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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Homework: HW 17.2
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Type an exact answer when finding the answer for fx(2,-1) and fy(-2,-2)
![### Finding Partial Derivatives and Evaluating at Specific Points
Given the function:
\[ f(x,y) = -7 e^{8x - 5y} \]
1. **Find \( f_x(x,y) \) and \( f_y(x,y) \)**:
- \( f_x(x,y) \) is the partial derivative of \( f \) with respect to \( x \).
- \( f_y(x,y) \) is the partial derivative of \( f \) with respect to \( y \).
2. **Evaluate \( f_x \) and \( f_y \) at specific points**:
- Find \( f_x(2, -1) \).
- Find \( f_y(-2, -2) \).
#### Steps for Finding Partial Derivatives
1. **Partial Derivative with respect to \( x \) (\( f_x \))**:
\[ f_x(x, y) = \frac{\partial}{\partial x} \left( -7 e^{8x - 5y} \right) \]
Using the chain rule:
\[ f_x(x, y) = -7 \cdot e^{8x - 5y} \cdot (8) \]
\[ f_x(x, y) = -56 e^{8x - 5y} \]
2. **Partial Derivative with respect to \( y \) (\( f_y \))**:
\[ f_y(x, y) = \frac{\partial}{\partial y} \left( -7 e^{8x - 5y} \right) \]
Using the chain rule:
\[ f_y(x, y) = -7 \cdot e^{8x - 5y} \cdot (-5) \]
\[ f_y(x, y) = 35 e^{8x - 5y} \]
#### Evaluating at Specific Points
1. **Evaluate \( f_x(2, -1) \)**:
\[ f_x(2, -1) = -56 e^{8(2) - 5(-1)} = -56 e^{16 + 5} = -56 e^{21} \]
2. **Evaluate \( f_y(-2, -2) \)**:
\[ f_y(-2, -2)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0c1127c9-3767-40bf-bfb8-794124c4dac3%2F3c77c561-1531-4a15-b3b0-2eee7a7e878c%2Fdkdgwv_processed.png&w=3840&q=75)
Transcribed Image Text:### Finding Partial Derivatives and Evaluating at Specific Points
Given the function:
\[ f(x,y) = -7 e^{8x - 5y} \]
1. **Find \( f_x(x,y) \) and \( f_y(x,y) \)**:
- \( f_x(x,y) \) is the partial derivative of \( f \) with respect to \( x \).
- \( f_y(x,y) \) is the partial derivative of \( f \) with respect to \( y \).
2. **Evaluate \( f_x \) and \( f_y \) at specific points**:
- Find \( f_x(2, -1) \).
- Find \( f_y(-2, -2) \).
#### Steps for Finding Partial Derivatives
1. **Partial Derivative with respect to \( x \) (\( f_x \))**:
\[ f_x(x, y) = \frac{\partial}{\partial x} \left( -7 e^{8x - 5y} \right) \]
Using the chain rule:
\[ f_x(x, y) = -7 \cdot e^{8x - 5y} \cdot (8) \]
\[ f_x(x, y) = -56 e^{8x - 5y} \]
2. **Partial Derivative with respect to \( y \) (\( f_y \))**:
\[ f_y(x, y) = \frac{\partial}{\partial y} \left( -7 e^{8x - 5y} \right) \]
Using the chain rule:
\[ f_y(x, y) = -7 \cdot e^{8x - 5y} \cdot (-5) \]
\[ f_y(x, y) = 35 e^{8x - 5y} \]
#### Evaluating at Specific Points
1. **Evaluate \( f_x(2, -1) \)**:
\[ f_x(2, -1) = -56 e^{8(2) - 5(-1)} = -56 e^{16 + 5} = -56 e^{21} \]
2. **Evaluate \( f_y(-2, -2) \)**:
\[ f_y(-2, -2)
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