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\title{Models using Differential Equations}
\date{February 1, 2021}

\begin{document}
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\vspace{-.2in}

\maketitle

\vspace{-.2in}

\prob{[Graded]
We saw that the exponential model for population growth $P' = aP$ predicts unbounded population sizes.
The logistic model resolves this problem: $P' = aP(1- b P)$ where $a$ and $b$ are positive constants.
If it helps you to solve the problem, you may use $a = 1$ and $b=0.5$.
\begin{enumerate}[a)]
\item Draw a phase line and use it to predict the long-term population trends.

\vspace{1.5in}

\item Use separation of variables to find a solution $P(t)$.
(I recommend using partial fractions in the integral; you'll also need to use laws of logarithms to solve for $P$).

\vfill

\item Calculate $\lim_{t \to \infty}P(t)$ and compare with your answer for part a.

\vspace{1.5in}

\end{enumerate}
}

\newpage

\prob{[Completion]
Newton's law of cooling states that the rate of change of an object's temperature is proportional to the difference between its temperature and the temperature of its environment.
As a differential equation: $T' = -k(T - T_m)$ where $k$ is a positive constant of proportionality and $T_m$ is the (constant) temperature of the environment.
\begin{enumerate}[a)]
  \item What is the eventual temperature $T$ of the object? Use your understanding of cooling (or heating) to answer (not math).

\item Draw a phase line for the model and verify that it agrees with your answer for part a.

%\vspace{1in}

\item Find the general solution to the differential equation.

\vfill

\end{enumerate}
}

\prob{[Completion]
A  cup of boiling water ($212^{\circ}$ F) is placed outside.
One minute later the temperature of the water is $152^{\circ}$ F.
After another minute the temperature is $112^{\circ}$ F.
What is the outside temperature?

\vfill

}

% \prob{[CSI Gonzaga]
%   A body is found outside on a $5^{\circ}$C day.
%   At 12:15 its temperature is $35^{\circ}$C and at 12:45 its temperature is $33^{\circ}$C.

%   \begin{enumerate}[a)]
%   \item Use the two points to find a formula for $T(t)$, the temperature of the body $t$ hours after 12:15.
    
%     \item Human body temperature is $37^{\circ}$C. Use this to find the time the body started cooling.

%     \vfill

%     \end{enumerate}
% }


\newpage

\prob{[Graded]
An object with a mass of $m = 10$ kg is launched upward with initial velocity of $v_0= 60\ \text{m}/\text{s}$.
The atmosphere resists the object's motion with a force of $5\ \text{Ns/m}$ (that's 5 Newtons for each m/s of speed).
The only other force acting on the object is gravity (the acceleration of which is $g = 9.8\ \text{m}/\text{s}^2$ downward).
This means that the total force on the object is $F = -mg - 5v$.
Newton tells us that $F = ma = mv'$.
We now have a differential equation (in the variable $v$): $\displaystyle \boxed{ mv'= -mg - 5v}$

\begin{enumerate}[a)]
\item  Find the terminal velocity of the object (the equilibrium solution to the equation).
\item Find a formula for the velocity of the object.
\end{enumerate}
}

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