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) =1
x;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 the relative rate of change of f(x) =25x +2x ln x
3)
4)
Find x’ for x = x(t) defined implicitly by t3–5x2= ln t and evaluate x’ at (t, x) = (0, –1).
4)
1
5)
Write composite function y =e3x2+ x – 1 in the form y = f(u) and u = g(x).
5)
6)
Find the relative rate of change of f(x) = 150x –0.08x2.
6)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Use the price–demand equation to find the values of p which meet the given condition of elasticity.
7)
x = f(p)=165 – 3p; determine the values of p for which demand has unit elasticity. Round to two
decimal places if necessary.
7)
A)
Inelastic at p =13.75
B)
Inelastic at p =27.5
C)
Inelastic at p =25.7
D)
Inelastic at p =12.85
Differentiate.
8)
Find dy
dx for y =x3
x – 1 .
8)
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
9)
Find f'(t) for f(x) =2x – 7
3x – 2 .
9)
A)
–17
(2x – 7)2
B)
–17
(3x – 2)2
C)
17
(3x – 2)2
D)
17
(2x – 7)2
Use the demand equation to find the revenue function.
10)
x = f(p) = 55 – 7 p
10)
A)
R(p) = 55p – 7p
B)
R(p) = 55p – 7p3/2
C)
R(p) = 55p – 7p2
D)
R(p) = 55p – 49p3/2
Solve the problem.
11)
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
12 r)60. Find the rate of change of the amount A with respect
to the interest rate r.
11)
A)
10,000(1 +1
12 r)59
B)
1000(1 +1
12 r)59
C)
120,000(1 +1
12 r)59
D)
12,000(1 +1
12 r)59
Find f(x).
12)
f(x) = ln x3–4ex+3x2
12)
A)
3
x–4ex+3x
B)
3
x2–4ex+6
C)
3
x–4ex+6x
D)
3
x–4xex–1+6x
Find the equation of the line tangent to the graph of f at the indicated value of x.
13)
f(x) =4ex– 1; x = 0
13)
A)
y =4x +3
B)
y =4x –3
C)
y =3x +3
D)
y =4x +4
Provide an appropriate response.
14)
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.
14)
A)
4
B)
–4
C)
6
D)
–6
Find the elasticity of the demand function as a function of p.
15)
x = D(p) =200
(p +6)2
15)
A)
E(p) =400p(p +6)
B)
E(p) =400p
(p +6)3
C)
E(p) =2p
p +6
D)
E(p) =2
p +6
Solve the problem.
16)
How long will it take money to double if it is invested at 5.25%, compounded continuously? Round
your answer to the nearest tenth.
16)
A)
0.13 yr
B)
14 yr
C)
13.2 yr
D)
26.4 yr
Provide an appropriate response.
17)
Find dy
dt for y = (5t2– 4t)2.
17)
A)
(5t2– 4t)(10t – 4)
B)
2(5t2– 4t)(10t – 4)
C)
2(5t2– 4t) + (10t – 4)
D)
2(10t – 4)
Find dy
dx for the indicated function y.
18)
y = – 8 ln x +9log5x
18)
A)
8
x+9
x ln 5
B)
–8
x+1
x9 ln 5
C)
–8
x+1
x ln 5
D)
–8
x+9
x ln 5
Provide an appropriate response.
19)
Find the equation(s) of the tangent line(s) to the graph of y2– xy + 3 = 0 at x = – 4.
19)
A)
y =3
2x +1
2
B)
y =3
2x – 3
C)
y = – 3
2x – 3
D)
y =3
2x + 3 and y = – 1
2x – 3
20)
Evaluate dy/dt for the function at the point.
x + y
x – y = x2+ y2; dx/dt = 12, x = 1, y = 0
20)
A)
–1
12
B)
1
12
C)
– 12
D)
12
21)
Find dy
dt for y = (5t2– 4t)2.
21)
A)
2(5t2– 4t)(10t – 4)
B)
2(10t – 4)
C)
(5t2– 4t)(10t – 4)
D)
2(5t2– 4t) + (10t – 4)
Find the equation of the tangent line to the graph of the given function at the given value of x.
22)
f(x) =(x2+12)3/4; x =2
22)
A)
y =
3
2x +5
B)
y =
3
4x +5
C)
y =
3
2x
D)
y =
3
2x +11
Provide an appropriate response.
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)
(0.44, 0.67), (4.18, 2.04)
C)
(1.23, 3.41)
D)
(0.87, 0.42), (1.23, 3.41)
B
Differentiate.
24)
Find f'(t) for f(x) = (5x – 5)(4x3– x2+ 1)
24)
A)
f'(x) =80x3– 75x2+ 10x + 5
B)
f'(x) =60x3+ 75x2– 25x + 5
C)
f'(x) =20x3+ 25x2– 75x + 5
D)
f'(x) =80x3– 25x2+ 75x + 5
A
Use the price–demand equation to find the values of p which meet the given condition of elasticity.
25)
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..
25)
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)
D
A
Provide an appropriate response.
26)
Find dy
dx for y =
–5x3– 5x2+ 3
–5x4+ 2 . Do not simplify.
26)
A)
(–5x3– 5x2+ 3)(–20x3) – (–5x4+ 2)(–15x2– 10x)
(–5x4+ 2)2
B)
(–5x4+ 2)(–15x2– 10x) – (–5x3– 5x2+ 3)(–20x3)
(–5x4+ 2)2
C)
(–5x4+ 2)(–15x2– 10x) – (–5x3– 5x2+ 3)(–20x3)
(–5x3– 5x2+ 3)2
D)
(–5x3– 5x2+ 3)(–20x3) – (–5x4+ 2)(–15x2– 10x)
(–5x3– 5x2+ 3)2
Find the equation of the line tangent to the graph of f at the indicated value of x.
27)
f(x) =6+ ln x; x = 1
27)
A)
y = x –7
B)
y = x +7
C)
y = x +5
D)
y = x –5
Provide an appropriate response.
28)
Find: lim
x
5000e–0.07t
28)
A)
1
B)
5000
C)
D)
0
Find f(x).
29)
f(x) =4 ln x + ln x8+2ex
29)
A)
4
x+8
x7+2xex–1
B)
4
x+8
x7+2ex
C)
12
x+2xex–1
D)
12
x+2ex
Provide an appropriate response.
30)
Find y’ and the slope of the tangent line to the graph of ln (xy) =y3+ 1 at (1, –1).
30)
A)
y
3xy3– x ; 1
4
B)
y
3xy3; –1
4
C)
y
3xy3– x ; –1
4
D)
x
3xy3– x ; –1
4
Differentiate.
31)
Find f'(x) for f(x) = (2x – 4)(2x3–x2+ 1).
31)
A)
f'(x) = 4x3– 10x2– 30x + 2
B)
f'(x) = 16x3– 10x2+ 30x + 2
C)
f'(x) = 12x3+ 30x2– 10x + 2
D)
f'(x) = 16x3– 30x2+ 8x + 2
Provide an appropriate response.
32)
Find f'(x) for f(x) =(x2+ 2)3 .
32)
A)
f'(x) = 6x5+ 12x3+ 12x
B)
f'(x) = 3x5+ 24x3+ 24x
C)
f'(x) = 6x5+ 24x3+ 24x
D)
f'(x) = 6x5+ 20x3+ 24x
8
Solve.
33)
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?
33)
A)
–$30,585/yr
B)
–$43,693/yr
C)
–$85,750/yr
D)
–$94,206/yr
Find dy
dx for the indicated function y.
34)
y =6log3x
34)
A)
1
x ln 3
B)
6
x ln 3
C)
6
3 ln x
D)
1
x6 ln 3
Solve the problem.
35)
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.
35)
A)
$10 per day
B)
$175 per day
C)
$5 per day
D)
$75 per day
Find f(x).
36)
f(x) = ln x7–4x2
36)
A)
7
x6–8x
B)
7
x–8x
C)
7
x–4x
D)
1
7x –8x
Solve the problem.
37)
The demand equation for a certain product is 2p2+ q2=1500, where p is the price per unit in
dollars and q is the number of units demanded. Find dq/dp.
37)
A)
dq/dp =-2q/p
B)
dq/dp = – q/2p
C)
dq/dp =-2p/q
D)
dq/dp = – p/2q
38)
A beverage company works out a demand function for its sale of soda and finds it to be
x = D(p) =3800 –30p
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?
38)
A)
For p >253 cents
B)
For p >57,000 cents
C)
For p <63 cents
D)
For p <127 cents
Provide an appropriate response.
39)
Find f'(x) for f(x) =(ln x)5
39)
A)
1
(ln x)5
B)
1
x5
C)
5ln4 x
D)
5ln4 x
x
E)
40)
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.
40)
10
A)
B)
C)
D)
Solve the problem.
41)
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.)
41)
A)
20,032 yr
B)
124,027 yr
C)
470 yr
D)
22,689 yr
Provide an appropriate response.
42)
Find dy
dx for y =16x–1
42)
A)
16x–1 ln(x)
B)
16x–1 ln(16)
C)
16 ln(16)
D)
16x–1 ln(16x–1)
43)
Find dy/dx by implicit differentiation.
x3+y3= 5
43)
A)
dy
dx =y2
x2
B)
dy
dx =x2
y2
C)
dy
dx = – y2
x2
D)
dy
dx = – x2
y2
Find dy
dx for the indicated function y.
44)
y =7x
44)
A)
7x ln x
B)
7 ln 7
C)
7x
ln 7
D)
7x ln 7
Provide an appropriate response.
45)
Find y’ for y = y(x) defined implicitly by 5y2– 8x4+ 3 = 0, and evaluate y’ at (x, y) = (1, 1).
45)
A)
y’ =11x2
5y2; y’ (1, 1) =11
5
B)
y’ =16x3
5y ; y’ (1, 1) =16
5
C)
y’ =16x2
5y2; y’ (1, 1) =16
5
D)
y’ =11x3
5y ; y’ (1, 1) =11
5
12
Find f(x).
46)
f(x) = ln x6
46)
A)
6
x5
B)
6
x
C)
1
6x
D)
6 ln x5
Provide an appropriate response.
47)
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?
47)
A)
4.9 ft/sec
B)
–8.1 ft/sec
C)
5.4 ft/sec
D)
8.1 ft/sec
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) = 4500 – 4x2; x = 20
48)
A)
5.5%
B)
3.6%
C)
–5.5%
D)
–1.0%
Differentiate.
49)
Find dy
dx for y =x2– 3x + 2
x7– 2 .
49)
A)
dy
dx =
– 5x8+ 19x7– 14x6– 4x + 6
(x7– 2)2
B)
dy
dx =
– 5x8+ 18x7– 14x6– 3x + 6
(x7– 2)2
C)
dy
dx =
– 5x8+ 18x7– 14x6– 4x + 6
(x7– 2)2
D)
dy
dx =
– 5x8+ 18x7– 13x6– 4x + 6
(x7– 2)2
Provide an appropriate response.
50)
Write composite function y =(2x4+ 3x + 1)3 in the form y = f(u) and u = g(x).
50)
A)
y = f(u) = 2x4+ 3x + 1 and u = g(x) =u3
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) = u and u = g(x) =(2x4+ 3x + 1)3
Use the demand equation to find the revenue function.
51)
x = f(p) = 30(15 – p)
51)
A)
R(p) = 30 – 30p2
B)
R(p) = 450p – 30p2
C)
R(p) = 450 – 450p2
D)
R(p) = 450 – 30p2
Provide an appropriate response.
52)
Find: d
dx 3 –e(2x2+ x) 4
52)
A)
2(3 –e(2x2+ x))3e(2x2+ x)(–4x – 1)
B)
4(3 –e(2x2+ x))3e(2x2+ x)(–4x – 1)
C)
4(3 –e(2x2+ x))3e(2x2+ x)(–4x2– 1)
D)
4(3 –e(2x2+ x))3e(2x2+ x)(–4x+ 1)
53)
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?
53)
A)
125 mph
B)
50 mph
C)
75 mph
D)
150 mph
54)
Find the composition f[g(x)] if f(u) =u5 and g(x) = 2 –3x2.
54)
A)
(2 –3x2)2
B)
(2 –3x2)5
C)
2 –3u10
D)
(10 –15x2)5
Find f(x).
55)
f(x) = 8ex+ 4 ln x3
55)
A)
8ex+4
x2
B)
8ex+12
x3
C)
8ex+12
x
D)
8ex+12
x2
Solve the problem.
56)
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.
56)
A)
0.06 yr
B)
5.88 yr
C)
58.81 yr
D)
0.59 yr
Differentiate.
57)
Find f'(t) if f(t) = 0.4t(5t2+ 1) and simplify.
57)
A)
f'(t) =6t2+ 0.4
B)
f'(t) =6t2+ 40
C)
f'(t) =6t2+ 4
D)
f'(t) =6t2– 0.4
58)
Find f'(x) for f(x) = (5x3+ 4)(3x7– 5).
58)
A)
f'(x) = 20x9+ 84x6– 75x
B)
f'(x) = 150x9+ 84x6– 75x2
C)
f'(x) = 150x9+ 84x6– 75x
D)
f'(x) = 20x9+ 84x6– 75x2
15
Provide an appropriate response.
59)
Find the derivative of the function f(x) =2x – 7
3x – 2 at x = 2.
59)
A)
–17
16
B)
–17
4
C)
17
4
D)
17
16
60)
Find dy
dx for y = ln (6x3– x2)
60)
A)
18x – 2
6x3– x
B)
18x – 2
6x2
C)
18x – 2
6x2– x
D)
6x – 2
6x2– x
Solve the problem.
61)
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
73t + 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 =16 minutes.
61)
A)
–
3
98 units/min
B)
–1
4802 units/min
C)
–
3
4802 units/min
D)
–1
49 units/min
62)
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.
62)
A)
$250 per day
B)
$150 per day
C)
$75 per day
D)
$175 per day
Provide an appropriate response.
63)
Find dy/dx by implicit differentiation.
x3+ 3x2y + y3= 8
63)
A)
dy
dx =x2+ 3xy
x2+ y2
B)
dy
dx =x2+ 2xy
x2+ y2
C)
dy
dx = – x2+ 3xy
x2+ y2
D)
dy
dx = – x2+ 2xy
x2+ y2
64)
Find x’ for x = x(t) defined implicitly by 3t + 4tx = 3e4x and evaluate x’ at (t, x) = (1, 0).
64)
A)
x’ =4 + 4x
12e4x – 4t ; x’ (1, 0) =1
2
B)
x’ =3 + 4x
12e4x – 4t ; x’ (1, 0) =7
8
C)
x’ =3 + 4x
12e4x – 4t ; x’ (1, 0) =1
4
D)
x’ =3 + 4x
12e4x – 4t ; x’ (1, 0) =3
8
65)
Find the values of x where the tangent line is horizontal for the graph of f(x) =4x2
x + 2 .
65)
A)
x = – 2, x = 0, x = – 4
B)
x = 0, x = – 4
C)
x = – 2
D)
x = 0, x = – 2
Differentiate.
66)
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.
66)
A)
5
B)
–5
C)
6
D)
–6
Provide an appropriate response.
67)
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.
67)
A)
13
20
B)
–13
30
C)
–20
13
D)
20
13
68)
Find f'(x) for f(x) =(x2+ 2)3 .
68)
A)
f'(x) = 6x5+ 20x3+ 24x
B)
f'(x) = 6x5+ 12x3+ 12x
C)
f'(x) = 3x5+ 24x3+ 24x
D)
f'(x) = 6x5+ 24x3+ 24x
D)
Solve the problem.
69)
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?
69)
A)
$161.61
B)
$175.32
C)
$849.47
D)
$159.38
D)
Provide an appropriate response.
70)
Find the composite g[f(–k)] if f(x) = 8x2– 5x and g(x) = 7x + 9.
70)
A)
392k2– 973k + 603
B)
392k2+ 973k + 603
C)
56k2– 35k + 9
D)
56k2+ 35k + 9
D)
18
D)