Researchers measured the average blood alcohol concentration...

Researchers measured the average blood alcohol concentration $C(t)$ of eight men starting one hour after consumption of $30 \mathrm{mL}$ of ethanol (corresponding to two alcoholic drinks). $$\begin{array}{|c|c|c|c|c|c|} \hline t \text { (hours) } & 1.0 & 1.5 & 2.0 & 2.5 & 3.0 \\ \hline C(t)(\mathrm{g} / \mathrm{dL}) & 0.033 & 0.024 & 0.018 & 0.012 & 0.007 \\ \hline \end{array}$$ (a) Find the average rate of change of $C$ with respect to $t$ over each time interval: (i) [1.0,2.0] (ii) [1.5,2.0] (iii) [2.0,2.5] (iv) [2.0,3.0] In each case, include the units. (b) Estimate the instantaneous rate of change at $t=2$ and interpret your result. What are the units?

Researchers measured the average blood alcohol concentration $C(t)$ of eight men starting one hour after consumption of $30 \mathrm{mL}$ of ethanol (corresponding to two alcoholic drinks).
$$\begin{array}{|c|c|c|c|c|c|}
\hline t \text { (hours) } & 1.0 & 1.5 & 2.0 & 2.5 & 3.0 \\
\hline C(t)(\mathrm{g} / \mathrm{dL}) & 0.033 & 0.024 & 0.018 & 0.012 & 0.007 \\
\hline
\end{array}$$
(a) Find the average rate of change of $C$ with respect to $t$ over each time interval:
(i) [1.0,2.0]
(ii) [1.5,2.0]
(iii) [2.0,2.5]
(iv) [2.0,3.0] In each case, include the units.
(b) Estimate the instantaneous rate of change at $t=2$ and interpret your result. What are the units?

Key Concepts

Difference Quotient

The difference quotient is the algebraic expression that forms the basis for both average and instantaneous rates of change. It is given by (f(b) - f(a)) / (b - a) for a finite interval, and as the interval shrinks to zero, it leads to the derivative. This concept underpins the fundamental notions of calculus and is essential for transitioning from a discrete to a continuous understanding of change.

Units in Rate of Change

When calculating rates of change, it is fundamental to maintain clarity by including the appropriate units. The units of an average or instantaneous rate of change are derived from the ratio of the units of the function’s output to the units of the independent variable. This ensures that the computed rates have meaningful real-world interpretations, such as grams per deciliter per hour in a study related to concentration changes over time.

Instantaneous Rate of Change

This key concept refers to the derivative of a function at a specific point, which represents the exact rate at which the function is changing at that moment. It is obtained as the limit of the average rate of change as the interval between the two points shrinks to zero. This measure is crucial in many applications, such as determining momentary speed in motion or the precise rate of change in a concentration, and it is graphically interpreted as the slope of the tangent line to the curve at that point.

Average Rate of Change

This concept involves calculating the change in a function’s value between two points divided by the change in the independent variable over that same interval. It is typically expressed as (f(b) - f(a)) / (b - a) and provides a measure of the overall change rate of the function over a specified interval. In many fields, including the analysis of physical phenomena or concentration levels over time, it helps in understanding how a quantity evolves on average between two moments.

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