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Math Help - Graph sketching, continuity, differentiation - Is this right?

  1. #1
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    Graph sketching, continuity, differentiation - Is this right?

    (I would appreciate either an explanation as to how to solve the problems, or the correct answer so I could figure out what I did wrong.)

    1. Let f(x) =
    4-x^2 if x < 2
    x^2 - 5x + 6 if 2 < x < 3
    x if x > 3
    a) Sketch the graph of f.

    http://img293.imageshack.us/my.php?image=calcgraphgk1.png


    b) At what numbers is f continuous?

    - I'm not sure if I did this right, are you supposed to do something with limits to show what numbers are continuous?
    (-∞, 2], [2, 3], [3, ∞)

    c) At what numbers is f differentiable?

    - Again, I've seen in some places where your answer has to include limits. Do you have to use limits in your work or something?
    (-∞ ,∞)

    d) Give a formula for f'.
    f'(x) =
    2x if x < 2
    2x - 5 if 2 < x < 3
    1 if x > 3
    2. Let f(x) = x^3 - 9x^2 + 24x + 3
    a) For what values of x does the graph of f have a horizontal tangent
    - Okay I have no idea what to do for this one. I tried looking through my textbook and couldn't find anything. If someone could show me how to do this, that would be great.

    b)
    f''(x) = 9
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  2. #2
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    1
    2. Let f(x) = x^3 - 9x^2 + 24x + 3
    a) For what values of x does the graph of f have a horizontal tangent
    - Okay I have no idea what to do for this one. I tried looking through my textbook and couldn't find anything. If someone could show me how to do this, that would be great.
    Since it is a cubic, it'll have 2 horizontal tangents.

    Differentiate, set to 0 and solve for x.

    The derivative will be a quadratic, which is easy to solve.

    For that matter, just graph it.
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  3. #3
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    So the rest of what I did is correct?
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  4. #4
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    I apologize for double posting, but I really need to know whether or not I am correct for at least number 1.

    Also, I would really like to know if I showed enough work, if not, how do I show more work?
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