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\title{Applications of the definite integral}
\date{November 20, 2023}

\begin{document}
\ \vspace{-.25in}

\maketitle

\vspace{-.15in}

\prob{
  The rate of memorizing information increases over time to a maximum, then decreases.
  Suppose this information memorization rate is modeled by $\displaystyle m(t) = -0.009t^2 + 0.18t$ words per minute (time $t$ is measured in minutes).
  \begin{enumerate}[a)]
  \item When is the memorization rate at a maximum?
  \item Make an educated guess (use your answer for part a) about whether more words are memorized from time $t_1=0$ to time $t_2=10$ or from time $t_1=5$ to time $t_2=15$.
  \item The number of words memorized from time $t_1$ to time $t_2$ is $\int_{t_1}^{t_2} m(t)\ dt$. Use this to find exact answers for the number of words memorized over the two intervals in the previous part. Were you right?
  \end{enumerate}
  }

\vfill

\centering{{\color{gray} \small{(continued on the reverse)}}}

\newpage

\prob{
For this problem, use the fact that $a(t) = v' (t) = s''(t)$.
For both parts, assume constant acceleration.
\begin{enumerate}[a)]
\item A car accelerates from 0 to 100 kph in 10 seconds. How far did it travel over this time?  
\item A car decelerates from 100 kph to a stop in 4 seconds. How far did it travel over this time?
\end{enumerate}
}

\vfill

\prob{
After WWII ended in 1945, the birth rate in the US increased dramatically (the baby boom).
Suppose the birth rate $b(t)$ (in millions of births per year) for 1945 to 1965 was \[ b(t) = 4 + 0.025t \]
where $t$ is the number of years after 1945.
\begin{enumerate}[a)]
\item According to this model, how many babies were born from 1945 to 1965?
\item The average value of $y=f(x)$ for $a \leq x \leq b$ is $\frac{1}{b-a} \int_a^b f(x)\ dx$. 
Calculate the average number of births per year over the period 1945 to 1965.
\item Find the accumulation function $B(t)$ for the total number of births in the $t$ years since 1945.
\item Starting after the war, how long did it take for 20 million births?  
\end{enumerate}
}


%\prob{
%  Solve the initial value problems.
%  It may help to know that there are 5280 feet per mile.
%  \begin{enumerate}[a)]
%  \item A car accelerates at a constant rate from 0 mph to 60 mph in 30 sec.
%    How far did it travel?
%  \item A car decelerates at a constant rate from 60 mph to 0 mph in 5 sec.
%    How far did it travel?
%  \end{enumerate}
%  }

\end{document}
