MTH 120 Practice Test 3
Sections 7.1 - 7.5, 8.2, 9.1 - 9.4, 9.6
Find the integral.
1)
∫ 4x2/3 dx
2)
∫ 8 dx
3)
∫
4)
∫7
5)
∫ (3x8 - 7x3 + 9) dx
6)
∫
36
dx
x2
3
x
∫
16)
∫
17)
∫
18)
∫ te-7t2 dt
∫
8)
∫
3 x - 5
dx
x2
x
(7x2 + 3)5
x9
dx
ex10
∫
25
∫
e
∫
5
3 x dx
0
21)
1
∫ 8e4y dy
22)
2
10)
dx
∫
20)
7)
8 dy
(y - 9)3
Evaluate the definite integral.
4
19)
(x + 5) dx
-1
5
4
- dx
2
x
x
(x4/3 - 3x5/2) dx
9)
15)
∫ (t6 + e3t) dt
9
dx
x
dt
1 + t
Evaluate the function.
23) Find g(6, 7) when g(x, y) = 2y2 - 8xy.
11)
12)
13)
14)
∫
∫
∫
x5 + 1
dx
x
24) Find g(3, 4) when g(x, y) = 2
+ 2ex dx
x
x - 6y
x2 + y 2
.
Find the partial derivative.
25) Find fx(3, 5) when f(x, y) = 6x3 - 5xy - y.
1
2
3
+ + dx
x x 2 x3
26) f(x, y) = x3 - 9x2 y + 7xy3 . Find fy(x, y).
∫ 4(2x + 5)3 dx
Find fx (x, y).
27) f(x, y) = e-2x - 4y
1
37) After a new firm starts in business, it finds that
its rate of profit (in hundreds of dollars) after t
years of operation is given by
Pʹ(t) = 3t2 + 2t + 11.
Find the second-order partial derivative.
28) Find fyy if f(x, y) = x4 y4 - 2 x6 y8 + 17x - y.
29) Find fxx when f(x,y) = 8x 3 y - 7y2 + 2x.
Find the profit in year 2 of the operation.
Solve the problem.
30) The slope of the tangent line of a curve is
given by
fʹ(x) = x2 - 5x + 7.
Find the area of the shaded region.
38)
If the point (0, 10) is on the curve, find an
equation of the curve.
31) Find C(x) if Cʹ(x) = x and C(9) = 40.
32) Find the cost function if the marginal cost
function is Cʹ(x) = 4x - 9 and the fixed cost is $
12.
39)
33) The rate at which an assembly line workerʹs
efficiency E (expressed as a percent) changes
with respect to time t is given by Eʹ(t) = 60 - 6t,
where t is the number of hours since the
workerʹs shift began. Assuming that E(1) = 92,
find E(t).
Find the area bounded by the given curves.
40) y = x3 , y = 4x
34) The rate of expenditure for maintenance of a
particular machine is given by
Mʹ(x) = 12x x2 + 5,
Find the area between the curves.
41) x = 0, x = 1, y = x2 + 6, y = x2 + 2
where x is time measured in years. Total
maintenance costs through the second year are
$106. Find the total maintenance function.
Use the definite integral to find the area between the
x-axis and f(x) over the indicated interval.
42) f(x) = -x2 + 9; [0, 5]
35) A company has found that the marginal cost
of a new production line (in thousands) is
9
,
Cʹ(x) = x + e
Find the average value of the function on the given
interval.
43) f(x) = x + 4; [1, 21]
where x is the number of years the line is in
use. Find the total cost function for the
production line (in thousands). The fixed cost
is $20,000.
44) f(x) = (7x + 1)1/2 ; [0, 5]
36) A company has found that its rate of
expenditure (in hundreds of dollars) on a
certain type of job is given by
Eʹ(x) = 6x + 11,
where x is the number of days since the start of
the job. Find the total expenditure if the job
takes 4 days.
2
Solve the problem.
45) Production of television sets is given by P(x,y)
= 100 2 x-2/3 + 2 y -1/3 -2 , where x is work
3
5
52) A computer firm markets two kinds of
electronic calculator that compete with one
another. The total revenue function is
R p, q = 80p - 6p2 - 4pq + 68q - 2q2 , where p
hours and y is the amount of capital. If 8 work
hours and 8 units of capital are used, what is
the production output?
is the price of the first calculator (in multiples
of $10), and q is the price of the second
calculator (in multiples of $10). What prices
should be charged in order to maximize the
total revenue? What is the maximum revenue?
(answer to this not on answer sheet )
46) The production function z for an industrial
country was estimated as z = x4 y6 , where x is
the amount of labor and y the amount of
capital. Find the marginal productivity of
labor.
Find the indicated relative minimum or maximum.
53) Minimum of f(x,y) = x2 + y 2 - xy,
subject to x - y = 10
47) A company has the following production
function for a certain product:
p(x, y) = 22x0.8y0.2 .
54) Maximum of f(x, y, z) = xy + z,
subject to x2 + y 2 + z2 = 1
Find the marginal productivity with fixed
capital, px .
Solve the problem.
55) The total cost to hand-produce x large dolls
and y small dolls is given by
C(x,y) = 2x 2 + 3y2 + 4xy + 40. If a total of 40
Find all points where the function has any relative
extrema or saddle points and identify the type of relative
extremum.
48) f(x,y) = x2 - y 2
dolls must be made, how should production
be allocated so that the total cost is
minimized?
49) f(x, y) = x2 + y2 - 11x - 2y
56) The production level P of a factory during one
time period is modeled by P(x, y) = Kx1/2y1/2
50) f(x,y) = 4xy - x2 y - xy2
where K is a positive integer, x is the number
of units of labor scheduled and y is the
number of units of capital invested. If labor
costs $2500/unit, capital costs $700/unit and
the owner has $1,600,000 available for one
time period, what amount of labor and capital
would maximize production?
Solve the problem.
51) Suppose that the labor cost for a building is
approximated by
C(x,y) = 2x2 + 5y2 - 320x - 60y + 28,000, where
x is the number of days of skilled labor and y
is the number of days of semiskilled labor
required. Find the x and y that minimize cost
C.
B) x = 30, y = 18
A) x = 6, y = 30
D) x = 160, y = 12
C) x = 80, y = 6
Evaluate the iterated integral.
-9
-8
57)
xy2 dx dy
-1
-4
∫ ∫
2
58)
∫ ∫
1
∫ ∫
-3
3
(x3 + 2x 2 y - y 3 + xy) dy dx
-3
0
59)
2
0
4
x2 + y2 dx dy
Answer Key
Testname: 120PRACTICETEST3
1)
12 5/3
x + C
5
2) 8x + C
36
3) - + C
x
4)
21 4/3
x
4
5)
1 9 7 4
x - x + 9x + C
4
3
5
6) - - 8 x + C
x
7)
3 7/3 6 7/2
x - x + C
7
7
8) - 5
6
+ + C
x x
9) 2e4y + C
t7 e3t
10) + + C
7
3
11)
1 5
x + ln x + C
5
12) 2 ln x + 2ex + C
2
3
+ C
13) ln x - - x 2x2
14)
15)
16)
1
(2x + 5)4 + C
2
-4
(y - 9)2
+ C
-1
+ C
56(7x2 + 3)4
17) - 18) - 1
10ex10
+ C
1 -7t2
e
+ C
14
19) 32.5
20) 250
21) 9
22) ln 2
23) -238
21
24) - 25
25) 137
26) -9x2 + 21xy2
27) fx(x, y) = -2e-2x - 4y
28) 12x4 y2 - 56 2 x6 y6
4
Answer Key
Testname: 120PRACTICETEST3
29) 48xy
1
5
30) f(x) = x3 - x2 + 7x + 10
3
2
2
31) C(x) = x3/2 + 22
3
32) C(x) = 2x2 - 9x + 12
33) E(t) = 60t - 3t2 + 35
34) M(x) = 4(x2 + 5)3/2 - 2
35) C(x) = 9 ln(x + e) + 11
36) $9200
37) $2100
38
38)
3
39)
2
3
40) 8
41) 4
10
42)
3
43) 15
86
44)
21
45) 744
46) 4x3 y6
47) 17.6
y 0.2
x
48) Saddle point at (0, 0)
49) Relative minimum at 11
, 1
2
50) Relative maximum at 4 4
, and saddle point at (0, 0)
3 3
51) C
52) $15 and $155
53) f(5, -5) = 75
54) f(0, 0, 1) = 1
55) Make 40 large dolls and 0 small ones
56) 320.0 units of labor and 1142.9 units of capital
57) - 5824
205
58)
6
59)
368
3
5