Calc 2 References for Winter Break?

[in the rye]

Hey everyone, I am winding down of finals next week, and will be off of school for a month. I was planning on beginning Calculus 2 over winter break to get a head start, and I don't want to lose the information I've already learned. We left off on U-Substitution on some basic integrals. I was thinking of using my book (Stewarts SV Calculus) in conjunction with MIT ocw, with use of various YouTube people that helped me get through Cal 1. Do you think this will work, or do you have any suggestions? Our book, to me, isn't very readable, so it's not a reliable for me to being my single source, so I'm trying to see what else is out there. Thanks.

 

[Dr. Courtney]

Coursera Calculus Two

https://www.coursera.org/learn/advanced-calculus

https://www.coursera.org/learn/calculus1

 

[MidgetDwarf]

I find Stewart to be very readable, until chapter 16 (Multi-Variable Calculus). Maybe you are reading the book like a novel, or it is your first attempt reading a mathematical text?

 

[in the rye]

I find Stewart to be very readable, until chapter 16 (Multi-Variable Calculus). Maybe you are reading the book like a novel, or it is your first attempt reading a mathematical text?

He is readable up to a point. However, the proofs are where I get lost, and on some of the examples. He skips a lot of steps and I don't know where he's getting numbers, and sometimes lack explanations that I like.

The most recent example was the chain rule for integrals. He literally just says, "use the chain rule," without any explanation on how to really apply it to an integral since we've never done it before. His method of applying it to an integral is with using 'u' as a substitute variable for 'x⁴' which actually seemed to confuse me more. When I watched PatrickJMT and KhanAcademy, they never apply the substitution and it made seeing the chain rule infinitely easier for me.

I feel he just makes simple topics rather tedious. I have another book called the "Calculus Lifesaver" and it can explain what Stewart does in less pages, and easier to read, while still applying the same theory and proofs. Another example I can think of off the top of my head was in Related Rates or Optimization. He is doing a substitution and applies it without explaining that he did a substitution, it is almost as if he pulls a number out of thin air.

Another one was in the beginning of our integral section. He shows the properties of integrals and basically says, "These are hard to prove," so he doesn't really explain them. I had to sit on my own and draw graphs etc. to see why they were true. Maybe it's just that he lacks the depth I like. I'm not sure. I like to know a lot of detail behind the theorems and properties so that I understand what I'm doing, rather than applying the algorithm. And if I only used his book alone, I feel like I'd be doing the latter which seems to be the case with a lot of my classmates because I've found myself explaining things to them that I've gotten from other books.

I don't know. I didn't mind him for derivatives or limits, but beyond that I always found myself using other resources to explain the material.

 

[MidgetDwarf]

He is readable up to a point. However, the proofs are where I get lost, and on some of the examples. He skips a lot of steps and I don't know where he's getting numbers, and sometimes lack explanations that I like.

The most recent example was the chain rule for integrals. He literally just says, "use the chain rule," without any explanation on how to really apply it to an integral since we've never done it before. His method of applying it to an integral is with using 'u' as a substitute variable for 'x⁴' which actually seemed to confuse me more. When I watched PatrickJMT and KhanAcademy, they never apply the substitution and it made seeing the chain rule infinitely easier for me.

I feel he just makes simple topics rather tedious. I have another book called the "Calculus Lifesaver" and it can explain what Stewart does in less pages, and easier to read, while still applying the same theory and proofs. Another example I can think of off the top of my head was in Related Rates or Optimization. He is doing a substitution and applies it without explaining that he did a substitution, it is almost as if he pulls a number out of thin air.

Another one was in the beginning of our integral section. He shows the properties of integrals and basically says, "These are hard to prove," so he doesn't really explain them. I had to sit on my own and draw graphs etc. to see why they were true. Maybe it's just that he lacks the depth I like. I'm not sure. I like to know a lot of detail behind the theorems and properties so that I understand what I'm doing, rather than applying the algorithm. And if I only used his book alone, I feel like I'd be doing the latter which seems to be the case with a lot of my classmates because I've found myself explaining things to them that I've gotten from other books.

I don't know. I didn't mind him for derivatives or limits, but beyond that I always found myself using other resources to explain the material.

He applies the definition of the chain rule to differentiate the polynomial. The way how Patrick does it, is he does it by mental math. Patricks attempt works. The problem is that you want to know how to apply the definition of the chain rule, in order to differentiate harder problems. i.e. y=sin^2(tan(Sec^2(cos(root 1+x^4+sin X over x^2+root x))

try doing that problem, without applying the actual definition of the chain rule.

Yes, the book is not the best, however, I think the problem comes from you not reading it enough and thinking about what you read. Most students tend to just open up the book for 5 min, find what they need to solve a problem and move on. Math does not work like this. Countless back and fourth.

I would recommended Thomas Calculus With Analytical Geometry 3rd edition and Serg Lang. However, if you have trouble reading Stewart, you will have trouble reading the ones i mentioned. The books I recommended are superior books.

The related rates, in defense of the author, that chapter mostly deals with proportions, something that students learn about in Geometry and Trigonometry.

 

[in the rye]

He applies the definition of the chain rule to differentiate the polynomial. The way how Patrick does it, is he does it by mental math. Patricks attempt works. The problem is that you want to know how to apply the definition of the chain rule, in order to differentiate harder problems. i.e. y=sin^2(tan(Sec^2(cos(root 1+x^4+sin X over x^2+root x))

try doing that problem, without applying the actual definition of the chain rule.

Yes, the book is not the best, however, I think the problem comes from you not reading it enough and thinking about what you read. Most students tend to just open up the book for 5 min, find what they need to solve a problem and move on. Math does not work like this. Countless back and fourth.

I would recommended Thomas Calculus With Analytical Geometry 3rd edition and Serg Lang. However, if you have trouble reading Stewart, you will have trouble reading the ones i mentioned. The books I recommended are superior books.

The related rates, in defense of the author, that chapter mostly deals with proportions, something that students learn about in Geometry and Trigonometry.

I understood his section on the chain rule, just not in the application of the integral. His substituting 'u' on the upper limit is what confused me. Patrick showed why the chain rule works on the integral by using similar terminology covered in Stewarts derivative section. Which is why it worked for me. I've never seen an integral used as a functoon, and then immediately we were exposed to using the chain rule on it without any explanation. My professor actually wrote a few pages on WHY we do it since he felt the same way about the explanation. So, when I contacted the TA, they gave me his notes.

The related rates makes sense. My school didn't require geometry, so I've never actually had a geometry course. I had algebra, trig, precalc.

I guess I'll stick to Stewart if the others are harder. I typically spend about 15-20min reading the section before class. A lot of times I just don't follow his train of thought. I also work better when he uses graphical representations rather than just notation. I can see it better through graphs than symbolically. Which is why Patrick has been helpful this semester. He represents a lot of the ideas using graphs before hand.

It's clear it's not a bad book. I've heard a lot of positive things. I've just seemed to have taken more out of my other references than from Stewart's. The Calculus Lifesaver book I mentioned has been my goto. But you're right in that a lot of times I don't sit and workout what he's saying. I think I just get frustrated with it, so I stop and use the other things I've mentioned. I've also notice the concepts tend to take a day or so to really sink in, which could be why I have trouble with it as well.

Thanks for the recommendations though.

 

[MidgetDwarf]

I understood his section on the chain rule, just not in the application of the integral. His substituting 'u' on the upper limit is what confused me. Patrick showed why the chain rule works on the integral by using similar terminology covered in Stewarts derivative section. Which is why it worked for me. I've never seen an integral used as a functoon, and then immediately we were exposed to using the chain rule on it without any explanation. My professor actually wrote a few pages on WHY we do it since he felt the same way about the explanation. So, when I contacted the TA, they gave me his notes.

The related rates makes sense. My school didn't require geometry, so I've never actually had a geometry course. I had algebra, trig, precalc.

I guess I'll stick to Stewart if the others are harder. I typically spend about 15-20min reading the section before class. A lot of times I just don't follow his train of thought. I also work better when he uses graphical representations rather than just notation. I can see it better through graphs than symbolically. Which is why Patrick has been helpful this semester. He represents a lot of the ideas using graphs before hand.

It's clear it's not a bad book. I've heard a lot of positive things. I've just seemed to have taken more out of my other references than from Stewart's. The Calculus Lifesaver book I mentioned has been my goto. But you're right in that a lot of times I don't sit and workout what he's saying. I think I just get frustrated with it, so I stop and use the other things I've mentioned. I've also notice the concepts tend to take a day or so to really sink in, which could be why I have trouble with it as well.

Thanks for the recommendations though.

Try Thomas Calculus 3rd ed with analytical Geometry. There is a lot of geometric intuition. Its not dated at all, and proves everything in the actual text and not the appendix. I loved the area under the curve and solid of revolution section. The proof of the revolutions is really nice. The book really shines in the Calculus 2 department. He explains the different integration techniques beautifully. I was also in your situation, when I first started Calculus. I had a lot of logical gaps. However, I reviewed all the material after the class and got a better understanding.

Maybe you are like me, I have go through two books, in order to understand the previous book. Which is fine. It's great that you found a book that works for you. Many students do not have a supplemental text, which is a bad decision.

I always, reading a book requires time, patience, and effort. We are currently doing Stoke's Theorem in my calculus 3 class. I did not understand it for 2 days. I woke up today, reviewed previous sections, read the Stoke's Chapter and Poof, I understood.

 

[MidgetDwarf]

And the section that was giving you trouble, was the fundamental theorem of calculus part 1?

 

[MidgetDwarf]

and an integral is a transformation, that changes a function into another function, via the process of integration and vise-versa. That is what the Fundamental Theorem of Calculus is intuitively. However, transformations will be seen in Linear Algebra for the first time. It will make more sense later on.

 

[in the rye]

Yeah, it was FTC Part 1. Coincidentally, I understood the theorem, but I didn't understand the chain rule in one of his examples. That is, I knew that the integral and derivatives were inverses, it made sense to me from the antiderivative section. It seemed obvious. Same with Part 2 of the theorem, it's simply a matter of seeing that the integral gives you a position formula and you're simply subtracting the distances between two points to get how far apart they are from each other. It seemed really intuitive. But there's an example where he uses an upper limit on part 1, I believe, where the upper limit is x⁴, he says make the x⁴ = 'u' and apply the chain rule. Literally gives no explanation on what's going on, and it is IMMEDIATELY after being exposed to the function notation of an integral. It just didn't make any sense to me. But, after watching PatrickJMT work with them, it was kind of like, oh, duh... It was just the 'u' notation on the limits that threw me off. They seem kind of redundant.

Regardless, I've noticed a lot of the integral parts take me about a day/night to really get what's going on. I kind of think about it before bed, and wake up realizing what I didn't understand, whereas throughout Calc 1 I was able to just go to class, do my homework, no issues. It never took any time to stick because the concepts seemed really intuitive. The integral concepts are, too, but they're a little more abstract, in my opinion.

I'll see if I can find the book on the lower end. Thanks for the suggestion.

 

[in the rye]

I found the 7th edition for 6$, so I went ahead and purchased that. Hopefully it does that matter too much.

 

[WWGD]

OP: What kind of applications of Calculus to Winter are you looking for ? :)

 

[Student100]

Hey everyone, I am winding down of finals next week, and will be off of school for a month. I was planning on beginning Calculus 2 over winter break to get a head start, and I don't want to lose the information I've already learned. We left off on U-Substitution on some basic integrals. I was thinking of using my book (Stewarts SV Calculus) in conjunction with MIT ocw, with use of various YouTube people that helped me get through Cal 1. Do you think this will work, or do you have any suggestions? Our book, to me, isn't very readable, so it's not a reliable for me to being my single source, so I'm trying to see what else is out there. Thanks.

I still prefer Anton as the first introduction to calculus, 6th editions are also cheap. A lot of people hate on Anton's book here, but I like his style. Maybe it's nostalgia for me, if such a thing is possible for a text.

MIT OCW is kind of garbage, in my opinion, what I've seen from the courses I've watched don't lead me to believe they're all that valuable as a learning tool. Same as Khan (especially) and some of the other video tutorials. Anyway, it's good to start getting used to reading math texts and understanding what's going on in the text without relying on videos and memorization of techniques.

Edit: https://www.amazon.com/dp/0471153060/?tag=pfamazon01-20

Here is the book I refer to. Most of the gripes are about the third volume (Calc3), and I agree with some of them, but the single variable stuff is well done (again, just my opinion.)

If you know any Matlab/Maple/Mathematica there are also exercises you can grind out basically built for them.

 

[Amrator]

The two best non-rigorous (i.e. non-Spivak/Apostol) calculus books in my opinion are Thomas and Simmons:

https://www.amazon.com/dp/0070576424/?tag=pfamazon01-20

https://www.amazon.com/dp/0201531747/?tag=pfamazon01-20

Larson and Anton are plug and chug book. Avoid them at all costs.

Another fantastic resource is Paul's Online Math Notes: http://tutorial.math.lamar.edu/

Edit: If you are simply looking for a reference then you may also want to check out this book: https://www.amazon.com/dp/0992001005/?tag=pfamazon01-20

Very well done explanations.

 

[bacte2013]

The two best non-rigorous (i.e. non-Spivak/Apostol) calculus books in my opinion are Thomas and Simmons:

https://www.amazon.com/dp/0070576424/?tag=pfamazon01-20

https://www.amazon.com/dp/0201531747/?tag=pfamazon01-20

Larson and Anton are plug and chug book. Avoid them at all costs.

Another fantastic resource is Paul's Online Math Notes: http://tutorial.math.lamar.edu/

Edit: If you are simply looking for a reference then you may also want to check out this book: https://www.amazon.com/dp/0992001005/?tag=pfamazon01-20

Very well done explanations.

I second George Simmons's book. His explanation is incredibly clear, and I did like his proofs listed in the Appendix. Only problem I have with his book is the price...

 

[WWGD]

I always give the same advice to those that have access to a good Math library. Spend a couple of hours in the Math section browsing through different books, to see which of them feel right to you, and select the one(s) you believe is/are the best among these.

 

[MidgetDwarf]

I second George Simmons's book. His explanation is incredibly clear, and I did like his proofs listed in the Appendix. Only problem I have with his book is the price...

I did not like his text. He is relentless to provide proofs that are trivially proved. Excellent problem sets.

I still recommend Thomas 3rd edition with Analytical Geometry and the 9th edition, together. The 3rd ed, is bare to bones, straight to the point, and concise explanations. The 9th has very good problem sets and excellent source of applications.

 

[WWGD]

I did not like his text. He is relentless to provide proofs that are trivially proved. Excellent problem sets.

I still recommend Thomas 3rd edition with Analytical Geometry and the 9th edition, together. The 3rd ed, is bare to bones, straight to the point, and concise explanations. The 9th has very good problem sets and excellent source of applications.

I think the fact that there is a 9th edition alone suggests there must be something to the book.

 

[micromass]

I think the fact that there is a 9th edition alone suggests there must be something to the book.

On the contrary, I always say that any STEM book that has over 8 editions must be really bad. I have not yet seen a counterexample to that claim. It might be that Thomas used to be good in its first editions, but after 9 editions they have watered it down incredibly. That is very usual.

 

[Student100]

... Anton are plug and chug book. Avoid them at all costs.

Very plug and chug.

Whatever that means in math.

If you're confusing "plug and chug" (I don't remember any calculus problems where I simply put numbers in for variables into an equation to solve) with focusing on technique and clarity over rigorous proofs of every theorem, sure Anton's guilty (he even says so in the preface). Sure, something like say, Apostol (The most rigorous calculus volumes I've personally worked through, and some people would even say that's not very rigorous! Dirty real world applications and all), goes into more depth on topics and justifies the majority of the theorems with proofs, but I got a lot more out of the reading (and working) after understanding the techniques of solving calculus problems.

This is my rub, and it bothers me to my core the older I get, this whole rigor at all cost mentality is silly. It isn't like you can only read one calculus book and then - that's it! You can't ever expand on what you've learned, ever again! Nevermind that the majority of students learning calculus for the first time are looking to apply it, not study it's underpinning, and you've got one confused dude on the hate Anton gets (Or even Larson, which I personally hate from the narrative style of the text and him completely throwing out the baby with the bathwater.)

To a physics student studying calculus for the first time, which is more immediately apply-able? Technique, or theory?

Have you actually bothered to work through the text before critiquing it?

I stand by my recommendation, it's cheap, so if he doesn't like the narrative style or level of rigor he can toss it in the recycling bin.

 

[bacte2013]

On the contrary, I always say that any STEM book that has over 8 editions must be really bad. I have not yet seen a counterexample to that claim. It might be that Thomas used to be good in its first editions, but after 9 editions they have watered it down incredibly. That is very usual.

Not in the field of biological science (part of STEM). Many excellent books in biological science, from general biology to microbiology, molecular biology, etc., are above 9th edition because of anecessity of introducing new discoveries, new and improved integration of the topics, etc.

 

[in the rye]

So, silly me, I purchased the Calculus and Analytical Geometry by Thomas, but it covers Multivariable and Vector Calculus, not single variable. I'll be keeping it regardless for when I get there. However, I found a Thomas' Calculus https://www.amazon.com/dp/0321185587/?tag=pfamazon01-20 for $6.99. Is there a difference between this and his analytical geometry? The 11th edition I saw covered Cal 1-4. I saw a a few complaints on the Thomas Calculus', but hardly anyone his analytic geometry. Thanks everyone for the recommendations.

 

[MidgetDwarf]

It is dumbed down, and not worth a purchase. go with both 3rd and 9th ed.

 

[Amrator]

Very plug and chug.

Whatever that means in math.

If you're confusing "plug and chug" (I don't remember any calculus problems where I simply put numbers in for variables into an equation to solve) with focusing on technique and clarity over rigorous proofs of every theorem, sure Anton's guilty (he even says so in the preface). Sure, something like say, Apostol (The most rigorous calculus volumes I've personally worked through, and some people would even say that's not very rigorous! Dirty real world applications and all), goes into more depth on topics and justifies the majority of the theorems with proofs, but I got a lot more out of the reading (and working) after understanding the techniques of solving calculus problems.

This is my rub, and it bothers me to my core the older I get, this whole rigor at all cost mentality is silly. It isn't like you can only read one calculus book and then - that's it! You can't ever expand on what you've learned, ever again! Nevermind that the majority of students learning calculus for the first time are looking to apply it, not study it's underpinning, and you've got one confused dude on the hate Anton gets (Or even Larson, which I personally hate from the narrative style of the text and him completely throwing out the baby with the bathwater.)

To a physics student studying calculus for the first time, which is more immediately apply-able? Technique, or theory?

Have you actually bothered to work through the text before critiquing it?

I stand by my recommendation, it's cheap, so if he doesn't like the narrative style or level of rigor he can toss it in the recycling bin.

If Anton is as you describe then I may be thinking of another book.

Either way, you are right; I apologize. No, I have not fully worked through the book. Perhaps "plug and chug" wasn't the best phrase either. By "plug and chug", I mean giving a formula and then proceeding to do a bunch of examples.

Again, I apologize.

 

[MidgetDwarf]

https://www.amazon.com/dp/0201531747/?tag=pfamazon01-20

 

[MidgetDwarf]

here is the the 9th ed, which cost the same as the inferior edition you were gone buy.

 

[Thewindyfan]

I guess I'll stick to Stewart if the others are harder. I typically spend about 15-20min reading the section before class. A lot of times I just don't follow his train of thought. I also work better when he uses graphical representations rather than just notation. I can see it better through graphs than symbolically. Which is why Patrick has been helpful this semester. He represents a lot of the ideas using graphs before hand.

It's clear it's not a bad book. I've heard a lot of positive things. I've just seemed to have taken more out of my other references than from Stewart's. The Calculus Lifesaver book I mentioned has been my goto. But you're right in that a lot of times I don't sit and workout what he's saying. I think I just get frustrated with it, so I stop and use the other things I've mentioned. I've also notice the concepts tend to take a day or so to really sink in, which could be why I have trouble with it as well.

Thanks for the recommendations though.

I second this as well, most of the time it's easy to read through his explanations but sometimes he glosses over steps in his explanations like when he goes over FTC 2 and the such. I usually only read to get a general idea of the material from the chapter but I work through the examples presented in the chapter to get a better understanding overall. For me, it's a lot easier to talk to someone like my professor with questions regarding the material works and why certain things work and etc.

 

[MidgetDwarf]

I second this as well, most of the time it's easy to read through his explanations but sometimes he glosses over steps in his explanations like when he goes over FTC 2 and the such. I usually only read to get a general idea of the material from the chapter but I work through the examples presented in the chapter to get a better understanding overall. For me, it's a lot easier to talk to someone like my professor with questions regarding the material works and why certain things work and etc.

Thats a good method, but not fool proof. There is gone be a time where you have an incompetent professor. So it comes down to, pass or fail. The only way to pass is to cross reference atleast 2 sources. Mathematics takes time to digest, and it is no wonder things are hard to grasp on the 1st or even 3rd reading. The good thing is that, mathematics can be learned. You just have to attack it, re-attack it, and go over it again multiple times. I think Stewart is one of the easier books to read in my opinion. I may have found it easy, because I always read my assigned textbook front to back, in every class, including stuff like Art History.

The value of a book does not usually correlate with its market price. We live in age that books can be bought for under 5 dollars, quality books I may add.

Always have atleast 2 books for every science and math course. Ask around what people think of books.

I usually take Micromass advice regarding books. He has given me a few gems, such as Edwin E. Moise; " Geometry and Ross, "Ordinary Differential Equations".

I usually understand the material once the course is over and review from page 1. My geometry teacher in college once told me, to understand a subject you have too look at it from a higher vantage point. The first exposure, is just to get survey the land. The second is to send in the army. The third is to conquer.

Tutoring at the college also helps.

 

[MidgetDwarf]

I second this as well, most of the time it's easy to read through his explanations but sometimes he glosses over steps in his explanations like when he goes over FTC 2 and the such. I usually only read to get a general idea of the material from the chapter but I work through the examples presented in the chapter to get a better understanding overall. For me, it's a lot easier to talk to someone like my professor with questions regarding the material works and why certain things work and etc.

Maybe try Thomas 3rd edition? Or Serge Lang. I like how every proof in Thomas is easy to replicate and read. The proofs do not go onto the back page, rather he keeps it on the first page, something I prefer. Nice margins to check calculations and right notes. I really enjoyed his section on the Theorem of Pappus, and how we can make a problem regarding a Tourous, extremely trivial.

The Shell Method proof was really elegant. I also Like Lang's books.

 

[Amrator]

Maybe try Thomas 3rd edition? Or Serge Lang. I like how every proof in Thomas is easy to replicate and read. The proofs do not go onto the back page, rather he keeps it on the first page, something I prefer. Nice margins to check calculations and right notes. I really enjoyed his section on the Theorem of Pappus, and how we can make a problem regarding a Tourous, extremely trivial.

The Shell Method proof was really elegant. I also Like Lang's books.

I hear Lang's books are quite advanced though.