Welcome to the HappyCal Zone

(Honors AP Calculus BC, Period A)

Are you nervous when you see NCWEE? concerned when you see CIRC? perturbed when you see PBC? Visit Mr. Hansen’s fabled abbreviations page to make sense of those cryptic markings you see on your papers.


Schedule at a Glance (see archives for older entries)
Written assignments should follow the HW guidelines. Enter your scores here.


W 5/15/13

(Relaxed start at 8:20 today, tomorrow, and Friday.)

HW due:

1. Yesterday’s assignment was a disappointment. Apparently almost everyone except Heein thought that question #3 was not worth a solid effort. There were a number of “I don’t know” responses, which were at least honest, as well as several “history-class baloney” responses that I wish I hadn’t read. You have studied the calculus, you know about limits, and you have tools at your disposal that go far beyond what the people of any earlier generation had to work with. You can do this. Everyone, including Heein (whose response was correct but somewhat vague), needs to redo #3 from Tuesday’s assignment. Giant hint: There are 1016 – 10 possible starting positions for the word SHAKESPEARE. This is essentially the same as 1016. At each starting position, the probability of finding S is

the probability of finding the digraph SH is  and so on, until the probability of finding SHAKESPEARE at that

starting position is  The probability of finding SHAKESPEARE in at least one location is 1 minus the

probability of finding SHAKESPEARE nowhere. (Why?) Therefore, one way of solving the problem is to write an expression for the probability of finding SHAKESPEARE nowhere; then use your knowledge of limits to evaluate that expression, and subtract from 1.

2. Design an original game involving dice, cards, pieces of paper, or other common objects, such that the winning probability is slightly less than 0.5 but close enough that people might be willing to wager even money (1:1 odds) to play your game. Describe your game so that the rules are clear, unambiguous, and compact.

3. Compute, by any means you wish, the expected value to the player in the game you invented in #2. (Your answer should be negative.) In other words, for each $1 that someone wagers in order to play the game against you, how much profit do you expect to earn? Show your work, and document any simplifying assumptions you may have to make.


Th 5/16/13

Friday schedule is in effect today.

HW due: Practice your skills on the Excelcise. (Click here and look for the 10/28/010 calendar entry.) In class, you will be expected to produce the final spreadsheet within 5 minutes without using any macros or copied-and-pasted text from external files. Copying and pasting within your spreadsheet (e.g., Edit, Paste Special, Transpose) is permitted.

If you do not pass today, you will be allowed to try again tomorrow.


F 5/17/13

Thursday schedule is in effect today.

HW due:

1. Practice your skills on the Excelcise (see 10/28/010 calendar entry). In class, there will be no practice time; we will cut straight to the 5-minute competition.

2. Determine, using any means available to you, the probability of selecting at least one ace in 8 draws (without replacement) from a well-shuffled standard deck of cards. Show your work, and give your answer with m.o.e. and confidence level.


M 5/20/13

15th Annual Field Trip to the National Cryptologic Museum, Fort Meade, MD. Bus departs at 8 a.m. from near the intersection of Garfield St. and the service road near Grant Meadow. We will return at approximately 1 p.m. after a guided tour of the museum and an interactive lecture by a working NSA mathematician. Regular school dress code is required.

If you prefer not to go on the field trip for some reason, be sure to say so. In that case, you are expected to attend all your other classes, A through F periods.


T 5/21/13

7:25 a.m.: It’s JBAM a at McDonald’s Week!

In class: Excelcise, penultimate day. Relaxed start at 8:20 a.m. for anyone who brings a McDonald’s receipt dated 5/21/2013.


W 5/22/13

7:25 a.m.: It’s JBAM b at McDonald’s Week!

In class: Excelcise, final day (partial period only). Relaxed start at 8:20 a.m. for anyone who brings a McDonald’s receipt dated 5/22/2013.


Th 5/23/13

No additional HW due, except for Ian, James, and Vasisht, who still need to pass the Excelcise.

In class: Card-deck simulations, distribution of Painting With Numbers book.


F 5/24/13

Last day of classes.

HW due: Read all of Chapter 1 (pp. 13-24) of Painting With Numbers. A quiz is possible.


T 5/28/13

Final Exam, Steuart 201-202, 2:00 p.m.

The exam will consist of a single question, as follows:

1. Write, in your own words, the story of the calculus.

You have 90 minutes (2 hours permitted, but 90 minutes is plenty of time, rest assured) to link the important topics together, in your own words. Up to 2 blank blue books will be provided, although in almost all cases a single blue book will suffice.

All seniors with less than a B average, as well as all sophomores and juniors regardless of average, are required to take the exam. The exam will count as 20% of your semester grade.

For seniors with a B average or above, this exam is optional and will count only if it helps you.

Some additional notes:

·         This is not supposed to be a regurgitation of a comprehensive treatment of the subject. In other words, don’t try to emulate Wikipedia.

·         The purpose of this exercise is to give you the chance to think deeply on the connections you have formed in your own brain over the course of the year. It is, therefore, a personal document. It should be coherent, yes, but it’s less a “teaching someone else about the calculus” document than it is a “documenting what you know and how you know it” document.

·         Please write legibly. It doesn’t need to be overly neat, and you should feel free to write quickly so that your ideas flow well, but avoid sloppiness. Don’t be fussy: the digit “1” is a simple downstroke, for example, and you don’t need to label your x- and y-axes unless you switch them for some unusual reason.

·         Good, clean, focused content is better than large volumes of random thoughts.

·         Pencil is preferred, but if your ideas flow better in ink, you may use a pen.

·         Here are a few suggested topics you might want to include, since they give you a chance to describe your personal thought process in some detail:

                   - Integration by parts
- Derivative of an inverse (how do you make sense of the formula?)
- Steps for optimizing a function (finding relative max/min or absolute max/min on some interval)
- Relationship among slope fields, linear approximators, and Euler’s Method
- Relationship between MVT and the Lagrange remainder term
- What is meant by “locally linear,” “locally quadratic,” “locally cubic,” etc.?
- Relationship between differentiation and integration (FTC)
- Adaptive quadrature, as a concept
- EVT, IVT, and how you might use them in tandem to design new ways of optimizing (finding min/max) via a computer program.

You can write a perfectly good, high-scoing essay without hitting these topics. (Well, scratch that, I think we all know you need to mention FTC at some point. But the other topics listed above are a matter of personal taste. There are, of course, some major topics that nearly everyone will want to include: variable-factor products, diff. eqs., limits, Riemann sums, and the like.)

·         You will write your exam without notes. You can’t even use a copy of this set of instructions! You may use a calculator if you wish to include a few tables or graphs, but that is not required. It’s mostly just you and your brain and a pencil, working to produce a document that cements your knowledge of the course for all time.


Essential Links:
-- STA School Handbook
-- College Board: AP Calculus BC Course Description
-- Eric Weisstein’s World of Mathematics, the Web’s most extensive mathematics resource (no kidding!)
-- WolframAlpha.com, a site that I possibly shouldn’t tell you about . . .

Extra Help:
-- Karl’s Calculus Tutor for first-year students
-- Calc101.com, another site I might not want to tell you about (click it and you’ll see why)
-- Temple University: Calculus on the Web (COW)

Links Based on Class Discussions:
-- Troy’s Integral Approximation Thingy: a neat JavaScript application for Midpoint Rule, Trapezoid Rule, Simpson’s Rule, etc.
-- The “RiemannSums Applet” found by John S. (actually shows you the rectangles or trapezoids)
-- Chris and Andrew’s proof that Simpson’s Rule is a weighted average of the Midpoint and Trapezoid Rules
-- Braxton’s direct proof of FTC2
-- Proof that FTC1 implies FTC2 and conversely
-- Related rates tutorial and practice problems
-- Partial fraction decomposition tutorial with sample problems and solutions, courtesy of the University of California at Davis

Links for AP Preparation:
-- Real sample AP questions from the College Board
-- AB Calculus Cram Sheet
-- BC Calculus Cram Sheet (courtesy of Will Felder and Mr. Hansen)
-- “Stuff you MUST know cold” (link to another AP calculus teacher’s site; requires Adobe Acrobat reader)
-- Review question logsheet (requires Microsoft Excel); also available are old versions for 2003, 2009, 2010, 2011, and 2012.
-- Permitted features for graphing calculators on the AP examination
-- Actual college tests from Mr. Hansen’s alma mater (great practice!) Note: MATH 121 is the calculus course I took back in the (ulp!) 1970s.
-- Multiple choice practice #1 with answer key
-- Multiple choice practice #2 with answer key
-- First semester recap (recycled from my 2006-07 IntroCal class, for which this handout served as a full-year recap)

Fun Links:
-- Homemade “Segway”-like balancing scooter uses a fair amount of calculus!
-- Mathematicians as depicted in the movies (Good Will Hunting, etc.)
-- An Algebra II problem that has a calculus flavor to it. (This is problem #26 from §11-7 of Foerster’s Algebra and Trigonometry: Functions and Applications.) The problem is to determine which sweepstakes prize is better: a $20,000 lump sum or $100 a month for life. Assume 4% annual interest compounded monthly. In part (d), the challenge is to determine how the answer changes if the interest rate changes to 7%.
-- The Mt. Sinai problem and two variations
-- The astonishing Bailey-Borwein-Plouffe algorithm for calculating pi to any desired decimal place
-- Sound wave analysis (harmonics, Doppler shift, etc.), excellent site developed by students at TJHSST in Virginia
-- Good problems (some calculus, some not)
-- More fun links on Mr. Hansen’s home page

Serious Links:
-- Summer math camps for talented high school students
-- Click here for other serious links

Return to Mr. Hansen’s home page

Return to Mathematics Department home page

Return to St. Albans home page

Last updated: 28 May 2013