Functions of a single variable

When the value of a quantity depends on that of a second quantity, it is said to be a function of the second quantity or variable. For example, the mass of a standard piece of A4 paper will be a function of its thickness. 

In mathematical notation, we would describe a function / of a single variable x as /(x). The function can then be defined by setting /(x) equal to some expression in x. An example would be 

In this case the function is obtained by taking the variable x and adding 3 to it. Clearly the function in this case will have a different value for every value of x. To obtain the value of a function for a given value of x, we simply replace x by that variable in the above expression. Thus if x = 2, we substitute directly to give 

Functions can be more complicated than this. A second example is 

Also note that /(O) isn t defined since this would involve the calculation of 1/0. 

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For functions of a single variable (e.g., energy, momentum or time) the projector Prz)(x) is simply 0(a ), the Heaviside step function, or a combination thereof. When also replacing x, k by the variables , t, the Fourier transform in Eq. (5) is given by... 

From basic calculus, it is known that a function of a single variable is analytic at a given interval if and only if it has well-defined derivatives, to any order, at any point in that interval. In the same way, a function of several variables is analytic in a region if at any point in this region, in addition to having well-defined derivatives for all variables to any order, the result of the differentiation with respect to any two different variables does not depend on the order of the differentiation. 

Figure B-3 Maxima, Minima, and Inflection Points of a Function of a Single Variable...

A good place to start our study- is with a function of a single variable f(x). Consider the function... 

The golden section search is the optimization analog of a binary search. It is used for functions of a single variable, F a). It is faster than a random search, but the difference in computing time will be trivial unless the objective function is extremely hard to evaluate. 

In addition to the programs to select the optimum discussed previously, graphic approaches are also available and graphic output is provided by a plotter from computer tapes. The output includes plots of a given response as a function of a single variable (Fig. 11) or as a function of all five variables (Fig. 12). The abscissa for both types is produced in experimental units, rather than physical units, so that it extends from—1.547 to + 1.547 (see Table 5). Use of the experimental units allows the superpositioning of the single plots (see Fig. 11) to obtain the composite plots (see Fig. 12). 

Graphical presentation of data assists in determining the form of the function of a single variable (or two variables). The response y versus the independent variable x can be plotted and the resulting form of the model evaluated visually. Figure 2.4 shows experimental heat transfer data plotted on log-log coordinates. The plot... 

Functions of a single variable and their corresponding trajectories. (Continues)... 

We define the property of continuity as follows. A function of a single variable x is continuous at a point x0 if... 

Figure 4.16 illustrates the character of ffx) if the objective function is a function of a single variable. Usually we are concerned with finding the minimum or maximum of a multivariable function fix)- The problem can be interpreted geometrically as finding the point in an -dimension space at which the function has an extremum. Examine Figure 4.17 in which the contours of a function of two variables are displayed. 

If both first and second derivatives vanish at the stationary point, then further analysis is required to evaluate the nature of the function. For functions of a single variable, take successively higher derivatives and evaluate them at the stationary point. Continue this procedure until one of the higher derivatives is not zero (the nth one) hence,/ (jc ),/"(jc ),. . ., /(w-1)(jc ) all vanish. Two cases must be analyzed ... 

As an example consider the following function of a single variable x (see Figure 5.1). 

One method of optimization for a function of a single variable is to set up as fine a grid as you wish for the values of x and calculate the function value for every point on the grid. An approximation to the optimum is the best value of /(x). Although this is not a very efficient method for finding the optimum, it can yield acceptable results. On the other hand, if we were to utilize this approach in optimizing a multivariable function of more than, say, five variables, the computer time is quite likely to become prohibitive, and the accuracy is usually not satisfactory. 

In optimization of a function of a single variable, we recognize (as for general multivariable problems) that there is no substitute for a good first guess for the starting point in the search. Insight into the problem as well as previous experience... 

In minimizing a function /(x) of several variables, the general procedure is to (a) calculate a search direction and (b) reduce the value of/(x) by taking one or more steps in that search direction. Chapter 6 describes in detail how to select search directions. Here we explain how to take steps in the search direction as a function of a single variable, the step length a. The process of choosing a is called a unidimensional search or line search. 

A set of nonlinear equations can be solved by combining a Taylor series linearization with the linear equation-solving approach discussed above. For solving a single nonlinear equation, h(x) = 0, Newton s method applied to a function of a single variable is the well-known iterative procedure... 

Even this diagram does not give a clear impression of the relative proportions of the various copper compounds present in solution. However, provided no polynuclear species are present, it is a relatively simple matter to use the values of x to evaluate these proportions and to plot them as a function of a single variable. Figure 6 shows a diagram of this kind using the same data as figure 5 calculated for pE = 10 and variable chloride activity under the assumption that all compounds have the same activity coefficient. It would not be difficult to allow for different values of activity coefficients if these were known. 

We are approximating the real function by a linear function. The process is sketched graphically in Fig. 6.2 for a function of a single variable. The method is best illustrated in some common examples. 

The functions Y 6,) are tabulated and can be represented as in Fig. 20.13. A series of spherical harmonics can be used to represent an arbitrary perturbation of a sphere, much the same as a Fourier series can represent an arbitrary function of a single variable. 

Functions of a single variable, involving a relation between two sets of numbers, may be expressed in terms of a table (expressing an association), formula, prescription or graphical plot. For functions of two independent variables (see below), the preferred representations are formula, prescription or graphical plot for three or more variables, a formula or prescription is the only realistic representation. 

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