1. The natural base
Chapter 4.3

2. 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.

3. The natural base e

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

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

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

7. Natural Base Functions – Pg. 245

8. 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

9. Graph
1. 𝑦=3 𝑒 0.25𝑥 X -2 -1 1 2 y

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

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

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