A powerful feature of mathematics is that it can be studied both as an abstract discipline and as an applied one. For instance, calculus can be developed almost entirely as an abstract collection of ideas that focus on properties of functions. At the same time, if we consider functions that represent meaningful processes, calculus can describe our experience of physical reality. We have already learned that for the position function \(y = s(t)\) of a ball being tossed straight up in the air, the derivative of the position function, \(v(t) = s'(t)\text{,}\) gives the ball’s velocity at time \(t\text{.}\)
In this section, we investigate several functions with contextual physical meaning and consider how the units on the independent variable, dependent variable, and the derivative function add to our understanding.
Suppose that you are driving a car on the highway using cruise control so that you drive at a constant speed. The time, \(T\) (measured in hours), it takes to drive 50 miles depends on the constant rate, \(s\) (measured in miles per hour) at which you are driving. Since
For instance, observe that \(T(75) = \frac{50}{75} = \frac{2}{3}\text{,}\) so it takes \(\frac{2}{3}\) of an hour to drive \(50\) miles at \(75\) miles per hour, and the point \((75, \frac{2}{3})\) lies on the graph of \(T(s)\text{.}\)
The function is decreasing and decreasing more quickly, because the travel time decreases as speed increases, and it does so in a way that the decreases get larger.
The function is decreasing but decreasing more slowly, because the travel time decreases as speed increases, but the amount of decrease is getting smaller.
and that this value has several different interpretations. For instance, if we set \(x = a\text{,}\)\(f'(a)\) is the slope of the tangent line to the graph of \(y = f(x)\) at the point \((a,f(a))\text{.}\)
We sometimes write \(\frac{df}{dx}\) or \(\frac{dy}{dx}\) instead of \(f'(x)\text{,}\) and these alternate notations emphasize the meaning of the derivative as the instantaneous rate of change of \(f\) with respect to \(x\). To understand the units on the derivative function, we use the fact that the instantaneous rate of change is the limit of the average rate of change. Recall that the average rate of change of \(f\) on the interval \([a,a+h]\) is
has the same units: units of \(f\) per unit of \(x\text{.}\) Said slightly differently, the units on the derivative function are “units of output per unit of input” for the variables and context of the original function, \(f\text{.}\)
Suppose that the function \(y = P(t)\) measures the population of a city (in thousands) at the start of year \(t\) (where \(t = 0\) corresponds to the year 2020). What does
The value \(P'(2) = 21.37\) tells us that the instantaneous rate of change of the city’s population with respect to time at the start of 2022 is \(21.37\) thousand people per year, since \(P\) is measured in thousands of people and \(t\) is measured in years. In addition, because the derivative value tells us how fast the population is changing, we can also infer about how much the population will change: we expect that in the year from 2022 to 2023, about \(21,370\) people will be added to the city’s population.
Consider the function \(T(s) = \frac{50}{s}\) from Preview Activity 1.5.1, whose output tells us the amount of time (in hours) it takes to drive 50 miles at \(s\) miles per hour. What is the meaning of
The value \(T'(60) = -0.0139\) means that the instantaneous rate of change of travel time, \(T\text{,}\) when \(s = 60\) is \(-0.0139\) hours per (mile per hour). This also tells us about how much the function value will change if we change the speed at which we drive: if we are driving at a speed of \(60\) miles per hour and we increase our speed by one mile per hour, we expect the time it takes us to drive 50 miles to drop by about \(0.0139\) hours. Similarly, if we increase our speed by two miles per hour, we’d expect the time it takes us to drive 50 miles to drop by about \(2 \cdot 0.0139 = 0.0278\) hours.
The units on \(T'(60)\) in Example 1.5.2 are “hours per (mile per hour)”, since the number of hours it takes to drive 50 miles depends on the speed (in miles per hour) at which we travel. If we tried to simplify those units, we lose the meaning inherent to the context of the value of \(T'(60)\text{.}\)
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.
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.
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.
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.
Subsection1.5.3Toward more accurate derivative estimates
Recall that to estimate the value of \(f'(a)\) at a given value of \(x = a\text{,}\) we can compute a difference quotient\(\frac{f(a+h)-f(a)}{h}\) with a relatively small value of \(h\text{.}\) We usually use both positive and negative values of \(h\) in order to account for the behavior of the function on both sides of the point of interest. In the setting where we have limited data, we introduce the notion of a central difference for estimating the value of \(f'(a)\text{.}\)
Suppose that \(y = f(x)\) is a function for which three values are known: \(f(1) = 2.5\text{,}\)\(f(2) = 3.25\text{,}\) and \(f(3) = 3.625\text{.}\) Estimate \(f'(2)\text{.}\)
We know that \(f'(2) = \lim_{h \to 0} \frac{f(2+h) - f(2)}{h}\text{.}\) But since we don’t have a graph or a formula for the function, we can neither sketch a tangent line nor evaluate the limit algebraically, nor can we use smaller and smaller values of \(h\) to estimate the limit.
Because the first approximation looks backward from the point \((2,3.25)\) and the second approximation looks forward, it makes sense to average these two estimates in order to account for behavior on both sides of \(x=2\text{.}\) Doing so, we find that
When we have two function values for \(f\) at equally spaced locations on opposite sides of \(x=a\text{,}\) the approach of averaging the two estimates — as we did in Example 1.5.4 — results in the best possible estimate we can find for \(f'(a)\) using the two known function values.
To see why, let’s think about how we look both “backward” and “forward” from \(x=a\text{,}\) like we did for the specific data in Example 1.5.4. Using a backward difference, \(BD\text{,}\) one estimate for \(f'(a)\) is
Because these two estimates take into account the function’s behavior on both sides of \(x=a\text{,}\) it makes sense to average them to include the most possible information for an even better estimate for the value of \(f'(a)\text{.}\) Doing so, we find that
In Figure 1.5.6, we see a function that passes through the three data points given in Example 1.5.4: \((1,2.5)\text{,}\)\((2,3.25)\text{,}\) and \((3,3.265)\text{.}\) The top line in the figure is the tangent line to the curve at \((2,f(2))\text{;}\) for the function shown, the slope of that tangent line turns out to be \(f'(2) = 0.5199\text{.}\) We found in Example 1.5.4 that the central difference approximation is \(0.5625\text{,}\) which is the slope of the secant line through the points \((1,f(1))\) and \((3,f(3))\) in Figure 1.5.6.
A potato is placed in an oven whose temperature is 350 degrees Fahrenheit, and the potato’s temperature \(F\) (in degrees Fahrenheit) is recorded in the following table. Time \(t\) is measured in minutes. 1
Thanks to Nick Owad of Hood College for conducting experiments with actual potatoes in his oven in order to generate the data for this activity.
Suppose we know that \(F(46) = 192.5\) and \(F'(46) = 1.39\text{.}\) What are the respective units on these two quantities? What do you expect the temperature of the potato to be when \(t = 47\text{?}\) when \(t = 48\text{?}\) Why?
Write a couple of careful sentences that describe the behavior of the temperature of the potato on the time interval \([10,40]\text{,}\) as well as the behavior of the instantaneous rate of change of the potato’s temperature on the same time interval.
In Section 1.4, we learned how use to the graph of a given function \(f\) to plot the graph of its derivative, \(f'\text{.}\) It is important to remember that when we do so, the scale (often) and the units (always) on the vertical axis have to change to represent \(f'\text{.}\) For example, suppose that \(P(t) = 212-148e^{-0.05t}\) tells us the temperature in degrees Fahrenheit of a potato in an oven at time \(t\) in minutes. In Figure 1.5.8, we sketch the graph of \(P\text{,}\) and in Figure 1.5.9 the graph of \(P'\text{.}\) Notice that the vertical scales are different in size and different in units, as the units of \(P\) are °F, while those of \(P'\) are °F/min.
Researchers at a major car company have found a function that relates gasoline consumption to speed for a particular model of car. In particular, they have determined that the consumption \(C\text{,}\) in liters per kilometer, at a given speed \(s\text{,}\) is given by a function \(C = f(s)\text{,}\) where \(s\) is the car’s speed in kilometers per hour.
Data provided by the car company tells us that \(f(80) = 0.015\text{,}\)\(f(90) = 0.02\text{,}\) and \(f(100) = 0.027\text{.}\) Use this information to estimate the instantaneous rate of change of fuel consumption with respect to speed at \(s = 90\text{.}\) Be as accurate as possible, use proper notation, and include units on your answer.
The derivative of a given function \(y=f(x)\) measures the instantaneous rate of change of the output variable with respect to the input variable. It also predicts about how much change we can expect when the input variable changes a small amount.
The units on the derivative function \(y = f'(x)\) are units of \(f\) per unit of \(x\text{.}\) Again, this measures how fast the output of the function \(f\) changes when the input variable changes.
This quantity measures the slope of the secant line to \(y = f(x)\) through the points \((a-h, f(a-h))\) and \((a+h, f(a+h))\text{.}\) The central difference generates a good approximation of the derivative’s value at \(x = a\text{.}\)
The temperature, \(H\text{,}\) in degrees Celsius, of a cup of coffee placed on the kitchen counter is given by \(H = f(t)\text{,}\) where \(t\) is in minutes since the coffee was put on the counter.
Suppose that \(|f'(15)| = 1\) and \(f(15) = 59\text{.}\) Fill in the blanks and select the appropriate units and terms to complete the following statement about the temperature of the coffee in this case.
A laboratory study investigating the relationship between diet and weight in adult humans found that the weight of a subject, \(W\text{,}\) in pounds, was a function, \(W=f(c)\text{,}\) of the average number of Calories, \(c\text{,}\) consumed by the subject in a day.
(e) Suppose that Sam reads about \(f'\) in this study and draws the following conclusion: If Sam increases his average calorie intake from 2600 to 2630 calories per day, then his weight will increase by approximately 0.3 pounds.
Let \(f(t)\) be the number of centimeters of rainfall that has fallen since midnight, where \(t\) is the time in hours. Match the following statements to their interpretations, given below.
A cup of coffee has its temperature \(F\) (in degrees Fahrenheit) at time \(t\) given by the function \(F(t) = 75 + 110 e^{-0.05t}\text{,}\) where time is measured in minutes.
Use a central difference with \(h = 0.01\) to estimate the value of \(F'(10)\text{.}\)
Write a sentence that describes the behavior of the function \(y = F'(t)\) on the time interval \(0 \le t \le 30\text{.}\) How do you think its graph will look? Why?
The temperature change \(T\) (in Fahrenheit degrees), in a patient, that is generated by a dose \(q\) (in milliliters), of a drug, is given by the function \(T = f(q)\text{.}\)
What does it mean to say \(f(50) = 0.75\text{?}\) Write a complete sentence to explain, using correct units.
Suppose that \(f'(50) = -0.02\text{.}\) Write a complete sentence to explain the meaning of this value. Include in your response the information given in (a).
The velocity of a ball that has been tossed vertically in the air is given by \(v(t) = 16 - 32t\text{,}\) where \(v\) is measured in feet per second, and \(t\) is measured in seconds. The ball is in the air from \(t = 0\) until \(t = 2\text{.}\)
What are the units on the value of \(v'(1)\text{?}\) What does this value and the corresponding units tell you about the behavior of the ball at time \(t = 1\text{?}\)
The value, \(V\text{,}\) of a particular automobile (in dollars) depends on the number of miles, \(m\text{,}\) the car has been driven, according to the function \(V = h(m)\text{.}\)
Suppose that \(h(40000) = 15500\) and \(h(55000) = 13200\text{.}\) What is the average rate of change of \(h\) on the interval \([40000,55000]\text{,}\) and what are the units on this value?
In addition to the information given in (a), say that \(h(70000) = 11100\text{.}\) Determine the best possible estimate of \(h'(55000)\) and write one sentence to explain the meaning of your result, including units on your answer.
Write a sentence to describe the long-term behavior of the function \(V = h(m)\text{,}\) plus another sentence to describe the long-term behavior of \(h'(m)\text{.}\) Provide your discussion in practical terms regarding the value of the car and the rate at which that value is changing.
