Unit 5
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Study/Retake Times
◦ Thursday Afterschool
◦ Friday Afterschool
◦ Saturday 12-3PM
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Final Exam: Study supplies coming Friday
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MC with one FRQ
◦ I might add a second FRQ, haven’t decided…
2005
Ex 3: Find the solution to the differential equation
through the point (0, 5).
𝑑𝑦
𝑑𝑥
= 4𝑦𝑥 that goes
2008
2004
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As one amount increases, the other increases
by a constant rate.
y=kx.
k is the constant of proportionality.
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What do you think?
Differential Equations (equations that contain derivatives) are good for
modeling how a quantity is changing over time. We need to be able to
both solve & write the differential equations that describe how a
quantity is changing.
Write (but don’t solve) a differential equation that models the following
descriptions:
Ex 1:The rate of change of the surface area, A of an object with respect
to time t, is directly proportional to the square of the surface area.
Ex 2: The rate of change of the volume, V of an object with respect to
time t, is directly proportional to the natural log of the volume.
Ex 3: The population of a country is growing at a rate proportional to its
population. The growth rate per year (constant of variation) is 4% of the
current population.
Ex 4: Advertisers generally assume that the rate at which people hear
about a product is proportional to the number of people who have not
year heard about it. Suppose that the size of a community is N and p
denotes the number of people who have heard about the product.
Ex 5: A secret spreads among a population of N people at a rate
proportional to the product of the number of people who have not
heard the secret and two times the number of people who have heard
the secret. If p denotes the number of people who have heard the
secret, write a differential equation that could be used to model this
situation with respect to time t, where k is a positive constant.
Now that we can write differential equations for how quantities change,
we can use them to answer questions about that quantity.
Ex 7: At the beginning of the summer, the population of a hive of baldfaced hornets is growing at a rate proportional to the population. From
a population of 10 on the morning of May 1, the number of hornets
grows to 50 in 30 days. If the growth continues to follow the same
model, how many days after May 1 will the population reach 100?
Ex 8: An isotope of neptunium (Np-240) has a half-life of 65 minutes
(half-life is the amount of time it takes for the sample to have half of its
original amount). If the decay of Np-240 is modeled by the differential
𝑑𝑦
equation
= −𝑘𝑦, where t is measured in minutes, what is the decay
𝑑𝑡
constant k?
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Come and get a
whiteboard and a
marker.
We will review
derivatives through the
chain rule.
Copy down the
function, show the
work, answer on the
board.
HWK Part 1: Write differential equations to model the following situations:
1. The rate of change of the number of ants in a person’s house, A, with
respect to time t, is directly proportional to the natural log of the number of
ants in a person’s house. Write a differential equation that describes this
relationship.
2.
A new idea spreads among a population of N people at a rate proportional
to the product of the number of people who have not heard the idea and
the number of people who heard the idea. If p denotes the number of
people who have heard the idea, write a differential equation that could be
used to model this situation with respect to time t, where k is a positive
constant.
3.
The rate of change of the number of people inside an amusement park, P
with respect to time t, is directly proportional to the cube root of the
number of people inside the park. Write a differential equation that
describes this relationship.
4.
A secret spreads among a population of N people at a rate proportional to
the number of people who have not heard the secret. If p denotes the
number of people who have heard the secret, write a differential equation
that could be used to model this situation with respect to time t, where k is
a positive constant.
HWK Part 2: Set up and Solve the Differential Equations
1. Population y grows according to the equation dy/dt =ky where k is a
constant and t is measured in years. If the population doubles every 8 years,
what is the value of k?
2. The population of a country is growing at a rate proportional to its
population. If the growth rate per year is 10% of the current population, how
long will it take for the population to triple?
3. The radioactive decay of Sm-151 can be modeled by the differential
equation dy/dt=-.0077y, where t is measured in years. Find the half-life of
samarium.