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Advanced calculus

  1. Mar 19, 2013 #1
    1. The problem statement, all variables and given/known data

    Show that x=cosx, for some xε(0,∏/2).

    2. Relevant equations



    3. The attempt at a solution

    Define f(x)=x-cosx, i want to show that for some aε(0,∏/2), limx→af(x)=0. is this correct?
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Mar 19, 2013 #2

    Mentallic

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    Homework Helper

    No, that won't help you.
    If the y value at x=0 is negative, and the y value at x = pi/2 is positive (these values can be shown because it's easy to compute them), then what can we conclude from this?

    Does this logic extend to every function? Think about y=1/x, at x=-1 we have y=-1, and at x=1 we have y=1, but the function doesn't cross the x-axis at all.
     
  4. Mar 19, 2013 #3

    do you have a better explanation sir?
     
  5. Mar 19, 2013 #4

    Mentallic

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    Homework Helper

    I suppose.

    Take the function y=2x. How do we show it crosses the x-axis between x=-1 and x=1?

    Well, what is the y value at x=-1? y=2(-1)=-2. So at x=-1, the function is below the x-axis.
    What about at x=1? y=2(1)=2, which is above the x-axis. So since the function went from below the x-axis at x=-1 to above the x-axis at x=1, does this mean we can conclude that it must've crossed the x-axis somewhere in between? Yes!

    Why? Well again, think about the function y=1/x and try using the same procedure I just showed you. Everything seems to be the same, except that this function doesn't cross the x-axis. What's different?
     
  6. Mar 19, 2013 #5

    Borek

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    Staff: Mentor

    It is a direct application of a known theorem - I guess it was discussed during lecture or is mentioned in your book.

    Mentallic tries to guide you to the intuitive understanding behind this theorem.
     
  7. Mar 19, 2013 #6

    Mentallic

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    Right, it was silly of me not to mention the theorem involved in solving this problem.

    kimkibun, the Intermediate Value Theorem is what you're looking for.
     
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