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Differential Equations and their Slope Fields Part A. Modeling the Growth of a Population using First Order Differential Equations. Let y(t) denote the size of a population (in millions) at time t (in years since start of monitoring). Thus, y(0) represents the initial population size. The population growth is then denoted as $frac{dy(t)}{dt}$ and the relative population growth as $frac{frac{dy}{dt}}{y(t)}$. Model 1: the Exponential Model Statement: ""The relative growth of the population is constant"": $frac{frac{dy}{dt}}{y(t)} = k$ or, equivalently, ""The population growth is proportional to the size of the population"": $frac{dy(t)}{dt} = ky(t)$ 1. Consider the problem: $frac{dy(t)}{dt} = ky(t)$, k = 0.2, subject to the condition: y(0) = 1 a. Compute the values of the slope dy/dt for the y values in the table: y dy/dt 0 2 4 6 8 10 12 b. Use the results in the table to sketch the slopefield for the problem and sketch the population trajectory:

          Differential Equations and their Slope Fields
Part A. Modeling the Growth of a Population using First Order Differential Equations.
Let y(t) denote the size of a population (in millions) at time t (in years since start of monitoring).
Thus, y(0) represents the initial population size. The population growth is then denoted as $frac{dy(t)}{dt}$
and the relative population growth as $frac{frac{dy}{dt}}{y(t)}$.
Model 1: the Exponential Model
Statement: ""The relative growth of the population is constant"": $frac{frac{dy}{dt}}{y(t)} = k$
or, equivalently,
""The population growth is proportional to the size of the population"": $frac{dy(t)}{dt} = ky(t)$
1. Consider the problem: $frac{dy(t)}{dt} = ky(t)$, k = 0.2, subject to the condition: y(0) = 1
a. Compute the values of the slope dy/dt for the y values in the table:
y
dy/dt
0
2
4
6
8
10
12
b. Use the results in the table to sketch the slopefield for the problem
and sketch the population trajectory:

$Differential Equations and their Slope Fields Part A. Modeling the Growth of a Population using First Order Differential Equations. Let y(t) denote the size of a population (in millions) at time t (in years since start of monitoring). Thus, y(0) represents the initial population size. The population growth is then denoted as fracdy(t)dt and the relative population growth as fracfracdydty(t). Model 1: the Exponential Model Statement: ""The relative growth of the population is constant"": fracfracdydty(t) = k or, equivalently, ""The population growth is proportional to the size of the population"": fracdy(t)dt = ky(t) 1. Consider the problem: fracdy(t)dt = ky(t), k = 0.2, subject to the condition: y(0) = 1 a. Compute the values of the slope dy/dt for the y values in the table: y dy/dt 0 2 4 6 8 10 12 b. Use the results in the table to sketch the slopefield for the problem and sketch the population trajectory:$

Added by Laura M.

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