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Old 07-24-2007   #7
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The concept is relatively easy. If you place a straight line on the circumference of a circle, it must be a tangent. It is not so easy if the curve is not regular but rather arbitrary. You could place the line at some point on the curve and estimate its position to be exactly tangent. You could then draw a right triangle with three points (one on the straight line and two on the curve). As the triangle is made smaller and smaller, it will force the line to be more exactly a tangent to the curve. As the triangle becomes vanishingly small (by mathematics) the limit of the lines position is in fact a tangent to the curve. Often, the tangent to a curve can represent an instantaneous slope (or ratio) that is useful (to scientists or engineers). For example the slope may represent distance traveled relative to time which of course is velocity. Differentiating such a curve at any given point on the curve could represent the momentary velocity of a racket on a nonlinear path. Differentiating a new curve representing all the velocities (change of velocity with time) represents acceleration (and therefore momentary force due to g's, etc.). The formulas are in books, but it is good to know what it is all about first.

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