Tennessee State University
College of Engineering, Technology, Computer and Mathematical Sciences
Department of Mathematical Sciences
Basic Calculus – Math 1830 – Final Exam Review Package
1. Find an equation of the tangent line to the
parabola y = x2 + 7x at the point (2, 18).
(a) y = 11x − 4
(b) y = −11x + 4
(c) y = −13x − 4
price at which the supplier will make 3 units of
the commodity available in the market.
(d) y = 11x − 6
(a) p = $99
(b) p = $93
(c) p = $63
(e) y = 12x + 4
2. Find the domain of the function f (x) = x2 − x − 7.
(a) (−∞, −7) ∪ (−7, 7) ∪ (7, ∞)
(b) [−7, 7]
(c) (−∞, 7) ∪ (7, ∞)
(d) p = $9
(e) p = $45
8. For the supply equations p = x3 + 2x2 + 2 where
x is the quantity supplied in units of a thousand
and p is the unit price in dollars, determine the
price at which the supplier will make 2 units of
the commodity available in the market.
(d) (−7, 7)
(a) p = $16
(b) p = $18
(c) p = $12
(e) (−∞, ∞)
3. Find the range of the function f (x) = 2x2 + 2.
(a) [−2, 2]
(b) (−2, 2)
(c) (2, ∞)
(d) [2, ∞)
(d) p = $10
(e) p = $14
9. Use the graph of the given function f to determine
lim f (x) at the indicated value of a, if it exists.
(e) (−∞, ∞)
x→a
4. Let f (x) = x3 + 9, g(x) = x2 − 3. Find the rule
for the function f + g.
(a) f + g = 2x3 + 15
(b) f + g = x3 + 9
(c) f + g = 2x2 + 6
(d) f + g = x2 − 3
(e) f + g = x3 + x2 + 6
5. Evaluate h(3), where h = g ◦ f ,
1
; g(x) = x2 + 8
f (x) =
x−1
(a) 14
1
(b) 17
(c) 8 41
(d)
1
8
(e)
1
16
(a) lim f (x) = 1
(c) lim f (x) = 3
(b) lim f (x) = −2
(d) The limit does not
exist.
x→−2
6. Find f (a+h)−f (a) for the function f (x) = 9x+7.
Simplify your answer.
x→−2
(a) 9h + 7
(b) 9h
(c) 9a − 9h
(d) 7a + 9h
x→−2
10. Select the correct graph of the function f and evaluate lim f (x), if it exists.
(e) 7h + 9
x→2
7. For the supply equations p = x2 + 18x + 36 where
x is the quantity supplied in units of a thousand
and p is the unit price in dollars, determine the
Page 1 of 15

 x
0
f (x) =

−x + 4
if x < 2
if x = 2
if x > 2
(1)
Review Package · Basic Calculus
Department of Mathematical Sciences
(a)
(d)
the limit does not exist
lim f (x) = 0
x→a
11. Find the limit. lim (7s2 )(6s + 1)
s→0
(d) −4
(a) −7
(b) −8
(c) −2
(b)
(e) the limit does not
exist
12. Find the limit. lim
√
x→78
(d) 9
(a) 2
(b) 16
(c) 11
lim f (x) = −2
x→a
x+3
(e) the limit does not
exist
13. Find the limit if it exists. lim
x→8
(a) 87
(b) 8
(c) 81
x
x−8
(d) the limit does not
exist
x6 − x3 + x − 4
x→∞ x7 + 2x4 + 4
14. Find the limit if it exists. lim
(c)
(a) − 16
(b) −5
(c) −∞
(d) ∞
(e) 0
15. Use the graph of the function f to find lim f (x),
x→1−
lim f (x) = 2
x→a
lim f (x) and lim f (x), if the limit exists.
x→1+
Page 2 of 15
x→1
Review Package · Basic Calculus
Department of Mathematical Sciences
x2 − 25
18. Find the one-sided limit, if it exists. lim
x→5− x − 5
(a) 2
(b) −5
(c) 10
(a) lim f (x) = 1; lim f (x) = 3; lim f (x) = 2
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) does
x→1−
x→1+
x→1
not exist
(d) lim f (x) = 1; lim f (x) = 3; lim f (x) does
x→1−
x→1+
(d) −4
(e) The limit does not
exist
19. 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.limx→a f (x) exists
3.limx→a f (x) = f (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
x→1
not exist
16. Refer to the graph of the function f
and determine whether the statement below is true or false.
lim f (x) = 1
x→2−
(a) x = 3; conditions 1, 2
(b) x = 1; conditions 2, 3
(c) x = 3; conditions 1, 2, 3
(d) x = 5; condition 2
(a) true
(b) false
17. Find the one-sided limit, if it exists. lim
x→3−
(a) −9
(b) − 12
(c) −3
x−6
x+3
20. Find the values of x for which the function is con4
tinuous. f (x) =
4x − 1
(a) x ∈ (−∞, 5) ∪ (8, ∞)
(d) 6
(b) x ∈ (−∞, 4) ∪ (4, ∞)
(e) The limit does not
exist
(d) x ∈ (−∞, ∞)
Page 3 of 15
(c) x ∈ (−∞, 14 ) ∪ ( 41 , ∞)
Review Package · Basic Calculus
21. The function P, whose graph follows, gives the
prime rate (the interest rate banks charge their
best corporate customers) as a function of time
for the first 32 wk during a year. Determine the
values of t for which P is discontinuous.
Department of Mathematical Sciences
1
16 1
(d) f ′ (x) = x 7 − x 4 + 2x − 9
7
26. If f (t) =
9
5
2
− 3 + find f ′ (t).
4
t
t
t
(a) f ′ (t) = 3t3
(b) f ′ (t) = 3t3 + 2t5 + 8
16 1
(c) f ′ (t) = t 7 + 2t + 1
7
1
1
16
(d) f ′ (t) = t 7 − t 4 + 2t − 9
7
27. Let f (x) = 5x3 − 6x. Find f ′ (4)
(a) f ′ (4) = −87
(b) f ′ (4) = −356
(c) f ′ (4) = 97
(d) f ′ (4) = 234
28. If f (x) = (x + 2)(5x2 − 8x + 1). Find f ′ (x)
(a) t = 9
(c) t = 20, 11, 15
(b) t = 12, 16, 28
(d) t = 12, 11, 15
22. Find the slope of the tangent line to the graph of
f (x) = 14x − 10
(a) f ′ (x) = 7x2 − 4x + 9
(b) f ′ (x) = 9x2 + 19x − 2
(c) f ′ (x) = 2x2 + 3x + 7
(d) f ′ (x) = 15x2 + 4x − 15
29. If f (x) = (x3 − 3)(x + 8). Find f ′ (x)
(a) 14
(c) 0
(a) f ′ (x) = 6x2 + 7x + 5
(b) 7x2 − 10x
(d) 14x
(b) f ′ (x) = 5x2 + 7
(c) f ′ (x) = 7x3 − 15x − 6
23. Find the slope and equation of the tangent given:
f (x) = 5x + 9 (3, 24)
(d) f ′ (x) = 4x3 + 24x2 − 3
30. If f (x) =
(a) 15; y = 15x
(c) 5; y = 5x
(b) 5; y = 5x + 9
(d) 15; y = 15x + 9
24. Let f (x) = 3x2 + 2. Find the tangent line at the
point (−1, 5).
8
4 5
25. If f (x) = 2x 7 − x 4 + x2 − 9x + 1 find f ′ (x).
5
(a) f ′ (x) =
1
16 1
x 8 − x 7 + 2x − 2
7
9
1
(c) f ′ (x) = (x − 5)2
(b) f ′ (x) = −
(d) f ′ (x) = −
1
(x − 5)2
1
x−5
x−7
. Find f ′ (x)
7x + 2
51
(c) f ′ (x) = 7x + 2
(a) f ′ (x) =
2
(7x + 2)
1
(7x + 2)2
(b) f ′ (x) = −
(d) f ′ (x) = −
7x + 2
51
(d) y = −12x − 1
(b) y = −12x − 6
(a) f ′ (x) = x − 5
31. If f (x) =
(c) y = −6x − 1
(a) y = −6x − 6
1
. Find f ′ (x)
x−5
32. If f (x) = (3x − 4)3 . Find f ′ (x)
(b) f ′ (x) = x 8 − x 4 − 9
16 1
(c) f ′ (x) = x 7 + 2x + 1
7
(a) 3x(3x − 4)3
(b) 9(3x − 4)3
Page 4 of 15
(c) 9(3x − 4)2
(d) 3(3x − 4)2
Review Package · Basic Calculus
33. If f (x) = 4(x3 − x)7 . Find f ′ (x)
(a) 28(3x2 −1)(x3 −x)6
(b) 4(3x2 − 1)(x3 − x)7
(c) 28(3x2 −x)(x3 −1)7
(d) 28(x3 − x)6
7
34. If f (x) = (x2 − 3) 2 . Find f ′ (x)
Department of Mathematical Sciences
40. The management of ThermoMaster Company,
whose Mexican subsidiary manufactures an
indoor-outdoor thermometer, has estimated that
the total weekly cost (in dollars) for producing x
thermometers is C(x) = 3000 + 7x dollars/year.
Find the average cost function C.
(a) C = 7
5
7
7
(c) 7x(x2 − 3) 2
x(x2 − 3) 2
2
5
5
7 2
7
(b) (x − 3) 2
(d) (2x) 2
2
2
√
35. If f (x) = 6x2 − 2x + 7. Find f ′ (x)
(a)
6x − 1
6x2 − 2x + 7
1
(b) √
2 12x − 2
(a) √
(b) C =
1
√
2
2 6x − 2x + 7
12x − 2
(d) √
6x2 − 2x + 7
(c)
36. If f (u) = (u−5 + 4u−2 )6 . Find f ′ (u)
(a) −6(u−5 + 4u−2 )5 (5u−6 + 8u−3 )
(d) 6(u−5 + 4u−2 )5
42. Williams Commuter Air Service realizes a monthly
revenue of R(x) = 18000x−100x2 dollars when the
price charged per passenger is x dollars. Find the
marginal revenue R’
37. If f (x) = 6x2 (9 − 8x)4 . Find f ′ (x)
(a) (−36x)(9 − 8x)3 (8x − 3)
(b) (−18x2 )(9 − 8x)2 (8x − 9)
(a) 18000 − 100x
8x)3
(b) −100x2
(d) (−12x)(9 − 8x)3 (6x − 9)
38. 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
(b) y = 11x + 15
(c) y = 11x + 14
39. The total weekly cost (in dollars) incurred by a
company in pressing x compact discs is C(x) where
C(x) = 4000 + 5x − 0.0001x2 for 0 ≤ x ≤ 9000.
What is the marginal cost when x = 2000 and
x = 3000?
(c) $5.00, $2.70
(b) $4.60, $4.40
(d) $1.30, $4.90
200
x
(d) 18000 − 200x
(c) 18000 −
5
43. The demand equation is x = − p + 10. Compute
6
the elasticity of demand and determine whether
the demand is elastic, unitary, or inelastic at the
p = 40.
5
; inelastic
6
1
(b) ; inelastic
4
(d) y = −11x + 15
(a) $4.70, $3.90
3000
x2
(c) C = 2 − 0.0001x, C’ = 0.0001
2000
2000
(d) C =
+2−0.0001x, C’ = − 2 −0.0001
x
x
(c) 6(5u−6 + 8u−3 )5
(a) y = −11x + 14
(d) C = −
7
x
41. Custom Office makes a line of executive desks. It
is estimated that the total cost for making x units
of their Senior Executive model is
C(x) = 3000 + 2x − 0.0001x2
0 ≤ x ≤ 8000 dollars/year. Find the average cost
function C. Find the marginal average cost function C’.
3000
3000
+2−0.0001x, C’ = − 2 −0.0001
(a) C =
x
x
(b) C = 2 − 0.0002x, C’ = 0.0001
(b) −6(u−5 + 4u−2 )(5u−6 + 8u−3 )
(c) 48x(9 −
3000
+7
x
(c) C = 3000 +
(a)
(c) 1; unitary
(d)
10
; elastic
7
44. Find the second derivatives of the function:
f (x) = 2x3 − 5x2 + 1
(a) f ′′ (x) = 12
(b) f ′′ (x) = 16x − 10
(c) f ′′ (x) = 16x
Page 5 of 15
(d) f ′′ (x) = 2x − 5
(e) f ′′ (x) = 12x − 10
Review Package · Basic Calculus
45. Find the first and second derivatives of the funcs−8
tion: f (s) =
s+8
(a) f ′ (s) =
(b) f ′ (s) =
(c)
f ′ (s)
=
(d) f ′ (s) =
(e) f ′ (s) =
16
32
, f ”(s) = −
(s + 8)
(s + 8)2
32
12
, f ”(s) = −
2
(S + 8)
(s + 8)3
24
12
, f ”(s) = −
2
(s + 8)
(s + 8)3
32
16
, f ”(s) = −
2
(s + 8)
(s + 8)3
16
24
, f ”(s) = −
2
(s + 8)
(s + 8)3
Department of Mathematical Sciences
51. Find an equation of the tangent line to the graph
of the function f defined by the given equation at
the point (0, 4).
7x2 + 3y 2 = 26
(a) y = x
(c) y = 4x
(b) y = 4
(d) y = 0
52. You are given the graph of a function f. Determine
the intervals where f is increasing, constant, or de-
46. Find the third derivative of the function:
f (x) = 5x4 − 7x3
(a) 60x − 7
(b) 15x − 42
(c) 120x − 42
(d) 5x − 7
creasing.
(e) x − 7
47. Find the third derivative of the function:
3
f (x) = 4
x
540
120
(a) 7
(d) − 7
x
x
3
360
(b) − 7
(e) − 7
x
x
720
(c) 7
x
48. Find the derivative
(a) −
1
7
1
(b)
7
49. Find the derivative
2
x2
(b) 2
(a)
dy
from x + 7y = 4
dx
7
(c) −
4
4
(d)
7
(a) Decreasing on (0, ∞) and increasing on
(−∞, 0)
(b) Decreasing on (−∞, 0) and increasing on
(0, ∞)
(c) Decreasing on (−∞, −5)∪(5, ∞) and increasing on (−5, 5)
53. You are given the graph of a function f. Determine
the intervals where f is increasing, constant, or de-
dy
from xy = 2
dx
(c) 2x
(d) −
2
x2
creasing.
dy
50. Find
from x5 + y 5 = 9
dx
y4
y4
(c) − 4
(a) 4
x
x
4
4
x
x
(b) − 4
(d) 4
y
y
Page 6 of 15
Review Package · Basic Calculus
(a) Decreasing on (−∞, −2), increasing on
(2, ∞), and constant on (−2, 2)
Department of Mathematical Sciences
√
59. Find the domain of the function f (x) = x + 3.
(c) Decreasing on (2, ∞),
increasing
(−∞, −2), and constant on (2, −2)
on
54. Find the interval(s) where the function is increasing and the interval(s) where it is decreasing.
g(x) = x3 + 6x2 + 12
3x − 6
x2 − 3
3x + 6
(b) 2
x −3
(c) (x3 + 6x2 − 3)
(a)
(b) Increasing on (−∞, 0), decreasing on (0, ∞)
(c) Increasing on (−∞, −4), and (0, ∞), decreasing on (−4, 0)
decreasing
on
55. Find the relative maxima and relative minima, if
any, of the function.
f (x) = x2 − 10x
61. Let f (x) = x − 8, g(x) =
for the function f − g.
(d) No relative maxima or minima.
(e) f − g = x − 8 +
(a) 29
(b) 20
(c) 36
(c) x ∈ (−∞, 0)
(d) x ∈ φ
(a) 6a2 − 6h2
(b) 5 − 6(a − h)2
(c) −6h(2a + h)
(a) x ∈ (−∞, 3)
(b) x ∈ (0, 3)
(c) x ∈ (−∞, 0) ∪ (3, ∞)
√
x + 8. Find the rule
√
x+8
(d) 215
(e) 225
(d) 5 − 6h2
(e) −6(a2 − h2 )
f (a + h) − f (a)
(h 6= 0) for the function:
h
f (x) = 20 − 6x2 .
64. Find
(d) x ∈ (−3, 3)
58. Find the inflection points, if any, of the function.
f (x) = 3x3 − 9x2 + 14x − 20
(b) (0, −20)
x3 + 6
x2 − 3
63. Find f (a+h)−f (a) for the function f (x) = 5−6x2 .
57. Determine where the function is concave upward.
f (x) = x4 − 6x3 + 4x + 5
(a) (1, −12)
(e)
62. Evaluate h(2), where h = f ◦ g given: f (x) =
x2 + x + 9; g(x) = x2
56. Determine where the function is concave upward.
f (x) = −3x2 − 8x + 4
(b) x ∈ (−4, 4)
3x − 6
x2 + 3
√
(d) f − g = x − 8
(b) No relative maximum; Relative minimum
f (5) = −25
(c) Relative maximum: f (5) = 25; Relative minimum f (0) = −5
(d)
x+8
√
(b) f − g = x − 8 − x + 8
√
(c) f − g = x + 8 − x + 8
(a) f − g = x −
(a) Relative maximum: f (5) = −25; No relative
minumum
(a) x ∈ (−∞, ∞)
(e) (−∞, ∞)
60. Let f (x) = x3 + 6, g(x) = x2 − 3. Find the rule
f
for the function .
g
(a) Increasing on (−∞, ∞)
(d) Increasing on (−4, 0),
(−∞, −4, and (0, ∞)
(d) [−3, ∞)
(a) (−3, 3)
(b) (−∞, 3]
(c) [−3, 3]
(b) Decreasing on (−∞, −2)∪(2, ∞) and increasing on (−2, 2)
(c) (8, −12)
(d) none
Page 7 of 15
(a) 6a − 6h
(b) 20 − 6h
(c) −6(a − h)
(d) −6(2a + h)
(e) 20 − 6(a − h)
Review Package · Basic Calculus
65. Find the limit: lim 2
Department of Mathematical Sciences
72. Find the derivative of: f (x) = −x3 + 5x2 − 9
x→7
(a) 7
(b) 8
(c) 2
(a) f ′ (x) = −3x2 +10x
(d) the limit does not
exist
x2 − 1
x→1 x − 1
(d) 6
(e) the limit does not
exist
67. Find the limit if it exists.
x−4
lim 3
x→4 x − 2x2 − 8x
1
(c) 24
(a)
24
(d) the limit does not
(b) 0
exist
68. Determine all values of x at which f (x) =
is discontinuous.
(a) −9 and 9
(b) −4 and 4
(d) 4
70. Find the slope of the tangent line to the graph of:
f (x) = −11x2 − 9x
(a) −11x
(b) −22x
(b) 4; y = 4x − 4
3
√
2 x
(d) f ′ (x) = 5 − 6x
m = 20;
m = 10;
m = 44;
m = 13;
= 20x + 44
= 10x
= 44x + 10
= 13x − 23
y
y
y
y
75. Find the derivative of: f (x) =
2x
+ 7)2
2
(b) f ′ (x) = − 2
x +7
(a) f ′ (x) = −
(x2
x2
1
+7
(c) f ′ (x) =
(x2 + 7)2
x
3
(a) f ′ (x) = x 2 (x2 + 5)2
5 − 3x2
(b) f ′ (x) = √
2 x(x2 + 5)2
(x2 + 5)2
(c) f ′ (x) =
5−x
√
2 x
′
(d) f (x) = − 2
x +5
77. Find the derivative and f ′ (5) given: f (x) = (3x −
5)(x2 + 5); x = 5
(c) −22x − 9
(a) f ′ (x) = 5x + 36; 17
(b) f ′ (x) = 38x2 − 5x;
−36
(c) f ′ (x) = 9x2 −10x+
(d) −11x − 9
71. Find the slope of the tangent line to the graph of
the function at the given point and determine an
equation of the tangent line.
f (x) = 4x2 at (1, 4)
(a) 5; y = 5x + 5
(c) f ′ (x) = 4 −
(a)
(b)
(c)
(d)
69. Determine all values of x at which the function is
discontinuous.
x2 − 4x
f (x) = 2
x − 9x + 20
(b) −4 and −5
(a) f ′ (x) = 4 − 3x
√
(b) f ′ (x) = 4 − 3 x
(d) f ′ (x) = (x2 + 7)2
√
x
76. Find the derivative of: f (x) = 2
x +5
(d) −2
(c) 4 and 5
(d) f ′ (x) = 3x2 + 18x
74. Find the slope m and the equation of the tangent
line to the graph of the function
f (x) = 2x2 − 3x + 9 at the point (4, 29).
4x
x2 − 81
(c) 3
(a) 5
(b) f ′ (x) = −x2 + 6x
√
73. Find the derivative of: f (x) = 4x − 3 x
66. Find the limit if it exists: lim
(a) 11
(b) 3
(c) 2
(c) f ′ (x) = −3x2 − 9
(c) 8; y = 8x − 4
15; 190
(d) f ′ (x) = 3x2 − 17;
−38
78. Find the derivative and f ′ (5) given:
3x + 1
;x=5
3x − 1
(d) 10; y = 10x + 5
Page 8 of 15
(a) f ′ (x) = −
10
3x
;
(3x + 1)2 11
f (x) =
Review Package · Basic Calculus
6
1
(b) f ′ (x) =
;
3x − 1 39
18
(c) f ′ (x) = −3(3x − 1)2 ; −
10
3
6
;−
(d) f ′ (x) = −
2
(3x − 1)
98
79. Find the derivative: f (x) =
(7x2
Department of Mathematical Sciences
83. Find the first and second derivatives of: f (x) =
5x2 − 4x + 10
(a) f ′ (x) = 10x − 4, f ”(x) = 10
(b) f ′ (x) = 5x − 4, f ”(x) = 5
+ 2x +
(c) f ′ (x) = 8x − 7, f ”(x) = 8
6)−5
(d) f ′ (x) = x − 4, f ”(x) = 1
(a) −10(7x + 1)(7x2 + 2x + 6)−4
(e) f ′ (x) = 3, f ”(x) = 0
(b) −5(7x2 + 2x + 6)−6
84. Find the second derivative of: h(t) = t4 − 2t3 +
6t2 − 3t + 4
(c) −5(14x + 2)−6
(d) −10(7x + 1)(7x2 + 2x + 6)−6
80. The management of Acrosonic plans to market
the ElectroStat, an electrostatic speaker system.
The marketing department has determined that
the demand for these speaker is, p = −0.04x + 600
0 ≤ x ≤ 30000, where p denotes the speaker’s unit
price (in dollars) and x denotes the quantity demanded.
Find the marginal revenue function R′ (x).
(a) −0.08x + 600
(b) 0.04 − 600x
(a) 2t2 − 4t + 12
(b) 12t2 − 12t + 12
(c) 12t2 + 12t + 12
(a) 7(x2 + 2)5 (13x2 + 2)
(b) 14(x2 + 2)5 (13x2 + 2)
600
x
(d) −0.04x + 600
(a) 0.000002x2 − 0.06x + 560
80000
(b) 0.000006x2 − 0.06x +
x
80000
2
(c) 0.000004x − 0.03 +
x
80000
(d) 0.000004x − 0.03 −
x2
(c) 3(x2 + 3)5 (13x + 2)
(d) 7(x2 + 2)5 (5x2 + 2)
(e) 7(x2 + 2)5 (5x + 2)
86. Find the second derivative of: g(t) = (2t2 −1)2 (5t2)
(a) 10(10t4 − 5t2 + 5)
(b) 10(72t4 − 10t2 + 1)
(c) 5(60t4 − 24t2 + 1)
(b) 3; elastic
(d) 5(60t4 − 24t2 + 1)
(e) 10(60t4 − 24t2 + 1)
87. Find the first and second derivatives of: f (x) =
x
9x + 1
(a) f ′ (x) =
1
82. Where the demand equation is x + p − 20 =
8
0, compute elasticity of demand and determine
whether the demand is elastic, unitary, or inelastic
at p = 70.
5
(a) ; inelastic
8
(e) t2 − 2t + 6
85. Find the second derivative of: f (x) = x2 + 2)7
(c) −0.04 +
81. The weekly total cost function associated with
manufacturing the Pulsar 25 color console television is given by
C(x) = 0.000002x3 − 0.03x2 + 560x + 80000
What is the marginal average cost function C’ ?
(d) 2t2 + 4t + 12
(b) f ′ (x) =
(c) f ′ (x) =
(d) f ′ (x) =
1
(c) ; inelastic
7
7
(d) ; inelastic
9
(e) f ′ (x) =
Page 9 of 15
18
2
, f ”(x) = −
2
9x + 1)
(9x + 1)3
2
8
, f ”(x) = −
2
(9x + 1)
(9x + 1)3
1
8
, f ”(x) =
(9x + 1)2
(9x + 1)3
18
1
, f ”(x) = −
2
(9x + 1)
(9x + 1)3
1
18
, f ”(x) = −
(9x + 1)
(9x + 1)2
Review Package · Basic Calculus
88. √
Find the first derivative of the function: f (u) =
3 − 7u
(a) f ′ (u) = −
(b) f ′ (u) = −
(c)
f ′ (u)
=
9
2(3 − 7u)
1
1
(3 − 7u) 2
7
2(3 − 7u)
1
2
(d) f ′ (u) = −
(e) f ′ (u) =
7
2(3 − 7u)
7
9(3 − 7u)
(c) Relative maximum: g(2) = −5 No relative
minima
1
2
(d) No relative maxima or minima
1
2
1
2
92. Find the relative extrema for: f (x) = 7x3 + 2
(a) Relative maximum: f (0) = 2
89. Find the third derivative of:
f (x) = 2x5 − 7x4 + 6x2 − 12x + 16
(a) 10x2 − 168x
(b) 60x2 − 140x
(c) 120x2 − 168x
Department of Mathematical Sciences
(b) No relative maxima Relative minimum
g(2) = −5
(b) Relative minimum: f (0) = 2
(d) 2x2 − 7x
(c) Relative maximum: f (1) = 9
(d) Relative minimum: f (1) = 9
(e) 80x2 − 7x
(e) none
90. The graph of the function f shown in the accompanying figure gives the elevation of that part of
the Boston Marathon course that includes the notorious Heartbreak Hill. Determine the intervals
(stretches of the course) where the function f is
increasing (the runner is laboring), where it is constant (the runner is taking a breather), and where
it is decreasing (the runner is coasting).
93. Find the absolute maximum value and the absolute minimum value for: f (x) = 4x2 + 9x − 1
97
; absolute min(a) Absolute maximum value:
16
imum value: none
97
(b) Absolute maximum value: − ; absolute
16
minimum value: none
(c) Absolute maximum value: none; absolute
97
minimum value: −
16
(d) No absolute extrema
(a) Decreasing on (15.7, 16.1) and (17.2, 17.3), increasing on (15.1, 15.7) and (16.1, 16.6), and
constant on (16.6, 17.2) and (17.3, 18.2)
(b) Decreasing on (16.6, 17.2) and (17.3, 18.2), increasing on (15.7, 16.1) and (17.2, 17.3), and
constant on (15.1, 15.7) and (16.1, 16.6)
(c) Decreasing on (15.1, 15.7) and (16.1, 16.6), increasing on (16.6, 17.2) and (17.3, 18.2), and
constant on (15.7, 16.1) and (17.2, 17.3)
91. Find the relative maxima and relative minima for:
g(x) = x3 − 3x2 + 5.
(a) Relative minimum: g(2) = 1 Relative maximum g(0) = 5
94. Find the absolute maximum value and the absolute minimum value for: f (x) = x3 + 6x2 − 8 on
[−7, 3]
(a) Absolute maximum value: 73; absolute minimum value: −57
(b) Absolute maximum value: 24; absolute minimum value: −8
(c) Absolute maximum value: 24; absolute minimum value: −57
(d) Absolute maximum value: 73; absolute minimum value: −8
Page 10 of 15
(e) No absolute extrema
Review Package · Basic Calculus
95. Find the absolute maximum value and the absolute minimum value for:
1
2
f (x) = x4 − x3 − 2x2 + 2 on [−3, 3]
2
3
13
; absolute min(a) Absolute maximum value:
2
10
imum value: −
3
85
(b) Absolute maximum value:
; absolute min2
10
imum value: −
3
10
(c) Absolute maximum value:
; absolute min3
85
imum value: −
2
(d) No absolute extrema
Department of Mathematical Sciences
100. Find the present value of $55561 due in 4 yr at
an interest rate of 7%/year compounded continuously.
101. Find the derivative of: f (x) = x6 ex
(a) x5 ex (x + 1)
(c) 6x5 ex
(b) x5 ex (x + 6)
(d) x5 e6x (x + 6)
5ex
x2
x
5e (x − 1)
(b)
x2
(a) −
103. Find the derivative of: f (x) = 5(ex + e−x )
(a) 5(ex + e−x )
96. The estimated monthly profit (in dollars) realizable by Cannon Precision Instruments for manufacturing and selling x units of its model M1 camera is
P (x) = −0.05x2 + 490x − 8000
To maximize its profits, how many cameras should
Cannon produce each month?
(a) 4875 cameras
(c) 5000 cameras
(b) 4900 cameras
(d) 4950 cameras
97. Simplify: log x(x + 5)7
5e
x
5ex
(c) 2
x
5ex (x − 1)
(d) −
x2
102. Find the derivative of: f (x) =
(b) 5(e−x − ex )
(c) 5(ex − e−x )
(d) 10ex
104. Find the derivative of: f (x) = 9e3x−7
(a) 9(3x − 7)e3x−7
(b) 3e3x−7
(c) 27e3x−7
(d) 27e3x
105. Find the derivative of: f (x) = (ex + 1)19
(a) 19ex (ex + 1)18
(c) 19(ex + 1)19
(b) 19(ex + 1)18
(d) 19ex (ex + 1)19
106. Find the derivative of: f (x) = (x − 1)e9x+8
(a) log x + 7 log(x + 5)
(c) log x7 + log(x + 5)
(a) e9x+8 (9x + 8)
(b) log x + log(x + 5)7
(d) 7 log x − log(x + 5)
(b) e9x+8 (9x − 8)(9x + 8)
(c) 9e9x+8 (x − 1)
(d) e9x+8 (9x − 8)
98. Simplify: ln x(x + 1)(x + 7)
107. Find the second derivative of: f (x) = e−9x + 3e4x
(a) 7 ln x − ln(x + 1) + ln(x + 7)
(b) − ln x + ln(x + 1) + ln(x + 7)
(c) ln x + ln(x + 1) + ln(x + 7)
(a) 9e−9x + 16e4x
(c) 9e−9x + 12e4x
(b) −81e−9x + 16e4x
(d) 81e−9x + 48e4x
108. Find the derivative of: f (x) = 5 ln x
(d) ln 1x + ln(x + 1) − ln(x + 7)
99. Find the interest rate needed for an investment of
$5000 to grow to an amount of $8500 in 4 yr if
interest is compounded monthly.
4
x
5
(b)
x
(a)
Page 11 of 15
(c)
1
x5
(d) −
1
x4
Review Package · Basic Calculus
109. Find the derivative of: h(t) = 6 ln t7
(a) 42 ln t6
(b)
42
t
Department of Mathematical Sciences
117. Find the derivative of: f (t) = e6t ln (t + 9)
13
t
13
(d) 7
t
(c)
√
110. Find the derivative of: f (x) = ln ( x + 1)
√
1
x
(a) √ √
(c) √
2 x( x + 1)
x+1
3
√
√
(b) √ √
(d) x ln ( x + 1)
x( x + 1)
1
7x6
42
(c)
x
1
(d) − 7
x
111. Find the derivative of: f (x) = ln
7
x6
6
(b) −
x
(a)
(b)
x+4
9x
2x
(x + 4)2
1
(b) −
x+4
(d)
18x − 8
(c)
2
9x − 8x + 3
1
(d)
16x − 8
9x
x+4
4
(c)
9x(x + 4)
4
(d)
x(x + 4)
(c) (x + 4)2 (6x + 20)(x + 5)
(d) (x + 4)2 (6x + 22)(x + 5)2
Z
120. Find the indefinite integral.
4x7 dx
1 7
x +C
7
1
(d) x7 + C
2
Z
1
121. Find the indefinite integral.
7x− 2 dx
1 8
x +C
2
4
(b) x8 + C
7
(c)
(a)
(c) 3x(2 ln 2x + 1)
(d) 6 ln 2x + 3x
115. Find the derivative of: f (x) =
(a) 6(x8 − 8)5
1
(b)
(x8 − 8)6
6(ln x)5
x
(b) 4x(ln x)5
−
8x7
−8
48x7
(d) 8
x −8
(c)
116. Find the derivative of: f (x) =
(a)
ln (x8
x8
(ln x)6
(c) 6(ln x)5
(d)
x2
(x + 4)3
(b) (x + 4)(5x + 22)(x + 5)2
1
(b) 3x(ln 2x + 2)
1
(x + 4)2
(a) (x + 4)(5x + 20)(x + 5)
114. Find the derivative of: f (x) = 3x2 ln 2x
(a) 6x
(c) −
119. Find the derivative of: y = (x + 4)2 (x + 5)3
113. Find the derivative of: f (x) = ln
4
(a)
x+4
118. Find the second derivative of: f (x) = ln (x + 4)
(a) −
112. Find the derivative of the function:
f (x) = ln (9x2 − 8x + 3)
16x3 − 8
(a)
9x2 − 8x + 3
1
(b)
2
9x − 8x + 3
e6t [6(t + 9) ln (t + 9) + 1]
t+9
6t
e [9 ln (t + 9) + 6]
(b)
t+9
6t
e [9(t + 9) + ln (t + 9)]
(c)
t+9
e6t
(d)
t+9
(a)
(ln x)5
x5
8)6
1
(a) 12x 2 + C
(c) 14x 2 + C
(b) 12x1 + C
(d) 14x 2 + C
1
122. Find the indefinite integral.
(a) 7x − 4x2 + C
(b) 7x −
8x2
+C
Z
(c) 7 − 4x + C
(d) 7 − 4x2 + C
123. Find the indefinite integral.
Z
1
7
(a) 2 + x2 + x8 + xex + C
2
8
1
8
(b) 2x + x2 + x8 + ex + C
2
7
Page 12 of 15
(7 − 8x)dx
(2 + x + 7x7 + ex )dx
Review Package · Basic Calculus
1
7
(c) 2x + x2 + x8 + ex + C
2
8
7
(d) 2x + x2 + x8 + ex + C
8
Z
1
1
3
124. Find the indefinite integral. (t 2 + 8t 8 − 4t− 2 )dt
2 5
t2
5
2 5
(b) t 2
5
2 5
(c) t 2
5
2 5
(d) t 2
5
(a)
1
(2x − 7)3 + C
2
(b) (2x + 7)3 + C
(a)
(c) x + C
1
(d) (2x + 7)3 + C
3
129. Find the indefinite integral.
Z
(3x2 − 2x + 1)(x3 − x2 + x)6 dx
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
125. Find the indefinite integral.
Department of Mathematical
Sciences
Z
128. Find the indefinite integral.
2(2x + 7)2 dx
1 3
(x − x2 + x)7 + C
7
(b) (x3 − x2 + x)7 + C
(a)
Z
√
3
( 3 x + − 6ex )dx
x
3 4
3
x 3 + 2 − 6xex + C
4
x
3
3 4
(b) x 3 + 2 − 6ex + C
4
x
3 4
(c) x 3 + 3 ln |x| − 6ex + C
4
3 4
(d) x 3 + 3 ln |x| − 6xex + C
4
126. Lorimar Watch Company manufactures travel
clocks. The daily marginal cost function associated with producing these clocks is
C(x) = 0.000003x2 − 0.007x + 5
where C ′ (x) is measured in dollars/unit and x denotes the number of units produced. Management
has determined that the daily fixed cost incurred
in producing these clocks is $150. Find the total
cost incurred by Lorimar in producing the first 600
travel clocks/day.
(a)
127. Cannon Precision Instruments makes an automatic electronic flash with Thyrister circuitry. The
estimated marginal profit associated with producing and selling these electronic flashes is −0.004x+
20 dollars/unit/month when the production level
is x units per month. Cannon’s fixed cost for
producing and selling these electronic flashes is
$20,000/month. What is the maximum monthly
profit?
1 3
(x + x2 + x) + C
7
(d) x + C
(c)
130. Find the indefinite integral.
3
2 3
(t + 9) 2 + C
3
(b) (t3 + 9) + C
(a)
(b) x + C
x5
dx
1 − x6
1
(c) −
+C
ln |1 + 6x6 |
1
(ln 6|x|)2 + C
6
(b) 6x2 + C
e2x
dx
4 + e2x
1
(c) ln (4 + e2x ) + C
2
(d) x + C
1
(ln |u|)6 + C
6
1
(b) u6 + C
6
Page 13 of 15
Z
ln 6x
dx
x
1
(c) (ln 6|x|)2 + C
2
(d) (ln 6|x|)2 + C
Z
(ln u)5
du
u
1
(c) ln 6|u| + C
6
134. Find the indefinite integral.
(a)
Z
(d) − ln |1 − x6 | + C
133. Find the indefinite integral.
(a)
p
t3 + 9dt
2
3 3
(t + 9) 3 + C
2
(d) t + C
132. Find the indefinite integral.
(a) ln (4 + e2x ) + C
1
(b) ln (x) + C
2
3t2
(c)
131. Find the indefinite integral.
1
(a) − ln |1 − x6 | + C
6
Z
Z
(d) 6(ln |u|)6 + C
Review Package · Basic Calculus
Department of Mathematical Sciences
Answer Key
Problem
Answer
Problem
Answer
Problem
Answer
Problem
Answer
Problem
Answer
1
A
28
D
55
B
82
D
109
B
2
E
29
D
56
A
83
A
110
A
3
D
30
B
57
C
84
B
111
B
4
E
31
A
58
A
85
B
112
C
5
C
32
D
59
D
86
E
113
C
6
B
33
A
60
E
87
D
114
C
7
A
34
C
61
B
88
C
115
8
B
35
62
A
89
C
116
9
C
36
63
C
90
B
117
10
C
37
64
D
91
A
118
11
C
38
65
C
92
B
119
12
D
39
66
C
93
C
120
13
D
40
67
A
94
B
121
C
14
E
41
A
68
A
95
D
122
A
15
C
42
D
69
A
96
B
123
C
16
A
43
D
70
C
97
A
124
D
17
B
44
E
71
C
98
C
125
C
18
C
45
D
72
A
99
126
19
B
46
C
73
C
100
127
20
C
47
E
74
D
101
B
128
D
21
B
48
A
75
102
B
129
C
22
A
49
D
76
103
C
130
A
Page 14 of 15
Review Package · Basic Calculus
Department of Mathematical Sciences
Problem
Answer
Problem
Answer
Problem
23
B
50
B
24
C
51
25
D
26
27
Answer
Problem
Answer
Problem
Answer
77
104
C
131
A
B
78
105
A
132
A
52
B
79
106
D
133
C
D
53
A
80
107
C
134
C
D
54
C
81
108
B
D
Last Version: November 20, 2013
Page 15 of 15