• Letter from the Editor
• Letter from the President
• Labs from Trish Morris
• Thoughts from Teachers
• NC state adoption list for calculus textbooks
• Math Careers
• Mathematicians
• AP Calculus Exam 2004
• AP Central^(TM) E-newsletter

LETTER FROM THE EDITOR
Deb Britt - Asheville High School - Asheville, NC

It has been a busy year for me -- recovering from the virus that attacked my heart and lungs last April -- starting our first BC Calculus course at Asheville High -- having an emergency appendectomy the first month of school and getting my own children raised. The Fall NCCTM conference was good and our business meeting and session was well attended. Please volunteer to be on next year's program and do a Calculus session. NCDPI and College Board are sponsoring an AP Spring Forum at Duke University on March 10, 2004. I am doing a session on slope fields and scoring the 2003 exam. Sam Morris is doing a session on differential equations. There are Statistics, English, and Administrative sessions scheduled. Contact Teresa A. Smith, Special Projects Coordinator, Dept. of Public Instruction, 6343 Mail Service Center, Raleigh, NC 27699, (919) 807-3820, (919) 807-3821 FAX, tsmith@dpi.state.nc.us

The summer newsletter may be a little late or very early as I will not be going to the AP Reading. I am starting a Ph.D. program in Mathematics Education this summer and will be in Ohio for 5--6 weeks. So, I'll get it in your hands as quickly as possible after all the writers get their commentaries to me. We also have to keep getting special permission to print the Standards each year and this takes time with College Board.

Thanks to Trish Morris for sharing the 9 or 10 labs that are included here in the first few pages. I have purposely put these on separate pages [in the printed newsletter] so that you can run these off without having to cut and paste. Most of the rest of the newsletter is from things I save and put in my "Things math teachers should know file." I apologize to anyone who did not get credit for their idea---I try to document where things come from but sometimes it gets lost. A lot of these ideas have come from people who write the calculus listserv or in emails that I receive.

LETTER FROM THE PRESIDENT
Sue Sams - Providence High School - Charlotte, NC

It's that time again! Where has the last year gone? The AP Bulletins are in the hands of students! Already we are posting the dates for the Advanced Placement Examinations, May 4--15, 2004! AP Calculus is on Wednesday, May 5, the morning examination. At our school, the Human Geography Examination will be the late administration since it is scheduled at the same time as Calculus.

As I teach the remaining topics on the syllabus, I am trying to remind students to do those little things that get a good score on the Examination: (1) Read carefully and be aware of initial conditions; (2) Don't forget units; (3) Label figures carefully; (4) Refer to previous mathematics by drawing arrows; and (5) Don't make assumptions about the problem. On all free responses, use calculus that has clarity and precision. Notation should be the best possible.

We were just made aware of the "AP Potential" web site through our testing coordinator. The program correlates a student's PSAT score with his "projected" AP score. Each AP teacher can get an access code from the principal or the testing coordinator. The program is really easy to use and I highly recommend it. It previously cost school systems $250, but is now free. It is a great instrument to help increase numbers in gender and ethnicity and will perhaps open the door to students who might not think of themselves as AP qualified. Used correctly, the precalculus teachers can use the program to guide minority students into the calculus program. For more information on the program, use www.appotential.collegeboard.com

I hope the remainder of the year is good for you and your students, and best wishes on the Examination! Please call on any of your Board Members if we can help you in any way.     sue.sams@cms.k12.nc.us or Sue Sams (980) 343-5390.

LABS
Trish Morris - Greensboro Day School - Greensboro, NC

Use the supplied link to download a Microsoft Word document containing the indicated lab project.

• Function Output---Limits
• Cake Project
• Lunch: MVT
• Lunch Again: Average Value
• Mean Value Theorems
• Volumes by Cross Section
• Riemann Sums: Group Project #1;   Group Project #2
• Integral as Accumulator

THOUGHTS FROM TEACHERS

An interesting way to put in equations
While I greatly appreciate the "advanced techniques" of polar and parametric methods, I must suggest using the power of visualization offered by the graphing calculator handhelds. Sometimes we use the graphical method to support the analytical, and sometimes we do some analysis to support our graph. Occasionally, we must resort to the graphing calculator because we have no other way. On your TI-83, use Y1=solve(X^3+Z^3-6XZ, Z, 3) and change the style to be disconnected (to avoid false connections). Zoom 4 gives a square window that is almost perfect for this problem. You can obviously tweak it some. The CALC tool 4:maximum works nicely (I used Left Bound = 2, RIght Bound = 3, and guess = 2.5) to get the max to be 3.1748021 at X = 2.5298404   Convince yourself that the tangent is horizontal at this max point. This "solve(" function is also integrable. Try 2*fnInt(Y1(X)-X, X, 0, 3) for the area of the loop. I get A = 5.5955 in about 9.1 minutes. I realize I lost a point (on a AP scoring rubric) because of the two-digit accuracy! Use Y1 = solve(X^3+Z^3-6XZ,z,{-3,0,3}) to see the whole folium. I recommend the increased visualization of implicitly defined functions by using the solve function this way.

A problem discussed on the AP Listserv
Jonathan Ring (ring_jonathan@yahoo.com) writes: Last week I discussed a math problem in class which raised a question that I did not know how to answer. The problem: find the horizontal and vertical tangents to x^3 + y^3 = 6xy. One of the answers is (0, 0). The difficulty is the implicit derivative dy/dx does not exist at (0,0).

Some help with your TI-89
I have posted several pages on my website for my students to access when they need help with their TI-89. You might find them a good reference: www.jamesrahn.com then click on Calculator Pages and just browse.

Calculator Help
If you go to www.pballew.net and look for the link that says "TI calculator guide" it will take you to a site that has a wonderful guide for ALL the new TI calculators. You click on the calculator you want and then click on a function you want to perform and there is a brief tutorial. A great job by someone who should get more credit than I am giving them now....(sorry whoever you are, and thanks). - Pat Ballew, Lakenheath HS, U.K.

An interesting discussion about APStats and AP Calculus
I am one of those lucky teachers who teach both AP Calculus and AP Statistics. AP Stat is a very different course than calculus, and much less "mathematical." Much more of an English course or logic course. Most of the math we do in the course is barely algebra 2. And yet, in some ways it is more difficult than calculus because solutions are much less "cookie cutter" than in calculus. I belong to the AP Stat listserv and over the past few days a question came across: If a triangle is chosen at random, what is the probability that it is acute? The thrust of the discussion was how poor the question is. What is a randomly chosen triangle? There were several methods used to define it, each yielding a different solution.

I find that when the word probability is mentioned to many math teachers, let alone students, they run for the hills. In some ways, probability is the least understood math topic I have encountered, both among teachers and students. The question you might have is: what in the world does this have to do with calculus? Read on.

Kurt is a student of mine, the smartest math student I have ever encountered in 33 years of teaching. He is in 10th grade, having completed Calculus AB in 8th grade, BC in 9th. He is now taking multivariable calculus on line given by Stanford. This morning he came into my stat class. I casually said to him, "Hey Kurt, what is the probability that a triangle chosen at random is acute?" He appeared disinterested.

So when he stopped into my room for a study hall 2 hours later, I asked him if he had a solution. He said that he didn't work long on it but he believed the answer was 19.3%. I was stunned because when I simulated the problem using Fathom (a terrific statistical program put out by Key Curriculum Press), I suspected the answer was 20%. How in the world did this kid do the problem? First---my definition of a random triangle: A = a random number between 0 and 180; B = a random number between 0 and 180-A; C = 180 - A - B.

Here is Kurt's logic:
First, using the definition above, the chance of choosing angle A to be obtuse is 0.5. So the chance of all three angles being acute must be less than 0.5.
If A=30, then the other two angles must add to 150. That means that one of the other two angles must be between 60 and 90 in order for the triangle to be acute (neglecting right triangles whose probability is zero). So the probability of the triangle being acute is (90-60)/150 = 30/150.
If A=70, the other two angles must add to 110. That means that one of the other two angles must be between 20 and 90 in order for the triangle to be acute. So the probability of the triangle being acute is (90-20)/150 = 70/150.
Suppose that A is chosen and is acute. That will leave (180-A)-90 and 90. 90-[(180-A)-90] = A. So the probability of the triangle with acute angle A being acute is A/(180-A).
Obviously there are infinite possibilities of angle A giving infinite probabilities of the angle being acute, each infinitely small. So Kurt decides to sum these probabilities up and takes the average probability. So he integrates the expression x/(180-x) from 0 to 90 and divides by 90. This, of course, is the average value of a function. He ends up with 0.386. He then multiplies by 0.5 because he must multiply the result by the probability that A was acute to begin with. He gets 0.193. I have simulated this in Fathom 10,000 times and my results are just about the same. A terrific use of probability theory and calculus. This from a 10th grader. Still has me in shock.    - Stu Schwartz, Wissahickon High

Suggestion for faster graphing
For the more extensive derivative graphs the 89 can take a while to graph d(y1(x), x). So what I usually do is find the equation of the derivative on the home screen using d(y1(x), x) then copy the answer and paste it in y2. This usually graphs much quicker.    - David Smith, Palm Bay High School

Another calculator hint
I put the function in y1. I go to the catalog and enter d(differentiate then type in the function, x) into y2.
Ellen Spiegle, Wantagh NY answers this question:
I need help on the TI-89. What is the proper syntax to ask the TI-89 to graph the derivative of a function? For instance, on the TI-83 if you place y1=x^2, you can then place y2=NDERIV(y1,x,x) and it will graph the derivative of y1. This does not work the same way on the '89. Help!

Software for solids of revolutions
Winplot: math.exeter.edu/rparris
Calculus in Motion: www.calculusinmotion.com

An idea for scheduling
I am teaching a second semester AP Calculus AB, and I 'm teaching it on a 4x4 block schedule. The woman with whom I'm teaching it with has been teaching it for about 8 years on a block and has a 92% pass rate. In our district we play with the titles of the classes a little bit. The fall semester is called "Calculus" and the spring is called "Calculus AB." The kids who take BC Calculus we put them in Calculus AB in the fall, and Calculus BC in the spring. It works REALLY well and we have plenty of time to get through the material.

Another one of those ideas to emphasize
x^2 = (x + x + x + ... + x) [add up x, x times]
Take the derivative of both sides...
2x = (1 + 1 + 1 + ... + 1) = x
What is wrong???
The addition rule for derivatives was derived for a fixed number of terms being added together, not a variable number of terms. I suggest you have the student re-read the proof of the addition rule, and, if possible, rewrite the proof for a variable number of terms.    - Gerry Ashton

Dave Renfro posed the following which I thought was different. & a response from Lin McMullin
We have x + x + ... + x (x-times) and we want to show why incorrectly differentiating this to get 1 + 1 + ... + 1 (x-times), which is x, is off by a factor of two (the correct derivative of x^2 is 2x).

For illustrative purposes, suppose x = 10 and we represent
x + x + ... + x (x-times)
using a 10 x 10 array.

 * * * * * * * * * *
 * * * * * * * * * *
 * * * * * * * * * *
 * * * * * * * * * *
 * * * * * * * * * *
 * * * * * * * * * *
 * * * * * * * * * *
 * * * * * * * * * *
 * * * * * * * * * *
 * * * * * * * * * *

 x + x + x + x + x + x + x + x + x + x

Here's how this works for f(x) = x. Suppose x is represented by one of the vertical columns above and we increment x by h=3. Then we'll get this:

         * |
         * h=3
         * |
 -------------
 *       *
 *       *
 *       *
 *       *
 *       *
 *       *
 *       *
 *       *
 *       *
 *       *

 f(x) f(x+3)

Differentiating x + x + ... + x (x-times) incorrectly to get 1 + 1 + ... + 1 (x-times) corresponds to increasing the lengths of each of the 10 columns of *'s, but keeping the same number of columns (x=10 many columns):

 * * * * * * * * * * |
 * * * * * * * * * * h=3
 * * * * * * * * * * |
 -------------------------
 * * * * * * * * * *
 * * * * * * * * * *
 * * * * * * * * * *
 * * * * * * * * * *
 * * * * * * * * * *
 * * * * * * * * * *
 * * * * * * * * * *
 * * * * * * * * * *
 * * * * * * * * * *
 * * * * * * * * * *

However, since the number of columns is also x, the number of columns should also be appropriately increased. The true situation when we increment x=10 by h=3 will be 13 columns of 13 *'s each:

 * * * * * * * * * * | * * *
 * * * * * * * * * * | * * *
 * * * * * * * * * * | * * *
 -------------------------------
 * * * * * * * * * * | * * *
 * * * * * * * * * * | * * *
 * * * * * * * * * * | * * *
 * * * * * * * * * * | * * *
 * * * * * * * * * * | * * *
 * * * * * * * * * * | * * *
 * * * * * * * * * * | * * *
 * * * * * * * * * * | * * *
 * * * * * * * * * * | * * *
 * * * * * * * * * * | * * *

Nestled within this explanation are at least three things of interest:

1. A geometric way to see that (x+h)^2 - x^2 is 2xh + h^2.
2. A geometric way of viewing the product rule.
3. Why discrete derivatives have the correct asymptotic behavior but aren't exactly the same (because they represent the behavior when the higher order terms, those involving h^2, h^3, etc. are not ignored).

Lin McMullin teaches me something else
Lin responds to Dave's post:  What if x = 3.4567? How do you add 3.4567 x's? The idea only works for positive integers. This is not a continuous function and hence not differentiable.    - Lin McMullin, Niantic, CT

Free-Response now available
AP Calculus free-response questions for 1989--1997 with official solutions are now available in the AP Download Store: apcentral.collegeboard.com/store/edoc The collection will soon include questions and solutions starting in 1969.

Problems on the Silver edition
I have gotten an error on calculators from both my new classroom set of TI-83 Plus Silver edition and also it has been spontaneously occurring on student-purchased calculators of the same make. The problem is that when they go to graph from the standard Y= menu, the 7 normal options of methods for graphs (dot-connected, dot-intermittent, shade-above, shade-below, etc...) are not available, and in their place are a rightward-pointing triangle followed by a vertical rectangle. this seems to cause the calculator to attempt to go to the graph screen when pressing [GRAPH], but then immediately, it jumps the calculator to a blank home screen. What is this, and how do I get it to stop (that is, how do I get it to return to the normal 7 options)? I have tried a number of menu and function key parameters, but to no avail. I've read the manual cover to cover and did not notice any such symbol. Thank you for your help regarding this matter.    - Jason Mutford, Coxsackie-Athens C.S.D.

Answers for the Silver edition problem from above
I believe you are running the "Transfrm" Application. Run the application again and choose to UNINSTALL and you will be in regular function mode.    - G. Ogden

The Calculus Archive Project Online!
hometown.aol.com/calcpage

The derivative of the Susquehanna River
From: Evan Romer, Conklin, NY 13748
The USGS has many of its river gauges online, and you can get graphs of the past 31 days for each site. The other day I noticed an interesting pair of gauges on the Susquehanna River:

(a) waterdata.usgs.gov/nwis/uv?format=gif&period=31&site_no=01553990
shows the height of the water behind the dam at Sunbury, PA;