Review for Test 2

Review for Test 2
Calculus I
Find the absolute extreme values of the function on the interval.
1) f(x) = 2x - 3, -2 ≤ x ≤ 3
2) g(x) = -x2 + 8x - 16, 4 ≤ x ≤ 4
3) F(x) = - 3 x2 , 0.5 ≤ x ≤ 2
4) F(x) = 3 x, -2 ≤ x ≤ 64
5) g(x) = 7 - 8x2, -2 ≤ x ≤ 3
Find the absolute extreme values of the function on the interval.
6) f(x) = x2/3, -1 ≤ x ≤ 27
Determine all critical points for the function.
7) f(x) = x2 + 4x + 4
Find the extreme values of the function and where they occur.
8) y = x2 + 2x - 3
Find the derivative at each critical point and determine the local extreme values.
9) y = x(9 - x2)
Find the value or values of c that satisfy the equation
the function and interval.
10) f(x) = x2 + 2x + 2, [-2, 1]
f(b) - f(a)
b - a
= f′(c) in the conclusion of the Mean Value Theorem for
Solve the problem.
11) Given the velocity and initial position of a body moving along a coordinate line at time t, find the bodyʹs
position at time t.
v = -19t + 1, s(0) = 8
Find the largest open interval where the function is changing as requested.
12) Increasing y = 7x - 5
Identify the functionʹs local and absolute extreme values, if any, saying where they occur.
13) f(x) = -x3+ 6x2 - 9x - 1

Graph the equation. Include the coordinates of any local and absolute extreme points and inflection points.
6x
14) y =
x2 + 9
y
x
Sketch the graph and show all local extrema and inflection points.
15) y = -x4 + 2x2 - 9
y
10
5
-10 -5 5 10
x
-5
-10
16) y = x x 2 - 3 4 x
500
y
250
-10 10
x
-250
-500

Graph the equation. Include the coordinates of any local and absolute extreme points and inflection points.
17) y = 2x3 - 3x2 - 12x
y
x
Identify the functionʹs local and absolute extreme values, if any, saying where they occur.
18) f(x) = x3 + 6.5x2 + 12x + 2
19) f(x) = x3- 9x2 + 27x + 3
20) f(r) = (r - 8) 3
x - 1
21) h(x) =
x2 + 3x + 5
Find the largest open interval where the function is changing as requested.
22) Increasing f(x) = 1 4 x 2 - 1 2 x
23) Decreasing f(x) = 4 - x
24) Decreasing f(x) = ∣x - 8∣
25) Decreasing y = 1 x2 + 7
Solve the problem.
26) Given the acceleration, initial velocity, and initial position of a body moving along a coordinate line at time t,
find the bodyʹs position at time t.
a = 20, v(0) = 11, s(0) = 11
Find the value or values of c that satisfy the equation
the function and interval.
27) f(x) = x + 75 , [3, 25]
x
f(b) - f(a)
b - a
= f′(c) in the conclusion of the Mean Value Theorem for

Find the derivative at each critical point and determine the local extreme values.
28) y = x2 25 - x
Find the extreme values of the function and where they occur.
29) y = x3 - 3x2 + 1
1
30) y =
x2 - 1
5x
31) y =
x2 + 1
32) y = (x + 1)2/3
33) y = x3 - 3x2 + 7x - 10
Determine all critical points for the function.
34) f(x) = x3 - 12x + 2
35) f(x) = x3 - 3x2 + 8
36) f(x) = 20x3 - 3x5
37) f(x) =
2x
x + 8
38) f(x) = (x - 4)3
39) y = 2x2 - 64 x
Find the absolute extreme values of the function on the interval.
40) f(x) = 5x2/3, -27 ≤ x ≤ 8

Graph the rational function.
x - 4
41) y =
x2 - 7x + 12
y
8
4
-8 -4 4 8
x
-4
-8
Solve the problem.
42) A company is constructing an open-top, square-based, rectangular metal tank that will have a volume of
64 ft3. What dimensions yield the minimum surface area? Round to the nearest tenth, if necessary.
43) 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:
Rx = 4x
Cx = 0.01x2 + 0.7x + 60.
Find the most general antiderivative.
∫
44) 7t2 + t 5
dt
45) ∫ (7x3 - 10x + 5) dx
46)
∫
1
x5 - x 5 - 1 4
dx
47) ∫ ( t - 6 t) dt
48)
∫
x x + x
x2
dx
A) -
x
2 - 3 x + C B)
2
2x - 2 x + C C) 2 x - 2
x + C
D) C
49) ∫ (-8 cos t) dt
50) ∫ (-7 sec2 x) dx

Solve the problem.
51) Suppose c(x) = x3 - 18x2 + 10,000x is the cost of manufacturing x items. Find a production level that will
minimize the average cost of making x items.
52) Suppose that c(x) = 4x3 - 22x2 + 6849x is the cost of manufacturing x items. Find a production level that will
minimize the average cost of making x items.
53) You are planning to close off a corner of the first quadrant with a line segment 21 units long running from (x, 0)
to (0, y). Show that the area of the triangle enclosed by the segment is largest when x = y.
54) A private shipping company will accept a box for domestic shipment only if the sum of its length and girth
(distance around) does not exceed 108 in. What dimensions will give a box with a square end the largest
possible volume?

55) A trough is to be made with an end of the dimensions shown. The length of the trough is to be 20 feet long.
Only the angle θ can be varied. What value of θ will maximize the troughʹs volume?
θ
θ
Graph the rational function.
x2
56) y =
x2 + 10
y
1.5
1
0.5
-3 -2 -1 1 2 3
x
-0.5
Solve the initial value problem.
57) dy
dx = 1 + x, x > 0; y(2) = 0
x3
Write the sum without sigma notation and evaluate it.
2
18k
58) ∑
k + 37
k = 1
Express the sum in sigma notation.
59) 1 - 2 + 4 - 8 + 16
Evaluate the sum.
11
60) k
∑
k = 1

61)
8
∑
k = 1
k3
62)
6
∑
k = 1
k2 - 4
63)
21
∑
k = 3
5
Express the sum in sigma notation.
64) 5 + 6 + 7 + 8 + 9 + 10
65) 1 3 + 1 9 + 1 27 + 1 81
Write the sum without sigma notation and evaluate it.
3
k + 4
66) ∑
k
k = 1
67)
3
∑
k = 1
(-1)k (k - 5)2
68)
4
∑
k = 1
k2
2
Solve the initial value problem.
69) dy
dx = 2x -3/4, y(1) = 10
70) ds
dt = cos t - sin t, s π 2 = 5
Evaluate the integral.
71)
∫
3
11
x dx
Use a definite integral to find an expression that represents the area of the region between the given curve and the x -axis
on the interval [0, b].
72) y = 21x2

Evaluate the integral.
9
73) 5 x dx
∫
0
74)
75)
∫
∫
4
6x5 dx
-2
0
1
(x + 2)3 dx
76)
∫
1
2
t + 1 t
2
dx
77)
∫
1
4
t2 + 1
t
dt
78)
79)
∫
0
∫
π/2
12 sin x dx
3π/2
8 cos x dx
π/2
Use a definite integral to find an expression that represents the area of the region between the given curve and the x -axis
on the interval [0, b].
80) y = 18πx2
81) y = 10x
82) y = x 4 + 4
Evaluate the integral.
3π/2
83) θ dθ
∫
π
84)
∫
0
1/9
t2 dt

85)
∫
0
3 9
x2 dx
86)
87)
88)
∫
0
∫
5
∫
1
π/2
θ2 dθ
7
11 dx
7
6 dx
89)
∫
2
14
z- 14 dz
90)
∫
0
5
2x2 + x + 9 dx
91)
∫
6
0
3x2 + x + 3 dx
Find the derivative.
d x3
92)
dx ∫ sin t dt
0
Find the total area of the region between the curve and the x -axis.
93) y = 2x + 7; 1 ≤ x ≤ 5
Find the area of the shaded region.
94)

95)
Find the total area of the region between the curve and the x -axis.
96) y = x2 - 6x + 9; 2 ≤ x ≤ 4
97) y = 3 x3 ; 1 ≤ x ≤ 3
98) y = -x2 + 9; 0 ≤ x ≤ 5
99) y =
1
x ; 1 ≤ x ≤ 4
Find the derivative.
100)
d
dx
∫
1
x
18t5 dt
101) d dt
∫
0
sin t
1
9 - u2 du
102) y =
∫
0
x
8x + 7 dt
103) y =
∫
0
x8
cos
t dt
104) y =
∫
0
tan x
t dt
Evaluate the integral.
x dx
105) ∫
(7x2 + 3)5

Solve the initial value problem.
106) dy
dx = 18(6x - 5) -6, y(0) = 1
107) dy
dx = x 4(x5 - 1) 5 , y(1) = 5
Evaluate the integral.
108) ∫ x2(x3 - 6)4 dx
109) ∫ x2
x3 + 3 dx
110)
∫
dx
xln x6
111) ∫ 10x2
4
8 + 2x 3 dx
112) ∫ sin (9x - 2) dx
Use the substitution formula to evaluate the integral.
1
113) x + 1 dx
∫
0
Find the area of the shaded region.
114) f(x) = x3 + x2 - 6x
30
y
25
20
15
(3, 18)
10
5
(0, 0)
-5 -4 -3 -2 -1 -5
1 2 3 4 5
-10
-15
-20
(-4, -24)
-25
-30
x
g(x) = 6x
Find the area enclosed by the given curves.
115) y = 2x - x2, y = 2x - 4
116) y = x3, y = 4x

117) y = x, y = x2
118) y = 1 2 x 2, y = -x2 + 6
119) Find the area of the region in the first quadrant bounded by the line y = 8x, the line x = 1, the curve y =
and the x-axis.
1
x ,
120) Find the area of the region in the first quadrant bounded on the left by the y-axis, below by the line y = 1 3 x,
above left by y = x + 4, and above right by y = - x2 + 10.
Find the area of the shaded region.
121) f(x) = -x3 + x2 + 16x
g(x) = 4x
30
y
25
20
15
(4, 16)
10
5
(0, 0)
-5 -4 -3 -2 -1 -5
1 2 3 4 5 6
x
-10
(-3, -12)
-15
-20
-25
-30
122) y = x2 - 4x + 3
y
10 987654321
y = x - 1
-3 -2 -1 -1
-2
1 2 3 4 5
-3
-4
-5
-6
-7
-8
-9
-10
x

123)
1
y
-1
1 2
x
y = - x4
-2
A) 7 15
y = x2 - 2x
B) 22
15
C) 2 D) 76
15
124) y = 2x2 + x - 6 y = x2 - 4
y
5
4
3
2
1
-4 -3 -2 -1 -1
1 2 3 4
-2
-3
-4
-5
-6
-7
-8
x
A) 9 2
B) 11
6
C) 19
3
D) 8 3
Use the substitution formula to evaluate the integral.
0
2t
125) ∫
-1
3 + t2 3 dt
A) - 7 7
B)
72
288
C) -
7
144
D) -
7
288
126)
∫
0
1
6 r dr
16 + 3r2
A) 19 - 4 B)
19
2
- 2 C) 2 19 - 8 D) - 2 19 + 8
127)
∫
0
1
(8y2 - y + 1) -1/3 (64y - 4) dy
A) 9 B) 24 C) 8 D) 18
2

128)
∫
1
4
9 - x
A) 15
2
x
dx
B) 15 C) 30 D) - 15
2
129)
2π
4 cos3 x sin x dx
π/3
A) - 15
16
∫
B) - 32769
524288
C) - 15
4
D) 15
16

Answer Key
Testname: REVIEW 2
1) absolute maximum is 3 at x = 3; absolute minimum is - 7 at x = -2
2) absolute maximum is 0 at x = 4; absolute minimum is 0 at 4 and 0 at x = 4
3) absolute maximum is - 3 4 at x = 2; absolute minimum is -12 at x = 1 2
4) absolute maximum is 4 at x = 64; absolute minimum is 0 at x =0
5) absolute maximum is 7 at x = 0; absolute minimum is -65 at x = 3
6) absolute maximum is 9 at x = 27; absolute minimum is 0 at x = 01
7) x = -2
8) The minimum is -4 at x = -1.
9)
Critical Pt. derivative Extremum Value
x = 1.73 0 local max 10.39
x = -1.73 0 local min -10.39
10) - 1 2
11) s = - 19
2 t 2 + 1t + 8
12) (-∞, ∞)
13) local maximum at x = 3; local minimum at x = 1
14) local minimum: (-3, -1)
local maximum: (3, 1)
inflection points: (0, 0), (-3 3, - 3 2
3),
(3 3, 3 2
3)
6
y
4
2
-4 -2 2 4
-2
x
-4
-6

Answer Key
Testname: REVIEW 2
15) Absolute maxima: (-1, -8), (1, -8)
Local minimum: (0, -9)
Inflection points: -
1
3 , 8 9 , 1
3 , 8 9
10
y
5
-10 -5 5 10
x
-5
-10
16) Local maximum: 6 5 , 124,416
3125
Local minimum: (6, 0)
Inflection point: 12
5 , 78,732
3125
500
y
250
-10 10
x
-250
-500
17) local minimum: (2, -20)
local maximum: (-1, 7)
1
inflection point:
2 , - 13
2
24
y
12
-8 -4 4 8
x
-12
-24

Answer Key
Testname: REVIEW 2
18) local maximum at x = -3; local minimum at x = - 4 3
19) no local extrema
20) no local extrema
21) local minimum at x = -2; local maximum at x = 4
22) (1, ∞)
23) (-∞, 4)
24) (-∞, 8)
25) (0, ∞)
26) s = 10t2 + 11t + 11
27) 5 3
28)
Critical Pt. derivative Extremum Value
0 min 0
x = 0
undefined min 0
x = 25
x = 20 0 local max 400 5
29) Local maximum at (0, 1), local minimum at (2, -3).
30) Local maximum at (0, -1).
31) The minimum value is - 5 2 at x = -1. The maximum value is 5 at x = 1.
2
32) The minimum value is 0 at x = -1.
33) None
34) x = -2 and x = 2
35) x = 0 and x = 2
36) x = 0, x = -2, and x = 2
37) x = -8
38) x = 4
39) x = 0 and x = 4
40) absolute maximum is 45 at x = -27 ; absolute minimum is 0 at x = 0
41)
8
y
4
-8 -4 4 8
x
-4
-8
42) 5 ft × 5 ft × 2.5 ft
43) 165 units
44) 7 3 t 3 + t 2
10 + C

Answer Key
Testname: REVIEW 2
45) 7 4 x 4 - 5x2 + 5x + C
46)
-1
4x4 - x 6
6 - x 4 + C
47) 2 3 t 3/2 - 6 7 t 7/6 + C
48) C
49) -8sin t + C
50) -7 tan x + C
51) 9 items
52) There is not a production level that will minimize average cost.
53) If x , y represent the legs of the triangle, then x2 + y2 = 212.
Solving for y, y = 441 - x2
A(x) = xy = x
Aʹ(x) = -
441 - x2
x2
2 441 - x2 + 441 - x2
2
Solving Aʹ(x) = 0, x = ± 21 2
2
Substitute and solve for y: ( 21 2 ) 2 + y2 = 441 ; y = 21 2
2
2
54) 18 in. × 18 in. × 36 in.
55) 30°
56)
1.5
y
∴ x = y.
1
0.5
-3 -2 -1 1 2 3
x
-0.5
1
57) y = -
2x2 + x 2
2 - 15
8
18
58)
1 + 37 + 36
2 + 37 = 345
247
4
59) ∑ (-1)k 2k
k = 0
60) 66
61) 1296
62) 67
63) 95

Answer Key
Testname: REVIEW 2
64)
5
∑
k + 5
k = 0
4
1
65) ∑
3
k = 1
66) 1 + 4
1
k
+ 2 + 4
2
+ 3 + 4
3
= 31
3
67) -(1 - 5)2 + (2 - 5)2 - (3 - 5)2 = -11
68) 1 2
2 + 2 2
2 + 3 2
2 + 4 2
2 = 15
69) y = 8x1/4 + 2
70) s = sin t + cos t + 4
71) 1
72) 7b3
73) 90
74) 4032
75) 65
4
76) 29
6
77) 72
5
78) 12
79) -16
80) 6πb3
81) 5b2
82) b 2
8 + 4b
83) 5π 2
8
1
84)
2187
85) 3
86) π 3
24
87) 22
88) 36
89) - 9 + 2 14
90) 845
6
91) - 252
92) 3x2 sin (x3)
93) 52
94) 10

Answer Key
Testname: REVIEW 2
95) 26
3
96) 2 3
97) 4 3
98) 10
3
99) 2
100) 9x2
cos t
101)
9 - sin2 t
102) 8x + 7
103) 8x7 cos (x4)
104) sec2 x tan x
105) - 1 56 (7x 2 + 3)-4 + C
106) y = - 3 5 (6x - 5) -5 + 1
107) y = 1 30 (x 5 - 1) 6 + 5
108)
x3 - 6 5
15
+ C
109) 2 9 x 3 + 3 3/2 + C
110) 1 6 ln ln x 6 + C
111) 4 3 8 + 2x 3 5/4 + C
112) - 1 9
cos (9x - 2) + C
113) 4 3
2 - 2 3
114) 937
12
115) 32
3
116) 8
117) 1 6
118) 16
119) 5 4
120) 73
6

Answer Key
Testname: REVIEW 2
121) 937
12
122) 19
2
123) A
124) C
125) D
126) C
127) D
128) B
129) A