example of Taylor polynomials for $\sin x$ExampleOfTaylorPolynomialsForSinX2005-02-18 15:39:022005-03-23 08:40:43ExampleTaylor seriesTaylor polynomialpretty graphssine function% this is the default PlanetMath preamble. as your knowledge
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\newcommand{\Rats}{\mathbb{Q}}In this entry we compute several Taylor polynomials for the function $\sin x$ 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 $n$ ($n\geq 0$) is given by:
$$T_n(x)=f(a)+f'(a)(x-a)+\frac{f''(a)}{2!}(x-a)^2+\ldots+\frac{f^{(n)}(a)}{n!}(x-a)^n$$
where $f^{(n)}$ denotes the $n$th derivative of $f(x)$. \\
From now on we assume $f(x)=\sin x$ and $a=0$. Notice that the derivatives of $\sin x$ are cyclic:
$$f'(x)=\cos x,\quad f''(x)=-\sin x, \quad f'''(x)=-\cos x, \quad f^{(4)}(x)=\sin x = f(x).$$
Therefore, the Taylor polynomials are easy to compute. In fact:
$$f^{(2n)}(0)=0, \quad f^{(2n+1)}(0)=(-1)^n$$
Thus, the first Taylor polynomial is given by:
$$T_1(x)= 0 + 1\cdot x = x$$
In the following graph one can compare the function $T_1(x)=x$ and $\sin x$.
\begin{center}
\includegraphics[scale=0.7]{taylor1}
The function $y=\sin x$ and the first Taylor polynomial.
\end{center}
Notice that $T_2(x)=T_1(x)$. 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:
$$T_3(x)=x-\frac{x^3}{3!}=x - \frac{x^3}{6}$$
\begin{center}
\includegraphics[scale=0.7]{taylor3}
The function $y=\sin x$ and the third Taylor polynomial.
\end{center}
The Taylor polynomial of degree $5$ is given by:
$$T_5(x)=x-\frac{x^3}{3!}+\frac{x^5}{5!}$$
\begin{center}
\includegraphics[scale=0.7]{taylor5}
The function $y=\sin x$ and the fifth Taylor polynomial.
\end{center}
Next, we compute some Taylor polynomials of higher degree. In particular, the Taylor polynomial of degree $15$ has the form:
$$T_{15}(x)=x-\frac{x^3}{6} + \frac{x^5}{120} - \frac{x^7}{5040} + \frac{x^9}{362880} - \frac{x^{11}}{39916800} + \frac{x^{13}}{6227020800} - \frac{x^{15}}{1307674368000}$$
\begin{center}
\includegraphics[scale=0.7]{taylor15}
The function $y=\sin x$ and the Taylor polynomial of degree $15$.
\end{center}
Finally, we produce a detailed view of the Taylor polynomial of degree $99$. In particular, notice that the graphs are very close until $x=34$ or so, but after that $T_{99}(x)$ behaves rather jittery and wildly.
\begin{center}
\includegraphics[scale=0.7]{detailtaylor100}
A detail of the Taylor polynomial of degree $99$ (the interval $(34,39)$).
\end{center}