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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

5-8 Verify that the function satisfies the three hypotheses of Rolle’s Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Rolle’s Theorem.

f ( x ) = sin ( x / 2 ) ,    [ π / 2 , 3 π / 2 ]

To determine

To verify:

(i) Whether the function satisfies the hypothesis of Rolle’s Theorem on the given interval.

(ii) Find all numbers c that satisfy the conclusion of Rolle’s theorem.

Explanation

1) Concept:

Using the Rolle’s Theorem verify the result and find the value of c

2) Theorem:

Rolle’s Theorem – Let f  be a function that satisfies the following 3 hypotheses:

i. f is continuous on the closed interval [a, b]

ii. f is differentiable on the open interval (a, b)

iii. f (a)=f (b)

Then there is a number c in (a, b) such that f'c=0

3) Given:

fx=sin(x2), [π2, 3π2]

4) Calculation:

(i) Consider the given function fx=sin(x2)

The given function is continuous and differentiable on the given interval

Therefore, f(x) is continuous on [π2, 3π2] and differentiable on (π2, 3π2)

Hence, it satisfies hypothesis i) and ii) of Rolle’s Theorem.

Now, fx=sin(x2)

At x=π2, fπ2=sinπ4=0.7071

At x=3π2, f3π2=sin3π4=0

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