Theorem. Suppose that
is a function for which

all exists. Let

and define

Then

Proof. Define

and
.
We prove that

Notice that

Thus

and

We may therefore apply l'H\opital's Rule
times to obtain

Since
is a polynomial of degree
,
its
st
derivative is constant; in fact,
.
Thus

Therefore

Definition. The polynomial
is called the **Taylor polynomial** of degree
for
at
.

Two functions
and
are said to be **equal up to order**
at
if

The above theorem says that the Taylor polynomial equals
up to order
near
.

Theorem. Let and be two polynomials in , of degree , and suppose that and are equal up to order neat . Then

Proof. Let
.
Then
is a polynomial of degree
.
Write

The given condition

implies that

For
,
we have

Thus
and

Therefore

For
,
we have

Thus
and

Continuing this way we see that

Corollary. Let be -times differentiable at , and suppose that is a polynomial in of degree , which equals up to order at . Then , the Taylor polynomial of degree for near a.

Write

Adding up these identities, we have

Theorem. Suppose that
,
are defined on
and
is defined by

Then

(1) there exists some
in
such that

(2) there exists some
in
such that

Moreover, if
is integrable on
then

(3)

Proof. Fix
and
.
Define

for
in
.
Then

Applying the Mean Value Theorem to
on the interval
there
exists some
in
such that

Note that

Therefore

This is called the **Cauchy form of remainder**.

Set
.
Now apply the Cauchy Mean value Theorem to
and
there exists some
in
such that

Therefore

or

This is called the **Lagrange form of the remainder.**

If
is integrable on
,
it follows from the Fundamental Theorem of Calculus that

Therefore

This is called the **integral form of the remainder.**

From the equation

we have

for all

From the equation

we
have