Cumulative Review Exercises 573

Cumulative Review Exercises

In Exercises 1-8, determine the limit.

2x2 - x - I 2. lim sin 3x1. lim----;:---- ­x-.I x 2 + X - 12 x-.o 4x

I ' ~l I + x

311m 4. Jim~• - ­x~o X .\"---.700 x - eX

t (I - cos t) 6. lim In(ex.- 1)5. lim .

1-.0 t - SIO r x-.o· In x 7. lim (eX + x)llx 8. lim (3X + I __._1_)x-.o x-.o x SIO X

29. Letf(x) = {2X - x , x::S I

2 - x, x> I.

(a) Find Iimx-.I- f(x).

(b) Find limxc..,l-f(x).

(c) Find limx-.l f(x).

(d) Is f continuous at x = I?

(e) Is f differentiable at x = J?

10. Find all the points of discontinuity of

f(x) =

11. Identify all horizontal and vertical asymptotes of

cos x y=-­

2x 2 - x·

12. Sketch a possible graph for a function y = f(x) that satisfies:

lim f(x) = 00, lim f(x) = -I,x----72- x----72~

lim f(x) = -3, lim f(x) = 3. X----7-oo .\"----700

13. Find the average rate of change of the function

f(x) = ~ over the interval [0, 5].

In Exercises 14-28, find dy/dx.

x+114. y=-- 15. y = cos (~)

x-2

16. y = sinx tanx 17. y = In (x 2 + 1)

18. y = ex2 -, 19. y = x 2 tan-I x

20. y = x-3ex 21. y = ( csc x )3 I + cos x

22. y = cos- J X - cot- J x 23. cos (xy) + y2 - In x = I

24. y = \ITJ 25. x = I + cos t, Y = I - sin r

26. y = (cos x V. _!!. < x < !!. 2 2

x

27.y= r~dt 28. J2

sin r dr 2x

29. Find d 2y/dx2 if y2 + 2y = sec x.

30. Suppose u and v are differentiable functions of x and that u(O) = 2, u'(O) = -I, v(O) = -3, and v'(O) = 3. Find

:x(~)lx=o' 31. A particle moves along the x-axis with its position at time I in

seconds given by x = t 3 - 6t 2 + 9t, O::S t ::S 5, in meters.

(a) Determine the velocity and acceleration of the particle at time t.

(b) When is the particle at rest?

(c) When is the particle moving to the right? left?

(d) What is the velocity when the acceleration is zero?

In Exercises 32-36, find an equation for (a) the tangent line and (b) the normal line to the curve at the indicated point.

32. y = 2x 3 - 6x2 + 4x - I at x = I

33. y = xcosx at x = 71'/3

2 x y2 (3V3 )

34. '4 + 9 = I at I, -2­

35. x = 2 cos r, y = 3 sin t, at t = 71'/3

36. r(l) = (sec I)i + (tan I)j, at 1=71'/4

37. Sketch the graph of a continuous function f with

-I, x < 3f(3) = 1 and !'(x) = ( 2, x> 3.

38. The graph of the function f over the interval [ - 2, 3] is given. At what domain points does f appear to be

(a) differentiable?

(b) continuous but not differentiable?

(c) neither continuous nor differentiable?

(d) Identify any extreme values and where they occur

y

y =j(x)

Ii' :-. :-. .x -2 2 3

39. A driver handed in a ticket at a toll booth showing that in 1.5 h he had covered 1.11 mi on a toll road with speed limit of 65 mph. The driver was cited for speeding. Why?

574 Cumulmive Review Exercises

40. Assume that 1 is continuous and differentiable on the

interval [ -2.2]. The table gives some values off'.

x f'(xJ x 1'(x)

-2 ~11 025 363 -1.75 -8.38 0.5 4

-15 -6 0.75 4.13 -1.25 -388 I 4

1 2 125 3.63

-0.75 -0.38 15 3 -05 I 1.75 213 -0.25 2.13 2 1

0 3

(a) Estimate where 1 is increasing, decreasing, and has local

extrema.

(b) Find a quadratic regression equation for the data in the

table and superimpose its graph on a scatter plot of the data.

(e) Use the equation in (b) to find a formula for 1 that

satisfies 1(0) = I.

41. Find the function 1 with 1'(x) = 2x - 3 + sin x whose

graph passes through the point prO, - 2).

42. Suppose l(x) = x 2\/4- xl. Find the intervals on which

the graph of1 is (a) increasing, (b) decreasing, (e) concave up, (d) concave down. Then find any (e) local extreme

values and where they occur, and (I') any inflection points.

43. A function 1 is continuous on its domain [ - 1,3], 1(- I) = J, 1(3) = -2, and l' and r have the following properties.

x -I <x< 1 x= I 1<x<2 x=2

l' + 0 -does not

exist

1" - -I -does not

exist

2<x<3

-

+I

(a) Find where all absolute extrema of f occur.

(b) Find where the points of inflection off occur.

(e) Sketch a possible graph of f 44. A rectangle with base on the x-axis is to be inscribed under

the upper half of the ellipse

x 2 1'2 -+~= 16 4

What are the dimensions of the rectangle with largest area,

and what is the largest area?

45. Find the linearization of l(x) = sec x at x = 7T 14.

46. The edge of a cube is measured as 8 cm with an error of 1%. The cube's volume is to be calculated from this measurement.

Estimate the percentage error in the volume calculation.

47. A dinghy is pulled toward a dock by a rope from the bow

through a ring on the dock 5 ft above the bow as shown in the figure. The rope is hauled in at the rate of 1.5 ft/sec.

(a) How fast is the boat approaching the dock when 8 ft of

rope are out?

(b) At what rate is the angle echanging at that moment?

48. Coffee is draining from a conical filter into a cylindrical coffeepot at the rate of 9 inJ/min as suggested in the figure.

(a) How fast is the level in the pot rising when the coffee in

the cone is 5 in. deep?

(b) How fast is the level in the cone falling at that moment')

49. The table below shows the velocity of a model train engine

moving along a track for to sec. Estimate the distance

trilveled by the engine, using 10 subintervals of length 1 with (a) LRAM and (b) RRAM.

Time Velocity Time Velocity (sec) (in./sec) (sec) (in./sec)

0 0 6 28.8 I 1.8 7 29.4 2 6.4 8 25.6 3 126 9 16.2 4 19.2 10 0 5 25.0

Cumulalive Review Exercises 575

In Exercises 50-61, evaluate the integral analytically. 2

f50. f21xl dx 51. L2 ~X2 dx

Tr/4

252. f (X2+~) dx 53. 0 sec x dx

2 + v:;- 2e dx

54. J ~ dx 55. e x(lnx)2J4

J 56. f [(3 - 21)i + (J/t)jJ dl

J ds 57. JeX cot2 (eX + I) dx 58. S2 + 4

59. J sin (x - 3) dx 60. Je-'\ cos 2x dx cos 3 (x - 3)

x + 2 61. 2 6 dxJx - 5x­

62. A rectangular swimming pool is 25 ft wide and 40 ft long. The table below shows the depth h(x) of the water at 5- ft intervals from one end of the pool to the other. Estimate the volume of the water in the pool using the Trapezoidal Rule with 11 = 8.

Position (ft) Depth (ft) Position (ft) Depth (ft) x h(x) x h(x)

0 3 25 10.7

5 8.3 30 99 10 9.9 35 8.3

15 107 40 3

20 11

In Exercises 63 and 64, solve the initial value problem.

63. ~ = (I + 1)-2 + e- 2" y(O) = 2

64. ~;~ = sin 2e - cos e, Y(7T/2) = Y'(7T/2) = 0

65. Evaluate Jx 2 sin xdx. Support your answer by super­imposing the graph of one of the antiderivatives on a slope field of the integrand.

66. Evaluate Jxe X fix. Confirm your answer by differentiation.

67. A colony of bacteria is grown under ideal conditions in a laboratory so that the population increases exponentially with time. At the end of 2 h there are 6.000 bacteria. At the end of 5 h there are 10,000 bacteria.

(a) Find a formula for the number of bacteria present at any time I.

(b) How many bacteria were present initially?

68. The temperature of an ingot of silver is 50°C above room temperature right now. Fifteen minutes ago, it was 65°C above room temperature.

(a) How far above room temperature wiJl the silver be 2 hours from now?

(b) When will the silver be 5°C above room temperature?

In Exercises 69 and 70, solve the differential equation.

dydy (_y ) 70. - = (y - 4)(x + 3)69. - = 0.08y I - 500dx dx

71. Use Euler's method to solve the initial value problem

y' = y + cosx, y(O) = 0,

on the interval 0 'S X 'S I with dx = 0.1.

In Exercises 72- 75, find the area of the region enclosed by the curves.

72. y = sin 2x, y = 0, x = -7T, X = 7T

x 2 273. y = 5 - , Y = x - 3

74. x = y2 - 3, y = x - 2

75. r = 3([ + cos e)

In Exercises 76 and 77, find the volume of the solid generated by revolving the region bounded by the curves about the indi­cated axis.

76. y = x 3/2, y = 0, x = -J, x = 1; x-axis

77. y = 4x - x 2, Y = 0; y-axis

78. Find the average value of ~ on the interval [0, 7T].

In Exercises 79-81, find the length of the curve.

79. y = tan x. -7T/4 'S x:5 7T/4

80. x = sin I, y = 1 + cos I, -7T/2'S 1 :5 7T/2

81. r = e, 0:5 e :5 7T

In Exercises 82-84, find the area of the surface generated by revolving the curve about the indicated axis.

x1282. y = e- , 0:5 X 'S 2; x-axis

83. x = sin I, y = 1 + cos I, 0:5 1 'S 7T/2; y-axis

84. r = e. 7T /2 'S e :5 7T; x-axis

85. A solid lies between planes perpendicular to the x-axis at x = 0 and x = I. The cross sections of the solid perpen­dicular to the x-axis between these planes are circular disks with diameters running from the parabola y = x 2 to the parabola y = v;.. Find the volume of the solid.

86, Find the volume of the solid generated by revolving about the x-axis the region bounded by y = 2 tariX', y = 0, x = -7T/4, and x = 7T/4. (The region lies in the first and third quadrants and resembles a bow tie.)

576 Cumulative Review Exercises

87. A force of 200 N will stretch a garage door spring 0.8 m beyond its unstressed length.

(a) How far will a 300-N force stretch the spring from its unstressed length?

(b) In (a), how much work was done in stretching the spring that far?

88. A right circular conical tank, point down, with top radius 5 ft and height 10ft is filled with a liquid whose weight­density is 60 Ib/ft 3

(a) To the nearest foot-pound, how much work will it take to pump the liquid to a point 2 ft above the tank')

(ll) If the pump is driven by a motor rated at 275 ft .Ib/sec O/2-hp), about how long wlll it take it to empty the tank')

89. You plan to store mercury (weight density 849 Ib/ft 3) in a vertical right circular cylindrical tank of inside radius I ft whose interior side wall can withstand a total fluid force of 40,000 lb. About how many cubic feet of mercury can you store al anyone time')

90. Does f(x) = In x grow faster than, at the same rate as. or slower than g(x) = ~ as X-'7oo')

In Exercises 91-96, determine whether the integral converges or diverges.

91. 1'''' 92. L dxell - ­

3 12 - 4 ln x

1 4r dr

93. L~ e-I'I dx 94.1 () I - r2

95. fO ~ 96, f dx o 1 - x o~

97. Find a power series to represent

I

1+ 2x

and identify its interval of convergence.

98. (a) Find a power series for

F(x) = lXcos (1 2) ell.

(b) What is the interval of convergence of the senes') Explain.

99. Find the Maclaurin series generated by In (2 + 2x). What is its interval of convergence')

100. Find the Taylor series generated by sin x at x = 27f.

101. Find a polynomial that you know will approximate e-"

throughout the interval [0, 1] with an error of magnitude Jess than 10-3 Explain.

102. Find the Taylor series generated by f(x) = ~ at x = 0 and identify its radius of convergence.

In Exercises 103-106, determine whether the series converges or diverges.

= 2 103. f ;" 104. L V;;

11=0 11=1 n

105. f (_I)" 106. f~ 11 I n.

11=0 n~O

In Exercises 107 and 108, (a) find the radius and interval of convergence. For what values of x is the convergence (b) absolute? (c) conditonal?

2)"107. f (-I)"(x 11

1/=1

109. Find the unit vector in the direction of (2. -3).

110. Find the component form of the unit vector that makes an angle of 7f/3 with the positive x-axis.

111. Find the unit vectors (four vectors in all) that are tangent and normal to the curve x = 4 sin I, y = 3 cos I, at 1= 37f/4.

112. The position of a particle in the plane is given by

r(l) = (J - sin I)i + (I - cos I)j.

(a) Find the velocity and acceleration of the particle.

(b) Find the distance the particle travels along the path from I = 7f12 to 1= 37f12.

113. A golf ball leaves the ground at a 45° angle at a speed of 100 ft/sec Will it clear the top of a 35-ft tree 130ft away? Explain.

114. Replace the polar equation r cos () - r sin () = 2 by an equivalent Cartesian equation. Then identify the graph.

115. Graph the polar curve r = J + 2 sin e. What is the shortest length a e-interval can have and still produce the graph?

116. Find equations for the horizontal and vertical tangents to the curve r = I - cos (), O::s: () :oS 27f.