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\author{Elizabeth Hunt}
\date{\today}
\title{}
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\begin{document}

\tableofcontents

\section{Taylor Series Approx.}
\label{sec:orgcc72ed1}
Suppose f has \(\infty\) many derivatives near a point a. Then the taylor series is given by

\(f(x) = \Sigma_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n\)

For increment notation we can write

\(f(a + h) = f(a) + f'(a)(a+h - a) + \dots\)

\(= \Sigma_{n=0}^{\infty} \frac{f^{(n)}(a)}{h!} (h^n)\)

Consider the approximation

\(e = |f'(a) - \frac{f(a + h) - f(a)}{h}| = |f'(a) - \frac{1}{h}(f(a + h) - f(a))|\)

Substituting\ldots{}

\(= |f'(a) - \frac{1}{h}((f(a) + f'(a) h + \frac{f''(a)}{2} h^2 + \cdots) - f(a))|\)

\(f(a) - f(a) = 0\)\ldots{} and \(distribute the h\)

\(= |-1/2 f''(a) h + \frac{1}{6}f'''(a)h^2 \cdots|\)

\subsection{With Remainder}
\label{sec:org7dfd6c7}
We can determine for some u \(f(a + h) = f(a) + f'(a)h + \frac{1}{2}f''(u)h^2\)

and so the error is \(e = |f'(a) - \frac{f(a + h) - f(a)}{h}| = |\frac{h}{2}f''(u)|\)

\begin{itemize}
\item\relax [\url{https://openstax.org/books/calculus-volume-2/pages/6-3-taylor-and-maclaurin-series}]
\begin{itemize}
\item > Taylor's Theorem w/ Remainder
\end{itemize}
\end{itemize}


\subsection{Of Deriviatives}
\label{sec:org1ec7c9b}

Again, \(f'(a) \approx \frac{f(a+h) - f(a)}{h}\),

\(e = |\frac{1}{2} f''(a) + \frac{1}{3!}h^2 f'''(a) + \cdots\)

\(R_2 = \frac{h}{2} f''(u)\)

\(|\frac{h}{2} f''(u)| \leq M h^1\)

\(M = \frac{1}{2}|f'(u)|\)

\subsubsection{Another approximation}
\label{sec:org16193b9}

\(\text{err} = |f'(a) - \frac{f(a) - f(a - h)}{h}|\)

\(= f'(a) - \frac{1}{h}(f(a) - (f(a) + f'(a)(a - (a - h)) + \frac{1}{2}f''(a)(a-(a-h))^2 + \cdots))\)

\(= |f'(a) - \frac{1}{h}(f'(a) + \frac{1}{2}f''(a)h)|\)
\end{document}