Calculus? Which Calculus?

In high school and University I learned Calculus, by which I mean the Differential and Integral Calculus of Newton and Leibniz, and their extensions. I do not think that I really grasped the meaning of a continuous function until I studied metric topology.

I like calculus, I have used it professionally, and it is a glory of the modern age, but I do question the cost/benefit of teaching it to millions (billions?) of children on account of its difficulty.

What makes calculus hard?
I would say the relatively high level of abstraction, and in particular becoming comfortable with with either limits or infinitesimals (hello: non-standard Analysis). These things strain our intuition.

Why is calculus taught in high-school? My guesses:

• It is essential for physics and engineering, and other quantitative fields
  
• It is very beautiful and powerful (but you will not see that at high school)
• It is challenging
  
• It has a filtering effect on students
  
• Tradition

Since it has to be re-taught at College / University, I wonder whether it might be time to start teaching other Calculi in high school, just to mix things up a bit.

Perhaps a progressive school could hold a "calculus bake-off" and try to gauge the suitability and broad benefits of teaching and learning the various calculi?

Which other calculi could be taught?
Some might opt for the predicate calculus (also known as first-order logic), but my vote goes to the lambda calculus, which is fundamental to the theory of computing, and useful in practice. In fact there are computer languages such as Scheme (a Lisp dialect), ML, and Haskell that are essentially souped-up forms of the lambda calculus.

And if high-school sounds a bit late to get started with such an important subject, here is a game designed for eight year-olds that introduces the essential ideas.