What property of the exponential function do these curves illustrate? 36. Find graphically the product of 32i by 2 + i. 37. Find all the values of: (cos + i sin 0)3; (cos + i sin 0); - 0. 1; VI. 38. Write a short theme on operators, making mention of (a) the integers; (b) ( 1); (c) √ −1; (d) cis e. Develop the rules for addition, subtraction, multiplication, and division of vectors, and state them in systematic form. 39. Show that = sin a cos b cos c + cos a sin b cos c + cos a cos b sin c sin a sin b sin c. sin (a+b+c) - 2 y = 3x + 0.63. on the same sheet of paper. What property of the exponential function do these curves illustrate? 41. Draw upon squared paper, using 2 cm. = 1, the curve y2 = x. By counting the small squares of the paper find the area bounded by the curve and the ordinates x = 1/2, 1, 11, 2, 21, 3, 31, 4, By plotting these points upon some form of coördinate paper, find the functional relation existing between the x coördinate and the area under the curve. They are 7 miles Find their difference in 42. The latitude of two towns is 27° 31'. apart measured on the parallel of latitude. longitude. 43. Solve 3x2-1 = 2x+1. Be very careful to take account of all questionable operations. There are two solutions. 44. Find (three problems) the equation connecting: for 45. Find the wave length, period, frequency, amplitude and velocity 47. Find the parametric equations of the cycloid. 48. Find the equation of the ellipse, center at the origin, axes coinciding with coördinate axes, passing through the point (— 3, 5) and having eccentricity 3/5. 49. Define the "logarithm of a number." Write its equation if oscillation. State the 51. A S.H.M. has amplitude 6, period 3. time be measured from the negative end of the difference between a S.H.M. and a wave. 52. Find by inspection one value of x satisfying the following equa tions: 57. A point moves so that the product of its distance from two fixed points is a constant. Find the equation of the locus. Discuss the 59. Find the velocity and frequency of the wave of problem 30. 60. Find the coördinates of the center, the eccentricity, and the lengths of the semi-axes of: (a) x2 + 3x + y2 (b) x2 + 2x+4y2 3y 0, (c) x2 x y2 (d) x2 + x + y +3 = 0. = = 7, У = 0, 61. Find the amplitude, period, frequency and epoch of the following S.H.M.: 63. Find graphically (on form M3) the fifth roots of 25 cis 35°. 64. Complete the following equations: 65. Change the equations of exercises 33 and 40 to logarithmic form. What properties of logarithms are illustrated by these equations? 67. Show that the sum of the two focal radii of the ellipse is constant. 68. y = -312 + 4t 5 and x = 5t are the parametric equations of a curve. Discuss the curve. 69. Show that [r(cos + i sin )] [r'(cos 0' + i sin (')] rr'[cos (0 +0') + i sin (0 + 0')]. = 70. Two S.H.M. have amplitude 6 and period two seconds. The point executing the first motion is one-fourth of a second in advance of the point executing the second motion. Write the equations of 73. Show that the difference of the two focal radii for the hyperbola is constant. 74. Find graphically the quotient of 6 75. Solve by inspection, for y: 2i by 3+75i. sin (90° + y) cos (90° y) + cos (90° + y) sin (90° y) = sin y. 76. Write the parametric equations for the circle, the ellipse, the hyperbola. CHAPTER XIV A REVIEW OF SECONDARY SCHOOL ALGEBRA 300. Only the most important topics are included in this review. From five to ten recitations should be given to this work before beginning regular work in Chapter I. With the kind permission of Professor Hart, a number of the exercises have been taken from the Second Course in Algebra, by Wells and Hart. 301. Special Products. The following products are fundamental: (1) The product of the sum and difference of any two numbers: If the second term of the binomial has the sign (−), then the middle term of the square has the sign (−). (3) The product of two binomials having a common term: (4) The product of two general binomials: (ax+b) (cx + d) = acx2 + (bc + ad)x + bd |