Name: ______________________________

AP Calculus Test #7
Follow the directions for each section. Take your time, and answer the questions you feel most
confident in first. Good luck!

Multiple Choice
Choose the best answer for each question. Indicate your choice by circling the letter of the
option you believe is correct.

1) Which of the following is an example of a differential equation?

2 2
a) 𝑧 = 𝑥 − 𝑦
2 2
b) 𝑥 = 𝑦
𝑑𝑦 2 2
c) 𝑑𝑥
=𝑥 −𝑦
d) None of the above

𝑑𝑦 𝑥
2) Which of the following is the general solution to: 𝑑𝑥
= 𝑒 + 4𝑥?
𝑥 2
a) 𝑦 = 𝑒 + 2𝑥 + 𝐶
𝑥 2
b) 𝑦 = 𝑒 + 2𝑥
𝑥
c) 𝑦 = 𝑒 + 4 + 𝐶
𝑥
d) 𝑦 = 𝑒 + 4

𝑑𝑦
3) Which of the following is the particular solution to: 𝑑𝑥
= 10𝑠𝑖𝑛(𝑥) that passes through
(0, -5)?
a) 𝑦 =− 10𝑐𝑜𝑠(𝑥) + 𝐶
b) 𝑦 = 10𝑠𝑖𝑛(𝑥) + 5
c) 𝑦 = 10𝑠𝑖𝑛(𝑥) + 𝐶
d) 𝑦 =− 10𝑐𝑜𝑠(𝑥) + 5

2
𝑑𝑦 𝑥
4) Which of the following is the general solution to: 𝑑𝑥
= 2𝑦
?
2 1 3
a) 𝑦 = 3
𝑥
1 2
b) 𝑦 = 2
𝑥
1 2
c) 𝑦 = 2
𝑥 + 𝐶
2 1 3
d) 𝑦 = 3
𝑥 + 𝐶
 𝑑𝑦
5) Which of the following is the particular solution to: 𝑑𝑡
= 51𝑦 when 𝑦 = 0. 5 at 𝑥 = 0?
51𝑡
a) 𝑦 = 0. 5𝑒
b) 𝑦 = 51𝑥 + 0. 5
c) 𝑦 = 𝑙𝑛(𝑥) + 0. 5
d) None of the above

6) The slope field to the right could represent a solution to

which of the following equations?
𝑑𝑦
a) 𝑑𝑥
= 𝑥 − 1
𝑑𝑦
b) 𝑑𝑥
= 𝑥 − 𝑦
𝑑𝑦
c) 𝑑𝑥
= 𝑥𝑦
d) None of the above

2 3
7) Solve: ∫ 𝑥(𝑥 + 3) 𝑑𝑥

1 3
a) 3
𝑥 + 3𝑥 + 𝐶
1 2 4
b) 8
(𝑥 + 3) + 𝐶
2 3
c) (𝑥 + 3) + 𝐶
1 2 2
d) 2
(𝑥 + 3) + 𝐶

8) A particle moves on a straight line with velocity represented by the equation

𝑣(𝑡) = 2𝑥 − 8. Which of the following COULD represent the position of the particle?
a) 𝑥(𝑡) = 2
2 3
b) 𝑥(𝑡) = 𝑥 − 8𝑥
2
c) 𝑥(𝑡) = 𝑥 − 8𝑥 + 1
d) None of the above
For questions 9-10, consider the graph to the right.
9)​ Which of the following integrals represents the shaded area?
3
a)​ ∫( 𝑥 − (− 𝑥))𝑑𝑥
0
3
b)​ ∫( 𝑥)𝑑𝑥
0

c)​ ∫( 𝑥 − ( 𝑥))𝑑𝑥

d)​ None of the above

10)​Which of the following integrals represents a solid with the shaded area as its base
consisting of square cross-sections?
2
2
a)​ ∫ ( 𝑥 − (− 𝑥)) 𝑑𝑥
−2
2
b)​ ∫ 2( 𝑥 − (− 𝑥)) 𝑑𝑥
−2
3
c)​ ∫ 2( 𝑥 − (− 𝑥)) 𝑑𝑥
0
3
2
d)​ ∫( 𝑥 − (− 𝑥)) 𝑑𝑥
0

Free Response​
Show any work you use to complete these problems. Partial credit will be given for answers that
show on-task work. Please put a circle or box around your final answer.

3
𝑥
11)​Solve: ∫ 4 3 𝑑𝑥
(5𝑥 +2)

−1
4 2
40(5𝑥 +2)
Name: ______________________________

Calculus Test #6
Follow the directions for each section. Take your time, and answer the questions you feel most
confident in first. Good luck!

Multiple Choice
Choose the best answer for each question. Indicate your choice by circling the letter of the
option you believe is correct.

For questions 1-4, consider the image to the right, which is the graph of the function f(x). Let the
𝑥
function g(t) be defined as follows: 𝑔(𝑥) = ∫ 𝑓(𝑡)𝑑𝑡.
0
1) On which of the following intervals is g(x) decreasing?
a) (1, 2)
b) (2, 9)
c) (9, 10)
d) (4, 8)

2) At which of the following x-values does g(x) have

an inflection point?
a) 𝑥 = 2
b) 𝑥 = 6
c) 𝑥 = 9
d) 𝑥 = 4

3) At which of the following x-values does g(x) have a relative maximum?

a) 𝑥 = 4
b) 𝑥 = 9
c) 𝑥 = 6
d) 𝑥 = 2
4
4) Which of the following is the value of ∫ 𝑓(𝑡)𝑑𝑡?
0
a) ½
b) 7
c) 3
d) -½
 8
5) Which of the following is equivalent to ∫ ℎ(𝑥)𝑑𝑥?
1
1
a) − ∫ ℎ(𝑥)𝑑𝑥
8
b) 8
c) 1
7
d) ∫ ℎ(𝑥)𝑑𝑥
0

4 1 4
6) Given that ∫ 𝑔(𝑥)𝑑𝑥 = 8 and ∫ 𝑔(𝑥)𝑑𝑥 = 3, which of the following is ∫ 𝑔(𝑥)𝑑𝑥?
1 −3 −3
a) 5
b) -11
c) 11
d) -5

7) Which of the following is shown in the graph to the right?

a) Right Riemann sum with 6 subintervals
b) Left Riemann sum with 6 subintervals
c) Midpoint Riemann sum with 3 subintervals
d) Trapezoidal Riemann sum with 3 subintervals

3𝑥
2
8) Given 𝐹(𝑥) = ∫(𝑡 − 4)𝑑𝑡, find F’(x).
0
2
a) 𝐹'(𝑥) = 𝑥 − 4
2
b) 𝐹'(𝑥) = 3(9𝑥 − 4)
c) 𝐹'(𝑥) = 2𝑡
d) 𝐹'(𝑥) = 3

9) Which of the following is part of the Fundamental Theorem of Calculus?

a) Derivatives and integrals are inverse functions
𝑏
b) ∫ 𝑓(𝑥)𝑑𝑥 = 𝐹(𝑎) − 𝐹(𝑏)
𝑎
c) Both (a) and (b)
d) None of the above
 10) Which of the following describes an integral?
a) The area under a curve
b) The net (or accumulated) change of a function over a certain period
c) The inverse function of a derivative
d) All of the above

Free Response
Show any work you use to complete these problems. Partial credit will be given for answers that
show on-task work. Please put a circle or box around your final answer.

11) Using the table below, calculate a right Riemann sum approximation with 4 subintervals
11
for ∫ 𝑓(𝑥)𝑑𝑥.
2

x 2 4 7 8 11

f(x) 14 12 18 14 17

4
12) Evaluate the following definite integral: ∫(4𝑥 + 5)𝑑𝑥.
0
Name: ______________________________

AP Calculus AB Test #5
Follow the directions for each section. Take your time, and answer the questions you feel most
confident in first. Good luck!

Multiple Choice
Choose the best answer for each question. Indicate your choice by circling the letter of the
option you believe is correct.

1) On the interval [1, 6], must there be a point where the instantaneous rate of change of the
2
function 𝑦 = 𝑥 − 8𝑥 + 14 equals − 1? Why or why not?
a) Yes, due to the Intermediate Value Theorem
b) Yes, due to the Mean Value Theorem
c) No, due to the Intermediate Value Theorem
d) No, due to the Mean Value Theorem

2) The function 𝑓(𝑥) is differentiable on the interval [-3, 14]. Which of the following
guarantees that there is at least one maximum and one minimum on this interval?
a) Mean Value Theorem
b) Intermediate Value Theorem
c) Extreme Value Theorem
d) None of the above

3) Which of the following are the x-values of the global

extrema of the function graphed to the right?
a) Max: 2, Min: 4
b) Max: -1, Min: -4
c) Max: 2, Min: None
d) Max: None, Min: -4

4) On which of the following intervals is the function

5 5 4 40 3
𝑓(𝑥) =− 𝑥 + 2
𝑥 + 3
𝑥 + 5 increasing?
a) (− ∞, − 2)
b) (− 2, 4)
c) (4, ∞)
d) Both (a) and (c)
 2
5) The derivative of a certain function is 𝑔'(𝑥) = 3𝑥 − 12. What are the x-values of the
relative maximum and minimum of this function?
a) Max: -2, Min: 2
b) Max: 2, Min: -2
c) Max: 0, Min: None
d) Max: None, Min: 0

6) Which of the following are the y-values of the absolute maximum and minimum of the
2
function ℎ(𝑥) =− 𝑥 + 3𝑥 on the interval [0, 3]?
a) Max: 0, Min: 2.25
b) Max: 3, Min: 0
c) Max: 0, Min: 3
d) Max: 2.25, Min: 0

3 2
7) On which of the following intervals is 𝑓(𝑥) = 𝑥 − 12𝑥 + 45𝑥 + 7 concave down?
a) (− ∞, 4)
b) (4, ∞)
c) Both (a) and (b)
d) None of the above

8) Which of the following graphs could be the derivative of the graph

to the right?

a)

b)

c)

d)
Name: ______________________________

AP Calculus AB Test #4
Follow the directions for each section. Take your time, and answer the questions you feel most
confident in first. Good luck!

Multiple Choice
Choose the best answer for each question. Indicate your choice by circling the letter of the
option you believe is correct.

1) If 𝑤(𝑡) represents the amount of water, in gallons, added to a bottle at a given time, in
seconds, what are the units of the first and second derivatives?
a) 1st: gallons/second, 2nd: gallons/(second)2
b) 1st: gallons, 2nd: seconds
c) 1st: seconds, 2nd: gallons
d) 1st: seconds/gallon, 2nd: (seconds)2/gallon

2) Which of the following represents the derivative with respect to time of the volume of a
4 3
sphere, 𝑉 = 3
π𝑟 ?
2
a) 𝐴 = π𝑟
b) 𝑉 = 3π𝑟
𝑑𝑉 2 𝑑𝑟
c) 𝑑𝑡
= 4π𝑟 𝑑𝑡
𝑑𝑉 𝑑𝑟
d) 𝑑𝑡
= 𝑑𝑡

3) For which of the following forms can you use L’Hospital’s Rule to simplify a limit?
0
a) 0
∞
b) ∞
c) Both (a) and (b)
d) None of the above

4) In which of the following situations do we expect the local linear approximation to

underestimate the value of a function?
a) Function is concave down
b) Function is concave up
c) Function is increasing
d) Function is decreasing
For questions 5-7, use the graph to the right, which represents the velocity, v(t) in m/s, of a
particle in motion on a straight line.
5) At what time(s) does the particle change direction?
a) 2 s
b) 4 s only
c) 8 s only
d) 4 and 8 s

6) During what interval(s) is the particle slowing down?

a) 1-3 s
b) 2-4 and 7-8 s
c) 6-7 s
d) 8-9 s

7) When is the particle moving at a constant speed?

a) 1-2 s
b) 2-3 s
c) 5-6 s
d) 6-7 s

8) Local linearity uses what to approximate the value of a function?

a) Tangent line
b) Integral
c) Product Rule
d) Limits

9) The number of tests Mr. Heibeck grades as a function of the number of minutes he has
spent grading is given by 𝑔(𝑡). Which of the following does 𝑔'(45) represent?
a) The number of tests he has graded at 45 min
b) The rate of change (or speed) in tests per minute at which he grades at 45 min
c) The number of tests he has left to grade at 45 min
d) The point at which he stops grading

𝑑𝑙
10) If the length of one side of a rectangle, given as 𝑙, is constant, what is 𝑑𝑡
?
a) undefined
b) 𝑙
c) 4
d) 0
Name: ______________________________

AP Calculus AB Test #3
Follow the directions for each section. Take your time, and answer the questions you feel most
confident in first. Good luck!

Multiple Choice
Choose the best answer for each question. Indicate your choice by circling the letter of the
option you believe is correct.

1) Which of the following functions would require the Chain Rule to determine the
derivative?
3
a) 𝑓(𝑥) = 𝑥 − 18
2
b) 𝑓(𝑥) = 𝑠𝑖𝑛(3𝑥 + 4)
c) 𝑓(𝑥) = 𝑐𝑜𝑠(𝑥)
d) 𝑓(𝑥) = 𝑙𝑛(𝑥)

2) Which of the following functions would require the process of implicit differentiation to
determine the derivative?
3 2
a) 𝑥 + 3𝑥𝑦 = 44
2
b) 𝑦 = 13𝑥 − 14
2
c) 𝑓(𝑥) = 𝑠𝑖𝑛 (𝑥)
d) None of the above

3) Which of the following tells you that 𝑔(𝑥) is the inverse of 𝑓(𝑥)?
−1
a) 𝑔(𝑥) = 𝑓 (𝑥)
b) 𝑓(𝑔(𝑥)) = 𝑥
c) Both (a) and (b)
d) None of the above

4) Which of the following represents the second derivative of the function 𝑦 = 𝑓(𝑥)?
a) 𝑓''(𝑥)
2
𝑑𝑦
b) 2
𝑑𝑥
c) Both (a) and (b)
d) None of the above
 5
5) Given that 𝑔(𝑥) is the inverse of 𝑓(𝑥) = 𝑥 + 2, where 𝑓(1) = 3, what is 𝑔'(3)?
a) -1
b) 1
1
c) 5
d) 5

3
6) Which of the following represents the derivative of 6𝑦 ?
2
a) 18𝑥
3
b) 6𝑦
𝑑𝑦 3
c) 6( 𝑑𝑥 )
2 𝑑𝑦
d) 18𝑦 𝑑𝑥

(5) 5
7) What is the fifth derivative, 𝑓 (𝑥), of the function 𝑓(𝑥) = 𝑥 ?
a) 5
b) 120
c) 5𝑥
5
d) 120𝑥

For numbers 8-9, use the table below.

𝑥 𝑔(𝑥) 𝑔'(𝑥) ℎ(𝑥) ℎ'(𝑥)

5 9 6 5 -4

8) Find 𝑓'(5) if 𝑓(𝑥) = 𝑔(ℎ(𝑥)).

a) 5
b) 12
c) -6
d) -24

2
9) Find 𝑓'(5) if 𝑓(𝑥) = (ℎ(𝑥)) .
a) -40
b) -20
c) 10
d) 25
 10) Which of the following represents the derivative of 𝑓(𝑥) = 𝑎𝑟𝑐𝑡𝑎𝑛(3𝑥 − 1)?
a) 3
2
b) 𝑠𝑒𝑐 (3𝑥 − 1)
1
c) 3( 2 )
1+(3𝑥−1)
d) None of the above

Free Response
Show any work you use to complete these problems. Partial credit will be given for answers that
show on-task work. Please put a circle or box around your final answer.

3 2
11) Find the derivative of the function 𝑓(𝑥) = 𝑙𝑛(𝑥 + 2𝑥 − 8).

3
12) Find the equation of the line tangent to the graph of 𝑓(𝑥) = (𝑡𝑎𝑛(𝑥)) at 𝑥 = 0.
Name: ______________________________

AP Calculus AB Test #2
Follow the directions for each section. Take your time, and answer the questions you feel most
confident in first. Good luck!

Multiple Choice
Choose the best answer for each question. Indicate your choice by circling the letter of the
option you believe is correct.

1) Using the table below, which of the following is the average rate of change of this
function between x=2 and x=7?
x 2 4 5 7

f(x) 5 -8 30 15

a) 1
b) 2
c) 3
d) 4

4
2) Which of the following represents the derivative of 9𝑥 ?
3
a) 9𝑥
4
b) 4𝑥
c) 36
3
d) 36𝑥

3) Which of the following is NOT one of the rules we have learned to evaluate derivatives?
a) Exponent Rule
b) Power Rule
c) Product Rule
d) Quotient Rule

4) Which of the following does the derivative of a function represent?

𝑓(𝑥+∆𝑥)−𝑓(𝑥)
a) lim ∆𝑥
∆𝑥 → 0
b) Slope of a tangent line to a point on the graph of that function
c) Instantaneous rate of change of the function
d) All of the above
Use the table below to answer questions 5-6.
x f(x) f’(x) g(x) g’(x)

2 1 2 3 4

𝑓(𝑥)
5) Which of the following is the derivative of 𝑔(𝑥)
?
9
a) 2
2
b) 9
c) -2
d) -9

6) Which of the following is the derivative of 𝑓(𝑥)𝑔(𝑥)?

a) 4
b) 18
c) -3
d) 10

7) Consider the graph of a function to the right. At which of the

following points is this function NOT differentiable?
a) 0
b) 1.5
c) Both (a) and (b)
d) The function is differentiable everywhere

2
8) Which of the following represents the derivative of 6𝑥 𝑐𝑜𝑠(𝑥)?
2
a) 6𝑥 (− 𝑠𝑖𝑛(𝑥)) + 12𝑥𝑐𝑜𝑠(𝑥)
b) 12𝑥𝑐𝑜𝑠(𝑥)
2
c) 6𝑥 (− 𝑠𝑖𝑛(𝑥))
d) 12𝑥 − 𝑠𝑖𝑛(𝑥)

3𝑥
9) Which of the following represents the derivative of 𝑥 ?
𝑒
3
a) 𝑥
𝑒
𝑥
b) 3 − 𝑒
𝑥 𝑥
3𝑒 −3𝑥𝑒
c) 𝑥 2
(𝑒 )
𝑥
d) 3 + 𝑒
 3
10) Which of the following expressions could be used to find the derivative of 𝑓(𝑥) = 𝑥 ?
3 3
(𝑥+∆𝑥) −𝑥
a) lim ∆𝑥
∆𝑥 → 0
3 2
b) 𝑥 + 𝑥 + 𝑥
3
c) (𝑥)
3
d) lim 𝑥
𝑥→3

Free Response
Show any work you use to complete these problems. Partial credit will be given for answers that
show on-task work. Please put a circle or box around your final answer.

3 4
11) Find the derivative of the following function: 𝑓(𝑥) = 2 − 𝑥+ 𝑥
𝑥

−6 1
3 − + 2𝑥
𝑥 2 𝑥

12) Find the derivative of the following function: 𝑓(𝑥) = (𝑙𝑛 𝑥)(𝑡𝑎𝑛 𝑥)

𝑡𝑎𝑛 𝑥 2
𝑥
+ (𝑙𝑛 𝑥)(𝑠𝑒𝑐 𝑥)
Name: ______________________________

AP Calculus AB Test #1
Follow the directions for each section. Take your time, and answer the questions you feel most
confident in first. Good luck!

Multiple Choice
Choose the best answer for each question. Indicate your choice by circling the letter of the
option you believe is correct. Use the graph below to answer questions 1-3.

1) Which of the following is lim ?

𝑥→3
a) 0
b) 1
c) 2
d) 3

2) Which of the following is lim ?

𝑥→5
a) -0.2
b) 2
c) 1
d) Does Not Exist (DNE)

3) Is the graph continuous at x=3?

a) Yes
b) No, because the function has no value at that point (f(3) DNE)
c) No, because the function has no limit at that point
d) No, because there is no function greater than this function

4) Which of the following must be true for a function to be continuous at a point?

a) A value of the function must exist at that point
b) A limit of the function must exist at that point
c) The value and limit of the function at that point must be equal to one another
d) All of the above
5) Based on the values in the table below, what is lim 𝑓(𝑥)?
𝑥→3

x 2.9 2.95 2.99 3.01 3.05 3.1

f(x) 6.1 6.05 6.01 6.01 6.05 6.1

a) 6
b) 3
c) 7
d) -6

6) What type of discontinuity is shown in the graph to the right?

a) Removable (Hole)
b) Jump
c) Infinite (Vertical Asymptote)
d) None of the above

2
7) Which of the following is lim 𝑥 − 2𝑥 + 6?
𝑥→4
a) 0
b) DNE
c) 4
d) 14

8) Which of the following represents the limit of f(x) as x approaches 1 from the right?
a) lim
+
𝑥→1
b) lim
−
𝑥→1
c) lim
𝑥→1
d) lim
𝑥→2

9) Which of the following functions has a removable discontinuity?

a) 𝑓(𝑥) = 𝑥 + 4
b) 𝑓(𝑥) = 𝑥 − 18
2
𝑥 −9
c) 𝑓(𝑥) = 𝑥+3
𝑥
d) 𝑓(𝑥) = 2
𝑥 −4
 10) Which of the following must be true for the Squeeze Theorem to apply?
a) Two functions must exist such that 𝑔(𝑥) ≤ 𝑓(𝑥) ≤ ℎ(𝑥) on an interval
containing the point of interest
b) The limit of functions g(x) and h(x) must equal one another at the point of interest
c) Both (a) and (b)
d) None of the above

Free Response
Show any work you use to complete these problems. Partial credit will be given for answers that
show on-task work. Please put a circle or box around your final answer.

2
𝑥 −𝑥−12
11) Evaluate the limit: lim 2
𝑥→4 𝑥 −16

lim = ⅞
𝑥→4

12) Use the graph below to determine both the right- and left-sided limits of this function as x
approaches 4. In giving your answer, make use of limit notation. Does a two-sided limit exist for
this function? Why or why not?

lim = 8 lim = 5
+ −
𝑥→4 𝑥→4
No, a two-sided limit does not exist because the one-sided
limits are not equal to one another.