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 – 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 § th 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. 2 2 – 1 = – 1 = – 1 = – 1 = 131, – 1 = – 1 = 524, – 1 = – 1 = 8,388, – 1 = – 1 = 536,870,911

14 § th Century in Germany and the Low Countries Student Discussion.

15 §10-7 Willebrord Snell Improvement on the classical method of . and if r = 1, NSnSn N(S n )N(S n )/ *

16 § 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 ~ And 180 · AP = Which is accurate to   A P O T

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

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

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

21 Assignment Discussion of Chapter 11.