In fact, in my one numerical analysis course, Fortran programming was a prerequisite. We were expected to code up our exercises. In fact, all computer computations involve substituting the calculus with more conventional algebra-type calculations. Computers cannot do differentiation or integration, all they can do is substitute simpler iterative operations to arrive at a numerical result.

All numerical analysis solutions involve simplifications of the original calculus so it is not necessary to formally carry out the integral or derivative.
Its like linear interpolation when using trig or log tables. Sometimes the approximation works, but sometimes it doesn't. You actually have to try it out with an example where the answer is already known.

Before digital computers, numerical analysis was a technique used to substitute very complex mathematical operations with simpler ones that could be done in a reasonable amount of time, or assigned to students who actually did the number crunching. These guys, by the way, were also called "computers", and followed step-by-step instructions printed on paper forms called "programs".

The major specialty of astronomical interest where I went to school was positional astronomy, or astrometry. Its dull, repetitive work, but it is the fundamental sky survey work on which everything else, including the more glamorous astrophysics and cosmology, relies. The actual work involved carefully measuring stellar positions with powerful microscopes off photographic plates, and subjecting the results to multiple corrections and adjustments to remove instrumental errors and variations introduced by precession, aberration and other geodynamical and orbital movements. The statistics and math were well understood, and not all that complex, but they were too tedious and involved to be employed on thousands of stars that might be on even one photographic plate. It was so labor-intensive that astrometrists relied on approximations to these equations (numerical analysis) to speed things up. Their skill and reputations were based on finding clever ways to do this.

When computers came along, these numerical approximations became unnecessary. You could just drop the entire equations in the Fortran code and calculate them out--brute force. And suddenly, these old time PhDs were forced to rely on undergraduates to code up their calculations, not just toil away as human computers. Their clever numerical approximations and shortcuts, and the proofs needed to guarantee they were equivalent to the full formal calculations, were no longer necessary. Anybody with computer skills and a knowledge of basic algebra could automate the calculations.

I should have learned my lesson then, but the young are arrogant and cocky. The day would come when the skills painfully honed over a lifetime would be automated, and suddenly everything you know becomes, if not obsolete, then superfluous.