MATH 140 Homework Problems
Contents
1 Limits and Continuity 2
2 Derivatives 9The Limit Definition of the Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Sum, Difference, Constant Multiple, and Reciprocal Rule. Derivatives of Power, Exponential, and
Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Product and Quotient Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Derivatives of Inverse Functions. Derivatives of Logarithms and Inverse Trigonometric Functions . 14Implicit and Logarithmic Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Higher Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3 Applications of the Derivative 18Curve Sketching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Answers
Answers Section 1 21
Answers Section 2 23
Answers Section 3 30
2 MATH 140 HOMEWORK PROBLEMS
1 Limits and Continuity
For the function y = f(x) whose graph is shown find all specified quantities (if a quantity does not existwrite “DNE”).
1. (a) f(0)
(b) limx→0
f(x)
(c) f(1)
(d) limx→1−
f(x)
(e) limx→1+
f(x)
(f) limx→1
f(x)
x
y
−1 1 2 3
1
2
3
2. (a) f(0)
(b) limx→0
f(x)
(c) f(2)
(d) limx→2−
f(x)
(e) limx→2+
f(x)
(f) limx→2
f(x)
x
y
−1 1 2 3
1
2
3
3. (a) f(2)
(b) limx→2−
f(x)
(c) limx→2+
f(x)
(d) limx→2
f(x)
x
y
−1 1 2 3
−1
1
2
3
4
4. (a) f(1) (b) limx→1
f(x)
x
y
−1 1 2 3
−1
1
2
3
4
1 LIMITS AND CONTINUITY 3
5. Sketch the graph of a function that has all the specified properties.
(a) limx→0
f(x) =∞
(b) f(0) = 1
(c) limx→1+
f(x) = −∞
(d) limx→1−
f(x) = 2
(e) limx→(−1)−
f(x) = −2
(f) limx→(−1)+
f(x) = 0
(g) f(−1) is undefined.
Find the limit, using the limit computation laws.
6. limx→2
(x2 + x− 2)
7. limx→−3
(x+ 2
x+ 1
)58. lim
x→−1(√x2 + 3− 3
√x− 7)
9. limx→√2(x2 +
√x4 + 21)
10. limx→5
x+ |x− 5|x2 − 1
11. limx→π
sin(x/3) cos(x/6)
1 + (x/π)2
12. limx→ln(2)
(3ex − e2x)
13. limx→e3
2 ln(x) + ln(√x)
x+ 1
14. limx→2
2x − 3x2
log2(x) + 1
15. limx→1/2
(x arcsin(x) + arctan(−2x))
Suppose that limx→2
f(x) = 3, limx→2
g(x) = 5, and limx→3
h(x) = 7. Find the limit.
16. limx→2
h(f(x))
17. limx→2
f(x)g(x)
18. limx→2
f(x)
g(x)
19. limx→1
f(x2 + 2x− 1)− xx2 + h(x+ 2)
20. limx→1
f(x2 − 1
x− 1
)h(x2 + x− 2
x2 − x
)21. lim
x→22 arctan
(√f(x)
)Find the indeterminate limit.
22. limx→1
x2 − 1
x− 1
23. limx→−2
3x+ 6
x2 + x− 2
24. limx→3
x2 + 2x− 15
x2 − 11x+ 24
25. limx→−1
√x+ 10− 3
x+ 1
26. limx→7
√43− x−
√29 + x√
2x+ 2−√
23− x
27. limx→−12
√13 + x−
√2x+ 25
x2 + 15x+ 36
28. limx→2
√x2 + x+ 10−
√2x2 + 8
x2 + x− 6
29. limx→2
12x+6 −
110
x− 2
4 MATH 140 HOMEWORK PROBLEMS
30. limx→3
2x2+x −
13x−3
x2 + 4x− 21
31. limx→−4
1x+2 −
13x+10√
x2 − 3−√
17 + x
32. limx→0
ex+1(x3 + x2)
x4 + 3x2
33. limx→2
x sin(πx2 − 3πx+ 2π
x2 − x− 2
)34. lim
x→−3e(√4+x−1)/(x2−9)
35. limx→6
x3 − 4x2 − 11x− 6
cos(πx/2)(x2 − 36)
36. limx→2
(x4 − 5x3 + 9x2 − 8x+ 4
x4 − 2x3 + x− 2
)2
37. limx→e5
ln3(x)− 4 ln2(x)− 4 ln(x)− 5
ln2(x)− 25
38. limx→ln(3)
√ex + 1−
√2ex − 2
e2x + 8ex − 33
39. limx→ln(2)
e2x − 2ex − xex + 2x
e2x − ex − 2
40. limx→π/6
2 sin2(x) + sin(x)− 1
2 sin2(x)− 7 sin(x) + 3
41. limx→π/4
cos(2x)
cos(x)− sin(x)
42. limx→0
cos(6x)− 1
cos(3x)− 1
43. limx→√3
9 arcsin(x/2)− 3π
3 arcsin(x/2)2 + (15− π) arcsin(x/2)− 5π
Find the one-sided limit.
44. limx→1+
2x+ |x− 1| − 2
2|1− x|
45. limx→1−
2x+ |x− 1| − 2
2|1− x|
46. limx→3+
|x2 − 9|x− 3
47. limx→3−
|x2 − 9|x− 3
48. limx→(−2)+
|x2 − x− 6|−2x2 + x+ 10
49. limx→(−2)−
|x2 − x− 6|−2x2 + x+ 10
Find the indicated limits for the piecewise defined function.
50.
f(x) =
x2 + 3x− 2 x ≤ 1
(x+ 1)2 − 4
2x− 2x > 1
(a) limx→1−
f(x)
(b) limx→1+
f(x)
(c) limx→1
f(x)
(d) limx→0
f(x)
(e) limx→2
f(x)
51.
f(x) =
√8− x+ x− 2
x2 − 1x < −1
ln(x2 + 1) −1 ≤ x ≤ 2|2− x|(3− x)
x2 − 4x > 2
(a) limx→(−1)−
f(x)
(b) limx→2−
f(x)
(c) limx→2+
f(x)
(d) limx→3
f(x)
1 LIMITS AND CONTINUITY 5
Use the Squeeze Theorem to find the limit.
52. limx→0
x2 cos(
1x2
)53. lim
x→0sin(x) cos(2/x3)
54. limx→0+
√xesin(1/x)
55. limx→1
cos(
(x− 1) sin2(
1x2−1
))56. Let f be a function such that x2 ≤ f(x) ≤ x2 − 8x+ 12
|x− 2|for all 0 < x < 2. Find lim
x→2−f(x). Carefully
explain your reasoning!
Find the trigonometric limit.
57. limx→0
sin(2x)
x
58. limx→π
sin(x)
x
59. limx→0
x cot(x)
60. limx→1
sin(x− 1)
3x− 3
61. limx→0
tan(x)
sin(2x)
62. limx→π
cosx− 1
x
63. limx→0
cos(x)− 1
5 sin(x)
64. limx→0
sin(3x) + tan(6x)
sin(8x)− tan(5x)
65. limx→3
sin(x2 − 9)
x− 3
66. limx→1
tan(√x− 1)
x2 − 1
67. limx→2
√2−√x
sin(x− 2)
68. limx→1
sin(√x+ 3− 2)
tan(√x2 + 5x+ 3− 3)
69. limx→0
x cot(2x) csc(3x) tan(5x)
70. limx→0
tan(5x) sin(x2) csc(x)
x3 csc(2x)
71. limx→0
cot(2x) cot(π2 − x)
72. limx→π/2
cosx
2x− π
73. limx→π/2
tan(2x)
x− π2
74. limx→0
arcsinx
x
75. limx→0
3x
arctan(2x)
Find the infinite limit.
76. limx→1+
ln(x− 1)
77. limx→2
5
(x− 2)2
78. limx→(−2)−
x+ 6x+ 8
x2 + 4x+ 4
79. limx→1+
√x2 − 1
x2 + x− 2
80. limx→0−
sin(x)
x2
81. limx→0+
cot(x) csc(x) tan(x− π/2)
6 MATH 140 HOMEWORK PROBLEMS
82. limx→(π/2)+
tanx+ 2x
1 + x2
83. limx→0
ln |x|x3 arcsin(x)
84. limx→1−
x2 + x
π − 2 arcsin(x)
85. limx→(√3)−
ln(x)
3 arctan(x)− π
Find the limit as x→∞ or x→ −∞, respectively.
86. limx→∞
x+ 1
x2 + 3x+ 5
87. limx→−∞
3x3 − 5x2 + x− 3
7x3 − 2x2 + x+ 5
88. limx→−∞
(2x2 + 5x+ 1)2(5x+ 1)
(x+ 1)(2x4 + x− 3)
89. limx→∞
5x√x+ 2x1/3 + 1√4x3 − x+ 5
90. limx→∞
3x+ 1√x2 − 1
91. limx→−∞
3x+ 1√x2 − 1
92. limx→−∞
2x+ 3√x2 + 2x+ 1 +
√4x2 + 3
93. limx→∞
(x−√x2 + x+ 2
)94. lim
x→−∞
(√4x2 + 5x+ 1−
√4x2 − 4x+ 5
)95. lim
x→−∞
x2 + 5√x4 + x+ 1
96. limx→∞
x2 + 1
x2 +√x4 + 3 + x
√9x2 + 5
97. limx→−∞
x2 + 1
x2 +√x4 + 3 + x
√9x2 + 5
98. limx→−∞
2x+ 33√
27x3 − 2x2 + x+ 15
99. limx→∞
arctan(x)
ex + 2
100. limx→−∞
arctan(x)
ex + 2
101. limx→∞
3e2x + 5ex + 1 + 2e−x
9e2x − ex + e−x
102. limx→−∞
3e2x + 5ex + 1 + 2e−x
9e2x − ex + e−x
103. limx→∞
5 ln2(x)− 2 ln(x) + 1√16 ln4(x) + ln2(x) + 1
104. limx→∞
sinx
x
105. limx→−∞
ex cos(x2 + 1)
106. limx→−∞
cos(x)− 1
x
107. limx→∞
2x2 + x sin(x) + 1
4x2 + cos(x)
108. limx→−∞
sin(x)ex + 3 + e−2x
5ex + cos(x2) + 3e−2x
Carefully explain why the following limits do not exist.
109. limx→4
|x2 − 16|x− 4
110. limx→0
sin(1/x)
111. limx→0
cos(x) sin(1/x2)
112. limx→0
x cot(x) csc(x)
113. limx→∞
arccos(x)
x2
114. limx→∞
tan(x)
1 LIMITS AND CONTINUITY 7
115. limx→∞
ex cos(x2 + 2x+ 5)
116. limx→∞
cos(x)
arctan(x)
117. limx→−∞
cos(x)x2 + x+ 5
3x2 + 15
Find the vertical asymptotes of the function (if any).
118. f(x) = ln(1− x2)
119. f(x) =x− 1√x2 − 1
120. f(x) = x cotx
121. f(x) =sinx
x2 + x
122. f(x) =x3 + x2 + x
sin(x/2) cos(x), −π < x < π
123. f(x) = e−1/x2
124. f(x) = e−1/(x+1)
125. f(x) = 3
(−x− 1
x4 − x2)
Find the horizontal asymptotes of the function (if any).
126. f(x) =5√x+ 3− x2
(5x)2
127. f(x) =√x2 + 5x+ 1−
√x2 + x+ 6
128. f(x) =arctan(x)x2 + 5
2x2 + 1
129. f(x) =x−2 + 3x−1
2x−2 + x−1
130. f(x) =5 ln3(x) + ln(x) + 1
10 ln3(x) + ln2(x) + 5
131. f(x) = e−1/x2
132. f(x) = cos(πe2x + ex − πe−x
3e2x + ex + 6e−x
)133. f(x) = e−x
3+50x+1 sin(x3)
134. Sketch the graph of a function y = f(x) that has all the specified properties:
(a) y = 1 is a horizontal asymptote as x→ −∞
(b) y = 0 is a horizontal asymptote as x→∞
(c) x = 1 is a vertical asymptote
(d) f(1) = 2
(e) f is discontinuous at x = 2
(f) limx→3
f(x) = 1
(g) f(3) is undefined
135. Find the value of a such that the function
f(x) =
x+ sinx
3x2 + xfor x 6= 0,
a for x = 0
is continuous at x = 0.
8 MATH 140 HOMEWORK PROBLEMS
136. Find the value of a such that the function
f(x) =
{e− 1
(x−1)4 · cos(
πx−1
)for x 6= 1,
a for x = 1
is continuous at x = 1.
137. Find the values of a and b such that the function
f(x) =
sinx
x+ a for x < 0,
3 sec(x) + b for 0 ≤ x < π/3,
5ax2 + x for x ≥ π/3
is continuous at x = 0 and x = π/3.
138. Find the values of a and b such that the function
f(x) =
a · x
2 + x− 6
x2 − 6x+ 8for x < 2,
bx2 − x+ 1 for 2 ≤ x ≤ 3,√x+ 6−
√x2 − x+ 3
x2 − 9for x > 3
is continuous on the entire real number line.
139. Find the value of b such that the function
f(x) =
{logb
(x2+5x−14
x−2
)for x > 2,
2 sin(πx
)for 0 < x ≤ 2
is continuous at x = 2.
Show that the equation has a real solution.
140. cosx = x
141. 2 sinx+ 1 = x2
142. arctanx = 2− x2
143. arcsinx = 1− x
144. ex + 1 = secx
145. 5x7 − πx3 + 2 = 3x2 + ex
146. Show that every polynomial function of odd degree has at least one real root. Is this also true forpolynomial functions of even degree? If yes, show that it is true, and if not give a counterexample.
147. Let f be a continuous function such that 0 ≤ f(x) ≤ 1 for all 0 ≤ x ≤ 1. Show that there is a solution tothe equation f(x) = x in the interval [0, 1].
148. Let f be a continuous function on [0, 2], and suppose that f(0) = f(2). Show that there is a solution tothe equation f(x) = f(x− 1) in the interval [1, 2].
149. Let f be a continuous function that satisfies f(x) · f(f(x)) = 1 for all −∞ < x < ∞. Suppose thatf(10) = 9. What is f(5)?
150. Show that on every circle of latitude on Earth there are two diametrically opposite locations where thetemperature is the same.
2 DERIVATIVES 9
2 Derivatives
The Limit Definition of the Derivative
Use the limit definition of the derivative to find f ′(a) at the given point a.
1. f(x) = 2x+ 3, a = 5
2. f(x) = x2 + 2, a = −1
3. f(x) =17π
5π3 − 1, a = e
4. f(x) =√π − 3x, a = 0
5. f(x) = sinx, a = 0
6. f(x) = tanx, a = 0
Find an equation for the tangent line to the graph of the function y = f(x) at the point where x = a. Youshould use the limit definition of the derivative when finding the slope!
7. f(x) =√
5x2 + 4x, a = 1
8. f(x) =3
2x+ 5, a = −3
9. f(x) =1
x2 + x, a = 2
10. f(x) =5√
2x2 + 7, a = 3
Use the limit definition of the derivative to finddf
dx.
11. f(x) = 1 +4
x− x2
12. f(x) = (πx− ln 2)2
13. f(x) = sin(2x)
14. f(x) = tan(πx)
Use the limit definition of the derivative to find y′.
15. y =5s
s2 + 2
16. y =u+ 1√3u− 10
17. y = 5x3 − 2x+ 1
18. y =√x+√x
19. y = 3√x
Hint : (u− a)(u2 + au+ a2) = u3 − a3
20. y = 2x
Hint : Use that limu→0
eu − 1
u= 1.
21. y = x cos(x)
Show that the function is not differentiable, yet continuous, at the given point a.
22. f(x) = 2|x|+ 1, a = 0
23. f(x) = |x− 3|+ 2x, a = 3
24. a = 0, and
f(x) =
{x sin(1/x) for x 6= 0,
0 for x = 0,
25. Show that
f(x) =
{x2 cos(1/x) for x 6= 0,
0 for x = 0
is differentiable at x = 0, and find f ′(0).
10 MATH 140 HOMEWORK PROBLEMS
26. Consider the function y = f(x) whose graph is shown below.
x
y
−1 1 2 3 4 5 6 7 8 9
−2
−1
1
2
3
4
(a) Find all x-values where the function f is discontinuous.
(b) Find all x-values where the function f is not differentiable.
(c) Find all x-values where f ′(x) = 0.
(d) Find f ′(4).
Sum, Difference, Constant Multiple, and Reciprocal Rule. Derivatives of Power,Exponential, and Trigonometric Functions
Differentiate the function.
27. f(x) = 3x11 − 3x7 + 8x3 + 1
28. g(t) = 5√t− 3
3√t− 15t3/2
29. ϕ(s) = (2s3 + 1)2
30. T (ω) = (√ω − 0.1)(1.5ω8/7 − 5ω)2
31. f(γ) = 3ln(5)
32. G(x) =
√17
x5/3− ln(15)x−7.385 + π2x
33. h(u) =5u√u− 3u2
u33√
8u2
34. y =1
3x+ 1
35. u =17
8x3 − 5.2x− e
36. f(x) = 5ex + x3.127
37. f(x) = 3ex + (3e)x
38. f(t) = 3t+1 + tln(7)
39. y =
√7es − 1.93e2s
8e3s+2
40. H(u) = 3u√2 + 5ue − 172u + e3u−2
Find the indicated derivative.
41.d
dx
(2 sinx− 3 tanx
)
42.d
du
(eu + 5u3 + cos(u)
)43.
d
ds
(3s−1 + 5 cos s− 3.8 csc s
)
44.d
dα
(5 sin(α)− 3
17 cos(α)
)
2 DERIVATIVES 11
45.d
dx
((5 cotx− 2 secx
)−1)
46.d
dx
(2 sin(x) + cos(x)
)∣∣∣x=π/3
47.d
du
(5u2 + sec(u)− tan(u))
)∣∣∣u=11π/6
48.d
d`
( π
3 sin `+ 5 tan `
)∣∣∣`=28π/3
Evaluate the limit by recognizing it as a derivative of some function.
49. limx→2
x10 − 1024
x− 2
50. limx→3
2x − 8
x− 3
51. limh→0
sec(π+3h
3
)− 2
h
52. limx→3
1x3−3x+1 −
119
2x− 6
Find an equation of the tangent line to the graph of the function y = f(x) at the given point.
53. y = 5x2 − x3, (1, 4)
54. y = x+ 2 3√x, (1, 3)
55. y = 5x− 2ex, (ln(3), 5 ln(3)− 6)
56. y = 3 cos(x)− 2 sin(x), (π/2,−2)
57. Find the x-coordinates of all points on the curve y = 8x3 + 12x2 − 48x − 10 where the tangent line ishorizontal.
58. Find the points on the curve y = 2ex − 5x where the tangent line is horizontal.
59. Find all x-values where f ′(x) = 0 for f(x) = 2 cos(x)− x.
Let f and g be differentiable functions such that f(−1) = −7, f ′(−1) = 2, and g(−1) = 17, g′(−1) = −3.Find h′(−1) for the function h.
60. h(x) = 3f(x)− 2g(x).
61. h(x) = 3x2 + f(x).
62. h(x) = 5f(x)− 1
g(x)
63. h(x) = 5 sin(πx) +2
f(x)
Note: You will need to use the limit definition of
the derivative to findd
dxsin(πx).
Product and Quotient Rule
Differentiate the function.
64. f(x) = (x2 + 3)(x3 − 15)
65. F (u) = (u+ 3√u)(15 3
√u− 5)
66. h(x) =(
5x3/2 − 2
x
)(2xπ − 3xln(5))
67. Q(v) = (3v + 5)(17v2 −√
3)(5v3 − 2v2 + 5v − 1)
68. g(q) = (√
2q3 − ln(3))(πq3 + 0.1q − 3)2
69. f(y) = yey + (y2 + 1)(√y − 5y)
70. s = (5et + t)(3t2 + 1)− t2 csc(t)
71. a = (4v3 − 5 sin v)(2 cos v + v tan v)
72. f(x) = (ex sin(x)− x2 cos(x))(tan(x)− 3x cot(x))
73. F (α) = sin(α) cos(α)5α
74. u = sec2(z)(sin(z)− 5ze6z+1)
12 MATH 140 HOMEWORK PROBLEMS
Finddy
dx.
75. y =x− 3
x+ 5
76. y =2x3 −
√6
x7 + 15
77. y =3√
2 · x3 + x
− x2
5x3 − 3.56x2 − x
78. y =x
(2x2 − 5)(3x3 − 2x+ 16)
79. y =(x3 − 2)2 − x
(x+√x)(x+ 1)2
80. y =3πx+ sin(1)
(√x+ x)2(x5/6 + ln(17))
81. y =5ex − 3x
2 · 7x − 15
82. y =x2ex − x2x
(5x2 − 7x)2
83. y =ex + 3 sin(x)
x2 + 5
84. y =x cos(x)
x2ex − 1
85. y =(3x2 +
√2) sin(x)
5 + 23x−5
86. y =5 csc(x)ex − 2x+cos(x)
13 sin(x)+x2
x3 sec(x) + 5 tan(x)
Find the x-coordinates of all points on the graph y = f(x) where the tangent line is horizontal, i.e., find allx-values where f ′(x) = 0 (if any).
87. f(x) = ex(x2 − 3x+ 1)
88. f(x) =x
x+ 2
89. f(x) =x2
x+ 1
90. f(x) = x2 − 4 sinx+ 4x cosx
91. f(x) = ex cos(x)
92. f(x) = x sinx, −π2 < x < π2
Let f and g be differentiable functions such that f(3) = −2, g(3) = 2, f ′(3) = −1 and g′(3) = −3. Find anequation of the tangent line to the curve y = h(x) at the point where x = 3.
93. h(x) = f(x)g(x)
94. h(x) =f(x)
g(x)
95. h(x) = exf(x) + x2g(x)
96. h(x) = sin(πx)(f(x) + x)
97. h(x) =xf(x) + cos(πx)
g(x)ex + x2
Chain Rule
Find y′.
98. y = (x2 + 5x+ 1)3
99. y = (2s4 − 5s3 − 2s+ 1)11
100. y = t(3t2 − 5t)3
101. y = (x2 + 5)3(7x3 − 8x2 + π)7
102. y =( u+ 5
2u3 + 7u− 1
)2103. y =
((x5 − 2x)3 + (7x− 5)5 + 1
)10104. y =
2(5 + (v + 1)3
)5
2 DERIVATIVES 13
105. y =20w(
w + (w + 1)2)2
106. y =7x2 − (3x− 2)7
1 + (x2 + 5)3
107. y =√
5x7 − 3x2
108. y =√ω +√ω
109. y =1
4√
5x7 + 2x− 1
110. y =(2x1/3 − 7x8/7
)1/13111. y =
s1/9
(3s6/7 + 2s)3
112. y = 3
√5x7 + 3x+ 1√
x+ 5x2
113. y =7
√2x3 + 3
√5x−
√x2 +
√x
114. y = 8 cos θ − 2 sin 5θ
115. y = sin(x)(cos(x2) + 5x)
116. y = 5 sin3(α)− cos(3α2 + α)
117. y = x sin3(5x)
118. y = (4t5 − 1)3 sin(√t)
119. y =sin(x+
√x)
1 + tan(x2)
120. y =(sec2(ex)− tan2(ex)
)−5121. y =
sec(5ex) + x
cos(x) + csc(x2 + 1)
122. y =√
(x2 + 1) tan(5x)
123. y = cot( θ√
θ2 + 2
)tan(θ +
2
θ2
)124. y = sec(sin(csc(tan(2r))))
125. y = ex2+1
126. y = e3 sin(5x)+1
127. y = (x2 + 1)3ecsc(x)+x
128. y = 3x−1 + sin(x)51−x2
129. y = tan(x32x+1
)130. y =
√esin(x) + 3
√72x+1 + cos(x2 + 1)
131. y = e
(x+2 cos(x)
sin(2x)ex+1
)Find an equation of the tangent line to the curve at the given point.
132. y = (1 + 3x)8, (0, 1)
133. y = x2 sin(π
3x2
), at the point where x = 1.
134. y = sin(sinx), at the point where x = π.
135. y = sin(2x) + sin2(x), (0, 0)
Find the x-coordinates of all points on the curve where the tangent line is horizontal.
136. y = x√
18− x2
137. y = x2√
9− x2
138. y = xa(1− x)b, where a, b > 0,0 < x < 1
139. y = cos(x)− cos2(x)
140. y = sec3(x)− 3 tan2(x)
141. y = sin(2x)− 2 sin(x)
14 MATH 140 HOMEWORK PROBLEMS
Let f and g be differentiable functions. Find an expression for h′(x). Simplify as much as possible.
142. h(x) = xf(x2)
143. h(x) = g(3x2 + 1)− f(1− x)
144. h(x) = sin(f(x) + x)
145. h(x) = f(3x)g(cos(x2) + 1)
146. h(x) = g(ex + 1
)√(f(x))2 + ex
147. h(x) = f( x
g(2x− 1)
)148. h(x) = ef(x)+x
2
149. h(x) = g(x) · 52f(x)
150. Let f be a differentiable function, and let g(x) = f(2√x). If f ′(6) = 12, find g′(9).
151. Let f be a differentiable function, and let g(x) = f(f(x)). Given that f(1) = 2, f(2) = 3, f(3) = 4,f ′(1) = −2, f ′(2) = 5, and f ′(3) = 2, find g′(2).
Derivatives of Inverse Functions. Derivatives of Logarithms and Inverse Trigono-metric Functions
Show that the following functions y = f(x) are one-to-one on the given interval I by finding the inversefunction f−1 explicitly.
152. f(x) = x2 − 3x+ 5, I = (−∞, 32 )
153. f(x) =2x√x2 − 1
, I = (−∞,−1)
154. f(x) =ex − e−x
2, I = (−∞,∞)
155. f(x) =2ex
ex + 1, I = (−∞,∞)
Show that the following functions y = f(x) are one-to-one on the given interval I by showing that f ′(x) 6= 0for all x in I. Do not attempt to find f−1 explicitly!
156. f(x) = 2x3 − 2x2 + 10x− 3,I = (−∞,∞)
157. f(x) = tan(x) + sec(x),I = (−π2 ,
π2 )
158. f(x) = 7x+ sin(3x), I = (−∞,∞)
159. f(x) = x2ex, I = (0,∞)
The following functions y = f(x) are one-to-one on the given interval I. Find (f−1)′(a) at the point(s) a.
160. f(x) = xex, I = (−1,∞).Find (f−1)′(0).
161. f(x) =√x+ tan(x), I = (π2 ,
3π2 ).
Find (f−1)′(√π).
162. f(x) = x3 + x2 + 2x+ 3, I = (−∞,∞).Find (f−1)′(7).
163. f(x) =√
3 sin(3x)− 3 cos(3x),I = (− π
18 ,5π18 ). Find (f−1)′(
√3) and (f−1)′(3).
2 DERIVATIVES 15
164. The following diagram shows the graph of an invertible function y = f(x) (drawn solid), and the line thatis tangent to the graph at the point (2, 1) (drawn dashed). What is (f−1)′(1)?
x
y
1 2 3 4 5
−1
1
2
3
4
Differentiate the function.
165. f(x) = 5 ln(x) + log2(x)
166. f(x) =√x ln(x)
167. f(s) = ln(s+√s2 − 4
)168. h(u) =
ln(u)
u+ ln(3 + 2u)
169. g(x) = ln(1 + ln(ln(x))
)170. f(x) = x2 ln(x) + arcsin(x)
171. h(t) = sin(t) arccos(t)
172. m(r) =er + ln(r2 + 1)
sin(r) + 1
173. g(s) = arcsin(ln(s))
174. k(u) = arctan(u2) + u arccot(u)
175. f(x) =csc2 x
1 + arcsec(x2)
176. h(q) = arctan(√q)− ln(arcsin(q))
177. f(x) = ln(xx)
178. f(x) = ln(ex(x2 + 1)2
xsin x
)Find an equation for the tangent line to the graph of the function at point where x = a.
179. f(x) = arcsin(2x), a = 14 .
180. f(x) =1
x√
lnx, a = e.
181. f(x) =1
(1 + lnx)2, a = 1
e2 .
182. f(x) = ln(x) arctan(x), a = 1.
183. f(x) = x2 log7(x), a = 17 .
16 MATH 140 HOMEWORK PROBLEMS
Implicit and Logarithmic Differentiation
Find y′.
184. y2 = 1 + x4
185. 2xy + 3y3 = 15− x2
186.y
1 + y2 + x4+
1
y2 + 1=
1
2
187. 5x sin(y) cos(x) + y = 1
188.x2
1 + x+ y2= 1− sin(x2y2)
189. x2 + arctan(x+ y4) = y
190. sin(x2 + 3x2y3 + 15y2) + cos(5x2 + 8xy+ 2y2) = 0
2 DERIVATIVES 17
Finddy
dx.
191. tan(x/y
)= 2x+ y
192. x2 − 2x+ y4 = ln(x+ y3)− 1
193. 2xy = cot(x/y2
)194. arctan
(x/y
)= x2 + y3
Find an equation of the tangent line to the curve at the given point.
195. xy2 + x sin(y) = π2, (1, π)
196. x2 + 4xy + y − y2 = 4, (2, 0)
197. x2 + y2 = (x2 + y2 − x)2, (0, 1)
198. sin(πy) + x2 + 1 = exy, (0, 1)
199. x2 − 4y2 = cos(πxy),(1, 12).
200. arctan(x2y) + x =y + 2
x+ 1, (1, 0)
Differentiate the function.
201. f(x) = x2x
202. f(x) =(tanx
)x2
203. f(x) =(x2 + lnx
5x4 + 1
)sin x204. f(x) = (2x2 + 1)
ln x+1ex+cos x
205. f(x) = x+ xx
206. f(x) = x2 −(cosx
)x+ xcos x
207. f(x) =(x4 + x)5(x2 + x+ 1)10(cosx+ 1)3
(x− 1)3(tan(2x) + x)5(lnx+ 1)7
208. f(x) =(x+ 1)x(x5 − 1)3ex
(x2 + 1)2(sinx)x
Higher Derivatives
Find y′′.
209. y = 5x3 − 2x+ 1
210. y = ex2
211. y = x2 sin(x) + 5x7
212. y =x+ 1
cos(x) + x
213. y = arcsin(2x) + cos(x2)
214. y = e2x + ln(x+ 1)
215. y = arctan(x+ ln(x))
216. y = xarccos x
Differentiate the function several times and identify a pattern in the resulting formulas. Then use the patternto determine the indicated derivatives.
217. f (203)(x) for f(x) = xex
218.d105f
dx105for f(x) =
1
(1 + x)2
219. f (3204)(x) for f(x) = x sin(x)
220.d46y
dx46for y = sin(x) cos(x)
18 MATH 140 HOMEWORK PROBLEMS
Findd2y
dx2.
221. x3 + 2y3 = 8xy
222.√x+ y = 1 + x2y2
223. 2 cos(x) sin(y) = 3x+ y
224. sin(xy2) = ln(x+ y)
3 Applications of the Derivative
Curve Sketching
The derivative of the function y = f(x) is given. Find the intervals on which the function is increasing ordecreasing, and find the x-coordinates of the points where local maxima or minima occur.
1. f ′(x) = (x+ 3)2(x+ 1)(x− 5)3
2. f ′(x) = (2 cos(x)− 1)(sin(x) + 1)3,0 ≤ x ≤ 2π.
3. f ′(x) = sin2(x)− 3 sin(x) + 2− cos2(x),0 ≤ x ≤ π.
4. f ′(x) = cos2(x)− sin2(x) + sin(2x),0 < x < π
2
The second derivative of the function y = f(x) is given. Find the intervals on which the function is concaveupward or downward, and find the x-coordinates of any inflection points.
5. f ′′(x) = (2x+ 1)(sec2(x)− 2),−π2 < x < π
2 .
6. f ′′(x) = ln3(x)− 2 ln2(x)− 3 ln(x)
7. f ′′(x) = (ln(x)− 1)(e2x+5 − 1)
8. f ′′(x) = 2 · 25x − 5x+1 − 3
9. The function y = f(x) is continuous on (−∞,∞), and the graph of its derivative is shown.
x
y
−2 −1 1 2 3 4 5 6
−1
1
2
y = f ′(x)
(a) Find the critical points of f .
(b) Find the intervals where f is increasing or decreasing.
(c) Find the x-coordinates of the points where f has any local maxima or minima.
(d) Find the intervals where f is concave upward or downward.
(e) Find the x-coordinates of any inflection points.
3 APPLICATIONS OF THE DERIVATIVE 19
10. Sketch the graph of a function y = f(x) that satisfies all the criteria.
• f(−3) = f(0) = 0
• limx→1−
f(x) = −∞, limx→1+
f(x) =∞
• limx→∞
f(x) = 0
• f ′(−2) = f ′(0) = 0, and f ′(3) does not exist
• f ′ > 0 on (−2, 0), and on (2, 3)
• f ′ < 0 on (−∞,−2), (0, 1), (1, 2), and (3,∞)
• f ′′ > 0 on (−∞,−1), (1, 3), and (3,∞)
• f ′′ < 0 on (−1, 1)
11. Let T (t) be the temperature at time t. In (a)–(d) below, data about the derivatives of T (t) at time t = 2are given. In each case, determine whether the temperature at time t = 2 is increasing or decreasing.Moreover, determine whether it continues to increase/decrease more rapidly or more gently (over a brieftime period).
(a) T ′(2) = 5, T ′′(2) = 3.
(b) T ′(2) = 5, T ′′(2) = −3.
(c) T ′(2) = −5, T ′′(2) = 3.
(d) T ′(2) = −5, T ′′(2) = −3.
12. Suppose that f is increasing with f ′(x) > 0 on (−∞,∞).
(a) Show that g(x) = ef(x) is increasing on (−∞,∞)
(b) Is g(x) = (f(x))2 increasing? What if f(x) > 0 for all x? Explain!
For each of the following functions, do the following:
(a) Find the domain.
(b) Find the x- and y-intercepts (if any).
(c) Determine whether the function is even, odd, or neither.
(d) Find the horizontal and vertical asymptotes (if any).
(e) Find the intervals on which the function is increasing or decreasing.
(f) Find the local maxima and minima (if any).
(g) Find the intervals of concavity.
(h) Find the inflection points (if any).
(i) Sketch a qualitative graph of the function based on the above information.
13. f(x) = x3 − 6x2 + 9x
14. f(x) = x3 − 3x2 + 3x− 9
15. f(x) = x4 − 6x2
16. f(x) = x5 − 5x4
17. f(x) =x2 + 2x+ 2
x− 1
18. f(x) =2x
x2 + 2
19. f(x) =x
4x3 + 1
20. f(x) =x2 + 2x+ 1
x2 + 2x− 15
21. f(x) = x2 +1
x2
20 MATH 140 HOMEWORK PROBLEMS
22. f(x) = 3x2/3 − 2x
23. f(x) = x(8− x)3/5
24. f(x) = (x− 3)2(x+ 1)2/3
25. f(x) =√x2 − 2x+ 2− x
26. f(x) = (x2 + 5x+ 4)4/5
27. f(x) =x+ 4√x2 + 8
28. f(x) =x− 1√x2 − 1
29. f(x) =√
3 sin(x) + cos(x)−π ≤ x ≤ π
30. f(x) = 2 cos(x) + x0 ≤ x ≤ 4π
31. f(x) = 2x− tan(x)−π2 < x < π
2
Note: You may use that the x-intercepts of f are0 and ±1.166 (approximately).
32. f(x) = 2 cos(x) + cos2(x)0 ≤ x ≤ 2π
33. f(x) = 3 sin(x)− 4 sin3(x)−π ≤ x ≤ π
34. f(x) =cos(x)
2 + sin(x)0 ≤ x ≤ 2π
35. f(x) = x− sin(x) cos(x) + 2 cos(x)0 < x < π
36. f(x) = x2 − 2 lnx
37. f(x) = ln(3− 2x− x2)
38. f(x) = ln(x4 + 3)
39. f(x) = ln(x+√x2 + 1)
40. f(x) =lnx
x2
41. f(x) =x
lnx
42. f(x) = ln(
1 +1
x
)43. f(x) = ln(1− lnx)
44. f(x) = xe−x
45. f(x) = (2x2 + x+ 1)ex
46. f(x) = e−x2
47. f(x) =e2x
1− ex
48. f(x) = sin(x)ex
−π ≤ x ≤ π
49. f(x) = e1/x
50. f(x) = x arctan(x)
51. f(x) = earctan(x)
52. f(x) = xx
ANSWERS SECTION 1 21
Answers
Note: Many of these answers were generated by a computer. The format of computer-generated answerscan be quite different from the format that you get by applying the rules of Calculus manually (the answersare equivalent, though, just like f(x) = 2x(x2 + 3)5 and f(x) = 10x3 + 30x are two equivalent mathematicalexpressions that define the same function f). When in doubt whether a stated answer here is equivalent tothe answer that you have found, ask!
Answers Section 1
1a DNE
1b 12
1c DNE
1d 32
1e 2
1f DNE
2a 1
2b 1
2c 2
2d 1
2e DNE
2f DNE
3a 1
3b 2
3c ∞
3d DNE
4a 3
4b −∞
6 4
7 132
8 4
9 7
10 524
11 38
12 2
13 152e3+2
14 −4
15 −π616 7
17 15
18 35
19 14
20 21
21 2π3
22 2
23 −1
24 − 85
25 16
26 − 49
27 118
28 − 340
29 − 150
30 − 1720
31 −√139
32 e3
33√
3
34 e−112
22 MATH 140 HOMEWORK PROBLEMS
35 − 4912
36 0
37 3110
38 − 156
39 2−ln(2)3
40 − 35
41√
2
42 4
43 9π+15
44 32
45 − 12
46 6
47 −6
48 59
49 − 59
50a 2
50b 2
50c 2
50d −2
50e 52
51a − 512
51b ln(5)
51c 14
51d 0
52 0
53 0
54 0
55 1
56 4
57 2
58 0
59 1
60 13
61 12
62 − 2π
63 0
64 3
65 6
66 14
67 − 12√2
68 314
69 56
70 10
71 12
72 − 12
73 2
74 1
75 32
76 −∞
77 ∞
78 −∞
79 ∞
80 −∞
81 −∞
82 −∞
83 −∞
84 ∞
85 −∞
86 0
ANSWERS SECTION 2 23
87 37
88 10
89 52
90 3
91 −3
92 − 23
93 − 12
94 − 94
95 1
96 15
97 −1
98 23
99 0
100 −π4
101 13
102 2
103 54
104 0
105 0
106 0
107 12
108 13
118 x = −1 and x = 1
119 x = −1
120 x = nπ, n ∈ Z, n 6= 0
121 x = −1
122 x = −π2 and x = π2
123 None
124 x = −1
125 x = −1
126 y = − 125 as x→∞, none as x→ −∞
127 y = 2 as x→∞, y = −2 as x→ −∞
128 y = π4 as x→∞, y = −π4 as x→ −∞.
129 y = 3 both as x→ ±∞
130 y = 12 as x→∞, none as x→ −∞
131 y = 1 both as x→ ±∞
132 y = 12 as x→∞, y =
√32 as x→ −∞.
133 y = 0 as x→∞, none as x→ −∞
135 a = 2
136 a = 0
137 a = − 3(π−12)5π2−9 , b = − 3π+10π2−54
5π2−9
138 a = 26405 , b = 17
81
139 b = 3
149 f(5) = 15
Answers Section 2
1 f ′(5) = 2
2 f ′(−1) = −2
3 f ′(e) = 0
4 f ′(0) = − 32√π
5 f ′(0) = 1
6 f ′(0) = 1
7 y = 73x+ 2
3
8 y = −6x− 21
9 y = − 536x+ 4
9
24 MATH 140 HOMEWORK PROBLEMS
10 y = − 625x+ 43
25
11 dfdx = − 4
x2 − 2x
12 dfdx = 2π(πx− ln 2)
13 dfdx = 2 cos(2x)
14 dfdx = π sec2(πx)
15 y′ = − 5(s2−2)(s2+2)2
16 y′ = 3u−232(3u−10)3/2
17 y′ = 15x2 − 2
18 y′ = 2√x+1
4√x+√x√x
19 y′ = 1
33√x2
20 y′ = ln(2)2x
21 y′ = cos(x)− x sin(x)
25 f ′(0) = 0
26a x = 2
26b x = 2 and x = 5.
26c x = 1
26d f ′(4) = − 13
27 f ′(x) = 33x10 − 21x6 + 24x2
28 g′(t) = 52 t−1/2 + t−4/3 − 45
2 t1/2
29 ϕ′(s) = 12s4 + 6s
30 T ′(ω) = 35156 ω
2514 − 555
14 ω2314 + 125
2 ω32 − 18
35 ω97 +
4514 ω
87 − 5ω
31 f ′(γ) = 0
32 G′(x) = π2 − 5√17
3 x83
+ 7.385 ln(15)x8.385
33 h′(u) = 5
2u83− 65
12u196
34 y′ = − 3(3x+1)2
35 u′ = − 17(24x2−5.2)(8x3−5.2x−e)2
36 f ′(x) = 5ex + 3.127x2.127
37 f ′(x) = 3ex + ln(3e)(3e)x
38 f ′(t) = 3 ln(3)3t + ln(7)tln(7)−1
39 y′ = −√74 e−2s−2 + 193
800e−s−2
40 H ′(u) = 3√
2u√2−1 + 5eue−1 − 2 ln(17)172u +
3e3u−2
41 2 cosx− 3 sec2 x
42 eu + 15u2 − sin(u)
43 ln(3)3s−1 − 5 sin s+ 3.8 csc s cot s
44 517 sec2(α)− 3
17 sec(α) tan(α)
45 5 csc2(x)+2 sec(x) tan(x)(5 cot x−2 sec x)2
46 −√32 + 1
47 55π3 − 2
48 − 74π147
49 5120
50 8 ln(2)
51 2√
3
52 − 12361
53 y = 7(x− 1) + 4
54 y = 53 (x− 1) + 3
55 y = (−1)(x− ln(3)) + 5 ln(3)− 6
56 y = −3(x− π/2)− 2
57 x = −2 and x = 1
58 (ln(5/2), 5− 5 ln(5/2))
59 x = 7π6 + (2π)n, x = 11π
6 + (2π)n, n ∈ Z
60 12
61 −13
62 2887289
63 −5π − 449
64 f ′(x) = 5x4 + 9x2 − 30x
65 F ′(u) = 20u13 + 75
2u16− 15
2√u− 5
ANSWERS SECTION 2 25
66 h′(x) = − 12
(15√x+ 4
x2
)(3xln(5) − 2xπ
)+(5x
32 − 2
x
)(2πxπ−1 − 3xln(5)−1 ln (5)
)67 Q′(v) = 1530 v5 + 1615 v4 − 20
(3√
3− 17)v3 − 3
(19√
3− 374)v2 − 10
(√3 + 17
)v − 22
√3
68 g′(q) = 3100 (10πq3 + q − 30)
2√2q2 + 1
50 (30πq2 + 1)(√
2q3 − ln(3))(10πq3 + q − 30)
69 f ′(y) = −15 y2 + yey + 52 y
32 + 1
2√y + ey − 5
70 s′(t) = (5 et + 1)(3 t2 + 1
)+ (t+ 5 et)6t+ t2 csc (t) cot (t)− 2 t csc (t)
71 a′(v) = (v tan (v) + 2 cos (v))(12 v2 − 5 cos (v)) + (4 v3 − 5 sin (v))(sec2 (v)v + tan (v)− 2 sin (v))
72 f ′(x) = (tan (x)− 3x cot (x)) · (x2 sin (x)− 2x cos (x) + ex sin (x) + ex cos (x))− (x2 cos (x)− ex sin (x)) ·(3x csc2 (x) + sec2(x)− 3 cot (x))
73 F ′(α) = 5α ln (5) sin (α) cos (α)− 5α sin2 (α) + 5α cos2 (α)
74 u′(z) = −2(5ze6 z+1 − sin (z)
)· tan (z) sec2 (z)−
(5ze6 z+1 ln (5) + 5z · 6 · e6 z+1 − cos (z)
)sec2 (z)
75 8(x+5)2
76 − (8 x7−7√6x4+90)x2
(x7−15)2
77 25 (375 x2−178 x−25)x2
(125 x3−89 x2−25 x)2 −50 x
125 x3−89 x2−25 x + 213
x+3 −2
13 x
(x+3)2
78 − 2 (12 x5−19 x3+16 x2+40)
(2 x2−5)2(3 x3−2 x+16)2
79 −( 1√
x+2)((x3−2)2−x)
2 (x+1)2(x+√x)2
+ 6 (x3−2)x2−1(x+1)2(x+
√x)− 2 ((x3−2)2−x)
(x+1)3(x+√x)
80 −( 1√
x+2)(3πx+sin(1))
(x56 +ln(17))(x+
√x)3
+ 3π
(x56 +ln(17))(x+
√x)2− 5 (3πx+sin(1))
6 (x56 +ln(17))
2(x+√x)2x
16
81 2 (3x−5 ex)7x ln(7)
(2·7x−15)2 − 3x ln(3)−5 ex2·7x−15
82 − 2xx ln(2)−x2ex−2 xex+2x
(7x−5 x2)2− 2 (7x ln(7)−10 x)(x2ex−2xx)
(7x−5 x2)3
83 ex+3 cos(x)x2+5 − 2 (ex+3 sin(x))x
(x2+5)2
84 − x sin(x)x2ex−1 −
(x2ex+2 xex)x cos(x)
(x2ex−1)2 + cos(x)x2ex−1
85 − 3 (√2+3 x2)23 x−5 ln(2) sin(x)
(23 x−5+5)2+
6 x sin(x)+(√2+3 x2) cos(x)
23 x−5+5
86 −(5 ex csc(x)− 2 x+cos(x)
x2+13 sin(x))(x3 tan(x) sec(x)+3 x2 sec(x)+5 tan(x)2+5)
(x3 sec(x)+5 tan(x))2
−5 ex csc(x) cot(x)−5 ex csc(x)− sin(x)−2
x2+13 sin(x)− (2 x+cos(x))(2 x+13 cos(x))
(x2+13 sin(x))2
x3 sec(x)+5 tan(x)
26 MATH 140 HOMEWORK PROBLEMS
87 x = −1 and x = 2
88 None
89 x = −2 and x = 0
90 x = 0, x = π6 + (2π)n, x = 5π
6 + (2π)n, n ∈ Z
91 x = π4 + nπ, n ∈ Z
92 x = 0
93 y = 4(x− 3) + (−4)
94 y = (−2)(x− 3) + (−1)
95 y = (−3e3 − 15)(x− 3) + (18− 2e3)
96 y = −π(x− 3)
97 y = −17e3−3(2e3+9)2 (x− 3)− 7
2e3+9
98 3(x2 + 5x+ 1)2(2x+ 5)
99 11(2s4 − 5s3 − 2s+ 1)10(8s3 − 15s2 − 2)
100 (3t2 − 5t)3 + t · 3(3t2 − 5t)2(6t− 5)
101 7(x2 + 5
)3(21x2 − 16x
)(π + 7x3 − 8x2
)6+ 6
(x2 + 5
)2(π + 7x3 − 8x2
)7x
102 − 2 (u+5)2(6u2+7)(2u3+7u−1)3 + 2 (u+5)
(2u3+7u−1)2
103 10(
35 (7x− 5)4
+ 3(5x4 − 2
)(x5 − 2x
)2)((7x− 5)
5+(x5 − 2x
)3+ 1)9
104 − 30 (v+1)2
((v+1)3+5)6
105 − 20 (3w2+3w−1)(w2+3w+1)3
1066 (x2+5)
2((3 x−2)7−7 x2)x
((x2+5)3+1)2 − 7 (3 (3 x−2)6−2 x)
(x2+5)3+1
107 35 x6−6 x2√5 x7−3 x2
1081√ω+2
4√ω+√ω
109 − 35 x6+2
4 (5 x7+2 x−1)(54 )
110 −2
(12 x(
17 )− 1
x23
)
39
(−7 x(
87 )+2 x(
13 ))( 12
13 )
111 −6
(9
s17
+7
)s(
19 )
7
(2 s+3 s(
67 ))4 + 1
9
(2 s+3 s(
67 ))3
s(89 )
112
2 (35 x6+3)5 x2+
√x−
(20 x+ 1√
x
)(5 x7+3 x+1)
(5 x2+√x)2
(5 x2+√x)(
23 )
6 (5 x7+3 x+1)(23 )
ANSWERS SECTION 2 27
113
72 x2−
4 x+ 1√x√
x2+√x−20
(5 x−√x2+√x)
23
84
((5 x−√x2+√x)( 1
3 )+2 x3
)( 67 )
114 −8 sin θ − 10 cos 5θ
115 −2x sin(x2)
sin (x) + 5x cos (x) + cos(x2)
cos (x) + 5 sin (x)
116 15 sin (α)2
cos (α) + 6α sin ((3α+ 1)x) + sin ((3α+ 1)α)
117 15x sin (5x)2
cos (5x) + sin (5x)3
118 60(4 t5 − 1
)2t4 sin
(√t)
+(4 t5−1)
3cos(√t)
2√t
119
(1√x+2)cos(x+
√x)
2 (tan(x2)+1) − 2 sec2(x2)x sin(x+√x)
(tan(x2)+1)2
120 0
121 5 ex tan(5 ex) sec(5 ex)+1cos(x)+csc(x2+1) +
(x+sec(5 ex))(2 x csc(x2+1) cot(x2+1)+sin(x))(cos(x)+csc(x2+1))2
1225 sec2(5x)(x2+1)+2 x tan(5 x)
2√
(x2+1) tan(5 x)
123
θ2(θ2+2
) 32
− 1√θ2+2
tan
(θ + 2
θ2
)csc
θ√θ2+2
2
−(tan
(θ + 2
θ2
)2+ 1
)(4θ3− 1
)cot
θ√θ2+2
124 −2 sec2(2r) cos (csc (tan (2 r))) tan (sin (csc (tan (2 r)))) sec (sin (csc (tan (2 r)))) csc (tan (2 r)) cot (tan (2 r))
125 ex2+12x
126 e3 sin(5x)+115 cos(5x)
127 −(csc (x) cot (x)− 1)(x2 + 1
)3e(x+csc(x)) + 6
(x2 + 1
)2xe(x+csc(x))
128 −2x5(−x2+1) ln (5) sin (x) + 3(x−1) ln (3) + 5(−x2+1) cos (x)
129 sec2(x3(2 x+1)
) (3(2 x+1) + 2x3(2 x+1) ln (3)
)
130
3 esin(x) cos(x)−2 (x sin(x2+1)−7(2 x+1) ln(7))
(7(2 x+1)+cos(x2+1))23
6
√(7(2 x+1)+cos(x2+1))(
13 )+esin(x)
131 e
(x+2 cos(x)
sin(2x)ex+1
)(− 2 sin(x)−1ex sin(2 x)+1 −
(x+2 cos(x))(ex sin(2 x)+2 ex cos(2 x))
(ex sin(2 x)+1)2
)132 y = 24x+ 1
133 y =(√
3− π3
)(x− 1
)+√32
134 y = −(x− π)
135 y = 2x
136 x = −3 and x = 3
137 x = −√
6, x = 0, x =√
6
28 MATH 140 HOMEWORK PROBLEMS
138 x = aa+b
139 x = nπ, x = π3 + (2π)n, x = −π3 + (2π)n, n ∈ Z
140 x = nπ, x = π3 + (2π)n, x = −π3 + (2π)n, n ∈ Z
141 x = (2π)n, x = 2π3 + (2π)n, x = − 2π
3 + (2π)n,n ∈ Z
142 h′(x) = 2x2f ′(x2) + f(x2)
143 h′(x) = 6xg′(3x2 + 1) + f ′(1− x)
144 h′(x) = cos(f(x) + x)(f ′(x) + 1)
145 h′(x) = −2x sin(x2)f (3x) · g′
(cos(x2)
+ 1)
+
3 g(cos(x2)
+ 1)· f ′ (3x)
146 h′(x) =
√f (x)
2+ exexg′ (ex + 1) +
(2 f(x)f ′(x)+ex)g(ex+1)
2√f(x)2+ex
147 h′(x) = −(
2 xg′(2 x−1)g(2 x−1)2 −
1g(2 x−1)
)· f ′(
xg(2 x−1)
)148 h′(x) = (2x+ f ′ (x))e(x
2+f(x))
149 h′(x) = 2 · 5(2 f(x)) ln (5) g (x) f ′ (x) +5(2 f(x))g′ (x)
150 4
151 10
152 f−1(x) = 3−√4x−112 , −∞ < x < 11
4
153 f−1(x) = x√x2−4 , −∞ < x < −2
154 f−1(x) = ln(x+√x2 + 1
), −∞ < x <∞
155 f−1(x) = ln(
x2−x
), 0 < x < 2
160 (f−1)′(0) = 1
161 (f−1)′(√π) = 2
√π
2√π+1
162 (f−1)′(7) = 17
163 (f−1)′(√
3) = 19 and (f−1)′(3) =
√39
164 23
165 f ′(x) = 5x + 1
x ln(2)
166 f ′(x) = ln x+22√x
167 f ′(s) =1+ s√
s2−4
s+√s2−4
168 h′(u) =(u+ln(3+2u)) 1
u−ln(u)(
22u+3+1
)(u+ln(3+2u))2
169 g′(x) = 1(1+ln(ln(x))) ln(x)x
170 f ′(x) = 2x ln(x) + x+ 1√1−x2
171 h′(t) = cos(t) arccos(t)− sin(t)√1−t2
172 m′(r) =2 rr2+1
+er
sin(r)+1 −(er+ln(r2+1)) cos(r)
(sin(r)+1)2
173 g′(s) = 1
s√
1−ln2(s)
174 k′(u) = − uu2+1 + 2u
u4+1 + arccot (u)
175 f ′(x) = − 2 csc2(x) cot(x)arcsec(x2)+1 −
2x csc2(x)
(arcsec(x2)+1)2x2√x4−1
176 h′(q) = 12 (q+1)
√q −
1
arcsin(q)√
1−q2
177 f ′(x) = ln(x) + 1
178 f ′(x) = − ln (x) cos (x) + 4 xx2+1 −
sin(x)x + 1
179 y = 4√3
(x− 1
4
)+ π
6
180 y = − 32e2 (x− e) + 1
e
181 y = 2e2(x− 1
e2
)+ 1
182 y = π4 (x− 1)
183 y = − 2 ln(7)−17 ln(7)
(x− 1
7
)− 1
49
184 2 x3
y
185 − 2 (x+y)9 y2+2 x
186 − 4 x3y
(x4+y2+1)2(
2 y2
(x4+y2+1)2+ 2 y
(y2+1)2− 1x4+y2+1
)
ANSWERS SECTION 2 29
187 5 (x sin(x) sin(y)−sin(y) cos(x))5 x cos(x) cos(y)+1
188 −2 xy2 cos(x2y2)+ 2 x
y2+x+1− x2
(y2+x+1)2
2
(x2y cos(x2y2)− x2y
(y2+x+1)2
)
189 −2 x+ 1
(y4+x)2+1
4 y3
(y4+x)2+1−1
190 − 2 ((5 x+4 y) sin(5 x2+8 xy+2 y2)−(3 xy3+x) cos(3 x2y3+x2+15 y2))4 (2 x+y) sin(5 x2+8 xy+2 y2)−3 (3 x2y2+10 y) cos(3 x2y3+x2+15 y2)
191sec2(x/y)
y −2sec2(x/y)x
y2+1
192 −2 x− 1
y3+x−2
4 y3− 3 y2
y3+x
193 −2 y+
csc
(xy2
)2y2
2
x− x csc
(xy2
)2y3
194 −2 x− 1(
x2
y2+1
)y
3 y2+ x(x2
y2+1
)y2
195 y = − π2
2π−1 (x− 1) + π
196 y = − 49 (x− 2)
197 y = x+ 1
198 y = − 1πx+ 1
199 y = 4+π2(4−π) (x− 1) + 1
2
200 y = −3(x− 1)
201 f ′(x) = 2(ln(x) + 1)x2x
202 f ′(x) =(
sec2(x)x2
tan(x) + 2x ln(tan(x)))(
tanx)x2
203 f ′(x) = −
(5 x4+1)
(20 (x2+ln(x))x3
(5 x4+1)2−
2 x+ 1x
5 x4+1
)sin(x)
x2+ln(x) − ln(x2+ln(x)5 x4+1
)cos (x)
(x2+ln(x)5 x4+1
)sin(x)
204 f ′(x) = −((
(ln(x)+1)(ex−sin(x))(ex+cos(x))2
− 1(ex+cos(x))x
)ln(2x2 + 1
)− 4 (ln(x)+1)x
(2 x2+1)(ex+cos(x))
)·(2x2 + 1
)( ln(x)+1ex+cos(x) )
205 f ′(x) = 1 + (ln(x) + 1)xx
206 f ′(x) =(x sin(x)cos(x) − ln (cos (x))
)cos (x)
x −(
ln (x) sin (x)− cos(x)x
)xcos(x) + 2x
207 f ′(x) =(x4 + x)5(x2 + x+ 1)10(cosx+ 1)3
(x− 1)3(tan(2x) + x)5(lnx+ 1)7
(− 3 sin(x)
cos(x)+1 + 10 (2 x+1)x2+x+1 −
5 (2 sec(2 x)2+1)x+tan(2 x) +
5 (4 x3+1)x4+x − 3
x−1 −
7(ln(x)+1)x
)208 f ′(x) =
(x+ 1)x(x5 − 1)3ex
(x2 + 1)2(sinx)x
(15 x4
x5−1 −x cos(x)sin(x) + x
x+1 −4 xx2+1 + ln (x+ 1)−
ln (sin (x)) + 1)
209 30x
210 4x2e(x2) + 2 e(x
2)
30 MATH 140 HOMEWORK PROBLEMS
211 210x5 − x2 sin (x) + 4x cos (x) + 2 sin (x)
212 2 (sin(x)−1)2(x+1)
(x+cos(x))3+ (x+1) cos(x)
(x+cos(x))2+ 2 (sin(x)−1)
(x+cos(x))2
213 −4x2 cos(x2)
+ 8 x
(−4 x2+1)32− 2 sin
(x2)
214 − 1(x+1)2
+ 4 e(2 x)
215 − 2 ( 1x+1)
2(x+ln(x))
((x+ln(x))2+1)2 − 1
((x+ln(x))2+1)x2
216(
ln(x)√−x2+1
− arccos(x)x
)2xarccos(x) −
(x ln(x)
(−x2+1)32
+ arccos(x)x2 + 2√
−x2+1x
)xarccos(x)
217 f (203)(x) = (x+ 203)ex
218 d105fdx105 = − 2·3·...·105·106
(x+1)107
219 f (3204)(x) = x sin(x)− 3204 cos(x)
220d46y
dx46= −246 sin(x) cos(x)
221 −3 (3 x2−8 y)
2y
(3 y2−4 x)2+6 x+
8 (3 x2−8 y)3 y2−4 x
2 (3 y2−4 x)
222 −8 (x+y)(
32 )y2+
(4√x+yxy2−1)
28 (x+y)
( 32 )x2+1
(4√x+yx2y−1)2
−2 (4√x+yxy2−1)
16 (x+y)( 3
2 )xy+1
4√x+yx2y−1
+1
2
(4 (x+y)(
32 )x2y−x−y
)
2232
(2 (2 sin(x) sin(y)+3) sin(x) cos(y)
2 cos(x) cos(y)−1+
(2 sin(x) sin(y)+3)2 sin(y) cos(x)
(2 cos(x) cos(y)−1)2+sin(y) cos(x)
)2 cos(x) cos(y)−1
224(
2
(2 ((xy2+y3) cos(xy2)−1)(x2y+2 xy2+y3)
2 (x2y+xy2) cos(xy2)−1 − ((xy2+y3) cos(xy2)−1)2(x3+2 x2y+xy2)
(2 (x2y+xy2) cos(xy2)−1)2
)cos(xy2)
+(x2y4 + 2xy5 + y6 − 4 ((xy2+y3) cos(xy2)−1)(x3y3+2 x2y4+xy5)
2 (x2y+xy2) cos(xy2)−1 +4 ((xy2+y3) cos(xy2)−1)
2(x4y2+2 x3y3+x2y4)
(2 (x2y+xy2) cos(xy2)−1)2
)sin(xy2)
+2 ((xy2+y3) cos(xy2)−1)2 (x2y+xy2) cos(xy2)−1 −
((xy2+y3) cos(xy2)−1)2
(2 (x2y+xy2) cos(xy2)−1)2 − 1)(
2(x3y + 2x2y2 + xy3
)cos(xy2)− x− y
)−1Answers Section 3
1 Increasing on: (−∞,−1), (5,∞)Decreasing on: (−1, 5)Local maxima: f(−1)Local minima: f(5)
2 Increasing on: [0, π/3), (5π/3, 2π]Decreasing on: (π/3, 5π/3)Local maxima: f(π/3)
Local minima: f(5π/3)
3 Increasing on: [0, π/6), (5π/6, π]Decreasing on: (π/6, 5π/6)Local maxima: f(π/6)Local minima: f(5π/6)
4 Increasing on: (0, 3π/8)
ANSWERS SECTION 3 31
Decreasing on: (3π/8, π/2)Local maxima: f(3π/8)Local minima: None
5 Concave up on: (−π/4,−1/2), (π/4, π/2)Concave down on: (−π/2,−π/4), (−1/2, π/4)Inflection point at: x = ±π/4, x = −1/2
6 Concave up on: (e−1, 1), (e3,∞)Concave down on: (0, e−1), (1, e3)Inflection point at: x = e−1, x = 1, x = e3
7 Concave up on: (e,∞)Concave down on: (0, e)Inflection point at: x = e
8 Concave up on: (log5(3),∞)Concave down on: (−∞, log5(3))Inflection point at: x = log5(3)
9a x = −2, x = 0, x = 1, x = 2, x = 4
9b Increasing on: (−∞,−2), (0, 1), (2, 4)Decreasing on: (−2, 0), (1, 2), (4,∞)
9c Local maxima at x = −2, x = 1, and x = 4, localminima at x = 0 and x = 2
9d Concave up on (−1, 1), (1, 3)Concave down on (−∞,−1), (3,∞)
9e x = −1, x = 3
10 There are infinitely many correct graphs. Here isone:
x
y
−4 −3 −2 −1 1 2 3 4 5 6
−3
−2
−1
1
2
3
4
11a Increases rapidly
11b Increases gently
11c Decreases gently
11d Decreases rapidly
13 Domain: (−∞,∞)
f(x) = (x− 3)2x
f ′(x) = 3 (x− 3)(x− 1)
f ′′(x) = 6x− 12
y-intercept: (0, 0)
x-intercept(s): (0, 0), (3, 0)
Symmetry: None
Vertical asymptotes: None
Horizontal asymptotes: None
Increasing on: (−∞, 1), (3,∞)
Decreasing on: (1, 3)
Local maxima: f(1) = 4
Local minima: f(3) = 0
Concave up on: (2,∞)
Concave down on: (−∞, 2)
Inflection points: (2, 2)
x
y
1
1
2
2
3
3
4
4
14 Domain: (−∞,∞)
f(x) = (x− 3)(x2 + 3
)f ′(x) = 3 (x− 1)
2
f ′′(x) = 6x− 6
y-intercept: (0,−9)
x-intercept(s): (3, 0)
Symmetry: None.
Vertical asymptotes: None
Horizontal asymptotes: None
Increasing on: (−∞,∞)
Decreasing on: Nowhere
Local maxima: None
Local minima: None
Concave up on: (1,∞)
Concave down on: (−∞, 1)
Inflection points: (1,−8)
32 MATH 140 HOMEWORK PROBLEMS
x
y
1 2 3
−10
−8
−6
−4
−2
2
15 Domain: (−∞,∞)
f(x) =(x2 − 6
)x2
f ′(x) = 4(x2 − 3
)x
f ′′(x) = 12 (x− 1)(x+ 1)
y-intercept: (0, 0)x-intercept(s): (±
√6, 0), (0, 0)
Symmetry: EvenVertical asymptotes: NoneHorizontal asymptotes: NoneIncreasing on: (−
√3, 0), (
√3,∞)
Decreasing on: (−∞,−√
3), (0,√
3)Local maxima: f(0) = 0Local minima: f(−
√3) = −9, f(
√3) = −9
Concave up on: (−∞,−1), (1,∞)Concave down on: (−1, 1)Inflection points: (−1,−5), (1,−5)
x
y
−3 −2 −1 1 2 3
−9
−7
−5
−3
−1
1
3
16 Domain: (−∞,∞)
f(x) = (x− 5)x4
f ′(x) = 5 (x− 4)x3
f ′′(x) = 20 (x− 3)x2
y-intercept: (0, 0)x-intercept(s): (0, 0), (5, 0)Symmetry: NoneVertical asymptotes: NoneHorizontal asymptotes: NoneIncreasing on: (−∞, 0), (4,∞)Decreasing on: (0, 4)Local maxima: f(0) = 0Local minima: f(4) = −256Concave up on: (3,∞)Concave down on: (−∞, 3)Inflection points: (3,−162)
x
y
−3−2−1 1 2 3 4 5
−250
−200
−150
−100
−50
50
17 Domain: (−∞, 1) ∪ (1,∞)
f(x) =x2 + 2x+ 2
x− 1
f ′(x) =x2 − 2x− 4
(x− 1)2
f ′′(x) =10
(x− 1)3
y-intercept: (0,−2)x-intercept(s): NoneSymmetry: NoneVertical asymptotes: x = 1Horizontal asymptotes: NoneIncreasing on: (−∞, 1−
√5), (1 +
√5,∞)
Decreasing on: (1−√
5, 1), (1, 1 +√
5)Local maxima: f(1−
√5) = 4− 2
√5
Local minima: f(1 +√
5) = 4 + 2√
5Concave up on: (1,∞)Concave down on: (−∞, 1)Inflection points: None
ANSWERS SECTION 3 33
x
y
−8 −6 −4 −2 2 4 6 8
−12
−9
−6
−3
3
6
9
12
15
18
18 Domain: (−∞,∞)
f(x) =2x
x2 + 2
f ′(x) = −2(x2 − 2
)(x2 + 2)
2
f ′′(x) =4(x2 − 6
)x
(x2 + 2)3
y-intercept: (0, 0)
x-intercept(s): (0, 0)
Symmetry: Odd
Vertical asymptotes: None
Horizontal asymptotes: y = 0 both as x→ ±∞Increasing on: (−
√2,√
2)
Decreasing on: (−∞,−√
2), (√
2,∞)
Local maxima: f(√
2) =√22
Local minima: f(−√
2) = −√22
Concave up on: (−√
6, 0), (√
6,∞)
Concave down on: (−∞,−√
6), (0,√
6)
Inflection points: (−√
6,−√64 ), (0, 0), (
√6,√64 )
x
y
−10−8−6−4−2 2 4 6 8 10
−1
1
19 Domain: (−∞,− 13√4
) ∪ (− 13√4,∞)
f(x) =x
4x3 + 1
f ′(x) = −(2x− 1)
(4x2 + 2x+ 1
)(4x3 + 1)
2
f ′′(x) =48(2x3 − 1
)x2
(4x3 + 1)3
y-intercept: (0, 0)x-intercept(s): (0, 0)Symmetry: NoneVertical asymptotes: x = − 1
3√4
Horizontal asymptotes: y = 0 both as x→ ±∞Increasing on: (−∞,− 1
3√4), (− 1
3√4, 12 )
Decreasing on: ( 12 ,∞)
Local maxima: f(1/2) = 1/3Local minima: NoneConcave up on: (−∞,− 1
3√4), ( 1
3√2,∞)
Concave down on: (− 13√4, 1
3√2)
Inflection points: ( 13√2, 13 3√2
)
x
y
−4 −3 −2 −1 1 2 3 4
−2
−1
1
2
20 Domain: (−∞,−5) ∪ (−5, 3) ∪ (3,∞)
f(x) =(x+ 1)
2
(x− 3)(x+ 5)
f ′(x) = − 32 (x+ 1)
(x− 3)2(x+ 5)
2
f ′′(x) =32(3x2 + 6x+ 19
)(x− 3)
3(x+ 5)
3
y-intercept: (0,− 115 )
x-intercept(s): (−1, 0)Symmetry: NoneVertical asymptotes: x = −5 and x = 3
34 MATH 140 HOMEWORK PROBLEMS
Horizontal asymptotes: y = 1 both as x→ ±∞Increasing on: (−∞,−5), (−5,−1)
Decreasing on: (−1, 3), (3,∞)
Local maxima: f(−1) = 0
Local minima: None
Concave up on: (−∞,−5), (3,∞)
Concave down on: (−5, 3)
Inflection points: None
x
y
−10 −8 −6 −4 −2 2 4 6 8 10
−10
−8
−6
−4
−2
2
4
6
8
10
21 Domain: (−∞, 0) ∪ (0,∞)
f(x) =x4 + 1
x2
f ′(x) =2 (x− 1)(x+ 1)
(x2 + 1
)x3
f ′′(x) =2(x4 + 3
)x4
y-intercept: None
x-intercept(s): None
Symmetry: Even
Vertical asymptotes: x = 0
Horizontal asymptotes: None
Increasing on: (−1, 0), (1,∞)
Decreasing on: (−∞,−1), (0, 1)
Local maxima: None
Local minima: f(−1) = 2, f(1) = 2
Concave up on: (−∞, 0), (0,∞)
Concave down on: Nowhere
Inflection points: None
x
y
−3 −2 −1 1 2 3
1
3
5
7
9
11
22 Domain: (−∞,∞)
f(x) = −2x+ 3x( 23 )
f ′(x) = −2(x( 1
3 ) − 1)
x13
f ′′(x) = − 2
3x( 43 )
y-intercept: (0, 0)x-intercept(s): (0, 0), (27/8, 0)Symmetry: NoneVertical asymptotes: NoneHorizontal asymptotes: NoneIncreasing on: (0, 1)Decreasing on: (−∞, 0), (1,∞)Local maxima: f(1) = 1Local minima: f(0) = 0Concave up on: NowhereConcave down on: (−∞, 0), (0,∞)Inflection points: None
x
y
−2 −1 1 2 3 4 5
13579
11
23 Domain: (−∞,∞)
f(x) = (−x+ 8)(35 )x
f ′(x) = − 8 (x− 5)
5 (−x+ 8)(25 )
f ′′(x) = − 24 (x− 10)
25 (−x+ 8)(25 )(x− 8)
y-intercept: (0, 0)x-intercept(s): (0, 0), (8, 0)
ANSWERS SECTION 3 35
Symmetry: None
Vertical asymptotes: None
Horizontal asymptotes: None
Increasing on: (−∞, 5)
Decreasing on: (5,∞)
Local maxima: f(5) = 5 · 33/5Local minima: None
Concave up on: (8, 10)
Concave down on: (−∞, 8), (10,∞)
Inflection points: (8, 0), (10,−10 · 23/5)
x
y
2 4 6 8 10
−15
−12
−9
−6
−3
3
6
9
12
24 Domain: (−∞,∞)
f(x) = (x− 3)2(x+ 1)(
23 )
f ′(x) =8 (x− 3)x
3 (x+ 1)(13 )
f ′′(x) =8(5x2 − 9
)9 (x+ 1)(
43 )
y-intercept: (0, 9)
x-intercept(s): (−1, 0) and (3, 0)
Symmetry: None
Vertical asymptotes: None
Horizontal asymptotes: None
Increasing on: (−1, 0), (3,∞)
Decreasing on: (−∞,−1), (0, 3)
Local maxima: f(0) = 9
Local minima: f(−1) = 0, f(3) = 0
Concave up on: (−∞,− 3√5
5 ), ( 3√5
5 ,∞)
Concave down on: (− 3√5
5 ,−1), (−1, 3√5
5 )
Inflection points: (− 3√5
5 , ( 5−3√5
5 )2/3 725 ),
( 3√5
5 , ( 5+3√5
5 )2/3 365 )
x
y
−2 −1 1 2 3 4 5
1
3
5
7
9
11
25 Domain: (−∞,∞)
f(x) = −x+√x2 − 2x+ 2
f ′(x) =x−√x2 − 2x+ 2− 1√x2 − 2x+ 2
f ′′(x) =1
(x2 − 2x+ 2)32
y-intercept: (0,√
2)x-intercept(s): (1, 0)Symmetry: NoneVertical asymptotes: NoneHorizontal asymptotes: y = −1 as x→∞Increasing on: NowhereDecreasing on: (−∞,∞)Local maxima: NoneLocal minima: NoneConcave up on: (−∞,∞)Concave down on: NowhereInflection points: None
x
y
−2 −1 1 2 3 4 5−1
1
3
5
26 Domain: (−∞,∞)
f(x) =(x2 + 5x+ 4
) 45
f ′(x) =4 (2x+ 5)
5 (x2 + 5x+ 4)(15 )
f ′′(x) =12(2x2 + 10x+ 5
)25 (x2 + 5x+ 4)(
65 )
36 MATH 140 HOMEWORK PROBLEMS
y-intercept: (0, 44/5)
x-intercept(s): (−4, 0), (−1, 0)
Symmetry: None
Vertical asymptotes: None
Horizontal asymptotes: None
Increasing on: (−4,−5/2), (−1,∞)
Decreasing on: (−∞,−4), (−5/2,−1)
Local maxima: f(−5/2) = (9/4)4/5
Local minima: f(−4) = 0, f(−1) = 0
Concave up on: (−∞, −5−√15
2 ), (−5+√15
2 ,∞)
Concave down on: (−5−√15
2 ,−4), (−4,−1),
(−1, −5+√15
2 )
Inflection points: (−5−√15
2 , (3/2)4/5), (−5+√15
2 , (3/2)4/5)
x
y
−6 −5 −4 −3 −2 −1 1
2
4
6
27 Domain: (−∞,∞)
f(x) =x+ 4√x2 + 8
f ′(x) = − 4 (x− 2)
(x2 + 8)32
f ′′(x) =8 (x− 4)(x+ 1)
(x2 + 8)52
y-intercept: (0,√
2)
x-intercept(s): (−4, 0)
Symmetry: None
Vertical asymptotes: None
Horizontal asymptotes: y = 1 as x → ∞, y = −1 asx→ −∞Increasing on: (−∞, 2)
Decreasing on: (2,∞)
Local maxima: f(2) =√
3
Local minima: None
Concave up on: (−∞,−1), (4,∞)
Concave down on: (−1, 4)
Inflection points: (−1, 1), (4, 2√6
3 )
x
y
−20 −15 −10 −5 5 10 15 20
−1
1
2
28 Domain: (−∞,−1) ∪ (1,∞)
f(x) =x− 1√x2 − 1
f ′(x) =1
(x+ 1)√x2 − 1
f ′′(x) = − 2x− 1
(x− 1)(x+ 1)2√x2 − 1
y-intercept: Nonex-intercept(s): NoneSymmetry: NoneVertical asymptotes: x = −1Horizontal asymptotes: y = 1 as x → ∞, y = −1 asx→ −∞Increasing on: (1,∞)Decreasing on: (−∞,−1)Local maxima: NoneLocal minima: NoneConcave up on: NowhereConcave down on: (−∞,−1), (1,∞)Inflection points: None
x
y
−5 −3 −1 1 3 5
−3
−1
1
29 Domain: [−π, π]
f(x) =√
3 sin (x) + cos (x)
f ′(x) =√
3 cos (x)− sin (x)
f ′′(x) = −√
3 sin (x)− cos (x)
y-intercept: (0, 1)x-intercept(s): (−π6 , 0), ( 5π
6 , 0)Symmetry: None
ANSWERS SECTION 3 37
Vertical asymptotes: NoneHorizontal asymptotes: NoneIncreasing on: (− 2π
3 ,π3 )
Decreasing on: [−π,− 2π3 ), (π3 , π]
Local maxima: f(−π) = −1, f(π/3) = 2Local minima: f(−2π/3) = −2, f(π) = −1Concave up on: [−π,−π6 ), ( 5π
6 , π]Concave down on: (−π6 ,
5π6 )
Inflection points: (−π6 , 0), ( 5π6 , 0)
x
y
−π− 2π3−π3
π3
2π3
π
−2
−1
1
2
30 Domain: [0, 4π]
f(x) = x+ 2 cos (x)
f ′(x) = −2 sin (x) + 1
f ′′(x) = −2 cos (x)
y-intercept: (0, 2)x-intercept(s): NoneSymmetry: NoneVertical asymptotes: NoneHorizontal asymptotes: NoneIncreasing on: [0, π/6), (5π/6, 13π/6), (17π, 6, 4π]Decreasing on: (π/6, 5π/6), (13π/6, 17π/6)Local maxima: f(π/6) = π/6 +
√3, f(13π/6) =
13π/6 +√
3, f(4π) = 4π + 2Local minima: f(0) = 2, f(5π/6) = 5π/6 −
√3,
f(17π/6) = 17π/6−√
3Concave up on: (π/2, 3π/2), (5π/2, 7π/2)Concave down on: [0, π/2), (3π/2, 5π/2), (7π/2, 4π]Inflection points: (π/2, π/2), (3π/2, 3π/2),(5π/2, 5π/2), (7π/2, 7π/2)
x
y
π3
2π3
π 4π3
5π3
2π 7π3
8π3
3π 10π3
11π3
4π
2
4
6
8
10
12
14
31 Domain: (−π2 ,π2 )
f(x) = 2x− tan (x)
f ′(x) = −(tan (x)− 1)(tan (x) + 1)
f ′′(x) = −2 sec (x)2
tan (x)
y-intercept: (0, 0)x-intercept(s): (0, 0), (±1.166, 0)Symmetry: NoneVertical asymptotes: x = ±π2Horizontal asymptotes: NoneIncreasing on: (−π4 ,
π4 )
Decreasing on: (−π2 ,−π4 ), (π4 ,
π2 )
Local maxima: f(π/4) = π/2− 1Local minima: f(−π/4) = 1− π/2Concave up on: (−π2 , 0)Concave down on: (0, π2 )Inflection points: (0, 0)
x
y
−π2 −π4π4
π2
−6
−4
−2
2
4
6
32 Domain: [0, 2π]
f(x) = (cos (x) + 2) cos (x)
f ′(x) = −2 (cos (x) + 1) sin (x)
f ′′(x) = 2 sin (x)2 − 2 cos (x)
2 − 2 cos (x)
y-intercept: (0, 3)x-intercept(s): (π/2, 0), (3π/2, 0)Symmetry: NoneVertical asymptotes: NoneHorizontal asymptotes: NoneIncreasing on: (π, 2π]Decreasing on: [0, π)Local maxima: f(0) = 3, f(2π) = 3Local minima: f(π) = −1
38 MATH 140 HOMEWORK PROBLEMS
Concave up on: (π/3, 5π/3)
Concave down on: [0, π/3), (5π3, 2π)Inflection points: (π/3, 5/4), (5π/3, 5/4)
x
y
π3
2π3
π 4π3
5π3
2π−1
1
2
3
33 Domain: [−π, π]
f(x) = −(
4 sin (x)2 − 3
)sin (x)
f ′(x) = −3 (2 sin (x)− 1)(2 sin (x) + 1)
· cos (x)
f ′′(x) = 3(
4 sin (x)2 − 8 cos (x)
2 − 1)
· sin (x)
y-intercept: (0, 0)
x-intercept(s): (−π, 0), (−2π/3, 0), (−π/3, 0), (0, 0),(π/3, 0), (2π/3, 0), (π, 0)
Symmetry: Odd
Vertical asymptotes: None
Horizontal asymptotes: None
Increasing on: (−5π/6,−π/2), (−π/6, π/6),(π/2, 5π/6)
Decreasing on: [−π,−5π/6), (−π/2,−π/6),(π/6, π/2), (5π/6, π]
Local maxima: f(−π) = 0, f(−π/2) = f(π/6) =f(5π/6) = 1
Local minima: f(−5π/6) = f(−π/6) = f(π/2) =−1, f(π) = 0
Concave up on: [−π,−2π/3), (−π/3, 0), (π/3, 2π/3)
Concave down on: (−2π/3,−π/3), (0, π/3), (2π/3, π]
Inflection points: (−2π/3, 0), (−π/3, 0), (0, 0),(π/3, 0), (2π/3, 0)
x
y
−π − 2π3−π3
π3
2π3
π
−1
1
Note: In fact, f(x) = sin(3x), which – if recognized atthe beginning – substantially simplifies the graphingdiscussion for this function.
34 Domain: [0, 2π]
f(x) =cos (x)
sin (x) + 2
f ′(x) = − 2 sin (x) + 1
(sin (x) + 2)2
f ′′(x) =2 (sin (x)− 1) cos (x)
(sin (x) + 2)3
y-intercept: (0, 1/2)x-intercept(s): (π/2, 0), (3π/2, 0)Symmetry: NoneVertical asymptotes: NoneHorizontal asymptotes: NoneIncreasing on: (7π/6, 11π/6)Decreasing on: [0, 7π/6), (11π/6, 2π]Local maxima: f(0) = 1/2, f(11π/6) =
√3/3
Local minima: f(7π/6) = −√
3/3, f(2π) = 1/2Concave up on: (π/2, 3π/2)Concave down on: [0, π/2), (3π/2, 2π]Inflection points: (π/2, 0), (3π/2, 0)
x
y
π3
2π3
π 4π3
5π3
2π
−1
1
35 Domain: (0, π)
f(x) = − sin (x) cos (x) + x+ 2 cos (x)
f ′(x) = sin (x)2 − cos (x)
2
− 2 sin (x) + 1
f ′′(x) = 4 sin (x) cos (x)− 2 cos (x)
y-intercept: Nonex-intercept(s): NoneSymmetry: NoneVertical asymptotes: NoneHorizontal asymptotes: NoneIncreasing on: NowhereDecreasing on: (0, π)Local maxima: NoneLocal minima: NoneConcave up on: (π/6, π/2), (5π/6, π)Concave down on: (0, π/6), (π/2, 5π/6)
Inflection points: (π/6, π/6 + 3√3
4 ), (π/2, π/2),
(5π/6, 5π/6− 3√3
4 )
ANSWERS SECTION 3 39
x
y
π3
2π3
π
1
2
36 Domain: (0,∞)
f(x) = x2 − 2 ln (x)
f ′(x) = 2x− 2
x
f ′′(x) =2
x2+ 2
y-intercept: Nonex-intercept(s): NoneSymmetry: NoneVertical asymptotes: x = 0Horizontal asymptotes: NoneIncreasing on: (1,∞)Decreasing on: (0, 1)Local maxima: NoneLocal minima: f(1) = 1Concave up on: (0,∞)Concave down on: NowhereInflection points: None
x
y
1 2 3
1
3
5
7
37 Domain: (−3, 1)
f(x) = ln(−x2 − 2x+ 3
)f ′(x) =
2 (x+ 1)
(x− 1)(x+ 3)
f ′′(x) = −2(x2 + 2x+ 5
)(x− 1)
2(x+ 3)
2
y-intercept: (0, ln(3))x-intercept(s): (−1−
√3, 0), (
√3− 1, 0)
Symmetry: None
Vertical asymptotes: x = −3, x = 1
Horizontal asymptotes: None
Increasing on: (−3,−1)
Decreasing on: (−1, 1)
Local maxima: f(−1) = ln(4)
Local minima: None
Concave up on: Nowhere
Concave down on: (−3, 1)
Inflection points: None
x
y
−3 −2 −1 1
−6
−5
−4
−3
−2
−1
1
2
38 Domain: (−∞,∞)
f(x) = ln(x4 + 3
)f ′(x) =
4x3
x4 + 3
f ′′(x) = −4(x2 − 3
)(x2 + 3
)x2
(x4 + 3)2
y-intercept: (0, ln(3))
x-intercept(s): None
Symmetry: Even
Vertical asymptotes: None
Horizontal asymptotes: None
Increasing on: (0,∞)
Decreasing on: (−∞, 0)
Local maxima: None
Local minima: f(0) = ln(3)
Concave up on: (−√
3,√
3)
Concave down on: (−∞,−√
3), (√
3,∞)
Inflection points: (−√
3, ln(12)), (√
3, ln(12))
40 MATH 140 HOMEWORK PROBLEMS
x
y
−3 −2 −1 1 2 3
1
2
3
4
39 Domain: (−∞,∞)
f(x) = ln(x+
√x2 + 1
)f ′(x) =
1√x2 + 1
f ′′(x) = − x
(x2 + 1)32
y-intercept: (0, 0)x-intercept(s): (0, 0)Symmetry: OddVertical asymptotes: NoneHorizontal asymptotes: NoneIncreasing on: (−∞,∞)Decreasing on: NowhereLocal maxima: NoneLocal minima: NoneConcave up on: (−∞, 0)Concave down on: (0,∞)Inflection points: (0, 0)
x
y
−3 −2 −1 1 2 3
−2
−1
1
2
40 Domain: (0,∞)
f(x) =ln (x)
x2
f ′(x) = −2 ln (x)− 1
x3
f ′′(x) =6 ln (x)− 5
x4
y-intercept: Nonex-intercept(s): (1, 0)
Symmetry: None
Vertical asymptotes: x = 0
Horizontal asymptotes: y = 0 as x→∞Increasing on: (0,
√e)
Decreasing on: (√e,∞)
Local maxima: f(√e) = 1/(2e)
Local minima: None
Concave up on: (e5/6,∞)
Concave down on: (0, e5/6)
Inflection points: (e5/6, 56e−5/3)
x
y
1 2 3
−11
−9
−7
−5
−3
−1
1
41 Domain: (0,∞)
f(x) =x
ln (x)
f ′(x) =ln (x)− 1
ln (x)2
f ′′(x) = − ln (x)− 2
x ln (x)3
y-intercept: None
x-intercept(s): None
Symmetry: None
Vertical asymptotes: x = 1
Horizontal asymptotes: None
Increasing on: (e,∞)
Decreasing on: (0, 1), (1, e)
Local maxima: None
Local minima: f(e) = e
Concave up on: (1, e2)
Concave down on: (0, 1), (e2,∞)
Inflection points: (e2, e2
2 )
ANSWERS SECTION 3 41
x
y
2 4 6 8 10 12 14
−11
−9
−7
−5
−3
−1
1
3
5
7
9
11
42 Domain: (−∞,−1) ∪ (0,∞)
f(x) = ln
(x+ 1
x
)f ′(x) = − 1
(x+ 1)x
f ′′(x) =2x+ 1
(x+ 1)2x2
y-intercept: Nonex-intercept(s): NoneSymmetry: NoneVertical asymptotes: x = −1, x = 0Horizontal asymptotes: y = 0 both as x→ ±∞Increasing on: NowhereDecreasing on: (−∞,−1), (0,∞)Local maxima: NoneLocal minima: NoneConcave up on: (0,∞)Concave down on: (−∞,−1)Inflection points: None
x
y
−3 −2 −1 1 2 3
−3
−2
−1
1
2
3
43 Domain: (0, e)
f(x) = ln (− ln (x) + 1)
f ′(x) =1
(ln (x)− 1)x
f ′′(x) = − ln (x)
(ln (x)− 1)2x2
y-intercept: None
x-intercept(s): (1, 0)
Symmetry: None
Vertical asymptotes: x = 0, x = e
Horizontal asymptotes: None
Increasing on: Nowhere
Decreasing on: (0, e)
Local maxima: None
Local minima: None
Concave up on: (0, 1)
Concave down on: (1, e)
Inflection points: (1, 0)
x
y
1 2 3
−5−4−3−2−1
12
44 Domain: (−∞,∞)
f(x) = xe(−x)
f ′(x) = −(x− 1)e(−x)
f ′′(x) = (x− 2)e(−x)
y-intercept: (0, 0)
x-intercept(s): (0, 0)
Symmetry: None
Vertical asymptotes: None
Horizontal asymptotes: y = 0 as x→∞Increasing on: (−∞, 1)
Decreasing on: (1,∞)
Local maxima: f(1) = 1/e
Local minima: None
Concave up on: (2,∞)
Concave down on: (−∞, 2)
Inflection points: (2, 2/e2)
42 MATH 140 HOMEWORK PROBLEMS
x
y
−1 1 2 3 4
−3
−2
−1
1
45 Domain: (−∞,∞)
f(x) =(2x2 + x+ 1
)ex
f ′(x) = (x+ 2)(2x+ 1)ex
f ′′(x) = (x+ 1)(2x+ 7)ex
y-intercept: (0, 1)x-intercept(s): NoneSymmetry: NoneVertical asymptotes: NoneHorizontal asymptotes: y = 0 as x→ −∞Increasing on: (−∞,−2), (−1/2,∞)Decreasing on: (−2,−1/2)Local maxima: f(−2) = 7e−2
Local minima: f(−1/2) = e−1/2
Concave up on: (−∞,−7/2), (−1,∞)Concave down on: (−7/2,−1)Inflection points: (−7/2, 22e−7/2), (−1, 2/e)
x
y
−6 −5 −4 −3 −2 −1 1
1
3
5
7
9
11
46 Domain: (−∞,∞)
f(x) = e(−x2)
f ′(x) = −2xe(−x2)
f ′′(x) = 2(2x2 − 1
)e(−x
2)
y-intercept: (0, 1)x-intercept(s): None
Symmetry: Even
Vertical asymptotes: None
Horizontal asymptotes: y = 0 both as x→ ±∞Increasing on: (−∞, 0)
Decreasing on: (0,∞)
Local maxima: f(0) = 1
Local minima: None
Concave up on: (−∞,−1/√
2), (1/√
2,∞)
Concave down on: (−1/√
2, 1/√
2)
Inflection points: (−1/√
2, e−1/2), (1/√
2, e−1/2)
x
y
−2 −1 1 2
1
47 Domain: (−∞, 0) ∪ (0,∞)
f(x) = − e(2 x)
ex − 1
f ′(x) = − (ex − 2)e(2 x)
(ex − 1)2
f ′′(x) = −(e(2 x) − 3 ex + 4
)e(2 x)
(ex − 1)3
y-intercept: None
x-intercept(s): None
Symmetry: None
Vertical asymptotes: x = 0
Horizontal asymptotes: y = 0 as x→ −∞Increasing on: (−∞, 0), (0, ln(2))
Decreasing on: (ln(2),∞)
Local maxima: f(ln(2)) = −4
Local minima: None
Concave up on: (−∞, 0)
Concave down on: (0,∞)
Inflection points: None
ANSWERS SECTION 3 43
x
y
−2 −1 1 2
−9
−7
−5
−3
−1
1
3
5
7
48 Domain: [−π, π]
f(x) = ex sin (x)
f ′(x) = (sin (x) + cos (x))ex
f ′′(x) = 2 ex cos (x)
y-intercept: (0, 0)x-intercept(s): (−π, 0), (0, 0), (π, 0)Symmetry: NoneVertical asymptotes: NoneHorizontal asymptotes: NoneIncreasing on: (−π/4, 3π/4)Decreasing on: [−π,−π/4), (3π/4, π]
Local maxima: f(−π) = 0, f(3π/4) =√22 e
3π/4
Local minima: f(π) = 0Concave up on: (−π/2, π/2)Concave down on: [−π,−π/2), (π/2, π]Inflection points: (−π/2,−e−π/2), (π/2, eπ/2)
x
y
−π− 3π4−π2−
π4
π4
π2
3π4
π−1
1
2
3
4
5
6
7
8
49 Domain: (−∞, 0) ∪ (0,∞)
f(x) = e1x
f ′(x) = −e1x
x2
f ′′(x) =(2x+ 1)e
1x
x4
y-intercept: Nonex-intercept(s): NoneSymmetry: NoneVertical asymptotes: x = 0Horizontal asymptotes: y = 1 both as x→ ±∞Increasing on: NowhereDecreasing on: (−∞, 0), (0,∞)Local maxima: NoneLocal minima: NoneConcave up on: (−1/2, 0), (0,∞)Concave down on: (−∞,−1/2)Inflection points: (−1/2, e−2)
x
y
−5 −3 −1 1 3 5
2
4
6
8
50 Domain: (−∞,∞)
f(x) = x arctan (x)
f ′(x) =x2 arctan (x) + x+ arctan (x)
x2 + 1
f ′′(x) =2
(x2 + 1)2
y-intercept: (0, 0)x-intercept(s): (0, 0)Symmetry: EvenVertical asymptotes: NoneHorizontal asymptotes: NoneIncreasing on: (0,∞)Decreasing on: (−∞, 0)Local maxima: None
44 MATH 140 HOMEWORK PROBLEMS
Local minima: f(0) = 0Concave up on: (−∞,∞)Concave down on: NowhereInflection points: None
x
y
−3 −2 −1 1 2 3
1
2
3
4
51 Domain: (−∞,∞)
f(x) = earctan(x)
f ′(x) =earctan(x)
x2 + 1
f ′′(x) = − (2x− 1)earctan(x)
(x2 + 1)2
y-intercept: (0, 1)x-intercept(s): NoneSymmetry: NoneVertical asymptotes: NoneHorizontal asymptotes: y = e−π/2 as x → −∞,y = eπ/2 as x→∞Increasing on: (−∞,∞)Decreasing on: NowhereLocal maxima: NoneLocal minima: NoneConcave up on: (−∞, 1/2)Concave down on: (1/2,∞)Inflection points: (1/2, earctan(1/2))
x
y
−3 −1 1 3 5 7 9
1
2
3
4
5
52 Domain: (0,∞)
f(x) = xx
f ′(x) = (ln (x) + 1)xx
f ′′(x) =(x ln (x)
2+ 2x ln (x) + x+ 1
)x(x−1)
y-intercept: Nonex-intercept(s): NoneSymmetry: NoneVertical asymptotes: NoneHorizontal asymptotes: NoneIncreasing on: (1/e,∞)Decreasing on: (0, 1/e)Local maxima: NoneLocal minima: f(1/e) = (1/e)1/e
Concave up on: (0,∞)Concave down on: NowhereInflection points: None
x
y
1 2
1
2
3
4
5