I have in mind a course taken by liberal-arts students who will probably never take another math course. I would like such a course to convey some of the way mathematical thinking is done (i.e. not a cookbook course) without getting "rigorous" (since scuh students can be assumed to understand that and can learn rather little of it during a one-year calculus course). Apparently I am not the first to think of excluding the mean value theorem, since one of James Stewart's books does that. I would also like to include some of the ways in which differential and integral calculus have played a role in the history of science.
I'm teaching a course in which I began with this and I use that idea repeatedly in exercises. It will of course be used in explaining the fundamental theorem. One of various places where I've used that proposition so far is #3 in this assignment, where I was told by multiple students that no one else who teaches math ever asks students to think through steps like this. They "know" very well that that's not at all how math is done. Hence, they say, it is quite confusing. I do some topics that might normally be done only in "rigorous" course, such as things like #1 in this, but as you see, I don't do it in the way in which rigorous arguments are written.
I'd like to see skills taught in such a course only to the extent to which they aid thinking, and I like to have students write carefully about that thinking. This contrasts with a practice that perhaps few if any mathematicians intend to do, but which is widespread, and that is that students in such courses are taught that mathematics consists entirely of skills. This leaves no place for things like one that I like to include: What is "natural" about the number $e$? (Here is how I begin the treatment of that question.)
It seems as if mathematical thinking is often reserved for advanced courses rather than freshman calculus or the like, despite what is probably overwhelming empirical evidence that it can be done even at the most elementary levels, e.g. teaching graph theory to 4th-graders.
The question here is: Which specific topics should be included in a course consistent with the ideas sketch above and why? In particular, which that are now customarily not included should be there, and vice-versa?