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Do Now:

Algebra II 6.2: The Natural Base e Objective: To introduce the number e, while also using e to extend our knowledge of functions, growth, decay, and compounded interest.

The Euler Number “e” The history of mathematics is marked by the discovery of special numbers such as π and i e is another special number and it is approximately 2.71828182846 While this is a long and confusing number, we can treat this number like any other.

Example 1 Simplify each expression. a. b. c.

Simplify the expression.

Simplify the expression.

Exponential Growth and Decay with e The Euler Function Exponential Growth: a > 0 and r > 0 Exponential Decay: a > 0 and r < 0

Continuously compounded interest

Example 3 You and your friend each have accounts that earn annual interest compounded continuously. The balance A (in dollars) of your account after t years can be modeled by A = 4500e 0.04t . The graph shows the balance of your friend’s account over time. Which account has a greater principal? Which has a greater balance after 10 years?

Example 3 Continued Find the average rate of change for the two accounts. Does your result make sense?