Equations of a hyperbola

Symmetry 50. Intercepts 50. Asymptotes 50. Equations of Slope 51. Tangents 51. Equations of a Straight Line 52. Equations of a Circle 53. Equations of a Parabola 53. Equations of an Ellipse of Eccentricity e 54. Equations of a Hyperbola 55. Equations of Three-Dimensional Coordinate Systems 56. Equations of a Plane 56. Equations of a Line 57. Equations of Angles 57. Equation of a Sphere 57. Equation of an Ellipsoid 57. Equations of Hyperboloids and Paraboloids 58. Equation of an Elliptic Cone 59. Equation of an Elliptic Cylinder 59. 

GRAPHICAL REPRESENTATION. The above expression represents the equation of a hyperbola (i.e., f(x) = axl(b + x) where a and b are constants) and a plot of the initial velocity as a function of [S] will result in a rectangular hyperbola. Another way for representing the Michaelis-Menten equation is by using the doublereciprocal (or, Lineweaver-Burk ) transformation ... 

The name of the catastrophe derives from the last equation in (2.46), which is the equation of a hyperbola. 

Note that for any given value of stress level cr the hrst member of Eq. (6.31) is a constant, also known as Neuber s constant, so that (6.31) is the equation of a hyperbola. The intersection of the Neuber s hyperbola with the stress-strain curve of the material yields the solution of the elastic-plastic problem. This is schematically shown in Eig. 6.21 where the coordinates of point A = (s.cr) are the notch tip strain and stress, respectively. Moving from monotonic loading to fatigue cycles Neuber s rule applies substituting the theoretical stress concentration factor ki with the notch factor fey (see Sect. 7.1), as suggested by Wetzel [31] in 1968 and Topper et al. [32] in 1969, obtaining what is known as the Neuber modihed equation... 

Mathematically, the Michaelis-Menten equation is the equation of a rectangular hyperbola. Sometimes you ll here reference to hyperbolic kinetics, this means it follows the Michaelis-Menten equation. A number of other names also imply that a particular enzyme obeys the Michaelis-Menten equation Michaelis-Menten behavior, saturation kinetics, and hyperbolic kinetics. 

A representation of this equation is in Figure 2.8, which shows the effect of h with changes in linear gas velocity. Equation 2.81 is that of a hyperbola having a minimum at velocity u =... 

The validity of Boyle s law can be demonstrated by making a simple series of pressure-volume measurements on a gas sample (Table 9.2) and plotting them as in Figure 9.6. When V is plotted versus P, as in Figure 9.6a, the result is a curve in the form of a hyperbola. When V is plotted versus 1/P, as in Figure 9.6b, the result is a straight line. Such graphical behavior is characteristic of mathematical equations of the form y = mx + b. In this case, y = V,m = the slope of the line (the constant k in the present instance), x = 1 /P, and b = the y-intercept (a constant 0 in the present instance). (See Appendix A.3 for a review of linear equations.)... 

C] /. While (2) is the equation for a hyperbola, it can be linearized by plotting / versus T and C = (slope)" of such a plot. The effective moment (or Curie constant) is thus obtained from analysis of data in the high-temperature limit. 

Obviously, this is mainly a mathematical description of a hyperbola, the first member of the equation being a constant, the second member a straight line and the third a hyperbola. Based on the examinations carried out, the formula was generalised as follows ... 

The numerical calculations of Fig. 12 show the predicted superpositions of states at Ap = 0 and As = 0. They also show that a final superposition between 1) and 3) is possible on some pieces of the hyperbolas (dashed lines). However, they are not robust since the equation of these hyperbolas (297) depend on the peak field amplitudes. 

Examples.—(1) Show that the equation of the hyperbola whose origin is at its vertex is dhj2 = 2afe2 + fc2 2. Substitute x + a for x in the regular -j equation. Note that y does not change. 

The equation of a rectangular hyperbola referred to its asymptotes as coordinate axes, is best obtained by passing from one set of coordinates to another inclined at an angle of - 45° to the old set, but having the same origin, as indicated on page 96. In this way it is found that the equation of the rectangular hyperbola is... 

As this is the equation for a hyperbola, the shape of the dose-response curve is explained if the response is directly proportional to A/f], Unfortunately, this simple theory does not explain another experimental finding—some agonists, called partial agoiiisls. cannot elicit the same maximum re.sponse as full agonists even if they have the same... 

In this case, v is the velocity of the reaction, [S] is the substrate concentration, Vmax (also known as V or Vj ) is the maximum velocity of the reaction, and is the Michaelis constant. From this equation quantitative descriptions of enzyme-catalyzed reactions, in terms of rate and concentration, can be made. As can be surmised by the form of the equation, data that is described by the Michaelis-Menten equation takes the shape of a hyperbola when plotted in two-dimensional fashion with velocity as the y-axis and substrate concentration as the x-axis (Fig. 4.1). Use of the Michaelis-Menten equation is based on the assumption that the enzyme reaction is operating under both steady state and rapid equilibrium conditions (i.e., that the concentration of all of the enzyme-substrate intermediates (see Scheme 4.1) become constant soon after initiation of the reaction). The assumption is also made that the active site of the enzyme contains only one binding site at which catalysis occurs and that only one substrate molecule at a time is interacting with the binding site. As will be discussed below, this latter assumption is not always valid when considering the kinetics of drug metabolizing enzymes. 

Figure 5.7 of the phase diagram T versus rpi) shows the hourglass behavior. The phase diagram can take the form of a hyperbola when Equation (5.21), the criteria for the equilibrium state, assumes the form... 

The phase diagram can assume the form of a hyperbola when Equation (5.21) assumes the form... 

Equation (2.2-5a) is the mathematical expression of a hyperbola. An alternate way to write (2.2-5a) is in terms of the fractional coverage ... 

In Figure 1.7 the schematic diagram plots V = (V/V-critical) and P = (F/F-critical) to yield an equation independent of the (a, b) parameters as we wUl soon show. The main point is that the isotherms at higher temperature appear as one branch of a hyperbola as expected but at the point marked as the critical point the curve has an inflection point and then at stUl lower temperatures the curve shows its behavior as a cubic curve in V. Careful measurements within the conditions at temperatures lower than the critical point actually confirm the presence of liquid droplets. This leads to a definition of the critical temperature as that temperature above which a gas cannot be compressed into a liquid phase at any pressure. At temperatures lower than the critical temperature of a material the gas can be compressed (squeezed) into the liquid form by applying higher pressure. 

Tangent to a point P of a hyperbola. The general equation of a hyper-... 

In this form, Alhas the units of torr.) The relationship defined by Equation (A15.4) plots as a hyperbola. That is, the MbOg saturation curve resembles an enzyme substrate saturation curve. For myoglobin, a partial pressure of 1 torr for jbOg is sufficient for half-saturation (Figure A15.1). We can define as the partial pressure of Og at which 50% of the myoglobin molecules have a molecule of Og bound (that is, F= 0.5), then... 

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