Calculus Without Tears - Vol. 3 - Nature's Favorite Functions

Polynomials are engineers' favorite functions because they are easy to calculate. Nature, however, has other ideas. In Vol. 3 we begin our study of nature's favorite functions, exponentials and trig functions. One problem is that they can't be calculated directly! Fortunately, they can be calculated using polynomials and Taylor's theorem. There is a very easy and direct demonstration of Taylor's theorem that you won't find in any current calculus text. It's probably the one Taylor came up with. The next hurdle is a demonstration of the Fundamental Theorem of Calculus. This turns out to be an easy generalization of the formula distance = velocity * time, and we went through all the steps of proving it in Vol. 2 Chapter 2 when we calculated numerical solutions to differential equations. The demonstration is intuitive and short, and I think better than that in any other calculus book. Why are exponentials and trig functions nature's favorites? Because they occur as the solutions to differential equations describing unforced physical systems. We calculate their derivatives, using easy tricks (as almost nothing is entirely straightforward once you're beyond polynomials), and finally arrive at the point where we can write and solve differential equations that engineers and physicists use to describe real systems. We give examples from mechanics and electrical circuits, analyzing the systems, writing the differential equations, and calculating and graphing their solutions.