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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![**Exercise 3: Verification of the Mean Value Theorem (MVT)**
Verify the Mean Value Theorem (MVT) for the function \( f(x) = \sqrt{7x} \) over the interval \([a, b]\) where \( a = 1 \) and \( b = 16 \).
In this exercise, you are tasked with proving that there exists at least one point \( c \) in the open interval \((1, 16)\) such that the derivative of the function \( f'(c) \) is equal to the average rate of change of the function over the interval \([1, 16]\).
To solve this problem:
1. **Compute the derivative** of the function \( f(x) \).
2. **Calculate** the average rate of change of \( f(x) \) from \( x = 1 \) to \( x = 16 \).
3. **Find a point \( c \)** in the interval \((1, 16)\) such that \( f'(c) \) matches the average rate of change.
This exercise focuses on applying the MVT, which is fundamental in understanding the behavior of differentiable functions over a closed interval.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdd27f7e0-834e-40ed-ac12-9ff71783ae4a%2Fd6cb1f39-4f69-4ab9-ab6d-64b213bf27da%2Fwyz7a7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Exercise 3: Verification of the Mean Value Theorem (MVT)**
Verify the Mean Value Theorem (MVT) for the function \( f(x) = \sqrt{7x} \) over the interval \([a, b]\) where \( a = 1 \) and \( b = 16 \).
In this exercise, you are tasked with proving that there exists at least one point \( c \) in the open interval \((1, 16)\) such that the derivative of the function \( f'(c) \) is equal to the average rate of change of the function over the interval \([1, 16]\).
To solve this problem:
1. **Compute the derivative** of the function \( f(x) \).
2. **Calculate** the average rate of change of \( f(x) \) from \( x = 1 \) to \( x = 16 \).
3. **Find a point \( c \)** in the interval \((1, 16)\) such that \( f'(c) \) matches the average rate of change.
This exercise focuses on applying the MVT, which is fundamental in understanding the behavior of differentiable functions over a closed interval.
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