1829 trace the curve which is the locus of Q; find its maximum and minimum ordinates, and the angles made by its two extremities with the axes. 72. Shew generally how to find the evolute of any curve whose equation is given, and find that of the common parabola. 73. In the expression y = 2x3 – 15x2 + 36x, find for what values of x, y is a maximum or minimum, and in each case which. 74. From the equation z = ƒ (y2 == f x to eliminate by ; trace the curve and draw its asymptotes. 76. At points of greatest and least curvature, the osculating circle will have with the curve a contact of a higher than the second order. 77. Shew how to determine the value of a vanishing fraction in all cases; and find the value of 78. Trace the curve, of which the equation is y = аха + mx5; draw its asymptote, and determine its singular points. 79. Find the equation between the angle and radius vector in a spiral, in which the radius vector is always equal to n times the chord of curvature drawn through the pole. Find also the value of the radius of curvature in such a spiral. 80. To inscribe the greatest ellipse in a given semi-circle, one axis of the ellipse being parallel to the diameter of the semi-circle. 81. Find the limiting ratio of the corresponding increments of √e2 + x2 and ; and those of and x. ex + 1 82. Expand sin x and cos x by Taylor's theorem; and find limits of the value of the terms after the nth term in each and find its maximum and minimum ordinates. 84. The chord of curvature at any point (x, y) of a curve, drawn through a point whose coordinates are a, ß, 2 (1 + p2) {y — ß − p (x − a)}, q √ (x − a)2 + (y — ß)2 85. If b, represent the coefficient of x" in the expansion of any function of ex by Maclaurin's theorem, and a, in a similar expansion of the hyperbolic logarithm of that function, prove that an = 1 nb. {---(n − 1)b ̧ an—1— (n—2)b ̧ɑn—2— - ... − b2-1α, +nbn}, and apply this theorem to determine the relation between the coefficients in the expansion of hyp. log cos x. 86. Trace the curve of which the equation is y2 (x - a) x3- b3, when a > b and when a <b. Find its asymptotes and singular points. 87. Apply Lagrange's theorem to determine the least root of the equation X3 5+7 = 0. 88. Investigate the conditions requisite in order that a function of two variables x, y may be a maximum or minimum. Apply them to find when 89. Trace the curve whose equation is y = find the number and nature of its singular points. 90. Eliminate by differentiation the constants from the equation y2 = ax + bx2, and shew how many derivatives of y2 the mth order there are to an equation containing n arbitrary constants. 1830 91. Define the differential coefficient of any function, and from that definition find the differential coefficient of uv, u and v being functions of x. 92. Find the differential of the surface of a solid of revolution. 93. Prove Maclaurin's theorem, and thence expand tan-1x to 5 terms. 94. Define the radius of curvature, and shew that in curves also that in general the circle of curvature at once touches and and determine the number and nature of its singular points. 96. Eliminate by differentiation ƒ (2) and 4 (xy) from the equation f 97. Of all spherical triangles which have the same base and equal perpendiculars from the vertex to the base, shew that the isosceles has the greatest vertical angle; and from the result prove that the same is true in plane triangles. 98. Expand f(x + h, y + k) in a series ascending by powers of h and k. 99. Prove Lagrange's theorem. 100. In each of the conic sections, the radius of curvature 101. Explain the method of finding whether a curve has multiple points, and find the number and nature of the multiple points of the curve the equation of which is y1- 2a2y2 - 2ax3 3a2x2 + a1 = 0. 102. Explain the transformation of the independent variable, and transform the equation where x is the independent variable, into one where is the independent variable, Ø being equal to cos-1x. 103. The radius of curvature of a spiral = rdr dp 104. Investigate the differential coefficients of Va2 + x2, log x, and sin x. 105. If u be a function of y, and y a function of x, then 106. Find the equation to a straight line touching a given curve at a given point; and apply it to draw a tangent to the ellipse at the extremity of the latus rectum. 107. If AP be any curve referred to a pole S, and if u be the solid generated by the revolution of the area ASP about AS, SP r, and the angle ASP = 0, = 108. Trace the curve of which amy = x (x a)m is the equation, m being an even number; find its maximum and minimum ordinates, point of contrary flexure, and the angle at which it cuts the axis of x. 109. Prove Maclaurin's theorem, and apply it to expand u as far as 3 when u3 6ux 8 0. 110. Find generally an expression for the radius of curvature of a spiral curve, and apply it to determine the radius of curvature of the reciprocal spiral, the equation to which is 111. Find the equation to a curve of the parabolic kind, that will pass through four given points. 1831 112. Investigate Lagrange's theorem. 113. Expand f(x + h, y + k), and if n = ƒ (x, y), shew that each term involving differential coefficients of the pth order will be of the form 1.2.3..p dx” dyn (n + 1)th term of expansion of (h + k)3. 114. Trace the curve whose equation is ay2 = x3 — bx2. 115. Let x and y be functions of a third variable t; it is required to determine what substitutions must be made for dy d2y d3y dx' dx2' dx31 in an expression in which is the independent variable, to change it to one in which ƒ is the independent variable. 116. Trace the curve whose equation is y X- - 3 (x − 1) (x -—2)' determine its greatest and least ordinates, and shew that it has a point of contrary flexure corresponding to an abscissa between 5 and 6. 117. Investigate the differential coefficients of √2ax + x2, a*, and sin mx. 118. Prove Taylor's theorem. 119. In determining trigonometrically the height of an object standing on a horizontal plane, where should the angle of elevation be observed, that for a given error in the observation the corresponding error in the height may be a minimum ? 120. Prove that the sectorial differential of the area of any r2d0 curve is in polar coordinates, and lar coordinates. 2 121. Obtain a general expression for the radius of curvature of a plane curve referred to rectangular coordinates, and apply it to find the radius of curvature at the extremity of the latus rectum of a parabola. |