Tangent to a curve

Asymptotes The hmiting position of the tangent to a curve as the point of contact tends to an infinite distance from the origin is called an asymptote. If the equation of a given curve can be expanded in a Laurent power series such that... 

A tangent is a straight line that meets a curve at a point, but not does cross it. If the heat capacity changes slightly with temperature, then we obtain the value of Cv as the gradient of the tangent to a curve of AU (as y) against T (as x). 

If only about the first 1% of a slow reaction is monitored, a plot of the concentration versus time plot is virtually linear, so the initial rate can be measured with little of the uncertainty associated with fitting a tangent to a curve. 

Arrhenius, 1889). It should be remembered that E and A are experimentally determined from the slope and intercept of the tangent to a curve (Christiansen, 1949 Benson, 1960b). In Table 7, data are given for hydrogen abstraction reactions some results obtained by techniques other than polymerization are also included. Some of the data of Table 7 (Nos. 2-9) satisfy fairly well the correlation... 

When t]i tends toward rj2, these two equations tends toward limits that give the coordinates of the intersection point of two characteristic lines that are infinitely close to each other, hence that are tangent to a curve that is defined as the envelope of these characteristic lines. These limits are... 

We sometimes define a tangent to a curve as a straight line which touches the curve at two coincident points.1 If, in Fig. 24, P and Q are two points on a curve such that MP = NB = y BQ = dy OM = x MN = PB = dx the straight line PQ = ds. Otherwise, the diagram explains itself. Now let the line APQ revolve about the point P. We have already shown, on page 15, that the chord PQ becomes more and more nearly equal to the arc PQ as Q approaches P when Q... 

Fig. 2,6. A plane tangent to a curved surface, illustrating the meaning of partial derivatives and total differentials.

Calculating the slope of the tangent to a curve at a particular point is undoubtedly easier using computer software. However, this should not be taken to imply that the result is free from uncertainty. 

The values of Api and A/ij. however, are given as ordinate intercepts of the tangents to the AG " = /(< 2) curve (see Section 6.3.5). The required equivalence of the chemical potentials in both phases can only be fulfilled if two points of the curve possess a common tangent (see Figure 6-9). But there is only one common tangent to a curve with two minima. The points of contaa A and B of this tangent to the AG" = curve determine the compositions (p and (p" of the two phases. 

When an enzyme is added to a reaction mixture, there is first a short lag period before a steady rate of reaction is attained. Once established, the reaction rate remains constant for a finite period of time. This can be as long as several hours, but in other cases it may be as short as only a few minutes. The rate of reaction begins to fall after this period because of factors like reduced substrate concentration or accumulation of end-products. The decrease in reaction rate is not easily analysed mathematically and enzyme studies are thus usually confined to studies of the constant reaction rate. Because the lag phase is extremely short, this reaction rate is computed as the initial reaction rate. In practice, it can be evaluated as the tangent to a curve relating substrate transformed to time. 

Graphs can also be used in a variety of measurements, for example, finding tangents to a curve, finding the intercept of a line or curve or determining the gradient of a line. Drawn graphs can also be used for interpolation and extrapolation. 

Asymptotic Refers to not meeting. An example is a straight line that is a limiting position of a tangent to a curve as its point of contact recedes indefinitely along an infinite branch of the curve. 

Values of a or may be determined in different ways. Since proton transfer in the transition state is expected to vary with the strength of the Bronsted acid or base, a and p should in general represent tangents to a curved log /c-log K plot. For small variations in pK systematic deviations from linearity are usually too small to be observed and only an average value of a or can be obtained, but for larger changes the curvature may be sufficient to evaluate a quadratic term in the dependence (equation 33) ... 

It is, however, always somewhat subjective when placing a tangent to a curve. An alternative method is to use the following procedure. Firstly, when posing a drop on the flat, horizontal surface, ensure fi-om the top that its contact area is circular. (This maybe done with a transparent slide, onto which are printed circles of various diameters. By adjustment of drop-to-transparent distance, it is possible to find a position at which a circle should coincide with drop circumference.) Any deviation of the triple line fi om a circle is a clear proof that the solid surface is either rough or inhomogeneous. You will need the surface tension, yLv> liquid [anyway, to exploit... 

The Newton-Raphson method is based on the premise that a line drawn tangent to a curve described by y =/(x) at the point Xi will intersect the jc-axis at a point jca having a value closer to the root than Xj. This is illustrated in Fig. 11-4. The slope of a line tangent to a curve at a point Xi is the derivative y (x,) = dy/dx. From the diagram we see that... 

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