Md. Mizanur Rahaman presented on the Mean Value Theorem of differential calculus. Some key points: 1) The Mean Value Theorem states that if a function f is continuous on the closed interval [a,b] and differentiable on the open interval (a,b), then there exists a number c in (a,b) such that the slope of the tangent line at (c,f(c)) equals the slope of the secant line between (a,f(a)) and (b,f(b)). 2) A restricted form of the theorem was proved by Rolle in 1691 for polynomials, while the modern form was stated and proved by Cauchy in 18

1. WELCOME TO
MY
PRESENTATION

2. NAME: Md. MIzANuR RAhAMAN
dEPARTMENT: CSE
STudENT Id: 171028005
COuRSE NAME: dIffERENTIAL
CALCuLuS
COuRSE COdE: MATh-101

3. The Mean Value Theorem
In this section, we will learn about:
The significance of the mean value theorem.

4. MEAN VALUE THEOREM
A special case of this theorem was first described by Parameshvara
(1370–1460), from the Kerala school of astronomy and mathematics in
India, in his commentaries on Govindasvāmi and Bhāskara II .
A restricted form of the theorem was proved by Rolle’s in 1691; the
result was what is now known as Rolle's theorem, and was proved only
for polynomials, without the techniques of calculus.
The mean value theorem in its modern form was stated and proved
by Cauchy in 1823.

5. MEAN VALUE THEOREM
Let f be a function that fulfills two hypotheses:
1. f is continuous on the closed interval [a, b].
2. f is differentiable on the open interval (a, b).
Then, there is a number c in (a, b) such that
( ) ( )
'( )
f b f a
f c
b a
−
=
−

6. MEAN VALUE THEOREM
f '(c) is the slope of the tangent line at (c, f(c)).
 The figures show the points A(a, f(a)) and
B(b, f(b)) on the graphs of two differentiable
functions.
 So, the Mean Value Theorem—in the form
given by Equation —states that there is at least
one point P(c, f(c)) on the graph where the slope
of the tangent line is the same as the slope of
the secant line AB.

7. MEAN VALUE THEOREM
To illustrate the Mean Value Theorem with a specific function, let’s consider
f(x) = x3
– x, a = 0, b = 2.
Example
 Since f is a polynomial, it is continuous and differentiable for all x. So, it is
certainly continuous on [0, 2] and differentiable on (0, 2).
Therefore, by the Mean Value Theorem, there is a number c in (0,2) such
that:
4
3
2 / 3±
2 / 3 1.15≈

8. MEAN VALUE THEOREM
The figure illustrates this
calculation.
The tangent line at this value
of c is parallel to the secant
line OB.
Example

9. END OF
MY
PRESENTATION

11. ANY QUESTION