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Calculus Chapter 9 The beginnings of calculus Introduction In this unit, you will learn how, before the introduction of calculus, mathematicians found tangents and areas enclosed by some of the simpler curves. You will then see what was distinctive about the work of Newton and Leibniz. This unit is designed to fake about 10 hours of your learning time. About half of this time will be outside the classroom. There are summaries and further practice exercises in Chapter 12. Mathematical knowledge assumed • knowledge of calculus will help you with this unit • you should have studied the work of Archimedes in Activity 4.7. 777
772 The beginnings of calculus 9.1 Euclid's definition of tangents 9.2 Archimedes's spiral 9.3 Fermat and maxima 9.4 Fermat and I constructing tangents I 9.5 Descartes's approach 9.6 Using indivisibles 9.7 Leibniz and area 9.8 Relating gradients and integration 9.9 Newton and differentiation Calculus was developed to solve two major problems with which many mathematicians had worked for centuries. One was the problem of finding extreme values of curves - the problem now called finding maxima and minima. The other was the problem of quadrature, or finding a systematic way to calculate the areas enclosed by curves. Until Leibniz and Newton applied themselves to the problems, several different geometrical methods existed for .solving these problems in specific cases. There was no general method which would work for any curve. The chapter begins with some of these geometrical methods, so that you can see how difficult they are and realise why there was such a need for a general method. It goes on to show something of the pioneering work of Fermat, Descartes, Cavalieri, Leibniz and Newton as they developed their ideas. Activities 9.1 and 9.2 show how Euclid and Archimedes drew tangents to two very special curves. In Activities 9.3 to 9.5 you see how Fermat and then Descartes approached the idea of drawing a tangent to a curve. Activities 9.6 and 9.7 show the beginnings of the modern ideas and notations for integrals. Activity 9.8 shows how Leibniz thought of the relation between differentiation and integration, and Activity 9.9 shows how Newton developed a system for differentiation. All the activities are suitable for working in a small group. Work through the activities in sequence. Early methods for finding tangents To find an extreme value, you have to find a tangent to a curve. For the Greeks, this was a problem of pure geometry. Later, in the 17th century, it became important for the science of optics studied by Fermat, Descartes, Huygens and Newton, and in the science of motion studied by Newton and Galileo. The work that the Greeks were able to do was limited by three factors: • their unsatisfactory ideas of angle - they tried to include angles in which at least one of the sides was curved (see Figure 9.1)
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Nuffield Advanced Mathematics Histo
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Contents Introduction 7 Unit 1 The
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Introduction This book, about the h
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The Babylonians Chapter 1 Introduct
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Introduction to the Babylonians On
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Figure 1.2 Stele inscribed with Ham
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Token V Figure 1.5 7 Introduction t
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There are no practice exercises for
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A sexagesimal number system is a sy
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ff m yfr < n Figure 2.5 «« Activi
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Activity 2.5 Babylonian fractions 2
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Activity 2.6 Babylonian arithmetic
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Figure 2.10 Activity 2.9 2 Babyloni
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Type Quadratic Table 2.1 x +ax = b
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Activity 2.12 Geometry Figure 2.13a
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Practice exercises for this chapter
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3 An introduction to Euclid fthese
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The Greeks Activity 3.3 B Figure 3.
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34 The Greeks Activity 3.4 g Now us
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36 The Greeks inscribes another wit
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38 The Greeks Activity 3.7 therefor
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The Greeks Interpret postulate 3 in
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42 More Greek mathematics 4.1 Some
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The Greeks Q.E.D. stands for 'quod
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Figure 4.3 46 The Greeks • treat
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Figure 4.6 48 The Greeks together w
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The Greeks Activity 4.8 The cone Th
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The Greeks Practice exercises for t
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5 Arab mathematics You should think
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The Arabs Western Arabic, or Gobar,
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The Arabs Activity 5.2 Figure 5.3a
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Page 73 and 74: 68 The Arabs Figure 5.10 Activity 5
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Page 77 and 78: 72 6.1 Read about Descartes The app
Page 79 and 80: Descartes Activity 6.1 Head about D
Page 81 and 82: 76 Descartes Figure 6.5 Figure 6.5
Page 83 and 84: Descartes fViete (Vieta in Latin) w
Page 85 and 86: Descartes The next two sections con
Page 87 and 88: 82 Descartes Activity 6.9 a 3 - b 3
Page 89 and 90: i——— Figure 6.12 Figure 6.13
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Page 93 and 94: Figure 7.2 Descartes Activity 7.1 i
Page 95 and 96: N- i- L Descartes Thus I have In th
Page 97 and 98: A Descartes Activity 7.7 Reflecting
Page 99 and 100: 94 Descartes 23 It is true that the
Page 101 and 102: 96 Descartes Figure 7.11 shows a ge
Page 103 and 104: 98 Descartes It is clear that the p
Page 105 and 106: Descartes ractice exercises lor thi
Page 107 and 108: 102 Descartes Activity 8.1 Activity
Page 109 and 110: 104 Descartes In the next section y
Page 111 and 112: Descartes Today we call these numbe
Page 113 and 114: Descartes Activity 8.9 Normals Figu
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Page 119 and 120: Figure 9.4 774 Calculus you can wri
Page 121 and 122: Calculus including Descartes, under
Page 123 and 124: 118 Calculus Activity 9.7 Cavalieri
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Page 129 and 130: 10.1 Visualising negative numbers 1
Page 131 and 132: 126 Searching for the abstract For
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Page 157 and 158: 752 12 Summaries and exercises Two
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Page 161 and 162: 156 12 Summaries and exercises usin
Page 163 and 164: 158 Hints Activity 2.3, page 15 2 F
Page 165 and 166: 13 Hints Activity 7.3, page 89 1 Wr
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outcome 1 step to the right Figure
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14 Answers 3 0;6,40 and 0;2,13,20 4
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14 Answers 3 If AD intersects the l
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74 Answers 2 Suppose that you can f
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14 Answers 4 If he had a daughter t
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14 Answers Activity 6.5, page 79 1
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14 Answers 6 The equation z 2 + az
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14 Answers b This meets the ellipse
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14 Answers To get an interpretation
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14 Answers of multiplication by neg
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14 Answers b The required number is
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14 Answers operator has the effect
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14 Answers A i- B 5 E %2 5)-25 = 39
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Index Ptolemy 49, 55, 58 Pythagoras
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