Take-Home Test on Chapter 8, plus §§9-2, 9-3, 9-8, and 9-9

Honors AP Calculus / Mr. Hansen	Name: _______________________________________
2/14/2007 (due 2/15/2007)	Signature to affirm compliance with rules 1-10 (see below): ___________________________________________

	Elapsed time (for informational purposes only): ____________

Carefully read the following rules regarding collaboration and outside resources:

1. No conferring with anyone other than a HappyCal classmate or Mr. Hansen.
2. (That means no parent help, no tutoring help, no help from the smart BCC kid down the street, etc.)
3. Discussions with HappyCal classmates are limited to numeric comparison only (no discussion of methods, no sharing of URLs, no sharing of textbook pages or other printed or on-line resources).
4. Do not let classmates fish an answer out of you unless you are certain they did the work.
5. (If a classmate says he “has not done the problem yet but will soon,” politely decline to share your answer.)
6. Be willing to provide your guess of a number if a classmate gives you his bona fide guess.
7. If your classmate is puzzled as to why his answer disagrees with yours, do not give any hints. Simply say, “Hmmm, well, I guess at least one of us is wrong.”
8. For questions that do not have a numeric answer, there is to be no substantive discussion whatsoever.
9. (It is permitted, however, to indicate how difficult you found one of the proofs to be.)
10. You may consult the Internet and written sources, but if you do, you must say so in your writeup. List your source(s) with URL and page numbers as appropriate. Even if the only Internet link that you consult is the MathWorld URL provided herein, you must list it. Even if the only written resources you consult are your own notes and the textbook, you must list them.
Neatness and clarity of presentation count. In fact, since you have the ability to compare numeric answers until you have a correct answer, and since the conclusions in all the proofs are straightforward, there will be little other basis upon which to make grading decisions.
Except for the blanks to be filled in above, do all of your writing on separate sheets of paper.
Calculator is permitted throughout. Rules concerning calculator usage are the same as those for the AP exam. If you wish, consult Mr. Hansen’s summary (modd.net/23calc/handouts/calcfeat.htm).

1.	A limaçon (see mathworld.com/Limacon.html) is a polar curve having the general form . Let us assume wlog that a, b > 0.

(a)	Sketch the curve , accurately showing the inner and outer loops.

(b)	Compute the length of the inner loop. As on the AP exam, you must show your setup and must justify any limits of integration that are not immediately obvious. (For example, computing the total arc length of the cardioid with polar equation would involve integrating from 0 to . That is considered obvious, but nearly anything else would require justification.)

(c)	Compute the area of the inner loop. The comments from part (b) apply here too, as well as to the rest of the test. However, in this case only, you may re-use your limits of integration without re-justifying them.

(d)	Use equation (4) found at the MathWorld limaçon page (see above) to double-check your answer to part (c). Show formula, plug-ins, answer.

(e)	Prove equation (4) at the MathWorld limaçon page, wlog. Show all steps, including an explanation of why the formula is valid only if b < a. Hint: You will find the green box on p. 452 of your textbook useful at one point.

(f)	Find the typo in equation (3). It is fairly easy to spot.

(g)	Sketch the limaçon , and compute the area to the left of the y-axis.

(h)	Find the total length of the limaçon , and demonstrate that your answer is in the proper ballpark.

2.(a)	State the parametric arc length formula. This was given in class but was never explicitly presented by your text.

(b)	Your textbook (p. 420) uses the Pythagorean Theorem to derive the polar arc length formula. However, if you let r be a function of parameter t, which is really all you are doing when you make a polar function, you can use part (a) to derive a polar arc length formula by calculus methods alone. Please do so. (Your answer should be virtually identical to the rightmost expression in the green box on p. 420, except with replaced by t.) Hint: Consider converting to rectangular coordinates first.

3.	Let R be the bounded region in the xy-plane between curves and .

(a)	Compute the area of R.

(b)	Compute the perimeter of R.

(c)	Upon region R we will pile modeling clay in the z dimension so that all cross sections formed by slicing perpendicular to the x-axis will be semicircular regions. What fruit shape is half-formed in this way?

(d)	Compute the volume of the “half fruit” solid described in part (c).

(e)	The portion of R for which (note: not the entire region R) is revolved about the line in order to form a different solid, a solid of revolution. Compute this volume in two different ways, using (i) the method of washers and (ii) the method of cylindrical shells. Note: It is extremely difficult to get all the numbers to work out perfectly. You can earn most of the points if your setups are valid and your two answers are both in the same ballpark.

4.	Compute , showing your work. If your answer is wrong, but if you can at least verify that it is wrong because it does not check, then you will earn more partial credit than if you simply circle a wrong answer.

5.	Consider the function on the interval .

(a)	Find the coordinates of the absolute maximum on the interval. Justification is required.

(b)	Find the coordinates of the absolute minimum on the interval. Justification is required.

(c)	Is there a plateau point anywhere on the interval? Justify your answer.

	Extra Credit Section (Optional)

	Extra Credit ALPHA: After solving #1(f), why not submit an error report to the site maintainer? I found a bug on a different MathWorld page about a year and a half ago, and my correction was accepted after a review period. Here is an opportunity to influence the world’s leading on-line mathematical resource. Extra Credit BETA: Use an alternate method to double-check your answer to #1(g). Show your steps. Extra Credit GAMMA: In #5(c), the term “plateau point” was used. What other name is sometimes given to a plateau point?