MTH203SP05 (Exam 2) - w American University of Sharjah...

Unformatted text preview: w' American University of Sharjah Department of Computer Science, Mathematics, and Statistics MTH 203 - EXAM II Spring 2005 ID# Instructor Name: Name: Section# Q1. (18%) Let f(x,y, z) = sinOIz) + ln(x2). Find the directional derivative of f at the pointl(1,1,7r) in the direction of u=<1,1,—1>. / afiyxi £49525 “’Tl g «chewflga 125'? ' ,2 4" W324 2/,n,~/> 45‘; (UL—“Jr? 6)]2 V ’r- I ,L‘ J— I; Dfiwmé<24~><ew)ef) ' M “’3”??? s % b)Find the direction in which f increases most rapidly at the point (1,1, 7:). What is the maximum rate of change of at point. b\YC’€(/w;\ ‘6 V€£<ZJQ7AP Z, maxim}: :— ULWWM/ {W c) Find the equation for the tangent plane to the level surface offatthe point(1,1,7z). (mm): o i V -’\T c“: 0 Pl (a,/T1yl>~<x"/ '4 )7 ) Qé.(18%) Find all critical values of the function j(x,y)=x(y2 a 4)ey and classify each as a local maximum, local minimum or saddle points K? : artwe‘j :0 //’5 9 71,2 y . 03. (18%) Use Lagrangian Multiplier to find the maximum and minimum of the function fix, y) '= x2 +y2 subject to 9m) a x? — 2x+y2 —' 4y =0. "”?st3)( 2%; MM”) "k 2>dI~A>3 “ZN A. X” A~I j Ev):de 26:”(“3’”) (a 9290"“:qu Q4. (12%) a) Set up double integral in both dxdy and dydx to find the area enclosed by the curves y = J)? and J7 = x and above the line y .= 1/2.(Do not evaluate the integrals) Z 9“ 4 %7/ l? a! fig olvwbw 0% fig \M V» "/ (12%) b) Find the center of mass of the lamina bounded by y‘ = 6x—x2 andy = x. The density function p(x,y) = p0. (Hint: m = p(x,y)dA, Mx =Hy p(x,y)dA,My =Hx p(x,y)dA, M D a _ 2 05. (12%) Consider the integral I: I W ex2+y2dxdy y (a) Sketch the region of integration (b) Change the Cartesian integral in to equivalent polar integral.(Do not evaluate the integral) ‘ 4‘! (Z 2 ngiM’La » \/’ o 3/ 7%0/6. (12%) Set up a double integral to findthe volume of the solid bounded by the paraboloid z = x2+y2 and the plane z = 2y. ( Do not evaluate the integral) ...
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