Get answer: Approximation Consider the following approximations for a function centered

Chapter 13, Problem 64

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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:

\(P_2(x,\ y)=f(0,\ 0)+f_x(0,\ 0)x+f_y(0,\ 0)y+\frac{1}{2}f_{xx}(0,\ 0)x^2+f_{xy}(0,\ 0)xy+\frac{1}{2}f_{yy}(0,\ 0)y^2\)

[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) If 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_xx (0, 0)x^2 + f_xy (0, 0)xy + 1/2 f_yy (0, 0)y^2

z = P_1 (x, y)

z = P_2 (x, y)