71)
A point is moving on the graph of xy = 24. When the point is at (4, 6), its x coordinate is increasing
at the rate of 9 units per second. How fast is the y coordinate changing at that moment?
71)
A)
decreasing at 27
2 units per second
B)
increasing at 27
2 units per second
C)
increasing at 9 units per second
D)
decreasing at 9 units per second
Find the derivative.
72)
Find f'(x) for f(x) = (4x2+ 3x)2.
72)
A)
f'(x) = 64x3+ 72x2+ 18x
B)
f'(x) = 64x3+ 36x2+ 18x
C)
f'(x) = 32x3+ 36x2+ 9x
D)
f'(x) = 32x3+ 36x2+ 18x
Find f(x).
73)
f(x) =9ex–2x4
73)
A)
9xex–1–8x4
B)
9ex–8x4
C)
9ex–8x3
D)
9ex–4x3
Solve.
74)
The resale value R (in dollars) of a company car after t years is estimated to be given by
R(t) = 22,500(0.84)t. What is the rate of depreciation (in dollars per year) after 4 years?
74)
A)
–$1641/yr
B)
–$1953/yr
C)
–$15,529/yr
D)
–$1378/yr
Solve the problem.
75)
Given the revenue and cost functions R = 26x – 0.3x2 and C = 3x + 10, where x is the daily
production, find the rate of change of profit with respect to time when 20 units are produced and
the rate of change of production is 7 units per day per day.
75)
A)
$149.00 per day
B)
$77.00 per day
C)
$156.80 per day
D)
$98.00 per day
Provide an appropriate response.
76)
Find f'(x) for f(x) = (4x2+ 3x)2.
76)
A)
f'(x) = 32x3+ 36x2+ 18x
B)
f'(x) = 32x3+ 36x2+ 9x
C)
f'(x) = 64x3+ 72x2+ 18x
D)
f'(x) = 64x3+ 36x2+ 18x
C
77)
Use graphical approximation methods to find the point(s) of intersection of f(x) =ex and g(x) =x6
to two decimal places.
77)
A)
(0.87, 0.42), (1.23, 3.41)
B)
(1.23, 3.41)
C)
(–0.87, 0.42)
D)
(–0.87, 0.42), (1.23, 3.41)
D
Find the equation of the line tangent to the graph of f at the indicated value of x.
78)
f(x) =5ex; x = 0
78)
A)
y =5x –5
B)
y = x +5
C)
y =5x
D)
y =5x +5
D
B
D)
Provide an appropriate response.
79)
Find f'(x) for f(x) =log4(x6+ 1)
79)
A)
1
(ln 4)(x6+ 1)
+6x5
B)
6x5
x6+ 1
C)
6x5
(ln 4)(x6+ 1)
D)
6x5(ln 4)
x6+ 1
Solve.
80)
A single bacterium divides every 0.5 hour to produce two complete bacteria. If we start with a
colony of 6000 bacteria, after t hours there will be A(t) = 6000 ·22t = 6000 ·4t bacteria. Find A'(3).
80)
A)
133,084 bacteria
B)
33,271 bacteria
C)
532,337 bacteria
D)
2,129,348 bacteria
Find the percentage rate of change of f(x) at the indicated value of x. Round to the nearest tenth of a percent.
81)
f(x) = 200 + 50x; x = 3
81)
A)
14.3%
B)
33.3%
C)
57.1%
D)
–14.3%
Use the price–demand equation to determine whether demand is elastic, is inelastic, or has unit elasticity at the indicated
values of p.
82)
x = f(p) =301 – 3p; p =72.
82)
A)
Inelastic
B)
Elastic
C)
Unit elasticity
Find dy
dx for the indicated function y.
83)
y =3x2–log5x
83)
A)
6x –1
5 ln x
B)
6x –1
x ln 5
C)
6x +1
5 ln x
D)
3x –1
x ln 5
Find the derivative.
84)
y = (x–2+ x)–3
84)
A)
dy
dx =3x4(2 – x3)
(1 + x3)3
B)
dy
dx =3x4(2 – x3)
(1 + x3)4
C)
dy
dx =3x5(2 – x3)
(1 + x3)4
D)
dy
dx =3x5(2 – x3)
(1 + x3)3
Find all values of x for the given function where the tangent line is horizontal.
85)
f(x) =x
(x2+3)3
85)
A)
±15
5
B)
0
C)
0, ±15
5
D)
±3
5
Differentiate.
86)
Find f'(t) for f(x) =x
9x –3
86)
A)
–3
(9x –3)2
B)
–3x
(9x –3)2
C)
18x –3
(9x –3)2
D)
–3
9x –3
Provide an appropriate response.
87)
Evaluate dy/dt for the function at the point.
x3+ y3= 9; dx/dt = – 3, x = 1, y = 2
87)
A)
–4
3
B)
–3
4
C)
3
4
D)
4
3
Find f(x).
88)
f(x) =x6+3ex
88)
A)
6x5+3ex
B)
6x5+ex
C)
6x +3ex
D)
6x5+3xex–1
Find the equation of the line tangent to the graph of f at the indicated value of x.
89)
f(x) = 2 + ln x4; x = e
89)
A)
y =4x
e+ 2
B)
y =4x
e– 2
C)
y =4x
e
D)
y =4x
e+6
Find the derivative.
90)
Find: dy
dx
88x7– 10
90)
A)
56x6
(8x7–10)7/8
B)
8 78x7– 10
C)
448x678x7– 10
D)
7x6
(8x7–10)7/8
Find the elasticity of the demand function as a function of p.
91)
x = D(p) =300 – p
91)
A)
E(p) =p
600 – 2p
B)
E(p) =p
300 – p
C)
E(p) =p
2p –600
D)
E(p) =1
600 – 2p
Provide an appropriate response.
92)
Find y’ for y = y(x) defined implicitly by 3xy –x2– 4 = 0.
92)
A)
y’ =3x –2y
4
B)
y’ =2
3x
C)
y’ =3y – 2x
3x
D)
y’ =2x – 3y
3x
Differentiate.
93)
Find dy
dx for y =3x –1
5x2+1
93)
A)
dy
dx =–15x2+10x +3
(5x2+1)2
B)
dy
dx =15x3–30x2+13x
(5x2+1)2
C)
dy
dx =45x2–10x +3
(5x2+1)2
D)
dy
dx =–15x2+ 7x +4
(5x2+1)2
Use appropriate properties of logarithms to rewrite f(x), and then find f(x).
94)
f(x) = 1 + ln 5
x4
94)
A)
5
x
B)
–5
x
C)
–4
x
D)
1 –4
x
24
D)
Find f(x).
95)
f(x) =-8ex+5x – 4
95)
A)
-8ex+5x
B)
-8xex–1+5
C)
-8ex+5
D)
-8ex+1
Find the elasticity of the demand function as a function of p.
96)
x = D(p) =800 – p
96)
A)
E(p) =p
800 – p
B)
E(p) = p(800 – p)
C)
E(p) =1
800 – p
D)
E(p) =p
p –800
Find dy
dx for the indicated function y.
97)
y =6x–e5
97)
A)
6x ln 6–e5
B)
6x ln 6
C)
6x ln x –e5
D)
6x
ln 6
Solve the problem.
98)
Suppose that $8000 is invested at an interest rate of 5.5% per year, compounded continuously. How
long would it take to double the investment?
98)
A)
11.6 yr
B)
2 yr
C)
12.6 yr
D)
13.6 yr
Find f(x).
99)
f(x) =-7 ln x –x5+3
99)
A)
–7
x–5x4
B)
7
x–5x4
C)
–7
x–5x
D)
–1
7x –5x4
Provide an appropriate response.
100)
Find t to four decimal places.
e–0.07t = 0.05
100)
A)
–70.1312
B)
44.321
C)
–66.4815
D)
42.7962
Find dy
dx for the indicated function y.
101)
y =5+2x3–7x
101)
A)
6x2–7 ln 7
B)
6x2–7x ln 7
C)
6x2+7x ln 7
D)
6x2–7x ln x
Solve the problem.
102)
The position of a particle at time t is given by s, where s3+ 3st + 4t3– 4t = 0. Find the velocity ds/dt.
102)
A)
ds/dt =4+ 3s – 12t2
3s2– 3t
B)
ds/dt =4– 3s – 12t2
3s2– 3t
C)
ds/dt =4+ 3s – 12t2
3s2+ 3t
D)
ds/dt =4– 3s – 12t2
3s2+ 3t
26
Provide an appropriate response.
103)
A 26–foot ladder is placed against a wall. If the top of the ladder is sliding down the wall at 2 feet
per second, at what rate is the bottom of the ladder moving away from the wall when the bottom of
the ladder is 10 feet away from the wall?
103)
A)
9.6 ft/sec
B)
2.4 ft/sec
C)
5.2 ft/sec
D)
4.8 ft/sec
104)
Find t to four decimal places.
e–t= 0.06
104)
A)
2.6134
B)
2.9134
C)
–2.8134
D)
2.8134
105)
Find f’x for f(x) =(3x +4)2
x3–x2+ 3x . Do not simplify.
105)
A)
(3x + 4)2(3x2– 2x + 3) – 6(x3–x2+ 3x)(3x + 4)
(x3–x2+ 3x)2
B)
(3x + 4)2(3x2– 2x + 3) – 6(x3–x2+ 3x)(3x + 4)
(3x + 4)4
C)
6(x3–x2+ 3x)(3x + 4) –(3x + 4)2(3x2– 2x + 3)
(3x + 4)4
D)
6(x3–x2+ 3x)(3x + 4) – (3x +4)2(3x2– 2x + 3)
(x3–x2+ 3x)2
Find the equation of the line tangent to the graph of f at the indicated value of x.
106)
f(x) =7 ln x; x = 1
106)
A)
y =7x
B)
y =7x – 1
C)
y =7x +7
D)
y =7x –7
Provide an appropriate response.
27
107)
A man with $9000 to invest puts the money into an account that earns 8% compounded
continuously. Graph the corresponding present value function and calculate the number of years
before the $9000 will be due in order for its present value to be $7000. Use the formula P = Ae–rt.
107)
A)
3.14 years
B)
6.28 years
C)
0.84 years
D)
7.45 years
Solve the problem.
108)
How long will it take for the value of an account to be $890 if $350 is deposited at 11% interest
compounded continuously? Round your answer to the nearest hundredth.
108)
A)
0.93 yr
B)
9.33 yr
C)
8.48 yr
D)
10.41 yr
Provide an appropriate response.
109)
A man 6 ft tall walks at a rate of 5 ft/sec away from a lamppost that is 13 ft high. At what rate is the
length of his shadow changing when he is 65 ft away from the lamppost?
109)
A)
30
19 ft/sec
B)
15
19 ft/sec
C)
30
7 ft/sec
D)
325
6 ft/sec
110)
Find dy/dx by implicit differentiation.
2xy –y2= 1
110)
A)
dy
dx =y
y – x
B)
dy
dx =y
x – y
C)
dy
dx =x
x – y
D)
dy
dx =x
y – x
Differentiate.
111)
Find y’ for y =x2
3–2x
111)
A)
–2x2+6x
(3 –2x)2
B)
–6x2+6x
(3 –2x)2
C)
2x3–4x2+6x
(3 –2x)2
D)
3x
(3 –2x)2
Use appropriate properties of logarithms to rewrite f(x), and then find f(x).
112)
f(x) = 5x + 4 ln 2x
112)
A)
5 +2
x
B)
7 +4
x
C)
5 +4
x
D)
5 +8
x
Provide an appropriate response.
113)
Find x to two decimal places.
x = 7,000e0.11
113)
A)
7813.95
B)
8320.50
C)
7831.95
D)
7975.01
Find the derivative.
114)
Find f'(x) for f(x) = (8x – 9)–4.
114)
A)
–32
(8x–9)3
B)
–4
(8x–9)5
C)
–4
(8x–9)3
D)
–32
(8x –9)5
Find f(x).
115)
f(x) =3ex–6x + 2
115)
A)
3ex–4
B)
3xex–1–6
C)
3ex–6
D)
3ex–6x
Solve the problem.
116)
If $5000 is invested at 5.25% compounded continuously, what is the amount in the account after 10
years?
116)
A)
$7420.65
B)
$8442.52
C)
$7625.00
D)
$8452.29
Find the derivative.
117)
Find d
d
4
(2+ 3)5
117)
A)
– 40
(2+ 3)5
B)
40
(2+ 3)6
C)
– 40
(2+ 3)6
D)
– 40
(2+ 3)6
Solve.
118)
A single bacterium divides every 0.5 hour to produce two complete bacteria. If we start with a
colony of 6000 bacteria, after t hours there will be A(t) = 6000 ·22t = 6000 ·4t bacteria. Find A'(t)
and A‘(1).
118)
A)
A'(t) = 6000(ln 2)4t; A'(1) = 16,635 bacteria
B)
A'(t) = 6000(ln 4)2t; A'(1) = 16,635 bacteria
C)
A'(t) = 6000(ln 2)2t; A'(1) = 8317 bacteria
D)
A'(t) = 6000(ln 4)4t; A'(1) = 33,271 bacteria
119)
An experiment was set up to find a relationship between weight and systolic blood pressure in
normal children. Using hospital records for 5000 normal children, the experimenters found that the
systolic blood pressure was given approximately by P(x) = 17.5(1 + ln x), 10
x
100, where P(x) is
measured in millimeters of mercury and x is measured in pounds. What is the rate of change of
blood pressure with respect to weight at the 60–pound weight level?
119)
A)
0.02 mm of mercury per pound of weight gain
B)
0.25 mm of mercury per pound of weight gain
C)
0.29 mm of mercury per pound of weight gain
D)
0.35 mm of mercury per pound of weight gain
Find the equation of the line tangent to the graph of f at the indicated value of x.
120)
f(x) = 1 +4ex; x = 1
120)
A)
y =8ex – 1
B)
y =4ex –8e + 1
C)
y =4ex + 1
D)
y =4ex +8e + 1
31
Use the price–demand equation to determine whether demand is elastic, is inelastic, or has unit elasticity at the indicated
values of p.
121)
x = f(p) = 1500 – 5p2; p = 10
121)
A)
Elastic
B)
Inelastic
C)
Unit elasticity
Solve the problem.
122)
A publishing company has published a new magazine for young adults. The monthly sales S (in
thousands) is given by S(t) =800t
t + 2 , where t is the number of months since the first issue was
published. Find S(3) and S'(3) and interpret the results.
122)
A)
At three months, the monthly sales are $2,400,000 and increasing at 800,000 magazines per
month.
B)
At three months, the monthly sales are $2, 400,000 and increasing at 64,000 magazines per
month.
C)
At three months, the monthly sales are $480,000 and decreasing at 64,000 magazines per
month.
D)
At three months, the monthly sales are $480,000 and increasing at 64,000 magazines per
month.
123)
One hour after x milligrams of a particular drug are given to a person, the change in body
temperature T(x), in degrees Celsius, is given approximately by:
T(x) =5x2
91 –x
9–160
9,0
x
6
Find the sensitivity, T'(x), of the body to a dosage of three milligrams.
123)
A)
–10
9 degrees per mg
B)
10
3 degrees per mg
C)
5
3 degrees per mg
D)
–5
3 degrees per mg
Use the price–demand equation to determine whether demand is elastic, is inelastic, or has unit elasticity at the indicated
values of p.
124)
x = f(p) = 2005 –p2; p = 13
124)
A)
Elastic
B)
Unit elasticity
C)
Inelastic
Solve the problem.
125)
An investor buys 100 shares of a stock for $20,000. After 5 years the stock is sold for $32,000. If
interest is compounded continuously, what annual nominal rate of interest did the original $20,000
investment earn? (Represent the answer as a percent to three decimal places.)
125)
A)
0.094%
B)
1.200%
C)
8.470%
D)
9.400%
Solve.
126)
A mathematical model for the average of a group of people learning to type is given by
N(t) = 10 + 6 ln t, t 1, where N(t) is the number of words per minute typed after t hours of
instruction and practice (2 hours per day, 5 days per week). What is the rate of learning after 80
hours of instruction and practice?
126)
A)
0.075 words per minute typed per hour of instruction and practice
B)
0.8 words per minute typed per hour of instruction and practice
C)
0.06 words per minute typed per hour of instruction and practice
D)
0.09 words per minute typed per hour of instruction and practice
Answer Key
Testname: C11
Answer Key
Testname: C11
35
Answer Key
Testname: C11