Teaching calculus in AP without the limit definition

Years ago as a college freshman I was taking my first calculus course. Another freshman skipped it because he had calculus in Advanced Placement in high school. I mentioned we were learning the limit definition of a derivative $$\lim_{\Delta x\to 0} \frac{f(x+\Delta x) -f(x)}{\Delta x}$$ and he didn't know what I was talking about.

Maybe I misunderstood (I hope I did!). But is it even possible to teach calculus without this? If so is that common in AP classes?

• This is core and is most definitely taught in AP Calculus. Perhaps your peer had a poor understanding of limits (and algebra, I guess) and memorized formulas, strictly, and learned the definition as $$\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$$ instead. Oct 24 '20 at 15:24
• Maybe that was it. Or maybe he was messing with me, since he did fine in the next calculus class. Oct 24 '20 at 23:13
• It's in Unit 2 of the syllabus here: apstudents.collegeboard.org/courses/ap-calculus-ab. OTOH, some high schools teach a course they call "AP Calculus" or sometimes just "Calculus" and they abridge the syllabus. They usually discourage the students from taking the AP test. This I've gleaned from conversations with first-year college students about their high school experiences, but I've never confirmed it with their high schools. Students generally are unsure whether their high school courses covered the complete syllabus specified by the College Board. Oct 25 '20 at 4:03
• I suspect your classmate forgot the derivative has a limit definition that was taught, due to (1) not being strongly interested in math and (2) never using the limit definition after learning the standard rules of differentiation. Students remember best what they actually use, which in calculus means formulas. I am sure if you asked a class of 100 freshman for the definition of the derivative, most would say “slope of tangent” and essentially none would give the limit definition.
– KCd
Oct 25 '20 at 7:37
• It's actually relatively common here on ME.SE for people to claim that they don't teach any limit definition in their Calculus courses, and that such would by inappropriate, and should be relegated a higher-level Analysis course. However, it's a core part of any textbook I've ever seen, and should be fundamental. Ben Crowell's answer below is a good one. Oct 26 '20 at 6:02

The definition of a limit, and of the derivative in terms of a limit, are standard material that should be covered in all freshman calculus classes. A student who is never exposed to these definitions is being shortchanged educationally.

Realistically, however, very few students in this type of class are intellectually capable of understanding these definitions or of doing even the most trivial epsilon-delta proofs, although they may understand the general concept at some more vague level, like "if you make the deltas small, you get the derivative." They also tend to ignore this material or forget it because it is never used again once rules for differentiation are introduced.

The problem with skipping this material would be that if you have 10% of the class who are going to be math majors or who are intellectually capable of doing this material, then you're not serving them well.

It is possible to use other frameworks for calculus. It's relatively straightforward to do derivatives of polynomials without any fancy foundational definitions. This was done historically before Newton and Leibniz. I do this in sections 1.2.3-5 in my book Fundamentals of Calculus. However, it's hard to use this approach to get off the ground with differentiation rules and derivatives of transcendental functions. That's what Newton and Leibniz accomplished. Until about 1900, students in the English-speaking world learned calculus using Newton's fluxions, and others learned it using Leibniz's infinitesimals. Infinitesimals were given a more secure logical foundation by Abraham Robinson and others ca. 1950 in non-standard analysis, and something similar can be done using nonclassical logic. The best-known freshman calc text to use NSA is one by Keisler, which is free online these days. For previous discussion of this, see Would teaching nonstandard calculus in an introduction calculus course make it easier to learn? .

• (Genuine ethical question - of course with broader implications): Is it okay to waste the time of 90% of students on material they will never understand in order to serve the 10%? Oct 26 '20 at 0:48
• @AlexanderWoo Counter argument: most of the classes taught at the university are directed at majors. Even the general education / diversity classes are taught in a particular department with (potential) majors in mind. Is it really fair to the 10% who want to be math majors to dumb down the class so that the other 90% don't get lost for a little bit? Personally (and I recognize that this might be a controversial opinion), I teach the the top 10-20% of the class, and expect the rest of the class to work their asses off to catch up. Oct 26 '20 at 1:28
• Ben: I might replace "intellectually capable" with "mathematically mature" (making the appropriate grammatical adjustments, as necessary). The issue is not one of intrinsic intelligence or capability, but one of background and preparedness. Oct 26 '20 at 1:32
• @Xander: "intellectually capable" and "mathematically mature" are entangled more than we like to think. We have first-year students coming in with similar background and preparedness and who, as far as we can tell, study equally hard and by similar methods, but one learns to grasp definitions like that of a limit in a semester or two, while another fails to do so, even taking many math courses as a major, before they graduate. It may not be measurable a priori, and it is not an absolute barrier, but very large differences in aptitude do exist. Oct 26 '20 at 3:05
• Regarding your second paragraph, there are two aspects that seem to be conflated. There's the definition of a derivative as a limit, for which students can work with at a purely algebraic level (simplify the difference quotient until you can plug in $\Delta x = 0,$ and sometimes work with piecewise defined functions), and there's the $\epsilon$-$\delta$ definition (or sequence definition) of a limit. The former is even done in business calculus classes (at least those I've taught), whereas the latter is often not formally covered unless an honors calculus class or until a real analysis class. Oct 26 '20 at 17:20

As a new AP Calculus teacher who just went through the certification process with College Board, I can expand on Ben Crowell's answer. Not only is it pretty irresponsible to teach introductory calculus without this formal definition of the limit, but such a course would not be allowed to call itself "AP Calculus AB" on student transcripts. Teachers and principals agree to follow the posted syllabus even when it covers material that is not covered on the Calculus AB exam.

But the formal limit definition of derivatives is material that definitely is covered on the Calculus AB exam. For instance, a student may be asked to calculate $$\lim_{x\to e}\frac{\ln x-1}{x-e}$$ without a calculator and should be able to apply the difference quotient definition to see that this is $$f'(e)$$ where $$f(x)=\ln x$$ and understand from there that the limit is $$1/e$$ in a few seconds.

• +1 for answering what I perceived to be the question: “Can AP Calculus be taught without the limit definition, and if yes, how commonly is it omitted?” This answer contributes new information beyond the usual discussion of the value of rigor/epsilonics in an intro calculus course. Oct 26 '20 at 21:56

Rate of change definition of derivative is still covered as part of AP calculus. (Your question is actually "someone told me something that sounded strange" and...well...yes, you were to think it sounded strange.)

See the official description of the AP course:

https://apcentral.collegeboard.org/courses/ap-calculus-bc/course

[In particular, see the second link on that page, a pdf document that gives an overview of the course and shows coverage of the rate of change limit definition of a derivative.]

The next question (that you didn't ask, but some people are interested to discuss) is what about epsilon-delta proofs. Those are very lightly covered in BC calculus. And arguably have been rather lightly covered for a long time in traditional engineering school calculus courses. But that's actually not what you asked about.

• I didn't see your answer until I wrote a comment to @Ben Crowell's answer, if you're wondering why I seem to have repeated some of what you've said. I suppose (+1) since you've actually specifically addressed what's (supposed to be) done in AP-calculus. Oct 26 '20 at 17:23
• You're cool. You are the class of this forum. Even though it doesn't want to admit it is a forum. ;-) Oct 26 '20 at 17:25

Disclaimer: At first, I want to highlight that I am not familiar with AP Calculus and the corresponding syllabus, so I will cover solely the mathematical dimension of this question.

So, teaching calculus without limits, in general, means the you have at hand some other means of talking about approximations etc. The only way that comes to my mind to do so is by using infinitesimals - i.e. positive quantities $$\varepsilon$$ such that $$\varepsilon for any real $$x>0$$.

So, using this and standard parts of hyperreal numbers, one may define the derivative of a function as follows:

The derivative of a function $$f$$ at some point $$a\in D_f$$ is defined as the standard part of the following hyperreal: $$\frac{f(x+dx)-f(x)}{dx},$$ whenever this standard part exists/has sense.

However, I find it mostly improbable that this was the one preferred in an introductory calculus class. However, one may "define" a notion as e.g. the derivative, procedurally by describing the way one may find a functions derivative - in terms of algebraic manipulations.

For instance, one way to do so is to take some $$h\neq0$$ and then calculate the rate of change fraction:

$$\frac{f(x+h)-f(x)}{h}.$$

Then, once the above fraction has been simplified enough, by setting $$h=0$$ - just to avoid to mention any limits - you get the so wanted derivative.

Even if the above approach has several - and from my viewpoint, enormous - faults in terms of sacrificing mathematical rigour in favour of a more procedural understanding, it still seems to me the only "reasonable" way in which one may avoid mentioning limits in a calculus class.

• The other option is with syntethic infinitesimal analysis, where you have infinitesimals that satisfy $\epsilon^2=0$. See for instance Bell "A primer of infinitesimal analysis." Oct 25 '20 at 12:01