example of Taylor polynomials for
In this entry we compute several Taylor polynomials for the function around and we produce graphs to compare the function with the corresponding Taylor polynomial. Recall that for a given function (here we suppose is infinitely differentiable) and a point , the Taylor polynomial of degree () is given by:
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.
A detail of the Taylor polynomial of degree (the interval ).
|Title||example of Taylor polynomials for|
|Date of creation||2013-03-22 15:03:43|
|Last modified on||2013-03-22 15:03:43|
|Last modified by||alozano (2414)|