8.3 The Number e

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

The Natural Base e The natural base e is irrational. It is defined as follows: As n approaches +∞, approaches e≈2.718281828459…

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

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

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

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.

Graphing Exponential Growth Exponential Decay

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

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)

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

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

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?

Examples: a) Graph the model. Estimate your height above sea level if you experience 12.23 lb/in2 of pressure.

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?