Slope Fields AP Calculus AB Mr Reed What

Slope Fields AP Calculus AB Mr. Reed

What is a slope field? • Let’s take a look • With your grapher, input the function into Y 1:

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?

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

Motivation for slope fields 1. Helps visualize the solutions to the differential equation (graphical approach vs. algebraic) - the solutions are “hiding” in the slope field 2. Helps up solve differential equations that can not be solved analytically 3. Helps us appropriately choose initial conditions (more to come on this idea)

What you will be required to do 1. Construct a slope field given a differential equation (FR) 2. Draw particular solutions within a slope field (FR) 3. Match a differential equation with correct slope field (MC) 4. Match a slope field with correct differential equation (MC)

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

Graphing slope fields on the TI -89 1. Go through instructions on handout 2. Plot slope fields for the following differential equations: **Notice trends in slpe fields when dy/dx depends on x, y, or both x & y**

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

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