Calculus

cookymonkey

Over summer I wanted to try and get a leg up on learning calculus before taking a class on it. Can anyone recommend anything? Maybe a specific book or website?

Clipse

"How to Ace Calculus: The Streetwise Guide" is supposed to be quite good. A few of my friends used it as a supplement while they were taking calculus, and all of them thought it was great. Also, if you're a university student, and mathematically inclined, you could pick up the textbook for the class now and skim through it over the summer. (If you're a high school student, you presumably don't have to buy the book ever, so this is probably a waste of money.)

Positronics

For all your basic math needs, this website is your best friend: Paul's Online Math Notes.

mrcheesypants

Doing the same thing and found some E-books to read in the summer here.

Big Dookie

Honestly, the thing that will most prepare you for taking a Calculus course will be brushing up on Math skills you've already obtained. Specifically, go back over your Algebra, Geometry, and Trigonometry as much as possible. Calculus is simply an extension of these disciplines, and having a deep understanding of them will help you not only learn how to perform Calculus operations, but to understand the underlying theory behind them.

That said, if you're simply trying to get a head start and learn how to perform some basic Calculus operations (saving the theory stuff for when you take the actual class), sites like Pauls online math notes is a good place to start. Solving basic differentiation and integration problems is actually fairly easy once you've learned the "rules" to do it, so just about any math site out there can help you learn them. Some specific topics you will want to cover are:

• Limits at Infinity
• Continuity
• Finding Limits
• The concept behind the Derivative
• Basic Derivative Rules (Derivative of a Constant, Power Rule, etc)
• Product and Quotient Rules for finding Derivatives
• The Chain Rule (VERY IMPORTANT)
• Memorizing Derivatives of certain fuctions such as Log and Trig Functions
• L'Hopital's Rule for Limits
• The concept behind the Integral
• Riemann Integrals
• The difference between Definite Integrals and Anti-Derivatives
• Basic Integral Rules (Power Rule, etc)
• Integral Substitution
• The Fundamental Theorem of Calculus

If you can burn through all of that (a Herculean effort), you'll be sitting pretty when you get to the actual class. The main thing you need to know about Calculus is that there are really only two "big ideas" behind it - Differentiation and Integration. And the really cool thing about it is that they are actually two sides of the same coin. The Fundamental Theorem of Calculus will show you that the two are closely related, and the implications of that relationship.

thatnerdyguy

I'll second going back and reviewing lots of your old algebra and trig, as well as basic arithmetic if it's been a while since you took a math class. I've got a number of calculus students that I tutor that can integrate complicated functions in their head, but make horrible mistakes adding fractions and such.

Also, if you want a nice, non-technical overview of what this whole calculus thing is about and what the big deal is, I'd recommend reading A Tour of the Calculus. I read that the summer before I took calc and I think it gave me a nice perspective on the whole subject when I was actually studying it.

Abels

You can easily learn the essentials of single-variable calculus in the summer (you probably won't learn the applications, calculus in non-cartesian coordinate systems, etc. but the important stuff that you can use to derive it from). My advice: (1) don't use ANY theorem unless you understand its proof, (2) do the excercises, and (3) I'M NOT KIDDING DO THE FUCKING EXCERCISES. The only way to learn math is to do math.

A superb, superb book for calculus is Michael Spivak's "Calculus". It starts assuming a couple basic theorems and axioms of field theory (as in, a + b = b + a) and quickly builds everything else on that.

If you don't go with Spivak (which is a really great [but difficult] book), here are a few things to make sure you understand: a little bit of axiomatic set theory, rigorous definitions of "function", "limit" (the delta-epsilon definition), "derivative", and "intergral", and the very nice proof of the FTC ("rigorous" means that the proof or definition is essentially air-tight and unambiguous).

Unknown

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