Chapter 3
1.
Given
() x
f x e x
−
=+
, find
‘( )fx
.
A)
‘( ) 1
x
f x e−
= − +
B)
‘( ) 1
x
f x e−
=+
C)
‘( ) 1
x
f x xe−
= − +
D)
‘( ) 1
x
f x xe−
=+
2.
Given
() x
f x e x
−
=+
, find
‘‘( )fx
.
A)
‘‘( ) x
f x e−
=
B)
‘‘( ) x
f x e−
=−
C)
‘‘( ) x
f x xe−
=
D)
‘‘( ) x
f x xe−
=−
3.
If
( ) cos
x
f x e x
−
=
for
02πx
, what is
‘( )fx
?
A)
cos sin
xx
e x e x
−−
−−
B)
cos sin
xx
e x e x
−−
−+
C)
sin
x
ex
−
−
D)
sin
x
ex
−
4.
If
32
( ) 5 3 7f x x x x= + − +
, find
3
3
dy
dx
Chapter 3
5.
Find the first derivative of
7yx=
.
A)
7
2x
B)
7
x
C)
7
2x
D)
7
2
x
6.
Find the first derivative of
32
yy
+
.
A)
2
2
2
3yy
+
B)
2
2
2
3yy
−
C)
2
32y+
D)
2
32y−
7.
Find the first derivative of
22
23t z z
z
−
= + +
.
A)
3
6
22zz
+−
B)
23
26
2zzz
+−
C)
23
26
2zzz
−−
D)
6
22zz
+−
Chapter 3
8.
Find the first derivative of
π6s t t=+
.
A)
26t
+
B)
6
C)
π1 1
π6
t−−
D)
π1
π6t−+
functions and sums of power functions. difficulty: medium section: 3.1
9.
Find the first derivative of w = x2 + ax.
A)
2x + a
B)
2x
C)
(2+ a)x
functions and sums of power functions. difficulty: easy section: 3.1
10.
Find the first derivative of
23
1
3y x x x
= + −
.
A)
22
92x x x−
++
B)
2
9 2 1xx+−
C)
22
92x x x−
+−
D)
2
9 2 1xx++
functions and sums of power functions. difficulty: easy section: 3.1
11.
Find the first derivative of
33
2
63t z z
z
−
= + +
.
A)
2
34
12 9
3zzz
++
B)
2
34
12 9
3zzz
−−
C)
2
4
69
3zzz
+−
D)
2
2
69
3zzz
+−
functions and sums of power functions. difficulty: easy section: 3.1
Chapter 3
Page 4
12.
Find the first derivative of
68s t t=+
A)
68t+
B)
68+
C)
61
68t−+
D)
1/ 2 6 8t−+
13.
Find the first derivative of w(x) = bx2 + xu.
A)
x2 + uxu-1
B)
2bx + ux
C)
2bx + u
D)
2bx + uxu-1
14.
The total cost, in dollars, to produce q units is given by
2
( ) 500 3C q q q= + +
. Find
‘(15)C
Ans:
$33
of power functions. difficulty: easy section: 3.1
15.
The equation for the tangent line to the curve
24xy+=
when x = 3 is y = _____x +
_____.
Part A:
Part B:
13
difficulty: easy section: 3.1
16.
A. The equation for the tangent line to the curve
36xy+=
when x = 4 is y = _____x
+ _____.
B. The tangent line meets the y-axis at y = _____ and the x-axis at x = _____.
Part A:
Part B:
134, 2.792
difficulty: medium section: 3.1
functions and sums of power functions. difficulty: medium section: 3.1
Chapter 3
17.
The equation for the tangent line to the function
42
( ) 9 2f x x x= − +
at x = 1 is y =
_____x + _____.
Part A:
Part B:
8
difficulty: easy section: 3.1
18.
The curve
42
( ) 25 7f x x x= − +
has a horizontal tangent at which of the following
points? Select all that apply.
A)
1
B)
-1
C)
52
2
D)
52
2
−
E)
0
a tangent line. difficulty: hard section: 3.1
19.
Find the derivative of
31
26
yx x
=−
A)
2
2
1
66
xx
+
B)
2
2
1
66
xx
−
C)
21
66
x−
D)
21
66
xx
+
functions and sums of power functions. difficulty: medium section: 3.1
Chapter 3
Page 6
20.
Find the derivative of
3
( ) 10f x x x=+
.
A)
2
53x
x
−+
B)
2
53x
x+
C)
2
53xx+
D)
2
10 3x+
21.
Find the derivative of
32
2
82
() tt
gt t
−+
=
.
A)
12 1t−
B)
1
12 1tt
−+
C)
3
4
8t
−
D)
3
4
8t
+
22.
A tomato is thrown from the top of a tomato cart its distance from the ground, in feet, is
modeled by the equation
2
( ) 16 32 3.4d t t x= − + +
where t is measured in seconds and
the initial height of the cart is
3.4
feet.
(A) At what time is the tomato at its maximum height?
(B) What is the maximum height?
(C) What is the initial velocity of the tomato (at t = 0)?
Chapter 3
23.
A landlord rents an apartment building with 250 apartments. The monthly profit, in
dollars, can be modeled by
2
( ) –10 4000 –100000P x x x=+
, where x is the number of
apartments rented. How many apartments should be rented to maximize profit?
Ans:
200
difficulty: medium section: 3.1
24.
Let
32
( ) 3 3g t t t t= − +
. Find
‘( )gt
.
of power functions. difficulty: easy section: 3.1
25.
Let
32
( ) 2 4g t t t t= − +
. Find
‘‘( )gt
.
of power functions. difficulty: easy section: 3.1
26.
Consider the function
43
( ) 4 5 3f x x x= − +
. We know that
()fx
is increasing when x
> _____.
Ans:
0.9375
27.
Consider the function
43
( ) 4 5 1f x x x= − +
. We know that
()fx
is concave down
when _____ < x < _____.
Part A:
0
Part B:
0.625
difficulty: medium section: 3.1
28.
A power function of the form
() n
f x ax=
has
‘(2)f=
-1/2 and
‘(4)f=
-1/8. What is
n?
Ans:
of power functions. difficulty: medium section: 3.1
Chapter 3
29.
Given
2
( ) 3f x x x=−
and
32
( ) 3 7g x x x= + −
, find
( )
( ) 4 ( )
dg x f x
dx −
.
Learning Objectives: Use formulas to compute derivatives of power functions and sums
of power functions. difficulty: medium section: 3.1
30.
A. Find the equation of the line tangent to the graph of
x
ye=
at
xa=
.
B. Find the x-intercept of this line.
C. Find the y-intercept of this line.
Learning Objectives: Use formulas to compute derivatives of exponential functions and
y = ln x. difficulty: medium section: 3.2
31.
The population of a town is approximated by the function
94, 000(1.02)t
, where t is the
number of years since 1980. Find
‘(20)P
. Round to the nearest whole number.
Ans:
2766
32.
The value of a car is falling at 10% per year so that if
0
C
is the purchase price of the
car in dollars, its value after t years is given by
0
( ) (0.9)t
V t C=
. How fast is the car
depreciating after 4 years?
A)
4
0(0.9)C−
dollars per year
B)
4
0ln(0.9)(0.9)C−
dollars per year
C)
0.9 4
0(0.9)Ce−
dollars per year
D)
3
0(4)(0.9)C−
dollars per year
Ans: B Learning Objectives: Interpret the meaning of derivatives of exponential
and logarithmic functions. difficulty: medium section: 3.2
Chapter 3
33.
Find the equation of the tangent line to the curve
x
ye=
which passes through the
origin.
y = ln x. difficulty: hard section: 3.2
34.
Find the derivative of
66
x−
.
A)
ln(6)6x
B)
1
ln(6)6x−
C)
1
6x
x−
D)
ln(5)5x
functions and y = ln x. difficulty: medium section: 3.2
35.
Find the derivative of
( ) 4 3
xx
f x e=−
.
A)
ln(4) ln(3)3
xx
e−
B)
4 ln(3)3
xx
e−
C)
1
43
xx
ex
−
−
D)
11
43
xx
xe x
−−
−
36.
Find the derivative of
π
( ) 3 x
g x e=
.
A)
π
3x
e
B)
π1
3πx
xe −
C)
π
3πx
e
D)
π
3ln(π) x
e
functions and y = ln x. difficulty: medium section: 3.2
Chapter 3
37.
Find the derivative of
2
( ) (ln 2) (ln 2) x
f x x e=+
A)
2(ln 2) (ln 2)
x
xe+
B)
(ln 4) (ln 2) x
xe+
C)
1
2(ln 2) (ln 2) x
x xe −
+
D)
1
(ln 4) (ln 2) x
x xe −
+
38.
Find the derivative of
6
1
( ) 6 6x
g x x x
= − +
A)
1
7 (6 )
x
x−
+
B)
7 / 6
1
6 ln 6(6 )
6
x
x
++
C)
7 / 6
6
6 ln 6(6 )
x
x
++
D)
1
1/ 6
1
6 (6 )
6
x
x
x
−
++
39.
Find the derivative of
5
π 5 5
( ) (π ) π
t
h t t t= + +
.
A)
5
π 5 5 4
ln( ) ln(π )(π ) 5π
t
t t t++
B)
5
5(π 1) 5 1 4
π (π ) 5π
t
t t t
−−
++
C)
5
5(π 1) 5 5 4
π ln(π )(π ) 5π
t
tt
−++
D)
5
4(π 1) 5 5 4 5
5π ln(π )(π ) 5π
t
t t t
−+ + +
Chapter 3
40.
Find the derivative of
6
( ) 2
t
t
g t e e
e
= + +
.
A)
11
(6 / )tt
t e te
−−
+
B)
6
ln t
te
e
+
C)
(6 / )tt
ee+
D)
6t
te
e
−+
functions and y = ln x. difficulty: medium section: 3.2
41.
( )
10
e
d
dx
Ans:
0
y = ln x. difficulty: easy section: 3.2
42.
( )
4x
d
dx
43.
The population of Ghostport has been declining since the beginning of 1800. The
population, in thousands, is modeled by
–0.005
( ) 20 t
P t e=
, where t is measured in years.
At what rate was the population declining at the beginning of 2000?
Ans:
0.036788 thousand people per year or 36.8 people per year.
y = ln x. difficulty: medium section: 3.2
44.
Consider the function
3
( ) 3 3x
g x x=+
. Give the equation of the tangent line at x = 2.
Round the coefficients to 2 decimal places.
y = ln x. difficulty: medium section: 3.2
Chapter 3
Page 12
45.
With a yearly inflation rate of 3%, prices are described by
0(1.03)t
PP=
, where
0
P
is
the price in dollars when t = 0 and t is time in years. If
0
P
= 1.2, how many cents per
year are prices rising when t = 12? Round to the nearest tenth of a cent.
46.
The equation of the tangent line to the curve
( ) 5 x
g x x e=−
at the point where it
touches the y-axis is y = _____x + _____.
Part A:
Part B:
y = ln x. difficulty: medium section: 3.2
47.
Given
23
xx
y=+
, find
‘y
.
y = ln x. difficulty: medium section: 3.2
48.
Given
56
xx
y=+
, find
”y
.
y = ln x. difficulty: medium section: 3.2
49.
Find the first derivative of
ln( 4)yx=+
.
A)
1
4x+
B)
4
4x+
C)
ln( 4)x+
D)
4ln( 4)x+
composition of functions. difficulty: easy section: 3.3
Ans:
5.1
logarithmic functions. difficulty: medium section: 3.2
Chapter 3
50.
Find the first derivative of
4
lnsz=
.
A)
4
1
z
B)
4
z
C)
4
4
z
D)
3
4ln z
51.
Find the first derivative of
3
ln( 4)tx=+
.
A)
3
1
4x+
B)
2
1
3x
C)
2
3
3
4
x
x+
D)
23
3 ln( 4)xx+
52.
Find the first derivative of
x
ya=
.
A)
1x
xa −
B)
x
a
C)
ln a
a
D)
ln
x
aa
Chapter 3
53.
Find the first derivative of
2
x
ye
−
=
A)
2
2x
xe−
−
B)
2
x
e−
C)
2
2x
xe
−
−
D)
2
2x
xe
−
−+
54.
Find the first derivative of
xe
s e x=+
.
A)
xe
ex+
B)
1xe
e ex −
+
C)
1xe
xe x
−+
D)
11xe
xe ex
−−
+
55.
Find the first derivative of
3x
te
+
=
.
A)
3
3x
e+
B)
2
( 3) x
xe
+
+
C)
3
( 3) x
xe
+
+
D)
3x
e+
56.
Find the first derivative of
ln( 3)
x
ye=+
.
A)
1
3
x
e+
B)
3
3
x
e+
C)
3
x
x
e
e+
D)
( 3) ln( 2)
xx
ee++
Chapter 3
57.
If $100 is invested at r % interest per year, compounded yearly, then the yield after 15
years is given by
15
100 1 100
r
F
=+
. Find
8r
dF
dr =
.
58.
Find the derivative of
33
( ) (4 )
x
h x x e=+
.
A)
3 2 2
3(4 ) (12 )
xx
x e x e++
B)
32
3(4 )
x
xe+
C)
22
3(12 )
x
xe+
D)
3 2 2
3(4 ) (12 )(24 )
x x x
x e x e x e+ + +
59.
Find the derivative of
3
( ) 6 x
g x x e=+
.
A)
3
1
26 x
xe+
B)
2
3
18
26
x
x
xe
xe
+
+
C)
2
18 x
xe+
D)
32
16 (18 )
2
xx
x e x e++
Chapter 3
60.
Find the derivative of
33
3
(3 )
()
3
x
x
xe
hx
xe
+
=+
.
A)
22
3
3(3 )
3
x
x
xe
xe
+
+
B)
3 5 / 2 2
7(3 ) (9 )
2
xx
x e x e++
C)
3 3/ 2 2
5(3 ) (9 )
2
xx
x e x e++
D)
23
2
(9 )
9
x
x
xe
xe
+
+
61.
Find the derivative of
35
( ) 5 x
fx −
=
.
A)
35
3(ln 5)5 x−
B)
35
5(ln 3)5 x+
C)
35
3(5 )
x−
D)
34
(3 5)5 x
x−
−
62.
Is the derivative of
2
() xx
g x e e=+
given by
( ) ( )
22
–1/ 2
1
‘( ) 2
2
x x x x
g x e e e xe= + +
?
63.
Consider the function
22
( ) ( )g x ax b=+
, where a and b are constants. Find
‘( )gx
.
64.
Consider the function
22
( ) ( )g x ax b=+
, where a and b are constants. Find
‘‘( )gx
.
Chapter 3
65.
Consider the function
22
( ) ( )g x ax b=+
, where a and b are constants. Find
‘‘‘( )gx
.
functions. difficulty: medium section: 3.3
66.
The population of Mexico in millions is described by the formula
0.027
( ) 67 t
P t e=
,
where t is the number of years after 1980. In the year 2025, the population will be
increasing at the rate of _____ million people per year. Round to 2 decimal places.
Ans:
6.10
67.
The population of Mexico in millions is described by the formula
0.027
( ) 67 t
P t e=
,
where t is the number of years after 1980. How many years will it take for the
population to triple? Round to 2 decimal places.
Ans:
40.69
68.
The population of Mexico in millions is described by the formula
0.027
( ) 67 t
P t e=
,
where t is the number of years after 1980. How many years will it take before the
population is increasing at a rate of 8 million people per year? Round to 2 decimal
places.
Ans:
55.06
functions. difficulty: medium section: 3.3
69.
What is the equation of the tangent line to
6
( ) 4 xx
f x e e−
=+
at the point where x = 0?
Chapter 3
70.
What is the equation of the line tangent to the curve
37yx=+
at the point above x =
3? Leave your coefficients in fraction form, such as “a/b”. They can be improper
fractions.
Learning Objectives: Use the chain rule to find the derivative of a composition of
functions. difficulty: hard section: 3.3
71.
The price in dollars of a house during a period of mild inflation is described by the
formula
0.05
( ) (80,000) t
P t e=
, where t is the number of years after 1990. By how many
dollars per year will the value of the house be increasing in the year 2015? Round to
the nearest dollar.
Ans:
$13,961
Learning Objectives: Use the chain rule to find the derivative of a composition of
functions. difficulty: medium section: 3.3
72.
The price in dollars of a house during a period of mild inflation is described by the
formula
0.05
( ) (90,000) t
P t e=
, where t is the number of years after 1990. How many
years will it take for the house to triple in value? Round to the nearest year.
Ans:
22
Learning Objectives: Use the chain rule to find the derivative of a composition of
functions. difficulty: medium section: 3.3
73.
The price in dollars of a house during a period of mild inflation is described by the
formula
0.05
( ) (120,000) t
P t e=
, where t is the number of years after 1990. How many
years will it be before the house in increasing in value at a rate of $15,000 per year?
Round to the nearest year.
Ans:
18
Learning Objectives: Use the chain rule to find the derivative of a composition of
functions. difficulty: medium section: 3.3
74.
The cost to produce q aircraft in South Africa is given by the function
0.848
( ) 2.5C q q=
,
with C in millions of rands. At a production level of 60 aircraft, how many million
rands will it cost to produce one more aircraft? Round to 2 decimal places.
Ans:
1.14
Learning Objectives: Use the chain rule to find the derivative of a composition of
functions. difficulty: medium section: 3.3
Chapter 3
75.
A drug has a concentration in the body given in ng/ml by the function
0.25
( ) 15 t
f t e−
=
,
where t is the number of hours after it was administered. By how many ng/ml per hour
is the amount of the drug in the body changing after 7 hours? Round to 2 decimal
places.
Ans:
Learning Objectives: Use the chain rule to find the derivative of a composition of
functions. difficulty: medium section: 3.3
76.
What is
(2 ln )
x
dx
dx
?
A)
2
ln 2 2 ln
x
xxx
+
B)
1
ln 2 2x
x
+
C)
1
ln 2 2x
x
D)
12
2 ln
x
x
xx
x
−+
Ans: A Learning Objectives: Use the product rule to find the derivative of a
product of functions. difficulty: medium section: 3.4
77.
What is
9
18x
d
dx e−
+
?
A)
0
B)
2
72
(1 8 )
x
x
e
e
−
−
+
C)
2
72
(1 8 )
x
x
e
e
−
−
−
+
D)
9
8x
e−
−
Chapter 3
78.
Compute the derivative of
2
4
3
7
4xx
yx x
−
=+
.
A)
3 3 2
16 14x x x
−−
++
B)
3 3 2
16 14x x x
−−
+−
C)
3 3 2
16 14x x x
−−
−+
D)
3 3 2
16 14x x x
−−
−−
79.
Compute the derivative of
2
3.5x
yx
=+
.
A)
x-1
x(3.5)
B)
x
(ln3.5)(3.5 ) 4 x−
C)
( )
x
3
2
(ln3.5)(3.5 )
x
−
D)
( )
x
3
1
(ln3.5)(3.5 )
x
−
80.
Find
dy
dx
for
(1.01)
ex
yx=
.
A)
11.01 ln(1.01)(1.01)
e x e x
ex x
− +
B)
11
1.01 (1.01)
e x e x
ex x x
−−
+
C)
1.01 ln(1.01)(1.01)
e x e x
xx +
D)
1
1.01 (1.01)
e x e x
x x x −
+