The natural base Chapter 4.3

Some history of mathematics The history of mathematics is marked by the discovery of special numbers such as π and i. Another special number is denoted by the letter e. The number is called the natural base e or the Euler number after its discoverer, Leonhard Euler. The expression (1+ 1 𝑛 ) 𝑛 approaches e as n increases.

The natural base e

Simplify each expression 𝑒 2 ∙ 𝑒 5 12 𝑒 4 3 𝑒 3 5 𝑒 −3𝑥 2

Simplify each expression 4. 𝑒 9 ∙ 𝑒 6 5. 60 𝑒 8 12 𝑒 3 6. −10 𝑒 −5𝑥 3

Evaluate each expression 𝑒 4 𝑒 −0.09 𝑒 6 𝑒 −0.28

Natural Base Functions – Pg. 245

Steps for Graphing: Step 1 – Decide if an exp. growth or decay Step 2 - Make a table of values, xy chart (y intercept, neg, postitive) Step 3 - Plot points from the table Step 4 – Identify and shade asymptote Step 5 - Draw a smooth curve line that approaches the asymptote on the one end, and extends to infinity on the other

Graph 1. 𝑦=3 𝑒 0.25𝑥 X -2 -1 1 2 y

Graph 2. 𝑦= 𝑒 −0.75(𝑥−2) +1 X -2 -1 1 2 y

Graph each function. State the Domain and Range 1. 𝑓(𝑥)= 2𝑒 0.5𝑥 X -2 -1 1 2 y

Graph each function. State the Domain and Range 2. 𝑓 𝑥 = 1 2 𝑒 −𝑥 +1 X -2 -1 1 2 y