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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

Prove that Mean value Theorem for Integrals by applying the Mean Value Theorem for derivatives (see Sections 3.2) to the function.

F ( x ) = a x f ( t )   d t

To determine

To prove:

The mean value theorem for integrals by applying the mean value theorem for derivatives to the function,

Fx=axft dt

Explanation

1) Concept:

Use the mean value theorem for derivatives and the fundamental theorem of calculus part 2 to provethe mean value theorem for integrals.

2) Theorem:

i) The mean value theorem: If f be a function satisfies,

1) f is continuous on [a,b]

2) f is differentiable on (a,b)

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

ii) The fundamental theorem of calculus part 2

If f is continuous on a,b, then

abfxdx=Fb-F(a)

Where F is any anti derivative of f, that is, a function F such that F'=f

4) Calculation:

Let Fx=axft dt

for x in a,b

F is continuous on a,b

F is differentiable on a,b and F=f

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