Section 64a 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 ID: 495448
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Slide1
Exponential Growth and Decay
Section 6.4aSlide2
Law of Exponential Change
Suppose we are interested in a quantity
that increases ordecreases 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 thissituation with the following initial value problem:
Differential Equation:
Initial Condition:
Note:
k
can be either
positive or negative
What happens in eachof these instances?Slide3
Law of Exponential Change
Let’s solve this differential equation:
Separate variables
Integrate
Exponentiate
Laws of Logs/ExpsSlide4
Law of Exponential Change
Let’s solve this differential equation:
Def. of Abs. Value
Let A = + e
C
–
Apply the
Initial Cond.
Solution:Slide5
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
0
where
k
> 0 represents growth and k < 0 representsdecay. The number k is the rate constant of theequation.Slide6
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
0
Interest can be compounded monthly (
k
= 12), weekly (k = 52),daily (k = 365), etc…Slide7
Compounding Interest
What if we compound interest
continuously
at a rate proportionalto 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
.Slide8
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 tothe 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.Slide9
Guided Practice
Find the solution to the differential equation
dy
/dt = ky
, k aconstant, that satisfies the given conditions.
1. k = – 0.5, y(0) = 200
Solution:Slide10
Guided Practice
Find the solution to the differential equation
dy
/dt = ky
, k aconstant, that satisfies the given conditions.
2. y
(0) = 60, y(10) = 30
Solution:
orSlide11
Guided Practice
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)Slide12
Guided Practice
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)!!!Slide13
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
866.418
years oldSlide14
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 presentinitially?
There were 1250 bacteria initiallySlide15
Guided Practice
The number of radioactive atoms remaining after
t
days in asample of polonium-210 that starts with y radioactive atoms
is
0
(a) Find the element’s half-life.
days
Half-life =Slide16
Guided Practice
The number of radioactive atoms remaining after
t
days in asample of polonium-210 that starts with y radioactive atoms
is
0
(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 thesample?
days