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Sachin Kaushal, Lovely Professional University Unit 24: Fundamental Theorem of Calculus

Unit 24: Fundamental Theorem of Calculus Notes

CONTENTS
Objectives
Introduction
24.1 Fundamental Theorem of Calculus
24.2 Primitive of a Function
24.3 Summary
24.4 Keywords
24.5 Review Questions
24.6 Further Readings

Objectives

After studying this unit, you will be able to:
 Discuss the fundamental theorem of calculus
 Explain the primitive of a function

Introduction

In this unit we will discuss about, what is the relationship between the two notions of
differentiation and integration? Now we shall try to find an answer to this question. In fact, we
shall show that differentiation and Integration are intimately related in the sense that they are
inverse operations of each other.
Let us begin by asking the following question: "when is a function f : [a, b]  R, the derivative
of some function F : [a, b]  R?"
For example consider the function f : [–1, 1]  R defined by
x
1
ì 0 if - £ < 0
f(x) = í
x
î i if 0 £ < 1
This function is not the derivative of any function F : [–1, 1]  R. Indeed if f is the derivative of
a function F : [–1, 1]  R then f must have the intermediate value property. But clearly, the
function f given above does not have the intermediate value property.
Hence f cannot be the derivative of any function F : [–1, 1]  R.
However if f : [–1, 1]  R is continuous, then f is the derivative of a function F : [–1, 1]  R. This
leads us to the following general theorem.

24.1 Fundamental Theorem of Calculus

Theorem 1: Let f be integrable on [a, b]. Define a function P on [a, b] as
x
Î
F(x) = ò f(t) dt," x [a,b].
a
Then F is continuous on [a, b]. Furthermore, if f is continuous at a point x, of [a, b], then F is
differentiable at x and F’1(x ) = f(x ).
0 0 0

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