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Math Help - How should you tackle the proofs in Apostol's "Calculus"?

  1. #1
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    How should you tackle the proofs in Apostol's "Calculus"?

    I'm trying to teach myself Calculus with Tom Apostol's first volume, and I love it but I am having trouble with all the exercises which require proofs. The only time I've done proofs myself was in high school geometry, and while I can follow complicated proofs by others I am confused as to how one "proves" some of the very basic statements arising from the field axioms. Is there a book or website which would prepare me for this kind of thing, or some general principles to guide me?

    In addition I am a little confused as to what structure the proofs should take. Are they just a series of numbered statements? How much non-mathematical language can you use? Thanks!
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  2. #2
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    Proofs are a logical sequence of conclusions. Unless you are writing a book, there is no need to concern yourself with much formality.

    Here's a silly example of sort-of induction and sort-of contradiction.

    My intent is to prove that all Whole Numbers are interesting.

    0 - is an interesting number, since it has unusual additive and multiplicative properties
    1 - is an interesting number, at least since it is odd but is not prime.
    2 - is an interesting number at least because it is even and it is prime.
    3 - is an interesting number at least because it is the lowest odd prime.

    This demonstrates that there are interesting Whole Numbers and that there are no non-interesting Whole Numbers less than these.

    If we assume, then, that there is a non-interesting Whole Number, there must be a "least" non-interesting Whole Number. However, since being the Least non-interesting Whole Number would be very interesting, indeed, we must conclude that there are no non-interesting Whole Numbers.

    Finally, since there are plenty of Whole Numbers, and none of them is non-intereting, we must conclude that all Whole Numbers are interesting. QEF

    That is not a very useful proof, but the style and conclusions are quite appropriate. Of course, the definition of "interesting" is not real clear - or should I say it is not Wholly clear.
    Last edited by TKHunny; February 15th 2010 at 07:05 PM.
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  3. #3
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    Thank you!
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  4. #4
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    Here are very good introductions, tips, exercises, and solutions for the sort of thing you're interested in. i recalled a professor of mine who was brilliant back from school and looked him up. on his site are these documents which ive attached.
    Attached Files Attached Files
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    Wow, thank you so much, Vince! Those help me so much!
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  6. #6
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    Re: How should you tackle the proofs in Apostol's "Calculus"?

    One question I keep bumping up against: what can you assume?

    Eg., I'm working on Apostol's exercises 1.7, proving properties of area based only on the area axioms. "Prove that every triangular region is measurable and its area is 1/2*b*h."

    I can easily see that you can define a right triangle as the intersection of two rectangles by setting an edge of the second rectangle as the diagonal of the first, thereby creating the hypotenuse of a right triangle. And I know from Euclidian geometry that the diagonal of a rectangle divides it neatly in two. If I can assume that fact, then the area of a right triangle formed from half a rectangle is half the area of the rectangle, ie, 1/2*b*h.

    And once you have such a right triangle defined, you create any other triangle by setting two right triangles of equal height side by side so that the common height forms the altitude of the non-right triangle. Its area would then be 1/2*b1*h+1/2*b2*h = 1/2*(b1+b2)*h = 1/2*b*h.

    But are you allowed to assume anything from Euclidian geometry? And if not, how on earth do you get what you need from just the area axioms?
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