Mrs. McConaughy Honors Algebra 2 1

Objectives: To use the natural base e as a base of an exponential functionTo use the natural base e in real-life problems

The Natural Base e

Mrs. McConaughy Honors Algebra 2 2

VOCABULARYThe Natural Base e

Natural base e ____________________________________________________

≈ __________ The number is_______________

(non-terminating, non-repeating decimal).

e = 1/0! + 1/1! + ½! + 1/3! + ¼! + …

= 1 + 1 + ½ + 1/6 + 1/24 + ….

2.718281828459. irrational.

Mrs. McConaughy Honors Algebra 2 3

Most scientific calculators have a special key for evaluating natural base e functions of the form f(x) = Ce bx. For example, to evaluate e 2 on the TI- calculators, you can use the key strokes:

2 = ______

2nd ex enter

Evaluating Functions of the Form f(x) = Ce bx

7.38905

Mrs. McConaughy Honors Algebra 2 4

SIMPLIFYING NATURAL BASE e EXPRESSIONS

EXAMPLE 1

SIMPLIFYING NATURAL BASE e EXPRESSIONS

A. e2 ∙ e3 = _____B. 6 e2/2 e = _____C. (e-1)2 = _____ The graph of the function f(x) = Ce bx

represents exponential ______ if b is positive and exponential _______ if b is negative.

e5

3e1/e2

growthdecay

SIMPLIFYING NATURAL BASE e EXPRESSIONS

Mrs. McConaughy Honors Algebra 2 5

EXAMPLE 2 GRAPHING THE NATURAL BASE e FUNCTION

In general, the graph of f(x) = Cebx + d has the line _____as its horizontal

asymptote.

GRAPHING NATURAL BASE e FUNCTIONS

f(x) = e x g(x) = 0.5 e 0.6x h(x) = 0.5 e -0.6x

y = d

Mrs. McConaughy Honors Algebra 2 6

FORMULA:

CONTINUOUSLY COMPOUNDED INTEREST

Formula for Compounding Interest n times per year: ________________________________

Formula for Compounding Interest Continuously: ________________________________

A = p (1 + r/n) nt

A = P e r t

Mrs. McConaughy Honors Algebra 2 7

EXAMPLE 3 Finding Continuously Compounded InterestYou deposit $10,000 in

an account that pays 6% interest. Find the balance after 10 years if the interest is compounded

a.quarterly b.continuously

Finding Continuously Compounded Interest

$18,140.18

$18,221.19

A = P e r t

Mrs. McConaughy Honors Algebra 2 8

Final Checks for Understanding

1. Simplify 5e3/20e -2

2. Sketch the graph of f(x) = ½ e 3x. Is this function an example of exponential growth or decay? Explain.

3. State the horizontal asymptote of the graph of f(x) = 2ex – 2.

Exit Ticket

1. Condense: 2/3log a 27 - 2 log a 3

2. Expand: log (x + 3 ) 2

3. Use properties of logarithms to solve for x:

5log 5 2 + log 5 x = log 5 64

Mrs. McConaughy Honors Algebra 2 9

Mrs. McConaughy Honors Algebra 2 10

Homework Assignment:

Natural Base e WS

1-32 all; 33, 37, 41 (calculator); 47, 50.