Section I - AP Calculus AB – McDermott
Name_________________________________
AP CALCULUS AB
Chapter 7 Review
Section I: Multiple-Choice Questions
A CALCULATOR SHOULD ONLY BE USED ON PROBLEMS NUMBERED 76 OR HIGHER
Instructions:
Section I of the AP exam contains 28 non-calculator multiple-choice questions (numbered 1-28) and 17
calculator multiple-choice questions (numbered 76-92). This review contains problems specific to the
current unit, however, any topic that has been covered thus far this year is a fair question to put on any
unit test (It is therefore suggested that you also look at past Unit Reviews). There is no penalty for getting
a multiple-choice question incorrect and therefore you should answer every question. The directions on
the AP Exam are as follows:
Solve each of the following problems, using the available space for scratch work. After examining the
form of the choices, decide which is the best of the choices given and place the letter of your choice in the
corresponding box on the student answer sheet. Do not spend too much time on any one problem.
In this exam:

Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers
x for which f (x) is a real number.

The inverse of a trigonometric function f may be indicated using the inverse function notation f 1
or with the prefix “arc” (e.g., sin1 x  arcsin x ).
Section I - AP Calculus AB – McDermott
1.
 cos  3x  dx 
(A) 3sin  3x   C
(E) 3sin  3x   C
1
(B)  sin  3 x   C
3
1
sin  3 x   C
(C)
3
(D) sin  3x   C
2.

x
e
x
dx 
(A) 2e x  C
1 x
e C
(B)
2
(C) e
x
(D) 2 x e
x
C
x
(E)
C
e
C
2 x
3. The slope field for a certain differential equation is shown to the right.
Which of the following could be a solution to the differential equation
with the initial condition y  0  1 ?
(A)
y  cos x
(D) y  1  x 2
1
(E) y 
1  x2
y  1  x2
(C) y  e x
(B)
4.
Which of the following is the solution to the differential equation
dy x 2

with the initial condition
dx y
y(3)  2 ?
A.


y  2e 9x
3
/3
B.
y  2e 9x
C.
y
2x 3
3
3
D.

/3
E.




2x 3
 14
3
2x 3
y
 14
3
y
Section I - AP Calculus AB – McDermott
5.
 sin( 2x)  cos(2x)dx 
d. 2cos(2x)  2sin( 2x)  C
e. 2cos(2x)  2sin( 2x)  C
1
1
cos(2x)  sin( 2x)  C
2
2
1
1
b.  cos(2x)  sin( 2x)  C
2
2
c. 2cos(2x)  2sin( 2x)  C
a.






6.
Shown to the right is a slope field for which of the following
differential equations?
dy
 xy
a.
dx
dy
 xy  y
b.
dx
dy
 xy  y
c.

dx
dy
 xy  x
d.

dx
dy
3
 x  1
e.

dx


7.
The acceleration of a particle moving along the x-axis at time t is given by a(t) = 6t  2. If the
velocity is 25 when t = 3 and the position is 10 when t = 1, then the position x(t) =
a. 9t 2 1
b. 3t 2  2t  4
c. t 3  t 2  4t  6
-1
8.
òe
-4 x
d. t 3  t 2  9t  20
e. 36t 3  4t 2  77t  55
dx
0
a.
-e-4
4
b. -4e-4
c. e-4 -1
d.
1 e-4
4 4
e. 4 - 4e-4
Section I - AP Calculus AB – McDermott
9. Using u = 2x +1,
2
ò
2x +1dx is equivalent to
0
1
1 2
a.
ò u du
2 -1
b.
2
10.
òx
2
( )
( )
x3
sin ( x 3 ) + C
3
11. If
2
ò
0
u du
c.
1
2
5
ò
2
u du
1
d.
ò
0
cos ( x 3 ) dx
1
a. - sin x 3 + C
3
1
b. sin x 3 + C
3
c. -
1
2
dy
= cos ( 2x ) , then y =
dx
1
a. - cos ( 2x ) + C
2
1
b. - cos 2 ( 2x ) + C
2
1
sin ( 2x ) + C
c.
2
1 2
sin ( 2x ) + C
d.
2
1
e. - sin ( 2x ) + C
2
x3
sin ( x 3 ) + C
3
x3 æ x4 ö
e.
sin ç ÷ + C
3
è4ø
d.
5
u du
e.
ò
1
u du
Section II - AP Calculus AB – McDermott
Name_________________________________
AP CALCULUS AB
Chapter 7 Review
Section II: Free-Response Questions
A CALCULATOR SHOULD ONLY BE USED ON PROBLEMS NUMBERED 3 OR LOWER
Instructions
For the free response portion of this exam, please write your answers in the space below the problem or in
the space provided within the problem. Keep the following in mind while completing these problems.

Show all of your work. Clearly label any functions, graphs, tables, or other objects that you
use. Your work will be graded on correctness and completeness of your methods as well as
your answers. Answers without supporting work may not receive credit. Justifications require
that you give mathematical (noncalculator) reasons.

Unless otherwise specified, answers (numerical or algebraic) need not be simplified. If you
use decimal approximations in calculations, your work will be graded on accuracy. Unless
otherwise specified, your final answers should be accurate to three places after the decimal
point.

Unless otherwise specified, the domain of a function f is assumed to be the set of all real
numbers x for which f (x) is a real number.
Section II - AP Calculus AB – McDermott
1. Consider the differential equation
dy x
 , where y  0.
dx y
(a) The slope field for the given differential equation is shown below. Sketch the solution curve that
passes through the point (3, –1), and sketch the solution curve that passes through the point (1, 2).
(Note: The points (3, –1) and (1, 2) are indicated in the figure.)
(b) Write an equation for the line tangent to the solution curve that passes through the point (1, 2).
(c) Find the particular solution y  f ( x) to the differential equation with the initial condition
f (3)  1, and state its domain.
Section II - AP Calculus AB – McDermott
2. The rate at which a baby bird gains weight is proportional to the difference between its adult weight
and its current weight. At time t = 0, when the bird is first weighed, its weight is 20 grams. If B(t) is the
weight of the bird, in grams, at time t days after it is first weighed, then
dB 1
 100  B  .
dt 5
Let y = B(t) be the solution to the differential equation above with initial condition B(0) = 20.
(a) Is the bird gaining weight faster when it weighs 40 grams or when it weighs 70 grams? Explain
your reasoning.
d 2B
d 2B
(b) Find
in terms of B. Use
to explain why the graph of B cannot resemble the following
dt 2
dt 2
graph.
(c) Use separation of variables to find y = B(t), the particular solution to the differential equation with
initial condition B(0) = 20.
Section II - AP Calculus AB – McDermott
3.
Consider the differential equation
dy 3  x
.

dx
y
(a) Let y = f (x) be the particular solution to the given differential equation for 1 < x < 5 such that the
line y = 2 is tangent to the graph of f. Find the x-coordinate of the point of tangency, and
determine whether f has a local maximum, local minimum, or neither at this point. Justify your
answer.
(b) Let y = g(x) be the particular solution to the given differential equation for 2 < x < 8, with the
initial condition g(6) = 4. Find y = g(x).
Section III - AP Calculus AB – Dickman
Answers
Multiple Choice
1. C
2. A
3. E
Free Response
4.