Calculus Good textbooks for really learning calculus?

1. Apr 19, 2017

ScienceMan

I'm going to take Calc I in the fall and Calc II and III later on and I want to actually understand the stuff intuitively instead of just trying to memorize formulas and then having trouble with the applications, like optimization.

I have James Stewart's Essential Calculus Early Transcendentals which is hard to understand. It seems kind of incomplete to me, but I don't really know since I'm no calculus expert. The book my instructor will be using in my Calc I class just came in the mail and it's called Calculus for Scientists and Engineers Early Transcendentals by William Briggs. I haven't looked through it yet but I'm not really optimistic seeing Pearson is the publisher and my algebra and trig books from them were terrible.

What texts do you guys recommend that might help me understand calculus at an intuitive level so my knowledge of the subject doesn't just disappear once I forget the formulas?

2. Apr 19, 2017

smodak

1. https://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098918 along with the https://www.amazon.com/Combined-Answer-Calculus-Fourth-Editions/dp/0914098926
2. Apostol https://www.amazon.com/Calculus-Vol-One-Variable-Introduction-Algebra/dp/0471000051 and https://www.amazon.com/Calculus-Vol-Multi-Variable-Applications-Differential/dp/0471000078
3. Courant and John Volume 1 and Volume 2
4. https://www.amazon.com/gp/product/1461479452
After that jump into good Analysis Books.

Last edited by a moderator: May 8, 2017
3. Apr 20, 2017

IGU

Well, much depends on what you mean by understand. If you want to understand like a mathematician, then you'll have to spend your time figuring out proofs. Understanding consists of being able to prove everything forwards and backwards. You'll want to be able to do things like understand how modifying one of the premises would change the meaning and applicability of a theorem.

If you want to understand like a scientist or engineer, you'll have to spend your time solving lots of problems using the various tools you are given. You'll want to be able to, given a problem, know what tool to use and how to use it. Lots of practice is essential, but you won't really care about proving anything.

So which are you after? Or is it something else? In any case, true understanding requires putting in substantial effort.

Your books are more useful for the latter than the former. Even so, they aren't very good. They are best for people who just want to memorize formulas and get on to the next thing.

4. Apr 26, 2017

alan2

Last edited by a moderator: May 8, 2017
5. Apr 26, 2017

Aufbauwerk 2045

I think this is like asking how can I understand trigonometry intuitively, as opposed to just memorizing formulas and working lots of problems. I think the process is first we memorize formulas and work lots of problems, and as we do, our intuition, whatever that means, develops.

The only difficulty I found in calculus was with limits. I think that's the real "Pons Asinorum" for calculus. Once you understand limits, the rest of Calc I is easy if you just relax and take it step by step. Just realize there is a ton of work and no shortcuts. The goal should be to solve every problem in the textbook, and be able to solve similar problems without consulting the textbook. Then the tests should go well.

For a quick start in Calculus, I recommend Bob Miller's Calc for the Clueless. He actually has books on Calc I, II, and III. He is a very good teacher and his books are fun to read.

Then for an actual textbook, I really like Ellis and Gulic, Calculus with Analytic Geometry. I have the 3rd edition in my library. If I had to select only one beginning calculus book, this would be the one.

Concerning memorization, I found it much better to memorize too much than too little. Even if you have lots of open book tests, there is a huge advantage in having things burned into your subconscious so you can't forget if you want to. I think our intuition comes in part from the subconscious, which processes, while we sleep, all the material we've memorized.

Last edited: Apr 26, 2017
6. May 9, 2017

ScienceMan

I'm looking to be a statistician, so I don't know what that falls under. In any case the grad program I want to get into requires real analysis and a lot of other upper division math so I'll have to figure out how to deal with proofs at some point.

I'm taking calculus because I need to be able to A) get into the statistics master's program I want and B) know what I'm doing both in grad school and as a statistician. It's nice to hear that calculus is 99% intuition but the way my book (the Stewart one anyway) appears to go at it is just teach the formulas (power rule, multiplication/division rules, chain rule, etc) then throw you at the applications with very little guidance. It's easy enough to take a derivative, it's just that knowing what to take the derivative of and when to take it isn't at all clear from the book's approach.

7. May 10, 2017

MidgetDwarf

Edwin E Moise: Calculus (Buy the Green Book). It is the perfect balance between intuition and rigor. It's level is higher than Stewart, but a bit lower than Spivak.

Explains the Why's and How's. I learned a lot of little tidbits from it. The author is very charming with his writing, and shows the power of MVT.

It also gives you Theorem on how to Check if Functions are invertible. I remember knew about this before this book!

8. May 10, 2017

Buffu

In proofs or in applications ?
The book is good but the proofs are not so basic as you may think, especially in third chapter. other than that it is a really nice book.

Last edited: May 10, 2017
9. May 13, 2017

alan2

Not sure what you mean by basic.