Math 125 Practice Problems for Test #3
Also study the assigned homework problems from the book. Donʹt forget to look over Test #1 and Test #2!
Find the derivative of the function.
1) Know the derivatives of all of the
trigonometric functions.
Solve the problem.
16) A certain radioactive isotope decays at a rate
of 3% per 100 years. If t represents time in
years and y represents the amount of the
isotope left, use the condition that y = 0.97y0
Find the derivative of y with respect to x, t, or θ, as
appropriate.
2) y = e7 - 9x
to find the value of k in the equation y = y0 ekt .
17) In a chemical reaction, the rate at which the
amount of a reactant changes with time is
proportional to the amount present, such that
dy
= -0.7y, when t is measured in hours. If
dt
3) y = e(10 x + x4 )
4) y = 8xex - 8ex
there are 61 g of reactant present when t = 0,
how many grams will be left after 3 hours?
Give your answer to the nearest tenth of a
gram.
5) y = (x2 - 2x + 5) ex
6) y = 3eθ(sin θ - cos θ)
Find the angle.
7) y = sin e-θ7
Find the derivative of y with respect to x, t, or θ, as
appropriate.
8) y = ln 7x
22) cos sin-1
1 - x
12
13
(x + 2)4
1
23) sin cos-1
2
1 + x
x2
24) sec cos-1
Find the inverse of the function.
14) f(x) = x3 + 7
15) f(x) = 2
2
Evaluate exactly.
ln x
x6
13) y = ln 19) cos-1 21) sec -1 2
10) y = ln 5x2
12) y = ln 3
2
20) tan-1 -1
9) y = ln (x - 2)
11) y = 18) sin-1 3
2
Find the derivative of the function.
25) Know the derivatives of all of the inverse
trigonometric functions.
5
x + 6
1
33) Minimum
Find the derivative of y with respect to x.
6x
26) y = tan-1 5
5
h(x)
4
3
27) y = cos-1 (5x2 + 4)
2
1
1
28) y = sin-1 x3
-5
-4
-3
-2
-1
-1
1
2
3
4
5 x
-2
29) y = sin-1 (e5t)
-3
-4
30) y = tan-1 7x
-5
Find the location of the indicated absolute extremum for
the function.
31) Minimum
Find the absolute extreme values of each function on the
interval.
34) f(x) = 2x - 3; -2 ≤ x ≤ 4
f(x)
6
5
35) f(x) = sin x + 4
7π
π
, 0 ≤ x ≤ 4
2
3
Find the derivative at each critical point and determine
the local extreme values.
36) y = x2/3(x2 - 4); x ≥ 0
2
1
-7
-6
-5
-4
-3
-2
-1
1
-1
2
3
4
5
37) y = x(1 - x2 )
-2
-3
38) y = x2 16 - x
-4
-5
-6
Determine whether the function satisfies the hypotheses
of the Mean Value Theorem for the given interval.
39) f(x) = x1/3,
-1,1
32) Maximum
g(x)
5
40) g(x) = x3/4,
0,3
41) s(t) = t(1 - t),
-1,5
4
3
2
1
-6
-5
-4
-3
-2
-1
-1
1
2
3
4
5
Find any relative extrema for the function.
42) f(x) = 0.1x3 -15x2 + 8x - 10
6
-2
43) f(x) = 0.1x4 - x3 - 15x2 + 59x + 14
-3
-4
44) f(x) = x4 - 3x3 - 21x2 + 74x - 87
-5
2
Determine the location of each local extremum of the
function.
45) f(x) = -x3 - 4.5x2 - 6x + 2
Determine the intervals where f is concave up or concave
down. Find the x-coordinate of the point of inflection.
53) f(x)=x2 -2x+5
46) f(x) = x3 - 12x2 + 48x - 2
54) f(x)=x3 -9x2 +2
55) f(x)=x4 -4x3 +10
Find the critical numbers of f. Determine the intervals on
which f is increasing or decreasing and find the local
extrema.
47) f(x)=x2 -x-2
Find the Horizontal and Veritcal asymptotes, if any.
x
56) f(x) = x + 3
48) f(x)=2x3 +3x2 +4
57) f(x) = 49) f(x)=x3 +6x2 +12x+1
Find a function that satisfies the given conditions and
sketch its graph.
58) lim f(x) = 0, lim f(x) = ∞, lim f(x) = ∞.
x→±∞
x→3 +
x→3 -
50) f(x)=3x4 -12x3 +5
Use the graph of the function f(x) to locate the local
extrema and identify the intervals where the function is
concave up and concave down.
51)
10
x
2
x + x + 2
10
8
y
6
4
y
2
5
-10 -8 -6 -4 -2-2
2 4
6 8 10 x
-4
-6
-10
-5
5
10 x
-8
-10
-5
59)
-10
lim g(x) = -4, lim g(x) = 4, lim g(x) =
x→-∞
x→∞
x→0 +
4, lim g(x) = -4.
x→0 -
52)
10
10
y
8
6
4
2
5
-10
-5
y
5
-10 -8 -6 -4 -2
-2
-4
10 x
-6
-8
-5
-10
-10
3
2 4
6 8 10 x
63) f(x) = x1/3(x2 - 175)
Sketch the graph and show all local extrema and
inflection points.
60) f(x) = 4x2 + 24x
200
y
y
100
x
-10
-5
5
10
x
-100
-200
64) f(x) = 61) f(x) = 16x
x2 + 16
x2
x2 + 10
y
y
x
x
65) f(x) = x + cos 2x, 0 ≤ x ≤ π
y
62) f(x) = 2x 3 - 15x2 + 24x
4
y
3
2
x
1
1
-1
Solve the problem.
4
2
3
x
66) Using the following properties of a
twice-differentiable function y = f(x), select a
possible graph of f.
D)
24
y
16
x
y
x < 2
11
-2
-2 < x < 0
0
-5
0 < x < 2
2
-21
x > 2
Derivatives
y′ > 0,y′′ < 0
y′ = 0,y′′ < 0
y′ < 0,y′′ < 0
y′ < 0,y′′ = 0
y′ < 0,y′′ > 0
y′ = 0,y′′ > 0
y′ > 0,y′′ > 0
8
-4 -3 -2 -1
-8
8
1
2
3
4
68) A company is constructing an open-top,
square-based, rectangular metal tank that will
have a volume of 51 ft 3 . What dimensions
-24
yield the minimum surface area? Round to the
nearest tenth, if necessary.
B)
y
69) Find two numbers whose difference is 100 and
whose product is a minimum.
16
8
-4 -3 -2 -1
-8
1
2
3
4
70) Find two positive numbers whose product is
100 and whose sum in a minimum.
x
71) Find a positive number such that the sum of
the number and its reciprocal is as small as
possible.
-16
-24
C)
Solve the problem.
72) A rectangular field is to be enclosed on four
sides with a fence. Fencing costs $5 per foot for
two opposite sides, and $7 per foot for the
other two sides. Find the dimensions of the
field of area 880 ft2 that would be the cheapest
y
16
8
-4 -3 -2 -1
-8
x
x
-16
24
4
67) From a thin piece of cardboard 50 in. by 50 in.,
square corners are cut out so that the sides can
be folded up to make a box. What dimensions
will yield a box of maximum volume? What is
the maximum volume? Round to the nearest
tenth, if necessary.
16
24
3
-24
y
-4 -3 -2 -1
-8
2
-16
A)
24
1
1
2
3
4
x
to enclose.
-16
-24
5
Answer Key
Testname: REVIEWT3FALL2008
1)
2) -9e7 - 9x
5
+ 4x3 e(10 x + x4 )
3)
x
4) 8xex
5) (x2 + 3) ex
6) 6eθ sin θ
7) (-7θ6 e-θ7 ) cos e-θ7
1
8)
x
9)
1
x - 2
10)
2
x
11)
1 - 6ln x
x7
12)
3x - 6
(x + 2)(1 - x)
13)
-4 - 3 x
2x(1 + x )
3
14) f-1 (x) = x - 7
-6x + 5
15) f-1 (x) = x
16) -0.00030
17) 7.5 g
π
18)
3
19)
π
4
20)
-π
4
21)
π
4
22)
5
13
23)
3
2
24)
2 3
3
25)
26)
30
36x2 + 25
6
Answer Key
Testname: REVIEWT3FALL2008
27)
28)
-10x
1 - (5x2 + 4)2
-3
x x6 - 1
29)
5 e5t
1 - e10t
30)
7
2(1 + 7x) 7x
31) x = 3
32) x = 0
33) No minimum
34) Maximum value is 5 at x = 4; minimum value is - 7 at x = -2
35) Maximum value of 1 at x = 0; minimum value of -1 at x = π
36)
Critical Pt. derivative Extremum Value
x = 0
Undefined local max 0
x = 1
0
minimum -3
37)
Critical Pt. derivative Extremum Value
x = 0.58
0
local max 0.38
x = -0.58 0
local min -0.38
38)
Critical Pt. derivative Extremum Value
0
0
min
x =0
0
undefined min
x = 16
x = 64
5
0
local max
16384
125
5
39) No
40) Yes
41) No
42) Approximate local maximum at 0.267; approximate local minimum at 99.733
43) Approximate local maximum at 1.735; approximate local minima at -6.777 and 12.542
44) Approximate local maximum at 1.604; approximate local minima at -3.089 and 3.735
45) Local maximum at -1; local minimum at -2
46) No local extrema
1
1
1
1
47) , decreasing on -∞ , , increasing on , ∞ , local minimum at 2
2
2
2
48) -1, 0; increasing on -∞ , -1 and 0 , ∞ ; decreasing on -1 , 0 ; local max at -1, local min at 0
49) -2, increasing for all real numbers, no relative extrema
50) 0 , 3; decreasing on (-∞ , 3 ); increasing on (3 , ∞ ); local minimum at 3
51) Local minimum at x = +1; local maximum at x = -1; concave up on (0, ∞); concave down on (-∞, 0)
52) Local minimum at x = 3; local maximum at x = -3 ; concave up on (0, ∞); concave down on (-∞, 0)
7
Answer Key
Testname: REVIEWT3FALL2008
53) Concave up for all x; no inflection points
54) Concave up on (3 , ∞) ; Concave down on (-∞ , 3) ; inflection point at 3
55) Concave up on (-∞ , 0) and (2 , ∞) ; Concave down on (0 , 2); inflection points at 0 and 2
56) Asymptotes: x = -3, y = 1
3
y
2
1
-10
-5
5
10 x
5
10 x
-1
-2
-3
57) Asymptote: y = 0
2
y
1
-10
-5
-1
-2
58) (Answers may vary.) Possible answer: f(x) = 10
8
1
.
x - 3
y
6
4
2
-10 -8 -6 -4 -2-2
2 4 6
8 10 x
-4
-6
-8
-10
8
Answer Key
Testname: REVIEWT3FALL2008
59) (Answers may vary.) Possible answer: f(x) = 4, x > 0
-4, x < 0
y
10
8
6
4
2
-10 -8 -6 -4 -2-2
2 4 6
8 10 x
-4
-6
-8
-10
60)
200
y
100
-10
-5
5
10
x
-100
-200
Minimum: (-3,-36)
No inflection points
61) Local minimum: (-4,-2)
Local maximum: (4,2)
Inflection point: (0,0), (-4 3, -4 3),(4 3, 4 3)
6
y
4
2
-4
-2
2
4
x
-2
-4
-6
9
Answer Key
Testname: REVIEWT3FALL2008
62) Local max: 1,11 , min: 4,-16
5 5
Inflection point: ,- 2 2
y
24
12
-8
-4
4
8
x
-12
-24
3
3
63) Local max: -5, 150 5 , min: 5, -150 5
Inflection point: (0,0)
400
y
300
200
100
-20
-10
20 x
10
-100
-200
-300
-400
64) Min: (0,0)
Inflection points: - 0.75
30 1
, , 4
3
30 1
, 4
3
y
0.5
0.25
-3
-2
-1
1
2
3 x
-0.25
-0.5
-0.75
10
Answer Key
Testname: REVIEWT3FALL2008
65) Local minimum: Inflection points: 5π 5π - 6 3
π π + 6 3
, ; local maximum: , 12
12
12
12
π π
3π 3π
, and , 4 4
4
4
y
4
3
2
1
1
2
3
x
-1
66) A
67) 33.3 in. by 33.3 in. by 8.3 in.; 9259.3 in. 3
68) 4.7 ft by 4.7 ft by 2.3 ft
69) -50 , 50
70) 10 , 10
71) 1
72) 35.1 ft @ $5 by 25.1 ft @ $7
11