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\title{Applied Calculus}
\author{Homework 2}
\date{Due in class, September 22, 2015}
\maketitle

\begin{enumerate}

\item The radioactive gas Radon-222 has a half life of $3.8$ days.
Initially, there are $2 \cdot 10^9$ atoms in a sample.  How many atoms
remain after $10$ days?

\item (Partially from MLS, p.~277--278.) Compute the following limits,
if they exist.
\begin{enumerate}
\item $\displaystyle \lim_{x\to1} \frac{x^2-1}{x-1}$
\item $\displaystyle \lim_{x\to-3} \frac{x^2-x+12}{x+3}$
\item $\displaystyle \lim_{x\to-3} \frac{x^2-x-12}{x+3}$
\item $\displaystyle \lim_{h\to0} \frac{(3+h)^2-9}h$
\item $\displaystyle \lim_{t\to0} \frac{\sqrt{2-t}-\sqrt2}t$
\item $\displaystyle \lim_{x\to1} \biggl( \frac2{1-x^2} +
\frac1{x-1} \biggr)$
\item $\displaystyle \lim_{s\to0} \biggl( \frac1{s\sqrt{s+1}} - \frac1s
\biggr)$
\item $\displaystyle \lim_{x\to\infty} \frac{3x-2}{x+1}$
\item $\displaystyle \lim_{x\to\infty} \frac{x+\ln x}{x}$
\item $\displaystyle \lim_{s\to-\infty} s^{100} \, \e^s$

\end{enumerate}
\hfill(2 Points each)


\item (Partially from MLS, p.~295.) Determine where $f(x)$ is
continuous.  If $f(x)$ has a discontinuity, state the type of
discontinuity (removable discontinuity, jump discontinuity, vertical
asymptote, or other).
\begin{enumerate}
\item $\displaystyle f(x) = x^2-x-6$
\item $\displaystyle f(x) = \frac{x^2-x-6}{x+2}$
\item $\displaystyle f(x) = \frac{3^x}x$
\item $\displaystyle f(x) =
\begin{cases}
  x & \text{for } x \leq 1 \\
  x+2 & \text{for } x > 1 
\end{cases}$
\item $\displaystyle f(x) =
\begin{cases}
  \frac{x^2 -1}{x-1} & \text{for } x < 1 \\
  2 x^2 & \text{for } x \geq 1 
\end{cases}$
\end{enumerate}
\hfill(2 Points each)

\item (From MLS, p.~296.) To be continuous or not: that is the
question.

\begin{enumerate}
\item When a patient receives an injection, the amount of the injected
substance in the body immediately goes up.   Comment on whether the
function that describes the amount of the drug with respect to time is
a continuous function.  Justify your reasoning.

\item When a person takes a pill, the amount of the drug contained in
the pill is immediately inside the body.  Comment on whether the
function that describes the amount of the drug in the blood stream
with respect to time is a continuous function.
\end{enumerate}


\item \emph{Note:} The following two problems problem require the use
of \emph{Scientific Python}.  An Ipython noteboook from the class
session September 3 is available on my web page, it already contains
the solution for part (a) below.

(From MLS, p.~45.) Use the data from the example shown in class.
\begin{enumerate}
\item Find the equation for the least-squares fit of the $y$ data in
terms of the $x$ data.

\item Find the equation for the least-squares fit of the $x$ data in
terms of the $y$ data.

\item Solve the equation in (b) for $y$ in terms of $x$.  Is this the
same equation that you obtained in (a)?  Explain why you think the
answers were the same or different.

\item Compute the $\rho^2$ value in each case and interpret the
results. 
\end{enumerate}


\item (MLS, Problem~4.14) See attached scan with data.

\end{enumerate}




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