Introduction to Slope Fields E. Alexander Burt Potomac School Something to keep you busy on your impromptu February break!
Differential Equations <ul><li>Scary math term! It means an equation that contains a function and its derivatives. </li><ul><li>(sometimes it just contains the derivatives)
Here is an easy example: </li></ul><li>dy/dx = 3x 2
As you can see, this equation is the derivative of y=x 3 – OR IS IT?! </li></ul>
Constants of Integration <ul><li>Let's look at that again: dy/dx=3x 2 </li><ul><li>It might be the derivative of y=x 3
But it's also the derivative of y=x 3 +5 and y=x 3 -2 and y=x 3 + p .
In fact, it's the derivative of y=x3+C where C is any constant. </li></ul><li>So the differential equation actually describes a whole “family” of curves that have the derivative dy/dx=3x 2 </li></ul>
Slope Fields: Visualizing that “family” of curves <ul><li>We'll chose some points to make a grid in the x-y plane... for example we could choose every integer point, so (1,1) (1,2) (2,1) (2,2) etc.
At each point we'll draw a short line segment at the correct slope. </li><ul><li>And how will we know the correct slope?
The derivative gives us the slope of the tangent line. </li></ul></ul>
Time to try it out <ul><li>Go to: http://www.math.lsa.umich.edu/courses/116/slopefields.html
“trust content” if necessary to get the applet to run.
In the box next to “eqn 1 dy/dx” type the equation we have been talking about: “3*x^2” </li><ul><li>Use the same format that you would use to put it in your calculator
Hopefully you'll get a set of lines that look like maybe you could put a y=x 3 +C curve in there. </li></ul></ul>
It will be hard to get a circle on that graph... it's not a function! </li></ul><li>Try some trig: dy/dx = sec 2 x </li><ul><li>You'll have to enter that as (1/cos(x))^2
Try it with Min. x set to -6.28 and Max x set to 6.28
Click in a few places to see tan (x) + C </li></ul></ul>
Time to try one for real... <ul><li>On the 2006 AP you can now do question 5 parts a and b. (Can't do part c just yet, but soon)
Part b is a brain teaser. Look for the solution in part a – or just ask yourself “what's the slope of a horizontal line?” </li><ul><li>You might try using the same slope field calculator as before. </li></ul></ul>