##### Citation preview

2 LIMITS 2.1 The Limit Idea: Instantaneous Velocity and Tangent Lines Preliminary Questions 1. Average velocity is equal to the slope of a secant line through two points on a graph. Which graph? Average velocity is the slope of a secant line through two points on the graph of position as a function of

SOLUTION

time.

2. Can instantaneous velocity be defined as a ratio? If not, how is instantaneous velocity computed? Instantaneous velocity cannot be defined as a ratio. It is defined as the limit of average velocity as time elapsed shrinks to zero.

SOLUTION

= 0 Dale entered Highway 1. At t = 2 he was 126 miles down the highway, on the side of the road with a flat tire. At t = 3 he was still on the side of the road, waiting for road assistance. What was Dale's average velocity over each of the time intervals:

3. With tin hours, at t

= 0 to t = 2

(a) From t

(b) From t = 0 tot= 3 (c) From t

= 2 tot= 3

SOLUTION

(a) Over the time interval from t = 0 tot= 2, Dale traveled 126 miles. His average velocity was therefore

126

- = 63 miles/hour 2-0 (b) Over the time interval from t

= 0 tot= 3, Dale traveled 126 miles. His average velocity was therefore 126 _ 3 0

= 42 miles/hour

(c) Over the time interval from t = 2 tot= 3, Dale traveled 0 mile~. His average velocity was therefore

0

-3 -2

. = 0 rmles/hour

4. What is the graphical interpretation of instantaneous velocity at a specific time t SOLUTION

time at t =

= t0 ?

Instantaneous velocity at time t = to is the slope of the line tangent to the graph of position as a function of t0 .

Exercises 1. A ball dropped from a state of rest at time t = 0 travels a distance s(t) (a) How far does the ball travel during the time interval [2, 2.5]?

= 4.9t2 min t seconds.

(b) Compute the average velocity over [2, 2.5].

(c) Compute the average velocity for the time intervals in the table and estimate the ball's instantaneous velocity at t = 2. Interval

[2, 2.01]

[2, 2.005]

[2, 2.001]

[2, 2.00001]

Average velocity SOLUTION

(a) Given s(t) = 4.9t2, the ball travels tu = s(2.5) - s(2) = 4.9(2.5)2

-

4.9(2)2 = 11.025 m during the time interval

[2, 2.5]. (b) The average velocity over [2, 2.5] is

tis s(2.5) - s(2) tit = 2 .5 _ 2

11.025

= -0.S = 22.05 m/s 49

50

CHAPTER 2

I

LIMITS

(c) time interval

[2, 2.01]

[2, 2.005]

[2,2.001]

[2, 2.00001]

average velocity

19.649

19.6245

19.6049

19.600049

The instantaneous velocity at t = 2 is approximately 19.6 m/s. 3. On her bicycle ride Fabiana's position (in km) as a function of time (in hours) is s(t) average velocity between t = 2 and t = 3? What was her instantaneous velocity at t = 2.5? SOLUTION

= 22t + 17. What was her

Fabiana's average velocity between t = 2 and t = 3 was s(3) - s(2) 3-2

= 83 -

61

1

= 22 km/hour

To estimate the instantaneous velocity, we compute the average velocities: time interval

[2.5, 2.51]

[2.5, 2.501]

[2.5, 2.5001]

[2.5, 2.50001]

average velocity

22

22

22

22

The instantaneous velocity is 22 km/hour.

In Exercises 5-6, a ball is dropped on Mars where the distance traveled is s(t) = l.9t2 meters int seconds. S. Compute the ball's average velocity over the time interval [3, 6] and estimate the instantaneous velocity at t = 3. SOLUTION

The ball's average velocity over the time interval [3, 6] is s(6) - s(3) = 68.4 - 17.1 = _ m/s 17 1 6-3 3

To estimate the instantaneous velocity, we compute the average velocities: time interval

[3,3.1]

[3, 3.01]

[3, 3.001]

[3, 3.0001]

average velocity

11.59

11.419

11.4019

11.40019

The instantaneous velocity is approximately 11.4 m/ s.

In Exercises 7-8, a stone is tossed vertically into the air from ground level with an initial velocity of 15 mis. Its height at time tis h(t) = l5t - 4.9t2 m. 7. Compute the stone's average velocity over the time interval [0.5, 2.5] and indicate the corresponding secant line on a sketch of the graph of h. SOLUTION

The average velocity is equal to h(2.5) - h(0.5) =

2

I

0 .3 ms

The secant line is plotted with h(t) below. h

10 8 6 4 2

0.511.522.53

9. The position of a particle at time t is s(t) estimate the instantaneous velocity at t = 2. SOLUTION

= 2t3.

Compute the average velocity over the time interval [2, 4] and

The average velocity over the time interval [2, 4] is s(4) - s(2) _ 128 - 16 _ 4-2 2 - 56

To estimate the instantaneous velocity at t

= 2, we examine the following table.

time interval

[2, 2.01]

[2,2.001]

[2, 2.0001]

[1.99, 2]

[l.999, 2]

[1.9999, 2]

average velocity

24.1202

24.012

24.0012

23.8802

23.988

23.9988

The instantaneous velocity at t = 2 is approximately 24.0.

s E c T 1o N

2.1

I

The Limit Idea: Instantaneous Velocity and Tangent Lines

51

In Exercises 11-20, estimate the slope of the tangent line at the point indicated.

x2 + x;

11. f(x) =

x =0

SOLUTION

x interval

[0,0.01]

[0,0.001]

[0,0.0001]

[-0.01,0]

[-0.001,0]

[-0.0001, O]

slope of secant

1.01

1.001

1.0001

0.99

0.999

0.9999

The slope of the tangent line at x = 0 is approximately 1.0.

13. f(t)

= 12t -

7;

= -4

t

SOLUTION

t interval

[-4, -3.99]

[-4,-3.999]

[-4, -3.9999]

slope of secant

12

12

12

t interval

[-4.01,-4]

[-4.001,-4]

[-4.0001,-4]

slope of secant

12

12

12

= -4 is 12, coinciding with the graph ofy = f(t).

The slope of the tangent line at t 1s. y(t) = v3t+ l;

t=1

SOLUTION

t interval

[l, 1.01]

[l, 1.001]

[l, 1.0001]

[0.99, l]

[0.999, l]

[0.9999, l]

slope of secant

.7486

.7499

.7500

.7514

.7501

.7500

= 1 is approximately 0. 75.

The slope of the tangent line at t

17. f(x) = ex;

x = e

SOLUTION

[e - 0.01, e] [e - 0.001, e] [e - 0.0001, e] [e, e + 0.01] [e, e + 0.001] [e, e + 0.0001]

x interval

15.0787

slope of secant

15.1467

15.1535

15.2303

15.1550

15.1618

The slope of the tangent line at x = e is approximately 15.15. 19. f (x) = tan x;

7r

x= -

4

SOLUTION

x interval slope of secant

[~

- 0.01,

~]

1.98026

[~

- 0.001,

~]

[~

1.99800

The slope of the tangent line at x

- 0.0001,

~]

1.99980

[~, ~

+ 0.01]

[~, ~

2.02027

[~, ~

+ 0.001]

+ 0.0001]

2.00020

2.00200

= ~ is approximately 2.00.

21. The height (in centimeters) at time t (in seconds) of a small mass oscillating at the end of a spring is h(t) = 3 sin(27rt). Estimate its instantaneous velocity at t = 4. SOLUTION

To estimate the instantaneous velocity at t

= 4, we examine the following table.

time interval

[4, 4.01]

[4,4.001]

[4,4.0001]

[3.99,4]

[3.999,4]

[3.9999, 4]

average velocity

18.8732

18.8494

18.8496

18.8732

18.8494

18.8496

The instantaneous velocity at t;;;:; 4 is approximately 18.85 cm/s. 23. Consider the function f(x) = yx. (a) Compute the slope of the secant lines from (0, 0) to (x,f(x)) for x = 1, 0.1, 0.01, 0.001, 0.0001. (b) Discuss what the secant-line slopes in (a) suggest happens to the tangent line at 0.

(c)

iGU[I

Plot the graph off near x = 0 and verify your observation from (b).

52

CHAPTER 2

I

LIMITS SOLUTION

(a) The slope of the secant line from (0, 0) to (x, f(x)) for the function f(x) f(x) -f(O)

Yx

x-0

x

= Yx is

1

-vx·

Thus, for x = 1, 0.1, 0.01, 0.001, 0.0001, the slope of the secant line is

x

1

0.1

0.01

0.001

0.0001

10

31.62278

100

slope of secant 1 3.16228

(b) The secant line slopes from part (a) suggest that the slope of the tangent line at x = 0 grows without bound; that is, the tangent line at x = 0 is a vertical line. (c) The graph off shown below confirms that at x = 0 the tangent line is vertical line.

0.01

0.00

0.005

0.010

0.015

0.020

25. ~ If an object in linear motion (but with changing velocity) covers tis meters in M seconds, then its average velocity is v0 = tis/ M mis. Show that it would cover the same distance if it traveled at constant velocity v0 over the same time interval. This justifies our calling tis IM the average velocity. SOLUTION At constant velocity, the distance traveled is equal to velocity times time, so an object moving at constant velocity v0 for flt seconds travels vo& meters. Since Vo= tis/tit, we find

distance traveled = v06t = (

~;) M = tis

So the object covers the same distance tis by traveling at constant velocity v0 • 27. ~ Which graph in Figure 5 has the following property: For all x, the slope of the secant line over [0,x] is greater than the slope of the tangent line at x. Explain.

(A)

(B)

FIGURE 5 SOLUTION The graph in (B) bends downward, so the slope of the secant line through (0, 0) and (x, f(x)) is larger than the slope of the tangent line at (x,f(x)). On the other hand, the graph in (A) bends upward, so the slope of the tangent line at (x, f(x)) is larger than the slope of the secant line through (0, 0) and (x, f(x)). Thus, the graph in (B) has the desired property.

29. Let Q(t) = t2. Find a formula for the slope of the secant line over the interval [1, t] and use it to estimate the slope of the tangent line at t = 1. Repeat for the interval [2, t] and for the slope of the tangent line at t = 2. SOLUTION

Let Q(t)

= t2. The slope of the secant line over the interval [1, t] is Q(t) - Q(l) t2 - 1 (t - l)(t + 1) ----=--= =t+l t-1 t-1 t-1

provided t if:. 1. To estimate the slope of the tangent line at t

= 1, examine the values in the table below.

sEc T Io N

2.1

I

The Limit Idea: Instantaneous Velocity and Tangent Lines

53

0.99 0.999 0.9999 1.01 1.001 1.0001

t

slope of secant 1.99 1.999 1.9999 2.01 2.001 2.0001 The slope of the tangent line at t = 1 is approximately 2.0. The slope of the secant line over the interval [2, t] is Q(t) - Q(2) =

provided t

t2 - 4 = (t - 2)(t + 2) = t + 2, t-2

t-2

t-2

* 2. To estimate the slope of the tangent line at t = 2, examine the values in the table below. t

1.99 1.999 1.9999 2.01 2.001 2.0001

slope of secant 3.99 3.999 3.9999 4.01 4.001 4.0001 The slope of the tangent line at t = 2 is approximately 4.0. 31. For f(x) = x3, show that the slope of the secant line over [-3, x] is :i2 - 3x + 9, and use this to estimate the slope of the tangent line at x = -3. SOLUTION

Let f(x)

=.x3. The slope of the secant line over the interval [-3, x] is f(x)-/(-3) = .x3+27 = (x+3)(i2-3x+9) x-(-3) x+3 x+3

provided x

=x2- 3x+ 9

* -3. To estimate the slope of the tangent line at x = -3, examine the values in the table below. x

-3.01

-3.001

-3.0001

-2.99

-2.999

-2.9999

slope of secant 27.0901 27.009001 27.000900 26.9101 26.991001 26.999100 The slope of the tangent line at x = -3 is approximately 27.0.

Further Insights and Challenges The next two exercises involve limit estimates related to the definite integral, an important topic introduced in Chapter 5. 33. Let A represent the area under the graph of y = x3 between x = 0 amd x = 1. In this problem, we will follow the process in Exercise 32 to approximate A. (a) As in (a)-(d) in Exercise 32, separately divide [O, l] into 2, 3, 5, and 10 equal-width subintervals, and in each case compute an overestimate of A using rectangles on each subinterval whose height is the value of .x3 at the right end of the subinterval. In this case, it can be shown that if we use n equal-width subintervals, then the total area A(n) of then rectangles is A(n) = (n

+ 1)2 4n2

(b) Compute A(n) for n

= 2, 3, 5, 10 to verify your results from (a).

(c) Compute A(n) for n = 100, 1000, and 10,000. Use your results to conjecture what the area A equals. SOLUTION

(a)

• Dividing [O, 1] into 2 equal-width subintervals produces two rectangles with width 1/2 and heights 1/8 and 1. The combined area of the two rectangles is then 1 1 1 9 -·-+-·l=2 8 2 16

• Dividing [O, 1] into 3 equal-width subintervals produces three rectangles with width 1/3 and heights 1/27, 8/27 and 1. The combined area of the three rectangles is then 1 1 I 8 1 4 -·-+-·-+-·1=3 27 3 27 3 9

• Dividing [0, 1] into 5 equal-width subintervals produces five rectangles with width 1/5 and heights

54

CHAPTER 2

I

LIMITS

respectively. The combined area of the five rectangles is then

(~)3 + !5 (~)3 + ! (~)3 + ! . 13 = _2._ 5 5 5 5 25

! (!)3 + ! 5 5 5 5

• Dividing [O, 1] into 10 equal-width subintervals produces 10 rectangles with width 1/10 and heights 3 3 3 3 3 3 3 3 3 1 ) ( 1 ) ( 3 ) ( 2 ) ( 1 ) ( 3 ) ( 7 ) (4 ) ( 9 ) 3 ( 10 ' S ' 10 ' S ' 2 ' S ' 10 ' S ' 10 ' and l The combined area of the five rectangles is then

(b) Let

= (n;;>2

A(n) Then,

9

9

16

4

A( 2)

= 4(4) = 16

A( 3 )

= 4(9) = 9

A(5 )

= 4(25) = 25'

36

9

121)

A(lO)

and

121

= 4(100) = 400

confirming the results from (a). (c) We find A(lOO) = A(lOOO)

4

101 2 (100)2 = 0.255025 1001 2

= 4 (1000)2 = 0.25050025,

A(lO, 000) =

4

and

10,001 2 (l0, 000) 2 = 0.2500500025

Based on these results, we conjecture that the area A equals ~.

2.2 Investigating Limits Preliminary Questions 1. What is the limit of f(x) = 1 as x SOLUTION

limx~"

n?

1 = 1.

2. What is the limit of g(t) SOLUTION

~

lim14,, t =

= t as t ~ n?

7r.

3. Is lim 20 equal to 10 or 20? x~lO

SOLUTION

limx~w20

= 20.

4. Can f(x) approach a limit as x

~

c if /(c) is undefined? If so, give an example.

Yes. The limit of a function fas x ~ c does not depend on what happens at x c. As an example, consider the function

SOLUTION

f

as x

~

f(x)

= x2 -1 x-1

= c, only on the behavior of

s Ec T Io N The function is clearly not defined at x

I

2.2

Investigating Limits

55

= 1 but limf(x) = lim i2- X--+1

x--+l

1

x- I

= lim(x +I)= 2 x--+1

5. What does the following table suggest about lim f(x) and lim f(x)? x--+I-

x f(x) SOLUTION

x--+t+

0.9

0.99

0.999

1.001

1.01

1.1

7

25

4317

3.00011

3.0047

3.0126

The values in the table suggest that limx_, 1- f(x)

= oo and limx_, 1+ f(x) = 3.

6. Can you tell whether limf(x) exists from a plot off for x > 5? Explain. x->5

No. By examining values of f(x) for x close to but greater than 5, we can determine whether the one-sided limit limx_, 5+ f(x) exists. To determine whether limx--.5 f(x) exists, we must examine value of f(x) on both sides of x = 5.

SOLUTION

7. If you know in advance that limf(x) exists, can you determine its value from a plot off for all x > 5? x->5

SOLUTION

Yes. If limx....5 f(x) exists, then both one-sided limits must exist and be equal.

Exercises In Exercises 1-5,fill in the table and guess the value of the limit .

= -.x3 X2 -

1. limf(x), where f(x) x--+1

1 1

x

f(x)

x

f(x)

1.002

0.998

1.001

0.999

1.0005

0.9995

1.00001

0.99999

SOLUTION

x

0.998

0.999

0.9995

0.99999

1.00001

1.0005

1.001

1.002

f(x)

1.498501

1.499250

1.499625

1.499993

1.500008

1.500375

1.500750

1.501500

The limit as x

--t

I is ~.

3. limf(y), where f(y) y->2

=~ - y -

2 y + y- 6

y

y

f(y)

f(y)

2.002

1.998

2.001

1.999

2.0001

1.9999

SOLUTION

The limit as y

--t

y

1.998

1.999

1.9999

2.0001

2.001

2.002

f(y)

0.59984

0.59992

0.599992

0.600008

0.60008

0.60016

2 is ~.

56

CHAPTER 2

I

LIMITS

5. limf(t), where f(t) =

l -cos2t

t

t-->0

f(t)

t

t

0.002

-0.002

0.001

-0.001

0.0005

-0.0005

0.00001

-0.00001

f(t)

SOLUTION

f(t)

t

t

f(t)

0.002

0.004

-0.002

-0.004

0.001

0.002

-0.001

-0.002

0.0005

0.001

-0.0005

-0.001

0.00001

0.00002

-0.00001

-0.00002

The limit as t ~ 0 is 0. 7. Determine lim f(x) for fas in Figure 10. x--+0.5

y

FIGURE 10 SOLUTION

The graph suggests that f(x)

~

1.5 as x

~

0.5.

In Exercises 9-IO, evaluate the limit. 9. lim x x--+21

As x ~ 21, f(x) = x ~ 21. You can see this, for example, on the graph of f(x) = x. 11. Show, via illustration, that the limits limx and lima are equal but the functions in each limit are different.

SOLUTION

X--+a

x--+a

SOLUTION

The figure above displays the graphs of f(x) = x and g(x) = a. Clearly, the two functions are different. It is also clear that as x approaches a, both graphs approach the point (a, a); that is, limx = lima =a x--+a

x--+a

In Exercises I3-20, verify each limit using the limit definition. For example, in Exercise 13, show that l3x - 121 can be made as small as desired by taking x close to 4.

13. lim3x = 12 x~4

SOLUTION

Ix - 41 small.

13x - 121 = 31x - 41. 13x - 121 can be made arbitrarily small by making x close enough to 4, thus making

S E CT I o N 2.2

15. lim(5x + 2) x .... 3

I

Investigating Limits

57

= 17

1(5x + 2) - 171 = 15x - 151 1(5x + 2) - 171 as small as desired. SOLUTION

51x - 31. Therefore, if you make Ix - 31 small enough, you can make

17. limx2 = 0 x ....o As x - t 0, we have lx2 - 01 = Ix+ Ollx - OI. To simplify things, suppose that lxl < 1, so that Ix+ Ollx - 01 = lxllxl < lxl. By making lxl sufficiently small, so that Ix+ Ollx - OI = x2 is even smaller, you can make lx2 - 01 as small as desired.

SOLUTION

19. lim(4x2 + 2x + 5) x->0

=5

As x - t 0, we have 14x2 + 2x + 5 - 51 = 14x2 + 2xl = lxll4x + 21. If lxl < 1, 14x + 21 can be no bigger than 6, so lxll4x + 21 < 61xl. Therefore, by making Ix - 01 = lxl sufficiently small, you can make 14x2 + 2x + 5 - 51 = lxll4x + 21 as small as desired.

SOLUTION

In Exercises 21-38, estimate the limit numerically or state that the limit does not exist. If infinite, state whether the one-sided limits are oo or -oo.

. yx-1

21. hm--x~i X- 1 SOLUTION

The limit as x

23. lim

-t

1 is

x

0.9995

0.99999

1.00001

1.0005

f(x)

0.500063

0.500001

0.49999

0.499938

x

1.999

1.99999

2.00001

2.001

f(x)

1.666889

1.666669

1.666664

1.666445

x

-0.Ql

-0.005

0.005

0.01

f(x)

1.999867

1.999967

1.999967

1.999867

x

-0.01

-0.001

0.001

0.01

f(x)

0.999850

0.999999

0.999999

0.999850

x

-0.05

-0.001

0.001

0.05

f (x)

0.0249948

0.0005

-0.0005

-0.0249948

t.

x2+x-6 -x- 2

x->2 X 2

SOLUTION

The limit as x--+ 2 is ~-

25. lim sin 2x x~o x SOLUTION

The limit as x --+ 0 is 2.

27• lim sin3x x->0 3x SOLUTION

The limit as x --+ 0 is 1. 29 . Jim cosB-1 o-.o B SOLUTION

The limit as x --+ 0 is 0.

58

CHAPTER 2

I

LIMITS

. 1 31. h ) m-( x-+4 X- 4 3 SOLUTION

The limit does not exist. As x

x

3.9

3.99

3.999

4.001

4.01

4.1

f (x)

-1000

-106

-109

109

106

1000

4+, f(x)

~

~

~

4-, f(x)

-oo; similarly, as x

~

oo.

x+3 33. l i m --x--+-3 x 2 + x-6 SOLUTION

x

-3.1

-3.01

-3.001

-2.999

-2.99

-2.9

f(x)

-0.196078

-0.199601

-0.199960

-0.200040

-0.200401

-0.204082

The limit as x ~ -3 is

-k-

x- 4 35. Jim ---zx-+3+ X - 9 SOLUTION

The limit does not exist. As x

~

x

3.001

3.01

3.1

f(x)

-166.472

-16.473

-1.475

3+, f(x)

~

-oo.

1

37. limsinhcos -h h--+0

SOLUTION

~

The limit as x

h

-0.01

-0.001

-0.0001

0.0001

0.001

0.01

f(h)

-0.008623

-0.000562

0.000095

-0.000095

0.000562

0.008623

0 is 0.

39. limlxl-" x--+0

SOLUTION

The limit as x

~

x

-0.05

-0.001

-0.00001

0.00001

0.001

0.05

f(x)

1.161586

1.006932

1.000115

0.999885

0.993116

0.860892

0 is 1.

t-e

41. l i m - He lnt-1 SOLUTION

The limit as t

~

t

e - 0.01

e - 0.001

e -0.0001

e + 0.0001

e + 0.001

e+O.Ql

f(t)

2.713279

2.717782

2.718232

2.718332

2.718782

2.723279

e is approximately 2.718. (The exact answer is e.)

1

. tan- x 43. h m - x--+1-

cos-IX

SOLUTION

x

0.9

0.99

0.999

0.9999

f(x)

1.624771

5.513466

17.549388

55.532038

The limit does not exist. As x--> 1-, f(x)

~

oo.

s Ec T I o N

2.2

(b) lim LxJ

x-+c-

SOLUTION

Investigating Limits

59

= n, where n is the unique integer

45. The greatest integer function, also known as the floor function, is defined by LxJ such that n ::; x < n + 1. Sketch the graph of y = LxJ. Calculate for c an integer:

(a) lim LxJ

I

(c) lim LxJ

x-+c+

x-+2.6

The graph ofy = LxJ is shown below. y

3 2

---0 ---0

--1--+---+---+---o---+--- 0 for any positive integer n. Moreover, as x -+

o+, x!'

-+

o+, so

. 1 1Im - =oo

x-+O+

x"

for any positive integer n. On the other hand, if x < 0, then x!' < 0 when n is an odd positive integer and x!' > 0 when n is an even positive integer. Accordingly,

. -1 = { -oo, IIm

x--+O-

xn

oo,

n is an odd positive integer n is an even positive integer.

Thus, the two infinite one-sided limits lim 1/ x!' are equal when n is an even positive integer. x--+O±

62

CHAPTER 2

I

LIMITS

71. !Gull

In some cases, numerical investigations can be misleading. Plot f(x) =cos~­

(a) Does Iimf(x) exist? x-+0

(b) Show, by evaluating f(x) at x = ±~, ±~, ±k, ... , that you might be able to trick your friends into believing that the limit exists and is equal to L = 1. (c) Which sequence of evaluations might trick them into believing that the limit is L = -1? SOLUTION

A graph off is shown below: y

(a) Based on the graph off, it appears that the function values oscillate more and more rapidly between +1 and -1 as x-+ 0. Accordingly, it appears that limf(x) does not exist. x-+0

(b) Evaluating f(x) at x = ±~, ±~, ±k, ... , we find

t(±~) = cos(±2:ir) = 1

t(±~) = cos(±4:ir) = 1

t(±~) = cos(±6:ir) = 1 and so on. (c) To trick your friends into believing that L = -1, evaluate f(x) at x = ±1, ±i, ±~, ....

Further Insights and Challenges . 1·im sin nB numenc . all y "ior severaI pos1t:J.ve . . mteger . . general. vaI ues of n. Then guess the value m 73• Invest:J.gate 9... 0

e

SOLUTION

• For n = 3, we have

e

-0.l

-0.Ql

-0.001

0.001

0.01

0.1

sinnB --

2.955202

2.999550

2.999996

2.999996

2.999550

2.955202

e

-0.1

-0.01

-0.001

0.001

O.Ql

0.1

sinnB --

4.794255

4.997917

4.999979

4.999979

4.997917

4.794255

e

The limit as e-+ 0 is 3. • For n = 5, we have

e

The limit as B -+ 0 is 5. ,.,

.

• ne sunnise

th

. . sinne at, m general, hm - 9... 0

e = n.

x" - 1

75. Investigate Iim - - for (m,n) equal to (2, 1), (1, 2), (2,3), and (3,2). Then guess the value of the limit in general x-+1 x"' - 1 and check your guess for two additional pairs. SOLUTION

The limit as x-+ 1 is!-

x

0.99

0.9999

1.0001

1.01

x-l x 2 -1

0.502513

0.500025

0.499975

0.497512

s EC T I O N x

x2- l

x-1

0.99

0.9999

1.0001

1.01

1.99

1.9999

2.0001

2.01

2.2

I

The limit as x --) 1 is 2.

x

0.99

0.9999

1.0001

1.01

x2- l x 3 -1

0.670011

0.666700

0.666633

0.663344

x

0.99

0.9999

1.0001

1.01

1.492513

1.499925

1.500075

1.507512

The limit as x --) 1 is ~.

x3- l x 2 -1 The limit as x --) 1 is ~.

x"-1 • For general m and n, we have lim - x .... 1 :!!"- 1

= -mn .

x

0.99

0.9999

1.0001

1.01

x-1 x 3 -1

0.336689

0.333367

0.333300

0.330022

The limit as x--) 1 is ~x

x3- l

x-1

0.99

0.9999

1.0001

1.01

2.9701

2.9997

3.0003

3.0301

The limit as x --) 1 is 3.

x

x3- l

x1 -1

0.99

0.9999

1.0001

1.01

0.437200

0.428657

0.428486

0.420058

The limit as x--) 1 is ~ ~ 0.428571. 2x 8 Plot the graph of f(x) = x ~ . 3 (a) Zoom in on the graph to estimate L = Iimf(x).

77.

~

!Gq

~

x .... 3

(b) Explain why

f(2.99999) :s; L :s; f(3.00001) Use this to determine L to three decimal places. SOLUTION

(a)

Y

5.525

x=3

Investigating Limits

63

64

CHAPTER 2

I

LIMITS (b) It is clear that the graph off rises as we move to the right. Mathematically, we may express this observation as: whenever u < v, f(u) < f(v). Because

2.99999 < 3 == limx < 3.00001 x->3

it follows that f(2.99999) < L == Iimf(x) < f(3.00001) x->3

With f(2.99999) "' 5.54516 and f(3.00001) "' 5.545195, the above inequality becomes 5.54516 < L < 5.545195; hence, to three decimal places, L == 5.545.

2.3 Basic Limit Laws Preliminary Questions 1. State the Sum Law and Quotient Law. SOLUTION

Suppose limx_,c f(x) and limx->c g(x) both exist. The Sum Law states that lim(f(x) + g(x)) == Iimf(x) + Iimg(x) X->C

Provided limx_,c g(x)

X->C

X->C

* 0, the Quotient Law states that limf(x) f( X ) X->C Iim - - = = - - x->c g(x) limg(x) X->C

2. Which of the following is a verbal version of the Product Law (assuming the limits exist)? (a) The product of two functions has a limit.

(b) The limit of the product is the product of the limits.

(c) The product of a limit is a product of functions. (d) A limit produces a product of functions. SOLUTION

The verbal version of the Product Law is (b): The limit of the product is the product of the limits.

3. Which statement is correct? The Quotient Law does not hold if (a) The limit of the denominator is zero

(b) The limit of the numerator is zero SOLUTION

Statement (a) is correct.

Exercises In Exercises 1-26, evaluate the limit using the Basic Limit Laws and the limits Iim xPfq == cpfq and lim k == k. X->C

1. limx x->9

SOLUTION

limx == 9 x->9

3. lim x

4

X-7~

SOLUTION

!~11 x

4

==

Gr

16

5. limr 1 1-->2

SOLUTION

We apply the definition of r

1

,

and then the Quotient Law: lim 1

. -I . 1 t->2 1 1Im t == 1Im - == - - == 1-->2

1-->2

t

lim t t->2

7. Jim (3x + 4) X->0.2

2

X->C

sEc T Io N SOLUTION

2.3

We apply the Laws for Sums and Constant multiples: lim(3x+4)

x--+0.2

= x--+0.2 lim 3x+ lim 4 x--+0.2 = 3 x--+0.2 lim x + lim 4 = 3(0.2) + 4 = 4.6 x--+0.2

9. lim (3x4

-

2.x3 + 4x)

x--+-1

SOLUTION

We apply the Laws for Sums, Constant multiples, and Powers: lim (3x4

x--+-1

-

2.x3 + 4x)

= x--+-1 lim 3x4 -

lim 2.x3 + lim

x--+-1

= 3 x--+-I lim x4 = 3(-1

4

) -

4x

x--+-1

2 lim .x3 + 4 lim x x--+-1

x--+-l

2(-1

3

) -

4

=3 + 2 -

4

=1

11. lim(x + 1)(3x2 - 9) x--+2

SOLUTION

We apply the Laws for Products, Sums, Constant multiples and Powers: lim (x + 1) (3x2 x--+2

9) = {limx + lim 1) {3 limx2 - lim 9) x--+2

x--+2

x--+2

x--+2

= (2 + 1) ( 3(2)2 - 9) = 3 . 3 = 9 .

1

13. h m t--+4 t + 4 SOLUTION

We apply the Laws for Quotients and Sums: lim 1

lim-1H4f+4

= ~ = _1_ = ! limt+4

4+4

t--+4

8

3t- 14 15. lim--t--+4 t+1 SOLUTION

We apply the Laws for Quotients, Sums, and Constant multiples: 3t - _ 14 lim _ 1--+4 t + 1

=

3limt-lim14 t--+4

3. 4- 14

t--+4

4+1

limt +Jim 1 t--+4

t--+4

2 =--

5

17. lim(l6y + 1)(2y112 + 1)

y--+i

SOLUTION

We apply the Laws for Products, Sums, Constant multiples, and Roots:

li~(l6y + 1)(2y

112

+ 1) = (16

y--+4

19. Jim y--+ 4

li~ y + Jim 1) · (2 lim y 1

12

y--+4

y--+!

y--+!

+Jim 1) y--+!

l ..j6y + 1

SOLUTION

We apply the Laws for Quotients, Roots, Sums, and Constant multiples: . 11m y--+ 4

1

..j6y + 1

liml y--+4

61imy +Jim 1 y--+4

21. Jim __ x_ x--+-1x3 +4x

y--+4

1 y6(4) + 1

1

1

= .../25 = 5

I

Basic Limit Laws

65

66

CHAPTER 2

I

LIMITS

We apply the Laws for Quotients, Sums, Powers, and Constant multiples:

SOLUTION

lim x

lim __x_ = x-+-1

x3

-1 (-1)3 +4(-1)

x-+-1

+ 4x

lim X-+-1

x3 + 4

lim x

-1

1

= -1-4 = 5

X-+-1

3 Vi- !t 23. t-+25 lim (t ~2

20

We apply the Laws for Quotients, Sums, Constant multiples, and Powers and Roots:

SOLUTION

1 3limVt-!limt 3 -r. . yt- 5f H25 5 H25 hm =------""'"" H25 (f - 20)2 ( )2 lim t - lim 20 f-425

3 VE - kC

Since limf(x) is nonzero, we can apply the Quotient Law:

SOLUTION

X->C

r

x~

1 ) (

f(x)

(lim 1) X->C

1

= ( ~f(x)) = 1~f(x)

1

In Exercises 29-32, evaluate the limit assuming that lim f(x) X-+-4

= 3 and X--+-4 lim g(x) = 1

29. lim f (x)g(x) x--+-4

lim f(x)g(x)

SOLUTION

X-+-4

= x-+-4 lim f (x)

lim g(x) X-+-4

=3 · 1 =3

31. lim g(x) x-+-4 x2

Since lim x2

SOLUTION

X-+-4

* 0, we may apply the Quotient Law, followed by the Law for Powers: lim

( ) = ~

x-+-4 x 2

lim g(x) _x->_-_4 _ _

lim x 2

= (-4)2 = 16

x->-4

33. Can the Quotient Law be applied to evaluate lim sin x? Explain. x--+0

X

SOLUTION The limit Quotient Law cannot be applied to evaluate lim sinx since limx = 0. This violates a condition x-+0 X x--+0 of the Quotient Law. Accordingly, the rule cannot be employed.

35. Assume that iflimf(x) x--+a not exist. (a) lim sin ( ~ 1 ) x--+0

(b) lim X-+7r/2

x

sinx X

(c) lim . (3x l x--+1 SID 1-x (d)

limx2 sin(JTx2) x-+I

= L, then limsinf(x) = sinL. In each case evaluate the limit or indicate that the limit does x-+a

s E c T 1o N 2.3 I Basic Limit Laws 67 SOLUTION

(a) Because limo

x=I

X->

= O~I = 0, it follows that limsin(-x-) x->O x-1

= sinO = 0

(b) By the Quotient Law,

sinx lim

x-m/2

X

. 7r sm 2 = ---yr-

2

= :;;: ~'

2 (c) Because lim(l - x) x->I

= 1 - 1 = 0, it follows that = sinO = 0

limsin(l-x) x->0

Now, lim 3x = 3(1) x->1

= 3 * 0, so . hm x->I

3x d . oes not exist sin(l - x)

(d) Because lim7rx2 = 7r(l) 2 = 7r, it follows that x->I

limsin(7rx2) x->1

= sin7r = 0

Then, by the Product Law, limx2 sin(7rx2) = limx2 · limsin(nx2) = 12 • 0 = 0 x--+1

x--+1

x--+1

37. Give an example where lim(f(x) + g(x)) exists but neither limf(x) nor limg(x) exists. x-+0

SOLUTION

limg(x) x-+0

x--+0

x-+0

= lim-1/x do not exist. x-+0

39. Give an example where Jim

x--+0

x-+0

x-+O

x-ioO

x--+0

Let f(x)

Then lim f(x)

x--+0

exists but neither limf(x) nor limg(x) exists.

f((x))

x--+0 g x

SOLUTION

x-+0

Let f(x) = 1/x and g(x) = -1/x. Then lim(f(x) + g(x)) = limO = 0 However, limf(x) = lim 1/x and

g(x)

={

x 2, f(x) = - - = 1. For x < 2, f(x)

SOLUTION

~-~

2.4

I

Limits and Continuity

71

(x - 2) . h . d' . . = --= -1. The f unction as a Jump iscontmmty at

0-zj

x = 2. Because lim f (x)

x--+2-

= -1 = f (2)

the function is left-continuous at x = 2.

31. f(x) =

2x;;;o = 2~;;0 is discontinuous at x = -5. Now,

The function f(x)

SOLUTION

. 2(x - 5)(x + 5) . 2x2 - 50 hm = 1im X +5 x-+-5 X +5

x-+-5

. ( ) 1im 2 x- 5 = - 20 = x-+-5

Because lim f(x) exists but f(-5) is not defined, f has a removable discontinuity at x x--+-5

left-continuous nor right-continuous at x

33. g(t)

= -5. The function is neither

= -5.

= tan 2t

SOLUTION

The function g(t)

sin2t is . d'iscontmuous . . = tan 2t = w henever cos 2 t = 0 ; 1.e., whenever

cos2t

(2n+ l)7r 2t = ,;___--'2

or

t=

(2n + l)7r 4

where n is an integer. At every such value oft there is an infinite discontinuity. The function is neither left-continuous nor right-continuous at any of these locations. 35. f(x)

= tan(sin x)

SOLUTION

The function f (x) = tan(sin x) is continuous everywhere. Reason: sin xis continuous everywhere and tan u

is continuous on (- ~, ~)-and in particular on -1 composite functions.

:6 X - 6

When we substitute x = 6 into ~=~6 , we obtain the indeterminate form and denominator and then simplifying, we find

SOLUTION

. x2 - 36 . (x - 6)(x + 6) . h m - - =hm =lim(x+6)= 12 X - 6 x-+6 X - 6 x-+6

x-+6

§. Upon factoring the numerator

78

c H APTER

2

I

LIMITS

x2+2x+l 3. lim - - - x--+-1 X+ 1 SOLUTION When we substitute x = -1 into x2::~+ 1 , we obtain the indeterminate form and simplifying, we find

g. Upon factoring the numerator

2

. x2 + x + 1 lIm X--+-1 X+ 1

. (x + 1) li ( 1Im - - - = m x + 1) = 0 = X--+-1 X+ 1 X--+-1

In Exercises 5-34, evaluate the limit, if it exists. If not, determine whether the one-sided limits exist (finite or infinite).

5.

x-7

lim~ X--41

49

X -

. x- 7 l Im -2 - x--+1 x - 49

SOLUTION

x- 7 l" 1 l = l"x--+7 Im = x--+7 Im - - = (x - 7)(x + 7) x+7 14

x2 + 3x+2 7. lim - - - x--+-2 X+ 2

. x2 + 3x + 2 l Im X+2

SOLUTION

x--+-2

=

li

m

(x + 2)(x + 1) X+ 2

x--+-2

.

= 1Im(x+ 1) = -1 x--+-2

2x2-9x-5

9. lim X-45

X

2

25

-

lim 2.x2 - 9x - 5 x2 - 25

SOLUTION

= lim (2x +

x--+5

l)(x - 5)

= lim 2x + 1 = .!..!..

x--+5 (x - 5)(x + 5)

10

x--+5 x + 5

2x+ 1 11. lim - 2- - - x--+-t 2x + 3x + 1 SOLUTION

13. lim X-42

2x + 1 2

lim

x--+-i 2x + 3x + 1

=

2x + 1

. lim

=

x--+-t (2x + l)(x + 1)

. 1 hm - -

x--+-~ x + 1

=-

1

1/2

=2

3.x2-4x-4 2 X2 - 8

SOLUTION

. 3.x2 - 4x - 4 1Im x--+2 2x2 - 8

(3x + 2)(x - 2) . 3x + 2 8 = Iix--+2 m = hm --- = - = 1 2(x - 2)(x + 2) x--+2 2(x + 2) 8

15. lim 42r - 1 t--+O 4' - 1 SOLUTION

4 2' - 1 lim - 1--+04'-1

17. lim ../X-

= lim t--+O

(4' - 1)(4' + 1) 4'-1

= lim(4' + 1) = 4° + 1 = 2 t--+O

4

x--+16 x-16

SOLUTION

. ..(X-4 hm - - -

x--+16 x-16

19 r (h + 2)2 • h1:!!J h

. = hm

../X-4

x--+16(..fX+4)(..fX-4)

. = hm

1

---

x--+16 ..(X+4

= -1 8

4

SOLUTION 1

1

lim (h+2)2 - 4 h--+0 h

4-(h+2J2 li 4(h+2)2 = h!!J --h-

= lim

h

h--+0

21. lim h--+O

-h-4

4(h+2i2 h

4-(h2+4h+4) -h2 -4h 4(h+2)2 . 4(h+2)2 1 = I m -= h1:!!J h h--+O h

r

= lim

-h - 4 h--+0 4(h + 2) 2

= -4 = _.!. 16

4

Yif+h - 2

SOLUTION

h

Observe that ash ~ 0, V2 + h - 2 ~

V2 - 2 * 0 and h -4 0. Accordingly,

. v2+h-2 hm h does not exist.

h--+0

S E c TI 0 N 2.5

As for the one-sided limits, Y2 + h - 2 ---)

I

Indeterminate Forms

79

.../2 - 2 < 0 and h ---) o- as h ---) o-; therefore lim

h->O-

.../2 + h-2 = oo h

.../2 - 2 < 0 and h---) o+ ash---) o+; therefore

On the other hand, .../2 + h - 2---)

lim h->O+

.../2+h-2 = -oo h

x-4 23. lim - - - = = x->4 yx - y8 - X SOLUTION

lim x->4

x-4

. (

Vx _ y8 _ X

= hm x->4

x-4

. ..:....__.;,..;....;.._-=---(x-4)(yx+ .../8-x) · vx+ .../8-x) = hm .../8 _ x x->4 2x - 8

Vx _ y8 _ x yx +

=limvx+~= 2

x->4

V4+V4=2 2

1 - - -4- ) 25. lim ( - x->4 yx-2 x-4 . ( 1 4 ) . lim - - - - - - = hm

SOLUTION

x->4

yx- 2

X -

4

x->4 (

yx+2-4

yx- 2)( yx + 2)

. = hm x->4 (

yx-2

yx- 2)( yx + 2)

1 = -

4

cotx 27. lim-x->O CSC X

cotx COSX • Jim--= Jim-.-· smx = cosO = 1

SOLUTION

x->O CSC X

x->0 Slil X

29. Jim ( -1- - -2-2 ) x->I 1 - x 1- x x-1 x-1 -1 1 1 2 ) Jim ( - - - - -2 =Jim - -2 =Jim =Jim - - = -HI 1 - x 1- x HI 1 - x x->I (1 - x)(l + x) x->I 1 + x 2

SOLUTION

221+21 -20 31. l i m -1- - - H2 2 -4 221 + 21 - 20 (21 + 5)(21 - 4) SOLUTION lim = lim 4 = lim(21 + 5) = 9 1 1->2 2 - 4 1->2 21 1->2 2 1 - ~--) tanB-1 tan2 8-1

33. lim ( O->i

lim (

SOLUTION

0->i

1 2 ) . tanB-1 tanB-1 1 = hm = Jim = Jim - - 2 tanB-1 tan2 8-1 O->i tan 8-1 o->~ (tanB- l)(tanB+ 1) O->~ tanB+ 1

1

1+1

2

35. The following limits all have the indeterminate form 0/0. One of the limits does not exist, one is equal to 0, and one is a nonzero limit. Evaluate each limit algebraically if you can or investigate it numerically if you cannot.

. x2 + 3x+ 2 • hm---x->-2 X +2 l -x- 1

• lim---x->I X - 2 + x-1 . x2 •hm-x->O

1-

ex

SOLUTION

.

• l1m X->-2

}"

x2+3x+2 . (x+2)(x+l) . = 11m = 11m(x+l)=-l X + 2 X->-2 X + 2 X->-2

1-

X-I

Ji

X -

1

}"

X -

1

}"

1

hi h d

• ).!!} x _ 2 + x-I = x.!!t x2 _ 2x + 1 = x1!!1 (x _ l) 2 = x1!!1 x _ 1 , w c proaches 1

* 0 while the denominator approaches 0.

. b h oes not exist ecause t e numerator ap-

• We will investigate this limit numerically. From the table below, it appears that lim _!____ = O. x->O

1-

ex

80

c HA p T E R

2

I

LIMITS

x

-0.01

-0.001

-0.0001

0.0001

0.001

0.01

X1

0.0100501

0.0010005

0.0001000

-0.000099995

-0.0009995

-0.00995

1-eX

In Exercises 37 and 38, a projectile is launched straight up in the air and is acted on by air resistance and gravity as in Example 7. The function H gives the maximum height that the ball attains as a function of the air-resistance parameter .

~

37. If tne mass of the ball is one kilogram and it is launched upward with an initial velocity of 60 m/sec, then

=

H(k)

60k - 9.8 ln ( JOOk +

1)

49

k2

Estimate the maximum height without air resistance by investigating lim H(k) numerically. k->0

Based on the values in the table below, we estimate that the maximum height with no air resistance is 183.67 meters.

SOLUTION

39. !GUii

k

0.001

0.0001

0.00001

0.000001

H(k)

182.9272

183.5985

183.6660

183.6729

Use a plot of f(x) =

x

Vi-

~

8-x

to estimate limf(x) to two decimal places. Compare with the answer x->4

obtained algebraically in Exercise 23. SOLUTION Let f(x) = vx~~- From the plot of f(x) shown below, we estimate ~~f(x) "" 2.00; to two decimal places, this matches the value of2 obtained in Exercise 23. y

3.6

3.8

1

41. Show numerically that for b = 3 and b = 5, lim bx x--+0

SOLUTION

4.0

4.2

4.4

appears to equal ln 3 and ln 5, respectively.

X

Based on the values in the table below, Y-1

lim - - "" 1.0986 x--+0

X

To four decimal places, ln 3 = 1.0986, so it appears that lim x--+O

x

3x_l

3x-i x

= ln 3.

x

x

JX-1 x

0.01

1.104669

-0.01

1.092600

0.001

1.099216

-0.001

1.098009

0.0001

1.098673

-0.0001

1.098552

0.00001

1.098618

-0.00001

1.098606

Based on the values in the table below, Y-1

lim - - "" 1.6094 x--+0

To four decimal places, In 5

X

= 1.6094, so it appears that Jim sx- 1 = ln 5. x--+O x x

5x_I

x

x

5x_l

x

0.001

1.610734

-0.001

1.608143

0.0001

1.609567

-0.0001

1.609308

0.00001

1.609451

-0.00001

1.609425

0.00,0001

1.609439

-0.000001

1.609437

s E c TI o N 2.5 I Indeterminate Forms

81

In Exercises 43-48, evaluate using the identity a 3 -b3 = (a-b)(a2 +ab+ b2 )

x3 - 8

43. lim-x-+2 x- 2 SOLUTION

45.

x3 - 8 (x - 2)(x2 + 2x + 4) . 2 lim - - = lim = hm(r + 2x + 4) = 12 x->2 X - 2 x->2 X - 2 x->2

x2-5x+4

lim-~-­ x3 - 1

x->I

SOLUTION

. x2-5x+4 hm x->1 x3 - 1

(x- l)(x-4) x-4 __ -3 ___ 1 = lim - - - l)(x2 + x + 1) x->l x 2 + x + 1 3

.

=1x->l lffi (x -

x4 -1 47. lim - 3- x->I X - 1 SOLUTION

x"-1 . (x2- l)(x2+1) . (x- l)(x+ l)(x2 +1) . (x+ l)(x2+1) 4 lim-- =hm =hm =hm = x->IX3-1 x->I(x-l)(x2+x+·l) x->I (x-l)(x2 +x+l) x->I x 2 +x+l 3

In Exercises 49-56, evaluate in terms of the constant a. 49. lim(2a + x) x->0

SOLUTION

lim(2a + x) x->0

= 2a

51. lim(4t - 2at + 3a) t-+-1

SOLUTION

lim(4t- 2at + 3a) = -4 + 5a

t-+-l

53. lim x-+a

Yx - Va x- a

SOLUTION

lim

..rx-va = hm. X -

x->a

a

X->a (

..rx-va

Yx - va)( Yx + va)

.

= hm x->a

1

Yx + Va

1

= -2 Va

(x+a) 3 -a3 55. l i m - - - x--+O

SOLUTION

X

lim

(x + a)3 - a 3

x--+0

X

= lim

x3 + 3x2a + 3xa2 + a 3 - a 3 X

X-i-0

= lim(x2 + 3xa + 3a2) = 3a2 x-+0

. of x, and rewnte . as a l"1rmt . as x . {'l + h - l . n·mt: Set x = -·~1 57. Evaluate I1m v1 + h n, express h as a fu nctlon h->0 h SOLUTION

Let x = {'1 + h. Then h = x4

-

-7

1.

l, and

. {'1 + h - 1 l" x - 1 l" x- 1 l" 1 = lffi - - = lffi = lffi - - - : : - - - : - : Ilffi h->0 h x->1 x4 - 1 x->l (X - l)(X + l)(x2 + 1) x->I (X + l)(x2 + 1)

4

Further Insights and Challenges In Exercises 59-62,find all values of c such that the limit exists. 59. lim x2 - Sx - 6 x--+c

SOLUTION

X- C

lim

x2 -5x-6 X- C

x--+c

will exist provided that x - c is a factor of the numerator. (Otherwise there will be an

infinite discontinuity at x = c.) Since x2 - 5x - 6 = (x + l)(x - 6), this occurs for c = -1 and c = 6.

c- ) 61. lim ( -l - - x->I x - 1 x3 - 1 SOLUTION

Because x3

- 1 = (x- l)(x2 + x + 1), c x - 1

---3 x- 1

1

x2+x+l-c (x - l)(x2 + x + 1)

Therefore, lim (~ - -/--- ) exists as long as x- 1 is a factor of x2 + x + 1 - c. Now, if x2 + x + 1 - c = (x - l)(x + q), x-+I XX - 1 then q - 1 = 1 and -q = 1 - c. Hence, q = 2 and c = 3.

s E CT I o N SOLUTION

2.6

I

The Squeeze Theorem and Trigonometric Limits

The sine function takes on values between -1 and l; therefore, \sin x~i I:::; 1 for all x

83

* 1. Multiplying by

Ix - 11 yields

l(x- l)sin~I:::; lx-11

or

x-l

-lx-11:::; (x-

l)sin~ :::; lx-11 x-

1

Now lim( -Ix - 11) = lim Ix - 11 = 0, so we can apply the Squeeze Theorem to conclude that x-+1

X-+1

lim(x- l)sin~ =0 x-->I X- 1 1

5. lim(2' - 1) cos t-->O t SOLUTION The cosine function takes on values between -1 and l; therefore, \cos~\ :::; 1 for all t 12' - 11 yields 1O t

7. lim(t2 - 4) cos __!_ 1-->2

SOLUTION

2 The cosine function takes on values between -1 and 1; therefore, \cos 1 ~ 2 \ (-

:::;

1 for all t

* 2. Multiplying by

1t2 - 41 yields 1 (t2 -4)cos - -1:::; 1t2 -41 t-2

l

or

- lt2 - 41 :::; (t2

-

4) cos __!_2 :::; 1t2 - 41 t-

Now lim(-lt2 - 41) = lim 1t2 - 41 = 0, so we can apply the Squeeze Theorem to conclude that t-+2

t-+2

1 lim(t2 - 4) cos - - = O t-->2 t-2

9. limcosBcos(tanB) 9_,.~

The cosine function takes on values between -1 and 1; therefore, lcos (tan B)I :::; 1 for all B near plying by Icos Bl yields SOLUTION

or

lcos Bcos (tan B)I :::; Icos Bl

~.

Multi-

- Icos Bl :::; cos Bcos (tan B) :::; Icos Bl

Now lim(-1 cos Bl) = lim I cos Bl = 0, so we can apply the Squeeze Theorem to conclude that 0--+~

8-+~

lim cos Bcos (tan B) = 0

8--+~

11. State precisely the hypothesis and conclusions of the Squeeze Theorem for the situation in Figure 6. y

y=u(x)

2

1 FIGURE 6 SOLUTION

Because there is an open interval containing x

= 1 on which l(x) :::; f(x)

it follows that limf(x) exists and x-->I

limf(x)

x-->1

2

=2

:::; u(x) and lim l(x) X-+l

= lim u(x) = 2, X-+]

84

C H A PT E R 2

I

LIMITS

13. What does the Squeeze Theorem say about limf(x) ifthe limits x-->1

lim l(x) = lim u(x) = 6 and f, u, and l are related as in Figure 8? The inequality f(x) :5 u(x) is not satisfied for all x. Does X_.7

X-i-7

this affect the validity of your conclusion?

6

7 FIGURE 8 SOLUTION The Squeeze Theorem does not require that the inequalities l(x) :5 f(x) :5 u(x) hold for all x, only that the inequalities hold on some open interval containing x = c. In Figure 8, it is clear that l(x) :5 f(x) :5 u(x) on some open interval containing x = 7. Because lim u(x) = lim l(x) = 6, the Squeeze Theorem guarantees that Iimf(x) = 6. x-+7

x-t?

x--+1

15. State whether the inequality provides sufficient information to determine lim f(x), and if so, find the limit. X-->1

(a) 4x - 5 :5 f (x) :5 (b)

(c)

x2 2x - 1 :5 f(x) :5 x2 4x - x2 :5 f(x) :5 x 2 + 2

SOLUTION

(a) Because lim(4x - 5) = -1 x--+1

* 1 = lim x2, the given inequality does not provide sufficient information to determine x--+1

limf(x).

X-->1

(b) Because lim(2x - 1) = 1 = limx2, it follows from the Squeeze Theorem that limf(x) = 1. x--+1

x--+1

x-+I

(c) Because lim(4x - x2) = 3 = lim(x2 + 2), it follows from the Squeeze Theorem that limf(x) = 3. x-+1

x-+1

x-+1

In Exercises 17-26, evaluate using Theorem 2 as necessary.

. tanx 17• hm-x-->O

X

. tanx . sinx 1 sinx 1 hm - - = hm - - - - = lim - - . lim - - = 1 . 1 = 1

SOLUTION

x-->O

X

x-->O

X

COS X

x-->O

X

x-->O COS X

. Yt3 + 9 sin t 19. h m - - - - t-->O t

.

hm

SOLUTION

-../r3 +9sint t

t-->O

. .~sint -~ sint = lim vt3 + 9 - = lim vt3 + 9 · lim = t-->O t t-->O t-->O t

V9. 1 = 3

21. lim_±_ 2 x-->O sin X

x2 Iim-.--.1 l" 1 l"Im-.-= 1 . 1Im--= =Im-.-. -1 . -1 = 1

SOLUTION

X-i-0 Sifl2 X

X--+0 ~ ~

x

x

X-+0

SlDX

x

X--+0 ~

x

1 1

23. lim sec () - 1 8-->0

()

SOLUTION

. sec e - I _ . sec e - I cos e _ . I - cos e I . I - cos() . I 1 lim 1 1 8-->o e -Im 8-->o e cos e -Im 8-->o e --=hm cos e 8-->o e ·hm--=0·-=0 8-->o cos e I

. sint 25. h m r-~

t

SOLUTION

~nt

-

t

. n is continuous at t = - . Hence, by substitution 4

27. Evaluate lim sin llx using a substitution() = llx. x-i-0

x

s E c T 1o N SOLUTION

2.6

Let () = 1lx. Then () -+ 0 as x -+ 0, and x =

I

The Squeeze Theorem and Trigonometric Limits

()I 11. Hence,

. sin llx . sin() . sin() h m - - = lim-- = llhm- = 11(1) = 11 x--+0 x 0--+0()111 0--+0 0 In Exercises 29-48, evaluate the limit.

. sin9h 29. hm-hh--+O

SOLUTION

sin 9h . sin 9h lim - - = hm 9 -9h = 9. h--+0 h h--+O

sink 5

31. limh h--+0

SOLUTION

. sink 1 . 1 sink hm - - = 1im - - - = 5h h--+0 5 h 5

h--+0

sin 1() 33. lim -.- 0--+0 sm 30 SOLUTION

We have sin 7e = 2 (sin 70) (~) sin 30 3 1() sin 30

Therefore, lim sin 1() O--+O 3()

= 2(um sin 7()) (1im ~) = 2(1) 3

O--+O

1()

O--+O

sin 3()

3

1 ( )

=2 3

35. lim x csc 25x x--+0

SOLUTION

37

r

• h1!!/i

. . x 1 . 25x 1 hmxcsc25x = hm-.-- = - h m - - - = x--+0 x--+0 sm 25x 25 x--+0 sin 25x 25

sin 2h sin 3h h2

SOLUTION

Iim

h--+O

sin 2h sin 3h h2

sin 2h sin 3h = Iih--+0 m h·h

. sin 2h sin 3h =lim-- - h--+O

h

h

sin 2h sin3h . sin 2h . sin3h . = 1im 2 - - 3 - - = 1im 2 - - hm3-- =2·3 =6 2h

h--+0

39

r

• /-To

3h

h--+0

2h

h--+0

sin(-30) sin40

SOLUTION

lim sin(- 30) O--+O sin(40)

= lim -sin(3(J) · ~. ~ 4

30

O--+O

sin(40)

= -~ 4

41. lim csc 8t HO csc4t SOLUTION

lim csc 8t = lim sin4t . ~ . ~ = ~ HO csc4t HO sin8t 4t 2 2

. sin 3x sin 2x 43. 1i m - - - x--+O xsin5x SOLUTION

45

lim sin 3~ sin 2x x--+o x sm 5x

= lim (3 sin 3x . ~ x--+o

3x

(sin 2x) I (2x)) = ~ 5 (sin 5x) / (5x) s

. lim sin(2h)(l - cos h) h--+0 h2

SOLUTION

. sin(2h)(l - cos h) _ . sin(2h) . 1 - cos h 1lffi h2 - 1lffi 2 - - - 1lffi =2·0=0 h--+0

. cos 2() - cos () 47. 1i m - - - - O--+O

()

h--+0

2h

h--+O

h

3h

85

86

CHAPTER 2

I

LIMITS SOLUTION

lim 9_,o

cos 2B - cos e (cos 2B - 1) + (1 - cos e) cos 2B - 1 1 - cos e = lim = lim + lim - - e 9_,o e 9_,o e 8_,o e . 1 - cos e . 1 - cos 2B O O O 2 ·+= 1 =-Im 21 +Im =9_,o 2e 8_,o e

. sin 2B - 2 sine 49. Use the identity sin 2B = 2 sine cos e to evaluate hm ez 8--'>0

SOLUTION

Using the identity sin 2B = 2 sine cos e, sinW - 2 sine= 2 sin ecos e - 2 sine= 2 sin e(cos e - 1) = -2 sine(l - cos e)

Then . sin2B-2sine_ . sine(cose-1)_ . sine 1-cose (l)(O)-O hm - - 211m - - 21 Im - - · =-2 8--'>0 ez 8--'>0 e2 8--'>0 e e

51. Explain why lim(csc e- cot 8) involves an indeterminate form, and then prove that the limit equals 0. 8_,o

SOLUTION Ase approaches 0 from the right, csce ~ oo and cote ~ oo, and as e approaches 0 from the left, csc e ~ -oo and cote ~ -oo. Thus, csc e - cote has the indeterminate form oo - oo as e ~ 0.

Now, 1 cose 1-cose csce-cote= - - - - - = - - sine sine sine

so . . 1-cose . 1-cose e hm( csc e - cote) = hm . = lim -.- = 0 · 1 = 0 8--'>0 8--'>0 sm e 8--'>0 e sm e imTii

.

.

53. ~ . Investigate l~

1 - cos 2h . h2 numerically or graphically. Then evaluate the limit using the double angle formula

cos2h = 1 - 2sin2 h. SOLUTION

h

-0.1

-0.01

O.Ql

0.1

1.993342

1.999933

1.999933

1.993342

l-cos2h h2

1.5

Both the numerical estimates and the graph suggest that the value of the limit is 2. • Using the double angle formula cos 2h = 1 - 2 sin2 h, l -cos2h = 1- (l -2sin2 h) = 2sin2 h Then . 1 - cos 2h 2 sin2 h sin h sin h hm h2 = lim-- = 2lim-- · - - = 2(1)(1) = 2 h--'>0 h--'>0 h2 h--'>0 h h In Exercises 55-57, evaluate using the result of Exercise 54.

. cos3h- 1 55. hm h2 h--'>0

S E CT I 0 N 2.6

SOLUTION

I

The Squeeze Theorem and Trigonometric Limits

We make the substitution B = 3h. Then h = 0/3, and . cos 3h - i hm h->O

h1

cos e - i = Ii8->0 m o

.Yl -cost 57. l i m - - - ,_,o t SOLUTION

lim

.Yl - cost

=

t

t->0

Further Insights and Challenges 59. Use the result of Exercise 54 to prove that form

* 0,

. cosmx- 1 m2 = 11m 1 x->O

SOLUTION

x

. x . . cos mx - 1 • u' vve obtam Sub stltute u = mx mto X2 . cos mx - 1 11m x1

x->O

2

= -mu . As x --+ 0 , u --+ 0 ; therefiore,

. cos u - 1 . cos u = hm = 1u->O 1mm2 u->O (ujm)1 u1

1

1

m = m2 ( --21 ) = -2

. . sinx-sine . 61. (a) Investigate hm numencally for the five values e = 0, ~, ~, } , ~. x-+c

X -

C

(b) Can you guess the answer for general c? (c) Check numerically that your answer to (b) works for two other values of e. SOLUTION

(a) Here e

= 0 and cose = 1.

Here e = ~ and cos e =

Here e

e -0.01

e -0.001

e + 0.001

c + 0.01

sinx - sine x-e

0.999983

0.99999983

0.99999983

0.999983

f , , .866025. x

c -0.01

e -0.001

e +0.001

e + 0.01

sinx- sine x-e

0.868511

0.866275

0.865775

0.863511

x

e - 0.01

e - 0.001

e + 0.001

e + 0.01

sinx - sine x-e

0.504322

0.500433

0.499567

0.495662

x

c -0.01

e-0.001

c + 0.001

c + 0.01

sinx - sine x-e

0.710631

0.707460

0.706753

0.703559

= } and cose = ~·

Here e = ~ and cos e =

Here e

x

Y{ ,,, 0.707107.

= ~ and cos e = 0.

. sinx-sine (b)l1m x-+c

X -

C

x

e- 0.01

e -0.001

e + 0.001

e + 0.01

sinx-sinc x-e

0.005000

0.000500

-0.000500

-0.005000

=cose.

87

88

cHAp TER

2

I

LIMITS

(c) Here c = 2 and cos c =cos 2"" -.416147. x

c -0.01

c- 0.001

c+0.001

c + 0.01

sinx - sine x-c

-0.411593

-0.415692

-0.416601

-0.420686

Here c = -~ andcosc = "{ "".866025. x

c-0.01

c -0.001

c+0.001

c + 0.01

sinx - sin c x-c

0.863511

0.865775

0.866275

0.868511

2. 7 Limits at Infinity Preliminary Questions 1. Assume that Jim f(x) = L

and

x--+oo

limg(x) = oo

x--+L

Which of the following statements are correct? (a) x =Lis a vertical asymptote of g.

(b) y =Lis a horizontal asymptote of g. (c) x =Lis a vertical asymptote off.

(d) y =Lis a horizontal asymptote off. SOLUTION

(a) Because lim g(x) = oo, x = L is a vertical asymptote of g(x). This statement is correct. x->L

(b) This statement is not correct. (c) This statement is not correct. (d) Because Jim f(x) = L, y =Lis a horizontal asymptote of f(x). This statement is correct.

x--+oo

2. What are the following limits? (a) lim x3 X--+OO

(b) Jim

x3

(c) Jim

X--+-DO

x--+-oo

.x4

SOLUTION

(a) limx--+oo

x3 = oo

(b) limx--+-oo x3 = -oo (c) limx--+-oo x 4 = oo

3. Sketch the graph of a function that approaches a limit as x ~ oo but does not approach a limit (either finite or infinite) as x ~ -oo. SOLUTION y

4. What is the sign of a if f(x) = ax3 + x + 1 satisfies lim f(x) = oo? X--+-00

SOLUTION

Because lim x--+-oo

x3 = -oo, a must be negative to have x--+-oo Jim f(x) == oo.

5. What is the sign of the coefficient multiplying x7 if f is a polynomial of degree 7 such that lim f(x) "" oo? x--+-oo

s Ec T Io N

2. 7

I

Limits at Infinity

89

The behavior of f(x) as x ~ -oo is controlled by the leading term; that is, limx--+-oo f(x) = limx--+-oo a1x1. Because x 1 ~ -oo as x ~ -oo, d7 must be negative to have limx--+-oo f(x) = oo.

SOLUTION

6. Explain why lim sin l exists but lim sin l does not exist. What is lim sin l? x--+oo

SOLUTION

x--+0

x

x

x--+oo

x

As x ~ oo, ; ~ 0, so

. sm. 1 lim

. 0 =0 = sm

X

X-HX)

On the other hand, ; ~ ±oo as x ~ 0, and as ; ~ ±oo, sin ; oscillates infinitely often. Thus 1 limsin X

X--+0

does not exist.

Exercises 1. What are the horizontal asymptotes of the function in Figure 7? y 2

-20

20

40

60

80

FIGURE 7 SOLUTION

Because lim f(x) X--+-oo

the function f has horizontal asymptotes of y

=1

and

lim f(x) X--+OO

=2

= 1 and y = 2.

3. Sketch the graph of a function f with a single horizontal asymptote y = 3. SOLUTION y

---4-r--_+z--~l..-.......-,__x

-5 -9

-13

5. !GUii Investigate the asymptotic behavior off(x) = _£___ numerically and graphically: x2 + 1 (a) Make a table of values of f(x) for x = ±50, ±100, ±500, ±1000. (b) Plot the graph off. (c) What are the horizontal asymptotes off? SOLUTION

(a) From the table below, it appears that . x2 1im - - = 1 2

x--+±00

x ! 00 x + 9 X->OO x- (x + 9) X->OO 1 + ~ 1+0 3.x2 + 20x 9. lim ---,,..-x->oo 2x4 + 3x3 - 29 SOLUTION

. hm x->oo

3.x2 + 20x . x-4 (3.x2 + 20x) ~ + ~ ___ o __ 0 . = hm = 1Im - - - 2x4 + 3x3 - 29 x->oo x- 4 (2x4 + 3x3 - 29) x->oo 2 + ~ - ~ 2

7x-9 11. l i m - x->oo 4x + 3 SOLUTION

7 . 7x-9 . x- 1(7x-9) . 7-~ 1I m - - = 1Im 1 =Im--=1 x->oo 4x + 3 X->OO x- (4x + 3) X->OO 4 + ~ 4 7.x2-9 4X + 3

13. lim x-+-oo

SOLUTION

. . 7.x2-9 x- 1(7x2-9) 7x-~ . 1Im - - - = 1Im = 1Im - - = -oo x->-oo 4x + 3 x->-oo x- 1(4x + 3) x->-oo 4 + ~

15. lim

3x3 - 10

x-+-oo

X

+4

SOLUTION

lim x->-oo

3x3 - 10

x +4

= lim

x- 1(3x3 - 10)

x->-oo

x- 1(x + 4)

= lim x->-oo

3.x2 -

x

In Exercises 17-24,find the horizontal asymptotes. 2.x2 - 3x 17. f(x) = 8x2 + 8 SOLUTION

First calculate the limits as x

~

lim x->oo

±oo. For x 2.x2 - 3x

8x2 + 8

~

oo, 2 - :!

2

1

=lim--x-=-=x->oo 8 + 8 4

!x

Similarly,

r

2x2

-

3x

x-l~oo 8x2 + 8 Thus, the horizontal asymptote off is y =

~.

2 8

lQ x

1 + '.!

4

= oo

s E c TI o N 2.7 I Limits at Infinity 91

19.

v36x2 +1 x+4 9

f (x) =

For x > 0, x- 1

SOLUTION

= lx- 11= R, so

V36

. v36x2 + 7 l" hm = im X-->OO 9x + 4 X-->OO

On the other hand, for x < 0, x- 1

v36x2 + 7 - 36 + :;, - V36 2 = lim = - - = -9x + 4 x-->-oo 9 + ix 9 3

~

x-->-oo

Thus, the horizontal asymptotes off are y

f (t) =

1

SOLUTION

2 3

= -lx- 11 = - R, so lim

21.

9

= ~ and y = - ~.

e' + e-i With e1

l i m - -1 = oo 1 + e-

1-->oo

and

e' lim - -1 =0 I + e-

1-->-oo

the function f has one horizontal asymptote, y = 0.

23. g(t) = 1:~-1 Because

SOLUTION

lim t--+-oo

r' =

and

00

lim 3-I t--+oo

=0

it follows that

. 10 hm - -1 =0

1-->-00

Thus, the horizontal asymptotes of g are y

1 + 3-

and

.

10

hm - -1 1 + 3-

1-->-00

= 10

= 0 and y = 10.

The following statement is incorrect: "If f has a horizontal asymptote y = Lat oo, then the graph off approaches the line y = Las x gets greater and greater, but never touches it." In Exercises 25 and 26, determine lim f(x) and indicate x-->oo

how f demonstrates that the statement is incorrect. 25• f(x) = SOLUTION

2x;lxl

For x > 0, lxl = x and 2x+x x

f(x)= - - =3

Thus, lim f (x) = 3, so f has a horizontal asymptote of y = 3. The statement that the graph of f never touches this X-'>00

horizontal asymptote is incorrect because, for all x > 0, the graph off coincides with the horizontal asymptote y = 3. In Exercises 27-34, evaluate the limit. . v9.x4 + 3x + 2 27. hm x--+oo 4 X3 + 1 SOLUTION

For x > 0, x- 3 = ix- 31 =

-..!r6, so

lim

x-->oo

8x2 + 7x1t3

29. lim --;::=== x+oo v16.x4 + 6

v9.x4 + 3x + 2 = lim 4x3 + 1 x--.oo

~ !r + !s + ~ 4 + J.. x3

=0

92

CH APT E R 2

I

LIMITS

For x < 0, x- 2 =

SOLUTION

IX-21= V?, so 8.x2 + 7x 1i 3 8 8 + )/3 =lim =--=2 lim x+oo Yl6x4 + 6 x+oo ~16 + -3i -{[6

t4f3

+ tlf3

31. Hoo lim (4 f 2/3 + 1)2 t4f3

+ tlf3

l~1!! (4t2/3 + 1)2

SOLUTION

l +l

=

1 16

l.To! (4 + ,2;3 )2

. lxl +x 33. hm - x~-oo X + 1 For x < 0, lxl = -x. Therefore, for all x < 0,

SOLUTION

lxl + x _ -x + x _ x+l -7+1- 0 consequently, lim lxl + x = 0 X->-oo X + 1 35. ~ SOLUTION

Determine lim tan- 1 x. Explain geometrically. X->OO As an angle(} increases from 0 to

~,

its tangent x = tan(} approaches oo. Therefore, lim tan- 1 x = ~ X->OO

2

Geometrically, this means that the graph of y = tan- x has a horizontal asymptote at y = ~. 1

37. In 2009, 2012, and 2015, the number (in millions) of smart phones sold in the world was 172.4, 680.1, and 1423.9, respectively. (a) CR 5 Lett represent time in years since 2009, and let S represent the number of smart phones sold in millions. Determine M, A, and k for a logistic model, S (t) = i+:!-k,, that fits the given data points. (b) What is the long-term expected maximum number of smart phones sold annually? That is, what is lim S (t)? t->oo (c) In what year does the model predict that smart-phone sales will reach 98% of the expected maximum? SOLUTION

(a) With t representing time in years since 2009, 2009 corresponds to t = 0, 2012 corresponds to t = 3, and 2015 corresponds tot = 6. To fit the given data to the logistic model S (t) = i+:!-k,, we must solve the system of equations M 172.4 = l +A'

M 680.1 = 1 + Ae-3k,

and

1423.9 =

M 6k 1 +Ae-

for M, A, and k. Using a computer algebra system to solve this system yields M"" 1863.3, A"" 9.808, and k "" 0.576. Hence, the model for the number of smart phones sold, in millions, is 1863.3 S (t) = 1 + 9.808e-o.s161 (b) Because lim e-0576' = 0, it follows that 1-->00

. 1863.3 1863.3 11m S(t) = lim = - - = 1863.3 Hoo 1 + 9.808e-0.576t 1+ 0

t tooo

The long-term expected maximum number of smart phones sold annually is approximately 1863.3 million. (c) To determine the year in which the model predicts that smart-phone sales will reach 98% of the expected maximum, we must solve the equation 1863.3 (0.98)1863.3 = 1 + 9.808e-0.576r for t. This equation is equivalent to 1 + 9.808e-0.576r =

1863.3 (0.98)1863.3

0.98

and then

1 0 02 9.808e-0·576' = - - - 1 = · = ..!.._ 0.98 0.98 49

s Ec T Io N

2. 7

I

Limits at Infinity

93

Finally,

1

1

t = -0.576 ln 49(9.808) ""

10 12 ·

Thus, the year in which the model predicts that smart-phone sales will reach 98% of the expected maximum is approximately 10.72 years after 2009, or sometime in 2019.

In Exercises 39-46, calculate the limit. 39. lim( ../4x4 + 9x - 2x2) X-+OO

Write

SOLUTION

4 ../4x4 + 9x - 2x2 = ( ../4x4 + 9x - 2x2) ../ x4 + 9x + 2x2 ../4x4 + 9x + 2x2 (4x4 + 9x) - 4x4

9x

../4x4 + 9x + 2x

V4x4 + 9x + 2xz

2

Thus,

9 x

lim( ../4x4 + 9x - 2x2) = lim x-+oo

../4x4 + 9x + 2x2

x-+oo

=

41. lim(2..;i- .Yx+ 2) X-+OO

Write

SOLUTION

_r::. _r--;; _r::. _r--;; 2..;x+ vx+2 2v .... - vx+2=(2vx- v x + 2 ) - - - - 2..;i+ ../x+2 4x- (x+ 2)

3x-2

2..;i+.Yx+2 Thus, lim(2../i- ../x+2) = lim x-+oo

x-+oo

2 3 x-

2 ..jX + .../X + 2

43. lim (ln(3x + 1) - ln(2x + 1)) x-+oo

Because

SOLUTION

3x+ 1 ln(3x + 1) - ln(2x + 1) = ln - 2x + 1

and lim 3x+ 1 2X + 1

=~ 2

X-+OO

it follows that lim (ln(3x + 1) - ln(2x + 1))

=

X-+OO

ln ~ 2

9)

45. lim tan- 1 (x2+ x-+oo 9- X SOLUTION

Because . x2+9 . x+~ oo hm - - = hm - - = - = -oo X-+oo

9-

X

x-+oo

~

-

}

-1

it follows that

. tan- l(x2+9) hm - - = - -1f 9- X 2

x-+oo

=

00

o

94

CHAPTER 2

I

LIMITS

47.

($J Let P(n) be the perimeter of an n-gon inscribed in a unit circle (Figure 8). (a) Explain, intuitively, why P(n) approaches 2n as n ~ oo. (b) Show that P(n) = 2n sin(~). (c) Combine (a) and (b) to conclude that lim !! n--+oo Tr sin(!!)= 1. n sinB (d) Use this to give another argument that lim -() = 1. · . " · . · . · . 00. e~o '' ' ~' ' " " ' OO X - 1 1->0+ 1- ( 2.8 The Intermediate Value Theorem Preliminary Questions x2 takes on the value 0.5 in the interval [O, l]. Observe that f(x) = x2 is continuous on [O, 1] with f(O) = 0 and f(l) = 1. Because f(O) < 0.5 < f(l), the 1. Prove that f(x) = SOLUTION Intermediate Value Theorem guarantees there is a c E [0, 1] such that f(c) = 0.5. 2. The temperature in Vancouver was 8°C at 6 AM and rose to 20°C at noon. Which assumption about temperature allows us to conclude that the temperature was 15°C at some moment of time between 6 AM and noon? SOLUTION We must assume that temperature is a continuous function of time. 3. What is the graphical interpretation of the IVT? SOLUTION Iff is continuous on [a, b], then the horizontal line y = k for every k between f(a) and f(b) intersects the graph of y = f(x) at least once. 96 CHAPTER 2 I LIMITS 4. Show that the following statement is false by drawing a graph that provides a counterexample: If f is continuous and has a root in [a, b], then f(a) and f(b) have opposite signs. SOLUTION y 5. Assume that f is continuous on [1,5] and that f(l) statements is always true, never true, or sometimes true. (a) f(c) = 3 has a solution with c E [l, 5]. (b) f(c) = 75 has a solution with c E [1, 5]. = 20, f(5) = 100. Determine whether each of the following (c) f(c) = 50 has no solution with c E [l, 5]. (d) f(c) = 30 has exactly one solution with c E [1, 5]. SOLUTION (a) This statement is sometimes true. Because 3 does not lie between 20 and 100, the IVT cannot be used to guarantee that the function takes on the value 3 but it may still do so. (b) This statement is always true. Because f is continuous on [l, 5] and 20 = f(l) < 75 < f(5) = 100, the IVT guarantees there exists a c E [1, 5] such that f(c) = 75. (c) This statement is never true. Because f is continuous on [1, 5] and 20 = f(l) < 50 < f(5) = 100, the IVT guarantees there exists a c E [l, 5] such that f(c) = 50. (d) This statement is sometimes true. Because f is continuous on [1, 5] and 20 = f(l) < 30 < f(5) = 100, the IVT guarantees there exists a c E [l, 5] such that f(c) = 30 but there may be more than one such value for c. Exercises 1. Use the IVT to show that f(x) = x3 + x takes on the value 9 for some x in [1, 2]. Observe that f(l) = 2 and f(2) = 10. Since f is a polynomial, it is continuous everywhere; in particular on [1,2]. Therefore, by the IVTthereis ac E [1,2] such thatf(c) = 9. SOLUTION 3. Show that g(t) SOLUTION there is a c E = r tan t takes on the value t for some tin [O, H g(O) = 0 and g(~) = ~- g(t) is continuous for all t between 0 and~. and O < [O, ~]such that g(c) = t- t < ~;therefore, by the IVT, = x has a solution in the interval [0, l]. Hint: Show that f(x) = x - cos x has a zero in [O, l]. SOLUTION Let f(x) = x - cos x. Observe that f is continuous with f(O) = -1 and f(l) = 1 - cos 1 ~ .46. Therefore, by the IVT there is a c E [0, l] such that f(c) = c - cos c = 0. Thus c = cos c and hence the equation cos x = x has a 5. Show that cos x solution c in [O, 1]. In Exercises 7-16, prove using the /VT. .ye+ -Ve+ 2 = 3 has a solution. SOLUTION Let f(x) = Vx + -Vx + 2 - 3. Note that f is continuous on [0,2] with f(O) = VO+ -.../2- 3 ~ -1.59 and f(2) = V2 + '14- 3 ~ 0.41. Therefore, by the IVT there is a c E [0,2] such that f(c) = .ye+ Ve+ 2 - 3 = 0. Thus .ye+ Ve+ 2 = 3, and the equation .ye+ vc + 2 = 3 has a solution c in [O, 2]. 9. V2 exists. Hint: Consider f(x) = x2. 7. Let f(x) = x2. Observe that f is continuous with f(l) = 1 and f(2) = 4. Therefore, by the IVT there is a [l, 2] such that f(c) = c2 = 2. This proves the existence of -.../2, a number whose square is 2. SOLUTION c E 11. For all positive integers k, cos x SOLUTION = :x!' has a solution. For each positive integer k, let f (x) = :x!' - cos x. Observe that f is continuous on [O, ~] with f(O) and f(~) = (~t > 0. Therefore, by the IVT there is a c E ~]such that f(c) hence the equation cos x = :x!' has a solution c in the interval [0, ~]. [o, 13. 2x + 3x = 4x has a solution. = c" - cos(c) = -1 = O. Thus cosc = c" and s E c TI o N 2.8 I The Intermediate Value Theorem 97 Let f(x) = 2x + Y - 4x_ Observe that f is continuous on [0, 2] with f(O) = 2° + 3° - 4° = 1 + 1 - 1 = 1 and f(2) = 22 + 32 - 42 = 4 + 9 - 16 = -3. Therefore, by the IVT, there is a c E [O, 2] such that f(c) = 2c + 3c - 4c = 0. Hence, the equation 2x + 3x = 4x has a solution. 15. ex + In x = 0 has a solution. 2 SOLUTION Let f(x) =ex+ Inx. Observe that f is continuous on [e- 2, 1] with f(e- 2 ) = e'- - 2 < 0 and f(l) = e > 0. 2 Therefore, by thelVT, there is ac E [e- , 1] such thatf(c) = ec + lnc = 0. 17. Use the Intermediate Value Theorem to show that the equation x 6 - 8.x4 + 10x2 - 1 = 0 has at least six distinct SOLUTION solutions. Let f(x) = x 6 - 8x4 + IOx2 - 1. Then f(O) = -1,f(±l) = 2,f(±2) = -25 and f(±3) = 170. Hence as we move along the number line from left to right through the points -3, -2, -1, 0, 1, 2, 3, the function changes sign at least 6 times. Hence there must be a zero of the function between any two of these integers, and therefore, there must be at least six distinct solutions to the equation x 6 - 8x4 + IOx2 - 1 = 0. SOLUTION In Exercises 18-20, determine whether or not the /VT applies to show that the given function takes on all values between f(a) and f(b) for x E (a, b). If it does not apply, determine any values between f(a) and f(b) that the function does not take on for x E (a,b). 19. -x f(x) = { x3 + 1 for x < 0 for x ~ 0 for the interval [-1, 1]. SOLUTION The graph off over the interval [-1, 1] is shown below. From the graph, we see that f is not continuous on [-1, 1] because of a jump discontinuity at x = 0. Therefore, the IVT does not apply. However, from the graph, we see that f does take on every value between f(-1) = 1 and f(l) = 2 for x E (-1, 1). y -1.0 -0.5 0.5 1.0 Carry out three steps of the Bisection Method for f(x) = 2x -x3 as follows: Show that f has a zero in [l, 1.5]. Show that f has a zero in [1.25, 1.5]. Determine whether [1.25, 1.375] or [1.375, 1.5] contains a zero. SOLUTION Note that f(x) is continuous for all x. (a) f(l) = 1, f(l.5) = 21. 5 - (1.5) 3 < 3 - 3.375 < 0. Hence, f(x) = 0 for some x between 1 and 1.5. (b) f(l.25) "'0.4253 > 0 and f(l.5) < 0. Hence, f(x) = 0 for some x between 1.25 and 1.5. (c) f(l.375)"' -0.0059. Hence, f(x) = 0 for some x between 1.25 and 1.375. 21. (a) (b) (c) 23. Find an interval of length ~ in [1, 2] containing a root of the equation x7 + 3x - 10 = 0. Let f(x) = x 1 + 3x - 10. Observe that f is continuous on [1, 2] with f(l) = -6 and f(2) = 124, so the IVT guarantees that the equation x7 + 3x - 10 = 0 has a root on the interval [l, 2]. The midpoint of the interval [1, 2] is 1.5 and f(l.5) = 11.585938 > 0, so we can conclude that the equation x7 + 3x - 10 = O has a root on the interval [1, 1.5]. Finally, the midpoint of the interval [1, 1.5] is 1.25 and f(l.25) = -1.481628 < 0, so we can conclude that the equation x1 + 3x - 10 = 0 has a root on the interval [l.25, 1.5]. SOLUTION In Exercises 25-28, draw the graph of a function f on [O, 4] with the given property. 25. Jump discontinuity at x = 2 and does not satisfy the conclusion of the IVT SOLUTION The function graphed below has a jump discontinuity at x = 2. Note that while f(O) = 2 and f(4) = 4, there is no point c in the interval [O, 4] such that f(c) = 3. Accordingly, the conclusion of the IVT is not satisfied. y 4 - 2 0 -+-+--+---+-+-+-x 2 4 98 CH APT E R 2 I LIMITS = 2 and does not satisfy the conclusion of the IVT The function graphed below has infinite one-sided limits at x = 2. Note that while f(O) = 2 and f(4) = 4, 27. Infinite one-sided limits at x SOLUTION there is no point c in the interval [0, 4] such that f(c) = 3. Accordingly, the conclusion of the IVT is not satisfied. y 6 -~ 5 4 2 x 2 4 -1 29. ~ Can Corollary 2 be applied to f(x) = x- 1 on (-1, 1]? Does f have any roots? SOLUTION Although f(-1) = -1 < 0 and f(l) = 1 > 0 are of opposite sign, Corollary 2 cannot be applied because f is not continuous on the interval [-1, 1]. This function does not have any roots. 31. At 1:00 PM. Jacqueline began to climb up Waterpail Hill from the bottom. At the same time Giles began to climb down from the top. Giles reached the bottom at 2:20 PM, when Jacqueline was 85% of the way up. Jacqueline reached the top at 2:50. Use the result in Exercise 30 to prove that there was a time when they were at the same elevation on the hill. SOLUTION Lett represent time in minutes since 1:00 PM, and let J(t) and G(t) represent the elevation in percentage of the way up the hill at time t of Jacqueline and Giles, respectively. Then J(O) = 0 < 1 = G(O) and G(80) = 0 < 0.85 = J(80). Assuming that J and G are continuous on (0, 80], the result in Exercise 30 implies there exists c E (0, 80] such that J(c) = G(c). At time c, Jacqueline and Giles were at the same elevation on the hill. Further Insights and Challenges Exercises 33 and 34 address the I-Dimensional Brouwer Fixed Point Theorem. It indicates that every continuous function f mapping the closed interval [0, 1] to itself must have a fixed point; that is, a point c such that f(c) = c. 33. ~ Show that if f is continuous and 0 ~ f(x) ~ 1 for 0 ~ x ~ 1, then f(c) y 1 ___________/ = c for some c in (0, 1] (Figure 7). Y=X /I / / / /I Y=f(x) I I I I I I I I c FIGURE 7 A function satisfying 0 ~ f(x) ~ 1for0 ~ x ~ 1. If f(O) = 0, the proof is done with c = 0. We may assume that f(O) > 0. Let g(x) = f(x) - x. g(O) = f (0) - 0 = f (0) > 0. Since f (x) is continuous, the Rule of Differences dictates that g(x) is continuous. We need to prove that g(c) = 0 for some c E [0, I]. Since f(l) ~ 1, g(l) = f(l) - 1 ~ 0. If g(l) = 0, the proof is done with c = 1, so let's assume that g(l) < 0. We now have a continuous function g(x) on the interval [O, l] such that g(O) > 0 and g(l) < 0. From the IVT, there must be some c E [O, 1] so that g(c) = 0, so f(c) - c = 0 and so f(c) = c. SOLUTION 35. Use the IVT to show that if f is continuous and one-to-one on an interval [a, b], then f is either an increasing or a decreasing function. Let f(x) be a continuous, one-to-one function on the interval [a, b]. Suppose for sake of contradiction that f(x) is neither increasing nor decreasing on [a, b]. Now, f(x) cannot be constant, for that would contradict the condition that f(x) is one-to-one. It follows that somewhere on [a, b], f(x) must transition from increasing to decreasing or from decreasing to increasing. To be specific, suppose f(x) is increasing for x 1 < x < x 2 and decreasing for x 2 < x < x 3 • Let k be any number between max{f(x1), f(x 3 )) and f(x 2 ). Because f(x) is continuous, the IVT guarantees there exists a c1 E (x1, X2) such that f(c1) = k; moreover, there exists a c2 E (x2 , X3) such that f(c 2 ) = k. However, this contradicts the condition that f(x) is one-to-one. A similar analysis for the case when f(x) is decreasing for x 1 < x < x 2 and increasing for Xz < x < X3 again leads to a contradiction. Therefore, f(x) must either be increasing or decreasing on [a, b ]. SOLUTION SECT Io N 2.9 I The Formal Definition of a Limit 99 37. ($1 Figure 8(B) shows a slice of ham on a piece of bread. Prove that it is possible to slice this open-faced sandwich so that each part has equal amounts of ham and bread. Hint: By Exercise 36, for all 0 :5 (} :5 n there is a line L((}) of incline(} (which we assume is unique) that divides the ham into two equal pieces. Let B((}) denote the amount of bread to the left of (or above) L((}) minus the amount to the right (or below). Notice that L(n) and L(O) are the same line, but B(n) = -B(O) since left and right get interchanged as the angle moves from 0 ton. Assume that Bis continuous and apply the IVT. (By a further extension of this argument, one can prove the full Ham Sandwich Theorem, which states that if you allow the knife to cut at a slant, then it is possible to cut a sandwich consisting of a slice of ham and two slices of bread so that all three layers are divided in half.) y

L(8)

L(O) = L(lf)

(B) A slice of ham on top

(A) Cutting a slice of ham

at an angle (J

of a slice of bread FIGURE 8

For each angle(}, 0 :5 (} < rr, let L((}) be the line at angle(} to the x-axis that slices the ham exactly in half, as shown in Figure 8. Let L(O) = L(rr) be the horizontal line cutting the ham in half, also as shown. For(} and L((}) thus defined, let B((}) = the amount of bread to the left of L((}) minus that to the right of L((}). To understand this argument, one must understand what we mean by "to the left" or "to the right". Here, we mean to the left or right of the line as viewed in the direction e. Imagine you are walking along the line in direction(} (directly right if(} = 0, directly left if(} = rr, etc). We will further accept the fact that B is continuous as a function of(}, which seems intuitively obvious. We need to prove that B(c) = 0 for some angle c. Since L(O) and L(rr) are drawn in opposite direction, B(O) = -B(rr). If B(O) > 0, we apply the IVT on [0, rr] with B(O) > 0, B(rr) < 0, and B continuous on [O, 7r]; by IVT, B(c) = 0 for some c E [0, rr]. On the other hand, if B(O) < 0, then we apply the IVT with B(O) < 0 and B(rr) > 0. If B(O) = 0, we are also done; L(O) is the appropriate line.

SOLUTION

2.9 The Formal Definition of a Limit Preliminary Questions 1. Given that lim cos x = 1, which of the following statements is true? x->O

(a) If lcos x - 11 is very small, then xis close to 0.

(b) There is an E > 0 such that if if 0 < lcos x - 11 < E, then lxl < 10-5 • (c) There is a 8 > 0 such that ifO < lxl < 8, then lcos x - 11 < 10-5 • (d) There is a 8 > 0 such that if 0 < Ix - 11 < 8, then lcos xi < 10-5 • SOLUTION

The true statement is (c): There is a 8 > 0 such that if O < lxl < 8, then lcosx - 11 < 10-5.

2. Suppose it is known that for a given E and 8, if 0 < Ix - 31 < 8, then lf(x) - 21 < E. Which of the following statements must also be true? (a) IfO a

= c to have the

gap be small enough. SOLUTION

See the figure below. With 8 = c, the gap is within c of a. f(x)=x

5. Consider limf(x), where f(x) = 8x + 3. x->4

(a) Show that lf(x) - 3SI = 81x - 41. (b) Show that for any E > 0, if 0 < Ix - 41 < 8, then If(x) - 3SI < c, where 8 that limf(x) = 3S.

= ~. Explain how this proves rigorously

x->4

SOLUTION

(a) lf(x) - 3SI = 18x + 3 - 3SI = 18x - 321 = 18(x - 4)1 = 8 Ix - 41. (Remember that the last step is justified because 8 > 0.) (b) Let c > 0. Let 8 = c/8 and suppose Ix - 41 < 8. By part (a), lf(x) - 3SI = 81x - 41 < 88. Substituting 8 = c/8, we see lf(x) - 3SI < 8c/8 = c. We see that, for any c > 0, we found an appropriate 8 so that Ix - 41 < 8 implies lf(x) - 3SI < c. Hence limf(x) = 3S. x->4

7. Consider lim x2 = 4 (refer to Example 2). x->2

(a) Show that if 0 < Ix - 21 < 0.01, then lx2 - 41 < O.OS. (b) Show that if 0 < Ix - 21 < 0.0002, then lx2 - 41 < 0.0009. (c) Find a value of 8 such that ifO C

forces lf(x) - LI < c/lal. Suppose Ix - cl < 6. laf(x) - aLI

= lallf(x) -

= L, we know there is a 6 > 0 such that Ix aLI < lal(c/lal)

cl < 6

= E, so the rule is proven.

35. The Product Law [Theorem 1, part (iii) in Section 2.3]. Hint: Use the identity. f(x)g(x) - LM

= (f(x) -

L) g(x) + L(g(x) - M)

Before we can prove the Product Law, we need to establish one preliminary result. We are given that Consequently, if we set E = 1, then the definition of a limit guarantees the existence of a 6 1 > 0 such that whenever 0 < Ix - cl < 61, lg(x) - Ml < 1. Applying the inequality lg(x)l - IMI ~ lg(x) - Ml, it follows that lg(x)I < 1 + IMI. In other words, because limx-->c g(x) = M, there exists a 61 > 0 such that lg(x)I < 1 + IMI whenever 0 < Ix - cl < 61. We can now prove the Product Law. Let E > 0. As proven above, because limx-->c g(x) = M, there exists a 6 1 > 0 such that lg(x)I < 1 + IMI whenever 0 < Ix - cl < 61. Furthermore, by the definition of a limit, limx--+c g(x) = M implies there exists a 62 > 0 such that lg(x) - Ml < Z(l:ILll whenever 0 < Ix - cl < 62. We have included the "1 +" in the denominator to avoid division by zero in case L = 0. The reason for including the factor of 2 in the denominator will become clear shortly. Finally, because limx--+c f(x) = L, there exists a 03 > 0 such that lf(x) - LI < Z(l:IMll whenever 0 < Ix - cl < 03. Now, let 6 = min(6 1, 62, 63 ). Then, for all x satisfying 0 < Ix - cl < 6, we have SOLUTION

limx-->c g(x)

= M.

lf(x)g(x) - LMI

= l(f(x) ~

L)g(x) + L(g(x) - M)I

lf(x) - LI lg(x)I + ILi lg(x) - Ml €

< 2(1 + IMD (l + IMI) + ILi 2(1 + ILD

Hence, limf(x)g(x) x-+c

37.

~

= LM = limf(x) · limg(x) x-+c x-+c

Here is a function with strange continuity properties:

f(x) =

(

~ q

if xis the rational number p/q in lowest terms

0

if x is an irrational number

(a) Show that f is discontinuous at c if c is rational. Hint: There exist irrational numbers arbitrarily close to c. (b) Show that f is continuous at c if c is irrational. Hint: Let I be the interval {x: Ix - cl < 1). Show that for any Q > O, I contains at most finitely many fractions p/q with q < Q. Conclude that there is a6 such that all fractions in {x: Ix- cl < 6) have a denominator larger than Q. SOLUTION

(a) Let c be any rational number and suppose that, in lowest terms, c = p/q, where p and q are integers. To prove the discontinuity off at c, we must show there is an E > 0 such that for any 6 > 0 there is an x for which Ix - cl < 6, but that lf(x) - f(c)I > E. Let E = -dq and 6 > 0. Since there is at least one irrational number between any two distinct real numbers, there is some irrational x between c and c + 6. Hence, Ix - cl < 6, but lf(x) - f(c)I = 10 - 11 = l > 1._ = E q

q

2q

Chapter Review Exercises

105

(b) Let c be irrational, let E > 0 be given, and let N > 0 be a prime integer sufficiently large so that~ < E. Let~, ... ,:: be all rational numbers 2q in lowest terms such that 12q - cl < 1 and q < N. Since N is finite, this is a finite list; hence, one number !!1. in the list must be closest to c. Let 6 = -21 11!.i. - cl. By construction, 11!.i. - cl > 6 for all i = 1 ... m. Therefore, for

.

~

.

any rational number 2 such that 12 - cl< 6, q > N, so! < ~ < E. Therefore, for an; rational n~mber x such that Ix '!_ cl < 6, lf(x) - f(c)I < number x, so Ix - cl < 6 implies that lf(x) - f(c)I < E for any number x. 39.

E.

lf(x) - f(c)I

= 0 for any irrational

~ Write a formal definition of the following: limf(x) =

oo

x-+a

SOLUTION

= oo if, for any M

limf(x) x-+a

> 0, there exists an 6 > 0 such that f(x) > M whenever 0 < Ix - al < 6.

CHAPTER REVIEW EXERCISES 1. The position of a particle at time t (s) is s(t) = Vt2 + 1 m. Compute its average velocity over [2, 5] and estimate its instantaneous velocity at t = 2. SOLUTION

Let s(t)

= ..fi2+1. The average velocity over [2, 5] is

s(5~ =;1

X

-

Let f(x) =

SOLUTION

2x1 - 1!x3. The data in the table below suggests that

1

. ( -7- - -3- ) ,,,,2.00 hm x-->I l -x7 l -x3 x

0.9

0.99

0.999

1.001

1.01

1.1

f(x)

2.347483

2.033498

2.003335

1.996668

1.966835

1.685059

(The exact value is 2.) In Exercises I I-50, evaluate the limit if it exists. If not, determine whether the one-sided limits exist. For limits that don't exist indicate whether they can be expressed as "= -oo" or "= oo ".

11. lim(3 + x 112 ) x-->4

lim(3 + x 112 ) = 3 +

SOLUTION

x-->4

.../4 = 5

4 13. lim 3 X--+-2 X 4 4 1 lim - = - - = --

SOLUTION

( -2)3

x-->-2 x3

2

. Yi-3 15. hm--1--.9 t - 9

Yt-3 Yt-3 1 1 1 lim - - = lim = lim - - = - - - = H9 t - 9 H9 (Yi- 3)( Yi+ 3) H9 Yi+ 3 -../§ + 3 6

SOLUTION

x3 -x 17. lim-x--+1 X- 1 x3 -

x

lim - - = lim x-->1 X - 1 x-->I t-6 19. lim~ H9 yt - 3 SOLUTION

As

SOLUTION

t-+

x(x - l)(x + 1) X -

9, the numerator t

1

-

. = hmx(x + 1) = 1(1+1) = 2 x-->I

6 -+ 3

* 0 while the denominator Yi - 3 -+ 0. Accordingly,

t- 6 lim - H9 Yt-3

does not exist.

Similarly, the one-sided limits as t -+ 9- and as t -+ 9+ also do not exist. Let's take a closer look at the limit as t -+ 9-. The numerator approaches a positive number while the denominator Yi - 3 -+ o-. We may therefore express this one-sided limit as

t-6 lim - - =-oo Yt-3

H9-

On the other hand, as t -+ 9+, the numerator approaches a positive number while the denominator can express this one-sided limit as

Yi - 3 -+ o+, so we

t-6 l i m - - =oo Yt-3

H9+

Because one of the one-sided limits approaches -oo and the other approaches oo, the two-sided limit can be expressed neither as"= -oo" nor as"= oo". .

1

21. hm - x--.-1+ X + 1 SOLUTION

As x -+ -1 +, the numerator remains constant at 1 . 1 hm - x--+-I + X

+}

* 0 while the denominator x + 1 -+ 0. Accordingly,

does not exist.

Taking a closer look at the denominator, we see that x + 1 -+ o+ as x -+ -1 +. Because the numerator is also approaching a positive number, we may express this limit as Jim - - = oo x+I

x--.-1•

Chapter Review Exercises

x3-2x 23. lim--x-+I X - 1 SOLUTION

As x---+ 1, the numerator x3 - 2x---+ -1

107

* 0 while the denominator x - 1 ---+ 0. Accordingly,

x3-2x lim--x.-.1 x-1

does not exist.

Similarly, the one-sided limits as x ---+ 1- and as x ---+ 1+ also do not exist. Let's take a closer look at the limit as x ---+ i - . The numerator approaches a negative number while the denominator x - 1 ---+ o-. We may therefore express this one-sided limit as . x3 - 2x 11m

x.-.J-

---=oo x- l

On the other hand, as x ---+ 1+, the numerator approaches a negative number while the denominator x - 1 ---+ o+, so we can express this one-sided limit as x3 -2x lim - - x-1

x.-.1+

Because one of the one-sided limits approaches neither as "= -oo" nor as "= oo".

=-oo

-oo and the other approaches oo, the two-sided limit can be expressed

e3x - ex

25. lim--x-+O ex - 1 SOLUTION

27. x.-.1.5 lim

l!J X

SOLUTION

29. lim z->-3

lim

x-.1.5

l!J =l-J=l~J = X

1 1.5

3

0

z+3 z2 + 4z + 3

SOLUTION

Jim z->-3

z + 3 = lim z+ 3 = lim _l_ z2 + 4z + 3 z->-3 (z + 3)(z + 1) z->-3 z + 1

=_! 2

x3 -b3 31. lim--bx-+b

X -

SOLUTION

33. lim x->O

. x3-b 3 (x-b)(x2+xb+b 2 ) . hm - - = hm = lim(x2 + xb + b1 ) = b1 + b(b) + b 2 = 3b1 x-.b X - b x->b X - b x->b

(J_ --

1 -) x(x + 3)

3x

SOLUTION

lim x-.O

35. lim x-+O-

(J_ ---1+- ) = x-.o lim 3x

x(x

3)

3 (x + ) - 3 = lim __l _ = __l _ 3x(x + 3)

x-.o 3(x + 3)

!

3(0 + 3) - 9

LxJ X

SOLUTION For x sufficiently close to zero but negative, LxJ = -1. Therefore, as x ---+ o-, the numerator LxJ ---+ -1 if= O while the denominator x ---+ 0. Accordingly,

. LxJ 11m-

x-+o-

x

Taking a closer look at the denominator, we see that x ---+ negative number, we may express this limit as lim x-+O-

37. lim esec e 8-+~

does not exist.

o- as x ---+ o-. Because the numerator is also approaching a LxJ = oo X

108

CHAPTER 2

I

LIMITS SOLUTION

First note that (}

(}sec (} = - cos 0

As (} ->

~,

the numerator (} ->

~

* 0 while the denominator cos (} -> 0. Accordingly, . (}sec (} = l"1m -(}11m

O-> ~

does not exist.

0-->~ COS(}

Similarly, the one-sided limits as (} -> ~- and as (} -> ~+ also do not exist. Let's take a closer look at the limit as (} -> ~-.The numerator approaches a positive number while the denominator cos(} -> o+. We may therefore express this one-sided limit as

On the other hand, as (} -> ~ +, the numerator approaches a positive number while the denominator cos (} -> can express this one-sided limit as

. OsecO 1im B-41+

=

l"im -(}-

8-4~+ COS 8

o-, so we

= -oo

Because one of the one-sided limits approaches -oo and the other approaches oo, the two-sided limit can be expressed neither as "= -oo" nor as "= oo". . cos(} - 2 39• 1 o~ (} SOLUTION

As(}-> 0, the numerator cos(} - 2 -> -1

* 0 while the denominator(} -> O. Accordingly,

. cos(}- 2 1i m - - - o_,o

does not exist.

(}

Similarly, the one-sided limits as (} -> o- and as(}-> o+ also do not exist. Let's take a closer look at the limit as(} -> o-. The numerator approaches a negative number while the denominator (} -> o-. We may therefore express this one-sided limit as . cos(} - 2 1tm

0-->0-

(}

= 00

On the other hand, as (} -> o+, the numerator approaches a negative number while the denominator e -> express this one-sided limit as

. cose- 2 1tm

0-->0+

(}

o+, so we can

= -oo

Because one of the one-sided limits approaches -oo and the other approaches oo, the two-sided limit can be expressed neither as"= -oo" nor as"= oo".

41. lim x- 3 x->2- X- 2 SOLUTION

As x -> 2-, the numerator x - 3 -> -1

* 0 while the denominator x - 2 -> O. Accordingly,

. x-3 1I m - 2

does not exist.

x->2+ X-

Taking a closer look at the denominator, we see that x - 2 -> a negative number, we may express this limit as

o- as x -> 2-. Because the numerator is also approaching

x-3

lim - - = oo 2

x->2- X -

1 43. lim ( -x->J+

~

SOLUTION

1 - - -) VX2 - 1

First note that

1 vx-1- vx2 -1

1

= vx-l

0

.y;:+f l vx+l - vx2 -l

=

.y;:+f - 1 vxz-1

Chapter Review Exercises

As x ~ 1+, the numerator

-.fX+1 -

1 ~ ..(2, - 1 ::f:. 0 while the denominator 1 lim ( - - ~

x--+i+

1 -)

Vx2 -

1~

1 ~ 0. Accordingly,

does not exist.

vxz-1

Taking a closer look at the denominator, we see that a positive number, we may express this limit as

Vx 2 -

109

o+ as x ~ 1+. Because the numerator is also approaching

1 - - -1 - ) = oo lim ( - ~

x--+1+

vxz-1

45. lim tanx

x-+1

SOLUTION

First note that sinx tanx = - cosx

As x

~ ~,

the numerator sin x

~

1 ::f:. 0 while the denominator cos x . . sinx 1im tanx = 1im - -

x-+1

x-+!

~

0. Accordingly,

does not exist.

COS X

Similarly, the one-sided limits as x ~ ~- and as x ~ f also do not exist. Let's take a closer look at the limit as x ~ ~ - . The numerator approaches a positive number while the denominator cos x ~ o+. We may therefore express this one-sided limit as

. . sinx 11m tanx = 11m - - =

x-+~-

oo

x-i-l- COS X

On the other hand, as x ~ ~ +, the numerator approaches a positive number while the denominator cos x ~ can express this one-sided limit as . 1im tanx x-+1+

=

Ii

sinx

m --

x-+~+ COS X

o-, so we

= -oo

Because one of the one-sided limits approaches -oo and the other approaches oo, the two-sided limit can be expressed neither as "= -oo" nor as "= oo".

SOLUTION

Fort > 0,

so

Because lim-Vt= lim

t-+O+

t-+O+

Vt=O

it follows from the Squeeze Theorem that lim

1--.0+

Vt cos(~)= 0 t

. cosx- 1 49 • 1lffi . x->O

smx

SOLUTION

Ii

x-To

cosx-1 r cosx-1 sin x = x~ sin x

cosx+ 1

.

-sin2 x

.

sinx

O

---= 1im = - hm = --=0 cos x + 1 x--.o sin x( cos x + 1) x--+O cos x + 1 1+ 1

110

c HAPTE R

2

I

LIMITS

51. Find the left- and right-hand limits of the function continuous (or both) at these points.

f in Figure 1 at x = 0, 2, 4. State whether f is left- or right-

y

FIGURE 1 SOLUTION

According to the graph of f(x), lim f(x) = lim f(x) = l

x--+O-

x--+O+

lim f(x) = lim f(x) = oo

x-.2-

x--+2+

lim f(x) = -oo x--+4-

lim f(x) = oo

x--+4+

The function is both left- and right-continuous at x = 0 and neither left- nor right-continuous at x = 2 and x = 4. 53. Graph h and describe the discontinuity: h(x) =

ex for x ~ 0 { lnx forx>O

Is h left- or right-continuous? SOLUTION

The graph of h(x) is shown below. At x = 0, the function has an infinite discontinuity but is left-continuous. y

55. Find the points of discontinuity of

g(x) =

cos (nx) 2 { Ix - 11

for lxl < 1 for lxl :::: 1

Determine the type of discontinuity and whether g is left- or right-continuous. First note that cos ( ¥) is continuous for -1 < x < 1 and that Ix - 11 is continuous for x ~ -1 and for x :::: 1. Thus, the only points at which g(x) might be discontinuous are x = ±1. At x = 1, we have SOLUTION

lim g(x) = lim cos(nx) =cos(:!.)= 0 x--.12 2

x--.1-

and lim g(x) = lim lx-11=11-11=0

X--+I+

x--+l+

so g(x) is continuous at x = 1. On the other hand, at x = -1, lim g(x) = Jim cos (nx) X->-1+ 2

X->-[+

=cos(-~)= O 2

and lim g(x) = lim Ix- 11 =I - 1 - 11=2

X--+-1-

X--+-}-

so g(x) has a jump discontinuity at x = -1. Since g(-1) = 2, g(x) is left-continuous at x = -1.

Chapter Review Exercises

111

57. Find a constant b such that his continuous at x = 2, where x+1

h(x)

= { b- .x-~'

for lxl < 2 for lxl ~ 2

With this choice of b, find all points of discontinuity. To make h(x) continuous at x = 2, we must have the two one-sided limits as x approaches 2 be equal. With

SOLUTION

lim h(x) = lim (x + 1) = 2 + 1 = 3

X-+2-

X-+2-

and lim h(x) = lim(b-.x2) = b-4

x-+2+

x-+2+

it follows that we must choose b = 7. Because x + 1 is continuous for -2 < x < 2 and 7 - x2 is continuous for x :s; -2 and for x ~ 2, the only possible point of discontinuity is x = -2. At x = -2, limh(x)= lim(x+l)=-2+1=-1

X-+-2+

X-+-2+

and lim h(x) = lim (7 -.x2) = 7 - (-2)2 = 3

X-+-2-

X-+-2-

so h(x) has a jump discontinuity at x = -2. In Exercises 58-65,find the horizontal asymptotes of the function by computing the limits at infinity.

x2- 3x4 59. f(x) = - -x- 1 SOLUTION Because .x2 - 3x4 x - 3.x3 lim - - - = lim - - - = X-+00 X - 1 X->OO 1 - 1/x

-oo

and

x2 - 3x4 x - 3.x3 lim - - - = lim - - - = X - 1 x-+-oo 1 - 1/x

oo

x-+-oo

it follows that the graph of y 61. f(u)

=

SOLUTION

x2 - 3x4

= -x-1 - - does not have any horizontal asymptotes.

2u 2 -1

-~

v6 + u4 Because

lim 2u2 - 1

= lim

V6 + u4

U->OO

U->OO

2- l/u2 =

2_ = 2

--/6Ju4 + 1

VI

and . . 2u2 -1 lI m - - - = 1Im u+oo

V6 + u4

2-1/u2

--/6Ju4 + 1

u+oo

2 =-=2

VI

2u 2 -1 it follows that the graph of y = - - - has a horizontal asymptote of y = 2.

v6+u4

t1;3 _ rl/3

63. f(t)

= (t -

SOLUTION

rI )1/3

Because 1 _ r213

tlf3 _ rl/3

lim

t tooo

(t - r l )1/3

=lim Hoo

(1 - r2)1/3

1

=-=1 11/3

and tlf3 _ rl/3

lim

t to-oo

.

It follows that the graph of y =

(t- r l )1/3

= lim

H-oo

1_

r213

(1 - r2)1/3

1

= ]173 =

t1;3 _ r1t3

(t - rl)l/3

has a horizontal asymptote of y = 1

.

1

112

C HA PT E R 2

I

LIMITS

65. g(x) =

7r

+ 2 tan- 1 x Because

SOLUTION

lim (7r + 2 tan- 1 x) =

7r

x---+-oo

it follows that the graph of y =

7r

+ 2 (-~) = 0 and

lim(7r + 2 tan- 1 x) =

2

7r

x---+oo

+ 2 (~) = 2

27r

+ 2 tan- 1 x has horizontal asymptotes of y = 0 and y = 27r.

67. CR S Determine M, A, and k for a logistic function p(t) = i+.7.-ki satisfying p(O) = 10, p(4) = 35, and p(lO) = 60. What are the horizontal asymptotes of p? To fit a logistic function p(t) = i+:.-k, to the data p(O) = 10, p(4) = 35, and p(lO) = 60, we must solve the system of equations

SOLUTION

10 =

__!!__

1 +A'

35

M = ----::1 + Ae-4k'

and

M

60= ---...,.,,..,. 1 + Ae-IOk

for M, A, and k. Using a computer algebra system to solve this system yields M,,, 62.777, A ,,, 5.278, and k,,, 0.474. Thus, 62.777 p(t) = 1 + 5.278e-0 .474'

Because lim e- 0.474' =

oo

and lim e- 0.474' = 0, it follows that

t---+-oo

t---+oo

lim

1->-00

62.777 1 + 5.278e-0.474t

=0

and

. 1Im 1->oo

62.777 62.777 = - - = 62.777 1 + 5.278e-0·4741 1+ 0

The horizontal asymptotes of p are y = 0 and y = 62. 777.

69. Assume that the following limits exist:

A= lim/(x),

B = limg(x),

x->a

X->a

L = lim f(x) x->a

g(x)

Prove that if L = 1, then A= B. Hint: You cannot use the Quotient Law if B = 0, so apply the Product Law to Land B instead. SOLUTION

Suppose the limits A, B, and Lall exist and L = 1. Then

B = B · 1 = B · L = limg(x) · lim f(x) = limg(x/(x) = lim/(x) =A x->a

71.

lSJ

SOLUTION

x->a

g(x)

x->a

g(x)

x->a

In the notation of Exercise 69, give an example where L exists but neither A nor B exists. Suppose 1 f(x)=--

(x - a) 3

I g(x)=--

and

(x - a) 5

Then, neither A nor B exists, but .

L = hm x---+a

(x - a)- 3

(x - a)-5

.

= hm(x - a) 2 = O x--+a

Chapter Review Exercises

73.

L$J Let f(x) = 113 lH where LxJ is the greatest integer function. Show that for x * 0, Then use the Squeeze Theorem to prove that limxl!J=l x-+0 X Hint: Treat the one-sided limits separately. SOLUTION Lety be any real number. From the definition of the greatest integer function, it follows thaty- 1 < LYJ ~ y, with equality holding if and only if y is an integer. If x 0, then ~ is a real number, so * Upon multiplying this inequality through by x, we find l-x 0. Therefore, by the Intermediate Value Theorem, there exists a c E (0, ~) such that f(c) = O; consequently, the curves y = x2 and y = cos x intersect. 77. Use the IVT to show that e-x2 = x has a solution on (0, 1). SOLUTION Let f(x) = e-x2 - x. Observe that f is continuous on [0, 1] with f(O) = e 0 - 0 = 1 > 0 and f(l) = e- 1 - 1 0. On the interval xE [--a-,'.:] c [-a,a] 2 + 2rra 2 ~ runs from ~ to ~ + 27f, so the sine function covers one full period and clearly takes on every value from - sin a through sin a. 81. ~Guij 0.05. Plot the function f(x) = x 113 • Use the zoom feature to find a o > 0 such that if Ix - 81 O h->0 h h->0 Alternately, f'(O) = lim f(x)- f(O) = lim x2 + 9 x - O = lim(x + 9) = 9 x->0 X - 0 x->O X x->0 5. f(x) = 3x2 + 4x + 2, SOLUTION a = -1 Let f(x) = 3.x2 + 4x + 2. Then f' (_1) = lim f (-1 + h) - f (-1) = lim 3(-1 + h)2 + 4(-1 h h->0 . = !rm h-+0 3h2 -2h h = h-+0 Iim(3h - + h) + 2 - 1 h h->0 2) = -2 Alternately, f'(-l)= lim f(x)-f(-1) = lim 3x2+4x+2-I x->-1 x-(-1) X->-1 x+ I . (3x+ I)(x+ I) . = hm = hm(3x+ I)= -2 x->-1 X + 1 x->-1 7. f(x) = x3 + 2x, SOLUTION Let f(x) a= 1 = x3 + 2x. Then f'(l) 3 = lim f(l + h) - f(l) = lim (I + h) + 2(1 + h) - 3 h->O h h->0 h 2 3 . I + 3h + 3h + h + 2 + 2h - 3 = Iim h = lim(S + 3h + h 2 ) = S h->0 h->O Alternately, f'(I) = lim f(x) x->I . = hm X-+l X - f(I) = lim I x->1 (x - I)(x2 X- + x + 3) 1 x3 + 2x - 3 X - I = lim(x2 + x + 3) = 5 X-+1 sEcT IoN 3.1 I Definition of the Derivative 117 In Exercises 9-I 2, refer to Figure I 3. 2.5 +···"·······+-············-;···············i·· 2.0k········•···············,.········•·····•·· 1.5.j....... +~~=1" 1.0+············;·-,,-... ,. ............. , .. 0.5 ..._.r....... ; ...............;................. . 0.5 1.0 1.5 2.0 2.5 3.0 FIGURE 13 9. ~ Find the slope of the secant line through (2,f(2)) and (2.5,f(2.5)). Is it largeror smaller than f'(2)? Explain. From the graph, it appears that f(2.5) = 2.5 and f (2) = 2. Thus, the slope of the secant line through (2,f(2)) and (2.5,f(2.5)) is SOLUTION f(2.5) - f(2) 2.5 -2 = 2.5 - =I 2 2.5-2 From the graph, it is also clear that the secant line through (2, f(2)) and (2.5, f(2.5)) has a larger slope than the tangent line at x = 2. In other words, the slope of the secant line through (2, f(2)) and (2.5, f (2.5)) is larger than f' (2). 11. Estimate f' (I) and f' (2). SOLUTION From the graph, it appears that the tangent line at x = I would be horizontal. Thus, f'(I)"" 0. The tangent line at x = 2 appears to pass through the points (0.5, 0.8) and (2, 2). Thus /'(2)"" 2 - 0.8 2-0.5 = 0.8 In Exercises 13-16, refer to Figure I4. 5 4 3 2 I 2 3 4 5 6 7 8 9 FIGURE 14 Graph off. 13. Determine f'(a) for a= I, 2, 4, 7. Remember that the value of the derivative off at x = a can be interpreted as the slope of the line tangent to the graph of y = f(x) at x = a. From Figure 14, we see that the graph of y = f(x) is a horizontal line (that is, a line with zero slope) on the interval 0 ~ x ~ 3. Accordingly, f'(l) = f'(2) = 0. On the interval 3 ~ x ~ 5, the graph of y = f(x) is a line of slope }; thus, f'(4) = }. Finally, the line tangent to the graph of y = f(x) at x = 7 is horizontal, so f'(7) = 0. SOLUTION 15. Which is larger, f'(5.5) or f'(6.5)? SOLUTION The line tangent to the graph of y = f(x) at x = 5.5 has a larger slope than the line tangent to the graph of y = f(x) at x = 6.5. Therefore, f'(5.5) is larger than f'(6.5). In Exercises 17-20, use the limit definition to calculate the derivative of the linear function. 17. f(x) = 1x-9 SOLUTION . f(a + h) - f(a) . 7(a + h) - 9 - (1a - 9) . I1m = 11m =hm7=7 o .h h ..... o h h ..... o h ..... 19. g(t) = 8 - 3t SOLUTION . g(a + h) - g(a) . 8 - 3(a. + h) - (8 - 3a) . -3h . I1m = 11m =hm-=bm(-3)=-3 o h h ..... o h h .....o h h ..... o h ..... 118 CH APT E R 3 I DIFFERENTIATION 21. Find an equation of the tangent line at x == 3, assuming that f(3) = 5 and f'(3) == 2. SOLUTION By definition, the equation of the tangent line to the graph of f(x) at x == 3 is y == f(3) + f'(3)(x - 3) = 5 + 2(x - 3) == 2x - I. 23. Describe the tangent line at an arbitrary point on the graph of y == 2x + 8. SOLUTION Since y == 2x + 8 represents a straight line, the tangent line at any point is the line itself, y == 2x + 8. I I 1 1 . . . 25. Let f(x) = -. Does f(-2 + h) equal - - or - + -h? Compute the difference quotient at a == -2 with h = 0.5. x -2+h -2 SOLUTION Letf(x) ==~-Then 1 f(-2 + h) == - -2 + h With a== -2 and h == 0.5, the difference quotient is f(-1.5)- f(-2) 0.5 f(a + h) - f(a) h 27. Let f(x) == 1/ I I =I5 - =z 0.5 3 yx. Compute f' (5) by showing that f(5 + h)- f(5) h SOLUTION Let f(x) == 1 / yx. Then I I ~--;rs f(5 + h) - f(5) h h VS+h) Ys - V5+h ( Ys + h Ys -../5 + h Ys + -../5 + h 5 - (5 + h) h Ys -../5 + h( -./5 + h + Ys) Ys -./5 + h(-../5 + h + Ys) Thus, f'(5) ==Jim f(5 + h) h->0 1 f(5) ==Jim- h h->0 1 Ys -../5 + h( -../5 + h + Ys) 1 Ys Ys(Ys + Ys) == - lOYs In Exercises 29-46, use the limit definition to compute f'(a) and.find an equation of the tangent line. 29. f(x) == 2x2 + lOx, SOLUTION a=3 Let f(x) == 3x2 + 2x. Then f'(2 ) =Jim f(2 + h) ~ h f(2) =Jim 3(2 + h)2 + 2(2 + h) - 16 ~ h . 12 + l2h + 3h2 + 4 + 2h - 16 = hm h->0 h = h->O Iim(14 + 3h) = 14 At a = 2, the tangent line is y = f'(2)(x - 2) + f(2) = I4(x - 2) + 16 = l4x - 12. 31. f(t) = t - 2t2, SOLUTION a= 3 Let f(t) = t - 2t2. Then f'( 3 ) =Jim f(3 + h) - f(3) =Jim (3 + h)- 2(3 + h) h->O h h->O h 2 - (-15) . 3 + h - 18 - 12h - 2h 2 + 15 = I1 m - - - - - - - - h->o h = h->O lim(-11 - 2h) = -11 At a = 3, the tangent line is y = f'(3)(t -3) + f(3) = -Il(t- 3)- 15 = -Ilt+ 18 I S E c T I 0 N 3.1 33. f(x) = x3 + x, SOLUTION a Definition of the Derivative =0 Let f(x) = x3 + x. Then f'(O) = lim f(h) - f(O) = lim h3 + h - 0 h->0 h h->0 h = lim(h2 + 1) = I h->0 At a = 0, the tangent line is y = f'(O)(x - 0) + /(0) = x 35. f(x) = x- 1 , SOLUTION a =8 Let f(x) = x- 1• Then 1 (I) /'(8) = lim /( 8 + h) - /( 8) = lim 8+h - 8 = lim h->O h h->O h h->0 The tangent at a 8-8-h 8O -h = _ _!_ (64 + 8h)h 64 = 8 is , I I I I y = f (8)(x- 8) + /(8) = --(x- 8) + - = - - x + 64 8 64 4 I 37. f(x) = - - , a= -2 x+ 3 SOLUTION Let f(x) = x!3 . Then '( 2) 1· f(-2 +h)-/(-2) =Im 1· ! - =Im h->O h h->O =z:!w -1 h t!h -h 1·Im--=-1 I =1·I m --1 - = 1·Im---= h->O h h->O h(I + h) h->O 1 + h The tangent line at a = - 2 is y = f'(-2)(x + 2) + /(-2) = -l(x + 2) + 1 = -x- 1 39. f(x) = .Yx+4, a= 1 SOLUTION Let f(x) = vx + 4. Then f 'ci) _ . 1c1 + h) - 1c1) _ . - 1Im - 11m h->O h h->0 . =1m 1 h->0 The tangent line at a h h( vh + s + -Yh+5 h SOLUTION I vx' a VS 1 h->O = 1 is ' ) = Vh + 5 + h vs) =hm vh + s + vs -- 2 vs y= ! (1 (x-1)+/(1)= 41. f(x) h->0 1 . vs -_Iim vh + s - vs ·vh + s + vs ---- 1 _,-; 1 - x +9.k(x-1)+ v5= 2 vs 2-VS 2-VS =4 Let f(x) = 1 -vx· Then I I 2- -./4+h 2+ -./4+h /'(4) = lim /(4 + h) - /(4) = lim 74.:'t. - 2 = lim ~ . ~ h->0 h h->O h h->O h -1 1 4V4 + h +2(4 + h) 16 = Iim - - - - - h->O 4-4-h = lim 4 -.f4+h+2O At a = 4 the tangent line is 1 1 I 3 y = f'(4)(x - 4) + /(4) = --(x- 4) + - = - - x + 16 2 16 4 h 119 120 CHAPTER 3 I DIFFERENTIATION 43. f (t) = ViZ+T, SOLUTION =3 a Let f(t) = ..fi2+T. Then j'(3) = lim f(3 + h) - f(3) = lim VIO + 6h + h h->O h h->0 h . = hm VIO + 6h + h2 h->O v'IO - h The tangent line at a v'IO) 1 = x2 + 1 , SOLUTION VfO =lim 6+h h....0 vIO + 6h + h 2 + 3 vw = vw = 3 is 3 y=f'(3)(t-3)+f(3)= _r;-nOh = lim h->O- 1- 1 h = 0, but Iim h->O+ f(l + h) - f(l) h . (1 + h) 11m = h->O+ h 2 - 1 . 11m = h->O+ 1 + 2h + h 2 - 1 h . (Z h) 11m + = 2 = h->O+ Because the two one-sided limits are not equal, the two-sided limit r J(l +h)-f(l) h'!R, h does not exist. Therefore, f is not differentiable at x = 1. Also, because the two one-sided limits exist but are not equal, the graph off has a corner at x = 1 (see the figure below). y 4 -+-+-+-+--+-+-++-X -3 3 S E CT I 0 N 3.1 I Definition of the Derivative 121 In Exercises 49-5I, sketch a graph off and identify the points c such that f'(c) does not exist. In which cases is there a corneratc? 49. f(x) = Ix+ 31 The graph of y not differentiable at x = -3. SOLUTION = Ix+ 31 is shown below. From the graph, we see that there is a corner at x = -3, so f is y -6 -3 51. f(x) = lx2 - 41 The graph of y = lx2 - 41 is shown below. From the graph, we see that there is a corner at x another at x = 2, so f is not differentiable at either x = -2 or x = 2. SOLUTION = -2 and y -4 53. !Gq Let f(x) the slope f'(O). = 1 : 2 x. 4 Plot f over [-2, 2]. Then zoom in near x = 0 until the graph appears straight, and estimate :zx The figure at the left shows the graph of f(x) = 1 over [-2, 2]. The figure below at the right is a close-up near x = 0. From the close-up, we see that the graph is nearly straight and passes through the points (-0.22, 2.15) and (0.22, 1.85). We therefore estimate SOLUTION /'(O)"'" 1.85 - 2.15 0.22 - (-0.22) = -0.3 = _0 _68 y 0.44 y 0.5 -2 -I 2 55. Determine the intervals along the x-axis on which the derivative in Figure 16 is positive. y 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 FIGURE 16 SOLUTION The derivative (that is, the slope of the tangent line) is positive when the height of the graph increases as we move to the right. From Figure 16, this appears to be true for 1 < x < 2.5 and for x > 3.5. 122 CHAPTER 3 I DIFFERENTIATION Jn Exercises 57-62, each limit represents a derivative f'(a). Find f(x) and a. (5 + h) 3 h 57. lim h--+0 - 125 . . (5 + h)3 - 125 f(a + h) - f(a) The difference quot:J.ent h has the form h where f(x) SOLUTION sin(~+ 59. lim = x3 and a = 5. h)-0.5 h h--+0 The difference quotient SOLUTION sin(~+ h) - h 0.5 has the form f(a + h) - f(a) h where f(x) = sin x and a = ~. -25 h 52+h 61. lim h--+0 SOLUTION The difference quotient 5(2+h) - 25 f(a + h) - f(a) h has the form h where f(x) =Y and a = 2. 63. Apply the method of Example 6 to f(x) = sin x to determine f' ( ~) accurately to four decimal places. SOLUTION We know that f'(7r/ 4) = lim f(7r/4 + h) - Creating a table of values with h close to zero: h sin(~ + h) - ( V2/2) h = lim sin(7r/4 + h) - f(7r/4) h - V2/2 h -0.001 -0.0001 -0.00001 0.00001 0.0001 0.001 0.7074602 0.7071421 0.7071103 0.7071033 0.7070714 0.7067531 Accurate up to four decimal places, f' ( ~) "" 0. 7071 . . 65. ~ between x For each graph in Figure 17, determine whether f' (1) is larger or smaller than the slope of the secant line = I and x = I + h for h > 0. Explain. y (B) (A) FIGURE 17 SOLUTION • On curve (A),f'(I) is larger than f(l + h) - f(I) h the curve is bending downwards, so that the secant line to the right is at a lower angle than the tangent line. We say such a curve is concave down, and that its derivative is decreasing. • On curve (B), f'(I) is smaller than f(I + h) - f(I) h the curve is bending upwards, so that the secant line to the right is at a steeper angle than the tangent line. We say such a curve is concave up, and that its derivative is increasing. 67. jGuij Sketch the graph of f(x) = x51 2 on (0, 6]. (a) Use the sketch to justify the inequalities for h > 0: /(4) - f(4 - h) < f'(4) < f(4 + h) - /(4) h - - h s Ec T Io N I 3.1 Definition of the Derivative 123 (b) Use (a) to compute f'(4) to four decimal places. (c) Use a graphing utility to plot y SOLUTION = f(x) and the tangent line at x = 4, utilizing your estimate for f'(4). A sketch of the graph off(x) = x512 on [O, 6] is shown below in the answer to part (c). (a) The slope of the secant line between points (4,f(4)) and (4 + h,f(4 + h)) is f(4 + h) - f(4) h x512 is a smooth curve increasing at a faster rate as x ~ oo. Therefore, if h > 0, then the slope of the secant line is greater than the slope of the tangent line at f(4), which happens to be f'(4). Likewise, if h < 0, the slope of the secant line is less than the slope of the tangent line at f(4), which happens to be f'(4). (b) We know that f'(4) = lim f(4 + h)- f(4) = lim (4 + h)s12 h h-->0 32 h h-->0 Creating a table with values of h close to zero: h -0.0001 -0.00001 0.00001 0.0001 (4 + h)512 - 32 h 19.999625 19.99999 20.0000 20.0000375 Thus, f'(4)"" 20.0000. (c) Using the estimate for f'(4) obtained in part (b), the equation of the line tangent to f(x) y =f'(4)(x-4) + f(4) =20(x - = x512 at x = 4 is 4) + 32 = 20x- 48 y 80 60 40 20 -20 -40 --{;0 69. !GU[I Use a plot of f(x) = x' to estimate the value c such that f'(c) = 0. Find c to sufficient accuracy so that lf(c + ~ - f(c)I::; 0.006 SOLUTION Here is a graph of f(x) for h = ±0.001 = x' over the interval [0, 1.5]. y 2 1.5 0.5 0.2 0.4 0.6 0.8 I 1.2 1.4 The graph shows one location with a horizontal tangent line. The figure below at the left shows the graph of f(x) together with the horizontal lines y = 0.6, y = 0.7 and y = 0.8. The line y = 0.7 is very close to being tangent to the graph of f(x). The figure below at the right refines this estimate by graphing f (x) and y = 0.69 on the same set of axes. The point of tangency has an x-coordinate ofroughly 0.37, soc"" 0.37. y y 0.2 0.4 0.6 0.8 1 1.2 1.4 0.2 0.4 0.6 0.8 1 1.2 1.4 124 CHAPTER 3 I DIFFERENTIATION We note that lf(0.37 + o~~g~i- f(0.37)' ""o.00491 < o.006 and f(0.37 - 0.001) - f(0.37)' "" 0.00304 < 0.006 0.001 I so we have determined c to the desired accuracy. The vapor pressure of water at temperature T (in kelvins) is the atmospheric pressure Pat which no net evaporation takes place. In Exercises 71 and 72, use the following table to estimate the indicated derivatives using the difference quotient approximation. T (K) 293 303 323 313 333 343 353 P (atm) 0.0278 0.0482 0.0808 0.1311 0.2067 0.3173 0.4754 71. Estimate P' (T) for T = 293, 313, 333. (Include proper units on the derivative.) SOLUTION P' _ P(303) - P(293) _ 0.0482 - 0.0278 _ O OO O K 303 - 293 10 - . 2 4 atm/ (2 93 ) - P'( 3 l 3)"" P(323)- P(313) 323 - 313 P' = 0.1311 - 0.0808 10 = 0.00503 tm/K a _ P(343) - P(333) _ 0.3173 - 0.2067 _ O Ol O K (3 33 ) 343 - 333 10 - . 1 6 atm/ Let P(t) represent the U.S. ethanol production as shown in Figure 19. In Exercises 73 and 74, estimate the indicated derivatives using the difference quotient approximation. P (billions of gallons) Pl- ~ f'j'b ~ f'jO h h-->0 h h-->0 h 3. f(x) = i3 SOLUTION Let f(x) = i3. Then, f '( ) X . f(x + h) - f(x) . (x + h) 3 = 1!ID = 1!ID h h-->0 2 - h h-->0 x3 . i3 + 3x2h + 3xh2 + h 3 - i3 = 1! I D - - - - - - - - - h h-->0 . 3x2h + 3xh + h = hm h = lim(3x2 + 3xh + h 2 ) = 3x2 3 h-->O S. f(x) = x SOLUTION ! h-->0 Yx Let f(x) = x - yx. Then, '(x-1m ) _ 1. f(x + h) h h-->0 . = 1 - hm h-->O f(x) _ . x + h - Vx + h - (x 1 -1m h-->0 h ~+~-x h( Vx + h + yx) . = 1 - hm h-->O 1 yx) . =1-hm h-->O 1 = 1 - -VX + h + yx 2 Yx vx + h - yx ( vx + h + yx) . h Vx+h+yx 128 CH APT E R 3 I DIFFERENTIATION In Exercises 7-14, use the Power Rule to compute the derivative. 7 . .!!:_ x4 \ dx x=-2 .!!:_ (x4 ) = 4.x3 SOLUTION dx = 4(-2)3 = -32 !!:_x4 \ so dx =-2 9. !!:_ t213\ dt t=S !!:_ (t2t3) dt SOLUTION = ~r113 so 3 !!:_t2t3\ dt t=S = ~( 8 )-113 = ! 3 3 d 11. dx X0.35 SOLUTION 13. ~ t-m SOLUTION In Exercises 15-18, compute f'(x) and find an equation of the tangent line to the graph at x =a. 15. f(x) = x4, a = 2 Let f(x) SOLUTION at x = x4. Then, by the Power Rule, f'(x) = 4.x3. The equation of the tangent line to the graph of f(x) = 2 is y 17. f(x) = 5x - 32 yx, Let f(x) SOLUTION = f'(2)(x - 2) + f(2) = 32(x - 2) + 16 = 32x - 48 a= 4 = 5x- 32x112 . Then f'(x) = 5 y 16x- 1t 2 . In particular, f'(4) = f'(4)(x-4) + f(4) = -3(x - 4) - 44 = -3x - = -3. The tangent line at x = 4 is 32 19. Calculate: d (a) -12ex d t d r-3 (c) dte (b) dt(25t - 8e) dx Hint for (c): Write e'- 3 as e- 3e'. SOLUTION d d (a) -9ex = 9-ex = 9ex dx dx d t d d t t (b) d/3t-4e) = 3 d/- 4";]/ = 3 - 4e d (c) -e'-3 dt d = e-3-e' = e-3 . e' = e'-3 dt In Exercises 21-32, calculate the derivative. 21. f(x) = 2x3 - 3x2 + 5 SOLUTION 23. f(x) = SOLUTION 25. g(z) d dx (2.x3 - 3x2 + 5) = 6x2 - 6x 4x5 13 - 3x-2 - 12 d ( 20 dx 4x513 - 3x-2 - 12) = 3x213 + 6x- 3 = 1z-5t 14 + z-5 + 9 SOLUTION d (1z- 5/1 4 + z-5 + 9) = dz - 5 2 --z- 19114 - 5z- 6 fS + Vs SOLUTION f(s) = Vs+ Vs= s 114 + s 113 . In this form, we can apply the Sum and Power Rules . 27. f(s) = .!!:_ (s1f4 + s113) ds = !cs -3, there are precisely two points where f' (x) = m. Indicate the position of these points and the corresponding tangent lines for one value of m in a sketch of the graph off. SOLUTION Let P = (a, b) be a point on the graph off(x) = x3 - 3x + 1. • The derivative satisfies f'(x) = 3x2 - 3 ~ -3 since 3x2 is nonnegative. • Suppose the slope m of the tangent line is greater than -3. Then f'(a) = 3a2 a 2 m+3 =->0 3 and thus a - 3 = m, whence = ± ~+3 -3 • The two parallel tangent lines with slope 2 are shown with the graph of f(x) here. y 61. Compute the derivative of f(x) = x31 2 using the limit definition. Hint: Show that f(x + h) - f(x) = (x + h) 3 - x3 ( h h 1 -V0 h 1 .,,/(x + h)3 + W ) .,,/(x + h)3 + W The first factor of the expression in the last line is clearly the limit definition of the derivative of x3, which is 3x2. The second factor can be evaluated, so .!!_x312 dx = 3x2_1_ = ~x112 2w 2 63. In each case use the limit definition to compute f' (0), and then find the equation of the tangent line at x = 0 (a) f (x) = (b) f(x) xex = x2ex SOLUTION (a) Let f(x) = xex. Then f(O) = 0, and f'(O) = lim f(O + h)- f(O) = lim heh - 0 = Iimeh = 1 h-->0 h h-->0 h h-->0 The equation of the tangent line is y = f' (O)(x - 0) + f(O) = 1(x - 0) +0 = x (b) Letf(x) = x2ex. Then f(O) = 0, and J'(O) = lim f(O + h)- f(O) = lim h2eh - 0 = limheh = 0 h-->0 h h-->0 h h-->0 The equation of the tangent line is y= f' (O)(x - 0) + f (0) = O(x - 0) + 0 = 0 65. The brightness b of the sun (in watts per square meter) at a distance of d meters from the sun is expressed as an inverse-square law in the form b = 4 where Lis the luminosity of the sun and equals 3.9 x 1026 watts. What is the derivative of b with respect to d at the earth's distance from the sun (1.5 x 1011 m)? /:az, SOLUTION Let b = 41r1;p = f,;d- 2• Then , L _ L b =-(-2d 3) = - 411' 2nd3 and b'(l.5 x 10") = 3.9 x 1026 Zn(l. 5 x 1011 )3 "" - 18 -8 3 watts/m • 4 x 10 67. The Clausius-Clapeyron Law relates the vapor pressure of water P (in atmospheres) to the temperature T (in kelvins): dP =k.!_ dT T2 where k is a constant. Estimate dP/dT for T = 303, 313, 323, 333, 343 using the data and the symmetric difference approximation dP P(T + 10) - P(T - 10) dT"" 20 T (K) 293 303 313 323 P(atm) 0.0278 0.0482 0.0808 0.1311 333 343 0.2067 0.3173 353 0.4754 Do your estimates seem to confirm the Clausius-Clapeyron Law? What is the approximate value of k? s E c T to N SOLUTION 3.2 I The Derivative as a Function 133 Using the indicated approximation to the first derivative, we calculate P'(303),,., P(313) - P(293) 20 = 0.0808 - 0.0278 = O.00265 atm /K P'(313),,., P(323)- P(303) 20 = 0.1311 - 0.0482 = O.004145 atm /K P'(323),,., P(333)- P(313) 20 - 0.0808 O = 0.2067 20 = .006295 atm /K 20 20 P(343)-P(323) 0.3173-0.1311 /K , P (333) ,,., = = 000931 . atm 20 20 P'( 343 ),,., P(353)- P(333) 20 = 0.4754 - 0.2067 20 = 0 _013435 atm/K 2 If the Clausius-Clapeyron law is valid, then T dP should remain constant as T varies. Using the data for vapor P dT pressure and temperature and the approximate derivative values calculated above, we find T (K) 303 313 323 333 343 T 2 dP P dT 5047.59 5025.76 5009.54 4994.57 4981.45 These values are roughly constant, suggesting that the Clausius-Clapeyron law is valid, and that k ,,., 5000. 69. In the setting of Exercise 68, show that the point of tangency is the midpoint of the segment of L lying in the first quadrant. SOLUTION In the previous exercise, we saw that the tangent line to the hyperbola xy = 1 or y = ~ at x = a has y-intercept P = (0, ~) and x-intercept Q = (2a, 0). The midpoint of the line segment connecting P and Q is thus (0+2 2a ' ~ 2+ 0) = ( !) a, a which is the point of tangency. 71. Make a rough sketch of the graph of the derivative of the function in Figure l 7(A). The graph has a tangent line with negative slope approximately on the interval (1,3.6) and has a tangent line with a positive slope elsewhere. This implies that the derivative must be negative on the interval (1, 3.6) and positive elsewhere. The graph may therefore look like this: SOLUTION y 73. Sketch the graph of f(x) = x Ix!. Then show that f' (0) exists. SOLUTION For x < 0, f(x) = -x2, and f'(x) = -2x. For x > 0, f(x) = x2, and f'(x) = 2x. At x = 0, we find lim f(O + h) - f(O) h->O+ h = lim h->O+ hz h =0 and 2 Jim f(O + h) - f(O) = Jim -h = 0 h h->0- h h->0- 134 CHAPTER 3 I DIFFERENTIATION Because the two one-sided limits exist and are equal, it follows that f' (0) exists and is equal to zero. Here is the graph of f(x) = xlxl. !GU[i In Exercises 75-80, zoom in on a plot off at the point (a,f(a)) and state whether or not f appears to be differentiable at x =a. If it is nondifferentiable, state whether the tangent line appears to be vertical or does not exist. 75. f(x) = (x - l)lxl, a= 0 The graph of f(x) = (x - l)lxl for x near 0 is shown below. Because the graph has a sharp corner at x it appears that f is not differentiable at x = 0. Moreover, the tangent line does not exist at this point. SOLUTION = 0, y -0.2 -0.3 77. f(x) = (x - 3) 113 , a =3 The graph of f(x) = (x - 3) 113 for x near 3 is shown below. From this graph, it appears that differentiable at x = 3. Moreover, the tangent line appears to be vertical. f is not SOLUTION The graph of f(x) = I sin xi for x near 0 is shown below. Because the graph has a sharp corner at x appears that f is not differentiable at x = 0. Moreover, the tangent line does not exist at this point. = 0, it SOLUTION 79. f(x) =I sin xi, a= 0 y 0.10 -0.10 -0.05 0.05 0.10 81. Find the coordinates of the point Pin Figure 19 at which the tangent line passes through (5, 0). y FIGURE 19 s E c TI 0 N 3.2 I The Derivative as a Function 135 Let f(x) = 9 - x2, and suppose P has coordinates (a, 9 - a2 ). Because f'(x) = -2x, the slope of the line tangent to the graph of f(x) at Pis -2a, and the equation of the tangent line is SOLUTION y = f' (a)(x - a)+ f(a) = -2a(x - a)+ 9 - a 2 = -2ax + 9 + a2 In order for this line to pass through the point (5, 0), we must have 0 = -lOa +9 +a2 = (a-9)(a- I) Thus, a= 1 or a= 9. We exclude a= 9 because from Figure 19, we are looking for an x-coordinate between 0 and 5. Thus, the point P has coordinates (1, 8). Exercises 83-86 refer to Figure 20. Length QR is called the subtangent at P, and length RT is called the subnormal. y R Q T FIGURE 20 83. Calculate the subtangent of f (x) = x2 + 3x SOLUTION Let f(x) at x =2 = x2 + 3x. Then f'(x) = 2x + 3, and the equation of the tangent line at x = 2 is y This line intersects the x-axis at x = f'(2)(x - 2) + /(2) = 7(x- 2) + 10 = 7x- 4 = ~·Thus Q has coordinates(~, 0), R has coordinates (2, 0) and the subtangent is 85. Prove in general that the subnormal at Pis lf'(x)f(x)I. SOLUTION The slope of the tangent line at P is f'(x). The slope of the line normal to the graph at Pis then - I/ f' (x), and the normal line intersects the x-axis at the point T with coordinates (x + f(x)f'(x), 0). The point R has coordinates (x, 0), so the subnormal is Ix+ f(x)f'(x)- xi= lf(x)f'(x)I 87. Prove the following theorem of Apollonius of Perga (the Greek mathematician born in 262 BCE who gave the parabola, ellipse, and hyperbola their names): The subtangent of the parabolay = x2 at x =a is equal to a/2. SOLUTION Let f(x) = x2. The tangent line to fat x = a is Y = f'(a)(x- a)+ f(a) = 2a(x - a)+ a 2 The x-intercept of this line (where y = 0) is ~ as claimed. y = 2ax- a 2 • 136 CH APTER 3 I DIFFERENTIATION !SJ = x". SOLUTION The generalized statement is: The subtangent toy = x" at x = a is equal to ~a. Proof: Let f(x) = x". Then f'(x) = nx"- 1, and the equation of the tangent line t x = a is 89. Formulate and prove a generalization of Exercise 88 for y y = f'(a)(x- a)+ f(a) = na"-\x - This line intersects the x-axis at x subtangent is = (n - a)+ a"= na"- 1x - (n - l)a" I)a/n. Thus, Q has coordinates ((n - l)a/n, 0), R has coordinates (a, 0), and the n- I 1 a- --a= -a n n Further Insights and Challenges 91. A vase is formed by rotating y = x2 around the y-axis. If we drop in a marble, it will either touch the bottom point of the vase or be suspended above the bottom by touching the sides (Figure 22). How small must the marble be to touch the bottom? FIGURE 22 SOLUTION Suppose a circle is tangent to the parabola y = x2 at the point (t, t2). The slope of the parabola at this point is 2t, so the slope of the radius of the circle at this point is -f, (since it is perpendicular to the tangent line of the circle). Thus the center of the circle must be where the line given by y = -f,(x - t) + t2 crosses the y-axis. We can find the y-coordinate by setting x = 0: We get y = + t2. Thus, the radius extends from (0, + t2) to (t, t2) and ! ! This radius is greater than ~ whenever t > O; so, if a marble has radius > 1/2 it sits on the edge of the vase, but if it has radius ::; I /2 it rolls all the way to the bottom. 93. Negative Exponents Let n be a whole number. Calculate the derivative off(x) = x-n by showing that f(x + h) - f(x) -1 (x + h)" - x" x"(x+h)" h h SOLUTION Let f(x) = x-n where n is a positive integer. • The difference quotient for f is f(x + h) - f(x) h (x + h)-n - x-n X'-(x+h)" x"(x+h)" h h -1 (x + h)" - x" x"(x + h)" h • Therefore, f'(x) = lim f(x + h)- f(x) = lim -I h->O h h->O x"(x + h)" . -1 = Ih->O !ID x"(x + h )" . (x+h)n 1Im h->O h -~ = -x _2 d " - (x") dx = -x-2" :x (x") = -x-2" • nx"- 1 = -nx-n- 1 • Since n is a positive integer, k = -n is a negative integer and we have .!!:.._ (x!') = .!!:.._ (X-") = -nx-n-I = kx!'- 1 ·that is .!!:.... (x!') = kx!'- 1 fornegative integers k dx dx ' 'dx · • From above, we continue: f' (x) s E c TI o N 95. Infinitely Rapid Oscillations SOLUTION I Product and Quotient Rules 137 Define h(x) = Show that his continuous at x 3.3 l xsin! x 0 x;tO x=O = 0 but h'(O) does not exist (see Figure 8). * 0 if x . As x if x = 0 Let h(x) = {xo sin 0) lh(x)- h(o)1 = --+ 0, ixsin(~)- oi = 1x1 lsinG)i-+ o since the values of the sine lie between -1and1. Hence, by the Squeeze Theorem, limh(x) x-->O = h(O) and thus his contin- uous atx = 0. As x --+ 0, the difference quotient at x = 0, h(x)-h(O) = xsin0)-0 =sin(!) x-0 x-0 x does not converge to a limit since it oscillates infinitely through every value between -1and1. Accordingly, h'(O) does not exist. 97. If lim f(c + hh- f(c) and Jim h-.-,O- ex'ist but are not equal, then f is not differentiable at c, and the graph off has f(c + hh- f(c) h--tO+ a comer at c. Prove that f is continuous at c. SOLUTION The function f is continuous at x = c if limf(x) = f(c), or equivalently if lim(f(c + h) - f(c)) = 0. Note /z-+Q X--+C that . (f( c + h) - f( c)) 1im h-->O- Now Jim h = 0, and by assumption Jim h-+O- . (hf(c + h)- f(c)) = 1Im . h · 1Im . f(c + h)- f(c) 1im = h-->0h h-->Oh-->Oh f(c+htf(c) h-+O- exists. It follows that Jim (f(c + h) - f(c)) = 0. Similarly, Jim (f(c + h--+O- h)- f(c)) = 0. Because the two one-sided limits exist and are equal to 0, lim(f(c + h) - f(c)) = 0 h-->O therefore, f is continuous at x = c. 3.3 Product and Quotient Rules Preliminary Questions 1. Are the following statements true or false? If false, state the correct version. (a) f g denotes the function whose value at xis f(g(x)). (b) f /g denotes the function whose value at xis f(x)/g(x). (c) The derivative of the product is the product of the derivatives. (d) (e) :x ! (fg)lx=4 = f(4)g'(4)- g(4)J'(4) (fg)lx=o = f'(O)g(O) + f(O)g'(O) SOLUTION (a) False. The notation f g denotes the function whose value at xis f(x)g(x). (b) True. (c) False. The derivative of a product f g is f'(x)g(x) + f(x)g'(x). (d) False. dd (fg)I = f'(4)g(4) x x=4 + f(4)g'(4). (e) True. 2. Find (f/g)'(l) if f(l) = f'(l) = g(l) = 2 and g'(l) = 4. SOLUTION d dx (f /g)lx=I = [g(l)J'(l) - f(l)g'(l))/g(1) 2 = [2(2) - 2(4))/22 = -1. = 0, f'(l) = 2, and (fg)'(l) = 10. (fg)'(l) = f(l)g'(l) + f'(l)g(l), so 10 = 0. g'(l) + 2g(l) and g(l) = 5. 3. Find g(l) if f(l) SOLUTION h-+O+ 138 CHAPTER 3 I DIFFERENTIATION Exercises In Exercises 1-6, use the Product Rule to calculate the derivative. 1. f(x) = .x3(2x2 + 1) SOLUTION Let f(x) = .x3(2x2 + 1). Then f'(x) = 3. f(x) • ds (2x2 + 1) + x3 :x (2x2 + 1) = (3x2)(2x2 + 1) + x3(4x) = 10x4 + 3x2 = x2ex SOLUTION 5 dhl (! x3) s=4' SOLUTION Letf(x) = x2ex. Then h(s) = (s- 112 + 2s)(7 - s- 1) Let h(s) = (s- 112 + 2s)(7 - s- 1). Then dh = (.!!:._cs- 112 + 2s)) (7 - s- 1) + (s- 112 + 2s) .!!:._(7 - s- 1) ds ds ds = 312 + 2) (7 (-~s2 s- 1) + (s- 112 + 2s)(s- 2) = 312 + ~s- 5 12 + 14 -~s2 2 Therefore, dh ds I = -~(4)-3/2 + ~(4)-5/2 + 14 = 871 2 s=4 2 64 In Exercises 7-12, use the Quotient Rule to calculate the derivative. x 7. f(x)= x-2 SOLUTION Let f(x) = x~ 2 • Then f' (x) = 9 dgl • dt t=-2' SOLUTION g(t) (x-2)..4..x-x..4..(x-2) dx dx (x - 2) 2 = (x-2)-x (x - 2) 2 -2 = __ (x - 2)2 t2 + 1 =t2 - 1 Let g(t) = t2 + 1 Then t - 1 -2-· dg - (t2 - l)fi(t2 + 1) (t2 - dt - (t2 + 1)2 l)fi(t2 - l) (t2 - 1)(2t) - (t2 + 1)(2t) (t2 - 1)2 Therefore, 4(-2) dgl dt 11. g(x) t=-2 8 = - ((-2)2 -1)2 = 9 1 =1 + eX SOLUTION 1 Letg(x) =--.Then 1 + eX dg (1 + ex)fx 1 - 1 ;f;O +ex) dx (1 + ex)2 (1 + ~)(0) - ex = 4t =- (t2 - 1)2 S E CT I 0 N 3.3 I Product and Quotient Rules 139 In Exercises I3-I8, calculate the derivative in two ways. First use the Product or Quotient Rule; then rewrite thefanction algebraically and directly calculate the derivative. x3 x- 3 13. f(x) = SOLUTION Let f(x) = x3 x- 3. By the Product Rule, J'(x) = (x3)'x-3 + x3 (x-3)' = 3.x2x-3 + x\-3x-4 ) = 3x- 1 - Alternatively, simplifying first, we find f(x) 15. f(t) = (2t + l)(t2 - SOLUTION = 1, so f'(x) = 0. 2) Let f(t) = (2t + l)(t2 - 2). Using the Product Rule, = (2t + l)'(t2 - f'(t) 2) + (2t + l)(t2 - 2)' = 2(t2 - 2) + (2t + 1)(2t) Alternatively, multiplying out first, we find f(t) = 2t3 + t2 - 4t - 2. Therefore, f'(t) 17. h(t) 3x- 1 = 0 = 6t2 + 2t- 4 = 6t2 + 2t - 4. t2 - 1 = --1 (- SOLUTION Let h(t) = ~~/.Using the Quotient Rule, , h (t) = (t - l)(t2 - l)' - (t2 - l)(t - 1)' (t - 1)2 = (t- 1)(2t) - (t2 - 1)(1) (t - 1)2 = t2 - 2t + 1 (t - 1)2 =1 fort* 1. Alternatively, simplifying first, we find fort* 1, h(t) Hence h'(t) = 1 fort* = (t - 1)(t + 1) (t- 1) =t + 1 1. In Exercises I9-40, calculate the derivative. 19. f(x) = (.x3 + 5)(.x3 + x + 1) SOLUTION Let f(x) = (.x3 + 5)(.x3 + x + 1). Using the Product Rule, f'(x) 21. dy' , dxx=3 1 y=x+lO SOLUTION Let y = (3.x2)(x3 + x + 1) + (x3 + 5)(3.x2 + 1) = 6.x5 + 4x3 + 1s.x2 + 5 = xlw. Using the Quotient Rule, (x + 10)(0) - 1(1) dy dx (x + 10)2 (x + 10)2 Therefore, 169 = ( -Vx + 1)( -Vx- 1) SOLUTION Let f(x) = ( -Vx + 1)( -Vx- 1). Multiplying through first yields f(x) = x for x ~ 0. If we carry out the Product Rule on f(x) = (x 1i 2 + l)(x 1i 2 - 1), we find 23. f(x) f'(x) = (!x-112) (x112 _ l) + (x1;2 + 1) (!cx-112)) 2 I, 25. dy dxx=2 x4 - 4 y=x2 -5 SOLUTION Let y 2 =! 2 _ !x-112 + ! + !x-112 2 2 2 x4 -4 = -x 2--5 - . Using the Quotient Rule, dy dx 4 {x2-5){4.x3)-(x -4)(2x) = (x2 -5)2 2.x5-20.x3+8x = --(x_2___5_)-=-2- Therefore, I dy dx x=2 = 2(2)5 _ 1 for x ~ 0. Therefore, f'(x) 20(2) 3 + 8(2) __ 80 (2 2 - 5) 2 =1 =1 140 CHAPTER 3 I DIFFERENTIATION 27. dzl dx x=I , z= 3 x +1 Let z = )+ 1 • Using the Quotient Rule, SOLUTION dz (x3 + 1)(0) - 1(3x2) 3.x2 + (x3 + 1)2 (x3 dx 1)2 Therefore, 3(1)2 (l3 + 1)2 3 4 t 29. h(t) = (t + l)(t2 + 1) SOLUTION t Let h(t) = -(t_+_l-)(...,,t2_+_l_) ' h (t) = t . Using the Quotient Rule, t3 +t2 +t+l (t3 + t2 + t + 1) (1) - t (3t2 + 2t + 1) (t3 + t2 + t + l)2 -2t3 - t2 + 1 =- - - - (t3+t2+t+1)2 31. f(x) = x2e2 SOLUTION Letf(x) = x2e 2 • Thenf'(x) = 2e2 x. 33. f(x) = (x + 3)(x - l)(x - 5) SOLUTION Let f(x) = (x + 3)(x - l)(x - 5). Using the Product Rule inside the Product Rule with a first factor of (x + 3) and a second factor of (x - l)(x - 5), we find f'(x) = (l)(x - l)(x - 5) + (x + 3) ((l)(x- 5) + (x- 1)(1)) = 3x2 - 6x - 13 Alternatively, f(x) = (x + 3) {x2 - 6x + 5) = x3 - 3x2 - 13x + 15 Therefore, f'(x) = 3x2 - 6x - 13. ex 35. f(x) = x+ 1 SOLUTION Let f(x) x+l (zz - 14)(zz + 21) 2 37• g(z) = ex = - -. Using the Quotient Rule, SOLUTION 2 Hint: Simplify first. Let z2 - 4) (z- -1) = ((z + 2)(z - 2) (z + l)(z - 1)) = (z - 2)(z + 1) 2 g (z) = ( - - z-1 for z - z+2 z-1 z+2 * -2 and z * 1. Then, g'(z) d (xt-4) 39. -d - t t2 -x SOLUTION = (z + 1)(1) + (z - 2)(1) = 2z - 1 (x constant) Let f(t) = :I=!. Using the Quotient Rule, f'(t) = (t2 - x)(x)- (xt - 4)(2t) (t2 - x)2 = xt2 -x2 - 2xt2 + 8t = -xt2 + 8t-x2 (t2 - x)2 (t2 - x)2 s Ec T Io N 3.3 I Product and Quotient Rules Jn Exercises 41-44, calculate f'(x) in terms of P(x), Q(x), and R(x), assuming that P'(x) R'(x) = P(x). 41. f(x) Q'(x) = -R(x), and xR(x) + Q(x) = SOLUTION Let f(x) = xR(x) + Q(x). Using the Product Rule on the first term, we find f'(x) 43. f(x) = Q(x), 141 = ~i:~ - x SOLUTION Let f(x) = ~i~ , f = (l)R(x) + xR'(x) + Q'(x) = R(x) + xP(x) -R(x) = xP(x) - x. Using the Quotient Rule on the first term, we find _ Q(x)P'(x)- P(x)Q'(x) _ _ Q(x)Q(x) + P(x)R(x) _ _ P(x)R(x) 11 - (Q(x))2 (x) (Q(x))2 (Q(x))2 In Exercises 45-48, calculate the derivative using the values: /(4) f'(4) g(4) g'(4) 10 -2 5 -1 45. (fg)'(4) and (f /g)'(4) SOLUTION Leth= Jg andH =fig. Then h' h'(4) = f'g+ Jg' andH' = gf'1g'. Finally, g = f'(4)g(4) + f(4)g'(4) = (-2)(5) + (10)(-1) = -20 and H'( ) 4 = g(4)/'(4)- f(4)g'(4) = (5)(-2)- (10)(-1) (g(4))2 =O (5)2 47. G'(4), where G(x) = (g(x))2 SOLUTION Let G(x) = g(x) 2 = g(x)g(x). Then G'(x) = g'(x)g(x) + g(x)g'(x) = 2g(x)g'(x), and G'(4) = 2g(4)g'(4) = 2(5)(-1) = -10 In Exercises 49 and 50, a rectangle's length L(t) and width W(t) (measured in inches) are varying in time (t, in minutes). Determine A'(t) in each case. ls the area increasing or decreasing at that time? = 6, L'(3) = -4, and W'(3) = 5. Because A(t) = L(t)W(t), it follows that A'(t) = L'(t)W(t) + L(t)W'(t). Then 49. At t = 3, we have L(3) = 4, W(3) SOLUTION A'(3) = L'(3)W(3) + L(3)W'(3) = (-4)(6) + 4(5) = -4 inches 2/minute Because A'(3) is negative, the area is decreasing at t = 3. 51. Calculate F'(O), where F(x) = x 9 + xs + 4.x5 - 7x .x4 - 3x2 + 2x + 1 Hint: Do not calculate F'(x). Instead, write F(x) = f(x)/g(x) and express F'(O) directly in terms of f(O),f'(O), g(O), g'(O). SOLUTION Taking the hint, let f(x) = x 9 + xs + 4.x5 - 7x and let g(x) = x4 Then F(x) - 3x2 + 2x + 1 = ~g~. Now, = 9x8 + 8x7 + 20x4 Moreover, f(O) = 0, f'(O) = -7, g(O) = 1, and g'(O) = 2. f'(x) 7 and g'(x) = 4x3 - 6x + 2 Using the Quotient Rule, F' (O) = g(O)f' (0) - f (O)g' (0) = - 7 - 0 = _ (g(0))2 1 7 142 CHAPTER 3 I DIFFERENTIATION d 2x 53. Use the Product Rule to calculate dx e . SOLUTION Note that e2x =ex·~. Therefore d d -e2x = -(ex. ex) =ex. ex+ ex. ex= 2e2x dx dx 55. ~Gq Plot f(x) = x/(x2 - 1) (in a suitably bounded viewing box). Use the plot to determine whether f'(x) is positive or negative on its domain {x: xi= ±1). Then compute f'(x) and confirm your conclusion algebraically. SOLUTION domain {x: x Let f(x) = _x_. The graph of f(x) is shown below. From this plot, we see that f(x) is decreasing on its x 2 -1 ±1). Consequently, f'(x) must be negative. Using the quotient rule, we find * , f (x) = (x2 - 1)(1) - x(2x) (x2 _ l)2 = x2 + 1 (x2 _ 1)2 which is negative for all x i= ± 1. y 57. Find a> 0 such that the tangent line to the graph of = x2e-x f(x) at x =a passes through the origin (Figure 5). y a FIGURE 5 SOLUTION Let f(x) = x2e-x. Then f(a) = a 2e-a, f'(x) = 2xe-x +x2(-e-x) f' (a) = (2a - a 2)e-a, and the equation of the tangent line to fat x y = f'(a)(x - = (2x-x2)e-x = a is a)+ f(a) = (2a - a 2)e-a(x- a)+ a 2e-a For this line to pass through the origin, we must have 0 Thus, a = 0 or a therefore a = 1. = (2a - a 2)e-a(-a) + a 2e-a = 1. The only value a = e-a (a 2 - 2a 2 + a 3) = a 2e-0 (a - 1) > 0 such that the tangent line to f(x) = x 2e-x passes through the origin is 59. The revenue per month earned by the Couture clothing chain at time tis R(t) = N(t)S (t), where N(t) is the number of stores and S (t) is average revenue per store per month. Couture embarks on a two-part campaign: (A) to build new stores at a rate of five stores per month, and (B) to use advertising to increase average revenue per store at a rate of$10,000 per month. Assume that N(O) = 50 and S (0) = $150,000. (a) Show that total revenue will increase at the rate dR - dt = SS (t) + 10,000N(t) Note that the two terms in the Product Rule correspond to the separate effects of increasing the riumber of stores on the one hand and the average revenue per store on the other. s E c TI o N 3.3 I Product and Quotient Rules (b) Calculate (idR 143 I. t t=O (c) If Couture can implement only one leg (A or B) of its expansion at t = 0, which choice will grow revenue most rapidly? SOLUTION (a) Given R(t) = N(t)S (t), it follows that dR - = N'(t)S (t) + N(t)S'(t) dt We are told that N' (t) = 5 stores per month and S '(t) = 10,000 dollars per month. Therefore, dR dt = 5S (t) + 10,000N(t) (b) Using part (a) and the given values of N(O) and S(O), we find ddRI t t=O = 5(150,000) + 10.000(50) = 1,250,000 (c) From part (b), we see that of the two terms contributing to total revenue growth, the term SS (0) is larger than the term 1O,OOON(O). Thus, if only one leg of the campaign can be implemented, it should be part A: increase the number of stores by 5 per month. 61. The curve y = l/(.x2 + 1) is called the witch ofAgnesi (Figure 6) afterthe Italian mathematician Maria Agnesi (17181799). This strange name is the result of a mistranslation of the Italian word la versiera, meaning "that which turns." Find equations of the tangent lines at x = ± 1. - ~ • x FIGURE 6 The witch of Agnesi. SOLUTION Let f(x) = _I_. Then f'(x) = (x2 + 1)(0)- 1(2x) = x2 + 1 (x2 + 1)2 2x (x2 + 1)2. • At x = -1, the tangent line is y = f'(-I)(x+ 1) + f(-1) = .!_(x+ I)+.!_= .!_x+ 1 2 2 2 • At x = 1, the tangent line is y = f'(l)(x - 1) + f(l) = _.!_(x - 1) + .!_ = _.!_x + 1 2 2 2 63. Use the Product Rule to show that (f2 )' = 2ff'. SOLUTION Let g = J2 =ff. Then g' = (f2)' =(ff)' = f' f +ff' = 2ff'. Further Insights and Challenges 65. Let f, g, h be differentiable functions. Show that (fgh)'(x) is equal to f'(x)g(x)h(x) + f(x)g'(x)h(x) + f(x)g(x)h'(x) Hint: Write f gh as f(gh). SOLUTION Let p = f gh. Then p' = (fgh)' =f'gh+ f(g'h+gh') = f'gh+ fg'h+ fgh' 67. Derivative of the Reciprocal Use the limit definition to prove ~(-1 )-_f'(x) dx f(x) - J2(x) Hint: Show that the difference quotient for 1/ f(x) is equal to f(x) - f(x + h) hf(x)f(x + h) 144 CH APT E R 3 I DIFFERENTIATION SOLUTION Let g(x) = f 1 x). We then compute the derivative of g(x) using the difference quotient: , x = lim g(x + h)- g(x) g ( ) h-.o h = lim ~ h->O (--1- __1_) h f(x + h) f(x) = lim ~ (f(x)- f(x + h)) h-.o h f(x)f(x + h) --lim(f(x+h)-f(x))( 1 ) h-.O h f(x)f(x + h) We can apply the rule of products for limits. The first parenthetical expression is the difference quotient definition of f'(x). The second can be evaluated at h = 0 to give (f(~))2 • Hence , g (x) d ( I ) f(x) = dx f'(x) = - f1-(x) 69. Use the limit definition of the derivative to prove the following special case of the Product Rule: !:_(xf(x)) dx SOLUTION = f(x) + xf'(x) First note that because f(x) is differentiable, it is also continuous. It follows that limf(x + h) = f(x) h->0 Now we tackle the derivative: !:_(xf(x)) = lim (x + h)f(x + h) - xf(x) = Iim(xf(x + h)- f(x) + f(x + h)) dx h->O h h->0 h . = x lIm h->O f(x + h) - f(x) h) . f( + 1Im x+ h h->0 = xf'(x) + f(x) 71. The Power Rule Revisited If you are familiar with proof by induction, use induction to prove the Power Rule for all whole numbers n. Show that the Power Rule holds for n = 1; then write x" as x · x"- 1 and use the Product Rule. SOLUTION Let k be a positive integer. If k = I, then X' = x. Note that d d dx (xi)= dx (x) = I = lxo Hence the Power Rule holds fork= 1. Assume it holds fork= n where n ~ 2. Then fork= n + 1, we have !:..._ (:Y!) = !:..._ (x"+ 1) = !:..._ (x · x") = x!:_ (x") + x" !:..._ (x) dx dx dx 1 = x · nx"- + x" · I dx = (n + l)x" = k:Y!- dx 1 Accordingly, the Power Rule holds for all positive integers by induction. Exercises 72 and 73: A basic fact of algebra states that c is a root of a polynomial f if and only if f(x) = (x - c)g(x)for some polynomial g. We say that c is a multiple root if f(x) = (x - c)2h(x), where his a polynomial. 73. Use Exercise 72 to determine whether c (a) x5 + 2x4 - 4.x3 - 8x2 - x + 2 (b) x4 = -1 is a multiple root. +x3 -5x2-3x+ 2 SOLUTION (a) To show that -1 is a multiple root of f(x) = x5 + 2x4 - 4x3 - 8.x2 - x + 2 it suffices to check that/( -1) = f' (-1) = 0. We have f (-I) = - I + 2 + 4 - 8 + 1 + 2 = O and f'(x) = 5x4 + 8x3 - 12.x2 - 16x- 1 f'(-1) = 5 - 8- 12 + 16- 1=0 (b) Let f(x) =x 4 + x3 - 5.x2 - 3x + 2. Then f' (x) = 4.x3 + 3x2 - lOx - 3. Because f(-l)= 1-1-5+3+2=0 but f'(-l) it follows that x = -4 + 3 + IO - 3 =6 :;e 0 = - l is a root off, but not a multiple root. s E c TI o N 3.4 I Rates of Change 145 3.4 Rates of Change Preliminary Questions 1. Which units might be used for each rate of change? (a) Pressure (in atmospheres) in a water tank with respect to depth (b) The rate of a chemical reaction (change in concentration with respect to time with concentration in moles per liter) SOLUTION (a) The rate of change of pressure with respect to depth might be measured in atmospheres/meter. (b) The reaction rate of a chemical reaction might be measured in moles/(liter·hour). 2. Two trains travel from New Orleans to Memphis in 4 h. The first train travels at a constant velocity of 90 mph, but the velocity of the second train varies. What was the second train's average velocity during the trip? SOLUTION Since both trains travel the same distance in the same amount of time, they have the same average velocity: 90mph. 3. Discuss how it is possible to be speeding up with a velocity that is decreasing. If the velocity of an object in motion is negative and decreasing, then the velocity is becoming more negative. This means that the magnitude of the velocity is increasing. But the magnitude of the velocity is the speed of the object, so the speed is increasing. Thus, if the velocity of an object in motion is negative and decreasing, then the object is speeding up. SOLUTION 4. Sketch the graph of a function that has an average rate of change equal to zero over the interval [O, l] but has instantaneous rates of change at 0 and 1 that are positive. Consider the graph of y = f(x) shown in the figure below. Because f(O) = f(l), the average rate of change off over the interval [0, l] is zero. On the other hand, the tangent lines to the graph off at x = 0 and x = 1 are positive, so the instantaneous rates of change at 0 and 1 are positive. SOLUTION y Exercises In Exercises 1-8,.find the rate of change. 1. Area of a square with respect to its side s when s =5 2 SOLUTION Let the area be A= f(s) = s • Then the rate of change of A with respect to sis d/ds(s 2 ) = 2s. Whens= 5, the area changes at a rate of 10 square units per unit increase. (Draw a 5 x 5 square on graph paper and trace the area added by increasing each side length by 1, excluding the corner, to see what this means.) Vx with respect to x when x = 1, 8, 27 Let f(x) = fi. Writing f(x) = x 1l 3 , we see the rate of change of f(x) 3. Cube root SOLUTION f'(x) = ~x- 2 1 3 • The requested rates of change are given in the table that follows; c ROC of f(x) with respect to x at x = c. 1 t(l) = t f'C8) = tc8-213) = KD = -&. 8 27 with respect to x is given by f'(l) = f'(27) = 1(27-2/3) = l( l) = l_ 3 3 9 27 5. The diameter of a circle with respect to radius SOLUTION The relationship between the diameter d of a circle and its radius r is d = 2r. The rate of change of the diameter with respect to the radius is then d' = 2. 146 CH APT E R 3 I DIFFERENTIATION 7. Volume V of a cylinder with respect to radius if the height is equal to the radius SOLUTION The volume of the cylinder is V = 7rr2 h = 7rr3 • Thus dV/dr = 37rr2 . In Exercises 9-11, refer to Figure I l, the graph of distance sfrom the origin as afunction of time for a car trip. 9. Find the average velocity over each interval. (b) [0.5, l] (c) [l, 1.5] (a) [0, 0.5] (d) [1, 2] SOLUTION (a) The average velocity over the interval (0, 0.5] is 50-0 - - = 100 km/hour 0.5-0 (b) The average velocity over the interval [0.5, 1] is lOO - 50 1-0.5 = 100 km/hour (c) The average velocity over the interval [l, 1.5] is 1OO - lOO = 0 km/hour 1.5 - 1 (d) The average velocity over the interval [1, 2] is 50 - lOO 2-1 = -50 km/hour 11. Match the descriptions (i)-(iii) with the intervals (a)-(c) in Figure 11. (i) Velocity increasing (ii) Velocity decreasing (iii) Velocity negative (a) [0, 0.5] (b) [2.5, 3] (c) [1.5, 2] Distance (km) 50 0.5 1.0 l.5 2.0 2.5 3.0 FIGURE 11 Distance from the origin versus time for a car trip. SOLUTION (a) (i) The distance curve is increasing, and is also bending upward, so that distance is increasing at an increasing rate. (b) (ii) Over the interval [2.5, 3], the distance curve is flattening, showing that the car is slowing down; that is, the velocity is decreasing. (c) (iii) The distance curve is decreasing, so the tangent line has negative slope; this means the velocity is negative. Exercises 12 and 13 refer to the data in Example 1. Approximate the derivative with the symmetric difference quotient . T(t + 20) - T(t - 20) (SDQ) approximation: T'(t)"" . 40 13. At what t does the SDQ approximation give the smallest (i.e., closest to 0) rate of change of temperature? What is the rate of change? SOLUTION From the values in the table in the solution to Exercise 12, the SDQ approximation gives the smallest (i.e., closest to 0) rate of change at t = 180 minutes and at t = 200 minutes. The rate of change at both of these times is approximately-0.10 °C/min. Exercises 14-16, refer to the four graphs of s as a function oft in Figure 7. 15. Match each situation with the graph that best represents it. (a) Rocky slowed down his car as it approached the moose in the road. The distance from the car to the moose is s and the time since he spotted the moose is t. S E CT I O N 3.4 I Rates of Change 147 (b) The rocket's speed increased after liftoff until the fuel was used up. The distance from the rocket to the launchpad is s and the time since liftoff is t. (c) The increase in college costs slowed for the fourth year in a row. The cost of college is sand the time since the start of the 4-year period is t. SOLUTION (a) Because the car is getting closer to the moose, the graph of s should be decreasing. Moreover, because the car is slowing down, the slope of the graph of s should be getting closer to zero. This matches the graph in Figure 7(C). (b) Because the rocket is getting farther from the launchpad, the graph of s should be increasing. Moreover, because the speed of the rocket is increasing, the slope of the graph of s should be increasing. This matches the graph in Figure 7(A). (c) Because college costs are increasing, the graph of s should be increasip.g. Moreover, because the rate of increase in college costs is slowing down, the slope of the graph of s should be decreasing. This matches the graph in Figure 7(B). 17. Sketch a graph of velocity as a function of time for the shuttle train in Example 6. SOLUTION Over the interval [0, 2], the velocity is positive and increasing, while over the interval [2, 4], the velocity is positive and decreasing. Over the interval [4, 6], the velocity is zero. Finally, over the interval [6, 8], the velocity is negative and decreasing, while over the interval [8, 10], the velocity is negative and increasing. A possible graph of the velocity as a function of time is shown in the figure below. v 2 4 6 19. Fred X has to make a book delivery from his warehouse, 15 mi north of the city, to the Amazing Book Store 10 mi south of the city. Traffic is usually congested within 5 mi of the city. He leaves at noon, traveling due south through the city, and arrives the store at 12:50. After 15 min at the store, he makes the return trip north to his warehouse, arriving at 2:00. Let s represent the distance from the warehouse in miles and t represent time in minutes since noon. Make sketches of the graphs of sand s' as functions oft for Fred's trip. SOLUTION Possible graphs of sand s' for Fred's trip are shown below. s' 30 20 120 10 40 40 80 80 120 21. The velocity (in centimeters per second) of blood molecules flowing through a capillary of radius 0.008 cm is v = 6.4 x 10-s - 0.00lr2 , where r is the distance from the molecule to the center of the capillary. Find the rate of change of velocity with respect to r when r = 0.004 cm. SOLUTION The rate of change of the velocity of the blood molecules is v'(r) is -8 x 10-6 1/s. = -0.002r. When r = 0.004 cm, this rate 23. Use Figure 14 to estimate dT /dh at h = 30 and 70, where Tis atmospheric temperature (in degrees Celsius) and his altitude (in kilometers). Where is dT /dh equal to zero? T(•C) 250 200 150 100 50 -50 -100 '--+10'-+-+- I and less if F'(r) < 1. (d) For the Lorenz curves L 1 and Li in Figure 15(B), what percentage of households have above-average income? SOLUTION (a) The total income among households in the bottom rth part is F(r)T and there are rN households in this part of the population. Thus, the average income among households in the bottom rth part is equal to F(r)T rN = F(r) . !.._ = F(r) A r N r (b) Consider the interval [r,r + l'lr]. The total income among households between the bottom rth part and the bottom r + l'lrth part is F(r + l'lr)T - F(r)T. Moreover, the number of households covered by this interval is (r + l'lr)N - rN = l'lrN. Thus, the average income of households belonging to an interval [r, r + l'lr] is equal to F(r + l'lr)T - F(r)T MN = F(r + l'lr) - F(r) T F(r + l'lr) - F(r) l'lr · N= l'lr A 152 CHAPTER 3 I DIFFERENTIATION (c) Take the result from part (b) and let flr---) 0. Because . F(r + flr) - F(r) F'( ) hm = r flr t.r~o we find that a household in the lOOrth percentile has income F'(r)A. (d) The point Pin Figure 15(B) has an r-coordinate of0.6, while the point Q has an r-coordinate of roughly 0.75. Thus, on curve L 1 , 40% of households have F' (r) > 1 and therefore have above-average income. On curve Lz, roughly 25% of households have above-average income. 49. Use Exercise 47(c) to prove: (a) F'(r) is an increasing function of r. (b) Income is distributed equally (all households have the same income) if and only if F(r) = r for 0$

r $1. SOLUTION (a) Recall from Exercise 47 (c) that F'(r)A is the income of a household in the lOOrth percentile. Suppose 0$ r1 < rz \$ 1. Because r 2 > r 1, a household in the 100r2 th percentile must have income at least as large as a household in the 1OOr1 th percentile. Thus, F'(r2 )A;:::: F'(r 1)A, or F'(r2 );:::: F'(r 1). This implies F'(r) is an increasing function of r. (b) If F(r) = r for O::;; r::;; 1, then F'(r) = I and households in all percentiles have income equal to the national average; that is, income is distributed equally. Alternately, if income is distributed equally (all households have the same income), then F'(r) = I for 0 ::;; r ::;; 1. Thus, F must be a linear function in r with slope l. Moreover, the condition F(O) == 0 requires the F intercept of the line to be 0. Hence, F(r) = I· r + 0 == r.

In Exercises 50 and 51, the average cost per unit at production level xis defined as Cavg(x) = C(x)/ x, where C(x) is the cost of producing x units. Average cost is a measure of the efficiency of the production process. 51. Show that Cavg(x) is equal to the slope of the line through the origin and the point (x, C(x)) on the graph of y = C(x). Using this interpretation, determine whether average cost or marginal cost is greater at points A, B, C, D in Figure 17.

c

Production level

FIGURE 17 Graph of y = C(x). SOLUTION

By definition, the slope of the line through the origin and (x, C(x)), that is, between (0, 0) and (x, C(x)) is C(x) - 0 == C(x) == Cav

x-0

x

At point A, average cost is greater than marginal cost, as the line from the origin to A is steeper than the curve at this point (we see this because the line, tracing from the origin, crosses the curve from below). At point B, the average cost is still greater than the marginal cost. At the point C, the average cost and the marginal cost are nearly the same, since the tangent line and the line from the origin are nearly the same. The line from the origin to D crosses the cost curve from above, and so is less steep than the tangent line to the curve at D; the average cost at this point is less than the marginal cost.

3.5 Higher Derivatives Preliminary Questions 1. For each headline, rephrase as a statement about first and second derivatives and sketch a possible graph. • "Stocks Go Higher, Though the Pace of Their Gains Slows" • "Recent Rains Slow Roland Reservoir Water Level Drop" • ''Asteroid Approaching Earth at Rapidly Increasing Rate!!"

SECTI

oN

3.5

I

Higher Derivatives

153

SOLUTION

• Because stocks are going higher, stock prices are increasing and the first derivative of stock prices must therefore be positive. On the other hand, because the pace of gains is slowing, the second derivative of stock prices must be negative. Stock price

'--------Time

• Because the water level is dropping, the water level is decreasing and the first derivative of the water level must therefore be negative. On the other hand, because the drop in water level is being slowed by the recent rains (that is, the rate of decrease in the water level is getting closer to zero), the second derivative of the water level must be positive. Water level

Time

• Because the asteroid is approaching Earth, the distance between the asteroid and Earth is decreasing and the first derivative of the distance must therefore be negative. On the other hand, because the rate at which the asteroid is approaching Earth is increasing (that is, the rate of decrease in the distance is becoming more negative), the second derivative of the water level must also be negative. Distance

Time

2. Sketch a graph of position as a function of time for an object that is slowing down and has positive acceleration. Because the object is slowing down but has positive acceleration, the velocity of the object must be negative. Thus, the graph of position as a function of time should be decreasing and bending upward.

SOLUTION

3. Sketch a graph of position as a function of time for an object that is speeding up and has negative acceleration. Because the object is speeding up but has negative acceleration, the velocity of the object must be negative. Thus, the graph of position as a function of time should be decreasing and bending downward.

SOLUTION

154

CH APT E R 3

I

DIFFERENTIATION

4. True or false? The third derivative of position with respect to time is zero for an object falling to Earth under the influence of gravity. Explain. This statement is true. The acceleration of an object falling to earth under the influence of gravity is constant; hence, the second derivative of position with respect to time is constant. Because the third derivative is just the derivative of the second derivative and the derivative of a constant is zero, it follows that the third derivative is zero.

SOLUTION

5. Which type of polynomial satisfies f'"(x) = 0 for all x? SOLUTION The third derivative of all quadratic polynomials (polynomials of the form ax2 +bx+ c for some constants a, band c) is equal to 0 for all x.

6. What is the millionth derivative of f(x) =ex? Every derivative of f(x) =ex is ex.

SOLUTION

7. What are the seventh and eighth derivatives of f(x) = x 7 ? Let f(x) = x 7. Then

SOLUTION

f'(x)

= 7x6 ,

J"(x)

= 42x5,

J"'(x)

= 2l0x4, f 4l(x) = 840.x3, f

5l(x)

= 2520x2,

and

f

6

l(x)

= 5040x

Finally,

f

7

l(x)

= 5040

and

f8l(x)

=0

Exercises In Exercises I-I 6, calculate y" and y"'.

1. y

= 14x2 Let y

SOLUTION

= 14x2. Then y' = 28x, y" = 28, and y"' = 0.

3. y = x 4 - 25x2 + 2x SOLUTION

5. y =

4

7rr

= x4 -

Let y

= ~JTr

25x2 + 2x. Then y'

= 4.x3 -

50x + 2, y"

= 12x2 -

50, and y"'

= 24x.

3

3

SOLUTION

7. y

Let y

= 20t4/5 -

SOLUTION

3

. Then y'

= 4JTr2 , y" = 8JTr, and y"' = 8JT.

6t213

Let y

= 20t415 -

Let y

= z - 4z- 1. Then y' = I + 4z-2, y" = -sz-3, and y"' = 24z-4.

6t2 13. Then y'

= 16r 115 -

4r 113 , y"

= -lf-r615 + ~r4!3, and y"' = ~runs -11r7f3.

4 9. y=z- -

z

SOLUTION

11. y = 02(28 + 7) Let y

SOLUTION

= 82 (28 + 7) = 2B3 + 7e2. Then y' = 6e2 + 148, y" = 12e + 14, and y"' = 12.

x-4

13. y= - x SOLUTION

15. y

Let y

= x~ = 1 4

4x-I. Then y'

= 4x-2 , y" = -Sx-3, and y'" = 24x-4.

= x5ex

SOLUTION

Lety = x5ex. Then y'

y" y"'

= 5x4ex + x5 ex = (x5 + 5x4)ex = (5x4 + 20x3)ex + (x5 + 5x4)ex = (x5 + l0x4 + 20x3)ex = (5x4 + 40.x3 + 60x2)ex + (x5 + l0x4 + 20.x3V = (x5 + l5x4 + 60x3 + 60x2)ex

In Exercises I7-26, calculate the derivative indicated.

17.

f

4

l(l),

SOLUTION

f(x)

= x4

Letf(x)

= x4. Then f'(x) = 4x3, f"(x) = 12x2, f'"(x) = 24x, and J(x) for 1 :5 k :5 4. Can you find a

x-1

general formula for fk>(x)? SOLUTION

Let f(x)

2 u· x+-. =smg a computer alb ge ra system,

x-1

f

3

1

_

I

3•l

(x) = - (x-1)2 - (-1) (x-1)1+1

6

3. 2·1

f"(x)= (x-1)3 =(-1)2(x-l)2+1 18

f"'(x)

= - (x-1)4 = (-1)

3

3·3! (x-1)3+1 and

4 3 . 4! (x - 1)5 - ( 1) (x - 1)4+1

f 4) (x) --

___}_}:____ -

From the pattern observed above, we conjecture

fk>(x)=(-l)k

3 ·k'

. (X - l)k+I

Further Insights and Challenges 51. What is p(x) for p(x) as in Exercise 50? SOLUTION

First note that for any integer n :5 98,

cf9 dx99x"

=0

Now, if we expand p(x), we find

p(x) =

x99 +

terms of degree at most 98

therefore,

cf9 cf9 cf9 99 dx99 p(x) = dx99 (x + terms of degree at most 98) = dx99 x 99 Using logic similar to that used to compute the derivative in Example (3), we compute:

cf9 99 dx99 (x )

= 99 x 98 x ... 1

so that :{;9 p(x) = 99 !. 53. Use the Product Rule to find a formula for (Jg)'" and compare your result with the expansion of (a+ b) 3 • Then try to guess the general formula for (fg)(x).

Let f(x) = cos x.

• Then f'(x) = - sin x, f" (x) = - cos x, f"' (x) = sin x, f 4>(x) = cos x, and f 5l(x) = - sin x. • Accordingly, the successive derivatives off cycle among {- sinx, -cos x, sinx, cosx) in that order. Since 8 is a multiple of 4, we have jb u-+b U - b i.e., limF(u) = F(b). Therefore, Fis continuous at u = b. u-+b

(b) Let g be a· differentiable function and take b = g(a). Let x be a number distinct from a. If we substitute u = g(a) into Eq. (1), both sides evaluate to 0, so equality is satisfied. On the other hand, if u g(a), then

*

f(u) - f(g(a)) = f(u) - f(g(a)) u - g(a) = f(u) - f(b) u - g(a) = F(u) u - g(a) x-a u-g(a) x-a u-b x-a x-a

(c) Hence for all u, we have f(u) - f(g(a)) = F(u) u - g(a) x-a x-a

Substituting u = g(x) in Eq. (1), we have f(g(x)) - f(g(a)) = F(g(x)) g(x) - g(a) x-a x-a Letting x -> a gives .

~~ f(g(x)~ = ~(g(a)) = ~~(F(g(x)) g(x~ =!(a))= F(g(a))g'(a) = F(b)g'(a) = J'(b)g'(a) = f'(g(a))g'(a) Therefore (fag)' (a) = f'(g(a))g'(a), which is the Chain Rule.

s EcT Io N

3.8

I

Implicit Differentiation

175

3.8 Implicit Differentiation Preliminary Questions . used to show dx d sm . y = cos Y dx dy?· . . ru1e is . h d""" 1• Whic iuerentlat10n

fx

The chain rule is used to show that siny =cosy~. 2. One of (a)-(c) is incorrect. Find and correct the mistake.

SOLUTION

(a) !!:..._ sin(y2) = 2y cos(y2)

dy

(b) !!:..._ sin(x2) = 2x cos(x2) dx (c) !!:..._ sin(y 2 ) = 2y cos(y2 ) dx SOLUTION

(a) This is correct. Note that the differentiation is with respect to the variable y. (b) This is correct. Note that the differentiation is with respect to the variable x. (c) This is incorrect. Because the differentiation is with respect to the variable x, the chain rule is needed to obtain

!!:..._ sin(y2) = 2y cos(y2) dy dx dx

3. On an exam, Jason was asked to differentiate the equation x2+2.xy+y3 =7

Find the errors in Jason's answer: 2x + 2.xy' + 3y2 = 0. There are two mistakes in Jason's answer. First, Jason should have applied the product rule to the second term to obtain

SOLUTION

!!:_(2.xy) = 2xddy + 2y dx x

Second, he should have applied the general power rule to the third term to obtain !!:_y3 = 3y2 dy dx dx

4. Which of (a) or (b) is equal to

:x (x sin

t)?

dt (a) (x cost) dx SOLUTION

(b) (xcos t) dt +sin t dx Using the product rule and the chain rule we see that d ( . ) dt -d xsmt = xcost+ sint x

dx

so the correct answer is (b).

5. Determine which inverse trigonometric function g has the derivative

1 g '( x)= -2-1 x +

SOLUTION

g(x) = tan- 1 x has the derivative

g '(x)= -1x2 + 1

6. What does the following identity tell us about the derivatives of sin- 1 x and cos- 1 x? sin- 1 x + cos- 1 x = ~ 2 SOLUTION

Differentiating both sides of the identity with respect to x yields d . I d I - sm- x + - cos- x = 0 dx dx

or

d . -1 d l -d sm x= --cos- x dx x

In other words, the derivatives of sin- 1 x and cos- 1 x are negatives of each other.

176

CH APT E R 3

I

DIFFERENTIATION

7. Assume that a is a constant and that y is implicitly a function of x. Compute the derivative with respect to x of each of a 2 , X2, andy2. SOLUTION

Because a is a constant, !!:._a2 = 0 dx

On the other hand, d dx

-x2

= 2x

Finally, because y is implicitly a function of x,

Exercises 1. Show that if you differentiate both sides of X2 + 2y 3

= 6, the result is 2x + 6y2 ~ = 0. Then solve for dy/dx and

evaluate it at the point (2, l ). SOLUTION

Let X2 + 2y3 = 6. Then d

2

-(xdx

2x+

d dx

3

+ 2y ) = -6

6y2~~ = 0

Solving for dy/dx yields 2dY dx

= -2x

dy dx

-2x

6y -

6y2

dy - -4 2 A t (2 , 1), dx - 6 - -3.

In Exercises 3-10, differentiate the expression with respect to x, assuming that y is implicitly a function of x.

3. :(ly3 SOLUTION

Assuming that y is implicitly a function of x, then !!:._

dx

(x2y3) = x2 · 3y2y' + y3 · 2x = 3x2y2y' + 2xy3

s. 0 and consider

the limit lim f (x) - f (0) = lim Jxr x-->0 x-0 x-->0 x IfO0

X

so f' (0) does exist. In summary, f (x) = Jxr is differentiable at x = 0 when a = 0 and when a > 1. Jn Exercises 83-88, use the following table of values to calculate the derivative of the given function at x = 2: x

f (x)

g(x)

f'(x)

g'(x)

2

5

4

-3

9

4

3

2

-2

3

83. S(x) = 3f(x) - 2g(x) Let S (x) = 3f(x)- 2g(x). Then S'(x) = 3f'(x) - 2g'(x) and

SOLUTION

S'(2) = 3f'(2) - 2g'(2) = 3(-3)- 2(9) = -27 f(x)

85. R(x)

= g(x) LetR(x) = f(x)/g(x). Then

SOLUTION

R'(x) = g(x)f'(x) - f(x)g'(x) g(x)2

and / 2) = g(2)f'(2) - f(2)g'(2) = 4(-3)- 5(9) = - 57 R( g(2)2 42 16 87. F(x) = f(g(2x))

Let F(x) = f(g(2x)). Then F'(x) = 2f'(g(2x))g'(2x) and

SOLUTION

F'(2) = 2f'(g(4))g'(4) = 2f'(2)g'(4) = 2(-3)(3) = -18

89. Find the points on the graph of f(x) = x3 - 3x2 + x + 4 where the tangent line has slope 10. Let f(x) = x3 - 3x2 + x + 4. Then f'(x) = 3x2 - 6x + 1. The tangent line to the graph off will have slope 10 when f'(x) = 10. Solving the quadratic equation 3.x2 - 6x + 1 = 10 yields x = -1 and x = 3. Thus, the points on the graph off where the tangent line has slope 10 are (-1, -1) and (3, 7). SOLUTION

91. Find a such that the tangent lines to y = SOLUTION

f'(a) = 3a2

-

x3 - 2x2 + x + 1 at x

= a and x = a + 1 are parallel.

Let f(x) = x3 - 2x2 + x + 1. Then f'(x) = 3x2 - 4x + 1 and the slope of the tangent line at x = a is 4a + 1, while the slope of the tangent line at x =a+ 1 is

f' (a+ 1) = 3(a + 1)2 -

4(a + 1) + 1 = 3(a2 + 2a + 1) - 4a - 4 + 1 = 3a2 + 2a

In order for the tangent lines at x =a and x =a+ 1 to have the same slope, we must have f'(a) = f'(a + 1), or 3a2 -4a+ 1=3a2 +2a The only solution to this equation is a

= ~.

212

CH APT E R 3

I

DIFFERENTIATION

In Exercises 93-98, calculate y". 93. y

= 12x3 -

SOLUTION

5x2 + 3x

Let y

= 12x3 -

5x2 + 3x. Then y'

= 36x2 -

lOx + 3 and y"

= nx -

10

95. y = V2x + 3 SOLUTION

y'

97. y

Let y

= -../2x + 3 = (2x + 3) 1' 2. Then

1

d

= -(2x + 3)- 112 -(2x + 3) = (2x + 3)- 112 2 ~

and

y"

1

d

= --(2x + 3)-3 ' 2 -(2x + 3) = -(2x + 3)-3' 2 2 ~

= tan(x2)

SOLUTION

Let y

= tan(x2). Then y' = 2x sec2 (x2 ) y"

.

and

= 2x (2 sec(x2) ~ sec(x2)) + 2 sec2 (x2) = 8x2 sec2 (x2) tan(x2) + 2 sec2 (x2) d

In Exercises 99-104, computed~. 99. x 3 -y3 = 4 SOLUTION

Consider the equation x3

- y3

= 4. Differentiating with respect to x yields

3x2 - 3y2 dy = 0 dx Therefore,

x2

dy dx

y2

101. y = xy2 + 2x2 SOLUTION

Consider the equation y

= xy2 + 2x2. Differentiating with respect to x yields dy dx

dy dx

- = 2.xy- + y 2 + 4x Therefore,

dy dx 103. y

y2 +4x 1-2.xy

= sin(x + y)

SOLUTION

Consider the equation y

= sin(x + y). Differentiating with respect to x yields dy = cos(x + y) ( 1 + -dy) -dx dx

Therefore,

dy dx

cos(x + y)

1-cos(x+y)

Chapter Review Exercises

213

d2 y

dy

In Exercises I 05 and 106 compute dx and dx2 • 105.

:x2 - 4y2

SOLUTION

= 8

Let :x2

- 4y2 = 8. Then dy 2x-8ydx=0

so

dy x -=dx 4y

Moreover,

f

d2y 4y(l) - x(4~) 4y 4y2 - :x2 -8 dx2 = l6y2 = 16y2 = ~ = 16y3 = - 2y3 107. In Figure 7, for the three graphs on the left, identify f, f', and f". Do the same for the three graphs on the right. y

FIGURE 7 SOLUTION First consider the plot on the left. Observe that the green curve is nonnegative whereas the red curve is increasing, suggesting that the green curve is the derivative of the red curve. Moreover, the green curve is linear with negative slope for x < 0 and linear with positive slope for x > 0 while the blue curve is a negative constant for x < 0 and a positive constant for x > 0, suggesting the blue curve is the derivative of the green curve. Thus, the red, green, and blue curves, respectively, are the graphs off, f' and f". Now consider the plot on the right. Because the red curve is decreasing when the blue curve is negative and increasing when the blue curve is positive and the green curve is decreasing when the red curve is negative and increasing when the red curve is positive, it follows that the green, red, and blue curves, respectively, are the graphs off, f', and f".

In Exercises 109-114, use logarithmic differentiation to find the derivative. 109

•y

= (x+1)3 (4x-2)2

SOLUTION

(x + 1)3 Let y = (4x - 2)2. Then 3

(x + 1) ) lny =In ( = ln(x+ 1)3 -ln(4x-2)2 = 3ln(x+ l)-2ln(4x-2) (4x - 2) 2

By logarithmic differentiation,

y' 3 2 3 4 -=-----·4=----y x + 1 4x - 2 x + 1 2x - 1 so 3

(x+l) ( 3 4) y = (4x-2) 2 x+ 1 - 2x- l I

111. y

= e