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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the second picture is part of the first picture, i could not fit it all in one screenshot.
![### Graph of the Derivative of a Function
#### Description:
The figure represents the graph of the derivative, \( f' \), of a function \( f \) on the interval \([-3, 3]\).
#### Graph Explanation:
The graph features a sinusoidal curve that crosses the x-axis at three points: approximately \( x = -2 \), \( x = 0 \), and \( x = 2 \). The curve reaches its maximum point at \( x = -1 \) and its minimum point at \( x = 1 \).
#### Determining Intervals of Increase and Decrease:
To determine where \( f \) is increasing or decreasing, we observe where \( f' \), the derivative, is positive or negative, respectively.
- \( f \) is increasing when \( f' \) is above the x-axis (positive).
- \( f \) is decreasing when \( f' \) is below the x-axis (negative).
#### Tasks:
1. Determine the intervals on which \( f \) is increasing.
2. Determine the intervals on which \( f \) is decreasing.
(Use the symbol \( \cup \) for combining intervals, and the appropriate type of parenthesis "(", ")", "[", "]" depending on whether the interval is open or closed.)
---
**Find the intervals where \( f \) is increasing:**
\[ f \text{ is increasing on} \quad \_\_\_\_\_\_\_\_\_\_\_\_\_\ \]
**Find the intervals where \( f \) is decreasing:**
\[ f \text{ is decreasing on} \quad \_\_\_\_\_\_\_\_\_\_\_\_\_\ \]
---
### Solution Explanation:
Use the graph to identify the intervals where the derivative is positive (above the x-axis) and negative (below the x-axis). From the graph:
- \( f \) is increasing on intervals where \( f' \) > 0:
- \( (-2, 0) \cup (2, 3) \)
- \( f \) is decreasing on intervals where \( f' \) < 0:
- \( (-3, -2) \cup (0, 2) \)
Therefore,
\[ f \text{ is increasing on} \quad (-2, 0) \cup (2, 3) \]
\[ f \text{ is decreasing on} \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F95425ac5-3f57-4b1e-926d-91b4503a2e10%2Fc4941b04-05bc-4f5a-b0b4-d28ce1a9e7b0%2F6vz3fkl.png&w=3840&q=75)
Transcribed Image Text:### Graph of the Derivative of a Function
#### Description:
The figure represents the graph of the derivative, \( f' \), of a function \( f \) on the interval \([-3, 3]\).
#### Graph Explanation:
The graph features a sinusoidal curve that crosses the x-axis at three points: approximately \( x = -2 \), \( x = 0 \), and \( x = 2 \). The curve reaches its maximum point at \( x = -1 \) and its minimum point at \( x = 1 \).
#### Determining Intervals of Increase and Decrease:
To determine where \( f \) is increasing or decreasing, we observe where \( f' \), the derivative, is positive or negative, respectively.
- \( f \) is increasing when \( f' \) is above the x-axis (positive).
- \( f \) is decreasing when \( f' \) is below the x-axis (negative).
#### Tasks:
1. Determine the intervals on which \( f \) is increasing.
2. Determine the intervals on which \( f \) is decreasing.
(Use the symbol \( \cup \) for combining intervals, and the appropriate type of parenthesis "(", ")", "[", "]" depending on whether the interval is open or closed.)
---
**Find the intervals where \( f \) is increasing:**
\[ f \text{ is increasing on} \quad \_\_\_\_\_\_\_\_\_\_\_\_\_\ \]
**Find the intervals where \( f \) is decreasing:**
\[ f \text{ is decreasing on} \quad \_\_\_\_\_\_\_\_\_\_\_\_\_\ \]
---
### Solution Explanation:
Use the graph to identify the intervals where the derivative is positive (above the x-axis) and negative (below the x-axis). From the graph:
- \( f \) is increasing on intervals where \( f' \) > 0:
- \( (-2, 0) \cup (2, 3) \)
- \( f \) is decreasing on intervals where \( f' \) < 0:
- \( (-3, -2) \cup (0, 2) \)
Therefore,
\[ f \text{ is increasing on} \quad (-2, 0) \cup (2, 3) \]
\[ f \text{ is decreasing on} \
![---
### Determining Concavity Intervals
#### Determine the intervals on which \( f \) is concave down.
(Use the symbol ∪ for combining intervals, and an appropriate type of parenthesis “\( ( \)”, “\[ \]” depending on whether the interval is open or closed.)
\[ f \text{ is concave down on } \underline{\hspace{400px}} \]
---
#### Determine the intervals on which \( f \) is concave up.
(Use the symbol ∪ for combining intervals, and an appropriate type of parenthesis “\( ( \)”, “\[ \]” depending on whether the interval is open or closed.)
\[ f \text{ is concave up on } \underline{\hspace{400px}} \]
---
### Explanation:
Concavity refers to the direction of the curvature of a graph:
- **Concave Down**: The graph of \( f \) is concave down where it curves downwards like an upside-down cup. This occurs where the second derivative \( f''(x) \) is less than zero (\( f''(x) < 0 \)).
- **Concave Up**: The graph of \( f \) is concave up where it curves upwards like a right-side-up cup. This occurs where the second derivative \( f''(x) \) is greater than zero (\( f''(x) > 0 \)).
When determining concavity intervals, use:
- **Parentheses \( (\ ) \)**: To denote open intervals (where the endpoint is not included).
- **Brackets \([ ]\)**: To denote closed intervals (where the endpoint is included).
- **Union \( \cup \)**: To combine multiple intervals.
---
This page is designed to help students understand and practice how to find and express the intervals where a function \( f \) is concave up or concave down.
For additional assistance, review resources on calculus and concavity concepts.
---](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F95425ac5-3f57-4b1e-926d-91b4503a2e10%2Fc4941b04-05bc-4f5a-b0b4-d28ce1a9e7b0%2Fg7ai7jz.png&w=3840&q=75)
Transcribed Image Text:---
### Determining Concavity Intervals
#### Determine the intervals on which \( f \) is concave down.
(Use the symbol ∪ for combining intervals, and an appropriate type of parenthesis “\( ( \)”, “\[ \]” depending on whether the interval is open or closed.)
\[ f \text{ is concave down on } \underline{\hspace{400px}} \]
---
#### Determine the intervals on which \( f \) is concave up.
(Use the symbol ∪ for combining intervals, and an appropriate type of parenthesis “\( ( \)”, “\[ \]” depending on whether the interval is open or closed.)
\[ f \text{ is concave up on } \underline{\hspace{400px}} \]
---
### Explanation:
Concavity refers to the direction of the curvature of a graph:
- **Concave Down**: The graph of \( f \) is concave down where it curves downwards like an upside-down cup. This occurs where the second derivative \( f''(x) \) is less than zero (\( f''(x) < 0 \)).
- **Concave Up**: The graph of \( f \) is concave up where it curves upwards like a right-side-up cup. This occurs where the second derivative \( f''(x) \) is greater than zero (\( f''(x) > 0 \)).
When determining concavity intervals, use:
- **Parentheses \( (\ ) \)**: To denote open intervals (where the endpoint is not included).
- **Brackets \([ ]\)**: To denote closed intervals (where the endpoint is included).
- **Union \( \cup \)**: To combine multiple intervals.
---
This page is designed to help students understand and practice how to find and express the intervals where a function \( f \) is concave up or concave down.
For additional assistance, review resources on calculus and concavity concepts.
---
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