The Calculus Diaries - Jennifer Ouellette [12]
Once he plotted a curve, Newton drew on Fermat’s prior work and figured out how to find the slope of the tangent line for any point along that curve—the derivative, which he called the fluxion. Then he realized that finding the area under the curve (the integral) represented the process in reverse. Newton’s key insight was the connection between the derivative and integral. Finding the area under a curve (integration) is the reverse of finding the slope of a tangent line (differentiation). That is the fundamental theorem of calculus.
Newton noticed other intriguing connections: The apple’s velocity is the derivative of its position, while its acceleration is the derivative of its velocity. This also works for the integral. Add up the accumulated rate of acceleration over time, and you get the apple’s velocity; add up the accumulated velocity over time, and you get the apple’s position. Thanks to the fundamental theorem of calculus, it is possible to change one problem into another problem. If we have an equation that tells us the position of a falling apple, from that we can deduce the equation for the velocity of the apple at any given moment of its fall.
What made Newton’s method so revolutionary was its universality: The same equations that can be applied to the speed and position of a falling apple are also applicable to the planets orbiting the sun, the rate at which a cup of coffee cools, how interest accumulates in a savings account—any system in which one quantity is changing with respect to another. So calculus is a nimble beast, a flexible tool that, with lots of practice and a bit of creativity, can take you from a situation where you only have a little bit of information, to one where you have deduced a lot more information.
In modern calculus, these quantities—position, velocity, acceleration, and so forth—are known as functions, a concept that didn’t exist in Newton’s time. Here’s the kind of textbook definition that, while technically correct, conveys very little actual meaning to the beginning calculus student: “A function is a set of ordered pairs where, for every value of x, there is only one corresponding value for y.” But another way to think of the function is as a link between cause and effect. The variables x and y, for instance, are wholly interdependent, such that, if a change occurs in one of them (cause, or the independent variable), the other changes in response (effect, or the dependent variable). Calculus describes this rate of change. In economics, price is a function of market supply and demand, rising and falling with the whims of consumer appetites. In physics, potential energy is a function of height: The apple’s potential energy is dependent on how high it is in the tree’s branches, and as the apple falls, that potential energy is converted into kinetic energy.
In the case of Newton’s apple, the position function is the entire collection of points that, taken together, describe the apple’s position at every single instant during its fall. A similar set of points plotted out for the apple’s velocity at any given moment in time comprises the velocity function. But a function is far more than the sum of its parts: It transcends them.
Functions are powerful tools because they confer the power of prediction. You no longer need to perform a new calculation to determine the position or velocity of that apple at each moment in time. With the function, you know the apple’s position or velocity at every possible moment in time.
Historians generally agree that Newton was the first to state the fundamental theorem of calculus and was also the first to apply derivatives and integrals in a single work (although he didn’t use those terms). The problem is that like Fermat, he suffered from publication procrastination. Fermat’s dilly-dallying left the field wide open for Descartes