Expert: Scotto - 1/12/2011

Question
Greetings Sir,
When taking the taylor expansion of certain functions,for example e^x using  polynomials different orders to approximate it.For example if we use 3 polynomials; taking the first 5 terms as the  4th order polynomial,the first 6 terms as the 5th order polynomial,the first 7 terms as the 6th order polynomial(ALL EXPANSIONS ABOUT THE SAME POINT).If one finds that they are all accurate for about one unit off that point,this would mean that at that portion/section of their graphs these polynomials will have the same shape.Sorry for taking so long to ask the question.But this motivates me to ask if you could please show me how to find an example of two polynomials of different degrees such that in a certain range the shape of their graphs is EXACTLY the same shape though out of this range they differ in shape.Or show me where/how to find them.Thanks.

Answer
The overall formula for approximating a function is sum(n=0..N)[f^n(x0)(x-x0)^n/n!].

That is, N is how far the polymonial goes, n is all numbers from 0 to N, f^n(x0) is the nth derivative evaluated at x0, { note that the 0th derivative is the function itself }, (x-x0)^n is the difference between the x value and the null value to the n power, and n! is n factorial.
For factorial, 1! = 1, 2! = 2*1! = 2, 3! = 3*2! = 6, (n+1)! = (n+1)n!, etc.
In other words, 4!=4*6=24, 5! = 5*24=120, 6!=6*120=720, 7!=7*720=5040, etc.
It can be seen the factorial terms quickly grow large and not very many terms are needed as long as (x-x0) is small and the derivatives don't get large quick.

A good place to look on factorials is http://en.wikipedia.org/wiki/Factorial

A good place to look on the aprroximation of a function by polynomials is
http://en.wikipedia.org/wiki/Newton's_method