A slope field for a differentiable function (measured in hundreds of rabbits) is…
MATHEMATICS · Multi-part · AP · Calculus BC · Differential Equations · Differential Equations
Problem
A slope field for a differentiable function P (measured in hundreds of rabbits) is shown to a student. Vertical axis values range from P=0 to P=12 and time t (in months) is shown on the horizontal axis. The slope marks satisfy the following qualitative observations.
• Along the horizontal line P=12, every segment is horizontal.
• For 0<P<12, the slope marks are positive; for P>12, the slope marks are negative.
• The slope marks appear steepest when P≈6 and become flatter as P moves farther away from 6.
Answer the questions that follow. (No sketch is required; use the verbal description.)
(a) Determine the limiting value approached by P(t) as t→∞. Justify your conclusion using specific features of the described slope field.
(b) Explain why the population of rabbits changes most rapidly when P≈6. Your explanation should cite the relationship between slope magnitude and rate of change.
(c) If P(0)=2, describe the long-term behavior of P(t). Does the population ever exceed the value you found in part (a)? Provide reasoning based on the slope field description.