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So I'm currently a calculus student, next semester I'll take calculus 2. I'm wondering if I should go to a book like the one by Spivak which is entirely different from the book used for my course, and learn or in a way re-learn it the way it's presented in that book?

Reason I ask this is, I know how to carry out the different steps for the calculus problems I've thus far been exposed and how to apply them to different problems but I don't really understand what I'm doing, if that's the appropriate way to word it.

Would a more proof based approach help me in this understanding? Will I always lack some aspect of understanding if I don't know how to prove these problems?

To quote one of the comments on this question, the question can also be put

"will studying calculus in a proof based manner help in understanding the techniques I've already learned"

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  • $\begingroup$ It would be helpful if you could give some details on which steps in what sort of arguments it is that you "don't really understand". This would make it easier for editors here to help you. $\endgroup$ – Mikhail Katz Dec 10 '14 at 15:55
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By learning a proof based version of the course, you will gain an understanding of how the mechanisms themselves work, rather than simply how to use them. It is the hope that once this is mastered, you will be able to develop new mechanisms for solving new problems - this is called research. It is very similar to knowing how to use a calculator versus knowing how to program a calculator.

That being said, I recommend not putting off learning new things because you feel you don't have enough of a grasp of the old. This will typically stunt your development rather than help it. I believe you will find that in the process of learning new subjects, you will learn more about the old ones than you would have if only studying the old.

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  • $\begingroup$ Of course knowing more about the subject will always further his understanding, but jumping from Calc 1 to an Advanced Calculus course is no easy transition. It requires some back ground on set theory, and I don't think he'd be finding the boost in understanding he is looking for by the time he gets started in his Calc 2 course next semester $\endgroup$ – jameselmore Dec 10 '14 at 16:11
  • $\begingroup$ Do you read my answer as suggesting he take an advanced calculus course? Also, the prerequisites for an advanced calculus course will vary with how it is taught. You seem to be assuming that it will be taught precisely as the one you may have taken, although this is only a guess. $\endgroup$ – RghtHndSd Dec 10 '14 at 16:16
  • $\begingroup$ That's fair. You are correct that I am making an assumption there. Please excuse my previous comment, I was not trying to imply that he should move straight to an Advanced Calc course, but I think we can agree that if he's looking for something to help him next semester, trying to teach yourself proof based calculus is probably not the most efficient approach in such a small time range. He is better off focusing on material which could help him in the course $\endgroup$ – jameselmore Dec 10 '14 at 16:21
  • $\begingroup$ @jameselmore: Without knowing a lot more information about the user, I can not make such a recommendation. $\endgroup$ – RghtHndSd Dec 10 '14 at 16:28
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Having more proof writing experience is very useful but not really 'necessary' until you get to more Advanced calculus topics. Any further research will certainly help build an understanding although studying proofs may not be the foundation that you are looking for. If you are in introductory level calculus, you may be better off finding texts and drills on carrying out derivatives / integrals. Probably best not to get caught up in all the epsilons and deltas until you have to.

You may want to check out: Counterexamples in Calculus By: Sergiy Klymchuk

It poses a bunch of propositions in calculus which at face value may appear true but are actually false. Finding counter examples can be a strong way to build understanding and this book also offers solutions to all the problems (it's designed to be a problem set reference for professors).

Hope that helps

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    $\begingroup$ I believe the question here is "will studying calculus in a proof based manner help in understanding the techniques I've already learned". If this is indeed the question, then I believe your answer does not address it. $\endgroup$ – RghtHndSd Dec 10 '14 at 16:00
  • $\begingroup$ "Having more proof writing experience is very useful but not really 'necessary' until you get to more Advanced calculus topics." $\endgroup$ – jameselmore Dec 10 '14 at 16:00
  • $\begingroup$ You claim it is "very useful" - what precisely does this mean? Useful how? Does it help one understand techniques learned in a non-proof based version of Calculus? Why is this? $\endgroup$ – RghtHndSd Dec 10 '14 at 16:03
  • $\begingroup$ Moving from practice based calculus (intro levels) to proof based calculus usually requires a shift in how you approach thinking about the problems. In a proof based course you would be spending more time defining and proving that a derivative exists for families of functions instead of actually finding what that derivative is $\endgroup$ – jameselmore Dec 10 '14 at 16:07
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Mathematics in general and calculus in particular are not about proof. Calculus is the study of the aspects and properties and implications of instantaneous rate of change of a function with its variable(s). Proof is a major concern of mathematicians and is necessary to protect us from untruth kust as other specialties that protect us from dangerous food and water and air and houses and streets and cars and planes etc. It is not clear what you mean by not understanding. I was told that in defense of my master's thesis and I have found it to be a very worthwhile question.

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