Transcript 6 2 Growth Decay

6-2: Growth and Decay
Objectives:
Assignment:
1. To solve differential
equations by separating
variables
• P. 418: 1-17 odd
2. To use differential
equations to model
growth and decay in
applied problems
• P. 418-420: 21, 25, 27,
33-37 odd, 42, 49, 57, 59,
63, 69, 71, 73-76
Warm Up
The variable 𝑦 is directly proportional to 𝑥
such that 𝑦 = 12 when 𝑥 = 15. Write an
equation that relates 𝑥 and 𝑦. What is the
constant of proportionality?
Objective 1
You will be able to solve
differential equations by
separating variables
Separation of Variables
The differential equation
𝑑𝑦
𝑑𝑥
= 𝑓(𝑥) can be written as
Differential Form
𝑑𝑦 =
𝑑𝑦 = 𝑓 𝑥 𝑑𝑥
Integrate
with
respect to
𝑦
Integrate
with
respect to
𝑥
𝑓 𝑥 𝑑𝑥
𝑦 =𝐹 𝑥 +𝐶
Separation of Variables
𝑑𝑦
If our differential equation is of the form 𝑑𝑥 = 𝐹 𝑥, 𝑦 , we employ the same
process, but we’ll also have rewrite the equation so that each variable
occurs on only one side of the equation corresponding to its differential.
Exercise 1
Solve
𝑑𝑦
𝑑𝑥
=
2𝑥
.
𝑦
𝑦 𝑑𝑦 = 2𝑥 𝑑𝑥
Sure, this is a different
constant, but it’s still a
constant…
𝑦 𝑑𝑦 =
2𝑥 𝑑𝑥
1 2
𝑦 = 𝑥2 + 𝐶
2
𝑦 2 − 2𝑥 2 = 𝐶
Objective 2
You will be able to use differential
equations to model growth and
decay in applied problems
Modeling Stuff
Mathematical
models often
take the form
of differential
equations
since realworld
quantities
change over
time.
(Like populations)
Populations of Stuff
It is not unreasonable to assume that the rate at
which a population* increases is directly
proportional its current population, at least under
ideal conditions. (As individuals in a population
reproduce, there are more individuals to
reproduce…)
(*of “bacteria”)
(*of “bugs”)
(*of “bears”)
Populations of Stuff
Based on this assumption:
Constant of proportionality
The rate at which
the population* is
changing
𝑑𝑃
=𝑘𝑃
𝑑𝑡
The population*
𝑑𝑃
= 𝑘 𝑑𝑡
𝑃
𝑑𝑃
=
𝑃
𝑘 𝑑𝑡
(Math)
𝑃 = 𝐶𝑒 𝑘𝑡
Exercise 2
For 𝑃 = 𝐶𝑒 𝑘𝑡 , find 𝑃 0 . Explain what this
means within the context of a population
growth scenario.
Exponential Growth & Decay
If 𝑦 is a differentiable function of 𝑡 such that
𝑦 > 0 and 𝑦 ′ = 𝑘𝑦 for some constant 𝑘, then
𝑦 = 𝐶𝑒 𝑘𝑡
Legend:
𝐶 = initial value of 𝑦
𝑘 = constant of proportionality
Exponential Growth: 𝑘 > 0
Extinction?: 𝑘 = 0
Exponential Decay: 𝑘 < 0
Exercise 3
The rate of change of 𝑦 is proportional to 𝑦.
When 𝑡 = 0, 𝑦 = 2. When 𝑡 = 2, 𝑦 = 4.
What is 𝑦 when 𝑡 = 3?
Half-Life
Hi, I’m Gordon Freeman, and I know
all about half-life. In fact, I probably
know twice as much about half-life
than most other people. Just try to do
a bit of research on half-life, and you
will more than likely find me swinging a
crowbar, wielding a gravity gun, or
dealing death to headcrabs.
None of that really has anything to do
with half-life, but it’s still really fun.
Half-Life 2
Anyway… You see, certain
substances decay over time. If the
decay rate is constant, it is
represented by the Greek letter λ
(lambda) and the decay can be
modeled exponentially:
𝑁 𝑡 = 𝑁0 𝑒 −𝜆𝑡
A substance’s half-life (t1/2) is the
amount of time it takes to decay to
½ of the original mass.
Half-Life 3 (Yeah, Right)
Let’s say you have a limited-edition
silver crowbar made from Ag-111,
which has a half-life of 7.45 days.
This means that in about a week, your
crowbar would only have ½ its original
mass. In a fortnight, your precious
crowbar would have ¼ of its original
mass.
My suggestion is to get a crowbar
made from steel or titanium. Those
will last a bit longer.
Half-Life
Half-life is the time it takes for a given amount of a
radioactive material to decay to 1/2 its original
mass.
According to nuclear
physicists (like Gordon
Freeman), the rate at which a
radioactive substance decays
is proportional to the mass of
the substance.
𝑦 = 𝐶𝑒 𝑘𝑡
Half-Lives
Half-life is the time it takes for a given amount of a
radioactive material to decay to 1/2 its original
mass.
Exercise 4
Suppose that 10
grams of the
plutonium isotope
Pu-239 was
released in the
Chernobyl nuclear
accident. How long
will it take for the 10
grams to decay to 1
gram?
Exercise 5
Suppose an experimental
population of fruit flies
increases according to the
law of exponential growth.
There were 100 flies after the
second day of the experiment
and 300 flies after the fourth
day. Approximately how
many flies were in the
original population?
Banana!
Exercise 6
Four months after it stops
advertising, a manufacturing
company notices that its
sales have dropped from
100,000 units per month to
80,000 units per month. If
the sales follow an
exponential pattern of
decline, what will they be
after another 2 months?
Metascore
Generally unfavorable reviews
Newton’s Law of Cooling
According to Newton, the rate at which the
temperature of an object changes is directly
proportional to the difference between the
object’s temperature and the ambient
temperature:
𝑑𝑇
= 𝑘 𝑇 − 𝑇𝑅
𝑑𝑡
(Math)
𝑇 = 𝑇𝑅 + 𝑇0 − 𝑇𝑅 𝑒 𝑘𝑡
“Research”
Exercise 7
Let 𝑦 represent the
temperature (in °F) of an
object in a room whose
temperature is kept at a
constant 60°. If the
object cools from 100° to
90° in 10 minutes, how
much longer will it take
for its temperature to
decrease to 80°.
6-2: Growth and Decay
Objectives:
Assignment:
1. To solve differential
equations by separating
variables
• P. 418: 1-17 odd
2. To use differential
equations to model
growth and decay in
applied problems
• P. 418-420: 21, 25, 27,
33-37 odd, 42, 49, 57, 59,
63, 69, 71, 73-76