calculus:resources:calculus_flipped_resources:applications:mean_value_theorem_tex [Department of Mathematics and Statistics, Binghamton University]

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\begin{document}

\begin{frame}
    Verify that the function satisfies the Mean Value Theorem on the given interval. Then
     find all numbers $c$ which satisfy the conclusion of the Mean Value Theorem.
    \vskip 10pt

    \begin{itemize}
        \item[\bf a)] $f(x) = 3x^2+2x+5$ on $[-1,1]$.
        \vskip 10pt

        \item[\bf b)] $g(x) = x^3+x-1$ on $[0,2]$.
        \vskip 10pt

        \item[\bf c)] $h(x) = \dfrac{x}{x+2}$ on $[1,4]$.
        \vskip 10pt

        \item[\bf d)] $i(x) = (x-2)^{-2}$ on $[1,4]$.
    \end{itemize}
\end{frame}

\begin{frame}
    On a toll road a driver takes a time stamped toll-card from the starting booth and 
    drives directly to the end of the toll section. After paying the required toll, the 
    driver is surprised to receive a speeding ticket along with the toll receipt. Which 
    of the following describes the situation?
    \vskip 5pt

    \begin{itemize}
        \item[\bf a)] The booth attendant does not have enough information to prove that 
        the driver was speeding.
            \vskip 5pt

        \item[\bf b)] The booth attendant can prove that the driver was speeding during 
        their trip.
            \vskip 5pt

        \item[\bf c)] The driver will get a ticker for a lower speed than their actual 
        maximum speed.
    \end{itemize}
\end{frame}

\begin{frame}
    {\bf True or False}
    \vskip 5pt

    An athlete is running back and forth along a straight path. She finishes her run at
     the place where she began. There must be at least one moment, other than the end of
      the race, where she was at a complete stop.
\end{frame}

\begin{frame}
    Two runners start a race at the same moment and finish in a tie. What must be true?
    \vskip 10pt

    \begin{itemize}
        \item[\bf a)] At some point during the race the two runners were not tied.
            \vskip 5pt

        \item[\bf b)] The runners' speeds at the end of the race must have been 
        exactly the same.
            \vskip 5pt

        \item[\bf c)] The runners must have had the same speed at exactly the same
         time at some point in the race.
            \vskip 5pt

        \item[\bf d)] The runners had to have the same speed at some moment, but not
         necessarily at exactly the same time.
    \end{itemize}
\end{frame}

\begin{frame}
    Show that for all values $a$ and $b$ $$|\sin(a)-\sin(b)| \leq |a-b|$$
    \vskip 35pt

    Suppose that $3\leq f'(x) \leq 5$ for all values of $x$. Show that 
    $$18 \leq f(8)-f(2) \leq 30$$
    \vskip 35pt

    Show that the polynomial $$f(x)=1+2x+x^3+4x^5$$ has exactly one real root.
\end{frame}


\end{document}
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