Free Printable Calculus Semester 2 Final Study

1.

If the position function for a particle is s(t) = -t² - t, what is the instantaneous velocity function for the particle?

a)

v(t) = -2

b)

v(t) -2t - 1

c)

v(t) = t³

d)

v(t) = -t

2.

a)

b)

c)

d)

3.

a)

b)

c)

d)

4.

a)

b)

c)

d)

5.

a)

b)

c)

d)

6.

a)

b)

c)

d)

7.

If the position of a particle is represented by s(t) = -t² + 1, what is its position at t = 1?

a)

Position = 0

b)

Position = 1

c)

Position = 2

d)

Position = -1

8.

If the position of a particle is represented by s(t) = -t² + 1, what is its instantaneous velocity at t = 1?

a)

Velocity = 0

b)

Velocity = 1

c)

Velocity = -1

d)

Velocity = -2

9.

The acceleration function is the first derivative of...

a)

position

b)

velocity

c)

calculus

d)

particle motion

10.

The position function x(t)=7t²-18t-7 is given. What is the velocity function?

a)

v(t)=14t-18

b)

v(t)=14t+18

c)

v(t)=18t-14

d)

v(t)=18t+14

11.

The position function x(t)=t³+6t²+16t-18 is given on the interval of 0<x<9. What is the velocity at t=6?

a)

197

b)

190

c)

18

d)

196

12.

A particle has positive velocity if it's position graph is

a)

negative

b)

positive

c)

decreasing

d)

increasing

13.

A particle has negative velocity if it's position graph is

a)

negative

b)

positive

c)

decreasing

d)

increasing

14.

Which of the following can be used to determine when a particle is at rest?

a)

x(t)=0

b)

v(t)=0

c)

a(t)=0

15.

Brandon is starting to clean up after a birthday party. He begins deflating each spherical balloon by puncturing a hole in each. The air leaves the balloon at a constant rate of 2 cm³/sec. How fast is the diameter changing when the diameter is 8 cm?

a)

-1/(16pi) cm/sec

b)

-1/16 cm/sec

c)

-1/(4pi) cm/sec

d)

1/(4pi) cm/sec

16.

Louisa and Karis were each dropped off at the same bus stop. Louisa’s bus drops her off at 3:30 whereas Karis is dropped off ten minutes later. Louisa runs home at a constant rate of 6 mph and Karis runs home at 3 mph. Louisa lives north of the bus stop and Karis lives to the east. How fast is the distance between them changing at 4:00?

a)

6.512 mph

b)

7.115 mph

c)

6.708 mph

d)

6.641 mph

17.

(3 min) The radius r of a sphere is increasing at the uniform rate of 0.3 inches per second. At the instant when the surface area S becomes 100 π square inches, what is the rate of increase, in cubic inches per second, in the volume V? SA = 4 π r² and V = (4/3) π r³

a)

30π

b)

10π

c)

12π

d)

25π

18.

A certain medical procedure requires that a balloon be inserted into the stomach and then inflated. Model the shape of the balloon by a sphere of radius r. If r is increasing at the rate of 0.3 cm/min, how fast is the volume changing when the radius is 4 cm?

a)

15.08 cm³/min

b)

268.08 cm³/min

c)

60.32 cm³/min

d)

6.03 cm³/min

19.

A conical tank is 10 feet across the top and 12 feet deep. If water is flowing into the tank at a rate of 10 cubic feet per minute, find the rate of change of the depth of the water when the water if 8 feet deep.

a)

V=(1/3)πr²h

b)

V=(4/3)πr³

c)

d=√(x²+y²)

d)

S=2lh+2lw+2hw

20.

Find

a)

5/7

b)

Undefined

c)

0

d)

-2/3

21.

a)

0

b)

10

c)

-10

d)

5

22.

a)

0

b)

1

c)

-32/3

d)

32/3

23.

a)

1

b)

-1

c)

0

d)

DNE

24.

a)

0

b)

5/3

c)

e^5x

d)

DNE

25.

The slope of the line tangent to the graph y = cos(x) at the point x = π/6 is –1/2. What is the linear approximation of cos(x) at π/6 ?

a)

L(x) = –1/2(x – π/6) + √(2)/2

b)

L(x) = –1/2(x – π/6) + 1/2

c)

L(x) = –1/2(x – π/6) + √(3)/2

d)

L(x) = –1/2(x – π/6) – 1/2

26.

Which of the following represents the linear approximation to f (x) at the point (c, f(c)) ?

a)

L(x) = f'(c) + f(c)(x - c)

b)

L(x) = f(c) + f'(c)(x - c)

c)

L(x) = f'(c) + f(c)(x + c)

d)

L(x) = f(c) - f'(c)(x - c)

27.

All the following are steps necessary for writing equation of the tangent line to a curve at the point (a, f(a)) except:

a)

Find the value of f (a)

b)

Evaluate the second derivative to obtain f''(a)

c)

Write equation of the tangent line in point slope form:

T (x) – f (a) = f ' (a) (x – a)

d)

Evaluate f ' (a) to obtain the slope of the tangent line

28.

Will a linear approximation to the graph of f (x) = –x²+3x overestimate the actual function values of f ?

Explain why or why not.

a)

No, it will not overestimate actual function values

b)

Yes, it will overestimate actual function values because f is concave upward

c)

Yes, it will overestimate actual function values because due to the concavity of f, any tangent drawn will be above the curve

d)

Cannot be determined

29.

Suppose f(x) = x³ – x.

Use a linear approximation at x = 2 to estimate f(2.5).

a)

10.5

b)

11

c)

11.5

d)

12

30.

Rachel is standing atop a 13 ft ladder. The ladder is leaning against a vertical wall. The ladder starts sliding away from the wall at a rate of 3 ft/sec. How fast is the ladder sliding down the wall when the tip of the ladder is 5 ft high?

a)

3 ft/sec

b)

-7.2 ft/sec

c)

7.2 ft/sec

d)

12

31.

Based on the table, use a Right Riemann sum and 4 sub-intervals to estimate the Area under the curve. (Choose the correct set-up)

a)

5(3)+1(4)+2(5)+1(7)

b)

0(3)+5(4)+6(5)+8(7)

c)

5(4)+1(5)+2(7)+1(6)

d)

5(3)+6(4)+8(5)+9(7)

32.

Based on the table, use a trapezoidal sum of 4 sub-intervals to estimate the area under the curve.

(calculator allowed for arithmetic)

a)

43

b)

40.5

c)

40

d)

45

33.

What does this picture represent?

a)

Left Riemann Sum

b)

Right Riemann Sum

c)

Middle Riemann Sum

d)

Trapezoidal Sum

34.

What does this picture represent?

a)

Left Riemann Sum

b)

Right Riemann Sum

c)

Middle Riemann Sum

d)

Trapezoidal Sum

35.

What does picture represent?

a)

Left Riemann Sum

b)

Right Riemann Sum

c)

Middle Riemann Sum

d)

Trapezoidal Sum

36.

For a function that is strictly decreasing, a right hand Riemann Sum is which of the following:

a)

Overestimate

b)

Underestimate

c)

Exact Solution

d)

Unable to Determine

37.

For a function that is strictly increasing, a right hand Riemann Sum is which of the following:

a)

Overestimate

b)

Underestimate

c)

Unable to Determine

d)

Exact Solution

38.

Based on the table, use a left Riemann sum and 4 sub-intervals to estimate the Area under the curve. (Choose the correct set-up.)

a)

5(3) + 1(4) + 2(5) + 1(7)

b)

5(4) + 1(5) + 2(7) + 1(6)

c)

5(3) + 6(4) + 8(5) + 9(7)

d)

0(3) + 5(4) + 6(5) + 8(7)

39.

Based on the table, use a Right Riemann sum and 4 sub-intervals to estimate the Area under the curve. (Choose the correct set-up.)

a)

5(3) + 1(4) + 2(5) + 1(7)

b)

5(4) + 1(5) + 2(7) + 1(6)

c)

5(3) + 6(4) + 8(5) + 9(7)

d)

0(3) + 5(4) + 6(5) + 8(7)

40.

A Riemann Sum uses rectangles to

a)

approximate the area under a curve. The more rectangles, the better the approximation.

b)

approximate the area under a curve. The less rectangles, the better the approximation.

c)

approximate the area under a curve. The more rectangles, the worse the approximation.

41.

The graph shows the rate of change of water, measured in gallons per day, in a lake over a 10 day period. How much water has the lake gained or lost in the first 6 days?

a)

10 gallons

b)

14 gallons

c)

6 gallons

d)

28 gallons

42.

The graph shows the rate of change of the number of people in a movie theater. Assume no one is in the theater at t = 0 hours. How many people are in the theater after 10 hours?

(a)

43.

$\int_{-4}^0\left(2x+2\right)dx$

a)

-6

b)

-17

c)

-9

d)

-8

44.

$\int_1^5\left(-x^2+6x-10\right)dx$

a)

$\frac{28}{3}$

b)

$\frac{-28}{3}$

c)

28

d)

$-\frac{7}{2}$

45.

\int_{-1}^3\left(-x^3+4x^2-4\right)dx

a)

$\frac{1}{3}$

b)

$\frac{4}{3}$

c)

$-\frac{2}{13}$

d)

$-\frac{2}{3}$

46.

Evaluate in terms of area
$\int_0^2f\left(x\right)dx$

a)

4

b)

-4

c)

4 + 2π

d)

-4 - 2π

47.

Using the areas of each region given
$\int_a^df\left(x\right)=$

a)

6

b)

20

c)

2

d)

24

48.

What is the meaning of the following symbol:

$\int_a^bf\left(x\right)dx$

a)

The area from the curve to the x-axis from x = a to x = b

b)

The rate at which f(x) is changing from x = a to x = b

c)

The area from the curve to the y-axis from x = a to x = b

d)

The volume when the curve is rotated about the x-axis from x = a to x = b

49.

If $\int_{-4}^3f\left(x\right)dx=9,\ \ \int_3^5f\left(x\right)dx=-11,\ \ \&\ \ \int_{-4}^3h\left(x\right)dx=14$ , then evaluate $\int_5^3f\left(x\right)dx$ if possible.

a)

-11

b)

11

c)

9

d)

not enough information to determine

50.

If $\int_{-4}^3f\left(x\right)dx=9,\ \ \int_3^5f\left(x\right)dx=-11,\ \ \&\ \ \int_{-4}^3h\left(x\right)dx=14$ , then evaluate $\int_{-4}^5f\left(x\right)dx$ if possible.

a)

-11

b)

-2

c)

20

d)

not enough information to determine

51.

If $\int_{-4}^3f\left(x\right)dx=9,\ \ \int_3^5f\left(x\right)dx=-11,\ \ \&\ \ \int_{-4}^3h\left(x\right)dx=14$ , then evaluate $\int_{-4}^32h\left(x\right)dx$ if possible.

a)

7

b)

14

c)

28

d)

not enough information to determine

52.

If $\int_{-4}^3f\left(x\right)dx=9,\ \ \int_3^5f\left(x\right)dx=-11,\ \ \&\ \ \int_{-4}^3h\left(x\right)dx=14$ , then evaluate $\int_{-4}^3\left[f\left(x\right)-h\left(x\right)\right]dx$ if possible.

a)

-5

b)

5

c)

23

d)

not enough information to determine

53.

If $\int_{-4}^3f\left(x\right)dx=9,\ \ \int_3^5f\left(x\right)dx=-11,\ \ \&\ \ \int_{-4}^3h\left(x\right)dx=14$ , then evaluate $\int_{-4}^3\left[4h\left(x\right)-3f\left(x\right)\right]dx$ if possible.

a)

-19

b)

19

c)

29

d)

not enough information to determine

54.

If $\int_{-4}^3f\left(x\right)dx=9,\ \ \int_3^5f\left(x\right)dx=-11,\ \ \&\ \ \int_{-4}^3h\left(x\right)dx=14$ , then evaluate $\int_3^4f\left(x\right)dx$ if possible.

a)

-11

b)

-5.5

c)

5.5

d)

not enough information to determine

55.

∫ t⁶ dt

a)

6t⁵

b)

6t⁵ + C

c)

1/7 t⁷ + C

d)

t⁷ + C

56.

∫ 4 dx

a)

0

b)

4t + C

c)

4x + C

d)

2x² + C

57.

a)

-(24/4)x⁴+C

b)

-4x⁶+C

c)

-4x⁵+C

d)

-24x⁶+C

58.

∫(4 - 18x)dx

a)

F(x) = -18

b)

F(x) = 4x - 9x²

c)

F(x) = 4x - 9x²+ C

d)

F(x) = (4 - 18x)² /2 + C

59.

a)

-4x⁶-5x²+C

b)

-24x⁶-10x²+C

c)

-4x⁵-5x+C

d)

-4x⁶-10+C

60.

$\int_{ }^{ }-15x^4\left(-3x^5-1\right)^5dx$

a)

$\frac{1}{5}\left(-3x^5-1\right)^6+C$

b)

$\frac{1}{5}\left(-15x^4\right)^6+C$

c)

$\frac{1}{6}\left(-3x^5-1\right)^6+C$

d)

$\frac{1}{6}\left(-15x^4\right)^6+C$

61.

Identify the u for the following:
$\int_{ }^{ }\frac{16x^3}{\left(-4x^4-1\right)^5}dx$

a)

$u=-16x^3$

b)

$u=-16x$

c)

$u=\left(-4x^4-1\right)^5$

d)

$u=-4x^4-1$

62.

$\int_{ }^{ }\ \frac{\left(5+\ln x\right)^5}{x}dx$

a)

$\frac{\left(5+\ln x\right)^6}{6}+C$

b)

$\frac{x^5}{5}+C$

c)

$\frac{e^{6x}}{6}+C$

d)

$\frac{e^x\left(5+\ln x\right)^6}{6x}$

63.

$\int_{ }^{ }6e^{3x}\cos\left(e^{3x}-5\right)dx$

a)

$-6\sin\left(e^{3x}-5\right)+C$

b)

$6\sin\left(e^{3x}-5\right)+C$

c)

$2\sin\left(e^{3x}-5\right)+C$

d)

$-2\sin\left(e^{3x}-5\right)+C$

64.

$\int^{ }\ \frac{-5\cos\left(-4+\ln4x\right)}{x}dx$

a)

$5\sin\left(-4+\ln4x\right)+C$

b)

$-5\sin\left(-4+\ln4x\right)+C$

c)

$-5\ln\left|\sin\left(-4+\ln4x\right)\right|+C$

d)

$-20\sin\left(-4+\ln4x\right)+C$

65.

7. Solve for "c" given the initial condition.

a)

C=4

b)

C=Pi/4

c)

C=8

d)

C=-4

66.

10.Solve for "c" given the initial condition.

a)

C=3

b)

C=5

c)

C=0

d)

C=-1

67.

Integrate:

a)

x^-2 + C

b)

^-2/_x³ + C

c)

^-1/_x + C

d)

²/_x + C

68.

∫₀²⁰C(n)dn =

a)

1000

b)

250

c)

750

d)

1125

69.

∫sec²x dx

a)

tanx +c

b)

-cotx +c

c)

secx +c

d)

-cscx +c

70.

a)

sin x + C

b)

tan x + C

c)

sin² x + C

d)

-sin x + C

71.

Y is proportional to the difference of x and z

a)

y= x-z

b)

y= k(z-x)

c)

y= k(x-z)

d)

y= z-x

72.

y is proportional to the product of z and the square root of x

a)

$y=zx^2$

b)

$y=kzx^2$

c)

$y=kz\sqrt{x}$

d)

$y=zkx$

73.

∫dx

a)

x+c

b)

2x+c

c)

3x+c

d)

4x+c

74.

Find the mistake if possible:

a)

The Diff EQ is solved correctly

b)

Step 1 is incorrect. The separation of variables wasn't done correctly.

c)

Step 2 is incorrect. They didn't integrate x² correctly.

d)

Step 3 is incorrect. They didn't take the reciprocal of x³/3+C correctly.

75.

Of the following, which is a solution to the differential equation, select all that apply:
$y''-6y'+8y=0$

a)

$y=2\sin\left(4x\right)$

b)

$y=3e^{2x}$

c)

$y=Ce^{4x},\$ where C is a constant

76.

In drawing the slope field for the differential equation , I would place short slope lines of _________ at the points (0,1), (1,-1), and (2,-2)
$\frac{dy}{dx}=2x-3y$

Select all that apply

a)

-3

b)

-1

c)

2

d)

5

e)

10

77.

Which choice below represents this slope field?

a)

dy/dx = 2x

b)

dy/dx = -x

c)

dy/dx = yx

d)

dy/dx = x²

78.

Which equation below represents the slope field?

a)

dy/dx = x - 2

b)

dy/dx = 1/2x + 1

c)

dy/dx =1/2 y - 2

d)

dy/dx = y + 2

79.

Which slope field is represented by dy/dx = x²?

a)

b)

c)

d)

80.

A puppy gains weight, w, at a rate approximately inversely proportional to its age, t, in months.

a)

A

b)

B

c)

C

d)

D

81.

dy/dx = 4x/y. Suppose y(0)=1
The particular solution is

a)

B

b)

C

c)

D

d)

E

82.

Find f(1) given f'(x)

a)

-12

b)

0

c)

14

d)

20

83.

Evaluate:

a)

4

b)

6

c)

8

d)

10

84.

Find the Particular Solution

a)

$y=2+e^{\left(\frac{x^2}{2}+x\right)}$

b)

$y=2e^{\left(\frac{x^2}{2}+x\right)}$

c)

$y=\ln\left|\frac{x^2}{2}+x+1\right|+2$

d)

$y=\ln\left|\frac{x^2}{2}+x+e^2\right|$

85.

Solve the Initial Value Problem

a)

$y=-\frac{1}{x}$

b)

$y=-x^2$

c)

$y=\frac{-1}{x+1}$

d)

$y=\frac{1}{x+1}$

86.

Water flows continuously from a large tank at a rate proportional to the amount of water in the tank, modeled by dy/dt = ky. There was initially 10,000 ft^3 at t=0. After 4 hours there were 8000 ft^3 remaining.
What is the value of k in the differential equation (calculator)

a)

-0.050

b)

-0.056

c)

-.169

d)

-.200

87.

2. Solve the differential equation

𝑑𝑦/𝑑𝑡 =3𝑡²/𝑦 with initial condition

𝑦(2) = 0.

a)

$y=\ln\left|15t\right|$

b)

$y=16t^3$

c)

$y\ =\ \sqrt{2t^3-16}$

d)

$y=2t^3-16$

88.

Determine the value of "c" that satisfies the differential equation $\frac{dy}{dx}=\frac{x+1}{y+2}$ if the curve goes through the point (0, -1).

a)

5/2

b)

-3/2

c)

-1/2

d)

1

89.

Which of the following is the solution to the differential equation $\frac{dy}{dx}=\frac{x^2}{y}$ with the initial condition y(3) = -2?

a)

$y=-2e^{\left(-9+\frac{x^3}{3}\right)}$

b)

$y=\sqrt{\frac{2x^3}{3}}$

c)

$y=\sqrt{\frac{2x^3}{3}-14}$

d)

$y=-\sqrt{\frac{2x^3}{3}-14}$

90.

The following Differential Equation is
$\frac{\text{d}y}{\text{d}x}=\frac{y}{x}$

a)

Separable.

b)

Non Separable

91.

The following Differential Equation is
$\frac{\text{d}y}{\text{d}x}=x^2y+x$

a)

Separable.

b)

Non Separable.

92.

The following Differential Equation is
$x\frac{\text{d}y}{\text{d}x}=yx^2$

a)

Separable.

b)

Non Separable.

93.

The following Differential Equation is
$\frac{\text{d}y}{\text{d}x}=y^2+xy^2$

a)

Separable.

b)

Non Separable.

94.

The following Differential Equation is
$\frac{\text{d}y}{\text{d}x}=\frac{e^y}{e^x}$

a)

Separable.

b)

Non Separable.

95.

Solve the following differential equations:
$\frac{\text{d}y}{\text{d}x}=\frac{x}{y}$

a)

$\ln y=\frac{x^2}{2}+C$

b)

$y=\frac{x^2}{2}+C$

c)

$x^2=y^2+C$

d)

$y^2=x^2+2C$

96.

Solve the following differential equations:
$\frac{\text{d}y}{\text{d}x}=\frac{1}{\cos y}$

a)

$\sin y=1+C$

b)

$\sin y=x+C$

c)

$-\sin y=\frac{x^2}{2}+C$

d)

$\sin y=0+C$

97.

Solve the following differential equations:
$\frac{\text{d}y}{\text{d}x}=e^x$

a)

$1=e^x+C$

b)

$y=\frac{e^x}{2}+C$

c)

$y=e^x+C$

d)

$0=e^x+C$

98.

Solve the following differential equations:
$\frac{\text{d}y}{\text{d}x}=3y$

a)

$y=Ce^{3x}$

b)

$y=3x+C$

c)

$\frac{y^2}{2}=3x+C$

d)

$\ln y=x+C$

99.

Solve the equation

$\frac{dx}{dt}=3xt^2$

a)

$x=e^{t^3}+\ C$

b)

$x=Ce+t^3$

c)

$x=Ce^{t^3}$

d)

$x=e^t+3C$

100.

The population of the little town of Scorpion Gulch is now 1000 people. The population is presently growing at about 5% per year.

Find the general solution.

a)

P=.05e^1000t

b)

P=1000e^5t

c)

P=1000e^.05t

d)

p=lne^1000t

Calculus Semester 2 Final Study

Total questions: 100

Worksheet time: 5hrs 30mins

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