Download presentation

Presentation is loading. Please wait.

Published byJonathan Joseph Modified over 5 years ago

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

Similar presentations

© 2021 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google