1.3 The Average Rate of Change of a Function (continued)

Summary

A cold can of soda is removed from a refrigerator. Its temperature $F$ in degrees Fahrenheit is measured at $5$-minute intervals, as recorded in the following table.

$t$ (minutes) | $0$ | $5$ | $10$ | $15$ | $20$ | $25$ | $30$ | $35$

$F$ (Fahrenheit temp) | $37.00$ | $44.74$ | $50.77$ | $55.47$ | $59.12$ | $61.97$ | $64.19$ | $65.92$

• Determine $AV_{[0,5]}$, $AV_{[5,10]}$, and $AV_{[10,15]}$, including appropriate units. Choose one of these quantities and write a careful sentence to explain its meaning. Your sentence might look something like “On the interval $\ldots$, the temperature of the soda is $\ldots$ on average by $\ldots$ for each $1$-unit increase in $\ldots$”. - On which interval is there more total change in the soda's temperature: $[10,20]$ or $[25,35]$? - What can you observe about when the soda's temperature appears to be changing most rapidly? - Estimate the soda's temperature when $t = 37$ minutes. Write at least one sentence to explain your thinking.

The position of a car driving along a straight road at time $t$ in minutes is given by the function $y = s(t)$ that is pictured in Figure. The car's position function has units measured in thousands of feet. For instance, the point $(2,4)$ on the graph indicates that after 2 minutes, the car has traveled 4000 feet.

• In everyday language, describe the behavior of the car over the provided time interval. In particular, carefully discuss what is happening on each of the time intervals $[0,1]$, $[1,2]$, $[2,3]$, $[3,4]$, and $[4,5]$, plus provide commentary overall on what the car is doing on the interval $[0,12]$. - Compute the average rate of change of $s$ on the intervals $[3,4]$, $[4,6]$, and $[5,8]$. Label your results using the notation “$AV_{[a,b]}$” appropriately, and include units on each quantity. - On the graph of $s$, sketch the three lines whose slope corresponds to the values of $AV_{[3,4]}$, $AV_{[4,6]}$, and $AV_{[5,8]}$ that you computed in (b). - Is there a time interval on which the car's average velocity is $5000$ feet per minute? Why or why not? - Is there ever a time interval when the car is going in reverse? Why or why not?

Consider an inverted conical tank (point down) whose top has a radius of $3$ feet and that is $2$ feet deep. The tank is initially empty and then is filled at a constant rate of $0.75$ cubic feet per minute. Let $V=f(t)$ denote the volume of water (in cubic feet) at time $t$ in minutes, and let $h= g(t)$ denote the depth of the water (in feet) at time $t$. It turns out that the formula for the function $g$ is $g(t) = \left( \frac{t}{\pi} \right)^{1/3}$.

• In everyday language, describe how you expect the height function $h = g(t)$ to behave as time increases. - For the height function $h = g(t) = \left( \frac{t}{\pi} \right)^{1/3}$, compute $AV_{[0,2]}$, $AV_{[2,4]}$, and $AV_{[4,6]}$. Include units on your results. - Again working with the height function, can you determine an interval $[a,b]$ on which $AV_{[a,b]} = 2$ feet per minute? If yes, state the interval; if not, explain why there is no such interval. - Now consider the volume function, $V = f(t)$. Even though we don't have a formula for $f$, is it possible to determine the average rate of change of the volume function on the intervals $[0,2]$, $[2,4]$, and $[4,6]$? Why or why not?