1. Exponential Growth and Decay
Section 6.4a

2. Law of Exponential Change
Suppose we are interested in a quantity that increases or
decreases at a rate proportional to the amount present…
Can you think of any examples???
If we also know the initial amount of , we can model this
situation with the following initial value problem:
Differential Equation:
Note: k can be either
positive or negative
What happens in each
of these instances?
Initial Condition:

3. Law of Exponential Change
Let’s solve this differential equation:
Separate variables
Integrate
Exponentiate
Laws of Logs/Exps

4. Law of Exponential Change
Let’s solve this differential equation:
Def. of Abs. Value
Let A = + e C –
Apply the
Initial Cond.
Solution:

5. Law of Exponential Change
If y changes at a rate proportional to the amount present
(dy/dt = ky) and y = y when t = 0, then
where k > 0 represents growth and k < 0 represents
decay. The number k is the rate constant of the
equation.

6. Compounding Interest
Suppose that A dollars are invested at a fixed annual interest
rate r. If interest is added to the account k times a year, the
amount of money present after t years is
Interest can be compounded monthly (k = 12), weekly (k = 52),
daily (k = 365), etc…

7. Compounding Interest Solution:
What if we compound interest continuously at a rate proportional
to the amount in the account?
We have another initial value problem!!!
Differential Equation:
Initial Condition:
Look familiar???
Solution:
Interest paid according to this formula is compounded
continuously. The number r is the continuous interest rate.

8. Radioactivity
Radioactive Decay – the process of a radioactive substance
emitting some of its mass as it changes forms.
Important Point: It has been shown that the rate at which a
radioactive substance decays is approximately proportional to
the number of radioactive nuclei present…
So we can use our
familiar equation!!!
Half-Life – the time required for half of the radioactive nuclei
present in a sample to decay.

9. Guided Practice 1. k = – 0.5, y(0) = 200 Solution:
Find the solution to the differential equation dy/dt = ky, k a
constant, that satisfies the given conditions.
1. k = – 0.5, y(0) = 200
Solution:

10. Guided Practice 2. y(0) = 60, y(10) = 30 Solution: or
Find the solution to the differential equation dy/dt = ky, k a
constant, that satisfies the given conditions.
2. y(0) = 60, y(10) = 30
Solution: or

11. Guided Practice (a) (b)
Suppose you deposit $800 in an account that pays 6.3% annual
interest. How much will you have 8 years later if the interest is
(a) compounded continuously? (b) compounded quarterly?
(a)
(b)

12. Guided Practice This is always the half-life
Find the half-life of a radioactive substance with the given decay
equation, and show that the half-life depends only on k.
Need to solve:
This is always the half-life
of a radioactive substance
with rate constant k (k > 0)!!!

13. The sample is about 866.418 years old
Guided Practice
Scientists who do carbon-14 dating use 5700 years for its half-
life. Find the age of the sample in which 10% of the radioactive
nuclei originally present have decayed.
Half-Life =
The sample is about years old

14. There were 1250 bacteria initially
Guided Practice
A colony of bacteria is increasing exponentially with time. At the
end of 3 hours there are 10,000 bacteria. At the end of 5 hours
there are 40,000 bacteria. How many bacteria were present
initially?
There were 1250 bacteria initially

15. Guided Practice
The number of radioactive atoms remaining after t days in a
sample of polonium-210 that starts with y radioactive atoms is
(a) Find the element’s half-life.
Half-life =
days

16. Guided Practice
The number of radioactive atoms remaining after t days in a
sample of polonium-210 that starts with y radioactive atoms is
(b) Your sample is no longer useful after 95% of the initial
radioactive atoms have disintegrated. For about how many
days after the sample arrives will you be able to use the
sample?
days