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

2. Introduction.
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)

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:

19. Sample 5:

20. Sample 6:

21. Sample 7:

22. Sample 8:

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

24. Tangent Line Approximation
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. 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
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