Math / Calculus

QuestionUse difference quotients with $\Delta x=0.1$ and $\Delta y=0.1$ to estimate $f_{x}(4,2)$ and $f_{y}(4,2)$ where $f(x, y)=e^{-x} \sin (y) .$ $\begin{array}{l} f_{x}(4,2) \approx \\ f_{y}(4,2) \approx \end{array}$ $\square$ $\square$ Then give better estimates by using $\Delta x=0.01$ and $\Delta y=0.01$ . $\begin{array}{l} f_{x}(4,2) \approx \square \\ f_{y}(4,2) \approx \square \end{array}$ Submit answer Next item

Studdy Solution

STEP 1

1. We are using difference quotients to approximate partial derivatives.
2. The function given is $f(x, y) = e^{-x} \sin(y)$ .
3. We will use the given increments $\Delta x$ and $\Delta y$ to estimate the partial derivatives at the point $(4, 2)$ .

STEP 2

1. Calculate $f_x(4, 2)$ using $\Delta x = 0.1$ .
2. Calculate $f_y(4, 2)$ using $\Delta y = 0.1$ .
3. Calculate $f_x(4, 2)$ using $\Delta x = 0.01$ .
4. Calculate $f_y(4, 2)$ using $\Delta y = 0.01$ .

STEP 3

To estimate $f_x(4, 2)$ using $\Delta x = 0.1$ , we use the forward difference quotient:
$f_x(4, 2) \approx \frac{f(4 + 0.1, 2) - f(4, 2)}{0.1}$
Calculate $f(4 + 0.1, 2)$ :
$f(4.1, 2) = e^{-4.1} \sin(2)$
Calculate $f(4, 2)$ :
$f(4, 2) = e^{-4} \sin(2)$
Substitute these into the difference quotient:
$f_x(4, 2) \approx \frac{e^{-4.1} \sin(2) - e^{-4} \sin(2)}{0.1}$

STEP 4

To estimate $f_y(4, 2)$ using $\Delta y = 0.1$ , we use the forward difference quotient:
$f_y(4, 2) \approx \frac{f(4, 2 + 0.1) - f(4, 2)}{0.1}$
Calculate $f(4, 2 + 0.1)$ :
$f(4, 2.1) = e^{-4} \sin(2.1)$
Substitute these into the difference quotient:
$f_y(4, 2) \approx \frac{e^{-4} \sin(2.1) - e^{-4} \sin(2)}{0.1}$

STEP 5

To get a better estimate of $f_x(4, 2)$ using $\Delta x = 0.01$ , use the forward difference quotient:
$f_x(4, 2) \approx \frac{f(4 + 0.01, 2) - f(4, 2)}{0.01}$
Calculate $f(4 + 0.01, 2)$ :
$f(4.01, 2) = e^{-4.01} \sin(2)$
Substitute these into the difference quotient:
$f_x(4, 2) \approx \frac{e^{-4.01} \sin(2) - e^{-4} \sin(2)}{0.01}$

STEP 6

To get a better estimate of $f_y(4, 2)$ using $\Delta y = 0.01$ , use the forward difference quotient:
$f_y(4, 2) \approx \frac{f(4, 2 + 0.01) - f(4, 2)}{0.01}$
Calculate $f(4, 2 + 0.01)$ :
$f(4, 2.01) = e^{-4} \sin(2.01)$
Substitute these into the difference quotient:
$f_y(4, 2) \approx \frac{e^{-4} \sin(2.01) - e^{-4} \sin(2)}{0.01}$
The approximate values for the partial derivatives are: $f_x(4, 2) \approx \text{calculated value with } \Delta x = 0.1$ $f_y(4, 2) \approx \text{calculated value with } \Delta y = 0.1$
And the better estimates are: $f_x(4, 2) \approx \text{calculated value with } \Delta x = 0.01$ $f_y(4, 2) \approx \text{calculated value with } \Delta y = 0.01$

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