## A Semester in Review

This semester in calculus has taught me many new skills, both concept wise and also problem solving skills. Two of such skills are generating ideas, and my newfound understanding of using derivatives to find local maxima and minima of functions.

My ability to find local minima and maxima of functions using derivatives is a concept I first learned of many years ago when my brother was asking for help with his calculus homework from my dad (albeit I had no idea what he was talking about; I still didn’t know the difference between a bar graph and a line graph.) Although most of what was said was as good as jibberish to me, one thing rang clear. The fact that if a line is first increasing, and then decreasing, at some point on the line, it will be moving sideways. He said this while using a pencil to trace a curve and stopped when the pencil was parallel to the “long line on the bottom” as I knew it as back then.

Now it wasn’t until many years later that Kyle expanded upon this idea, and while he may have used tangent lines instead of a pencil, it made perfect sense. He used tangent lines to find the derivative of a function, and showed that at a vertex, the tangent is zero. This means that these points are either maxima or minima of a curve, because they exist only where the line transitions from a positive to a negative, or negative to positive, slope.

One example of my understanding of this Idea is seen in my P.O.W. #5. In this problem we were asked to find the shortest amount of road needed to connect four cities (local minimum of a function) using derivatives. I did extremely well on this problem and got a perfect score.

My ability to find local minima and maxima of functions using derivatives is a concept I first learned of many years ago when my brother was asking for help with his calculus homework from my dad (albeit I had no idea what he was talking about; I still didn’t know the difference between a bar graph and a line graph.) Although most of what was said was as good as jibberish to me, one thing rang clear. The fact that if a line is first increasing, and then decreasing, at some point on the line, it will be moving sideways. He said this while using a pencil to trace a curve and stopped when the pencil was parallel to the “long line on the bottom” as I knew it as back then.

Now it wasn’t until many years later that Kyle expanded upon this idea, and while he may have used tangent lines instead of a pencil, it made perfect sense. He used tangent lines to find the derivative of a function, and showed that at a vertex, the tangent is zero. This means that these points are either maxima or minima of a curve, because they exist only where the line transitions from a positive to a negative, or negative to positive, slope.

One example of my understanding of this Idea is seen in my P.O.W. #5. In this problem we were asked to find the shortest amount of road needed to connect four cities (local minimum of a function) using derivatives. I did extremely well on this problem and got a perfect score.

One problem solving skill I have developed during this semester has been Idea generation. One instance I have demonstrated this was during our First problem of the week. In this problem we were given a situation in which a group of eight people were sitting in a circle, and each had a coin. Each person flips their coin, and anyone whose coin flip resulted in heads stands up. We were asked to find the probability that after everyone had flipped their coin, no two neighbors were standing up.

As soon as I heard this problem and was set loose upon it, I came up with the idea “Eight is a really big number, Let’s try it with four people and then just scale it up!” Which was done, and we immediately got an answer. Even though it was incorrect, it was a start.

Idea generation is broken up into three parts, Formulating a plan / anticipate major intermediate steps, Generating multiple means of approaching a problem; brainstorm plans, and Identifying and applying appropriate mathematical tools (formulae, equations, diagrams, graphs). I can always come up with an idea. It may be way out of left field, and cause confusion and disapproving looks from my teacher, but I can find a way to approach a problem. I can also model these ideas and fashion equations to fit them. But I do not like generating multiple Ideas for a problem. I’ll be the first to admit, I am bigoted and bull headed when it comes to math. I like to make my ideas work, and will sink hours into a problem that can be easily circumvented by another method. But I am working on this by thinking broader. I am trying to find multiple methods to use on a problem. In order to help myself overcome my urge to stick with a problem, I force myself to think of multiple ways to approach it, and then pick the one I believe will work the best.