1. 7.3 Use Functions Involving e
Algebra II

2. 7.3 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, Л, and imaginary numbers.
Like Л and ‘i’, ‘e’ denotes a number.
Called The Euler Number after Leonhard Euler ( )
It can be defined by:
e= …
0! 1! 2! 3! 4! 5!
= ½ + 1/6 + 1/24 + 1/
≈ ….
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→∞), (1+1/n)n gets closer and closer to …..
Which is the value of e.

3. Ex. 1 Simplify the expression.
a.)
e3 · e4 = e7
c.)
(3e-4x)2
9e(-4x)2 9
e8x
b.)
10e3 =
5e2
2e3-2 = 2e

4. Ex.2 Simplify the expression.
b.)
2-2e10x=
e10x 4
a.)
24e8 =
8e5
3e3

5. Using a calculator Evaluate e2 using a scientific calculator
7.389
Evaluate e2 using a scientific calculator
Locate the ex button
you need to use the second button

6. Evaluate e-.06 with a calculator

7. f(x) = aerx is a natural base exponential function
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

8. Graphing Ex. 3a Graph y=ex
Remember the rules for graphing exponential functions!
The graph goes thru (0,a) and (1,e)
(1,2.7)
(0,1)

9. Graphing 3b
Graph y=e-x
(1,.368)
(0,1)

10. Graphing Ex. 4 Graph y=2e0.75x State the Domain & Range
Because a=2 is positive and r=0.75, the function is exponential growth.
Plot (0,2)&(1,4.23) and draw the curve.
(1,4.23)
(0,2)

12. A = Pert Using e in real life.
In 7.1 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

13. Ex. 6) Continuously Compounded Interest
You deposit $ into an account that pays 8% annual interest compounded continuously. What is the balance after 1 year?

14. Ex. 6) Continuously Compounded Interest
You deposit $ 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 ≈ $

15. Assignment