## Mathematics & Science
Learning Center |
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## A Theoretical Introduction to Slope FieldsLong ago, in a class room not so very far away, you learned about a mathematical
idea called the the derivative of a function gives its slopeWhen we work with dy/dx = x^{2} If we simply replace the variable slope = x^{2} So what? Well, let's remind ourselves of our usual find the function whose derivative appears in the equation In our example this means that our goal is to find Now you may be one of those clever students who's always one step ahead of the
instructor. If so you're probably already having thoughts about how you could
easily solve the current example (i.e. find Anyway, let's get back to our analysis of slope. We've established that our goal is to find the function which satisfies: slope = x^{2} In other words, we're seeking a function whose slope at any point in the
( - At the point (1,2) the slope would be 1
^{2}= 1. - At the point (5,3) the slope would be 5
^{2}= 25. - At the point (-3,11) the slope would be (-3)
^{2}= 9.
(Notice that the Hmmm...maybe we could use these slopes to get a picture of what the function we seek—the function which has these slopes—looks like? Well, for starters we'd need to be a bit more systematic about how we choose our points. (The choices I made above were somewhat random.) We could divide the entire plane into a grid, kind of like the squares on a piece of graph paper, and at each grid intersection we could make a slope calculation like we were doing above. Obviously doing this for the entire plane is actually impossible, since it's infinite, so we'll have to be satisfied with some "reasonable" subset of the plane. This is starting to sound like a lot of work. We may be talking about slope calculations at literally thousands of points, here. Sounds like a job for someone who doesn't mind doing myriads of mind-numbingly repetitive tasks. Someone who can maintain accuracy despite the mountain of (admittedly trivial) calculations involved. Well, you knew you were sitting at a computer for a reason, didn't you? OK, so we'll have the computer do the calculations, but there's still
something we haven't decided on yet! What do we do with all those thousands of
slopes once we've found them? We mentioned earlier that we'd use the slopes to
get a picture of what the function . direction fieldThe picture produced by a computer program may look a little like this: A sample slope field made with Mathematica
If you feel that you followed the above description of how a slope field is formed then carry on down the page. If you're still a little lost you can either read a summary of the procedure, or you can even return to the beginning of this Introduction and read it through again. ## Making Slope Fields with the ComputerLet's continue to use the example of finding a slope field for the differential equation: dy/dx = x^{2} As I mentioned above, it would be impossible to produce a slope field covering the entire, infinite, Cartesian plane. Instead, for our example, let's restrict the section of the plane we consider to: x ≤ 2, and -2 ≤ y ≤ 2 To see how we would create a slope field for this example with |
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ODE Laboratories: A Sabbatical Project by Christopher A. Barker©2017 San Joaquin Delta College, 5151 Pacific Ave., Stockton, CA 95207, USA e-mail: cbarker@deltacollege.edu |