Consider the following approximations for a function

Chapter 13, Problem 60

(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) = cos x + sin y

centered at (0,0).

(b) Find the quadratic approximation of

f(x, y) = cos x + sin y

centered at (0,0).

(c) When y = 0 in the quadratic approximation, you obtain the second-degree Taylor polynomial for what function?

(d) Complete the table.

(e) Use a computer algebra system to graph the surfaces  \(z=f(x, y), z=P_{1}(x, y)\),  and  \(z=P_{2}(x, y)\).  How does the accuracy of the approximations change as the distance from (0, 0) increases?

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

z = f(x, y), z = P_1 (x, y)

z = P_2 (x, y)