I've been working through Spivak's Calculus for a few weeks; however, I am stuck on Chapter 5 (limits, epsilon-delta, etc.). I'm finding that I need to reference his earlier chapters (on inequalities) to follow the introductory proofs. Furthermore, I don't understand how he comes up with his methods for proving the theorems. Ideally, I would like to just 'grind' through the difficulties, but I am pressed for time--2 months+other daily activities--to learn the topics covered in high school BC calculus (AP curriculum). Due to these reasons, I am considering switching to another text, Art of Problem Solving Calculus, and coming back to Spivak later down the road. Based on my situation, would you recommend staying with Spivak or switching to the latter text?

A bit of auxiliary information: Spivak is my first introduction to Calculus; I need to learn BC calculus because the course I'm taking following the 2 months is multi-variable calculus.


closed as primarily opinion-based by Claude Leibovici, José Carlos Santos, user91500, C. Falcon, Shailesh Jun 19 '17 at 0:13

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    $\begingroup$ No one should study in university using only one book (per subject). $\endgroup$ – OR. Jun 18 '17 at 2:26
  • $\begingroup$ Sadly, I'm in high school and self-studying, so I have a limited budget. $\endgroup$ – guest Jun 18 '17 at 2:32
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    $\begingroup$ If you're pressed for time, then it's reasonable to study calculus from a less rigorous book. Lang's A First Course in Calculus is concise (even more so in the first few editions than the later ones) and is practically as rigorous as a book can be if it leaves out all "epsilonics." I haven't seen the Art of Problem Solving book, but it seems to be contest-oriented, so it is also likely to have parts that are more difficult than they need to be if all you want to do is meet the prerequisites for a non-rigorous multivariable calculus class. $\endgroup$ – user49640 Jun 18 '17 at 2:32
  • $\begingroup$ Just to add to what I said, I've seen the first, third and fifth editions of Lang's book. The first edition is missing certain parts I liked in the third edition, especially a section on practical optimization problems. On the other hand, some good parts have been removed from the fifth edition (such as the exercises on trigonometric limits), and the fifth edition also has a lot of peripheral topics added. So of the editions I know, I think the third is the best, and it's still quite concise. Including only the single-variable parts of the book, it's just 345 pages. $\endgroup$ – user49640 Jun 18 '17 at 3:13

In my opinion, I think people should stick with the common college texts to learn calculus: either go for Stewart or Thomas. They are always my go-to if I have to remember something.

Spivak is nice but it feels like a prop for the advanced student. Many of the analysis (epsilon-delta, inequalities, etc.) are used in a course in advanced calculus (typically in your third year of college) so you can ignore it.

For a typical multivariable calculus class, you should know

  • how to evaluate limits by squeezing, evaluation, or l'Hopital.

  • what continuity is.

  • what derivatives are and how to compute them.

  • how to use derivatives in optimization.

  • the chain rule.

  • how to integrate.

  • how to find areas and volumes via integration.

Most, if not, all of this, is used in Calculus BC.

TL;DR, find easier texts. The proofs can wait. Your local bookstore should have some decently priced material - a shop like Half Price Books is good for low cost calculus texts.

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    $\begingroup$ I diagree with this. I think with books like Stewart's that are a thousand pages long, it's too easy to get lost in the material. Books like that are very bad at really bringing out what's important. Also, they include a lot of so-called "real-world applications" that add little or nothing to mathematical understanding. In Stewart's book, you might see something like "Here's the formula for blood flow in an artery" followed by an optimization problem that gives zero insight into the physical model. Very few students have the patience to read through a book like that, and by the time they do... $\endgroup$ – user49640 Jun 18 '17 at 3:18
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    $\begingroup$ they're usually exhausted. If you're going to learn calculus from a non-rigorous book, do it from one that really emphasizes the important mathematical points, that does include and emphasize proofs when they're not too difficult for students at that level, and that does it in 600 pages or less. $\endgroup$ – user49640 Jun 18 '17 at 3:18
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    $\begingroup$ I should add that it's very good if the book's "real-world applications" require the student to have some understanding of the mathematical model he is using. For example, a question in Lang asks: "Find the length of the longest rod which can be carried horizontally around a corner from a corridor 8 ft wide into one 4 ft wide." Stewart tends to avoid problems like this where there is significant difficulty in setting up the mathematical model. $\endgroup$ – user49640 Jun 18 '17 at 3:35

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