1. 4.4 The Fundamental Theorem of Calculus (Part 2) Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 1998 Morro Rock, California

2. Objectives Understand and use the Mean Value Theorem for Integrals. Find the average value of a function over a closed interval. Understand and use the Second Fundamental Theorem of Calculus.

3. The average value of a function is the value that would give the same area if the function were a constant:

4. The mean value theorem for definite integrals says that for a continuous function, at some point on the interval the actual value will equal to the average value. Mean Value Theorem (for definite integrals) If f is continuous on then there exists a number c in, such that

6. Example 5: At different altitudes in the Earth’s atmosphere, sound travels at different speeds. The speed of sound S(x) in m/s can be modeled by

7. Example 5: Total distance sound traveled: Average Speed:

8. Other examples: Total distance traveled from t=a to t=b Antiderivative of v is distance Total cola consumption in billions of gallons from 1980 to 1990

9. Definite integral as a number: Definite integrals as a function: constant f is a function of t F is a function of x

10. Evaluate

11. If you were being sent to a desert island and could take only one equation with you, might well be your choice. Here is my favorite calculus textbook quote of all time, from CALCULUS by Ross L. Finney and George B. Thomas, Jr., ©1990.

12. The Second Fundamental Theorem of Calculus If f is continuous on an open interval I containing a, then for every x in the interval Also,

13. Second Fundamental Theorem: 1. Derivative of an integral.

14. 2. Derivative matches upper limit of integration. Second Fundamental Theorem: 1. Derivative of an integral.

15. 2. Derivative matches upper limit of integration. 3. Lower limit of integration is a constant. Second Fundamental Theorem:

16. 1. Derivative of an integral. 2. Derivative matches upper limit of integration. 3. Lower limit of integration is a constant. New variable. Second Fundamental Theorem:

17. 1. Derivative of an integral. 2. Derivative matches upper limit of integration. 3. Lower limit of integration is a constant. The long way: Second Fundamental Theorem:

18. 1. Derivative of an integral. 2. Derivative matches upper limit of integration. 3. Lower limit of integration is a constant.

19. The upper limit of integration does not match the derivative, but we could use the chain rule.

20. Homework 4.4 (page 285) #45-51 odd, 54-60 all, 69-91 odd, 103, 105