1. Slope Fields
AP Calculus AB
Mr. Reed

2. What is a slope field? Let’s take a look
With your grapher, input the function into Y1:

3. Questions to consider What do the short line segments represent?
Are the slopes the same for all given values of x?
Are the slopes the same for all given values of y?
What is the slope equation (differential equation) for all of the functions?
What variable controls the slope?
How does this show up on the whiteboard?

4. Slope Field (defined)
A graphical representation of a differential equation. A graph of short line segments whose slope is determined by evaluating the derivative at the midpoint of the segment

5. Motivation for slope fields
Helps visualize the solutions to the differential equation (graphical approach vs. algebraic) - the solutions are “hiding” in the slope field
Helps up solve differential equations that can not be solved analytically
Helps us appropriately choose initial conditions (more to come on this idea)

6. What you will be required to do
Construct a slope field given a differential equation (FR)
Draw particular solutions within a slope field (FR)
Match a differential equation with correct slope field (MC)
Match a slope field with correct differential equation (MC)

7. Examples Ex’s [1&2]  p.337-339: 2,4,6,8,10 Ex’s [3]  9 (packet)
Es’s [4]  5,10 (packet)

8. Graphing slope fields on the TI-89
Go through instructions on handout
Plot slope fields for the following differential equations:
**Notice trends in slpe fields when dy/dx depends on x, y, or both x & y**

9. Check your understanding
Work 2004 FR #5 (Form B) problem

10. Homework From textbook  p.336-339: Q1-Q10, 1-11 (odd)
From packet  all remaining problems