Practice 3 Math 242
Multiple Choice
Identify the choice that best completes the statement or answers the question.
1. Evaluate the integral.
4. Differentiate the function.
4
!x
2
+6
1
x
y=x
dx
2
x " #$
x #9
2x # 6
5. Calculate y%.
Select the correct answer.
a. #1/2
b. 3
c. #3
d. 0
e. limit does not exist
3. Consider the function f(x) =
y=
x cos
x
Select the correct answer.
1
2+e
1 /x
. Find the
value of lim f(x).
x"0
7x
Select the correct answer.
7x Ê
e Á
1 ˆ˜
a. y% = x ÁÁÁÁ 7 ln x + ˜˜˜˜
ÁË
x ˜¯
Ê
ˆ
7x
7x e Á
Á
1 ˜˜
b. y% = e x ÁÁÁ 7 ln x + 2 ˜˜˜
Á
x ˜¯
Ë
7x Ê
7x e Á
1 ˆ˜
c. y% = e x ÁÁÁÁ 7 ln x # ˜˜˜˜
ÁË
x ˜¯
Ê
7x
7x e Á
1 ˆ˜
d. y% = e x ÁÁÁÁ 7 ln x + ˜˜˜˜
ÁË
x ˜¯
7x Ê
e Á
1 ˜ˆ
e. y% = 7x ÁÁÁÁ 7 ln x + ˜˜˜˜
ÁË
x ˜¯
Select the correct answer.
a. 24.4
b. 37
c. 74
d. 49.2
e. 21.4
2. Find the limit.
lim
e
#
Select the correct answer.
a. 1.5
b. #0.1
c. 0.1
d. 0.9
e. 0.5
1
a.
y% = cos
x#
b.
y% =
sin
x#
c.
y% = #
1
2
d.
y% = #
1
2
e.
1
y% = #
2
sin
2
x
x
x cos
x
2 x
ÊÁ
ˆ
ÁÁ cos x # 1 ˜˜˜
ÁÁ
˜˜
ÁÁ
˜˜
ÁÁ
˜˜
x
Ë
¯
ÊÁ
ˆ
˜
ÁÁ sin x # 1 ˜˜
ÁÁ
˜˜
ÁÁ
˜˜
ÁÁ
˜˜
x
Ë
¯
ÊÁ
ÁÁ x sin x # cos
ÁÁ
ÁÁ
ÁÁ
x
Ë
ˆ˜
x ˜˜˜
˜˜
˜˜
˜¯
#1
6. Find the limit.
lim
x"0
+
10. If f(x) = 3x + ln x, find f (3).
ln 4x
x
Select the correct answer.
#1
a. f (3) = #3
Select the correct answer.
a. 4
b. 0
c. &
d. #$
e. $
7. Evaluate the limit.
c.
f (3) = 1
d.
f (3) = #1
e.
f (3) = 0
#1
#1
#1
! 2 sin x cos(cos x)dx
lim |x + 2|
Select the correct answer.
1
a.
sin(cos x) + C
2
b. #2 cos(sin x) + C
c. 2 sin(cos x) + C
d. #2 sin(cos x) + C
1
e. 2 cos x(sin x) # + C
2
12. Find f %(x).
Select the correct answer.
a. 2
b. 4
c. #2
d. 0
e. The limit does not exist
4t + 1 , find f ' (2).
Select the correct answer.
4
a. #
27
b. 3
2
c. #
3
2
d.
3
4
e.
27
9. Evaluate the limit.
3
f (3) = 3
11. Evaluate the integral.
x " #2
8. If f(t) =
#1
b.
x
2
f(x) = 7e # 3x arctan(x)
Select the correct answer.
2
lim (x + 2) (x # 6)
f %(x) = 7e #
b.
f %(x) = 7e #
c.
f %(x) = 7e #
d.
f %(x) = 7e #
x"1
e.
Select the correct answer.
a. 27
b. #45
c. #135
d. 29
e. #125
2
x
a.
x
x
x
3x
2
1+x
3x
2
# 6x(arctan(x))
2
# 6x(arcsin(x))
2
+ 6x(arctan(x))
2
1+x
3x
# 2x(arctan(x))
2
1+x
3x
2
2
1+x
none of these
13. If f(x) =
x
4
, find f %(e ).
ln x
16. Evaluate the integral.
!6
Select the correct answer.
3
a. #
16
3
b. #
4
3
c.
4
3
d.
16
1
e. #
2
14. Evaluate the integral.
#2
2
cos (
a.
6
ln(sin ()
b.
6
+C
ln(6)
c.
6
+C
ln(6)
cos (
sin (
sin (
6
+C
ln(sin ()
e. none of these
17. Find the limit.
d.
|
|| dx
lim
x"$
Select the correct answer. The choices are
rounded to the nearest hundredth.
a. 109.00
b. 123.67
c. 102.33
d. 209.00
e. 23.67
15. Evaluate the integral if it exists.
!
cos ( d (
Select the correct answer.
5
! ||| 4x # x
sin (
6x
x
ln(2 + 5e )
Select the correct answer.
a. 6
b. #$
c. 0
d. $
e. 2/5
18. Evaluate the integral.
1
2
0
3
3x cos(x )dx
!
Select the correct answer.
a. sin(0.1)
b. 3sin(0.1)
c. 3 sin(1)
d. cos(1)
e. none of these
5
0
1
25 + t
2
dt
Select the correct answer.
a. & / 20
b. & / 10
c. & / 2
d. 10 &
e. 4 & / 5
3
19. Differentiate.
22. Evaluate the definite integral.
3
4
&/2
K(x) = (2x + 1)(x # 2x)
!
Select the correct answer.
2
3
a. 6x + 4x # 2
6
3
2
b. 14x + 4x # 12x # 2x
2
4
3
3
c. x (x + 2x) + (2x + 1)(4x + 2)
2
4
2
4
3
#& / 2
6x (x # 2) + 4x (2x + 1)
e.
6x (x # 2x) + (2x + 1)(4x # 2)
3
3
20. Evaluate the integral.
!
2
ÁÊÁ 1 # y ˜ˆ˜
ÁÁ
˜˜ dy
ÁÁ
˜
ÁË y ˜˜¯
! cos
21. Find the derivative of the following function and
calculate it for x = 25 to the nearest tenth.
x+
x+
6
dx
7
x sin x dx
Select the correct answer.
1
8
a.
cos x + C
8
1
8
b. # sin x + C
8
1
7
c. # cos x + C
8
1
8
d.
sin x + C
8
1
8
e. # cos x + C
8
24. Evaluate the definite integral.
Select the correct answer.
a. 2 y ln |y|+ C
b. y # 2 ln |y|# 1 / y + C
c. 2 y ln |y|# 1 / y + C
d. y # 2 ln |y|+ 1 / y + C
e. 2 y + ln |y|+ 1 / y + C
y(x) =
4+x
Select the correct answer.
a. #2
b. 1
c. 3
d. 0
e. #0.5
23. Evaluate the indefinite integral.
3
d.
2
x sin x
x
Select the correct answer.
a. 1.1
b. #0.9
c. 0.1
d. 0.2
e. 0.3
&/8
! sin 8t dt
0
Select the correct answer.
a. 0.25
b. 3.25
c. 0.35
d. 1.25
e. #0.25
4
25. Find the limit, if it exists.
28. Evaluate the integral.
5#x
|5 # x|
lim
x"5
3
! (6 + 6y # y
2
) dy
0
Select the correct answer.
a. 5
b. #1
c. 1
d. 0
e. limit does not exist
26. Evaluate the indefinite integral.
!
4x sin(1 + x
3 /2
Select the correct answer.
a. #18
b. 45
c. 54
d. 36
e. #12
29. Find the limit.
) dx
lim [ln(8 + x) # ln(5 + x)]
x"$
Select the correct answer.
Ê
ˆ3
4
a.
cos(1) + ÁÁÁÁ cos x ˜˜˜˜
3
Ë
¯
ÊÁ
ˆ
Ê
ˆ
˜
3
Á
˜˜
2
Á
b.
cos ÁÁÁ 1 + ÁÁÁÁ x ˜˜˜˜ ˜˜˜
Á
3
Ë
¯ ˜¯
Ë
ÊÁ
Ê
ˆ 3 ˆ˜˜
Á
4
c.
sin ÁÁÁÁ 1 + ÁÁÁÁ x ˜˜˜˜ ˜˜˜˜
3
ÁË
Ë
¯ ˜¯
Ê
ˆ ˆ˜˜ 3
4 ÊÁÁÁ
Á
˜
Á
d. # ÁÁ cos ÁÁ x ˜˜˜ ˜˜˜
3 Ë
Ë
¯¯
ÊÁ
Ê
ˆ 3 ˆ˜˜
Á
4
e. # cos ÁÁÁÁ 1 + ÁÁÁÁ x ˜˜˜˜ ˜˜˜˜
3
ÁË
Ë
¯ ˜¯
27. Find the limit.
Select the correct answer.
a. #3
b. 3
c. 0
d. 8/5
e. #2
30. Find the limit.
lim
("0
Select the correct answer.
a. 1
b. sin 1
c. cos 1
d. 0
e. 2
2
x # 2x
lim
x"2
#
cos(cos ()
sec (
2
x # 4x + 4
Select the correct answer.
a. #$
b. $
c. 2
d. #2
e. 0
5
31. Differentiate the function.
3
34. Determine the infinite limit.
2
y = ln(x sin x)
lim
x"0
Select the correct answer.
3 sin x + x cos x
a. y% =
3
2
x sin x
3 sin x # 2x
b. y% =
x sin x
3 sin x + 2x cos x
c. y% =
x sin x
3 cos x + 2x sin x
d. y% =
x cos x
20t
! sin
dt
sin 10t
2
!
y% =
2
0
xe
#2x
2
Select the correct answer.
sin 10t
a.
+C
5
cos10t
b.
+C
5
sin 10t
c.
+C
10
cos10t
d. #
+C
5
sin 10t
e. #
+C
5
dx
Choose all correct answers.
1 #8 1
a. # e +
4
4
1
b. #
4
1 #4
c. # e
4
d.
x (x + 7)
Select the correct answer.
a. 0
b. 1/7
c. #1/7
d. #$
e. $
35. Find the general indefinite integral.
6x + cos x sin x
cos x
32. Evaluate the integral.
e.
x+1
2
#4
4e
1 1 /8
e. # e
4
33. Use continuity to evaluate the limit.
lim sin(x + 4 sin x)
x " 13 &
Select the correct answer.
a. 13&
b. #1
c. 0
d. $
e. 1
6
ID: A
Practice 3 Math 242
Answer Section
MULTIPLE CHOICE
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
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24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
A
A
E
D
E
D
D
A
C
C
D
B
D
E
E
C
A
A
E
B
C
D
E
A
E
E
A
D
C
C
C
A
C
E
A
1