The graph shows the influence of the temperature T on the...

The graph shows the influence of the temperature $T$ on the maximum sustainable swimming speed $S$ of Coho salmon.
(a) What is the meaning of the derivative $S^{\prime}(T) ?$ What are its units?
(b) Estimate the values of $S^{\prime}(15)$ and $S^{\prime}(25)$ and interpret them.

Key Concepts

Graphical Interpretation of Slope

Estimating the derivative at a specific point involves determining the slope of the tangent to the graph at that point. This graphical technique allows for understanding how the swimming speed changes as temperature varies. At temperatures where the tangent is steep and positive, the function is increasing rapidly, while a tangent that is shallow or negative indicates slower increase or a decrease in speed, respectively. This method of estimating slopes from the graph is a practical application of the derivative concept, aiding in the interpretation of physical relationships.

Instantaneous Rate of Change

The derivative of a function, in this case S'(T), represents the instantaneous rate of change of the swimming speed with respect to the temperature. This means it gives the rate at which the swimming speed is increasing or decreasing at a specific temperature. It is a fundamental concept in calculus that helps in understanding how a small change in the independent variable (temperature) affects the dependent variable (swimming speed) at any given point.

Units of a Derivative

The units of the derivative are determined by the units of the dependent variable divided by the units of the independent variable. Here, if the swimming speed S is measured in meters per second (m/s) and the temperature T in degrees Celsius (°C) or another temperature unit, then S'(T) has units of m/s per degree. This ratio quantifies how many meters per second the speed changes for every one unit change in temperature.

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