Average value of a function
We've been working with functions throughout high school- adding them, subtracting them, multiplying, dividing them, and even composing them- but we've never actually talked about a function's average value... until today! Over an interval of values x = a to x = b, a function can give out a number of different values. With your table partner, please think about this question: How do you think we might define a function's average value? For example, if we looked at the function f(x) = x – 1 from x = 1 to x = 5, what might it's average value be? Use the sketch below to answer the questions following it in your notebook (part of IW #11).
1. What is the average value of this function from x = 2 to x = 5? Can you find a pattern for the average value? 2. Change the function to a different linear function- does your observation hold? 3. What if the function is constant? Try f(x) = 3. 4. Try the function f(x) = x^2 – 1 for x = 1 to x = 3. Does your explanation still hold? Use the check boxes- see if that helps you explain what's going on. 5. Play with the sketch- try a sinusoidal function, etc. You can find the definition of the average value of the function f on the interval [a, b] on page 291 in your text book. Write it down. 6. Read about the Mean Value Theorem for Definite Integrals on p. 291-2. Do Exploration 1 on p. 292 on your Geogebra sketch first (using different values of r) and then by setting up integrals.