1. Slope Fields

2. Determining Slope Fields Graphs
Plug in numbers for x and y to find the slope at that point.
Point
Slope
At every point on the y-axis (x = 0): the slope = 0, tangent is horizontal
At every point on the x-axis (y = 0): the slope doesn’t exist, tangent is vertical
The graph is a family of semicircles.

3. Tips for Differential Equations and Slope Fields
Does not have y  all dashes on the same vertical axis are
Horizontal
Dashes _____ 
Does not have x  all dashes on the same horizontal axis are
 Forward Slashes
 Back Slashes
is periodic  dashes look
like a periodic function

4. Describe the Slope Field 1
All dashes on vertical axis are parallel – no y
Slope is zero at x = 0 and x = 2
Slope > 0 at x < 0 and x > 2
4. Slope < 0 at 0 < x < 2
Guess the Differential
Equation !!!

5. Describe the Slope Field 2
All dashes on same horizontal axis are parallel: no x
Periodic function

6. Guess the Slope Field A B C D E F

7. Correct! Guess the Slope Field
Since 2x2 is always positive, the dashes should always be /.
This was the only slope field with such conditions.
Next

8. Guess the Slope Field A B C  D E F

9. Correct! Guess the Slope Field
Slope should be 0 when y=-3. (field should show –)
This was the only slope field with an asymptote at
y=-3.
Slope changes from negative to positive
Next

10. Guess the Slope Field  A B C  D E F

11. Correct! Guess the Slope Field
When x=y, the slope = 0, and the dashes should
be —. This was the only slope field to fit such description.
Next

12. Guess the Slope Field  A B C   D E F

13. Correct! Guess the Slope Field Next
Remember, when there is no x, the lines are || on the horizontal. Plus there is an asymptote at
y = -2.This was the only graph that fit such descriptions.
Next

14. Guess the Slope Field   A B C   D E F

15. Correct! Guess the Slope Field
Remember, when there is no y, the lines are || on the vertical. When x>0, dy/dx>0.
When x is increasing, the slope becomes steeper.
Next

16.       Guess the Slope Field A B C D E F
Since there is only one slope field left to pick, guess what the differential equation instead.
(Hint: remember the Slope Field Rules and look at the shape of the slope field.   A B C     D E F

17. Correct! Guess the Slope Field Next
Since the slope field is periodic, we usually think of trig equations. This slope field resembles –cos x. Remember that the derivative of –cos x is sin x, which is the given differential equation.
Next

18. Slope Fields on TI-89 Push “Mode”
For Graph, select “6: Diff Equations”
Press “Enter” twice to save
Select “”, then “F1”
Set tØ = initial x
(If you don’t know, set 0 for now)
y1’= dy/dx
Instead of x, type in t
Instead of y, type in y1, y2, or y# (# is the number in: y_’)
Select “F2”
Select 4:ZoomDec (“F2” + 4”) for best graph
Select “”, then “F3” to graph
To draw a sample line through a point in slope field
Set tØ = initial x
Set yi1 = initial y

19. Solution curve. Example 1
Draw the direction
field going through:
(0, 0)
(0, 2)
(3,0)

20. Solution curve. Example 2
Draw the direction field going through:
(-2, 0)
(0, 1)
(0, 0)

21. Solution curve. Example 3
Draw the direction field going through:
(0, 0)
(2, 0)
(0, 1)