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Private Equity Operational Due Diligence: Tools to Evaluate Liquidity, Valuation, and Documentation by Jason A. Scharfman

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APPENDIX 9A

Mathematical Concepts

The purpose of this Appendix is to clarify certain mathematical concepts introduced earlier in this chapter. Each of these concepts is outlined in more detail in the following sections.

THE DERIVATIVE

A derivative is defined as the rate of change of one variable with respect to another. For instance, in physics, the derivative, or rate of change, of the position of an object is its velocity. In other words, velocity is the rate at which position varies with time. When one launches a rock vertically into the air, it will stop moving when it reaches the peak of its trajectory (see Exhibit 9A.1). At this point, its vertical speed is zero. In other words, at the instant that it reaches its peak, its position is not changing, and hence the derivative of position is zero at that moment.

Therefore, it becomes apparent that by setting the velocity of an object equal to zero, it is possible to find the time at which the object reaches its maximum vertical position. The following set of equations define this problem:

In the following, x denotes height,  is velocity, and  is acceleration (an overdot indicates a derivative with respect to time). First, let the acceleration, = a and the initial velocity of the projectile be . Then,

Setting the last equation ...

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