change sign if x increases through a value for which the function becomes infinite. Thus 1 x 4' whose graph is given on page 27, changes sign, from negative to positive, if x increases through the value 4. The intercepts on the x-axis and the vertical asymptotes should therefore be determined before considering the sign of a function. In the example just cited, the vertical asymptote does not separate intervals in which the function increases or decreases, but it may do so. For example, the function 1/2 becomes infinite if x approaches zero, and hence the y-axis is a vertical asymptote. As x increases, this function increases if x is negative, and decreases if x is positive. This function also shows that a vertical asymptote need not separate intervals in which the function has opposite signs, for 1/2 is positive for all real values of x. A good order for considering the elements of the variation of a function is as follows: Zeros of the function, Function becomes infinite, Sign of the function, Maxima and minima, Changes of the function. As the zeros and asymptotes are taken up in discussing the table of values, only the last three are new ideas. These properties of a function may be determined approximately by inspection of the graph of the function. This procedure is especially useful if the table of values or the graph be given and the functional relation itself is unknown. The accuracy of the results will depend upon the choice of units on the axes, the care with which the graph is drawn, and the closeness with which it is read. EXAMPLE 2. A thermograph is an instrument which records the temperature continuously by means of such a curve as in the figure. Discuss the variation of the temperature as a function of the time. Zeros of the function. The graph cuts the time axis at B and D, hence the temperature was zero at 3 A.M. and at 8:30 a.m. Sign of the function. The graph is above the time axis from A to B and from D to F and below from B to D. Hence the temperature was above zero from 12 Mid. to 3 A.M. and from 8.30 A.M. to 12 Mid. the next night, and below zero from 3 A.M. to 8:30 A.M. Maximum and minimum values. At C the ordinates cease to decrease and begin to increase. Hence, the temperature had a minimum value of about 14° at 6 A.M. At E the ordinates cease to increase and begin to decrease. Hence, the temperature had a maximum value of about +45° at 3 P.M. Changes of the function. The graph rises from C to E and falls from A to C and from E to F. Hence, the temperature increased from 6 a.m. to 3 P.M. and decreased from 12 Mid. to 6 A.M. and from 3 P.M. to 12 Mid. If it is not desired to treat each topic separately, the results might be stated as follows: The temperature at midnight was 10°. It decreased until it became zero at 3 A.M. and continued to decrease until 6 A.M., when it reached a minimum value of about 14°. It then increased, becoming zero at 8:30 A.M., until 3 P.M., when its maximum value was about 45°. From that time on it decreased to about 20° at midnight. The discussion of this function is continued in the next section. EXERCISES 1. Discuss the table of values, plot the graph, and determine the variation of each of the functions: 2. The surplus and shortage of railroad cars in thousands for the months of the years 1915 and 1916 were as follows. Plot the graph and discuss the function. 26, 1915. 180, 279, 321, 327, 291, 299, 275, 265, 185, 78, 38. 1916. 46, 21, -20, 3, 33, 57, 52, 9, 19, -60, -114, -107. 3. The following are the data for the surplus reserves of New York banks in millions for the months of the years indicated. Plot the graph and discuss the function. The Federal Reserve System was inaugurated November 16, 1914. 1914. 29, 32, 20, 17, 38, 40, 15, -29, -28, -0.4, 72, 117. 1915. 121, 135, 131, 156, 168, 185, 159, 177, 197, 179, 168, 155. 4. One side of a rectangle whose perimeter is 12 inches is x. Find the area as a function of x. Construct the graph of the function, discussing the table of values, and find the value of x if the area is a maximum. What is the maximum area? 5. A farmer wishes to fence off a poultry yard whose area is to be 6 square rods. If one dimension is x, express the perimeter (the amount of fencing needed) as a function of x. Discuss the table of values and plot the graph of the function. What will the dimensions be to require the least amount of fencing? How much should he purchase? 6. There are a number of diseases with continued fever in which the course of the temperature is sufficiently characteristic to furnish the diagnosis. Croupous pneumonia is one of these. (See Fig. 22.) Discuss the three functions. The zero for temperature would be the normal temperature 98°.4. By comparison of the three functions state in words some of the symptoms of the disease. 7. The data for the temperature curves of measles and scarlet fever 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. 6 7 2 3 4 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. Let y denote the temperature Ax x at any time x, and ▲y the change in temperature during an inter y Ay Ay/Ax 12 Mid. 10 3 -10 -3.3 3 A.M. 0 6 A.M. - 14 9 3 P.M. +45 +59 +6.5 8 A.M. -6 2 10 A.M. +19 +25 +12.5 val 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 = x2 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 ▲y be the change in y produced by a change of Ax in x. Then x + Ax and y + Ay are cor |
