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General Discussion => Other Discussions => Math and Science => Topic started by: pimathbrainiac on December 10, 2013, 04:57:06 pm

Title: The Complete N00bs Guide to Calculus I (AP Calc AB)
Post by: pimathbrainiac on December 10, 2013, 04:57:06 pm
So I realized something: A lot of us Omni members in the US are sophomores in HS or younger, which means that a lot of us US high school members have no calculus experience (excluding Legimet, myself, and probably one or two other people that I forgot about). Since I think it is a very interesting subject, I'm going to make a "Complete N00bs Guide to Calculus I" (aka AP Calc AB) and post all the lessons and stuffs here.

So, without further ado: the first lesson.

Spoiler For Lesson I:
Lesson I: Limits Part I
Prerequisites: Algebra I

Let's start by defining a limit: A limit is what a dependent variable (of a function) (f(x)) approaches as an independent variable (x) approaches n, where n is a constant so long as it is the same number from the positive and negative side (not entirely correct, but it will work for the purposes of this lesson).
Or, in other words: "the limit of f(x) as x approaches n is [answer]"
So you can say that the "limit of x2 as x approaches 2 is 4"

Or, in proper notation (-> is an arrow)
lim(x->2) x2 = 4

Is this the same thing as plugging x in and solving?
No, well, not exactly
A limit is what x approaches, not what x is.

Take the piece-wise function f(x) = {x2 if x !=0 | 1 if x=0}
lim(x->0) {x2 if x !=0 | 1 if x=0} = 0

Why?
Because even though f(x) is 1 when x = 0, as x gets closer and closer to 0, f(x) gets closer and closer to 0.

Pretty cool, right?

So how do I solve a limit?
Ready?
You simplify the equation as much as possible, cancelling out anything and everything, then you plug in n for x.

So what if you have a function where "f(x) = 1/x", so that the number as x approaches 0 is different coming from the positive side than coming from the negative side?
That's where one-sided limits come into play. One-sided limits are the same as normal limits, but they are specifically as x->n from either the positive side or the negative side. A normal limit is when the two one-sided limits are the same.
one-sided limits are denoted as follows:

lim(x->0)+ f(x)
if being approached from the positive (right) side, and
lim(x->0)- f(x)
if being approached from the negative (left) side.

so

lim(x->0)+ 1/x = infinity
and
lim(x->0)- 1/x = negative infinity

These are useful with holes and other things you find in graphical algebra.

So how do I solve a one-sided limit?

The best way, as I have found, is to look at the graph and see if has an asymptote. If it is, is it going towards infinity, or negative infinity? There's your answer. If there is a jump discontinuity, you have to get an x value as close as you can to n from the correct side, and estimate based on that.

In the next lesson: Limits II: Derivatives by Limits

Spoiler For Lesson II:
Lesson II: Derivatives by Limits
Prerequisites: Algebra I and Lessons I

So we now understand what a limit is, right? Why are they so important? Because (at least for now) they help you find derivatives.

First, we must understand what a derivative is.
A derivative is simply a rate of change, or a change in one variable divided by a change in another. This is the same as slope, kind of.

Slope is defined as the change in f(x) in relation to change in x. This can be denoted as (http://latex.codecogs.com/gif.download?%5Cfrac%7B%5CDelta%20y%7D%7B%5CDelta%20x%7D), where (http://latex.codecogs.com/gif.download?%5CDelta) means change in [variable].

(http://latex.codecogs.com/gif.download?%5CDelta%20y) is also denoted as (http://latex.codecogs.com/gif.download?f%28x%20+%20%5CDelta%20x%29%20-%20f%28x%29)
(http://latex.codecogs.com/gif.download?%5CDelta%20x) is denoted as (http://latex.codecogs.com/gif.download?%5CDelta%20x) because it is arbitrary in many cases. X is arbitrary as well.

So, slope is the rate of change (http://latex.codecogs.com/gif.download?%5Cfrac%7Bf%28x%20+%20%5CDelta%20x%29%7D%7B%5CDelta%20x%7D) where X and (http://latex.codecogs.com/gif.download?%5CDelta%20x) are arbitrary.

A derivative is a slope at a point. what is (http://latex.codecogs.com/gif.download?%5CDelta%20x), then? 0, right? But that can't happen because we can't devide by 0. That's where limits come into play.

Oh, I get it! It must be (http://latex.codecogs.com/gif.download?%5Clim_%7B%5CDelta%20x%5Crightarrow%200%7D), right?

Exactly! In other words, (http://latex.codecogs.com/gif.download?%5Clim_%7B%5CDelta%20x%5Crightarrow%200%7D%20%5Cfrac%7Bf%28x%20+%20%5CDelta%20x%29%7D%7B%5CDelta%20x%7D)

So how do I use this?

Take the function f(x)=x2

(http://latex.codecogs.com/gif.download?%5Clim_%7B%5CDelta%20x%5Crightarrow%200%7D%5Cfrac%7B%28x%20+%20%5CDelta%20x%29%5E2%20-%20%28x%5E2%29%7D%7B%5CDelta%20x%7D) simplifies to

(http://latex.codecogs.com/gif.download?%5Clim_%7B%5CDelta%20x%5Crightarrow%200%7D%5Cfrac%7Bx%5E2%20+%202x%5CDelta%20x%20+%20%5CDelta%20x%5E2%20-%20x%5E2%7D%7B%5CDelta%20x%7D) and then to

(http://latex.codecogs.com/gif.download?%5Clim_%7B%5CDelta%20x%5Crightarrow%200%7D%5Cfrac%7B2x%5CDelta%20x%20+%20%5CDelta%20x%5E2%7D%7B%5CDelta%20x%7D) and then

(http://latex.codecogs.com/gif.download?%5Clim_%7B%5CDelta%20x%5Crightarrow%200%7D2x%20+%20%5CDelta%20x) and then we plug in "0" for (http://latex.codecogs.com/gif.download?%5CDelta%20x)

= 2x

We can then plug in the "x" coordinate of the point of which we want the derivative!

There you have it! That's the derivative of x2

Derivatives are denoted as follows:

(http://latex.codecogs.com/gif.download?%5Cfrac%7Bd%20y%7D%7Bd%20x%7D%20f%28x%29)

where (http://latex.codecogs.com/gif.download?%5Cfrac%7Bd%20y%7D%7Bd%20x%7D) means the derivative of y with respect to x and f(x) is the function.

That is all I have to teach you in this lesson! Here are a few practice problems!

find:

(http://latex.codecogs.com/gif.download?%5Cfrac%7Bd%20y%7D%7Bd%20x%7D%20%28x%5E2%20+%202x%20+%201%29)

Spoiler For solution:
2x + 2

(http://latex.codecogs.com/gif.download?%5Cfrac%7Bd%20y%7D%7Bd%20x%7D%20%28x%5E3%20+%205x%5E2%20+%202x%29)

Spoiler For solution:
3x2 + 10x + 2

I'm not that creative with practice problems. You should go do these: https://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/defderdirectory/DefDer.html

Next up: The "Cheater" Way to do Derivatives!
Title: Re: The Complete N00bs Guide to Calculus I (AP Calc AB)
Post by: Sorunome on December 10, 2013, 05:37:57 pm
skimming it, it looks nice, this might help out some people :)
Title: Re: The Complete N00bs Guide to Calculus I (AP Calc AB)
Post by: AssemblyBandit on December 10, 2013, 05:38:40 pm
Awesome, thanks! I'm well beyond high school but I don't know a thing about calculus. Thanks to my programming ability and my trusty calc, I had my college instructors fooled into thinking I was good at math, enough that once I was pulled out of class to verify an equation. My response, 'Yup, that looks about right.' hehehe.
Title: Re: The Complete N00bs Guide to Calculus I (AP Calc AB)
Post by: chickendude on December 10, 2013, 08:49:12 pm
It's a nice idea, but some of it isn't all that clear to me. The first sentence, for example, made absolutely no sense to me until i looked at the following explanation. What is a dependent/independent variable? And what is a piecewise function? Just a function divided up depending on what value x has? Thinking about it more your example makes sense now, though you say that f(x) = 0 when x = 0, i think that should be f(x) = 1. Also, you wrote "closer ans closer".

The rest is pretty straightforward to me.

Like AssemblyBandit, i'm well out of high school. I took AP Calc in high school but have long since forgotten it all. I'll keep reading on as calculus and physics were always really interesting to me, i just never stuck with them after dropping out of high school and it's all vanished.

Thanks for putting the lessons together, pimathbrainiac!
Title: Re: The Complete N00bs Guide to Calculus I (AP Calc AB)
Post by: pimathbrainiac on December 10, 2013, 09:11:19 pm
Thanks for the feedback!

Quote from: chickendude on December 10, 2013, 08:49:12 pm
It's a nice idea, but some of it isn't all that clear to me. The first sentence, for example, made absolutely no sense to me until i looked at the following explanation. What is a dependent/independent variable? And what is a piecewise function? Just a function divided up depending on what value x has? Thinking about it more your example makes sense now, though you say that f(x) = 0 when x = 0, i think that should be f(x) = 1. Also, you wrote "closer ans closer".

The rest is pretty straightforward to me.

Like AssemblyBandit, i'm well out of high school. I took AP Calc in high school but have long since forgotten it all. I'll keep reading on as calculus and physics were always really interesting to me, i just never stuck with them after dropping out of high school and it's all vanished.

Thanks for putting the lessons together, pimathbrainiac!

The "ans" and the f(x) = 0 were typos... changed to "and" and "f(x) = 1"

Thanks for that.
As for piecewise, it is taught in Algebra I, so that is probably something you forgot. Your guess as to what it is is correct, though.

EDIT: It is notated incorrectly, though, as I don't have a tool to make the equations be images and that was the best I could do with the resources I have.
Title: Re: The Complete N00bs Guide to Calculus I (AP Calc AB)
Post by: chickendude on December 10, 2013, 09:23:46 pm
I just forgot, then. Oh well. Thanks for clearing it up, though.

Also, if you really think it's worth the effort you could look for an online LateX editor. I found this: http://www.codecogs.com/latex/eqneditor.php which will let you do things like (http://latex.codecogs.com/gif.download?%5Clim_%7Bx%5E%5Cto%202%7D%5Cleft%20x%5E%7B2%7D%20%3D%204) relatively easily.
Title: Re: The Complete N00bs Guide to Calculus I (AP Calc AB)
Post by: pimathbrainiac on December 10, 2013, 09:24:55 pm
okay. Thanks for the tip!
Title: Re: The Complete N00bs Guide to Calculus I (AP Calc AB)
Post by: pimathbrainiac on December 11, 2013, 10:21:24 am
2nd lesson up!
Title: Re: The Complete N00bs Guide to Calculus I (AP Calc AB)
Post by: Link on December 11, 2013, 05:06:01 pm
Ah, I remember those days... Calculus I was one of my favourite classes, and it only gets worse from here on :( So possible warning for some of you. Anyhow, I liked this and wanted to post this:

(I did not actually write this, but it is useful!)
Calculus I notes. (http://tutorial.math.lamar.edu/Classes/CalcI/CalcI.aspx)

And for future classes:
Calculus II notes. (http://tutorial.math.lamar.edu/Classes/CalcII/CalcII.aspx)
Calculus III notes. (http://tutorial.math.lamar.edu/Classes/CalcIII/CalcIII.aspx)

Finally, I'd like to offer my help if anyone has questions, just ask away! (here in the forum, or on chat).
Title: Re: The Complete N00bs Guide to Calculus I (AP Calc AB)
Post by: Deep Toaster on December 17, 2013, 10:22:10 pm
Yay math tutorials! I have a suggestion—since most all of the people here are pretty good with calculators, these lessons could be centered around them, as in teaching people calculus using calculators as a tool.
Title: Re: The Complete N00bs Guide to Calculus I (AP Calc AB)
Post by: bb010g on December 17, 2013, 10:22:52 pm
What about hyperreals?