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I'm finding myself falling into the trap of memorizing some sort of cookbook recipes in calculus class instead of using my brain. I'd really like to find some conceptually challenging calculus 1 (freshman undergrad) problems to break this. The problem is that the "challenge problems" in pretty much any textbook I've found are just really tedious calculations or algebraic manipulations, aka "number crunching".

Thus, I'm looking for some sort of book or website (preferably with answers) that had truly creative, stroke-of-genius required calculus problems.

To further refine what I'm looking for, I'll add that the problems:

  • shouldn't be just asking for a proof. I have no idea how to even go about a formal proof. (learning this is a separate quest)

  • shouldn't require any theorems or facts outside of a pure math freshman calculus course (like some weird law of physics or statistics).

  • shouldn't require overly complicated integration or derivative formulas. I'll probably just crunch any resulting equations with Wolfram Alpha.

  • isn't just a very long problem asking to find 75 different things from 75 unsightfull equations.

  • doesn't ask me to hand-draw some stupid graph.

I've had some minor success in learning how to think more in depth by reading very old calculus books from like 1962, but they either don't have practice problems or keep asking me to write proofs or do something super mathematically rigorous. I'm really more after a brain-teaser puzzle...but that works my calculus muscles a bit more.

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closed as too broad by user223391, C. Falcon, Daniel W. Farlow, JMP, R_D Jul 1 '16 at 4:32

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ Can you give us an example of a problem which you liked? $\endgroup$ – user 170039 Jun 30 '16 at 4:19
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    $\begingroup$ Maybe you should get a copy of Spivak? Apostol, as you say anything from before about 1970 might be useful. There does exist a book in the vein you indicate for gifted highschoolers, alas, I can't find the link... $\endgroup$ – James S. Cook Jun 30 '16 at 4:20
  • $\begingroup$ You've told us a lot about what you don't want to learn (solving complicated integrals or derivatives, writing proofs, etc.) What is it that you do want to learn? Calculus (as opposed to what we teach in the US as Analysis) is traditionally largely about evaluating derivatives and integrals by hand. You can go beyond that by focusing on how to apply calculus (e.g. to problems in physics) or you can dive deeper into the mathematics by proving the theorems, but you don't seem to want to do either of those. It's very unclear what you are asking for. $\endgroup$ – Brian Borchers Jun 30 '16 at 4:21
  • $\begingroup$ pretty much anything where I'd need to at least come up with my own equation. In my textbooks, literally every problem is 1. copy given equation. 2. take derivative/integral, 3. plug and chug. The tiniest variation from that would be fantastic. I'm looking for an example problem now.. $\endgroup$ – Flurpy Jun 30 '16 at 4:27
  • $\begingroup$ And I AM learning the stuff I said to exclude here, it is just that I already have good resources for that. Now I see how weird my question looks! "teacher, can I have math, but without the numbers?" $\endgroup$ – Flurpy Jun 30 '16 at 4:28
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There are really endless conceptual questions which are not formula based. You need only ask yourself to find examples of functions which are with or without certain properties. For example, find a function which is once, but, not twice differentiable. Or, to take it up a notch, find a function which is everywhere continuous, but, nowhere differentiable.

You can look at a few given sign-charts for a potential $f(x), f'(x), f''(x)$ etc... then ask; is it possible? To answer this you must have a complete understanding of the first, second and higher derivative tests.

Anyway, I think you want a book, so, I'd guess this is more or less on target (but, I think reading Apostol, Spivak and older texts, really anything without color might be good) would be:

https://www.artofproblemsolving.com/store/item/calculus

I think if you ask this question with more tact it might get a lot of interesting answers.

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