Activity 12.
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 \(0.89\).
Hint.
Answer.
\(V'(10250) = -0.89\) means that the instantaneous rate of change of the car’s value when the car has been driven \(10250\) miles is \(-0.89\) dollars per mile. In addition, when the car has been driven \(10250\) miles, if the car is driven one more mile, we expect that the value of the car will decrease by about \(0.89\) dollars.
Solution.
\(V'(10250) = -0.89\) means that the instantaneous rate of change of the car’s value when the car has been driven \(10250\) miles is \(-0.89\) dollars per mile. This tells us that when the car has been driven \(10250\) miles, if the car is driven one more mile, we expect that the value of the car will decrease by about \(0.89\) dollars.
(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.
Hint.
Answer.
\(W'(0.75) = 3.43\) means that the instantaneous rate of change of volume of water in the tank when the water is \(0.75\) meters deep is \(3.43\) liters per meter. This tells us that when the water is \(0.75\) meters deep, if the water rises one more meter, we expect that the volume of water in the tank will increase by about \(3.43\) liters.
Solution.
\(W'(0.75) = 3.43\) means that the instantaneous rate of change of volume of water in the tank when the water is \(0.75\) meters deep is \(3.43\) liters per meter. This tells us that when the water is \(0.75\) meters deep, if the water rises one more meter, we expect that the volume of water in the tank will increase by about \(3.43\) liters.
(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.
Hint.
Answer.
\(S'(20) = -0.527\) means that the instantaneous rate of change of the soda’s temperature at the instant \(t = 20\) minutes is \(-0.527\) degrees Celsius per minute. This tells us that after \(20\) minutes have elapsed, if one more minute passes, we expect that the soda’s temperature will drop by about \(0.527\) degrees Celsius.
Solution.
\(S'(20) = -0.527\) means that the instantaneous rate of change of the soda’s temperature at the instant \(t = 20\) minutes is \(-0.527\) degrees Celsius per minute. This tells us that after \(20\) minutes have elapsed, if one more minute passes, we expect that the soda’s temperature will drop by about \(0.527\) degrees Celsius.
(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.
Hint.
Answer.
\(C'(19) = 52.1\) means that the instantaneous rate of change of the rate at which the biker is burning calories when traveling at a speed of \(19\) kilometers per hour is \(52.1\) calories per hour per kilometer per hour. This tells us that when the person is riding at \(19\) kilometers per hour, if they increase their speed by \(1\) kilometer per hour, we expect that they will burn about \(52.1\) additional calories over the next hour.
Solution.
\(C'(19) = 52.1\) means that the instantaneous rate of change of the rate at which the biker is burning calories when traveling at a speed of \(19\) kilometers per hour is \(52.1\) calories per hour per kilometer per hour. This tells us that when the person is riding at \(19\) kilometers per hour, if they increase their speed by \(1\) kilometer per hour, we expect that they will burn about \(52.1\) additional calories over the next hour.
