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Q. 59

Found in: Page 249


Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861

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Short Answer

Determine whether or not each function f satisfies the hypotheses of the Mean Value Theorem on the given interval [a,b]. For those that do, use derivatives and algebra to find the exact values of all c(a,b) that satisfy the conclusion of the Mean Value Theorem.

f(x)=sinx, [a,b]=[0,π2].

The function f(x)=sinx satisfies the Mean Value Theorem and the value is, c=cos-12π.

See the step by step solution

Step by Step Solution

Step 1. Given Information.

The given function is,

f(x)=sinx, [a,b]=[0,π2].

Step 2. Proving Mean Value Theorem.

The function f(x)=sinx is continuous and differentiable on [0,π2]. The Mean Value Theorem applies to this function on the interval [0,π2].

The slope of the line from (0,f(0)) to (π2,f(π2)) is:

f(π2)-f(0)π2-0=sinπ2-sin0π2-0 =1-0π2-0 =1π2 =2π

By the Mean Value Theorem, there must exist at least one point c(0,π2) with f'(c)=2π

We have to find the value of c with f'(c)=2π we solve it:


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