10 terms, by how much does their sum differ from ? 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 may differ from by less than 9 43 ? 18562 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+h4 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 xy 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 ac2 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 inches; find the increase in the area, when the radius is increased by 001 inch. [Area of circle of radius r=πr2 VII. and 31416.] T= 28. Show, by constructing a table of spaces fallen through in hundredths of seconds (1, 09, 08. ... 01 sec.) and then taking differences, that the space fallen through in the interval between any two consecutive hundredths of a second is '0032 ft. 29. If the interval were between two consecutive 1 ths of seconds, what would the space fallen 100000000 through be? 30. If the intervals were seconds what would the spaces be? VIII. 31. Show, by forming a table, that if 2 be a variable and receive small successive increments of '001, the differential co-efficient of 22=2×2. 32. If 5 be a variable and receive small increments of 0001, show, by forming a table, that the differential co-efficient of 52=2 × 5. 33. Find, by constructing a table, the second differential co-efficient of 32, supposing 3 to receive small increments of '001. 34. Supposing 25 to be a variable and to receive small increments of 0000001, what is the first differential co-efficient of 252 ? What is the second differential co-efficient? 35. If the numbers, whose squares are the functions, be supposed to vary, give the first and second differential coefficients of 192, 372, 10012. IX. 36. If 2 be supposed to vary, and to receive small increments of 0001, find, by constructing a table, the first, second, and third differential co-efficients of 23. 37. If 2 be supposed to vary and to receive small increments of 0001, find, by forming a table, the first, second, third, and fourth differential co-efficients of 2*. 38. What is the germ or essence of the 7th power? 39. What is the germ or essence of the (n−1)th power? 40. What is the germ or essence of the (p-q)th power? Prove the truth of your answer by substituting 225 for p and 220 for q. 41. Give the first differential co-efficients of (1) x2, (2)x3 (3) x (4) x17 (5) 245 (6) x100 42. Find the second, third, fourth, fifth, ninth, and twentieth differential coefficients of 20 43. Give the differential coefficients of (see Arts. 27 and 46) (1)x+x2, (2) ax2+c, 3x2- a2 (5) b 44. A cube of metal, whose edge is 12 inches, has this edge increased by 001 inch. Find the cubical expansion. XI. 45. Show, by forming a table, that, if 3 be a variable and receive small increments of 0001, the differential co-efficient of 1 3 32 46. Find, by forming a table, the differential co 1 efficient of as 5 varies and receives small increments of '001. 52 1 47. Find the differential co-efficient of 4 being 43' a variable and receiving small increments of '00001. 48. Find the differential co-efficients of 49. By constructing a table, find the second differential co-efficient of when 2 is the variable, and 23 receives small increments of '001. 50. What is the second differential co-efficient of when 3 receives small increments? 51. Find the second differential co-efficients of 52. Find the differential co-efficient of the sine of an angle, which lies between 180° and 270° (geometrically). 53. Find the differential co-efficient of the cosine of an angle, which lies between 90° and 180° (geometrically). |