1

Cultural ConnectionThe Revolt of the Middle Class

Student led discussion.

The Eighteenth Century in Europe and America.

2

12 – The 18th Century and Exploitation of the Calculus

The student will learn about

More great mathematicians and the advances of applied mathematics.

3

§12-1 Introduction

Student Discussion.

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 1700 17 periodicals containing some math.

18th Century 210 periodicals containing some math.

19th Century 950 periodicals containing only math.

20th Century 2600 periodicals containing only math.

Growth of insurance, economics, technology problems, industrialization, world wars, computers, space program – applied mathematics.

5

§12-2 The Bernoulli’s

Student Discussion.

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?

Event P Value Expect

H ½ 1 ¢ ½ ¢

TH ¼ 2 ¢ ½ ¢

TTH 1/8 4 ¢ ½ ¢

TTTH 1/16 8 ¢ ½ ¢

Σ =

7

§12-3 De Moivre

Student Discussion.

8

§12-3 De Moivre’s Error Function

0

x

2dxe

2

9

§12-3 Sterling’s Formula

for n large. nn2/1 nen2!n n Actual Sterling Accuracy

13 6227020800 6187239475 0.6%

20 2.432902008 1018 2.422786847 1018 0.4%

50 3.041409320 10 64 3.036344594 10 64 0.17%

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

11

§12- 4 Taylor and Maclaurin

Student Discussion.

12

§12- 4 Taylor & Maclaurin SeriesThe Maclaurin and Taylor series are polynomials used to approximate functions.

Taylor Series -

n

1k

kk

)cx(!k

)c(f)c(f)x(f

Maclaurin Series –

c = 0 in Taylor

n

1k

kk

)x(!k

)0(f)0(f)x(f

OR...)0(f

!3

x)0(f

!2

x)0(f

!1

x)0(f)x(f iii

3ii

2i

Note that this diverges rather quickly because of the denominator of k!.

13

§12- 4 MaclaurinConsider the following three functions:

f (x) f i (x) f ii (x) f iii (x) . . .

sin x cos x - sin x - cos x . . .

cos x - sin x - cos x sin x . . .

ex ex ex ex . . .

f (x) f (0) f i (0) f ii (0) f iii (0) . . .

sin x 0 1 0 - 1 . . .

cos x 1 0 - 1 0 . . .

ex 1 1 1 1 . . .

14

§12- 4 MaclaurinThus:

...0!4

x1

!3

x0

2

x1x0)x(sin

432

Example on a graphing calculator.

OR...

!7

x

!5

x

!3

xx)x(sin

753

and...

!6

x

!4

x

!2

x1)x(cos

642

and...

!4

x

!3

x

!2

xx1e

432x

15

§12-5 Euler

Student Discussion.

16

§12-5 Euler e ix = cos x – i sin x

e ix = cos x + i sin x

...!4

x

!3

x

!2

xx1e

432x

...

!4

ix

!3

ix

!2

ixix1e

432xi

...!4

x

!3

xi

!2

xxi1

432

...!5

x

!3

xxi

!4

x

!2

x1

5342

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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§12-5 Euler

For any polyhedron the following holds:

v + f = e + 2

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§12 - 6 Clairaut, d’Alembert and Lambert

Student Comment

20

§12 - 6 is irrationalIf 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

21

§12-7 Agnesi and du Châtelet

Student Discussion.

22

§12-7 “Witch of Agnesi”y (x2 + a2) = a 3

23

§12 – 8 Lagrange

Student Discussion.

24

§12 – 9 Laplace and Legendre

Student Discussion.

25

§12 – 10 Monge and Carnot

Student Discussion.

26

§12 – 11 The Metric System

Student Discussion.

27

§12 – 12 Summary

Student Discussion.

28

Assignment

Continued discussion of Chapters 10, 11, and 12.