 1 7

=⇒

f −1

y 5 = 1 7 ...

x = f −1 (y) 2 3 8

❚❤❡s❡ ❛r❡ t❤❡ s❛♠❡ ♣❛✐rs✦ ❚❤❡r❡❢♦r❡✱ t❤❡② ❛r❡ r❡♣r❡s❡♥t❡❞ ❜② t❤❡ s❛♠❡ ♣♦✐♥ts ♦♥ t❤❡ xy ✲♣❧❛♥❡✳ ■t ✐s ❝♦♠♠♦♥✱ ❤♦✇❡✈❡r✱ t♦ ♣✉t t❤❡ ✐♥♣✉t ✈❛r✐❛❜❧❡ ✐♥ t❤❡ ❤♦r✐③♦♥t❛❧ ❛①✐s ❛♥❞ t❤❡ ♦✉t♣✉t ✐♥ t❤❡ ✈❡rt✐❝❛❧✳

✸✳✻✳

❚❤❡ ✐♥✈❡rs❡ ♦❢ ❛ ❢✉♥❝t✐♦♥

✷✺✹

❚❤✐s ♠❛❦❡s ✉s r❡♣❧❛❝❡ t❤❡ ♣♦✐♥ts ✐♥ t❤❡ xy ✲♣❧❛♥❡ ✇✐t❤ ♥❡✇ ♣♦✐♥ts ✐♥ t❤❡ yx✲♣❧❛♥❡✿ (x, y) −→ (y, x)

(2, 5) −→ (5, 2)

(3, 1) −→ (1, 3)

(8, 7) −→ (7, 8) ...

...

❚❤❡ t✇♦ ❝♦♦r❞✐♥❛t❡s ❛r❡ ✐♥t❡r❝❤❛♥❣❡❞✿

❲❡ r❡❛❧✐③❡ t❤❛t ❡❛❝❤ ♣♦✐♥t ❥✉♠♣s ❛❝r♦ss t❤❡ ❞✐❛❣♦♥❛❧ ❧✐♥❡ y = x✦ ❙♦✱ ✇❡ ❤❛✈❡ ❛ ♠❛t❝❤✿ ◮ ❊✈❡r② ♣♦✐♥t (x, y) ✐♥ t❤❡ xy ✲♣❧❛♥❡ ❝♦rr❡s♣♦♥❞s t♦ t❤❡ ♣♦✐♥t (y, x) ✐♥ t❤❡ yx✲♣❧❛♥❡✳ ❆❜♦✈❡ ✇❡ ♠❛❞❡ ❛ ❝♦♣② ♦❢ t❤❡ ❣r❛♣❤ ♦❢ f ✱ ✢✐♣♣❡❞ ✐t✱ ❛♥❞ t❤❡♥ ♦♥ t♦♣ ♦❢ t❤❡ ♦r✐❣✐♥❛❧✳ ❊①❛♠♣❧❡ ✸✳✻✳✷✶✿ ✐♥✈❡rs❡ ❣r❛♣❤ ♣♦✐♥t ❜② ♣♦✐♥t

❙✉♣♣♦s❡✱ ❛❣❛✐♥✱ ❛ ❢✉♥❝t✐♦♥ ✐s ❣✐✈❡♥ ♦♥❧② ❜② ✐ts ❣r❛♣❤ ❛♥❞ ✇❡ ♥❡❡❞ t♦ ❝♦♥str✉❝t t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ✐♥✈❡rs❡ x = f −1 (y)✳ ❚❤✐s t✐♠❡ ✇❡ ❛r❡ t♦ ❞♦ t❤✐s ✇✐t❤♦✉t ❛♥② ❞❛t❛✿

❙t❛rt ✇✐t❤ ❝❤♦♦s✐♥❣ ❛ ❢❡✇ ♣♦✐♥ts ♦♥ t❤❡ ❣r❛♣❤✳ ❊❛❝❤ ♦❢ t❤❡♠ ✇✐❧❧ ❥✉♠♣ ❛❝r♦ss t❤❡ ❞✐❛❣♦♥❛❧ ✉♥❞❡r t❤✐s ✢✐♣✳ ❍♦✇ ❡①❛❝t❧②❄ ❚❤❡ ❣❡♥❡r❛❧ r✉❧❡ ❢♦r ♣❧♦tt✐♥❣ ❛ ❝♦✉♥t❡r♣❛rt ♦❢ ❛ ♣♦✐♥t ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✿ ◮ ❋r♦♠ t❤❡ ♣♦✐♥t ❣♦ ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ ❞✐❛❣♦♥❛❧ ❛♥❞ t❤❡♥ ♠❡❛s✉r❡ t❤❡ s❛♠❡ ❞✐st❛♥❝❡ ♦♥ t❤❡ ♦t❤❡r s✐❞❡✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ♣❧♦t ❛ ❧✐♥❡ t❤r♦✉❣❤ ♦✉r ♣♦✐♥t ✇✐t❤ s❧♦♣❡ −1✳ ◆♦✇✱ ✇❡ ❝❛♥ s✐♠♣❧✐❢② ♦✉r ❥♦❜ ❜② ❝❤♦♦s✐♥❣ t❤❡ ♣♦✐♥ts ♠♦r❡ ❥✉❞✐❝✐♦✉s❧②❀ ✇❡ ❝❤♦♦s❡ ♦♥❡s ✇✐t❤ ❡❛s②✲t♦✲✜♥❞ ❝♦✉♥t❡r♣❛rts✳ ❋✐rst✱ ♣♦✐♥ts ♦♥ t❤❡ ❞✐❛❣♦♥❛❧ ❞♦♥✬t ♠♦✈❡ ❜② t❤❡ ✢✐♣ ❛❜♦✉t t❤❡ ❞✐❛❣♦♥❛❧✳ ❙❡❝♦♥❞✱ ♣♦✐♥ts ♦♥ ♦♥❡ ♦❢ t❤❡ ❛①❡s ❥✉♠♣ t♦ t❤❡ ♦t❤❡r ❛①✐s ✇✐t❤ ♥♦ ♥❡❡❞ ❢♦r ♠❡❛s✉r✐♥❣✳ ❋✐♥❛❧❧②✱ ♦♥❝❡ ❛❧❧ ♣♦✐♥ts ❛r❡ ✐♥ ♣❧❛❝❡✱ ✜♥❛❧❧②✱ ❞r❛✇ ❛ ❝✉r✈❡ t❤❛t ❝♦♥♥❡❝ts t❤❡♠✳

✸✳✻✳

❚❤❡ ✐♥✈❡rs❡ ♦❢ ❛ ❢✉♥❝t✐♦♥

✷✺✺

❊①❛♠♣❧❡ ✸✳✻✳✷✷✿ ❣r❛♣❤ ♦❢ ✐♥✈❡rs❡ ♣♦✐♥t ❜② ♣♦✐♥t P❧♦t t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ✐♥✈❡rs❡ ♦❢ y = f (x) s❤♦✇♥ ❜❡❧♦✇✿

❚❤❡s❡ ❛r❡ t❤❡ st❡♣s✿ • ❉r❛✇ t❤❡ ❞✐❛❣♦♥❛❧ y = x✳ • P✐❝❦ ❛ ❢❡✇ ♣♦✐♥ts ♦♥ t❤❡ ❣r❛♣❤ ♦❢ f ✭✇❡ ❝❤♦♦s❡ ❢♦✉r✮✳ • P❧♦t ❛ ❝♦rr❡s♣♦♥❞✐♥❣ ♣♦✐♥t ❢♦r ❡❛❝❤ ♦❢ t❤❡♠✿ ✕ ♦♥ t❤❡ ❧✐♥❡ t❤r♦✉❣❤ ♣♦✐♥t A t❤❛t ✐s ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ ❞✐❛❣♦♥❛❧ ✭✐✳❡✳✱ ✐ts s❧♦♣❡ ✐s 45 ❞❡❣r❡❡s ❞♦✇♥✮ ✕ ♦♥ t❤❡ ♦t❤❡r s✐❞❡ ♦❢ t❤❡ ❞✐❛❣♦♥❛❧ ❢r♦♠ A ✕ ❛t t❤❡ s❛♠❡ ❞✐st❛♥❝❡ ❢r♦♠ t❤❡ ❞✐❛❣♦♥❛❧ ❛s A • ❉r❛✇ ❜② ❤❛♥❞ ❛ ❝✉r✈❡ ❢r♦♠ ♣♦✐♥t t♦ ♣♦✐♥t✳

❊①❛♠♣❧❡ ✸✳✻✳✷✸✿ ✢✐♣ ❣r❛♣❤ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ❝❛♥ ❜❡ ❞♦♥❡ ✇✐t❤♦✉t ❛ ♣❡♥✳ ■❢ ✇❡ ❤❛✈❡ ❛ ♣✐❡❝❡ ♦❢ ♣❛♣❡r ✇✐t❤ t❤❡ xy ✲❛①✐s ❛♥❞ t❤❡ ❣r❛♣❤ ♦❢ y = f (x) ♦♥ ✐t✱ ✇❡ ✢✐♣ ✐t ❜② ❣r❛❜❜✐♥❣ t❤❡ ❡♥❞ ♦❢ t❤❡ x✲❛①✐s ✇✐t❤ t❤❡ r✐❣❤t ❤❛♥❞ ❛♥❞ ❣r❛❜❜✐♥❣ t❤❡ ❡♥❞ ♦❢ t❤❡ y ✲❛①✐s ✇✐t❤ t❤❡ ❧❡❢t ❤❛♥❞ t❤❡♥ ✐♥t❡r❝❤❛♥❣✐♥❣ t❤❡♠✿

❲❡ ❢❛❝❡ t❤❡ ♦♣♣♦s✐t❡ s✐❞❡ ♦❢ t❤❡ s❤❡❡t t❤❡♥✱ ❜✉t t❤❡ ❣r❛♣❤ ✐s st✐❧❧ ✈✐s✐❜❧❡✿ t❤❡ x✲❛①✐s ✐s ♥♦✇ ♣♦✐♥t✐♥❣ ✉♣ ❛♥❞ t❤❡ y ✲❛①✐s r✐❣❤t✱ ❛s ✐♥t❡♥❞❡❞✳ ❆ tr❛♥s♣❛r❡♥t s❤❡❡t ♦❢ ♣❧❛st✐❝ ✇♦✉❧❞ ✇♦r❦ ❡✈❡♥ ❜❡tt❡r✳

❊①❛♠♣❧❡ ✸✳✻✳✷✹✿ ❢♦❧❞ ❣r❛♣❤ ❆❧t❡r♥❛t✐✈❡❧②✱ ✇❡ ❝❛♥ ❛❧s♦ ❢♦❧❞✿

❚❤❡ s❤❛♣❡s ♦❢ t❤❡ ❣r❛♣❤s ❛r❡ t❤❡ s❛♠❡ ❜✉t t❤❡② ❛r❡ ♠✐rr♦r

✐♠❛❣❡s ♦❢ ❡❛❝❤ ♦t❤❡r✿

✸✳✻✳

❚❤❡ ✐♥✈❡rs❡ ♦❢ ❛ ❢✉♥❝t✐♦♥

✷✺✻

❊①❡r❝✐s❡ ✸✳✻✳✷✺

❲❤❛t ❣r❛♣❤s ✇✐❧❧ ❧❛♥❞ ♦♥ t❤❡♠s❡❧✈❡s ✉♥❞❡r t❤✐s tr❛♥s❢♦r♠❛t✐♦♥❄ ❊①❡r❝✐s❡ ✸✳✻✳✷✻

❲❤❛t ❧❡tt❡rs ❤❛✈❡ t❤✐s ❦✐♥❞ ♦❢ s②♠♠❡tr②❄ ❊①❛♠♣❧❡ ✸✳✻✳✷✼✿ s❧♦♣❡ ♦❢ ✐♥✈❡rs❡

❚❤✐s ✐s ✇❤❛t ❤❛♣♣❡♥s ✇❤❡♥ ✇❡ ❛♣♣❧② t❤✐s ✢✐♣ t♦ t❤❡ ❣r❛♣❤ ♦❢ ❛ ❧✐♥❡❛r ❢✉♥❝t✐♦♥✿

❚❤❡♥✱ ✇❡ ❝❛♥ ❝♦♠♣❛r❡✿ s❧♦♣❡ ♦❢ f =

r✐s❡ B r✐s❡ A = ❛♥❞ s❧♦♣❡ ♦❢ f −1 = = . r✉♥ B r✉♥ A

❚❤❡② ❛r❡✱ ❛s ✇❡ ❛❧r❡❛❞② ❦♥♦✇✱ t❤❡ r❡❝✐♣r♦❝❛❧s ♦❢ ❡❛❝❤ ♦t❤❡r✦ ❊①❛♠♣❧❡ ✸✳✻✳✷✽✿ ✐♥✈❡rs❡ ❣r❛♣❤ ✇✐t❤ ❝♦♠♣✉t❡r ❣r❛♣❤✐❝s

❙✉❝❤ ❛ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ t❤❡ ♣❧❛♥❡ ❝❛♥ ❜❡ ❛❝❝♦♠♣❧✐s❤❡❞ ✇✐t❤ s✐♠♣❧❡ ✐♠❛❣❡ ❡❞✐t✐♥❣ s♦❢t✇❛r❡ ❜② ✜rst r♦t❛t✐♥❣ t❤❡ ✐♠❛❣❡ ❝❧♦❝❦✇✐s❡ 90 ❞❡❣r❡❡s ❛♥❞ t❤❡♥ ✢✐♣♣✐♥❣ ✐t ✈❡rt✐❝❛❧❧②✿

❚❤✐s ✐s ❤♦✇ st❛rt✐♥❣ ❢r♦♠ ❛ ❣r❛♣❤ ✭✜rst ❜❡❧♦✇✮✱ ✇❡ ✜♥❞ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ✐♥✈❡rs❡ ✭s❡❝♦♥❞✮✱ ❛♥❞ t❤❡♥ ❜r✐♥❣ t❤❡♠ t♦❣❡t❤❡r ❢♦r ❝♦♠♣❛r✐s♦♥ ✭t❤✐r❞✮✿

❲❛r♥✐♥❣✦ ■t✬s ✐❧❧✲❛❞✈✐s❡❞ t♦ tr② t♦ ❣✉❡ss ✇❤❛t t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ✐♥✈❡rs❡ ❧♦♦❦s ❧✐❦❡✳

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

✸✳✻✳

❚❤❡ ✐♥✈❡rs❡ ♦❢ ❛ ❢✉♥❝t✐♦♥

✷✺✼

◆❡①t✱ s♦♠❡ ♠♦r❡ ♣r♦❢♦✉♥❞ ✐ss✉❡s✳✳✳

❊①❛♠♣❧❡ ✸✳✻✳✷✾✿ sq✉❛r❡ ✈s✳ sq✉❛r❡ r♦♦t ❍❡r❡ ✐s ❛ ❢❛♠✐❧✐❛r ♣❛✐r ♦❢ ❢✉♥❝t✐♦♥s t❤❛t ✉♥❞♦ ❡❛❝❤ ♦t❤❡r✿

x →

→ y →

sq✉❛r❡

◆♦✇✱ t❤❡ ❞✐❛❣r❛♠ ❢❛✐❧s ✐❢ ✇❡ ♣❧✉❣ ✐♥

−2 →

→ x✱

s❛♠❡❄

x = −2✿ → 4 →

sq✉❛r❡

sq✉❛r❡ r♦♦t

sq✉❛r❡ r♦♦t

→ 2✱

♥♦t t❤❡ s❛♠❡✦

❆s ✇❡ ❦♥♦✇✱ ♥♦t ❛❧❧ ❢✉♥❝t✐♦♥s ❤❛✈❡ ✐♥✈❡rs❡s✳✳✳

❍♦✇❡✈❡r✱ ✇❡ ❝❛♥ ♠❛❦❡ ✐t ✇♦r❦ ❜②

•

❚❤❡ ♦❧❞ ❢✉♥❝t✐♦♥ ✐s

•

❚❤❡ ♥❡✇ ❢✉♥❝t✐♦♥ ✐s

y = x

2

r❡str✐❝t✐♥❣ t❤❡ ❞♦♠❛✐♥ ❛♥❞ t❤❡ ❝♦❞♦♠❛✐♥

♦❢ t❤❡ ❢✉♥❝t✐♦♥✿

✱ ✇✐t❤ t❤❡ ❞♦♠❛✐♥ ❛♥❞ t❤❡ ❝♦❞♦♠❛✐♥ ❛ss✉♠❡❞ t♦ ❜❡

✐♥✈❡rs❡✮✳

y = x2 ✱

✇✐t❤ t❤❡ ❞♦♠❛✐♥ ❛♥❞ t❤❡ ❝♦❞♦♠❛✐♥ ❝❤♦s❡♥ t♦ ❜❡

❚❤❡♥✱ t❤❡ ♥❡✇ ❢✉♥❝t✐♦♥ ❤❛s t❤❡ ✐♥✈❡rs❡✿

(−∞, ∞)

✭♥♦

[0, ∞)✳

❲❡ ❤❛✈❡ ♠❛❞❡ ✐t ♣♦ss✐❜❧❡ ❜② r❡♠♦✈✐♥❣ t❤❡ s❡❝♦♥❞ ♣♦ss✐❜✐❧✐t②✿

22 = 4, (−2)2 = 4 . ❚❤❡ ❢♦❧❧♦✇✐♥❣ ✇✐❧❧ ❜❡ ✈❡r② ✉s❡❢✉❧✳

❚❤❡♦r❡♠ ✸✳✻✳✸✵✿ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ P♦✇❡r ❋✉♥❝t✐♦♥s ✶✳ ❚❤❡ ♦❞❞ ♣♦✇❡rs ❛r❡ ♦♥❡✲t♦✲♦♥❡❀ ❝♦♥s❡q✉❡♥t❧②✱ t❤❡② ❛r❡ ✐♥✈❡rt✐❜❧❡✳

❚❤❡

❡✈❡♥ ♣♦✇❡rs ❛r❡♥✬t ♦♥❡✲t♦✲♦♥❡❀ ❝♦♥s❡q✉❡♥t❧②✱ t❤❡② ❛r❡ ♥♦t ✐♥✈❡rt✐❜❧❡✳ ✷✳ ❚❤❡ ❡✈❡♥ ♣♦✇❡rs ✇✐t❤ ❞♦♠❛✐♥s ❛♥❞ ❝♦❞♦♠❛✐♥s r❡❞✉❝❡❞ t♦

(−∞, 0]✮

❛r❡ ♦♥❡✲t♦✲♦♥❡❀ ❝♦♥s❡q✉❡♥t❧②✱ t❤❡② ❛r❡ ✐♥✈❡rt✐❜❧❡✳

[0, +∞)

✭♦r

✸✳✻✳

❚❤❡ ✐♥✈❡rs❡ ♦❢ ❛ ❢✉♥❝t✐♦♥

❲❡ ❝❛♥ s❡❡ ❜❡❧♦✇ ❤♦✇ t❤❡

✷✺✽

❍♦r✐③♦♥t❛❧ ▲✐♥❡ ❚❡st

✐s s❛t✐s✜❡❞ ❢♦r t❤❡ ♦❞❞ ♣♦✇❡rs✱ ❢❛✐❧s ❢♦r t❤❡ ❡✈❡♥ ♣♦✇❡rs✱

❛♥❞ ✐s s❛t✐s✜❡❞ ❢♦r t❤❡ ✏r❡❞✉❝❡❞✑ ❡✈❡♥ ♣♦✇❡rs✿

❲❛r♥✐♥❣✦

❲❡ ✇✐❧❧ ❤❛✈❡ t♦ ❦❡❡♣ tr❛❝❦ ♦❢ ❜♦t❤ ❜r❛♥❝❤❡s ✭s❡♣❛✲ r❛t❡❧②✮ ✇❤❡♥ ✇❡ s♦❧✈❡ ❡q✉❛t✐♦♥s✿

x2 = 1 =⇒ x = 1 ❖❘ x = −1 . ❊①❛♠♣❧❡ ✸✳✻✳✸✶✿ ✐♥✈❡rs❡ ❢r♦♠ ❢♦r♠✉❧❛ ❙✉♣♣♦s❡ t❤✐s t✐♠❡ t❤❛t ❛ ❢✉♥❝t✐♦♥ ✐s ❣✐✈❡♥ ♦♥❧② ❜② ✐ts x = f −1 (y)✳ ❋♦r ❡①❛♠♣❧❡✱ ❧❡t 3

❢♦r♠✉❧❛✳

❋✐♥❞ t❤❡ ❢♦r♠✉❧❛ ♦❢ t❤❡ ✐♥✈❡rs❡

f (x) = x − 3 .

❲❡ s✐♠♣❧② r❡✇r✐t❡

y = x3 − 3 , ❛♥❞ t❤❡♥ s♦❧✈❡ ❢♦r

x✿ x=

❚❤✉s✱ t❤❡ ❛♥s✇❡r ✐s✿

p 3 y + 3.

f −1 (y) = ▼♦r❡ ♦♥ s♦❧✈✐♥❣ ❡q✉❛t✐♦♥s ✐♥ ❈❤❛♣t❡r ✺✳

p 3 y + 3.

❊①❡r❝✐s❡ ✸✳✻✳✸✷ ❋✉♥❝t✐♦♥

y = f (x)

t❛❜❧❡✳

✐s ❣✐✈❡♥ ❜❡❧♦✇ ❜② ❛ ❧✐st ✐ts ✈❛❧✉❡s✳ ❋✐♥❞ ✐ts ✐♥✈❡rs❡ ❛♥❞ r❡♣r❡s❡♥t ✐t ❜② ❛ s✐♠✐❧❛r

x 0 1 2 3 4 y = f (x) 1 2 0 4 3

❊①❡r❝✐s❡ ✸✳✻✳✸✸ ❲❤❛t ❦✐♥❞ ♦❢ ❢✉♥❝t✐♦♥ ✐s ✐ts ♦✇♥ ✐♥✈❡rs❡❄

❊①❡r❝✐s❡ ✸✳✻✳✸✹ P❧♦t t❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ s❤♦✇♥ ❜❡❧♦✇✱ ✐❢ ♣♦ss✐❜❧❡✿

✸✳✼✳

❯♥✐ts ❝♦♥✈❡rs✐♦♥s ❛♥❞ ❝❤❛♥❣❡s ♦❢ ✈❛r✐❛❜❧❡s

✷✺✾

❊①❡r❝✐s❡ ✸✳✻✳✸✺ P❧♦t t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ✐♥✈❡rs❡ ♦❢ t❤✐s ❢✉♥❝t✐♦♥✿

❊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) 7 5 3 4 6

❊①❡r❝✐s❡ ✸✳✻✳✸✼

P❧♦t t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥

f (x) =

❢❡❛t✉r❡s✳

1 x−1

❛♥❞ t❤❡ ❣r❛♣❤ ♦❢ ✐ts ✐♥✈❡rs❡✳

■❞❡♥t✐❢② ✐ts ✐♠♣♦rt❛♥t

✸✳✼✳ ❯♥✐ts ❝♦♥✈❡rs✐♦♥s ❛♥❞ ❝❤❛♥❣❡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❝✳

✸✳✼✳

❯♥✐ts ❝♦♥✈❡rs✐♦♥s ❛♥❞ ❝❤❛♥❣❡s ♦❢ ✈❛r✐❛❜❧❡s

✷✻✵

❆❧♠♦st ❛❧❧ ❝♦♥✈❡rs✐♦♥ ❢♦r♠✉❧❛s ❛r❡ ❥✉st ♠✉❧t✐♣❧✐❝❛t✐♦♥s✱ s✉❝❤ ❛s t❤✐s ♦♥❡✿ # ♦❢ ♠❡t❡rs = # ♦❢ ❦✐❧♦♠❡t❡rs · 1000 . ❲❛r♥✐♥❣✦ ❲❡ ❞♦♥✬t ❝♦♥✈❡rt ✏♣♦✉♥❞s t♦ ❦✐❧♦s✑✱ ✇❡ ❝♦♥✈❡rt t❤❡

♥✉♠❜❡r ♦❢

♣♦✉♥❞s t♦ t❤❡

♥✉♠❜❡r ♦❢

❦✐❧♦s✳

❚❤❡ ♦♥❧② ❡①❝❡♣t✐♦♥ ♦❢ t❤❡ t❡♠♣❡r❛t✉r❡✱ ❜❡❝❛✉s❡ 0 ❞❡❣r❡❡s ♦❢ ❈❡❧s✐✉s ❞♦❡s♥✬t ❝♦rr❡s♣♦♥❞ t♦ 0 ❞❡❣r❡❡s ♦❢ ❋❛❤r❡♥❤❡✐t✳ ❚❤✐s ✐s t❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ ❞❡❣r❡❡s ❛♥❞ r❛❞✐❛♥s✿ π r❛❞✐❛♥s = 180 ❞❡❣r❡❡s .

■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ❝♦♥✈❡rs✐♦♥ ❜❡t✇❡❡♥ t❤❡ ♥✉♠❜❡r ♦❢ ❞❡❣r❡❡s d ❛♥❞ t❤❡ ♥✉♠❜❡r ♦❢ r❛❞✐❛♥s r ✐s t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥✿ πr = 180d .

❚❤❡r❡❢♦r❡✱ ✇❡ ❝♦♥✈❡rt ❢r♦♠ ❞❡❣r❡❡s t♦ r❛❞✐❛♥s ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢✉♥❝t✐♦♥✿ r=

π d. 180

❚❤❡♥✱ ✇❡ ❝♦♥✈❡rt ❢r♦♠ r❛❞✐❛♥s ❞❡❣r❡❡s ✇✐t❤ t❤❡ ✐♥✈❡rs❡ ♦❢ t❤✐s ❢✉♥❝t✐♦♥✿ d=

180 r. π

❲✐t❤✐♥ ❡❛❝❤ ♦❢ t❤❡ ❝❛t❡❣♦r✐❡s✱ t❤❡r❡ ♠❛② ❜❡ ❝♦♠♣❧❡①✱ ❡✈❡♥ ❝✐r❝✉❧❛r✱ r❡❧❛t✐♦♥s✳ ❋♦r ❡①❛♠♣❧❡✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛♠♦♥❣ t❤❡s❡ ❝✉rr❡♥❝✐❡s✿ # ♦❢ x ❞♦❧❧❛rs  ×1.3 

×.9

−−−−−→

# ♦❢ ♣♦✉♥❞s ←−−−−−−− ×0.007

# ♦❢ ❡✉r♦s  ×122 y

♦r

# ♦❢ ②❡♥

U SD  /1.3 y

/.9

←−−−−− /0.007

EU x R  /122 

GBP −−−−−−−→ JP Y

❚❤❡ ❛rr♦✇s✱ ♦❢ ❝♦✉rs❡✱ ✐♥❞✐❝❛t❡ ❢✉♥❝t✐♦♥s✱ t✇♦ ✐♥ ❛ r♦✇ ✐♥❞✐❝❛t❡ ❝♦♠♣♦s✐t✐♦♥s✱ ❛♥❞ t❤❡ r❡✈❡rs❡❞ ❛rr♦✇s ❛r❡ t❤❡ ✐♥✈❡rs❡s✦ ❊①❡r❝✐s❡ ✸✳✼✳✶

▼❛❦❡ ②♦✉r ✇❛② ❢r♦♠ ♠✐♥✉t❡s t♦ ②❡❛rs✳ ❲❡ ❞♦♥✬t ❞❡❛❧ ✇✐t❤ t❤❡s❡ q✉❛♥t✐t✐❡s ♦♥❡ ❜② ♦♥❡ ♥♦r ❡✈❡♥ ✐♥ t❤❡s❡ ♣❛✐rs✳ ❲❡ ✇✐❧❧ st✉❞② t❤❡ ❤❛✈❡ t❤❡♠ ❛s ✈❛r✐❛❜❧❡s✳

❢✉♥❝t✐♦♥s

t❤❛t

❲❡ ✇✐❧❧ ✜rst ❝♦♥s✐❞❡r t❤❡ ❝♦♠♣♦s✐t✐♦♥s ♦❢ t❤❡s❡ ❢✉♥❝t✐♦♥s ✇✐t❤ t❤❡ ❢✉♥❝t✐♦♥s t❤❛t r❡♣r❡s❡♥t t❤❡ ✉♥✐t ❝♦♥✈❡r✲ s✐♦♥s✳ ❊①❛♠♣❧❡ ✸✳✼✳✷✿ ✉♥✐ts ♦❢ ❞✐st❛♥❝❡

❙✉♣♣♦s❡ t ✐s t❤❡ t✐♠❡ ❛♥❞ x ✐s t❤❡ ❧♦❝❛t✐♦♥✳ ❙✉♣♣♦s❡ ❛❧s♦ t❤❛t ❛ ❢✉♥❝t✐♦♥ g r❡♣r❡s❡♥ts t❤❡ ❝❤❛♥❣❡ ♦❢ ✉♥✐ts ♦❢ ❧❡♥❣t❤✱ s✉❝❤ ❛s ❢r♦♠ ♠✐❧❡s t♦ ❦✐❧♦♠❡t❡rs✿ z = g(x) = 1.6x .

❚❤❡♥✱ t❤❡ ❝❤❛♥❣❡ ♦❢ t❤❡ ✉♥✐ts ✇✐❧❧ ♠❛❦❡ ✈❡r② ❧✐tt❧❡ ❞✐✛❡r❡♥❝❡❀ t❤❡ ❝♦❡✣❝✐❡♥t✱ m = 1.6✱ ✐s t❤❡ ♦♥❧② ❛❞❥✉st♠❡♥t ♥❡❝❡ss❛r②✳ ■❢ f ✐s t❤❡ ❞✐st❛♥❝❡ ✐♥ ♠✐❧❡s✱ t❤❡♥ h ✐s t❤❡ ❞✐st❛♥❝❡ ✐♥ ❦✐❧♦♠❡t❡rs✿ h(t) = 1.6f (t)✳ ❚❤✉s✱ ❛❧❧ t❤❡ ❢✉♥❝t✐♦♥s ❛r❡ r❡♣❧❛❝❡❞ ✇✐t❤ t❤❡✐r ♠✉❧t✐♣❧❡s✳ ❚❤❡ ❣r❛♣❤s ❛r❡ str❡t❝❤❡❞✦ ❲❡ ❝❛❧❧ s✉❝❤ ❛ ✉♥✐t ❝♦♥✈❡rs✐♦♥ ❛ ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s✳ ❯s✉❛❧❧②✱ ✐t ✐s ❞♦♥❡ ♦♥❡ ❛t ❛ t✐♠❡✿ ❡✐t❤❡r t❤❡ ❞❡♣❡♥❞❡♥t ♦r t❤❡ ✐♥❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡✳

✸✳✼✳

❯♥✐ts ❝♦♥✈❡rs✐♦♥s ❛♥❞ ❝❤❛♥❣❡s ♦❢ ✈❛r✐❛❜❧❡s

✷✻✶

❊①❛♠♣❧❡ ✸✳✼✳✸✿ ♠♦t✐♦♥ ❛♥❞ ✉♥✐ts ❙✉♣♣♦s❡ ✇❡ st✉❞②

♠♦t✐♦♥

❛♥❞ ✇❡ ❤❛✈❡ ❛ ❢✉♥❝t✐♦♥

y = f (x)

t❤❛t r❡❧❛t❡s

• x✱ t✐♠❡ ✐♥ ♠✐♥✉t❡s✱ t♦ • y ✱ ❧♦❝❛t✐♦♥ ✐♥ ✐♥❝❤❡s✳

❲❤❛t ✐❢ ✇❡ ♥❡❡❞ t♦ s✇✐t❝❤ t♦

• t✱ t✐♠❡ ✐♥ s❡❝♦♥❞s✱ ♦r • z ✱ ❧♦❝❛t✐♦♥ ✐♥ ❢❡❡t❄

❚❤❡ ❛❧❣❡❜r❛ ✐s ❝❧❡❛r✿

x = t/60

❛♥❞

z = y/12 .

❛♥❞

z = f (x)/12 .

❚❤❡♥ ✇❡ ♠✐❣❤t ❤❛✈❡ t✇♦ ♥❡✇ ❢✉♥❝t✐♦♥s✿

y = f (t/60) ◆♦✇✱ ✇❤❛t ✇✐❧❧ t❤❡ ♥❡✇ ❣r❛♣❤s ❧♦♦❦ ❧✐❦❡❄

❚♦ ❛♥s✇❡r✱ ✇❡ ❝♦♠❜✐♥❡ t❤❡ ❣r❛♣❤ ♦❢

f

✇✐t❤ t❤❡ t✇♦

tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ t✇♦ ❛①❡s✱ ❛s ❢♦❧❧♦✇s✿

❚❤❡ r❡s✉❧t ✐s ❛ ✈❡rt✐❝❛❧ ❛♥❞ ❛ ❤♦r✐③♦♥t❛❧ str❡t❝❤✴s❤r✐♥❦✳ ❍♦✇❡✈❡r✱ ✐t✬s ❡♥t✐r❡❧② ✉♣ t♦ ✉s t♦ ❝❤♦♦s❡ t❤❡ ✉♥✐ts ♦♥ t❤❡ ♥❡✇ ❛①❡s t♦ ♠❛t❝❤ t❤❡ ♦❧❞✿ t❤❡ ❣r❛♣❤ ✇✐❧❧ r❡♠❛✐♥ t❤❡ s❛♠❡✦

❊①❡r❝✐s❡ ✸✳✼✳✹ ❲❤❛t ✐s t❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ s❡❝♦♥❞s ❛♥❞ ❢❡❡t❄

❊①❛♠♣❧❡ ✸✳✼✳✺✿ t✐♠❡ ❛♥❞ t❡♠♣❡r❛t✉r❡ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ❢✉♥❝t✐♦♥

f

t❤❛t r❡❝♦r❞s t❤❡ t❡♠♣❡r❛t✉r❡ ✭✐♥ ❋❛❤r❡♥❤❡✐t✮ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✐♠❡ ✭✐♥

♠✐♥✉t❡s✮✳

◗✉❡st✐♦♥✿

◮

❲❤❛t s❤♦✉❧❞

f

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

❢✉♥❝t✐♦♥ ♦❢ t✐♠❡ ✐♥ s❡❝♦♥❞s❄

▲❡t✬s ♥❛♠❡ t❤❡ ✈❛r✐❛❜❧❡s✿

• • • •

s ✐s t❤❡ t✐♠❡ ✐♥ s❡❝♦♥❞s✱ m ✐s t❤❡ t✐♠❡ ✐♥ ♠✐♥✉t❡s✱ F ✐s t❤❡ t❡♠♣❡r❛t✉r❡ ✐♥ ❋❛❤r❡♥❤❡✐t✱ C ✐s t❤❡ t❡♠♣❡r❛t✉r❡ ✐♥ ❈❡❧s✐✉s✳

✸✳✼✳

❯♥✐ts ❝♦♥✈❡rs✐♦♥s ❛♥❞ ❝❤❛♥❣❡s ♦❢ ✈❛r✐❛❜❧❡s

✷✻✷

❙✉♣♣♦s❡ t❤❡ ♦r✐❣✐♥❛❧ ❢✉♥❝t✐♦♥✱ s❛②✱

F = f (m) , ✐s t♦ ❜❡ r❡♣❧❛❝❡❞ ✇✐t❤ s♦♠❡ ♥❡✇ ❢✉♥❝t✐♦♥✱

C = g(s) .

❋✐rst✱ ✇❡ ♥❡❡❞ t❤❡

❝♦♥✈❡rs✐♦♥ ❢♦r♠✉❧❛s

❢♦r t❤❡s❡ ✉♥✐ts✳ ❋✐rst✱ t❤❡ t✐♠❡✳ ❚❤✐s ✐s ✇❤❛t ✇❡ ❦♥♦✇✿

1

♠✐♥✉t❡

= 60

❍♦✇❡✈❡r✱ t❤✐s ✐s ♥♦t t❤❡ ❢♦r♠✉❧❛ t♦ ❜❡ ✉s❡❞ t♦ ❝♦♥✈❡rt ❛♥❞ t❤❡

♥✉♠❜❡r

s❡❝♦♥❞s✳

s t♦ m ❜❡❝❛✉s❡ t❤❡s❡ ❛r❡ t❤❡ ♥✉♠❜❡r

♦❢ s❡❝♦♥❞s

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

m = s/60 . ❲❡ r❡♣r❡s❡♥t t❤❡ ❢✉♥❝t✐♦♥ ❜② ✐ts ❣r❛♣❤ ❛♥❞ ❛s ❛ tr❛♥s❢♦r♠❛t✐♦♥✿

❙❡❝♦♥❞✱ t❤❡ t❡♠♣❡r❛t✉r❡✳ ❚❤✐s ✐s ✇❤❛t ✇❡ ❦♥♦✇✿

C = (F − 32)/1.8 . ❲❡ r❡♣r❡s❡♥t t❤❡ ❢✉♥❝t✐♦♥ ❜② ✐ts ❣r❛♣❤ ❛♥❞ ❛s ❛ tr❛♥s❢♦r♠❛t✐♦♥✿

❚❤❡s❡ ❛r❡ t❤❡ r❡❧❛t✐♦♥s ❜❡t✇❡❡♥ t❤❡ ❢♦✉r q✉❛♥t✐t✐❡s✿

g:

s/60

(F −32)/1.8

f

s −−−−−−→ m −−−−→ F −−−−−−−−−−→ C

■♥st❡❛❞ ♦❢ tr❛♥s❢♦r♠✐♥❣ t❤❡ ❛①❡s ❛♥❞✱ t❤❡r❡❢♦r❡✱ t❤❡ ♣❧❛♥❡✱ ✇❡ ❝❤♦♦s❡ t♦ s✐♠♣❧②

❚❤❡ ❛♥s✇❡r t♦ ♦✉r q✉❡st✐♦♥ ✐s✱ ✇❡ r❡♣❧❛❝❡

f

✇✐t❤

g✱

t❤❡

❝♦♠♣♦s✐t✐♦♥


F = g(s) = f (s/60) − 32 /1.8 .

r❡❧❛❜❡❧

t❤❡♠✿

♦❢ t❤❡ ❛❜♦✈❡ ❢✉♥❝t✐♦♥s✿

✸✳✼✳

❯♥✐ts ❝♦♥✈❡rs✐♦♥s ❛♥❞ ❝❤❛♥❣❡s ♦❢ ✈❛r✐❛❜❧❡s

✷✻✸

◆♦t❡ t❤❛t ❜♦t❤ ♦❢ t❤❡ ❝♦♥✈❡rs✐♦♥ ❢♦r♠✉❧❛s ❛r❡ ♦♥❡✲t♦✲♦♥❡ ❢✉♥❝t✐♦♥s✦ ❚❤❛t✬s ✇❤❛t ❣✉❛r❛♥t❡❡s t❤❛t t❤❡ ❝♦♥✈❡rs✐♦♥s ❛r❡ ✉♥❛♠❜✐❣✉♦✉s ❛♥❞ r❡✈❡rs✐❜❧❡✳ ▼♦r❡ ♣r❡❝✐s❡❧②✱ ✇❡ s❛② t❤❛t t❤❡s❡ ❢✉♥❝t✐♦♥s ❛r❡ ✐♥✈❡rt✐❜❧❡✳ ■♥❞❡❡❞✱ t❤❡s❡ ❛r❡ t❤❡ ✐♥✈❡rs❡s✱ ❢♦r t❤❡ t✐♠❡✿ s = 60m ,

❛♥❞ ❢♦r t❤❡ t❡♠♣❡r❛t✉r❡✿ F = 1.8C + 32 .

◆♦t❡ t❤❛t ❛❧❧ ♦❢ t❤❡ ❝♦♥✈❡rs✐♦♥ ❢♦r♠✉❧❛s ❤❛✈❡ ❜❡❡♥ ♣r♦✈✐❞❡❞ ❜② ❧✐♥❡❛r ❢✉♥❝t✐♦♥s✳ ❚❤❡♥✱ ❛ ❧✐♥❡❛r ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✇✐❧❧ ❝❛✉s❡ t❤❡ x✲❛①✐s ♦r t❤❡ y ✲❛①✐s t♦ s❤✐❢t✱ str❡t❝❤✱ ♦r ✢✐♣ ✭✈❡rt✐❝❛❧❧② ♦r ❤♦r✐③♦♥t❛❧❧②✮✿

❲❡ ❝♦♥❝❧✉❞❡✿ ◮ ❆ ❧✐♥❡❛r ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✇✐❧❧ ❝❛✉s❡ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ t♦ s❤✐❢t✱ str❡t❝❤✱ ♦r ✢✐♣✳

❙♦♠❡ ♥♦♥❧✐♥❡❛r ❝❤❛♥❣❡s ♦❢ ✈❛r✐❛❜❧❡s ❛r❡ ❛❧s♦ ❦♥♦✇♥✳ ❊①❛♠♣❧❡ ✸✳✼✳✻✿ ♥♦♥✲❧✐♥❡❛r ❝❤❛♥❣❡ ♦❢ ✉♥✐ts

❚❤❡ ❧♦✉❞♥❡ss ♦❢ s♦✉♥❞ ✭t❤❡ ❞❡❝✐❜❡❧✮ ❛♥❞ t❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ ❡❛rt❤q✉❛❦❡s ✭t❤❡ ❘✐❝❤t❡r s❝❛❧❡✮ ❛r❡ ♠❡❛s✉r❡❞ ♦♥ ❛ ❧♦❣❛r✐t❤♠✐❝ s❝❛❧❡✳ ❚❤✐s s❝❛❧❡ ✐s ❜❛s❡❞ ♦♥ ♦r❞❡rs ♦❢ ♠❛❣♥✐t✉❞❡ r❛t❤❡r t❤❛♥ ❛ ❧✐♥❡❛r s❝❛❧❡✱ ✐✳❡✳✱ t❤❡ ♥❡①t ♠❛r❦ ♦♥ t❤❡ s❝❛❧❡ ❝♦rr❡s♣♦♥❞s t♦ t❤❡ ♠❛❣♥✐t✉❞❡ ❛t t❤❡ ♣r❡✈✐♦✉s ♠❛r❦ ♠✉❧t✐♣❧✐❡❞ ❜② ❛ ♣r❡❞❡t❡r♠✐♥❡❞ ❝♦❡✣❝✐❡♥t✱ s✉❝❤ ❛s 10✿

❙♣r❡❛❞s❤❡❡t s♦❢t✇❛r❡ ♠❛② ❤❛✈❡ ❛♥ ♦♣t✐♦♥ t♦ s✇✐t❝❤ t❤❡ s❝❛❧❡ ✇✐t❤ ❥✉st ❛ ❝♦✉♣❧❡ ♦❢ ❝❧✐❝❦s✳ ❋♦r ❡①❛♠♣❧❡✱ ✇❡ ❝❛♥ s❡❡ ❤♦✇ t❤❡ ❣r❛♣❤ ♦❢ ❛ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥ ❜❡❝♦♠❡s ❛ str❛✐❣❤t ❧✐♥❡ ✇❤❡♥ ✇❡ s✇✐t❝❤ t❤❡ y ✲❛①✐s t♦ t❤❡ ❧♦❣❛r✐t❤♠✐❝ s❝❛❧❡✿

❊①❡r❝✐s❡ ✸✳✼✳✼

❊st❛❜❧✐s❤ ❛❧❣❡❜r❛✐❝ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ q✉❛♥t✐t✐❡s ❡①♣r❡ss❡❞ ✐♥ t❤❡ ✉♥✐ts ❧✐st❡❞ ✐♥ t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤❡ s❡❝t✐♦♥✳ ❊①❡r❝✐s❡ ✸✳✼✳✽

❲❤❛t ✐s t❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ ♠✐❧❡s ♣❡r ❣❛❧❧♦♥ ❛♥❞ ❦✐❧♦♠❡t❡rs ♣❡r ❧✐t❡r❄

✸✳✽✳

❚r❛♥s❢♦r♠✐♥❣ t❤❡ ❛①❡s tr❛♥s❢♦r♠s t❤❡ ♣❧❛♥❡

✷✻✹

❊①❡r❝✐s❡ ✸✳✼✳✾

❲❤❛t ✐s t❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ ♠✐❧❡s ♣❡r ❣❛❧❧♦♥ ❛♥❞ ❣❛❧❧♦♥s ♣❡r ♠✐❧❡❄ ❊①❛♠♣❧❡ ✸✳✼✳✶✵✿ ✇❤❛t ❤❛♣♣❡♥s t♦ ❢♦r♠✉❧❛s ✉♥❞❡r ❝♦♥✈❡rs✐♦♥s ♦❢ ✉♥✐ts

❲❤❛t ❤❛♣♣❡♥s t♦ t❤❡ ❢♦r♠✉❧❛ ♦❢ ❛ ❢✉♥❝t✐♦♥ ✉♥❞❡r ❝♦♥✈❡rs✐♦♥s ♦❢ ✉♥✐ts ❛♥❞ tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ❛①❡s❄ ❲❡ ❥✉st ❡①❡❝✉t❡ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s✉❜st✐t✉t✐♦♥s ✐♥ t❤❡ ❢♦r♠✉❧❛✳ ❋♦r ❡①❛♠♣❧❡✱ ❝♦♥s✐❞❡r ❛ ❢✉♥❝t✐♦♥

y = f (x) . ❲❤❛t ✇✐❧❧ ❜❡ t❤❡ ❢♦r♠✉❧❛ ♦❢ t❤✐s ❢✉♥❝t✐♦♥ ✐❢ ✇❡ • s❤✐❢t ✐t 5 ✉♥✐ts r✐❣❤t✱ ❛♥❞ t❤❡♥ • s❤✐❢t ✐t 2 ✉♥✐ts ✉♣❄ ❲❡ ❤❛✈❡ ♥❡✇ ✈❛r✐❛❜❧❡s✿ u = x + 5, v = y + 2 . ❲❡ ♥❡❡❞ t♦ ❡①❡❝✉t❡ t❤❡ ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✈✐❛ ❛ s✉❜st✐t✉t✐♦♥✳ ❋♦r t❤❛t✱ ✇❡ s♦❧✈❡ ❢♦r x ❛♥❞ y ✿

x = u − 5, y = v − 2 . ❲❡ ♥♦✇ s✉❜st✐t✉t❡ t❤❡s❡ ✐♥t♦ t❤❡ ❢✉♥❝t✐♦♥

v − 2 = f (u − 5) .

❲❡ ❤❛✈❡ ❛ ♥❡✇ ❲❡ s♦❧✈❡ ❢♦r v ✿

r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ♥❡✇ ✈❛r✐❛❜❧❡s✱ ❜✉t ❢♦r ❛ ❢✉♥❝t✐♦♥✱ ✇❡ ♥❡❡❞ ❛♥ ❡①♣❧✐❝✐t ❢♦r♠✉❧❛✳ v = f (u − 5) + 2 .

❚❤❛t✬s t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ♥❡✇ ❝✉r✈❡ ♦♥ t❤❡ ♥❡✇ uv ✲♣❧❛♥❡✳ ❈♦♠♣❛r❡ t❤❡s❡✱ ❤♦✇❡✈❡r✿ • ❚❤❡ s❤✐❢t 2 ✉♥✐ts ✉♣ ❤❛s ♣r♦❞✉❝❡❞ +2 ✐♥ t❤❡ ❢♦r♠✉❧❛✱ ❛s ❡①♣❡❝t❡❞✳ • ❚❤❡ s❤✐❢t 5 ✉♥✐ts r✐❣❤t ❤❛s ♣r♦❞✉❝❡❞ −5 ✐♥ t❤❡ ❢♦r♠✉❧❛✱ t❤❡ ♦♣♣♦s✐t❡ ♦❢ ✇❤❛t✬s ❡①♣❡❝t❡❞✳ ❚❤✐s ♠✐♥✉s s✐❣♥ ❢♦r t❤❡ ✐♥❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡✱ ✉♥❧✐❦❡ t❤❡ ❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡✱ ✐s s♦♠❡t❤✐♥❣ t♦ ✇❛t❝❤ ♦✉t ❢♦r ✇❤❡♥ ❞❡❛❧✐♥❣ ✇✐t❤ tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ ❢♦r♠✉❧❛s ♦❢ ❢✉♥❝t✐♦♥s✳

✸✳✽✳ ❚r❛♥s❢♦r♠✐♥❣ t❤❡ ❛①❡s tr❛♥s❢♦r♠s t❤❡ ♣❧❛♥❡

■♥ t❤❡ ❝♦♥t❡①t ♦❢ ♦✉r st✉❞② ♦❢ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s✱ ✇❤② ❞♦ ✇❡ ❝❛r❡ ❛❜♦✉t tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ♣❧❛♥❡❄ ❇❡❝❛✉s❡ t❤❡✐r ❣r❛♣❤s ❛r❡ ❞r❛✇♥ ♦♥ t❤❡ xy ✲♣❧❛♥❡✿

❲❡ ❛❧s♦ ❦♥♦✇ t❤❛t ❢✉♥❝t✐♦♥s tr❛♥s❢♦r♠ t❤❡ r❡❛❧ ❧✐♥❡ ❛♥❞✱ t❤❡r❡❢♦r❡✱ tr❛♥s❢♦r♠ t❤❡

❛①❡s ♦❢ t❤❡ xy✲♣❧❛♥❡✳

❲❡ ♥❛rr♦✇ t❤✐s ❞♦✇♥✿

◮ ❍♦✇ ❞♦ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ❛①❡s ✕ ❤♦r✐③♦♥t❛❧ ❛♥❞ ✈❡rt✐❝❛❧ ✕ ❛✛❡❝t t❤❡ xy ✲♣❧❛♥❡❄

✸✳✽✳

❚r❛♥s❢♦r♠✐♥❣ t❤❡ ❛①❡s tr❛♥s❢♦r♠s t❤❡ ♣❧❛♥❡

✷✻✺

▲❡t✬s r❡✈✐❡✇ ✇❤❛t ✇❡ ❦♥♦✇✳ ❚❤❡s❡ ❛r❡ t❤❡ t❤r❡❡ ❧✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥s ♦❢

❛♥ ❛①✐s✿

s❤✐❢t✱ ✢✐♣✱ ❛♥❞ str❡t❝❤✿

◆♦✇✱ ❧❡t✬s ✐♠❛❣✐♥❡ t❤❛t tr❛♥s❢♦r♠✐♥❣ ❛♥ ❛①✐s tr❛♥s❢♦r♠s ✕ ✐♥ ✉♥✐s♦♥ ✕ ❛❧❧ t❤❡ ❧✐♥❡s ♦♥ t❤❡ ♣❧❛♥❡ ♣❛r❛❧❧❡❧ t♦ ✐t✳ ❚❤❡ x✲❛①✐s ❛♥❞ ✐ts tr❛♥s❢♦r♠❛t✐♦♥s ❛r❡

❤♦r✐③♦♥t❛❧

❛♥❞ s♦ ❛r❡ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ xy ✲♣❧❛♥❡✿

❚❤❡♥✱ t❤❡ s❤✐❢t ♦❢ t❤❡ x✲❛①✐s ❜❡❝♦♠❡s ❛ ❤♦r✐③♦♥t❛❧ s❤✐❢t ♦❢ t❤❡ xy ✲♣❧❛♥❡✱ t❤❡ ✢✐♣ ♦❢ t❤❡ x✲❛①✐s ❜❡❝♦♠❡s ❛ ❤♦r✐③♦♥t❛❧ ✢✐♣ ♦❢ t❤❡ xy ✲♣❧❛♥❡ ✭❛r♦✉♥❞ t❤❡ ✈❡rt✐❝❛❧ ❛①✐s✮✱ ❛♥❞ t❤❡ str❡t❝❤ ♦❢ t❤❡ x✲❛①✐s ❜❡❝♦♠❡s ❛ ❤♦r✐③♦♥t❛❧ str❡t❝❤ ♦❢ t❤❡ xy ✲♣❧❛♥❡ ✭❛✇❛② ❢r♦♠ t❤❡ ✈❡rt✐❝❛❧ ❛①✐s✮✳ ❋♦r ❛♥ ❛❧❣❡❜r❛✐❝ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡s❡ tr❛♥s❢♦r♠❛t✐♦♥s✱ ✇❡ ❥✉st ❛❞❞ y ✱ t❤❛t r❡♠❛✐♥s ✉♥❝❤❛♥❣❡❞✱ t♦ t❤❡ ❢♦r♠✉❧❛✱ ❛s ❢♦❧❧♦✇s✿

x x

s❤✐❢t ❜② s

−−−−−−−−−→ ✢✐♣

−−−−−−→ str❡t❝❤ ❜② k

x −−−−−−−−−−→

x + s =⇒ (x, y) −x

x·k

=⇒ (x, y)

❤♦r✐③♦♥t❛❧ s❤✐❢t ❜② s

−−−−−−−−−−−−−−−→ ❤♦r✐③♦♥t❛❧ ✢✐♣

−−−−−−−−−−−−→ ❤♦r✐③♦♥t❛❧ str❡t❝❤ ❜② k

=⇒ (x, y) −−−−−−−−−−−−−−−−→

❲❤❛t ❛❜♦✉t t❤❡ y ✲❛①✐s❄ ❚❤❡ y ✲❛①✐s ❛♥❞ ✐ts tr❛♥s❢♦r♠❛t✐♦♥s ❛r❡ t❤❡ xy ✲♣❧❛♥❡✿

✈❡rt✐❝❛❧

(x + s, y) (−x, y) (x · k, y)

❛♥❞ s♦ ❛r❡ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥s ♦❢

✸✳✽✳

❚r❛♥s❢♦r♠✐♥❣ t❤❡ ❛①❡s tr❛♥s❢♦r♠s t❤❡ ♣❧❛♥❡

✷✻✻

❚❤❡ s❤✐❢t ♦❢ t❤❡ y ✲❛①✐s ♣r♦❞✉❝❡s ❛ ✈❡rt✐❝❛❧ s❤✐❢t ♦❢ t❤❡ xy ✲♣❧❛♥❡✱ t❤❡ ✢✐♣ ♦❢ t❤❡ y ✲❛①✐s ♣r♦❞✉❝❡s ❛ ✈❡rt✐❝❛❧ ✢✐♣ ♦❢ t❤❡ xy ✲♣❧❛♥❡ ✭❛r♦✉♥❞ t❤❡ ❤♦r✐③♦♥t❛❧ ❛①✐s✮✱ ❛♥❞ t❤❡ str❡t❝❤ ♦❢ t❤❡ y ✲❛①✐s ♣r♦❞✉❝❡s ❛ ✈❡rt✐❝❛❧ str❡t❝❤ ♦❢ t❤❡ xy ✲♣❧❛♥❡ ✭❛✇❛② ❢r♦♠ t❤❡ ❤♦r✐③♦♥t❛❧ ❛①✐s✮✳ ❋♦r ❛♥ ❛❧❣❡❜r❛✐❝ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡s❡ tr❛♥s❢♦r♠❛t✐♦♥s✱ ✇❡ ❥✉st ❛❞❞ x✱ t❤❛t r❡♠❛✐♥s ✉♥❝❤❛♥❣❡❞✱ t♦ t❤❡ ❢♦r♠✉❧❛✿ y y

s❤✐❢t ❜② s

−−−−−−−−−→ ✢✐♣

−−−−−−→ str❡t❝❤ ❜② k

y −−−−−−−−−−→

y + s =⇒ (x, y) −y

y·k

=⇒ (x, y)

✈❡rt✐❝❛❧ s❤✐❢t ❜② s

−−−−−−−−−−−−−→ ✈❡rt✐❝❛❧ ✢✐♣

−−−−−−−−−−→ ✈❡rt✐❝❛❧ str❡t❝❤ ❜② k

(x, y + s) (x, −y)

=⇒ (x, y) −−−−−−−−−−−−−−−→ (x, y · k)

❍♦r✐③♦♥t❛❧ tr❛♥s❢♦r♠❛t✐♦♥s ❞♦♥✬t ❝❤❛♥❣❡ y ❛♥❞ ✈❡rt✐❝❛❧ ❞♦♥✬t ❝❤❛♥❣❡ x✦ ❊①❛♠♣❧❡ ✸✳✽✳✶✿ tr❛♥s❢♦r♠❛t✐♦♥s ✇✐t❤ ❝♦♠♣✉t❡r ❣r❛♣❤✐❝s

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

❚❤❡ ✜rst t✇♦ r♦✇s s❤♦✇ r✐❣✐❞

♠♦t✐♦♥s✱ ✇❤✐❧❡ t❤❡ ❧❛st ✐s r❡✲s❝❛❧✐♥❣✳

❙♦✱ t❤❡ ❛❧❣❡❜r❛ ♦❢ t❤❡ r❡❛❧ ❧✐♥❡ ❝r❡❛t❡s ❛ ♥❡✇ ❛❧❣❡❜r❛ ♦❢ t❤❡ ❈❛rt❡s✐❛♥ ♣❧❛♥❡✳ ▲❡t✬s r❡✈✐s✐t t❤❡s❡ s✐① tr❛♥s❢♦r✲ ♠❛t✐♦♥s ♦♥❡ ❜② ♦♥❡✳ ❲❡ st❛rt ✇✐t❤ ❛ ✈❡rt✐❝❛❧ s❤✐❢t✳ ❲❡ s❤✐❢t t❤❡ ✇❤♦❧❡ xy ✲♣❧❛♥❡ ❛s ✐❢ ✐t ✐s ♣r✐♥t❡❞ ♦♥ ❛ s❤❡❡t ♦❢ ♣❛♣❡r✳ ❋✉rt❤❡r♠♦r❡✱ t❤❡r❡ ✐s ❛♥♦t❤❡r s❤❡❡t ♦❢ ♣❛♣❡r ✉♥❞❡r♥❡❛t❤ ✉s❡❞ ❢♦r r❡❢❡r❡♥❝❡✳ ■t ✐s t♦ t❤❡ s❡❝♦♥❞ s❤❡❡t t❤❛t ✇❡ tr❛♥s❢❡r t❤❡ r❡s✉❧t✐♥❣ ♣♦✐♥ts✳ ❲❡ t❤❡♥ ✉s❡ ✐ts ❝♦♦r❞✐♥❛t❡ s②st❡♠ t♦ r❡❝♦r❞ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ♥❡✇ ♣♦✐♥t✳ ❋♦r ❡①❛♠♣❧❡✱ ❛ s❤✐❢t ♦❢ 3 ✉♥✐ts ✉♣✇❛r❞ ✐s s❤♦✇♥ ❜❡❧♦✇✿

✸✳✽✳ ❚r❛♥s❢♦r♠✐♥❣ t❤❡ ❛①❡s tr❛♥s❢♦r♠s t❤❡ ♣❧❛♥❡

✷✻✼

❙♦✱ ❛❧❧ ✈❡rt✐❝❛❧ ❧✐♥❡s ❛r❡ s❤✐❢t❡❞ ✉♣ ❜② s✳ ❚❤❡♥✱ t❤❡ ✇❤♦❧❡ ♣❧❛♥❡ ✐s s❤✐❢t❡❞ s > 0 ✉♥✐ts ✉♣✳ ❆ ❣❡♥❡r✐❝ ♣♦✐♥t (x, y) ♠❛❦❡s ❛ st❡♣ ✉♣✴❞♦✇♥ ❜② s ❛♥❞ ❜❡❝♦♠❡s (x, y + s)✳ ❚❤✐s ✐s ❛♥♦t❤❡r ❛❧❣❡❜r❛✐❝ ✇❛② t♦ ♣r❡s❡♥t t❤❡ tr❛♥s❢♦r♠❛t✐♦♥✿ ✉♣ s

(x, y) −−−−−−→ (x, y + s)

■t ✐s ❛s ✐❢ t❤❡ ❛❧❣❡❜r❛ ♦❢ t❤❡ ✢✐♣ ♦❢ t❤❡ y ✲❛①✐s ❣✐✈❡♥ ♣r❡✈✐♦✉s❧②✱ y 7→ y + s✱ ✐s ❝♦♣✐❡❞ ❛♥❞ t❤❡♥ ♣❛✐r❡❞ ✉♣ ✇✐t❤ x✳ ❊①❡r❝✐s❡ ✸✳✽✳✷

❲❤❛t ✐s t❤❡ ❡✛❡❝t ♦❢ t✇♦ ✈❡rt✐❝❛❧ str❡t❝❤❡s ❡①❡❝✉t❡❞ ❝♦♥s❡❝✉t✐✈❡❧②❄ ❲❤❛t ❛❜♦✉t t❤❡ ❤♦r✐③♦♥t❛❧ s❤✐❢t ❄ ❋♦r ❡①❛♠♣❧❡✱ ❛ s❤✐❢t ♦❢ 2 ✉♥✐ts r✐❣❤t ✐s s❤♦✇♥ ❜❡❧♦✇✿

❙♦✱ ❛❧❧ ❤♦r✐③♦♥t❛❧ ❧✐♥❡s ❛r❡ s❤✐❢t❡❞ r✐❣❤t ❜② s✳ ❚❤❡♥✱ t❤❡ ✇❤♦❧❡ ♣❧❛♥❡ ✐s s❤✐❢t❡❞ s > 0 ✉♥✐ts r✐❣❤t✳ ❆ ❣❡♥❡r✐❝ ♣♦✐♥t (x, y) ♠❛❦❡s ❛ st❡♣ r✐❣❤t✴❧❡❢t ❜② s ❛♥❞ ❜❡❝♦♠❡s (x + s, y)✳ ❚❤✐s ✐s ❛♥ ❛❧❣❡❜r❛✐❝ ✇❛② t♦ ♣r❡s❡♥t t❤❡ tr❛♥s❢♦r♠❛t✐♦♥✿ r✐❣❤t s

(x, y) −−−−−−−→ (x + s, y)

■t ✐s ❛s ✐❢ t❤❡ ❛❧❣❡❜r❛ ♦❢ t❤❡ ✢✐♣ ♦❢ t❤❡ x✲❛①✐s ❣✐✈❡♥ ♣r❡✈✐♦✉s❧②✱ x → x + s✱ ✐s ❝♦♣✐❡❞ ❛♥❞ t❤❡♥ ♣❛✐r❡❞ ✉♣ ✇✐t❤ y✳ ❊①❡r❝✐s❡ ✸✳✽✳✸

❲❤❛t ✐s t❤❡ ❡✛❡❝t ♦❢ ❛ ✈❡rt✐❝❛❧ str❡t❝❤ ❛♥❞ ❛ ❤♦r✐③♦♥t❛❧ str❡t❝❤ ❡①❡❝✉t❡❞ ❝♦♥s❡❝✉t✐✈❡❧②❄ ❲❤❛t ✐❢ ✇❡ ❝❤❛♥❣❡ t❤❡ ♦r❞❡r❄ ❚❤❡s❡ s❤✐❢ts ❝❛♥ ❛❧s♦ ❜❡ ❞❡s❝r✐❜❡❞ ❛s ❛ tr❛♥s❧❛t✐♦♥ ❛❧♦♥❣ t❤❡ y ✲❛①✐s ❛♥❞ ❛ tr❛♥s❧❛t✐♦♥ ❛❧♦♥❣ t❤❡ x✲❛①✐s✱ r❡s♣❡❝t✐✈❡❧②✳ ◆♦✇ ❛ ✈❡rt✐❝❛❧ ✢✐♣✳ ❲❡ ❧✐❢t✱ t❤❡♥ ✢✐♣ t❤❡ s❤❡❡t ♦❢ ♣❛♣❡r ✇✐t❤ t❤❡ xy ✲♣❧❛♥❡ ♦♥ ✐t✱ ❛♥❞ ✜♥❛❧❧② ♣❧❛❝❡ ✐t ♦♥ t♦♣ ♦❢ ❛♥♦t❤❡r s✉❝❤ s❤❡❡t s♦ t❤❛t t❤❡ x✲❛①❡s ❛❧✐❣♥✳ ❚❤✐s ✢✐♣ ✐s s❤♦✇♥ ❜❡❧♦✇✿

✸✳✽✳ ❚r❛♥s❢♦r♠✐♥❣ t❤❡ ❛①❡s tr❛♥s❢♦r♠s t❤❡ ♣❧❛♥❡

✷✻✽

❙♦✱ ❛❧❧ ✈❡rt✐❝❛❧ ❧✐♥❡s ❛r❡ ✢✐♣♣❡❞ ❛❜♦✉t t❤❡✐r ♦r✐❣✐♥s✳ ❚❤❡♥✱ t❤❡ ✇❤♦❧❡ ♣❧❛♥❡ ✐s ✢✐♣♣❡❞ ❛❜♦✉t t❤❡ x✲❛①✐s✳ ❆ ❣❡♥❡r✐❝ ♣♦✐♥t (x, y) ❥✉♠♣s ❛❝r♦ss t❤❡ x✲❛①✐s ❛♥❞ ❜❡❝♦♠❡s (x, −y)✳ ❚❤✐s ✐s t❤❡ ❛❧❣❡❜r❛✐❝ ♦✉t❝♦♠❡✿ ✈❡rt✐❝❛❧ ✢✐♣

(x, y) −−−−−−−−−−→ (x, −y) ❊①❡r❝✐s❡ ✸✳✽✳✹

❲❤❛t ✐s t❤❡ ❡✛❡❝t ♦❢ t✇♦ ✈❡rt✐❝❛❧ ✢✐♣s ❡①❡❝✉t❡❞ ❝♦♥s❡❝✉t✐✈❡❧②❄ ❋♦r t❤❡ ❤♦r✐③♦♥t❛❧ ✢✐♣✱ ✇❡ ❧✐❢t✱ t❤❡♥ ✢✐♣ t❤❡ s❤❡❡t ♦❢ ♣❛♣❡r ✇✐t❤ t❤❡ xy ✲♣❧❛♥❡ ♦♥ ✐t✱ ❛♥❞ ✜♥❛❧❧② ♣❧❛❝❡ ✐t ♦♥ t♦♣ ♦❢ ❛♥♦t❤❡r s✉❝❤ s❤❡❡t s♦ t❤❛t t❤❡ y ✲❛①❡s ❛❧✐❣♥✳ ❚❤✐s ✢✐♣ ✐s s❤♦✇♥ ❜❡❧♦✇✿

❙♦✱ ❛❧❧ ❤♦r✐③♦♥t❛❧ ❧✐♥❡s ❛r❡ ✢✐♣♣❡❞ ❛❜♦✉t t❤❡✐r ♦r✐❣✐♥s✳ ❚❤❡♥✱ t❤❡ ✇❤♦❧❡ ♣❧❛♥❡ ✐s ✢✐♣♣❡❞ ❛❜♦✉t t❤❡ y ✲❛①✐s✳ ❆ ❣❡♥❡r✐❝ ♣♦✐♥t (x, y) ❥✉♠♣s ❛❝r♦ss t❤❡ y ✲❛①✐s ❛♥❞ ❜❡❝♦♠❡s (−x, y)✳ ❚❤✐s ✐s t❤❡ ❛❧❣❡❜r❛✐❝ ♦✉t❝♦♠❡✿ ❤♦r✐③♦♥t❛❧ ✢✐♣

(x, y) −−−−−−−−−−−−→ (−x, y) ❊①❡r❝✐s❡ ✸✳✽✳✺

❲❤❛t ✐s t❤❡ ❡✛❡❝t ♦❢ ❛ ✈❡rt✐❝❛❧ ✢✐♣ ❛♥❞ ❛ ❤♦r✐③♦♥t❛❧ ✢✐♣ ❡①❡❝✉t❡❞ ❝♦♥s❡❝✉t✐✈❡❧②❄ ❲❤❛t ✐❢ ✇❡ ❝❤❛♥❣❡ t❤❡ ♦r❞❡r❄ ❚❤❡s❡ ✢✐♣s ❝❛♥ ❛❧s♦ ❜❡ ❞❡s❝r✐❜❡❞ ❛s ❛ ♠✐rr♦r r❡✢❡❝t✐♦♥ ❛❜♦✉t t❤❡ x✲❛①✐s ❛♥❞ ❛ ♠✐rr♦r r❡✢❡❝t✐♦♥ ❛❜♦✉t t❤❡ y ✲❛①✐s✱ r❡s♣❡❝t✐✈❡❧②✳ ◆❡①t✱ ❛ ✈❡rt✐❝❛❧ str❡t❝❤✳ ❚❤❡ ❝♦♦r❞✐♥❛t❡ s②st❡♠ ✐s♥✬t ♦♥ ❛ ♣✐❡❝❡ ♦❢ ♣❛♣❡r ❛♥②♠♦r❡✦ ■t ✐s ♦♥ ❛ r✉❜❜❡r s❤❡❡t✳ ❲❡ ❣r❛❜ ✐t ❜② t❤❡ t♦♣ ❛♥❞ t❤❡ ❜♦tt♦♠ ❛♥❞ ♣✉❧❧ t❤❡♠ ❛♣❛rt ✐♥ s✉❝❤ ❛ ✇❛② t❤❛t t❤❡ x✲❛①✐s ❞♦❡s♥✬t ♠♦✈❡✳ ❋♦r ❡①❛♠♣❧❡✱ ❛ str❡t❝❤ ❜② ❛ ❢❛❝t♦r ♦❢ 2 ✐s s❤♦✇♥ ❜❡❧♦✇✿

❙♦✱ ❛❧❧ ✈❡rt✐❝❛❧ ❧✐♥❡s ❛r❡ str❡t❝❤❡❞ ❜② k > 0 ❛✇❛② ❢r♦♠ t❤❡✐r ♦r✐❣✐♥s✳ ❚❤❡♥✱ t❤❡ ✇❤♦❧❡ ♣❧❛♥❡ ✐s str❡t❝❤❡❞ ❜② ❛ ❢❛❝t♦r k ❛✇❛② ❢r♦♠ t❤❡ x✲❛①✐s✳ ❚❤❡ ❞✐st❛♥❝❡ ♦❢ ❛ ❣❡♥❡r✐❝ ♣♦✐♥t (x, y) ❢r♦♠ t❤❡ x✲❛①✐s ❣r♦✇s ♣r♦♣♦rt✐♦♥❛❧❧② t♦ k ❛♥❞ t❤❡ ♣♦✐♥t ❜❡❝♦♠❡s (x, y · k)✳ ❚❤✐s ✐s t❤❡ ❛❧❣❡❜r❛ t♦ ❞❡s❝r✐❜❡ ✐t✿ ✈❡rt✐❝❛❧ str❡t❝❤ ❜② k

(x, y) −−−−−−−−−−−−−−−→ (x, y · k)

✸✳✽✳

✷✻✾

❚r❛♥s❢♦r♠✐♥❣ t❤❡ ❛①❡s tr❛♥s❢♦r♠s t❤❡ ♣❧❛♥❡

❊✈❡♥ t❤♦✉❣❤ t❤❡ str❡t❝❤ ✐s t❤❡ s❛♠❡ ❢♦r ❛❧❧ s✉❜s❡ts ♦❢ t❤❡ ♣❧❛♥❡✱ t❤❡ ♥❡✇ ❧♦❝❛t✐♦♥ ✇✐❧❧ ✈❛r② ❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ ❧♦❝❛t✐♦♥ ♦❢ t❤❡ s✉❜s❡t r❡❧❛t✐✈❡ t♦ t❤❡ x✲❛①✐s✿

❊①❡r❝✐s❡ ✸✳✽✳✻

❲❤❛t ✐s t❤❡ ❡✛❡❝t ♦❢ ❛ ✈❡rt✐❝❛❧ ✢✐♣ ❛♥❞ ❛ ❤♦r✐③♦♥t❛❧ s❤✐❢t ❡①❡❝✉t❡❞ ❝♦♥s❡❝✉t✐✈❡❧②❄ ❲❤❛t ✐❢ ✇❡ ❝❤❛♥❣❡ t❤❡ ♦r❞❡r❄ ❚❤❡ ❝❛s❡ k = 0 ✐s ✈❡r② s♣❡❝✐❛❧✳ ❆s ❡❛❝❤ ✈❡rt✐❝❛❧ ❧✐♥❡ ❝♦❧❧❛♣s❡s ♦♥ ✐ts x✲✐♥t❡r❝❡♣t✱ t❤❡ ✇❤♦❧❡ ♣❧❛♥❡ ❧❛♥❞s ♦♥ t❤❡ x✲❛①✐s✳ ■t ✐s ❝❛❧❧❡❞ t❤❡ ♣r♦❥❡❝t✐♦♥ ♦♥ t❤❡ x✲❛①✐s ✿

❚❤✐s ✐s t❤❡ ❛❧❣❡❜r❛✿ ♣r♦❥❡❝t✐♦♥

(x, y) −−−−−−−−−−→ (x, 0) ❊①❡r❝✐s❡ ✸✳✽✳✼

❲❤❛t ✐s t❤❡ r❛♥❣❡ ♦❢ t❤❡ ♣r♦❥❡❝t✐♦♥❄ ❊①❡r❝✐s❡ ✸✳✽✳✽

❲❤❛t ✐s t❤❡ ❡✛❡❝t ♦❢ ❛ ✈❡rt✐❝❛❧ ✢✐♣ ❛♥❞ ❛ ♣r♦❥❡❝t✐♦♥ ♦♥ t❤❡ x✲❛①✐s ❡①❡❝✉t❡❞ ❝♦♥s❡❝✉t✐✈❡❧②❄ ❲❤❛t ✐❢ ✇❡ ❝❤❛♥❣❡ t❤❡ ♦r❞❡r❄ ❲❤❛t ❛❜♦✉t ❤♦r✐③♦♥t❛❧ str❡t❝❤ ❄ ❚❤✐s t✐♠❡✱ ✇❡ ❣r❛❜ ✐t ❜② t❤❡ ❧❡❢t ❛♥❞ r✐❣❤t ❡❞❣❡s ♦❢ t❤❡ r✉❜❜❡r s❤❡❡t ❛♥❞ ♣✉❧❧ t❤❡♠ ❛♣❛rt ✐♥ s✉❝❤ ❛ ✇❛② t❤❛t t❤❡ y ✲❛①✐s ❞♦❡s♥✬t ♠♦✈❡✳ ❋♦r ❡①❛♠♣❧❡✱ ❛ str❡t❝❤ ❜② ❛ ❢❛❝t♦r ♦❢ 2 ✐s s❤♦✇♥ ❜❡❧♦✇✿

❙♦✱ ❛❧❧ ❤♦r✐③♦♥t❛❧ ❧✐♥❡s ❛r❡ str❡t❝❤❡❞ ❜② k > 0 ❛✇❛② ❢r♦♠ t❤❡✐r ♦r✐❣✐♥s✳ ❚❤❡♥✱ t❤❡ ✇❤♦❧❡ ♣❧❛♥❡ ✐s str❡t❝❤❡❞ ❜② ❛ ❢❛❝t♦r k ❛✇❛② ❢r♦♠ t❤❡ y ✲❛①✐s✳ ❚❤❡ ❞✐st❛♥❝❡ ♦❢ ❛ ❣❡♥❡r✐❝ ♣♦✐♥t (x, y) ❢r♦♠ t❤❡ y ✲❛①✐s ❣r♦✇s ♣r♦♣♦rt✐♦♥❛❧❧②

✸✳✽✳

✷✼✵

❚r❛♥s❢♦r♠✐♥❣ t❤❡ ❛①❡s tr❛♥s❢♦r♠s t❤❡ ♣❧❛♥❡

t♦ k ❛♥❞ t❤❡ ♣♦✐♥t ❜❡❝♦♠❡s (kx, y)✳ ❚❤✐s ✐s ❛ ✇❛② ❞❡s❝r✐❜❡ ❛ ❤♦r✐③♦♥t❛❧ str❡t❝❤✿ ❤♦r✐③♦♥t❛❧ str❡t❝❤ ❜② k

(x, y) −−−−−−−−−−−−−−−−→ (x · k, y) ❊①❡r❝✐s❡ ✸✳✽✳✾

❲❤❛t ✐s t❤❡ ❡✛❡❝t ♦❢ ❛ ✈❡rt✐❝❛❧ ✢✐♣ ❛♥❞ ❛ ❤♦r✐③♦♥t❛❧ ✢✐♣ ❡①❡❝✉t❡❞ ❝♦♥s❡❝✉t✐✈❡❧②❄ ❲❤❛t ✐❢ ✇❡ ❝❤❛♥❣❡ t❤❡ ♦r❞❡r❄ ❚❤❡ ❝❛s❡ k = 0 ✐s ✈❡r② s♣❡❝✐❛❧✳ ❆s ❡❛❝❤ ❤♦r✐③♦♥t❛❧ ❧✐♥❡ ❝♦❧❧❛♣s❡s ♦♥ ✐ts y ✲✐♥t❡r❝❡♣t✱ t❤❡ ✇❤♦❧❡ ♣❧❛♥❡ ❧❛♥❞s ♦♥ t❤❡ y ✲❛①✐s✳ ■t ✐s ❝❛❧❧❡❞ t❤❡ ♣r♦❥❡❝t✐♦♥ ♦♥ t❤❡ y ✲❛①✐s ✿

❚❤✐s ✐s t❤❡ ❛❧❣❡❜r❛✿ ♣r♦❥❡❝t✐♦♥

(x, y) −−−−−−−−−−→ (0, y) ❊①❡r❝✐s❡ ✸✳✽✳✶✵

❲❤❛t ✐s t❤❡ ❡✛❡❝t ♦❢ ❛ ✈❡rt✐❝❛❧ ♣r♦❥❡❝t✐♦♥ ❛♥❞ ❛ ❤♦r✐③♦♥t❛❧ ♣r♦❥❡❝t✐♦♥ ❡①❡❝✉t❡❞ ❝♦♥s❡❝✉t✐✈❡❧②❄ ❚❤❡s❡ str❡t❝❤❡s ❝❛♥ ❛❧s♦ ❜❡ ❞❡s❝r✐❜❡❞ ❛s ❛ ✉♥✐❢♦r♠ ❞❡❢♦r♠❛t✐♦♥ ❛✇❛② ❢r♦♠ t❤❡ y ✲❛①✐s ❛♥❞ ❛ ✉♥✐❢♦r♠ ❞❡❢♦r✲ y ✲❛①✐s✱ r❡s♣❡❝t✐✈❡❧②✳

♠❛t✐♦♥ ❛✇❛② ❢r♦♠ t❤❡

❚❤❡s❡ ❛r❡ ♦✉r s✐① ❜❛s✐❝ tr❛♥s❢♦r♠❛t✐♦♥s✿

❚❤❡ ❛❧❣❡❜r❛ ❜❡❧♦✇ r❡✢❡❝ts t❤❡ ❣❡♦♠❡tr② ❛❜♦✈❡✳ ❚❤❡♦r❡♠ ✸✳✽✳✶✶✿ ❋♦r♠✉❧❛s ♦❢ ❚r❛♥s❢♦r♠❛t✐♦♥s ♦❢ P❧❛♥❡

❚❤❡ ❢♦❧❧♦✇✐♥❣ tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ♣❧❛♥❡ ❛r❡ ❣✐✈❡♥ ❜② t❤❡✐r ❢♦r♠✉❧❛s✿ ✈❡rt✐❝❛❧ s❤✐❢t✿ ( x , y ) 7 ( x , y+k ) →

✢✐♣✿ ( x , y ) 7 ( x , y · (−1) ) →

str❡t❝❤✿ ( x , y ) 7 ( x , y·k ) →

✸✳✽✳

❚r❛♥s❢♦r♠✐♥❣ t❤❡ ❛①❡s tr❛♥s❢♦r♠s t❤❡ ♣❧❛♥❡ ❤♦r✐③♦♥t❛❧ s❤✐❢t✿

✷✼✶

✢✐♣✿

( x , y ) 7→ ( x + k , y )

str❡t❝❤✿

( x , y ) 7→ ( x · (−1) , y )

■t ✇✐❧❧ ❜❡ ✐♠♣♦rt❛♥t ❧❛t❡r t❤❛t t❤❡s❡ tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ♣❧❛♥❡ ❛r❡

( x , y ) 7→ ( x · k , y )

❢✉♥❝t✐♦♥s ♦❢ t❤❡ ♣❧❛♥❡ t♦ ✐ts❡❧❢✱ ✐✳❡✳✱

F : R2 → R2 . ❊①❡r❝✐s❡ ✸✳✽✳✶✷

❲❤❛t ❛r❡ t❤❡ ✐♠❛❣❡s ♦❢ t❤❡s❡ s✐① ❢✉♥❝t✐♦♥s❄ ❲❤❛t ❛❜♦✉t t❤❡ ♣r♦❥❡❝t✐♦♥s❄ ❋♦r ♥♦✇✱ ❡❛❝❤ ♦❢ t❤❡s❡ s✐① ♦♣❡r❛t✐♦♥s ✐s ❧✐♠✐t❡❞ t♦ ♦♥❡ ♦❢ t❤❡ t✇♦ ❞✐r❡❝t✐♦♥s✿ ❛❧♦♥❣ t❤❡ x✲❛①✐s ♦r ❛❧♦♥❣ t❤❡ y ✲❛①✐s✳ ❲❡ ❝♦♠❜✐♥❡ t❤❡♠ ❛s ❝♦♠♣♦s✐t✐♦♥s✳ ❋♦r ❡①❛♠♣❧❡✱ ♣♦✐♥t →

str❡t❝❤ ✈❡rt✐❝❛❧❧② ❜② k

(x, y) →

♠✉❧t✐♣❧② y ❜② k

→

♣♦✐♥t

→

→ (x, yk) →

✢✐♣ ❤♦r✐③♦♥t❛❧❧② ♠✉❧t✐♣❧② x ❜② (−1)

→

→ (−x, yk)

❲❡ ♣r♦❞✉❝❡ ❛ ✈❛r✐❡t② ♦❢ r❡s✉❧ts✿

❊①❡r❝✐s❡ ✸✳✽✳✶✸

❊①❡❝✉t❡ ✕ ❜♦t❤ ❣❡♦♠❡tr✐❝❛❧❧② ❛♥❞ ❛❧❣❡❜r❛✐❝❛❧❧② ✕ t❤❡ ❢♦❧❧♦✇✐♥❣ tr❛♥s❢♦r♠❛t✐♦♥s✿ ✶✳ ❚r❛♥s❧❛t❡ ✉♣ ❜② 2✱ t❤❡♥ r❡✢❡❝t ❛❜♦✉t t❤❡ x✲❛①✐s✱ t❤❡♥ tr❛♥s❧❛t❡ ❧❡❢t ❜② 3✳ ✷✳ P✉❧❧ ❛✇❛② ❢r♦♠ t❤❡ y ✲❛①✐s ❜② ❛ ❢❛❝t♦r ♦❢ 3✱ t❤❡♥ ♣✉❧❧ t♦✇❛r❞ t❤❡ x✲❛①✐s ❜② ❛ ❢❛❝t♦r ♦❢ 2✳ ❊①❡r❝✐s❡ ✸✳✽✳✶✹

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

♣♦✐♥t

✸✳✽✳

❚r❛♥s❢♦r♠✐♥❣ t❤❡ ❛①❡s tr❛♥s❢♦r♠s t❤❡ ♣❧❛♥❡

✷✼✷

❊①❡r❝✐s❡ ✸✳✽✳✶✺

❉❡s❝r✐❜❡ ✕ ❜♦t❤ ❣❡♦♠❡tr✐❝❛❧❧② ❛♥❞ ❛❧❣❡❜r❛✐❝❛❧❧② ✕ ❛ tr❛♥s❢♦r♠❛t✐♦♥ t❤❛t ♠❛❦❡s ❛ 1 × 1 sq✉❛r❡ ✐♥t♦ ❛ 2 × 1 r❡❝t❛♥❣❧❡✳ ❊①❡r❝✐s❡ ✸✳✽✳✶✻

❲❤❛t tr❛♥s❢♦r♠❛t✐♦♥s ✐♥❝r❡❛s❡✴❞❡❝r❡❛s❡ s❧♦♣❡s ♦❢ ❧✐♥❡s❄ ❊①❡r❝✐s❡ ✸✳✽✳✶✼

P♦✐♥t ♦✉t t❤❡ ✐♥✈❡rs❡s ♦❢ ❡❛❝❤ ♦❢ t❤❡ s✐① tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ♣❧❛♥❡ ♦♥ t❤❡ ❧✐st✳ ❊①❡r❝✐s❡ ✸✳✽✳✶✽

❍❛s t❤✐s ♣❛r❛❜♦❧❛ ❜❡❡♥ s❤r✉♥❦ ✈❡rt✐❝❛❧❧② ♦r str❡t❝❤❡❞ ❤♦r✐③♦♥t❛❧❧②❄

❊①❛♠♣❧❡ ✸✳✽✳✶✾✿ ♠♦r❡ tr❛♥s❢♦r♠❛t✐♦♥s ❛s s②♠♠❡tr✐❡s

❘❡❝❛❧❧ s♦♠❡ ♦❢ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ♣❧❛♥❡ ✇❡ s❛✇ ✐♥ ❈❤❛♣t❡r ✷ ✇❤❡♥ ✇❡ ❞✐s❝✉ss❡❞ s②♠♠❡tr②✳ ❋♦r ❡①❛♠♣❧❡✱ t❤❡ ❢❛❝t t❤❛t t❤❡ ♣❛r❛❜♦❧❛✬s ❧❡❢t ❜r❛♥❝❤ ✐s ❛ ♠✐rr♦r ✐♠❛❣❡ ♦❢ ✐ts r✐❣❤t ❜r❛♥❝❤ ✐s r❡✈❡❛❧❡❞ ✈✐❛ ❛ ❤♦r✐③♦♥t❛❧ ✢✐♣✿ (x, y) 7→ (−x, y) ●❡♥❡r❛❧❧②✱ ❛ s✉❜s❡t A ♦❢ t❤❡ ♣❧❛♥❡ ✐s

♠✐rr♦r s②♠♠❡tr✐❝ ❛❜♦✉t t❤❡ y✲❛①✐s ✇❤❡♥ ✇❡ ❤❛✈❡✿

(x, y) ❜❡❧♦♥❣s t♦ A =⇒ (−x, y) ❜❡❧♦♥❣s t♦ A . ■t ✐s ❛ r❡s✉❧t ♦❢ ❛ ❢♦❧❞✿

▲❡t✬s ✜♥❞ ❛ ❢♦r♠✉❧❛ ❢♦r ❛ r♦t❛t✐♦♥ 180 ❞❡❣r❡❡s ❛r♦✉♥❞ t❤❡ ♦r✐❣✐♥✱ ❛❧s♦ ❝❛❧❧❡❞ t❤❡

❝❡♥tr❛❧ s②♠♠❡tr② ✿

❲❡ ❛❝❤✐❡✈❡ t❤❡ s❛♠❡ ❡✛❡❝t ✐❢ ✇❡ ✐♥st❡❛❞ ✢✐♣ t❤❡ ♣❧❛♥❡ ❛❜♦✉t t❤❡ y ✲❛①✐s ❛♥❞ t❤❡♥ ❛❜♦✉t t❤❡ x✲❛①✐s ✭♦r

✸✳✾✳

❈❤❛♥❣✐♥❣ ❛ ✈❛r✐❛❜❧❡ tr❛♥s❢♦r♠s t❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥

✷✼✸

✈✐❝❡ ✈❡rs❛✮✿

(x, y) 7→ (−x, y) 7→ (−x, −y)

❚❤❡♥ ❛ s✉❜s❡t

A

❝❡♥tr❛❧❧② s②♠♠❡tr✐❝ ✇❤❡♥ ✇❡ ❤❛✈❡✿

♦❢ t❤❡ ♣❧❛♥❡ ✐s

(x, y)

❜❡❧♦♥❣s t♦

❍♦✇❡✈❡r✱ s♦♠❡ tr❛♥s❢♦r♠❛t✐♦♥s

❜❡❧♦♥❣s t♦

A.

❝❛♥♥♦t ❜❡ ❞❡❝♦♠♣♦s❡❞ ✐♥t♦ ❛ ❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤♦s❡ s✐① ❜❛s✐❝ tr❛♥s❢♦r✲

♠❛t✐♦♥s✦ ❍❡r❡ ✐s✱ ❢♦r ❡①❛♠♣❧❡✱ ❛

❆♥♦t❤❡r ✐s ❛ ✢✐♣ ❛❜♦✉t t❤❡ ❧✐♥❡

A =⇒ (−x, −y)

90✲❞❡❣r❡❡

x=y

r♦t❛t✐♦♥✿

t❤❛t ❛♣♣❡❛r❡❞ ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥✿

■t ✐s ❣✐✈❡♥ ❜②

(x, y) 7→ (y, x) . ✭❚r❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ♣❧❛♥❡ ❛r❡ ❞✐s❝✉ss❡❞ ✐♥ ❈❤❛♣t❡r ✹❍❉✲✸ ❛♥❞ ❈❤❛♣t❡r ✺❉❊✲✷✳✮

❊①❡r❝✐s❡ ✸✳✽✳✷✵

❋✐♥❞ ❛ ❢♦r♠✉❧❛ ❢♦r ❛

90✲❞❡❣r❡❡

❝❧♦❝❦✇✐s❡ r♦t❛t✐♦♥✳

❊①❡r❝✐s❡ ✸✳✽✳✷✶

❲❤❛t ❛r❡ t❤❡ ✐♥✈❡rs❡s ♦❢ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥s ♣r❡s❡♥t❡❞ ✐♥ t❤✐s s❡❝t✐♦♥❄

✸✳✾✳ ❈❤❛♥❣✐♥❣ ❛ ✈❛r✐❛❜❧❡ tr❛♥s❢♦r♠s t❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥

❈♦♥s✐❞❡r t❤❡s❡ t❤r❡❡ ❢❛❝ts✳ ❋✐rst✱ ❛ ❢✉♥❝t✐♦♥ ✐s r❡♣r❡s❡♥t❡❞ ❜② ✐ts ❣r❛♣❤✱ ✇❤✐❝❤ ✐s ❛ ❝❡rt❛✐♥ s✉❜s❡t ♦❢ t❤❡

xy ✲♣❧❛♥❡✳

❙❡❝♦♥❞✱ ❛ ❢✉♥❝t✐♦♥ ✐s s❡❡♥ ❛s ❛ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ t❤❡ r❡❛❧ ♥✉♠❜❡r ❧✐♥❡✳

♣r♦♠✐♥❡♥t ♥✉♠❜❡r ❧✐♥❡s ♦♥ t❤❡ t❤❡

xy ✲♣❧❛♥❡

xy ✲♣❧❛♥❡✿

t❤❡

x✲❛①✐s ❛♥❞ t❤❡ y ✲❛①✐s✦ x ♦r y ✳

❚❤✐r❞✱ t❤❡r❡ ❛r❡ t✇♦

❙♦✱ t❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥ ✏❞r❛✇♥✑ ♦♥

✐s ❜❡✐♥❣ tr❛♥s❢♦r♠❡❞ ❜② ❢✉♥❝t✐♦♥s ♦❢

y = f (x)✳ s✇✐t❝❤ ❢r♦♠ x✱

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

z✱

❧♦❝❛t✐♦♥ ✐♥ ❢❡❡t✿

❙✉♣♣♦s❡ ✇❡ ❢♦r♠ t❤❡ ❝♦♠♣♦s✐t✐♦♥s ♦❢ t✐♠❡ ✐♥ ♠✐♥✉t❡s✱ ❛♥❞

y✱

f

✇✐t❤ ♦t❤❡r

❧♦❝❛t✐♦♥ ✐♥ ✐♥❝❤❡s✱ t♦

t✱

t✐♠❡

✸✳✾✳

❈❤❛♥❣✐♥❣ ❛ ✈❛r✐❛❜❧❡ tr❛♥s❢♦r♠s t❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥

✷✼✹

❲❡ ❦♥♦✇ ❢r♦♠ t❤❡ ❧❛st s❡❝t✐♦♥ t❤❛t ✇❡ ❢❛❝❡ tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ t✇♦ ❛①❡s✳ ❊✈❡♥ t❤♦✉❣❤ ✇❡ ❝❛♥ ❢♦❧❧♦✇ t❤❡ ❛rr♦✇s ❛♥❞ ✜♥❞ t❤❡ ✈❛❧✉❡s ♦❢ t❤✐s ❝♦♠♣♦s✐t✐♦♥✱ ✇❤❛t ❞♦❡s ✐ts

❣r❛♣❤ ❧♦♦❦ ❧✐❦❡❄

❲❡ ❛s❦ t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ q✉❡st✐♦♥s✿ ✶✳ ❲❤❛t ✇✐❧❧ ❤❛♣♣❡♥ t♦ t❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥ ✷✳ ❲❤❛t ✇✐❧❧ ❤❛♣♣❡♥ t♦ t❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥

y = f (x) ✐❢ t❤❡ x✲❛①✐s ✐s tr❛♥s❢♦r♠❡❞ ❜② ❛♥♦t❤❡r y = f (x) ✐❢ t❤❡ y ✲❛①✐s ✐s tr❛♥s❢♦r♠❡❞ ❜② ❛♥♦t❤❡r

❆ s❤♦rt ❛♥s✇❡r t♦ ❜♦t❤ ✐s✿ ❚❤❡ ❣r❛♣❤ ✇✐❧❧ tr❛♥s❢♦r♠ ✐♥t♦ t❤❛t ♦❢ t❤❡ ❚❤❡

♦r❞❡r✱ ❤♦✇❡✈❡r✱ ✐s ❞✐✛❡r❡♥t✳ 1. t →

■t✬s ✏❜❡❢♦r❡

g

f✑

✈s✳ ✏❛❢t❡r

→ x →

x →

2.

f f

f ✑✿

❝♦♠♣♦s✐t✐♦♥

→ y

→ y →

♦❢

f

❢✉♥❝t✐♦♥❄ ❢✉♥❝t✐♦♥❄

✇✐t❤ t❤✐s ❢✉♥❝t✐♦♥✳

→ z

h

❚❤✐s ✐s ✇❤② t❤❡ ❛♥s✇❡rs t♦ t❤❡ t✇♦ q✉❡st✐♦♥s ✇✐❧❧ ❜❡ ❞✐✛❡r❡♥t✳

❲❛r♥✐♥❣✦ ●r❛♣❤s

❛r❡♥✬t

❢✉♥❝t✐♦♥s

❛♥❞

❢✉♥❝t✐♦♥s

❛r❡♥✬t

❣r❛♣❤s❀ t❤✐s ✐s ❛❧❧ ❛❜♦✉t ✈✐s✉❛❧✐③❛t✐♦♥✳

■❢ t❤❡ ♣r♦❜❧❡♠ ✐s tr✉❧② ❛❜♦✉t ✉♥✐ts ❝♦♥✈❡rs✐♦♥✱ ✇❡✬❞ ❜❡tt❡r ❤❛✈❡ t❤❡ ✈❛r✐❛❜❧❡s ❛♥❞ t❤❡ ❛①❡s ♥❛♠❡❞ ❞✐✛❡r❡♥t❧②✳ ❍♦✇❡✈❡r✱ ✐♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ ♣✉rs✉❡ t❤❡ ✐❞❡❛ ♦❢ ♣r♦❞✉❝✐♥❣ ❛ ♥❡✇ ❢✉♥❝t✐♦♥ ♦♥ t❤❡

s❛♠❡ xy✲♣❧❛♥❡✳

❋♦r t❤❡ s✐① ❜❛s✐❝ tr❛♥s❢♦r♠❛t✐♦♥s ❞✐s❝✉ss❡❞ ♣r❡✈✐♦✉s❧②✱ t❤❡r❡ ✇✐❧❧ ❜❡ s✐① r✉❧❡s ❣♦✈❡r♥✐♥❣ t❤❡ ❛❧❣❡❜r❛ ♦❢ t❤❡ ❢✉♥❝t✐♦♥s ❛✛❡❝t❡❞ ❜② t❤❡♠✳ ❲❡ st❛rt ✇✐t❤ t❤❡

✈❡rt✐❝❛❧ s❤✐❢t✳

❙✐♥❝❡ t❤❡ ❣r❛♣❤ ✐s ❞r❛✇♥ ♦♥ t❤❡ s❛♠❡ ♣✐❡❝❡ ♦❢ ♣❛♣❡r✱ ✐t ✐s s❤✐❢t❡❞ ❡①❛❝t❧②

t❤❡ s❛♠❡ ✇❛②✿

❲❡ s❡❡ t❤❡ ♦r✐❣✐♥❛❧ ❢✉♥❝t✐♦♥

y = f (x)

y = F (x) ♦♥ t❤❡ ❢❛r r✐❣❤t✳ ❲❤❛t y = x2 ❛♥❞ s❤✐❢t ✉♣ ❜② 5✱ ✇❡ ❤❛✈❡

♦♥ t❤❡ ❢❛r ❧❡❢t ❛♥❞ t❤❡ ♥❡✇ ❢✉♥❝t✐♦♥

✐s t❤❡ ❢♦r♠✉❧❛ ♦❢ t❤❡ ♥❡✇ ❣r❛♣❤❄ ■❢ ✇❡ t❛❦❡ ❛ ♣♦✐♥t (x, y) ♦♥ t❤❡ ❣r❛♣❤ ♦❢ 2 ♥❡✇ ♣♦✐♥t (x, y + 5)✳ ■t ❧✐❡s ♦♥ t❤❡ ❣r❛♣❤ ♦❢ y = x + 5✳

✸✳✾✳

❈❤❛♥❣✐♥❣ ❛ ✈❛r✐❛❜❧❡ tr❛♥s❢♦r♠s t❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥

✷✼✺

❚❤❡♦r❡♠ ✸✳✾✳✶✿ ❱❡rt✐❝❛❧ ❙❤✐❢t ■❢ t❤❡ ❣r❛♣❤

y = F (x)

✐s t❤❡ ❣r❛♣❤ ♦❢

y = f (x)

s❤✐❢t❡❞

s

✉♥✐ts ✉♣✱ t❤❡♥

F (x) = f (x) + s ❛♥❞ ✈✐❝❡ ✈❡rs❛✳

Pr♦♦❢✳

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

• •

❆ ♣♦✐♥t

(x, y)

❧✐❡s ♦♥ t❤❡ ❣r❛♣❤ ♦❢

❚❤❡ s❤✐❢t❡❞ ♣♦✐♥t

(x, y + s)

f✱

❤❡♥❝❡

f (x) = y ✳ F ✱ ❤❡♥❝❡ F (x) = y + s✳ y = f (x) ❛♥❞ ❝♦♥❝❧✉❞❡✿

❧✐❡s ♦♥ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ♥❡✇ ❢✉♥❝t✐♦♥

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

F (x) = f (x) + s . ❇② ✏✈✐❝❡ ✈❡rs❛✑✱ ✇❡ ♠❡❛♥✱ ♦❢ ❝♦✉rs❡✱ t❤❡

◮

■❢ ❛ ❢✉♥❝t✐♦♥

g

s❛t✐s✜❡s

❝♦♥✈❡rs❡ ✿

F (x) = f (x) + s✱

t❤❡♥ ✐ts ❣r❛♣❤ ✐s t❤❡ ❣r❛♣❤ ♦❢

f

s❤✐❢t❡❞

s

✉♥✐ts ✉♣✳

❲❡ ❝❛♥ s❡❡ ❤♦✇ t❤❡ ❣r❡❡♥ ♣♦✐♥ts ❛r❡ tr❛♥s❢❡rr❡❞ t♦ t❤❡ r❡❞ ♦♥❡s ♦♥❡ ❜② ♦♥❡✿

❚❤❡ ❝❤❛♥❣❡ ✐s t❤❡ s❛♠❡ ❢♦r ❛❧❧ ♣♦✐♥ts✳ ❚❤❡ ♠♦st ✐♠♣♦rt❛♥t ❝♦♥❝❧✉s✐♦♥ ✐s t❤❛t t❤❡ ♥❡✇ ❢✉♥❝t✐♦♥ ❤❛s t❤❡ ❡①❛❝t s❛♠❡

s❤❛♣❡ ♦❢ t❤❡ ❣r❛♣❤ ❛s t❤❡ ♦❧❞ ♦♥❡✳

❊①❛♠♣❧❡ ✸✳✾✳✷✿ ❢♦r♠✉❧❛ ❢♦r ✈❡rt✐❝❛❧ s❤✐❢t

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

♦r✐❣✐♥❛❧✿ s❤✐❢t❡❞ ✉♣ ❜②

√ x2 + x f (x) = 7   2x −√ x + x 3 : F (x) = +3 x−7

❲❡ ❥✉st ♣✉t t❤❡ ♦r✐❣✐♥❛❧ ✐♥s✐❞❡ ❧❛r❣❡ ♣❛r❡♥t❤❡s❡s ❛♥❞ t❤❡♥ ❛❞❞ ✏ +3✑ ❛❢t❡r✳

◆❡①t ✐s t❤❡

❤♦r✐③♦♥t❛❧ s❤✐❢t ✿

✸✳✾✳

❈❤❛♥❣✐♥❣ ❛ ✈❛r✐❛❜❧❡ tr❛♥s❢♦r♠s t❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥

✷✼✻

❊①❛♠♣❧❡ ✸✳✾✳✸✿ ♣❛r❛❜♦❧❛ s❤✐❢t❡❞

❚♦ ❣✉❡ss t❤❡ ❢♦r♠✉❧❛✱ ❧❡t✬s t❛❦❡ f (x) = x2 ❛♥❞ s❤✐❢t 2 ✉♥✐ts t♦ t❤❡ ❧❡❢t ❛♥❞ t♦ t❤❡ r✐❣❤t✿ f (x + 2) = (x + 2)2 ❛♥❞ f (x − 2) = (x − 2)2 .

❚❤❡s❡ ❛r❡ t❤❡ ❣r❛♣❤s ♦❢ t❤❡ t❤r❡❡ ❢✉♥❝t✐♦♥s✿

❚♦ ❝♦♥✜r♠ t❤❡ ♠❛t❝❤✱ ✇❤❛t ❛r❡ t❤❡ x✲✐♥t❡r❝❡♣ts ♦❢ t❤❡s❡ t✇♦ ♥❡✇ ❢✉♥❝t✐♦♥s❄ ❙❡t ❡✐t❤❡r ❡q✉❛❧ t♦ 0 ❛♥❞ s♦❧✈❡✿ 2 2 (x + 2) = 0 (x − 2) = 0 x+2=0 x−2=0 x = −2 x = 2

❧❡❢t s❤✐❢t

r✐❣❤t s❤✐❢t

❚❤❡ s❤✐❢t ✐s ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦♣♣♦s✐t❡ t♦ t❤❡ s✐❣♥ ♦❢ ✇❤❛t ✐s ❛❞❞❡❞✦ ❚❤❡♦r❡♠ ✸✳✾✳✹✿ ❍♦r✐③♦♥t❛❧ ❙❤✐❢t ■❢ t❤❡ ❣r❛♣❤ ♦❢

y = F (x)

✐s t❤❡ ❣r❛♣❤ ♦❢

y = f (x)

s❤✐❢t❡❞

s

✉♥✐ts t♦ t❤❡ r✐❣❤t✱

t❤❡♥

F (x) = f (x − s) ❛♥❞ ✈✐❝❡ ✈❡rs❛✳

Pr♦♦❢✳

■♥ ♦r❞❡r t♦ ✜♥❞ t❤❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ ♥❡✇ ❢✉♥❝t✐♦♥✱ ✇❡ ❝♦♠♣❛r❡ t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ ❢❛❝ts ❛❜♦✉t t❤❡ t✇♦ ❣r❛♣❤s✿ • ❆ ♣♦✐♥t (x, y) ❧✐❡s ♦♥ t❤❡ ❣r❛♣❤ ♦❢ f ✱ t❤❡♥ f (x) = y ✳ • ❚❤❡ s❤✐❢t❡❞ ♣♦✐♥t (x + s, y) ❧✐❡s ♦♥ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ♥❡✇ ❢✉♥❝t✐♦♥ F ✱ ❤❡♥❝❡ F (x + s) = y ✳ ❙✉❜st✐t✉t❡ y = f (x) ❛♥❞ ❝♦♥❝❧✉❞❡✿ F (x + s) = f (x) .

❚❤❡r❡❢♦r❡✱ t❤❡ ♥❡✇ ❢✉♥❝t✐♦♥ ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✿ F (x) = f (x − s) .

❲❡ ❝❛♥ s❡❡ ❤♦✇ t❤❡ ❣r❡❡♥ ♣♦✐♥ts ❛r❡ tr❛♥s❢❡rr❡❞ t♦ t❤❡ r❡❞ ♦♥❡s ♦♥❡ ❜② ♦♥❡✿

❚❤❡ ❝❤❛♥❣❡ ✐s t❤❡ s❛♠❡ ❢♦r ❛❧❧ ♣♦✐♥ts✳ ❖♥❝❡ ❛❣❛✐♥✱ t❤❡ ♥❡✇ ❢✉♥❝t✐♦♥ ❤❛s t❤❡ ❡①❛❝t s❛♠❡ s❤❛♣❡ ♦❢ t❤❡ ❣r❛♣❤ ❛s t❤❡ ♦❧❞ ♦♥❡✳

✸✳✾✳

❈❤❛♥❣✐♥❣ ❛ ✈❛r✐❛❜❧❡ tr❛♥s❢♦r♠s t❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥

✷✼✼

❊①❛♠♣❧❡ ✸✳✾✳✺✿ ❢♦r♠✉❧❛ ❢♦r ❤♦r✐③♦♥t❛❧ s❤✐❢t

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

♦r✐❣✐♥❛❧✿ s❤✐❢t❡❞ r✐❣❤t ❜② ✸✿

❲❡ ❥✉st r❡♣❧❛❝❡ ❡❛❝❤

x

√ x2 + x f (x) = x −p 7 (x−3)2 + (x−3) F (x) = (x−3) − 7

✐♥ t❤❡ ♦r✐❣✐♥❛❧ ✇✐t❤ ✏ (x−3)✑✳

❊①❛♠♣❧❡ ✸✳✾✳✻✿ t✇♦ s❤✐❢ts ❝♦♠♣❛r❡❞

❘❡❝❛❧❧ t❤❡ ❛❧❣♦r✐t❤♠✐❝ ✐♥t❡r♣r❡t❛t✐♦♥ ♦❢ t❤❡s❡ t✇♦ t②♣❡s ♦❢ s❤✐❢ts✿ ✈❡rt✐❝❛❧ s❤✐❢t✱ ✉♣

x →

3:

❤♦r✐③♦♥t❛❧ s❤✐❢t✱ r✐❣❤t

3: x →

s✉❜tr❛❝t

→ x →

3

f f

→ y →

❛❞❞

3

→ y

→ y

❊①❛♠♣❧❡ ✸✳✾✳✼✿ t✇♦ s❤✐❢ts ❝♦♠❜✐♥❡❞

❲❤❛t ❛❜♦✉t ✏❝♦♠❜✐♥❛t✐♦♥✑ ♦❢ t❤❡s❡ t✇♦ t②♣❡s ♦❢ tr❛♥s❢♦r♠❛t✐♦♥s❄ ▲❡t

Q(x) = (x + 2)2 + 4 . ❲❤❛t ✐s t❤❡ ❣r❛♣❤❄ ❲❡ ❝❛♥ ♣❧♦t ✐t ♣♦✐♥t ❜② ♣♦✐♥t✱ ❜✉t ❛s ❛ ♣r❡✈✐❡✇ ♦❢ t❤✐♥❣s t♦ ❝♦♠❡✱ ❧❡t✬s ❞❡❝♦♠♣♦s❡ t❤❡ ❢✉♥❝t✐♦♥✿

x → ❚❤✐s ✐s ❛

2✲✉♥✐t

x+2

→ u →

❧❡❢t✇❛r❞ s❤✐❢t ❢♦❧❧♦✇❡❞ ❜② ❛

u2

4✲✉♥✐t

→ z →

z+4

→ y

✉♣✇❛r❞ s❤✐❢t✿

❊①❡r❝✐s❡ ✸✳✾✳✽

❲❤❛t ❤❛♣♣❡♥s t♦ t❤❡

x✲

❛♥❞

y ✲✐♥t❡r❝❡♣ts

♦❢ ❛ ❢✉♥❝t✐♦♥ ✉♥❞❡r ✈❡rt✐❝❛❧ ❛♥❞ ❤♦r✐③♦♥t❛❧ s❤✐❢ts❄

❊①❡r❝✐s❡ ✸✳✾✳✾

❲❤❛t ❤❛♣♣❡♥s t♦ t❤❡ ❞♦♠❛✐♥ ❛♥❞ t❤❡ r❛♥❣❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ✉♥❞❡r ✈❡rt✐❝❛❧ ❛♥❞ ❤♦r✐③♦♥t❛❧ s❤✐❢ts❄

◆❡①t ✐s t❤❡

✈❡rt✐❝❛❧ ✢✐♣ ✿

✸✳✾✳

❈❤❛♥❣✐♥❣ ❛ ✈❛r✐❛❜❧❡ tr❛♥s❢♦r♠s t❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥ (x, y) ♦♥ y = −x2 ✳

■❢ ✇❡ t❛❦❡ ❛ ♣♦✐♥t ♦♥ t❤❡ ❣r❛♣❤ ♦❢

t❤❡ ❣r❛♣❤ ♦❢

y = x2

✷✼✽

❛♥❞ ✢✐♣ ❛❜♦✉t t❤❡

x✲❛①✐s✱

✇❡ ❤❛✈❡ ♥❡✇ ♣♦✐♥t

(x, −y)✳

■t ❧✐❡s

❚❤❡♦r❡♠ ✸✳✾✳✶✵✿ ❱❡rt✐❝❛❧ ❋❧✐♣ ■❢ t❤❡ ❣r❛♣❤ ♦❢ y = F (x) ✐s t❤❡ ❣r❛♣❤ ♦❢ y = f (x) ✢✐♣♣❡❞ ✈❡rt✐❝❛❧❧②✱ t❤❡♥ F (x) = −f (x)

❛♥❞ ✈✐❝❡ ✈❡rs❛✳ ❲❡ ❝❛♥ s❡❡ ❤♦✇ t❤❡ ❣r❡❡♥ ♣♦✐♥ts ❛r❡ tr❛♥s❢❡rr❡❞ t♦ t❤❡ r❡❞ ♦♥❡s ♦♥❡ ❜② ♦♥❡✿

❚❤❡ ❞✐st❛♥❝❡ ❢r♦♠ ❡✈❡r② ♥❡✇ ♣♦✐♥t t♦ t❤❡

x✲❛①✐s

✐s t❤❛t s❛♠❡ ❛s t❤❛t ♦❢ t❤❡ ♦r✐❣✐♥❛❧ ♣♦✐♥t ❜✉t ♦♥ t❤❡ ♦t❤❡r

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

❊①❡r❝✐s❡ ✸✳✾✳✶✶ Pr♦✈❡ t❤❡ t❤❡♦r❡♠✳

❊①❛♠♣❧❡ ✸✳✾✳✶✷✿ ❢♦r♠✉❧❛ ❢♦r ✈❡rt✐❝❛❧ ✢✐♣ ❚❤✐s ✐s ❤♦✇ ❡❛s② ✐t ✐s t♦ ✇r✐t❡ t❤❡ ❢♦r♠✉❧❛ ❢♦r ❛ ♥❡✇ ❢✉♥❝t✐♦♥✿

√ x2 + x f (x) = 7   2x −√ x + x F (x) = − x−7

♦r✐❣✐♥❛❧✿ ✢✐♣♣❡❞ ✈❡rt✐❝❛❧❧②✿

❲❡ ❥✉st ♣✉t t❤❡ ♦r✐❣✐♥❛❧ ✐♥s✐❞❡ ❧❛r❣❡ ♣❛r❡♥t❤❡s❡s ❛♥❞ t❤❡♥ ❛❞❞ ✏ −✑ ✐♥ ❢r♦♥t✳

❊①❛♠♣❧❡ ✸✳✾✳✶✸✿ ♦r❞❡r ♠✐❣❤t ♠❛tt❡r ▲❡t

Q(x) = −f (x) + 3 . ❍♦✇ ❞♦ ✇❡ ❣❡t t❤❡ ❣r❛♣❤ ♦❢

h❄

▲❡t✬s ❞❡❝♦♠♣♦s❡ ❛♥❞ ♣r♦✈✐❞❡ ♠❛t❝❤✐♥❣ tr❛♥s❢♦r♠❛t✐♦♥s✿

x 7−→ f (x)

7−→ |{z}

✈❡rt✐❝❛❧ ✢✐♣

−f (x)

❚❤❡ ❣r❛♣❤s ❛r❡ ❛❧s♦ ❞✐✛❡r❡♥t✿

7−→ |{z}

✈❡rt✐❝❛❧ s❤✐❢t

−f (x) + 3

7−→ |{z}

−(f (x) + 3)

✈❡rt✐❝❛❧ s❤✐❢t

❚❤❡ ♦r❞❡r ♠❛tt❡rs✦ ▲❡t✬s ✐♥t❡r❝❤❛♥❣❡ t❤❡s❡ t✇♦✿

x 7−→ f (x)

7−→ |{z}

f (x) + 3

✈❡rt✐❝❛❧ ✢✐♣

✸✳✾✳ ❈❤❛♥❣✐♥❣ ❛ ✈❛r✐❛❜❧❡ tr❛♥s❢♦r♠s t❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥

✷✼✾

◆❡①t ✐s t❤❡ ❤♦r✐③♦♥t❛❧ ✢✐♣ ✿

■❢ ✇❡ t❛❦❡ ❛ ♣♦✐♥t (x, y) ♦♥ t❤❡ ❣r❛♣❤ ♦❢ y = x2 ❛♥❞ ✢✐♣ ❛❜♦✉t t❤❡ y ✲❛①✐s✱ ✇❡ ❤❛✈❡ ♥❡✇ ♣♦✐♥t (−x, y)✳ ■t ❧✐❡s ♦♥ t❤❡ ❣r❛♣❤ ♦❢ y = (−x)2 ✳ ■t✬s t❤❡ s❛♠❡ ❣r❛♣❤ ❜❡❝❛✉s❡ ♦❢ t❤❡ ♠✐rr♦r s②♠♠❡tr②✳ ❚❤❡♦r❡♠ ✸✳✾✳✶✹✿ ❍♦r✐③♦♥t❛❧ ❋❧✐♣

■❢ t❤❡ ❣r❛♣❤ ♦❢ y = F (x) ✐s t❤❡ ❣r❛♣❤ ♦❢ y = f (x) ✢✐♣♣❡❞ ❤♦r✐③♦♥t❛❧❧②✱ t❤❡♥ F (x) = f (−x)

❛♥❞ ✈✐❝❡ ✈❡rs❛✳ ❲❡ ❝❛♥ s❡❡ ❤♦✇ t❤❡ ❣r❡❡♥ ♣♦✐♥ts ❛r❡ tr❛♥s❢❡rr❡❞ t♦ t❤❡ r❡❞ ♦♥❡s ♦♥❡ ❜② ♦♥❡✿

❚❤❡ ❞✐st❛♥❝❡ ❢r♦♠ ❡✈❡r② ♥❡✇ ♣♦✐♥t t♦ t❤❡ y ✲❛①✐s ✐s t❤❛t s❛♠❡ ❛s t❤❛t ♦❢ t❤❡ ♦r✐❣✐♥❛❧ ♣♦✐♥t ❜✉t ♦♥ t❤❡ ♦t❤❡r s✐❞❡✳ ❖♥❝❡ ❛❣❛✐♥✱ t❤❡ ♥❡✇ ❢✉♥❝t✐♦♥ ❤❛s t❤❡ ❡①❛❝t s❛♠❡ s❤❛♣❡ ♦❢ t❤❡ ❣r❛♣❤ ❛s t❤❡ ♦❧❞ ♦♥❡✳ ■♥ ❢❛❝t✱ ❛❧❧ ❢♦✉r tr❛♥s❢♦r♠❛t✐♦♥s ❛r❡ r✐❣✐❞ ♠♦t✐♦♥s✳ ❊①❡r❝✐s❡ ✸✳✾✳✶✺

Pr♦✈❡ t❤❡ t❤❡♦r❡♠✳

✸✳✾✳

❈❤❛♥❣✐♥❣ ❛ ✈❛r✐❛❜❧❡ tr❛♥s❢♦r♠s t❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥

✷✽✵

❊①❛♠♣❧❡ ✸✳✾✳✶✻✿ ❢♦r♠✉❧❛ ❢♦r ❤♦r✐③♦♥t❛❧ ✢✐♣ ❚❤✐s ✐s ❤♦✇ ❡❛s② ✐t ✐s t♦ ✇r✐t❡ t❤❡ ❢♦r♠✉❧❛ ❢♦r ❛ ♥❡✇ ❢✉♥❝t✐♦♥✿

♦r✐❣✐♥❛❧✿

✢✐♣♣❡❞ ❤♦r✐③♦♥t❛❧❧②✿

❲❡ ❥✉st r❡♣❧❛❝❡ ❡❛❝❤

x

√ x2 + x f (x) = x −p 7 2 (−x) + (−x) F (x) = (−x) − 7

✐♥ t❤❡ ♦r✐❣✐♥❛❧ ✇✐t❤ ✏ (−x)✑✳

❊①❡r❝✐s❡ ✸✳✾✳✶✼ ❉♦❡s t❤❡ ♦r❞❡r ♠❛tt❡r ✇❤❡♥ t❤❡ ❤♦r✐③♦♥t❛❧ ✢✐♣ ✐s ❝♦♠❜✐♥❡❞ ✇✐t❤ ❛ ❤♦r✐③♦♥t❛❧ s❤✐❢t❄ ❲❤❛t ❛❜♦✉t t❤❡ ❤♦r✐③♦♥t❛❧ ✢✐♣ ✇✐t❤ ❛

✈❡rt✐❝❛❧

s❤✐❢t❄

❊①❛♠♣❧❡ ✸✳✾✳✶✽✿ t✇♦ ✢✐♣s ❝♦♠♣❛r❡❞ ❚❤❡s❡ ❛r❡ t❤❡ ❞✐❛❣r❛♠s✿ ✈❡rt✐❝❛❧ ✢✐♣✿ ❤♦r✐③♦♥t❛❧ ✢✐♣✿

x →

♠✉❧t✐♣❧② ❜②

−1

x →

→ x →

f f

→ y →

♠✉❧t✐♣❧② ❜②

→ y

−1

→ y

❊①❡r❝✐s❡ ✸✳✾✳✶✾ ❲❤❛t ❤❛♣♣❡♥s t♦ t❤❡

x✲

❛♥❞

y ✲✐♥t❡r❝❡♣ts

♦❢ ❛ ❢✉♥❝t✐♦♥ ✉♥❞❡r ✈❡rt✐❝❛❧ ❛♥❞ ❤♦r✐③♦♥t❛❧ ✢✐♣s❄

❊①❡r❝✐s❡ ✸✳✾✳✷✵ ❲❤❛t ❤❛♣♣❡♥s t♦ t❤❡ ❞♦♠❛✐♥ ❛♥❞ t❤❡ r❛♥❣❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ✉♥❞❡r ✈❡rt✐❝❛❧ ❛♥❞ ❤♦r✐③♦♥t❛❧ ✢✐♣s❄

◆❡①t ✐s t❤❡

✈❡rt✐❝❛❧ str❡t❝❤ ✿

(x, y) ♦♥ t❤❡ 2 ♦❢ y = 3x ✳

■❢ ✇❡ t❛❦❡ ❛ ♣♦✐♥t ❧✐❡s ♦♥ t❤❡ ❣r❛♣❤

❣r❛♣❤ ♦❢

y = x2

❛♥❞ str❡t❝❤ ✈❡rt✐❝❛❧❧② ❜②

3✱

✇❡ ❤❛✈❡ ♥❡✇ ♣♦✐♥t

(x, 3y)✳

❚❤❡♦r❡♠ ✸✳✾✳✷✶✿ ❱❡rt✐❝❛❧ ❙tr❡t❝❤ ■❢ t❤❡ ❣r❛♣❤ ♦❢ ♦❢

k > 0✱

y = F (x) ✐s t❤❡ ❣r❛♣❤ ♦❢ y = f (x) str❡t❝❤❡❞ ✈❡rt✐❝❛❧❧② ❜② ❛ ❢❛❝t♦r

t❤❡♥

F (x) = kf (x) ❛♥❞ ✈✐❝❡ ✈❡rs❛✳

❲❡ ❝❛♥ s❡❡ ❤♦✇ t❤❡ ❣r❡❡♥ ♣♦✐♥ts ❛r❡ tr❛♥s❢❡rr❡❞ t♦ t❤❡ r❡❞ ♦♥❡s ♦♥❡ ❜② ♦♥❡✿

■t

✸✳✾✳

❈❤❛♥❣✐♥❣ ❛ ✈❛r✐❛❜❧❡ tr❛♥s❢♦r♠s t❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥

✷✽✶

k = 2✱ t❤❡ ❞✐st❛♥❝❡ ❢r♦♠ ❡✈❡r② ♥❡✇ ♣♦✐♥t t♦ t❤❡ ♦r✐❣✐♥❛❧ ♣♦✐♥t ✐s t❤❡ s❛♠❡ ❛s ❢r♦♠ t❤❡ ♦r✐❣✐♥❛❧ t♦ x✲❛①✐s✳ ■t ✐s ❝❧❡❛r t❤❛t t❤❡ ♥❡✇ ❢✉♥❝t✐♦♥ ❤❛s ❛ s♦♠❡✇❤❛t ❞✐✛❡r❡♥t s❤❛♣❡ ♦❢ t❤❡ ❣r❛♣❤ t❤❛♥ t❤❡ ♦❧❞ ♦♥❡✳

■♥ ❝❛s❡ ♦❢ t❤❡

❊①❡r❝✐s❡ ✸✳✾✳✷✷

Pr♦✈❡ t❤❡ t❤❡♦r❡♠✳ ❊①❛♠♣❧❡ ✸✳✾✳✷✸✿ ❢♦r♠✉❧❛ ❢♦r ✈❡rt✐❝❛❧ str❡t❝❤

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

♦r✐❣✐♥❛❧✿

f (x) =

str❡t❝❤❡❞ ✈❡rt✐❝❛❧❧② ❜② ✺✿

F (x) = 5·

√ x2 + x x−7 ! √ 2 x + x x−7

❲❡ ❥✉st ♣✉t t❤❡ ♦r✐❣✐♥❛❧ ✐♥s✐❞❡ ❧❛r❣❡ ♣❛r❡♥t❤❡s❡s ❛♥❞ t❤❡♥ ❛❞❞ ✏ 5✑ ✐♥ ❢r♦♥t✳

◆❡①t ✐s t❤❡

❤♦r✐③♦♥t❛❧ str❡t❝❤ ✿

❲❡ ❝❛♥ ❣✉❡ss t❤❛t

y = x2

str❡t❝❤❡❞ ❤♦r✐③♦♥t❛❧❧② ❜②

3

❜❡❝♦♠❡s

y = (x/3)2 ✳

❚❤❡♦r❡♠ ✸✳✾✳✷✹✿ ❍♦r✐③♦♥t❛❧ ❙tr❡t❝❤ ■❢ ❣r❛♣❤

y = F (x)

✐s t❤❡ ❣r❛♣❤ ♦❢

y = f (x)

str❡t❝❤❡❞ ❜② t❤❡ ❢❛❝t♦r

❤♦r✐③♦♥t❛❧❧②✱ t❤❡♥

F (x) = f (x/k) ❛♥❞ ✈✐❝❡ ✈❡rs❛✳

❊①❡r❝✐s❡ ✸✳✾✳✷✺

Pr♦✈❡ t❤❡ t❤❡♦r❡♠✳

❲❡ ❝❛♥ s❡❡ ❤♦✇ t❤❡ ❣r❡❡♥ ♣♦✐♥ts ❛r❡ tr❛♥s❢❡rr❡❞ t♦ t❤❡ r❡❞ ♦♥❡s ♦♥❡ ❜② ♦♥❡✿

k > 0

✸✳✾✳

❈❤❛♥❣✐♥❣ ❛ ✈❛r✐❛❜❧❡ tr❛♥s❢♦r♠s t❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥

✷✽✷

k = 2✱ t❤❡ ❞✐st❛♥❝❡ ❢r♦♠ ❡✈❡r② ♥❡✇ ♣♦✐♥t t♦ t❤❡ ♦r✐❣✐♥❛❧ ♣♦✐♥t ✐s t❤❡ s❛♠❡ ❛s ❢r♦♠ t❤❡ ♦r✐❣✐♥❛❧ t♦ y ✲❛①✐s✳ ■t ✐s ❝❧❡❛r t❤❛t t❤❡ ♥❡✇ ❢✉♥❝t✐♦♥ ❤❛s ❛ s♦♠❡✇❤❛t ❞✐✛❡r❡♥t s❤❛♣❡ ♦❢ t❤❡ ❣r❛♣❤ t❤❛♥ t❤❡ ♦❧❞ ♦♥❡✳

■♥ ❝❛s❡ ♦❢ t❤❡

❚❤❡ ❧❛st t✇♦ tr❛♥s❢♦r♠❛t✐♦♥s ❛r❡♥✬t r✐❣✐❞ ♠♦t✐♦♥s✳ ❊①❛♠♣❧❡ ✸✳✾✳✷✻✿ tr❛♥s❢♦r♠❛t✐♦♥s ✇✐t❤ ❝♦♠♣✉t❡r ❣r❛♣❤✐❝s

❖♥❡ ❝❛♥ str❡t❝❤ ❛♥❞ s❤r✐♥❦ ✇✐t❤ ✐♠❛❣❡ ❡❞✐t✐♥❣ s♦❢t✇❛r❡✿

❊①❛♠♣❧❡ ✸✳✾✳✷✼✿ ❢♦r♠✉❧❛ ❢♦r ❤♦r✐③♦♥t❛❧ str❡t❝❤

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

♦r✐❣✐♥❛❧✿ str❡t❝❤❡❞ ❤♦r✐③♦♥t❛❧❧② ❜②

❲❡ ❥✉st r❡♣❧❛❝❡ ❡❛❝❤

x

√ x2 + x f (x) = x −p 7 (x/5)2 + (x/5) 5 : F (x) = (x/5) − 7

✐♥ t❤❡ ♦r✐❣✐♥❛❧ ✇✐t❤ ✏ (x/5)✑✳

❊①❛♠♣❧❡ ✸✳✾✳✷✽✿ t✇♦ str❡t❝❤❡s ❝♦♠♣❛r❡❞

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

x→ f →y →

3:

❤♦r✐③♦♥t❛❧ str❡t❝❤ ❜②

3: t→

❞✐✈✐❞❡ ❜②

♠✉❧t✐♣❧② ❜②

3 → x→ f →y

3 → z

❚❤❡ t❤❡♦r❡♠ ❜❡❧♦✇ s✉♠♠❛r✐③❡s ♦✉r ❛♥❛❧②s✐s✳ ❚❤❡♦r❡♠ ✸✳✾✳✷✾✿ ❚r❛♥s❢♦r♠❛t✐♦♥s ❛s ❈♦♠♣♦s✐t✐♦♥s

y = f (x) r❡s✉❧ts ❢r♦♠ ✐ts ❝♦♠♣♦s✐t✐♦♥ ✇✐t❤ ❛ ❢✉♥❝t✐♦♥ t❤❛t ❢♦❧❧♦✇s f ✱ ✐✳❡✳✱ h ◦ f ✳ ❆ ❤♦r✐③♦♥t❛❧ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ t❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥ f r❡s✉❧ts ❢r♦♠ ✐ts ❝♦♠♣♦s✐t✐♦♥ ✇✐t❤ ❛ ❢✉♥❝t✐♦♥ t❤❛t ♣r❡❝❡❞❡s f ✱ ✐✳❡✳✱ f ◦ g ✳

✶✳ ❆ ✈❡rt✐❝❛❧ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ t❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥

✷✳

✸✳✾✳

❈❤❛♥❣✐♥❣ ❛ ✈❛r✐❛❜❧❡ tr❛♥s❢♦r♠s t❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥

✷✽✸

❋♦r ♣❛rt ✶✱ t❤✐s ✐s ✇❤❛t z = 2y ❞♦❡s t♦ t❤❡ ❣r❛♣❤ ♦❢ y = f (x)✿

❋♦r ♣❛rt ✷✱ t❤✐s ✐s ✇❤❛t x = 2t ❞♦❡s t♦ t❤❡ ❣r❛♣❤ ♦❢ y = f (x)✿

❆s ✇❡ ❝❛♥ s❡❡✱ s✉❝❤ ❛ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ ❛♥ ❛①✐s ✐s ❛ t✇♦ ✿ ✐♥♣✉t ❛♥❞ ♦✉t♣✉t✳ ❚❤❡♥✿

❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s

✭♦r ✉♥✐ts✮✳ ❊✈❡r② ❢✉♥❝t✐♦♥ f ❤❛s

• ■♥ t❤❡ ❢♦r♠❡r ❝❛s❡✱ ✇❡ ❝❤❛♥❣❡ t❤❡ ♦✉t♣✉t ✈❛r✐❛❜❧❡✿ ❢r♦♠ y t♦ z = h(y)✳

• ■♥ t❤❡ ❧❛tt❡r ❝❛s❡✱ ✇❡ ❝❤❛♥❣❡ t❤❡ ✐♥♣✉t ✈❛r✐❛❜❧❡✿ ❢r♦♠ x t♦ t = g −1 (x)✳

❚❤✐s ❞✐✛❡r❡♥❝❡ ✐s t❤❡ r❡❛s♦♥ ✇❤② t❤❡ ❡✛❡❝t ♦♥ t❤❡ ❣r❛♣❤ ♦❢ f ✐s s♦ ❞✐✛❡r❡♥t✳ ❊①❡r❝✐s❡ ✸✳✾✳✸✵

❲❤❛t ❤❛♣♣❡♥s t♦ t❤❡ x✲ ❛♥❞ y ✲✐♥t❡r❝❡♣ts ♦❢ ❛ ❢✉♥❝t✐♦♥ ✉♥❞❡r ✈❡rt✐❝❛❧ ❛♥❞ ❤♦r✐③♦♥t❛❧ str❡t❝❤❡s❄ ❊①❡r❝✐s❡ ✸✳✾✳✸✶

❲❤❛t ❤❛♣♣❡♥s t♦ t❤❡ ❞♦♠❛✐♥ ❛♥❞ t❤❡ r❛♥❣❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ✉♥❞❡r ✈❡rt✐❝❛❧ ❛♥❞ ❤♦r✐③♦♥t❛❧ str❡t❝❤❡s❄ ❊①❡r❝✐s❡ ✸✳✾✳✸✷

✭❛✮ ❍♦✇ ❞♦ t❤❡s❡ tr❛♥s❢♦r♠❛t✐♦♥s ❛✛❡❝t t❤❡ ♠♦♥♦t♦♥✐❝✐t② ♦❢ ❛ ❢✉♥❝t✐♦♥❄ ✭❜✮ ■♥✈❡st✐❣❛t❡ t❤❡ ♠♦♥♦✲ t♦♥✐❝✐t② ♦❢ q✉❛❞r❛t✐❝ ❢✉♥❝t✐♦♥s✳ ❊①❡r❝✐s❡ ✸✳✾✳✸✸

❇② tr❛♥s❢♦r♠✐♥❣ t❤❡ ❣r❛♣❤ ♦❢ y = x2 ✱ ♣❧♦t t❤❡ ❣r❛♣❤s ♦❢ t❤❡ ❢✉♥❝t✐♦♥s✿ ✭❛✮ y =

√

x ❛♥❞ ✭❜✮ y =

√

x + 3✳

■♥ ❝♦♥❝❧✉s✐♦♥✱ ❝♦♠♣♦s✐♥❣ ❛ ❢✉♥❝t✐♦♥ ✇✐t❤ ❛ ❧✐♥❡❛r ❢✉♥❝t✐♦♥ ✕ ❜❡❢♦r❡ ♦r ❛❢t❡r ✕ ✇✐❧❧ tr❛♥s❢♦r♠ ✐ts ❣r❛♣❤ ✐♥ t❤❡s❡ s✐① ✇❛②s✿ s❤✐❢t✱ ✢✐♣✱ ❛♥❞ str❡t❝❤✱ ✈❡rt✐❝❛❧ ♦r ❤♦r✐③♦♥t❛❧✳

✸✳✶✵✳

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

✷✽✹

❙✉♠♠❛r② ♦❢ t❤❡ r✉❧❡s✿

y

✈❡rt✐❝❛❧✱ s❤✐❢t ❜②

s:

:

✢✐♣

str❡t❝❤ ❜②

y = f (x) y = f (x) k : y = f (x)

❤♦r✐③♦♥t❛❧✱

+s ·(−1) ·k

■♥ t❤❡ ✏❤♦r✐③♦♥t❛❧✑ ❝♦❧✉♠♥✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦r♠✉❧❛s ♦❢ t❤❡

•

❆❞❞

•

▼✉❧t✐♣❧② ❜②

(−1)

✈s✳ ♠✉❧t✐♣❧② ❜②

•

▼✉❧t✐♣❧② ❜②

k>0

✈s✳ ❞✐✈✐❞❡ ❜②

s

✈s✳ s✉❜tr❛❝t

x −s) ·(−1)) /k)

y = f (x y = f (x y = f (x

✐♥✈❡rs❡s ✿

s✳ (−1)✳

k✳

F ✱ ❛❢t❡r ❛ ❝❤❛♥❣❡ ♦❢ F (x+s) = f (x) ❛❢t❡r ❛ ❤♦r✐③♦♥t❛❧ s❤✐❢t✱ ✇❡ s✉❜st✐t✉t❡ u = x + s ❛♥❞ ✜♥❞ x ✐♥ t❡r♠s ♦❢ u✱ ✐✳❡✳✱ x = u − s✱ r❡s✉❧t✐♥❣ ✐♥ F (u) = f (u − s)✳ ❆❢t❡r ❛♥ ♦♣t✐♦♥❛❧ r❡♥❛♠✐♥❣✱ u ❢♦r x✱ t❤❡ ❢♦r♠✉❧❛ t❛❦❡s ✐ts ✜♥❛❧ ❢♦r♠✱ F (x) = f (x − s)✳ ■♥✈❡rs❡s ❛♣♣❡❛r ❡✈❡r② t✐♠❡ ✇❡ s♦❧✈❡ ❛♥ ❡q✉❛t✐♦♥ t♦ ✜♥❞ t❤❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ ♥❡✇ ❢✉♥❝t✐♦♥✱ t❤❡ ✐♥♣✉t ✈❛r✐❛❜❧❡ ✭✐✳❡✳✱ ❛ ❤♦r✐③♦♥t❛❧ tr❛♥s❢♦r♠❛t✐♦♥✮ ♦❢ ❛ ❢✉♥❝t✐♦♥

f✳

❋♦r ❡①❛♠♣❧❡✱ ✐❢ ✇❡ ❤❛✈❡

❊①❛♠♣❧❡ ✸✳✾✳✸✹✿ ♦r❞❡r ♦❢ ♦♣❡r❛t✐♦♥s ✈s✳ ♦r❞❡r ♦❢ tr❛♥s❢♦r♠❛t✐♦♥s

❚❤❡r❡ ✐s ❛ ❜✐t ♠♦r❡ ❝♦♠♣❧❡①✐t② ❤❡r❡✿ ❚❤❡ ♦r❞❡r ✐♥ ✇❤✐❝❤ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥s ❛r❡ ❝❛rr✐❡❞ ♦✉t ❢♦❧❧♦✇s t❤❡ ❞✐r❡❝t✐♦♥

❛✇❛②

❢r♦♠

f✳

❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ t❤✐s ❢✉♥❝t✐♦♥✿

y

x ... →

❛❞❞

5 →

−1 →x→

♠✉❧t✐♣❧② ❜②

→y→

f

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

3

❛♥❞ t❤❡♥ str❡t❝❤ ❜②

5

✢✐♣ ❤♦r✐③♦♥t❛❧❧② ❛♥❞ t❤❡♥ s❤✐❢t ❧❡❢t ❜②

3 →

♠✉❧t✐♣❧②

5 → ...

✈❡rt✐❝❛❧

❤♦r✐③♦♥t❛❧

• •

s✉❜tr❛❝t

f✿

✭❧❡❢t t♦ r✐❣❤t✮❀

5

✭r✐❣❤t t♦ ❧❡❢t✮✳

✸✳✶✵✳ ❚❤❡ ❣r❛♣❤ ♦❢ ❛ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧ ✐s ❛ ♣❛r❛❜♦❧❛

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

y = x2

s♣❡❝✐✜❝

❝❧❛ss ♦❢ ❢✉♥❝t✐♦♥s✳

❛ ✏♣❛r❛❜♦❧❛✑✳ ❆r❡ t❤❡r❡ ♦t❤❡rs❄

❚❤❡ ❣r❛♣❤ ♦❢ ❡✈❡r② q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧ ✐s ❛ ♣❛r❛❜♦❧❛✳

❚❤✐s ✐s ✇❤❛t ✐t ♠❡❛♥s✿

◮

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

t❤❡

♣❛r❛❜♦❧❛ ♦❢

♦✉r s✐① ❜❛s✐❝ tr❛♥s❢♦r♠❛t✐♦♥s ✭♦r ❡✈❡♥ ❢❡✇❡r✮✳ ❲❡ ❛r❡ ❛❜♦✉t t♦ ❝❛rr② ♦✉t t❤✐s ♣❧❛♥✿

tr❛♥s❢♦r♠❛t✐♦♥s

y = x2 −−−−−−−−−−−−−→ y = ax2 + bx + c, a 6= 0

f (x) = x2

✈✐❛

✸✳✶✵✳

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

✷✽✺

❊①❛♠♣❧❡ ✸✳✶✵✳✶✿ tr❛♥s❢♦r♠✐♥❣ ♣❛r❛❜♦❧❛

❙✉♣♣♦s❡ t❤❡ ❣r❛♣❤ ♦❢ g ✐s ❣✐✈❡♥ ♦♥ t❤❡ r✐❣❤t✳ ❲❡ ♥❡❡❞ t♦ tr❛♥s❢♦r♠ t❤❡ ❣r❛♣❤ ♦❢ y = x2 ✐♥t♦ t❤✐s ♣❛r❛❜♦❧❛✿

❲❡ st❛rt ✇✐t❤ ❛ ❝♦♠♣❛r✐s♦♥✳ • ❚❤❡ ♠♦st ♦❜✈✐♦✉s ❢❡❛t✉r❡ ✐s t❤❛t t❤❡ r✐❣❤t ♦♥❡ ♦♣❡♥s ❞♦✇♥✳ ❲❡ ✇✐❧❧✱ t❤❡r❡❢♦r❡✱ ❤❛✈❡ t♦ ❞♦ ❛ ✈❡rt✐❝❛❧ ✢✐♣✳ • ❚❤❡ s❡❝♦♥❞ ♠♦st ♣r♦♠✐♥❡♥t ❢❡❛t✉r❡ ✐s t❤❛t t❤❡ r✐❣❤t ♦♥❡ ✐s s❧✐♠♠❡r✳ ❲❡ ✇✐❧❧ ❤❛✈❡ t♦ ❞♦ ❛ ✈❡rt✐❝❛❧ str❡t❝❤ ♦r ❛ ❤♦r✐③♦♥t❛❧ s❤r✐♥❦✳ • ❚❤❡ ❧❛st ♦♥❡ ✐s t❤❡ ❧♦❝❛t✐♦♥ ✿ ❚❤❡ ✈❡rt❡① ♦❢ t❤❡ ♣❛r❛❜♦❧❛ ✐s ❛✇❛② ❢r♦♠ t❤❡ ♦r✐❣✐♥✳ ❲❡ ✇✐❧❧ ❤❛✈❡ t♦ ❞♦ ❜♦t❤ ✈❡rt✐❝❛❧ ❛♥❞ ❤♦r✐③♦♥t❛❧ s❤✐❢ts✳ ◆♦t❡ t❤❛t ❛ ❤♦r✐③♦♥t❛❧ ✢✐♣ ✐s ✉s❡❧❡ss ❤❡r❡ ❜❡❝❛✉s❡ t❤❡ ♣❛r❛❜♦❧❛ ❤❛s ❛ ✈❡rt✐❝❛❧ ♠✐rr♦r s②♠♠❡tr②✳ ❆❧s♦ ♥♦t❡ t❤❛t ✇❡ ♣✐❝❦ ❛ ✈❡rt✐❝❛❧ str❡t❝❤ ♦✈❡r ❛ ❤♦r✐③♦♥t❛❧ s❤r✐♥❦ ❛s t❤❡ s✐♠♣❧❡r ♦♥❡✳ ❲❡ ♥♦✇ t✉r♥ t♦ t❤❡ ❛❝t✉❛❧ ❛❧❣❡❜r❛✳ ❲❤❛t ✐s t❤❡ ♦r❞❡r ♦❢ t❤❡ ♦♣❡r❛t✐♦♥s t❤❛t ✇❡ ❤❛✈❡ ♦✉t❧✐♥❡❞❄ ❲❡ ❞♦ ❛❧❧ ♦❢ t❤❡ ✈❡rt✐❝❛❧ ✜rst✱ t❤❡♥ ❤♦r✐③♦♥t❛❧✿ ♦r✐❣✐♥❛❧✿ x2 ✈❡rt✐❝❛❧ ✢✐♣✿ x2 ·(−1) 2 ✈❡rt✐❝❛❧ str❡t❝❤✿ x · (−1)·5 ✈❡rt✐❝❛❧ s❤✐❢t✿ x2 · (−1) · 5+4 ❤♦r✐③♦♥t❛❧ s❤✐❢t✿ (x−3)2 · (−1) · 5 + 4

= −x2 = −5x2 = −5x2 + 4 = −5(x − 3)2 + 4

❚❤✐s ❛❧❣❡❜r❛ ❝♦♠❡s ❢r♦♠ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡q✉❡♥❝❡ ♦❢ ❝♦♠♣♦s✐t✐♦♥s✿ x ↓ x+3=r

→ −→

x2 = y −→

→ −→

−y = u −→

→ −→

5u = v −→

→ −→

v+4=w || w=z

❚❤❡ ❜♦tt♦♠ r♦✇ ✐s t❤❡ ♥❡✇ ❢✉♥❝t✐♦♥ g : r 7→ z ✳ ❚❤❡s❡ tr❛♥s❢♦r♠❛t✐♦♥s ♣r♦❞✉❝❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡✛❡❝t✿

❖♥❡ ❝❛♥ ❛❧s♦ s❡❡ ❤♦✇ t❤❡ ❞❛t❛ ✐s ❝❤❛♥❣✐♥❣ ❛s ✇❡ ♣r♦❣r❡ss t❤r♦✉❣❤ t❤❡ s❡q✉❡♥❝❡✿

✸✳✶✵✳

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

✷✽✻

❚❤❡ ✜rst t✇♦ ❝♦❧✉♠♥s ❢♦r♠ t❤❡ ♦r✐❣✐♥❛❧ ❢✉♥❝t✐♦♥ ❛♥❞ t❤❡ ✜♥❛❧ ❢✉♥❝t✐♦♥ ✐s ✐♥ t❤❡ ❧❛st t✇♦ ❝♦❧✉♠♥s✳ ❇✉t ❝❛♥ ✇❡ ❣♦ ❜❛❝❦✇❛r❞s ❛♥❞ ✜♥❞ ❤♦✇ t♦ r❡♣r❡s❡♥t ❛ q✉❛❞r❛t✐❝ ❢✉♥❝t✐♦♥ ❣✐✈❡♥ t♦ ✉s✱ ❛s ❛ ❢♦r♠✉❧❛✱ ❛s ❛ r❡s✉❧t ♦❢ s✉❝❤ ❛ s❡q✉❡♥❝❡ ♦❢ tr❛♥s❢♦r♠❛t✐♦♥s❄ ❚♦ ❜❡❣✐♥ ✇✐t❤✱ ❧❡t✬s ❧❡❛r♥ ❤♦✇ t♦ ✜♥❞ t❤❡ ✈❡rt❡① ♦❢ ❛ ♣❛r❛❜♦❧❛✿

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

r❡♣r❡s❡♥t❛t✐♦♥ ✿

f (x) = ax2 + bx + c, a 6= 0 .

❏✉❞❣✐♥❣ ❜② t❤❡ ❡①❛♠♣❧❡✱ ✇❡ ♥❡❡❞ t♦ ♠♦r♣❤ ✐t ✐♥t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✿ f (x) = a(x − h)2 + k

❲❤✐❧❡ a ❝♦♥t❛✐♥s ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t t❤❡ str❡t❝❤✴s❤r✐♥❦ ❛♥❞ t❤❡ ✢✐♣ ♦❢ t❤❡ ❣r❛♣❤✱ h ❛♥❞ k ❛r❡ t❤❡ s❤✐❢ts✳ ■♥ ❢❛❝t✱ t❤❡ ♣♦✐♥t (h, k) ✐s t❤❡ ✈❡rt❡① ♦❢ t❤❡ ♣❛r❛❜♦❧❛✳ ❲❡ ✇✐❧❧ ✉s❡ t❤❡

❝♦♠♣❧❡t❡ sq✉❛r❡ ❢♦r♠✉❧❛ ✿

(u + v)2 = u2 + 2uv + v 2

❊①❛♠♣❧❡ ✸✳✶✵✳✷✿ ❝♦♠♣❧❡t✐♥❣ ❛ sq✉❛r❡

▲❡t✬s s❤♦✇ ❤♦✇ t❤✐s ✐s ❞♦♥❡ ❛❧❣❡❜r❛✐❝❛❧❧②✳ ❙✉♣♣♦s❡ f (x) = 2x2 + 8x + 3 .

✸✳✶✵✳

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

✷✽✼

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

f (x) = 2x2 + 8x + 3 = (2x2 + 8x) + 3 = 2(x2 + 4x) + 3 2

2

❙t❛rt ✇✐t❤ t❤❡ ♦r✐❣✐♥❛❧✳ ❇r✐♥❣ t♦❣❡t❤❡r t❤❡ t✇♦ t❡r♠s ✇✐t❤ 2 ❋❛❝t♦r✱ s♦ t❤❛t ②♦✉ ❤❛✈❡ x .

2

= 2(x + 4x+2 − 2 ) + 3 = 2(x2 + 4x + 22 ) − 2 · 22 + 3 = 2(x + 2)2 − 8 + 3 = 2(x + 2)2 − 5 ❘❡❛❞✐♥❣ ❢r♦♠ t❤❡ ✐♥s✐❞❡ ♦✉t✿ ❙❤✐❢t ❧❡❢t ❜②

(−2, −5)✳

x.

❚❤❡ ♦t❤❡r t❡r♠ ✐s ❤❛❧❢ t❤❡ ❝♦❡✣❝✐❡♥t ♦❢ x, 4/2 = 2 . 2 ❆❞❞ t❤❡ ♠✐ss✐♥❣ t❡r♠✱ 2 , ♦❢ t❤❡ ❝♦♠♣❧❡t❡ sq✉❛r❡✳ P✉❧❧ ♦✉t t❤❡ ❡①tr❛ t❡r♠✳ ❈♦♠♣❧❡t❡ t❤❡ sq✉❛r❡✳ ❆❝q✉✐r❡ t❤❡ ✜♥❛❧ ❢♦r♠✳

2✱

str❡t❝❤ ✈❡rt✐❝❛❧❧② ❜②

2✱

s❤✐❢t ❞♦✇♥ ❜②

5✳

❚❤❡ ✈❡rt❡① ✐s ❛t

❚♦ ❝♦♥✜r♠✱ ✇❡ ❝❛rr② ♦✉t t❤❡s❡ ❝♦♠♣✉t❛t✐♦♥s ✭❛♥❞ ♣❧♦tt✐♥❣✮ ✇✐t❤ ❛ s♣r❡❛❞s❤❡❡t✿

❊①❡r❝✐s❡ ✸✳✶✵✳✸

❍♦✇ ❝❛♥ ②♦✉ tr❛♥s❢♦r♠

❛♥② ♣❛r❛❜♦❧❛ ✐♥t♦ ❛♥② ♦t❤❡r ♣❛r❛❜♦❧❛❄

❆ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧ ♠❛② ❜❡ ❝❛❧❧❡❞ ❛ ✏❝♦♠♣❧❡t❡ sq✉❛r❡✑ ✐❢ ✐t ❝❛♥ ❜❡ ♣✉t ✐♥ t❤✐s ❢♦r♠✿

f (x) = (x − h)2 = x2 + 2xh + h2 , ❢♦r s♦♠❡ ♥✉♠❜❡r

h✳

■t ✐s ❡❛s② t♦ ♣❧♦t❀ ❥✉st s❤✐❢t t❤❡ ♦r✐❣✐♥❛❧

❛♣♣❧✐❝❛❜✐❧✐t②✱ ❜✉t t❤❡ ❝❤❛❧❧❡♥❣❡ ✐s t♦

h

✉♥✐ts r✐❣❤t✳

❚❤✐s ✐❞❡❛ ❤❛s ❛ ❜r♦❛❞❡r

r❡❝♦❣♥✐③❡ ❝♦♠♣❧❡t❡ sq✉❛r❡s ✐♥ ✭♦r ❡①tr❛❝t ❢r♦♠✮ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧s✳

❚❤❡♦r❡♠ ✸✳✶✵✳✹✿ ❱❡rt❡① ❋♦r♠ ♦❢ ◗✉❛❞r❛t✐❝ P♦❧②♥♦♠✐❛❧

❆♥② q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧✱

f (x) = ax2 + bx + c, a 6= 0 , ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❛s ❛♥ ✏✐♥❝♦♠♣❧❡t❡ sq✉❛r❡✑ ♦r ❛ ✏✈❡rt❡① ❢♦r♠✑✿

f (x) = a(x − h)2 + k ✇❤❡r❡

h, k

❛r❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♥✉♠❜❡rs✿

h=−

b , 2a

k = c − ah2

✸✳✶✵✳

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

✷✽✽

Pr♦♦❢✳ ❆❧❧ ✇❡ ♥❡❡❞ ✐s t♦ ✜♥❞ t❤❡s❡ ♣❛r❛♠❡t❡rs✿ h ❛♥❞ k ✳ ❋✐rst✱ ✇❡ s❡t t❤❡ t✇♦ r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ t❤❡ s❛♠❡ ❢✉♥❝t✐♦♥ ❡q✉❛❧ t♦ ❡❛❝❤ ♦t❤❡r✿ f (x) = ax2 + bx + c = a(x − h)2 + k .

❲❡ t❤❡♥ ❡①♣❛♥❞ t❤❡ ❧❛tt❡r ❛♥❞ ❛❧✐❣♥ ✐ts t❡r♠s ✇✐t❤ t❤❡ ❢♦r♠❡r✿ a ·x2 + b ·x +c = a ·x2 + 2ah ·x +(ah2 + k) .

❚❤❡♥ ✇❡ ♠❛t❝❤ ✉♣ t❤❡ ❝♦❡✣❝✐❡♥ts ♦❢ t❤❡ t❡r♠s✿ b = 2ah, c = ah2 + k .

❆♥❞ ✜♥❛❧❧② ✇❡ s♦❧✈❡ ❢♦r h ❛♥❞ k ✳

❊①❡r❝✐s❡ ✸✳✶✵✳✺ ❏✉st✐❢② ♠❛t❝❤✐♥❣ t❤❡ ❝♦❡✣❝✐❡♥ts ✐♥ t❤❡ ♣r♦♦❢✳ ❈♦♥s❡q✉❡♥t❧②✱ t❤❡ ❣r❛♣❤ ♦❢ f ❝❛♥ ❜❡ ❛❝q✉✐r❡❞ ❢r♦♠ t❤❡ ❣r❛♣❤ ♦❢ y = x2 ✈✐❛ t❤❡ s✐① tr❛♥s❢♦r♠❛t✐♦♥s✱ ♦r ❥✉st ❢♦✉r ✐❢ ✇❡ ❡①❝❧✉❞❡ t❤❡ ✈❡rt✐❝❛❧ ✢✐♣ ❛♥❞ t❤❡ ❤♦r✐③♦♥t❛❧ s❤r✐♥❦✳ ❲❡ ❝❛♥ s❛② t❤❛t t❤❡r❡ ✐s ♦♥❧② ♦♥❡ ♣❛r❛❜♦❧❛ ✿

❊①❡r❝✐s❡ ✸✳✶✵✳✻ ❍♦✇ ❝❛♥ t❤❡ ❢♦r♠✉❧❛ ❢♦r h ❛❜♦✈❡ ❜❡ ❡①tr❛❝t❡❞ ❢r♦♠ t❤❡ ◗✉❛❞r❛t✐❝ ❋♦r♠✉❧❛❄ ■♥ s✉♠♠❛r②✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿

❉❡✜♥✐t✐♦♥ ✸✳✶✵✳✼✿ ♣❛r❛❜♦❧❛ ❆ ♣❛r❛❜♦❧❛ ✐s t❤❡ ❣r❛♣❤ ♦❢ ❛ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧✳

✸✳✶✵✳

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

✷✽✾

❊①❡r❝✐s❡ ✸✳✶✵✳✽ ❚❤❡ ❣r❛♣❤s ❜❡❧♦✇ ❛r❡ ♣❛r❛❜♦❧❛s✳ ❖♥❡ ✐s

■♥ ❝♦♥❝❧✉s✐♦♥✱ ❝♦♠♣♦s✐♥❣ ❛ ❢✉♥❝t✐♦♥

f

y = x2 ✳

❲❤❛t ✐s t❤❡ ♦t❤❡r❄

✇✐t❤ ❛♥♦t❤❡r ❢✉♥❝t✐♦♥✱

h◦f

f ◦g,

♦r

tr❛♥s❢♦r♠s ✐ts ❣r❛♣❤ ✈❡rt✐❝❛❧❧② ♦r ❤♦r✐③♦♥t❛❧❧②✱ r❡s♣❡❝t✐✈❡❧②✳

❚❤❡♦r❡♠ ✸✳✶✵✳✾✿ ❚r❛♥s❢♦r♠❛t✐♦♥s ♦❢ ●r❛♣❤s

❚❤❡ ❣r❛♣❤ ♦❢ y = F (x) ✐s t❤❡ ❣r❛♣❤ ♦❢ y = f (x) tr❛♥s❢♦r♠❡❞ ❛s ✐♥❞✐❝❛t❡❞ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐ts ❢♦r♠✉❧❛ ✐s ♣r♦✈✐❞❡❞ ✐♥ t❤❡ t❛❜❧❡ ❜❡❧♦✇✿ ✈❡rt✐❝❛❧❧②

❤♦r✐③♦♥t❛❧❧②

s❤✐❢t❡❞ ❜② s : y = f (x) +s y = f (x ✢✐♣♣❡❞ : y = f (x) ·(−1) y = f (x y = f (x str❡t❝❤❡❞ ❜② k : y = f (x) ·k

−s) ·(−1)) /k)

Pr♦♦❢✳ ❚❤❡ tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡

y ✲❛①✐s

❝♦♠❡ ❢r♦♠ ❛ ❢✉♥❝t✐♦♥ ❛♣♣❧✐❡❞

✈❡rt✐❝❛❧✿

s❤✐❢t ❜② k

y −−−−−−−−−→ ✢✐♣

y −−−−−−→

str❡t❝❤ ❜② k

z = h(y),

s✉❜st✐t✉t❡

x✲❛①✐s

z = f (x) + k

z = −y

z = −f (x)

=⇒ =⇒

❝♦♠❡ ❢r♦♠ ❛ ❢✉♥❝t✐♦♥ ❛♣♣❧✐❡❞

❤♦r✐③♦♥t❛❧✿

s❤✐❢t ❜② k

t −−−−−−−−−→ ✢✐♣

t −−−−−−→

str❡t❝❤ ❜② k

y = f (x), z = h(f (x))

z = y + k =⇒

y −−−−−−−−−−→ z = y · k

❚❤❡ tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡

❛❢t❡r f ✿

x = g(t)

z = f (x)k

❜❡❢♦r❡ f ✿

s✉❜st✐t✉t❡❞ ✐♥t♦

y = f (x), y = f (g(t))

x = t − k =⇒

x = −t

t −−−−−−−−−−→ x = t/k

y = f (t − k)

=⇒

y = f (−t)

=⇒

y = f (t/k)

❊①❛♠♣❧❡ ✸✳✶✵✳✶✵✿ ✈❡rt✐❝❛❧ str❡t❝❤✐♥❣ ✈s✳ ❤♦r✐③♦♥t❛❧ s❤r✐♥❦✐♥❣ ■t ♠✐❣❤t s❡❡♠ t❤❛t ✈❡rt✐❝❛❧ str❡t❝❤✐♥❣ ✐s s♦♠❡❤♦✇ ❡q✉✐✈❛❧❡♥t t♦ ❤♦r✐③♦♥t❛❧ s❤r✐♥❦✐♥❣✳ ❲❡ ✇♦✉❧❞ ♥❡✈❡r ♠❛❦❡ t❤❛t ♠✐st❛❦❡ ✇❤❡♥ ❞❡❛❧✐♥❣ ✇✐t❤ ❛ r❡❛❧✲❧✐❢❡ ♦❜ ❥❡❝t✱ ❜✉t ✇❡ ♠✐❣❤t ❜❡ ❝♦♥❢✉s❡❞ ❜② ❛♥ s✉❝❤ ❛s ❛ ❧✐♥❡ ♦r ❛ ♣❛r❛❜♦❧❛✳✳✳ ❈♦♥s✐❞❡r

f (x) = x . ❙tr❡t❝❤❡❞ ❜② ❛ ❢❛❝t♦r

2

✈❡rt✐❝❛❧❧②✱ ✐t ❜❡❝♦♠❡s✿

F (x) = 2x . ❙❤r✉♥❦ ❜② ❛ ❢❛❝t♦r

2

❤♦r✐③♦♥t❛❧❧②✱ ✐t ❜❡❝♦♠❡s✿

Q(x) = x



1 = 2x . 2

✐♥✜♥✐t❡

s❤❛♣❡✱

✸✳✶✵✳

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

❚❤✐s

✐s

t❤❡ s❛♠❡ ❢✉♥❝t✐♦♥✦

❍♦✇❡✈❡r✱ ✐❢ ✇❡ ❞♦ t❤❡ s❛♠❡ t♦

❚❤✐s ✐s

✷✾✵

♥♦t

f (x) = x2 ✱

✇❡ ❞✐s❝♦✈❡r t❤❛t

2x2 6= (2x)2 . t❤❡ s❛♠❡ ❢✉♥❝t✐♦♥✦

▲❡t✬s tr② t❤❡ ❤♦r✐③♦♥t❛❧ str❡t❝❤ ❜②

√

2✳

❚❤❡♥

√ H(x) = ( 2x)2 . ◆♦✇ t❤❡r❡ ✐s ❛

♠❛t❝❤

✿

√

2x


2

= 2x2 .

❲❡ ❞♦♥✬t ❡①♣❡❝t s✉❝❤ ❛ ♠❛t❝❤ ❢♦r ♠♦st ❢✉♥❝t✐♦♥s❀ ❥✉st tr②

f (x) = 1 ✦

❈❤❛♣t❡r ✹✿ ❚❤❡ ♠❛✐♥ ❝❧❛ss❡s ♦❢ ❢✉♥❝t✐♦♥s

❈♦♥t❡♥ts ✹✳✶ ❚❤❡ s✐♠♣❧❡st ❢✉♥❝t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✳✷ ▼♦♥♦t♦♥✐❝✐t② ❛♥❞ t❤❡ ❡①tr❡♠❡ ✈❛❧✉❡s ♦❢ ❢✉♥❝t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✳✸ ❋✉♥❝t✐♦♥s ✇✐t❤ s②♠♠❡tr✐❡s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✳✹ ◗✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✳✺ ❚❤❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✳✻ ❚❤❡ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✳✼ ❚❤❡ r♦♦t ❢✉♥❝t✐♦♥s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✳✽ ❚❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✳✾ ❚❤❡ ❧♦❣❛r✐t❤♠✐❝ ❢✉♥❝t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✳✶✵ ❚❤❡ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

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

✹✳✶✳ ❚❤❡ s✐♠♣❧❡st ❢✉♥❝t✐♦♥s

■♥ t❤✐s ❝❤❛♣t❡r✱ ✇❡ ✇✐❧❧ st✉❞② ♠❛♥② s♣❡❝✐✜❝ ❢✉♥❝t✐♦♥s ❛s ✇❡❧❧ ❛s s♦♠❡ ❜r♦❛❞ ❝❛t❡❣♦r✐❡s ♦❢ ❢✉♥❝t✐♦♥s✳ ❲❡ st❛rt ✇✐t❤ t❤❡ ❢♦r♠❡r✳ ❊✈❡♥ ✐♥ t❤❡ ♠♦st ❣❡♥❡r❛❧ s✐t✉❛t✐♦♥ ✕ ♥♦t❤✐♥❣ ❜✉t s❡ts ✕ t❤❡r❡ ❛r❡ ❛❧✇❛②s t✇♦ ❢✉♥❝t✐♦♥s t❤❛t ❛r❡ ✈❡r② s✐♠♣❧❡✳ ▲❡t✬s t✉r♥ t♦ t❤❡ ❡①❛♠♣❧❡ ♦❢ t❤❡ t✇♦ s❡ts ✇❡ ❝♦♥s✐❞❡r❡❞ ✐♥ ❈❤❛♣t❡r ✷✿

• X

✐s t❤❡ ✜✈❡ ❜♦②s❀ ❛♥❞

• Y

✐s t❤❡ ❢♦✉r ❜❛❧❧s✳

◆♦✇✱ ✇❤❛t ✐❢

❛❧❧

❜♦②s ♣r❡❢❡r ❜❛s❦❡t❜❛❧❧❄ ❚❤❡♥ t❤❡ ✏♣r❡❢❡r❡♥❝❡ ❢✉♥❝t✐♦♥✑✱

F✱

❝❛♥♥♦t ❜❡ s✐♠♣❧❡r✿ ❆❧❧ ♦❢ ✐ts

✈❛❧✉❡s ❛r❡ ❡q✉❛❧ ❛♥❞ ❛❧❧ t❤❡ ❛rr♦✇s ♣♦✐♥t t♦ t❤❡ ❜❛s❦❡t❜❛❧❧✿

❚❤❡ t❛❜❧❡ ♦❢ t❤✐s ❢✉♥❝t✐♦♥

F

✐s ❛❧s♦ ✈❡r② s✐♠♣❧❡✿ ❆❧❧ ❝r♦ss❡s ❛r❡ ✐♥ t❤❡ s❛♠❡ ❝♦❧✉♠♥✳ ❚❤❡ ❣r❛♣❤ ✐s ❥✉st ❛s

s✐♠♣❧❡✿ ❆❧❧ ❞♦ts ❛r❡ ♦♥ t❤❡ s❛♠❡ ❤♦r✐③♦♥t❛❧ ❧✐♥❡✳ ❚❤❡ ✈❛❧✉❡ ♦❢

y = F (x)

❞♦❡s♥✬t ✈❛r② ❛s

x

✈❛r✐❡s❀ ✐t ✐s

❝♦♥st❛♥t✳

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

✹✳✶✳

❚❤❡ s✐♠♣❧❡st ❢✉♥❝t✐♦♥s

✷✾✷

❉❡✜♥✐t✐♦♥ ✹✳✶✳✶✿ ❝♦♥st❛♥t ❢✉♥❝t✐♦♥ ❙✉♣♣♦s❡ s❡ts

X

❛♥❞

Y

❛r❡ ❣✐✈❡♥✳ ❆ ❢✉♥❝t✐♦♥

f :X →Y

❢✉♥❝t✐♦♥ ✐❢✱ ❢♦r s♦♠❡ s♣❡❝✐✜❡❞ ❡❧❡♠❡♥t b ♦❢ Y ✱ ✇❡ s❡t✿

✐s ❝❛❧❧❡❞ ❛

❝♦♥st❛♥t

f (x) = b ❋❖❘ ❊❆❈❍ x

❚❤❡ ♣r♦❝❡ss ✐s ✐❞❡♥t✐❝❛❧ ❢♦r ❡✈❡r② ✐♥♣✉t✿

x →

❝❤♦♦s❡

3

→ y

■♥ t❤❡ ❣❡♥❡r✐❝ ✐❧❧✉str❛t✐♦♥ ❜❡❧♦✇✱ ❛❧❧ ❛rr♦✇s ❝♦♥✈❡r❣❡ ♦♥ ❛ s✐♥❣❧❡ ♦✉t♣✉t✿

❊①❡r❝✐s❡ ✹✳✶✳✷ ❲❤❛t ✐s t❤❡ r❛♥❣❡ ♦❢ ❛ ❝♦♥st❛♥t ❢✉♥❝t✐♦♥❄

❚❤❡ ❣r❛♣❤ ♦❢ ❛ ❝♦♥st❛♥t

♥✉♠❡r✐❝❛❧

❢✉♥❝t✐♦♥ ✐s ❛ ❤♦r✐③♦♥t❛❧ ❧✐♥❡✿

❖❢ ❝♦✉rs❡✱ ✐❢ t❤❡ ❞♦♠❛✐♥ ✐s ❞✐s❝♦♥♥❡❝t❡❞✱ t❤❡♥ s♦ ✐s t❤❡ ❣r❛♣❤✳

❊①❛♠♣❧❡ ✹✳✶✳✸✿ st❡♣ ❢✉♥❝t✐♦♥ ❈♦♥st❛♥t ❢✉♥❝t✐♦♥s ❛r❡ ❝♦♥✈❡♥✐❡♥t ❜✉✐❧❞✐♥❣ ❜❧♦❝❦s ❢♦r ♠♦r❡ ❝♦♠♣❧❡① ❢✉♥❝t✐♦♥s✳

❍❡r❡ ✐s ❛ ❢❛♠✐❧✐❛r

❡①❛♠♣❧❡ ♦❢ ❤♦✇ ✇❡ ❜✉✐❧❞ ❢r♦♠ t❤r❡❡ ❝♦♥st❛♥t ❢✉♥❝t✐♦♥s ❛ s✐♥❣❧❡ ✏♣✐❡❝❡✇✐s❡ ❝♦♥st❛♥t✑ ❢✉♥❝t✐♦♥✿

y = sign(x) : x → ■t ✐s ❛❧s♦ ♦❢t❡♥ ❝❛❧❧❡❞ ❛

st❡♣ ❢✉♥❝t✐♦♥ ✿

x

ր ✐❢ x>0

❝❤♦♦s❡

1

❝❤♦♦s❡

0

ց ✐❢ x 0✱ t❤❡♥ f ✐s str✐❝t❧② ❞❡❝r❡❛s✐♥❣ ♦♥ (−∞, h) ❛♥❞ str✐❝t❧② ✐♥❝r❡❛s✐♥❣ ♦♥ (h, +∞)✳ • ■❢ a < 0✱ t❤❡♥ f ✐s str✐❝t❧② ✐♥❝r❡❛s✐♥❣ ♦♥ (−∞, h) ❛♥❞ str✐❝t❧② ❞❡❝r❡❛s✐♥❣ ♦♥ (h, +∞)✳ ❚❤✐♥❦✐♥❣ ❣❡♦♠❡tr✐❝❛❧❧②✱ t❤❡ ❣r❛♣❤ ♦❢ ❛ str✐❝t❧② ✐♥❝r❡❛s✐♥❣ ❢✉♥❝t✐♦♥ ❝❛♥✬t ❝♦♠❡ ❜❛❝❦ ❛♥❞ ❝r♦ss ❛ ❤♦r✐③♦♥t❛❧ ❧✐♥❡ ❢♦r t❤❡ s❡❝♦♥❞ t✐♠❡✿

■t✱ t❤❡r❡❢♦r❡✱ ♣❛ss❡s t❤❡ ❍♦r✐③♦♥t❛❧

▲✐♥❡ ❚❡st✳ ❚❤❡ r❡s✉❧t ❜❡❧♦✇ ❢♦❧❧♦✇s✳

❚❤❡♦r❡♠ ✹✳✷✳✶✾✿ ▼♦♥♦t♦♥❡ ✈s✳ ❖♥❡✲t♦✲♦♥❡ ❆❧❧ str✐❝t❧② ♠♦♥♦t♦♥❡ ❢✉♥❝t✐♦♥s ❛r❡ ♦♥❡✲t♦✲♦♥❡✳

Pr♦♦❢✳

❆❧❣❡❜r❛✐❝❛❧❧②✱ ✇❡ st❛rt ✇✐t❤ t❤❡ ❞❡✜♥✐t✐♦♥ ❛♥❞ ❝♦♥❝❧✉s✐♦♥ ❢♦❧❧♦✇s✿ f ր ✇❤❡♥✿

❍❡♥❝❡ f ✐s ♦♥❡✲t♦✲♦♥❡✳

x 0 t❤❛t f (x + T ) = f (x) ❋❖❘ ❊❆❈❍ x

❚❤❡ s♠❛❧❧❡st s✉❝❤ T ✐s ❝❛❧❧❡❞ t❤❡ ♣❡r✐♦❞ ♦❢ f ✳ ❙✉❝❤ ❛ ❢✉♥❝t✐♦♥ ✐s r❡♣❡t✐t✐✈❡✿ ✐t ❣♦❡s ♦✈❡r t❤❡ s❛♠❡ ✈❛❧✉❡ ✐♥ t❤❡ s❛♠❡ ♠❛♥♥❡r✳ ❚❤❛t ✐s ✇❤② t❤❡ ♠❛✐♥ r❡❛s♦♥ t♦ st✉❞② ♣❡r✐♦❞✐❝ ❢✉♥❝t✐♦♥s ✐s t♦ r❡♣r❡s❡♥t ✐❞❡❛❧✐③❡❞ r❡♣❡t✐t✐✈❡ ♣r♦❝❡ss❡s✱ s✉❝❤ ❛s t❤❡s❡✿ • t❤❡ t❡♠♣❡r❛t✉r❡ ❞✉r✐♥❣ t❤❡ ②❡❛r ✐♥ ❛ ♣❛rt✐❝✉❧❛r ❧♦❝❛t✐♦♥

• t❤❡ ❞❛②❧✐❣❤t ♦✈❡r t❤❡ ②❡❛r✱ t❤❡ t✐♠❡ ♦❢ t❤❡ s✉♥s❡t ❛♥❞ t❤❡ s✉♥r✐s❡ • t❤❡ ♦s❝✐❧❧❛t✐♦♥ ♦❢ ❛ str✐♥❣ • s♦✉♥❞ ✇❛✈❡s✱ ❡t❝✳

❊①❡r❝✐s❡ ✹✳✸✳✶✺

❉❡s❝r✐❜❡ s✉❝❤ ❛ ❢✉♥❝t✐♦♥ ❛s ❛ tr❛♥s❢♦r♠❛t✐♦♥✳

✹✳✸✳

❋✉♥❝t✐♦♥s ✇✐t❤ s②♠♠❡tr✐❡s

✸✶✺

❊①❛♠♣❧❡ ✹✳✸✳✶✻✿ ❝♦♥st❛♥t ✐s ♣❡r✐♦❞✐❝

❈♦♥st❛♥t ❢✉♥❝t✐♦♥s ❛r❡ ✕ tr✐✈✐❛❧❧② ✕ ♣❡r✐♦❞✐❝✱ ❢♦r ❛♥② ❛♥❞ ❡✈❡r②

T✿

f (x) = c = f (x + T ) . ❊①❡r❝✐s❡ ✹✳✸✳✶✼

❲❤❛t ✐s ✐ts ♣❡r✐♦❞❄

❊①❛♠♣❧❡s ❜❡②♦♥❞ t❤❡ tr✐✈✐❛❧ ❛r❡ ❤❛r❞❡r t♦ ❝♦♠❡ ❜②✱ ❢♦r ♥♦✇✳ ❋♦r ❛ ❢✉♥❝t✐♦♥

f (x + T ) ❛♥❞ f (x) = mx + b, m 6= 0✱ ✐s ♣❡r✐♦❞✐❝✿

✇❡ ♥❡❡❞ t♦ st❛rt ✇✐t❤

✇♦r❦ ♦✉r ✇❛② t♦

f (x)✱

y = f (x)

❣✐✈❡♥ ❜② ❛ ❢♦r♠✉❧❛✱

✐❞❡♥t✐❝❛❧❧②✳ ❋♦r ❡①❛♠♣❧❡✱ ♥♦ ❧✐♥❡❛r ♣♦❧②♥♦♠✐❛❧

f (x + T ) = m(x + T ) + b = mx + mT + b = f (x) + mT 6= f (x) , ♥♦ ♠❛tt❡r t❤❡ ❝❤♦✐❝❡ ♦❢

T✳

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

❲❡ ❝❛♥✱ ❤♦✇❡✈❡r✱ ♣✐❝❦ t❤❡

❛❧t❡r♥❛t✐♥❣ s❡q✉❡♥❝❡ ✿ an = (−1)n ,

❞❡✜♥❡❞ ♦✈❡r ❛❧❧ ✐♥t❡❣❡rs✿

■t ✐s ♣❡r✐♦❞✐❝ ✇✐t❤ ♣❡r✐♦❞

T = 2✳

❊①❛♠♣❧❡ ✹✳✸✳✶✾✿ ❢r❛❝t✐♦♥❛❧ ♣❛rt

❆♥♦t❤❡r s✉❝❤ ❢✉♥❝t✐♦♥ ✕ ✇✐t❤ t❤❡ ❞♦♠❛✐♥ ❛❧❧ r❡❛❧s ✕ ✐s t❤❡ s♦✲❝❛❧❧❡❞ ✏❢r❛❝t✐♦♥❛❧ ♣❛rt✑✿

{x} = x − [x] . ■ts ❣r❛♣❤ ❝♦♥s✐sts ♦❢ ❞✐❛❣♦♥❛❧ s❡❣♠❡♥ts✿

■ts ♣❡r✐♦❞ ✐s

1✳

▼♦r❡ ❡①❛♠♣❧❡s ✕ t❤❡ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s ✕ ✇✐❧❧ ❛♣♣❡❛r ❧❛t❡r ✐♥ t❤❡ ❝❤❛♣t❡r✳

❲❤❛t ❞♦❡s t❤❡ ✈❡rt✐❝❛❧ ✢✐♣ ❞♦ t♦ ❣r❛♣❤s❄ ❲✐t❤ t❤❡ ❡①❝❡♣t✐♦♥ ♦❢ ❜❡ tr❛♥s❢♦r♠❡❞ ✐♥t♦ ✐ts❡❧❢ ✇✐t❤ t❤✐s ✢✐♣ ❛s ✐t ✇♦✉❧❞ ✈✐♦❧❛t❡ t❤❡

f (x) = 0✱ t❤❡ ❣r❛♣❤ ♦❢ ♥♦ ❢✉♥❝t✐♦♥ ✇✐❧❧ ❡✈❡r

❱❡rt✐❝❛❧ ▲✐♥❡ ❚❡st ✿

✹✳✸✳ ❋✉♥❝t✐♦♥s ✇✐t❤ s②♠♠❡tr✐❡s

✸✶✻

❚❤❡ ❤♦r✐③♦♥t❛❧ ✢✐♣ ❧♦♦❦s ♠♦r❡ ♣r♦♠✐s✐♥❣✿

■t ♣r❡s❡r✈❡s y ❛♥❞ ✢✐♣s t❤❡ s✐❣♥ ♦❢ x✿ ❤♦r✐③♦♥t❛❧ ✢✐♣

(x, y) −−−−−−−−−−−−→ (−x, y)

❙♦✱ ✐❢ t❤❡ ❣r❛♣❤ ♦❢ y = G(x) ✐s t❤❡ ❣r❛♣❤ ♦❢ y = f (x) ✢✐♣♣❡❞ ❤♦r✐③♦♥t❛❧❧②✱ t❤❡♥ G(x) = f (−x) .

❚❤❡ ♥❡✇ ❢✉♥❝t✐♦♥ ♠✉st ❜❡ t❤❡ ♦r✐❣✐♥❛❧ ♦♥❡✱ f ✱ ❢♦r t❤❡r❡ t♦ ❜❡ s②♠♠❡tr②✿

❲❡ ❝❛♥ ❛❧s♦ t❛❦❡ ❛❞✈❛♥t❛❣❡ ♦❢ t❤❡ ❢❛❝t t❤❛t ♦✉r ❜♦❞✐❡s ❤❛✈❡ ❛ ♠✐rr♦r s②♠♠❡tr②❀ ♦♥❡ ❝❛♥ tr❛❝❡ t❤❡ t✇♦ ❤❛❧✈❡s ♦❢ t❤❡ ❣r❛♣❤ ✇✐t❤ ❤✐s t✇♦ ❤❛♥❞s ♠♦✈✐♥❣ ❛✇❛② ❢r♦♠ t❤❡ ❝❡♥t❡r✿

❲❡ ❤❛✈❡ ❛♥ ✐❞❡♥t✐t② t♦ ❜❡ ✈❡r✐✜❡❞✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ✇✐❧❧ ❜❡ r♦✉t✐♥❡❧② ✉s❡❞✳

❉❡✜♥✐t✐♦♥ ✹✳✸✳✷✵✿ ❡✈❡♥ ❢✉♥❝t✐♦♥ ❆ ❢✉♥❝t✐♦♥ f ✐s ❝❛❧❧❡❞ ❡✈❡♥ ✐❢ ✐t s❛t✐s✜❡s✿ f (−x) = f (x) ❋❖❘ ❊❆❈❍ x

✐♥ ✐ts ❞♦♠❛✐♥✳ ❚❤❡ ❞❡✜♥✐t✐♦♥ st❛t❡s t❤❛t t❤❡s❡ t✇♦ ❢✉♥❝t✐♦♥s ❛r❡ t❤❡ s❛♠❡✦

❊①❛♠♣❧❡ ✹✳✸✳✷✶✿ s②♠♠❡tr② ♦❢ sq✉❛r✐♥❣ ❢✉♥❝t✐♦♥ ❚❤❡ ❣r❛♣❤ ♦❢ f (x) = x2 s❡❡♠s t♦ ❡①❤✐❜✐t t❤❡ ❤♦r✐③♦♥t❛❧ ♠✐rr♦r s②♠♠❡tr②✿

✹✳✸✳

❋✉♥❝t✐♦♥s ✇✐t❤ s②♠♠❡tr✐❡s

❍♦✇❡✈❡r✱ ❧❡t✬s ❡①❛♠✐♥❡ t❤❡

f (−x)

✸✶✼

❛❧❣❡❜r❛✳

❛♥❞ ✇♦r❦ ♦✉r ✇❛② t♦

f (x)✱

❋♦r ❛ ❢✉♥❝t✐♦♥

❢♦r ❛❧❧

x✬s

y = f (x)

❣✐✈❡♥ ❜② ❛ ❢♦r♠✉❧❛✱ ✇❡ ♥❡❡❞ t♦ st❛rt ✇✐t❤

❛t t❤❡ s❛♠❡ t✐♠❡✳ ❚❤❡ ❢❛❝t t❤❛t t❤❡ sq✉❛r✐♥❣ ❢✉♥❝t✐♦♥ ✐s

❡✈❡♥ ✐s ♣r♦✈❡♥ ❛❧❣❡❜r❛✐❝❛❧❧② ❛s ❢♦❧❧♦✇s✿

f (−x) = (−x)2 = (−x) · (−x) = (−1)x · (−1)x = (−1)(−1)x2 = x2 = f (x) .

❨❡s✦

❊①❡r❝✐s❡ ✹✳✸✳✷✷ Pr♦✈❡ t❤❡ s❛♠❡ ❢♦r ♦t❤❡r ❡✈❡♥ ♣♦✇❡rs✳

❊①❡r❝✐s❡ ✹✳✸✳✷✸ Pr♦✈❡ t❤❡ s❛♠❡ ❢♦r t❤❡ ❛❜s♦❧✉t❡ ✈❛❧✉❡ ❢✉♥❝t✐♦♥

| − x| = x . ❊①❡r❝✐s❡ ✹✳✸✳✷✹ ❉❡s❝r✐❜❡ ❛♥ ❡✈❡♥ ❢✉♥❝t✐♦♥ ❛s ❛ tr❛♥s❢♦r♠❛t✐♦♥✳

❊①❛♠♣❧❡ ✹✳✸✳✷✺✿ ♥♦♥✲s②♠♠❡tr② ♦❢ ❝✉❜✐❝ ♣♦✇❡r ■♥ t❤❡ ❛❜♦✈❡ ❡①❛♠♣❧❡✱ t❤❡ ♥❡❣❛t✐✈❡ s✐❣♥ ❞✐s❛♣♣❡❛rs✳ ❚❤✐s ✇♦♥✬t ❤❛♣♣❡♥ ✇✐t❤ t❤❡ ❝✉❜✐❝ ♣♦✇❡r✳ ■♥❞❡❡❞✱ ✇❡ ❤❛✈❡ ❛ ♠✐s♠❛t❝❤ ❢♦r ❡✈❡r②

x 6= 0✿

f (−x) = (−x)3 = (−x) · (−x) · (−x) = (−1)3 x3 = −x3 6= x3 = f (x) . ◆❡①t✱ t❤❡

❝❡♥tr❛❧ s②♠♠❡tr②

t❤❡ ♣❧❛♥❡ ✐♥ ❤❛❧❢ ❛❧♦♥❣ t❤❡

❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ❛s

y ✲❛①✐s

t✇♦ ❝♦♥s❡❝✉t✐✈❡ ♠✐rr♦r s②♠♠❡tr✐❡s✳

❛♥❞ t❤❡♥ ❛❧♦♥❣ t❤❡

x✲❛①✐s

◆♦✦

■♥❞❡❡❞✱ ✇❡ ❥✉st ❢♦❧❞

t♦ ♠❛❦❡ ♦♥❡ ❜r❛♥❝❤ ❧❛♥❞ ♦♥ t♦♣ ♦❢ t❤❡ ♦t❤❡r✿

✹✳✸✳

❋✉♥❝t✐♦♥s ✇✐t❤ s②♠♠❡tr✐❡s

✸✶✽

❚❤❡ t✇♦ ♦❢ t❤❡♠ ❝♦♠❜✐♥❡❞ ❞♦♥✬t ❣✐✈❡ ❛ ♠✐rr♦r ✐♠❛❣❡ ❛♥②♠♦r❡❀ ✐t✬s ✉♥❞❡rst♦♦❞ ❛❧s♦ ❛s ❛

180

❞❡❣r❡❡s

r♦t❛t✐♦♥ t❤r♦✉❣❤

❛r♦✉♥❞ t❤❡ ♦r✐❣✐♥✿

❚❤❡ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ✈❡rt✐❝❛❧ ❛♥❞ ❤♦r✐③♦♥t❛❧ ✢✐♣s ✏✢✐♣s✑ t❤❡ s✐❣♥s ♦❢ ❜♦t❤

x

❛♥❞

y✿

❝❡♥tr❛❧ s②♠♠❡tr②

(x, y) −−−−−−−−−−−−−−→ (−x, −y) ❇❡❧♦✇✱ ✇❡ ❣♦ ❢r♦♠ ❣r❡❡♥ t♦ r❡❞ ❛♥❞ t❤❡♥ ❢r♦♠ r❡❞ t♦ ❜❧✉❡✿

❙♦✱ ✐❢ t❤❡ ❣r❛♣❤ ♦❢

y = G(x)

✐s t❤❡ ❣r❛♣❤ ♦❢

y = f (x)

✢✐♣♣❡❞ ❤♦r✐③♦♥t❛❧❧② ❛♥❞ ✈❡rt✐❝❛❧❧②✱ t❤❡♥

G(x) = −f (−x) . ❚❤❡ ♥❡✇ ❢✉♥❝t✐♦♥ ♠✉st ❜❡ t❤❡ ♦r✐❣✐♥❛❧ ♦♥❡✱

f✱

❢♦r t❤❡r❡ t♦ ❜❡ s②♠♠❡tr②✿

❲❡ ❤❛✈❡ ❛♥ ✐❞❡♥t✐t② t♦ ❜❡ ✈❡r✐✜❡❞✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ✇✐❧❧ ❜❡ r♦✉t✐♥❡❧② ✉s❡❞✳

❉❡✜♥✐t✐♦♥ ✹✳✸✳✷✻✿ ♦❞❞ ❢✉♥❝t✐♦♥ ❆ ❢✉♥❝t✐♦♥ f ✐s ❝❛❧❧❡❞ ♦❞❞ ✐❢ ✐t s❛t✐s✜❡s✿ f (−x) = −f (x) ❋❖❘ ❊❆❈❍ x ✐♥ ✐ts ❞♦♠❛✐♥✳

❊①❛♠♣❧❡ ✹✳✸✳✷✼✿ s②♠♠❡tr② ♦❢ ❝✉❜✐❝ ❢✉♥❝t✐♦♥ ❚❤❡ ❣r❛♣❤ ♦❢

f (x) = x3

s❡❡♠s t♦ ❡①❤✐❜✐t ❝❡♥tr❛❧ s②♠♠❡tr②✿

✹✳✸✳

❋✉♥❝t✐♦♥s ✇✐t❤ s②♠♠❡tr✐❡s

❍♦✇❡✈❡r✱ ❧❡t✬s ❡①❛♠✐♥❡ t❤❡ t♦ st❛rt ✇✐t❤

−f (−x)

✸✶✾

❛❧❣❡❜r❛✳

❖♥❝❡ ❛❣❛✐♥✱ ✐❢ ✇❡ ❤❛✈❡ ❛ ❢♦r♠✉❧❛ ❢♦r ❛ ❢✉♥❝t✐♦♥

❛♥❞ ✇♦r❦ ♦✉r ✇❛② t♦

f (x)✱

y = f (x)✱

✇❡ ♥❡❡❞

✐❞❡♥t✐❝❛❧❧②✳ ❚❤❡ ❝✉❜✐❝ ♣♦✇❡r ✐s ♦❞❞✿

−f (−x) = −(−x)3 = −(−x) · (−x) · (−x) = −(−1)3 x3 = x3 = f (x) .

❨❡s✦

❊①❡r❝✐s❡ ✹✳✸✳✷✽ Pr♦✈❡ t❤❡ s❛♠❡ ❢♦r ♦t❤❡r

♦❞❞

♣♦✇❡rs✳

❊①❡r❝✐s❡ ✹✳✸✳✷✾ Pr♦✈❡ t❤❛t t❤❡ s✐❣♥ ❢✉♥❝t✐♦♥ ✐s ❛❧s♦ ♦❞❞✿

− sign(−x) = sign(x) . ❊①❡r❝✐s❡ ✹✳✸✳✸✵ ❉❡s❝r✐❜❡ ❛♥ ♦❞❞ ❢✉♥❝t✐♦♥ ❛s ❛ tr❛♥s❢♦r♠❛t✐♦♥✳

❊①❛♠♣❧❡ ✹✳✸✳✸✶✿ ♥♦♥✲s②♠♠❡tr② ♦❢ sq✉❛r❡ ❢✉♥❝t✐♦♥ ■♥ t❤❡ ❛❜♦✈❡ ♣r♦♦❢✱ t❤❡ ♥❡❣❛t✐✈❡ s✐❣♥ ✐s ♣r❡s❡r✈❡❞✱ ❜② ❞❡s✐❣♥✳ ❚❤✐s ❝❛♥✬t ❤❛♣♣❡♥ ✇✐t❤ t❤❡ sq✉❛r❡ ♣♦✇❡r✳ ■♥❞❡❡❞✱ ✇❡ ❤❛✈❡ ❛ ♠✐s♠❛t❝❤ ❢♦r ❡✈❡r②

x 6= 0✿

−f (−x) = −(−x)2 = −(−x) · (−x) = −(−1)x · (−1)x = −(−1)(−1)x2 = −x2 6= x2 = f (x). ❙♦✱ ❛♥ ❡✈❡♥ ♥✉♠❜❡r ♦❢ ♦❢

(−1)✬s

1✿

♣r♦❞✉❝❡ ❛

(−1)(−1) = 1 . ❆♥ ♦❞❞ ♥✉♠❜❡r ♦❢ ♦❢

(−1)✬s

♣r♦❞✉❝❡ ❛

−1✿ (−1)(−1)(−1) = −1 .

❊①❛♠♣❧❡ ✹✳✸✳✸✷✿ s②♠♠❡tr② ❛♥❞ ♥♦♥✲s②♠♠❡tr② ♦❢ r❡❝✐♣r♦❝❛❧ ❚❤❡ ❢✉♥❝t✐♦♥

1/x

s❡❡♠s t♦ ❤❛✈❡ t❤✐s s②♠♠❡tr②✿

◆♦✦

✹✳✸✳ ❋✉♥❝t✐♦♥s ✇✐t❤ s②♠♠❡tr✐❡s

✸✷✵

❲❡ ❝❛♥ s❡❡ ✐t ✐♥ t❤❡ ❝❛s❡ ♦❢ t❤❡ r❡st ♦❢ t❤❡ r❡❝✐♣r♦❝❛❧s ♦❢ t❤❡ ♣♦✇❡rs✱ ✇❤✐❝❤ ❛❧s♦ ❝♦♥✜r♠s t❤❡ r❡❛s♦♥ ❢♦r t❤❡ ♥❛♠❡s✿

❲❛r♥✐♥❣✦ ❆ r❛♥❞♦♠❧② ❝❤♦s❡♥✱ ♦r ❛ t②♣✐❝❛❧✱ ❢✉♥❝t✐♦♥ ✐s ♥❡✐t❤❡r ❡✈❡♥ ♥♦r ♦❞❞✳

❊①❡r❝✐s❡ ✹✳✸✳✸✸

❉❡♠♦♥str❛t❡✿ ✭❛✮ ❆ ✈❡rt✐❝❛❧ str❡t❝❤ ✇✐❧❧ ♥♦t ♣r❡s❡r✈❡ t❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥ ❡①❝❡♣t ❢♦r ❴❴❴ ✳ ✭❜✮ ❆ ❤♦r✐③♦♥t❛❧ str❡t❝❤ ✇✐❧❧ ♥♦t ♣r❡s❡r✈❡ t❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥ ❡①❝❡♣t ❢♦r ❴❴❴ ✳ ❊①❡r❝✐s❡ ✹✳✸✳✸✹

■s t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ t✇♦ ♦❞❞✴❡✈❡♥ ❢✉♥❝t✐♦♥s ♦❞❞✴❡✈❡♥❄ ❊①❡r❝✐s❡ ✹✳✸✳✸✺

❍❛❧❢ ♦❢ t❤❡ ❣r❛♣❤ ♦❢ ❛♥ ❡✈❡♥ ❢✉♥❝t✐♦♥ ✐s s❤♦✇♥ ❜❡❧♦✇❀ ♣r♦✈✐❞❡ t❤❡ ♦t❤❡r ❤❛❧❢✿

❍✐♥t✿ ❚r② t♦ tr❛❝❡ ✇✐t❤ ②♦✉r ❧❡❢t ❤❛♥❞ ✇❤❛t ②♦✉r r✐❣❤t ❤❛♥❞ ✐s tr❛❝✐♥❣✳ ❊①❡r❝✐s❡ ✹✳✸✳✸✻

❍❛❧❢ ♦❢ t❤❡ ❣r❛♣❤ ♦❢ ❛♥ ♦❞❞ ❢✉♥❝t✐♦♥ ✐s s❤♦✇♥ ❛❜♦✈❡❀ ♣r♦✈✐❞❡ t❤❡ ♦t❤❡r ❤❛❧❢✳

✹✳✹✳

◗✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧s

✸✷✶

❊①❡r❝✐s❡ ✹✳✸✳✸✼

P❧♦t t❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥ t❤❛t ✐s ❜♦t❤ ♦❞❞ ❛♥❞ ❡✈❡♥✳

❊①❡r❝✐s❡ ✹✳✸✳✸✽

■s t❤❡ ✐♥✈❡rs❡ ♦❢ ❛♥ ♦❞❞✴❡✈❡♥ ❢✉♥❝t✐♦♥ ♦❞❞✴❡✈❡♥❄

❊①❡r❝✐s❡ ✹✳✸✳✸✾

❲❤❛t ✐s t❤❡ r❡❧❛t✐♦♥✱ ✐❢ ❛♥②✱ ❜❡t✇❡❡♥ t❤❡s❡ t✇♦ ♣❛✐rs ♦❢ ♣r♦♣❡rt✐❡s✿ ✭❛✮ ❡✈❡♥ ❛♥❞ ♦♥❡✲t♦✲♦♥❡✱ ✭❜✮ ♦❞❞ ❛♥❞ ♦♥❡✲t♦✲♦♥❡❄ ❊①❡r❝✐s❡ ✹✳✸✳✹✵

❲❤❛t tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ♣❧❛♥❡ ♣r❡s❡r✈❡ t❤❡ ✏❡✈❡♥♥❡ss✑ ❛♥❞ ✏♦❞❞♥❡ss✑ ♦❢ ❢✉♥❝t✐♦♥s❄

❖✉r ❧✐st ♦❢ t❤❡ ♠♦st ❢✉♥❞❛♠❡♥t❛❧ ❝❧❛ss❡s ♦❢ ❢✉♥❝t✐♦♥s ❤❛s ❣r♦✇♥ t♦ t❤❡ ❢♦❧❧♦✇✐♥❣✿ ✶✳ ❜♦✉♥❞❡❞ ❛♥❞ ✉♥❜♦✉♥❞❡❞ ✷✳ ♦♥❡✲t♦✲♦♥❡ ❛♥❞ ♦♥t♦ ✸✳ ✐♥❝r❡❛s✐♥❣ ❛♥❞ ❞❡❝r❡❛s✐♥❣ ✹✳ ♦❞❞ ❛♥❞ ❡✈❡♥ ✺✳ ♣❡r✐♦❞✐❝ ❚❤❡ ❢✉♥❝t✐♦♥s ✐♥tr♦❞✉❝❡❞ ❜❡❧♦✇ ✭❛♥❞ t❤❡✐r r❡str✐❝t✐♦♥s✮ ✇✐❧❧ ❜❡ ♠❛t❝❤❡❞ ❛❣❛✐♥st t❤✐s ❧✐st✳

✹✳✹✳ ◗✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧s

■♥ ❈❤❛♣t❡r ✷✱ ✇❡ t❤♦r♦✉❣❤❧② st✉❞✐❡❞ t❤❡ ✏sq✉❛r❡ ❢✉♥❝t✐♦♥✑

f (x) = x2 ✳

❲❡ ♥♦✇ t❛❦❡ t❤❡ ✐❞❡❛ ♦❢ sq✉❛r✐♥❣ t❤❡

✐♥♣✉t t♦ t❤❡ ♥❡①t ❧❡✈❡❧ ❛♥❞ ❝♦♠❜✐♥❡ ✐t ✇✐t❤ ♦t❤❡r ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s✳ ❆

q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧ ✐s ❛ ❢✉♥❝t✐♦♥

❣✐✈❡♥ ❜②

f (x) = ax2 + bx + c, a 6= 0 ❲❡ s❛② t❤❛t ✐t ✐s ♣r❡s❡♥t❡❞ ❤❡r❡ ✐♥ t❤❡

st❛♥❞❛r❞ ❢♦r♠✳ ❚❤❡ ♥✉♠❜❡rs a, b, c ❛r❡ t❤❡ ♣❛r❛♠❡t❡rs t❤❛t ❞❡t❡r♠✐♥❡

❛ ❧♦t ♦❢ ❤♦✇ t❤❡ ❢✉♥❝t✐♦♥ ❜❡❤❛✈❡s✳ ❲❡ r❡q✉✐r❡ t❤❡ ✜rst ❝♦❡✣❝✐❡♥t

a t♦ ❜❡ ♥♦♥✲③❡r♦ t♦ s❡♣❛r❛t❡ t❤❡ q✉❛❞r❛t✐❝

♣♦❧②♥♦♠✐❛❧s ❢r♦♠ t❤❡ ❧✐♥❡❛r ♣♦❧②♥♦♠✐❛❧s✳ ❆s ✇❡ s❛✇ ✐♥ ❈❤❛♣t❡r ✸✱ t❤❡ ❣r❛♣❤ ♦❢ ❡✈❡r② q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧ ✐s ♠❛❞❡ ❢r♦♠ t❤❡ ✏♦r✐❣✐♥❛❧ ♣❛r❛❜♦❧❛✑ ♦❢ y = x2 ✈✐❛ s❤✐❢t✐♥❣✱ ✢✐♣♣✐♥❣✱ ❛♥❞ str❡t❝❤✐♥❣✳ ❙♦♠❡ t❤✐♥❣s ❛❜♦✉t t❤✐s ♣♦❧②♥♦♠✐❛❧ ✇✐❧❧ ❜❡ ❡❛s✐❧② ❞❡❞✉❝❡❞ ❢r♦♠ t❤♦s❡ ❛❜♦✉t t❤❡ ♦r✐❣✐♥❛❧ t❤❛t ✇❡ ❧❡❛r♥❡❞ ✐♥ ❈❤❛♣t❡r ✷ ❜✉t ♦t❤❡rs ✇✐❧❧ r❡q✉✐r❡ s♦♠❡ ✇♦r❦✳ ❚❤❡ ❞♦♠❛✐♥✱ t❤❡ ♣♦ss✐❜❧❡ ✐♥♣✉t ✈❛❧✉❡s✱ r❡♠❛✐♥s t❤❡ s❛♠❡✱ ❛❧❧ r❡❛❧s✱ ❜❡ ❜♦t❤ ♣♦s✐t✐✈❡ ❛♥❞ ♥❡❣❛t✐✈❡✿

(−∞, +∞)✳

❚❤❡ ♦✉t♣✉t ✈❛❧✉❡s ❝❛♥ ♥♦✇

✹✳✹✳

◗✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧s

✸✷✷

❏✉st ❛s ✇✐t❤ t❤❡ ♦r✐❣✐♥❛❧ ♣❛r❛❜♦❧❛✱ ✇❡ ♥♦t✐❝❡ t❤❛t t❤❡ ❣r❛♣❤ ❛❜♦✈❡ s❤♦✇s t❤❡ ❧❛❝❦ ♦❢ t❤❡ ✉♥❞❡s✐r❛❜❧❡ ❢❡❛t✉r❡s✿ ❣❛♣s ❛♥❞ ❜r❡❛❦s✱ ❝♦r♥❡rs ❛♥❞ ❝✉s♣s✳✳✳ ❉✐✛❡r❡♥t ✐♥♣✉ts ❝❛♥ ♣r♦❞✉❝❡✱ ❛❣❛✐♥✱ s❛♠❡ ♦✉t♣✉ts❀ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧s ❛r❡♥✬t ♦♥❡✲t♦✲♦♥❡✦

■♥ ❢❛❝t✱ ❛

♣❛tt❡r♥ st❛rts t♦ ❡♠❡r❣❡ ✐❢ ✇❡ ♣✉t t❤❡ ✈❛❧✉❡s ♦❢✱ s❛②✱

f (x) = x2 − 2x + 2 , ✐♥ ❛ ❧✐st✿

✖✕ ❞✐✛❡r❡♥t ✐♥♣✉ts

                  

          

−2 10  −1 5   02  11   22 35 4 10

          

         

s❛♠❡ ♦✉t♣✉ts✳✳✳ ✖✕ ❚❤❛t✬s t❤❡ ❛①✐s✦

        

❚❤❡ ✐♥♣✉ts✴♦✉t♣✉ts ❛r❡ ♣❛✐r❡❞ ✉♣ ❛s t❤❡ ❧❛tt❡r st❛rt t♦ r❡♣❡❛t t❤❡♠s❡❧✈❡s ✕ ✐♥ r❡✈❡rs❡ ♦r❞❡r ✕ ❛❢t❡r ✇❡ ♣❛ss

x = 1✳

❚❤❡r❡ ✐s✱ t❤❡r❡❢♦r❡✱ ❛

♠✐rr♦r s②♠♠❡tr② ❡✈❡♥ t❤♦✉❣❤ ♥♦t ❛t t❤❡ y✲❛①✐s✳✳✳

❋✉rt❤❡r♠♦r❡✱ ✇❡ ❦♥♦✇ t❤❛t t❤❡ ✏❧❡❛❞✐♥❣ ❝♦❡✣❝✐❡♥t✑

a

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

•

■❢

a > 0✱

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

•

■❢

a < 0✱

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

❊①❡r❝✐s❡ ✹✳✹✳✶

❉❡s❝r✐❜❡ s✉❝❤ ❛ ❢✉♥❝t✐♦♥ ❛s ❛ tr❛♥s❢♦r♠❛t✐♦♥✳

❆♥♦t❤❡r ❢❛♠✐❧✐❛r ♦❜s❡r✈❛t✐♦♥ ✐s t❤❛t t❤❡ s❧♦♣❡s ✈❛r② ❢r♦♠ ❧♦❝❛t✐♦♥ t♦ ❧♦❝❛t✐♦♥✳ ❊①❛♠♣❧❡ ✹✳✹✳✷✿ s❤♦♦t✐♥❣ ❛ ❝❛♥♥♦♥

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

50

100✲♠❡t❡r

t❛❧❧ ❜✉✐❧❞✐♥❣ ✈❡rt✐❝❛❧❧② ✉♣ ✐♥ t❤❡ ❛✐r ✇✐t❤ ❛ s♣❡❡❞

♠❡t❡rs ♣❡r s❡❝♦♥❞ ✇✐t❤ ❛ ❣r❛✈✐t②✲❝❛✉s❡❞ ❛❝❝❡❧❡r❛t✐♦♥ ♦❢

−9.8

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

y = f (x) =

−9.8 2 x + 50x + 100 . 2

♠❡t❡rs ♣❡r s❡❝♦♥❞ sq✉❛r❡❞✳ ❚❤❡

✹✳✹✳

◗✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧s

✸✷✸

❲❡ ♣❧♦t ❜♦t❤ t❤❡ ❛❧t✐t✉❞❡ ✭✜rst ♣❧♦t✮ ❛♥❞ t❤❡ ✈❡❧♦❝✐t② ♦❢ t❤✐s ♣r♦❥❡❝t✐❧❡ ✭s❡❝♦♥❞ ♣❧♦t✮✿

❖❢ ❝♦✉rs❡✱ t❤❡ ✈❡❧♦❝✐t② ✐s ❢♦✉♥❞ ❛s t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ♦❢ t❤❡ ❛❧t✐t✉❞❡✳

❊①❡r❝✐s❡ ✹✳✹✳✸ ✭❛✮ ❲❤❡♥ ❞♦❡s t❤❡ ♣r♦❥❡❝t✐❧❡ r❡❛❝❤ t❤❡ ❤✐❣❤❡st ♣♦✐♥t❄ ✭❜✮ ❍♦✇ ❢❛st ❞♦❡s ✐t ❣♦ ❛t t❤❛t ♠♦♠❡♥t❄ ✭❝✮ ❍♦✇ ❢❛st ❞♦❡s ✐t ❤✐t t❤❡ ❣r♦✉♥❞❄

❊①❡r❝✐s❡ ✹✳✹✳✹ ✭❛✮ P❧♦t t❤❡ ❛❧t✐t✉❞❡ ❛♥❞ t❤❡ ✈❡❧♦❝✐t② ♦❢ ❛ ♣r♦❥❡❝t✐❧❡ ❧❛✉♥❝❤❡❞ ♦✉t ♦❢ ❛ 10✲♠❡t❡r ❞❡❡♣ tr❡♥❝❤ ✇✐t❤ ❛ s♣❡❡❞ ♦❢ 20 ♠❡t❡rs ♣❡r s❡❝♦♥❞✳ ✭❜✮ ❆♥s✇❡r t❤❡ q✉❡st✐♦♥s ❛s❦❡❞ ✐♥ t❤❡ ❧❛st ❡①❡r❝✐s❡✳ ❚❤❡ ✐♥❢♦r♠❛t✐♦♥ ✇❡ ❤❛✈❡ ❝♦❧❧❡❝t❡❞ ♣r❡✈✐♦✉s❧② ❛❧❧♦✇s ✉s t♦ ❡❛s✐❧② ❛♥s✇❡r t❤❡ q✉❡st✐♦♥s ♦♥ ♦✉r ❧✐st✿ ✶✳ ❇♦✉♥❞❡❞ ♦r ✉♥❜♦✉♥❞❡❞❄ ❯♥❜♦✉♥❞❡❞✦ ✷✳ ❖♥❡✲t♦✲♦♥❡ ♦r ♦♥t♦❄ ◆♦ ❛♥❞ ◆♦✦

✸✳ ■♥❝r❡❛s✐♥❣ ♦r ❞❡❝r❡❛s✐♥❣❄ ■♥❝r❡❛s✐♥❣ t❤❡♥ ❞❡❝r❡❛s✐♥❣ ♦r ✈✐❝❡ ✈❡rs❛✦

✹✳ ❖❞❞ ♦r ❡✈❡♥❄ ❊✈❡♥ ❜✉t ♦♥❧② ✐❢ ✇❡ s❤✐❢t ✐t✦ ✺✳ P❡r✐♦❞✐❝❄ ◆♦✦

❊①❡r❝✐s❡ ✹✳✹✳✺ ❏✉st✐❢② ❛♥❞ ❡❧❛❜♦r❛t❡ ♦♥ t❤❡ ❛♥s✇❡rs✳ ❇❡❧♦✇ ✇❡ ✇✐❧❧ ❧♦♦❦ ❛t ♠♦r❡ s✉❜t❧❡ ❢❡❛t✉r❡s ♦❢ t❤✐s ❢✉♥❝t✐♦♥✳ ❘❡❝❛❧❧ t❤❛t t❤❡ x✲✐♥t❡r❝❡♣ts ♦❢ ❛ ❢✉♥❝t✐♦♥ y = f (x) ❛r❡ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❡q✉❛t✐♦♥ f (x) = 0✳ ■♥ t❤❡ ♣r❡s❡♥t ❝♦♥t❡①t✱ ✇❡ ♦❢t❡♥ ✉s❡ ❛ ❞✐✛❡r❡♥t ❧❛♥❣✉❛❣❡✳

❉❡✜♥✐t✐♦♥ ✹✳✹✳✻✿ r♦♦ts ♦❢ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧ ❚❤❡ r♦♦ts ♦❢ ❛ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧ f ❛r❡ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ q✉❛❞r❛t✐❝ ❡q✉❛t✐♦♥✿ f (x) = 0✳

✹✳✹✳

◗✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧s

✸✷✹

❆ ❢❛♠✐❧✐❛r✱ ❛♥❞ ✈❡r② ✐♠♣♦rt❛♥t✱ ❢♦r♠✉❧❛ ✐s ❜❡❧♦✇✳ ❚❤❡♦r❡♠ ✹✳✹✳✼✿ ◗✉❛❞r❛t✐❝ ❋♦r♠✉❧❛ ❚❤❡ r♦♦ts ♦❢ ❛ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧

f (x) = ax2 + bx + c, a 6= 0 , ❛r❡ ❣✐✈❡♥ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣✿

x=

−b ±

√

b2 − 4ac 2a

❊①❡r❝✐s❡ ✹✳✹✳✽

❙t❛t❡ t❤❡ t❤❡♦r❡♠ ❛s ❛♥ ✐♠♣❧✐❝❛t✐♦♥ ✭✏✐❢ ✳✳✳ t❤❡♥ ✳✳✳✑✮✳ ■s t❤❡ ❝♦♥✈❡rs❡ tr✉❡❄

❊①❡r❝✐s❡ ✹✳✹✳✾

Pr♦✈❡ t❤❡ t❤❡♦r❡♠✳

❚❤❡ ❢♦❧❧♦✇✐♥❣ s✐♠♣❧❡ ❡①❛♠♣❧❡ s❤♦✇s t❤❡ ❧✐♠✐t❛t✐♦♥s ♦❢ t❤❡ ❛♣♣❧✐❝❛❜✐❧✐t② ♦❢ t❤❡ ❢♦r♠✉❧❛✿

◮

❚❤❡ ❢✉♥❝t✐♦♥

f (x) = x2 + 1

❤❛s ♥♦

x✲✐♥t❡r❝❡♣ts✳

❚❤❡ ❢♦r♠✉❧❛ ✐s ❢r❡q✉❡♥t❧② ✇r✐tt❡♥ ❛s ❢♦❧❧♦✇s✿

x1,2 =

−b ±

√

b2 − 4ac , 2a

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

x1 =

−b −

√

b2 − 4ac 2a

❛♥❞

x2 =

−b +

√

b2 − 4ac 2a

❲❡ ❝❛♥ ✐♠❛❣✐♥❡ t❤❛t t❤❡ t✇♦ ❝♦♠❡ ❢r♦♠ t❤❡ t✇♦ ✐♥✈❡rs❡s ♦❢ t❤❡ ❢✉♥❝t✐♦♥✳✳✳ ❚❤❡♦r❡♠ ✹✳✹✳✶✵✿ ❱✐❡t❛✬s ❋♦r♠✉❧❛s ❚❤❡ r♦♦ts

x 1 , x2

♦❢ t❤❡ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧

f (x) = ax2 + bx + c

❢♦❧❧♦✇✐♥❣ ❡q✉❛t✐♦♥s✿

x1 + x2 = − ❊①❡r❝✐s❡ ✹✳✹✳✶✶

Pr♦✈❡ t❤❡ ❢♦r♠✉❧❛s✳

b a

❛♥❞

x1 · x2 =

c a

s❛t✐s❢② t❤❡

✹✳✹✳

✸✷✺

◗✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧s

❆ ✈❡r② ✐♠♣♦rt❛♥t r❡s✉❧t ❜❡❧♦✇ ❣✐✈❡s ✉s ❛ ♥❡✇ ❢♦r♠ ♦❢ t❤❡ q✉❛❞r❛t✐❝ ❢✉♥❝t✐♦♥✳

❚❤❡♦r❡♠ ✹✳✹✳✶✷✿ ❋❛❝t♦r❡❞ ❋♦r♠ ♦❢ ◗✉❛❞r❛t✐❝ P♦❧②♥♦♠✐❛❧ ■❢

x 1 , x2

❛r❡ t❤❡ r♦♦ts ♦❢ ❛ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧

f (x) = ax2 + bx + c✱

t❤❡♥ ✇❡

❤❛✈❡✿

f (x) = a(x − x1 )(x − x2 )

Pr♦♦❢✳ ❲❡ s✐♠♣❧② s✉❜st✐t✉t❡ ❛♥❞ t❤❡♥ ♥♦t✐❝❡ t❤❛t ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❜② 0 ❛❧✇❛②s ♣r♦❞✉❝❡s 0✿ f (x1 ) = a(x − x1 )(x − x2 ) = a(x1 − x1 )(x1 − x2 ) = a · 0 · (x1 − x2 ) = 0 . x=x1 f (x2 ) = a(x − x1 )(x − x2 ) = a(x2 − x1 )(x2 − x2 ) = a · (x2 − x1 ) · 0 = 0 . x=x2

❊①❡r❝✐s❡ ✹✳✹✳✶✸

❙t❛t❡ t❤❡ ❝♦♥✈❡rs❡ ♦❢ t❤❡ t❤❡♦r❡♠✳ ■s ✐t tr✉❡❄ ❙♦✱ t❤❡r❡ ♠❛② ❜❡ t✇♦ ❧✐♥❡❛r ❢❛❝t♦rs✦ ❆s t❤❡ s♦❧✉t✐♦♥s t♦ t❤❡ ❡q✉❛t✐♦♥

f (x) = ax2 + bx + c = 0 ,

t❤❡s❡ ♥✉♠❜❡rs ♠❛② ♦r ♠❛② ♥♦t ❡①✐st ♦r t❤❡② ♠✐❣❤t ❝♦✐♥❝✐❞❡✳ ❚❤❡ ❢♦r♠✉❧❛ ♦♥❧② ♣r♦❞✉❝❡s x✲✐♥t❡r❝❡♣ts ✇❤❡♥ ✇❤❛t✬s ✐♥s✐❞❡ t❤❡ sq✉❛r❡ r♦♦t ✐s ♥♦♥✲♥❡❣❛t✐✈❡✳

❉❡✜♥✐t✐♦♥ ✹✳✹✳✶✹✿ ❞✐s❝r✐♠✐♥❛♥t ❚❤❡

❞✐s❝r✐♠✐♥❛♥t

♦❢ ❛ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧ f (x) = ax2 + bx + c, a 6= 0 ,

✐s t❤❡ ♥✉♠❜❡r ❞❡✜♥❡❞ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣✿ D = b2 − 4ac

◆♦✇✱ ❧❡t✬s s❡❡ ✇❤❛t ❞❡❝r❡❛s✐♥❣ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❞✐s❝r✐♠✐♥❛♥t D ✭✇❤✐❝❤ ♠❛② ❜❡ ❛❝❤✐❡✈❡❞ ❜② ✐♥❝r❡❛s✐♥❣ t❤❡ ✈❛❧✉❡ ♦❢ c✱ ❢♦r ❡①❛♠♣❧❡✮ ❞♦❡s t♦ t❤❡ ❣r❛♣❤ ♦❢ y = f (x)✳ ❚❤✐s ♣r♦❝❡ss tr❛♥s❢♦r♠s t❤❡ ❣r❛♣❤❀ ✐t ♠♦✈❡s ✉♣✇❛r❞✳ ■♥✐t✐❛❧❧②✱ t❤❡ ❣r❛♣❤ ❤❛s t✇♦ x✲✐♥t❡r❝❡♣ts ❜✉t ✇❤❡♥ D✱ ❢♦r ❛ ♠♦♠❡♥t✱ r❡❛❝❤❡s 0✱ t❤❡ ❣r❛♣❤ t♦✉❝❤❡s t❤❡ x✲❛①✐s✳ ❆s D ❜❡❝♦♠❡s ♥❡❣❛t✐✈❡✱ t❤❡ ❣r❛♣❤ ♣❛ss❡s t❤❡ x✲❛①✐s ❡♥t✐r❡❧② s♦ t❤❛t t❤❡ x✲✐♥t❡r❝❡♣ts ❞✐s❛♣♣❡❛r✿

■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ t✇♦ x✲✐♥t❡r❝❡♣ts st❛rt t♦ ❣❡t ❝❧♦s❡r t♦ ❡❛❝❤ ♦t❤❡r✱ t❤❡♥ ♠❡r❣❡✱ ❛♥❞ ✜♥❛❧❧② ❞✐s❛♣♣❡❛r✳

✹✳✹✳

◗✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧s

✸✷✻

❲❛r♥✐♥❣✦ ❚❤❡ ❝❛s❡

D=0

✐s ❛ ✏❜♦r❞❡r❧✐♥❡✑ ❝❛s❡✳

■t ♠❛❦❡s s❡♥s❡ t♦ ✜♥❞ t❤❡ ❞✐s❝r✐♠✐♥❛♥t ✜rst ❛♥❞ ❝❧❛ss✐❢② t❤❡ ❡q✉❛t✐♦♥✿ ❙t❡♣ ✶✿

❙t❡♣ ✷✿

D

#

D>0

2

D=0

1

D 0✱ ❛♥❞ ♦❢ f ✇❤❡♥ a < 0✳

✐s t❤❡ ❣❧♦❜❛❧ ♠✐♥✐♠✉♠ ♣♦✐♥t ♦❢ ✐s t❤❡ ❣❧♦❜❛❧ ♠❛①✐♠✉♠ ♣♦✐♥t

❲❛r♥✐♥❣✦ ❚❤❡ ❢♦r♠✉❧❛ ✭❛♥❞ t❤❡ ♣r♦♦❢ ✮ ✇♦r❦s ❡✈❡♥ ✇❤❡♥

0

❛♥❞ t❤❡r❡ ❛r❡ ♥♦

D
0✱ ♦r • (−∞, M ] ✇❤❡♥ a < 0✱ ✇❤❡r❡ m ❛♥❞ M ❛r❡ t❤❡ ♠✐♥✐♠✉♠

y = ax2 + bx + c

✐s ❛ ❝❧♦s❡❞ r❛②✿

❛♥❞ t❤❡ ♠❛①✐♠✉♠ ✈❛❧✉❡s ♦❢ t❤❡ ❢✉♥❝t✐♦♥✱

r❡s♣❡❝t✐✈❡❧②❀ t❤❡② ❛r❡ ❡q✉❛❧ t♦ t❤❡ ✈❛❧✉❡ ♦❢

m = f (h)

♦r

f

❛t t❤❡ ✈❡rt❡①✿

M = f (h) .

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

❊①❡r❝✐s❡ ✹✳✹✳✷✷ ■s ✐t ♣♦ss✐❜❧❡ ❢♦r ❛ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧ t♦ ❤❛✈❡ ❡①❛❝t❧② ♦♥❡ ❧✐♥❡❛r ❢❛❝t♦r❄

❊①❡r❝✐s❡ ✹✳✹✳✷✸ ❙✐♠♣❧✐❢②✿



a x−

−b +

√

b2 − 4ac 2a

√   −b − b2 − 4ac x− =? 2a

❚❤✐s ✐s t❤❡ s✉♠♠❛r② ♦❢ t❤❡ t❤r❡❡ ❛❧❣❡❜r❛✐❝ r❡♣r❡s❡♥t❛t✐♦♥s ✭❢♦r♠s✮ ♦❢ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧s ❛♥❞ t❤❡ tr❛♥✲ s✐t✐♦♥s ❛♠♦♥❣ t❤❡♠✿

st❛♥❞❛r❞ ❢♦r♠✿

ax2 + bx + c ւ◗❋ ❢❛❝t♦r❡❞ ❢♦r♠✿

a(x − x1 )(x − x2 )

ւ −→

h=(x1 +x2 )/2

−−−−−−−−−−−→

ց −→

h=−b/(2a)

ց

✈❡rt❡① ❢♦r♠✿

a(x − h)2 + k

✹✳✺✳ ❚❤❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s

❲❤❛t ❞♦ ❧✐♥❡❛r ♣♦❧②♥♦♠✐❛❧s

f (x) = mx + b ❛♥❞ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧s g(x) = ax2 + bx + c ❤❛✈❡ ✐♥ ❝♦♠♠♦♥❄

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

❉❡✜♥✐t✐♦♥ ✹✳✺✳✶✿ ♣♦❧②♥♦♠✐❛❧ ❆

♣♦❧②♥♦♠✐❛❧

✐s ❛ ✭♥✉♠❡r✐❝❛❧✮ ❢✉♥❝t✐♦♥ ❝♦♠♣✉t❡❞ ❢r♦♠ ✐ts ✐♥♣✉t ✈✐❛ ❛❞❞✐t✐♦♥✱

s✉❜tr❛❝t✐♦♥✱ ❛♥❞ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦♥❧②✳

✹✳✺✳

❚❤❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s

✸✷✾

◆♦ ❞✐✈✐s✐♦♥ ✕ ♥♦ ❝❤❛♥❝❡ ♦❢ ❞✐✈✐❞✐♥❣ ❜② ③❡r♦✦ ❲❡ ❝♦♥❝❧✉❞❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿ ❚❤❡♦r❡♠ ✹✳✺✳✷✿ ❉♦♠❛✐♥ ♦❢ P♦❧②♥♦♠✐❛❧ ❚❤❡ ❞♦♠❛✐♥ ♦❢ ❛ ♣♦❧②♥♦♠✐❛❧ ✐s t❤❡ s❡t ♦❢ ❛❧❧ r❡❛❧ ♥✉♠❜❡rs✱

R = (−∞, +∞)✳

▼✉❧t✐♣❧✐❝❛t✐♦♥ ✐♥❝❧✉❞❡s ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❜② ❛ ❝♦♥st❛♥t r❡❛❧ ♥✉♠❜❡r ❛s ✇❡❧❧ ❛s ❜② t❤❡ ✐♥♣✉t✱

x✱

✐ts❡❧❢✳

❋♦r

❡①❛♠♣❧❡✱ ✇❡ ♠✐❣❤t ❤❛✈❡✿

x →

❛❞❞

→

2

♠✉❧t✐♣❧② ❜②

2

x →

→

❛❞❞

→

♠✉❧t✐♣❧② ❜②

3

↓

x

→ y

❚❤✐s ✐s t❤❡ ❢♦r♠✉❧❛ ♦❢ t❤✐s ❢✉♥❝t✐♦♥✿



f (x) = x

❆s ❛♥ ❡①tr❡♠❡ ❝❛s❡✱ ✐t ✐s ♣♦ss✐❜❧❡ t❤❛t ♣♦❧②♥♦♠✐❛❧s✱ ✇❡ ❞✐s❝♦✈❡r ❛ ♥❡✇ ❝❧❛ss✿

♣♦✇❡r ❢✉♥❝t✐♦♥s

✳






(x + 2) · 2 + 3 · x .

♠✐❣❤t ♥♦t ❡✈❡♥ ❛♣♣❡❛r✦ ❚❤❡♥✱ ✐♥ ❛❞❞✐t✐♦♥ t♦ ❧✐♥❡❛r ❛♥❞ q✉❛❞r❛t✐❝

❝♦♥st❛♥t ♣♦❧②♥♦♠✐❛❧s

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

❚❤❡ ✇❛② ✇❡ ❧❡❛r♥❡❞ t♦ ✉♥❞❡rst❛♥❞ t❤❡ ❧✐♥❡❛r✱ f (x) = mx + b✱ ❛♥❞ t❤❡ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧s✱ ax2 + bx + c✱ ✇❛s t♦ ❞❡r✐✈❡ ♠❛♥② ♦❢ t❤❡✐r ♣r♦♣❡rt✐❡s ❢r♦♠ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡✐r ♣❛r❛♠❡t❡rs✱ ♦r ❛♥❞

a, b, c✱ r❡s♣❡❝t✐✈❡❧②✳

❚❤❡r❡ ✐s ❛

g(x) = ✿ m, b

❝♦❡✣❝✐❡♥ts

st❛♥❞❛r❞

✇❛② t♦ r❡♣r❡s❡♥t ❛ ♣♦❧②♥♦♠✐❛❧✳ ❋♦r ❛ ❧✐♥❡❛r ♣♦❧②♥♦♠✐❛❧✱ ✐t ✐s ✐ts g(x) = ax2 + bx + c ✐s t❤❡ st❛♥❞❛r❞ ❢♦r♠✱

✏s❧♦♣❡✲✐♥t❡r❝❡♣t ❢♦r♠✑✳ ❋♦r ❛ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧✱ t❤❡ ❢♦r♠✉❧❛ ❛s ♦♣♣♦s❡❞ t♦✱ s❛②✱ t❤❡ ❢❛❝t♦r❡❞ ❢♦r♠✿

g(x) = a(x − x1 )(x − x2 )✳

❲❡ tr② t♦ ❛♣♣r♦❛❝❤ ❛❧❧ ♣♦❧②♥♦♠✐❛❧s t❤✐s ✇❛② ❛♥❞ ♣✉t t❤❡♠ ✐♥ ❛ s✐♠✐❧❛r✱ ♠❛♥❛❣❡❛❜❧❡ ❢♦r♠✳ ❋♦r ❡①❛♠♣❧❡✱ t❤✐s ✐s ❤♦✇ ✇❡ s✐♠♣❧✐❢② t❤❡ ♦♥❡ ❛❜♦✈❡✿

f (x) =








    (x + 2) · 2 + 3 · x = (2x + 4) + 3 · x = 2x + 7 · x = 2x2 + 7x .

❲❡ ❝❛♥ ❞♦ t❤❡ s❛♠❡ ✇✐t❤ ❛♥② ♣♦❧②♥♦♠✐❛❧✱ ♥♦ ♠❛tt❡r ❤♦✇ ♠❛♥② st❡♣s ❛r❡ ✐♥✈♦❧✈❡❞✳ ❲❤❛t ✐s t❤❡

st❛♥❞❛r❞

✇❛② t♦ r❡♣r❡s❡♥t ❛ ♣♦❧②♥♦♠✐❛❧❄ ▲❡t✬s ❢♦❧❧♦✇ t❤❡ tr❡❛t♠❡♥t ♦❢ t❤❡ ❧✐♥❡❛r ❛♥❞ q✉❛❞r❛t✐❝

♣♦❧②♥♦♠✐❛❧s ❛♥❞ ❛s❦✿ ❲❤❛t ❞♦ t❤❡② ❤❛✈❡ ✐♥ ❝♦♠♠♦♥❄ ❚♦ ♠❛❦❡ t❤❡ ♣❛tt❡r♥ ❝❧❡❛r✱ ❧❡t✬s ❛❞❞ ❛ ❝♦♥st❛♥t t♦ t❤❡ ❧✐st✿ ♣♦❧②♥♦♠✐❛❧s

t❡r♠s

❝♦♥st❛♥t✿ ❧✐♥❡❛r✿ q✉❛❞r❛t✐❝✿

3x2

5 2x + 5 + 2x + 5

❙✐♠✐❧❛r t❡r♠s ❛r❡ ❛❧✐❣♥❡❞ ✈❡rt✐❝❛❧❧②✦ ❆❝❝♦r❞✐♥❣❧②✱ t❤❡② ❤❛✈❡ t❤❡s❡ s♣❡❝✐❛❧ ♥❛♠❡s✱ ♥♦ ♠❛tt❡r ✐♥ ✇❤❛t ♣♦❧②♥♦✲ ♠✐❛❧s t❤❡② ❛♣♣❡❛r✿ t❡r♠s✿

q✉❛❞r❛t✐❝ 2

3x

❧✐♥❡❛r

+

2x

❝♦♥st❛♥t

+

5

❲❡ ❛❞❞ t❤❡ ❝✉❜✐❝ t♦ t❤❡ ❧✐st✳ ❲❡ ❝❛♥ s❡❡ t❤❡ ♣❛tt❡r♥ ❡✈❡♥ ❝❧❡❛r❡r ✐❢ ✇❡ ✐❞❡♥t✐❢② t❤❡ ♣♦✇❡rs ♦❢ 0 ♠✐♥❞ t❤❛t x = 1✮✿ ❝♦♥st❛♥t✿ 5x0 1 ❧✐♥❡❛r✿ 2x + 5x0 q✉❛❞r❛t✐❝✿ 3x2 + 2x1 + 5x0 ❝✉❜✐❝✿ −2x3 + 3x2 + 2x1 + 5x0

❚❤❡ ♣♦✇❡rs ♦❢

x

❣♦ ❞♦✇♥ ❛❧❧ t❤❡ ✇❛② t♦

0✳

x

✭❦❡❡♣✐♥❣ ✐♥

❋✉rt❤❡r♠♦r❡✱ ✇❡ ♣r♦❣r❡ss t♦ t❤❡ ♥❡①t ❧✐♥❡ ❜② ❥✉st ❛❞❞✐♥❣ ❛♥

❡①tr❛ t❡r♠ ✕ ❛ t❡r♠ ♦❢ ❛ ❤✐❣❤❡r ♣♦✇❡r✦ ❲❤❛t ❞♦❡s ✐t ❞♦ t♦ t❤❡ ❢✉♥❝t✐♦♥❄

❏✉st t❛❦❡ ❛ ❧♦♦❦ ❛t t❤❡ ❣r❛♣❤s✳

❆❧❧ ❧✐♥❡❛r ♣♦❧②♥♦♠✐❛❧s ❛r❡ ❧✐♥❡s ❛♥❞

❛❧❧ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧s ❛r❡ ♣❛r❛❜♦❧❛s✳ ❈♦♥s✐❞❡r✱ ❢♦r ❝♦♥tr❛st✱ t❤❡ ❡①tr❛ ❞❡❣r❡❡ ♦❢ ❝♦♠♣❧❡①✐t② ✕ s❡✈❡r❛❧ ❞✐st✐♥❝t❧② ❞✐✛❡r❡♥t s❤❛♣❡s ♦❢ t❤❡ ❣r❛♣❤ ✕ t❤❛t t❤❡ ❝✉❜✐❝ ♣♦❧②♥♦♠✐❛❧s ♠✐❣❤t ❤❛✈❡✿

✹✳✺✳

❚❤❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s

✸✸✵

■♥❞❡❡❞✱ t❤❡ ♥✉♠❜❡r ♦❢ ✇❛②s t❤❡ ♠♦♥♦t♦♥✐❝✐t② ♠❛② ❝❤❛♥❣❡ ❤❛s ✐♥❝r❡❛s❡❞✿ • ❧✐♥❡❛r✿ ց ♦r ր

• q✉❛❞r❛t✐❝✿ ցր ♦r րց

• ❝✉❜✐❝✿ ց ♦r ր ♦r ցրց ♦r րցր

❊①❡r❝✐s❡ ✹✳✺✳✸

❉❡s❝r✐❜❡ t❤❡s❡ ❢✉♥❝t✐♦♥s ❛s tr❛♥s❢♦r♠❛t✐♦♥s✳

❉❡✜♥✐t✐♦♥ ✹✳✺✳✹✿ ❞❡❣r❡❡ ♦❢ ♣♦❧②♥♦♠✐❛❧ ❚❤❡ ❞❡❣r❡❡ ♦❢ ❛ ♣♦❧②♥♦♠✐❛❧ ✐s t❤❡ ♥✉♠❜❡r n t❤❛t ✐s t❤❡ ❤✐❣❤❡st ♣♦✇❡r ♦❢ x t❤❛t ♥❡❡❞s t♦ ❜❡ ❝♦♠♣✉t❡❞✳ ❚❤❡ ❞❡❣r❡❡ ♦❢ ❛ ♣♦❧②♥♦♠✐❛❧ P ✐s ❞❡♥♦t❡❞ ❜② deg P

❋r♦♠ ♥♦✇ ♦♥✱ ✇❡ ✇✐❧❧ ♣r❡❢❡r t♦ ✐❞❡♥t✐❢② ♣♦❧②♥♦♠✐❛❧s ❛❝❝♦r❞✐♥❣ t❤❡✐r ❞❡❣r❡❡s✿ ❞❡❣r❡❡s

3

2

1

0 5x0

0 2x

1 2 3

✳✳ ✳

−2x3

✳✳ ✳

+

1

1 t❡r♠

+

5x

0

2 t❡r♠s

3x2

+

2x1

+

5x0

3 t❡r♠s

3x2

+

2x1

+

5x0

4 t❡r♠s

✳✳ ✳

✳✳ ✳

✳✳ ✳

✳✳ ✳

❆s ②♦✉ ❝❛♥ s❡❡✱ t❤❡ t❡r♠s ❛r❡ ❛❧s♦ ❛ss✐❣♥❡❞ ❞❡❣r❡❡s✱ ✐♥st❡❛❞ ♦❢ ♥❛♠❡s✳

❲❛r♥✐♥❣✦ ❆❧❧ t❡r♠s ❝❛♥ ❜❡ ③❡r♦ ❡①❝❡♣t t❤❡ ❧❡❢t♠♦st✳

❊①❡r❝✐s❡ ✹✳✺✳✺ ❯s❡ t❤❡ ❇✐♥♦♠✐❛❧ ❋♦r♠✉❧❛ t♦ ✜♥❞ t❤❡ st❛♥❞❛r❞ ❢♦r♠ ♦❢ (2x + 3)6 ✳ ❙❤♦✇ t❤❡ ❝♦rr❡s♣♦♥❞❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ tr✐❛♥❣❧❡ ✐♥ t❤❡ t❛❜❧❡ ❛❜♦✈❡ ❛♥❞ t❤❡ P❛s❝❛❧ tr✐❛♥❣❧❡✳ ❙♦✱ t❤❡ nt❤ ❞❡❣r❡❡ ♣♦❧②♥♦♠✐❛❧ ✇✐❧❧ ❜❡ ❧♦❝❛t❡❞ ✐♥ t❤❡ (n + 1)st r♦✇ ♦❢ t❤❡ t❛❜❧❡✳ ■t ✇✐❧❧ ❤❛✈❡ n + 1 t❡r♠s✱ s❛②✿ f (x) = 2xn − 11xn−1 + ... − 2x3 + 3x2 + 2x1 + 5x0 .

✹✳✺✳

❚❤❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s

✸✸✶

❍❡r❡ ✏✳✳✳✑ ✐♥❞✐❝❛t❡s t❤❡ ❝♦♥t✐♥✉❛t✐♦♥ ♦❢ t❤❡ ♣❛tt❡r♥✿ ❞❡❝❧✐♥✐♥❣✱ ❜② ❢r♦♠ t❤❡ ❢♦r♠✉❧❛✱ ✇❡ r❡❛❧✐③❡ t❤❛t ✇❡ ❤❛✈❡ ❛ ✜♥✐t❡

s❡q✉❡♥❝❡

1✱ ❞❡❣r❡❡s ♦❢ x✳

■❢ ✇❡ ❡①tr❛❝t t❤❡ ❝♦❡✣❝✐❡♥ts

✭❈❤❛♣t❡r ✶✮✿

2, −11, ..., −2, 3, 2, 5 . ❋♦r ❛ ❣❡♥❡r❛❧ ❢♦r♠✉❧❛✱ ✇❡ ❝❤♦♦s❡ ❧❡tt❡rs t♦ r❡♣r❡s❡♥t t❤❡ ❝♦❡✣❝✐❡♥ts ♦❢ t❤❡ ♣♦✇❡rs ♦❢

x

✇✐t❤ t❤❡ s✉❜s❝r✐♣ts

✐♥❞✐❝❛t✐♥❣ t❤❡ ♣♦✇❡r ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ t❡r♠✿

k = 0 1 2 ... n − 1 n ak = a0 a1 a2 ... an−1 an ❋♦r t❤❡ ❛❜♦✈❡ ♣♦❧②♥♦♠✐❛❧✱ ✇❡ ❤❛✈❡✿ t❤❡ ❞❡❣r❡❡s ♦❢ t❤❡ t❡r♠s✿ t❤❡ ♣♦❧②♥♦♠✐❛❧✿ t❤❡ t❡r♠s✿ t❤❡ ❝♦❡✣❝✐❡♥ts✿

n n−1 2xn −11xn−1 2xn , −11xn−1 , an = 2, an−1 = −11,

... 3 2 1 0 ... −2x3 +3x2 +2x1 +5x0 ... −2x3 , 3x2 , 2x1 , 5x0 ... a3 = −2, a2 = 3, a1 = 2, a0 = 5

❉❡✜♥✐t✐♦♥ ✹✳✺✳✻✿ st❛♥❞❛r❞ ❢♦r♠ ♦❢ ♣♦❧②♥♦♠✐❛❧ ❆ ♣♦❧②♥♦♠✐❛❧ ✐♥ ❛

st❛♥❞❛r❞ ❢♦r♠

✐s ❣✐✈❡♥ ❜② ❛ ❢♦r♠✉❧❛✿

f (x) = an xn + an−1 xn−1 + ... + a2 x2 + a1 x + a0 ✇❤❡r❡

an , an−1 , ..., a2 , a1 , a0

❛r❡ r❡❛❧ ♥✉♠❜❡rs ❝❛❧❧❡❞ t❤❡

❝♦❡✣❝✐❡♥ts

♦❢ t❤❡ ♣♦❧②♥♦♠✐❛❧✳

❲❡ ❝❛♥ r❡✲❛ss❡♠❜❧❡ t❤❡ ♣♦❧②♥♦♠✐❛❧ ❢r♦♠ ✐ts s❡q✉❡♥❝❡ ♦❢ ❝♦❡✣❝✐❡♥ts✳

❊①❡r❝✐s❡ ✹✳✺✳✼ ❲r✐t❡ t❤❡ st❛♥❞❛r❞ ❢♦r♠ ♦❢ ♣♦❧②♥♦♠✐❛❧ ✐♥ t❤❡ s✐❣♠❛ ♥♦t❛t✐♦♥✳

❉❡✜♥✐t✐♦♥ ✹✳✺✳✽✿ t❡r♠s ♦❢ ♣♦❧②♥♦♠✐❛❧ ❚❤❡ t❡r♠s ♦❢ ❛ ♣♦❧②♥♦♠✐❛❧ ❛r❡ ♥❛♠❡❞ ❛s ❢♦❧❧♦✇s✿

f (x) = + ... + + +

an xn an−1 xn−1 ... a2 x2 a1 x1 a0

nt❤ (n − 1)t❤ ... 2♥❞ 1st 0t❤

❧❡❛❞✐♥❣ t❡r♠ q✉❛❞r❛t✐❝ t❡r♠ ❧✐♥❡❛r t❡r♠ ❝♦♥st❛♥t t❡r♠

❚❤❡ ❝♦❡✣❝✐❡♥ts ❛r❡ ♥❛♠❡❞ ❛❝❝♦r❞✐♥❣❧②✳

❖❢ ❝♦✉rs❡✱ s♦♠❡ ♦❢ t❤❡s❡ ❝♦❡✣❝✐❡♥ts ❝❛♥ ❜❡ ③❡r♦✳ ❚❤❡ ❡①❝❡♣t✐♦♥ ✐s t❤❡ ❧❡❛❞✐♥❣ ♦♥❡✱

an ✳

❚❤❡♦r❡♠ ✹✳✺✳✾✿ ▲❡❛❞✐♥❣ ❚❡r♠ ✈s✳ ❉❡❣r❡❡ ♦❢ P♦❧②♥♦♠✐❛❧

❚❤❡ ❞❡❣r❡❡ ♦❢ ❛ ♣♦❧②♥♦♠✐❛❧ ✐s t❤❡ ♥✉♠❜❡r n t❤❛t ✐s t❤❡ ❤✐❣❤❡st ♣♦✇❡r ♦❢ x ♣r❡s❡♥t ✐♥ ✐ts st❛♥❞❛r❞ ❢♦r♠✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ❛♥② ♣♦❧②♥♦♠✐❛❧ ✇✐t❤ ❝♦❡✣❝✐❡♥ts a0 , a1 , a2 , ...,

✹✳✺✳ ❚❤❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s

✸✸✷

s❛t✐s✜❡s✿ deg f = n =⇒ an 6= 0, an+1 = 0, an+2 = 0, ... ❊①❡r❝✐s❡ ✹✳✺✳✶✵ ❲❤❛t ❛❜♦✉t t❤❡ ❝♦♥✈❡rs❡❄

❆s ✇❡ ✇✐❧❧ s❡❡ ✐♥ ❈❤❛♣t❡r ✷❉❈✲✶✱ t❤❡ ❧❡❛❞✐♥❣ t❡r♠ ❞❡t❡r♠✐♥❡s t❤❡ ❧❛r❣❡✲s❝❛❧❡ ❜❡❤❛✈✐♦r ♦❢ t❤❡ ♣♦❧②♥♦♠✐❛❧✳ ❇❡❧♦✇✱ ✇❡ ❛r❡ ♠♦st❧② ❝♦♥❝❡r♥❡❞ ✇✐t❤ t❤❡ s♠❛❧❧✲s❝❛❧❡ ✐ss✉❡s✳ ❋♦r ❡①❛♠♣❧❡✱ ✇❤❛t ✐s t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ ❝♦♥st❛♥t t❡r♠❄

❚❤❡♦r❡♠ ✹✳✺✳✶✶✿ ❈♦♥st❛♥t ❚❡r♠ ■s

y ✲■♥t❡r❝❡♣t

❚❤❡ ❝♦♥st❛♥t t❡r♠ a0 ♦❢ ❛ ♣♦❧②♥♦♠✐❛❧ f (x) = an xn +an−1 xn−1 +...+a2 x2 +a1 x+a0 ✐s ✐ts y ✲✐♥t❡r❝❡♣t✳ Pr♦♦❢✳

f (0) = an x + an−1 x + ... + a2 x + a1 x + a0 x=0 = an 0n + an−1 0n−1 + ... + a2 02 + a1 0 + a0 = a0 . n

n−1

2

❙♦✱ ✐♥ ❛❞❞✐t✐♦♥ t♦ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ♣♦❧②♥♦♠✐❛❧s ✐♥ t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤✐s s❡❝t✐♦♥✱ ✇❡ ❝❛♥ s❛② t❤❛t ♣♦❧②♥♦♠✐❛❧s ❛r❡ ✏♠❛❞❡✑ ♦❢ ♣♦✇❡rs ♦❢

x

✭♣r❡s❡♥t❡❞ ✐♥ ❈❤❛♣t❡r ✷✮✿

❚❤❡② ❛r❡ ❝♦♠❜✐♥❡❞ ✈✐❛ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❜② ❝♦♥st❛♥t ❝♦❡✣❝✐❡♥ts ❢♦❧❧♦✇❡❞ ❜② ❛❞❞✐t✐♦♥ ✭❝❛❧❧❡❞ ✏❧✐♥❡❛r ❝♦♠❜✐♥❛✲

✹✳✺✳

❚❤❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s

✸✸✸

t✐♦♥s✑✮✿

x3 x2 x2 3 2 2·x 4·x (−1) · x2 f (x) = 2x3 + 4x2 − x2

x1 0 · x1

x0 = 1 10 · 1 + 10

■t ✐s t❤❡ ♣r❡s❡♥❝❡ ♦❢ ✈❛r✐♦✉s ❞❡❣r❡❡ t❤❛t ❡①♣❧❛✐♥ t❤❡ ✈❛r✐❡t② ♦❢ ❜❡❤❛✈✐♦rs ♦❢ ♣♦❧②♥♦♠✐❛❧s✿

❍♦✇❡✈❡r✱ ✇❡ ✇✐❧❧ s❡❡ t❤❛t✱ ✇❤❡♥ ✇❡ ③♦♦♠ ♦✉t ❢r♦♠ t❤❡♠✱ t❤❡ ❣r❛♣❤s ♦❢ ♣♦❧②♥♦♠✐❛❧s r❡s❡♠❜❧❡ t❤♦s❡ ♦❢ t❤❡ ♣♦✇❡r ❢✉♥❝t✐♦♥s✦ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ✇❤❡♥ ③♦♦♠❡❞ ✐♥ ♦♥ t❤❡

x✲❛①✐s✱

✇❡ ♠✐❣❤t s❡❡ ♠❛♥②

x✲✐♥t❡r❝❡♣ts✿

❍♦✇ ♠❛♥②❄ ❲❡ ❤❛✈❡ ❧❡❛r♥❡❞ ❢r♦♠ ♦✉r ❡①♣❡r✐❡♥❝❡ ✇✐t❤ ❧✐♥❡❛r ❛♥❞✱ ❡s♣❡❝✐❛❧❧②✱ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧s t❤❛t t❤❡r❡ ✐s ❛♥♦t❤❡r✱ ❥✉st ❛s ✐♠♣♦rt❛♥t ❢♦r♠ ❢♦r ❛ ♣♦❧②♥♦♠✐❛❧✿ t❤❡

❧✐♥❡❛r ♣♦❧②♥♦♠✐❛❧

❋♦r ❡①❛♠♣❧❡✱ ❛

❢❛❝t♦r❡❞ ❢♦r♠✳

✐s ❢❛❝t♦r❡❞ ❛s ❢♦❧❧♦✇s✿



b f (x) = mx + b = m x + m



, m 6= 0 .

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

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

✐t✿

◮x=−

b m

✐s t❤❡

◆❡①t✱ s✉♣♣♦s❡ ❛

x✲✐♥t❡r❝❡♣t

♦❢

f✳

q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧

❤❛♣♣❡♥s t♦ ❤❛✈❡ t✇♦ r❡❛❧ r♦♦ts

◗✉❛❞r❛t✐❝ ❋♦r♠✉❧❛✮✳ ❚❤❡② ❣✐✈❡ ✉s t✇♦✱ ♦r ♦♥❡ ✐❢ ❡q✉❛❧✱

x✲✐♥t❡r❝❡♣ts

x1

❛♥❞

x2

✭t❤❡② ❝♦♠❡ ❢r♦♠ t❤❡

❛♥❞ t✇♦ ✭♣♦ss✐❜❧② ❡q✉❛❧✮ ❢❛❝t♦rs✿

f (x) = ax2 + bx + c = a(x − x1 )(x − x2 ) . ❲❡ s❡❡ ❤♦✇ t❤❡ st❛♥❞❛r❞ ❢♦r♠ ✐s ❝♦♥✈❡rt❡❞ t♦ t❤❡ ❢❛❝t♦r❡❞ ❢♦r♠ ✇✐t❤ t✇♦ ❧✐♥❡❛r ❢❛❝t♦rs✳ ❖♥❝❡ ❛❣❛✐♥✱ t❤❡ ❢❛❝t♦rs ❤❛✈❡ ❛ s♣❡❝✐❛❧ ✐♠♣♦rt❛♥❝❡ ❜❡❝❛✉s❡ ✇❡ ❝❛♥

◮ x1

❛♥❞

❲❤❡♥ t❤❡ t✇♦

x2

❛r❡ ✐s t❤❡

x✲✐♥t❡r❝❡♣ts

x✲✐♥t❡r❝❡♣ts

❛r❡ ❡q✉❛❧✱

♦❢

r❡❛❞

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

f✳

x1 = x2 ✱

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

f (x) = a(x − x1 )(x − x1 ) = a(x − x1 )2 .

✹✳✺✳

❚❤❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s

✸✸✹

❊✈❡♥ t❤♦✉❣❤ t❤❡r❡ ❛r❡ st✐❧❧ t✇♦ ❢❛❝t♦rs t❤♦✉❣❤✱ t❤❡r❡ ✐s ♦♥❧② ♦♥❡

x✲✐♥t❡r❝❡♣t✦

❆ tr✉❧② ❞✐✛❡r❡♥t s✐t✉❛t✐♦♥ ✐s t❤❛t ♦❢ ❛♥ ✐rr❡❞✉❝✐❜❧❡ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧✱ s✉❝❤ ❛s

x2 +1❀ ✐t ❤❛s ♥♦ x✲✐♥t❡r❝❡♣ts

♥♦r ❧✐♥❡❛r ❢❛❝t♦rs✦ ❲❡ s❛② t❤❛t t❤❡r❡ ✐s ♦♥❡ ✕ ✐rr❡❞✉❝✐❜❧❡ q✉❛❞r❛t✐❝ ✕ ❢❛❝t♦r✳ ❲❤❛t ♠❛❦❡s ✐t ✏✐rr❡❞✉❝✐❜❧❡✑ ✐s t❤❛t ❢❛st t❤❛t ✐t ❝❛♥♥♦t ❜❡ r❡❞✉❝❡❞ t♦ ❢❛❝t♦rs ♦❢ ❧♦✇❡r ❞❡❣r❡❡✱ ✐✳❡✳✱ ❧✐♥❡❛r ❢❛❝t♦rs✳ ❊①❡r❝✐s❡ ✹✳✺✳✶✷

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

❲❤❛t ❞♦ ✇❡ ❦♥♦✇ ❛❜♦✉t

❢❛❝t♦r✐③❛t✐♦♥ ♦❢ ♣♦❧②♥♦♠✐❛❧s ✐♥ ❣❡♥❡r❛❧❄

❖♥❡ ❝❛♥ t❛❦❡ ❛❞✈❛♥t❛❣❡ ♦❢ ♦♥❡✬s ♣r✐♦r ❡①♣❡r✐❡♥❝❡ ✇✐t❤ ✐♥t❡❣❡rs ✭❈❤❛♣t❡r ✶✮ ❜② ❢♦❧❧♦✇✐♥❣ t❤❡s❡ ♣❛✐rs ♦❢ ♠❛t❝❤✐♥❣ ♦❜s❡r✈❛t✐♦♥s✳ ❆♥❛❧♦❣②✿ ✐♥t❡❣❡rs ✈s✳ ♣♦❧②♥♦♠✐❛❧s

❚❤❡

s✉♠ ❛♥❞ t❤❡ ♣r♦❞✉❝t

■♥t❡❣❡rs P♦❧②♥♦♠✐❛❧s ♦❢ t✇♦ ✐♥t❡❣❡rs ✐s ❛♥ ✐♥t❡❣❡r✳

❆

♣r✐♠❡ ♥✉♠❜❡r ❝❛♥♥♦t ❜❡ ❢✉rt❤❡r

❢❛❝t♦r❡❞ ✐♥t♦ s♠❛❧❧❡r ✐♥t❡❣❡rs

> 1.

❚❤❡

s✉♠ ❛♥❞ t❤❡ ♣r♦❞✉❝t

♦❢ t✇♦ ♣♦❧②♥♦♠✐❛❧s

✐s ❛ ♣♦❧②♥♦♠✐❛❧✳ ❆♥

✐rr❡❞✉❝✐❜❧❡ ♣♦❧②♥♦♠✐❛❧ ❝❛♥♥♦t ❜❡ ❢✉rt❤❡r

❢❛❝t♦r❡❞ ✐♥t♦ ❧♦✇❡r ❞❡❣r❡❡ ♣♦❧②♥♦♠✐❛❧s✳

2, 3, 5, 7, 11, 13, ... x − 1, x + 2, x2 + 1, ... ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❆r✐t❤♠❡t✐❝✿

❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❆❧❣❡❜r❛✿

❊✈❡r② ✐♥t❡❣❡r ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❛s

❊✈❡r② ♣♦❧②♥♦♠✐❛❧ ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❛s

t❤❡

♣r♦❞✉❝t

♦❢ ♣r✐♠❡s✳

❚❤✐s r❡♣r❡s❡♥t❛t✐♦♥ ✐s

✉♥✐q✉❡✱

✉♣ t♦ t❤❡ ♦r❞❡r ♦❢ t❤❡ ❢❛❝t♦rs✳

5

❚❤❡

2

q =2 ·3 ·5

1

♠✉❧t✐♣❧✐❝✐t② ✐s ❤♦✇ ♠❛♥② t✐♠❡s ✐t ❛♣♣❡❛rs✳

t❤❡

♣r♦❞✉❝t

♦❢ ✐rr❡❞✉❝✐❜❧❡ ❢❛❝t♦rs✳

❚❤✐s r❡♣r❡s❡♥t❛t✐♦♥ ✐s

✉♥✐q✉❡✱

✉♣ t♦ t❤❡ ♦r❞❡r ♦❢ t❤❡ ❢❛❝t♦rs✳

Q = (x − 1)5 · (x + 2)2 · (x2 + 1) ❚❤❡

♠✉❧t✐♣❧✐❝✐t② ✐s ❤♦✇ ♠❛♥② t✐♠❡s ✐t ❛♣♣❡❛rs✳

■rr❡❞✉❝✐❜❧❡ ♣♦❧②♥♦♠✐❛❧s ❤❛✈❡ ❞❡❣r❡❡s

1

♦r

✐s

n✳

2.

❊①❡r❝✐s❡ ✹✳✺✳✶✸

❲❤❛t ✐s t❤❡ ❞❡❣r❡❡ ♦❢ t❤❡ ♣♦❧②♥♦♠✐❛❧

Q

❛❜♦✈❡❄

❈♦r♦❧❧❛r② ✹✳✺✳✶✹✿ ❉❡❣r❡❡ ♦❢ ❋❛❝t♦r❡❞ P♦❧②♥♦♠✐❛❧ ❚❤❡ s✉♠s ♦❢ t❤❡ ❞❡❣r❡❡s ♦❢ t❤❡ ❢❛❝t♦rs ♦❢ ❛ ♣♦❧②♥♦♠✐❛❧ ♦❢ ❞❡❣r❡❡

n

❊①❡r❝✐s❡ ✹✳✺✳✶✺

Pr♦✈❡ t❤❡ ❝♦r♦❧❧❛r②✳

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

x✲✐♥t❡r❝❡♣ts✳

❚❤❡ ❢❛❝t t❤❛t ✇❡ ❝❛♥ s❡❡ t❤❡♠ ❢♦❧❧♦✇s

❢r♦♠ t❤✐s ❢✉♥❞❛♠❡♥t❛❧ r❡s✉❧t ❛❜♦✉t ♥✉♠❜❡rs✳ ❚❤❡♦r❡♠ ✹✳✺✳✶✻✿ ❩❡r♦ ❋❛❝t♦r Pr♦♣❡rt② ■❢ t❤❡ ♣r♦❞✉❝t ♦❢ t✇♦ ♥✉♠❜❡rs ✐s ③❡r♦✱ t❤❡♥ ❡✐t❤❡r ♦♥❡ ♦❢ t❤❡♠ ♦r ❜♦t❤ ✐s ③❡r♦ t♦♦✳ ❈♦♥✈❡rs❡❧②✱ ✐❢ ❡✐t❤❡r ♦♥❡ ♦❢ t❤❡ t✇♦ ♥✉♠❜❡rs ✐s ③❡r♦✱ t❤❡♥ s♦ ✐s t❤❡✐r ♣r♦❞✉❝t✳

✹✳✺✳

✸✸✺

❚❤❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s

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

a = 0 ❖❘ b = 0 ⇐⇒ ab = 0 ❚❤❡ ❝♦♥❝❡♣t ♦❢ x✲✐♥t❡r❝❡♣t r❡❢❡rs t♦ t❤❡ ❣❡♦♠❡tr② ♦❢ t❤❡ ❣r❛♣❤❀ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐s ❛♥ ❛❧❣❡❜r❛✐❝ s✉❜st✐t✉t❡✳

❉❡✜♥✐t✐♦♥ ✹✳✺✳✶✼✿ r♦♦ts ♦❢ ♣♦❧②♥♦♠✐❛❧ ❚❤❡

r♦♦ts

♦❢ ❛ ♣♦❧②♥♦♠✐❛❧ P ❛r❡ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ♣♦❧②♥♦♠✐❛❧ ❡q✉❛t✐♦♥✿

P (x) = 0 . ❚❤❡♥✱ s✉❜st✐t✉t✐♥❣ ✐ts r♦♦t x = x1 ✐♥t♦ t❤❡ ❧✐♥❡❛r ♣♦❧②♥♦♠✐❛❧ L(x) = x − x1 ♠❛❦❡s ✐t ❡q✉❛❧ t♦ ③❡r♦ ✕ ❛s ✇❡❧❧ ❛s ❛♥② ♣♦❧②♥♦♠✐❛❧ t❤❛t ❤❛s ✐t ❛s ❛ ❢❛❝t♦r✦ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ✐s ❝r✉❝✐❛❧✳

❚❤❡♦r❡♠ ✹✳✺✳✶✽✿ ▲✐♥❡❛r ❋❛❝t♦r ❚❤❡♦r❡♠ (x − x1 ) ✐s ♣r❡s❡♥t ✐♥ x = x1 ✐s ❛♥ x✲✐♥t❡r❝❡♣t ♦❢ P ✳

❆ ❧✐♥❡❛r ❢❛❝t♦r ♦♥❧② ✐❢

t❤❡ ❢❛❝t♦r✐③❛t✐♦♥ ♦❢ ❛ ♣♦❧②♥♦♠✐❛❧

P

✐❢ ❛♥❞

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿

(x − x1 )

✐s ❛ ❢❛❝t♦r ♦❢

P ⇐⇒ P (x1 ) = 0 .

❲❛r♥✐♥❣✦ Pr♦✈❡ t❤❡

=⇒

♣❛rt ♦❢ t❤❡ t❤❡♦r❡♠✳

❆♥ ✐♠♣♦rt❛♥t ❢❛❝t ❢♦❧❧♦✇s✳

❈♦r♦❧❧❛r② ✹✳✺✳✶✾✿ ◆✉♠❜❡r ♦❢ x✲■♥t❡r❝❡♣ts ❆ ♣♦❧②♥♦♠✐❛❧ ❝❛♥♥♦t ❤❛✈❡ ♠♦r❡

x✲✐♥t❡r❝❡♣ts

t❤❛♥ ✐ts ❞❡❣r❡❡✳

❚❤❡ t❤❡♦r❡♠ ❡st❛❜❧✐s❤❡s ❛ ❝♦rr❡s♣♦♥❞❡♥❝❡✿ ❆❧❣❡❜r❛ ←→ ●❡♦♠❡tr② ❢❛❝t♦r (x − x1 ) ←→ x✲✐♥t❡r❝❡♣t x = x1 ■t✬s t❤❡ s❛♠❡ t❤✐♥❣✳✳✳ ❜✉t t❤❡r❡ ✐s ♠♦r❡✦

❉❡✜♥✐t✐♦♥ ✹✳✺✳✷✵✿ ♠✉❧t✐♣❧✐❝✐t② ♦❢ ❢❛❝t♦rs ❛♥❞ r♦♦ts ✶✳ ❚❤❡ ♠✉❧t✐♣❧✐❝✐t② ♦❢ ❛ ❧✐♥❡❛r ❢❛❝t♦r ♦❢ ❛ ♣♦❧②♥♦♠✐❛❧ P ✐s ❤♦✇ ♠❛♥② t✐♠❡s ✐t ❛♣♣❡❛rs ✐♥ t❤❡ ❢❛❝t♦r✐③❛t✐♦♥ ♦❢ P ✳ ✷✳ ❚❤❡ ♠✉❧t✐♣❧✐❝✐t② ♦❢ ❛ r♦♦t ❛♥❞✱ ❝♦♥❝♦♠✐t❛♥t❧②✱ t❤❡ ♠✉❧t✐♣❧✐❝✐t② ♦❢ ❛♥ x✲ ✐♥t❡r❝❡♣t ♦❢ P ✐s t❤❛t ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❧✐♥❡❛r ❢❛❝t♦r✱ ✐✳❡✳✱ ❤♦✇ ♠❛♥② t✐♠❡s ✐t ❛♣♣❡❛rs ✐♥ t❤❡ ❢❛❝t♦r✐③❛t✐♦♥ ♦❢ P ✳ ❲❡ ❤❛✈❡ ❛ ♠♦r❡ ♣r❡❝✐s❡ ❝♦rr❡s♣♦♥❞❡♥❝❡ ♥♦✇✿ ❆❧❣❡❜r❛ ←→ ●❡♦♠❡tr② ❢❛❝t♦r (x − x1 )m ←→ x✲✐♥t❡r❝❡♣t x = x1 ✇✐t❤ ♠✉❧t✐♣❧✐❝✐t② ❛t ❧❡❛st m

✹✳✺✳

✸✸✻

❚❤❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s

❊①❡r❝✐s❡ ✹✳✺✳✷✶

■♥ ❤♦✇ ♠❛♥② ✇❛②s ❝❛♥ t❤❡ ❣r❛♣❤ t♦ ❝r♦ss t❤❡ x✲❛①✐s❄ ❲❡ ✇✐❧❧ ❛ss✉♠❡ ❜❡❧♦✇ t❤❛t s♦♠❡♦♥❡ ❤❛s ❛❧r❡❛❞② ❢❛❝t♦r❡❞ t❤❡ ♣♦❧②♥♦♠✐❛❧ ❢♦r ✉s✳✳✳ ❊①❛♠♣❧❡ ✹✳✺✳✷✷✿ ❢❛❝t♦r❡❞ ♣♦❧②♥♦♠✐❛❧

❈♦♥s✐❞❡r t❤❡ ♣♦❧②♥♦♠✐❛❧

Q(x) = x(x − 1)5 · (x + 2)2 · (x − .5) · (x2 + 1) · ... ❍❡r❡✱ ✇❡ ❛ss✉♠❡ t❤❛t t❤❡ r❡st ♦❢ t❤❡ ❢❛❝t♦rs ❛r❡ ✐rr❡❞✉❝✐❜❧❡✳ ❚❤❡ ♣♦❧②♥♦♠✐❛❧ ❤❛s x✲✐♥t❡r❝❡♣ts ✭r♦♦ts✮ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ❡❛❝❤ ♦❢ ✐ts ❧✐♥❡❛r ❢❛❝t♦rs✱ ❛s ❢♦❧❧♦✇s✿

P = x (x − 1)5 · (x + 2)2 · (x − .5)· (x2 + 1) · ... ❧✐♥❡❛r ❢❛❝t♦rs✿ (x − 0) (x − 1) (x + 2) (x − .5) ♠✉❧t✐♣❧✐❝✐t✐❡s✿ 1 5 2 1 x✲✐♥t❡r❝❡♣ts✿ 0 1 −2 .5 ♣♦✐♥ts ♦♥ t❤❡ ❣r❛♣❤✿ (0, 0) (1, 0) (−2, 0) (.5, 0) ❚❤❡ q✉❛❞r❛t✐❝ ❢❛❝t♦r✱ ❛♥❞ t❤❡ r❡st✱ ❞♦❡s♥✬t ❝♦♥tr✐❜✉t❡ ❛♥②t❤✐♥❣ ❜❡❝❛✉s❡ ✐t r❡♠❛✐♥s ♣♦s✐t✐✈❡ ❢♦r ❡✈❡r② x✳ ■❢ ✇❡ ❛rr❛♥❣❡ t❤❡ ✈❛❧✉❡s ♦❢ x✱ ✇❡ ❝❛♥ s❡❡ ✇❤❛t t❤❡ x✲❛①✐s ❧♦♦❦s ❧✐❦❡ ✇✐t❤ t❤❡ ❢♦✉r ♣♦✐♥ts ♦❢ t❤❡ ❣r❛♣❤ s❤♦✇♥✿ x✲❛①✐s✿ − • • • • ... −→ x −2 0 .5 1 ... ❚♦ ❜❡ ❝♦♥t✐♥✉❡❞✳✳✳

■♥ ♦t❤❡r ✇♦r❞s✱ s♦❧✈✐♥❣ ❛ ♣♦❧②♥♦♠✐❛❧ ❡q✉❛t✐♦♥ P (x) = 0 ❛♥❞ ❢❛❝t♦r✐♥❣ t❤❡ ♣♦❧②♥♦♠✐❛❧ P ✐s t❤❡ s❛♠❡ t❤✐♥❣✳✳✳ ❡①❝❡♣t t❤❡ ❢♦r♠❡r ❞♦❡s♥✬t ❝❛r❡ ❛❜♦✉t t❤❡ ♠✉❧t✐♣❧✐❝✐t✐❡s ✦ ❇✉t ✇❡ ❞♦✳ ❊①❛♠♣❧❡ ✹✳✺✳✷✸✿ ♣♦✇❡r ❢✉♥❝t✐♦♥s

❘❡❝❛❧❧ ✇❤❛t ✇❡ ❦♥♦✇ ❛❜♦✉t t❤❡ ♣♦✇❡r ❢✉♥❝t✐♦♥s✱ x1 , x2 , x3 , ... ✭❈❤❛♣t❡r ✷✮ ❜✉t t❤✐s t✐♠❡ ✇❡ ❛❧s♦ ♥♦t✐❝❡ ❤♦✇ ❞✐✛❡r❡♥t❧② t❤❡ ❣r❛♣❤s ✐♥t❡rs❡❝t t❤❡ x✲❛①✐s❀ s♦♠❡ ❝r♦ss ❛♥❞ s♦♠❡ t♦✉❝❤✿

❇✉t t❤❡ ❡①♣♦♥❡♥ts ♦❢ t❤❡s❡ ♣♦✇❡rs ✕ 1, 2, 3, ... ✕ ❛r❡ ❥✉st t❤❡ x = (x − 0) ❛♥❞ t❤❡ r♦♦t x = 0✳ ❚❤❡ ❡①❛♠♣❧❡ s✉❣❣❡sts t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥✈❡♥✐❡♥t r✉❧❡ ❛❜♦✉t t❤❡ ❜② ✐ts ♠✉❧t✐♣❧✐❝✐t②✳

♠✉❧t✐♣❧✐❝✐t✐❡s ✦

♥❛t✉r❡ ♦❢ ❛♥

❖❢ ✇❤❛t❄ ❚❤❡ ❢❛❝t♦r

x✲✐♥t❡r❝❡♣t ❛s ✐t ✐s ❞❡t❡r♠✐♥❡❞

❚❤❡♦r❡♠ ✹✳✺✳✷✹✿ ▼✉❧t✐♣❧✐❝✐t② ❘✉❧❡ ❢♦r P♦❧②♥♦♠✐❛❧s ❋♦r ❡✈❡r② ♣♦❧②♥♦♠✐❛❧✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿

•

■❢ t❤❡ ♠✉❧t✐♣❧✐❝✐t② ♦❢ ❛ ❧✐♥❡❛r ❢❛❝t♦r ✐s

t❤❡

•

♦❞❞✱

❛♥❞ ♥❡❣❛t✐✈❡ ✈❛❧✉❡s ✐♥ t❤❡ ✈✐❝✐♥✐t② ♦❢ t❤❡

x✲❛①✐s✿

t❤❡ ❢✉♥❝t✐♦♥ t❛❦❡s ❜♦t❤ ♣♦s✐t✐✈❡

x✲✐♥t❡r❝❡♣t❀ ✐✳❡✳✱ t❤❡ ❣r❛♣❤ ❝r♦ss❡s

✉♣✇❛r❞ ♦r ❞♦✇♥✇❛r❞✳

■❢ t❤❡ ♠✉❧t✐♣❧✐❝✐t② ♦❢ ❛ ❧✐♥❡❛r ❢❛❝t♦r ✐s

❡✈❡♥✱

t❤❡ ❢✉♥❝t✐♦♥ t❛❦❡s ❡✐t❤❡r ♦♥❧②

✹✳✺✳

❚❤❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s

✸✸✼

♣♦s✐t✐✈❡ ♦r ♦♥❧② ♥❡❣❛t✐✈❡ ✈❛❧✉❡s ✐♥ t❤❡ ✈✐❝✐♥✐t② ♦❢ t❤❡ ❣r❛♣❤

t♦✉❝❤❡s

t❤❡

x✲❛①✐s✿

x✲✐♥t❡r❝❡♣t❀

✐✳❡✳✱ t❤❡

❢r♦♠ ❛❜♦✈❡ ♦r ❢r♦♠ ❜❡❧♦✇✳

❊①❡r❝✐s❡ ✹✳✺✳✷✺

❙t❛t❡ t❤❡ ❝♦♥✈❡rs❡ ♦❢ ❡✐t❤❡r ♣❛rt ♦❢ t❤❡ t❤❡♦r❡♠✳ ■s ✐t tr✉❡❄

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

y✿

✐t ✐s ♣♦s✐t✐✈❡ ✐♥ t❤❡ ✉♣♣❡r ❤❛❧❢ ♦❢ t❤❡ ♣❧❛♥❡ ❛♥❞

♥❡❣❛t✐✈❡ ✐♥ t❤❡ ❧♦✇❡r ❤❛❧❢✳ ❚❤❡♥✱ ✇❡ ✉s❡ t❤❡ ❧❛♥❣✉❛❣❡ ♦❢ ✏❝❤❛♥❣❡ ♦❢ s✐❣♥✑ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ✐♥ t❤❡ ✈✐❝✐♥✐t② ♦❢ t❤❡

x✲✐♥t❡r❝❡♣t✿

❆❜❜r❡✈✐❛t❡❞✱ t❤❡ t❤❡♦r❡♠ ❜❡❝♦♠❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥✈❡♥✐❡♥t r✉❧❡✿ ▲✐♥❡❛r ❢❛❝t♦rs ❛♥❞ t❤❡✐r ❆❧❣❡❜r❛

x✲✐♥t❡r❝❡♣ts

●❡♦♠❡tr②

♠✉❧t✐♣❧✐❝✐t②

s✐❣♥

♦❞❞

❝❤❛♥❣❡ ♦❢ s✐❣♥

❡✈❡♥

♥♦ ❝❤❛♥❣❡ ♦❢ s✐❣♥

x✲✐♥t❡r❝❡♣t ր ♦r ց ⌣ ♦r ⌢

❝r♦ss t♦✉❝❤

❊①❛♠♣❧❡ ✹✳✺✳✷✻✿ ❢❛❝t♦r❡❞ ♣♦❧②♥♦♠✐❛❧✱ ❝♦♥t✐♥✉❡❞

❲❡ ❝♦♥t✐♥✉❡ ✇✐t❤ t❤❡ ❧❛st ❡①❛♠♣❧❡✳

❲❡ ❛ss✉♠❡ t❤❛t t❤❡ r❡st ♦❢ t❤❡ ❢❛❝t♦rs ♦❢ ♦✉r ♣♦❧②♥♦♠✐❛❧ ❛r❡

✐rr❡❞✉❝✐❜❧❡ ❛♥❞ ♣♦s✐t✐✈❡✿

Q(x) = x(x − 1)5 · (x + 2)2 · (x − .5) · (x2 + 1) · ... ❚❤❡ t❤❡

x✲✐♥t❡r❝❡♣ts ❛♥❞ t❤❡✐r ♠✉❧t✐♣❧✐❝✐t✐❡s ❛r❡ ❧✐st❡❞ ❜❡❧♦✇ ❛♥❞ t❤❡② t❡❧❧ ✉s ❤♦✇ t❤❡ ❣r❛♣❤ ✐♥t❡r❛❝ts ✇✐t❤ x✲❛①✐s✿ Q(x) = x✲✐♥t❡r❝❡♣ts✿ ♠✉❧t✐♣❧✐❝✐t✐❡s✿ ♠✉❧t✐♣❧✐❝✐t✐❡s ❛r❡✿ ❣r❛♣❤✿ ♦♣t✐♦♥s✿ ♦♣t✐♦♥s✿

x· 0 1

(x − 1)5 · (x + 2)2 · (x − .5) 1 −2 .5 5 2 1

♦❞❞

♦❞❞

❡✈❡♥

♦❞❞

❝r♦ss

❝r♦ss

t♦✉❝❤

❝r♦ss

ց ր

ց ր

⌣ ⌢

ց ր

❚❤❡r❡ ❛r❡ t✇♦ ♣♦ss✐❜✐❧✐t✐❡s ❢♦r ❡❛❝❤

x✲✐♥t❡r❝❡♣t

t❤♦✉❣❤✳

❧✐♥❡❛r ❢❛❝t♦rs r❡❛❞ ❢r♦♠ t❤❡ ❢❛❝t♦r✐③❛t✐♦♥ r❡❛❞ ❢r♦♠ t❤❡ ❢❛❝t♦r✐③❛t✐♦♥ r❡❛❞ ❢r♦♠ t❤❡ ❢❛❝t♦r✐③❛t✐♦♥ ❤♦✇ ✐t ♠❡❡ts t❤❡ t✇♦ ❝❤♦✐❝❡s ♣❡r t✇♦ ❝❤♦✐❝❡s ♣❡r

✜rst ♦♣t✐♦♥✿ s❡❝♦♥❞ ♦♣t✐♦♥✿ ❚♦ ✜♥❞ ✇❤✐❝❤ ♦♥❡ ♦❝❝✉rs✱ ✇❡ ♣✐❝❦ ❛

x✲✐♥t❡r❝❡♣t x✲✐♥t❡r❝❡♣t

❉♦❡s ✐t ♠❡❛♥ t❤❛t ✇❡ ❤❛✈❡ ❛ t♦t❛❧ ♦❢

♣♦ss✐❜✐❧✐t✐❡s❄ ◆♦✿ ✐❢ t❤❡ ❣r❛♣❤ ❝r♦ss❡s ✉♣✱ ✐t ❝❛♥✬t ❝r♦ss ✉♣ ❛❣❛✐♥✳ ❲❡ ❤❛✈❡

x✲✐♥t❡r❝❡♣ts✿

x✲❛①✐s

t✇♦ ♣♦ss✐❜✐❧✐t✐❡s✿

16

0 1 −2 .5 ց ր ⌣ ց ր ց ⌢ ր

st❛rt✐♥❣ ♣♦✐♥t

❢♦r t❤❡ ❣r❛♣❤❀ ✐t s❡r✈❡s ❛s ❛ ✏❞❡❝✐❞❡r✑✳ ❍♦✇ ❛❜♦✉t

x = −10❄ ❲❡ s✉❜st✐t✉t❡ ❛♥❞ ❞✐s❝♦✈❡r t❤❛t Q(−10) < 0✳ ❚❤❡r❡❢♦r❡✱ t❤❡r❡ ✐s ❛ ♣♦✐♥t t♦ t❤❡ ❧❡❢t ♦❢ t❤❡ ✜rst x✲✐♥t❡r❝❡♣t✱ x = −2✱ t❤❛t ❧✐❡s ❜❡❧♦✇ t❤❡ x✲❛①✐s✳ ■t ❢♦❧❧♦✇s t❤❛t ✇❡ ♠✉st ❝r♦ss t❤❡ ❛①✐s ✉♣✇❛r❞ ❛t

✹✳✺✳

❚❤❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s

x = −2✦

◆♦✇ ✇❡ ❛r❡

♦♥✳ ❲❡ ♦r❞❡r t❤❡

❛t ❡❛❝❤

❛❜♦✈❡

✸✸✽

x✲✐♥t❡r❝❡♣t

x✲❛①✐s✳

t❤❡

x✲✐♥t❡r❝❡♣ts✱

❚❤❡r❡❢♦r❡✱ ✇❡ ♠✉st ❝r♦ss t❤❡ ❛①✐s

❞♦✇♥✇❛r❞

❛t

x = 0✦

❆♥❞ s♦

❛♥❞ t❤❡♥ ❣♦ ❢r♦♠ ❧❡❢t t♦ r✐❣❤t✱ ❞✐s❝♦✈❡r✐♥❣ ✇❤✐❝❤ ✇❛② ✇❡ ❝r♦ss t❤❡ ❛①✐s

❛s s♦♦♥ ❛s ✇❡ r❡❛❝❤ ✐t✿

x✲✈❛❧✉❡s✿ −10 y ✲✈❛❧✉❡s✿ − ❣r❛♣❤✿

y ✲✈❛❧✉❡s✿

−

−2 0

0 0

?

.5 0

?

?

1 0

❝r♦ss

❝r♦ss

t♦✉❝❤

❝r♦ss

ր 0

ց 0

⌢ 0

ր 0

+

−

−

x✲✐♥t❡r❝❡♣ts ?

❛❜♦✈❡ ♦r ❜❡❧♦✇❄ ❤♦✇ ✐t ♠❡❡ts t❤❡

+

x✲❛①✐s

t❤❡ ❛♥s✇❡r

❲❡ ❝❛♥ ♥♦✇ ❞r❛✇ ❛ r♦✉❣❤ s❦❡t❝❤ ♦❢ t❤❡ ❣r❛♣❤✿

■♠♣❧✐❝✐t❧②✱ ✇❡ ❤❛✈❡ r❡❧✐❡❞ ♦♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢❛❝t t❤❛t ♦♥❡ ❝❛♥♥♦t ❝r♦ss ❛ r✐✈❡r ❢r♦♠ s♦✉t❤ t♦ ♥♦rt❤ t✇✐❝❡ ✐♥ ❛ r♦✇✳ ❚♦ ♣✉t ✐t ❞✐✛❡r❡♥t❧②✱ ✐❢ ♣♦✐♥t

A

✐s ♦♥ t❤❡ s♦✉t❤❡r♥ ❜❛♥❦ ♦❢ ❛ r✐✈❡r ❛♥❞ ♣♦✐♥t

B

♦♥ t❤❡ ♥♦rt❤❡r♥ ❜❛♥❦✱

✇❡✬❧❧ ❤❛✈❡ t♦ s✇✐♠✿

❲❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢❛❝t ✭t♦ ❜❡ ❛❞❞r❡ss❡❞ ✐♥ ❈❤❛♣t❡r ✷❉❈✲✷✮✳ ❚❤❡♦r❡♠ ✹✳✺✳✷✼✿ ❈♦♥t✐♥✉✐t② ♦❢ P♦❧②♥♦♠✐❛❧s

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

x✲✐♥t❡r❝❡♣t

x✲❛①✐s

❛♥❞ ❛ ♣♦✐♥t ❛❜♦✈❡

❜❡t✇❡❡♥ t❤❡♠✳

❊①❡r❝✐s❡ ✹✳✺✳✷✽

❙t❛t❡ t❤❡ t❤❡♦r❡♠ ❛s ❛♥ ✐♠♣❧✐❝❛t✐♦♥✱ ❛♥ ✏✐❢✲t❤❡♥✑ st❛t❡♠❡♥t✳

❊①❡r❝✐s❡ ✹✳✺✳✷✾

❆♣♣❧② t❤❡ ❛♥❛❧②s✐s ✐♥ t❤❡ ❧❛st ❡①❛♠♣❧❡ t♦ t❤❡ ♣♦❧②♥♦♠✐❛❧

P (x) = x2 (2x − 1)3 · (x − 1)3 · (x2 − 1) . ❲❡ ❝❛♥ ❛✈♦✐❞ ✉s✐♥❣ t❤❡

▼✉❧t✐♣❧✐❝✐t② ❘✉❧❡

❞✐r❡❝t❧② ✐❢ ✇❡ ❝♦♥❝❡♥tr❛t❡ ♦♥ t❤❡

♦✉r ❛♥❛❧②s✐s ♦♥ ❛ ❢❛♠✐❧✐❛r t❛❜❧❡✿

s✐❣♥s

♦❢ t❤❡ ❢❛❝t♦rs✳ ❲❡ ✇✐❧❧ ❜❛s❡

❘✉❧❡ ♦❢ ❙✐❣♥s ❢♦r ▼✉❧t✐♣❧✐❝❛t✐♦♥

(+) (+) (−) (−)

· · · ·

(+) (−) (+) (−)

= = = =

(+) (−) (−) (+)

❲❡ ✇✐❧❧ ❛❧s♦ ✉s❡ t❤❡ t❤❡♦r❡♠ ❛❜♦✈❡ t❤❛t st❛t❡s t❤❛t t❤❡ s✐❣♥ ♦❢ ❛ ♣♦❧②♥♦♠✐❛❧ ❝❛♥ ♦♥❧② ❝❤❛♥❣❡ ❛t ❛♥ ✐♥t❡r❝❡♣t✳

x✲

❚❤❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s

✹✳✺✳

✸✸✾

❊①❛♠♣❧❡ ✹✳✺✳✸✵✿ ♣❧♦tt✐♥❣ ♣♦❧②♥♦♠✐❛❧s

▲❡t✬s ❛♥❛❧②③❡ t❤✐s ♣♦❧②♥♦♠✐❛❧ 2 ❲❡ ♥❡❡❞ t♦ ❢✉rt❤❡r ❢❛❝t♦r ✐t✳ f (x) = (x4 − 9x2 )(x − 4)2 (3x2 + 2)
2 2 2 2 2 2 2 = x x − 3 (x − 2 ) (3x + 2) ❲❡ ✉s❡ t❤❡ ❉✐✛❡r❡♥❝❡ ♦❢ ❙q✉❛r❡s ❋♦r♠✉❧❛✳ 2 2 2 2 = x (x − 3)(x + 3)(x − 2) (x + 2) · (3x + 2) .

❚❤❡ x✲✐♥t❡r❝❡♣ts ❛r❡ s✐♠♣❧② r❡❛❞ ♦✛ t❤❡ ❧✐st ♦❢ ❧✐♥❡❛r ❢❛❝t♦rs✿

x = 0, 3, −3, −2, 2 .

❚❤✐s t✐♠❡✱ ✇❡ ✇✐❧❧ r❡❧② ♦♥ t❤❡ ❢❛❝t t❤❛t ❛t t❤❡s❡ ♣♦✐♥ts ❛♥❞ ❛t t❤❡s❡ ♣♦✐♥ts ♦♥❧② ❝❛♥ t❤❡ ❢✉♥❝t✐♦♥ ❝❤❛♥❣❡ ✐ts s✐❣♥✳ ❲❡ ♥♦✇ ❧✐st ❛❧❧ t❤❡ ❢❛❝t♦rs✳ ❚❤❡② ❛r❡ s✐♠♣❧❡ ❡♥♦✉❣❤ ❢♦r ✉s t♦ ❞❡t❡r♠✐♥❡ ✇❤❡r❡ ❛♥❞ ✇❤❡t❤❡r ❡❛❝❤ ❝❤❛♥❣❡s ✐ts s✐❣♥✿ t❤❡ ♣♦✐♥ts ❢❛❝t♦rs

−3

−2

0

2

3

s✐❣♥s

x2

+

+

+

+

+

0

+

+

+

+

+

x−3

−

−

−

−

−

−

−

−

−

0

+

x+3

−

0

+

+

+

+

+

+

+

+

+

(x − 2)2

+

+

+

+

+

+

+

+

+

0

+

(x + 2)2

+

0

+

+

+

+

+

+

+

+

+

❞♦♠❛✐♥

···

•

···

•

··· • ··· • ··· • ··· → x

−3

x= f (x) =

−2

0

2

3

+

0

−

0

−

0

−

0

−

0

+

ց

0

⌣

0

⌣

0

⌣

0

⌣

0

ր

❍❡r❡ ✇❡ ❣♦ ✈❡rt✐❝❛❧❧② ✐♥ ❡❛❝❤ ❝♦❧✉♠♥ ❛♥❞ ❞❡t❡r♠✐♥❡ t❤❡ s✐❣♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ✉s✐♥❣ t❤❡ ❛❜♦✈❡ r✉❧❡✳ ❆♥ ✐❞❡❛ ♦❢ ✇❤❛t t❤❡ ❣r❛♣❤ ♦❢ f ❧♦♦❦s ❧✐❦❡ ✐s s❡❡♥ ✐♥ t❤❡ ❧❛st r♦✇✿ ■t ❝r♦ss❡s t❤❡ x✲❛①✐s ❞♦✇♥✇❛r❞✱ t♦✉❝❤❡s ✐t t❤r❡❡ t✐♠❡s ❢r♦♠ ❜❡❧♦✇✱ ❛♥❞ t❤❡♥ ❝r♦ss❡s ✐t ✉♣✇❛r❞✳ ❲❡ ❝❛♥ s❦❡t❝❤ t❤❡ ❣r❛♣❤ ❜❛s❡❞ ♦♥ t❤❡ ❧❛st r♦✇ ♦♥❧②✿

❚❤❡ t❛❜❧❡ s♦❧✈❡s ❢♦r ✉s t❤✐s ❡q✉❛t✐♦♥ ❛♥❞ t❤❡s❡ ✐♥❡q✉❛❧✐t✐❡s✿ f (x) = 0, f (x) > 0, f (x) ≥ 0, f (x) < 0, f (x) ≤ 0 .

❋♦r ❡①❛♠♣❧❡✱ t❤❡ s♦❧✉t✐♦♥ s❡t ♦❢ t❤❡ ✐♥❡q✉❛❧✐t②✿ ✐s

f (x) > 0, ♦r (x4 − 9x2 )(x2 − 4)2 (3x2 + 2) > 0 , (−∞, −3) ∪ (3, +∞) .

❲❡ ❝♦♥✜r♠ ♦✉r ❛♥❛❧②s✐s ✇✐t❤ ❛ ❣r❛♣❤✐❝ ✉t✐❧✐t②✿

✹✳✺✳

❚❤❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s

✸✹✵

❊①❡r❝✐s❡ ✹✳✺✳✸✶

❘❡♣❡❛t t❤❡ ❛♥❛❧②s✐s ❜✉t ✉s❡ t❤❡

y ✲✐♥t❡r❝❡♣t

❛s t❤❡ ✏❞❡❝✐❞❡r✑✳

❊①❡r❝✐s❡ ✹✳✺✳✸✷

❘❡❞♦ t❤❡ ♣r♦❜❧❡♠ ✉s✐♥❣ t❤❡ ▼✉❧t✐♣❧✐❝✐t② ❘✉❧❡✳

❊①❡r❝✐s❡ ✹✳✺✳✸✸

❙♦❧✈❡

f (x) ≤ 0✳

❊①❡r❝✐s❡ ✹✳✺✳✸✹

❆♣♣❧② t❤❡ ❛♥❛❧②s✐s t♦ t❤❡ ♣♦❧②♥♦♠✐❛❧

f (x) = x3 (2x − 2) · (x − 1)2 · (x2 + 2x + 1) . ❊①❛♠♣❧❡ ✹✳✺✳✸✺✿ ✐♥❡q✉❛❧✐t✐❡s ❛s ❜②♣r♦❞✉❝t

❚❤✐s ✐s ✇❤❛t ❤❛♣♣❡♥s ✇❤❡♥ ♦♥❡ s♦❧✈❡s ❛ ♣♦❧②♥♦♠✐❛❧ ❡q✉❛t✐♦♥ ♦r ✐♥❡q✉❛❧✐t② ✇❤❡♥ t❤❡ ♣♦❧②♥♦♠✐❛❧ ✐s ❛❧r❡❛❞② ❢❛❝t♦r❡❞❀ ✇❡ ❤❛✈❡ ✐t ♣❧♦tt❡❞ ❛♥❞ t❤❡ ❛♥s✇❡rs ❛r❡ r❡❛❞ ❢r♦♠ t❤❡ ❣r❛♣❤✿

❋♦r ❡①❛♠♣❧❡✱ t❤❡ s♦❧✉t✐♦♥ s❡t ♦❢ t❤❡ ❡q✉❛t✐♦♥✿

f (x) = 0 , ✐s

{−2, 0, 1, 2} . ❚❤❡ s♦❧✉t✐♦♥ s❡t ♦❢ t❤❡ ✐♥❡q✉❛❧✐t②✿

f (x) ≤ 0 , ✐s ✭✐♥ ❜❧✉❡✮

(−∞, 0] ∪ [1, 2] . ❊①❡r❝✐s❡ ✹✳✺✳✸✻

❙♦❧✈❡

f (x) > 0✳

✹✳✻✳

❚❤❡ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥s

✸✹✶

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

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

❋✐rst✱ t❤❡r❡ ❛r❡ t❤r❡❡ x✲✐♥t❡r❝❡♣ts✿ 1, 2, 3✳ ❚❤❡✐r ♠✉❧t✐♣❧✐❝✐t✐❡s ❛r❡✱ r❡s♣❡❝t✐✈❡❧②✱ ♦❞❞✱ ❡✈❡♥✱ ♦❞❞✳ ❚❤❡ s✐♠♣❧❡st ❝❤♦✐❝❡s ♦❢ t❤❡ ♠✉❧t✐♣❧✐❝✐t✐❡s ❛r❡ 1, 2, 1✳ ❚❤❡♥ t❤❡ ❢❛❝t♦rs ❛r❡ (x − 1), (x − 2), (x − 3)✳ ❚❤❡r❡ ❛r❡ ♥♦ ♦t❤❡r ❧✐♥❡❛r ❢❛❝t♦rs✦ ❚❤❡♥ ♦✉r ❢✉♥❝t✐♦♥ ❝♦✉❧❞ ❜❡ ❛t ✐ts s✐♠♣❧❡st t❤❡ ❢♦❧❧♦✇✐♥❣✿

f (x) = −(x − 1)(x − 2)(x − 3) . ❚❤❡ ♥❡❣❛t✐✈❡ s✐❣♥ ❝♦♠❡s ❢r♦♠ t❤❡ ❢❛❝t t❤❛t t❤❡ ❣r❛♣❤ ♦♣❡♥s

❞♦✇♥✳

❊①❡r❝✐s❡ ✹✳✺✳✸✽

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

✹✳✻✳ ❚❤❡ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥s

❊①❛♠♣❧❡ ✹✳✻✳✶✿ r❡❝✐♣r♦❝❛❧ ❢✉♥❝t✐♦♥

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

y=

1 . x

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

❋✐rst✱

❤♦❧❡s ✐♥ t❤❡ ❞♦♠❛✐♥ ❛♣♣❡❛r✳

❆s ✇❡ s❡❡ ❛❜♦✈❡✱ t❤❡ ❤♦❧❡ ✐♥ t❤❡ ❞♦♠❛✐♥ ❝♦rr❡s♣♦♥❞s t♦ ❛ ✈❡rt✐❝❛❧

✹✳✻✳

❚❤❡ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥s

❧✐♥❡ ♦♥ t❤❡ t♦

xy ✲♣❧❛♥❡

✸✹✷

t❤❛t t❤❡ ❣r❛♣❤ ❝❛♥✬t ✐♥t❡rs❡❝t✦ ❆s t❤❡ ❝✉r✈❡ ❛♣♣r♦❛❝❤❡s t❤✐s ❧✐♥❡✱ ✐t ❤❛s t♦ st❛rt

❝❧✐♠❜✱ ❢❛st❡r ❛♥❞ ❢❛st❡r✱ ✉♣ ♦r ❞♦✇♥✿

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

x 1 1/2 1/3 ... 1/100 ... 1/x 1 2 3 ... 100 ...

▲✐♥❡s ❧✐❦❡ t❤✐s ❛r❡ ❝❛❧❧❡❞

✈❡rt✐❝❛❧ ❛s②♠♣t♦t❡s✳

❆♥ ♦♣♣♦s✐t❡ ✕ ❜✉t ❛❧s♦ ✈❡r② s✐♠✐❧❛r ✕ ❜❡❤❛✈✐♦r ✐s s❡❡♥ ❛❧♦♥❣ ❛ ❝❡rt❛✐♥ ❤♦r✐③♦♥t❛❧ ❧✐♥❡✳ ■❢ 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 ... x 1 2 1/x 1 1/2 1/3 ... 1/100 ... ❙✉❝❤ ❛ ❧✐♥❡ ✐s ❝❛❧❧❡❞ ❛

❤♦r✐③♦♥t❛❧ ❛s②♠♣t♦t❡✳

■❢ ✇❡ ③♦♦♠ ♦✉t✱ t❤❡ ❡♥❞s ♦❢ t❤❡ ❣r❛♣❤ ♠❡r❣❡ ✇✐t❤ t❤❡ ❛s②♠♣t♦t❡s✳ ✭❚❤❡ t♦♣✐❝ ♦❢ ❛s②♠♣t♦t❡s ✐s ❢✉❧❧② ❛❞❞r❡ss❡❞ ✐♥ ❈❤❛♣t❡r ✷❉❈✲✷✳✮

❊①❡r❝✐s❡ ✹✳✻✳✷

❉❡s❝r✐❜❡ t❤❡ s②♠♠❡tr② ❛❧❧✉❞❡❞ t♦ ✐♥ t❤❡ ❡①❛♠♣❧❡✳

❊①❡r❝✐s❡ ✹✳✻✳✸

❉❡s❝r✐❜❡ t❤✐s ❢✉♥❝t✐♦♥ ❛s ❛ tr❛♥s❢♦r♠❛t✐♦♥✳

❲❡ ❛r❡ ❛❞❞✐♥❣

❞✐✈✐s✐♦♥ t♦ t❤❡ ❛❧❧♦✇❡❞ ♦♣❡r❛t✐♦♥s ♥♦✇✦

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

15

❛♥❞

23

♣r♦❞✉❝❡

❢r❛❝t✐♦♥s ✿

15 . 23

❙✐♠✐❧❛r❧②✱ ✇❡ ✉s❡ ❥✉st ♣♦❧②♥♦♠✐❛❧s ✐♥✐t✐❛❧❧②✱ ❜✉t ♦♥❝❡ ✇❡ st❛rt ❞✐✈✐❞✐♥❣✱ ✇❡ ❤❛✈❡ t♦ ❢❛❝❡

x2 − 1

❛♥❞

x3 − x

♣r♦❞✉❝❡

x2 − 1 . x3 − x

❆♥♦t❤❡r ✇♦r❞ ❢♦r ✏❢r❛❝t✐♦♥✑ ✐s ✏r❛t✐♦✑✳ ❚❤❛t ✐s ✇❤② ✇❡ ✉s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ t❡r♠✐♥♦❧♦❣②✿

❢r❛❝t✐♦♥s ✿

✹✳✻✳

❚❤❡ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥s

✸✹✸

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

• ❚❤❡ ❢r❛❝t✐♦♥s ♦❢ ✐♥t❡❣❡rs ❛r❡ ❝❛❧❧❡❞ •

❚❤❡ ❢♦❧❧♦✇✐♥❣ ✇✐❧❧ ❜❡ r♦✉t✐♥❡❧② ✉s❡❞✳

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

f (x) =

r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥ ✿ P (x) . Q(x)

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

❚❤❡ ♣♦ss✐❜✐❧✐t② ♦❢ ❞✐✈✐s✐♦♥ ♠❛❦❡s t❤❡ ✐ss✉❡ ♦❢ t❤❡ ✐♠♣❧✐❡❞ ❞♦♠❛✐♥✱ ✐♥ ❝♦♥tr❛st t♦ ♣♦❧②♥♦♠✐❛❧✱ ❛ ♥♦♥✲tr✐✈✐❛❧ ♠❛tt❡r✳ ❲❡ ❢❛❝t♦r t❤❡ ❞❡♥♦♠✐♥❛t♦r ❛♥❞ ✉s❡ t❤❡ ▲✐♥❡❛r ❋❛❝t♦r ❚❤❡♦r❡♠ ❢r♦♠ t❤❡ ❧❛st s❡❝t✐♦♥✳

❊①❛♠♣❧❡ ✹✳✻✳✺✿ ❞♦♠❛✐♥ ♦❢ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥ ❲❤❛t ✐s t❤❡ ❞♦♠❛✐♥ ♦❢

x−1 ? x+1 ❲❡ ❧♦♦❦ ❛t t❤❡ ❞❡♥♦♠✐♥❛t♦r✱ s❡t ✐t ❡q✉❛❧ t♦ ③❡r♦✱ x + 1 = 0✱ ❛♥❞ s♦❧✈❡ ❢♦r x✳ ❚❤❡♥✱ x = −1 ✐s ♥♦t ✐♥ t❤❡ ❞♦♠❛✐♥ ❛♥❞ t❤❡ r❡st ♦❢ t❤❡ r❡❛❧ ♥✉♠❜❡rs ❛r❡✳ f (x) =

❊①❡r❝✐s❡ ✹✳✻✳✻ ❲❤❛t ✐s t❤❡ ❞♦♠❛✐♥ ♦❢ t❤✐s ❢✉♥❝t✐♦♥✿

f (x) =

x+1 ? x+1

❊①❛♠♣❧❡ ✹✳✻✳✼✿ ♠♦r❡ ❞♦♠❛✐♥s ♦❢ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥s ❲❤❛t ✐s t❤❡ ❞♦♠❛✐♥ ♦❢

f (x) = ❙❡t t❤❡ ❞❡♥♦♠✐♥❛t♦r t♦ 0 ❛♥❞ s♦❧✈❡✿

x−1 ? x2 − 4

x2 − 4 = 0 ⇐⇒ 2 x =4 ⇐⇒ x = ±2 .

❙♦✱ t❤❡ ❞♦♠❛✐♥ ❝♦♥s✐sts ♦❢ ❛❧❧ r❡❛❧ ♥✉♠❜❡rs ❡①❝❡♣t ±2✿

❞♦♠❛✐♥ = {x : x 6= ±2} = (−∞, −2) ∪ (−2, 2) ∪ (2, +∞) . ❚❤❡ s♦❧✉t✐♦♥s t❤❛t ✇❡ ❤❛✈❡ ❢♦✉♥❞✱ x = −2 ❛♥❞ x = 2 ❛r❡✱ ✐♥ ❢❛❝t✱ t❤❡ ❡q✉❛t✐♦♥s ♦❢ t✇♦ ✈❡rt✐❝❛❧ ❧✐♥❡s t❤❛t ❝✉t t❤❡ xy ✲♣❧❛♥❡ s♦ t❤❛t t❤❡ ❣r❛♣❤ ❝❛♥✬t ❝r♦ss t❤❡♠✦ ❚❤❡r❡❢♦r❡✱ t❤❡ ❣r❛♣❤ ✇✐❧❧ ❤❛✈❡ t♦ ❤❛✈❡ t❤r❡❡ ❜r❛♥❝❤❡s ✕ ❡❛❝❤ ✇✐t❤✐♥ ♦♥❡ ♦❢ t❤❡s❡ t❤r❡❡ ♣❛rts ♦❢ t❤❡ ♣❧❛♥❡✿

✹✳✻✳

❚❤❡ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥s

✸✹✹

❊①❡r❝✐s❡ ✹✳✻✳✽

❲❤❛t ✐s t❤❡ ❞♦♠❛✐♥ ♦❢

f (x) =

x2 − 1 ? (x2 + 4x + 4)(x4 + 2)

❙♦✱ ✇❡ ❞✐s❝♦✈❡r❡❞ ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥ t❤❛t t❤❡ ❛❧❣❡❜r❛ ♦❢ ♣♦❧②♥♦♠✐❛❧s ♠✐♠✐❝s t❤❡ ❛❧❣❡❜r❛ ♦❢ ✐♥t❡❣❡rs✳ ❲❡ t❛❦❡ ❛♥♦t❤❡r st❡♣ ✐♥ t❤❛t ❞✐r❡❝t✐♦♥ ❛♥❞ ♦❜s❡r✈❡ t❤❛t t❤❡ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥s ❛♣♣❡❛r ❢r♦♠ ♣♦❧②♥♦♠✐❛❧s ✕ ❥✉st ❧✐❦❡ t❤❡ r❛t✐♦♥❛❧ ♥✉♠❜❡rs ❢r♦♠ ✐♥t❡❣❡rs ✕ ✈✐❛ ❞✐✈✐s✐♦♥✿

❲❡ ♥♦✇ ❝♦♥t✐♥✉❡ ✇✐t❤ t❤❡

❛♥❛❧♦❣②

✇❡ st❛rt❡❞ ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥✳

❆♥❛❧♦❣②✿ ✐♥t❡❣❡rs ✈s✳ ♣♦❧②♥♦♠✐❛❧s

■♥t❡❣❡rs P♦❧②♥♦♠✐❛❧s

❚❤❡

s✉♠

❛♥❞ t❤❡

♣r♦❞✉❝t

♦❢ t✇♦ ✐♥t❡❣❡rs

❚❤❡

s✉♠

❛♥❞ t❤❡

♣r♦❞✉❝t

♦❢ t✇♦ ♣♦❧②♥♦♠✐❛❧s

✐s ❛♥ ✐♥t❡❣❡r✳ ✐s ❛ ♣♦❧②♥♦♠✐❛❧✳ ❚❤❡

r❛t✐♦

♦❢ t✇♦ ✐♥t❡❣❡rs ❚❤❡

r❛t✐♦

♦❢ t✇♦ ♣♦❧②♥♦♠✐❛❧s

✐s ❝❛❧❧❡❞ ❛ r❛t✐♦♥❛❧ ♥✉♠❜❡r✳ ✐s ❝❛❧❧❡❞ ❛ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥✳ ❘❛t✐♦♥❛❧ ♥✉♠❜❡rs ❘❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥s

❚❤❡ s✉♠✱ t❤❡ ♣r♦❞✉❝t✱ ❛♥❞ t❤❡ r❛t✐♦ ❚❤❡ s✉♠✱ t❤❡ ♣r♦❞✉❝t✱ ❛♥❞ t❤❡ r❛t✐♦ ♦❢ t✇♦ r❛t✐♦♥❛❧ ♥✉♠❜❡rs ✐s ❛ r❛t✐♦♥❛❧ ♥✉♠❜❡r✳ ♦❢ t✇♦ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥s ✐s ❛ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥✳ 2 1 7 1 x2 + 2x − 1 x + = + = 3 2 3·2 x−1 x+1 (x − 1)(x + 1) ❚❤❡ ♥✉♠❡r❛t♦r ❛♥❞ t❤❡ ❞❡♥♦♠✐♥❛t♦r ♦❢ ❡✈❡r② ❚❤❡ ♥✉♠❡r❛t♦r ❛♥❞ t❤❡ ❞❡♥♦♠✐♥❛t♦r ♦❢ ❡✈❡r② r❛t✐♦♥❛❧ ♥✉♠❜❡r ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❛s r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥ ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❛s t❤❡ ♣r♦❞✉❝ts ♦❢ ❞✐✛❡r❡♥t ♣r✐♠❡s✳ t❤❡ ♣r♦❞✉❝ts ♦❢ ❞✐✛❡r❡♥t ✐rr❡❞✉❝✐❜❧❡ ❢❛❝t♦rs✳ 25 · 52 · 71 · ... P (x − 1)5 · (x + 2)2 · (x2 + 1) · ... p = 2 = q 3 · 112 · ... Q (x + 1)2 · (x − 3)1 · (2x2 + 1) · ... ❲❛r♥✐♥❣✦ ❈❛♥❝❡❧✐♥❣ ❢❛❝t♦rs ✇✐❧❧ s✐♠♣❧✐❢② ❛ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥✱ ❜✉t ✐t ♠✐❣❤t ❛❧s♦ ❝❤❛♥❣❡ ✐ts ❞♦♠❛✐♥✳

❍♦✇ ❞♦ ✇❡ ✜♥❞ t❤❡ x✲✐♥t❡r❝❡♣ts ♦❢ ❛ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥❄ ❏✉st ❛s ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥✱ ✇❡ ✉s❡ t❤❡ ▲✐♥❡❛r ❋❛❝t♦r t❤❛t st❛t❡s t❤❛t s✉❜st✐t✉t✐♥❣ x = x1 ✐♥t♦ t❤❡ ❧✐♥❡❛r ♣♦❧②♥♦♠✐❛❧ x − x1 ♠❛❦❡s ✐t ❡q✉❛❧ t♦ ③❡r♦ ✕ ❛s ✇❡❧❧ ❛s ❛♥② ♣♦❧②♥♦♠✐❛❧ t❤❛t ❤❛s ✐t ❛s ❛ ❢❛❝t♦r✳

❚❤❡♦r❡♠

❊①❛♠♣❧❡ ✹✳✻✳✾✿

x✲✐♥t❡r❝❡♣ts

♦❢ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥s

❖♥❝❡ ❜♦t❤ ♣♦❧②♥♦♠✐❛❧s ❛r❡ ❢❛❝t♦r❡❞✱ ✇❡ ❥✉st ♥❡❡❞ t♦ ✇❛t❝❤ ❢♦r ♣♦ss✐❜❧❡ ❝❛♥❝❡❧❧❛t✐♦♥s✿

x(x − 1) x = ❢♦r x 6= 1 . (x + 1)(x − 1) x+1

✹✳✻✳

❚❤❡ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥s

❙♦✱

x1 = −1

✐s ❛♥

✸✹✺

x✲✐♥t❡r❝❡♣t

❜✉t

x2 = 1

✐s ♥♦t✦

❚❤✐s ❛♥❛❧②s✐s ✐♠♣❧✐❡s t❤❡ ❢♦❧❧♦✇✐♥❣✿

❚❤❡♦r❡♠ ✹✳✻✳✶✵✿ ❘❛t✐♦♥❛❧ ❋✉♥❝t✐♦♥✬s

x✲✐♥t❡r❝❡♣t

❆ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥✬s

• •

x✲■♥t❡r❝❡♣ts

✐s ❛ r♦♦t ♦❢ ✐ts ♥✉♠❡r❛t♦r✱ ♣r♦✈✐❞❡❞ ✐t ✐s♥✬t ❛❧s♦ ❛♠♦♥❣ t❤❡ r♦♦ts ♦❢ t❤❡ ❞❡♥♦♠✐♥❛t♦r✳

Pr♦♦❢✳

P (x) = 0 ❆◆❉ Q(x) 6= 0 =⇒

P (x) = 0. Q(x)

❊①❡r❝✐s❡ ✹✳✻✳✶✶ ❙t❛t❡ t❤❡ t❤❡♦r❡♠ ✐♥ t❤❡ ❢♦r♠ ♦❢ ❛♥ ❡q✉✐✈❛❧❡♥❝❡✳

❲❛r♥✐♥❣✦ ❇❡❧♦✇ ✇❡ ❛ss✉♠❡ t❤❛t t❤❡ ♥✉♠❡r❛t♦r ❛♥❞ t❤❡ ❞❡✲ ♥♦♠✐♥❛t♦r ❞♦♥✬t s❤❛r❡ ❢❛❝t♦rs✳

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

s❡♣❛r❛t❡ ❜r❛♥❝❤❡s

✳

❊①❛♠♣❧❡ ✹✳✻✳✶✷✿ ♣❧♦t r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥ ❲❤❛t ❞♦❡s t❤❡ ♣❧♦t ♦❢ t❤✐s ❢✉♥❝t✐♦♥ ❧♦♦❦ ❧✐❦❡✿

f (x) = ❲❡ ❧♦♦❦ ❛t t❤❡ ❞❡♥♦♠✐♥❛t♦r✱ s❡t ✐t ❡q✉❛❧ t♦ ③❡r♦✱

x ? x2 − 1 x 2 − 1 = 0✱

❛♥❞ s♦❧✈❡ ❢♦r

x✳

t❤❡ ❞♦♠❛✐♥ ❛♥❞ t❤❡ r❡st ♦❢ t❤❡ r❡❛❧ ♥✉♠❜❡rs ❛r❡✳ ❲❡ t❤❡♥ ❞r❛✇ t❤❡s❡ ✈❡rt✐❝❛❧ ❧✐♥❡s

x = −1 ❚❤❡ ❣r❛♣❤ ❝❛♥✬t ❝r♦ss t❤❡s❡ ❧✐♥❡s✦

❛♥❞

x = ±1 ✐s ♥♦t ✐♥ ♦♥ t❤❡ xy ✲♣❧❛♥❡✿

❚❤❡♥✱

x = 1.

❚❤❡r❡ ❛r❡✱ t❤❡r❡❢♦r❡✱ t❤r❡❡ ❜r❛♥❝❤❡s✳

❇❛s❡❞ ♦♥ t❤❡ ❛✈❛✐❧❛❜❧❡

✐♥❢♦r♠❛t✐♦♥✱ t❤✐s ✐s ♦♥❡ ♦❢ t❤❡ ♣♦ss✐❜✐❧✐t✐❡s✿

❊①❡r❝✐s❡ ✹✳✻✳✶✸ Pr❡s❡♥t ♦t❤❡r ❡①❛♠♣❧❡s ♦❢ ❣r❛♣❤s ♦❢ ❢✉♥❝t✐♦♥s ✇✐t❤ t❤✐s ❞♦♠❛✐♥ ❛♥❞ t❤✐s ♦✇♥ ❡①❛♠♣❧❡✳ ❘❡♣❡❛t✳

x✲✐♥t❡r❝❡♣t✳

▼❛❦❡ ✉♣ ②♦✉r

✹✳✻✳

❚❤❡ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥s

✸✹✻

❚❤✐s ✐s t❤❡ s✉♠♠❛r② ♦❢ ✇❤❡r❡ t❤❡ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t t❤❡ ❣r❛♣❤ ♦❢ ❛ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥ ❝♦♠❡s ❢r♦♠✿ ▲✐♥❡❛r ❢❛❝t♦r (x − x1 ) ✐♥ ♥✉♠❡r❛t♦r → x = x1 ✐s ❛♥ x✲✐♥t❡r❝❡♣t✳ −−•−− ▲✐♥❡❛r ❢❛❝t♦r (x − x2 ) ✐♥ ❞❡♥♦♠✐♥❛t♦r → x = x2 ✐s ❛ ✈❡rt✐❝❛❧ ❛s②♠♣t♦t❡✳ − − ◦ − − ▼✉❧t✐♣❧②✐♥❣ ❜② 0 ❛♥❞ ❞✐✈✐❞✐♥❣ ❜② 0 ♣r♦❞✉❝❡ ✈❡r② ❞✐✛❡r❡♥t r❡s✉❧ts✿

❲❛r♥✐♥❣✦ ❆ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥ ❞♦❡s♥✬t ❤❛✈❡ t♦ ❤❛✈❡ ❛s②♠♣✲ t♦t❡s✱ ✈❡rt✐❝❛❧ ♦r ❤♦r✐③♦♥t❛❧✳

❏✉st ❛s ✐♥ t❤❡ ❝❛s❡ ♦❢ ♣♦❧②♥♦♠✐❛❧s✱ ✇❡ ❤❛✈❡ t♦ ❧♦♦❦ ❛t t❤❡ ♠✉❧t✐♣❧✐❝✐t✐❡s ♦❢ t❤❡s❡ ❧✐♥❡❛r ❢❛❝t♦rs✳ ❚❤❡ ❡✛❡❝t ♦❢ t❤❡ ❧✐♥❡❛r ❢❛❝t♦rs ♦❢ t❤❡ ♥✉♠❡r❛t♦r r❡♠❛✐♥s t❤❡ s❛♠❡ ❛s ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥❀ ✐t ♣r♦❞✉❝❡s x✲✐♥t❡r❝❡♣ts ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡✛❡❝t✿ • ❚❤❡ ❢✉♥❝t✐♦♥ ✐s ❝❤❛♥❣✐♥❣ t❤❡ s✐❣♥ ✭❛♥❞ t❤❡ ❣r❛♣❤ ✐s ❝r♦ss✐♥❣ t❤❡ x✲❛①✐s✮ ✇❤❡♥ t❤❡ ♠✉❧t✐♣❧✐❝✐t② ✐s ♦❞❞✳

• ❚❤❡ ❢✉♥❝t✐♦♥ ✐s ♥♦t ❝❤❛♥❣✐♥❣ t❤❡ s✐❣♥ ✭❛♥❞ t❤❡ ❣r❛♣❤ ✐s t♦✉❝❤✐♥❣ t❤❡ x✲❛①✐s✮ ✇❤❡♥ t❤❡ ♠✉❧t✐♣❧✐❝✐t② ✐s

❡✈❡♥✳

❚❤❡ ❝❛s❡ ♦❢ ❛ ❧✐♥❡❛r ❢❛❝t♦r ✐♥ t❤❡ ❞❡♥♦♠✐♥❛t♦r ✐s ✈❡r② s✐♠✐❧❛r ❛s ✇❡ ❝❛♥ ❝♦♥t✐♥✉❡ t♦ ✇❛t❝❤ ❤♦✇ t❤❡ s✐❣♥s ❝❤❛♥❣❡✳ ◆♦ ❝❤❛♥❣❡ ♦♥ t❤❡ ❧❡❢t✱ ❝❤❛♥❣❡ ♦♥ t❤❡ r✐❣❤t✿

✹✳✻✳ ❚❤❡ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥s

✸✹✼

❚❤❡ ❞✐✛❡r❡♥❝❡ ✐s t❤❛t t❤❡ s✐❣♥s ❝❤❛♥❣❡ ♥♦t ❜② ❝r♦ss✐♥❣ t❤❡ x✲❛①✐s ❜✉t ❜② ✭✐♥✜♥✐t❡✮ ❥✉♠♣✐♥❣✳ ❲❡ s✉♠♠❛r✐③❡ t❤❡ r❡s✉❧t ❜❡❧♦✇✳ ❚❤❡♦r❡♠ ✹✳✻✳✶✹✿ ▼✉❧t✐♣❧✐❝✐t② ❘✉❧❡ ❢♦r ❘❛t✐♦♥❛❧ ❋✉♥❝t✐♦♥s ❋♦r ❛ ❧✐♥❡❛r ❢❛❝t♦r ✐♥ t❤❡ ❞❡♥♦♠✐♥❛t♦r ♦❢ ❛ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♥ t❤❡ ✈✐❝✐♥✐t② ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ✈❛❧✉❡ ♦❢

• •

■❢ t❤❡ ♠✉❧t✐♣❧✐❝✐t② ♦❢ t❤❡ ❢❛❝t♦r ✐s t❤❡

x✲❛①✐s✿

♦❞❞✱ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❥✉♠♣s ♦✈❡r

✉♣✇❛r❞ ♦r ❞♦✇♥✇❛r❞✳

■❢ t❤❡ ♠✉❧t✐♣❧✐❝✐t② ♦❢ t❤❡ ❢❛❝t♦r ✐s t❤❡ s❛♠❡ s✐❞❡ ♦❢ t❤❡

x✲❛①✐s✿

x✿

❡✈❡♥✱ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ st❛②s ♦♥

❛❜♦✈❡ ♦r ❜❡❧♦✇✳

❲❡ ✐♥t❡r♣r❡t t❤❡ t❤❡♦r❡♠ ✐♥ t❡r♠s ♦❢ t❤❡ ❝❤❛♥❣❡ ♦❢ s✐❣♥ ✿

❆❜❜r❡✈✐❛t❡❞✱ t❤❡ t❤❡♦r❡♠ ❜❡❝♦♠❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥✈❡♥✐❡♥t r✉❧❡✿ ▲✐♥❡❛r ❢❛❝t♦rs ♦❢ ❞❡♥♦♠✐♥❛t♦r ❛♥❞ ❛s②♠♣t♦t❡s ❆❧❣❡❜r❛

●❡♦♠❡tr②

♠✉❧t✐♣❧✐❝✐t② s✐❣♥ ✈❡rt✐❝❛❧ ❛s②♠♣t♦t❡ ✳ ✳ ♦❞❞ ❝❤❛♥❣❡ ♦❢ s✐❣♥ ր ✳✳ ր ♦r ց ✳✳ ց ✳ ✳ ❡✈❡♥ ♥♦ ❝❤❛♥❣❡ ♦❢ s✐❣♥ ր ✳✳ ց ♦r ց ✳✳ ր ■♥ ❝♦♥tr❛st t♦ t❤❡ ❝❛s❡ ♦❢ ❛ ♣♦❧②♥♦♠✐❛❧✱ t❤❡ ❥✉♠♣ ♦✈❡r ❛ ✈❡rt✐❝❛❧ ❛s②♠♣t♦t❡ ♠✐❣❤t ❜❡ ✐♥✜♥✐t❡ ✦ ❊①❛♠♣❧❡ ✹✳✻✳✶✺✿ ❣r❛♣❤ ♦❢ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥

❈♦♥s✐❞❡r t❤❡ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥ f (x) =

x2 · (x + 2) · (x − 1) · (x2 + 1) . (x + 1) · (x − 3)2 · (x2 + 2)

❲❡ ❛♥❛❧②③❡ t❤❡s❡ t✇♦ ♣♦❧②♥♦♠✐❛❧s s❡♣❛r❛t❡❧② ❛♥❞ ❞❡r✐✈❡ t✇♦ s❡♣❛r❛t❡ s❡ts ♦❢ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t t❤❡

✹✳✻✳

❚❤❡ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥s

✸✹✽

❢✉♥❝t✐♦♥✿

x✲✐♥t❡r❝❡♣ts✿ x = −2 0 1 x= −1 3

❢r♦♠ t❤❡ ♥✉♠❡r❛t♦r✳✳✳ ❢r♦♠ t❤❡ ❞❡♥♦♠✐♥❛t♦r✳✳✳

✈❡rt✐❝❛❧ ❛s②♠♣t♦t❡s✿

❚❤❡ t✇♦ ✐rr❡❞✉❝✐❜❧❡ q✉❛❞r❛t✐❝ ❢❛❝t♦rs ❞♦♥✬t ❝♦♥tr✐❜✉t❡ ❛♥②t❤✐♥❣✳ ❚❤✐s ✐s ✇❤❛t t❤❡

x✲❛①✐s

❧♦♦❦s ❧✐❦❡

✇✐t❤ t❤❡ t❤r❡❡ ♣♦✐♥ts ♦❢ t❤❡ ❣r❛♣❤ ❛♥❞ t✇♦ ❤♦❧❡s ✐♥ t❤❡ ❞♦♠❛✐♥ s❤♦✇♥✿

x✲❛①✐s✿ −

• ◦ • • ◦ ... −→ x −2 −1 0 1 3

❚❤❡ ♥✉♠❡r❛t♦r✬s r♦♦ts ❛♥❞ t❤❡✐r ♠✉❧t✐♣❧✐❝✐t✐❡s t❡❧❧ ✉s ✇❤❡t❤❡r t❤❡ ❣r❛♣❤

♣❛ss❡s ❛❝r♦ss

t❤❡

x✲❛①✐s

♦r

❥✉♠♣s ♦✈❡r

t❤❡

x✲❛①✐s

♦r

st❛②s ♦♥ t❤❡ s❛♠❡ s✐❞❡✿

−2 1

0 2

1 1

❝r♦ss

t♦✉❝❤

❝r♦ss

x✲✐♥t❡r❝❡♣ts✿ ♠✉❧t✐♣❧✐❝✐t✐❡s✿ ❣r❛♣❤✿

❚❤❡ ❞❡♥♦♠✐♥❛t♦r✬s r♦♦ts ❛♥❞ t❤❡✐r ♠✉❧t✐♣❧✐❝✐t✐❡s t❡❧❧ ✉s ✇❤❡t❤❡r t❤❡ ❣r❛♣❤ st❛②s ♦♥ t❤❡ s❛♠❡ s✐❞❡✿ ❛s②♠♣t♦t❡s✿ ♠✉❧t✐♣❧✐❝✐t✐❡s✿ ❣r❛♣❤✿

−1 1

3 2

❥✉♠♣

st❛②

❚❤❡r❡ ❛r❡ st✐❧❧ t✇♦ ♣♦ss✐❜✐❧✐t✐❡s ❢♦r ❡❛❝❤ ♦❢ t❤❡s❡ ♣♦✐♥ts✳ ❏✉st ❛s ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥✱ ✇❡ ♥❡❡❞ ❛

st❛rt✐♥❣

♣♦✐♥t ❛s ❛ ✏❞❡❝✐❞❡r✑✳ ❲❡ ❝❤♦♦s❡ x = −10✳ ❙✐♥❝❡ f (−10) < 0✱ ✇❡ ❤❛✈❡ ❛ ♣♦✐♥t t♦ t❤❡ ❧❡❢t ♦❢ t❤❡ ❧❡❢t✲♠♦st x✲✐♥t❡r❝❡♣t✱ x = −2✱ t❤❛t ❧✐❡s ❜❡❧♦✇ t❤❡ x✲❛①✐s✳ ■t ❢♦❧❧♦✇s t❤❛t ✇❡ ♠✉st ❝r♦ss t❤❡ x✲❛①✐s ✉♣✇❛r❞ ❛t x = −2✳ ❆❢t❡r t❤❛t✱ ✇❡ ❛r❡ ❛❜♦✈❡ t❤❡ x✲❛①✐s ❛♥❞ t❤❡ ♣r♦❝❡ss ❝♦♥t✐♥✉❡s✿ x✲✈❛❧✉❡s✿ −10 ❣r❛♣❤✿ − ❣r❛♣❤✿

x − ❛①✐s✿

−2 •

❝r♦ss

ր

−1 ◦

❥✉♠♣ ✳ ր ✳✳ ✳ ✳ ✳ ✳ ✳ ✳ ր

0 •

1 •

t♦✉❝❤

❝r♦ss

⌢

ր

3 ◦

st❛② ✳ ր ✳✳ ց ✳ ✳ ✳ ✳ ✳ ✳

−→ x

−→ x

❚❤❡ ❜♦tt♦♠ ♦❢ t❤❡ t❛❜❧❡ ✐s ♠❡❛♥t t♦ ✈✐s✉❛❧✐③❡ t❤❡ ❞❛t❛ t❤❛t ✇❡ ❤❛✈❡ ❝♦❧❧❡❝t❡❞✳ ❋✉rt❤❡r♠♦r❡✱ ❤❡r❡ ✐s ❛ r♦✉❣❤ s❦❡t❝❤ ♦❢ t❤❡ ❣r❛♣❤ ❜❛s❡❞ ♦♥ t❤✐s ❞❛t❛✿

❲❡ ❝❛♥ ❛❧s♦ ❞❡r✐✈❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ✐♥❡q✉❛❧✐t✐❡s ❢♦r t❤❡ ❢✉♥❝t✐♦♥ ❢r♦♠ t❤✐s s❦❡t❝❤✳ ❋♦r ❡①❛♠♣❧❡✱ ❢♦r t❤❡ ✐♥❡q✉❛❧✐t②

f (x) > 0 , t❤❡ s♦❧✉t✐♦♥ s❡t ✐s

(−2, −1) ∪ (1, 3) ∪ (3, +∞) . ❆♥❞ s♦ ♦♥✳

❯♥❧✐❦❡ ♣♦❧②♥♦♠✐❛❧s✱ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥s

❝❛♥

❥✉♠♣ ♦✈❡r t❤❡

x✲❛①✐s✳

❍♦✇❡✈❡r✱ ✐t ❝❛♥ ♦♥❧② ❤❛♣♣❡♥ ✉♥❞❡r ✈❡r②

s♣❡❝✐✜❝ ❝✐r❝✉♠st❛♥❝❡s✿ ✈❡rt✐❝❛❧ ❛s②♠♣t♦t❡s✳ ■♠♣❧✐❝✐t❧②✱ ✇❡ ❤❛✈❡ r❡❧✐❡❞ ♦♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢❛❝t ✭t♦ ❜❡ ❛❞❞r❡ss❡❞ ✐♥ ❈❤❛♣t❡r ✷❉❈✲✷✮✳

✹✳✻✳

❚❤❡ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥s

✸✹✾

❚❤❡♦r❡♠ ✹✳✻✳✶✻✿ ❈♦♥t✐♥✉✐t② ♦❢ ❘❛t✐♦♥❛❧ ❋✉♥❝t✐♦♥s

❚❤❡ ❣r❛♣❤ ♦❢ ❛ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥ ❝❛♥ ❤❛✈❡ ❛ ♣♦✐♥t ❜❡❧♦✇ t❤❡ ❛❜♦✈❡ t❤❡

x✲❛①✐s

❛t t❤❡ s❛♠❡ t✐♠❡ ♦♥❧② ✐❢ t❤❡r❡ ✐s ❛♥

x✲❛①✐s

x✲✐♥t❡r❝❡♣t

❛♥❞ ❛ ♣♦✐♥t

♦r ❛ ✈❡rt✐❝❛❧

❛s②♠♣t♦t❡ ❜❡t✇❡❡♥ t❤❡♠✳

❊①❡r❝✐s❡ ✹✳✻✳✶✼

❙t❛t❡ t❤❡ t❤❡♦r❡♠ ❛s ❛♥ ✐♠♣❧✐❝❛t✐♦♥✱ ❛♥ ✏✐❢✲t❤❡♥✑ st❛t❡♠❡♥t✳

❊①❡r❝✐s❡ ✹✳✻✳✶✽

❲r✐t❡ t❤❡ s♦❧✉t✐♦♥ s❡ts ❢♦r t❤❡ ✐♥❡q✉❛❧✐t✐❡s ✇✐t❤

f

❢r♦♠ t❤❡ ❧❛st ❡①❛♠♣❧❡✿

f (x) ≥ 0, f (x) < 0, f (x) ≥ 0 . ❲❡ ❝❛♥ ❛✈♦✐❞ ✉s✐♥❣ t❤❡ ▼✉❧t✐♣❧✐❝✐t② ❘✉❧❡ ❞✐r❡❝t❧② ✐❢ ✇❡ ❝♦♥❝❡♥tr❛t❡ ♦♥ t❤❡

s✐❣♥s ♦❢ t❤❡ ❢❛❝t♦rs✳ ❲❡ ✇✐❧❧ ❜❛s❡

♦✉r ❛♥❛❧②s✐s ♦♥ t❤❡ ❢♦❧❧♦✇✐♥❣ r✉❧❡s ❢♦r ❞✐✈✐s✐♦♥ t❤❛t ❛r❡ ✐❞❡♥t✐❝❛❧ t♦ t❤♦s❡ ❢♦r ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✉s❡❞ ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥✿ ❘✉❧❡ ♦❢ ❙✐❣♥s ❢♦r ❉✐✈✐s✐♦♥

(+) (+) (−) (−)

/ / / /

(+) (−) (+) (−)

= = = =

(+) (−) (−) (+)

■♥❞❡❡❞✱ ✇❡❛t❤❡r ✇❡ ♠✉❧t✐♣❧② ♦r ❞✐✈✐❞❡✱ t❤❡ s✐❣♥ ♦❢ t❤❡ t❡r♠ ❛✛❡❝ts t❤❡ r❡s✉❧t t❤❡ s❛♠❡✳ ❲❡ ✇✐❧❧ ❛❧s♦ ✉s❡ t❤❡ ❢❛❝t t❤❛t t❤❡ s✐❣♥ ♦❢ ❛ ♣♦❧②♥♦♠✐❛❧ ❝❛♥ ♦♥❧② ❝❤❛♥❣❡ ❛t ❛♥

x✲✐♥t❡r❝❡♣t✳

❊①❛♠♣❧❡ ✹✳✻✳✶✾✿ ❛♥♦t❤❡r ❣r❛♣❤ ♦❢ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥

▲❡t✬s ❛♥❛❧②③❡ t❤✐s r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥

f (x) = ❋✐rst✱ t❤❡ ❞♦♠❛✐♥ ✐s ❛❧❧ r❡❛❧s ❡①❝❡♣t ♦♥

x✮

x = ±1✳

x(3x2 + 1) . (x − 1)(x + 1)

❲❡ ♥♦✇ ♥❡❡❞ t♦ ✜♥❞ t❤❡ s✐❣♥s ♦❢ t❤❡ ❢❛❝t♦rs ✭❞❡♣❡♥❞❡♥t

❛♥❞ t❤❡♥ t❤❡ s✐❣♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥✳ ❲❡ t❛❦❡ ♥♦t❡ ♦❢ t❤❡ ❞♦♠❛✐♥ ❛♥❞ ❧✐st ❛❧❧ t❤❡ ❢❛❝t♦rs ♣r❡s❡♥t✳

x ❛s ✐t ♣❛ss❡s f (x) ❛s ✐t ✈❛r✐❡s ✇✐t❤ x✱ ♦♥❡ ✐♥t❡r✈❛❧ ❛t ❛ t✐♠❡ ✉s✐♥❣ t❤❡ ❘✉❧❡

❲❡ t❤❡♥ ❞❡t❡r♠✐♥❡ ✇❤❡r❡ ❡❛❝❤ ♦❢ t❤❡♠ ✐s ❡q✉❛❧ t♦ ③❡r♦ ❛♥❞ ❤♦✇ ✐ts s✐❣♥ ❝❤❛♥❣❡s ✇✐t❤ t❤✐s ❧♦❝❛t✐♦♥✳ ❋✐♥❛❧❧②✱ ✇❡ ✜♥❞ t❤❡ s✐❣♥ ♦❢

♦❢ ❙✐❣♥s ✭❢♦r ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❛♥❞ ❢♦r ❞✐✈✐s✐♦♥✮ ♣r❡s❡♥t❡❞ ❛❜♦✈❡✳ t❤❡ ♣♦✐♥ts ❢❛❝t♦rs

s✐❣♥s

x 3x + 1 x−1 x+1 x✲❛①✐s x=

− + − −

f (x)

−

2

−1 − + − 0 ◦ −1 ✳ ✳ ✳

0 − + − +

0 + − + • 0

❚❤❡s❡ ❛r❡ t❤❡ r❡s✉❧ts✿

1 + + − +

+ 0 −

❲❡ ❝♦♥✜r♠ t❤❡ r❡s✉❧t ❜② ✉s✐♥❣ ❛ ❣r❛♣❤✲♣❧♦tt✐♥❣ ❛♣♣❧✐❝❛t✐♦♥✿

+ + 0 + ◦ 1 ✳ ✳ ✳

+ + + + −→ x +

✹✳✻✳

❚❤❡ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥s

✸✺✵

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

f (x) ≤ 0 , t❤❡ s♦❧✉t✐♦♥ s❡t ✐s

(−∞, −1) ∪ [0, 1) . ❆♥❞ s♦ ♦♥✳ ❊①❡r❝✐s❡ ✹✳✻✳✷✵

❙♦❧✈❡

f (x) > 0✳

❊①❡r❝✐s❡ ✹✳✻✳✷✶

❘❡♣❡❛t t❤❡ ❛♥❛❧②s✐s ❜✉t ✉s❡ t❤❡

y ✲✐♥t❡r❝❡♣t

❛s t❤❡ ✏❞❡❝✐❞❡r✑✳

❊①❡r❝✐s❡ ✹✳✻✳✷✷

❘❡❞♦ t❤❡ ♣r♦❜❧❡♠ ✉s✐♥❣ t❤❡ ▼✉❧t✐♣❧✐❝✐t② ❘✉❧❡✳

❊①❡r❝✐s❡ ✹✳✻✳✷✸

❉❡s❝r✐❜❡ s✉❝❤ ❛ ❢✉♥❝t✐♦♥ ❛s ❛ tr❛♥s❢♦r♠❛t✐♦♥✳ ❊①❡r❝✐s❡ ✹✳✻✳✷✹

❆♣♣❧② t❤❡ ❛♥❛❧②s✐s t♦ t❤❡ r❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t②✿

x3 (2x − 2) . f (x) = (x − 1)2 · (x2 + 2x + 1) ❆s ✇❡ ❧♦♦❦ ❛t t❤❡ ❣r❛♣❤s✱ ✇❡ r❡❛❧✐③❡ t❤❛t t❤❡ ❡✛❡❝t ♦❢ ❛ ❧✐♥❡❛r ❢❛❝t♦r ✐♥ t❤❡ ♥✉♠❡r❛t♦r ❛♥❞ t❤❡ ❞❡♥♦♠✐♥❛t♦r ♦❢ ❛ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥ ✐s✱ ✐♥ ❛ s❡♥s❡✱

❧✐♥❡❛r ❢❛❝t♦r ❧✐♥❡❛r ❢❛❝t♦r

(x − x1 ) (x − x2 )

♦♣♣♦s✐t❡ ✿

✐♥ ♥✉♠❡r❛t♦r ✐♥ ❞❡♥♦♠✐♥❛t♦r

→ →

✈❛❧✉❡ ♦❢

y

❛t

x = x1

✐s

0

(x✲✐♥t❡r❝❡♣t✮✳

✈❛❧✉❡ ♦❢

y

❛t

x = x2

✐s

∞

✭✈❡rt✐❝❛❧ ❛s②♠♣t♦t❡✮✳

❊①❛♠♣❧❡ ✹✳✻✳✷✺✿ r❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t② ❛s ❜②♣r♦❞✉❝t

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

r❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t② ✭❛♥❞ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❡q✉❛t✐♦♥✮ ✇❤❡♥ t❤❡

✹✳✻✳

❚❤❡ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥s

✸✺✶

❋♦r ❡①❛♠♣❧❡✱ t❤❡ s♦❧✉t✐♦♥ s❡t ♦❢ t❤❡ ❡q✉❛t✐♦♥✿ f (x) = 0 ,

✐s ❚❤❡ s♦❧✉t✐♦♥ s❡t ♦❢ t❤❡ ✐♥❡q✉❛❧✐t②✿

{−1.5, 1.5} . f (x) ≤ 0

✐s

[−1.5, −1) ∪ [1.5, 2) . ❊①❡r❝✐s❡ ✹✳✻✳✷✻

❙♦❧✈❡ f (x) > 0✳ ❊①❛♠♣❧❡ ✹✳✻✳✷✼✿ r❛♥❣❡ ♦❢ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥

❲❡ ❤❛✈❡ ❧❡❛r♥❡❞ t❤❛t ✐ts ❞♦♠❛✐♥ ✐s ❡♥t✐r❡❧② ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ ❧✐♥❡❛r ❢❛❝t♦rs ♦❢ t❤❡ ❞❡♥♦♠✐♥❛t♦r ♦❢ t❤❡ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥✱ f (x) = P (x)/Q(x)✳ ❲❤❛t ❛❜♦✉t t❤❡ r❛♥❣❡ ❄ ❚❤❡ ❡①❛♠♣❧❡ ♦❢ t❤❡ ♥❡❣❛t✐✈❡ ♣♦✇❡rs✱ ✐✳❡✳✱ P (x) = 1, Q(x) = xn ✱ s❤♦✇s t❤❡ s❝♦♣❡ ♦❢ ♣♦ss✐❜✐❧✐t✐❡s✿

❙♦✱ ✇❡ ❤❛✈❡✿ 1 • ■❢ n ✐s ♦❞❞✱ t❤❡ r❛♥❣❡ ♦❢ n ✐s (−∞, 0) ∪ (0, +∞)✳ x

• ■❢ n ✐s ❡✈❡♥✱ t❤❡ r❛♥❣❡ ♦❢

❚❤❡ ❡①❛♠♣❧❡ ♦❢ f (x) =

x2

1 ✐s (0, +∞)✳ xn

1 ❜❡❧♦✇ s❤♦✇s t❤❛t t❤❡ r❛♥❣❡ ❞♦❡s♥✬t ❤❛✈❡ t♦ ❜❡ ✐♥✜♥✐t❡✿ +1

✹✳✻✳

❚❤❡ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥s

✸✺✷

❊①❡r❝✐s❡ ✹✳✻✳✷✽

❋✐♥❞ t❤❡ r❛♥❣❡ ♦❢ t❤❡ ❧❛st ❢✉♥❝t✐♦♥ ❛♥❞ t❤❡♥ ❞❡s❝r✐❜❡ t❤❡ ❢✉♥❝t✐♦♥ ❛s ❛ tr❛♥s❢♦r♠❛t✐♦♥✳ ❊①❛♠♣❧❡ ✹✳✻✳✷✾✿ r❡❝♦♥str✉❝t ❢✉♥❝t✐♦♥ ❢r♦♠ ❣r❛♣❤

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

❋✐rst✱ t❤❡r❡ ❛r❡ t❤r❡❡ x✲✐♥t❡r❝❡♣ts✿ x = −2, 2, 4 ✇✐t❤ ♠✉❧t✐♣❧✐❝✐t✐❡s✱ r❡s♣❡❝t✐✈❡❧②✱ ♦❞❞✱ ❡✈❡♥✱ ♦❞❞✳ ❚❤❡♥ t❤❡ ❢❛❝t♦rs ❢♦r t❤❡ ♥✉♠❡r❛t♦r ❛r❡ ❝❤♦s❡♥ t♦ ❜❡✿ (x + 2), (x − 2)2 , (x − 4)✳ ❙❡❝♦♥❞✱ t❤❡r❡ ❛r❡ t✇♦ ✈❡rt✐❝❛❧ ❛s②♠♣t♦t❡s✿ x = −1, 3 ✇✐t❤ ♠✉❧t✐♣❧✐❝✐t✐❡s✱ r❡s♣❡❝t✐✈❡❧②✱ ♦❞❞✱ ❡✈❡♥✳ ❚❤❡♥ t❤❡ ❢❛❝t♦rs ❢♦r t❤❡ ❞❡♥♦♠✐♥❛t♦r ❛r❡ ❝❤♦s❡♥ t♦ ❜❡✿ (x + 1), (x − 3)2 ✳ ❋✐♥❛❧❧②✱ ♦✉r ❢✉♥❝t✐♦♥ ❝♦✉❧❞ ❜❡ ❛t ✐ts s✐♠♣❧❡st t❤❡ ❢♦❧❧♦✇✐♥❣✿ f (x) =

(x + 2)(x − 2)2 (x − 4) . (x + 1)(x − 3)2

❚❤❡ ♣♦s✐t✐✈❡ s✐❣♥ ❝♦♠❡s ❢r♦♠ t❤❡ ❢❛❝t t❤❛t t❤❡ ❣r❛♣❤ ❡♥❞ ❛t +∞✳ ❊①❡r❝✐s❡ ✹✳✻✳✸✵

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

✹✳✼✳

❚❤❡ r♦♦t ❢✉♥❝t✐♦♥s

✸✺✸

✹✳✼✳ ❚❤❡ r♦♦t ❢✉♥❝t✐♦♥s

❊①❛♠♣❧❡ ✹✳✼✳✶✿ ❡q✉❛t✐♦♥s ✇✐t❤ ♣♦✇❡rs

✶✳ ❲❤❛t s❤♦✉❧❞ ❜❡ t❤❡ s✐❞❡ ♦❢ ❛ sq✉❛r❡ ♣♦♦❧ ✇✐t❤ ❛♥ ❛r❡❛ ♦❢ 200 sq✉❛r❡ ❢❡❡t❄ ❲❡ ♥❡❡❞ t♦ s♦❧✈❡ t❤❡ ❡q✉❛t✐♦♥✿ x2 = 200 . ✷✳ ❲❤❛t s❤♦✉❧❞ ❜❡ t❤❡ s✐❞❡ ♦❢ ❛ ❝✉❜✐❝ t❛♥❦ t♦ ❝♦♥t❛✐♥ 200 ❝✉❜✐❝ ❢❡❡t ♦❢ ✇❛t❡r❄ ❲❡ ♥❡❡❞ t♦ s♦❧✈❡ t❤❡ ❡q✉❛t✐♦♥✿ x3 = 200 . ■❢ ✇❡ ❛r❡ t♦ ❞♦ t❤✐s r❡♣❡t✐t✐✈❡❧②✱ ✇❡✬❧❧ ♥❡❡❞ t♦ ✜❣✉r❡ ♦✉t t❤❡ ✐♥✈❡rs❡s ♦❢ t❤❡s❡ t✇♦ ♣♦✇❡r ❢✉♥❝t✐♦♥s✳ ❲❡ ❤❛✈❡ ♣r♦❞✉❝❡❞ ♥❡✇ ❢✉♥❝t✐♦♥s st❛rt✐♥❣ ❢r♦♠

♣♦✇❡r ❢✉♥❝t✐♦♥s✱

x, x2 , x3 , . . . , ❜② ❛♣♣❧②✐♥❣ s♦♠❡ ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s✿ ♣♦✇❡r ❢✉♥❝t✐♦♥s ❛❞❞✐t✐♦♥✱ s✉❜tr❛❝t✐♦♥✱ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ↓ ♣♦❧②♥♦♠✐❛❧s

ւ

ց −→

❞✐✈✐s✐♦♥ ↓ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥s

✐♥✈❡rs❡s ♦❢ t❤❡ ♣♦✇❡r ❢✉♥❝t✐♦♥s ❄ ❖✉r ❡①♣❡r✐❡♥❝❡ ✇✐t❤ y = x2 t❡❧❧s ✉s ❤♦✇ t♦ ❞♦ t❤✐s ❛❧❧ ❛t ♦♥❝❡✳ ❲❡ ✢✐♣ t❤❡ ❣r❛♣❤s ♦❢ ❛❧❧ ♣♦✇❡r ❢✉♥❝t✐♦♥s

❲❤❛t ✐❢ ✕ ✐♥ ❛❞❞✐t✐♦♥ t♦ t❤❡s❡ ♦♣❡r❛t✐♦♥s ✕ ✇❡ ❛❧s♦ ❝♦♥s✐❞❡r t❤❡ ❛❜♦✉t t❤❡ ❧✐♥❡ y = x ❛t ♦♥❝❡✿

✹✳✼✳

❚❤❡ r♦♦t ❢✉♥❝t✐♦♥s

✸✺✹

❖❢ ❝♦✉rs❡✱ t❤❡ ❡✈❡♥ ❞❡❣r❡❡ ♣♦✇❡rs ❛r❡♥✬t ♦♥❡✲t♦✲♦♥❡✦ ❚♦ ♠❛❦❡ t❤❡♠ ✐♥✈❡rt✐❜❧❡✱ ✇❡ ❝✉t t❤❡✐r ❞♦♠❛✐♥s✱ ❛♥❞ ❝♦❞♦♠❛✐♥s✱ t♦

[0, +∞)✿

♥❛♠❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥

sq✉❛r❡ r♦♦t✱ ❚❤❡ ❝✉❜✐❝ r♦♦t✱ ❚❤❡ ❢♦✉rt❤ ❞❡❣r❡❡ r♦♦t✱ ❚❤❡

❢♦r♠✉❧❛

y=

x✱ √ y= 3x✱ √ y= 4x✱

✳✳✳ ❚❤❡

nt❤

❞❡❣r❡❡ r♦♦t✱

❞♦♠❛✐♥

√

y=

√ n

x

✱

✐s t❤❡ ✐♥✈❡rs❡ ♦❢

x = y2

✱

✐s t❤❡ ✐♥✈❡rs❡ ♦❢

x=y

3

✱

✐s t❤❡ ✐♥✈❡rs❡ ♦❢

x=y

4

✱

x, y ≥ 0 ✳

✐s t❤❡ ✐♥✈❡rs❡ ♦❢

x = yn

✱

❛❧❧

x, y ≥ 0 ✳ ❛❧❧

x, y ✳

x, y

✇❤❡♥

x, y ≥ 0

✳✳✳

n

✇❤❡♥

✐s ❛❞❞ ❛♥❞

n

✐s ❡✈❡♥✳

❚❤❡ ❞❡✜♥✐t✐♦♥ ❜❡❧♦✇ ✐s ❥✉st ❛♥♦t❤❡r ✇❛② t♦ s❛② t❤❡ s❛♠❡✳

❉❡✜♥✐t✐♦♥ ✹✳✼✳✷✿ nt❤ r♦♦t ❋♦r ❛ r❡❛❧ ♥✉♠❜❡r

x✱

t❤❡

nt❤

r♦♦t ♦❢ x ✐s s✉❝❤ ❛ ♥✉♠❜❡r y t❤❛t yn = x✱ ❞❡♥♦t❡❞

❜②

y= ❲❡ r❡q✉✐r❡

■♥ ♦t❤❡r ✇♦r❞s✱

y

x≥0

✇❤❡♥

✐s ❛ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❡q✉❛t✐♦♥

n

√ n

x

✐s ❡✈❡♥✳

y n = x✳

❚❤❡r❡ ✐s ♦♥❧② ♦♥❡ s✉❝❤ s♦❧✉t✐♦♥✦

❚❤✐s ✐s t❤❡ ✇❛② t♦ ❧♦♦❦ ❛t t❤❡s❡ ❡①♣r❡ss✐♦♥s ❛s ❢✉♥❝t✐♦♥s✳

❘♦♦ts p n

✐♥♣✉t

❚❤❡ s②♠❜♦❧

= √

♦✉t♣✉t ✐s ❝❛❧❧❡❞ t❤❡

✏r❛❞✐❝❛❧✑✳

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

x✿

❚❤❡ ✐❧❧✉str❛t✐♦♥ s✉❣❣❡sts t❤❡ ❢♦❧❧♦✇✐♥❣✿

x✱

❛♥❞ ❝r❡❛t❡❞ ❛ ✇❤♦❧❡ s❡q✉❡♥❝❡ ♦❢

✹✳✼✳

❚❤❡ r♦♦t ❢✉♥❝t✐♦♥s

✸✺✺

❚❤❡♦r❡♠ ✹✳✼✳✸✿ ❖❞❞ ❉❡❣r❡❡ ❘♦♦ts ❚❤❡ r♦♦ts ♦❢ ♦❞❞ ❞❡❣r❡❡s

y= ❤❛✈❡ t❤❡ ❞♦♠❛✐♥ ❛♥❞ r❛♥❣❡ ❡q✉❛❧ t♦

√ n

x, n

,

♦❞❞

(−∞, +∞)✳

❚❤❡② ❛r❡ ❛❧s♦ str✐❝t❧② ✐♥❝r❡❛s✐♥❣✱

♦♥❡✲t♦✲♦♥❡✱ ❛♥❞ ♦❞❞✳

❚❤❡♦r❡♠ ✹✳✼✳✹✿ ❊✈❡♥ ❉❡❣r❡❡ ❘♦♦ts ❚❤❡ r♦♦ts ♦❢ ❡✈❡♥ ❞❡❣r❡❡s

y=

√ n

❤❛✈❡ t❤❡ ❞♦♠❛✐♥ ❛♥❞ r❛♥❣❡ ❡q✉❛❧ t♦ ♦♥❡✲t♦✲♦♥❡✱ ❛♥❞

♥♦t ❡✈❡♥✳

x, n

,

❡✈❡♥

[0, +∞)✳

❚❤❡② ❛r❡ ❛❧s♦ str✐❝t❧② ✐♥❝r❡❛s✐♥❣✱

❇❡❝❛✉s❡ ♦❢ t❤❡ ❞r❛st✐❝ st❡♣ ♦❢ ❝✉tt✐♥❣ t❤❡ ❞♦♠❛✐♥✱ t❤❡ s②♠♠❡tr② ♦❢ t❤❡ ❡✈❡♥ ♣♦✇❡rs ✐s ❧♦st✳✳✳ ❊①❡r❝✐s❡ ✹✳✼✳✺

Pr♦✈❡ t❤❡ t❤❡♦r❡♠s✳

❲❡ ❤❛✈❡ ❝❧❛ss✐✜❡❞ t❤❡ r♦♦t ❢✉♥❝t✐♦♥s ❛❝❝♦r❞✐♥❣ t♦ t❤❡s❡ ❜r♦❛❞ ❝❛t❡❣♦r✐❡s✳ ❚❤✐s ✐s ✇❤❛t t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ ❛ r♦♦t ❢✉♥❝t✐♦♥ ❧♦♦❦s ❧✐❦❡✿

❊①❡r❝✐s❡ ✹✳✼✳✻

❉♦❡s t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❧♦♦❦ ❧✐❦❡ ❛♥② ❢❛♠✐❧✐❛r ❢✉♥❝t✐♦♥❄

◆♦✇ ✇❡ t✉r♥ t♦

❛❧❣❡❜r❛✳

❇❡❝❛✉s❡ t❤❡ r♦♦t ❢✉♥❝t✐♦♥s ❛r❡ t❤❡ ✐♥✈❡rs❡s ♦❢ t❤❡ ♣♦✇❡r ❢✉♥❝t✐♦♥s✱ t❤❡✐r ❛❧❣❡❜r❛✐❝ ♣r♦♣❡rt✐❡s ❛r❡ ♣❛r❛❧❧❡❧✳ ❋♦r ❡①❛♠♣❧❡✱ ✇❡ ❦♥♦✇ t❤❛t ✇❡ ❝❛♥ ✏❞✐str✐❜✉t❡✑ ♣♦✇❡rs ✭✐♥ ♦t❤❡r ✇♦r❞s✱ ❡①♣♦♥❡♥ts✱ ❛s ❞✐s❝✉ss❡❞ ✐♥ ❈❤❛♣t❡r ✶✮ ♦✈❡r ♠✉❧t✐♣❧✐❝❛t✐♦♥✿

(A · B)n = An · B n . ❆s ✐t t✉r♥s ♦✉t✱ ✇❡ ❝❛♥ ❛❧s♦ ✏❞✐str✐❜✉t❡✑ r♦♦ts ♦✈❡r ♠✉❧t✐♣❧✐❝❛t✐♦♥✱ ❛s ❢♦❧❧♦✇s✳ ❚❤❡♦r❡♠ ✹✳✼✳✼✿ Pr♦❞✉❝t ♦❢ ❘♦♦ts ❋♦r ❛♥② ✐♥t❡❣❡r

n✱

✇❡ ❤❛✈❡✿

√ n ♣r♦✈✐❞❡❞

a, b ≥ 0

✇❤❡♥❡✈❡r

n

a·b=

✐s ❡✈❡♥✳

√ n

a·

√ n

b

✹✳✼✳

❚❤❡ r♦♦t ❢✉♥❝t✐♦♥s

✸✺✻

Pr♦♦❢✳

❲❡ ✉s❡ t❤❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ✐♥t❡❣❡r ❡①♣♦♥❡♥ts ✭r❡♣❡❛t❡❞ ♠✉❧t✐♣❧✐❝❛t✐♦♥✮ ♣r❡s❡♥t❡❞ ✐♥ ❈❤❛♣t❡r ✶✳ ❙✉♣♣♦s❡

A=

√ n

a,

B=

√ n

b.

❚❤❡♥✿

a = An ,

b = Bn .

❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡✿

ab = An · B n = (AB)n . ❚❤❡♥✿

√ n

❋♦r ♣♦✇❡rs ♦r r♦♦ts✱ t❤❡ ❡①♣r❡ss✐♦♥

ab = AB .

s♣❧✐ts✳

❊①❛♠♣❧❡ ✹✳✼✳✽✿ ❢♦r♠✉❧❛s ❛r❡ s❤♦rt❝✉ts

❲❡ ✉s❡ t❤❡ r✉❧❡ ❛s ❛ s❤♦rt❝✉t✱ t♦ ❡①♣❛♥❞✿

√

20 =

❛♥❞ t♦ ❝♦♥tr❛❝t✿

√

4·5=

√

4·

√

√ 5 = 2 5.

√ √ √ √ 5 5 5 5 2 2 = 2 · 2 = 4.

❊①❡r❝✐s❡ ✹✳✼✳✾

❊①♣❛♥❞✿

√ 4

400 = ?

❊①❡r❝✐s❡ ✹✳✼✳✶✵

❈♦♥tr❛❝t✿

√ √ √ 6 2 3 2 3= ? ❲❛r♥✐♥❣✦ ❲❡ ❝❛♥✬t ✏❞✐str✐❜✉t❡✑ r♦♦ts ✭♥♦r ♣♦✇❡rs✮ ♦✈❡r ❛❞❞✐✲ t✐♦♥✿

√ n

❊①❡r❝✐s❡ ✹✳✼✳✶✶

■s t❤❡r❡ ❛ ✇❛② t♦ s✐♠♣❧✐❢② t❤✐s✿

(A + B)n ❄

◆❡①t✱ ✇❡ ❦♥♦✇ t❤❛t t❤❡ ♣♦✇❡rs ✭❡①♣♦♥❡♥ts✮ ❛r❡ ♠✉❧t✐♣❧✐❡❞ ✇❤❡♥ ❝♦♠♣♦s❡❞✿

(Qn )m = Qn·m . ❚❤❡ s❛♠❡ ❤❛♣♣❡♥s ✇✐t❤ t❤❡ ❞❡❣r❡❡s ♦❢ r♦♦ts✳ ❚❤❡♦r❡♠ ✹✳✼✳✶✷✿ ❈♦♠♣♦s✐t✐♦♥ ♦❢ ❘♦♦ts ❋♦r ❛♥② ✐♥t❡❣❡rs

n

❛♥❞

m✱

✇❡ ❤❛✈❡✿

q n

♣r♦✈✐❞❡❞

a≥0

✇❤❡♥❡✈❡r

n

♦r

m

√ m

a=

✐s ❡✈❡♥✳

√

nm

a

a + b 6= ...

✹✳✼✳

❚❤❡ r♦♦t ❢✉♥❝t✐♦♥s

✸✺✼

Pr♦♦❢✳

❲❡ ✉s❡ t❤❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ✐♥t❡❣❡r ❡①♣♦♥❡♥ts ✭r❡♣❡❛t❡❞ ♠✉❧t✐♣❧✐❝❛t✐♦♥✮ ♣r❡s❡♥t❡❞ ✐♥ ❈❤❛♣t❡r ✶✳ ❙✉♣♣♦s❡ √ √ A=

❚❤❡♥✿

a,

m

Q=

a = Am ,

❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡✿

n

A.

A = Qn .

Qnm = (Qn )m = (A)m = a .

❚❤❡♥✿

√

nm

a = Q.

❋♦r ♣♦✇❡rs ♦r r♦♦ts✱ t❤❡ ❝♦♠♣♦s✐t✐♦♥ ✐s r❡♣❧❛❝❡❞ ✇✐t❤ ♠✉❧t✐♣❧✐❝❛t✐♦♥✳ ❚❤❡ ❞❡❣r❡❡s m ❛♥❞ n ❛r❡ ♣r❡s❡♥t❡❞ ✐♥ t❤❡ ♦♣♣♦s✐t❡ ♦r❞❡r t♦ ❢❛❝✐❧✐t❛t❡ t❤❡ ♣r♦♦❢ ✭t❤❡ ❧❛tt❡r ❢r♦♠ t❤❡ ❢♦r♠❡r✮✳ ❲❡ ❝❛♥ ✇❛t❝❤ t❤❡♠ ❝❛♥❝❡❧ ❜❡❧♦✇✿ x → nt❤ ♣♦✇❡r → mt❤ ♣♦✇❡r → mt❤ r♦♦t → nt❤ r♦♦t → y

■♥❞❡❡❞✱ t❤❡ t✇♦ ✐♥ t❤❡ ♠✐❞❞❧❡ ❛r❡ ✐♥✈❡rs❡s ❛♥❞ ✉♥❞♦ ❡❛❝❤ ♦t❤❡r✿ x → nt❤ ♣♦✇❡r →

→

→

→ nt❤ r♦♦t → y

→

❚❤❡ t✇♦ ❧❡❢t ❛r❡ ❛❧s♦ ✐♥✈❡rs❡s ❛♥❞ ❛❧s♦ ✉♥❞♦ ❡❛❝❤ ♦t❤❡r✿ x→

→

→

→

→

→

→

→

→y

❚❤❡r❡❢♦r❡✱ t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ ✜rst t✇♦ ❢✉♥❝t✐♦♥s ✐s t❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ s❡❝♦♥❞ t✇♦✳ ❊①❛♠♣❧❡ ✹✳✼✳✶✸✿ ❢♦r♠✉❧❛s ❛r❡ s❤♦rt❝✉ts

❲❡ ✉s❡ t❤❡ r✉❧❡ ❛s ❛ s❤♦rt❝✉t✱ t♦ ❡①♣❛♥❞✿ √ 6

❛♥❞ t♦ ❝♦♥tr❛❝t✿

64 = q 5 √

√

3·2

2=

64 =

q

q

2=

5

√ 2

3

√ 2

64 = √

5·2

√ 3

2=

8 = 2;

√

10

2.

❊①❡r❝✐s❡ ✹✳✼✳✶✹

❊①♣❛♥❞✿

√ 4

100 = ?

❊①❡r❝✐s❡ ✹✳✼✳✶✺

❈♦♥tr❛❝t✿

q 3

√ 3

3=?

❚❤❡s❡ t❤❡♦r❡♠s ✇✐❧❧ ❜❡ ✉s❡❞ ❧❛t❡r ✐♥ t❤✐s ❝❤❛♣t❡r t♦ ♣r♦✈❡ t❤❛t ❡✈❡r② r♦♦t ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ ✭❢r❛❝t✐♦♥❛❧✮ ♣♦✇❡r✱ t❤❡ r❡❝✐♣r♦❝❛❧ ♦❢ t❤❡ ♣♦✇❡r t❤❡ r♦♦t ❝❛♠❡ ❢r♦♠✳ ❲❛r♥✐♥❣✦ ❚❤♦✉❣❤

❝♦♥✈❡♥✐❡♥t✱

t❤❡s❡

t✇♦

r✉❧❡s

s♦r❜❡❞ ✐♥t♦ t❤❡ r✉❧❡s ♦❢ ❡①♣♦♥❡♥ts✳

❖♥❡ ❝❛♥ t❤✐♥❦ ♦❢ ♦t❤❡r r✉❧❡s✿

√ m

1 = 1,

√ m

0 = 0 , ❡t❝✳

✇✐❧❧

❜❡

❛❜✲

✹✳✼✳

❚❤❡ r♦♦t ❢✉♥❝t✐♦♥s

✸✺✽

❊①❛♠♣❧❡ ✹✳✼✳✶✻✿ ❞♦♠❛✐♥ ❢r♦♠ ✢♦✇❝❤❛rt ❈♦♥s✐❞❡r t❤❡ ❢✉♥❝t✐♦♥

q √ x − x.

❚❤✐s ✐s ✐ts ✢♦✇❝❤❛rt✿

ր x →

f: x →

x √

ց x →

→ x ց → u ր

−

→ z

❋✐♥❞ t❤❡ ❞♦♠❛✐♥✳ ❲❤❛t❡✈❡r ✐s ✐♥s✐❞❡ t❤❡ sq✉❛r❡ r♦♦t ❝❛♥✬t ❜❡ ♥❡❣❛t✐✈❡✳ ❙♦✱ ✇❡ ♥❡❡❞ t♦ ✜♥❞ ❛❧❧ ✇❤✐❝❤ ✇❡ ❤❛✈❡✿

x ≥ 0 ❆◆❉ x −

√

x

❢♦r

x ≥ 0.

❙♦❧✈❡ t❤❡ ❧❛tt❡r ❛ss✉♠✐♥❣ t❤❡ ❢♦r♠❡r✿

x≥

√

x =⇒ x2 ≥ x =⇒ x ≥ 1 .

❚❤❡ ❞♦♠❛✐♥ ✐s

D = {x : x ≥ 0} ∩ {x : x ≥ 1} = [1, ∞) .

❊①❡r❝✐s❡ ✹✳✼✳✶✼ ❋✐♥❞ t❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❛❜♦✈❡ ❛♥❞ ✐ts ❞♦♠❛✐♥✳

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

▼♦❞❡❧s

❉✐r❡❝t ♣r♦❜❧❡♠s

■♥✈❡rs❡ ♣r♦❜❧❡♠s

✶✳ ❙q✉❛r❡ ♣♦♦❧✿

◗✿ ❲❤❛t ❛r❡❛ ✐❢ s✐❞❡ ✶✷ ❢❡❡t❄

◗✿ ❍♦✇ ❧❛r❣❡ s✐❞❡ t♦ ❤❛✈❡

x×x

❙✉❜st✐t✉t❡✿

sq✉❛r❡ ✇✐t❤ ❛r❡❛

f (x) = x2

f (12) = 122

x2 = 300 . √ =⇒ x = 300 ≈ 17

✷✳ ❈✉❜✐❝ t❛♥❦✿

◗✿ ❲❤❛t ✈♦❧✉♠❡ ✐❢ s✐❞❡ ✶✷ ❢❡❡t❄

x×x×x

❙✉❜st✐t✉t❡✿

f (x) = x3

sq ❢❡❡t❄

300

❝✉ ❢❡❡t❄

❙♦❧✈❡✿

= 144 .

❝✉❜❡ ✇✐t❤ ✈♦❧✉♠❡

300

f (12) = 123

❢❡❡t✳

◗✿ ❍♦✇ ❧❛r❣❡ s✐❞❡ t♦ ❤❛✈❡

x3 = 300 . √ 3 =⇒ x = 300 ≈ 6.69 ❙♦❧✈❡✿

= 1728 .

❢❡❡t✳

❚❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ✐♥✈❡rs❡ ♣r♦❜❧❡♠s ❛r❡ s✐♠♣❧② ❛♣♣❧✐❝❛t✐♦♥s ♦❢ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ r♦♦t✦

❊①❛♠♣❧❡ ✹✳✼✳✶✽✿ ❛✈❡r❛❣❡ r❛t❡ ♦❢ ❣r♦✇t❤ ❲❤❡♥ ✇❡ s❛② t❤❛t ❛ ♣♦♣✉❧❛t✐♦♥ ♦❢ ❜❛❝t❡r✐❛ ❣r♦✇s ❛t r❛t❡

r

♣❡r ❞❛②✱ ✇❡ ❥✉st ♠✉❧t✐♣❧② ✐t ❜②

❙✉♣♣♦s❡ ♥♦✇ t❤❛t t❤❡ ♣♦♣✉❧❛t✐♦♥ ❞♦✉❜❧❡❞ ②❡st❡r❞❛② ❛♥❞ q✉❛❞r✉♣❧❡❞ t♦❞❛②✳ ❲❤❛t ✐s ✐ts ♦❢ ❣r♦✇t❤❄ ■t ✇♦✉❧❞ ❜❡ ♥❛✐✈❡ t♦ ❛♥s✇❡r t❤❛t t❤❡ r❛t❡ ❤❛s ❜❡❡♥

2+4 = 3✳ 2

r

❡✈❡r② ❞❛②✳

❛✈❡r❛❣❡ r❛t❡

❚❤❡ ❛♥s✇❡r ✇♦✉❧❞ s✉❣❣❡st

t❤❛t t❤❡ ♣♦♣✉❧❛t✐♦♥ ❤❛s tr✐♣❧❡❞ ❡✈❡r② ❞❛②✦ ❇✉t tr✐♣❧❡❞ t✇✐❝❡ ♠❡❛♥s t❤❛t ✐t ❤❛s ❣♦♥❡ ✉♣ ❜② ❛ ❢❛❝t♦r ♦❢

3 · 3 = 9✱

♥♦t

2 · 4 = 8✦

❚❤❡ ❝♦rr❡❝t ❛♥s✇❡r ✐s s✉❝❤ ❛ ♥✉♠❜❡r

x

t❤❛t ✇❤❡♥ ♠✉❧t✐♣❧✐❡❞ t✇✐❝❡ ✭sq✉❛r❡❞✮ ♣r♦❞✉❝❡

x · x = 8 =⇒ x = ❚❤❡ ❞✐✛❡r❡♥❝❡ ✐s ✈✐s✐❜❧❡ ❜❡❧♦✇✿

√

8 ≈ 2.83 .

8✿

✹✳✼✳

❚❤❡ r♦♦t ❢✉♥❝t✐♦♥s

✸✺✾

❊①❛♠♣❧❡ ✹✳✼✳✶✾✿ ❛✈❡r❛❣❡ r❡t✉r♥ ♦❢ ✐♥✈❡st♠❡♥t

❙✉♣♣♦s❡ ❛ ♠✉t✉❛❧ ❢✉♥❞ r❡♣♦rts ❛

10%

r❡t✉r♥ ❧❛st ②❡❛r ❛♥❞

5%

❛t❡ ✐ts ♣❡r❢♦r♠❛♥❝❡✱ ✇❡ ♥❡❡❞ t♦ ✉♥❞❡rst❛♥❞ t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡

❢♦r t❤❡ ②❡❛r ❜❡❢♦r❡✳ ■♥ ♦r❞❡r t♦ ❡✈❛❧✉✲

❛✈❡r❛❣❡ r❡t✉r♥✳ ■t✬s ♥♦t 10 2+ 5 = 7.5%✦

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

1.05 · 1.10✳

❲❡✱ t❤❡r❡❢♦r❡✱ ♥❡❡❞ s✉❝❤ ❛♥

x

t❤❛t

x · x = 1.05 · 1.10 = 1.155 . ❲❡ s♦❧✈❡✿

x= ❙♦✱ t❤❡ ❛♥s✇❡r ✐s ✏❛❜♦✉t

√

1.155 ≈ 1.0747 .

7.47%✑✳

❊①❛♠♣❧❡ ✹✳✼✳✷✵✿ ❛✈❡r❛❣❡ ♦❢ t✇♦ ♥✉♠❜❡rs

❙✉♣♣♦s❡ ❛ q✉❛♥t✐t② ❤❛s ❣r♦✇♥ ❢r♦♠

1

t♦

9

♦✈❡r ❛ ♣❡r✐♦❞ ♦❢ t✐♠❡✳ ◆♦✇✱ ✇❤❛t ✇❛s ✐t ❤❛❧❢✲✇❛② t❤r♦✉❣❤

t❤✐s ♣❡r✐♦❞❄ ■t ❞❡♣❡♥❞s✦ ❈♦♥s✐❞❡r t❤❡s❡ t✇♦ ♣♦ss✐❜✐❧✐t✐❡s✿

•

■t ❤❛s ✏❣r♦✇♥ ❜②

•

■t ❤❛s ✏❣r♦✇♥

8✑✳

❚❤❡♥ ❤❛❧❢✲✇❛② ✐t ❤❛s ❣r♦✇♥ ❜② t❤✐s ❛♠♦✉♥t✿

1 + 4 = 5✳ 9✲❢♦❧❞✑✳

❚❤❡♥ ❤❛❧❢✲✇❛② ✐t ❤❛s ❣r♦✇♥ ❜② t❤✐s ❢❛❝t♦r✿

8/2 = 4✳

❚❤❡♥✱ ✇❡ ❤❛✈❡✿

√

❚❤❡♥✱ ✇❡ ❤❛✈❡✿

9 = 3✳

1 · 3 = 3✳

■♥ s✉♠♠❛r②✱ t❤❡ q✉❡st✐♦♥ ✏✇❤❛t ✐s t❤❡ ❛✈❡r❛❣❡ ♦❢ t✇♦ ♥✉♠❜❡rs❄✑ ❞❡♣❡♥❞s ♦♥ t❤❡ ❛❧❣❡❜r❛ t❤❡② ❛r❡ ✐♥✈♦❧✈❡❞ ✐♥✦ ❚❤❡ ❞✐✛❡r❡♥❝❡ ✐s ❜❡t✇❡❡♥ ❛♥ ❛r✐t❤♠❡t✐❝ ♣r♦❣r❡ss✐♦♥ ❛♥❞ ❛ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥✿

❆♥s✇❡r

◗✉❡st✐♦♥ ❍♦✇ ❞♦ ②♦✉ ❛❞❞

a

✐♥ t✇♦ st❡♣s ❄

❍♦✇ ❞♦ ②♦✉ ♠✉❧t✐♣❧② ❜②

a

✐♥ t✇♦ st❡♣s ❄

❆❞❞

a 2

t✇✐❝❡✳

▼✉❧t✐♣❧② ❜②

√

a

t✇✐❝❡✳

❊①❡r❝✐s❡ ✹✳✼✳✷✶

❙❤♦✇ t❤❛t t❤❡ ❧❛tt❡r ✐s ❛❧✇❛②s s♠❛❧❧❡r t❤❛♥ t❤❡ ❢♦r♠❡r✳

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

✹✳✽✳

❚❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥s

✸✻✵

❆♥❛❧♦❣②✿ r❡♣❡❛t❡❞ ❛❞❞✐t✐♦♥ ✈s✳ r❡♣❡❛t❡❞ ♠✉❧t✐♣❧✐❝❛t✐♦♥

▼✉❧t✐♣❧✐❝❛t✐♦♥✿

❊①♣♦♥❡♥t✐❛t✐♦♥✿

r❡♣❡❛t❡❞ ❛❞❞✐t✐♦♥

r❡♣❡❛t❡❞ ♠✉❧t✐♣❧✐❝❛t✐♦♥

x ✐s t❤❡ ❛✈❡r❛❣❡ ♦❢ a ❛♥❞ b : a + b = x + x a·b=x·x √ a+b =⇒ x = =⇒ x = a · b 2 ♥❛♠❡✿ ❛r✐t❤♠❡t✐❝ ♠❡❛♥ ❣❡♦♠❡tr✐❝ ♠❡❛♥
√ 1 n a1 + a2 + ... + an ❣❡♥❡r❛❧ ❢♦r♠✉❧❛✿ a1 · a2 · ... · an n

✹✳✽✳ ❚❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥s

❘❡❝❛❧❧ ✇❤❛t ✇❡ ❧❡❛r♥❡❞ ❛❜♦✉t t❤❡ ❛❧❣❡❜r❛ ♦❢ ❡①♣♦♥❡♥ts ✐♥ ❈❤❛♣t❡r ✶✳ ❇❡❧♦✇ ✐s t❤❡ s✉♠♠❛r② ♦❢ t❤❡ ❝♦♥✈❡♥t✐♦♥s ✇❡ ❤❛✈❡ s❡t ✉♣ ✭❢♦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 t✐♠❡s

x 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 x = −1, −2, ...

a·0 =0

ax = (−a)(−x)

axy = (ax )y

❚❤✐s ✐s ❥✉st ❛ r❡✈✐❡✇ ♦❢ ❤♦✇ t❤❡ ♣♦✇❡r ✕ ✐♥❝❧✉❞✐♥❣ t❤❡ r❡❝✐♣r♦❝❛❧ ♣♦✇❡rs ✕ ❢✉♥❝t✐♦♥s ♦♣❡r❛t❡✳ ❍♦✇❡✈❡r✱ ♦✉r ❝✉rr❡♥t ✐♥t❡r❡st ✐s ❛ ❞✐✛❡r❡♥t ❦✐♥❞ ♦❢ ❢✉♥❝t✐♦♥✳✳✳ ❘❡❝❛❧❧ t❤❡ ♥♦t❛t✐♦♥✿ ❇❛s❡ ❛♥❞ ❡①♣♦♥❡♥t

a

↑ ❜❛s❡ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ✐s t❤❡ ♥❡✇ t❡r♠✐♥♦❧♦❣② ✇❡ ✇✐❧❧ ✉s❡✳

❡①♣♦♥❡♥t ↓

x

✹✳✽✳

❚❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥s

✸✻✶

❉❡✜♥✐t✐♦♥ ✹✳✽✳✶✿ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥ ❚❤❡

❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥ ♦❢ ❜❛s❡ a > 0 ✐s ❞❡✜♥❡❞ t♦ ❜❡ f (x) = ax

✇✐t❤ t❤❡

❞♦♠❛✐♥ t❤❡ s❡t ♦❢ ❛❧❧ ✐♥t❡❣❡rs✿ Z = {..., −3, −2, −1, 0, 1, 2, 3, ...} .

❲❛r♥✐♥❣✦ ❯♥❧✐❦❡ ❛ ♣♦✇❡r ❢✉♥❝t✐♦♥✱ t❤✐s ✐s ❛ ❢✉♥❝t✐♦♥✱

2x ✱

x2 ✱

❜❛s❡✱ ❡①♣♦♥❡♥t✳

✇✐t❤ ❛ ✈❛r✐❛❜❧❡

✇✐t❤ ❛ ✈❛r✐❛❜❧❡

❯♥❢♦rt✉♥❛t❡❧②✱ t❤❡ ❞♦♠❛✐♥ ♠✐ss❡s s♦♠❡ ♦❢ t❤❡ ♥✉♠❜❡rs t❤❛t ✐♥t❡r❡st ✉s✦

❊①❛♠♣❧❡ ✹✳✽✳✷✿ ❜❛❝t❡r✐❛ st✐❧❧ ♠✉❧t✐♣❧②✐♥❣ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ♣♦♣✉❧❛t✐♦♥ ♦❢ ❜❛❝t❡r✐❛ t❤❛t ❞♦✉❜❧❡s ❡✈❡r② ❞❛② st❛rt✐♥❣ ✇✐t❤ p0 = 1✳ ❲❡ ❞❡s❝r✐❜❡ t❤✐s ❛s ❛ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥ ✜rst ✭❈❤❛♣t❡r ✶✮✿ pn+1 |{z}

♣♦♣✉❧❛t✐♦♥✿ ❛t t✐♠❡

=2· n+1

pn |{z}

❛t t✐♠❡

❚❤❡ ❣r❛♣❤ ✐s ♠❛❞❡ ♦❢ ❞✐s❝♦♥♥❡❝t❡❞ ♣♦✐♥ts✿

=⇒ pn = 2n . n

▲❡t✬s t❤✐♥❦ ♦❢ ✐t ❛s ❛ ❢✉♥❝t✐♦♥✳ ■t ✐s ❣✐✈❡♥ ❜② t❤❡ s❛♠❡ ❢♦r♠✉❧❛✱ ✇✐t❤ x✬s st✐❧❧ ❧✐♠✐t❡❞ t♦ t❤❡ ✐♥t❡❣❡rs✿ p(x) = 2x .

◆♦✇✱ ✇❤❛t ✐s t❤❡ ♣♦♣✉❧❛t✐♦♥ ✐♥ t❤❡ ♠✐❞❞❧❡ ♦❢ t❤❡ ✜rst ❞❛②❄ ❚❤❡ ❢✉♥❝t✐♦♥ ❞♦❡s♥✬t t❡❧❧ √ ✉s ✭✐t ✐s ✉♥❞❡✜♥❡❞ ❛t x = .5✮ ❜✉t✱ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❛♥❛❧②s✐s ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥✱ t❤❡ ❛♥s✇❡r s❤♦✉❧❞ ❜❡ 2✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❝❤♦♦s❡ t❤❡ ❣❡♦♠❡tr✐❝ ♠❡❛♥ ♦✈❡r t❤❡ ❛r✐t❤♠❡t✐❝ ♠❡❛♥✿

❉♦❡s t❤✐s ♠❡❛♥ t❤❛t p(1/2) =

√

2 ❛♥❞ 21/2 =

√

2?

✹✳✽✳

❚❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥s

✸✻✷

❚❤❡ ❞♦♠❛✐♥ ♦❢ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥

y = ax

✇✐t❤ ❜❛s❡

a>0

✐s ❛❧❧ ✐♥t❡❣❡rs✳ ❲❤❛t ❞♦ ✇❡ ❞♦ ❛❜♦✉t t❤❡s❡

❣❛♣s❄ ❈❛♥✱ ♦r s❤♦✉❧❞✱ ✇❡ ✜❧❧ t❤❡♠✱ ✐♥ ❛ ♠❡❛♥✐♥❣❢✉❧ ♠❛♥♥❡r❄

❲❡ ✜❣✉r❡❞ ♦✉t ✐♥ ❈❤❛♣t❡r ✶ t❤❡ ♠❡❛♥✐♥❣ ♦❢ ✏♠✉❧t✐♣❧✐❝❛t✐♦♥ ❜② ♠❡❛♥✐♥❣ ♦❢ ✏♠✉❧t✐♣❧✐❝❛t✐♦♥ ❜②

a✱ ❛ ♥❡❣❛t✐✈❡ ♥✉♠❜❡r ♦❢ t✐♠❡s✑✳

a✱

③❡r♦ t✐♠❡s✑✳

❲❡ ❛❧s♦ ✜❣✉r❡❞ ♦✉t t❤❡

❇✉t ✇❤❛t ✏♠✉❧t✐♣❧✐❝❛t✐♦♥ ❜②

a ♦♥❡ ❤❛❧❢ t✐♠❡s✑

❝❛♥ ♣♦ss✐❜❧② ♠❡❛♥❄✦ ❊①❛♠♣❧❡ ✹✳✽✳✸✿ ❝♦♠♣♦✉♥❞❡❞ ✐♥t❡r❡st✱ r❡✈✐s✐t❡❞

❈♦♥s✐❞❡r ✇❤❛t ❤❛♣♣❡♥s t♦ ❛

$1000 ❞❡♣♦s✐t ✇✐t❤ 10% ❛♥♥✉❛❧ ✐♥t❡r❡st✱ ❝♦♠♣♦✉♥❞❡❞ ②❡❛r❧② ✭❈❤❛♣t❡r ✶✮✿ $1000, 1000 · 1.10 = 1000 + 10% ♦❢ 1000 ❜❡❣✐♥✱ ❛❢t❡r ✶ ②❡❛r = ♣r✐♥❝✐♣❛❧ + ✐♥t❡r❡st = 1000 + 1000 · .10 = 1000(1 + 0.1) = 1000 · 1.1 .

■t✬s ❛ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥ ❜✉t t❤✐s t✐♠❡ ✇❡ ✇r✐t❡ ✐s ❛s ❛ ❢✉♥❝t✐♦♥❀ ❛❢t❡r

x

②❡❛rs ✇❡ ❤❛✈❡✿

f (x) = 1000 · 1.1x . ✇❤❡r❡

x

✐s ❛ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r✳

◆♦✇✱ ✇❤❛t ✐❢ ■ ✇❛♥t t♦ ✇✐t❤❞r❛✇ ♠② ♠♦♥❡② ✐♥ t❤❡

♠✐❞❞❧❡

♦❢ t❤❡ ②❡❛r❄ ■t ✇♦✉❧❞ ❜❡ ❢❛✐r t♦ r❡q✉❡st

❢r♦♠ t❤❡ ❜❛♥❦ ❢♦r t❤❡ ✐♥t❡r❡st t♦ ❜❡ ❝♦♠♣♦✉♥❞❡❞ ♥♦✇✳ ■t ✇♦✉❧❞ ❛❧s♦ ❜❡ ❢❛✐r ❢♦r t❤❡ ❜❛♥❦ t♦ ✇❛♥t t♦ ❞♦ ✐t ✐♥ s✉❝❤ ❛ ✇❛② t❤❛t t❤❡ ❛♥♥✉❛❧ r❡t✉r♥ r❡♠❛✐♥s t❤❡ s❛♠❡ ❡✈❡♥ ✐❢ ✇❡ ❝♦♠♣♦✉♥❞ t✇✐❝❡✳ ❆❝❝❡♣t✐♥❣ 5% ✐♥t❡r❡st ✇♦✉❧❞ ♣r♦❞✉❝❡ 1.052 = 1.1025✱ ♦r 10.25% ❛♥♥✉❛❧ ✐♥t❡r❡st✳ ❚❤❡♥✱ ✇❤❛t s❤♦✉❧❞ ❜❡ t❤✐s s❡♠✐✲❛♥♥✉❛❧ ✐♥t❡r❡st r❛t❡❄ ❙✉♣♣♦s❡ t❤❡ ❛♠♦✉♥t ❤❛s ❣r♦✇♥ ❜② ❛ ♣r♦♣♦rt✐♦♥✱

r✱

s♦ t❤❛t✱ ✐❢ ❛♣♣❧✐❡❞ ❛❣❛✐♥✱ ✐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= ♦r ❛❜♦✉t

4.9

√

1.1 ≈ 1.0488 ,

♣❡r❝❡♥t✳

❊①❡r❝✐s❡ ✹✳✽✳✹

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

❊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❡✈✐s✐t❡❞ ❘❡❝❛❧❧ ❛♥ ❡①❛♠♣❧❡ ❢r♦♠ ❈❤❛♣t❡r ✶✳ ❚❤❡ r❛❞✐♦❛❝t✐✈❡ ❝❛r❜♦♥ ❧♦s❡s ❤❛❧❢ ♦❢ ✐ts ♠❛ss ♦✈❡r ❛ ❝❡rt❛✐♥ ♣❡r✐♦❞ ♦❢ t✐♠❡ ❝❛❧❧❡❞ t❤❡

❤❛❧❢✲❧✐❢❡

♦❢ t❤❡ ❡❧❡♠❡♥t✳ ■t✬s ❛ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥ ❛❣❛✐♥✿

an+1 = an · ❯♥❢♦rt✉♥❛t❡❧②✱

1 . 2

n ✐s ♥♦t t❤❡ ♥✉♠❜❡r ♦❢ ②❡❛rs ❜✉t t❤❡ ♥✉♠❜❡r ♦❢ ❤❛❧❢✲❧✐✈❡s✦

♦❢ t❤✐s ❡❧❡♠❡♥t✱

14

❍♦✇❡✈❡r✱ ✇❡ ♦♥❧② ❦♥♦✇ t❛❦❡s t♦ ❣♦ ❢r♦♠

❋♦r ❡①❛♠♣❧❡✱ t❤❡ ♣❡r❝❡♥t❛❣❡

❈✱ ❧❡❢t ♣❧♦tt❡❞ ❜❡❧♦✇ ❛❣❛✐♥st t✐♠❡ ♠❛② ❧♦♦❦ ❧✐❦❡ t❤✐s✿

t✇♦

♣♦✐♥ts ♦♥ t❤❡ ❣r❛♣❤✦ ❙✉♣♣♦s❡ t❤❡ ❤❛❧❢✲❧✐❢❡ ✐s

100% t♦ 50%✮✳

5730

②❡❛rs ✭✐✳❡✳✱ t❤❡ t✐♠❡ ✐t

❚❤❡ ♠♦❞❡❧ ♠❡❛s✉r❡s t✐♠❡ ✐♥ ♠✉❧t✐♣❧❡s ♦❢ t❤❡ ❤❛❧❢✲❧✐❢❡✱

5730 ②❡❛rs✱ ❛♥❞

❛♥② ♣❡r✐♦❞ s❤♦rt❡r t❤❛♥ t❤❛t ✇✐❧❧ r❡q✉✐r❡ ❛ ♥❡✇ ✐♥s✐❣❤t✳ ❇❡❢♦r❡ ✇❡ ❡✈❡♥ tr② t♦ ❞❛t❡ ❛ ♣❛r❝❤♠❡♥t ✇✐t❤✱ s❛②✱

75%

♦❢

14

❈ ❧❡❢t✱ ❧❡t✬s tr② t♦ ❛s❦ ❛ s✐♠♣❧❡r q✉❡st✐♦♥✿ ❍♦✇ ♠✉❝❤ ✐s ❧❡❢t ❛❢t❡r

❚❤❡ ❧✐♥❡❛r ✭♦r ❛r✐t❤♠❡t✐❝✮ ❛♥s✇❡r ✐s t❤❛t t❤❡ ❧♦ss ✐s ❤❛❧❢ ♦❢ t❤❡ ❤❛❧❢✳ ✇❤❛t✬s ❧❡❢t ✐s

75%✦

5730/2 = 2865

②❡❛rs❄

■t ✐s✱ t❤❡r❡❢♦r❡✱ ❛ q✉❛rt❡r✱ ❛♥❞

❚❤❡ r❡❛❧ ♥✉♠❜❡r ✐s ❧♦✇❡r✿

r

1 ≈ .707 . 2

❊①❛♠♣❧❡ ✹✳✽✳✼✿ sq✉❛r❡ r♦♦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❛♣❤✿

✹✳✽✳

❚❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥s

❖♥❡ ❝❛♥ ❛❧s♦ ✐♠❛❣✐♥❡ t❤❛t t❤❡

✸✻✹

x✲❛①✐s✱

❛s t❤❡ ❞♦♠❛✐♥✱ ✐s ❜❡❝♦♠✐♥❣ ❞❡♥s❡r ❛♥❞ ❞❡♥s❡r ❝♦✈❡r❡❞✳ ❚❤❡s❡

✐♥✐t✐❛❧❧② ❧♦♦s❡ ♣♦✐♥ts s❡❡♠ t♦ st❛rt t♦ ❢♦r♠ ❛ ❝✉r✈❡✿

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

❚❤❡ ❣r❛♣❤ ♦❢ ♦✉r ❢✉♥❝t✐♦♥ ✭❧❡❢t✮ ❧✐❡s ❜❡❧♦✇ ❛♥② ❝❤♦r❞ t❤❛t ❝♦♥♥❡❝ts t✇♦ ♣♦✐♥ts ♦♥ t❤❡ ❣r❛♣❤ ✭r✐❣❤t✮✳ ❙✉❝❤ ❛ ❢✉♥❝t✐♦♥ ✐s ❝❛❧❧❡❞ ✏❝♦♥❝❛✈❡ ✉♣✑ ✭❈❤❛♣t❡r ✷❉❈✲✸✮✳ ❊①❛♠♣❧❡ ✹✳✽✳✽✿

nt❤

r♦♦t

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

3 ♣❛rts ✐♥st❡❛❞ ♦❢ 2❄ 1

23 =

√ 3

❲❤❛t ✐s t❤❡ ✈❛❧✉❡ ♦❢

2x

✐❢

x = 1/3❄

❲❡ ❢♦❧❧♦✇

2.

❋✉rt❤❡r♠♦r❡✱ ✇❡ ❝❛♥ ❞✐✈✐❞❡ ✐♥t♦ s♠❛❧❧❡r ❛♥❞ s♠❛❧❧❡r ♣❛rts❀ t❤❡♥✿ 1

2n =

√ n

2.

❆s ❛ ♥❡✇ ❝♦♥✈❡♥t✐♦♥✱ ✇❡ ❞❡✜♥❡ t❤❡ r❡❝✐♣r♦❝❛❧ ❡①♣♦♥❡♥t✱ ❛s ❢♦❧❧♦✇s✳ ❋♦r ❛♥② ♣♦s✐t✐✈❡ ✐♥t❡❣❡r

n✱

✇❡ s❡t t❤❡

✹✳✽✳

❚❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥s

1 t❤ n

♣♦✇❡r

♦❢

a > 0✱

♦r

a

✸✻✺

t❛❦❡♥ t♦ t❤❡ ♣♦✇❡r 1/n✱ t♦ ❜❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿ a1/n =

√ n

a

❊①❡r❝✐s❡ ✹✳✽✳✾ ❙❤♦✇ t❤❛t r✉❧❡s ✷ ❛♥❞ ✸ ✐♥ t❤❡ ❛❜♦✈❡ t❛❜❧❡ ❛❜♦✈❡ st✐❧❧ ❤♦❧❞✿

ax bx = (ab)x , axy = (ax )y . ❍❛✈✐♥❣ t❤❡ r❡❝✐♣r♦❝❛❧s ❛s ❡①♣♦♥❡♥ts ✐s♥✬t ❡♥♦✉❣❤ ✭✇❤❛t ✐s ♥❡✇ ❝♦♥✈❡♥t✐♦♥ ✐s t♦ ✐♥❝❧✉❞❡ ❛❧❧ t❤❡ ❞♦♠❛✐♥ t♦ ❜❡

(−∞, ∞)✳

210.5 ❄✮

❜✉t ✐t ✐s ❛ st❡♣ ✐♥ t❤❡ r✐❣❤t ❞✐r❡❝t✐♦♥✳ ❖✉r

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

❉❡✜♥✐t✐♦♥ ✹✳✽✳✶✵✿ r❛t✐♦♥❛❧ ❡①♣♦♥❡♥t ■❢

x =

a>0

m ✱ n

✇❤❡r❡

m, n

❛r❡ ✐♥t❡❣❡rs ✇✐t❤

n > 0✱

t❤❡♥ ✇❡ s❡t t❤❡

xt❤

♣♦✇❡r

♦❢

t♦ ❜❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿ m

an = ■t ❛❧s♦ r❡❛❞s ✏ a t♦ t❤❡ ♣♦✇❡r ♦❢

√ n

am

x✑✳

❲❛r♥✐♥❣✦ ❯♥❧✐❦❡ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥s✱ ❛♥ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝✲ t✐♦♥ ❝❛♥ ♦♥❧② ❤❛✈❡ ❛ ♣♦s✐t✐✈❡ ❜❛s❡✱

a > 0❀

❛♥②♠♦r❡✦

❲❡ ♥♦✇ ♥❡❡❞ t♦ s❤♦✇ t❤❛t t❤❡ r✉❧❡s st✐❧❧ ❛♣♣❧② t♦

❛❧❧

r❛t✐♦♥❛❧ ♥✉♠❜❡rs✳

❚❤❡♦r❡♠ ✹✳✽✳✶✶✿ ❆❞❞✐t✐♦♥✲▼✉❧t✐♣❧✐❝❛t✐♦♥ ❘✉❧❡ ♦❢ ❊①♣♦♥❡♥ts ❋♦r ❡✈❡r② r❡❛❧

a>0

❛♥❞ ❡✈❡r② r❛t✐♦♥❛❧

x

❛♥❞

y✱

✇❡ ❤❛✈❡✿

ax+y = ax ay

❊①❡r❝✐s❡ ✹✳✽✳✶✷ Pr♦✈❡ t❤❡ ❢♦r♠✉❧❛✳

❚❤❡♦r❡♠ ✹✳✽✳✶✸✿ ❉✐str✐❜✉t✐✈❡ ❘✉❧❡ ♦❢ ❊①♣♦♥❡♥ts ❋♦r ❡✈❡r② r❡❛❧

a, b > 0

❛♥❞ ❡✈❡r② r❛t✐♦♥❛❧

x✱

✇❡ ❤❛✈❡✿

ax bx = (ab)x

Pr♦♦❢✳ ❙✉♣♣♦s❡

x=

1 ✱ n

✇❤❡r❡

n

✐s ❛ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r✳ ❚❤❡♥✿ 1

1

ax bx = a n b n =

√ √ √ 1 n n n a b = ab = (ab) n = (ab)x ,

♥♦

(−1)x

✹✳✽✳

❚❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥s

❛❝❝♦r❞✐♥❣ t♦ t❤❡

✸✻✻

Pr♦❞✉❝t ♦❢ ❘♦♦ts

t❤❡♦r❡♠ ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥✳ ❲❡ ❤❛✈❡ t❤❡ ❢♦r♠✉❧❛ ♣r♦✈❡♥ ❢♦r t❤❡

r❡❝✐♣r♦❝❛❧s✳

◆❡①t✱ s✉♣♣♦s❡

x=

m ✱ n

✇❤❡r❡

m n

x x

a b =a b

m n

m

✐s ❛♥ ✐♥t❡❣❡r ❛♥❞

=

√ n

am

√ n

bm

=

√ n

n

✐s ❛ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r✳ ❚❤❡♥✿

am bm



=

ab

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


m

 n1

= ab


mn

= (ab)x ,

❚❤❡♦r❡♠ ✹✳✽✳✶✹✿ ▼✉❧t✐♣❧✐❝❛t✐♦♥✲❊①♣♦♥❡♥t✐❛t✐♦♥ ❘✉❧❡ ♦❢ ❊①♣♦♥❡♥ts ❋♦r ❡✈❡r② r❡❛❧

a>0

❛♥❞ ❡✈❡r② r❛t✐♦♥❛❧

x

y✱

❛♥❞

✇❡ ❤❛✈❡✿

axy = (ax )y

Pr♦♦❢✳ ❙✉♣♣♦s❡

x=

1 n

❛♥❞

y=

1 ✱ m

✇❤❡r❡



x y

(a ) = a ❛❝❝♦r❞✐♥❣ t♦ t❤❡

n, m

1 n

 m1

❛r❡ ♣♦s✐t✐✈❡ ✐♥t❡❣❡rs✳ ❚❤❡♥✿

=

❈♦♠♣♦s✐t✐♦♥ ♦❢ ❘♦♦ts

q m

√ n

a=

√

nm

1 1

1

a = a nm = a n m = axy ,

t❤❡♦r❡♠ ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥✳ ❲❡ ❤❛✈❡ t❤❡ ❢♦r♠✉❧❛ ♣r♦✈❡♥ ❢♦r

t❤❡ r❡❝✐♣r♦❝❛❧s✳

❊①❡r❝✐s❡ ✹✳✽✳✶✺ Pr♦✈✐❞❡ t❤❡ ♠✐ss✐♥❣ ♣❛rts ♦❢ t❤❡ ♣r♦♦❢✳

❚❤✐s ✐s t❤❡ ✐❞❡❛ ♦❢ ♦✉r ❢♦r♠✉❧❛✿

❘❛t✐♦♥❛❧ ❡①♣♦♥❡♥t

m an

=

√ n

am

■s t❤❡ ❣r❛♣❤ ♦❢ t❤✐s ♥❡✇ ❢✉♥❝t✐♦♥ ❛ ✏❝♦♠♣❧❡t❡✑ ❝✉r✈❡ s✉❝❤ t❤❛t ♦❢

y = x2 ❄

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

Q

❛♥❞ t❤❡ r❛t✐♦♥❛❧ ♥✉♠❜❡rs ❛r❡ s♦

♠❛♥② ♦❢ t❤❡♠ ✭❛❢t❡r ❛❧❧✱ ✐❢

p

❛♥❞

q

❣r❛♣❤✦ ❍♦✇❡✈❡r✱ ✐ts ♣♦✐♥ts r❡♠❛✐♥

❞❡♥s❡

t❤❛t ❛♥② ✐♥t❡r✈❛❧✱ ♥♦ ♠❛tt❡r ❤♦✇ s♠❛❧❧✱ ✇✐❧❧ ❝♦♥t❛✐♥ ✐♥✜♥✐t❡❧②

❛r❡ r❛t✐♦♥❛❧✱ t❤❡♥ s♦ ✐s

❞✐s❝♦♥♥❡❝t❡❞

(p + q)/2✮✳

❚❤❡r❡❢♦r❡✱ t❤❡r❡ ❛r❡

♥♦ ❣❛♣s

✐♥ t❤❡

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

❚❤❡ r❡❛s♦♥ ✐s t❤❛t t❤❡ ♣♦✐♥ts ♦♥ t❤❡ ❣r❛♣❤✱ ✉♥❧✐❦❡ t❤❛t ♦❢

x=

√

2,

√

y = x2 ✱

3, π .

✇✐t❤

✐rr❛t✐♦♥❛❧ x✲❝♦♦r❞✐♥❛t❡s ❛r❡ ♠✐ss✐♥❣✿

✹✳✽✳

❚❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥s

✸✻✼

❆♥❞ t❤❡ ✐rr❛t✐♦♥❛❧ ♥✉♠❜❡rs ❛r❡✱ t♦♦✱ s♦

❞❡♥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❡✦ ❲❡ ❝❛♥ ❞❡✜♥❡ t❤❡ ❢✉♥❝t✐♦♥ ❢♦r t❤❡ ✐rr❛t✐♦♥❛❧ ❡①♣♦♥❡♥ts ❜② ✏❛♣♣r♦①✐♠❛t✐♥❣✑ t❤❡♠ ✇✐t❤ r❛t✐♦♥❛❧ ♥✉♠❜❡rs ✭t♦ ❜❡ 3 3.1 3.14 , ... ✇✐❧❧ ❛♣♣r♦①✐♠❛t❡ 2π ✳ ❚❤❡ t❤r❡❡ ♣r❡s❡♥t❡❞ ✐♥ ❈❤❛♣t❡r ✷❉❈✲✶✮✳ ❋♦r ❡①❛♠♣❧❡✱ t❤❡ s❡q✉❡♥❝❡ 2 , 2 , 2 ❛❧❣❡❜r❛✐❝ r✉❧❡s ❛r❡ st✐❧❧ t♦ ❜❡ ♦❜❡②❡❞✳

❲❛r♥✐♥❣✦

❊✈❡♥ t❤♦✉❣❤ t❤✐s ❢✉♥❝t✐♦♥ ❤❛s ❜❡❡♥ ❜✉✐❧t ❢r♦♠ t❤❡ ♣♦✇❡r ❢✉♥❝t✐♦♥s ❛♥❞ t❤❡ r♦♦ts✱ ✐t✬s ✈❡r② ❞✐✛❡r❡♥t ❢r♦♠ t❤♦s❡✳ ❲❡ st❛rt tr❡❛t✐♥❣ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥ ❛s ✐❢ ✐t ❤❛s ❜❡❡♥ ❛❧r❡❛❞② ❢✉❧❧② ❝♦♥str✉❝t❡❞✳

❊①❡r❝✐s❡ ✹✳✽✳✶✻ ❉❡s❝r✐❜❡ s✉❝❤ ❛ ❢✉♥❝t✐♦♥ ❛s ❛ tr❛♥s❢♦r♠❛t✐♦♥✳

❇❡❧♦✇ ✇❡ ♣❧♦t t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t

∆f ✱ ∆x

✐✳❡✳✱ t❤❡ s❛♠♣❧❡❞ s❧♦♣❡s✱ ♦❢

f (x) = 2x ✿

■♥ ❝♦♥tr❛st t♦ ❛❧❧ ♦t❤❡r ❢✉♥❝t✐♦♥s ✇❡ ❤❛✈❡ s❡❡♥✱ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❛♣♣❡❛rs t♦ ❡①❤✐❜✐t ❛ ❜❡❤❛✈✐♦r s✐♠✐❧❛r t♦ t❤❡ ♦r✐❣✐♥❛❧ ❢✉♥❝t✐♦♥✦ ❲❡ ✇✐❧❧ s❤♦✇ ✭❈❤❛♣t❡r ✷❉❈✲✸✮ t❤❛t t❤❡ r❛t❡ ♦❢ ❣r♦✇t❤ ♦❢ ❛♥ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥ ❣r♦✇s ❡①♣♦♥❡♥t✐❛❧❧②✳

❊①❡r❝✐s❡ ✹✳✽✳✶✼ P❧♦t t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ t❤❡ ❧♦❣✐st✐❝ ❢✉♥❝t✐♦♥ ✭✐✳❡✳✱ r❡str✐❝t❡❞ ❣r♦✇t❤✮✿

f (x) =

1 . 1 + 2−x

❲❤❛t ♣❛tt❡r♥ ❞♦ ②♦✉ s❡❡❄

❇❡❧♦✇ ✐s t❤❡ s✉♠♠❛r② ♦❢ ♦✉r

❝♦♥✈❡♥t✐♦♥s ✭a > 0 r❡❛❧✱ m, n ♣♦s✐t✐✈❡ ✐♥t❡❣❡rs✮✿

✹✳✽✳

❚❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥s

✸✻✽

❆♥❛❧♦❣②✿ r❡♣❡❛t❡❞ ❛❞❞✐t✐♦♥ ✈s✳ r❡♣❡❛t❡❞ ♠✉❧t✐♣❧✐❝❛t✐♦♥

▼✉❧t✐♣❧✐❝❛t✐♦♥✿

1. x = 1, 2, ...

ax = |a + a + {z a + ... + a} x t✐♠❡s

a·0 =0

2. x = 0 3. x = −1, −2, ... 4. x =

m n

❊①♣♦♥❡♥t✐❛t✐♦♥✿

ax = |a · a · {z a · ... · a} x t✐♠❡s

a

0

ax = (−a)(−x) = −a(−x) ax ax =

a am = n n m

ax

5. x r❡❛❧

=1  −x 1 1 = = −x a a √
m √ = n am = n a

❈❤❛♣t❡r ✷❉❈✲✶

❆s ✇❡ ♣r♦❣r❡ss ❢r♦♠ t❤❡ ♥❛t✉r❛❧ ♥✉♠❜❡rs t♦ t❤❡ r❡❛❧✱ ✇❡ ❛❧s♦ ♣r♦❣r❡ss ❢r♦♠ s❡q✉❡♥❝❡s t♦ ❢✉♥❝t✐♦♥s ✐♥ ♦✉r ❛♥❛❧♦❣②✿ ❆♥❛❧♦❣②✿ r❡♣❡❛t❡❞ ❛❞❞✐t✐♦♥ ✈s✳ r❡♣❡❛t❡❞ ♠✉❧t✐♣❧✐❝❛t✐♦♥

▼✉❧t✐♣❧✐❝❛t✐♦♥✿

❊①♣♦♥❡♥t✐❛t✐♦♥✿

❛r✐t❤♠❡t✐❝ ♣r♦❣r❡ss✐♦♥✱ an+1 = an + a ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥✱ an+1 = an · a ❧✐♥❡❛r ❢✉♥❝t✐♦♥✱ f (x) = mx + b ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥✱ f (x) = ax ▲❡t✬s ❝♦❧❧❡❝t s♦♠❡ ❢❛❝ts ❛❜♦✉t t❤✐s ❢✉♥❝t✐♦♥ ✈✐s✐❜❧❡ ✐♥ t❤❡ ❣r❛♣❤ ♦❢ y = 2x ✿

❚❤❡♦r❡♠ ✹✳✽✳✶✽✿ ❋❛❝ts ❆❜♦✉t ❊①♣♦♥❡♥t✐❛❧ ❋✉♥❝t✐♦♥

❚❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥ y = ax ✇✐t❤ ❜❛s❡ a > 0 s❛t✐s✜❡s t❤❡ ❢♦❧❧♦✇✐♥❣✿ ✶✳ ❚❤❡ ❞♦♠❛✐♥ ✐s (−∞, +∞)✳ ✷✳ ❚❤❡ r❛♥❣❡ ✐s (0, ∞)✳ ✸✳ ❚❤❡ y ✲✐♥t❡r❝❡♣t ✐s (0, 1)✳ ✹✳ ❚❤❡r❡ ❛r❡ ♥♦ x✲✐♥t❡r❝❡♣ts✳ ❊①❡r❝✐s❡ ✹✳✽✳✶✾

Pr♦✈❡ ♣❛rts ✷✲✹✳ ❚❤❡r❡ ❛r❡ t✇♦ ✈❡r② ❞✐✛❡r❡♥t ❦✐♥❞s ♦❢ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥s ❤♦✇❡✈❡r✿

✹✳✽✳

❚❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥s

✸✻✾

❚❤❡ ❢♦❧❧♦✇✐♥❣ ✐s ❛♥ ❡①t❡♥s✐♦♥ ♦❢ t❤❡ t❤❡♦r❡♠ ❛❜♦✉t t❤❡ ♠♦♥♦t♦♥✐❝✐t② ♦❢ t❤❡ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥✳ ❚❤❡♦r❡♠ ✹✳✽✳✷✵✿ ▼♦♥♦t♦♥✐❝✐t② ♦❢ ❊①♣♦♥❡♥t

❚❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥ ✇✐t❤ ❜❛s❡ a > 0 s❛t✐s✜❡s t❤❡ ❢♦❧❧♦✇✐♥❣✿ y = ax ✐s str✐❝t❧② ✐♥❝r❡❛s✐♥❣ ✐❢ a > 1 , y = ax ✐s str✐❝t❧② ❞❡❝r❡❛s✐♥❣ ✐❢ a < 1 . Pr♦♦❢✳

❙✉♣♣♦s❡

y

a > 1✳

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

x✳

◆♦✇✱ s✉♣♣♦s❡

x

❛♥❞

x < y ✳ ❙✉♣♣♦s❡ • x = n/m ❢♦r t✇♦ ✐♥t❡❣❡rs m, n✱ ❛♥❞ • y = p/q ❢♦r t✇♦ ✐♥t❡❣❡rs p, q ✳

❛r❡ r❛t✐♦♥❛❧ ♥✉♠❜❡rs ❛♥❞

❚❤❡♥✿

n/m < p/q

nq < mp .

✱ ♦r

❚❤❡♥✿

anq < amp , ❜❡❝❛✉s❡ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥ ✐s ✐♥❝r❡❛s✐♥❣ ❢♦r ✐♥t❡❣❡r ✐♥♣✉ts✳ ❚❤❡r❡❢♦r❡✿

√ q

anq
0✱ t❤❡ ❧♦❣❛r✐t❤♠ ❜❛s❡ a ✐s t❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥ ❜❛s❡ a✱ ✐✳❡✳✱ y = ax ✱ ✇✐t❤ ❝♦❞♦♠❛✐♥ (0, +∞)✳ ❚❤❡ ❢✉♥❝t✐♦♥ ✐s ❞❡♥♦t❡❞ ❜②

x = loga y

❊①❛♠♣❧❡ ✹✳✾✳✸✿ ♣♦♣✉❧❛t✐♦♥ ❣r♦✇t❤✱ r❡✈✐s✐t❡❞ ❆r♠❡❞ ✇✐t❤ ♥♦t❤✐♥❣ ❜✉t t❤✐s ❞❡✜♥✐t✐♦♥ ✕ ❛♥❞ ❛ ❝❛❧❝✉❧❛t♦r ✕ ✇❡ ❝❛♥ s♦❧✈❡ t❤❡ ♣r♦❜❧❡♠ ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥✳ ■❢ t❤❡ ♣♦♣✉❧❛t✐♦♥ ✐s ♣r❡❞✐❝t❡❞ t♦ ❜❡ ❛❢t❡r x ②❡❛rs✿

1, 000, 000 · 1.005x , ❤♦✇ ❧♦♥❣ ✇✐❧❧ ✐t t❛❦❡ ❢♦r t❤❡ ♣♦♣✉❧❛t✐♦♥ t♦

❞♦✉❜❧❡ ❄

❙♦❧✈✐♥❣ t❤❡ ❡q✉❛t✐♦♥✿

1, 000, 000 · 1.005x = 2, 000, 000 , ♣r♦❞✉❝❡s✿

1.005x = 2 =⇒ x = log1.005 2 ≈ 139 ②❡❛rs✳

❊①❡r❝✐s❡ ✹✳✾✳✹ ❯s❡ t❤❡ ❞❡✜♥✐t✐♦♥ ❛♥❞ ❛ ❝❛❧❝✉❧❛t♦r t♦ ✜♥❞ ❤♦✇ ❧♦♥❣ ✐t t❛❦❡s t♦ tr✐♣❧❡ ②♦✉r ♠♦♥❡② ✐❢ ②♦✉r ✐♥t❡r❡st ✐s 1.5%✳

✹✳✾✳

❚❤❡ ❧♦❣❛r✐t❤♠✐❝ ❢✉♥❝t✐♦♥s

✸✼✹

❚❤❡ ❧✐♥❦ ❜❡t✇❡❡♥ t❤❡ t✇♦ ❢✉♥❝t✐♦♥s ✐♥✈❡rs❡ ♦❢ ❡❛❝❤ ♦t❤❡r ✇✐❧❧ r❡♠❛✐♥ ✉♥❜r♦❦❡♥✿

ax = y ⇐⇒ loga y = x ❊①❛♠♣❧❡ ✹✳✾✳✺✿ ❞❡✜♥✐t✐♦♥ ♦❢ log ❊✈❡r② ❝♦♠♣✉t❡❞ ❡①♣♦♥❡♥t ❣❡♥❡r❛t❡s ❛ ❝♦♠♣✉t❡❞ ❧♦❣❛r✐t❤♠✿ ⇐⇒ log2 8 = 3 . 23 = 8 102 = 100 ⇐⇒ log10 100 = 2 .

❚❤❡ ❜❛s❡s r❡♠❛✐♥ ✉♥❝❤❛♥❣❡❞ ❜✉t t❤❡ ✐♥♣✉ts ❜❡❝♦♠❡ ♦✉t♣✉ts ❛♥❞ ✈✐❝❡ ✈❡rs❛✳

❊①❡r❝✐s❡ ✹✳✾✳✻ ❈♦♠♣✉t❡✿ ✭❛✮ log2 1024 ✱ ✭❜✮ log2 .25 ✱ ✭❝✮ log1 1 ✳ ❏✉st ❛s ✇✐t❤ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥✱ ✐♥ t❤❡ ♥♦t❛t✐♦♥ ❢♦r t❤❡ ❧♦❣❛r✐t❤♠ ♦❢ t❤❡ ❢✉♥❝t✐♦♥✿

t❤❡ ❜❛s❡ ✐s ♣❧❛❝❡❞ ❜❡❧♦✇ t❤❡ ✐♥♣✉t

❜❛s❡x ❛♥❞ log❜❛s❡ y ❚❤❡ ✐♥♣✉t ✐s ❛ s✉♣❡rs❝r✐♣t ♦❢ t❤❡ ❜❛s❡ ✐♥ t❤❡ ❢♦r♠❡r ❝❛s❡ ❛♥❞ t❤❡ ❜❛s❡ ✐s ❛ s✉❜s❝r✐♣t ♦❢ t❤❡ ✐♥♣✉t ✐♥ t❤❡ ❧❛tt❡r ❝❛s❡✳

❲❛r♥✐♥❣✦ ❋♦❧❧♦✇✐♥❣ ♦✉r ❝♦♥✈❡♥t✐♦♥s ❛❜♦✉t ❢✉♥❝t✐♦♥s✱ ✇❡ ❝❛♥ ❛❧t❡r♥❛t✐✈❡❧② ✇r✐t❡✿

a(x)

❛♥❞

loga (x)

❲❡ ❤❛✈❡ ❛❝q✉✐r❡❞✱ ✐♥ ♦♥❡ str♦❦❡✱ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡✇ ❢✉♥❝t✐♦♥s✳ ◆♦✇ t❤❡ ❣r❛♣❤s✳ ❚❤✐s ✐s ❤♦✇ ✇❡ ❣❡t t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❧♦❣❛r✐t❤♠ ✇✐t❤ ❛ ✢✐♣ ❛❜♦✉t t❤❡ ❞✐❛❣♦♥❛❧ y = x✿

■♥ ❢❛❝t✱ ✇❡ ❝❛♥ ❤❛✈❡ ❛❧❧ ♦❢ t❤❡ ❣r❛♣❤s ❛t ♦♥❝❡✿

✹✳✾✳

❚❤❡ ❧♦❣❛r✐t❤♠✐❝ ❢✉♥❝t✐♦♥s

✸✼✺

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

x

❛♥❞

y✿ y = ax

y = loga x

❚❤❡ ❞♦♠❛✐♥ ✐s ❚❤❡ r❛♥❣❡ ✐s ❚❤❡

(−∞, +∞).

(0, ∞).

y ✲✐♥t❡r❝❡♣t

❚❤❡r❡ ❛r❡ ♥♦

✐s

(0, 1).

x✲✐♥t❡r❝❡♣ts✳

❚❤❡ r❛♥❣❡ ✐s

(−∞, +∞).

❚❤❡ ❞♦♠❛✐♥ ✐s ❚❤❡

(0, ∞).

x✲✐♥t❡r❝❡♣t

❚❤❡r❡ ❛r❡ ♥♦

✐s

(0, 1).

y ✲✐♥t❡r❝❡♣ts✳

❚❤❡② ❛r❡ ❛❧s♦ ❛❧❧ ♦♥❡✲t♦✲♦♥❡✳ ❊①❡r❝✐s❡ ✹✳✾✳✼

❉♦ ②♦✉ s❡❡ ❛♥② ❛s②♠♣t♦t❡s ✐♥ t❤❡ ❣r❛♣❤s❄

❏✉st ❛s ✇✐t❤ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥✱ t❤❡r❡ ❛r❡ t✇♦ ✈❡r② ❞✐✛❡r❡♥t ❦✐♥❞s ♦❢ ❧♦❣❛r✐t❤♠s✿

❋r♦♠ t❤❡

▼♦♥♦t♦♥✐❝✐t② ♦❢ ❊①♣♦♥❡♥t t❤❡♦r❡♠ ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥ ✇❡ ❞❡r✐✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛❜♦✉t t❤❡ ❧♦❣❛r✐t❤♠✳ ❚❤❡♦r❡♠ ✹✳✾✳✽✿ ▼♦♥♦t♦♥✐❝✐t② ♦❢ ▲♦❣❛r✐t❤♠

❚❤❡ ❧♦❣❛r✐t❤♠ ❢✉♥❝t✐♦♥ ✇✐t❤ ❜❛s❡ a > 0 s❛t✐s✜❡s t❤❡ ❢♦❧❧♦✇✐♥❣✿ y = loga x ✐s str✐❝t❧② ✐♥❝r❡❛s✐♥❣ ✐❢ a > 1 . y = loga x ✐s str✐❝t❧② ❞❡❝r❡❛s✐♥❣ ✐❢ a < 1 . ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ st❛t❡♠❡♥t ✏ x ❛♥❞ ❊①❡r❝✐s❡ ✹✳✾✳✾

Pr♦✈❡ t❤❡ t❤❡♦r❡♠✳

y

✐♥❝r❡❛s❡ t♦❣❡t❤❡r✑ r❡♠❛✐♥s tr✉❡ ✐❢ ✇❡ ✐♥t❡r❝❤❛♥❣❡

x

❛♥❞

y✳

✹✳✾✳

❚❤❡ ❧♦❣❛r✐t❤♠✐❝ ❢✉♥❝t✐♦♥s

✸✼✻

❊①❡r❝✐s❡ ✹✳✾✳✶✵

❉❡s❝r✐❜❡ s✉❝❤ ❛ ❢✉♥❝t✐♦♥ ❛s ❛ tr❛♥s❢♦r♠❛t✐♦♥✳ ❚❤✐s ✐s t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ t❤❡ ❧♦❣❛r✐t❤♠✿

❊①❡r❝✐s❡ ✹✳✾✳✶✶

❉♦❡s ✐t ❧♦♦❦ ❢❛♠✐❧✐❛r❄ ◆♦✇ s♦♠❡ ❛❧❣❡❜r❛✳✳✳

❚❤❡ ❧♦❣❛r✐t❤♠ ❜❛s❡ a ♦❢ y ✐s t❤❡ ♥✉♠❜❡r t❤❛t ❝❛♥ ❜❡ ❞❡✜♥❡❞ ✈❡r❜❛❧❧② ❛s ❢♦❧❧♦✇s✿ loga y ✐s t❤❡ ♣♦✇❡r t♦ ✇❤✐❝❤ ②♦✉ ❤❛✈❡ t♦ r❛✐s❡ a t♦ ❣❡t y ✳

❚❤❡r❡❢♦r❡✱ ✇❡ ❝❛♥ s❛② t❤❛t t❤❡s❡ t✇♦ ❜❡❧♦✇ ❛r❡ t❤❡ s❛♠❡ st❛t❡♠❡♥t ✇r✐tt❡♥ ✐♥ t✇♦ ❞✐✛❡r❡♥t ✇❛②s✿ ax = y ✈s✳ loga y = x .

❆❢t❡r ❛❧❧✱ t❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ x ❛♥❞ y ✭❈❤❛♣t❡r ✷✮ r❡♠❛✐♥s t❤❡ s❛♠❡✳ ❊①❛♠♣❧❡ ✹✳✾✳✶✷✿ ❧♦❣❛r✐t❤♠s ❝♦♠❡ ❢r♦♠ ♣r✐♦r ❡①♣♦♥❡♥ts

❋♦r ❛ ✜①❡❞ ❜❛s❡✱ ❡①♣♦♥❡♥ts ❛♥❞ ❧♦❣❛r✐t❤♠s ❝♦♠❡ ✐♥ ♣❛✐rs✿ ♠❡❛♥s 23 = 8 ♠❡❛♥s 102 = 100 ♠❡❛♥s ax = y

1 ♠❡❛♥s 2−1 = 2√ 31/2 = 3 ♠❡❛♥s

loga y = x . log2 8 = 3 . log10 100 = 2 . 1 log2 = −1 . 2 √ 1 log3 3 = . 2

■♥ ❛ s❡♥s❡✱ ❡✈❡r② ❧♦❣❛r✐t❤♠✐❝ ❡①♣r❡ss✐♦♥ ✉s❡❞ t♦ ❜❡ ❛♥ ❡①♣♦♥❡♥t✐❛❧ ❡①♣r❡ss✐♦♥✳ ❚❤✐s ✐s ✇❤②✱ ✐♥✐t✐❛❧❧②✱ ❝♦♠♣✉t✐♥❣ ❧♦❣❛r✐t❤♠s ♠❡❛♥s r❡♠❡♠❜❡r✐♥❣ ❡①♣♦♥❡♥ts ❝♦♠♣✉t❡❞ ✐♥ t❤❡ ♣❛st✱ ❥✉st ❧✐❦❡ ✇✐t❤ t❤❡ r♦♦ts ❡❛r❧✐❡r ✐♥ t❤✐s ❝❤❛♣t❡r✿ √ 4 16❄ ■ s❡❡♠ t♦ r❡❝❛❧❧ ❝♦♠♣✉t✐♥❣ t❤❡ ♣♦✇❡rs ♦❢ 2 ❛♥❞ ♣r♦❞✉❝✐♥❣ 16 ❛❢t❡r 4 r❡♣❡t✐t✐♦♥s✳ • ❲❤❛t ✐s √ ❙♦✱ 4 16 = 2✳ • ❲❤❛t ✐s log3 27❄ ■ s❡❡♠ t♦ r❡❝❛❧❧ ❝♦♠♣✉t✐♥❣ t❤❡ ♣♦✇❡rs ♦❢ 3 ❛♥❞ ♣r♦❞✉❝✐♥❣ 27 ✐♥ 3 st❡♣s✳ ❙♦✱ log3 27 = 3✳ ■♥ ♠♦r❡ ❝♦♠♣❧❡① s✐t✉❛t✐♦♥s✱ ✇❡ ❤❛✈❡ t♦ ✇♦r❦ ♦✉r ✇❛② t♦ ❛ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ✐♥♣✉t ♦❢ t❤❡ ❧♦❣❛r✐t❤♠ ❛s ❛ ♣♦✇❡r ♦❢ ✐ts ❜❛s❡✳ 1 1 • ❲❤❛t ✐s log10 .01❄ ■ ♥❡❡❞ t♦ r❡♣r❡s❡♥t .01 ❛s ❛ ♣♦✇❡r ♦❢ 10✳ ❍❡r❡ ✐t ✐s✿ .01 = = 2 = 10−2 ✳ 100 10 ❙♦✱ log10 .01 = −2✳

✹✳✾✳

❚❤❡ ❧♦❣❛r✐t❤♠✐❝ ❢✉♥❝t✐♦♥s •

❲❤❛t ✐s

•

❲❤❛t ✐s

log4 2❄

✸✼✼

■ ♥❡❡❞ t♦ r❡♣r❡s❡♥t 5 ❛s ❛ ♣♦✇❡r ♦❢ log 5 ❦♥♦✇❀ ❤❡r❡ ✐t ✐s✿ 5 = 4 4 ✦ ❙♦✱ log4 5 = log4 5✳

4✳

❚❤❡r❡ s❡❡♠s t♦ ❜❡ ♥♦ ❡❛s② ✇❛② t♦ ❞♦ ✐t✳ ■

log4 5❄

❚❤✐s ✏❛♥s✇❡r✑ ✐s ❥✉st ❛s ❛❝❝❡♣t❛❜❧❡ ❛s t❤❡ ♦♥❡s ❜❡❢♦r❡❀ ✇❡ ❛r❡ ❥✉st ✉♥❛❜❧❡ t♦

❙♦✱

log4 2 =

1 ✳ 2

❍❡r❡ ✐t ✐s✿

♠❡t❤♦❞✳

2=

4 = 41/2 ✳

4✳

❚❤✐s✱ ❤♦✇❡✈❡r✱ ✐s ❤❛r❞❧② ❛

2

√

❛s ❛ ♣♦✇❡r ♦❢

■ ♥❡❡❞ t♦ r❡♣r❡s❡♥t

s✐♠♣❧✐❢② t❤✐s ❡①♣r❡ss✐♦♥✳

❲❛r♥✐♥❣✦ ❚❤❡r❡ ✐s ♥♦ ✏❢♦r♠✉❧❛✑ ❢♦r t❤❡ ❧♦❣❛r✐t❤♠✱ ❥✉st ❛s t❤❡r❡ ✐s ♥♦ ✏❢♦r♠✉❧❛✑ ❢♦r t❤❡ sq✉❛r❡ r♦♦t✳

❚❤❡ ❢❛❝t t❤❛t t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥ ❛♥❞ t❤❡ ❧♦❣❛r✐t❤♠ ✕ ♦❢ t❤❡ s❛♠❡ ❜❛s❡ ✕ ❛r❡ ✐♥✈❡rs❡s ♦❢ ❡❛❝❤ ♦t❤❡r ♠❡❛♥s t❤❛t t❤❡②

✉♥❞♦ ❡❛❝❤ ♦t❤❡r ✇❤❡♥ ❝♦♠♣♦s❡❞✱ ✐♥ ❡✐t❤❡r ♦r❞❡r✳ y = ax

❛♥❞

❋♦r t❤❡ ♣❛✐r

x = loga y ,

✇❡ ❤❛✈❡ t❤❡s❡ t✇♦ r✉❧❡s✳

❚❤❡♦r❡♠ ✹✳✾✳✶✸✿ ❈❛♥❝❡❧❧❛t✐♦♥ ▲❛✇s ♦❢ ▲♦❣❛r✐t❤♠s ❙✉♣♣♦s❡

a > 0✳

x

❚❤❡♥ ❢♦r ❛♥② r❡❛❧

❛♥❞ ❛♥②

aloga y

y > 0✱

✇❡ ❤❛✈❡✿

=y

loga ax = x

■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ♦✉t♣✉ts ♦❢ t❤❡s❡ ❝♦♠♣♦s✐t✐♦♥s ❛r❡ t❤❡ s❛♠❡ ❛s t❤❡ ✐♥♣✉ts✿

x →

ax = y

y →

loga y = x

→ y →

loga y = x

→ x

ax = y

→y

❛♥❞

→ x →

❚❤❡s❡ t✇♦ ✢♦✇❝❤❛rts ❝❛♥ ❜❡ s❡❡♥ ✐♥ t❤❡ ❛❜♦✈❡ ❢♦r♠✉❧❛s✱ ❛s ❢♦❧❧♦✇s✿

1.

x (x)

2. 3. loga (

a

1. 2.

ax )

3.

y loga ( y ) (loga ( y ))

a

❲❛r♥✐♥❣✦ ❚❤❡ ❡①♣♦♥❡♥t ❛♥❞ t❤❡ ❧♦❣❛r✐t❤♠ ❝❛♥❝❡❧ ❡❛❝❤ ♦t❤❡r ♦♥❧② ✐❢ t❤❡② ❛r❡ ♦❢ t❤❡

❲❡ ✉s❡ t❤❡s❡ ❢♦r♠✉❧❛s t♦ s♦❧✈❡

❡①♣♦♥❡♥t✐❛❧ ❡q✉❛t✐♦♥s✳

❊①❛♠♣❧❡ ✹✳✾✳✶✹✿ s♦❧✈✐♥❣ ❡①♣ ❡q✉❛t✐♦♥s ✇✐t❤ ❙♦❧✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r

s❛♠❡ ❜❛s❡✳

log

x✿ 2x−5 = 3 .

❚♦ ✏❦✐❧❧✑ t❤❡ ❡①♣♦♥❡♥t ❛♥❞ ❣❡t t♦

x✱

❛♣♣❧② ✐ts

✐♥✈❡rs❡✱ t❤❡ ❧♦❣❛r✐t❤♠ ♦❢ t❤❡ s❛♠❡ ❜❛s❡✱ t♦ ❜♦t❤ s✐❞❡s ♦❢

✹✳✶✵✳

❚❤❡ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s

✸✼✽

t❤❡ ❡q✉❛t✐♦♥✿

log2 (2x−5 )= log2 (3) . ❲❡ ❛r❡ ❛❢t❡r t❤✐s ❝♦♠♣♦s✐t✐♦♥✿

log2 (2x−5 ) = log2 3 . ■t ♥♦✇ ❞✐s❛♣♣❡❛rs ❛❢t❡r t❤❡ s❡❝♦♥❞ ❝❛♥❝❡❧❧❛t✐♦♥ ❧❛✇✿

x − 5 = log2 3 =⇒ x = log2 3 + 5 . ❊①❡r❝✐s❡ ✹✳✾✳✶✺

❙♦❧✈❡ t❤❡ ❡q✉❛t✐♦♥ ❜❡❧♦✇✳

3x

2 +1

= 2.

▼❛❦❡ ✉♣ ②♦✉r ♦✇♥ ❡q✉❛t✐♦♥ ❛♥❞ s♦❧✈❡ ✐t✳ ❘❡♣❡❛t✳

❊①❡r❝✐s❡ ✹✳✾✳✶✻

❙✉♣♣♦s❡ ❢✉♥❝t✐♦♥

f

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

g

♣❡r❢♦r♠s ✏t❛❦❡ t❤❡

sq✉❛r❡ r♦♦t ♦❢ ✑✳ ✭❛✮ ❋✐♥❞ t❤❡ ❢♦r♠✉❧❛s ❢♦r t❤❡ t✇♦ ♣♦ss✐❜❧❡ ❝♦♠♣♦s✐t✐♦♥s✳ ✭❜✮ ❋✐♥❞ t❤❡✐r ❞♦♠❛✐♥s✳

❊①❛♠♣❧❡ ✹✳✾✳✶✼✿ s♦❧✈✐♥❣

❙♦❧✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r

log

❡q✉❛t✐♦♥s ✇✐t❤ ❡①♣

x✿ log2 (x − 5) = 3 .

❚♦ ✏❦✐❧❧✑ t❤❡ ❧♦❣❛r✐t❤♠ ❛♥❞ ❣❡t t♦

x✱

❛♣♣❧② ✐ts

✐♥✈❡rs❡✱ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ s❛♠❡ ❜❛s❡✱ t♦

❜♦t❤ s✐❞❡s ♦❢ t❤❡ ❡q✉❛t✐♦♥✿

2(log2 (x−5)) =2(3) . ❲❡ ❛r❡ ❛❢t❡r t❤✐s ❝♦♠♣♦s✐t✐♦♥✿

2log2 (x−5) = 8 . ■t ♥♦✇ ❞✐s❛♣♣❡❛rs ❛❢t❡r t❤❡ ✜rst ❝❛♥❝❡❧❧❛t✐♦♥ ❧❛✇✿

x−5 =8 =⇒ x = 13 . ❚❤✐s ❦✐♥❞ ♦❢ ✐♥t❡r❛❝t✐♦♥ ❜❡t✇❡❡♥ ❛ ❢✉♥❝t✐♦♥ ❛♥❞ ✐ts ✐♥✈❡rs❡ ✐s ❝♦♠♠♦♥✿

✹✳✶✵✳ ❚❤❡ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s

❖♥❡ ❡♥❝♦✉♥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✿

✹✳✶✵✳

❚❤❡ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s

✸✼✾

❲❡ ❝❛❧❧ s✉❝❤ ❢✉♥❝t✐♦♥s ♣❡r✐♦❞✐❝✳ ❍♦✇❡✈❡r✱ ♥♦♥❡ ♦❢ ♠❛✐♥ ❝❧❛ss❡s ♦❢ ❢✉♥❝t✐♦♥s ✇❡ ❤❛✈❡ s♦ ❢❛r ✐♥tr♦❞✉❝❡❞ ❡①❤✐❜✐t t❤✐s ❜❡❤❛✈✐♦r✳ ❋♦r ❡①❛♠♣❧❡✱ ❛ ✇❡❧❧✲❝❤♦s❡♥ ♣♦❧②♥♦♠✐❛❧ ❝❛♥ ♠✐♠✐❝ ♣❡r✐♦❞✐❝✐t② ❜✉t ❡✈❡♥t✉❛❧❧② ✇✐❧❧ ❤❛✈❡ t♦ r✉♥ ❛✇❛② t♦ ✐♥✜♥✐t②✿

❊①❛♠♣❧❡ ✹✳✶✵✳✶✿ ♣❡r✐♦❞✐❝ ❜❡❤❛✈✐♦r ❚❤❡ s✐♠♣❧❡st ♣❡r✐♦❞✐❝ ❜❡❤❛✈✐♦r ✐s ♦s❝✐❧❧❛t✐♦♥ ♦❢ ❛♥ ♦❜❥❡❝t ♦♥ ❛ s♣r✐♥❣ ♦r ❛ str✐♥❣ ♦❢ ❛ ♠✉s✐❝❛❧ ✐♥str✉♠❡♥t ♦r ❛♥ ♦r❜✐t✐♥❣ ♣❧❛♥❡t✿

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

❣❡♦♠❡tr② ✿

❉❡✜♥✐t✐♦♥ ✹✳✶✵✳✷✿ s✐♥❡✱ ❝♦s✐♥❡✱ ❛♥❞ t❛♥❣❡♥t ❙✉♣♣♦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 cos α =

✹✳✶✵✳

❚❤❡ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s

✸✽✵

❲❡ ❛❧s♦ ❞❡✜♥❡ ✐ts t❛♥❣❡♥t ✿ tan α =

sin α b = a cos α

❚❤❡ ✐♠♣♦rt❛♥❝❡ ♦❢ t❤❡ t❛♥❣❡♥t ✐s s❡❡♥ ✐♥ t❤✐s ❢♦r♠✉❧❛✿ b r✐s❡ tan α = = = slope ♦❢ t❤❡ ❤②♣♦t❡♥✉s❡ c, a r✉♥ ✐❢ s✐❞❡ a ❢♦❧❧♦✇s t❤❡ x✲❛①✐s✿

❍♦✇❡✈❡r✱

s✐♠✐❧❛r tr✐❛♥❣❧❡s ❤❛✈❡ ❡q✉❛❧ ❛♥❣❧❡s✿

❚❤❡♥✱ ❧❡t✬s ♣✐❝❦ t❤❡ s✐♠♣❧❡st✱ t❤❡ ♦♥❡ ✇✐t❤ ❛ ❤②♣♦t❡♥✉s❡ ♦❢ ❧❡♥❣t❤ 1✦ ❚❤❡♥ ♦✉r ❞❡✜♥✐t✐♦♥s ❞♦♥✬t ♥❡❡❞ ❢r❛❝t✐♦♥s ❛♥②♠♦r❡✿ cos α = a sin α = b

❇✉t ✇❤❛t ❞♦❡s tr✐❣♦♥♦♠❡tr② ❤❛✈❡ t♦ ❞♦ ✇✐t❤ ♣❡r✐♦❞✐❝✐t② ❛♥❞ r❡♣❡t✐t✐✈❡♥❡ss❄ ❘♦t❛t✐♦♥✳ ❊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❤ 1 ✐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❄

❚❤❡ ♠♦✈✐♥❣ ❡♥❞ ♦❢ t❤❡ r♦❞✱ ♦❢ ❝♦✉rs❡✱ tr❛❝❡s ♦✉t ❛

❝✐r❝❧❡ ✿

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

x = cos θ ❛♥❞ y = sin θ . ❇✉t ❞♦ t❤❡ ❢♦r♠✉❧❛s ❣✐✈❡ ✉s t❤❡ ✇❤♦❧❡ ❝✐r❝❧❡❄ ◆♦✳ ❇❡❝❛✉s❡ ✐❢ ✇❡ ❦❡❡♣ r♦t❛t✐♥❣ ❜❡②♦♥❞ 90 ❞❡❣r❡❡s✱ t❤❡r❡ ✐s ✕ ❛♥❞ t❤❡r❡ ❝❛♥ ❜❡ ✕ ♥♦ r✐❣❤t tr✐❛♥❣❧❡ ✇✐t❤ t❤✐s ❛♥❣❧❡✳ ❙♦✱ ✐♥ ❛ r✐❣❤t tr✐❛♥❣❧❡✱ ❛♥ ❛♥❣❧❡ ❝❛♥♥♦t ❣♦ ❜❡②♦♥❞ 90 ❞❡❣r❡❡s✳ ❚❤✐s ✐s♥✬t ❣♦♦❞ ❡♥♦✉❣❤ ❛♥②♠♦r❡✦ ❲❡ ♥❡❡❞ ❛ ♥❡✇ ✇❛② t♦ ❞❡✜♥❡ t❤❡ ❛♥❣❧❡ ❛♥❞ t❤❡ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s ♦❢ t❤✐s ❛♥❣❧❡✳ ❚❤✐s ✐s t❤❡ ✐❞❡❛✿

◮ ❚❤❡ ❛♥❣❧❡ ✐s♥✬t ❛♥ ❛♥❣❧❡ ♦❢ ❛ tr✐❛♥❣❧❡ ❛♥②♠♦r❡ ❜✉t t❤❡ ❛♥❣❧❡ ♦❢

r♦t❛t✐♦♥✳

❋✐rst✱ ✇❡ ♥❡❡❞ t♦ ❝❤♦♦s❡ ❛ ♥❛♠❡ ❢♦r t❤❡ ✐♥❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡ t❤❛t r❡♣r❡s❡♥ts t❤❡ ❛♥❣❧❡ ❛♥❞ r✉♥s t❤r♦✉❣❤♦✉t (−∞, +∞)✳ ❲❡ ❝❛❧❧ ✐t✱ ❛❣❛✐♥✱ x✳ ❲❡ t❤❡♥ ❛r❡ ❢♦r❝❡❞✱ s❡❝♦♥❞✱ t♦ ✉s❡ ❛❧t❡r♥❛t✐✈❡ ♥❛♠❡s ❢♦r t❤❡ ❛①❡s ♦❢ t❤❡ ❝♦♦r❞✐♥❛t❡ ♣❧❛♥❡ ✇❤❡r❡ t❤❡ ❝♦♥✲ str✉❝t✐♦♥ ✐s t♦ ❤❛♣♣❡♥❀ ✇❡ ❝❛❧❧ ✐t t❤❡ uv ✲♣❧❛♥❡✿

✹✳✶✵✳

❚❤❡ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s

✸✽✷

❚❤✐r❞✱ ✇❤❛t ✉♥✐ts ❞♦ ✇❡ ✉s❡ ❢♦r x❄ Pr❛❝t✐❝❛❧❧②✱ ❛♥② ✉♥✐t ♦❢ ❛♥❣❧❡ ✐s ❛❝❝❡♣t❛❜❧❡✿ ❞❡❣r❡❡s✱ ♠✐♥✉t❡s✱ ❡t❝✳ ■♥ ❢❛❝t✱ t❤❡ ❢✉♥❝t✐♦♥ t❤❡② ♣r♦❞✉❝❡ ✇✐❧❧ ❞✐✛❡r ♦♥❧② ❜② ❛ ❤♦r✐③♦♥t❛❧ str❡t❝❤ ✭❈❤❛♣t❡r ✸✮✳ ❍♦✇❡✈❡r✱ ❥✉st ❛s e ✐s t❤❡ ♠♦st ♥❛t✉r❛❧ ❝❤♦✐❝❡ ♦❢ t❤❡ ❜❛s❡ ♦❢ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥✱ t❤❡r❡ ✐s t❤❡ ❜❡st ❝❤♦✐❝❡ ❢♦r t❤❡ ✉♥✐t ♦❢ t❤❡ ❛♥❣❧❡ x✳ ❚❤❡ ❝❤♦✐❝❡ ✐s ❜❛s❡❞ ♦♥ t❤❡ ❢❛❝t ✭t♦ ❜❡ ❞❡♠♦♥str❛t❡❞ ✐♥ ❈❤❛♣t❡r ✸■❈✲✸✮ t❤❛t t❤❡ ❧❡♥❣t❤ ♦❢ t❤✐s ✉♥✐t ❝✐r❝❧❡ ✐s 2π ❛♥❞ t❤❡ ❧❡♥❣t❤ ♦❢ ✐ts ❛r❝ ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤✐s ♥✉♠❜❡r ❛♥❞ t❤❡ ❛♥❣❧❡ ✐t ✐s ❜❛s❡❞ ♦♥✳ ❚❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❛r❝ t❤❡♥ ❝❛♥ ❜❡ ✉s❡❞ t♦ ♠❡❛s✉r❡ t❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❛♥❣❧❡✳ ■♥ ❛❞❞✐t✐♦♥✱ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ y = sin x ❝r❡❛t❡❞ t❤✐s ✇❛② ✕ ❥✉st ❧✐❦❡ t❤❛t ♦❢ y = ex ✕ ❝r♦ss❡s t❤❡ y ✲❛①✐s ❛t 45 ❞❡❣r❡❡s ✭❈❤❛♣t❡r ✷❉❈✲✸✮✦ ❲❡ ❛❞♦♣t t❤❡ ❝♦♥✈❡♥t✐♦♥ t❤❛t t❤❡ s✐③❡ ♦❢ t❤❡ ❤❛❧❢✲t✉r♥ ❛♥❣❧❡ ✕ ❛♥❞ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❤❛❧❢✲❝✐r❝❧❡ ✕ ✐s ❡q✉❛❧ t♦ π r❛❞✐❛♥s✳ ❚❤❡♥✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥✿

❉❡✜♥✐t✐♦♥ ✹✳✶✵✳✺✿ r❛❞✐❛♥s 180 ❞❡❣r❡❡s = π r❛❞✐❛♥s✳ ❲❡✱ t❤❡r❡❢♦r❡✱ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥✈❡rs✐♦♥ ❢♦r♠✉❧❛s ❢♦r t❤❡s❡ ✉♥✐ts✿

# ❞❡❣r❡❡s = # ♦❢ r❛❞✐❛♥s · ❚❤❡ ♣♦s✐t✐✈❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ ❛♥❣❧❡ ✐s

θ x

180 π

❝♦✉♥t❡r❝❧♦❝❦✇✐s❡ ✿

❞❡❣r❡❡s✿ 0 90 180 270 360 ... r❛❞✐❛♥s✿ 0 π/2 π 3π/2 2π ...

■❢ t❤❡ ❞♦♠❛✐♥ ✐s♥✬t t❤❡ x✲❛①✐s✱ ❝❛♥ ✇❡ st✐❧❧ ✈✐s✉❛❧✐③❡ ✐t❄ ❖♥❡ ✇❛② t♦ s❡❡ t❤❡ x✬s ✐s ♦♥ t❤❡ ❝✐r❝❧❡ ✐ts❡❧❢✳ ❚♦ ❜❡❣✐♥ ✇✐t❤✱ ♣✐❡❝❡s ♦❢ t❤❡ ✐♥t❡r✈❛❧ [0, 2π] ❛r❡ ✇r❛♣♣❡❞ ❛r♦✉♥❞ t❤✐s ❝✐r❝❧❡✿

❍♦✇❡✈❡r✱ x ✐s st✐❧❧ t❤❡ ❛♥❣❧❡ ♦❢ r♦t❛t✐♦♥✿

✹✳✶✵✳

❚❤❡ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s

❚❤❡♥✱ ♦❢ ❝♦✉rs❡✱ ✐t ❝❛♥ ❣♦ ♣❛st

360

✸✽✸

❞❡❣r❡❡s✱ ❧✐❦❡ t❤✐s✿

❚❤❛t✬s ❝❧♦❝❦✇✐s❡❀ ✐t ❝❛♥ ❛❧s♦ ❣♦ ❝♦✉♥t❡r❝❧♦❝❦✇✐s❡✱ ✇✐t❤

❚❤❡

x✲❛①✐s

x

♥❡❣❛t✐✈❡✿

✐s t❤❡♥ s❡❡♥ ❛s ❛ s♣✐r❛❧ ✇r❛♣♣❡❞ ♦♥ t❤✐s ✉♥✐t ❝✐r❝❧❡✱ ❧✐❦❡ t❤✐s✿

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

♥❡❣❛t✐✈❡

❚♦ ❛❝❝♦♠♠♦❞❛t❡ ❛❧❧ ♣♦ss✐❜❧❡ ✐♥♣✉ts✱ ✇❡

t♦♦❀ ✇❡ ✉s❡ ❝♦♦r❞✐♥❛t❡s✱ ❛s ❢♦❧❧♦✇s✳

❉❡✜♥✐t✐♦♥ ✹✳✶✵✳✻✿ s✐♥❡ ❛♥❞ ❝♦s✐♥❡ ❢✉♥❝t✐♦♥s ❙✉♣♣♦s❡ ❛ r❡❛❧ ♥✉♠❜❡r

x

✐s ❣✐✈❡♥✳

❲❡ ❝♦♥str✉❝t ❛ ❧✐♥❡ s❡❣♠❡♥t ♦❢ ❧❡♥❣t❤

t❤❡ ❈❛rt❡s✐❛♥ ♣❧❛♥❡ ✭✇✐t❤ t❤❡ ❤♦r✐③♦♥t❛❧ ❛①✐s ❛♥❣❧❡

• •

x

♥♦t

♠❛r❦❡❞

x✮

st❛rt✐♥❣ ❛t

0

1

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

❚❤❡ ❚❤❡

❝♦s✐♥❡ s✐♥❡

♦❢

♦❢

x

x

✐s t❤❡ ❤♦r✐③♦♥t❛❧ ❝♦♦r❞✐♥❛t❡ ♦❢ t❤❡ ❡♥❞ ♦❢ t❤❡ s❡❣♠❡♥t✳

✐s t❤❡ ✈❡rt✐❝❛❧ ❝♦♦r❞✐♥❛t❡ ♦❢ t❤❡ ❡♥❞ ♦❢ t❤❡ s❡❣♠❡♥t✳

❚❤❡② ❛r❡ ❞❡♥♦t❡❞✱ r❡s♣❡❝t✐✈❡❧②✱ ❜②✿

cos x

❛♥❞

sin x

♦♥

✇✐t❤

✹✳✶✵✳

❚❤❡ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s

✸✽✹

■♥ ♦t❤❡r ✇♦r❞s✱ t❤✐s ✐s ❛ t✇♦✲st❡♣ ♣r♦❝❡❞✉r❡✿

x →

❛♥❣❧❡

→

❧♦❝❛t✐♦♥ ♦♥ uv ✲♣❧❛♥❡

→

u✲❝♦♦r❞✐♥❛t❡

→ y = cos x

x →

❛♥❣❧❡

→

❧♦❝❛t✐♦♥ ♦♥ uv ✲♣❧❛♥❡

→

v ✲❝♦♦r❞✐♥❛t❡

→ y = sin x

❚❤❡ ♦✉t❧✐♥❡ ♦❢ t❤❡ ❝♦♥str✉❝t✐♦♥ ✐s s❤♦✇♥ ❜❡❧♦✇✿

❲❛r♥✐♥❣✦ ❚❤❡ ✏ x✑ ✐♥

cos x ❛♥❞ sin x ❞♦❡s♥✬t r❡❢❡r t♦ t❤❡ x✲❛①✐s

♦❢ t❤❡ ♣❧❛♥❡ ✇❤❡r❡ t❤❡ ❝✐r❝❧❡ ✐s ♣❧♦tt❡❞✳

❲❡ ❝❛♥ ♥♦✇ ✜♥❞ t❤❡ ✈❛❧✉❡s ♦❢ y = sin x ❜② ❡①❛♠✐♥✐♥❣ t❤❡ ✈❛❧✉❡s ♦❢ x✱ ❛s ❛♥❣❧❡s✱ ♣✐❝t✉r❡❞ ❛❜♦✈❡✿

❲❡ ❣❡t ♠♦r❡ ❛♥❞ ♠❛❦❡ ❛ t❛❜❧❡✿ ✐♥♣✉t x ... −2π −3π/2 −π −π/2 0 π/2 π 3π/2 2π ... ❤♦r✐③♦♥t❛❧ ♦✉t♣✉t y ... 0 1 0 −1 0 1 0 −1 0 ... ✈❡rt✐❝❛❧

❲❡ ♣❧♦t t❤❡ ❣r❛♣❤ ♦❢ y = sin x ✕ ♣♦✐♥t ❜② ♣♦✐♥t ❢♦❧❧♦✇✐♥❣ t❤❡ t❛❜❧❡ ❛❜♦✈❡ ✕ ✇✐t❤ t❤❡ ❞♦♠❛✐♥ ♣r❡s❡♥t❡❞ ❛s t❤❡ x✲❛①✐s ♦♥ t❤❡ xy ✲♣❧❛♥❡✱ ❛s ✉s✉❛❧✿

❍♦✇ ❞♦ ✇❡ ✜❧❧ t❤❡ ❣❛♣s ✐♥ t❤❡ ❣r❛♣❤❄ ❆ ♠❡t❤♦❞ ♠❛② ❜❡ ❡♠♣❧♦②❡❞ t❤❛t ✐s s✐♠✐❧❛r t♦ t❤❡ ♦♥❡ ✇❡ ✉s❡❞ ❢♦r t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥✿

◮ ❲❡ ❞✐✈✐❞❡ ✐♥t❡r✈❛❧s ✐♥ ❤❛❧❢✿ ✐❢ ✇❡ ❦♥♦✇ t❤❡ ❢✉♥❝t✐♦♥ ❛t t✇♦ ♣♦✐♥ts✱ t❤❡r❡ ✐s ❛ ❢♦r♠✉❧❛ t♦ ✜♥❞ t❤❡ ✈❛❧✉❡ ✐♥ t❤❡ ♠✐❞❞❧❡✳

❊①❛♠♣❧❡ ✹✳✶✵✳✼✿ ✜❧❧✐♥❣ ❣❛♣s ✐♥ ❣r❛♣❤s ❚❤❡ tr✐❣ ❢♦r♠✉❧❛ ✇❡ ♥❡❡❞ ✐s ♣r♦✈❡♥ ✐♥ t❤❡ ♥❡①t ❝❤❛♣t❡r✿ r   1 − cos α cos β − sin α sin β α+β =± . sin 2 2

✹✳✶✵✳

❚❤❡ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s

✸✽✺

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

❋♦r ❡①❛♠♣❧❡✱

π/4

t❤❡ ❢♦r♠✉❧❛✿

✐s ❤❛❧❢ ✇❛② ❜❡t✇❡❡♥

0

❛♥❞

π/2✳

❲❡ ❥✉st s✉❜st✐t✉t❡ t❤❡s❡ ✈❛❧✉❡s ❢r♦♠ t❤❡ t❛❜❧❡ ✐♥t♦

  π 0 + π/2 sin = sin 4 2 r 1 − cos 0 cos π/2 − sin 0 sin π/2 = 2 r 1 − cos 0 cos π/2 − sin 0 sin π/2 = 2 r 1 = . 2

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

◆❡①t✱

π/8 ✐s ❤❛❧❢ ✇❛② ❜❡t✇❡❡♥ 0 ❛♥❞ π/4✳

❲❡ ❥✉st s✉❜st✐t✉t❡ t❤❡ ✈❛❧✉❡s ✇❡ ❥✉st ❢♦✉♥❞ ✐♥t♦ t❤❡ ❢♦r♠✉❧❛✿

π sin = sin 8



0 + π/4 2



= ...

❆♥❞ s♦ ♦♥✳

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

❚❤❡ 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✐♥❡✿

❊①❡r❝✐s❡ ✹✳✶✵✳✽ ❲❤❛t s✐♠✐❧❛r✐t✐❡s ❜❡t✇❡❡♥ t❤❡ t✇♦ ❣r❛♣❤s ❞♦ ②♦✉ s❡❡❄

❊①❡r❝✐s❡ ✹✳✶✵✳✾ ❉❡s❝r✐❜❡ t❤❡s❡ ❢✉♥❝t✐♦♥s ❛s tr❛♥s❢♦r♠❛t✐♦♥s✳

✹✳✶✵✳

❚❤❡ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s

✸✽✻

❉❡✜♥✐t✐♦♥ ✹✳✶✵✳✶✵✿ t❛♥❣❡♥t ❚❤❡

t❛♥❣❡♥t

❢✉♥❝t✐♦♥ ✐s ❞❡✜♥❡❞ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣✿ tan x =

sin x cos x

❲❛r♥✐♥❣✦ ❋♦❧❧♦✇✐♥❣ ♦✉r ❝♦♥✈❡♥t✐♦♥s ❛❜♦✉t ❢✉♥❝t✐♦♥s✱ ✇❡ ❝❛♥

sin(x)✱ cos(x)✱ ❛♥❞ tan(x)✱ ❜❡✲ sin✱ cos✱ ❛♥❞ tan ❛r❡ t❤❡ ♥❛♠❡s ♦❢ ❛♥❞ x ✐s t❤❡ ✐♥❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡✳

❛❧t❡r♥❛t✐✈❡❧② ✇r✐t❡✿ ❝❛✉s❡✱ ❛❢t❡r ❛❧❧✱ t❤❡ ❢✉♥❝t✐♦♥s

❚♦ ✜♥❞ t❤❡ ❞♦♠❛✐♥ ♦❢ ❛ ❢r❛❝t✐♦♥✱ ✇❡ s❡t t❤❡ ❞❡♥♦♠✐♥❛t♦r ❡q✉❛❧ t♦ 0 ❛♥❞ s♦❧✈❡✳ ❋✐♥❞ ❛❧❧ x✬s t❤❛t s❛t✐s❢②✿ cos x = 0 .

❚❤❡ s♦❧✉t✐♦♥ s❡t ♦❢ t❤✐s ❡q✉❛t✐♦♥ ✐s ✇❤❛t ✐s ❡①❝❧✉❞❡❞ ❢r♦♠ t❤❡ ❞♦♠❛✐♥✿ t❤❡ ♠✉❧t✐♣❧❡s ♦❢ π st❛rt✐♥❣ ❢r♦♠ π/2✳

❚❤❡♦r❡♠ ✹✳✶✵✳✶✶✿ ❉♦♠❛✐♥ ♦❢ ❚❛♥❣❡♥t ❚❤❡ ❞♦♠❛✐♥ ♦❢ t❤❡ t❛♥❣❡♥t✱

y = tan x✱

✐s t❤❡ ❢♦❧❧♦✇✐♥❣ s❡t✿

  π 3π 5π , , ... . x : x 6= ..., , 2 2 2

❚❤✐s ✐s ✐ts ❣r❛♣❤✿

❏✉st ❛s ✇✐t❤ t❤❡ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥s✱ t❤❡ ❤♦❧❡s ✐♥ t❤❡ ❞♦♠❛✐♥ ❝♦rr❡s♣♦♥❞ t♦ ✇❤❛t ❛♣♣❡❛rs t♦ ❜❡ ✈❡rt✐❝❛❧ ❛s②♠♣t♦t❡s✳ ▲❡t✬s ❛s❦ ❛♥❞ ❛♥s✇❡r s♦♠❡ ♦❧❞ q✉❡st✐♦♥s ❛❜♦✉t t❤❡s❡ ♥❡✇ ❢✉♥❝t✐♦♥s✳ ❲❡ t❛❦❡ ❛♥♦t❤❡r ❧♦♦❦ ❛t t❤❡ ❣r❛♣❤s ♦❢ t❤❡ t✇♦ ❢✉♥❝t✐♦♥s✿

❚❤❡② ✜t ✐♥s✐❞❡ ❛ ❤♦r✐③♦♥t❛❧ ❜❛♥❞✦ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ✇✐❧❧ r❡✲❛♣♣❡❛r ♠❛♥② t✐♠❡s ✐♥ ♦✉r st✉❞②✳

✹✳✶✵✳

❚❤❡ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s

✸✽✼

❚❤❡♦r❡♠ ✹✳✶✵✳✶✷✿ ❇♦✉♥❞❡❞♥❡ss ♦❢ ❚r✐❣♦♥♦♠❡tr✐❝ ❋✉♥❝t✐♦♥s

❚❤❡ s✐♥❡ ❛♥❞ t❤❡ ❝♦s✐♥❡ ❛r❡ ❜♦✉♥❞❡❞ ❢✉♥❝t✐♦♥s ✇✐t❤ t❤❡s❡ ❜♦✉♥❞s✿ −1 ≤ sin x ≤ 1 −1 ≤ cos x ≤ 1

s❛t✐s✜❡❞ ❢♦r ❡❛❝❤ x✱ ✇❤✐❧❡ t❤❡ t❛♥❣❡♥t ✐s ✉♥❜♦✉♥❞❡❞✳ ❊①❡r❝✐s❡ ✹✳✶✵✳✶✸ Pr♦✈❡ t❤❡ ❧❛st ♣❛rt✳

❊①❡r❝✐s❡ ✹✳✶✵✳✶✹ ❲❤❛t ❛r❡ t❤❡ ✐♠❛❣❡s ♦❢ t❤❡s❡ ❢✉♥❝t✐♦♥s❄ ❲❤❛t ❛r❡ t❤❡ ♣r❡✐♠❛❣❡s❄

❆❢t❡r ❛ ❢✉❧❧ t✉r♥✱ ✇❡ ❛rr✐✈❡ t♦ t❤❡ s❛♠❡ s♣♦t✱ ♥♦ ♠❛tt❡r ✇❤❡r❡ ✇❡ st❛rt✿

❇✉t t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s ❛r❡ ❞❡t❡r♠✐♥❡❞ ❜② t❤✐s ❧♦❝❛t✐♦♥ ♦♥❧②❀ t❤❛t✬s

2π ✲♣❡r✐♦❞✐❝✐t②✳

❲❡ ✇✐❧❧ r❡❧② ♦♥ t❤❡ ❢♦❧❧♦✇✐♥❣✿

❚❤❡♦r❡♠ ✹✳✶✵✳✶✺✿ P❡r✐♦❞✐❝✐t② ♦❢ ❚r✐❣♦♥♦♠❡tr✐❝ ❋✉♥❝t✐♦♥s

❚❤❡ ❢✉♥❝t✐♦♥s sin ❛♥❞ cos ❛r❡ ♣❡r✐♦❞✐❝ ✇✐t❤ ♣❡r✐♦❞ 2π ✱ ✇❤✐❧❡ t❤❡ ❢✉♥❝t✐♦♥ tan ✐s ♣❡r✐♦❞✐❝ ✇✐t❤ ♣❡r✐♦❞ π ❀ ✐✳❡✳✱ sin(x + 2π) = sin x cos(x + 2π) = cos x tan(x + π) = tan x

s❛t✐s✜❡❞ ❢♦r ❡❛❝❤ x✳ ❊①❡r❝✐s❡ ✹✳✶✵✳✶✻ Pr♦✈❡ t❤❡ ❧❛st ♣❛rt✳

❖♥❡ ❝❛♥ s❡❡ t❤❡ ♣❡r✐♦❞✐❝✐t② ✐♥ t❤❡ ❣r❛♣❤s✳ ❚❤❡r❡ ❛r❡ t✇♦ ✇❛②s✳ ❋✐rst ✐s t❤❡ s❤✐❢t✿

❚❤❡ s❡❝♦♥❞ ✐s ❝♦♣②✲❛♥❞✲♣❛st❡✿

✹✳✶✵✳

❚❤❡ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s

✸✽✽

◆♦✇✱ ✇❤❛t ❛❜♦✉t t❤❡ s②♠♠❡tr✐❡s ❄ ▲❡✬s r♦t❛t❡ ♦✉r r♦❞ t❤❡ s❛♠❡ ❛♠♦✉♥t✱ x✱ ❝❧♦❝❦✇✐s❡ ❛♥❞ ❝♦✉♥t❡r✲❝❧♦❝❦✇✐s❡ ❛♥❞ ❝♦♠♣❛r❡✿

❚❤❡ ❡✛❡❝t ✐s ❞✐✛❡r❡♥t ❢♦r t❤❡ t✇♦ ❢✉♥❝t✐♦♥s✿ ✶✳ ■t ♠❛❦❡s t❤❡ ✈❡rt✐❝❛❧ ♣r♦❣r❡ss✱ sin x✱ ♥❡❣❛t✐✈❡✳ ✷✳ ■t ❞♦❡s♥✬t ❝❤❛♥❣❡ t❤❡ ❤♦r✐③♦♥t❛❧ ♣r♦❣r❡ss✱ cos x✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ✐s ❛ ❝♦♥✈❡♥✐❡♥t ♦❜s❡r✈❛t✐♦♥✳ ❚❤❡♦r❡♠ ✹✳✶✵✳✶✼✿ ❖❞❞✲❊✈❡♥ ❚r✐❣ ❋✉♥❝t✐♦♥s

❚❤❡ s✐♥❡ ❛♥❞ t❤❡ t❛♥❣❡♥t ❛r❡ ♦❞❞ ❢✉♥❝t✐♦♥s✱ ✇❤✐❧❡ t❤❡ ❝♦s✐♥❡ ✐s ❡✈❡♥✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿ sin(−x) = − sin x cos x cos(−x) = tan(−x) = − tan x

s❛t✐s✜❡❞ ❢♦r ❡❛❝❤ x✳ ❊①❡r❝✐s❡ ✹✳✶✵✳✶✽

Pr♦✈❡ t❤❡ ♣❛rt ❛❜♦✉t t❛♥❣❡♥t✳ ❚❤❡ ❣r❛♣❤s ❞♦ ❡①❤✐❜✐t t❤❡s❡ s②♠♠❡tr✐❡s✿

✹✳✶✵✳

❚❤❡ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s

✸✽✾

❲❡ ❝❛♥ s❡❡ t❤✐s ❢❛❝t ❛❜♦✉t t❤❡ s✐♥❡ ✈✐❛ ❝♦♣②✲❛♥❞✲♣❛st❡✿

■♥ ❢❛❝t✱ ✇❤❡♥ ③♦♦♠❡❞ ✐♥ ♦♥ t❤❡ y ✲✐♥t❡r❝❡♣t✱ t❤❡ ❣r❛♣❤ ♦❢ y = sin x ❧♦♦❦s ❧✐❦❡ y = x ❛♥❞ y = cos x ❧✐❦❡ y = 1✦ ❊①❡r❝✐s❡ ✹✳✶✵✳✶✾

❲❤❛t ♣❛r❛❜♦❧❛ ❞♦❡s t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❝♦s✐♥❡ ❧♦♦❦ ❧✐❦❡ ❛r♦✉♥❞ t❤❡ y ✲✐♥t❡r❝❡♣t❄ ❊①❡r❝✐s❡ ✹✳✶✵✳✷✵

❲❤❛t ❞♦❡s t❤❡ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ t❛♥❣❡♥t ❧♦♦❦ ❧✐❦❡ ✐❢ ✇❡ ③♦♦♠ ✐♥ ♦♥ ✐t ❛r♦✉♥❞ t❤❡ ♦r✐❣✐♥❄ ❚❤✐s ✐s t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t

∆f ✱ ✐✳❡✳✱ t❤❡ s❛♠♣❧❡❞ s❧♦♣❡s✱ ♦❢ sin x ✭❜♦tt♦♠ r♦✇✮✿ ∆x

❊①❡r❝✐s❡ ✹✳✶✵✳✷✶

❲❤❛t ♣❛tt❡r♥ ❞♦ t❤❡ s❧♦♣❡s ❡①❤✐❜✐t❄ ❊①❡r❝✐s❡ ✹✳✶✵✳✷✷

P❧♦t t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❢♦r ❝♦s✐♥❡ ❛♥❞ ❛♥s✇❡r t❤❡ s❛♠❡ q✉❡st✐♦♥✳ ◆♦♥❡ ♦❢ t❤❡s❡ ❢✉♥❝t✐♦♥s ✐s ♦♥❡✲t♦✲♦♥❡✳ ◆♦♥❡✱ t❤❡r❡❢♦r❡✱ ❝❛♥ ❤❛✈❡ ❛♥ ✐♥✈❡rs❡✳

√

❍♦✇❡✈❡r✱ ✇❤❛t ✇❡ ❞✐❞ ✇✐t❤ y = x2 ✐♥ ♦r❞❡r t♦ ❣❡t x = y ❝❛♥ ❜❡ r❡♣❡❛t❡❞ ❤❡r❡✿ ◮ ❘❡str✐❝t t❤❡ ❞♦♠❛✐♥s ♦❢ t❤❡ ❢✉♥❝t✐♦♥ t♦ ♠❛❦❡ ✐t ♦♥❡✲t♦✲♦♥❡✳

❚❤❡ ❝♦❞♦♠❛✐♥ ✐s ❛❧s♦ r❡str✐❝t❡❞ s♦ t❤❛t t❤❡ ❢✉♥❝t✐♦♥ ✐s ❛❧s♦ ♦♥t♦✳

✹✳✶✵✳

❚❤❡ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s

✸✾✵

❚❤❡♦r❡♠ ✹✳✶✵✳✷✸✿ ❘❡str✐❝t❡❞ ❙✐♥❡ • •

❲✐t❤ t❤❡ ❞♦♠❛✐♥ r❡str✐❝t❡❞ t♦

[−π/2, π/2]✱

t❤❡ s✐♥❡✱

y = sin x✱

✐s ❛ ♦♥❡✲t♦✲

y = sin x✱

✐s ❛♥ ♦♥t♦

♦♥❡ ❢✉♥❝t✐♦♥✳ ❲✐t❤ t❤❡ ❝♦❞♦♠❛✐♥ r❡str✐❝t❡❞ t♦

[−1, 1]✱

t❤❡ s✐♥❡✱

❢✉♥❝t✐♦♥✳

❲❤② t❤✐s ✐♥t❡r✈❛❧ ❢♦r t❤❡ ❞♦♠❛✐♥❄ ❋✐rst✱ ✐t ✐s✱ ✐♥ ❛ s❡♥s❡✱ t❤❡ ❧❛r❣❡st ♣♦ss✐❜❧❡✿ ✇❡ ❝❛♥✬t ❡①t❡♥❞ t❤❡ ✐♥t❡r✈❛❧ ✕ t♦ t❤❡ ❧❡❢t ♦r t♦ t❤❡ r✐❣❤t ✕ ✇✐t❤♦✉t ♠❛❦✐♥❣ t❤❡ ❢✉♥❝t✐♦♥ ♥♦t ♦♥❡✲t♦✲♦♥❡✳ ❇✉t✱ s❡❝♦♥❞✱ ✇❤❡♥ r❡str✐❝t❡❞ t❤✐s ✇❛②✱ t❤❡ s✐♥❡ ✐s st✐❧❧ ❛♥ ♦❞❞ ❢✉♥❝t✐♦♥✳ ❆♥❞ ♥♦✇ s♦ ✐s ✐ts ✐♥✈❡rs❡✳

❉❡✜♥✐t✐♦♥ ✹✳✶✵✳✷✹✿ ❛r❝s✐♥❡

❚❤❡ ❛r❝s✐♥❡ ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ s✐♥❡ ❢✉♥❝t✐♦♥ r❡str✐❝t❡❞ t♦ [−π/2, π/2]✱ ❞❡♥♦t❡❞ ❜② ❡✐t❤❡r✿

arcsin y = sin−1 y

❲❛r♥✐♥❣✦ ❏✉st ❛s t❤❡r❡ ✐s ♥♦ ✏❢♦r♠✉❧❛✑ ❢♦r t❤❡ ❧♦❣❛r✐t❤♠ ♦r ❢♦r t❤❡ sq✉❛r❡ r♦♦t✱ t❤❡r❡ ✐s ♥♦♥❡ ❢♦r

❊①❡r❝✐s❡ ✹✳✶✵✳✷✺ ❙❤♦✇ t❤❛t t❤❡② ❛r❡ ❜♦t❤

arcsin✳

✐♥❝r❡❛s✐♥❣✳

❚❤✉s✱ ✇❡ ❤❛✈❡ ❛ ♣❛✐r ♦❢ ✐♥✈❡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 x ❛♥❞ y ✐♥t❡r❝❤❛♥❣❡❞✿

❊①❡r❝✐s❡ ✹✳✶✵✳✷✻ Pr♦✈❡ t❤❛t arcsin ✐s ♦❞❞✳ ❙✐♠✐❧❛r❧②✱ ✇❡ ❝❤♦♦s❡ t❤❡ t♦ r❡str✐❝t t❤❡ ❞♦♠❛✐♥ ♦❢ t❤❡ ♥❡✇ ❝♦s✐♥❡ t♦ t❤❡ ✐♥t❡r✈❛❧ [0, π]✳ ❲❤② t❤✐s ✐♥t❡r✈❛❧❄ ❋✐rst✱ ✇❡ ❝❛♥✬t ❡①t❡♥❞ ❜❡②♦♥❞ t❤✐s ✐♥t❡r✈❛❧ ❜❡❝❛✉s❡ t❤❛t ✇♦✉❧❞ ♠❛❦❡ t❤❡ ❢✉♥❝t✐♦♥ ♥♦t ♦♥❡✲t♦✲♦♥❡✳ ❇✉t s❡❝♦♥❞✱ ✇❡ t❛❦❡ t❤❡ ❤❛❧❢ ♦❢ t❤❡ ❣r❛♣❤ t❤❛t ✐s r❡♣❡❛t❡❞ ♦♥ t❤❡ ♦t❤❡r s✐❞❡ ♦❢ t❤❡ y ✲❛①✐s ❜❡❝❛✉s❡ t❤❡ ❝♦s✐♥❡ ✐s ❡✈❡♥✳

✹✳✶✵✳

❚❤❡ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s

✸✾✶

❚❤❡♦r❡♠ ✹✳✶✵✳✷✼✿ ❘❡str✐❝t❡❞ ❈♦s✐♥❡ • •

❲✐t❤ t❤❡ ❞♦♠❛✐♥ r❡str✐❝t❡❞ t♦

[0, π]✱

t❤❡ ❝♦s✐♥❡✱

y = cos x✱

✐s ❛ ♦♥❡✲t♦✲♦♥❡

❢✉♥❝t✐♦♥✳ ❲✐t❤ t❤❡ ❝♦❞♦♠❛✐♥ r❡str✐❝t❡❞ t♦

[−1, 1]✱

t❤❡ ❝♦s✐♥❡✱

y = cos x✱

✐s ❛♥ ♦♥t♦

❢✉♥❝t✐♦♥✳

❉❡✜♥✐t✐♦♥ ✹✳✶✵✳✷✽✿ ❛r❝❝♦s✐♥❡ ❚❤❡ ❛r❝❝♦s✐♥❡ ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ ❝♦s✐♥❡ ❢✉♥❝t✐♦♥ ♦♥ [0, π]✱ ❞❡♥♦t❡❞ ❜② ❡✐t❤❡r✿ arccos y = cos−1 y

❚❤✉s✱ ✇❡ ❤❛✈❡ ❛ ♣❛✐r ♦❢ ✐♥✈❡rs❡ ❢✉♥❝t✐♦♥s✿ y = cos x x = cos−1 y ❞♦♠❛✐♥✿ [0, π] ❞♦♠❛✐♥✿ [−1, 1] r❛♥❣❡✿ [−1, 1] r❛♥❣❡✿ [0, π]

❚❤❡ ❣r❛♣❤s ❛r❡✱ ♦❢ ❝♦✉rs❡✱ t❤❡ s❛♠❡ ✇✐t❤ ❥✉st x ❛♥❞ y ✐♥t❡r❝❤❛♥❣❡❞✿

❊①❡r❝✐s❡ ✹✳✶✵✳✷✾ ❙❤♦✇ t❤❛t ❜♦t❤ s✐♥❡ ❛♥❞ ❝♦s✐♥❡ ❛r❡ ♦♥❡✲t♦✲♦♥❡ ♦♥ ❛♥② ✐♥t❡r✈❛❧ ♦❢ ❧❡♥❣t❤ π ✳

❊①❛♠♣❧❡ ✹✳✶✵✳✸✵✿ ❛♥❣❧❡ ♦❢ s✉♥ ❆r♠❡❞ ♦♥❧② ✇✐t❤ t❤✐s ❦♥♦✇❧❡❞❣❡ ❛♥❞ ❛ ❝❛❧❝✉❧❛t♦r✱ ✇❡ ❝❛♥ tr② t♦ s♦❧✈❡ s♦♠❡ ♣r❛❝t✐❝❛❧ ♣r♦❜❧❡♠s✳ ❋♦r ❡①❛♠♣❧❡✱ ❤♦✇ ❤✐❣❤ ✐s t❤❡ s✉♥ ❛❜♦✈❡ t❤❡ ❤♦r✐③♦♥ ✇❤❡♥ ❛ 1✲✐♥❝❤ st✐❝❦ ❝❛sts ❛ s❤❛❞♦✇ .5 ✐♥❝❤ ❧♦♥❣❄

❲❡ ♠❡❛s✉r❡ t❤❡ ❤❡✐❣❤t ♦❢ t❤❡ s✉♥ ✐♥ t❡r♠s ♦❢ t❤❡ ❛♥❣❧❡ x t❤❡ s✉♥❧✐❣❤t ❤✐ts t❤❡ ❣r♦✉♥❞✳ ❚❤❡♥✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿ ❚❤✐s ✐s✱ ✐♥ ❢❛❝t✱ 60 ❞❡❣r❡❡s✳

cos x = 1/2 =⇒ x = arccos(1/2) ≈ 1.05 .

❚❤❡ t❛♥❣❡♥t ❢✉♥❝t✐♦♥ ✐s π ✲♣❡r✐♦❞✐❝ ❛♥❞✱ t❤❡r❡❢♦r❡✱ ♥♦t ♦♥❡✲t♦✲♦♥❡✳ ❚♦ ❜✉✐❧❞ ✐ts ✐♥✈❡rs❡✱ ✇❡✱ ❛❣❛✐♥✱ r❡str✐❝t t❤❡ ❢✉♥❝t✐♦♥✬s ❞♦♠❛✐♥✳ ❲❡ s✐♠♣❧② ❝❤♦♦s❡ t❤❡ ❜r❛♥❝❤ ♦❢ tan ♦✈❡r −π/2 < x < π/2✳

✹✳✶✵✳

❚❤❡ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s

✸✾✷

❚❤❡♦r❡♠ ✹✳✶✵✳✸✶✿ ❘❡str✐❝t❡❞ ❚❛♥❣❡♥t ❲✐t❤ t❤❡ ❞♦♠❛✐♥ r❡str✐❝t❡❞ t♦

(−π/2, π/2)✱

t❤❡ t❛♥❣❡♥t✱

y = tan x✱

✐s ❛ ♦♥❡✲t♦✲

♦♥❡ ❢✉♥❝t✐♦♥✳

■t✬s st✐❧❧ ♦❞❞ ❛♥❞ s♦ ✐s ✐ts ✐♥✈❡rs❡✳

❉❡✜♥✐t✐♦♥ ✹✳✶✵✳✸✷✿ ❛r❝t❛♥❣❡♥t ❚❤❡

❛r❝t❛♥❣❡♥t ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ t❛♥❣❡♥t ❢✉♥❝t✐♦♥ r❡str✐❝t❡❞ t♦

t❤❡ ❞♦♠❛✐♥

(−π/2, π/2)✱

❞❡♥♦t❡❞ ❜② ❡✐t❤❡r✿

arctan y = tan−1 y

❚❤❡♥✱ ✇❡ ❛❣❛✐♥ ❤❛✈❡ ❛ ♣❛✐r ♦❢ ✐♥✈❡rs❡ ❢✉♥❝t✐♦♥s✿

y = tan x x = tan−1 y ❞♦♠❛✐♥✿ (−π/2, π/2) ❞♦♠❛✐♥✿ (−∞, +∞) r❛♥❣❡✿ (−∞, +∞) r❛♥❣❡✿ (−π/2, π/2) ❚❤❡ ❣r❛♣❤s ❛r❡✱ ♦❢ ❝♦✉rs❡✱ t❤❡ s❛♠❡ ✇✐t❤ ❥✉st

x

❛♥❞

y

✐♥t❡r❝❤❛♥❣❡❞✿

tr✐❣♦♥♦♠❡tr✐❝ ❡q✉❛t✐♦♥s✳

❲❡ ✉s❡ t❤❡s❡ ❢✉♥❝t✐♦♥s t♦ s♦❧✈❡

❊①❛♠♣❧❡ ✹✳✶✵✳✸✸✿ s♦❧✈✐♥❣ tr✐❣ ❡q✉❛t✐♦♥s ❙♦❧✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r

x✿ sin(x − 5) = .3 .

❚♦ ✏❦✐❧❧✑ t❤❡ s✐♥❡ ❛♥❞ ❣❡t t♦

x✱

❲❡ ❝❛♥❝❡❧ ❛s ❜❡❢♦r❡ ❛♥❞ ✜♥✐s❤✿

❛♣♣❧② ✐ts

✐♥✈❡rs❡✱ t❤❡ ❛r❝s✐♥❡✱ t♦ ❜♦t❤ s✐❞❡s ♦❢ t❤❡ ❡q✉❛t✐♦♥✿


arcsin sin(x − 5) = arcsin(.3) . x − 5 = arcsin .3 =⇒ x = arcsin .3 + 5 .

❊①❡r❝✐s❡ ✹✳✶✵✳✸✹ ❍❛✈❡ ✇❡ ❢♦✉♥❞ ❛❧❧ s♦❧✉t✐♦♥s❄

❇❡❧♦✇✱ ✇❡ s✉♠♠❛r✐③❡ ❤♦✇✱ ❤②♣♦t❤❡t✐❝❛❧❧②✱ t❤❡s❡ ❝❧❛ss❡s ♦❢ ❢✉♥❝t✐♦♥s ❝♦✉❧❞ ❤❛✈❡ ❛♣♣❡❛r❡❞✿

✹✳✶✵✳ ❚❤❡ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s

✸✾✸

❍✐st♦r② ♦❢ ❢✉♥❝t✐♦♥s P❤❡♥♦♠❡♥❛ ❘❡q✉✐r❡♠❡♥ts ❢♦r ♥❡✇ ❢✉♥❝t✐♦♥s r♦❧❧✐♥❣ ❜❛❧❧

t❤r♦✇♥ ❜❛❧❧

❣r❛✈✐t② ✐♥ s♣❛❝❡

♣♦♣✉❧❛t✐♦♥ ❣r♦✇t❤✴❞❡❝❧✐♥❡✱ ❝♦♦❧✐♥❣✴❤❡❛t✐♥❣ ✇❛✈❡s✱ ♣❧❛♥❡t❛r② ♠♦t✐♦♥

♥❡✇✱ ✉♥❦♥♦✇♥ ♣❤❡♥♦♠❡♥❛

■♥✈❡rs❡s

◆❡❡❞s t♦ ✈❛r②✳

−→

❧✐♥❡❛r ❢✉♥❝t✐♦♥s

−→

♣♦❧②♥♦♠✐❛❧s

−→

r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥s

−→

❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥s

−→

tr✐❣ ❢✉♥❝t✐♦♥s

−→

❛❜str❛❝t ❢✉♥❝t✐♦♥s

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

−→

❧✐♥❡❛r ❢✉♥❝t✐♦♥s

−→

✏❛❧❣❡❜r❛✐❝ ❢✉♥❝t✐♦♥s✑

−→

✏❛❧❣❡❜r❛✐❝ ❢✉♥❝t✐♦♥s✑

−→

❧♦❣❛r✐t❤♠s

−→

✐♥✈❡rs❡ tr✐❣ ❢✉♥❝t✐♦♥s

❈❤❛♣t❡r ✺✿ ❆❧❣❡❜r❛ ❛♥❞ ❣❡♦♠❡tr②

❈♦♥t❡♥ts ✺✳✶ ❚❤❡ ❛r✐t❤♠❡t✐❝ ♦♣❡r❛t✐♦♥s ♦♥ ❢✉♥❝t✐♦♥s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺✳✷ ❚❤❡ ❛❧❣❡❜r❛ ♦❢ ❝♦♠♣♦s✐t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✳✸ ❙♦❧✈✐♥❣ ❡q✉❛t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✳✹ ❚❤❡ ❛❧❣❡❜r❛ ♦❢ ❧♦❣❛r✐t❤♠s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✳✺ ❚❤❡ ❈❛rt❡s✐❛♥ s②st❡♠ ❢♦r t❤❡ ❊✉❝❧✐❞❡❛♥ ♣❧❛♥❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✳✻ ❚❤❡ ❊✉❝❧✐❞❡❛♥ ♣❧❛♥❡✿ ❞✐st❛♥❝❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✳✼ ❚r✐❣♦♥♦♠❡tr② ❛♥❞ t❤❡ ✇❛✈❡ ❢✉♥❝t✐♦♥ ✺✳✽ ❚❤❡ ❊✉❝❧✐❞❡❛♥ ♣❧❛♥❡✿ ❛♥❣❧❡s ✺✳✾ ❋r♦♠ ❣❡♦♠❡tr② t♦ ❝❛❧❝✉❧✉s ✺✳✶✵ ❙♦❧✈✐♥❣ ✐♥❡q✉❛❧✐t✐❡s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

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

✺✳✶✳ ❚❤❡ ❛r✐t❤♠❡t✐❝ ♦♣❡r❛t✐♦♥s ♦♥ ❢✉♥❝t✐♦♥s

❲❡ ✇♦✉❧❞ ❧✐❦❡ t♦ tr❡❛t ❛❧❧ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s ❛s ❛ s✐♥❣❧❡ ❣r♦✉♣✳ ❲❡ ✜♥❞ ✐♥s♣✐r❛t✐♦♥ ✐♥ ❤♦✇ ✇❡ ❤❛✈❡ ❤❛♥❞❧❡❞ t❤❡ r❡❛❧ ♥✉♠❜❡rs✳ ❲❡ ♣✉t t❤❡♠ t♦❣❡t❤❡r ✐♥ t❤❡ r❡❛❧ ♥✉♠❜❡r ❧✐♥❡✱ ✇❤✐❝❤ ♣r♦✈✐❞❡s ✉s ✇✐t❤ ❛ ❜✐r❞✬s✲❡②❡ ✈✐❡✇✿

❲❡ ❛❧s♦ r❡❝♦❣♥✐③❡ t❤❛t t❤❡s❡ ❡♥t✐t✐❡s ❛r❡ ✐♥t❡r❛❝t✐♥❣ ✇✐t❤ ❡❛❝❤ ♦t❤❡r✱ ♣r♦❞✉❝✐♥❣ ♦✛s♣r✐♥❣ ✈✐❛ ❛r✐t❤♠❡t✐❝✿ 3 + 6 = 9, 5 · 7 = 35, ❡t❝✳

❯♥❞❡rst❛♥❞✐♥❣ t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡s❡ ❝♦♠♣✉t❛t✐♦♥s r❡q✉✐r❡s ✉♥❞❡rst❛♥❞✐♥❣ t❤❛t t❤❡ ❜❡❣✐♥♥✐♥❣ ❛♥❞ t❤❡ ❡♥❞ ♦❢ s✉❝❤ ❛ ❝♦♠♣✉t❛t✐♦♥ ❛r❡ ❥✉st t✇♦ ❞✐✛❡r❡♥t r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ t❤❡ s❛♠❡ ♥✉♠❜❡r✿ 1 + 1 = 2 · 1 = 2.

❚❤❡r❡ ✐s ❛ s✐♥❣❧❡ ❧♦❝❛t✐♦♥ ❢♦r ❡❛❝❤ ♦❢ t❤❡s❡ ❡①♣r❡ss✐♦♥s ♦♥ t❤❡ r❡❛❧ ♥✉♠❜❡r ❧✐♥❡✦ ❙✐♠✐❧❛r❧②✱ x + x ❛♥❞ 2x ❝♦rr❡s♣♦♥❞ t♦ t❤❡ s❛♠❡ ✭❛❧❜❡✐t ✉♥s♣❡❝✐✜❡❞✮ ❧♦❝❛t✐♦♥ ♦♥ t❤❡ ♥✉♠❜❡r ❧✐♥❡✳ ❲❡ s✐♠♣❧② s❛② t❤❛t t❤❡② ❛r❡ ❡q✉❛❧✳ ■t ✐s ♠✉❝❤ ♠♦r❡ ❝❤❛❧❧❡♥❣✐♥❣ t♦ ✜♥❞ s✉❝❤ ❛ ❜✐r❞✬s✲❡②❡ ✈✐❡✇ ❢♦r ❢✉♥❝t✐♦♥s ✦ ❋♦r ❡①❛♠♣❧❡✱ t❤✐s ✐s ✇❤❛t ❛♥ ❛tt❡♠♣t t♦ ✈✐s✉❛❧✐③❡ ❛❧❧ ❧✐♥❡❛r ❢✉♥❝t✐♦♥s ✇♦✉❧❞ ❧♦♦❦ ❧✐❦❡✿

✺✳✶✳

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

✸✾✺

✐♥t❡r❛❝t✐♦♥s ❜❡t✇❡❡♥ ❢✉♥❝t✐♦♥s t❤❛t ♠❛❦❡ t❤❡♠ ♠❛♥❛❣❡❛❜❧❡ ❛s ❛ ✇❤♦❧❡✳ ❋♦r ❡❛❝❤ ♦❢ t❤❡ ❢♦✉r ❛r✐t❤♠❡t✐❝ ♦♣❡r❛t✐♦♥s ♦♥ ♥✉♠❜❡rs ✕ ❛❞❞✐t✐♦♥✱ s✉❜tr❛❝t✐♦♥✱ ♠✉❧t✐♣❧✐❝❛t✐♦♥✱ ❛♥❞ ❞✐✈✐s✐♦♥ ✕ t❤❡r❡ ✐s ❛♥ ♦♣❡r❛t✐♦♥ ♦♥ ✭♥✉♠❡r✐❝❛❧✮ ❢✉♥❝t✐♦♥s✳

❏✉st ❛s ✇✐t❤ ♥✉♠❜❡rs✱ ✐t ✐s

❇✉t ✜rst ❧❡t✬s ♠❛❦❡ s✉r❡ t❤❛t ✇❡ ❤❛✈❡ ❛ ❝❧❡❛r ✉♥❞❡rst❛♥❞✐♥❣ ♦❢ ✇❤❛t ✐t ♠❡❛♥s ❢♦r t✇♦ ❢✉♥❝t✐♦♥s t♦ ❜❡ t❤❡ s❛♠❡✳ ❋♦r ❡①❛♠♣❧❡✱ t❤❡s❡ t✇♦ ❢✉♥❝t✐♦♥s ❛r❡ r❡♣r❡s❡♥t❡❞ ❜② t✇♦ ❞✐✛❡r❡♥t ❢♦r♠✉❧❛s✿

x + x ❛♥❞ 2x . ❆r❡ t❤❡② t❤❡ s❛♠❡ ❢✉♥❝t✐♦♥❄ ❖❢ ❝♦✉rs❡✦ ❚❤❡s❡ t✇♦ ❢✉♥❝t✐♦♥s ❛r❡ r❡♣r❡s❡♥t❡❞ ❜② t✇♦ s✐♠✐❧❛r ❢♦r♠✉❧❛s✿

x − 1 ❛♥❞ 1 − x . ❆r❡ t❤❡② t❤❡ s❛♠❡ ❢✉♥❝t✐♦♥❄ ❖❢ ❝♦✉rs❡ ♥♦t✦ ❍♦✇ ❞♦ ✇❡ ❦♥♦✇❄ ❚❤❡ ❛♥s✇❡r ✭❛s ✐s t❤❡ q✉❡st✐♦♥ ✐ts❡❧❢✮ ✐s ❞❡♣❡♥❞❡♥t ♦♥ ♦✉r ❞❡✜♥✐t✐♦♥ ♦❢ ❢✉♥❝t✐♦♥✿

❆ ❢✉♥❝t✐♦♥ ✐s ❛ ❧✐st ♦❢ ✐♥♣✉ts ❛♥❞ ♦✉t♣✉ts✳ ❚♦ ❛♥s✇❡r t❤❡ q✉❡st✐♦♥✱ ✇❡ ❝❛♥ s✐♠♣❧② t❡st t❤❡ ❢♦r♠✉❧❛s ❜② ♣❧✉❣❣✐♥❣ ✐♥♣✉t ✈❛❧✉❡s ❛♥❞ ✇❛t❝❤✐♥❣ t❤❡ ♦✉t♣✉ts✿ =0 s❛♠❡✦ = 0, 2x x + x x=0 x=0 = 2 s❛♠❡✦ = 2, x + x x + x x=1

x=1

...

...

???

■t s❡❡♠s t❤❡ s❛♠❡✳ ◆♦✇ t❤❡ s❡❝♦♥❞ ♣❛✐r✿ = 1 ❞✐✛❡r❡♥t✦ = −1, 1 − x x − 1 x=0 x=0 x − 1 = 0, 1 − x = 0 s❛♠❡✦ x=1

x=1

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

2x2 + 2x ❛♥❞ x2 + x , 2

♦r t❤♦s❡✿

2x2 + 2x ❛♥❞ 2x + 2 ? x

❲❡ ❛❣❛✐♥ ♣❧✉❣ ✐♥ t❤❡ ✈❛❧✉❡s✿

2x2 + 2x 2 = 0 s❛♠❡✦ = 0, x + x 2 x=0 x=0 2x2 + 2x 2 = 2, x + x = 2 s❛♠❡✦ 2 x=1 x=1

...

...

???

✺✳✶✳

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

✸✾✻

❚❤❡ r❡s✉❧ts ❛r❡ t❤❡ s❛♠❡ ❢♦r ❡✈❡r② x✦ ❲❤❛t ❛❜♦✉t t❤❡ ❧❛tt❡r❄ ■t ❜r❡❛❦s ❞♦✇♥✿ 2x2 + 2x ✉♥❞❡✜♥❡❞ , 2x + 2 = 2 ❞✐✛❡r❡♥t✦ x x=0 x=0

P❧✉❣❣✐♥❣ ✐♥ x = 0 ✇✐❧❧ ♣r♦❞✉❝❡ ❞✐✈✐s✐♦♥ ❜② 0 ❢♦r t❤❡ ✜rst ❢✉♥❝t✐♦♥ ✐♥ t❤❡ ♣❛✐r ❜✉t ♥♦t ❢♦r t❤❡ s❡❝♦♥❞✳ ■t ✐s ❝❧❡❛r t❤❡♥ t❤❛t t✇♦ ❢✉♥❝t✐♦♥s ❝❛♥✬t ❜❡ t❤❡ s❛♠❡ ✉♥❧❡ss t❤❡✐r ❞♦♠❛✐♥s ❛r❡ ❡q✉❛❧ t♦♦ ✭❛s s❡ts✮✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ✐s t❤❡ t❡st t✇♦ ❢✉♥❝t✐♦♥ ❢✉♥❝t✐♦♥s f ❛♥❞ g ❛r❡ s✉❜❥❡❝t❡❞ t♦✿

f x →

❙♦✱ f ❛♥❞ g ❛r❡ ❝❛❧❧❡❞

ր

ց

ց

ր

g

s❛♠❡❄

❡q✉❛❧✱ ♦r ✇❡ s❛② ✐t✬s t❤❡ s❛♠❡ ❢✉♥❝t✐♦♥✱ ✐❢ t❤❡② ❤❛✈❡ t❤❡ s❛♠❡ ❞♦♠❛✐♥ ❛♥❞ f (x) = g(x) ❋❖❘ ❊❆❈❍ x

✐♥ t❤❡ ❞♦♠❛✐♥✳ ❚❤❡s❡ ❛r❡ ♦✉r ❛♥s✇❡rs t♦ t❤❡ ❛❜♦✈❡ q✉❡st✐♦♥s✳ ❆r❡ t❤❡s❡ t✇♦ ❢✉♥❝t✐♦♥s t❤❡ s❛♠❡✿

f (x) = ❨❡s✱ ❜❡❝❛✉s❡ ■t ✐s ❝r✉❝✐❛❧ t❤❛t t❤❡

✐♠♣❧✐❡❞

2x2 + 2x ❛♥❞ g(x) = x2 + x ? 2

2x2 + 2x = x2 + x ❢♦r ❡✈❡r② x. 2 ❞♦♠❛✐♥s ✕ ❢♦r ❡✈❡r② x ✕ ♦❢ t❤❡ t✇♦ ❢✉♥❝t✐♦♥s ❛r❡ t❤❡ s❛♠❡✳

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

❊q✉❛❧ ❢✉♥❝t✐♦♥s f =g ❆r❡ t❤❡s❡ t✇♦ ❢✉♥❝t✐♦♥s t❤❡ s❛♠❡✿

2x2 + 2x ❛♥❞ g(x) = 2x + 2 ? f (x) = x ◆♦✱ ❜❡❝❛✉s❡ t❤❡ ✐♠♣❧✐❡❞ ❞♦♠❛✐♥ ♦❢ t❤❡ ❢♦r♠❡r ❞♦❡s♥✬t ✐♥❝❧✉❞❡ 0 ✇❤✐❧❡ t❤❛t ♦❢ t❤❡ ❧❛tt❡r ❞♦❡s✳ ❚❤❡ ❞✐✛❡r❡♥❝❡ ✐s ✐♥ ❛ s✐♥❣❧❡ ✈❛❧✉❡✦ ❲❡ ❛❧s♦ ✉s❡ t❤✐s s✐♠♣❧❡ ♥♦t❛t✐♦♥✿

◆♦t ❡q✉❛❧ ❢✉♥❝t✐♦♥s f 6= g ❆s ②♦✉ ❝❛♥ s❡❡✱ ♦♥❝❡ ✇❡ ❞✐s❝♦✈❡r t❤❛t t❤❡ ❞♦♠❛✐♥s ❞♦♥✬t ♠❛t❝❤✱ ✇❡ ❛r❡ ❞♦♥❡✳ ❍♦✇❡✈❡r✱ ❝❤♦♦s✐♥❣ ❛♥♦t❤❡r ❞♦♠❛✐♥ ✇✐❧❧ ✜① t❤❡ ♣r♦❜❧❡♠✿

2x2 + 2x ❛♥❞ g(x) = 2x + 2 ❛r❡ t❤❡ s❛♠❡ ❢✉♥❝t✐♦♥ ♦♥ t❤❡ ❞♦♠❛✐♥ {x : x 6= 0} . x ❆s ❛♥♦t❤❡r r❡❧❡✈❛♥t ❡①❛♠♣❧❡✱ t❤❡s❡ ❛r❡ t✇♦ ❞✐✛❡r❡♥t ❢✉♥❝t✐♦♥s✿ f (x) =

• x2 ✇✐t❤ ❞♦♠❛✐♥ (−∞, ∞)❀

• x2 ✇✐t❤ ❞♦♠❛✐♥ [0, ∞)✳

✺✳✶✳

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

✸✾✼

❊①❡r❝✐s❡ ✺✳✶✳✶ ❈♦♥s✐❞❡r✿

x x2

✈s✳

1 . x

❊①❡r❝✐s❡ ✺✳✶✳✷ ❙✉❣❣❡st ②♦✉r ♦✇♥ ❡①❛♠♣❧❡s ♦❢ ❢✉♥❝t✐♦♥s t❤❛t ❞✐✛❡r ❜② ❛ s✐♥❣❧❡ ✈❛❧✉❡✳

❚❤❡ st❛t❡♠❡♥t ✐♥ t❤❡ ❞❡✜♥✐t✐♦♥✱ s✉❝❤ ❛s

2x2 + 2x = x2 + x 2 ✐s ❝❛❧❧❡❞ ❛♥

❢♦r ❡✈❡r② r❡❛❧

x,

✐❞❡♥t✐t②✳ ❚❤❡ ❧❛st ♣❛rt ✐s ♦❢t❡♥ ❛ss✉♠❡❞ ❛♥❞ ♦♠✐tt❡❞ ❢r♦♠ ❝♦♠♣✉t❛t✐♦♥s✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ st❛t❡♠❡♥t

✐s ❛❧s♦ ❛♥ ✐❞❡♥t✐t②✿

2x2 + 2x = 2x + 2 x

❍♦✇❡✈❡r✱ t❤❡ ❧❛st ♣❛rt ✐s ❛ ❝❛✈❡❛t t❤❛t

❝❛♥♥♦t

❢♦r ❡✈❡r② r❡❛❧

x 6= 0 .

❜❡ ♦♠✐tt❡❞✦ ■♥ ♦t❤❡r ✇♦r❞s✱ ❛♥ ✐❞❡♥t✐t② ✐s ❥✉st ❛ st❛t❡♠❡♥t

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

t✇✐♥

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

✐♥❢♦r♠❛❧❧② ❝❛❧❧❡❞ ✏s✐♠♣❧✐✜❝❛t✐♦♥s✑ ♦r ✏❝❛♥❝❡❧❧❛t✐♦♥s✑✳ ◆♦✇✱ t❤❡ ♦✉t♣✉ts ♦❢ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s ❛r❡

♥✉♠❜❡rs✳

❚❤❡r❡❢♦r❡✱ ❛♥② ❛r✐t❤♠❡t✐❝ ♦♣❡r❛t✐♦♥ ♦♥ ♥✉♠❜❡rs ✕

❛❞❞✐t✐♦♥✱ s✉❜tr❛❝t✐♦♥✱ ♠✉❧t✐♣❧✐❝❛t✐♦♥✱ ❛♥❞ ❞✐✈✐s✐♦♥ ✕ ❝❛♥ ♥♦✇ ❜❡ ❛♣♣❧✐❡❞ t♦ ❢✉♥❝t✐♦♥s✱ ♦♥❡ ✐♥♣✉t ❛t ❛ t✐♠❡✳ ❖♥❝❡ ❛❣❛✐♥✱ ❢✉♥❝t✐♦♥s ✐♥t❡r❛❝t ❛♥❞ ♣r♦❞✉❝❡ ♦✛s♣r✐♥❣✱ ♥❡✇ ❢✉♥❝t✐♦♥s✳ ❚❤❡ ❞❡✜♥✐t✐♦♥s ♦❢ t❤❡s❡ ♥❡✇ ❢✉♥❝t✐♦♥s ❛r❡ s✐♠♣❧❡✳

❉❡✜♥✐t✐♦♥ ✺✳✶✳✸✿ s✉♠ ♦❢ ❢✉♥❝t✐♦♥s ●✐✈❡♥ t✇♦ ❢✉♥❝t✐♦♥s

f

❛♥❞

g✱

t❤❡

s✉♠✱ f + g✱ ♦❢ f

❛♥❞

g

✐s t❤❡ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞

❜② t❤❡ ❢♦❧❧♦✇✐♥❣✿

(f + g)(x) = f (x) + g(x) ❋❖❘ ❊❆❈❍ x ✐♥ t❤❡ ✐♥t❡rs❡❝t✐♦♥ ♦❢ t❤❡ ❞♦♠❛✐♥s ♦❢

f

❛♥❞

g✳

◆♦t❡ ❤♦✇ t❤❡ t✇♦ ♣❧✉s s✐❣♥s ✐♥ t❤❡ ❢♦r♠✉❧❛ ❛r❡ ❞✐✛❡r❡♥t✿ ❚❤❡ ✜rst ♦♥❡ ✐s ❛ ♣❛rt ♦❢ t❤❡

♥❛♠❡

♦❢ t❤❡ ♥❡✇

❢✉♥❝t✐♦♥ ✇❤✐❧❡ t❤❡ s❡❝♦♥❞ ✐s t❤❡ ❛❝t✉❛❧ s✐❣♥ ♦❢ s✉♠♠❛t✐♦♥ ♦❢ t✇♦ r❡❛❧ ♥✉♠❜❡rs✳ ❚❤✐s ✐s t❤❡ ✏❞❡❝♦♥str✉❝t✐♦♥✑ ♦❢ t❤❡ ♥♦t❛t✐♦♥✿

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

f +g ↑




(x)

♥❛♠❡ ♦❢ t❤❡ ♥❡✇ ❢✉♥❝t✐♦♥

↓ ↓ = f (x) + g (x) ↑ ↑ ↑

♦♣❡r❛t✐♦♥ ♦♥ ♥✉♠❜❡rs

❋✉rt❤❡r♠♦r❡✱ ✇❡ ♥♦✇ ❤❛✈❡ ❛♥ ♦♣❡r❛t✐♦♥ ♦♥ ❢✉♥❝t✐♦♥s✿

f +g

✐s ❛ ♥❡✇ ❢✉♥❝t✐♦♥✳

❊①❛♠♣❧❡ ✺✳✶✳✹✿ ❛❧❣❡❜r❛ ♦❢ ❢✉♥❝t✐♦♥s ❚❤❡ s✉♠ ♦❢

g(x) = x2

❛♥❞

f (x) = x + 2

✺✳✶✳

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

✐s

✸✾✽



(g + f )(x) = g(x) + f (x) = x2 + x + 2 .

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

❚❤✐s ✐s ❛♥ ✐❧❧✉str❛t✐♦♥ ♦❢ t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ s✉♠ ♦❢ t✇♦ ❢✉♥❝t✐♦♥s✿

❖♥❡ ❝❛♥ s❡❡ ❤♦✇ t❤❡ ✈❛❧✉❡s ❛r❡ ❛❞❞❡❞✱ ❧♦❝❛t✐♦♥ ❜② ❧♦❝❛t✐♦♥✳ ❲❡ r❡♣r❡s❡♥t ❛ ❢✉♥❝t✐♦♥

f

❞✐❛❣r❛♠♠❛t✐❝❛❧❧② ❛s ❛ ✐♥♣✉t

❢✉♥❝t✐♦♥

→

x ◆♦✇✱ s✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛♥♦t❤❡r ❢✉♥❝t✐♦♥

→

f

y

❢✉♥❝t✐♦♥

→

t f + g❄

♦✉t♣✉t

g✿

✐♥♣✉t

❍♦✇ ❞♦ ✇❡ r❡♣r❡s❡♥t t❤❡✐r s✉♠

❜❧❛❝❦ ❜♦① t❤❛t ♣r♦❝❡ss❡s t❤❡ ✐♥♣✉t ❛♥❞ ♣r♦❞✉❝❡s t❤❡ ♦✉t♣✉t✿

♦✉t♣✉t

→

g

u

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

t♦❣❡t❤❡r s✐❞❡ ❜② s✐❞❡✿

❇✉t ✐t✬s ♦♥❧② ♣♦ss✐❜❧❡ ✇❤❡♥ t❤❡ ✐♥♣✉t ♦❢ ♦❢

g✳

❲❡ r❡♣❧❛❝❡

t

✇✐t❤

x✳

f

x → || t →

→ y l → u

f g

❝♦✐♥❝✐❞❡s ✇✐t❤ t❤❡ ✐♥♣✉t ♦❢

g✳

❲❡ ♠❛② ❤❛✈❡ t♦

r❡♥❛♠❡ t❤❡ ✈❛r✐❛❜❧❡

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

f +g : x → ❲❡ s❡❡ ❤♦✇ t❤❡ ✐♥♣✉t ✈❛r✐❛❜❧❡

x

x

ր x →

f

ց x →

g

→ y ց

❛❞❞

→ u ր

→ z

✐s ❝♦♣✐❡❞ ✐♥t♦ t❤❡ t✇♦ ❢✉♥❝t✐♦♥s✱ ♣r♦❝❡ss❡❞ ❜② t❤❡♠

→ z

✐♥ ♣❛r❛❧❧❡❧✱ ❛♥❞ ✜♥❛❧❧②

t❤❡ t✇♦ ♦✉t♣✉ts ❛r❡ ❛❞❞❡❞ t♦❣❡t❤❡r t♦ ♣r♦❞✉❝❡ ❛ s✐♥❣❧❡ ♦✉t♣✉t✳ ❚❤❡ r❡s✉❧t ❝❛♥ ❜❡ s❡❡♥ ❛s ❥✉st ❛ ♥❡✇ ❜❧❛❝❦ ❜♦①✿

x →

f +g

→ y

❲❛r♥✐♥❣✦ ❲❤❡♥ ✉♥✐ts ❛r❡ ✐♥✈♦❧✈❡❞✱ ✇❡ ♠✉st ♠❛❦❡ s✉r❡ t❤❛t t❤❡ ♦✉t♣✉ts ♠❛t❝❤ s♦ t❤❛t ✇❡ ❝❛♥ ❛❞❞ t❤❡♠✳

❊①❛♠♣❧❡ ✺✳✶✳✺✿ ❛❧❣❡❜r❛ ♦❢ ❢✉♥❝t✐♦♥s ❲❡ ❤❛✈❡ ❝♦♠❜✐♥❡❞ t✇♦ ❢✉♥❝t✐♦♥s ✐♥t♦ ♦♥❡ ❜✉t ✇❡ ♦❢t❡♥ ♥❡❡❞ t♦ ❣♦ t❤❡ ♦t❤❡r ✇❛② ❛♥❞ ❜r❡❛❦ ❛ ❝♦♠♣❧❡① ❢✉♥❝t✐♦♥ ✐♥t♦ s✐♠♣❧❡r ♣❛rts t❤❛t ❝❛♥ t❤❡♥ ❜❡ st✉❞✐❡❞ s❡♣❛r❛t❡❧②✳ ❘❡♣r❡s❡♥t

z = h(x) = x2 +

√ 3

x

❛s t❤❡ s✉♠ ♦❢ t✇♦ ❢✉♥❝t✐♦♥s✳ ❍❡r❡ ✐s t❤❡ ❛♥s✇❡r✿

x 7→ y = x2

❛♥❞

❙✉❜tr❛❝t✐♦♥ ❛❧s♦ ❣✐✈❡s ✉s ❛♥ ♦♣❡r❛t✐♦♥ ♦♥ ❢✉♥❝t✐♦♥s✳

x 7→ y =

√ 3

x.

✺✳✶✳

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

✸✾✾

❉❡✜♥✐t✐♦♥ ✺✳✶✳✻✿ ❞✐✛❡r❡♥❝❡ ♦❢ ❢✉♥❝t✐♦♥s ●✐✈❡♥ t✇♦ ❢✉♥❝t✐♦♥s

f

❛♥❞

g✱

t❤❡

❞❡✜♥❡❞ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣✿

❞✐✛❡r❡♥❝❡✱ g − f ✱

♦❢

f

❛♥❞

g

✐s t❤❡ ❢✉♥❝t✐♦♥

(g − f )(x) = g(x) − f (x) ❋❖❘ ❊❆❈❍ x ✐♥ t❤❡ ✐♥t❡rs❡❝t✐♦♥ ♦❢ t❤❡ ❞♦♠❛✐♥s ♦❢

f

❛♥❞

g✳

❇❡❢♦r❡ ✇❡ ❣❡t t♦ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ ❢✉♥❝t✐♦♥s✱ t❤❡r❡ ✐s ❛ s✐♠♣❧❡r ❜✉t ✈❡r② ✐♠♣♦rt❛♥t ✈❡rs✐♦♥ ♦❢ t❤✐s ♦♣❡r❛t✐♦♥✳

❉❡✜♥✐t✐♦♥ ✺✳✶✳✼✿ ❝♦♥st❛♥t ♠✉❧t✐♣❧❡ ♦❢ ❢✉♥❝t✐♦♥ ●✐✈❡♥ ❛ ❢✉♥❝t✐♦♥

f✱

t❤❡

❝♦♥st❛♥t ♠✉❧t✐♣❧❡ cf

♦❢

f✱

❢♦r s♦♠❡ r❡❛❧ ♥✉♠❜❡r

c✱

✐s t❤❡

❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣✿

(cf )(x) = cf (x) ❋❖❘ ❊❆❈❍ x ✐♥ t❤❡ ❞♦♠❛✐♥ ♦❢

f✳

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

c = 1.3

♦♥❡ ❧♦❝❛t✐♦♥ ❛t ❛ t✐♠❡✿

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

❊①❛♠♣❧❡ ✺✳✶✳✽✿ ❛❧❣❡❜r❛ ♦❢ ❢✉♥❝t✐♦♥s ❙✉♠ ❝♦♠❜✐♥❡❞ ✇✐t❤ ❞✐✛❡r❡♥❝❡s✿

h(x) = 2x3 −

5 + 3x − 4 . x

❚❤❡ ❢✉♥❝t✐♦♥ ✐s ❛❧s♦ s❡❡♥ ❛s t❤❡ s✉♠ ♦❢ ❝♦♥st❛♥t ♠✉❧t✐♣❧❡s✱ ❝❛❧❧❡❞ ❛ ✏❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥✑✿


1 h(x) = 2 · x3 + (−5) · + 3 · x + (−4) · 1 . x

❊①❛♠♣❧❡ ✺✳✶✳✾✿ ❛❧❣❡❜r❛ ♦❢ ❢✉♥❝t✐♦♥s ❣✐✈❡♥ ❜② t❛❜❧❡s

❲❤❡♥ t✇♦ ❢✉♥❝t✐♦♥s ❛r❡ r❡♣r❡s❡♥t❡❞ ❜② t❤❡✐r ❧✐sts ♦❢ ✈❛❧✉❡s✱ t❤❡✐r s✉♠ ✭❞✐✛❡r❡♥❝❡✱ ❡t❝✳✮ ❝❛♥ ❜❡ ❡❛s✐❧② ❝♦♠♣✉t❡❞✳ ❲❡ s✐♠♣❧② ❣♦ r♦✇ ❜② r♦✇ ❛❞❞✐♥❣ t❤❡ ✈❛❧✉❡s✳ ❙✉♣♣♦s❡ ✇❡ ♥❡❡❞ t♦ ❛❞❞ t❤❡s❡ t✇♦ ❢✉♥❝t✐♦♥s✱

f

❛♥❞

g ✱ ❛♥❞ ❝r❡❛t❡ ❛ ♥❡✇ ♦♥❡✱ h✱ r❡♣r❡s❡♥t❡❞ ❜② ❛ s✐♠✐❧❛r

✺✳✶✳

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

❧✐st✿

✹✵✵

x y = f (x) 0 1 2 1 3 2 0 3 4 1

+

x y = g(x) 0 5 −1 1 2 2 3 3 4 0

= ?

❲❡ s✐♠♣❧② ❛❞❞ t❤❡ ♦✉t♣✉t ♦❢ t❤❡ t✇♦ ❢✉♥❝t✐♦♥s ❢♦r t❤❡ s❛♠❡ ✐♥♣✉t✳ ❋✐rst r♦✇✿ f : 0 7→ 1,

g : 0 7→ 5

=⇒ h : 0 7→ 1 + 5 = 6 .

❙❡❝♦♥❞ r♦✇✿ f : 1 7→ 2,

g : 1 7→ −1

❆♥❞ s♦ ♦♥✳ ❚❤✐s ✐s t❤❡ ✇❤♦❧❡ s♦❧✉t✐♦♥✿ x y = f (x) 0 1 2 1 3 2 0 3 4 1

+

x y = g(x) 0 5 −1 1 2 2 3 3 4 0

=

=⇒ h : 1 7→ 2 + (−1) = 1 . x 0 1 2 3 4

y = f (x) + g(x) 1+5=6 2 + (−1) = 1 = 3+2=5 0+3=3 1+0=1

x 0 1 2 3 4

y = h(x) 6 1 . 5 3 1

❊①❛♠♣❧❡ ✺✳✶✳✶✵✿ ❛❧❣❡❜r❛ ♦❢ ❢✉♥❝t✐♦♥s ✇✐t❤ s♣r❡❛❞s❤❡❡t ❚❤✐s ✐s ❤♦✇ t❤❡ s✉♠ ♦❢ t✇♦ ❢✉♥❝t✐♦♥s ✐s ❝♦♠♣✉t❡❞ ✇✐t❤ ❛ s♣r❡❛❞s❤❡❡t✿

❚❤❡ ❢♦r♠✉❧❛ ✐s ✈❡r② s✐♠♣❧❡✿

❂❘❈❬✲✻❪✰❘❈❬✲✸❪

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

❉❡✜♥✐t✐♦♥ ✺✳✶✳✶✶✿ ♣r♦❞✉❝t ♦❢ ❢✉♥❝t✐♦♥s ●✐✈❡♥ t✇♦ ❢✉♥❝t✐♦♥s f ❛♥❞ g ✱ t❤❡ ♣r♦❞✉❝t✱ f · g ✱ ♦❢ f ❛♥❞ g ✐s t❤❡ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣✿ (f · g)(x) = f (x) · g(x) ❋❖❘ ❊❆❈❍ x

✐♥ ✐♥t❡rs❡❝t✐♦♥ ♦❢ t❤❡ ❞♦♠❛✐♥s ♦❢ f ❛♥❞ g ✳ ❋♦r ❡❛❝❤ ✈❛❧✉❡ ♦❢ x✱ ✇❡ ❝❛♥ ✉s❡ t❤❡ ♣❛✐r f (x) ❛♥❞ g(x) ❛s t❤❡ s✐❞❡s ♦❢ ❛ r❡❝t❛♥❣❧❡✳ ❚❤❡♥ t❤❡ ♣r♦❞✉❝t f (x)·g(x) ✐s s❡❡♥ ❛s t❤❡ ❛r❡❛ ♦❢ t❤✐s r❡❝t❛♥❣❧❡✿

✺✳✶✳

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

✹✵✶

■❢ ✇❡ t❤✐♥❦ ♦❢ x ❛s t✐♠❡✱ ✇❡ ❝❛♥ s❡❡ t❤✐s ❝♦♥str✉❝t✐♦♥ ❛s ❛ s❤♦rt ❝❧✐♣ ♦❢ s♣r❡❛❞✐♥❣ ❛ t❛r♣✿

❉❡✜♥✐t✐♦♥ ✺✳✶✳✶✷✿ q✉♦t✐❡♥t ♦❢ ❢✉♥❝t✐♦♥s ●✐✈❡♥ t✇♦ ❢✉♥❝t✐♦♥s f ❛♥❞ g ✱ t❤❡ q✉♦t✐❡♥t✱ f /g ✱ ♦❢ f ❛♥❞ g ✐s t❤❡ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣✿
f /g (x) = f (x)/g(x) ❋❖❘ ❊❆❈❍ x

✐♥ t❤❡ ✐♥t❡rs❡❝t✐♦♥ ♦❢ t❤❡ ❞♦♠❛✐♥s ♦❢ f ❛♥❞ g ✇✐t❤ g(x) 6= 0✳

❋♦r ❡❛❝❤ ✈❛❧✉❡ ♦❢ x✱ ✇❡ ❝❛♥ ✉s❡ t❤❡ ♣❛✐r f (x) ❛♥❞ g(x) ❛s t❤❡ s✐❞❡s ♦❢ ❛ r✐❣❤t tr✐❛♥❣❧❡✳ ❚❤❡② ❛r❡ ❤♦r✐③♦♥t❛❧ ❛♥❞ ✈❡rt✐❝❛❧✱ r❡s♣❡❝t✐✈❡❧②✱ t❤❡ r✉♥ ❛♥❞ t❤❡ r✐s❡✳ ❚❤❡♥ t❤❡ q✉♦t✐❡♥t f (x)/g(x) ✐s t❤❡ s❧♦♣❡ ♦❢ t❤✐s ❧✐♥❡✿

❊①❡r❝✐s❡ ✺✳✶✳✶✸ ❊①♣❧❛✐♥ t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡s❡ t✇♦ ❢✉♥❝t✐♦♥s✿ r

√ x−1 x−1 . ❛♥❞ √ x+1 x+1

■♥ s✉♠♠❛r②✱ t❤❡ ♦✉t♣✉ts ❛r❡ ♦♥ t❤❡ y ✲❛①✐s ❛♥❞ ✐ts ❛❧❣❡❜r❛ ❛❧❧♦✇s ✉s t♦ ✜♥❞ t❤❡ ♦✉t♣✉ts ❢♦r t❤❡ s✉♠ ❛♥❞ ♦t❤❡r ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s ♦♥ f, g ✿

✺✳✷✳ ❚❤❡ ❛❧❣❡❜r❛ ♦❢ ❝♦♠♣♦s✐t✐♦♥s

✹✵✷

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

x

ր x →

f

ց x →

g

→ y ց

+ − ·÷

→ u ր

→ z

❊①❡r❝✐s❡ ✺✳✶✳✶✹

❍❛✈❡ ✇❡ ❢♦✉♥❞ t❤❡ ❞♦♠❛✐♥s ♦❢ t❤❡s❡ ♥❡✇ ❢✉♥❝t✐♦♥s❄ ❲❛r♥✐♥❣✦ ❚❤❡ ❛❧❣❡❜r❛ ♦❢ ❢✉♥❝t✐♦♥s ❝♦♠❡s ❢r♦♠ t❤❡ ❛❧❣❡❜r❛ ♦❢

♦✉t♣✉ts ❀ t❤❡ ✐♥♣✉ts ❞♦♥✬t ❡✈❡♥ ❤❛✈❡ t♦ ❜❡ ♥✉♠❜❡rs✳

❈♦♠♣♦s✐t✐♦♥✱ ❤♦✇❡✈❡r✱ ✐s t❤❡ ♠♦st ✐♠♣♦rt❛♥t ♦♣❡r❛t✐♦♥ ♦♥ ❢✉♥❝t✐♦♥s✳ ❚❤❡r❡ ✐s ♥♦ ♠❛t❝❤✐♥❣ ♦♣❡r❛t✐♦♥ ❢♦r ♥✉♠❜❡rs✳

✺✳✷✳ ❚❤❡ ❛❧❣❡❜r❛ ♦❢ ❝♦♠♣♦s✐t✐♦♥s

❋✉♥❝t✐♦♥s ❝❛♥ ❜❡ ✈✐s✉❛❧✐③❡❞ ❛s ✢♦✇❝❤❛rts ❛♥❞ s♦ ❝❛♥ t❤❡✐r ❝♦♠♣♦s✐t✐♦♥s✿

■❢ ✇❡ ♥❛♠❡ t❤❡ ✈❛r✐❛❜❧❡s ❛♥❞ ✉s❡ t❤❡ ❛❧❣❡❜r❛✐❝ ♥♦t❛t✐♦♥✱ ✇❡ ♣r♦❞✉❝❡ ❛ ♠♦r❡ ❝♦♠♣❛❝t ✈❡rs✐♦♥ ♦❢ t❤✐s ✢♦✇❝❤❛rt✿ x →

x+3

→ y →

y·2

→ z →

z2

→ u

◆♦t❡ ❤♦✇ t❤❡ ♥❛♠❡s ♦❢ t❤❡ ✈❛r✐❛❜❧❡s ♠❛t❝❤ s♦ t❤❛t ✇❡ ❝❛♥ ♣r♦❝❡❡❞ t♦ t❤❡ ♥❡①t st❡♣✳ ❆ ♣✉r❡❧② ❛❧❣❡❜r❛✐❝ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ❞✐❛❣r❛♠ ✐s ❜❡❧♦✇✿ x + 3 = y,

y · 2 = z,

z2 = u .

■t ✐s ❛❧s♦ ♣♦ss✐❜❧❡✱ ❜✉t ♥♦t r❡q✉✐r❡❞✱ t♦ ♥❛♠❡ t❤❡ ❢✉♥❝t✐♦♥s✱ s❛② f, g, h✳ ❚❤❡♥ ✇❡ ❤❛✈❡✿ y = f (x) = x + 3,

z = g(y) = y · 2,

u = h(z) = z 2 .

❆s ✇❡ s❡❡✱ ✇✐t❤ t❤❡ ✈❛r✐❛❜❧❡s ♣r♦♣❡r❧② ♥❛♠❡❞✱

❝♦♠♣♦s✐t✐♦♥ ✐s s✉❜st✐t✉t✐♦♥✳ ■♥ t❤❡ ❛❜♦✈❡ ❝♦♠♣♦s✐t✐♦♥✱ ✇❡ ❝❛♥ ❝❛rr② ♦✉t t❤❡s❡ t✇♦ s✉❜st✐t✉t✐♦♥s✿ • ❲❡ s✉❜st✐t✉t❡ z = g(y) = y · 2 ✐♥t♦ u = h(z) = z 2 ✱ ✇❤✐❝❤ r❡s✉❧ts ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣✿ u = h(z) = h(g(y)),

u = z 2 = (y · 2)2 .

• ❲❡ s✉❜st✐t✉t❡ y = f (x) = x + 3 ✐♥t♦ z = g(y) = y · 2✱ ✇❤✐❝❤ r❡s✉❧ts ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣✿ z = g(y) = g(f (x)),

z = y · 2 = (x + 3) · 2 .

✺✳✷✳

❚❤❡ ❛❧❣❡❜r❛ ♦❢ ❝♦♠♣♦s✐t✐♦♥s

✹✵✸

■♥ ❣❡♥❡r❛❧✱ ✇❡ r❡♣r❡s❡♥t ❛ ❢✉♥❝t✐♦♥

f

❞✐❛❣r❛♠♠❛t✐❝❛❧❧② ❛s ❛

❜❧❛❝❦ ❜♦① t❤❛t ♣r♦❝❡ss❡s t❤❡ ✐♥♣✉t ❛♥❞ ♣r♦❞✉❝❡s

t❤❡ ♦✉t♣✉t✿ ✐♥♣✉t

❢✉♥❝t✐♦♥

→

x

✐♥♣✉t

y

❢✉♥❝t✐♦♥

→

x

❞✐❛❣r❛♠s t♦❣❡t❤❡r

→

f

g✿

◆♦✇✱ s✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛♥♦t❤❡r ❢✉♥❝t✐♦♥

❍♦✇ ❞♦ ✇❡ r❡♣r❡s❡♥t t❤❡✐r ❝♦♠♣♦s✐t✐♦♥

♦✉t♣✉t

g ◦ f❄

g

♦✉t♣✉t

→

y

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

❝♦♥s❡❝✉t✐✈❡❧② ✭✐♥st❡❛❞ ♦❢ ✐♥ ♣❛r❛❧❧❡❧✱ ❛s ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥✮✿ x →

→ y → ???

f

❇✉t ✐t✬s ♦♥❧② ♣♦ss✐❜❧❡ ✇❤❡♥ t❤❡ ♦✉t♣✉t ♦❢

f

→ x →

♠❛t❝❤❡s ✇✐t❤ t❤❡ ✐♥♣✉t ♦❢

❋♦r ❡①❛♠♣❧❡✱ ✇❡ ❝❛♥ ♠❛❦❡ t❤✐s s✇✐t❝❤✿

→ y

g g✳

❲❡ ❝❛♥

r❡♥❛♠❡ t❤❡ ✈❛r✐❛❜❧❡ ♦❢ g✳

y2 − 1 x2 − 1 → . x+2 y+2 ❲❛r♥✐♥❣✦ ■❢ t❤❡ ♥❛♠❡s ♦❢ t❤❡ ✈❛r✐❛❜❧❡s ❞♦♥✬t ♠❛t❝❤✱ ✐t ♠✐❣❤t ❜❡ ❢♦r ❛ ❣♦♦❞ r❡❛s♦♥✳

❚❤✐s ✐s ✇❤❛t ✇❡ ❤❛✈❡ ❛❢t❡r r❡♥❛♠✐♥❣✿

x →

→ y →

f

→ z

g

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

g◦f : x →

x →

→ y →

f

g

→ z

→ z

■t✬s ❥✉st ❛♥♦t❤❡r ❜❧❛❝❦ ❜♦①✿

x →

g◦f

→ z

❈♦♠♣♦s✐t✐♦♥s ❛r❡ ♠❡❛♥t t♦ r❡♣r❡s❡♥t t❛s❦s t❤❛t ❝❛♥♥♦t ❜❡ ❝❛rr✐❡❞ ♦✉t ✐♥ ♣❛r❛❧❧❡❧✳ ■♠❛❣✐♥❡ t❤❛t ②♦✉ ❤❛✈❡ t✇♦ ♣❡rs♦♥s ✇♦r❦✐♥❣ ❢♦r ②♦✉✱ ❜✉t ②♦✉ ❝❛♥✬t s♣❧✐t t❤❡ ✇♦r❦ ✐♥ ❤❛❧❢ t♦ ❤❛✈❡ t❤❡♠ ✇♦r❦ ♦♥ ✐t ❛t t❤❡ s❛♠❡ t✐♠❡ ❜❡❝❛✉s❡ t❤❡ s❡❝♦♥❞ t❛s❦ ❝❛♥♥♦t ❜❡ st❛rt❡❞ ✉♥t✐❧ t❤❡ ✜rst ✐s ✜♥✐s❤❡❞✳

❊①❛♠♣❧❡ ✺✳✷✳✶✿ ♦r❞❡r ♠❛tt❡rs ❋♦r ❡①❛♠♣❧❡✱ ②♦✉ ❛r❡ ♠❛❦✐♥❣ ❛

❝❤❛✐r✳

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

t❤❡♠ ❛t t❤❡ s❛♠❡ t✐♠❡✿ ❝❤❛✐r

→

♣♦❧✐s❤✐♥❣

→

♣❛✐♥t✐♥❣

→

✜♥✐s❤❡❞ ❝❤❛✐r

❨♦✉ ❝❛♥✬t ❝❤❛♥❣❡ t❤❡ ♦r❞❡r ❡✐t❤❡r✦

❊①❛♠♣❧❡ ✺✳✷✳✷✿ ❝♦♠♣✉t✐♥❣ ✇✐t❤ ❝❛❧❝✉❧❛t♦r ❇❡❧♦✇✱ t❤❡ ✐♥str✉❝t✐♦♥s ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ✐s ✏♣✉s❤ t❤❡s❡ ❜✉tt♦♥s✑ ✭✐♥ t❤❛t ♦r❞❡r✮✿

✺✳✷✳

❚❤❡ ❛❧❣❡❜r❛ ♦❢ ❝♦♠♣♦s✐t✐♦♥s

✹✵✹

❚❤❡s❡ ❛r❡ t❤❡ ❢✉♥❝t✐♦♥s ❝r❡❛t❡❞✿

1 , x2

√

− x2

x + 6,


2

.

❏✉st ❛s ✇✐t❤ t❤❡ r❡st ♦❢ t❤❡ ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s✱ ✇❡ s♦♠❡t✐♠❡s ✇❛♥t t♦ ✏✉♥❞♦✑ ❝♦♠♣♦s✐t✐♦♥s✳ ❇② ❞♦✐♥❣ s♦✱ ✇❡

❞❡❝♦♠♣♦s❡ t❤❡ ❣✐✈❡♥ ❢✉♥❝t✐♦♥ ✐♥t♦ t✇♦ ✭♦r ♠♦r❡✮ s✐♠♣❧❡r ♣❛rts t❤❛t ❝❛♥ t❤❡♥ ❜❡ ❛❞❞r❡ss❡❞ s❡♣❛r❛t❡❧②✳

❊①❛♠♣❧❡ ✺✳✷✳✸✿ ❞❡❝♦♠♣♦s✐t✐♦♥

❘❡♣r❡s❡♥t

z = h(x) =

√ 3

x2 + 1

❛s t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ t✇♦ ❢✉♥❝t✐♦♥s✳

❲❡ ♥❡❡❞ t♦ ✜♥❞ ❛♥ ❛♣♣r♦♣r✐❛t❡ ♣❧❛❝❡ t♦ st♦♣ ✐♥ t❤❡ ♠✐❞❞❧❡ ♦❢ ✐ts ❝♦♠♣✉t❛t✐♦♥✳ ❖♥❡ ♦❢ ❝❧✉❡s ❢♦r s✉❝❤ ❛ s♣♦t ♠✐❣❤t t❤❡

r❛❞✐❝❛❧ s✐❣♥✳

❲❤❛t✬s ✐♥s✐❞❡ ✐s t♦ ❜❡ ❝♦♠♣✉t❡❞ ✜rst✿

√ 3 ■t✬s ♦✉r ✐♥t❡r♠❡❞✐❛t❡ ✈❛r✐❛❜❧❡✱ s❛②

x2 + 1 .

y✿ y = x2 + 1 .

◆♦✇ t♦ t❤❡ s❡❝♦♥❞ st❛❣❡✳ ❲❡ ❥✉st r❡♣❧❛❝❡

x2 + 1

✐♥ ♦✉r ❢♦r♠✉❧❛ ✇✐t❤

z=

√ 3

y.

❚❤✐s ✐s t❤❡ ❞❡❝♦♠♣♦s✐t✐♦♥✿

x 7→ x2 + 1 = y 7→

❚♦ ❝♦♥✜r♠✱

y✿

√ 3

y = z.

s✉❜st✐t✉t❡ y ❜❛❝❦ ✐♥t♦ t❤❡ ❡①♣r❡ss✐♦♥ ❢♦r z ✳

❊①❛♠♣❧❡ ✺✳✷✳✹✿ ❛♥♦t❤❡r ❞❡❝♦♠♣♦s✐t✐♦♥

❉❡❝♦♠♣♦s❡✿

y = sin(x2 + 1) . ❆♥♦t❤❡r ❝❧✉❡ ♣♦✐♥t✐♥❣ ❛t t❤❡ ✐♥t❡r♠❡❞✐❛t❡ ✈❛r✐❛❜❧❡ ♠✐❣❤t ❜❡

♣❛r❡♥t❤❡s❡s ✿

y = sin (x2 + 1) . ❲❤❛t✬s ✐♥s✐❞❡ ✇✐❧❧ ❜❡ ♦✉r ♥❡✇ ✈❛r✐❛❜❧❡✿

u = x2 + 1 . ❲❡ s✉❜st✐t✉t❡ t❤❛t ✐♥t♦ t❤❡ ♦r✐❣✐♥❛❧✿

y = sin(u) . ❉♦♥❡✦ ❋♦r ❝♦♠♣❧❡t❡♥❡ss✱ ✇❡ ❝❛♥

♥❛♠❡ t❤❡ ❢✉♥❝t✐♦♥s✿

u = f (x) = x2 + 1, y = g(u) = sin u =⇒ h(x) = (g ◦ f )(x) = sin(x2 + 1) .

✺✳✷✳

❚❤❡ ❛❧❣❡❜r❛ ♦❢ ❝♦♠♣♦s✐t✐♦♥s

✹✵✺

❊①❡r❝✐s❡ ✺✳✷✳✺

❉❡❝♦♠♣♦s❡ y = (x + 1)3 ✳ ❊①❡r❝✐s❡ ✺✳✷✳✻

❉❡❝♦♠♣♦s❡ y = 2x−1 ✳ ❊①❛♠♣❧❡ ✺✳✷✳✼✿ ❝♦♠♣❧✐❝❛t❡❞ ❞❡❝♦♠♣♦s✐t✐♦♥

❉❡❝♦♠♣♦s❡✿ y=

t2 + 1 . t2 − 1

❚❤❡r❡ ❛r❡ ♥♦ ♣❛r❡♥t❤❡s❡s t♦ ✉s❡ ❛s ❛ ❝❧✉❡ ❤❡r❡✳ ❚❤❡ ❝❧✉❡ ♠✐❣❤t ❜❡ t❤❡ r❡♣❡t✐t✐♦♥✳ ❚❤❡ ❢♦r♠✉❧❛ s✉❣❣❡sts t❤❛t ✇❡ ❝❛♥ ❝♦♠♣✉t❡ ♦♥❝❡ ❛♥❞ s✉❜st✐t✉t❡ t✇✐❝❡ s❛✈✐♥❣ ❝♦♠♣✉t✐♥❣ t✐♠❡✿ y=

❙♦✱ ❧❡t✬s tr②✿

t2 + 1 . t2 − 1

x = t2 .

❲❡ s✉❜st✐t✉t❡ t❤❛t ✐♥t♦ t❤❡ ♦r✐❣✐♥❛❧✿ y=

❏✉st ❛s ✈❛❧✐❞ ❛ ❝❤♦✐❝❡ ✐s

x+1 . x−1

u = t2 + 1 .

❚❤❡♥ y=

u . u−2

x=

t2 + 1 ? t2 − 1

❲❤❛t ❝❛♥ ✇❡ s❛② ❛❜♦✉t t❤❡ ❝❤♦✐❝❡

❚❤❡ ♥❡✇ ❢✉♥❝t✐♦♥ ✐s t❤❡ s❛♠❡ ❛s t❤❡ ♦r✐❣✐♥❛❧✳ ❙✉❝❤ ❛ ❞❡❝♦♠♣♦s✐t✐♦♥ ✐s✱ t❤♦✉❣❤ t❡❝❤♥✐❝❛❧❧② ❝♦rr❡❝t✱ ✐s ♣♦✐♥t❧❡ss ❛s ✐t ♣r♦✈✐❞❡s ♥♦ s✐♠♣❧✐✜❝❛t✐♦♥✳ ❊①❡r❝✐s❡ ✺✳✷✳✽

❉❡❝♦♠♣♦s❡ y =

1 ✳ t2

❚❤❡r❡ ♠❛② ❜❡ ♠♦r❡ t❤❛♥ t✇♦ ❢✉♥❝t✐♦♥s ✐♥✈♦❧✈❡❞ ✐♥ ❝♦♠♣♦s✐t✐♦♥s✳ ❊①❛♠♣❧❡ ✺✳✷✳✾✿ ❣❛s ♠✐❧❡❛❣❡

❙✉♣♣♦s❡ ❛ ❝❛r ✐s ❞r✐✈❡♥ ❛t 60 ♠✐✴❤✳ ❙✉♣♣♦s❡ ✇❡ ❛❧s♦ ❦♥♦✇ t❤❛t t❤❡ ❝❛r ✉s❡s 30 ♠✐✴❣❛❧✱ ✇❤✐❧❡ t❤❡ ❝♦st ♣❡r ❣❛❧❧♦♥ ✐s $5✳ ❘❡♣r❡s❡♥t t❤❡ ❡①♣❡♥s❡ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✐♠❡✳ ❈♦♥s✐❞❡r t❤❡ ✢♦✇❝❤❛rt✿ 60 ♠✐✴❤

30 ♠✐✴❣❛❧

5$/ ❣❛❧

t✐♠❡ ✭❤✮ −−−−−−→ ❞✐st❛♥❝❡ ✭♠✐✮ −−−−−−−→ ❣❛s ✉s❡❞ ✭❣❛❧✮ −−−−−→ ❡①♣❡♥s❡ ($) t

f

−−→

d

k

−−→

g

❚❤❡s❡ ❛r❡ t❤❡ ♣❛rt✐❝✐♣❛t✐♥❣ ❢✉♥❝t✐♦♥s✿ 60t = d

d =g 30

5g = e

h

−−→

e

✺✳✷✳

❚❤❡ ❛❧❣❡❜r❛ ♦❢ ❝♦♠♣♦s✐t✐♦♥s

✹✵✻

❚♦ ❛♥s✇❡r t❤❡ q✉❡st✐♦♥✱ ✇❡ s✉❜st✐t✉t❡ ❢r♦♠ r✐❣❤t t♦ ❧❡❢t✿     d 60t e = 5g = 5 =5 . 30 30 ❙✐♠♣❧✐✜❡❞✿

e = 10t . ❊①❡r❝✐s❡ ✺✳✷✳✶✵

❘❡❞♦ t❤❡ ❡①❛♠♣❧❡ ✉s✐♥❣ t❤❡ ❢✉♥❝t✐♦♥s f, k, h✳ ❊①❛♠♣❧❡ ✺✳✷✳✶✶✿ ♦r❞❡r ♠❛tt❡rs

❋✐♥❞ f (g(x)) ❛♥❞ g(f (x)) ✇✐t❤✿

f (x) = x2 ❛♥❞ g(x) = cos x .

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

x → y → z ❚♦ ✜♥❞ f (g(x))✱ ✜rst r❡✇r✐t❡✿

f (y) = y 2 ❛♥❞ y = g(x) = cos x .

❚❤❡♥ r❡♣❧❛❝❡ ✭s✉❜st✐t✉t❡✮ y ✐♥ f ✇✐t❤ (cos x)✱ ❛❧✇❛②s ✇✐t❤ ♣❛r❡♥t❤❡s❡s✿

y 2 −→ (cos x)2 , ♦❢t❡♥ ✇r✐tt❡♥ ❛s cos2 x✳ ❚♦ ✜♥❞ g(f (x))✱ ✜rst r❡✇r✐t❡✿ ❚❤❡♥ r❡♣❧❛❝❡ y ✐♥ g ✇✐t❤ (x2 )✿

y = f (x) = x2 ❛♥❞ g(y) = cos y . cos y −→ cos(x2 ) .

❊①❛♠♣❧❡ ✺✳✷✳✶✷✿ r❡❝✉rs✐✈❡ s❡q✉❡♥❝❡s

❈♦♠♣♦s✐t✐♦♥s s❤✐♥❡ ❛ ♥❡✇ ❧✐❣❤t ♦♥ s♦♠❡ ♦❧❞ ✐❞❡❛s✳ ❆♥ ❛r✐t❤♠❡t✐❝ ♣r♦❣r❡ss✐♦♥ ✐s ❞❡✜♥❡❞ ❛s ❢♦❧❧♦✇s✿

n= 0 1 2 3 4 ... 2, ❛❞❞ 3, ❛❞❞ 3, ❛❞❞ 3, ❛❞❞ 3, ... an = 2 5 8 11 14 ... ■t ✐s ❥✉st ❛ r❡♣❡❛t❡❞ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ t❤❡ s❛♠❡ ❢✉♥❝t✐♦♥✦ ❲❡ ❤❛✈❡ ❝❛❧❧❡❞ ❛ s❡q✉❡♥❝❡ an r❡❝✉rs✐✈❡ ✇❤❡♥ ✐ts ♥❡①t t❡r♠ ✐s ❢♦✉♥❞ ❢r♦♠ t❤❡ ❝✉rr❡♥t t❡r♠ ❜② ❛ ❢♦r♠✉❧❛✳ ❆t ✐ts s✐♠♣❧❡st✱ ✐ts ♥❡①t t❡r♠ ✐s ❢♦✉♥❞ ❢r♦♠ t❤❡ ❝✉rr❡♥t t❡r♠ ❜② ❛♣♣❧②✐♥❣ t❤❡ s❛♠❡ ❢✉♥❝t✐♦♥✱ ✇❤✐❝❤ ❞♦❡s♥✬t ❞❡♣❡♥❞ ♦♥ n✿

an+1 = F (an ), n = 1, 2, 3, ... ❙♦✱ t❤❡ t❡r♠s ♦❢ t❤❡ s❡q✉❡♥❝❡ ❛r❡ ❝♦♠♣✉t❡❞ ✕ ♦♥❡ ❛t ❛ t✐♠❡ ✕ ❜② ❛♣♣❧②✐♥❣ t❤❡ ❢✉♥❝t✐♦♥ F r❡♣❡t✐t✐✈❡❧②✿

n=

0 1 2 3 4 ... a0 , ❛♣♣❧② F, ❛♣♣❧② F, ❛♣♣❧② F, ❛♣♣❧② F, ... an = a0 a1 = F (a0 ) a2 = F (a1 ) a3 = F (a2 ) a4 = F (a3 ) ... ■t✬s ❥✉st ♦♥❡ ❧♦♥❣ ❝♦♠♣♦s✐t✐♦♥✳

✺✳✷✳

❚❤❡ ❛❧❣❡❜r❛ ♦❢ ❝♦♠♣♦s✐t✐♦♥s

✹✵✼

❊①❡r❝✐s❡ ✺✳✷✳✶✸

❲❤❛t ✐s t❤❡ ❢✉♥❝t✐♦♥ t❤❛t ❞❡✜♥❡s ❛ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥❄ ❲❤❛t ✐s ❞✐✛❡r❡♥t ❛❜♦✉t t❤❡ ❢❛❝t♦r✐❛❧❄ ❊①❛♠♣❧❡ ✺✳✷✳✶✹✿ ❝♦♠♣♦s✐t✐♦♥s ♦❢ ❢✉♥❝t✐♦♥s ❣✐✈❡♥ ❜② ❧✐sts

❲❤❡♥ t❤❡ t✇♦ ❢✉♥❝t✐♦♥s ❛r❡ r❡♣r❡s❡♥t❡❞ ❜② t❤❡✐r t❛❜❧❡s ♦❢ ✈❛❧✉❡s✱ t❤❡ ❝♦♠♣♦s✐t✐♦♥ ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ❥✉st ❛s ✇✐t❤ t❤❡ r❡st ♦❢ ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s ✭❛s ❞✐s❝✉ss❡❞ ❡❛r❧✐❡r✮✳ ■t ✐s ♠♦r❡ ❝♦♠♣❧❡① ❛s ✇❡ ❝❛♥♥♦t s✐♠♣❧② ❣♦ ✇✐t❤ t❤❡ s❛♠❡ r♦✇ ❢♦r ❛❧❧ ❢✉♥❝t✐♦♥s✳ ❖♥❡ ❤❛s t♦ ✜♥❞ t❤❡ r✐❣❤t ❡♥tr② ✐♥ t❤❡ ♥❡①t ❢✉♥❝t✐♦♥✳ ❙✉♣♣♦s❡ ✇❡ ♥❡❡❞ t♦ ❝♦♠♣♦s❡ t❤❡s❡ t✇♦ ❢✉♥❝t✐♦♥s✿ t x = f (t) 0 1 2 1 2 3 0 3 1 4

❢♦❧❧♦✇❡❞ ❜②

x y = g(x) 0 5 −1 1 . 2 2 3 3 0 4

❚❤❡ r❡s✉❧t s❤♦✉❧❞ ❜❡ ❛♥♦t❤❡r t❛❜❧❡ ♦❢ ✈❛❧✉❡s ❢♦r t❤❡ ❢✉♥❝t✐♦♥ h = g ◦ f ✳ ❚♦ ✜❧❧ t❤✐s t❛❜❧❡✱ ✇❡ ✇❛t❝❤ ✇❤❡r❡ ❡✈❡r② t ❣♦❡s✳ ❲❤❛t ❤❛♣♣❡♥s t♦ 0 ❛❢t❡r t✇♦ st❡♣s❄ ❲❡ ❧♦♦❦ ❛t t❤❡ ✜rst t❛❜❧❡✿ ❯♥❞❡r f ✱ ✇❡ ❤❛✈❡ 0 7→ 1✳ ◆❡①t✱ ✇❤❡r❡ ❞♦❡s 1 ❣♦ ✉♥❞❡r g ❄ ❲❡ ❧♦♦❦ ❛t t❤❡ s❡❝♦♥❞ t❛❜❧❡✿ ❯♥❞❡r g ✱ ✇❡ ❤❛✈❡ 1 7→ −1✳ ❚❤❡r❡❢♦r❡✱ ✉♥❞❡r h✱ ✇❡ ❤❛✈❡ 0 7→ −1✳ ❚❤❛t ❣✐✈❡s ✉s t❤❡ ✜rst r♦✇ ✐♥ t❤❡ ♥❡✇ t❛❜❧❡✿ f : 0 7→ 1 g : 1 7→ −1

=⇒ h : 0 7→ −1 .

❋✉rt❤❡r♠♦r❡✿ f : 1 7→ 2 g : 2 7→ 2

=⇒ h : 1 7→ 2 .

❆♥❞ s♦ ♦♥✳ ❙♦♠❡ ♦❢ t❤❡ ❝♦♠♣✉t❛t✐♦♥s ❛r❡ s❤♦✇♥ ✐♥ t❤✐s t❛❜❧❡✿ t 0 1 2 3 4

x x y 1 ց 0 5 2 ց 1 −1 =⇒ 0 → −1 3 ց 2 2 =⇒ 1 → 2 0 3 3 =⇒ 2 → 3 1 4 0

❆♥❞ t❤✐s ✐s t❤❡ ❛♥s✇❡r✿ g◦f =

t y = g(f (t)) 0 −1 2 1 . 3 2 3 5 −1 4

❊①❛♠♣❧❡ ✺✳✷✳✶✺✿ ❞r✐✈✐♥❣ t❤r♦✉❣❤ t❡rr❛✐♥

❋✉♥❝t✐♦♥s r❡♣r❡s❡♥t❡❞ ❜② ❣r❛♣❤s ❝❛♥ ❛❧s♦ ❜❡ ❝♦♠♣♦s❡❞✳ ❚❤❡ ♣r♦❝❡❞✉r❡ ✐s✱ ❤♦✇❡✈❡r✱ ♠♦r❡ ❝♦♥✈♦❧✉t❡❞ t❤❛♥ t❤♦s❡ ❢♦r t❤❡ r❡st ♦❢ t❤❡ ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s✳ ▲❡t✬s r❡✈✐❡✇✳ ❙✉♣♣♦s❡ ❛ ❝❛r ✐s ❞r✐✈❡♥ t❤r♦✉❣❤ ❛ ♠♦✉♥t❛✐♥ t❡rr❛✐♥✳ ❚❤❡ ❧♦❝❛t✐♦♥✱ ❛s s❡❡♥ ♦♥ ❛ ♠❛♣ ✭❧❡❢t✮✱ ✐s ❦♥♦✇♥ ❛♥❞ s♦ ✐s t❤❡ ❛❧t✐t✉❞❡ ❢♦r ❡❛❝❤ ❧♦❝❛t✐♦♥ ✭r✐❣❤t✮✿

✺✳✷✳

❚❤❡ ❛❧❣❡❜r❛ ♦❢ ❝♦♠♣♦s✐t✐♦♥s

✹✵✽

❚❤❡s❡ ❛r❡ t❤❡ t❤r❡❡ ✈❛r✐❛❜❧❡s✿ • t ✐s t✐♠❡ ✕ ♠❡❛s✉r❡❞ ✐♥ ❤r✳ • x ✐s t❤❡ ❧♦❝❛t✐♦♥ ♦❢ t❤❡ ❝❛r ✕ ♠❡❛s✉r❡❞ ✐♥ ♠✐✳ • y ✐s t❤❡ ❡❧❡✈❛t✐♦♥ ♦❢ t❤❡ r♦❛❞ ✕ ♠❡❛s✉r❡❞ ✐♥ ❢t✳ ❚❤✐s ✐s t❤❡✐r r❡❧❛t✐♦♥✿ t → x → y

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

❚❤❡ s❡❝♦♥❞ ❢✉♥❝t✐♦♥ ✐s ❧✐t❡r❛❧❧② t❤❡ ♣r♦✜❧❡ ♦❢ t❤❡ r♦❛❞✳ ❍❡r❡✱ ✇❡ ❤❛✈❡✿ • x = f (t) ✐s t❤❡ ❧♦❝❛t✐♦♥ ♦❢ t❤❡ ❝❛r ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✐♠❡✳ • y = g(x) ✐s t❤❡ ❡❧❡✈❛t✐♦♥ ♦❢ t❤❡ r♦❛❞ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ ✭❤♦r✐③♦♥t❛❧✮ ❧♦❝❛t✐♦♥✳ • y = h(t) = g(f (t)) ✐s t❤❡ ❛❧t✐t✉❞❡ ♦❢ t❤❡ r♦❛❞ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✐♠❡✳ ❚❤✐s ✐s t❤❡ ❢❛♠✐❧✐❛r ✇❛② t♦ ❡✈❛❧✉❛t❡ ❛ ❢✉♥❝t✐♦♥✿

f (x) = x2 − x =⇒ f (3) = 32 − 3 . ❚❤❡r❡ ✐s ❛❧s♦ ❛♥ ❛❧t❡r♥❛t✐✈❡ ♥♦t❛t✐♦♥✿

❙✉❜st✐t✉t✐♦♥ ♥♦t❛t✐♦♥ f (x) = x2 − x =⇒ f (3) = x2 − x

x=3

= 32 − 3

❚❤✐s ♥♦t❛t✐♦♥ ❛❧s♦ ❛♣♣❧✐❡s t♦ ❝♦♠♣♦s✐t✐♦♥s✳ ❋♦r ❡①❛♠♣❧❡✿

• ❲❡ s✉❜st✐t✉t❡ z = g(y) = y · 2 ✐♥t♦ u = h(z) = z 2 ✱ ✇❤✐❝❤ r❡s✉❧ts ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣✿ 2 u=z = (y · 2)2 . z=y·2

• ❲❡ s✉❜st✐t✉t❡ y = f (x) = x + 3 ✐♥t♦ z = g(y) = y · 2✱ ✇❤✐❝❤ r❡s✉❧ts ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣✿ = (x + 3) · 2 . z = y · 2 y=x+3

✺✳✸✳

❙♦❧✈✐♥❣ ❡q✉❛t✐♦♥s

✹✵✾

✺✳✸✳ ❙♦❧✈✐♥❣ ❡q✉❛t✐♦♥s

▲❡t✬s r❡✈✐❡✇ ✇❤❛t ✐t ♠❡❛♥s t♦ s♦❧✈❡ ❛♥ ❡q✉❛t✐♦♥✳ ❲❡ ❣♦ ❜❛❝❦ t♦ ♦✉r ❡①❛♠♣❧❡ ♦❢ ❜♦②s ❛♥❞ ❜❛❧❧s✳ ❚❤✐s ✐s ♦✉r ❢✉♥❝t✐♦♥ t❤❛t t❡❧❧s ✇❤❛t ❜❛❧❧ ❡❛❝❤ ❜♦② ♣r❡❢❡rs✿

F( F( F( F( F(

) ◆❡❞ ) ❇❡♥ ) ❑❡♥ ) ❙✐❞ ) ❚♦♠

= = = = =

❜❛s❦❡t❜❛❧❧ t❡♥♥✐s ❜❛s❦❡t❜❛❧❧ ❢♦♦t❜❛❧❧ ❢♦♦t❜❛❧❧

❙♦✱ ♦✉r ❢✉♥❝t✐♦♥ ✕ ✐♥ t❤❡ ❢♦r♠ ♦❢ t❤✐s ❧✐st ✕ ❛♥s✇❡rs t❤❡ q✉❡st✐♦♥✿

◮

❲❤✐❝❤ ❜❛❧❧ ✐s t❤✐s ❜♦② ♣❧❛②✐♥❣ ✇✐t❤❄

❍♦✇❡✈❡r✱ ✇❤❛t ✐❢ ✇❡ t✉r♥ t❤✐s q✉❡st✐♦♥ ❛r♦✉♥❞✿

◮

❲❤✐❝❤ ❜♦② ✐s ♣❧❛②✐♥❣ ✇✐t❤ t❤✐s ❜❛❧❧❄

▲❡t✬s tr② ❛♥ ❡①❛♠♣❧❡✿ ❲❤♦ ✐s ♣❧❛②✐♥❣ ✇✐t❤ t❤❡ ❜❛s❦❡t❜❛❧❧❄ ❇❡❢♦r❡ ❛♥s✇❡r✐♥❣ ✐t✱ ✇❡ ❝❛♥ ❣✐✈❡ t❤✐s q✉❡st✐♦♥ ❛ ♠♦r❡ ❝♦♠♣❛❝t ❢♦r♠✱ t❤❡ ❢♦r♠ ♦❢ ❛♥

❡q✉❛t✐♦♥ ✿ F(

❜♦②

■♥❞❡❡❞✱ ✇❡ ♥❡❡❞ t♦ ✜♥❞ t❤❡ ✐♥♣✉ts t❤❛t✱ ✉♥❞❡r

F✱

)=

❜❛s❦❡t❜❛❧❧

.

♣r♦❞✉❝❡ t❤✐s s♣❡❝✐✜❝ ♦✉t♣✉t✳ ❲❡ ✈✐s✉❛❧✐③❡ ❛♥❞ ❛♥s✇❡r t❤❡

q✉❡st✐♦♥ ❜② ❡r❛s✐♥❣ ❛❧❧ ✐rr❡❧❡✈❛♥t ❛rr♦✇s✿

❚❤❡s❡ ❛r❡ ❛ ❢❡✇ ♦❢ ♣♦ss✐❜❧❡ q✉❡st✐♦♥s ♦❢ t❤✐s ❦✐♥❞ ❛❧♦♥❣ ✇✐t❤ t❤❡ ❛♥s✇❡rs✿

•

❲❤♦ ✐s ♣❧❛②✐♥❣ ✇✐t❤ t❤❡ ❜❛s❦❡t❜❛❧❧❄ ❚♦♠ ❛♥❞ ❇❡♥✦

•

❲❤♦ ✐s ♣❧❛②✐♥❣ ✇✐t❤ t❤❡ t❡♥♥✐s ❜❛❧❧❄ ◆❡❞✦

•

❲❤♦ ✐s ♣❧❛②✐♥❣ ✇✐t❤ t❤❡ ❜❛s❡❜❛❧❧❄ ◆♦ ♦♥❡✦

•

❲❤♦ ✐s ♣❧❛②✐♥❣ ✇✐t❤ t❤❡ ❢♦♦t❜❛❧❧❄ ❑❡♥ ❛♥❞ ❙✐❞✦

■t s❡❡♠s t❤❛t t❤❡r❡ ❛r❡ s❡✈❡r❛❧ ❛♥s✇❡rs t♦ ❡❛❝❤ ♦❢ t❤❡s❡ q✉❡st✐♦♥s✳ ❖r ❛r❡ t❤❡r❡❄ ✏❚♦♠✑ ❛♥❞ ✏❇❡♥✑ ❛r❡♥✬t

t✇♦

❛♥s✇❡rs❀ ✐t✬s ♦♥❡✿ ✏❚♦♠ ❛♥❞ ❇❡♥✑✦ ■♥❞❡❡❞✱ ✐❢ ✇❡ ♣r♦✈✐❞❡ ♦♥❡ ♥❛♠❡ ❛♥❞ ♥♦t t❤❡ ♦t❤❡r✱ ✇❡ ❤❛✈❡♥✬t ❢✉❧❧②

❛♥s✇❡r❡❞ t❤❡ q✉❡st✐♦♥✳ ❲❡ ❦♥♦✇ t❤❛t ✇❡ s❤♦✉❧❞ ✇r✐t❡ t❤❡ ❛♥s✇❡r ❛s

{

❚♦♠✱ ❇❡♥

}.

■t✬s ❛ s❡t✦ ▲❡t✬s r❡✈✐❡✇✳ ❚❤❡ s♦❧✉t✐♦♥ ♦❢ ❛♥ ❡q✉❛t✐♦♥ ♠❛② ❝♦♥t❛✐♥

❛❧❧

✈❛❧✉❡s ♦❢

❛♥②

x

f (x) = y

✇✐t❤

f :X→Y

✐s ❛❧✇❛②s ❛ s❡t ✭❛ s✉❜s❡t ♦❢

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

t❤❛t s❛t✐s❢② t❤❡ ❡q✉❛t✐♦♥✳ ■♥ ♦t❤❡r ✇♦r❞s✿

✶✳ ❲❤❡♥ ✇❡ s✉❜st✐t✉t❡ ❛♥② ♦❢ t❤♦s❡ ✷✳ ❚❤❡r❡ ❛r❡ ♥♦ ♦t❤❡r s✉❝❤

x✬s✳

x✬s

✐♥t♦ t❤❡ ❡q✉❛t✐♦♥✱ ✇❡ ❤❛✈❡ ❛ tr✉❡ st❛t❡♠❡♥t✳

x

X✮

❛♥❞ ✐t

♠❡❛♥s t♦ ✜♥❞

✺✳✸✳

❙♦❧✈✐♥❣ ❡q✉❛t✐♦♥s

✹✶✵

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

x + 2 = 5 =⇒ x = 3 . ❚❤❛t✬s ❛♥ ❛❜❜r❡✈✐❛t❡❞ ✈❡rs✐♦♥ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ st❛t❡♠❡♥t✿

◮

■❢

x

s❛t✐s✜❡s t❤❡ ❡q✉❛t✐♦♥

x + 2 = 5✱

t❤❡♥

x

s❛t✐s✜❡s t❤❡ ❡q✉❛t✐♦♥

x = 3✳

P❧✉❣ ✐♥✿

x + 2 = (3) + 2 = 5 . ■t ❝❤❡❝❦s ♦✉t✦ ❲❡ ❝♦✉❧❞

tr②

♦t❤❡rs ❛♥❞ t❤❡② ✇♦♥✬t ❝❤❡❝❦ ♦✉t✿

x=0 x=1 x=2 x=3 x=4 ...

(x) + 2 = (0) + 2 = 2 (1) + 2 = 3 (2) + 2 = 4 (3) + 2 = 5 (4) + 2 = 6 ...

=? 5 6= 5 6= 5 6= 5 =5 6= 5

❚❘❯❊✴❋❆▲❙❊ ❋❆▲❙❊ ❋❆▲❙❊ ❋❆▲❙❊ ❚❘❯❊ ❋❆▲❙❊ ...

❆❞❞ ✐t t♦ t❤❡ ❧✐st✦

❖❢ ❝♦✉rs❡✱ t❤✐s tr✐❛❧✲❛♥❞✲❡rr♦r ♠❡t❤♦❞ ✐s ✉♥❢❡❛s✐❜❧❡ ❜❡❝❛✉s❡ t❤❡r❡ ❛r❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♣♦ss✐❜✐❧✐t✐❡s✳

❤♦✇

❆ ♠❡t❤♦❞ ♦❢

✇❡ ♠❛② ❛rr✐✈❡ t♦ t❤❡ ❛♥s✇❡r ✐s ❞✐s❝✉ss❡❞ ✐♥ t❤✐s s❡❝t✐♦♥✳

❘❡❝❛❧❧ t❤❡ ❜❛s✐❝ ♠❡t❤♦❞s ✭✏r✉❧❡s✑✮ ♦❢

❤❛♥❞❧✐♥❣

❡q✉❛t✐♦♥s✳

❊①❛♠♣❧❡ ✺✳✸✳✶✿ s✐♠♣❧❡ ❡q✉❛t✐♦♥s ■♥ ♦r❞❡r t♦ s♦❧✈❡ t❤❡ ❡q✉❛t✐♦♥

x + 2 = 5, s✉❜tr❛❝t

2

❢r♦♠ ❜♦t❤ s✐❞❡s ♣r♦❞✉❝✐♥❣

x + 2 − 2 = 5 − 2 =⇒ x = 3 . ■♥ ♦r❞❡r t♦ s♦❧✈❡ t❤❡ ❡q✉❛t✐♦♥

3x = 2 , ❞✐✈✐❞❡ ❜②

3

❜♦t❤ s✐❞❡s ♣r♦❞✉❝✐♥❣

3x/3 = 2/3 =⇒ x = 2/3 . ❚❤✐s ✐s t❤❡ s✉♠♠❛r②✳

❚❤❡♦r❡♠ ✺✳✸✳✷✿ ❇❛s✐❝ ❆❧❣❡❜r❛ ♦❢ ❊q✉❛t✐♦♥s

•

▼✉❧t✐♣❧②✐♥❣ ❜♦t❤ s✐❞❡s ♦❢ ❛♥ ❡q✉❛t✐♦♥ ❜② ❛ ♥✉♠❜❡r ♣r❡s❡r✈❡s ✐t✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿

a = b =⇒ ka = ka •

❢♦r ❛♥②

k.

❆❞❞✐♥❣ ❛♥② ♥✉♠❜❡r t♦ ❜♦t❤ s✐❞❡s ♦❢ ❛♥ ❡q✉❛t✐♦♥ ♣r❡s❡r✈❡s ✐t✳ ✇♦r❞s✱ ✇❡ ❤❛✈❡✿

a = b =⇒ a + s = b + s

❢♦r ❛♥②

s.

❲❛r♥✐♥❣✦ ▼✉❧t✐♣❧②✐♥❣ ❛♥ ❡q✉❛t✐♦♥ ❜②

0

✐s ♣♦✐♥t❧❡ss✳

■♥ ♦t❤❡r

✺✳✸✳

❙♦❧✈✐♥❣ ❡q✉❛t✐♦♥s

✹✶✶

❊①❡r❝✐s❡ ✺✳✸✳✸ ❲❤❛t ❛❜♦✉t t❤❡ ❝♦♥✈❡rs❡s❄

❆s ❛ r❡♠✐♥❞❡r✱ ✐♥ t❤❡ s♣❡❝✐❛❧ ❝❛s❡ ✇❤❡♥ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ♦❢ t❤❡ ❡q✉❛t✐♦♥ ✐s ③❡r♦✱ t❤❡ s♦❧✉t✐♦♥ s❡t t♦ t❤✐s ❡q✉❛t✐♦♥ ❤❛s ❛ ❝❧❡❛r ❡q✉❛t✐♦♥

x✲❛①✐s

f (x) = 0✳

❣❡♦♠❡tr✐❝

♠❡❛♥✐♥❣✿

❆♥

x✲✐♥t❡r❝❡♣t ♦❢ ❛ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥ f x✲❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ✐♥t❡rs❡❝t✐♦♥s

■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡s❡ ❛r❡ t❤❡

✐s ❛♥② s♦❧✉t✐♦♥ t♦ t❤❡ ♦❢ t❤❡ ❣r❛♣❤ ✇✐t❤ t❤❡

✭t♦♣✮✿

❋✉rt❤❡r♠♦r❡✱ ✇❤❡♥ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ✐s ❛ ♥✉♠❜❡r✱ s❛②✱ t❤❡ ✐♥t❡rs❡❝t✐♦♥ ♦❢ t❤❡ ❣r❛♣❤ ✇✐t❤ t❤❡ ❧✐♥❡

y=k

k✱

t❤❡ s♦❧✉t✐♦♥ t♦ t❤✐s ❡q✉❛t✐♦♥

f (x) = k

❣✐✈❡s ✉s

✭❜♦tt♦♠✮✳

❊①❛♠♣❧❡ ✺✳✸✳✹✿ ❝♦✉♥t✐♥❣ s♦❧✉t✐♦♥s ❖✉r ❦♥♦✇❧❡❞❣❡ ♦❢ t❤❡ s❤❛♣❡ ♦❢ t❤❡ ❣r❛♣❤ ♦❢ ❛ ♣❛rt✐❝✉❧❛r ❢✉♥❝t✐♦♥ ✇✐❧❧ t❡❧❧ ✉s ❤♦✇ ♠❛♥② s♦❧✉t✐♦♥s s✉❝❤ ❛♥ ❡q✉❛t✐♦♥ ♠✐❣❤t ❤❛✈❡✳

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

❆♥ ❡q✉❛t✐♦♥ ✇✐t❤ ❛♥

nt❤

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

n

s♦❧✉t✐♦♥s✿

❇✉t ❛♥ ❡q✉❛t✐♦♥ ✇✐t❤ ❛♥ ♦❞❞ ❞❡❣r❡❡ ♣♦❧②♥♦♠✐❛❧ ✇✐❧❧ ❤❛✈❡ ❛t ❧❡❛st ♦♥❡ s♦❧✉t✐♦♥✦

❆♥ ❡q✉❛t✐♦♥ ✇✐t❤ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥ ❝❛♥ ❤❛✈❡ ♦♥❡ s♦❧✉t✐♦♥ ♦r ♥♦♥❡✿

✺✳✸✳ ❙♦❧✈✐♥❣ ❡q✉❛t✐♦♥s

✹✶✷

❆♥ ❡q✉❛t✐♦♥ ✇✐t❤ ❛ ♣❡r✐♦❞✐❝ ❢✉♥❝t✐♦♥ ✭s✉❝❤ ❛s t❤❡ s✐♥❡✮ ❝❛♥ ❤❛✈❡ ✐♥✜♥✐t❡❧② ♠❛♥② s♦❧✉t✐♦♥s ♦r ♥♦♥❡✿

❖✉r ✐♥t❡r❡st ✐♥ t❤✐s s❡❝t✐♦♥✱ t❤♦✉❣❤✱ ✐s ❛❧❣❡❜r❛✳ ❲❡ ✇✐❧❧ ❞❡❛❧ ✇✐t❤ ❛ s✐♠♣❧❡ ❦✐♥❞ ♦❢ ❡q✉❛t✐♦♥✿

◮ x ✐s ♣r❡s❡♥t ♦♥❧② ♦♥❝❡ ✭✐♥ t❤❡ ❧❡❢t✲❤❛♥❞ s✐❞❡✮✳ ▲✐❦❡ t❤✐s✿

x2 = 17 .

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

◮ x ✐s ✐s♦❧❛t❡❞ ✭✐♥ t❤❡ ❧❡❢t✲❤❛♥❞ s✐❞❡✮✳ ▲✐❦❡ t❤✐s✿

x=

√

17 . ❲❛r♥✐♥❣✦ ❚❤✐s ✐s ❛♥ ❡q✉❛t✐♦♥✳

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ✇✐❧❧ tr② t♦ ✜♥❞ ✇❛②s t♦ ❣❡t ❢r♦♠ ❧❡❢t t♦ r✐❣❤t ❤❡r❡✿  1  √ ...... sin e x = 5 −→ ??? −→ x = |{z}

♥♦ x ❤❡r❡

❚❤❡ ♠❛✐♥ ✐❞❡❛ ♦❢ ❤♦✇ t♦ ♠❛♥✐♣✉❧❛t❡ ❡q✉❛t✐♦♥s ✐s ❛s ❢♦❧❧♦✇s✿

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

x + 2 = 5, ✇❡ tr❡❛t ✐t ❛s ❛ ♥✉♠❜❡r✱ ❝❛❧❧ ✐t y ✳ ❚❤❡♥ ✇❡ ❞❡❛❧ ✇✐t❤ t❤✐s ♥✉♠❜❡r✿

y = x + 2 = 5, ❛♣♣❧② z = y − 2 =⇒ (x + 2) − 2 = 3 − 2 =⇒ x = 3 ❙♦❧✈❡❞✦ ❚❤❡ ✐❞❡❛ ✐s t♦ ♣r♦❞✉❝❡ ✕ ❢r♦♠ ❛♥ ❡q✉❛t✐♦♥ s❛t✐s✜❡❞ ❜② x ✕ ❛♥♦t❤❡r ❡q✉❛t✐♦♥ s❛t✐s✜❡❞ ❜② x✳ ❍♦✇❡✈❡r✱ t❤❡ ❝❤❛❧❧❡♥❣❡ ✭❛♥❞ ❛♥ ♦♣♣♦rt✉♥✐t②✮ ✐s t❤❛t ❛♣♣❧②✐♥❣ ❛♥② ❢✉♥❝t✐♦♥ ✐♥ t❤✐s ♠❛♥♥❡r ✇✐❧❧ ♣r♦❞✉❝❡ ❛ ♥❡✇ ❡q✉❛t✐♦♥ s❛t✐s✜❡❞ ❜② x✦ ❋♦r ❡①❛♠♣❧❡✿

y = x + 2 = 5, ❛♣♣❧② z = y + 2 =⇒ (x + 2) + 2 = 5 + 2 =⇒ x + 4 = 7 ◆♦t s♦❧✈❡❞✦ y = x + 2 = 5, ❛♣♣❧② z = y 2 =⇒ (x + 2)2 = 52 ◆♦t s♦❧✈❡❞✦ y = x + 2 = 5, ❛♣♣❧② z = sin y =⇒ sin(x + 2) = sin 5 ◆♦t s♦❧✈❡❞✦

✺✳✸✳ ❙♦❧✈✐♥❣ ❡q✉❛t✐♦♥s

✹✶✸

■♥❞❡❡❞✱ ✐t ✐s t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❛ ❢✉♥❝t✐♦♥ t❤❛t ❡✈❡r② ✐♥♣✉t ❤❛s ❡①❛❝t❧② ♦♥❡ ♦✉t♣✉t✳ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ❢✉♥❝t✐♦♥ g ✳ ❚❤❡♥✱ t✇♦ ❡q✉❛❧ ✭❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❡q✉❛t✐♦♥✮ ✐♥♣✉ts ♦❢ g ✇✐❧❧ ♣r♦❞✉❝❡ t✇♦ ❡q✉❛❧ ♦✉t♣✉ts ✭❛♥♦t❤❡r ❡q✉❛t✐♦♥✮✱ ❛❧✇❛②s✿ ♦❧❞ ❡q✉❛t✐♦♥✿ a = b   g g y y ♥❡✇ ❡q✉❛t✐♦♥✿ g(a) = g(b) ■♥ ♣❛rt✐❝✉❧❛r✱ ✇❡ ❤❛✈❡✱ ❢♦r ❛♥② ❢✉♥❝t✐♦♥ g ✿

x + 2 = 5 =⇒ g(x + 2) = g(5) . ❚❤❛t ✐s ✇❤② ✐❢ t❤❡ ✜rst ❡q✉❛t✐♦♥ ✐s s❛t✐s✜❡❞ ❜② x✱ t❤❡♥ s♦ ✐s t❤❡ s❡❝♦♥❞✳ ❚❤❡r❡ ❛r❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♣♦ss✐❜✐❧✐t✐❡s ❢♦r t❤✐s ❢✉♥❝t✐♦♥ g ✿

(x + 2)2 = 52 ↑ x+2=5 ↓ (x + 2)3 = 53

x+4=x+2+2=7 տ x+5=x+2+3=8 ← ւ x=x+2−2=3

sin(x + 2) = sin 5 ր → 2x+2 = 25 ց √ √ x+2= 5

■❢ x s❛t✐s✜❡s t❤❡ ❡q✉❛t✐♦♥ ✐♥ t❤❡ ♠✐❞❞❧❡✱ ✐t ❛❧s♦ s❛t✐s✜❡s t❤❡ r❡st ♦❢ t❤❡ ❡q✉❛t✐♦♥s✳ ■❢ ✇❡ ✇❛♥t t♦ s♦❧✈❡ t❤❡ ♦r✐❣✐♥❛❧ ❡q✉❛t✐♦♥✱ ✇❤✐❝❤ ❢✉♥❝t✐♦♥ ✕ ♦✉t ♦❢ ✐♥✜♥✐t❡❧② ♠❛♥② ✕ ❞♦ ✇❡ ♣✐❝❦❄ ❙♦♠❡ ♦❢ t❤❡♠ ❝❧❡❛r❧② ♠❛❦❡ t❤❡ ❡q✉❛t✐♦♥ ♠♦r❡ ❝♦♠♣❧❡① t❤❛♥ t❤❡ ♦r✐❣✐♥❛❧✦ ■t ✐s t❤❡ ❝❤❛❧❧❡♥❣❡ ❢♦r t❤❡ ❡q✉❛t✐♦♥ s♦❧✈❡r t♦ ❤❛✈❡ ❡♥♦✉❣❤ ❢♦r❡s✐❣❤t t♦ ❝❤♦♦s❡ ❛ ❢✉♥❝t✐♦♥ t♦ ❛♣♣❧② t❤❛t ✇✐❧❧ ♠❛❦❡ t❤❡ ❡q✉❛t✐♦♥ s✐♠♣❧❡r✳

❊①❛♠♣❧❡ ✺✳✸✳✺✿ s♦❧✈✐♥❣ ❡q✉❛t✐♦♥ ✇✐t❤ ❢✉♥❝t✐♦♥ ❣✐✈❡♥ ❜② ✢♦✇❝❤❛rt ▲❡t✬s ❝♦♥s✐❞❡r t❤✐s ❡q✉❛t✐♦♥✿

√

 x 3· + 1 = 6. 4 ❍❡r❡✱ s❡✈❡r❛❧ ❢✉♥❝t✐♦♥s ❛r❡ ❝♦♥s❡❝✉t✐✈❡❧② ❛♣♣❧✐❡❞ t♦ x✳ ❚❤✐s ✐s t❤❡ ✢♦✇❝❤❛rt ♦❢ t❤✐s ❢✉♥❝t✐♦♥✿ r♦♦t

f: x →

→

❞✐✈✐❞❡ ❜② 4

→

❛❞❞ 1

→0→

❛❞❞ 1

→

♠✉❧t✐♣❧② ❜② 3

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

0 →

❞✐✈✐❞❡ ❜② 4

→0→

❲❡ t❡st ♣♦ss✐❜❧❡ ✐♥♣✉ts t❤✐s ✇❛②✳ ❚❤❛t ♦♥❡ ❤❛s ❢❛✐❧❡❞❀ ✐t✬s ♥♦t 6✦

→ y

♠✉❧t✐♣❧② ❜② 3

→1→

→ 3

■s t❤❡r❡ ❛ ❜❡tt❡r ♠❡t❤♦❞❄ ❲❡ ✇♦✉❧❞ ❧✐❦❡ t♦ ❣❡t t♦ x✳ ❚♦ ❣❡t t♦ ✐t✱ ✇❡ ✇✐❧❧ ♥❡❡❞ t♦ ✉♥❞♦ t❤❡s❡ ❢✉♥❝t✐♦♥s ♦♥❡ ❜② ♦♥❡✳ ■♥ ✇❤❛t ♦r❞❡r❄ ❘✐❣❤t t♦ ❧❡❢t✱ ♦❢ ❝♦✉rs❡✳ ❲❡ r❡✈❡rs❡ t❤❡ ✢♦✇ ♦❢ t❤❡ ✢♦✇❝❤❛rt✿

f:

x →

f −1 : x ←

r♦♦t

→

sq✉❛r❡

←

❞✐✈✐❞❡ ❜② 4 ♠✉❧t✐♣❧② ❜② 4

→ ←

❛❞❞ 1 s✉❜tr❛❝t 1

→ ←

♠✉❧t✐♣❧② ❜② 3 ❞✐✈✐❞❡ ❜② 3

❖❢ ❝♦✉rs❡✱ t❤❡ ♣❛✐rs ♦❢ ❢✉♥❝t✐♦♥s ❛❧✐❣♥❡❞ ✈❡rt✐❝❛❧❧② ❛r❡ t❤❡ ✐♥✈❡rs❡s ♦❢ ❡❛❝❤ ♦t❤❡r✿

x → x ←

r♦♦t ✐♥✈❡rs❡❄ sq✉❛r❡

→ ←

❞✐✈✐❞❡ ❜② 4 ✐♥✈❡rs❡ ♠✉❧t✐♣❧② ❜② 4

→ ←

❛❞❞ 1 ✐♥✈❡rs❡ s✉❜tr❛❝t 1

❚❤❡②✱ t❤❡r❡❢♦r❡✱ ❝❛♥ ❜❡ ❝❛♥❝❡❧❡❞ ♦✉t ♦♥❡ ♣❛✐r ❛t ❛ t✐♠❡✿

x →

r♦♦t

→

❞✐✈✐❞❡ ❜② 4

→

x ←

sq✉❛r❡

←

♠✉❧t✐♣❧② ❜② 4

←

→ ←

❛❞❞ 1 ↓ s✉❜tr❛❝t 1

♠✉❧t✐♣❧② ❜② 3 ✐♥✈❡rs❡ ❞✐✈✐❞❡ ❜② 3

 

→ 6 ← 6

→ y ↓ ← y

✺✳✸✳

❙♦❧✈✐♥❣ ❡q✉❛t✐♦♥s

✹✶✹

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

x →

r♦♦t

x ←

sq✉❛r❡

→

❞✐✈✐❞❡ ❜②

←

♠✉❧t✐♣❧② ❜②

4

↓

4









❆♥❞ s♦ ♦♥✳ ❲❡ ❤❛✈❡ ❞❡♠♦♥str❛t❡❞ t❤❛t t❤❡ s❡❝♦♥❞ r♦✇ ✐s ✐♥❞❡❡❞ t❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ ✜rst✳ ❲✐t❤ t❤✐s ❢❛❝t ✉♥❞❡rst♦♦❞✱ ✇❡ ✜♥❞

±16 ←

sq✉❛r❡

←4←

x

❜② st❛rt✐♥❣ ♦♥ t❤❡ ❧❡❢t ✇✐t❤

←1←

4

♠✉❧t✐♣❧② ❜②

❙✉❝❤ ❡q✉❛t✐♦♥s ❝❛♥ ❜❡ ✈✐s✉❛❧✐③❡❞ ❛s ❢♦❧❧♦✇s❀ ✇❡ s❡❡ ❛ s✐♥❣❧❡

6✿

s✉❜tr❛❝t

x

←2←

1

❞✐✈✐❞❡ ❜②

3

←6

✏✇r❛♣♣❡❞✑ ✐♥ s❡✈❡r❛❧ ❧❛②❡rs ♦❢ ❢✉♥❝t✐♦♥s✱ ❛s ✐❢

❛ ❣✐❢t✿

❚♦ ❣❡t t♦ t❤❡ ❣✐❢t✱ t❤❡ ♦♥❧② ♠❡t❤♦❞ ✐s t♦ r❡♠♦✈❡ ♦♥❡ ✇r❛♣♣❡r ❛t ❛ t✐♠❡✱ ❢r♦♠ t❤❡ ♦✉ts✐❞❡ ✐♥✳ ■♥ ❢❛❝t✱ ②♦✉ ❞♦♥✬t ❡✈❡♥ ❦♥♦✇ ✇❤❛t ❦✐♥❞ ♦❢ ✇r❛♣♣❡r ✐s t❤❡ ♥❡①t ✉♥t✐❧ ②♦✉✬✈❡ r❡♠♦✈❡❞ t❤❡ ❧❛st ♦♥❡✦

❲❛r♥✐♥❣✦ ❆s ✇❡ ❛r❡ ✉♥✇r❛♣♣✐♥❣ t❤❡ ❧❡❢t✲❤❛♥❞ s✐❞❡✱ ✇❡ ❛r❡

✇r❛♣♣✐♥❣

t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ✐♥ t❤❡ ❧❛②❡rs ♦❢ t❤❡

✐♥✈❡rs❡s ♦❢ t❤❡s❡ ❢✉♥❝t✐♦♥s❀ ♠♦r❡ ✇♦r❦ ✐❢ t❤❡r❡ ✐s ❛♥

3·

√

x

✇❡ ❛r❡ ♦♥❧② ❝r❡❛t✐♥❣

t❤❡r❡✱ ❛s ✐♥✿

x +1 4



= x.

❊①❛♠♣❧❡ ✺✳✸✳✻✿ ✉♥✇r❛♣♣✐♥❣ ❧❛②❡rs ♦❢ ❢✉♥❝t✐♦♥s ❲❤❛t ✐❢ ✇❡ ❤❛✈❡

s❡✈❡r❛❧

❢✉♥❝t✐♦♥s ❛♣♣❧✐❡❞ ❝♦♥s❡❝✉t✐✈❡❧② t♦

❈♦♥s✐❞❡r✿ ❊q✉❛t✐♦♥ ✶✿ ❚❤❡ ❧❛st ♦♣❡r❛t✐♦♥ ♦♥ t❤❡ ❧❡❢t ✐s

−17✳

5·

 x 2

+1

2

x❄

❲❤✐❝❤ ❢✉♥❝t✐♦♥ ❞♦ ✇❡ ❝❤♦♦s❡ t♦ ❛♣♣❧②❄

 + 3 − 17 = 3

❚❤❛t✬s t❤❡ ❢✉♥❝t✐♦♥ ✇❡ ❢❛❝❡✱ ❛♥❞ t❤❡ ✐♥✈❡rs❡ ♦❢ t❤✐s ❢✉♥❝t✐♦♥ ✐s

t♦ ❜❡ ❛♣♣❧✐❡❞✳ ❲❡ ❝❤♦♦s❡✱ t❤❡r❡❢♦r❡✿

f (z) = z−17 , ✇❤❡r❡

z =5· ❚❤❡♥ ✇❡ ❛♣♣❧②

 x 2

+1

2



+3 .

g(y) = f −1 (y) = y+17 . ❲❡ ❝♦♥❝❧✉❞❡✿

z−17 = 3 =⇒ (z−17)+17 = 3+17 =⇒ z = 20 . ❲❡ ❤❛✈❡ ❛ ♥❡✇ ❡q✉❛t✐♦♥ ♥♦✇✿ ❊q✉❛t✐♦♥ ✷✿

5·

 x 2

+1

2

 + 3 = 20

✺✳✸✳

❙♦❧✈✐♥❣ ❡q✉❛t✐♦♥s

✹✶✺

❲❡ ♣❛✉s❡ t♦ ❛♣♣r❡❝✐❛t❡ t❤❡ ❢❛❝t t❤❛t ✇❡ ❢❛❝❡ ❛

s✐♠♣❧❡r

❡q✉❛t✐♦♥ t❤❛♥ t❤❡ ♦r✐❣✐♥❛❧✦

❲❡ ❝♦♥t✐♥✉❡ ✐♥ t❤✐s ❢❛s❤✐♦♥✳ ❆s ✇❡ ♣r♦❣r❡ss✱ ✇❡ ❛♣♣❧② t❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ t❤❛t ❛♣♣❡❛rs ✜rst ✐♥ t❤❡ ❡q✉❛t✐♦♥✱ ✐✳❡✳✱ ✐t ✐s ❛♣♣❧✐❡❞ ❧❛st✳

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

❢✉♥❝t✐♦♥s ♦♥❡ ❛t ❛ t✐♠❡✿

❊q✉❛t✐♦♥ ✷✿

❊q✉❛t✐♦♥ ✸✿

❊q✉❛t✐♦♥ ✹✿ ❊q✉❛t✐♦♥ ✺✿

❊q✉❛t✐♦♥ ✻✿ ❊q✉❛t✐♦♥ ✼✿

!2

  x 2  + 3 = 20 =⇒ 5 · + 1 + 3 /5 = 20/5 2 x 2 x =⇒ + 1 +3 = 4 + 1 +3 − 3 = 4−3 2 2 !2 r 2 √ x x =⇒ +1 =1 +1 = 1 2 2 x x +1 = 1 =⇒ +1 − 1 = 1−1 2 2 x x · 2 = 0·2 =0 =⇒ 2 2 x = 0. 5·



x +1 2 !2

=⇒ =⇒ ?

=⇒ =⇒ =⇒

❲❡ ❛r❡ ✜♥✐s❤❡❞✦ ❇✉t ✇❛✐t ❛ ♠✐♥✉t❡✱ t❤❡r❡ ❛r❡

t✇♦

x = −4 ✐s ❛❧s♦ ❛ s♦❧✉t✐♦♥✦ ❲❤❛t ❤❛♣♣❡♥❡❞❄ ❍♦✇ ❞✐❞ ✇❡ ❧♦s❡ y = x2 ❛♥❞ ❞✐sr❡❣❛r❞❡❞ t❤❡ ❧❛tt❡r ♦❢ t❤❡s❡ t✇♦ ❝❛s❡s✿ √ ❝❛s❡ ✶✿ z ≥ 0 =⇒ √z 2 = z . ❝❛s❡ ✷✿ z ≤ 0 =⇒ z 2 = −z .

✐t❄ ❲❡ ❢♦r❣♦t t❤❛t

✐♥✈❡rs❡s ♦❢

❲❡ ♥❡❡❞ t♦ r❡✲❞♦ t❤❡ s♦❧✉t✐♦♥ st❛rt✐♥❣ ❛t t❤❡ q✉❡st✐♦♥ ♠❛r❦✳ ❚❤❡ ❡q✉❛t✐♦♥ ♣r♦❞✉❝❡s t✇♦ ❝❛s❡s ✭t❤❡ ❧❛tt❡r ✐s ✜♥✐s❤❡❞ ❛❜♦✈❡✮✿

ւ

r

x z = +1≥0 2  x +1 = 1 =⇒ 2 x +1 = 1 =⇒ 2 x = 0 =⇒ 2 x ❇♦t❤

2 √ x +1 = 1 2

❖❘ ❖❘ ❖❘ ❖❘ ❖❘

=0

ց

x z = +1≤0 2 x  − +1 =1 =⇒ 2 x +1 = −1 =⇒ 2 x = −2 =⇒ 2 = −4

x

x = 1 ❛♥❞ x = −4 s❛t✐s❢② t❤❡✐r r❡s♣❡❝t✐✈❡ ❝♦♥❞✐t✐♦♥s✳

❚❤❡r❡❢♦r❡✱ t❤❡ s♦❧✉t✐♦♥ s❡t ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✿

{x : x = 0 ❖❘ x = −4} = {0} ∪ {−4} = {0, −4} . ❊①❡r❝✐s❡ ✺✳✸✳✼

❙♦❧✈❡ t❤❡ ❡q✉❛t✐♦♥✿

5·

  x2 2

+1

2

 + 3 − 17 = 3 .

✺✳✸✳ ❙♦❧✈✐♥❣ ❡q✉❛t✐♦♥s

✹✶✻

❊①❡r❝✐s❡ ✺✳✸✳✽ ❙♦❧✈❡ t❤❡ ❡q✉❛t✐♦♥✿



5·

 x 2

+1

2



+ 3 − 17

▼❛❦❡ ✉♣ ②♦✉r ♦✇♥ ❡q✉❛t✐♦♥ ❛♥❞ s♦❧✈❡ ✐t✳ ❘❡♣❡❛t✳

3

= 27 .

❊①❛♠♣❧❡ ✺✳✸✳✾✿ ♥♦t ♦♥❡✲t♦✲♦♥❡❄ ■❢ ❜❡t✇❡❡♥ ✉s ❛♥❞

x

t❤❡r❡ ✐s ❛ ❢✉♥❝t✐♦♥✱ ✇❡ ✇♦✉❧❞ ❧✐❦❡ t♦ r❡♠♦✈❡ ✐t✳ ❍♦✇❄ ❇❛s❡❞ ♦♥ ♦✉r ❡①♣❡r✐❡♥❝❡✱

t❤❡ ❛♥s✇❡r ✐s✿ ❆♣♣❧② t❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ t❤❛t ✇❡ ❢❛❝❡✦ ❲❡ ❞✐❞ t❤❛t ❛❜♦✈❡✿

x 7→ f (x) = x + 2 =⇒ y 7→ g(y) = f −1 (y) = y − 2 . ■s ❛♣♣❧②✐♥❣ t❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❛ ❢♦♦❧✲♣r♦♦❢ ♣❧❛♥ ❄ ◆♦✳ ❚r② t❤✐s✿

x2 = 1, ❲❡ ❤❛✈❡ ❧♦st

x = −1✦

z=

❛♣♣❧②

√

y =⇒

√

x2 =

√

1 =⇒ x = 1 ??

■t s❛t✐s✜❡s t❤❡ ✜rst ❡q✉❛t✐♦♥ ❜✉t ♥♦t t❤❡ ❧❛st✳ ❲❤❛t ❤❛♣♣❡♥❡❞❄ ❚❤❡r❡ ✐s ♥♦t❤✐♥❣

✇r♦♥❣ ✇✐t❤ t❤❡ ❧♦❣✐❝ ♦❢ ❛♣♣❧②✐♥❣ t❤❡ ✐♥✈❡rs❡ ❛♥❞ ♣r♦❞✉❝✐♥❣ ❛ ♥❡✇ ❡q✉❛t✐♦♥ s❛t✐s✜❡❞ ❜②

x❀

❤♦✇❡✈❡r✱

t❤❡ ❝❛♥❝❡❧❧❛t✐♦♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❛♥❞ ✐ts ✏✐♥✈❡rs❡✑ ✇❛s ✐♥❝♦rr❡❝t✳ ❚❤❡ sq✉❛r❡ ❛♥❞ t❤❡ sq✉❛r❡ r♦♦t ❛r❡ ✐♥✈❡rs❡s ♦❢ ❡❛❝❤ ♦t❤❡r ✭❛♥❞ ❝❛♥❝❡❧✮ ♦♥❧②✱ s❡♣❛r❛t❡❧②✱ ♦♥ t❤❡ r❛②s

[0, +∞)

❛♥❞

(−∞, 0]✳

❲❡ ❞✐sr❡❣❛r❞❡❞

t❤❡ ❧❛tt❡r ♦❢ t❤❡s❡ t✇♦ ❝❛s❡s✿ ❝❛s❡ ✶✿

√

x ≥ 0 =⇒

x2 = x,

x ≤ 0 =⇒

❝❛s❡ ✷✿

√

x2 = −x .

❲❡ ❝❛♥ ♣♦✐♥t ♦✉t ❡①❛❝t❧② ✇❤② t❤✐s ❤❛s ❢❛✐❧❡❞❀ t❤❡ ❢✉♥❝t✐♦♥ ✐s♥✬t ♦♥❡✲t♦✲♦♥❡✦

❊①❛♠♣❧❡ ✺✳✸✳✶✵✿ ♥♦t ♦♥t♦❄ ■s ❛♣♣❧②✐♥❣ t❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ✕ ♠❛❦✐♥❣ s✉r❡ t❤❛t ✐t ✐s ♦♥❡✲t♦✲♦♥❡ ✕ ❛ ❢♦♦❧✲♣r♦♦❢ ♣❧❛♥ ❄ ◆♦✳ ❚r② t❤✐s✿

√

x = −1,

❛♣♣❧②

z = y 2 =⇒

√
2 x = (−1)2 =⇒ x = 1 ??

❲❡ ❤❛✈❡ ❛ s♦❧✉t✐♦♥ ✇❤❡r❡ t❤❡r❡ ✐s ♥♦♥❡✦ ■t s❛t✐s✜❡s t❤❡ ❧❛st ❡q✉❛t✐♦♥ ❜✉t ♥♦t t❤❡ ✜rst✳ ❲❤❛t ❤❛♣♣❡♥❡❞❄ ❚❤❡r❡ ✐s ♥♦t❤✐♥❣ ✇r♦♥❣ ✇✐t❤ ♦✉r ❧♦❣✐❝❀ ❤♦✇❡✈❡r✱ t❤❡ ❝❛♥❝❡❧❧❛t✐♦♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❛♥❞ ✐ts ✏✐♥✈❡rs❡✑ ✇❛s ✐♥❝♦rr❡❝t✱ ♦♥❝❡ ❛❣❛✐♥✳ ❚❤❡ sq✉❛r❡ ❛♥❞ t❤❡ sq✉❛r❡ r♦♦t ❛r❡ ✐♥✈❡rs❡s ♦❢ ❡❛❝❤ ♦t❤❡r ✭❛♥❞ ❝❛♥❝❡❧✮ ♦♥❧②✱ s❡♣❛r❛t❡❧②✱ ♦♥ t❤❡ r❛②s

[0, +∞)

❛♥❞

(−∞, 0]✳

❲❡ ❝❛♥ ♣♦✐♥t ♦✉t ❡①❛❝t❧② ✇❤② t❤✐s ❢❛✐❧❡❞❀ t❤❡ ❢✉♥❝t✐♦♥

✐s♥✬t ♦♥t♦✦

❊①❛♠♣❧❡ ✺✳✸✳✶✶✿ s♦❧✈✐♥❣ ❡q✉❛t✐♦♥ ❈♦♥s✐❞❡r t❤❡ ❡q✉❛t✐♦♥✿

√

❊①❛♠✐♥✐♥❣ t❤❡ ❡q✉❛t✐♦♥✱ ✇❡ s❡❡

x

x + 1 = 3.

♦♥ t❤❡ ❧❡❢t ♦♥❧② ❛♥❞ ✐t ✐s s✉❜❥❡❝t❡❞ t♦ t✇♦ ❢✉♥❝t✐♦♥s✱ t❤❡ ❧❛st ♦❢

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

√

x+1

2

x✱

sq✉❛r❡ ❜♦t❤ s✐❞❡s✿

= 32 .

■❢ ✇❡ ✇❡r❡ t♦ ❝❛♥❝❡❧ t❤❡ t✇♦ ❢✉♥❝t✐♦♥s ♦♥ t❤❡ ❧❡❢t ❛s ✐♥✈❡rs❡s✱ ✇❡ ❣❡t t❤✐s ♥❡✇ ❡q✉❛t✐♦♥✿

x + 1 = 9. ❚❤❡ s♦❧✉t✐♦♥ ✐s

x = 8✳

❇✉t ✇❛s t❤❡ ❝❛♥❝❡❧❧❛t✐♦♥ ✈❛❧✐❞❄ ■♥st❡❛❞ ♦❢ ❝♦♥s✐❞❡r✐♥❣ t❤❡ ❢✉♥❝t✐♦♥✱ ✇❡ ❥✉st

❝♦♥✜r♠ t❤❛t t❤❡ s♦❧✉t✐♦♥ ❢❛❧❧s ✇✐t❤✐♥ t❤❡ ❞♦♠❛✐♥✱ ✇❤✐❝❤ ✐s

x + 1 ≥ 0✱ ♦❢ t❤❡ ♦r✐❣✐♥❛❧ ❡q✉❛t✐♦♥✳

■t ❞♦❡s✳

✺✳✸✳

❙♦❧✈✐♥❣ ❡q✉❛t✐♦♥s

✹✶✼

❊①❡r❝✐s❡ ✺✳✸✳✶✷

❙♦❧✈❡ t❤❡ ❡q✉❛t✐♦♥✿

√

x2 + 1 = 3 .

❊①❡r❝✐s❡ ✺✳✸✳✶✸

❙♦❧✈❡ t❤❡ ❡q✉❛t✐♦♥✿

√ 5 x + 1 = 3.

▼❛❦❡ ✉♣ ②♦✉r ♦✇♥ ❡q✉❛t✐♦♥ ❛♥❞ s♦❧✈❡ ✐t✳ ❘❡♣❡❛t✳

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

• ✏ ■❋ x • ✏x

s❛t✐s✜❡s ❛♥ ❡q✉❛t✐♦♥✱

s❛t✐s✜❡s ❛♥ ❡q✉❛t✐♦♥

❚❍❊◆

✐t s❛t✐s✜❡s t❤❡ ♥❡①t✑ t♦

■❋ ❆◆❉ ❖◆▲❨ ■❋

✐t s❛t✐s✜❡s t❤❡ ♥❡①t✑✳

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

t❤❡ s❛♠❡ s♦❧✉t✐♦♥ s❡t

❛s ✇❡ ♣r♦❣r❡ss t❤r♦✉❣❤

t❤❡ st❛❣❡s✳ ❋♦r ❡①❛♠♣❧❡✱ t❤✐s ✐s ❤♦✇ ✇❡ ✇♦✉❧❞ r❛t❤❡r ♣r❡s❡♥t t❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ ✈❡r② ✜rst ❡q✉❛t✐♦♥ ✐♥ t❤✐s s❡❝t✐♦♥✿

x + 2 = 5 ⇐⇒ x = 3 . ❚❤❛t✬s ❛♥ ❛❜❜r❡✈✐❛t❡❞ ✈❡rs✐♦♥ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ st❛t❡♠❡♥t✿

◮x

s❛t✐s✜❡s

x + 2 = 5 ■❋ ❆◆❉ ❖◆▲❨ ■❋ x

s❛t✐s✜❡s

x = 3✳

❘❡♣❡❛t❡❞ ❛s ♠❛♥② t✐♠❡s ❛s ♥❡❝❡ss❛r②✱ t❤❡ s♦❧✉t✐♦♥ s❡t ♦❢ ❡❛❝❤ ❡q✉❛t✐♦♥ ✐s t❤❡ s❛♠❡ ❛s t❤❛t ♦❢ t❤❡ ♦r✐❣✐♥❛❧ ❡q✉❛t✐♦♥✦ ❊①❛♠♣❧❡ ✺✳✸✳✶✹✿ s♦❧✈✐♥❣ ❡q✉❛t✐♦♥s

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

(1) 2(x + 2) − 3 = 5x ⇐⇒ 2x + 4 − 3 = 5x ⇐⇒ −3x = −1 ⇐⇒ x = 1/3 . (2) x2 + 1 = 0 ⇐⇒ x2 = −1 ⇐⇒ ∅ . ❙♦✱ t♦ s♦❧✈❡ t❤❡ t②♣❡ ♦❢ ❡q✉❛t✐♦♥ ✇❡ ❢❛❝❡ ✕ t❤❡ ✈❛r✐❛❜❧❡

x

s✉❜❥❡❝t❡❞ t♦ ❛ s❡q✉❡♥❝❡ ♦❢ ❢✉♥❝t✐♦♥s ✕ ✇❡ ❛♣♣❧②

t❤❡ ✐♥✈❡rs❡s ♦❢ t❤❡s❡ ❢✉♥❝t✐♦♥s ✐♥ t❤❡ r❡✈❡rs❡❞ ♦r❞❡r✳ ❲❡ ♠❛② ❤❛✈❡ t♦ s♣❧✐t t❤❡ ❞♦♠❛✐♥s ♦❢ t❤❡ ❢✉♥❝t✐♦♥s s♦ t❤❛t t❤❡② ❛r❡ ❜♦t❤ ♦♥❡✲t♦✲♦♥❡ ❛♥❞ ♦♥t♦✳ ❆s ✇❡ ❤❛✈❡ s❡❡♥✱ t❤✐s st❡♣ ❛❧s♦ s♣❧✐ts ♦✉r ❡q✉❛t✐♦♥✳ ❊①❛♠♣❧❡ ✺✳✸✳✶✺✿ s♣❧✐t ❞♦♠❛✐♥

❆ s♦❧✉t✐♦♥ ✐s s❤♦✇♥ ❜❡❧♦✇✿

x2 = 1 ⇐⇒ x = −1 ❖❘ x = 1 .

❊①❡r❝✐s❡ ✺✳✸✳✶✻

❙♦❧✈❡ t❤❡ ❡q✉❛t✐♦♥ ✐♥ t❤✐s ♠❛♥♥❡r

√

x + 1 = 3.

❊①❡r❝✐s❡ ✺✳✸✳✶✼

❙♦❧✈❡ t❤❡ ❡q✉❛t✐♦♥ ✐♥ t❤✐s ♠❛♥♥❡r

√

x2 + 1 = 3 .

▼❛❦❡ ✉♣ ②♦✉r ♦✇♥ ❡q✉❛t✐♦♥ ❛♥❞ s♦❧✈❡ ✐t✳ ❘❡♣❡❛t✳

✺✳✹✳

❚❤❡ ❛❧❣❡❜r❛ ♦❢ ❧♦❣❛r✐t❤♠s

✹✶✽

❊①❛♠♣❧❡ ✺✳✸✳✶✽✿ s♣❧✐t ❞♦♠❛✐♥

❲❤❛t ✐❢✱ ✐♥st❡❛❞ ♦❢ ❛ s✐♥❣❧❡

x✱

t❤✐s✿

5·



x2 + x + 1


2

❋♦r ❡①❛♠♣❧❡✱ ✇❡ ♠✐❣❤t ❢❛❝❡

y = x2 + x + 1 ✱

r❡♣r❡s❡♥t❡❞ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢

 + 3 − 17 = 3 .

❇✉t ✇❤❛t ✐s ❛♥ ✏❡①♣r❡ss✐♦♥✑ ✐❢ ♥♦t ❥✉st ❛♥♦t❤❡r ✈❛r✐❛❜❧❡✱

x✳

x❄

✇❡ ❢❛❝❡ ❛♥ ❡①♣r❡ss✐♦♥ t❤❛t ❞❡♣❡♥❞s ♦♥

❚♦ ♠❛❦❡ t❤✐s ❝❤♦✐❝❡ ❝❧❡❛r✱ ✇❡ s✉❜st✐t✉t❡✿

  5 · (y)2 + 3 − 17 = 3 .

■♥ t❤❛t ✇❡ ❝❛s❡✱ ✇❡ ❣♦ ❛❢t❡r t❤✐s ✈❛r✐❛❜❧❡✱ ♦♥❡ ❜② ♦♥❡✳ ❖♥❝❡

y

y✱

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

✐s ❢♦✉♥❞✱ ❛s ✐♥ t❤❡ ❡①❛♠♣❧❡✮✿

y = 0 ❖❘ y = −4 . ✇❡ ❤❛✈❡ ❛ ♠✉❝❤ s✐♠♣❧❡r ❡q✉❛t✐♦♥✱ ♦r ❛ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ❡q✉❛t✐♦♥s✱ ❢♦r

x✿

x2 + x + 1 = 0 ❖❘ x2 + x + 1 = −4 . ❊①❡r❝✐s❡ ✺✳✸✳✶✾

❋✐♥✐s❤ t❤❡ s♦❧✉t✐♦♥✳ ▼❛❦❡ ✉♣ ②♦✉r ♦✇♥ ❡q✉❛t✐♦♥ ❛♥❞ s♦❧✈❡ ✐t✳ ❘❡♣❡❛t✳

❆s ❛ s✉♠♠❛r②✱ ✇❡ ❤❛✈❡ ❞❡✈❡❧♦♣❡❞ t❤❡ ❢♦❧❧♦✇✐♥❣ ♠❡t❤♦❞ ♦❢ s♦❧✈✐♥❣ ❡q✉❛t✐♦♥s✳ ❚❤❡♦r❡♠ ✺✳✸✳✷✵✿ ●❡♥❡r❛❧ ❆❧❣❡❜r❛ ♦❢ ❊q✉❛t✐♦♥s ❙✉♣♣♦s❡

g ✐s ❛♥ ✐♥✈❡rt✐❜❧❡ ❢✉♥❝t✐♦♥✳

❚❤❡♥ ❛♣♣❧②✐♥❣

g t♦ ❜♦t❤ s✐❞❡s ♦❢ ❛♥ ❡q✉❛t✐♦♥

❝r❡❛t❡s ❛♥ ❡q✉✐✈❛❧❡♥t ❡q✉❛t✐♦♥✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿

a = b ⇐⇒ g(a) = g(a) . ❋✉rt❤❡r♠♦r❡✱ t❤❡ t✇♦ ❡q✉❛t✐♦♥s ❤❛✈❡ t❤❡ s❛♠❡ s♦❧✉t✐♦♥ s❡t✳

❊①❡r❝✐s❡ ✺✳✸✳✷✶

❲❤❛t ✐s t❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ t✇♦ s♦❧✉t✐♦♥ s❡ts ✐❢ ✐♥st❡❛❞ ♦❢

⇐⇒✱

✇❡ ❤❛✈❡ ✭❛✮

⇐= ✱

✭❜✮

=⇒ ❄

✺✳✹✳ ❚❤❡ ❛❧❣❡❜r❛ ♦❢ ❧♦❣❛r✐t❤♠s

▲❡t✬s ❣♦ ❛❧❧ t❤❡ ✇❛② ❜❛❝❦ t♦ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❡①♣♦♥❡♥t ❛s r❡♣❡❛t❡❞ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✭❈❤❛♣t❡r ✶✮✿

· ... · a} = ax , |a · a {z x t✐♠❡s

❢♦r ❛♥②

x

x = 1, 2, 3, ...✳

❲❡ ❞❡✜♥❡❞ t❤❡ ❧♦❣❛r✐t❤♠ ❜❛s❡

a ♦❢ y

✐s ❛ ♥❛t✉r❛❧ ♥✉♠❜❡r✱ t❤✐s s✐♠♣❧② ♠❡❛♥s ✏t❤❡ ♥✉♠❜❡r ♦❢ t✐♠❡s ■ ♠✉❧t✐♣❧②

a

❜② ✐ts❡❧❢ t♦




loga |a · a {z · ... · a} = x . x t✐♠❡s

a t♦ ❣❡t y ✑✳ ❣❡t y ✑✿

t♦ ❜❡ ✏t❤❡ ♣♦✇❡r ■ ❛♠ t♦ r❛✐s❡

❘✉❧❡s ♦❢ ❊①♣♦♥❡♥ts t❤❡♦r❡♠✱ ✇❡ ✇✐❧❧ ❞❡r✐✈❡ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ r✉❧❡s ❛❜♦✉t t❤❡ ❧♦❣❛r✐t❤♠✳ ❲❡ ✇✐❧❧ ❝♦♥t✐♥✉❡ t♦ ✉s❡ t❤❡ ❈❛♥❝❡❧❧❛t✐♦♥ ▲❛✇s ✿ ❋r♦♠ t❤❡

❲❤❡♥

✺✳✹✳ ❚❤❡ ❛❧❣❡❜r❛ ♦❢ ❧♦❣❛r✐t❤♠s

✹✶✾

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

a0 = 1 ⇐⇒ loga 1 = 0 ❛♥❞

a1 = a ⇐⇒ loga a = 1 ❲❡ t✉r♥ t♦ t❤❡ ❆❞❞✐t✐♦♥✲▼✉❧t✐♣❧✐❝❛t✐♦♥ ❘✉❧❡ ✭#1✮ ♥♦✇✿

ax+y = ax ·ay . ■♥ t❤❡ ❝❛s❡ ♦❢ ♥❛t✉r❛❧

x

❛♥❞

y✱

✐t ❝♦♠❡s ❢r♦♠ ❝♦✉♥t✐♥❣ t✇♦ str✐♥❣s ♦❢

x+y t✐♠❡s

a✬s✿

ax+y

z }| { }| { z · ... · a} = |a · a {z · ... · a} = ax+y . · ... · a} · |a · a {z · ... · a} · |a · a {z a ·a =a | · a {z x

y

x t✐♠❡s

y t✐♠❡s

ax

ay

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

loga

  x+y t✐♠❡s
z }| { ax · ay = loga |a · a {z · ... · a} = x + y . · ... · a} · a | · a {z x t✐♠❡s

▲❡t✬s ❛♣♣❧②

loga

t♦ t❤✐s ❡q✉❛t✐♦♥ ❛♥❞ t❤❡♥ ❝♦♥❝❡♥tr❛t❡ ♦♥

y t✐♠❡s

X = ax

❛♥❞

Y = ay ✳

❚❤❡♦r❡♠ ✺✳✹✳✶✿ ❆❞❞✐t✐♦♥✲▼✉❧t✐♣❧✐❝❛t✐♦♥ ❘✉❧❡ ♦❢ ▲♦❣❛r✐t❤♠s ❋♦r ❛♥②

a>0

❛♥❞ ❛♥②

X, Y > 0✱

✇❡ ❤❛✈❡✿

loga (X · Y ) = loga X + loga Y Pr♦♦❢✳ ❲❡ ❛✐♠ t♦ ❛♣♣❧② ❆❞❞✐t✐♦♥✲▼✉❧t✐♣❧✐❝❛t✐♦♥ ❘✉❧❡ ♦❢ ❊①♣♦♥❡♥ts✳

❇✉t ✇❤❛t ❛r❡ t❤♦s❡ ❡①♣♦♥❡♥ts❄

❲❡

❞❡✜♥❡✿

x = loga X, y = loga Y ⇐⇒ X = ax , Y = ay . ❲❡ ♥♦✇ ❛♣♣❧② t❤❡ ❧♦❣❛r✐t❤♠ t♦ ❜♦t❤ s✐❞❡s ♦❢ t❤❡ r✉❧❡✿

ax+y ⇐⇒ loga (ax+y ) ⇐⇒ x+y ⇐⇒ loga X + loga Y

= ax = loga (ax = loga (ax = loga (X

· · · ·

ay ay ) ay ) Y ).

loga . ◆♦✇ ❝❛♥❝❡❧✳ ❆♣♣❧②

◆♦✇ s✉❜st✐t✉t❡✳

❊①❛♠♣❧❡ ✺✳✹✳✷✿ ❡①♣❛♥❞✲❝♦♥tr❛❝t ■♥ ❛♥ ❛tt❡♠♣t t♦ ✏s✐♠♣❧✐❢②✑ ❛♥ ❡①♣r❡ss✐♦♥✱ ✇❡ ♠✐❣❤t ❤❛✈❡ t♦ ❝❛rr② ♦✉t t❤❡ ❢♦❧❧♦✇✐♥❣ ♦♣♣♦s✐t❡ t❛s❦s✳

✺✳✹✳

❚❤❡ ❛❧❣❡❜r❛ ♦❢ ❧♦❣❛r✐t❤♠s

✹✷✵

❋✐rst✱ ✇❡ ♠❛② ❤❛✈❡ t♦ ❡①♣❛♥❞ ❛ ❧♦❣❛r✐t❤♠ ✐♥t♦ s❡✈❡r❛❧ ❜② r❡❛❞✐♥❣ t❤❡ ❧❛st ❢♦r♠✉❧❛ ❢r♦♠ ❧❡❢t t♦ r✐❣❤t✿ log2 (24) = log2 (23 · 3) = log2 (23 ) + log2 (3) = 3 + log2 3 .

❚❤✐s r❡q✉✐r❡s ❢❛❝t♦r✐♥❣ t❤❡ ♥✉♠❜❡r ✐♥s✐❞❡ t❤❡ ❧♦❣❛r✐t❤♠✳ ❍❛✈✐♥❣ ♦♥❡ ♦❢ t❤❡ ❢❛❝t♦rs ❡q✉❛❧ t♦ t❤❡ ❜❛s❡ ❤❡❧♣s t♦ s✐♠♣❧✐❢② ❢✉rt❤❡r✳ ❙❡❝♦♥❞✱ ✇❡ ♠❛② ❤❛✈❡ t♦ ❝♦♥tr❛❝t s❡✈❡r❛❧ ❧♦❣❛r✐t❤♠s ✐♥t♦ ♦♥❡ ❜② r❡❛❞✐♥❣ t❤❡ ❢♦r♠✉❧❛ ❢r♦♠ r✐❣❤t t♦ ❧❡❢t✿ log3 (7) + log3 (2) + 1 = log3 (7 · 2) + log3 (3) = log3 (7 · 2 · 3) = log3 (56) .

❚❤❡ ♥❡❡❞ ❢♦r ♦♥❡ ♦r t❤❡ ♦t❤❡r ❞❡♣❡♥❞s ♦♥ t❤❡ ❝♦♥t❡①t ♦r t❤❡ ❣♦❛❧ ✇❡ ❛r❡ ♣✉rs✉✐♥❣✳ ❚❤✐s ✐s ✐❞❡♥t✐❝❛❧ t♦ ✇❤❛t ✇❡ ❞✐❞ ✐♥ t❤❡ ♣❛st ✇✐t❤ t❤❡ ❇✐♥♦♠✐❛❧

❋♦r♠✉❧❛ ✿

= x2 + 2x + 1, ❡①♣❛♥❞ ❝♦♥tr❛❝t, 9x2 + 6x + 1 = (3x + 1)2

(x + 1)2

❊①❡r❝✐s❡ ✺✳✹✳✸

❊①♣❛♥❞ t❤❡ ❧♦❣❛r✐t❤♠ log3 (24)✳ ▼❛❦❡ ②♦✉r ♦✇♥ ❡①♣r❡ss✐♦♥ ❛♥❞ ❡①♣❛♥❞ ✐t✳ ❘❡♣❡❛t✳ ❊①❡r❝✐s❡ ✺✳✹✳✹

❈♦♥tr❛❝t t❤✐s t♦ ❛ s✐♥❣❧❡ ❧♦❣❛r✐t❤♠✿ log2 (27) + log2 (2) + 0✳ ▼❛❦❡ ②♦✉r ♦✇♥ ❡①♣r❡ss✐♦♥ ❛♥❞ ❝♦♥tr❛❝t ✐t✳ ❘❡♣❡❛t✳ ❊①❛♠♣❧❡ ✺✳✹✳✺✿ t❛❜❧❡s ♦❢ ❧♦❣❛r✐t❤♠s

❚❤❡ ❧♦❣❛r✐t❤♠s ✇❡r❡ ✉s❡❞ ❢♦r ❝♦♠♣✉t❛t✐♦♥s ✐♥ t❤❡ t✐♠❡s ❜❡❢♦r❡ ❝♦♠♣✉t❡rs✳ ❙♣❡❝✐✜❝❛❧❧②✱ t❤❡ ❆❞❞✐t✐♦♥✲ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❘✉❧❡ ♦❢ ❊①♣♦♥❡♥ts ✇❛s ✉s❡❞ t♦ ♠✉❧t✐♣❧② ❧❛r❣❡ ♥✉♠❜❡rs✳ ❚❤❡ ❦❡② ✐❞❡❛ ✐s t❤❛t ✐t ❝♦♥✈❡rts ♠✉❧t✐♣❧✐❝❛t✐♦♥ t♦ ❛❞❞✐t✐♦♥✱ ✇❤✐❝❤ ✐s ❡❛s✐❡r✿ loga X

× |{z}

♠✉❧t✐♣❧✐❝❛t✐♦♥

=

loga X

+ |{z}

Y




loga Y

❛❞❞✐t✐♦♥

❚❤❡② ✉s❡❞ ❧❛r❣❡ t❛❜❧❡s ♦❢ ♣r❡✲❝♦♠♣✉t❡❞ ❧♦❣❛r✐t❤♠s ❛♥❞ ❡①♣♦♥❡♥ts✿

✺✳✹✳

❚❤❡ ❛❧❣❡❜r❛ ♦❢ ❧♦❣❛r✐t❤♠s

✹✷✶

❚♦ ♠✉❧t✐♣❧② X ❜② Y ✱ ❢♦❧❧♦✇ t❤❡s❡ st❡♣s✿

X→

Y →

t❛❜❧❡

t❛❜❧❡

→ loga X = x

→ loga Y = y

ց ր

x+y =z →

t❛❜❧❡

→ az = X · Y, ❞♦♥❡✳

❚❤❡ ❝♦♠♣✉t❛t✐♦♥ ❛♠♦✉♥ts t♦ ✉s✐♥❣ t❤❡ t❛❜❧❡s t❤r❡❡ t✐♠❡s ❛♥❞ ❞♦✐♥❣ ❛❞❞✐t✐♦♥ ✭✐♥st❡❛❞ ♦❢ ♠✉❧t✐♣❧✐❝❛t✐♦♥✮ ♦♥❝❡✳ ❋♦r ❡①❛♠♣❧❡✱ t♦ ♠✉❧t✐♣❧② 1234 ❛♥❞ 4321 ✇❡ ❢♦❧❧♦✇ t❤❡s❡ st❡♣s✿

1234 → t❛❜❧❡ → ln 1234 4321 → t❛❜❧❡ → ln 4321 → t❛❜❧❡ → e15.48925834

= 7.118016204 + = 8.371242136 = 15.48925834 = 5332114

❚❤❡ s❛♠❡ ♠❡t❤♦❞ ♦❢ ❛❞❞✐t✐♦♥ ✐s ✉s❡❞ ✐♥ t❤❡ ❞❡s✐❣♥ ♦❢ t❤❡ s❧✐❞❡ r✉❧❡✳ ❚❤❡ t✇♦ r✉❧❡s ✕ ♦❢ ❡①♣♦♥❡♥ts ❛♥❞ ❧♦❣❛r✐t❤♠s ✕ ♠❛t❝❤✿

ax+y loga X+ loga Y

= ax ·ay , = loga (X·Y )

■♥ ❢❛❝t✱ t❤❡s❡ ❛❞❞✐t✐♦♥ s✐❣♥s ♦♥ t❤❡ ❧❡❢t ❛♥❞ t❤❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥ s✐❣♥s ♦♥ t❤❡ r✐❣❤t ♠❛t❝❤ ✉♣ t♦♦✦ ❙♦✱ t❤❡ ❛❞❞✐t✐♦♥ t❤❛t ✐s ❤❛♣♣❡♥✐♥❣ ✐♥ t❤❡ x✲❛①✐s ✐s tr❛♥s❢♦r♠❡❞ ❜② t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥ t♦ t❤❡ y ✲❛①✐s ✇❤❡r❡ ✐t ❜❡❝♦♠❡s ♠✉❧t✐♣❧✐❝❛t✐♦♥❀ t❤❡ ❧♦❣❛r✐t❤♠ ❞♦❡s t❤❡ ♦♣♣♦s✐t❡✿

◆❡①t✱ ✐❢

• t❤❡ ❧♦❣❛r✐t❤♠s t✉r♥ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✐♥t♦ ❛❞❞✐t✐♦♥✱ t❤❡♥ • t❤❡ ❧♦❣❛r✐t❤♠s t✉r♥ ❞✐✈✐s✐♦♥ ✐♥t♦ s✉❜tr❛❝t✐♦♥✳

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

ax−y loga X− loga Y

= ax ÷ay = loga (X÷Y )

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

loga

1 = − loga Y Y

✺✳✹✳

✹✷✷

❚❤❡ ❛❧❣❡❜r❛ ♦❢ ❧♦❣❛r✐t❤♠s

❊①❛♠♣❧❡ ✺✳✹✳✻✿ ❡①♣❛♥❞ ❛♥❞ ❝♦♥tr❛❝t

❚❤❡s❡ ❢♦r♠✉❧❛s ❝❛♥ ❜❡ r❡❛❞ ❢r♦♠ ❧❡❢t t♦ r✐❣❤t✿ log3 6 − log3 18 = log3

♦r ❢r♦♠ r✐❣❤t t♦ ❧❡❢t✿ log5

1 6 = log3 = log3 3−1 = −1 ; 18 3

1 = log5 1 − log5 25 = −2 . 25

❊①❡r❝✐s❡ ✺✳✹✳✼

❊①♣❛♥❞ t❤❡ ❧♦❣❛r✐t❤♠ log2 (.125)✳ ▼❛❦❡ ②♦✉r ♦✇♥ ❡①♣r❡ss✐♦♥ ❛♥❞ ❡①♣❛♥❞ ✐t✳ ❘❡♣❡❛t✳ ❊①❡r❝✐s❡ ✺✳✹✳✽

❈♦♥tr❛❝t t❤✐s t♦ ❛ s✐♥❣❧❡ ❧♦❣❛r✐t❤♠✿ log2 (27) − log2 (2) + 1✳ ▼❛❦❡ ②♦✉r ♦✇♥ ❡①♣r❡ss✐♦♥ ❛♥❞ ❝♦♥tr❛❝t ✐t✳ ❘❡♣❡❛t✳ ◆♦✇ t❤❡ ▼✉❧t✐♣❧✐❝❛t✐♦♥✲❊①♣♦♥❡♥t✐❛t✐♦♥ ❘✉❧❡ ✭#3✮✿ axy = (ax )y .

■♥ t❤❡ ❝❛s❡ ♦❢ ♥❛t✉r❛❧ x ❛♥❞ y ✱ ✐t ❝♦♠❡s ❢r♦♠ ❝♦✉♥t✐♥❣ r❡♣❡❛t❡❞ str✐♥❣s ♦❢ a✬s✿ ax
y ax ax = , ... ax

   

y t✐♠❡s

  

 a · a · ... · a  | {z }    x t✐♠❡s    a · a · ... · a | {z }  = x t✐♠❡s   ...     a · a · ... · a | {z }  

y t✐♠❡s

x t✐♠❡s

=a · ... · a} = axy . | · a {z xy t✐♠❡s

❙♦✱ ✇❡ ❝❛♥ s❡❡ t❤❛t ✇❤❡♥ ❛ ♣♦✇❡r ✐s t❛❦❡♥ t♦ ❛♥♦t❤❡r ♣♦✇❡r✱ t❤❡ ❢♦r♠❡r ♣♦✇❡r✱ ✐✳❡✳✱ t❤❡ ❧♦❣❛r✐t❤♠✱ ✐s ♠✉❧t✐♣❧✐❡❞ ❜② t❤❡ ❧❛tt❡r ♣♦✇❡r✿ loga

   

y · ... · a}x  = loga a = loga |ax · ax{z ax · ... · a} = xy . | · a {z y t✐♠❡s

xy t✐♠❡s

▲❡t✬s ❝♦♥❝❡♥tr❛t❡ ♦♥ X = ax ✳

❚❤❡♦r❡♠ ✺✳✹✳✾✿ ▼✉❧t✐♣❧✐❝❛t✐♦♥✲❊①♣♦♥❡♥t✐❛t✐♦♥ ❘✉❧❡ ♦❢ ▲♦❣❛r✐t❤♠s ❋♦r ❛♥②

a>0

❛♥❞ ❛♥②

x, y > 0✱

✇❡ ❤❛✈❡✿

loga (X y ) = y · loga X Pr♦♦❢✳

❲❡ ❛✐♠ t♦ ❛♣♣❧② t❤❡ ❲❡ ❞❡✜♥❡✿

▼✉❧t✐♣❧✐❝❛t✐♦♥✲❊①♣♦♥❡♥t✐❛t✐♦♥ ❘✉❧❡ ♦❢ ❊①♣♦♥❡♥ts✳

x = loga X ⇐⇒ X = ax .

❇✉t ✇❤❛t ✐s t❤✐s ❡①♣♦♥❡♥t❄

✺✳✹✳

❚❤❡ ❛❧❣❡❜r❛ ♦❢ ❧♦❣❛r✐t❤♠s

✹✷✸

❲❡ ♥♦✇ ❛♣♣❧② t❤❡ ❧♦❣❛r✐t❤♠ t♦ ❜♦t❤ s✐❞❡s ♦❢ t❤❡ ✐❞❡♥t✐t②✿

axy = x·y

⇐⇒ loga (a ⇐⇒

) = loga

x · y = loga

(ax )y
(ax )y
(ax )y

❆♣♣❧②

loga .

◆♦✇ ❝❛♥❝❡❧✳ ◆♦✇ s✉❜st✐t✉t❡✳

loga (X y ) .

⇐⇒ loga X · y =

❚❤❡ ❢♦r♠✉❧❛ ❛♥s✇❡rs t❤❡ q✉❡st✐♦♥ ❛❜♦✉t ✇❤❛t ❤❛♣♣❡♥s ✐❢ ✇❡ ❛♣♣❧② ❛ ❧♦❣❛r✐t❤♠ t♦ ❛♥ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥ ✇✐t❤ ❛

❞✐✛❡r❡♥t

❜❛s❡✳

❝♦♠♣♦s✐t✐♦♥ ✐♥t♦ ❛

❚❤♦✉❣❤ ♥♦t ❛ ❝♦♠♣❧❡t❡ ❝❛♥❝❡❧❧❛t✐♦♥ ✭❛s ✐♥ t❤❡ ❝❛s❡ ♦❢ ❡q✉❛❧ ❜❛s❡s✮✱ ✐t t✉r♥s t❤❡

❧✐♥❡❛r ❢✉♥❝t✐♦♥ ✿ b=a

❖❢ ❝♦✉rs❡✱ ❝❤♦♦s✐♥❣

loga (by ) = loga b · y .

❜r✐♥❣s ✉s ❜❛❝❦ t♦ t❤❡

❈❛♥❝❡❧❧❛t✐♦♥ ▲❛✇✱ ✐✳❡✳✱ t❤❡ ❞❡✜♥✐t✐♦♥✿

loga (ay ) = y loga a = y . ❊①❛♠♣❧❡ ✺✳✹✳✶✵✿ s✐♠♣❧✐❢②

♦♣♣♦s✐t❡

■♥ ❛♥ ❛tt❡♠♣t t♦ ✏s✐♠♣❧✐❢②✑ ❛♥ ❡①♣r❡ss✐♦♥✱ ✇❡ ♠✐❣❤t ❤❛✈❡ t♦ ❝❛rr② ♦✉t t❤❡ ❢♦❧❧♦✇✐♥❣ ❋✐rst✱ ✇❡ ♠❛② ❤❛✈❡ t♦

❡①♣❛♥❞

t❛s❦s✳

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

❢r♦♠ ❧❡❢t t♦ r✐❣❤t✿

log5 (32) = log5 (25 ) = 5 log5 2 . ❙❡❝♦♥❞✱ ✇❡ ♠❛② ❤❛✈❡ t♦

❝♦♥tr❛❝t

r✐❣❤t t♦ ❧❡❢t✿

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

√ 1 log3 7 = log3 (71/2 ) = log3 7 . 2

❚❤❡ ♥❡❡❞ ❢♦r ♦♥❡ ♦r t❤❡ ♦t❤❡r ❞❡♣❡♥❞s ♦♥ t❤❡ ❝♦♥t❡①t ♦r t❤❡ ❣♦❛❧ ✇❡ ❛r❡ ♣✉rs✉✐♥❣✳

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

❊①❡r❝✐s❡ ✺✳✹✳✶✶

❊①♣❛♥❞ t❤❡ ❧♦❣❛r✐t❤♠

log2 (100)✳

▼❛❦❡ ②♦✉r ♦✇♥ ❡①♣r❡ss✐♦♥ ❛♥❞ ❡①♣❛♥❞ ✐t✳ ❘❡♣❡❛t✳

❊①❡r❝✐s❡ ✺✳✹✳✶✷

❈♦♥tr❛❝t t❤✐s t♦ ❛ s✐♥❣❧❡ ❧♦❣❛r✐t❤♠✿

▲❡t✬s ❝♦♥s✐❞❡r ❛❣❛✐♥ t❤❡

1 log2 (125)✳ 3

▼❛❦❡ ②♦✉r ♦✇♥ ❡①♣r❡ss✐♦♥ ❛♥❞ ❝♦♥tr❛❝t ✐t✳ ❘❡♣❡❛t✳

❡①♣♦♥❡♥t✐❛❧ ♠♦❞❡❧s ✭❈❤❛♣t❡r ✹✮✿ f (x) = Cax .

❍❡r❡

x

✐s t✐♠❡ ❛♥❞

f (x)

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

a✳

❲✐t❤ t❤❡ ❤❡❧♣ ♦❢ t❤❡ ❧♦❣❛r✐t❤♠✱ ✇❡ ❛r❡ ❛❜❧❡ t❤✐s t✐♠❡ t♦ s♦❧✈❡ ❛ ❧♦t ♠♦r❡ ♣r♦❜❧❡♠s ❛❜♦✉t t❤❡s❡ ♠♦❞❡❧s✳ ❚❤❡ ♥❛t✉r❡ ♦❢ t❤❡s❡ ♣r♦❜❧❡♠s ✐s ✏✐♥✈❡rs❡✑ t♦ t❤♦s❡ ✇❡ ❝♦♥s✐❞❡r❡❞ ♣r❡✈✐♦✉s❧②✳ ❇❡❧♦✇ ✇❡ ❧✐st s♦♠❡ ❢❛♠✐❧✐❛r ♠♦❞❡❧s

✺✳✹✳ ❚❤❡ ❛❧❣❡❜r❛ ♦❢ ❧♦❣❛r✐t❤♠s

✹✷✹

❛❧♦♥❣ ✇✐t❤ t❤❡ ♦❧❞ ❛♥❞ t❤❡ ♥❡✇ ♣r♦❜❧❡♠s✿ ▼♦❞❡❧s

❉✐r❡❝t ♣r♦❜❧❡♠s✿ s✉❜st✐t✉t❡

■♥✈❡rs❡ ♣r♦❜❧❡♠s✿ s♦❧✈❡ ❡q✉❛t✐♦♥

❯s❡ ❛♥ ❡♥❣✐♥❡❡r✐♥❣ ❝❛❧❝✉❧❛t♦r✳

❯s❡ ❛ s♣r❡❛❞s❤❡❡t✳

✶✳ ❈♦♠♣♦✉♥❞❡❞ ✐♥t❡r❡st✿ ◗✿ ❍♦✇ ♠✉❝❤ ❛❢t❡r ✶✵ ②❡❛rs❄ $✶✵✵✵ ❞❡♣♦s✐t✱ ✶✵✪ ❆P❘✳ ❙✉❜st✐t✉t❡✿ f (10) = 1000 · 1.110 f (x) = 1000 · 1.1x

✷✳ P♦♣✉❧❛t✐♦♥ ❞❡❝❧✐♥❡✿ ✶▼ ❝✉rr❡♥t✱ ✺✪ ❞❡❝❧✐♥❡✳ f (x) = 1 · .95x

✸✳ ❘❛❞✐♦❝❛r❜♦♥ ❞❡❝❛②✿ ✺✵✵✵ ②❡❛rs ❤❛❧❢✲❧✐❢❡✳ f (x) = .5x/5000

✹✳ ❍❡❛t✐♥❣ ❛♥❞ ❝♦♦❧✐♥❣✿ ❴❴ f (x) =

◗✿ ❍♦✇ ❧♦♥❣ t♦ r❡❛❝❤ $2000? ❙♦❧✈❡✿ 1000 · 1.1x = 2000 ⇒ 1.1x = 2 ≈ $2, 593 . ⇒ x = log1.1 2 ≈ 7.27 ②❡❛rs✳ ◗✿ ❍♦✇ ❧❛r❣❡ ❛❢t❡r ✺ ②❡❛rs❄ ◗✿ ❍♦✇ ❧♦♥❣ t♦ r❡❛❝❤ ✺✵✵❑❄ ❙✉❜st✐t✉t❡✿ f (5) = .955 ❙♦❧✈❡✿ 1 · .95x = .5 ⇒ .95x = .5 ≈ 774K. ⇒ x = log.95 .5 ≈ 13.51 ②❡❛rs✳ ◗✿ ❍♦✇ ♠✉❝❤ ❧❡❢t ❛❢t❡r ✶✵✵✵ ②❡❛rs❄ ◗✿ ❍♦✇ ❧♦♥❣ t♦ ❧♦s❡ ✾✵✪❄ ❙✉❜st✐t✉t❡✿ f (5) = .51000/5000 ❙♦❧✈❡✿ .5x/5000 = .1 ⇒ x/5000 = log.5 .1 ≈ 87%. ⇒ x = 5000 log.5 .1 ≈ 16610 ②❡❛rs✳ ◗✿ ❍♦✇ ✇❛r♠ ❛❢t❡r ❴❴ ♠✐♥✉t❡s❄ ◗✿ ❍♦✇ ❧♦♥❣ t♦ r❡❛❝❤ ❴❴ ❞❡❣r❡❡s❄ ❙✉❜st✐t✉t❡✿ ❙♦❧✈❡✿

❚❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ✐♥✈❡rs❡ ♣r♦❜❧❡♠s ❛r❡ s✐♠♣❧② ❛♣♣❧✐❝❛t✐♦♥s ♦❢ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❧♦❣❛r✐t❤♠✦ ❊①❡r❝✐s❡ ✺✳✹✳✶✸

❈♦♠♣❧❡t❡ t❤❡ r♦✇ ❢♦r t❤❡ ❤❡❛t✐♥❣✴❝♦♦❧✐♥❣ ♠♦❞❡❧✳ ❆ ❞✐✛❡r❡♥t✱ ❜✉t r❡❧❛t❡❞✱ s❡t ♦❢ ♣r♦❜❧❡♠s ✐s ❛❜♦✉t ✜♥❞✐♥❣ t❤❡ ♣❛r❛♠❡t❡rs ♦❢ ❛♥ ❡①♣♦♥❡♥t✐❛❧ ♠♦❞❡❧ ❞❡s❝r✐❜❡❞ ✐♥❞✐r❡❝t❧②✳ ❙♣❡❝✐✜❝❛❧❧②✱ t❤❡r❡ ❛r❡ ❢♦✉r q✉❛♥t✐t✐❡s ✐♥ t❤❡ ❡q✉❛t✐♦♥✿ y = Cax .

■❢ t❤r❡❡ ♦❢ t❤❡♠ ❛r❡ ❦♥♦✇♥✱ ✜♥❞ t❤❡ ❢♦✉rt❤✳ ❊①❛♠♣❧❡ ✺✳✹✳✶✹✿ ❝♦♠♣♦✉♥❞❡❞ ✐♥t❡r❡st✱ ♦r✐❣✐♥❛❧ ❞❡♣♦s✐t ❂ ❄

❙✉♣♣♦s❡ ■ ♥❡❡❞ $10, 000 ✐♥ 5 ②❡❛rs✳ ❍♦✇ ♠✉❝❤ ❞♦ ■ ♥❡❡❞ t♦ ❞❡♣♦s✐t ❛ss✉♠✐♥❣ t❤❛t t❤❡ ❆P❘ ✇✐❧❧ ❜❡ 10%❄ ❖✉r ♠♦❞❡❧ ♣r♦❞✉❝❡s t❤❡ ❢♦❧❧♦✇✐♥❣✿ 10, 000 = C · 1.15 .

❲❡ s♦❧✈❡ ❢♦r C ✿ C=

10, 000 ≈ $6, 209 . 1.15

❊①❛♠♣❧❡ ✺✳✹✳✶✺✿ ❝♦♠♣♦✉♥❞❡❞ ✐♥t❡r❡st✱ ♣❛st ❜❛❧❛♥❝❡❂❄

❙✉♣♣♦s❡ ■ ❤❛✈❡ $1000✳ ❍♦✇ ♠✉❝❤ ❞✐❞ ■ ❤❛✈❡ 3 ②❡❛rs ❛❣♦ ✐❢ t❤❡ ❆P❘ ❤❛s ❜❡❡♥ 10%❄ ■♥st❡❛❞ ♦❢ t❤✐♥❦✐♥❣ ♦❢ t❤❡ ✉♥❦♥♦✇♥ t♦ ❜❡ t❤❡ ♦r✐❣✐♥❛❧ ❞❡♣♦s✐t ❛t t✐♠❡ x = 0✱ ✇❡ ❛ss✉♠❡ t❤❛t t♦❞❛② ✐s x = 0 ❛♥❞ ✇❡ ❧♦♦❦ ❜❛❝❦✇❛r❞ ✐♥ t✐♠❡✱ x = −3✳ ❚❤❡♥ ♦✉r ♠♦❞❡❧ ♣r♦❞✉❝❡s t❤❡ ❢♦❧❧♦✇✐♥❣✿ 1000 = C · 1.1−3 .

❲❡ s♦❧✈❡ ❢♦r C ✿

C = 1000 · 1.1−3 ≈ $750 .

❊①❛♠♣❧❡ ✺✳✹✳✶✻✿ ❝♦♠♣♦✉♥❞❡❞ ✐♥t❡r❡st✱ ❆P❘ ❂ ❄

❲❤❛t ❆P❘ ✇✐❧❧ ❞♦✉❜❧❡ ♠② ♠♦♥❡② ✐♥ 5 ②❡❛rs❄ ■❢ a ✐s t❤❡ r❛t❡ ♦❢ ❣r♦✇t❤✱ ♦✉r ♠♦❞❡❧ ♣r♦❞✉❝❡s t❤❡ ❢♦❧❧♦✇✐♥❣✿ 2 = 1 · a5 .

✺✳✹✳

❚❤❡ ❛❧❣❡❜r❛ ♦❢ ❧♦❣❛r✐t❤♠s

❲❡ s♦❧✈❡ ❢♦r

✹✷✺

a✿ a=

❚❤❡ ❆P❘ t❤❡♥ s❤♦✉❧❞ ❜❡ ❛t ❧❡❛st

√ 5

2 = 21/5 ≈ 1.15 .

15%✳

❊①❡r❝✐s❡ ✺✳✹✳✶✼ ❲❤❛t ❤❛s ❜❡❡♥ ♠② ❆P❘ ✐❢ ■ ❤❛✈❡ tr✐♣❧❡❞ ♠② ♠♦♥❡② ✐♥

20

②❡❛rs❄

❊①❛♠♣❧❡ ✺✳✹✳✶✽✿ r❛❞✐♦❝❛r❜♦♥ ❞❛t✐♥❣✱ ❤❛❧❢✲❧✐❢❡ ❂ ❄ ❲❤❛t ✐s t❤❡ ❤❛❧❢✲❧✐❢❡ ♦❢ ❛ r❛❞✐♦❛❝t✐✈❡ ❡❧❡♠❡♥t ✐❢ ❛♥ ❡①♣❡r✐♠❡♥t ❤❛s s❤♦✇♥ t❤❛t ✐t ❧♦s❡s ✐♥

1

.1% ♦❢ ✐ts ✇❡✐❣❤t

②❡❛r❄ ❚❤❡ ♠♦❞❡❧ ✐s ❛s ❢♦❧❧♦✇s✿

.999 = 1 · a1 . ❚❤❡r❡❢♦r❡✱

a = .999 . ❲❡ ❤❛✈❡ ♥♦✇ ❛ ❝♦♠♣❧❡t❡ ♠♦❞❡❧✿

y = C · .999x . ■❢

x

✐s t❤❡ ❤❛❧❢✲❧✐❢❡✱ t❤❡♥✿

.5 = 1 · .999x =⇒ ln .5 = ln(.999x ) = x ln .999 . ❙♦❧✈❡ ❢♦r

x✿ x=

ln .5 ≈ 690 ln .999

②❡❛rs✳

❊①❡r❝✐s❡ ✺✳✹✳✶✾ ❲❤❛t ✐s t❤❡ ❤❛❧❢✲❧✐❢❡ ♦❢ ❛ r❛❞✐♦❛❝t✐✈❡ ❡❧❡♠❡♥t ✐❢ ❛♥ ❡①♣❡r✐♠❡♥t ❤❛s s❤♦✇♥ t❤❛t ✐t ❧♦s❡s ✐♥

10

1% ♦❢ ✐ts ✇❡✐❣❤t

②❡❛rs❄

■t ✐s ❛❧s♦ ♣♦ss✐❜❧❡ t❤❛t ♦♥❧②

t✇♦ ♦✉t ♦❢ ❢♦✉r

q✉❛♥t✐t✐❡s ✐♥ t❤❡ ❡q✉❛t✐♦♥ ❛r❡ ❦♥♦✇♥✱ ❜✉t t❤❡② ❛r❡ ❦♥♦✇ t✇✐❝❡✿

y1 = Cax1

❛♥❞

y2 = Cax2 .

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

❊①❛♠♣❧❡ ✺✳✹✳✷✵✿ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥ ❢r♦♠ ❣r❛♣❤ ❋✐♥❞ ❛ r❡♣r❡s❡♥t❛t✐♦♥

y = Cax

❢♦r t❤✐s ❣r❛♣❤✿

f (x) = Cax ✿ (0, 2) ✐s ♦♥ t❤❡ ❣r❛♣❤✳ =⇒ x = 0, y = 2 =⇒ Ca = 2 =⇒ C = 2 ✳ √ (2, .25) ✐s ♦♥ t❤❡ ❣r❛♣❤✳ =⇒ x = 2, y = .25 =⇒ 2a2 = .25 =⇒ a = .125 ✳

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

• •

❚❤❡ ♣♦✐♥t ❚❤❡ ♣♦✐♥t

❏✉st ❧✐❦❡ ✇✐t❤ ❛ str❛✐❣❤t ❧✐♥❡✱ ✇❡ ♦♥❧② ♥❡❡❞ t✇♦ ♣♦✐♥ts✦

0

✺✳✹✳

❚❤❡ ❛❧❣❡❜r❛ ♦❢ ❧♦❣❛r✐t❤♠s

✹✷✻

❲❡ ♦❜s❡r✈❡❞ ✐♥ t❤❡ ❧❛st ❝❤❛♣t❡r t❤❛t ✇❡ ❝❛♥ ❣❡t ❛❧❧ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥s ❜② ❤♦r✐③♦♥t❛❧❧② str❡t❝❤✐♥❣ ❛♥❞ ✢✐♣♣✐♥❣ ❛ s✐♥❣❧❡ ♦♥❡ ♦❢ t❤❡♠✿

■t ❢♦❧❧♦✇s t❤❛t ✇❡ ❝❛♥ ❣❡t ❛❧❧ ❧♦❣❛r✐t❤♠s ❜② ✈❡rt✐❝❛❧❧② str❡t❝❤✐♥❣ ❛♥❞ ✢✐♣♣✐♥❣ ❛ s✐♥❣❧❡ ♦♥❡ ♦❢ t❤❡♠✳ ▲❡t✬s ♣r♦✈❡ t❤✐s ❢❛❝t ✇✐t❤ ❛❧❣❡❜r❛✳ ❙✉♣♣♦s❡ a > 0 ❛♥❞ b > 0 ❛r❡ t✇♦ ❜❛s❡s t❤❛t ✇❡ ✇❛♥t t♦ ♠❛t❝❤✳ ▲❡t✬s tr❛♥s❢♦r♠ t❤❡ ❢♦r♠❡r ✐♥t♦ t❤❡ ❧❛tt❡r✿ ❯s❡ ♦♥❡ ♦❢ t❤❡ ❈❛♥❝❡❧❧❛t✐♦♥ ▲❛✇s t♦ ✐♥s❡rt blogb . y = ax x = blogb a ◆♦✇ t❤❡ ▼✉❧t✐♣❧✐❝❛t✐♦♥✲❊①♣♦♥❡♥t✐❛t✐♦♥ ❘✉❧❡ ♦❢ ▲♦❣❛r✐t❤♠s✳ (logb a)·x =b ❇✉t logb a ✐s ❥✉st ❛ ❝♦♥st❛♥t ❝♦❡✣❝✐❡♥t✦ ❲❡ ❝♦♥❝❧✉❞❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿

❤♦r✐③♦♥t❛❧❧② ❜② ❛ ❢❛❝t♦r ♦❢ logb a✳ ❚❤❡ ❣r❛♣❤ ♦❢ y = loga x ✐s t❤❛t ♦❢ y = logb x s❤r✉♥❦ ✈❡rt✐❝❛❧❧② ❜② ❛ ❢❛❝t♦r ♦❢ logb a✳

• ❚❤❡ ❣r❛♣❤ ♦❢ y = ax ✐s t❤❛t ♦❢ y = bx str❡t❝❤❡❞ •

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

❚❤❡♦r❡♠ ✺✳✹✳✷✶✿ ❊①♣♦♥❡♥t ❛♥❞ ▲♦❣❛r✐t❤♠ ❇❛s❡ ❈♦♥✈❡rs✐♦♥ ❋♦r♠✉❧❛s ❋♦r ❡✈❡r② ♣❛✐r

a, b > 0

❛♥❞ ❡✈❡r②

x > 0✱

✇❡ ❤❛✈❡✿

ax

= b(logb a)·x 1 loga x = logb x logb a

❊①❡r❝✐s❡ ✺✳✹✳✷✷ ❉❡r✐✈❡ t❤❡ s❡❝♦♥❞ ❢♦r♠✉❧❛✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ✐s ❛♥ ✐♠♣♦rt❛♥t ❝♦♥❝❧✉s✐♦♥✿

❡❛s✐❧② ❡①♣r❡ss❡❞ ✐♥ t❡r♠s ♦❢ ❛♥② ♦t❤❡r✳ • ❊✈❡r② ❧♦❣❛r✐t❤♠ ❢✉♥❝t✐♦♥ ❝❛♥ ❜❡ ❡❛s✐❧② ❡①♣r❡ss❡❞ ✐♥ t❡r♠s ♦❢ ❛♥② ♦t❤❡r✳ ❙♦✱ ❥✉st ❛s ✇❡ ♥❡❡❞ ♦♥❧② ♦♥❡ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧✱ y = x2 ✱ ✇❡ ♦♥❧② ♥❡❡❞ ♦♥❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥ ❛♥❞ ♦♥❡ • ❊✈❡r② ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥ ❝❛♥ ❜❡

❧♦❣❛r✐t❤♠✦

❉❡✜♥✐t✐♦♥ ✺✳✹✳✷✸✿ ♥❛t✉r❛❧ ❧♦❣❛r✐t❤♠ ❚❤❡ ❧♦❣❛r✐t❤♠ ❜❛s❡ e ✐s ❝❛❧❧❡❞ t❤❡ ♥❛t✉r❛❧

❧♦❣❛r✐t❤♠ ❛♥❞ ❞❡♥♦t❡❞ ❜②

y = ln x

✺✳✹✳

❚❤❡ ❛❧❣❡❜r❛ ♦❢ ❧♦❣❛r✐t❤♠s

✹✷✼

■ts ❣r❛♣❤ ❝r♦ss❡s t❤❡ x✲❛①✐s ❛t 45 ❞❡❣r❡❡s ✭t♦ ❜❡ ❞✐s❝✉ss❡❞ ✐♥ ❈❤❛♣t❡r ✷❉❈✲✸✮✿

❲❤❛t ✇❡ ❤❛✈❡ ❞✐s❝♦✈❡r❡❞ ✐s t❤❛t ✐t s✉✣❝❡s t♦ ❝♦♥s✐❞❡r ❛ s✐♥❣❧❡ ❜❛s❡ ✕ t❤❡ st❛♥❞❛r❞ ❜❛s❡ ✕ t♦ ❣❡t ❛❧❧ ♦❢ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ♠♦❞❡❧s✦ ■♥❞❡❡❞✱ ❜② ♠❛❦✐♥❣ ♠✉❧t✐♣❧❡s ♦❢ y = ex ✱ ✇❡ ❝❛♥ ❣❡t ❛❧❧ ♣♦ss✐❜❧❡ ♣❛tt❡r♥s ♦❢ ❡①♣♦♥❡♥t✐❛❧ ❣r♦✇t❤✿

❊①❡r❝✐s❡ ✺✳✹✳✷✹ ❙❤♦✇ t❤❛t t❤❡ ✈❡rt✐❝❛❧ str❡t❝❤❡s ♦❢ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥s ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ❛s ❤♦r✐③♦♥t❛❧ s❤r✐♥❦s✳ ❙❛♠❡ ❢♦r t❤❡ ❧♦❣❛r✐t❤♠s✳

❉❡✜♥✐t✐♦♥ ✺✳✹✳✷✺✿ ❡①♣♦♥❡♥t✐❛❧ ♠♦❞❡❧ ❚❤❡

❡①♣♦♥❡♥t✐❛❧ ♠♦❞❡❧

✭✐♥ t❤❡ st❛♥❞❛r❞ ❢♦r♠✮ ✐s ❛ ❢✉♥❝t✐♦♥✿

y = Cekx ✇✐t❤ k ❝❛❧❧❡❞ ✐ts r❛t❡✳ ❋✉rt❤❡r♠♦r❡✱ • y = Cekx , k > 0✱ ✐s ❝❛❧❧❡❞ t❤❡ ❡①♣♦♥❡♥t✐❛❧ • y = Cekx , k < 0✱ ✐s ❝❛❧❧❡❞ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❚❤❡ tr❛♥s✐t✐♦♥ ✐s str❛✐❣❤t✲❢♦r✇❛r❞✿

❣r♦✇t❤ ♠♦❞❡❧✱ ❛♥❞ ❞❡❝❛② ♠♦❞❡❧✳

❢r♦♠ y = Cax t♦ y = Cekx ,

Cax = Cekx =⇒ ax = ekx = ek

❊①❛♠♣❧❡ ✺✳✹✳✷✻✿ ♣♦♣✉❧❛t✐♦♥ ❣r♦✇t❤


x

=⇒ a = ek .

❙✉♣♣♦s❡ t❤❛t t❤❡ ♣♦♣✉❧❛t✐♦♥ ❣r♦✇s ❜② 10% ❛ ②❡❛r✳ ❍♦✇ ❧♦♥❣ ✇✐❧❧ ✐t t❛❦❡ t♦ ❞♦✉❜❧❡❄ ■♥ ❝♦♥tr❛st t♦ t❤❡ ♣r❡✈✐♦✉s ❛♥❛❧②s✐s ♦❢ t❤❡ s❛♠❡ ♣r♦❜❧❡♠✱ ✇❡ ✇✐❧❧ ✜♥❞ t❤❡ st❛♥❞❛r❞✱ ❜❛s❡ e✱ ❡①♣♦♥❡♥t✐❛❧

✺✳✹✳

❚❤❡ ❛❧❣❡❜r❛ ♦❢ ❧♦❣❛r✐t❤♠s x

♠♦❞❡❧ ❢♦r t❤✐s s❡t✉♣ ✭

✹✷✽

t✐♠❡✮✿

f (x) = Cekx , ▲❡t✬s ❛ss✉♠❡

f (0) = 1✱

t❤❡♥

f (1) = 1.1✳

✇❤❛t ✐s

■t ❢♦❧❧♦✇s t❤❛t

y = ax

k?

C = 1✳

❙✉❜st✐t✉t❡ t❤❡s❡ ✐♥t♦ ❜♦t❤ ♠♦❞❡❧s✿

y = ekx ek·1 = 1.1

=⇒

a = 1.1 k = ln 1.1 . ❲❡ ❝❛♥ ✉s❡ ❡✐t❤❡r ♦❢ t❤❡ ♠♦❞❡❧s t♦ s♦❧✈❡ t❤❡ ♦r✐❣✐♥❛❧ ♣r♦❜❧❡♠✿

y = eln 1.1·x

y = 1.1x 1.1x = 2

eln 1.1·x = 2 =⇒ ln 2 ≈ 7.27 . x = log1.1 2 x = ln 1.1

❊①❛♠♣❧❡ ✺✳✹✳✷✼✿ ✜♥❞ ✐♥✈❡rs❡ ❢r♦♠ ❢♦r♠✉❧❛ ❋✐♥❞ t❤❡ ✐♥✈❡rs❡ ♦❢ 3

f (x) = ex . ❙♦❧✈❡ t❤❡ ❡q✉❛t✐♦♥✿

y = ex

3

x✳ ❚♦ ❣❡t t♦ x✱ ✇❡ ♥❡❡❞ y = ex ✱ ✐t✬s x = ln y ✿

✱ s♦❧✈❡ ❢♦r

❛♣♣❧② ✐ts ✐♥✈❡rs❡ t♦ ❜♦t❤ s✐❞❡s✳ ❋♦r

t♦ ❣❡t r✐❞ ♦❢ t❤❡ ❡①♣♦♥❡♥t✳ ❚♦ ❝❛♥❝❡❧ ✐t✱

3

ln y = ln ex ln y = x3 ❆♣♣❧② t❤❡ ✐♥✈❡rs❡ t♦ ❣❡t t♦

❆♥s✇❡r✿

f −1 (y) =

p 3 ln y ✳

x✿

p 3

ln y =

√ 3

x3 = x .

■♥ s✉♠♠❛r②✿

y = Cax →

a = ek k = ln a

→ y = Cekx

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

❚❤❡♦r❡♠ ✺✳✹✳✷✽✿ ◆❛t✉r❛❧ ▲♦❣ ❛♥❞ ❊①♣ ❋♦r♠✉❧❛s ❚❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✉❧❛s ❤♦❧❞ ❢♦r ❡✈❡r② ♣❛✐r x ❛♥❞ y ❢♦r ✇❤✐❝❤ t❤❡② ❛r❡ ❞❡✜♥❡❞✿ eln x = x e0 = 1 ex ey = ex+y ex = ex−y y e
y ex = exy

ln ex = x ln 1 = 0 ln x + ln y = ln xy x ln x − ln y = ln y ln xy = y ln x

✺✳✺✳

❚❤❡ ❈❛rt❡s✐❛♥ s②st❡♠ ❢♦r t❤❡ ❊✉❝❧✐❞❡❛♥ ♣❧❛♥❡

✹✷✾

✺✳✺✳ ❚❤❡ ❈❛rt❡s✐❛♥ s②st❡♠ ❢♦r t❤❡ ❊✉❝❧✐❞❡❛♥ ♣❧❛♥❡

■♥ ❈❤❛♣t❡r ✷✱ ✇❡ ✐♥tr♦❞✉❝❡❞ t❤❡ ❈❛rt❡s✐❛♥ ❝♦♦r❞✐♥❛t❡ ♣❧❛♥❡ ❛s ❛ ❞❡✈✐❝❡ ❢♦r ✈✐s✉❛❧✐③✐♥❣

❢✉♥❝t✐♦♥s ❀ ✐t✬s ✇❤❡r❡

t❤❡ ❣r❛♣❤s ❧✐✈❡ ✭t♦♣ r♦✇✮✿

❲❡ ♥♦✇ ❛♣♣r♦❛❝❤ ✐t ❢r♦♠ ❛♥♦t❤❡r ❞✐r❡❝t✐♦♥ ✭❜♦tt♦♠ r♦✇✮✿ ❲❡ ✇❛♥t t♦ st✉❞② t❤❡ ❊✉❝❧✐❞❡❛♥ ❣❡♦♠❡tr②✱ ❛❧❣❡❜r❛✐❝❛❧❧②✳ ❲❡

❊✉❝❧✐❞❡❛♥ ♣❧❛♥❡✱

❛♥❞

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

❈❛rt❡s✐❛♥ ❣r✐❞ ♦✈❡r t❤✐s ♣❧❛♥❡✳ ❚❤❡ ♠❛✐♥ ❞✐✛❡r❡♥❝❡ ✐s t❤❛t t❤❡ ❢♦r♠❡r ♣❧❛♥❡ ❞♦❡s♥✬t ♥❡❡❞ t♦ ❤❛✈❡ ❛ sq✉❛r❡ ❣r✐❞ ❛s t❤❡ ✉♥✐ts ♦❢ ❜❡ ✉♥r❡❧❛t❡❞ ✭❞♦❧❧❛rs ✈s✳ ❤♦✉rs✮✳ ❚❤❡ ❧❛tt❡r ❞♦❡s ❛♥❞ t❤❡ ✉♥✐ts ♦❢

x

❛♥❞

y

x ❛♥❞ y

♠✐❣❤t

❛r❡ ❜❡tt❡r ❜❡ t❤♦s❡ ♦❢ ❧❡♥❣t❤

✭♠✐❧❡s✱ ❢❡❡t✱ ❡t❝✳✮✳ ❚❤❡ ✐❞❡❛ ♦❢ ✏❛♥❛❧②t✐❝ ❣❡♦♠❡tr②✑ ✐s t♦ ✉s❡ ❛ ❝♦♦r❞✐♥❛t❡ s②st❡♠ t♦ tr❛♥s✐t✐♦♥

❣❡♦♠❡tr② ✿ ♣♦✐♥ts✱ t❤❡♥ ❧✐♥❡s✱ tr✐❛♥❣❧❡s✱ ❝✐r❝❧❡s✱ t❤❡♥ ♣❧❛♥❡s✱ ❝✉❜❡s✱ s♣❤❡r❡s✱ ❡t❝✳✱ • t♦ ❛❧❣❡❜r❛ ✿ ♥✉♠❜❡rs✱ t❤❡♥ ❝♦♠❜✐♥❛t✐♦♥s ♦❢ ♥✉♠❜❡rs✱ t❤❡♥ r❡❧❛t✐♦♥s ❛♥❞ ❢✉♥❝t✐♦♥s✱ ❡t❝✳ ❚❤✐s ✇✐❧❧ ❛❧❧♦✇ ✉s t♦ s♦❧✈❡ ❣❡♦♠❡tr✐❝ ♣r♦❜❧❡♠s ✇✐t❤♦✉t ♠❡❛s✉r✐♥❣ ✕ ❜❡❝❛✉s❡ ❡✈❡r②t❤✐♥❣ ✐s ♣r❡✲♠❡❛s✉r❡❞✦ •

❢r♦♠

❲❡

✇✐❧❧ ✐♥✐t✐❛❧❧② ❧✐♠✐t ♦✉rs❡❧✈❡s t♦ t❤❡ t✇♦ s✐♠♣❧❡st ❣❡♦♠❡tr✐❝ t❛s❦s✿ ✶✳ ✜♥❞✐♥❣ ❞✐st❛♥❝❡s✿ ❜❡t✇❡❡♥ t✇♦ ♣♦✐♥ts✱ ❜❡t✇❡❡♥ ❛ ♣♦✐♥t ❛♥❞ ❛ ❧✐♥❡ ♦r ❝✉r✈❡✱ ❡t❝✳✱ ❛♥❞ ✷✳ ✜♥❞✐♥❣ ❛♥❣❧❡s✿ ❜❡t✇❡❡♥ t✇♦ ❧✐♥❡s ♦r t✇♦ ❝✉r✈❡s✳

❞✐♠❡♥s✐♦♥ 1✳ ▲❡t✬s r❡♣❡❛t t❤❡ ❝♦♥str✉❝t✐♦♥ ❢r♦♠ ❈❤❛♣t❡r ✶ ✜rst✳ ❙✉♣♣♦s❡ ✇❡ ❧✐✈❡ ♦♥ ❛ r♦❛❞ s✉rr♦✉♥❞❡❞ ❜② ♥♦t❤✐♥❣♥❡ss✿ ❲❡ st❛rt ✇✐t❤

❚❤❡ ❝♦♦r❞✐♥❛t❡ s②st❡♠ ✐♥t❡♥❞❡❞ t♦ ❝❛♣t✉r❡ ✇❤❛t ❤❛♣♣❡♥s ♦♥ t❤✐s r♦❛❞ ✐s ❞❡✈✐s❡❞ t♦ ❜❡ s✉♣❡r✐♠♣♦s❡❞ ♦♥ t❤❡ r♦❛❞✳

✺✳✺✳

❚❤❡ ❈❛rt❡s✐❛♥ s②st❡♠ ❢♦r t❤❡ ❊✉❝❧✐❞❡❛♥ ♣❧❛♥❡

✹✸✵

■t ✐s ❜✉✐❧t ✐♥ s❡✈❡r❛❧ st❛❣❡s✿ ✶✳ ❉r❛✇ ❛ ❧✐♥❡✱ t❤❡ x✲❛①✐s✳ ✷✳ ❈❤♦♦s❡ ♦♥❡ ♦❢ t❤❡ t✇♦ ❞✐r❡❝t✐♦♥s ♦♥ t❤❡ ❧✐♥❡ ❛s ✸✳ ❈❤♦♦s❡ ❛ ♣♦✐♥t O ❛s t❤❡

♦r✐❣✐♥✳

✹✳ ❙❡t ❛ s❡❣♠❡♥t ♦❢ t❤❡ ❧✐♥❡ ✕ ♦❢ ❧❡♥❣t❤ 1 ✕ ❛s ❛

♣♦s✐t✐✈❡✱ t❤❡♥ t❤❡ ♦t❤❡r ✐s ♥❡❣❛t✐✈❡✳

✉♥✐t✳

✺✳ ❯s❡ t❤❡ s❡❣♠❡♥t t♦ ♠❡❛s✉r❡ ❞✐st❛♥❝❡s t♦ ❧♦❝❛t✐♦♥s ❢r♦♠ t❤❡ ♦r✐❣✐♥ O ✕ ♣♦s✐t✐✈❡ ✐♥ t❤❡ ♣♦s✐t✐✈❡ ❞✐r❡❝t✐♦♥ ❛♥❞ ♥❡❣❛t✐✈❡ ✐♥ t❤❡ ♥❡❣❛t✐✈❡ ❞✐r❡❝t✐♦♥ ✕ ❛♥❞ ❛❞❞ ♠❛r❦s t♦ t❤❡ ❧✐♥❡✱ t❤❡ ❝♦♦r❞✐♥❛t❡s ❀ ❧❛t❡r t❤❡ s❡❣♠❡♥ts ❛r❡ ❢✉rt❤❡r s✉❜❞✐✈✐❞❡❞ t♦ ❢r❛❝t✐♦♥s ♦❢ t❤❡ ✉♥✐t✱ ❡t❝✳ ✻✳ ❲❡ ❤❛✈❡ ❛ ❝♦♦r❞✐♥❛t❡ s②st❡♠ ♦♥ t❤❡ ❧✐♥❡✳ ❚❤❡ r❡s✉❧t ✐s ❛ ❝♦rr❡s♣♦♥❞❡♥❝❡✿ ❧♦❝❛t✐♦♥

P

←→ ♥✉♠❜❡r

①

❚❤✐s ✐s t❤❡ r❡❛❧ ❧✐♥❡✳ ❚❤❡ ❝♦rr❡s♣♦♥❞❡♥❝❡ ✇♦r❦s ✐♥ ❜♦t❤ ❞✐r❡❝t✐♦♥s✳ ❋♦r ❡①❛♠♣❧❡✱ s✉♣♣♦s❡ P ✐s ❛ ❧♦❝❛t✐♦♥ ♦♥ t❤❡ ❧✐♥❡✳ ❲❡ t❤❡♥ ✜♥❞ t❤❡ ❞✐st❛♥❝❡ ❢r♦♠ t❤❡ ♦r✐❣✐♥ ✕ ♣♦s✐t✐✈❡ ✐♥ t❤❡ ♣♦s✐t✐✈❡ ❞✐r❡❝t✐♦♥ ❛♥❞ ♥❡❣❛t✐✈❡ ✐♥ t❤❡ ♥❡❣❛t✐✈❡ ❞✐r❡❝t✐♦♥ ✕ ❛♥❞ t❤❡ r❡s✉❧t ✐s t❤❡ ❝♦♦r❞✐♥❛t❡ ♦❢ P ✱ s♦♠❡ ♥✉♠❜❡r x✳ ❲❡ ✉s❡ t❤❡ ♥❡❛r❡st ♠❛r❦ t♦ s✐♠♣❧✐❢② t❤❡ t❛s❦✳

❈♦♥✈❡rs❡❧②✱ s✉♣♣♦s❡ x ✐s ❛ ♥✉♠❜❡r✳ ❲❡ t❤❡♥ ♠❡❛s✉r❡ x ❛s t❤❡ ❞✐st❛♥❝❡ t♦ t❤❡ ♦r✐❣✐♥ ✕ ♣♦s✐t✐✈❡ ✐♥ t❤❡ ♣♦s✐t✐✈❡ ❞✐r❡❝t✐♦♥ ❛♥❞ ♥❡❣❛t✐✈❡ ✐♥ t❤❡ ♥❡❣❛t✐✈❡ ❞✐r❡❝t✐♦♥ ✕ ❛♥❞ t❤❡ r❡s✉❧t ✐s ❛ ❧♦❝❛t✐♦♥ P ♦♥ t❤❡ ❧✐♥❡✳ ❲❡ ✉s❡ t❤❡ ♥❡❛r❡st ♠❛r❦ t♦ s✐♠♣❧✐❢② t❤❡ t❛s❦✳

❊①❛♠♣❧❡ ✺✳✺✳✶✿ ❞✐✛❡r❡♥t ❝♦♦r❞✐♥❛t❡ s②st❡♠s✱ ❞✐♠❡♥s✐♦♥ 1 ❖❢ ❝♦✉rs❡✱ ✇❡ ❝❛♥ ♣❧❛❝❡ ❞✐✛❡r❡♥t ❝♦♦r❞✐♥❛t❡ s②st❡♠s ✭❢❡❡t ✐♥st❡❛❞ ♦❢ ✐♥❝❤❡s✮ ♦♥ t❤❡ s❛♠❡ ❧✐♥❡✿

❍❡r❡✱ t❤❡ ♣♦✐♥t P s❤♦✇♥ ❤❛s✿ • ❝♦♦r❞✐♥❛t❡ 1.5 ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ✜rst s②st❡♠ ❛♥❞

✺✳✺✳ ❚❤❡ ❈❛rt❡s✐❛♥ s②st❡♠ ❢♦r t❤❡ ❊✉❝❧✐❞❡❛♥ ♣❧❛♥❡

✹✸✶

• ❝♦♦r❞✐♥❛t❡ 2 ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ♦t❤❡r✳ ❲❡ ❦♥♦✇ ❢r♦♠ ❈❤❛♣t❡r ✸ t❤❛t ❛❧❧ t❤❡ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ♦♥ t❤❡ ✜rst ❛①✐s ❛r❡ tr❛♥s❢♦r♠❡❞ t♦ t❤❡ ♦♥❡s ♦♥ t❤❡ s❡❝♦♥❞ ❜② ❛ s✐♥❣❧❡ ❢✉♥❝t✐♦♥❀ ✐♥ t❤✐s ♣❛rt✐❝✉❧❛r ❝❛s❡✱ ✐t ✐s✿ x = (u + 1)/2 . ❖❢ ❝♦✉rs❡✱ ❛❧❧ t❤❡ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ♦♥ t❤❡ s❡❝♦♥❞ ❛①✐s ❛r❡ tr❛♥s❢♦r♠❡❞ t♦ t❤❡ ♦♥❡s ♦♥ t❤❡ ✜rst ❜② t❤❡ ✐♥✈❡rs❡ ♦❢ t❤✐s ❢✉♥❝t✐♦♥✿ u = 2x + 1 . ❲✐t❤ t❤❡s❡ t✇♦ ❢✉♥❝t✐♦♥s✱ ❛❧❧ t❤❡ q✉❛♥t✐t✐❡s ✕ ❣❡♦♠❡tr✐❝ ♦r ♣❤②s✐❝❛❧ ✕ ❞❡✜♥❡❞ ✇✐t❤✐♥ t❤❡ t✇♦ ❝♦♦r❞✐♥❛t❡ s②st❡♠s ❛r❡ tr❛♥s❢♦r♠❡❞ t♦ ❡❛❝❤ ♦t❤❡r✳ ❊①❡r❝✐s❡ ✺✳✺✳✷

❲❤❛t tr❛♥s❢♦r♠❛t✐♦♥ ✐s♥✬t ♠❡♥t✐♦♥❡❞ ❛❜♦✈❡❄ Pr♦✈✐❞❡ ❛♥ ✐❧❧✉str❛t✐♦♥ ❛♥❞ ❛ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ t❤r❡❡✳ ◆♦✇ t❤❡ ❝♦♦r❞✐♥❛t❡ s②st❡♠ ❢♦r ❞✐♠❡♥s✐♦♥ 2✱ t❤❡ ♣❧❛♥❡✳ ❚❤❡r❡ ✐s ♠✉❝❤ ♠♦r❡ ❣♦✐♥❣ ♦♥ t❤❛♥ ❜❡❢♦r❡✿

❚❤❡ ✐❞❡❛ ✐s t❤❡ s❛♠❡✿ s♦❧✈✐♥❣ ❣❡♦♠❡tr✐❝ ♣r♦❜❧❡♠s ✇✐t❤ ❛❧❣❡❜r❛✳ ▲❡t✬s r❡♣❡❛t ✕ ✇✐t❤ s♦♠❡ ♠✐♥♦r ❝❤❛♥❣❡s ✕ t❤❡ ❝♦♥str✉❝t✐♦♥ ❢r♦♠ ❈❤❛♣t❡r ✷ ✜rst✳ ❙✉♣♣♦s❡ ✇❡ ❧✐✈❡ ♦♥ ❛ ✜❡❧❞ ❛♥❞ ✇❡ ❜✉✐❧❞ t✇♦ r♦❛❞s ✐♥t❡rs❡❝t✐♥❣ ❛t 90 ❞❡❣r❡❡s✿

❲❡ ❝❛♥ t❤❡♥ tr❡❛t ❡✐t❤❡r ♦❢ t❤❡ t✇♦ r♦❛❞s ❛s ❛ 1✲❞✐♠❡♥s✐♦♥❛❧ ❈❛rt❡s✐❛♥ s②st❡♠✱ ❛s ❛❜♦✈❡✱ ❛♥❞ ✉s❡ t❤❡✐r ♠✐❧❡st♦♥❡s t♦ ♥❛✈✐❣❛t❡✳ ❇✉t ✇❤❛t ❛❜♦✉t t❤❡ r❡st ♦❢ t❤❡ ✜❡❧❞❄ ❍♦✇ ❞♦ ✇❡ ♥❛✈✐❣❛t❡ ✐t❄ ❲❡ ❝♦✉❧❞ ❜✉✐❧❞ ❛ ❝✐t② ✇✐t❤ ❛ ❣r✐❞ ♦❢ str❡❡ts✿

✺✳✺✳

❚❤❡ ❈❛rt❡s✐❛♥ s②st❡♠ ❢♦r t❤❡ ❊✉❝❧✐❞❡❛♥ ♣❧❛♥❡

✹✸✷

❲❡ ❛❧s♦ ♥✉♠❜❡r t❤❡ str❡❡ts✳ ■t✬s t✇♦✲❞✐♠❡♥s✐♦♥❛❧ ♥♦✇✦ ❆ ♥❡✇ ❝♦♦r❞✐♥❛t❡ s②st❡♠ ✐♥t❡♥❞❡❞ t♦ ❝❛♣t✉r❡ ✇❤❛t ❤❛♣♣❡♥s ✐♥ t❤✐s ❝✐t② ♦r ♦♥ t❤✐s ✜❡❧❞ ✐s ❞❡✈✐s❡❞ t♦ ❜❡ s✉♣❡r✐♠♣♦s❡❞ ♦♥ t❤❡ ✜❡❧❞✳ ■t✬s ❛ ❣r✐❞✿

■t ✐s ❜✉✐❧t ✐♥ s❡✈❡r❛❧ st❛❣❡s✿ ✶✳ ❈❤♦♦s❡ t✇♦ ✐❞❡♥t✐❝❛❧ ❝♦♦r❞✐♥❛t❡ ❛①❡s✱ t❤❡ x✲❛①✐s ✜rst ❛♥❞ t❤❡ y ✲❛①✐s s❡❝♦♥❞✱ ✇✐t❤ t❤❡ s❛♠❡ ✉♥✐ts✳ ✷✳ P✉t t❤❡ t✇♦ ❛①❡s t♦❣❡t❤❡r ❛t t❤❡✐r ♦r✐❣✐♥s s♦ t❤❛t ✐t ✐s ❛ 90✲❞❡❣r❡❡ t✉r♥ ❢r♦♠ t❤❡ ♣♦s✐t✐✈❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ x✲❛①✐s t♦ t❤❡ ♣♦s✐t✐✈❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ y ✲❛①✐s✳ ✸✳ ❯s❡ t❤❡ ♠❛r❦s ♦♥ t❤❡ ❛①❡s t♦ ❞r❛✇ ❛ ❣r✐❞✳ ❲❛r♥✐♥❣✦ ❚❤❡

xy ✲♣❧❛♥❡

✐s♥✬t t❤❡ s❛♠❡ ❛s t❤❡

❊①❛♠♣❧❡ ✺✳✺✳✸✿ ✉♥✐ts

■t ✐s ♣♦ss✐❜❧❡ t❤♦✉❣❤ ✉♥❝♦♠♠♦♥ t♦ ❤❛✈❡ ❞✐✛❡r❡♥t ✉♥✐ts ❢♦r t❤❡ t✇♦ ❛①❡s✿

yx✲♣❧❛♥❡✳

✺✳✺✳

❚❤❡ ❈❛rt❡s✐❛♥ s②st❡♠ ❢♦r t❤❡ ❊✉❝❧✐❞❡❛♥ ♣❧❛♥❡

❲❡ ❤❛✈❡ ❛ ❝♦rr❡s♣♦♥❞❡♥❝❡ t❤❛t ✇♦r❦s ✐♥ ❜♦t❤

✹✸✸

❞✐r❡❝t✐♦♥s ✿

❧♦❝❛t✐♦♥ P ←→ ❛ ♣❛✐r ♦❢ ♥✉♠❜❡rs ✭①✱②✮ ❊①❛♠♣❧❡ ✺✳✺✳✹✿ ❝♦♦r❞✐♥❛t❡s ❢r♦♠ ♣♦✐♥t ❛♥❞ ❜❛❝❦

❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ t❤❡ ❊✉❝❧✐❞❡❛♥ ♣❧❛♥❡ ❡q✉✐♣♣❡❞ ✇✐t❤ ❛ ❈❛rt❡s✐❛♥ s②st❡♠✳ ❙✉♣♣♦s❡ P ✐s ❛ ♣♦✐♥t ❛s s❤♦✇♥✿

✶✳ ❲❡ ❞r❛✇ ❛ ✈❡rt✐❝❛❧ ❧✐♥❡ t❤r♦✉❣❤ P ✉♥t✐❧ ✐t ✐♥t❡rs❡❝ts t❤❡ x✲❛①✐s✳ ❚❤❡ ♣♦✐♥t ♦❢ ✐♥t❡rs❡❝t✐♦♥ t❤❡♥ ❧✐❡s ♦♥ t❤✐s ❛①✐s✱ ✇❤✐❝❤ ✐s ❡q✉✐♣♣❡❞ ✇✐t❤ ❛ 1✲❞✐♠❡♥s✐♦♥❛❧ ❈❛rt❡s✐❛♥ s②st❡♠✳ ❚❤✐s ♣♦✐♥t ❤❛s ❛ ❝♦♦r❞✐♥❛t❡✱ s❛② 4✱ ✇✐t❤✐♥ t❤✐s s②st❡♠✳ ✷✳ ❲❡ ❞r❛✇ ❛ ❤♦r✐③♦♥t❛❧ ❧✐♥❡ t❤r♦✉❣❤ P ✉♥t✐❧ ✐t ✐♥t❡rs❡❝ts t❤❡ y ✲❛①✐s✳ ❚❤❡ ♣♦✐♥t ♦❢ ✐♥t❡rs❡❝t✐♦♥ t❤❡♥ ❧✐❡s ♦♥ t❤✐s ❛①✐s✱ ✇❤✐❝❤ ✐s ❡q✉✐♣♣❡❞ ✇✐t❤ ❛ 1✲❞✐♠❡♥s✐♦♥❛❧ ❈❛rt❡s✐❛♥ s②st❡♠✳ ❚❤❡ ♣♦✐♥t ❤❛s ❛ ❝♦♦r❞✐♥❛t❡✱ s❛② 1✱ ✇✐t❤✐♥ t❤✐s s②st❡♠✳ ✸✳ ❲❡ ❤❛✈❡ ❞✐s❝♦✈❡r❡❞ t❤❛t ♦✉r ♣♦✐♥t ❤❛s ❝♦♦r❞✐♥❛t❡s P = (4, 1)✦ ❖♥ t❤❡ ✢✐♣ s✐❞❡✱ s✉♣♣♦s❡ ✇❡ ❤❛✈❡ t✇♦ ♥✉♠❜❡rs✱ −2 ❛♥❞ 4✿

✶✳ ❲❡ ✜♥❞ t❤❡ ❧♦❝❛t✐♦♥ ♦♥ t❤❡ x✲❛①✐s ✇✐t❤ ❝♦♦r❞✐♥❛t❡ −2✳ ❲❡ t❤❡♥ ❞r❛✇ ❛ ✈❡rt✐❝❛❧ ❧✐♥❡ t❤r♦✉❣❤ t❤✐s ♣♦✐♥t✳

✺✳✺✳

❚❤❡ ❈❛rt❡s✐❛♥ s②st❡♠ ❢♦r t❤❡ ❊✉❝❧✐❞❡❛♥ ♣❧❛♥❡

✹✸✹

✷✳ ❲❡ ✜♥❞ t❤❡ ❧♦❝❛t✐♦♥ ♦♥ t❤❡ y ✲❛①✐s ✇✐t❤ ❝♦♦r❞✐♥❛t❡ 4✳ ❲❡ t❤❡♥ ❞r❛✇ ❛ ❤♦r✐③♦♥t❛❧ ❧✐♥❡ t❤r♦✉❣❤ t❤✐s ♣♦✐♥t✳ ✸✳ ❚❤❡ ✐♥t❡rs❡❝t✐♦♥ ♦❢ t❤❡s❡ t✇♦ ❧✐♥❡s ✐s t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♣♦✐♥t P = (4, 1) ♦♥ t❤❡ ♣❧❛♥❡✦ ■♥ s✉♠♠❛r②✿

• ■❢ P ✐s ❛ ❧♦❝❛t✐♦♥ ♦♥ t❤❡ ♣❧❛♥❡✱ ✇❡ ✜♥❞ t❤❡ ❞✐st❛♥❝❡s ❢r♦♠ ❡✐t❤❡r ♦❢ t❤❡ t✇♦ ❛①❡s t♦ t❤❛t ❧♦❝❛t✐♦♥ ✕ ♣♦s✐t✐✈❡ ✐♥ t❤❡ ♣♦s✐t✐✈❡ ❞✐r❡❝t✐♦♥ ❛♥❞ ♥❡❣❛t✐✈❡ ✐♥ t❤❡ ♥❡❣❛t✐✈❡ ❞✐r❡❝t✐♦♥ ✕ ❛♥❞ t❤❡ r❡s✉❧t ✐s t❤❡ t✇♦ ❝♦♦r❞✐♥❛t❡s ♦❢ P ✱ s♦♠❡ ♥✉♠❜❡rs x ❛♥❞ y ✳

• ■❢ x ❛♥❞ y ❛r❡ ♥✉♠❜❡rs✱ ✇❡ ♠❡❛s✉r❡ x ❛s t❤❡ ❞✐st❛♥❝❡ ❢r♦♠ t❤❡ y ✲❛①✐s ❛♥❞ y ❛s t❤❡ ❞✐st❛♥❝❡ ❢r♦♠ t❤❡ x✲❛①✐s ✕ ♣♦s✐t✐✈❡ ✐♥ t❤❡ ♣♦s✐t✐✈❡ ❞✐r❡❝t✐♦♥ ❛♥❞ ♥❡❣❛t✐✈❡ ✐♥ t❤❡ ♥❡❣❛t✐✈❡ ❞✐r❡❝t✐♦♥ ✕ ❛♥❞ s✉❝❤ ❧♦❝❛t✐♦♥s t♦❣❡t❤❡r ❢♦r♠ ❛ ✈❡rt✐❝❛❧ ❧✐♥❡ ❛♥❞ ❛ ❤♦r✐③♦♥t❛❧ ❧✐♥❡✱ ❛♥❞ t❤❡ ✐♥t❡rs❡❝t✐♦♥ ♦❢ t❤❡s❡ t✇♦ ✐s t❤❡ ❧♦❝❛t✐♦♥ P ♦♥ t❤❡ ♣❧❛♥❡✳ ❚❤❡ ♠❛✐♥ ❞✐✛❡r❡♥❝❡ ❢r♦♠ t❤❡ xy ✲♣❧❛♥❡ ✇❡ ❤❛✈❡ s❡❡♥ ♣r❡✈✐♦✉s❧② ❝♦♠❡s ❢r♦♠ ✐ts ♣✉r♣♦s❡✳ ■♥ ♦r❞❡r t♦ ❞♦ ❊✉❝❧✐❞❡❛♥ ❣❡♦♠❡tr② ♦♥ t❤✐s ❝♦♦r❞✐♥❛t❡ ♣❧❛♥❡✱ sq✉❛r❡s s❤♦✉❧❞ ❜❡ sq✉❛r❡s ❛♥❞ ♥♦t r❡❝t❛♥❣❧❡s❀ ❝✐r❝❧❡s s❤♦✉❧❞ ❜❡ ❝✐r❝❧❡s ❛♥❞ ♥♦t ♦✈❛❧s✱ ❡t❝✳✿

❊✈❡♥ t❤♦✉❣❤ ✇❡ ❝❛♥ ♠❛❦❡ ❧❛r❣❡r ♦r s♠❛❧❧❡r ♣❧♦ts✱ t❤❡ r❡❧❛t✐✈❡ ❞✐♠❡♥s✐♦♥s ✭❛♥❞✱ ❝♦♥s❡q✉❡♥t❧②✱ t❤❡ ❛♥❣❧❡s ✮ ✇✐❧❧ ❤❛✈❡ t♦ r❡♠❛✐♥ t❤❡ s❛♠❡✳ ❚♦ ❛✈♦✐❞ s✉❝❤ ❛ ❞✐s♣r♦♣♦rt✐♦♥❛❧ r❡s✐③✐♥❣✱ ✇❡ ♥❡❡❞ t♦ ♠❛❦❡ s✉r❡ t❤❛t ✇❡ ✉s❡ t❤❡ s❛♠❡ ✉♥✐ts ❢♦r t❤❡ x✲❛①✐s ❛♥❞ t❤❡ y ✲❛①✐s✳ ❚❤✐s ✇✐❧❧ ❤❛✈❡ t♦ ❜❡ ❛ sq✉❛r❡ ❣r✐❞✳ ❚❤✉s✱ ✇❡ ❛r❡ ❥✉st ♥❛rr♦✇✐♥❣ ❞♦✇♥ t❤❡ s❝♦♣❡ ♦❢ ♣♦ss✐❜✐❧✐t✐❡s ✐♥ ❝♦♠♣❛r✐s♦♥ t♦ t❤❡ ✇❛② t❤❡ ❈❛rt❡s✐❛♥ s②st❡♠ ✇❛s tr❡❛t❡❞ ♣r❡✈✐♦✉s❧②✳

❊①❛♠♣❧❡ ✺✳✺✳✺✿ ❝♦♦r❞✐♥❛t❡s ✉s❡❞ ✐♥ ❝♦♠♣✉t✐♥❣ ❚❤❡ 2✲❞✐♠❡♥s✐♦♥❛❧ ❈❛rt❡s✐❛♥ s②st❡♠ ✐s♥✬t ❛s ✇✐❞❡s♣r❡❛❞ ❛s t❤❡ ♦♥❡ ❢♦r ❞✐♠❡♥s✐♦♥ 1 ✭♥✉♠❜❡rs✮✳ ■t ✐s✱ ❤♦✇❡✈❡r✱ ❝♦♠♠♦♥ ✐♥ ❝❡rt❛✐♥ ❛r❡❛s ♦❢ ❝♦♠♣✉t✐♥❣✳ ❋♦r ❡①❛♠♣❧❡✱ ❞r❛✇✐♥❣ ❛♣♣❧✐❝❛t✐♦♥s ❛❧❧♦✇ ②♦✉ t♦ ♠❛❦❡ ✉s❡ ♦❢ t❤✐s s②st❡♠ ✕ ✐❢ ②♦✉ ✉♥❞❡rst❛♥❞ ✐t✳ ❚❤❡ ❧♦❝❛t✐♦♥ ♦❢ ②♦✉r ♠♦✉s❡ ✐s s❤♦✇♥ ✐♥ t❤❡ st❛t✉s ❜❛r ♦♥ t❤❡ ❧♦✇❡r ❧❡❢t✱ ❝♦♥st❛♥t❧② ✉♣❞❛t❡❞ ✐♥ r❡❛❧ t✐♠❡✳ ❚❤❡ ♠❛✐♥ ❞✐✛❡r❡♥❝❡ ✐s t❤❛t t❤❡ ♦r✐❣✐♥ ✐s ✐♥ t❤❡ ❧❡❢t ✉♣♣❡r ❝♦r♥❡r ♦❢ t❤❡ ✐♠❛❣❡ ❛♥❞ t❤❡ y ✲❛①✐s ✐s ♣♦✐♥t✐♥❣ ❞♦✇♥ ✿

❚❤❡ ❝❤♦✐❝❡ ✐s ❡①♣❧❛✐♥❡❞ ❜② t❤❡ ✇❛② ✇❡ ✇r✐t❡✿ ❞♦✇♥✇❛r❞✳

❊①❛♠♣❧❡ ✺✳✺✳✻✿ ❞✐✛❡r❡♥t ❝♦♦r❞✐♥❛t❡ s②st❡♠s✱ ❞✐♠❡♥s✐♦♥ 2 ❖❢ ❝♦✉rs❡✱ ✇❡ ❝❛♥ ♣❧❛❝❡ ❞✐✛❡r❡♥t ❝♦♦r❞✐♥❛t❡ s②st❡♠s ♦♥ t❤❡ s❛♠❡ ♣❧❛♥❡✳ ❋♦r ❡①❛♠♣❧❡✱ ✇❡ ❝❛♥ ❤❛✈❡ t✇♦ s②st❡♠s t❤❛t ❞✐✛❡r ♦♥❧② ❜② s❝❛❧❡✿

✺✳✺✳

❚❤❡ ❈❛rt❡s✐❛♥ s②st❡♠ ❢♦r t❤❡ ❊✉❝❧✐❞❡❛♥ ♣❧❛♥❡

✹✸✺

❚❤❛t✬s ❛ ✉♥✐❢♦r♠ str❡t❝❤✦ ❖r t❤❡② ❝❛♥ ❞✐✛❡r ♦♥❧② ❜② t❤❡ ❧♦❝❛t✐♦♥ ♦❢ t❤❡ ♦r✐❣✐♥✿

❚❤❛t✬s ❛ s❤✐❢t✦ ❲❡ s❛✇ ✐♥ ❈❤❛♣t❡r ✸ ❤♦✇ ✇❡ ❝❛♥ tr❛♥s❢❡r ✐♥❢♦r♠❛t✐♦♥ ✭♣♦✐♥ts✱ s❡t✱ ❢✉♥❝t✐♦♥s✱ ❡t❝✳✮ ❞❡✜♥❡❞ ♦♥ t❤❡ ✜rst ♣❧❛♥❡ t♦ t❤❡ s❡❝♦♥❞✳ ■t ♦♥❧② t❛❦❡s ❛ s✐♥❣❧❡ ❢✉♥❝t✐♦♥✳ ❋♦r t❤❡ ❢♦r♠❡r ❡①❛♠♣❧❡✱ ✐t ✐s

(x, y) 7→ (x/2, y/2) . ❋♦r t❤❡ ❧❛tt❡r ❡①❛♠♣❧❡✱ ✐t ✐s

(x, y) 7→ (x − 3, y − 4) .

❲✐t❤ t❤❡s❡ ❢✉♥❝t✐♦♥s✱ ❛❧❧ t❤❡ q✉❛♥t✐t✐❡s ✕ ❣❡♦♠❡tr✐❝ ♦r ♣❤②s✐❝❛❧ ✕ ❞❡✜♥❡❞ ✇✐t❤✐♥ t❤❡ t✇♦ ❝♦♦r❞✐♥❛t❡ s②st❡♠s ❛r❡ tr❛♥s❢♦r♠❡❞ t♦ ❡❛❝❤ ♦t❤❡r✳ ▼♦r❡♦✈❡r✱ t❤❡ ❝♦♦r❞✐♥❛t❡ s②st❡♠s ❝❛♥ ❛❧s♦ ✈❛r② ✐♥ t❡r♠s ♦❢ t❤❡

❞✐r❡❝t✐♦♥s ♦❢ t❤❡ ❛①❡s✿

❘♦t❛t✐♦♥s ❛♥❞ ♦t❤❡r tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ♣❧❛♥❡ ❛r❡ ❝♦♥s✐❞❡r❡❞ ✐♥ ❈❤❛♣t❡r ✺❉❊✲✷✳ ❊①❡r❝✐s❡ ✺✳✺✳✼

❲❤❛t tr❛♥s❢♦r♠❛t✐♦♥ ✐s♥✬t ♠❡♥t✐♦♥❡❞ ❛❜♦✈❡❄ Pr♦✈✐❞❡ ❛♥ ✐❧❧✉str❛t✐♦♥ ❛♥❞ ❛ ❢♦r♠✉❧❛ ❢♦r ✐ts ❝♦♠❜✐♥❛t✐♦♥ ✇✐t❤ t❤❡ ✜rst t✇♦✳ ❊①❛♠♣❧❡ ✺✳✺✳✽✿ ♠♦t✐♦♥ ♦♥ ♣❧❛♥❡✱ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s

❆ s❡q✉❡♥❝❡ ♦❢ x✬s✱ xn ✱ ❛♥❞ ❛ s❡q✉❡♥❝❡ ♦❢ y ✬s✱ yn ✱ ✇✐❧❧ ❝r❡❛t❡ ❛

s❡q✉❡♥❝❡ ♦❢ ♣♦✐♥ts ♦♥ t❤❡ ♣❧❛♥❡ ✿

✺✳✺✳

❚❤❡ ❈❛rt❡s✐❛♥ s②st❡♠ ❢♦r t❤❡ ❊✉❝❧✐❞❡❛♥ ♣❧❛♥❡

✹✸✻

■♥ ❝♦♥tr❛st t♦ t❤❡ st✉❞② ✐♥ ❈❤❛♣t❡r ✶✱ ✇❡ ❝❛♥ r❡♣r❡s❡♥t t❤❡ ♣❛t❤ ♦❢ ❛ ♠♦✈✐♥❣ ❜❛❧❧ ❜❡②♦♥❞ ❥✉st ✏✉♣ ♦r ❞♦✇♥✑✦ ❚♦ ❜❡❣✐♥ ✇✐t❤✱ ✇❤❡♥ ❜♦t❤ s❡q✉❡♥❝❡s ❛r❡ ❛r✐t❤♠❡t✐❝ ♣r♦❣r❡ss✐♦♥s✱ t❤❡ ♣❛t❤ ✐s ❛ str❛✐❣❤t ❧✐♥❡✿

❲✐t❤

n

❛s

t✐♠❡✱ ✇❡ ❤❛✈❡✱ r❡❝✉rs✐✈❡❧②✿ xn+1 = xn + p, yn+1 = yn + q , p ❛♥❞ q ✱ r❡♣r❡s❡♥t t❤❡ ❤♦r✐③♦♥t❛❧ ❛♥❞ t❤❡ ✈❡rt✐❝❛❧ q = 1✱ p = 2 ✇✐t❤ t❤❡ ✐♥✐t✐❛❧ ✈❛❧✉❡s −1 ❛♥❞ 0 ❛s ❛♥

✇❤❡r❡ t❤❡ ✐♥❝r❡♠❡♥ts ♦❢ t❤❡ ❛r✐t❤♠❡t✐❝ ♣r♦❣r❡ss✐♦♥s✱ ❝♦♠♣♦♥❡♥ts ♦❢ t❤❡

✈❡❧♦❝✐t②✱ r❡s♣❡❝t✐✈❡❧② ✭❛❜♦✈❡

❡①❛♠♣❧❡✮✳ ■❢ ✇❡ ❝❤♦♦s❡

yn

t♦ ❜❡ q✉❛❞r❛t✐❝ ✐♥st❡❛❞✱ ✇❡ ❤❛✈❡ ❛ ✏♣❛r❛❜♦❧❛✑✿

❲❡ ❝❛♥ ♥♦✇ r❡♣r❡s❡♥t t❤❡

♣❛t❤ ♦❢ ❛ t❤r♦✇♥ ❜❛❧❧✳

❛ ❝❛♥♥♦♥❜❛❧❧ str❛✐❣❤t ❢♦r✇❛r❞ ❛t

200

❋♦r ❡①❛♠♣❧❡✱ ❢r♦♠ t❤❡ ❡❧❡✈❛t✐♦♥ ♦❢

❢❡❡t ♣❡r s❡❝♦♥❞✿

xn = 200n, yn = 200 − 16n2 . ❈✐r❝✉❧❛r ♠♦t✐♦♥ ✐s ❝♦♥s✐❞❡r❡❞ ❧❛t❡r ✐♥ t❤✐s ❝❤❛♣t❡r✳

❊①❡r❝✐s❡ ✺✳✺✳✾

❙❤♦✇ t❤❛t t❤❡ ♣✐❝t✉r❡ ❞❡s❝r✐❜❡s t❤❡ ♣❛t❤ ❣✐✈❡♥ ❜② t❤❡ s❡q✉❡♥❝❡s✳

200

❢❡❡t✱ ✇❡ s❤♦♦t

✺✳✻✳

✹✸✼

❚❤❡ ❊✉❝❧✐❞❡❛♥ ♣❧❛♥❡✿ ❞✐st❛♥❝❡s

✺✳✻✳ ❚❤❡ ❊✉❝❧✐❞❡❛♥ ♣❧❛♥❡✿ ❞✐st❛♥❝❡s

❙✐♥❝❡ ❡✈❡r②t❤✐♥❣ ✐♥ t❤❡ ❈❛rt❡s✐❛♥ s②st❡♠ ✐s ♣r❡✲♠❡❛s✉r❡❞✱ ✇❡ ❝❛♥ s♦❧✈❡ s♦♠❡ ❣❡♦♠❡tr✐❝ ♣r♦❜❧❡♠s ❜② ❛❧❣❡✲ ❜r❛✐❝❛❧❧② ♠❛♥✐♣✉❧❛t✐♥❣ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ ♣♦✐♥ts✳ ■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ ❝♦♥s✐❞❡r t❤❡ ✈❡r② ❜❛s✐❝ ❣❡♦♠❡tr✐❝ t❛s❦ ♦❢ ❝♦♠♣✉t✐♥❣ ✕ ❛s ♦♣♣♦s❡❞ t♦ ♠❡❛s✉r✐♥❣ ✕ ❞✐st❛♥❝❡s✳ ❋✐rst✱ t❤❡

✳

❧✐♥❡

❚❤❡ ❞✐st❛♥❝❡ ✕ ❛ ♥✉♠❜❡r ✕ ❜❡t✇❡❡♥ t✇♦ ❧♦❝❛t✐♦♥s P ❛♥❞ Q ✐s ✐♥❤❡r✐t❡❞ ❢r♦♠ t❤❡ ❊✉❝❧✐❞❡❛♥ ❧✐♥❡ t❤❛t ✉♥❞❡r❧✐❡s t❤❡ ❈❛rt❡s✐❛♥ s②st❡♠✳ ❚❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t✇♦ ♣♦✐♥ts P ❛♥❞ Q ✐s ❞❡♥♦t❡❞ ❛s ❢♦❧❧♦✇s✿ ❉✐st❛♥❝❡ ♦♥ ❧✐♥❡

d(P, Q) ◆♦✇✱ ❤♦✇ ❞♦ ✇❡ ❡①♣r❡ss t❤✐s ♥✉♠❜❡r ✐♥ t❡r♠s ♦❢ t❤❡✐r ❝♦♦r❞✐♥❛t❡s✱ s❛② x ❛♥❞ x′ ❄ ❊①❛♠♣❧❡ ✺✳✻✳✶✿ ❞✐st❛♥❝❡ ❞✐♠ ✶

❖♥❡ ✜♥❞s t❤❡ ❞✐st❛♥❝❡ t❤❛t ❤❛s ❜❡❡♥ ❝♦✈❡r❡❞ ♦♥ t❤❡ r♦❛❞ ❜② s✉❜tr❛❝t✐♥❣ t❤❡ ♥✉♠❜❡r ♦♥ t❤❡ ♠✐❧❡st♦♥❡ ✐♥ t❤❡ ❜❡❣✐♥♥✐♥❣ ❛♥❞ t❤❡ ♥✉♠❜❡r ♦♥ t❤❡ ♠✐❧❡st♦♥❡ ❛t t❤❡ ❡♥❞✿ ❢r♦♠ P = 4 t♦ Q = 6 =⇒ ❞✐st❛♥❝❡ = Q − P = 6 − 4 = 2 . ❇✉t ✇❤❛t ✐❢ ✇❡ ❛r❡ ♠♦✈✐♥❣ ✐♥ t❤❡ ♦♣♣♦s✐t❡ ❞✐r❡❝t✐♦♥❄ ❚❤❡ ❞✐st❛♥❝❡ s❤♦✉❧❞ ❜❡ t❤❡ s❛♠❡✦ ❆♥❞ t❤❡ ❝♦♠♣✉t❛t✐♦♥ s❤♦✉❧❞ ❜❡ t❤❡ s❛♠❡✿ ❢r♦♠ Q = 6 t♦ P = 4 =⇒ ❞✐st❛♥❝❡ = Q − P = 6 − 4 = 2 . ■♥❞❡❡❞✿

■♥ ♦t❤❡r ✇♦r❞s✱ ♦♥❡ ♠✉st s✉❜tr❛❝t t❤❡ s♠❛❧❧❡r ♥✉♠❜❡r ❢r♦♠ t❤❡ ❧❛r❣❡r ♦♥❡ ❡✈❡r② t✐♠❡ ✐♥ ♦r❞❡r ❢♦r t❤❡ ❝♦♠♣✉t❛t✐♦♥ t♦ ♠❛❦❡ s❡♥s❡✳ ❚❤❡♦r❡♠ ✺✳✻✳✷✿ ❉✐st❛♥❝❡s ♦♥ ▲✐♥❡ ❚❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t✇♦ ❧♦❝❛t✐♦♥s

P

❛♥❞

Q

♦♥ t❤❡ r❡❛❧ ❧✐♥❡ ❣✐✈❡♥ ❜② t❤❡✐r

x ❛♥❞ x′ ✐s✿ • d(P, Q) = x′ − x ✇❤❡♥ x < x′ ✱ • d(P, Q) = x − x′ ✇❤❡♥ x > x′ ✱ • d(P, Q) = 0 ✇❤❡♥ x = x′ ✳

❝♦♦r❞✐♥❛t❡s

■s t❤❡r❡ ❛ s✐♥❣❧❡ ❢♦r♠✉❧❛ ❢♦r t❤✐s ❝♦♠♣✉t❛t✐♦♥❄ ❚❤❡ ✐❞❡❛ t❤❛t t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t✇♦ ❧♦❝❛t✐♦♥s ❝❛♥ ♥❡✈❡r ❜❡ ♥❡❣❛t✐✈❡ s✉❣❣❡sts t❤❛t t❤✐s ❤❛s s♦♠❡t❤✐♥❣ t♦ ❞♦ ✇✐t❤ t❤❡ ❛❜s♦❧✉t❡ ✈❛❧✉❡✿

✺✳✻✳ ❚❤❡ ❊✉❝❧✐❞❡❛♥ ♣❧❛♥❡✿ ❞✐st❛♥❝❡s

✹✸✽

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

❲❡ ❥✉st s✉❜st✐t✉t❡

a = x′ − x

  −a 0 |a| =  a

✇❤❡♥ ✇❤❡♥ ✇❤❡♥

a < 0, a = 0, a > 0.

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

❚❤❡♦r❡♠ ✺✳✻✳✸✿ ❉✐st❛♥❝❡ ❋♦r♠✉❧❛ ❢♦r ❉✐♠❡♥s✐♦♥

1

❚❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t✇♦ ♣♦✐♥ts ♦♥ t❤❡ r❡❛❧ ❧✐♥❡ ✇✐t❤ ❝♦♦r❞✐♥❛t❡s x ❛♥❞ x′ ✐s t❤❡ ❛❜s♦❧✉t❡ ✈❛❧✉❡s ♦❢ t❤❡✐r ❞✐✛❡r❡♥❝❡ ✭✐♥ ❡✐t❤❡r ♦r❞❡r✮✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿ d(x, x′ ) = |x − x′ | = |x′ − x| = d(x′ , x) ❊①❡r❝✐s❡ ✺✳✻✳✹

❉❡r✐✈❡ t❤❡ ❢♦r♠✉❧❛ ❢r♦♠ t❤❡ ❢♦❧❧♦✇✐♥❣✿

  x − x′ 0 d(P, Q) =  x − x′

❊①❡r❝✐s❡ ✺✳✻✳✺

Pr♦✈❡ t❤❛t ❢♦r ❛♥② t✇♦ ♥✉♠❜❡rs

x, x′ ✱

✇❡ ❤❛✈❡

x > x′ , x = x′ , x < x′ .

✇❤❡♥ ✇❤❡♥ ✇❤❡♥

|x + x′ | ≤ |x| + |x′ | ✳

❊①❡r❝✐s❡ ✺✳✻✳✻

Pr♦✈❡ t❤❛t t❤❡ ♣♦✐♥t ❤❛❧❢✲✇❛② ❜❡t✇❡❡♥ ♣♦✐♥ts ′ ❝♦♦r❞✐♥❛t❡

x+x 2

P = x

❛♥❞

Q = x′

✭❝❛❧❧❡❞ t❤❡✐r ✏♠✐❞♣♦✐♥t✑✮ ❤❛s t❤❡

✳

❚❤❡ ✇♦r❞ ✏str❡t❝❤✑ t❤❛t ✇❡ ❤❛✈❡ ✉s❡❞ ✐♥ t❤❡ ♣❛st ♥♦✇ t❛❦❡s ❛ ♠❡❛♥✐♥❣ t❤❛t r❡❧✐❡s ♦♥ t❤❡ ✐❞❡❛ ♦❢ ❞✐st❛♥❝❡✿ ❚❤❡ ❞✐st❛♥❝❡s ✐♥❝r❡❛s❡ ♣r♦♣♦rt✐♦♥❛❧❧②✳ ❚❤❡♦r❡♠ ✺✳✻✳✼✿ ▲✐♥❡❛r ❚r❛♥s❢♦r♠❛t✐♦♥s ✐♥ ❉✐♠❡♥s✐♦♥

1

❆ ❧✐♥❡❛r ❢✉♥❝t✐♦♥ str❡t❝❤❡s t❤❡ x✲❛①✐s ❜② ❛ ❢❛❝t♦r ♦❢ |m|✱ ✇❤❡r❡ m ✐s ✐ts s❧♦♣❡✳ Pr♦♦❢✳

❚❤✐s ✐s ✇❤❛t ❤❛♣♣❡♥s t♦ t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t✇♦ ♣♦✐♥ts ❢✉♥❝t✐♦♥

f (x) = mx + b

u

✐s ❛♣♣❧✐❡❞✿

❛♥❞

v✱

✇❤✐❝❤ ✐s

|v − u|✱

❛❢t❡r ❛ ❧✐♥❡❛r

|f (v) − f (u)| = |(mv + b) − (mu + b)| = |mv − mu| = |m| · |v − u| . ❚❤❡ ❞✐st❛♥❝❡ ❤❛s ✐♥❝r❡❛s❡❞ ❜② ❛ ❢❛❝t♦r ♦❢

|m| < 1✮✳

|m|✳

✭❲❡ s❛② t❤❛t ✐t ❤❛s ❞❡❝r❡❛s❡❞ ❜② ❛ ❢❛❝t♦r ♦❢

❚❤✐s str❡t❝❤✴s❤r✐♥❦ ❢❛❝t♦r ✐s t❤❡ s❛♠❡ ❡✈❡r②✇❤❡r❡✳

m

✇❤❡♥

✺✳✻✳

✹✸✾

❚❤❡ ❊✉❝❧✐❞❡❛♥ ♣❧❛♥❡✿ ❞✐st❛♥❝❡s

◆❡①t✱ t❤❡

♣❧❛♥❡✳

❲❡ st❛rt ✇✐t❤ ❛ r❡s✉❧t t❤❛t ✐s ♦♥❡ ♦❢ t❤❡ ♠♦st ✐♠♣♦rt❛♥t✳

❚❤❡♦r❡♠ ✺✳✻✳✽✿ P②t❤❛❣♦r❡❛♥ ❚❤❡♦r❡♠ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ r✐❣❤t tr✐❛♥❣❧❡ ✇✐t❤ s✐❞❡s

a, b, c✱

✇✐t❤

c

t❤❡ ❧♦♥❣❡st ♦♥❡ ❢❛❝✐♥❣

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

a2 + b2 = c2

Pr♦♦❢✳

❲❡ ✉s❡ ✇❤❛t ✇❡ ❦♥♦✇ ❛❜♦✉t s✐♠✐❧❛r tr✐❛♥❣❧❡s ✭✐✳❡✳✱ t❤❡ ♦♥❡s ✇✐t❤ ❡q✉❛❧ ❛♥❣❧❡s✮✿ ▲❡t

◮ ❚❤❡ r❛t✐♦ ♦❢ ❛♥② t✇♦ ❝♦rr❡s♣♦♥❞✐♥❣ ABC ❜❡ ♦✉r r✐❣❤t tr✐❛♥❣❧❡✱ ✇✐t❤✿ • ✈❡rt❡① A ♦♣♣♦s✐t❡ t♦ s✐❞❡ a✱ • ✈❡rt❡① B ♦♣♣♦s✐t❡ t♦ s✐❞❡ b✱ • ✈❡rt❡① C ♦♣♣♦s✐t❡ t♦ s✐❞❡ c✳

s✐❞❡s ♦❢ s✐♠✐❧❛r tr✐❛♥❣❧❡s ✐s t❤❡ s❛♠❡✳

❲❡ ❞r❛✇ t❤❡ ❤❡✐❣❤t ✭t❤❡ ❧✐♥❡ ♣❡r♣❡♥❞✐❝✉❧❛r t♦ c✮ ❢r♦♠

❚❤❡ ♥❡✇ tr✐❛♥❣❧❡

ACH

✐s s✐♠✐❧❛r t♦ ♦✉r ♦r✐❣✐♥❛❧ tr✐❛♥❣❧❡

❛♥❞ t❤❡② s❤❛r❡ t❤❡ ❛♥❣❧❡ ❛t

ABC ✳

C✱

A✱ α✳

❛♥❞ ❝❛❧❧

ABC ✱

H

✐ts ✐♥t❡rs❡❝t✐♦♥ ✇✐t❤ t❤❡ s✐❞❡ c✿

❜❡❝❛✉s❡ t❤❡② ❜♦t❤ ❤❛✈❡ ❛ r✐❣❤t ❛♥❣❧❡✱

■♥ t❤❡ s❛♠❡ ✇❛②✱ ✇❡ ♣r♦✈❡ t❤❛t t❤❡ tr✐❛♥❣❧❡

CBH

✐s ❛❧s♦ s✐♠✐❧❛r t♦

❚❤❡ s✐♠✐❧❛r✐t② ♦❢ t❤❡s❡ t✇♦ ♣❛✐rs ♦❢ tr✐❛♥❣❧❡s ❧❡❛❞s t♦ t❤❡ ❡q✉❛❧✐t② ♦❢ r❛t✐♦s ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣

s✐❞❡s✿

BH AH AC BC = = AB BC AB AC 2 2 BC = AB · BH AC = AB · AH

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

b2 + a2 = BC 2 + AC 2 = AB · BH + AB · AH = AB · (AH + BH) = AB 2 = c2 . ❊①❡r❝✐s❡ ✺✳✻✳✾

❙t❛t❡ t❤❡ ❝♦♥✈❡rs❡ ♦❢ t❤❡ t❤❡♦r❡♠✳ ■s ✐t tr✉❡❄ ❊①❡r❝✐s❡ ✺✳✻✳✶✵

❲❤❛t ❞♦ t❤❡ r❛t✐♦s ✐♥ t❤❡ ♣r♦♦❢ t❡❧❧ ✉s ❛❜♦✉t t❤❡ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s ♦❢ t❤❡ ❛♥❣❧❡ ❚❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t✇♦ ❧♦❝❛t✐♦♥s

P

t❤❡ ❈❛rt❡s✐❛♥ s②st❡♠✳ ❚❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t✇♦

α❄

Q ✐s ✐♥❤❡r✐t❡❞ ❢r♦♠ t❤❡ ❊✉❝❧✐❞❡❛♥ ♣❧❛♥❡ t❤❛t ✉♥❞❡r❧✐❡s ♣♦✐♥ts P ❛♥❞ Q ✐s ❞❡♥♦t❡❞ ❛s ❢♦❧❧♦✇s✿ ❛♥❞

✺✳✻✳

✹✹✵

❚❤❡ ❊✉❝❧✐❞❡❛♥ ♣❧❛♥❡✿ ❞✐st❛♥❝❡s

❉✐st❛♥❝❡ ♦♥ ♣❧❛♥❡

d(P, Q)

◆♦✇✱ ✇❤❛t ✐❢ t❤✐s t✐♠❡ ✇❡ ❤❛✈❡ ❛ ❈❛rt❡s✐❛♥ s②st❡♠ ♣❧❛❝❡❞ ♦♥ t♦♣ ♦❢ t❤✐s ♣✐❡❝❡ ♦❢ ♣❛♣❡r❄ ❍♦✇ ❞♦ ✇❡ ❡①♣r❡ss t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ P ❛♥❞ Q ✐♥ t❡r♠s ♦❢ t❤❡✐r ❝♦♦r❞✐♥❛t❡s (x, y) ❛♥❞ (x′ , y ′ )❄ ❚❤❡ ✐❞❡❛ ✐s t♦ ✜♥❞ t❤❡ ❞✐st❛♥❝❡s ❛❧♦♥❣ t❤❡ ❛①❡s ✜rst✿

❲❡ ❛♣♣❧✐❡❞ t❤❡ ❉✐st❛♥❝❡ ❋♦r♠✉❧❛ ❢♦r ❉✐♠❡♥s✐♦♥ 1 ❢♦r ❡✐t❤❡r ♦❢ t❤❡ t✇♦ ❛①❡s ❛♥❞ t❤❡♥ ✉s❡❞ t❤❡ P②t❤❛❣♦r❡❛♥ ✳

❚❤❡♦r❡♠

❚❤❡ ❢♦❧❧♦✇✐♥❣ ✐s ♦♥❡ ♦❢ t❤❡ ♠♦st ✉s❡❢✉❧ r❡s✉❧ts ✐♥ ❣❡♦♠❡tr② ♦❢ t❤❡ ❈❛rt❡s✐❛♥ ♣❧❛♥❡✳ ❚❤❡♦r❡♠ ✺✳✻✳✶✶✿ ❉✐st❛♥❝❡ ❋♦r♠✉❧❛ ❢♦r ❉✐♠❡♥s✐♦♥

2

❚❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t✇♦ ♣♦✐♥ts ✇✐t❤ ❝♦♦r❞✐♥❛t❡s P = (x, y) ❛♥❞ Q = (x′, y′) ✐s d(P, Q) =

p

(x − x′ )2 + (y − y ′ )2

Pr♦♦❢✳

❆❝❝♦r❞✐♥❣ t♦ t❤❡ ❢♦r♠✉❧❛✿ • ❚❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ x ❛♥❞ x′ ♦♥ t❤❡ x✲❛①✐s ✐s |x − x′ |✳ • ❚❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ y ❛♥❞ y ′ ♦♥ t❤❡ y ✲❛①✐s ✐s |y − y ′ |✳ ❚❤❡♥✱ t❤❡ s❡❣♠❡♥t ❜❡t✇❡❡♥ t❤❡ ♣♦✐♥ts P (x, y) ❛♥❞ Q = (x′ , y ′ ) ✐s t❤❡ ❤②♣♦t❡♥✉s❡ ♦❢ t❤❡ r✐❣❤t tr✐❛♥❣❧❡ ✇✐t❤ s✐❞❡s✿ |x − x′ | ❛♥❞ |y − y ′ |✳ ❚❤❡♥ ♦✉r ❝♦♥❝❧✉s✐♦♥ ❜❡❧♦✇ ❢♦❧❧♦✇s ❢r♦♠ t❤❡ P②t❤❛❣♦r❡❛♥ ❚❤❡♦r❡♠ ✿ d(P, Q)2 = |x − x′ |2 + |y − y ′ |2 .

❙✐♥❝❡ |z|2 = z 2 ❢♦r ❛♥② z ✱ ✇❡ ❝❛♥ r❡♠♦✈❡ t❤❡ ❛❜s♦❧✉t❡ ✈❛❧✉❡ s✐❣♥s✳ ❚❤❡ r❡s✉❧t ✐s s♦ ✐♠♣♦rt❛♥t t❤❛t ♦♥❡ ❝❛♥ ❡✈❡♥ s❛② t❤❛t t❤❡ 90✲❞❡❣r❡❡ ❛♥❣❧❡ ❜❡t✇❡❡♥ t❤❡ ❛①❡s ✇❛s ❝❤♦s❡♥ s♦ t❤❛t ✇❡ ❝❛♥ ♣r♦❞✉❝❡ t❤✐s ❢♦r♠✉❧❛ ❢r♦♠ t❤❡ P②t❤❛❣♦r❡❛♥ t❤❡♦r❡♠✳ ❊①❡r❝✐s❡ ✺✳✻✳✶✷

❋✐♥❞ t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ ♣♦✐♥ts (−5, 2) ❛♥❞ (2, −1)✳

✺✳✻✳ ❚❤❡ ❊✉❝❧✐❞❡❛♥ ♣❧❛♥❡✿ ❞✐st❛♥❝❡s

✹✹✶

❲❛r♥✐♥❣✦ ❈♦♠❜✐♥❡

x✬s

✇✐t❤

x✬s

❛♥❞

y ✬s

✇✐t❤

y ✬s✳

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

❞✐st❛♥❝❡s

x✲❛①✐s

❞✐♠❡♥s✐♦♥

1 d(P, Q)

2

2 d(P, Q)

2

y ✲❛①✐s

′ 2

= (x − x )

= (x − x′ )2 + (y − y ′ )2

■♥ ♦t❤❡r ✇♦r❞s✿

◮

❚❤❡ sq✉❛r❡ ♦❢ t❤❡ ❞✐st❛♥❝❡ ✐s t❤❡ s✉♠ ♦❢ t❤❡ sq✉❛r❡s ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡s ♦❢ t❤❡ ❝♦♦r❞✐♥❛t❡s✳

❲❡ ✇✐❧❧ s❡❡ ❛ ❝♦♥t✐♥✉❛t✐♦♥ ♦❢ t❤✐s ❧✐st ❛♥❞ ♦❢ t❤✐s ♣❛tt❡r♥ ✐♥ ❤✐❣❤❡r ❞✐♠❡♥s✐♦♥s ✭❈❤❛♣t❡r ✸■❈✲✹✮✳

❊①❛♠♣❧❡ ✺✳✻✳✶✸✿ ♠✐❧❡s ❛♥❞ ❦✐❧♦♠❡t❡rs ■t ✐s ♣♦ss✐❜❧❡ t♦ ❤❛✈❡ t❤❡

x✲❛①✐s

♠❡❛s✉r❡❞ ✐♥ ❞✐✛❡r❡♥t ✉♥✐ts ❢r♦♠ t❤❡

y ✲❛①✐s✳

❋♦r ❡①❛♠♣❧❡✱ ✐t ✐s t②♣✐❝❛❧

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

❆❧s♦✱ ♦♥❡ ❝❛♥ s♣❡❛❦✱ ❤②♣♦t❤❡t✐❝❛❧❧②✱ ♦❢ ❛ ♣♦✐♥t ❧♦❝❛t❡❞ ✏ 2 ♠✐❧❡s ❡❛st ❛♥❞

5 ❦✐❧♦♠❡t❡rs ♥♦rt❤✑

❢r♦♠ ❤❡r❡✳

❍♦✇❡✈❡r✱ ✇❤❡♥ t❤✐s ✐s t❤❡ ❝❛s❡✱ t❤❡ ❉✐st❛♥❝❡ ❋♦r♠✉❧❛ ✇♦♥✬t ❜❡ ❛♣♣❧✐❝❛❜❧❡ ❛♥②♠♦r❡✦

❍❡r❡ ✐s ❛ ❢❛♠✐❧✐❛r ♣r♦♣❡rt② ♦❢ t❤❡ ❧❡♥❣t❤s ♦❢ t❤❡ s✐❞❡s ♦❢ ❛ tr✐❛♥❣❧❡✿ ❚❤❡ ❧❡♥❣t❤ ♦❢ ❛♥② s✐❞❡ ✐s ❧❡ss t❤❛♥ t❤❡ s✉♠ ♦❢ t❤❡ ❧❡♥❣t❤s ♦❢ t❤❡ ♦t❤❡r t✇♦✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✐❢

a, b, c

❛r❡ t❤❡s❡ t❤r❡❡ s✐❞❡s✱ t❤❡♥✿

c < a + b.

❆s ②♦✉ ❝❛♥ s❡❡ ✭❜♦tt♦♠ r♦✇✮✱ ❡✈❡♥ ✇❤❡♥ t❤❡ tr✐❛♥❣❧❡ ✏❞❡❣❡♥❡r❛t❡s✑ t♦ ❛ s❡❣♠❡♥t✱ ✇❡ ❤❛✈❡ ❛♥ ❡q✉❛t✐♦♥✿

c = a + b. ❚❤✉s✱ t❤❡ ✐♥❡q✉❛❧✐t② r❡♠❛✐♥s tr✉❡ t❤♦✉❣❤ ♥♦♥✲str✐❝t✳ ❲❡ r❡st❛t❡ t❤✐s ❡①t❡♥❞❡❞ ✐♥❡q✉❛❧✐t② ❛s ❢♦❧❧♦✇s✳

❚❤❡♦r❡♠ ✺✳✻✳✶✹✿ ❚r✐❛♥❣❧❡ ■♥❡q✉❛❧✐t② ❋♦r ❛♥② t❤r❡❡ ♣♦✐♥ts

P, Q, R

♦♥ t❤❡ ♣❧❛♥❡✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿

d(P, R) ≤ d(P, Q) + d(Q, R)

✺✳✻✳

✹✹✷

❚❤❡ ❊✉❝❧✐❞❡❛♥ ♣❧❛♥❡✿ ❞✐st❛♥❝❡s

■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ str❛✐❣❤t ❧✐♥❡ ✐s t❤❡ s❤♦rt❡st✳

❊①❡r❝✐s❡ ✺✳✻✳✶✺ ❉❡r✐✈❡ t❤❡ t❤❡♦r❡♠ ❢r♦♠ t❤❡ ❉✐st❛♥❝❡ ❋♦r♠✉❧❛✳

❊①❡r❝✐s❡ ✺✳✻✳✶✻ ❉❡r✐✈❡ t❤❡ ✐♥❡q✉❛❧✐t② ❢♦r ❛ r✐❣❤t tr✐❛♥❣❧❡ ❢r♦♠ t❤❡ P②t❤❛❣♦r❡❛♥ ❚❤❡♦r❡♠✳ ❲❤❡♥ t❤❡ tr✐❛♥❣❧❡ ❞❡❣❡♥❡r❛t❡s ✐♥t♦ ❛ s❡❣♠❡♥t✱ ✇❡ ❤❛✈❡ t❤❡ tr✐❛♥❣❧❡ ✐♥❡q✉❛❧✐t② ❢♦r ❞✐♠❡♥s✐♦♥ 1 ♣r❡s❡♥t❡❞ ❛❜♦✈❡✳

❊①❡r❝✐s❡ ✺✳✻✳✶✼ Pr♦✈❡ t❤❛t t❤❡ ♣♦✐♥t M ❤❛❧❢✲✇❛② ❜❡t✇❡❡♥ ♣♦✐♥ts P = (x, y) ❛♥❞ Q = (x′ , y ′ ) ✕ ❝❛❧❧❡❞ t❤❡✐r ✐s ❣✐✈❡♥ ❜② t❤❡✐r ❛✈❡r❛❣❡ ❝♦♦r❞✐♥❛t❡s✿   x + x′ y + y ′ , . M= 2 2

♠✐❞♣♦✐♥t

✕

❚❤❡ s✐♠♣❧❡st ❣❡♦♠❡tr✐❝ ✜❣✉r❡ t❤❛t r❡❧✐❡s ♦♥ t❤❡ ✐❞❡❛ ♦❢ ❞✐st❛♥❝❡ ✐s t❤❡ ❝✐r❝❧❡✳ ■❢ t❤❡ ❝❡♥t❡r✱ ✇❤✐❝❤ ✐s ❛ ♣♦✐♥t ♦♥ t❤❡ ♣❧❛♥❡✱ ❛♥❞ t❤❡ r❛❞✐✉s✱ ✇❤✐❝❤ ✐s ❛ ♣♦s✐t✐✈❡ ♥✉♠❜❡r✱ ❛r❡ ❣✐✈❡♥✱ ✇❡ ❥✉st ❞r❛✇ s❡❣♠❡♥ts ♦❢ t❤✐s ❧❡♥❣t❤ ✐♥ ❛❧❧ ❞✐r❡❝t✐♦♥s ❢r♦♠ t❤❡ ❝❡♥t❡r ❛♥❞ t❤❡✐r ❡♥❞s ✇✐❧❧ ❝r❡❛t❡ t❤❡ ❝✐r❝❧❡✿

❆ ♠♦r❡ ♣r❡❝✐s❡ ✇❛② t♦ r❡♣r❡s❡♥t ❛ ❝✐r❝❧❡ ✐s ❛s ❛ s✉❜s❡t ♦❢ t❤❡ ♣❧❛♥❡ ❞❡✜♥❡❞ ✈✐❛ t❤❡ s❡t✲❜✉✐❧❞✐♥❣ ♥♦t❛t✐♦♥✱ ❛s ❢♦❧❧♦✇s✳

❉❡✜♥✐t✐♦♥ ✺✳✻✳✶✽✿ ❝✐r❝❧❡ ❚❤❡ ❝✐r❝❧❡ ♦❢ r❛❞✐✉s r > 0 ❝❡♥t❡r❡❞ ❛t ♣♦✐♥t P ♦♥ t❤❡ ♣❧❛♥❡ ✐s t❤❡ s❡t ♦❢ ❛❧❧ ♣♦✐♥ts r ✉♥✐ts ❛✇❛② ❢r♦♠ P ✿ {Q : d(P, Q) = r} . ❲❡ ❝❤❡❝❦ ♦♥❡ ♣♦✐♥t ❛t ❛ t✐♠❡✿

✺✳✻✳ ❚❤❡ ❊✉❝❧✐❞❡❛♥ ♣❧❛♥❡✿ ❞✐st❛♥❝❡s

✹✹✸

❲❛r♥✐♥❣✦ ❆s ❛ s❡t✱ t❤❡ ❝✐r❝❧❡ ❞♦❡s ♥♦t ❝♦♥t❛✐♥ t❤❡ ❝❡♥t❡r ♦❢ t❤❡ ❝✐r❝❧❡ ♥♦r t❤❡ r❡st ♦❢ ✐ts ✐♥t❡r✐♦r✳ ■t✬s ❛ ❝✉r✈❡✦

❊①❡r❝✐s❡ ✺✳✻✳✶✾ Pr♦✈❡ t❤❛t ✐❢ t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❝❡♥t❡rs ♦❢ t✇♦ ❝✐r❝❧❡s ✐s ❧❛r❣❡r t❤❛♥ t❤❡ s✉♠ ♦❢ t❤❡✐r r❛❞✐✐✱ t❤❡♥ t❤❡ ❝✐r❝❧❡s ❞♦♥✬t ✐♥t❡rs❡❝t✳ ❍✐♥t✿ ❚❤❡ ❚r✐❛♥❣❧❡ ■♥❡q✉❛❧✐t②✳

❲❤❛t ❛❜♦✉t ❛ ❝♦♦r❞✐♥❛t❡ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❝✐r❝❧❡s❄ ❙✉♣♣♦s❡

P = (h, k)

✐s t❤❡ ❝❡♥t❡r✳

❚❤❡♥✱ ✐❢ ❛ ♣♦✐♥t

Q = (x, y)

❧✐❡s ♦♥ t❤❡ ❝✐r❝❧❡ ♦❢ r❛❞✐✉s

r

❝❡♥t❡r❡❞ ❛t

t❤❡♥✱ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ P②t❤❛❣♦r❡❛♥ ❚❤❡♦r❡♠ ✱ ♦r t❤❡ ❉✐st❛♥❝❡ ❋♦r♠✉❧❛ ✱ ✐ts ❞✐st❛♥❝❡ t♦ t❤❡ ❝❡♥t❡r

P

P✱

✐s t❤❡

❢♦❧❧♦✇✐♥❣✿

(x − h)2 + (y − k)2 = d(P, Q)2 = r2 . ❚❤❡ t❤r❡❡ q✉❛♥t✐t✐❡s ❛r❡ ✈✐s✐❜❧❡ ❜❡❧♦✇ ❛s ✇❡ ❝❤❡❝❦ ♣♦✐♥t

Q✿

❈♦♥✈❡rs❡❧②✱ ✐❢ t❤❡ ♣♦✐♥t ✐s ♥♦t ♦♥ t❤❡ ❝✐r❝❧❡✱ t❤❡ ❡q✉❛t✐♦♥ ❢❛✐❧s✳ ❲❡ ❛rr✐✈❡ ❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❝❧✉s✐♦♥✳

❚❤❡♦r❡♠ ✺✳✻✳✷✵✿ ❊q✉❛t✐♦♥ ♦❢ ❈✐r❝❧❡ ❚❤❡ ❝✐r❝❧❡ ♦❢ r❛❞✐✉s

r > 0

❝❡♥t❡r❡❞ ❛t ❛ ♣♦✐♥t

P = (h, k)

✐s t❤❡ ❣r❛♣❤ ♦❢ t❤❡

❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥✿

(x − h)2 + (y − k)2 = r2 ■♥ ♦t❤❡r ✇♦r❞s✱ ❡✈❡r② ♣❛✐r ♦❢ ♥✉♠❜❡rs

x

❛♥❞

y

t❤❛t s❛t✐s❢② t❤✐s ❡q✉❛t✐♦♥ ♣r♦❞✉❝❡s ❛ ♣♦✐♥t

t❤✐s ❝✐r❝❧❡✱ ❛♥❞ ✈✐❝❡ ✈❡rs❛✳ ❚❤❡ ❝✐r❝❧❡ ✐s ❝♦♥str✉❝t❡❞ ❢r♦♠ ❛❧❧ ♣♦ss✐❜❧❡ r❡❝t❛♥❣❧❡s ❧✐❦❡ t❤✐s✿

(x, y)

t❤❛t ❧✐❡s ♦♥

✺✳✻✳

✹✹✹

❚❤❡ ❊✉❝❧✐❞❡❛♥ ♣❧❛♥❡✿ ❞✐st❛♥❝❡s

❙♦✱ t❤❡ ❝✐r❝❧❡ ♦❢ r❛❞✐✉s r > 0 ❝❡♥t❡r❡❞ ❛t ❛ ♣♦✐♥t P = (h, k) ✐s t❤❡ ❢♦❧❧♦✇✐♥❣ s❡t✿ {(x, y) : (x − h)2 + (y − k)2 = r2 } .

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

❉❡✜♥✐t✐♦♥ ✺✳✻✳✷✶✿ ♦♣❡♥ ❛♥❞ ❝❧♦s❡❞ ❞✐s❦s • ❚❤❡

♦❢ r❛❞✐✉s r > 0 ❝❡♥t❡r❡❞ ❛t ♣♦✐♥t P ♦♥ t❤❡ ♣❧❛♥❡ ✐s t❤❡ s❡t ♦❢ ❛❧❧ ♣♦✐♥ts ❧❡ss t❤❛♥ r ✉♥✐ts ❛✇❛② ❢r♦♠ P ✿ ♦♣❡♥ ❞✐s❦

{Q : d(P, Q) < r} . • ❚❤❡

♦❢ r❛❞✐✉s r > 0 ❝❡♥t❡r❡❞ ❛t ♣♦✐♥t P ♦♥ t❤❡ ♣❧❛♥❡ ✐s t❤❡ s❡t ♦❢ ❛❧❧ ♣♦✐♥ts ❧❡ss t❤❛♥ ♦r ❡q✉❛❧ t♦ r ✉♥✐ts ❛✇❛② ❢r♦♠ P ✿ ❝❧♦s❡❞ ❞✐s❦

{Q : d(P, Q) ≤ r} .

❲❡ ❛❧s♦ ❝♦♥❝❧✉❞❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿ • ❚❤❡ ♦♣❡♥ ❞✐s❦ ✐s t❤❡ s❡t ♦❢ ♣♦✐♥ts t❤❛t ❧✐❡ ✐♥s✐❞❡ t❤❡ ❝✐r❝❧❡ ♦❢ r❛❞✐✉s r > 0 ❝❡♥t❡r❡❞ ❛t ♣♦✐♥t P ✳

• ❚❤❡ ❝❧♦s❡❞ ❞✐s❦ ✐s s❡t ♦❢ ❛❧❧ ♣♦✐♥ts t❤❛t ❧✐❡ ✐♥s✐❞❡ t❤❡ ❝✐r❝❧❡ ♦❢ r❛❞✐✉s r > 0 ❝❡♥t❡r❡❞ ❛t ♣♦✐♥t P t♦❣❡t❤❡r

✇✐t❤ t❤❡ ❝✐r❝❧❡ ✐ts❡❧❢✳

❙♦✱ t❤❡ ❝❧♦s❡❞ ❞✐s❦ ✐s t❤❡ ✉♥✐♦♥ ♦❢ t❤❡ ♦♣❡♥ ❞✐s❦ ❛♥❞ t❤❡ ❝✐r❝❧❡ ✐ts❡❧❢✿

◆♦✇ t❤❡✐r ❝♦♦r❞✐♥❛t❡ r❡♣r❡s❡♥t❛t✐♦♥s✳

❈♦r♦❧❧❛r② ✺✳✻✳✷✷✿ ❉✐s❦ ✈✐❛ ■♥❡q✉❛❧✐t② ❆ ♣♦✐♥t (x, y) ❜❡❧♦♥❣s t♦ t❤❡ ♦♣❡♥ ❞✐s❦ ♦❢ r❛❞✐✉s r > 0 ❝❡♥t❡r❡❞ ❛t ♣♦✐♥t P = (h, k) ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐t s❛t✐s✜❡s t❤❡ ✐♥❡q✉❛❧✐t②✿ (x − h)2 + (y − k)2 < r2

❆ ♣♦✐♥t (x, y) ❜❡❧♦♥❣s t♦ t❤❡ ❝❧♦s❡❞ ❞✐s❦ ♦❢ r❛❞✐✉s r > 0 ❝❡♥t❡r❡❞ ❛t ♣♦✐♥t P = (h, k) ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐t s❛t✐s✜❡s t❤❡ ✐♥❡q✉❛❧✐t②✿ (x − h)2 + (y − k)2 ≤ r2

❚❤✐s ✐s t❤❡ ❝♦♥♥❡❝t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ❝✐r❝❧❡ ❛♥❞ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❝❧♦s❡❞ ❞✐s❦✿

✺✳✻✳

✹✹✺

❚❤❡ ❊✉❝❧✐❞❡❛♥ ♣❧❛♥❡✿ ❞✐st❛♥❝❡s

❊①❡r❝✐s❡ ✺✳✻✳✷✸

Pr♦✈❡ t❤❡ t❤❡♦r❡♠✳ ❊①❡r❝✐s❡ ✺✳✻✳✷✹

❉❡s❝r✐❜❡ t❤❡ r❡❧❛t✐♦♥s ♣r❡s❡♥t ✐♥ t❤❡ t❤❡♦r❡♠✳ ❊①❡r❝✐s❡ ✺✳✻✳✷✺

❲r✐t❡ t❤❡ t❤r❡❡ ✉s✐♥❣ t❤❡ s❡t✲❜✉✐❧❞✐♥❣ ♥♦t❛t✐♦♥✳ ❙♦✱ ✐t✬s ❥✉st ❛ ♠❛tt❡r ♦❢ t❤❡ ❞✐st❛♥❝❡ ❢r♦♠ t❤❡ ❝❡♥t❡r ❡q✉❛❧ t♦ r ♦r ❧❡ss t❤❛♥ ✭♦r ❡q✉❛❧✮ t♦ r✳ ❊①❛♠♣❧❡ ✺✳✻✳✷✻✿ ❝✐r❝❧❡ ✈✐❛ ❝❡♥t❡r ❛♥❞ r❛❞✐✉s

❲❤❛t ✐s t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ❝✐r❝❧❡ ❝❡♥t❡r❡❞ ❛t (2, 3) ❛♥❞ r❛❞✐✉s 5❄ ❚❤✐s ✐s ❥✉st ❛ ♠❛tt❡r ♦❢ s✉❜st✐t✉t✐♦♥✿ h = 2, k = 3, r = 5 .

❚❤❡♥✱ t❤❡ t❤❡♦r❡♠ ♣r♦❞✉❝❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉❛t✐♦♥✿ (x − 2)2 + (y − 3)2 = 52 .

❲❡ ♣❧♦t t♦ ❝♦♥✜r♠✿

■♥ ♦r❞❡r t♦ s❦❡t❝❤ ❛ ❝✐r❝❧❡ ❜② ❤❛♥❞✱ ✇❡ ❝❛♥ ❥✉st ♠❛❦❡ ❛ st❡♣ r ✉♥✐ts ❧♦♥❣ ✐♥ ❡❛❝❤ ♦❢ t❤❡ ❢♦✉r ❞✐r❡❝t✐♦♥s ❛♥❞ t❤❡♥ ❝♦♥♥❡❝t t❤❡s❡ ♣♦✐♥ts ❜② ❛ ❝✉r✈❡✳ ❊①❛♠♣❧❡ ✺✳✻✳✷✼✿ ❝✐r❝❧❡ ✈✐❛ ❝❡♥t❡r ❛♥❞ ♣♦✐♥t

❙♦♠❡t✐♠❡s t❤❡ ❞❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ ❝✐r❝❧❡ ✐s ❧❡ss ❡①♣❧✐❝✐t✿ ❲❤❛t ✐s t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ❝✐r❝❧❡ ❝❡♥t❡r❡❞ ❛t (2, 3) ❛♥❞ ♣❛ss✐♥❣ t❤r♦✉❣❤ t❤❡ ♦r✐❣✐♥❄ ❚❤❡ ❝❡♥t❡r ✐s ❦♥♦✇♥ ❜✉t t❤❡ r❛❞✐✉s ✐s ♠✐ss✐♥❣✦ ❚♦ ✜♥❞ ✐t✱ ✇❡ ♦❜s❡r✈❡ t❤❛t t❤❡ ❞✐st❛♥❝❡ ❢r♦♠ ❛♥② ♣♦✐♥t ♦♥ t❤❡ ❝✐r❝❧❡ t♦ ✐ts ❝❡♥t❡r ✐s ❡q✉❛❧ t♦ ✐ts r❛❞✐✉s✿

✺✳✻✳

✹✹✻

❚❤❡ ❊✉❝❧✐❞❡❛♥ ♣❧❛♥❡✿ ❞✐st❛♥❝❡s

❚❤❡r❡❢♦r❡✱ ✇❡ ❝❛♥ ♣❧♦t ✐t ❜② ♠❛❦✐♥❣ 3✲t❤❡♥✲2 ❛♥❞ 2✲t❤❡♥✲3 ✐♥ ❛❧❧ ❞✐r❡❝t✐♦♥s ❢r♦♠ t❤❡ ❝❡♥t❡r✳ ❚♦ ✜♥❞ t❤❡ ❡q✉❛t✐♦♥✱ ✇❡ ✉s❡ t❤❡ t✇♦ ♣♦✐♥ts t♦ ✜♥❞ ✐t✱ ♦r ✐ts sq✉❛r❡✱ ❜② t❤❡ ❉✐st❛♥❝❡

❋♦r♠✉❧❛

✿

r2 = 22 + 32 = 13 .

❚❤❡♥✱ t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ❝✐r❝❧❡ t❛❦❡♥ ❢r♦♠ t❤❡ t❤❡♦r❡♠ ✐s ❛s ❢♦❧❧♦✇s✿ (x − 2)2 + (y − 3)2 = 13 . ❊①❛♠♣❧❡ ✺✳✻✳✷✽✿ ❝✐r❝❧❡ ✈✐❛ ❝❡♥t❡r ❛♥❞ t❛♥❣❡♥t

■♥❞✐r❡❝t ❛♥❞ s✉❜t❧❡r✿ ❲❤❛t ✐s t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ❝✐r❝❧❡ ❝❡♥t❡r❡❞ ❛t (2, 3) ❛♥❞ t♦✉❝❤✐♥❣ t❤❡ x✲❛①✐s❄ ❲❡ ♥❡❡❞ ❛ s❦❡t❝❤✿

❲❡ r❡❛❧✐③❡ t❤❡♥ t❤❛t t❤❡ r❛❞✐✉s ♠✉st ❜❡ r = 3✳ ❚❤❡♥✱ t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ❝✐r❝❧❡ ✐s ❛s ❢♦❧❧♦✇s✿ (x − 2)2 + (y − 3)2 = 32 .

❲❡ s❛② t❤❛t t❤❡ ❛①✐s ✐s t❛♥❣❡♥t t♦ t❤❡ ❝✐r❝❧❡ ✭❈❤❛♣t❡r ✷❉❈✲✸✮✳ ❊①❡r❝✐s❡ ✺✳✻✳✷✾

❲❤❛t ✐s t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ❝✐r❝❧❡ ❝❡♥t❡r❡❞ ❛t (1, 1) ❛♥❞ ♣❛ss✐♥❣ t❤r♦✉❣❤ t❤❡ ♣♦✐♥t (2, 3)❄ ❊①❡r❝✐s❡ ✺✳✻✳✸✵

❲❤❛t ✐s t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ❝✐r❝❧❡ ❝❡♥t❡r❡❞ ❛t (2, 3) ❛♥❞ t♦✉❝❤✐♥❣ t❤❡ y ✲❛①✐s❄ ❍❡r❡ ✐s t❤❡ ❝✐r❝❧❡ ✇❡ ✉s❡❞ t♦ ❜✉✐❧❞ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s✳

✺✳✻✳ ❚❤❡ ❊✉❝❧✐❞❡❛♥ ♣❧❛♥❡✿ ❞✐st❛♥❝❡s

✹✹✼

❈♦r♦❧❧❛r② ✺✳✻✳✸✶✿ ❯♥✐t ❈✐r❝❧❡ ❊q✉❛t✐♦♥ ❚❤❡ ❝✐r❝❧❡ ♦❢ r❛❞✐✉s

1

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

x2 + y 2 = 1

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

•

❊✈❡r② ♣❛r❛❜♦❧❛ ❝❛♥ ❜❡ ❛❝q✉✐r❡❞ ❢r♦♠ t❤❡ st❛♥❞❛r❞ ♦♥❡✱

•

❚❤❡ ❣r❛♣❤ ♦❢ ❛♥② ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥

•

ax

y = x2 ✱

✈✐❛ ✢✐♣s✱ s❤✐❢ts✱ ❛♥❞ str❡t❝❤❡s✳

❝❛♥ ❜❡ ❛❝q✉✐r❡❞ ❢r♦♠ t❤❡ st❛♥❞❛r❞ ♦♥❡

ex

✈✐❛ ✢✐♣s ❛♥❞

str❡t❝❤❡s✳ ❙❛♠❡ ❢♦r t❤❡ ❧♦❣❛r✐t❤♠s✳ ❊✈❡r② ❝✐r❝❧❡ ❝❛♥ ❜❡ ❛❝q✉✐r❡❞ ❢r♦♠ t❤❡ st❛♥❞❛r❞ ♦♥❡✱

x 2 + y 2 = 1✱

✈✐❛ s❤✐❢ts ❛♥❞ str❡t❝❤❡s ✭✢✐♣s ❤❛✈❡

♥♦ ❡✛❡❝t✮✳

❋✐rst✱ ❛♥② r❛❞✐✉s ✐s ❛❝❤✐❡✈❡❞ ❜② ❛ ✉♥✐❢♦r♠ str❡t❝❤ ♦❢ t❤❡ ♣❧❛♥❡✿

(x, y) → (rx, ry) . ❆s ✇❡ str❡t❝❤ t❤❡ ❝✐r❝❧❡ ❛❣❛✐♥ ❛♥❞ ❛❣❛✐♥✱ t❤❡ ✏❝♦♥❝❡♥tr✐❝✑ ❝✐r❝❧❡s ♣r♦❞✉❝❡❞ ❝♦✈❡r t❤❡ ✇❤♦❧❡ ♣❧❛♥❡✿

❊①❡r❝✐s❡ ✺✳✻✳✸✷

❙❤♦✇ t❤❛t ✈❡rt✐❝❛❧ ♦r ❤♦r✐③♦♥t❛❧ str❡t❝❤❡s ♦❢ ❛ ♣❛r❛❜♦❧❛ ✇✐❧❧ ♣r♦❞✉❝❡ ❛♥♦t❤❡r ♣❛r❛❜♦❧❛✳ ❙❤♦✇ t❤❛t t❤✐s ✐s ♥♦t t❤❡ ❝❛s❡ ❢♦r ❝✐r❝❧❡s✳

❙❡❝♦♥❞✱ ❛♥② ❧♦❝❛t✐♦♥ ❝❛♥ ❜❡ ❛❝❤✐❡✈❡❞ ❜② ❛ s❤✐❢t ♦❢ t❤❡ ♣❧❛♥❡✿

(x, y) → (x + h, y + k) . ❲❡ r❡❧♦❝❛t❡ ♦✉r ❝✐r❝❧❡✿

❙♦✱ s✉♣♣♦s❡ ✇❡ ♥❡❡❞ t❤❡ ❝✐r❝❧❡ ❝❡♥t❡r❡❞ ❛t ❝❡♥t❡r❡❞ ❛t

0

❢♦❧❧♦✇✐♥❣ t❤❡s❡ st❡♣s✿

✶✳ ❙tr❡t❝❤ ✉♥✐❢♦r♠❧② ❜② ✷✳ ❙❤✐❢t r✐❣❤t ❜② ✸✳ ❙❤✐❢t ✉♣ ❜②

k✳

h✳

r✳

(h, k)

♦❢ r❛❞✐✉s

r✳

❲❡ ❛❝q✉✐r❡ ✐t ❢r♦♠ t❤❡ ❝✐r❝❧❡ ♦❢ r❛❞✐✉s

1

✺✳✻✳

✹✹✽

❚❤❡ ❊✉❝❧✐❞❡❛♥ ♣❧❛♥❡✿ ❞✐st❛♥❝❡s

❇❡❧♦✇ ✐s t❤❡ s✉♠♠❛r② ♦❢ t❤✐s ❝♦♥str✉❝t✐♦♥✿ x2 + y2 = 1 2 (x − h) + (y − k)2 = r2

❖♥❡ ❝❛♥ st✐❧❧ s❡❡ t❤❡ t❤r❡❡ tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ♣❧❛♥❡✳ ❊①❡r❝✐s❡ ✺✳✻✳✸✸

❈❛♥ ✇❡ ❝❤❛♥❣❡ t❤❡ ♦r❞❡r❄ ❋✉rt❤❡r♠♦r❡✱ ❜♦t❤ t❡r♠s ♦♥ t❤❡ ❧❡❢t ❛r❡ ❝♦♠♣❧❡t❡

sq✉❛r❡s

✿

(x − h)2 = x2 − 2xh + h2 (y − k)2 = y 2 − 2yk + k 2

❚♦ ✜♥❞ t❤✐s r❡♣r❡s❡♥t❛t✐♦♥✱ ♦♥❡ ✇♦✉❧❞ ♥❡❡❞ t♦ ♠❛t❝❤ t❤❡ t❡r♠s ♦♥ t❤❡ r✐❣❤t ✇✐t❤ t❤❡ t❡r♠s ✐♥ t❤❡ ❝♦♠♣❧❡t❡ sq✉❛r❡ ❢♦r♠✉❧❛✿ (a + b)2 = a2 + 2ab + b2 .

❊①❛♠♣❧❡ ✺✳✻✳✸✹✿ ❝♦♠♣❧❡t✐♥❣ sq✉❛r❡s

❋✐♥❞ t❤❡ ❝❡♥t❡r ❛♥❞ t❤❡ r❛❞✐✉s ♦❢ t❤✐s ❝✐r❝❧❡✿ x2 + 2x + y 2 − 4y + 1 = 0 .

❯♥❢♦rt✉♥❛t❡❧②✱ t❤❡ ❡q✉❛t✐♦♥ ❞♦❡s♥✬t ❝♦♥❢♦r♠ t♦ ♦✉r ❢♦r♠✉❧❛✦ ❲❡ ♥❡❡❞ t♦ ♠❛♥✐♣✉❧❛t❡ t❤❡ t✇♦ ♣❛rts ♦❢ t❤❡ ❡q✉❛t✐♦♥ t♦✇❛r❞s ✐t ❛♥❞ ✜♥❞ h, k, r✿ x2 + 2x = x2 + 2 · 1 · x+12 − 12 = (x2 + 2 · 1 · x + 12 ) − 1 = (x + 1)2 − 1

y 2 − 4y = y 2 − 2 · 2 · y+22 − 22 = (y 2 − 2 · 2 · y + 22 ) − 4 = (y − 2)2 − 4

❙t❛rt ✇✐t❤ t❤❡ ♦r✐❣✐♥❛❧✳ ❆❞❞ t❤❡ ♠✐ss✐♥❣ t❡r♠ ♦❢ t❤❡ ❝♦♠♣❧❡t❡ sq✉❛r❡✳ P✉❧❧ ♦✉t t❤❡ ❡①tr❛ t❡r♠✳ ❈♦♠♣❧❡t❡ t❤❡ sq✉❛r❡✳

❚❤❡ t❡r♠s ✐♥ r❡❞ ❛r❡ t❤❡ s❤✐❢ts✿ h = −1 ❛♥❞ k = 2✳ ❖✉r ❡q✉❛t✐♦♥ ❜❡❝♦♠❡s✿ x2 +2x+y 2 −4y +1 = (x2 +2x)+(y 2 −4y)−4 = ((x+1)2 −1)+((y −2)2 −4)+1 = (x+1)2 +(y −2)2 −4 .

❚❤❡ ♥❡✇ ❢♦r♠ ♦❢ t❤❡ ❡q✉❛t✐♦♥ ✐s ❛s ❢♦❧❧♦✇s✿ (x + 1)2 + (y − 2)2 = 22 .

❲❡ ❤❛✈❡ ❞✐s❝♦✈❡r❡❞ t❤❡ r❛❞✐✉s t♦♦✿ r = 2✦ ❚❤✐s ✐s ❛ ❝✐r❝❧❡ ❝❡♥t❡r❡❞ ❛t (−1, 2) ♦❢ r❛❞✐✉s 2 ❛❝q✉✐r❡❞ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣✿ ✉♥✐❢♦r♠ str❡t❝❤ ❜② 2✱ s❤✐❢t ❧❡❢t ❜② 1✱ s❤✐❢t ✉♣ ❜② 2✿

✺✳✻✳ ❚❤❡ ❊✉❝❧✐❞❡❛♥ ♣❧❛♥❡✿ ❞✐st❛♥❝❡s

✹✹✾

❊①❡r❝✐s❡ ✺✳✻✳✸✺ ❲❤❛t ✐❢ ✇❡ ❝❤❛♥❣❡

+1

✐♥ t❤❡ ❛❜♦✈❡ ❡q✉❛t✐♦♥ t♦

−3❄

❲❤❛t ✐❢ ✐t ✐s

−4❄

❊①❡r❝✐s❡ ✺✳✻✳✸✻

x2 − 6x + y 2 − 4y + 12 = 0❄

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

❲❛r♥✐♥❣✦

❆♥ ❡q✉❛t✐♦♥ ♦❢ t❤✐s ❦✐♥❞ ❝❛♥ ♣r♦❞✉❝❡ ✕ ✐♥ ❛❞❞✐t✐♦♥ t♦ ❛ ❝✐r❝❧❡ ✕ ❛ ♣❛r❛❜♦❧❛✱ ❛ ❤②♣❡r❜♦❧❛✱ ❛♥❞ s♦♠❡ ♦t❤❡r ❝✉r✈❡s✳ ❚❤❡♦r❡♠ ✺✳✻✳✸✼✿ ❈❡♥t❡r❡❞ ❋♦r♠ ♦❢ ❈✐r❝❧❡ ❆♥② ❝✐r❝❧❡ ♦♥ t❤❡ ♣❧❛♥❡ ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ✐♥ ❛ ✏❝❡♥t❡r❡❞ ❢♦r♠✑✿

(x − h)2 + (y − k)2 = r2 ✇❤❡r❡

h, k, r

❛r❡ s♦♠❡ ♥✉♠❜❡rs✳

❊①❡r❝✐s❡ ✺✳✻✳✸✽ ❙t❛t❡ t❤❡ ❝♦♥✈❡rs❡ ♦❢ t❤❡ t❤❡♦r❡♠✳

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

/

✏t❡♠♣❧❛t❡✑

❢✉♥❝t✐♦♥

❧✐♥❡✿

y = 2(x − 1) + 3

y=x ⌣

♣❛r❛❜♦❧❛✿

y = x2 ◦

❝✐r❝❧❡✿

r❡❧❛t✐♦♥

y − 3 = 2(x − 1) s❤✐❢t ✉♣ ❜②

3,

s❤✐❢t r✐❣❤t ❜②

1,

✈❡rt✐❝❛❧ str❡t❝❤ ❜②

2

1,

✈❡rt✐❝❛❧ str❡t❝❤ ❜②

2

1,

✉♥✐❢♦r♠ str❡t❝❤ ❜②

2

y = 2(x − 1)2 + 3 y − 3 = 2(x − 1)2 s❤✐❢t ✉♣ ❜② ◆✴❆

x2 + y 2 = 1

3,

s❤✐❢t r✐❣❤t ❜②

(y − 3)2 + (x − 1)2 = 22 s❤✐❢t ✉♣ ❜②

3,

s❤✐❢t r✐❣❤t ❜②

❆s ②♦✉ ❝❛♥ s❡❡✱ t❤❡s❡ s♣❡❝✐❛❧ r❡♣r❡s❡♥t❛t✐♦♥s r❡✈❡❛❧ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥s t❤❛t ❤❛✈❡ ♠❛❞❡ t❤❡ ❝✉r✈❡✳

❊①❡r❝✐s❡ ✺✳✻✳✸✾ ❲❤❡r❡ ✐s t❤❡ ✢✐♣ ✐♥ t❤✐s t❛❜❧❡❄

❊①❡r❝✐s❡ ✺✳✻✳✹✵ ❊①♣❧❛✐♥ ❤♦✇ t❤❡ ✈❡rt✐❝❛❧ str❡t❝❤ ❝❤❛♥❣❡s t❤❡ s❧♦♣❡ ♦❢ ❛ ❧✐♥❡✳

❊①❡r❝✐s❡ ✺✳✻✳✹✶ ❲❤❛t ✐❢ ✇❡ ❛♣♣❧② t❤❡ tr❛♥s❢♦r♠❛t✐♦♥✿ s❤✐❢t ✉♣ ❜②

3✱

s❤✐❢t r✐❣❤t ❜②

1✱

✈❡rt✐❝❛❧ str❡t❝❤ ❜②

2✱

t♦ t❤❡ ✉♥✐t

❝✐r❝❧❡❄

❚❤❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❝✐r❝❧❡ ✐s ❝♦♥✈❡♥✐❡♥t ❜✉t ♥♦t ❢♦r ♣❧♦tt✐♥❣✦ ❚❤❡ ♠✐ss✐♥❣ ♣❛rt ✐♥ t❤❡ ❧❛st r♦✇ ♦❢ t❤❡ t❛❜❧❡ ✐s ❡①♣❧❛✐♥❡❞ ❜② t❤❡ ❢❛❝t t❤❛t ✕ ✉♥❧✐❦❡ t❤❡ ♦t❤❡r t✇♦ ✕ t❤❡ ❝✐r❝❧❡ ❞♦❡s♥✬t ♣❛ss t❤❡ ❱❡rt✐❝❛❧ ▲✐♥❡ ❚❡st ❀ ✐t✬s ♥♦t t❤❡ ❣r❛♣❤ ♦❢ ❛♥② ❢✉♥❝t✐♦♥✦ ❲❡ ❣❡t ❛r♦✉♥❞ t❤✐s ♣r♦❜❧❡♠ ✐♥ ❛ ♠❛♥♥❡r s✐♠✐❧❛r t♦ ❤♦✇ ✇❡ ❤❛♥❞❧❡❞ ✜♥❞✐♥❣ t❤❡ 2 ✐♥✈❡rs❡ ♦❢ y = x ✕ ✇❡ s♣❧✐t ✐t✳

✺✳✻✳ ❚❤❡ ❊✉❝❧✐❞❡❛♥ ♣❧❛♥❡✿ ❞✐st❛♥❝❡s

✹✺✵

❲❡ ❥✉st tr❡❛t ♦✉r r❡❧❛t✐♦♥ ❛s ❛♥ ❡q✉❛t✐♦♥ ❛♥❞ s♦❧✈❡ ✐t✿

x2 + y 2 = R 2 y=

√

R2

−

ւ

x2

ց

√ y = − R 2 − x2

❖✉r ✐♥t❡r♣r❡t❛t✐♦♥ ♦❢ t❤❡ r❡s✉❧t ✐s ❜❡❧♦✇✳ ❚❤❡♦r❡♠ ✺✳✻✳✹✷✿ ❈✐r❝❧❡ ❛s ❚✇♦ ●r❛♣❤s ❚❤❡ ❝✐r❝❧❡ ♦❢ r❛❞✐✉s

R

❝❡♥t❡r❡❞ ❛t t❤❡ ♦r✐❣✐♥✱

x2 + y 2 = R 2 , ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❛s t❤❡ ✉♥✐♦♥ ♦❢ t❤❡ ❣r❛♣❤s ♦❢ t✇♦ ❢✉♥❝t✐♦♥s✿

y=

√

R 2 − x2

❛♥❞

√ y = − R 2 − x2

❲❡ ❤❛✈❡ t✇♦ s❡♠✐❝✐r❝❧❡s✿

❚❤❡ ❛♥❛❧②t✐❝ ❣❡♦♠❡tr② ♦❢ ❛♥❣❧❡s✱ ❢♦r ❜♦t❤ ❞✐♠❡♥s✐♦♥s✱ ✐s ❞✐s❝✉ss❡❞ ❧❛t❡r ✐♥ t❤❡ ❝❤❛♣t❡r✳ ❲❡ ✇✐❧❧ ♥♦t ❛❞❞r❡ss t❤❡

3✲❞✐♠❡♥s✐♦♥❛❧

❝❛s❡ ✭❈❤❛♣t❡r ✸■❈✲✸✮✱ ❜✉t ✐t s❤♦✉❧❞ ❜❡ ♦❜✈✐♦✉s t❤❛t t❤❡ ❝♦♥str✉❝t✐♦♥

✇✐❧❧ st❛rt t❤✐s ✇❛②✿ ✏❇✉✐❧❞ t❤r❡❡ ❝♦♦r❞✐♥❛t❡ ❛①❡s ❛♥❞ ❛rr❛♥❣❡ t❤❡♠ ❛❝❝♦r❞✐♥❣❧②✑✿

❲❡ t❤❡♥ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❣❡♦♠❡tr②✲❛❧❣❡❜r❛ ❝♦rr❡s♣♦♥❞❡♥❝❡✿

❧♦❝❛t✐♦♥

P ←→

tr✐♣❧❡

(x, y, z)

❖♥❡ ❝❛♥ ❛❧r❡❛❞② s❡❡ ❤♦✇ ❤❛r❞❡r ✐t ✐s t♦ ✈✐s✉❛❧✐③❡ t❤✐♥❣s ✐♥ t❤❡

3✲❞✐♠❡♥s✐♦♥❛❧

s♣❛❝❡✱ ✇❤✐❝❤ ❢✉rt❤❡r ❥✉st✐✜❡s

t❤❡ ♥❡❡❞ ❢♦r t❤❡ ❛❧❣❡❜r❛✐❝ ❛♣♣r♦❛❝❤ t♦ ❣❡♦♠❡tr② t❤❛t ✇❡ ❤❛✈❡ ♣r❡s❡♥t❡❞ ✐♥ t❤✐s ❝❤❛♣t❡r✳ ❈♦❧❧❡❝t✐✈❡❧②✱ t❤❡s❡ ❛r❡ ❝❛❧❧❡❞

1✲✱ 2✲✱

❛♥❞

3✲❞✐♠❡♥s✐♦♥❛❧

❊✉❝❧✐❞❡❛♥ s♣❛❝❡s✳

✺✳✼✳

❚r✐❣♦♥♦♠❡tr② ❛♥❞ t❤❡ ✇❛✈❡ ❢✉♥❝t✐♦♥

✹✺✶

✺✳✼✳ ❚r✐❣♦♥♦♠❡tr② ❛♥❞ t❤❡ ✇❛✈❡ ❢✉♥❝t✐♦♥

▲❡t✬s r❡✈✐❡✇ ❢r♦♠ t❤❡ ❧❛st ❝❤❛♣t❡r ❤♦✇ t❤❡ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s ❡♠❡r❣❡ ❢r♦♠ ♣❧❛♥❡

❣❡♦♠❡tr②✳

❋✐rst✱ t❤❡ P②t❤❛❣♦r❡❛♥ ❚❤❡♦r❡♠ st❛t❡s t❤❛t ✐❢ ✇❡ ❤❛✈❡ ❛ r✐❣❤t tr✐❛♥❣❧❡ ✇✐t❤ s✐❞❡s a, b, c✱ ✇✐t❤ c t❤❡ ❧♦♥❣❡st ♦♥❡ ❢❛❝✐♥❣ t❤❡ r✐❣❤t ❛♥❣❧❡✱ t❤❡♥ t❤❡② s❛t✐s❢② t❤❡ ❢♦❧❧♦✇✐♥❣✿ a2 + b2 = c2

◆❡①t✱ ✇❡ ❝♦♥❝❡♥tr❛t❡ ♦♥ α✱ t❤❡ ❛♥❣❧❡ ❜❡t✇❡❡♥ s✐❞❡ a ❛♥❞ s✐❞❡ c✿

❊①❡r❝✐s❡ ✺✳✼✳✶

❲❤❛t ❞♦ t❤❡ r❛t✐♦s t❡❧❧ ✉s ❛❜♦✉t t❤❡ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s ♦❢ t❤❡ ❛♥❣❧❡ α❄ ❚❤❡s❡ ❛r❡ t❤❡ ❞❡✜♥✐t✐♦♥s✿ a c b sin α = c cos α =

=⇒ tan α =

sin α b = a cos α

❚❤❡r❡ ❛r❡ ♠❛♥② ❛❧❣❡❜r❛✐❝ r❡❧❛t✐♦♥s ❜❡t✇❡❡♥ t❤❡ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s✳ ❇❡❧♦✇ ✇❡ ❡①❛♠✐♥❡ t❤❡ ♠♦r❡ ✐♠♣♦rt❛♥t ♦♥❡s✳ ❙✐♥❝❡ t❤❡ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s ❞❡♣❡♥❞ ♦♥❧② ♦♥ t❤❡ ♣r♦♣♦rt✐♦♥s ♦❢ t❤❡ tr✐❛♥❣❧❡✱ ✇❡ ❝❤♦♦s❡ t♦ ❞❡❛❧ ✇✐t❤ t❤❡ s✐♠♣❧❡st ❦✐♥❞ ♦♥❧②✿ ❛ r✐❣❤t tr✐❛♥❣❧❡ ✇✐t❤ t❤❡ ❤②♣♦t❡♥✉s❡ ✭t❤❡ ❧♦♥❣❡st ♦♥❡✱ ❢❛❝✐♥❣ t❤❡ r✐❣❤t ❛♥❣❧❡✮ ♦❢ ❧❡♥❣t❤ 1✿

❚❤❡ t❤❡♦r❡♠ t❤❡♥ t❛❦❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ❤✐❣❤❧② ✉s❡❢✉❧ ❢♦r♠✳ ❈♦r♦❧❧❛r② ✺✳✼✳✷✿ P②t❤❛❣♦r❡❛♥ ❚❤❡♦r❡♠ ♦❢ ❚r✐❣♦♥♦♠❡tr② ❋♦r ❛♥② r❡❛❧ ♥✉♠❜❡r

x✱

✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐❞❡♥t✐t②✿

sin2 x + cos2 x = 1

❲❛r♥✐♥❣✦

sin2 x

st❛♥❞s ❢♦r

(sin x)2 ✳

✺✳✼✳

❚r✐❣♦♥♦♠❡tr② ❛♥❞ t❤❡ ✇❛✈❡ ❢✉♥❝t✐♦♥

✹✺✷

Pr♦♦❢✳ ❲❡ ❝❛♥ ❛❧s♦ ❞❡r✐✈❡ t❤❡ ❢♦r♠✉❧❛ ❞✐r❡❝t❧②✳ ❲❡ t❛❦❡ t❤❡ P②t❤❛❣♦r❡❛♥ ❚❤❡♦r❡♠✿

a2 + b2 = c2 , ❛♥❞ ❞✐✈✐❞❡ ❜②

c2 ✿

♦r

❋✐♥❛❧❧②✱ ✇❡ s✉❜st✐t✉t❡ t❤❡ ❢♦r♠✉❧❛s ❢♦r

a2 b2 c2 + = , c2 c2 c2  a  2  b 2 + = 1. c c

sin

cos

❛♥❞

✐♥t♦ t❤✐s ❡q✉❛t✐♦♥✳

❊①❛♠♣❧❡ ✺✳✼✳✸✿ r♦t❛t✐♥❣ r♦❞✱ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ❇❛❝❦ t♦ ♦✉r ❡①❛♠♣❧❡ ♦❢ tr❛❝✐♥❣ t❤❡ ❡♥❞ ♦❢ ❛ r♦t❛t✐♥❣ r♦❞✿

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

❚❤❡ ❡q✉❛t✐♦♥s✱ ✇✐t❤ t❤❡ ❛♥❣❧❡ ❝✐r❝❧❡ ♦❢ r❛❞✐✉s

θ

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

R✿ x = R cos θ

❛♥❞

❲❡ ❤❛✈❡ t✇♦ ❢✉♥❝t✐♦♥s ✇✐t❤ t❤❡ s❛♠❡ ✐♥♣✉t❀ ✐t✬s ❛

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

y = R sin θ .

♣❛r❛♠❡tr✐❝ ❝✉r✈❡

y = f (x)

✳

✭❧❡❢t✮ ✇✐t❤ t❤❡ ✈❛❧✉❡s ♦❢

❖r ✐t ♠❛② ❝♦♠❡ ❢r♦♠ t❤❡ t✇♦ ❢✉♥❝t✐♦♥s ❛❜♦✈❡ ✭r✐❣❤t✮ ❛♥❞ t❤✐s t✐♠❡ ✐t ✐s ♥♦t

x

❞✐str✐❜✉t❡❞ ❡✈❡♥❧②✿

x t❤❛t ♣r♦❣r❡ss❡s ✉♥✐❢♦r♠❧②✱

❜✉t t❤❡ ❛♥❣❧❡✳ ❚❤❡ ❛❞✈❛♥t❛❣❡ ♦❢ t❤❡ ❧❛tt❡r r❡♣r❡s❡♥t❛t✐♦♥ ✐s ✈✐s✐❜❧❡✳ ❋✉rt❤❡r♠♦r❡✱ ✇✐t❤ t❤✐s ❛♣♣r♦❛❝❤ ♥♦t❤✐♥❣ st♦♣s ✉s ❢r♦♠ ♣r♦❣r❡ss✐♥❣ ❜❡②♦♥❞

180

❞❡❣r❡❡s✦

✺✳✼✳

❚r✐❣♦♥♦♠❡tr② ❛♥❞ t❤❡ ✇❛✈❡ ❢✉♥❝t✐♦♥

✹✺✸

❊①❛♠♣❧❡ ✺✳✼✳✹✿ r♦t❛t✐♥❣ r♦❞✱ ❝♦♥t✐♥✉❡❞

❆❧t❡r♥❛t✐✈❡❧②✱ ✇❡ ❝♦♥❝❡♥tr❛t❡ ♦♥ t❤❡ ❛❝t✉❛❧ r♦t❛t✐♦♥ ♦❢ t❤❡ r♦❞ ❛♥❞ r❡♣❧❛❝❡ t❤❡ ❛♥❣❧❡ θ ❛s t❤❡ ♣❛r❛♠❡t❡r ✇✐t❤ t✐♠❡ t✿ x = R cos t, y = R sin t .

❲✐t❤ ♥♦ r❡str✐❝t✐♦♥s ♦♥ t✱ ✇❡ ❤❛✈❡ ❛ ❝✐r❝❧❡ tr❛❝❡❞ ✐♥✜♥✐t❡❧② ♠❛♥② t✐♠❡s✿

▲❡t✬s ♠❛❦❡ ✐t s♣❡❝✐✜❝✱ R = 1✳ ❚❤❡r❡ ❛r❡ t✇♦ ❢✉♥❝t✐♦♥s✿ t❤❡ ❤♦r✐③♦♥t❛❧ ❛♥❞ t❤❡ ✈❡rt✐❝❛❧ ❧♦❝❛t✐♦♥s✱ ♣♦ss✐❜❧② s❤❛❞♦✇s ♦♥ t❤❡ ❣r♦✉♥❞ ❛♥❞ ❛ ✇❛❧❧✳ ❲❡ ♣❧♦t t❤❡♠ ❜❡❧♦✇✱ x ❛❣❛✐♥st t ❛♥❞ y ❛❣❛✐♥st t ✭t♦♣ r♦✇✮✿

❋✉rt❤❡r♠♦r❡✱ ❡✐t❤❡r ❢✉♥❝t✐♦♥ ❤❛s ✐ts ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✱ ✐✳❡✳✱ t❤❡ ❤♦r✐③♦♥t❛❧ ❛♥❞ t❤❡ ✈❡rt✐❝❛❧ ✈❡❧♦❝✐t✐❡s ✭❜♦tt♦♠ r♦✇✮✳ ■❢ t❤❡s❡ t✇♦ ❛r❡ ❝❤❛♥❣✐♥❣ ❞✐s♣r♦♣♦rt✐♦♥❛❧❧②✱ ✇❡ s❤♦✉❧❞ ❜❡ ❛❜❧❡ t♦ s❡❡ ❛ ❝❤❛♥❣❡ ✐♥ ❞✐r❡❝t✐♦♥ ✐❢ ✇❡ ❝♦♠❜✐♥❡ t❤❡♠✳ ❙♦✱ t✇♦ ❞✐✛❡r❡♥t ♦❜s❡r✈❡rs ✇✐❧❧ s❡❡ t✇♦ ❞✐✛❡r❡♥t s✐❞❡✲t♦✲s✐❞❡ ♠♦✈❡♠❡♥ts✳ ❍♦✇❡✈❡r✱ t❤❡ t❤❡ ❡♥❞ ♦❢ t❤❡ r♦❞ ✐s s❤♦✇♥ ♦♥ t❤❡ xy ✲♣❧❛♥❡✿

tr❛❥❡❝t♦r② ♦❢

❚❤❡r❡ ✐s ♥♦ t✲❛①✐s ❜✉t t❤❡ ❧♦❝❛t✐♦♥s ❛♥❞ t❤❡ ✈❡❧♦❝✐t✐❡s ❛r❡ ♠❛r❦❡❞ ❛❝❝♦r❞✐♥❣❧②✳ ❇② ♠❛t❝❤✐♥❣ t❤❡s❡ t✐♠❡ st❛♠♣s ✇❡ ❝❛♥ ✜♥❞ t❤❡ ✈❡❧♦❝✐t② ❢♦r ❡❛❝❤ ❧♦❝❛t✐♦♥✿

✺✳✼✳

❚r✐❣♦♥♦♠❡tr② ❛♥❞ t❤❡ ✇❛✈❡ ❢✉♥❝t✐♦♥

✹✺✹

❚❤✐s ❛rr♦✇ ❢♦✉♥❞ ♦♥ t❤❡ r✐❣❤t ✐s t❤❡♥ ❛tt❛❝❤❡❞ t♦ t❤❡ ❧♦❝❛t✐♦♥ ♦♥ t❤❡ ❧❡❢t✳ ❚❤✐s ✐s t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ ♠♦t✐♦♥ ❛t t❤✐s ♠♦♠❡♥t✦ ❊①❡r❝✐s❡ ✺✳✼✳✺

✭❛✮ ❲❤❡♥ ✐s t❤❡ ❤♦r✐③♦♥t❛❧ ✈❡❧♦❝✐t② t❤❡ ❤✐❣❤❡st❄ ✭❜✮ ❲❤❡r❡ ✐t t❤❡ ❤♦r✐③♦♥t❛❧ ✈❡❧♦❝✐t② t❤❡ ❤✐❣❤❡st❄ ✭❝✮ ❲❤❛t ♣❛tt❡r♥ ❞♦ t❤❡ ✈❡❧♦❝✐t✐❡s ❡①❤✐❜✐t❄

❊①❡r❝✐s❡ ✺✳✼✳✻

❆♥s✇❡r t❤❡ q✉❡st✐♦♥s ✐♥ t❤❡ ❧❛st ❡①❡r❝✐s❡ ❢♦r t❤❡s❡ t✇♦ ❛❧t❡r♥❛t✐✈❡ ✇❛②s t♦ ♠♦✈❡ ❛❧♦♥❣ t❤✐s ❝✐r❝❧❡✿ ✭❛✮ ❜❛❝❦✇❛r❞ ❞✐r❡❝t✐♦♥✿

x = cos(−t), y = sin(−t) ; ❛♥❞ ✭❜✮ t✇✐❝❡ ❛s ❢❛st✿

x = cos 2t, y = sin 2t . ❊①❡r❝✐s❡ ✺✳✼✳✼

❊①♣❧❛✐♥ t❤❡ ♣❧♦t ♦❢ t❤❡ ✈❡❧♦❝✐t✐❡s✳

❊①❡r❝✐s❡ ✺✳✼✳✽

❉❡s✐❣♥ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ✭♦r t✇♦✮ ❢♦r t❤❡ ❝❧♦❝❦✳

❊①❛♠♣❧❡ ✺✳✼✳✾✿ ✈❛❧✉❡s ♦❢

❲❤❡♥ t❤❡ ❛♥❣❧❡ ✐s

45

sin

❞❡❣r❡❡s✱ t❤❡ s✐♥❡ ❛♥❞ ❝♦s✐♥❡ ❛r❡ ❡q✉❛❧✿

sin ❚❤❡r❡❢♦r❡✱ t❤❡

P②t❤❛❣♦r❡❛♥ ❚❤❡♦r❡♠

π π = cos . 4 4

✐♠♣❧✐❡s t❤❛t

√ 2 π π sin = cos = . 4 4 2 ❊①❡r❝✐s❡ ✺✳✼✳✶✵

❋✐♥❞ ❛❧❧ ♣♦ss✐❜❧❡ ✈❛❧✉❡s ♦❢

x

✭✏♣r❡✐♠❛❣❡✑✮ ❢♦r ✇❤✐❝❤

sin x =

√

2 . 2

✺✳✼✳

❚r✐❣♦♥♦♠❡tr② ❛♥❞ t❤❡ ✇❛✈❡ ❢✉♥❝t✐♦♥

✹✺✺

❊①❡r❝✐s❡ ✺✳✼✳✶✶

❉❡r✐✈❡ t❤❡s❡ ❢♦r♠✉❧❛s✿ sin θ = ± √ cos θ = ± √

tan θ 1 + tan2 θ 1 1 + tan2 θ

; .

❚❤❡ P②t❤❛❣♦r❡❛♥ ❚❤❡♦r❡♠ ❛❜♦✈❡ ❡st❛❜❧✐s❤❡s ❛r❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ t✇♦ ♠❛✐♥ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s✱ sin x ❛♥❞ cos x❀ ✐❢ ✇❡ ❦♥♦✇ ♦♥❡✱ ✇❡ ❦♥♦✇ t❤❡ ♦t❤❡r✳ ❆r❡ t❤❡r❡ ♦t❤❡r s✉❝❤ r❡❧❛t✐♦♥s❄ ❚❤❡ t✇♦ ❢✉♥❝t✐♦♥s ❛r❡ 2π ✲♣❡r✐♦❞✐❝✱ ✇❤✐❝❤ ♠❡❛♥s t❤❛t ✇❤❡♥ s❤✐❢t❡❞ ❤♦r✐③♦♥t❛❧❧② ❜② 2π ✱ t❤❡ ❣r❛♣❤ ❧❛♥❞s ❡①❛❝t❧② ♦♥ ✐ts❡❧❢✿

❇✉t t❤❡② ❛❧s♦ ❛♣♣❡❛r t♦ ❜❡ s❤✐❢ts ♦❢ ❡❛❝❤ ♦t❤❡r✦ ▲❡t✬s ❝♦♥✜r♠ t❤✐s ♦❜s❡r✈❛t✐♦♥✳ ❲❡ ❣♦ ❜❛❝❦ t♦ t❤❡ ❞❡✜♥✐t✐♦♥✳ ■❢ t❤❡ ❤②♣♦t❡♥✉s❡ ✐s 1✱ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ s✐♥❡ ❛♥❞ t❤❡ ❝♦s✐♥❡ ♦❢ ❛♥ ❛♥❣❧❡ s✐♠♣❧✐✜❡s✿ cos( ❛♥❣❧❡ ) = ❛❞❥❛❝❡♥t s✐❞❡ sin( ❛♥❣❧❡ ) = ♦♣♣♦s✐t❡ s✐❞❡ ❇✉t t❤✐s ❞❡✜♥✐t✐♦♥ ❛♣♣❧✐❡s t♦ ❡✐t❤❡r ♦❢ t❤❡ t✇♦ ♦t❤❡r ❛♥❣❧❡s ♦❢ ♦✉r tr✐❛♥❣❧❡✱ x ❛♥ y ✿

❆s x ✐s r❡♣❧❛❝❡❞ ✇✐t❤ y ✱ ❛❞❥❛❝❡♥t ❜❡❝♦♠❡s ♦♣♣♦s✐t❡ ❛♥❞ ✈✐❝❡ ✈❡rs❛✳ ❲❡ ❤❛✈❡✱ ❝♦♥s❡q✉❡♥t❧②✿ sin x = cos y ❛♥❞ cos x = sin y .

❋✉rt❤❡r♠♦r❡✱ s✐♥❝❡ x ❛♥❞ y ❛r❡ t❤❡ t✇♦ ❛♥❣❧❡s ♦❢ t❤❡ r✐❣❤t tr✐❛♥❣❧❡✱ t❤❡② ❛❞❞ ✉♣ t♦ 90 ❞❡❣r❡❡s✿ x + y = π/2 .

❚❤❡♥✱ ✇❡ s✉❜st✐t✉t❡ y = π/2 − x ✐♥t♦ t❤❡ t✇♦ ❢♦r♠✉❧❛s ♣r♦✈✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣✳ ❚❤❡♦r❡♠ ✺✳✼✳✶✷✿ ❙✐♥❡ ❛♥❞ ❈♦s✐♥❡ ♦❢ ❈♦♠♣❧❡♠❡♥t❛r② ❆♥❣❧❡s ❋♦r ❛♥②

x✱

✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐❞❡♥t✐t✐❡s✿

sin x = cos(π/2 − x)

❛♥❞

cos x = sin(π/2 − x)

❚❤❡r❡❢♦r❡✱ t❤❡ ❣r❛♣❤ ♦❢ ♦♥❡ ❢✉♥❝t✐♦♥ ✐s t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ♦t❤❡r ❤♦r✐③♦♥t❛❧❧② ✢✐♣♣❡❞ ❛♥❞ t❤❡♥ s❤✐❢t❡❞ r✐❣❤t ❜② π/2✳ ❊①❡r❝✐s❡ ✺✳✼✳✶✸

❊①♣❧❛✐♥ t❤❡ ❧❛st st❛t❡♠❡♥t✳ ❆ ♠♦st ✐♠♣♦rt❛♥t ❝♦♥❝❧✉s✐♦♥ ✐s ❛s ❢♦❧❧♦✇s✿

✺✳✼✳

❚r✐❣♦♥♦♠❡tr② ❛♥❞ t❤❡ ✇❛✈❡ ❢✉♥❝t✐♦♥

✹✺✻

◮ ❚❤❡ s❤❛♣❡s ♦❢ ❣r❛♣❤s ♦❢ s✐♥❡ ❛♥❞ ❝♦s✐♥❡ ❛r❡ ✐❞❡♥t✐❝❛❧✳

❍♦✇❡✈❡r✱ ❛♥ ❡✈❡♥ ❝❧♦s❡r r❡❧❛t✐♦♥ ✐s ✈✐s✐❜❧❡ ✐♥ t❤❡ ❣r❛♣❤✿ ❚❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❝♦s✐♥❡ s❤✐❢t❡❞ r✐❣❤t ❜② π/2 ✐s t❤❡ ❣r❛♣❤ ♦❢ t❤❡ s✐♥❡✳ ❚♦ ♣r♦✈❡ t❤✐s✱ ✇❡ ❥✉st t❛❦❡ t❤❡ ❢♦r♠✉❧❛ ✐♥ t❤❡ t❤❡♦r❡♠ ❛♥❞ ✉s❡ t❤❡ ❢❛❝t t❤❛t t❤❡ ❝♦s✐♥❡ ✐s ❡✈❡♥✿ ❚❤❡s❡ ❛r❡ t❤❡ ❢♦r♠✉❧❛s✿

sin x = cos(π/2 − x) = cos(−(π/2 − x)) = cos(x − π/2) . ❈♦r♦❧❧❛r② ✺✳✼✳✶✹✿ ❙❤✐❢t ♦❢ ❙✐♥❡ ✐s ❈♦s✐♥❡ ❋♦r ❛♥②

x✱

✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐❞❡♥t✐t✐❡s✿

sin x = cos(x − π/2)

❛♥❞

cos x = sin(x + π/2)

❚❤❡ ❢♦❧❧♦✇✐♥❣ ❛r❡ t✇♦ ✐♠♣♦rt❛♥t tr✐❣♦♥♦♠❡tr✐❝ ❢♦r♠✉❧❛s t❤❛t ❛❧❧♦✇ ✉s t♦ r❡♣r❡s❡♥t t❤❡ s✐♥❡ ❛♥❞ t❤❡ ❝♦s✐♥❡ ♦❢ t❤❡ s✉♠ ♦❢ t✇♦ ❛♥❣❧❡s ✐♥ t❡r♠s ♦❢ t❤❡ t✇♦ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s ♦❢ t❤❡ t✇♦ ❛♥❣❧❡s✳ ❚❤❡♦r❡♠ ✺✳✼✳✶✺✿ ❙✐♥❡ ❛♥❞ ❈♦s✐♥❡ ♦❢ ❙✉♠ ❋♦r ❛♥② t✇♦ ❛♥❣❧❡s

α

❛♥❞

β✱

✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐❞❡♥t✐t✐❡s✿

sin(α + β) = sin α cos β + cos α sin β cos(α + β) = cos α cos β − sin α sin β Pr♦♦❢✳

❚❤❡ ♣r♦♦❢ ✐s ❢♦r t❤❡ ❛❝✉t❡ α✱ β ✱ α + β ✳ ❚❤❡ ❧❛❜❡❧s ❢♦r t❤❡ s✐❞❡s ♦❢ t❤❡ ✭r✐❣❤t✮ tr✐❛♥❣❧❡s ❛r❡ ❞❡❞✉❝❡❞ ❢r♦♠ t❤❡ ❞❡✜♥✐t✐♦♥s ♦❢ t❤❡ s✐♥❡ ❛♥❞ ❝♦s✐♥❡ st❛rt✐♥❣ ✇✐t❤ t❤❡ s❡❣♠❡♥t ♦❢ ❧❡♥❣t❤ 1 ✭r❡❞✮✿

❚❤❡ ❢❛❝t t❤❛t t❤❡ ❤♦r✐③♦♥t❛❧ s✐❞❡s ♦❢ t❤❡ r❡❝t❛♥❣❧❡ ❛r❡ ❡q✉❛❧ ♣r♦❞✉❝❡s t❤❡ ❢♦r♠❡r ❢♦r♠✉❧❛✱ ❛♥❞ t❤❡ ❢❛❝t t❤❛t t❤❡ ✈❡rt✐❝❛❧ s✐❞❡s ♦❢ t❤❡ r❡❝t❛♥❣❧❡ ❛r❡ ❡q✉❛❧ ♣r♦❞✉❝❡s t❤❡ ❧❛tt❡r✳ ▲❛t❡r ✐♥ t❤✐s ❝❤❛♣t❡r ✭❛♥❞ t❤❡♥ ✐♥ ❈❤❛♣t❡r ✷❉❈✲✸✮✱ ✇❡ ✇✐❧❧ ♥❡❡❞ ❛ tr✐❣♦♥♦♠❡tr✐❝ ❢♦r♠✉❧❛ t❤❛t ❛❧❧♦✇s ✉s t♦ r❡♣r❡s❡♥t t❤❡ ❝♦s✐♥❡ ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t✇♦ ❛♥❣❧❡s ✐♥ t❡r♠ ♦❢ t❤❡ t✇♦ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s ♦❢ t❤❡ t✇♦ ❛♥❣❧❡s✿

✺✳✼✳

❚r✐❣♦♥♦♠❡tr② ❛♥❞ t❤❡ ✇❛✈❡ ❢✉♥❝t✐♦♥

✹✺✼

❈♦r♦❧❧❛r② ✺✳✼✳✶✻✿ ❙✐♥❡ ❛♥❞ ❈♦s✐♥❡ ♦❢ ❉✐✛❡r❡♥❝❡ ❋♦r ❛♥② t✇♦ ❛♥❣❧❡s

α

❛♥❞

β✱

✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐❞❡♥t✐t✐❡s✿

sin(α − β) = sin α cos β − cos α sin β cos(α − β) = cos α cos β + sin α sin β

Pr♦♦❢✳ ❚♦ ❞❡r✐✈❡ t❤❡ r❡s✉❧t ❢r♦♠ t❤❡ t❤❡♦r❡♠✱ ✇❡ s✐♠♣❧② ✉s❡ t❤❡ ♦❞❞♥❡ss ♦❢ t❤❡ s✐♥❡ ❛♥❞ t❤❡ ❡✈❡♥♥❡ss ♦❢ t❤❡ ❝♦s✐♥❡✳

❈♦r♦❧❧❛r② ✺✳✼✳✶✼✿ ❙✐♥❡ ❛♥❞ ❈♦s✐♥❡ ♦❢ ❉♦✉❜❧❡ ❆♥❣❧❡ ❋♦r ❛♥② ❛♥❣❧❡

θ✱

✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐❞❡♥t✐t✐❡s✿

sin 2θ = 2 sin θ cos θ cos 2θ = cos2 θ − sin2 θ = 1 − 2 sin2 θ = 2 cos2 θ − 1

Pr♦♦❢✳ ❏✉st ❝❤♦♦s✐♥❣ α = β = θ ✐♥ t❤❡ t❤❡♦r❡♠ ✭❛♥❞ ✉s✐♥❣ t❤❡ P②t❤❛❣♦r❡❛♥ ❚❤❡♦r❡♠ ❢♦r t❤❡ s❡❝♦♥❞ ♣❛rt✮ ♣r♦❞✉❝❡s t❤❡ ❢♦r♠✉❧❛✳

❈♦r♦❧❧❛r② ✺✳✼✳✶✽✿ ❙✐♥❡ ❛♥❞ ❈♦s✐♥❡ ♦❢ ❍❛❧❢✲❆♥❣❧❡ ❋♦r ❛♥② ❛♥❣❧❡

sin

α✱

✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐❞❡♥t✐t✐❡s✿

α 2

=±

r

1 − cos α 2

❛♥❞

cos

α 2

=±

r

1 + cos α 2

Pr♦♦❢✳ ❋r♦♠ t❤❡ s❡❝♦♥❞ ✐❞❡♥t✐t② ✐♥ t❤❡ ❝♦r♦❧❧❛r②✱ ✇❡ t❛❦❡✿ cos 2θ = 1 − 2 sin2 θ ,

s❡t α = 2θ✱ ❛♥❞ s♦❧✈❡✱ ♣r♦❞✉❝✐♥❣ t❤❡ ✜rst ❢♦r♠✉❧❛✳ ❋r♦♠ t❤❡ s❡❝♦♥❞ ✐❞❡♥t✐t②✱ ✇❡ ❛❧s♦ t❛❦❡✿ cos 2θ = 2 cos2 θ − 1 ,

s❡t α = 2θ✱ ❛♥❞ s♦❧✈❡✱ ♣r♦❞✉❝✐♥❣ t❤❡ s❡❝♦♥❞ ❢♦r♠✉❧❛✳ ❚❤❡ ❢♦r♠✉❧❛ ❜❡❧♦✇ ❛❧❧♦✇❡❞ ✉s ✐♥ t❤❡ ❧❛st ❝❤❛♣t❡r t♦ ❝♦♠♣✉t❡ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ s✐♥❡ ❛♥❞ ❝♦s✐♥❡ ❢♦r ❞❡♥s❡r ❛♥❞ ❞❡♥s❡r ✈❛❧✉❡s ♦❢ x✿

✺✳✼✳

❚r✐❣♦♥♦♠❡tr② ❛♥❞ t❤❡ ✇❛✈❡ ❢✉♥❝t✐♦♥

✹✺✽

❈♦r♦❧❧❛r② ✺✳✼✳✶✾✿ ❙✐♥❡ ❛♥❞ ❈♦s✐♥❡ ♦❢ ❆✈❡r❛❣❡ ❆♥❣❧❡ ❋♦r ❛♥② ❛♥❣❧❡s

α

❛♥❞

β✱

✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐❞❡♥t✐t✐❡s✿

r  α+β 1 − cos α cos β − sin α sin β sin =± 2 2 r   1 + cos α cos β − sin α sin β α+β cos =± 2 2 

Pr♦♦❢✳

❲❡ s✐♠♣❧② ❝♦♠❜✐♥❡ t❤❡ ❧❛st t❤❡♦r❡♠ ❛♥❞ t❤❡ ❧❛st ❝♦r♦❧❧❛r②✳ ❲❡ ✇✐❧❧ ✉s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♥ t❤✐s s❡❝t✐♦♥✿ ❈♦r♦❧❧❛r② ✺✳✼✳✷✵✿ ❙✉♠ ♦❢ ❙✐♥❡s ❛♥❞ ❈♦s✐♥❡s ❋♦r ❛♥②

α

❛♥❞

β✱

✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐❞❡♥t✐t✐❡s✿



   α+β α−β sin α + sin β = 2 sin sin  2   2  α+β α−β cos α + cos β = 2 cos cos 2 2 Pr♦♦❢✳

▲❡t A = α + β, B = α − β .

❚❤❡r❡❢♦r❡✿

α = (A + B)/2, β = (A − B)/2 .

❲❡ ♥❡❡❞ t♦ s✐♠♣❧✐❢② t❤✐s✿

sin α + sin β = sin((A + B)/2) + sin((A − B)/2) .

❲❡ ❛♣♣❧② t❤❡ t❤❡♦r❡♠ t♦ ❡✐t❤❡r ♣❛rt✿ sin((A + B)/2) = sin A/2 cos B/2 + cos A/2 sin B/2 ; sin((A − B)/2) = sin A/2 cos B/2 − cos A/2 sin B/2 .

❆❞❞✐♥❣ t❤❡s❡ t✇♦✱ ✇❡ ✜♥❞✿ sin(A + B)/2 + sin(A − B)/2) = 2 sin A/2 cos B/2 .

❚❤❡ ❢♦r♠✉❧❛s ❢♦❧❧♦✇✳

✺✳✼✳

❚r✐❣♦♥♦♠❡tr② ❛♥❞ t❤❡ ✇❛✈❡ ❢✉♥❝t✐♦♥

✹✺✾

❊①❡r❝✐s❡ ✺✳✼✳✷✶

Pr♦✈✐❞❡ ❞❡t❛✐❧❡❞ ♣r♦♦❢s ♦❢ t❤❡s❡ r❡s✉❧ts✳ ❊①❡r❝✐s❡ ✺✳✼✳✷✷

❙♦❧✈❡ t❤❡ ❡q✉❛t✐♦♥ ❜❡❧♦✇✿

cos2 (x2 + 1) = .2 .

▼❛❦❡ ✉♣ ②♦✉r ♦✇♥ ❡q✉❛t✐♦♥ ❛♥❞ s♦❧✈❡ ✐t✳ ❘❡♣❡❛t✳ ❊①❛♠♣❧❡ ✺✳✼✳✷✸✿ s❡q✉❡♥❝❡

sin n

❈♦♥s✐❞❡r t❤❡ s❡q✉❡♥❝❡ an = sin n .

❇❡❝❛✉s❡ t❤❡ ✐♥❝r❡♠❡♥t ♦❢ t❤❡ ✐♥♣✉t ✐s♥✬t ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ♣❡r✐♦❞ ♦❢ t❤❡ ❢✉♥❝t✐♦♥✱ ✐t ❣✐✈❡s ❛♥ ❛♣♣❡❛r✲ ❛♥❝❡ ♦❢ r❛♥❞♦♠♥❡ss✳ ❚❤❡ ✈❛❧✉❡s ❧♦♦❦ ✉♥✐❢♦r♠❧② s♣r❡❛❞ ✭❜❡t✇❡❡♥ −1 ❛♥❞ 1✮✿

❚❤❡♥ t❤❡ ✈❛❧✉❡s ♦❢ ✐ts s✉♠ Σan ❛♥❞ ❞✐✛❡r❡♥❝❡ ∆an ✭❈❤❛♣t❡r ✶✮ ❛❧s♦ s❡❡♠ r❡str✐❝t❡❞ t♦ ❛ ❝❡rt❛✐♥ ✐♥t❡r✈❛❧✦ ❊①❡r❝✐s❡ ✺✳✼✳✷✹

Pr♦✈❡ t❤❡ ❧❛st st❛t❡♠❡♥t ✉s✐♥❣ t❤❡ ❛❜♦✈❡ r❡s✉❧ts✳ ❘❡❝❛❧❧ ❤♦✇ ✐♥ ❈❤❛♣t❡r ✸ ✇❡ ♣r♦❞✉❝❡❞ t❤❡ ❣r❛♣❤s ♦❢ ❛❧❧ q✉❛❞r❛t✐❝ ❢✉♥❝t✐♦♥s ❛s tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ ♦♥❡ ♣❛r❛❜♦❧❛✿ t❤❡ ❱❡rt❡① ❋♦r♠✉❧❛ ♦❢ ◗✉❛❞r❛t✐❝ P♦❧②♥♦♠✐❛❧✳ ❇❛s❡❞ ♦♥ t❤❡ ❛❜♦✈❡ r❡s✉❧ts✱ ✇❡ ❛❧s♦ ❤❛✈❡ ❛ ❝♦♠♠♦♥ ♥❛♠❡ ❢♦r ❛❧❧ tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ❣r❛♣❤ ♦❢ y = sin x✱ t❤❡ s✐♥✉s♦✐❞ ✿ t❡♠♣❧❛t❡ ❣❡♥❡r❛❧ y = x2

y = a(x − h)2 + k

y = sin x y = A sin(ωx + φ) y = sin x y = B cos(ωx + φ)

♥❛♠❡ ♦❢ t❤❡ ❝✉r✈❡

♣❛r❛❜♦❧❛ s✐♥✉s♦✐❞ s✐♥✉s♦✐❞

❲❡ ✇✐❧❧ ❧❡❛r♥ ❤♦✇ t♦ ✉s❡ t❤❡ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s t♦ r❡♣r❡s❡♥t ♥✉♠❡r♦✉s ❦✐♥❞s ♦❢ ♣❡r✐♦❞✐❝ ♣❤❡♥♦♠❡♥❛✳ ❚❤❡s❡ ♣❤❡♥♦♠❡♥❛ ❛r❡ ♦❢t❡♥ r❡♣r❡s❡♥t❡❞ ❜② ❛ s✐♥❣❧❡ q✉❛♥t✐t② t❤❛t ❝❤❛♥❣❡s ✐♥ ❛ r❡♣❡t✐t✐✈❡ ♠❛♥♥❡r✿

✺✳✼✳

❚r✐❣♦♥♦♠❡tr② ❛♥❞ t❤❡ ✇❛✈❡ ❢✉♥❝t✐♦♥

✹✻✵

❚❤❡ s✐♠♣❧❡st ❢✉♥❝t✐♦♥s ♦❢ t❤✐s ❦✐♥❞ ❛r❡ tr✐❣♦♥♦♠❡tr✐❝✳

❉❡✜♥✐t✐♦♥ ✺✳✼✳✷✺✿ ✇❛✈❡ ❢✉♥❝t✐♦♥ ❚❤❡ ❢✉♥❝t✐♦♥

f (x) = A sin(ωx + φ) ✐s ❝❛❧❧❡❞ ❛

✇❛✈❡ ❢✉♥❝t✐♦♥

✳ ❋✉rt❤❡r♠♦r❡✱

• A > 0 ✐s ❝❛❧❧❡❞ t❤❡ ❛♠♣❧✐t✉❞❡✱ • ω > 0 ✐s ❝❛❧❧❡❞ t❤❡ ❢r❡q✉❡♥❝②✱ ❛♥❞ • φ ✐s ❝❛❧❧❡❞ t❤❡ ♣❤❛s❡✱

♦❢ t❤❡ ✇❛✈❡ ❢✉♥❝t✐♦♥✳

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

❛♠♣❧✐t✉❞❡ ❢r❡q✉❡♥❝② ♣❤❛s❡

y = sin x

A y → Ay ω x → ωx φ x→x+φ

✐♥ ♣r❡❞✐❝t❛❜❧❡ ✇❛②s✿

✈❡rt✐❝❛❧ str❡t❝❤ ❤♦r✐③♦♥t❛❧ s❤r✐♥❦ ❤♦r✐③♦♥t❛❧ s❤✐❢t

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

❂❘✶❈❬✶❪✯❙■◆✭❘✷❈❬✶❪✯❘❈❬✲✶❪✰❘✸❈❬✶❪✮ ❲❡ ✈❛r② t❤❡s❡ ♣❛r❛♠❡t❡rs ❜❡❧♦✇✳

❊①❛♠♣❧❡ ✺✳✼✳✷✻✿ ✇❛✈❡ ❢✉♥❝t✐♦♥✱ ❛♠♣❧✐t✉❞❡ ✈❛r✐❡s ❆s t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ s✐♥❡ r✉♥ ❜❡t✇❡❡♥ ■t✬s ❛ ✈❡rt✐❝❛❧ str❡t❝❤✴s❤r✐♥❦✿

−1

❛♥❞

1✱

t❤❡ ✈❛❧✉❡s ♦❢ t❤✐s ❢✉♥❝t✐♦♥ r✉♥ ❜❡t✇❡❡♥

−A

❛♥❞

A✳

y✿

✺✳✼✳

❚r✐❣♦♥♦♠❡tr② ❛♥❞ t❤❡ ✇❛✈❡ ❢✉♥❝t✐♦♥

❆ str❡t❝❤ ❜② ❛ ❢❛❝t♦r ♦❢

2

✹✻✶

✐s s❤♦✇♥✳ ❲❡ ❝❛♥ s❡❡ t❤❛t t❤❡ ❢✉♥❝t✐♦♥ st✐❧❧ r❡♣❡❛ts ✐ts❡❧❢ ❡✈❡r②

s✇✐♥❣ ❤❛s ❞♦✉❜❧❡❞✳

❊①❛♠♣❧❡ ✺✳✼✳✷✼✿ ✇❛✈❡ ❢✉♥❝t✐♦♥✱ ♣❤❛s❡ ✈❛r✐❡s ❚❤❡ ♣❤❛s❡

❆ s❤✐❢t ♦❢

φ

✐s ❥✉st ❛ ❤♦r✐③♦♥t❛❧ s❤✐❢t✿

π/2

r✐❣❤t ✐s s❤♦✇♥✳ ❚❤✐s ✐s t❤❡ ❝♦s✐♥❡✦

❊①❛♠♣❧❡ ✺✳✼✳✷✽✿ ✇❛✈❡ ❢✉♥❝t✐♦♥✱ ❢r❡q✉❡♥❝② ✈❛r✐❡s ❆s t❤❡ s✐♥❡ ✐s

2π ✲♣❡r✐♦❞✐❝✱

t❤✐s ❢✉♥❝t✐♦♥✬s ♣❡r✐♦❞ ✐s

2π/ω ✳

■t✬s ❛ ❤♦r✐③♦♥t❛❧ str❡t❝❤✴s❤r✐♥❦✿

2π ❀

❥✉st t❤❡

✺✳✼✳

❚r✐❣♦♥♦♠❡tr② ❛♥❞ t❤❡ ✇❛✈❡ ❢✉♥❝t✐♦♥

❆ s❤r✐♥❦ ❜② ❛ ❢❛❝t♦r ♦❢ ✐♥t❡r✈❛❧ ♦❢

2π ❀

2

✐ts ♣❡r✐♦❞ ✐s

✐s s❤♦✇♥✳

❲❡ ❝❛♥ s❡❡ t❤❛t t❤❡ ❢✉♥❝t✐♦♥ r❡♣❡❛ts ✐ts❡❧❢

2π/2 = π ✳

✹✻✷

t✇✐❝❡

✇✐t❤✐♥ ❡✈❡r②

❊①❡r❝✐s❡ ✺✳✼✳✷✾ Pr♦✈❡ t❤❡ st❛t❡♠❡♥t ❛❜♦✉t t❤❡ ♣❡r✐♦❞s✳

❚❤❡ ❝♦♠♠♦♥ ♥❛♠❡ ✏s✐♥✉s♦✐❞✑ ✐s ❥✉st✐✜❡❞ ❜② t❤❡ ❢❛❝t t❤❛t ❛❧❧ t❤❡s❡ ❝✉r✈❡s ❧♦♦❦ t❤❡ s❛♠❡ r❡❣❛r❞❧❡ss ♦❢ t❤❡ ❝❤♦✐❝❡ ♦❢ t❤❡ ♣❛r❛♠❡t❡rs✳ ❲❤❛t ✐❢ ✇❡

❛❞❞

t✇♦ ✇❛✈❡ ❢✉♥❝t✐♦♥s ✇✐t❤ ❞✐✛❡r❡♥t ✈❛❧✉❡s ♦❢ t❤❡ s❛♠❡ ♣❛r❛♠❡t❡r❄ ❙✉❝❤ ❛♥ ❛❞❞✐t✐♦♥ ❝♦rr❡✲

s♣♦♥❞s t♦✱ ❢♦r ❡①❛♠♣❧❡✱ t✇♦ s♦✉♥❞s ❤❡❛r❞ ❛t t❤❡ s❛♠❡ t✐♠❡✳ ❲❡ ❦♥♦✇ ❤♦✇ t♦ ❛❞❞ ❢✉♥❝t✐♦♥s✱ ❜✉t ✇❡ ✇♦✉❧❞ ❧✐❦❡ t♦ ♣r❡❞✐❝t t❤❡ ❦✐♥❞ ♦❢ ❢✉♥❝t✐♦♥ ✇✐❧❧ r❡s✉❧t✿

f (x) = A sin(ωx + φ) g(x) = B sin(δx + ψ) f (x) + g(x) = ?

❊①❛♠♣❧❡ ✺✳✼✳✸✵✿ ❛❞❞✐♥❣ ✇❛✈❡ ❢✉♥❝t✐♦♥s✱ ❞✐✛❡r❡♥t ❛♠♣❧✐t✉❞❡s ❲❤❛t ✐❢ ✇❡ ❝♦♠❜✐♥❡ t✇♦ ✇❛✈❡ ❢✉♥❝t✐♦♥s ✇✐t❤ ❞✐✛❡r❡♥t ❛♠♣❧✐t✉❞❡s❄ ❈♦♥s✐❞❡r✿

y = sin x + 2 sin x = 3 sin x . ❆♠♣❧✐t✉❞❡s ❛r❡ ❛❞❞❡❞✱ t❤❛t✬s ❛❧❧✦ ■♥❞❡❡❞✿

❊①❡r❝✐s❡ ✺✳✼✳✸✶ ▼❛❦❡ ❛ ❣❡♥❡r❛❧ st❛t❡♠❡♥t ❛❜♦✉t ❛❞❞✐♥❣ ✇❛✈❡ ❢✉♥❝t✐♦♥s ✇✐t❤ ❞✐✛❡r❡♥t ❛♠♣❧✐t✉❞❡s ❛♥❞ ♣r♦✈❡ ✐t✳

✺✳✼✳

❚r✐❣♦♥♦♠❡tr② ❛♥❞ t❤❡ ✇❛✈❡ ❢✉♥❝t✐♦♥

✹✻✸

❊①❛♠♣❧❡ ✺✳✼✳✸✷✿ ❛❞❞✐♥❣ ✇❛✈❡ ❢✉♥❝t✐♦♥s✱ ❞✐✛❡r❡♥t ♣❤❛s❡s ❲❤❛t ✐❢ ✇❡ ❝♦♠❜✐♥❡ t✇♦ ✇❛✈❡ ❢✉♥❝t✐♦♥s ✇✐t❤ ❞✐✛❡r❡♥t ♣❤❛s❡s❄ ❈♦♥s✐❞❡r✿ y = sin(x) + sin(x + π/2)     x − (x + π/2) x + (x + π/2) sin , = 2 sin 2    2  −π/2) 2x + π/2 sin = 2 sin 2 2 √ = − 2 sin (x + π/4) .

❜② ❙✉♠ ♦❢ ❙✐♥❡s ❛♥❞ ❈♦s✐♥❡s ❢♦r♠✉❧❛

■t✬s st✐❧❧ ❛ s✐♥✉s♦✐❞✦ ❇✉t q✉✐t❡ ❛ ❞✐✛❡r❡♥t ♦♥❡✿

❊①❡r❝✐s❡ ✺✳✼✳✸✸ ▼❛❦❡ ❛ ❣❡♥❡r❛❧ st❛t❡♠❡♥t ❛❜♦✉t ❛❞❞✐♥❣ ✇❛✈❡ ❢✉♥❝t✐♦♥s ✇✐t❤ ❞✐✛❡r❡♥t ♣❤❛s❡s ❛♥❞ ♣r♦✈❡ ✐t✳

❊①❛♠♣❧❡ ✺✳✼✳✸✹✿ ❛❞❞✐♥❣ ✇❛✈❡ ❢✉♥❝t✐♦♥s✱ ❞✐✛❡r❡♥t ❢r❡q✉❡♥❝✐❡s ❲❤❛t ✐❢ ✇❡ ❝♦♠❜✐♥❡ t✇♦ ✇❛✈❡ ❢✉♥❝t✐♦♥s ✇✐t❤ ❞✐✛❡r❡♥t ❢r❡q✉❡♥❝✐❡s❄ ❈♦♥s✐❞❡r✿ y = sin(x) + .3 sin(7x) .

❚❤✐s ✐s ♥♦t ❛ s✐♥✉s♦✐❞ ❛♥②♠♦r❡✦ ■♥❞❡❡❞✿

■t ✐s st✐❧❧ 2π ✲♣❡r✐♦❞✐❝✦ ❆❧s♦✱ ❤✐❣❤❧✐❣❤t❡❞ ❜② t❤❡ ❞✐✛❡r❡♥❝❡ ✐♥ ❛♠♣❧✐t✉❞❡s✱ ❜♦t❤ ♦❢ t❤❡ ❢r❡q✉❡♥❝✐❡s ❛r❡ ❝❧❡❛r❧② ✈✐s✐❜❧❡✳ ■❢ t❤❡ ✇❛✈❡ ❢✉♥❝t✐♦♥s r❡♣r❡s❡♥t s♦✉♥❞✱ t❤❡ s❡❝♦♥❞ ❢✉♥❝t✐♦♥ g ♠❛② ❜❡ t❤❡ ♥♦✐s❡✳ ■t ✐s t❤❡♥ ♦✉r ❝❤❛❧❧❡♥❣❡ t♦ ❣♦ ✐♥ t❤❡ ❜❛❝❦✇❛r❞ ❞✐r❡❝t✐♦♥✱ ✐✳❡✳✱ ❢r♦♠ f + g t♦ f ❛♥❞ g ✱ ✐♥ ♦r❞❡r t♦ r❡♠♦✈❡ ✐t✳

❊①❡r❝✐s❡ ✺✳✼✳✸✺ ❙❤♦✇ t❤❛t t❤❡ s✉♠ ♦❢ t✇♦ ✇❛✈❡ ❢✉♥❝t✐♦♥s sin x + sin 3x ❤❛s ♣❡r✐♦❞ 2π ✳

❊①❡r❝✐s❡ ✺✳✼✳✸✻ ❲❤❛t ❤❛♣♣❡♥s ✇❤❡♥ t❤❡ ❢r❡q✉❡♥❝✐❡s ❛r❡♥✬t ♣r♦♣♦rt✐♦♥❛❧✱ s✉❝❤ ❛s sin x + sin πx❄ ❲❡ ♥♦✇ ❛♣♣❧② t❤✐s✱ t✐♠❡✲❞❡♣❡♥❞❡♥t✱ ✐♥t❡r♣r❡t❛t✐♦♥ ♦❢ ♣❡r✐♦❞✐❝ ♠♦t✐♦♥ t♦ t❤❡ r♦t❛t✐♦♥ ♣r♦❜❧❡♠✳ ❚❤❡r❡ ✇✐❧❧ ❜❡ t✇♦ ✇❛✈❡ ❢✉♥❝t✐♦♥s ♣r♦❞✉❝✐♥❣ ❛ ♣❛r❛♠❡tr✐❝ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ❝✐r❝❧❡✳

✺✳✼✳

❚r✐❣♦♥♦♠❡tr② ❛♥❞ t❤❡ ✇❛✈❡ ❢✉♥❝t✐♦♥

✹✻✹

❲❡ st❛rt ✇✐t❤ t❤❡ ❛♥❣❧❡✲❞❡♣❡♥❞❡♥t r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ❝✐r❝❧❡ ♦❢ r❛❞✐✉s R ❝❡♥t❡r❡❞ ❛t 0✿ (

x = R cos θ , y = R sin θ .

❇✉t ✐♥ t❤❡ ♥❡✇ ♠♦❞❡❧ t❤❡ ❛♥❣❧❡✱ θ✱ ❞❡♣❡♥❞s ♦♥ t✐♠❡✳ ❲❡ ❛ss✉♠❡ ❛ ❝♦♥st❛♥t s♣❡❡❞ ♦❢ r♦t❛t✐♦♥ ✭❛♥❣✉❧❛r ✈❡❧♦❝✐t②✮ ❛♥❞✱ t❤❡r❡❢♦r❡✱ ❛ ❝♦♥st❛♥t ❢r❡q✉❡♥❝② ✿ θ = ωt + φ .

❙✉❜st✐t✉t✐♦♥ ♣r♦❞✉❝❡s t❤✐s ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✿ (

x = R cos(ωt + φ) , y = R sin(ωt + φ) .

❊①❛♠♣❧❡ ✺✳✼✳✸✼✿ ❋❡rr✐s ✇❤❡❡❧

▲❡t✬s ❝♦♥s✐❞❡r ❛ s♣❡❝✐✜❝ ❝❛s❡ ♦❢ t❤❡ ❋❡rr✐s ✇❤❡❡❧✿ • ■t ❤❛s ❛ r❛❞✐✉s ♦❢ 100 ❢❡❡t ❛♥❞ • ■t ♠❛❦❡s ❛ ❢✉❧❧ t✉r♥ ✐♥ 2 ♠✐♥✉t❡s✳

❲❡ t❤❡♥ ❝❛♥ ❛s❦ q✉❡st✐♦♥s ❛❜♦✉t ✇❤❡r❡ ✭t❤❡ ✜rst t✇♦✮ ❛s ✇❡❧❧ ❛s ❛❜♦✉t ✇❤❡♥ ✭t❤✐r❞✮✿ ✶✳ ❍♦✇ ❢❛r ❛✇❛② ❢r♦♠ t❤❡ ❜❛s❡✱ ❤♦r✐③♦♥t❛❧❧②✱ ❛r❡ ②♦✉ ✐♥ 5 ♠✐♥✉t❡s❄ ✷✳ ❍♦✇ ❤✐❣❤ ❛r❡ ②♦✉ ✐♥ 20 s❡❝♦♥❞s❄ ✸✳ ❲❤❡♥ ❛r❡ ②♦✉ 30 ❢❡❡t ❛❜♦✈❡ t❤❡ ❣r♦✉♥❞❄ ❋✐rst✱ ✇❡ ♥❡❡❞ ❛ ❝♦♠♣❧❡t❡ ♠♦❞❡❧❀ ✇❡ ♥❡❡❞ t♦ ✜♥❞ t❤❡ ❖❢ ❝♦✉rs❡✱ t❤❡ r❛❞✐✉s ✐s

t❤r❡❡

♣❛r❛♠❡t❡rs ❢♦r t❤❡ t✇♦ ❢✉♥❝t✐♦♥s ❛❜♦✈❡✳

R = 100 .

❙❡❝♦♥❞✱ ♠❛❦✐♥❣ ❛ ❢✉❧❧ t✉r♥ ✭2π r❛❞✐❛♥s✮ ✐♥ 2 ♠✐♥✉t❡s ♠❡❛♥s t❤❛t t❤❡ ❢r❡q✉❡♥❝② ✐s ω=

2π = π. 2

❚❤✐r❞✱ t❤❡ st❛rt✐♥❣ ♣♦✐♥t ♦❢ t❤❡ tr✐♣ ✐s ❛t t❤❡ ❜♦tt♦♠ ♦❢ t❤❡ ✇❤❡❡❧✱ ✐✳❡✳✱ ✇✐t❤ t❤❡ r♦t❛t✐♥❣ s❡❣♠❡♥t ♣♦✐♥t✐♥❣ ❞♦✇♥❀ t❤❡r❡❢♦r❡✱ t❤❡ ♣❤❛s❡ ✐s φ = −π/2 .

❙♦✱ t❤❡ ♠♦❞❡❧✬s ❡q✉❛t✐♦♥s ♠❛❦❡ ✉♣ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣❛r❛♠❡tr✐❝ (

❝✉r✈❡ ✭t✐♠❡ ✐♥ ♠✐♥✉t❡s✮✿

x = 100 cos(πt − π/2) , y = 100 sin(πt − π/2) .

❚❤❡② ❝❛♥ ❛♥s✇❡r ❛❧❧ q✉❡st✐♦♥s ✇❡ ♠❛② ❤❛✈❡✳ ❋♦r t❤❡ ✜rst q✉❡st✐♦♥✱ ✇❡ ❥✉st s✉❜st✐t✉t❡ t = 5 ✐♥t♦ t❤❡ ✜rst ❡q✉❛t✐♦♥✿ x = 100 cos(π · 5 − π/2) = 100 cos(4π + π/2) = 100 cos(π/2) = 100 · 0 = 0 ❢❡❡t✳

✺✳✽✳

✹✻✺

❚❤❡ ❊✉❝❧✐❞❡❛♥ ♣❧❛♥❡✿ ❛♥❣❧❡s

❚❤❛t✬s ❤♦✇ ❤✐❣❤ ✇❡ ❛r❡ ❛t t❤❛t t✐♠❡✳ ❋♦r t❤❡ s❡❝♦♥❞ q✉❡st✐♦♥✱ ✇❡ s✉❜st✐t✉t❡ t = 1/3 ✭✐✳❡✳✱ 20 s❡❝♦♥❞s ❡①♣r❡ss❡❞ ✐♥ ♠✐♥✉t❡s✮ ✐♥t♦ t❤❡ s❡❝♦♥❞ ❡q✉❛t✐♦♥✿   y = 100 sin π ·

1 π − 3 2

 π = 100 sin − = 100 · (−.5) = −50 ❢❡❡t✳ 6

❚❤❛t✬s 100 − 50 = 50 ❢❡❡t ❛❜♦✈❡ t❤❡ ❣r♦✉♥❞✳

❋♦r t❤❡ t❤✐r❞ q✉❡st✐♦♥✱ ✇❡ s✉❜st✐t✉t❡ y = −100 + 30 = −70 ✐♥t♦ t❤❡ s❡❝♦♥❞ ❡q✉❛t✐♦♥✱ ❛♥❞ s♦❧✈❡✿ −70 = 100 sin(πt − π/2) =⇒ sin(πt − π/2) = −.7 =⇒ π(t − 1/2) = arcsin(−.7) =⇒ 1 1 t = arcsin(−.7) + π 2 ≈ .25 ♠✐♥✉t❡s.

❚❤❛t✬s t❤❡ t✐♠❡ ✇❤❡♥ ✇❡ r❡❛❝❤ t❤✐s ❤❡✐❣❤t ❢♦r t❤❡ ✜rst t✐♠❡✳ ✭❲❡ ✇✐❧❧ ❝♦♥t✐♥✉❡ ♦✉r st✉❞② ♦❢ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s ✐♥ ❈❤❛♣t❡r ✸■❈✲✹✳✮

❊①❡r❝✐s❡ ✺✳✼✳✸✽ ❋✐♥✐s❤ t❤❡ ❛♥s✇❡r t♦ t❤❡ t❤✐r❞ q✉❡st✐♦♥✳ ❲❡ ❤❛✈❡ s❡❡♥ ❛ ❧♦t ♦❢ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s✳ ■♥ s♣✐t❡ ♦❢ t❤❡ ✇✐❞❡ ❛♣♣❧✐❝❛❜✐❧✐t② ♦❢ t❤❡ ❙✐♥❡ ❛♥❞ ✐ts ❝♦r♦❧❧❛r✐❡s✱ ♦♥❧② t❤❡s❡ ❛r❡ ✐♠♣♦rt❛♥t ❡♥♦✉❣❤ t♦ ♠❡♠♦r✐③❡✿

♦❢ ❙✉♠

❢♦r♠✉❧❛

❚r✐❣♦♥♦♠❡tr✐❝ ■❞❡♥t✐t✐❡s ❉❡s❝r✐♣t✐♦♥s✿

❋♦r♠✉❧❛s✿

❙✐♥❡ ❛♥❞ ❝♦s✐♥❡ ❛r❡ ♣❡r✐♦❞✐❝✿

sin(x + 2π) = sin(x) ,

cos(x + 2π) = cos(x)

❙✐♥❡ ✐s ♦❞❞ ❛♥❞ ❝♦s✐♥❡ ❡✈❡♥✿

sin(−x) = − sin(x) ,

cos(−x) = cos(x)

❙✐♥❡ ❛♥❞ ❝♦s✐♥❡ ❛r❡ s❤✐❢ts ♦❢ ❡❛❝❤ ♦t❤❡r✿ P②t❤❛❣♦r❡❛♥ ❚❤❡♦r❡♠✿

sin(x + π/2) = cos(x) , cos(x − π/2) = sin(x) sin2 x + cos2 x = 1

✺✳✽✳ ❚❤❡ ❊✉❝❧✐❞❡❛♥ ♣❧❛♥❡✿ ❛♥❣❧❡s

■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ t❛❦❡ ✉♣ t❤❡ s❡❝♦♥❞ ❣❡♦♠❡tr✐❝ t❛s❦✱ ❡✈❛❧✉❛t✐♥❣ ❞✐r❡❝t✐♦♥s✱ ✐♥ t❤❡ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡ ❡q✉✐♣♣❡❞ ✇✐t❤ t❤❡ ❈❛rt❡s✐❛♥ ❝♦♦r❞✐♥❛t❡ s②st❡♠✳ ❋✐rst✱ t❤❡ 1✲❞✐♠❡♥s✐♦♥❛❧ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡✱ ✐✳❡✳✱ ♥♦t❤✐♥❣ ❜✉t t❤❡ x✲❛①✐s✳ ❲❤❛t ❞♦❡s ❛ ❞✐r❡❝t✐♦♥ ♦♥ t❤❡ r❡❛❧ ❧✐♥❡ ♠❡❛♥❄ ❲❡ ✇✐❧❧ ♣✉rs✉❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛♣♣r♦❛❝❤✳

❉❡✜♥✐t✐♦♥ ✺✳✽✳✶✿ ✈❡❝t♦r ✐♥ ❞✐♠❡♥s✐♦♥ 1 ■❢ ❛ ❧✐♥❡ s❡❣♠❡♥t✬s st❛rt✐♥❣ ♣♦✐♥t ✐s t❤❡ ♦r✐❣✐♥✱ ✐✳❡✳✱ ✐t✬s OP ❢♦r s♦♠❡ ♣♦✐♥t P 6= O✱ ✐t ✐s ❝❛❧❧❡❞ ❛ ✭1✲❞✐♠❡♥s✐♦♥❛❧✮ ✈❡❝t♦r ✐♥ R✳ ❱❡❝t♦rs ❛r❡ ✉s✉❛❧❧② ✈✐s✉❛❧✐③❡❞ ❛s ❛rr♦✇s✿

✺✳✽✳ ❚❤❡ ❊✉❝❧✐❞❡❛♥ ♣❧❛♥❡✿ ❛♥❣❧❡s

✹✻✻

❱❡❝t♦rs ❤❛✈❡ t✇♦ ♠❛✐♥ ❛ttr✐❜✉t❡s✿

•

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

•

❆ ✈❡❝t♦r ❤❛s ✐ts ❞✐r❡❝t✐♦♥✱ ✇❤✐❝❤ ✐s✱ ✐♥ t❤❡ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡✱ ❡✐t❤❡r ♣♦s✐t✐✈❡ ♦r ♥❡❣❛t✐✈❡✳

x

♦❢ ✐ts t❡r♠✐♥❛❧ ♣♦✐♥t✿

|x|✳

◆♦✇✱ ❤♦✇ ❞♦ ✇❡ ❝♦♠♣❛r❡ t❤❡ ❞✐r❡❝t✐♦♥s ♦❢ t✇♦ ✈❡❝t♦rs❄ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ t✇♦ ♣♦✐♥ts✿

P

❛♥❞

•

Q✳

■❢

P

P 6= O, Q 6= O✳

❲❡ ❞❡❛❧ ✇✐t❤ t❤❡ ❞✐r❡❝t✐♦♥s ❢r♦♠ t❤❡ ♦r✐❣✐♥

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

Q

❛r❡ ♦♥ t❤❡ s❛♠❡ s✐❞❡ ♦❢

O✱

t❤❡♥ t❤❡ ❞✐r❡❝t✐♦♥s ❛r❡ s❛♠❡ ✿

P Q ←O→ •

■❢

P

O t♦✇❛r❞ ❧♦❝❛t✐♦♥s

❛♥❞

Q

❛r❡ ♦♥ t❤❡ ♦♣♣♦s✐t❡ s✐❞❡s ♦❢

O✱

♦r

←O→ P Q

t❤❡♥ t❤❡ ❞✐r❡❝t✐♦♥s ❛r❡ ♦♣♣♦s✐t❡ ✿

P ←O→ Q

♦r

Q ←O→ P

▲❡t✬s ❡①❛♠✐♥❡ t❤❡ ❝♦♦r❞✐♥❛t❡s✳ ❚❤❡s❡ ❛r❡ t❤❡ ❢♦✉r ♣♦ss✐❜✐❧✐t✐❡s✿

❲❤❡♥ t❤❡ t✇♦ ✈❡❝t♦rs ❛r❡ r❡♣r❡s❡♥t❡❞ ❜② t❤❡✐r ❝♦♦r❞✐♥❛t❡s✱

x ❛♥❞ x′ ✱ t❤❡ ❛♥❛❧②s✐s ♦❢ t❤❡✐r ❞✐r❡❝t✐♦♥s ❜❡❝♦♠❡s

❛❧❣❡❜r❛✐❝✿

•

■❢

x > 0, x′ > 0

♦r

x < 0, x′ < 0✱

t❤❡♥ t❤❡ ❞✐r❡❝t✐♦♥s ❛r❡ t❤❡ s❛♠❡✳

•

■❢

x > 0, x′ < 0

♦r

x < 0, x′ > 0✱

t❤❡♥ t❤❡ ❞✐r❡❝t✐♦♥s ❛r❡ t❤❡ ♦♣♣♦s✐t❡✳

❋♦rt✉♥❛t❡❧②✱ t❤❡ ♣r♦❞✉❝t ♣r♦✈✐❞❡s ✉s ✇✐t❤ ❛ s✐♥❣❧❡ ❡①♣r❡ss✐♦♥ t❤❛t ♠❛❦❡s t❤✐s ❞❡t❡r♠✐♥❛t✐♦♥✿

♣♦✐♥ts✿

P Q ←O→ ←O→ P Q

s✐❣♥s✿

−·− = +

♣♦✐♥ts✿

P ←O→Q Q ←O→ P

s✐❣♥s✿

−·+ = −

+·+ = +

−·+ = −

s❛♠❡

+ ♦♣♣♦s✐t❡

−

✺✳✽✳

✹✻✼

❚❤❡ ❊✉❝❧✐❞❡❛♥ ♣❧❛♥❡✿ ❛♥❣❧❡s

❚❤❡♦r❡♠ ✺✳✽✳✷✿ ❉✐r❡❝t✐♦♥s ❢♦r ❉✐♠❡♥s✐♦♥ ✶ 0 t♦ x 6= 0 ❛♥❞ x′ 6= 0 ′ s❛♠❡ ✇❤❡♥ x · x > 0❀ ❛♥❞ ′ ♦♣♣♦s✐t❡ ✇❤❡♥ x · x < 0✳

❚❤❡ ❞✐r❡❝t✐♦♥s ❢r♦♠

• •

t❤❡ t❤❡

❛r❡

▲❡t✬s r❡st❛t❡ t❤❡ t❤❡♦r❡♠ ✐♥ t❡r♠s ♦❢ t❤❡ s✐❣♥ ❢✉♥❝t✐♦♥✿ s✐❣♥ ♦❢ t❤❡ ♣r♦❞✉❝t ❞✐r❡❝t✐♦♥s sign(x · x′ ) = 1 s❛♠❡ ′ sign(x · x ) = −1 ♦♣♣♦s✐t❡

→ → → ←

❛♥❣❧❡ 0 ❞❡❣r❡❡s 180 ❞❡❣r❡❡s

❲❡ ❛❧s♦ ♠❡❛s✉r❡ t❤❡ ❛❝t✉❛❧ ❛♥❣❧❡s ❜❡t✇❡❡♥ t❤❡ ✈❡❝t♦rs ✭❧❛st ❝♦❧✉♠♥✮✳

❊①❡r❝✐s❡ ✺✳✽✳✸ ❲❤❡r❡ ❡❧s❡ ❞✐❞ ✇❡ s❡❡ t❤❡ r❡❧❛t✐♦♥✿ 1 ↔ 0 ❞❡❣r❡❡s ❛♥❞ −1 ↔ 180 ❞❡❣r❡❡s❄ ❚❤✐s ❜r✐♥❣s ✉s t♦ t❤❡ ✐❞❡❛ ♦❢ ❛♥ ♦r✐❡♥t❡❞ ✐♥t❡r✈❛❧✱ ♦r s❡❣♠❡♥t✱ ✇✐t❤✐♥ t❤❡ ❛①✐s✳ ❆♥② t✇♦ ❞✐st✐♥❝t ♣♦✐♥ts P ❛♥❞ Q ♣r♦❞✉❝❡ t✇♦ ✐♥t❡r✈❛❧s✿ P Q ❛♥❞ QP ✳ ❚❤❡s❡ t✇♦ ✐♥t❡r✈❛❧s ❤❛✈❡ ♦♣♣♦s✐t❡ ❞✐r❡❝t✐♦♥s✱ ❛♥❞ ✇❡ s❛② t❤❛t t❤❡② ❤❛✈❡ ♦♣♣♦s✐t❡ ♦r✐❡♥t❛t✐♦♥s✳ ❋✉rt❤❡r♠♦r❡✱ s✐♥❝❡ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ x✲❛①✐s ✐s s❡t ❛❤❡❛❞ ♦❢ t✐♠❡✱ ✇❡ ❝❛♥ ♠❛t❝❤ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ ✐♥t❡r✈❛❧ ✇✐t❤ ✐t✱ ❛s ❢♦❧❧♦✇s✳

❉❡✜♥✐t✐♦♥ ✺✳✽✳✹✿ ♣♦s✐t✐✈❡❧② ❛♥❞ ♥❡❣❛t✐✈❡❧② ♦r✐❡♥t❡❞ s❡❣♠❡♥ts • ■❢ P < Q✱ t❤❡ s❡❣♠❡♥t P Q ✐s ❝❛❧❧❡❞ • ■❢ P > Q✱ t❤❡ s❡❣♠❡♥t P Q ✐s ❝❛❧❧❡❞

✳ ♦r✐❡♥t❡❞✳

♣♦s✐t✐✈❡❧② ♦r✐❡♥t❡❞ ♥❡❣❛t✐✈❡❧②

■♥ ❢❛❝t✱ ✇❡ ❛❞♦♣t t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥✈❡♥t✐♦♥✿ QP = −P Q .

❙♦✱ ❛ r✉❧❡r ❤❛s t✇♦ ❡♥❞s ❛♥❞ t❤❡r❡ ❛r❡ t✇♦ ✇❛②s t♦ ♣❧❛❝❡ ✐t ❛❧♦♥❣ ❛ ♠❡❛s✉r✐♥❣ t❛♣❡✳ ❙❡❝♦♥❞✱ t❤❡ 2✲❞✐♠❡♥s✐♦♥❛❧ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡✱ ✐✳❡✳✱ t❤❡ xy ✲♣❧❛♥❡✳ ❚❤❡r❡ ❛r❡ ✐♥✜♥✐t❡❧② ♠❛♥② ❞✐r❡❝t✐♦♥s ♥♦✇✿

❚❤❡ ✐ss✉❡ ♦❢ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ ❛ s✐♥❣❧❡ ❧✐♥❡ ✭♦r ❛ ✈❡❝t♦r✮ ❤❛s ❜❡❡♥ s♦❧✈❡❞✿ ■t✬s t❤❡ s❧♦♣❡✦ ▲❡t✬s r❡✈✐❡✇✳ ❚❤❡ q✉❡st✐♦♥ ✐s t❤❡ ♦♥❡ ❛❜♦✉t t❤❡ ❛♥❣❧❡ ❜❡t✇❡❡♥ t❤❡ ❧✐♥❡ OP ✇✐t❤ t❤❡ x✲❛①✐s✳ ■t ✐s ❞❡t❡r♠✐♥❡❞ ❢r♦♠ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ P = (x, y) ✈✐❛ t❤✐s s✐♠♣❧❡ tr✐❣♦♥♦♠❡tr②✿

✺✳✽✳

✹✻✽

❚❤❡ ❊✉❝❧✐❞❡❛♥ ♣❧❛♥❡✿ ❛♥❣❧❡s

❊①❡r❝✐s❡ ✺✳✽✳✺

❊①♣❧❛✐♥ t❤❡s❡ ❝♦♠♣✉t❛t✐♦♥s✳ ❖♥❡ ♦❢ t❤❡s❡ ❢♦r♠✉❧❛s ❤❛s ❜❡❡♥ ❡s♣❡❝✐❛❧❧② ✐♠♣♦rt❛♥t✳ ❚❤❡♦r❡♠ ✺✳✽✳✻✿ ❙❧♦♣❡ ✐s ❚❛♥❣❡♥t

❚❤❡ t❛♥❣❡♥t ♦❢ t❤❡ ❛♥❣❧❡

P = (x, y) 6= O

α

❜❡t✇❡❡♥ t❤❡

x✲❛①✐s

❛♥❞ t❤❡ ❧✐♥❡ ❢r♦♠

O

t♦ ❛ ♣♦✐♥t

✐s ❡q✉❛❧ t♦ t❤❡ s❧♦♣❡ ♦❢ t❤✐s ❧✐♥❡✿

tan α =

y x

❚❤❡s❡ ❢♦r♠✉❧❛s ✇✐❧❧ ❜❡ ✉s❡❞ ❧❛t❡r✳ ❚❤❡♦r❡♠ ✺✳✽✳✼✿ ❙✐♥❡ ❛♥❞ ❈♦s✐♥❡ ♦❢ ❉✐r❡❝t✐♦♥

❚❤❡ s✐♥❡ ❛♥❞ t❤❡ ❝♦s✐♥❡ ♦❢ t❤❡ ❛♥❣❧❡ t♦ ❛ ♣♦✐♥t

P = (x, y) 6= O

α

❜❡t✇❡❡♥ t❤❡

x✲❛①✐s

❛♥❞ t❤❡ ❧✐♥❡ ❢r♦♠

O

❛r❡ ❣✐✈❡♥ ❜②✿

cos α = p

x x2

y

+ y2

sin α = p x2 + y 2 ❊①❡r❝✐s❡ ✺✳✽✳✽

Pr♦✈❡ t❤❡ t❤❡♦r❡♠✳ ❘❡❝❛❧❧ ❢r♦♠ ❊✉❝❧✐❞❡❛♥ ❣❡♦♠❡tr② t❤❛t t✇♦ ❧✐♥❡s ❛r❡ ❝❛❧❧❡❞

♣❛r❛❧❧❡❧

✐❢ t❤❡② ❢♦r♠ t❤❡ s❛♠❡ ❛♥❣❧❡ ✇✐t❤ ❛ ❣✐✈❡♥

❧✐♥❡ ✭♠✐❞❞❧❡✮✿

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

x✲❛①✐s

✭r✐❣❤t✮✱ ✇❡ ❝❛♥ ❛♣♣❧② t❤❡ ❛❜♦✈❡ t❤❡♦r❡♠ t♦ t❤❡ ❛♥❣❧❡

α✳

❲❡

✺✳✽✳ ❚❤❡ ❊✉❝❧✐❞❡❛♥ ♣❧❛♥❡✿ ❛♥❣❧❡s

✹✻✾

❚❤❡♦r❡♠ ✺✳✽✳✾✿ P❛r❛❧❧❡❧ ▲✐♥❡s✱ ❙❛♠❡ ❙❧♦♣❡ ❚✇♦ ✭♥♦♥✲✈❡rt✐❝❛❧✮ ❧✐♥❡s ♦♥ t❤❡

xy ✲♣❧❛♥❡

❛r❡ ♣❛r❛❧❧❡❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡② ❤❛✈❡

❡q✉❛❧ s❧♦♣❡s✳

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

y = mx + b || y = m′ x + b′ ⇐⇒ m = m′ . ❊①❡r❝✐s❡ ✺✳✽✳✶✵

❋✐♥❞ t❤❡ ❧✐♥❡ ♣❛r❛❧❧❡❧ t♦

y = −3x t❤❛t ♣❛ss❡s t❤r♦✉❣❤ t❤❡ ♣♦✐♥t (1, 1)✳

❙✉❣❣❡st ❛♥♦t❤❡r ❧✐♥❡ ❛♥❞ r❡♣❡❛t✳

❊①❡r❝✐s❡ ✺✳✽✳✶✶

❙♣❧✐t t❤❡ t❤❡♦r❡♠ ✐♥t♦ ❛ st❛t❡♠❡♥t ❛♥❞ ✐ts ❝♦♥✈❡rs❡✳

❊①❡r❝✐s❡ ✺✳✽✳✶✷

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

▲✐♥❡s ♦♥ t❤❡ ♣❧❛♥❡ ❛r❡ ✐ts s✉❜s❡ts✳ ■t ✐s✱ t❤❡r❡❢♦r❡✱ ♥❛t✉r❛❧ t♦ ❛s❦ ❛❜♦✉t t❤❡✐r ✐♥t❡rs❡❝t✐♦♥s✳ ❲❡ ❦♥♦✇ ❢r♦♠ ❊✉❝❧✐❞❡❛♥ ❣❡♦♠❡tr② t❤❛t t✇♦ ♣❛r❛❧❧❡❧ ❧✐♥❡s ❞♦♥✬t ✐♥t❡rs❡❝t✳ ■❢ t❤❡② ❞✐❞✱ t❤❡②✬❞ ❢♦r♠ ❛ tr✐❛♥❣❧❡ ✇✐t❤ t❤❡ s✉♠ ♦❢ t❤❡ ❛♥❣❧❡s ❛❜♦✈❡

180

❞❡❣r❡❡s✿

❚❤❡ ❈❛rt❡s✐❛♥ s②st❡♠ ♠❛❦❡s ✐t ♣♦ss✐❜❧❡ t♦ ♣r♦✈❡ t❤✐s ❢❛❝t ❛❧❣❡❜r❛✐❝❛❧❧②✳ ❈♦r♦❧❧❛r② ✺✳✽✳✶✸✿ P❛r❛❧❧❡❧ ▲✐♥❡s✿ ❇❛s✐❝ ❋❛❝ts

• •

P❛r❛❧❧❡❧ ❧✐♥❡s ❞♦♥✬t ✐♥t❡rs❡❝t✳ ❈♦♥✈❡rs❡❧②✱ ♥♦♥✲♣❛r❛❧❧❡❧ ❧✐♥❡s ✐♥t❡rs❡❝t✳

Pr♦♦❢✳

❆❝❝♦r❞✐♥❣ t♦ t❤❡ ❧❛st t❤❡♦r❡♠✱ ✇❡ ❤❛✈❡ t✇♦ ❧✐♥❡s ✇✐t❤ t❤❡ s❛♠❡ s❧♦♣❡✱

b

❛♥❞

c✿

❋♦r ❛ ♣♦✐♥t

 (x, y)

y = mx +b y = mx +c .

m✱ ❛♥❞ t✇♦ ❞✐✛❡r❡♥t y ✲✐♥t❡r❝❡♣ts✱

❆◆❉

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

♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s✦

❙✉❜tr❛❝t✐♥❣ t❤❡ t✇♦ ❡q✉❛t✐♦♥s ♣r♦❞✉❝❡s✿

t❤❡r❡❢♦r❡✱ ♥♦ ✐♥t❡rs❡❝t✐♦♥✳

b − c = 0✳

❊①❡r❝✐s❡ ✺✳✽✳✶✹

Pr♦✈❡ t❤❡ ❝♦♥✈❡rs❡ ♣❛rt ♦❢ t❤❡ ❝♦r♦❧❧❛r②✳

❊①❡r❝✐s❡ ✺✳✽✳✶✺

❙t❛t❡ t❤❡ ❝♦r♦❧❧❛r② ❛s ❛♥ ❡q✉✐✈❛❧❡♥❝❡ ✭❛♥ ✏✐❢✲❛♥❞✲♦♥❧②✲✐❢ ✑ st❛t❡♠❡♥t✮✳

❲❡ ❤❛✈❡ ❛ s②st❡♠

❚❤❡r❡ ✐s ♥♦ s♦❧✉t✐♦♥ ❛♥❞✱

✺✳✽✳

❚❤❡ ❊✉❝❧✐❞❡❛♥ ♣❧❛♥❡✿ ❛♥❣❧❡s

✹✼✵

❊①❡r❝✐s❡ ✺✳✽✳✶✻

Pr♦✈❡ t❤❛t ✈❡rt✐❝❛❧ ❧✐♥❡s ❞♦♥✬t ✐♥t❡rs❡❝t ❡❛❝❤ ♦t❤❡r ❛♥❞ ❞♦ ✐♥t❡rs❡❝t ❛❧❧ ♦t❤❡r ❧✐♥❡s✳ ❊①❛♠♣❧❡ ✺✳✽✳✶✼✿ ♠✐①t✉r❡s✱ ✇❤❛t ❝❛♥ ❤❛♣♣❡♥

❘❡❝❛❧❧ ❛♥ ❡①❛♠♣❧❡ ❢r♦♠ ❈❤❛♣t❡r ✷✿ ■s ✐t ♣♦ss✐❜❧❡ t♦ ❝r❡❛t❡✱ ❢r♦♠ t❤❡ ❑❡♥②❛♥ ❝♦✛❡❡ ✭$2 ♣❡r ♣♦✉♥❞✮ ❛♥❞ t❤❡ ❈♦❧♦♠❜✐❛♥ ❝♦✛❡❡ ✭$3 ♣❡r ♣♦✉♥❞✮✱ 6 ♣♦✉♥❞s ♦❢ ❜❧❡♥❞ ✇✐t❤ ❛ t♦t❛❧ ♣r✐❝❡ ♦❢ $14❄ ❚❤❡ ♣r♦❜❧❡♠ ✐s s♦❧✈❡❞ ✈✐❛ ❛ s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s✿  x +y = 6 ❆◆❉ 2x +3y = 14 . ❲✐t❤♦✉t ❡✈❡♥ s♦❧✈✐♥❣ ✐t✱ ✇❡ ❢♦❧❧♦✇ t❤✐s ❧✐♥❡ ♦❢ t❤♦✉❣❤t✿ • ❚❤❡ s❧♦♣❡s ♦❢ t❤❡ t✇♦ ❧✐♥❡s ❛r❡ ❞✐✛❡r❡♥t❀ t❤❡r❡❢♦r❡✱ • t❤❡ ❧✐♥❡s ❛r❡ ♥♦t ♣❛r❛❧❧❡❧❀ t❤❡r❡❢♦r❡✱ • t❤❡ ❧✐♥❡s ✐♥t❡rs❡❝t❀ t❤❡r❡❢♦r❡✱ • t❤❡ s②st❡♠ ❤❛s ❛ s♦❧✉t✐♦♥✳ ❈♦♥✜r♠❡❞✿

❙♦✱ ✐t ✐s ♣♦ss✐❜❧❡ t♦ ❝r❡❛t❡ s✉❝❤ ❛ ❜❧❡♥❞✦ ■t ✇♦✉❧❞ ❜❡ ✐♠♣♦ss✐❜❧❡ ✐❢ ❜♦t❤ t②♣❡s ♦❢ ❝♦✛❡❡ ✇❡r❡ ♣r✐❝❡❞ ❛t $2 ♣❡r ♣♦✉♥❞✳ ❊①❡r❝✐s❡ ✺✳✽✳✶✽

❏✉st✐❢② t❤❡ ❧❛st st❛t❡♠❡♥t ✕ s❡❝♦♥❞ ♣♦ss✐❜✐❧✐t② ✕ ✉s✐♥❣ t❤❡ t❤❡♦r❡♠ ❛❜♦✉t ♣❛r❛❧❧❡❧ ❧✐♥❡s✳ ❲❤❛t ✐s t❤❡ t❤✐r❞ ♣♦ss✐❜✐❧✐t②❄ ❚❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❛♥❣❧❡ ❜❡t✇❡❡♥ t❤❡ ❞✐r❡❝t✐♦♥s ✭✐✳❡✳✱ t❤❡ ❧✐♥❡s✮ ❢r♦♠ O t♦ ♣♦✐♥t P ❛♥❞ ❢r♦♠ O t♦ ♣♦✐♥t Q ❝♦♠❡s ❢r♦♠ t❤❡ tr✐❛♥❣❧❡ OP Q✳ ❚❤✐s tr✐❛♥❣❧❡ ❛♥❞ ❛❧❧ ♦❢ ✐ts ♠❡❛s✉r❡♠❡♥ts ✐s ✐♥❤❡r✐t❡❞ ❢r♦♠ t❤❡ ❊✉❝❧✐❞❡❛♥ [ ✳ ◆♦✇✱ t❤❡ q✉❡st✐♦♥ ✐s✿ ♣❧❛♥❡ t❤❛t ✉♥❞❡r❧✐♥❡s t❤❡ ❈❛rt❡s✐❛♥ s②st❡♠✳ ❚❤❡ ❛♥❣❧❡ ✐s ❞❡♥♦t❡❞ ❜② QOP

[ ✐♥ t❡r♠s ♦❢ t❤❡✐r ❝♦♦r❞✐♥❛t❡s P = (x, y) ❛♥❞ Q = (x′ , y ′ )❄ ◮ ❍♦✇ ❞♦ ✇❡ ❡①♣r❡ss QOP ❆❜♦✈❡ ✇❡ ❝♦♥s✐❞❡r❡❞ ❛ s♣❡❝✐❛❧ ❝❛s❡✿ Q = (1, 0)✳ ❚❤❡ ❣❡♦♠❡tr② ✐s ✐❧❧✉str❛t❡❞ ❜❡❧♦✇✿

❲❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢r♦♠ t❤❡ t❤❡♦r❡♠ ❛❜♦✈❡✳ ❚❤❡ t✇♦ ❛♥❣❧❡s ❢r♦♠ t❤❡ x✲❛①✐s t♦ P = (x, y) ❛♥❞ t♦

✺✳✽✳

✹✼✶

❚❤❡ ❊✉❝❧✐❞❡❛♥ ♣❧❛♥❡✿ ❛♥❣❧❡s

Q = (x′ , y ′ )✱ r❡s♣❡❝t✐✈❡❧②✱ s❛t✐s❢② t❤❡ ❢♦❧❧♦✇✐♥❣✿ x′ x , cos β = p 2 cos α = p x2 + y 2 x′ + y ′ 2 y y′ sin α = p , sin β = p 2 x2 + y 2 x′ + y ′ 2

❚❤❡ ❢♦r♠✉❧❛s ❧♦♦❦ ❝♦♠♣❧✐❝❛t❡❞ ❜✉t ❦❡❡♣ ✐♥ ♠✐♥❞ t❤❛t t❤❡ ❞❡♥♦♠✐♥❛t♦rs ❛r❡ ❥✉st t❤❡ ❞✐st❛♥❝❡s ❢r♦♠ O t♦ P ❛♥❞ Q✱ r❡s♣❡❝t✐✈❡❧②✳ ❍♦✇❡✈❡r✱ ♦✉r ✐♥t❡r❡st ✐s♥✬t t❤❡s❡ t✇♦ ❛♥❣❧❡s ❜✉t t❤❡✐r ❞✐✛❡r❡♥❝❡✱

[ =α−β. QOP ❋♦rt✉♥❛t❡❧②✱ t❤❡r❡ ✐s ❛♥♦t❤❡r tr✐❣♦♥♦♠❡tr✐❝ ❢♦r♠✉❧❛ ✕ ❙✐♥❡ ❛♥❞ ❈♦s✐♥❡ ♦❢ ❉✐✛❡r❡♥❝❡ ✕ t❤❛t ❛❧❧♦✇s ✉s t♦ r❡♣r❡s❡♥t t❤❡ ❝♦s✐♥❡ ♦❢ t❤✐s ❛♥❣❧❡ ✐♥ t❡r♠s ♦❢ t❤❡ ❢♦✉r q✉❛♥t✐t✐❡s ❛❜♦✈❡✿

cos(α − β) = cos α cos β + sin α sin β

x x′ y′ y p p p =p ❲❡ s✉❜st✐t✉t❡✳ + x2 + y 2 x′ 2 + y ′ 2 x2 + y 2 x′ 2 + y ′ 2 xx′ + yy ′ p . =p ❆♥❞ s✐♠♣❧✐❢②✳ x2 + y 2 x′ 2 + y ′ 2

❚❤❡ t✇♦ ♣❛rts ♦❢ t❤❡ ❞❡♥♦♠✐♥❛t♦r ❛r❡ t❤❡ ❞✐st❛♥❝❡s t♦ P = (x, y) ❛♥❞ Q = (x′ , y ′ )✿ q p 2 2 d(O, P ) = x + y ❛♥❞ d(O, Q) = x′ 2 + y ′ 2 . ❇✉t ✇❤❛t ❛❜♦✉t t❤❡ ♥✉♠❡r❛t♦r❄

❲❡ t❛❦❡ t❤❡ ✈❡❝t♦r ❛♣♣r♦❛❝❤ ❛❣❛✐♥✿

❉❡✜♥✐t✐♦♥ ✺✳✽✳✶✾✿ ✈❡❝t♦r ✐♥ ❞✐♠❡♥s✐♦♥ 2 ■❢ ❛ s❡❣♠❡♥t✬s st❛rt✐♥❣ ♣♦✐♥t ✐s t❤❡ ♦r✐❣✐♥✱ ✐✳❡✳✱ ✐t✬s OP ❢♦r s♦♠❡ P 6= O✱ ✐t ✐s ❝❛❧❧❡❞ ❛ ✭2✲❞✐♠❡♥s✐♦♥❛❧✮ ✈❡❝t♦r ✐♥ R2 ✳ ■ts ❝♦♠♣♦♥❡♥ts ❛r❡ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ ✐ts t❡r♠✐♥❛❧ ♣♦✐♥t P ✱ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ ♥♦t❛t✐♦♥✿

P = (a, b) ⇐⇒ OP =< a, b > ❆ ✈❡❝t♦r ❤❛s ❛ ❞✐r❡❝t✐♦♥✱ ✇❤✐❝❤ ✐s ♦♥❡ ♦❢ t❤❡ t✇♦ ❞✐r❡❝t✐♦♥s ♦❢ t❤❡ ❧✐♥❡✱ ❛♥❞ ❛ ♠❛❣♥✐t✉❞❡✱ ❞❡✜♥❡❞ ❛s ❢♦❧❧♦✇s✳

❉❡✜♥✐t✐♦♥ ✺✳✽✳✷✵✿ ♠❛❣♥✐t✉❞❡ ♦❢ ✈❡❝t♦r ❚❤❡

♠❛❣♥✐t✉❞❡

♦❢ ❛ ✈❡❝t♦r OP =< a, b > ✐s ❞❡✜♥❡❞ ❛s t❤❡ ❞✐st❛♥❝❡ ❢r♦♠ O t♦

✺✳✽✳

✹✼✷

❚❤❡ ❊✉❝❧✐❞❡❛♥ ♣❧❛♥❡✿ ❛♥❣❧❡s

✐ts t✐♣ P ✱ ❞❡♥♦t❡❞ ❜② || < a, b > || =

√

a2 + b2

❊①❡r❝✐s❡ ✺✳✽✳✷✶ ❆♣♣❧② t❤❡ ❞❡✜♥✐t✐♦♥ t♦ ❛ ✈❡❝t♦r < x, 0 > ❛♥❞ ❡①♣❧❛✐♥ ✐ts r❡❧❛t✐♦♥ t♦ t❤❡ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡✳ ❚❤❡ ♥♦t❛t✐♦♥ ✇✐❧❧ ❤❡❧♣ ✉s ✇✐t❤ ♦✉r ❢♦r♠✉❧❛ ❢♦r t❤❡ ❛♥❣❧❡✿

■t ✐s r❡✇r✐tt❡♥ ❛s ❢♦❧❧♦✇s✿ cos(α − β) =

xx′ + yy ′ . || < x, y > || · || < x′ , y ′ > ||

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

❉❡✜♥✐t✐♦♥ ✺✳✽✳✷✷✿ ❞♦t ♣r♦❞✉❝t ❚❤❡

❞♦t ♣r♦❞✉❝t

♦❢ t✇♦ ✈❡❝t♦rs < a, b > ❛♥❞ < c, d > ✐s ❞❡✜♥❡❞ ❛s < a, b > · < c, d >= ac + bd

❲❛r♥✐♥❣✦ ❖❢t❡♥✱ t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ ❞♦t ✏ ·✑ ❤❛s t♦ ❜❡ ❞❡t❡r✲ ♠✐♥❡❞ ❢r♦♠ t❤❡ ❝♦♥t❡①t✳

❊①❡r❝✐s❡ ✺✳✽✳✷✸ ❙❤♦✇ t❤❛t t❤❡ ❞♦t ♣r♦❞✉❝t ✐s ❧✐♥❦❡❞ ❜❛❝❦ t♦ t❤❡ ♠❛❣♥✐t✉❞❡ ❜② t❤❡ ❢♦r♠✉❧❛✿ || < a, b > ||2 =< a, b > · < a, b > .

❚❤❡ ♥✉♠❡r❛t♦r ♦❢ t❤❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❛♥❣❧❡ cos(α − β) t❛❦❡s ❛ s✐♠♣❧❡r ❢♦r♠ ♥♦✇✿ < x, y > · < x′ , y ′ > .

❲❡ ❝♦♥❝❧✉❞❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿

❚❤❡♦r❡♠ ✺✳✽✳✷✹✿ ❉✐r❡❝t✐♦♥s ❢♦r ❉✐♠❡♥s✐♦♥ ✷ ❚❤❡ ❛♥❣❧❡ ❜❡t✇❡❡♥ t❤❡ ✈❡❝t♦rs

(x′ , y ′ ) 6= O✱

OP

❛♥❞

OQ✱

✇❤❡r❡

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

[ = cos QOP

P = (x, y) 6= O

< x, y > · < x′ , y ′ > || < x, y > || || < x′ , y ′ > ||

❛♥❞

Q=

✺✳✽✳ ❚❤❡ ❊✉❝❧✐❞❡❛♥ ♣❧❛♥❡✿ ❛♥❣❧❡s

✹✼✸

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

❊①❡r❝✐s❡ ✺✳✽✳✷✺ ❊①♣❧❛✐♥ t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ ✏t❤❡ ❛♥❣❧❡ ❜❡t✇❡❡♥ t✇♦ ✈❡❝t♦rs✑ ❛♥❞ ✏t❤❡ ❛♥❣❧❡ ❜❡t✇❡❡♥ t✇♦ ❧✐♥❡s✑✳

❊①❡r❝✐s❡ ✺✳✽✳✷✻ ❲❤❛t ✐s t❤❡ ❛♥❣❧❡ ❜❡t✇❡❡♥ t❤❡ ❧✐♥❡s r❡♣❡❛t✳

y = −2x + 3

❛♥❞

y = x − 1❄

❙✉❣❣❡st ❛♥♦t❤❡r ♣❛✐r ♦❢ ❧✐♥❡s ❛♥❞

❊①❡r❝✐s❡ ✺✳✽✳✷✼ ❋✐♥❞ ❛ ❧✐♥❡ t❤❛t ♠❛❦❡s ❛

30✲❞❡❣r❡❡

❛♥❣❧❡ ✇✐t❤ t❤❡ ❧✐♥❡

❛♥♦t❤❡r ❧✐♥❡ ❛♥❞ r❡♣❡❛t✳

y = −2x + 3❄

❙✉❣❣❡st ❛♥♦t❤❡r ❛♥❣❧❡ ❛♥❞

❊①❡r❝✐s❡ ✺✳✽✳✷✽ ❉❡r✐✈❡ ❛ ❢♦r♠✉❧❛ ❢♦r t❤❡ s✐♥❡ ♦❢ t❤✐s ❛♥❣❧❡✳

❊①❛♠♣❧❡ ✺✳✽✳✷✾✿ ❛♥❣❧❡ ✇✐t❤ ✐ts❡❧❢ ❲❤❡♥ t✇♦ ✈❡❝t♦rs ❛r❡ ❡q✉❛❧ t♦ ❡❛❝❤ ♦t❤❡r✱ ✇❡ ❤❛✈❡ ❢r♦♠ t❤❡ t❤❡♦r❡♠✿

cos P[ OP = ❚❤❡r❡❢♦r❡✱

P[ OP = 0✱

xx + yy < x, y > · < x, y > p = 1. =p 2 || < x, y > || || < x, y > || x + y 2 x2 + y 2

❛s ❡①♣❡❝t❡❞✳

❊①❛♠♣❧❡ ✺✳✽✳✸✵✿ ❛♥❣❧❡ ✇✐t❤✐♥

x✲❛①✐s

❲❤❡♥ t❤❡ t❡r♠✐♥❛❧ ♣♦✐♥ts ♦❢ ❜♦t❤ ✈❡❝t♦rs ❧✐❡ ♦♥ t❤❡ ❢♦❧❧♦✇✐♥❣✿

[ = cos QOP ❚❤❡r❡ ❛r❡ ♦♥❧② t✇♦ ♣♦ss✐❜✐❧✐t✐❡s ❤❡r❡✱

x✲❛①✐s✱

✐✳❡✳✱

y = y ′ = 0✱

t❤❡ ❢♦r♠✉❧❛ t✉r♥s ✐♥t♦ t❤❡

x x′ xx′ = = sign(x) · sign(x′ ) . ′ ′ |x| |x | |x| |x |

[ 1 ♦r −1✱ ❛♥❞✱ t❤❡r❡❢♦r❡✱ QOP

❝❛♥ ♦♥❧② ❜❡ ❡✐t❤❡r

0 ♦r 180 ❞❡❣r❡❡s✳

❊①❡r❝✐s❡ ✺✳✽✳✸✶ ❙❤♦✇ ❤♦✇ t❤✐s ❢❛❝t ❞❡♠♦♥str❛t❡s t❤❡ t❤❡♦r❡♠ ❛❜♦✉t t❤❡ ❞✐r❡❝t✐♦♥s ✐♥ ❞✐♠❡♥s✐♦♥

1✳

❆ ❝❛s❡ s♣❡❝✐❛❧ ✐♠♣♦rt❛♥❝❡ ✐s✿ ❲❤❡♥ ❛r❡ t✇♦ ✈❡❝t♦rs ♦r t✇♦ ❧✐♥❡s ♣❡r♣❡♥❞✐❝✉❧❛r ❄ ❆♥ ❡①❛♠♣❧❡ ♦❢ t❤❡s❡ t✇♦ ❧✐♥❡s✱

y = 2x

❛♥❞

1 y = − x✱ 2

s✉❣❣❡sts t❤❛t t❤❡ s❧♦♣❡s ✇✐❧❧ ❤❛✈❡ t♦ ❜❡ ♥❡❣❛t✐✈❡ r❡❝✐♣r♦❝❛❧s ♦❢ ❡❛❝❤ ♦t❤❡r✿

▲❡t✬s ♣r♦✈❡ t❤✐s ❢❛❝t ✉s✐♥❣ t❤❡ t❤❡♦r❡♠✳ ❲❡ ❤❛✈❡

0 = cos π/2 =

< x, y > · < x′ , y ′ > . || < x, y > || || < x′ , y ′ > ||

✺✳✽✳ ❚❤❡ ❊✉❝❧✐❞❡❛♥ ♣❧❛♥❡✿ ❛♥❣❧❡s

✹✼✹

❚❤❡r❡❢♦r❡✱

y y′ · = −1 . < x, y > · < x , y >= 0 ⇐⇒ xx + yy = 0 ⇐⇒ xx = −yy ⇐⇒ x x′ ′

′

′

′

′

′

❇✉t t❤❡s❡ t✇♦ ❡①♣r❡ss✐♦♥s ❛r❡ t❤❡ s❧♦♣❡s ♦❢ t❤❡ ❧✐♥❡s✦

❚❤❡♦r❡♠ ✺✳✽✳✸✷✿ ❙❧♦♣❡s ♦❢ P❡r♣❡♥❞✐❝✉❧❛r ▲✐♥❡s ❚✇♦ ❧✐♥❡s ✇✐t❤ s❧♦♣❡s

m

❛♥❞

m′

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

♥❡❣❛t✐✈❡ r❡❝✐♣r♦❝❛❧s ♦❢ ❡❛❝❤ ♦t❤❡r❀ ✐✳❡✳✱

mm′ = −1 . ❊①❡r❝✐s❡ ✺✳✽✳✸✸ ❙♣❧✐t t❤❡ t❤❡♦r❡♠ ✐♥t♦ ❛ st❛t❡♠❡♥t ❛♥❞ ✐ts ❝♦♥✈❡rs❡✳

❙✐♥❝❡ ❛♥② ✈❡rt✐❝❛❧ ❧✐♥❡ ✐s ♣❡r♣❡♥❞✐❝✉❧❛r t♦ ❛♥② ❤♦r✐③♦♥t❛❧ ❧✐♥❡ ❛♥❞ ✈✐❝❡ ✈❡rs❛✱ ✇❡ ❤❛✈❡ s♦❧✈❡❞ t❤❡ ♣r♦❜❧❡♠ ♦❢ ♣❡r♣❡♥❞✐❝✉❧❛r✐t②✳ ❚❤✐s ✐s t❤❡ s✉♠♠❛r②✿

❊①❡r❝✐s❡ ✺✳✽✳✸✹ ❋✐♥❞ t❤❡ ❧✐♥❡ ♣❡r♣❡♥❞✐❝✉❧❛r t♦ r❡♣❡❛t✳

y = −3x t❤❛t ♣❛ss❡s t❤r♦✉❣❤ t❤❡ ♣♦✐♥t (1, 1)✳

❙✉❣❣❡st ❛♥♦t❤❡r ❧✐♥❡ ❛♥❞

❚❤❡ t❤❡♦r② ♦❢ ✈❡❝t♦rs ✐s ❢✉rt❤❡r ❞❡✈❡❧♦♣❡❞ ✐♥ ❈❤❛♣t❡r ✹❍❉✲✶✳ ❚❤❡ ❛①❡s ❝♦♥t❛✐♥ ♦r✐❡♥t❡❞ s❡❣♠❡♥ts✳ ❚❤❡s❡ s❡❣♠❡♥ts t❤❡♥ ❢♦r♠ ♦r✐❡♥t❡❞ r❡❝t❛♥❣❧❡s ✐♥ t❤❡ ♣❧❛♥❡✿

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

❙♦✱ s✐♥❝❡ ❛ ♣✐❡❝❡ ♦❢ ❢❛❜r✐❝ ❤❛s t✇♦ s✐❞❡s✱ ✐♥s✐❞❡ ❛♥❞ ♦✉ts✐❞❡✱ ✐t ❝❛♥ ❜❡ ♣❧❛❝❡❞ ♦♥ t❤❡ t❛❜❧❡ ✐♥ t✇♦ ✇❛②s✳ ❚❤✐s ✐❞❡❛ ✐s ❢✉rt❤❡r ❞❡✈❡❧♦♣❡❞ ✐♥ ❈❤❛♣t❡r ✹❍❉✲✺✳

✺✳✾✳

❋r♦♠ ❣❡♦♠❡tr② t♦ ❝❛❧❝✉❧✉s

✹✼✺

✺✳✾✳ ❋r♦♠ ❣❡♦♠❡tr② t♦ ❝❛❧❝✉❧✉s

❚❤❡r❡ ❛r❡ t✇♦ ♠❛✐♥ ❡♥tr② ♣♦✐♥ts t♦ ❝❛❧❝✉❧✉s✳ ❚❤❡ ✜rst ♦♥❡ ✐s ✈✐❛ t❤❡ st✉❞② ♦❢ ✐♥ ❈❤❛♣t❡r ✶✳ ❚❤❡ s❡❝♦♥❞ ♣❛t❤ ✐s ✈✐❛

❣❡♦♠❡tr②✳

♠♦t✐♦♥✳

❲❡ ❢♦❧❧♦✇❡❞ t❤✐s ♣❛t❤

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

✇✐❧❧

•

❝♦♠♣✉t❡

s❧♦♣❡s

•

❝♦♠♣✉t❡

❛r❡❛s

♦❢ ❝✉r✈❡s ✉s✐♥❣

❞✐✛❡r❡♥❝❡s✱ ❛♥❞

♦❢ ❝✉r✈❡❞ r❡❣✐♦♥s ✉s✐♥❣

s✉♠s✳

▲❡t✬s ❤❛✈❡ ❛ ♣r❡✈✐❡✇✳ ❋✐rst✱ ✐♥ ✇❤❛t ❞✐r❡❝t✐♦♥ ✇✐❧❧ ❧✐❣❤t ❜♦✉♥❝❡ ♦✛ ❛ ❝✉r✈❡❞ ♠✐rr♦r❄ ❲❡ ❝❛♥ ❛♥s✇❡r t❤❡ q✉❡st✐♦♥ ✐❢ ✇❡ ❦♥♦✇ t❤❡ ❛♥❣❧❡ ♦❢ t❤❡ ♠✐rr♦r ❛t ❡✈❡r② ❧♦❝❛t✐♦♥✿

▲❡t✬s ③♦♦♠ ✐♥ ♦♥ t❤❡ ♣♦✐♥t ♦❢ ❝♦♥t❛❝t✳ ❲❡ ♠✐❣❤t s❡❡ ❛ ❝✉r✈❡ ♠❛❞❡ ♦❢ ❞♦ts✱ s✐♠✐❧❛r t♦ ✇❤❛t ✇❡ s❛✇ ✐♥ ❈❤❛♣t❡r ✶✳ ❊✈❡r② ❛❞❥❛❝❡♥t ♣❛✐r ♦❢ ♣♦✐♥ts ❣✐✈❡s ✉s ❛ ❧✐♥❡✱ ❝❛❧❧❡❞ t❤❡

❚❤❡♥ t❤❡

s❧♦♣❡ ♦❢ t❤❡ s❡❝❛♥t ❧✐♥❡

✐s s❡❡♥ t♦ ❜❡✱ ❡①❛❝t❧② ♦r ❛♣♣r♦①✐♠❛t❡❧②✱ t❤❡ s❧♦♣❡ ♦❢ t❤❡ ❝✉r✈❡✳

●❡♥❡r❛❧❧②✱ s✉♣♣♦s❡ ✇❡ ❤❛✈❡ t✇♦ ♣♦✐♥ts ♦♥ t❤❡

xy ✲♣❧❛♥❡✿

(x1 , y1 ) ❚❤❡

s❧♦♣❡

s❡❝❛♥t ❧✐♥❡ ✿

❛♥❞

(x2 , y2 ) .

♦❢ t❤❡ ❧✐♥❡ ❜❡t✇❡❡♥ t❤❡♠ ✐s ❦♥♦✇♥ t♦ ❜❡ ✏r✐s❡ ♦✈❡r t❤❡ r✉♥✑✿ s❧♦♣❡

=

y2 − y1 . x2 − x1

✺✳✾✳ ❋r♦♠ ❣❡♦♠❡tr② t♦ ❝❛❧❝✉❧✉s

✹✼✻

❖❢ ❝♦✉rs❡✱ ✐❢ ✇❡ ❤❛✈❡ ♠♦r❡ ♣♦✐♥ts✱ ✇❡ ❤❛✈❡ ♠♦r❡ s❧♦♣❡s✳ ❙✉♣♣♦s❡ ❛ ❝✉r✈❡ ♦♥ t❤❡ xy ✲♣❧❛♥❡ ✐s ❝r❡❛t❡❞ ❢r♦♠ t❤❡s❡ s❡q✉❡♥❝❡s✿ • xn ♦♥ t❤❡ x✲❛①✐s✱

• yn ♦♥ t❤❡ y ✲❛①✐s✱ ❛♥❞

• (xn , yn ) ♦♥ t❤❡ ♣❧❛♥❡✳

❚❤❡ s❧♦♣❡s ♦❢ t❤❡ ❝✉r✈❡ ❛r❡ ❢♦✉♥❞ ❛s ❢♦❧❧♦✇s✳ ❘❡❝❛❧❧ t❤❡ ❞❡✜♥✐t✐♦♥ ❢r♦♠ ❈❤❛♣t❡r ✶✿ ■❢ xn ❛♥❞ yn ❛r❡ t✇♦ s❡q✉❡♥❝❡s✱ t❤❡♥ t❤❡✐r ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ yn ♦✈❡r t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ xn ❀ ✐✳❡✳✱ ∆yn yn+1 − yn = . ∆xn xn+1 − xn

■♥❞❡❡❞✱ t❤❡ ♥✉♠❡r❛t♦r✱ t❤❡ r✐s❡✱ ✐s t❤❡ ❝❤❛♥❣❡ ♦❢ y ✱ ✐✳❡✳✱ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ y ✿ ∆yn = yn+1 − yn .

❚❤❡ ❞❡♥♦♠✐♥❛t♦r✱ t❤❡ r✉♥✱ ✐s t❤❡ ❝❤❛♥❣❡ ♦❢ x✱ ✐✳❡✳✱ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ x✿ ∆xn = xn+1 − xn .

❋✉rt❤❡r♠♦r❡✱ r❡❝❛❧❧ t❤❡ ❞❡✜♥✐t✐♦♥ ❢r♦♠ ❈❤❛♣t❡r ✷✿ ■❢ y = f (x) ✐s ❛ ❢✉♥❝t✐♦♥ ❛♥❞ xn ✐s ❛ s❡q✉❡♥❝❡ ♦❢ ♣♦✐♥ts ✇✐t❤✐♥ ✐ts ❞♦♠❛✐♥✱ t❤❡♥ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ f ✇✐t❤ r❡s♣❡❝t t♦ t❤✐s s❡q✉❡♥❝❡ ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ s❡q✉❡♥❝❡ ♦❢ s❧♦♣❡s ♦❢ t❤❡ ❧✐♥❡s ❢r♦♠ (xn , f (xn )) t♦ (xn+1 , f (xn+1 )) ♦♥ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥✿ f (xn+1 ) − f (xn ) ∆f = ∆x xn+1 − xn

❚❤❡ s❡q✉❡♥❝❡ xn ✐s ♦❢t❡♥ ❝❤♦s❡♥ t♦ ❜❡ ❛♥ ❛r✐t❤♠❡t✐❝ ♣r♦❣r❡ss✐♦♥✳ ■♥ t❤❛t ❝❛s❡✱ ✐ts ❞✐✛❡r❡♥❝❡ ✐s t❤❡ ✐♥❝r❡♠❡♥t ♦❢ t❤❡ s❡q✉❡♥❝❡✿ h = ∆xn .

❚❤❡♥✱ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✐s s✐♠♣❧② ❛ ♠✉❧t✐♣❧❡ ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ yn = f (xn )✿
∆yn f (xn+1 ) − f (xn ) 1 f (xn+1 ) − f (xn ) . = = ∆xn h h ❊①❛♠♣❧❡ ✺✳✾✳✶✿ s❧♦♣❡s ♦❢ ♣❛r❛❜♦❧❛

❈♦♥s✐❞❡r ❛ ♣❛r❛❜♦❧❛✱ s❛②✱ y = f (x) = −(x − 1.5)2 + 3✳ ■t ✐s ❝♦♠♣✉t❡❞ ♦♥❡ ✈❛❧✉❡ ❛t ❛ t✐♠❡ ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ s♣r❡❛❞s❤❡❡t ❢♦r♠✉❧❛✿ ❂✭✲✶✮✯✭❘❈❬✲✶❪✲✶✳✺✮✂ ✷✰✸

❛♥❞ t❤❡♥ ♣❧♦tt❡❞ ♣♦✐♥t ❜② ♣♦✐♥t✿

✺✳✾✳

✹✼✼

❋r♦♠ ❣❡♦♠❡tr② t♦ ❝❛❧❝✉❧✉s

❋♦r ❡❛❝❤ t✇♦ ❝♦♥s❡❝✉t✐✈❡ ♣❛✐r ♦❢ ♣♦✐♥ts✱ t❤❡ s❧♦♣❡ ✐s ❢♦✉♥❞ ❜② t❤❡ ❢♦r♠✉❧❛✱ ❥✉st ❛s ✐♥ ❈❤❛♣t❡r ✶✿ ❂✭❘❈❬✲✷❪✲❘❬✲✶❪❈❬✲✷❪✮✴✭❘❈❬✲✶❪✲❘❬✲✶❪❈❬✲✶❪✮

❚❤❡ ❧✐♥❡ ✐s t❤❡♥ ❞r❛✇♥ ✈✐❛ t❤❡ ♣♦✐♥t✲s❧♦♣❡ ❛♥❞ ❞❡♥s❡r✿

✳ ■♥ s❡❛r❝❤ ♦❢ ❛ ♣❛tt❡r♥✱ ✇❡ ♠❛❦❡ t❤❡ ♣♦✐♥ts ❞❡♥s❡r

❢♦r♠✉❧❛

❆s ✇❡ ♣r♦❞✉❝❡ ♠♦r❡ ❛♥❞ ♠♦r❡ ♣♦✐♥ts ♦♥ t❤✐s ✐♥t❡r✈❛❧✱ ✇❡ r❡❛❧✐③❡ t❤❛t t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✐s ❛ str❛✐❣❤t ❧✐♥❡✦ ❊①❡r❝✐s❡ ✺✳✾✳✷

Pr♦✈❡ t❤❡ ❧❛st st❛t❡♠❡♥t✳ ❲❡ ❤❛✈❡ ❢♦❧❧♦✇❡❞ t❤✐s ♣❛tt❡r♥ s❡✈❡r❛❧ t✐♠❡s ✐♥ t❤❡ ♣❛st✳ ❲❡ ✇✐❧❧ 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✿

✺✳✾✳ ❋r♦♠ ❣❡♦♠❡tr② t♦ ❝❛❧❝✉❧✉s

✹✼✽

■t ❛♣♣❡❛rs t❤❛t t❤❡ ♣♦✇❡r ❣♦❡s ❞♦✇♥ ❜② ♦♥❡✦ ❊①❡r❝✐s❡ ✺✳✾✳✸

❙❦❡t❝❤ t❤❡ ♥❡①t ♣❛✐r✳ ❚❤❡ r❡st ♦❢ t❤❡ ❢✉♥❝t✐♦♥s ❛❧s♦ s❡❡♠ t♦ ❜❡ ❝♦♥♥❡❝t❡❞ t♦ ❡❛❝❤ ♦t❤❡r✿

❊①❡r❝✐s❡ ✺✳✾✳✹

❚r② t♦ ❛♥s✇❡r t❤❡s❡ q✉❡st✐♦♥s✳ ❊①❡r❝✐s❡ ✺✳✾✳✺

❯s❡ t❤❡ ❢♦r♠✉❧❛ ❛❜♦✈❡ t♦ ✜♥❞ ❛♥❞ ♣❧♦t t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❢♦r t❤❡ ❝✐r❝❧❡✳ ❚❤❡ ♣❛tt❡r♥s s✉❣❣❡st❡❞ ❜② t❤✐s ❞❛t❛ ❛♥❞ t❤❡s❡ ♣✐❝t✉r❡s ❛r❡ st✉❞✐❡❞ ❜② ❝❛❧❝✉❧✉s✳ ❇✉t✱ ❢✉rt❤❡r♠♦r❡✱ ❤♦✇ ❞♦ ✇❡ tr❡❛t ♠♦t✐♦♥ ✇❤❡♥ t❤❡ t✐♠❡ ✈❛r✐❡s ❝♦♥t✐♥✉♦✉s❧② ✐♥st❡❛❞ ♦❢ ✐♥❝r❡♠❡♥t❛❧❧②❄ ❲❤❛t ✐s t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ ✈❡❧♦❝✐t② t❤❡♥❄ ❈❛❧❝✉❧✉s ❛❧s♦ ❣✐✈❡s t❤❡ ❛♥s✇❡r ✭❈❤❛♣t❡r ✷❉❈✲✸✮✳ ❙❡❝♦♥❞✱ ❤♦✇ ❞♦ ✇❡ ✜♥❞ t❤❡ ❛r❡❛ ♦❢ ❛ ❝✉r✈❡❞ r❡❣✐♦♥❄

❲❤❡♥ ✇❡ ③♦♦♠ ✐♥✱ ✇❡ ♠✐❣❤t s❡❡ t❤❛t t❤❡ ❝✉r✈❡ ✐s ♠❛❞❡ ♦❢ ❛ s❡q✉❡♥❝❡ ♦❢ ♣♦✐♥ts✱ ❥✉st ❛s ❛❜♦✈❡✳ ❚❤❡ ❝✉r✈❡ ♦♥ t❤❡ xy ✲♣❧❛♥❡ ✐s ❝r❡❛t❡❞ ❜② t✇♦ s❡q✉❡♥❝❡s✱ (xn , yn )✳ ❆t ✐ts s✐♠♣❧❡st✱ t❤❡ r❡❣✐♦♥ ✐s ♠❛❞❡ ♦❢ r❡❝t❛♥❣❧❡s ✿

✺✳✾✳ ❋r♦♠ ❣❡♦♠❡tr② t♦ ❝❛❧❝✉❧✉s

✹✼✾

❚❤❡ ❛r❡❛ ♦❢ ❡❛❝❤ ♦❢ t❤❡♠ ✐s t❤❡ ✇✐❞t❤ t✐♠❡s t❤❡ ❤❡✐❣❤t✱ ✇❤✐❝❤ ✐s t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❛t t❤❛t x✿ ∆xn · yn = ∆xn · f (xn ) .

◆♦✇✱ t❤❡ t♦t❛❧ ❛r❡❛ ✐s t❤❡ s✉♠ ♦❢ t❤❡ ❛r❡❛s ♦❢ t❤❡ r❡❝t❛♥❣❧❡s❀ ✇❡ ✉s❡ t❤❡ s✉♠s ♦❢ s❡q✉❡♥❝❡s ❥✉st ❛s ✐♥ ❈❤❛♣t❡r ✶✿ A=

m X n=1

∆xn · f (xn ) =

m X

f (xn ) ∆xn

n=1

■t ✐s ❝❛❧❧❡❞ t❤❡ ❘✐❡♠❛♥♥ s✉♠✳ ❚❤❡ s❡q✉❡♥❝❡ xn ✐s ♦❢t❡♥ ❝❤♦s❡♥ t♦ ❜❡ ❛♥ ❛r✐t❤♠❡t✐❝ ♣r♦❣r❡ss✐♦♥✳ ■♥ t❤❛t ❝❛s❡✱ ✐ts ❞✐✛❡r❡♥❝❡ ✐s t❤❡ ✐♥❝r❡♠❡♥t ♦❢ t❤❡ s❡q✉❡♥❝❡✿ h = ∆xn .

❚❤❡♥✱ t❤❡ ❘✐❡♠❛♥♥ s✉♠ ✐s s✐♠♣❧② ❛ ♠✉❧t✐♣❧❡ ♦❢ t❤❡ s✉♠ ♦❢ t❤❡ s❡q✉❡♥❝❡ yn = f (xn )✿ A=

m X

f (xn ) ∆xn = h

n=1

m X

f (xn ) .

n=1

❊①❛♠♣❧❡ ✺✳✾✳✻✿ ❛r❡❛ ♦❢ ❝✐r❝❧❡

❲❡ ❦♥♦✇ t❤❛t t❤❡ ❛r❡❛ ♦❢ ❛ ❝✐r❝❧❡ ♦❢ r❛❞✐✉s r ✐s s✉♣♣♦s❡❞ t♦ ❜❡ A = πr2 ✳ ▲❡t✬s tr② t♦ ❝♦♥✜r♠ t❤✐s ✇✐t❤ ♥♦t❤✐♥❣ ❜✉t ❛ s♣r❡❛❞s❤❡❡t✳ ❋✐rst✱ ✇❡ ♣❧♦t t❤❡ ❣r❛♣❤ y=

√

1 − x2 ,

❜② ❧❡tt✐♥❣ t❤❡ ✈❛❧✉❡s ♦❢ x r✉♥ ❢r♦♠ −1 t♦ 1 ❡✈❡r② .1 ❛♥❞ ✜♥❞✐♥❣ t❤❡ ✈❛❧✉❡s ♦❢ y ✇✐t❤ t❤❡ s♣r❡❛❞s❤❡❡t ❢♦r♠✉❧❛✿ ❂❙◗❘❚✭✶✲❘❈❬✲✷❪✂ ✷✮

❲❡ ♣❧♦t t❤❡s❡ 20 ♣♦✐♥ts❀ t❤❡ r❡s✉❧t ✐s ❛ ❤❛❧❢✲❝✐r❝❧❡✿

❲❡ ♥❡①t ❝♦✈❡r✱ ❛s ❜❡st ✇❡ ❝❛♥✱ t❤✐s ❤❛❧❢✲❝✐r❝❧❡ ✇✐t❤ ✈❡rt✐❝❛❧ ❜❛rs t❤❛t st❛♥❞ ♦♥ t❤❡ ✐♥t❡r✈❛❧ [−1, 1]✳ ❲❡ r❡✲✉s❡ t❤❡ ❞❛t❛✿ ❚❤❡ ❜❛s❡s ♦❢ t❤❡ ❜❛rs ❛r❡ ♦✉r ✐♥t❡r✈❛❧s ✐♥ t❤❡ x✲❛①✐s✱ ❛♥❞ t❤❡ ❤❡✐❣❤ts ❛r❡ t❤❡ ✈❛❧✉❡s ♦❢ y ✳ ❚♦ s❡❡ t❤❡ ❜❛rs✱ ✇❡ s✐♠♣❧② ❝❤❛♥❣❡ t❤❡ t②♣❡ ♦❢ t❤❡ ❝❤❛rt ♣❧♦tt❡❞ ❜② t❤❡ s♣r❡❛❞s❤❡❡t✿

✺✳✶✵✳ ❙♦❧✈✐♥❣ ✐♥❡q✉❛❧✐t✐❡s

✹✽✵

❚❤❡♥✱ t❤❡ ❛r❡❛ ♦❢ t❤❡ ❝✐r❝❧❡ ✐s ❛♣♣r♦①✐♠❛t❡❞ ❜② t❤❡ s✉♠ ♦❢ t❤❡ ❛r❡❛s ♦❢ t❤❡ ❜❛rs✦ ❚♦ ❧❡t t❤❡ s♣r❡❛❞s❤❡❡t ❞♦ t❤❡ ✇♦r❦ ❢♦r ✉s✿ ▼✉❧t✐♣❧② t❤❡ ❤❡✐❣❤ts ❜② t❤❡ ✭❝♦♥st❛♥t✮ ✇✐❞t❤s ✐♥ t❤❡ ♥❡①t ❝♦❧✉♠♥ ❛♥❞ ❛❞❞ t❤❡♠ ✉♣✳ ❲❡ ✉s❡ t❤❡ ❢♦r♠✉❧❛s ❢♦r s✉♠s ❥✉st ❛s ✐♥ ❈❤❛♣t❡r ✶✿

❂❘❬✲✶❪❈✰❘❈❬✲✶❪✯✭❘❈❬✲✷❪✲❘❬✲✶❪❈❬✲✷❪✮ ❚❤❡ r❡s✉❧t ♣r♦❞✉❝❡❞ ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✿ ❆♣♣r♦①✐♠❛t❡ ❛r❡❛ ♦❢ t❤❡ s❡♠✐❝✐r❝❧❡ = 1.552 . ■t ✐s ❝❧♦s❡ t♦ ✇❤❛t ✇❡ ❦♥♦✇✳ ■♥ s✉♠♠❛r②✱ t❤❡ ❛r❡❛ A ✐s t❤❡ s✉♠ ♦❢ t❤❡ s❡q✉❡♥❝❡✿ p an = 1 − t2n · .1 , n = 1, 2, ..., 20 . ■♥ ♦t❤❡r ✇♦r❞s✿

A=

20 X n=1

p .1 · 1 − t2n =

20 X p 1 − t2n n=1

!

· .1 .

❚❤❡ t❤❡♦r❡t✐❝❛❧ r❡s✉❧t ✇✐❧❧ ❜❡ ♣r♦✈❡♥ ✐♥ ❈❤❛♣t❡r ✸■❈✲✶✳ ❊①❡r❝✐s❡ ✺✳✾✳✼

❯s❡ t❤❡ ❢♦r♠✉❧❛ ❛❜♦✈❡ t♦ ✜♥❞ t❤❡ ❛r❡❛ ✉♥❞❡r t❤❡ ♣❛r❛❜♦❧❛ ✐♥ t❤❡ ✜rst ❡①❛♠♣❧❡✳

✺✳✶✵✳ ❙♦❧✈✐♥❣ ✐♥❡q✉❛❧✐t✐❡s

❆s ❡①♣❧❛✐♥❡❞ ✐♥ ❈❤❛♣t❡r ✷✱ t♦ s♦❧✈❡ ❛♥ ✐♥❡q✉❛❧✐t② ✇✐t❤ r❡s♣❡❝t t♦ x ♠❡❛♥s t❤❡ s❛♠❡ ❛s s♦❧✈✐♥❣ ❛♥ ❡q✉❛t✐♦♥ ✕ t♦ ✜♥❞ ❛❧❧ ✈❛❧✉❡s ♦❢ x t❤❛t✱ ✇❤❡♥ s✉❜st✐t✉t❡❞✱ ♣r♦❞✉❝❡ ❛ tr✉❡ st❛t❡♠❡♥t✳ ❚❤❡s❡ ♥✉♠❜❡rs ❢♦r♠ ❛ s❡t ❝❛❧❧❡❞ t❤❡ s♦❧✉t✐♦♥ s❡t✳ ❚❤❡r❡ ✐s ❛ ❞✐✛❡r❡♥❝❡✱ ♦❢ ❝♦✉rs❡❀ ✐♥ t❤❡ ❝❛s❡ ♦❢ ❛♥ ❡q✉❛t✐♦♥✱ s✉❝❤ ❛ st❛t❡♠❡♥t ✐s ❧✐❦❡❧② t♦ ❜❡

0 = 0, 0 = 1, 100 = 100, ❡t❝✳ ♦r s✐♠✐❧❛r✱ ✇❤✐❧❡ ✐♥ t❤❡ ❝❛s❡ ♦❢ ❛♥ ✐♥❡q✉❛❧✐t②✱ t❤❡r❡ ♠✐❣❤t ❜❡ ❛ ✈❛r✐❡t② ♦❢ t❤❡s❡✿

0 < 1, 0 > 1, 0 < 100, 100 ≤ 100, ❡t❝✳ ❈♦♥s✐❞❡r ❤♦✇ t❤✐s ✐♥❡q✉❛❧✐t② ✐s s♦❧✈❡❞✿

x + 2 ≥ 5 =⇒ x ≥ 3 . ❚❤❛t✬s ❛♥ ❛❜❜r❡✈✐❛t❡❞ ✈❡rs✐♦♥ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ st❛t❡♠❡♥t✿

◮ ■❢ x s❛t✐s✜❡s t❤❡ ✐♥❡q✉❛❧✐t② x + 2 ≥ 5✱ t❤❡♥ x s❛t✐s✜❡s t❤❡ ✐♥❡q✉❛❧✐t② x ≥ 3✳

P❧✉❣ ✐♥ s♦♠❡ ✈❛❧✉❡s ❢♦r x ❛♥❞ s❡❡ ✐❢ t❤❡② ❝❤❡❝❦ ♦✉t✿

x=0 x=1 x=2 x=3 x=4

(x) + 2 = (0) + 2 = 2 (1) + 2 = 3 (2) + 2 = 4 (3) + 2 = 5 (4) + 2 = 6

≥? 5 6≥ 5 6≥ 5 6≥ 5 ≥5 ≥5

❚❘❯❊✴❋❆▲❙❊ ❋❆▲❙❊ ❋❆▲❙❊ ❋❆▲❙❊ ❚❘❯❊ ❆❞❞ ✐t t♦ t❤❡ ❧✐st✦ ❚❘❯❊ ❆❞❞ ✐t t♦ t❤❡ ❧✐st✦

✺✳✶✵✳

✹✽✶

❙♦❧✈✐♥❣ ✐♥❡q✉❛❧✐t✐❡s

❖❢ ❝♦✉rs❡✱ t❤✐s tr✐❛❧✲❛♥❞✲❡rr♦r ♠❡t❤♦❞ ✐s ✉♥❢❡❛s✐❜❧❡ ❜❡❝❛✉s❡ t❤❡r❡ ❛r❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♣♦ss✐❜✐❧✐t✐❡s✳ ❘❡❝❛❧❧ t❤❡ ❜❛s✐❝ ♠❡t❤♦❞s✱ ✐✳❡✳✱ r✉❧❡s✱ ♦❢ ❤❛♥❞❧✐♥❣ ✐♥❡q✉❛❧✐t✐❡s✳ ❚❤❡♦r❡♠ ✺✳✶✵✳✶✿ ❇❛s✐❝ ❆❧❣❡❜r❛ ♦❢ ■♥❡q✉❛❧✐t✐❡s

•

▼✉❧t✐♣❧②✐♥❣ ❜♦t❤ s✐❞❡s ♦❢ ❛♥ ✐♥❡q✉❛❧✐t② ❜② ❛ ♣♦s✐t✐✈❡ ♥✉♠❜❡r ♣r❡s❡r✈❡s ✐t✿

a < b =⇒ ka < kb •

k > 0.

▼✉❧t✐♣❧②✐♥❣ ❜♦t❤ s✐❞❡s ♦❢ ❛♥ ✐♥❡q✉❛❧✐t② ❜② ❛ ♥❡❣❛t✐✈❡ ♥✉♠❜❡r r❡✈❡rs❡s ✐t✿

a < b =⇒ ka > kb •

❢♦r ❛♥②

❢♦r ❛♥②

k < 0.

❆❞❞✐♥❣ ❛♥② ♥✉♠❜❡r t♦ ❜♦t❤ s✐❞❡s ♦❢ ❛♥ ✐♥❡q✉❛❧✐t② ♣r❡s❡r✈❡s ✐t✿

a < b =⇒ a + s < b + s

❢♦r ❛♥②

s.

❲❛r♥✐♥❣✦ ▼✉❧t✐♣❧②✐♥❣ ❛♥ ✐♥❡q✉❛❧✐t② ❜②

0

✐s ♣♦✐♥t❧❡ss✳

❆s ❛ r❡♠✐♥❞❡r✱ ✐♥ t❤❡ s♣❡❝✐❛❧ ❝❛s❡ ✇❤❡♥ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ♦❢ t❤❡ ✐♥❡q✉❛❧✐t② ✐s ③❡r♦✱ t❤❡ s♦❧✉t✐♦♥ s❡t t♦ t❤✐s ✐♥❡q✉❛❧✐t② ❤❛s ❛ ❝❧❡❛r ❣❡♦♠❡tr✐❝ ♠❡❛♥✐♥❣✿ ■t ✐s ❝♦♠♣r✐s❡❞ ♦❢ t❤❡ ✐♥t❡r✈❛❧s ❝✉t ❜② t❤❡ x✲✐♥t❡r❝❡♣ts ✿

❊①❡r❝✐s❡ ✺✳✶✵✳✷

❚❤❡s❡ s♦❧✉t✐♦♥ s❡ts ❛r❡ t❤❡ ✐♥t❡rs❡❝t✐♦♥s ♦❢ ❴❴❴❴❴ ✇✐t❤ t❤❡ x✲❛①✐s✳ ❊①❛♠♣❧❡ ✺✳✶✵✳✸✿ ❝♦✉♥t✐♥❣ ✐♥t❡r✈❛❧s

❖✉r ❦♥♦✇❧❡❞❣❡ ♦❢ t❤❡ s❤❛♣❡ ♦❢ t❤❡ ❣r❛♣❤ ♦❢ ❛ ♣❛rt✐❝✉❧❛r ❢✉♥❝t✐♦♥ ✇✐❧❧ t❡❧❧ ✉s ❤♦✇ ♠❛♥② s♦❧✉t✐♦♥s s✉❝❤ ❛♥ ✐♥❡q✉❛❧✐t② ♠✐❣❤t ❤❛✈❡✳ ❋♦r ❡①❛♠♣❧❡✱ ❛ q✉❛❞r❛t✐❝ ✐♥❡q✉❛❧✐t② ❝❛♥ ❤❛✈❡ t✇♦✱ ♦♥❡✱ ♦r ♥♦ ✐♥t❡r✈❛❧s✿

❆♥ ❡q✉❛t✐♦♥ ✇✐t❤ ❛♥ nt❤ ❞❡❣r❡❡ ♣♦❧②♥♦♠✐❛❧ ❝❛♥♥♦t ❤❛✈❡ ♠♦r❡ t❤❛♥ n + 1 ✐♥❡q✉❛❧✐t✐❡s✿

✺✳✶✵✳

❙♦❧✈✐♥❣ ✐♥❡q✉❛❧✐t✐❡s

✹✽✷

❆♥ ✐♥❡q✉❛❧✐t② ✇✐t❤ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥ ❝❛♥ ❤❛✈❡ ♦♥❡ ✐♥t❡r✈❛❧ ♦r ♥♦♥❡✿

❆♥ ✐♥❡q✉❛❧✐t② ✇✐t❤ ❛ ♣❡r✐♦❞✐❝ ❢✉♥❝t✐♦♥ ✭s✉❝❤ ❛s t❤❡ s✐♥❡✮ ❝❛♥ ❤❛✈❡ ✐♥✜♥✐t❡❧② ♠❛♥② ✐♥t❡r✈❛❧s ♦r ♥♦♥❡✿

❊①❡r❝✐s❡ ✺✳✶✵✳✹ ❙❦❡t❝❤ t❤❡ s♦❧✉t✐♦♥ s❡t ❢♦r t❤❡ ✐♥❡q✉❛❧✐t✐❡s

f (x) > 0

❛♥❞

f (x) < 0

✇✐t❤ t❤❡ ❢✉♥❝t✐♦♥s

f

❣✐✈❡♥ ✐♥ t❤❡

❧❛st ❡①❛♠♣❧❡✳

❤♦✇ ✇❡ ♠❛② ❛rr✐✈❡ t♦ t❤❡ ❛♥s✇❡r ✐s ❞✐s❝✉ss❡❞ ✐♥ t❤✐s s❡❝t✐♦♥✳ s♦❧✈✐♥❣ ❡q✉❛t✐♦♥s ♣r❡s❡♥t❡❞ ❡❛r❧✐❡r ✐♥ t❤✐s ❝❤❛♣t❡r✳

❆ ♠❡t❤♦❞ ♦❢

❚❤❡ ❛♥❛❧②s✐s ❢♦❧❧♦✇s t❤❡ ♦♥❡ ❢♦r

❲❡ ✇✐❧❧ ❛❞❞r❡ss ❛ s✐♠♣❧❡ ❦✐♥❞ ♦❢ ✐♥❡q✉❛❧✐t②✿

◮x

✐s ♣r❡s❡♥t ♦♥❧② ♦♥❝❡ ✭✐♥ t❤❡ ❧❡❢t✲❤❛♥❞ s✐❞❡✮✳

▲✐❦❡ t❤✐s✿

x2 ≥ 17 . ❙t❛rt✐♥❣ ✇✐t❤ s✉❝❤ ❛♥ ❡q✉❛t✐♦♥✱ ♦✉r ❣♦❛❧ ✐s ✕ t❤r♦✉❣❤ ❛ s❡r✐❡s ♦❢ ♠❛♥✐♣✉❧❛t✐♦♥s ✕ t♦ ❛rr✐✈❡ t♦ ❛♥ ❡✈❡♥ s✐♠♣❧❡r ❦✐♥❞ ♦❢ ✐♥❡q✉❛❧✐t②✿

◮x

✐s

✐s♦❧❛t❡❞

✭✐♥ t❤❡ ❧❡❢t✲❤❛♥❞ s✐❞❡✮✳

▲✐❦❡ t❤✐s✿

x≥ ❲❤❛t ✇❡ s❡❡ ❛❜♦✈❡ ✐s ❛ s✐♥❣❧❡ ❢✉♥❝t✐♦♥s ❛♣♣❧✐❡❞ t♦

x✳

x

√

17 .

✏✇r❛♣♣❡❞✑ ✐♥ s❡✈❡r❛❧ ❧❛②❡rs ♦❢ ❢✉♥❝t✐♦♥s✱ ✐✳❡✳✱ ❛ ❝♦♠♣♦s✐t✐♦♥ ♦❢ s❡✈❡r❛❧

❲❡ ✇✐❧❧ r❡♠♦✈❡ t❤❡s❡ ❧❛②❡rs ♦♥❡ ❜② ♦♥❡✱ ❢r♦♠ t❤❡ ♦✉ts✐❞❡ ✐♥✳

❲❛r♥✐♥❣✦ ❚❤✐s ♣❧❛♥ ❞♦❡s ♥♦t ✇♦r❦ ❢♦r ♠❛♥② ❢❛♠✐❧✐❛r t②♣❡s ♦❢ ✐♥❡q✉❛❧✐t✐❡s ❝♦♥s✐❞❡r❡❞ ✐♥ ❈❤❛♣t❡r ✹✳

❚❤❡ ♠❛✐♥ ✐❞❡❛ ✐s t❤❡ s❛♠❡ ❛s ❢♦r ❡q✉❛t✐♦♥s✿

✺✳✶✵✳ ❙♦❧✈✐♥❣ ✐♥❡q✉❛❧✐t✐❡s

✹✽✸

◮ ❲❡ ❛♣♣❧② ❛ ❢✉♥❝t✐♦♥ t♦ ❜♦t❤ s✐❞❡s ♦❢ t❤❡ ✐♥❡q✉❛❧✐t②✱ ♣r♦❞✉❝✐♥❣ ❛ ♥❡✇ ✐♥❡q✉❛❧✐t② ✇✐t❤ ❛ ♣♦ss✐❜❧②

r❡✈❡rs❡❞ s✐❣♥✳

❋♦r ❡①❛♠♣❧❡✱ ✐❢ ✇❡ ❤❛✈❡ ❛♥ ✐♥❡q✉❛❧✐t②✱ s❛②✱ x + 2 ≥ 5,

✇❡ tr❡❛t t❤❡ t✇♦ s✐❞❡s ♦❢ t❤❡ ✐♥❡q✉❛❧✐t✐❡s ❛s t✇♦ ✈❛❧✉❡s ♦❢ t❤❡ s❛♠❡ ✈❛r✐❛❜❧❡✱ s❛②✱ y ✳ ■❢ ✇❡ ❛❞❞ 2 t♦ ❜♦t❤✱ t❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡♠ r❡♠❛✐♥s✳ ▼♦r❡ ♣r❡❝✐s❡❧②✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿ x + 2 ≥ 5, ❛♣♣❧② z = y − 2 =⇒ (x + 2) − 2 ≥ 3 − 2 =⇒ x ≥ 3 .

❋r♦♠ ❛♥ ✐♥❡q✉❛❧✐t② s❛t✐s✜❡❞ ❜② x✱ ✇❡ ♣r♦❞✉❝❡ ❛♥♦t❤❡r ✐♥❡q✉❛❧✐t② s❛t✐s✜❡❞ ❜② x✳ ❚❤❡ s♦❧✉t✐♦♥ s❡t ✐s [3, +∞) ❍♦✇❡✈❡r✱ t❤❡ ❝❤❛❧❧❡♥❣❡ ✐s t❤❛t ❛♥② ❢✉♥❝t✐♦♥ ❛♣♣❧✐❡❞ t❤✐s ✇❛② ♠❛② ♣r♦❞✉❝❡ ❛ ♥❡✇ ✐♥❡q✉❛❧✐t②✦ ▼♦r❡ ♣r❡❝✐s❡❧②✱ ❛♥② ✐♥❝r❡❛s✐♥❣ ❢✉♥❝t✐♦♥ ✇✐❧❧ ❛✉t♦♠❛t✐❝❛❧❧② ♣r♦❞✉❝❡ ❛ ♥❡✇✱ ❝♦rr❡❝t✱ ✐♥❡q✉❛❧✐t②✿ x + 2 ≥ 5, ❛♣♣❧② z = y + 2 =⇒ (x + 2) + 2 ≥ 5 + 2 =⇒ x + 4 ≥ 7 . x + 2 ≥ 5, ❛♣♣❧② z = 7y =⇒ 7 · (x + 2) ≥ 7 · 5 . ◆♦t s♦❧✈❡❞✦ 3 3 3 x + 2 ≥ 5, ❛♣♣❧② z = y =⇒ (x + 2) ≥ 5 . ◆♦t s♦❧✈❡❞✦

❚❤✐s ✐s t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❛♥ ✐♥❝r❡❛s✐♥❣ ❢✉♥❝t✐♦♥✿ ▲❛r❣❡r ✐♥♣✉ts ✭❛♥ ✐♥❡q✉❛❧✐t②✮ ♣r♦❞✉❝❡✱ ✉♥❞❡r g ✱ ❧❛r❣❡r ♦✉t♣✉ts ✭❛♥♦t❤❡r ✐♥❡q✉❛❧✐t②✮✳ ❚❤❛t✬s ✇❤❛t ✇❡ ❤❛✈❡ ♦♥ t❤❡ ❧❡❢t✱ ❛♥❞ t❤❡ ♦♣♣♦s✐t❡ ♦♥ t❤❡ r✐❣❤t✿ ♦❧❞ ✐♥❡q✉❛❧✐t②✿

a ≥ b   g g gր y y ♥❡✇ ✐♥❡q✉❛❧✐t②✿ g(a) ≥ g(b)

♦❧❞ ✐♥❡q✉❛❧✐t②✿

a ≥ b   g g gց y y ♥❡✇ ✐♥❡q✉❛❧✐t②✿ g(a) ≤ g(b)

❙♦✱ ✇❡ ❤❛✈❡✱ ❢♦r ❛♥② ✐♥❝r❡❛s✐♥❣ ❢✉♥❝t✐♦♥ g ✿ x + 2 ≥ 5 =⇒ g(x + 2) ≥ g(5) .

❚❤❡ ♦♣♣♦s✐t❡ ✐s tr✉❡ ❢♦r ❛♥② ❞❡❝r❡❛s✐♥❣ ❢✉♥❝t✐♦♥ g ✿ x + 2 ≥ 5 =⇒ g(x + 2)≤g(5) .

❯♥❢♦rt✉♥❛t❡❧②✱ t❤❡r❡ ❛r❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♣♦ss✐❜✐❧✐t✐❡s ❢♦r t❤✐s ❢✉♥❝t✐♦♥ g ✿ (−2) · (x + 2)≤(−2) · 5 7 · (x + 2) ≥ 7 · 5 տ ↑ ր x+2≥5 → 2x+2 ≥ 25 x+5≥8 ← ւ ↓ ց √ √ 3 3 x+2≥ 5 x+6≥9 (x + 2) ≥ 5 x+4≥7

❋r♦♠ t❤❡s❡✱ t❤❡ ♦♥❧② ❞❡❝r❡❛s✐♥❣ ❢✉♥❝t✐♦♥ ✐s (−2)y ✳ ■❢ x s❛t✐s✜❡s t❤❡ ✐♥❡q✉❛❧✐t② ✐♥ t❤❡ ♠✐❞❞❧❡✱ ✐t ❛❧s♦ s❛t✐s✜❡s t❤❡ r❡st ♦❢ t❤❡ ✐♥❡q✉❛❧✐t✐❡s✳ ■❢ ✇❡ ✇❛♥t t♦ s♦❧✈❡ t❤❡ ♦r✐❣✐♥❛❧ ✐♥❡q✉❛❧✐t②✱ ✇❡ ✇✐❧❧ ♥❡❡❞ ❛ ❢♦r❡s✐❣❤t t♦ ❝❤♦♦s❡ ❛ ❢✉♥❝t✐♦♥ t♦ ❛♣♣❧② t❤❛t ✇✐❧❧ ♠❛❦❡ t❤❡ ✐♥❡q✉❛❧✐t② s✐♠♣❧❡r✳ ❏✉st ❛s ✇✐t❤ ❡q✉❛t✐♦♥s✱ ✐❢ ❜❡t✇❡❡♥ ✉s ❛♥❞ x t❤❡r❡ ✐s ❛ ❢✉♥❝t✐♦♥✱ ✇❡ r❡♠♦✈❡ ✐t ❜② ❛♣♣❧②✐♥❣ ✐ts ✐♥✈❡rs❡ t♦ t❤❡ ✐♥❡q✉❛❧✐t②✳ ❲❡ ✇♦r❦ ❢r♦♠ ♦✉ts✐❞❡ ✐♥ ✭✇❤✐❧❡ t❤❡ ❣✐❢t✲❣✐✈❡r ✇♦r❦❡❞ ❢r♦♠ t❤❡ ✐♥s✐❞❡ ♦✉t✮✿

✺✳✶✵✳ ❙♦❧✈✐♥❣ ✐♥❡q✉❛❧✐t✐❡s

✹✽✹

❊①❛♠♣❧❡ ✺✳✶✵✳✺✿ s♦❧✈✐♥❣ ✐♥❡q✉❛❧✐t②

❲❤❛t ✐❢ ✇❡ ❤❛✈❡ s❡✈❡r❛❧ ❢✉♥❝t✐♦♥s ❛♣♣❧✐❡❞ ❝♦♥s❡❝✉t✐✈❡❧② t♦ x❄ ❲❤✐❝❤ ❢✉♥❝t✐♦♥ ❞♦ ✇❡ ❝❤♦♦s❡ t♦ ❛♣♣❧②❄ ■♥ t❤❡ ✐♥❡q✉❛❧✐t②✱    x +1 2

5·

2

+ 3 − 17 ≥ 3 ,

t❤❡ ❧❛st ♦♣❡r❛t✐♦♥ ♦♥ t❤❡ ❧❡❢t ✐s −17✳ ❚❤❛t✬s t❤❡ ❢✉♥❝t✐♦♥ ✇❡ ❢❛❝❡✱ ❛♥❞ t❤❡ ✐♥✈❡rs❡ ♦❢ t❤✐s ❢✉♥❝t✐♦♥ ✐s t♦ ❜❡ ❛♣♣❧✐❡❞✳ ❲❡ ❝❤♦♦s❡✱ t❤❡r❡❢♦r❡✿ f (z) = z − 17 ,

✇❤❡r❡

z =5·

❚❤❡♥ ✇❡ ❛♣♣❧②

 x 2

+1

2

 +3 .

g(y) = f −1 (y) = y + 17 .

t♦ ❜♦t❤ s✐❞❡s✳ ❲❡ ❝♦♥❝❧✉❞❡✿ z − 17 ≥ 3 =⇒ (z − 17) + 17 ≥ 3 + 17 =⇒ z ≥ 20 .

❲❡ ❤❛✈❡ ❛ ♥❡✇ ✐♥❡q✉❛❧✐t② ♥♦✇✿

5·



x +1 2

!2

 + 3 ≥ 20 .

❆s ✇❡ ♣r♦❣r❡ss✱ ✇❡ ❛♣♣❧② t❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ t❤❛t ❛♣♣❡❛rs ✜rst ✐♥ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ♦❢ t❤❡ ✐♥❡q✉❛❧✐t②✳ ❲❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡q✉❡♥❝❡ ♦❢ st❡♣s r❡♠♦✈✐♥❣ ❢✉♥❝t✐♦♥s ♦♥❡ ❛t ❛ t✐♠❡✿ 5·



!2  x + 3 ≥ 20 =⇒ 5 · + 1 + 3 /5 ≥ 20/5 =⇒ 2 x 2 +1 +3−3≥4−3 =⇒ +3≥4 =⇒ 2

x +1 2 !2

x +1 2 x +1 2

!2

!2





≥ 1.

❚❤❡ ❧❛st ✐♥❡q✉❛❧✐t② ♣r♦❞✉❝❡s t✇♦ ❝❛s❡s ❞❡♣❡♥❞✐♥❣ ♦♥ ✇❤✐❝❤ ❜r❛♥❝❤ ♦❢ y = x2 ✇❡ ❝♦♥s✐❞❡r✿ x +1 2

!2

≥1

ւ z=

x +1≥0 2 !

x +1 2 x +1 2 x 2 x

ց

❖❘ ≥ 1 =⇒

❖❘

≥ 1 =⇒

❖❘

≥ 0 =⇒

❖❘

≥0

❖❘

x +1≤0 2 x  − +1 ≥1 2 z=

x +1 2 x 2

x

=⇒

≤ −1 =⇒ ≤ −2 =⇒ ≤ −4

❚❤❡ ✈❛❧✉❡s ♦❢ x✱ ❛♥❞ ♦♥❧② t❤❡②✱ t❤❛t ❜❡❧♦♥❣ t♦ [0, +∞) ♦r t♦ (−∞, −4] s❛t✐s❢② t❤✐s r❡str✐❝t✐♦♥✳ ❚❤❡r❡❢♦r❡✱ t❤❡ s♦❧✉t✐♦♥ s❡t ✐s t❤❡ ✉♥✐♦♥ ♦❢ t❤❡s❡ t✇♦ ✐♥t❡r✈❛❧s✿ (−∞, −4] ∪ [0, +∞) .

✺✳✶✵✳ ❙♦❧✈✐♥❣ ✐♥❡q✉❛❧✐t✐❡s

✹✽✺

❊①❡r❝✐s❡ ✺✳✶✵✳✻

❙♦❧✈❡ t❤❡ ✐♥❡q✉❛❧✐t②✿

5·

  x2

5·

 x

2

+1

2

 + 3 − 17 ≥ 3 .

❊①❡r❝✐s❡ ✺✳✶✵✳✼

❙♦❧✈❡ t❤❡ ✐♥❡q✉❛❧✐t②✿



2

+1

2



+ 3 − 17

▼❛❦❡ ✉♣ ②♦✉r ♦✇♥ ✐♥❡q✉❛❧✐t② ❛♥❞ s♦❧✈❡ ✐t✳ ❘❡♣❡❛t✳

3

≥ 27 .

❯s✐♥❣ ✐♥✈❡rt✐❜❧❡ ❢✉♥❝t✐♦♥s ❡♥s✉r❡s t❤❛t ✇❡ ✇✐❧❧ ❤❛✈❡ t❤❡ s❛♠❡ s♦❧✉t✐♦♥ s❡t ❛s ✇❡ ♣r♦❣r❡ss t❤r♦✉❣❤ t❤❡ st❛❣❡s✳ ❋♦r ❡①❛♠♣❧❡✱ t❤✐s ✐s ❤♦✇ ✇❡ ✇♦✉❧❞ r❛t❤❡r ♣r❡s❡♥t t❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ ✈❡r② ✜rst ✐♥❡q✉❛❧✐t② ✐♥ t❤✐s s❡❝t✐♦♥✿

x + 2 ≥ 5 ⇐⇒ x ≥ 3 . ❚❤❛t✬s ❛♥ ❛❜❜r❡✈✐❛t❡❞ ✈❡rs✐♦♥ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ st❛t❡♠❡♥t✿

x s❛t✐s✜❡s x + 2 ≥ 5 ✐❢ ❛♥❞ ♦♥❧② ✐❢ x s❛t✐s✜❡s x ≥ 3✳

❘❡♣❡❛t❡❞ ❛s ♠❛♥② t✐♠❡s ❛s ♥❡❝❡ss❛r②✱ t❤❡ s♦❧✉t✐♦♥ s❡t ♦❢ ❡❛❝❤ ✐♥❡q✉❛❧✐t② ✐s t❤❡ s❛♠❡ ❛s t❤❛t ♦❢ t❤❡ ♦r✐❣✐♥❛❧ ✐♥❡q✉❛❧✐t②✦ ❊①❛♠♣❧❡ ✺✳✶✵✳✽✿ s♣❧✐t ❞♦♠❛✐♥

❚✇♦ ❝♦♠♣❧❡t❡ s♦❧✉t✐♦♥s ❛r❡ ❜❡❧♦✇✿

(1) 2(x + 2) − 3 ≥ 5x ⇐⇒ 2x + 4 − 3 ≥ 5x ⇐⇒ −3x ≥ −1 ⇐⇒ x ≤ 1/3 . (2) x2 + 1 ≥ 0 ⇐⇒ x2 ≥ −1 ⇐⇒ R . ❙♦✱ t♦ s♦❧✈❡ t❤❡ t②♣❡ ♦❢ ✐♥❡q✉❛❧✐t② ✇❡ ❢❛❝❡ ✕ ❛ ✈❛r✐❛❜❧❡ x s✉❜❥❡❝t❡❞ t♦ ❛ s❡q✉❡♥❝❡ ♦❢ ❢✉♥❝t✐♦♥s ✕ ✇❡ ❛♣♣❧② t❤❡ ✐♥✈❡rs❡s ♦❢ t❤❡s❡ ❢✉♥❝t✐♦♥s ✐♥ t❤❡ r❡✈❡rs❡❞ ♦r❞❡r✳ ❲❡ ♠❛② ❤❛✈❡ t♦ s♣❧✐t t❤❡ ❞♦♠❛✐♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ s♦ t❤❛t ✐t ✐s ♦♥❡✲t♦✲♦♥❡ ♦♥ ❡❛❝❤ ♦❢ t❤❡ s✉❜s❡ts✳ ❆s ✇❡ ❤❛✈❡ s❡❡♥✱ t❤✐s st❡♣ ❛❧s♦ s♣❧✐ts ♦✉r ✐♥❡q✉❛❧✐t②✳ ❆s ❛ s✉♠♠❛r②✱ t❤✐s ✐s ♦✉r ♠❡t❤♦❞✳ ❚❤❡♦r❡♠ ✺✳✶✵✳✾✿ ●❡♥❡r❛❧ ❆❧❣❡❜r❛ ♦❢ ■♥❡q✉❛❧✐t✐❡s

•

■❢

g

✐s ❛ str✐❝t❧② ✐♥❝r❡❛s✐♥❣ ❢✉♥❝t✐♦♥✱ t❤❡♥ ❛♣♣❧②✐♥❣

g

t♦ ❜♦t❤ s✐❞❡s ♦❢ ❛♥

✐♥❡q✉❛❧✐t② ❝r❡❛t❡s ❛♥ ❡q✉✐✈❛❧❡♥t ✐♥❡q✉❛❧✐t②✿

a < b ⇐⇒ g(a) < g(a) . •

■❢

g

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

t♦ ❜♦t❤ s✐❞❡s ♦❢ ❛♥

✐♥❡q✉❛❧✐t② ❝r❡❛t❡s ❛♥ ❡q✉✐✈❛❧❡♥t ✭❜✉t r❡✈❡rs❡❞✮ ✐♥❡q✉❛❧✐t②✿

a < b ⇐⇒ g(a) > g(a) . ■♥ ❡✐t❤❡r ❝❛s❡✱ t❤❡ t✇♦ ✐♥❡q✉❛❧✐t✐❡s ❤❛✈❡ t❤❡ s❛♠❡ s♦❧✉t✐♦♥ s❡t✳ ❊①❛♠♣❧❡ ✺✳✶✵✳✶✵✿ s♣❧✐t ❞♦♠❛✐♥

❆ s♦❧✉t✐♦♥ ✐s ❜❡❧♦✇✿

g

x2 ≥ 1 ⇐⇒ x ≤ −1 ❖❘ x ≥ 1 .

✺✳✶✵✳

❙♦❧✈✐♥❣ ✐♥❡q✉❛❧✐t✐❡s

✹✽✻

❊①❡r❝✐s❡ ✺✳✶✵✳✶✶ ❙♦❧✈❡ t❤❡ ✐♥❡q✉❛❧✐t② ✐♥ t❤✐s ♠❛♥♥❡r✿

2x+1 ≥ 3 . ❊①❡r❝✐s❡ ✺✳✶✵✳✶✷ ❙♦❧✈❡ t❤❡ ✐♥❡q✉❛❧✐t② ✐♥ t❤✐s ♠❛♥♥❡r✿

2x

2 +1

≥ 3.

▼❛❦❡ ✉♣ ②♦✉r ♦✇♥ ✐♥❡q✉❛❧✐t② ❛♥❞ s♦❧✈❡ ✐t✳ ❘❡♣❡❛t✳

❊①❛♠♣❧❡ ✺✳✶✵✳✶✸✿ s♣❧✐t ❞♦♠❛✐♥ ■❢ ✇❡✱ ✐♥st❡❛❞ ♦❢ ❛ s✐♥❣❧❡

5· ❲❡ s✉❜st✐t✉t❡✿

❛♥❞ ❣♦ ❛❢t❡r

y



x✱

❢❛❝❡ ❛♥ ❡①♣r❡ss✐♦♥ t❤❛t ❞❡♣❡♥❞s ♦♥

x2 + x + 1


2

 + 3 − 17 ≥ 3 =⇒

x✱

❝❤♦♦s❡

✇❡ tr❡❛t ✐t ❛s ❥✉st ❛♥♦t❤❡r ✈❛r✐❛❜❧❡✿

y = x2 + x + 1 .

  5 · (y)2 + 3 − 17 ≥ 3 ,

❜② ❢♦❧❧♦✇✐♥❣ t❤❡ s❛♠❡ ♣❧❛♥ ❛s ❜❡❢♦r❡✳ ❖♥❝❡ t❤✐s ✐♥❡q✉❛❧✐t② ✐s s♦❧✈❡❞✱ ②♦✉ ❤❛✈❡ ❛ ♠✉❝❤

s✐♠♣❧❡r ✐♥❡q✉❛❧✐t②✱ ♦r ✐♥❡q✉❛❧✐t✐❡s✱ ❢♦r

x✿

x2 + x + 1 ≥ 1 ❖❘ x2 + x + 1 ≤ −4 . ❊①❡r❝✐s❡ ✺✳✶✵✳✶✹ ❋✐♥✐s❤ t❤❡ s♦❧✉t✐♦♥✳ ▼❛❦❡ ✉♣ ②♦✉r ♦✇♥ ✐♥❡q✉❛❧✐t② ❛♥❞ s♦❧✈❡ ✐t✳ ❘❡♣❡❛t✳

❈❤❛♣t❡r ✿

❊①❡r❝✐s❡s

❈♦♥t❡♥ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✽✼

✷ ❊①❡r❝✐s❡s✿ ❙❡q✉❡♥❝❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✾✵

✸ ❊①❡r❝✐s❡s✿ ❙❡ts ❛♥❞ ❧♦❣✐❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✾✶

✹ ❊①❡r❝✐s❡s✿ ❈♦♦r❞✐♥❛t❡ s②st❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✾✷

✺ ❊①❡r❝✐s❡s✿ ▲✐♥❡❛r ❛❧❣❡❜r❛

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✾✸

✻ ❊①❡r❝✐s❡s✿ P♦❧②♥♦♠✐❛❧s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✾✺

✼ ❊①❡r❝✐s❡s✿ ❘❡❧❛t✐♦♥s ❛♥❞ ❢✉♥❝t✐♦♥s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✾✼

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺✵✵

✾ ❊①❡r❝✐s❡s✿ ❈♦♠♣♦s✐t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺✵✷

✶✵ ❊①❡r❝✐s❡s✿ ❚r❛♥s❢♦r♠❛t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺✵✺

✶✶ ❊①❡r❝✐s❡s✿ ❇❛s✐❝ ♠♦❞❡❧s

✺✵✼

✶ ❊①❡r❝✐s❡s✿ ❆❧❣❡❜r❛

✽ ❊①❡r❝✐s❡s✿ ●r❛♣❤s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✳ ❊①❡r❝✐s❡s✿ ❆❧❣❡❜r❛

❊①❡r❝✐s❡ ✶✳✶ ❙♦❧✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉❛t✐♦♥✿

p x2 − 7 − 3 = 0✳

❊①❡r❝✐s❡ ✶✳✷ ❙♦❧✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉❛t✐♦♥✿

x2 − 7


3

− 8 = 0✳

❊①❡r❝✐s❡ ✶✳✹ ❊①❡r❝✐s❡ ✶✳✸ ❈♦♥s✐❞❡r t❤❡ ♣❛r❛❜♦❧❛ ❜❡❧♦✇✳

❋✐♥❞ t❤❡ ❡①❛❝t ✈❛❧✉❡s ♦❢ t❤❡ ❋✐♥❞ ✐ts ✈❡rt❡①✱ ✐ts

❛①✐s ♦❢ s②♠♠❡tr②✱ ❛♥❞ ✐ts ♠❛①✐♠✉♠ ♦r ♠✐♥✐♠✉♠✳

✐♥t❡rs❡❝t✐♦♥s s❤♦✇♥ ❜❡❧♦✇✿

❜❡t✇❡❡♥

t❤❡

x✲❝♦♦r❞✐♥❛t❡s

♣❛r❛❜♦❧❛

❛♥❞

♦❢ t❤❡

t❤❡

❧✐♥❡

✶✳ ❊①❡r❝✐s❡s✿ ❆❧❣❡❜r❛

✹✽✽

❊①❡r❝✐s❡ ✶✳✶✻

❙♦❧✈❡ t❤❡s❡ ❡q✉❛t✐♦♥s✿ x2 = x,

x = −x,

x = 0 · x.

❊①❡r❝✐s❡ ✶✳✶✼

❘❡♣r❡s❡♥t ❛s ❛ ♣♦✇❡r ♦❢ 5✿ 53 · .2, (125 · 5)5 . ❊①❡r❝✐s❡ ✶✳✺

❊①❡r❝✐s❡ ✶✳✶✽

❙♦❧✈❡ t❤❡ ❡q✉❛t✐♦♥✿ 2x = 3x+1 ✳ ❉♦♥✬t s✐♠♣❧✐❢②✳ ❊①❡r❝✐s❡ ✶✳✻

■❢ ❜❛❝t❡r✐❛ tr✐♣❧❡ ✐♥ ♥✉♠❜❡r ❡✈❡r② ❞❛② ❛♥❞ t❤❡ ❝✉r✲ r❡♥t ♣♦♣✉❧❛t✐♦♥ ✐s 9000✱ ❤♦✇ ♠❛♥② ✇❡r❡ t❤❡r❡ t❤r❡❡ ❞❛②s ❛❣♦❄

❙♦❧✈❡ t❤❡ ❡q✉❛t✐♦♥✿ 3x = 3x+1 ✳

❊①❡r❝✐s❡ ✶✳✶✾

❋❛❝t♦r 2a2 + 12ab + 18b2 ✳

❊①❡r❝✐s❡ ✶✳✼

❙♦❧✈❡ t❤❡ ❡q✉❛t✐♦♥✿ 3x = 2✳

❊①❡r❝✐s❡ ✶✳✷✵

❈♦♥tr❛❝t t❤✐s s✉♠♠❛t✐♦♥✿

❊①❡r❝✐s❡ ✶✳✽

2−

❙♦❧✈❡ t❤❡ ❡q✉❛t✐♦♥✿ 2x = 2 · 3x+1 ✳ ❉♦♥✬t s✐♠♣❧✐❢②✳

2 2 2 + − =? 2 3 4

❊①❡r❝✐s❡ ✶✳✾

❊①❡r❝✐s❡ ✶✳✷✶

❚♦ ✇❤❛t ♣♦✇❡r s❤♦✉❧❞ ②♦✉ r❛✐s❡ 3 t♦ ❣❡t 10❄

❊①♣❛♥❞ t❤✐s s✉♠♠❛t✐♦♥✿ 5 X

❊①❡r❝✐s❡ ✶✳✶✵

❋✐♥❞ t❤❡ ❞♦♠❛✐♥✱ t❤❡ r❛♥❣❡✱ ❛♥❞ t❤❡ ❛s②♠♣t♦t❡s ♦❢ t❤❡ ❢✉♥❝t✐♦♥ f (x) = ln(x − 3) + ln 3✳ ❊①❡r❝✐s❡ ✶✳✶✶

❚❤❡r❡ ❛r❡ 125 s❤❡❡♣ ❛♥❞ 5 ❞♦❣s ✐♥ ❛ ✢♦❝❦✳ ❍♦✇ ♦❧❞ ✐s t❤❡ s❤❡♣❤❡r❞❄

k=−1

k2 =? k+2

❊①❡r❝✐s❡ ✶✳✷✷

❚r✉❡ ♦r ❋❛❧s❡❄ ✭❛✮ x < r =⇒ |x| < r❀ ✭❜✮ x < r ⇐= |x| < r❀ ✭❝✮ x < r ⇐⇒ |x| < r✳

❊①❡r❝✐s❡ ✶✳✶✷

❊①❡r❝✐s❡ ✶✳✷✸

▲❡t h(x) = x + 3x − 10✳ ❋✐♥❞ t❤❡ x✲ ❛♥❞ y ✲ ✐♥t❡r❝❡♣ts ❛♥❞ s❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥✳

❋♦r 4y + 16x = 20✱ ♣r♦✈✐❞❡ t❤❡ s❧♦♣❡ ❛♥❞ t❤❡ y ✲ ✐♥t❡r❝❡♣t✳

❊①❡r❝✐s❡ ✶✳✶✸

❊①❡r❝✐s❡ ✶✳✷✹

❙✉❣❣❡st ❛♥ ❡q✉❛t✐♦♥ t❤❡ s♦❧✉t✐♦♥s ♦❢ ✇❤✐❝❤ ❛r❡ 1 ❛♥❞ 2✳

▲❡t f (x) = 4x2 + 2x + 2 ❛♥❞ ❧❡t

2

g(h) = ❊①❡r❝✐s❡ ✶✳✶✹

❊①♣❛♥❞✿

f (1 + h) − f (1) . h

❉❡t❡r♠✐♥❡ ❡❛❝❤ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣✿ (x + 1)5 = ...

✶✳ g(1) = ... ✷✳ g(0.1) = ...

❊①❡r❝✐s❡ ✶✳✶✺

✸✳ g(0.01) = ...

❙✐♠♣❧✐❢②✿ 2

20

·2


35 10

= ...

❲❤❛t ✐s t❤❡ tr❡♥❞❄

✶✳ ❊①❡r❝✐s❡s✿ ❆❧❣❡❜r❛

❊①❡r❝✐s❡ ✶✳✷✺

❆ ❞✐❛♠❡t❡r ♦❢ ❛ ❝✐r❝❧❡ r✉♥s ❜❡t✇❡❡♥ ♣♦✐♥ts R ❛♥❞ T ✳ ❚❤❡ ❝❡♥t❡r ♦❢ t❤❡ ❝✐r❝❧❡✱ P ✱ ❤❛s ❝♦♦r❞✐♥❛t❡s (−4, 1)✳ ❚❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ♣♦✐♥t R ❛r❡ (2, −3)✳ ❲❤❛t ❛r❡ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ T ❄ ❊①❡r❝✐s❡ ✶✳✷✻

▲❡t f (x) = 7+2x−x2 ✳ ❋✐♥❞ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t f (3 + h) − f (3) . h

❙✐♠♣❧✐❢② ②♦✉r ❛♥s✇❡r✳ ❊①❡r❝✐s❡ ✶✳✷✼

❋✐♥❞ ❛♥ ❡①♣r❡ss✐♦♥ ❢♦r f (x) ❛♥❞ st❛t❡ ✐ts ❞♦♠❛✐♥ ✐♥ ✐♥t❡r✈❛❧ ♥♦t❛t✐♦♥ ❣✐✈❡♥ t❤❛t f ✐s t❤❡ ❢✉♥❝t✐♦♥ t❤❛t t❛❦❡s ❛ r❡❛❧ ♥✉♠❜❡r x ❛♥❞ ♣❡r❢♦r♠s t❤❡ ❢♦❧❧♦✇✐♥❣ t❤r❡❡ st❡♣s ✐♥ ♦r❞❡r✿ ✶✳ ❞✐✈✐❞❡ ❜② 3✱ ✷✳ t❛❦❡ sq✉❛r❡ r♦♦t✱ ❛♥❞ t❤❡♥ ✸✳ ♠❛❦❡ t❤❡ q✉❛♥t✐t② t❤❡ ❞❡♥♦♠✐♥❛t♦r ♦❢ ❛ ❢r❛❝✲ t✐♦♥ ✇✐t❤ ♥✉♠❡r❛t♦r 13✳ ❊①❡r❝✐s❡ ✶✳✷✽

❙♦❧✈❡ t❤❡ ❡q✉❛t✐♦♥✿ sin x = cos x.

✹✽✾

✷✳ ❊①❡r❝✐s❡s✿ ❙❡q✉❡♥❝❡s

✹✾✵

✷✳ ❊①❡r❝✐s❡s✿ ❙❡q✉❡♥❝❡s

❊①❡r❝✐s❡ ✷✳✶

❈♦♠♣✉t❡

4 X

❊①❡r❝✐s❡ ✷✳✾

■♥ t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ ❡❛❝❤ ②❡❛r✱ ❛ ♣❡rs♦♥ ♣✉ts $5000 ✐♥ ❛ ❜❛♥❦ t❤❛t ♣❛②s 3% ❝♦♠♣♦✉♥❞❡❞ ❛♥♥✉❛❧❧②✳ ❍♦✇ ♠✉❝❤ ❞♦❡s ❤❡ ❤❛✈❡ ❛❢t❡r 15 ②❡❛rs❄

n2 ✳

n=1

❊①❡r❝✐s❡ ✷✳✷

❊①❡r❝✐s❡ ✷✳✶✵

Pr❡s❡♥t t❤❡ ✜rst 5 t❡r♠s ♦❢ t❤❡ s❡q✉❡♥❝❡✿

❆♥ ♦❜❥❡❝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❄

a1 = 1,

an+1 = −(an + 1).

❊①❡r❝✐s❡ ✷✳✸

❘❡♣r❡s❡♥t ✐♥ s✐❣♠❛ ♥♦t❛t✐♦♥✿ −1 − 2 − 3 − 4 − 5 − ... − 10. ❊①❡r❝✐s❡ ✷✳✹

❋✐♥❞ t❤❡ s✉♠ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣✿ −1 − 2 − 3 − 4 − 5 − ... − 10. ❊①❡r❝✐s❡ ✷✳✺

❋✐♥❞ t❤❡ s❡q✉❡♥❝❡ ♦❢ s✉♠s ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡✲ q✉❡♥❝❡✿ −1, 2, −4, 8, −5, ... ❊①❡r❝✐s❡ ✷✳✻

n ✐s ❛♥ ✐♥❝r❡❛s✐♥❣ s❡q✉❡♥❝❡✳ ❲❤❛t n+1 n+1 ❦✐♥❞ ♦❢ s❡q✉❡♥❝❡ ✐s ❄ ●✐✈❡ ❡①❛♠♣❧❡s ♦❢ ✐♥✲ n

❙❤♦✇ t❤❛t

❝r❡❛s✐♥❣ ❛♥❞ ❞❡❝r❡❛s✐♥❣ s❡q✉❡♥❝❡s✳ ❊①❡r❝✐s❡ ✷✳✼

❋✐♥❞ t❤❡ ♥❡①t ✐t❡♠ ✐♥ ❡❛❝❤ ❧✐st✿ ✶✳ 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❤❡ ♣✐❧❡❄

✸✳ ❊①❡r❝✐s❡s✿ ❙❡ts ❛♥❞ ❧♦❣✐❝

✹✾✶

✸✳ ❊①❡r❝✐s❡s✿ ❙❡ts ❛♥❞ ❧♦❣✐❝

❊①❡r❝✐s❡ ✸✳✶

✇❤❛t ❝❛♥ ②♦✉ ❝♦♥❝❧✉❞❡ ❛❜♦✉t

A❄

❘❡♣r❡s❡♥t t❤❡ ❢♦❧❧♦✇✐♥❣ s❡t ✐♥ t❤❡ s❡t✲❜✉✐❧❞✐♥❣ ♥♦✲ t❛t✐♦♥✿

❊①❡r❝✐s❡ ✸✳✶✵

X = [0, 1] ∪ [2, 3] = ...

❲❡ ❦♥♦✇ t❤❛t ✏■❢ ✐t r❛✐♥s✱ t❤❡ r♦❛❞ ❣❡ts ✇❡t✑✳ ❉♦❡s ✐t ♠❡❛♥ t❤❛t ✐❢ t❤❡ r♦❛❞ ✐s ✇❡t✱ ✐t ❤❛s r❛✐♥❡❞❄

❊①❡r❝✐s❡ ✸✳✷

❙✐♠♣❧✐❢②✿

❊①❡r❝✐s❡ ✸✳✶✶

{x > 0 : x

✐s ❛ ♥❡❣❛t✐✈❡ ✐♥t❡❣❡r

❆ ❣❛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 ❛r❡ t❤❡ ♠❛①✱ ♠✐♥✱ ❛♥❞ ❛♥② ❜♦✉♥❞s ♦❢ t❤❡ s❡t ♦❢ ✐♥t❡❣❡rs❄ ❲❤❛t ❛❜♦✉t

R❄

❊①❡r❝✐s❡ ✸✳✶✷

■❢ ❛♥ ❛❞✈❡rt✐s❡♠❡♥t ❝❧❛✐♠s t❤❛t ✏❆❧❧ ♦✉r s❡❝♦♥❞✲ ❊①❡r❝✐s❡ ✸✳✹

❤❛♥❞ ❝❛rs ❝♦♠❡ ✇✐t❤ ✇♦r❦✐♥❣ ❆❈✑✱ ✇❤❛t ✐s t❤❡ ❡❛s✲

■s t❤❡ ❝♦♥✈❡rs❡ ♦❢ t❤❡ ❝♦♥✈❡rs❡ ♦❢ ❛ tr✉❡ st❛t❡♠❡♥t

✐❡st ✇❛② t♦ ❞✐s♣r♦✈❡ t❤❡ s❡♥t❡♥❝❡❄

tr✉❡❄ ❊①❡r❝✐s❡ ✸✳✶✸ ❊①❡r❝✐s❡ ✸✳✺

❚❡❛❝❤❡rs ♦❢t❡♥ s❛② t♦ t❤❡ st✉❞❡♥t✬s ♣❛r❡♥ts✿ ✏■❢ ②♦✉r

❙t❛t❡ t❤❡ ❝♦♥✈❡rs❡ ♦❢ t❤✐s st❛t❡♠❡♥t✿ ✏t❤❡ ❝♦♥✈❡rs❡

st✉❞❡♥t ✇♦r❦s ❤❛r❞❡r✱ ❤❡✬❧❧ ✐♠♣r♦✈❡✑✳

♦❢ t❤❡ ❝♦♥✈❡rs❡ ♦❢ ❛ tr✉❡ st❛t❡♠❡♥t ✐s tr✉❡✑✳

✇♦♥✬t ✐♠♣r♦✈❡ ❛♥❞ t❤❡ ♣❛r❡♥ts ❝♦♠❡ ❜❛❝❦ t♦ t❤❡

❲❤❡♥ ❤❡

t❡❛❝❤❡r✱ ❤❡ ✇✐❧❧ ❛♥s✇❡r✿ ✏❍❡ ❞✐❞♥✬t ✐♠♣r♦✈❡✱ t❤❛t ♠❡❛♥s ❤❡ ❞✐❞♥✬t ✇♦r❦ ❤❛r❞❡r✑✳ ❆♥❛❧②③❡✳

❊①❡r❝✐s❡ ✸✳✻

❘❡♣r❡s❡♥t t❤❡s❡ s❡ts ❛s ✐♥t❡rs❡❝t✐♦♥s ❛♥❞ ✉♥✐♦♥s✿ ✶✳

(0, 5)

✷✳

{3}

✸✳

∅

✹✳

{x : x > 0 ❖❘ x

✺✳

{x : x

✐s ❛♥ ✐♥t❡❣❡r}

✐s ❞✐✈✐s✐❜❧❡ ❜②

6}

❊①❡r❝✐s❡ ✸✳✼

❚r✉❡ ♦r ❢❛❧s❡✿

0 = 1 =⇒ 0 = 1❄

❊①❡r❝✐s❡ ✸✳✽

Pr♦✈❡✿

max{max A, max B} = max(A ∪ B). ❊①❡r❝✐s❡ ✸✳✾

✭❛✮ ■❢✱ st❛rt✐♥❣ ✇✐t❤ ❛ st❛t❡♠❡♥t ❝♦♥❝❧✉s✐♦♥s ②♦✉ ❛rr✐✈❡ t♦ ❝❧✉❞❡ ❛❜♦✉t

A✱

A❄

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♦

0 = 0✱

✹✳ ❊①❡r❝✐s❡s✿ ❈♦♦r❞✐♥❛t❡ s②st❡♠

✹✾✷

✹✳ ❊①❡r❝✐s❡s✿ ❈♦♦r❞✐♥❛t❡ s②st❡♠

❊①❡r❝✐s❡ ✹✳✶

❊①❡r❝✐s❡ ✹✳✼

❋✐♥❞ t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ❧✐♥❡ ♣❛ss✐♥❣ t❤r♦✉❣❤ t❤❡ ♣♦✐♥ts (−1, 1) ❛♥❞ (−1, 5)✳

❋♦r t❤❡ ♣♦✐♥ts P = (0, 1)✱ Q = (1, 2)✱ ❛♥❞ R = (−1, 2)✱ ❞❡t❡r♠✐♥❡ t❤❡ ♣♦✐♥ts t❤❛t ❛r❡ s②♠♠❡tr✐❝

❊①❡r❝✐s❡ ✹✳✷

❲❤❛t ✐s t❤❡ ❞✐st❛♥❝❡ ❢r♦♠ t❤❡ ❝❡♥t❡r ♦❢ t❤❡ ❝✐r❝❧❡ 2

2

(x − 1) + (y + 3) = 5

t♦ t❤❡ ♦r✐❣✐♥❄ ❊①❡r❝✐s❡ ✹✳✸

❲❤❛t ✐s t❤❡ ❞✐st❛♥❝❡ ❢r♦♠ t❤❡ ❝✐r❝❧❡ 2

2

x + (y + 3) = 2

t♦ t❤❡ ♦r✐❣✐♥❄

✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❛①✐s ❛♥❞ t❤❡ ♦r✐❣✐♥✳ ❊①❡r❝✐s❡ ✹✳✽

❚❤❡ ❤②♣♦t❡♥✉s❡ ♦❢ ❛♥ ✐s♦s❝❡❧❡s r✐❣❤t tr✐❛♥❣❧❡ ✐s 10 ✐♥❝❤❡s✳ ❚❤❡ ♠✐❞♣♦✐♥ts ♦❢ ✐ts s✐❞❡s ❛r❡ ❝♦♥♥❡❝t❡❞ t♦ ❢♦r♠ ❛♥ ✐♥s❝r✐❜❡❞ tr✐❛♥❣❧❡✱ ❛♥❞ t❤✐s ♣r♦❝❡ss ✐s r❡✲ ♣❡❛t❡❞✳ ❋✐♥❞ t❤❡ s✉♠ ♦❢ t❤❡ ❛r❡❛s ♦❢ t❤❡s❡ tr✐❛♥❣❧❡s ❛s t❤✐s ♣r♦❝❡ss ✐s ❝♦♥t✐♥✉❡❞✳ ❊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❡ ✹✳✶✵

❊①❡r❝✐s❡ ✹✳✹

❋✐♥❞ t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ❝✐r❝❧❡ ❝❡♥t❡r❡❞ ❛t (−1, −1) ❛♥❞ ♣❛ss✐♥❣ t❤r♦✉❣❤ t❤❡ ♣♦✐♥t (−1, 1)✳

❙❦❡t❝❤ t❤❡ r❡❣✐♦♥ ❣✐✈❡♥ ❜② t❤❡ s❡t {(x, y) : xy < 0}✳ ❲❤✐❝❤ ❛①❡s ❛♥❞ ✇❤✐❝❤ q✉❛❞r❛♥ts ♦❢ t❤❡ ♣❧❛♥❡ ❛r❡ ✐♥❝❧✉❞❡❞ ✐♥ t❤❡ s❡t❄ ❊①❡r❝✐s❡ ✹✳✶✶

❊①❡r❝✐s❡ ✹✳✺

❚❤r❡❡ str❛✐❣❤t ❧✐♥❡s ❛r❡ s❤♦✇♥ ❜❡❧♦✇✳ ❋✐♥❞ t❤❡✐r s❧♦♣❡s✿

❋✐♥❞ ❛❧❧ x s✉❝❤ t❤❛t t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ ♣♦✐♥ts (3, −8) ❛♥❞ (x, −6) ✐s 5✳ ❊①❡r❝✐s❡ ✹✳✶✷

❚✇♦ ❝❛r❡ ❧❡❛✈❡ ❛ ❤✐❣❤✇❛② ❥✉♥❝t✐♦♥ ❛t t❤❡ s❛♠❡ t✐♠❡✳ ❚❤❡ ✜rst tr❛✈❡❧s ✇❡st ❛t 70 ♠✐❧❡s ♣❡r ❤♦✉r ❛♥❞ t❤❡ s❡❝♦♥❞ tr❛✈❡❧s ♥♦rt❤ ❛t 60 ♠✐❧❡s ♣❡r ❤♦✉r✳ ❍♦✇ ❢❛r ❛♣❛rt ❛r❡ t❤❡② ❛❢t❡r 1.5 ❤♦✉rs❄ ❊①❡r❝✐s❡ ✹✳✶✸

❊①❡r❝✐s❡ ✹✳✻

❚❤r❡❡ str❛✐❣❤t ❧✐♥❡s ❛r❡ s❤♦✇♥ ❜❡❧♦✇✳ ❋✐♥❞ t❤❡✐r ❡q✉❛t✐♦♥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❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❝✐r❝❧❡ ♦❢ r❛❞✐✉s R ❝❡♥✲ t❡r❡❞ ❛t ❛ ♣♦✐♥t C ✳ ✭❜✮ ❋✐♥❞ t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ❝✐r✲ ❝❧❡ ❝❡♥t❡r❡❞ ❛t t❤❡ ♣♦✐♥t (1, 1) t❤❛t ♣❛ss❡s t❤r♦✉❣❤ (0, 0)✳

✺✳ ❊①❡r❝✐s❡s✿ ▲✐♥❡❛r ❛❧❣❡❜r❛

✹✾✸

✺✳ ❊①❡r❝✐s❡s✿ ▲✐♥❡❛r ❛❧❣❡❜r❛

❊①❡r❝✐s❡ ✺✳✶

❊①❡r❝✐s❡ ✺✳✾

❲❤❛t ✐s t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ❧✐♥❡ t❤r♦✉❣❤ t❤❡ ♣♦✐♥ts

❚❤❡ t❛①✐ ❝❤❛r❣❡s

A = (−3, 2)

❛♥❞

❛♥❞

B = (2, 5)❄

$0.35

$1.75 ❢♦r t❤❡ ✜rst q✉❛rt❡r ♦❢ ❛ ♠✐❧❡

❢♦r ❡❛❝❤ ❛❞❞✐t✐♦♥❛❧ ✜❢t❤ ♦❢ ❛ ♠✐❧❡✳ ❋✐♥❞

f

❛s ❛

a =< 1, 2 >, b =< −2, 1 >✱

✜♥❞

❛ ❧✐♥❡❛r ❢✉♥❝t✐♦♥ ✇❤✐❝❤ ♠♦❞❡❧s t❤❡ t❛①✐ ❢❛r❡ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ♥✉♠❜❡r ♦❢ ♠✐❧❡s ❞r✐✈❡♥✱

❊①❡r❝✐s❡ ✺✳✷

x✳

❋✐♥❞ t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ ♣♦✐♥ts ♦❢ ✐♥t❡rs❡❝✲ t✐♦♥ ♦❢ t❤❡ ❝✐r❝❧❡ ❛①❡s✳

(x − 1)2 + (y − 2)2 = 6

✇✐t❤ t❤❡

❊①❡r❝✐s❡ ✺✳✶✵

●✐✈❡♥ ✈❡❝t♦rs

t❤❡✐r ♠❛❣♥✐t✉❞❡s ❛♥❞ t❤❡ ❛♥❣❧❡ ❜❡t✇❡❡♥ t❤❡♠✳ ❊①❡r❝✐s❡ ✺✳✸

❙❡t ✉♣✱ ❜✉t ❞♦ ♥♦t s♦❧✈❡✱ ❛ s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛✲ t✐♦♥s ❢♦r t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦❜❧❡♠✿

❛♥❞ ✐t ❝♦♥s✐sts ♦❢ t✇♦

❙❡t ✉♣ ❛ s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s ✕ ❜✉t ❞♦ ♥♦t

❚❤❡ st♦❝❦s ❛r❡ ♣r✐❝❡❞ ❛s ❢♦❧❧♦✇s✿

s♦❧✈❡ ✕ ❢♦r t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦❜❧❡♠✿ ✏❆ ♠✐① ♦❢ ❝♦❢✲

♣♦rt❢♦❧✐♦ ✐s ✇♦rt❤ st♦❝❦s

A $2.1

A

B✳

❛♥❞

♣❡r s❤❛r❡✱

❊①❡r❝✐s❡ ✺✳✶✶

✏❙✉♣♣♦s❡ ②♦✉r

$20, 000 B $1.5

♣❡r s❤❛r❡✳ ❙✉♣♣♦s❡ ❛❧s♦

t❤❛t ②♦✉ ❤❛✈❡ t✇✐❝❡ ❛s ♠✉❝❤ ♦❢ st♦❝❦

A

t❤❛♥

B✳

❢❡❡ ✐s t♦ ❜❡ ♣r❡♣❛r❡❞ ❢r♦♠✿ ❑❡♥②❛♥ ❝♦✛❡❡ ✲ ♣♦✉♥❞ ❛♥❞ ❈♦❧♦♠❜✐❛♥ ❝♦✛❡❡ ✲

$5

♠✉❝❤ ♦❢ ❡❛❝❤ ❞♦ ②♦✉ ♥❡❡❞ t♦ ❤❛✈❡

❍♦✇ ♠✉❝❤ ♦❢ ❡❛❝❤ ❞♦ ②♦✉ ❤❛✈❡❄✑

❜❧❡♥❞ ✇✐t❤

$3.50

$3

♣❡r

♣❡r ♣♦✉♥❞✳ ❍♦✇

10

♣♦✉♥❞s ♦❢

♣❡r ♣♦✉♥❞❄✑

❊①❡r❝✐s❡ ✺✳✹

■♥ ❛♥ ❡✛♦rt t♦ ✜♥❞ t❤❡ ♣♦✐♥t ✐♥ ✇❤✐❝❤ t❤❡ ❧✐♥❡s

2x − y = 2

−4x + 2y = 1 ✐♥t❡rs❡❝t✱ ❛ st✉❞❡♥t 2 ❛♥❞ t❤❡♥ ❛❞❞❡❞ t❤❡ s❡❝♦♥❞✳ ❍❡ ❣♦t 0 = 5✳ ❊①♣❧❛✐♥ t❤❡

❊①❡r❝✐s❡ ✺✳✶✷

❛♥❞

❙❡t ✉♣✱ ❞♦ ♥♦t s♦❧✈❡✱ t❤❡ s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s

♠✉❧t✐♣❧✐❡❞ t❤❡ ✜rst ♦♥❡ ❜②

❢♦r t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦❜❧❡♠✿ ✏❖♥❡ s❡r✈✐♥❣ ♦❢ t♦♠❛t♦

r❡s✉❧t t♦ t❤❡ r❡s✉❧t✳

s♦✉♣ ❝♦♥t❛✐♥s

100

❈❛❧ ❛♥❞

❣ ♦❢ ❝❛r❜♦❤②❞r❛t❡s✳

❋✐♥❞ t❤❡ ❛♥❣❧❡ ❜❡t✇❡❡♥ t❤❡ ❧✐♥❡s✿ ❛♥❞ ❢r♦♠

❣ ♦❢ ❝❛r❜♦❤②❞r❛t❡s✳

(0, 0)

t♦

(1, 2)✳

❢r♦♠

(0, 0)

t♦

70

❈❛❧ ❛♥❞

13

❍♦✇ ♠❛♥② s❡r✈✐♥❣s ♦❢ ❡❛❝❤

s❤♦✉❧❞ ❜❡ r❡q✉✐r❡❞ t♦ ♦❜t❛✐♥

❊①❡r❝✐s❡ ✺✳✺

(1, 1)

18

❖♥❡ s❧✐❝❡ ♦❢ ✇❤♦❧❡ ❜r❡❛❞ ❝♦♥t❛✐♥s

230

❈❛❧ ❛♥❞

42

❣ ♦❢

❝❛r❜♦❤②❞r❛t❡s❄✑

❉♦♥✬t s✐♠♣❧✐❢②✳ ❊①❡r❝✐s❡ ✺✳✶✸

❙♦❧✈❡ t❤❡ s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s✿

❊①❡r❝✐s❡ ✺✳✻



❙♦❧✈❡ t❤❡ s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s✿



x −y = −1, 2x +y = 0.

x − y = 2, x + 2y = 1.

❊①❡r❝✐s❡ ✺✳✶✹ ❊①❡r❝✐s❡ ✺✳✼

❙♦❧✈❡ t❤❡ s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s ❛♥❞ ❣❡♦♠❡tr✐✲

❙♦❧✈❡ t❤❡ s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s✿

❝❛❧❧② r❡♣r❡s❡♥t ✐ts s♦❧✉t✐♦♥✿





x −2y = 1, 2x +y = 0.

❊①❡r❝✐s❡ ✺✳✽

❆ ♠♦✈✐❡ t❤❡❛t❡r ❝❤❛r❣❡s

x − 2y = 1, x + 2y = −1.

❊①❡r❝✐s❡ ✺✳✶✺

$10

$6 ❢♦r 320 ♣❡♦♣❧❡ ♣❛✐❞ ✇❡r❡ $3120✳ ❍♦✇

❢♦r ❛❞✉❧ts ❛♥❞

❝❤✐❧❞r❡♥✳ ❖♥ ❛ ♣❛rt✐❝✉❧❛r ❞❛② ✇❤❡♥ ❛♥ ❛❞♠✐ss✐♦♥✱ t❤❡ t♦t❛❧ r❡❝❡✐♣ts

♠❛♥② ✇❡r❡ ❛❞✉❧ts ❛♥❞ ❤♦✇ ♠❛♥② ✇❡r❡ ❝❤✐❧❞r❡♥❄

●❡♦♠❡tr✐❝❛❧❧② r❡♣r❡s❡♥t t❤✐s s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛✲ t✐♦♥s✿



x − 2y = 1, x + 2y = 1.

✺✳ ❊①❡r❝✐s❡s✿ ▲✐♥❡❛r ❛❧❣❡❜r❛

✹✾✹

❊①❡r❝✐s❡ ✺✳✶✻

❲❤❛t ❛r❡ t❤❡ ♣♦ss✐❜❧❡ ♦✉t❝♦♠❡s ♦❢ ❛ s②st❡♠ ♦❢ ❧✐♥✲ ❡❛r ❡q✉❛t✐♦♥s❄

❊①❡r❝✐s❡ ✺✳✶✼

❋✐♥❞ t❤❡ ✈❛❧✉❡ ♦❢ ♣♦✐♥ts

(−6, 0)

❝♦♥t❛✐♥✐♥❣ t❤❡

k

s♦ t❤❛t t❤❡ ❧✐♥❡ ❝♦♥t❛✐♥✐♥❣ t❤❡

(k, −5) ♣♦✐♥ts (4, 3) ❛♥❞

✐s ♣❛r❛❧❧❡❧ t♦ t❤❡ ❧✐♥❡ ❛♥❞

(1, 7)✳

✻✳ ❊①❡r❝✐s❡s✿ P♦❧②♥♦♠✐❛❧s

✹✾✺

✻✳ ❊①❡r❝✐s❡s✿ P♦❧②♥♦♠✐❛❧s

❊①❡r❝✐s❡ ✻✳✶

❙✉♣♣♦s❡ f ✐s ❛ ♣♦❧②♥♦♠✐❛❧ ♦❢ ❞❡❣r❡❡ 55 ❛♥❞ ✐ts ❧❡❛❞✲ ✐♥❣ t❡r♠ ✐s −1✳ ❉❡s❝r✐❜❡ t❤❡ ❧♦♥❣ t❡r♠ ❜❡❤❛✈✐♦r ♦❢ t❤✐s ❢✉♥❝t✐♦♥✳ ❊①❡r❝✐s❡ ✻✳✷

❋♦r t❤❡ ♣♦❧②♥♦♠✐❛❧ f (x) = −2x2 (x + 2)2 (x2 + 1)✱ ✜♥❞ ✐ts x✲✐♥t❡r❝❡♣ts✳ ❊①❡r❝✐s❡ ✻✳✸

❊①❡r❝✐s❡ ✻✳✽

❋✐♥❞ ❛ ❢♦r♠✉❧❛ ❢♦r ❛ ♣♦❧②♥♦♠✐❛❧ ✇✐t❤ t❤❡s❡ r♦♦ts✿ 1✱ 2✱ ❛♥❞ 3✳

❋✐♥❞ ❛ ♣♦ss✐❜❧❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❢✉♥❝t✐♦♥ ♣❧♦tt❡❞ ❜❡✲ ❧♦✇✿

❊①❡r❝✐s❡ ✻✳✹

■s t❤✐s ❛ ♣❛r❛❜♦❧❛❄

❊①❡r❝✐s❡ ✻✳✾ ❊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✳

❋♦r t❤❡ ♣♦❧②♥♦♠✐❛❧ f (x) = −2x(x − 2)2 (x + 1)3 ✱ ✜♥❞ ✐ts x✲✐♥t❡r❝❡♣ts ❛♥❞ ✐ts ❧❛r❣❡ s❝❛❧❡ ❜❡❤❛✈✐♦r✱ ✐✳❡✳✱ f (x) →? ❛s x → ±∞✳ ❊①❡r❝✐s❡ ✻✳✶✵

❊①❡r❝✐s❡ ✻✳✻

❋♦r t❤❡ ♣♦❧②♥♦♠✐❛❧s ❣r❛♣❤❡❞ ❜❡❧♦✇✱ ✜♥❞ t❤❡ ❢♦❧✲ ❧♦✇✐♥❣✿ s♠❛❧❧❡st ♣♦ss✐❜❧❡ ❞❡❣r❡❡ s✐❣♥ ♦❢ t❤❡ ❧❡❛❞✐♥❣ ❝♦❡✣❝✐❡♥t ❞❡❣r❡❡ ✐s ♦❞❞✴❡✈❡♥

1 2 3

●✐✈❡♥ f (x) = −(x − 3)4 (x + 1)3 ✳ ❋✐♥❞ t❤❡ ❧❡❛❞✐♥❣ t❡r♠ ❛♥❞ ✉s❡ ✐t t♦ ❞❡s❝r✐❜❡ t❤❡ ❧♦♥❣ t❡r♠ ❜❡❤❛✈✐♦r ♦❢ t❤❡ ❢✉♥❝t✐♦♥✳ ❊①❡r❝✐s❡ ✻✳✶✶

❆ ❢❛❝t♦r② ✐s t♦ ❜❡ ❜✉✐❧t ♦♥ ❛ ❧♦t ♠❡❛s✉r✐♥❣ 240 ❢t ❜② 320 ❢t✳ ❆ ❜✉✐❧❞✐♥❣ ❝♦❞❡ 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❤❡ ❧❛✇♥ ❜❡❄ ❊①❡r❝✐s❡ ✻✳✶✷

❊①❡r❝✐s❡ ✻✳✼

❋✐♥❞ ❛ ♣♦ss✐❜❧❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❢✉♥❝t✐♦♥ ♣❧♦tt❡❞ ❜❡✲ ❧♦✇✿

❆ ❢❛❝t♦r② ♦❝❝✉♣✐❡s ❛ ❧♦t ♠❡❛s✉r✐♥❣ 240 ❢t ❜② 320 ❢t✳ ❆ ❜✉✐❧❞✐♥❣ ❝♦❞❡ 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❤❡ ❧❛✇♥ ❜❡❄

✻✳ ❊①❡r❝✐s❡s✿ P♦❧②♥♦♠✐❛❧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❡s✿ ❘❡❧❛t✐♦♥s ❛♥❞ ❢✉♥❝t✐♦♥s

✹✾✼

✼✳ ❊①❡r❝✐s❡s✿ ❘❡❧❛t✐♦♥s ❛♥❞ ❢✉♥❝t✐♦♥s

❊①❡r❝✐s❡ ✼✳✶

❊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❡ ♥❡❡❞❡❞✳

❆♥ ❛♠✉s❡♠❡♥t ♣❛r❦ s❡❧❧s ♠✉❧t✐✲❞❛② ♣❛ss❡s✳ ❚❤❡ ❢✉♥❝t✐♦♥ g(x) = 1/3x r❡♣r❡s❡♥t t❤❡ ♥✉♠❜❡r ♦❢ ❞❛②s ❛ ♣❛ss ✇✐❧❧ ✇♦r❦✱ ✇❤❡r❡ x ✐s t❤❡ ❛♠♦✉♥t ♦❢ ♠♦♥❡② ♣❛✐❞✱ ✐♥ ❞♦❧❧❛rs✳ ■♥t❡r♣r❡t t❤❡ ♠❡❛♥✐♥❣ ♦❢ g(6) = 3✳

❊①❡r❝✐s❡ ✼✳✻ ❊①❡r❝✐s❡ ✼✳✷

❱✐s✉❛❧✐③❡ t❤❡ r❡❧❛t✐♦♥✿ x2 y 2 + = 1. 4 9

❚❤❡ ♣❡r✐♠❡t❡r ♦❢ ❛ r❡❝t❛♥❣❧❡ ✐s 10 ❢❡❡t✳ ✭❛✮ ❊①♣r❡ss t❤❡ ❛r❡❛ ♦❢ t❤❡ r❡❝t❛♥❣❧❡ ✐♥ t❡r♠s ♦❢ ✐ts ✇✐❞t❤✳ ✭❜✮ ❋✐♥❞ t❤❡ ♠✐♥✐♠❛❧ ♣♦ss✐❜❧❡ ❛r❡❛✳ ✭❝✮ ❋✐♥❞ t❤❡ ♠❛①✲ ✐♠❛❧ ♣♦ss✐❜❧❡ ❛r❡❛✳

❉♦ ②♦✉ s❡❡ t❤❡s❡ 4 ❛♥❞ 9 ♦♥ t❤❡ ❣r❛♣❤❄ ❊①❡r❝✐s❡ ✼✳✼

❊①❡r❝✐s❡ ✼✳✸

❙✉♣♣♦s❡ t❤❡ ❝♦st ✐s f (x) ❞♦❧❧❛rs ❢♦r ❛ t❛①✐ tr✐♣ ♦❢ x ♠✐❧❡s✳ ■♥t❡r♣r❡t t❤❡ ❢♦❧❧♦✇✐♥❣ st♦r✐❡s ✐♥ t❡r♠s ♦❢ f ✳ ✶✳ ▼♦♥❞❛②✱ ■ t♦♦❦ ❛ t❛①✐ t♦ t❤❡ st❛t✐♦♥ 5 ♠✐❧❡s ❛✇❛②✳ ✷✳ ❚✉❡s❞❛②✱ ■ t♦♦❦ ❛ t❛①✐ t♦ t❤❡ st❛t✐♦♥ ❜✉t t❤❡♥ r❡❛❧✐③❡❞ t❤❛t ■ ❧❡❢t s♦♠❡t❤✐♥❣ ❛t ❤♦♠❡ ❛♥❞ ❤❛❞ t♦ ❝♦♠❡ ❜❛❝❦✳ ✸✳ ❲❡❞♥❡s❞❛②✱ ■ t♦♦❦ ❛ t❛①✐ t♦ t❤❡ st❛t✐♦♥ ❛♥❞ ■ ❣❛✈❡ ♠② ❞r✐✈❡r ❛ ✜✈❡ ❞♦❧❧❛r t✐♣✳ ✹✳ ❚❤✉rs❞❛②✱ ■ t♦♦❦ ❛ t❛①✐ t♦ t❤❡ st❛t✐♦♥ ❜✉t t❤❡ ❞r✐✈❡r ❣♦t ❧♦st ❛♥❞ ❞r♦✈❡ ✜✈❡ ❡①tr❛ ♠✐❧❡s✳ ✺✳ ❋r✐❞❛②✱ ■ ❤❛✈❡ ❜❡❡♥ t❛❦✐♥❣ ❛ t❛①✐ t♦ t❤❡ st❛✲ t✐♦♥ ❛❧❧ ✇❡❡❦ ♦♥ ❝r❡❞✐t❀ ■ ♣❛② ✇❤❛t ■ ♦✇❡ t♦❞❛②✳ ❲❤❛t ✐❢ t❤❡r❡ ✐s ❛♥ ❡①tr❛ ❝❤❛r❣❡ ♣❡r r✐❞❡ ♦❢ m ❞♦❧✲ ❧❛rs❄ ❊①❡r❝✐s❡ ✼✳✹

▲❡t f : A → B ❛♥❞ g : C → D ❜❡ t✇♦ ♣♦ss✐✲ ❜❧❡ ❢✉♥❝t✐♦♥s✳ ❋♦r ❡❛❝❤ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢✉♥❝t✐♦♥s✱ st❛t❡ ✇❤❡t❤❡r ♦r ♥♦t ②♦✉ ❝❛♥ ❝♦♠♣✉t❡ f ◦ g ✿ • D⊂B • C⊂A • B⊂D • B=C

▲❡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♦✲

✈✐❞❡s ❛ ♣r❛❝t✐❝❛❧ ✐♥t❡r♣r❡t❛t✐♦♥ ♦❢ t❤❡ ❝♦♠♣♦s✐t✐♦♥ f (h(t))❄ ✶✳ ❚❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ r❛❞✐✉s ❛t t✐♠❡ t✳ ✷✳ ❚❤❡ ❛r❡❛ ♦❢ t❤❡ ❝✐r❝❧❡ ❛t t✐♠❡ t✳ ✸✳ ❚❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ r❛❞✐✉s ♦❢ ❛ ❝✐r❝❧❡ ✇✐t❤ ❛r❡❛ A = f (r) ❛t t✐♠❡ t✳ ✹✳ ❚❤❡ ❛r❡❛ ♦❢ t❤❡ ❝✐r❝❧❡ ✇❤✐❝❤ ❛t t✐♠❡ t ❤❛s r❛❞✐✉s h(t)✳ ✺✳ ❚❤❡ t✐♠❡ t ✇❤❡♥ t❤❡ ❛r❡❛ ✇✐❧❧ ❜❡ A = f (r)✳ ✻✳ ❚❤❡ t✐♠❡ t ✇❤❡♥ t❤❡ r❛❞✐✉s ✇✐❧❧ ❜❡ r = h(t)✳

❊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✳ ✭❝✮ ❋✐♥❞ t❤❡ ♠❛①✐♠❛❧ ♣♦ss✐❜❧❡ ♣❡r✐♠❡t❡r✳

❊①❡r❝✐s❡ ✼✳✾

❚❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ 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❛✇✐♥❣✳

✼✳ ❊①❡r❝✐s❡s✿ ❘❡❧❛t✐♦♥s ❛♥❞ ❢✉♥❝t✐♦♥s

✹✾✽

❊①❡r❝✐s❡ ✼✳✶✼

❋✐♥❞ t❤❡ ✐♠♣❧✐❡❞ ❞♦♠❛✐♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥✿

(x − 1)(x2 + 1)2x . ❊①❡r❝✐s❡ ✼✳✶✽

❋✐♥✐s❤ t❤❡ s❡♥t❡♥❝❡✿ ✏■❢ ❛ ❢✉♥❝t✐♦♥ ❢❛✐❧s t❤❡ ❤♦r✐③♦♥✲ t❛❧ ❧✐♥❡ t❡st✱ t❤❡♥✳✳✳✑

❊①❡r❝✐s❡ ✼✳✶✾

❘❡st❛t❡ ✭❜✉t ❞♦ ♥♦t s♦❧✈❡✮ 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✐♦♥

❣❡❜r❛✐❝❛❧❧②✿ ✏❲❤❛t ❛r❡ t❤❡ ❞✐♠❡♥s✐♦♥s ♦❢ t❤❡ r❡❝t✲ ❛♥❣❧❡ ✇✐t❤ t❤❡ s♠❛❧❧❡st ♣♦ss✐❜❧❡ ♣❡r✐♠❡t❡r ❛♥❞ ❛r❡❛ ✜①❡❞ ❛t

100❄✑

❊①❡r❝✐s❡ ✼✳✷✵ ❊①❡r❝✐s❡ ✼✳✶✶

❆ s❦❡t❝❤ ♦❢ t❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥

❋✐♥❞ ❛❧❧ ♣♦ss✐❜❧❡ ✈❛❧✉❡s ♦❢

x

❢♦r ✇❤✐❝❤

f

❛♥❞ ✐ts t❛❜❧❡

♦❢ ✈❛❧✉❡s ❛r❡ ❣✐✈❡♥ ❜❡❧♦✇✳ ❈♦♠♣❧❡t❡ t❤❡ t❛❜❧❡✿

tan x = 0 .

3 1 x 0 y 2 4 5

❊①❡r❝✐s❡ ✼✳✶✷

▼❛❦❡ ❛ ❤❛♥❞✲❞r❛✇♥ s❦❡t❝❤ ♦❢ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝✲ t✐♦♥✿

  −3 f (x) = x2  x

✐❢ ✐❢ ✐❢

x < 0, 0 ≤ x < 1, x > 1.

❊①❡r❝✐s❡ ✼✳✶✸

❋✐♥❞ t❤❡ ✐♠♣❧✐❡❞ ❞♦♠❛✐♥s ♦❢ t❤❡ ❢✉♥❝t✐♦♥s ❣✐✈❡♥ ❜②✿

x+1 √ ; x2 − 1

✭❛✮

✭❜✮

√ 4

x + 1.

❊①❡r❝✐s❡ ✼✳✶✹

❋✐♥❞ t❤❡ ✐♠♣❧✐❡❞ ❞♦♠❛✐♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❣✐✈❡♥ ❜②✿

1 . (x − 1)(x2 + 1)

❊①❡r❝✐s❡ ✼✳✷✶

P❧♦t t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥

❋✐♥❞ t❤❡ ✐♠♣❧✐❡❞ ❞♦♠❛✐♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❣✐✈❡♥ ❜②✿

√

✇❤❡r❡

x

f (x)

✐s

t❤❡ t❛① ❜✐❧❧ ✭✐♥ t❤♦✉s❛♥❞s ♦❢ ❞♦❧❧❛rs✮ ❢♦r t❤❡ ✐♥❝♦♠❡ ♦❢

❊①❡r❝✐s❡ ✼✳✶✺

y = f (x)✱

✐s t❤❡ ✐♥❝♦♠❡ ✭✐♥ t❤♦✉s❛♥❞s ♦❢ ❞♦❧❧❛rs✮ ❛♥❞

x✱

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

✜rst

$10, 000✱

10%

❢♦r t❤❡ r❡st ♦❢ t❤❡ ✐♥❝♦♠❡✳

t❤❡♥

5%

❢♦r t❤❡ ♥❡①t

$10, 000✱

❛♥❞

1 . x+1 ❊①❡r❝✐s❡ ✼✳✷✷

❊①❡r❝✐s❡ ✼✳✶✻

❋✐♥❞ t❤❡ ✐♠♣❧✐❡❞ ❞♦♠❛✐♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥✿

x−1 ln(x2 + 1) sin x . x+1

P❧♦t t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ t✐♠❡ ✐♥ ❤♦✉rs ❛♥❞

x

✐s t❤❡ ♣❛r❦✐♥❣ ❢❡❡ ♦✈❡r

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

✜rst ❤♦✉r✱ t❤❡♥

3

y = f (x)

y = f (x)✱ ✇❤❡r❡ x ✐s

$1

♣❡r ❡✈❡r② ❢✉❧❧ ❤♦✉r ❢♦r t❤❡ ♥❡①t

❤♦✉rs✱ ❛♥❞ ❛ ✢❛t ❢❡❡ ♦❢

$5

❢♦r ❛♥②t❤✐♥❣ ❧♦♥❣❡r✳

✼✳ ❊①❡r❝✐s❡s✿ ❘❡❧❛t✐♦♥s ❛♥❞ ❢✉♥❝t✐♦♥s

✹✾✾

❊①❡r❝✐s❡ ✼✳✷✸

❊①♣❧❛✐♥ t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡s❡ t✇♦ ❢✉♥❝t✐♦♥s✿ r

√ x−1 x−1 ❛♥❞ √ . x+1 x+1

❊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❡ ✼✳✷✺

❉❡s❝r✐❜❡ t❤❡ ❢✉♥❝t✐♦♥ t❤❛t ❝♦♠♣✉t❡s t❤❡ ❝❛s❤✲❜❛❝❦ ♦❢ 5% ❢♦❧❧♦✇❡❞ ❜② t❤❡ ❞✐s❝♦✉♥t ♦❢ 10%✳ ❲❤❛t ✐❢ ✇❡ r❡✈❡rs❡ t❤❡ ♦r❞❡r❄ ❊①❡r❝✐s❡ ✼✳✷✻

❘❡♣r❡s❡♥t t❤✐s ❢✉♥❝t✐♦♥ ❛s ❛ ❧✐st ♦❢ ✐♥str✉❝t✐♦♥s✿ f (x) =

√ 3

sin x + 2


1/2

.

❊①❡r❝✐s❡ ✼✳✷✼

❋✐♥❞ ❛ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❢♦❧❧♦✇✐♥❣ ❢✉♥❝t✐♦♥✿ →

sq✉❛r❡ ✐t

→ t❛❦❡ ✐ts r❡❝✐♣r♦❝❛❧

→

❊①❡r❝✐s❡ ✼✳✷✽

P❧♦t t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❣✐✈❡♥ ❜② t❤❡ ❧✐st ♦❢ ✐♥str✉❝t✐♦♥s✿ ✶✳ ❛❞❞ −1❀ ✷✳ ❞✐✈✐❞❡ ❜② 0❀ ✸✳ sq✉❛r❡ t❤❡ ♦✉t❝♦♠❡✳ ❊①❡r❝✐s❡ ✼✳✷✾

❋✐♥❞ t❤❡ x✲ ❛♥❞ y ✲✐♥t❡r❝❡♣ts ❢♦r t❤❡ ❣r❛♣❤s ✐♥ t❤✐s s❡❝t✐♦♥✳

✽✳ ❊①❡r❝✐s❡s✿ ●r❛♣❤s

✺✵✵

✽✳ ❊①❡r❝✐s❡s✿ ●r❛♣❤s

❊①❡r❝✐s❡ ✽✳✶

❊①❡r❝✐s❡ ✽✳✼

f

✐s ❣✐✈❡♥ ❜❡✲

●✐✈❡ ❛♥ ❡①❛♠♣❧❡ ♦❢ ❛♥ ❡✈❡♥ ❢✉♥❝t✐♦♥✱ ❛♥ ♦❞❞ ❢✉♥❝✲

❧♦✇✳ ❉❡s❝r✐❜❡ ✐ts ❜❡❤❛✈✐♦r t❤❡ ❢✉♥❝t✐♦♥ ✉s✐♥❣ ✇♦r❞s

t✐♦♥✱ ❛♥❞ ❛ ❢✉♥❝t✐♦♥ t❤❛t✬s ♥❡✐t❤❡r✳ Pr♦✈✐❞❡ ❢♦r♠✉✲

✏❞❡❝r❡❛s✐♥❣✑ ❛♥❞ ✏✐♥❝r❡❛s✐♥❣✑✳

❧❛s✳

❆ s❦❡t❝❤ ♦❢ t❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥

❊①❡r❝✐s❡ ✽✳✽

❚❡st ✇❤❡t❤❡r t❤❡ ❢♦❧❧♦✇✐♥❣ t❤r❡❡ ❢✉♥❝t✐♦♥s ❛r❡ ❡✈❡♥✱ ♦❞❞✱ ♦r ♥❡t❤❡r✿ ✭❛✮

f (x) = x3 + 1❀

✭❜✮ t❤❡ ❢✉♥❝t✐♦♥

t❤❡ ❣r❛♣❤ ♦❢ ✇❤✐❝❤ ✐s ❛ ♣❛r❛❜♦❧❛ s❤✐❢t❡❞ ♦♥❡ ✉♥✐t ✉♣❀ ✭❝✮ t❤❡ ❢✉♥❝t✐♦♥ ✇✐t❤ t❤✐s ❣r❛♣❤✿

❊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❡ ✽✳✾

❋✐♥❞ ❤♦r✐③♦♥t❛❧ ❛s②♠♣t♦t❡s ♦❢ t❤❡s❡ ❢✉♥❝t✐♦♥s✿

❊①❡r❝✐s❡ ✽✳✸

❋✉♥❝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 0 5 0

❊①❡r❝✐s❡ ✽✳✶✵

f (x) ✐s ❣✐✈❡♥ ❜❡❧♦✇✳ ✭❛✮ f (4)✳ ✭❜✮ ❋✐♥❞ s✉❝❤ ❛♥ x

❚❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥ ❋✐♥❞ t❤❛t

f (−4)✱ f (0)✱ ❛♥❞ f (x) = 2✳ ✭❝✮ ■s t❤❡

❢✉♥❝t✐♦♥ ♦♥❡✲t♦✲♦♥❡❄

❊①❡r❝✐s❡ ✽✳✹

❲❤❛t ✐s t❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ ❜❡✐♥❣ ✭❛✮ ♦♥❡✲t♦✲♦♥❡ ♦r ♦♥t♦ ❛♥❞ ✭❜✮ ❤❛✈✐♥❣ ❛ ♠✐rr♦r s②♠♠❡tr② ♦r ❝❡♥✲ tr❛❧ s②♠♠❡tr②❄

❊①❡r❝✐s❡ ✽✳✺

❇② ❝❤❛♥❣✐♥❣ ✐ts ❞♦♠❛✐♥ ♦r ❝♦❞♦♠❛✐♥✱ ♠❛❦❡ t❤❡ ❢✉♥❝t✐♦♥

3

y =x −x

✭❛✮ ♦♥t♦✱ ❛♥❞ ✭❜✮ ♦♥❡✲t♦✲♦♥❡❄

❊①❡r❝✐s❡ ✽✳✶✶

■s

sin x/2

♣❡r✐♦❞✳

❛ ♣❡r✐♦❞✐❝ ❢✉♥❝t✐♦♥❄

■❢ ✐t ✐s✱ ✜♥❞ ✐ts

❨♦✉ ❤❛✈❡ t♦ ❥✉st✐❢② ②♦✉r ❝♦♥❝❧✉s✐♦♥ ❛❧❣❡✲

❜r❛✐❝❛❧❧②✳ ❊①❡r❝✐s❡ ✽✳✻

■s t❤❡ ❢✉♥❝t✐♦♥ ❜❡❧♦✇ ❡✈❡♥✱ ♦❞❞✱ ♦r ♥❡✐t❤❡r❄

f (x) =

ex

x 1 + x−1 −1 2

❊①❡r❝✐s❡ ✽✳✶✷

■s

sin x + cos πx

❛ ♣❡r✐♦❞✐❝ ❢✉♥❝t✐♦♥❄ ■❢ ✐t ✐s✱ ✜♥❞

✐ts ♣❡r✐♦❞✳ ❨♦✉ ❤❛✈❡ t♦ ❥✉st✐❢② ②♦✉r ❝♦♥❝❧✉s✐♦♥ ❛❧✲

✽✳ ❊①❡r❝✐s❡s✿ ●r❛♣❤s

✺✵✶

❣❡❜r❛✐❝❛❧❧②✳

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

❊①❡r❝✐s❡ ✽✳✶✸

1

■s sin x + sin 2x ♦r sin x + sin x ❛ ♣❡r✐♦❞✐❝ ❢✉♥❝✲ 2 t✐♦♥❄ ■❢ ✐t ✐s✱ ✜♥❞ ✐ts ♣❡r✐♦❞✳ ❨♦✉ ❤❛✈❡ t♦ ❥✉st✐❢② ②♦✉r ❝♦♥❝❧✉s✐♦♥ ❛❧❣❡❜r❛✐❝❛❧❧②✳ ❊①❡r❝✐s❡ ✽✳✶✹

✭❛✮ ❙t❛t❡ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❛ ♣❡r✐♦❞✐❝ ❢✉♥❝t✐♦♥✳ ✭❜✮ ●✐✈❡ ❛♥ ❡①❛♠♣❧❡ ♦❢ ❛ ♣❡r✐♦❞✐❝ ♣♦❧②♥♦♠✐❛❧✳ ❊①❡r❝✐s❡ ✽✳✶✺

Pr♦✈❡✱ ❢r♦♠ t❤❡ ❞❡✜♥✐t✐♦♥✱ t❤❛t t❤❡ ❢✉♥❝t✐♦♥ f (x) =

x2 + 1 ✐s ✐♥❝r❡❛s✐♥❣ ❢♦r x > 0✳

✹✳ ❚❤❡ ❢✉♥❝t✐♦♥ f (x) = x3 ✇✐t❤ ❞♦♠❛✐♥ [−3, 3] ❤❛s ❛t ❧❡❛st ♦♥❡ ✐♥♣✉t ✇❤✐❝❤ ♣r♦❞✉❝❡s ❛ s♠❛❧❧❡st ♦✉t♣✉t ✈❛❧✉❡✳ ✺✳ ❚❤❡ ❢✉♥❝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❡ ✽✳✷✶

●✐✈❡ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❛ ❝✐r❝❧❡✳

❊①❡r❝✐s❡ ✽✳✶✻

❚❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ y = f (x) ✐s ❣✐✈❡♥ ❜❡❧♦✇✳ ✭❛✮ ❋✐♥❞ ✐ts ❞♦♠❛✐♥✳ ✭❜✮ ❉❡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✳

❊①❡r❝✐s❡ ✽✳✶✼

■❢ ❛ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥ ❤❛s 10 ✈❡rt✐❝❛❧ ❛s②♠♣t♦t❡s✱ ❤♦✇ ♠❛♥② ❜r❛♥❝❤❡s ❞♦❡s ✐ts ❣r❛♣❤ ❤❛✈❡❄ ❊①❡r❝✐s❡ ✽✳✶✽

√

❋♦r t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ y = x + 8✱ ❛♥s✇❡r t❤❡ ❢♦❧❧♦✇✐♥❣ q✉❡st✐♦♥s✿ ■s t❤❡ ❣r❛♣❤ s②♠♠❡tr✐❝ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ x✲❛①✐s❄ ❚❤❡ y ✲❛①✐s❄ ❚❤❡ ♦r✐❣✐♥❄ ❊①❡r❝✐s❡ ✽✳✶✾

❉❡t❡r♠✐♥❡ ✇❤✐❝❤ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ st❛t❡♠❡♥ts ❛r❡ tr✉❡ ❛♥❞ ✇❤✐❝❤ ❛r❡ ❢❛❧s❡✳ ✶✳ ❚❤❡ ❢✉♥❝t✐♦♥ sin x ♦♥ t❤❡ ❞♦♠❛✐♥ (−π, π) ❤❛s ❛t ❧❡❛st ♦♥❡ ✐♥♣✉t ✇❤✐❝❤ ♣r♦❞✉❝❡s ❛ s♠❛❧❧❡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]

✾✳ ❊①❡r❝✐s❡s✿ ❈♦♠♣♦s✐t✐♦♥s

✺✵✷

✾✳ ❊①❡r❝✐s❡s✿ ❈♦♠♣♦s✐t✐♦♥s

❊①❡r❝✐s❡ ✾✳✶

❊①❡r❝✐s❡ ✾✳✼

h(x) = sin2 x + sin3 x ❛s g ◦ f ♦❢ t✇♦ ❢✉♥❝t✐♦♥s y = f (x)

❘❡♣r❡s❡♥t t❤❡ ❢✉♥❝t✐♦♥

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

t❤❡ ❝♦♠♣♦s✐t✐♦♥

♦❢ t✇♦ ❢✉♥❝t✐♦♥s✿

❛♥❞

z = g(y)✳

h(x) =

❊①❡r❝✐s❡ ✾✳✷

❋✉♥❝t✐♦♥

y = f (x)

2x3 + x.

❊①❡r❝✐s❡ ✾✳✽

✐s ❣✐✈❡♥ ❜❡❧♦✇ ❜② ❛ ❧✐st ✐ts ✈❛❧✲

✉❡s✳ ❋✐♥❞ ✐ts ✐♥✈❡rs❡ ❛♥❞ r❡♣r❡s❡♥t ✐t ❜② ❛ s✐♠✐❧❛r t❛❜❧❡✳

p

f ◦g

x 0 1 2 3 4 y = f (x) 0 1 2 4 3

h(x) = (g ◦ f )(x) ♦❢ t❤❡ ❢✉♥❝✲ t✐♦♥s y = f (x) = x −1 ❛♥❞ g(y) = 3y −1. ❊✈❛❧✉❛t❡ h(1)✳ ❋✐♥❞ t❤❡ ❝♦♠♣♦s✐t✐♦♥

2

❊①❡r❝✐s❡ ✾✳✾ ❊①❡r❝✐s❡ ✾✳✸

❘❡♣r❡s❡♥t t❤❡ ❢✉♥❝t✐♦♥

❋✐♥❞ t❤❡ ❢♦r♠✉❧❛s ♦❢ t❤❡ ✐♥✈❡rs❡s ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢✉♥❝t✐♦♥s✿ ✭❛✮

f (x) = (x + 1)3 ❀

✭❜✮

g(x) = ln(x3 )✳

❊①❡r❝✐s❡ ✾✳✹

❛s t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ t✇♦ ❢✉♥❝t✐♦♥s ♦♥❡ ♦❢ ✇❤✐❝❤ ✐s tr✐❣♦♥♦♠❡tr✐❝✳

❊①❡r❝✐s❡ ✾✳✶✵

❆r❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢✉♥❝t✐♦♥s ✐♥✈❡rt✐❜❧❡❄ ✶✳

f (n)

✐s

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

h(x) = 2 sin3 x + sin x + 5

nt❤

❞❛② ♦❢ t❤❡ ②❡❛r✳ ✷✳

t♦t❛❧ ❛❝❝✉♠✉❧❛t❡❞ r❛✐♥❢❛❧❧ ✐♥ ✐♥❝❤❡s

f (t)

✐s t❤❡

t ♦♥ ❛ ❣✐✈❡♥ ❞❛②

✐♥ ❛ ♣❛rt✐❝✉❧❛r ❧♦❝❛t✐♦♥✳

y = f (x) y = −f (x + 5) − 6✳

❝♦♠♣♦s✐t✐♦♥ ♦❢ t✇♦

❢♦r♠✉❧❛s ❢♦r t❤❡ t✇♦ ♣♦ss✐❜❧❡ ❝♦♠♣♦s✐t✐♦♥s ♦❢ t❤❡ t✇♦ ❢✉♥❝t✐♦♥s✿ ✏t❛❦❡ t❤❡ ❧♦❣❛r✐t❤♠ ❜❛s❡

2

♦❢ ✑ ❛♥❞

✏t❛❦❡ t❤❡ sq✉❛r❡ r♦♦t ♦❢ ✑✳

❊①❡r❝✐s❡ ✾✳✺

❚❤❡ ❣r❛♣❤ ♦❢

3

h(x) = ex −1 ✱ ❛s t❤❡ ❢✉♥❝t✐♦♥s f ❛♥❞ g ✳ ✭❜✮ Pr♦✈✐❞❡

✭❛✮ ❘❡♣r❡s❡♥t t❤❡ ❢✉♥❝t✐♦♥

❊①❡r❝✐s❡ ✾✳✶✶

✐s ♣❧♦tt❡❞ ❜❡❧♦✇✳

❙❦❡t❝❤

❙✉♣♣♦s❡ ❛ ❢✉♥❝t✐♦♥ t❤❡ ❧♦❣❛r✐t❤♠ ❜❛s❡

f ♣❡r❢♦r♠s t❤❡ ♦♣❡r❛t✐♦♥✿ ✏t❛❦❡ 2 ♦❢ ✑✱ ❛♥❞ ❢✉♥❝t✐♦♥ g ♣❡r❢♦r♠s✿

✏t❛❦❡ t❤❡ sq✉❛r❡ r♦♦t ♦❢ ✑✳ t❤❡ ✐♥✈❡rs❡s ♦❢

f

❛♥❞

t❤❡s❡ ❢♦✉r ❢✉♥❝t✐♦♥s✳

g✳

✭❛✮ ❱❡r❜❛❧❧② ❞❡s❝r✐❜❡

✭❜✮ ❋✐♥❞ t❤❡ ❢♦r♠✉❧❛s ❢♦r

✭❝✮ ●✐✈❡ t❤❡♠ ❞♦♠❛✐♥s ❛♥❞

❝♦❞♦♠❛✐♥s✳

❊①❡r❝✐s❡ ✾✳✶✷

✶✳ ❘❡♣r❡s❡♥t t❤❡ ❢✉♥❝t✐♦♥ t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ t✇♦ ❊①❡r❝✐s❡ ✾✳✻

f, g ✱

y = g(x) 0 4 3 0 1

y 0 1 2 3 4

f (g(x))

✜♥❞ t❤❡ t❛❜❧❡ ♦❢

f ◦ g✿ x 0 1 2 3 4

y = g(x) = 2x − 1✳

✷✳ Pr♦✈✐❞❡ ❛ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❝♦♠♣♦s✐t✐♦♥

●✐✈❡♥ t❤❡ t❛❜❧❡s ♦❢ ✈❛❧✉❡s ♦❢ ✈❛❧✉❡s ♦❢

p x2 − 1 ❛s ❢✉♥❝t✐♦♥s f ❛♥❞ g ✳ h(x) =

z = f (y) 4 4 0 1 2

❲❤❛t ✐❢ t❤❡ ❧❛st r♦✇s ✇❡r❡ ♠✐ss✐♥❣❄

♦❢

f (u) = u2 + u

❛♥❞

❊①❡r❝✐s❡ ✾✳✶✸

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

f (u) = sin u

❛♥❞

g(x) =

√

y = f (g(x))

x✳

❊①❡r❝✐s❡ ✾✳✶✹

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

2

f (u) = u − 3u + 2

❛♥❞

g(x) = x✳

y = f (g(x))

✾✳ ❊①❡r❝✐s❡s✿ ❈♦♠♣♦s✐t✐♦♥s

✺✵✸

❊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❡s❡♥t t❤❡ ❢✉♥❝t✐♦♥

h(x) =

√

❝♦♠♣♦s✐t✐♦♥ ♦❢ t✇♦ ❢✉♥❝t✐♦♥s✳

✷✳ ❘❡♣r❡s❡♥t t❤❡ ❢✉♥❝t✐♦♥

k(t) =

x − 1 ❛s t❤❡

p t2 − 1 ❛s t❤❡

❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤r❡❡ ❢✉♥❝t✐♦♥s✳

✸✳ ❘❡♣r❡s❡♥t t❤❡ ❢✉♥❝t✐♦♥

p(t) = sin

p

t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ ❢♦✉r ❢✉♥❝t✐♦♥s✳

t2 − 1

❊①❡r❝✐s❡ ✾✳✷✹ P❧♦t t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ✐♥✈❡rs❡ ♦❢ t❤✐s ❢✉♥❝t✐♦♥✿

❛s

❊①❡r❝✐s❡ ✾✳✶✼

f ◦ g ❢♦r t❤❡ ❢✉♥❝t✐♦♥s f (u) = u2 + 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❡ ✾✳✶✽ ■s t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ t✇♦ ❢✉♥❝t✐♦♥s t❤❛t ❛r❡ ♦❞❞✲ ✴❡✈❡♥ ♦❞❞✴❡✈❡♥❄

❊①❡r❝✐s❡ ✾✳✷✺

❊①❡r❝✐s❡ ✾✳✶✾

x3 + 1 ❘❡♣r❡s❡♥t t❤✐s ❢✉♥❝t✐♦♥✿ h(x) = , x3 − 1 ❝♦♠♣♦s✐t✐♦♥ ♦❢ t✇♦ ❢✉♥❝t✐♦♥s ♦❢ ✈❛r✐❛❜❧❡s x

h(x) = tan(2x) ❢✉♥❝t✐♦♥s ♦❢ ✈❛r✐❛❜❧❡s x

❘❡♣r❡s❡♥t t❤✐s ❢✉♥❝t✐♦♥✿

❛s t❤❡

❝♦♠♣♦s✐t✐♦♥ ♦❢ t✇♦

❛♥❞

y✳

❛s t❤❡ ❛♥❞

y✳

❊①❡r❝✐s❡ ✾✳✷✻

h(x) = (g ◦ f )(x) ♦❢ t❤❡ ❢✉♥❝✲ y−1 ✳ ❊✈❛❧✉❛t❡ t✐♦♥s y = f (x) = x −1 ❛♥❞ g(y) = y+1 h(0)✳ ❋✐♥❞ t❤❡ ❝♦♠♣♦s✐t✐♦♥

2

❊①❡r❝✐s❡ ✾✳✷✵ ❘❡♣r❡s❡♥t t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡s❡ t✇♦ ❢✉♥❝t✐♦♥s✿

p 1 f (x) = + 1 ❛♥❞ g(y) = y − 1✱ ❛s x t✐♦♥ h ♦❢ ✈❛r✐❛❜❧❡ x✳ ❉♦♥✬t s✐♠♣❧✐❢②✳

❛ s✐♥❣❧❡ ❢✉♥❝✲

❊①❡r❝✐s❡ ✾✳✷✼

h(x) = (g ◦ f )(x) ♦❢ t❤❡ ❢✉♥❝✲ y = f (x) = x2 −1 ❛♥❞ g(y) = 3y −1. ❊✈❛❧✉❛t❡

❋✐♥❞ t❤❡ ❝♦♠♣♦s✐t✐♦♥

❊①❡r❝✐s❡ ✾✳✷✶ ❋✉♥❝t✐♦♥ ✉❡s✳

y = f (x)

t✐♦♥s ✐s ❣✐✈❡♥ ❜❡❧♦✇ ❜② ❛ ❧✐st ✐ts ✈❛❧✲

h(1)✳

❋✐♥❞ ✐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❡ ✾✳✷✽ ❘❡♣r❡s❡♥t t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡s❡ t✇♦ ❢✉♥❝t✐♦♥s✿

f (x) = 1/x ❛♥❞ g(y) = ❊①❡r❝✐s❡ ✾✳✷✷

h

♦❢ ✈❛r✐❛❜❧❡

x✳

y2

y ✱ −3

❛s ❛ s✐♥❣❧❡ ❢✉♥❝t✐♦♥

❉♦♥✬t s✐♠♣❧✐❢②✳

●✐✈❡ ❡①❛♠♣❧❡s ♦❢ ❢✉♥❝t✐♦♥s t❤❛t ❛r❡ t❤❡✐r ♦✇♥ ✐♥✲ ✈❡rs❡s❄

❊①❡r❝✐s❡ ✾✳✷✾

❘❡♣r❡s❡♥t t❤✐s ❢✉♥❝t✐♦♥✿

P❧♦t t❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ s❤♦✇♥ ❜❡❧♦✇✱ ✐❢

❝♦♠♣♦s✐t✐♦♥ ♦❢ t✇♦ ❢✉♥❝t✐♦♥s ♦❢

♣♦ss✐❜❧❡✳

x3 + 1 , x3 − 1 ✈❛r✐❛❜❧❡s x

h(x) =

❊①❡r❝✐s❡ ✾✳✷✸

❛s t❤❡ ❛♥❞

y✳

✾✳ ❊①❡r❝✐s❡s✿ ❈♦♠♣♦s✐t✐♦♥s

✺✵✹

❊①❡r❝✐s❡ ✾✳✸✵

❊①❡r❝✐s❡ ✾✳✸✼

y = f (x)

❋✉♥❝t✐♦♥

✐s ❣✐✈❡♥ ❜❡❧♦✇ ❜② ❛ ❧✐st ♦❢ ✐ts

✈❛❧✉❡s✳ ■s t❤❡ ❢✉♥❝t✐♦♥ ♦♥❡✲t♦ ♦♥❡❄ ❲❤❛t ❛❜♦✉t ✐ts

❙❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❜❡✲ ❧♦✇✿

✐♥✈❡rs❡❄

x 0 1 2 3 4 y = f (x) 0 1 2 1 2

❊①❡r❝✐s❡ ✾✳✸✶

y = f (x)

❋✉♥❝t✐♦♥

✐s ❣✐✈❡♥ ❜❡❧♦✇ ❜② ❛ ❧✐st ♦❢ ✐ts

✈❛❧✉❡s✳ ■s t❤❡ ❢✉♥❝t✐♦♥ ♦♥❡✲t♦ ♦♥❡❄ ❲❤❛t ❛❜♦✉t ✐ts ❊①❡r❝✐s❡ ✾✳✸✽

✐♥✈❡rs❡❄

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

x 0 1 2 3 4 y = f (x) 7 5 3 4 6

♥♦t ♦♥❡✲t♦✲♦♥❡✿

f (x) = (x − 1)3

❛♥❞

g(x) = 2x−1 .

❊①❡r❝✐s❡ ✾✳✸✷

❋✉♥❝t✐♦♥s

y = f (x)

❛♥❞

u = g(y)

❛r❡ ❣✐✈❡♥ ❜❡❧♦✇

❜② t❛❜❧❡s ♦❢ s♦♠❡ ♦❢ t❤❡✐r ✈❛❧✉❡s✳ Pr❡s❡♥t t❤❡ ❝♦♠✲

u = h(x)

♣♦s✐t✐♦♥ t❛❜❧❡✿

♦❢ t❤❡s❡ ❢✉♥❝t✐♦♥s ❜② ❛ s✐♠✐❧❛r

x 0 1 2 3 4 y = f (x) 1 1 2 0 2 0 1 2 3 4 y u = g(y) 3 1 2 1 0

❊①❡r❝✐s❡ ✾✳✸✸

❋✉♥❝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❡ ✾✳✸✹

P❧♦t t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥

f (x) =

1 x−1

❛♥❞

t❤❡ ❣r❛♣❤ ♦❢ ✐ts ✐♥✈❡rs❡✳ ■❞❡♥t✐❢② ✐ts ✐♠♣♦rt❛♥t ❢❡❛✲ t✉r❡s✳

❊①❡r❝✐s❡ ✾✳✸✺

✭❛✮ ❆❧❣❡❜r❛✐❝❛❧❧②✱ s❤♦✇ t❤❛t t❤❡ ❢✉♥❝t✐♦♥

f (x) = x2

✐s ♥♦t ♦♥❡✲t♦✲♦♥❡✳ ✭❜✮ ●r❛♣❤✐❝❛❧❧②✱ s❤♦✇ t❤❛t t❤❡

g(x) = 2x+1 ♦❢ g ✳

❢✉♥❝t✐♦♥ ✐♥✈❡rs❡

✐s ♦♥❡✲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❡ ✾✳✸✾

❙❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ ❛❜♦✈❡ ❢✉♥❝t✐♦♥ ❛♥❞ ✭❛✮

y = x2 ✳

y = 2x❀

✭❜✮

y = x − 1❀

❛♥❞ ✭❝✮

✶✵✳ ❊①❡r❝✐s❡s✿ ❚r❛♥s❢♦r♠❛t✐♦♥s

✺✵✺

✶✵✳ ❊①❡r❝✐s❡s✿ ❚r❛♥s❢♦r♠❛t✐♦♥s

❊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❡ ✶✵✳✷

❖♥❡ ♦❢ t❤❡ ❣r❛♣❤s ❜❡❧♦✇ ✐s t❤❛t ♦❢ y = arctan x✳ ❲❤❛t ❛r❡ t❤❡ ♦t❤❡rs❄ ❊①❡r❝✐s❡ ✶✵✳✼

❚❤❡ ❣r❛♣❤ ♦❢ ♦♥❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥s ❜❡❧♦✇ ✐s y = ex ✳ ❲❤❛t ✐s t❤❡ ♦t❤❡r❄

❊①❡r❝✐s❡ ✶✵✳✸

❲❤❛t ❤❛♣♣❡♥s t♦ t❤❡ ❞♦♠❛✐♥ ❛♥❞ t❤❡ r❛♥❣❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ✉♥❞❡r t❤❡ s✐① ❜❛s✐❝ tr❛♥s❢♦r♠❛t✐♦♥s❄ ❊①❡r❝✐s❡ ✶✵✳✽ ❊①❡r❝✐s❡ ✶✵✳✹

❍♦✇ ❞♦ t❤❡ s✐① ❜❛s✐❝ tr❛♥s❢♦r♠❛t✐♦♥s ❛✛❡❝t ❛ ❢✉♥❝✲ t✐♦♥ ❜❡✐♥❣ ♦♥❡✲t♦✲♦♥❡ ♦r ♦♥t♦❄

❚❤❡ ❣r❛♣❤s ❜❡❧♦✇ ❛r❡ ♣❛r❛❜♦❧❛s✳ ❖♥❡ ✐s y = x2 ✳ ❲❤❛t ✐s t❤❡ ♦t❤❡r❄

❊①❡r❝✐s❡ ✶✵✳✺

❚❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ y = f (x) ✐s ❣✐✈❡♥ ❜❡✲ ❧♦✇✳ ❙❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢ y = 2f (3x) ❛♥❞ t❤❡♥ y = f (−x) − 1✳ ❊①❡r❝✐s❡ ✶✵✳✾

❚❤❡ ❣r❛♣❤ ❜❡❧♦✇ ✐s t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ f (x) = A sin x + B ❢♦r s♦♠❡ A ❛♥❞ B ✳ ❋✐♥❞ t❤❡s❡ ♥✉♠❜❡rs✳

❊①❡r❝✐s❡ ✶✵✳✻

❚❤❡ ❣r❛♣❤ ❞r❛✇♥ ✇✐t❤ ❛ s♦❧✐❞ ❧✐♥❡ ✐s y = x3 ✳ ❲❤❛t ❛r❡ t❤❡ ♦t❤❡r t✇♦❄

✶✵✳ ❊①❡r❝✐s❡s✿ ❚r❛♥s❢♦r♠❛t✐♦♥s

✺✵✻

❊①❡r❝✐s❡ ✶✵✳✶✵

❊①❡r❝✐s❡ ✶✵✳✶✻

f ✐s ❣✐✈❡♥ ❜❡❧♦✇✳ ❙❦❡t❝❤ y = 2f (x + 2) + 2✳ ❊①♣❧❛✐♥ ❤♦✇ ②♦✉

❚❤❡ ❣r❛♣❤ ♦❢ ❢✉♥❝t✐♦♥

t❤❡

❣r❛♣❤ ♦❢

❣❡t

✐t✳

❚❤❡ ❣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❡ ✶✵✳✶✶ ❇② tr❛♥s❢♦r♠✐♥❣ t❤❡ ❣r❛♣❤ ♦❢ ♦❢ t❤❡ ❢✉♥❝t✐♦♥

f (x) = 2e

x−3

y = ex ✱ ♣❧♦t t❤❡ ❣r❛♣❤ ✳ ■❞❡♥t✐❢② t❤❡ ❞♦♠❛✐♥✱

t❤❡ r❛♥❣❡✱ ❛♥❞ t❤❡ ❛s②♠♣t♦t❡s✳

❊①❡r❝✐s❡ ✶✵✳✶✼ ❲❤❛t ✐s t❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡s❡ t✇♦ ❢✉♥❝t✐♦♥s❄

❊①❡r❝✐s❡ ✶✵✳✶✷ ❍❛❧❢ ♦❢ t❤❡ ❣r❛♣❤ ♦❢ ❛♥ ❡✈❡♥ ❢✉♥❝t✐♦♥ ✐s s❤♦✇♥ ❜❡✲ ❧♦✇❀ ♣r♦✈✐❞❡ t❤❡ ♦t❤❡r ❤❛❧❢✿

❊①❡r❝✐s❡ ✶✵✳✶✽ P❧♦t t❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥ t❤❛t ✐s ❜♦t❤ ♦❞❞ ❛♥❞ ❡✈❡♥✳

❊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❡s✿ ❇❛s✐❝ ♠♦❞❡❧s

✺✵✼

✶✶✳ ❊①❡r❝✐s❡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✐♦♥ ♦❢ ❛ ❝✐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 ❧♦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❡s✿ ❇❛s✐❝ ♠♦❞❡❧s

✺✵✽

◆❡①t✳✳✳

✶✶✳ ❊①❡r❝✐s❡s✿ ❇❛s✐❝ ♠♦❞❡❧s

✺✵✾

■♥❞❡①

❛❜s♦❧✉t❡ ✈❛❧✉❡ ❢✉♥❝t✐♦♥✱ ✷✶✸ ❆❞❞✐t✐♦♥✲▼✉❧t✐♣❧✐❝❛t✐♦♥ ❘✉❧❡ ♦❢ ❊①♣♦♥❡♥ts✱ ✸✽✱ ✸✻✺ ❆❞❞✐t✐♦♥✲▼✉❧t✐♣❧✐❝❛t✐♦♥ ❘✉❧❡ ♦❢ ▲♦❣❛r✐t❤♠s✱ ✹✶✾ ❆❞❞✐t✐✈✐t② ❢♦r ❙✉♠s✱ ✽✹ ❛❧t❡r♥❛t✐♥❣ s❡q✉❡♥❝❡✱ ✹✹ ❛♠♣❧✐t✉❞❡✱ ✹✻✵ ❆◆❉✱ ✷✵✹ ❛r❝❝♦s✐♥❡✱ ✸✾✶ ❛r❝s✐♥❡✱ ✸✾✵ ❛r❝t❛♥❣❡♥t✱ ✸✾✷ ❆r✐t❤♠❡t✐❝ ❛♥❞ ●❡♦♠❡tr✐❝ Pr♦❣r❡ss✐♦♥s✿ ❋♦r♠✉❧❛s✱ ✸✼ ❛r✐t❤♠❡t✐❝ ♣r♦❣r❡ss✐♦♥✱ ✸✷✱ ✸✼✱ ✹✷✱ ✻✺ ❜❛s❡ ♦❢ ❡①♣♦♥❡♥t✱ ✸✻✵ ❜✐♥♦♠✐❛❧ ❝♦❡✣❝✐❡♥t✱ ✻✶✱ ✻✸ ❇✐♥♦♠✐❛❧ ❈♦❡✣❝✐❡♥t ❋♦r♠✉❧❛✱ ✻✷ ❜✐♥♦♠✐❛❧ ❡①♣❛♥s✐♦♥✱ ✺✽✱ ✻✵ ❇✐♥♦♠✐❛❧ ❚❤❡♦r❡♠✱ ✻✵✱ ✼✻ ❜♦✉♥❞ ♦❢ s❡t✱ ✶✸✶ ❜♦✉♥❞❡❞ ❢✉♥❝t✐♦♥✱ ✶✾✺ ❜♦✉♥❞❡❞ s❡t✱ ✶✸✶ ❇♦✉♥❞❡❞♥❡ss ♦❢ ❚r✐❣♦♥♦♠❡tr✐❝ ❋✉♥❝t✐♦♥s✱ ✸✽✼ ❈❛♥❝❡❧❧❛t✐♦♥ ▲❛✇s ♦❢ ▲♦❣❛r✐t❤♠s✱ ✸✼✼ ❈❛rt❡s✐❛♥ s②st❡♠✱ ✶✸✻✱ ✹✹✵✱ ✹✻✽✱ ✹✻✾✱ ✹✼✶✱ ✹✼✷✱ ✹✼✹ ❈❡♥t❡r❡❞ ❋♦r♠ ♦❢ ❈✐r❝❧❡✱ ✹✹✾ ❈❤♦♦s❡ ♥ ❋r♦♠ ♠✱ ✻✸ ❝✐r❝❧❡✱ ✹✹✷✕✹✹✹✱ ✹✹✾✱ ✹✺✵ ❈✐r❝❧❡ ❛s ❚✇♦ ●r❛♣❤s✱ ✹✺✵ ❝♦❞♦♠❛✐♥ ♦❢ ❢✉♥❝t✐♦♥✱ ✶✶✶ ❝♦♠♣❧❡t❡ sq✉❛r❡✱ ✷✽✻ ❝♦♠♣♦♥❡♥ts ♦❢ ✈❡❝t♦r✱ ✹✼✶ ❝♦♠♣♦s✐t✐♦♥✱ ✷✸✺✱ ✷✸✻✱ ✷✹✽✱ ✷✽✷✱ ✷✾✸✕✷✾✺✱ ✷✾✼✱ ✸✵✶ ❈♦♠♣♦s✐t✐♦♥ ♦❢ ❘♦♦ts✱ ✸✺✻ ❈♦♠♣♦s✐t✐♦♥ ✇✐t❤ ■❞❡♥t✐t② ❋✉♥❝t✐♦♥✱ ✷✾✺ ❈♦♠♣♦s✐t✐♦♥s ✇✐t❤ ❈♦♥st❛♥t ❋✉♥❝t✐♦♥✱ ✷✾✸ ❝♦♥st❛♥t ❢✉♥❝t✐♦♥✱ ✷✾✷✱ ✷✾✸ ❝♦♥st❛♥t ♠✉❧t✐♣❧❡ ♦❢ ❢✉♥❝t✐♦♥✱ ✸✾✾ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ❢♦r ❉✐✛❡r❡♥❝❡s✱ ✾✹ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ❢♦r ❙✉♠s✱ ✾✺ ❝♦♥st❛♥t s❡q✉❡♥❝❡✱ ✷✽✱ ✻✺✱ ✻✽✱ ✽✵ ❝♦♥st❛♥t t❡r♠✱ ✸✸✶ ❈♦♥st❛♥t ❚❡r♠ ■s ②✲■♥t❡r❝❡♣t✱ ✸✸✷ ❈♦♥t✐♥✉✐t② ♦❢ P♦❧②♥♦♠✐❛❧s✱ ✸✸✽ ❈♦♥t✐♥✉✐t② ♦❢ ❘❛t✐♦♥❛❧ ❋✉♥❝t✐♦♥s✱ ✸✹✾ ❝♦♥✈❡rs❡✱ ✶✹ ❝♦♦r❞✐♥❛t❡s✱ ✶✸✻ ❝♦s✐♥❡✱ ✸✼✾✱ ✸✽✸✱ ✸✽✼✱ ✸✽✽✱ ✸✾✶✱ ✹✺✶✱ ✹✺✺✕✹✺✽✱ ✹✻✺ ❞❡❝r❡❛s✐♥❣ ❢✉♥❝t✐♦♥✱ ✸✵✸✕✸✵✺✱ ✸✵✽ ❞❡❝r❡❛s✐♥❣ s❡q✉❡♥❝❡✱ ✸✵ ❉❡❣r❡❡ ♦❢ ❋❛❝t♦r❡❞ P♦❧②♥♦♠✐❛❧✱ ✸✸✹ ❞❡❣r❡❡ ♦❢ ♣♦❧②♥♦♠✐❛❧✱ ✸✸✵✱ ✸✸✹

❞❡❣r❡❡ ♦❢ ♣♦✇❡r✱ ✶✽✼ ❞❡❣r❡❡s✱ ✸✽✷ ❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡✱ ✶✶✶ ❉✐✛❡r❡♥❝❡ ♦❢ ❆r✐t❤♠❡t✐❝ Pr♦❣r❡ss✐♦♥✱ ✻✺ ❉✐✛❡r❡♥❝❡ ♦❢ ❈♦♥st❛♥t ❙❡q✉❡♥❝❡✱ ✻✽ ❞✐✛❡r❡♥❝❡ ♦❢ ❢✉♥❝t✐♦♥s✱ ✸✾✾ ❉✐✛❡r❡♥❝❡ ♦❢ ●❡♦♠❡tr✐❝ Pr♦❣r❡ss✐♦♥✱ ✻✻ ❞✐✛❡r❡♥❝❡ ♦❢ s❡q✉❡♥❝❡✱ ✻✹✕✻✻✱ ✻✽✕✼✵✱ ✾✷✱ ✾✹✱ ✾✻✱ ✾✼ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✱ ✶✽✸✱ ✶✾✵✱ ✸✻✼✱ ✸✽✾✱ ✹✼✻ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ s❡q✉❡♥❝❡✱ ✼✶ ❉✐✛❡r❡♥❝❡ ✉♥❞❡r ❙✉❜tr❛❝t✐♦♥✱ ✻✾ ❉✐r❡❝t✐♦♥s ❢♦r ❉✐♠❡♥s✐♦♥ ✶✱ ✹✻✼ ❉✐r❡❝t✐♦♥s ❢♦r ❉✐♠❡♥s✐♦♥ ✷✱ ✹✼✷ ❞✐s❝r✐♠✐♥❛♥t ♦❢ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧✱ ✸✷✺✱ ✸✷✻ ❞✐s❦✱ ✹✹✹ ❉✐s❦ ✈✐❛ ■♥❡q✉❛❧✐t②✱ ✹✹✹ ❞✐st❛♥❝❡✱ ✹✸✼✱ ✹✹✶✱ ✹✹✸✱ ✹✹✹ ❉✐st❛♥❝❡ ❋♦r♠✉❧❛ ❢♦r ❉✐♠❡♥s✐♦♥ ✶✱ ✹✸✽ ❉✐st❛♥❝❡ ❋♦r♠✉❧❛ ❢♦r ❉✐♠❡♥s✐♦♥ ✷✱ ✹✹✵ ❉✐st❛♥❝❡s ♦♥ ▲✐♥❡✱ ✹✸✼ ❉✐str✐❜✉t✐✈❡ ❘✉❧❡ ♦❢ ❊①♣♦♥❡♥ts✱ ✸✽✱ ✸✻✺ ❞♦♠❛✐♥ ♦❢ ❢✉♥❝t✐♦♥✱ ✶✶✶✱ ✷✾✽ ❉♦♠❛✐♥ ♦❢ P♦❧②♥♦♠✐❛❧✱ ✸✷✾ ❉♦♠❛✐♥ ♦❢ ❚❛♥❣❡♥t✱ ✸✽✻ ❞♦t ♣r♦❞✉❝t✱ ✹✼✷ ❡❧❡♠❡♥ts ♦❢ s❡t✱ ✶✵✵ ❡♠♣t② s❡t✱ ✶✷✻ ❡q✉❛❧ ❢✉♥❝t✐♦♥s✱ ✷✾✺✱ ✸✾✻ ❡q✉❛❧ s❡ts✱ ✶✵✶ ❡q✉❛t✐♦♥✱ ✶✷✻✱ ✹✶✵✱ ✹✶✽ ❊q✉❛t✐♦♥ ♦❢ ❈✐r❝❧❡✱ ✹✹✸ ❊q✉❛t✐♦♥s✿ ❇❛s✐❝ ❆❧❣❡❜r❛✱ ✹✶✵ ❊q✉❛t✐♦♥s✿ ●❡♥❡r❛❧ ❆❧❣❡❜r❛✱ ✹✶✽ ❡q✉✐✈❛❧❡♥❝❡✱ ✶✹ ❊✈❡♥ ❉❡❣r❡❡ ❘♦♦ts✱ ✸✺✺ ❡✈❡♥ ❢✉♥❝t✐♦♥✱ ✸✶✻✱ ✸✽✽ ❡①♣♦♥❡♥t✱ ✸✻✵✱ ✸✼✼✱ ✹✷✽ ❊①♣♦♥❡♥t ❛♥❞ ▲♦❣❛r✐t❤♠ ❇❛s❡ ❈♦♥✈❡rs✐♦♥ ❋♦r♠✉❧❛s✱ ✹✷✻ ❊①♣♦♥❡♥t ❊q✉❛❧ t♦ ✲✶✱ ✺✸ ❡①♣♦♥❡♥t✐❛❧ ❞❡❝❛②✱ ✹✷✼ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥✱ ✸✻✶✱ ✸✻✺✱ ✸✻✻✱ ✸✻✽✱ ✸✻✾✱ ✸✼✷✱ ✸✼✸✱ ✹✷✼ ❊①♣♦♥❡♥t✐❛❧ ❋✉♥❝t✐♦♥ ■s ■♥✈❡rt✐❜❧❡✱ ✸✻✾ ❊①♣♦♥❡♥t✐❛❧ ❋✉♥❝t✐♦♥✿ ❇❛s✐❝ ❋❛❝ts✱ ✸✻✽ ❡①♣♦♥❡♥t✐❛❧ ❣r♦✇t❤✱ ✹✷✼ ❡①♣♦♥❡♥t✐❛❧ ♠♦❞❡❧✱ ✹✷✼ ❡①tr❡♠✉♠ ♦❢ ❢✉♥❝t✐♦♥✱ ✸✵✾ ❋❛❝t♦r❡❞ ❋♦r♠ ♦❢ ◗✉❛❞r❛t✐❝ P♦❧②♥♦♠✐❛❧✱ ✸✷✺ ❢❛❝t♦r✐❛❧✱ ✸✹✱ ✸✻ ❢❛❝t♦r✐❛❧ ♦❢ ③❡r♦✱ ✹✷

✺✶✶ ❢❛❝t♦r✐♥❣ ✐♥t❡❣❡rs✱ ✺✶ ❢❛❝t♦r✐③❛t✐♦♥✱ ✸✸✹ ❢❛❝t♦rs✱ ✸✵✺✱ ✸✸✹✱ ✸✸✽✱ ✸✹✾ ❋♦r♠✉❧❛ ❢♦r ❆❧t❡r♥❛t✐♥❣ ❙❡q✉❡♥❝❡✱ ✹✹ ❋♦r♠✉❧❛ ❢♦r ❆r✐t❤♠❡t✐❝ Pr♦❣r❡ss✐♦♥✱ ✹✷ ❋♦r♠✉❧❛ ❢♦r ●❡♦♠❡tr✐❝ Pr♦❣r❡ss✐♦♥✱ ✹✷ ❋♦r♠✉❧❛s ♦❢ ❚r❛♥s❢♦r♠❛t✐♦♥s ♦❢ P❧❛♥❡✱ ✷✼✵ ❢r❡q✉❡♥❝②✱ ✹✻✶ ❢✉♥❝t✐♦♥✱ ✶✶✶✱ ✶✶✸✱ ✶✹✸✱ ✶✺✼✱ ✸✾✸✱ ✸✾✻ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❆❧❣❡❜r❛✱ ✸✸✹ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❆r✐t❤♠❡t✐❝✱ ✺✶✱ ✸✸✹ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s ♦❢ ❙❡q✉❡♥❝❡s ■✱ ✽✽ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s ♦❢ ❙❡q✉❡♥❝❡s ■■✱ ✽✽ ❣❡♦♠❡tr✐❝ ❣r♦✇t❤ ❛♥❞ ❞❡❝❛②✱ ✸✸ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥✱ ✸✸✱ ✸✼✱ ✹✷✱ ✺✺✱ ✻✻✱ ✼✽ ❣❧♦❜❛❧ ❡①tr❡♠❡ ♣♦✐♥ts✱ ✸✵✾ ❣r❛♣❤ ♦❢ ❢✉♥❝t✐♦♥✱ ✶✺✹✱ ✶✻✺✱ ✹✺✵ ●r❛♣❤ ♦❢ ▲✐♥❡❛r ❘❡❧❛t✐♦♥✱ ✶✹✵ ❣r❛♣❤ ♦❢ r❡❧❛t✐♦♥✱ ✶✸✾✱ ✹✺✵ ❍♦r✐③♦♥t❛❧ ❍♦r✐③♦♥t❛❧ ❍♦r✐③♦♥t❛❧ ❍♦r✐③♦♥t❛❧

❋❧✐♣✱ ✷✼✾ ▲✐♥❡ ❚❡st✱ ✶✺✼✱ ✷✸✵ ❙❤✐❢t✱ ✷✼✻ ❙tr❡t❝❤✱ ✷✽✶

✐❞❡♥t✐t② ❢✉♥❝t✐♦♥✱ ✷✾✹✱ ✷✾✺ ■❋✲❆◆❉✲❖◆▲❨✲■❋✱ ✶✹ ■❋✲❚❍❊◆✱ ✶✸ ✐♠❛❣❡ ♦❢ ❢✉♥❝t✐♦♥✱ ✶✾✸✱ ✷✸✷✱ ✸✷✽ ✐♠♣❧✐❝❛t✐♦♥✱ ✶✸✱ ✶✹ ✐♠♣❧✐❡❞ ❞♦♠❛✐♥ ♦❢ ❢✉♥❝t✐♦♥✱ ✶✻✷ ✐♥❝❧✉s✐♦♥ ❢✉♥❝t✐♦♥✱ ✸✵✵ ✐♥❝r❡❛s✐♥❣ ❢✉♥❝t✐♦♥✱ ✸✵✸✕✸✵✺✱ ✸✵✽ ✐♥❝r❡❛s✐♥❣ s❡q✉❡♥❝❡✱ ✸✵ ✐♥❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡✱ ✶✶✶ ✐♥❞❡① ♦❢ t❡r♠s ♦❢ s❡q✉❡♥❝❡✱ ✷✻ ■♥❡q✉❛❧✐t✐❡s✿ ❇❛s✐❝ ❆❧❣❡❜r❛✱ ✹✽✶ ■♥❡q✉❛❧✐t✐❡s✿ ●❡♥❡r❛❧ ❆❧❣❡❜r❛✱ ✹✽✺ ✐♥❡q✉❛❧✐t②✱ ✹✽✶✱ ✹✽✺ ✐♥t❡❣❡r ✈❛❧✉❡ ❢✉♥❝t✐♦♥✱ ✷✶✹ ✐♥t❡rs❡❝t✐♦♥ ♦❢ s❡ts✱ ✷✵✶✱ ✷✵✹ ✐♥t❡r✈❛❧✱ ✶✷✽✱ ✶✷✾ ✐♥t❡r✈❛❧ ♥♦t❛t✐♦♥✱ ✶✷✽✱ ✶✷✾ ✐♥✈❡rs❡ ❢✉♥❝t✐♦♥✱ ✷✹✺✱ ✷✹✻✱ ✷✹✽✱ ✷✾✼ ■♥✈❡rs❡ ♦❢ ▲✐♥❡❛r P♦❧②♥♦♠✐❛❧✱ ✷✺✶ ■♥✈❡rs❡ ✈✐❛ ❈♦♠♣♦s✐t✐♦♥s✱ ✷✹✽✱ ✷✾✼ ✐♥✈❡rt✐❜❧❡ ❢✉♥❝t✐♦♥✱ ✷✹✾✱ ✷✺✼ ✐rr❡❞✉❝✐❜❧❡ ❢❛❝t♦r✱ ✸✷✻✱ ✸✸✹ ■rr❡❞✉❝✐❜❧❡ ◗✉❛❞r❛t✐❝ P♦❧②♥♦♠✐❛❧✱ ✸✷✼ ❧❡❛❞✐♥❣ t❡r♠ ♦❢ ♣♦❧②♥♦♠✐❛❧✱ ✸✸✶ ▲❡❛❞✐♥❣ ❚❡r♠ ✈s✳ ❉❡❣r❡❡ ♦❢ P♦❧②♥♦♠✐❛❧✱ ✸✸✶ ▲✐♥❡❛r ❋❛❝t♦r ❚❤❡♦r❡♠✱ ✸✸✺ ❧✐♥❡❛r ❢✉♥❝t✐♦♥✱ ✶✹✸✱ ✶✼✻✱ ✸✵✺ ▲✐♥❡❛r ❋✉♥❝t✐♦♥s✱ ❖♥❡✲t♦✲♦♥❡ ❖♥t♦✱ ✷✸✶ ❧✐♥❡❛r ♣♦❧②♥♦♠✐❛❧✱ ✶✼✻✱ ✷✺✶ ❧✐♥❡❛r r❡❧❛t✐♦♥✱ ✶✹✵✱ ✶✹✸ ▲✐♥❡❛r ❚r❛♥s❢♦r♠❛t✐♦♥s ✐♥ ❉✐♠❡♥s✐♦♥ ✶✱ ✹✸✽

❧✐st ♥♦t❛t✐♦♥ ❢♦r s❡ts✱ ✶✵✵ ❧♦❣❛r✐t❤♠✱ ✸✼✸✱ ✸✼✺✱ ✸✼✼✱ ✹✶✾✱ ✹✷✷✱ ✹✷✻✱ ✹✷✽ ♠❛❣♥✐t✉❞❡ ♦❢ ✈❡❝t♦r✱ ✹✼✶ ♠❛①✐♠✉♠ ♦❢ ❢✉♥❝t✐♦♥✱ ✸✵✾✱ ✸✷✼ ♠❛①✐♠✉♠ ♦❢ s❡t✱ ✶✸✵ ▼❛①✐♠✉♠✲▼✐♥✐♠✉♠ ♦❢ ◗✉❛❞r❛t✐❝ P♦❧②♥♦♠✐❛❧✱ ✸✷✼ ♠✐♥✐♠✉♠ ♦❢ ❢✉♥❝t✐♦♥✱ ✸✵✾✱ ✸✷✼ ♠✐♥✐♠✉♠ ♦❢ s❡t✱ ✶✸✵ ♠♦♥♦t♦♥❡ ❢✉♥❝t✐♦♥✱ ✸✵✸✕✸✵✺✱ ✸✵✽✱ ✸✻✾ ♠♦♥♦t♦♥❡ s❡q✉❡♥❝❡✱ ✸✵✱ ✻✽ ▼♦♥♦t♦♥❡ ✈s✳ ❖♥❡✲t♦✲♦♥❡✱ ✸✵✽ ♠♦♥♦t♦♥✐❝✐t②✱ ✸✵✸ ▼♦♥♦t♦♥✐❝✐t② ❛♥❞ ❙✉❜tr❛❝t✐♦♥✱ ✼✵ ▼♦♥♦t♦♥✐❝✐t② ♦❢ ❊①♣♦♥❡♥t✱ ✸✻✾ ▼♦♥♦t♦♥✐❝✐t② ♦❢ ●❡♦♠❡tr✐❝ Pr♦❣r❡ss✐♦♥✱ ✺✺ ▼♦♥♦t♦♥✐❝✐t② ♦❢ ▲✐♥❡❛r ❋✉♥❝t✐♦♥s✱ ✸✵✺ ▼♦♥♦t♦♥✐❝✐t② ♦❢ ▲♦❣❛r✐t❤♠✱ ✸✼✺ ▼♦♥♦t♦♥✐❝✐t② ♦❢ ❙✉♠✱ ✽✵ ▼♦♥♦t♦♥✐❝✐t② ❚❤❡♦r❡♠ ❢♦r ❙❡q✉❡♥❝❡s✱ ✻✽ ▼✉❧t✐♣❧✐❝❛t✐♦♥✲❊①♣♦♥❡♥t✐❛t✐♦♥ ❘✉❧❡ ♦❢ ❊①♣♦♥❡♥ts✱ ✸✾✱ ✸✻✻ ▼✉❧t✐♣❧✐❝❛t✐♦♥✲❊①♣♦♥❡♥t✐❛t✐♦♥ ❘✉❧❡ ♦❢ ▲♦❣❛r✐t❤♠s✱ ✹✷✷ ♠✉❧t✐♣❧✐❝✐t② ♦❢ ❢❛❝t♦r✱ ✺✶✱ ✸✸✺ ▼✉❧t✐♣❧✐❝✐t② ❘✉❧❡ ❢♦r P♦❧②♥♦♠✐❛❧s✱ ✸✸✻ ▼✉❧t✐♣❧✐❝✐t② ❘✉❧❡ ❢♦r ❘❛t✐♦♥❛❧ ❋✉♥❝t✐♦♥s✱ ✸✹✼ ♥❛t✉r❛❧ ❞♦♠❛✐♥ ♦❢ ❢✉♥❝t✐♦♥✱ ✶✻✷ ◆❛t✉r❛❧ ▲♦❣ ❛♥❞ ❊①♣ ❋♦r♠✉❧❛s✱ ✹✷✽ ♥❛t✉r❛❧ ❧♦❣❛r✐t❤♠✱ ✹✷✻ ♥❡❣❛t✐✈❡ ❡①♣♦♥❡♥t✱ ✺✸ ◆❡❣❛t✐✈❡ ❊①♣♦♥❡♥t ❘✉❧❡✱ ✺✹ ♥♦t ❡q✉❛❧ ❢✉♥❝t✐♦♥s✱ ✷✾✻✱ ✸✾✻ ♥t❤ r♦♦t✱ ✸✺✹✕✸✺✻✱ ✸✻✺✱ ✸✻✻✱ ✸✻✽ ♥✉♠❜❡r ❧✐♥❡✱ ✷✷ ◆✉♠❜❡r ♦❢ P❡r♠✉t❛t✐♦♥s✱ ✸✻ ◆✉♠❜❡r ♦❢ ❚❡r♠s ✐♥ ❇✐♥♦♠✐❛❧ ❊①♣❛♥s✐♦♥✱ ✺✽ ◆✉♠❜❡r ♦❢ ①✲■♥t❡r❝❡♣ts✱ ✸✸✺ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥✱ ✶✶✷ ❖❞❞ ❉❡❣r❡❡ ❘♦♦ts✱ ✸✺✺ ♦❞❞ ❢✉♥❝t✐♦♥✱ ✸✶✽✱ ✸✽✽ ❖❞❞✲❊✈❡♥ ❚r✐❣ ❋✉♥❝t✐♦♥s✱ ✸✽✽ ❖♥❡✲t♦✲♦♥❡ ❛♥❞ ❖♥t♦ ✈s✳ ■♠❛❣❡✱ ✷✸✷ ♦♥❡✲t♦✲♦♥❡ ❢✉♥❝t✐♦♥✱ ✷✷✻✱ ✷✸✵✱ ✷✸✷✱ ✷✹✻✱ ✷✺✼✱ ✸✵✽✱ ✸✻✾ ❖♥❡✲t♦✲♦♥❡ ❖♥t♦ ✈s✳ ■♥✈❡rs❡✱ ✷✹✻ ❖◆▲❨✲■❋✱ ✶✹ ♦♥t♦ ❢✉♥❝t✐♦♥✱ ✷✷✺✱ ✷✸✵✱ ✷✸✷✱ ✷✹✻✱ ✷✺✼ ❖❘✱ ✷✵✹ ♣❛r❛❜♦❧❛✱ ✷✽✽ P❛r❛❧❧❡❧ ▲✐♥❡s✱ ❙❛♠❡ ❙❧♦♣❡✱ ✹✻✾ P❛r❛❧❧❡❧ ▲✐♥❡s✿ ❇❛s✐❝ ❋❛❝ts✱ ✹✻✾ P❛s❝❛❧ ❚r✐❛♥❣❧❡✱ ✺✾ ♣❡r✐♦❞✱ ✸✶✹ ♣❡r✐♦❞✐❝ ❢✉♥❝t✐♦♥✱ ✸✶✹✱ ✸✽✼ P❡r✐♦❞✐❝✐t② ♦❢ ❚r✐❣♦♥♦♠❡tr✐❝ ❋✉♥❝t✐♦♥s✱ ✸✽✼ ♣❡r♠✉t❛t✐♦♥s✱ ✸✻

✺✶✷ ♣❤❛s❡✱ ✹✻✶ ♣✐❡❝❡✇✐s❡✲❞❡✜♥❡❞ ❢✉♥❝t✐♦♥✱ ✷✵✽ P♦✐♥t✲❙❧♦♣❡ ❋♦r♠ ♦❢ ▲✐♥❡✱ ✶✼✽ ♣♦❧②♥♦♠✐❛❧✱ ✸✷✽✕✸✸✷✱ ✸✸✹✕✸✸✽✱ ✸✹✸✱ ✸✹✹ ♣♦s✐t✐✈❡❧② ❛♥❞ ♥❡❣❛t✐✈❡❧② ♦r✐❡♥t❡❞ s❡❣♠❡♥ts✱ ✹✻✼ ♣♦✇❡r ❢✉♥❝t✐♦♥s✱ ✶✽✼✱ ✷✺✼✱ ✸✺✹ P♦✇❡r ❋✉♥❝t✐♦♥s✱ ❈❧❛ss✐✜❝❛t✐♦♥✱ ✷✺✼ ♣♦✇❡r s❡q✉❡♥❝❡✱ ✷✾ ♣r❡✐♠❛❣❡ ♦❢ ✈❛❧✉❡✱ ✶✾✻ ♣r✐♠❡ ♥✉♠❜❡r✱ ✺✶ ♣r♦❞✉❝t ♦❢ ❢✉♥❝t✐♦♥s✱ ✹✵✵ Pr♦❞✉❝t ♦❢ ❘♦♦ts✱ ✸✺✺ Pr♦❞✉❝t ❘✉❧❡ ❢♦r ❉✐✛❡r❡♥❝❡s✱ ✾✻ P②t❤❛❣♦r❡❛♥ ❚❤❡♦r❡♠✱ ✹✸✾ P②t❤❛❣♦r❡❛♥ ❚❤❡♦r❡♠ ♦❢ ❚r✐❣♦♥♦♠❡tr②✱ ✹✺✶ ◗✉❛❞r❛t✐❝ ❋♦r♠✉❧❛✱ ✸✷✹ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧✱ ✷✽✼✱ ✸✷✸✕✸✷✽ q✉♦t✐❡♥t ♦❢ ❢✉♥❝t✐♦♥s✱ ✹✵✶ ◗✉♦t✐❡♥t ❘✉❧❡ ❢♦r ❉✐✛❡r❡♥❝❡s✱ ✾✼ r❛❞✐❛♥s✱ ✸✽✷ r❛♥❣❡ ♦❢ ❢✉♥❝t✐♦♥✱ ✶✾✸✱ ✸✷✽ ❘❛♥❣❡ ♦❢ ▲✐♥❡❛r ❋✉♥❝t✐♦♥✱ ✶✾✸ ❘❛♥❣❡ ♦❢ ◗✉❛❞r❛t✐❝ P♦❧②♥♦♠✐❛❧✱ ✸✷✽ r❛t✐♦♥❛❧ ❡①♣♦♥❡♥t✱ ✸✻✺✱ ✸✻✻✱ ✸✻✽ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥✱ ✸✹✸✕✸✹✺✱ ✸✹✼✱ ✸✹✾ ❘❛t✐♦♥❛❧ ❋✉♥❝t✐♦♥✬s ①✲■♥t❡r❝❡♣ts✱ ✸✹✺ r❡❝✐♣r♦❝❛❧ ❡①♣♦♥❡♥t✱ ✸✻✹ r❡❝✉rs✐✈❡✱ ✸✷✕✸✹✱ ✹✷✱ ✹✽✱ ✽✼✱ ✹✸✻ r❡❧❛t✐♦♥✱ ✶✵✹✱ ✶✶✶✱ ✶✹✸✱ ✶✺✼ r❡♣❡❛t❡❞ ❛❞❞✐t✐♦♥✱ ✸✼✕✸✾✱ ✹✶✱ ✺✶✱ ✺✹✱ ✸✸✹✱ ✸✹✹✱ ✸✻✵✱ ✸✻✽ r❡♣❡❛t❡❞ ♠✉❧t✐♣❧✐❝❛t✐♦♥✱ ✸✼✕✸✾✱ ✹✶✱ ✺✶✱ ✺✹✱ ✸✸✹✱ ✸✹✹✱ ✸✻✵✱ ✸✻✽ ❘❡str✐❝t❡❞ ❈♦s✐♥❡✱ ✸✾✶ ❘❡str✐❝t❡❞ ❙✐♥❡✱ ✸✾✵ ❘❡str✐❝t❡❞ ❚❛♥❣❡♥t✱ ✸✾✷ r❡str✐❝t✐♦♥✱ ✸✵✶ r❡str✐❝t✐♦♥ ♦❢ ❢✉♥❝t✐♦♥✱ ✷✾✽ ❘❡str✐❝t✐♦♥ ✈✐❛ ❈♦♠♣♦s✐t✐♦♥s✱ ✸✵✶ r♦♦ts ♦❢ ♣♦❧②♥♦♠✐❛❧✱ ✸✸✺✱ ✸✹✺ r♦♦ts ♦❢ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧✱ ✸✷✸✕✸✷✻ ❘✉❧❡s ♦❢ ❊①♣♦♥❡♥ts✱ ✸✼✷ ❘✉❧❡s ♦❢ ❋❛❝t♦r✐♥❣ ■♥t❡❣❡rs✱ ✺✶ s❡q✉❡♥❝❡✱ ✷✻ s❡q✉❡♥❝❡ ❛s ❢✉♥❝t✐♦♥✱ ✶✶✷ s❡q✉❡♥❝❡ ♦❢ ❞✐✛❡r❡♥❝❡s✱ ✻✹✱ ✽✽✱ ✾✷✱ ✾✹✱ ✾✻✱ ✾✼ s❡q✉❡♥❝❡ ♦❢ s✉♠s✱ ✼✸✱ ✼✻✱ ✽✽✱ ✾✸✱ ✾✺ s❡t✱ ✶✵✶ s❡t✲❜✉✐❧❞✐♥❣ ♥♦t❛t✐♦♥✱ ✶✷✻✱ ✶✷✼✱ ✶✸✼✱ ✶✹✵✱ ✶✹✷✱ ✶✻✷✱ ✶✻✺✱ ✷✵✷✱ ✹✹✷ s✐❣♠❛ ♥♦t❛t✐♦♥✱ ✼✹ s✐❣♥ ❢✉♥❝t✐♦♥✱ ✷✶✷ s✐♥❡✱ ✸✼✾✱ ✸✽✸✱ ✸✽✼✱ ✸✽✽✱ ✸✾✵✱ ✹✺✶✱ ✹✺✺✕✹✺✽✱ ✹✻✵✱ ✹✻✺ ❙✐♥❡ ❛♥❞ ❈♦s✐♥❡ ♦❢ ❆✈❡r❛❣❡ ❆♥❣❧❡✱ ✹✺✽ ❙✐♥❡ ❛♥❞ ❈♦s✐♥❡ ♦❢ ❈♦♠♣❧❡♠❡♥t❛r② ❆♥❣❧❡s✱ ✹✺✺ ❙✐♥❡ ❛♥❞ ❈♦s✐♥❡ ♦❢ ❉✐✛❡r❡♥❝❡✱ ✹✺✼

❙✐♥❡ ❛♥❞ ❈♦s✐♥❡ ♦❢ ❉✐r❡❝t✐♦♥✱ ✹✻✽ ❙✐♥❡ ❛♥❞ ❈♦s✐♥❡ ♦❢ ❉♦✉❜❧❡ ❆♥❣❧❡✱ ✹✺✼ ❙✐♥❡ ❛♥❞ ❈♦s✐♥❡ ♦❢ ❍❛❧❢✲❆♥❣❧❡✱ ✹✺✼ ❙✐♥❡ ❛♥❞ ❈♦s✐♥❡ ♦❢ ❙✉♠✱ ✹✺✻ s✐♥✉s♦✐❞✱ ✹✺✾ s❧♦♣❡✱ ✶✼✷✱ ✶✼✹✱ ✶✼✽✱ ✸✵✺✱ ✹✻✽✱ ✹✻✾ ❙❧♦♣❡ ❇❛❝❦✇❛r❞s✱ ✶✼✹ ❙❧♦♣❡ ❋r♦♠ ❚✇♦ P♦✐♥ts✱ ✶✼✹ ❙❧♦♣❡ ✐s ❚❛♥❣❡♥t✱ ✹✻✽ s❧♦♣❡✲✐♥t❡r❝❡♣t ❢♦r♠ ♦❢ ❧✐♥❡✱ ✶✼✻ ❙❧♦♣❡s ♦❢ P❡r♣❡♥❞✐❝✉❧❛r ▲✐♥❡s✱ ✹✼✹ st❛♥❞❛r❞ ❢♦r♠ ♦❢ ♣♦❧②♥♦♠✐❛❧✱ ✸✸✶ str✐❝t❧② ❞❡❝r❡❛s✐♥❣ ❢✉♥❝t✐♦♥✱ ✸✵✹ str✐❝t❧② ✐♥❝r❡❛s✐♥❣ ❢✉♥❝t✐♦♥✱ ✸✵✹ str✐❝t❧② ♠♦♥♦t♦♥❡ ❢✉♥❝t✐♦♥s✱ ✸✵✹ s✉❜s❡t✱ ✶✵✶✱ ✷✾✽ s✉❜st✐t✉t✐♦♥✱ ✷✸✻ ❙✉❜tr❛❝t✐♥❣ ❙✉♠s ♦❢ ❙❡q✉❡♥❝❡s✱ ✽✶ ❙✉❜tr❛❝t✐♥❣ ❙✉♠s✿ ▼♦♥♦t♦♥✐❝✐t②✱ ✽✷ ❙✉♠ ■s ❈♦♥st❛♥t ❙❡q✉❡♥❝❡✱ ✽✵ ❙✉♠ ♦❢ ❆r✐t❤♠❡t✐❝ Pr♦❣r❡ss✐♦♥✱ ✼✼ ❙✉♠ ♦❢ ❇✐♥♦♠✐❛❧ ❈♦❡✣❝✐❡♥ts✱ ✻✶ s✉♠ ♦❢ ❢✉♥❝t✐♦♥s✱ ✸✾✼ ❙✉♠ ♦❢ ●❡♦♠❡tr✐❝ Pr♦❣r❡ss✐♦♥✱ ✼✽ s✉♠ ♦❢ s❡q✉❡♥❝❡✱ ✾✸✱ ✾✺ ❙✉♠ ♦❢ ❙✐♥❡s ❛♥❞ ❈♦s✐♥❡s✱ ✹✺✽ ❙✉♠ ❘✉❧❡ ❢♦r ❉✐✛❡r❡♥❝❡s✱ ✾✷ ❙✉♠ ❘✉❧❡ ❢♦r ❙✉♠s✱ ✾✸ t❛♥❣❡♥t✱ ✸✼✾✱ ✸✽✻✱ ✸✽✼✱ ✸✾✷✱ ✹✻✽ t❡r♠s ♦❢ ♣♦❧②♥♦♠✐❛❧✱ ✸✸✶ tr❛♥s❢♦r♠❛t✐♦♥s✱ ✷✼✵✱ ✷✼✺✱ ✷✼✻✱ ✷✼✽✕✷✽✷✱ ✷✽✾✱ ✹✸✽ ❚r❛♥s❢♦r♠❛t✐♦♥s ❛s ❈♦♠♣♦s✐t✐♦♥s✱ ✷✽✷ ❚r❛♥s❢♦r♠❛t✐♦♥s ♦❢ ●r❛♣❤s✱ ✷✽✾ ❚r✐❛♥❣❧❡ ■♥❡q✉❛❧✐t②✱ ✹✹✶ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s✱ ✸✼✾✱ ✸✽✸✱ ✸✽✻✕✸✽✽✱ ✹✻✵✱ ✹✻✶ ✉♥❜♦✉♥❞❡❞ ❢✉♥❝t✐♦♥✱ ✶✾✺ ✉♥✐♦♥ ♦❢ s❡ts✱ ✷✵✵✱ ✷✵✹ ❯♥✐q✉❡♥❡ss ♦❢ ■♥✈❡rs❡✱ ✷✹✻ ❯♥✐t ❈✐r❝❧❡ ❊q✉❛t✐♦♥✱ ✹✹✼ ✈❛r✐❛❜❧❡s✱ ✶✺✵ ✈❡❝t♦r ✐♥ ❞✐♠❡♥s✐♦♥ ✶✱ ✹✻✺ ✈❡❝t♦r ✐♥ ❞✐♠❡♥s✐♦♥ ✷ ✱ ✹✼✶ ❱❡rt❡① ❋♦r♠ ♦❢ ◗✉❛❞r❛t✐❝ P♦❧②♥♦♠✐❛❧✱ ✷✽✼ ✈❡rt❡① ♦❢ ♣❛r❛❜♦❧❛✱ ✸✷✼ ✈❡rt✐❝❛❧ ❛s②♠♣t♦t❡✱ ✸✹✾ ❱❡rt✐❝❛❧ ❋❧✐♣✱ ✷✼✽ ❱❡rt✐❝❛❧ ▲✐♥❡ ❚❡st✱ ✶✺✼ ❱❡rt✐❝❛❧ ❙❤✐❢t✱ ✷✼✺ ❱❡rt✐❝❛❧ ❙tr❡t❝❤✱ ✷✽✵ ❱✐❡t❛✬s ❋♦r♠✉❧❛s✱ ✸✷✹ ✇❛✈❡ ❢✉♥❝t✐♦♥✱ ✹✻✵✱ ✹✻✶ ❲❤❡♥ ▲✐♥❡❛r ❘❡❧❛t✐♦♥ ■s ❋✉♥❝t✐♦♥✱ ✶✹✸ ❲❤❡♥ ❘❡❧❛t✐♦♥ ■s ❋✉♥❝t✐♦♥✱ ✶✶✶ ①✲✐♥t❡r❝❡♣t✱ ✶✼✵✱ ✸✸✽✱ ✸✹✾ ①✲■♥t❡r❝❡♣ts ♦❢ P❛r❛❜♦❧❛✱ ✸✷✻

✺✶✸ ②✲✐♥t❡r❝❡♣t✱ ✶✼✵

❩❡r♦ ❋❛❝t♦r Pr♦♣❡rt②✱ ✸✸✹

❩❡r♦ ❊①♣♦♥❡♥t ❘✉❧❡✱ ✹✵

③❡r♦ ♣♦✇❡r✱ ✹✵