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Lang before Spivak

  1. Dec 19, 2013 #1
    I'm looking to begin an independent study of Calculus, having read Lang's Basic Mathematics. I would like to know if I should move straight on to Spivak's Calculus, or start with Lang's A First Course in Calculus? If it affects the answer in any way, I have had some rudimentary experience with calculus and proofs, but nothing truly concrete.
     
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  3. Dec 19, 2013 #2
    I strongly advice against starting with Spivak. The book is excellent and certainly something you should read, but not as a first encounter with calculus. Lang's first course in calculus is very suitable as introduction to calculus. It covers the topics intuitively, but not too rigorous. So I think that doing Lang first, and then Spivak or Apostol is a very good choice!
     
  4. Dec 19, 2013 #3

    Student100

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    I'd like to add, since I replied to up your other post on a similar topic, if you go through Lang's book try to find supplemental information on limits.
     
  5. Dec 21, 2013 #4

    QuantumCurt

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    I took Calculus I this last semester, and I'm currently working through the Lang book over break to keep it fresh and to get some different perspective. It's a very well written book with a lot of clear proofs and relevant examples. I aced calc I, but there are still some aspects and ideas in the Lang book that were never covered in the course. I'm also using it to help get a little bit ahead on some of the main ideas of Calculus II. From what I've seen of it so far, I highly recommend the Lang book.

    I've heard that the Spivak book is a lot more challenging, and not really practical as an introduction. I plan on picking it up sometime down the line though.
     
  6. Dec 22, 2013 #5

    Student100

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    Lang's book is fine, but it shows it's age.
     
  7. Dec 22, 2013 #6

    QuantumCurt

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    I'd also add that I agree regarding limits. The treatment of limits in Lang's book is fairly cursory. It definitely doesn't contain the rigorous treatment that I got in my calc I course.
     
  8. Dec 23, 2013 #7
    That is completely true, but that's Lang's intent. He finds that a rigorous treatment of limits is not something to waste time on in a first calc course. People were doing calculus for hundreds of years before they made sense of limits, so it's not that a rigorous understanding of limits is essential to calculus. People might disagree with this of course, but I think he has a point. More rigorous books like Spivak or Apostol do treat limits the correct way, but they're more second course calc books or even analysis books.
     
  9. Dec 23, 2013 #8

    QuantumCurt

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    Yes, this is quite true. We covered Epsilon-Delta in my calculus course, but never really did much with it. The first section of chapter one (the chapter on limits) was focused on the Epsilon-Delta definition, and it seemed rather confusing and unnecessary. It seems redundant to prove limits before you even really understand the concept of what a limit actually IS. That was one of the more abstract ideas out of the whole course. It doesn't seem very productive to start the course with it.

    Regardless, I still feel like Lang could have incorporated a slightly more comprehensive treatment of limits without getting into Epsilon-Delta. The first part of the book seems kind of backwards in some ways. He covers the limit definition of a derivative before he even formally covers limits. That struck me as odd. Perhaps other courses are different, but it seems more natural to me to start a calculus course with limits, rather than jumping right into derivatives. That may just be due to the fact that I learned it in that order though.
     
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