Objectives: 1. Be able to find the Euler Number.

Functions Involving "e" • Objectives: • 1. Be able to find the Euler Number. • Be simplify expressions using the Natural base (with a calculator also) • Be able to graph a Natural Base Exponential. Objectives: Euler Number, Natural base

I. The Euler Number Use the expression to fill in the table: Value of “n” Formula Value Significance 1 2 Annually 2 2.25 Semi-Annually 2.44140625 4 Quarterly 2.61303529 12 Monthly 2.714567482 365 Daily 2.718126691 Hourly 8,760 2.718279215 Minutely 525,600 2.718282473 31,536,000 Secondly

II. The Natural Base The Natural Base e: The natural base e is irrational. It is defined as follows: As n approaches ∞ approaches e ≈ 2.718281828 Directions: Simplify the natural base expressions 1. e2 • e5 2. 3. (5e-3x)2 52e-6x e2+5 4e4-3 e7 4e Directions: Evaluate each expression with a calculator 1. e7 1096.633158 2. e3/4 2.117000017

III. Graphing a Natural Base Exponential Function Form: f(x) = aerx If a > 0 and r > 0, then the function is an exponential growth If a > 0 and r < 0, then the function is an exponential decay Exponential Decay Exponential Growth

III. Graphing a Natural Base Exponential Function Graph: f(x) = 3e.25x • Exponential Growth 0 1 -2 -1 2 1.82 2.34 3 3.85 4.95 Domain: All Real Numbers Range: y > 0

III. Graphing a Natural Base Exponential Function Graph: f(x) = e.-0.75(x - 2) + 1 • Exponential Decay 2 3 0 1 4 5.48 3.12 2 1.47 1.22 Domain: All Real Numbers Range: y > 1

Page 495-496 #3-49 odd (24 problems) DOMAIN AND RANGE ON ALL GRAPHS!!!!!

I. The Euler Number Use the expression to fill in the table: Value of “n” Formula Value Significance 1 2 4 12 365 8,760 525,600 31,536,000

Objectives: 1. Be able to find the Euler Number.