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Math Help - How to understand this problem?

  1. #1
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    How to understand this problem?

    I don't understand how you get the answer to number 1 in the following link: http://batty.mullikin.org/shc_course..._solutions.pdf

    What's the thinking I need to be doing when trying to figure out which is least small to biggest?
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  2. #2
    A Plied Mathematician
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    Imagine a car driving along the graph from left to right. When the car is working the hardest (going the most uphill), then the derivative is greatest. When the car is braking the hardest, going downhill, to keep from going too fast, the derivative is most negative. Does that make sense?
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  3. #3
    MHF Contributor

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    The derivative of the function is the slope of the tangent line at each point. If the curve is rising steeply, the slope is a large positive number, if rising slowly, a small positive number, if declining slowly a "small" negative number, if declining sharply, a "large" negative number.
    Looking at the graph at x= -2, I see it is rising so the derivative is positive; 0< g'(-2). Looking at x= 0 it is decreasing so the derivative is negative. That, so far, gives g'(0)< 0< g'(-2). Looking at x= 2, I can see that the curve is increasing but it looks like it is not increasing as fast as at x= -2. so g'(0)< 0< g'(2)< g'(-2). Finally, looking at x= 4, I see the curve is still increasing but it looks like it is not increasing as fast as at either x= -2 or x= 2: g'(0)< 0< g'(4)< g'(2)< g'(-2).
    Last edited by HallsofIvy; February 20th 2011 at 07:50 AM.
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