4. Determine scales on the axes so that the graph of y = 3(100.0122) is a straight line.

5. Determine the constants in y = k10m2 by means of the pairs of values (2, 0.5) and (3, 0.8). Choose scales on the axes so that the graph of the equation is a straight line. From the graph read the value of y if x =2.5.

6. Find the radius of the moon in miles, given that the diameter of the moon subtends an angle of 32′ as seen from the earth, the distance of the moon from the earth being 240,000. Find the ratio of the mass of the moon to that of the earth if the density of the moon is 0.6 that of the earth.

7. If the distance of the sun from the earth is 92,800,000 miles and the diameter of the sun subtends an angle of 32.4' as seen from the earth, find the diameter of the sun, and the ratio of the volume of the sun to that of the earth.

8. If the length, l, of a range finder is 20 feet and the distances, d, are calculated by the formula d 7 tan 0, find the values of corresponding

to the extreme values 400 ft.

=

and 20,000 ft. on the dial of the instrument.

(Use the "log rad" table on page 28 of the Tables.)

9. The following table gives the collegiate running records, the distance

d being in yards and the time t in seconds. Find the law and the value of t if d 600. Find t if d

=

=

3520.

10. An observer on a destroyer moving at the rate of 35 miles an hour notes that the line of sight to a ship makes an angle of 41°.56 with the forward path of the destroyer and that one minute later the angle is 74°.83. He estimates that the ship is moving on a parallel path at the rate of 14 miles an hour. Find the distance between the paths.

11. The economic law of diminishing utility is stated as follows: The total utility of a thing to any one (that is, the total pleasure or other benefit the thing yields) increases with any increase in one's stock of it but not so fast as the stock increases. If one's stock increases at a uniform rate, the benefit derived from it increases at a diminishing rate.

Another way of stating the law is: The increase in the utility of a thing, or marginal utility, diminishes with every increase in the amount of it any one already has.

Plot the graphs of these two statements of the law.

12. Find approximate values of the real roots of the following equations, which can not be solved by the ordinary methods.

Hint. Plot the graph of the function on the left and from it locate an intersection with the x-axis as accurately as possible. Enlarge the table of values until the coördinates of two points are obtained such that the intersection lies between them and such that the part of the graph

between them may be assumed straight. Then determine an approximate value of the root by interpolation.

(a) f(x) = 2-x 3x0. Solution. The graph of f(x) shows that there is a root between 1 and 2, near 1.2. Enlarging the table of values, it is seen that the root is between 1.2 and 1.3, since the corresponding values of f(x) have opposite signs.

In order to find a value of x by interpolation, construct the graph between x

=

1.2 and x

=

€1.3 on the assumption that it is a straight line BD. To find AE we have, by similar triangles,

1

2

3

1.20.435 0.400 0.035

13. Some properties of many functions can be expressed entirely in terms of the notation f(x). If analogous properties of two functions can be so expressed an abstract point of view is obtained which gives a deeper insight into the differences between the functions. Establish the following relations:

14. The speed of an aeroplane is 80 miles an hour and the wind is blowing from the north with a velocity of 20 miles per hour. The pilot desires to move S. E. Find the direction in which he should head the machine and how fast he will move.

15. Solve the preceding exercise supposing that the pilot desires to sail N. W. If the aeroplane can stay in the air 4 hours, find the greatest distance the pilot can sail S. E. and be able to return to the starting point.

16. The speed of an aeroplane is 100 miles per hour and the wind is blowing from the west. The plane can stay in the air 5 hours. How far can the pilot sail in the direction 10° S. of W. and return to the starting point?

CHAPTER VI

DIFFERENTIATION OF ALGEBRAIC FUNCTIONS

91. Introduction. If y is a function of x, the average rate of change of y with respect to x in any interval Ax is ▲y /Ax (page 35). If the average rate of change is constant, its value is the rate of change of y with respect to x (Definition, page 48), and the graph of y is a straight line (Theorem, page 50) whose slope is the rate of change. Numerous applications of uniform rate of change were given in Chapter II.

If the average rate of change, Ay/Ax, is not constant, then the rate of change of y with respect to x is defined to be the limit of Ay/Ax as Ax approaches zero (page 94). It is represented graphically by the slope m of a line tangent to the graph of the function y.

In this chapter we shall consider this limit more formally than heretofore, and derive rules for finding it expeditiously if y is an algebraic function of x. The applications are based either on the geometric interpretation of the limit of Ay/Ax as the slope of the tangent line or on the physical interpretation of the limit as the rate of change of y with respect to x.

The ideas to be considered in this chapter, and the one following, are among the most fundamental and far-reaching concepts in mathematics. They were developed by the famous Englishman Sir Isaac Newton (1642-1727) and the noted German Gottfried Wilhelm von Liebnitz (1646–1716), and a bitter controversy lasting for many years was waged over the question as to which one of these men should be accorded the honor of the discovery. Leibnitz was the first to publish some of his results, in 1684, but Newton had written a paper on the subject and submitted it to some of his friends in 1669. The

264

followers of each claimed that the other had been guilty of claiming ideas not his own, but most historians of mathematics are agreed that the work of Newton and Leibnitz was independent. On the basis of the work of these men, there followed a period of rapid and extensive mathematical development.

Let us first consider the underlying concept of the limit of a variable.

92. Limits. By definition (footnote, page 93) the limit of a variable x is a constant such that the numerical value of the difference between x and a becomes and remains as small as we please. This explains what is meant by saying that "x approaches a as a limit," or in more compact form, "x approaches a." It is immaterial whether or not x becomes equal to a.

A function of x is a second variable y (Definition, page 5). The functions we have studied are such that if x approaches a limit, so also does y, provided that y does not become infinite as x approaches a. The notation

is used to indicate that "the limit of y, as x approaches a, is b." This means that the numerical value of the difference between y and b can be made as small as we please by taking x sufficiently near to a.

Graphically, the difference between the ordinates y and b can be made as small as we please by making the difference between the abscissas x and a sufficiently small. In other

words, Ay by approaches zero

=

if Ax = α x approaches zero.

For example, the average rate of

Ay

P

Ax

b

y

a

FIG. 157.

change of the function ax2 in an interval Ax beginning at a definite point x is (see (5), page 95)