Which calculus text should a 36-year-old use for self-study?

Question

I am 36 years old, and have forgotten a lot of math from high school, of which I only took up to Algebra 2. However I am teaching myself mathematics and am now, completely fascinated with the logic and beauty of it, but pitiably I am only now beginning to end Algebra 1. I've a long way to go yet, and have the Saxon method at home (Alg.1, Alg.2, Advanced Mathematics (trig and precalc), and Saxon Calculus). So far I like it, as it is what I am accustomed to, having used it in high school.

But my question is if a calc text like Spivak or Apostol would be better than the Saxon Calc. I have found that Saxon is not mentioned in the same circles as Apostol or Spivak.

Answer 1

I think one's first exposure to calculus-no matter how gifted or ambitious the student is-should be a physically and geometrically motivated approach that illustrates most of important applications of calculus. Sadly, many people think that means a "pencil-pushing" or "cookbook" approach where things are done sloppily and with no careful explanation of underlying theory. That's simply not true. You can certainly do calculus non-rigorously while still doing it carefully enough to give students the broad picture of the underlying theory. I think in the case of a late beginner like you, this is even more important since your mathematical instincts are either undeveloped or so rusty as to be near useless. You'll need this combination of intuition and careful rigor to truly learn the material correctly.

The best example of this kind of book, to me, is Gilbert Strang's Calculus. Strang's emphasis is clearly on applications and it has more applications then just about any other calculus text-including many kinds of differential equations in physics (mechanics), chemistry (first and second order kinetics), biology (modeling heart rythum) and economics and a basic introduction to probability. But Strang doesn't avoid a proof when it's called for and the book has many pictures to soften the blows of these careful proofs. This would be my first choice for a high school student just starting out with calculus and I think it'll work very well for someone in your situation.

Answer 2

I think Saxon's books can be good for fairly weak high school students who also do not intend to progress further than precalculus or a first semester calculus course. However, I do not think Saxon's books are appropriate for anyone who aspires to reach Spivak's calculus level at some future time. My suggestion would be to look at the first two references I cited yesterday at Preparing For University and Advanced Mathematics (the 4 Gelfand School Outreach Program books and Mary P. Dolciani's book), followed by Mathemagician1234's suggestion of Gilbert Strang's book.

Answer 3

Reading your situation, I can offer some advice from someone who is 33 years old and currently self-studying Apostol. I cannot speak to Spivak directly, but I imagine the things I have to say about Apostol will apply more or less equally to Spivak.

Apostol is a wonderful book, but would not be my choice as an introduction to either calculus or proof-based mathematics. My background differs somewhat from yours in that I took a full year of Calculus in high school, and have an engineering degree so took classes on ODE, linear algebra, and statistics (amongst some other courses that made significant use of mathematical techniques). That was years ago, so I've forgotten many things, but with that background I find Apostol challenging at times, and it certainly requires a lot of motivation to keep moving forward. Without someone to guide you, there are lots of places in a rigorous treatment of Calculus to get hopelessly confused. If you look at my profile you can see the questions (an expanding list) that I have asked from problems in Apostol to get a flavor for the sorts of exercises that the book contains. I think it would be very difficult to undertake the book without a good background in proving mathematical statements rigorously, and a strong notion of the ideas of Calculus. The theorems and definitions would just be too obtuse if you didn't already understand what was going. Apostol is not the place to learn these things.

That said, I would echo one of the comments that there is math to learn other than Calculus. Not to say Calculus shouldn't be on your agenda, just that there are beautiful and accessible areas of math that might appeal to you and are better for developing mathematical technique. I strongly recommend looking into books on elementary number theory (preferably something with complete solutions) or discrete math. These areas are completely accessible to you right now, provide a forum in which to apply some of the basic mathematical techniques you are learning, will really help you develop your skills with manipulating equations and "thinking mathematically," and will start you on the road to understanding the difference between solving equations at the algebra level and writing proofs.

As for actual book recommendations, a very straightforward number theory book with solutions is Jones & Jones. It's not a great book at all, but it has full solutions and is targeted at undergraduates. A very ambitious book might be Concrete Mathematics, much of which will be too advanced at the moment, but is a book that can grow with you. It contains wonderful discussions of very concrete (as the title might suggest) problems, and methods for working with sums and equations which are invaluable as you gain maturity.

Hope that helps.

Added: Also since you seem to be interested in mathematics generally, I would highly recommend taking the time to read some math history and "popular" math books. Math has a wonderfully rich and entertaining history. These can give great insight into problems that have been pivotal to the development of mathematics, and give glimpses to the sorts of things mathematicians are interested in. Also, many "pop" math books are about theorems or concepts that have fascinated mathematically minded people for centuries, so they are likely to fascinate you. Finally, they can be read right now while you are still developing your basic math skills, and can provide lots of motivation and inspiration.

Specific books I'd recommend are E.T. Bell's "Men of Mathematics" for math history, some of recent books on the Riemann hypothesis are good reads about an open problem (du Sautoy, The Music of the Primes and Derbyshire's "Prime Obsession" were both good), and I found "Symmetry and the Monster" by Ronan really inspiring (if you want to get a glimpse of what "modern" algebra is about).

Answer 4

My answer is not specific to calculus, but learning any subject from books:

Take any book that has a reasonable reputation. Start reading. You will sooner or later get stuck. Try reading some/figuring out what happens. If still stuck, get another book. Repeat. If this process begins to cycle a lot, then just switch to a different subject, read it for a while, come back later to the subject you start from.

This is what appears to work for me. I find it easier to learn about a subject when I know related this (even not directly related). Maybe it'll work for you.

(Regarding calculus text: a book I like is Hardy's "Course of Pure Mathematics")

Answer 5

1. I love Saxon's books. I'm actually at (or near by 1-2pts) the top of my college CalcII class, and know that Saxon's books have prepared me incredibly well.

2. As far as a recommendation for learning Calc, I suggest using http://www.khanacademy.org/.