6.6 The Natural Base, e
Mar 26, 2015
6.6 The Natural Base, e
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
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!!!
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
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
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
The Graph of the Exponential Function
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
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
Graph the function. State the domain and range.
y = 2e 0.5x
domain: all real numbers,range: y>0
Try This!
Graph the function. State the domain and range.12
f(x)= e -x +1
domain: all real numbers,range: y>1
Try This!
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
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
Find the value:
Of $500 after 4 years invested at an annual interest rate of 9% compounded continuously.
$716.66
Homework
Green Book!!Green Book!!Page 142 # 1-25odd
#27 and 28