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\begin{frame}
\frametitle{Linear Approximation and Differentials}

\begin{exampleblock}{}
What is the linear approximation of $f(x)= \cos x$ at $0$?\\
Use it to approximate $\cos 0.01$.
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We have:
\begin{talign}
f(0) &= \cos 0 = \mpause[1]{1} \\
\mpause[2]{f'(x) }&\mpause[2]{= }\mpause[3]{-\sin x} & \mpause[4]{f'(0) = }\mpause[5]{0}
\end{talign}
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Thus the linear approximation of $\cos x$ at $0$ is:
\begin{talign}
L(x) = \mpause[1]{1 + 0 (x-0)} \mpause[2]{= 1}
\end{talign}
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We use this to approximate $\cos 0.01$:
\begin{talign}
\cos 0.01 \approx \mpause[1]{L(0.01) = 1}
\end{talign}
\end{exampleblock}
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Approximations for $\sin$ and $\cos$ are often applied in physics
(e.g. optics).
\end{frame}