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I teach AP Calculus BC at my high school and we have AP Calculus AB as a pre-req for taking BC. So most of my students are coming in with a strong calculus foundation, and I can spend less time on the basics (like product and quotient rule). This made me think about some more challenging topics I could include for the class. So now I'm curious as to what topics could appear in a standard college level Calculus I and II course that are left out of the AP Calculus curriculum (pages 27-30 of this pdf). Here's a preliminary list I've already brainstormed, but I'd love to see other topics that AP Calculus is missing out on.

  1. Delta-Epsilon definition of limits
  2. Logarithmic Differentiation
  3. Newton's Method of finding roots
  4. Simpson's Rule
  5. More work with trig integrals (like integrating $\int\sin^{2}{(x)} dx$)
  6. Trig substitutions for integrals
  7. Volumes by the shell method
  8. Surface area of revolution
  9. Root Test for series convergence (I think this might have been part of the AP Curriculum in the past, but it's not currently).

Personally, I don't think I would show my high school class the delta-epsilon definition, but I think the other topics I listed would be fair to include. Any other good topics I left off?

EDIT: I teach in the US. "AP" calculus refers to "Advanced Placement," which means that at the end of the school year, the students take a standardized exam. Some colleges offer college credit for students that reach a certain score threshold. There are two AP Calculus classes: AB and BC (they don't stand for anything that I'm aware of). You can see specifically what AB and BC cover in the pdf I linked. But roughly speaking, AB covers all of Calc I plus a bit of calc II (like volumes of rotation and some easy DiffEQ stuff) while BC covers most of Calc I and most of Calc II.

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    $\begingroup$ All of that was in BC Calculus back in the 70's... Sigh, I'm old. $\endgroup$
    – Jon Custer
    May 23, 2023 at 19:13

2 Answers 2

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  1. It's a very good list already.

  2. I would add that a more fulsome differential equations introduction used to be part of BC (and may still be part of stronger college courses). In particular, the second order+ ODE with constant coefficients (including the heterogenous version and variation of parameters and undetermined coefficients). This was still part of the course when I took it in 1983. I think it was pulled out in the 90s along with other small diminishments of the course, along with the addition of "TI time" (which most colleges still think is not important). See this text, for example and its ODE coverage (which was written specifically for AP of the 80s).

https://www.youtube.com/watch?v=GvcRUUh_rtc

I used this and so did Escalante's students. It is (for a normal calc book, not Spivak and ilk) reasonably rigorous. It's the only text I know (ODE or calc) that gives a reasonable way to determine the method of repeated roots if you don't suspect the answer already (not the "we should try this", but a deductive method).

2.5 (edit) I guess hyperbolic trig functions. Did a find text on your pdf and didn't see it listed. By the way, that is a very hard document to look at with all the colors, three column lists, etc. Makes me wonder (harumph) about the ed reformers making that stuff.

  1. If you have not done so, I would take a look at this site (and leave a comment asking your question):

https://teachingcalculus.com/

Most of the participants here are not high school teachers--you will probably get a stronger answer there.

  1. I would consider carefully if you want to go down this road of increasing the difficulty. Are you at an elite school? In general, I find that students (and schools) that feel the need to stretch a year of college calculus out to two years of high school are not as strong as those that just can do BC in a year, straight up. Don't know your situation, but just a caution. It seems like your school's policy is to grant the kids the handicap of going slower, so you might think twice about (partially) reversing this by "enriching". Maybe just set things up for a win and help them get high scores on the BC. [And I say this as someone who likes the additional content, wish it were required, etc.]
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  • $\begingroup$ 4. is a good point you bring up. My school is pretty normal (not elite and not struggling). The reason why we make AB a prereq is that we're on an alternating A/B schedule, with reduced class time. (Instead of 45 minutes everyday or 90 minutes every other day, we only have 70 minutes every other day, which is roughly 70% of what a typical school has). So going straight from precal to BC is a stretch for most students because they got short changed in precal (and algebra II and geometry before that). (1/2) $\endgroup$
    – ruferd
    May 22, 2023 at 19:20
  • $\begingroup$ This year when I taught it, I only taught the AP topics. But the problem with that is that the first semester was basically all review for my students. It was nice to review related rates and optimization, but they didn't really see anything new. So they fell into the mind set of "oh this must be pretty easy, we saw all this last year" and then when we got to the harder stuff (polar, parametric, series, etc.) they had already checked out (senioritis). (2/3) $\endgroup$
    – ruferd
    May 22, 2023 at 19:24
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    $\begingroup$ Good luck. You sound thoughtful. $\endgroup$
    – guest
    May 22, 2023 at 19:33
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    $\begingroup$ (+1) for #4 regarding students spending 2 years doing AB and BC calculus. @ruferd -- You'll probably get more suggestions here than you can cover, and perhaps that's even true for the suggestions you've already come up with, so my advice (and my guess your intention anyway) would be to chose those topics that you are most interested in, or at least that you feel you can do the best job with (excitement, competence, testability, etc.). [I included "testability" because you'll likely find Delta-Epsilon stuff hard to test, and if it's not represented much on your tests, students will not . . .] $\endgroup$ May 23, 2023 at 12:43
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    $\begingroup$ Regarding what I just listed -- "excitement, competence, testability" -- another factor to base your choice of topics on is how well it reviews stuff they are supposed to know (for the AP test). For this, I would guess "Newton's Method of finding roots" rates low (but maybe high in something else?) and "Logarithmic Differentiation" rates high (reviews properties of logarithms and basic differentiation rules; also can be done quickly, half a class hour -- another plus -- and also allows differentiation of functions of the form $f(x)^{g(x)}).$ $\endgroup$ May 23, 2023 at 12:49
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I will tell my impressions based on the BC students I've taught.

Important weaknesses/gaps (skills that prepare them for vector calc. & diff. eq.):

  • Trig. & integrals (OP's ##5,6).
  • Setting up integrals $dx$ vs. $dy$ and applying an appropriate method (e.g. volume, OP's #7).
  • Setting up integrals in polar coordinates for the area of intersecting graphs.
  • Facility/confidence with power series (integrating, differentiating, finding sums, problem-solving skill in determining convergence). This seems a big weakness of BC at least among the cohort of students we get; a segment of them elects to retake calc II in large part because of this.

Less important:

  • Delta-epsilon.
  • Logarithmic differentiation. It's more important that when they see something of the form $a^b$ and do not know immediately what to do, they think, does rewriting it $a^b = e^{b \ln a}$ help?
  • Newton's method. It's not important for further work where I teach (because they get it from scratch in a later course), but I include it in calc. I because it's important that students know how to calculate, at least one method, to know calculators and computers are not beyond their understanding. One of my pet passions. Cum grano salis and all that.
  • Simpson's rule. Don't have time to explain quadratic interpolation, so I stick to the obvious one, the trapezoidal rule. Trapezoidal rule is good for measured data. Simpsons used to be used all over the place, I think or was told, but now adaptive Gauss-Kronrod is in the TIs. Simpsons's not as impressive as when I took calc. (1980s).
  • Surface area. I teach this in calc. II because we do volume in calc. I, and this is a way to refresh their understanding of integration. Otherwise, it helps but is not crucial for surface integrals in vector calc.
  • Root test.

Other things to consider:

  • More applications, problem-solving, multistep problems that have the students practice applying concepts and skills already learned in BC.
  • Multiple approaches to the same problem. It makes connections. Another pet passion. Sometimes an increase in excitement leads to an increase in knowledge and understanding.

One favorite application for power series from my college calc. class was the generating function (a power series) for the Fibonacci sequence (details courtesy of Google). You can (1) sum the series, which is a rational function, (2) use partial fractions to break it down into a sum of proper algebraic fractions with linear denominators, (3) expand the fractions as geometric series, and (4) obtain an explicit formula for the Fibonacci sequence; and, since one of the two terms in the formula is less than a half in magnitude, you can drop it and round the other term to the nearest integer, thus (5) teaching them a cool thing about approximating integers. So three major BC skills and a solution (4) with a cherry on top (5).

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    $\begingroup$ Regarding Taylor series, some of the things I've done are given in this MSE answer. $\endgroup$ May 23, 2023 at 19:18
  • $\begingroup$ @DaveLRenfro Thanks. Those should be helpful for the OP. I've done such things, too. $\endgroup$
    – user1815
    May 23, 2023 at 19:22
  • $\begingroup$ break it down into a sum of proper algebraic fractions with linear denominators -- Surprisingly, I had never seen this "trick" in a book (or heard of it) until the 2000s (when I saw it in advanced calculus texts, 1800s algebra and calculus texts, etc.), at least not that I'm aware of. I was the proctor of the Dec 1999 Putnam Exam where I was teaching at that time, and after the morning portion a student asked me about a question he spent a lot of time on and didn't get -- the calculations he tried kept "blowing up". (continued) $\endgroup$ May 24, 2023 at 10:42
  • $\begingroup$ Looking it up now, I see that it was A3. The students went to lunch and, after a bathroom break, I started playing with the problem. It took me at least 15-20 minutes before I thought of using a partial fraction expansion, after which it was mostly straightforward algebraic manipulation. I showed my work to the student when he came back (early). At the time I thought this was a neat and clever method that could be used in many other ways. In the months-years that followed, (continued) $\endgroup$ May 24, 2023 at 10:43
  • $\begingroup$ when looking through advanced calculus (and other) texts for various reasons, I saw this method in many of them, and I then realized that this was a fairly standard method, but somehow I had missed it. (I never took advanced calculus, skipping straight to graduate real analysis in the 1970s.) If I had known about this before Dec 1999 (the time by which all of those examples I posted were created, since in the late 1999s I was teaching very strong calculus students, but not so much afterwards), then that student (continued) $\endgroup$ May 24, 2023 at 10:43

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