1. Average Value of a Function and the Second Fundamental Theorem of Calculus Day 2 – Section 5.4 Nancy Powell 2008

2. Average Value of a Function The Mean Value Theorem for Integrals focuses on the fact that there is a “c” such that f(c) AVERAGE ValueThe value of f(c) given in the MVT is called the AVERAGE Value of f on the interval [a,b]. C

3. Average Value of a Function Average Value of a Function The Average Value of a Function on an interval states: If f is integrable on the closed interval [a,b], then the average value of f on the interval is

4. So let’s revisit a problem from yesterday… We found that c=1.6510 and that the height of our rectangle was f(c) = 2. average valueof f Let’s calculate the average value of f on [1,3] C

5. The Second Fundamental Theorem of Calculus The definite integral as a NUMBER The definite integral as a Function Constant f is a function of x F is a Function of x f is a function of t Getting ready for

6. Integrals: Integrals as Accumulated Area Functions You are given the graph of a function y = f (x) on a grid. Assuming that the grid lines are spaced 1 unit apart both horizontally and vertically, sketch the graph of the area function over the interval [1,5] for the function. Then sketch the graph of the derivative of each area function and compare it with the original function’s graph.

7. Accumulated Area under a curve You are given the graph of a function f - sketch the graph of the area function over the interval [1,5] for the function. Interval Approx. Area Accumulated Area 1 to 1 1 to 2 2 to 3 3 to 4 4 to 5 - 1.0 0.5 0.5 1.0 - 0.5 - 1.0 0 2.5 1.5 0 3.5 This is where the area = 0 (our initial condition!)

8. The Definite Integral as a Function Evaluate the function below at x = 0,  /6,  /4,  /3, and  /2 You could evaluate 5 different integrals or better yet, temporarily fix x as a constant and use the Fundamental Theorem as shown below So, what do we know?

9. The Definite Integral as a Function xsin(x) 00  /6.5  /4.7071  /3.8660  /2 1 So, what do we know? What if we changed the integral to

10. The Definite Integral as a Function What if we changed the integral to

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

12. The Second Fundamental Theorem Remember Find:

13. The Second Fundamental Theorem Find the derivative of Because x 3 is our upper limit of integration, we need to look at this integral with what we know about the chain rule. Chain Rule Definition of dF du Substitute

14. The Second Fundamental Theorem Find the derivative of Substitute u for x 3 Apply the 2 nd Fundamental Theorem of Calculus Rewrite as a function of x