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