Any good calculus places to start?

[mathwonk]

I still don't understand your objection. To prove that 1 is the lub of all the numbers of form .99...9,

all I need to do is show two things:
1) 1 is at least as great as all those finite decimals,
2) no smaller number than 1 is also as great as all of them, i.e. any number smaller than 1 is also smaller than one of those finite decimals, or equivalently no positive number is smaller than all the differences 1 - .999...9.
= .000...001.

But part 1) is true by definition of the lexicographic order on finite decimals. Part 2) is proved in post 30.

If you like, I am showing that 1 is less than or equal to .99999..., not directly, but by showing it is less than or equal to EVERY upper bound of all the finite numbers .9999...9. In particular, since .999... is such an upper bound, 1 is less than or equal to it too.

 

[mathwonk]

I have lots of notes just sitting on my computer, and some on my website, but not any of the calculus notes. The website has mostly abstract algebra notes and a few advanced algebraic geometry notes.

 

[mathwonk]

Ah I see your objection, you are assuming as an honest person would do, that I will give a direct proof. But I am giving instead a very roundabout one. To show 1 is the lub, I show that 1 is itself an upper bound, and then that no number smaller than 1 is an upper bound, hence any other upper bound is at least as large as 1, so 1 is the least upper bound. Tedious, but I claim correct.

 

[mathwonk]

It seems my high school notes on real numbers are only 21 pages long and broken into two files, so maybe they will fit on here. I'll try. I taught these to students as young as sophomores, but they had all taken AP calculus. I considered this honors precalculus.

 

[mathwonk]

Here is one of my most creative attempts at explaining limits. That is, I like them but the class did not seem to get it so much. What do you think of this approach to limits via continuity rather than the other way around? My hypothesis was that continuity is more intuitive than limits, so that makes it really hard to do things the other way around.

Moreover after defining limits first and then continuity in terms of limits, all our textbooks then immediately began to actually find limits of polynomials and roots, not by the definition, but by the factoring process that I try to justify here. I.e. our books did not actually do what they had taught. So it seemed to me hard for the students to understand.

Anytime a book says one thing, just parroting other books, and then does it differently in the examples, also just parroting other books, it always made me feel they were not thinking about what they were saying. Consequently the books seemed worthless to me for understanding the subject. Of course the rub is that some students do not seem to care deeply about understanding the subject, just passing it by following some rules. That could explain a negative reaction against even the most well intentioned attempts at explanation.

 

[TylerH]

The harder part is to show that no number smaller than 1 is as large as all those finite decimals. Note that those finite decimals of form .9999. differ from 1 by a finite decimal of form .0001. So if there were a number lying strictly between 1 and all those finite decimals, it would differ from 1 by a positive number which is less than every number of form .000000...0001. I quit there in an elementary class saying that cannot happen. But for you, here is a sketch of the slightly tedious argument:

It amounts to showing there is no positive number smaller than all those numbers of form
.000000...0001. Now any non zero finite decimal has a first non zero digit in some position, and if we put a 0 in that position and follow it by a 1, we have a smaller number of the form .000000...0001. Thus no finite decimal can be smaller than all those.

As for an infinite decimal, it is at least as large as all its finite truncations by definition, and we can find a number of form .000000...0001. that is strictly smaller than one of those truncations. Thus also every infinite decimal is larger than some number of form .000000...0001.

Would this not be the proof, rendering "sup{.9, .99, .999, ...} = 1" extraneous?

 

[mathwonk]

Lazernugget, if you follow the notes in post 23, you are well on your way to understanding integrals.

 

[mathwonk]

Lazernugget, I will post one more set of notes on finding areas and volumes by antiderivatives. I hope some of this gives you a start. This is an actual lecture from my second semester honors college calculus class.

 

[Lazernugget]

Wow! I'll look at the pdf things later, but the other posts you made were amazing! Thanks for the help! I got to go read my trigonometry book...bye!

 

[mathwonk]

You are extremely welcome. The pleasure is mine. Some of those pdf files are really elementary, and you might glance at one or two of them.

the ones in post 34 were written for honors high school students, and the ones in 23 start out really elementary, with area of a circle. These are easier to read than a trig book.

 

[FeDeX_LaTeX]

Hello;

I started learning calculus around your age too. While I was a bit younger I remember "edugratis" helped me a bit. Unfortunately the website is no longer active, but you can find some of his videos on YouTube. Search "edugratis".

KhanAcademy is invaluable, too.

 

[mathwonk]

forgive me for this post but i have some honors calc notes that may interest someone who always wondered why his prof did not prove the basic theorems on continuity. Here is the fundamental boundedness result. This is the tedious technical result, not the more fun ones, but it is late and I am tired now of making it at least mildly compatible with current technology.