Unformatted text preview: t , given by ( x,y,z ) = ( t 2 ,t sin t, cos t ). The temperature T = T ( x,y,z ) satisﬁes ∇ T ( π 2 , ,1) = (1 , 2 ,1). [Recall that ∇ T = ( T x ,T y ,T z ).] Compute the rate at which the ﬂy’s temperature is changing at t = π . State any assumptions required in your computation. 4. Let f ( x,y ) = xe xy , and suppose x = x ( s,t ) = ( s + 2 t ) 2 and y = y ( s,t ) = sin( st ). (a) Find f s and f t . (b) Find f ss and f tt . 5. Let f,g : R → R be twice diﬀeretiable, and α ∈ R . Show that u ( x,t ) = f ( xαt ) + g ( x + αt ) satisﬁes the wave equation u tt = α 2 u xx 6. Let g : R → R , and f ( u,v ) = g ( uv 2 ), where u = u ( x,y ) = ( x + y ) 3 and v = v ( x ) = 1 x . Calculate ∂ 2 f ∂y∂x . State any assumptions required in your computation....
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 Spring '08
 WOLCZUK
 Math, Approximation, Linear Approximation, Partial differential equation, contradict Theorem, approximate arctan, handy inequality, Remark. Observe

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