10

1
10 Analytic Geometry and Precalculus
Development
The student will learn about
Some European mathematics leading up to the
calculus.
2
10-1 Analytic Geometry
Student Discussion.
3
10-2 René Descartes
Student Discussion.
4
10-2 René Descartes
I think therefore I am.
In La géométrie part 2 he wrote on construction
of tangents to curves. A theme leading up to the
calculus.
In La géométrie part 3 he wrote on equations of
degree gt 2. The Rule of Signs, method of
undetermined coefficients and used our modern
notation of a 2, a 3, a 4, . . . .
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10-3 Pierre de Fermat
Student Discussion.
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10-3 Pierre de Fermat
Little Fermat Theorem If p is prime and a is
prime to p, then a p 1 1 is divisible by p.
Example Let p 7 and a 4. Show 4 7 1 1
is divisible by 7. 4 7 1 1 4096 1 4095
which is divisible by 7.
Every non-negative integer can be represented as
the sum of four or fewer squares.
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10-3 Pierre de Fermat
Fermats Last Theorem There do not exist
positive integers x, y, z such that x n y n z
n, when n gt 2.
Case when n 2..
To divide a cube into two cubes, a fourth power,
or general any power whatever into two powers of
the same denomination above the second is
impossible, and I have assuredly found an
admirable proof of this, but the margin is too
narrow to contain it.
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10-4 Roberval and Torricelli
Student Discussion.
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10-4 Torricelli
Found the area under and tangents to cycloids.
Visit Florence, Italy and view the bridge over
the Aarn river.
Isogonal center of a triangle. The point whose
distance to the vertices is minimal. This is
called the Fermat point in many texts.
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10-5 Christiaan Huygens
Student Discussion.
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10-5 Christiaan Huygens
Improved Snells trigonometric method for finding
?. More on this topic later.
Invented mathematical expectation.
Did much work in improving and perfecting clocks.
Why was this important?
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10-6 17th Century in France and Italy
Student Discussion.
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10-6 Marin Mersenne
Primes of the form 2 p 1.
2 2 1 3 2 13 1 8191
2 3 1 7 2 17 1 131,071
2 5 1 31 2 19 1 524,287
2 7 1 127 2 23 1 8,388,607
2 11 1 2039 2 29 1 536,870,911
If p 4253 the prime has more than 1000 digits.
Visit web sites to find the current largest
Mersenne prime number.
http//www.mersenne.org/prime.htm
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10-7 17th Century inGermany and the Low
Countries
Student Discussion.
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10-7 Willebrord Snell
Improvement on the classical method of ?.
and if r 1,
N Sn N(Sn) N(Sn)/2
6 1.0000000000 6.0000000000 3.0000000000
12 0.51763809 6.211657082 3.105828541
24 0.261052384 6.265257226 3.132628613
48 0.130806258 6.278700406 3.139350203
96 0.0654381 6.2820639 3.1410309
192 0.0327234 6.2829048 3.1414529
384 0.01636222 6.2831154 3.1415577
768 0.0081812 6.2831694 3.1415847
1536 0.0040906 6.2831788 3.1415894
3072 0.0020453 6.2831976 3.1415988
6144
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10 - 7 Huygens Improvement on Snell
AP AT if ? is small.
AP AT tan ? tan (?/3) sin ? /(2 cos
?)
If ? 1 (I.e. 360 sides) then AP 0.017453293
And 180 AP 3.141592652
Which is accurate to 0.000000002
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10 7 Nicolaus Mercator
Converges for - 1 lt x ? 1.
Show convergence on a graphing calculator.
Let x 1
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10 8 17th Century in Great Britain
Student Discussion.
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10 8 Viscount Brouncker
Area bounded by xy 1, x axis, x 1, and x 2,
is
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10 8 James Gregory
Which gives ? as 3.15786 for the first three
terms but which starts to converge more rapidly
as the denominators increase.
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Assignment
Discussion of Chapter 11.