By Isaac Todhunter

This Elibron Classics ebook is a facsimile reprint of a 1864 variation by means of Macmillan and Co., Cambridge and London.

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**Additional resources for A Treatise On The Differential Calculus with numerous examples**

**Example text**

X Ϫ y)1ր3 87. What is the result of entering 23 on a calculator? 2 88. Refer to Problem 87. What is the difference between 2(3 ) and 3 2 32 (2 ) ? Which agrees with the value of 2 obtained with a calculator? APPLICATIONS In Problems 33–38, change to rational exponent form. Do not simplify. 33. 1361 3 34. 2172 35. 4x 2y3 5 4 36. 27x3y2 3 37. 2x2 ϩ y2 3 3 38. 2x2 ϩ 2y2 In Problems 39–50, evaluate each expression that represents a real number. 39. 1001ր2 40. 1691ր2 41. 1121 42. 1361 1ր3 44. 27 3 45.

Z Integers Natural numbers, their negatives, and 0 (also called whole numbers) . . , Ϫ2, Ϫ1, 0, 1, 2, . . 414213 . . 71828182 . † R Real numbers Rational numbers and irrational numbers *The overbar indicates that the number (or block of numbers) repeats indefinitely. †Note that the ellipsis does not indicate that a number (or block of numbers) repeats indefinitely. qxd 10/12/09 4:38 PM Page 3 SECTION R–1 3 Algebra and Real Numbers so it is true that every element of the empty set is an element of any given set.

4,390 17. 027 76. 2 18. 11 19. 000 000 064 1 2 15 78. 20. 000 0325 In Problems 21–26, write each number in standard decimal form. 21. 4 ϫ 10Ϫ3 22. 5 ϫ 10Ϫ6 23. 99 ϫ 105 24. 75 ϫ 1011 25. 1 ϫ 10Ϫ7 26. 167 ϫ 10Ϫ4 12y 16y 5 6 7 11 81. x 2 3xy 84. 3 1y 2 1y Ϫ 3 79. 1 77. 3 1 7 4 16 Ϫ 2 3 82. 2a 2 8a8b13 85. 2 15 ϩ 312 5 15 ϩ 212 3 3 254 80. 12 16 ϩ 2 83. 12m15 120m 86. 312 Ϫ 213 313 Ϫ 212 2 In Problems 27–32, change to radical form. Do not simplify. 27. 321ր5 28. 6253ր4 29. 4xϪ1ր2 30. 32yϪ2ր5 31. x1ր3 Ϫ y1ր3 32.