Approximation Consider the following approximations for a function centered at Linear
Chapter 13, Problem 65(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:
\(P_{2}(x, y)=f(0,0)+f_{x}(0,0) x+f_{y}(0,0) y+ \frac{1}{2} f_{x x}(0,0) x^{2}+f_{x y}(0,0) x y+\frac{1}{2} f_{y y}(0,0) y^{2}\)
[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.
(e) Use a computer algebra system to graph the surfaces \(z=f(x, y), z=P_{1}(x, y)\), and \(z=P_{2}(x, y)\).
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 + 1 / 2 f_{x x}(0, 0) x^2 + f_{x y}(0, 0) xy + 1 / 2 f_{y y}(0,0) y^2
f(x, y) = e^(x - y)
z = f(x, y), z = P_1(x, y)
z = P_2(x, y)
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