Richard_Exam_3_Practice_Winter_2007

Richard_Exam_3_Practice_Winter_2007 - M ATH-203 E XAM 3 l....

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MATH-203 EXAM 3 PLEASE SHOW ALL WORK l. LetF(r. 1- z\=x>t^ rz' a. Find the sradient of'F at the point P(2. 2' -l )' b. Find the Jirectional derivative o{-F at 1he point P(2^2. -1) in the direction of the vector :i+ +i - zri . c. Find the equation of the tangent plane to the surl-ace *v,t + z-- -5 at the point P(2'2' 1)' 2. Use the method of Lagrange Multipliers to find the rnaximurn and minimum values of ri-. vi: xy3 subject to theionstraint xr + y2 :44' 3. Find and classif, all relative extema and saddle points of f(x' y): xy - *'- yt' -1. Let D be the region in the first quadrant borurded between the graphs of x: yt + y and y -- x ' a. Sketch thc region D. b. Write a double integral in rectangular coordinates which gives the area of D' Do not integrate' 5. Let S be the solid bouncled the graphs of z: | (*t +yt) ar-rd z: 0. the solid S. in rectangular coordinates which sives the vctlumc of S. Do not integrate' r Jrl 6. Consider the J j x'e*v dx dy' 0v/ the region ofintegration' b. Reverse (interchange) the order of intesration. Do not integrate. I lr 7. lrvaluate iterated f i (3xv' + 2x)dy dx '
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Answers -40 l. <-2,-2.13> ,o -#, 1c'-2x-2v+l3z---)1 2. Maximum value of f(2, 6) : 432'Minimum of f(2' -6) = 432 3. Saddle point at (0,0), Relative Maximum " [:';) I loy 4b J J axaY uy+y 5a. Solid is bounded above by the paraboloid z | -(*t * y' ) and below by the plane z: 0 overthe region bounded inside the circle x2 * yt 1' I Jr-t sb. [ ltt - x' - v'; dvdx ., i: -r-rl l-x' l3x 24x2 uo I i x2e-YdYdx + x'e"dYdx 00 I 0 43 1. 30
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MATH-203 EXAM 3 PLE.{SE SHOW ALL WORK I . Find and classif' all relative extema and saddre points of f(x, y) : xa * gxy + 2y2. 2. Let F(x, y, z): y'z' + xtz. a. Find a vector that is perpendicular to the surface y'23 + x'z: l0 at the point p(l, -1, 2). b- Find the directional derivative of F at the point P( l, -1, 2) in the direction of the vector i-zj+:L. c. Find the equation of the tangent plane to the surfac " yrr. + x3z: l0 at the point p( I , _l 2). 3. Use the T"lh.odr:f L,ugrange Multipliers to find the maximum value of (x, y): xy2 subject to the constraintx- + y' -- 12. consider integrat J "f .rt"1r* + y) dy dx . 02 Sketch the region of integration. Rer erse (interchange) the order of integration. Do not integrate. Let D be the region bounded between the graphs ofy = l0 - 2x, y 2, y= 4, and x Q. the region D. write a double integral in rectangular coordinates that gives the area of D. Do not integrate. Let S be the solid in the first octant bounded of 2x + y + z: 4 and z: 0. the solid S. Write a double in rectansular coordinates which gives the volume of S. Do not intesrate. 7. Evaluate the iterated intesral y' i (*y' + y)dx dy. 2 2 b. 5. a. 6. 2 j 0
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Answers 1 . Relative minimums of f(2, 4) : -16 f(-2, -4), Saddle point at (0. 0)' 2a. 6i- l6i + 13k 71 'n "ltq 2c. 6r -161'+ 132:48 3. Maximum value of f is l6 at the points Q, Jg ) and (2, - "6 I Il l ll 3 4b I J sin(2x+Y)dxdY 20 t!:r 12 sb.A:l J dxdy z0 / fv 6b.v:j I fq-2x-y)dydx 00 1.8
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PLEASE SHOW ALL WORK NIATH-203 EXAM 3 l. a. b. Let F(x, y, z): yz3 + xz.
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Richard_Exam_3_Practice_Winter_2007 - M ATH-203 E XAM 3 l....

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