Reddit reviews Calculus

From my experience, Calculus in America is taught in 2 different ways: rigorous/mathematical in nature like Calculus by Spivak and applied/simplified like the one by Larson.

Looking at the link, I dont think you need to know sets and math induction unless you are about to start learning Rigorous Calculus or Real Analysis. Also, real numbers are usually introduced in Real Analysis that comes after one's exposure to Applied/Non-Rigorous Calculus. Complex numbers are, I assume, needed in Complex Analysis that follows Real Analysis, so I wouldn't worry about sets, real/complex numbers beyond the very basics. Math induction is not needed in non-proof based/regular/non-rigorous Calculus.

From the link:

For Calc 1(applied)- again, in my experience, this is the bulk of what's usually tested in Calculus placement exams:

Solving inequalities and equations

Properties of functions

Composite functions

Polynomial functions

Rational functions

Trigonometry

Trigonometric functions and their inverses

Trigonometric identities

Conic sections

Exponential functions

Logarithmic functions

For Calc 2(applied) - I think some Calc placement exams dont even contain problems related to the concepts below, but to be sure, you, probably, should know something about them:

Sequences and series

Binomial theorem

In Calc 2(leading up to multivariate Calculus (Calc 3)). You can pick these topics up while studying pre-calc, but they are typically re-introduced in Calc 2 again:

Vectors

Parametric equations

Polar coordinates

Matrices and determinants

As for limits, I dont think they are terribly important in pre-calc. I think, some pre-calc books include them just for good measure.

Here's mine

To understand life, I'd highly recommend this textbook that we used at university http://www.amazon.com/Campbell-Biology-Edition-Jane-Reece/dp/0321558235/ That covers cell biology and basic biology, you'll understand how the cells in your body work, how nutrition works, how medicine works, how viruses work, where biotech is today, and every page will confront you with what we "don't yet" understand too with neat little excerpts of current science every chapter. It'll give you the foundation to start seeing how life is nothing special and just machinery (maybe you should do some basic chemistry/biology stuff on KhanAcademy first though to fully appreciate what you'll read).

For math I'd recommend doing KhanAcademy aswell https://www.khanacademy.org/ and maybe a good Algebra workbook like http://www.amazon.com/The-Humongous-Book-Algebra-Problems/dp/1592577229/ and after you're comfortable with Algebra/Trig then go for calc, I like this book http://www.amazon.com/Calculus-Ron-Larson/dp/0547167024/ Don't forget the 2 workbooks so you can dig yourself out when you get stuck http://www.amazon.com/Student-Solutions-Chapters-Edwards-Calculus/dp/0547213093/ http://www.amazon.com/Student-Solutions-Chapters-Edwards-Calculus/dp/0547213107/ That covers calc1 calc2 and calc3.

Once you're getting into calc Physics is a must of course, Math can describe an infinite amount of universes but when you use it to describe our universe now you have Physics, http://www.amazon.com/University-Physics-Modern-12th/dp/0321501217/ has workbooks too that you'll definitely need since you're learning on your own.

At this point you'll have your answers and a foundation to go into advanced topics in all technical fields, this is why every university student who does a technical degree must take courses in all those 3 disciplines.

If anything at least read that biology textbook, you really won't ever have a true appreciation for the living world and you can't believe how often you'll start noticing people around you spouting terrible science. If you could actually get through all the work I mentioned above, college would be a breeze for you.

I would say that the best way to start is to pick a single book in Calculus, such as this or this or even this, and work all the way through it.

Then it is up to you; you could go straight towards Real Analysis; I recommend starting with a book that bears Intro in the name.

Or you could pursue a more collegiate curriculum and move onto Differential Equations and Linear Algebra, then Real Analysis.

I assume you are doing this all independently, so you should look at college sequences for math majors and the likes. You can mimic those, and look for online syllabi of the courses to make sure you are covering the appropriate material. This helps because it gives a nice structure to your learning.

Whatever the case, work through a calculus book, then decide what further direction you wish to take.

My recommendation, get a Calculus book like this: Calculus by Larson. It explains everything from the beginning (analytical geometry) and with your algebra background you should understand it. Study section by section.

In one year you should have covered to Integral Calculus single variable. If you get stuck with some topic, post your question here!

Good luck!

Stewart, Larson, Rogawski

I found Calculus from Larson and Edwards pretty good.

You could try "Precalculus" by Stitz & Zeager. Chapters 10 and onwards is their Trigonometry book. This should be a very smooth book to work through.

Have you already picked out a Calculus textbook? Also, what are her plans as an MIT student? If she's going into engineering and the like, I would say Larson's "Calculus" (solutions manuals vol 1, 2) would be good enough.

If she plans on being a math student, though, I would say give her a a few months with Velleman's "How To Prove It". Afterwards, I can't recommend Spivak's Calculus (Answer Book) and Jim Hefferon's Linear Algebra (solutions manual on same page) enough. This is a good time to introduce mathematical rigor as a normal thing in mathematics because, really, this is what math is about.