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Thread: Relationship between the graph lines/points of a Function, its Derivative, & Integral

  1. #1
    Jan 2011

    Relationship between the graph lines/points of a Function, its Derivative, & Integral

    I was using this applet, looking at a line, it's derivative, and it's integral.

    So say you have a graph of the functions: f(x), f'(x), and F(x). What is the relationship between the lines and the points? Now, I know to calculate these things using calculus, but when I look at the graphs I can't seem to correlate anything from them. I see the original line, but I don't really see what the Integral of a line means. I know how to use it to calculate area, but the line itself seems just weird. What data is it showing us?

    I know that when you have a function like y=x^2, the derivative is simply 2x, and can be plotted rather easily. It's more or less a graph of the rate of change of x and y, right? So we have a linear line...

    So when you have bigger equations like X^3 + 4x^2 +x.... what exactly does the graph of the derivative even mean? Is it useful? It's really weird looking and it just looks like another function. Do you have to keep breaking it down till you get the slope?

    Here is the applet I was playing with. What are transformations I should do to it for study purposes? I am trying to learn more about calculus:

    Function, Derivative and Integral
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  2. #2
    MHF Contributor
    Aug 2007
    You seem to be trying to see things that may not be observable.

    1) There is no such thing as "the integral of a line". Remember that arbitrary constant? Which is it that you think should mean something to you?
    2) Constant terms vanish under differentiation. This might give some insight. If you hae not already encountered it, you will find it useful in simple motion problems with uniform acceleration.

    2a) x(t) might be the displacement of some object at time t.
    2b) x'(t) = v(t) might be the velocity of some object at time t.
    2c) x"(t) = v'(t) = a(t) might be the acceleration of some object at time t.
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  3. #3
    Member integral's Avatar
    Dec 2009
    Graph the natural log of x, ln(x) and then graph it's derivative 1/x.

    1/x is the slope of the tangent line on ln(x). Notice how as x goes to infinity, ln(x) becomes more and more flat ( slope goes to zero) and 1/x gets closer and closer to zero.

    Also notice that as x goes to 0, ln(x) becomes more and more vertical (i.e. slope goes to infinity) and 1/x tends to infinity.

    Now take the integral of ln(x), xln(x)-x.
    Graph these two functions.
    Notice how xln(x)-x starts off as a negative value, and becomes more negative until the graph of ln(x) becomes positive. The area under ln(x) is becoming more negative until x=1 at which point the area begins to increase. Also, on the graph on xln(x)-x, you can see as x goes to infinity the graph becomes more linear. This reflects the fact that the slope of ln(x) tends to zero. Why? Because integral of a constant 'a' (which has a slope of zero) is represented by a linear function.

    I could not really think of any other better examples, hope this helps.
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