Mathematics 241 – Multivariable Calculus Discussion 2 – Applications of Parametric Curves September 22, 2010

Background

Today we will look at several more examples with parametric curves. At the end of the period we will reconvene and discuss your solutions.

I. A slide has the shape of a helix–the curve

α(s) = (a cos(bs), a sin(bs), cs)

for s ∈ R and some constants a, b, c. The radius is of the slide is 5 feet and it has height 50 feet, making 5 complete turns about the z axis as it descends to the ground. To get to the top of the slide, you climb a ladder that goes straight up the vertical axis of the slide to a platform at the top. Suppose that it takes 2 minutes to climb the 50 feet, 15 seconds to walk to the edge of the platform, then 30 seconds to slide to the bottom. Assume that each portion of the trip is made at a constant speed (different for each portion). Construct a compound (that is, piecewise) parametrization of the whole motion.

II. Anoter way parametric curves often come into play is this: When we have several related quantities changing with respect to time, it is often instructive to look both at individual plots versus time, and to look at what is sometimes called a phase portrait that describes the evolution of the system together. If we have two quantities x, y changing with respect to time, then the phase portrait will be the curve x = x(t), y = y(t), an example of a parametric curve, like the lines we have studied before. Here is a realistic example, adapted from p. 65 of our book. Part of the human immune system are specialized cells called killer T-cells that attack other cells that have been infected by viruses. The points that are plotted in the graph on the back are from a simulation of the response to a viral infection in the bloodstream of a human patient. To interpret what this is saying,

1) The coordinates are pairs (I, K) where I = number of infected cells per ml of blood in hundreds and K = number of killer T-cells from the immune system per ml of blood, also in hundreds.

2) The initial measurement was (I, K) = (5, 2) made at time t = 0.

3) The subsequent points were measurements at equal time intervals of one hour.

4) As t increases, the later states of the system are given by points that appear to move

counterclockwise in the (I, K)-plane.

Questions

A) From the given (I, K) data points, sketch plots of K versus t and I versus t. Try to

“connect the dots” to get smooth curves here. B) When do I and K reach their absolute maximum and minimum values? When do

they have local maxima and minima? C) Where in the phase portrait (the (I, K) data plot) are the infected cell and T-cell

counts changing the fastest? Explain the reasoning behind your answer. D) What happens in the long run? Does the immune response kill off all of the infected

cells? Does the infection ”explode” and grow unchecked? Or is there a chronic infection (one that stays at a nonzero level over a long time period)?