| --- Log opened Mon Jul 30 00:00:39 2007 |
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| 03:33 | | Chalcedon is now known as NSGuest-310 |
| 03:34 | | NSGuest-310 is now known as Chalcedon |
| 03:35 | < Chalcedon> | is there anyone mathematically minded about? |
| 03:35 | | * Chalcedon would like to check that her brain is screwed on right |
| 03:37 | | * Vornicus is the mathman |
| 03:38 | < Chalcedon> | I am reviewing algebra |
| 03:39 | < Chalcedon> | actually it's a review test, but it doesn't say whether it's open or closed book or anything helpful. |
| 03:39 | < Chalcedon> | (it doesn't count for marks though) |
| 03:39 | < Chalcedon> | A lot of it I'm confident about, but could I run a couple of things by you to see if I'm remembering it correctly? |
| 03:40 | < Vornicus> | sure |
| 03:41 | | * Chalcedon tries to work out how to convert stuff to text... |
| 03:42 | < Chalcedon> | 2x^5y^2(x^2y)^-2 would be = 2x ? |
| 03:42 | < Vornicus> | 2 * x^5 * y^2 * (x^2 * y) ^ -2 ? |
| 03:43 | < Chalcedon> | yes |
| 03:43 | < Chalcedon> | (thats better, thanks :)) |
| 03:43 | | * Vornicus does the fiddling. |
| 03:43 | < Vornicus> | 2 * x^5 * y^2 * x^-4 * y^-2 |
| 03:43 | < Vornicus> | 2 * x^(5-4) * y(2-2) |
| 03:43 | < Vornicus> | 2 * x * 1 |
| 03:43 | < Vornicus> | 2 * x |
| 03:44 | < Chalcedon> | win :) |
| 03:45 | < Chalcedon> | ok, next? |
| 03:45 | < Chalcedon> | 2 * x^4 / sqrt(x) |
| 03:45 | < Vornicus> | chalcysproblems.next() |
| 03:45 | < Chalcedon> | (I got 2 * x ^-2) |
| 03:45 | < Chalcedon> | heh :) |
| 03:45 | < Vornicus> | yeah, that's wrong |
| 03:45 | < Chalcedon> | ok. |
| 03:46 | < Vornicus> | 2 * x^4 / x^(1/2) |
| 03:46 | < Vornicus> | 2 * x^4 * x^(-1/2) |
| 03:46 | < Chalcedon> | I seemed to recall that sqrt(x) = x^ 1/2 |
| 03:46 | < Vornicus> | 2 * x^(7/2) |
| 03:46 | | * Chalcedon thwap self |
| 03:47 | < Chalcedon> | I got to the 2 * x^4 * x ^ (-1/2) stage |
| 03:47 | < Chalcedon> | I just failed my exponents check |
| 03:47 | < Vornicus> | and multiplied instead of added |
| 03:49 | < Chalcedon> | one more for now I think |
| 03:51 | < Chalcedon> | This one I couldn't figure out how to factorise, thus I assumed a straight combine like terms/simplify |
| 03:51 | < Chalcedon> | 7*x^2 + x/2 + 5 - (4*x^2 - x + 1) |
| 03:52 | < Vornicus> | Combine like terms, simplify, then figure out the factors |
| 03:52 | < Vornicus> | Which isn't too hard, if you know the trick. |
| 03:52 | < Chalcedon> | I used to, but I forgot it |
| 03:52 | < Vornicus> | Remember the quadratic formula? |
| 03:53 | < Chalcedon> | no? |
| 03:53 | < Chalcedon> | (I got 3 *x^2 +3x/2 + 4) |
| 03:53 | < Vornicus> | Right, so far so good |
| 03:53 | < Chalcedon> | which actually looks like it'll factorise |
| 03:53 | < Vornicus> | Yeah, it ought to be able to |
| 03:53 | < Vornicus> | ( -b ? sqrt(b^2 - 4ac) ) / 2a |
| 03:54 | < Vornicus> | the quadratic formula, when applied to a quadratic expression like that, gets you the roots or zeroes of the expression |
| 03:54 | | * Chalcedon writes this down |
| 03:54 | < Chalcedon> | ...actually |
| 03:54 | < Chalcedon> | I remember that |
| 03:56 | | * Chalcedon applies this to her current answer |
| 03:56 | < Vornicus> | and then you can take those and plug them into the "by-zeroes" form of polynomials: k(x - z1)(x - z2), where zn are the zeroes of the polynomial |
| 03:56 | < Vornicus> | and k is the coefficient of the highest order term |
| 03:59 | < Chalcedon> | remind me where the b a and c come from? |
| 04:00 | < Vornicus> | ax^2 + bx + c |
| 04:02 | < Chalcedon> | that's what I thought |
| 04:03 | < Chalcedon> | that formula doesn't always work iirc, because, as in this case (-1/2)^2 = 1/4. 1/4 - 4*7*5 = a negative number (which can't then be square rooted) |
| 04:03 | < Chalcedon> | (am I remembering correctly?) |
| 04:03 | < Vornicus> | er |
| 04:04 | < Vornicus> | use the thing where you've cleaned it up to just three numbers |
| 04:04 | < Vornicus> | But you're right that it doesn't always work. |
| 04:04 | < Chalcedon> | you mean k(x - z1)(x - z2)? |
| 04:04 | < Vornicus> | Well, unless you're crazy and throw around complex numbers like there's no tomorrow. |
| 04:05 | < Vornicus> | The one that goes x*2^2 + 3x/2 + 4 |
| 04:05 | | * Chalcedon isn't this crazy |
| 04:05 | | * Chalcedon duh |
| 04:05 | < Chalcedon> | sorry, that was thick |
| 04:05 | < Vornicus> | only, copied right |
| 04:06 | < Chalcedon> | it's still negative |
| 04:06 | < Vornicus> | Yeah, I noticed. Looks like you can't factor it. |
| 04:07 | < Chalcedon> | ok I am happy for now, thanks for your help :) |
| 04:07 | | * Chalcedon cookies |
| 04:07 | | * Vornicus nrom. |
| 04:07 | | * Vornicus stoatburgers |
| 04:08 | | * Chalcedon nrom |
| 04:09 | < Chalcedon> | oh, while you're still about? |
| 04:09 | < Vornicus> | yeeeees? |
| 04:09 | < Chalcedon> | can you give me a quick differentiation lesson? |
| 04:10 | < Chalcedon> | I did this at school but my text book teaches it differently and I'm horribly confused |
| 04:10 | < Vornicus> | Ooh fun. |
| 04:10 | < Chalcedon> | or point me in the direction of one online? |
| 04:10 | < Vornicus> | Would you like the underlying goodies, or would you prefer "tell me how this particular expression differentiates" |
| 04:11 | < Chalcedon> | both would probably be useful |
| 04:11 | < Vornicus> | Okay. |
| 04:11 | < Chalcedon> | but can we work through an example first? |
| 04:11 | < Vornicus> | Sure. |
| 04:11 | < Vornicus> | Let's take, well, 3x^2 + 3x/2 + 4 |
| 04:11 | < Chalcedon> | ok |
| 04:12 | < Vornicus> | So, you have the addition rule - it goes f(x) + g(x) -> f'(x) + g'(x), which is pretty boring |
| 04:12 | | * Chalcedon nods |
| 04:13 | < Vornicus> | so we can go "what's the derivative of 3x^2?" and you get 6x. Specifically, the derivative of something like cx^e -> ecx^(e-1) |
| 04:14 | < Chalcedon> | that's right.... |
| 04:14 | < Chalcedon> | *lights start going on in Chalcy's brain* |
| 04:14 | < Vornicus> | THen you look at 3x/2 and you get 3/2 |
| 04:15 | < Vornicus> | And then you look at 4 and you get... 0. |
| 04:15 | | * Chalcedon nods |
| 04:15 | < Vornicus> | So you end up with 6x + 3/2 |
| 04:16 | < Chalcedon> | is it differentiation or integration that uses the symbol that looks like an old f character? |
| 04:16 | < Vornicus> | Integration. |
| 04:16 | < Chalcedon> | thats right |
| 04:16 | < Vornicus> | Integration is differentiation backwards. |
| 04:16 | | * Chalcedon remembers :) |
| 04:17 | < Vornicus> | Usually you want to plug it into a calculator or have a table of integration floating around for that |
| 04:17 | | * Chalcedon nods |
| 04:17 | < Vornicus> | because it gets Crazy and you Don't Want To Have To Think About It. |
| 04:17 | < Chalcedon> | heh :) |
| 04:17 | < Chalcedon> | ok. So what're the underlying mechanics (now that I actually remember)? |
| 04:18 | < Vornicus> | My calc 3 teacher, someone asked him, "So, what do you do if the table doesn't have anything like what you've got?" "you.... guess." |
| 04:18 | < Chalcedon> | o.O |
| 04:18 | < Chalcedon> | why am I not surprised? |
| 04:19 | < Vornicus> | Because you know math. |
| 04:19 | < Vornicus> | Anyway. |
| 04:19 | < Vornicus> | Here is The Difference Quotient: (f(x + h) - f(x)) / h |
| 04:20 | | * Chalcedon nods |
| 04:20 | < Vornicus> | What it does, is it says "for a given h, f(x) changes by that amount over the next h units along x" |
| 04:21 | < Vornicus> | Now, here's the funny part. |
| 04:24 | | * Chalcedon cookies |
| 04:25 | < Vornicus> | 3x^2 + 3x/2 + 4 -> ((3(x+h)^2 + 3(x+h)/2 + 4) - (3x^2 + 3x/2 + 4)) / h -> ((3x^2 + 6xh + 3h^2 + 3x/2 + 3h/2 + 4) - (3x^2 + 3x/2 + 4)) / h -> (6xh + 3h^2 + 3h/2) / h -> 6x + 3h + 3/2 |
| 04:25 | < Chalcedon> | o.O |
| 04:25 | | * Chalcedon translates this |
| 04:25 | < Vornicus> | You push it all through, and you are very, very likely /not/ to have any h on the bottom anymore. |
| 04:26 | < Vornicus> | The first step is going from f(x) to the difference quotient of f(x); the rest is all simplification |
| 04:26 | | * Chalcedon nods |
| 04:26 | < Vornicus> | So then, given the difference quotient of a function... |
| 04:26 | < Vornicus> | all you have to do is pretend you can set h to 0, and do that, and you have the derivative. |
| 04:28 | < Vornicus> | Because what the derivative actually /is/, is the instantaneous rate of change of the function. |
| 04:29 | | * Chalcedon nods |
| 04:29 | < Vornicus> | and the difference quotient gives that to you. |
| 04:29 | | * Chalcedon nods |
| 04:29 | < Vornicus> | But it's kinda rare that you need to actually go through that. |
| 04:29 | < Chalcedon> | but knowing how is probably important |
| 04:29 | < Vornicus> | Generally because it's already been done for you. |
| 04:30 | < Vornicus> | e^x -> e^x; ln(x) -> 1/x; sin(x) -> cos(x) -> -sin(x) -> -cos(x) -> sin(x) |
| 04:30 | < Vornicus> | (assuming radians) |
| 04:31 | < Vornicus> | f(g(x)) -> f'(g(x))*g'(x); f(x) * g(x) -> f'(x)g(x) + f(x)g'(x)... |
| 04:32 | < Chalcedon> | ok... let me work through simplifying this |
| 04:33 | < Vornicus> | ok |
| 04:42 | < Vornicus> | how's it coming? |
| 04:42 | < Chalcedon> | how do you get rid of the last h? |
| 04:42 | < Vornicus> | set h to 0. |
| 04:43 | < Chalcedon> | ah! |
| 04:43 | < Chalcedon> | which you can do given it's no longer being divided by h |
| 04:43 | < Chalcedon> | right |
| 04:43 | < Vornicus> | Technically you're not allowed to do that, but when mathematicians aren't allowed to do something, they find a way around it. |
| 04:43 | < Chalcedon> | heh |
| 04:45 | < Vornicus> | Also they rant. |
| 04:45 | < Vornicus> | But. |
| 04:47 | | * Chalcedon hugs |
| 04:48 | < Chalcedon> | what's with this bit? sin(x) -> cos(x) -> -sin(x) -> -cos(x) -> sin(x) |
| 04:48 | < Vornicus> | The derivative of sin(x) is cos(x) |
| 04:48 | | * Chalcedon nods |
| 04:48 | < Vornicus> | the derivative of cos(x) is -sin(x) |
| 04:48 | < Vornicus> | etc etc |
| 04:49 | < Chalcedon> | ok :) |
| 04:49 | < Vornicus> | it, like beer, is proof that god loves us and wants us to be happy. |
| 04:49 | < Chalcedon> | hee |
| 04:50 | < Chalcedon> | ok |
| 04:50 | < Chalcedon> | I think I get all that |
| 04:50 | < Vornicus> | good. |
| 04:50 | < Chalcedon> | what is the f(g(x)) -> f'(g(x))*g'(x); f(x) * g(x) -> f'(x)g(x) + f(x)g'(x)... for? |
| 04:50 | < Vornicus> | That's the chain rule and the multiplication rule. |
| 04:51 | < Chalcedon> | what do they do? |
| 04:51 | < Vornicus> | If you have a complicated function that you can break down into smaller bits, you can use them to get the derivative of the whole function. |
| 04:51 | < Vornicus> | consider sin(2x) |
| 04:51 | < Chalcedon> | ah ha! |
| 04:51 | | * Chalcedon nods |
| 04:53 | < Vornicus> | f(u) is sin(u); g(x) is 2x. f'(u) is cos(u); g'(x) is 2. f'(g(x)) * g'(x) = cos(2x) * 2 |
| 04:53 | < Chalcedon> | I think I see |
| 04:54 | < Vornicus> | then the multiplication rule works the same way |
| 04:54 | | * Chalcedon nods |
| 04:54 | < Vornicus> | addition and subtraction are just f(x) ? g(x) -> f'(x) ? g'(x) |
| 04:54 | | * Chalcedon nods |
| 04:56 | | * Chalcedon hugs Vorn |
| 04:56 | < Vornicus> | and then division is complicated and I never manage to remember it, but: http://en.wikipedia.org/wiki/Quotient_rule |
| 04:56 | < Chalcedon> | I think I get it :) |
| 04:56 | < Chalcedon> | I'm fairly sure we'll cover the chain, multiplication and quotient rule during the course |
| 04:56 | < Chalcedon> | so grokking now isn't necessary |
| 04:59 | | * Vornicus hugs the Chalcy. |
| 05:00 | | * Chalcedon hugs Vorn |
| 05:00 | < Chalcedon> | thank you so much for your help! |
| 05:00 | < Chalcedon> | I feel like I can actually make progress now! |
| 05:00 | | * Chalcedon cookies |
| 05:00 | | * Vornicus nrom |
| 05:02 | < Vornicus> | Anyway really the first thing you'll probably talk about if it's a reasonably introductory calc course is limits, which is essentially "You can't do that!" "Piffle! I will do it anyway!" |
| 05:04 | < Chalcedon> | heh |
| 05:04 | < Chalcedon> | yeah limits are the first thing in the notes |
| 05:08 | < Vornicus> | Which is Why You Can Do That With h, Way Up There. |
| 05:08 | | * Chalcedon nods |
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