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Math derivative

  1. Oct 11, 2007 #1
    1. The problem statement, all variables and given/known data
    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!!!).

    2. Relevant 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.

    3. The attempt at a solution

    I dont even know how to begin this. SO if someone could give me some clues as to how to do this
  2. jcsd
  3. Oct 11, 2007 #2


    User Avatar
    Science Advisor
    Homework Helper

    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 [itex]e^x[/itex] and its derivatives? What restrictions does this give on the derivatives you found in a) ?
    For c, you should finish a and b first.
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