Solution: Approximation Consider the following approximations for a function centered at

Chapter 13, Problem 75

(choose chapter or problem)

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{aligned}

P_{2}(x, y)=& f(0,0)+f_{x}(0,0) x+f_{y}(0,0) y+\\

& \frac{1}{2} f_{x x}(0,0) x^{2}+f_{x y}(0,0) x y+\frac{1}{2} f_{y y}(0,0) y^{2}

\end{aligned}\)

[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) If 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.

 

(e) Use a computer algebra system to graph the surfaces

\(z=f(x, y), z=P_{1}(x, y), \text { and } z=P_{2}(x, y)\).

Text Transcription:

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

P_2(x,y)=&f(0,0)+f_x(0,0)x+f_y(0,0)y+

frac12f_xx(0,0)x^2+f_xy(0,0)xy+frac12f_yy(0,0)y^2

f(x,y)=e^(x-y)

z=f(x,y)

z=P_1(x,y)

z=P_2(x,y)