Optimization Problems

Hey everyone this is m@rk and i'm scribing for Wednesday's class. First of we took a quiz on visual calculus about what we've learned from the first and second derivative test. Second , Mr. K introduced a new topic called Optimization Problems. Optimization problems are problems that deal with either finding the maximum or minimum of the function in the problem.

Slide 1

First of all you need to do is draw a diagram to help you visualize the problem. Second, construct the constraint equation. The constraint equation is the equation is something that limits the problem. In this case it is 320=2L+2W. Third, find what you are trying to optimize and construct an optimization equation. In this case you are trying to maximize area, so you need to use the area formula A=LW. Fourth, you need to solve for one of the unknown variables on the constraint equation and then plug it in on the optimization equation. We choose to solve for L in this case and we plug it in on the optimization equation. Fifth, solve for the derivative of our new optimization equation. Sixth, solve for the roots of the derivative of our optimization equation. We found out that the root is located at W=80. Seventh, use the first derivative test and find out whether it is a maximum or a minimum. Lastly, answer the question in a complete sentence.

I'm not going to go over the other question since it is just almost the same problem but instead I'm going to give you an outline how to solve an optimization problem, so after digging up the apcalc blog here it is:

Step 1: Find what you are trying to maximize or minimize. This will be stated excplicitly (in the question).

Step 2: Write an equation for it. Use V = for volume and M = for material, A for area, etc. I must insist on using descriptive variables, because in optimization, if you are sloppy, you lose track of what's going on.

Step 3: Try to get the equation into a two variable form, so you can take the derivative.
- Step 3a: To do step 1, you will often have to create a second equation from additional information given in the problem. This may require ingenuity, but it should become natural.
- Step 3b: ISolate one of the two variables in the equation drawn in step 3a.
- Step 3c: In the original equation that you are trying to minimize or maximize, replace the variable you isolated in step 3b.

Step 4: Take the derivative of your two-variable equation.

Step 5: Set the derivative to 0, and solve for the value of the remaining variable.

Step 6: Plus the value of that variable into the first two equations to find all dimensions, including the final goal, such as the amount of material, cost, or volume.

NOTE: This is just an outline so follow it loosely. There is really no chronological way to answer this problems.

Thats it , the next scribe is haiyan.

BOB

Well, this unit about derivatives isn't too difficult considering that i learned this before. The only thing that i need to be careful on is working too fast for my own good. I usually make most of my mistakes on being too overconfident, but just to make sure I'll go over the orange and blue book tonight. Hopefully, i manage to get a good mark. Good luck everyone and study hard.

March 17, 2008

Today's class went similar to the class last Friday. Mr. K assigned people into different groups of three or four students, each group having at least one AP Calculus student. He wants the non-AP students to learn from their peers and the AP students to apply what they've learned before. The first proof that was done was the derivative of the power function, but before that Mr. K refreshed our memory about the binomial theorem. Next we worked on the proof of the derivative of a constant times a function. Then, we applied what we know about limits to prove the derivative of sum and difference of functions. Thats about everything that happened in class today. The next scribe will be rusz.l.

BOB

Well, this unit about limits went by really fast. I didn't even feel that we already finished a unit. This unit has been really easy for me ( thanks to AP Calculus). This unit has been sort of a review of the section that i learned not too long ago from calculus but is just discussed in a greater depth. The only thing that will pose problems for me in the test is evaluating using the limit theorems. I particularly don't like writing the limit notation in every step of the process, other than that i think that i should be fine. Good luck everyone on the test and study hard!!