Practice Test #2

MTH 120 Practice Test #2
Sections 4.3-4.5, 5.1-5.4, 6.1, 6.2
Find the derivative.
17) f(x) = 1) y = (4x + 3)5
x2
x2 + 5
A)
2) y = 4x + 2
y
3) f(x) = (x3 - 8)2/3
4) f(x) = 1
5
(2x - 3)4
-2
-1
Find all values of x for the given function where the
tangent line is horizontal.
5) f(x) = x2 + 4x + 15
x
1
2
1
2
3x
1
2
x
-1
B)
y
Find the derivative.
6) y = 4e-4x
1
7) y = e9x2 + x
-2
-1
8) y = 4ex2
-1
9) y = 5x 2 e3x
C)
10) y = 100
y
2 + 9e.3x
1
Find the derivative of the function.
11) y = ln (x - 8)
-2
-1
12) y = ln 8x
-1
13) y = ln 3x2
D)
y
14) y = ln (5 + x2 )
1
15) y = ln (x + 9)5
-2
16) y = (6x2 + 7) ln(x + 4)
-1
1
-1
Sketch the graph and show all extrema, inflection points,
and asymptotes where applicable.
1
2
x
B)
Find the derivative.
ex
18) y = ln x
y
3
2
19) y = e x3 ln x
1
-6
Find the open interval(s) where the function is changing
as requested.
20) Increasing; f(x) = x2 - 2x + 1
-4
-2
2
4
6
8x
2
4
6
8x
2
4
6
8x
-1
-2
-3
21) Decreasing; f(x) = x + 3
x - 5
C)
y
3
Find fʺ(x) for the function.
22) f(x) = 8x2 + 3x - 3
2
1
23) f(x) = 3x4 - 8x2 + 4
-6
-4
-2
-1
Find the requested value of the second derivative of the
function.
24) f(x) = x4 + 4x3 - 2x + 5; Find f′′ (1).
-2
-3
D)
Find the indicated derivative of the function.
25) f(4)(x) of f(x) = 2x6 - 3x4 + 2x2
y
3
Sketch the graph and show all extrema, inflection points,
and asymptotes where applicable.
1
26) f(x) = 2
x + 2x - 3
2
1
-6
-4
-2
-1
A)
-2
y
-3
3
2
Find the equation of the tangent line to the graph of the
given function at the given value of x.
27) f(x) = x3 x3 + 8; x = 1
1
-6
-4
-2
2
4
6
8x
-1
-2
Solve the problem.
28) The sales in thousands of a new type of
product are given by S(t) = 210 - 40e-0.1t,
-3
where t represents time in years. Find the rate
of change of sales at the time when t = 3.
2
29) A companyʹs total cost, in millions of dollars,
is given by C(t) = 120 - 60e-t where t = time in
36) Increasing
years. Find the marginal cost when t = 3.
30) The demand function for a certain book is
given by the function x = D(p) = 73e-0.006p.
Find the marginal demand Dʹ(p).
31) Assume the total revenue from the sale of x
items is given by R(x) = 24 ln (1x + 1), while
the total cost to produce x items is C(x) = x/4.
Find the approximate number of items that
should be manufactured so that profit,
R(x) - C(x), is maximum.
Find the open interval(s) where the function is changing
as requested.
1
37) Increasing; f(x) = x2 + 1
38) Decreasing; f(x) = - x + 3
32) Suppose that the demand function for x units
180 ln(x + 5)
,
of a certain item is p = 100 + x
Solve the problem.
39) A manufacturer sells telephones with cost
function C(x) = 7.22x - 0.0002x2 , 0 ≤ x ≤ 650
where p is the price per unit, in dollars. Find
the marginal revenue.
and revenue function R(x) = 9.2x - 0.002x2 , 0
≤ x ≤ 650. Determine the interval(s) on which
the profit function is increasing.
Identify the intervals where the function is changing as
requested.
33) Decreasing
40) Suppose the total cost C(x) to manufacture a
quantity x of insecticide (in hundreds of liters)
is given by C(x) = x3 - 27x2 + 240x + 650.
Where is C(x) decreasing?
Find the location and value of all relative extrema for the
function.
41)
34) Increasing
35) Decreasing
42)
3
43)
Find the largest open intervals where the function has the
indicated concavity.
54) Concave downward
Find the x-value of all points where the function has
relative extrema. Find the value(s) of any relative extrema.
x2 + 1
44) f(x) = x2
55) Concave upward
45) f(x) = x3 - 12x + 2
46) f(x) = 1
2
x + 1
Find the largest open intervals where the function is
concave upward.
56) f(x) = 4x 3 - 45x2 + 150x
47) f(x) = x2/5 - 1
48) f(x) = (ln 3x)2 , x > 0
57) f(x) = 49) f(x) = ln x - x, x > 0
50) f(x) = 2xe-x
6
x
Solve the problem.
58) Find the point of diminishing returns (x, y) for
the function R(x) = 5000 - x3 + 33x2 + 500x,
Solve the problem.
51) The annual revenue and cost functions for a
manufacturer of grandfather clocks are
approximately
R(x) = 480x - 0.01x2 and C(x) = 200x + 100,000,
0 ≤ x ≤ 20, where R(x) represents revenue in
thousands of dollars and x represents the
amount spent on advertising in tens of
thousands of dollars.
where x denotes the number of clocks made.
What is the maximum annual profit?
52) Find the number of units, x, that produces the
maximum profit P, if C(x) = 40 + 64x and
p = 68 - 2x.
53) Find the price p per unit that produces the
maximum profit P if C(x) = 90 + 44x and p = 84
- 2x.
4
Find the location of the indicated absolute extremum for
the function.
59) Minimum
6
Solve the problem.
65) P(x) = -x3 + 12x2 - 36x + 400, x ≥ 3 is an
approximation to the total profit (in thousands
of dollars) from the sale of x hundred
thousand tires. Find the number of hundred
thousands of tires that must be sold to
maximize profit.
f(x)
66) Of all numbers whose difference is 6, find the
two that have the minimum product.
6 x
-6
67) If the price charged for a bolt is p cents, then x
thousand bolts will be sold in a certain
x
hardware store, where p = 48 - . How many
16
-6
bolts must be sold to maximize revenue?
Find the indicated absolute extremum as well as all values
of x where it occurs on the specified domain.
60) f(x) = x3 - 3x 2 ; [0, 4]
68) A company wishes to manufacture a box with
a volume of 44 cubic feet that is open on top
and is twice as long as it is wide. Find the
width of the box that can be produced using
the minimum amount of material.
Minimum
61) f(x) = x2 e-0.25x ; [3,10]
Maximum
69) A piece of molding 164 cm long is to be cut to
form a rectangular picture frame. What
dimensions will enclose the largest area?
Find the location of the indicated absolute extremum for
the function.
62) Maximum
6
f(x)
6 x
-6
-6
Find the absolute extrema, on the given interval, if they
exist as well as where they occur.
x - 2
63) f(x) = on the interval (-4, 5)
x2 + 3x + 6
64) f(x) = -3x4 + 16x3 - 18x2 + 8 on (1, 10)
5
Answer Key
Testname: 120PRACTICETEST2
1)
dy
= 20(4x + 3)4
dx
2)
2
dy
= dx
4x + 2
3) fʹ(x) = 4) fʹ(x) = 2x2
3
x3 - 8
-40
(2x - 3)5
5) -2
6) -16e-4x
7) 18xe9x2 + 1
8) 8xe
9) 5xe 3x(3x + 2)
-270e.3x
10)
(2 + 9e.3x )2
11)
1
x - 8
12)
1
x
13)
2
x
14)
2x
2
x + 5
15)
5
x + 9
16)
6x2 + 7
+ 12x ln(x + 4)
x + 4
17) D
18)
x ex ln x - ex
x ln2 x
19)
ex3 + 3x3 e x3 ln x
x
20) (1, ∞)
21) (-∞, 5), (5, ∞)
22) 16
23) 36x2 - 16
24) 36
25) 720x2 - 72
26) B
13
19
27) y = x - 2
2
28) 3.0 thousand per year
6
Answer Key
Testname: 120PRACTICETEST2
29) 2.99 million dollars per year
30) Dʹ(p) = -0.438e-0.006p
31) 95 items
180
dR
= 100 + 32)
x + 5
dx
33) (-3, -2)
34) (0, 5)
35) (1, 2)
36) (3, ∞)
37) (-∞, 0)
38) (-3, ∞)
39) (0, 550)
40) (8, 10)
41) Relative maximum of 3 at -2 ; Relative minimum of 0 at 2.
42) Relative minimum of -2 at -3 ; Relative maximum of 2 at 3.
43) None
44) No relative extrema.
45) Relative maximum of 18 at -2; Relative minimum of -14 at 2.
46) Relative maximum of 1 at 0.
47) Relative minimum of -1 at 0.
1
48) , 0 , relative minimum
3
49) (1, -1), relative maximum
2
50) 1, , relative maximum
e
51) $1,860,000
52) 1 units
53) $64
54) (-∞, -2)
55) (0, ∞)
15
, ∞
56)
4
57) (0, ∞)
58) (11 , 13,162)
59) x = 3
60) -4 at x = 2
61) 8.6615 at x = 8
62) x = -1
63) Absolute minimum of - 1 at x = -2; absolute maximum of 64) Absolute maximum of 35 at x = 3; no absolute minima
65) 6 hundred thousand
66) 3 and -3
67) 384 thousand bolts
68) 3.4 ft
69) 41 cm × 41 cm
7
1
at x = 6
15

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