The problem I have is this:
Use suitable linear approximation to find the approximate values for given functions at the points indicated:
$f(x, y) = xe^{y+x^2}$ at $(2.05, -3.92)$
I know how to do linear approximation with just one variable (take the derivative and such), but with two variables (and later on in the assignment, three variables) I'm a bit lost. Do I take partial derivatives and combine then somehow? Can someone guide me through a problem of this type?
Thank you in advance.
$\endgroup$2 Answers
$\begingroup$Let $L(x,y)$=$f(x_0,y_0)+f_x(x_0,y_0)(x-x_0)+f_y(x_0,y_0)(y-y_0)$. Then $L(x,y) \approx f(x,y)$. Consider $(x_0,y_0)=(2,-4)$. Then, \begin{equation} L(x,y)=2+9(x-2)+2(y+4) \implies f(2.05,-3.92) \approx L(2.05, -3.92)=2.61 \end{equation} Notice, from a calculator, $f(2.05,-3.92)=2.7192$
$\endgroup$ 5 $\begingroup$Yes. Denoting the partial derivatives by $f'_x$ and $f'_y$, the formula is: $$f(x_0+h,y_0+k)=f(x_0,y_0)+f'_x(x_0,y_0)h+f'_y(x_0,y_0)k+o\bigl(\lVert(h,k)\rVert\bigr).$$
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