Free Printable Calculus Midyear Review

1.

Which of the following is true?

a)

the derivative is a way to show rate of change, that is - the amount by which a function is changing at one given point

b)

dx/dy is the derivative

2.

a)

cosx

b)

-cosx

c)

-sinx cosx

d)

sinx cosx

3.

a)

sinx

b)

-sinx

c)

-cscx cotx

d)

sec²x

4.

Find f(2).

a)

1

b)

-1

c)

5

d)

DNE

5.

Find the limit of the function as x approaches 2.

a)

1

b)

-1

c)

5

d)

DNE

6.

If a function has a derivative that is negative, what does that tell you?

a)

The function is increasing

b)

The function is decreasing

c)

The concavity of the function is up

d)

The concavity of the function is down

7.

The acceleration function is the derivative of...

a)

position

b)

velocity

c)

calculus

d)

particle motion

8.

The image displays the ______

a)

Limit definition of the first derivative

b)

The bane of my existance

9.

Find the derivative of the given equation
f(x) = 7

a)

7

b)

0

c)

7x

d)

14

10.

Differentiate y= 12x^-2

a)

24x^-1

b)

-24x^-3

c)

-24x^-1

d)

6x^-3

11.

If f'(x) = 0 what does that imply about the x value?

a)

It is a critical point, it is a possible max, min, or point of inflection.

b)

That the limit does not exist.

12.

When looking for critical points we did....

a)

took the limit of the function. 2. Graphed the critical points.

b)

Found f'(x). 2. Set f'(x) = 0 and solved for x. 3. Created a sign diagram. 4. Took the limit

c)

Found f'(x). 2. Set f'(x) = 0 and solved for x.

13.

When applying calculus. The second derivative helps find...

a)

the distance traveled by an object.

b)

The velocity of a particle at any given point

c)

acceleration of an object at any given time

14.

Differentiate means to:

a)

Find a differential equation

b)

Take the derivative

c)

Take the integral

d)

Write the equation of the tangent line

15.

Which of the following words indicates that you need to find the maximum?

a)

greatest

b)

least

c)

most

d)

highest

e)

smallest

16.

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

17.

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

18.

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

19.

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

20.

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

21.

The velocity of an object is given as v = 2t + 3t³. What is the acceleration of the object at t = 2 secs?

a)

38 m/s/s

b)

27 m/s/s

c)

16 m/s/s

d)

49 m/s/s

22.

Given x(t)=t³-4t²+7, what is the initial position?

a)

11

b)

-10

c)

7

d)

0

23.

A particle's speed is decreasing if

a)

it's acceleration is positive

b)

it's velocity is positive

c)

it's velocity and acceleration have the same sign

d)

it's velocity and acceleration have different signs

24.

v(t)>0 means

a)

the particle is moving to the right

b)

the particle is moving to the left

c)

the particle has positive position

d)

the particle is at rest

25.

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

26.

s(t) = t² - 20
Find the average velocity from t = 3 to t = 5.

a)

2

b)

4

c)

6

d)

8

27.

The acceleration function is the derivative of...

a)

position

b)

velocity

c)

calculus

d)

particle motion

28.

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

a)

negative

b)

positive

c)

decreasing

d)

increasing

29.

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

a)

negative

b)

positive

c)

decreasing

d)

increasing

30.

An objects distance from its starting point at time t is given by the equation

s(t) = t³ - 6t² - 4.

What is the speed of the object when its acceleration is 0?

a)

12

b)

-24

c)

22

d)

44

31.

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. (what formula would you use)

a)

V=(1/3)πr²h

b)

V=(4/3)πr³

c)

d=√(x²+y²)

d)

S=2lh+2lw+2hw

32.

Water is draining from the bottom of a cone-shaped funnel at the rate of 0.03 ft³ /sec. The height of the funnel is 2 ft and the radius at the top of the funnel is 1 ft. At what rate (in ft/s) is the height of the water in the funnel changing when the height of the water is 1/2 ft? Round off your answer to the nearest thousandths.

(a)

33.

A tank is in the form of an inverted cone having an altitude of 10 ft and a radius of 5 feet. Water is flowing into the tank at the rate of 1 ft³/min. How fast is the water level rising when the water is 3 ft deep?

a)

3.4 ft/min

b)

.14 ft/min

c)

1 ft/min

d)

1/2 ft/min

34.

A spherical snowball melts so that its radius decreases at a rate of 4 in/sec. At what rate is the volume of the snowball changing when the radius is 4 in?

a)

-262π in³/sec

b)

-247π in³/sec

c)

-256π in³/sec

d)

-263π in³/sec

35.

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

36.

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

37.

The radius r of a sphere is increasing at a rate of 2 inches per minute. Find the rate of change of the volume when r=6 inches. (which formula would you use?)

a)

C=2πr

b)

A=πr²

c)

V=(4/3)πr³

d)

S=4πr²

38.

A spherical balloon is being filled with air at the constant rate of 2 cm³ /sec (see figure above). How fast in cm/s is the radius increasing when the radius is 3 cm? Round off your answer to the nearest thousandths. (just number)

(a)

39.

Car A is traveling west at 50 mi/hr and car B is traveling north at 60 mi/hr. At what rate are the cars traveling away from each other when car A is 0.3 mi and car B is 0.4 mi from the intersection?

a)

78 mi/hr

b)

38 mi/hr

c)

68 mi/hr

d)

54 mi/hr

40.

A ladder 13 feet long rests against a vertical wall and is sliding down the wall at the rate of 3 ft/s at the instant the foot of the ladder is 5 feet from the base of the wall. At this instant, how fast is the foot of the ladder moving away from the wall?

a)

dz/dt = 36/5 ft/sec

b)

dx/dt = 36/5 ft/sec

c)

dy/dt = 36/5 ft/sec

d)

dx/dt = 36/5 ft²/sec

41.

Oil spilling from a ruptured tanker spreads in a circle on the surface of the ocean. The radius of the spill increases at a rate of 5 m/min. How fast is the area of the spill increasing when the radius is 5 m?

a)

50π m²/min

b)

47π m²/min

c)

52π m²/min

d)

40π m²/min

42.

Water leaking onto a floor forms a circular pool. The radius of the pool increases at a rate of 4 cm/min. How fast is the area of the pool increasing when the radius is 10 cm?

a)

86π cm²/min

b)

89π cm²/min

c)

80π cm²/min

d)

71π cm²/min

43.

Devin set up a toy rocket. For safety, he stands 6 meters from the rocket. He sets off the rocket and it heads straight up at a constant rate of 4 m/s. How fast is the distance between the rocket and Devin changing after 2s?

a)

-2.5 m/s

b)

2.5 m/s

c)

3.2 m/s

d)

-3.2 m/s

44.

What will be true at an inflection point? (select the best answer)

a)

f(x)=0

b)

f'(x)=0

c)

f''(x)=0

d)

The function is undefined

45.

Derivative means the same thing as

a)

slope of the tangent line

b)

slope of the normal line

c)

exponent

d)

potato

46.

The slope of a function is described by its ____________.

a)

first derivative

b)

second derivative

c)

third derivative

d)

expression

47.

If a function's FIRST derivative is negative at a certain point, what does that tell you?

a)

The function is increasing at that point

b)

The function is decreasing at that point

c)

The concavity of the function is up at that point

d)

The concavity of the function is down at that point

48.

Use the sign chart for f'(x).
There is(are) ...

a)

a local maximum at x = -2.

b)

a local maximum at x = 4.

c)

local maxima at x = -2 and x = 4.

d)

no extrema.

49.

When f'(x) changes from positive to negative, there is(are) ...

a)

a maximum.

b)

a minimum.

c)

no extrema.

50.

If (a,b) is a local minimum, then what will be true about f'(a)?

a)

It's positive

b)

It's negative

c)

It's zero

d)

Cannot be determined

51.

Use the sign chart for f'(x). There is(are) ...

a)

a local maximum at x = -2.

b)

a local minimum at
x = -2 and a local maximum at x = 4.

c)

a local maximum at
x = -2 and a local minimum at x = 4.

d)

no extrema.

52.

If f'(x) = 0 what does that imply about the x value?

a)

It is a critical point, it is a possible max, min, or point of inflection.

b)

That the limit does not exist.

53.

When f'(x) changes from negative to positive, there is(are) ...

a)

a maximum.

b)

a minimum.

c)

no extrema.

54.

The concavity of a function is described by its _______________.

a)

first derivative

b)

second derivative

c)

third derivative

d)

expression

55.

concavity

a)

concave up only

b)

concave down only

c)

both concave up and down

d)

none

56.

For a function g(x), g''(3)=-8 indicates that g(x) is ____________ at x=3.

a)

increasing

b)

decreasing

c)

concave up

d)

concave down

57.

For a function f(x), f''(4)=0 indicates that x=4 is _____________.

a)

an inflection point

b)

a critical point

c)

a relative maximum

d)

a relative minimum

58.

What is the product rule for the derivative of f = u⋅v?

a)

f' = u'v + uv'

b)

f' = uv' - vu'

c)

f' = u'u + v'v

d)

f' = v'u' - vu

59.

Given that y = (x+2)/(x-3), find y'.

a)

y' = -1/(x-3)²

b)

y' = -5/(x-3)²

c)

y' = (2x-5)/(x-3)²

d)

y' = (2x-1)/(x-3)²

60.

Use the product rule to find the derivative. $f\left(x\right)\ =\ -x^3\left(3x^4-2\right)$

a)

$-3x^2+12x^3$

b)

$-21x^3+6x$

c)

$-21x^6+6x^2$

d)

$-84x^6-6x^2$

61.

Given that the position function for a particular object is given by

s(t) = t³- 3t +5, find the velocity of the object at t = 2 seconds. (Plug 2 into the derivative of the function)

a)

51

b)

12

c)

9

d)

6

62.

Given that f(x) = 3x³ - x - 1, find f'(2). [plug 2 into the derivative]

a)

35

b)

18

c)

17

d)

12

63.

Find f'(x) if

f(x)=(2x-7)/(x+3)

[quotient rule, f/g]

a)

2

b)

13/(x+3)²

c)

-4/(x+3)²

d)

-13/(x+3)

64.

$x^2\left(-3x^2-2\right)$ [product rule]

a)

$-6x^3-4x$

b)

$-12x^3-4x$

c)

$12x^3+4x$

d)

$-36x^2-4$

65.

Let y = (x - 1)(x + 2). Find dy/dx. [product rule]

a)

2x + 1

b)

2x - 1

c)

4x - 1

d)

2x² + x - 1

66.

Let f(x) = (x - 1)(x + 2). Find f'(0)

[product rule, plug 0 into the derivative]

a)

0

b)

1

c)

2

d)

3

67.

Find the derivative of the given equation
f(x) = 7

a)

7

b)

0

c)

7x

d)

14

68.

Find the derivative of the given equation
f(x) = -4x

a)

4

b)

x

c)

-4

d)

0

69.

Find the derivative of the given equation
f(x) = x⁴

a)

x³

b)

4x

c)

4x³

d)

x⁴

70.

Find the derivative of the given equation
f(x) = 1 - x²

a)

1 - 2x

b)

-2x

c)

-2

d)

-1

71.

Find the derivative of the given equation
f(x) = x³ + x² + 3

a)

3x² + 2x

b)

3x + 2x

c)

3x + 2x + 3

d)

x³ + x²

72.

Find dy/dx for y = 4x³+ 3x + 1

a)

dy/dx = 4x² + 3

b)

dy/dx = 4x² + 3x

c)

dy/dx = 12x² + 3x

d)

dy/dx = 12x² + 3

73.

Find the derivative of the given equation

f(x)=12-8x

a)

-8x

b)

-8s

c)

-8

d)

12 - 8

74.

Find the derivative of the funtion. (hint: you will need to multiply first!)

a)

4x⁸+60x⁶+12x³

b)

12x⁷+36x²

c)

32x⁷+36x²

d)

20x⁶

75.

Find dy/dx by Implicit Differentiation
xy = 6

a)

-x/ y

b)

-y/ x

c)

-6y/ x

d)

y/x

76.

Find dy/dx
xy+y²=2

a)

-y/(x+2y)

b)

y/(x+2y)

c)

-3y/x

d)

-3x/y

77.

Find dy/dx by Implicit Differentiation
x³ +y³ = 36

a)

6 -x

b)

3x² +3y²

c)

−x²/y²

d)

0

78.

Plug (2, 1) into your derivative

a)

-2

b)

-2/3

c)

2/3

d)

2

79.

Find the derivative (normal product and e rule):

y=5x²e^3x

a)

y'=10xe^3x(2x+3)

b)

y'=5xe^3x(3x+2)

c)

y'=10ex^3x(3x+2)

d)

y'=5xe^3x(2x+3)

80.

Find the second derivative of

f(x) = x²+ e^x - cosx

(normal e and cos rules)

a)

f"(x) = 2 + e^x + cosx

b)

f"(x) = 2x + e^x + cosx

c)

f"(x) = 2x + xe^x - cosx

d)

f"(x) = 2x + e^x + sinx

81.

Evaluate the limit.

a)

5

b)

DNE

c)

3

d)

7

82.

Evaluate the limit.

a)

10

b)

DNE

c)

7

d)

∞

83.

$\lim_{x\rightarrow11}\ \ \frac{x^2-121}{x-11}\ =\ ....$

a)

22

b)

21

c)

20

d)

19

e)

18

84.

$\lim_{x\rightarrow-5}\ \ \ \frac{x+5}{2}\ =\ ....$

a)

0

b)

5

c)

$\infty$

d)

$\frac{5}{2}$

e)

$\frac{2}{5}$

85.

What is the limit?

a)

DNE

b)

Infinity

c)

6

d)

12

86.

lim_x→2f(x)=

a)

3

b)

2

c)

0

d)

1.7

87.

$\lim_{x\rightarrow-3}\ \ \frac{x+3}{x^2-9}\ =\ ....$

a)

0

b)

$-6$

c)

6

d)

$-\frac{1}{6}$

e)

$\frac{1}{6}$

88.

$\lim_{x\rightarrow-4}\ \ \ 17\ =\ ....$

a)

17

b)

$-68$

c)

$-4$

d)

$-17$

e)

0

89.

a)

2

b)

DNE

c)

7

d)

49

90.

a)

Infinity

b)

25

c)

DNE

d)

7

91.

\lim_{x\rightarrow1}^{ }x^3+3x^2-2x

(a)

92.

\lim_{x\rightarrow1}^{ }x^3+3x^2-2x

(a)

93.

\lim_{x\rightarrow-3}\frac{x^2+4x+3}{x^2-3}

(a)

94.

\lim_{x\rightarrow2}\frac{x^2+5x+6}{x+2}

(a)

95.

Given the graph, find $\lim_{x\rightarrow2^+}f\left(x\right)$ .

(a)

96.

The three situations where derivatives fail to exist are at corners or cusps, at a vertical tangent, and...

a)

horizontial tangent

b)

discontinuity

c)

curve

d)

intercepts

97.

The derivative of which function is itself?

a)

e^x

b)

x

c)

sinx

d)

tanx

98.

The derivative of

a)

y'=2x-x^-2

b)

y'=x^-1+8x

c)

y'=x^-2+8x

d)

y=8x-x^-2

99.

a)

A

b)

B

c)

C

d)

D

100.

Find the derivative of the given equation
f(x) = 7

a)

7

b)

0

c)

7x

d)

14

101.

Find the derivative of the given equation
f(x) = -4x

a)

4

b)

x

c)

-4

d)

0

102.

Find the derivative of the given equation
f(x) = x⁴

a)

x³

b)

4x

c)

4x³

d)

x⁴

103.

Find the derivative of the given equation
f(x) = 1 - x²

a)

1 - 2x

b)

-2x

c)

-2

d)

-1

104.

Find the derivative of the given equation
f(x) = x³ + x² + 3

a)

3x² + 2x

b)

3x + 2x

c)

3x + 2x + 3

d)

x³ + x²

105.

Find dy/dx for y = 4x³+ 3x + 1

a)

dy/dx = 4x² + 3

b)

dy/dx = 4x² + 3x

c)

dy/dx = 12x² + 3x

d)

dy/dx = 12x² + 3

106.

Differentiate y = $\frac{12}{x^2}$

a)

y' = 24x^-1

b)

y' = -24x^-3

c)

y' = -24x^-1

d)

y' = 6x^-3

107.

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

a)

y = 2x + 1 - 3x^-1

b)

y' = 4x + 1 - 1x^-2

c)

y' = 2 + 3x^-2

d)

y' = 4x + 1 - 3x^-2

108.

Find the derivative of the given equation

f(x)=12-8x

a)

-8x

b)

-8s

c)

-8

d)

12 - 8

109.

Where on this graph would you find a HORIZONTAL tangent line?

a)

x = -3

b)

x = 0

c)

x = 3

d)

There are no horizontal tangent lines on this graph

110.

At which point(s) will the slope(s) of the tangent line be POSITIVE?

a)

at C only

b)

at points A, C and E only

c)

at point B and D only

d)

at points A and E only

111.

Find the derivative of the given equation
f(x) = $\frac{1}{x^2}$ Hint: rewrite with a negative exponent

a)

f'(x) = 1/2x

b)

f'(x) = -2x^-3

c)

f'(x) = 2x

d)

f'(x) = -2x

112.

At which point(s) will the slopes of the tangent line be zero?

a)

at C only

b)

at points A, C and E only

c)

at point B and D only

d)

at points A and E only

113.

Find the derivative of the funtion. (hint: you will need to multiply first!)

a)

4x⁸+60x⁶+12x³

b)

12x⁷+36x²

c)

32x⁷+36x²

d)

20x⁶

114.

Find the fully simplified Derivative of y = 4x² + 3x + 6x^-2

a)

y' = 8x + 3 -12x^-3

b)

y' = 8x + 3 +12/x³

c)

y' = 8x + 3 -12/x³

d)

y' = 8x + 3 + 12/x

115.

Find the limit as x approaches 3 from the left

a)

4

b)

3

c)

2

d)

DNE

116.

What is the limit of the function as x approaches -4 from the left?

a)

2

b)

-4

c)

DNE

d)

-2

117.

What is the limit of the function as x approaches 1 from the left?

a)

DNE

b)

1

c)

4

d)

-2

118.

What is the limit of the function as x approaches 1 from the right?

a)

DNE

b)

1

c)

4

d)

-2

119.

Use interval notion to describe those intervals on which the function is positive.
y=-x(x+2)(x-1)

a)

(-2,0),(1,∞)

b)

(-2,0),(0,1)

c)

(-∞,-2),(0,1)

d)

(0,1),(1,∞)

120.

a)

Infinity

b)

Negative Infinity

c)

2

d)

Does not exist

121.

a)

3

b)

1

c)

Infinity

d)

Does not exist

122.

a)

Does not exist

b)

6

c)

4

d)

3

123.

a)

0

b)

2

c)

3

d)

4

124.

a)

0

b)

3

c)

4

d)

DNE

125.

a)

0

b)

1

c)

2

d)

DNE

126.

a)

0

b)

1

c)

2

d)

DNE

127.

a)

0

b)

1

c)

-1

d)

DNE

128.

a)

-1

b)

0

c)

1

d)

DNE

129.

a)

0

b)

∞

c)

- ∞

d)

DNE

130.

a)

0

b)

2

c)

3

d)

4

Calculus Midyear Review

Total questions: 130

Worksheet time: 9hrs 16mins

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