\input{ThisWeek.tex}
\newcommand{\classday}{\Tuesday}
\input{../TemplateWeek/InClassWork.tex}
\input{Exercises.tex}

%\booltrue{teacher}

%The first problem took more than half the class and required huge amounts of guidance.  The extensive breaking into steps hasn't helped nearly as much as I'd hoped, partially because students have trouble even with the smaller steps, and partially because even when they follow the suggested steps, they keep the information so disorganized that they can't put it together.
%It might help to use $s$ for seconds and $t$ for hours, or something like that, to help keep them separate.  Also to be more explicit about the fact that in the early steps, they're calculating $h(1)$, $h(2)$, and so on.

%Also, they really don't seem to grasp the distinction between problems and guidance to the problems.  That needs to be reiterated more.


%This worksheet also got adjusted to deal with overflow from other days and the exam

% \section*{Warm Up}

% \begin{enumerate}[label=(\alph*)]
% \item \Exercise{E3-2_WarmUp1}
% \item \Exercise{E3-2_WarmUp2}
% \end{enumerate}

% \pagebreak

\section*{Exercise 1}
\textit{A few groups started on this problem last week, but none of them got it right.  I've tweaked the instructions a bit, so even if you tried this before, start it over with a fresh eye.}

A car is traveling on a widing road at varying speed.  Let $g(t)$ represent the distance, in miles, that the object has traveled after $t$ seconds.  Let $h(t)$ represent the distance in miles the same object has traveled after $t$ hours.
\begin{enumerate}[label=(\alph*)]
\item Write an equation relating the functions $g$ and $h$.

\texttt{
This is a question we have plenty of intuition for, but it's hard to bring that intuition to bear on an abstract function like $g(t)$.  A useful technique is to first think about a concrete example.  We'll make up an example function which could be $g(t)$ and think about that example before trying to think about the general case.  We have to pick a concrete function to stand in for $g(t)$; if it's too complicated, it won't be any better than an unknown function, but if it's too simple, it won't be representative.  Picking good examples is an art that takes practice; in this case, I happen to know that a linear function will be too simple, but that anything more complicated will work.}

\texttt{
  \begin{enumerate}[label=(\roman*)]
  \item Make up an example function $g$.  It shouldn't be linear, but shouldn't be too much more complicated.  Some good options are a quadratic (if you want to think about the car as travelling in one direction and accelerating) or sin/cos (if you want to think about the car going back and forth).
  \item Use the function $g$ you chose to calculate how far the object travels in one hour
  \item Now use the function $g$ to calculate how far the object travels in two hours
  \item And one and a half hours...
  \item Try to generalize to find the relationship between $g$ and $h$
  \item Now try to come up with the general relationship between $g$ and $h$ which works no matter what $g$ is.
  \end{enumerate}
}
\item Write an equation relating $g'$ and $h'$.  

\texttt{
Again, a good way to start is by picking a concrete example which could be $g'$.  It doesn't have to be the derivative of $g$ you used above---this is a \emph{different} example.  This time it turns out that even a linear function is complicated enough.  When figuring out the relationship between $g'$ and $h'$, it may help to identify what the units of $g'$ and $h'$ are.
}
\item This equation should allow you to compute $h'(1)$ if you know 
some values of $g'(x)$.  Which values of $x$ do you need to know $g'(x)$ for in order
to compute $h'(1)$?  
\end{enumerate}

\section*{Exercise 2}
 \Exercise{E3-2_Main1}

\section*{Exercise 3}
\Exercise{E3-2_Main2}

\section*{Exercise 4}
\Exercise{E3-2_Main3}

% \section*{Exercise 4}
% \Exercise{E3-2_Main4}
\pagebreak
\section*{Exercise 5}
\Exercise{E3-2_Main5}

\section*{Exercise 6}
\Exercise{E3-2_Main6}
\end{document}