\DefineExercise{E3-4_WarmUp1}{
Let $s(t)$ be the speed of a train in MPH at time $t$ 
seconds after it leaves $30^{th}$ Street Station.  What is the
interpretation of $\frac{ds}{dt}$ and what are its units?
}

\DefineExercise{E3-4_WarmUp2}{
Water is leaking out of a tank at a variable rate, leaking
at rate $f(t)$ at time $t$.
\begin{enumerate}[label=(\alph*)]
\item What are reasonable units for $f(t)$ and $t$?
\item Suppose that $g'(t) = f(t)$.  What is an interpretation for 
$g(t)$ and what are its units?
\end{enumerate}
}

\DefineExercise{E3-5_WarmUp1}{
Find $\frac{d}{dz}\frac{\tan z+e^z}{\sin z}$.
}

\DefineExercise{E3-4_Main1}{
In a simple economic supply and demand model for the price of gasoline, there is a {\bf demand 
function} $g$ such that setting the price of gas at $x$ per gallon
results in total consumption of gas equal to $g(x)$ gallons.  
What is an interpretation of the quantity $\frac{d}{dx} (x \cdot g(x))$?  (This doesn't have a pithy name; just describe it in words.)
}

\DefineExercise{E3-4_Main2}{
At constant temperature, the ideal gas law says that $PV=c$ where $c$ is a constant, $P$ is the pressure of the gas, and $V$ is the volume taken up by the gas.  (Pressure is measured in $kg/m\cdot s^2$, volume in $m^3$, and the constant has units $kg\cdot m^2/s^2$.)

  \begin{enumerate}[label=(\alph*)]
  \item Find $\frac{dP}{dV}$ (including units)
  \item Find $\frac{dV}{dP}$ (including units)
  \end{enumerate}
}

\DefineExercise{E3-4_Main3}{
Write an equation representing the following Malthusian
population law.  

\begin{quote}
``The rate of population growth is always proportional to the amount of population.''
\end{quote}

\texttt{
  \begin{enumerate}[label=(\roman*)]
  \item Start by picking a variable to represent the population at time $t$ (that is, as a function of $t$)
  \item Figure out how to express ``the rate of population growth''
  \item Figure out how to express ``the amount of population''
  \item Write down the equation connecting the two.  (You may have forgotten the definition of ``proportional''.  It's way back on page 7!)
  \end{enumerate}
}
}

\DefineExercise{E3-4_Main4}{
\begin{quote}
``The downward acceleration of a falling object is equal to the gravitational
constant minus an air resitance proportional to the speed of the object.''
\end{quote}

\begin{enumerate}[label=(\alph*)]
\item Write an equation for this modified gravitational law, giving 
units and an interpretation of all functions, variables and constants. 
\item In the notation just established, write an equation stating that
``The falling object is not accelerating.''
\item The speed of the object when it is no longer accelerating is the
{\bf terminal velocity}.  What is the terminal velocity in terms of 
the constants you have defined?
\end{enumerate}

}

\DefineExercise{E3-5_Main1}{
What is $\frac{d^{191}}{dx^{191}}\sin x$?
}

\DefineExercise{E3-5_Main2}{
\begin{enumerate}[label=(\alph*)]
\item Write down a trigonometric identity relating $\sin^2 x$ to $\cos^2x$.
\item Take the derivatives of both sides of this identity to obtain an identity involving $\frac{d}{dx}\sin^2 x$ and $\frac{d}{dx}\cos^2x$.  What do you learn about the relationship between $\frac{d}{dx}\sin^2x$ and $\frac{d}{dx}\cos^2x$?

\texttt{Leave $\frac{d}{dx}\sin^2 x$ and $\frac{d}{dx}\cos^2 x$ alone---we want an equation that includes those two derivatives.}
\item Calculate the derivatives of $\sin^2x$ and $\cos^2x$ and verify that the identity you discovered above holds.
\end{enumerate}
}

\DefineExercise{E3-5_Main3}{
The limit
\[\lim_{z \to \pi / 6} \frac{\sec (z) - \sec (\pi/6)}{z - \pi/6} \, \]
is equal to the derivative of some function at some point.    Use this definition and the formula for the corresponding derivative to calculate this limit.
}

\DefineExercise{E3-6_WarmUp1}{
 The chain rule is used to find the derivative of functions which are computed in multiple steps.  For example, the function $e^{\sin x}$ is computed by first finding the intermediate value $u=\sin x$, and then calculating $e^u$.  Break each of the following functions into a composition of two separate simpler functions.  (There's no need to take the derivative.)
  \begin{enumerate}[label=(\roman*)]
  \item $e^{\tan x}$
  \item $\sin e^x$
  \item $\tan^{-1}(1+x^3)$
  \item $\ln\cos^{-1}\sqrt{x}$
  \item $\ln\tan e^{x^2}$
  \end{enumerate}
}

\DefineExercise{E3-6_WarmUp2}{
  \begin{enumerate}[label=(\roman*)]
  \item Find $\frac{d}{dx}e^{\tan x}$.
  \item Find $\frac{d}{dx}\sin e^x$.
\end{enumerate}
}

\DefineExercise{E3-6_Main1}{
  \begin{enumerate}[label=(\alph*)]
\item What is $\frac{d}{dx}\sin e^{x^2}$?\\
\texttt{
Anything written in this font is only there to provide hints to get through a problem.  You're always free to ignore them if you don't find them necessary.  i, ii, and iii aren't separate questions, they're just steps you might want to take while solving a.
  \begin{enumerate}[label=(\roman*)]
  \item First write this as a composition of two functions: $\sin e^{x^2}=f(g(x))$
  \item Find $f'(u)$ and $g'(x)$
  \item Use the chain rule to find the answer.
  \end{enumerate}
}
\item What is the derivative with respect to $x$ of $e^{\sqrt{1 + x^2}}$?
  \end{enumerate}
}

\DefineExercise{E3-6_Main2}{
  \begin{enumerate}[label=(\alph*)]
  \item What is $\frac{d}{dx}\ln \sin x$?
  \item What is $\frac{d}{dx}\sqrt{\ln \sin x}$?
  \item What is $\frac{d}{dx}\tan\sqrt{\ln \sin x}$?
  \item What is $\frac{d}{dx}e^{\tan\sqrt{\ln \sin x}}$?
  \end{enumerate}
}

\DefineExercise{E3-6_Main3}{
A car is traveling on a winding road at a varying speed.  Let $g(s)$ represent the distance, in miles, that the object has traveled after $s$ 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 $h$.
\item Take the derivative to find an equation relating $g'$ and $h'$.  
\end{enumerate}
}

% \DefineExercise{E3-6_Main4}{
% You are in a room with a heater in one end and an open window at the other.  This causes a temperature gradient: when you are $d$ feet from the heater, the temperature is $F(d)=75-d^2$ degrees Fahrenheit.  You roll a thermometer across the room; it gets slowed by the rug, so its position after $t$ seconds is $d(t)=\ln(t+1)$ feet.

% Include units on every part.

%   \begin{enumerate}
%   \item What is the temperature as a function of time?
%   \item What is the derivative of temperature with respect to distance from the heater?
%   \item What is the derivative of distance from the heater with respect to time?
%   \item What is the derivative of temperature with respect to time?
%   \end{enumerate}

% }

\DefineExercise{E3-6_Main4}{
A lamp is on the ground 4ft away from a wall and angled upwards at angle $\theta$ as in this picture
  \begin{tikzpicture}[scale=0.5]
    \draw (0,0)--(0,-4)--(-3,-4);
\draw[dashed] (-3,-4)--(0,0);
\draw [<->,dashed] (-1.6,-4)  arc (0:45:1.6);
\node at (-2.1,-3.5) {\small$\theta$};
\draw[<->,dashed] (-3,-4.2)--(0,-4.2);
\node at (-1.5,-4.5) {\small 4ft};
\draw[<->,dashed] (0.2,0)--(0.2,.-4);
\node at (1.0,-2) {\small$h(\theta)$};
  \end{tikzpicture}
It casts a beam on the wall at height $h(\theta)$.

\begin{enumerate}[label=(\alph*)]
\item What is $h(\theta)$ as a function of $\theta$?
\item Give the units of $\frac{dh}{d\theta}$ and an interpretation.
\item What is $\frac{dh}{d\theta}$?
\item What is $\theta(d)$, the angle in radians as a function of the angle in degrees?
\item What is $\frac{d}{dd}h(\theta(d))$?
\item When the angle is at $\pi/4$ radians (equivalently, $45$ degrees), is $\frac{d}{dd}h(\theta(d))$ the same number as $\frac{d}{d\theta}h(\theta)$?  If not, which is bigger?
\end{enumerate}
}

\DefineExercise{E3-6_Main5}{
If $f, g$ and $h$ are differentiable functions, compute the
following quantities (your answer might include the functions $f, g, h$ and $f', g', h'$, as well as other functions).
\begin{enumerate}[label=(\alph*)]
\item $[f(x)\cdot\sin g(x)]'$

\texttt{Make sure to test your answer: try an example where you know $f, g, f', g'$ and where the derivative of $f(x)\cdot \sin g(x)$ is easy to find, and make sure your formula gives the same answer you get by hand.}
\item $[f(g(h(x)))]'$
\item $[\ln (f(x)\cdot g(x))]'$
\end{enumerate}
}

\DefineExercise{E3-6_Main6}{
What is the derivative of $3^x$?
}

\DefineExercise{E3-6_Main7}{
There is a function called $\mathrm{erf}$ with the property that $\frac{d}{dx}\mathrm{erf}(x)=\frac{2}{\sqrt{\pi}}e^{-x^2}$.  What is $\frac{d}{dz}\mathrm{erf}(\sin z)$?
}

\DefineExercise{E3-6_Main8}{
You know the following values of $f, g, f', g'$:
\begin{tabular}{l|l|l|l|l}
&1&2&3&4\\
\hline
  f(x)&1&2&3&4\\
g(x)&2&1&3&4\\
f'(x)&4&3&2&1\\
g'(x)&3&2&4&1\\
\end{tabular}

What is $[f\circ g]'(2)$?
}