Question

use difference quotients with $delta x = 0.1$ and $delta y = 0.1$ to estimate $f_x(2,3)$ and $f_y(2,3)$ where $f(x,y)=e^{-x}sin(y)$. $f_x(2,3)approx - 0.01816$ $f_y(2,3)approx - 0.1347$ then give better estimates by using $delta x = 0.01$ and $delta y = 0.01$. $f_x(2,3)approx - 0.0124$ $f_y(2,3)approx - 0.1299$

Explanation:

Step1: Recall difference - quotient formula for partial derivative

The difference - quotient formula for the partial derivative of $z = f(x,y)$ with respect to $x$ is $f_x(x,y)\approx\frac{f(x + \Delta x,y)-f(x,y)}{\Delta x}$, and with respect to $y$ is $f_y(x,y)\approx\frac{f(x,y+\Delta y)-f(x,y)}{\Delta y}$.

Step2: Calculate $f(x,y)$ at relevant points for $\Delta x = 0.1$ and $\Delta y = 0.1$

First, find $f(2,3)=e^{-2}\sin(3)$, $f(2 + 0.1,3)=e^{-2.1}\sin(3)$, $f(2,3 + 0.1)=e^{-2}\sin(3.1)$.
Then $f_x(2,3)\approx\frac{e^{-2.1}\sin(3)-e^{-2}\sin(3)}{0.1}=\frac{\sin(3)(e^{-2.1}-e^{-2})}{0.1}$.
Since $e^{-2}\approx0.1353$ and $e^{-2.1}\approx0.1225$, $\sin(3)\approx0.1411$, we have $f_x(2,3)\approx\frac{0.1411\times(0.1225 - 0.1353)}{0.1}=\frac{0.1411\times(- 0.0128)}{0.1}\approx - 0.01816$.
And $f_y(2,3)\approx\frac{e^{-2}\sin(3.1)-e^{-2}\sin(3)}{0.1}=e^{-2}\frac{\sin(3.1)-\sin(3)}{0.1}$.
Since $\sin(3.1)\approx0.0467$, we have $f_y(2,3)\approx0.1353\times\frac{0.0467 - 0.1411}{0.1}=0.1353\times\frac{-0.0944}{0.1}\approx - 0.1347$.

Step3: Calculate $f(x,y)$ at relevant points for $\Delta x = 0.01$ and $\Delta y = 0.01$

Find $f(2 + 0.01,3)=e^{-2.01}\sin(3)$, $f(2,3 + 0.01)=e^{-2}\sin(3.01)$.
$f_x(2,3)\approx\frac{e^{-2.01}\sin(3)-e^{-2}\sin(3)}{0.01}=\frac{\sin(3)(e^{-2.01}-e^{-2})}{0.01}$.
Since $e^{-2.01}\approx0.1339$, we have $f_x(2,3)\approx\frac{0.1411\times(0.1339 - 0.1353)}{0.01}=\frac{0.1411\times(-0.0014)}{0.01}\approx - 0.0124$.
$f_y(2,3)\approx\frac{e^{-2}\sin(3.01)-e^{-2}\sin(3)}{0.01}=e^{-2}\frac{\sin(3.01)-\sin(3)}{0.01}$.
Since $\sin(3.01)\approx0.1282$, we have $f_y(2,3)\approx0.1353\times\frac{0.1282 - 0.1411}{0.01}=0.1353\times\frac{-0.0129}{0.01}\approx - 0.1299$.

Answer:

When $\Delta x = 0.1$ and $\Delta y = 0.1$, $f_x(2,3)\approx - 0.01816$, $f_y(2,3)\approx - 0.1347$. When $\Delta x = 0.01$ and $\Delta y = 0.01$, $f_x(2,3)\approx - 0.0124$, $f_y(2,3)\approx - 0.1299$.