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All Textbook Solutions for Calculus Volume 3
Find the indicated higher-order partial derivatives. 139. Given f(x,yz)=xyz , find fxyz , fyxy and fyxyFind the indicated higher-order partial derivatives. 140. Given f(x,y,z)=z2xsin(z2y) , show that fxyy=fyxy .Find the indicated higher-order partial derivatives. 141. Show that z=12(eyey)sinx is a solution of the differential equation 2zx2+2zy2=0Find the indicated higher-order partial derivatives. 142. Find fxx(x,y) for fxx(x,y)=4x2y+y22x.Find the indicated higher-order partial derivatives. 143. Let f(x,y,z)=x2y2z3xy2z3+5x2zy3z. Find fxyz .Find the indicated higher-order partial derivatives. 144. Let F(x,y,z)=x3yz22x2yz+3xz2y3z Find Fxyz .Find the indicated higher-order partial derivatives. 145. Given f(x,y)=x2+x3xy+y35 , find all points at which fx=fy=0 simultaneously.Find the indicated higher-order partial derivatives. 146. Given f(x,y)=2x2+2xy+y2+2x3 , find all points at which fx=0 and fy=0 simultaneously.Find the indicated higher-order partial derivatives. 147. Given f(x,y)=y33yx33y23x2+1 , find all points at which fy=0 simultaneously.Find the indicated higher-order partial derivatives. 148. Given f(x,y)=15x33xy+15y3 , find all points on f at which fx(x,y)=fy(x,y)=0 simultaneously.Find the indicated higher-order partial derivatives. 149. Show that z=exsiny satisfies the equation 2zx2+2xy2=0 .Find the indicated higher-order partial derivatives. 150. Show that f(x,y)ln(x2+y2) solves Laplace’s equation 2zx2+2zy2=0.Find the indicated higher-order partial derivatives. 151. Show that z=etcos(xc) satisfies the heta equation zt=etcos(xc) .Find the indicated higher-order partial derivatives. 152. Find 0limf(x+x)f(x,y)x for f(x,y)=7x2xy+7yFind the indicated higher-order partial derivatives. 153. Find 0limf(x+x)f(x,y)x f(x,y)=7x2xy+7y.Find the indicated higher-order partial derivatives. 154. Find x0limf(x,y+y)f(x,y)y for f(x,y)=x2y2+xy+yFind the indicated higher-order partial derivatives. 155. Find x0limfx=x0limf(x+x,y)f(x,y)x for f(x,y)=sin(xy) .Find the indicated higher-order partial derivatives. 156. Show that z=et(xc) satisfies the heat equation zt=etcos(xc) .Find the indicated higher-order partial derivatives. 157. The equation for heat flow in the xy-plane is ft=2fx2+2fy2 . Show that f(x,y,t)=e2t sin x sin v is a solution.Find the indicated higher-order partial derivatives. 158. The basic wave equation is ftt=fxx. Verify that f(x,t)=sin(x+t) and f(x. t) = sin(x — t) are solutions.Find the indicated higher-order partial derivatives. 159. The law of cosines can be thought of as a function of three variables. Let x. v and be two sides of any triangle where the angle is the included angle between the two sides. Then, F(x,y,)=x2+y22xycosgives the square of the third side of the triangle. Find F and Fx when x = 2. y = 3, and =6 .Suppose the sides of a rectangle are changing with respect to time. The first side is changing at a rate of 2 in./sec whereas the second side is changing at the rate of 4 in/sec. How fast is the diagonal of the rectangle changing when the first side measures 16 in. and the second side measures 20 in.? (Round answer to three decimal places.)A Cobb-Douglas production function f(x,y)=200x0.7y0.3 where x and y represent the amount of labor and capital available. Let x = 500 and y=1000. Find fx and fy at these values, which represent the marginal productivity of labor and capital, respectively.The apparent temperature index is a measure of how the temperature feels, and it is based on two variables: h. which is relative humidity, and t, which is the air temperature. A = 0.885t — 22.41t+ 1.20th — 0.544. Find At and Ahwhen t = 20°F and h = 0.90.For the following exercises, find a unit normal vector to the surface at the indicated point. 163. f(x,y)=x3,(2,1,8)For the following exercises, find a unit normal vector to the surface at the indicated point. 164. ln(xyz)=0whenx=y=1For the following exercises, as a useful review for techniques used in this section, find a normal vector and a tangent vector at point P. 165. x2+xy+y2=3,P(1,1)For the following exercises, as a useful review for techniques used in this section, find a normal vector and a tangent vector at point P. 166. (x2+y2)=9(x2y2)P(2,1)For the following exercises, as a useful review for techniques used in this section, find a normal vector and a tangent vector at point P. 167. xy2x2y2=3xy7,P(12)For the following exercises, as a useful review for techniques used in this section, find a normal vector and a tangent vector at point P. 168. 2x3x2y2=3xy7,P(1,2)For the following exercises, as a useful review for techniques used in this section, find a normal vector and a tangent vector at point P. 169. zex2y23=0,P(2,2,3)For the following exercises, find the equation for the tangent plane to the surface at the indicated point. (Hint: Solve for z in terms of x and y.) 170. 8x3y7z=19,P(1,1,2)For the following exercises, find the equation for the tangent plane to the surface at the indicated point. (Hint: Solve for z in terms of x and y.) 171. z=9x23y2,P(2,1,39)For the following exercises, find the equation for the tangent plane to the surface at the indicated point. (Hint: Solve for z in terms of x and y.) 172. x2+10xyz+y2+8z2=0,P(1,1,1)For the following exercises, find the equation for the tangent plane to the surface at the indicated point. (Hint: Solve for z in terms of x and y.) 173. z=ln(10x2+2y3+1),P(0,0,0)For the following exercises, find the equation for the tangent plane to the surface at the indicated point. (Hint: Solve for z in terms of x and y.) 174. z=e7x2+4y2,P(0,0,1)For the following exercises, find the equation for the tangent plane to the surface at the indicated point. (Hint: Solve for z in terms of x and y.) 175. xy+yz+zx=11,P(1,2,3)For the following exercises, find the equation for the tangent plane to the surface at the indicated point. (Hint: Solve for z in terms of x and y.) 176. x2+4y2=z2,P(3,2,5)For the following exercises, find the equation for the tangent plane to the surface at the indicated point. (Hint: Solve for z in terms of x and y.) 177. x3+y3=3xyz,P(1,232)For the following exercises, find the equation for the tangent plane to the surface at the indicated point. (Hint: Solve for z in terms of x and y.) 178. z=axy,P(1,1a,1)For the following exercises, find the equation for the tangent plane to the surface at the indicated point. (Hint: Solve for z in terms of x and y.) 179. z=sinx+siny+sin(x+y),P(0,0,0)For the following exercises, find the equation for the tangent plane to the surface at the indicated point. (Hint: Solve for z in terms of x and y.) 180. f(x,y)=lnx2+y2,P(3,4)the following exercises, find the equation for the tangent plane to the surface at the indicated point. (Hint: Solve for z in terms of x and y.) 181. z=x22xy+y2,P(1,2,1)For the following exercises, find parametric equations for the normal line to the surface at the indicated point. (Recall that to find the equation of a line in space, you need a point on the line, P0(x0,y0,z0)and a vector n = ( a. b. c ) that is parallel to the line. Then the equation of the line is xx0=at,yy0=bt,zz0=ct.) 182. 3x+9y+4z=4,P(1,1,2)For the following exercises, find parametric equations for the normal line to the surface at the indicated point. (Recall that to find the equation of a line in space, you need a point on the line, P0(x0,y0,z0)and a vector n = ( a. b. c ) that is parallel to the line. Then the equation of the line is xx0=at,yy0=bt,zz0=ct.) 183 z=5x22y2,P(2,1,18)For the following exercises, find parametric equations for the normal line to the surface at the indicated point. (Recall that to find the equation of a line in space, you need a point on the line, P0(x0,y0,z0)and a vector n = ( a. b. c ) that is parallel to the line. Then the equation of the line is xx0=at,yy0=bt,zz0=ct.) 184. x28xyz+y2+6z2=0,P(1,1,1)For the following exercises, find parametric equations for the normal line to the surface at the indicated point. (Recall that to find the equation of a line in space, you need a point on the line, P0(x0,y0,z0)and a vector n = ( a. b. c ) that is parallel to the line. Then the equation of the line is xx0=at,yy0=bt,zz0=ct.) 185. z=ln(3x2+7y2+1),P(0,0,0)For the following exercises, find parametric equations for the normal line to the surface at the indicated point. (Recall that to find the equation of a line in space, you need a point on the line, P0(x0,y0,z0)and a vector n = ( a. b. c ) that is parallel to the line. Then the equation of the line is xx0=at,yy0=bt,zz0=ct.) 186. z=e4x2+6y2,P(0,0,1)For the following exercises, find parametric equations for the normal line to the surface at the indicated point. (Recall that to find the equation of a line in space, you need a point on the line, P0(x0,y0,z0)and a vector n = ( a. b. c ) that is parallel to the line. Then the equation of the line is xx0=at,yy0=bt,zz0=ct.) 187. z=x22xy+y2atpointP(1,2,1)For the following exercises, use the figure shown here. 188. The length of line segment AC is equal to whatFor the following exercises, use the figure shown here. 189. The length of line segment BC is equal to what mathematical expression?For the following exercises, use the figure shown here. 190. Using the figure, explain what the length of line segment AB represents.For the following exercises, complete each task. 191. Show that f(x,y)=exyxis differentiable at point (1, 0).For the following exercises, complete each task. 192. Find the total differential of the function w=eycos(x)+z2For the following exercises, complete each task. 193. Show that f(x,y)=x2+3y is differentiable at every point. In other words, show = fix + i. v + v) — f(x. y) = f x + f Y + e1 -‘ + &‘2 Y. where both E1 and F2 approach zero as iv. y) approaches (0. 0).For the following exercises, complete each task. 194. Find the total differential of the function z=xyy+x where x changes from 10 to 10.5 and y changes from 15 to 13.For the following exercises, complete each task. 195. Let z=f(x,y)=xey. Compute zfrom P( 1 2) to Q(1.05. 2.1) and then find the approximate change in z from point P to point Q. Recall z=f(x+x,y+y)f(x,y). and dz and z are approximately equal.For the following exercises, complete each task. 196. The volume of a right circular cylinder is given by V(r. h) = r2h. Find the differential dV. Interpret the formula geometrically.For the following exercises, complete each task. 197. See the preceding problem. Use differentials to estimate the amount of aluminum in an enclosed aluminum can with diameter 8.0 cm and height 1 2 cm if the aluminum is 0.04 cm thick.For the following exercises, complete each task. 198. Use the differential dz to approximate the change in z=4x2y2 as (x. y) moves from point (1. 1) to point (1 .01. 0.97). Compare this approximation with the actual change in the function.For the following exercises, complete each task. 199. Let z=f(x,y)=x2+3xyy2. Find the exact change in the function and the approximate change in the function as x changes from 2.00 to 2.05 and ‘ changes from 3.00 to 2.96.The centripetal acceleration of a particle moving in a circle is given by a(r,v)=y2r where v is the velocity and r is the radius of the circle. Approximate the maximum percent error in measuring the acceleration resulting from errors of 3 in v and 2% in r. (Recall that the percentage error is the ratio of the amount of error over the original amount. So, in this case, the percentage error in a is given by -)The radius r and height h of a tight circular cylinder ate measured with possible errors of 4% and 5%. respectively. Approximate the maximum possible percentage error in measuring the volume (Recall that the percentage error is the ratio of the amount of error over the original amount. So, in this case, the percentage error in is given by dvV .)The base radius and height of a right circular cone are measured as 10 in. and 25 in., respectively, with a possible error in measurement of as much as 0. 1 in. each. Use differentials to estimate the maximum error in the calculated volume of the cone.The electrical resistance R produced by wiring resistors R and R, in parallel can be calculated from the formula 1R=1R1+1R2 • If R1and R2are measured to beThe area of an ellipse with axes of length 2a and 2b is given by the formula A = ab. Approximate the percent change in the area when a increases by 2% and b increases by 1 .5%.The period T of a simple pendulum with small oscillations is calculated from the formula T=2Lg . where L is the length of the pendulum and g is the acceleration resulting from gravity. Suppose that L and g have errors of, at most, 0.5% and 0.1 %. respectively. Use differentials to approximate the maximum percentage error in the calculated value of T.Electrical power P is given by P = V2R where V is the voltage and R is the resistance. Approximate the maximum percentage error in calculating power if 120 V is applied to a 2000 — resistor and the possible percent errors in measuring V and R are 3% and 4%. respectively.For the following exercises, find the linear approximation of each function at the indicated point. 207. f(x,y)=xy,P(1,4)For the following exercises, find the linear approximation of each function at the indicated point. 208. f(x,y)=excosy;P(0,0)For the following exercises, find the linear approximation of each function at the indicated point. 209. f(x,y)=arctan(x+2y),P(1,0)For the following exercises, find the linear approximation of each function at the indicated point. 210. f(x,y)=20x27y2,P(2,1)For the following exercises, find the linear approximation of each function at the indicated point. 211. f(x,y,z)=x2+y2+z2,P(3,2,6)[T] Find the equation of the tangent plane to the surface f(x,y)=x2+y2 at point (1. 2. 5), and graph the surface and the tangent plane at the point.[T] Find the equation for the tangent plane to the surface at the indicated point, and graph the surface and the tangent plane: z=ln(10x2+2y2+1),P(0,0,0) .[T] Find the equation of the tangent plane to the surface z = f(x. y) = sin(x + y2) at point (4,0,22) . and graph the surface and the tangent plane.
For the following exercises, use the information provided to solve the problem. 215. Let w(x,y,z)=xycosz,wherex=t,y=t2,andz=arcsint.FinddwdtFor the following exercises, use the information provided to solve the problem. 217. If w=5x2+2y2,x=3s+tandy=s-4tfindwtandwtFor the following exercises, use the information provided to solve the problem. 218. If w=xy2,x=5cos(2t)andy=5sin(2t)findwtFor the following exercises, use the information provided to solve the problem. 219. If f(x,y)=xy,x=rcos,andy=rsinfindfrandexpresstheanswerintermsofrandFor the following exercises, use the information provided to solve the problem. 220. Suppose f(x,y)=x+y,u=exsiny,x=t2 and y=t , where x=rcos and y=rsin . Find f .For the following exercises, find dfdtusing the chain rule and direct substitution. 221. f(x,y)=x2+y2,x=t,y=t2For the following exercises, find dfdtusing the chain rule and direct substitution. 222. f(x,y)=x2+y2,y=t2,x=tFor the following exercises, find dfdtusing the chain rule and direct substitution. 223. f(x,y)=xy,x=1t,y=1+tFor the following exercises, find dfdtusing the chain rule and direct substitution. 224. f(x,y)=xy,x=et,y=2etFor the following exercises, find dfdtusing the chain rule and direct substitution. 225. f(x,y)=ln(x+y),x=et,y=etFor the following exercises, find dfdtusing the chain rule and direct substitution. 226. f(x,y)=x4,x=t,y=tLet w(x. v, z) = x + y + z, x = cos t, v = sin t, and z = e’. Express w as a function of t and find d31’ directly. Then, find using the chain rule.Let z=x2y. where x=t2 and v=t2. Find dzdt .Let u=exsiny, where x=t2 and y=t . Find dudtwhen x = in 2 and y=4For the following exercises, find dv using partial dx derivatives. 230. Sin sin(6x)+tan(8y)+5=0For the following exercises, find dydx using partial derivatives. 231. x3+y2x3=0For the following exercises, find dydx using partial derivatives. 232. sin(x+y)+cos(xy)=4For the following exercises, find dydx using partial derivatives. 233. x22xy+y4=4For the following exercises, find dydx using partial derivatives. 234. xey+yex2x2y=0For the following exercises, find dydx using partial derivatives. 235. x2/3+y2/3=a2/3For the following exercises, find dydx using partial derivatives. 236. xcos(xy)+ycosx=2For the following exercises, find dydx using partial derivatives. 237. exy+yey=1For the following exercises, find dydx using partial derivatives. 238. x2y3+cosy=0For the following exercises, find dydx using partial derivatives. 239. Find dxdt using the chain rule where z=3x2y3,x=t4,andy=t2For the following exercises, find dydx using partial derivatives. 240. Let z=3cosxsin(xy),x=1tandy=3t.FinddzdtFor the following exercises, find dydx using partial derivatives. 241. Let z=e1xy,x=t1/3,andy=t3.Finddzdt.For the following exercises, find dydx using partial derivatives. 242. Find dzdt by the chain rule where z=cosh2(xy),x=12t , and y=etFor the following exercises, find dydx using partial derivatives. 243.Let z=xy,x=2cosu,andy=3sinv.Findzuandzv.For the following exercises, find dydx using partial derivatives. 224. Let z=ex2y ,where x=uv and y=1v . Find zu and zvFor the following exercises, find dydx using partial derivatives. 245. If z=xyex/y,x=rcos , and y=rsin , find zr and zv when r=2 and =6 .For the following exercises, find dydx using partial derivatives. 246. Find ws if w=4x+y2+z3,x=ers2,y=ln(r+st) and z=rst2 .For the following exercises, find dydx using partial derivatives. 247. If w=sin(xyz),x=13t,y=e1t , and z=4t , find wt .For the following exercises, use this information: A function f(x. y) is said to be homogeneous of degree n if f(tx. ty) = tnf(x. v). For all homogeneous functions of degree n. the following equation is true: xfx+yfy=nf(x,y) . Show that the given function is homogeneous and verify that xfx+yfy=nf(x,y). 248. f(x,y)=3x2+y2For the following exercises, use this information: A function f(x. y) is said to be homogeneous of degree n if f(tx. ty) = tnf(x. v). For all homogeneous functions of degree n. the following equation is true: xfx+yfy=nf(x,y) . Show that the given function is homogeneous and verify that xfx+yfy=nf(x,y). 249. f(x,y)=x2+y2For the following exercises, use this information: A function f(x. y) is said to be homogeneous of degree n if f(tx. ty) = tnf(x. v). For all homogeneous functions of degree n. the following equation is true: xfx+yfy=nf(x,y) . Show that the given function is homogeneous and verify that xfx+yfy=nf(x,y). 250. f(x,y)=x2y2y3The volume of a right circular cylinder is given by V(x. y) = x2y where x is the radius of the cylinder and y is the cylinder height. Suppose x and y are functions of i given by x=12t and y=13t so that .t and are both increasing with time. How fast is the volume increasing when x = 2 and y = 5?The pressure P of a gas is related to the volume and temperature by the formula PV = kT. where temperature is expressed in kelvins. Express the pressure of the gas as a function of both V and T. Find dpdt when k = 1, dvdt=2cm3/min dVdt=2cm3/min , and T = 20°F.The radius of a tight circular cone is increasing at 3 cm/min mm whereas the height of the cone is decreasing at 2 cm/min. Find the rate of change of the volume of the cone when the radius is 1 3 cm and the height is 1 8 cm.The volume of a frustum of a cone is given by the formula V=13z(x2+y2+xy), where x is the radius of the smaller circle, y is the radius of the larger circle, and is the height of the frustum (see figure). Find the rate of change of the volume of this frustum when x =10 in. y=12 in.. and z= 18 in.A closed box is in the shape of a rectangular solid with dimensions x. v, and z. (Dimensions are in inches.) Suppose each dimension is changing at the rate of 0.5 in. mm. Find the rate of change of the toal surface area of the box when x = 2in.,v = 3 in.. and z = I in.The total resistance in a circuit that has three individual resistances represented by x. y. and z is given by the formula R(x,y,z)=xyzyz+xz+xy . Suppose at a given time the x resistance is 100 . the y resistance is 200 . and the z resistance is 300 . Also, suppose the v resistance is changing at a rate of 2 /min. the y resistance is changing at the rate of 1 /min. and the z resistance has no change. Find the rate of change of the total resistance in this circuit at this time.The temperature T at a point (x. y) is T(x. y) and is measured using the Celsius scale. A fly crawls so that its position after a’ seconds is given by x=1+t and y=2+13t where x and y are measured in centimeters. The temperature function satisfies Tx(2,3)=4 and Ty(2,3)=3 . How fast is the temperature increasing on the fly’s path after 3 sec?The x and y components of a fluid moving in two dimensions are given by the following functions: u(x. y) = 2y and v(x,y)=2x;x0;y0 . The speed of the fluid at the point (x. y) is s(x,y)=u(x,y)2+v(x,y)2. Find sx and sy using the chain rule.Let u = u(x. y, z). where x = x(w, t). y = (w. t). z = z(w, t). w = w(r, s), and t = t(r, s). Use a tree diagram and the chain rule to find an expression for ur .For the following exercises, find the directional derivative using the limit definition only. 260. f(x,y)=52x212y2 at point P(3. 4) in the direction of u = u=(cos4)i+(sin4)JFor the following exercises, find the directional derivative using the limit definition only. 261. f(x,y)=y2cos(2x)atpointP(3,2)inthedirectionofu=(cos4)i+(sin4)jFor the following exercises, find the directional derivative using the limit definition only. 262. Find the directional derivative of f(x,y)=y2sin(2x) at point P(4,2) in the direction of u=5i+12j.For the following exercises, find the directional derivative of the function at point P in the direction of v. 263. f(x,y)=xy,P(02),v=12i+32jFor the following exercises, find the directional derivative of the function at point P in the direction of v. 264. h(x,y)=exsiny,P(1,2)v=iFor the following exercises, find the directional derivative of the function at point P in the direction of v. 265. h(x,y)=xyz,P(2,1,1),v=2i+jkFor the following exercises, find the directional derivative of the function at point P in the direction of v. 266. f(x,y)=xy,P(2,1,1)v=2i+j-kFor the following exercises, find the directional derivative of the function at point P in the direction of v. 267. f(x,y)=x2y2,u=(32,12),P(1,0)For the following exercises, find the directional derivative of the function at point P in the direction of v. 268. f(x,y)=3x+4y+7,u=(35,45),P(0,2)For the following exercises, find the directional derivative of the function at point P in the direction of v. 269. f(x,y)=excosy,u=(0,1),P=(0,2)For the following exercises, find the directional derivative of the function at point P in the direction of v. 270. f(x,y)=y10,u=(0,1),P=(11)For the following exercises, find the directional derivative of the function at point P in the direction of v. 271. f(x,y)=ln(x2+y2),u=(35,45),P(1,2)For the following exercises, find the directional derivative of the function at point P in the direction of v. 272. f(x,y)=x2y,P(5,5),v=3i-4jFor the following exercises, find the directional derivative of the function at point P in the direction of v. 273. f(x,y)=y2+xz,P(1,2,2),v=(2,1,2)For the following exercises, find the directional derivative of the function in the direction of the unit vector u cos i + sin j . 274. f(x,y)=x2+2y2,=6For the following exercises, find the directional derivative of the function in the direction of the unit vector u cos i + sin j . 275. f(x,y)=yx+2y,=4For the following exercises, find the directional derivative of the function in the direction of the unit vector u cos i + sin j 276. f(x,y)=cos(3x+y)=4For the following exercises, find the directional derivative of the function in the direction of the unit vector u cos i + sin j 277. w(x,y)=yex,=3For the following exercises, find the directional derivative of the function in the direction of the unit vector u cos i + sin j 278. f(x,y)=xarctan(y),=2For the following exercises, find the directional derivative of the function in the direction of the unit vector u cos i + sin j 279. f(x,y)=ln(x+2y),=3For the following exercises, find the gradient. 280. Find the gradient of f(x,y)=14x2y23 . Than, find the gradient a point P(1,2) .For the following exercises, find the gradient. 281. Find the gradient of f(x,y,z)=xy+yz+xz , at point P(1,2,3)For the following exercises, find the gradient. 282. Find the gradient of f(x,y,z) at P and in the direction of u:f(x,y,z)=ln(x2+2y2+3z2),P(2,1,4)u=313i-413j-1213kFor the following exercises, find the gradient. 283. f(x,y,z)=4x5y2z3,P(2,1,1),u=13i+23j-23kFor the following exercises, find the directional derivative of the function at point P in the direction of Q. 284. f(x,y)=x2+3y2,P(1,1),Q(4,5)For the following exercises, find the directional derivative of the function at point P in the direction of Q. 285. f(x,y,z)=yx+z,P(2,1,1),Q(1,2,0)For the following exercises, find the derivative of the function at P in the direction of u. 286. f(x,y)=7x+2y,P(24),u=4i-3jFor the following exercises, find the derivative of the function at P in the direction of u. 287. f(x,y)=ln(5x+4y),P(3,9),u=6i+8j[T] Use technology to sketch the level curve of (x. y) = 4x — 2y + 3 that passes through P(1. 2) and draw the gradient vector at P.[T] Use technology to sketch the level curve of f(x,y)=x2+4y2that passes through P(—2. 0) and draw the gradient vector at P.For the following exercises, find the gradient vector at the indicated point. 290. f(x,y)=xy2yx2P(1,1)For the following exercises, find the gradient vector at the indicated point. 291. f(x,y)=xeyln(x),P(3,0)For the following exercises, find the gradient vector at the indicated point. 292. f(x,y,z)=xyln(z),P(2,2,2)For the following exercises, find the gradient vector at the indicated point. 293. f(x,y,z)=xy2+z2,P(2,1,1)For the following exercises, find the derivative of the function. 294. f(x,y)=x2+xy+y2at point (-5, -4) in the direction the function increases most rapidlyFor the following exercises, find the derivative of the function. 295. f(x,y)=exy at point (6. 7) in the direction the function increases most rapidlyFor the following exercises, find the derivative of the function. 296. f(x,y)=arctan(yx) at point (—9. 9) in the direction the function increases most rapidlyFor the following exercises, find the derivative of the function. 297. f(x,y,z)=ln(xy+yz+zx)at point (-9, -1 8. -27) in the direction the function increases most rapidlyFor the following exercises, find the derivative of the function. 298. f(x,y,z)=xy+yz+zx at point (5, —5, 5) in the direction the function increases most rapidlyFor the following exercises, find the maximum rate of change of f at the given point and the direction in which it occurs. 299. f(x,y)=xey,(1,0)For the following exercises, find the maximum rate of change of f at the given point and the direction in which it occurs. 300. f(x,y)=x2+2y,(4,10)For the following exercises, find the maximum rate of change of f at the given point and the direction in which it occurs. 301. f(x,y)=cos(3x+2y),(6,8)For the following exercises, find equations of a. the tangent plane and b. the normal line to the given surface at the given point. 302. The level curve f(x,y,z)=12 for f(x,y,z)=4x22y2+z2at point (2. 2. 2).For the following exercises, find equations of a. the tangent plane and b. the normal line to the given surface at the given point. 303. f(x,y,z)=xy+yz+xz=3 at point (1,1,1)For the following exercises, find equations of a. the tangent plane and b. the normal line to the given surface at the given point. 304. f(x,y,z)=xyz=6 at point (1,2,3)For the following exercises, find equations of a. the tangent plane and b. the normal line to the given surface at the given point. 305. f(x,y,z)=xeycoszz=1 at point (1,0,0)For the following exercises, solve the problem. 306. The temperature T in a metal sphere is inversely proportional to the distance from the center of the sphere (the origin: (0. 0. 0)). The temperature at point (1. 2. 2) is 120°C. a. Find the rate of change of the temperature at point (1 2. 2) in the direction toward point (2. 1. 3). b. Show that, at any point in the sphere, the direction of greatest increase in temperature is given by a vector that points toward the origin.For the following exercises, solve the problem. 307. The electrical potential (voltage) in a certain region of space is given by the function V(x,y,z)=5x23xy+xyz. a. Find the rate of change of the voltage at point (3. 4. 5) in the direction of the vector ( 1. 1. —1) b. In which direction does the voltage change most rapidly at point (3. 4. 5)? c. What is the maximum rate of change of the voltage at point (3, 4. 5)?For the following exercises, solve the problem. 308. If the electric potential at a point (x. y) in the xy-plane is V(x,y)=e2xcos(2y) . then the electric intensity vector at E=-V(x,y). a. Find the electric intensity vector at ( 4,0 ). b. Show that, at each point in the plane, the electric potential decreases most rapidly in the direction of the vector E. 309. In two dimensions, the motion of an ideal fluid is governed by a velocity potential q. The velocity components of the fluid u in the x—direction and v in the y—direction, are given by ( u. v ) = V q. Find the velocity components associated with the velocity potential p(x, y) = sin .zx sin 2irv.For the following exercises, solve the problem. 309. In two dimensions, the motion of an ideal fluid is governed by a velocity potential q. The velocity components of the fluid u in the x—direction and v in the y—direction, are given by ( u. v ) = V q. Find the velocity components associated with the velocity potential p(x, y) = sin .zx sin 2irv.For the following exercises, find all critical points. 310. f(x,y)=1+x2+y2For the following exercises, find all critical points. 311. f(x,y)=(3x2)2+(y4)2For the following exercises, find all critical points. 312. f(x,y)=x4+y416xyFor the following exercises, find all critical points. 313. f(x,y)=15x33xy+15y3For the following exercises, find the critical points of the function by using algebraic techniques (completing the square) or by examining the form of the equation. Verify your results using the partial derivatives test. 314. f(x,y)=x2+y2+1For the following exercises, find the critical points of the function by using algebraic techniques (completing the square) or by examining the form of the equation. Verify your results using the partial derivatives test. 315. f(x,y)=x25y2+8x10y13For the following exercises, find the critical points of the function by using algebraic techniques (completing the square) or by examining the form of the equation. Verify your results using the partial derivatives test. 316. f(x,y)=x2+y2+2x6y+6For the following exercises, find the critical points of the function by using algebraic techniques (completing the square) or by examining the form of the equation. Verify your results using the partial derivatives test. 317. f(x,y)=x2+y2+1For the following exercises, use the second derivative test to identify any critical points and determine whether each critical point is a maximum, minimum, saddle point, or none of these. 318. f(x,y)=x2+4xy2y2+1For the following exercises, use the second derivative test to identify any critical points and determine whether each critical point is a maximum, minimum, saddle point, or none of these. 319. f(x,y)=x2y2For the following exercises, use the second derivative test to identify any critical points and determine whether each critical point is a maximum, minimum, saddle point, or none of these. 320. f(x,y)=x26x+y24y8For the following exercises, use the second derivative test to identify any critical points and determine whether each critical point is a maximum, minimum, saddle point, or none of these. 321. f(x,y)=2xy+3x+4yFor the following exercises, use the second derivative test to identify any critical points and determine whether each critical point is a maximum, minimum, saddle point, or none of these. 322. f(x,y)=8xy(x+y)+7For the following exercises, use the second derivative test to identify any critical points and determine whether each critical point is a maximum, minimum, saddle point, or none of these. 323. f(x,y)=x2+4xy+y2For the following exercises, use the second derivative test to identify any critical points and determine whether each critical point is a maximum, minimum, saddle point, or none of these. 324. f(x,y)=x3+y3300x75y3For the following exercises, use the second derivative test to identify any critical points and determine whether each critical point is a maximum, minimum, saddle point, or none of these. 325. d(x,y)=9x4y4For the following exercises, use the second derivative test to identify any critical points and determine whether each critical point is a maximum, minimum, saddle point, or none of these. 326. f(x,y)=7x2y+9xy2For the following exercises, use the second derivative test to identify any critical points and determine whether each critical point is a maximum, minimum, saddle point, or none of these. 327. f(x,y)=3x22xy+y28yFor the following exercises, use the second derivative test to identify any critical points and determine whether each critical point is a maximum, minimum, saddle point, or none of these. 328. f(x,y)=3x2+2xy+y2For the following exercises, use the second derivative test to identify any critical points and determine whether each critical point is a maximum, minimum, saddle point, or none of these. 329. f(x,y)=y2+xy+3y+2x+3For the following exercises, use the second derivative test to identify any critical points and determine whether each critical point is a maximum, minimum, saddle point, or none of these. 330. f(x,y)=x2+xy+y23xFor the following exercises, use the second derivative test to identify any critical points and determine whether each critical point is a maximum, minimum, saddle point, or none of these. 331. f(x,y)=x2+2y2x2yFor the following exercises, use the second derivative test to identify any critical points and determine whether each critical point is a maximum, minimum, saddle point, or none of these. 332. f(x,y)=x2+yeyFor the following exercises, use the second derivative test to identify any critical points and determine whether each critical point is a maximum, minimum, saddle point, or none of these. 333. f(x,y)=e(x2+y2+2x)For the following exercises, use the second derivative test to identify any critical points and determine whether each critical point is a maximum, minimum, saddle point, or none of these. 334. f(x,y)=x2+xy+y2xy+1For the following exercises, use the second derivative test to identify any critical points and determine whether each critical point is a maximum, minimum, saddle point, or none of these. 335. f(x,y)=x2+10xy+y2For the following exercises, use the second derivative test to identify any critical points and determine whether each critical point is a maximum, minimum, saddle point, or none of these. 336. f(x,y)=x25y2+10x30y62For the following exercises, use the second derivative test to identify any critical points and determine whether each critical point is a maximum, minimum, saddle point, or none of these. 337. f(x,y)=120x+120yxyx2y2For the following exercises, use the second derivative test to identify any critical points and determine whether each critical point is a maximum, minimum, saddle point, or none of these. 338. f(x,y)=2x2+2xy+y2+2x3For the following exercises, use the second derivative test to identify any critical points and determine whether each critical point is a maximum, minimum, saddle point, or none of these. 339. f(x,y)=x2+x3xy+y35For the following exercises, use the second derivative test to identify any critical points and determine whether each critical point is a maximum, minimum, saddle point, or none of these. 340. f(x,y)=2xyex2y2For the following exercises, determine the extreme values and the saddle points. Use a CAS to graph the function. 341. [T]f(x,y)=yexeyFor the following exercises, determine the extreme values and the saddle points. Use a CAS to graph the function. 342. [T]f(x,y)=xsin(y)For the following exercises, determine the extreme values and the saddle points. Use a CAS to graph the function. 343. [T]f(x,y)=sin(x)sin(y),x(0,2),y(0,2)Find the absolute extrema of the given function on the indicated closed and bounded set R. 344. f(x. v) = xy — x — 3y: R is the triangular region with vertices (0. 0). (0. 4). and (5. 0).Find the absolute extrema of the given function on the indicated closed and bounded set R. 344. f(x,y)=xyx3y;R is the triangular region with vertices (0,0)(0,4), and (5,0)Find the absolute extrema of the given function on the indicated closed and bounded set R. 346. f(x,y)=x33xyy3onR=(x,y):2x2,2y2Find the absolute extrema of the given function on the indicated closed and bounded set R. 347. f(x,y)=2yx2+y2+1onR=(x,y):x2+y24Find three positive numbers the sum of which is 27, such that the sum of their squares is as small as possible.Find the points on the surface x2yz=5that are closest to the origin.Find the maximum volume of a rectangular box with three faces in the coordinate planes and a vertex in the first octant on the plane x + y + z = 1.The sum of the length and the girth (perimeter of a cross-section) of a package carried by a delivery service cannot exceed 108 in. Find the dimensions of the rectangular package of largest volume that can be sent.A cardboard box without a lid is to be made with a volume of 4 ft3. Find the dimensions of the box that requires the least amount of cardboard.Find the point on the surface f(x,y)=x2+y2+10 nearest the plane x+2yz=0 . Identify the point on the plane.Find the point in the plane 2xy+2z=16 that is closest to the origin.A company manufactures two types of athletic shoes: jogging shoes and cross-trainers. The total revenue from x units of jogging shoes and y units of cross—trainers is given by R(x,y)=5x28y22xy+42x+102y . where x and v are in thousands of units. Find the values of x and y to maximize the total revenue.A shipping company handles rectangular boxes provided the sum of the length, width, and height of the box does not exceed 96 in. Find the dimensions of the box that meets this condition and has the largest volume.Find the maximum volume of a cylindrical soda can such that the sum of its height and circumference is 1 20 cm.For the following exercises, use the method of Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraints. 358. f(x,y)=x2y;x2+2y2=6For the following exercises, use the method of Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraints. 359. f(x,y,z)=xyz,x2+2y2+3z2=6For the following exercises, use the method of Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraints. 360. f(x,y)=xy;4x2+8y2=16For the following exercises, use the method of Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraints. 361. f(x,y)=4x3+y2+z2,x4+y4+z4=1For the following exercises, use the method of Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraints. 362. f(x,y,z)=x2+y2+z2,x4+y4+z4=1For the following exercises, use the method of Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraints. 363. f(x,y,z)=yz+xy,xy=1,y2+z2=1For the following exercises, use the method of Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraints. 364. f(x,y)=x2+y2,(x1)2+4y2=4For the following exercises, use the method of Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraints. 365. f(x,y)=4xy,x29+y216=1For the following exercises, use the method of Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraints. 366. f(x,y,z)=x+y+z,1x+1y+1z=1For the following exercises, use the method of Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraints. 367. f(x,y,z)=x+3yz,x2+y2+z4=4For the following exercises, use the method of Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraints. 368. f(x,y,z)=x2+y2+z2,xyz=4For the following exercises, use the method of Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraints. 369. Minimize f(x,y)=x2+y2 on the hyperbola xy=1For the following exercises, use the method of Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraints. 370. Maximize f(x,y)=xy on the ellipse b2x2+a2y2=a2b2 .For the following exercises, use the method of Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraints. 371. Maximize f(x,y,z)=2x+3y+5z on the sphere x2+y2+z2=19 .For the following exercises, use the method of Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraints. 372. Maximze g(x,y)=yx2=0f(x,y)=x2y2;x0,y0;For the following exercises, use the method of Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraints. 373. The curve x3y3=1 is asymptotic to the line y=x, Find the point(s) on the curve x3y3=1 farthest from the line y=x .For the following exercises, use the method of Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraints. 374. Maximize U(x,y)=8x4/5y1/5;4x+2y=12For the following exercises, use the method of Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraints. 375. Minimize f(x,y)=x2+y2,x+2y5=0For the following exercises, use the method of Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraints. 376. Maximize f(x,y)=6x2y2,x+y2=0For the following exercises, use the method of Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraints. 377. Minimize f(x,y,z)=x2+y2+z2+x+y+z=1For the following exercises, use the method of Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraints. 378. Minimize f(x,y)=x2y2,x+y+z=1 bFor the following exercises, use the method of Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraints. 379. Minimize f(x,y,z)=x2+y2+z2 when x+y+z=9 and x+2y+3z=20For the next group of exercises, use the method of Lagrange multipliers to solve the following applied problems. 380. A pentagon is formed by placing an isosceles triangle on a rectangle, as shown in the diagram. If the perimeter of the pentagon is 10 in., find the lengths of the sides of theFor the next group of exercises, use the method of Lagrange multipliers to solve the following applied problems. 381. A rectangular box without a top (a topless box) is to be made from 12 ft2 of cardboard. Find the maximum volume of such a box.For the next group of exercises, use the method of Lagrange multipliers to solve the following applied problems. 382. Find the minimum and maximum distances between the ellipse x2+xy+2y2=1For the next group of exercises, use the method of Lagrange multipliers to solve the following applied problems. 383. Find the point on the surface x22xy+y2x+y=0 closest to the point (1,2,-3)Show that, of all the triangles inscribed in a circle of radius R (see diagram), the equilateral triangle has the largest perimeter.For the next group of exercises, use the method of Lagrange multipliers to solve the following applied problems. 385. Find the minimum distance from point (0. 1) to the parabola x2= 4y.For the next group of exercises, use the method of Lagrange multipliers to solve the following applied problems. 386. Find the minimum distance from the parabola y=x2 to point (0. 3).For the next group of exercises, use the method of Lagrange multipliers to solve the following applied problems. 387. Find the minimum distance from the plane x+y+z= 1 to point (2 1. 1).For the next group of exercises, use the method of Lagrange multipliers to solve the following applied problems. 388. A large container in the shape of a rectangular solid must have a volume of 480 m3. The bottom of the container costs $5m2 to construct whereas the top and sides cost $3/rn2 to construct. Use Lagrange multipliers to find the dimensions of the container of this size that has the minimum cost.For the next group of exercises, use the method of Lagrange multipliers to solve the following applied problems. 389. Find the point on the line y = 2x + 3 that is closest to point (4. 2).For the next group of exercises, use the method of Lagrange multipliers to solve the following applied problems. 390. Find the point on the plane 4x + 3y + z = 2 that is closest to the point (1, — 1, 1).For the next group of exercises, use the method of Lagrange multipliers to solve the following applied problems. 391. Find the maximum value of f(x. v) = sin x sin v, where x and y denote the acute angles of a right triangle. Draw the contours of the function using a CAS.For the next group of exercises, use the method of Lagrange multipliers to solve the following applied problems. 392. A rectangular solid is contained within a tetrahedron with vertices at (I, 0. 0). (0. 1. 0). (0. 0. 1). and the origin. The base of the box has dimensions x, y and the height of the box is z. If the sum of x. y and z is 1.0, find the dimensions that maximizes the volume of the rectangular solid.For the next group of exercises, use the method of Lagrange multipliers to solve the following applied problems. 393. [T] By investing x units of labor and y units of capital, a watch manufacturer can produce P(x,y)=50x0.4,y0.6 watches. Find the maximum number of watches that can be produced on a budget of $20,000 if labor costs $100/unit and capital costs $200/ unit. Use a CAS to sketch a contour plot of the function.For the following exercises, determine whether the statement is true or false. Justify your answer with a proof or a counterexample. 394. The domain of f(x,y)=x3sin1 is x= all real numbers, and y.For the following exercises, determine whether the statement is true or false. Justify your answer with a proof or a counterexample. 395. If the function f(x,y) is continuous ever where, then fxy=fyxFor the following exercises, determine whether the statement is true or false. Justify your answer with a proof or a counterexample. 396. The linear approximation to the function of f(x,y)=5x2+xtan(y) at (2,) is given by L(x,y)=22+21(x2)+(y) .For the following exercises, determine whether the statement is true or false. Justify your answer with a proof or a counterexample. 397. (34,916) is a critical point of g(x,y)=4x32x2y+y22For the following exercises, sketch the function in one graph and, in a second, sketch several level curves. 398. f(x,y)=e(x2+2y2)For the following exercises, sketch the function in one graph and, in a second, sketch several level curves. 399. f(x,y)=x+4y2For the following exercises, evaluate the following limits, if they exist. If they do not exist, prove it. 400. (x,y)(1,1)lim4xyx2y2For the following exercises, evaluate the following limits, if they exist. If they do not exist, prove it. 401. (x,y)(0,0)lim4xyx2y2For the following exercises, find the largest interval of continuity for the function. 402. f(x,y)=x3sin1(y)For the following exercises, find the largest interval of continuity for the function. 403. g(x,y)=ln(4x2y2)For the following exercises, find the largest interval of continuity for the function. 404. f(x,y)=x22For the following exercises, find the largest interval of continuity for the function. 405. u(x,y)=x43xy+1,x=2t,y=t3For the following exercises, find all second partial derivatives. 406. g(t,x)=3t2sin(x+t)For the following exercises, find all second partial derivatives. 407. h(x,y,z)=x3e2yzFor the following exercises, find the equation of the tangent plane to the specified surface at the given point. 408. z=x32y2+y1atpoint(1,1,1)For the following exercises, find the equation of the tangent plane to the specified surface at the given point. 409. 3z3=ex+2y at point (0,1,3)For the following exercises, find the equation of the tangent plane to the specified surface at the given point. 408. z=x32y2+y1 at point (1,1,1)For the following exercises, find the equation of the tangent plane to the specified surface at the given point. 411. Find the differential dz of h(x,y)=4x2+2xy3y and approximate z at the point (1,2) . Let x=0.1 and x=0.01For the following exercises, find the equation of the tangent plane to the specified surface at the given point. 412. Find the differential dz of f(x,y)=x2+6xyy2 in the direction v=i+4j .For the following exercises, find the equation of the tangent plane to the specified surface at the given point. 413. Find the maximal directional derivative magnitude direction for the function f(x,y)=x3+2xycos(y) at point (3. 0).For the following exercises, find the gradient. 414. c(x,t)=e(tx)2+3cos(t)For the following exercises, find the gradient. 415. f(x,y)=x+y2xyFor the following exercises, find and classify the critical points. 416. z=x3xy+y21For the following exercises, find and classify the critical points. 417. f(x,y)=x2y,x2+y2=4For the following exercises, use Lagrange multipliers to find the maximum and minimum values for the functions with the given constraints. 418. f(x,y)=x2y2,6y=4A machinist is constructing a right circular cone out of a block of aluminum. The machine gives an error of 59 in height and 2% in radius. Find the maximum error in the volume of the cone if the machinist creates a cone of height 6 cm and radius 2 cm.A trash compactor is in the shape of a cuboid. Assume the trash compactor is filled with incompressible liquid. The length and width are decreasing at rates of 2 ft/sec and 3 ft/sec, respectively. Find the rate at which the liquid level is rising when the length is 1 4 ft, the width is 1 0 ft, and the height is 4 ft.In the following exercises, use the midpoint rule with m = 4 and m = 2 to estimate the volume of the solid bounded by the surface = f(x. y),the vertical planes x = 1. x = 2. y =1. and y = 2. and the horizontal plane z = 0. 1. f(x. y) = 4x + 2’ + 8xyIn the following exercises, use the midpoint rule with m = 4 and m = 2 to estimate the volume of the solid bounded by the surface = f(x. y),the vertical planes x = 1. x = 2. y =1. and y = 2. and the horizontal plane z = 0. 2. f(x,y)=16x2+y2In the following exercises, estimate the volume of the solid under the surface z= f(x. y) and above the rectangular legion R by using a Riemann sum with m = n = 2 and the sample points to be the lower left corners of the subrectangles of the partition. 3. f(x,y)=sinxcosy,R[0,][0,]In the following exercises, estimate the volume of the solid under the surface z= f(x. y) and above the rectangular legion R by using a Riemann sum with m = n = 2 and the sample points to be the lower left corners of the subrectangles of the partition. 4. f(x,y)=cosx+cosy,R=[0,][0,2]In the following exercises, estimate the volume of the solid under the surface z= f(x. y) and above the rectangular legion R by using a Riemann sum with m = n = 2 and the sample points to be the lower left corners of the subrectangles of the partition. 5. Use the midpoint rule with m=n=2 to estimate Rf(x,y)dA, where the values of the function f on R=[8,10][9,11] are given in the following table. y x 9 9.5 10 10.5 11 8 9.8 5 6.7 5 5.6 8.5 9.4 4.5 8 5.4 3.4 9 8.7 4.6 6 5.5 3.4 9.5 6.7 6 4.5 5.4 6.7 10 6.8 6.4 5.5 5.7 6.8In the following exercises, estimate the volume of the solid under the surface z= f(x, y) and above the rectangular legion R by using a Riemann sum with m = n = 2 and the sample points to be the lower left corners of the subrectangles of the partition. 6. The values of the function f on the rectangle R=[0,2][7,9] are given in the following table. Estimate the double integral Rf(x,y)dA by using a Riemann sum with m = n = 2. Select the sample points to be the upper right corners of the subsquares of R.In the following exercises, estimate the volume of the solid under the surface z= f(x. y) and above the rectangular legion R by using a Riemann sum with m = n = 2 and the sample points to be the lower left corners of the subrectangles of the partition. 7. The depth of a children’s 4-ft by 4-ft swimming pool, measured at 1—ft intervals, is given in the following table. a. Estimate the volume of water in the swimming pool by using a Riemann sum with m = n = 2. Select the sample points using the midpoint rule on R=[0,4][0,4] . b. Find the average depth of the swimming pool. y x 0 1 2 3 4 0 1 1.5 2 2.5 3 1 1 i. 2 2.5 3 1 1.5 1.5 2.5 3 1 1 1.5 2 2.5 4 1 1 1 1.5 2In the following exercises, estimate the volume of the solid under the surface z= f(x. y) and above the rectangular legion R by using a Riemann sum with m = n = 2 and the sample points to be the lower left corners of the subrectangles of the partition. 8. The depth of a 3-ft by 3-ft hole in the ground, measured at 1—ft intervals, is given in the following table. a. Estimate the volume of the hole by using a Riemann sum with m =n = 3 and the sample points to be the upper left conwrs of the subsquares of R. b. Find the average depth of the hole. y x 0 1 2 3 0 6 6.5 6.4 6 1 6.5 7 7.5 6.5 2 6.5 6.7 6.5 6 3 6 6.5 5 5.6In the following exercises, estimate the volume of the solid under the surface z= f(x. y) and above the rectangular legion R by using a Riemann sum with m = n = 2 and the sample points to be the lower left corners of the subrectangles of the partition. 9. The level curves f(x. y) = k of the function fare given in the following graph, where k is a constant. a. Apply the midpoint rule with in = ii = 2 to estimate the double integral if f(x. y)dA. whereIn the following exercises, estimate the volume of the solid under the surface z= f(x. y) and above the rectangular legion R by using a Riemann sum with m = n = 2 and the sample points to be the lower left corners of the subrectangles of the partition. 10. The level curves f(x. y) = k given in the following graph, where a. Apply the midpoint rule with m=n=2 to estimate the double integral Rf(x,y)dA , where R=[0,1,0.5][0,1,0.5] b. Estimate the average value of the function f on R.In the following exercises, estimate the volume of the solid under the surface z= f(x. y) and above the rectangular legion R by using a Riemann sum with m = n = 2 and the sample points to be the lower left corners of the subrectangles of the partition. 11. The solid lying under the surface z=4y2 and above the rectangular region R = [0. 2] [0, 2] is illustrated in the following graph. Evaluate the double integral Rf(x,y)dA. where f(x,y)=4y2 . by finding the volume of the corresponding solid.In the following exercises, estimate the volume of the solid under the surface z= f(x. y) and above the rectangular legion R by using a Riemann sum with m = n = 2 and the sample points to be the lower left corners of the subrectangles of the partition. 12. The solid lying under the plane z=y+4 the rectangular region R=[0,2][0,4] in the following graph. Evaluate the Rf(x,y)dA. where f(x,y)=y+4. by finding the volume of the corresponding solid.In the following exercises, calculate the integrals by interchanging the order of integration. 13. 11( 22( 2x+3y+5)dx)dyIn the following exercises, calculate the integrals by interchanging the order of integration. 14. 02(01( x+2ey 3)dx)dyIn the following exercises, calculate the integrals by interchanging the order of integration. 15. 127(12(3 x+3y)dy)dxIn the following exercises, calculate the integrals by interchanging the order of integration. 16. 116(18( 4x +23y )dy)dxIn the following exercises, calculate the integrals by interchanging the order of integration. 17. ln2ln3(01 e x+ydy)dxIn the following exercises, calculate the integrals by interchanging the order of integration. 18. 02(01 3 x+ydy)dx