Optimization

Google's definitions:

Make the best or most effective use of (a situation, opportunity, or resource)

• to optimize viewing conditions the microscope should be correctly adjusted

Rearrange or rewrite data to improve efficiency of retrieval or processing

Let's look at the second one. In calculus, I am learning how to solve word problems that have to do with optimizing - finding the maximum or minimum values of something.

Here are the steps:

1. From the word problem, derive two equations (for my level anyway so far). This means two variables, usually one given constant and one unknown constant.

2. Rearrange the equation with the given constant so that one unknown is isolated.

3. Sub that equation into the second equation, so that there is only one variable in the "new" equation.

4. Now, since the "new" equation is equivalent to an unknown constant, we can take the derivative of the equation, which will give you the slope of the underived equation.

This is the tricky part that I didn't understand until today =)

Why do we take the derivative of the equation? To find the slope.

Why do we want to know what the slope is? - we don't (in some sense). we already know that the slope is zero at a maximum and minimum value for x, so we just need to find the unknown variable (x).  Don't get it? Take a look at this graph:

f(x) is the graph of the "new" equation, underived. (the one with both equations combined)

f '(x) is the graph of the derivative of the above equation. there are two because you can have a question asking for the maximum or the minimum.

As you should remember from previous years of doing math, the slope of a straight line, y = b, is always zero.  By looking for the slope of the tangent at a maximum or minimum point, we can have the equation with one variable on one side equal to zero, on the other side.    (ex.  4x+1 = 0)

Also, another reason why you can just equal your new equation to zero is because the unknown constant is a constant (pretty redundant, but still helpful to know). What does this mean? When you find the derivative of a constant, it is zero. Isn't that a coincidence? Not really, it's supposed to be zero, or else there'd be something wrong with this kind of math (also called calculus)!

Let's move on now...

5. With the derivative of the equation, we should now be able to solve for x. You should only have one unknown now (since the constant turned into zero).

6. If you got a quadratic equation, just factor it or use the quadratic equation if you can't (your scientific calculator should be able to do this too). If it was a monomial, then that's easy!

7. Now, to make sure that you have the right value (s), plug them into the equation you made before you took the derivative of it. If you have multiple values, choose the value for your variable (x) where f (x) = largest/smallest.

8. You can also use the sign chart to check your solution. (if it's negative, then you have a negative slope = going down,  positive = going up. going from left to right, a max value should have + going to -, and vice versa for min values)

9. If you got it right, YAY CELEBRATEEEEEEEEEEEEEEE!

If not, you probably did some funky algebra! Go back and check your steps (tedious task, but necessary!).

* this post is mainly for helping me understand and study.... and also good for me to look back at later when studying for final exam. I understand better and things "click" when I "teach" stuff. =) thanks for helping me learn by helping you learn!