Section Summary

In this section, we used sine and cosine functions to model periodic behavior in nature, in particular, phenomena with yearly periods: minutes of daylight and monthly average temperatures.

We also set up and solved an initial value problem that represents the repetitive motion of a simple spring-mass system. We studied this problem numerically and graphically before arriving at a symbolic solution. In the course of that study, we saw that we could apply Euler's Method by turning our second-order differential equation into two first-order equations, one for position and one for velocity. The graphical solutions for the two functions resembled graphs of cosine and sine functions, respectively.

We used this information to solve the problem symbolically and found that the position function is indeed a constant multiple of a cosine function, so the velocity function is a constant multiple of a sine function.