x

y

Graph22 3 1xy

Horizontal Asymptote: y = -1

Domain:

Range: y > -1

all real numbers

a = ____2

b= ____3

h = ____2

k = ____-1

Objective: (1) Using The Natural Base e (2) Using e in Real Life

CA Standards: 12.0 Students know the laws of fractional exponents, understand exponential functions, and use theses functions in problems involving exponential growth and decay.

Agenda: 02/26/15Agenda: 02/26/151.) Warm-up

2.) Questions:

WS Exponential Decay Practice A/B

3.) Lesson: 8.3 The Number e

4.) Class/Homework: TB TB 8.3

5.) Work with Your Neighbor

STAY ON TASK!!!

6.) Quiz: 8.1, 8.2, & 8.3

Friday 02/28/15

8.3 The Number “e”

• Named after its discoverer, Leonhard Euler (Oiler)• He lived from (1707 – 1783)• History has taught us special numbers: Counting

Numbers, Zero, Negative numbers, π, and Imaginary numbers.

• One of the most famous numbers of modern times is “e.”• This number is called the “Natural Base e.”• Let us explore this number using the calculator.

1. Let us analyze the following the table.

n 10 100 1000 10000 100000

1000000

2. What value does seem to be approaching

as “n” becomes larger?

11

n

n

2.59374

2.70481

2.71692

2.71815

2.71827

2.71828

11

n

n

2.718281828459…

Ex. 1 Simplify.

1) 3)

4)

36e9

18e63e 3x 3

27e12x32) 22

4

6

2

e

e

3 3e 3x 3

e9x

33 e9x

27

11

12 33 27 xe

1 1

3 123 3 3 xe4

3e4 x

2e9 6 2e3

6 2e 4

2e4 1

72e8

You Try!

You Try!

x

yEx. 2 Graph y 3e2x

x y

1 22

10 3

0.4 Y – Int.: (0,3)

HA: y = 0

Domain: All Real Numbers

Range: y > 0

Exponential Growth because the coefficient of the x is POSITIVE.

x

yEx. 3 Graph y 4e x

x y

1 1.5

10 4

10.8Y – Int.: (0,4)

HA: y = 0

Domain: All Real Numbers

Range: y > 0

Exponential Decay because the coefficient of the x is NEGATIVE.

rincipalP Pinitial interest ratert time in years

Ex. 4 You deposit $1,000,000 in a savings account that pays 7% annual interest compounded continuously. What is the balance after 5 years?

0.07(5)1,000,000A e

$1,419,067.55A

Continuously Compounded

Interest

A Pert

Write Write thisthis

HOMEWORK

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