Math 121
Final Review
1
Pre-Calculus
1. Let f (x) = (x + 2)2 and g(x) = x3 , find:
a. (f + g)(x)
b. (f · g)(x)
c. (f /g)(x)
d. (f ◦ g)(x)
e. (g ◦ f )(x)
2. Given f (x) = x3 + 3x2 − 20 and g(x) = 2x2 − 15x − 50, find the domain of
a. f + g
b. f − g
c. f /g
d. f · g
3. Given f (x) = 3 sin x + x3 + 14x + 3 and g(x) = 7x2 + 3x find f ◦ g and g ◦ f .
x and g(x) = sin2 x
4. Find f ◦ g for f (x) = 1 −
x
√
√
5. For f (x) = x4 − x2 + x, find f (−a), f (a−1 ), f ( a), and f (a2 ).
6. If f (x) = x2 + sec x and g(x) = sin x − cos x, find (f ◦ g)(x) and (g ◦ f )(x)
7. Find the equation of the line that passes through the point (1, 5) and is parallel to the line 2x + y = 10
8. For the function f (x) =
1
find the domain and f (5).
x2 − 1
9. Find the equation of the line through the point (1,5) perpendicular to the line through the points (9, 10)
and (11, 7).
10. If f (x) = |x − 3| − 5, find f (1) − f (5).
11. Find the line perpendicular to 2x + 3y = 5 through the point (1, 1).
+2
1
12. For f (x) = x
x − 4 and g(x) = x2 find f ◦ g(x) and g ◦ f (x)
Simplify the expressions in Problems 13 through 15.
13. 2 ln e2
14. 3e5 ln 2
15. blogb 3+logb 5
2
Limits
x4 − x2
x→1 x4 − 1
16. lim
17. limπ (1 − sin θ cos θ cot θ)2 + θ
θ→ 2
18. lim
√
4
x→0
19. lim
x→0
x cos
1
x
tan x
sin 2x
x2 + 3x + 14
x→∞
2x2 + 5
3
x + 2x2 − 5x + 7 x ≥ 1
21. Find c so f (x) =
is continuous.
2x + c
x < 1
cx2
x≤2
22. Find c so f (x) =
is continuous.
2x + c x > 2
x−2 x≤5
23. Find c so f (x) =
is continuous.
cx − 3 x > 5
q
+5
24. Find where the function f (x) = 3 x
x − 5 is continuous.
20. lim
25. Find the points of discontinuity for f (x) = x2 − 1 and are they removable?
x −1
2
2
26. If lim f (x) = 6 and lim g(x) = 3, find lim [f (x)] + 2f (x)g(x) + [g(x)]
x→c
x→c
x→c
3
Derivatives
Use the definition of derivative to find the derivatives of the functions in problems 27 through 32.
27. f (x) = x2 − 2x.
30. f (x) = 3x + 2
31. f (x) = 1 −x 2x
√
32. f (x) = x + 1.
28. f (x) = x3 + 1.
1 .
29. f (x) = x +
5
In problems 33 through 76, find the derivative of the function indicated with respect to x.
33. f (x) = sin x cos x.
54. f (x) = x ln x
+ sin x .
34. f (x) = 11 −
cos x
3
55. f (x) = x
3x
56. f (x) = x sin 3x
2
35. f (x) = x −3 3x − 1
x +1
57. g(x) = x2 e2x
36. f (x) = x tan x.
9
37. f (x) = x tan(2x)
2
(x + 4)3
38. f (x) = ln
cos x
39. f (x) = esin x
40. f (x) = 3 −
−4
2
x1/3
√
3
41. f (x) = 3 cos 3x +
x√− 5x
x
58. h(x) = ln(x +
59. f (x) =
√
x2 + 1)
x2 +x3
x2
p
(1 + x2 )5
4
61. f (x) = 1 + x3
4
62. f (x) = 1+2x
1+3x
60. f (x) =
63. f (x) = x√+ 1
x
42. f (x) = sin(x2 ).
64. f (x) = (3 sin 4x)(cos 7x2 )
43. f (x) = (x + tan x)2 .
65. f (x) = (1 − x3 )(2x + 4)4/3
44. f (x) = (x2 + 2x − 15)100 .
66. f (x) = x2 2x
tan2 x cos2 x + cos(5x)
sin(x12 ) csc(x12 )
√
3 x − 8x3
46. f (x) =
1 + 5x2
67. f (x) = (2x)(cos x − 43x )3/2
45. f (x) =
47. f (x) = ln(ln(5x2 − csc x))
48. f (x) = e
x2 cos x
49. f (x) = x2 +
√
ln e
3
51. f (x) = log5 (7ln x )
√
ln x
52. f (x) = x2
e
53. f (x) = (cosh x)3
69. f (x) =
3x
(4 + 3x + x2 )4
70. f (x) = (4 + 5x2 )1/4 (6 − x)6
x2 sin x
x + sin x
3
50. f (x) = x 1+ cos x
x +3
68. f (x) = (6x2 + 5)(3x2 − 4)
71. f (x) =
sin(x2 )
ln(x2 + 3x)
72. f (x) = ln(x2 + 1)
73. f (x) = (x2 + 1)5
74. f (x) = √ 1 2
1−x
3
sin 2x
75. f (x) = 1 +
tan 3x
√
76. f (x) = 1 + 3x2 .
Find the derivative of the following functions:
81. y = x2 arctan x2
77. y = arcsin (ln (x))
78. y = arctan ex
79. y = tan(arcsin x)
80. y = arcsin x + sin1 x
82. y = earcsin x
83. y = ex arctan x2
In problems 84 through 89, use implicit differentiation for find
84. x2 y 2 + x2 y + xy 2 + x + y + 1 = xy
85. x sin y = ln x2 (9y)
86. x sin y − cos y = 0
dy
for:
dx
87. x3 − x2 y + y 5 = 8
88. x3 + y 3 = 3xy
√
89. x + xy 2 − 2x3 + 12 y 3 = 4x2 y
90. Given the curve xy 2 − x3 y = 6
dy
a. Find
dx
b. Find the tangent line(s) to the curve where x = 1.
c. Find the x-coordinate of each point on the curve where the tangent line is vertical.
91. Find the equation of the tangent line to x2 y 3 + 3xy 2 + y = 5 at (1, 1).
In problems 92 through 99, use logarithmic differentiation to find f 0 (x) for:
√
√
√
(sin(2x + 1))3
92. f (x) = x + 1 3 x + 2 5 x + 3
96. f (x) =
(4x3 + 6x)2
3
(x + 3)
h 2
i4
93. f (x) = 2
97. f (x) = ln 7x3 + 1
(x − 2)4
3x − 2
2
2 3
3/4
(x + 2)
3
2
2 2x
√ (x − 5)
94. f (x) = ln
√ − 5x + 4x
98.
f
(x)
=
x
2x + 1
x + 2x + 3
r
3
(3x2 + 2x − 7)2
x2 + 1
95. f (x) =
99. f (x) =
3
5
2
(7x − 8x)
x −1
In Problems 100 to 103, given f (x), find g 0 (b) where g(x) is the inverse of f (x).
100. f (x) = 3x + 4
101. f (x) = x3 + 1
b=3
b=2
102. f (x) = 41 x3 + x − 1
b=3
103. f (x) = 3x5 − 5x3 + 12x
b = 80
104. Find the equation of the tangent line to f (x) = x2 + 1 at the point where x = 3.
105. Find the equation of the tangent line to y = e2x ln(2x) at x = 2
106. Find the equation of the tangent line to f (x) = 43 x3 + 6x at the point (2, 18).
107. Find the equation of the tangent line to y = x3 − 2x2 at (1, −1).
108. Find the values of x for all points on the graph of f (x) = x3 − 2x2 + 5x − 16 at which the slope of the
tangent line is 4.
109. Find the point(s) on the graph of y = x2 where the tangent line passes through the point (2, −12).
4
4.1
Applications of Derivatives
Related Rates
110. A rectangle has a width that is 1/4 its length. At what rate is the area increasing if its width is 20 cm
and is increasing at 0.5 cm/s?
111. Air is leaking out of a spherical balloon at the rate of 3 cubic inches per minute. When the radius is 5
inches, how fast is the radius decreasing?
112. Sand is being emptied from a hopper at the rate of 10 ft3 /s. The sand forms a conical pile whose
height is always twice its radius. At what rate is the radius of the pile increasing when the height is 5ft?
113. A balloon is 200 feet off the ground and rising vertically at the constant rate of 15 feet per second. An
automobile passed beneath it traveling along a straight line at the constant rate of 66 miles per hour. How
fast is the distance between them changing one second later?
114. The top of a 25-foot ladder leaning against a vertical wall is slipping down the wall at the rate of one
foot per minute. How fast is the bottom of the ladder slipping along the ground when the bottom of the
ladder is 7 feet away from the base of the wall?
115. A plane flying parallel to the ground at the height of four kilometers passes directly over a radar
station. A short time later, the radar reveals that the plane is 5 km away and the distance between the
plane and the station is increasing at the rate of 300 km/hr. (The distance is straight line distance from
ground level at the station to the plane four kilometers high.) At that moment, how fast is the plane moving
horizontally?
116. A spotlight is on the ground 100 feet from the vertical side of a very tall building. A person six feet
tall stands at the spot light and walks directly toward the building at a constant rate of 5 feet per second.
How fast is the top of the person’s shadow moving down the building when the person is 50 feet away from
it?
117. Two balloons are attached so air can flow freely between them. If the radius of balloon A is decreasing
at 2 in/min, what is the rate of change of balloon B. The radius of balloon A is 3 in and the radius of balloon
B is 5 in.
118. A girl is flying a kite which is 120 feet above the ground. The wind is carrying the kite horizontally
away from the girl at a speed of 10 feet/second. How fast must the string be let out when the string from
the girl to the kite is 150 feet long (and taut)?
119. A rocket is rising straight up from the ground at the rate of 1000 km per hour. An observer 2 km
from the launching site is photographing the rocket. How fast is the angle θ of the camera with the ground
changing when the rocket is 1.5 km above the ground?
120. What is the radius of an expanding circle at the moment when the rate of change of its area is
numerically twice as large as the rate of change of its radius?
121. A woman standing on a cliff is watching a motor boat though a telescope as the boat approaches the
shoreline directly below her. If the telescope is 250 feet above the water and if the boat is approaching at
20 feet per second, at what rate is the angle of the telescope changing when the boat is 250 feet from shore?
122. Suppose water is leaking out of a cone shaped funnel at a rate of 2 in3 /min. Assume the height of the
cone is 16 in and the radius of the cone is 4 in. How fast is the depth of the water decreasing when the level
of water is 8 in. deep?
4.2
Graphing
123. Find where f (x) = 3x4 + 4x3 − 12x2 is increasing and decreasing.
124. Find where f (x) = 3x2 − 6x is increasing or decreasing
125. Find where f (x) = 4x5 − 5x4 is increasing or decreasing and classify all critical points.
126. Find the concavity of f (x) = 3x4 − 12x3 + 1
127. Find the concavity of f (x) = x4 − 4x3 and any inflection points.
For the functions in problems 128 to 132 find the domain, range, x-intercepts, y-intercepts,
where y is increasing, decreasing, critical points, where y is concave up, concave down, inflection
points, vertical asymptotes, horizontal asymptotes, and sketch the graph of y.
x
x2 + 1
129. y = 2 1
x −9
7x2 − 4
x2 + 4x + 4
132. y = 2 x
x −1
128. y =
130. y =
4.3
131. y =
2(x2 − 9)
x2 − 4
Max/Min
133. Find the maximum and minimum values of f (x) = 4x3 − 8x2 + 5x for −1 ≤ x ≤ 2.
134. What positive number exceeds its cube by the greatest amount? (Hint: x > x3 only if 0 < x < 1).
135. Find two positive numbers whose sum is 20 and whose product is as large as possible.
136. Among all rectangles with corners on the ellipse
y2
x2
+
=1
9
4
which has the largest area?
137. Find the absolute and local maximums and minimums for f (x) = 6x7 − 2x3 + 2 on [−1, 1].
138. Find the maximum and minimum of f (x) = 2 x
on [0, 3]
x +1
139. Which points on the graph of y = 4 − x2 are closest to (0, 2)?
140. Find the maximum and minimum of f (x) = 6x2 − 4x − 10 on the interval [−4, 5].
141. A student see Professor Butler standing down stream on the other side of a straight river, 3 km wide.
He then remembers that they agreed to go to Baker’s Square in hopes that they haven’t run out of pie. He
wants to reach Prof. Butler as quickly as possible and recalls that he can swim at 6 km/h and run at 8
km/h. Where should the student reach the other side of the shore relative to where he began if Professor
Butler is 8 km down stream to minimize time?
142. A donkey owner has 750 ft of fencing. He wants to make a rectangular area for them bound by a wall
on one side. What dimensions will enable him to fit the largest number of asses inside the fenced area?
143. Milo has a pet elephant named Tiny. Since his pet elephant has an affinity for trampling the neighbors,
Milo has to build a fence to keep Tiny in. He’s building the fence up against his house so it only needs to
have three sides. Fencing costs $3 a foot. The recent price gouging of elephant food has Milo in a financial
bind, so he only has $3300 to spend on the fence. What are the dimensions that give Tiny the most room?
144. You want to make an open-topped rectangular box with square base. You want its volume to be 1000
cubic centimeters. The material for the base costs 10 cents per square centimeter and the material for the
sides costs 7 cents per square centimeter. Find the dimensions of the cheapest box to build.
145. Among the all the pairs of nonnegative numbers that add up to 5, find the pair that maximizes the
product of the square of the first number and the cube of the second number.
146. A rectangular yard is to be laid out and fenced in, and divided into 10 enclosures by fences parallel to
one side of the yard. If 22 miles of fencing is available, what dimensions will maximize the area?
147. A motorist is stranded in a desert 5 miles from a point A, which is the point on a long straight road
nearest him. He wishes to get to a point B on the road that is 10 miles from point A. He can travel at 15
miles an hour in the desert and 39 miles an hour on the road. Find the point where he must meet the road
to get to B in the shortest possible time. Assume he travels in the desert in a straight line.
148. A rectangular area with fence all around is to be divide into three smaller areas by running two lengths
of fence parallel to one side. If you have 800 yds total of fence, what is the largest area that can be enclosed?
149. You work for an ice cream cone company and you are given the task of finding the greatest possible
volume of a cone given a slant height of 4 inches.
150. A certain company owns two buildings. The first building is located on one side of a river that is
1500 m wide. The second building is located on the opposite side of the river, but 5500 m down stream.
The company owns a computer systems on each of the buildings and wants to network the two buildings
with cables. Of course, they want to minimize the cost of cabling since it will cost twice as much to lay
cable underwater as compared to laying cable above ground. What path should the cabling take in order to
minimize the total cost?
151. A peanut vendor sells bags of peanuts. He sells them for $5 a bag. At this price he sells 500 bags a
day. The vendor observes that for every $0.50 he takes off his price per bag, he sells 100 more bags per day.
At what price should the peanut man sell his peanuts to maximize revenue.
152. A window is to be made in the shape of a rectangle surmounted by a semicircle with diameter equal
to the width of the rectangle. If the perimeter of the window is 22 feet, what dimensions will emit the most
light (i.e. maximize the area)?
153. A greeny bus caries 10 passengers from North side to Schmitt Lecture Hall. The cost to ride is $0.50
per person. Market Research reveals that 2 fewer people will ride the bus for each $0.02 increase in fare.
What fare should be charged to get the largest possible revenue?
154. An oil can is to be made in the form of a right circular cylinder and will contain 1000 cm3 . What are
the dimensions of the can that requires the least amount of material?
155. A rectangular swimming pool is to be built with a 6 foot wide deck at the north and south ends, and
a 10 foot wide deck at the east and west ends. If the total area available is 6000 square feet, what are the
dimensions of the largest possible water area? Be sure to draw a picture of your set-up.
156. A flower bed will be in the shape of a sector of a circle (a pie-shaped region) of radius r and vertex
angle θ. Find r and θ if its area is a constant A and the perimeter is a minimum.
157. A cylindrical container is to be produced that will have a capacity of 10 cubic meters. The top and
bottom of the container are to be made of a material that costs $2 per square meter, while the side of the
container is to be made of material costing $1.50 per square meter. Find the dimensions that will minimize
the total cost of the container. What is this minimum cost?
4.4
Newton’s Method
158. Use Newton’s method to find where cos x = x to 4 decimal places.
159. Use Newton’s method to find the root of x3 − 3x2 + 1 = 0 between 2 and 3. (4 decimal places)
√
160. Find 3 19 by using Newton’s method on f (x) = x3 − 19.
161. It is a dark and stormy night. You are drinking hot coffee as you do your chemistry lab. You are
inspecting the graph of your data when a loud clap of thunder startles you. Your coffee spills- a drop
falls on to the graph and since the lab print-out is so thin the paper dissolves! There is now a hole right
where the graph crosses the x-axis. You need the x-intercept for you experiment. You know the equation is
f (x) = x3/2 − 10. Use Newton’s method to find the x-intercept between 4.2 and 4.8.
4.5
Linear Approximaitons
162. Use√a linear approximation L(x) to an appropriate function f (x), with an appropriate value of a to
estimate 257.
163. Use√a linear approximation L(x) to an appropriate function f (x), with an appropriate value of a to
estimate 16.5.
164. Use√a linear approximation L(x) to an appropriate function f (x), with an appropriate value of a to
estimate 25.3.
4.6
l’Hospital’s Rule
3x2 + 10x + 3
x→−3 x2 + 2x − 3
√
x−2
166. lim 2
x→4 x − 16
165. lim
x3 + 1
x→1 x2 + 1
167. lim
168.
169.
170.
171.
172.
173.
x2 cos x + 2x
lim
x→0
sin x
√
√
2+x− 2
lim
x→0
x
sin x
lim
.
x→0 5x
1+x
lim
.
x→∞ 5x2
1 − cos x
.
lim
x→0
3x
x
lim
.
x→3− x − 3
x2 + 6x + 5
.
x→−1 x2 − 3x − 4
p
175. lim x − x2 − 3x
174. lim
x→∞
sin x
3x
sin x
177. lim
x→∞ 3x
p
p
178. lim ( x2 + 2x + 3 − x2 − 2x + 3)
176. lim
x→0
x→∞
179. lim+ x ln x
x→0
180. lim x sin
x→∞
π
x
181. lim xx
x→0+
x
2
182. lim 1 +
x→∞
x
1
x
183. lim (cos x)
x→0
1
1
− x
x→0 x
e −1
1
185. lim 1 + x2 ln x
184. lim
x→∞
(2 + x)2 − 4
x→0
x
186. lim
5
Integration
187. Compute the Riemann Sum using the right-hand end-points over the indicated interval divided into
“n” sub-intervals. Also, compute the integral and compare the results. f (x) = 13 x3 + 1; [−1, 2], n = 6.
Z 3
Z 3
Z 3
188. If
f (x) dx = 4 and
g(x) = 2, find
4f (x) − g(x) dx
1
1
1
Compute the derivative of the following functions:
Zx
189. F (x) =
1
dt
1 + t3
Zln x
191. F (x) =
2
2
Z2
190. F (x) =
1
dt
1 + t3
Z
Z
205.
Z
206.
3
(x2 + 3)(x3 − 1) dx
Z
207.
(2 cos 3t + 5 sin 4t + 3e7t )dt
2
51/x
dx
x3
√
2
Z
196.
Z
197.
Z
(2x3 − 1)5 x2 dx
x + 3x2 dx
1
198.
√
(t2 + 42t + 42)5 dt
e
Compute the following integrals:
Z
sin x
√
193.
dx
1 + cos x
Z
194.
sin x cos x dx
Z
x
192. F (x) =
x
195.
1
dt
1 + t3
Z
ex ln(sin ex )
dx
tan ex
Z
e2x
dx
1 + e2x
208.
209.
Z
2
x(x + 1) dx
ex
dx
ex + 1
210.
x3 9x
4
+2
dx
0
Z
199.
Z 1
3
211.
x 1 + x dx
212.
1
dx
x(ln x)2
213.
2
x(x + 1) dx
0
Z
200.
Z
201.
Z
202.
2 x3
x e
dx
Z
2x + 1
dx
x+1
Z
p
x 3 − 7x2 dx
203.
204.
√
dx
Z
sec2 2x
dx
tan 2x
Z
√
3
Z
√
1+ x
√
dx
x
214.
x + x−3 dx
Z
2x + 1
dx
x2 + x + 1
Z
6x2
dx
cos2 (4 + x3 )
215.
216.
x3 + 1
x5
Z
dx
x ln x
217.
Z
Z
p
x 1 − x2 dx
0
cos2 x sin3 x dx
218.
1
232.
Z
4
233.
√
1
dx
2x + 1
√
1
√ dx
x(1 + x)
0
2
Z
9
5
(10x + 3x ) dx
219.
−2
4
Z
Z
x dx
16 + x
dx
x2
221.
4
Z
222.
√
Z
238.
2
|x − 4x + 3| dx
239.
4
0
Z
π/2
Z
225.
240.
cos x dx
0
Z
241.
1
Z
x3 − x2
226.
dx
0
Z
4
Z
1
√ dx
x
227.
1
0
cos x dx
sec2 x dx
π/4
(2x + 1)4 dx
231.
0
6
6.1
√
247.
1
dx
1 − 9x2
Z
5
dx
x + 6x + 13
√
2
Z
cos x
dx
9 + sin2 x
Z
x2
dx
1 + x2
246.
1
Z
1/6
1
dx
x 4x2 − 1
π/3
Z
230.
10x
dx
10x + 1
Z
245.
0
2
x 3x dx
x+3
dx
x2 + 9
244.
229.
1
dx
x
Z
π
Z
20
0
e2x dx
228.
√
x 1 + x dx
242.
243.
3
Z
8
0
Z
5
224.
x sin x2 dx
0
Z
6
dx
x2
1
Z
π
237.
π/4
223.
1
√ dx
ex
0
sin x dx
3
1
236.
3π/4
Z
1
dx
x ln x
e
9
Z
e2
235.
0
Z
9
234.
1
√
220.
Z
Applications of Integration
Area
248. Find the area of the region R bounded by the line y = x + 2 and the parabola y = x2 − 4.
249. Find the area of the region bounded by the graphs of f (x) = x3 − 6x and g(x) = −2x.
250. Find the area of the region enclosed by y = 3 − x2 and y = −x + 1 between x = 0 and x = 2.
251. Find the area of the region enclosed by y = x + 4/x2 , the x-axis, x = 2, and x = 4.
252. Find the area of the region enclosed by y = x + 5 and y = x2 − 1.
253. Find the area of the region enclosed by x = y 2 − 4y + 2 and x = y − 2.
254. Find the area of the region enclosed by y = 6x − x2 and y = x2 − 2x
6.2
Volume
255. Set-up the integral for the volume of revolution, if the region bounded by y = 4x − x2 , and y = x2 is
spun about x = 4
256. Set up the
√ integral that represents the volume of the formed by revolving the region bounded by the
graphs of y = 25 − x2 and y = 3 about the x-axis. (Do not evaluate the integral.)
1 , y = 0, x = 1and x = 4 is spun about
257. Find the volume generated when the region bounded by y = x
the x-axis.
258. Using the method of cylindrical shells, find the volume if the solid generated by rotating around the
line x = 2 the region bounded by y = x2 + 1, y = 0 and x = 2.
259. Find the volume if the area enclosed by f (x) = x2 , x = 4, x = 1, y = −2 is rotated about the line
y = −2
260. Find the volume if the region bounded by y = x2 , y = −2x + 3, and the y-axis in the first quadrant, is
rotated about the y-axis.
261. Find the volume of the solid that results when the area of the smaller region enclosed by y 2 = 4x,
y = 2, and x = 4 is revolved about the y-axis.
262. Find the volume of the solid that results when the first quadrant region enclosed by y = x3 and y = x
is revolved about the y-axis.
263. Find the volume of the solid that results when the area enclosed by y 2 = 4x, y = 2, and x = 4 is
revolved about the x-axis.
264. Find the volume of the solid that results when the area of the region enclosed by y = 2x, x = 0, and
y = 2 is revolved about x = 1.
265. Find the volume of the solid that results when the area of the region enclosed by y 2 = x3 , x = 1, and
y = 0 is revolved about the x-axis.
266. Find the volume of the solid that results when the area of the region enclosed by y 2 = 4x and y = x
is revolved about the line x = 4.
267. A storage tank is designed by rotating y = −x2 + 1, −1 ≤ x ≤ 1, about the x-axis where x and y are
measured in meters. Use cylindrical shell to determine how many cubic meters the tank will hold.
6.3
Work
268. A spring has a natural length of 4 feet. A force of 30 lbs. is required to compress that spring to a
length of 2.5 feet. How much work is done to stretch the spring from its natural length to 6 feet?
269. A 50 foot chain weighing 10 pounds per foot supports a beam weighing 1000 pounds. How much work
is done in winding 40 feet of the chain onto a drum?
270. A conical tank has a diameter of 9 feet and is 12 feet deep. If the tank is filled with water of density
62.4 lbs/ft3 , how much work is required to pump the water over the top?
271. A cylindrical tank 8 feet in diameter and 10 feet high is filled with water weighing 62.4 lbs/ft3 . How
much work is required to pump the water over the top of the tank?
272. Set up the integral for the work need to empty a right circular conical tank of altitude 20 ft and radius
of base 5 ft has its vertex at ground level and axis vertical. If the tank is full of orange marmalade weighting
100 lb/ft3 pumping all the orange marmalade over the top of the tank.
273. Water is drawn from a well 50 feet deep using a bucket that scoops up 200 lbs of water. The bucket
is pulled up at the rate of 3 ft/s, but it has a hole in the bottom through which water leaks out at a rate
of 3/4 lb/s. How much work is done in pulling the bucket to the top of the well. Neglect the weight of the
bucket and rope and work done to overcome friction.
274. You are a secret agent. You and your partner, who is kind of annoying, are trying to escape an evil
super villain. His hit men chase you on foot for several dozen miles before your partner collapses and is
captured. Even though you are really tried and you don’t like your partner that much, you infiltrate the
villain’s lair to try to rescue him. You find your partner has been knocked out and secured at the bottom of
a circular pool of water 10 foot in radius and 9 feet in height and the pool is filled with water (density ρ=
62.4 lb/ft3 ). Since you are so tired and you wouldn’t really be that upset if your partner died, you decide
that you are only willing to expend 1 million ft pounds of work. Calculate the amount of work required to
lift all the water out of the pool to decide if your partner will be rescued.
275. A 20 foot chain weighing 5 pounds per foot is lying coiled on the ground. How much work is required
o raise one end of the chain to a height of 20 feet?
276. You are at an ice cream shop. You order your favorite ice cream, chocolate chip cookie dough. You
haven’t eaten all day, and begin to gobble down the ice cream. If the ice cream cone is 2 feet long and
has a radius of 1 foot, how much work is required to eat all the ice cream. (The density of ice cream is
ρ = 55.5lb/ft3 .)
Math 121
Final Review
Answers
1. a. x3 + x2 + 4x + 4
b. x5 + 4x4 + 4x3
1+ 4 + 4
c. x
x2
x3
d. x6 + 4x3 + 4
e. (x2 + 4x + 4)3
2. a. all reals.
b. all reals.
c. x 6= 10, −5/2
d. all reals
3. (f ◦ g)(x) = 3 sin(7x2 + 3x) + (7x2 + 3x)3 + 14(7x2 + 3x) + 3 and (g ◦ f )(x) = 7(3 sin x + x3 + 14x +
3)2 + 13(3 sin x + x3 + 14x + 3)
4.
5.
(f ◦ g)(x) = tan2 x
√
√
√
√
√
f (−a) = a4 − a2 − a, f (a−1 ) = a−4 − a−2 + a−1 , f ( a) = a2 − a + a1/2 , and f (a2 ) = a8 − a4 + a2 .
6. (f ◦ g)(x) = (sin x − cos x)2 + sec(sin x − cos x), (g ◦ f )(x) = sin(x2 + sec x) − cos(x2 + sec x)
7. 2x + y = 7
1
8. Domain x 6= ±1, f (5) = 24
9. y = 2
3 (x − 1) + 5
10. 0
11. y = 23 x − 1
2
1 +2
2
x
12. f ◦ g(x) = 1
g ◦ f (x) = h 1 i2
x+2
−
4
x2
x−4
13.
4
14.
96
15.
15
1
2
16.
17.
18.
19.
20.
21.
22.
1+ π
2
0
1
2
1/2
c=3
k=4
3
24.
c = 56
Continuous everywhere except where x = 5
25.
x = 1 is removable, x = −1 is not removable.
26.
81
27.
f 0 (x) = 2x − 2
28.
f 0 (x) = 3x2
29.
f 0 (x) = −
30.
f 0 (x) = 3
31.
f 0 (x) =
32.
f 0 (x) = √ 1
2 x+1
33.
f 0 (x) = cos2 x − sin2 x
34.
1
f 0 (x) = cos x − sin x −
(1 − cos x)2
35.
f 0 (x) =
36.
f 0 (x) = x sec2 x + tan x
37.
f 0 (x) = 2x9 sec2 (2x) + 9x8 tan(2x)
38.
f 0 (x) =
39.
y 0 = cos xesin x
f 0 (x) = (−4) 3 −
23.
1
(x + 5)2
1
(1 − 2x)2
(x3 + 1)(2x − 3) − (x2 − 3x − 1)(3x2 )
(x3 + 1)2
6x + tan x
x2 + 4
−5 42.
2
x1/3
3x4/3
√ 1 −2/3
√
1 )
( x)( 3 x
− 5) − ( 3 x − 5x)( √
2 x
0
f (x) == −9 sin 3x +
x
0
2
f (x) = 2x cos(x )
43.
f 0 (x) = 2(x + tan x)(1 + sec2 x)
44.
f 0 (x) = 100(x2 + 2x − 15)99 (2x + 2)
45.
f 0 (x) = 2 sin x cos x − 5 sin(5x)
46.
f 0 (x) =
√
( 32 x−1/2 − 24x2 )(1 + 5x2 ) − (3 x − 8x3 )(10x)
(1 + 5x2 )2
47.
f 0 (x) =
10x + csc x cot x
(5x2 − csc x) ln(5x2 − csc x)
48.
f 0 (x) = (2x sin x + x2 cos x)ex
49.
1 + cos x
f 0 (x) = 2x + √
2 x
50.
1 + 3)(3x2 − sin x) − (x3 + cos x)(− 1 )
(x
x2
f (x) =
1
2
( x + 3)
40.
41.
2
2
0
cos x
+ x2 sin xex
2
cos x
(2x cos x − x2 sin x)
51.
52.
3)
f 0 (x) = 1 (ln 7)( x
ln 5
2
1 − 1 (ln x)2xex2
ex 2x
0
22
f (x) =
(ex )2
53.
f 0 (x) = 3(cosh x)2 sinh x
54.
f 0 (x) = 1 + ln x
55.
f 0 (x) =
56.
x2 [3 − x(ln 3)]
3x
0
f (x) = 3x cos 3x + sin 3x
57.
g 0 (x) = 2x2 e2x + 2xe2x
58.
1
x
√
h (x) =
1+ √ 2
(x + x2 + 1)
x +1
59.
f 0 (x) = 1
60.
f 0 (x) = 5x(1 + x2 )3/2
3
f 0 (x) = − x122 1 + x3
3
f 0 (x) = −4 (1+2x)
5
(1+3x)
61.
62.
0
√
63.
64.
1 (x + 1)
x− √
2 x
f (x) =
x
0
f (x) = (3 sin 4x)(−14x sin 7x2 ) + (12 cos 4x)(cos 7x2 )
0
65.
1/3
4/3
2
f 0 (x) = (1 − x3 ) 4
3 (2x + 4) (2) + (2x + 4) (−3x )
66.
f 0 (x) = 2x(2x ) + x2 (2x ) ln 2
67.
f (x) = 2(cos x − 43x )3/2 + (2x) 32 (cos x − 43x )1/2 (− sin x − (43x (ln 4)(3))
68.
f 0 (x) = (12x)(3x2 − 4) + (6x2 + 5)(6x)
69.
f 0 (x) =
70.
f 0 (x) = 14 (10x)(4 + 5x2 )−3/4 (6 − x)6 − (4 + 5x2 )1/4 6(6 − x)5
+3
ln(x2 + 3x)(2x cos(x2 )) − sin(x2 ) 2x
x2 + 3x
f 0 (x) =
(ln(x2 + 3x))2
71.
(4 + 3x + x2 )4 (3) − (3x)4(4 + 3x + x2 )3 (3 + 2x)
(4 + 3x + x2 )8
2x
x2 + 1
72.
f 0 (x) =
73.
f 0 (x) = 10x(x2 + 1)4
x
f 0 (x) =
(1 − x2 )3/2
2 (1 + tan 3x)(2 cos 2x) − 3(sin 2x)(sec2 3x)
sin 2x
f 0 (x) = 3 1 +
tan 3x
(1 + tan 3x)2
74.
75.
76.
77.
f 0 (x) = √ 3x 2
1 + 3x
0
y = p 1
x 1 − (ln x)2
ex
1 + e2x
78.
y0 =
79.
y 0 = (1 − x2 )−3/2
80.
y 0 = √ 1 2 − csc x cot x
1−x
81.
y0 =
82.
83.
2x3 + 2x arctan x2
1 + x4
earcsin x
y0 = √
1 − x2
y 0 = ex 2x 4 + ex arctan x2
1+x
84.
dy
−2xy 2 − 2xy − y 2 + y − 1
=
dx
2x2 y + x2 + 2xy + 1 − x
85.
2
sin y − x
dy
= 1
dx
y − x cos y
86.
dy
− sin y
= x cos y + sin y
dx
87.
dy
2xy − 3x2
=
dx
5y 4 − x2
88.
dy
y − x2
= 2
dx
y −x
89.
1 + y 2 − 6x2 − 8xy
√
2 x
dy
=
dx
−x2y − 32 y 2 + 4x2
90. a.
dy
3x2 y − y 2
=
dx
2xy − x3
b. At (1, 3) the y = 3, at (1, −2) then y + 2 = 2(x − 1)
√
c. x = 5 −24
91.
92.
93.
y − 1 = −1
2 (x − 1)
√
√
√
1 +1 1 +1 1
f 0 (x) = ( x + 1 3 x + 2 5 x + 3) 12 x +
1 3x+2 5x+3
h
i
(x + 3)3
3 − 8x
f 0 (x) = x +
3 x2 − 2 (x2 − 2)4
94.
2 +
1
9x2
f 0 (x) = x +
2 4(x3 − 5) − 2x + 1
95.
f 0 (x) =
96.
97.
98.
(7x3 − 8x)5 )(2)(3x2 + 2x − 7)(6x + 2) − (3x2 + 2x − 7)2 (5)(7x3 − 8x)4 (21x2 − 8)
(7x3 − 8x)10
(4x3 + 6x)2 3(sin(2x + 1))2 cos(2x + 1)2 − (sin(2x + 1))3 2(4x3 + 6x)(12x2 + 6)
(4x3 + 6x)4
h
i
14x − 9x2
f 0 (x) = 4
2
3
7x + 1 3x − 2
#
" √
2
1 x−1/2 + 2)
2
3
2
(
x
+
2x
+
3)(6x
−
10x
+
4)
−
(2x
−
5x
+
4x)(
3
2
3
2
2x
−
5x
+
4x
2x
−
5x
+
4x
0
2
2
√
√
√
f (x) = x 2
+2x
x + 2x + 3
x + 2x + 3
( x + 2x + 3)2
f 0 (x) =
99.
f (x) = 3
2
0
r
x2 + 1
x2 − 1
−4x
(x2 − 1)2
104.
1
3
1
3
1
4
1
192
y = 6x − 8
105.
y − (e4 ln 4) = 178.7(x − 2)
106.
y = 15x − 12
107.
y = −x
108.
109.
1, 1
3
x = −2, 6
110.
80 cm2 /s
111.
dr = 3
100π
dt
dr = 4
5π
dt
53 ft/sec
100.
101.
102.
103.
112.
113.
115.
Bottom is slipping away at 24
7 ft/min.
500 km/hr
116.
-1.2 ft per second.
117.
18
25
118.
6 ft/sec
119.
320 rad/hr
120.
1
r= π
1
25 radian per second
114.
121.
in/min.
dy
= −0.16in/min.
dt
123. Inc (−2, 0) ∪ (1, ∞) dec: (−∞, −2) ∪ (0, 1)
122.
124.
Inc: x > 1, dec: x < 1.
125.
Inc: (−∞, 0) ∪ (1, ∞), Dec: (0, 1) C.P. (0, 0) max (1, −1) min.
126.
Concave up: (−∞, 0) ∪ (2, ∞), concave down (0, 2).
127.
Con up: (−∞, 0) ∪ (2, ∞), Con dn: (0, 2) I.P. (0, 0) and (2, −16)
133. max. = 10 (at x = 2), min. = -17 (at x = −1)
q
134. x = 13
135.
10 and 10
136.
137.
√
√
Corners at x = ± 3 2 2 and y = ± 2
p
p
(−1, −6) abs min, (− 4 1/7, 2.27) local max, ( 4 1/7, 1.73) local min, (1, 6) abs max.
138. Maximum is .5 at x = 1, the minimum is 0 at x = 0.
q
q
5 ) and (− 3 , 5 )
139. ( 3
,
2 2
2 2
140.
Max: 120 min: -32/3
141.
He should reach shore √9 km down stream from where he started.
7
142.
375 by 187.5
143.
550 by 275
144.
11.2 X 11.2 X 8
145.
First is 2, second is 3
146.
1 X 5.5
147.
2.08 miles A
148.
20,000
149.
25.8 in3
150.
866 m downstream.
151.
$3.75
152.
r = π 22
+4
153.
$0.30
q
q
3 500
r = 3 2000
and
h
=
2
π
4π
154.
155.
157.
48 by 80
√
r = A, θ = 2
q
30
r = 3 8π
158.
0.7391
159.
2.8793
√
3
19 ≈ 2.67
156.
160.
161.
162.
163.
164.
165.
x = 4.641589
√
√
f (x) = x, x0 = 256 so 257 ≈ 513
32
√
√
f (x) = x, a = 16 so 16.5 ≈ 4.0620
√
√
f (x) = x, a = 25 so 25.3 ≈ 5.03
167.
2
1
16
1
168.
2
166.
171.
1
√
2 2
1
5
0
172.
0
173.
−∞
−4
5
3
2
169.
170.
174.
175.
176.
1
3
177.
0
178.
2
179.
0
180.
π
181.
1
182.
e2
183.
184.
1
1
2
185.
e2
186.
4
187.
5.0625 and 4.25
188.
14
189.
1
1 + x3
F 0 (x) = − 1 3
1+x
190.
F 0 (x) =
1
1 + (ln x)3
1
x
191.
F 0 (x) =
192.
F 0 (x) = (x2 + 42x + 42)5
√
−2 1 + cos x + C
193.
1 sin2 x + C
2
195. 150.25
194.
196.
197.
198.
199.
200.
(2x3 − 1)6
+C
36
x2 + x3 + C
2
20
21
15
8
2x
3 (1
+ x)3/2 −
4
15 (1
+ x)5/2 + C
203.
− 1 +C
ln x
1 ex3 + C
3
2x − ln |x + 1| + C
204.
1 (3 − 7x2 )3/2 + C
− 21
205.
2 sin 3t − 5 cos 4t + 3 e7t + C
3
4
7
206.
−5
+C
2 ln 5
√ x
2 e +1+C
201.
202.
207.
208.
209.
1/x2
1 (ln(sin ex ))2 + C
2
1 ln |1 + e2x | + C
2
4
210.
211.
9x +2 + C
4 ln 9
1 − 1 +C
−x
4x4
214.
1 ln | tan 2x| + C
2
3 x4/3 − 1 + C
4
2x2
√ 2
(1 + x) + C
215.
ln |x2 + x + 1| + C
216.
2 tan(4 + x3 ) + C
217.
ln(ln x) + C
212.
213.
218.
219.
220.
221.
222.
223.
cos5 x − cos3 x + C
5
3
0
16
3
20 + ln 9
9
4
√
2
225.
4
28
3
1
226.
1
− 12
227.
2
1 (e6 − 1)
2
0
√
3−1
121
5
224.
228.
229.
230.
231.
233.
1
3
2
234.
ln 4
235.
ln 2
236.
2 − 2e−1/2
237.
1
238.
1192
15
ln 5
232.
239.
2
240.
241.
242.
243.
244.
245.
246.
3x + C
2 ln 3
ln(10x + 1)
+C
ln 10
π
18
1 ln(x2 + 9) + arctan x + C
2
3
arcsec(2x) + C
5 arctan x + 3 + C
2
2
1 arctan sin x + C
3
3
249.
x − arctan x + C
125
6
8
250.
10/3
251.
7
252.
125/6
253.
9/2
254.
64
3
247.
248.
Z
255.
2π
2
(4 − x) (4x − x2 ) − x2 dx
0
Z4
(16 − x2 ) dx
256.
π
257.
3π
4
20π
3
V = 300.6π
V = 14π
12
−4
258.
259.
260.
261.
98π
5
262.
4π
15
263.
18π
264.
4π
3
265.
π
4
266.
64π
5
267.
16π
15
268.
40 ft-pounds
269.
52,000 ft lbs
270.
15,163 π
271.
49,920 π ft lbs
Z 20
x2
W =
100π(20 − x)
dx
16
0
272.
273.
9687.5 ft pounds.
274.
252,720 π ft pounds.
275.
1000 ft-pounds
276.
58.1195 foot-pounds.