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\centerline{MATH 135 -- Calculus 1}
\centerline{The MVT and Consequences}
\centerline{November 13, 2019}
\vskip 10pt
\noindent
{\it Background}
\vskip 10pt
Recall from today's video, recall that we now know a statement called the
Mean Value Theorem (MVT):
\begin{theorem} (MVT)  Let $f(x)$ be continuous on $[a,b]$ and differentiable on 
$(a,b)$.  Then there exists $c \in (a,b)$ such that 
$$f'(c) = \frac{f(b) - f(a)}{b - a}.$$
\end{theorem}
The MVT lets us show in a ``rigorous'' (i.e. complete and convincing) way that
some of the patterns we have discussed intuitively actually always hold.  For instance
\begin{itemize}
\item If $f'(x) > 0$ on an interval $(a,b)$, then $f$ is increasing on that interval.
\item If $f'(x) < 0$ on an interval $(a,b)$, then $f$ is decreasing on that interval.
\item If $f'(x) = 0$ on an interval $(a,b)$, then $f$ is constant on that interval.
\end{itemize}
In addition, we get the following way to determine whether a critical point $c$
of $f$ is a local maximum or local minimum:
\begin{theorem} (First Derivative Test)  Let $c$ be a critical point of a function 
$f$.  
\begin{itemize} 
\item If $f'$ changes sign from positive to negative at $c$, then 
$f(c)$ is a local maximum of $f$.
\item If $f'$ changes sign from negative to positive at $c$, then 
$f(c)$ is a local minimum of $f$.
\item If $f'$ does not change sign at $c$, then $f(c)$ is neither a local 
maximum nor a local minimum.
\end{itemize}
\end{theorem}
\vskip 10pt
\noindent
{\it Questions}
\vskip 10pt
\noindent
\begin{enumerate}
\item[1.]  Show that the conclusion of the Mean Value Theorem is true
for $f(x) = x^3 - 12x^2 + 21x$ on the interval $[-1,10]$.  That is, find all the $c\in (-1,10)$
for which $f'(c) = \frac{f(10) - f(-1)}{10 - (-1)}$.  There is a plot of the graph $y = f(x)$
on the back of this page.  Draw in lines illustrating the conclusion of the Mean Value Theorem.
\item[2.]  Slim Shady drove through the EZPass lane and got on the Mass Pike at Allston 
at 11:00am one morning.  He drove west to Auburn and exited through
the EZPass lane there at 11:37am.  The distance between the two 
exits is 42 miles.  Two weeks later, Slim received a speeding ticket from the Mass State
Police for \$150 in the mail.  Should he try to fight this in court, or is the ticket justified? 
Explain.  (Note:  The maximum posted speed is 65 miles per hour the whole way.)
\item[3.]  Now suppose that the graph on the back is $y = g'(x)$ for some other function $g(x)$.
What are the critical points of $g(x)$?  Classify each of them as a local maximum or local
minimum using the First Derivative Test above.  
\end{enumerate}
\vfill\eject
\begin{figure}
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\caption{Plot for questions 1 and 3}
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\end{document}
