Of all of the AP Calc methods you will learn this is one of the more difficult ones to give you a cut and dry “this is what you always do” method. The reason is not that the process is that difficult, it is because each different type of problem can require a different starting equation . Once you have that starting equation, then you are really just going to run a standard 1 ^{st} Derivative Test or Global Extrema Process using that equation. This, again, is why doing a large diversity of problems is the best way to practice. The more you see the more you will recognize a similar setup equation just wrapped in different language.
These are the general steps that I follow when trying to solve an optimization problem.
Step 1: Draw a picture of the situation.
Look for key words in the problem to help you decide on shapes. If they are talking about a spherical this or a circular that. That is telling you what shape you should be trying to draw.
Keep in mind that most pictures are going to be made up of your standard geometric shapes (i.e., triangles, rectangles, circles, spheres, cones, cylinders).
Step 2: Label the picture you drew.
Step 3: Create a constraint equation (if provided).
You won’t always have or need a constraint equation , but you should be on the lookout for it as it does happen often. It is often this constraint equation that you will use to get your optimization equation to talk just one variable .
Step 4: Create your optimization equation .
The most important piece to keep in mind when creating your optimization equation is that you need a single equation that only talks about one variable (i.e., all x ’s, or all r ’s). This is really the part of the process that requires you to know your area, volume, and other formulas.
You might need to optimize a volume of a cylinder, $V=\pi {r}^{2}h$, which includes two variables a radius and a height. You will need to find another relationship (equation) between the radius and height, which will allow you to solve for one, radius or height (depending on what you need), and then plug it back into the equation you are trying to optimize . This is usually going to be your constraint equation .
Keep in mind there is one equation you are trying to optimize . It is that one equation you are trying to optimize, and it is that one equation that you must do whatever it takes to get it into one variable .
Step 5: Once your optimization equation is in one variable , you will start either a 1 ^{st} Derivative Test or Global Extrema Process depending on if you have endpoints on your interval.
This is the most often occurrence.
When trying to figure out whether or not you would have endpoints, it is usually best to think about what would happen in extreme situations. Could you have zero as one of your variables (i.e., no side length), or could you have all of something or none of something (i.e., all on land or all in the ocean).