174 computer-based assignment

These problems will account for 30% of Monday's test. 90 minutes' work should be plenty — great formality is not required.

From the author's website, this page, please choose Browse visuals on the left, then section 10.3 for Polar curves, then Exercises at top right, then problems 5, 6, 7, and 8. These can be done with a graphing calculator as well as with the online instrument gives a lot more information.

For those who are unable to see the problems online, here they are:

5. Investigate the family of curves given by r = 1 + b cos (theta). Describe how the shape of the curve changes as b varies.

6. Graph r = b cos(theta) + p sin(theta) for several values of b and p. What geometrical figure do you conjecture all polar graphs of the form r = b cos(theta) + p sin(theta) look like? [You should support this conjecture, of course.]

7. Consider the function r = 2 cos (theta) + sin (2 theta).

(a) By looking at the Cartesian graph [of y = r(t), as on page 643], where is r <= 0?

(b) Can you explain why Quadrants II and III of the polar graph are empty?

(c) How many values of theta for 0 <= theta <= 2 pi satisfy r = 1?

(d) The polar graph intersects the unit circle 4 times. Explain the discrepancy between this and part (c).

8. Graph the function r = 0.5 + 2 sin(3 theta) for 0 <= theta <= 2 pi. Looking at the Cartesian graph, can you explain the different-sized loops in the polar graph?

Here are links to some online graphers:

http://www.webmath.com/polar.html

http://fooplot.com/ (use dropdown box where you put in function to change Function to Polar

http://www.teachers.ash.org.au/mikemath/calculators.html

Not that you don't already own a grapher, of course, and not to mention the always-valuable Winplot — but the last of these links is actually to a French grapher Tracés animés that's a real prize.