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Math Help - Is it normal to have two differing gradients for one point?

  1. #1
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    Question Is it normal to have two differing gradients for one point?

    I have an exam tomorrow and part of it will consist of dervitives...

    I was given a revision sheet and this is one of the questions:

    Find the average rate of change of f(x) = x^3 - 4x^2 + x +2 from x=1 to x=4

    So, I found the derivitive: f'(x) = 3x^2 + 8x + 1

    Subbed in 1: f'(1) = -4
    Subbed in 4: f'(4) = 17

    So, average rate of change is delta y over delta x
    17-(-4)/4-1
    = 21/3
    = 7
    Therefore, average rate of change is 7...

    Looked at the revision sheet answer and the answer was 2.
    I realise how they've gotten it... Just subbed in the values into the original equation.

    What I don't get is how there's two different averages. Am I doing something wrong? -It's a test on derivitives so it defeats the purpose if you don't work the derivitive out... Hmmm.

    Which method should I use??
    Help would be appreciated.
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  2. #2
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    Quote Originally Posted by Lucille View Post
    Find the average rate of change of f(x) = x^3 - 4x^2 + x +2 from x=1 to x=4

    So, I found the derivitive: f'(x) = 3x^2 + 8x + 1
    You messed up here.
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  3. #3
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    Can you please tell me how I messed it up?
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  4. #4
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    Do you mean with the derivitive?

    f'(x) = 3x^2 + 8x + 1

    --- I'm sorry, it was a typing error. My working out says it's"

    f'(x) = 3x^2 - 8x + 1
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  5. #5
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Lucille View Post
    Can you please tell me how I messed it up?
    the derivative gives the instantaneous rate of change, we do not need f'(x) here.

    for a function f(x), the average rate of change between x = a and x = b is given by:

    \mbox {Average Rate of Change} = \frac {f(b) - f(a)}{b - a} ......that is, the slope of the secant line connecting the two points
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    Thankyou so much. I was wondering why it wasn't working. That makes sense.

    I just wondered why they would put something like that in a derivitive test.
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  7. #7
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Lucille View Post
    Thankyou so much. I was wondering why it wasn't working. That makes sense.

    I just wondered why they would put something like that in a derivitive test.
    well, it is the basic structure from which the derivative evolved. remember, the derivative is actually the limit as a gets close to b of the secant line

    that is, f'(a) = \lim_{x \to a} \frac {f(x) - f(a)}{x - a}

    or equivalently, f'(x) = \lim_{h \to 0} \frac {f(x + h) - f(x)}{h}

    this may help
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