PRACTICE SEMESTER EXAM - MATH 120 Spring 2008
Problems 1-27 could be on the No Derive/No Graphing Calculator part of the test.
Find the derivative of the function.
16) y = ln (5x3 - x2 )
Find the derivative.
1) y = 10x-2 + 5x3 - 4x
Evaluate.
4
2) y = x3
17)
5
∫
25
∫
e
5x4 dx
-1
Differentiate. Show use of product rule.
3) f(x) = (2x3 + 4)(4x7 - 4)
18)
Find the derivative.
4) y = 3x6 -2x4 + 5 x
5 x dx
0
Differentiate. Show use of the quotient rule.
x2 - 3x + 2
5) y = x7 - 2
6) y = ∫
19)
1
2x - 9
2x2 + 8
20)
∫ (7x2 + 1) dx
21)
∫
22)
∫ 8x1/3 dx
23)
∫ (5
Differentiate using the chain rule.
7) f(x) = 11x - x5
Differentiate.
7
dx
x
162
dx
x
8) f(x) = (x3 - 8)2/3
Find the derivative.
9) y = (4x2 + 5x)2
3
3
x2 + - 4x3 -2)dx
x
Evaluate the integral.
6 4
24)
2x dy dx
-3 -10
Find the second-order partial derivative.
10) Find fxy when f(x,y) = 8x 3 y - 7y2 + 2x.
∫ ∫
11) Find fyy when f(x,y) = 8x 3 y - 7y2 + 2x.
1
25)
12) Find fyx when f(x, y) = ln(2x + 9y).
∫ ∫
0
13) Find fxy when f(x, y) = 8xexy.
1
(10x - 1y) dy dx
0
Solve the problem.
26) Find the average rate of change for the
function over the given interval.
y = 6t2 as t changes from t1 = 3 to t2 = 5.
Find the partial derivative.
14) Let f(x, y) = x3 - 10x2 y + 3xy3 . Find fx.
Differentiate.
15) f(x) = 9e-6x
1
Decide whether the limit exists. If it exists, find its value.
27)
7
6
5
4
3
2
1
-7 -6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
-7
Determine the continuity of the function at the given
points.
33)
for x = 1,
-1,
f(x) =
1 3
for x ≠ 1
2 - x ,
3
y
at x = 1 and x = 2
1 2 3 4 5 6 7x
4
y
3
2
1
Find lim f(x) and lim f(x).
x→0 x→0 +
-1
-1
1
2
3
x
-2
28) Find: lim f(x) , lim f(x) , lim f(x)
x→1
x→1 +
x→1 -
-3
Solve the problem.
34) The graph shows the total sales in thousands
of dollars from the distribution of x thousand
catalogs. Find the average rate of change of
sales with respect to the number of catalogs
distributed for the change in x.
29) lim f(x)
x→1
20 to 30
Find f ′(a) for the given value of a.
30) f(x) = x4 + 4x3 + 2x - 2, a = -2
For the given function, find the points on the graph at
which the tangent line has slope 1.
1
1
35) y = x3 - x2 + x
3
2
Find the equation of the line tangent to the graph of the
function at the indicated point.
31) f(x) = x2 - 2 at (-4, 14)
Determine where the given function is concave up and
where it is concave down.
36) f(x) = x3 + 3x2 - x - 24
Find all values of x (if any) where the tangent line to the
graph of the function is horizontal.
32) y = x3 - 3x 2 + 1
Determine where the given function is increasing and
where it is decreasing.
37) f(x) = x4 - 8x2 - 6
2
45) Assume that the temperature of a person
during an illness is given by:
Find the relative extrema of the function, if they exist.
38) f(x) = x3 - 12x + 4
Solve the problem.
39) If the price (in dollars) of a product is given by
1024
P(x) = + 2200, where x represents the
x
T(t) = where T = the temperature, in degrees
Fahrenheit, at time t, in hours. Find the rate of
change of the temperature with respect to
time.
demand for the product, find the rate of
change of price when the demand is 8 units.
40) Find the number of units that must be
produced and sold in order to yield the
maximum profit, given the following
equations for revenue and cost:
R(x) = 7x
C(x) = 0.001x2 + 0.8x + 80.
Find the equation of the line tangent to the graph of the
function at the indicated point.
(x3 - 3x)4
at the point (2, 16)
46) y = (4x - 9)2
Determine the horizontal asymptote of the given
function. If none exists, state that fact.
4x3 - 3x - 7
47) h(x) = 9x2 + 5
Write an equation of the tangent line to the graph of y =
f(x) at the point on the graph where x has the indicated
value.
41) f(x) = (-3x2 + 3x + 2)(-2x + 5), x = 0
Find the absolute maximum and absolute minimum
values of the function, if they exist, on the indicated
interval.
42) f(x) = 6 - x2/3; [-1, 1]
Solve the problem.
43) The total profit from selling x units of
cookbooks is P(x) = (8x - 9)(4x - 3). Find the
marginal average profit function.
48) h(x) = 4x4 - 7x2 - 3
2x5 - 9x + 9
49) h(x) = 6x2 - 3x - 4
8x2 - 7x + 9
Determine the vertical asymptote(s) of the given function.
If none exists, state that fact.
x - 2
50) f(x) = 4x - x3
44) The demand function for a certain product is
given by:
D(p) = 6t
+ 98.6,
2
t + 1
51) g(x) = 4p + 130
.
11p + 19
Find the marginal demand Dʹ(p).
52) f(x) = Find x + 11
x2 - 9x
x + 9
x2 + 1
d2 y
.
dx 2
53) y = 3x4 - 5x2 + 7
Find the relative extrema of the function and classify each
as a maximum or minimum.
54) f(x) = 3x2 - 24x + 49
3
61) Suppose that the daily cost, in dollars, of
producing x televisions is
C(x) = 0.003x3 + 0.1x2 + 62x + 620,
55) f(x) = 2x3 + 3x2 - 12x - 2
Find the points of inflection.
56) f(x) = 2x 3 - 12x2 + 18x
and currently 60 televisions are produced
daily. Use C(60) and the marginal cost to
estimate the daily cost of increasing
production to 63 televisions daily. Round to
the nearest dollar.
Solve the problem.
57) The annual revenue and cost functions for a
manufacturer of grandfather clocks are
approximately R(x) = 500x - 0.01x2 and
Find the derivative.
62) y = (x2 - 2x + 1) ex
C(x) = 160x + 100,000, where x denotes the
number of clocks made. What is the maximum
annual profit?
63) y = Find the absolute maximum and absolute minimum
values of the function, if they exist, over the indicated
interval, and indicate the x-values at which they occur.
58) f(x) = x3 - x2 - x + 3; [0, 2]
10
ex
2
6x + 6
Find the indicated tangent line.
64) Find the tangent line to the graph of
f(x) = 5e6x at the point (0, 5).
y
8
Solve the problem.
65) A companyʹs total cost, in millions of dollars,
is given by C(t) = 300 - 70e-t where t = time in
6
years. Find the marginal cost when t = 2.
4
2
Find the derivative.
66) y = e x ln x
1
2 x
67) y = e x3 ln x
Find the absolute maximum and absolute minimum
values of the function, if they exist, over the indicated
interval. When no interval is specified, use the real line ( ∞, ∞).
16
59) f(x) = x + ; [-7, -1]
x
Find the derivative of the function.
68) y = ln (8x3 - x2 )
Solve the problem.
69) A special-events promoter sells x tickets and
has a marginal-revenue function given by
Rʹ(x) = 4x - 1240, where Rʹ(x) is in dollars per
ticket. This means that the rate of change of
total revenue with respect to the number of
tickets sold, x, is Rʹ(x). Find the total revenue
from the sale of the first 340 tickets.
Solve the problem.
60) A grocery store estimates that the weekly
profit (in dollars) from the production and sale
of x cases of soup is given by
P(x) = -5600 + 9.5x - 0.0017x2
and currently 1300 cases are produced and
sold per week. Use the marginal profit to
estimate the increase in profit if the store
prodcues and sells one additional case of soup
per week.
Find f such that the given conditions are satisfied.
70) fʹ(x) = x - 3, f(5) = 11
Evaluate using the substitution method.
71)
4
∫ x6(x7 - 9)4 dx
Solve the problem.
72) A manufacturer determined that its marginal
cost per unit produced is given by the function
C′(x) = 0.0006x2 - 0.4x + 88.
Find the area of the shaded region.
75) f(x) = -x3 + x2 + 16x, g(x) = 4x
y
30
Find the total cost of producing the 101st unit
through the 200th unit.
20
(4, 16)
10
Find the area of the shaded region.
73) f(x) = -x3 - x2 + 20x, g(x) = 0
50
(0, 0)
-4
y
40
(-3, -12)
-2
2
4
6 x
-10
30
-20
20
10
-30
1 2 3 4 5 6 x
-6 -5 -4 -3 -2 -1
-10
-20
Find the average value over the given interval.
76) y = 5x + 3; [2, 9]
-30
-40
-50
Solve the problem.
77) A company determines that its marginal
revenue per day is given by Rʹ(t) = 50et,
Find the area under the given curve over the indicated
interval.
74) y = x2 + 3; [0, 2]
9
R(0) = 0, where R(t) = the revenue, in dollars,
on the tth day. The companyʹs marginal cost
per day is given by Cʹ(t) = 60 - 0.1t, C(0) = 0,
where C(t) = the cost, in dollars, on the tth day.
y
8
Find the total profit from t = 0 to t = 8 (the first
8 days).
Note:
T
P(T) = R(T) - C(T) = [Rʹ(t) - Cʹ(t)] dt.
0
7
6
5
∫
4
3
2
Find the equilibrium point.
78) D(x) = (x - 4)2 , S(x) = x2 + x + 7
1
1
2
3
4
x
Find the consumer surplus at the equilibrium point.
79) D(x) = (x - 2)2 ; x = 2
Find the producer surplus at the equilibrium point.
80) S(x) = -3x2 ; x = 1
Solve the problem.
81) Find the amount in a savings account after 9
yr from an initial investment of $1360 at
interest rate 6% compounded continuously.
5
82) Find the amount of a continuous money flow
in which $1500 per year is being invested at
8.5%, compounded continuously for 10 years.
91) A farmer has 300 m of fencing. Find the
dimensions of the rectangular field of
maximum area that can be enclosed by this
amount of fencing.
83) What should P0 be so that the amount of a
Evaluate the integral.
1
x2
92)
x dy dx
0
0
continuous money flow over 30 years at
interest rate 8.5%, compounded continuously,
will be $60,000?
∫ ∫
Evaluate the function.
84) Find f(2, 0, 9) when f(x, y, z) = 2x2 + 2y2 - z 2 .
Find the indicated relative minimum or maximum of f
subject to the given constraint.
93) Minimum of f(x, y) = x2 - 14x + y 2 - 16y,
Find the requested partial derivative.
∂z
85) z = 2x3 - 4xy - y; ∂y (4, -3)
subject to 2x + 3y = 12
94) Maximum of f(x, y) = xy,
subject to x + y = 100
Find any relative extrema.
86) f(x, y) = x3 + y 3 - 9xy
87) f(x, y) = x2 + xy + y 2 - 3x + 2
Solve the problem.
88) Suppose that the labor cost for a building is
approximated by
C(x, y) = 10x2 + 2y2 - 40x - 60y + 24,000,
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.
89) A firm produces two kinds of tennis balls, one
for recreational players which sells for $2.50
per can, and one for serious players which
sells for $4.00 per can. The total revenue from
the sale of x thousand cans of the first ball and
y thousand cans of the second ball is given by
R(x, y) = 2.5x + 4y .
The company determines that the total cost, in
thousands of dollars, of producing x thousand
cans of the first ball and y thousand cans of the
second ball is given by C(x, y) = x2 - 2xy + 2y2
. Find the number of each type of ball which
must be produced and sold in order to
maximize the profit.
90) Find two numbers x and y such that x + y =
144 and xy 2 is maximized.
6
Answer Key
Testname: 120PRACTICESEMESTEREXAM
1)
dy
= -20x-3 + 15x2 - 4
dx
2)
3
dy
= dx
4
4 x
3) fʹ(x) = 80x9 + 112x6 - 24x2
dy
5
4)
= 18x5 - 8x3 + dx
2 x
5)
dy -5x8 + 18x7 - 14x6 - 4x + 6
= dx
(x7 - 2)2
6)
dy -4x2 + 36x + 16
= dx
(2x2 + 8)2
7) fʹ(x) = 8) fʹ(x) = 11 - 5x4
2 11x - x5
2x2
3
x3 - 8
9) 64x3 + 120x2 + 50x
10) 24x2
11) -14
12)
-18
(2x + 9y)2
13) 8(2xexy + x2 yexy)
14) 3x2 - 20xy + 3y3
15) -54e-6x
15x - 2
16)
5x2 - x
17) 3126
1250
18)
3
19) 7
7
20) x3 + x + C
3
21) 162 ln x + C
22) 6x4/3 + C
23) 3x5/3 + C
24) 378
9
25)
2
26) 48
27) 2; -1
28) 1, 1, 1
29) Does not exist
30) 18
7
Answer Key
Testname: 120PRACTICESEMESTEREXAM
31) y = -8x - 18
32) 0, 2
33) The function f is continuous at x = 2 but not at x = 1.
34) 1
5
35) (0, 0) and 1, 6
36) Concave up on (-1, ∞), concave down on (-∞, -1)
37) Decreasing on (-∞, -2] and [0, 2], increasing on [-2, 0] and [2, ∞)
38) Relative maximum at (-2, 20); relative minimum at (2, -12)
39) -$16/unit
40) 3100 units
41) y = 11x + 10
42) Absolute maximum: 6, absolute minimum: 5
27
43) 32 - x2
44) Dʹ(p) = 45)
-1354
(11p + 19)2
dT 6(1 - t2 )
= dt
(t2 + 1)2
46) y = 416(x - 2) + 16
47) no horizontal asymptotes
48) y = 0
3
49) y = 4
50) x = 0, x = -2
51) x = 0, x = 9
52) none
53) 36x2 - 10
54) Relative minimum: (4, 1)
55) Relative maximum: -2, 18 , relative minimum: 1, -9
56) 2,4
57) $2,790,000
58) Absolute maximum = 5 at x = 2; absolute minimum = 2 at x = 1
59) Absolute maximum: -8, absolute minimum: -17
60) $5.08
61) $5667
62) (x2 - 1) ex
63)
ex(6x2 - 12x + 6)
(6x2 + 6)2
64) y = 30x + 5
65) 9.47 million dollars per year
ex(x ln x + 1)
66)
x
67)
ex3 + 3x3 e x3 ln x
x
8
Answer Key
Testname: 120PRACTICESEMESTEREXAM
68)
24x - 2
8x2 - x
69) -$190,400
x2
27
70) f(x) = - 3x + 2
2
(x7 - 9)5
71)
+ C
35
72) $4146.14
2137
73)
12
74)
26
3
75)
937
12
76)
61
2
77) $148,521
78) 1, $9.00
79) $2.67
80) -$2
81) $2333.77
82) $23,640.83
83) $431.94
84) -73
85) -17
86) f(3, 3) = -27, relative minimum
87) f(2, -1) = -1, relative minimum
88) x = 2, y = 15
89) 4500 of the $2.50 cans and 3250 of the $4.00 cans
90) x = 48 and y = 96
91) 75 m by 75 m
1
92)
4
93) Minimum = -61 at (3, 2)
94) Maximum = 2500 at (50, 50)
9