4.3 – Use Functions involving e

GPS: MM3A2e, MM3A2f, MM3A2d

GPS MM3A2e – Investigate and explain characteristics of

exponential and logarithmic functions including domain and range, asymptotes, zeros, intercepts, intervals of increase and decrease, and rate of change.

MM3A2f – Graph functions as transformations of f(x) = ax, f(x) = logax, f(x) = ex, f(x) = ln x.

MM3A2g - Explore real phenomena related to exponential and logarithmic functions including half-life and doubling time.

MM3A3d –Solve a variety of types of equations by appropriate means, choosing among mental calculation, pencil and paper, or appropriate technology.

The number denoted by the letter e is called the natural base e.

It is an irrational number defined as follows: As n approaches +∞, (1 + )n approaches e≈2.718281828.

A function of the form f(x)=aerx is called a natural base exponential function.

Continuously Compounded Interest – When interest is compounded continuously, the amount A in an account after t years is given by the formula A=Pert, where P is the principal and r is the annual interest rate expressed as a decimal.

Vocabulary

Remember Exponent Properties? Property Statement of Property

Product of Power Property

Power of a Power Property

Power of a Product Property

Negative Exponent Property

Zero Exponent Property

Quotient of Powers Property

Power of a Quotient Property

Simplify the expression◦ e6 * e4 = e6+4= e10

◦ (-2ex)3 = (-2)3(ex)3 = -8e3x

◦ 1

Simplify natural base expressions

Graph

Steps: ◦ Make a table◦ Plot your points◦ Asymptote?◦ Domain & Range

Graph natural base functions

X -2 -1 0 1 2

Y

Graph

Steps: ◦ Make a table◦ Plot your points◦ Asymptote?◦ Domain & Range

Graph natural base functions

X -2 -1 0 1 2

Y

Graph

Steps: ◦ Make a table◦ Plot your points◦ Asymptote?◦ Domain & Range

Graph y=abx-h +k for 0 < b < 1

X -2 -1 0 1 2

Y

You deposit $500 in an account that pays 3% annual interest compounded continuously. What is the balance after 2 years?

Use the formula for continuously compounded interest.

A = Pert

Model continuously compounded interest

Continuously Compounded Interest

Try page 141: 1-8

Guided Practice Page

x y

-2

-1

0

1

2