# Week 4

### Assignment:

1. Computing e: Read Tutorial 1 in PHASER for computing e using the initial-value problem x' = x, x(0) = 1. Recall that the exact solution of the initial-value problem x' = x, x(0) = x_0 is x(t) = x_0 e^t. Using the method in this tutorial, compute sqrt(e), 3.4*sqrt(e), 4.4e^2, and -1/e. Check your numbers against a calculator (Google calculator, for example). Report any differences. (Hints: stop your computations at the appropriate time; you can solve for negative time; you can change the initial condition.)

2. Radioisotopes used in nuclear medicine: Investiagate which radioisotopes are used in nuclear medicine. Select one used in Positron Emission Tomography (PET) imaging, and another one in radiotheraphy for cancer treatment. Look up their half-lives. Use Phaser and the ODE x1' = k*x1, determine the decay constants k of the two radioisotopes you have selected. Use at least two initial conditions for each to make sure that k does not depend on the initial amount. Here is a link to get you started on radioisotopes in nuclear medicine.

3. C-14 dating: The half life of radiactive carbon C-14 is known to be appproximately 5730 years. Using Phaser, determine the decay constant k in the decay differential equation x1' = -k*x1. Do the calculations with two different initial conditions, say 195 and 490. Does the decay constant depend on the initial amount?

In 1989, fibres from the Shroud of Turin were found to contain about 92% of the level of C-14 in living matter. Determine the age of the shroud using PHASER. Suppose that there was 0.15% error in the determination of the percentage of C-14 in the sample of the shroud. What is the range of possible dates for the sample?

Some people do not agree with the results of the C-14 dating of the Shroud. Do some research on the Web (see the links above, for example) and identify an objection or raise one of your own. Argue for or against the objection.

4. Libby's Nobel Lecture: Read the Nobel lecture of Willard F. Libby (Chemistry Prize in 1960) linked above. Why was he awarded the Nobel prize? This problem is for your edificiation; you do not have to turn in an answer, if you do not want to.

5. Another form of Logistic: Consider the continuous analog of the logistic model using the ODE:
x' = r x (1 - x/k)
where x(t) is the population size at time t, r (growth rate) and k (carrying capacity) are positive parameters. This form of the Logistic ODE is preferred by ecologists.
• Find the equilibrium points and determine their stability types using the Linearization Theorem for the positive values of the parameters.
• For r = 1 and k = 1.5, several solutions of this equation with various initial conditions are displayed in the picture below. Describe what happens to the population as the initial population size varies.
Download the following phaser Project file logisticODE.ppf by just clicking on it ( or by right-click and save it to our computer. Now load this file into Phaser). You should see the following XivsTime view:

• Fix the parameter r = 1 and vary the parameter k from 0.5 to 3.0. Next fix k = 1.5 and vary r from 0.5 to 4.0. Describe the results of your experiments from the biological viewpoint.