Chapter 3
81.
Find
dy
dx
for
4
ln 4
x
yx
=+
.
A)
2
14
4x
−
B)
2
3
16
16
x
xx
−
+
C)
2
3
16
16
x
xx
+
+
D)
2
3
16
16
x
xx
+
−
82.
Find
dy
dx
for
xx
y e e−
=+
.
A)
2
xx
xx
ee
ee
−
−
+
+
B)
xx
ee
−
−
C)
2
xx
xx
ee
ee
−
−
−
+
D)
1
2xx
ee
−
+
composition of functions. difficulty: medium section: 3.3
83.
A linear approximation of
28
() 2
x
fx x
+
=−
valid for x near 3 is given by
( ) –12 50f x x+
A)
True
B)
False
Chapter 3
84.
The following table gives values for two functions f and g and their derivatives. What
is
–1
[ ( ) ( )]
x
df x g x
dx =
?
x
-1
0
1
2
3
f
3
3
1
0
1
g
1
2
2.5
3
4
f ‘
-3
-2
-1.5
-1
1
g‘
2
3
2
2.5
3
85.
The following table gives values for two functions f and g and their derivatives. What
is
1
()
() x
d f x
dx g x =
? Round to 2 decimal places.
x
-1
0
1
2
3
f
3
3
1
0
1
g
1
2
2.5
3
4
f ‘
-3
-2
-1.5
-1
1
g‘
2
3
2
2.5
3
86.
The following table gives values for two functions f and g and their derivatives. What
is
2
[ ( ) ( )]
x
df x g x
dx =
?
x
-1
0
1
2
3
f
3
3
1
0
1
g
1
2
2.5
3
4
f ‘
-3
-2
-1.5
-1
1
g‘
2
3
2
2.5
3
Chapter 3
87.
Find and simplify the derivative of
34
( ) (2 8)f x x x
−
=+
.
A)
4
2 24x−
−
B)
4
2 24x−
+
C)
1
24x−
D)
1
12x−
−
88.
The equation of the tangent line to the graph of
22
() 1
x
gx x
−
=+
at the point at which x =
0 is y = _____
x−
_____. Enter fractions in the form “a/b”.
89.
Differentiate
4
( ) 4
tt
g t e e
−−
=+
.
A)
4
16
tt
ee
−−
+
B)
4
4
tt
ee
−−
−−
C)
4
16
tt
ee
−−
−−
D)
1 4 1
4
tt
te e
− − − −
−−
90.
Given
() x
f x e=
,
( ) 5x
gx=
, and
( ) ( ) ( )h x f x g x=
, what is
‘‘( )hx
?
A)
2
5 (ln 5 1)
xx
e+
B)
5 (ln 5 1)
xx
e+
C)
2
5 (ln 5)
xx
e
D)
1
( 3 3 )
x x x
ex −
+
Chapter 3
91.
Given
() x
f x e=
,
( ) 5x
gx=
, and
()
() ()
gx
jx fx
=
, find
‘‘( )jx
.
A)
5(ln 5 1)
x
x
e−
B)
2
5(ln 5 1)
x
x
e−
C)
2
(ln 5) 5x
x
e
D)
( )
2
5 (ln ) 5
xx
x
A
e
−
92.
Given
3
() 21
x
fx x
=+
,
2
2
2
() 3
x
gx x
+
=
, and
( ) ( ) ( )h x f x g x=
, find
‘(2)h
. Enter
fractions in the form “a/b”.
93.
If
() ()()j x g x h x=
and
32
‘( ) 2 3 (2 1)j x x x x= + +
, then which two of the following are
()gx
and
()hx
?
A)
3
x
B)
2
3x
C)
3
2x
D)
21x+
E)
2
xx+
94.
Differentiate
5
3x
e
.
A)
5
3x
e
B)
51
15 x
xe −
C)
5( 1)
15 x
xe −
D)
5
15 x
e
Chapter 3
95.
Differentiate
2
3
8
1
x
x+
.
A)
16
3x
B)
3
32
8 ( 2)
( 1)
xx
x
−
+
C)
3
32
8 ( 2)
( 1)
xx
x
−+
+
D)
3
8
( 1)
x
x+
96.
A demand curve for a product has the equation
60(0.95)q
p=
, where p is price and q is
quantity. What is the marginal revenue as a function of the quantity sold?
A)
60 ln 0.95(0.95)q
B)
(0.95) (60 60 ln 0.95)
qq+
C)
60(0.95)q
D)
21
60 (0.95) 60(0.95)
qq
q−+
97.
The quantity, q, of tickets sold for a certain flight is a function of the selling price, p.
Thus
()q f p=
. You are given the information that
(250) 170f=
and
‘(250) 1f=−
. Revenue is given by
R pq=
. When tickets are being sold at a price of $250, an
increase of $1 in the sales price will cause revenue to go down by how much?
Ans:
$80
functions. difficulty: medium section: 3.4
98.
The concentration, in
g
ml
, of a drug introduced gradually into the body can be
modeled by
2
7
() 0.01 3.5
t
Ct t
=+
minutes. At what time does the concentration reach its
maximum?
Ans:
18.7
Chapter 3
99.
The concentration, in
g
ml
, of a drug introduced gradually into the body can be
modeled by
2
5
() 0.01 4.4
t
Ct t
=+
. What is the rate of change in the concentration at t =
120?
100.
A drug’s concentration is modeled by
–0.03
( ) 15 t
C t te=
with C in mg/ml and t in minutes.
Is C'(t) positive or negative when t = 35 ? Find and interpret C'(35) in terms of drug
concentration
decreasing in the next minute by 0.262 mg/ml.
101.
A college savings account is opened the day baby Brad is born. The initial amount
deposited is $2500. The account is compounded quarterly at a nominal rate of 3 %.
Assuming no other money is deposited in the account, find A(18) and also A‘(18).
102.
Find
dy
dx
if
–0.07 ln
x
y e x=+
.
A)
–0.07 1
–0.07 x
ex
+
B)
–0.07 1
x
ex
+
C)
–0.07
–0.07 ln
x
ex+
D)
–0.07 1 1
–0.07 x
xe x
−+
103.
Find
dy
dx
if
2
4 sin(π)y x x=
.
A)
8π cos(π )xx
B)
2
8 sin(π ) 4π cos(π )x x x x+
C)
2
8 sin(π ) 4π cos(π )x x x x−
D)
2
8 sin(π ) 4 cos(π )x x x x+
104.
Find
dy
dx
if
5
(2 )
x
yx=+
A)
4
5(2 )
xx+
B)
4
1
5 ln 2 2 2
x
x
+
C)
41
5(2 ) ln 2 2 2
xx
xx
+ +
D)
41
1
5(2 ) 2 2
xx
xx x
−
+ +
105.
Find the derivative of
23
( 3 )y x x=+
.
A)
22 3
3( 3 ) 2 2
x x x x
+ +
B)
2
3
32 2
xx
+
C)
22 3
3( 3 ) 2 2
x x x x
+ −
D)
22
3( 3 )xx+
Chapter 3
106.
Find the derivative of
8sin(4 )
x
y−
=
.
A)
8ln 4 4 cos(4 )
xx−−
B)
8ln 4 4 cos(4 )
xx−−
−
C)
1
8 4 cos(4 )
xx
x− − −
−
D)
8cos(4 )
x−
107.
Find the derivative of
cos(3 )y x x=
.
A)
cos(3 ) 3 sin(3 )x x x+
B)
cos(3 ) sin(3 )x x x−
C)
cos(3 ) 3 sin(3 )x x x−
D)
3sin(3 )x−
108.
Find the derivative of
2
ln 5
6
x
yx
+
=+
.
A)
1
2
x
x
B)
22
6ln 2 ln 9
( 6)
x x x x
x
−−
+
C)
22
22
6 2 ln 9
( 6)
x x x
xx
−+
+
D)
22
22
6 2 ln 9
( 6)
x x x
xx
−−
+
Chapter 3
109.
Compute
dy
dx
for
cos(ln )y x x=
.
A)
cos(ln ) sin(ln )xx−
B)
cos(ln ) sin(ln )xx+
C)
cos(ln ) sin(ln )x x x−
D)
cos(ln ) sin(ln )x x x+
functions. difficulty: medium section: 3.5
110.
Compute
dy
dx
for
28
sin(2 ) 3
x
xe
yx
+
=+
.
A)
8 2 8
2
(2 8 )(sin(2 ) 3) ( )(2cos(2 ))
(sin(2 ) 3)
xx
x e x x e x
x
+ + + +
+
B)
8 2 8
2
(2 8 )(sin(2 ) 3) ( )(2cos(2 ))
(sin(2 ) 3)
xx
x e x x e x
x
+ + − +
+
C)
8 1 2 8
2
(2 )(sin(2 ) 3) ( )(2cos(2 ))
(sin(2 ) 3)
xx
x xe x x e x
x
−
+ + − +
+
D)
8 2 8
2
(2 8 )(sin(2 ) 3) ( )(2cos(2 ) 3)
(sin(2 ) 3)
xx
x e x x e x
x
+ + − + +
+
111.
The equation of the tangent line to
cos2yx=
at the point where
π / 6x=
is given by
3π3
36
yx +
=+
.
A)
True
B)
False
functions. difficulty: medium section: 3.5
112.
The height off the ground of a person riding a Ferris wheel is represented by the
function
( ) 16 14cos πh t t=−
. What is
‘( )ht
?
difficulty: easy section: 3.5
Chapter 3
113.
The height off the ground of a person riding a Ferris wheel is represented by the
function
( ) 16 14cos πh t t=−
. On which interval(s) is
‘( )ht
increasing?
A)
01t
B)
12t
C)
23t
D)
34t
114.
The size of an impala population is represented by the function
( ) 10 2cos(π / 6)R t t=+
,
where t is time in months since the beginning of the year and
()Rt
is measured in
thousands. After 7 months, the population is __________(increasing/decreasing) at a
rate of _____ thousand per month. Round to 2 decimal places.
115.
The number of hours, H, of daylight in Madrid as a function of the date is given by the
formula
12 2.4sin(0.0172( 80))Ht= + −
, where t is the number of days since the
beginning of the year. What are the units of
dH
dt
?
116.
The number of hours, H, of daylight in Madrid as a function of the date is given by the
formula
12 2.4sin(0.0172( 80))Ht= + −
, where t is the number of days since the
beginning of the year. What is
150t
dH
dt =
? Round to 3 decimal places.
difficulty: medium section: 3.5
Chapter 3
117.
Find the derivative of
22
( ) sin(2 ) cos(2 )f w w w=+
.
A)
22
4 cos(2 ) 4 sin(2 )w w w w−
B)
22
4 sin(2 ) 4 cos(2 )w w w w−
C)
22
2cos(2 ) 2sin(2 )ww−
D)
22
cos(2 ) sin(2 )ww−
functions. difficulty: medium section: 3.5
118.
Find the derivative of
( ) cos(sin(7 ))g x x=
.
A)
7cos(7 )sin(sin(7 ))xx
B)
7cos(7 )sin(sin(7 ))xx−
C)
7sin(cos(7 ))x−
D)
7sin(cos(7 ))x
functions. difficulty: medium section: 3.5
119.
Find the derivative of
cos sin
() zz
h z e e=+
.
A)
cos sinzz
ee
+
B)
cos sin
cos sin
zz
ze ze+
C)
cos sin
sin cos
zz
ze ze
−+
D)
cos sin
sin cos
zz
ze ze−
functions. difficulty: medium section: 3.5
120.
When
5w=
, the graph of
6
( ) cos( )f w w=
is
A)
increasing and concave up
B)
increasing and concave down
C)
decreasing and concave up
D)
decreasing and concave down
Chapter 3
121.
The height off the ground, in meters, of a person riding a Ferris wheel is represented by
the function
2
( ) 250cos 55
5
h t t
= − +
, where time is in seconds. Find
(1)h
.
Explain what this represents in terms of the passenger on the Ferris wheel.
Part A:
298.783
difficulty: medium section: 3.5
122.
( )
4
sin ( )
dx
dx
is
sin(4 )x
A)
True
B)
False
functions. difficulty: easy section: 3.5
123.
True or False?
( )
sin(x) cos(x)
d
dx
is equivalent to
cos(2x)
Ans:
True
difficulty: easy section: 3.5
124.
cos(x)
sin(x)
d
dx
difficulty: medium section: 3.5
125.
Differentiate
siny a bx=−
. Assume a and b are positive constants.
A)
cosab bx−
B)
cosa bx−
C)
cosab bx
D)
cosa bx
functions. difficulty: easy section: 3.5