I have been reading a lovely 18th century book by John Rowe called An Introduction to the Doctrine of Fluxions. It presents calculus completely motivated by geometric questions  finding tangents to curves; finding and maximising lengths, areas, and volumes; finding radii of curvature and evolutes, etc. etc. There are many more of these books, by L'Hospital, Hayes, Emerson, MacLaurin, Simpson, etc., always illustrated with copious figures. Are there any modern books on calculus written in this spirit?
I like Analysis by its History, by Hairer and Wanner. Chapter I "Introduction to Analysis of the Infinite" covers precalculus as Euler would. Chapter II "Differential and Integral Calculus" proceeds essentially as you desire (through envelopes, caustics, curvature, and differential equations). Then Chapters III and IV 'redo' everything with Weierstrassian rigor, and move on to several variables.
Try Visual Complex Analysis by Tristan Needham. It is not Calculus per se, but surely the type of book you are interested in.
Have a look at Sylvanus P. Thompson's "Calculus Made Easy" subtitled, What One Fool Can Do, Another Can. It was written at the end of the 19h Century but takes a pretty much 18th Century approach (and rigor). It was brought back into print by Dover, and the latest Dover edition has an Introduction by Martin Gardner. The Kindle edition is under $10.

3$\begingroup$ I agree! The book can also be downloaded legally for free: gutenberg.org/files/33283/33283pdf.pdf $\endgroup$ – Frank Thorne Jun 14 '13 at 14:10
Here is an outstanding modern example: http://www.pdmi.ras.ru/~olegviro/Shchepin/index.html On Euler's footsteps, by Evgeny Shchepin.
On a more advanced level, MR1656255 Stalker, John Complex analysis, Fundamentals of the classical theory of functions. Birkhäuser Boston, Inc., Boston, MA, 1998. It is an interesting, unusual book, though I strongly disagree with many statements in it.