58. Write down the general expression for the equation to the normal at any point of a curve, explaining what each symbol indicates. 59. Find the equation to the tangent at any point of the curve x3+y3=a3, and show that the portion of the tangent intercepted between the axes of coordinates is constant. 60. Find the equation to the tangent, at the point (x'y'), to the hyperbola 21-22=1. 62. A right circular cylinder is inscribed in a given right cone; find the radius of the cylinder when the whole surface of the solid is a maximum. 63. Find the radius of curvature at any point of an ellipse, and express it in terms of the focal distances of the point. 64. Find in the following examples : du dx du dx 65. Show that u= (x) will have a maximum value when a root of =0 substituted in gives a negative result. d2u dx2 Can there be a maximum value of u if d2u =0, for dx2 du the same value of x that makes =0? dx 66. Given the volume of a right cone, find its altitude when the surface is a maximum. 67. Find the equation of the tangent to the curve (*)*+ (1⁄2)"=2, and show that at the point (a, ¿) it is the same, whatever be the value of n. 68. Investigate the differential expression for determining the area of a plane curve referred to polar coordinates, and apply it to determine the whole area of the looped curve r2a2 cos 2 0. 69. Expand sin-i (x+h) to three terms by Taylor's theorem. 70. If u=: f(x) • (x)' deduce the rule for finding the differen tial coefficient of u with respect to x. 71. Investigate Maclaurin's theorem, and expand by means of it (1) etan ̄1x and (2) ɛa tan-1x involving x3. as far as the term 72. Determine the numerical value of the radius of cur vature at the origin of the curve y=x-4x3- 18x2. 73. Inscribe the greatest ellipse in a given semicircle. 74. Expand (1) tan-1 x, (2) log. (1+x), by Maclaurin's 76. Find the least isosceles triangle which can circumscribe a given circle. 77. Find the radius of curvature, in polar coordinates, at the extremity of the latus rectum of a parabola. 78. Find the asymptotes and singular points of the curves 79. Two ships are sailing uniformly with velocities u, v, along lines inclined at an angle 0. Given that at a certain time the ships are distant respectively a and b from the point of intersection of their courses, find the least distance between the ships. du dx dy_y 80. If y3—3x2y+x3=0, prove dx 81. If u=f(x) has a maximum value, show that generally d2u =0, and that is finite and negative. dx2 82. Of all triangles on the same base and having the same perimeter, show that the isosceles triangle is the greatest. 83. If r be the radius of curvature at any point of a plane curve, p the corresponding perpendicular on the tangent, show that the radius of curvature at the point where r In the ellipse the equation from the focus being b2r p2 = = where a, b are the semi-axes major and 9 2a -r minor, find from the above expression the radius of curvature at the extremity of the axis minor. 84. Find the maxima and mimima values of the function x2+log (x-3). 85. Prove that the locus of the feet of the perpendiculars from the origin on the tangent to the curve ay2=x3 is given by the equation 27y2 (x2+y2)=4ax3; and that of the locus of the feet of the perpendiculars from the origin on the normals is 27x3 (x2+y2)=4ay2 (2y2+3x2). 86. Enunciate and prove Taylor's theorem, and explain what is meant by the failure of Taylor's theorem. Deduce Maclaurin's theorem from Taylor's theorem, and apply it to find the coefficient of x in the expansion of ex cos x in powers of x. 87. If y3—xy2—1=0, find the first four terms of the expansion of y in terms of x. 88. Explain any method of finding the asymptotes of a curve given by a rational algebraic equation in x and y. Find the asymptotes of the curves (1) (y-x) (y2-4xy+3x2)=4x2+5xy+1. 89. Define the circle of curvature at any point of a curve, and prove that the radius of curvature 90. If s be the arc of a curve, measured from the point where it cuts the axis of y, and if y2-s2 be constant, prove that the radius of curvature varies as y2. 91. Enunciate and prove Leibnitz's theorem for the differentiation of the product of two functions. If y=ax3+bx2+cx, prove that 92. If ƒ (a)=0, and 4(a)=0, show how to find the ulti mate value of the fraction f(x) as x approaches the value a. (x) Examine the case in which ƒ(a) and ø(a) are both infinite. 93. What is meant by contact of the nth order between two curves? Prove that the coordinates of the vertex of a parabola having contact of the second order with a curve at the point (x, y), and having its axis parallel to the axis of x, are Find the locus of the vertex when the given curve is the circle x2+y=a2. INTEGRAL CALCULUS. CXXVI. 1. Explain the meaning of Integration; and find the ეს value of xdx, regarding it as the limit of a certain summation. 2. Show how to integrate the fraction F(x) f(x)' when the roots of the equation ƒ (x)=0 are real and unequal. 3. Integrate the following functions of x : (1) sin ax sin bx; (2) x2 cos x; (3) x2 (x-a)(x-b) 4. Integrate, with respect to x, the functions : |