Transcript 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 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
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square/day
1
2
3
4
5
6
7
...
64
rice
1
2
4
8
16
32
64
Sum
1
3
7
15
31
63
127
_____
_____
How much
wheat did the
King owe for
64th day?
How much
wheat in all?
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 Growth
There is a function related to this
story: f(x)=2^x
day
1
2
3
4
...
64
day rice
rice 2^(day-1)
1
2
4
8
sum rice
2^day-1
2^0 = ____
2^1 = ____
2^2 = ____
2^3 = ____
____ 2^63 = ____
2^1 - 1 = ____
2^2 - 1 = ____
2^3 - 1 = ____
2^4 - 1 = ____
2^64-1 = ____
Copy and fill out this chart.
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Grains of Rice
10000000000000000000.00
9000000000000000000.00
8000000000000000000.00
Grains
7000000000000000000.00
6000000000000000000.00
5000000000000000000.00
4000000000000000000.00
3000000000000000000.00
2000000000000000000.00
1000000000000000000.00
0.00
0
10
20
30
40
Day
50
60
70
Exponential Growth
Moore's Law (from the intel website):
http://www.intel.com/research/silicon/mo
oreslaw.htm
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 Growth
chip
Year
Transistors
4004
8008
8080
8086
286
386 processor
486 DX processor
Pentium® processor
Pentium II processor
Pentium III processor
Pentium 4 processor
1971
1972
1974
1978
1982
1985
1989
1993
1997
1999
2000
2,250
2,500
5,000
29,000
120,000
275,000
1,180,000
3,100,000
7,500,000
24,000,000
42,000,000
Produce a plot of year vs. transistors
Exponential Growth- Moore’s Law
chip
Year
Transistors
4004
8008
8080
8086
286
386 processor
486 DX processor
Pentium® processor
Pentium II processor
Pentium III processor
Pentium 4 processor
1971
1972
1974
1978
1982
1985
1989
1993
1997
1999
2000
2,250
2,500
5,000
29,000
120,000
275,000
1,180,000
3,100,000
7,500,000
24,000,000
42,000,000
Produce a plot of year vs. transistors
(from the intel website): http://www.intel.com/research/silicon/mooreslaw.htm
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
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?)