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I would say it is not the Fundamental Theorem of Calculus, but rather some notion connecting limits and continuity, perhaps the $(\epsilon,\delta)$-definitions of limits and continuity. But I would be interested to learn from experienced calculus instructors, as it would help me in courses that have Calculus as a prerequisite, to understand where to anticipate weaknesses.

For the usual meaning of "Calculus 1" (or "Intro to Calculus"), see Is there a more telling name for “Calculus 2”? See also Wikipedia's List of calculus topics.

Summary by 16Aug2019: I did not anticipate such lack of consensus.
  • @vonbrand: "the whole $\epsilon/\delta$ dance."
  • @SueVanHattum: implicit derivatives and optimization
  • @guest2: "multistep problem solving"
  • @Maesumi: "the concept of a variable and that of a function"
  • @JamesSCook: "the chain rule is most troublesome"
  • @user1527: "the point of the mean value theorem"
Joseph O'Rourke
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    Do you want calculus-specific concepts, or would you accept the application of non-calculus topics, such as writing a function that performs a particular task (distance between two points, area of a basic shape, etc.)? For me, teaching calculus always implies teaching prerequisite material. – Nick C Aug 14 '19 at 13:38
  • @NickC: I was thinking of topics, but I would be interested as well to learn of skills that seem difficult to master. – Joseph O'Rourke Aug 14 '19 at 13:39
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    My guess is the point of the mean value theorem. Or based on impact, perhaps the Chain Rule. If the limit definition is not taught, it can hardly be difficult. If it not used in the course you're teaching, it's hardly important. Otherwise limits are probably one of the more difficult concepts, although it seems less difficult in the third or fourth semester of learning limits (calc 1 -> 2 -> 3 -> transition-to-proofs course) – user1815 Aug 14 '19 at 14:14
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    Mechanically, I think the chain rule is most troublesome. However, logically, anything involving non-standard algebra and multi-step logic is an opportunity to fail; optimization, graphing with calculus, related rates, conceptual problems which illustrate theorems... anything not algorithmic. – James S. Cook Aug 14 '19 at 14:20
  • This really depends on the level at which the course is being taught. If you're teaching the completeness axiom of the reals, with real proofs of things like the intermediate value theorem, then those are extremely deep logically, and not realistically something that any but the most talented 18-year-olds will be able to undrstand. If the course is being taught at a low level, then basically students are not learning any concepts at all. They're just learning how to calculate derivatives and integrals, without any understanding of what it means. –  Aug 15 '19 at 00:47
  • I don't know how it's done now, but 35 years ago, University Calc 1 was derivatives, and Calc II was integrals. – RonJohn Aug 15 '19 at 00:51
  • @BenCrowell: "If you're teaching the completeness axiom of the reals," then I would say you are teaching Analysis and not Calculus 1. Although I get your point, I hope that in introductory courses these days, students are learning some concepts. – Joseph O'Rourke Aug 15 '19 at 12:24
  • The difficulty most students have with the chain rule is that they do not understand the variable and function concepts. – Dan Fox Aug 21 '19 at 09:34
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    In my experience, few students understand that the fundamental theorem of calculus is not a tautology. – Dan Fox Aug 21 '19 at 09:35
  • As a student: In my opinion "Limit" is the most difficult concept to grasp... – Ashish Shukla Sep 02 '19 at 05:50

5 Answers5

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At my (community) college, I believe many teachers skip the (,)-definitions of limits. I give the main definition, and teach it with an example I call the cookie crispness index. But I don't assign problems in that section, and I don't test on it. I figure actually working with it is for an analysis course.

Of the topics that I do test on, implicit derivatives and related rates seem to be the hardest for students to understand. They also have trouble with optimization, because too many of them can't actually do anything with math other than follow the steps. (Hmm, I wonder if I can somehow have them optimize throughout the course. It seems like the most important topic that they do badly on.)

Sue VanHattum
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    Optimization requires a lot of algebraic skill and it involves words. I also find graphing with calculus gets a lot of students, I think the logic and nuance flies in the face of the cookie-cutter problem solving mode so many of them are in before calculus. I used to test on $\epsilon-\delta$... it is even worse, which makes sense as it is even more removed from canned-problem solving. – James S. Cook Aug 14 '19 at 14:17
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    Once you learn how to do those (epsilon delta) problems, they don't seem so removed from canned procedures to me. What I think makes them hard is how symbolic and distant from practical concerns they are. They feel, to me, like the terrain of mathematicians. I like that terrain, but my students usually need to be more grounded. – Sue VanHattum Aug 14 '19 at 15:16
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    @SueVanHattum: "terrain" and "grounded"---kinda clashing metaphors :-) – Joseph O'Rourke Aug 14 '19 at 16:01
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    (I was thinking of the mathematical terrain as mountains in the sky, sort of, and grounded referring to down on the flatter ground.) – Sue VanHattum Aug 14 '19 at 17:48
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    What is calculus if not analysis? In my language the name for the calculus translates literally to "mathematical analysis"... – Džuris Aug 14 '19 at 23:50
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    In the U.S., the beginning calculus courses are not so rigorous. Students learn how to use calculus to solve problems. In analysis, everything is carefully proved. (Does someone else have a more precise way of describing the difference? My first calculus course, honors at UMich, was actually an analysis course.) – Sue VanHattum Aug 15 '19 at 01:18
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    @Džuris Etymologically, the English word "calculus" originally mean simply "a system or method of calculation." Indeed, it is still possible to use the word this way, though it is not terribly common. Remember that 17th Century mathematicians (principally Newton and Leibniz, but others were involved) developed integration and differentiation in large part as a means for performing computations. In the US educational system, classes called "calculus" typically focus on computation, while "analysis" courses focus on rigour. – Xander Henderson Aug 15 '19 at 12:19
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I'd say the whole $\epsilon/\delta$ dance. Just because it involves inequalities, while students just have trained equalities all their life. To just work with (sometimes very rough) bounds, when you were drilled to get exact answers goes against the grain.

vonbrand
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    In many universities, $\epsilon/\delta$ is not used in introductory Calc I - III. It may appear in Advanced Calculus (for Honor students with AP experience in calc.). – amWhy Aug 16 '19 at 19:07
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    @Namaste, the above observation is from my experience (now long ago). – vonbrand Aug 16 '19 at 20:30
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Related rates problems show the most issues on AP grading. (No reference, just my own observation) Topic involves word problems, geometry and some multistep problem solving.

I don't think the concept of epsilon delta is so difficult as the tediousness of the algebra along with students openly or unconsciously questioning the value add for science and engineering. Even for the calculus class itself there is an issue that you do it st the beginning and set it aside. Not really a tool you are building with. These create an engagement issue.

Edit. Just noticed your question on prereqs. I think for calc 2, 3, and diffy Qs, the most important issue is just having solid drilled ability to work problems. And having solidified algebra. Calculus tends to do that. For real analysis, the course itself tends to cover the dive into theory versus assuming it. So even here main benefit of calculus is the muscle building. Although of course having seen definitions and theorems before once has a benefit in terms of repetition when doing a theoretical calc course.

JTP - Apologise to Monica
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guest2
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I'd say the concept of zero, infinity, and the infinitesimal. Often they're thought of like numbers and this leads to calculations which don't make sense where, even if they work numerically, it's not actually correct since the process is wrong.

With calculus, I think this would be the idea of the limit and everything that revolves around it, which is basically everything.

Sat
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In terms of issues affecting most students I believe the concept of a variable and that of a function are still the most difficult concepts for calculus 1 students, even though the concepts are introduced in precalculus. Writing a full and correct mathematical sentence is a topic most students struggle with.

Maesumi
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