10 – analytic geometry and precalculus development
DESCRIPTION
10 – Analytic Geometry and Precalculus Development. The student will learn about. Some European mathematics leading up to the calculus. §10-1 Analytic Geometry. Student Discussion. §10-2 René Descartes. Student Discussion. §10-2 René Descartes. I think therefore I am. - PowerPoint PPT PresentationTRANSCRIPT
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10 – Analytic Geometry and Precalculus Development
The student will learn about
Some European mathematics leading up to the calculus.
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§10-1 Analytic Geometry
Student Discussion.
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§10-2 René Descartes
Student Discussion.
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§10-2 René DescartesI 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 > 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
Fermat’s Last Theorem – There do not exist positive integers x, y, z such that x n + y n = z n, when n > 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 Snell’s 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 MersennePrimes of the form 2 p – 1.
If p = 4253 the prime has more than 1000 digits. Visit web sites to find the current largest Mersenne prime number.
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
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 SnellImprovement on the classical method of .
2n
22n2 Sr4rr2S and if r = 1,
2nn2 S42S
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
A
P
O
T
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§10 – 7 Nicolaus Mercator
Converges for - 1 < x 1.
...4
x
3
x
2
xx)x1(ln
432
Show convergence on a graphing calculator.
Let x = 1 ...4
1
3
1
2
112ln
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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
...65
1
43
1
21
1
Notice the relations ship with Mercator’s work on the previous slide.
...4
1
3
1
2
112ln
OR 2lndxx
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1
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§10 – 8 James Gregory
...99
2
35
2
3
2
...7
x
5
x
3
xxxarctan
753
For x = 1
...7
1
5
1
3
11
4
...
119
1
75
1
31
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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.