Yes I’m an Optimist…but Don’t Blame Me, Blame the Integral!

While I may be a self-proclaimed optimist and the proud owner of an internal locus of control, I maintain that not every day is (or should be) a good one. As Hannah Montana taught us, “Everybody has those days,” when things kind of just suck. But rather than wallowing in misfortune, I find that it’s these times when I most need to remember the calculus I’ve learned.

In integral application problems, definite anti-derivatives are often expressed as the total change in a function between two endpoints with respect to a variable--they are the essence of accumulation.

The ideas of accumulation, however, are most easily represented in the form of a Riemann Sum. When I take a midpoint Riemann Sum of a function using three sub-intervals, the accuracy of the calculation is low compared to the actual value of the function's integral.

If I use 10 sub-intervals, the error decreases significantly, and would decrease even more if I were to use 50 sub-intervals. Through the use of infinitely many (and therefore infinitely small) sub-intervals, I have essentially calculated the exact value of the integral.

A small change may seem immeasurable because it is, in fact, infinitesimal. Regardless, over time when we add up many infinitesimals, we see a quantifiable difference; the integral from a to b of dt = b-a...the fundamental theorem! If I strive to always make a positive change, I know I will eventually see the positive effects of that change.

Following that same train of thought, through the tenants of the extreme value theorem (EVT), it is guaranteed that life has both high points and low points. The EVT states that for a function continuous on the interval [a,b], there must be both a minimum and a maximum value of the function on the interval [a,b]. Even when the derivative of the function that is my life seems to be perpetually below zero, I remember that by the EVT, a return to positivity is assured.

If you’re anything like me, then you have a tendency to overthink simple things. If someone doesn’t wave back to me in the hallway I’m more likely to jump straight to assuming that they’re mad at me rather than assuming they just didn’t see me. When I feel myself beginning to overanalyze a situation, considering the nature of a Taylor Series keeps me grounded.

A Taylor Series is a representation of a function as an infinite sum of terms that are calculated from the values of the function’s derivatives at a single point. By constructing a function’s Taylor Series, I am taking something simple and turning it into something needlessly complicated. For example, the Taylor Series for the simple function, cos(x) is as follows:

1-(x^2)/2!+(x^4)/4!-(x^6)/6!+(x^8)/8!-(x^10)/10!+...

Essentially, by overreacting, I am creating a Taylor Series for the situation when I could more easily create the function and still have properly appraised my circumstances.

My last mathematical metaphor involves life's inevitable duality. For example, anyone who reads this story would readily agree that it was written in black font. However, who's to say that the way I experience black is the same way you experience black? Everything in life can be viewed from a multitude of perspectives, as is exemplified by the logarithm.

Upon being asked to find the base of a logarithm such that log(7)=18, one might be initially stumped. Nonetheless, by remembering just two manipulations of the same formula the answer is easily reached:

1. The logarithm with base B of the numerical value N is equal to L

2. N=B^L

The application of these formulas yields the following, much more manageable equation, 7=X^18, which is easily solved by taking the 18th root of 7 to find out that X=1.1. Even though the two permutations of the formula represent the same scenario, one was much more helpful in terms of solving than the other.

When life presents a problem I can't seem to solve or I'm forced into a crappy situation, remembering the rules of logarithms reminds me that thinking in negative way won't get me anywhere-- it's best to try and re-frame my thinking so that it's more helpful to the situation.