# Linear Approximation

by KingBigness
Tags: approximation, linear
 P: 94 1. The problem statement, all variables and given/known data Let f(x,y) = $(xe^y)^8$ i) Find $\frac{∂f}{∂x}$ $\frac{∂f}{∂y}$ $\frac{∂^2f}{∂x^2}$ ii) Using a tangent plane of f(x,y) find an approximate value of (0.98e^0.01)^8 2. Relevant equations 3. The attempt at a solution i) $\frac{∂f}{∂x}$ =$8e^{8y}x^{7}$ $\frac{∂f}{∂y}$ = $8x^{8}e^{8y}$ $\frac{∂^2f}{∂x^2}$ = $56e^{8y}x^{6}$ ii) I have done many questions on finding linear approximations but I have always had a function, a point to evaluate the function at and points to approximate it at. In this I have the function Let f(x,y) = $(xe^y)^8$ and want to use it to approximate f(0.98,0.01) but I'm not sure at what point I should evaluate it at. Can anyone help out?
 Math Emeritus Sci Advisor Thanks PF Gold P: 38,705 Since .98 is reasonably close to 1 and 0.01 close to 0, I think x= 1, y= 0 would be a good try.
 P: 94 Was leaning towards that, just wanted to make sure. That all worked out nicely, thanks for your advice =D

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