rectangle (a-x)x dc1 = a -2x, therefore there is a maximum, since the dc, is negative, and this maximum is given by equating de1 to 0. a-2x=0, that is to say, the line must be bisected. 5. Let AB be the diameter of a given circle, it is required to find a point C in the diameter, so that the rectangle formed by the chord DE, which is perpendicular to AB, and the part AC may be the greatest possible. and this is to be a maximum; if it is, its square will also be, viz., or and but :: or i.e., 4x2(ax-x2), 12αx2-16x3=0, 16x3=12αx2, Therefore there is a maximum, and it is given by the value ga―i.e., we must take of a to find C. 6. To approximate to the roots of an equation. Let the equation be x3-3x+1=0, x3-3x+1 being a function of x. But and function = (function) + (dc) xh, since the function is the left-hand side of the equation. :: (function)+(de1) × h=0, Now, by trial, 15 is found to be near one of the roots. Let h be the difference between 15 and the root, so that x=1.5+h, which is of the form (a+h). .. (function) = a3-3a+1 = (1·5)3 — 3 × 1·5+1 The Roman numbers refer to the sections in the body of the magnitude, and state the value of each when b=a. 4. Develop into a series, by actual division, the and show, by this means, that its value 10 terms, by how much does their sum differ from 9 ? 6. How many terms of the series must be taken in order that their sum may differ from by less than 9 7. How many terms must be taken that their sum 12. Find the value of the fraction in (8) by substituting a-h for b. 13. Show how in (8) the value of the fraction becomes more and more nearly the value of the limit, as b approaches a, by means of numerical illustrations. 16. What is the limit to which the ratio of h2: 3x2h2+3xh3+h1 approaches, as h diminishes and ultimately vanishes ? V. 17. Define Differential Co-efficient. 18. State what you mean by a function; and give 5 examples of a function of x, 5 of a function of y, and 5 of a function of z. 19. If y be a variable quantity and receive small increments of 1, show that the corresponding values of '01 x y increase uniformly. 20. If px-C be a function of x, show that it increases uniformly as the variable receives successive increments of a+b. 21. Find the differential co-efficient of 5x. 22. Give the differential co-efficients of 23. If the side of a square increase uniformly at the rate of 3 feet per second, at what rate is the area of the square increasing when the side becomes 10 feet? 24. If x increase uniformly at the rate of 2 per unit of time, at what rate does ax2 increase when a= = 4, and x=10? 25. If x increase uniformly at the rate of 1 per unit of time, at what rate does the value of the function a+2x2 increase when a=4, and x=6. 26. If x increase uniformly at the rate of 1 per second, at what rate does increase when x becomes x2 α 4, the constant a being equal to 10? 27. The radius of a circular plate of metal is 12 |