Top Calculus Books for High Schoolers | Summer Reading Recommendations

[pierce15]

Hello,

I'm in high school and I'm wrapping up calc this year. Does anyone know of any good math books (particularly focused on calculus- single or multivariable) for me to read this summer?

Thanks

 

[slider142]

Since you have already been introduced to how to use calculus and the general ideas involved, it is a good time to get a solid grounding in the theory of calculus: I would recommend Spivak's Calculus. To really understand multivariable calculus, you should understand basic linear algebra. So you should pick up a textbook on basic linear algebra as well. Or you can challenge yourself into learning the necessary linear algebra at the same time as multivariable calculus, and pick up Hubbard's Vector Calculus, Linear Algebra and Differential Forms . If you like challenges and learning new things, you should find this book a lot of fun.

 

[pierce15]

Thanks for the reply,

Is Spivak's Calculus a textbook? Also, I'd like to read about linear algebra and multivariable calculus separately, as I'm taking them as separate courses next year. Do you know of anything good that concentrates on these subjects separately?

By the way, I'd like to add to my original post that I'm interested in reading almost anything math-related, as long as it has some realistic applications (for example, I'm not really interested in number theory, it seems kind of pointless to me).

 

[Mark44]

Thanks for the reply,

Is Spivak's Calculus a textbook? Also, I'd like to read about linear algebra and multivariable calculus separately, as I'm taking them as separate courses next year. Do you know of anything good that concentrates on these subjects separately?

By the way, I'd like to add to my original post that I'm interested in reading almost anything math-related, as long as it has some realistic applications (for example, I'm not really interested in number theory, it seems kind of pointless to me).

Number theory used to be thought of as "pure" mathematics (i.e., not applied), but no more. Computer encryption algorithms are heavily dependent on number theory.

 

[WannabeNewton]

If you are not interested in pure mathematics, I doubt you will like books like Spivak's single and multivariable calculus texts.

 

[ktheo]

I really don't know why one would recommend Spivak to a Calculus major in high school... I remember years ago I googled "best calculus book" and ended up ordering Spivak's book all doe eyed to get it and start working it out... might as well have just bought a real analysis book. It's way too hard for anyone who isn't advanced in their mathematical maturity or who isn't extremely gifted in mathematics. I'd probably recommend Stewart... it's used universally, is slightly more gauged towards engineering in general, but it's still a good book to be used with outside resources like this forum.

 

[micromass]

I really don't know why one would recommend Spivak to a Calculus major in high school... I remember years ago I googled "best calculus book" and ended up ordering Spivak's book all doe eyed to get it and start working it out... might as well have just bought a real analysis book. It's way too hard for anyone who isn't advanced in their mathematical maturity or who isn't extremely gifted in mathematics. I'd probably recommend Stewart... it's used universally, is slightly more gauged towards engineering in general, but it's still a good book to be used with outside resources like this forum.

Spivak is written for motivated students who already finished calculus I and II. It is absolutely not written for people new to calculus. That said, Spivak really is a very difficult book with very difficult exercises. It's only meant for those who are up for a tough challenge.

So yes, maybe Spivak is too difficult for the OP, and since he's not interested in pure math, he might not be interested in Spivak. But he might still check the book out to see for himself.

I think Stewart is a horrible book. I absolutely don't recommend it. I do recommend the excellent book by Lang: "a first course in calculus". The OP already finished a calculus sequence, so maybe it's too easy for him.

 

[ktheo]

Spivak is written for motivated students who already finished calculus I and II. It is absolutely not written for people new to calculus. That said, Spivak really is a very difficult book with very difficult exercises. It's only meant for those who are up for a tough challenge.

So yes, maybe Spivak is too difficult for the OP, and since he's not interested in pure math, he might not be interested in Spivak. But he might still check the book out to see for himself.

I think Stewart is a horrible book. I absolutely don't recommend it. I do recommend the excellent book by Lang: "a first course in calculus". The OP already finished a calculus sequence, so maybe it's too easy for him.

I agree that the Langs book are good, I used it for linear algebra (if we're both assuming it's the yellow-covered one with the publisher Springer that does all those graduate work-books on topology and geometry). I said Stewart because that's what I used and I got a pretty good understanding of fundamental calculus from it.

 

[pierce15]

I really don't know why one would recommend Spivak to a Calculus major in high school... I remember years ago I googled "best calculus book" and ended up ordering Spivak's book all doe eyed to get it and start working it out... might as well have just bought a real analysis book. It's way too hard for anyone who isn't advanced in their mathematical maturity or who isn't extremely gifted in mathematics. I'd probably recommend Stewart... it's used universally, is slightly more gauged towards engineering in general, but it's still a good book to be used with outside resources like this forum.

I actually used Stewart's Calculus textbook for my math class this year (well, technically it wasn't a class). I think I'm going to try to read Spivak's book.

 

[pierce15]

I think Stewart is a horrible book. I absolutely don't recommend it.

What didn't you like about it? Is it because the problems were too easy?

By the way, when I said that I wasn't interested in pure math, I mostly meant number theory. I love doing challenging integrals and I realize that this has no application to the real world. However, I am not particularly interested in number theory and I doubt that I will enjoy studying abstract algebra.

 

[[Quadratic]]

I rather liked Rogawski's book, Early Transcendentals. Pretty straight forward with good explanations of concepts, and the problem sets have a decent range of difficulty.

 

[dustbin]

If you want a harder (more theoretical) textbook that has applications as well, I'd recommend Apostol's Calculus. It's sufficiently rigorous, has some great problems, and really balances technique and theory. You can get an international edition of this book on abebooks.com for about $15. The book also has a nice treatment of linear algebra. There is a second volume that goes further into linear algebra and then multivariable calculus.

I highly recommend them.

 

[pierce15]

Thanks for the suggestions. I'm also interested in reading a little about complex analysis, because I'm taking it next year. Any suggestions for books about that?

 

[micromass]

Thanks for the suggestions. I'm also interested in reading a little about complex analysis, because I'm taking it next year. Any suggestions for books about that?

I doubt you actually mean a very rigorous complex analysis course since that requires quite some math prereqs. But here are two nice books which do not require much knowledge:

https://www.amazon.com/dp/0198534469/?tag=pfamazon01-20
https://www.amazon.com/dp/0486613887/?tag=pfamazon01-20

 

[dustbin]

+1 for visual complex analysis. If you ever want a similar book for abstract algebra (specifically group theory), then there is a comparable book called Visual Group Theory.

 

[pierce15]

I doubt you actually mean a very rigorous complex analysis course since that requires quite some math prereqs. But here are two nice books which do not require much knowledge:

https://www.amazon.com/dp/0198534469/?tag=pfamazon01-20
https://www.amazon.com/dp/0486613887/?tag=pfamazon01-20

You're right, it probably won't be very rigorous because I haven't taken real analysis. I'm going to take linear alg/multivariable at the same time, the head of the math department thought that would be OK.

 

[micromass]

You're right, it probably won't be very rigorous because I haven't taken real analysis. I'm going to take linear alg/multivariable at the same time, the head of the math department thought that would be OK.

Be sure you know about partial derivatives and line integrals before the complex analysis course. If you do, you'll probably be ok.

 

[mathwonk]

Spivak and Apostol are very good but very hard. They will suit only a small percentage of people. If they appeal to you after inspection and experimentation, go for it. If you would like other options, I like the Lectures on freshman calculus by cruse and granberg. Lang's books are good for clear explanations of basic ideas but insufficient in examples for mastery. There are many editions of Stewart, some of the earlier ones of which I liked, maybe the second. The first edition of Edwards and Penney was also nice, as are the early editions of Thomas from the 1950's, (very non theoretical, engineering oriented).

I recommend you go to a university library and browse the calculus section to see what appeals to you.

See if you can understand this pdf file I am attaching. It contains actual proofs of the three main theorems not usually proved in differential calculus, but it will take work to understand them. You need to know that a real number is represented by an infinite decimal, and you need to know the definition of continuity.

These notes prove what Spivak calls "three hard theorems" in an early chapter of his book. After teaching for almost 50 years, I have decided they are hard partly because Mike makes them look hard by adopting the abstract axiomatic and set theoretic approach that is so common today.

There is no difficulty in giving rigorous proofs that are closer to intuition and might be more easily understood, but we never try because we just follow the tradition of the textbooks that are commonly used today. I have tried to offer an alternative. However my notes are very brief, and you may or may not need help filing in the details.

E.g. a lemma I omit that you need is that, if a function f is continuous at p, then there is some interval centered at p on which f is bounded. This is an easy consequence of the definition of continuity, but it will be a good test of whether you have understood that definition.

 

[pierce15]

I can't open your attachment

 

[mathwonk]

apparently it hasn't been "approved" by the server. i will try to append a simple text version.


I've tried to give elementary proofs of the “three big theorems” on continuity used in elementary calculus (taken together they say the continuous image of a closed bounded interval is again a closed bounded interval). I suggest this can be presented at least in a typical first year honors calculus class. First they should know the epsilon – delta definition of continuity.

1) Intermediate value theorem: Assume f continuous on [0,1], and assume f(0)<0<f(1). We claim that f(x) = 0 at some x between 0 and 1. By looking at the values f(0), f(0.1), f(0.2),…,f(.9), f(1.0), there is some integer a1 between 0 and 9 so that f(.a1) ≤ 0 ≤ f(.a1+.1). If one of these two values is zero we stop.

If not, then there is some integer a2 so that f(.a1a2) ≤ 0 ≤ f(.a1a2 +.01).
Continue…..

Either we find at some stage a finite decimal x where f(x) = 0, or else we find a sequence of decimals xn = .a1a2….an, and xn+1 = = .a1a2….an + 1/10^n, so that f(xn) < 0 < f(xn+1) for all n, and |xn – xn+1| < 1/10^n.

Since both sequences {xn} and {xn + 1/10^n} converge to the same decimal x = a1a2a3……., and since all f(xn)<0 while all
f(xn +1/10^n) >0, it follows that 0 ≤ f(x) ≤ 0. QED.

Here are two more such arguments along the same lines.

2) Every function f continuous on [0,1] is bounded there.
proof: if not then it is unbounded on some interval of form
[.a1, .a1+.1],
hence also on some interval of form [.a1a2, .a1a2 +.01].

Continuing we find an infinite decimal x = .a1a2a3... in [0,1], such that f is unbounded on every interval containing x. But if f is continuous at x, then f is bounded on some neighborhood of x. QED.

3) Claim: A continuous f takes on a maximum on [0,1].
proof: By theorem 1 above (IVT) the set of values of f form an interval, and by theorem 2), they form a bounded interval. If that interval is not closed on the right it has form say (c,d), but then the continuous function 1/(f(x)-d) would be unbounded on [0,1]. QED.

 

[pierce15]

Ok... Is the notation " $.a_1a_2...$" accepted in rigorous proofs? Does Spivak use it?

Wouldn't it be more rigorous to write:

$$ x_n = \sum_{k=1}^n a_k, \quad a_k \in \mathbb{N}:[1,9] $$

(I'm not sure that this is the correct notation for denoting $x$ in the set of natural numbers on the closed interval [1,9], but you get the point).

I guess these proofs make sense. I thought that you were going to present a rigorous proof of the chain rule or something.

By the way, in the first proof, the $x$ value for which $f(x) = C$ doesn't necessarily have to occur on [0,1], right? Shouldn't the proof assume that $f(x)$ is continuous on [a,b]?

 

[mathwonk]

In 1), yes the x does lie in [0,1]. That is a corollary of the proof.

A rigorous proof of the chain rule is also very easy, but these three theorems are usually thought to be the hardest fundamental theorems in calculus.

As to whether the proof is done on [0,1] or [a,b], I have presented the hard part. Generalizing it to [a,b] is trivial. (Just compose f:[a,b]-->R, with the continuous map [0,1] --> [a,b], taking x to a + x(b-a), and then apply this version.)

After reading your post I think you might be pleased by Spivak's version of things rather than mine. I am trying to strip away the abstraction, whereas you seem to enjoy that abstraction. You are probably a strong student, for whom my concern as to the difficulty of understanding is unnecessary.

But your notation for a decimal seems to be a bit off. You seem to need some denominators which are powers of 10, and you need to allow a_k to equal zero.. This is why I try to avoid needlessly complicated notation - it is so hard to understand that even its own user may be confused by it.

 

[mathwonk]

The chain rule:If we have a composite function y(u(x)), and both component functions y(u) and u(x) are differentiable, then we claim that also y(u(x)) is differentiable, and that its derivative equals y’(u).u’(x) = y’(u(x)).u’(x).

I.e. we claim the limit of (∆y/∆x), as ∆x -->0, equals y’(u).u’(x).

We are assuming that (∆u/∆x)-->u’(x) as ∆x -->0, and also that (∆y/∆u)-->y’(u) as ∆u -->0.
Moreover, by definition of continuity, and since differentiable functions are continuous, we know ∆u -->0 whenever ∆x -->0, hence both (∆u/∆x)-->u’(x) and (∆y/∆u)-->y’(u), as ∆x -->0.

Thus by the product rule for limits, (∆y/∆x) = (∆y/∆u)( ∆u/∆x) --> y’(u).u’(x), as ∆x -->0,
as long as the factorization (∆y/∆x) = (∆y/∆u)( ∆u/∆x) makes sense for all small values of (∆x).

The only way it does not, is if there are arbitrarily small values of ∆x for which ∆u = 0.
Now in that case, it follows that the fraction ∆u/∆x equals zero for all those ∆x’s, and since this fraction has limit u'(x), the only possible value of that limit is zero. I.e. in the only case where the proof does not work, we know that u’(x) = 0.

Thus for the theorem to hold also in that case, we only need to prove that y’(x) = y’(u).u’(x) = y’(u).0 = 0. I.e. all we have to do is prove that in this case the fraction ∆y/∆x still approaches zero even though we cannot always factor it into a product of fractions.

The secret is to notice that we can still factor it as ∆y/∆x = (∆y/∆u)(∆u/∆x), at those points where ∆u ≠ 0. I.e. there are two kinds of ∆x’s, those for which ∆u = 0, and those for which ∆u ≠ 0.
But when ∆u = 0, it is trivial that the fraction ∆y/∆x = 0, since ∆y is the difference of the value of y at the same two values of u, so of course it equals zero. And in the case where ∆u ≠ 0, we can still factor the fraction as ∆y/∆x = (∆y/∆u)(∆u/∆x), and use the other product argument.

Thus for both types of values of ∆x in this case, we have ∆y/∆x --> 0 = y’(u).u’(x), and so the chain rule holds in all cases.

This result was traditionally proved correctly in turn of
the century English language books, such as Pierpont's Theory of functions
of a real variable, and in 19th century European books such as that of
Tannery [see the article by Carslaw, in vol XXIX of B.A.M.S.], but
unfortunately not in the first three editions of the influential book Pure
Mathematics, by G.H.Hardy. Although Hardy reinstated the classical proof in later editions, modern books usually deal with the problem by giving the slightly more sophisticated linear approximation proof, or making what to me are somewhat artificial constructions.

Summary:
The point is simply that in proving a function has limit L, one only needs
to prove it at points where the function does not already have value L.
Thus to someone who says that the usual argument for the chain rule for
y(u(x)), does not work for x's where ∆u = 0, one can simply reply that
these points are irrelevant.

 

[pierce15]

Sorry, I meant to write:

$$ x_n = \sum_{k=1}^n \frac{a_k}{10^k}, \quad a_k \in \mathbb{N}:[1,9] $$

I'm not going to lie, I didn't follow your chain rule proof completely. I think that I could sit down and decipher it if I had some time lying around, though.

 

[mathwonk]

why don't you try when you get a few free moments, or do as i do and just try to read it and get in mind,a nd then while driving to work or doing something boring, think about it.

 

[theoristo]

Calculus for the Practical Man J. E. Thompson

 

[Billioville]

From my experience, the best books on Calculus are in order,
James Stewart, Soo Tan, followed by Thomas' Calculus and Howard Anton. The first 2 are excellent in that the proofs are presented well with no need to read in between.
I would mention Robert Minton, then Robert Adams'&Essex book as good reference books.
The point is to understand the material and I would not recommend stuff like Spivak, Lang, Apostol or any other old school books as they might even discourage you as they are a bit cumbersome and to most people overwhelming.

 

[thankz]

I got an older version of stewart (6e) simply because it was cheap, but it shows how to solve problems (from the preface) geometrically, numerically, algebraically and descriptively. to make sure I understand everything correctly I bought a back up book (swokowski 2nd ed) not only that I wanted to have someone explain so I got a 36 hour dvd course by Larson along with calc 1 and calc 2 for dummies an extra multivariable book by stewart and some videos from the teaching company.

that's how I'm doing it, 2 cents.