we have the rule at the top of the table: "Moving the decimal point two places in n is equivalent to moving it one place in √n."

If n is greater than 10 or less than 0.1, by moving the point two places at a time, to the left or right, respectively, we will ultimately obtain a number in which the decimal point either precedes or follows the first significant figure. In the former case the square root is found in the upper part of the table, and in the latter case in the lower part. The square root of the original number is then found by re-shifting the decimal point in accordance with the rule.

For example, to find the square root of 35,700, we shift the point two places to the left twice in succession, obtaining 3.57. From the lower part of the table we obtain √3.57 1.889. Then applying the rule twice, shifting the point to the right, √35700 = 188.9.

To find the square root of 0.0024, we shift the point two places to the right, getting 0.24. From the upper part of the table, v/0.24 = 0.490, whence, by the rule, 0.0024 = 0.0490. Notice that the zero following the nine is retained in order to show the value of the radical to three figures.

The table of cube roots on page 5 is divided into three parts because in moving the decimal point in any number three places at a time (see rule at the top of the table), we ultimately obtain a number in which the point either follows the first significant figure, precedes it immediately, or is followed by a single cipher.

Table of reciprocals. The reciprocal of n, 1/n, differs from n2, n3, √ñ, vñ, in that as n increases the reciprocal decreases. For a reason which will appear in Section 42, it is desirable to have all the tables so arranged that the numbers in the body of the table increase as we read down the page, and from left to right. The table of reciprocals on page 7 is so arranged. The values of n, given in the border of the table, increase as we read up and to the left.

The rule for shifting the decimal point at the top of the table follows from the fact that the product of n and its reciprocal is unity. Hence if one is multiplied by 10, the other must be divided by 10.

To find 1/436, for example, we read up the table, on the right, until we come to 4.3, then over to the left until we are in the column headed 6, where we find 0.2294, which is the value of 1/4.36. Then by the rule for shifting the point, 1/436 = 0.002294.

In using any one of these tables the position of the decimal point may frequently be determined by inspection, and it is well to check the result obtained from the table.

EXERCISES

1. Find 3.242, √/0.47, √4.72, 7.43, 0.0235, 0.24, 1.84, 1/24.

2. Find the squares of 25.4, 0.86, 3540, 0.0043.

3. Find the square roots of 59, 590, 4300, 0.000382.

4. Find the cubes of 54, 0.317, 53200, 0.0000371.

5 Find the cube roots of 1540, 470, 18.3, 0.0048, 0.0000259.

6. Find the reciprocals of 23.4, 0.478, 532, 0.0074.

7. Find the value of 0.074, 0.066, √14.23, 1/23-42, 0.02472.

8. Find the hypothenuse of a right triangle whose sides are 61 and 74. 9. Find the mean proportional between 6 and 34.

38. Graph of x", n>1. It will be seen in this section, and those following, that a part of the graph of x" lies in the first quadrant, and that the remaining part may be found by means of the symmetry of the curve. Hence particular attention should be made to fix in mind the various forms of the part of the graph in the first quadrant. The general appearance of this part of the graph varies according as n>1, 0<n<1, or n<0, so that these cases are considered separately. These groups of values include all values of n except n = O and n 1, which separate the groups, and which are exceptional in that they are the only values for which the graph is a straight line. n = 0. If n = 0, y = x" becomes y = 1, since x° = 1, whose graph is the straight line parallel to the x-axis and one unit above it. 1. For n = 1 we have y x, whose graph is the straight line through the origin whose slope is unity, that is, the bisector of the first and third quadrants.

n>1. In this case the graph of

Y = xn

(1)

is tangent to the x-axis at the origin (0, 0), rises to the right and passes through the point (1, 1), at which the slope of the tangent line is n. That it passes through these points is seen by substituting their coördinates in (1).

To find the slope of the line tangent to the graph at any point, when n is a positive integer, replace x by x + Ax and y by y + Ay in (1), which gives y + Ay = (x + Ax)".

Expanding the right hand member by the binomial theorem,

Passing to the limit as Ax approaches zero, the slope of the tangent line at any point is

m = 0 if n >1. Hence the tangent at the origin is the x-axis.

At the point (1, 1) the slope of the tangent line is m=n. Hence the larger the value of n the steeper the curve is at this point, from which it follows that it must be flatter near the origin.

If n is a positive integer greater than unity, the part of the graph of x in the first quadrant is then very much like the graph of x2.

If n is a positive fraction greater than unity the part of the graph in the first quadrant has the same general appearance. To see this, let r, s, and t be three values of n such that r<s<t. For a positive value of x the value of xs will lie between x and xt, and hence the part of the graph of x in the first quadrant lies between the graphs of x" and at (this holds for all values of r, s, and t). For example, the graph of lies between the graphs of x and x2; that of x between those of x2 and x3; etc. The figures show the graphs of x2, x3, and x2, which are symmetrical with respect to the y-axis, the origin, and the x-axis

respectively. The symmetry of the last curve is seen by writing y = x in the form y2 = 23, and applying Theorem 3B, page 24. The graph of x", n>1, always resembles one of these curves. Thus the graph of 24 is very much like a parabola, but it is flatter near the origin and steeper elsewhere.

The x-axis is tangent to the graph of x3 at the origin. This tangent differs from any we have encountered hitherto in that it crosses the curve at the point of tangency.

The graph of x is remarkable in that it has a sharp point at the origin. It differs from other curves we have studied in detail in that vertical lines to the right of the y-axis cut it

y

Si

(y, x)

twice, corresponding to the fact that x = √ has two values for each positive value of x. 39. Graph of x", 0<n<1. Graphs of Inverse Functions. Let us first consider n 1, or the function x. If we set y = x2, and solve for x, we get x = y2. This differs from the equation y = x2 only in that x and y have. been interchanged. Hence if (x, y) is a point on the graph of either equation, the point (y, x) is on the graph of the other. But these points are symmetrical with respect to the bisector of the first and third quadrants, as may be established from the figure. Hence the graphs

N

P (x,y)

R

M x

FIG. 59.

of the two equations, that is, the graphs of x and x2, are symmetrical to each other with respect to this bisector.

The graph of x may therefore be constructed as follows: Construct the graph of x2 and the bisector of the first and

김수

2

·6

H

1 2 3

FIG. 60.

6 x

third quadrants. Choose a number of points on the graph of 2, and construct the points symmetrical to them with respect to this bisector. Draw a smooth curve through the points so obtained.

We may now get properties of the function by interpreting its graph. For example, since the graph of x2 is symmetrical with respect to the y-axis that of x is symmetrical with respect to the x-axis, and hence to each value of x there correspond two values of x which are equal numerically but differ in sign. And since no part of the graph lies to the left of the y-axis (why?), the function is imaginary if x is negative. What other properties may be obtained in this way?

If two curves are symmetrical to each other with respect to a line they are congruent, for one may be brought into co