The number of Drosophila fruit flies in a controlled experiment assumes the following growth pattern when the food source is limited:
N(t) = 120/(1+11e^(−0.4t))
D. At what rate is the number of fruit flies increasing after 4 days?
To find the rate at which the number of fruit flies is increasing after 4 days, we need to find the first derivative of the function N(t) with respect to t and evaluate it at t = 4.
Step 1: Differentiate the function N(t) with respect to t:
N'(t) = d/dt [120/(1+11e^(-0.4t))]
Step 2: Apply the chain rule:
N'(t) = (d/dt)[120] * (d/dt)[(1+11e^(-0.4t))^(-1)]
N'(t) = 0 * (d/dt)[(1+11e^(-0.4t))^(-1)]
Step 3: Now, we only need to differentiate the expression inside the square brackets using the chain rule:
(d/dt)[(1+11e^(-0.4t))^(-1)] = -(1+11e^(-0.4t))^(-2) * (d/dt)[1+11e^(-0.4t)]
= -(1+11e^(-0.4t))^(-2) * (d/dt)[11e^(-0.4t)]
Step 4: Differentiate 11e^(-0.4t):
(d/dt)[11e^(-0.4t)] = (-0.4) * 11 * e^(-0.4t) = -4.4e^(-0.4t)
Step 5: Plug in the values into the derived equation:
N'(t) = 0 * (-4.4e^(-0.4t))
N'(t) = 0
Step 6: Evaluate N'(t) at t = 4:
N'(4) = 0
Therefore, the rate at which the number of fruit flies is increasing after 4 days is 0.
To find the rate at which the number of fruit flies is increasing after 4 days, we need to calculate the derivative of the function.
The function given is: N(t) = 120 / (1 + 11e^(-0.4t))
To find the derivative of N(t), we will use the chain rule. The derivative of N(t) can be expressed as follows:
dN(t)/dt = (dN(t)/dt)(d(t)) / dt
We need to find the derivative of N(t) with respect to t:
dN(t)/dt = (120(0) - (1 + 11e^(-0.4t))(0.4)e^(-0.4t)) / (1 + 11e^(-0.4t))^2
Simplifying this expression further, we have:
dN(t)/dt = (-4.4e^(-0.4t)) / (1 + 11e^(-0.4t))^2
Now, we can substitute t = 4 into this expression to find the rate of increase after 4 days:
dN(4)/dt = (-4.4e^(-0.4(4))) / (1 + 11e^(-0.4(4)))^2
Simplifying further:
dN(4)/dt = (-4.4e^(-1.6)) / (1 + 11e^(-1.6))^2
This is the rate at which the number of fruit flies is increasing after 4 days.
dN/dt = 528e^0.4t / (11+e^0.4t)^2
Now just plug in t=4