I’m having trouble getting started on my blog this week. Coincidentally, I just read the blog, “Starting Trouble,” by &. It explores the similarities between writer’s block and programmer’s block. I wonder if the article inspired me. One tonic to relieve the block, suggested in the blog, was to just write, anything at all if necessary, even if gibberish. So here it goes.
I am tired tonight. I spent several hours today preparing worksheets and reviewing a lesson plan on the topic of optimization in calculus. Optimizing is finding a value that minimizes or maximizes a model (mathematical function). A great deal of vocabulary surrounds the procedure for finding the values. I found it difficult to get to the heart of the matter after spending much of my time defining terms.
The minimum and maximum values on a particular domain are values of a function that stand-out; they are extreme values that are notable by being higher or lower than any other values in their neighborhood. What exactly is a neighborhood? Informally, a neighborhood is an interval around an independent value, usually a small, symmetric interval, as small as you need it to be to prove a point (which in this case is to identify a point). An extremum can be a relative minimum or a relative maximum, also called local minimum or local maximum, the connotation of local fitting with the terminology neighborhood more closely than the word relative, but relative gives the sense of judgement tempered by location, so it too suggests neighborhood. When you have more than one extremum, you have extrema; English showing its roots in older languages. Anywhere you are to find an extremum you must also find a critical point or an endpoint (but finding a critical point or an endpoint does not guarantee an extremum). Critical points come in two flavors, singular or stationary. Endpoints are ending points of closed or semi-closed intervals. Singular points are internal points (not endpoints) where the derivative of the function evaluated at the point is not defined (then how did you evaluate it you might ask). Stationary points are internal points where the derivative of the function evaluated at the point is zero (hence the idea of stationary since the derivative is all about rate of change). The absolute maxima or minima, which in the general case may not exist, are relative maxima or minima which have the highest or lowest values of the function, respectively. But they are guaranteed to exist if you constrain the function to continuity and the domain to being closed on both sides. The name of the guaranteeing theorem is the Extreme Value Theorem (back to that term again). You test critical values to see if they are relative minima or maxima by looking at sign changes in the derivative function from one side of the critical point to the other (they are internal so they must have two sides). The endpoints, where they exist, must be tested individually in half neighborhoods, usually at the end of the procedure, but that is not why they are end points, as was noted earlier.
That should be enough to get started, certainly enough to meet the mark of gibberish. Unfortunately, this gibberish is what must be gibbered to introduce optimization using techniques of differentiation. Is it any wonder that students develop a mental calculus in the study of calculus?