The Calculus is made up of a few basic principles that anyone can understand. If looked at in the right way, it’s easy to apply these principles to the world around you and to see how the real world works in their terms. Of the two main ideas of The Calculus — the derivative and the integral — today we’ll focus on the derivative.
You can enjoy this article by itself, but it is also worth looking back at the previous installment in this series. We went over the history of The Calculus and saw how it arose from two paradoxes put forth by a 4th century philosopher named Zeno of Elea. These paradoxes lead to the derivative/integral ideas that revolutionized mankind’s understanding of motion.
The derivative is what is necessary to solve Zeno’s “The Arrow” paradox. In this paradox, Zeno asks how an arrow flying through the air can have a velocity at an instant in time. For at each instant, the arrow is still. Before we begin with our process of discovering the answer — a process that took civilization over two thousand years to figure out — let us refresh our static memory with our previous description of the derivative:
The Derivative – The derivative is a technique that will allow us to calculate the velocity of the arrow in “The Arrow” paradox. We will do this by looking at positions of the arrow through incrementally smaller amounts of time, such that the precise velocity will be known when the time between measurements is infinitely small.
Is It Really This Simple?
I think you will find that it is, and even wonder why it took civilization so long to figure it out. Zeno wants the velocity of the arrow. In order to calculate velocity we need two things – time and position. Velocity is just distance divided by time, such as miles per hour or feet per second. If we measure the distance traveled by the arrow and divide it by the time it took to move the distance, that is by definition the velocity of the arrow.
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Sources
All images are from The Teaching Company course, Lecture 2.