Suppose N(t) denotes a population size at time t where the dN/dt=0.03 N(t).
If the population size at time t=4 is equal to 100, use a linear approximation to estimate the size of the population at time t=4.1.
Well, let's break this down! We have the equation dN/dt = 0.03 N(t). This equation describes the rate of change of the population size over time.
Now, to estimate the population size at t = 4.1 using linear approximation, we can assume that the rate of change remains constant over a small time interval.
So, let's find the rate of change at t = 4:
dN/dt = 0.03 N(4) = 0.03 * 100 = 3
Now, we can use this rate of change to approximate the population size at t = 4.1 by adding the rate of change times the time interval to the population size at t = 4:
N(4.1) ≈ N(4) + (dN/dt) * (t2 - t1)
N(4.1) ≈ 100 + 3 * (4.1 - 4)
N(4.1) ≈ 100 + 3 * 0.1
N(4.1) ≈ 100 + 0.3
N(4.1) ≈ 100.3
So, using linear approximation, we estimate that the population size at t = 4.1 is approximately 100.3.
To estimate the size of the population at t = 4.1 using linear approximation, we can use the formula:
N(t + Δt) ≈ N(t) + Δt * dN/dt
Given that dN/dt = 0.03 N(t), we can substitute this value into the formula:
N(t + Δt) ≈ N(t) + Δt * 0.03 N(t)
Since we know that the population size at t = 4 is equal to 100, we can substitute this value into the formula:
N(4 + Δt) ≈ 100 + Δt * 0.03 * 100
Now, to estimate the size of the population at t = 4.1, we need to find the value of N(4 + 0.1):
N(4.1) ≈ 100 + 0.1 * 0.03 * 100
Simplifying the equation:
N(4.1) ≈ 100 + 0.003 * 100
N(4.1) ≈ 100 + 3
Therefore, the estimated size of the population at t = 4.1 is approximately 103.
To estimate the size of the population at time t=4.1 using a linear approximation, we need to find the rate of change of the population size at time t=4, which will give us the slope of the linear approximation.
Given that dN/dt = 0.03N(t), we can substitute t=4 into the equation to find the rate of change at time t=4:
dN/dt = 0.03N(t)
dN/dt = 0.03N(4)
Since we know that the population size at time t=4 is 100, we can substitute N(4) = 100:
dN/dt = 0.03 * 100
dN/dt = 3
Therefore, the slope of the linear approximation is 3.
Now, we can use this slope to approximate the change in population size from t=4 to t=4.1. Since the time difference is 0.1, we multiply the slope by the time difference:
Change in population size = slope * time difference
Change in population size = 3 * 0.1
Change in population size = 0.3
To estimate the population size at t=4.1, we add the change in population size to the size at t=4:
Population size at t=4.1 = Population size at t=4 + Change in population size
Population size at t=4.1 = 100 + 0.3
Population size at t=4.1 = 100.3
Therefore, the estimated size of the population at time t=4.1 is approximately 100.3.
dN = .03N dt
you have N=100 and dt=0.1
plug and chug
N(4.1) ≈ 100 + dN