Calculus Best textbook to truly understand single variable calculus?

Varma21

I am currently self learning high school level AP Calculus BC over the summer. From a mathematical perspective, I have heard that high school level calculus is just pure and shallow computational work. For this reason, I seek a deeper understanding in calculus. So far, the BC curriculum has been a joke in my eyes. So, what textbooks (Spivak, Apostol etc.) would you recommend so I could learn calculus at a true college level. Once I complete calculus BC I plan on taking a Calc III course at a college when my school year starts again. Also, any additional tips for self-study would be humbly appreciated. :) (I'm going into 11th grade when the next school year starts).

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Wrichik Basu

Gold Member
2018 Award
It would be helpful if you write out the syllabus of "Calculus BC". From India, I cannot get the intuition of your syllabus, as I'm not used to US education.

Although your title says "textbook", I hope you will consider looking at other resources. If you do so, then there are three courses that I will refer to:
I haven't gone through each course, but NPTEL courses are generally designed for at least UG level, and they go into some depth. Go through at least one or two lectures in each course and see if they satisfy your needs.

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MidgetDwarf

Are you familiar with proofs? Do you understand pre-calculus level material well?

Varma21

Yes, I found pre-calculus material to be fairly easy. It just seems that the AP test from which the AP calculus course's curriculum is centered around focuses explicitly on computations. Even the epsilon-delta definition of a limit is seldom taught in AP courses. Students scoring the highest on the AP test don't even know what the epsilon delta definition is :(
• Here is the syllabus for AP Calculus AB and BC
• It's also important to note that Calculus AB corresponds to a single semester Calculus 1 course in college and Calculus BC corresponds to both Calculus 1 and 2. Thus, the test for BC covers all the topics of AB in addition to BC-specific topics.
• Wrichik Basu: Those courses do seem helpful, I'll definitely consider using those as study aids.
A mere glance over some of the sample test questions in the above link makes me question how one is to be qualified for an upper level college math course because they scored better than a 70% on the AP test (the college board has a scoring system from 1-5. Getting higher that a 65-70% in most cases equates to a score of 5 in the BC exam). Textbook-wise I was looking at either Calculus by Michael Spivak or Calculus Vol. 1 by Tom M. Apostol. I have heard that the books I just stated are a good rigorous intro to calculus. Do you guys have a preference toward any specific calculus textbooks?

MidgetDwarf

Hmmm. When I learned Calculus, I learned from Thomas: Calculus with Analytic Geometry 3rd ed. It is of the computational variety, but it shows how to apply the Calculus.

When I wanted a better understanding, I looked at Spivak and Courant. I think Spivak is more user friendly, and will prepare you for Pure Mathematics. I found Courant funner, because of the writing and physics applications in it. However, the language can be termed and the exercises are harder.

Are you familiar with formal logic?
I think Spivak would be a better choice, but my preference is Courant. Have you done proofs before? Ie., induction principle, direct proof, proof by contradiction, proof by contrapositive..,?

Spivak may be too hard. But there is no money lost if it is too hard. You can always revisit it later. Do not use the solution manuals... It defeats the purpose of using books such as these.

You can always try Moise: Calculus. A balance between theory and application. It is a bit easier than Spivak, and has more in common with Courant. Correct proofs are presented. I would buy both... If Spivak is too hard, you can learn from Moise...

Varma21

Thank you for your insight. I do have some experience with formal logic and mathematical proofs, so I feel inclined to give Spivak a read. Personally, I feel that I am proficient with the application and computation of the calculus so far (I'm almost done with the last chapter of the stewart book), so I feel that Spivak's proof-based dabble into the analysis of calculus would suit me the best. I plan on using math for physics, but out of respect for mathematics I oblige myself to gain a deeper understanding of calculus and analysis.

MidgetDwarf

I would also recommend to maybe self study Linear Algebra. It is a great way to practice proofs, while learning some very important math.

I like Linear Algebra by Berberian.
Proof based mathbook, that is very well written. Very nice insights.

MidgetDwarf

Since you want to apply math to physics.

I like the 3 book series, Alonso and Finn: Fundamental University Physics.

Very heavy use of calculus from the onset. Derivations can be mathy.

ConfusedMonkey

The only thing that I don't like about Courant is that he doesn't use set-builder notation. Other than that, it's an excellent book.

Varma21

I am most likely going to be learning linear algebra and multivariate calculus concurrently as of next school year. Quick follow-up question: is the differential eqs. class hard? I have been told that differential equations comes after calculus III and linear algebra.

MidgetDwarf

I am most likely going to be learning linear algebra and multivariate calculus concurrently as of next school year. Quick follow-up question: is the differential eqs. class hard? I have been told that differential equations comes after calculus III and linear algebra.
No it is not hard. I took right away after Calculus 1.

I was concurrently enrolled in calculus 2 and ODE.

It is mostly is a computational class with some proofs sprinkled here and there. The hardest thing for me was knowing how to preform the methods to solve certain ODE. Once you learn how to do it, the majority of the problems become trivial.

Power series solutions and the Laplace Transform/ Inverse Laplace Transform can give you a bit of challenge.

A second course at the upper division level becomes more interesting.

Their is a really nice cheap book.
Ross:Ordinary Differential Equations.

Thanks.

smodak

I just finished teaching AP Calculus AB to my daughter; AP Calculus BC is just a few extra topics (series etc.). It definitely focuses on understanding, calculation, and mainly applications rather than formalism. For a student of physics, I would think the application is pretty important. If you think you have mastered the concepts taught in your calculus BC course, you are definitely ready for Spivak which is the book I would recommend. Other books that I would highly recommend that you look at are How to Think About Analysis by Alcock and Analysis: With an Introduction to Proof by Lay. Hope this helps. The Alcock book especially is superb.

Math_QED

Homework Helper
Are you familiar with functions, calculus and proofs? Then I won't recommend Spivak's calculus.

(NOTE: I'm talking about Spivak's calculus, not his book "Calculus on manifolds")

Although it is a good book, there are often hypotheses not stated (example: limit exists iff left limit and right limit exist and are equal -> the limit must be taken at an interior point of the domain, but this not mentioned, otherwise the theorem is false for cases like $\lim_{x \to 0} \sqrt{x}$).

Another reason why I dislike the book is the definition of function it gives. There is no such thing mentioned as "codomain"; which is essential, and the word surjectivity is not used once in the entire book!

So, my conclusion is: If you are already familiar with formal treatment of functions, sets, proofs, ... Don't use Spivak's book for the theory, but use something more advanced (E.g. Tao's Analysis book, but he expects the reader to do a lot of work, so be warned in advance!). The exercises of Spivak's book can be very interesting though, and I do recommend these.

Mondayman

I've perused through Apostol's book and quite like it; however, I find the cost of his calculus books to be appalling: $295 for volume 1 and$220 for volume 2 on Amazon.

Mondayman

I've heard far too many bad things about international editions to waste money on them. Pages out of order and occasionally entire chapters missing? How do you mess that up?

Math_QED

Homework Helper
Are you familiar with proofs? Do you understand pre-calculus level material well?
See post #4.

alan2

I don’t know if this thread is stale but I just saw it. For what it’s worth, I have some observations from almost 30 years of teaching physics and calculus.

1) AP calc is almost worthless. Unless you are the unusually gifted student who self taught everything missing from your AP course you should plan on starting at calc 1 at a decent university. Going from BC to calc 3 virtually guarantees you will spend most of the semester catching up.

2) If you want to learn calculus for physics skip Apostol and Spivak, you won’t learn any physical intuition. Get a copy of Thomas’ Calculus, edition 1, 2, or 3. He died after those and later editions were reworked by other people. The usefulness for physics is beyond compare. I never use a textbook anymore, I just post my lecture notes but if students want a book I recommend Thomas. They are reasonably priced used. All current calculus books suck. Something like 9798% of students who take calculus don’t want to be mathematicians but calculus books are all written by pure mathematicians who don’t seem to understand who their audience is. I can’t think of who does it at the moment but some engineering schools are starting to teach their own math courses because the math dept can’t do it in a useful manner.

3) Don’t become enamored with epsilon-delta proofs. When you are in a physics class your professor will say “assume dt is infinitesimal....” Jerome Keisler has a good book for free on his website that teaches calculus using infinitesimals. That’s the way every physicist and engineer uses calculus and it’s very intuitive. Not only do people not use epsilon-delta proofs in application, they are not intuitively helpful.

4) I don’t know where other people took their ODE courses but you can not pass mine without a solid understanding of calc 2. Even calc 3 is prerequisite but I do let exceptional students take it concurrently.

MidgetDwarf

I don’t know if this thread is stale but I just saw it. For what it’s worth, I have some observations from almost 30 years of teaching physics and calculus.

1) AP calc is almost worthless. Unless you are the unusually gifted student who self taught everything missing from your AP course you should plan on starting at calc 1 at a decent university. Going from BC to calc 3 virtually guarantees you will spend most of the semester catching up.

2) If you want to learn calculus for physics skip Apostol and Spivak, you won’t learn any physical intuition. Get a copy of Thomas’ Calculus, edition 1, 2, or 3. He died after those and later editions were reworked by other people. The usefulness for physics is beyond compare. I never use a textbook anymore, I just post my lecture notes but if students want a book I recommend Thomas. They are reasonably priced used. All current calculus books suck. Something like 9798% of students who take calculus don’t want to be mathematicians but calculus books are all written by pure mathematicians who don’t seem to understand who their audience is. I can’t think of who does it at the moment but some engineering schools are starting to teach their own math courses because the math dept can’t do it in a useful manner.

3) Don’t become enamored with epsilon-delta proofs. When you are in a physics class your professor will say “assume dt is infinitesimal....” Jerome Keisler has a good book for free on his website that teaches calculus using infinitesimals. That’s the way every physicist and engineer uses calculus and it’s very intuitive. Not only do people not use epsilon-delta proofs in application, they are not intuitively helpful.

4) I don’t know where other people took their ODE courses but you can not pass mine without a solid understanding of calc 2. Even calc 3 is prerequisite but I do let exceptional students take it concurrently.
Just curious. Why is Calculus 3 a requirement for you're ODE course? Is it because you give problems out problems that require parameterizarion, requirement to change from Cartesian coordinates to spherical coordinate/cylindrical coordinate for integration purposes? Heavy use of facts regarding the Taylor Series and particular expansions. Ie e^x, cosx, sinx, etc?

Jordan C. Forms?

"Best textbook to truly understand single variable calculus?"

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