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Given f(x, y) = values: fz(3, 2) = fy(3, 2) = - 5x³ + 5xy¹ + 2y2, find the following numerical

Given f(x, y) = values: fz(3, 2) = fy(3, 2) = - 5x³ + 5xy¹ + 2y2, find the following numerical

Calculus: Early Transcendentals
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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### Problem Statement Given the function \( f(x, y) = -5x^3 + 5xy^4 + 2y^2 \), find the following numerical values: - \( f_x(3, 2) = \) [Input Box] - \( f_y(3, 2) = \) [Input Box] ### Instructions 1. **Partial Derivative with Respect to \( x \):** Calculate the partial derivative of the function \( f \) with respect to \( x \) to obtain \( f_x(x, y) \). 2. **Evaluate \( f_x \):** Substitute \( x = 3 \) and \( y = 2 \) into \( f_x(x, y) \) to find \( f_x(3, 2) \). 3. **Partial Derivative with Respect to \( y \):** Calculate the partial derivative of the function \( f \) with respect to \( y \) to obtain \( f_y(x, y) \). 4. **Evaluate \( f_y \):** Substitute \( x = 3 \) and \( y = 2 \) into \( f_y(x, y) \) to find \( f_y(3, 2) \). ### Explanation - \[ f_x(x, y) \] represents the rate of change of the function \( f(x, y) \) in the \( x \)-direction, holding \( y \) constant. - \[ f_y(x, y) \] represents the rate of change of the function \( f(x, y) \) in the \( y \)-direction, holding \( x \) constant. By finding these derivatives and then evaluating them at the given point, you'll obtain the required numerical rates of change at \( (3, 2) \).
Transcribed Image Text:### Problem Statement Given the function \( f(x, y) = -5x^3 + 5xy^4 + 2y^2 \), find the following numerical values: - \( f_x(3, 2) = \) [Input Box] - \( f_y(3, 2) = \) [Input Box] ### Instructions 1. **Partial Derivative with Respect to \( x \):** Calculate the partial derivative of the function \( f \) with respect to \( x \) to obtain \( f_x(x, y) \). 2. **Evaluate \( f_x \):** Substitute \( x = 3 \) and \( y = 2 \) into \( f_x(x, y) \) to find \( f_x(3, 2) \). 3. **Partial Derivative with Respect to \( y \):** Calculate the partial derivative of the function \( f \) with respect to \( y \) to obtain \( f_y(x, y) \). 4. **Evaluate \( f_y \):** Substitute \( x = 3 \) and \( y = 2 \) into \( f_y(x, y) \) to find \( f_y(3, 2) \). ### Explanation - \[ f_x(x, y) \] represents the rate of change of the function \( f(x, y) \) in the \( x \)-direction, holding \( y \) constant. - \[ f_y(x, y) \] represents the rate of change of the function \( f(x, y) \) in the \( y \)-direction, holding \( x \) constant. By finding these derivatives and then evaluating them at the given point, you'll obtain the required numerical rates of change at \( (3, 2) \).
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