The 0.0001 Time Rate Approach, an Extended Application

In the previous post, I introduced to you the 0.0001 Time Rate approach This post is an extended application of the said technique on curvatures and radius of curvatures.

Curvature is defined (by The Facts on File Dictionary of Mathematics) as the rate of change of the slope of the tangent to a curve. For each point on a smooth curve there is a circle that has the same tangent and the same curvature at that point. The radius of this circle called the “radius of curvature”, is the reciprocal of the curvature, and its center is known as the center of curvature. If the graph of a function y=f(x) is a continuous curve, the slope of the tangent at any point is given by the derivative dy/dx and the curvature is given by

C = \displaystyle\frac{\displaystyle\frac{d^2y}{dx^2}}{(1+(\displaystyle\frac{dy}{dx})^2)^{3/2}}

and the radius of curvature is given by

RC = \frac{1}{C} = \displaystyle \frac{(1+(\displaystyle \frac{dy}{dx})^2)^{3/2}}{\displaystyle \frac{d^2y}{dx^2}}

where: \frac{dy}{dx} is the derivative of y with respect to x and \frac{d^2x}{dx^2}  is the second derivative of y with respect to x or in simpler terms, it is the derivative of \frac{dy}{dx}. » Read more