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Exponential Growth. According to legend, chess was invented by Grand Vizier Sissa Ben Dahir, and given to King Shirham of India. The king offered him a reward, and he requested the following: .

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exponential growth
Exponential Growth

According to legend, chess was invented by Grand Vizier Sissa Ben Dahir, and given to King Shirham of India. The king offered him a reward, and he requested the following:

  • "Just one grain of wheat on the first square of the chessboard. Then put two on the second square, four on the next, then eight, and continue, doubling the number of grains on each successive square, until every square on the chessboard is reached."
exponential growth2
Exponential Growth

You may give me the wheat or its equal value on the 64th day. This is all I require for my services. The king agreed, but he lost his entire kingdom to Sissa Ben Dahir. Why?

Sources:

http://muller.lbl.gov/teaching/Physics10/physics%2010%20notes/DoublingRule.html

http://www.blarg.net/~math/y2l2p.html

http://members.nbci.com/Templarser/chess.html

exponential growth3
Exponential Growth

square/day rice Sum

1 1 1

2 2 3

3 4 7

4 8 15

5 16 31

6 32 63

7 64 127

...

64 _____ _____

How much wheat did the King owe for 64th day?

How much wheat in all?

exponential growth4
Exponential Growth

In all, the king owed about 18,000,000,000,000,000,000 grains of wheat. This was more than the worth of his entire kingdom!

exponential growth5
Exponential Growth

There is a function related to this story: f(x)=2^x

day rice sum rice

day rice 2^(day-1) 2^day-1

1 1 2^0 = ____ 2^1 - 1 = ____

2 2 2^1 = ____ 2^2 - 1 = ____

3 4 2^2 = ____ 2^3 - 1 = ____

4 8 2^3 = ____ 2^4 - 1 = ____

...

64 ____ 2^63 = ____ 2^64-1 = ____

Copy and fill out this chart.

exponential growth7
Exponential Growth

Moore's Law (from the intel website):

http://www.intel.com/research/silicon/mooreslaw.htm

exponential growth8
Exponential Growth

Gordon Moore (co-founded Intel in 1968) made his famous observation in 1965, just four years after the first planar integrated circuit was discovered. The press called it "Moore's Law" and the name has stuck. In his original paper, Moore predicted that the number of transistors per integrated circuit would double every 18 months. He forecast that this trend would continue through 1975. Through Intel's technology, Moore's Law has been maintained for far longer, and still holds true as we enter the new century. The mission of Intel's technology development team is to continue to break down barriers to Moore's Law.

exponential growth9
Exponential Growth

chip Year Transistors

4004 1971 2,250

8008 1972 2,500

8080 1974 5,000

8086 1978 29,000

286 1982 120,000

386 processor 1985 275,000

486 DX processor 1989 1,180,000

Pentium® processor 1993 3,100,000

Pentium II processor 1997 7,500,000

Pentium III processor 1999 24,000,000

Pentium 4 processor 2000 42,000,000

Produce a plot of year vs. transistors

exponential growth moore s law
Exponential Growth- Moore’s Law

chip Year Transistors

4004 1971 2,250

8008 1972 2,500

8080 1974 5,000

8086 1978 29,000

286 1982 120,000

386 processor 1985 275,000

486 DX processor 1989 1,180,000

Pentium® processor 1993 3,100,000

Pentium II processor 1997 7,500,000

Pentium III processor 1999 24,000,000

Pentium 4 processor 2000 42,000,000

Produce a plot of year vs. transistors

(from the intel website): http://www.intel.com/research/silicon/mooreslaw.htm

exponential growth11
Exponential Growth

Review of how to do a point plot:

"STAT" "Edit" enter year in L1 and transistors in L2.

"2nd" "Y=" "Plotsoff" "Enter" “Enter"

"2nd" "Y="

Choose Plot1 {On, Scatterplot, L1, L2, mark}

"Zoom" 9

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Exponential Growth

1. Describe the graph:

2. How does this relate to the rice problem?

3. Can you think of other things that “grow” this way (ie. Doubling over a constant period of time?)