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...
Related questions
Question
t3
![**Problem Statement:**
Let \( z(x, y) = -4x^2 + 3y^2 \) where \( x = -s - 9t \) and \( y = -8s - 6t \).
**Task:**
Calculate \(\frac{\partial z}{\partial s}\) and \(\frac{\partial z}{\partial t}\) by first finding:
- \(\frac{\partial x}{\partial s}\), \(\frac{\partial y}{\partial s}\)
- \(\frac{\partial x}{\partial t}\), \(\frac{\partial y}{\partial t}\)
and using the chain rule.
**Expressions to Find:**
\[
\frac{\partial z}{\partial s} = \text{(Use this space for your answer)}
\]
\[
\frac{\partial z}{\partial t} = \text{(Use this space for your answer)}
\]
**Instructions:**
- Determine the partial derivatives of \( x \) and \( y \) with respect to \( s \) and \( t \).
- Apply the chain rule to find \(\frac{\partial z}{\partial s}\) and \(\frac{\partial z}{\partial t}\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4e39a1d0-143c-41f0-ae59-e967e0535bad%2F408c0feb-cb2d-45de-bda4-d5cf54fcd9a0%2Fbw3ms8d_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Let \( z(x, y) = -4x^2 + 3y^2 \) where \( x = -s - 9t \) and \( y = -8s - 6t \).
**Task:**
Calculate \(\frac{\partial z}{\partial s}\) and \(\frac{\partial z}{\partial t}\) by first finding:
- \(\frac{\partial x}{\partial s}\), \(\frac{\partial y}{\partial s}\)
- \(\frac{\partial x}{\partial t}\), \(\frac{\partial y}{\partial t}\)
and using the chain rule.
**Expressions to Find:**
\[
\frac{\partial z}{\partial s} = \text{(Use this space for your answer)}
\]
\[
\frac{\partial z}{\partial t} = \text{(Use this space for your answer)}
\]
**Instructions:**
- Determine the partial derivatives of \( x \) and \( y \) with respect to \( s \) and \( t \).
- Apply the chain rule to find \(\frac{\partial z}{\partial s}\) and \(\frac{\partial z}{\partial t}\).
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