For the logistic differential equation, dy/dt=2y(50-y), what is the limiting value for y?
a) 100
b) 50
c) 25
d) 2
see problem 5b. https://secure-media.collegeboard.org/apc/ap06_calcBC_formB_samples_q5.pdf
To find the limiting value for y, we need to find the equilibrium points of the logistic differential equation.
Equilibrium points are the values of y for which dy/dt equals zero. In other words, equilibrium points are the values of y where the rate of change of y is neither positive nor negative, meaning y is not changing.
In this case, we have the logistic differential equation dy/dt = 2y(50-y).
Setting dy/dt equal to zero, we get:
0 = 2y(50-y)
To solve for y, we can set each factor equal to zero:
2y = 0 or 50-y = 0
The first equation, 2y = 0, implies that y = 0.
The second equation, 50-y = 0, implies that y = 50.
Therefore, the equilibrium points are y = 0 and y = 50.
Now, to decide which value is the limiting value for y, we need to consider the behavior of the logistic differential equation.
In the equation dy/dt = 2y(50-y), when y is close to 0, the term 2y dominates and causes dy/dt to increase. This means y will increase.
When y is close to 50, the term (50-y) dominates and causes dy/dt to decrease. This means y will decrease.
Since we are looking for the limiting value, we are interested in the value to which y approaches as t goes to infinity.
When y approaches 0, dy/dt increases and y will increase without bound.
When y approaches 50, dy/dt decreases and y will decrease without bound.
Therefore, the limiting value for y is 0, not 50.
Thus, the correct answer is option a) 0