Do I need a new book? (Using Calculus: The elements by Comenetz)

[CuriousBanker]

Hello all,

I am re-teaching myself calculus (I don't remember anything from Calculus in college, and I only took calc I in college)

I am using the book stated in the title. I got up to page 65...so far so good.

Until it started asking me to compute definite integrals. Nowhere so far has it told me HOW to do so...it just explained the intuition behind it (fairly simple to grasp) and then starts giving questions...without telling me how to answer them! So I decided to just google how to do it, assuming maybe I missed something somehow even though I read it over multiple times...but all of the things I looked up when I googled, had not been introduced in the book yet!

So far in this book, there has been no mention of 1) What a derivative is 2) What an antiderivative is 3) what the central theorem of calculus is 4) how to integrate a function

So far, all I have learned about is some basic physics equations (f=ma, p=mv, w=fd, etc), the concept of an infinitesimal, the concept of a differentiation, and the concept of an integral...so far I understand all of these

So I skipped ahead a few pages thinking maybe I can just go back to it later, but the book just keeps going on talking about things that it has not explained. I am getting extremely discouraged and frustrated.

The only other book I have is Calculus by Spivak, and I think my girlfriend has the stewart book, but the stewart book seems to be mostly questions, and needs a teacher to guide, and I am self-studying so this does not help me.

Will I be able to work through Spivak with absolutely no prior knowledge of calculus (I know basic algebra and basic trig and some geometry)? Also, what levels of calculus does Spivak cover (in terms of calc 1-3 taught in school)? I am looking at Spivak table of contents and I don't see taylor series, chain rule, etc (I have no idea what these things mean, but I know I see them in other books)

What book should I be reading so that I know what the heck is going on? I want to 1) be able to compute the questions in the book (doesn't seem too much to ask) 2) Actually understand the concepts and understand why the formulas are the way they are. So far I was understanding everything in my current book until this, and now I am so lost and confused.

Thanks in advance for all of the help.

 

[sshai45]

I managed to find the page you're asking about on Google's book preview. On page 65 of the book, it says "Use the area interpretation" to find the integrals. So the functions being integrated should correspond to simple shapes you can find the areas of (and thus the integrals) using formulas from basic Euclidean geometry.

 

[CuriousBanker]

Well how would I find the integral from -1 to 1 (1-x^2)^1/2 dx, without using anything but geometry?

 

[sshai45]

Well how would I find the integral from -1 to 1 (1-x^2)^1/2 dx, without using anything but geometry?

1. What shape does the region bounded by the graph of that function -- f(x) = (1 - x^2)^(1/2), the x-axis, and the vertical lines at x = -1 and x = 1 from the x-axis to the graph, correspond to? Is it one that is familiar to you or not?

2. Do you know the formula for the area of such a shape?

3. Can you calculate its area, given 1) and 2)?

 

[CuriousBanker]

Is that the unit circle? Unit circle area is pi r ^2...so the area would be pi 1^2 = 1pi=pi...but the answer in the back of the book says the answer is pi/2

 

[CuriousBanker]

Oh wait no that is a semi circle

 

[CuriousBanker]

ahhh, now I get it

Should be simple, but what about the integral from 0 to 1 for xdx?

 

[CuriousBanker]

I guess I just find the area for that triangle huh...

Doh! I guess it was so simple all along! Thanks so much!

But normally aren't derivatives taught before integration?

 

[yungman]

Hello all,

Until it started asking me to compute definite integrals. Nowhere so far has it told me HOW to do so...it just explained the intuition behind it (fairly simple to grasp) and then starts giving questions...without telling me how to answer them! So I decided to just google how to do it, assuming maybe I missed something somehow even though I read it over multiple times...but all of the things I looked up when I googled, had not been introduced in the book yet!

So far in this book, there has been no mention of 1) What a derivative is 2) What an antiderivative is 3) what the central theorem of calculus is 4) how to integrate a functionThe only other book I have is Calculus by Spivak, and I think my girlfriend has the stewart book, but the stewart book seems to be mostly questions, and needs a teacher to guide, and I am self-studying so this does not help me.

Will I be able to work through Spivak with absolutely no prior knowledge of calculus (I know basic algebra and basic trig and some geometry)? Also, what levels of calculus does Spivak cover (in terms of calc 1-3 taught in school)? I am looking at Spivak table of contents and I don't see taylor series, chain rule, etc (I have no idea what these things mean, but I know I see them in other books)

First, all calculus book should explain what is a derivative, anti derivative, theorem of calculus and how to integrate a function. I don't know why you say your book does not have it.

Regarding to what book you should use, the first question is what is your goal? How far do you want to go? If you just want to get the basic calculus...which is the first two semesters of calculus, one book I used is by Sherman Stein and Anthony Barcellos.

https://www.amazon.com/Calculus-Analytic-Geometry-Stein-Sherman/dp/0070611750/ref=sr_1_1?s=books&ie=UTF8&qid=1368517182&sr=1-1&keywords=sherman+stein

It is an easy book to read and contain all the materials for this. But if your goal is to go beyond this and go into multi variables. I strongly suggest you to look into the one by Howard Anton.

https://www.amazon.com/Calculus-Late-Transcendentals-Single-Multivariable/dp/0471482730/ref=sr_1_10?s=books&ie=UTF8&qid=1368517053&sr=1-10&keywords=howard+anton+8th+edition+calculus

You can see the used ones are less than $5! I would get both used. Anton is harder, but it is a lot more precised. The definitions are very straight, which make it harder at the beginning, but you'll appreciate it later on. Stein is very easy and good for the beginning part, but fell apart in multiple variables.

I am a self studier also, I have over 5 books just on the basic calculus. Each book presents the topics a little differently, I would get a few so if you stuck with one, go to the other book. Some books are good in one section and suck in the other, only Anton is perfect...after study from cover to cover.

Good thing is the used calculus books are so cheap on Amazon and most of the used book I bought from Amazon are like new. If you are serious about calculus, get the solution manual. They have solution manual for both books I mentioned. Both definitely have Taylor, Macl. Series. Both have definitions on derivatives, fundamental theorem of calculus etc.

 

[DrewD]

I'm not going to disagree with yungman's book advice, but if you already have three calculus books, I would buy any more. The books you have are fine. Spivak covers everything and well beyond the average college calculus (if you go to a hard school or your school offers an honors calc or separate calc for math majors, Spivak is at that level). If you are having trouble self-studying, it is probably because self study is hard. There may be better books, but the three books you have cover everything. Just ask questions when things don't make sense.

If you decide to get another book, get rid of one of the ones you have (or at least hide it). In my experience having too many books on a single subject just makes it easy to put one down and switch when I'm getting confused. After realizing that the second book isn't helping, I go to the third... and repeat. Sometimes math is just hard. Ask questions and get used to being frustrated.

This is just my opinion from my personal experience.

 

[yungman]

Please note that I am not trying to start a debate, I just want to share my experience with the OP from total self study for the last 10 years on my own. Lessons I learned along the way. I have a tall books shelf of books to show for.

If Spivak is more advanced book, I strongly suggest getting another book. I am a self studier, I studied all the way to PDE without going to school. I also studied classical physics, electromagnetics on my own. Having more books is about the most important thing. I generally have 5 books for each subject.

One thing I learn. Easier book usually explain the simple stuff better and easy to understand. BUT they tend to be more relax and cut corners. Case in point the beginners calculus by Sherman Stein and the PDE book by Asmar. Both are the easiest book for the subject. They are easy to read and get started. BUT both fall apart on the more difficult part. Both get sloppy in the definitions. Sometimes, they are getting close to miss leading. At the same time, a more advanced book might be hard to get into. Without an instructor, it is so easy to get confuse when you read a very precise definition like in Anton. You really need multiple books.

From the collection of books I use, I can tell you, every book has certain style of explaining that suit individual better or worst. You'll find you understand certain part of a book better, other part with another book. For people that don't have the benefit of an instructor like OP, it is vitally important to have a few books, just like different opinions. Yes, you have to change your mind set and learn the terms of different books. But believe me, if you are stuck on one subject, it's worth your while to learn the terms of another book if that can get you over the hump.

This is basic calculus books, usually there is no mistake in the book. Wait until you get into the more advanced books! In the more advanced subjects, you really have to watch out typos, mistakes in the textbooks. You cannot be sure until you see the see formulas in both books. Case in point, when I was studying Phase Lock Loop, I actually challenged Roland Best in his book and he even offered to sent me the new copy! It is that bad. Having more books is the utmost important thing for me if you are self studier like I am.

To the OP, Sherman Stein might just be the ticket if your book is an advanced book.

 

[CuriousBanker]

Thanks all. Yeah, I will try to get through this one, I was just confused, because I was talking to my girlfriend who is a chemist and she was confused how my book started teaching integrals without first talking about derivatives...I know SPivak does derivatives first

 

[tfr000]

My favorite calc book. The original is really old - 1910 or something. It's not complete - it doesn't cover things like implicit differentiation - but for the basics, it's great.
https://www.amazon.com/dp/0312185480/?tag=pfamazon01-20

 

[mathwonk]

If you don't want comenetz's book, i'd like to have it. it is not a cookbook, but one you need to think about.

 

[DrewD]

Integrals and derivatives can be studied without reference to each other. It just turns out that their relationship to each other make a lot of problems easier. I don't know if the book is any good, but I would not discount it for this reason.