The aim of the paper is to present some mean value theorems obtained as consequences of the intermediate value property. First, we will prove that any nonextremum value of a Darboux function can be represented as an arithmetic, geometric or harmonic mean of some different values of this function. Then, we will present some extensions of the Cauchy or Lagrange Theorem in classical or integral form. Also, we include similar results involving divided differences. The paper was motivated by some problems published in mathematical journals.
The most frequently used instrument for measuring velocity distribution in the cross-section of small rivers is the propeller-type current meter. Output of measuring using this instrument is point data of a tiny bulk. Spatial interpolation of measured data should produce a dense velocity profile, which is not available from the measuring itself. This paper describes the preparation of interpolation models. Measuring campaign was realized to obtain operational data. It took place on real streams with different velocity distributions. Seven data sets were obtained from four cross-sections varying in the number of measuring points, 24-82. Following methods of interpolation of the data were used in the same context: methods of geometric interpolation arithmetic mean and inverse distance weighted, the method of fitting the trend to the data thin-plate spline and the geostatistical method of ordinary kriging. Calibration of interpolation models carried out in the computational program Scilab is presented. The models were tested with error criteria by cross-validation. Ordinary kriging was proposed to be the most suitable interpolation method, giving the lowest values of used error criteria among the rest of the interpolation methods.