Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Provide an appropriate response.
1)
Find x to two decimal places.
x = 7,000e0.11
A)
8320.50
B)
7975.01
C)
7813.95
D)
7831.95
Answer:
D
2)
Find t to four decimal places.
e–t= 0.06
A)
2.6134
B)
2.8134
C)
–2.8134
D)
2.9134
Answer:
B
3)
Find t to four decimal places.
e–0.07t = 0.05
A)
–66.4815
B)
42.7962
C)
–70.1312
D)
44.321
Answer:
B
4)
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.
A)
B)
1
C)
D)
Answer:
B
5)
Find: lim
x 5000e–0.07t
A)
1
B)
5000
C)
D)
0
Answer:
D
6)
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.
A)
3.14 years
B)
6.28 years
2
C)
7.45 years
D)
0.84 years
Answer:
A
Solve the problem.
7)
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?
A)
$159.38
B)
$161.61
C)
$849.47
D)
$175.32
Answer:
B
8)
If $5000 is invested at 5.25% compounded continuously, what is the amount in the account after 10 years?
A)
$7420.65
B)
$7625.00
C)
$8452.29
D)
$8442.52
Answer:
C
9)
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.
A)
8.48 yr
B)
0.93 yr
C)
9.33 yr
D)
10.41 yr
Answer:
A
10)
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.
A)
5.88 yr
B)
58.81 yr
C)
0.59 yr
D)
0.06 yr
Answer:
A
11)
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?
A)
13.6 yr
B)
2 yr
C)
12.6 yr
D)
11.6 yr
Answer:
C
12)
How long will it take money to double if it is invested at 5.25%, compounded continuously? Round your
answer to the nearest tenth.
A)
0.13 yr
B)
26.4 yr
C)
13.2 yr
D)
14 yr
Answer:
C
13)
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.)
A)
0.094%
B)
8.470%
C)
9.400%
D)
1.200%
Answer:
C
3
14)
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.)
A)
22,689 yr
B)
470 yr
C)
124,027 yr
D)
20,032 yr
Answer:
A
Find f(x).
15)
f(x) =9ex–4x + 2
A)
9ex–4
B)
9xex–1–4
C)
9ex–4x
D)
9ex–2
Answer:
A
16)
f(x) = –6ex+7x – 4
A)
–6xex–1+7
B)
–6ex+3
C)
–6ex+7x
D)
–6ex+7
Answer:
D
17)
f(x) =x8+5ex
A)
8x7+5ex
B)
8x +5ex
C)
8x7+ex
D)
8x7+5xex–1
Answer:
A
18)
f(x) =7ex–3x2
A)
7ex–6x2
B)
7ex–2x
C)
7xex–1–6x2
D)
7ex–6x
Answer:
D
19)
f(x) = –9 ln x –x4+1
A)
9
x–4x3
B)
–1
9x –4x3
C)
–9
x–4x
D)
–9
x–4x3
Answer:
D
20)
f(x) = ln x4
A)
1
4x
B)
4
x
C)
4
x3
D)
4 ln x3
Answer:
B
21)
f(x) = ln x4–4x2
A)
4
x–8x
B)
4
x–4x
C)
4
x3–8x
D)
1
4x –8x
Answer:
A
22)
f(x) = ln x5–4ex+2x2
A)
5
x–4xex–1+4x
B)
5
x–4ex+2x
C)
5
x–4ex+4x
D)
5
x4–4ex+4
Answer:
C
4
23)
f(x) =4 ln x + ln x7+2ex
A)
11
x+2xex–1
B)
11
x+2ex
C)
4
x+7
x6+2ex
D)
4
x+7
x6+2xex–1
Answer:
B
24)
f(x) = 8ex+ 4 ln x3
A)
8ex+4
x2
B)
8ex+12
x
C)
8ex+12
x2
D)
8ex+12
x3
Answer:
B
Use appropriate properties of logarithms to rewrite f(x), and then find f(x).
25)
f(x) = 5x + 4 ln 2x
A)
5 +4
x
B)
7 +4
x
C)
5 +8
x
D)
5 +2
x
Answer:
A
26)
f(x) = 1 + ln 5
x4
A)
–5
x
B)
1 –4
x
C)
–4
x
D)
5
x
Answer:
C
Find dy
dx for the indicated function y.
27)
y =4log2x
A)
4
2 ln x
B)
1
x ln 2
C)
1
x4 ln 2
D)
4
x ln 2
Answer:
D
28)
y =5x
A)
5x ln 5
B)
5 ln 5
C)
5x
ln 5
D)
5x ln x
Answer:
A
29)
y =3x3–log x
A)
3x2–1
x ln 10
B)
9x2–1
x ln 10
C)
9x2+1
10 ln x
D)
9x2–1
10 ln x
Answer:
B
30)
y =10 +3x2–4x
A)
6x –4 ln 4
B)
6x –4x ln 4
C)
6x +4x ln 4
D)
6x –4x ln x
Answer:
B
5
31)
y = –8 ln x +7log3x
A)
–8
x+7
x ln 3
B)
8
x+7
x ln 3
C)
–8
x+1
x7 ln 3
D)
–8
x+1
x ln 3
Answer:
A
32)
y =2x–e5
A)
2x ln x –e5
B)
2x ln 2
C)
2x
ln 2
D)
2x ln 2–e5
Answer:
B
Find the equation of the line tangent to the graph of f at the indicated value of x.
33)
f(x) =5ex; x = 0
A)
y =5x
B)
y =5x +5
C)
y = x +5
D)
y =5x –5
Answer:
B
34)
f(x) =5ex– 1; x = 0
A)
y =5x –4
B)
y =4x +4
C)
y =5x +4
D)
y =5x +5
Answer:
C
35)
f(x) = 1 +2ex; x = 1
A)
y =2ex –4e + 1
B)
y =2ex +4e + 1
C)
y =2ex + 1
D)
y =4ex – 1
Answer:
C
36)
f(x) =6+ ln x; x = 1
A)
y = x –5
B)
y = x –7
C)
y = x +7
D)
y = x +5
Answer:
D
37)
f(x) =9 ln x; x = 1
A)
y =9x +9
B)
y =9x
C)
y =9x –9
D)
y =9x – 1
Answer:
C
38)
f(x) = 2 + ln x6; x = e
A)
y =6x
e+8
B)
y =6x
e+ 2
C)
y =6x
e– 2
D)
y =6x
e
Answer:
B
Provide an appropriate response.
39)
Use graphical approximation methods to find the point(s) of intersection of f(x) =ex and g(x) =x6 to two
decimal places.
A)
(0.87, 0.42), (1.23, 3.41)
B)
(1.23, 3.41)
C)
(–0.87, 0.42), (1.23, 3.41)
D)
(–0.87, 0.42)
Answer:
C
6
40)
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.
A)
(–0.87, 0.42)
B)
(1.23, 3.41)
C)
(0.87, 0.42), (1.23, 3.41)
D)
(0.44, 0.67), (4.18, 2.04)
Answer:
D
Solve.
41)
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 3 years?
A)
–$43,693/yr
B)
–$30,585/yr
C)
–$85,750/yr
D)
–$94,206/yr
Answer:
B
42)
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?
A)
–$1641/yr
B)
–$1953/yr
C)
–$15,529/yr
D)
–$1378/yr
Answer:
B
43)
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).
A)
A'(t) = 6000(ln 4)4t; A'(1) = 33,271 bacteria
B)
A'(t) = 6000(ln 2)2t; A'(1) = 8317 bacteria
C)
A'(t) = 6000(ln 4)2t; A'(1) = 16,635 bacteria
D)
A'(t) = 6000(ln 2)4t; A'(1) = 16,635 bacteria
Answer:
A
44)
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).
A)
2,129,348 bacteria
B)
33,271 bacteria
C)
532,337 bacteria
D)
133,084 bacteria
Answer:
C
45)
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 80–pound
weight level?
A)
0.19 mm of mercury per pound of weight gain
B)
0.01 mm of mercury per pound of weight gain
C)
0.22 mm of mercury per pound of weight gain
D)
0.25 mm of mercury per pound of weight gain
Answer:
C
46)
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?
A)
0.5 words per minute typed per hour of instruction and practice
B)
0.12 words per minute typed per hour of instruction and practice
C)
0.1 words per minute typed per hour of instruction and practice
D)
0.15 words per minute typed per hour of instruction and practice
Answer:
B
7
Differentiate.
47)
Find f'(t) for f(x) = (3x – 3)(3x3– x2+ 1)
A)
f'(x) =27x3+ 36x2– 12x + 3
B)
f'(x) =9x3+ 12x2– 36x + 3
C)
f'(x) =36x3– 36x2+ 6x + 3
D)
f'(x) =36x3– 12x2+ 36x + 3
Answer:
C
48)
Find f'(x) for f(x) = (2x – 4)(2x3–x2+ 1).
A)
f'(x) = 16x3– 10x2+ 30x + 2
B)
f'(x) = 12x3+ 30x2– 10x + 2
C)
f'(x) = 16x3– 30x2+ 8x + 2
D)
f'(x) = 4x3– 10x2– 30x + 2
Answer:
C
49)
Find f'(x) for f(x) = (5x3+ 4)(3x7– 5).
A)
f'(x) = 150x9+ 84x6– 75x2
B)
f'(x) = 20x9+ 84x6– 75x
C)
f'(x) = 150x9+ 84x6– 75x
D)
f'(x) = 20x9+ 84x6– 75x2
Answer:
A
50)
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.
A)
6
B)
–5
C)
5
D)
–6
Answer:
C
51)
Find f'(t) if f(t) = 0.4t(5t2+ 1) and simplify.
A)
f'(t) =6t2+ 40
B)
f'(t) =6t2+ 0.4
C)
f'(t) =6t2– 0.4
D)
f'(t) =6t2+ 4
Answer:
B
52)
Find f'(t) for f(x) =x
8x –3
A)
–3
(8x –3)2
B)
16x –3
(8x –3)2
C)
–3x
(8x –3)2
D)
–3
8x –3
Answer:
A
53)
Find f'(t) for f(x) =2x – 7
3x – 2.
A)
17
(3x – 2)2
B)
–17
(2x – 7)2
C)
–17
(3x – 2)2
D)
17
(2x – 7)2
Answer:
A
54)
Find y’ for y =x2
4–8x
A)
8x3–16x2+8x
(4 –8x)2
B)
–8x2+8x
(4 –8x)2
C)
–24x2+8x
(4 –8x)2
D)
4x
(4 –8x)2
Answer:
B
8
55)
Find dy
dx for y =6x –7
5x2+8
A)
dy
dx =–30x2+ 22x +104
(5x2+8)2
B)
dy
dx =30x3–60x2+118x
(5x2+8)2
C)
dy
dx =90x2–70x +48
(5x2+8)2
D)
dy
dx =–30x2+70x +48
(5x2+8)2
Answer:
D
56)
Find dy
dx for y =x3
x – 1 .
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
Answer:
B
57)
Find dy
dx for y =x2– 3x + 2
x7– 2 .
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
Answer:
D
Provide an appropriate response.
58)
Find the derivative of the function f(x) =2x – 7
3x – 2 at x = 2.
A)
17
4
B)
–17
4
C)
17
16
D)
–17
16
Answer:
C
59)
Find dy
dx for y =–5x3– 5x2+ 3
–5x4+ 2 . Do not simplify.
A)
(–5x3– 5x2+ 3)(–20x3) – (–5x4+ 2)(–15x2– 10x)
(–5x3– 5x2+ 3)2
B)
(–5x3– 5x2+ 3)(–20x3) – (–5x4+ 2)(–15x2– 10x)
(–5x4+ 2)2
C)
(–5x4+ 2)(–15x2– 10x) – (–5x3– 5x2+ 3)(–20x3)
(–5x3– 5x2+ 3)2
D)
(–5x4+ 2)(–15x2– 10x) – (–5x3– 5x2+ 3)(–20x3)
(–5x4+ 2)2
Answer:
D
9
60)
Find f’x for f(x) =(3x +4)2
x3–x2+ 3x. Do not simplify.
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
Answer:
D
61)
Find the values of x where the tangent line is horizontal for the graph of f(x) =4x2
x + 2.
A)
x = –2, x = 0, x = –4
B)
x = 0, x = –4
C)
x = –2
D)
x = 0, x = –2
Answer:
B
Solve the problem.
62)
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.
A)
5
3 degrees per mg
B)
–10
9 degrees per mg
C)
–5
3 degrees per mg
D)
10
3 degrees per mg
Answer:
A
63)
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.
A)
At three months, the monthly sales are $2, 400,000 and increasing at 64,000 magazines per month.
B)
At three months, the monthly sales are $2,400,000 and increasing at 800,000 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.
Answer:
C
Provide an appropriate response.
64)
Write composite function y =(2x4+ 3x + 1)3 in the form y = f(u) and u = g(x).
A)
y = f(u) = u and u = g(x) =(2x4+ 3x + 1)3
B)
y = f(u) =(2x4+ 3x + 1 )3and u = g(x) = u
C)
y = f(u) =u3 and u = g(x) =2x4+ 3x + 1.
D)
y = f(u) = 2x4+ 3x + 1 and u = g(x) =u3
Answer:
C
10
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
65)
Write composite function y =e3x2+ x – 1 in the form y = f(u) and u = g(x).
Answer:
y = f(u) =ex and u = g(x) =3x2+ x – 1.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
66)
Find the composition f[g(x)] if f(u) =u5 and g(x) = 2 –3x2.
A)
(2 –3x2)2
B)
(10 –15x2)5
C)
(2 –3x2)5
D)
2 –3u10
Answer:
C
67)
Find the composite g[f(–k)] if f(x) = 8x2– 5x and g(x) = 7x + 9.
A)
392k2– 973k + 603
B)
56k2– 35k + 9
C)
392k2+ 973k + 603
D)
56k2+ 35k + 9
Answer:
D
68)
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)).
A)
g(x) =x2;h(x) =3–x
x
B)
g(x) =9–x
x;h(x)=x2
C)
g(x) =x ; h(x) =9 –x2
x2
D)
Both A and B
Answer:
D
69)
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)).
A)
g(x) =11 – x ; h(x) =1
x2
B)
g(x) =1x;h(x) = 11 –x2
C)
g(x) =1
11 – x;h(x) =x2
D)
Both A and C
Answer:
D
Find the derivative.
70)
Find d
d
4
(2+ 3)5
A)
– 40
(2+ 3)6
B)
40
(2+ 3)6
C)
– 40
(2+ 3)5
D)
– 40
(2+ 3)6
Answer:
D
11
71)
Find f'(x) for f(x) = (8x – 9)–4.
A)
–4
(8x–9)3
B)
–32
(8x –9)5
C)
–4
(8x–9)5
D)
–32
(8x–9)3
Answer:
B
72)
Find: d
dx
88x7– 10
A)
878x7– 10
B)
448x678x7– 10
C)
7x6
(8x7–10)7/8
D)
56x6
(8x7–10)7/8
Answer:
C
73)
y = (x–2+ x)–3
A)
dy
dx =3x4(2 – x3)
(1 + x3)4
B)
dy
dx =3x5(2 – x3)
(1 + x3)3
C)
dy
dx =3x5(2 – x3)
(1 + x3)4
D)
dy
dx =3x4(2 – x3)
(1 + x3)3
Answer:
C
74)
Find f'(x) for f(x) = (4x2+ 3x)2.
A)
f'(x) = 64x3+ 36x2+ 18x
B)
f'(x) = 64x3+ 72x2+ 18x
C)
f'(x) = 32x3+ 36x2+ 18x
D)
f'(x) = 32x3+ 36x2+ 9x
Answer:
B
Provide an appropriate response.
75)
Find f'(x) for f(x) =(x2+ 2)3 .
A)
f'(x) = 3x5+ 24x3+ 24x
B)
f'(x) = 6x5+ 24x3+ 24x
C)
f'(x) = 6x5+ 12x3+ 12x
D)
f'(x) = 6x5+ 20x3+ 24x
Answer:
B
76)
Find f'(x) for f(x) =(x2+ 2)3 .
A)
f'(x) = 6x5+ 20x3+ 24x
B)
f'(x) = 3x5+ 24x3+ 24x
C)
f'(x) = 6x5+ 24x3+ 24x
D)
f'(x) = 6x5+ 12x3+ 12x
Answer:
C
77)
Find dy
dt for y = (5t2– 4t)2.
A)
2(10t – 4)
B)
(5t2– 4t)(10t – 4)
C)
2(5t2– 4t) + (10t – 4)
D)
2(5t2– 4t)(10t – 4)
Answer:
D
78)
Find f'(x) for f(x) = (4x2+ 3x)2.
A)
f'(x) = 64x3+ 36x2+ 18x
B)
f'(x) = 32x3+ 36x2+ 18x
C)
f'(x) = 64x3+ 72x2+ 18x
D)
f'(x) = 32x3+ 36x2+ 9x
Answer:
C
12
79)
Find dy
dt for y = (5t2– 4t)2.
A)
(5t2– 4t)(10t – 4)
B)
2(5t2– 4t)(10t – 4)
C)
2(5t2– 4t) + (10t – 4)
D)
2(10t – 4)
Answer:
B
80)
Find dy
dx for y = ln (7x3– x2)
A)
21x – 2
7x2
B)
21x – 2
7x2– x
C)
21x – 2
7x3– x
D)
7x – 2
7x2– x
Answer:
B
81)
Find f'(x) for f(x) =(ln x)8
A)
1
x8
B)
8ln7 x
C)
8ln7 x
x
D)
E)
1
(ln x)8
Answer:
C
82)
Find dy
dx for y =21x–1
A)
21 ln(21)
B)
21x–1 ln(x)
C)
21x–1 ln(21)
D)
21x–1 ln(21x–1)
Answer:
C
83)
Find f'(x) for f(x) =log6(x9+ 1)
A)
9x8(ln 6)
x9+ 1
B)
9x8
x9+ 1
C)
9x8
(ln 6)(x9+ 1)
D)
1
(ln 6)(x9+ 1) +9x8
Answer:
C
Find the equation of the tangent line to the graph of the given function at the given value of x.
84)
f(x) =(x2+12)3/4; x =2
A)
y =3
2x
B)
y =3
4x +5
C)
y =3
2x +11
D)
y =3
2x +5
Answer:
D
Find all values of x for the given function where the tangent line is horizontal.
85)
f(x) =x
(x2+7)3
A)
±7
5
B)
0
C)
0, ±35
5
D)
±35
5
Answer:
D
13
Solve the problem.
86)
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.
A)
1000(1 +1
12r)59
B)
12,000(1 +1
12r)59
C)
10,000(1 +1
12r)59
D)
120,000(1 +1
12r)59
Answer:
C
87)
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
62t + 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 =4 minutes.
A)
–1
324 units/min
B)
–1
18 units/min
C)
–1
162 units/min
D)
–1
18 units/min
Answer:
C
Provide an appropriate response.
88)
Find y’ for y = y(x) defined implicitly by 5y2– 8x4+ 3 = 0, and evaluate y‘ at (x, y) = (1, 1).
A)
y’ =16x3
5y ; y’ (1, 1) =16
5
B)
y’ =11x3
5y ; y’ (1, 1) =11
5
C)
y’ =11x2
5y2; y’ (1, 1) =11
5
D)
y’ =16x2
5y2; y’ (1, 1) =16
5
Answer:
A
89)
Find y’ for y = y(x) defined implicitly by 3xy –x2– 4 = 0.
A)
y’ =3y – 2x
3x
B)
y’ =3x –2y
4
C)
y’ =2
3x
D)
y’ =2x – 3y
3x
Answer:
D
90)
Find: d
dx 3 –e(2x2+ x) 4
A)
4(3 –e(2x2+ x))3e(2x2+ x)(–4x2– 1)
B)
2(3 –e(2x2+ x))3e(2x2+ x)(–4x – 1)
C)
4(3 –e(2x2+ x))3e(2x2+ x)(–4x+ 1)
D)
4(3 –e(2x2+ x))3e(2x2+ x)(–4x – 1)
Answer:
D
91)
Find x’ for x = x(t) defined implicitly by 3t + 4tx = 3e4x and evaluate x‘ at (t, x) = (1, 0).
A)
x’ =3 + 4x
12e4x – 4t; x’ (1, 0) =7
8
B)
x’ =3 + 4x
12e4x – 4t; x’ (1, 0) =3
8
C)
x’ =3 + 4x
12e4x – 4t; x’ (1, 0) =1
4
D)
x’ =4 + 4x
12e4x – 4t; x’ (1, 0) =1
2
Answer:
B
14
92)
Find dy/dx by implicit differentiation.
x3+y3= 5
A)
dy
dx = – y2
x2
B)
dy
dx = – x2
y2
C)
dy
dx =x2
y2
D)
dy
dx =y2
x2
Answer:
B
93)
Find dy/dx by implicit differentiation.
2xy –y2= 1
A)
dy
dx =y
x – y
B)
dy
dx =y
y – x
C)
dy
dx =x
y – x
D)
dy
dx =x
x – y
Answer:
B
94)
Find dy/dx by implicit differentiation.
x3+ 3x2y + y3= 8
A)
dy
dx = – x2+ 3xy
x2+ y2
B)
dy
dx = – x2+ 2xy
x2+ y2
C)
dy
dx =x2+ 2xy
x2+ y2
D)
dy
dx =x2+ 3xy
x2+ y2
Answer:
B
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
95)
Find x’ for x = x(t) defined implicitly by t3–5x2= ln t and evaluate x’ at (t, x) = (0, –1).
Answer:
x’ =3t3– 1
10x ; 1
10
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
96)
Find the equation(s) of the tangent line(s) to the graph of y2– xy + 3 = 0 at x = –4.
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
Answer:
C
97)
Find y’ and the slope of the tangent line to the graph of ln (xy) =y3+ 1 at (1, –1).
A)
y
3xy3– x; 1
4
B)
x
3xy3– x; –1
4
C)
y
3xy3– x; –1
4
D)
y
3xy3; –1
4
Answer:
A
Solve the problem.
98)
The demand equation for a certain product is 9p2+ q2=1700, where p is the price per unit in dollars and q is
the number of units demanded. Find dq/dp.
A)
dq/dp = –p/9q
B)
dq/dp = –9p/q
C)
dq/dp = –q/9p
D)
dq/dp = –9q/p
Answer:
B
15
99)
The position of a particle at time t is given by s, where s3+ 4st + 4t3– 12t = 0. Find the velocity ds/dt.
A)
ds/dt =12 + 4s – 12t2
3s2– 4t
B)
ds/dt =12 + 4s – 12t2
3s2+ 4t
C)
ds/dt =12 – 4s – 12t2
3s2– 4t
D)
ds/dt =12 – 4s – 12t2
3s2+ 4t
Answer:
D
Provide an appropriate response.
100)
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.
A)
4
B)
6
C)
–4
D)
–6
Answer:
C
101)
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.
A)
–20
13
B)
13
20
C)
–13
30
D)
20
13
Answer:
C
102)
Evaluate dy/dt for the function at the point.
x3+ y3= 9; dx/dt = –3, x = 1, y = 2
A)
–3
4
B)
–4
3
C)
4
3
D)
3
4
Answer:
D
103)
Evaluate dy/dt for the function at the point.
x + y
x – y = x2+ y2; dx/dt = 12, x = 1, y = 0
A)
1
12
B)
–1
12
C)
12
D)
– 12
Answer:
C
104)
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?
A)
decreasing at 9 units per second
B)
increasing at 9 units per second
C)
decreasing at 27
2 units per second
D)
increasing at 27
2 units per second
Answer:
C
105)
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?
A)
50 mph
B)
150 mph
C)
75 mph
D)
125 mph
Answer:
C
16
106)
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?
A)
9.6 ft/sec
B)
4.8 ft/sec
C)
2.4 ft/sec
D)
5.2 ft/sec
Answer:
B
107)
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?
A)
4.9 ft/sec
B)
–8.1 ft/sec
C)
5.4 ft/sec
D)
8.1 ft/sec
Answer:
D
108)
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?
A)
325
6 ft/sec
B)
15
19 ft/sec
C)
30
7 ft/sec
D)
30
19 ft/sec
Answer:
C
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
109)
Find the relative rate of change of f(x) = 150x –0.08x2.
Answer:
150 – 0.16x
150x –0.08x2
110)
Find the relative rate of change of f(x) =15x +2x ln x
Answer:
17 +2ln x
15x +2x ln x
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the percentage rate of change of f(x) at the indicated value of x. Round to the nearest tenth of a percent.
111)
f(x) = 200 + 50x; x = 3
A)
14.3%
B)
57.1%
C)
33.3%
D)
–14.3%
Answer:
A
112)
f(x) = 4500 – 4x2; x = 20
A)
5.5%
B)
3.6%
C)
–5.5%
D)
–1.0%
Answer:
C
Find the elasticity of the demand function as a function of p.
113)
x = D(p) =900– p
A)
E(p) =p
900– p
B)
E(p) =1
900– p
C)
E(p) =p
p –900
D)
E(p) = p(900– p)
Answer:
A
114)
x = D(p) =700– p
A)
E(p) =p
1400 – 2p
B)
E(p) =p
2p –1400
C)
E(p) =1
1400 – 2p
D)
E(p) =p
700– p
Answer:
A
17
115)
x = D(p) =700
(p +8)2
A)
E(p) =1400p(p +8)
B)
E(p) =1400p
(p +8)3
C)
E(p) =2p
p +8
D)
E(p) =2
p +8
Answer:
C
Use the price–demand equation to determine whether demand is elastic, is inelastic, or has unit elasticity at the indicated
values of p.
116)
x = f(p) =257– 5p; p =39.
A)
Inelastic
B)
Unit elasticity
C)
Elastic
Answer:
C
117)
x = f(p) = 2005 –p2; p = 13
A)
Elastic
B)
Inelastic
C)
Unit elasticity
Answer:
B
118)
x = f(p) = 1500 – 5p2; p = 10
A)
Elastic
B)
Unit elasticity
C)
Inelastic
Answer:
B
Use the price–demand equation to find the values of p which meet the given condition of elasticity.
119)
x = f(p)=182– 9p; determine the values of p for which demand has unit elasticity. Round to two decimal places
if necessary.
A)
Inelastic at p =26.98
B)
Inelastic at p =13.49
C)
Inelastic at p =5.06
D)
Inelastic at p =10.11
Answer:
D
120)
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..
A)
Elastic on (6, 108), inelastic on (0, 6)
B)
Elastic on (0, 6), inelastic on (6, 6 3)
C)
Elastic on (0, 6), inelastic on (6, 108)
D)
Elastic on (6, 6 3), inelastic on (0, 6)
Answer:
D
Use the demand equation to find the revenue function.
121)
x = f(p) = 30(15 – p)
A)
R(p) = 450 – 30p2
B)
R(p) = 450 – 450p2
C)
R(p) = 450p – 30p2
D)
R(p) = 30 – 30p2
Answer:
C
122)
x = f(p) = 55 – 7 p
A)
R(p) = 55p – 7p
B)
R(p) = 55p – 49p3/2
C)
R(p) = 55p – 7p2
D)
R(p) = 55p – 7p3/2
Answer:
D
18
Solve the problem.
123)
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.
A)
$5 per day
B)
$75 per day
C)
$10 per day
D)
$175 per day
Answer:
C
124)
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.
A)
$175 per day
B)
$250 per day
C)
$75 per day
D)
$150 per day
Answer:
A
125)
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.
A)
$77.00 per day
B)
$149.00 per day
C)
$156.80 per day
D)
$98.00 per day
Answer:
A
126)
A beverage company works out a demand function for its sale of soda and finds it to be
x = D(p) =2900 –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?
A)
For p <121 cents
B)
For p >242 cents
C)
For p <60 cents
D)
For p >34,800 cents
Answer:
C
19