8.3 The number e p. 480. The Natural base e Much of the history of mathematics is marked by the...
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Transcript of 8.3 The number e p. 480. The Natural base e Much of the history of mathematics is marked by the...
![Page 1: 8.3 The number e p. 480. The Natural base e Much of the history of mathematics is marked by the discovery of special types of numbers like counting numbers,](https://reader036.fdocuments.in/reader036/viewer/2022062618/55140aa9550346ec488b4da1/html5/thumbnails/1.jpg)
8.3 The number e
p. 480
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The Natural base e
• Much of the history of mathematics is marked by the discovery of special types of numbers like counting numbers, zero, negative numbers, Л, and imaginary numbers.
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Natural Base e• Like Л and ‘i’, ‘e’ denotes a number.• Called The Euler Number after Leonhard
Euler (1707-1783)• It can be defined by:
e= 1 + 1 + 1 + 1 + 1 + 1 +…
0! 1! 2! 3! 4! 5!
= 1 + 1 + ½ + 1/6 + 1/24 + 1/120+...
≈ 2.718281828459….
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• The number e is irrational – its’ decimal representation does not terminate or follow a repeating pattern.
• The previous sequence of e can also be represented:
• As n gets larger (n→∞), (1+1/n)n gets closer and closer to 2.71828…..
• Which is the value of e.
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Examples
• e3 · e4 =
• e7
•10e3 = 5e2
•2e3-2 =
•2e
•(3e-4x)2
•9e(-4x)2
•9e-8x
• 9 e8x
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More Examples!
• 24e8 =
8e5
• 3e3
•(2e-5x)-2=•2-2e10x=• e10x
4
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Using a calculator
• Evaluate e2 using a graphing calculator
• Locate the ex button
• you need to use the second button
7.389
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Evaluate e-.06 with a calculator
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Graphing
• f(x) = aerx is a natural base exponential function
• If a>0 & r>0 it is a growth function
• If a>0 & r<0 it is a decay function
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Graphing examples
• Graph y=ex
• Remember the rules for graphing exponential functions!
• The graph goes thru (0,a) and (1,e)
(0,1)
(1,2.7)
y=0
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Graphing cont.
• Graph y=e-x
(0,1) (1,.368)
y=0
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Graphing Example
• Graph y=2e0.75x
• State the Domain & Range
• Because a=2 is positive and r=0.75, the function is exponential growth.
• Plot (0,2)&(1,4.23) and draw the curve.
(0,2)
(1,4.23)
y=0
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Using e in real life.
• In 8.1 we learned the formula for compounding interest n times a year.
• In that equation, as n approaches infinity, the compound interest formula approaches the formula for continuously compounded interest:
•A = Pert
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Example of continuously compounded interest
• You deposit $1000.00 into an account that pays 8% annual interest compounded continuously. What is the balance after 1 year?
• P = 1000, r = .08, and t = 1
•A=Pert = 1000e.08*1 ≈ $1083.29
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Homework
•P. 483 (17-73) odd