1. Continuous Growth and the Number e
Lesson 3.4

2. Compounding Multiple Times Per Year
Given the following formula for compounding
P = initial investment
r = yearly rate
n = number of compounding periods
t = number of years

3. Compounding Multiple Times Per Year
What if we invested $1000 for 5 years at 4% interest
Try the formula for different numbers of compounding periods
Monthly
Weekly (n = 52)
Daily (n = 365)
Hourly (n = 365 * 24)
What phenomenon do you notice?

4. Compounding Multiple Times Per Year
You should see that we seem to reach a limit as to how much multiple compounding periods increase the final amount
So we come up with continuous compounding

5. Using e As the Base We have used y = A * Bt Consider letting B = ek
Then by substitution y = A * (ek)t
Recall B = (1 + r) (the growth factor)
It turns out that

6. Continuous Growth
The constant k is called the continuous percent growth rate
For Q = a bt
k can be found by solving ek = b
Then Q = a ek*t
For positive a
if k > 0 then Q is an increasing function
if k < 0 then Q is a decreasing function

7. Continuous Growth
For Q = a ek*t Assume a > 0
k > 0
k < 0

8. Continuous Growth For the function what is the continuous growth rate?
The growth rate is the coefficient of t
Growth rate = 0.4 or 40%
Graph the function (predict what it looks like)

9. Converting Between Forms
Change to the form Q = A*Bt
We know B = ek
Change to the form Q = A*ek*t
We will eventually discover that k = ln B

10. Assignment
Lesson 3.4
Page 133
Exercises 1 – 25 odd