MATH1020 Calculus II	Maple Lab 4
Section 1-4	Due: Tuesday, October 12, 1999,
	by 5pm in your TA's mailbox.

This lab explores polar coordinates. Be sure to follow the lab guidelines as described on the class handouts.

1.

 Let $r=1+a\cos(2\theta)$.

(a)

Plot the graph of this function for $0 \leq \theta \leq \pi$, for several values of a between 0 and 2. Describe how the graph changes as a changes, paying particular attention to values of a close to 1. You can use the Maple commands:

r := 1 + a*cos(2*t);
plot([subs(a=2,r),t,t=0..Pi],coords=polar);

to get the curve for different values of a.

(b)

For the rest of this question, set a=2.

i.

For what values of $\theta$ is the curve in the positive orthant? (That is, for what values of $\theta$ do we have $x \geq 0$ and $y \geq 0$?)

ii.

What is the area under the curve for the part of the curve that is in the positive orthant? (Note: In both this part and the next one, the range of $\theta$ is not from 0 to $\pi/2$.)

iii.

What is the length of the part of the curve in the positive orthant?

2.

Let $r=a \cos(\theta) + \sin(b \theta)$.

(a)

Plot the graph of this function for $0 \leq \theta \leq \pi$, with a=2 and b=12.

(b)

With b=12, what is the effect of using values of a larger than 2? What happens if 0 < a < 2? What about if a < 0?

(c)

With a=2, plot graphs corresponding to b=11, $b=\frac{23}{2}$, and $b=\frac{23}{4}$. In the last two cases, the graph is not a closed curve. How can you choose the $\theta$ interval to give a closed curve?

3.

Let $r=1+a \cos(b \theta)$.

(a)

Plot the graph of this function for $0 \leq \theta \leq 2 \pi$, with a=0.2 and b=6.

(b)

With b=6, let a take on increasing positive values and plot the graph in each case. Describe how the graph changes as a increases. What happens as a passes through one?

(c)

Let a=0.6 and $b=\frac{13}{2}$. Plot the graph, choosing the $\theta$ interval to give a closed curve. Now let a=3 and replot the graph. What are the differences between the graphs for the two different values of a?