December 2 Blog Post

First, we started off class with our Quiz. OB was nice, and gave us extra time to do it, so we ran until 10:10.
After the Quiz was over, we pulled out the "1999 AP Calculus AB" Multiple Choice Questions (just the first five) that we worked on.

Then, we pulled up the "Exploration 8-3: Maximal Cylinder in a Cone Problem" and started it on our own. As usual, we worked through it for a while, before OB decided it was time for him to step in and help us work through the rest of it.

This Exploration was all about "Optimization." Optimization means you're looking for the best, worst, or optimal result of a problem. For our Exploration, we needed to find the Optimal Volume. We started out by making of a table relating radius, height, and Volume.



We filled in this chart by using the Volume of a Cylinder (). So, we have something "tres bizarre" here. The Maximum volume is actually up in the Area of 3. But we can't just take Nickname's word for it, and use 3 as our Optimal Height. We need to check other values that aren't just 1, 2, 3, or 4.
Then, we rewrote the Volume of a Cylinder Equation in terms of x, which resulted in . This is our "Primary Equation". We then found an equation for the relation to x and y. We can either look at this as a Number Pattern, or use point/slope to create a y=12-3x. Now that we have both of these equations, we can combine them to create our "Secondary Equation", so we just have an x: .
For the third part, we just used implicit differentiation, and got
Now we can just set V' equal to Zero, and find our answer for 4. Hopefully we all know how to simplify by now, so I won't write the steps. Our resulting Zeros were x=0, and x=-8/3. We can just take out Zero, from Context (we can't have x be zero, or our Volume is nonexistent), so our only resulting answer is x=-8/3! Ta-Da!

But, we have to check it in our Calclulators, so we graphed our Function, and made the Calculator find the Maximum. However, we're smarter than our silly calculators, because we can find the real answer, not just some rounded decimal approximation.

Now, we need to make sure that even through all of our Smart Moments, we need to answer the Question. So in summary, we should write something along the lines of:










And for future Reference, the thing that makes Optimization problems really hard isn't the Concept schtuff, it's the fact that they are the dreaded……………

So, we'll work on these for the next couple of days.

Here's Paul's Explanation of Optimization.

And at the End of Class, OB told us to make sure to tell everyone else that the Quiz was CRAZY hard, and our brains have turned to Jell-O.