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Math Help - Calculus Help Needed!

  1. #1
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    Calculus Help Needed!

    If f(2)=4 and f'(x)>2 for 2<x<11, how small can f(11) be?

    Thanks!
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by asnxbbyx113 View Post
    If f(2)=4 and f'(x)>2 for 2<x<11, how small can f(11) be?

    Thanks!
    Note that f satisfies the conditions for the mean value theorem to be
    applicable on [2,11].

    Suppose f(11)<22, then by the mean value theorem there exists a c in (2,11)
    such that:

    f'(c)=(f(11)-f(2))/9 = (f(11)-4)/9,

    but if f(11)<22 this is <2, which contradicts the assumption that f'(x)>=2.

    Now let f(x)=4+2(x-2), then f'(x)=2 for x in (2,11), and f(11)=22.

    Therefore f(11) can be as small as 22 and no smaller.

    RonL
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  3. #3
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    Hello, asnxbbyx113!

    Did you make a sketch?


    If f(2)=4 and f'(x) \geq 2 for 2 \leq x \leq 11, how small can f(11) be?

    We consider the graph of f(x) from x = 2 to x = 11.

    SInce f(2) = 4, the graph starts at (2,4).
    Code:
            | (2,4)
            |   *
            |
            |
          - + - + - - - - - - -
            |   2

    From x = 2 to x = 11, the slope is \text{at least 2.}

    The very "lowest" graph would be straight line from (2,4) with slope 2.
    . . Its equation is: . f(x) \:=\:2x

    It would look like this:
    Code:
            |
            |                 * (11,22)
            |               * :
            |             *   :
            |           *     :
            |         *       :
            |       *         :
            |     *           :
            |   *             :
            |   :             :
            |   :             :
          - | - + - - - - - - + - -
            |   2            11

    Hence, the least value for f(11) is: . f(11) = 22

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  4. #4
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    Quote Originally Posted by CaptainBlank View Post
    Note that f satisfies the conditions for the mean value theorem to be
    applicable on [2,11].
    I am honored that I taught you something. I remember somebody asked a similar question. You answered it in some other way. I answered it elegantly via MVT. Now it seems you liked mine more.
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  5. #5
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    Quote Originally Posted by ThePerfectHacker View Post
    I am honored that I taught you something. I remember somebody asked a similar question. You answered it in some other way. I answered it elegantly via MVT. Now it seems you liked mine more.
    Well I've always known about the use of the MVT, but I thought it
    inappropriate for the level of knowlege of the student involved, as
    it probably is here also, which is why Soroban is thought a better helper
    than both of us .

    A MVT proof is more appropriate to a first Real Analysis course than to
    any course calling itself Calculus X.


    It seems some of your tendency to answer questions with methods
    above the heads of the asker has rubbed of on me

    RonL
    Last edited by CaptainBlack; January 21st 2007 at 02:48 AM.
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  6. #6
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    Quote Originally Posted by CaptainBlank View Post
    .
    It seems some of your tendency to answer questions with methods
    above the heads of the asker
    That is what you think, I do not see how I make myself complicated. I know that I do not try to be simple I consider sme simple approaches mathematically inappropriate and need to use a better one.
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  7. #7
    Grand Panjandrum
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    Quote Originally Posted by ThePerfectHacker View Post
    That is what you think, I do not see how I make myself complicated. I know that I do not try to be simple I consider sme simple approaches mathematically inappropriate and need to use a better one.
    If they haven't met the MVT, then such a solution is beyond their experience. Such an answere would have to in effect prove (or otherwise justify the MVT).

    RonL
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