54. Find the differential co-efficient of the tangent of an angle, which lies between 270° and 360° (geometrically). 55. Find the differential co-efficient of the cotangent of an angle, which lies between 90° and 180° (geomettrically). 56. Find the differential co-efficient of the secant of an angle, which lies between 180° and 270° (geometrically). 57. Find the differential co-efficient of the cosecant of an angle, which lies between 270° and 360° (geometrically). 58. Find that angle which increases twice as fast as its cosine. XIV. 59. Find the differential co-efficient of cos-1x. 60. Find the differential co-efficient of cot-1x. 61. Find the differential co-efficient of cosec -1x. XV. 62. Establish, by taking successive cube roots of 1000, the principle laid down in Section XV. 63. What is the value of 95an-x when x=n? 64. What is the value of 1000° -1° ? natural cosine of 30°8660254, and difference and natural sine of 30°='5000000; let 30° receive small increments of 001 and show, by constructing a table, that the differential co-efficient of cos x= - sin x approximately. 66. Having given arc= angle × 3·1416; 180° natural cot 14°=40107809, difference for and natural cosec 14° = 4·1335655; let 14° receive small increments of 1 and show, by constructing a table, that the differential co-efficient of cot 14° - cosec214°. 67. Find the angle which increases at the rate of 2 times the rate of its sine. Convert these into Naperian logarithms, and show that differential co-efficient of Nap. log 62300= 69. Given Common log 33.863=1.5297254, 1 62300 log 33.864-1.5297383, Convert these into Naperian logarithms, and show that differential co-efficient of Nap. log 33.863= XVIII. 1 33.863 70. Show by successive differentiating that the fourth differential co-efficient of x4+x3+x2+x+1=2x 3 x 4. 71. Find the fourth differential co-efficient of 72. Find the eighth differential co-efficient of x”. 73. Find the third differential co-efficient of 75. Find the fifth differential co-efficient of x-x-4. 75. Required the seventh and eighth differential coefficients of cosx. 77. Expand cosx, by Maclaurin's theorem, in terms of x. 78. Differentiate the series in (77), and show that the result is the expression for sinx. 79. Approximate to the roots of the equation x3-12x-28=0. 80. Approximate to the roots of the equation x2+x-3=0. XIX. 81. Find when 16x - x2 will be a maximum or a minimum. 82. Find when the function 2x3-9αx2+12a2x - 4a3 will be a maximum or minimum, and give the value of the function which is a maximum or minimum. is a maximum and a minimum. 85. Give the maximum and minimum values of the function 4x3- x2 - 2x+1. 86. Give the maximum and minimum values of 23-7x2+8x+32. 87. Find the fraction which exceeds its second power by the greatest possible quantity. 88. Divide the quantity a into two such parts that their product shall be the greatest possible. 89. Divide a given line AB into two parts so that the sum of the areas of the squares described on the parts shall be the least possible. 90. A gentleman has a plot of ground in the form of a triangle, the base of which is 400 feet and the perpendicular 300 feet, in which he wishes to make the greatest rectangular garden possible, one of the sides of which is in the base. It is required to find how many feet from the vertex the other side must be drawn. MISCELLANEOUS EXERCISES. 91. Upon AB describe a semi-circle, draw a chord AP; draw PN perpendicular to AB; then prove that AP=PN ultimately-i.e., at the moment when the arc AP vanishes. AP= √2ax and PN= √2ax - a2. 92. Develop into a series, by Maclaurin's theorem, √a+x. 93. In (92) put a=1, then √a+x= √1+x. Now, by putting x=1, find the value of/2, correct to three decimal places. 94. Expand into a series, by Maclaurin's theorem, 3/1+; and, by substituting 8 for x, give the series for the calculation of 3/9. 95. Find the differential co-efficient of 97. Find the value, when x=2, of the fraction x3- x2-8x+12 x-9x2+4x+12 98. If x increase uniformly at the rate of 1 per 4x3+ a second, at what rate is the expression increasing b when a becomes 10, a being equal to 4 and b to 6? |