The Manga Guide to Calculus

72 Chapter 2 Let’s Learn Dierentiation Techniques!

Using the Mean Value Theorem

We saw before that the derivative is the coefficient of x in the approximate

linear function that imitates function f(x) in the vicinity of x = a.

That is,

f x f a x a f a

( )

≈

′

( )

−

( )

+

( )

(when x is very close to a)

But the linear function only “pretends to be” or “imitates” f(x), and for b,

which is near a, we generally have

u f b f a b a f a

( )

≠

′

( )

−

( )

+

( )

So, this is not exactly an equation.

In other words, we can make expression u hold with an equal sign not

with f ′(a) but with f ′(c), where c is a value existing somewhere between a

and b.

*

* That is, there must ...