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Unformatted text preview: MAT1332 Spring/Summer 2009 Assignment 5 Solutions 1. Consider the function f ( x , y ) : = x 2 y 2 x 3 xy 3 (a) What is the gradient of f ? (b) What is ∂ 2 f ∂ yx ? (c) What is the tangent plane to f at the point (3,5) ? First, note that f ( x , y ) simplifies to f ( x , y ) = x y 2 3 xy 3 . For (a), we observe that the gradient ∇ f of f is given by ∇ f ( x , y ) = £ y 2 3 y 3 x 2 9 xy 2 / For (b), we have that, since ∂ f ∂ x = y 2 3 y 3 , ∂ 2 f ∂ yx = 1 2 9 y 2 . For (c), we have that the tangent plane at (3,5) is given by z f (3,5) = ( 5 2 375)( x 3) + ( 3 2 675)( y 5) = 745 2 ( x 3) + 1347 2 ( y 5). Since f (3,5) = 2235 2 we have z = ( 5 2 375)( x 3) + ( 3 2 675)( y 5) 2235 2 = 745 2 ( x 3) + 1347 2 ( y 5) 2235 2 2. Solve the following linear system of differential equations ‰ dv dt = 6 v + w dw dt = 9 v 2 w with intial conditions v = 7 and w = 2 . First we observe that the the coefficient matrix A = • 6 1 9 2 ‚ 1 has characteristic polynomial ( A λ I ) equal to λ 2 4 λ 21. This factors as ( λ 7)( λ + 3) and therefore the eigenvalues are λ = 7, 3. We now solve for the corresponding eigenvectors.3....
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This note was uploaded on 01/16/2011 for the course MAT 1332 taught by Professor Munteanu during the Fall '07 term at University of Ottawa.
 Fall '07
 MUNTEANU
 Calculus

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