8.3 The number e

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8.3 The number e

• p. 480
• What is the Euler number?
• How is it defined?
• Do laws of exponents apply to e number?
• How do you use e on your calculator?
• When graphing base e, how do you know if you have
  growth or decay?
• What is the formula for continuously compounded
  interest?

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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, ?, and
  imaginary numbers.

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Natural Base e

• Like ? 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.

4

• 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?8), (11/n)n gets closer and
  closer to 2.71828..
• Which is the value of e.

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Examples

• e3 e4
• e7

• (3e-4x)2
• 9e(-4x)2
• 9e-8x
• 9
• e8x

• 10e3
• 5e2
• 2e3-2
• 2e

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More Examples!

• (2e-5x)-2
• 2-2e10x
• e10x
• 4

• 24e8
• 8e5
• 3e3

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Using a calculator
7.389

• Evaluate e2 using a graphing calculator
• Locate the ex button
• you need to use the second button

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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 agt0 rgt0 it is a growth function
• If agt0 rlt0 it is a decay function

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Graphing examples

• Graph yex
• Remember the rules for graphing exponential
  functions!
• The graph goes thru (0,a) and (1,e)

(1,2.7)
(0,1)
y0
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Graphing cont.

• Graph ye-x

(1,.368)
(0,1)
y0
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Graphing Example

• Graph y2e0.75x
• State the Domain Range
• Because a2 is positive and r0.75, the function
  is exponential growth.
• Plot (0,2)(1,4.23) and draw the curve.

(1,4.23)
(0,2)
y0
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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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Continuously Compounded Interest
A Pert
Shampoo Problems
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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
• APert 1000e.081 1083.29

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• What is the Euler number?
• Natural base e
• How is it defined?
• 2.718 - - it is an irrational number like pi
• Do laws of exponents apply to e number?
• Yes- - all of them.
• How do When graphing base e, how do you know if
  you have growth or decay?
• Growth rises on the right and decay rises on the
  left.
• What is the formula for continuously compounded
  interest?
• Pert

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Assignment

• Page 483, 17-65 odd, 76