Growth and Decay: Integral Exponents

1
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)
Lets 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 0.09(Cost of Previous
Year)
Suppose that a hamburger currently costs 4.
Lets look at some projected future costs by
using a table
Initial Cost
Time (years from now)
Cost (dollars)
0
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)
0
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 gt 0, the function gives future costs
When t lt 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.
Lets 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 - 0.09(Cost
of Previous Year)
OR
Each year we multiply the previous years cost by
0.91
Current Cost 0.91(Cost of Previous Year)
Lets 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
Lets look at the graphs of these functions
Exponential Growth
Exponential Decay
Growth and decay can be modeled by
If r gt 0 (1r gt 1) ? Exponential Growth
If 0 gt r gt -1 (0 gt 1r gt 1) ? Exp. Decay
6
Exponential Growth and Exponential Decay
R is between zero and negative one (0 gt -0.15 gt -1
So Exponential Decay
Lets 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 106.1 kg 6 days ago.
7
Review of Laws of Exponents
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

• P173-174 1-21(odd) Day 2 23-45 (odd)