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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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.
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….
• 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.
Using a calculator
• Evaluate e2 using a graphing calculator
• Locate the ex button
• you need to use the second button
7.389
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
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)
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)
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
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