Blog Post Monday, February 4th

Hi guys-

Little Lu Rocket has the stomach bug, so I'm home with her today.



To begin class, I'd like everyone to quietly read Thursday's blog post. There's a lot going on here...



After, I'd like everyone to take a look at IW #7. There are a couple questions posted online- I've done my best to help with these. If others arise that you're unable to sort through as a class, please be sure to post them so that we can look them through together.



Next, I'd like you to think about something. 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?

Your work today in class is part of IW #8. Jot your thoughts on a fresh sheet of paper (entitled IW #8), and then open this sketch in Geogebra. Please answer the following questions on your IW #8 paper:

• What is the average value of this function from x = 2 to x = 5? Can you find a pattern for the average value?
• Change the function to a different linear function- does your observation hold?
• What if the function is constant? Try f(x) = 3.
• 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.
• Play with the sketch- try a sinusoidal function, etc. Use Command-E to see how I created the average value of the function f on the interval [a, b]. Check your definition with the one on page 291 in your text book.
• 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.

Changing gears, please do this exploration of definite integral properties with your table partner. If you hate trees, feel free to print a copy. Otherwise, use the screen copy, and write your answers on IW #8. Check your answers on p. 289 and here.

You may now do the text book problems from IW #8 (see iCal).