What is the derivative of (12,000) / 1+39e^(-1.2t)
The full quesiton is "The population of a certain species of fish introduced into a lake is described by the logistic equation G(t) = (12,000) / 1+39e^(-1.2t) where G(t) is the population after t years. Find the point at which the growth rate of this population begins to decline."
To solve- I know you should take the derivate and set equal to zero. I have (3.97, 9000) as the answer but not sure it is correct and I cannot recreate the answer.
g = 12000/(1+e^(-1.2t))
= 12000 * (1+e^(-1.2t))^-1
g' = (12000)(-1)((1+e^(-1.2t))^-2 * (-1.2 e^(-1.2t))
= 14400e^(-1.2t)/((1+e^(-1.2t))^2
Now, g' is the growth rate. It starts to decline when g'' changes from positive to negative.
g'' = 17280*e^1.2t*(1 - e^1.2t)/((1+e^(-1.2t))^3
g'' = 0 where 1 = e^1.2t, or t=0
Hmmm. There's something wrong here. g' is always positive, meaning that the population is always growing.
However, g'' is negative for t>0, meaning that the growth rate is always slowing down. In fact, as t grows large, g(t) approaches 12000.
In fact the graph of g(t) displays this behavior, with fastest growth at t=0. Is there a typo somewhere?
Steve,
The problem has a 39 in front of the "e".
To find the derivative of the function (12,000) / (1 + 39e^(-1.2t)), we can use the quotient rule. The quotient rule states that if you have a function f(x) divided by a function g(x), then the derivative of the quotient is given by:
(f'(x) * g(x) - f(x) * g'(x)) / (g(x))^2
Let's apply the quotient rule to the given function f(t) = (12,000) / (1 + 39e^(-1.2t)):
Step 1: Find f'(t)
To find f'(t), the derivative of the numerator, we can treat it as a constant:
f'(t) = 0
Step 2: Find g(t)
g(t) = 1 + 39e^(-1.2t)
Step 3: Find g'(t)
To find g'(t), the derivative of the denominator, we can apply the chain rule:
g'(t) = -39 * (-1.2) * e^(-1.2t)
= 46.8e^(-1.2t)
Step 4: Apply the quotient rule
Using the quotient rule formula, we can substitute the values we found into the formula:
(f'(t) * g(t) - f(t) * g'(t)) / (g(t))^2
(0 * (1 + 39e^(-1.2t)) - (12,000) * (46.8e^(-1.2t))) / ((1 + 39e^(-1.2t))^2)
Simplifying this expression would give you the derivative of the function (12,000) / (1 + 39e^(-1.2t)).