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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.
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§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.
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§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 =
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§12-3 De Moivre
Student Discussion.
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§12-3 De Moivre’s Error Function
0
x
2dxe
2
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§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%
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§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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§12- 4 Taylor and Maclaurin
Student Discussion.
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§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!.
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§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 . . .
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§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
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§12-5 Euler
Student Discussion.
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§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
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§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
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§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
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§12-7 Agnesi and du Châtelet
Student Discussion.
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§12-7 “Witch of Agnesi”y (x2 + a2) = a 3
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§12 – 8 Lagrange
Student Discussion.
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§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.
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