Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Transcription for Educational Website
#### Problem Description
3. Given to the right is a table of values for the function \( z = f(x, y) \). Assuming that \( f \) is differentiable, estimate \( f_x(3,2) \) and \( f_y(3,2) \).
#### Data Table
The table provides the values of the function \( f(x, y) \) at specific points:
\[
\begin{array}{|c|c|c|c|c|}
\hline
x \, \backslash \, y & 0 & 2 & 5 \\
\hline
1 & 3 & 1 & 2 \\
\hline
3 & 6 & 1 & 4 \\
\hline
6 & -3 & 0 & 2 \\
\hline
\end{array}
\]
#### Instructions for Estimation
To estimate the partial derivatives \( f_x(3,2) \) and \( f_y(3,2) \):
- \( f_x(3,2) \): Use the change in function values along the \( x \)-axis for \( y = 2 \).
- \( f_y(3,2) \): Use the change in function values along the \( y \)-axis for \( x = 3 \).
#### Next Steps
4. To the right is the graph of the function \( f(x, y) = \frac{1}{2} y e^{-x/2} \) on the domain \( 0 \leq x \leq 4 \) and \( 0 \leq y \leq 4 \).
The graph is not provided in this transcription, but it is important for visualizing the behavior of the function across the specified domain.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F65d8350d-ebfd-4934-96e8-9cf40cb8cc35%2Ffc9c9a46-edd8-4c71-939f-17bc6415e60b%2Fx7wgh7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Transcription for Educational Website
#### Problem Description
3. Given to the right is a table of values for the function \( z = f(x, y) \). Assuming that \( f \) is differentiable, estimate \( f_x(3,2) \) and \( f_y(3,2) \).
#### Data Table
The table provides the values of the function \( f(x, y) \) at specific points:
\[
\begin{array}{|c|c|c|c|c|}
\hline
x \, \backslash \, y & 0 & 2 & 5 \\
\hline
1 & 3 & 1 & 2 \\
\hline
3 & 6 & 1 & 4 \\
\hline
6 & -3 & 0 & 2 \\
\hline
\end{array}
\]
#### Instructions for Estimation
To estimate the partial derivatives \( f_x(3,2) \) and \( f_y(3,2) \):
- \( f_x(3,2) \): Use the change in function values along the \( x \)-axis for \( y = 2 \).
- \( f_y(3,2) \): Use the change in function values along the \( y \)-axis for \( x = 3 \).
#### Next Steps
4. To the right is the graph of the function \( f(x, y) = \frac{1}{2} y e^{-x/2} \) on the domain \( 0 \leq x \leq 4 \) and \( 0 \leq y \leq 4 \).
The graph is not provided in this transcription, but it is important for visualizing the behavior of the function across the specified domain.
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