Chapter 13, Problem 60RE

(a)

To determine

To calculate: The linear approximation of $f\left(x,y\right)=\cos x+\sin y$ centered at $\left(0,0\right)$; where the linear approximation for the function $f\left(x,y\right)$ centered at $\left(0,0\right)$ is given by

$P_1\left(x,y\right)=f\left(0,0\right)+f_x\left(0,0\right)x+f_y\left(0,0\right)y$.

(b)

To determine

To calculate: The quadratic approximation of $f\left(x,y\right)=\cos x+\sin y$ centered at $\left(0,0\right)$; where the quadratic approximation for the function $f\left(x,y\right)$ centered at $\left(0,0\right)$ is given by:

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

(c)

To determine

The function of the second-degree Taylor polynomial, when $y=0$ in the quadratic approximation $P_2\left(x,y\right)=1+y-\frac{1}{2}{x}^2$.

(d)

To determine

To fill: To complete the table

$x$	$y$	$f\left(x,y\right)$	$P_1\left(x,y\right)$	$P_2\left(x,y\right)$
$0$	$0$
$0$	$0.1$
$0.2$	$0.1$
$0.5$	$0.3$
$1$	$0.5$

Where; $f\left(x,y\right)=\cos x+\sin y$, $P_1\left(x,y\right)=1+y$ and $P_2\left(x,y\right)=1+y-\frac{1}{2}{x}^2$.

(e)

To determine

To graph: The surface $z=\cos x+\sin y$, $z=1+y\text{ and }z=1+y-\frac{1}{2}{x}^2$ and to determine the accuracy of the approximations changes as the distance of the surfaces increases from the point $\left(0,0\right)$.