# There are only 40 calculus problems

There is a saying in mathematics that there are really only 40 calculus problems, and all the other problems are just permutations of these using different numbers or stories attached to them. This is obviously an oversimplification, but it is true that there are many specific problems asked here daily that are simply an altered version of one of the prototypical problems of calculus.

To that end, I feel it would be useful to have a list of typical calculus (I, II, and III) problems which are explained and solved in their abstract form in a manner more thorough than is reasonable to give on any one specific question, in order to serve as a "problem solving guide," and which can be used to quickly link a user to a more thorough, but general answer to their question. I have made an example problem about one sided limits, which I think is illustrative of how well suited the self-answering feature of MSE is to this project, furthermore, many users could contribute to the given general answer making it more clear and thorough, as opposed to contributing individual less thorough answers.

Ideally, I think that this might be used in addition to specific answers, for example if a user asks

What is $\lim \limits_{h \rightarrow 0^+} \frac{h+1}{2}$?

a possible response might be:

Using the graphical method shown here, we can see that the graph (Insert graph here) of the function approaches $\frac{1}{2}$ as $h$ approaches $0$ from the positive side.

What do you think about this idea? I have also started an answer, which you can feel free to edit, showing the possible areas and the specific questions, as quick reference to anyone that wants to use this list.

• I didn't sleep too well, so I upvote this as one more attempt to put the lazy askers and rep wh&%¤ answerers under control. Calculus is probably the worst offending tag here, but similar things happen in all high volume tags. But, did you search Meta? I think there already is a list of commonly asked questions. Martin Sleziak will find it faster than I ever could, so... – Jyrki Lahtonen Jun 1 '15 at 6:26
• @JyrkiLahtonen Here is the largest list, but it seems to be a repository of common duplicates more than anything. – Juan Sebastian Lozano Jun 1 '15 at 6:37
• @JyrkiLahtonen Did you mean List of Generalizations of Common Questions. (I had to comment, since I was explicitly called out.) – Martin Sleziak Jun 1 '15 at 7:00
• Thank you, Martin. That's the one :-) – Jyrki Lahtonen Jun 1 '15 at 7:04
• – Asaf Karagila Jun 1 '15 at 18:17
• Oh, so like a reference for duplicates? – Alexander Gruber Jun 3 '15 at 1:41
• Also related (especially to calculus): Catalog of standard exercises. – Martin Sleziak Dec 29 '16 at 20:24

While this may be true - and certainly, many problems we solve within the tag may seem like deja vu - attempts to effectively shut down the tag in the manner described by the OP will do nothing to stem calculus questions or the need for the tag. The reason is that outlines such as that provided by the OP for the one-sided limit differ very little from the textbooks that most people posting questions find confusing.

While I applaud the OP in acknowledging that the tag is quite overpopulated with seemingly repetitive questions, I believe that such an overpopulation is the price we pay for distinguishing over a textbook. Here we have dozens of examples of one-sided limits. So what? This makes Math.SE a very valuable and unique resource.

Cutting off the flow of new examples and replacing them with a one-size-fits-all tutorial, which might seem like a very satisfying exercise for the OP, will make Math.SE little more than yet another online calculus textbook. Believe me, the powers that be which invest a lot of capital in hosting us and letting us play for free will not be so excited to devote so many resources to something that is an effective clone of a lot of other free stuff on the web.

• I think that this is a fair critique of my proposal, as it could actually be very detrimental to MSE as a whole. However, I feel that perhaps it might actually serve to better the quality of answers by providing a general overview of the topic, and therefore allowing answerers to focus more specifically on the parts which the asker is having trouble with. Furthermore, I recognize that this is only going to be effective in a subset of the calculus tag, but there will be plenty of cases where this would not be effective, if the question is not cookie cutter homework, so to speak. – Juan Sebastian Lozano Jun 1 '15 at 11:48
• @JuanSebastianLozanoMuñoz: thank you for your fair-minded, non-defensive response. – Ron Gordon Jun 1 '15 at 22:02
• If I got it right I find the sentiment behind ".. such an overpopulation is the price we pay for distinguishing over a textbook" interesting. Never thought about it that way! As long as the principle is used responsibly (= not as an excuse to operate a homework mill), which I'm sure enough people do, it has a lot of merit. +1 for introducing this POV - non-confrontationally to boot :-) – Jyrki Lahtonen Jun 2 '15 at 5:57
• (cont'd) My opposition to such an overpopulation comes from the belief that it is at odds with the principle that question & answers should have some permanent value (in order to be kept on the site). Elementary number theory and calculus (and a few other tags) have so many essential duplicates that many users apparently find searching for the best previous match too taxing. Which only exacerbates the problem. – Jyrki Lahtonen Jun 2 '15 at 6:05
• BUt Q&A's should have some permanent value. If they don't have any value, if they're worthless, then we can simply delete them. I believe that's the whole point. – Najib Idrissi Jun 2 '15 at 7:17
• I feel that the difference between "do this problem ... solution for the problem" and "I'm having trouble with this particular aspect of a problem ... explanation about that aspect" is being overlooked somewhat here. – user14972 Jun 2 '15 at 21:19

There is a legitimate need for figuring out the gaps and/or errors in questioners' knowledge in a much more individualized way than simply saying, "Here, read this." On the other hand, often one would like to say, "Here are some ideas about where I think you went astray: _________. Read this for more details about _____ and ______." And it might be handy if "read this" pointed to a question (and a variety of answers) on MSE itself rather than to some other site, useful as some of those sites might be.

Sometimes I find that in fact someone has asked a question that was asked (or at least answered) in general enough terms that I can point to it as the "read this." It's nice when that happens; it can make it easier to answer questions that way than when one has to worry about how much one has to cover of the information already available elsewhere (and how to cover it here).

I'd likely use the list of 40 (or whatever) questions at least for this purpose, though possibly not exactly the way it is shown in the original question.

• Should this have been a comment instead of an answer? The question did ask, "What do you think," but maybe other kinds of thoughts were intended. – David K Jun 1 '15 at 19:53
• No, I think it is fine. If you wonder because of the down-vote it is likely just that somebody disagrees, which is a reason to dv on meta. (Although I for one find it often not really useful when not explained as there are typically many and contradictory ways to disagree with something.) – quid Jun 1 '15 at 19:56
• OK. I do also wonder about the point of disagreement, but ultimately it is up to others whether they want to describe specific criticisms. As long as the higher-voted answers are deservedly so (and I think they are), it's not really a problem. – David K Jun 1 '15 at 20:03
• +1 Although originally I did not intend to use this list like you speak of above, after reading the answer by Ron Gordon, I think that the use you proposed is actually much more beneficial to the site than my original intention. – Juan Sebastian Lozano Jun 2 '15 at 1:10

As a "problem solving guide" reference list:

# Calculus I

### Limits and Continuity

• One sided limits
• $$\epsilon-\delta$$ proofs
• Special Trig Limits
• Limits of Rational Functions
• Proving Continuity
• Identifying types of Discontinuities

### Differentiation

• Derivative of a polynomial
• Derivative of a composite function (Chain Rule)
• Derivative of two functions multiplied together (Product Rule)
• Derivative of a rational function (Quotient Rule)
• Derivative of an exponential function
• Logarithmic differentiation

### Applications of Differentiation

• Related Rates
• Slope Fields
• Separable differential equations