Give an appropriate answer.
82)
If g (–2) = – 12 and h (–2) =2, find f (–2) for f(x) =3g(x) +3h(x) +1.
82)
A)
–30
B)
–29
C)
–41
D)
–42
Let f(x) = 8x2– 5x and g(x) = 7x +
9.
Find the composite.
83)
g[f(k)]
83)
A)
56k2+ 35k + 9
B)
56k2– 35k + 9
C)
392k2+ 973k + 603
D)
392k2– 973k + 603
Write an equation of the tangent line to the graph of y = f(x) at the point on the graph where x has the indicated value.
84)
f(x) =
–5x2+ 10
–3x – 2 , x = 0
84)
A)
y = – 15
2x + 5
B)
y =15
2x + 5
C)
y = – 15
2x – 5
D)
y =15
2x – 5
Solve the problem.
85)
The polynomial C(x) = – 0.006x4+ 0.140x3– 0.53x2+ 1.79x measures the concentration of a dye in
the bloodstream x seconds after it is injected. Find the rate of change of concentration with respect
to time.
85)
A)
dC
dt = – 0.018x3+ 0.280x2– 0.53x + 1.79
B)
dC
dt = – 0.024x4+ 0.420x3– 1.06x2+ 1.79x
C)
dC
dt = – 0.024x3+ 0.420x2– 1.06x + 1.79
D)
dC
dt = – 0.006x3+ 0.140x2– 0.53x + 1.79
21
Find the equation of the tangent line to the graph of the given function at the given value of x.
86)
f(x) =(x2+12)3/4; x =2
86)
A)
y =3
4x +5
B)
y =3
2x
C)
y =3
2x +5
D)
y =3
2x +11
Find the slope of the line tangent to the graph of the function at the given value of x.
87)
y = 4x3/2 – 5x1/2; x = 16
87)
A)
197
8
B)
101
4
C)
91
4
D)
187
8
Use the product rule to find the derivative.
88)
f(x) = (3x4+ 8)2
88)
A)
f'(x) = 144x15 + 96x3
B)
f'(x) = 6x4+ 16
C)
f'(x) = 9x16 + 64
D)
f'(x) = 72x7+ 192x3
Provide an appropriate response.
89)
A quantity Q is increasing by 900 per year at the present time. This means that . (Provide a
statement that is always true, involving either Q‘(0), Q(0), Q(900), or Q'(900).)
89)
A)
Q'(900) = 0
B)
Q(0) =900
C)
Q(900) = 0
D)
Q'(0) =900
Find the derivative.
90)
y =6x – 1
90)
A)
6x – 1 ln x
B)
6x – 1 ln 6
C)
6 ln 6
D)
6x – 1 ln 6x – 1
91)
y = (3x2+ 5x + 1)3/2
91)
A)
dy
dx =3
2(6x + 5)(3x2+ 5x + 1)1/2
B)
dy
dx =3
2(3x2+ 5x + 1)1/2
C)
dy
dx = (6x + 5)(3x2+ 5x + 1)1/2
D)
dy
dx = (3x2+ 5x + 1)1/2
92)
y =4ex
2ex+ 1
92)
A)
ex
(2ex+ 1)2
B)
4ex
(2ex+ 1)2
C)
4ex
(2ex+ 1)
D)
4ex
(2ex+ 1)3
Solve the problem.
93)
The natural resources of an island limit the growth of the population to a limiting value of 3230.
The population of the island is given by the logistic equation
P(t) =3230
1 +4.01e–0.36t,
where t is the number of years after 1980. What is the population of the island in 1984?
93)
A)
1574
B)
1491
C)
851
D)
1656
Find the derivative.
94)
y =9x–2+ 4x3+ 2x, find f'(x)
94)
A)
–18x–3+ 12x2+ 2
B)
–18x–1+ 12x2
C)
–18x–1+ 12x2+ 2
D)
–18x–3+ 12x2
23
Solve the problem.
95)
The nationwide attendance per day for a certain motion picture can be approximated using the
equation A(t) =12t2e–t, where A is the attendance per day in thousands of persons and t is the
number of months since the release of the film. Find and interpret the rate of change of the daily
attendance after 4 months.
95)
A)
3.517 thousand persons/day · month; the daily attendance is increasing.
B)
–3.517 thousand persons/day · month; the change in daily attendance is decreasing.
C)
1.758 thousand persons/day · month; the change in the daily attendance is increasing.
D)
–1.758 thousand persons/day · month; the daily attendance is decreasing.
The table lists the values of the functions f and g and their derivatives at several points. Use the table to find the
indicated derivative.
96)
x 1 2 3 4
f(x) 4 3 1 2
f'(x) –5 3 –5–6
g(x) 3 4 1 2
g'(x) 1 6 4 –2
Find Dx(g[f(x)]) at x = 2.
96)
A)
12
B)
–30
C)
4
D)
3
Use the quotient rule to find the derivative.
97)
y =x2– 4
x
97)
A)
dy
dx = 1 +4
x2
B)
dy
dx = 1 –4
x2
C)
dy
dx = 1 +4
x
D)
dy
dx = x +4
x2
Find the derivative of the function.
98)
y = log |–9x|
98)
A)
x
ln 10
B)
–ln 10
x
C)
1
x(ln 10)
D)
–1
x (ln 10)
Find the derivative.
99)
y = 4ex2
99)
A)
8xe2x
B)
8xex2
C)
8xe
D)
8xe4x2
Find the derivative of the function.
100)
y = ln (4+x2)
100)
A)
8
x
B)
2x
x2+ 4
C)
1
2x + 4
D)
2
x
Find the slope of the line tangent to the graph of the function at the given value of x.
101)
y = – 8x–1+ 5x–2; x = 2
101)
A)
–3
4
B)
13
4
C)
3
4
D)
–13
4
25
Find f[g(x)] and g[f(x)].
102)
f(x) = 7/(x)4; g(x) = 2x3
102)
A)
f[g(x)] = 7x12/686
g[f(x)] = x12/16
B)
f[g(x)] = 7x12/16
g[f(x)] = x12/686
C)
f[g(x)] = 686/7x12
g[f(x)] = 16/(x)12
D)
f[g(x)] = 7/16x12
g[f(x)] = 686/(x)12
Solve the problem.
103)
The velocity of water in ft/s at the point of discharge is given by v =11.12 P, where P is the
pressure in lb/in.2of the water at the point of discharge. Find the rate of change of the velocity with
respect to pressure if the pressure is 10.00 lb/in.2.
103)
A)
.5560 ft/s per lb/in.2
B)
3.52 ft/s per lb/in.2
C)
17.58 ft/s per lb/in.2
D)
1.7582 ft/s per lb/in.2
104)
Exposure to ionizing radiation is known to increase the incidence of cancer. One thousand
laboratory rats are exposed to identical doses of ionizing radiation, and the incidence of cancer is
recorded during subsequent days. The researchers find that the total number of rats that have
developed cancer t months after the initial exposure is modeled by N(t) =1.04t2.3 for 0 t
10
months. Find the rate of growth of the number of cancer cases at the 7th month. Round your
answer to the nearest tenth, if necessary.
104)
A)
25 cases/month
B)
34 cases/month
C)
30 cases/month
D)
210.1 cases/month
105)
Suppose that the demand function for x units of a certain item is p =100 +180 ln(x + 5)
x, where p is
the price per unit, in dollars. Find the marginal revenue.
105)
A)
dR
dx =180 [x – (x + 5) ln(x + 5)]
x2(x + 5)
B)
dR
dx =180[x –[ln(x + 5) ]2]
x2 ln(x + 5)
C)
dR
dx =100 +180
x + 5
D)
dR
dx =100 +180
ln(x + 5)
Find the derivative.
106)
y =5–x
106)
A)
– ln 5 (5–x)
B)
–5–x
C)
ln 5 (5–x)
D)
5–x
Let f(x) = 8x2– 5x and g(x) = 7x +
9.
Find the composite.
107)
f[g(–4)]
107)
A)
765
B)
2983
C)
148
D)
37
Solve the problem.
108)
The sales in thousands of a new type of product are given by S(t) =140 – 30e–0.5t, where t
represents time in years. Find the rate of change of sales at the time when t =3.
108)
A)
–66.9 thousand per year
B)
3.4 thousand per year
C)
–3.4 thousand per year
D)
66.9 thousand per year
Write the function as the composition of two functions f and g such that y = f[g(x)]).
109)
y =1
x2– 8
109)
A)
f(x) =1
x, g(x) =x2– 8
B)
f(x) =1
8, g(x) =x2– 8
C)
f(x) =1
x2, g(x) = x – 8
D)
f(x) =1
x2, g(x) = – 1
8
Solve the problem.
110)
The energy loss E (in joules/kilogram) due to friction when water flows through a pipe is given by
E = 0.020(L/D)v2. In the formula, L is the pipe length (in m), D is the pipe diameter (in m), and v is
the water velocity (in m/s). Find a formula for the instantaneous rate of change of energy with
respect to velocity.
110)
A)
dE/dv = 0.04(L/D)v
B)
dE/dv = (L/D)v
C)
dE/dv = 0.04(L/D)v2
D)
dE/dv = 0.02(L/D)v
111)
Murrel‘s formula for calculating the total amount of rest, in minutes, required after performing a
particular type of work activity for 30 minutes is given by the formula R(w) =30(w – 4)
w – 1.5 , where w
is the work expended in kilocalories per min. A bicyclist expends 9 kcal/min as she cycles home
from work. Find R'(w) for the cyclist; that is, find R'(9).
111)
A)
1.33 min2/kcal
B)
1.6 min2/kcal
C)
20 min2/kcal
D)
1.07 min2/kcal
Solve the following.
112)
Find all points of the graph of f(x) =2x2+ 6x whose tangent lines are parallel to the line y – 22x = 0.
112)
A)
(7, 140)
B)
(5, 80)
C)
(4, 56)
D)
(6, 108)
28
Write the function as the composition of two functions f and g such that y = f[g(x)]).
113)
y =2
3x + 2
113)
A)
f(x) =2
x, g(x) =3x + 2
B)
f(x) =2, g(x) =3+ 2
C)
f(x) =2
x, g(x) =3x + 2
D)
f(x) =3x + 2, g(x) =2
Find the derivative.
114)
y = (2x – 1)3(x + 7)–3
114)
A)
dy
dx = 45(2x – 1)2(x + 7)–4
B)
dy
dx = 45(2x – 1)3(x + 7)–2
C)
dy
dx = 45(2x – 1)3(x + 7)–4
D)
dy
dx = 45(2x – 1)2(x + 7)–3
Find f[g(x)] and g[f(x)].
115)
f(x) = x2+ 2x + 3; g(x) = x – 4
115)
A)
f[g(x)] = x2+ 2x – 1
g[f(x)] = x2+ 6x + 11
B)
f[g(x)] = x2+ 6x + 11
g[f(x)] = x2+ 2x – 1
C)
f[g(x)] = x2+ 2x – 1
g[f(x)] = x2– 6x + 11
D)
f[g(x)] = x2– 6x + 11
g[f(x)] = x2+ 2x – 1
Find the derivative of the function.
116)
y = log |3– x|
116)
A)
–1
ln 10 (3– x)
B)
–3– x
ln 10
C)
–1
ln 10
D)
1
ln 10 (3– x)
117)
y = log (2x)
117)
A)
1
x
B)
1
x(ln 2)
C)
1
ln 10
D)
1
x(ln 10)
Find the derivative.
118)
y = x x2+ 1
118)
A)
dy
dx =2x2+ 1
x2+ 1
B)
dy
dx =x2+ 1
2x2+ 1
C)
dy
dx =x2+ 1
x2+ 1
D)
dy
dx =x2+ 1
x2+ 1
Find the derivative of the function.
119)
y = ln 5x2
119)
A)
1
2x + 5
B)
10
x
C)
2
x
D)
2x
x2+ 5
Provide an appropriate response.
120)
If Q =89 – e–0.5t what happens to Q and to Q’ as t increases?
120)
A)
Q decreases and Q’ increases.
B)
Q increases and Q‘ increases.
C)
Q increases and Q’ decreases.
D)
Q decreases and Q’ decreases.
121)
What rule is applied first to find the derivative of the function
f(x) =(4x3– 4) 3x3– 4
3x + 3 ?
121)
A)
Quotient rule
B)
Product rule
C)
Power rule
D)
Square root rule
Write the function as the composition of two functions f and g such that y = f[g(x)]).
122)
y =1
x2+ 8
122)
A)
f(x) =1
x2, g(x) =8
B)
f(x) = x + 8, g(x) =1
x2
C)
f(x) =1
x, g(x) =1
x+ 8
D)
f(x) = x, g(x) =1
x+ 8
B
Find an equation for the line tangent to given curve at the given value of x.
123)
y =x2+4; x =3
123)
A)
y =6x – 10
B)
y =3x – 5
C)
y =6x – 5
D)
y =6x – 14
C
Using a graphing calculator, find the values of x for which f(x) = 0, to three decimal places.
124)
f(x) =7–x2x2–7
124)
A)
0
B)
0, –1.685, 1.685
C)
0, –2.196, 2.196
D)
There are no real values of x for which f(x) = 0.
C
A
Provide an appropriate response.
125)
True or false? The derivative of the product of two functions is the product of their derivatives.
125)
A)
True
B)
False
Solve the problem.
126)
The profit in dollars from the sale of x thousand compact disc players is P(x) = x3– 9x2+ 12x + 8.
Find the marginal profit when the value of x is 10.
126)
A)
$190
B)
$198
C)
$140
D)
$132
127)
The total revenue from the sale of x stereos is given by R(x) =4000 1 –x
700 2. Find the marginal
average revenue.
127)
A)
5.71 –4000
x2
B)
0.008–700
x2
C)
5.71 –700
x2
D)
0.008–4000
x2
Find the derivative.
128)
y =97x
128)
A)
63 (ln 9) 97x
B)
9 (ln 7) 97x
C)
63 (ln 7) 97x
D)
7 (ln 9) 97x
Find the derivative of the given function.
129)
y = (x2+4)3
129)
A)
3x5+48x3+96x
B)
6x5+40x3+96x
C)
6x5+24x3+48x
D)
6x5+48x3+96x
32