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\begin{document}

\begin{frame}
    Find the linearization of each function:
    \vskip 5pt

        \begin{itemize}
            \item[\bf a)] $h(x) = x^4-3x^2-1$ at $a=-1$.
            \vskip 20pt

            \item[\bf b)] $f(x)=\sin^2(x)$ at $a=\frac{\pi}{2}$.
            \vskip 20pt

            \item[\bf c)] $g(x) = \dfrac{1}{(1+3x)^4}$ at $a=0$.
            \vskip 20pt

            \item[\bf d)] $r(t) = t^{\frac{3}{4}}$ at $a=16$.
        \end{itemize}
\end{frame}

\begin{frame}
    \large Use a linear approximation to estimate the value of $\sqrt[3]{9}$.
    \vskip 30pt

    Use a linear approximation to estimate the value of $\tan(44^o)$.
\end{frame}

\begin{frame}
  \large The line tangent to the graph of $f(x)=\sin(x)$ at the point $(0,0)$ is 
  $y=x$. This implies that
    \vskip 10pt

    \begin{enumerate}[a)]
        \item $\sin(0.0005) \approx 0.0005$
        \vskip 10pt

        \item The line $y=x$ touches the graph of $f(x)=\sin(x)$ at exactly one 
        point, $(0,0)$.
        \vskip 10pt

        \item $y=x$ is the best straight line approximation to the graph of $f$ 
        for all $x$.
    \end{enumerate}
\end{frame}

\begin{frame}
  \large Peeling an orange changes its volume V. What does $\Delta V$ represent?
  \vskip 10pt

    \begin{enumerate}[a)]
        \item the volume of the rind.
        \vskip 10pt

        \item the surface area of the orange.
        \vskip 10pt

        \item the volume of the "edible part" of the orange.
        \vskip 10pt

        \item $-1\times $(the volume of the rind).
    \end{enumerate}
\end{frame}

\begin{frame}
  \large Imagine that you increase the dimensions of a square with side $x_1$ to a 
  square with side length $x_2$. The change in the area of the square, $\Delta A$, 
  is approximated by the differential $dA$. Find $dA$:
  \vskip 10pt

    \begin{enumerate}[a)]
        \item $2x_1(x_2-x_1)$
        \vskip 10pt

        \item $2x_2(x_2-x_1)$
        \vskip 10pt

        \item $x_1^2-x_2^2$
        \vskip 10pt

        \item $(x_2-x_1)^2$
    \end{enumerate}
\end{frame}

\begin{frame}
  \large Imagine that you increase the dimensions of a square with side $x_1$ to a 
  square with side length $x_2$. The change in the area of the square, $\Delta A$, 
  is approximated by the differential $$dA=2x_1(x_2-x_1)$$ This approximation will 
  result in an
  \vskip 5pt

    \begin{enumerate}[a)]
        \item overestimate
        \vskip 10pt

        \item underestimate
        \vskip 10pt

        \item exactly equal
    \end{enumerate}
\end{frame}

\begin{frame}
    Find the differential of each function:
    \begin{columns}
    \begin{column}{0.5\textwidth}
        \begin{itemize}
            \item[\bf a)] $y=\sqrt{1+x^2}$
            \vskip 20pt

            \item[\bf b)] $y=x^2\sin(x)$
        \end{itemize}
    \end{column}

    \begin{column}{0.5\textwidth}
        \begin{itemize}
            \item[\bf c)] $y=\sec\left(\sqrt{7x}\right)$
            \vskip 20pt

            \item[\bf d)] $y=\dfrac{3-t^2}{3+t^2}$
        \end{itemize}
    \end{column}
    \end{columns}
\end{frame}

\begin{frame}
    \large The radius of a sphere is measured to be $84$ inches with a possible 
    error of $0.5$ inches.
        \begin{itemize}
            \item[\bf a)] Use differentials to estimate the maximum error in the 
            calculated surface area. What is the relative error?
            \vskip 20pt

            \item[\bf b)] Use differentials to estimate the maximum error in the 
            calculated volume. What is the relative error?
        \end{itemize}
\end{frame}

\begin{frame}
    \large Use differentials to estimate the amount of paint needed to apply a 
    coat of paint $0.1$ cm thick to hemispherical dome with diameter $50$ meters.
\end{frame}

\begin{frame}
    \large A window has the shape of a square surmounted by a semicircle.
    \vskip 15pt

    The base of the window is measured as having width $50$ inches with a possible 
    error in measurement of $0.1$ inches.
    \vskip 15pt

    Use differentials to estimate the maximum error possible in computing the area 
    of the window. What is the maximum relative error?
\end{frame}


\end{document} 
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