MAT331
Exercise 4: More Maple

1.

Make several pictures of the n-leaved rose given by the polar equation for various integer values of n. How is the number of ``petals'' related to n?

2.

This problem is related to the ``dog on a leash'' example discussed in class.

2a.

Consider the set of points p in the plane for which

where c is a constant greater than 1. Make a good picture of this set for various values of c, in particular, for . If you like, you may put all the graphs on one set of axes. You should probably use either implicitplot or contourplot.

2b.

Show that the equation for the set of points in the plane p which satisfy

is a parabola of the form , for an appropriate choice of a and b.

2c.

What do you get if you replace the y-axis with an arbitrary line of the form y=mx+b? Can you get a nice form (that is, without square roots) for the relationship between x and y?

3.

Consider a train moving at a speed of 60 mph directly north. The center of each wheels of the train is always 1 foot above the track, and extend 1.2" below the track. The engine is red, and it is a Tuesday. At exactly 4 pm, there is a point on the train moving due south. Where is it, and what is its velocity? (Hints: One mile is 5,280 feet. Cycloids and Trochoids are interesting objects. Ben Franklin wanted a turkey for the national bird, because he thought eagles were nasty. Maple is better at arithmetic than I am. One foot equals 12". Most of the words in this problem are irrelvant.)

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