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\begin{frame}
  \frametitle{Linear Approximation and Differentials}
  
  \begin{exampleblock}{}
    What is the linear approximation of $f(x)= \sin x$ at $0$?\\
    Use it to approximate $\sin 0.01$. 
    \pause\bigskip
    
    We have:
    \begin{talign}
      f(0) &= \sin 0 = \mpause[1]{0} \\
      \mpause[2]{f'(x) }&\mpause[2]{= }\mpause[3]{\cos x} & \mpause[4]{f'(0) = }\mpause[5]{1}
    \end{talign}
    \pause\pause\pause\pause\pause\pause
    Thus the linear approximation of $\sin x$ at $0$ is:
    \begin{talign}
      L(x) = \mpause[1]{0 + 1 (x-0)} \mpause[2]{= x}
    \end{talign}
    \pause\pause\pause
    We use this to approximate $\sin 0.01$:
    \begin{talign}
      \sin 0.01 \approx \mpause[1]{L(0.01) = 0.01}
    \end{talign}
  \end{exampleblock}
\end{frame}