Scribe Post 1/3/13

Today we started off class with the unit three followup test.

Then we moved onto some new and exciting things. To start this we google image searched slope fields. This gave us an idea of what we would be drawing, since for now we were working mostly with images tables and what not. If you have forgotten what slope fields look like here is a link to the
Google Image Search.

Slope fields are made up of differential equations. A differential equation is an equation with a derivative in it.

To start drawing a slope field you look at a grid of points and look at lattice points, which are points with integer coefficients. This means they are where the grid lines intersect. ex/ (1,1) (2,2) (1,2)
Then you pick a lattice point and calculate the slope, but plugging the x and y coordinates into the equation. Then at that lattice point draw a small dash with that slope.

A good example equation is
 
Then we created a table so we could easily view what the slope is at each specific point when we plugged in the x and y coordinates.


Here is a picture where i have plotted these table coordinates on the slope field. 

 


.

Here is a completed image of this slope field:









 Then we started to learn how to solve this euqation
We have to start with separation of the variables, cross multiplication.



 That funky unknown symbol simply means take the anti derivative of.

You then get:



 


Then you plug in your point: (2,2)


 For this specific function you can see that this slope field forms circles. If you can't see that then here is a picture that should help you

 

 Since the formula for a circle is
 


r is the radius so the radius of this circle must be the square root of 8. 






So the square root of 8 is the radius at the point (2,2)

That's all. If you want more help you can talk to Nate Catell about DiffEqu's (4th semester calc) or here is a nice applet to help you visualize slope fields:

Applet