Monthly Schedule

(Honors AP Calculus, Period A)

M 11/2/09

No additional HW due.

In class: Vote on policies and points for the second quarter. If you have specific suggestions for changes, be prepared to propose them in a clear manner so that they can be discussed.

 

T 11/3/09

Quiz on inverse functions and recent material discussed in class.

 

W 11/4/09

HW due: Read all of Chapter 4, skipping over the parts that are redundant and paying special attention to those (such as the bottom of p. 149) that involve tricky explanations. Then write §4-5 #21, 22, 23, §4-6 #31-34 all.

In the reading, you may safely skip over things that you are already good at. For example, most (though not all) of you are good at implicit differentiation, which means that you may be able to skip reading §4-8.

 

Th 11/5/09

HW due: Read §5-3; write §4-7 #3, 7, 12, §4-8 #12.

 

F 11/6/09

No school (teacher work day).

 

M 11/9/09

HW due: Review problems on pp. 173-175 #R2-R8 all, plus your choice of problems #T7-T17 on pp. 176-177. (Do as many as time permits. Please especially try to do #T15-T17.)

 

T 11/10/09

Test (100 pts.), cumulative through Chapter 4. By your vote, this test will be held in MH-001, where there is room to spread out. Bring your HW binder for a spot check to occur during the test.

 

W 11/11/09

HW due: Read §5-4; write §5-3 #1-4 all, 6-36 mo3, 37, 38.

 

Th 11/12/09

HW due: Read §5-5; write §5-4 #3-42 mo3, 43, 44, 46, and the exercise below.

99. In your own words, explain why local linearity (with finite slope) is equivalent to differentiability at a point.

 

F 11/13/09

HW due: Read §5-6; write §5-5 #4, 7-12 all. You may wonder what other sort of integral could possibly exist other than the Riemann integral. Here is one: the Lebesgue integral.

 

M 11/16/09

Announcement: Class today will be held in MH-001.

HW due: Read §5-7; write §5-6 #1, 1, 3, 4, 5, 6. In other words, do #1 twice so that you have practice writing out the MVT. Here is one format I can recommend:




 

T 11/17/09

HW due: Read §5-8 and this handout. You will eventually be responsible for showing that FTC1 implies FTC2 and conversely. There are no additional written problems for today, but you must be prepared to give a reason for each step in the proofs shown in the handout.

 

W 11/18/09

HW due: Read §5-9; write §5-8 #4-9 all. If you cannot do #4 and #5, then do all the rest. (Note that #6 uses #4 as a lemma.)

 

Th 11/19/09

HW due: Read §5-10; write §5-10 #3, 4, 7. Also choose any 2 problems from §5-9.

 

F 11/20/09

HW due: Work any two additional problems of your choice from §5-10. If you would rather sleep, then sleep. (I was a couple of minutes late in posting the assignment.)

 

M 11/23/09

HW due: Please do all of the following in preparation for your test on Tuesday. I will not collect them, since the first one has a solution key and the others are not practical to collect and grade, but you should do them for your own benefit.

1. Prove that Simpson’s Rule is a weighted average of midpoint and trapezoid rules, namely (2M + T)/3. Compare your solution against this student proof from 2002.

2. Work a selection of “R” and “T” problems at the end of Chapter 5.

3. Prepare a list of review questions to ask during the first half of class today.

4. Prepare a Simpson’s Rule program for your calculator. I will not collect this (how could I?), but you can use it during tomorrow’s test. A small prize will be awarded for the best program.

Hint: If the number of intervals in your Simpson’s Rule program is a constant (say, n = 10), you can pre-populate the list of weights (which we called L3 last Friday) with 1, 4, 2, 4, 2, 4, 2, 4, 2, 4, 1. Otherwise, if you let the user specify the value for n, you should include logic to force n to be an even integer. The following line should do the trick as long as the user specifies n to be greater than or equal to 1:

int((N+1)/2)*2N

Then, you need to fill list L3 with 1, 4, 2, 4, 2, etc. My suggestion would be either to

(1) write a loop (you’ll need your calculator manual for this) that puts 4 into all the even-numbered list entries and 2 into all the odd-numbered list entries, or to

(2) write a clever seq function that returns a 4 for even-numbered entries and 2 for odd-numbered entries, after which you would STO the result of the seq function into L3.

Regardless of which approach you take, you would then reset L3(1) and L3(N+1) to 1.

In class: Review for test (first half of period), followed by guest speaker, Mr. Joseph Morris (STA ’62) from MITRE Corporation.

 

T 11/24/09

Test/CFU through Chapter 5. This test will be slightly shorter than a typical test so that there is time for feedback at the end. The score will be recorded for my information but will not play any role in your quarter average.

 

 

Thanksgiving break. See answers below.

 

M 11/30/09

HW due: Rework your CFU from last Tuesday, solving each problem completely. If you have a good solution to one or more problems (or a solution you think is pretty good), e-mail it to me. I will post the best ones here on the website. Before you take your test, you will need to show me that you finished your CFU. It is definitely to your advantage to do this work, since the test will be quite similar to the CFU, with additional problems pulled from the “R” and “T” problems at the end of Chapter 5. Don’t wait until Sunday to start. Work a little on Friday, a little on Saturday, and you will be in good shape for the test.

In class: Test (100 pts.) through Chapter 5.

Update: As of 11:00 a.m. on Sunday, 11/29, nobody had sent me anything, except for one anonymous request to post answers. That is not how it is supposed to work. However, here are the answers with some of the supporting details.

1. All previously posted online.
2.(a) 4.307
   (b) Using 4 intervals, , since  Therefore, 1.123 additional accumulation occurs from x = 2 to x = 2.2. Answer: 3.184 + 1.123 = 4.307.
   (c) , provided  is Riemann integrable.

[Note: It is also acceptable to use x for the dummy variable here.]

3. Total difficulty = 1531.315 difficulty units
4. Done in class: If y = exp(x2), then  on the positive reals [work must be provided], meaning that concavity is positive, meaning that the slope is increasing, meaning that there is upward curvature. Make a sketch showing how the upward-curving graph of the integrand always leaves some unused area beneath the bounding trapezoid, for each subinterval. For full credit, the fact that the integrand is positive should also be mentioned. You can stop there, assuming you explain your sketch coherently. A rigorous proof was not requested.

5. Done in class:

Note to Anonymous: You did not state what part of question 2 was confusing you. I will try to provide more explanation here. If this helps (or if it doesn’t), please let me know so that I’m not stabbing in the dark.

2.(a) This is a differential equation with an initial condition for f. The general problem type is common on AP multiple-choice tests (and usually heavily missed by students). Given: The derivative of an unknown function f (x) is exp(2 sin x), and f (2) = 3.184. By FTC1, the integral of the derivative from 2 to 2.2 equals f (2.2) − f (2), since f is clearly an antiderivative of . (We note in passing that since  is continuous, an antiderivative, f, is guaranteed to exist.) Use algebra to write what we know, and then solve for f (2.2) as follows:





   (b) Having solved part (a), you should hopefully find the work given above for part (b) to be adequate. If not, please send me another anonymous message.

   (c) Repeat the work for part (a), replacing all occurrences of f with g in the second line. This produces a legitimate expression as long as  is continuous or at least integrable (a weaker condition).*

Second note to Anonymous: In #2(a), there is no way that I know of to find the definite integral without using MATH 9. That is why you were allowed to use your calculator on this CFU. Please be aware, most of the integrals you will see in your life are like this: impossible to compute using a “closed form” and FTC1. It is only in calculus class where many of the problems have a cookbook feel to them: e.g., . Real life is rarely so clean.


*Follow-up note regarding #2(c): It is possible to construct pathological functions that are differentiable but not integrable. Such a thing seems impossible at first glance—but remember the Weierstrass function, which also seemed impossible at first glance. (That was a function that was continuous everywhere but differentiable nowhere.)

The FTC says that every continuous function has an antiderivative, and that the definite integral of any continuous function equals the difference of antiderivatives evaluated at the upper and lower endpoints. Fine. However, the FTC makes no converse claim that every function g that is an antiderivative of  serves as an accumulator function (in the sense of being able to compute the integral of ). Why not? The reason is that continuity of  is not guaranteed simply by the existence of . (Continuity of g is guaranteed, of course, since , but that is a different matter.) Consider, for example, the following bizarre function:


It is an excellent exercise, and probably something that will appear on a future test, to prove that g is differentiable on  However,  is not Riemann integrable on any interval that passes through 0 or has 0 as an endpoint, a fact that you may be able to prove currently (though I would not hold you for it just yet). Thus the answer posted above for #2(c) has now been modified slightly to include the words, “. . . provided  is Riemann integrable.”

Third note to Anonymous: In #5, I could provide the work a second time, but the work was already shown on the board in class. If I provide the work again, am I going to tick off the other students who took notes? Or am I depriving you of the chance to learn collaboratively by calling another student? Or am I running the risk of ticking you off by asking you all these questions? Since I don’t want to lose you as a happy, involved HappyCal student, I will quickly re-post the solution:

Let . Then




 

 


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Last updated: 02 Dec 2009