1. Growth and Decay: Integral Exponents
Section 5-1
Growth and Decay: Integral Exponents
Objective:
To define and apply integral exponents

2. Introduction Exponential Growth and Exponential Decay
These are functions where we have variables for exponents
Ex)
Ex)
Let’s look at an EXAMPLE:
Suppose that the cost of a hamburger has been increasing at the rate of 9% per year.
That means each year the cost is 1.09 times the cost of the previous year.
Or Current Cost = Cost of Previous Year (Cost of Previous Year)
Suppose that a hamburger currently costs $4.
Let’s look at some projected future costs by using a table
Initial Cost
Time (years from now)
Cost (dollars) 1 2 3 t 4
4(1.09)
4(1.09)2
4(1.09)3
4(1.09)t
X 1.09
X 1.09
X 1.09
X 1.09

3. Exponential Growth and Exponential Decay
Initial Cost
Cost at time t
Time (years from now)
Cost (dollars) 1 2 3 t 4
4(1.09)
4(1.09)2
4(1.09)3
4(1.09)t
X 1.09
X 1.09
X 1.09
X 1.09
This table suggests that cost is a function of time t.
We can not only predict cost in the future, but we could predict costs in the past.
When t > 0, the function gives future costs
When t < 0, the function gives past costs
a) 5 years from now
b) 5 years ago
The cost will be about $6.15
The cost was about $2.60

4. Exponential Growth and Exponential Decay
The previous example (cost of hamburgers) was an example of exponential growth.
Let’s look at an example of Exponential Decay:
Say Arielle bought her graphing calculator for $70, and its value depreciates (decreases) by 9% each year. How much will the calculator be worth after t years?
Current Cost = Cost of Previous Year (Cost of Previous Year) OR
Each year we multiply the previous year’s cost by 0.91
Current Cost = 0.91(Cost of Previous Year)
Let’s briefly compare the two situations:
Hamburger cost: $4 now
Calculator Cost: $70 now
Cost increasing at 9% each year
Cost decreasing at 9% each year

5. Exponential Growth and Exponential Decay
Hamburger cost: $4 now
Calculator Cost: $70 now
Cost increasing at 9% each year
Cost decreasing at 9% each year
Let’s look at the graphs of these functions
Exponential Growth
Exponential Decay
Growth and decay can be modeled by:
If r > 0 (1+r > 1)  Exponential Growth
If 0 > r > -1 (0 > 1+r > 1)  Exp. Decay

6. Exponential Growth and Exponential Decay
R is between zero and negative one (0 > > -1
So Exponential Decay
Let’s use this formula in an example:
Suppose that a radioactive isotope decays so that the radioactivity present decreases by 15% per day. If 40 kg are present now, find the amount present:
a) 6 days from now
b) 6 days ago
There will be about 15.1 kg left after 6 days
There was about kg 6 days ago.

7. Review of Laws of Exponents

8. Apply the of Laws of Exponents
Example:
Distribute the exponents
Solution
Distribute the exponents
Subtract the exponents

9. Apply the of Laws of Exponents
Example:
Get rid of the negative exponents
Solution
Get a common base
Get rid of the negative exponents

10. Where do we see Exponential Growth?
Wikipedia

11. Homework
P : 1-21(odd) Day 2: (odd)