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CONTENTS
ix
SECTION PAGE 61. Differentiation of................................................................................................................................................................................................................................................................................................................................................................................................65 62. Implicit functions................................................................................................................................................................................................................................................................................................................................................................................................69 63. Differentiation of implicit functions................................................................................................................................................................................................................................................................................................................................................................................................69 CHAPTER VI
SIMPLE APPLICATIONS OF THE DERIVATIVE64. Direction of a curve................................................................................................................................................................................................................................................................................................................................................................................................73 65. Equations of tangent and normal, lengths of subtangent and subnormal. Rectangular coördinates................................................................................................................................................................................................................................................................................................................................................................................................76 66. Parametric equations of a curve................................................................................................................................................................................................................................................................................................................................................................................................79 67. Angle between the radius vector drawn to a point on a curve and the tangent to the curve at that point................................................................................................................................................................................................................................................................................................................................................................................................83 68. Lengths of polar subtangent and polar subnormal................................................................................................................................................................................................................................................................................................................................................................................................86 69. Solution of equations having multiple roots................................................................................................................................................................................................................................................................................................................................................................................................88 70. Applications of the derivative in mechanics. Velocity................................................................................................................................................................................................................................................................................................................................................................................................90 71. Component velocities................................................................................................................................................................................................................................................................................................................................................................................................91 72. Acceleration................................................................................................................................................................................................................................................................................................................................................................................................92 73. Component accelerations................................................................................................................................................................................................................................................................................................................................................................................................93 CHAPTER VII
SUCCESSIVE DIFFERENTIATION74. Definition of successive derivatives................................................................................................................................................................................................................................................................................................................................................................................................97 75. Notation................................................................................................................................................................................................................................................................................................................................................................................................97 76. Thederivative
................................................................................................................................................................................................................................................................................................................................................................................................98 77. Leibnitz's formula for the................................................................................................................................................................................................................................................................................................................................................................................................98 78. Successive differentiation of implicit functions................................................................................................................................................................................................................................................................................................................................................................................................100 CHAPTER VIII
MAXIMA AND MINIMA. POINTS OF INFLECTION. CURVE TRACING79. Introduction................................................................................................................................................................................................................................................................................................................................................................................................103 80. Increasing and decreasing functions................................................................................................................................................................................................................................................................................................................................................................................................106 81. Tests for determining when a function is increasing and when decreasing................................................................................................................................................................................................................................................................................................................................................................................................108 82. Maximum and minimum values of a function................................................................................................................................................................................................................................................................................................................................................................................................109 83. First method for examining a function for maximum and minimum values................................................................................................................................................................................................................................................................................................................................................................................................111 84. Second method for examining a function for maximum and minimum values................................................................................................................................................................................................................................................................................................................................................................................................112 85. Definition of points of inflection and rule for finding points of inflection................................................................................................................................................................................................................................................................................................................................................................................................125 86. Curve tracing................................................................................................................................................................................................................................................................................................................................................................................................128
