Good textbooks for really learning calculus?

[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?

 

[smodak]

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?

1. https://www.amazon.com/dp/0914098918/?tag=pfamazon01-20 along with the https://www.amazon.com/dp/0914098926/?tag=pfamazon01-20
2. Apostol https://www.amazon.com/dp/0471000051/?tag=pfamazon01-20 and https://www.amazon.com/dp/0471000078/?tag=pfamazon01-20
3. Courant and John Volume 1 and Volume 2
4. https://www.amazon.com/gp/product/1461479452/?tag=pfamazon01-20
After that jump into good Analysis Books.

And do not forget to read https://www.amazon.com/dp/0198723539/?tag=pfamazon01-20

 

[IGU]

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.

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.

 

[alan2]

Why are you taking calculus? That makes a big difference. I find it very unfortunate that you would mention memorizing formulas because calculus is 99% intuition, there are no formulas to memorize. Unfortunately, at least in the US, that's how students have been taught to do math. The biggest problem with calculus texts is that they're written by mathematicians who think they need to teach calculus the way they were taught analysis but it's rare to get a student in a calculus class, particularly calc 1, that wants to be a mathematician. It depends on how you categorize things but roughly 1/3 of undergrad degrees are in some field of science and engineering and another 1/3 are in a business related field while only about 1% are in the combined fields of math, statistics, and applied math so probably 99% of the students in a math class want to learn how to use it. Fortunately, calculus is the most intuitive branch of mathematics you will find but books never focus on how it is actually used. I tell my students that I don't care if they have the book. I write everything up for them and post the notes. Sometimes it takes some work to get particular students to think instead of looking for a formula but they do come around. Please skip books like Spivak or Apostol if you are one of the 99%. You will give up pretty quickly. Jerome Keisler has made his book available for free. https://www.math.wisc.edu/~keisler/calc.html You can also get it cheap because it has been reprinted by Dover. https://www.amazon.com/dp/0486484521/?tag=pfamazon01-20. Keisler does calculus the way that physicists and engineers use calculus so you don't sit through a math class thinking that you understand and then go into a physics class and not have a clue what they're talking about. Interestingly, after decades of wondering why students struggle in calculus, some universities are now actually offering 2 calculus sequences, one for people who need to use it and one for people who want to do proofs. In fact, the University of Michigan offers about 100 sections of calc 1 per semester, several sections of "honors applied calculus", and 1-2 sections of the theoretical calc sequence which is only taken by math majors. The people who need to learn how to use it learn how to use it. And, by the way, learning to use it doesn't mean not learning why it's true or not learning proofs. I prove every single thing that I use, beginning with limits, but in a way that engineers understand it. Pure mathematicians get their panties in a wad protesting that you can't teach infinitesimals to first year students but that's complete BS. Abraham Robinson proved that standard and non-standard analysis are entirely equivalent and I can do a proof in 3 lines that students understand which your average textbook puts in an appendix and is usually skipped, because it's several pages that nobody with a HS background in math can even understand. "No proof, just memorize the result" is not a way to teach math and it cheats students out of the basic intuition that they will need when they actually have to use calculus. Every single result and every single line has meaning in calculus and if you learn it the way it was created, with intuition, you will be able to use it.

 

[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.

 

[ScienceMan]

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.

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.

Why are you taking calculus? That makes a big difference. I find it very unfortunate that you would mention memorizing formulas because calculus is 99% intuition, there are no formulas to memorize. Unfortunately, at least in the US, that's how students have been taught to do math. The biggest problem with calculus texts is that they're written by mathematicians who think they need to teach calculus the way they were taught analysis but it's rare to get a student in a calculus class, particularly calc 1, that wants to be a mathematician. It depends on how you categorize things but roughly 1/3 of undergrad degrees are in some field of science and engineering and another 1/3 are in a business related field while only about 1% are in the combined fields of math, statistics, and applied math so probably 99% of the students in a math class want to learn how to use it. Fortunately, calculus is the most intuitive branch of mathematics you will find but books never focus on how it is actually used. I tell my students that I don't care if they have the book. I write everything up for them and post the notes. Sometimes it takes some work to get particular students to think instead of looking for a formula but they do come around. Please skip books like Spivak or Apostol if you are one of the 99%. You will give up pretty quickly. Jerome Keisler has made his book available for free. https://www.math.wisc.edu/~keisler/calc.html You can also get it cheap because it has been reprinted by Dover. https://www.amazon.com/dp/0486484521/?tag=pfamazon01-20. Keisler does calculus the way that physicists and engineers use calculus so you don't sit through a math class thinking that you understand and then go into a physics class and not have a clue what they're talking about. Interestingly, after decades of wondering why students struggle in calculus, some universities are now actually offering 2 calculus sequences, one for people who need to use it and one for people who want to do proofs. In fact, the University of Michigan offers about 100 sections of calc 1 per semester, several sections of "honors applied calculus", and 1-2 sections of the theoretical calc sequence which is only taken by math majors. The people who need to learn how to use it learn how to use it. And, by the way, learning to use it doesn't mean not learning why it's true or not learning proofs. I prove every single thing that I use, beginning with limits, but in a way that engineers understand it. Pure mathematicians get their panties in a wad protesting that you can't teach infinitesimals to first year students but that's complete BS. Abraham Robinson proved that standard and non-standard analysis are entirely equivalent and I can do a proof in 3 lines that students understand which your average textbook puts in an appendix and is usually skipped, because it's several pages that nobody with a HS background in math can even understand. "No proof, just memorize the result" is not a way to teach math and it cheats students out of the basic intuition that they will need when they actually have to use calculus. Every single result and every single line has meaning in calculus and if you learn it the way it was created, with intuition, you will be able to use it.

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.

 

[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!

 

[Buffu]

what to take the derivative of

In proofs or in applications ?

rome Keisler has made his book available for free. https://www.math.wisc.edu/~keisler/calc.html

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.

 

[alan2]

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.

Not sure what you mean by basic.