\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 $\cos 0.01$. 
    \pause\bigskip
    
    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}
    \pause\pause\pause\pause\pause\pause
    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}
    \pause\pause\pause
    We use this to approximate $\cos 0.01$:
    \begin{talign}
      \cos 0.01 \approx \mpause[1]{L(0.01) = 1}
    \end{talign}
  \end{exampleblock}
  \pause\pause\bigskip
  
  Approximations for $\sin$ and $\cos$ are often applied in physics
  (e.g. optics).
\end{frame}