Math 121Final 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 ofa. f + gb. f − gc. f/gd. f · g

3. Given f(x) = 3 sinx+ x3 + 14x+ 3 and g(x) = 7x2 + 3x find f ◦ g and g ◦ f .

4. Find f ◦ g for f(x) = x1− x and g(x) = sin2 x

5. For f(x) =√x4 − x2 + x, find f(−a), f(a−1), f(

√a), and f(a2).

6. If f(x) = x2 + secx and g(x) = sinx− cosx, 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) = 1x2 − 1

find the domain and f(5).

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).

12. For f(x) = x+ 2x− 4 and g(x) = 1

x2find 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

16. limx→1

x4 − x2

x4 − 1

17. limθ→π

2

(1− sin θ cos θ cot θ)2 + θ

18. limx→0

4√x cos

(1

x

)19. lim

x→0

tanx

sin 2x

20. limx→∞

x2 + 3x+ 14

2x2 + 5

21. Find c so f(x) =

{x3 + 2x2 − 5x+ 7 x ≥ 12x+ c x < 1

is continuous.

22. Find c so f(x) =

{cx2 x ≤ 22x+ c x > 2

is continuous.

23. Find c so f(x) =

{x− 2 x ≤ 5cx− 3 x > 5

is continuous.

24. Find where the function f(x) = 3

√x+ 5x− 5 is continuous.

25. Find the points of discontinuity for f(x) = x− 1x2 − 1

and are they removable?

26. If limx→c

f(x) = 6 and limx→c

g(x) = 3, find limx→c

([f(x)]

2+ 2f(x)g(x) + [g(x)]

2)

3 Derivatives

Use the definition of derivative to find the derivatives of the functions in problems 27 through 32.

27. f(x) = x2 − 2x.

28. f(x) = x3 + 1.

29. f(x) = 1x+ 5.

30. f(x) = 3x+ 2

31. f(x) = x1− 2x

32. f(x) =√x+ 1.

In problems 33 through 76, find the derivative of the function indicated with respect to x.

33. f(x) = sinx cosx.

34. f(x) = 1 + sinx1− cosx .

35. f(x) = x2 − 3x− 1x3 + 1

36. f(x) = x tanx.

37. f(x) = x9 tan(2x)

38. f(x) = ln

[(x2 + 4)3

cosx

]39. f(x) = esin x

40. f(x) =(

3− 2x1/3

)−441. f(x) = 3 cos 3x+

3√x− 5x√x

42. f(x) = sin(x2).

43. f(x) = (x+ tanx)2.

44. f(x) = (x2 + 2x− 15)100.

45. f(x) = tan2 x cos2 xsin(x12) csc(x12)

+ cos(5x)

46. f(x) =3√x− 8x3

1 + 5x2

47. f(x) = ln(ln(5x2 − cscx))

48. f(x) = ex2 cos x ln ex

2 sin x

49. f(x) = x2 +√x+ sinx

50. f(x) = x3 + cosx1x + 3

51. f(x) = log5(7ln x3

)

52. f(x) =ln√x

ex2

53. f(x) = (coshx)3

54. f(x) = x lnx

55. f(x) = x33x

56. f(x) = x sin 3x

57. g(x) = x2e2x

58. h(x) = ln(x+√x2 + 1)

59. f(x) = x2+x3

x2

60. f(x) =√

(1 + x2)5

61. f(x) =(1 + 3

x

)462. f(x) =

(1+2x1+3x

)463. f(x) = x+ 1√

x

64. f(x) = (3 sin 4x)(cos 7x2)

65. f(x) = (1− x3)(2x+ 4)4/3

66. f(x) = x22x

67. f(x) = (2x)(cosx− 43x)3/2

68. f(x) = (6x2 + 5)(3x2 − 4)

69. f(x) = 3x(4 + 3x+ x2)4

70. f(x) = (4 + 5x2)1/4(6− x)6

71. f(x) =sin(x2)

ln(x2 + 3x)

72. f(x) = ln(x2 + 1)

73. f(x) = (x2 + 1)5

74. f(x) = 1√1− x2

75. f(x) =(

sin 2x1 + tan 3x

)376. f(x) =

√1 + 3x2.

Find the derivative of the following functions:

77. y = arcsin (ln (x))

78. y = arctan ex

79. y = tan(arcsinx)

80. y =(

arcsinx+ 1sinx

)81. y = x2 arctanx2

82. y = earcsin x

83. y = ex arctanx2

In problems 84 through 89, use implicit differentiation for finddydx

for:

84. x2y2 + x2y + xy2 + x+ y + 1 = xy

85. x sin y = ln[x2(9y)

]86. x sin y − cos y = 0

87. x3 − x2y + y5 = 8

88. x3 + y3 = 3xy

89.√x+ xy2 − 2x3 + 1

2y3 = 4x2y

90. Given the curve xy2 − x3y = 6

a. Finddydx

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 x2y3 + 3xy2 + y = 5 at (1, 1).

In problems 92 through 99, use logarithmic differentiation to find f ′(x) for:

92. f(x) =√x+ 1 3

√x+ 2 5

√x+ 3

93. f(x) =(x+ 3)3

(x2 − 2)4

94. f(x) = ln

((x+ 2)2(x3 − 5)3/4√

2x+ 1

)

95. f(x) =(3x2 + 2x− 7)2

(7x3 − 8x)5

96. f(x) =(sin(2x+ 1))3

(4x3 + 6x)2

97. f(x) = ln[

7x2 + 13x3 − 2

]498. f(x) = x2

[2x3 − 5x2 + 4x√

x+ 2x+ 3

]299. f(x) =

√(x2 + 1x2 − 1

)3100. Find the equation of the tangent line to f(x) = x2 + 1 at the point where x = 3.

101. Find the equation of the tangent line to y = e2x ln(2x) at x = 2

102. Find the equation of the tangent line to f(x) = 34x

3 + 6x at the point (2, 18).

103. Find the equation of the tangent line to y = x3 − 2x2 at (1,−1).

104. Find the values of x for all points on the graph of f(x) = x3 − 2x2 + 5x− 16 at which the slope of thetangent line is 4.

105. Find the point(s) on the graph of y = x2 where the tangent line passes through the point (2,−12).

4 Applications of Derivatives

4.1 Related Rates

106. A rectangle has a width that is 1/4 its length. At what rate is the area increasing if its width is 20 cmand is increasing at 0.5 cm/s?

107. Air is leaking out of a spherical balloon at the rate of 3 cubic inches per minute. When the radius is 5inches, how fast is the radius decreasing?

108. Sand is being emptied from a hopper at the rate of 10 ft3/s. The sand forms a conical pile whoseheight is always twice its radius. At what rate is the radius of the pile increasing when the height is 5ft?

109. A balloon is 200 feet off the ground and rising vertically at the constant rate of 15 feet per second. Anautomobile passed beneath it traveling along a straight line at the constant rate of 66 miles per hour. Howfast is the distance between them changing one second later?

110. The top of a 25-foot ladder leaning against a vertical wall is slipping down the wall at the rate of onefoot per minute. How fast is the bottom of the ladder slipping along the ground when the bottom of theladder is 7 feet away from the base of the wall?

111. A plane flying parallel to the ground at the height of four kilometers passes directly over a radarstation. A short time later, the radar reveals that the plane is 5 km away and the distance between theplane and the station is increasing at the rate of 300 km/hr. (The distance is straight line distance fromground level at the station to the plane four kilometers high.) At that moment, how fast is the plane movinghorizontally?

112. A spotlight is on the ground 100 feet from the vertical side of a very tall building. A person six feettall 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 fromit?

113. Two balloons are attached so air can flow freely between them. If the radius of balloon A is decreasingat 2 in/min, what is the rate of change of balloon B. The radius of balloon A is 3 in and the radius of balloonB is 5 in.

114. A girl is flying a kite which is 120 feet above the ground. The wind is carrying the kite horizontallyaway from the girl at a speed of 10 feet/second. How fast must the string be let out when the string fromthe girl to the kite is 150 feet long (and taut)?

115. A rocket is rising straight up from the ground at the rate of 1000 km per hour. An observer 2 kmfrom the launching site is photographing the rocket. How fast is the angle θ of the camera with the groundchanging when the rocket is 1.5 km above the ground?

116. What is the radius of an expanding circle at the moment when the rate of change of its area isnumerically twice as large as the rate of change of its radius?

117. A woman standing on a cliff is watching a motor boat though a telescope as the boat approaches theshoreline directly below her. If the telescope is 250 feet above the water and if the boat is approaching at20 feet per second, at what rate is the angle of the telescope changing when the boat is 250 feet from shore?

118. Suppose water is leaking out of a cone shaped funnel at a rate of 2 in3/min. Assume the height of thecone is 16 in and the radius of the cone is 4 in. How fast is the depth of the water decreasing when the levelof water is 8 in. deep?

4.2 Graphing

119. Find where f(x) = 3x4 + 4x3 − 12x2 is increasing and decreasing.

120. Find where f(x) = 3x2 − 6x is increasing or decreasing

121. Find where f(x) = 4x5 − 5x4 is increasing or decreasing and classify all critical points.

122. Find the concavity of f(x) = 3x4 − 12x3 + 1

123. Find the concavity of f(x) = x4 − 4x3 and any inflection points.

For the functions in problems 124 to 128 find the domain, range, x-intercepts, y-intercepts,where y is increasing, decreasing, critical points, where y is concave up, concave down, inflectionpoints, vertical asymptotes, horizontal asymptotes, and sketch the graph of y.

124. y = xx2 + 1

125. y = 1x2 − 9

126. y =2(x2 − 9)x2 − 4

127. y = 7x2 − 4x2 + 4x+ 4

128. y = xx2 − 1

4.3 Max/Min

129. Find the maximum and minimum values of f(x) = 4x3 − 8x2 + 5x for −1 ≤ x ≤ 2.

130. What positive number exceeds its cube by the greatest amount? (Hint: x > x3 only if 0 < x < 1).

131. Find two positive numbers whose sum is 20 and whose product is as large as possible.

132. Among all rectangles with corners on the ellipse

x2

9+y2

4= 1

which has the largest area?

133. Find the absolute and local maximums and minimums for f(x) = 6x7 − 2x3 + 2 on [−1, 1].

134. Find the maximum and minimum of f(x) = xx2 + 1

on [0, 3]

135. Which points on the graph of y = 4− x2 are closest to (0, 2)?

136. Find the maximum and minimum of f(x) = 6x2 − 4x− 10 on the interval [−4, 5].

137. 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. Hewants to reach Prof. Butler as quickly as possible and recalls that he can swim at 6 km/h and run at 8km/h. Where should the student reach the other side of the shore relative to where he began if ProfessorButler is 8 km down stream to minimize time?

138. A donkey owner has 750 ft of fencing. He wants to make a rectangular area for them bound by a wallon one side. What dimensions will enable him to fit the largest number of asses inside the fenced area?

139. 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 tohave three sides. Fencing costs $3 a foot. The recent price gouging of elephant food has Milo in a financialbind, so he only has $3300 to spend on the fence. What are the dimensions that give Tiny the most room?

140. You want to make an open-topped rectangular box with square base. You want its volume to be 1000cubic centimeters. The material for the base costs 10 cents per square centimeter and the material for thesides costs 7 cents per square centimeter. Find the dimensions of the cheapest box to build.

141. Among the all the pairs of nonnegative numbers that add up to 5, find the pair that maximizes theproduct of the square of the first number and the cube of the second number.

142. A rectangular yard is to be laid out and fenced in, and divided into 10 enclosures by fences parallel toone side of the yard. If 22 miles of fencing is available, what dimensions will maximize the area?

143. A motorist is stranded in a desert 5 miles from a point A, which is the point on a long straight roadnearest him. He wishes to get to a point B on the road that is 10 miles from point A. He can travel at 15miles an hour in the desert and 39 miles an hour on the road. Find the point where he must meet the roadto get to B in the shortest possible time. Assume he travels in the desert in a straight line.

144. A rectangular area with fence all around is to be divide into three smaller areas by running two lengthsof fence parallel to one side. If you have 800 yds total of fence, what is the largest area that can be enclosed?

145. You work for an ice cream cone company and you are given the task of finding the greatest possiblevolume of a cone given a slant height of 4 inches.

146. A certain company owns two buildings. The first building is located on one side of a river that is1500 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 buildingswith cables. Of course, they want to minimize the cost of cabling since it will cost twice as much to laycable underwater as compared to laying cable above ground. What path should the cabling take in order tominimize the total cost?

147. A peanut vendor sells bags of peanuts. He sells them for $5 a bag. At this price he sells 500 bags aday. 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.

148. A window is to be made in the shape of a rectangle surmounted by a semicircle with diameter equalto the width of the rectangle. If the perimeter of the window is 22 feet, what dimensions will emit the mostlight (i.e. maximize the area)?

149. A greeny bus caries 10 passengers from North side to Schmitt Lecture Hall. The cost to ride is $0.50per 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?

150. An oil can is to be made in the form of a right circular cylinder and will contain 1000 cm3. What arethe dimensions of the can that requires the least amount of material?

151. A rectangular swimming pool is to be built with a 6 foot wide deck at the north and south ends, anda 10 foot wide deck at the east and west ends. If the total area available is 6000 square feet, what are thedimensions of the largest possible water area? Be sure to draw a picture of your set-up.

152. A flower bed will be in the shape of a sector of a circle (a pie-shaped region) of radius r and vertexangle θ. Find r and θ if its area is a constant A and the perimeter is a minimum.

153. A cylindrical container is to be produced that will have a capacity of 10 cubic meters. The top andbottom of the container are to be made of a material that costs $2 per square meter, while the side of thecontainer is to be made of material costing $1.50 per square meter. Find the dimensions that will minimizethe total cost of the container. What is this minimum cost?

4.4 Newton’s Method

154. Use Newton’s method to find where cosx = x to 4 decimal places.

155. Use Newton’s method to find the root of x3 − 3x2 + 1 = 0 between 2 and 3. (4 decimal places)

156. Find 3√

19 by using Newton’s method on f(x) = x3 − 19.

157. It is a dark and stormy night. You are drinking hot coffee as you do your chemistry lab. You areinspecting the graph of your data when a loud clap of thunder startles you. Your coffee spills- a dropfalls on to the graph and since the lab print-out is so thin the paper dissolves! There is now a hole rightwhere the graph crosses the x-axis. You need the x-intercept for you experiment. You know the equation isf(x) = x3/2 − 10. Use Newton’s method to find the x-intercept between 4.2 and 4.8.

4.5 Linear Approximaitons

158. Use a linear approximation L(x) to an appropriate function f(x), with an appropriate value of a toestimate

√257.

159. Use a linear approximation L(x) to an appropriate function f(x), with an appropriate value of a toestimate

√16.5.

160. Use a linear approximation L(x) to an appropriate function f(x), with an appropriate value of a toestimate

√25.3.

4.6 l’Hospital’s Rule

161. limx→−3

3x2 + 10x+ 3

x2 + 2x− 3

162. limx→4

√x− 2

x2 − 16

163. limx→1

x3 + 1

x2 + 1

164. limx→0

x2 cosx+ 2x

sinx

165. limx→0

√2 + x−

√2

x

166. limx→0

sinx

5x.

167. limx→∞

1 + x

5x2.

168. limx→0

1− cosx

3x.

169. limx→3−

x

x− 3.

170. limx→−1

x2 + 6x+ 5

x2 − 3x− 4.

171. limx→∞

x−√x2 − 3x

172. limx→0

sinx

3x

173. limx→∞

sinx

3x

174. limx→∞

(√x2 + 2x+ 3−

√x2 − 2x+ 3)

175. limx→0+

x lnx

176. limx→∞

x sinπ

x

177. limx→0+

xx

178. limx→∞

(1 +

2

x

)x

179. limx→0

(cosx)

(1x

)

180. limx→0

(1

x− 1

ex − 1

)

181. limx→∞

(1 + x2

)( 1lnx

)

182. limx→0

(2 + x)2 − 4

x

5 Integration

183. 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) = 13x

3 + 1; [−1, 2], n = 6.

184. If

∫ 3

1

f(x) dx = 4 and

∫ 3

1

g(x) = 2, find

∫ 3

1

4f(x)− g(x) dx

Compute the derivative of the following functions:

185. F (x) =

x∫2

1

1 + t3dt

186. F (x) =

2∫x

1

1 + t3dt

187. F (x) =

ln x∫2

1

1 + t3dt

188. F (x) =

∫ x

e

(t2 + 42t+ 42)5dt

Compute the following integrals:

189.

∫sinx√

1 + cosxdx

190.

∫sinx cosx dx

191.

∫ 3

2

(x2 + 3)(x3 − 1) dx

192.

∫(2x3 − 1)5x2 dx

193.

∫x+ 3x2 dx

194.

∫ 1

0

√x(x2 + 1) dx

195.

∫ 1

0

x(x2 + 1)3 dx

196.

∫x√

1 + x dx

197.

∫1

x(lnx)2dx

198.

∫x2ex

3

dx

199.

∫2x+ 1

x+ 1dx

200.

∫x√

3− 7x2 dx

201.

∫(2 cos 3t+ 5 sin 4t+ 3e7t)dt

202.

∫51/x

2

x3dx

203.

∫ex√ex + 1

dx

204.

∫ex ln(sin ex)

tan exdx

205.

∫e2x

1 + e2xdx

206.

∫x39x

4+2 dx

207.

∫ (x3 + 1

x5

)dx

208.

∫sec2 2x

tan 2xdx

209.

∫3√x+ x−3 dx

210.

∫1 +√x√

xdx

211.

∫2x+ 1

x2 + x+ 1dx

212.

∫6x2

cos2(4 + x3)dx

213.

∫dx

x lnx

214.

∫cos2 x sin3 x dx

215.

∫ 2

−2(10x9 + 3x5) dx

216.

∫ 4

0

√x dx

217.

∫ 9

4

16 + x

x2dx

218.

∫ 3π/4

π/4

sinx dx

219.

∫ 3

1

6

x2dx

220.

∫ 5

0

|x2 − 4x+ 3| dx

221.

∫ π/2

0

cosx dx

222.

∫ 1

0

(x3 − x2

)dx

223.

∫ 4

1

1√xdx

224.

∫ 3

0

e2x dx

225.

∫ π

0

cosx dx

226.

∫ π/3

π/4

sec2 x dx

227.

∫ 1

0

(2x+ 1)4 dx

228.

∫ 1

0

x√

1− x2 dx

229.

∫ 4

0

1√2x+ 1

dx

230.

∫ 9

1

1√x(1 +

√x)

dx

231.

∫ e2

e

1

x lnxdx

232.

∫ 1

0

1√ex

dx

233.

∫ √π0

x sinx2 dx

234.

∫ 8

0

x√

1 + x dx

235.

∫ 20

4

1

xdx

236.

∫x 3x

2

dx

237.

∫10x

10x + 1dx

238.

∫ 1/6

0

1√1− 9x2

dx

239.

∫x+ 3

x2 + 9dx

240.

∫1

x√

4x2 − 1dx

241.

∫5

x2 + 6x+ 13dx

242.

∫cosx

9 + sin2 xdx

243.

∫x2

1 + x2dx

6 Applications of Integration

6.1 Area

244. Find the area of the region R bounded by the line y = x+ 2 and the parabola y = x2 − 4.

245. Find the area of the region bounded by the graphs of f(x) = x3 − 6x and g(x) = −2x.

246. Find the area of the region enclosed by y = 3− x2 and y = −x+ 1 between x = 0 and x = 2.

247. Find the area of the region enclosed by y = x+ 4/x2, the x-axis, x = 2, and x = 4.

248. Find the area of the region enclosed by y = x+ 5 and y = x2 − 1.

249. Find the area of the region enclosed by x = y2 − 4y + 2 and x = y − 2.

250. Find the area of the region enclosed by y = 6x− x2 and y = x2 − 2x

6.2 Volume

251. Set-up the integral for the volume of revolution, if the region bounded by y = 4x− x2, and y = x2 isspun about x = 4

252. Set up the integral that represents the volume of the formed by revolving the region bounded by thegraphs of y =

√25− x2 and y = 3 about the x-axis. (Do not evaluate the integral.)

253. Find the volume generated when the region bounded by y = 1x, y = 0, x = 1and x = 4 is spun about

the x-axis.

254. Using the method of cylindrical shells, find the volume if the solid generated by rotating around theline x = 2 the region bounded by y = x2 + 1, y = 0 and x = 2.

255. Find the volume if the area enclosed by f(x) = x2, x = 4, x = 1, y = −2 is rotated about the liney = −2

256. Find the volume if the region bounded by y = x2, y = −2x+ 3, and the y-axis in the first quadrant, isrotated about the y-axis.

257. Find the volume of the solid that results when the area of the smaller region enclosed by y2 = 4x,y = 2, and x = 4 is revolved about the y-axis.

258. Find the volume of the solid that results when the first quadrant region enclosed by y = x3 and y = xis revolved about the y-axis.

259. Find the volume of the solid that results when the area enclosed by y2 = 4x, y = 2, and x = 4 isrevolved about the x-axis.

260. Find the volume of the solid that results when the area of the region enclosed by y = 2x, x = 0, andy = 2 is revolved about x = 1.

261. Find the volume of the solid that results when the area of the region enclosed by y2 = x3, x = 1, andy = 0 is revolved about the x-axis.

262. Find the volume of the solid that results when the area of the region enclosed by y2 = 4x and y = xis revolved about the line x = 4.

263. A storage tank is designed by rotating y = −x2 + 1, −1 ≤ x ≤ 1, about the x-axis where x and y aremeasured in meters. Use cylindrical shell to determine how many cubic meters the tank will hold.

6.3 Work

264. A spring has a natural length of 4 feet. A force of 30 lbs. is required to compress that spring to alength of 2.5 feet. How much work is done to stretch the spring from its natural length to 6 feet?

265. A 50 foot chain weighing 10 pounds per foot supports a beam weighing 1000 pounds. How much workis done in winding 40 feet of the chain onto a drum?

266. A conical tank has a diameter of 9 feet and is 12 feet deep. If the tank is filled with water of density62.4 lbs/ft3, how much work is required to pump the water over the top?

267. A cylindrical tank 8 feet in diameter and 10 feet high is filled with water weighing 62.4 lbs/ft3. Howmuch work is required to pump the water over the top of the tank?

268. Set up the integral for the work need to empty a right circular conical tank of altitude 20 ft and radiusof base 5 ft has its vertex at ground level and axis vertical. If the tank is full of orange marmalade weighting100 lb/ft3 pumping all the orange marmalade over the top of the tank.

269. Water is drawn from a well 50 feet deep using a bucket that scoops up 200 lbs of water. The bucketis pulled up at the rate of 3 ft/s, but it has a hole in the bottom through which water leaks out at a rateof 3/4 lb/s. How much work is done in pulling the bucket to the top of the well. Neglect the weight of thebucket and rope and work done to overcome friction.

270. You are a secret agent. You and your partner, who is kind of annoying, are trying to escape an evilsuper villain. His hit men chase you on foot for several dozen miles before your partner collapses and iscaptured. Even though you are really tried and you don’t like your partner that much, you infiltrate thevillain’s lair to try to rescue him. You find your partner has been knocked out and secured at the bottom ofa 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 decidethat you are only willing to expend 1 million ft pounds of work. Calculate the amount of work required tolift all the water out of the pool to decide if your partner will be rescued.

271. A 20 foot chain weighing 5 pounds per foot is lying coiled on the ground. How much work is requiredo raise one end of the chain to a height of 20 feet?

272. You are at an ice cream shop. You order your favorite ice cream, chocolate chip cookie dough. Youhaven’t eaten all day, and begin to gobble down the ice cream. If the ice cream cone is 2 feet long andhas a radius of 1 foot, how much work is required to eat all the ice cream. (The density of ice cream isρ = 55.5lb/ft3.)

Math 121Final Review

Answers

1. a. x3 + x2 + 4x+ 4

b. x5 + 4x4 + 4x3

c. 1x + 4

x2+ 4x3

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 sinx + x3 + 14x +3)2 + 13(3 sinx+ x3 + 14x+ 3)

4. (f ◦ g)(x) = tan2 x

5. 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) = (sinx− cosx)2 + sec(sinx− cosx), (g ◦ f)(x) = sin(x2 + secx)− cos(x2 + secx)

7. 2x+ y = 7

8. Domain x 6= ±1, f(5) = 124

9. y = 23(x− 1) + 5

10. 0

11. y = 32x−

12

12. f ◦ g(x) =

1x2

+ 2

1x2− 4

g ◦ f(x) = 1[x+ 2x− 4

]213. 4

14. 96

15. 15

16. 12

17. 1 + π2

18. 0

19. 12

20. 1/2

21. c = 3

22. k = 43

23. c = 65

24. Continuous everywhere except where x = 5

25. x = 1 is removable, x = −1 is not removable.

26. 81

27. f ′(x) = 2x− 2

28. f ′(x) = 3x2

29. f ′(x) = − 1(x+ 5)2

30. f ′(x) = 3

31. f ′(x) = 1(1− 2x)2

32. f ′(x) = 12√x+ 1

33. f ′(x) = cos2 x− sin2 x

34. f ′(x) = cosx− sinx− 1(1− cosx)2

35. f ′(x) =(x3 + 1)(2x− 3)− (x2 − 3x− 1)(3x2)

(x3 + 1)2

36. f ′(x) = x sec2 x+ tanx

37. f ′(x) = 2x9 sec2(2x) + 9x8 tan(2x)

38. f ′(x) = 6xx2 + 4

+ tanx

39. y′ = cosxesin x

40. f ′(x) = (−4)(

3− 2x1/3

)−5 (2

3x4/3

)

41. f ′(x) == −9 sin 3x+(√x)(1

3x−2/3 − 5)− ( 3

√x− 5x)( 1

2√x

)

x

42. f ′(x) = 2x cos(x2)

43. f ′(x) = 2(x+ tanx)(1 + sec2 x)

44. f ′(x) = 100(x2 + 2x− 15)99(2x+ 2)

45. f ′(x) = 2 sinx cosx− 5 sin(5x)

46. f ′(x) =( 32x−1/2 − 24x2)(1 + 5x2)− (3

√x− 8x3)(10x)

(1 + 5x2)2

47. f ′(x) = 10x+ cscx cotx(5x2 − cscx) ln(5x2 − cscx)

48. f ′(x) = (2x sinx+ x2 cosx)ex2 cos x + x2 sinxex

2 cos x(2x cosx− x2 sinx)

49. f ′(x) = 2x+ 12√x

+ cosx

50. f ′(x) =( 1x + 3)(3x2 − sinx)− (x3 + cosx)(− 1

x2)

( 1x + 3)2

51. f ′(x) = 1ln 5

(ln 7)( 3x )

52. f ′(x) =ex

2 12x −

12(lnx)2xex

2

(ex2

)2

53. f ′(x) = 3(coshx)2 sinhx

54. f ′(x) = 1 + lnx

55. f ′(x) =x2[3− x(ln 3)]

3x

56. f ′(x) = 3x cos 3x+ sin 3x

57. g′(x) = 2x2e2x + 2xe2x

58. h′(x) = 1(x+

√x2 + 1)

[1 + x√

x2 + 1

]59. f ′(x) = 1

60. f ′(x) = 5x(1 + x2)3/2

61. f ′(x) = − 12x2

(1 + 3

x

)362. f ′(x) = −4

((1+2x)3

(1+3x)5

)63. f ′(x) =

√x− 1

2√x

(x+ 1)

x

64. f ′(x) = (3 sin 4x)(−14x sin 7x2) + (12 cos 4x)(cos 7x2)

65. f ′(x) = (1− x3)43(2x+ 4)1/3(2) + (2x+ 4)4/3(−3x2)

66. f ′(x) = 2x(2x) + x2(2x) ln 2

67. f(x) = 2(cosx− 43x)3/2 + (2x)32(cosx− 43x)1/2(− sinx− (43x(ln 4)(3))

68. f ′(x) = (12x)(3x2 − 4) + (6x2 + 5)(6x)

69. f ′(x) =(4 + 3x+ x2)4(3)− (3x)4(4 + 3x+ x2)3(3 + 2x)

(4 + 3x+ x2)8

70. f ′(x) = 14(10x)(4 + 5x2)−3/4(6− x)6 − (4 + 5x2)1/46(6− x)5

71. f ′(x) =ln(x2 + 3x)(2x cos(x2))− sin(x2) 2x+ 3

x2 + 3x(ln(x2 + 3x))2

72. f ′(x) = 2xx2 + 1

73. f ′(x) = 10x(x2 + 1)4

74. f ′(x) = x(1− x2)3/2

75. f ′(x) = 3(

sin 2x1 + tan 3x

)2 [ (1 + tan 3x)(2 cos 2x)− 3(sin 2x)(sec2 3x)(1 + tan 3x)2

]76. f ′(x) = 3x√

1 + 3x2

77. y′ = 1x√

1− (lnx)2

78. y′ = ex

1 + e2x

79. y′ = (1− x2)−3/2

80. y′ = 1√1− x2

− cscx cotx

81. y′ = 2x3

1 + x4+ 2x arctanx2

82. y′ = earcsin x√1− x2

83. y′ = ex 2x1 + x4

+ ex arctanx2

84.dydx

=−2xy2 − 2xy − y2 + y − 12x2y + x2 + 2xy + 1− x

85.dydx

=sin y − 2

x1y − x cos y

86.dydx

=− sin y

x cos y + sin y

87.dydx

=2xy − 3x2

5y4 − x2

88.dydx

=y − x2y2 − x

89.dydx

=

12√x

+ y2 − 6x2 − 8xy

−x2y − 32y

2 + 4x2

90. a.dydx

=3x2y − y22xy − x3

b. At (1, 3) the y = 3, at (1,−2) then y + 2 = 2(x− 1)

c. x = 5√−24

91. y − 1 = −12(x− 1)

92. f ′(x) = (√x+ 1 3

√x+ 2 5

√x+ 3)

(12

1x+ 1 + 1

31

x+ 2 + 15

1x+ 3

)93. f ′(x) =

[3

x+ 3 −8x

x2 − 2

] [(x+ 3)3

(x2 − 2)4

]94. f ′(x) = 2

x+ 2 + 9x2

4(x3 − 5)− 1

2x+ 1

95. f ′(x) =(7x3 − 8x)5)(2)(3x2 + 2x− 7)(6x+ 2)− (3x2 + 2x− 7)2(5)(7x3 − 8x)4(21x2 − 8)

(7x3 − 8x)10

96. f ′(x) =(4x3 + 6x)23(sin(2x+ 1))2 cos(2x+ 1)2− (sin(2x+ 1))32(4x3 + 6x)(12x2 + 6)

(4x3 + 6x)4

97. f ′(x) = 4[

14x7x2 + 1

− 9x2

3x3 − 2

]98. f ′(x) = x22

[2x3 − 5x2 + 4x√

x+ 2x+ 3

] [(√x+ 2x+ 3)(6x2 − 10x+ 4)− (2x3 − 5x2 + 4x)(1

2x−1/2 + 2)

(√x+ 2x+ 3)2

]+2x

[2x3 − 5x2 + 4x√

x+ 2x+ 3

]2

99. f ′(x) = 32

√(x2 + 1x2 − 1

)(−4x

(x2 − 1)2

)100. y = 6x− 8

101. y − (e4 ln 4) = 178.7(x− 2)

102. y = 15x− 12

103. y = −x

104. 13 , 1

105. x = −2, 6

106. 80 cm2/s

107. drdt

= 3100π

108. drdt

= 45π

109. 53 ft/sec

110. Bottom is slipping away at 247 ft/min.

111. 500 km/hr

112. -1.2 ft per second.

113. 1825 in/min.

114. 6 ft/sec

115. 320 rad/hr

116. r = 1π

117. 125 radian per second

118.dydt

= −0.16in/min.

119. Inc (−2, 0) ∪ (1,∞) dec: (−∞,−2) ∪ (0, 1)

120. Inc: x > 1, dec: x < 1.

121. Inc: (−∞, 0) ∪ (1,∞), Dec: (0, 1) C.P. (0, 0) max (1,−1) min.

122. Concave up: (−∞, 0) ∪ (2,∞), concave down (0, 2).

123. Con up: (−∞, 0) ∪ (2,∞), Con dn: (0, 2) I.P. (0, 0) and (2,−16)

129. max. = 10 (at x = 2), min. = -17 (at x = −1)

130. x =√

13

131. 10 and 10

132. Corners at x = ±3√

22 and y = ±

√2

133. (−1,−6) abs min, (− 4√

1/7, 2.27) local max, ( 4√

1/7, 1.73) local min, (1, 6) abs max.

134. Maximum is .5 at x = 1, the minimum is 0 at x = 0.

135. (√

32 ,

52) and (−

√32 ,

52)

136. Max: 120 min: -32/3

137. He should reach shore 9√7

km down stream from where he started.

138. 375 by 187.5

139. 550 by 275

140. 11.2 X 11.2 X 8

141. First is 2, second is 3

142. 1 X 5.5

143. 2.08 miles A

144. 20,000

145. 25.8 in3

146. 866 m downstream.

147. $3.75

148. r = 22π + 4

149. $0.30

150. r = 3

√20004π and h = 2 3

√500π

151. 48 by 80

152. r =√A, θ = 2

153. r = 3

√308π

154. 0.7391

155. 2.8793

156. 3√

19 ≈ 2.67

157. x = 4.641589

158. f(x) =√x, x0 = 256 so

√257 ≈ 513

32

159. f(x) =√x, a = 16 so

√16.5 ≈ 4.0620

160. f(x) =√x, a = 25 so

√25.3 ≈ 5.03

161. 2

162. 132

163. 1

164. 2

165. 12√

2

166. 15

167. 0

168. 0

169. −∞

170. −45

171. 32

172. 13

173. 0

174. 2

175. 0

176. π

177. 1

178. e2

179. 1

180. 12

181. e2

182. 4

183. 5.0625 and 4.25

184. 14

185. F ′(x) = 11 + x3

186. F ′(x) = − 11 + x3

187. F ′(x) = 11 + (lnx)3

(1x

)188. F ′(x) = (x2 + 42x+ 42)5

189. −2√

1 + cosx+ C

190. 12 sin2 x+ C

191. 150.25

192.(2x3 − 1)6

36 + C

193. x22 + x3 + C

194. 2021

195. 158

196. 2x3 (1 + x)3/2 − 4

15 (1 + x)5/2 + C

197. − 1lnx

+ C

198. 13e

x3

+ C

199. 2x− ln |x+ 1|+ C

200. − 121(3− 7x2)3/2 + C

201. 23 sin 3t− 5

4 cos 4t+ 37e

7t + C

202. −51/x2

2 ln 5+ C

203. 2√ex + 1 + C

204. 12(ln(sin ex))2 + C

205. 12 ln |1 + e2x|+ C

206. 9x4+2

4 ln 9+ C

207. − 1x −

14x4

+ C

208. 12 ln | tan 2x|+ C

209. 34x

4/3 − 12x2

+ C

210. (1 +√x)2 + C

211. ln |x2 + x+ 1|+ C

212. 2 tan(4 + x3) + C

213. ln(lnx) + C

214. cos5 x5 − cos3 x

3 + C

215. 0

216. 163

217. 209 + ln 9

4

218.√

2

219. 4

220. 283

221. 1

222. − 112

223. 2

224. 12(e6 − 1)

225. 0

226.√

3− 1

227. 1215

228. 13

229. 2

230. ln 4

231. ln 2

232. 2− 2e−1/2

233. 1

234. 119215

235. ln 5

236. 3x2

2 ln 3+ C

237.ln(10x + 1)

ln 10+ C

238. π18

239. 12 ln(x2 + 9) + arctan x3 + C

240. arcsec(2x) + C

241. 52 arctan x+ 3

2 + C

242. 13 arctan

(sinx

3

)+ C

243. x− arctanx+ C

244. 1256

245. 8

246. 10/3

247. 7

248. 125/6

249. 9/2

250. 643

251. 2π

∫ 2

0

(4− x)[(4x− x2)− x2

]dx

252. π

4∫−4

(16− x2) dx

253. 34π

254. 20π3

255. V = 300.6π

256. V = 14π12

257. 98π5

258. 4π15

259. 18π

260. 4π3

261. π4

262. 64π5

263. 16π15

264. 40 ft-pounds

265. 52,000 ft lbs

266. 15,163 π

267. 49,920 π ft lbs

268. W =

∫ 20

0

100π(20− x)x2

16dx

269. 9687.5 ft pounds.

270. 252,720 π ft pounds.

271. 1000 ft-pounds

272. 58.1195 foot-pounds.