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\large 
\underline
{MA2160 \hspace{5mm}Calculus II with Technology \hspace{50mm} Ch. 10-4}

Now that we can construct Taylor polynomials, how can we analyze how close to the actual function is the polynomial?  


\Large
Goal
\large
\begin{enumerate}
\item Find a method to determine the error of the Taylor polynomial in relation to its corresponding function.
\item Apply this method to find a large enough degree polynomial to ensure accuracy to a given degree.
\item Apply this method to find a range of values to which a given degree polynomial is accurate to a given tolerance.
\end{enumerate}


If we define the error function to be the difference between the Taylor polynomial of degree $n$ and the function, it is easy to see that all derivatives of this error function from the first to the $n$th must be 0.  If we take the ($n+1$)th derivative, we get our first opportunity for a non-zero value.  Therefore, if we can find a bound on the ($n+1$)th derivative, we can find a function which will bound our error for us (LaGrange Error Bound).  

Given a Taylor polynomial of degree $n$ about $x=a$ and $M$ equal to a bound of $f^{(n+1)}(x)$ on a given interval of $x$ about $a$, the function for absolute value of the error is as follows:

$$|E_n(x)|\leq \frac{M}{(n+1)!}(|x-a|)^{n+1}$$

Therefore, the bound on the error is based on the next term of the Taylor series of the Taylor polynomial. Of course, in some instances we can calculate the error directly.
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Examples:

\begin{enumerate}
\item What is the error on the Taylor polynomial of degree 6 centered at $x=0$ for $e^1$?
\vspace{10mm}
\item What degree Taylor polynomial centered at $x=0$ is necessary to guarantee $e^1$ to the nearest .0000001?
\vspace{10mm}
\item At what interval $[-x,0]$ is the degree 6 Taylor polynomial for $f(x)=e^x$ centered at $x=0$ guaranteed to be accurate to the nearest .0001?
\vspace{10mm}
\item What is the error on the Taylor polynomial of degree 6 centered at $x=0$ for $ \ln 2$?     (Note:  We use the function $\ln (1+x)=f(x)$)
\vspace{10mm}
\item What degree Taylor polynomial centered at $x=0$ is necessary to guarantee $ \ln 2$ to the nearest .000001?
\vspace{10mm}
\item At what interval $[0,x]$ is the degree 6 Taylor polynomial for $f(x)= \ln (x+1)$ centered at $x=0$ guaranteed to be accurate to the nearest .0001?
\end{enumerate}


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