Taylor's Theorem

Let is differentiable on . Let . Then :

Mean value theorem can be derived from taylor’s theorem when .

Proof Hint

• Define as mentioned above
• Consider the interval
• Use Cauchy’s mean value theorem for after making sure the conditions are met.

The above equation can be written like:

Taylor Polynomial

This part of the above equation is called the Taylor polynomial. Denoted by .

Remainder

Denoted by .

Integral form of the remainder

Proof Hint

• Method 1: Use integration by parts and mathematical induction.
• Method 2: Use Generalized MVT for Riemann Integrals where:

Note

When :

From this: a local minimum is at . Converse is not true.