Antiderivatives and Slope Fields

1. Antiderivatives and Slope Fields Greg Kelly, Hanford High School, Richland, Washington

2. Consider: or then: Given: find First, a little review: It doesn’t matter whether the constant was 3 or -5, since when we take the derivative the constant disappears. However, when we try to reverse the operation: We don’t know what the constant is, so we put “C” in the answer to remind us that there might have been a constant.

3. Given: and when , find the equation for . This is called an initial value problem. We need the initial values to find the constant. An equation containing a derivative is called a differential equation. It becomes an initial value problem when you are given the initial condition and asked to find the original equation. If we have some more information we can find C.

4. Slope fields are mostly used as a learning tool and are mostly done on a computer or graphing calculator, but a recent AP test asked students to draw a simple one by hand. Initial value problems and differential equations can be illustrated with a slope field.

5. 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

6. 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.

7. For more challenging differential equations, we will use the calculator to draw the slope field.