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Unit 5 Study/Retake Times ◦ Thursday Afterschool ◦ Friday Afterschool ◦ Saturday 12-3PM Final Exam: Study supplies coming Friday 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 As one amount increases, the other increases by a constant rate. y=kx. k is the constant of proportionality. 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? 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.