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MIT18_02SC_pb_30_comb

MIT18_02SC_pb_30_comb - Chain rule and total dierentials 1...

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Chain rule and total differentials the total 1 . Find differential of w = z e ( x + y ) at (0 , 0 , 1). Answer: The total differential at the point ( x 0 , y 0 , z 0 ) is dw = w x ( x 0 , y 0 , z 0 ) dx + w y ( x 0 , y 0 , z 0 ) dy + w z ( x 0 , y 0 , z 0 ) dz. In our case, w = ( x + y ) = z e ( x + y ) x z e , w y , w z = e ( x + y ) Substituting in the point (0 , 0 , 1) we get: w x (0 , 0 , 1) = 1 , w y (0 , 0 , 1) = 1 , w z (0 , 0 , 1) = 1 . Thus, dw = dx + dy + dz. 2 . dw Suppose w = z e ( x + y ) and x = t , y = t 2 , z = t 3 . Compute and evaluate it when dt t = 2. Answer: We use the chain rule: dw ∂w dx ∂w dy ∂w dz = + + dt ∂x dt ∂y dt ∂z dt = ( z e ( x + y ) )(1) + ( z e ( x + y ) )(2 t ) + (e ( x + y ) )(3 t 2 ) . At t = 2 we have x = 2 , y = 4 , z = 8 . Thus, dw
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