To create a good model we first decide what kind of function to use. What sort of function has the right shape to describe the process we want to model? Should it be increasing or decreasing, or some combination of both? Is the slope constant or is it changing?
Forrest leaves his house to go to school. For each of the following situations, sketch a possible graph of Forrest’s distance from home as a function of time.
Forrest walks at a constant speed until he reaches the bus stop.
Forrest walks at a constant speed until he reaches the bus stop, waits there until the bus arrives, and then the bus drives him to school at a constant speed.
The graph is a straight-line segment, as shown in figure (a). It begins at the origin because at the instant Forrest leaves the house, his distance from home is 0. (In other words, when \(t = 0, y = 0\text{.}\)) The graph is a straight line because Forrest has a constant speed. The slope of the line is equal to Forrest’s walking speed.
The first part of the graph is the same as part (a). But while Forrest waits for the bus, his distance from home remains constant, so the graph at that time is a horizontal line, as shown in figure (b). The line has slope \(0\) because while Forrest is waiting for the bus, his speed is \(0\text{.}\)
The graph begins like the graph in part (b). The last section of the graph represents the bus ride. It has a constant slope because the bus is moving at a constant speed. Because the bus (probably) moves faster than Forrest walks, the slope of this segment is greater than the slope for the walking section. The graph is shown in figure (c).
Erin walks from her home to a convenience store, where she buys some cat food, and then walks back home. Sketch a possible graph of her distance from home as a function of time.
The graphs in Example 5.6.1 are portions of straight lines. We can also consider graphs that bend upward or downward. The bend is called the concavity of the graph.
The two functions described in this example are both increasing functions, but they increase in different ways. Match each function to its graph and to the appropriate table of values.
The number of flu cases reported at an urban medical center during an epidemic is an increasing function of time, and it is growing at a faster and faster rate.
The number of flu cases is described by graph (A) and table (1). The function values in table (1) increase at an increasing rate. We can see this by computing the rate of change over successive time intervals.
\begin{gather*}
x = 0 \text{ to } x = 5: ~~~~~~~~m = \frac{\Delta y}{\Delta x}=\frac{123-70}{5-0} = 10.6\\
\\
x = 5\text{ to } x = 10:~~~~~~~~ m = \frac{\Delta y}{\Delta x}=\frac{217-123}{10-5} = 18.8\\
\\
x = 10 \text{ to } x = 15:~~~~~~~~m = \frac{\Delta y}{\Delta x}
=\frac{383 - 217}{15 - 10} = 33.2
\end{gather*}
The increasing rates can be seen inthe figure below; the graph bends upward as the slopes increase.
Francine bought a cup of cocoa at the cafeteria. The cocoa cooled off rapidly at first, and then gradually approached room temperature. Which graph more accurately reflects the temperature of the cocoa as a function of time? Explain why. Is the graph you chose concave up or concave down?
Graph (a): The graph has a steep negative slope at first, corresponding to an initial rapid drop in the temperature of the cocoa. The graph becomes closer to a horizontal line, corresponding to the cocoa approaching room temperature. The graph is concave up.
Subsection5.6.2Using the Basic Functions as Models
We have considered situations that can be modeled by linear or quadratic functions. In this section we’ll look at a few of the other basic functions.
Choose one of the eight basic functions to model each situation, and sketch a possible graph. \(~\alert{\text{[TK]}}~~\)
The number of board-feet, \(B\text{,}\) that can be cut from a Ponderosa pine is a function of the cube of the circumference, \(c\text{,}\) of the tree at a standard height.
The manager of an appliance store must decide how many coffee-makers to order every quarter. The optimal order size, \(Q\text{,}\) is a function of the square root of the annual demand for coffee-makers, \(D\text{.}\)
The loudness, or intensity, \(I\text{,}\) of the music at a concert is a function of the reciprocal of the square of your distance, \(d\text{,}\) from the speakers.
The contractor for a new hotel is estimating the cost of the marble tile for a circular lobby. The cost, \(C\text{,}\) is a function of the square of the diameter, \(D\text{,}\) of the lobby.
Investors are deciding whether to support a windmill farm. The wind speed, \(v\text{,}\) needed to generate a given amount of power is a function of the cube root of the power, \(P\text{.}\)
The speed of sound is a function of the temperature of the air in kelvins. (The temperature, \(T\text{,}\) in kelvins is given by \(T = C + 273\text{,}\) where \(C\) is the temperature in degrees Celsius.) The table shows the speed of sound, \(s\text{,}\) in meters per second, at various temperatures, \(T\text{.}\)
On a summer night when the temperature is \(20\degree\) Celsius, you see a flash of lightning, and 6 seconds later you hear the thunderclap. Use your function to estimate your distance from the thunderstorm.
We are looking for a value of \(k\) so that the function \(f(T) = k \sqrt{T}\) fits the data. We substitute one of the data points into the formula and solve for \(k\text{.}\) If we choose the point \((100, 200.6)\text{,}\) we obtain
\begin{equation*}
200.6 = k \sqrt{100}
\end{equation*}
and solving for \(k\) yields \(k = 20.06\text{.}\) We can check that the formula \(s = 20.06 \sqrt{T} \) is a good fit for the rest of the data points as well. Thus, we suggest the function
\begin{equation*}
f (T) = 20.06\sqrt{T}
\end{equation*}
The lightning and the thunderclap occur simultaneously, and the speed of light is so fast (about 30,000,000 meters per second) that we see the lightning flash as it occurs. So if the sound of the thunderclap takes \(6\) seconds after the flash to reach us, we can use our calculated speed of sound to find our distance from the storm.
The ultraviolet index (UVI) is issued by the National Weather Service as a forecast of the amount of ultraviolet radiation expected to reach Earth around noon on a given day. The data show how much exposure to the sun people can take before risking sunburn.
Plot \(m\text{,}\) the minutes to burn, against \(u\text{,}\) the UVI, to obtain two graphs, one for people who are more sensitive to sunburn, and another for people less sensitive to sunburn. Which of the basic functions do your graphs most resemble?
At this point, a word of caution is in order. There is more to choosing a model than finding a curve that fits the data. A model based purely on the data is called an empirical model. However, many functions have similar shapes over small intervals of their input variables, and there may be several candidates that model the data. Such a model simply describes the general shape of the data set; the parameters of the model do not necessarily correspond to any actual process.
In contrast, mechanistic models provide insight into the biological, chemical, or physical process that is thought to govern the phenomenon under study. Parameters derived from mechanistic models are quantitative estimates of real system properties. Here is what GraphPad Software has to say about modeling:
"Choosing a model is a scientific decision. You should base your choice on your understanding of chemistry or physiology (or genetics, etc.). The choice should not be based solely on the shape of the graph.
"Some programs . . . automatically fit data to hundreds or thousands of equations and then present you with the equation(s) that fit the data best. Using such a program is appealing because it frees you from the need to choose an equation. The problem is that the program has no understanding of the scientific context of your experiment. The equations that fit the data best are unlikely to correspond to scientifically meaningful models. You will not be able to interpret the best-fit values of the variables, and the results are unlikely to be useful for data analysis."
The absolute value function is used to model problems involving distance. Recall that the absolute value of a number gives the distance from the origin to that number on the number line.
We can find the distance from a number \(x\) to some point other than the origin, say \(a\text{,}\) by computing \(\abs{x - a}\text{.}\) For instance, the distance on the number line from \(x=-2\) to \(a=5\) is
For example, the equation \(\abs{x - 2} = 6\) means "the distance between \(x\) and \(2\) is \(6\) units." The number \(x\) could be to the left or the right of \(2\) on the number line. Thus, the equation has two solutions, \(8\) and \(-4\text{,}\) as shown below.
The distance between \(p\) and \(5\) is two units, or \(\abs{p - 5} = 2\text{.}\) If we count two units on either side of \(5\text{,}\) we see that \(p\) can be \(3\) or \(7\text{.}\)
The distance between \(a\) and \(-2\) is less than four units, or \(\abs{a - (-2)} \lt 4\text{,}\) or \(\abs{a + 2} \lt 4\text{.}\) Count four units on either side of \(-2\text{,}\) to find \(-6\) and \(2\text{.}\) Then \(a\) is between \(-6\) and \(2\text{,}\) or \(-6 \lt a \lt 2\text{.}\)
A graph can help us analyze an absolute value equation. For example, we know that the simple equation \(\abs{x} = 5\) has two solutions, \(x = 5\) and \(x = -5\text{.}\)
In fact, we can see from the graph at right that the equation \(\abs{x} = k\) has two solutions if \(k \gt 0\text{,}\) one solution if \(k = 0\text{,}\) and no solution if \(k \lt 0\text{.}\)
The graph shows the graphs of \(y = \abs{3x - 6}\) and \(y = 9\text{.}\) We see that there are two points on the graph of \(y = \abs{3x - 6}\) that have \(y = 9\text{,}\) and those points have \(x\)-coordinates \(x = -1\) and \(x = 5\text{.}\) We can verify algebraically that the solutions are \(-1\) and \(5\text{.}\)\(~\alert{\text{[TK]}}\)
Because of the V-shape of the graph, all points with \(y\)-values less than \(9\) lie between the two solutions of \(~\abs{3x - 6} = 9~\text{,}\) that is, between \(-1\) and \(5\text{.}\) Thus, the solutions of the inequality \(~\abs{3x - 6} \lt 9~\) are \(-1 \lt x \lt 5, \) as shown in figure (a).
On the other hand, to solve the inequality \(~\abs{3x - 6} \gt 9~\text{,}\) we look for points on the graph with \(y\)-values greater than \(9\text{.}\) In figure (b), we see that these points have \(x\)-values outside the interval between \(-1\) and \(5\text{.}\) In other words, the solutions of the inequality \(~\abs{3x - 6} \gt 9~\) are \(x \lt -1\) or \(x \gt 5\text{.}\)
Because the inequality symbol is \(\lt\text{,}\) the solutions of the inequality are between these two values: \(3.7475 \lt x \lt 3.7525\text{.}\) In interval notation, the solutions are \((3.7475, 3.7525)\text{.}\)\(~\alert{\text{[TK]}}~~\)
The graph defines a function, \(h\text{,}\) that shows the height, \(s\text{,}\) in meters, of a duck \(t\) seconds after it is flushed out of the bushes.
Four different functions are described below. Match each description with the appropriate table of values and with its graph.
As a chemical pollutant pours into a lake, its concentration is a function of time. The concentration of the pollutant initially increases quite rapidly, but due to the natural mixing and self-cleansing action of the lake, the concentration levels off and stabilizes at some saturation level.
The population of a small suburb of a Florida city is a function of time. The population began increasing rather slowly, but it has continued to grow at a faster and faster rate.
The level of production at a manufacturing plant is a function of capital outlay, that is, the amount of money invested in the plant. At first, small increases in capital outlay result in large increases in production, but eventually the investors begin to experience diminishing returns on their money, so that although production continues to increase, it is at a disappointingly slow rate.
The number of bacteria in a person during the course of an illness is a function of time. It increases rapidly at first, then decreases slowly as the patient recovers.
A squirrel drops a pine cone from the top of a California redwood. The height of the pine cone is a function of time, decreasing ever more rapidly as gravity accelerates its descent.
Enrollment in Ginny’s Weight Reduction program is a function of time. It began declining last fall. After the holidays, enrollment stabilized for a while but soon began to fall off again.
The table shows how the amount of water, \(A\text{,}\) flowing past a point on a river is related to the width, \(W\text{,}\) of the river at that point.
As the global population increases, many scientists believe it is approaching, or has already exceeded, the maximum number the Earth can sustain. This maximum number, or carrying capacity, depends on the finite natural resources of the planet -- water, land, air, and materials -- but also on how people use and preserve the resources. The graphs show four different ways that a growing population can approach its carrying capacity over time. (Source: Meadows, Randers, and Meadows, 2004)
Overshoot and collapse: the population exceeds the carrying capacity with severe damage to the resource base, and is forced to decline rapidly to achieve a new balance with a reduced carrying capacity.
Overshoot and oscillation: the population exceeds the carrying capacity without inflicting permanent damage, then oscillates around the limit before leveling off.
Use absolute value notation to write each expression as an equation or an inequality. (It may be helpful to restate each sentence using the word distance.)
