p4_drugs.tex

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{tufte-handout}

\usepackage{amsmath}

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\usepackage{kmath,kerkis} % The order of the packages matters; kmath changes the default text font
\usepackage[T1]{fontenc}

\usepackage{enumitem}


% USEFUL SHORTCUTS FOR MATH
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\begin{document}
\begin{fullwidth}
    \mbox{\LARGE PreCalculus BC: Project Four - \today }\hfill
\end{fullwidth}
\section*{Introduction}
Many real-world processes happen in stages. When a population is growing,
each new generation represents a new stage of population growth. Compound
interest is another example, where interest is paid in stages and creates
a new account balance. Many things that change continuously are more easily
measured in discrete stages. Temperature, which is a continuous quantity,
can be measured with regular intervals producing a list of temperatures
such as 67, 63, 58, 55, and so on. Although the temperature is changing
continuously, we actually only care about the discrete steps from one
measurement point to the next.

Recursive sequences can be used to model this types of situations. In some
cases, we can get an explicit formula for a sequence from the recursion
relation that defines it be finding a pattern in the terms of a sequence.

\section*{Recursive Sequences as Models}
Suppose you deposit some money in an account that pays 6\% interest compounded
monthly. The bank has a definite rule for paying interest. At the end of each
month the bank adds to your account $\frac{1}{2}$\% (or 0.005) of the amount
in your account at that time. As an equation, we have:

$$
    \text{amount at end of month} = \text{amount at end of last month}
        + 0.005 (\text{amount at end of last month})
$$

We can convert this into a more expressive formula, by realizing that if we
let $A_0$ be the amount of the original deposit, $A_1$ is the amount at the end
of the first month, $A_2$ is the amount at the end of the second month, and so
on. $A_n$ represents the amount at the end of $n$ months.

$$
    A_n = 1.005A_{n-1}
$$

We recognize this as a recursively defined sequence since it gives us the amount
at each stage in terms of the amount at the preceding stage.

\section*{Daily Drug Dose - Preliminary Work not included in your report}
In this project, you will construct a recursive model for the amount of a drug
that is present in a patient's bloodstream. There are two important things
to consider. First, the amount of drug in a patient's bloodstream must be above
a certain threshold level for the drug to be effective. Second, the amount must
not exceed some maximum level beyond which the drug would do damage to the patient
in some way, such as side effects of perhaps death.

Consider first a patient who takes a 50-mg pill of a certain drug every morning.
It is known that the body eliminates 40\% of the drug every 24 hours. Find
a recursive sequence that models the amount $A_n$ of the drug in the patient's
body after each pill is taken. List out at least the first four terms of the
sequence $A_n$. Find a formula for $A_n$. How much of the drug remains in the patient's
body after 5 days? How much will accumulate in his system after prolonged use?

\section*{Prednisone}
Prednisone is often prescribed for acute asthma attacks and suppresses
the immune system. For 5 mg tablets, typical instructions are: ``Take 8
tablets the first day, 7 the second, and decrease by one tablet each day until
all tablets are gone.'' Prednisone decays exponentially in the body. This
looks like regular exponential decay that we have studied previously. However,
in a medical context we will develop the idea of biological half-life. The
biological half-life of a substance is the time it takes for the substance to
lose half of its pharmacologic activity. The Wikipedia page for biological
half-life provides some additional detail, if you're interested. For
prednisone, the biological half-life is one hour.

Let $x(t)$ represent the amount of prednisone (in mg) in the body at time $t$.

\begin{enumerate}
  \item Write formulas involving $x$, for the amount of prednisone in the body:
  \begin{enumerate}
      \item 24 hours after taking the first dose (of 8 tablets), right before taking the second dose (of 7 tablets).
      \item Immediately after taking the second dose (of 7 tablets).
      \item Immediately after taking the third dose (of 6 tablets).
      \item Immediately after taking the eighth dose (of 1 tablet).
      \item 24 hours after taking the eighth dose.
      \item $n$ days after taking the eighth dose.
  \end{enumerate}
  \item If a patient takes all the prednisone tablets as prescribed, how many
  days after taking the eighth dose is there less than 3\% of a prednisone
  tablet in the patient's body?
  \item A patient is prescribed $n$ tablets of prednisone the first day, $n-1$
  the second, and one tablet fewer each day until all the tablets are gone. Write
  a formula that represents $T_n$, the number of prednisone tablets in the body
  immediately after taking the final dose.
  \item If a patient accidently takes all the prednisone tablets at once,
  what percentage of a prednisone tablet will be present in the patient's body?
  How long will it take for there to be less than 3\% of a prednisone tablet
  in the patient's body?
\end{enumerate}

\section*{What I will be looking for in your report}

\begin{itemize}
    \item Your report includes a proper introduction that explains to the reader
    that you will be investigating how drugs are processed in the body.
    \item Your report gives a brief explanation of prednisone and how it is used.
    \item Your report describes biological half-life and its importance in
    studying how drugs are processed by the human body.
    \item You have answered and discussed questions 1, 2, 3, and 4 as they
    appear in this project description. This should not appear as simply a list
    of answers, but instead should take on a more natural paragraph form where
    you walk the reader through the development of your answers.
    \item Strictly speaking, this report does not require any graphs, but when
    explaining how the concentration of prednisone changes over time, a graph can
    be very useful. If you decide to use a graph, it must appear in a figure
    environment with appropriate caption, label, and references from the text.
    \item Properly-formatted LaTeX style, as we have discussed in class. One
    of the items I will check for here is that any variables or equations are
    wrapped in dollar signs so that they appear typeset correctly.
    \item Please remember that your report's filename should follow the convention
    of being in all lower-case, no spaces, and should contain your last name
    followed by \verb|_drug.tex|.
\end{itemize}

\end{document}