Page 300 - DMTH401_REAL ANALYSIS

Real Analysis Sachin Kaushal, Lovely Professional University

Notes Unit 25: Mean Value Theorem

CONTENTS
Objectives
Introduction
25.1 First Mean Value Theorem
25.2 The Generalised First Mean Value Theorem
25.3 Second Mean Value Theorem

25.4 Summary
25.5 Keywords
25.6 Review Questions
25.7 Further Readings

Objectives

After studying this unit, you will be able to:
 Discuss the first mean value theorem
 Explain the generalized first mean value theorem

 Describe the second mean value theorem
Introduction

In last unit, we discussed some mean-value theorems concerning the differentiability of a function.
Quite analogous, we have two mean value theorems of integrability which we intend to discuss
here. You are quite familiar with the two well-known techniques of integration namely the
integration by parts and integration by substitution which you must have studied in your
earlier classes.
25.1 First Mean Value Theorem

Let f : [a, b]  R be a continuous function. Then there exists c  [a, b] such that
b
=
ò f(x) dx (b a)f(c).
-
Proof: We know that
h
-
£
-
m(b a) £ ò f(x) dx M(b a), thus
a
b
ò f(x) dx
m £ a £ M, where
(b a)
-
m = glb {f(x) : x [a,b]}, and
M = lub {f(x) : x  [a,b]}.

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