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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![**Mathematical Analysis on Functions**
### Problem Statement
Consider the function \( f(x) = -3x^3 + 3x^2 - 3x - 1 \).
1. Find the average rate of change on the interval \([-4, -1]\).
2. Find the \( c \) in the interval which works.
### Detailed Explanation
To find the average rate of change of the function \( f(x) \) over the interval \([-4, -1]\), we use the formula:
\[
\text{Average rate of change} = \frac{f(b) - f(a)}{b - a}
\]
where \( a \) and \( b \) are the endpoints of the interval \([-4, -1]\).
1. **Calculate \( f(-4) \) and \( f(-1) \)**:
- \( f(-4) = -3(-4)^3 + 3(-4)^2 - 3(-4) - 1 \)
- \( f(-1) = -3(-1)^3 + 3(-1)^2 - 3(-1) - 1 \)
2. **Plug the values into the average rate of change formula**:
- \( \text{Average rate of change} = \frac{f(-1) - f(-4)}{-1 - (-4)} \)
Next, identify the value of \( c \) within the interval \([-4, -1]\) that satisfies the Mean Value Theorem, if applicable. For this, ensure the first derivative \( f'(x) \) exists and is continuous over the interval. Verify \( f'(c) = \text{average rate of change} \).
### Diagram Explanation
While this problem does not include specific graphs or diagrams, plotting \( f(x) \) along with its derivative \( f'(x) \) over the interval \([-4, -1]\) would greatly aid in visualizing changes and confirming the solutions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffb2133c9-e1e5-4d56-9c72-044227328930%2F8b258e18-c66b-462a-8c47-6347bcbe2775%2Ft4vlf3g_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Mathematical Analysis on Functions**
### Problem Statement
Consider the function \( f(x) = -3x^3 + 3x^2 - 3x - 1 \).
1. Find the average rate of change on the interval \([-4, -1]\).
2. Find the \( c \) in the interval which works.
### Detailed Explanation
To find the average rate of change of the function \( f(x) \) over the interval \([-4, -1]\), we use the formula:
\[
\text{Average rate of change} = \frac{f(b) - f(a)}{b - a}
\]
where \( a \) and \( b \) are the endpoints of the interval \([-4, -1]\).
1. **Calculate \( f(-4) \) and \( f(-1) \)**:
- \( f(-4) = -3(-4)^3 + 3(-4)^2 - 3(-4) - 1 \)
- \( f(-1) = -3(-1)^3 + 3(-1)^2 - 3(-1) - 1 \)
2. **Plug the values into the average rate of change formula**:
- \( \text{Average rate of change} = \frac{f(-1) - f(-4)}{-1 - (-4)} \)
Next, identify the value of \( c \) within the interval \([-4, -1]\) that satisfies the Mean Value Theorem, if applicable. For this, ensure the first derivative \( f'(x) \) exists and is continuous over the interval. Verify \( f'(c) = \text{average rate of change} \).
### Diagram Explanation
While this problem does not include specific graphs or diagrams, plotting \( f(x) \) along with its derivative \( f'(x) \) over the interval \([-4, -1]\) would greatly aid in visualizing changes and confirming the solutions.
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