For everyone except mathematicians (and, prior to 1820 or so, everyone except George Berkeley), the ultimate reason for believing calculus is that it helps engineers build bridges that don't collapse. Analysis does not justify calculus; applications do. Do some naive manipulations with series get one into trouble? Yes, but it is not necessary to develop analysis to deal with the problem; one just has to learn by experience not to do those manipulations. (This is not a new point of view. I am just trying to explain what Wittgenstein tried to explain to Turing in 1946.)
In the same sense that literature is an unnecessary, parasitic phenomenon upon ordinary language, "higher mathematics" is an unnecessary, parasitic phenomenon upon ordinary calculation. English majors study Shakespeare because it is a great historical achievement of our civilization and its study teaches us various useful skills. Math majors study analysis and algebra for the same reason. (EDIT: I reread this, and realized that it's possible for people to misread it. I have the greatest respect both for the study of literature and the study of higher mathematics and think both are worthwhile pursuits. I think the viewpoint that denigrates these pursuits is a bad viewpoint, but at the same time I don't think it is an irrational viewpoint.)
In my experience, although students may ask for motivation, what they are really looking for is something with which they are familiar to which they can compare the new stuff they are learning, so that they can build a context for the new concepts. I hope you will get some answers answering this implied question, but since I believe in brutal honesty with students, I think my above points needed mentioning.
