Avon High School
Name ____________________________________________
AP Calculus AB
UNIT 7 EXAM REVIEW
Sections 4.1, 4.4, 4.5: Antiderivatives, Fundamental Theorems of Calculus, & Integration
by Substitution
You should not use your TI-Nspire for any part of this review unless otherwise stated.
Evaluate each of the following definite integrals. Use either u-substitution or change of variable methods
where applicable.
1
dx
1.)  x 9  5 x 2 dx
2.) 
2x  1

dx
6.)
 x sec x  dx
dx
8.)
 x  1
3
 sin 3x  cos 3x dx
5.)

cos x
7.)

5x
x
x2
sec 2 x
4.)
3.)
tan x
dx
2
2
x
3
dx

9.) Consider y  4 x  x .
2


a. Sketch the curve y  4 x  x 2 .

b. Shade the region that represents the area of the region
enclosed by y  4 x  x 2 and the x-axis.









c. Write the definite integral that represents the area of the region enclosed by
y  4 x  x 2 and the x-axis.


d. Find the average value of the function y  4 x  x 2 on the interval [0, 4].
e. Find the value(s) of c guaranteed by the Mean Value Theorem for Integrals for y  4 x  x 2
on the interval [0,4].
2
dP
 5t 3  200t 3 , where P is the population size and
dt
t is the time in years. The initial population is 800. (when t  0 )
10.) The rate of growth of a certain population is
a. Determine the population function.
b. Use the population function to calculate the population after 10 years. You may use your TI-Nspire.
(Round your answer to the nearest integer.)
c. Find the number of years it takes the population reaches 25,000. (Round your answer to one
decimal place.) You may use your TI-Nspire.
d. What is the average population during the time from t  0 to t  10 ?
e. During which year does the population equal the average found in part d?


2
dE
 20t  15t 3 ,
dt
where E(t) is the school’s enrollment t is the time in years. The initial enrollment is 2,500. (when t  0 )
11.) The rate of growth of the enrollment of McKinley High School is characterized by
a. Determine the enrollment function, E(t).
b. Use the enrollment function to calculate the enrollment after 8 years.
c. In which year does year will the enrollment reach 5,000 students?
d. What is the average enrollment during the time from t  0 to t  8 ?
e. During which year does the enrollment equal the average found in part d?
y

12.) The graph of f is given in the figure to the right.

7
a. Evaluate
 f ( x) dx .

1

x
b. Determine the average value of f on the interval [1,7].




c. Determine the answers to both parts a and b if f (x) is translated two units upward.





The following problem appeared on the 2014 AP Calculus AB Exam.
Graph of f
13.) The function f is defined on the closed interval  5, 4 . The graph of f consists of three line segments and
x
is shown in the figure to the above. Let g be the function defined by g ( x) 
 f (t ) dt.
3
a.
Find g (3).
b. On what open intervals contained in 5  x  4 is the graph of g both increasing and concave up?
c.
The function h(x) is defined by h( x) 
g ( x)
. Find h(3) .
5x
The following problem appeared on the 2012 AP Calculus AB Exam.
14.) Let f be the continuous function defined on  4,3 whose graph, consisting of three line segments and
x
a semicircle centered at the origin, is given above. Let g be the function given by g ( x)   f (t ) dt.
1
a.
Find the values of g (2) and g ( 2) .
b. For each of g (3) and g (3) , find the value or state that it doesn’t exist.
c. Find the x-coordinate of each point at which the graph of g has a horizontal tangent line. For each of
these points, determine whether g has a relative minimum, relative maximum, or neither a minimum
nor a maximum at the point. Justify your answers.
d. For 4  x  3 , find all values of x for which the graph of g has a point of inflection. Explain your
reasoning.
3
15.)
 x( x
2
4
 1) dx
3
16.)
1
17.)

9

4  x dx
3
0
3  x 
3
3
18.)
x

2
( x  cos x) dx
0
19.) Find the average value of the function f ( x)  x 2  x over 1, 4 .
20.) Find the value of c guaranteed by the Mean Value Theorem for Integrals for f ( x)  4  x over 1, 4 .
21.) Use the Second Fundamental Theorem of Calculus to find F ( x) .
4
2x
a.) F ( x) 

5t 3  17 dt
8
22.) The velocity of a moving object, v (t ) , measured in feet
per second is represented by the graph to the right which
consists entirely of line segments along the time interval
4  t  3 seconds.
a.
Find the total distance the object traveled on the
interval 4  t  3 .
b. Find the displacement of the object on the
interval 4  t  3 .
b.) F ( x)   cos5  3t  4  dt
x
23.) The velocity of a moving object, v (t ) , measured in inches per second is represented by the
continuous function y  ( x  1)2  1 on the time interval 4  t  3 seconds. Calculate the total
distance the object traveled on the interval 4  t  3 .
20
24.) Let f ( x) be a continuous function. What is the value of
 f ( x) dx
4
(A) 6
(C) 20
(E) 25
5
(B)
4
5
(D)
2
5
if it is known that
 f (4 x) dx  5 ?
1