But this is not symbolic differentiation, it's numerical differentiation. A true symbolic integrator would not give 48.00120000993502 when asked for the value of 3x^2 at 4.

This kind of approximation exercise is given to calculus students all the time ("Use the definition of derivative to approximate the derivative of f(x) at 3 by taking dx=.001," etc.).

edit: I misread! See reply below.

Right, I was referencing later lessons. Section 2.3 of the book. Surprisingly early.

Granted Knuth's work also did symbolic rather early. A bit harder to work with in his.

Edit: To be fair, I could have worded that first post more clearly! :D