Link to MIT Press Site
You know that feeling you get when it's the second half of January and you put on new clothes that have just come out of the dryer? This book is like a cross between that and a kick in the face.
The warm fuzzy-clothes-out-of-the-dryer feeling will come from the realization that you can wield unsurmountable power. The kick in the face will come when you realize you're not doing it yet.
The premise of the book is something along the lines of: We've all been taught how to solve math problem exactly. Science isn't exact. Turns out when you realize this, you can do a heck of a lot. Let Sanjoy show you how.
As an undergrad, I had the supreme fortune of taking some life changing courses. One of the ones that has struck me the deepest was Ph 101: Order of Magnitude Physics. It did a remarkable job building my confidence. It's one thing to go through your classes and complete the homework assignments. It's another thing entirely to feel as you can take a stab at just about any question anyone can ask.
This book is the handbook that will introduce you to the techniques and ways of thinking you'll need in order to tackle the most general of questions. The first chapter is Dimensional Analysis, something that every high school student should be exposed to. I love Dimensional Analysis. The rest of the book goes on to estimate Integrals, Sums and Differential Equations, thinking about limiting cases and scaling, and thinking pictorially.
The best part: it's available in a creative commons version, i.e. for free. Just follow the creative common pdf link in the left sidebar.
One of the biggest flaws I see in modern physics teaching is that physics courses have a tendency of being reduced to plugging numbers into highlighted and yellow boxed equations. That's not physics! Physics is a way of thinking about the world. It's the delight you obtain when you understand something for the first time. It's the power you can wield by being able to properly predict phenomenon that only minutes ago you found baffling. In a word: it's awesome. In order to be able to see past all of the equations, you need to have an appreciation for how powerful intelligent approximations can be.
The amazing fact is that with a proper introductory physics course, you are capable of understanding a huge deal of the world around you.
If physics classes were taught the way Sanjoy would like them to be taught, if they relied fundamentally on the kinds of techniques he discusses, I think students would like physics a lot more. I think the world would be a better place.

What one fool can do, another can.All it really takes to understand calculus is the ability to imagine a very little bit of something. That and a caring and skilled tutor to lead you on your way. What name can you think of that sounds more caring and skillful than Silvanus Phillips Thompson. I can think of no legitimate reason this book isn't used in each and every high school calculus in America. Seriously.