Slope Fields For Snowy Days

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A lesson for when we can't get to school

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Slope Fields For Snowy Days

  1. 1. Introduction to Slope Fields E. Alexander Burt Potomac School Something to keep you busy on your impromptu February break!
  2. 2. 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)
  3. 3. Here is an easy example: </li></ul><li>dy/dx = 3x 2
  4. 4. As you can see, this equation is the derivative of y=x 3 – OR IS IT?! </li></ul>
  5. 5. 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
  6. 6. But it's also the derivative of y=x 3 +5 and y=x 3 -2 and y=x 3 + p .
  7. 7. 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>
  8. 8. 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.
  9. 9. 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?
  10. 10. The derivative gives us the slope of the tangent line. </li></ul></ul>
  11. 11. Time to try it out <ul><li>Go to: http://www.math.lsa.umich.edu/courses/116/slopefields.html
  12. 12. “trust content” if necessary to get the applet to run.
  13. 13. 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
  14. 14. 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>
  15. 15. It should look like this
  16. 16. Suppose we want a specific curve <ul><li>Again with our differential equation:
  17. 17. dy/dx = 3x 2
  18. 18. y=x 3 +C is the solution
  19. 19. If we knew one point, we could find a specific solution – in other words, we could find C
  20. 20. For example if we take the point (1,2) we find that C=1 </li><ul><li>If you click on a point in the slope field applet, it will draw a curve through that point.
  21. 21. You can also enter the point in the “add init cond” boxes – try it! </li></ul></ul>
  22. 22. Slope Field with One Curve
  23. 23. Try some curves you know: <ul><li>Take the circle: x 2 +y 2 =r 2 </li><ul><li>By implicit differentiation, 2x dx+2y dy=0
  24. 24. Solving: dy/dx = -x/y
  25. 25. 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
  26. 26. Try it with Min. x set to -6.28 and Max x set to 6.28
  27. 27. Click in a few places to see tan (x) + C </li></ul></ul>
  28. 28. 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)
  29. 29. 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>

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