Activity 1.5.2.
In each of the following contexts, explain the meaning of the given derivative values in two ways that are similar to the sentence structures and discussions of units in the two examples preceding this activity in the text.
First, write a sentence that explicitly describes what we know about the instantaneous rate of change of the given function at a particular instant, with units. Then, write a second sentence that explains the amount of change we expect in the function value if the input variable changes by one unit at the known instant.
Remember not to try to simplify the units on the derivative value, but rather keep the units in the form “units of output per unit of input”.
(a)
Suppose that \(V(m)\) measures the value of a car (in dollars) after the car has been driven \(m\) miles. Explain the meaning of the statement
\begin{equation*}
V'(10250) = -0.89
\end{equation*}
by writing two sentences that address the instantaneous rate of change and the predicted change in the function, respectively.
Explicitly, your first sentence might have structure like
\(V'(10250) = -0.89\) means that the instantaneous rate of change of when the car has been driven miles is \(-0.89\),
where the final blank should be completed using the units on the derivative. Your second sentence might have form
When the car has been driven miles, if the car is driven one more mile, we expect that the value of the car will by about \(0.89\).
(b)
Suppose that \(W(h)\) measures the amount of water (in liters) in a tank that is filled with water that is \(h\) meters deep. Explain the meaning of the statement
\begin{equation*}
W'(0.75) = 3.43
\end{equation*}
by writing two sentences in the structure described above.
(c)
Suppose that \(S(t)\) measures the temperature of a can of soda (in degrees Celsius) in a refrigerator at time \(t\) in minutes. Explain the meaning of the statement
\begin{equation*}
S'(20) = -0.527
\end{equation*}
by writing two sentences in the structure described above.
(d)
Suppose that \(C(s)\) measures the rate at which a person burns calories (in calories per hour) when riding a bike at a speed of \(s\) kilometers per hour. Explain the meaning of the statement
\begin{equation*}
C'(19) = 52.1
\end{equation*}
by writing two sentences in the structure described above.
