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Thread: Drawing graphs - CALCULUS

  1. #1
    Nov 2012

    Unhappy Drawing graphs - CALCULUS


    This is the link to the question:

    I'm having problems with di and diii.

    I do not know how to draw those graphs without a calculator (it's a non-calculator paper).

    Please help me do those two questiojns and guide me on the general methods needed to be done when facing with drawing graphs with weird absolutes like this.

    The answers are here:

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  2. #2
    MHF Contributor
    Mar 2010

    Re: Drawing graphs - CALCULUS

    I'm going to assume that you can draw the graph of $\displaystyle \frac{\ln{x}}{x}$ when x is positive. If that's part of what you're having trouble with, post again and I (or someone else) will help you with that.

    It usually helps to consider what happens when x is positive and when x is negative separately. Or in the general case, look at what's inside the absolute value and figure out when it's positive and what happens then, then when it's negative and what happens then.

    So in (d)(i), you're asked to graph $\displaystyle \frac{\ln{|x|}}{x}$. When x is positive, this is just $\displaystyle \frac{\ln{x}}{x}$. And when x is negative, the numerator, $\displaystyle \ln{|x|}$, is the same as when it's positive, while the denominator, x, changes sign. So the function has the opposite sign to the left of the y-axis.

    In (d)(iii), when x is positive, the functions g and h are the same, since $\displaystyle |x|=x$. And of course, g(x) is the same as in part (i), but for h(x), both the numerator and denominator are the same sign as when x is positive, so you have the same y-value for -x as you did for x. So you can see the graphs of g(x) and h(x) in the answers.

    And since when x is negative, g(x) = -h(x), h(x) > g(x) whenever h(x) is positive, which you can also see in the graph.

    - Hollywood
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