1.6
Derivatives: Problem Solving
Consider that water temperature has affected the growth rate of brook trout. The amount of weight it gained over a fixed period at various water temperatures is tabulated. The response of the growth rate to temperature changes is estimated using derivatives.
At the first data point, the derivative is approximated using the slope of a forward secant line drawn between the first and second points.
At the second point, the derivative is estimated using a central difference by calculating the slope of the secant line connecting the previous and next points, which provides a more accurate estimate of the derivative.
At the third point, again use a central difference to calculate the slope of the secant line connecting the points before and after it.
Similarly, at the fourth point, the derivative is calculated using the slope of the secant line between the neighboring data points.
Finally, at the last point, the derivative is approximated using the slope of the backward secant line drawn between the final two data points.
These derivative estimates are compiled in a table. The corresponding plot shows a consistent decline in growth rate, showing a negative correlation between temperature and weight gain.
Temperature-Dependent Growth of Brook Trout
The growth of brook trout is closely influenced by water temperature. Experimental data demonstrate how trout weight changes over a 24-day period in response to varying water temperatures. At lower temperatures, such as 15.5 degrees Celsius, brook trout show significant weight gain. However, as the temperature increases, the amount of weight gained steadily decreases. At the highest temperature measured, 24.4 degrees Celsius, trout experience a net loss in weight. This pattern suggests that brook trout thrive in cooler water and face metabolic stress or reduced growth efficiency in warmer conditions.
Estimating Changes in Growth Rate
To understand how the rate of weight gain changes with temperature, scientists estimate the rate at which weight changes between successive temperature values. This is done by comparing the change in weight over small temperature intervals. Where only one adjacent temperature point is available, the estimate uses the change between those two points. In cases where a data point lies between two others, an average of the two rates of change on either side provides a better estimate. These values offer insight into how rapidly the trout’s growth response shifts with temperature.
Interpretation of Growth Rate Trends
The resulting estimates show that as temperature increases, the rate of weight gain becomes more negative. This indicates that brook trout not only grow less efficiently but eventually begin losing weight under higher temperature conditions. The rate of change is expressed in units of grams per degree Celsius, representing how much the trout's weight changes for each degree increase in temperature. A graph of these values shows a consistent decline, supporting the conclusion that higher water temperatures are increasingly unfavorable for brook trout growth.
Consider that water temperature has affected the growth rate of brook trout. The amount of weight it gained over a fixed period at various water temperatures is tabulated. The response of the growth rate to temperature changes is estimated using derivatives.
At the first data point, the derivative is approximated using the slope of a forward secant line drawn between the first and second points.
At the second point, the derivative is estimated using a central difference by calculating the slope of the secant line connecting the previous and next points, which provides a more accurate estimate of the derivative.
At the third point, again use a central difference to calculate the slope of the secant line connecting the points before and after it.
Similarly, at the fourth point, the derivative is calculated using the slope of the secant line between the neighboring data points.
Finally, at the last point, the derivative is approximated using the slope of the backward secant line drawn between the final two data points.
These derivative estimates are compiled in a table. The corresponding plot shows a consistent decline in growth rate, showing a negative correlation between temperature and weight gain.
From Chapter 1:
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