The Mean Value Theorem

Using the mean value theorem for definite integrals... 

The function QG occurring in the integrand is represented by two straight line segments below the axis. Xf the mean value theorem is applied to the separate integrals, the result is... 

Then, applying the mean value theorem, the surface integral around the observation point p can be represented as... 

Note that since the mean value theorem involves an integration over all allowed values of the independent variable, the integration relevant to average speed is from zero to infinity. In terms of the mean value theorem stated in Eq. (25), the probability of a negative speed is zero, so the integral in Eq. (37) is only over positive values of v. The integral in Eq. (37) may be evaluated by making the... 

Also, if j x) is any function which is intcgrable in the interval (— a, ) then, hv using the mean value theorem of the integrul calculus, we see that... 

Provided that level shifts resulting from the configuration interaction are neglected we can apply the mean value theorem and factor out of the integrand of eq. (2-23) the average value of the matrix element... 

An examination of Eq. (230), which is similar in form to Eq. (81), shows that, for melting, d20i/dX2 > 0. From the mean value theorem it follows immediately that an upper bound for the temperature in melting is given by... 

For a moment, let us suppose a special situation where a branched molecule is Gaussian so that (p jO j-l- Then it is obvious that the nth derivative, Z(n), diverges when n>l and d<2. Thus, the expansion works only if the series is truncated to the first two terms together with d>3. The manipulation is equivalent to applying the mean value theorem ... 

The time, t, to reach the STEL of 0.02 ppm is obtained by using the mean value theorem. 

In the multiphase flow literature these integral terms are generally approximated as the product of the interfacial area concentration and a mean interfacial flux using the mean value theorem for integrals. 

Using the mean value theorem the interfacial mass transfer rate due to phase change (3.144) becomes [61, 72, 12] ... 

In an attempt to find an exact formula for the integral, we may resort to the mean value theorem of calculus. This theorem states that if the integrand is evaluated at a particular known instant t = t between tn and tn+i, the integral is equal to f(T,tp(Ty)At. However, in the present case the theorem is of little use since the instant r is unknown. 

The above expressions contain the continuous bubble number probability density, f d,t). The source terms must be expressed entirely in terms of the dependent variable IVj. This can be achieved by using the mean value theorem [151]. Note, as mentioned before, at this point the discrete method of Ramkrishna [151] may deviate slightly from the multi-group method. 

The mean value theorem is used to cast the equations (12.267) to (12.270) entirely in terms of Ni and Nj. In each interval the variables are replaced by a mean value. 

For example, using the discrete method the mean value theorem can be used to express that... 

The RHS of equation (12.272) is zero for d Xi and equals Ni when d = Xj. Applying the mean value theorem on the source terms yields ... 

The restoration of autonomy relies on our ability to write the integrals in Eq. (7.11) as functions of Ni. This can be done by using the mean-value theorem. For example, the second integral corresponds to the following equality ... 

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