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\begin{document}


\title{On-line Math 21}


\email{david.johnson@lehigh.edu}


\urladdr{http://www.lehigh.edu/\textasciitilde{}dlj0}

\maketitle

\subsection{Applications of the derivative.}

\begin{example}
Find an approximate value of \[
\sqrt{4.2}.\]
 
\end{example}

\subsubsection*{Solution:}

\( f(x)=\sqrt{x} \) in this example (you have to figure out the function in
these examples). Then, \( a=4 \), \( \Delta x=0.2 \),  and we have: \begin{eqnarray*}
f(a+\Delta x) & = & f(a)+f'(a)\Delta x\\
\sqrt{4.2} & = & \sqrt{4}+\frac{1}{2\sqrt{4}}0.2\\
 & = & 2+\frac{1}{4}0.2=2.05.
\end{eqnarray*}


By the way, my computer said \[
\sqrt{4.2}=2.04939015319192.\]
 

Of course, the computer also gives an approximation. We usually assume that
the computer approximation is closer to the real answer than some by-hand linear
approximation. But it isn't, necessarily. The computer uses a fancy approximation
scheme for any of the functions it is programmed to compute. Many of those approximations
are based on things like linear approximation. So, the computer approximation
has the same sources of error as these methods. In addition, the computer does
not indicate how accurate the approximation is that it is using. Often the digits
displayed are not all accurate. 
\end{document}
