4 slope fields
Download
Skip this Video
Download Presentation
4. Slope Fields

Loading in 2 Seconds...

play fullscreen
1 / 14

4. Slope Fields - PowerPoint PPT Presentation


  • 112 Views
  • Uploaded on

4. Slope Fields. Slope Fields. We know that antidifferentiation , indefinite integration, and solving differential equations all imply the same process

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about '4. Slope Fields' - nydia


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
slope fields
Slope Fields
  • We know that antidifferentiation, indefinite integration, and solving differential equations all imply the same process
  • The differential equations we’ve seen so far have been explicit functions of a single variable, like dy/dx = 3x3+4x or f’(x)=sin(x) or h”(t)=5t
  • Solving these equations meant getting back to y = or f(x)= or h(t)=.
  • Many times, differential equations are NOT explicit functions of a single variable, and sometimes they are not solvable by analytic methods.
  • Fear not! There are ways to solve such differential equations. Today we will look at how to solve them graphically.
slope fields1
Slope Fields
  • 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”
  • If we have the differential equation dy/dx = x2, if we replace the dy/dx in this equation with what it represents we get

slope at any point (x,y) = x2

slope fields2
Slope Fields
  • Consider the following:

http://www.hippocampus.org/course_locator;jsessionid=9449EC23D16F51693C3E640FCB76BEB1?course=AP Calculus AB II&lesson=33&topic=2&width=800&height=684&topicTitle=Slope%20Fields&skinPath=http://www.hippocampus.org/hippocampus.skins/default

slope fields3
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
  • Draw a small segment at that point with that approximate slope. Make sure your slopes of 0,1,-1 and infinity are correct. All other slopes must be a steepness relative to others around it.
  • It is impossible to draw a slope field at every point in the x,y plane, so it is restricted to points around the origin
slide6
Example 1

Draw a segment with slope of 2.

Draw a segment with slope of 0.

Draw a segment with slope of 4.

0

0

0

0

1

0

0

2

0

0

3

0

2

1

0

1

1

2

2

0

4

-1

-2

0

0

-4

-2

slide7
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.

example 2
Example 2

Construct a slope field for y’ = x + y and draw a solution through y(0)=1

slide9
The more tangent lines we draw, the better the picture of the solutions. There are computer programs and programs for your calculator that will construct them for you. Below is a slope field done on a computer. Notice how we can now, with more confidence and accuracy, draw particular solutions, such as those passing through (0,-2), (0,-1), (0,0), (0,1), and (0,2)
online slope field grapher
Online slope field grapher
  • http://mathplotter.lawrenceville.org/mathplotter/mathPage/slopeField.htm?inputField=2x+-+y
ad