Calculus 120 Exam Review

Find the equation of the tangent line.

1) y = 7x + 1

at x = 3

Use the definition f'(a) = limh->0

f(a + h) - f(a)h

to find the derivative of the given function at the given value of a.

2) f(x) = x3 + 4, a = 8

Find y .3) y = 6 sin(2x + 12)

Suppose that the functions f and g and their derivatives with respect to x have the following values at the given values ofx. Find the derivative with respect to x of the given combination at the given value of x.

4)x f(x) g(x) f (x) g (x)3 1 9 6 34 -3 3 2 -5

f(g(x)) at x = 4

5)x f(x) g(x) f (x) g (x)3 1 16 8 34 -3 3 2 -6

f(x) + g(x) at x = 3

Solve the problem.6) The position of a particle moving along a coordinate line is s = 5 + 4t, with s in meters and t in seconds. Find

the particle's velocity at t = 1 sec.

Use implicit differentiation to find dy/dx and d2y/dx2.7) x2 + y2 = 8

Use l'Hopital's Rule to evaluate the limit.

8) limx 0

cos 9x - 1x2

9) limt 0

cos 7t - 1et - t - 1

10) lim/6

1 + cos 61 - sin 3

1

^Use l'Hopital's rule to find the limit.

11) limx 0+

ln (x3 + 10x)ln 3x

12) limt 1

t - 1ln t - tan t

Use logarithmic differentiation to find dy/dx.13) y = (cos x)x

Find the local extrema.14) y = x4 - 18x2 + 9

15) h(x) = 3xx2 + 1

Find the intervals on which the function is increasing and the intervals on which the function is decreasing.16) f(x) = 48x - x3

17) f(x) = x 9 - x

Use the First Derivative Test to determine the local extrema of the function, and identify any absolute extrema.18) y = xe5x

Find all points of inflection of the function.

19) y = 12

x4 - 2x3 + 12

20) y = xe17x

2

Sketch a graph of a single function that has these properties.21) a) Continuous and differentiable for all real numbers

b) f (x) > 0 on (-3 , -1) and ( 2 , ) c) f (x) < 0 on (- , -3) and ( -1 , 2) d) f (x) > 0 on (- , -2) and ( 1 , ) e) f (x) < 0 on (-2 , 1) f) f (-3) = f (-1) = f (2) = 0g) f (x) = 0 at (-2 , 0) and (1, 1)

22) a) Continuous for all real numbers b) Differentiable everywhere except x = 0c) f (x) < 0 on (- , 0) d) f (x) > 0 on ( 0 , ) e) f (x) < 0 on (- , 0) and (0, ) f) f(-2) = f (2) = 5g) y-intercept and x-intercept at (0,0)

Solve the problem.23) A rectangular sheet of perimeter 39 cm and dimensions x cm by y cm is to be rolled into a cylinder as shown in

part (a) of the figure. What values of x and y give the largest volume?

3

24) From a thin piece of cardboard 30 in. by 30 in., square corners are cut out so that the sides can be folded up tomake a box. What dimensions will yield a box of maximum volume? What is the maximum volume? Roundto the nearest tenth, if necessary.

Find the linearization L(x) of f(x) at x = a.25) f(x) = 8x + 9, a = 0

Use the linearization (1 + x)k 1 + kx to approximate the value. Give your answer in the form indicated.26) (1.0004)50

Give your answer as a fraction.

27)3

1.003Give your answer as a decimal.

Use the linear approximation (1 + x)k 1 + kx, as specified.28) Find an approximation for the function f(x) = (1 - x)4 for values of x near zero.

Approximate the root by using a linearization centered at an appropriate nearby number.29) 99

Solve.30) Find dy given y = x 7x + 2

31) Given y = esin 3x, find dy and evaluate dy for x = 2 , and dx = 0.1.

32) Given y = 6 sec 1 - x6

, find dy and evaluate dy for x = 2 and dx = 0.1.

(Round to five decimal places when appropriate.)

33) Given 5y = x2 - 3xy, find dy and evaluate dy for x = 2 and dx = -0.1.(Round to four decimal places when appropriate.)

Solve the problem.34) Water is falling on a surface, wetting a circular area that is expanding at a rate of 5 mm2/s. How fast is the

radius of the wetted area expanding when the radius is 145 mm? (Round approximations to four decimalplaces.)

35) The radius of a right circular cylinder is increasing at the rate of 7 in./s, while the height is decreasing at the rateof 3 in./s. At what rate is the volume of the cylinder changing when the radius is 17 in. and the height is 5 in.?

36) A ladder is slipping down a vertical wall. If the ladder is 15 ft long and the top of it is slipping at the constantrate of 2 ft/s, how fast is the bottom of the ladder moving along the ground when the bottom is 12 ft from thewall?

4

Find dy/dx.

37)x

116t9 dt

Find the general solution to the exact differential equation.

38) dydx

= csc2x - 36x5

39) dydt

= cos t - e-t - 35t6

40) dydu

= u5 -1

u5

Solve the initial value problem explicitly.

41) dydx

= sin (4x + ), y = 5 when x = 0

42) dvdt

= 4 sec t tan t + 9t - 9et and v = 7 when t = 0

43) dydx

= 6x2 - 4x + 18; y = 25 when x = 1

44) y =y

9 + x and y = 18 when x = 0

45) y =9x2

y and y = 1 when x = 0

Use Riemann Sums to evaluate the definite integral.

46)2

-1x2 + 2x - 4 dx

47)5

3(x2-4x) dx

48)4

0x3 - x dx

5

Express the limit as a definite integral.

49) limnn

k = 1c 7

k xk, [-1, 5]

50) limnn

k = 1(3 c 2

k - 11ck + 19) xk, [-5, 4]

51) limnn

k = 1

4

11 - 9 c 2k

xk, [3, 5]

Solve the problem.

52) Suppose that 5

3f(x) dx = -4. Find

5

5f(x) dx and

3

5f(x) dx .

53) Suppose that f and g are continuous and that 10

6f(x) dx = -6 and

10

6g(x) dx = 9.

Find 10

64f(x) + g(x) dx .

54) Suppose that f is continuous and that 4

-4f(z) dz = 0 and

7

-4f(z) dz = 3. Find

4

7f(x) dx .

Evaluate the integral.

55)b

a7x dx , 0 < a < b

56)4

-416 - x2 dx

57)-1

-23x-4 dx

58)/2

- /2(10 + cos x) dx

6

Evaluate the integral.

59) (3 x + 6) dx

60) (13x-7 + csc2 x) dx

61) x-4 +1

3 x dx

62) x5x2 - 6

dx

63) dxx2 + 25

64) sin t(3 + cos t)3

dt

65) 6x - 24x2 - 4x - 32

dx

66) x + 9x2 + 2x

dx

Use the given trig identity to set up a u-substitution and then evaluate the indefinite integral.

67) tan23x dx, tan23x = sec23x - 1

68) dxsin28x

, csc 8x = 1sin 8x

69) cos38x dx , cos28x = 1 - sin28x

Solve the problem.70) A car accelerates from rest at the rate of 1.2 + 3 t mph per second for 16 seconds. What is its velocity after 16

seconds?

71) A car accelerates from rest at the rate of 1.3 + 2 t mph per second for 4 seconds. How far does the car travel inthose 4 seconds?

7

72) In town A , the birth rate is given by b'(t) = 50e0.30t (births per year),

where t is the number of years since 1990. In town B , the birth rate is given by B'(t) = 89e0.40t (births per year),

where t is the number of years since 1990. How many more births are there in town B than in town A duringthe 1990s (from t = 0 to t = 10)? Round your answer to the nearest whole number.

73) Population density measures the number of people per square mile inhabiting a given living area. Thepopulation density of a certain city decreases as you move away from the city center and can be approximatedby the function 8100(3.9 - r) at a distance r miles from the city center. The radius of the city is 3.9 miles. Set upan integral that can be used to estimate the total population of the city.

74) The rate at which your home consumes electricity is measured in kilowatts. If your home consumes electricity atthe rate of 1 kilowatt for 1 hour, you will be charged for 1 "kilowatt-hour" of electricity. Suppose that theaverage consumption rate for a certain home is modeled by the function C(t) = 4.3 - 2.1sin( t/12), where C(t) ismeasured in kilowatts and t is the number of hours past midnight. Find the average daily consumption for thishome, measured in kilowatt-hours.

75) A force of 4 N will stretch a rubber band 5 cm. Assuming Hooke's law applies, how much work is done on therubber band by a 12 N force?

Find the work done by the force of F(x) newtons along the x-axis from x = a meters to x = b meters.

76) F(x) = x 4 - x2, a = 0, b = 2

Find the area of the shaded region.77) f(x) = -x3 + x2 + 16x, g(x) = 4x

8

78) y = 7 - 5x y = 2x2

]

Find the area enclosed by the given curves.

79) Find the area of the region in the first quadrant bounded on the left by the y-axis, below by the line y = 13

x,

above left by y = x + 4, and above right by y = - x2 + 10.

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis.80) y = x, y = 0, x = 0, x = 6

81) y = 1x

, y = 0, x = 1, x = 5

Find the volume of the solid generated by revolving the region about the given line.

82) The region in the first quadrant bounded above by the line y = 3, below by the line y = 3x2

, and on the left by the

y-axis, about the line y = 3

83) The region in the first quadrant bounded above by the line y = 3, below by the curve y = 3x, and on the left bythe y-axis, about the line x = -1

Find the volume of the solid generated by revolving the region about the y-axis.

84) The region enclosed by x = y24

, x = 0, y = - 4, y = 4

85) The region enclosed by x = 2tan y5

, x = 0, y = -54

9

Answer KeyTestname: EXAM REVIEW

1) y = -716

x + 4916

2) 1923) - 24 sin(2x + 12)4) -30

5) 112 17

6) 23

m/sec

7) dydx

= -xy

; d2ydx2

= -x2 + y2

y3

8) -812

9) -4910) 411) 1

12) 11 -

13) (cos x)x (ln cos x - x tan x)14) Local minima at (3, -72), (-3, -72); local maximum at (0, 9)

15) Local minimum at (-1, - 32

); local maximum at (1, 32

)

16) Increasing on (-4, 4), decreasing on (- , -4) and (4, )17) Increasing on , 6 , decreasing on 6, 9

18) Absolute minimum at -15

, - 15e

19) (0, 12) and (2, 4)

20) -217

, - 217e2

21)

22)

10

Answer KeyTestname: EXAM REVIEW

23) x = 13 cm; y = 132

cm

24) 20 in. × 20 in. × 5 in.; 2000 in.3

25) L(x) = 43

x + 3

26) 5150

27) 1.00128) f(x) 1 - 4x29) 9.9500

30) 21x + 42 7x + 2

dx

31) (3cos 3x)esin 3xdx; 0.3

32) -sec 1 - x6

tan 1 - x6

dx; -0.10012

33) dy = 2x - 3y5 + 3x

dx; -0.0264

34) 0.0055 mm/s35) 323 in.3/s36) 1.5 ft/s37) 8x4

38) y = -cot x - 6x6 + C39) y = sin t + e-t - 5t7 + C

40) y = u66

+1

4u4+ C

41) y = -14

cos (4x + ) + 194

42) v = 4 sec t + 92

t2 - 9et + 12 for - /2 < t < /2

43) y = 2x3 - 2x2 + 18x + 744) y = 2x + 18

45) y = 9x3 + 22

2/3

46) -6

47) 23

48) 56

49)5

-1x7 dx

50)4

-5(3x2 - 11x + 19) dx

11

Answer KeyTestname: EXAM REVIEW

51)5

3

411 - 9x2

dx

52) 0; 453) -1554) -3

55) 72

(b2 - a2)

56) 8

57) 78

58) 2 + 1059) 2x3/2 + 6x + C

60) -136

x-6 - cot x + C

61) -1

3x3+

23

x1/2 + C

62) 110

ln 5x2 - 6 + C

63) 15

tan-1 x5

+ C

64) 12(3 + cos t)2

+ C

65) ln (x - 8)2(x + 4)4 + C

66) 12

ln x9

(x + 2)7+ C

67) 13

tan 3x - x + C

68) - 18

cot 8x + C

69) 18

sin 8x - 124

sin38x + C

70) 147.2 mph71) 0.008 mi72) 8745

73)3.9

08100(3.9 - r)(2 r) dr

74) 103.2 kilowatt-hours75) 0.9 J76) 2.667 J

77) 93712

78) 1615

12

Answer KeyTestname: EXAM REVIEW

79) 736

80) 18

81) 45

82) 6

83) 575

84) 1285

85) 20 - 5 2

13