Mean value theorem of differential calculus

Application of the mean value theorem of differential calculus (see Theorem A-2 of Appendix A) to the first two terms of Eq. (13-20) and the mean value theorem of... 

Application of the mean value theorem of differential calculus to the first two terms and the mean value theorem of integral calculus to each of the integrals of Eq. (13-82) followed by the limiting process whereby AZ is allowed to go to zero yields the following result upon recognition that Z and Z + AZ were selected arbitrarily in the domain (0 < Z < ZT) of interest... 

The difference [f(a) — f(Xfc)] of the two column vectors may be replaced by its equivalent as given by the mean value theorem of differential calculus for multivariable functions (Theorem A-7), since the continuity requirements of the theorem are satisfied by suppositions stated above. Then, each element / of [f(Xk) — f(a)] may be stated as follows... 

Theorem A-2-2 Mean value theorem of differential calculus If the function f(x) is continuous in the interval a [Pg.594]

Theorem A-2-7 Mean value theorem of differential calculus for multivariable functions Let/(x, y, z) be continuous and have continuous first partial derivatives in a domain D. Furthermore, let (x0, y0> zo) and (x0 4 h, y0 + k, z0 + /) be points in D such that the line segment joining these points lies in D. Then... 

Applying the mean value theorem of differential calculus for compound fuction [6,7], we obtain the following equation for an intermediate value C in the open interval ( cx/dx, 0)... 

Under certain circumstances, there is a point, in the interval a, h) for which the derivative can be calculated exactly from Eq. (3.3). This is confirmed by the mean-value theorem of differential calculus ... 

Application of the mean value theorems of integral and differential calculus followed by the limiting process wherein AZ and AW are allowed to go to zero yields... 

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