How Do You Find the k-th Derivative of a Polynomial at x=0?

[luju]

Homework Statement

Consider the following polynomial of degree n > 1,
P(x) = anxn + an¡1xn¡1 + ¢ ¢ ¢ + a1x + a0;
where a0; : : : ; an are some non-zero constants (don't give them values!).

Homework Equations

a. We know that P0(x) = nanxn¡1+(n¡1)anxn¡2+¢ ¢ ¢+2a2x+a1, and then P0(0) = a1. Keep
di®erentiating and ¯nd a formula for P(k)(0), the k-th derivative of P at x = 0, for k = 1; : : : ; n.
Recall that the factorial of n is de¯ned as n! := 1 £ ¢ ¢ ¢ £ (n ¡ 1) £ n, 0! := 1.
b. Consider f(x) = ex. Find the polynomial P of degree n that \best approximates" f around
x = 0. That is, use the result in part a to ¯nd a0; a1; : : : an in the formula of P, such that:
f(0) = P(0); f0(0) = P0(0); f00(0) = P00(0); : : : ; f(n)(0) = P(n)(0):
c. Use P to give an approximate formula for the number e. This is the n-th order approximation
to e.

The Attempt at a Solution

I don't even know how to begin this. SO if someone could give me some clues as to how to do this

 

[CompuChip]

Welcome to PF luju.
Unfortunately your post is not very readable. Do you know a little bit about LaTeX? Then you can just [ t e x ] and [ / t e x ] tags (remove all the spaces from the tags) to use TeX code to make it clearer. Otherwise, please try to use only basic ASCII characters, without stuff like ¡ £ ¢

For a: Just start as the question suggests: differentiate the polynomial. And differentiate again. And again ...
For b: What do you know about the function $e^x$ and its derivatives? What restrictions does this give on the derivatives you found in a) ?
For c, you should finish a and b first.