In Calculus, use of approximation techniques in an age of calculators?

I am teaching a secondary school calculus course in which topics such as Newton's method, Simpson's rule, and other approximation techniques are presented. But the students not only have powerful calculators, these calculators are even obligatory on their external exam. They can find approximations to zeros and areas very quickly on these calculators. There are many mathematical techniques which are intermediaries in other processes, justifying their inclusion in a course even when the techniques can be done by calculator, but I cannot find any such justification for Newton's method or Simpson's rule. Nor can they be justified with the argument that the techniques used for their development are good to know: there are lots of other more worthwhile recursive procedures along the lines of Newton's method, and the proofs for Simpson's rule are simply tedious. However, I have to teach these topics, as there is an external exam which will put questions which require the use of these techniques. So, my question is: is there some justification which is not artificial which I can present to the students for learning these techniques, besides "it's on the exam"?