Chapter 13.7, Problem 61E

Approximation Consider the following approximations for a function f(x, y) centered at (0, 0).

Linear Approximation:

$P_1(x,y)=f(0,0)+f_x(0,0)x+f_y(0,0)y$

Quadratic Approximation:

$\begin{array}{l}{P}_2(x,y)=f(0,0)+f_x(0,0)x+f_y(0,0)y+\\ \frac{1}{2}{f}_{xx}(0,0)x^2+f_{xy}(0,0)xy+\frac{1}{2}{f}_{yy}(0,0)y^2\end{array}$

Note that the linear approximation is the tangent plane to the surface at (0, 0, f (0, 0)).]

(a) Find the linear approximation of $f(x,y)=e^{x-y}$ centered at (0, 0).

(b) Find the quadratic approximation of $f(x,y)=e^{x-y}$ centered at (0, 0).

(c) When $x=0$ in the quadratic approximation, you obtain the second-degree Taylor polynomial for what function? Answer the same question for $y=0$.

(d) Complete the table.

x	y	f(x, y)	$P_1(x,y)$	$P_2(x,y)$
0	0
0	0.1
0.2	0.1
0.2	0.5
1	0.5

(e) Use a computer algebra system to graph the surfaces $z=f(x,y),z=P_1(x,y)$ and $z=P_2(x,y)$