A wildlife conservation agency models the population of a fish species in thousands…
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
A wildlife conservation agency models the population N of a fish species in thousands as a differentiable function of time t (in years) with derivative
N′(t)=3te−0.2t−5cos(πt).
Seasonal policies allow fishing quotas to be set once per year, and quotas must be justified with evidence about population trends.
(a) Find all critical points of N for 0≤t≤8. Clearly show how you solve N′(t)=0 and verify that the points lie in the stated interval.
(b) Construct a complete sign analysis for N′(t) on 0≤t≤8. Identify every interval on which the fish population is increasing and every interval on which it is decreasing.
(c) Explain, in ecological terms with correct units, what the sign of N′(t) indicates about the fish population during each interval found in part (b).
(d) The agency may legally open the fishery only during periods when the population is not at risk of decline. Using your work above, recommend all years in 0≤t≤8 during which opening the fishery is ecologically justifiable. Support your answer with a concise calculus-based argument.
(e) Suppose unexpected climate conditions introduce an additional mortality term, modifying the derivative to N′(t)=3te−0.2t−5cos(πt)−0.4. Without recalculating every critical point, describe qualitatively—using first-derivative reasoning—how the intervals of increase and decrease would shift. Provide clear justification.