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All Textbook Solutions for Calculus Volume 2

For the following exercises, convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form. 32. x=4t+3,y=16t29 For the following exercises, convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form. 33. x=t2,y=2Int,t1 For the following exercises, convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form. 34. x=t3,y=3Int,t1 For the following exercises, convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form. 35. x=tn,y=nInt,t1 where n is a natural number For the following exercises, convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form. 36. x=In( 5t)x=In( t 2 )For the following exercises, convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form. 37. x=2sin( 8t)x=2cos( 8t)For the following exercises, convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form. 38. x=tanty=sec2t1 For the following exercises, the pairs of parametric equations represent tines, parabolas, circles, ellipses, or hyperbolas. Name the type of basic curve that each pair of equations represents. 39. x=3t+4y=5t2 For the following exercises, the pairs of parametric equations represent tines, parabolas, circles, ellipses, or hyperbolas. Name the type of basic curve that each pair of equations represents. 40. x4=5ty+2=t For the following exercises, the pairs of parametric equations represent tines, parabolas, circles, ellipses, or hyperbolas. Name the type of basic curve that each pair of equations represents. 41. x=2t+1y=t23 For the following exercises, the pairs of parametric equations represent tines, parabolas, circles, ellipses, or hyperbolas. Name the type of basic curve that each pair of equations represents. 42. x=3costy=3sint For the following exercises, the pairs of parametric equations represent tines, parabolas, circles, ellipses, or hyperbolas. Name the type of basic curve that each pair of equations represents. 43. x=2cos(3t)y=2sin(3t)For the following exercises, the pairs of parametric equations represent tines, parabolas, circles, ellipses, or hyperbolas. Name the type of basic curve that each pair of equations represents. 44. x=coshty=sinht For the following exercises, the pairs of parametric equations represent tines, parabolas, circles, ellipses, or hyperbolas. Name the type of basic curve that each pair of equations represents. 45. x=3costy=4sint For the following exercises, the pairs of parametric equations represent tines, parabolas, circles, ellipses, or hyperbolas. Name the type of basic curve that each pair of equations represents. 46. x=2cos(3t)y=5sin(3t)For the following exercises, the pairs of parametric equations represent tines, parabolas, circles, ellipses, or hyperbolas. Name the type of basic curve that each pair of equations represents. 47. x=3cosh(4t)y=4sinh(4t)For the following exercises, the pairs of parametric equations represent tines, parabolas, circles, ellipses, or hyperbolas. Name the type of basic curve that each pair of equations represents. 48. x=2coshty=2sinht Show that x=h+rcosy=k+rsin represents the equation of a Circle.Use the equations in the preceding problem to find a set of parametric equations for a circle whose radius is 5 and whose center is (2, 3).For the following Exercises, use a graphing utility to graph the curve represented by the parametric equations and identify the curve from its equation. 51. [T] x=+siny=1cos For the following Exercises, use a graphing utility to graph the curve represented by the parametric equations and identify the curve from its equation. 52. [T] x=2t2sinty=22cost For the following Exercises, use a graphing utility to graph the curve represented by the parametric equations and identify the curve from its equation. 53. [T] x=t0.5sinty=11.5cost An airplane traveling horizontally at 100 m/s over ?at ground at an elevation of 4000 meters must drop an emergency package on a target on the ground. The trajectory of the package is given by x=100t,y=4.9t2+4000,t0 where the origin is the point on the ground directly beneath the plane at the moment of release. How many horizontal meters before the target should the package be released in order to hit the target?The trajectory of a bullet is given by x=v0(cos)ty=v0(sin)t12gt2 where v0=500m/s,g=9.8=9.8m/s2 , and =30 degrees. When will the bullet hit the ground? How far from the gun will the bullet hit the ground?[T] Use technology to sketch the curve represented by x=sin(4t),y=sin(3t),0t2 .[T] Use technology to sketch x=2tan(t),y=3sec(t),t .Sketch the curve known as all epitrochoid, which gives the path of a paint on a circle of radius b as it rolls on the outside of a circle of radius a. The equations are x=(a+b)costccos[( a+b)tb]y=(a+b)sintcsin[( a+b)tb] Let a=1,b=2,c=1 .[T] Use technology to sketch the spiral curve given by x=tcos(t),y=tsin(t) from 2t2 ..[T] Use technology to graph the curve given by the parametric equations x=2cot(t),y=1cos(2t),/2t/2 . This curve is known as the witch of Agnesi.[T] Sketch the curve given by parametric equations x=cosh(t)y=sinh(t) , Where 2t2 .For the following exercises, each set of parametric equations represents a line. Without eliminating the parameter, find the slope of each line. 62. x=3+t,y=1t For the following exercises, each set of parametric equations represents a line. Without eliminating the parameter, find the slope of each line. 63. x=8+2t,y=1 For the following exercises, each set of parametric equations represents a line. Without eliminating the parameter, find the slope of each line. 64. x=43t,y=2+6t For the following exercises, each set of parametric equations represents a line. Without eliminating the parameter, find the slope of each line. 65. x=5t+7,y=3t1 For the following exercises, determine the slope of the tangent line, then find the equation of the tangent line at the given value of the parameter. 66. x=3sint,y=3cost,t=4 For the following exercises, determine the slope of the tangent line, then find the equation of the tangent line at the given value of the parameter. 67. x=cost,y=8sint,t=2 For the following exercises, determine the slope of the tangent line, then find the equation of the tangent line at the given value of the parameter. 68. x=2t,y=t3,t=1 For the following exercises, determine the slope of the tangent line, then find the equation of the tangent line at the given value of the parameter. 69. x=t+1t,y=t1t,t=1 For the following exercises, determine the slope of the tangent line, then find the equation of the tangent line at the given value of the parameter. 70. x=t,y=2t,t=4 For the following exercises, find all points on the curve that have the given slope. 71. x=4cost,y=4sint,slope=0.5 For the following exercises, find all points on the curve that have the given slope. 72. x=2cost,y=8sint,slope=1 For the following exercises, find all points on the curve that have the given slope. 73. x=t+1t,y=t1t,slope=1 For the following exercises, find all points on the curve that have the given slope. 74. x=2+t,y=24t,slope=0 For the following exercises, write the equation of the tangent line in Cartesian coordinates for the given parameter t. 75. x=et,y=1Int2,t=1 For the following exercises, write the equation of the tangent line in Cartesian coordinates for the given parameter t. 76. x=tInt,y=sin2t,t=4 For the following exercises, write the equation of the tangent line in Cartesian coordinates for the given parameter t. 77. x=et,y=(t1)2 , at (1, 1)For x=sin(2t),y=2sint where 0t2 . Find all values of t at which a horizontal tangent line exists.For x=sin(2t),y=2sint where 0t2 . Find all values of t at which a vertical tangent line exists.Find all points on the curve x=4cos(t),y=4sin(t) that have the slope of 12 .Find dydx for x=sin(t),y=cos(t) .Find the equation of the tangent line to x=sin(t),y=cos(t) at t=4 .For the curve x=4t,y=3t2 , find the slope and concavity of the curve at t = 3.For the parametric curve whose equation is x=4cos,y=4sin , find the slope and concavity of ?le curve at =4 .Find the slope and concavity for the curve whose equation is x=2+sec,y=1+2tan at =6 .Find all points on the curve x=t+4,y=t33t at which there are vertical and horizontal tangents.Find all points on the curve x=sec,y=tan at which horizontal and vertical tangents exist.For the following exercises, find d2y/dx2 . 88. x=t41,y=tt2 For the following exercises, find d2y/dx2 . 89. x=sin(t),y=cos(t)For the following exercises, find d2y/dx2 . 90. x=et,y=te2t For the following exercises, find points on the curve at which tangent line is horizontal or vertical. 91. x=t(t23),y=3(t23)For the following exercises, find points on the curve at which tangent line is horizontal or vertical. 92. x=3t1+t3,y=3t21+t3 For the following exercises, find dy/dx at the value of the parameter. 93. x=cost,y=sint,t=4 For the following exercises, find dy/dx at the value of the parameter. 94. x=t,y=2t+4,t=9 For the following exercises, find dy/dx at the value of the parameter. 95. x=4cos(2s),y=3sin(2s),s=14 For the following exercises, ?nd d2y/dx2 at the given point without eliminating the parameter. 96. x=12t2,y=13t3,t=2 For the following exercises, ?nd d2y/dx2 at the given point without eliminating the parameter. 97. x=t,y=2t+4,t=1 For the following exercises, ?nd d2y/dx2 at the given point without eliminating the parameter. 98. Find t intervals on which the curve x=3t2,y=t3t is concave up as well as concave down.For the following exercises, ?nd d2y/dx2 at the given point without eliminating the parameter. 99. Determine the concavity of the curve x=2t+Int,y=2tInt .For the following exercises, ?nd d2y/dx2 at the given point without eliminating the parameter. 100. Sketch and find the area under one arch of the cycloid x=r(sin),y=r(1cos) .For the following exercises, ?nd d2y/dx2 at the given point without eliminating the parameter. 101. Find the area bounded by the curve x=cost,y=et,0t2 and the1i11es y = 1 and x = 0.For the following exercises, ?nd d2y/dx2 at the given point without eliminating the parameter. 102. Find the area enclosed by the ellipse x=acos,y=bsin,02 .For the following exercises, find d2y/dx2 at the given point without eliminating the parameter. Find the area of the region bounded by x=2sin2,y=2sin2tan for 02.For the following exercises, find the area of the regions bounded by the parametric curves and the indicated values of the parameter. 104. x=2cot,y=2sin2,0 For the following exercises, find the area of the regions bounded by the parametric curves and the indicated values of the parameter. 105. [T] x=2acostacos(2t),y=2asintasin(2t),0t2 For the following exercises, find the area of the regions bounded by the parametric curves and the indicated values of the parameter. 106. [T] x=asin(2t),y=bsin(t),0t2 (the “hourglass”)For the following exercises, find the area of the regions bounded by the parametric curves and the indicated values of the parameter. 107. [T] x=2acostasin(2t),y=bsint,0t2 (the “teardrop”)For the following exercises, find the arc length of the curve on the indicated interva1 of the parameter. 108. x=4t+3,y=3t2,0t2 For the following exercises, find the arc length of the curve on the indicated interva1 of the parameter. 109. x=13t3,y=12t2,0t1 For the following exercises, find the arc length of the curve on the indicated interva1 of the parameter. 110. x=cos(2t),y=sin(2t),0t2 For the following exercises, find the arc length of the curve on the indicated interva1 of the parameter. 111. x=1+t2,y=(1+t)3,0t1 For the following exercises, find the arc length of the curve on the indicated interva1 of the parameter. 112. x=etcost,y=etsint,0t2 (express answer as a decimal rounded to three places)For the following exercises, find the arc length of the curve on the indicated interva1 of the parameter. 113. x=acos3,y=asin3 on the interval [0,2) (the hypocycloid)For the following exercises, find the arc length of the curve on the indicated interva1 of the parameter. 114. Find the length of one arch of the cycloid x=4(tsint),y=4(1cost) .For the following exercises, find the arc length of the curve on the indicated interva1 of the parameter. 115. Find the distance traveled by a particle with position (x, y) as t varies in the given time interval: x=sin2t,y=cos2t,0t3 For the following exercises, find the arc length of the curve on the indicated interva1 of the parameter. 116. Find the length 0f one arch of the cycloid x=sin,y=1cos .For the following exercises, find the arc length of the curve on the indicated interva1 of the parameter. 117. Show that the total length of the ellipse x=4sin,y=3cos is L=160/21 e 2 sin 2d , where e=ca and c=a2b2 .For the following exercises, find the arc length of the curve on the indicated interva1 of the parameter. 118. Find the length of the curve x=ett,y=4et/2,8t3 .For the following exercises, find the area of the surface obtained by rotating the given curve about the x—axis. 119. x=t3,y=t2,0t1 For the following exercises, find the area of the surface obtained by rotating the given curve about the x—axis. 120. x=acos3,y=asin3,0t2 For the following exercises, find the area of the surface obtained by rotating the given curve about the x—axis. 121. [T] Use a CAS to find the area of the surface generated by rotating x=t+t3,y=t1t2,1t2 about the x—axis. (Answer to three decimal places.)For the following exercises, find the area of the surface obtained by rotating the given curve about the x—axis. 122. Find the surface area obtained by rotating x=3t2,y=2t3,0t5 about the y-axis.For the following exercises, find the area of the surface obtained by rotating the given curve about the x—axis. 123. Find the area of the surface generated by revolving x=t2,y=2t,0t4 about the x-axis.For the following exercises, find the area of the surface obtained by rotating the given curve about the x—axis. 124. Find the surface area generated by revolving x=t2y=2t2,0t1 about the y—axis.In the following exercises, plot the point whose polar coordinates are given by first constructing the angle ( and then marking off the distance r along the ray. 125. (3,6)In the following exercises, plot the point whose polar coordinates are given by first constructing the angle ( and then marking off the distance r along the ray. 126. (2,53)In the following exercises, plot the point whose polar coordinates are given by first constructing the angle ( and then marking off the distance r along the ray. 127. (0,76)In the following exercises, plot the point whose polar coordinates are given by first constructing the angle ( and then marking off the distance r along the ray. 128. (4,34)In the following exercises, plot the point whose polar coordinates are given by first constructing the angle ( and then marking off the distance r along the ray. 129. (1,4)In the following exercises, plot the point whose polar coordinates are given by first constructing the angle ( and then marking off the distance r along the ray. 130. (2,56)In the following exercises, plot the point whose polar coordinates are given by first constructing the angle ( and then marking off the distance r along the ray. 131. (1,2)For the following exercises, consider the polar graph below. Give two sets of polar coordinates for each point. 132. Coordinates of point A.For the following exercises, consider the polar graph below. Give two sets of polar coordinates for each point. 133. Coordinates of point B.For the following exercises, consider the polar graph below. Give two sets of polar coordinates for each point. 134. Coordinates of point C.For the following exercises, consider the polar graph below. Give two sets of polar coordinates for each point. 135. Coordinates of point D.For the following exercises, the rectangular coordinates of a point are given. Find two sets of polar coordinates for the point in (0, 2(]. Round to three decimal places. 136. (2, 2)For the following exercises, the rectangular coordinates of a point are given. Find two sets of polar coordinates for the point in (0, 2(]. Round to three decimal places. 137. (3, 4) (3, 4)For the following exercises, the rectangular coordinates of a point are given. Find two sets of polar coordinates for the point in (0, 2(]. Round to three decimal places. 138. (8, 15)For the following exercises, the rectangular coordinates of a point are given. Find two sets of polar coordinates for the point in (0, 2(]. Round to three decimal places. 139. (6, 8)For the following exercises, the rectangular coordinates of a point are given. Find two sets of polar coordinates for the point in (0, 2(]. Round to three decimal places. 140. (4, 3)For the following exercises, the rectangular coordinates of a point are given. Find two sets of polar coordinates for the point in (0, 2(]. Round to three decimal places. 141. (3,3)For the following exercises, find rectangular coordinates for the given point in polar coordinates. 142. (2,54)For the following exercises, find rectangular coordinates for the given point in polar coordinates. 143. (2,6)For the following exercises, find rectangular coordinates for the given point in polar coordinates. 144. (5,3)For the following exercises, find rectangular coordinates for the given point in polar coordinates. 145. (1,76)For the following exercises, find rectangular coordinates for the given point in polar coordinates. 146. (3,34)For the following exercises, find rectangular coordinates for the given point in polar coordinates. 147. (0,2)For the following exercises, find rectangular coordinates for the given point in polar coordinates. 148. (4.5, 6.5)For the following exercises, determine whether the graphs of the paler equation are symmetric with respect to the xaxis, the yaxis, or the origin. 149. r=3sin(2)For the following exercises, determine whether the graphs of the paler equation are symmetric with respect to the xaxis, the yaxis, or the origin. 150. r2=9cos For the following exercises, determine whether the graphs of the paler equation are symmetric with respect to the xaxis, the yaxis, or the origin. 151. r=cos(5)For the following exercises, determine whether the graphs of the paler equation are symmetric with respect to the xaxis, the yaxis, or the origin. 152. r=2sec For the following exercises, determine whether the graphs of the paler equation are symmetric with respect to the xaxis, the yaxis, or the origin. 153. r=1+cos For the following exercises, describe the graph of each polar equation. Confirm each description by converting into a rectangular equation. 154. r = 3 For the following exercises, describe the graph of each polar equation. Confirm each description by converting into a rectangular equation. 155. =4 For the following exercises, describe the graph of each polar equation. Confirm each description by converting into a rectangular equation. 156. r=sec For the following exercises, describe the graph of each polar equation. Confirm each description by converting into a rectangular equation. 157. r=csc For the following exercises, convert the rectangular equation to polar form and sketch its graph. 158. x2+y2=16 For the following exercises, convert the rectangular equation to polar form and sketch its graph. 159. x2y2=16 For the following exercises, convert the rectangular equation to polar form and sketch its graph. 160. x = 8 For the following exercises, convert the rectangular equation to polar form and sketch its graph. 161. 3xy=2 For the following exercises, convert the rectangular equation to polar form and sketch its graph. 162. y2=4x For the following exercises, convert the polar equation to rectangular form and sketch its graph. 163. r=4sin For the following exercises, convert the polar equation to rectangular form and sketch its graph. 164. r=6cos For the following exercises, convert the polar equation to rectangular form and sketch its graph. 165. r=For the following exercises, convert the polar equation to rectangular form and sketch its graph. 166. r=cotcsc For the following exercises, sketch a graph of the polar equation and identify any symmetry. 167. r=1+sin For the following exercises, sketch a graph of the polar equation and identify any symmetry. 168. r=32cos For the following exercises, sketch a graph of the polar equation and identify any symmetry. 169. r=22sin For the following exercises, sketch a graph of the polar equation and identify any symmetry. 170. r=54sin For the following exercises, sketch a graph of the polar equation and identify any symmetry. 171. r=3cos(2)For the following exercises, sketch a graph of the polar equation and identify any symmetry. 172. r=3sin(2)For the following exercises, sketch a graph of the polar equation and identify any symmetry. 173. r=2cos(3)For the following exercises, sketch a graph of the polar equation and identify any symmetry. 174. r=3cos(2)For the following exercises, sketch a graph of the polar equation and identify any symmetry. 175. r2=4cos(2)For the following exercises, sketch a graph of the polar equation and identify any symmetry. 176. r2=4sin For the following exercises, sketch a graph of the polar equation and identify any symmetry. 177. r=2 [T] Use a graphing utility and sketch the graph of r=62sin3cos .[T] Use a graphing utility to graph r=11cos .[T] Use technology to graph r=esin()2cos(4) .[T] Use technology to plot r=sin(37) (use the interval 014 Without using technology, sketch the polar curve =23 .[T] Use a graphing utility to plot r=sin for .[T] Use technology to plot r=e0.1 for 1010 .[T] There is a curve known as the “Black Hole.” Use technology to plot r=e0.01 for 100100 .[T] Use the results of the preceding two problems to explore the graphs of r=e0.001 and r=e0.0001 for ||100 .For the fallowing exercises, determine a definite integral that represents the area. 188. Region enclosed by r = 4 For the fallowing exercises, determine a definite integral that represents the area. 189. Region enclosed by r=3sin For the fallowing exercises, determine a definite integral that represents the area. 190. Region in the ?rst quadrant within the cardioid r=1+sin For the fallowing exercises, determine a definite integral that represents the area. 191. Region enclosed by one petal of r=8sin(2)For the fallowing exercises, determine a definite integral that represents the area. 192. Region enclosed by one petal of r=cos(3)For the fallowing exercises, determine a definite integral that represents the area. 193. Region below the polar axis and enclosed by r=1sin For the fallowing exercises, determine a definite integral that represents the area. 194. Region in the first quadrant enclosed by r=2cos For the fallowing exercises, determine a definite integral that represents the area. 195. Region enclosed by the inner loop of r=23sin For the fallowing exercises, determine a definite integral that represents the area. 196. Region enclosed by the inner loop of r=34cos For the fallowing exercises, determine a definite integral that represents the area. 197. Region enclosed by r=12cos and outside the inner loop For the fallowing exercises, determine a definite integral that represents the area. 198. Region common to r=3sin and r=2sin For the fallowing exercises, determine a definite integral that represents the area. 199. Region common to r = 2 and r=4cos For the fallowing exercises, determine a definite integral that represents the area. 200. Region common to r=3cos and r=3sin For the following exercises, find the area of the described region. 201. Enclosed by r=6sin For the following exercises, find the area of the described region. 202. Above the polar axis enclosed by r=2+sin For the following exercises, find the area of the described region. 203. Below the polar axis and enclosed by r=2cos For the following exercises, find the area of the described region. 204. Enclosed by one petal of r=4cos(3)For the following exercises, find the area of the described region. 205. Enclosed by one petal of r=3cos(2)For the following exercises, find the area of the described region. 206. Enclosed by r=1+sin For the following exercises, find the area of the described region. 207. Enclosed by the inner loop of r=3+6cos For the following exercises, find the area of the described region. 208. Enclosed by r=2+4cos and outside the inner loop For the following exercises, find the area of the described region. 209. Common interior of r=4sin(2) and r = 2 For the following exercises, find the area of the described region. 210. Common interior of r=32sin and r=3+2sin For the following exercises, find the area of the described region. 211. Common interior of r=6sin and r = 3 For the following exercises, find the area of the described region. 212. Inside r=1+cos and outside r=cos For the following exercises, find the area of the described region. 213. Common interior of r=2+2cos and r=2sin For the following exercises, find a definite integral that represents the arc length. 214. r=4cos on the interval 02 For the following exercises, find a definite integral that represents the arc length. 215. r=1+sin on the interval 02 For the following exercises, find a definite integral that represents the arc length. 216. r=2sec on the interval 03 For the following exercises, find a definite integral that represents the arc length. 217. r=e on the interval 01 For the following exercises, find the length of the curve over the given interval. 218. r = 6 on the interval 02 For the following exercises, find the length of the curve over the given interval. 219. r=e3 on the interval 02 For the following exercises, find the length of the curve over the given interval. 220. r=6cos on the interval 02 For the following exercises, find the length of the curve over the given interval. 221. r=8+8cos on the interval 0 For the following exercises, find the length of the curve over the given interval. 222. r=1sin on the interval 02 For the following exercises, use the integration capabilities of a calculator to approximate the length of the curve. 223. [T] r=3 on the interval 02 For the following exercises, use the integration capabilities of a calculator to approximate the length of the curve. 224. [T] r=2 on the interval 02 For the following exercises, use the integration capabilities of a calculator to approximate the length of the curve. 225. [T] r=sin2(2) on the interval 0 For the following exercises, use the integration capabilities of a calculator to approximate the length of the curve. 226. [T] r=22 on the interval 0 For the following exercises, use the integration capabilities of a calculator to approximate the length of the curve. 227. [T] r=sin(3cos) on the interval 0 For the fallowing exercises, use the familiar formula from geometry to find the area of the region described and then confirm by using the definite integral. 228. r=3sin on the interval 0 For the fallowing exercises, use the familiar formula from geometry to find the area of the region described and then confirm by using the definite integral. 229. r=sin+cos on the interval 0 For the fallowing exercises, use the familiar formula from geometry to find the area of the region described and then confirm by using the definite integral. 230. r=6sin+8cos on the interval 0 For the following exercises, use the familiar formula from geometry to find the length of the curve and then confirm using the de?nite integral. 231. r=3sin on the interval 0 For the following exercises, use the familiar formula from geometry to find the length of the curve and then confirm using the de?nite integral. 232. r=sin+cos on the interval 0 For the following exercises, use the familiar formula from geometry to find the length of the curve and then confirm using the de?nite integral. 233. r=6sin+8cos on the interval 0 For the following exercises, use the familiar formula from geometry to find the length of the curve and then confirm using the de?nite integral. 234. Verify that if y=rsin=f()sin then dyd=f()sin+f()cos .For the following exercises, ?nd the slope of a tangent line to a polar curve r=f() . Let x=rcos=f()cos and y=rsin=f()sin , so the polar equation r=f() is now written in parametric form. 235. Use the definition 0f the derivative dydx=dy/ddx/d and the product rule to derive the derivative of a polar equation.For the following exercises, ?nd the slope of a tangent line to a polar curve r=f() . Let x=rcos=f()cos and y=rsin=f()sin , so the polar equation r=f() is now written in parametric form. 236. r=1sin;(12,6)For the following exercises, ?nd the slope of a tangent line to a polar curve r=f() . Let x=rcos=f()cos and y=rsin=f()sin , so the polar equation r=f() is now written in parametric form. 237. r=4cos,(2,3)For the following exercises, ?nd the slope of a tangent line to a polar curve r=f() . Let x=rcos=f()cos and y=rsin=f()sin , so the polar equation r=f() is now written in parametric form. 238. r=8sin;(4,56)For the following exercises, ?nd the slope of a tangent line to a polar curve r=f() . Let x=rcos=f()cos and y=rsin=f()sin , so the polar equation r=f() is now written in parametric form. 239. r=4+sin;(3,32)For the following exercises, ?nd the slope of a tangent line to a polar curve r=f() . Let x=rcos=f()cos and y=rsin=f()sin , so the polar equation r=f() is now written in parametric form. 240. r=6+3cos;(3,)For the following exercises, ?nd the slope of a tangent line to a polar curve r=f() . Let x=rcos=f()cos and y=rsin=f()sin , so the polar equation r=f() is now written in parametric form. 241. r=4cos(2) ; tips of the leaves For the following exercises, ?nd the slope of a tangent line to a polar curve r=f() . Let x=rcos=f()cos and y=rsin=f()sin , so the polar equation r=f() is now written in parametric form. 242. r=2sin(3) ; tips of the leaves For the following exercises, ?nd the slope of a tangent line to a polar curve r=f() . Let x=rcos=f()cos and y=rsin=f()sin , so the polar equation r=f() is now written in parametric form. 243. r=2;(2,4)Find the paints on the interval at which the cardioid r=1cos has a vertical or horizontal tangent line.For the cardioid r=1+sin , find the slope of the tangent line when =3 .For the following exercises, find the slope of the tangent line to the given polar curve at the point given by the value of (. 246. r=3cos,=3 For the following exercises, find the slope of the tangent line to the given polar curve at the point given by the value of (. 247. r=,=2 For the following exercises, find the slope of the tangent line to the given polar curve at the point given by the value of (. 248. r=In,=e For the following exercises, find the slope of the tangent line to the given polar curve at the point given by the value of (. 249. [T] Use technology: r=2+4cos at =6 For the following exercises, find the points at which the following polar curves have a horizontal or vertical tangent line. 250. r=4cos For the following exercises, find the points at which the following polar curves have a horizontal or vertical tangent line. 251. r2=4cos(2)For the following exercises, find the points at which the following polar curves have a horizontal or vertical tangent line. 252. r=2sin(2)For the following exercises, find the points at which the following polar curves have a horizontal or vertical tangent line. 253. The cardioid r=1+sin For the following exercises, find the points at which the following polar curves have a horizontal or vertical tangent line. 254. Show that the curve r=sintan (called a cissoids of Diodes) has the line x = 1 as a vertical asymptote.For the following exercises, determine the equation of the parabola using the information given. 255. Focus (4, 0) and directrix x = 4 For the following exercises, determine the equation of the parabola using the information given. 256. Focus (0, 3) and directrix y = 3 For the following exercises, determine the equation of the parabola using the information given. 257. Focus (0, 0.5) and directrix y = 0.5 For the following exercises, determine the equation of the parabola using the information given. 258. Focus (2, 3) and directrix x = 2 For the following exercises, determine the equation of the parabola using the information given. 259. Focus (0, 2) and directrix y = 4 For the following exercises, determine the equation of the parabola using the information given. 260. Focus (1, 4) and directrix x = 5 For the following exercises, determine the equation of the parabola using the information given. 261. Focus (3, 5) and directrix y = 1 For the following exercises, determine the equation of the parabola using the information given. 262. Focus (52,4) and directrix x=72 For the following exercises, determine the equation of the ellipse using the information given. 263. Endpoints of major ands at (4, 0), (4, 0) and foci located at (2, 0), (2, 0)For the following exercises, determine the equation of the ellipse using the information given. 264. Endpoints of major ads at (0, 5), (0, 5) and foci located at (0, 3), (0, 3)For the following exercises, determine the equation of the ellipse using the information given. 265. Endpoints of major ands at (0, 2), (0, 2) and foci located at (3, 0), (3, 0)For the following exercises, determine the equation of the ellipse using the information given. 266. Endpoints of major ands at (3, 3), (7, 3) and foci located at (2, 3), (6, 3)For the following exercises, determine the equation of the ellipse using the information given. 267. Endpoints of major axis at (3, 5), (3, 3) and foci located at (3, 3), (3, l)For the following exercises, determine the equation of the ellipse using the information given. 268. Endpoints of major axis at (0, 0), (0, 4) and fed located at (5, 2), (5, 2)For the following exercises, determine the equation of the ellipse using the information given. 269. Foci located at (2, 0), (2, 0) and eccentricity of 12 For the following exercises, determine the equation of the ellipse using the information given. 270. Foci located at (0, 3), (0, 3) and eccentricity of 34 For the following exercises, determine the equation of the hyperbola using the information given. 271. Vertices located at (5, 0), (5, 0) and foci located at (6, 0), (6, 0)For the following exercises, determine the equation of the hyperbola using the information given. 272. Vertices located at (0, 2), (O, 2) and foci located at (0, 3), (0, 3)For the following exercises, determine the equation of the hyperbola using the information given. 273. Endpoints of the conjugate axis located at (0, 3), (0, 3) and foci located (4, 0), (4, 0)For the following exercises, determine the equation of the hyperbola using the information given. 274. Vertices located at (0, l), (6, l) and focus located at (8, 1)For the following exercises, determine the equation of the hyperbola using the information given. 275. Vertices located at (2, 0), (2, 4) and focus located at (2, 8)For the following exercises, determine the equation of the hyperbola using the information given. 276. Endpoints of the conjugate axis located at (3, 2), (3, 4) and focus located at (3, 7)For the following exercises, determine the equation of the hyperbola using the information given. 277. Foci located at (6, 0), (6, 0) and eccentricity of 3 For the following exercises, determine the equation of the hyperbola using the information given. 278. (0, 10), (O, 10) and eccentricity of 2.5 For the following exercises, consider the following polar equations of comics. Determine the eccentricity and identify the conic. 279. r=11+cos For the following exercises, consider the following polar equations of comics. Determine the eccentricity and identify the conic. 280. r=82sin For the following exercises, consider the following polar equations of comics. Determine the eccentricity and identify the conic. 281. r=52+sin For the following exercises, consider the following polar equations of comics. Determine the eccentricity and identify the conic. 282. r=51+2sin For the following exercises, consider the following polar equations of comics. Determine the eccentricity and identify the conic. 283. r=326sin For the following exercises, consider the following polar equations of comics. Determine the eccentricity and identify the conic. 284. r=34+3sin For the following exercises, ?nd a polar equation of the conic with focus at the origin and eccentricity and directrix as given. 285. Directrix x=4;e=15 For the following exercises, ?nd a polar equation of the conic with focus at the origin and eccentricity and directrix as given. 286. Directrix x = 4; e = 5 For the following exercises, ?nd a polar equation of the conic with focus at the origin and eccentricity and directrix as given. 287. Directrix: y = 2; e = 2 For the following exercises, ?nd a polar equation of the conic with focus at the origin and eccentricity and directrix as given. 288. Directrix: y=2;e=12 For the following exercises, sketch the graph of each conic. 289. r=11+sin For the following exercises, sketch the graph of each conic. 290. r=11cos For the following exercises, sketch the graph of each conic. 291. r=41+cos For the following exercises, sketch the graph of each conic. 292. r=105+4sin For the following exercises, sketch the graph of each conic. 293. r=1532cos For the following exercises, sketch the graph of each conic. 294. r=323+5sin For the following exercises, sketch the graph of each conic. 295. r(2+sin)=4 For the following exercises, sketch the graph of each conic. 296. r=32+6sin For the following exercises, sketch the graph of each conic. 297. r=34+2sin For the following exercises, sketch the graph of each conic. 298. x29+y24=1 For the following exercises, sketch the graph of each conic. 299. x24+y216=1 For the following exercises, sketch the graph of each conic. 300. 4x2+9y2=36 For the following exercises, sketch the graph of each conic. 301. 25x24y2=100 For the following exercises, sketch the graph of each conic. 302. x216y29=1 For the following exercises, sketch the graph of each conic. 303. x2=12y For the following exercises, sketch the graph of each conic. 304. y2=20x For the following exercises, sketch the graph of each conic. 305. 12x=5y2 For the following equations, determine which of the conic sections is described. 306. xy = 4 For the following equations, determine which of the conic sections is described. 307. x2+4xy2y26=0 For the following equations, determine which of the conic sections is described. 308. x2+23xy+3y26=0 For the following equations, determine which of the conic sections is described. 309. x2xy+y22=0 For the following equations, determine which of the conic sections is described. 310. 34x224xy+41y225=0 For the following equations, determine which of the conic sections is described. 311. 52x272xy+73y2+40x+30y75=0 The mirror in an automobile headlight has a parabolic cross section, with the lightbulb at the focus. On a schematic, the equation of the parabola is given as x2=4y . At what coordinates should you place the lightbulb?A satellite dish is shaped like a paraboloid of revolution. The receiver is to be located at the focus. If the dish is 12 feet across at its opening and 4 feet deep at its center, where should the receiver be placed?Consider the satellite dish of the preceding problem. If the dish is 8 feet across at the opening and 2 feet deep, where should we place the receiver?A searchlight is shaped like a paraboloid of revolution. A light source is located 1 foot from the base along the axis of symmetry. If the opening of the searchlight is 3 feet across, ?nd the depth.Whispering galleries are rooms designed with elliptical ceilings. A person standing at one focus can whisper and be heard by a person standing at the other focus because all the sound waves that reach the ceiling are re?ected to the other person. If a whispering gallery has a length of 120 feet and the foci are located 30 feet from the center, find the height of the ceiling at the center.A person is standing 3 feet from the nearest wall in a whispering gallery. If that person is at one focus and the other focus is 80 feet away, what is the length and the height at the center of the gallery?For the following exercises, determine the polar equation form of the orbit given the length of the major axis and eccentricity for the orbits of the comets or planets. Distance is given in astronomical units (AU). 318. Halley’s Comet: length of major axis = 35.88, eccentricity = 0.967 For the following exercises, determine the pelar equaticm farm 0f the Orbit given the length of the major axis and eccentrieity fer the orbits of the comets (1r planets. Distance is given in astronomical units (AU). .318. Halley’s Comet: length of major axis = 35.88, eccentricity = 0.967 319. Hale—Bopp Comet: length of major axis 2 525.91, eccentricity = 0.995 320. Mars: length of major axis = 3.049, eccentricity = 0.0934 321. Jupiter: length of major axis = 10.408, eccentricity = 0.0484 For the following exercises, determine the pelar equaticm farm 0f the Orbit given the length of the major axis and eccentrieity fer the orbits of the comets (1r planets. Distance is given in astronomical units (AU). .318. Halley’s Comet: length of major axis = 35.88, eccentricity = 0.967 319. Hale—Bopp Comet: length of major axis 2 525.91, eccentricity = 0.995 320. Mars: length of major axis = 3.049, eccentricity = 0.0934 321. Jupiter: length of major axis = 10.408, eccentricity = 0.0484 For the following exercises, determine the polar equation farm 0f the Orbit given the length of the major axis and eccentrieity fer the orbits of the comets or planets. Distance is given in astronomical units (AU). Jupiter: length of major axis = 10.408, eccentricity = 0.0484.True or False? Justify your answer with a proof or a counterexample. 322. The rectangular coordinates of the point (4,56) are (23,2) .True or False? Justify your answer with a proof or a counterexample. 323. The equations x=cosh(3t),y=2sinh(3t) represent a hyperbola.True or False? Justify your answer with a proof or a counterexample. 324. The arc length of the spiral given by r=2 for 03 is 943 .True or False? Justify your answer with a proof or a counterexample. 325. Given x = f(t) and y = g(t), if dxdy=dydx , then f(t)=g(t)+C , where C is a constant.For the following exercises, sketch the parametric curve and eliminate the parameter to find the Cartesian equation of the curve. 326. x=1+t,y=t21,1t1 For the following exercises, sketch the parametric curve and eliminate the parameter to find the Cartesian equation of the curve. 327. x=et,y=1e3t,0t1 For the following exercises, sketch the parametric curve and eliminate the parameter to find the Cartesian equation of the curve. 328. x=sin,y=1csc,02 For the following exercises, sketch the parametric curve and eliminate the parameter to find the Cartesian equation of the curve. 329. x=4cos=y=1sin,02 For the following exercises, sketch the polar curve and determine what type of symmetry exists, if any. 330. r=4sin(3)For the following exercises, sketch the polar curve and determine what type of symmetry exists, if any. 331. r=5cos(5)For the following exercises, find the polar equation for the curve given as a Cartesian equation. 332. x+y=5

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