1. 40. Section 9.3 Slope Fields and Euler’s Method

2. Section 9.3 Slope Fields
Essential Question – Why do we use slope fields?

3. Slope Fields
Many differential equations cannot be solved explicitly. A slope field is a graphical approach to give a visual understanding of equations.
We will also look at Euler’s method which is a numerical approach to differential equations.
Slope fields show the general “flow” of a differential equation’s solution. They are an array of small segments which tell the slope of the equation or “tell the equation which direction to go in”

4. Slope Fields Consider the following:
449EC23D16F51693C3E640FCB76BEB1?course=AP Calculus AB II&lesson=33&topic=2&width=800&height=684&topicTitl e=Slope%20Fields&skinPath= g/hippocampus.skins/default

5. Slope Fields
To construct a slope field, start with a differential equation. We’ll use
Rather than solving the differential equation, we’ll construct a slope field
Pick points in the coordinate plane
Plug in the x and y values
The result is the slope of the tangent line at that point

6. 1 2 3 1 2 1 1 2 2 4 -1 -2 -2 -4 Draw a segment with slope of 2.
Draw a segment with slope of 2. 1 2 3 1 2
Draw a segment with slope of 0. 1 1 2
Draw a segment with slope of 4. 2 4 -1 -2 -2 -4

7. If you know an initial condition, such as (1,-2), you can sketch the curve.
By following the slope field, you get a rough picture of what the curve looks like.
In this case, it is a parabola.

8. Example
Construct a slope field for y’ = 2x – y.

9. Construct a slope field for y’ = 2x – y.

10. Example
Construct the slope field for

11. Online slope field grapher
e/slopeField.htm?inputField=2x+-+y

12. Euler’s Method (Oiler)

13. Euler
Leonhard Euler made a huge number of contributions to mathematics, almost half after he was totally blind.
(When this portrait was made he had already lost most of the sight in his right eye.)
Leonhard Euler

14. It was Euler who originated the following notations:
(function notation)
(base of natural log)
(pi)
(summation)
(finite change)
Leonhard Euler

15. There are many differential equations that can not be solved.
We can still find an approximate solution using a numerical method. Euler came up with a method based on tangent line approximations

16. The error gets worse as you get further away from initial value
The error gets better if you use a smaller x
If the curve is concave down, Euler overestimates the y value, if the curve is concave up Euler underestimates it

17. We will practice with an easy one that can be solved.
Initial value:
Use steps of 0.5

20. Exact Solution:

21. Euler’s Method (x,y) ∆x (x + ∆x, y + ∆y)
Use Euler’s Method for dy/dx= y – 1 with increments of ∆x = .1 to approximate the value of y when x = 1.3.
y = 3 when x = 1.
(x,y) ∆x
(x + ∆x, y + ∆y)

22. Euler’s Method (x,y) ∆x (x+∆x,y+∆y) (1,3) 2 .1 .2 (1.1,3.2) 2.2 .22
Use Euler’s Method with increments of ∆x = .1 to approximate the value of y when x = 1.3 and y = 3 when x = 1.
(x,y) ∆x
(x+∆x,y+∆y)
(1,3) 2 .1 .2
(1.1,3.2)
2.2
.22
(1.2,3.42)
2.42
.242
(1.3,3.662)

23. Euler’s Method (x,y) ∆x (x + ∆x, y + ∆y)
Use Euler’s Method for dy/dx= 2x – 7 and f(2) = 3 with five equal steps to approximate f(1.5).
(x,y) ∆x
(x + ∆x, y + ∆y)

24. Euler’s Method (x,y) ∆x (x+∆x,y+∆y) (2,3) 1 -.1 (1.9,2.9) .9 -0.09
Use Euler’s Method for dy/dx= 2x – 7 and f(2) = 3 with five equal steps to approximate f(1.5).
(x,y) ∆x
(x+∆x,y+∆y)
(2,3) 1
-.1
(1.9,2.9)
(1.9, 2.9) .9
-0.09
(1.8, 2.81)
(1.8,2.81)
.79
-0.079
(1.7,2.731)
.669
-.0667
(1.6, 2.664)
.536
-.0536
(1.5, 2.611)

25. Assignment
Worksheet