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I am seeking a list of topics that students taking calculus should memorise. Some topics from Calculus I might include:

  1. $\varepsilon-\delta$ definition of limit;
  2. Definition of the derivative of a function;
  3. Product rule, quotient rule and chain rule for derivative;
  4. Fundamental theorem of calculus;
  5. Definition of Riemann sum.

This list is far from being complete. Here I am asking for a more comprehensive list that covers topics in Calculus I, II and III. Thank you!

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    $\begingroup$ Have you tried, e.g., going through the a table of contents (for example: Stewart's tome) or looking over the US's AP Calculus Curriculum (at least for AB and BC, which approximate Calc I and II). It might help, too, to indicate what this list is intended to accomplish; e.g., I don't want students to only memorise the product rule and quotient rule - I'd want them to know how to derive each of them in addition to being able to recall their statements. $\endgroup$
    – Benjamin Dickman
    Dec 26, 2018 at 17:21
  • $\begingroup$ @BenCrowell Why did you de-capitalise Theorem and Calculus? It is the name of a theorem, so should be capitalised. $\endgroup$
    – Jessica B
    Mar 13, 2019 at 6:58

4 Answers 4

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An argument can be made that one should memorize almost nothing—or at least as little as possible—and instead derive the results needed on the fly. I do this myself with rotation matrices (although I have to admit this is hardly necessary, as what needs to be memorized is so little).

          enter image description here
          Brian McLogan YouTube: Why you should never memorize the unit circle.

For example, many trig identities can be derived easily from a few core relationships: See MSE @BrL or MSE @zahbaz. Similar arguments could be made for calculus equations.

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    $\begingroup$ So you're saying they should memorize the core relationships? Which ones? Of course students shouldn't memorize everything, that wasn't the question. The question is what should students memorize and what can (and should) be rederived. $\endgroup$
    – Nate Bade
    Mar 13, 2019 at 17:21
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I'm thinking certain mental tricks or rules of thumb mght be worth memorising too: for example

  • to work out what trig substitution to use, draw a triangle and check which side is in the integral.
  • when struggling to remember or follow an $\epsilon$–$\delta$ proof or definition, try mentally inserting "however small" after "for any $\epsilon >0$", "small enough" in "a number $\delta$", and " large enough" in "a number $N$".

These aren't theorems or strict mathematical procedures, but they can help a lot.

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    $\begingroup$ My reason for the $\epsilon$–$\delta$ one: I think the most confusing thing about the $\epsilon$–$\delta$ definitions is that the central idea of the definition doesn't appear in it: instead of making $\epsilon$ arbitrarily snall we see $\epsilon>0$ which looks like making it large enough, instead of making $\delta$ small enough to suit $\epsilon$ we see "there exists a number $\delta$ and so on. $\endgroup$
    – timtfj
    Dec 29, 2018 at 16:25
  • $\begingroup$ I did fine with epsilon delta and was exposed to it moderately. But going on about it as if it is so important is te typical math major blindness. 95% of your students taing calc as a service course won;t do any more of that but will do lots of basic deriving and integrasting in science and engineering (and even easier for doctors, nurses, b-majors, etc.) $\endgroup$
    – guest
    Dec 29, 2018 at 23:02
  • $\begingroup$ @guest Really I'm suggesting a category: things that aren't compulsory knowledge, but which are so helpful that it's a really good idea to memorise them. (You're right about the $ε–δ$ stuff—I don't think I encountered them once in the whole of an engineering degree. $ε$ was the permittivity of a dielectric.) $\endgroup$
    – timtfj
    Dec 30, 2018 at 0:51
  • $\begingroup$ I looooove it. Dielectric uber real analysis. $\endgroup$
    – guest
    Dec 30, 2018 at 7:07
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  1. Derivatives of common functions (sin, cos, polynomial, exponential, log, etc.) These should be at your fingertip versus needing to rederive them or refer to a table. Since you will use them in harder problems (e.g. with chain rule).

  2. Most of the common function antiderivatives follow from (1), but there may be a few that is worth memorizing or have used so often they are in working memory. Sorry, can't recall which.

  3. Formula for integration by parts. (Yes, you could derive it but should have it in working memory like you do quadratic formula versus completing the square.)

  4. Formula for arc length. Personally, I struggled to derive it under duress. But you should probably force yourself to derive it occasionally also so it is not just a magical formula.

  5. The concept of using tan theta over two substitution.

  6. There is no formula to remember with partial fractions, per se, (I think it is intuitive) but just having done enough problems to be confident to rock out the algebra.

  7. Formula for integration of volume by discs. Personally, this formula is rather intuititive so should not be hard to keep in memory.

  8. [During a course and shortly after it)] formulas for common quadratic and radical forms. Don't think these need to be retained for the rest of your engineering career though. Can refer to tables. But worth having had them in working memory once. Makes using tables easier in the future and they do crop up. But at least you can say "oh I remember that now" having once memorized them.

  9. Definitions of div, grad, curl (and all that) [Sorry can't remember much more from calc 3. Liked calc 2 and ODE more.]

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    $\begingroup$ The formula for arc length doesn't need to be memorized; it's just the integral of sqrt((dx)^2 + (dy)^2), that is, an infinitesimal version of the Pythagorean theorem; this can be deduced straightforwardly by reasoning about approximation by secant lines. When you put everything in terms of a common parameter t, you get the familiar sqrt((dx/dt)^2 + (dy/dt)^2) dt. $\endgroup$
    – Daniel Hast
    Dec 26, 2018 at 21:53
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    $\begingroup$ Of the items on this list, the only one I agree with is #1. $\endgroup$
    – user507
    Dec 27, 2018 at 1:37
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    $\begingroup$ Several of those listed are clearly better NOT memorized: 3. Integration by parts follows by integrating the derivative of a product of functions (moreover, thinking of it this way helps avoid sign errors); 4. arc length is obtained by approximating a curve by polygonal curves and considering their lengths as Riemann sums; 7. This is better deduced as a special case of the Cavalieri principle - volumes are obtained by integrating cross-sectional areas; 5. memorizing when to use particular substitutions is precisely the sort of thing to be avoided. $\endgroup$
    – Dan Fox
    Dec 28, 2018 at 9:59
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    $\begingroup$ Ben: " Here I am asking for a more comprehensive list that covers topics in Calculus I, II and III." $\endgroup$
    – guest
    Dec 28, 2018 at 17:57
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    $\begingroup$ Ben: You are actually "anti memorization" in general. Zuriel needs to get different points of view. But he should weigh yours with this realization. See this thread of yours: matheducators.stackexchange.com/questions/1026/… [As I commented there, the benefit to memorizing things, hopefully more by drill, is that one internalizes concepts more and has more instant tools at ones ready command when dealing with trickier problems. This is a different form of "memorize" than night before cram. Capisce? $\endgroup$
    – guest
    Dec 28, 2018 at 18:08
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They should know by heart how to set up, solve, and check problems. I discuss this in my answers to "What are non-math majors supposed to get out of an undergraduate calculus class?" and "How can you be perfect at maths (highschool)?"

The list in the original post is a good start for calculus techniques that the students should know. I would add to it the Reynolds Transport Theorem. In particular, they should be able to convert "what goes in, either stays in, or comes out" into a mathematical description. To do this, they need to know how to set up:

  • An area integral
  • A volume integral
  • A dot product (to calculate the component of a vector that is perpendicular to a surface in the outward direction)
  • A surface integral
  • The time derivative of a surface integral (a.k.a. flux)
  • The time derivative of a volume integral
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