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Quote of the Day: "One should always generalize." -- Carl Jacobi Objectives: The student will sketch a slope field for a given differential equation. Given a slope field, the student will sketch a solution curve through a given point. The student will be able to match a slope field to a differential equation. The student will be able to match a slope field to a solution of a differential equation. 1. Collect Homework. 2. Today we are going to study a concept called slope fields. Slope fields provide an excellent way to visualize a family of solutions of differential equations. Some differential equations can be solved algebraically, and the slope field for those differential equations can be used to verify that the algebraic solution matches the graphical solution. Other differential equations cannot be solved algebraically, and the slope field for those differential equations provides a way to solve the differential equation graphically. Slope fields also give us a great way to visualize a family of antiderivatives. 3. Recall how to graph a line given a point and the slope. Graph a line passing through (-2, 3) with a slope of 2.
4. Hand out rulers and slope field packets.
Start with a differential equation such as dy/dx = x + 1. Pick a starting point on our grid and substitute it into the differential equation to determine the slope at that point. Then we draw a tiny segment that passes through our point and has the slope that we found. Next ask the students to name other points that have the same slope. They notice that all of the points that have the same x-coordinate will have the same slope because our differential equation has an x-term but no y-term. After completing the slope field for dy/dx = x + 1, try another differential equation, such as dy/dx = 2y. This time the students notice that all of the points that have the same y-coordinate will have the same slope because our differential equation contains a y-term but no x-term. This knowledge helps the students when they are asked to match a differential equation to a slope field. The student looks at the slope field to see if all of the segments in the vertical direction have the same slope; if they do, then the differential equation contains an x-term but no y-term. If all of the segments in the horizontal direction on a slope field have the same slope, then the differential equation contains a y-term but no x-term. After making these observations, move on to differential equations that contain both an x-term and a y-term, such as dy/dx = x +y and look for points that have the same slope as we draw the slope field for this differential equation. The students like to use a ruler at first to help draw their segments so that they have the correct slope, but soon they are able to draw them without using a ruler. 5. Let students work on the slope field packets. 6. Assignment p. 364 (53, 54, 55, 56)Finish slope field packets. |