Water temperature affects the growth rate of brook trout....

Water temperature affects the growth rate of brook trout. The table shows the amount of weight gained by brook trout after 24 days in various water temperatures. $$ \begin{array}{|c|c|c|c|c|c|} \hline \text { Temperature }\left({ }^{\circ} \mathrm{C}\right) & 15.5 & 17.7 & 20.0 & 22.4 & 24.4 \\ \hline \text { Weight gained }(\mathrm{g}) & 37.2 & 31.0 & 19.8 & 9.7 & -9.8 \\ \hline \end{array} $$ If $W(x)$ is the weight gain at temperature $x$, construct a table of estimated values for $W^{\prime}$ and sketch its graph. What are the units for $W^{\prime}(x) ?$ Souree: Adapted from J. Chsdwick Jr., "Temperature Effects on Growth and Stress Physiology of Brook Trout: Implications for Climate Change Impacts on an leonic Cold-Water Fish." Masters Theses. Paper 897. $2012 .$ scholarworks.umass.edu/theses/897.

Water temperature affects the growth rate of brook trout. The table shows the amount of weight gained by brook trout after 24 days in various water temperatures.
$$
\begin{array}{|c|c|c|c|c|c|}
\hline \text { Temperature }\left({ }^{\circ} \mathrm{C}\right) & 15.5 & 17.7 & 20.0 & 22.4 & 24.4 \\
\hline \text { Weight gained }(\mathrm{g}) & 37.2 & 31.0 & 19.8 & 9.7 & -9.8 \\
\hline
\end{array}
$$
If $W(x)$ is the weight gain at temperature $x$, construct a table of estimated values for $W^{\prime}$ and sketch its graph. What are the units for $W^{\prime}(x) ?$
Souree: Adapted from J. Chsdwick Jr., "Temperature Effects on Growth and Stress Physiology of Brook Trout: Implications for Climate Change Impacts on an leonic Cold-Water Fish." Masters Theses. Paper 897. $2012 .$ scholarworks.umass.edu/theses/897.

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Key Concepts

Derivative (Rate of Change)

The derivative of a function represents the instantaneous rate at which the function’s value changes with respect to changes in the independent variable. In this problem, W'(x) provides the rate of weight gain per degree Celsius change in temperature, summarizing how sensitive the trout’s weight gain is to temperature variations.

Finite Difference Approximation

When only discrete data points are available, we can approximate the derivative by using finite differences. This involves computing the change in weight gain over a change in temperature between adjacent data points, giving an estimate of the slope of the tangent line to the curve at a given point.

Units of the Derivative

The derivative of a function assumes the units of the function divided by the units of the independent variable. Here, since weight gain is measured in grams and temperature in degrees Celsius, the derivative W'(x) will have units of grams per degree Celsius.

Graphical Representation of Functions and Derivatives

Plotting the estimated derivative values helps visualize how the rate of weight gain changes as water temperature varies. This graph illustrates the sensitivity and trends of the function, offering a clear picture of the underlying behavior of the relationship between temperature and weight gain.

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