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Rolles Theorem

  1. Mar 11, 2012 #1
    1. The problem statement, all variables and given/known data

    Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Rolle's Theorem. (Enter your answers as a comma-separated list.)
    f(x) = cos 5x, [π/20, 7π/20]

    2. Relevant equations

    Rolles Theorem states:
    Let f be a function that satisfies the following three hypotheses:
    1. f is continuous on the closed interval [a,b]
    2. f is differentiable on the open interval (a,b)
    3. f(a)=f(b)

    Then there is a number c in (a,b) such that f'(c)=0




    3. The attempt at a solution

    1. F is continuous on the closed interval because there is no where on the interval that f(x) is undefined.
    2. f is differentiable on the open interval because it is continuous on the closed interval
    3. This is where I get confused, these are not a part of the unit circle, do I just plug them in to the calculator to determine that f(a)=f(b)?

    Then, to find C I must first find the derivative-
    f(x)=cos5x
    f'(x)=-5sin5x
    then I set this equal to 0 to find c
    0=-5sin5x
    so f'(x)= 0 when x is 0, when the sin5x=0. I don't even know how to go about finding this with numbers that are not on the unit circle.
     
  2. jcsd
  3. Mar 12, 2012 #2
    First off, 2 isn't entirely correct - there are functions which are continuous but not differentiable (e.g. absolute value).

    As for the third part, it might be easier to make a change of variables: y = 5x. Then you're looking at cos y. What's the interval this would be on? It'd be [5*(pi/20), 5*(7 pi/20)] = [pi/4, 7 pi/4] (note that 8 pi/4 = 2 pi, so you're close)

    As for finding C, you're almost there. sin 5x = 0 is the correct equation. When is sin y = 0? How does that translate into x?
     
  4. Mar 12, 2012 #3
    So I am looking for a value between [pi/4, 7pi/4] rather than [pi/20,7pi/20]?
    This makes sense and I understand that.
    As for finding C, sin y= 0 at pi/2 and 3pi/2. I'm not sure how to translate this into x. I thought I would have to multiply it by 5 in order to translate it into x getting 5pi/2 and 15pi/2, but this was wrong. I then looked again and thought I realized my error and divided by 5 instead, getting pi/10 and 3pi/10, and I was still wrong. Where is my error translating this in to x?
     
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