1. 1 10 – Analytic Geometry and Precalculus Development The student will learn about Some European mathematics leading up to the calculus.

2. 2 §10-1 Analytic Geometry Student Discussion.

3. 3 §10-2 René Descartes Student Discussion.

4. 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. 5 §10-3 Pierre de Fermat Student Discussion.

6. 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. 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. 8 §10-4 Roberval and Torricelli Student Discussion.

9. 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 §10-5 Christiaan Huygens Student Discussion.

11. 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. 12 §10-6 17 th Century in France and Italy Student Discussion.

13. 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. 2 2 – 1 = 32 13 – 1 = 8191 2 3 – 1 = 72 17 – 1 = 131,071 2 5 – 1 = 312 19 – 1 = 524,287 2 7 – 1 = 1272 23 – 1 = 8,388,607 2 11 – 1 = 20392 29 – 1 = 536,870,911 http://www.mersenne.org/prime.htm

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

15. 15 §10-7 Willebrord Snell Improvement on the classical method of . and if r = 1, NSnSn N(S n )N(S n )/2 61.00000000006.00000000003.0000000000 120.517638096.2116570823.105828541 240.2610523846.2652572263.132628613 480.1308062586.2787004063.139350203 96*0.06543816.28206393.1410309 1920.03272346.28290483.1414529 3840.016362226.28311543.1415577 7680.00818126.28316943.1415847 15360.00409066.28317883.1415894 30720.00204536.28319763.1415988 6144

16. 16 §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

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

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

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

20. 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. 21 Assignment Discussion of Chapter 11.