CHAPTER II

LINEAR FUNCTIONS

16. Uniform Rate of Change. The position of a body moving on a line, straight or curved, is commonly indicated by its distance s from a given point on the line. The body is said to move uniformly, or at a constant rate, along the line if the ratio of any change in s to the corresponding change in time is constant (that is, if this ratio has the same value for all intervals of time). If As is the change in s during the time At, then for uniform motion As/At = v, a constant called the velocity. The value of v gives the change in s during a unit of time.

Since As/At is the average rate of change of s during the interval of time At (see Section 13), the above definition is equivalent to the statement that a body moves uniformly at the rate of v units of distance per unit of time if the average rate of change of s is constant and always equal to v.

EXAMPLE. A farmer lives 10 miles from town. He starts from home and drives away from town at the uniform rate of 5 miles an hour. Construct and interpret a graph showling his distance from town at any time.

30

D

E

3

(1)

FIG. 27.

S = 5t + 10.

In plotting the graph of equation (1) we take values of t as abscissas ac

cording to a well-established custom. The pairs of values of t and s in the table are represented by the points A, B, C, D, E, which appear to lie on

a straight line, l. We assume that the graph is indeed straight, an assumption which will be justified in Section 20.

The interpretation of the coördinates of any point on the graph is that the ordinate represents the man's distance from town at the time represented by the abscissa.

The interpretation of the velocity is worthy of detailed consideration. The successive values of At given in the table are represented by AF

=

= BG

=

CH = DI = 1, and those of As by FB = GC = HD = IE = 5. Hence v = As/At is represented by any one of the ratios

Now consider any interval of time beginning at t1 and ending at t2. The distances from town at these times are respectively, from (1),

with water, the rate at which a manufacturer turns out his product, etc.

A generalization of uniform rate of change with respect to time is given in the

DEFINITION. It is said that any variable y changes uniformly with respect to a second variable x, of which y is a function, if the ratio of any change in y to the change in x producing it, Ay/Ax, is equal to a constant, which is called the rate of change of y with respect to x. This rate gives the change in y due to a

unit change in x.

For example, if mercury is poured into a vertical tube, the weight of the mercury in the tube is a function of the height of the column of mercury. The ratio of any change in the weight to the change in height is constant, and equal to the weight of a column of unit height. Hence the weight changes uniformly with respect to the height.

It follows from the definition that if y changes uniformly with respect to x, equal changes in x produce equal changes in y.

EXERCISES

It is assumed that the graphs in the following exercises are straight lines. 1. Solve the example in Section 16 if the man starts toward town, considering the motion for 4 hours.

2. A through freight train was 90 miles from a city at 2 o'clock, and 150 miles at 4 o'clock. Construct the graph of its motion. Find its velocity, and interpret it geometrically. From the graph find how far the train was from the city at noon, and when it passed through the city. 3. A tank full of water is being emptied. After 5 minutes it contains 150 gallons, and after 12 minutes 108 gallons. Construct a graph showing the amount of water in the tank at any time. Find the capacity of the tank, the time required to empty it, and the rate at which it is being emptied. What represents each of these quantities graphically? 4. A man walks up a hill inclined at 30° to the horizon. Find the rate at which his altitude increases with respect to the distance he walks. Is it essential that he walk with a constant velocity?

5. A coal wagon is being filled with coal. Would the rate of change of the weight of the coal in the wagon with respect to the volume of the coal be constant? If so, how would it be measured?

6. Mention some quantity which is changing uniformly with respect to some other quantity, different from time, and state what the rate of change would be.

7. On the same axes, plot graphs showing the number of minute spaces the hands of a clock pass over in t minutes, starting at noon and ending at 1 o'clock. Determine from them how many spaces the hands are apart at 12:30.

17. Characteristic Property of a Straight Line. Let P1 and P2 be any two points on a straight line, M1P1 and M2P2 their ordinates, and let PiQ be perpendicular to M2P2. Then Ax PIQ is the difference of the abscissas of P1 and P2, and

=

Since the triangles P1QP2 and P3RP are similar (why?), we have

The ratio of the difference of the ordinates of any two points on a straight line to the difference of the abscissas, ▲y/Ax, is the same no matter what two points on the line be chosen.

This fact has been illustrated in the preceding section. Conversely, a line is a straight line if the ratio of Ay/Ax is always the same for any two points on the line.

Let P1 and P2 be two definite points and Pз any third point on the given line which is to be proved straight. That the given line is straight will be proved by showing that P3 lies on the straight line determined by P1 and P2. Draw the line through P1 parallel to the x-axis cutting the ordinates of P2 and P3 at Q and R respectively. Since, by hypothesis, the values of Ay/Ax computed for P1 and P2 and for P1 and P3 are equal, it follows that

Hence the triangles PiQP2 and P1RP, are similar (why?),

m, this definition may be expressed in the symbolic form

For any graph, the ratio of the difference of the ordinates to the difference of the abscissas, Ay/Ax, represents the average rate of change of a function. Hence the facts proved above may be stated as the

Theorem. If y is a function of x, the graph of y is a straight line if, and only if, the average rate of change of y with respect to x is constant.

To plot the graph of a function we usually express the functional relation in the form of an equation, build the table of values, and plot the curve. This process is unnecessary if it is known that the average rate of change of the function is constant, for the theorem just proved shows that the graph is a straight line. To draw the graph it is sufficient to obtain two pairs of values, plot the points representing them, and draw the straight line through these points.

EXAMPLE. Commercial alcohol, 95% pure, is poured into a bottle. Construct a graph showing the amount of alcohol in the bottle as a function of the amount of liquid in the bottle. Interpret the slope.

Plot the amount of liquid, l, on the horizontal axis, and the amount of alcohol, a, on the vertical axis. If Aa is the amount of alcohol in Al units of the liquid, we have ▲a = 0.95Al, since 95% of the liquid is alcohol. Hence Aa/Al = 0.95, so that the rate of change of a with respect to l is