Jim Belk Cornell University

Takehome Midterm

Due Date: Sunday, October 23 at 11:59 pm

Instructions: Your solutions must be typed, and should be e-mailed to me sometime before the due date. In addition to your solutions, please e-mail me any Excel or Mathematica notebooks that you use for solving the problems.

Rules: This is a takehome exam, not a homework assignment. You must solve the problems entirely on your own, and you should not discuss the problems with any other students in the class, or with anyone on the internet. You should feel free to use Excel or Mathematica as much as you like, and you may consult internet resources such as Wikipedia articles, online Mathematica documentation, or Wolfram alpha, but you cannot post to any forums or solicit help from anyone on the internet.

  1. A hospital patient was administered 50 mg of a certain drug intravenously. The following table shows the measured concentration of the drug in the patient's bloodstream over the following twelve hours.\(\require{unicode}\) \[ \begin{array}{rccc} \hline \text{time after dosage (hours)} & 2 & 4 & 6 & 8 & 10 & 12 \\ \hline \text{concentration (}\unicode{x03BC}\text{g/mL)} & 7.0 & 4.8 & 3.2 & 2.2 & 1.5 & 1.0 \\ \hline \end{array} \]
    1. Find an approximate formula for the concentration of the drug in the patient's bloodstream after \(t\) hours. Try to be as accurate as possible, and describe the evidence you have that your model is correct.
    2. Estimate the total volume of the patient's blood as accurately as possible. Explain how you arrived at your answer.
  2. The next time the same patient from problem #1 visits the hospital, she is given an oral tablet that contains 100 mg of the same drug. The tablet is roughly spherical in shape, and is designed to dissolve slowly in the patient's stomach over the course of four hours.
    1. Assuming the radius of the tablet decreases linearly with time, find an approximate formula for the rate \(R(t)\) at which the drug is absorbed into the bloodstream. Explain the reasoning behind your answer.
    2. Write a differential equation for the concentration \(C(t)\) of the drug in the patient's bloodstream during the time that the tablet is dissolving. Explain the reasoning behind each term in your answer.
    3. Solve your differential equation from part (b) to find a formula for the concentration \(C(t)\).
    4. Based on your model, when will the concentration of the drug in the patient's bloodstream attain its maximum value? What will this maximum concentration be?
  3. It is possible to use Euler's method to approximate the solution to a system of differential equations. Given a system of differential equations of the form \[ \frac{dy}{dt} \,=\, f(y,z)\qquad\text{and}\qquad \frac{dz}{dt} \,=\, g(y,z), \] we choose a sequence \(\{t_n\}\) of times (typically equally spaced in intervals of \(\Delta t\)), and then we compute the sequences \(\{y_n\}\) and \(\{z_n\}\) defined recursively by the formulas \[ y_n \,=\, y_{n-1} + (t_n-t_{n-1})\,f(y_{n-1},z_{n-1})\qquad\text{and}\qquad z_n \,=\, z_{n-1} + (t_n-t_{n-1})\,g(y_{n-1},z_{n-1}). \]
    1. Implement Euler's method in Mathematica with step size \(\Delta t = 0.1\) to approximate the solution to the initial value problem \[ \frac{dy}{dt} \,=\, y-yz,\qquad \frac{dz}{dt} \,=\, yz-z,\qquad y(0)=z(0)=2, \] for \(0\leq t\leq 20\).
    2. Use ListPlot to make three different plots of your data from part (a): one showing \(\{y_n\}\) by itself, one showing \(\{z_n\}\) by itself, and one showing the combined sequence \(\{(y_n,z_n)\}\) of ordered pairs.
    3. Repeat parts (a) and (b) for step sizes \(\Delta t = 0.01\) and \(\Delta t = 0.001\).
    4. As you can see from the plots, the solutions \(y(t)\) and \(z(t)\) to this initial value problem are periodic. Use your data from part (c) to estimate the period as accurately as you can.