5013 - Slope Fields and Euler’s Method

1. 5013 - Slope Fields and Euler’s Method AP Calculus

2. Anti-derivatives find families of Accumulation (position) functions from given Rate of Change (velocity) functions. • However, 97.8% of Rate of Change functions do not have elementary Accumulation functions. • NEED A METHOD TO APPROXIMATE THE • Accumulation FUNCTION • Slope Fields or Direction Fields – graphical (gives the impression of the family of curves) • Euler’s Method – numerical (finds the approximate next value on a particular curve) Introduction.

3. Slope Fields Slope Fields or Direction Fields – graphical (gives the impression of the family of curves)

4. Slope Fields: Sketch To Sketch: Evaluate each point in and sketch a small slope segment at that point. ( 0 , -1 )  ( 0 , 0)  ( 0 , 1 )  ( 0 , 2 )  ( 1 , 0) 

5. Slope Fields: Sketch To Sketch: Choose an initial position and sketch the curve trapped by the slope segments This is a Solution Curve or Integral Curve.

6. Slope Fields: Sketch To Sketch: Choose an initial position and sketch the curve trapped by the slope segments This is a Solution Curve or Integral Curve.

9. Slope Fields: Identify A Family of Curves To Identify a Solution Function: if vertically parallel, f(x,y) is in terms of x only if horizontally parallel, f (x,y) is in terms of y only. if not parallel, f(x,y) is in terms of both x and y. I. ………………………… * II. May have to test the slope at points to differentiate between possibilities. Choose an extreme point.

10. Slope Fields : Identify A Family of Curves To Identify a solution function: if vertically parallel, f(x,y) is in terms of x only if horizontally parallel, f (x,y) is in terms of y only. if not parallel, f(x,y) is in terms of both x and y.

11. Slope Fields : Identify A Family of Curves To Identify a solution function: if vertically parallel, f(x,y) is in terms of x only if horizontally parallel, f (x,y) is in terms of y only. if not parallel, f(x,y) is in terms of both x and y.

12. Slope Fields : Identify A Family of Curves To Identify a solution function: if vertically parallel, f(x,y) is in terms sof x only if horizontally parallel, f (x,y) is in termw of y only. if not parallel, f(x,y) is in terms of both x and y.

13. Slope Fields : Identify End Behavior : For some functions in terms of BOTH x and y you must look at the local and end behaviors: large x / small x large y / small y

15. Sample 1:

16. Sample 2:

17. Sample 3:

18. Sample 4:

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22. Sample 8:

23. EULER’S Method • Euler’s Method – numerical (finds the approximate next value on a particular curve)

24. EULER’S Method • Euler’s Method – numerical (finds the approximate next value on a particular curve) Euler’s method is Tangent Line Approximation

25. Euler’s Given and initial condition ( 0 , 1 ), Use Euler’s Method with step size to estimate the value of y at x = 2.

26. Euler’s Method: Approximate a value Given and initial condition ( 0 , 1 ), Use Euler’s Method with step size to estimate the value of y at x = 2. 0 1 .5 1 1.5 At x = 2, y

27. Euler’s Method: Graph Given and initial condition ( 1 , 1 ), Use Euler’s Method with step size to approximate f (1.3)

28. Last Update • 2/16/10 p.328 41-47 odd