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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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BuyFindarrow_forward

Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

If f is a continuous function, what is the limit as h 0 of the average value of f on the interval [ x , x + h ] ?

To determine

To find:

The limit as h0 of the average value of f on the interval x, x+h.

Explanation

1) Concept:

Use the mean value theorem for integrals

2) Theorem:

If f is continuous on  [a, b], then there exists a number c in [a, b] such that

fc=fave =1b-aabfxdx In other words,

abfxdx=f(c)(b-a)

3) Calculations:

Since f is continuous,

By the mean value theorem for integrals,

fave =1b-aabfxdx

Limit as h0 of the average value of f on interval x, x+h,

limh0fave=limh01x+h-x

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