Name: ______________________
Class: _________________
Date: _________
Review for Test 2
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____
1. For the cost function, find the marginal cost at the given production level x.
C(x) = 10,000 + 5x −
a.
b.
c.
d.
e.
____
$4.80
$5.00
$4.81
$5.20
$4.78
per
per
per
per
per
item
item
item
item
item
2. Find the value of x for which the marginal profit is zero.
C(x) = 2x, R(x) = 7x −
a.
b.
c.
d.
e.
____
2
x
, x =1,000
10,000
2
x
2,000
x = 2,500
x = 10,000
x = –5,000
x = 5,000
x = 7,000
3. The cost, in thousands of dollars, of airing x television commercials during a Super Bowl game is given by
the formula
2
C(x) = 250 + 1,500x − 0.002x .
Estimate how fast (in dollars per television commercial) the cost is going up when x = 8.
a.
b.
c.
d.
e.
$1,499,968
$1,500,032
$1,500
$1,500
$1,499,984
1
____
4. The cost of producing x teddy bears per day at the Cuddly Companion Company is calculated by their
marketing staff to be given by the formula
2
C(x) = 250 + 30x − 0.005x .
Evaluate the average cost C(100).
a.
b.
c.
d.
e.
____
5. Your monthly profit (in dollars) from selling magazines is given by P (x) = 4x + 3 x where x is the
number of magazines you sell in a month. If you are currently selling x = 100 magazines per month, find
your profit and your marginal profit.
a.
b.
c.
d.
e.
____
$29.50
$3,249.50
$28.00
$3,200.00
$32.00
P(100)
P(100)
P(100)
P(100)
P(100)
=
=
=
=
=
$430, P'(100) = $4.15
$417.32, P'(100) = $0.35
$860, P'(100) = $4.65
$417.32, P'(100) = $4.15
$215, P'(100) = $2.08
6. According to the equation given, a toy manufacturer calculates its daily profit P (in dollars) based on the
number of workers n it employs.
P = 400n − 0.4n
2
Calculate the marginal product at an employment level of 100 workers.
a.
b.
c.
d.
e.
$320
$36,000
$440
$39,960
$3,999,960
2
____
7. Your company is planning to air a number of television commercials during the ABC television network's
presentation of the Academy Awards. ABC is charging your company $795,000 per 30-second spot.
Additional fixed costs (development and personnel costs) amount to $400,000, and the network has
agreed to provide a discount of D(x) = 5,000 x for x television spots. Compute marginal cost C'(3) and
average cost C(3).
____
a.
C'(3) = $925,447 per spot; C(3) = $859,502 per spot
b.
C'(3) = $925,447 per spot; C(3) = $793,557 per spot
c.
C'(3) = $793,557 per spot; C(3) = $925,447 per spot
d.
C'(3) = $792,557 per spot; C(3) = $925,447 per spot
e.
C'(3) = $793,557 per spot; C(3) = $925,347 per spot
8. The cost of controlling emissions at a firm goes up rapidly as the amount of emissions reduced goes up.
Here is a possible model:
C(q) = 5,000 + 200q
2
where q is the reduction in emissions (in pounds of pollutant per day) and C is the daily cost (in dollars)
of this reduction. If a firm is currently reducing its emissions by 7 pounds each day, what is the marginal
cost of reducing emissions even further?
a.
b.
c.
d.
e.
____
$700
$7,800
$2,800
$2,100
$1,400
9. Your Porsche's gas mileage (in miles per gallon) is given as a function M(x) of speed x in miles per hour.
It is found that
M'(x) =
−2
4,900x
−1
.
−1
2
(4,900
+ x)
Find M '(30), M '(70) and M '(95).
a.
b.
c.
d.
e.
M '(30) = 0.0001, M '(70) = –0.0049, M '(95) = 0
M '(30) = 0.0001, M '(70) = 0, M '(95) = –0.0049
M '(70) = 0.0049, M '(30) = 0, M '(95) = –0.000051
M '(30) = 0.0049, M '(70) = 0, M '(95) = –0.000051
M '(70) = 0.0049, M '(70) = 0, M '(95) = –0.000051
3
____ 10. Calculate
dy
. You need not expand your answer.
dx
ÊÁ x
ˆ˜
3.6 ˆ˜˜˜ ÊÁÁÁ 2
˜
y = ÁÁÁÁ
+
˜˜ ÁÁ x + 4 ˜˜˜
ÁË 3.6
˜¯
x ˜¯ ÁË
a.
b.
c.
d.
e.
ÊÁ
ÁÁ 1
2x ÁÁÁ
ÁÁ 3.6
Ë
ÊÁ
ÁÁ 1
ÁÁ
ÁÁ 3.6 −
Á
Ë
ÊÁ
ÁÁ 1
ÁÁ
ÁÁ 3.6 −
Á
Ë
ÁÊÁ
Á 1
2x ÁÁÁ
ÁÁ 3.6
Ë
2x
____ 11. Calculate
ˆ˜
3.6 ˜˜˜
−
˜
2 ˜˜
x ˜¯
ˆ˜
ˆ˜
ÊÁ x
3.6 ˜˜˜ ÊÁÁÁ 2
3.6 ˆ˜˜˜
˜˜ ÁÁ x + 4 ˜˜˜˜ − 2x ÁÁÁÁ
+
˜
˜¯
ÁË 3.6
2 ˜Á
x ˜˜¯
x ˜¯ Ë
ˆ˜
ˆ˜
ÊÁ x
3.6 ˜˜˜ ÊÁÁÁ 2
3.6 ˆ˜˜˜
˜˜ ÁÁ x + 4 ˜˜˜˜ + 2x ÁÁÁÁ
+
˜
˜¯
ÁË 3.6
2 ˜˜ ÁË
x ˜˜¯
x ¯
˜ˆ
˜ˆ
3.6 ˜˜˜ ÁÊÁÁ x
3.6 ˜ˆ˜˜ ÁÊÁÁ 2
−
˜˜ + ÁÁ
+
x + 4 ˜˜˜˜
˜
Á
˜
Á
˜¯
2 ˜ Á 3.6
x ˜¯ ÁË
x ˜¯ Ë
dy
.
dx
2
y = x ( 2x + 3 ) ( 5x + 5 )
a.
b.
c.
d.
e.
2
+ 75x + 30x
2
25x + (2x + 3) (5x + 5 )
2
2x + 75x + 30
2
65x + (2x + 75) (5x + 5 )
3
2
40x + 75x + 30x
2x
3
4
____ 12. Calculate
Ê
y = ÁÁÁÁ
Ë
a.
b.
c.
d.
e.
Ê
ˆ˜ ÁÁÁÁ
x + 4 ˜˜˜ ÁÁ
¯ ÁÁ
Ë
a.
b.
c.
d.
e.
4
x +
2
x
ˆ˜
˜˜
˜˜
˜˜
˜
¯
ÊÁ
ˆ˜ ÊÁ
ˆ˜
ÁÁ
˜ Á
˜
ÁÁ x + 4 ˜˜˜ + ÁÁÁ 1 + 8 ˜˜˜ ÁÊÁÁ x + 4 ˜ˆ˜˜
ÁÁ
˜
Á
˜¯
2 ˜ Á
3 ˜˜ ÁË
2 x ÁË
x ˜¯ ÁË 2 x
x ˜¯
ÊÁ
ˆ˜ Ê
˜ˆ˜ Ê
ˆ
1 ÁÁÁ
4 ˜˜˜ ÁÁÁÁ 1
ÁÁ x +
˜˜ + ÁÁ
− 8x ˜˜˜˜ ÁÁÁÁ x + 4 ˜˜˜˜
Á
˜
2
Á
˜¯ Ë
¯
2 x ÁË
x ˜¯ Ë 2 x
ˆ˜
Ê
˜ˆ ÊÁ
ˆ˜
x ÁÁÁÁ
4 ˜˜˜ ÁÁÁÁ x
8 ˜˜˜˜ ÊÁÁ
˜˜
ÁÁ x +
˜˜ + ÁÁ
−
x
+
4
Á
˜
Á
˜¯
2 ÁÁ
2 ˜˜ ÁÁ 2
3 ˜˜˜ Ë
x ¯ Ë
x ¯
Ë
ÊÁ
ˆ˜ ÊÁ
ˆ˜
ˆ
1 ÁÁÁ
4 ˜˜˜ ÁÁÁ 1
8 ˜˜˜ ÊÁÁ
ÁÁ x +
˜˜ + ÁÁ
−
˜˜ ÁÁ x + 4 ˜˜˜˜
Á
˜
Á
˜
2
3 Ë
¯
2 x ÁË
x ˜¯ ÁË 2 x
x ˜¯
ÊÁ
ˆ˜ Ê
ˆ
ˆ
1 ÁÁÁ
4 ˜˜˜ ÁÁÁÁ 1
8 ˜˜ Ê
ÁÁ x +
˜˜ + ÁÁ
− ˜˜˜˜ ÁÁÁÁ x + 4 ˜˜˜˜
2 ˜ Á
x ˜¯ Ë
¯
x ÁÁË
x ˜¯ Ë x
1
____ 13. Calculate
y=
dy
.
dx
dy
. You need not expand your answer.
dx
2x − 3
(x − 5)(x − 1)(x − 4)
2(x − 5)(x − 4) − (3x
2
− 20x + 29)
2
((x − 5)(x − 4))
2
2(x − 5)(x − 1)(x − 4) − (3x − 20x + 10)(2x − 3)
(x − 5)(x − 1)(x − 4)
2
2
3x − 20x + 10
2
2(x − 5)(x − 1)(x − 4) + (3x − 20x + 10)(2x − 3)
2
((x − 5)(x − 1)(x − 4))
2(x − 5)(x − 1)(x − 4) − (3x
2
− 20x + 29)(2x − 3)
2
((x − 5)(x − 1)(x − 4))
5
____ 14. Compute the derivative.
d
dx
a.
b.
c.
d.
e.
ÈÍ
˘˙ |
ÍÍ 3
˙
ÍÍ (x + 3x)(x 2 − x) ˙˙˙ |
ÍÍ
˙˙ |
Î
˚|
x=2
78
59
92
72
36
____ 15. Find the equation of the line tangent to the graph of the given function at the point with x = 0.
f(x) =
a.
b.
c.
d.
e.
x+1
x+2
y = -0.5
y = -0.25x + 0.5
y = --0.25x + 0.5
y = --0.25x
y = 0.5
____ 16. The Thoroughbred Bus Company finds that its monthly costs for one particular year were given by
2
2
C( t) = 50 + t dollars after t months. After t months, the company had P( t) = 500 + t passengers per
month. How fast was its cost per passenger changing after 3 months?
a.
b.
c.
d.
e.
-$0.18 per month
$0.29 per month
$0.01 per month
$0.10 per month
$0.39 per month
6
____ 17. The "Verhulst model" for population growth specifies the reproductive rate of an organism as a function
of the total population according to the following formula:
R(p) =
r
1 + kp
where p is the total population in thousands of organisms, r and k are constants that depend on the
particular circumstances and organism being studied, and R(p) is the reproduction rate in thousands of
organisms per hour. If k = 0.05 and r = 40, find R '(p).
2
a.
2
(1 + 0.05p)
2
2
(1 + 0.05p)
2
1 + 0.05p
40
2
1 + 0.05p
2
2
1 + 0.05p
b.
c.
d.
e.
____ 18. Find
x
2
a.
b.
c.
d.
e.
dy
using implicit differentiation.
dx
− 18y = 8
x
18
1
18
x
9
-9
8
7
____ 19. Calculate the derivative of the function.
2
-3
g (x) =(2x + 2x + 3)
-4
+ 6x + 9 )
2
= -3 (4x + 2)(2x + 2x + 3)
2
-4
= -3( 2x + 2x + 3 )
2
-4
= -12 ( 2x + 2x + 3 )
2
-4
= -3 ( 4x + 2 ) ( 2x + 2x + 3 )
a.
g '(x) = ( -6x
b.
g '(x)
c.
g '(x)
d.
g '(x)
e.
g '(x)
2
____ 20. Calculate the derivative of the function.
ÊÁ 6x + 7
s(x) = ÁÁÁÁ
ÁË 5x − 2
ˆ˜ 5
˜˜
˜˜
˜¯
a.
ÁÊ 6x + 7
s'(x) = ÁÁÁÁ
ÁË 5x − 2
b.
s'(x)
c.
s'(x)
d.
s'(x)
e.
s'(x)
˜ˆ˜ 4
˜˜
˜˜
¯
47
2
(5x − 2)
ÊÁ 6x + 7 ˆ˜ 4
12
˜˜
= -5 ÁÁÁÁ
˜
ÁË 5x − 2 ˜˜¯
2
(5x − 2)
ÊÁ 6x + 7 ˆ˜ 4
47x
˜˜
= -5 ÁÁÁÁ
˜˜
ÁË 5x − 2 ˜¯
2
(5x − 2)
ÊÁ 6x + 7 ˆ˜ 4
˜˜
= 5 ÁÁÁÁ
˜
ÁË 5x − 2 ˜˜¯
ÊÁ 6x + 7 ˆ˜ 4
47
˜˜
= -5 ÁÁÁÁ
˜
ÁË 5x − 2 ˜˜¯
2
(5x − 2)
8
____ 21. Find the indicated derivative. The independent variable is a function of t.
y=x
a.
b.
c.
d.
e.
0.5
dy
dt
dy
dt
dy
dt
dy
dt
dy
dt
(1 + x);
dy
=?
dt
-0.5 dx
)
dt
0.5 dx
= (1.5x
)
dt
-0.5
0.5 dx
= (0.5x
+ 1.5x
)
dt
-0.5
0.5 dx
= (0.5x
+ 2.5x
)
dt
0.5
0.5 dx
= (0.5x
+ 2.5x
)
dt
= (0.5x
____ 22. Find the indicated derivative.
y = 8x
3
+
11
dx |
, x = 5 when t = 1,
|
x
dt |
= 11;
t =1
dy
dt
Please round the answer to the nearest hundredth.
|
|
||
a.
dy
dt
b.
dy |
|
dt ||
|
|
||
c.
dy
dt
d.
dy |
|
dt ||
e.
dy |
|
dt ||
= 1315.16
t=1
= 13151.60
t=1
= 6595.16
t=1
= 599.56
t=1
= 2175.80
t=1
9
|
|
||
=?
t =1
____ 23. Find the indicated derivative.
y=6
x +
6
x
, x = 10 when t = 1,
dx |
|
dt |
= 15;
t=1
Please round the answer to the nearest hundredth.
|
|
||
a.
dy
dt
b.
dy |
|
dt ||
|
|
||
c.
dy
dt
d.
dy |
|
dt ||
e.
dy |
|
dt ||
= 128.07
t=1
= 18.73
t=1
= 0.85
t=1
= 25.61
t=1
= 12.81
t=1
____ 24. Compute the indicated derivative using the chain rule.
y = 8x + 2;
a.
b.
c.
d.
e.
dx
dy
1
2
-4
-2
1
8
8
10
dy
dt
|
|
||
=?
t=1
____ 25. Compute the indicated derivative using the chain rule.
y = 10x
a.
b.
c.
d.
e.
2
− 7x ;
|
dx |
dy ||
x=2
7
10
2
1
33
1
13
10
7
____ 26. Find the derivative of the following function.
f(x) = log 7 4x
a.
b.
c.
d.
e.
7
x ln(4)
1
4x ln(7)
4
x ln(7)
1
x ln(7)
none of these
11
____ 27. Find the derivative of the function.
f(x) = (x
a.
b.
c.
d.
e.
9
+ 8) ln x
8
x (1 + 9 ln x) + 8
x
9
x (1 + ln x) + 8
x
9
x (9 + 9 ln x) + 8
x
9
x (1 + 9 ln x) + 8
x
none of these
____ 28. Find the derivative of the function.
|
6
|
(5x + 3)
f(x) = ln |
| (4x + 2) 9 (8x + 9)
|
a.
b.
c.
d.
e.
|
|
|
|
|
30
36
8
−
−
5x + 3
4x + 2
8x + 9
30
36
8
+
+
5x + 3
4x + 2
8x + 9
5
4
8
−
−
6
9
8x + 9
(5x + 3)
(4x + 2)
5
4
8
+
+
6
9
8x + 9
(5x + 3)
(4x + 2)
none of these
12
____ 29. Find the derivative of the function.
ÈÍ
˘4
Í
7 ˙˙
r(x) = ÍÍÍÍ ln (x ) ˙˙˙˙
ÍÎ
˙˚
a.
b.
c.
d.
e.
6 3
28[ln (x )]
7
x
7 3
28[ln (x )]
7
x
7 3
28[ln (x )]
x
7 4
28[ln (x )]
7
x
none of these
____ 30. Find the derivative of the function.
|
2x |
r(x) = ln | 2x + e |
|
|
a.
b.
c.
d.
e.
2 + 2e
2x
2x
2x + 2e
2x
2 + 2e
2x
2x + e
2x
2+e
2x
2x + 2e
2x
2+e
2x
2x + e
none of these
13
____ 31. Find the derivative of the function.
f(x) = e
5x 7
ln 4x
a.
5x 7 7
4e
35e
x ln 4x +
b.
5x 7 6
e
35e
x ln 4x +
c.
5x 7 6
4e
7e
x ln 4x +
d.
5x 7 6
e
35e
x ln 4x +
e.
5x 6 6
e
35e
x ln 4x +
5x 7
x
5x 7
4
5x 7
x
5x 7
x
5x 7
x
____ 32. Find the derivative of the function.
h (x) = e
a.
b.
c.
d.
e.
5x 2 − 2x + 1x
10x
5x
5x
3
3
10x
3
− 2x
2
x
− 4x
2
x
− 4x
x
2
2
2
2
−1
−1
−1
e
e
e
− 2x − 1
e
x
none of these
5x 2 − 2x + 1x
5x 2 − 2x + 1x
5x 2 − 2x + 1x
5x 2 − 2x + 1x
14
____ 33. Find the equation of the straight line, tangent to y = e
a.
b.
c.
d.
e.
3
e
x−
ln 8
8
e
y (x) =
x−
ln 3
8
e
y (x) =
x+
ln 3
3
e
y (x) =
x+
ln 8
none of these
y (x) =
3
e
ln 8
8
e
ln 3
8
e
ln 3
3
e
ln 8
dy
using implicit differentiation.
dx
____ 34. Find
x
e y=9
a.
b.
c.
d.
e.
ye
x
x
xe
9
1
y
-y
1−y
dy
using implicit differentiation.
dx
____ 35. Find
y lnx + y = 10
a.
b.
c.
d.
e.
y
x (ln x + 1)
y
x ln x
y
x (ln x + 1)
1
x (ln x + 1)
x
y (ln y + 1)
-
15
3x
log 8 x at the point (1, 0).
dy
using implicit differentiation.
dx
____ 36. Find
x
2
+y
a.
b.
c.
d.
e.
a.
b.
c.
d.
e.
=6
2y
y
x
x
y
2x + 2y
2x
dx
using implicit differentiation.
dy
____ 37. Find
( xy )
2
2
+y
2
=3
2y + 2x
x
y
2
+ 1)
xy
xy
2
(x + 1)
xy
2
x +1
-
(x
16
____ 38. Find
dy
using implicit differentiation.
dx
y
x
x e − y e = 10
y−1
x−1
y
x
ye − e
y
x
xe − e
y
x
ye + e
y
x
xe − e
y
x
xe + e
y
x
ye + e
y
x
xe − e
y
x
ye − e
a.
b.
c.
d.
e.
____ 39. Find
e
st
ds
using implicit differentiation.
dt
=s
5
s
a.
5s
b.
c.
d.
e.
4
se
4
− te
st
st
st
5s − te
st
se
4
st
5s − te
st
e
st
5−e
st
5−e
17
____ 40. Find
e
y
x
2
a.
dy
using implicit differentiation.
dx
= 12 + e
ye
c.
d.
e.
____ 41. Find
x
dy
using implicit differentiation.
dx
ln (20 + e
a.
b.
xy
c.
e.
)=y
y
1−x
xy
ye
20 + e
d.
x
2 y
+ 3y e
x
3 y
2e + y e
x
ye
y
3
2+y
x
2ye
y
x
e + ye
x
ye
x
3 y
2e + y e
12e
b.
y
20 + e
x+y
xy
1
xy
(1 − x)
xy
ye
xy
20 + e (1 − x)
18
____ 42. Find the equation of the tangent line for (xy)
a.
b.
c.
d.
e.
1
x−1
11
y = -11x − 1
y = -11x − 2
1
y=x+1
11
1
y=x−2
11
y = (x
b.
c.
d.
e.
+ (xy) − x = 11 at the point (-11 , 0).
y=-
____ 43. Use logarithmic differentiation to find
a.
2
3
+ x)
x
3
dy
.
dx
+6
ÊÁ
ˆ˜
ÁÁ 2
2
˜˜
ÁÁ 3x + 1
x
˜˜
3x
x + 12 ÁÁ
+
˜˜
ÁÁ 3
˜
3
ÁË x + x
2x + 6 ˜˜¯
ÊÁ
ˆ˜
ÁÁ 2
2
˜˜
ÁÁ 3x + 1
3x
˜˜
ÁÁ
+
˜˜
ÁÁ 3
˜
3
Á x +x
2x + 12 ˜˜¯
Ë
ÁÊÁ
˜ˆ˜
ÁÁ 1
˜˜
3
3
1
Á
˜˜
(x + x) x + 6 ÁÁ
+
˜˜
ÁÁ 3
3
ÁË x + x 2 (x + 6) ˜˜¯
ÊÁ
ˆ˜
ÁÁ 2
2
˜˜
3
3
ÁÁ 3x + 1
3x
˜˜
(x + x) x + 6 ÁÁ
+
˜˜
ÁÁ 3
˜
3
Á x +x
2 (x + 6) ˜˜¯
Ë
ÁÊÁ 2
˜ˆ˜
2
ÁÁ 3x + 1
˜˜
2
3x
Á
˜˜
(3x + 1) ÁÁ
+
˜˜
ÁÁ
x
3
˜˜
Á
2x
+
12
Ë
¯
2
3
19
____ 44. The number P of CDs the Snappy Hardware Co. can manufacture at its plant in one day is given by
P=x
0.8 0.6
y
where x is the number of workers at the plant and y is the annual expenditure at the plant (in dollars).
dy
Compute
at a production level of 30,000 CDs per day and x = 104.
dx
a.
b.
c.
d.
e.
-759.27
-771.39
-787.72
771.39
-792.93
____ 45. Find the exact location of all the absolute extrema of the function with domain (-5, ∞).
4
3
f(x) =x −12x
a.
b.
c.
d.
e.
(-9, –2187) - relative maximum
(9, –2187) - relative minimum
(9, 0) - absolute maximum
(9, 2187) - absolute minimum
(9, –2187) - absolute minimum
____ 46. Find the exact location of all the relative and absolute extrema of the function.
f(x) =
x
x
a.
b.
c.
d.
e.
(0,
(0,
(0,
(0,
(0,
2
2
+ 49
; -61 ≤ x ≤ 61, x ≠ ± 7
− 49
-1) - relative minimum
-1) - relative maximum
-49) - absolute minimum
49) - absolute maximum
-1) - absolute maximum
____ 47. Find the exact location of all the relative and absolute extrema of the function with domain (-∞, ∞).
f(x) = e
a.
b.
c.
d.
e.
(0,
(0,
(1,
(1,
(0,
-5x 8
1) - absolute maximum
8) - relative minimum
0) - absolute maximum
5) - absolute maximum
1) - absolute minimum
20
____ 48. Find the exact location of all the relative and absolute extrema of the function with domain ( 0, ∞ ).
f(x) = x ln x
a.
(
b.
(
c.
(
d.
(
e.
(
1
e
1
e
3
e
1
e
1
e
3
1
) - relative minimum
3e
3
, - ) - absolute minimum
e
1
,) - absolute minimum
3e
3
, - ) - relative maximum
e
3
, ) - absolute minimum
e
,-
____ 49. The graph of the derivative of a function f is shown. Determine the x-coordinates of all stationary and
singular points of f. (Assume that f (x) is defined and continuous everywhere in [-10, 1].)
a.
b.
c.
d.
e.
x=9
x = -1
x=3
x = -4
x = -3
____ 50. Maximize P =xy with 4x + 5y = 160.
a.
b.
c.
d.
e.
P = 2,000
P=5
P = 160
P = 640
P = 320
21
____ 51. Minimize S = x + y with x y = 16 and both x and y > 0.
a.
b.
c.
d.
e.
S
S
S
S
S
=8
= 16
=6
=4
= –8
____ 52. Minimize F = x
a.
b.
c.
d.
e.
2
+y
2
with x + 4y = 68.
F = 68
F = 256
F = 204
F=4
F = 272
____ 53. For a rectangle with perimeter 8 to have the largest area, what dimensions should it have?
a.
b.
c.
d.
e.
2 ×1
2 ×3
1 ×3
4 ×4
2 ×2
____ 54. For a rectangle with area 81 to have the smallest perimeter, what dimensions should it have?
a.
b.
c.
d.
e.
8 ×10
81 ×1
9 ×8
9 ×10
9 ×9
____ 55. The cost of controlling emissions at a firm goes up rapidly as the amount of emissions reduced goes up.
Here is a possible model:
C(q) = 1,000 + 150q
2
where q is the reduction in emissions (in pounds of pollutant per day) and C is the daily cost to the firm
(in dollars) of this reduction. Government clean air subsidies amount to $750 per pound of pollutant
removed. How many pounds of pollutant should the firm remove each day to minimize net cost (cost
minus subsidy)?
a.
b.
c.
d.
e.
2.5 pounds
10 pounds
75 pounds
2 pounds
5 pounds
22
____ 56. I want to fence in a rectangular vegetable patch. The fencing for the east and west sides costs $5 per
foot, while the fencing for the north and south sides costs only $3 per foot. I have a budget of $150 for
the project. What is the largest area I can enclose?
a.
b.
c.
d.
e.
375 square feet
15 square feet
187.5 square feet
93.75 square feet
150 square feet
____ 57. A packaging company is going to make closed boxes, with square bases, that hold 216 cubic centimeters.
What are the dimensions of the box that can be built with the least material?
a.
b.
c.
d.
e.
12 ×12 ×1.5 cm
6 ×6 ×6 cm
1 ×1 ×216 cm
3 ×3 ×24 cm
1.5 ×1.5 ×96 cm
2
d y
____ 58. Calculate
.
2
dx
y = -14x
a.
b.
c.
d.
e.
2
+ 11x
-28
28x
3
2
-7x + 5.5x
2
-14x + 11x
3
2
-7x − 154x
23
____ 59. Calculate
y=
a.
b.
c.
d.
e.
18
x
2
d y
2
dx
2
d y
2
dx
2
d y
2
dx
2
d y
2
dx
2
d y
2
dx
____ 60. Calculate
y = 5x
a.
b.
c.
d.
e.
2
d y
.
2
dx
-5
=-
2
x
18
=-
18
2
x
=
3
x
18
=
18
3
x
=
36
3
x
2
d y
.
2
dx
+ 2 lnx
7
150x
5
2
−
7
x
x
125
1
−
2
2
x
x
150
2
+
7
2
x
x
150
2
−
7
2
x
x
24
____ 61. The position s of a point (in feet) is given as a function of time t (in seconds).
2
s = -11 + 2t − 10t ; t = 2
a. Find its acceleration as a function of t.
b. Find its acceleration at the specified time.
a.
b.
c.
d.
e.
a(t) = 24 ft/sec2 , a(t = 2) = -24 ft/sec2
a(t) = 5 ft/sec2 , a(t = 2) = 5 ft/sec2
a(t) = 20 ft/sec2 , a(t = 2) = 20 ft/sec2
a(t) = -2 ft/sec2 , a(t = 2) = 2 ft/sec2
a(t) = -20 ft/sec2 , a(t = 2) = -20 ft/sec2
____ 62. The position s of a point (in feet) is given as a function of time t (in seconds).
s = 4116
3
t + 6t ; t = 49
a. Find its acceleration as a function of t.
b. Find its acceleration at the specified time.
a.
b.
c.
d.
e.
-1029t
2058
t
2058
t t
1029
t t
1029
t
2
2
t + 36t ft/sec , 1,761 ft/sec
2
2
+ 12t ft/sec , 1,764 ft/sec
2
2
+ 18t ft/sec , 1,761 ft/sec
2
2
+ 36t ft/sec , 1,761 ft/sec
2
2
+ 18t ft/sec , 1,764 ft/sec
25
2 2
____ 63. The graph of a function f(x) = x (x − 24) is given.
Find the coordinates of all points of inflection of this function (if any).
a.
(0, 0)
b.
c.
d.
(0, 0), (- 24 , 0), (
(-2, -80), (2, -80)
(-2, 80), (2, -80)
e.
(-
12 , -144), (
24 , 0)
12 , -144)
____ 64. In 1965 the economist F.M. Scherer modeled the number, n, of patens produced by a firm as a function
of the size, s, of the firm (measured in annual sales in millions of dollars). He came up with the following
equation based on a study of 448 large firms.
2
3
n = -3.53 + 134.08s − 24.66s + 1.428s
2
d n
Find
2
ds
a.
b.
c.
d.
e.
|
|
|
||
.
s=2
–27.18
–30.18
–32.18
17.14
–35.18
26
____ 65. A company finds that the number of new products it develops per year depends on the size if its annual
R&D budget, x (in thousands of dollars), according to the following formula.
n = -1 + 11x + 2x
2
− 0.2x
3
Find n"(2).
a.
b.
c.
d.
e.
1.6
6.6
2.6
–2.4
4.6
Multiple Response
Identify one or more choices that best complete the statement or answer the question.
____ 66. Locate all maxima in the graph.
Select all correct answers.
a.
b.
c.
d.
e.
(9, 9)
(0, 0)
(4, 0)
(-6, -7)
(-3, 6)
27
____ 67. Find the exact location of all relative and absolute extrema of the function f (x) = 6x
with domain [-6, 6].
Select all correct answers.
a.
b.
c.
d.
e.
f.
(–6, 361) - absolute maximum
(–2, –361) - relative minimum
(–6, 73) - relative maximum
(6, 361) - absolute maximum
(6, 73) - relative maximum
(–2, –23) - absolute minimum
____ 68. Find the exact location of all the relative and absolute extrema of the function.
f(x) =
x (x − 21), x ≥ 0
Select all correct answers.
a.
b.
(7, 7 ) - relative minimum
(0, 0) - absolute maximum
c.
(7, -14
d.
e.
(7, -14 7 ) - relative minimum
(0, 0) - relative maximum
7 ) - absolute minimum
____ 69. Find the exact location of all the relative and absolute extrema of the function.
f(x) =
x
2
− 32
x+6
Select all correct answers.
a.
b.
c.
d.
e.
(–4, –8) - relative minimum
(8, 16) - relative maximum
(–8, –16) - relative minimum
(–8, –16) - relative maximum
(–4, –8) - relative maximum
28
2
+ 24x + 1
____ 70. Find the exact location of all absolute extrema of the function.
f(x) =
x−2
2
x + 32
Select all correct answers.
a.
b.
c.
d.
e.
(1, 0.0625) - relative minimum
(8, 0.0625) - absolute maximum
(0, –0.1250) - relative minimum
(8, 0.0625) - absolute minimum
(–4, –0.1250) - absolute minimum
Numeric Response
2
71. The monthly sales of Sunny Electronics' new stereo system is given by S(x) = 30x − x hundred units per
2
month, x months after its introduction. The price Sunny charges is p(x) = 800 − x dollars per stereo
system, x months after its introduction. The revenue Sunny earns then must be R(x) = 100p(x)S(x). Find
the rate of change of revenue 10 months after introduction.
Please enter your answer in dollars/month without the units.
72. The number P of CDs the Snappy Hardware Co. can manufacture at its plant in one day is given by
P=x
0.8 0.6
y
where x is the number of workers at the plant and y is the annual expenditure at the plant (in dollars).
dy
Compute
at a production level of 24,000 CDs per day and x = 94.
dx
73. Maximize P = x y z with x + y = 12 and z + y = 12, and x, y, and z > 0.
74. The demand for rubies at Royal Ruby Retailers is given by
q=-
5
p + 100
4
where p is the price RRR charges (in dollars) and q is the number of rubies RRR sells per week. At what
price should RRR sell its rubies to maximize its weekly revenue?
Please enter your answer in dollars without the units.
29
75. Calculate
y = -4x
2
2
d y
.
2
dx
+ 3x
Matching
Calculate the derivatives of the functions.
Match each function with the corresponding derivative.
a.
f(x) = (19x + 63)
b.
f(x) = (34x − 81)
c.
f(x) = (31x + 74)
8
13
0.6
____ 76. f '(x) = 152(19x + 63)
____ 77. f '(x) = 442(34x − 81)
7
12
____ 78. f '(x) = 18.6(31x + 74)
−0.4
Short Answer
79. Find the equation of the line tangent to the graph of the given function at the point x = 1.
f(x) = (x
2
+ 2) (x
3
+ x)
80. Use logarithmic differentiation to find
y=x
dy
.
dx
4x
81. Find the exact location of all relative and absolute extrema of the function f(x) = 8x
domain [-6, 6].
30
2
+ 80x + 1 with
82. The graph of the derivative of a function f is shown. Determine the x-coordinates of all stationary and
singular points of f. (Assume that f (x) is defined and continuous everywhere in [-10, 10].)
Please enter your answer in the form "x = a, x = b, ..." where a, b, ... are the x-coordinates of stationary
or singular points of f.
83. Calculate the derivative of the function.
2
5
f(x) =(x + 4x)
84. Calculate the derivative of the function.
˘7
ÍÈÍ
8
5 ˙˙
f(x) = ÍÍÍÍ (4.5x −6) +(7.7x +4) ˙˙˙˙
ÍÎ
˙˚
Please enter your answer as an expression.
85. Find the derivative of the following function.
f(x) = log 4 2x
86. Find the derivative of the function.
È
˘
h (x) = ln ÍÍÎ (7x + 7) (-9x + 1) ˙˙˚
31
ID: A
Review for Test 2
Answer Section
MULTIPLE CHOICE
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
A
D
A
E
A
A
C
C
D
C
E
D
E
D
C
C
B
C
E
E
C
C
E
D
C
D
D
A
C
B
D
A
A
D
A
C
C
B
C
E
PTS:
PTS:
PTS:
PTS:
PTS:
PTS:
PTS:
PTS:
PTS:
PTS:
PTS:
PTS:
PTS:
PTS:
PTS:
PTS:
PTS:
PTS:
PTS:
PTS:
PTS:
PTS:
PTS:
PTS:
PTS:
PTS:
PTS:
PTS:
PTS:
PTS:
PTS:
PTS:
PTS:
PTS:
PTS:
PTS:
PTS:
PTS:
PTS:
PTS:
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
ID: A
41.
42.
43.
44.
45.
46.
47.
48.
49.
50.
51.
52.
53.
54.
55.
56.
57.
58.
59.
60.
61.
62.
63.
64.
65.
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
E
A
D
A
E
B
A
B
D
E
A
E
E
E
A
D
B
A
E
E
E
D
C
C
A
PTS:
PTS:
PTS:
PTS:
PTS:
PTS:
PTS:
PTS:
PTS:
PTS:
PTS:
PTS:
PTS:
PTS:
PTS:
PTS:
PTS:
PTS:
PTS:
PTS:
PTS:
PTS:
PTS:
PTS:
PTS:
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
PTS:
PTS:
PTS:
PTS:
PTS:
1
1
1
1
1
MULTIPLE RESPONSE
66.
67.
68.
69.
70.
ANS:
ANS:
ANS:
ANS:
ANS:
A, E
C, D, F
C, E
A, D
B, E
NUMERIC RESPONSE
71. ANS: 300,000
PTS: 1
72. ANS: -662.71
PTS: 1
73. ANS: 256
PTS: 1
2
ID: A
74. ANS: 40
PTS: 1
75. ANS: -8
PTS: 1
MATCHING
76. ANS: A
77. ANS: B
78. ANS: C
PTS: 1
PTS: 1
PTS: 1
SHORT ANSWER
79. ANS:
y = 16x − 10
PTS: 1
80. ANS:
4x
4x (ln x + 1)
PTS: 1
81. ANS:
(-5, -199), (-6, -191), (6, 769)
PTS: 1
82. ANS:
x=2
PTS: 1
83. ANS:
2
4
5(2x + 4) ⋅ (x + 4x)
PTS: 1
84. ANS:
8
5 6
7
4
7((4.5x − 6) + (7.7x + 4) ) ⋅ (36(4.5x − 6) + 38.5(7.7x + 4) )
PTS: 1
85. ANS:
1
(x ⋅ ln(4))
PTS: 1
3
ID: A
86. ANS:
7
-9
+
(7x + 7) (-9x + 1)
PTS: 1
4