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6.6 The Natural Base, e
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Natural Base, e
The natural base, e, is used to estimate the ages of artifacts and to calculate interest that is compounded continuously.
2.718e
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Natural Exponential Function The exponential function with base e, f(x)=ex,
is called the natural exponential function.
And e is called the natural base.You calculator has a FANCY ex button you can use!!!
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Evaluate f(x)=ex to the nearest thousandth for each value of x below: x = 2
x = -1/3
x = 0.25
2(2) 7.389f e
0.25(0.25) 1.284f e
1
310.717
3f e
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Simplify the expression.
a. e2 e5 = e2 + 5
= e7
b. 12e4
3e3 = e4 – 34
= 4e
(5 )c. e –3x 2 = 52 (e –3x )2
= 25e –6x
= 25e6x
Exponential Properties
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Simplify the expression.
1. e7 e4 e11
3. 24e8
4e5
12e22. 2e–3 6e5
6 e3
Try This!
4. (10e –4x )3 1000e12x
2.117
5. Use a calculator to evaluate 4
3e
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The Graph of the Exponential Function
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Graph the function. State the domain and range.
a. y = 3e 0.25x
SOLUTION
Because a = 3 is positive and r = 0.25 is positive, the function is an exponential growth function. Plug in 0 and 1.
The domain is all real numbers, and the range is y > 0.
Graph natural base functions
x y
01
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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
Graph natural base functions
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Graph the function. State the domain and range.
y = 2e 0.5x
domain: all real numbers,range: y>0
Try This!
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Graph the function. State the domain and range.12
f(x)= e -x +1
domain: all real numbers,range: y>1
Try This!
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Continuous Compounding Formula If P dollars are invested at an interest rate, r,
that is compounding continuously, then the amount, A, of the investment at time t is given by:
A = Pert
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An investment of $1000 earns an annual interest rate of 7.6% Compare the final amounts after 8 years for interest
compounded quarterly and for interest compounded continuously.
Compounded Quarterly Continuously
1nt
rtrA P A Pe
n
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Find the value:
Of $500 after 4 years invested at an annual interest rate of 9% compounded continuously.
$716.66
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Homework
Green Book!!Green Book!!Page 142 # 1-25odd
#27 and 28