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Thursday, March 19, 2009

A New Unit: Differential Equations

So today was a new unit.

Gundam Unit Exia

But apparently, it's something new, but something old. Like a nice pair of chucks. :)

We started with an equation to see if we could grasp the concept before he formally told us.

So Joyce did this question for us. Basically what she did is she anti differentiated a(t) to get v(t). She determined C to be ten by setting v(0) = 10 so then the 0 made the first and second term in v(t) to be 0 so then C = 10.

She then found the roots at 5 and 1 by factoring the function. v(t) = 2(t-5)(t-1). She stated the particle (Why are particles moving along x-axis' again??), changes direction at one and five, but Mr. K asked her to verify it. So she drew a graph, but he then again asked her how she could back that up, and she used the first derivative test.


Because velocity is the derivative of position. So from negative infinity to 1, the function is positive, from one to five the function is negative and finally from five to infinity, the graph is positive. Therefore, the particle changes direction at one and five.

This is when thing's started clicking. We realized that..

Velocity is the derivative of Position
Acceleration is the derivative of Velocity.

We realized this because due to the given values, they were all related to each other and they are rates of change of each other.

So we tried the theory and antidifferentiated velocity.

This is when he introduced that these were differential equations, and introduced five words to us.
Differential Equation
General Solution
Initial Conditions
Initial Value Problem

So if there is a derivative of the equation and is represented by a variable, we call it differential equations. But wait! There's more. There's flavours but we'll get into that later.

Next are the order.

Obsessive Pencil Ordering by mimi.amoure

Order tells you which derivative is in the equation. For example. v(t) would be the first order, and a(t) would be the second order. We could talk about the degree (here comes the ripples...), in which it would be an exponent on the function for example: v(t)^2.

The general solution is the family of functions, which is the function plus the vertical shift [ + C]

Initial conditions are the one value that picks a certain function from the family. Using conditions like v(0) = 10, which would pick out that one function from the whole family, like Tito Boy (Filipino Joke :P).

The initial value is determined by the initial conditions of the differential equations.

So learning all tha simple stuff, we were given a differential equation to solve given the intial conditions.

So Justus did the courtesy of doing this question for us (which I assume everyone did correctly).
First the anti derivative of f'(x) was found and then we subbed in f(1) to find C to find the parent function. Pretty simple right? Yeah, I liked it too.

Anyways, zeph's the new Scribe. :)

Now to end it off with the new trend... YouTube Videos!!! I don't know how old school some of you people are, but I'm upset that they don't do re-runs of "Whose Line Is it Anyway?". Best Improv show I've watched.

One of my favourite episodes. XD


.:. J + ME .:. said...

second. I totally calc-rolled myself.

Never gonna give you up ♪

Rence said...

LOL I accidentally posted before I even started.

.:. J + ME .:. said...

why did my comment show up cut off...wt double hockey sticks, man.

Not Paul said...

Nicely done, also one of my favourite episodes.

The letter "h" still haunts me to this day...

Benofschool said...

Man I miss that show....

Skyline said...

Best show haha