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10 – Analytic Geometry and Precalculus Development

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.

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10 – Analytic Geometry and Precalculus Development

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  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 > 2. The Rule of Signs, method of undetermined coefficients and used our modern notation of a 2, a 3, a 4, . . . .

  5. §10-3 Pierre de Fermat Student Discussion.

  6. §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.

  7. §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.”

  8. §10-4 Roberval and Torricelli Student Discussion.

  9. §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.

  10. §10-5 Christiaan Huygens Student Discussion.

  11. §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?

  12. §10-6 17th Century in France and Italy Student Discussion.

  13. §10-6 Marin Mersenne Primes 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. http://www.mersenne.org/prime.htm

  14. §10-7 17th Century inGermany and the Low Countries Student Discussion.

  15. §10-7 Willebrord Snell Improvement on the classical method of . and if r = 1,

  16. §10 - 7 Huygens Improvement on Snell P T   A O 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

  17. §10 – 7 Nicolaus Mercator Converges for - 1 < x  1. Show convergence on a graphing calculator. Let x = 1

  18. §10 – 8 17th Century in Great Britain Student Discussion.

  19. §10 – 8 Viscount Brouncker OR Notice the relations ship with Mercator’s work on the previous slide. Area bounded by xy = 1, x axis, x = 1, and x = 2, is

  20. §10 – 8 James Gregory For x = 1 Which gives  as 3.15786 for the first three terms but which starts to converge more rapidly as the denominators increase.

  21. Assignment Discussion of Chapter 11.

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