- #1
luju
- 11
- 0
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