1) Consider the differential equation Warm Up – NO CALCULATOR 1) Consider the differential equation b) Let y = f(x) be a particular solution to the differential equation with the initial condition f(0) = -2. Use Euler’s Methods, starting at x = 0 with step size of ½, to approximate f(1). Show the work that leads to your answer.

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Homework

Logistic Curves

Exponential growth is unrestricted Exponential growth is unrestricted. An exponential growth function increases at an ever increasing rate and is not bounded above. In many growth situations, however, there is a limit to the possible growth. A plant can only grow so tall, the number of fish in an aquarium is limited by the size of the aquarium, etc. In such situations, the growth begins exponentially, but eventually slows and the graph levels out. The associated growth function is bounded both above and below by horizontal asymptotes. Use transparency to cross out numbers as they come up (1 – 50) Randint on calculator

Days since the rumor began # of people that have heard the rumor AP High School has 1200 students. A group of students start a rumor that spreads through the school. The number of students that know the rumor after t days is shown in the chart below. Days since the rumor began # of people that have heard the rumor 30 1 71 2 161 3 331 5 837 6 1020 8 1166 9 1186 11 1198 12 1199

L is the carrying capacity Logistics Curve equation: L is the carrying capacity

Write a logistics model for the rumor data collected in class. When was the rumor spreading the fastest? Maximum rate occurs when P = ½ L

The differential equation of a logistic curve looks like… or Write a differential equation for the rumor data collected in class.

The growth rate of a population of bears in a newly established wildlife preserve is modeled by dP/dt = 0.008P(100 – P), where t is measured in years. What was the carrying capacity for bears in this wildlife preserve? What is the bear population when the population is growing fastest? What is the rate of change of the population when it is growing the fastest?

The growth rate of a population of bears in a newly established wildlife preserve is modeled by dP/dt = 0.008P(100 – P),where t is measured in years. Solve the differential equation. If the preserve started with 2 bears, find the logistic model for population of bears in the preserve.

The growth rate of a population of bears in a newly established wildlife preserve is modeled by dP/dt = 0.008P(100 – P), where t is measured in years. With a calculator, generate the slope field and show that your solution conforms to the slope field. Remember to “fix” the Window before running the program.

A 2000-gallon tank can support no more than 150 guppies A 2000-gallon tank can support no more than 150 guppies. Six guppies are introduced into the tank. Assume that the rate of growth of the population is dP/dt = 0.0015P ( 150 – P ), where t is time in weeks. Find a formula for the guppy population in terms of t. How long will it take for the guppy population to be 125?