for example, when analyzing where a hydraulic jump will occur in a given open channel of water, there are sooooo many variables and sub-equations that must first be manipulated. You need to understand the math behind the equations so that you can understand how certain adjustments, should affect the final output.
Another example would be one of the tests i took in one of my electives. i hadnt done a dericative or integration in 2+years, but on the test we were given a problem(most engineering problems are "real world") were the way we were taught to do that kind of problem didnt work. so we had to manipulate some equations, by integrating to solve the problem in a new way that hadnt been taught to us. if we had not learned how to look at problems and analyze them the way that was engrained into us, we would have been screwed.
Computers are awsome cause they itterate, and model super quickly. Other than that, i can do it almost as fast with my calc. You have to know what your plugging in and how it works if you are going to be a leader in a field. Situations pop up all the time where a project will be done in such a way that has never been done before, so you are creating all your analytic equations from scratch, and you better be damn sure you are modeling it correctly or people will die.
thats the difference between a engineers,docters and the rest of the work force. if we fuck up, and it can be a tiny fuck up, people die.