Tennessee State University
College of Engineering
Department of Mathematical Sciences
Calculus I – Math 1910 – Final Exam Review
1. For the function f whose graph is shown, state the following.
6. Evaluate the limit: lim (9x2 + 10x + 8)
x→5
(c) 283
(a) 275
(b) 233
7. If f and g are continuous functions with f (3) = 2 and
lim [6f (x) − f (X)] = 2, find g(3)
lim f (x)
x→−4
x→3
(a) g(3) = 10
(b) g(3) = 2
(a) −9
(b) 1
(c) −∞
(d) 0
8. Sketch the graph of the function.
f (x) =
(e) ∞

 x2 − 3x − 28
f (x) =
 5 x−7
x−1
2. Consider the function f (x) = 3 3
. Use the results
x −1
to guess the value of the limit lim f (x).
x→1
(a)
(b)
(c)
(d)
1.784405
1
0.033335
0.539356
(a)
x−1
x2 (x + 7)
(d) − 17
3. Determine the infinite limit lim
x→0
(a) −∞
(b) ∞
(c) 71
(e) 0
4. Given that lim = −6 and lim g(x) = 8
x→9
x→9
Evaluate the limit: limx→9 (f (x) + g(x))
(a) 15
(c) 14
(b) 17
(d) 2
5. Given that,
lim f (x) = −2 and lim g(x) = 5,
x→2
(b)
x→2
2f (x)
x→2 g(x) − f (x)
4
(a)
−3
4
(b) −
7
then lim
(c)
(c) g(3) = 14
4
3
Page 1 of 12
if x 6= 7
if x = 7
Review Package · Calculus I
Department of Mathematical Sciences
12. Select the correct graph of the function f and evaluate
lim f (x), if it exists.
x→2

if x < 2
 x
0
if x = 2
f (x) =

−x + 4 if x > 2
(c)
√
55 + x
9. Use continuity to evaluate the limit. lim √
x→9
55 + x
(a)
29
4
(a)
52
8
(c) ∞
(b)
the limit does not exist
10. Find the slope of the tangent line to the parabola
y = x2 + 6x at the point (9, 135)
(a) 29
(c) 34
(b) 21
(d) 24
(b)
11. Use the graph of the given function f to determine
lim f (x) at the indicated value of a, if it exists.
x→a
lim f (x) = −2
x→2
(c)
(a) lim f (x) = 1
x→−2
(b) lim f (x) = −2
x→−2
(c) lim f (x) = 3
x→−2
(d) the limit does not exist
Page 2 of 12
lim f (x) = 2
x→2
Review Package · Calculus I
Department of Mathematical Sciences
x6 − x3 + x − 4
18. Find the limit if it exists. lim
x→∞ x7 + 2x4 + 4
1
(c) −∞
(a) −
6
(d) ∞
(b) −5
(d)
(e) 0
19. Use the graph of the function f to find lim f (x),
x→1−
lim f (x) and lim f (x), if the limit exists.
x→1+
x→1
lim f (x) = 0
x→2
13. Find the limit. lim (8 − 5x2 )
x→6
(a) −179
(b) −162
(c) −167
(d) −172
(e) the limit does not exist
14. Find the limit. lim (7s2 − 2)(6s + 1)
s→0
(a) −7
(b) −8
(c) −2
(d) −4
15. Find the limit. lim
x→7
1
(a)
17
1
(b)
9
17
(c)
9
(a) lim f (x) = 1; lim f (x) = 3; lim f (x) = 2
(e) the limit does not exist
x→1−
x→1+
x→1
(b) lim f (x) = 3; lim f (x) = 1; lim f (x) = 2
x→1−
√
x→1+
x→1
(c) lim f (x) = 3; lim f (x) = 1; lim f (x) = DN E
x2 + 32
2x + 3
x→1−
x→1+
x→1
(d) lim f (x) = 1; lim f (x) = 3; lim f (x) = DN E
9
(d)
17
(e) the limit does not exist
x→1−
x→1+
x→1
20. Refer to the graph of the function lim f (x)
x→0−
x2 − 1
x→1 x − 1
16. Find the limit if it exists. lim
(a) 11
(b) 3
(c) 2
(d) 6
(e) the limit does not exist
17. Find the limit if it exists. lim
x→4 x3
(a)
1
24
(b) 0
x−4
− 2x2 − 8x
(c) 24
(d) the limit does not exist
Page 3 of 12
(a) 2
(c) 4
(b) 0
(d) DNE
Review Package · Calculus I
x2 − 25
21. Find the one-sided limit, if it exists. lim
x→5− x − 5
(a) 2
(b) −5
(c) 10
(d) −4
(e) the limit does not exist
22. Continuity at a Point. A function f is continuous at
the point x = a if the following conditions are satisfied.
1. f (a) is defined.
2. lim f (x) exists.
x→a
3. lim f (x) = f (a)
x→a
Determine the values of x, if any, at which the function
is discontinuous. At each point of discontinuity, state
the condition(s) for continuity that are violated.
2x − 4 if x ≥ 1
f (x) =
3
if x > 1
Department of Mathematical Sciences
24. Determine all values of x at which the function is discontinuous.
x2 − 4x
f (x) = 2
x − 9x + 20
(a) 5
(c) 4 and 5
(b) −4 and −5
(d) 4
25. Find the derivative of the function by using the rules
of differentiation.
2
9
5
f (t) = 4 − 3 +
t
t
t
(a) f ′ (t) = 3t3
(b) f ′ (t) = 3t3 + 2t5 + 8
8
8
27
(c) f ′ (t) = − 5 + 4 − 2
t
t
t
8
27
5
′
(d) f (t) = − 5 + 4 − 2
t
t
t
2
26. Let f (x) = x3 + x2 − 12x + 9 Find f ′ (x) = 12
3
1
(a) x = 19
(c) x = −4,
14
(d) x = 14, 11
(b) x = −4, 3
27. Find the derivative of the function.
f (x) = (x3 − 3)(x + 8)
(a) f ′ (x) = 6x2 + 7x + 5
(b) f ′ (x) = 5x2 + 7
(c) f ′ (x) = 7x3 − 15x − 6
(d) f ′ (x) = 4x3 + 24x2 − 3
(a) x = 3; conditions 1, 2
28. Find the derivative of the function. f (x) =
(b) x = 1; conditions 2, 3
(c) x = 3; conditions 1, 2
(d) x = 5; condition 2
(e) Continuous everywhere
23. Find the values of x for which the function is continu4
ous. f (x) =
4x − 1
(a) x ∈ (−∞, 5) ∪ (8, ∞)
1
x−5
(a) f ′ (x) = x − 5
(c) f ′ (x) = (x − 5)2
(b) f ′ (x) = −
(d) f ′ (x) = −
1
(x − 5)2
1
x−5
29. Find the derivative of the function and evaluate f ′ (x)
at the given value of x.
f (x) = (3x − 5)(x2 + 5); x = 5
(a) f ′ (x) = 5x + 36; 17
(b) x ∈ (−∞, 4) ∪ (4, ∞)
1
1
(c) x ∈ (−∞, ) ∪ ( , ∞)
4
4
(d) x ∈ (−∞, ∞)
(b) f ′ (x) = 38x2 − 5x; −36
(c) f ′ (x) = 9x2 − 10x + 15; 190
(d) f ′ (x) = 3x2 − 17; −38
Page 4 of 12
Review Package · Calculus I
30. Find the derivative of the function.
f (x) = (3x − 4)3
(a) 3x(3x − 4)3
(b) 9(3x − 4)3
Department of Mathematical Sciences
(b)
= 14x(x2 + 2)6 ,
f ”(x) = 14(x2 + 2)5 (13x2 + 2)
f ′ (x)
(c) f ′ (x) = 14(x2 + 2)6 ,
f ”(x) = 3(x2 + 3)5 (13x + 2)
(c) 9(3x − 4)2
(d) 3(3x − 4)2
(d) f ′ (x) = 3(x2 + 2)6 , f ”(x) = 7(x2 + 2)5 (5x2 + 2)
(e) f ′ (x) = 6x(x2 + 2)5 , f ”(x) = 7(x2 + 2)5 (5x + 2)
31. Find the derivative of the function.
7
f (x) = (x2 − 3) 2
7
7
(a) x(x2 − 3) 2
2
5
7 2
(b) (x − 3) 2
2
(c)
7x(x2
(d)
5
7
(2x) 2
2
− 3)
37. Find the third derivative of the function.
f (x) = 2x5 − 7x4 + 6x2 − 12x + 16
5
2
(a) f ′′′ (x) = 10x2 − 168x
(b) f ′′′ (x) = 60x2 − 140x
(c) f ′′′ (x) = 120x2 − 168x
32. Find the derivative of the function.
1
f (t) = √
2t − 9
√
−1
(c) 2 2t − 9
(a)
3
(2t − 9) 2
−1
1
(b)
(d)
3
1
2(2t − 9) 2
2(2t − 9) 2
(d) f ′′′ (x) = 2x2 − 7x
(e) f ′′′ (x) = 80x2 − 7x
38. Find
y4
x4
x4
(b) − 4
y
(a)
33. Find the derivative of the function.
f (x) = 6x2 (9 − 8x)4
(a) (−36x)(9 − 8x)3 (8x − 3)
(c) 48x(9 − 8x)3
34. Find the derivative of the function.
x−5 3
)
f (x) = (
x+4
3(x + 5)2
27(x − 5)4
(a)
(c)
−
(x − 4)4
(x + 4)2
(d)
40. Differentiate: h(x) =
(b) y = 11x + 15
x+2
x−8
10
(x − 8)2
10
(b) h′ (x) =
(x − 8)2
(a) h′ (x) = −
27(x − 5)2
(x + 4)4
35. Find an equation of the tangent line to the graph of
the function at the point (−1, 3).
f (x) = (2 − x)(x2 − 2)2
(a) y = −11x + 14
(d)
5
(a) f ′ (x) = 15 + √
x
′
(b) f (x) = 15 + 20x
(d) (−12x)(9 − 8x)3 (6x − 9)
x−5 2
)
x+4
(c) −
y4
x4
x4
y4
√
39. Find the derivative of f (x) = 15x + 10 x
(b) (−18x2 )(9 − 8x)2 (8x − 9)
(b) 3(
dy
by implicit differentiation. x5 + y 5 = 9
dx
√
(c) f ′ (x) = 15 + 5 x
10
(d) f ′ (x) = 15 + √
x
2
(x − 8)2
2
(d) h′ (x) =
(x − 8)2
(c) h′ (x) = −
41. If f is a differentiable function, find the derivative of
y = x7 f (x).
d 7
(x f (x)) = 7x7 f (x) − x6 f ′ (x)
dx
d 7
(b)
(x f (x)) = 6x6 f (x) + x7 f ′ (x)
dx
d 7
(x f (x)) = 7x6 f (x) + x7 f ′ (x)
(c)
dx
d 7
(x f (x)) = 7x7 f (x) + x6 f ′ (x)
(d)
dx
(a)
(c) y = 11x + 14
(d) y = −11x + 15
36. Find the first and second derivatives of the function.
f (x) = (x2 + 2)7
(a) f ′ (x) = 7(x2 + 2)6 ,
f ”(x) = 7(x2 + 2)5 (13x2 + 2)
Page 5 of 12
Review Package · Calculus I
Department of Mathematical Sciences
(a) 30(5x + 7)5 + (6x2 + 8x − 7)10 (132x + 88)
dg
42. Let g(x) = 2 sec x + tan x. Find
dx
dg
dx
dg
(b)
dx
dg
(c)
dx
dg
(d)
dx
(a)
(b) 30(5x + 7)5 (6x2 + 8x − 7)11 + (5x + 7)6 (6x2 + 8x −
7)10 (132x + 88)
= 2 sec x tan x + 1 − tan2 x
(c) 25(6x + 6)5 (8x3 + 11x − 7)11 + (5x + 7)6 (6x2 + 8x −
7)10 (120x + 80)
2
= 2 sec x tan x + sec x
(d) 30(5x + 7)5 + (6x2 + 8x − 7)11 + (5x + 7)6 + (6x2 +
8x − 7)10 (132x + 88)
= 2 sec x tan x + 1 + tan x
= 2 sec x tan x + 1 − sec x
sin x
1 + cos x
dy
1 cos x − 1
1 cos x + 2
dy
(c)
=
=
(a)
2
dx
(1 + cos x)
dx
(1 − cos x)2
1 cos x + 1
1 cos x − 1
dy
dy
=
=
(b)
(d)
dx
(1 + cos x)2
dx
(1 − cos x)2
43. Differentiate. y =
9x
44. Differentiate. y =
sin x + cos x
dy
dx
dy
(b)
dx
dy
(c)
dx
dy
(d)
dx
(a)
9
sin x + cos x
9
=
sin x + cos x
9
=
sin x + cos x
9
=
sin x + cos x
=
45. Differentiate. y =
50. For the function f whose graph is given, state the limit.
lim f (x)
x→−3−
cos x + sin x
(sin x + cos x)2
cos x − sin x
−x
(sin x + cos x)2
cos x − sin x
− 9x
(sin x + cos x)2
cos x + sin x
− 9x
(sin x + cos x)2
dy
= cos x − sin x
dx
dy
= cos x + 2 sin x
(d)
dx
(c)
46. Find an equation of the tangent line to the curve
y = x8 cos x at the point (π, −π 8 ).
(a) y = −15π 7 x + 9π 8
+
(e) ∞
+ 9x
dy
= cos x + sin x
dx
dy
= 2 cos x + sin x
(b)
dx
(b) y =
(d) 3
(a) 3πft
(b) 24πft
(c) −∞
tanx − 2
sec x
(a)
−8π 7 x
49. A sperical balloon is being inflated. Find the rate of
increase of the surface area S = 4πr2 with respect to
the radius r when 4 = 3ft
7π 8
(d) y =
−7π 7 x
+
(b) f ′ (x) = 12 − cot2 x
(d) 3
lim (x7 − 3x6 )
51. Find the limit.
x→−∞
11π 8
(c) f ′ (x) = 8 − cot2 x
(c) −∞
(b) ∞
(c) y = −16π 7 x + 2π 8
47. If f (x) = 8x + cot x, find f ′ (x).
(a) f ′ (x) = 13 − cot2 x
(a) 4
(a) 0
(c) 3
(b) −∞
(d) ∞
52. Find the most general anti-derivative of the function:
1
1
f (x) = 6x 5 − 8x 7
(d) f ′ (x) = 8 − csc2 x
48. Find the derivative of the following function:
G(x) = (5x + 7)6 (6x2 + 8x − 7)11
6
8
6
8
4
6
(a) F (x) = 5x 5 − 7x 7 + C
(b) F (x) = 6x 5 − 8x 7 + C
(c) F (x) = 5x 5 − 7x 7 + C
Page 6 of 12
Review Package · Calculus I
53. Find the most general anti-derivative of the function:
f (x) = 6 cos x − 9 sin x
(a) F (x) = 6 sin (x) − 9 cos (x) + C
(b) F (x) = 6 sin (x) + 9 cos (x) + C
(c) F (x) = −6 sin (x) + 9 cos (x) + C
54. Find g ′ (x) by evaluating the integral using Part 2 of
the Fundamental
Theorem and then differentiating.
Z
x
(3 + cos (t))dt
g(x) =
Department of Mathematical Sciences
59. Z
Evaluate the integral by making the given substitution:
12
dx
(1 + 3x)3
1
1
(a) −2
(c) −2
+C
4
(1 + 3x)
(1 + 3x)4
1
1
+C
+C
(d) −2
(b) 2
2
(1 + 3x)
(1 + 3x)2
Z
60. Evaluate the indefinite integral:
4x(x2 + 2)4 dx
2 2
(x + 2)3 + C
5
2
(b) (x2 + 2)3
5
π
dg(x)
= 3x + sin (x)
dx
dg(x)
= 3 + cos (x)
(b)
dx
dg(x)
(c)
= − sin (x)
dx
Z 6
55. Evaluate the integral.
(3 + 2y − y 2 )dy
(a)
2 2
(x + 2)5 + C
5
Z
61. Evaluate the indefinite integral:
cos6 x sin xdx
1
cos7 x + C
7
1
(b) sin6 x + C
7
1
(c) − sin7 x + C
7
1
(d) − cos7 x + C
7
Z
62. Evaluate the indefinite integral:
sec8 x tan xdx
(c) −90
(b) −18
56. Evaluate the integral.
(d) −486
Z
7π
1
(a) − sec8 x + C
8
1
(b) tan8 x + C
8
cos θdθ
5π
(c) −1
(a) 1
(b) 2
(d) 0
t4
4
t4
(b) 60t − 5t2 − 2t3 −
4
4
t
(c) 60t − 5t2 + 2t3 +
4
4
t
(d) 60t − 5t2 + 2t3 −
4
+C
+C
+C
(a) removable
(b) jump
+C
58. Z
Evaluate the integral by making the given substitution:
p
x2 x3 + 5dx
3
2 3
(x + 5) 2 + C
9
1
1 3
(b) (x + 5) 2 + C
9
(a)
1
csc8 x + C
8
1
(d) sec8 x + C
8
(c)
63. For x = −9, what type of discontinuous exists
57. Z
Find the general indefinite integral.
(6 − t)(10 + t2 )dt
(a) 60t + 2t2 + 5t3 −
(d)
(a)
0
(a) 126
(c) (x2 + 2)5 + C
(a)
(c) infinite
64. Which of the given functions is discontinuous?
3
2
(c) − (x3 + 5) 2 + C
9
3
2 3
(d) (x + 5) 2
9
Page 7 of 12
(a)
f (x) =


1
, x 6= 5
x−5
 4,
x=5
Review Package · Calculus I
(b)

1


, x≥5
x
−
2
f (x) =
1

 ,
x<5
3
Department of Mathematical Sciences
71. If a ball is thrown vertically upward with a velocity of 216 ft/s, then its height after t seconds is s =
216t − 18t2 . What is the maximum height reached by
the ball?
65. Use continuity to evaluate the limit. lim sin (x + 3 sin x)
(a) 36 ft
(b) 6 ft
x→6π
(d) −1
(a) 6π
(b) ∞
(c) 1
72. Find the average rate of change of the area of a circle
with respect to its radius r as r changes from 2 to 3.
(e) 0
66. Find the points at which f is discontinuous. At which
of these points is f continuous from the right, from the
left, or neither?
(x − 4)3 if x < 3
f (x) =
(x + 7)3 if x ≥ 3
(a) 5π
(b) 9π
(a) dy = (5x − 4)dx
(b) dy = (5x4 + 4)dx
(b) x = −7, continuous from the left
(c) x = 3, continuous from the right
67. Find a function g that agrees with f for x 6= 4 and is
continuous on
√ [0, ∞).
2− x
f (x) =
4−x
1
1
(a) g(x) =
(c) g(x) =
2+x
2−x
1
1
√
(d) g(x) =
(b) g(x) =
2+ x
4−x
(d) −3
(d) 32
x700 − 1
x→1 x − 1
(b) −3
(d) 60
3
,0
7
3
(b) 3, , 0
7
(a) 64
(b) −126
12
7
6
(d) 3, , 0
7
(c) 3,
(c) −128
(d) 4
77. Find the exact values of the numbers c that satisfy the
conclusion of The Mean Value Theorem for the function f (x) = x3 − 4x for the interval [−4, 4].
√
(a) c = ±4 3
70. Evaluate lim
(a) 700
(b) Does not exist
(c) 0
(a)
69. If f (−3) = −3, f ′ (−3) = 4, g(−3) = 5 and g ′ (−3) =
−4, find (f g)′ (−3).
(b) 31
(a) 10
76. Find the absolute minumum values of
y = 8x2 − 64x + 2 on the interval [0, 5].
(c) −6
(c) 27
(d) dy = (5x5 + 4)dx
75. Find the critical numbers of the function:
4
F (x) = x 5 (x − 3)2
68. If f (4) = 1, f ′ (4) = 2, g(4) = 3 and g ′ (4) = −5, find
(f + g)′ (4).
(a) 39
(c) dy = (x4 + 4)dx
74. Find the critical numbers of the function:
y = 10x2 + 60x
(d) x = 4, continuous from the right
(b) −2
(c) 3π
73. Find the differential of the function. y = x5 + 4x
(a) x = 3, continuous from the left
(a) 7
(c) 648 ft
√
4 3
(b) c = −
3
(c) 1
Page 8 of 12
√
4 3
(c) c =
3
√
4 3
(d) c = ±
3
Review Package · Calculus I
78. Verify that the function satisfies the hypothesis of The
Mean Value Theorem on the given interval. Then find
all numbers c that satisfy the conclusion of The Mean
Value Theorem.
√
f (x) = 4 x, [0, 1]
√
3
1
3
(a) c =
(c)
c
=
16
16
√
√
3
16
16
(d) c =
(b) c =
16
16
79. Verify that the function satisfies the hypothesis of The
Mean Value Theorem on the given interval. Then find
all numbers c that satisfy the conclusion of The Mean
Value Theorem.
x
f (x) =
, [0, 1]
x+3
√
√
(a) c = 3
(c) c = 12
√
√
(b) c = −3 + 12
(d) c = −12 + 3
80. Find the intervals on which the following function f is
increasing:
f (x) = x3 − 108x + 10
(a) (−∞, −18), (18, ∞)
(b) (−∞, 6)
(c) (−6, 6)
Department of Mathematical Sciences
84. Consider the following problem: A farmer with 850ft
of fencing wants to enclose a rectangular area and then
divide it into four pens with fencing parallel to one side
of the rectangle. What is the largest possible total area
of the four pens¿
(c) 18061.5ft2
(b) 18051.5ft2
(d) 18085.5ft2
85. Find the most general anti-derivative of the function:
f (x) = 21x2 − 8x + 3
(a) F (x) = 35x5 − 16x4 + 3x + C
(b) F (x) = 21x3 − 8x2 + 3x + C
(c) F (x) = 7x3 − 4x2 + 3x + C
86. Find f : f ′ (x) = 12x + 36x2
(a) f (x) = 6x3 + 6x4 + Cx + D
(b) f (x) = 2x3 + 3x4 + Cx + D
(c) f (x) = 4x3 + 12x4 + Cx + D
87. Find f : f ′ (x) = 49 cos (7x)
(a) f (x) = − cos (7x) + Cx + D
(d) (−∞, −6), (6, ∞)
(b) f (x) = y = 49 cos (x) + Cx + D
(e) (−6, ∞)
(c) f (x) = y = − cos (7x) + Cx2 + D
81. Find the inflection points of the following function:
f (x) = −2x + 2 − 2 sin x
0 < x < 3π
88. Find f : f ′ (x) = 3 cos (x) + 10 sin (x) f (0) = 3
(a) f (x) = 3 sin (x) − 10 cos (x) + 13
(b) f (x) = 3 sin (x) + 10 cos (x) + 13
(c) f (x) = −3 sin (x) − 10 cos (x) + 3
(a) (π, −2π), (2π, −4π + 2)
(b) (π, 2), (2π, −4π + 2)
89. A particle moves along a straight line with velocity
function v(t) = 4 sin (t) − 9 cos (t) and its initial displacement is s(0) = 5 Find its position function.
(c) (π, −2π), (2π, −4π)
(d) (π, −2π + 2), (2π, −4π + 2)
(a) s(t) = 4 − 4 cos (t) + 9 sin (t)
82. How many points of inflection are on the graph of the
function:
f (x) = 15x3 + 9x2 − 6x − 2
(a) 3
(c) 4
(b) 2
(d) 1
83. Find two positive numbers whose product is 144 and
whose sum is a minumum.
(a) 4, 36
(b) 2, 72
(a) 18062.5ft2
(b) s(t) = 5 − 4 cos (t) − 9 sin (t)
(c) s(t) = 9 + 4 cos (t) − 9 sin (t)
(d) s(t) = 9 − 4 cos (t) − 9 sin (t)
90. A stone is dropped from the upper observation deck
(the Space Deck) of a tower, 360m above the ground.
Find the distance of the stone above ground level at
time t.
(a) s(t) = 360 − 9.8t2
(c) 12, 12
(b) s(t) = 360 + 9.8t2
Page 9 of 12
(c) s(t) = 360 + 4.9t2
(d) s(t) = 360 − 4.9t2
Review Package · Calculus I
91. A stone was dropped off a cliff and hit the ground with
a speed of 192ft/s. What is the height of the cliff?
(a) 18421 ft
(c) 1152 ft
(b) 36864 ft
(d) 576 ft
(a) −
92. What constant acceleration is required to increase the
speed of a car from 30 ft/s to 50 ft/s in 10 s?
ft
s2
ft
(b) 2 2
s
(a) 8
(c) 800
(d) 4
ft
s2
ft
s2
1
sin 5x + C
5
1
(b) sin x + C
5
1
(c) sin 5x
5
1
(d) − sin 5x + C
5
(a)
(d)
2
x2
(b) 2
7
4
4
7
(c) 2x
(a)
(d) −
9 − x2
9 + x2
9 + 3x2
(b)
(9 + x2 )2
2
x2
0
242
33
242
(d)
30
(c)
(c)
9 − x2
(9 + x2 )2
(d)
9 + 3x2
9 + x2
dy
by implicit differentiation. x3 − 3xy = 7
dx
y
y
(a)
(c) x −
−x
x
x
x
x
(b)
−x
(d) x −
y
y
100. Find
dy
by implicit differentiation. x3 y 3 − xy = 41
dx
x
y
(a) −
(c)
y
x
x
y
(b)
(d) −
y
x
101. Find
1
(c) − sin (8 − t3 ) + C
3
1
(d) − sin (8 − t3 ) + C
2
96. Z
Evaluate the indefinite integral:
1
x2 (1 + 2x3 )4 dx
247
30
242
(b)
31
1
7
(a)
95. Z
Evaluate the indefinite integral:
t2 cos (8 − t3 )dt
(a)
(c) −
dy
by solving the given implicit
99. Find the derivative
dx
equation for y explicity in terms of x.
x
− x2 = 9
y
94. Z
Evaluate the indefinite integral:
2 + 16x
√
dx
6 + 2x + 8x2
√
(a) 6 + 2x + 8x2 + C
√
(b) −2 6 + 2x + 8x2 + C
√
(c) 3 6 + 2x + 8x2 + C
√
(d) 2 6 + 2x + 8x2 + C
1
sin (8 − t3 ) + C
3
1
(b) sin (8 − t3 )
3
(b)
1
7
dy
by solving the given implicit
98. Find the derivative
dx
equation for y explicity in terms of x.
xy = 2
93. Z
Evaluate the integral by making the given substitution:
cos 5xdx, u = 5x
(a)
Department of Mathematical Sciences
dy
97. Find the derivative
by solving the given implicit
dx
equation for y explicity in terms of x.
x + 7y = 4
102. Find
1
1
dy
by implicit differentiation. x 2 + y 2 = 6
dx
1
(a) −
x2
1
y2
1
(c)
x2
1
y2
1
(b)
Page 10 of 12
y2
1
x2
1
(d) −
y2
1
x2
Review Package · Calculus I
dy
1
1
103. Find
by implicit differentiation. 2 + 2 = 25
dx
x
y
y3
x3
(c) − 3
(a) − 3
y
x
3
x
y3
(b) 3
(d)
y
x3
dy
√
by implicit differentiation. xy = 3x + y 5
dx
√
√
3x xy − y
6x xy − y
(a)
(c)
√
√
x − 5y 4 xy
x − 5y 4 xy
√
√
3 xy − y
6 xy − y
(b)
(d)
√
√
x − 5y 4 xy
x − 10y 4 xy
104. Find
105. Find an equation of the tangent line to the graph of the
function f defined by the given equation at the point
(1, 3).
x2 y 3 − y 2 + xy − 8 = 0
57
x+
22
22
(b) y = − x +
57
(a) y = −
123
22
1
22
22
123
x+
57
22
123
57
(d) y = x −
22
22
(c) y = −
d2 y
of the function defined
106. Find the second derivative
dx2
implicity by the given equation.
y 2 − xy = 7
2y(y + x)
(2y − x)2
y−x
(b)
(2y − x)3
(a)
2y(y − x)
(2y − x)3
y+x
(d)
(2y − x)2
(b) 11.7ft/sec
(d) 12ft/sec
1
8 9
+ t 8 − 8t 2 + C
9
1
64 9
+ t 8 − 4t 2 + C
9
1
8 8
+ t 9 − 8t 2 + C
9
1
64 9
+ t 8 − 8t 2 + C
9
109. Find the indefinite integral.
x7 − 1
= x5 − x−2
x2
1
1
(a) x5 + + C
5
x
1 6 1
(b) x + + C
6
x
Z
x7 − 1
dx
x2
Hint:
1 6 1
x − +C
6
x
1 6 1
(d) x − + C
5
x
Z
110. Find the indefinite integral:
(4t + 3)(t − 4)dt
4 3
t −
3
4
(b) t3 −
3
(a)
13 2
t − 11t + C
2
15 2
t − 12t + C
2
1 3
(x − x2 + x)7 + C
7
(b) (x3 − x2 + x)7 + C
(a)
107. The base of a 80-ft ladder leaning against a wall begins to slide away from the wall. At the instant of time
when the base is 64 ft from the wall, the base is moving
at the rate of 9 ft/sec. How fast is the top of the ladder
sliding down the wall at that instant of time?
(c) 13.6ft/sec
2 5
t2
5
2 5
(b) t 2
5
2 5
(c) t 2
5
2 5
(d) t 2
5
(a)
(c)
4 3
t −
3
4
(d) t3 −
3
(c)
15 2
t − 11t + C
2
13 2
t − 12t + C
2
111. Z
Find the indefinite integral.
(3x2 − 2x + 1)(x3 − x2 + x)6 dx
(c)
(a) 7.8ft/sec
Department of Mathematical
Sciences
Z
1
3
1
108. Find the indefinite integral.
(t 2 + 8t 8 − 4t− )dt
2
112. Find the indefinite integral.
3
2 3
(t + 9) 2 + C
3
(b) (t3 + 9) + C
(a)
Last Version: November 18, 2013
Page 11 of 12
1 3
(x + x2 + x) + C
7
(d) x + C
(c)
Z
3t2
p
t3 + 9dt
2
3 3
(t + 9) 3 + C
2
(d) t + C
(c)
Review Package · Calculus I
Department of Mathematical Sciences
Problem
Answer
Problem
Answer
Problem
Answer
Problem
Answer
Problem
Answer
1
C
24
C
47
D
70
B
93
A
2
B
25
D
48
B
71
B
94
D
3
A
26
B
49
B
72
A
95
A
4
D
27
D
50
B
73
B
96
D
5
B
28
B
51
B
74
B
97
B
6
C
29
C
52
A
75
D
98
A
7
A
30
C
53
B
76
B
99
C
8
B
31
C
54
B
77
D
100
A
9
B
32
A
55
B
78
C
101
D
10
D
33
A
56
D
79
B
102
D
11
C
34
D
57
D
80
D
103
A
12
C
35
C
58
A
81
D
104
C
13
D
36
B
59
D
82
D
105
A
14
C
37
C
60
D
83
C
106
C
15
D
38
B
61
D
84
A
107
D
16
C
39
A
62
D
85
C
108
A
17
A
40
A
63
A
86
B
109
B
18
E
41
C
64
A
87
A
110
D
19
C
42
B
65
E
88
A
111
A
20
C
43
B
66
C
89
D
112
A
21
C
44
C
67
B
90
D
22
B
45
D
68
D
91
D
23
C
46
B
69
D
92
B
Page 12 of 12