I see no reason to suppose that this visualization would change. Whatever we take as the unit of area or the unit of volume, we would, sooner or later, need to show that the area of a rectangle is proportional to the product of its two dimensions, while the volume of a right parallelopiped is proportional to the product of its three.
On Sun, 01 Sep 2013 10:37:41 -0600, Joe Niederberger <firstname.lastname@example.org> wrote:
> Its easy to visualize what's going on with the derivatives of x**2 and > x**3 with the usual square and cube representations of those functions: > a square can be enlarged by "building out" along two edges, a cube can > be likewise by "building out" on three faces -- the "error" artifacts > are the little dx corner square in the 2D case, the corner cube as well > as the three edge "lines" in the 3D case. Its just a cute way of seeing > where the derivatives d/dx(x**2) = 2x, and d/dx(x**3) = 3(x**2) come > from. > > But I don't see how to do anything similar with triangles or > tetrahedrons. Perhaps Kirby will show us. Or does this simple exercise > point to something a bit more fundamental than simple "cultural choice"? > > Cheers, > Joe N
- -- - --Louis A. Talman Department of Mathematical and Computer Sciences Metropolitan State University of Denver