1. 8.3 The Number e

2. Investigation
Copy the table and use a calculator to complete the table.
Do the values in the table appear to be approaching a fixed decimal number? If so, what is the number rounded to three decimal places? n
101
102
103
104
105
106
2.594

3. The Natural Base e
The natural base e is irrational. It is defined as follows:
As n approaches +∞, approaches e≈ …

4. Examples:
Simplify the expression. 1) 2) 3)

5. Examples:
Simplify the expression. 4) 5) 6)

6. Examples:
Use a graphing calculator to evaluate to expression 7) 8) 9)

7. Graphing Natural base exponential function-
If a>0 and r>0, the function is an exponential growth function.
If a>0 and r<0, the function is an exponential decay function.

8. Graphing
Exponential
Growth
Exponential
Decay

9. Examples:
Graph the function. State the domain and range. 1) 2) 3)

10. Real Life e
A is an account earning interest compounded n times per year for t years
P- principal
r- annual interest rate (as a decimal)

11. Continuously Compounded Interest
A-Amount in an account earning interest
P-Principal
r-rate (in decimal form)
t-time

12. Examples
1) You deposit $1500 in an account that pays 7.5% annual interest compounded continuously. What is the balance after 1 year?

13. Examples:
The atmospheric pressure P(in pounds per square inch) of an object d miles above sea level can be modeled by
a) How much pressure per square
inch would you experience at the
summit of Mount Washington, 6288
feet above sea level?

14. Examples:
a) Graph the model. Estimate your height above sea level if you experience lb/in2 of pressure.

15. Examples:
3) The radioactive decay of radon-222 can be modeled by the function , where A is the amount remaining, C is the original amount, and t is the time in days. If there are 15 mg of radon-222 sealed in a glass tube, how much will remain after 8 days? If 10 mg of radon-222 remains after 5 days, how much was originally there?