Exercises

Use Maple (pencil or pen and paper too!) to work these problems.

1.

Consider the parametric curve:

$$x = 2 \cot \theta,\;\;\; y = 2 \sin^2 \theta$$

(a)

What is the maximum domain for the parameter?

(b)

What is the period?

(c)

Plot the curve using either the ``plot'' command or the ``ParamPlot'' command.

(d)

Eliminate the parameter and show that the x-axis is an asymptote.

(e)

Write the curve in Cartesian form y=y(x) and plot it.

2.

Consider the curve:

$$x = \cos t,\;\;\; y = \sin^2 t$$

(a)

Plot the curve for t between 0 and Pi.

(b)

Plot the curve for t between 0 and 1,000. Why do you think the plots are different. Should there be a difference?

(c)

Determine the period.

(d)

Eliminate the parameter to obtain the Cartesian form y=y(x) and plot the curve.

3.

The equation for the asteroid is:

$$x = \cos^3 t, \;\;\; y = \sin^3 t, \;\;\; 0\leq t\leq 2\pi$$

(a)

Plot the graph of the asteroid.

(b)

Find the length of the part of the curve that is in the first quadrant.

(c)

Find the area bounded by the curve and the positive part of the x-axis and y-axis.

4.

Describe the hypocycloid defined by:

$$x = (a-r) \cos t + r \cos ((a-r)t/r), \;\; y = (a-r) \sin t - r \sin ((a-r)t/r, \;\; 0\leq t\leq 2\pi r$$

Try out various values for a and r. Try a=28 and r=13 and decrease from there.