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By F. P. Agterberg

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This operation is a translation (see Fig. 7A). A further transformation υ = v/c would yield υ = u and all quadratic polynomials can be reduced to this form. The equation v = cu represents a parabola. It is symmetrical with respect to the V· axis; the parabola shown in Fig. 7A has c equal to a negative constant, indicating that the curve is concave upwards. The absolute value of c determines the rate at which the curve falls off. Symmetry The functions x2, x 4 , x 6 , . . are even powers of x\ x,x 3 , x 5 , .

It is calculated in the same manner as the standard deviation, which is a more general concept from the theory of statistics (see Chapter 6). Estimates for the standard error and the standard deviation will both be indicated as s (x). x: is provided by the mean or average : x = (xx + x 2 + . . 23] * = 0/«)Σ *,. /=1 44 2 REVIEW OF CALCULUS Approximate errors in individual observations are obtained by subtracting x from the data which yields the deviations: The standard error then is calculated as : m (Xi-X)2 *« = V f w - i I 2 · 24 ] The square of the standard error is called the variance s2(x) which is approximately equal to the average square deviation from the mean 3c.

It represents the change in slope and is negative around a maximum and positive for a minimum. A function f(x) has an inflection point at JC = 0 if the second derivative f"(a) = 0. At such a point the curve changes from concave upward to concave downward or vice versa. Example The probability curve has f(x) = c e _ a x . Putting u - - ax2 gives / ' ( J C ) = ceudu/âx = - 2acx e~'ax2. Further,/"(JC) = 2ac (2ax2 - 1) e~fl*2. For the ex tremum/'(JC) = 0 orjc = 0. At this point/"(0) is negative which shows that the extremum is a maximum.

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