The second differential coefficient is positive, and the walling BCDA is a minimum. 11. Construct a cylinder whose total surface may be equal to a given area, and its volume a maximum. a2 Let the radius of the base of the cylinder, y = its height, and x = the given surface; then = = area of the base, and 2x = the circumference of base; Consequently the cylinder is equilateral, having its altitude equal to the diameter of its base, and its volume is obviously a maximum, since d2 u dx2 12. A small plane surface is placed horizontally upon a table, and illuminated by a lamp placed at a given horizontal distance; find the height of the flame so that the plane shall receive the greatest illumination from it. = Let a = the given distance P B of the small plane or disk from the foot of the lamp B F, and let 0 the variable angle FPB; therefore we have PB a cos FPB cos Ꮎ tance from F to P. = a sec = the dis Now the degree of illumination varies inversely as the square of the distance FP, and directly as the sine of the inclination FPB; hence we have cos 20 1 sin e X a2 = sin cos 20 = a maximum. O, which gives tan 0 = Differentiating a second time, we get d 02 sin cos 20 (2 tan 20 - 7) which vanishes for cos = 0. √2, then both sin 0 and cos 0 are positive, and - 7 is negative; hence this value renders the function a maxi 13. A and B are two known objects in the straight line A B C; find a point P in the given line CD from which these objects shall subtend the greatest angle. Let A C = a, BC = b, CP = x, and draw P A, P B, and also P M perpendicular to A B C; then by trigo nometry we have (angle ACP being = 0) Now if the angle APB be a maximum, its tangent will also be a maximum; hence suppressing the constant factor (a - b) sin 0, and taking the reciprocal of the expression, we get 14. Find the greatest area that can be included by four given straight = b, CD = c, DA = d, and draw the diagonal ABC = 0, and angle ADC; then, by 2 ab cos 0 A C2 = c2 + d2 2 cd cosp; = (1). Again, the double area of the quadrilateral gives u = ab sin cd sin = a maximum . (2). Differentiating (1) and (2) with respect to as the independent variable, we get ab sin c d sin du = d Ꮎ аф = 0 (3); d Ꮎ ab cos 0 + c d cos p Multiply (3) by cos p, and A (4) by sin p, and add; then we get *This is not generally true, but if the angle be less than 90°, it may be employed with much advantage. The following method is preferable, though not quite so simple: ab (sin cos + cos e sin ) = = sin ; 0, or sin (+6) = 0 hence + = T, and the quadrilateral may be inscribed in a circle. Also the area of the quadrilateral is found by (1) and (2) to be EXERCISES IN MAXIMA AND MINIMA. 1. Divide 21 into two parts so that the less multiplied by the square of the greater may be a maximum. Ans. 14 and 7. 2. Inscribe the greatest rectangle in a triangle whose base is a feet and perpendicular b feet. Ans. The sides area and b. 2x + 5 be a maximum or minimum? Ans. When x = u = 41, a minimum, 3. When will u 4 23 хо 4. Of all triangles upon the same base, and having the same perimeter, the isosceles has the greatest area. 5. Inscribe the greatest isosceles triangle in a circle whose radius is a. Ans. The triangle is equilateral, side = a√√3, area = a2√3. 6. Of all the squares inscribed in a square whose side is a, which is the least? Ans. The side is a√2. 7. To divide the number a into two parts, such that the product of the power of the one by the nth power of the other shall be a maximum. mth 8. To describe the least isosceles triangle about a circle whose radius is a. Ans. The triangle is equilateral, side 9. Bisect a triangle by the shortest line. Ans. The length of the line = {} (a + b − c) (a − b + c)}. 10. The edges of a rectangular piece of tin are to be turned up so as to form the greatest rectangular vessel; the length of the piece of tin is a inches and the breadth b inches; find how much of the edge must be turned up. Ans. (a+b — √ (a2 — a b + b2)}. 11. Find the dimensions of the greatest cone which can be cut out of a sphere whose radius is a. r, height ja. 12. Find the dimensions of the greatest cylinder that can be formed out of a cone whose altitude = = 13. A cistern, open at top, is to be formed with a square base; find its dimensions so that a superficial feet of lead may cover it. 14. What is the height of the greatest cylinder that can be cut out of a sphere whose radius is a. Ans. Height a/3. 15. Through a given point, between the lines containing a given angle, to draw a straight line which shall cut off the least triangle, the distances of the point from the lines being a and b. VOL. I. Ans. The line is bisected in the given point. 2 A 16. Inscribe the greatest rectangle in an ellipse, the major axis being = 2a and the minor axis = 2 b. 17. Through a given point between the lines containing a right angle, to draw the shortest line terminated by the given lines, the distances of the point from the lines being a and b. Ans. The segments of the lines cut off by the shortest line are a3 (a* + b3) and b3 (a* + b3); and the line itself is (a* + b3)3. 18. Cut the greatest parabola from a given cone. Ans. Axis is ths of the slant side of the cone. 19. Inscribe the greatest parabola in a given isosceles triangle. Ans. Axis is ths of the altitude of the triangle. 20. Given the whole surface of a cylinder, to find its form when its volume is a maximum. Ans. Altitude the diameter of the base. = 21. Given the length of a circular arc = 2a, to find what part of a circle it must be that the area of the corresponding segment may be a maximum. Ans. It must constitute a semicircle, whose radius is 2 a =3 T 22. Of all the cones whose convex surface is a, find that whose volume is a maximum. 23. If the whole surface of a cone be = a, what are its dimensions when the volume is a maximum. 24. To determine the maximum and minimum values of y in the equation y 2axy + x2 = m2. 25. Given u = sin me sin "(0), to find 0 that u may be a maximum or minimum. 26. The height of an inclined plane is a feet; find its length, so that a given weight P, acting freely upon another weight W, in a direction paralle to the plane, may draw it up the plane in the least time. 27. P and Q are two weights suspended over a fixed pulley, where P is given and greater than Q; find the weight of Q, so that P, acting time. freely upon Q, may communicate to it the greatest momentum in a given Ans. Q = P(√ 2 − 1). 28. A, B, C are three elastic bodies in a straight line of which A and C are given; find B so that the motion communicated by A to C through B may be a maximum. Ans. B = (AC). 29. Find the straight line of quickest descent from a given point to a given circle in the same vertical plane. TANGENTS TO CURVES. 50. Let A PB be a curve referred to rectangular coordinates O X and O Y, and let it be required to draw a tangent to the curve at a given point P in the curve. Let P and Q be two points of the curve, and S P Q an indefinite straight line drawn through these two points. Through P and Q draw PM,QN parallel to the axis of y, and PV parallel to the axis of x, meeting QN in V. Conceive the point Q to move towards P; then the direction of the secant S P Q will approach a certain limiting position as Q approaches P. Let PT be the limiting position, then PT is called the tangent to the curve at the point P. Draw PR at right angles to PT, then PR is called the normal to the curve at the point P. Also M T is called the subtangent, and M R the subnormal. = y'; then will tan PSX Q V y' - y = = ; V P h = y, Let y = fx be the equation to the curve, O M = x, M P ON = x + h, NQ and since PT is the limiting position of QPS, when Q approaches P, that is, when y'. y and h approach zero; therefore Angle T PR being a right angle, we have angle M PR= angle PTX; The lengths of the portions of the tangent and normal intercepted between the point of contact and the axis of x are given by the equations |