0
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Explanation:
Let's say that point A is at (0,0) and B is somewhere else on the parabola.
I'll make point B go to the right of point A.
For now, let's say B is at (4,16).
If we compute the slope of line AB, then we find the average rate of change (AROC). The AROC in this case is (y2-y1)/(x2-x1) = (16-0)/(4-0) = 16/4 = 4. Because point A is at (0,0), we're really just computing y/x where the x,y values come directly from point B.
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Now let's move B to (3,9). If we used the slope formula again, we would get the slope of 3. Note how y/x = 9/3 = 3.
Then let's move B to (2,4). The AROC is now y/x = 4/2 = 2
As B gets closer to A, the AROC is decreasing. The AROC is slowly approaching the IROC (instantaneous rate of change).
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Point B is generally located at (x,x^2) for any real number x. Keeping A always fixed at the origin, the slope of line AB is y/x = (x^2)/x = x.
What does this all mean? It means that if x = 0, then the IROC is 0. You might be quick to notice that we cannot divide by zero. So instead of letting x be zero itself, we'll just get closer and closer to it. This is where the concept of limits come into use. This is what calculus is based on (both integral and differential calculus).
Anyway, when calculating the IROC, we're really calculating the slope of the tangent line to the f(x) curve. Refer to the diagram below.
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In short, the slope of the tangent line at x = 0 is m = 0. We have a flat horizontal line that touches the parabola at (0,0).
