PlanetMath: example of Taylor polynomials for $\sin x$

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example of Taylor polynomials for $\sin x$

(Example)

In this entry we compute several Taylor polynomials for the function $\sin x$ 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 ( $n\geq 0$ ) is given by:

$\displaystyle T_n(x)=f(a)+f'(a)(x-a)+\frac{f''(a)}{2!}(x-a)^2+\ldots+\frac{f^{(n)}(a)}{n!}(x-a)^n$

where $f^{(n)}$ denotes the

th derivative of

From now on we assume $f(x)=\sin x$ and . Notice that the derivatives of $\sin x$ are cyclic:

$\displaystyle f'(x)=\cos x,\quad f''(x)=-\sin x, \quad f'''(x)=-\cos x, \quad f^{(4)}(x)=\sin x = f(x).$

Therefore, the Taylor polynomials are easy to compute. In fact:

$\displaystyle f^{(2n)}(0)=0, \quad f^{(2n+1)}(0)=(-1)^n$

Thus, the first Taylor polynomial is given by:

$\displaystyle T_1(x)= 0 + 1\cdot x = x$

In the following graph one can compare the function

and $\sin x$ .

$\includegraphics[scale=0.7]{taylor1}$

The function $y=\sin x$ and the first Taylor polynomial.

Notice that . More generally, $T_{2n}(x)=T_{2n-1}(x)$ so we will not compute any other even order Taylor polynomials. However, the third degree Taylor polynomial is given by the formula:

$\displaystyle T_3(x)=x-\frac{x^3}{3!}=x - \frac{x^3}{6}$

$\includegraphics[scale=0.7]{taylor3}$

The function $y=\sin x$ and the third Taylor polynomial.

The Taylor polynomial of degree is given by:

$\displaystyle T_5(x)=x-\frac{x^3}{3!}+\frac{x^5}{5!}$

$\includegraphics[scale=0.7]{taylor5}$

The function $y=\sin x$ and the fifth Taylor polynomial.

Next, we compute some Taylor polynomials of higher degree. In particular, the Taylor polynomial of degree has the form:

$\displaystyle T_{15}(x)=x-\frac{x^3}{6} + \frac{x^5}{120} - \frac{x^7}{5040} + ... ...ac{x^{11}}{39916800} + \frac{x^{13}}{6227020800} - \frac{x^{15}}{1307674368000}$

$\includegraphics[scale=0.7]{taylor15}$

The function $y=\sin x$ 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 $T_{99}(x)$ behaves rather jittery and wildly.