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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.