What is the most difficult concept to grasp in Calculus 1?

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"
• 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. Commented Aug 14, 2019 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. Commented Aug 14, 2019 at 13:39
• 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) Commented Aug 14, 2019 at 14:14
• 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. Commented Aug 14, 2019 at 14:20
• In my experience, few students understand that the fundamental theorem of calculus is not a tautology. Commented Aug 21, 2019 at 9:35

At my (community) college, I believe many teachers skip the (𝜖,𝛿)-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.)

• 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. Commented Aug 14, 2019 at 14:17
• @SueVanHattum: "terrain" and "grounded"---kinda clashing metaphors :-) Commented Aug 14, 2019 at 16:01
• (I was thinking of the mathematical terrain as mountains in the sky, sort of, and grounded referring to down on the flatter ground.) Commented Aug 14, 2019 at 17:48
• What is calculus if not analysis? In my language the name for the calculus translates literally to "mathematical analysis"... Commented Aug 14, 2019 at 23:50
• 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.) Commented Aug 15, 2019 at 1:18

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.

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.

• 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.). Commented Aug 16, 2019 at 19:07
• @Namaste, the above observation is from my experience (now long ago). Commented Aug 16, 2019 at 20:30

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.

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.