4.3 use functions involving e p. 244 what is the euler number? how is it defined? do laws of...
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4.3 Use Functions Involving e
p. 244What 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, π, and imaginary numbers.
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Natural Base e• Like π 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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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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Use a calculator to evaluate the expression.
a. e4
b. e –0.09
Expression Keystrokes Display
54.59815003
0.9139311853
[ ] ex 4
[ ]ex 0.09
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Simplify the expression.4. (10e –4x )3 = 103(e –4x ) 3
= 1000e12x
Use a calculator to evaluate5. e 3/4 .
2.117e 3/4 =
= 1000e –12x
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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 & RangeBecause 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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Graph the function. State the domain and range.
SOLUTION
The domain is all real numbers, and the range is y > 1.
b. y = e –0.75(x – 2) + 1
a = 1 is positive and r = –0.75 is negative, so the function is an exponential decay function. Translate the graph of y = right 2 units and up 1 unit.
e –0.75x
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Using e in real life.
• In 4.1 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
•A=Pert = 1000e.08*1 ≈ $1083.29
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A = Pert
SOLUTION
FINANCE: You deposit $2500 in an account that pays 5% annual interest compounded continuously. Find the balance after each amount of time?
Use the formula for continuously compounded interest.
Write formula.
Substitute 2500 for P, 0.05 for r, and 2 for t.
≈2762.9
The balance at the end of 2 years is $2762.90.ANSWER
a. 2 years
= 2500e0.10
= 2500e (0.05 •2)
= 2500 •1.105
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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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4.3 Assignment
Page 247, 3-48 every third problem