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How should I approach self teaching myself Calculus?

  1. Jun 22, 2009 #1
    I've been trying to start teaching myself Calculus recently out of interest and the amount of useful info on PF is awesome but I can't seem to find a good way to actually start it off. As far as books are concerned, it seems that Stewart's is the standard but supposedly it's too watered down. Then I found Spivak's and Apostol's books which seem to be regarded as the best if you want to gain a deep understanding of Calculus. I was able to find a copy of Spivak's but it seems that it's mainly proving theorems and analysis than actual Calculus. As far as I know, Apostol's requires some previous calculus knowledge and even then it sounded like it was mainly proofs and analysis. I don't want to flood my brain with anything too advanced, I just wanna teach myself single variable calculus but at the same time I want to understand it really well, not just be able to solve the problems by memorizing steps...

    I might not have a strong mathematical background since I just finished grade 10 and we didn't learn any calculus yet, except for fundamental differentiation which was easy and useless by itself in a way. I'm a pretty good self learner, guess I'm just overwhelmed by all the information available and just looking for a place to start :)

    Sorry for the long post and thanks in advance for any advice or suggestions!
  2. jcsd
  3. Jun 22, 2009 #2


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    I think "deep understanding" to many people means the proofs and analysis you're finding in books like Spivak and Apostol. The idea is that when you understand why the theorems of calculus are true, you're able to apply them better. But if you're getting confused by the proofs etc., then just forget about it for now - you can always learn calculus in a simple practical way the first time and then go back through the proofs.
  4. Jun 22, 2009 #3
    There is a book that I have been using called
    Calculus: An Intuitive and Physical Approach

    It has been a great source, it teaches calculus by using physics and physical concepts so that as you work through it the calculus comes naturally. The book also lets you practice and apply what you learn to physical problems.
    Last edited by a moderator: May 4, 2017
  5. Jun 22, 2009 #4


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    Don't worry about learning the theory too much for now. Just make sure you know how to do the problems.

    As far as the question:

    "but I can't seem to find a good way to actually start it off"

    What exactly do you mean by that?
  6. Jun 23, 2009 #5
    Yeah, just go through the standard exercises first, and don't worry about proofs immediately. Read and try to understand them, but don't linger on them or get discouraged if they're elusive. Just being exposed to the machinery of it will put you in a better frame of mind to understand the essence of it.
  7. Jun 23, 2009 #6
    Last edited by a moderator: May 4, 2017
  8. Jun 23, 2009 #7
    Hmmm, sounds interesting, especially since I want to learn Calculus to understand Physics, will check it out. Just hope it proves challenging! Thanks.

    Well, I didn't want to study from a watered down or easy book fearing that it might ruin the experience. I want to learn it correctly the first time around and was just looking for advice on getting it right the first time.

    Not very rigorous... Guess it depends on the person but I'll check out the preview :) Thanks!

    And thanks for the advice guys, seems that the best way to go is to learn the basics and how to solve the problems first then the proofs.
  9. Jun 23, 2009 #8
    Well, calculus is one of those things where you need to have a clear idea of why you are learning it in the first place. If its purely mathematical, then you probably should have a more deeper understanding of it than say a physics student who may need it purely to get from point A to point B. If you are interested in calculus on its own, you may want to pick a more mathematically inclined textbook or if you are interested in its applications, you might want to try Calculus: An intuitive and Physical Approach by Morris Kline (2nd edition), quite a wonderful book!
  10. Jun 23, 2009 #9
    Yeah well I want to learn it so I can apply it to Physics but at the same time I don't want to breeze by it and only know how to use it, I also want to understand it deeply.

    Thanks for the book suggestion, more the reason to check it out!
  11. Jun 23, 2009 #10
    That is a good approach. Sometimes, when you solve problems, you will realize some "hidden ideas" and later on when you learn the proofs and theorems proper, these hidden concepts will be presented in a more "concrete" manner.
  12. Jun 23, 2009 #11
    I wouldn't worry about "rigor" too much if you're learning it for the first time. If you know intuitively what a limit is, for example, then you can understand the epsilon-delta definition, and even then it takes some getting used to. I can't imagine not having any idea of what a limit is and then going right into the formal definition. I'm not sure that's beneficial.

    I'd say get something like Stewart or Thomas, get a good understanding of a concept, and then go over the same concept in something like Apostol to get the rigor.
  13. Jun 23, 2009 #12
    Apostol is quite advanced. You might get discouraged by it. George Thomas was my first exposure to Calc, but that was in the pre-calculator days.

    There's a truly wonderful book called "Calculus in Context" by Callahan, Cox, Hoffman, O'Shea, Pollatsek and Senechal. Graphing calculators are central to it, and it's presentation style is a bit unorthodox. However, I think it's great for self-study, (as long as you have a graphing calculator, of course).
  14. Jun 23, 2009 #13


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    Calculus was invented purely for the reason that we need to model and deal with changes in various systems. If you realize that by understanding change in a system, you understand possibly how that system functions at some other level, then you have the basic idea of why calculus was the "natural next step" in the evolution of mathematics.

    We can extend it to vectors, differential forms, whatever. The point is that we are using some framework that allows us to analyze in some form how a system changes. It can change with respect to time, to other variables, to its past historical state: basically anything relative to some variable.

    Now we treat the physical world as Newton did as simply a system that changes. We theorize that the change is based on certain rules that seem to approximate reality. After defining firstly what the system is modelling, and describing then the mechanics of the system, we then move to postulate how it changes with respect to other quantitive measures of physical phenomena.

    For example we define acceleration as the rate of change of velocity with respect to time. But through calculus we also know that velocity is linked to the "anti-derivative" of acceleration. Basically this means that if start at an initial velocity and add up all the "changes in acceleration" we will end up with some final velocity. The fundamental theorem of calculus is quite a powerful result that reflects that property. In this case the fundamental theorem will refer to an antiderivative with two endpoints saying that if we integrate the function of acceleration over some domain: that will be an antiderivative corresponding to a final velocity value minus a initial velocity value.

    This is very powerful. It means that in a system if we know how something changes we have a very powerful shortcut method of predicting things like velocities, distances, volumes, basically anything quantitative in a way that is so beautifully simple and yet so very powerful.

    If you realize what I'm saying then I think that you will understand why calculus is taught as the "fundamental" way to analyze things or simply as the topic of "analysis".
  15. Jun 26, 2009 #14
    I self taught differential and much of integral calculus.
    I later decided to return to school to study math and physics. I still have never taken Calc I and the math department has waived my requirement for that.

    I would recommend learning the absolute basics first. You can learn how to differentiate basic powers, logs, and exponentials in an afternoon. You can learn to differentiate trigonometric identities in about the same amount of time.

    I still don't know all the "rules." I suppose I know what the chain rule is, and that's all I've ever used.
    If you can learn logarithmic differentiation, you can do that to cover all the difficult mix of functions.

    After you can do the basics (it really should only take a few days before you can differentiate almost any basic function) work through as many problems as you can until you feel comfortable differentiating any problem you'd come across with any regularity.

    Then go back and learn the theory....In my opinion, you'll understand the theory MUCH more when you're not trying to learn how to use the rules at the same time.

    I taught my 8 year old niece differential calculus in a few hours on Easter. She has no idea WHY she's doing what she's doing....but give her any basic power or exponent and she can give you the first differential of it.
    In my opinion, she'll get much more out of her Calculus courses later on with that basic knowledge.
  16. Jun 26, 2009 #15
    I'm not exactly sure how differentiating a lot of functions helps in learning the theory all that much. I mean the very basic algebraic rules for differentiation are easy to prove, and I can't imagine anyone skipping over them for the sake of computational fortitude. The proof of the chain rule (a rigorous one) is considerably more difficult it does not really have that much to do with learning to take derivatives.

    While I think learning the basics without being too rigorous is a fine approach, it is perhaps a bit overemphasized in this thread. For instance, you do not need to know what a limit is before you read Spivak. In fact, the very first thing he explains in the chapter on limits is the intuitive notion of the limit. Although Spivak does not proceed to provide many computations of limits, he provides many graphs to help the reader visualize the idea of a limit, but he is doing so with the specific aim of presenting the rigorous definition later on. But before he gives the definition, he essentially builds up to the definition step by step. He provides examples that show you how you would eventually go about proving a limit, but are just informal enough to help you gain the intuition.

    I guess what I am saying is that if the OP is motivated enough to learn calculus, he can learn it from a text such as Spivak or Apostol. Apparently, he managed to post on this forum, so there is no excuse for not having other resources.
  17. Jun 28, 2009 #16
    I think you are both right. Perhaps the "best" approach in a given context depends largely on the learning styles of each individual. Some of us learn well by going from the concrete to the abstract, learning specific examples first and then embracing the general patterns that they share. Others prefer the satisfying rigor that a proof-based approach takes.

    My two cents worth of experience tends to favor the skill-acquisition-first approach, and then get down to the theorem proving later. I've had high school students who had real trouble with factoring, because they didn't "get" what multiplication really was. It wasn't clear to them for example that the word "of" means "times" when you talk about two thirds of a cake. I think if they had to memorize the times tables when they were younger, it would have come easier to them. Unfortunately, all things rote are considered evil nowadays.
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