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\title{Math 1b.  Lecture 25 \\
Applications of Taylor Polynomials}

\author{T. Judson}

\date{Spring 2006}


\begin{document}

\maketitle


\section{Goals}

\begin{itemize}

\item
To understand how Taylor polynomials can be used to approximate functions.



\end{itemize} 

\section*{Taylor's Remainder Theorem and Taylor's Inequality\protect\footnote{This material is different than what is presented in Stewart.}}

\begin{itemize}

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


\item
If $f$ has continuous derivatives of orders $1, 2, \ldots, n+1$ between $a$ and $x$, then  
$$
f(x) = T_n(x) + \frac{f^{(n+1)} (c)}{(n + 1)!} (x - a)^{n + 1}
$$
where $c$ lies between $a$ and $x$.

Notice that the Mean Value Theorem is a special case of Taylor's Remainder Theorem.

\end{itemize}


\section{Worksheet Problems}

\begin{enumerate}


\item
Compute the Taylor polynomial of $f(x) = \sqrt{x}$ for $n = 2$ and $a = 4$.  Estimate the accuracy of the approximation $f(x) \approx T_n(x)$ when $x$ lies in the interval $4 \leq x \leq 4.2$.


\item
Compute the Taylor polynomial of $f(x) = \cos x$ for $n = 4$ and $a = \pi/3$.  Estimate the accuracy of the approximation $f(x) \approx T_n(x)$ when $x$ lies in the interval $0 \leq x \leq 2 \pi / 3$.

\item
Let $f$ be a function having derivatives of all orders for all real numbers.  The third-degree Taylor polynomial for $f$ about $x = 2$ is given by
$$
T_3(x) = 7 - 9(x - 2)^2 - 3(x - 2)^3.
$$
\begin{enumerate}

\item
Find $f(2)$ and $f'(2)$.

\item
Is there enough information given to determine whether $f$ has a critical point at $x = 2$?  If not, explain why not.  If so, determine whether $f(2)$ is a relative minimum, a relative maximum, or neither.  Justify your answer.

\item
Use $T_3(x)$ to find an approximation for $f(0)$.  Is there enough information given to determine whether $f$ has a critical point at $x = 0$?  If not, explain why not.  If so, determine whether $f(0)$ is a relative minimum, a relative maximum, or neither.  Justify your answer.

\item
The fourth derivative of $f$ satisfies the inequality 
$$
\left| f^{(4)}(x) \right| \leq 6
$$
for all $x$ in the closed interval $[0,2]$.  Find an error bound on the approximation for $f(0)$ that you found in part (c).  Explain why $f(0)$ must be negative.

\end{enumerate}


\end{enumerate}





\section*{References}

\begin{itemize}

\item
\S 8.9 in 
James Stewart.
{\it Single Variable Calculus: Concepts \& Context}, third edition.
Brooks/Cole, Belmont CA, 2005.
ISBN 0--534--41022--7.


\end{itemize}



\section*{Notes}

\subsection*{April 12, 2006}



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