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# AP Calculus AB

An interactive log for students and parents in my AP Calculus class. This ongoing dialogue is as rich as YOU make it. Visit often and post your comments freely.

## Wednesday, December 14, 2005

### M V T (Mean Value Theorem)

Now that the plague of Related Rates problems has ended (i don't know with the others but it was for me... laughs!) , a fresh new topic has emerged from its shell for us students to focus on for the next couple of days. I found Optimization problems much easier than related rates... or is it too early to judge?

This morning was the start of the Solving sessions in groups with the new topic Optimization. We had to work on Scuba Steve's Shark Cages. Given the total perimeter of the rectangle 450 feet, we had to divide the whole area into two equal parts so that the sharks wouldn't kill each other. The question is, what is the maximum area of each section of the cage that Scuba Steve can build?

First, know your constraint and optimization equations. From the constraint, solve for one variable, whichever is easier, the length (L) or the width (W). Use this to solve for the Area of the rectangle. Then, get the derivative of that function. After that, do the first derivative test and there you have it, a good start on Optimization problems. ;)

We also started on a new one today, the M V T. Most Valuable Tlayer? (laughs!) Nope, it's the MEAN VALUE THEOREM. M V T states that if a function is continuous on the closed interval [a , b] and differentiable on (a , b), then there exists at least one number c in the open interval (a , b) such that:

f ' ( c ) = f ( b ) - f ( a )
b - a

So given a function and its interval, get the values for intervals a and b. Then use the MVT to get the slope of that interval. Next, get the derivative of that function and equate it to its slope. So at X = c , the slope at c is then equal to the slope of the secant line in the interval a , b.

Sounds easy? Let's just wait and see. :D
Meanwhile, the next scribe is Xun. ;)