They were asked to draw the solution curves through the two points (0, 1) and (0, –1) shown here in blue. Notice that this equation is not separable students were not expected to solve it. The example below is from the 2002 BC exam question 5. Be sure you are in a square window (CTRL+Q). Enter the differential equation in the box and adjust the other settings as necessary. In Winplot, follow the path Window > 2-dim > Equa > differential > dy/dx. Here is a brief example that shows how powerful an animated graph can be. (I have not found a good slope field generator app for iPads if anyone makes apps, consider this a hint.) The good ones let you draw and animate solution curves over the slope field. The figures in this post were done with Winplot (of course, my favorite). There are many websites that will draw slope fields and solution curves for you. Calculator screens are not the best for seeing slope fields they are too small, and you should be sure to always be in a square window (or the slopes will not look right). There are various graphing calculator programs available on the internet. Slope fields are tedious and time-consuming to draw by hand. The solution will touch a segment only if the midpoint of the segment happens to be on the solution – this is not usually the case. Notice how the solution graph follows the slope field, but does not necessarily hit any of the segments. This is shown drawn on the slope field in the next graph. In the previous post (example 2) we found the particular solution of with the initial condition point (4, –3) to be. After plotting the initial condition (a point) students should draw a curve through the point that follows the slope field from edge to edge in both directions. Since the solution graphs are lurking in the slope field, the next thing to do is to use the slope field to sketch a particular solution. We solved this differential equation in the last post. What do they look like in the figure? Circles of course. The solutions are lurking in the slope field. The big idea with slope fields is to use them to get an idea of what the solutions look like, especially if the differential equation cannot be solved. Students are given a graph with 9 – 12 points plotted and they are asked to use them to draw a slope field for a given differential equation.) (This is, in fact, a common free-response question on the AP exams. Have them calculate the derivative at their point(s) and then come to the board and draw a short segment through their point(s) with the slope they calculated. Here is the slope field for the differential equationĪ good way to introduce slope fields to your class is to put or project a coordinate system on the board. These segments graphed together form the slope field. At each point the value of the derivative is calculated and a short segment with that slope is drawn. Slope fields make use of this by imposing a grid of points evenly spaced across the Cartesian plane. A first derivative expressed as a function of x and y gives the slope of the tangent line to the solution curve that goes through any point in the plane. The graph of a differential equation is a slope field. Of course, we always want to see the graph of an equation we are studying.
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