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Slope Fields For Snowy Days
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Slope Fields For Snowy Days

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A lesson for when we can't get to school

A lesson for when we can't get to school

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Transcript

  • 1. Introduction to Slope Fields E. Alexander Burt Potomac School Something to keep you busy on your impromptu February break!
  • 2. Differential Equations
    • Scary math term! It means an equation that contains a function and its derivatives.
      • (sometimes it just contains the derivatives)
      • 3. Here is an easy example:
    • dy/dx = 3x 2
    • 4. As you can see, this equation is the derivative of y=x 3 – OR IS IT?!
  • 5. Constants of Integration
    • Let's look at that again: dy/dx=3x 2
      • It might be the derivative of y=x 3
      • 6. But it's also the derivative of y=x 3 +5 and y=x 3 -2 and y=x 3 + p .
      • 7. In fact, it's the derivative of y=x3+C where C is any constant.
    • So the differential equation actually describes a whole “family” of curves that have the derivative dy/dx=3x 2
  • 8. Slope Fields: Visualizing that “family” of curves
    • 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. At each point we'll draw a short line segment at the correct slope.
      • And how will we know the correct slope?
      • 10. The derivative gives us the slope of the tangent line.
  • 11. Time to try it out
    • Go to: http://www.math.lsa.umich.edu/courses/116/slopefields.html
    • 12. “trust content” if necessary to get the applet to run.
    • 13. In the box next to “eqn 1 dy/dx” type the equation we have been talking about: “3*x^2”
      • Use the same format that you would use to put it in your calculator
      • 14. Hopefully you'll get a set of lines that look like maybe you could put a y=x 3 +C curve in there.
  • 15. It should look like this
  • 16. Suppose we want a specific curve
    • Again with our differential equation:
    • 17. dy/dx = 3x 2
    • 18. y=x 3 +C is the solution
    • 19. If we knew one point, we could find a specific solution – in other words, we could find C
    • 20. For example if we take the point (1,2) we find that C=1
      • If you click on a point in the slope field applet, it will draw a curve through that point.
      • 21. You can also enter the point in the “add init cond” boxes – try it!
  • 22. Slope Field with One Curve
  • 23. Try some curves you know:
    • Take the circle: x 2 +y 2 =r 2
      • By implicit differentiation, 2x dx+2y dy=0
      • 24. Solving: dy/dx = -x/y
      • 25. It will be hard to get a circle on that graph... it's not a function!
    • Try some trig: dy/dx = sec 2 x
      • You'll have to enter that as (1/cos(x))^2
      • 26. Try it with Min. x set to -6.28 and Max x set to 6.28
      • 27. Click in a few places to see tan (x) + C
  • 28. Time to try one for real...
    • On the 2006 AP you can now do question 5 parts a and b. (Can't do part c just yet, but soon)
    • 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?”
      • You might try using the same slope field calculator as before.