Here's the basic question: Is there a straightedge and compass construction of the derivative and the integral?

When I came up with this question, I was trying to figure out how I would explain why kinetic energy ($\frac{mv^2}{2}$ has anything to do with the area of a square (the [mathv^2[/math] part) to someone who didn't know algebra, but could knew things about geometry like length and area.

I came up with this question while thinking about how mathematicians in Euclid's day knew nothing about algebra as we have it today, but they could construct any rational or square-root based number using a compass and straightedge.