1. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Antiderivatives and Slope Fields Section 6.1

2. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 2 What you’ll learn about Differential Equations Slope Fields Antiderivative rules … and why Differential equations have been a prime motivation for the study of calculus and remain so to this day.

3. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 3 Differential Equation An equation involving a derivative is called a differential equation. The order of a differential equation is the order of the highest derivative involved in the equation.

4. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 4 Example Solving a Differential Equation

5. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 5 First-order Differential Equation If the general solution to a first-order differential equation is continuous, the only additional information needed to find a unique solution is the value of the function at a single point, called an initial condition. A differential equation with an initial condition is called an initial-value problem. It has a unique solution, called the particular solution to the differential equation.

6. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 6 Example Solving an Initial Value Problem

7. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 7 Example Using the Fundamental Theorem to Solve an Initial Value Problem

8. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 8 Slope Field The differential equation gives the slope at any point (x, y). This information can be used to draw a small piece of the linearization at that point, which approximates the solution curve that passes through that point. Repeating that process at many points yields an approximation called a slope field.

9. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 9 Example Constructing a Slope Field

10. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 10 Indefinite Integral

11. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 11 Example Evaluating an Indefinite Integral

12. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 12 Properties of Indefinite Integrals

13. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Pages 312-313 (7-23 odd, 27, 29) ========================== Page 313 (31-43 odd) Slide 6- 13