Bartleby Sitemap - Textbook Solutions
All Textbook Solutions for Calculus Volume 2
For the following exercises, use shells to find the volume generated by rotating the regions between the given curve and y=0 around the x-axis. 131. y=x2,x=0 , and x=2For the following exercises, use shells to find the volume generated by rotating the regions between the given curve and y=0 around the x-axis. 132. y=ex,x=0 , and x=1For the following exercises, use shells to find the volume generated by rotating the regions between the given curve and y=0 around the x-axis. 133. y=ln(x),x=1 , and x=eFor the following exercises, use shells to find the volume generated by rotating the regions between the given curve and y=0 around the x-axis. 134. x=11+y2,y=1 , and y=4For the following exercises, use shells to find the volume generated by rotating the regions between the given curve and y=0 around the x-axis. 135. x=1+y21,y=0 , and y=2For the following exercises, use shells to find the volume generated by rotating the regions between the given curve and y=0 around the x-axis. 136. x=cosy,y=0 , and y=For the following exercises, use shells to find the volume generated by rotating the regions between the given curve and y=0 around the x-axis. 137. x=y34y2,x=1 , and x=2For the following exercises, use shells to find the volume generated by rotating the regions between the given curve and y=0 around the x-axis. 138. x=yey,x=1 , and x=2For the following exercises, use shells to find the volume generated by rotating the regions between the given curve and y=0 around the x-axis. 139. x=cosyey,x=0 , and x=For the following exercises, find the volume generated when the region between the curves is rotated around the given axis. 140.y=3-x, y=0, x=0, and x=2 rotated around the y-axis.For the following exercises, find the volume generated when the region between the curves is rotated around the given axis. 141. y=x3, y=0, and y=8 rotated around the y-axis.For the following exercises, find the volume generated when the region between the curves is rotated around the given axis. 142.y=x2, y=x, rotated around the y-axis.For the following exercises, find the volume generated when the region between the curves is rotated around the given axis. 143. y=x , x=0, and x=1 rotated around the line x=2.For the following exercises, find the volume generated when the region between the curves is rotated around the given axis. 144. y=14-x , x=1, and x=2 rotated around the line x=4.For the following exercises, find the volume generated when the region between the curves is rotated around the given axis. 145.y=x and y=x2rotated around the y-axis.For the following exercises, find the volume generated when the region between the curves is rotated around the given axis. 146.y=x and y = x2rotated around the line x=2.For the following exercises, find the volume generated when the region between the curves is rotated around the given axis. 147. x=y3, y=,x=1, and y=2 rotated around the x-axis.For the following exercises, find the volume generated when the region between the curves is rotated around the given axis. 148.x=y2 and y = x rotated around the line y=2.For the following exercises, find the volume generated when the region between the curves is rotated around the given axis. 149.[T] Left of x=sin(y) , right of y=x, around the y-axis.For the following exercises, use technology to graph the region. Determine which method you think would be easiest to use to calculate the volume generated when the function is rotated around the specified axis. Then, use your chosen method to find the volume. 150.[T] y = x2and y=4x rotated around the y-axis.For the following exercises, use technology to graph the region. Determine which method you think would be easiest to use to calculate the volume generated when the function is rotated around the specified axis. Then, use your chosen method to find the volume. 151. [T] y=cos(x) , y=sin(x) x=14 , and x=54 rotated around the y-axis.For the following exercises, use technology to graph the region. Determine which method you think would be easiest to use to calculate the volume generated when the function is rotated around the specified axis. Then, use your chosen method to find the volume. 152.[T] y=x2- 2x, x=2, and x = 4 rotated around the y-axis.For the following exercises, use technology to graph the region. Determine which method you think would be easiest to use to calculate the volume generated when the function is rotated around the specified axis. Then, use your chosen method to find the volume. 153.[T] y = x2- 2x, x = 2, and x = 4 rotated around the x-axis.For the following exercises, use technology to graph the region. Determine which method you think would be easiest to use to calculate the volume generated when the function is rotated around the specified axis. Then, use your chosen method to find the volume. 154.[T] y = 3x3-2, y = x, and x = 2 rotated around the x-axis.For the following exercises, use technology to graph the region. Determine which method you think would be easiest to use to calculate the volume generated when the function is rotated around the specified axis. Then, use your chosen method to find the volume. 155.[T] y = 3x3— 2, y = x, and x = 2 rotated around the y-axis.For the following exercises, use technology to graph the region. Determine which method you think would be easiest to use to calculate the volume generated when the function is rotated around the specified axis. Then, use your chosen method to find the volume. 156.[T] x=sin(y2) and x=2y rotated around the x-axis.For the following exercises, use technology to graph the region. Determine which method you think would be easiest to use to calculate the volume generated when the function is rotated around the specified axis. Then, use your chosen method to find the volume. 157. [T] x = y2, x = y2- 2y + 1, and x = 2 rotated around the y-axis.For the following exercises, use the method of shells to approximate the volumes of some common objects, which are pictured in accompanying figures. 158. Use the method of shells to find the volume of a sphere of radius r.For the following exercises, use the method of shells to approximate the volumes of some common objects, which are pictured in accompanying figures. 159. Use the method of shells to find the volume of a cone with radius r and height h.For the following exercises, use the method of shells to approximate the volumes of some common objects, which are pictured in accompanying figures. 160. Use the method of shells to find the volume of an ellipse (x2/a2)+(y2/b2)=1 rotated around the x-axis.For the following exercises, use the method of shells to approximate the volumes of some common objects, which are pictured in accompanying figures. 161. Use the method of shells to find the volume of a cylinder with radius r and height h.For the following exercises, use the method of shells to approximate the volumes of some common objects, which are pictured in accompanying figures. 162. Use the method of shells to find the volume of the donut created when the circle x2+y2=4 is rotated around the line x=4.Consider the region enclosed by the graphs of y = f(x), y = 1 + f(x), x = 0, y = 0,and x = a >0. What is the volume of the solid generated when this region is rotated around the v-axis ? Assume that the function is defined over the interval [0, a].Consider the function y=f(x), which decreases from f(0)=b to f(1) = 0. Set up the integrals for determining the volume, using both the shell method and the disk method, of the solid generated when this region, with x = 0 and y = 0,is rotated around the y-axis. Prove that both methods approximate the same volume. Which method is easier to apply? (Hint: Since f(x) is one-to-one, there exists an inverse f-1 (y).)For the following exercises, find the Length of the functions over the given interval. 165. y=5x from x=0 to x=2For the following exercises, find the Length of the functions over the given interval. 166. y=12x+25 from x=1 to x=4For the following exercises, find the Length of the functions over the given interval. 167. x=4y from y=1 to y=1Pick an arbitrary linear function x = g(y) over any interval of your choice (y1,y2). Determine the length of the function and then prove the length is correct by U5ing geometry.Find the surface area of the volume generated when the curve y=x revolves around the x-axis from (1, 1) to (4,2), as seen here.Find the surface area of the volume generated when the curve y = x2revolves around the y-axis from (1, 1) to (3, 9).For the following exercises, find the lengths of the functions of x over the given interval. If you cannot evaluate the integral exactly, use technology to approximate it. 171. y=x3/2 from (0,0) to (1,1)For the following exercises, find the lengths of the functions of x over the given interval. If you cannot evaluate the integral exactly, use technology to approximate it. 172. y=x2/3 from (1,1) to (8,4)For the following exercises, find the lengths of the functions of x over the given interval. If you cannot evaluate the integral exactly, use technology to approximate it. 173. y=13(x2+2)2/3 from x=0 to x=1For the following exercises, find the lengths of the functions of x over the given interval. If you cannot evaluate the integral exactly, use technology to approximate it. 174. y=13(x22)2/3 from x=2 to x=4For the following exercises, find the lengths of the functions of x over the given interval. If you cannot evaluate the integral exactly, use technology to approximate it. 175. [T] y=ex on x=0 to x=1For the following exercises, find the lengths of the functions of x over the given interval. If you cannot evaluate the integral exactly, use technology to approximate it. 176. y=x33+14x from x=1 to x=3For the following exercises, find the lengths of the functions of x over the given interval. If you cannot evaluate the integral exactly, use technology to approximate it. 177. y=x44+18x2 from x=1 to x=2For the following exercises, find the lengths of the functions of x over the given interval. If you cannot evaluate the integral exactly, use technology to approximate it. 178. y=2x3/23x3/22 from x=1 to x=4For the following exercises, find the lengths of the functions of x over the given interval. If you cannot evaluate the integral exactly, use technology to approximate it. 179. y=127(9x2+6)3/2 from x=0 to x=2For the following exercises, find the lengths of the functions of x over the given interval. If you cannot evaluate the integral exactly, use technology to approximate it. 180. [T] y=sinx on x=0 to x=For the following exercises, find the Lengths of the functions of y over the given interval. If you cannot evaluatetheintegralexactly,usetechnologyto approximate it. 181. y=53x4 from y=0 to y=4For the following exercises, find the Lengths of the functions of y over the given interval. If you cannot evaluatetheintegralexactly,usetechnologyto approximate it. 182. x=12(ey+ey) from y=1 to y=1For the following exercises, find the Lengths of the functions of y over the given interval. If you cannot evaluatetheintegralexactly,usetechnologyto approximate it. 183. x=5ye3/2 from y=0 to y=1For the following exercises, find the Lengths of the functions of y over the given interval. If you cannot evaluatetheintegralexactly,usetechnologyto approximate it. 184. [T] x=y2 from y=0 to y=1For the following exercises, find the Lengths of the functions of y over the given interval. If you cannot evaluatetheintegralexactly,usetechnologyto approximate it. 185. x=y from y=0 to y=1For the following exercises, find the Lengths of the functions of y over the given interval. If you cannot evaluatetheintegralexactly,usetechnologyto approximate it. 186. x=23(y2+1)3/2 from y=1 to y=3For the following exercises, find the Lengths of the functions of y over the given interval. If you cannot evaluatetheintegralexactly,usetechnologyto approximate it. 187. [T] x=tan y from y=0 to y=34For the following exercises, find the Lengths of the functions of y over the given interval. If you cannot evaluatetheintegralexactly,usetechnologyto approximate it. 188. [T] x=cos2y from y=2 to y=2For the following exercises, find the Lengths of the functions of y over the given interval. If you cannot evaluatetheintegralexactly,usetechnologyto approximate it. 189. [T] x=cos2y from y=0 to y=2For the following exercises, find the Lengths of the functions of y over the given interval. If you cannot evaluatetheintegralexactly,usetechnologyto approximate it. 190. [T] x=ln(y) on y=1e to y=eFor the following exercises, find the surface area of the volume generated when the following curves revolve around the x-axis. If you cannot evaluate the integral exactly, use your calculator to approximate it. 191. y=x from x=2 to x=6For the following exercises, find the surface area of the volume generated when the following curves revolve around the x-axis. If you cannot evaluate the integral exactly, use your calculator to approximate it. 192. y=x3 from x=0 to x=1For the following exercises, find the surface area of the volume generated when the following curves revolve around the x-axis. If you cannot evaluate the integral exactly, use your calculator to approximate it. 193. y=7x from x=1 to x=1For the following exercises, find the surface area of the volume generated when the following curves revolve around the x-axis. If you cannot evaluate the integral exactly, use your calculator to approximate it. 194. [T] y=1x2 from x=1 to x=3For the following exercises, find the surface area of the volume generated when the following curves revolve around the x-axis. If you cannot evaluate the integral exactly, use your calculator to approximate it. 195. y=4x2 from x=0 to x=2For the following exercises, find the surface area of the volume generated when the following curves revolve around the x-axis. If you cannot evaluate the integral exactly, use your calculator to approximate it. 196. y=4x2 from x=1 to x=1For the following exercises, find the surface area of the volume generated when the following curves revolve around the x-axis. If you cannot evaluate the integral exactly, use your calculator to approximate it. 197. y=5x from x=1 to x=5For the following exercises, find the surface area of the volume generated when the following curves revolve around the x-axis. If you cannot evaluate the integral exactly, use your calculator to approximate it. 198. [T] y=tan x from x=4 to x=4For the following exercises, find the surface area of the volume generated when the following curves revolve around the y-axis. If you cannot evaluate the integral exactly, use your calculator to approximate it. 199. y=x2 from x=0 to x=2For the following exercises, find the surface area of the volume generated when the following curves revolve around the y-axis. If you cannot evaluate the integral exactly, use your calculator to approximate it. 200. y=12x2+12 from x=0 to x=1For the following exercises, find the surface area of the volume generated when the following curves revolve around the y-axis. If you cannot evaluate the integral exactly, use your calculator to approximate it. 201. y=x+1 from x=0 to x=3For the following exercises, find the surface area of the volume generated when the following curves revolve around the y-axis. If you cannot evaluate the integral exactly, use your calculator to approximate it. 202. [T] y=1x from x=12 to x=1For the following exercises, find the surface area of the volume generated when the following curves revolve around the y-axis. If you cannot evaluate the integral exactly, use your calculator to approximate it. 203. y=x3 from x=1 to x=27For the following exercises, find the surface area of the volume generated when the following curves revolve around the y-axis. If you cannot evaluate the integral exactly, use your calculator to approximate it. 204. [T] y=3x4 from x=0 to x=1For the following exercises, find the surface area of the volume generated when the following curves revolve around the y-axis. If you cannot evaluate the integral exactly, use your calculator to approximate it. 205. [T] y=1x from x=1 to x=3For the following exercises, find the surface area of the volume generated when the following curves revolve around the y-axis. If you cannot evaluate the integral exactly, use your calculator to approximate it. 206. y=cosx from x=0 to x=2The base of a lamp is constructed by revolving a quarter circle y=2xx2 around the y-axis from x=1 to x = 2, as seen here. Create an integral for the surface area of this curve and compute it.[T] A lampshade is constructed by rotating y =1/x around the x-axis from y = 1 to y=2, as seen here. Determine how much material you would need to construct this lampshade—that is, the surface area—accurate to four decimal places.[T] An anchor drags behind a boat according to the function y=24ex/224 , where y represents the depth beneath the boat and x is the horizontal distance of the anchor from the back of the boat. If the anchor is 23 ft below the boat, how much rope do you have to pull to reach the anchor? Round your answer to three decimal places.[T] You are building a bridge that will span 10 ft. You intend to add decorative rope in the shape of y=5|sin((x)/5)| , where x is the distance in feet from one end of the bridge. Find out how much rope you need to buy, rounded to the nearest foot.For the following exercises, find the exact are length for the following problems over the given interval. 212.y=ln(sinx) from x=/4 Tox=(3)/4 . (Hint: Recall trigonometric identities.)For the following exercises, find the exact are length for the following problems over the given interval. 213.Draw graphs of y = x2, y = x6, and y = x10. For y = xn, as n increases, formulate a prediction on the are length from (0, 0) to (1, 1). Now, compute the lengths of these three functions and determine whether your prediction is correct.For the following exercises, find the exact are length for the following problems over the given interval. 214. Compare the lengths of the parabola x=y2and the line x=by from (0, 0) to (b2, b) as b increases. What do you notice?For the following exercises, find the exact are length for the following problems over the given interval. 215. Solve for the length of x =y2 from (0, 0) to (l, 1). Show that x=(1/2)y2 from (0, 0) to (2, 2) is twice as long. Graph both functions and explain why this is so.For the following exercises, find the exact are length for the following problems over the given interval. 216.[T] Which is longer between (1, 1) and (2, 1/2): the hyperbola y =1/x or the graph of x+2y =3?For the following exercises, find the exact are length for the following problems over the given interval. 217. Explain why the surface area is infinite when y=1/x is rotated around the x-axis for 1x , but the volume is finite.For the following exercises, find the work done. 218. Find the work done when a constant force F = 12 lb moves a chair from x = 0.9 to x = 1.1 ft.For the following exercises, find the work done. 219. How much work is done when a person lifts a 50 lb box of comics onto a truck that is 3 ft off the ground?For the following exercises, find the work done. 220. What is the work done lifting a 20 kg child from the floor to a height of 2 m? (Note that 1 kg equates to 9.8 N)For the following exercises, find the work done. 221.Find the work done when you push a box along the floor 2 m, when you apply a constant force of F= 100 N.For the following exercises, find the work done. 222.Compute the work done for a force F = 12/x2 N from x = 1 to x = 2 m.For the following exercises, find the work done. 223. What is the work done moving a particle from x = 0 to x = 1 m if the force acting on it is F = 3x2 N?For the following exercises, find the mass of the one-dimensional object. 224. A wire that is 2 ft long (starting at x = 0) and has a density function of (x)=x2+2xlb/ftFor the following exercises, find the mass of the one-dimensional object. 225.A car antenna that is 3 ft long (starting at x = 0) and has a density function of (x)=3x+2 lb/ftFor the following exercises, find the mass of the one-dimensional object. 226. A metal rod that is 8 in. long (starting at x = 0) and has a density function of (x)=e1/2xlb/in.For the following exercises, find the mass of the one-dimensional object. 227.A pencil that is 4 in. long (starting at x = 2) and has a density function of (x)=5/x oz/in.For the following exercises, find the mass of the one-dimensional object. 228.A ruler that is 12 in. long (starting at x = 5) and has a density function of (x)=ln(x)+(1/2)x2 oz/in.For the following exercises, find the mass of the two-dimensional object that is centered at the origin. 229.An oversized hockey puck of radius 2 in. with density function density function (x)=x32x+5For the following exercises, find the mass of the two-dimensional object that is centered at the origin. 230.A frisbee of radius 6 in. with density function (x)=exFor the following exercises, find the mass of the two-dimensional object that is centered at the origin. 231.A plate of radius 10 in. with density function (x)=1+cos(x)For the following exercises, find the mass of the two-dimensional object that is centered at the origin. 232.A jar lid of radius 3 in. with density function (x)=ln(x+1)For the following exercises, find the mass of the two-dimensional object that is centered at the origin. 233. A disk of radius 5 cm with density function (x)=3xFor the following exercises, find the mass of the two-dimensional object that is centered at the origin. 234. A 12 -in. spring is stretched to 15 in. by a force of 75 lb. What is the spring constant?For the following exercises, find the mass of the two-dimensional object that is centered at the origin. 235. A spring has a natural length of 10 cm. It takes 2 J to stretch the spring to 15 cm. How much work would it take to stretch the spring from 15 cm to 20 cm?For the following exercises, find the mass of the two-dimensional object that is centered at the origin. 236. A 1 -m spring requires 10 J to stretch the spring to 1.1m. How much work would it take to stretch the spring from 1 into 1.2 m?For the following exercises, find the mass of the two-dimensional object that is centered at the origin. 237. A spring requires 5 J to stretch the spring from 8 cm to 12 cm, and an additional 4 J to stretch the spring from 12 cm to 14 cm. What is the natural length of the spring?For the following exercises, find the mass of the two-dimensional object that is centered at the origin. 238.A shock absorber is compressed 1 in. by a weight of 1 t. What is the spring constant?For the following exercises, find the mass of the two-dimensional object that is centered at the origin. 239.A force of F=20xx3 N stretches a nonlinear spring by x meters. What work is required to stretch the spring from x = 0 to x = 2 m?For the following exercises, find the mass of the two-dimensional object that is centered at the origin. 240. Find the work done by winding up a hanging cable of length 100 ft and weight-density 5 lb/ft.For the following exercises, find the mass of the two-dimensional object that is centered at the origin. 241.For the cable in the preceding exercise, how much work is done to lift the cable 50 ft?For the following exercises, find the mass of the two-dimensional object that is centered at the origin. 242.For the cable in the preceding exercise, how much additional work is done by hanging a 200 lb weight at the end of the cable?For the following exercises, find the mass of the two-dimensional object that is centered at the origin. 243.[T] A pyramid of height 500 ft has a square base 800 ft by 800 ft. Find the area A at height h. If the rock used to build the pyramid weighs approximately w=100 lb/ft3, how much work did it take to lift all the rock?For the following exercises, find the mass of the two-dimensional object that is centered at the origin. 244. [T] For the pyramid in the preceding exercise, assume there were 1000 workers each working 10 hours a day, 5 days a week, 50 weeks a year. If the workers, on average, lifted 10 100 lb rocks 2 ft/hr, how long did it take to build the pyramid?For the following exercises, find the mass of the two-dimensional object that is centered at the origin. 245. [T] The force of gravity on a mass m is F=((GMm)/x2) newtons. For a rocket of mass m = 1000 kg, compute the work to lift the rocket from x = 6400 to x = 6500 km. (Note: G=61017 N m2/kg2 and M=61024 kg.)For the following exercises, find the mass of the two-dimensional object that is centered at the origin. 246.[T] For the rocket in the preceding exercise, find the work to lift the rocket from x = 6400 to x= .For the following exercises, find the mass of the two-dimensional object that is centered at the origin. 247.[T] A rectangular dam is 40 ft high and 60 ft wide. Compute the total force F on the dam when a.the surface of the water is at the top of the dam and b.the surface of the water is halfway down the dam.For the following exercises, find the mass of the two-dimensional object that is centered at the origin. 248.[T] Find the work required to pump all the water out of a cylinder that has a circular base of radius 5 ft and height 200 ft. Use the fact that the density of water is 62 lb/ft3.For the following exercises, find the mass of the two-dimensional object that is centered at the origin. 249.[T] Find the work required to pump all the water out of the cylinder in the preceding exercise if the cylinder is only half full.For the following exercises, find the mass of the two-dimensional object that is centered at the origin. 250. [T] How much work is required to pump out a swimming pool if the area of the base is 800 ft2, the water is 4 ft deep, and the top is 1 ft above the water level? Assume that the density of water is 62 lb/ft3.For the following exercises, find the mass of the two-dimensional object that is centered at the origin. 251.A cylinder of depth H and cross-sectional area A stands full of water at density . Compute the work to pump all the water to the top.For the following exercises, find the mass of the two-dimensional object that is centered at the origin. 252.For the cylinder in the preceding exercise, compute the work to pump all the water to the top if the cylinder is only half full.For the following exercises, find the mass of the two-dimensional object that is centered at the origin. 253.A cone-shaped tank has a cross-sectional area that increases with its depth: A=(zr2h2)/H3 . Show that the work to empty it is half the work for a cylinder with the same height and base.The Grand Canyon Skywalk opened to the public on March 28, 2007. This engineering marvel is a horseshoe-shaped observation platform suspended 4000 ft above the Colorado River on the West Rim of the Grand Canyon. Its crystal-clear glass floor allows stunning views of the canyon below (see the following figure). The Skywalk is a cantilever design, meaning that the observation platform extends over the rim of the canyon, with no visible means of support below it. Despite the lack of visible support posts or struts, cantilever structures are engineered to be very stable and the Skywalk is no exception. The observation platform is attached firmly to support posts that extend 46 ft down into bedrock. The structure was built to withstand 100-mph winds and an 8.0-magnitude earthquake within 50 mi, and is capable of supporting more than 70,000,000 lb. One factor affecting the stability of the Skywalk is the center of gravity of the structure. We are going to calculate the center of gravity of the Skywalk, and examine how the center of gravity changes when tourists walk out onto the observation platform. The observation platform is U-shaped. The legs of the U are 10 ft wide and begin on land, under the visitors' center, 48 ft from the edge of the canyon. The platform extends 70 ft over the edge of the canyon. To calculate the center of mass of the structure, we treat it as a lamina and use a two-dimensional region in the xy-plane to represent the platform. We begin by dividing the region into three subregions so we can consider each subregion separately. The first region, denoted R1, consists of the curved part of the U. We model R1as a semicircular annulus, with inner radius 25 ft and outer radius 35 ft, centered at the origin (see the following figure). The legs of the platform, extending 35 ft between R1and the canyon wall, comprise the second sub-region, R2, Last, the ends of the legs, which extend 48 ft under the visitor center, comprise the third sub-region, R3. Assume the density of the lamina is constant and assume the total weight of the platform is 1,200,000 lb (not including the weight of the visitor center; we will consider that later). Use g = 32 ft/sec2. 1.Compute the area of each of the three sub-regions. Note that the areas of regions R2and R3should include the areas of the legs only, not the open space between them. Round answers to the nearest square footThe Grand Canyon Skywalk opened to the public on March 28, 2007. This engineering marvel is a horseshoe-shaped observation platform suspended 4000 ft above the Colorado River on the West Rim of the Grand Canyon. Its crystal-clear glass floor allows stunning views of the canyon below (see the following figure). The Skywalk is a cantilever design, meaning that the observation platform extends over the rim of the canyon, with no visible means of support below it. Despite the lack of visible support posts or struts, cantilever structures are engineered to be very stable and the Skywalk is no exception. The observation platform is attached firmly to support posts that extend 46 ft down into bedrock. The structure was built to withstand 100-mph winds and an 8.0-magnitude earthquake within 50 mi, and is capable of supporting more than 70,000,000 lb. One factor affecting the stability of the Skywalk is the center of gravity of the structure. We are going to calculate the center of gravity of the Skywalk, and examine how the center of gravity changes when tourists walk out onto the observation platform. The observation platform is U-shaped. The legs of the U are 10 ft wide and begin on land, under the visitors' center, 48 ft from the edge of the canyon. The platform extends 70 ft over the edge of the canyon. To calculate the center of mass of the structure, we treat it as a lamina and use a two-dimensional region in the xy-plane to represent the platform. We begin by dividing the region into three subregions so we can consider each subregion separately. The first region, denoted R1, consists of the curved part of the U. We model R1as a semicircular annulus, with inner radius 25 ft and outer radius 35 ft, centered at the origin (see the following figure). The legs of the platform, extending 35 ft between R1and the canyon wall, comprise the second sub-region, R2, Last, the ends of the legs, which extend 48 ft under the visitor center, comprise the third sub-region, R3. Assume the density of the lamina is constant and assume the total weight of the platform is 1,200,000 lb (not including the weight of the visitor center; we will consider that later). Use g = 32 ft/sec2. 2.Determine the mass associated with each of the three sub-regions.The Grand Canyon Skywalk opened to the public on March 28, 2007. This engineering marvel is a horseshoe-shaped observation platform suspended 4000 ft above the Colorado River on the West Rim of the Grand Canyon. Its crystal-clear glass floor allows stunning views of the canyon below (see the following figure). The Skywalk is a cantilever design, meaning that the observation platform extends over the rim of the canyon, with no visible means of support below it. Despite the lack of visible support posts or struts, cantilever structures are engineered to be very stable and the Skywalk is no exception. The observation platform is attached firmly to support posts that extend 46 ft down into bedrock. The structure was built to withstand 100-mph winds and an 8.0-magnitude earthquake within 50 mi, and is capable of supporting more than 70,000,000 lb. One factor affecting the stability of the Skywalk is the center of gravity of the structure. We are going to calculate the center of gravity of the Skywalk, and examine how the center of gravity changes when tourists walk out onto the observation platform. The observation platform is U-shaped. The legs of the U are 10 ft wide and begin on land, under the visitors' center, 48 ft from the edge of the canyon. The platform extends 70 ft over the edge of the canyon. To calculate the center of mass of the structure, we treat it as a lamina and use a two-dimensional region in the xy-plane to represent the platform. We begin by dividing the region into three subregions so we can consider each subregion separately. The first region, denoted R1, consists of the curved part of the U. We model R1as a semicircular annulus, with inner radius 25 ft and outer radius 35 ft, centered at the origin (see the following figure). The legs of the platform, extending 35 ft between R1and the canyon wall, comprise the second sub-region, R2, Last, the ends of the legs, which extend 48 ft under the visitor center, comprise the third sub-region, R3. Assume the density of the lamina is constant and assume the total weight of the platform is 1,200,000 lb (not including the weight of the visitor center; we will consider that later). Use g = 32 ft/sec2. 3.Calculate the center of mass of each of the three sub-regions.The Grand Canyon Skywalk opened to the public on March 28, 2007. This engineering marvel is a horseshoe-shaped observation platform suspended 4000 ft above the Colorado River on the West Rim of the Grand Canyon. Its crystal-clear glass floor allows stunning views of the canyon below (see the following figure). The Skywalk is a cantilever design, meaning that the observation platform extends over the rim of the canyon, with no visible means of support below it. Despite the lack of visible support posts or struts, cantilever structures are engineered to be very stable and the Skywalk is no exception. The observation platform is attached firmly to support posts that extend 46 ft down into bedrock. The structure was built to withstand 100-mph winds and an 8.0-magnitude earthquake within 50 mi, and is capable of supporting more than 70,000,000 lb. One factor affecting the stability of the Skywalk is the center of gravity of the structure. We are going to calculate the center of gravity of the Skywalk, and examine how the center of gravity changes when tourists walk out onto the observation platform. The observation platform is U-shaped. The legs of the U are 10 ft wide and begin on land, under the visitors' center, 48 ft from the edge of the canyon. The platform extends 70 ft over the edge of the canyon. To calculate the center of mass of the structure, we treat it as a lamina and use a two-dimensional region in the xy-plane to represent the platform. We begin by dividing the region into three subregions so we can consider each subregion separately. The first region, denoted R1, consists of the curved part of the U. We model R1as a semicircular annulus, with inner radius 25 ft and outer radius 35 ft, centered at the origin (see the following figure). The legs of the platform, extending 35 ft between R1and the canyon wall, comprise the second sub-region, R2, Last, the ends of the legs, which extend 48 ft under the visitor center, comprise the third sub-region, R3. Assume the density of the lamina is constant and assume the total weight of the platform is 1,200,000 lb (not including the weight of the visitor center; we will consider that later). Use g = 32 ft/sec2. 4.Now, treat each of the three sub-regions as a point mass located at the center of mass of the corresponding sub-region. Using this representation, calculate the center of mass of the entire platform.The Grand Canyon Skywalk opened to the public on March 28, 2007. This engineering marvel is a horseshoe-shaped observation platform suspended 4000 ft above the Colorado River on the West Rim of the Grand Canyon. Its crystal-clear glass floor allows stunning views of the canyon below (see the following figure). The Skywalk is a cantilever design, meaning that the observation platform extends over the rim of the canyon, with no visible means of support below it. Despite the lack of visible support posts or struts, cantilever structures are engineered to be very stable and the Skywalk is no exception. The observation platform is attached firmly to support posts that extend 46 ft down into bedrock. The structure was built to withstand 100-mph winds and an 8.0-magnitude earthquake within 50 mi, and is capable of supporting more than 70,000,000 lb. One factor affecting the stability of the Skywalk is the center of gravity of the structure. We are going to calculate the center of gravity of the Skywalk, and examine how the center of gravity changes when tourists walk out onto the observation platform. The observation platform is U-shaped. The legs of the U are 10 ft wide and begin on land, under the visitors' center, 48 ft from the edge of the canyon. The platform extends 70 ft over the edge of the canyon. To calculate the center of mass of the structure, we treat it as a lamina and use a two-dimensional region in the xy-plane to represent the platform. We begin by dividing the region into three subregions so we can consider each subregion separately. The first region, denoted R1, consists of the curved part of the U. We model R1as a semicircular annulus, with inner radius 25 ft and outer radius 35 ft, centered at the origin (see the following figure). The legs of the platform, extending 35 ft between R1and the canyon wall, comprise the second sub-region, R2, Last, the ends of the legs, which extend 48 ft under the visitor center, comprise the third sub-region, R3. Assume the density of the lamina is constant and assume the total weight of the platform is 1,200,000 lb (not including the weight of the visitor center; we will consider that later). Use g = 32 ft/sec2. 5.Assume the visitor center weighs 2,200,000 lb, with a center of mass corresponding to the center of mass of R3. Treating the visitor center as a point mass, recalculate the center of mass of the system. How does the center of mass change?The Grand Canyon Skywalk opened to the public on March 28, 2007. This engineering marvel is a horseshoe-shaped observation platform suspended 4000 ft above the Colorado River on the West Rim of the Grand Canyon. Its crystal-clear glass floor allows stunning views of the canyon below (see the following figure). The Skywalk is a cantilever design, meaning that the observation platform extends over the rim of the canyon, with no visible means of support below it. Despite the lack of visible support posts or struts, cantilever structures are engineered to be very stable and the Skywalk is no exception. The observation platform is attached firmly to support posts that extend 46 ft down into bedrock. The structure was built to withstand 100-mph winds and an 8.0-magnitude earthquake within 50 mi, and is capable of supporting more than 70,000,000 lb. One factor affecting the stability of the Skywalk is the center of gravity of the structure. We are going to calculate the center of gravity of the Skywalk, and examine how the center of gravity changes when tourists walk out onto the observation platform. The observation platform is U-shaped. The legs of the U are 10 ft wide and begin on land, under the visitors' center, 48 ft from the edge of the canyon. The platform extends 70 ft over the edge of the canyon. To calculate the center of mass of the structure, we treat it as a lamina and use a two-dimensional region in the xy-plane to represent the platform. We begin by dividing the region into three subregions so we can consider each subregion separately. The first region, denoted R1, consists of the curved part of the U. We model R1as a semicircular annulus, with inner radius 25 ft and outer radius 35 ft, centered at the origin (see the following figure). The legs of the platform, extending 35 ft between R1and the canyon wall, comprise the second sub-region, R2, Last, the ends of the legs, which extend 48 ft under the visitor center, comprise the third sub-region, R3. Assume the density of the lamina is constant and assume the total weight of the platform is 1,200,000 lb (not including the weight of the visitor center; we will consider that later). Use g = 32 ft/sec2. 6.Although the Skywalk was built to limit the number of people on the observation platform to 120, the platform is capable of supporting up to 800 people weighing 200 lb each. If all 800 people were allowed on the platform, and all of them went to the farthest end of the platform, how would the center of gravity of the system be affected? (Include the visitor center in the calculations and represent the people by a point mass located at the farthest edge of the platform, 70 fit from the canyon wall.)For the following exercises, calculate the center of mass for the collection of masses given. 254.m1= 2 at x1=1 and m2=4 at x2=2For the following exercises, calculate the center of mass for the collection of masses given. 255.m1=1 at x1=-1 and m2=3 at x2=2For the following exercises, calculate the center of mass for the collection of masses given. 256.m=3 at x = 0, 1, 2, 6For the following exercises, calculate the center of mass for the collection of masses given. 257. Unit masses at (a, y) = (1, 0), (0, 1), (1, 1)For the following exercises, calculate the center of mass for the collection of masses given. 258.m1=1 at (1, 0) and m2 = 4 at (0, 1)For the following exercises, calculate the center of mass for the collection of masses given. 259.m1=l at (1, 0) and m2= 3 at (2, 2)For the following exercises, compute the center of mass x. 260. =1 for x(1,3)For the following exercises, compute the center of mass x. 261. =x2 for x(0,L)For the following exercises, compute the center of mass x. 262. =1 for x(0,1) and =2 for x(0,2)For the following exercises, compute the center of mass x. 263. =sinx for x(0,2)For the following exercises, compute the center of mass x. 264. =cosx for x(0,2)For the following exercises, compute the center of mass x. 265. =ex for x(0,2)For the following exercises, compute the center of mass x. 266. =x3+xex for x(0,1)For the following exercises, compute the center of mass x. 267. =x sin x for x(0,)For the following exercises, compute the center of mass x. 268. =x for x(1,4)For the following exercises, compute the center of mass x. 269. =lnx for x(1,e)For the following exercises, compute the center of mass (x,y). Use symmetry to help locate the center of mass whenever possible. 270. =7 in the square 0x1 , 0y1For the following exercises, compute the center of mass (x,y). Use symmetry to help locate the center of mass whenever possible. 271.=3 in the triangle with vertices (0, 0), (a, 0), and (0, b)For the following exercises, compute the center of mass (x,y). Use symmetry to help locate the center of mass whenever possible. 272. =2 for the region bounded by y=cos(x), y =-cos(x), x=2 , and x=2For the following exercises, use a calculator to draw the region, then compute the center of mass (x,y) . Use symmetry to help locate the center of mass whenever possible. 273. [T] The region bounded by y = cos(2x), x=4 and x=4For the following exercises, use a calculator to draw the region, then compute the center of mass (x,y) . Use symmetry to help locate the center of mass whenever possible. 274.[T] The region between y = 2x2,y = 0, x = 0, and x = 1For the following exercises, use a calculator to draw the region, then compute the center of mass (x,y) . Use symmetry to help locate the center of mass whenever possible. 275.[T] The region between y=54x2 and y = 5For the following exercises, use a calculator to draw the region, then compute the center of mass (x,y) . Use symmetry to help locate the center of mass whenever possible. 276.[T] Region between y=x , y=ln(x), x = 1, and x = 4For the following exercises, use a calculator to draw the region, then compute the center of mass (x,y) . Use symmetry to help locate the center of mass whenever possible. 277. [T] The region bounded by y = 0, x=x24+y29=1For the following exercises, use a calculator to draw the region, then compute the center of mass (x,y) . Use symmetry to help locate the center of mass whenever possible. 278. [T] The region bounded by y = 0, x = 0, and x24+y29=1For the following exercises, use a calculator to draw the region, then compute the center of mass (x,y) . Use symmetry to help locate the center of mass whenever possible. 279.[T] The region bounded by y = x and y = x4in the first quadrantFor the following exercises, use the theorem of Pappus to determine the volume of the shape. 280. Rotating y = mx around the x -axis between x = 0 and x = 1For the following exercises, use the theorem of Pappus to determine the volume of the shape. 281. Rotating y = mx around the y -axis between x = 0 and x = 1For the following exercises, use the theorem of Pappus to determine the volume of the shape. 282.A general cone created by rotating a triangle with vertices (0, 0), (a, 0), and (0, b) around the y -axis. Does your answer agree with the volume of a cone?For the following exercises, use the theorem of Pappus to determine the volume of the shape. 283.A general cylinder created by rotating a rectangle with vertices (0, 0), (a, 0), (0, b), and (a, b) around the y -axis. Does your answer agree with the volume of a cylinder?For the following exercises, use the theorem of Pappus to determine the volume of the shape. 284. A sphere created by rotating a semicircle with radius a around the y-axis. Does your answer agree with the volume of a sphere?For the following exercises, use a calculator to draw the region enclosed by the curve. Find the area M and the centroid (x,y) for the given shapes. Use symmetry to help locate the center of mass whenever possible. 285. [T] Quarter-circle: y=1x2, y = 0, and x=0For the following exercises, use a calculator to draw the region enclosed by the curve. Find the area M and the centroid (x,y) for the given shapes. Use symmetry to help locate the center of mass whenever possible. 286.[T] Triangle: y = x, y = 2 - x, and y = 0For the following exercises, use a calculator to draw the region enclosed by the curve. Find the area M and the centroid (x,y) for the given shapes. Use symmetry to help locate the center of mass whenever possible. 287.[T] Lens: y = x2and y = xFor the following exercises, use a calculator to draw the region enclosed by the curve. Find the area M and the centroid (x,y) for the given shapes. Use symmetry to help locate the center of mass whenever possible. 288.[T] Ring: y2+x2= 1 and y2+ x2= 4For the following exercises, use a calculator to draw the region enclosed by the curve. Find the area M and the centroid (x,y) for the given shapes. Use symmetry to help locate the center of mass whenever possible. 289.[T] Half-ring: y2+x2=1 , y2+x2=4 , and y=0Find the generalized center of mass in the sliver between y = xaand y = xbwith a >b. Then, use the Pappus theorem to find the volume of the solid generated when revolving around the y-axis.Find the generalized center of mass between y = a2- x2, x = 0, and y = 0. Then, use the Pappus theorem to find the volume of the solid generated when revolving around the y-axis.Find the generalized center of mass between y = b sin(ax), x = 0, and x=a . Then, use the Pappus theorem to find the volume of the solid generated when revolving around the y-axis.Use the theorem of Pappus to find the volume of a torus (pictured here). Assume that a disk of radius a is positioned with the left end of the circle at x = b, b >0, and is rotated around the y-axis.Find the center of mass (x,y) for a thin wire along the semicircle y=1x2 with unit mass. (Hint: Use the theorem of Pappus.)For the following exercises, find the derivative dydx . 295. y=ln(2x)For the following exercises, find the derivative dydx . 296. y=ln(2x+1)For the following exercises, find the derivative dydx . 297. y=1lnxFor the following exercises, find the indefinite integral. 298. dt3tFor the following exercises, find the indefinite integral. 299. dx1+xFor the following exercises, find the derivative dy/dx. (You can use a calculator to plot the function and the derivative to confirm that it is correct.) 300. [T] y=ln(x)xFor the following exercises, find the derivative dy/dx. (You can use a calculator to plot the function and the derivative to confirm that it is correct.) 301. [T] y=xln(x)For the following exercises, find the derivative dy/dx. (You can use a calculator to plot the function and the derivative to confirm that it is correct.) 302. [T] y=log10xFor the following exercises, find the derivative dy/dx. (You can use a calculator to plot the function and the derivative to confirm that it is correct.) 303. [T] y=ln(sinx)For the following exercises, find the derivative dy/dx. (You can use a calculator to plot the function and the derivative to confirm that it is correct.) 304. [T] y=ln(lnx)For the following exercises, find the derivative dy/dx. (You can use a calculator to plot the function and the derivative to confirm that it is correct.) 305. [T] y=7ln(4x)For the following exercises, find the derivative dy/dx. (You can use a calculator to plot the function and the derivative to confirm that it is correct.) 306. [T] y=7ln((4x)7)For the following exercises, find the derivative dy/dx. (You can use a calculator to plot the function and the derivative to confirm that it is correct.) 307. [T] y=ln(tanx)For the following exercises, find the derivative dy/dx. (You can use a calculator to plot the function and the derivative to confirm that it is correct.) 308. [T] y=ln(tan(3x))For the following exercises, find the derivative dy/dx. (You can use a calculator to plot the function and the derivative to confirm that it is correct.) 309. [T] y=ln(cos2x)For the following exercises, find the definite or indefinite integral. 310. 01dx3+xFor the following exercises, find the definite or indefinite integral. 311. 01dt3+2tFor the following exercises, find the definite or indefinite integral. 312. 02xdx x 2+1For the following exercises, find the definite or indefinite integral. 313. 02 x 3dx x 2+1For the following exercises, find the definite or indefinite integral. 314. 2edxxlnxFor the following exercises, find the definite or indefinite integral. 315. 2edx (xln(x)) 2For the following exercises, find the definite or indefinite integral. 316. cosxdxsinxFor the following exercises, find the definite or indefinite integral. 317. 0/4tanxdxFor the following exercises, find the definite or indefinite integral. 318. cot(3x)dxFor the following exercises, find the definite or indefinite integral. 319. (lnx) 2 dx xForthe following exercises, compute dy/dx by differentiating ln y. 320. y=x2+1Forthe following exercises, compute dy/dx by differentiating ln y. 321. y=x2+1x21Forthe following exercises, compute dy/dx by differentiating ln y. 322. y=esinxForthe following exercises, compute dy/dx by differentiating ln y. 323. y=x1/xForthe following exercises, compute dy/dx by differentiating ln y. 324. y=e(ex)Forthe following exercises, compute dy/dx by differentiating ln y. 325. y=xeForthe following exercises, compute dy/dx by differentiating ln y. 326. y=x(ex)Forthe following exercises, compute dy/dx by differentiating ln y. 327. y=xx3x6Forthe following exercises, compute dy/dx by differentiating ln y. 328. y=x1/lnxForthe following exercises, compute dy/dx by differentiating ln y. 329. y=elnxFor the following exercises, evaluate by any method. 330. 510dtt5x10xdttFor the following exercises, evaluate by any method. 331. 1edxx+21dxxFor the following exercises, evaluate by any method. 332. ddxx1dttFor the following exercises, evaluate by any method. 333. ddxxx2dttFor the following exercises, evaluate by any method. 334. ddxln(secx+tanx)For the following exercises, use the function ln x. If you are unable to find intersection points analytically, use a calculator. 335.Find the area of the region enclosed by x = 1 and y = 5 above y = ln x.For the following exercises, use the function ln x. If you are unable to find intersection points analytically, use a calculator. 336. [T] Find the arc length of ln x from x = 1 to x = 2.For the following exercises, use the function ln x. If you are unable to find intersection points analytically, use a calculator. 337.Find the area between ln x and the x-axis from x = 1 to x = 2.For the following exercises, use the function ln x. If you are unable to find intersection points analytically, use a calculator. 338. Find the volume of the shape created when rotating this curve from x = 1 to x = 2 around the x-axis, as pictured here.For the following exercises, use the function ln x. If you are unable to find intersection points analytically, use a calculator. 339.[T] Find the surface area of the shape created when rotating the curve in the previous exercise from x = 1 to x = 2 around the x-axis.Find the area of the hyperbolic quarter-circle enclosed by x=2andy=2abovey=1xIf you are unable to find intersection points analytically in the following exercises, use a calculator. 341. [T] Find the arc length of y = 1/x from x = 1 to x = 4.If you are unable to find intersection points analytically in the following exercises, use a calculator. 342.Find the area under y = 1/x and above the x-axis from x = 1 to x = 4.For the following exercises, verify the derivatives and antiderivatives. 343. ddxln(x+x2+1)=11+x2For the following exercises, verify the derivatives and antiderivatives. 344. ddxln(xax+a)=2a(x2a2)For the following exercises, verify the derivatives and antiderivatives. 345. ddxln(1+ x 2 +1x)=1x1x2For the following exercises, verify the derivatives and antiderivatives. 346. ddxln(x+x2a2)=1xx2a2For the following exercises, verify the derivatives and antiderivatives. 347. dxxln(x)ln(lnx)=ln(ln(lnx))+CTrue or False? If true, prove it. If false, find the true answer. 348. The doubling time for y = ectis (ln (2))/(ln (c)).True or False? If true, prove it. If false, find the true answer. 349.If you invest $500, an annual rate of interest of 3% yields more money in the first year than a 2.5% continuous rate of interest.True or False? If true, prove it. If false, find the true answer. 350. If you leave a 100°C pot of tea at room temperature (25°C) and an identical pot in the refrigerator (5°C), with k = 0.02, the tea in the refrigerator reaches a drinkable temperature (70°C) more than 5 minutes before the tea at room temperature.True or False? If true, prove it. If false, find the true answer. 351.If given a half-life of t years, the constant k for y = ektis calculated by k=ln(1/2)/t .For the following exercises, use y=y0ekt . 352.If a culture of bacteria doubles in 3 hours, how many hours does it take to multiply by 10?For the following exercises, use y=y0ekt . 353.If bacteria increase by a factor of 10 in 10 hours, how many hours does it take to increase by 100?For the following exercises, use y=y0ekt . 354.How old is a skull that contains one-fifth as much radiocarbon as a modern skull? Note that the half-life of radiocarbon is 5730 years.For the following exercises, use y=y0ekt . 355. If a relic contains 90% as much radiocarbon as new material, can it have come from the time of Christ (approximately 2000 years ago)? Note that the half-life of radiocarbon is 5730 years.For the following exercises, use y=y0ekt . 356. The population of Cairo grew from 5 million to 10 million in 20 years. Use an exponential model to find when the population was 8 million.For the following exercises, use y=y0ekt . 357.The populations of New York and Los Angeles are growing at 1% and 1.4% a year, respectively. Starting from 8 million (New York) and 6 million (Los Angeles), when are the populations equal?For the following exercises, use y=y0ekt . 358.Suppose the value of $1 in Japanese yen decreases at 2% per year. Starting from $1 = ¥250, when will $1=¥1?For the following exercises, use y=y0ekt . 359. The effect of advertising decays exponentially. If 40% of the population remembers a new product after 3 days, how long will 20% remember it?For the following exercises, use y=y0ekt . 360.If y= 1000 at t = 3 and y = 3000 at t = 4, what was y0 at t = 0?For the following exercises, use y=y0ekt . 361.If y = 100 at t = 4 and y = 10 at t = 8, when does y = 1?For the following exercises, use y=y0ekt . 362. If a bank offers annual interest of 7.5% or continuous interest of 7.25%, which has a better annual yield?For the following exercises, use y=y0ekt . 363.What continuous interest rate has the same yield as an annual rate of 9%?For the following exercises, use y=y0ekt . 364.If you deposit $5000 at 8% annual interest, how many years can you withdraw $500 (starling after the first year) without running out of money?For the following exercises, use y=y0ekt . 365.You are trying to save $50,000 in 20 years for college tuition for your child. If interest is a continuous 10%, how much do you need to invest initially?For the following exercises, use y=y0ekt . 366.You are cooling a turkey that was taken out of the oven with an internal temperature of 165°F. After 10 minutes of resting the turkey in a 70°F apartment, the temperature has reached 155°F. What is the temperature of the turkey 20 minutes after taking it out of the oven?For the following exercises, use y=y0ekt . 367.You are trying to thaw some vegetables that are at a temperature of 1°F. To thaw vegetables safely, you must put them in the refrigerator, which has an ambient temperature of 44°F. You check on your vegetables 2 hours after putting them in the refrigerator to find that they are now 12°F Plot the resulting temperature curve and use it to determine when the vegetables reach 33 °F.For the following exercises, use y=y0ekt . 368. You are an archaeologist and are given a bone that is claimed to be from a Tyrannosaurus Rex. You know these dinosaurs lived during the Cretaceous Era (146 million years to 65 million years ago), and you find by radiocarbon dating that there is 0.000001% the amount of radiocarbon. Is this bone from the Cretaceous?For the following exercises, use y=y0ekt . 369. The spent fuel of a nuclear reactor contains plutonium-239, which has a half-life of 24,000 years. If 1 barrel containing 10 kg of plutonium-239 is sealed, how many years must pass until only 10g of plutonium-239 is left?For the next set of exercises, use the following table, which features the world population by decade. Years since 1950 Population (millions) 0 2,556 10 3,039 20 3,706 30 4,453 40 5,279 50 6,083 60 6,849 Source: http://www.factmonster.com/ipka/A0762181.html. 370.[T] The best-fit exponential curve to the data of the form P(t) = aebtis given by P(t) = 2686e0.01604t. Use a graphing calculator to graph the data and the exponential curve together.For the next set of exercises, use the following table, which features the world population by decade. Years since 1950 Population (millions) 0 2,556 10 3,039 20 3,706 30 4,453 40 5,279 50 6,083 60 6,849 Source: http://www.factmonster.com/ipka/A0762181.html. 371. [T] Find and graph the derivative y' of your equation. Where is it increasing and what is the meaning of this increase?For the next set of exercises, use the following table, which features the world population by decade. Years since 1950 Population (millions) 0 2,556 10 3,039 20 3,706 30 4,453 40 5,279 50 6,083 60 6,849 Source: http://www.factmonster.com/ipka/A0762181.html. 372.[T] Find and graph the second derivative of your equation. Where is it increasing and what is the meaning of this increase?For the next set of exercises, use the following table, which features the world population by decade. Years since 1950 Population (millions) 0 2,556 10 3,039 20 3,706 30 4,453 40 5,279 50 6,083 60 6,849 373. [T] Find the predicted date when the population reaches 10 billion. Using your previous answers about the first and second derivatives, explain why exponential growth is unsuccessful in predicting the future.For the next set of exercises, use the following table, which shows the population of San Francisco during the 19th century. Years since 1850 Population (thousands) 0 21.00 10 56.80 20 149.5 30 234.0 374.[T] The best-fit exponential curve to the data of the form P(t) = aebtis given by P(t) = 35.26e0.06407t. Use a graphing calculator to graph the data and the exponential curve together.For the next set of exercises, use the following table, which shows the population of San Francisco during the 19th century. Years since 1850 Population (thousands) 0 21.00 10 56.80 20 149.5 30 234.0 375. [T] Find and graph the derivative y' of your equation. Where is it increasing? What is the meaning of this increase? Is there a value where the increase is maximal?For the next set of exercises, use the following table, which shows the population of San Francisco during the 19th century. Years since 1850 Population (thousands) 0 21.00 10 56.80 20 149.5 30 234.0 376.[T] Find and graph the second derivative of your equation. Where is it increasing? What is the meaning of this increase?[T] Find expressions for cosh x + sinh x and cosh x - sinh x. Use a calculator to graph these functions and ensure your expression is correct.From the definitions of cosh(x) and sinh(x), find their antiderivatives.Show that cosh(x) and sinh(x) satisfy y" = y.Use the quotient rule to verify that tanh(x)' = sech2(x).Derive cosh2(x) + sinh2(x) = cosh(2x) from the definition.Take the derivative of the previous expression to find an expression for sinh(2x).Prove sinh(x + y) = sinh(x)cosh(y) + cosh(x)sinh(y) by changing the expreS5ion to exponentials.Take the derivative of the previous expression to find an expression for cosh(x + y).For the following exercises, find the derivatives of the given functions and graph along with the function to ensure your answer is correct. 385.[T] cosh(3x+ 1)For the following exercises, find the derivatives of the given functions and graph along with the function to ensure your answer is correct. 386.[T] sinh(x2)For the following exercises, find the derivatives of the given functions and graph along with the function to ensure your answer is correct. 387.[T] 1cosh(x)For the following exercises, find the derivatives of the given functions and graph along with the function to ensure your answer is correct. 388.[T] sinh(ln(x))For the following exercises, find the derivatives of the given functions and graph along with the function to ensure your answer is correct. 389.[T] cosh2(x) + sinh2(x)For the following exercises, find the derivatives of the given functions and graph along with the function to ensure your answer is correct. 390.[T] cosh2(x)- sinh2(x)For the following exercises, find the derivatives of the given functions and graph along with the function to ensure your answer is correct. 391. [T] tanh(x2+1)For the following exercises, find the derivatives of the given functions and graph along with the function to ensure your answer is correct. 392. [T] 1+tanh(x)1tanh(x)For the following exercises, find the derivatives of the given functions and graph along with the function to ensure your answer is correct. 393.[T] sinh6 (x)For the following exercises, find the derivatives of the given functions and graph along with the function to ensure your answer is correct. 394.[T] ln(sech(x) + tanh(x))For the following exercises, find the antiderivatives for the given functions. 395.cosh(2x+1)For the following exercises, find the antiderivatives for the given functions. 396.tanh(3x + 2)For the following exercises, find the antiderivatives for the given functions. 397.x cosh(x2)For the following exercises, find the antiderivatives for the given functions. 398.3x3 tanh(x4)For the following exercises, find the antiderivatives for the given functions. 399.cosh2(x)sinh(x)For the following exercises, find the antiderivatives for the given functions. 400.tanh2(x)sech2(x)For the following exercises, find the antiderivatives for the given functions. 401. sinh(x)1+cosh(x)For the following exercises, find the antiderivatives for the given functions. 402.coth(x)For the following exercises, find the antiderivatives for the given functions. 403.cosh(x) + sinh(x)For the following exercises, find the antiderivatives for the given functions. 404.(cosh(x) + sinh(x))nFor the following exercises, find the derivatives for the functions. 405.tanh_1(4x)For the following exercises, find the derivatives for the functions. 406.sinh_1(x2)For the following exercises, find the derivatives for the functions. 407.sinh-1(cosh(x))For the following exercises, find the derivatives for the functions. 408.cosh-1(x3)For the following exercises, find the derivatives for the functions. 409.tanh-1(cos(x))For the following exercises, find the derivatives for the functions. 410.esinh-1(x)For the following exercises, find the derivatives for the functions. 411.ln(tanh-1(x))For the following exercises, find the antiderivatives for the functions. 412. dx4 x 2For the following exercises, find the antiderivatives for the functions. 413. dx a 2 x 2For the following exercises, find the antiderivatives for the functions. 414. dx x 2 +1For the following exercises, find the antiderivatives for the functions. 415. xdx x 2 +1For the following exercises, find the antiderivatives for the functions. 416. dxx 1 x 2For the following exercises, find the antiderivatives for the functions. 417. e x e 2x 1For the following exercises, find the antiderivatives for the functions. 418. 2x x 41For the following exercises, use the fact that a falling body with friction equal to velocity squared obeys the equation dv/dt = g — v2. 419. Show that v(t)=gtanh(gt) satisfies this equation.For the following exercises, use the fact that a falling body with friction equal to velocity squared obeys the equation dv/dt = g — v2. 420. Derive the previous expression for v(t) by integrating dvgv2=dt.For the following exercises, use the fact that a falling body with friction equal to velocity squared obeys the equation dv/dt = g — v2. 421. [T] Estimate how far a body has fallen in 12 seconds by finding the area underneath the curve of v(t).For the following exercises, use this scenario: A cable hanging under its own weight has a slope S=dy/dx that satisfies dS/dx=c1+S2. The constant c is the ratio of cable density to tension. 422.Show that S = sinh(xx) satisfies this equation.For the following exercises, use this scenario: A cable hanging under its own weight has a slope S=dy/dx that satisfies dS/dx=c1+S2. The constant c is the ratio of cable density to tension. 423.Integrate dy/dx = sinh(cx) to find the cable height y(x) if y(0) = 1/c.For the following exercises, use this scenario: A cable hanging under its own weight has a slope S=dy/dx that satisfies dS/dx=c1+S2. The constant c is the ratio of cable density to tension. 424.Sketch the cable and determine how far down it sags at x = 0.For the following exercises, solve each problem. 425.[T] A chain hangs from two posts 2 m apart to form a catenary described by the equation y = 2 cosh(x/2) — 1. Find the slope of the catenary at the left fence post.