Graphical Gradients and Intercepts

1 Staight Line (Linear) Graphs Gradients and Derivatives 

Any equation of first degree (i.e. involving only x (andhaving no x2, x3 etc. terms)) can be represented by a straight line graph. The general equation takes the form  

The line has a gradient or slope equal to m and an intercept (i.e. meets the y-axis) at a distance c from the origin (0,0). It should be noted that if a graph is plotted which does not actually include the origin then the value of c is no longer equal to the intercept and adjustment must then be made in its calculation to take account of the shift in origin necessary to include the point (0,0). 

The gradient of the line, m, can also be equated to the differential coefficient (written as dy/dx)) and can be said in turn to be the result of differentiating y with respect to x. The intercept on the y-axis (the ordinate) when x = 0 is c, another constant (subject to the caveat above), m can also be determined from the coordinates, (xi, yi) and (xj, yi) of two arbitrary points (best separated as far apart as is possible to improve the accuracy of this calculation of gradient) on the straight line. 

We can also define a derivative of a derivative (or second derivative) written as d2y/dx2 and equalling the differential of dy/dx with respect to x (= d/dx dy/dx ). This represents the change in the gradient (= dy/dx) as x is increased. 

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Graphical Gradients and Intercepts Differentiation in Thermodynamics Equation of State for an Ideal Gas... 

C, is the solution of the system of Equations (65) and (66). Graphically this means that C, is the intercept of the kinetics curve with the straight line of gradient (F/F,) and the A -intercept is the entrance concentration of the / -th CSTR, i.e. the exit concentration of the (i — l)th CSTR. The estimation of the concentrations in the CSTRs one after the other is shown in Figure 5. 

Hofstee plot A graphical method used in enzyme kinetics to obtain a straight line fiom experimental data. It involves forming aplotofV/SversusVinwhichSis the substrate concentration at which the velocity v is observed. The gradient of the line is equal to -K and the intercept on the y-axis is equal to the maximum velocity V. Also known as the Eadie-Hofstee plot, it is named after Canadian biochemist George Sharp Eadie (1895-1976) and B. H. J. Hofstee who developed the plot in 1942 and 1959, respectively. See michaelis-MENTEN KINETICS. 

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