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

  \begin{exampleblock}{Final Exam 2003 (Spring)}
    Use differentials or the linearization approximation method to approximate $\ln(0.9)$.
    \pause\bigskip
    
    We have $f(x) = \ln x$.\pause\bigskip
    
    We need to choose where to compute the linearization:
    $a = \pause 1$.\pause
    \begin{talign}
      f(1) &= \mpause[1]{0} \\
      \mpause[2]{f'(x) }&\mpause[2]{= }\mpause[3]{\frac{1}{x}} & 
      \mpause[4]{f'(1) = 1}
    \end{talign}
    \pause\pause\pause\pause\pause
    The linearization of $f$ at $1$ is:\vspace{-.7ex}
    \begin{talign}
      L(x) = \mpause[1]{0 + 1(x-1)} \mpause[2]{= x-1}
    \end{talign}
    \pause\pause\pause
    Then the approximation of $\ln(0.9)$ is:\vspace{-.7ex}
    \begin{talign}
      \ln(0.9) \approx
      \mpause[1]{L(0.9)}
      &\mpause[2]{= 0.9 - 1}
      \mpause[3]{= -0.1}
    \end{talign}
  \end{exampleblock}  
\end{frame}