4. Determine if the Mean Value Theorem can be applied to the following function on the the given closed interval. If so, find all possible values of c: a b on [1, 3]. f(3) = 3 1+3 FC) - Hb) -f(a) = 6-9 1 + x
4. Determine if the Mean Value Theorem can be applied to the following function on the the given closed interval. If so, find all possible values of c: a b on [1, 3]. f(3) = 3 1+3 FC) - Hb) -f(a) = 6-9 1 + x
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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What are the values of C if the mean theorem value is applied?
![**Educational Website Content: Understanding the Mean Value Theorem**
---
### Example Problems Using the Mean Value Theorem
#### Problem 4:
**Instruction:** Determine if the Mean Value Theorem can be applied to the following function on the given closed interval. If so, find all possible values of \( c \).
**Function:**
\[ f(x) = \frac{x}{1+x} \]
**Interval:** \([1, 3]\)
**Steps:**
1. Evaluate \( f(x) \) at the endpoints of the interval:
\[
f(3) = \frac{3}{1+3} = \frac{3}{4}
\]
\[
f(1) = \frac{1}{1+1} = \frac{1}{2}
\]
2. Apply the Mean Value Theorem formula:
\[
f'(c) = \frac{f(b) - f(a)}{b-a}
\]
Where \( a = 1 \) and \( b = 3 \).
---
#### Problem 5:
**Instruction:** Determine if the Mean Value Theorem can be applied to the following function on the given closed interval. If so, find all possible values of \( c \).
**Function:**
\[ f(x) = \sin(2x) \]
**Interval:** \([0, \pi]\)
**Steps:**
1. Evaluate \( f(x) \) at the endpoints of the interval:
\[
f(\pi) = \sin(2\pi) = 0
\]
\[
f(0) = \sin(2 \cdot 0) = \sin 0 = 0
\]
**Conclusion:** In this case, further steps would involve computing the derivative \( f'(x) \) and checking for values in the interval where the derivative equals the average rate of change over \([0, \pi]\).
---
**Note:** This content delves into applying the Mean Value Theorem on two distinct functions, providing examples of calculating the function's values at specified points and applying relevant formulas.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2910e309-4023-42bb-9437-090123672c12%2F95a6b301-df82-4514-a3b9-e7d9687bb22f%2F7chfrw6_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Educational Website Content: Understanding the Mean Value Theorem**
---
### Example Problems Using the Mean Value Theorem
#### Problem 4:
**Instruction:** Determine if the Mean Value Theorem can be applied to the following function on the given closed interval. If so, find all possible values of \( c \).
**Function:**
\[ f(x) = \frac{x}{1+x} \]
**Interval:** \([1, 3]\)
**Steps:**
1. Evaluate \( f(x) \) at the endpoints of the interval:
\[
f(3) = \frac{3}{1+3} = \frac{3}{4}
\]
\[
f(1) = \frac{1}{1+1} = \frac{1}{2}
\]
2. Apply the Mean Value Theorem formula:
\[
f'(c) = \frac{f(b) - f(a)}{b-a}
\]
Where \( a = 1 \) and \( b = 3 \).
---
#### Problem 5:
**Instruction:** Determine if the Mean Value Theorem can be applied to the following function on the given closed interval. If so, find all possible values of \( c \).
**Function:**
\[ f(x) = \sin(2x) \]
**Interval:** \([0, \pi]\)
**Steps:**
1. Evaluate \( f(x) \) at the endpoints of the interval:
\[
f(\pi) = \sin(2\pi) = 0
\]
\[
f(0) = \sin(2 \cdot 0) = \sin 0 = 0
\]
**Conclusion:** In this case, further steps would involve computing the derivative \( f'(x) \) and checking for values in the interval where the derivative equals the average rate of change over \([0, \pi]\).
---
**Note:** This content delves into applying the Mean Value Theorem on two distinct functions, providing examples of calculating the function's values at specified points and applying relevant formulas.
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