52 Chapter 1 Prerequisites for Calculus

~ T • -- _~. ~~;

~CJ'lapter~:t-~:: - - r~='- -. --

Review ExercisesIn Exercises 1-14, write an equation for the specified line.

1. through (1, -6) with slope 3

2. through (-1, 2) with slope,-1/2

3. the vertica11ine through (0, -3)

4. through (-3,6) and (1, -2)

5. the horizontal line through (0, 2)

6. through (3,3) and (-2, 5)

7. with slope -3 and y-intereept 3

8. through (3, 1) and parallel to 2x - y = -2

9. through (4, -12) and parallel to 4x + 3y = 12

10. through (-2, -3) and perpendicular to 3x - 5y = 1

11. through (-1,2) and perpendicular to ±x + ty = 1

12. with x-intercept 3 and y-intercept -5

13. the line y = f(x), where f has the following values:

x -2 2 4

f(x) 4 2

14. through (4, -2) with x-intercept -3

In Exercises 15-18, determine whether the graph of the functionis symmetric about the y-axis, the origin, or neither.

15. y = xl/5 16. y = x2/5

17. y = x2 - 2x - 1 18. y = e-x2

In Exercises 19-26, determine whether the function is even, odd,or neither.

19. y = x2 + 1

21. y = 1 - cas xX4 + 1

23. y = -3--2-x - x

25. y = x + cos x

20. y = x5 - x3 - x

22. Y = see x tan x

24. y = 1 - sin x

26.y=~

In Exercises 27-38, find the (a) domain and (b) range, and (c)graph the function.

27. y = Ix! - 2

29.y=~

31. y = 2e-x - 3

33. y = 2 sin (3x + 7T)- 1

35. y = In (x - 3) + 1

37.y=l~x-,x, -4:5x:50\ vx 0<xS4

28. y= -2+~30. y = 32-x + 1

32. y = tan (2x - 7T)

34. y = x2/5

36. y = - 1 + V2 - x

(

-x - 238. y = x, '

-x + 2,

-2:5x:s:-1

-1<xS11<x:52

•Take it to the NETOn the Internet, visitwww.phschool.comfor a chapter self-test.

In Exercises 39 and 40, write a piecewise formula for thefunction.

39. y 40. y(2,5)

5

-o~--~----~-7X

--~------4----7X

In Exercises 41 and 42, find

(a) (f 0 g)(- 1) (b) (g 0 f)(2)

(c) (f 0 f)(x) (d}(g a g)(x)

1 141. f(x) = -, g(x) = , r=t=":

X vx + 2

42. f(x) = 2 - X, g(x) = Vx+l

In Exercises 43 and 44, (a) write a formula for fog and go fand find the (b) domain and (c) range of each.

43. f(x) ~ 2 - X2, g(x) = ~44. f(x) = \IX, g(x) = ~

In Exercises 45-48, a parametrization is given for a curve.

(a) Graph the curve. Identify the initial and terminal points,if any. Indicate the direction in which the curve is traced.

(b) Find a Cartesian equation for a curve that contains theparametrized curve. What portion of the graph of theCartesian equation is traced by the parametrized curve?

45. x = 5 cas t, y = 2 sin t, 0:5 is: 27T

46. x = 4 cas t , y = 4 sin t, 7T/2:S: t < .37T/2

47. x = 2 - t, Y = 11- 2t, -2:5 i s: 4

48. x = 1 + t , Y =~, t s: 2

In Exercises 49-52, give a parametrization for the curve.

49. the line segment with endpoints (-2, 5) and (4,3)

50. the line through (-3, -2) and (4, -1)

51. the ray with initial point (2, 5) that passes through (-1, 0)

52. y = x (x - 4), x:5 2

In Exercises 53 and 54, work in groups of two or three to(a) fmdf-I and show that (f 0 rl)(x) = (f-I,o f)(x) = x.(b) graph f and f -I in the same viewing window.

53. f(x) = 2 - 3x

54. f(x) = (x + 2)2, x::2:-2

In Exercises 55 and 56, find the measure of the angle in radiansand degrees.

55. sin-I (0.6) .56. tan"" (-2.3)

57. Find the six trigonometric values of () = cas -I (3/7). Giveexact answers.

58. Solve the equation sin x ~ -0.2 in the following intervals.

(a) 0 -s x < 27T (b) -00 < x < 00

59. Solve for x: e-O.2x = 4

60. The graph of f is shown. Draw the graph of each function.

(a)y=f(-x)

(c) y = -2f(x + 1) + 1.

(b) y = -f(x)

(d) y = 3f(x - 2) - 2

y

-I

61. A portion of the graph of a function defined on [ - 3, 3] isshown. Complete the graph assuming that the function is

(a) even. (b) odd.y

(3,2)

62. Depreciation Smith Hauling purchased an 18-wheel truckfor $100,000. The truck depreciates at the constant rate of$10,000 per year for 10 years.

(3) Write an expression that gives the value y after x years.

(b) When is the value of the truck $55,OOO?

63. Drug Absorption A drug is administered intravenously forpain. The function

f( t) = 90 - 52 In (1 + z}, O::s t :s; 4

gives the number of units of the drug in the body after t hours.

(a) What was the initial number of units of the drugadministered?

(b) How much is present after 2 hours?

(c) Draw the graph of f64. Finding Time If Joenita invests $ISOO in a retirement

account that earns 8% compounded annually, how long willit take this single payment to grow to $5000?

Review Exercises 53

65. Guppy Population The number of guppies in Susan'saquarium doubles every day. There are four guppies initially.

(a) Write the number of guppies as a function of time t.

(b) How many guppies were present after 4 days? afterIweek?

(c) When will there be 2000 guppies?

(d) Writing to Learn Give reasons why this might not bea good model for the growth of Susan's guppy population.

66. Doctoral Degrees Table 1.22 shows the number ofdoctoral degrees earned in the given academic year byHispanic students. Let x = 0 represent 1970-71, x = 1represent 1971-72, and so forth.

Table .1,22 Doctorates Earned by HispanicAmericans

Year Number of Degrees

1976-771980-811984-851988-891990-911991-921992-93

520460680630730810830

Source: u.s. Department of Education, as reported in the Chronicle ofHigher Education, April 28, 1995.

(a) Find the linear regression equation for the data andsuperimpose its graph on a scatter plot of the data.

(b) Use the regression equation to 'predict the number of. doctoral degrees that will be earned by Hispanic Americansin the academic year 2000-01. .

(c) Writing to learn Find the slope of the regression line.What does the slope represent?

67. Estimating Population Growth Use the data in Table 1.23about the population of New York State. Let x = 60represent 1960, x = 70 represent 1970, and so forth.

Table 1.23 Population of New York State

Year Population (millions)

1960 16.781980 17.561990 17.99

Source: The Statesman's Yearbook, 129th ed. (London: The MacmillanPress, Ltd., 1992).

(a) Find the exponential regression equation for the data.

(b) Use the regression equation to predict when the .population will be 25 million.

(c) What annual rate of growth can we infer from theregression equation?

62 Chapter 2 Limits and Continuity

Quick Review 2B1

In Exercises 1-4, find [(2).

1. [(x) = 2x3 - 5x2 + 44X2 - 5

2. f(x) = 3 4x +

3. [(x) = sin (7T~)

(

3X - 14. [(x) = _1_'

x2 - 1 '

x<2x2:2

Section 2.1 Exercises

In Exercises 5-8, write the inequality in the form a < x < b.

5. Ixi < 4 6. Ixl < c2

7.lx-21<3 8.lx-cl<d2

In Exercises 9 and 10, write the fraction in reduced form.X2 - 3x - 18 2x2 - x

9. x + 3 10. -=2--,x2"..--,-+-x----=-1

I YI lr- -----±~ ! I

I-r !

-~-- r-r-i-i-tJ I IS, .-i! I ir-

I - II vi= ( )

!- c--+- 1/-+--1--! I II

In Exercises 1-6, use the graph to estimate the limits and value 4.of the function, or explain why the limits do not exist.

1. fT-f- y I Ii- -f I

H-: I,I I iv!= I x)I I! I I 1 xg-li :1 :~+t--,- I:

(a) Jim [(x) (b) lim [(x) (c) lim[(x) (d) [(3)X-73- x---t3+ x---t3

2. H- i- - -j-. 'Yc-r-.,..-

! !

tt ~l'-.I

~g.t)I -1 II 1 II , I .J ! t

f I I - 1 1rr I ! I I

(a) lim get) (b) lim get) (c) Jim get) (d) g(-4)t--7-4- (--7-4+ (---7-4

3. Fr-TT-' - Y I I]-'- ! i> -,It{ ) -1-t-r{-. ~ I I

i I if \if I t I I I t- - I r\ II \ ,-r- III---H- I r i

c:rr 1\ J I II _L_Lk ____L

(a) lim [(h) (b) lim [(h) (c) lim f(h) (d) f(O)h->O- h->O+ h->O

(a) lim pes) (b) Jim pes) (c) lim pes) (d) p(-2)s-t-2- S--7-2+ s4-2

(a) Jim F(x) (b) Jim F(x) (c) Jim F(x) (d) F(O)x---tO- x---tO+ ,X---tO

6. L I ,_1:: .- t-r---,-j-!II _\ -IY c. (x . ~ L

---\<,

I it f--- x

I A1 t :---r_..L:J.

(a) lim G(x) (b) lim G(x) (c) lim G(x) (d) G(2)x---t2- x---t2+ x-+2 ,

In Exercises 7-16, determine the limit by substitution. -Supportgraphically.

7. lim 3x2(2x - 1) 8. lim (x + 3)1998x->-l/2 x->-4 _"

9. Jim (x3 + 3x2 - 2x - 17)x->l

y2 + Sy + 610. lim + 2y-->2 Y

12. lim int xx-->1/2

14. limYx+3x ...•2

16. lim In (sin x)X-->7r12

11 I· yZ + 4y + 3.lm

y-->-3 y2 - 3

13. Jim (x - 6) 2/3x~-2

15. Jim e' cos xx ....•a

In Exercises 17-20, explain why you cannot use substitution todetermine the limit. Find the limit if it exists.

17. lim v:;-=2 18. Jim ~x...-.+-2 x~o x-

19. lim M 20. lim (4 + x)2 - 16_ox _0 x

In Exercises 21-30, determine the limit graphically. Confirmalgebraically.

x-I21. lim -2 --1

x~l x -

5x3 + 8x2

23. lim 3 4 16 2x ...•o x - x

(2 + x)3 - 825. lim -'----'---

x...•o x. sin x

27. lim -2-2--x...•0 x - x

. 229. lim sm x

.r•.••0 X

22. Jim (2 - 3( + 2I...•Z t2 - 4

1 1----2 + x 2

24. Jimx...•o x

26. lim sin 2xx->o X

28. lim x + sin xx ...• 0 x

30. Jim 3 ~in 4xx ...• 0 sm 3x

In Exercises 31 and 32, which of the statements are true aboutthe function Y = f(x) graphed there, and which are false?

31. yy =f(x)

(a) lim f(x) = 1x--t-].f.

(c) lim I(x) = 1X-70-

(e) lirn f(x) existsx...•o

(g) lim f(x) = 1X-70

(i) Jimf(x) = 0.1HI

32. Y

(b) lim f(x) = 0x-?o-

(d) lim I(x) = lim I(x)X-70- X-?O+

(f) Jim f(x) = 0x ...• 0

(h) limf(x) = 1x...•I

(j) Jim f(x) = 2x-?2-

y = f(x)

(a) lim f(x) = 1.l:-4-1+

2

(b) Jimf(x) does not exist.x ...•z

Section 2.1 Rates of Change and Limits 63

(c) lirn j'(x) = 2 (d) lim f(x) = 2X-72 x41-

(e) Jim f(x) == 1 (f) lirnf(x) does not exist.x--....tl+ X-71

(g) Jim f(x) == lim I(x)x-->o· x ....•o-

(h) lim f(x) exists at every c in (-1, 1).x.•.•c

(i) lim f(x) exists at every c in (1,3).x....•c

In Exercises 33-36, match the function with the table.X2 + x - 2 x2 - X - 2

33. Yl = 34. YI = 1x-I x-, x2 - 2x + 1 x2 + x - 2

35. YI = 36. YI = 1x-I x+

X YI- -.4765.8 -.3111.9 -.1526 ~I 01.1 .147621.2 .290911.3 .43043

X= .7

(a)

X YI- 7.3667.8 10.8.9 . 20.9I ERROR1.1 -18.9····\.2 -8.8· .1.3 -5.367

X= .7,

(b)

X YI . "- -.3.8 -.2.9 -.1 -I ERROR1.1 .11.2 .21.3 .3

X~ .7

(d)

X YI

f.IIII!II 2.7.8 2.8.9 2.9I ERROR1.1 3.11.2 3.21.3 3.~

X= .7

(c)

In Exercises 37-44, determine the limit.

37. Jim intxx-?o+

38. Jim intxx ...•o- 39. Jim intx

x...•0.01

40.lim intxX---72-

. x41. lim -1-1

X-7O+ X42. lim -ixi

x-?o- x

43. Assume that Jimf(x) = 0 and Jim g(x) = 3.X---74 x--t4

(a) lim (g(x) + 3)X"" 4

(b) lim xf(x)X-.74

(d) lim g(x)x ...•4 f(x) - 1

44. Assume that Jimf(x) = 7 and lim g(x) = -3.x...•b x->b

(a) Jim (f(x) + g(x»x ...•b (b) Jim (f(x) • g(x»

x->b

(d) Jim f(x)x-eb g(x)

(c) Jim 4 g(x)x...•b

In Exercises 45-48, complete parts (a), (b), and (c) for thepiecewise-defined function.

(a) Draw the graph of f(b) Determine lim, ...•c+ f(x) and lim; ...•c- f(x).

(c) Writing to Learn Does lim, ...•c!(x) exist? If so,what is it? If not, explain .

64 Chapter 2 Limits and Continuity

(

3 - x45. c = 2,f(x) = ~ + I

2 '

[

3 - x,46. c = 2,f(x) = 2,

x/2,1

x<2

x>2

x<2x=2x>2

x<l47. c = l,f(x) = x-I'

x3 - 2x + 5, x ~ 1

{1 - x2, x*--l

48.c=-1,f(x)= 2, x=-I

In Exercises 49-52, complete parts (a)-(d) for the piecewise-defined function.

(a) Draw the graph of f(b) At what points c in the domain of j does limHcf(x)exist?

(e) At what points c does only the left-hand limit exist?

(d) At what points c does only the right-hand limit exist?

-2Tr:S:: x < 0o :s:: x :s:: 2Tr

49. j(x) = {sin x,cas x,

{cas X -Tr :s:: x < 0

50. j(x) = 'see x, 0 :s::x :s:: -tt

. {~' Q:s::x< 151. j(x) = 1, 1 :s:: x < 2

2, x = 2

-I :s:: x < 0, or 0 < x :s:: 1x=Ox < -1, or x> 1

[

X,

52. j(x) = 1,0,

In Exercises 53-56, find the limit graphically. Use the SandwichTheorem to confirm your answer.

53. lim x sin xx->o 54. lim x2 sin xx->o

. 2' 155. lim x sm2x->o X. 2 156. lim x cas 2x->o X

57. Free Fall A water balloon dropped from a window highabove the ground falls y = 4.9t2 m in t sec. Find theballoon's

(a) average speed during the first 3 see of fall.

(b) speed at the instant t = 3.

58. Free Fall on a Small Airless Planet A rock released fromrest to fall on a small airless planet falls y = gt? m in t see,g a constant. Suppose that the rock falls to the bottom of acrevasse 20 m below and reaches the bottom in 4 sec.

(a) Find the value of g.

(b) Find the average speed for the fall.

(e) With what speed did the rock hit the bottom?

ExplorationsIn Exercises 59-62, complete the following tables and statewhat you believe lim j(x) to be.

x->o(a). x I -0.1 -0.01 -0.001 -0.00017Wl ? ? ? ?

(b)

x 1~~O.~1__ ~O~.O~1 0~.0~0~1 0~.~OO~0~1__ ~ _7Wl ? ? ? ?

59. j(x) = x sin l. xl O" - I61. j(x) = ----

x

60. j(x) = sin lx

62. j(x) = x sin (In Ixl)

63. Proof that lime->o (sin e)le = I when e is Measured inRadians Work in groups oj two or three. The plan is toshow that the right- and left-hand limits are both 1.

(a) To show that the right-hand limit is 1, explain why wecan restrict our attention to 0< e < Tr/2.

(b) Use the figure to show that

area of "'OAP = t sin B,

earea of sector OAP = 2'

1area of "'OAT = 2 tan e.

y

T

(e) Use (b) and the figure to show that for 0< e < Tr/2,

1 . e 1 e I e2sm <2 <2tan.,

(d) Show that for 0 < e < 7f/2 the inequality of (c) can bewritten in the form

e 11<---<--sin () cos ().

(e) Show that for 0 < e < 'Tf/2 the inequality of (d) can bewritten in the form

sin ecose<-e-< 1.

(f) Use the Sandwich Theorem to show that

lirn sin e = 1.9-->0+ e

(g) Show that (sin e)/e is an even function.

(h) Use (g) to show that

lirn sin e = 1.0-70- e

(i) Finally, show thatr sin e - 19~-e-- .

Extending the Ideas64. Controlling Outputs Let f(x) = "\I3x"-=2.

(a) Show that limx-72f(x) = 2 = f(2).

(b) Use a graph to estimate values for a and b so that1.8 <f(x) < 2.2 provided a < x < b.

(c) Use a graph to estimate values for a and b so that1.99 <f(x) < 2.01 provided a < x < b.

[-6,6] by [-4, 4]

Figure 2.9 The graph off (x) = l/x.

Section 2.2 Limits Involving Infinity 65

65. Controlling Outputs Let f(x) = sin x.

(a) Find f(rr/6).

(b) Use a graph to estimate an interval (a, b) aboutx = 'Tf/6 so that 0.3 < f(x) < 0.7 provided a < x < b.

(c) Use a graph to estimate an interval (a, b) aboutx = 1i/6 so that 0.49 < f(x) < 0.51 provided a < x < b.

66. Limits and Geometry Let Pea, a2) be a point on theparabola y = X2, a > O. Let 0 be the origin and (0, b) they-intercept of the perpendicular bisector of line segment OPFind Jim p-->o b.

Limits Involving InfinityFinite Limits as X~:±:OO • Sandwich Theorem Revisited CD InfiniteLimits as x~a • End Behavior Models III "Seeing" Limits as x~±oo

Finite Limits asx~±ooThe symbol for infinity (00) does not represent a real number. We use 00 todescribe the behavior of a function when the values in its domain or range out-grow all finite bounds. For example, when we say "the limit of f as x approachesinfinity" we mean the limit of j as x moves increasingly far to the right on thenumber line. When we say "the limit of j as x approaches negative infinity(- 00)" we mean the limit of j as x moves increasingly far to the left. (The limitin each case mayor may not exist.)

Looking at j(x) = l/x (Figure 2.9), we observe

(a) as x~oo, (1/x)~O and we write lim (l/x) = 0,x"""OO

(b) as x~-oo, (l/x)~O and we write lim (l/x) = O.X~-CX)

We say that the line y = 0 is a horizontal asymptote of the graph of f .

Definition --Horiz;;nta'-Asymptot~---------------lThe line y = b is a horizontal asymptote of the graph of a function \'y = j(x) if either .

lim j(x) = b or lim j(x) = b. IL- z-eee x.....,-oo !

[-10, 10) by [-1, 1)

(a)

[-21T, 21T) by [-2,2)

(b)

Figure 2.15 The graphs of (a) f(x) =sin (l/x) and (b) g(x) = f(1/x) = sin x.(Example 10)

Quick Review 2,,2

Section 2.2 Limits Involving Infinity 71

Solution An end behavior model for f is4x2 22x3 X

So, limx--7:,::=f(x) = lirnX-7:'::=2/x = 0 and y = 0 is the horizontalasymptote of f

We can see from Examples 8 and 9 that a rational function always has asimple power function as an end behavior model.

"Seeing" Limits as X--7+oo

We can investigate the graph of y = f(x) as x~±oo by investigating the graphof y = f(l/x) as x~O.

Example 10 USING SUBS~rnUTION

Find lim sin (llx).X--700

Solution Figure 2.15a suggests that the limit is O.Indeed, replacinglimX--7= sin (I Ix) by the equivalent limx--7o+sin x = 0 (Figure 2.15b),we find

In Exercises 1-4, find j'"! and graphf, f-1, and y = x in thesame square viewing window.

1. f(x) = 2x - 3

3. I(x) = tan-1 x

2. f(x) = eX

4. I(x) = coe! x

In Exercises 5 and 6, find the quotient q(x) and remainder rex)when I(x) is divided by g(x).

5. f(x) = 2x3 - 3:;2 + x-I, g(x) = 3x3 + '4x - 5

Section 2.2 Exercises

In Exercises 1-8, use graphs and tables to find (a) limx-->oof(x)and (b) Jimx-->-oof(x). (e) Identify all horizontal asymptotes.

1. f(x) = cos (~ ) 2. f(x) = sin 2xx

5. f(x) = 3x + 1lr] + 2

4. f(x) = 3x3,- X + 1x+3

2x - 16. f(x) = Ix! _ 3

e-X3. f(x) =-

x

lim sin l/x = lim sin x = O.X-7C'O X--7O+

6. f(x) = 2x5 - x3 + x-I, g(x) = x3 - x2 + 1

'In Exercises 7-10, write a formula for (a)f(-x) and (b)f(l/x).Simplify where possible.

7. f(x) = cos x

9. f(x) =: lnxx

8. f(x) = e-X

10. f(x) = (x + ~) sin x

x7. f(x) =: N [x]

8.f(x)=:~

In Exercises 9-16, use graphs and tables to find the limits.

9. Jim _1_ 10. lim ~2x...•2+ x - 2 x...•2- X -

ll.lim ~3X-7-3- x + 12.lim _x_

X-7-3+ X + ::l

72 Chapter 2 Limits and Continuity

intx13. lim

x......,O+ X

15. lim csc xX-7O+

14. lim intxx->o- x

16.lim secxX-4(ll-l2)+

In Exercises 17-22, (a) find the vertical asymptotes of the graphoff(x). (b) Describe the behavior off(x) to the left and right ofeach vertical asymptote.

I17. f(x) = X2 - 4

19. f(x) = x2 - 2xx+I

21. f(x) = cotx

In Exercises 23-28, find 'Jimx-->=y and Jimx->-= y.

23. y= (2 __ x )(~)x + I S + x2

25 = cos (l/x)• y I + (l/x)

27 = sin x• y 2x2 + X

X2 - 118·f(x) =--

2x + 4I-x

20. f(x) = 2x2 - Sx - 3

22. f(x) = see x

26. y = 2x + sin xx

28 = x sin x + 2 sin x. y 2x2

In Exercises 29-32, match the function with the graph of its endbehavior model.

2x3 - 3x2 + I29.y='=':':"'-~~':"X'+3

2X4 - x3 + X2 - 131. y = ---'-'-~~---=-2-x

(a)

(el

xS - x4 + X + 130. y = ~--=-?=---~~.:..2x- + x - 3

X4 - 3x3 + x2 ~ 132. y = 2

I-x

(b)

(d)

In Exercises 33-38, (a) find a power function end behaviormodel for f (b) Identify any horizontal asymptotes.

33. f(x) = 3x2 - 2x + 1

34. f(x) = -4x3 + x2 - 2x - 1

x-235. f(x) = -2---'x2::-+-3-x---S- 36. f(x) = 3x2

- X + Sx2 - 4

37. f(x) = 4x3- 2x + 1x-2

38. f(x) = -x4 + ~x2 : x - 3

x -

In Exercises 39-42, find (a) a simple basic function as a rightend behavior model and (b) a simple basic function as a left endbehavior model for the function.

39. y = eX - 2x

41. y = x + In Ixl

40. y = x2 + e-X

42. y = x2 + sin x

In Exercises 43-46, use the graph of y = f(lIx) to findlimx->oof(x) and limx->-oof(x).

43. f(x) = xe?

45. f(x) = In Ixlx

44. f(x) = x2 e-x

46. f(x) = x sin .lx

In Exercises 47 and 48, find the limit of f(x) as (a) X-7-00,

(b) X-700, (c) X-70-, and (d) X-70+,

47. f(x) = (lIx, x < 0-I, x2:0

Ix- 2-- x<O

48. f(x) = x-I' -I/x2, x> 0

In Exercises 49 and SO,work in groups of two or three andsketch a graph of a function' y = f(x) that satisfies the statedconditions. Include any asymptotes,

49. lim f(x) = 2, Jim f(x) = 00, lim f(x) = 00,x~l x~5- x~5+

lim f(x) = -I, Iim f(x) =' -00,x---too X--7-2+

lim f(x) = 00, lim f(x) = 0x---t-2- X-7-00

50. Jim f(x) = -1, lim f(x) = -00, lim f(x) = 00,x---t2 X---74+ x---t4-

lim f(x) = 00, lim f(x) = 2x---too X--7-00

51. End Behavior Models Work in groups of two or three,Suppose that gl (x) is a right end behavior model for I,(x)and that g2(X) is a right end behavior model forf2(x),Explain why this makes gl (X)/g2(X) a right end behaviormodel for fl(X)/h(x).

52. Writing to Learn Let L be a real number, limx->cf(x) = i,and limx->cg(x) = 00 or -00. Can limx->c (J(x) + g(x» bedetermined? Explain,

53. Table 2.2 gives the average amount per grant of emer-gency financial assistance for veterans in Franklin County,Ohio. Let .r = ° represent 1980, .r = 1 represent 1981,and so forth.

Table 2.2 Emergency Financial Assistance

. Average AmountYear (dollars)

1986 145.381987 155.001988 230.431989 420.701990 494.551991 555.001992 508.771993 460.921994 453.77

Source: Franklin County Veterans Service Commission as reported byMary Stephens in The Columbus [Ohio} Dispatch, May 31, 1995.

(a) Find a cubic regression equation and superimpose itsgraph on a scatter plot of the data.

(b) Find a quartic regression equation and superimpose itsgraph on a scatter plot of the data.

(c) Use each regression equation to predict the averageamount per grant of financial assistance for veterans in the •year 2000.

(d) Writing to Learn Find an end behavior model foreach regression equation. What does each predict about thefuture sizes of grants? Does this seem reasonable? Explain.

200190180

~ 170cE 160?l 150'""e 140oe 130'fd 120"::c: 110

1009080

Continuity

Section 2.3 Continuity 73

Exploration54. Exploring Properties of Limits Find the limits of f, g, and

fg as X--7C.

1(a)f(x) = -, g(x) = x, C = °

x2

(b)f(x) = -3' g(x) = 4x3, C = 0x3

(c)f(x)= x-2' g(x)=(x-2)3, c=2

(d)f(x) = (3 ~ X)4' g(x) = (x - 3)2, C = 3

(e) Writing to Learn Suppose that limx->c!(x) = ° andlim,He g(x) = 00. Based on your observations in (a)-(d),what can you say about limx->e (J(x) • g(x»? III

Extending the Ideas55. The Greatest Integer Function

(a) Show that

X 1 int x x-I int x )---<--:5 1 (x>O) and -->--;::=: 1 (x<O.

x x x x. . intx

(b) Determine lim --.x-+oo x

. . intx(c) Determine lim --.

x-+-oo X

56. Sandwich Theorem Use the Sandwich Theorem toconfirm the limit as X--7OO found in Exercise 3.

57. Writing to Learn Explain 'why there is no value L forwhich limx-->~sin x = L.

In Exercises 58-60, find the limit. Give a convincing argumentthat the value is correct.

58. lim In x2x-->~ lnx

59. lim In xx-->"" log x

60. lim In (x + 1)x-->~ lnx

Continuity at a Point e Continuous Functions • AlgebraicCombinations " Composites 0 Intermediate Value Theorem forContinuous Functions

Continuity at a PointWhen we plot function values generated in the laboratory or collected in thefield, we often connect the plotted points with an unbroken curve to showwhat the function's values are likely to have been at the times we did not mea-sure (Figure 2.16). In doing so, we are assuming that we are working witha continuous function, a function whose outputs vary continuously with theinputs and do not jump from one value to another without taking on thevalues in between. Any function y = f(x) whose graph can be sketched in one.continuous motion without lifting the pencil is an example of a continuousfunction.

oMinutes after exercise

Figure 2.16 How the heartbeat returns toa normal rate after running.

80 Chapter 2 Limits and Continuity

Quick Review 2.33x2 - 2x + 1

1. Find lim 3 + 4 .x....•-l X

2. Let f(x) = int x. Find each limit.

(a) lim f(x) (b) Jim f(x) (c) Jim f(x) (d) f(-I)x....•-I- x....•-I+ . x--.-I

f( ) - {X2 - 4x + 5, x < 23. Let x - 4 _ > 2x, x_ .

Find each limit.

(a) lim f(x) (b) lim f(x) (c) Urnf(x) (d)f(2)x~2- x--t2+ x-+2

In Exercises 4-6, find the remaining functions in the list offunctions: t. g, fog, g 0 f

2x - 1 14. f(x) = --, g(x) = - + 1x+5 x

Section 2.3 Exercises

In Exercises 1-10, find the points of discontinuity of thefunction. Identify each type of discontinuity.

1 x + 11. y = (x + 2)2' 2. y = x2 - 4x + 3

13. y = x2 + 1

5.y=~

7. y = Ixl/x

4. y = Ix - 116. y = V2x - 1

8. y = cotx

9. y = e1/x 10. y = In (x + 1)

In Exercises 11-18, use the functionfdefined and graphedbelow to answer the questions.

1

x2 - 1,2x,

f(x) = 1,-2x + 4,0,

-l:O;x<OO<x<lx=1l<x<22<x<3

y

Y=f(X;~

y=2x1

--~--~--~----~--~--~x

-1

11. (a) Does f(-l) exist?

(b) Does limx->-l+ f(x) exist?

(c) Does lim,....•-I+ f(x) = f(-I)?

(d) Is f continuous at x = -I?

5. f(x) = x2, (g 0 f) (x) = sin x2, domain of g =' [0,00)

6. g(x) =~, (g 0 f)(x) = l/x, x> 0

7. Use factoring to solve 2x2 + 9x - 5 = O.

8. Use graphing to solve x3 + 2x - 1 = O.

In Exercises 9 and 10, let

f(x) = (~:zx~ 6x - 8, ~ ~ ~.

9. -Solve the equation f(x) = 4.

10. Find a value of c for which the equation f(x) = c has nosolution.

12. (a) Does f(l) exist?

(b) Does lim,....•1f(x) exist?

(c) Does lirnx->d(x) = f(1)?

(d) Is f continuous at x =·1?

13. (a) Is f defined at x = 2? (Look at the definition of f)(b) Is f continuous at x = 2?

14. At what values of x is f continuous?15. What value should be assigned to f(2) to make the extended

function continuous at x = 2?

16. What new value should be assigned to f(l) to make the newfunction continuous at x = I?

17. Writing to Learn Is it possible to extendfto becontinuous at x = O? If so, what value should the extendedfunction have there? If not, why not?

18. Writing to Learn Is it possible to extend f to becontinuous at x = 3? If so, what value should the extendedfunction have there? If not, why not?

In Exercises 19-24, (a) find each point of discontinuity.(b) Which of the discontinuities are removable? not removable?Give reasons for your answers.

19. f(x) = (~~ Xl : : ~2 '

20. f(x) = {;,- xx/2,

21. f(x) = ( x ~ 1 'x3 - 2x + 5,

x<2x=2x>2

x<1

X 2:: 1

{I- X2

22. f(x) = 2, '

23. y

x =1--1x =-1

y;f(x)

24. yy =f(x)

__J-__~ ~ __-L__~ ~x-1

In Exercises 25-30, give a formula for the extended function thatis continuous at the indicated point.

X2 - 9 x3 - 125. f(x) = --.;-:t3' x = -3 26. f(x) = x2 _ I ' x = 1

sin x sin 4x27. f(x) = -, x = 0 28. f(x) = --, x = 0

x xx-4

29. f(x) = , I ' ' X = 4vx - 2

x3 - 4x2 - llx + 3030. f(x) = x2 _ 4 ,x = 2

In Exercises 31-34, work in groups of two or three. Verify thatthe function is continuous and state its domain. Indicate whichtheorems you are using, and which functions you are assumingto be continuous.

31. Y = , r=r":vx + 2

32. y = x2 + V4 - x

{

x2 - 134. y = --.;=T' x =I- 1

2, x = I33. y = Ix2 - 4xl

In Exercises 35-38, sketch a possible graph for a function I thathas the stated properties.

35. f(3) exists but limx->3f(x) does not.

36. f(-2) exists, limx->-2+ f(x) = f(-2), but limx---+_2f(x)does not exist.

37. f(4) exists, limx->4f(x) exists, butfis not continuous atx = 4.

38. f(x) is continuous for all x except x = 1, where f has anonremovable discontinuity.

39. Solving Equations Is any real number exactly 1 less thanits fourth power? Give any such values accurate to 3 decimalplaces.

40. Solving Equations Is any real number exactly 2 more thanits cube? Give any such values accurate to 3 decimal places.

Section 2.3 Continuity 81

41. Continuous Function Find a value for a so that thefunction

I(x) = {X2 - 1,2ax,

x<3X 2::: 3

is continuous.

42. Writing to learn Explain why the equation e-X = x hasat least one solution.

43. Salary Negotiation A welder's contract promises a 3.5%salary increase each year for 4 years and Luisa has an initialsalary of $36,500.

(a) Show that Luisa's salary is given by

y = 36,500(1.035)intt,

where t is the time, measured in years, since Luisa signedthe contract.

(b) Graph Luisa's salary function. At what values of t is itcontinuous?

44. Airport Parking Valuepark charge $1.10 per hour orfraction of an hour for airport parking. The maximum chargeper day is $7.25.

(a) Write a formula that gives the charge for x hours witho ::; x ::; 24. (Hint: See Exercise 43.)

(b) Graph the function in (a). At what values of x is itcontinuous?

Exploration

45. Let f(x) = (1 + ~ r-(a) Find the domain of f(b) Draw the graph of f(e) Writing to learn Explain why x = -1 and x = 0are points of discontinuity of f(d) Writing to learn Are either of the discontinuitiesin (c) removable? Explain.

(e) Use graphs and tables to estimate lirnx->= f(x). Ii;l!

Extending the Ideas46. Continuity at a Point Show that j'(x) is continuous at

x = a if and only if

lim f(a + h) = f(a).h..•o

47. Continuity on Closed Intervals Letfbe continuous andnever zero on [a, b]. Show that either f(x) > 0 for all x in[a, b] or f(x) < 0 for all x in [a, b].

48. Properties of Continuity Prove that if f is continuous onan interval, then so is If I.

49. Everywhere Discontinuous Give a convincing argumentthat the following function is not continuous at any realnumber.

f(x) = {I, if x ~s~atio.nal0, if x 1S irrational

Section 2.4 Rates of Change and Tangent Lines 87

of change of its position y = f(t). Its instantaneous speed at any time t is theinstantaneous rate of change of position with respect to time at time t, or

l' f(t + h) - f(t)h~ h .

We saw in Example 1, Section 2.1, that the rock's instantaneous speed att = 2 see was 64 ft/sec.

Particle Motion

We only have considered objects moving in

one direction in this chapter. In Chapter 3,we will deal with more complicated motion.

Example 6 INVESTIGATING FREE FALL

Find the speed of the falling rock in Example 1, Section 2.1, at t = 1 sec.

Solution The position function of the rock is f(t) = 16t2. The averagespeed of the rock over the interval between t = 1 and t = 1 + h sec was

f(1 + h) - f(1)h

16(1 + h)2 - 16(1)2h

The rock's speed at the instant t = 1 was

In Exercises 1 and 2, find the increments ~x and ~y from point'A to point B.

1. A(-5, 2), B(3,5) 2. A(l, 3), B(a, b)

In Exercises 3 and 4, find the slope of the line determined by thepoints.

3. (-2,3), (5, -I) 4. (-3, -1), (3,3)

In Exercises 5-9, write an equation for the specified line.

S. through (-2, 3) with slope = 3/2

Section 2.4 ExercisesIn Exercises 1-6, find the average rate of change of the functionover each interval.

1. f(x) = x3 + I

(a)[2,3) (b)[-I, I)

2·f(x)=~

(a) [0, 2} (b) uo, 12}

3. f(x) = e'

(a) [ -2, a} (b) [I, 3}

lim 16(h + 2) = 32 ft/sec.h-..;O

6. through (1, 6) and (4, - I)

37. through (I, 4) and parallel to y = -"4x + 2

8. through (1, 4) and perpendicular to' y = - ~x + 2

9. through (-1, 3) and parallel to 2x + 3y = 5

10. For what value of b will the slope of the line through (2, 3)and (4, b) be 5/3?

4·f(x)=lnx

(a) [I, 4} (b) nco, 103}

S. f(x) = cot t

(a) [7T/4, h/4} (b) [7T/6, 7T/2}

6. f(x) = 2 + cas t

(a)[O,7T) (b)[-7T,7T)

88 Chapter 2 Limits and Continuity

In Exercises 7 and 8, a distance-time graph is shown.

(a) Estimate the slopes of the secants PQI. PQ2. PQ3. andPQ4. arranging them in order in a table. What is theappropriate unit for these slopes?

(b) Estimate the speed at point P7. Accelerating from a Standstill The figure shows the

distance-time graph for a 1994 Ford® Mustang Cobra"accelerating from a standstill.

s

650600

500

:§: 400<luc:~ 300is

200 f---+--.,<i

100

5 10 15 20Elapsed time (see)

8. Lunar Data The figure shows a distance-time graph for awrench that fell from the top platform of a communicationmast on the moon to the station roof 80 m below.

y

80

:§: 60c:~::§ 40<loc:S 20'"is

0

I c,ljPI V

f--1-- ~ ! I !~

f--

rr:Tffz ~ \--./.

I i ./ "- \--I IQd/ .!y r- f- j--

L.-- 1---1 I5

Elapsed time (see)

10

In Exercises 9-12, at the indicated point find

(a) the slope of the curve,

(b) an equation of the tangent, and

(c) an equation of the normal.

(d) Then draw a graph of the curve, tangent line, and normalline in the same square viewing window.

9. y = x2 at x = -2

10. y = x2 - 4x at x = 1

111. Y = --1 at x = 2x-12. y = x2 - 3x - 1 at x = 0

In Exercises 13 and 14,find the slope of the curve at theindicated point.

13. f(x) = [x] at (a) x = 2 (b)x = -3

14.f(x)=ix-2i at x=l

In Exercises 15-18, determine whether the curve has a tangentat the indicated point. If it does, give its slope. If not, explainwhy not.

{2 - 2x - x2 X < 0

15. f(x) = 2x + 2. 'x 2: 0 at x = 0

{-x

16. f(x) = 2'• x - x,

II/x,17. f(x) = 4 - x

4 '18. f(x) = {Sin x,

cas x,

x<Oat x = 0x2:0

x:o;2

x> 2 at x = 2

O:o;x<3wM ~ x=3wM3w/4 :0; x :0; 2w

In Exercises 19-22, (a) find the slope of the curve at x = a and(b)·Writing to Learn describe what happens to the tangent atx = a as a changes.

19.y = X2 + 2 20. Y = 2/x1

21 y = -- 22. Y = 9 - X2. x-I

23. Free Fall An object is dropped from the top of a lOO-mtower. Its height above ground after t sec is 100 - 4.9t2 m,How fast is it falling 2 see after it is dropped?

24. Rocket Launch At t see after lift-off, the height of a rocketis 3t2 ft. How fast is the rocket climbing after 10 see?

25. Area of Circle What is the rate of change of the area of acircle with respect to the radius when the radius is r = 3 in.?

26. Volume of Sphere What is the rate of change of thevolume of a sphere with respect to the radius when theradius is r = 2 in.?

27. Free Fall on Mars The equation for free fall at the surfaceof Mars is s = 1.86t2 m with t in seconds. Assume a rockis dropped from the top of a 200-m cliff. Find the speed ofthe rock at t = 1 sec.

28. Free Fall on Jupiter The equation for free fall at thesurface of Jupiter is s = 11.44t2 m with t in seconds.Assume a rock is dropped from the top of a 500-m cliff.Find the speed of the rock at t = 2 sec.

29. Horizontal Tangent At what point is the tangent tof(x) = x2 + 4x - 1 horizontal?

30. Horizontal Tangent At what point is the tangent tof(x) = 3 - 4x - X2 horizontal?

31. Finding Tangents and Normals

(a) Find an equation for each tangent to the curvey = l/(x - 1) that has slope -1. (See Exercise 21.)

(b) Find an equation for each normal to the curvey = l/(x - 1) that has slope 1.

32. Finding Tangents Find the equations of alllines tangentto y = 9 - X2 that pass through the point (1, 12).

33. Table 2.3 gives the amount of federal funding for theImmigration and Naturalization Service (INS) forseveral years.

(a) Find the average rate of change in funding from 1993to 1995.

(b) Find the average rate of change from 1995 to 1997.

(c) Let x = 0 represent 1990, x = 1 represent 1991, andso forth. Find the quadratic regression equation for the dataand superimpose its graph on a scatter plot of the data.

(d) Compute the average rates of change in (a) and (b) usingthe regression equation.

(e) Use the regression equation to find how fast the fundingwas growing in 1997.

Section 2.4 Rates of Change and Tangent Lines 89

Table 2.3 Federal Funding

INS FundingYear ($ billions)

1993 1.51994 1.61995 2.11996 2.61997 3.1

Source: Immigration and Naturalization Service as reported by Bob Lairdin USA Today, February 18, 1997.

34. Table 2.4 gives the amounts of money earmarked by the U.S.Congress for collegiate academic programs for several years.

(a) Let x = 0 represent 1980, x = 1 represent 1981, andso forth. Make a scatter plot of the data.

(b) Let P represent the point corresponding to 1997, and Qthe point for anyone of the previous years. Make a table ofthe slopes possible for the secant line PQ.(c) Writing to Learn Based on the computations, explainwhy someone might be hesitant to make a prediction aboutthe rate of change of congressional funding in 1997.

Table 2.4 Congressional Academic Funding

YearFunding

($ millions)

1988198919901991199219931994199519961997

225289

·270493684763651600296446

Source: The Chronicle of Higher Education, March 28, 1997.

ExplorationsIn Exercises 35 and 36, complete the following for the function.

(a) Compute the difference quotientf(1 + h) - f(1)

h(b) Use graphs and tables to estimate the limit of thedifference quotient in (a) as h~O.

(c) Compare your estimate in (b) with the given number.

(d) Writing to Learn Based on your computations,do you think the graph of f has a tangent at x = I? If so,estimate its slope. If not, explain why not.

36. f(x) = 2X,' In 4 II

90 Chapter 2 Limits and Continuity

In Exercises 37-40, work in groups of two or three.

The curve y = f(x) has a vertical tangent at x = a ifI f(a + h) - f(a) = OQ

hl!Po hor if

lim f(a + h) - f(a) = -00.

h-->O hIn each case, the right- and left-hand limits are required to be thesame: both +00 or both -00.

Use graphs to investigate whether the curve has a vertical tangentat x = O.

37. y = x2/5

39. y = xll3

38. y = x3/5

40. y = X2/3

Key Termsaverage rate of change (p. 82)average speed (p. 55)connected graph (p. 79)Constant Multiple Rule for Limits (p. 67)continuity at a point (p. 75)continuous at an endpoint (p. 75)continuous at an interior point (p. 75)continuous extension (p. 77)continuous function (p. 77)continuous on an interval (p. 77)difference quotient (p. 86)Difference Rule for Limits (p. 67)discontinuous (p. 75)end behavior model (p. 70)free fall (p. 55)horizontal asymptote (p. 65)infinite discontinuity (p. 76)instantaneous rate of change (p. 87)instantaneous speed (p. 87)intermediate value property (p. 79)Intermediate Value Theorem for

Continuous Functions (p, 79)

Extending the IdeasIn Exercises 41 and 42, determine whether the graph of thefunction has a tangent at the origin. Explain your answer.

[

2 . 141. f(x) = x sm~,

0,

x=t-o

x=o

42. f(x) = [x sin ±, x =t-00, x = 0

43. Sine Function Estimate the slope of the curve y = sin x'at x = 1. (Hint: See Exercises 35 and 36.)

jump discontinuity (p. 76)left end behavior model (p. 69)left-hand limit (p. 60)limit of a function (p. 57)normal to a curve (p. 86)oscillating discontinuity (p. 76jpoint of discontinuity (p. 75)Power Rule for Limits (p. 67)Product Rule for Limits (p. 67)Properties of Continuous Functions (p, 78)Quotient Rule for Limits (p. 67)removable discontinuity (p. 76)right end behavior model (p. 69)right-hand limit (p. 60)Sandwich Theorem (p, 61)secant to a curve (p, 82)slope of a curve (p. 85)Sum Rule for Limits (p. 67)tangent line to a curve (p. 85)two-sided limit (p. 60)vertical asymptote (p. 68)vertical tangent (p. 90)

Review ExercisesIn Exercises 1-14, find the limits.

1. lim (X3 - 2X2 + 1)x...•;-2

2.1im x2 + 1x->-2 3x2 - 2x + 5

4. lim "V"9 - X2x-t5

3. lim V1="2x".r•.• 4

1 1----2 + x 2 6 li 2X2 + 3

'x-!~oo 5x2 + 7

sin 2x8.lim-4-

.<->0 x

5.limx->o X

X4 + x3

7'x~~oo 12x3 + 128

x csc x + 19.lim----

x...•o x csc x 10. lim eX sin xx->o

11. lim int (2x - I)x->7/2+

12. lim int (2x - 1)x->712-

14. lim x + sin xx...•oo x + cas x13. lim e-X cas x

x-too

In Exercises IS-20, determine whether the limit exists on thebasis of the graph of y = f(x). The domain of f is the set of realnumbers.

15. lim f(x)x...•d

17. lim f(x)x-tC-

16. Iim f(x)X-4C+

18. timf(x).t-4C

19. limf(x)x...•b 20. limf(x)

x->a

y

ey=fcx

---r-L---~-L--~--~xa

In Exercises 21-24, determine whether the functionfused inExercises 15-20 is continuous at the indicated point.

21. x = a 22. x = b

23. x = c 24. x = d

Review Exercises 91Take it to the NETOn the Internet. visitwww.phschool.comfor a chapter self-test.

In Exercises 25 and 26, use the graph of the function withdomain - 1 ::s x ::s 3.

25. Determine

(a) lim g(x).x~3-

(b) g(3).

(c) whether g(x) is continuous at x = 3.

(d) the points of discontinuity of g(x).

(e) Writing to Learn whether any points of discontinuityare removable. If so, describe the extended function. If not,explain why not.

y

2

--'---,-+---'----'----'--+x

26. Determine

(a) lim k(x). (b) lim k(x). (c) k(l).X-41- X---7I+

(d) whether k(x) is continuous at x = 1.

(e) the points of discontinuity of k(x).

(f) Writing to Learn whether any points of discontinuityare removable. If so, describe the extended function. If not,explain why not.

y

y = k(x)

(1. l.5)2

In Exercises 27 and 28, (a) find the vertical asymptotes of thegraph of y = f(x), and (b) describe the behavior of f(x) to theleft and right of any vertical asymptote.

x+3 x-I27. f(x) = x + 2 28. f(x) = x2 (x + 2)

92 Chapter 2 Limits and Continuity.

In Exercises 29 and 30, answer the questions for the piecewise-defined function.

\

1.-x

29. f(x) = ~x~

1,

x:O;-l-l<x<Ox=OO<x<lx~l

(a) Find the right-hand and left-hand limits of j at x = -1,0, and 1.

(b) Does f have a limit as x approaches -I? O? I? If so,what is it? If not, why not?

(c) Is f continuous at x = -I? O? I? Explain.

30. f(x) = {lx3 - 4xl, x < 1X2 - 2x - 2, x 2: 1

(a) Find the right-hand and left-hand limits of f at x = 1.

(b) Does f have a limit as x--7l ? If so, what is it? If not,why not?

(c) At what points is I continuous?

(d) At what points is I discontinuous?

In Exercises 31 and 32, find all points of discontinuity of thefunction.

31 f( )=~• x 4 _ x2 32. g(x) = V3x + 2

In Exercises 33-36, find (a) a power function end behaviormodel and (b) any horizontal asymptotes.

33. f(x) = 2x + 1 34. f(x) = 2x2 + 5x - 1x2 - 2x + 1 x2 + 2x

35. f(x) = x3- 4x2 + 3x + 3

x-336. f(x) = x4 - 3x2 + x-I

x3 - X + 1

In Exercises 37 and 38, find (a) a right end behavior model and'(b) a left end behavior model for the function.

37. f(x) = x + eX 38. f(x) = In Ixl + sin x

In Exercises 39 and 40, work in groups of two or three. Whatvalue should be assigned to k to make f a continuous function?

(

X2 + 2x - 1539. f(x) = x - 3 ,x '* 3

k, x = 3

(

SIll X

40. f(x) = lx' xr 0k, x = 0

In Exercises 41 and 42, work in groups of two or three. Sketcha graph of a function f that satisfies the given conditions.

41. lim f(x) = 3, lim f(x) = 00,x~oo X--7-00

lim f(x) = 00, lim f(x) = -00X--73+ X--73-

42. lim f(x) does not exist, lim f(x) =1(2) = 3X--72 X---72+

43. Average Rate of Change Find the average rate of changeof I(x) = 1 + sin x over the interval [0, 1T/2].

44. Rate of Change Find the instantaneous rate of change ofthe volume V = (l/3)1Tr2H of a cone with respect to theradius r at r = a if the height H does not change.

45. Rate of Change Find the instantaneous rate of change ofthe surface area S = 6x2 of a cube with respect to the edgelength x at x = a.

46. Slope of a Curve Find the slope of the curvey = x2 - X - 2 at x = a.

47. Tangent and Normal Let I(x) = X2 - 3x andP = (1, f(1». Find (a) the slope of the curve y = f(x)at P, (b) an equation of the tangent at P, and (c) an equationof the normal at P

48. Horizontal Tangents At what points, if any, are thetangents to the graph of I(x) = x2 - 3x horizontal?(See Exercise 47)

49. Bear Population The number of bears in a federal wildlifereserve is given by the population .equation

200pet) = 1 + 7e-01t'

where t is in years.

(a) Writing to Learn Find p (0). Give a possibleinterpretation of this number.

(b) Find lim pet).1-'0=

(c) Writing to Learn Give a possible interpretation of theresult in (b).

50. Taxi Fares Bluetop Cab charges $3.20 for the first mile and$1.35 for each additional mile or pan of a mile.

(a) Write a formula that gives the charge for x miles witho :s; x :s; 20.

(b) Graph the function in (a). At what values of x is itdiscontinuous?

51. Congressional Academic Funding Consider Table 2.4 inExercise 34 of Section 2.4.

(a) Let x = 0 represent 1980, x = 1 represent 1981, andso forth. Find a cubic and a quartic regression equation forthe data.

(b) Find an end behavior model for each regressionequation. What does each predict' about future funding?

52. Limit Properties Assume that

lim [J(x) + g(x)] = 2,x->c

lim [J(x) - g(x)] = 1,x->c

and that limx->cf(x) and limx->cg(x) exist. Findlimx->cf(x) and limx->c g(x).

Review Exercises 93

53. Table 2.5 gives the spending for software, equipment, andservices to create and run intranets-private, Internet-likenetworks that link a company's operations.

(a) Let x = a represent 1990, x = 1 represent 1991, andso forth. Make a scatter plot of the data.

(b) Let P represent the point corresponding to 2000, and Qthe point for anyone of the previous years. Make a table ofthe slopes possible for the secant line PQ.

(c) Predict the rate of change of spending in 2000.

(d) Find a quadratic regression equation for the data, anduse it to calculate the rate of change of spending in 2000.

Table 2.5 Intranet Spending

YearSpending

($ billions)

199519961997199819992000

2.74.87.8

11.2

15.220.1

Source: Killen & Associates as reported by Anne R. Carey and Elys A.McLean in USA Today. April 14, 1997.