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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![**Partial Derivative Analysis**
**Problem Statement:**
Calculate the sign of the partial derivative using the graph of the surface.
\[ f_y(1, 1) \]
**Graph Description:**
- This is an interactive 3D graph depicting a 3D surface.
- The graph consists of three axes labeled \(x\), \(y\), and \(z\).
- The \(x\)- and \(y\)-axes are positioned horizontally, while the \(z\)-axis is vertical.
- The surface shown in the graph has a parabolic shape, opening upwards.
- The surface intersects the \(z\)-axis at \(z = 0\), and the values increase as we move outward from the origin along the surface.
**Diagram Analysis:**
- The \(z\)-axis has markings indicating positive and negative values.
- The \(x\)- and \(y\)-axes also have their respective scales, indicating positive and negative directions.
- The \(3D\) surface suggests how \(z\) changes with respect to \(x\) and \(y\).
**Question Options:**
- \( \bigcirc \) positive
- \( \bigcirc \) zero
- \( \bigcirc \) negative
**Explanation:**
To determine the sign of the partial derivative \( f_y(1, 1) \), we need to look at how the surface is changing as we move along the \(y\)-axis when \(x = 1\).
- If the surface rises as \( y \) increases, \( f_y \) is positive.
- If the surface falls as \( y \) increases, \( f_y \) is negative.
- If the height of the surface remains unchanged as \( y \) increases, \( f_y \) is zero.
In the graph, when we move along the \(y\)-axis at \(x = 1\), the surface appears to be declining. Thus, we can conclude that at the point \((1, 1)\), the partial derivative \( f_y(1, 1) \) is:
\[ \bigcirc \] negative](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fea4f7893-f945-4258-8f97-b3ee0b0ad980%2F2941f1c9-9ed1-4377-83ce-49bd17476849%2Fb55lwyp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Partial Derivative Analysis**
**Problem Statement:**
Calculate the sign of the partial derivative using the graph of the surface.
\[ f_y(1, 1) \]
**Graph Description:**
- This is an interactive 3D graph depicting a 3D surface.
- The graph consists of three axes labeled \(x\), \(y\), and \(z\).
- The \(x\)- and \(y\)-axes are positioned horizontally, while the \(z\)-axis is vertical.
- The surface shown in the graph has a parabolic shape, opening upwards.
- The surface intersects the \(z\)-axis at \(z = 0\), and the values increase as we move outward from the origin along the surface.
**Diagram Analysis:**
- The \(z\)-axis has markings indicating positive and negative values.
- The \(x\)- and \(y\)-axes also have their respective scales, indicating positive and negative directions.
- The \(3D\) surface suggests how \(z\) changes with respect to \(x\) and \(y\).
**Question Options:**
- \( \bigcirc \) positive
- \( \bigcirc \) zero
- \( \bigcirc \) negative
**Explanation:**
To determine the sign of the partial derivative \( f_y(1, 1) \), we need to look at how the surface is changing as we move along the \(y\)-axis when \(x = 1\).
- If the surface rises as \( y \) increases, \( f_y \) is positive.
- If the surface falls as \( y \) increases, \( f_y \) is negative.
- If the height of the surface remains unchanged as \( y \) increases, \( f_y \) is zero.
In the graph, when we move along the \(y\)-axis at \(x = 1\), the surface appears to be declining. Thus, we can conclude that at the point \((1, 1)\), the partial derivative \( f_y(1, 1) \) is:
\[ \bigcirc \] negative
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