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Math Help - A Clarification: Sketching the Derivative

  1. #1
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    A Clarification: Sketching the Derivative



    The graph (in black) is f(x). When asked to for the interval of increase/decrease, would it be correct to say that f(x) is increasing for all values of x? However, f(x) appears to plateau at 1.5<x<2.5. I know a point of inflection occurs at (2, f(2)).

    If f(x) is increasing for all values of x, how would I begin to sketch the first derivative (I would need an equation since I cannot draw free-hand curves in the program I use)?

    The pencil lines indicate a guess: The quadratic first derivative and linear second derivative.

    Could anyone please provide clarification?
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  2. #2
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    Keep in mind that the derivative of a function is always one order of x lower than the function itself. To me, the function you posted resembles a cubic which is shifted left by 2 units and shifted up 5/2 units. Giving the function f(x) = (x-2)^3 + 5/2 and deriving that would give you something of the form 3(x-2)^2.

    The derivative of a cubic is a quadratic and your graph should look something like the attachment.
    Attached Thumbnails Attached Thumbnails A Clarification: Sketching the Derivative-picture-7.png  
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  3. #3
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    Quote Originally Posted by jameselmore91 View Post
    Keep in mind that the derivative of a function is always one order of x lower than the function itself. To me, the function you posted resembles a cubic which is shifted left by 2 units and shifted up 5/2 units. Giving the function f(x) = (x-2)^3 + 5/2 and deriving that would give you something of the form 3(x-2)^2.

    The derivative of a cubic is a quadratic and your graph should look something like the attachment.
    Thank you for the clarification.
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