1. 7.1 Antiderivatives OBJECTIVES * Find an antiderivative of a function. *Evaluate indefinite integrals using the basic integration formulas. *Use initial conditions, or boundary conditions, to determine an antiderivative. Slide 4.2 - 1

2. Who comes up with the ready-made functions we find derivatives for? Isn’t it hard sometimes to find a function for total cost, profit, etc.? Sometimes it is easier to calculate the rate of change of something and get the function for the total from it. This process, the reverse of finding a derivative, is antidifferentiation. Slide 4.2 - 2 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

3. Slide 4.2 - 3 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 1: Can you think of a function that would have x 2 as its derivative? Antiderivatives

4. Slide 4.2 - 4 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley One antiderivative is x 3 /3. All other antiderivatives differ from this by a constant. So, we can represent any one of them as follows: To check this, we differentiate.

5. Slide 4.2 - 5 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley THEOREM If two functions F and G have the same derivative over an interval, then F(x) = G(x) + C, where C is a constant. Antiderivatives

6. Slide 4.2 - 6 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Antiderivatives Integrals and Integration Antidifferentiating is often called integration. To indicate the antiderivative of x 2 is x 3 /3 +C, we write

7. Slide 4.2 - 7 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Antiderivatives The notation is used to represent the antiderivative of f (x). More generally, where F(x) + C is the general form of the antiderivative of f (x).

8. Slide 4.2 - 8 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley THEOREM : Basic Integration Formulas

9. Slide 4.2 - 9 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 2: Evaluate 4.2 Area, Antiderivatives, and Integrals

10. Slide 4.2 - 10 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Check:

11. Slide 4.2 - 11 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 3: Evaluate 4.2 Area, Antiderivatives, and Integrals

12. Slide 4.2 - 12 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Check:

13. Slide 4.2 - 13 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley THEOREM 4 (The integral of a constant times a function is the constant times the integral of the function.) (The integral of a sum or difference is the sum or difference of the integrals.) 4.2 Area, Antiderivatives, and Integrals

14. Slide 4.2 - 14 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 4: Evaluate 4.2 Area, Antiderivatives, and Integrals

15. Slide 4.2 - 15 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

16. Slide 4.2 - 16 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 5: Evaluate and check by differentiation: 4.2 Area, Antiderivatives, and Integrals a) 7 e 6 x  x   dx ; b) 1  3 x  1 x 4        dx ;

17. Slide 4.2 - 17 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Antiderivatives Example 5 (concluded):

18. Slide 4.2 - 18 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 5 (continued): Check: Antiderivatives

19. Slide 4.2 - 19 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 5 (continued): Antiderivatives

20. Slide 4.2 - 20 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 5 (concluded): Check: Antiderivatives

21. Slide 4.2 - 21 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 6: Find the function f such that First find f (x) by integrating. Antiderivatives

22. Slide 4.2 - 22 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 6 (concluded): Then, the initial condition allows us to find C. Thus, Antiderivatives

23. Slide 4.2 - 23 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

24. Slide 4.2 - 24 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

25. Slide 4.2 - 25 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley