A favorite saying of my old English teacher went along the lines of, “Close only counts in horseshoes and hand-grenades.” In quantitative studies, especially, the idea of mere guesswork or a kinda-sorta-correct answer is frowned upon. Nonetheless, this project essentially consisted of exploring better ways of estimating (which is the coward's way to say “guessing”). The class was given a function and asked to estimate its root, using the Newton-Raphson method and three different initial guesses, then analyze which initial guess was most efficient. Of course, the process proceeded more like a game of darts than horseshoes. If the initial shot was too far left of the target, the next shot was corrected 180 degrees to the right and aimed directly at the person behind us (behavior only suited to either a degenerate or a mathematician – whichever you prefer). Fun, games, and injuries aside, I ultimately concluded that the usefulness of an initial estimate was more dependent on the tangent line's location relative to trends in the function and its behavior at that location.
Returning to the idea of over-correcting and throwing darts backwards, this project provided some examples of how a shallow slope is particularly ineffective. For both C0 = -1 and C0= 3, the initial tangent line provided a new root estimate that was a significant distance from the location where the tangent line was actually tangent to the function. In both cases, they provided a negative root when the actual root was clearly positive (determined graphically). This resulted in the next tangent line occurring at that new location on the function (resulting in another negative root). Eventually, after jumping between positive and negative roots for a few estimates, the tangent lines would get close enough to the first estimate (C0=0) and behave from then on. For comparison, when aiming for a target, one usually stands directly in front of the bulls-eye and fires straight forward, making only very small adjustments to get closer each try. If one (again, a “one” of very questionable capacity) chose to start by standing five feet to the left of the target, their first try, ideally, would be at an angle, toward the right. If their next standing position could only be at the location of their previous try, they would have to take several large steps to the right, and then have to fire at an angle to the left – and so on until they were standing basically in front of the target like their more practical comrade. As has been established in class, mathematicians/students are lazy. It is more ideal to be firing “directly forward” in as few “steps” as possible.
In this case, the mathematical equivalent of firing “directly forward” would be to find where the tangent line has a steeper slope, as that would require less initial side-stepping. Intervals that could provide steep tangent lines can be found using the first and second derivatives of the function. In places where the sign of the first and second derivative are the same, the slope is steeper. More specifically, when the function is increasing at an increasing rate (concave up, positive second derivative), or decreasing at an increasing rate (concave down, negative second derivative), slope is steep. In this function, the root was conveniently between two extrema, a maximum and a minimum. The fact that there must have been an inflection point “close to” the root hinted that the desired interval was also “close.” Extrema themselves, by nature of having the shallowest tangents of all, should obviously be avoided. As the tangent line approaches the location of an extremum, it becomes more shallow, then horizontal, and then finally changes sign. Of course, by then, ??????, who has the alarming tendency to stand right behind incompetent people who are trying to concentrate, has been hit by a few darts. No one wants that (not really, anyway).