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Problems Recommendation in Spivak's Calculus

  1. Jul 22, 2013 #1
    Hello guys,

    I'm currently working through Spivak's Calculus book and I've just reached the chapter "Number of Various Sorts." I find several of the problem quite challenging at the first place. I wonder if you guys could perhaps recommend me, based on your personal experience, both as students or even as instructors, which of the problems to do first after first reading. The one that tests my basic understanding of the chapter, so I can proceed to another chapter.

    I will try to do some of the more challenging problem perhaps later after I become more mature or more advanced in Math. Thankfully I've no great problem in the first chapter, and I'm quite amazed that I can already prove simple statements.

    The problem I've done so far for your consideration in "Number of Various Sorts" chapter are:
    Problems 1, 2, 3 with the exception of (c). Problem 4, 8, 12, and problem 27, the one with the professors.

    Thank You
  2. jcsd
  3. Jul 22, 2013 #2
    Apart from the ones you've already done, I recommend the following:

    5) This establishes a formula that is very important in mathematics. It is called a geometric series. Memorizing the formula is not a bad thing to do.

    9) This is a cute application of induction. You should be able to solve such things very quickly.

    10) This gives a proof of why induction works using the well-ordering theorem. It's worth doing.

    11) This gives a proof of strong induction. So you should do such things once.

    13) If you can handle contradiction proofs, then this shouldn't be difficult. It's very similar to proving ##\sqrt{2}## is irrational.

    19) Bernouilli's inquality is yet another fundamental inequality in mathematics. So this is important.

    21) The Cauchy-Schwarz inequality is very important in later mathematics.

    These are the minimum number of exercises you should solve in this chapter. If you want to solve some more, then be sure to ask!
  4. Jul 22, 2013 #3
    Thanks micromass, I'll certainly do those you marked, and ask if I've any problem with it. Hopefully this can also be useful for others who'd like to do Spivak's Calculus.

    Tbh, I'm quite worried about the theoretical abstract part of 9-10-11. For number 21, it looks a bit intimidating, but thankfully I've done the simpler Cauchy-Schwarz proof in the first chapter, so I'll try to tackle this general version as well.
  5. Jul 23, 2013 #4
    If you want a study buddy for Spivak, I'm looking for people to work with over the summer!
  6. Jul 25, 2013 #5
    Micromass, I'm going through the next chapter now, which is function. Surprisingly it seems less intimidating in comparison to the previous chapter. Do you have recommended problem here?

  7. Jul 25, 2013 #6
    Problem 1: Should be very easy

    Problem 2: Introduces you to one of the weirder functions in calculus, but it's a very important one!

    Problem 3: Should be easy.

    Problem 5: Is important for finding derivatives and stuff later on. Techniques such as these should be well-known

    Problem 9: Introduces you to characteristic functions (the ##C_A##). These are important in later mathematics

    Problem 12: Even and odd are very important notions. They can simplify integration a lot.

    Problem 21,22,23,24,25,26: Operations on functions such as these should be second nature. So do these exercises to get practice.
  8. Jul 25, 2013 #7
    Thanks for the explanations of why these problems are important, micromass! I've done a lot of the problems in the first half or so of Spivak, but sometimes I don't see the underlying importance.
  9. Aug 2, 2013 #8
    Hi micromass, I've finally reached chapter 5 on limit. I've finished the routine problem 1, 2, 3. Do you have any recommendation on top of that?
  10. Aug 2, 2013 #9
    Problem 7 and 8 contain nice statements that you need to show right/wrong. Very nice way of testing your intuition of the concepts.

    Problem 9 should be very easy if you grasp the concepts. Nevertheless, it's a handy result that is used a lot.

    Problem 11 gives one of the crucial properties of limits, that it's a local property. This property is so conceptually useful, that it requires some attention.

    Problem 12 tells you how limits interact with inequalities. This is crucial.

    Problem 13 is the squeeze theorem and an incredibly useful result.

    Problem 14 is something that people accept intuitively, but it is rarely proved in intro calc courses. Try to prove the result. Make one or two exercises from Problem 15 to see why it is nice.

    Problem 16 establishes continuity of some functions.

    Problem 18 involves an important technique for showing boundedness of functions. This is a very important technique that shows up a lot.

    Problem 19 is a standard problem in showing limits don't exist.

    Problem 29 is again something you see in a lot of intro calc courses. So it's worth doing.

    Problem 30 isn't particularly useful, but it's nice to work through.

    Problem 34 is again some substitution theorem. Many people accept it intuitively, but it's rarely proven.

    Problem 40 is cute.

    Problem 41 is fun. See if you can find the fallacy!
  11. Mar 30, 2016 #10
    Hey micromass, I've been somewhat following your problem recommendations in this thread and I find your brief descriptions of their purposes quite helpful. I just reached chapter 6 on continuity. Do you have any recommendations on that?
  12. Mar 30, 2016 #11
    Problem 1 is standard of course.

    Problem 4 can be a bit difficult, but is really easy once you know the trick.

    Problem 5 is really nice too.

    Problem 7 is a very well-known result. See if you can show it

    Problem 9 gives some alternative ways to see discontinuities

    Problem 10 establishes the continuity of some other functions

    Problem 12 is crucial. Part (a) is so very important, it's used all the time.

    Problem 13 should be easy but tests your familiarity with the definitions and concepts

    Problem 14 is a pasting result and is very important. This often shows up as a calculus problem, but here you need to use epsilon-deltas rigorously.

    Problem 16 has some really neat counterexample. Especially (a) is really worth knowing well. It is called the "topological sine function". Do think about (e) but don't waste too much time on a proof, it's very difficult.
  13. Apr 3, 2016 #12
    Thanks micromass. I went through those problems and finished reading chapter 7, which has to do with "three hard theorems" (it feels pretty intimidating compared to the previous ones!). Do you have recommendations for chapter 7?
  14. Apr 4, 2016 #13
    1,2,3 are standard but might be tricky sometimes
    5 should be easy
    6 is important, it parametrizes the unit circle
    11 is a fixed point theorem. It is actually the simplest case of the notorious Brouwer fixed point theorem.
    13 gives a neat counterexample called the topological sine function and gives a nice converse to the intermediate value theorem
    14 is of importance in functional analysis and basically in all of further analysis
    17 is also something you should be able to solve now
  15. Jun 3, 2016 #14
    Thanks a lot for recommending problems Micromass, I recently started studying from this book and have so far worked through all of your recommended problems (plus some more of the interesting problems in the book of course!) in the first three chapters.

    I'm now working through the problem set on chapter 4 and I was wondering if anyone who has studied from this book could give me a few tips on what problems and concepts are worth paying extra attention to in this chapter? It would be greatly appreciated! :)
  16. Jun 8, 2016 #15


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    in general there are specific graphs which are good comoputational practice, and there are conceptual relations more often with letters (unknowns) which express useful general facts, and these are also useful since you can use these facts again. In my "world student" edition from 1967, besides a selection of numerifal problems, i would suggest maybe 6,7,8,9,10,11 and 19 as interesting. in chapter 4, graphs.
  17. Sep 3, 2016 #16
    Thanks mathwonk!

    What about chapter 8 on least upper bounds? Does anyone have any recommendation on problems that are important to tackle?
  18. Sep 3, 2016 #17


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    for 16.e on continuity, try it assuming the fact that a function on [a,b] with limits at every point must be a uniform limit of step functions.

    in chap 8, maybe 3,4,6,8,14,15,16, and 11 looks interesting. but choose for yourself and see what happens.
  19. Sep 4, 2016 #18
    Thank you, sir!
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