1. AP Calculus AB Antiderivatives, Differential Equations, and Slope Fields

2. Solution Review Consider the equation Find

3. Antiderivatives What is an inverse operation? Examples include:  Addition and subtraction  Multiplication and division  Exponents and logarithms

4. Antiderivatives Differentiation also has an inverse… antidefferentiation

5. Antiderivatives Consider the function whose derivative is given by. What is ? Solution We say that is an antiderivative of.

6. Antiderivatives Notice that we say is an antiderivative and not the antiderivative. Why? Since is an antiderivative of, we can say that. If and, find and.

7. Differential Equations Recall the earlier equation. This is called a differential equation and could also be written as. We can think of solving a differential equation as being similar to solving any other equation.

8. Differential Equations Trying to find y as a function of x Can only find indefinite solutions

9. Differential Equations There are two basic steps to follow: 1. Isolate the differential 2.Invert both sides…in other words, find the antiderivative

10. Differential Equations Since we are only finding indefinite solutions, we must indicate the ambiguity of the constant. Normally, this is done through using a letter to represent any constant. Generally, we use C.

11. Solution Differential Equations Solve

12. Slope Fields Consider the following: HippoCampus

13. Slope Fields A slope field shows the general “flow” of a differential equation’s solution. Often, slope fields are used in lieu of actually solving differential equations.

14. Slope Fields To construct a slope field, start with a differential equation. For simplicity’s sake we’ll use Slope FieldsSlope Fields Rather than solving the differential equation, we’ll construct a slope field Pick points in the coordinate plane Plug in the x and y values The result is the slope of the tangent line at that point

15. Slope Fields Notice that since there is no y in our equation, horizontal rows all contain parallel segments. The same would be true for vertical columns if there were no x. Construct a slope field for.