Taylor Series

Lecture 9

Adha Imam Cahyadi, Dr.EngControl and Instruments Lab

JTETI, UGM

All continuous functions which vanish at x = aare approximately equal at x = a,but some are more approximately equal than others

Taylor, Brook (1715)

"Proposition VII, Theorem 3, Corollary 2" (in Latin).Methodus Incrementorum Directa et Inversa. London. pp. 21-23.

Without the aid of CALCULATOR..... , compute!

= 10

Now, how would you solve for this!

g = 2

Moral of the stories Not all of mathematical problems are

solvable! (including the ones in Engineering)!

In many cases it is unavoidable to computethe approximate

About our class today It is dedicated to give alternative solutions of

the both cases!

It is important for your live -> you have to dedicate yourselves too to this class!

Why study mathematics?

It is foundation of science

It has been used in so many areas: economics, engineering, social science, etc

Example?

Robots almost do nothing but math

(Robotics researchers)

And Also!

History

Zenos paradox by ancient GreekPhilosoper of Elea

Method of Exhaustion byArchimedes

Brook Taylor 1715 found theSeries

Collin Mc. Laurin found thespecific form of Taylor Series

Theorem

What does it mean?

Example

Illustration of Taylor polynomial

0-th order Taylor is just a constant

The first order is a linear function

How about higher order?

Taylor Expansion for exp(x)

Taylor Polynomial of Polynomial function

Consider the following function, f(x)=1+x+x^2+x^3

Find its Taylor Polynomial!

Here is the answer

Newton Binomial

How to compute (1+x)^n ?

Newton Binomial