6.6 The Natural Base, e

Warm-up

Learning Objective: To evaluate natural exponential and natural logarithmic functions and to model exponential growth and decay processes.

Learning Objective: To evaluate natural exponential and natural logarithmic functions and to model exponential growth and decay processes.

EX 1 – complete the table to investigate the growth of a $1 investment that earns 100% annual interest over one year.

Compounding Schedule

n Value, An

n

111

Annually 1 2.0000

Semiannually 2 2.2500

Quarterly 4 2.4414

Monthly 12 2.6130

Daily 365 2.7146

Hourly 8760 2.7181

Every minute 525600 2.7183

Every second 31536000 2.7183

1

1

111

2

2

111

4

4

111

12

12

111

365

365

111

8760

8760

111

525600

525600

111

31536000

31536000

111

e is called the natural base.

It was discovered by Leonhard Euler, and it’s irrational just like

2.71828...e

EX 2 – Simplify3 4a. e e

7e

3

2

10b.

5

e

e2e 24c. 3 xe

9 8xe

8

9xe

EX 3 – use the calculator to evaluate : (round to 3 decimal places)

3a. e

20.086

0.12b. e

0.887

Learning Objective: To evaluate natural exponential and natural logarithmic functions and to model exponential growth and decay processes.

b. ln 0.25 1.386

Learning Objective: To evaluate natural exponential and natural logarithmic functions and to model exponential growth and decay processes.

e is used to find continuously compounded interest.

Final amount

Initial amount

rate

time

the percent written as a decimal

EX 4 – You deposit $1500 in an account that pays 7.5% annual interest compounded continuously. What is the balance after 3 years? rty Pe

rty PeJust e

y 1500 e.075 3

y 1878.48

Out – How long does it take an investment to double at an annual interest rate of 7.5% compounded continuously?

Summary – I understand…