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# PS3_answers - Here are answers to most of the problems I...

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Unformatted text preview: Here are answers to most of the problems. I have, of course, not included ones in which the question asks to prove or verify something. Also, for now, I have not had a chance to include pictures. Math 237 Problem Set 3 Answers A1. dw dt (- 2) = 76. A2. a) If f , x ( t ), y ( t ) are differentiable then g prime ( t ) = f x ( x ( t ) , y ( t )) x prime ( t ) + f y ( x ( t ) , y ( t )) y prime ( t ) . b) If f is differentiable at (1 , 0) we get g prime (0) = 4. A3. If f is differentiable at (1 , 1 , 1) then we get g prime (1) =- 13 A4. If f is differentiable then ∇ g (1 , 2) = (14 ,- 8). A5. i) ∂w ∂t = ∂w ∂x ∂x ∂t + ∂w ∂y ∂y ∂t + ∂w ∂z ∂z ∂t . ii) dz dt = f x ( x, y ) + f y ( x, y ) g prime ( x ) iii) ∂z ∂u = f x ( x, y ) h u ( u, v ) + f y ( x, y ) g prime ( x ) h u ( u, v ) iv) dw dz = f x ( x, y, z ) + f y ( x, y, z )( g x ( x, z ) + g z ( x, z ) h prime ( x )) + f z ( x, y, z ) h prime ( x ) v) parenleftBig ∂w ∂p parenrightBig q = parenleftBig ∂F...
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