A fisheries biologist records the number of rainbow trout,
MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives
Problem
A fisheries biologist records the number of rainbow trout, P(t), living in a small research pond on several days during the first two weeks after spawning season. The data are shown in the table below, where t is the number of days after spawning.
Day t (days): 0 2 3 5 8 11 14
Population P(t) (fish): 480 552 590 660 755 826 865
Assume P(t) is differentiable for 0≤t≤14 and accurately models the trout population during this period.
(a) Estimate the average rate of change of the population on the interval [3,8]. Include appropriate units in your answer.
(b) Describe, in practical terms, what the instantaneous rate of change of P at t=8 days would represent for the biologist.
(c) Explain why there must be at least one time c in the interval 3<c<8 at which P′(c) equals the average rate you calculated in part (a). Justify your reasoning without calculating c.