Given the population functuon P(t)= 50/ (1 + 4e^(-.1t) , fine the maximum rate of growth for this population where t is measured in days. What is the actual population at this time.
so you want to find where the first derivative (rate of growth) is a maximum.
Isn't that where the second derivative is equal to zero ?
Give it a shot
yes
To find the maximum rate of growth for the population function P(t), we need to determine the derivative of P(t) with respect to t, and then find the value of t where the derivative is maximum.
First, let's find the derivative of P(t) using the chain rule:
P'(t) = (50 / (1 + 4e^(-0.1t)))'
= 50 * (-1) * (4e^(-0.1t))' / (1 + 4e^(-0.1t))^2
= -200e^(-0.1t) / (1 + 4e^(-0.1t))^2
To find the maximum rate of growth, we need to find the critical points of P'(t). In this case, we want to find the values of t where P'(t) = 0 or P'(t) is undefined.
Setting P'(t) = 0, we have:
-200e^(-0.1t) / (1 + 4e^(-0.1t))^2 = 0
Since a fraction is equal to zero when the numerator is zero, we can ignore the denominator:
-200e^(-0.1t) = 0
Since e^(-0.1t) is never zero, there are no solutions to this equation.
Next, let's check for any points where P'(t) is undefined:
-200e^(-0.1t) / (1 + 4e^(-0.1t))^2 is undefined when the denominator (1 + 4e^(-0.1t))^2 is equal to zero. So, we need to solve:
(1 + 4e^(-0.1t))^2 = 0
Taking the square root of both sides, we get:
1 + 4e^(-0.1t) = 0
Subtracting 1 from both sides, we have:
4e^(-0.1t) = -1
This equation has no real solutions, as the exponential function e^(-0.1t) is always positive.
So, there are no critical points for P'(t), which means there is no maximum rate of growth for the population.
To find the actual population at a given time, we can substitute the value of t into the population function P(t). Without a specific value of t provided, we cannot determine the actual population.