Table 1.21 shows the concentration, c, of creatinine in the...

Table 1.21 shows the concentration, $c,$ of creatinine in the bloodstream of a dog. $^{41}$ (a) Including units, find the average rate at which the concentration is changing between the (i) $6^{\mathrm{th}}$ and $8^{\mathrm{th}}$ minutes. (ii) $\quad 8^{\mathrm{th}}$ and $10^{\mathrm{th}}$ minutes. (b) Explain the sign and relative magnitudes of your results in terms of creatinine. $$\begin{array}{c|ccccc} \hline t \text { (minutes) } & 2 & 4 & 6 & 8 & 10 \\ \hline c(\mathrm{mg} / \mathrm{ml}) & 0.439 & 0.383 & 0.336 & 0.298 & 0.266 \\ \hline \end{array}$$

Table 1.21 shows the concentration, $c,$ of creatinine in the bloodstream of a dog. $^{41}$
(a) Including units, find the average rate at which the concentration is changing between the
(i) $6^{\mathrm{th}}$ and $8^{\mathrm{th}}$ minutes. (ii) $\quad 8^{\mathrm{th}}$ and $10^{\mathrm{th}}$ minutes.
(b) Explain the sign and relative magnitudes of your results in terms of creatinine.
$$\begin{array}{c|ccccc}
\hline t \text { (minutes) } & 2 & 4 & 6 & 8 & 10 \\
\hline c(\mathrm{mg} / \mathrm{ml}) & 0.439 & 0.383 & 0.336 & 0.298 & 0.266 \\
\hline
\end{array}$$

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Key Concepts

Average Rate of Change

This concept refers to the ratio of the change in the function’s output (in this case, the concentration of a substance) to the change in the input (time) over a specific interval. It is essentially a measure of how rapidly the function’s value is increasing or decreasing over that period, and is computed by finding the difference between the values of the function at the endpoints of the interval and dividing by the difference in the corresponding time points.

Units Consistency

When computing rates of change, it is crucial to include and properly analyze units. The units of the rate are determined by dividing the units of the dependent variable by the units of the independent variable, ensuring that the calculated rate carries physical meaning. This is especially important in scientific contexts to accurately track changes over time.

Interpretation of Negative Rates

A negative average rate of change indicates that the function is decreasing over the interval. Interpreting this sign is important in any analysis since it provides insight into the behavior of the system—in this case, that the substance's concentration is diminishing as time progresses. The magnitude reveals the speed of this decrease, with larger absolute values indicating a faster decline.

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