What's the best way for a self-learner to prepare for Richard Courant's Calculus Book? I'm a software developer looking to make a career switch to Mechanical Engineering (at least I think so). My interests lie in design, complex systems, maglev, rockets, and new forms of space propulsion. Before I really commit myself to University, I would like to get my feet wet in Physics/Engineering. Most likely I'd major in Applied Physics or Engineering Physics.
Ultimately, my goal is to get through Fundamental University Physics by Alonso & Finn and then to get through Shigley's Mechanical Engineering Design. Alonso's text seems to require some heavy duty calculus, and Courant's calculus book seems to have a lot of practical application problems so I think the two would complement one another.
The only problem is, I'm not exactly sure what kind of precalculus prep I would need for that. Principles of Mathematics by Allendoerfer looks to be a rigorous and proof-focused trek through precalculus, but I'm not sure that this would be the best book for me if my preferences lean towards the more practical.
Generally, to prep myself for calculus I've been following a rough outline of what many here have already suggested:
Algebra:

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*Elementary Algebra by Jacobs

*Algebra by Gelfand

*Functions and Graphs by Gelfand

Geometry:

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*Geometry (2nd Edition) by Jacobs

*Geometry Revisited by Coxeter

Trigonometry:

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*Trigonometry by Gelfand

General:

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*What is Mathematics? by Courant

Now, how should I handle this whole precalculus thing? I skimmed through Addison-Wesley's Algebra and Trigonometry and I love that it's replete with practical-looking problems and novel facts about the given subtopic but it feels like a "plug and chug" type book. And as cool as calculating the parabolic cross sections of a car's headlights seems, problems like these feel cheap and trivial.
So what's a good precalculus book for the aspiring Applied Physicist / Engineer? Should I just stick with Principles of Mathematics?. Or should I replace that with Precalculus Mathematics in a Nutshell or Serge Lang's Basic Mathematics? What would best prepare me for Courant's book and beyond, given my goals?
Thanks!
 A: Courant's book has proofs and theory in it, so you should read a better algebra book than Jacobs (except if you've never studied any algebra at all), perhaps Basic Mathematics by Lang as you suggest. 
Lang's book will give you enough geometry too, including some discussion of vectors. There's no need to read a whole geometry book. Geometry Revisited is difficult and is not useful for calculus. It covers material that would rarely if ever be needed for physics.
You don't really need to consider What is Mathematics? a prerequisite—read that for enjoyment only.
I don't particularly like the book by Allendoerfer (if it's the one I'm thinking of). It's not entirely rigorous. It introduces some advanced concepts, but then studies them very superficially. My recollection is that at times it also accepts some statements without proof but without even mentioning that fact.
The books by Gelfand are a great idea, particularly Algebra and Functions and Graphs. The Method of Coordinates is also good. The trigonometry book is okay but not terrific. It seems to have been written mainly by the American coauthor and is dumbed down compared to Gelfand's trig book in Russian. (The others are actual translations.) Again, Basic Mathematics is likely to be enough here, but if not, you also have other choices, such as Trigonometry by Nobbs (which you can read after the trigonometry sections of Lang).
One thing I would add to this list is an introductory book on inequalities, since those come up a lot in proofs in calculus. Introduction to Inequalities by Beckenbach and Bellman is good for that. 
Another possibility, though not strictly necessary, would be an introduction to vector geometry beyond what's contained in Lang's book. For example, Elementary Vector Algebra by Macbeath. This isn't really important for single-variable calculus, but it's important for physics and for multivariable calculus. You can put this off until later if you like.
One last point I'd make is that one possible consideration when choosing between Courant, Apsotol and Spivak is that Spivak has a complete solution manual. Even if these books differ in some ways, they're more alike than they are different, so the solution manual could make Spivak worth it.
