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1 Cultural Connection The Revolt of the Middle Class Student led discussion. The Eighteenth Century in Europe and America.

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2 12 – The 18 th Century and Exploitation of the Calculus The student will learn about More great mathematicians and the advances of applied mathematics.

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3 §12-1 Introduction Student Discussion.

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4 §12-1 Introduction Fifty percent of all known mathematics was created during the past fifty years, and fifty percent of all mathematicians who have ever lived are alive today. Before 170017 periodicals containing some math. 18 th Century210 periodicals containing some math. 19 th Century950 periodicals containing only math. 20 th Century2600 periodicals containing only math. Growth of insurance, economics, technology problems, industrialization, world wars, computers, space program – applied mathematics.

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5 §12-2 The Bernoulli’s Student Discussion.

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6 §12-2 The Petersburg Paradox Nicolaus Bernoulli. If A receives a penny when a head appears on the first toss of a coin, two pennies if a head does not appear until the second toss, four pennies if a head does not appear until the third toss, and so on, what is A’s expectation? EventPValueExpect H½1 ¢½ ¢ TH¼2 ¢½ ¢ TTH1/84 ¢½ ¢ TTTH1/168 ¢½ ¢ Σ =

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7 §12-3 De Moivre Student Discussion.

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8 §12-3 De Moivre’s Error Function

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9 §12-3 Sterling’s Formula for n large. nActualSterlingAccuracy 13622702080061872394750.6% 202.432902008 10 18 2.422786847 10 18 0.4% 503.041409320 10 64 3.036344594 10 64 0.17%

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10 §12-3 De Moivre’s Formula (cos x + i sin x) n = cos nx + i sin nx Contained in a typical high school trig class and can be shown through expansion. I.e. = cos 2 x – sin 2 x + 2 i cos x sin x = cos 2x + i sin 2x (cos x + i sin x) 2 = cos 2 x + 2 i cos x sin x + i 2 sin 2 x

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11 §12- 4 Taylor and Maclaurin Student Discussion.

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12 §12- 4 Taylor & Maclaurin Series The Maclaurin and Taylor series are polynomials used to approximate functions. Taylor Series - Maclaurin Series – c = 0 in Taylor OR Note that this diverges rather quickly because of the denominator of k!.

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13 §12- 4 Maclaurin Consider the following three functions: f (x)f i (x)f ii (x)f iii (x)... sin xcos x- sin x- cos x... cos x- sin x- cos xsin x... exex exex exex exex f (x)f (0)f i (0)f ii (0)f iii (0)... sin x010- 1... cos x10- 10... exex 1111

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14 §12- 4 Maclaurin Thus: Example on a graphing calculator. OR and

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15 §12-5 Euler Student Discussion.

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16 §12-5 Euler e ix = cos x – i sin x e ix = cos x + i sin x

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17 §12-5 Euler (e i = - 1 + 0) In the previous equation let x = . e i = - 1 + 0 e ix = cos x + i sin x Real number to an imaginary power is a real number! The five basic constants in mathematics in one neat formula. Hence, God must exist!

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18 §12-5 Euler For any polyhedron the following holds: v + f = e + 2

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19 §12 - 6 Clairaut, d’Alembert and Lambert Student Comment

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20 §12 - 6 is irrational If x 0 is rational, then tan x is irrational. The contra positive of the above is then logically true, or if tan x is rational then x is irrational. But tan /4 = 1 so /4 is irrational and hence is irrational. Johann Lambert

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21 §12-7 Agnesi and du Châtelet Student Discussion.

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22 §12-7 “Witch of Agnesi” y (x 2 + a 2 ) = a 3

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23 §12 – 8 Lagrange Student Discussion.

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24 §12 – 9 Laplace and Legendre Student Discussion.

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25 §12 – 10 Monge and Carnot Student Discussion.

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26 §12 – 11 The Metric System Student Discussion.

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27 §12 – 12 Summary Student Discussion.

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28 Assignment Continued discussion of Chapters 10, 11, and 12.

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Www.le.ac.uk Differentiation Department of Mathematics University of Leicester.

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