Assignment11

In the case k = 1, I draw the graphs of the equation as a = b = 1 (purple), a = b = 2 (red), and a = b = 3 (blue).

The inside area of the curve is getting greater as a and b get greater together.

Here is a movie indicating this behavior from a = b = 0 to a = b = 10.

In the case that k = 2, the graphs are shown below, as a = b = 1 (purple), a = b = 2 (red), and a = b = 3 (red).

The inside area is getting greater as the same way.

Here is a movie telling this from a = b = 0 to a = b = 10.

In the case k = 3, I draw three graphs as a = b = 1 (purple), a = b = 2 (red), and a = b = 3 (red).

I give a movie showing the cases that a = b = 1 and k is one integer of 1, 2, 3, ...9, and 10.

In summary, if a and b are equal and are getting greater, the inside area of the curve gets greater.

In addition, the number of leaves is to be k. For example, when k = 1 the number of leaves is 1 and when k = 4 the number of leaves is 4.

Consider .

I choose b = 1 and draw graphs for k = 1 (purple), 2 (red), 3 (blue), 4 (green), and 5 (sky blue).

The number of leafs are same as k only if k is an odd number. If k is an even number, the number of leaves is 2k.

In order to show these facts, I made a movie below.

You can click to stop the movie and then, click to change the shape of the curve (or the value of k).

While the number of step in this movie has been 10, it would be more interesting to change the number of step into 100.

See the graph below and click to stop the movie and then, click many times to change the shape of the curve.

All the graphs on the previous movie are the cases that k is an integer. The latest movie is telling us even the cases that k is not an integer.

Consider the what if cos is replaced with sin.

I show you the movie of , with k = 0, 1, 2, ...,9, 10.

It seems that the graphs (one I showed as a movie) is rotated into the graphs at 90 degrees.

Besides, here is a movie with 100 steps.

All the graphs in this case go over the point (0, 0), while the graps have been going over (1, 0).

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