limits. we realize now, that making h smaller and smaller, will allow the secant line to get closer...

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Limits

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

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Page 1: Limits. 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

Limits

Page 2: Limits. 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

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….

Page 3: Limits. 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

• An Ancient Greek mathematician, physicist, engineer, inventor, and astronomer.

287 BC – 212 BC

Page 4: Limits. 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

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

the area of a circle

http://en.wikipedia.org/wiki/Infinitesimals

Page 5: Limits. 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

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

Page 6: Limits. 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

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

Page 7: Limits. 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

So Archimedes decided to tackle the problem:

What is the area of a circle?...

Page 8: Limits. 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

They would find the area of the square (s2), and they would either ignore the light blue part, or they would add on “just a bit”…

4 sides

Page 9: Limits. 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

Archimedes had an idea…

Now known as the “Method of Exhaustion”

What if we inscribed shapes with more than 4 sides?....

Page 10: Limits. 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

There is less light blue area… …area that is not counted.

6 sides

Page 11: Limits. 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

The amount of blue space is less now…

8 sides

Page 12: Limits. 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

Page 13: Limits. 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

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?

Page 14: Limits. 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

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.

Page 15: Limits. 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

The area of the polygon “intends” to hit the area of

the circle

Page 16: Limits. 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

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

Page 17: Limits. 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

4 sides

Page 18: Limits. 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

The amount of blue space is less now…

8 sides

Page 19: Limits. 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

Polygon to Circle

Page 20: Limits. 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

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

Page 21: Limits. 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

Determine the following for the graph above:

a) lim 2xf(x) lim

2xf(x)b)

Page 22: Limits. 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

Make a note of continuous and discontinuous functions on 27

• Examine the key concepts on pg 29

Page 23: Limits. 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

Homework: Pg 29

1, 2, 3, 8, 9, 13, 14

Page 24: Limits. 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

Page 25: Limits. 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

Page 26: Limits. 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

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