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Math Help - help me with this intermediate theorem plz!!

  1. #1
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    help me with this intermediate theorem plz!!

    starting from 0 degree latitude and proceeding in a westerly direction along the equator, let T(x) be the temperature at the point x at any given time. Assuming T(x) is a continuous function of x, show that at any fixed time. there are at least 2 diametrically opposite points on the equator, say a and a+180, that have exactly the same temperature.
    Hint: construct a function F(x) using t(x) and use the intermediate value theorem.


    all i can come up is that we need to prove:
    T(x) = T(x+180)
    which is T(x) - T(x+180) = 0

    let f(x) = T(x) - T(x+180)

    if we can prove that f(x) has a root, then T(x) can = T(x+180)

    since T(x) and T(x+180) are continous function, so f(x) as well.

    that is as far as i can go

    how can i determine where f(x) is negative and positive ???

    thanks!!!!
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  2. #2
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    Quote Originally Posted by dull1234 View Post
    starting from 0 degree latitude and proceeding in a westerly direction along the equator, let T(x) be the temperature at the point x at any given time. Assuming T(x) is a continuous function of x, show that at any fixed time. there are at least 2 diametrically opposite points on the equator, say a and a+180, that have exactly the same temperature.
    Hint: construct a function F(x) using t(x) and use the intermediate value theorem.


    all i can come up is that we need to prove:
    T(x) = T(x+180)
    which is T(x) - T(x+180) = 0

    let f(x) = T(x) - T(x+180)

    if we can prove that f(x) has a root, then T(x) can = T(x+180)

    since T(x) and T(x+180) are continous function, so f(x) as well.

    that is as far as i can go

    how can i determine where f(x) is negative and positive ???

    thanks!!!!
    You are off to a good start. Consider f(0) and f(180):

    f(0) = T(0) - T(180)
    f(180) = T(180) - T(360) = T(180) - T(0) = -f(0)

    Think you can take it from there?
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  3. #3
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    yep, thx. i figure that out when i am having dinner as well XD
    but i sub in a and a + 180 instead of 0 and 180
    basically just the same thing

    so f(a) and -f(a) are two opposite number, since f(x) is a continous function, there should be a point within the interval [a, a+180] that makes f(x) = 0, which means T(x) = T(x+180)

    XDXD
    am i correct ?
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  4. #4
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    That's right.

    (You might want to consider the case f(a) = 0 separately, just to be sure you have all the possibilities covered. But what you have is correct.)
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