Method: Optimization

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.

• If you don’t have defined pieces like a height or radius , then I try to use x to note horizontal sides, y to note vertical sides, and z for diagonals . Try to keep you labeling scheme consistent throughout problems.
• Do not expect to label all of the pieces of your picture with actual values from the language of the problem. The picture is there to hopefully help you see all you have been given, see all you can find (i.e., you have two sides of a right triangle, and you could find the third side using the Pythagorean Theorem), and from that find a way to create your optimization equation .

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 .

• Your constraint equation is usually the equation you will use to get your optimization equation into one variable.
• The constraint equation will usually be based on one of your standard geometric shapes.
• In the language of your word problem, your constraint equation will usually be associated to a flat amount, or value given to you in the actual problem. You might have a limited area or volume for the situation. A common example would be that the perimeter is limited to a certain size. This would mean the constraint equation you have to create is a perimeter equation since it is the piece being constrained by an actual constraint value .
• Set your constraint equation equal to the constraint value (the flat amount given to you in the words of your problem), and solve the constraint equation for the variable you want to substitute into your optimization equation , and then substitute it into the optimization equation . Remember you want to solve for the variable you want to eliminate in the optimization equation , not the variable you want to keep.

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.

• If you have a do not have a domain with endpoints included, ( a , b ) , begin a 1 ^st Derivative Test.

This is the most often occurrence.

• If you have a domain with endpoints, [ a , b ], run a Global Extrema Process.

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).