In the context of a standard undergraduate Calculus sequence, I've noticed there is a big emphasis on teaching the algebra part of Calculus. What I mean by this is that a student may feel more comfortable finding, say, a limit by computing a bunch of values approaching some number, or by going from the graph. But yet, it seems we spend an inordinate amount of time on finding certain limits by factoring, or rationalizing, etc.
Same goes for being able to find a derivative from the definition, or with finding indefinite integrals by hand. Integration is an algorithmic task which we can only perform for hand picked cases at great pains, and there is a huge class of functions for which there is no antiderivative in elementary functions. Given that CAS can do it much more easily, why exactly do we require students to learn this?
While I understand some of these things are interesting in their own right to a mathematician (for example, partial fractions brings to a discussion of what polynomials are irreducible over the reals), I'm not sure who in the real world actually goes around trying to find an indefinite integral by hand, or calculating derivatives from their definition. Again, I do understand these are important skills, but I frequently hear the argument that these are the most important skills, as opposed to having an intuitive "feel" for limits, or for when derivatives come into models, for example.
So my question is:
Do engineers/scientists,etc. actually need to know the algebraic part of Calculus? How?
Concrete examples would be greatly appreciated. Thanks!