example of Taylor polynomials for sinx
In this entry we compute several Taylor polynomials for the function sinx around x=0 and we produce graphs to compare the function with the corresponding Taylor polynomial. Recall that for a given function y=f(x) (here we suppose f is infinitely differentiable
) and a point x=a, the Taylor polynomial of degree n (n≥0) is given by:
Tn(x)=f(a)+f′(a)(x-a)+f′′ |
where denotes the th derivative of .
From now on we assume and . Notice that the derivatives of are cyclic:
Therefore, the Taylor polynomials are easy to compute. In fact:
Thus, the first Taylor polynomial is given by:
In the following graph one can compare the function and .
The function and the first Taylor polynomial.
Notice that . More generally, so we will not compute any other even order Taylor polynomials. However, the third degree Taylor polynomial is given by the formula:
The function and the third Taylor polynomial.
The Taylor polynomial of degree is given by:
The function and the fifth Taylor polynomial.
Next, we compute some Taylor polynomials of higher degree. In particular, the Taylor polynomial of degree has the form:
The function and the Taylor polynomial of degree .
Finally, we produce a detailed view of the Taylor polynomial of degree . In particular, notice that the graphs are very close until or so, but after that behaves rather jittery and wildly.
Title | example of Taylor polynomials for |
---|---|
Canonical name | ExampleOfTaylorPolynomialsForsinX |
Date of creation | 2013-03-22 15:03:43 |
Last modified on | 2013-03-22 15:03:43 |
Owner | alozano (2414) |
Last modified by | alozano (2414) |
Numerical id | 7 |
Author | alozano (2414) |
Entry type | Example |
Classification | msc 41A58 |
Related topic | ComplexSineAndCosine |
Related topic | HigherOrderDerivativesOfSineAndCosine |