the natural base e section 6.2 beginning on page 304
TRANSCRIPT
The Natural Base e
Section 6.2 beginning on page 304
The Natural Base eThe Natural Base e, or the Euler number, is a number like or .
e is an irrational number and it is defined as follows:
As x approaches , approaches
Example 1: Simplify each expression.
a) b) c)
¿𝑒3+6
¿𝑒9¿ 4𝑒5−4
¿ 4𝑒¿32𝑒2 (− 4 𝑥)
¿9𝑒− 8𝑥
¿9
𝑒8𝑥
Graphing Natural Base FunctionsPg. 305
Graphing Natural Base FunctionsExample 2: Tell whether each function represents exponential growth or exponential decay, then graph the function.
a) ** Because a = 3 (it is positive) and r = 1 (also positive), this is an exponential growth function.
** Use a table of values to graph the function.
X -2 -1 0 1
Y 0.41 1 .10 3 8.15
Graphing Natural Base FunctionsExample 2: b)
** Because a = 1 (it is positive) and r = -0.5 (negative), this is an exponential decay function.
** Use a table of values to graph the function.
X -4 -2 0 2
Y 7.39 2.72 1 0.37
Why did I choose different x values t his time?
Solving Real Life Problems
𝐴=4500𝑒0.04 𝑡
Your friends account has a balance of about $7,300 after 10 years.
𝐴=4500𝑒0.04 (10 )
𝐴=4500𝑒4
𝐴≈$6,713.21
Since when t=0 the balance is 4000, your principal is greater than your friends because you started with 4500.
Your friend has a greater balance after 10 years.
Monitoring ProgressSimplify the expressions:
1) 2) 3)
Tell whether the function represents exponential growth or decay then graph the function.
4) 5) 6)
¿𝑒11 ¿3𝑒3 ¿1000
𝑒9𝑥
Monitoring Progress7) You deposit $4250 in an account that earns 5% annual interest compounded continuously. Compare the balance after 10 years with the accounts in example 3.
Friends account in example 3: Balance after 10 years $7,00Your account in example 3: Balance after 10 years
This account will have a balance of $7007.07, which is greater than your original account, but still less than your friends account.