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\begin{frame}
\frametitle{Finding a Tangent}

Alternative definition of the slope:
\begin{block}{}
The slope of $f$ at point $(a,f(a))$ is:
\begin{talign}
m = \lim_{h\to 0} \frac{f(a+h) - f(a)}{h}
\end{talign}
\end{block}
\pause
\begin{block}{}
The slope $m$ is also called the \emph{slope of the curve} at the point.
\end{block}
\pause\medskip

\begin{exampleblock}{}
Find an equation of the tangent to $f(x) = \frac{3}{x}$ at point $(3,1)$.
\pause\medskip

The slope is:
\begin{talign}
m &= \lim_{h\to 0} \frac{f(3+h) - f(3)}{h}
\mpause[1]{= \lim_{h\to 0} \frac{\frac{3}{3+h} - 1}{h}}
\mpause[2]{= \lim_{h\to 0} \frac{\frac{3 - (3+h)}{3+h}}{h}}\\
&\mpause[3]{= \lim_{h\to 0} \frac{-h}{h(3+h)}}
\mpause[4]{= \lim_{h\to 0} -\frac{1}{3+h}}
\mpause[5]{= -\frac{1}{3}}
\end{talign}
\pause[10]
Thus $y - 1 = -\frac{1}{3} (x-3)$\pause, that is, the tangent is $y = 2 - \frac{x}{3}$.
\end{exampleblock}
\end{frame}