1. 4009 Fundamental Theorem of Calculus (Part 2) BC 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 at Evaluate the Definite Integral for each of these points. The Definite Integral is actually finding points on the Accumulation graph. Evaluate the definite integral.

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

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

6. Demonstration: find In Words:

7. Example: Find and verify:

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:

11. Demonstration: Find: In Words:

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 without Integrating: break into two parts..... chose any number in domain of for a and rewrite into required form.

15. Last Update: 1/25/11 Worksheet