5. On the same axes, using as large a scale as possible, construct the graph of x" for n = 1 and 3. What can be said of the graph of x2 if n is an odd negative integer? 6. Construct the graph of x" if n = (a) By regarding the function as the reciprocal of the function xt; (b) By regarding it as the inverse of x-2. 7. Construct the graph of (a) x −1, (b) x −3, in each case using the properties of the graphs of reciprocal functions. What relation exists between these two functions? = 8. The quantity of water which flows from an exit pipe h feet below the surface of a reservoir is given by the equation q 8.02a√h, where a is the area of the cross section of the pipe. Assume a = 1, and plot the graph. If one outlet pipe is 10 feet below the surface and another is twice as deep, is the flow of one twice that of the other? Why? 9. The pressure p, the volume v, and the temperature t of a gas are connected by the relation pv = kt, where k is a constant. Assuming k plot the graph if (1) p is constant, (2) v is constant, (3) t is constant. = = 1, 10. As water runs out of a basin, a depression is formed near the outlet. A vertical section of this depression in the surface of the water is called the curve of whirl. It is given by the equation y h2r2/x2, where r is the radius of the orifice in the outlet pipe, and h is the head, or depth of water. The head changes from instant to instant, so that the curve of whirl constantly changes. Assuming r = 1, plot the graph for h 11. Plot the graphs of the functions following, and then sketch the graphs of the reciprocal functions. Find the reciprocal function in each = 1, 1, 2. 12. What is a simple way of sketching the graph of 1/(2x+3)? Of 3/(2x+3)? 42. Interpolation. The process of finding, for example, the cube of such a number as 2.647, which lies between two successive numbers 2.64 and 2.65 whose cubes are given in the table, is called interpolation. According to Huntington's Tables, 2.643 = 18.40 and 2.653 = 18.61. The difference between these two values of y x3 is ▲y = 0.21, and is called the tabular difference. In general, the tabular difference is the difference between two successive numbers in the body of a table. For the process of interpolation it is essential that the = successive values of the tabular difference should be very nearly equal. x y = x3 Ay The table gives one row of Huntington's table of cubes and the successive tabular differences. Since Ay changes only slightly while Ax is always equal to 0.01, the average rate of change, Ay/Ax, is nearly constant, and the part of the graph of x3 constructed from this table would be very nearly a straight line. Some of the successive values of Ay are equal, due to the fact that the values of x3 are approximate to four figures and not exact, and hence some of the points plotted from the table would actually lie on a straight line. 2.60 17.58 .20 2.61 17.78 2.62 17.98 2.63 18.19 .20 2.70 19.68 .21 For the process of interpolation we assume that the part of the graph lying between two successive points is straight. = To find 2.6473, consider that part of the graph lying between x 2.64 and 2.65, as given in Figure 65. We seek the value of the ordinate EF at the point E for which x 2.647. It may be obtained by adding GF as a correction to AB 2.643. To х = = find GF, we have the slope of BF is the same as that of BD, since the graph is assumed straight. Hence EF may also be obtained by subtracting ID as a correction from CD 2.653. To find ID, we have the slope of FD is equal to that of BD, since the graph is assumed straight, so that = 0.003 The latter procedure is usually preferable if the figure for which we are interpolating, 7 in the illustration, is greater than 5, while the former is used if it is less than, D F B A 2.64 0.21 18.61 The arithmetical operations involved in interpolation may be described as follows: To find 2.64a3, where a stands for the digit for which we are interpolating, apply a-tenths of the tabular difference as a correction to 2.643 if a<5, while if a>5, apply (10 − a)-tenths of the tabular difference as a correction to 2.653. In either case, the correction must be applied (added or subtracted) in such a way that the result lies between 2.643 and 2.653. In practice the decimal point is usually omitted in x, x3, and the tabular difference, and inserted in the proper place at the end of the operation. EXAMPLE 1. Find 26.023. The figure for which we must interpolate is 2, and the tabular difference is 20. Two-tenths of the tabular difference is 4, the correction to be applied to 1758, the cube of 260. In order to obtain a result between 1758 and 1778, the cube of 261, the correction must be added, giving 1762 as the first four digits in the required cube. From the rule at the top of the table, we see that 26.023 = 17,620. EXAMPLE 2. Find 2.6873. Instead of applying 7-tenths of the tabular difference as a correction to the cube of 2.68, we apply 3-tenths of it to the cube of 2.69. Three-tenths of the tabular difference is 6.6 = 7, and hence 2.6873 = 19.40. Except for purposes of explanation, there is no need of writing anything but the desired result. All the tables in Huntington's Tables are arranged so that the numbers in the body of any table increase as we read from left to right, for the reason that a uniform procedure is obtained for applying the correction. In interpolating, the desired result lies between two numbers in the same row; a correction applied to the left-hand number is always added, while a correction to be applied to the right-hand number is always subtracted. EXERCISES = 1. Plot the graph of x2 from x = 4.50 to x 4.60, using the values of x2 given in Huntington's Tables. 2. If we plot the graph of √x for values of x from 2.30 to 3.40, using the square roots given in the Tables, why will the graph be nearly straight? 3. Find the values of the numbers given below, illustrating the interpolation graphically. (a) 3.1722. (b) √3.478. (c) 6.493. (d) 0.02814. (e) 1/926. 4. Find the numbers following without illustrating the interpolation graphically. The tabular difference and the necessary number of tenths of that difference should be obtained mentally. (a) 1.5342, 0.468, 8.433, √2.46, 1/647. (b) Squares of 2.784, 3762, 0.01388, 3846000, 0.00003728. (c) Square roots of 0.634, 3.248, 42.7, 384.3, 279,000, 0.001876. (d) Cubes of 3.143, 0.774, 1683, 0.00004592, 4889000. (e) Cube roots of 0.02258, 0.226, 19.34, 0.00176, 328000. (f) Reciprocals of 31.76, 0.00647, 35990, 0.0004325, 647. 5. Find the square of 4732, and check the result by finding its square root. Find the cube root of 3479, and check the result by cubing it. Find the reciprocal of 25.63, and check the result. 6. Find 374.33, 0.024782, 1/73-262, 1/2438, 1/48-363. 7. Solve the following equations, using the Tables to simplify the computations: (a) x2 + 32x + 19 = 0. (c) 3x2 + 29x 40 = 0. (b) 2x2 47x270. 8. How long an umbrella will go into a trunk measuring 31.5 x 18.5 × 22.5 inches, inside measure, (1) if the umbrella is laid on the bottom? (2) if it is placed diagonally between opposite corners of the top and bottom? 9. A house with a gambrel roof is 28.5 feet wide, the first set of rafters has a slope of, and the top set a slope of 3. If the joint in the roof comes 7.8 feet from the side of the building measured horizontally, how long is each set of rafters? 10. If five individuals weigh 120, 124, 116, 112, 123 pounds, respectively, and five others 95, 150, 132, 105, 113 pounds, respectively, then the average weight M of either group is 119. But one group is distributed very closely around the mean M, whereas the other group exhibits marked deviations from it. A measure of the variability or tendency to deviation of measurements is given by the following formula, called the standard deviation. S.D. +dn2 = n The average M is found, and each individual measurement is subtracted algebraically from M, thus obtaining a series of deviations d. The sum of the squares of these deviations is divided by n, the number of measurements, and the square root of the quotient is the standard deviation from the average M. The coefficient of variability is defined as the ratio of the standard deviation from the average to the average. Compare the coefficients of variability for the two sets of measurements. 11. The strength of grip of right hand and left hand, in hectograms, for 10 boys is given in the following table. Compute the standard deviation and coefficient of variability for each. Right hand. 158, 200, 210, 226, 248, 270, 296, 320, 348, 403. Left hand. 138, 185, 200, 224, 244, 260, 282, 305, 336, 400. 43. Variation. DEFINITION. It is said that y varies as, or is proportional to, the nth power of x if y = kx", n being positive, while y varies inversely as, or is inversely proportional to, the nth power of x if y = k/x". If it is desired to contrast these two forms of variation, the former is called direct variation as opposed to inverse variation. The case of direct variation in which n = 1 has already been considered in Section 21, page 61. In any case, the value of the constant k may be determined from a given pair of values of x and y, as in that section, by substituting the given values of x and y. The graph of the relation y=kx" may be obtained from one of the curves considered in Sections 38 to 40, by means of the theorem in Section 30. The language of variation is used frequently in the applica |
