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Math Help - Derivatives exercises

  1. #1
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    Question area under the curve and derivatives

    Hello I need some help with my calculus revision. I have troubles with some of the exercises the teacher gave us. Could you please help me resolve this. My exam is coming up and I have a hard time with derivatives.
    I have to find the derivative of this function: f (x) = log2(cos(x)) + 2 arctan( x). EDIT: Ok I found for this one but I have a problem with another number :/

    This one:

    Area under the curve:

    a) Area under the curve of y= x^3 between x= 2 and x= 4
    b) Area under the curve of y= sin(x) between x= π / 3 and x= 2π / 3 (that's pi)

    And I have this one too:

    F(x) is defined by |x| ≤ 2 with:
    f (x) =
    −1/ x if − 2 ≤ x < −1
    x2 if −1 ≤ x ≤1
    2x −1 if 1 < x ≤ 2

    a. show continuity of f(x) on interval [-2, 2]
    b. Values of x in ]-2, 2[ if f'(x) does not exist? Clue: use f'(x) to decide if x exists at x = -1 et x = 1.
    c. Find all x values as f'(x) = 0. For all these values find f"(x) if it exists.
    d. Find all local maximums and minimums of f(x) in ]-2, 2[ and find the global max and min of f(x) in ]-2, 2[.

    Thank you!

    I may have some other questions but for now these are mine.
    Last edited by Emy28; December 11th 2012 at 08:55 AM.
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  2. #2
    MHF Contributor
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    Re: Derivatives exercises

    Hey Emy28.

    Hint: for the first one, try using the fact that all continuous functions in an interval satisfy lim x->a f(x) = f(a) and show that all the piecewise limits are the same from each side.

    This is establishing continuity for a piecewise function.
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