Monthly Schedule

(AP Calculus AB, Period A)

T 11/1/05

HW due: §4-4 #3-36 mo3, 41. Then read §4-5.

 

W 11/2/05

HW due: Read the green box in #29 on p. 153; write §4-5 #10-24 even, plus your choice of 26 or 28; then read #29 a second time.

 

Th 11/3/05

Quiz (cumulative through §4-6).

HW due: §4-6 #1-12 all (sloppy sketches OK), #13-20 all (better quality sketches needed).

 

F 11/4/05

No school.

 

M 11/7/05

HW due: Read §4-7; write §4-7 #1, 6, 7. You may find your earlier class notes on parametric functions (and the techniques of how to plot them on your calculator) to be helpful.

 

T 11/8/05

HW due: §4-6 #31-35 all; §4-7 #12.

Quiz on problems like §4-6 #31-35.

 

W 11/9/05

HW due: Finish yesterday’s HW, especially #12; read §4-8.

 

Th 11/10/05

HW due: §4-8 #2-8 even. Then, complete as much of the STA Mathcross #1 as possible. You may seek help from your classmates, from your parents, and from the Internet, but not from students of mine in other classes. There will be a prize for the best entry.

 

F 11/11/05

HW due: §4-8 #10-20 even, 25, 27.

 

M 11/14/05

HW due: Chapter Test on pp. 176-177, all problems. Also write one multiple-choice practice problem on a separate sheet of paper. Offer the reader a choice of A, B, C, D, or E. Do not copy a problem from somewhere else; actually write a problem of your own creation. Please specify in large letters whether the problem is supposed to be “WITH CALC” or “WITHOUT CALC.”

 

T 11/15/05

HW due: As mentioned in class yesterday (Monday), I would like you to prove that the tangent to a circle makes a right angle to a radius at the point of tangency. This proof takes only a couple of minutes using implicit differentiation. Then read §5-3; write §5-2 #3-15 mo3, 17abcd; §5-3 #7-24 all. See examples of #7-24 below.

7. y = 7x3
    dy = 21x2 dx
11. y = 3x2 + 5x – 9
    dy = (6x + 5) dx

Do you see the pattern? This is nothing more than computing dy/dx, except that you write the dy on the LHS and the dx on the RHS. Remember to include parentheses if appropriate.

 

W 11/16/05

Test on Chapter 4. The recent material from Chapter 5 will not be included. This test will be predominantly short answer and multiple choice, with one proof and approximately one “show your work” algebraic tangle. The test will be shortened so that we can spend the first 5-10 minutes of the period answering any pending questions you may have.

Here are the solutions to the Chapter Test problems on p. 177 that we did not cover during class yesterday:

T16. Cube both sides to get y3 = x7. Then implicitly differentiate to get 3y2y¢ = 7x6, which can be solved to get y¢ = 7x6/(3y2). But what is y2? Since y = x7/3 (given), y2 can be expressed more usefully as x14/3. Therefore, y¢ = 7x6/(3x14/3) = (7/3) x4/3. That, of course, is exactly what we would have found if we had simply differentiated the given equation using the power rule.

T17. Continuity requires x3 + 1 = a(x – 2)2 + b when x = 1, i.e., 2 = a(–1)2 + b, from which we conclude a + b = 2. Note that the converse is also true: a + b = 2 Þ f is continuous at x = 1.

Differentiability requires 3x2 = 2a(x – 2) when x = 1, i.e., 3 = 2a(–1), from which we conclude a = –3/2. The earlier equation a + b = 2 allows us to conclude a = –3/2, b = 7/2.

The last part of T17 asks us to show that the RH and LH limits of f (x) as x ® 1 can agree but still allow f to fail to be differentiable at x = 1. There is nothing to do, really, except to say that the weak condition a + b = 2 only gives us continuity. Any such choice of a and b adding up to 2, unless a = –3/2 and b = 7/2, will create continuity at x = 1 but with a cusp, since the LH and RH derivatives disagree.

Another topic that needs some more discussion is the derivative of an inverse. Here is the question that almost everyone bombed from the Nov. 3 quiz:

Let functions f and g be inverses, with additional facts as shown in the table below. Compute g¢(3) and explain your answer briefly.

 

 

x

f (x)

f ¢(x)

g(x)

 

 

0

3

–2

2

 

 

2

4

–1

3

 

 

4

7

1.5

2

 

 

6

11

6

3

 

 

8

13

5

2

 

 

Solution: Since (0, 3) is a point on the graph of f, the point (3, 0) must be on the graph of g. The first row of the table tells us that f ¢(0) = –2. [The remaining rows of the table are completely irrelevant, as it turns out.] In function g, the roles of x and y are interchanged, which means that the slope at the analogous point on the graph of g must be the reciprocal slope. Conclusion: g¢(3) = –½.

 

Th 11/17/05

No additional HW due. You are welcome to work on the STA Mathcross #2.

 

F 11/18/05

HW due: Take the Chapter 4 test again (under time pressure, if possible) and complete the entire test. For maximum learning benefit, write explanations of what you realize you had trouble with on the test when you took it the first time. For example, if you forgot the product rule on #1, you might write, “Forgot to apply PR.” If you applied the quotient rule on #4 but scrambled the order of the numerator, you might write, “Forgot to start with ho dhi when applying QR.” If you knew a technique but simply ran out of time, you might write, “I knew how to do this (as proved here) but couldn’t do it fast enough on the day of the test.”

This activity—reflecting on what you know and how well you know it—is an exercise in metaknowledge. In some ways, metaknowledge is more important than knowledge. For success in college and any other intellectual activity, metaknowledge is critical. A young man who knows when he needs to look something up, knows where to find it, and knows what he knows by memory is much more competent than a fellow who simply “knows a lot of stuff” (or worse yet, crams for tests and forgets everything afterward).

I am considering the possibility of applying a significant number of “test enhancement points” for people who do a good job with this metaknowledge exercise. (Hint, hint.)

Since this is homework, you may bounce ideas off other students, but only after you have finished the entire test on your own. I will consider it an honor code issue if you are working through the test together. Obviously, there is no copying permitted, and a copied metaknowledge statement would be ridiculous.

Near the end of class today, the champion of STA Mathcross #2 will be able to select an activity.

Update: Since there is no champion yet, we will try again on Monday.

 

M 11/21/05

HW due: §5-3 #35-40 all. Then complete the STA Mathcross #2 as a real assignment. It has more educational content than you probably realize. You may work with friends, parents, classmates, and even students in other classes (e.g., STAtistics).

 

T 11/22/05

HW due: Read §5-4; write §5-4 #1-44 all. If you can’t get all the way through #42 (because of slow writing, or whatever), do make sure to solve #43 and #44.

 

 

Thanksgiving break.

 

M 11/28/05

HW due: Read §5-5; write §5-5  #7, 8, 9. There will also be a Quiz on Chapter 5, through §5-5.

 

T 11/29/05

HW due: §5-5 #11. Use the Thingy (see “Links Based on Class Discussions”) to do the computations.

 

W 11/30/05

HW due: As stated in class, your assignment is to debug the program we were working on so that it produces correct results. (Mine already does, but that is mainly because I did not make any typos while entering it.) See if you can modify your program so that it can produce all of the answers you achieved yesterday by using the Thingy.

Bonus opportunity: By allowing the user to input a, b, and n (i.e., interval endpoints and number of intervals), you can have the program itself compute the step size (Dx, sometimes called h) as well as the mesh point values. The Prompt command is what you need for that purpose. Then, by adding “If” statements to test for Y1(M) as compared to Y1(M+H), and using whichever is larger (resp., smaller) as the rectangle height, you can also program something that comes close to an upper (resp., lower) Riemann sum.

Technical note: To get a true upper or lower sum, you would need to find the maximum (resp, minimum) value of the function on each subinterval. Frequently, the extreme values are found at interval endpoints, but that is not always the case. For now, we will set aside the knotty problem of programming the TI-83 to find the maximum or minimum function value on each subinterval.

 

 

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Last updated: 04 Dec 2005