In a freshwater reservoir

MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties

Problem

In a freshwater reservoir, the temperature TT (in ∘^{\circ}C) varies with depth dd (in meters) below the surface. A technician records two points that lie on a smooth temperature–depth curve near d=2.0 md=2.0\text{ m}: • (1.5 m, 22.0∘C)(1.5\text{ m},\,22.0^{\circ}\text{C}) • (2.5 m, 18.0∘C)(2.5\text{ m},\,18.0^{\circ}\text{C}) Assume the curve between these depths is differentiable. (a) Using the given information, approximate the instantaneous rate of change dTdd\dfrac{dT}{dd} at d=2.0 md=2.0\text{ m}. Show all work and state units. (b) Explain what the numerical value obtained in part (a) indicates about the temperature profile in the reservoir at that depth. (c) Aquatic biologists define a “strong thermocline” as a temperature change that exceeds 4∘C4^{\circ}\text{C} per meter. Based on your estimate, determine whether a strong thermocline exists at d=2.0 md=2.0\text{ m} and discuss one ecological implication of your conclusion.

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