1. 6.6 The Natural Base, e

2. Natural Base, e
The natural base, e, is used to estimate the ages of artifacts and to calculate interest that is compounded continuously.

3. 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. 1
You calculator has a FANCY ex button you can use!!!  2

4. Evaluate f(x)=ex to the nearest thousandth for each value of x below:

5. Exponential Properties
Are the same
for base e!!
Simplify the expression. a. e2 e5 =
e2 + 5 = e7 b. 12 e4 3 e3 =
e4 – 3 4 = 4e (5 ) c.
e –3x 2 = 52 (
e –3x ) 2 = 25
e –6x = 25
e6x

6. Try This! (10 ) Simplify the expression. 1. e7 e4 e11 4. e –4x 3 10002. 2
e–3
6e5 12 e2
Use a calculator to evaluate 3. 24 e8 4 e5 6 e3
2.117

7. The Graph of the Exponential Function1 3 2

8. Graph natural base functions
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.
x y 1
The domain is all real numbers, and the range is y > 0.

9. Graph natural base functions
Graph the function. State the domain and range. b. y =
e –0.75(x – 2)
+ 1
SOLUTION
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
The domain is all real numbers, and the range is y > 1.

10. Try This! Graph the function. State the domain and range. y 2e 0.5x =
domain: all real numbers, range: y>0

11. Try This! Graph the function. State the domain and range. 1 2 f(x) =
e -x +1
domain: all real numbers, range: y>1

12. 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

13. 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

14. Find the value:
Of $500 after 4 years invested at an annual interest rate of 9% compounded continuously.
$716.66

15. Homework
Green Book!!
Page 142 # 1-25odd
#27 and 28