Our sprint starts with **Lagrange’s Mean-Value theorem** which states:

A function is

(1) continous on closed interval

(2) differentiable on open interval

.

Let’s prove it.

The area of a triangle with vortices and is the absolute value of

where (See “Had Heron Known Analytic Geometry“)

Let

,

Since is differentiable, we have

.

Clearly,

Therefore, by Rolle’s Theorem (See “Rolle’s theorem”) , i.e.,

or

.

The geometric meaning of Lagrange’s Mean-Value theorem is illustrated for several functions in Fig. 1. It shows that the graph of a differentiable function has at least one tangent line parallel to the cord connecting and .

Fig. 1

The bottom curve falsifies the theorem due to its missing differentiability at one point.

Following Lagrange’s Mean-Value theorem are two corollaries. We have encountered and accepted the first one without proper proof in the past (See “Inching towards Definite Integral“)

Let’s state and prove them now.

**Corollary 1**. A function on an open interval has derivative zero at each point , a constant.

It is true due to the fact that is both continous on and differentiable on . By Lagrange’s Mean-Value theorem, . Since , We have . i.e., . Hence , a constant on .

**Corollary 2**. Two functions, and have the same derivative at each point on an open interval , a constant.

For . By corollary 1, , a constant.

Next, we define a set as follows:

i.e.,

is a set of function such that for all ‘s.

is certainly not empty for we have proved that by showing (See “Inching towards Definite Integral“). It follows that ,

By Corollary 2,

.

Let , we have

.

Hence

or

.

Let’s summarize the result of our exploration in a theorem:

On an open interval containing , a function is differentiable and, ,

This is FTC, **The Fundamental Theorem of Calculus**.

Let , (1) becomes the Newton-Leibniz formula

,

our old friend for evaluating (See “Inching towards Definite Integral“)