Tangent inflexional

III. Inflexional tangents. We can discuss the equations of a surface by the aid of the extension of Taylor s theorem, on page 292, and the methods described in 101. There are an infinite number of lines... 

These two lines cut the surface at three coincident points. These lines are called inflexional tangents. They are real and distinct, coincident, or imaginary, according as the quadratic in Z, and m,... 

Eaoh inflexional tangent to the surface will cut the ourve in three coincident points at the point of contact. The inflexional tangents have a closer contact with the surface than any of the other tangent lines. At the point of contact of the curve with the tangent plane there will be a cusp, conjugate point, or a node, according as the above expression is positive, zero, or negative. 

Ti is the initial temperature of the decomposition step at the intersection of the tangent drawn at the inflexion point of the mass loss curve with the quasi-horizontal inflexion tangent of the precedent plateau ... 

T2 is the half-way temperature of the decomposition step corresponding to the mid-point of the step on the TG curve which is vertically equidistant between the extended quasi-horizontal inflexion tangents of the precedent and subsequent plateau ... 

Allowing a step input disturbance of magnitude Co, as shown in Fig. 2.23, the constants x and to are derived by drawing a tangent to the step response curve at the point of inflexion. 

Figure 19.3. Chromatogram obtained by elution chromatography of a mixture of two solutes. The retention time // is the time taken by a solute to pass through the column. Im is the mobile-phase holdup and is measured as the retention time of a non-sorbed solute. t R is the adjusted retention time, the total time spent by the solute in the stationary phase it is equal to // — Im- tw is the width of a solute band at the baseline, i.e., the distance between the points of intersection of the baseline with tangents at the points of inflexion on the sides of the...

Tb = the breakthrough time referring to Fig. 26, this is defined as the intercept on the dimensionless time axis of the tangent to the curve at the point of inflexion. 

The controller is placed on manual control (i.e. effectively removing it from the control loop) and the response of the measured variable to a small step change in the manipulated variable is recorded as shown in Fig. 7.58a(21). This response is called the process reaction curve. A tangent is drawn to this curve at the point of inflexion (Fig. 7.586). The intercept of this tangent on the abscissa is termed the apparent dead time (rad) of the system. The gradient of the tangent is given by ... 

The dependence of XE on mole fraction can be analysed to obtain the dependence on mole fraction of the relative partial molar quantities, Xx — X and X2 — X2, which are calculated from the intercepts on the and X2-axes of the tangent to the XE -curve. Consequently the XE -data must be precise if these partial molar values are to be meaningful. For example, one of the features of some aqueous mixtures is a minimum in the value of V2 — V2 at low values of x2. This means that the VE -curve has a point of inflexion at this mole fraction. 

On peut encore tenter d identifier les paramfetres de ces expressions avec ceux obtenus precedemment, en se basant soit sur l approximation zero, soit sur la formule (88b) qui represente l equation de la tangente d inflexion des isothermes de contraction. En simplifiant la formule empirique (67) de ces isothermes, on peut ecrire, en effet, lorsque... 

En portant (114) dans l expression de (113), on obtient la valeur de la tangente d inflexion soit ... 

La tangente d inflexion /( des isothermes de contraction aux basses temperatures (T < T2) correspondant a l expression (A 12), s obtient en derivant deux fois cette formule par rapport a In (f — tt). Afin de... 

Tableau A 3. Valeurs de 10 fTjb correspondant a l amplitude initiate 6i et la tangente d inflexion fl des isothermes de contraction.

The isotherm numbered 3 in the figure represents the transition between isotherms corresponding to the gas phase only, and those including a horizontal portion corresponding to a liquid-gas equilibrium. In this isotherm the horizontal segment has contracted to a single point of inflexion G, This is the critical point of the system. It is characterized by the conditions for the existence of a point of inflexion with a horizontal tangent ... 

Hence in an equilibrium displacement at constant pressure of a binary system, the temperature of coexistence passes through an extreme value maximum, minimum or inflexion with a horizontal tangent) if the composition of the two phases is the same. 

We shall not repeat here any discussion of the properties already established for states of uniform composition. It will be recalled that a state of uniform composition corresponds to an extreme value (maximum, minimum, or inflexion with a horizontal tangent) of the equilibrium pressure at constant temperature, or of the equilibrium temperature at constant pressure (Gibbs-Konovalow theorems, chap. XVIII, 6 and 9). 

In fig. 8.9, we illustrate how one may distinguish between the two types of profile using experimental data by drawing tangents at the inflexion points, one can determine whether the lineshape curve rises above one of the tangents near the line centre (Fano shape) or whether it remains below both tangents near the line centre (sinusoidal shape). 

By reference to Fig. 71 it will be noticed that the tangent crosses the curve at the points B and S. Such a point is called a turning point or point of inflexion. You will get a point of inflexion by plotting y = x3. The point of inflexion marks the spot where the curve passes from a convex to a concave, or from a concave to a convex configuration with regard to one of. the coordinate axes. The terms concave and convex have here their ordinary meaning. 

Show that the tangent crosses a curve at a point of inflexion. Let the equation of the curve be y = f x) of the tangent, Ax + By + C = 0. The necessary condition for a point of inflexion in the ourve y = f(x) is that dPyjdx2 = 0. But for the equation of the tangent, d yfdx2 is also zero. Hence, there is a contact of the second order at the point of inflexion, and the tangent crosses the curve. 

Fig. 3. A sigmoidal dependence of rate upon substrate concentration. As in the case of a hyperbolic dependence, the curve is asymptotic to an upper limit. There is a point of inflexion in the curve, however, and in the middle region of the curve a tangent would cross the abscissa axis well to the right of the origin. This means that over that region a given fractional increase in [A] results in a greater fractional increase in V. Such amplification over a critical concentration range may often be of physiological value.

Determination of plate number is simple from a detector chart recording, since tangents to such a curve at the point of inflexion intercept the baseUne at a distance of 4o apart. Assuming that peak shapes are Gaussian the peak width is therefore given by 4ct. This can... 

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