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

\date{Spring 2006}

\begin{document}

\maketitle

\begin{itemize}

\item
The $n$th degree {\em Taylor Polynomial\/} of a function $f$ is
$$
T_n(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + \frac{f'''(a)}{3!}(x - a)^3 + \cdots + \frac{f^{(n)}(a)}{n!} (x - a)^n.
$$
If $f$ can be represented by a Taylor series, then $f(x) = T_n(x) + R_n(x)$, where $R_n$ is the remainder series.


\item
If $|f^{(n + 1)}| \leq M$ between $a$ and $x$, then the remainder of the Taylor series satisfies the inequality
$$
|R_n(x)| \leq \frac{M}{(n+1)!} |x - a|^{n+1}.
$$


\end{itemize}

\begin{enumerate}


\item
Let
$$
f(x) = \sum_{n = 0}^\infty \frac{(-1)^n x^{2n}}{(2n + 1)!} = 1 - \frac{x^2}{3!} + \frac{x^4}{5!} - \frac{x^6}{7!} + \cdots 
$$
for all real numbers $x$.
\begin{enumerate}

\item
Find $f'(0)$ and $f''(0)$.  Determine whether $f$ has a local minimum, a local maximum, or neither at $x = 0$.

\vfill



\item
Show that $y = f(x)$ is a solution to the differential equation 
$$
xy' + y = \cos x.
$$


\vfill


\end{enumerate}

\clearpage


\item
Let $f$ be the function given by 
$$
f(x) = \sin\left(  5x + \frac{\pi}{4} \right),
$$
and let $P(x)$ be the third-degree Taylor polynomial for $f$ centered about $x = 0$.
\begin{enumerate}

\item
Find $P(x)$.

\vfill

\item 
Find the coefficient of $x^{22}$ in the Taylor series for $f$ about $x = 0$.

\vfill

\item
Use the error bound to show that
$$
\left| f\left( \frac{1}{10} \right) - P\left(  \frac{1}{10} \right) \right| < \frac{1}{100}.
$$

\vfill

\item
Let
$$
G(x) = \int_0^x f(t) \, dt.
$$
Write the third-degree Taylor polynomial for $G$ about $x = 0$.

\vfill

\end{enumerate}

\item
If $f(x) = \cos x$ and $g(x) = \sin x$, find the first five terms of the power series for $f(x) g(x)$.


\vfill

\end{enumerate}







\end{document}