Introduction to Optimization
We we’ve seen, there are many useful applications of differential calculus. One that is very useful is to use the derivative of a function (and set it to 0) to find a minimum or maximum to find either the smallest something can be, or the largest it can be. We call this optimization, since we are typically finding the optimal or “best value” for something.
Some examples of optimization include finding the greatest profit, least cost, greatest volume, optimum size, greatest strength, and so on.
Let’s first revisit absolute extrema, which are the absolute minimum and maximum of a function on an interval. In the Curve Sketching section here, we learned that a critical number is where the function’s derivative is 0, or not defined. By the First Derivative Test, critical numbers exist at relative minimums or maximums (also called local minimums or maximums). And of the course the absolute minimums and maximums are the one number that is the smallest or largest, respectively.
With optimization, if we have more than one critical point, we need to check the candidates to see which one is the absolute extrema in the possible domain (values that the “x” can be). Typically, to get the domain of a problem, we find the x values that would “make sense”; for example, we can’t have a dimension of 0 or less.
With optimization, we also need to check the endpoints of an interval, since these points could also be the minimum or maximum to get the absolute extrema.
So when we solve optimization problems, we typically put everything in terms of one variable, take the derivative, and set to 0 to get the minimum or maximum. And in the case of a closed interval (for example, when we could have a 0 value for an amount), we need to check the endpoints of the interval to make sure they aren’t lower (in the case of a minimum) or higher (in the case of a maximum).
Let’s do some problems; the first have to do with area and volume. Remember to always draw a picture first!
Area and Volume Optimization Problems
Endpoint Candidates Problem
Here is a difficult optimization problem where we need to use the Pythagorean Theorem. Note that since the domain can actually contain endpoints, we have to check the endpoint candidates to make sure we have the minimum distance we need:
Learn these rules, and practice, practice, practice!
On to Differentials, Linear Approximation and Error Propagation – you are ready!