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Calculus: Differentiation and Integration, 1st Edition
book

Calculus: Differentiation and Integration, 1st Edition

by ICFAI
April 2024
Intermediate to advanced content levelIntermediate to advanced
710 pages
20h 50m
English
Pearson India
Content preview from Calculus: Differentiation and Integration, 1st Edition
9-6  Calculus – Differentiation and Integration
STATEMENT If a function f is
a. Continuous in the closed interval [a, b].
b. Derivable in the open interval (a, b).
Then there exists at least one value c Î (a, b) such that
f b f a
b a
f ' c
( ) ( )
( ).
=
Proof: Now, we define a new function f involving f and designed so as to satisfy the conditions of
Rolle’s theorem.
Consider, f(x) = f(x) + Ax, where a is a constant to be determined such that f(a) = f(b).
+ = +
=
=
f a Aa f b Ab
f b f a A a b
A
f b f a
b a
( ) ( )
( ) ( ) ( )
( ) ( )
.
Now the function f(x) is differentiable, x is differentiable and A being a constant is also differentiable.
Therefore, ...
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Publisher Resources

ISBN: 9789353948221