[pzhon]

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I've never understood the resistance to using calculus on these types of problems, as if it is some kind of failure. Finding different roads to a solution is often very interesting, but what's wrong with calculus?

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First, it makes the solution more opaque. I'm not so interested in solving any one problem. I'm interested in acquiring powerful techniques which will let me attack additional problems. Since I know calculus, seeing a calculus solution generally does not help me with future problems, but seeing something like my reflection argument very well might.

Second, the non-calculus arguments were much simpler and easier to remember, both to show A=B, and then A=B=C.

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All calculus problems? How would one frame a generic differential equation in language that does not involve calculus? Is there some kind of geometric trickery that will give me Bessel functions?

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Right, I don't buy that. Parts of calculus may be formally removable, but only by greatly increasing the complexity of calculations in general. I don't want to see someone try to remove calculus from a simple optimization with Lagrange multipliers, or a curvature calculation, or a deconvolution to sharpen a blurry image.