Infinite Powers: How Calculus Reveals the Secrets of the Universe

by Steven H. Strogatz

why should the universe respect the workings of any kind of logic, let alone the kind of logic that we puny humans can muster? This is what Einstein marveled at when he wrote, “The eternal mystery of the world is its comprehensibility.”

That’s the big idea behind calculus. Everything becomes simpler at infinity.

A limit is a subtle concept but a central one in calculus. It’s elusive because it’s not a common idea in daily life. Perhaps the closest analogy is the Riddle of the Wall. If you walk halfway to the wall, and then you walk half the remaining distance, and then you walk half of that, and on and on, will there ever be a step when you finally get to the wall?

In the limit as the divisor approaches zero, the answer approaches infinity. That’s the real reason why we can’t divide by zero. The faint of heart say the answer is undefined, but the truth is it’s infinite.

the thing that makes e special is that the rate of change of ex is ex. Hence, as the graph of this exponential function soars higher and higher, its slope always tilts to match its current height. The higher it gets, the steeper it climbs. In the jargon of calculus, ex is its own derivative.

when a variable follows a perfect sine-wave pattern, its rate of change is also a perfect sine wave timed a quarter of a cycle ahead.