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\title{\vspace*{-1.5in}\textbf{\S8.7: Taylor's Inequality}\\\normalsize (Created by Faan Tone Liu)}
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%\underline{Math 1300-400: Quiz 1}
\lhead{Math 2300-007: Calculus II}
\rhead{April 2, 2018}
\chead{\S8.7: Taylor's Inequality}
\vspace{-.8in}
\noindent
\textbf{Key Points:}
\begin{itemize}
\item \textbf{Goal:} Find the size of the error when using a Taylor Polynomial to estimate a function. The error (remainder) is $R_n(x) = f(x) - T_n(x)$, where $T_n(x)$ is the $n$th degree Taylor polynomial of $f$. 
\item The size of the error is influenced by the following:
\begin{itemize}
\item the degree $n$ of the Taylor Polynomial
\item the distance between $x$ and the point $a$ around which the Taylor Series is centered
\item the size of $\left|{f^{(n+1)}(x)}\right|$ in the interval between $x$ and $a$.
\end{itemize}

\item \textbf{Taylor's Inequality:} If $f^{(n+1)}$ is continuous and $\left|{f^{(n+1}(x)}\right| \leq M$ between $x$ and $a$, then the remainder $R_n(x)$ satisfies the inequality\[\left|R_n(x)\right| \leq \frac{M}{(n+1)!}\left|{x-a}\right|^{n+1}\]

\item Procedure: Find $M$ and fill in $a$. Then $R_n(x)$, $n$ and $x$ interact. You'll typically be given two of them and have to find the third.
\end{itemize}
\noindent
\textbf{Examples:}\vskip12pt
\begin{enumerate}
\item Use the $6$th degree Tayolor Polynomial for $e^x$, centered at $a=0$ to estimate $e$. How accurate is your estimate guaranteed to be? \vfill
\item How accurate is the $3$rd degree Taylor Polynomial centered at 0 guaranteed to be in estimating $\sin(9^\circ)$.\vfill\newpage
\item Show that the Taylor Series for $\cos(x)$ centered at $a=0$ converges to $\cos(x)$. \vfill
\item How large should $n$ be to estimate $e$ to within six decimal places (0.0000005) using a Taylor Polynomial centered at $x=0$.\vfill
\item Use a $5$th degree Taylor Polynomial to estimate $\ln(1.1)$. How accurate is your estimate?  \vfill
\end{enumerate}
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