Point-Slope Methods

The Newton-Raphson approach, being essentially a point-slope method, converges most rapidly for near linear objective functions. Thus it is helpful to note that tends to vary as 1/P and as exp(l/T). For bubble-point-temperature calculation, we can define an objective function... 

Point-Slope Methods. Euler s method follows directly from the initial condition as a starting point and the differential equation as the slope (Fig. 3). Consider the simple model of a single differential Eq. (13) with one first-order rate process ... 

Step 3. Use the point-slope method to predict temperature and conversion at a small distance downstream in the primary flow direction. 

The Runge-Kutta method takes the weighted average of the slope at the left end point of the interval and at some intermediate point. This method can be extended to a fourth-order procedure with error 0 (Ax) and is given by... 

In order to compress the measured data through a wavelet-based technique, it is necessary to perform a series of convolutions on the data Becau.se of the finite size of the convolution filters, the data may be decomposed only after enough data has been collected so as to allow convolution and decomposition on a wavelet basis. Therefore, point-bypoint data compression as done by the boxcar or backward slope methods is not possible using wavelets. Usually, a window of data of length 2" m e Z, is collected before decomposition and selection of the appropriate... 

By differentiating the titration curve twice and then equating the second derivative to zero, it can be shown that for a symmetrical titration curve ( i = the point of maximum slope theoretically coincides with the equivalence point. This conclusion is the basis for potentiometric end-point detection methods. On the other hand, if 2> the titration curve is asymmetrical in the vicinity of the equivalence point, and there is a small titration error if the end point is taken as the inflection point In practice the error from this source is usually insignificant compared with such errors as inexact stoichiometry, slowness of titration reaction, and slowness of attainment of electrode equilibria. 

In clinical laboratories, the vapor pressure osmolality technique has been reported to be less precise than the freezing point depression method. For serum samples, the coefficients of variation obtained for the vapor pressure osmometer are about twice those obtained for the freezing point depression osmometer. The lesser degree of precision is related to the lower slope of dew point decrease compared with freezing point decrease (i.e., 0.303 °C versus 1.86 °C per osmol/kg H2O). 

Thus, the slope of a ln(C) against l/T plot equals —E /2k. The data has minimal uncertainty so the slope can be calculated by the two-point difference method. Alternatively, a linear regression lit of ( /T, ln(G/S)) data points gives the slope. 

To get detailed data on soil properties, exploratory wells dug in the study area, made new loess, old loess and silty clay samples each 15 groups. Then, they were tested on the GDS advanced dynamic triaxial test system in State Key Laboratory of Continental Dynamics, Northwest University. Show in Figure 5. Finally, we got series of the effective cohesion and effective friction angle. Because of the variation of c which made the greater impact of slope stability than other parameters, we obtain the mean and standard deviation to used in Rosenblueth point estimate method. Table 1 summarizes the values of soil properties obtained from laboratory tests in the study area. 

The use of the corresponding underlying experimental data for model-based analysis frequently involves the computation of first-order derivatives of the actually measured data at multiple timepoints most popular are growth rates (derived from cell counts) and production/consumption rates (derived from intra- or extracellular metabolite concentrations). Trivially, experimental data is prone to noise and mostly sampled at nonequidistant timepoints. From these data, the corresponding rates at a certain timepoint are classically derived by a simple two-point slope calculation. Rates computed this way can be almost or completely unusable (see example in Figure 4.3). A suitable and well-defined method is the application of a band-pass filter to the measured data before the calculation of the derivative. 

Richmond developed a simple method for determining the HLB of the pseudo surfactant phase and the slope of the SAD = 0 line that required only a minimum of inversion points. This method is shown below. 

The scatter of the points around the calibration line or random errors are of importance since the best-fit line will be used to estimate the concentration of test samples by interpolation. The method used to calculate the random errors in the values for the slope and intercept is now considered. We must first calculate the standard deviation Sy/x, which is given by ... 

The most obvious sensor for an acid-base titration is a pH electrode.For example, Table 9.5 lists values for the pH and volume of titrant obtained during the titration of a weak acid with NaOH. The resulting titration curve, which is called a potentiometric titration curve, is shown in Figure 9.13a. The simplest method for finding the end point is to visually locate the inflection point of the titration curve. This is also the least accurate method, particularly if the titration curve s slope at the equivalence point is small. 

Another method for finding the end point is to plot the first or second derivative of the titration curve. The slope of a titration curve reaches its maximum value at the inflection point. The first derivative of a titration curve, therefore, shows a separate peak for each end point. The first derivative is approximated as ApH/AV, where ApH is the change in pH between successive additions of titrant. For example, the initial point in the first derivative titration curve for the data in Table 9.5 is... 

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