The table shows values of the viral load V(t) in HIV patient...

The table shows values of the viral load $V(t)$ in HIV patient $303,$ measured in RNA copies $/ \mathrm{mL}, t$ days after ABT-538 treatment was begun. $$\begin{array}{|c|c|c|c|c|c|} \hline t & 4 & 8 & 11 & 15 & 22 \\ \hline V(t) & 53 & 18 & 9.4 & 5.2 & 3.6 \\ \hline \end{array}$$ (a) Find the average rate of change of $V$ with respect to $t$ over each time interval: (i) [4,11] (ii) [8,11] (iii) [11,15] (iv) [11,22] What are the units? (b) Estimate and interpret the value of the derivative $V^{\prime}(11)$.

The table shows values of the viral load $V(t)$ in HIV patient $303,$ measured in RNA copies $/ \mathrm{mL}, t$ days after ABT-538 treatment was begun.
$$\begin{array}{|c|c|c|c|c|c|}
\hline t & 4 & 8 & 11 & 15 & 22 \\
\hline V(t) & 53 & 18 & 9.4 & 5.2 & 3.6 \\
\hline
\end{array}$$
(a) Find the average rate of change of $V$ with respect to $t$ over each time interval:
(i) [4,11]
(ii) [8,11]
(iii) [11,15]
(iv) [11,22]
What are the units?
(b) Estimate and interpret the value of the derivative $V^{\prime}(11)$.

Key Concepts

Average Rate of Change

The average rate of change of a function over an interval is calculated by dividing the change in the function’s value by the change in the independent variable. This provides an overall measure of a function’s behavior (increase or decrease) over that specific interval, offering insight into the general trend across the interval.

Derivative

The derivative of a function at a given point represents the instantaneous rate of change at that point. It is defined as the limit of the average rate of change as the interval approaches zero, giving a precise measure of how quickly the function is changing at that exact moment.

Difference Quotient

The difference quotient is the expression (f(c + h) - f(c)) / h, which approximates the derivative when h is small. It is the foundational concept underlying the computation of derivatives and allows one to estimate instantaneous rates of change using finite differences.

Units of Measurement

In the context of rate of change or derivative calculations, the units result from dividing the units of the dependent variable by the units of the independent variable. For example, if the function represents viral load measured in RNA copies per mL and the independent variable is time measured in days, then the average rate of change or derivative will have units of (RNA copies per mL)/day.

Interpretation in Context

When applied to a real-world scenario such as tracking viral load in HIV patients, the derivative at a specific time indicates the instantaneous rate at which the viral load is changing. A negative derivative suggests that the viral load is decreasing at that moment, which can be critical for understanding the effectiveness of treatments and the progression of the condition.

Transcript

00:01 Using a table to look at the average rate of change between different time intervals, and we will use the equation the average rate of change equals f of b minus f a over b minus a.

00:09 I have the table written out here for us where t is in days, and v of t is in r and a copies per milliliter.

00:23 So the first interval is 411.

00:26 So in this case a is 4 and b is 11.

00:29 So if you look back at the chart, f of b will be 9 .4.

00:33 And f of a will be 53 over 11 minus 4.

00:39 And 9 .4 minus 53 is negative 43 .6, and 11 minus 4 is 7.

00:46 So that divided by 7 will give us negative 6 .23 rna copies per milliliter per day.

01:03 And that is the unit for all four of these intervals.

01:07 Basically all four of these rate of changes.

01:10 The second interval we'll be looking at is 8 .11.

01:14 So again, at 11, f of b is 9 .4, and at 8, it is 18.

01:22 And 11 minus 8 will be the denominator.