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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![**Title: Exploring Level Curves of a Function**
---
**Instructions:**
Sketch the level curves of the function:
\[ f(x, y) = \frac{y + 5}{x} \]
**Objective:**
Identify the level curves for the function corresponding to different constant values:
- \( c = 4 \)
- \( c = -2 \)
- \( c = 2 \)
**Explanation:**
For the given function \( f(x, y) = \frac{y + 5}{x} \), a level curve for a constant \( c \) is obtained by setting:
\[ \frac{y + 5}{x} = c \]
Which simplifies to:
\[ y + 5 = cx \]
Thus:
\[ y = cx - 5 \]
Each level curve is a straight line with slope \( c \) and y-intercept \(-5\).
**Tasks:**
1. Plot the line \( y = 4x - 5 \) for \( c = 4 \).
2. Plot the line \( y = -2x - 5 \) for \( c = -2 \).
3. Plot the line \( y = 2x - 5 \) for \( c = 2 \).
**Graph Details:**
- Use an appropriate scale for the \( x \) and \( y \) axes.
- Clearly label each line with its corresponding \( c \) value.
- Highlight the intersections and slopes to distinguish between different curves.
**Conclusion:**
Understanding and sketching level curves help visualize how a function behaves for constant function values, offering insights into the geometry of the function.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb338c8f4-5a01-4673-bd1c-86a932cb5c80%2F35d5c3f4-acb1-469a-92c0-04cd9d1c1115%2Fab7fu2_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Exploring Level Curves of a Function**
---
**Instructions:**
Sketch the level curves of the function:
\[ f(x, y) = \frac{y + 5}{x} \]
**Objective:**
Identify the level curves for the function corresponding to different constant values:
- \( c = 4 \)
- \( c = -2 \)
- \( c = 2 \)
**Explanation:**
For the given function \( f(x, y) = \frac{y + 5}{x} \), a level curve for a constant \( c \) is obtained by setting:
\[ \frac{y + 5}{x} = c \]
Which simplifies to:
\[ y + 5 = cx \]
Thus:
\[ y = cx - 5 \]
Each level curve is a straight line with slope \( c \) and y-intercept \(-5\).
**Tasks:**
1. Plot the line \( y = 4x - 5 \) for \( c = 4 \).
2. Plot the line \( y = -2x - 5 \) for \( c = -2 \).
3. Plot the line \( y = 2x - 5 \) for \( c = 2 \).
**Graph Details:**
- Use an appropriate scale for the \( x \) and \( y \) axes.
- Clearly label each line with its corresponding \( c \) value.
- Highlight the intersections and slopes to distinguish between different curves.
**Conclusion:**
Understanding and sketching level curves help visualize how a function behaves for constant function values, offering insights into the geometry of the function.
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