Circle Time!

Circles are the idyllic shape. They’re symmetrical, they have no beginning or end, and their applications in the world are endless.  The definition of a circle is all points a set distance from a center point.  This symmetry and equivolency lets us use circles as a tool for constructing perfect right angles, they let us scope a likely perimeter when hunting for escaped criminals, they allow us to calculate the epicenter of an earthquake, and seriously, who doesn’t love a perfect pokkadot?  While learning basic calculus I realized a groundbreaking truth: that the circumference of a circle is the derivative of its area. The basic formula that we are taught to swear by starting in grade 5 had betrayed me.  The idea that formulas are just grabbed out of thin air and simply work was taken from me.  Teaching me the simple truth that there are no simple truths, my mind has not stopped buzzzing since this accident.  Everything is interconnected. 

Specifically about the circle: many secondary math students will realize this connection and wonder about it’s meaning.  Really there is no deep “meaning”, rather there is a common connection between this realization and the teachings of Calculus. An integral is the area under a curve or graph, or the antiderivative of the function.  Shown above is the estimation of the area of a circle using a Riemann Sum, which is the estimate of its integral. The area of a circle is the area under the curve of a circle ’s circumference (achieved graphically by breaking a circle into two equations) therefore making the area the antiderivative of the area, and the circumference the derivative of the area.  The chain rule for derivatives can then be applyed giving 2(pi)r as the derivative of (pi)r^2.  Everything is interconnected.

 The establishment of this fact is evident in many, or all, aspects of life.  The simple thought that if one spends one’s time eating, then one is not cleaning, working, or creating, thus rendering the act of eating connected to cleaning etc. In the world of technology great care is taken to create progress in the connectness of the disciplines.  Bioengineers need electrical engineers to formulate a way to start the heart.  Civil engineers need mechanical bouldozers to create their projects.  Buisnesses need these buildings to operate; society needs the product of the business in their daily life, and the economy needs the stimulation of consumption. You cannot isolate yourself in your one interest or discipline and expect to flourish alone.  Everything is interconnected.

Source: http://www.dictionary.com