1. Limits

2. We realize now, that making “h” smaller and smaller, will allow the secant line to get closer to the tangent line To formalize this process, we look to Archimedes….

3. An Ancient Greek mathematician, physicist, engineer, inventor, and astronomer. 287 BC – 212 BC

4. Generally considered the greatest mathematician of antiquity and one of the greatest of all time, Archimedes anticipated modern calculus and analysis by applying concepts of infinitesimals and the method of exhaustion to derive and rigorously prove a range of geometrical theorems, including infinitesimalsmethod of exhaustiongeometrical theorems the area of a circle

5. The Greeks we excellent at geometry…as long as there were no curves……

6. As time went on, their simple polygons became more complex, and they started to look more like circles…

7. So Archimedes decided to tackle the problem: What is the area of a circle?...

8. They would find the area of the square (s 2 ), and they would either ignore the light blue part, or they would add on “just a bit”… 4 sides

9. Archimedes had an idea… Now known as the “Method of Exhaustion” What if we inscribed shapes with more than 4 sides?....

10. There is less light blue area… …area that is not counted. 6 sides

11. The amount of blue space is less now… 8 sides

13. How about a myriagon (10000 sides) And then the big idea…… Will the area of an infinitely sided polygon be precisely the area of the circle that contains it?

14. Archimedes’ method of finding the area of a circle introduces the concept of a limit. The circle, that has a finite area, is the limiting shape of the polygon. As the number of sides gets larger, the area of the polygon approached its limit, the shape of the circle, without ever becoming an actual circle.

15. The area of the polygon “intends” to hit the area of the circle

16. Archimedes not only approached the area of a circle from the inside, but from the outside as well.

17. 4 sides

18. The amount of blue space is less now… 8 sides

19. Polygon to CircleCircle

20. We are going to use limits in 2 cases. 1.To help us examine the characteristics of various functions 2. To help us formalize the manipulation of the variable “h” in our difference quotient

21. Determine the following for the graph above: a) lim f(x) lim f(x) b)

22. Make a note of continuous and discontinuous functions on 27 Examine the key concepts on pg 29

23. Homework: Pg 29 1, 2, 3, 8, 9, 13, 14