A wildlife biologist models the population of beavers (in number of animals) living…
MATHEMATICS · Multi-part · AP · Calculus BC · Differential Equations · Differential Equations
Problem
A wildlife biologist models the population y of beavers (in number of animals) living along a stretch of river with the differential equation y′=0.02y(800−y),y(0)=50.
Time t is measured in months.
Throughout this problem you are NOT asked to solve the differential equation explicitly.
(1) Part 1: Sketch a plausible graph of the solution y=y(t) for 0≤t≤100. Clearly label the initial point, the horizontal asymptote(s), and the point at which the graph changes concavity.
(2) Part 2: Determine the beaver population at which the growth rate is greatest, and explain your reasoning using the differential equation.
(3) Part 3: For each interval 0<y<400 and 400<y<800, decide whether the solution curve is concave up or concave down. Justify your answer in terms of the signs and relative magnitudes of the slopes indicated by the differential equation.
(4) Part 4: Describe the long-term behavior of the beaver population as t→∞, including an explanation of how this behavior is reflected in your sketch from part (a).