6.1 Antiderivatives and Indefinite Integration

1. 6.1 Antiderivatives and Indefinite Integration Objectives: 1.) Understand the concept of a antiderivative 2.) Use differentiation rules to produce and use antidifferentiation rules.

3. Vocab • Differential: the differential is an equation that relates the change in y with respect to the change in x. dy = f’(x)dx

4. Vocabulary f(x)= x3 + 2x What was the function that WAS derived to get this? F(x) f’(x)= 3x2 + 2 f(x) The antiderivative function, notated BIG F, is the fuction that was derived to get a function f. F’(x) = f(x) for all x in I We are going to start going backwards now. We are going to UNDERIVE functions…

6. Integration and antidifferentiation mean the same thing • The process of underiving

7. Notation

8. Notation/Representation • We call G(x) the general antiderivative of f. G(x) = F(x) + C for all x in I the indefinite integral . • C is called the constant of integration. It is the constant number that could have been wiped out in differentiation. When we antidifferentiate, we need to consider a constant may have been there. • Consider f(x) = x2 + 1

9. General antiderivative and General Solution are synonomous.

10. Anyone of these graphs could have produced f’(x) = 2x

11. Basic Rules of Integration (pg. 390) Integration of ZERO Integration of a constant Integration of a power Integration with a scalar multiple Integration of sums and differences

12. Homework: • Pg 394 #1; 4; 9-13(odd); 19-23(odd); 26; 28-31; 37-39; 42; 43

13. Examples

15. Objectives 1.) Apply integration to vertical motion functions… 2.) Start thinking forwards… backwards.

16. Particular Solution vs. General Solution

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