2011-03-29, 12:10 PM
Well, I was aware of the meaning of d, which was why I put dx in quotation marks. I was also aware of the shift from:
a/ slope = Δy/Δx; to
b/ f'(x) = df(x)/dx
Which are essentially the same thing: finding the rate of change one way or another: a/ does it over a range, and b/ does it at any spontaneous spot, an infinitesimal small range, which can only be managed when we put the whole thing in a limit and let the distance approach zero.
What I pondered about was just the use of dx in the intuition behind computing the area under the curve, and eventually it gets messy when I get to the volume under a surface and so on so forth.
I see what confused me though: I forgot that f(x) on a graph does not refer to a point y units away from the x-axis, it actually refers to the distance from the x-axis to that point, a.k.a a line. This way, my two different intuitions behind this seems to merge into one.
- Given x=a, evaluating f(a) gives a line connecting x=a and y=f(a). Summing up an infinite number of those lines along the x direction gives the desired area, but we can't sum things with zero area, so we have to divide the x distance into infinitely many dx's to give those lines some width.
- The same is true for finding the slope of a curve. Since Δy can't be computed along a curve, it has to be reduced to the point where the curve becomes a line, then divided by the corresponding amount of x changes, which consequently is also infinitesimally small.
a/ slope = Δy/Δx; to
b/ f'(x) = df(x)/dx
Which are essentially the same thing: finding the rate of change one way or another: a/ does it over a range, and b/ does it at any spontaneous spot, an infinitesimal small range, which can only be managed when we put the whole thing in a limit and let the distance approach zero.
What I pondered about was just the use of dx in the intuition behind computing the area under the curve, and eventually it gets messy when I get to the volume under a surface and so on so forth.
I see what confused me though: I forgot that f(x) on a graph does not refer to a point y units away from the x-axis, it actually refers to the distance from the x-axis to that point, a.k.a a line. This way, my two different intuitions behind this seems to merge into one.
- Given x=a, evaluating f(a) gives a line connecting x=a and y=f(a). Summing up an infinite number of those lines along the x direction gives the desired area, but we can't sum things with zero area, so we have to divide the x distance into infinitely many dx's to give those lines some width.
- The same is true for finding the slope of a curve. Since Δy can't be computed along a curve, it has to be reduced to the point where the curve becomes a line, then divided by the corresponding amount of x changes, which consequently is also infinitesimally small.