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Math Help - Help with Derivatives.

  1. #1
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    Help with Derivatives.

    I have a mid-term tomorrow and still don't understand this specific concept.

    I only know how to do (a-b) I think. Please help with (c)-(f)

    Let f(x)=(x^4-18x^2)/5

    (a) Use the definition of a derivative or the derivative rules to find f'(x).

    My answer: f'(x)= (4x^3-36x)/5

    (b) Use the definition of a derivative or the derivative rules to find f' '(x)=

    My answer: f' '(x) = (12x^2-36)/5

    (c) On what interval is f increasing (include the endpoints in the interval)?
    interval of increasing =

    (d) On what interval is f decreasing (include the endpoints in the interval)?
    interval of increasing =

    (e) On what interval is f concave downward (include the endpoints in the interval)?
    interval of increasing =

    (f) On what interval is f concave upward (include the endpoints in the interval)?
    interval of increasing =
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  2. #2
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    Re: Help with Derivatives.

    I have figured out (c) and (d) for you. Maybe someone else that knows more can help you with (e) and (f).

    f' = (4x^3 - 36x)/5

    To figure out where its decreasing or increasing, we need to find the critical point. Which is found by setting the first dervative equal to 0, and then solving for x.

    0 = (4x^3 - 36x)/5

    0 = x(4x^2 - 36)
    0 = x(2x+6)(2x-6)

    So x = 0, x = -3, x = 3 are our options.
    To classify them as a critical point we need to test points within our critical points, and observe the behavior when we plug them into f'(x)... observing meaning just check whether they are + or -
    f'(-4) = -
    f'(-1) = +
    f'(1) = -
    f'(4) = +

    So we are decreasing from x = -∞ to x= -3
    Then increasing from x = -3 to x = 0
    Then decreasing from x= 0 to x = 3
    Then increasing from x = 3 to x = ∞

    So our interval of decreasing:
    (-∞, -3] U [0, 3]
    And our interval of increasing:
    [-3, 0] U [3, ∞)
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  3. #3
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    Re: Help with Derivatives.

    The function f is increasing if f'(x)>0, decreasing if f'(x)<0, concave upward if f''(x)>0, and concave downward if f''(x)<0. Since you've calculated the first and second derivatives already, it's only a matter of determining where they're positive and negative.

    - Hollywood
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  4. #4
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    Re: Help with Derivatives.

    Let f(t) = 9t^4 - 7t^2 + 12t -4 , Find (laplace) L[d^2f/dt^2]

    I get answer 108(2/s^3) - 14/s

    Is it my answer correct?
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