Infinite Powers: How Calculus Reveals the Secrets of the Universe

by Steven H. Strogatz

we zoom in closer and closer, that piece of the parabola should begin to look straighter and straighter. In the limit of infinite magnification (which amounts to zooming in on an infinitesimal piece of the curve around the point of interest), that magnified piece should approach a straight line. If it does, that limiting straight line is defined as the tangent line at that point on the curve, and its slope is defined as the derivative there. Notice that we are using the Infinity Principle here—we are trying to make a complicated curve simpler by chopping it into infinitesimal straight pieces. This is what we always do in calculus. Curved shapes are hard; straight shapes are easy, even if there are infinitely many of them and even if they are infinitesimally small. Calculating a derivative in this way is a quintessential calculus move and one of the most fundamental applications of the Infinity Principle.