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plugin-Math255-Pset-7 solutions

plugin-Math255-Pset-7 solutions - Winter 2010 Math 255...

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Winter 2010 Math 255 Problem Set 7 Due Tuesday, February 23 Section 15.3: 20, 24, 44, 60, 68, 71, 80 Section 15.4: 4, 24, 38, 42 Section 15.5: 4, 28, 47, 56 Solution and grading suggestion Section 15.3 20. f ( s, t ) = st 2 s 2 + t 2 . ∂f ( s,t ) ∂s = t 2 ( s 2 + t 2 ) - st 2 (2 s ) ( s 2 + t 2 ) 2 = t 4 - s 2 t 2 ( s 2 + t 2 ) 2 , ∂f ( s,t ) ∂t = 2 st ( s 2 + t 2 ) - st 2 (2 t ) ( s 2 + t 2 ) 2 = 2 s 3 t ( s 2 + t 2 ) 2 . 24. f ( x, y ) = R x y cos( t 2 ) dt . ∂f ( x,y ) ∂x = cos( x 2 ), ∂f ( x,y ) ∂y = - cos( y 2 ). 44. sin( xyz ) = x +2 y +3 z with x and y being variables. Take x partial derivative for both sides: cos( xyz ) · ( yz + xy ∂z ∂x ) = 1 + 3 ∂z ∂x , So ∂z ∂x = yz cos( xyz ) - 1 3 - xy cos( xyz ) . Take y partial derivative for both sides: cos( xyz ) · xz + xy ∂z ∂y = 2 + 3 ∂z ∂y , So ∂z ∂y = xz cos( xyz ) - 2 3 - xy cos( xyz ) . 1
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Winter 2010 Math 255 60. f ( r, s, t ) = r ln( rs 2 t 3 ) f r = ln( rs 2 t 3 ) + r · s 2 t 3 rs 2 t 3 = ln( rs 2 t 3 ) + 1 f rs = 2 rst 3 rs 2 t 3 = 2 s So f rss = - 2 s 2 and f rst = 0. 68. (a) u xx = u yy = 2. NO (b) u xx = 2, u yy = - 2. YES (c) u xx = u yy = 6 x . NO (d) u x = 2 x/ 2 x 2 + y 2 x 2 + y 2 = x x 2 + y 2 u xx = ( x 2 + y 2 ) - x (2 x ) ( x 2 + y 2 ) 2 = y 2 - x 2 ( x 2 + y 2 ) 2 By symmetry of x and y , we have u
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