Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Provide an appropriate response.
1)
Consider the function: f(x) =9 –x2
x2. Choose the answer choice that includes all of the pair(s) of
functions from the list so that f(x) can be written as a composition: f(x) = g(h(x)).
1)
A)
g(x) =x2;h(x) =3–x
x
B)
g(x) =x ; h(x) =9 –x2
x2
C)
g(x) =9–x
x;h(x)=x2
D)
Both A and B
2)
Consider the function: f(x) =1
11 –x2. Choose the answer choice that includes all of the pair(s) of
functions from the list so that f(x) can be written as a composition: f(x) = g(h(x)).
2)
A)
g(x) =1x;h(x) = 11 –x2
B)
g(x) =1
11 – x;h(x) =x2
C)
g(x) =11 – x ; h(x) =1
x2
D)
Both A and C
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
3)
Find x’ for x = x(t) defined implicitly by t3–5x2= ln t and evaluate x’ at (t, x) = (0, –1).
3)
4)
Find the relative rate of change of f(x) =25x + 4x ln x
4)
1
5)
Find the relative rate of change of f(x) = 150x –0.08x2.
5)
6)
Write composite function y =e3x2+ x – 1 in the form y = f(u) and u = g(x).
6)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Use the demand equation to find the revenue function.
7)
x = f(p) = 30(15 – p)
7)
A)
R(p) = 450 – 30p2
B)
R(p) = 30 – 30p2
C)
R(p) = 450 – 450p2
D)
R(p) = 450p – 30p2
Solve the problem.
8)
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.
8)
A)
0.93 yr
B)
10.41 yr
C)
8.48 yr
D)
9.33 yr
Provide an appropriate response.
9)
Find dy
dt for y = (5t2– 4t)2.
9)
A)
2(5t2– 4t) + (10t – 4)
B)
2(10t – 4)
C)
2(5t2– 4t)(10t – 4)
D)
(5t2– 4t)(10t – 4)
Find the equation of the line tangent to the graph of f at the indicated value of x.
10)
f(x) =5 ln x; x = 1
10)
A)
y =5x –5
B)
y =5x – 1
C)
y =5x +5
D)
y =5x
Solve the problem.
11)
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.
11)
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 increasing at 64,000 magazines per
month.
D)
At three months, the monthly sales are $480,000 and decreasing at 64,000 magazines per
month.
Differentiate.
12)
Find f'(t) for f(x) =2x – 7
3x – 2.
12)
A)
–17
(3x – 2)2
B)
17
(2x – 7)2
C)
17
(3x – 2)2
D)
–17
(2x – 7)2
Provide an appropriate response.
13)
Find dy
dt for y = (5t2– 4t)2.
13)
A)
2(5t2– 4t) + (10t – 4)
B)
2(10t – 4)
C)
(5t2– 4t)(10t – 4)
D)
2(5t2– 4t)(10t – 4)
14)
Find f’x for f(x) =(3x +4)2
x3–x2+ 3x. Do not simplify.
14)
A)
(3x + 4)2(3x2– 2x + 3) – 6(x3–x2+ 3x)(3x + 4)
(3x + 4)4
B)
(3x + 4)2(3x2– 2x + 3) – 6(x3–x2+ 3x)(3x + 4)
(x3–x2+ 3x)2
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
15)
Find f'(x) for f(x) =log4(x5+ 1)
15)
A)
5x4
x5+ 1
B)
5x4
(ln 4)(x5+ 1)
C)
5x4(ln 4)
x5+ 1
D)
1
(ln 4)(x5+ 1) +5x4
Solve the problem.
16)
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.
16)
A)
–10
9 degrees per mg
B)
5
3 degrees per mg
C)
–5
3 degrees per mg
D)
10
3 degrees per mg
Provide an appropriate response.
17)
Assume x = x(t) and y = y(t). Find dx
dt if x2+y2= 25 and dy
dt = 3 when x = 3 and y = 4.
17)
A)
–4
B)
–6
C)
6
D)
4
Solve the problem.
18)
How long will it take for $8400 to grow to $14.600 at an interest rate of 9.4% if the interest is
compounded continuously? Round the number of years to the nearest hundredth.
18)
A)
0.59 yr
B)
5.88 yr
C)
0.06 yr
D)
58.81 yr
19)
How long will it take money to double if it is invested at 5.25%, compounded continuously? Round
your answer to the nearest tenth.
19)
A)
26.4 yr
B)
14 yr
C)
13.2 yr
D)
0.13 yr
20)
The position of a particle at time t is given by s, where s3+ 7st + 4t3– 6t = 0. Find the velocity ds/dt.
20)
A)
ds/dt =6– 7s – 12t2
3s2– 7t
B)
ds/dt =6– 7s – 12t2
3s2+ 7t
C)
ds/dt =6+ 7s – 12t2
3s2– 7t
D)
ds/dt =6+ 7s – 12t2
3s2+ 7t
Find the derivative.
21)
Find f'(x) for f(x) = (4x2+ 3x)2.
21)
A)
f'(x) = 32x3+ 36x2+ 9x
B)
f'(x) = 32x3+ 36x2+ 18x
C)
f'(x) = 64x3+ 36x2+ 18x
D)
f'(x) = 64x3+ 72x2+ 18x
5
Provide an appropriate response.
22)
Assume x = x(t) and y = y(t). Find dx
dt if x2(y – 6) = 12y + 3 and dy
dt = 2 when x = 5 and y = 12.
22)
A)
–13
30
B)
20
13
C)
–20
13
D)
13
20
23)
Use graphical approximation methods to find the point(s) of intersection of f(x) =(ln x)2 and g(x) =
x to two decimal places.
23)
A)
(–0.87, 0.42)
B)
(0.87, 0.42), (1.23, 3.41)
C)
(0.44, 0.67), (4.18, 2.04)
D)
(1.23, 3.41)
C
24)
Find x’ for x = x(t) defined implicitly by 3t + 4tx = 3e4x and evaluate x’ at (t, x) = (1, 0).
24)
A)
x’ =3 + 4x
12e4x – 4t; x’ (1, 0) =1
4
B)
x’ =3 + 4x
12e4x – 4t; x’ (1, 0) =7
8
C)
x’ =3 + 4x
12e4x – 4t; x’ (1, 0) =3
8
D)
x’ =4 + 4x
12e4x – 4t; x’ (1, 0) =1
2
C
25)
Find f'(x) for f(x) =(x2+ 2)3 .
25)
A)
f'(x) = 6x5+ 12x3+ 12x
B)
f'(x) = 3x5+ 24x3+ 24x
C)
f'(x) = 6x5+ 20x3+ 24x
D)
f'(x) = 6x5+ 24x3+ 24x
D
6
A
Solve the problem.
26)
If $2000 is invested at an annual interest rate r compounded monthly, the amount in the account
after 5 years is given by A = 2,000(1 +1
12r)60. Find the rate of change of the amount A with respect
to the interest rate r.
26)
A)
1000(1 +1
12r)59
B)
120,000(1 +1
12r)59
C)
10,000(1 +1
12r)59
D)
12,000(1 +1
12r)59
Provide an appropriate response.
27)
Find the values of x where the tangent line is horizontal for the graph of f(x) =4x2
x + 2.
27)
A)
x = 0, x = –2
B)
x = 0, x = –4
C)
x = –2
D)
x = –2, x = 0, x = –4
28)
Find the composition f[g(x)] if f(u) =u5 and g(x) = 2 –3x2.
28)
A)
(2 –3x2)2
B)
(10 –15x2)5
C)
2 –3u10
D)
(2 –3x2)5
29)
Find the derivative of the function f(x) =2x – 7
3x – 2 at x = 2.
29)
A)
–17
4
B)
17
16
C)
17
4
D)
–17
16
Find the elasticity of the demand function as a function of p.
30)
x = D(p) =800
(p +2)2
30)
A)
E(p) =2
p +2
B)
E(p) =1600p(p +2)
C)
E(p) =2p
p +2
D)
E(p) =1600p
(p +2)3
Solve.
31)
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 50
hours of instruction and practice?
31)
A)
0.5 words per minute typed per hour of instruction and practice
B)
0.15 words per minute typed per hour of instruction and practice
C)
0.1 words per minute typed per hour of instruction and practice
D)
0.12 words per minute typed per hour of instruction and practice
Use the price–demand equation to determine whether demand is elastic, is inelastic, or has unit elasticity at the indicated
values of p.
32)
x = f(p) = 2005 –p2; p = 13
32)
A)
Elastic
B)
Unit elasticity
C)
Inelastic
Provide an appropriate response.
33)
Write composite function y =(2x4+ 3x + 1)3 in the form y = f(u) and u = g(x).
33)
A)
y = f(u) =(2x4+ 3x + 1 )3and u = g(x) = u
B)
y = f(u) =u3 and u = g(x) =2x4+ 3x + 1.
C)
y = f(u) = 2x4+ 3x + 1 and u = g(x) =u3
D)
y = f(u) = u and u = g(x) =(2x4+ 3x + 1)3
8
Find f(x).
34)
f(x) = ln x5–9ex+3x2
34)
A)
5
x–9xex–1+6x
B)
5
x–9ex+6x
C)
5
x–9ex+3x
D)
5
x4–9ex+6
Use the price–demand equation to determine whether demand is elastic, is inelastic, or has unit elasticity at the indicated
values of p.
35)
x = f(p) =276– 4p; p =48.
35)
A)
Inelastic
B)
Elastic
C)
Unit elasticity
Find f(x).
36)
f(x) = ln x9–5x2
36)
A)
9
x8–10x
B)
1
9x –10x
C)
9
x–5x
D)
9
x–10x
Provide an appropriate response.
37)
Evaluate dy/dt for the function at the point.
x3+ y3= 9; dx/dt = –3, x = 1, y = 2
37)
A)
–3
4
B)
4
3
C)
–4
3
D)
3
4
Differentiate.
38)
Find f'(x) for f(x) = (5x3+ 4)(3x7– 5).
38)
A)
f'(x) = 150x9+ 84x6– 75x2
B)
f'(x) = 150x9+ 84x6– 75x
C)
f'(x) = 20x9+ 84x6– 75x
D)
f'(x) = 20x9+ 84x6– 75x2
Find f(x).
39)
f(x) = ln x9
39)
A)
9
x
B)
9
x8
C)
1
9x
D)
9 ln x8
Find the equation of the line tangent to the graph of f at the indicated value of x.
40)
f(x) =6ex– 1; x = 0
40)
A)
y =5x +5
B)
y =6x +6
C)
y =6x +5
D)
y =6x –5
41)
f(x) = 2 + ln x4; x = e
41)
A)
y =4x
e– 2
B)
y =4x
e+ 2
C)
y =4x
e+ 6
D)
y =4x
e
Provide an appropriate response.
42)
Find f'(x) for f(x) = (4x2+ 3x)2.
42)
A)
f'(x) = 64x3+ 72x2+ 18x
B)
f'(x) = 32x3+ 36x2+ 18x
C)
f'(x) = 64x3+ 36x2+ 18x
D)
f'(x) = 32x3+ 36x2+ 9x
Find dy
dx for the indicated function y.
43)
y =8log2x
43)
A)
1
x ln 2
B)
1
x8 ln 2
C)
8
2 ln x
D)
8
x ln 2
Use appropriate properties of logarithms to rewrite f(x), and then find f(x).
44)
f(x) = 1 + ln 5
x4
44)
A)
1 –4
x
B)
–4
x
C)
–5
x
D)
5
x
Find f(x).
45)
f(x) =-7 ln x –x4+2
45)
A)
–7
x–4x3
B)
–7
x–4x
C)
7
x–4x3
D)
–1
7x –4x3
Provide an appropriate response.
46)
Find dy/dx by implicit differentiation.
2xy –y2= 1
46)
A)
dy
dx =x
y – x
B)
dy
dx =y
y – x
C)
dy
dx =y
x – y
D)
dy
dx =x
x – y
47)
Find t to four decimal places.
e–t= 0.06
47)
A)
2.6134
B)
2.8134
C)
2.9134
D)
–2.8134
Find the percentage rate of change of f(x) at the indicated value of x. Round to the nearest tenth of a percent.
48)
f(x) = 200 + 50x; x = 3
48)
A)
14.3%
B)
33.3%
C)
–14.3%
D)
57.1%
Provide an appropriate response.
49)
Find t to four decimal places.
e–0.07t = 0.05
49)
A)
–70.1312
B)
42.7962
C)
–66.4815
D)
44.321
Solve.
50)
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 2 years?
50)
A)
–$2768/yr
B)
–$2325/yr
C)
–$11,004/yr
D)
–$1953/yr
Solve the problem.
51)
A company is manufacturing a new digital watch and can sell all it manufactures. The revenue (in
dollars) is given by R(x) = 50x –x2
50 , where the production output in one day is x watches. If
production is increasing at 5 watches per day when production is 375 watches per day, find the rate
of increase in revenue.
51)
A)
$75 per day
B)
$150 per day
C)
$175 per day
D)
$250 per day
Find dy
dx for the indicated function y.
52)
y =3+2x2–7x
52)
A)
4x +7x ln 7
B)
4x –7x ln 7
C)
4x –7 ln 7
D)
4x –7x ln x
Differentiate.
53)
Find f'(t) for f(x) = (2x – 2)(3x3– x2+ 1)
53)
A)
f'(x) =18x3+ 24x2– 8x + 2
B)
f'(x) =24x3– 8x2+ 24x + 2
C)
f'(x) =24x3– 24x2+ 4x + 2
D)
f'(x) =6x3+ 8x2– 24x + 2
Use the price–demand equation to find the values of p which meet the given condition of elasticity.
54)
x = f(p) = 216–2p2; determine the values of p for which demand is elastic and the values of p for
which demand is inelastic..
54)
A)
Elastic on (6, 6 3), inelastic on (0, 6)
B)
Elastic on (0, 6), inelastic on (6, 108)
C)
Elastic on (6, 108), inelastic on (0, 6)
D)
Elastic on (0, 6), inelastic on (6, 6 3)
Differentiate.
55)
Find dy
dx for y =x3
x – 1 .
55)
A)
dy
dx =2x3– 3x2
(x – 1)2
B)
dy
dx =2x3+ 3x2
(x – 1)2
C)
dy
dx =–2x3– 3x2
(x – 1)2
D)
dy
dx =– 2x3+ 3x2
(x – 1)2
Find dy
dx for the indicated function y.
56)
y =2x4–log3x
56)
A)
8x3–1
3 ln x
B)
8x3–1
x ln 3
C)
2x3–1
x ln 3
D)
8x3+1
3 ln x
Find the derivative.
57)
Find d
d
4
(2+ 3)5
57)
A)
– 40
(2+ 3)6
B)
40
(2+ 3)6
C)
– 40
(2+ 3)5
D)
– 40
(2+ 3)6
Provide an appropriate response.
58)
Find x to two decimal places.
x = 7,000e0.11
58)
A)
8320.50
B)
7813.95
C)
7975.01
D)
7831.95
59)
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?
59)
A)
30
7 ft/sec
B)
15
19 ft/sec
C)
325
6 ft/sec
D)
30
19 ft/sec
Solve the problem.
60)
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.)
60)
A)
8.470%
B)
9.400%
C)
1.200%
D)
0.094%
61)
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.
61)
A)
$98.00 per day
B)
$149.00 per day
C)
$156.80 per day
D)
$77.00 per day
Answer:
D
A)
B)
C)
D)
Provide an appropriate response.
62)
Find dy
dx for y = ln (4x3– x2)
62)
A)
12x – 2
4x2
B)
12x – 2
4x2– x
C)
4x – 2
4x2– x
D)
12x – 2
4x3– x
Answer:
B
A)
B)
C)
D)
Differentiate.
63)
Find f'(t) if f(t) = 0.4t(5t2+ 1) and simplify.
63)
A)
f'(t) =6t2– 0.4
B)
f'(t) =6t2+ 4
C)
f'(t) =6t2+ 40
D)
f'(t) =6t2+ 0.4
D
A)
B)
C)
D)
64)
Let f and g be functions that satisfy: f(4) = –1, g(4) = 3, f’(4) = 2, and g'(4) = –3. Find h'(4) for
h(x) = f(x)g(x) – 2f(x) + 7.
64)
A)
–6
B)
6
C)
–5
D)
5
Answer:
D
A)
B)
C)
D)
Answer:
B
Explanation:
A)
B)
C)
D)
Find f(x).
65)
f(x) =x4+5ex
65)
A)
4x3+ex
B)
4x +5ex
C)
4x3+5xex–1
D)
4x3+5ex
Provide an appropriate response.
66)
Find f'(x) for f(x) =(x2+ 2)3 .
66)
A)
f'(x) = 6x5+ 20x3+ 24x
B)
f'(x) = 6x5+ 12x3+ 12x
C)
f'(x) = 6x5+ 24x3+ 24x
D)
f'(x) = 3x5+ 24x3+ 24x
67)
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?
67)
A)
5.2 ft/sec
B)
9.6 ft/sec
C)
2.4 ft/sec
D)
4.8 ft/sec
68)
Find dy
dx for y =x2– 3x + 2
x7– 2 .
68)
A)
dy
dx =– 5x8+ 18x7– 14x6– 3x + 6
(x7– 2)2
B)
dy
dx =– 5x8+ 18x7– 13x6– 4x + 6
(x7– 2)2
C)
dy
dx =– 5x8+ 19x7– 14x6– 4x + 6
(x7– 2)2
D)
dy
dx =– 5x8+ 18x7– 14x6– 4x + 6
(x7– 2)2
Solve.
69)
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).
69)
A)
A'(t) = 6000(ln 2)4t; A'(1) = 16,635 bacteria
B)
A'(t) = 6000(ln 2)2t; A'(1) = 8317 bacteria
C)
A'(t) = 6000(ln 4)4t; A'(1) = 33,271 bacteria
D)
A'(t) = 6000(ln 4)2t; A'(1) = 16,635 bacteria
Use the price–demand equation to find the values of p which meet the given condition of elasticity.
70)
x = f(p)=246– 8p; determine the values of p for which demand has unit elasticity. Round to two
decimal places if necessary.
70)
A)
Inelastic at p =31.36
B)
Inelastic at p =7.69
C)
Inelastic at p =15.68
D)
Inelastic at p =15.38
Provide an appropriate response.
71)
Find: lim
x 5000e–0.07t
71)
A)
5000
B)
C)
1
D)
0
Solve.
72)
The salvage value S (in dollars) of a company airplane after t years is estimated to be given by
S(t) = 250,000(0.7)t. What is the rate of depreciation (in dollars per year) after 6 years?
72)
A)
–$52,699/yr
B)
–$29,412/yr
C)
–$10,491/yr
D)
–$14,987/yr
Find f(x).
73)
f(x) =-8ex+7x – 4
73)
A)
-8ex+7
B)
-8xex–1+7
C)
-8ex+7x
D)
-8ex+3
Differentiate.
74)
Find f'(x) for f(x) = (2x – 4)(2x3–x2+ 1).
74)
A)
f'(x) = 16x3– 30x2+ 8x + 2
B)
f'(x) = 16x3– 10x2+ 30x + 2
C)
f'(x) = 12x3+ 30x2– 10x + 2
D)
f'(x) = 4x3– 10x2– 30x + 2
Solve the problem.
75)
If $5000 is invested at 5.25% compounded continuously, what is the amount in the account after 10
years?
75)
A)
$7420.65
B)
$8452.29
C)
$8442.52
D)
$7625.00
Use appropriate properties of logarithms to rewrite f(x), and then find f(x).
76)
f(x) = 5x + 4 ln 2x
76)
A)
5 +2
x
B)
5 +8
x
C)
7 +4
x
D)
5 +4
x
18
Provide an appropriate response.
77)
Find: d
dx 3 –e(2x2+ x) 4
77)
A)
4(3 –e(2x2+ x))3e(2x2+ x)(–4x2– 1)
B)
4(3 –e(2x2+ x))3e(2x2+ x)(–4x – 1)
C)
2(3 –e(2x2+ x))3e(2x2+ x)(–4x – 1)
D)
4(3 –e(2x2+ x))3e(2x2+ x)(–4x+ 1)
Find the equation of the line tangent to the graph of f at the indicated value of x.
78)
f(x) =5+ ln x; x = 1
78)
A)
y = x –4
B)
y = x +4
C)
y = x +6
D)
y = x –6
79)
f(x) = 1 +3ex; x = 1
79)
A)
y =6ex – 1
B)
y =3ex + 1
C)
y =3ex +6e + 1
D)
y =3ex –6e + 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)
33,271 bacteria
B)
133,084 bacteria
C)
532,337 bacteria
D)
2,129,348 bacteria
Provide an appropriate response.
81)
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?
81)
A)
increasing at 9 units per second
B)
decreasing at 27
2 units per second
C)
increasing at 27
2 units per second
D)
decreasing at 9 units per second
Find the derivative.
82)
Find: dy
dx
88x7– 10
82)
A)
56x6
(8x7–10)7/8
B)
878x7– 10
C)
7x6
(8x7–10)7/8
D)
448x678x7– 10
83)
y = (x–2+ x)–3
83)
A)
dy
dx =3x5(2 – x3)
(1 + x3)4
B)
dy
dx =3x5(2 – x3)
(1 + x3)3
C)
dy
dx =3x4(2 – x3)
(1 + x3)4
D)
dy
dx =3x4(2 – x3)
(1 + x3)3
20
Find the elasticity of the demand function as a function of p.
84)
x = D(p) =200– p
84)
A)
E(p) = p(200– p)
B)
E(p) =p
200– p
C)
E(p) =p
p –200
D)
E(p) =1
200– p
Find the percentage rate of change of f(x) at the indicated value of x. Round to the nearest tenth of a percent.
85)
f(x) = 4500 – 4x2; x = 20
85)
A)
3.6%
B)
–5.5%
C)
–1.0%
D)
5.5%
Solve the problem.
86)
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?
86)
A)
11.6 yr
B)
12.6 yr
C)
2 yr
D)
13.6 yr
Provide an appropriate response.
87)
Find dy
dx for y =23x–1
87)
A)
23x–1 ln(23x–1)
B)
23 ln(23)
C)
23x–1 ln(x)
D)
23x–1 ln(23)
88)
Find dy/dx by implicit differentiation.
x3+ 3x2y + y3= 8
88)
A)
dy
dx =x2+ 3xy
x2+ y2
B)
dy
dx = – x2+ 3xy
x2+ y2
C)
dy
dx = – x2+ 2xy
x2+ y2
D)
dy
dx =x2+ 2xy
x2+ y2
Differentiate.
89)
Find dy
dx for y =5x –5
9x2+1
89)
A)
dy
dx =–45x2+ 85x +10
(9x2+1)2
B)
dy
dx =135x2–90x +5
(9x2+1)2
C)
dy
dx =–45x2+90x +5
(9x2+1)2
D)
dy
dx =45x3–90x2+95x
(9x2+1)2
Solve the problem.
90)
Radioactive carbon–14 has a continuous compound rate of decay of r = –0.000124. Estimate the age
of a skull uncovered at an archaeological site if 6% of the original amount of carbon–14 is still
present. (Compute answer to the nearest year.)
90)
A)
20,032 yr
B)
124,027 yr
C)
470 yr
D)
22,689 yr
Find f(x).
91)
f(x) =9ex–6x + 2
91)
A)
9xex–1–6
B)
9ex–6
C)
9ex–6x
D)
9ex–4
Provide an appropriate response.
22
92)
Graph the function which calculates the present value of an amount of $5000 at an annual nominal
rate of 7% compounded continuously for 0
t
10. Use the formula P = Ae–rt.
92)
A)
B)
C)
D)
Solve the problem.
93)
What will the value of an account (to the nearest cent) be after 8 years if $100 is invested at 6.0%
interest compounded continuously?
93)
A)
$849.47
B)
$175.32
C)
$161.61
D)
$159.38
Provide an appropriate response.
94)
Find dy
dx for y =–5x3– 5x2+ 3
–5x4+ 2 . Do not simplify.
94)
A)
(–5x3– 5x2+ 3)(–20x3) – (–5x4+ 2)(–15x2– 10x)
(–5x3– 5x2+ 3)2
B)
(–5x4+ 2)(–15x2– 10x) – (–5x3– 5x2+ 3)(–20x3)
(–5x3– 5x2+ 3)2
C)
(–5x3– 5x2+ 3)(–20x3) – (–5x4+ 2)(–15x2– 10x)
(–5x4+ 2)2
D)
(–5x4+ 2)(–15x2– 10x) – (–5x3– 5x2+ 3)(–20x3)
(–5x4+ 2)2
Solve the problem.
95)
The concentration of a certain drug in the bloodstream t minutes after swallowing a pill containing
the drug can be approximated using the equation C(t) =1
64t + 1 –1/2, where C(t) is the
concentration in arbitrary units and t is in minutes. Find the rate of change of concentration with
respect to time at t =12 minutes.
95)
A)
–1
21 units/min
B)
–1
4116 units/min
C)
–1
1029 units/min
D)
–1
42 units/min
Use the demand equation to find the revenue function.
96)
x = f(p) = 55 – 7 p
96)
A)
R(p) = 55p – 49p3/2
B)
R(p) = 55p – 7p2
C)
R(p) = 55p – 7p
D)
R(p) = 55p – 7p3/2
Provide an appropriate response.
97)
Find y’ for y = y(x) defined implicitly by 3xy –x2– 4 = 0.
97)
A)
y’ =3x –2y
4
B)
y’ =3y – 2x
3x
C)
y’ =2
3x
D)
y’ =2x – 3y
3x
Use the price–demand equation to determine whether demand is elastic, is inelastic, or has unit elasticity at the indicated
values of p.
98)
x = f(p) = 1500 – 5p2; p = 10
98)
A)
Inelastic
B)
Unit elasticity
C)
Elastic
Provide an appropriate response.
99)
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.
99)
25
A)
7.45 years
B)
6.28 years
C)
0.84 years
D)
3.14 years
Find the elasticity of the demand function as a function of p.
100)
x = D(p) =700– p
100)
A)
E(p) =p
1400 – 2p
B)
E(p) =p
2p –1400
C)
E(p) =1
1400 – 2p
D)
E(p) =p
700– p
26
Find f(x).
101)
f(x) =4 ln x + ln x6+8ex
101)
A)
10
x+8ex
B)
4
x+6
x5+8xex–1
C)
4
x+6
x5+8ex
D)
10
x+8xex–1
Find the derivative.
102)
Find f'(x) for f(x) = (8x – 9)–4.
102)
A)
–4
(8x–9)5
B)
–32
(8x–9)3
C)
–32
(8x –9)5
D)
–4
(8x–9)3
Solve the problem.
103)
A beverage company works out a demand function for its sale of soda and finds it to be
x = D(p) =3000 –24p
where x = the quantity of sodas sold when the price per can, in cents, is p. At what prices, p, is the
elasticity of demand inelastic?
103)
A)
For p <63 cents
B)
For p >36,000 cents
C)
For p <125 cents
D)
For p >250 cents
Provide an appropriate response.
104)
A 26–foot ladder is placed against a wall. If the top of the ladder is sliding down the wall at 3 feet
per second, at what rate is the bottom of the ladder moving away from the wall when the bottom of
the ladder is 9 feet away from the wall?
104)
A)
5.4 ft/sec
B)
8.1 ft/sec
C)
4.9 ft/sec
D)
–8.1 ft/sec
Solve the problem.
105)
A company is manufacturing a new digital watch and can sell all it manufactures. The cost (in
dollars) is given by C(x) = 5000 + 2x, where the production output in one day is x watches. If
production is increasing at 5 watches per day when production is 375 watches per day, find the rate
of increase in cost.
105)
A)
$10 per day
B)
$175 per day
C)
$75 per day
D)
$5 per day
Provide an appropriate response.
106)
Use graphical approximation methods to find the point(s) of intersection of f(x) =ex and g(x) =x6
to two decimal places.
106)
A)
(1.23, 3.41)
B)
(–0.87, 0.42)
C)
(–0.87, 0.42), (1.23, 3.41)
D)
(0.87, 0.42), (1.23, 3.41)
C
107)
Evaluate dy/dt for the function at the point.
x + y
x – y = x2+ y2; dx/dt = 12, x = 1, y = 0
107)
A)
–1
12
B)
12
C)
– 12
D)
1
12
B
Solve the problem.
108)
The demand equation for a certain product is 8p2+ q2=1300, where p is the price per unit in
dollars and q is the number of units demanded. Find dq/dp.
108)
A)
dq/dp = –q/8p
B)
dq/dp =-8q/p
C)
dq/dp = –p/8q
D)
dq/dp =-8p/q
D
28
A
Differentiate.
109)
Find y’ for y =x2
8–6x
109)
A)
8x
(8 –6x)2
B)
–6x2+16x
(8 –6x)2
C)
6x3–12x2+16x
(8 –6x)2
D)
–18x2+16x
(8 –6x)2
110)
f(x) =5ex–3x2
110)
A)
5ex–6x2
B)
5ex–2x
C)
5ex–6x
D)
5xex–1–6x2
Provide an appropriate response.
111)
Find the equation(s) of the tangent line(s) to the graph of y2– xy + 3 = 0 at x = –4.
111)
A)
y =3
2x +1
2
B)
y = – 3
2x – 3
C)
y =3
2x + 3 and y = – 1
2x – 3
D)
y =3
2x – 3
Differentiate.
112)
Find f'(t) for f(x) =x
8x –8
112)
A)
–8
8x –8
B)
–8
(8x –8)2
C)
–8x
(8x –8)2
D)
16x –8
(8x –8)2
Find the equation of the tangent line to the graph of the given function at the given value of x.
113)
f(x) =(x2+28)4/5; x =2
113)
A)
y =8
5x
B)
y =8
5x +96
5
C)
y =4
5x +64
5
D)
y =8
5x +64
5
Find the equation of the line tangent to the graph of f at the indicated value of x.
114)
f(x) =5ex; x = 0
114)
A)
y =5x +5
B)
y =5x –5
C)
y = x +5
D)
y =5x
Provide an appropriate response.
115)
Find dy/dx by implicit differentiation.
x3+y3= 5
115)
A)
dy
dx = – y2
x2
B)
dy
dx =y2
x2
C)
dy
dx =x2
y2
D)
dy
dx = – x2
y2
116)
Find the composite g[f(–k)] if f(x) = 8x2– 5x and g(x) = 7x + 9.
116)
A)
56k2– 35k + 9
B)
392k2– 973k + 603
C)
56k2+ 35k + 9
D)
392k2+ 973k + 603
117)
Find y’ and the slope of the tangent line to the graph of ln (xy) =y3+ 1 at (1, –1).
117)
A)
y
3xy3– x; 1
4
B)
y
3xy3; –1
4
C)
x
3xy3– x; –1
4
D)
y
3xy3– x; –1
4
30
118)
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 70–pound weight level?
118)
A)
0.22 mm of mercury per pound of weight gain
B)
0.01 mm of mercury per pound of weight gain
C)
0.29 mm of mercury per pound of weight gain
D)
0.25 mm of mercury per pound of weight gain
Provide an appropriate response.
119)
Find f'(x) for f(x) =(ln x)4
119)
A)
4ln3 x
B)
1
x4
C)
1
(ln x)4
D)
E)
4ln3 x
x
Find dy
dx for the indicated function y.
120)
y = –9 ln x +7log x
120)
A)
–9
x+1
x ln 10
B)
9
x+7
x ln 10
C)
–9
x+1
x7 ln 10
D)
–9
x+7
x ln 10
31
Provide an appropriate response.
121)
Find y’ for y = y(x) defined implicitly by 5y2– 8x4+ 3 = 0, and evaluate y’ at (x, y) = (1, 1).
121)
A)
y’ =11x3
5y ; y’ (1, 1) =11
5
B)
y’ =16x2
5y2; y’ (1, 1) =16
5
C)
y’ =11x2
5y2; y’ (1, 1) =11
5
D)
y’ =16x3
5y ; y’ (1, 1) =16
5
Find f(x).
122)
f(x) = 8ex+ 4 ln x3
122)
A)
8ex+12
x
B)
8ex+4
x2
C)
8ex+12
x3
D)
8ex+12
x2
Find dy
dx for the indicated function y.
123)
y =2x
123)
A)
2x
ln 2
B)
2 ln 2
C)
2x ln x
D)
2x ln 2
Find all values of x for the given function where the tangent line is horizontal.
124)
f(x) =x
(x2+1)3
124)
A)
0
B)
0, ±5
5
C)
±5
5
D)
±1
5
32
Provide an appropriate response.
125)
Suppose two automobiles leave from the same point at the same time. If one travels north at 60
miles per hour and the other travels east at 45 miles per hour, how fast will the distance between
them be changing after three hours?
125)
A)
75 mph
B)
50 mph
C)
150 mph
D)
125 mph
Find dy
dx for the indicated function y.
126)
y =4x–e3
126)
A)
4x
ln 4
B)
4x ln 4
C)
4x ln 4–e3
D)
4x ln x –e3
Answer Key
Testname: C3
Answer Key
Testname: C3
Answer Key
Testname: C3