6034 Fundamental Theorem of Calculus (Part 2)

1. 6034 Fundamental Theorem of Calculus (Part 2) AB CALCULUS

2. The Indefinite Integral (Antiderivative) finds a Family of Functions whose derivative is given. Given an Initial Condition we find the Particular Function

3. The Definite Integral as a Particular Function: Evaluate the definite integral. Evaluate at 1 Evaluate the Definite Integral for each of these points. The Definite Integral is actually finding points on the Accumulation graph.

4. Since A(x) is a function, what then is the rate of change of that function? Take derivative In words, integration and differentiation are inverse operations

5. 2nd Fundamental Theorem of Calculus Given: , we want to find Note: a is a constant, u is a function of x; and the order matters! “a” is a constant 2nd Fundamental Theorem of Calculus: If fis continuous on an open interval, I, containing a point, a, then for every x in I :

6. Demonstration: < function x only > find In Words: Sub in the function u and multiply by derivative of u

7. Example: Find and verify: this Not this =

8. Example: Find without Integrating:

9. THE COMPOSITE FUNCTION If g(x) is given instead of x: In words: Substitute in g(x) for t and then multiply by the derivative of g(x)…exactly the chain rule (derivative of the outside * derivative of the inside)

10. THE COMPOSITE FUNCTION If , (a composite function) then In Words: Sub u in for t and multiply by u’

11. Demonstration: < The composite function > Find: Verify In Words: Sub in for t and multiply by the derivative of

12. Example : Find without Integrating: If , solve for

13. Example: Rewriting the Integral Find without integrating: Show middle step

14. Example: Rewriting the Integral - Two variable limits: Find withoutIntegrating: break into two parts . . . . . chose any number in domain of for a and rewrite into required form .

15. Last Update: • 1/25/11 • Worksheet