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Determine whether the function satisfies the hypotheses of the Mean Value Theorem for the given interval. s(t) = J(1 - ), [-1,5] %D O Yes O No

Determine whether the function satisfies the hypotheses of the Mean Value Theorem for the given interval. s(t) = J(1 - ), [-1,5] %D O Yes O No

Calculus: Early Transcendentals
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: Understanding the Application of the Mean Value Theorem** **Objective:** To determine whether the function satisfies the hypotheses of the Mean Value Theorem on the given interval. **Problem Statement:** Consider the function \( s(t) = \sqrt{t(1-t)} \) over the interval \([-1, 5]\). **Question:** Does the function \( s(t) \) satisfy the hypotheses of the Mean Value Theorem for the given interval? - [ ] Yes - [ ] No **Solution Approach:** 1. **Check Continuity:** - For the function to satisfy the Mean Value Theorem on the interval \([-1, 5]\), it must be continuous on the closed interval \([a, b]\). 2. **Check Differentiability:** - The function must also be differentiable on the open interval \((a, b)\). 3. **Evaluate the Hypotheses:** - Determine if the function's behavior on \([-1, 5]\) meets both the continuity and differentiability requirements. Understanding the properties of the function and the provided interval will help determine whether the conditions for the Mean Value Theorem are met.
Transcribed Image Text:**Title: Understanding the Application of the Mean Value Theorem** **Objective:** To determine whether the function satisfies the hypotheses of the Mean Value Theorem on the given interval. **Problem Statement:** Consider the function \( s(t) = \sqrt{t(1-t)} \) over the interval \([-1, 5]\). **Question:** Does the function \( s(t) \) satisfy the hypotheses of the Mean Value Theorem for the given interval? - [ ] Yes - [ ] No **Solution Approach:** 1. **Check Continuity:** - For the function to satisfy the Mean Value Theorem on the interval \([-1, 5]\), it must be continuous on the closed interval \([a, b]\). 2. **Check Differentiability:** - The function must also be differentiable on the open interval \((a, b)\). 3. **Evaluate the Hypotheses:** - Determine if the function's behavior on \([-1, 5]\) meets both the continuity and differentiability requirements. Understanding the properties of the function and the provided interval will help determine whether the conditions for the Mean Value Theorem are met.
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