Building Understanding for the Average Value of a Function

1. Building Understanding for the Average Value of a Function Jim Rahn www.jamesrahn.com James.rahn@verizon.net

2. Equity & Access The College Board and the Advanced Placement Program encourage teachers, AP Coordinators, and school administrators to make equitable access a guiding principle for their AP programs. The College Board is committed to the principle that all students deserve an opportunity to participate in rigorous and academically challenging courses and programs. All students who are willing to accept the challenge of a rigorous academic curriculum should be considered for admission to AP courses. The Board encourages the elimination of barriers that restrict access to AP courses for students from ethnic, racial, and socioeconomic groups that have been traditionally underrepresented in the AP Program. Schools should make every effort to ensure that their AP classes reflect the diversity of their student Population.

3. Philosophy • Students should develop an understanding of the concepts of calculus and have experience with its methods and applications. • The Rule of Four (multi-representational approach) is paramount in accomplishing this. • There are questions on every AP calculus exam that test the concepts in each of these four ways. • It is important to stress the Rule of Four in your teaching. • Knowing how to compute a derivative or a definite integral is not enough. • Students will be asked to recognize the derivative and the integral in written contexts, as it relates to graphs and tables of values. • Students who can only work with formulas and equations will have a difficult time on the AP Calculus exams.

4. What is an average value of a function? • If you were given a discrete set of values for a function, what would you normally do to find the average value of a function? • Can we apply any of our previous knowledge of average to find the average value of a function on a given interval?

5. This activity will help build understanding for what it means to find the average value of a function by studying four different, but similar functions. • Many students understand how to find an average when they are given a discrete or finite number of elements, but how do you find an average when there an infinite number of elements?

6. Function 1 Y=2x+5 on the interval [0,5]

7. In both cases the average value of f is 10. • The line y = f(x) = 2x + 5 takes on all the y values between and . • Predict the average value of y=2x+5 on [0,5] using 6 values and 11 values. 5 15

8. Area of Trapezoid = 50 • Graph y1=2x+5 and y2= 10 (or average value) • What is special about the point of intersection? • Find the bounded area between y1, x=0, x=5, and y=0. • Find the bounded area between y2, x=0, x=5, and y=0. • What do you notice about the two areas? • What is in common between the two regions? Area of Rectangle =50

9. How is the average value of the function of f(x) related to the height of the rectangle? • At how many x-values in the interval (0,5) does the function take on this average value of f(x)?

10. Let’s Study a Non-Linear Function Let on the interval [0,5].

11. The function takes on all the y values between and . • Calculate the average value for 6 and 11 points. 12.5 0 Second average is about 4.420 First average is about 4.58

12. Make a conjecture about the location of the average value of this function. • But we must remember that the domain [0,5] is not made up of a finite number of x-values but rather an infinite number of values. So, again we can involve some calculus to help us determine this average value of the function. • Use an integral to find the bounded area.

13. 20.833 • Remember that from Function 1 we were able to find the average value of f(x) by dividing the bounded area under f(x) by the width of the interval [0,5]. • What is the height of the rectangle whose width is 5 that has the same area? • Create the rectangle by entering a value in y2. • What is the significance of the point of intersection? • Was your estimate for the average close?

14. Try to think about why the average value of g(x) is located where it is. (You might want to think about the values in your two tables.) • Again, how is the average value of g(x) related to the height of the rectangle? • At how many x-values in the interval (0,5) does the function take on this average value of g(x)?

15. Let’s Study another Function • Think about the function for values of x in the interval on the interval [0,5].

16. 0 • This function takes on all values between and . 12.5

17. Estimate the average value of the function by using 6 and 11 points. First average is about 6.666 Second average is about 7.5 Make a conjecture about the location of the average value of this function.

18. Recall from the last two examples we need to find the bounded area. Again use Calculus to find this area. 41.666 • How tall is the rectangle?

19. Try to think about why the average value of h(x) is located where it is. (You might want to think about the values in your two tables.) • Again, how is the average value of h(x) related to the height of the rectangle? • At how many x-values in the interval (0,5) does the function take on this average value of g(x)?

20. Let’s Think about 1 More Function • Let

21. Create a graph of this function. • This function takes on all the y values between and . 6 0

22. Estimate the average value of this function using 6 and 11 points. First average is about 2.25 Second average is about 2.386

23. To find the average we’ll need to find the bounded area using calculus: 13 • To find the average value of j(x) over the interval [0,5] we need to find the height of the rectangle with the width of 5 that has an area of 13. • What is the average value of j(x) over the interval [0,5]?

24. Try to think about why the average value of j(x) is located where it is. (You might want to think about the values in your two tables.) • How is the average value of j(x) related to the height of the rectangle? • At how many x-values in the interval (0,5) does the function take on this average value of j(x)?

25. Bringing It Altogether • Sometimes geometric formulas can be used to determined the bounded area, but other times we need to write an integral. If a region is bounded by y = f(x), x = a, x = b, and the x-axis, what integral would find the area under y = k(x) between x = a and x = b?

26. The bounded area can then be used to find favg. Write a formula involving the integral that will yield favg.

27. Sometimes we are interested in finding the point c in the interval [0,5] that produces favg. This is f stated at favg=f(c) . Explain how you can determine the value of c?

28. From the four functions explored in the activity, find the value of c in [0,5] that yields favg.

29. The graph of function f is made up three line segments and a semi-circle of radius 2. Domain: [0,10] • Find the average value of this functions for the given domain. • Find the value of c in the domain where the function takes on .

30. 2011 Form B AB2

32. 2008 AB Test Using the numerical antiderivative on the graphing calculator yields