The population (P) of an island y years after colonization is given by the function
P=250/1+4e^-0.01y
c. After how many years was the population growing the fastest?
d. Using Curve Sketching methods, sketch the graph of the function. Make sure that you include all steps, charts, and derivations details.
growing the fastest means P' has a max, and thus P" = 0
P' = 10e^(.01y)/(1+4e^.01y)^2
P" = .1 e^.01y (4-e^.01y)/(1+4e^.01y)^3
So, P' is max when e^.01y = 4
for curve sketching, you know there are horizontal asymptotes at P=0 and y=250.
You can also avoid the calculus stuff by reviewing the properties of Logistic Growth curves.
Would you gladly please explain to me the curve sketching?
google is your friend. It will provide you with many examples, illustrations and videos. I'm sure your text also discusses the subject!
To find the years when the population was growing the fastest, we need to find the maximum rate of change. This can be determined by finding the derivative of the population function, setting it equal to zero, and solving for y.
Let's start by finding the derivative of the population function with respect to y.
dP/dy = d(250/(1+4e^(-0.01y)))/dy
To simplify the calculations, we can rewrite the population function as:
P = 250(1+4e^(-0.01y))^-1
Now, let's find the derivative:
dP/dy = 250 * d[(1+4e^(-0.01y))^(-1)]/dy
To find the derivative of the function inside the parentheses, we can use the chain rule. Let u = 1+4e^(-0.01y):
dP/dy = 250 * d(u^-1)/du * du/dy
Now, let's find the derivatives of the individual parts.
d(u^-1)/du = -1 * u^(-2) = -1/(1+4e^(-0.01y))^2
du/dy = (d(1+4e^(-0.01y))/dy) = -0.01 * 4e^(-0.01y)
Now, substitute these derivatives back into the original equation:
dP/dy = 250 * (-1/(1+4e^(-0.01y))^2) * (-0.01 * 4e^(-0.01y))
Simplifying further:
dP/dy = 250 * (0.04e^(-0.01y))/(1+4e^(-0.01y))^2
Now, we want to find the years when the population is growing the fastest, so we need to find the values of y that make dP/dy = 0.
Setting the derivative equal to zero:
0 = 250 * (0.04e^(-0.01y))/(1+4e^(-0.01y))^2
Since the derivative involves an exponential function, it's not easy to solve analytically. We can solve it numerically using computational methods or approximation techniques to get an estimate.
Using numerical methods, we can solve this equation to find the value of y when the population is growing the fastest.