MA16010 Exam 2 Practice Questions Nametrolling/16010_E2_Practice.pdfMA16010 Exam 2 Practice Questions A spherical balloon is inﬂated with gas at a rate of 5 cubic centimeters per

MA16010 Exam 2 Practice Questions

Name:

If h(t) = sin(3t) + cos(3t), find h(3)(t).

1.A© 27 sin(3t) + 27 cos(3t)

B© −27 sin(3t)− 27 cos(3t)

C© 27 sin(3t)− 27 cos(3t)

D© −27 sin(3t) + 27 cos(3t)

E© sin(3t)− cos(3t)

F© sin(3t) + cos(3t)

A toy rocket is launched from a platform on earth and flies straight up into the air.Its height during the first 10 seconds after launching is given by: s(t) = t3 + 3t2 + 4t + 100, where s is measured incentimeters, and t is in seconds.Find the velocity when the acceleration is 18 cm/s2.

2.A© 44 cm/s

B© 2 cm/s

C© 28 cm/s

D© 16 cm/s

E© 32 cm/s

F© 13 cm/s


Finddy

dxby implicit differentiation.

ln(xy) + 2x = ey

3.A© dydx

=−2− yx− ey

B© dydx

=−2y

1− yey

C© dydx

=−2xy − yx− xyey

D© dydx

=1 + 2xy

xyey

E© dydx

=−xy − y

2x− xyey

F© dydx

= yey − yx− 2y

Find the critical numbers of y = x2ex.

4.A© −2 and 0

B© 0 and 2

C© 1 and 2

D© 0 and 1

E© −2 and 2

F© -2 and 1


Given f(x) =2(3−x2)√

3x2 + 1. Find f ′(1).

5.A© −7

2

B© −1

2

C© −13

6

D© −3

2

E© −9

4

F© −3

4

Find the largest open interval where g(t) is increasing.

g(t) = −1

3t3 +

3

2t2

6.A© (−∞, 0)

B© (0,∞)

C© (−∞, 3)

D© (0, 3)

E© (3,∞)

F© (−∞, 0) and (3,∞)


A spherical balloon is inflated with gas at a rate of 5 cubic centimeters per minute. How fast is the radius of theballoon changing at the instant when the radius is 4 centimeters?

The volume V of a sphere with a radius r is V =4

3πr3.

7.A© 5π

64centimeters per minute

B© 5

4πcentimeters per minute

C© 5


D© 5


E© 256π

3centimeters per minute

F© 25


Find f ′(2).

f(t) =2t− 1

(2t+ 1)2

8.A© 22125

B© − 2125

C© − 110

D© 2125

E© − 225

F© 4124

If y = ( 2x−12x+1 )3, then dy

dx=

9.A© 12(2x−1)2

(2x+1)3

B© 48(2x+1)4

C© 12(2x−1)2

(2x+1)4

D© 3( 2x−12x+1 )2

E© 24x−12(2x+1)3

F© 6(2x−1)2

(2x+1)3


Given f(x) = e5x ln(7x+ e). Find f ′(0).

10.A© 35

e

B© 1 +7

e

C© 1

e

D© 5 +7

e

E© 5

e

F© 1 +1

e

The price of a commodity is given by p(t) = (t2 + 2t)2 + 100000, where p(t) is the price in dollars and t is years after2000. At what rate is the price changing in the year of 2010?

11.A© $5280/year

B© $900/year

C© $2640/year

D© $4800/year

E© $2400/year

F© $1680/year

Find g′(x) if g(x) = tan2(3x2 + 2).

12.A© 6x tan(3x2 + 2) sec2(3x2 + 2)

B© 12x tan(3x2 + 2) sec2(3x2 + 2)

C© 12x sec2(3x2 + 2)

D© 2 sec2(6x)

E© 2 tan(6x)

F© 12x tan(3x2 + 2)


Use implicit differentiation to find dy

dxif x2 + y2 = 2xy + 5.

13.A© 2x− 2y − 5

2x− 2y

B© x

x−y

C© 2y − 2x+ 5

2y − 2x

D© 1

E© 0

F© x

1−y

All edges of a cube are expanding at a rate of 2 centimeters per second. How fast is the surface area changing wheneach edge is 3 centimeters?

14.A© 36 cm2/sec

B© 12 cm2/sec

C© 72 cm2/sec

D© 48 cm2/sec

E© 54 cm2/sec

F© 46 cm2/sec

Water flows into a right cylindrical shaped swimming pool with a circular base at a rate of 4 m3/min. The radius ofthe base is 3 m. How fast is the water level rising inside the swimming pool? The volume of a right cylinder with acircular base is V = πr2h, where r is the radius of the base and h is the height of the cylinder.

15.A© 29π m/min

B© 49π m/min

C© 23π m/min

D© 43π m/min

E© 316π m/min

F© 38π m/min


A 10-ft ladder, whose base is sitting on level ground, is leaning at an angle against a vertical wall when its base startsto slide away from the vertical wall. When the base of the ladder is 6 ft away from the bottom of the vertical wall,the base is sliding away at a rate of 4 ft/sec. At what rate is the vertical distance from the top of the ladder to theground changing at this moment?

16.A© 14 ft/sec

B© 8 ft/sec

C© 4 ft/sec

D© −3 ft/sec

E© − 13 ft/sec

F© −34 ft/sec

Given f(x) = sin3(2x), find f ′( π12 ).

17.A© − 3√

38

B© 12

C© −√

34

D© 3√

34

E© 32

F© 94

Given f(x) = ln 3

√

3 + 3x

3− x , find f ′(1).

18.A© 13

B© 23

C© 12

D© 16

E© 14

F© 18


Use implicit differentiation to find the equation of the tangent line to the graph at (−2, 2).

x2 + xy = 4− y2

19.A© y = 2

B© y = −x+ 4

C© y = x+ 2

D© y = x+ 4

E© y = −x+ 2

F© y = −x

Find dy

dxby implicit differentiation.

exy = 8x− 8y

20.A© dy

dx=

8− xexy8 + yexy

B© dy

dx=

8

8 + xexy

C© dy

dx=

8 + yexy

8− xexy

D© dy

dx=

8

8− xexy

E© dy

dx=

8− yexy8 + xexy

F© dy

dx=

8 + xexy

8− yexy

The position of an object moving on a straight line is given by s(t) = 48 − 3t − 2t2 − 6t3, where t is in minutes ands(t) is in meters. What is the acceleration when t = 3 minutes?

21.A© -177 m/min2

B© -114 m/min2

C© -110 m/min2

D© -112 m/min2

E© -108 m/min2

F© -76 m/min2


The sides of an equilateral triangle are expanding at a rate of 2 cm per minute. Find the rate of change of the area

when the length of each side is 3 cm. Use the fact that the area of an equilateral triangle is A =√

34 x

2 , where x is thelength of a side.

22.A© 9√

34 cm2/min

B©√

3 cm2/min

C© 3√

32 cm2/min

D© 3√

34 cm2/min

E© 3√

3 cm2/min

F© 9√

32 cm2/min

Given f(x) =x3

3+ x+

√x3. Find f ′′(4).

23.A© 496

B© 498

C© 192

D© 678

E© 354

F© 263

Given y = x ln x, find y′′(e).

24.A© 0

B© 1

e

C© 2

D© 1

e+ 1

E© e+ 1

F© e


Find the relative extrema of g(x) =x

x2 + 9.

25.A© Relative maximum at x =√

3; Relative minimum at x = −√

3

B© Relative maximum at x = 3; Relative minimum at x = −3

C© Relative maximum at x = 3; Relative minimum at x = −√

3

D© Relative maximum at x = −√

3; Relative minimum at x =√

3

E© Relative maximum at x = −3; Relative minimum at x =√

3

F© Relative maximum at x = −3; Relative minimum at x = 3

Find the largest open interval(s) on whichf(x) = (3x− 4)(x+ 2)

is increasing.

26.A© (− 13 ,∞)

B© (−∞, 3)

C© (−∞, 3) and (3,∞)

D© (−∞,− 13 )

E© (−∞,−2) and (43 ,∞)

F© (−2, 43 )

If x and y are both functions of t andx+ y2 = 4ex,

find dydt

when dxdt

= 2, x = 0, and y = −2.

27.A© 1

B© − 32

C© 0

D© −1

E© − 12

F© 3


Find g′(1).

g(x) =

(

x2

x+ 2

)3

28.A© 13

B© 527

C© 53

D© 2527

E© 59

F© 253

The position of a particle on a straight line t seconds after it starts moving is s(t) = 2t3 − 3t2 + 6t+ 1 feet. Find theacceleration of the particle when its velocity is 78 ft/sec.

29.A© 46 ft/sec2

B© 30 ft/sec2

C© 84 ft/sec2

D© 42 ft/sec2

E© 105 ft/sec2

F© 258 ft/sec2

Find the relative maximum of f(x) = 2x3 − 6x.

30.A© (−1, 0)

B© (1,−4)

C© (1, 4)

D© (1, 0)

E© (0, 0)

F© (−1, 4)


Given thaty2x− x2 = y ln(x) + 3,

use implicit differentiation to find dydx at (1,−2).

31.A© − 25

B© 5

C© 1

D© −2

E© 2

F© −1

Find f ′(4) if f(x) = (x2 + 3)√x2 − 7.

32.A© 1103

B© 1636

C© 323

D© 1483

E© 43

F© 1423

Find the x value at which the function f(x) = x3 − 9x2 − 120x+ 3 has a relative minimum.

33.A© x = −4

B© x = −3

C© x = −10

D© x = 4

E© x = 10

F© x = 3


Which of the following is a critical number of

y =1

3sin(3x)− x

2?

34.A© π18

B© π3

C© π9

D© π6

E© 0

F© π12

An observer stands 400 feet away from the point where a hot air balloon is launched. If the balloon ascends verticallyat a (constant) rate of 30 feet per second, how fast is the balloon moving away from the observer 10 seconds after itis launched?

35.A© 24 ft/sec

B© 50 ft/sec

C© 37.5 ft/sec

D© 30 ft/sec

E© 18 ft/sec

F© 40 ft/sec

Find the second derivative of f(x) = ln(4x) + ex2

.

36.A© f ′′(x) = − 1x2 + ex

2

(4x2 + 2)

B© f ′′(x) = − 1x2 + 4x2ex

2

C© f ′′(x) = − 1x2 + 2ex

2

D© f ′′(x) = − 14x2 + ex

2

(4x2 + 2)

E© f ′′(x) = − 14x2 + 2ex

2

F© f ′′(x) = − 14x2 + 4x2ex

2


An airplane flies at an altitude of y = 2 miles straight towards a point directly over an observer. The speed of the

plane is 500 miles per hour. Find the rate at which the observer’s angle of elevation is changing when the angle isπ

3.

37.A© 125√

3

2radian per hour

B© 75

4radian per hour

C© 225

8radian per hour

D© 375

2radian per hour

E© 125

2radian per hour

F© 50√

3 radian per hour

If f(t) = e2t − sin(2t), find the 3rd derivative, f (3)(t).

38.A© 8e2t + 8 sin(2t)

B© 2e2t + 2 cos(2t)

C© 8e2t + 8 cos(2t)

D© e2t − cos(2t)

E© 8e2t − 8 cos(2t)

F© e2t + sin(2t)

The radius of a sphere changes at a rate of 2 inches per second. What is the rate of change of the surface area of thesphere, in in2/sec, when the radius is 3 inches? The surface area of the sphere is given by the formula

S = 4πr2

39.A© 36π

B© 108π

C© 16π

D© 72π

E© 48π

F© 24π


Find the derivative of y =√r2 − 10x2, where r is a constant.

40.A© r − 10x√r2 − 10x2

B© −10x√r2 − 10x2

C© r2 − 10x2

2√r2 − 10x2

D© −10x2

√2r − 20x

E© 10√r2 − 10x2

F© r2 − 10x2

√2r − 20x

Find f ′(x) given that f(x) = tan(cos(3x)).

41.A© sec2(cos(3x)) + 3 cos(3x)

B© −3 sin(3x) sec2(cos(3x))

C© −3 cot(3x)

D© 3 sin(cos(3x))

E© cos(3x)− 3 sec2(sin(3x))

F© − sec2(3 sin(3x))

Find the largest open interval where h(t) is decreasing.

h(t) = t3 − 15

2t2

42.A© (−∞, 5)

B© (15

2,∞)

C© (5,∞)

D© (0, 5)

E© (0,15

2)

F© (−∞, 0)


Use implicit differentiation to finddy

dxgiven y3 + xy = x4.

43.A© 4x3 − 3y2 − yx

B© 4x3 − y3y2 + x

C© 4x3 − 3y2

x

D© 4x3

3y2 − x

E© 4x3 − y − x3y2

F© 4x3 − y3y

A diver in midair has vertical position given by

h(t) = −16t2 + 5t+ 25

where h(t) is the diver’s height above the water, in feet, t seconds after beginning the dive. What is the diver’sacceleration, in ft/sec2, t seconds after the dive begins?

44.A© −28t

B© −28t+ 25

C© 25

D© −32t+ 5

E© −32

F© −16


Given f(x) = x3 − 3x, find the relative extrema.

45.A© relative maximum: (3,−1); relative minimum: (−1, 3)

B© relative maximum: (−1, 2); relative minimum: (1,−2)

C© relative maximum: (−2, 1); relative minimum: (2,−1)

D© relative maximum: (−1, 1); relative minimum: (−2, 2)

E© relative maximum: (2,−1); relative minimum: (−2, 1)

F© relative maximum: (1,−2); relative minimum: (−1, 2)

Find the critical number(s) of y = 3x2e2x.

46.A© 0 and 1

B© −1 only

C© −1 and 1

D© 1 only

E© 0 only

F© 0 and −1

What is the slope of the tangent line to the graph of y = ln(2x3 + 5x) at x = 1?

47.A© 77

B© 1

7

C© 1

2

D© 11

7

E© ln 7

F© 7

11


Given y ln(x) = x ln(y), use implicit differentiation to finddy

dxat the point of (2, 4). (Round to 4 decimal places.)

48.A© 0.8448

B© 2.0422

C© −3.1774

D© −5.2755

E© −9.6251

F© −2.7204

A bird sits on the ground eating acorns. A second bird is directly east of the first bird, and is flying straight east ata speed of 35 feet per second at a constant height of 20 feet above the ground. How fast is the distance between thetwo birds increasing when the distance is 25 feet?

49.A© 16 feet per second

B© 17 feet per second

C© 21 feet per second

D© 18 feet per second

E© 19 feet per second

F© 20 feet per second


Find the equation of the tangent line to the graph of y = ex2

at x = 1.

50.A© y = e2x− e2

B© y = ex

C© y = e2x− e2 + e

D© y = 2xex2

(x− 1) + e

E© y = 2xex2

(x− 1) + ex2

F© y = 2ex− e

If f(x) = x5 ln(10x), find the second derivative, f ′′(x).

51.A© 20x4 ln(10x) + 910x

4

B© 5x4 ln(10x) + x4

C© 20x3 ln(10x) + 4x3

D© 20x3 ln(10x) + 9x3

E© 5x4 ln(10x) + 110x

4

F© 20x3 ln(10x) + 5x3

A particle is moving along a straight line with the position function S(t) = 16t3 + 8t2 + 2t, where S(t) is in miles andt is in hours. What is the acceleration of the particle when its velocity is 66 miles/hour?

52.A© 26 miles/hour2

B© 74 miles/hour2

C© 112 miles/hour2

D© 230 miles/hour2

E© 154 miles/hour2

F© 1 mile/hour2


Find the slope of the line tangent to xy2 = x2 + y2 at the point (2,−2).

53.A© −2

B© 2

C© 1

D© −1

E© 0

F© 3

Use implicit differentiation to find dydx

if 2 cos(x) sin(y) = 3.

54.A© dy

dx= cos(x) cos(y)

sin(x) sin(y)

B© dy

dx= sin(x) sin(y)

cos(x) cos(y)

C© dy

dx= cos(xy)

sin(xy)

D© dy

dx= sin(xy)

cos(xy)

E© dy

dx= sin(x) cos(y)

cos(x) sin(y)

F© dy

dx= cos(x) sin(y)

sin(x) cos(y)

Air is being pumped into a spherical balloon at a rate of 5 cm3/min. Determine the rate at which the radius of theballoon is increasing when the DIAMETER of the balloon is 10 cm. The volume V of a sphere with a radius r isV = 4

3πr3.

55.A© 20π cm/min

B© 120π cm/min

C© 180π cm/min

D© 80π cm/min

E© 140π cm/min

F© 340π cm/min


The side of a cube is increasing at a rate of 2 cm/min. Find the rate of change of the volume of the cube when theside is 3 cm.

56.A© 6 cm3/min

B© 3 cm3/min

C© 54 cm3/min

D© 18 cm3/min

E© 27 cm3/min

F© 12 cm3/min

A boat is pulled into a dock by a rope attached to the front of the boat and passing through a pulley on the dock thatis 1 meter higher than the front of the boat. If the boat is pulled at a rate of 1 m/s, how fast is the boat approachingthe dock when it is 8 meters away from the dock?

57.A©√

658 m/s

B©√

638 m/s

C© 8√65

m/s

D© 8√

63 m/s

E© 8√63

m/s

F©√

65 m/s

Which of the following numbers IS a critical number of the function g(x) = 16x

6 + 115 x

5 + 7x4?

58.A© −4

B© 5

C© 7

D© 3

E© −6

F© −2


Find the largest open interval(s) where the function y =x5

5− x

3

3is increasing.

59.A© (−∞,−1) and (0, 1)

B© (−1, 0) and (1,∞)

C© (−1,∞)

D© (−∞, 1)

E© (−∞,−1) and (1,∞)

F© (−1, 1)

Find the x value at which the function f(x) = 3x4 − 4x3 − 12x2 + 3 has a relative maximum.

60.A© x = −2

B© x = 1

C© x = −1

D© x = 2

E© x = 0

F© x = 5

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MA16010 Exam 2 Practice Questions Nametrolling/16010_E2_Practice.pdfMA16010 Exam 2 Practice Questions A spherical balloon is inﬂated with gas at a rate of 5 cubic centimeters per

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