section 6-1 continued slope fields. definition a slope field or directional field for a...

Section 6-1 continuedSlope Fields

Definition

• A slope field or directional field for a differentiable equation

is a collection of short line segments with the same slope as the solution curve through a given point (x,y)

),(' yxFy

Definition• A slope field shows the general shape of all

the solutions of a differential equation

yxy '6. Sketch a slope field for

for the points (-1, 1) (0, 1) and (1, 1)

yxy 2'7. Sketch a slope field for

through the point (1, 1)x -2 -2 -1 -1 0 0 1 1 2 2

y -1 1 -1 1 -1 1 -1 1 -1 1

y’=2x+y

A slope field or direction field consists of line segments with slopes given by the differential equation. These line segments give a visual perspective of the slopes of the solutions of the differential equation.

a)Sketch using pencil two approximate solutions of the differential equation on the given slope field, one of which passes through the indicated pointb)Use integration to find the particular solution of the differential equation and use a graphing calculator to graph the solutionc)Compare results with part (a)

Instructions

a) Sketch two possible solutions, one through the point ( -1 , 3)

12 xdx

dy8.

3,112 xdx

dy

b) Integrate to find the general solution

c) Graph solution using graphing calculator

Assignment

Practice worksheet 6-1