A standard part of the calculus curriculum is learning exponential growth models. This paper, designed to serve as a teaching aid, extends the standard modeling by showing that simple exponential models, relying on two points to fit parameters do not do a good job in modeling population data of the distant past. Moreover, they provide a constant doubling time. Therefore, the student is introduced to hyperbolic modeling, and it is demonstrated that with only two population data points, an amazing amount of information can be obtained, such as reasonably accurate doubling times that are a function of t, as well as accurate estimates of such entertaining topics as the total number of people that have ever lived on earth.
Sine-generated curves belong to a class of intrinsic functions which describe a curve by specifying its "direction angle". The curve is determined by ω, the maximum angle which the curve makes with the horizontal, and the fact that the direction angle changes in a sinusoidal fashion along the path. Sine-generated curves are shown to be excellent approximations to the path of minimal average curvature, and expressions for radius of curvature and curve sinuosity are derived.
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