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\centerline{MATH 133 -- Calculus with Fundamentals 1}
\centerline{The MVT and Consequences}
\centerline{November 21, 2017}
\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.  Recall, saying $f(c)$ is a \emph{local maximum}
value if $f(c) \ge f(x)$ for all $x$ in some interval $(a,b)$ containing $c$.  Similarly,
$f(c)$ is a \emph{local minimum} value if $f(c) \le f(x)$ for all $x$ in some interval $(a,b)$
containing $c$.

\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}
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\noindent
{\it Questions}
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\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 $[-1,3]$.  That is, find all the $c\in (-1,3)$
for which $f'(c) = \frac{f(3) - f(-1)}{3 - (-1)}$.  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.]  Joe Guy 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 11:35am.  The distance between the two 
exits is 42 miles.  Two weeks later, Joe 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.  (The maximum posted speed is 65 miles per hour the whole way.)
\item[3.]  Suppose that the graph in Figure 1 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}
\begin{figure}
\begin{center}
\includegraphics[width=.6\linewidth]{cubic.png}
\caption{Plot for questions 1 and 3}
\end{center}
\end{figure}
\end{document}
