# Self-teaching Calculus?

I hope this is the correct forum for this thread. I know there are many threads on this but my situation is a little more specific.

I'll be a freshman in college this fall and will be taking Calc 1. In HS I took Alg 1,2,3 (useless class, same as alg 2... I had a scheduling conflict), pre-calc. I made A's in all those classes pretty easily but I wouldn't say I'm good at math. I guess I'm average. Anyways, I want to try to teach myself some calculus before I enter college in a couple months. I would like to really learn the subject so I would rather cover less material but learn it as well as possible. I'm basically looking for the best resources (online, preferably) to use to teach myself. I'll try to stop rambling and get to the point... How should I begin this process? What are some good websites/video lectures to use? I'm a little lost with what to do.

If it helps, I have a calculus book by Stewart (Concepts & Contexts, 2nd ed.) and also an Analytical Geometry and Calculus book from the late 50s.

I appreciate any suggestions!

I suggest that you stay away from multivariable calculus. Any arguments in support of taking m.v. calc could be used to justify teaching yourself all sorts of advanced courses, which is not advised.

I would study the following topics/ideas, whether in YouTube videos, books, wb resources, etc:
1) techniques for evaluating limits when direct substitution fails (avoid the delta-epsilon limit)
2) difference quotient/definition of the derivative
3) basic derivatives. These are on the inside cover of most calc books.
4) the CALCULUS definition of the (natural) logarithm.
5) fundamental theorem of calculus (of course!)

calculus is really not that bad. It gets a bad rep because it's heavy on algebra and courses usually move at a steady pace. Good luck!

Well, I took Calc I, II, and III my freshman year of college so I know what material you want to know for sure.
1. Basic derivatives (1st order, 2nd order, etc..)
2. Series (Taylor and Maclaurin)
3. Limits (need this before you do series)
4. Convergence/divergence
5. Trig identities
6. logarithms
7. trig substitutions
8. quotient/product rule

There are other topics but these are the most CRUCIAL. If you understand these, you'll do just fine. Like "The Chaz" said, Calc isn't really that bad. Have fun!!!

Would watching the MIT videos be sufficient or will they be more difficult to follow since it's MIT?

Also, has anyone read Calculus Made Easy (1914) by Silvanus P. Thompson?

I'm probably over thinking this....

Sylvanus Thompson is a bit dated but fun and easy to read.
It is more a book for Arts majors. Not very rigorous but good for many students.

You can find Gilbert Strang's book at books.google.com.
He teaches at MIT and its a first class book.
Several others you will encounter with a google search.

Integration is just multiplication with one of the elements changing.
And since multiplication is just repeated addition [ 4 x 3 = 4+4+4 ], it
comes down to addition. Area under the function from one x value to another x value.
Differentiation is just division and since division is repeated subtraction, it
comes down to subtraction. Instantaneous slope of the tangent line at a point on the function.

4) the CALCULUS definition of the (natural) logarithm.

What would that definition be? something other than the inverse of e^x?

Mark44
Mentor
What would that definition be? something other than the inverse of e^x?
$$ln(x) = \int_1^x \frac{dt}{t}$$

I think that the Stewart text should be enough for you. You could maybe find a tutor/mentor who can maybe provide you with guidance or create "deadlines" for things to keep you motivated. What others have listed is basically whats inside the Stewart text. And even if you don't FULLY understand what you're reading just the fact that you'll be familiar with it all once you take the course provides you with a head start. Calculus isn't hard. It may seem like it is at first, but if you just sleep on things it always helps and once you're done you'll look back and think that was easy!

Good luck!

My honest advice is to stay well away from the M.I.T. videos.

At first I thought they were really hard and was kind of scared away from them.

I went off and learned all of this stuff seperate from these videos & I would go back and watch videos on topics I now understand (having previously failed to get it) and no, it's just the videos that are at fault. Terrible!

I then tried an experiment, I watched ahead with those M.I.T. videos to see would I learn anything about, say, improper integrals etc... but no, the videos take about 5 seconds on the important thing (which is not given the importance it deserves) and then you go off on a tangent...

Stick with the books you have.

If you get stuck at any point, www.khanacademy.org[/url] , [url]www.justmathtutoring.com[/url] , [url]http://www.uccs.edu/~math/vidarchive.html[/URL] , are all video lectures based on the calculus course. The uccs ones are based off the Stewart textbook and are helpful but the other two are shorter and better, see what you think!

Basically, get your algebra, trigonometry and logarithms working perfectly and calculus will be extremely easy.

The important idea's in the course will be

multiplying by 1, i.e. $$\frac{\frac{1}{x}}{\frac{1}{x}}$$ is actually just equal to 1 but will be used for limits!

Never dividing by zero! i.e. if you're dividing by zero you're either doing the math wrong or are encountering a limit/derivative/l'hopital etc... Important to keep an eye on this as you might take a derivative but need to see where your original equation divides by zero - that can trip you up!

the pythagorean theorem

doing algebraic tricks!

some trigonometric substitutions

using the area formula's for triangles, squares, etc...

These are the things you'll be using as you learn all of the topics the guys/girls listed above in ths thread, enjoy! :biggrin:

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You just have to register on the site, takes a second and it's free.
I don't go there or anything :tongue2: I just registered and got access.
It'll ask you to give your pw and e-mail on every course you click.

Ahhh... I didn't get that far, thinking that it was a (paid) student-only resource!

Oh! I remembered something I saw a few years back, so I felt I had to share. There is a very, very good sequence of calculus videos that you can find at here. It has been a long time since I've watched them, but from what I recall they were excellent.

I have also read a majority of Calculus Made Easy. It's a lovely little book that tried valiantly to teach me calculus. Alas, I think I was too young at the time to understand everything, and indeed it does not go into enough depth. However, it is very well written, as it must be for me to have fond memories of it. If you can get your hands on a copy of the book, go for it. But if you are studying or preparing for a course, you will need more, such as the videos I recommended above.

Regarding MIT videos, they typically vary in difficulty and usefulness. I haven't watched the single variable calculus ones, but I have watched a few of the multivariable ones. I don't have much criticism for them, but I don't have much enthusiasm for them either. They were merely all right. The linear algebra ones, by Gilbert Strang, however, I found excellent, even if they are better appreciated after gaining a passing familiarity with the subject.

Check out this textbook Elementary Calculus: An infinitesimal Approach

http://www.math.wisc.edu/~keisler/calc.html

The approach is slightly different. I highly recommend reading the first chapter carefully before continuing with the rest of the text as instead of conventional limits he uses infinitesimals and hyper real numbers in a lot of his proofs. They are essentially the same thing.Personally, I find infinitesimals to be more convenient as it is less cumbersome than using limit notation over and over again.

I took Calculus AB my senior year of high school and I'm also just entering undergraduate. Currently, I'm teaching myself the BC material. I suggest buying Cracking the AP Calculus AB & BC Exams by Princeton Review for \$20. It's pretty helpful. Each chapter there are examples that are explained step by step, which helps for a beginner. Also, the book tries to get you to pattern recognize the problems so that you become familiar with the situations.

http://www.math.armstrong.edu/faculty/hollis/calcvideos/
http://ocw.mit.edu/courses/#mathematics

The Khan Academy has gentle introductions to the key topics in videos typically just under 10 minutes each. They're a nice way to get quickly familiar with some of the main ideas and techniques without getting bogged down in theory. The second link has videos by Selwyn Hollis, typically 20 minutes each, in a more formal style than the Khan Academy. At MIT, start on 18.01 Single Variable Calculus, Fall 2006, for which they now have a complete set of video lectures, 50 minutes each. Try them all, see what works best for you. As you can see from the YouTube comments, not everyone shares sponsoredwalk's negative opinion of the MIT series! I haven't watched them all yet. When I first started, I found them useful, but a struggle to keep up with. More recently. I've gone back to them, and am getting more out of them now I've had more practice.

The important idea's in the course will be

multiplying by 1, i.e. $$\frac{\frac{1}{x}}{\frac{1}{x}}$$ is actually just equal to 1 but will be used for limits!

Multiplying by 1 doesn't, in general, give a result equal to 1. Your equation, assuming it means (1/x)/(1/x) = x-1*(x-1)-1 = x-1*x = x/x, doesn't illustrate multiplication by 1, but rather multiplying a number by its reciprocal, in other words dividing a number by itself. This is equal to 1, except that division by zero is undefined. But these are basic definitions in algebra, which will probably be taken for granted in a calculus course.

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Check out this textbook Elementary Calculus: An infinitesimal Approach

http://www.math.wisc.edu/~keisler/calc.html

The approach is slightly different. I highly recommend reading the first chapter carefully before continuing with the rest of the text as instead of conventional limits he uses infinitesimals and hyper real numbers in a lot of his proofs. They are essentially the same thing.Personally, I find infinitesimals to be more convenient as it is less cumbersome than using limit notation over and over again.

I would avoid any introductory calculus book based on the non-standard reals like the plague. That is not a subject for an elementary class, and it will not be conducive to learning analysis at an intermediate level later. To really do non-standard analysis correctly one must first construct the non-standard real numbers and that in and of itself requires some sophistication -- the usual construction starting with ultrafilters (and if you don't know what an ultrafilter is, then that is just the beginning of the problem).

Limits are NOT a notation. They are an important concept, probably the single most important concept that you encounter in calculus, and if you think that infinitesimals obviate the need to understand limits then you are badly mistaken.

Far better to understand calculus and analysis as it is usually presented.

That said there are not many good calculus books out there, and there are a lot of rather poor ones. Spivak's book is pretty good, but not all that common.

The fundamental problem with calculus is that there are two levels of understanding. The deeper one is based on understanding the structure of the real numbers and the meaning of "completeness" at least in the form of the least upper bound property. The more common level is just an facility with symbol manipulation and an ability to calculate derivatives and integrals, without a great deal of understanding of what that actually means. What you get in a typical introductory calculus class ina university is the latter level of understanding, but with the advantage of an instructor who possesses the deeper understanding of the subject. This makes self-teaching of calculus from most texts a difficult undertaking, because the student lacks the depth of understanding to know the real basis of the theory, and it is not presented in the text, thus making the subject rather mysterious.

There is no good remedy for this situation short of learning the theory at a level of depth not ordinarily contained in a book with a title like "calculus". There are some good introductory books on real analysis, however, and for the gifted student reading one of those can be enlightening (and probably completely opaque to a less gifted student). One good introductory text at that level is Rosenlicht's Introduction to Analysis.

My recommendation is to simply learn your calculus from a competent mathematician at the university level. Take advantage of that opportunity, and learn more than just symbol manipulation.

Multiplying by 1 doesn't, in general, give a result equal to 1. Your equation, assuming it means (1/x)/(1/x) = x-1*(x-1)-1 = x-1*x = x/x, doesn't illustrate multiplication by 1, but rather multiplying a number by its reciprocal, in other words dividing a number by itself. This is equal to 1, except that division by zero is undefined. But these are basic definitions in algebra, which will probably be taken for granted in a calculus course.

What?

$$\frac{\frac{1}{x} }{\frac{1}{x} } \ = \ \frac{1}{x} \cdot \frac{x}{1} \ = \ \frac{x}{x} \ = \ 1$$

This is an example of multiplying by 1. If you're finding a limit of some rational polynomial & are looking to reduce the degree of both numerator & denominator in such a way that you're final answer will be a fixed answer, this is what you'd do.

For example;

$$\lim_{x \to \infty} \ \frac{x^2 \ + \ 2x \ + 2}{2x^2 \ - \ 3x \ + \ 4}$$

$$\lim_{x \to \infty} \ \frac{\frac{1}{x^2}(x^2 \ + \ 2x \ + 2)}{\frac{1}{x^2}(2x^2 \ - \ 3x \ + \ 4)}$$

$$\lim_{x \to \infty} \ \frac{1\ + \ \frac{2}{x} \ + \frac{2}{x^2}}{2 \ - \ \frac{3}{x} \ + \ \frac{4}{x^2}}$$

$$\frac{1}{2}$$

I don't know how you'd do this without multiplying the equation by 1 EDIT: I just think there are better, easier & more user friendly ways to learn calculus than by suffering with M.I.T.'s & Selwyn Hollis as they are not helpful when you're viewing these things for the first time. I really suffered wasting time with them. I just gave my opinion, enjoy them if you want - I wasted time with them & learned better from alternative sources - like the thinkwell videos, and khanacademy, and justmathtutoring. To each her/his own!

Ah, sorry sponsoredwalk, I see what you mean now: a useful way to simplify an expression (be it a limit or some other kind of equation) is to multiply the numerator (top) and denominator (bottom) by the same number, which we can do without changing the value of the expression because this is equivalent to multiplying the expression by x/x = 1.

When you wrote "multiplying by 1, i.e. (1/x)/(1/x) is actually just equal to 1" you intended the scope of the "i.e." to be "1", rather than (as I mistook it) the whole expression "multiplying by 1". The syntax was ambiguous and I thought you were saying that the operation of multiplying (1/x) by 1/(1/x) was called "multiplying by 1", which of course it isn't.

It might be a little embarrassing, but my first encounter with calculus was Mark Ryan's "Calculus for Dummies" which I found in grade 9. I read it through carefully, borrowed the companion problem-book from my library, and understood it. Then I went through some more rigourous calculus textbooks, like Michael Spivak's "Calculus". I'm off to my first year of university this fall where I'll actually learn calculus from a professor! Exciting!