2. In the following table the relation between x and y is y = k/x2. x 1, 2, Plot the graph and determine k. Check the y | 3, 0.78, 0.32, 0.19, accuracy of the result. 3, 4, 3. In the table V represents the volume of a vessel containing a gas and p the pressure on the walls. Find the relation between V and p. V | 0.1, 0.2, 1, 2, 10, p 10, 5.01, 0.99, 0.52, 0.13, 4. Determine the law of the attraction A of an electrified rod for a the table. What is the attraction pith ball at a distance d as given in when the distance is 1.2? d | 0.51, 0.98, 1.53, 1.97, 2.56, 5.98, 1.48, 0.66, 0.38, 0.23, A 5. In the following table W represents the effect of the blow when a weight of 16 pounds is dropped and strikes with a velocity v. Plot the data, determine the law, and find W if v = บ 0, 1, 2, 3, 4, W 0, 0.23, 1, 2.37, 4.01, 6. The discharge of water 2.5. through a large pipe line or conduit is measured by a Venturi meter, on which we read the difference in 0, 1.00, 1.98, 2.98, 3.98, Q | 0, 0.0442, 0.0637, 0.0780, 0.0909, height of a column of mercury in the two sides of a U-tube. In the table, a denotes the difference in height and Q the discharge. Determine the law. 7. From the following data determine the resistance R, in ohms per d | 0.083, 0.120, 0.148, 0.220, mile, of a telegraph wire of diameter d, in inches. 2018 R 51, 24.42, 16.1, 7.26, 5.03, 2.56, 1.21, 0.80, 0.68, pressure p and the volume V of a gas if the pressure on the walls of a con taining vessel for different volumes was found to be as in the table. 9. In the following table, x denotes half the distance between two tele graph poles, and y the amount the wire sags at the center. law. 20, 30, 40, 50, 60, 0.98, 2.21, 3.95, 6.19, 8.97, 10. The table gives the number of h 11.63, 62.8, 138.8, 174.5, R.P.M. 440, 1020, 1525, 1680, Find the revolutions per minute, R.P.M., at which a certain form of water wheel runs without any load, for different values of the head, h (the height of the surface of the mill pond above the bottom of the wheel). Find the relation. h H.P. = 11. The horse power developed by the water wheel in Exercise 10 is 69.8, 104.8, 139.8, 174.5, given in the table. Find the law 0.64, 1.18, 1.81, 2.56, in the form kx. 12. (a) Economists define supply as the amount of anything which the producers will offer for sale at a given price. The table gives an assumed set of values of the supply of wheat, in mil Price 0.60, 1.00, 1. 40, 1.80, Supply 1, 2, 3, 4, lion bushels, at various prices in dollars. Plot the graph and determine the law. What price per bushel will stimulate the producers to furnish 5 million bushels? (b) The term demand is used to denote the quantity of anything which the consumers will purchase at a given price. How does the demand fluctuate as the price rises? As the Price 0.40, 0.80, 1.00, 2.00, Demand 10, 5, 4, 2, price falls? The table gives an assumed set of values of the demand for wheat, in million bushels, at various prices, in dollars. Plot the graph and determine the law. What price will stimulate the consumers to demand 7 million bushels? (c) Assuming the conditions in the preceding parts of this exercise, plot both graphs on the same axes, and find the price and the quantity of wheat transferred from producer to consumer. 13. If the apple crop amounts to 3 million bushels, then, at the harvest season, the supply at any price high enough to pay to handle the crop is 3. The table gives assumed values for Price 0.50, 1, 1.25, 2.50, Demand 5, 2.5, 2, 1, the demand in million bushels. Plot both graphs on the same axes, and determine the equations giving the laws. At what price will apples sell under these conditions? Price 3, 4, 5, 6, 8, 10, Demand 6, 5, 4, 3, 2, 1, 14. Some industries develop approximately to a point where any amount, within certain limits, can be supplied at a constant price. Suppose that any number of shoes will be supplied at the price of $5 per pair, and that the demand, in million pairs, at a price, in dollars, is as given in the table. Plot both graphs on the same axes, and find the price and the number of shoes sold. Why cannot the price be less than $5? Assuming competition, why will it not exceed $5? (b) If a monopoly arose, with the object of making as much money as possible instead of serving the public to the best of its ability with reasonable profits, the output would be restricted to the point of maximum profits. Assuming the conditions of the preceding part of the problem, suppose that the cost of manufacturing a pair of shoes is $4. Construct a table giving the total profits at various prices, plot the graph, and determine the price and profits. 15. Plot the graphs and determine the laws for the data given in the tables. Determine the freight rate and the amount of freight handled. 45. The Linear Fractional Function ax + b In order that this function should really be fractional, it is essential that α b c≤ 0, for if c = 0, it reduces to the linear function + x d. The simplest linear fractional function is 1/x, obtained by setting b = c, and a = d = 0, whose graph has been considered in Section 40. If bc, while a = d = 0, we have the function b/cx, or k/x, = where k b/c. Its graph may be obtained by multiplying by k the ordinates of points on the graph of 1/x (Theorem, page 89). If we set y k/x, whence xy = 4 -3 -2 ย A k, it is seen that the graph is symmetrical with respect to the line y = x, since the equation is unchanged if x and y 3 are interchanged. The graph is also symmetrical with respect to the origin. The geometric significance of the constant k is obtained by solving the equations xy = k and y = x; the solutions are (√k, √k) and (− √k, √k), which are the coördinates of the points of intersection of the graphs of the equations. The distance from the origin to the point of intersection A is found from the right triangle OAB to be √2k. Hence, for a small value of k, the graph lies close to the origin and axes, while if k is large, it lies at some distance from them. If k is negative, the graph will lie in the second and fourth quadrants. The graph of xy= k is called a rectangular, or equilateral, hyperbola. The former term is derived from the fact that the asymptotes are perpendicular. In order to determine the form of the graph of the general linear fractional function, set Multiplying both sides by cx + b, and subtracting ax + b from both sides, cxy ax + dy − b = 0. (2) Let us now see if we can simplify this equation by translating the axes (Section 31, page 89). Setting we get x = x' + h and y = y' + k, (3) c(x' + h) (y' + k) − a(x' + h) + d(y' + k) − b = 0, (4) or, removing the parentheses and collecting like terms, cx'y' + (ck − a)x' + (ch + d)y' + chk - ah + dk - b = 0. (5) Equating to zero the coefficients of x' and y', we get Solving for h and k, which is always possible since c # 0, (6) (7) Simplifying, subtracting the constant term from both sides and dividing by c, x'y' bc - ad (8) This equation, referred to the new axes, has the same graph as (1). It is of the form x'y' = k, where k (bc ad)/c2, or y' = k/x'. The graph is therefore a rectangular hyperbola. Hence we have the Theorem. The graph of a linear fractional function (1), or of an equation of the form (2), is a rectangular hyperbola whose asymptotes are parallel to the axes of coördinates. EXERCISES 1. Plot the graph of y (2x − 4)/(x + 3), (a) by translating the axes; (b) by finding the asymptotes (by the method on page 27), plotting one branch of the curve from a table of values, and the other by means of the symmetry with respect to the point of intersection of the asymptotes. 3. The linear fractional function may be written in the form x(x+b/a)/c(x+d/c). Prove that the linear factors cancel, and the function is a constant, if and only if bc - ad 0. For this reason, it is always assumed that bc ad 0. 4. The relation between the radius of curvature of a concave mirror, R, the distance from the center of the mirror to the object, x, and the distance from the center to the reflected image, y, is If R = 2 feet, construct the graph of the equation. 46. Integral Rational Functions. The general form of an integral rational function or polynomial is (see page 39) f(x) = αox2 + α1פ−1 + . where n is an integer, and ao, a1, tive, negative or zero, except that ao degree n. + An-1X + an, an are constants, posi0. It is said to be of We have already studied polynomials of the first and second degree in considering linear and quadratic functions. The calculation of a table of values for a polynomial of higher degree than the second by direct substitution presents no new features, but because of the greater number of terms that may be present there is more labor involved. An alternative method which is simpler, in general, than that of direct substitution is developed in the next sections. |