Experience Mathematics # 24 -- The Calculus

Happy New Year. The earth has finished another revolution around the sun, taking a little more than 365 days to do so. Meanwhile, the moon continues to rotate around the earth, the planets around the sun, and the same forces that make these things move in an elliptical path ensure that a ball thrown up in the air always falls down, or that a ball thrown in the air (towards a friend) takes a parabolic path.

Over this and the next few columns, I will discuss these natural motivations that are behind the notions that you encounter as you study the Calculus.

The first concept is that of a function. Mathematicians were already familiar with curves from Euclidean and coordinate geometry by 1600 or so A.D. It was natural to begin modelling various physical phenomena with functions. For example, $y=1-x^2$ models the parabola. For each value of the input $x$, we get a unique output $y$. If you plot the curve in the coordinate plane, you obtain a parabola.

It was natural to do two things. To figure out laws that can explain why a ball thrown in the air follows a path traced by such a curve. This led to the laws of Gravitation. And the other thing is to use these laws to predict the answers to common questions that arise. For example: How high will the ball go? How far will the ball go? Given the curve, when does the curve go up (increase)? And when does it come down (decrease)? We will consider such questions and relate them to what you encounter in Calculus.

Curves such as the circle ($x^2+y^2=1$) are not functions since there is not one output $y$ for each input $x$. For example, for $x=0, y$ can be $1$ or $–1$. So, every curve does not give rise to a function.