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\title{Math 1b. Taylor and Maclaurin Series I}

\date{Spring 2006}

\begin{document}

\maketitle

\begin{itemize}

\item
A {\em Taylor Series\/} for $f$ centered at $x = a$ is given by
$$
f(x) =  \sum_{n = 0}^\infty \frac{f^{(n)}(a)}{n!} (x - a)^n \\
= f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + \frac{f'''(a)}{3!}(x - a)^3 + \cdots.
$$

\item
A Taylor series centered at $x = 0$ is called a {\em Maclaurin series},
$$
f(x) =  \sum_{n = 0}^\infty \frac{f^{(n)}(0)}{n!} x^n \\
= f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \cdots
$$

\item
The $n$th degree {\em Taylor Polynomial\/} of a function $f$ is
$$
T_n = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + \frac{f'''(a)}{3!}(x - a)^3 + \cdots + \frac{f^{(n)}(a)}{n!} (x - a)^n.
$$

\end{itemize}

\begin{enumerate}


\item
Find the Taylor polynomial for $f(x) = \cos x$, where $n = 4$ and $a = 0$

\vfill

\clearpage

\item
Find the Taylor polynomial for $f(x) = \cos x$, where $n = 4$ and $a = \pi/6$

\vfill

\item
Find the Taylor series for $f(x) = \cos x$, where $a = 0$.

\vfill


\item
Find the Taylor series for $f(x) = x^3 - 3x^2 + 2$, where $a = 2$.

\vfill

\item
Find the Taylor series for $f(x) = \ln x$, where $a = 1$.

\vfill

\end{enumerate}


\end{document}