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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
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' 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
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(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
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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!
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\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
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» \/’
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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This note was uploaded on 10/20/2008 for the course MATHS MTH 203 taught by Professor None during the Spring '08 term at American University of Sharjah.
 Spring '08
 none
 Calculus

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