are given in the following tables. Compare the graphs with the following graph. Temperature, pulse, and respiration tables in croupous pneumonia. Day of fever. 1 T. (Measles) A.M. T. (Scarlet fever) P.M. A.M. P.M. { FIG. 22. 4 6 7 2 3 5 8 99°.4 101°.2 101°.2 101°.8 103°.4 102°.8 99° 98°.4 103°.8 101°.8 102°.8 104°.8 105°.8 103°.4 99°.8 normal 98°.4 105°.2 103°.8 104°.8 104° 103°.6 102°.6 101°.8 100°.8 99°.98 103°.4 105°.8 105°.2 104°.4 103°.4 103°.2 101°.4 99°.4 Discuss these functions. How do the temperature curves distinguish pneumonia, measles, and scarlet fever, one from another? 13. Average Rate of Change of a Function. The average rate of change of temperature during a given period of time is a familiar idea. For instance, in Example 2 of the preceding section, the temperature rose from 14° to 45° between 6 A.M. and 3 P.M., a total rise of 59° in 9 hours. Dividing 59 by 9, we see that the average rate of change in this interval of time was about 6.5 degrees per hour. On the graph, CK represents the change in time from 6 A.M. to 3 P.M., and KE the corresponding change in temperature. Hence the average rate of change of temperature in this interval, 6.5 degrees per hour, is represented by the ratio KE/CK. The graph has its most abrupt rise from about the point G to the point H, which indicates that the average rate of change of temperature was greatest from about 8 A.M. to 10 A.M. This average rate of change is represented by the ratio JH/GJ. X 12 Mid. | Ay Ay/Ax 10 3 -10 -3.3 3 A.M. 0 6 A.M. -14 9 +59 +6.5 3 P.M. +45 8 A.M. -6 2 +25 +12.5 Let y denote the temperature Ax at any time x, and Ay the change in temperature during an interval of time Ax. Since the average rate of change is the ratio of the change of temperature to the corresponding change in time it is expressed by Ay/Ax. The computation of the average rate of change for several different intervals may be effected conveniently in tabular form, the values of y for the given values of x being obtained from the graph. The idea of the average rate of change of a function of the time has been extended to include a function of any variable. This generalization is given in the DEFINITION. The average rate of change of a function of x, for a particular change in x, is the ratio of the corresponding change in the function to the change in x. If y denotes the function, and Ay the change in y due to a change of Ax in x, then the average rate of change of y with respect to x is symbolized by Ay/Ax. If x increases from one given value to another, the average rate of change of y may be found as in the illustration above, except that the values of y would be computed from the given function instead of being read from the graph. The method of finding a general expression for the average rate of change for any interval is illustrated in the EXAMPLE. Find the average rate of change of the function y = 1x2 for the intervals of x from 1 to 3, from 2 to 4, and from 2 to 6. Find also the average rate of change for any interval. The details of the computation for the three given intervals are given in the table on page 36. To find the average rate of change of the function for any interval, we start with any pair of corresponding values x and y, and let Ay be the change in y produced by a change of Ax in x. Then x + Ax and y + Ay are cor responding values of the independent variable and the function, and hence these values satisfy the given equation. The average rate for any particular interval may be obtained from this result by substitution. For example: For the first given interval, from 1 to 3, x starts with the value x 1, and increases by Ax = 2. Substituting these values in (1), we get For the interval from 2 to 4, x = 2, and Ax 2. Hence And for the interval from 2 to 6, x = 2, and Ax = 4, so that As these results agree with those obtained above, we have a desirable check on the correctness of the entire procedure. The graphical representation of the average rate of change is important. The corresponding pairs of values (x, y) and (x + Ax, y + ▲y) are represented by two points P and Q on the graph, so that x = OM, y = MP, x + ▲x = ON, y + Ay Hence Ax MN PR, and Ay = RQ. the ratio RQ/PR. That is, the -2 This interpretation holds for any two points P and Q on the curve. For no matter what the relative positions of the points may be, we have, by the definition on page 13, OM + MN = ON and 8210 1 y 1 2 3 4 5 6 NR + RQ = NQ. EXERCISES 1. Find the average rate of change of each of the functions below if x changes from 0 to 1; from 1 to 3; from 1 to 4. What can be said of these average rates? Find the average rate for any interval, draw the graph, and interpret the average rate graphically (a) 2x - 3. (d) x + 2. (b) 2x. (c) x +3. (f) - 2x + 5. 2. Find the average rate of change of each of the following functions for the intervals from 1 to 3; from 1 to -1; from 0 to 3; for any interval. Check the results. Plot the graph and interpret the average rates. (a) - x2 + 16. (d) x2 + 2. (b) x2 - 2x. (c) x2 – 2x + 1. (f) x3. 3. For each of the following functions, discuss the table of values, plot the graph, and determine the variation, including the average rate of change 4. Discuss the function represented by the following temperature graph. Find the greatest and the least average rate for the three-hour intervals indicated on the time axis. 5. The daily variation of temperature for a person in normal condition is given in the following table. Plot the graph and discuss the function. For what hour of the day is the average rate the greatest? The least? The temperature from 2 A.M. to 6 A.M. is stationary. 14. Classification of Functions. In discussing functions it is convenient to separate them into classes according to the properties they possess or the character of the operations that are involved in calculating them. The following scheme indicates the important divisions and subdivisions of the functions which we shall study in this course. An algebraic function* is a function whose value may be computed when that of x is given by the application, a finite number of times, of the operations of algebra, namely, addition, subtraction, multiplication, division, involution. and evolution. The following are examples of algebraic functions: In the work of this course only real numbers will be used. Hence, for us, such an algebraic function as V16 x2 is defined only for values of x such that -4 ≤ x ≤ + 4 (read “x is greater = *This definition is sufficiently general for the purposes of this course. The definition used in higher mathematics, which is more inclusive is as follows: Given a polynomial in y, + An-1y + An, whose coefficients do, a1, a2, an-1, an are polynomials in x, tnen any solution of the equation ƒ(y) = 0 for y in terms of x is called an algebraic function. |