1. First, a little review: Consider: then: or 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: Given:findWe don’t know what the constant is, so we put “C” in the answer to remind us that there might have been a constant.

2. If we have some more information we can find C. 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.

3. General and Particular Solutions A function y = f(x) is called a solution of a differential equation if the equation is satisfied when y and its derivatives are replaced by f(x) and its derivatives. y' + 2y = 0 y = 5e -2x differential equation particular solution general solution y = Ce -2x

4. y' + 2y = 0 The order of a differential equation is determined by the highest-order derivative in the equation. First-order Second-order y'' = -32

5. Initial value problems and differential equations can be illustrated with a slope field. 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.

6. Draw a segment with slope of 2. Draw a segment with slope of 0. Draw a segment with slope of 4. 000 010 00 00 2 3 10 2 112 204 0 -2 0-4

7. 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.

8. Sketch the solution for the equation y' = 1/3 x 2 - 1/2 x that passes through (1, 1)

9. Sketch the solution for the equation y' = y + xy that passes through (0, 4)

10. y' = xy Sketch the solution that passes through the point (0, 1/2)

11. y' = xy Sketch the solution that passes through the point (0, 1/2)

12. The AP Board Expects you to be able to do the following with Slope Fields Sketch a slope field for a given differential equation Given a slope field, sketch a solution curve through a given point Match a slope field to a differential equation Match a slope field solution to a differential equation

13. Sketching a Slope Field Compute the slope for each point Make a small mark that approximates the slope through the point. When x = 0, dy/dx = -1 When x = 1, dy/dx = 0 When x = 2, dy/dx = 1 When x = -1, dy/dx = -2 When x = -2, dy/dx = -3

14. Sketching a Slope Field in x and y You may find it useful to make a table -2012 3-5-4-3-2 2-4-3-20 1-3-201 0-2012 0123 -201234 y x

15. Example based on prior year free response questions Graph the slope field. 01 3-1/301/3 2-1/201/2 101 0---- 10 y x

16. Let y = f(x) be the particular solution with the initial condition f(1) = 2 Write an equation of the tangent line to the curve at (1, 2). We have the point (1, 2) At (1, 2) we had a slope of 1/2 Use the line to approximate f(1.1) Find the particular solution to y = f(x) with the initial condition f(1) = 2. Plug in the point Keep only the positive root. Why?