Note: I've edited this question on October 9th, after establishing a bounty on it.
What are the best introductory calculus textbooks that
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explain why calculus is important in a broad intellectual and scientific context, justifying its inclusion in a liberal education. Necessarily, this means it puts great emphasis on applications to science and engineering, with specifics;
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are non-dogmatic, i.e. they justify what they say, in most cases in a serious way, i.e. heuristic rather than rigorous (logical rigor in mathematics, a beautiful thing in some contexts, can approach sarcasm in other contexts);
- Are at a level that can be understood by students adequately prepared in prerequisites but primarily interested in other things than mathematics and not inclined to develop their mathematical ability beyond what is needed to become liberally educated and to know that such a field as mathematics exists (in a way in which most educated people currently do not know there is such a field).
Such a book would not say "This differential equation arises in the study of fluid flow", but rather "This section will explain how to derive this differential equation from these physical principles, and some of the exercises are on understanding steps in this and related derivations, and some of them are on applying the techniques." That's a part of the "non-dogmatic" nature of the book. It would also explain why, or at least that, a particular equation is consequential beyond mathematics.
Such a book would necessarily be ruthless about refusing to include some topics that are part of the (far too) ritualistic standard course.
PS: Could answers describe the books' contents and say why they are judged to answer the points above?
Best Answer
Try looking at Agnew's book, which now seems to be freely available on the internet:
Ralph Palmer Agnew, Calculus. Analytic Geometry and Calculus, with Vectors, McGraw-Hill Book Company, 1962, xiv + 738 pages.
Agnew's book is probably not as "strongly oriented towards applications in science and engineering" as some books (still, it is probably above average in this regard, being written back during a time when such applications were more thorough than is the case today), but the writing is remarkably fresh and the exercises are among the best you can find in an elementary calculus text.
Another book worth looking at is below. This one is a bit more strongly oriented towards applications and has some novel approaches to certain topics.
James Callahan, et al, Calculus in Context. The Five College Calculus Project, W. H. Freeman, 1995/2008, xxi + 845 pages. [This also is online. See .pdf file for Chapters 1 through 6 and .pdf file for Chapter 7 through 12.]