College of the Holy Cross, Fall Semester, 2004 Math 131, Lab #3 (Professor Hwang) Friday, October 29

Purpose

A parametrized curve gives the position of a point as a function of time. The curve “swept out” by the particle depends on the component functions. On this lab you’ll use Maple plot several parametrized curves that would be difficult to draw by hand. You needn’t print out your plots; just answer the questions. (Not every number has questions to be handed in.) Start by entering the Maple command with(plots);

Once you’ve done this, commands such as

plot([cos(t), sin(t), t=0 .. 2*Pi], scaling=constrained);

will graph the indicated curve. (Note the square brackets around the component functions and time interval. As in the previous lab, the “scaling” option forces Maple to use the same horizontal and vertical scales.)

Plot the following curves, and use calculus to answer the accompanying questions. Don’t forget to use a “∗” for multiplication, and “Pi” (capital P, small i) for π.

1. x(t) = 2t/(t^{2} +1), y(t) = (t^{2} −1)/(t^{2} +1) for −100 ≤ t ≤ 100. Identify this curve visually, and use algebra to show that your guess is correct.

2. x(t) = t − sin t, y(t) = 1 − cos t for 0 ≤ t ≤ 6π. Find the values of t at which the velocity is zero. Where on the curve are these points located?

Explanation: This curve, a cycloid, is the path traced by a point on the rim of a rolling wheel of radius 1. By changing the multiplier on the trig functions, you can simulate motion on other wheels. For example, x(t) = t − 1.2 sin t, y(t) = 1 − 1.2 cos t might represent the path of a point on the rim of a train wheel, while x(t) = t − 0.9 sin t, y(t) = 1−0.9 cos t might be the path of a bicycle reflector. (Plot these curves as well to see the changes.)

Note that unlike a function graph, a parametrized curve can have cusps or corners—even at points where both component functions are differentiable—if the velocity is zero. Intuitively, the point can come to a stop, then proceed in a different direction, thereby tracing a curve with no tangent line.

3. x(t) = cos^{3} t, y(t) = sin^{3} t, 0 ≤ t ≤ 2π. Find all points on the curve where the velocity is zero. Where on the curve are these points located? Use algebra to show that this curve is given implicitly by x^{2/3} + y^{2/3} = 1.

4. x(t) = e^{t/2π} cos t, y(t) = e^{t/2π} sin t, −4 ≤ t ≤ 4. Try graphing x(t) = e^{kt} cos t, y(t) = e^{kt} sin t for a few values of k (both positive and negative). What happens as k → 0? How do the curves with k > 0 differ from those with k < 0? List four natural forms (from at least two kingdoms) that this curve resembles.

Explanation: Curves in this family are known as logarithmic spirals, and have the property that they meet every radial line through the origin at a constant angle. This property explains their appearance in biological growth.

5. x(t) = t cos t, y(t) = t sin t, 0 ≤ t ≤ 6π.

Explanation: This curve, a spiral of Archimedes, can be used to trisect angles using only a compass and straightedge. It is known that a general angle cannot be trisected using only a compass and straightedge.

6. x(t) = cos(3t) cos t, y(t) = cos(3t) sin t, for 0 ≤ t ≤ 2π. How many “petals” do you think x(t) = cos(5t) cos t, y(t) = cos(5t) sin t, 0 ≤ t ≤ 2π, has? Plot the curve and see if your guess is correct.

What about x(t) = cos(2t) cos t, y(t) = cos(2t) sin t, for 0 ≤ t ≤ 2π? Again, try it and see. If you can, explain the outcome of this experiment. How many petals do you think x(t) = cos(4t) cos t, y(t) = cos(4t) sin t has?

Explanation: These curves are often called “rosettes”, for the obvious reason that they look like stylized flowers.

7. Consider a circular ring of inner radius 96, inside of which is rolled a wheel of radius r < 96. (See figure.) A point on the wheel 80% of the way from the center to the edge traces the “epicycloid” parametrized by

x(t) = (96−r) cos t+(0.8r/96) cos(96t/r), y(t) = (96−r) sin t−(0.8r/96) sin(96t/r).

Plotting over 0 ≤ t ≤ 2π may not give the entire curve; if this happens, use a larger multiple of 2π for the plot interval. Plot the resulting curves for r = 24, 30, 35, 36, 40, 52, 56, 63, 64, 72, and 84.