8.4: Face-to-Face Courses are Superior to Online Courses
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- Cheryl E. Ball & Drew M. Loewe ed.
- West Virginia University via Digital Publishing Institute and West Virginia University Libraries
Author: Andrew Bourelle, English, University of New Mexico.
Academia has seen a shift to online education, with many courses being taught from a distance to accommodate students who cannot attend traditional face-to-face (f2f) classes on campus for various reasons, including familial obligations or work schedules. To accommodate these students, the environment in which classes are taught has changed to include correspondence courses taught via mail and television-broadcast lectures, to courses taught solely in a digital platform with students communicating remotely through asynchronous (i.e., discussion boards) and synchronous (i.e., video conferencing) methods. The method of ease associated with online education makes access possible for students from the comfort of home without actually setting foot in a classroom. But make no mistake: The easiness comes from convenience of access, not necessarily from the coursework itself. Online classes can be just as—if not more—rigorous, educational, and pedagogically sound than f2f classes, with the scholarship surrounding distance education reporting no significant difference between learning in online classes versus classes taught in f2f environments. However, there remains a common misconception among instructors and students alike that the online class will be less challenging or rigorous than its f2f counterpart. From students to the general public— and even among college instructors—most people continue to think that traditional classes held in a brick-and-mortar classroom are, simply put, better than online classes.
In fact, a survey of online education conducted by the Babson Research Group reported that many instructors question the value of online education’s worth. Words like validity and legitimacy often arise when instructors discuss the merits of online education, and when it comes to writing, they ask how and if students can learn to craft their writing without forming a community of peers or when f2f interaction with the professor is missing. This skepticism from the faculty trickles down to the students and even filters into societal thinking at large, with a 2013 Gallup poll suggesting that Americans remain “tepid at best” when rating their experiences or opinions of online classes and programs. Even businesses tend to prefer graduates with degrees earned in f2f programs over those who received solely online degrees.
How did this myth associated with online education take shape? In the past, some universities and the best faculty members within refused to teach online for various reasons, including the inability to see the student benefit from online education. When powerhouse universities such as Arizona State University led the charge in expanding education opportunities for those students who would not normally have access, critics thought the gain was only monetary, benefiting the university and not the students themselves. Many universities have joined the ranks of ASU, scrambling to keep the pace in order to secure their share of distance students. In addition, administrators may push departments to offer online classes because they do not require a physical space, thus costing less money to facilitate. Other concerns include the higher rate of withdrawals from online students and the tendency for students with a lower GPA to perform at an even lower level in the online environment. Perhaps one area of cynicism surrounds the increase of Massive Open Online Courses, or MOOCs, with thousands of students taking a course simultaneously. Some may wonder just how an instructor can interact with hundreds of students one at a time, and in terms of a writing course, how she can respond to thousands of pages of writing in one semester; however, these open classes are markedly different than the average writing course that strives to maintain low numbers and one-on-one interaction with students.
Certainly there are monetary benefits to universities adding online courses, and all students may not succeed in the online space; however, these factors are not necessarily synonymous with a diminished education. Within writing courses, students’ participation and writing practice improves when classes are shifted online. Additionally, the online classroom opens up a space for diversity, with more students being willing to share opinions in the safe space of an online classroom. For writing classes, the majority of the interactions among peers and between peers and the instructor is communicated in written form, suggesting that students have the opportunity to practice writing more than if they were taking an f2f class. In terms of simple word count, online students write a lot more than f2f students. Scholars also suggest that students think more critically about the discussion prompts they respond to because they are not required to think on the spot; instead, they can carefully craft their posts and even revise and edit afterward. Thus, numerous elements of the course encourage writing and thoughtful peer-to-peer and peer-to-teacher interaction. Those interactions are different than f2f but certainly not inferior.
While online education continues to grow at a rapid rate, the pedagogy within remains cutting-edge, comparable to f2f classes. For instance, societal changes and technological improvements have prompted writing instructors to encourage students to create multimodal texts, or texts that use more than one mode to communicate. These multimodal texts could be videos, podcasts, or websites, just to name a few. Instructors often encourage students to create these types of texts because students are interacting with similar media on a daily basis in their extracurricular lives; therefore, it seems natural to ask them to not only analyze these documents for rhetorical effectiveness, but to also create similar documents, in an effort to promote new literacies and effective communication skills necessary for the 21st century. Teachers of online classes have adapted their practices to include this type of curriculum. For instance, in any specific unit that corresponds to a writing prompt, an instructor may offer different media to analyze, including podcasts, videos, and written text, encouraging discussion among peers of the rhetorical effectiveness of such media. Not only do students have the ability to return to this media at any point in the online course, but they can also reread the archived discussion posts for further clarification of concepts. Successful practices of multimodal composition only solidify that writing, in its various forms, can be taught just as well, if not better, than in an f2f format.
The online classroom itself can also be a democratic space: The digital platform can further encourage inquiry and interaction, as ideas are constructed as a community, with community being society at large, not just peers in the class. When an instructor asks students to find artifacts, research, and other items of interest in pop or other cultures, those artifacts become representative of communities outside the classroom setting, thus widening the learning sphere. Students move from being merely passive consumers of media, including books, videos, advertisements, and so on, to becoming collaborators in knowledge-making who think critically about the plethora of media they come in contact with both in and outside the class. While this critical thinking can also be possible in f2f and blended (part f2f, part online) classes, it is perhaps not encouraged or acquired as seamlessly as in a solely online environment where students have access to such media at their fingertips, both in the classroom curriculum and on the Internet.
However, even with these advantages and enhancements that the online classroom promotes, an online course is only as effective as the instructor and the way she has designed the course. Instructor ambition and attention may cause student interest and motivation to wane. If an instructor isn’t active in the course, the students may put in little effort as well. Online writing courses can be successful at promoting writing skills, but only if the instructor re-envisions her pedagogical practices. For instance, instructors must create assignments that promote students taking control of their learning, challenging them to share ideas and collaborate with one another through the digital technology available in online courses. Indeed, we would argue that f2f writing instructors can learn from online pedagogy. As the world becomes increasingly influenced by digital technology, f2f instructors can learn a lot about enhancing their traditional classes with digital, multimodal enhancements and online writing tools.
Instead of using f2f classes as the barometer to measure online writing instruction, the time has come for instructors, administrators, students, and others interested in the quality of education offered in our universities to start recognizing that both f2f and online writing classes can provide challenging, intellectually stimulating educational experiences for students. Instead of viewing online and f2f courses in opposition to each other, we need to view them as parallel means of educating students. One fear may be that online classes will replace traditional f2f courses; however, as distance education continues to grow, universities remain committed to offering online classes in addition to f2f courses to accommodate both students who need access to remote courses and those who need the local constraints of a f2f classroom. Online and f2f classes are two paths available to students on their educational journeys, and both can lead students to the same destinations.
Further Reading
For popular, researched texts on online education, see “Chasing the Elusive ‘Quality’ in Online Education” by Anya Kamenetz, on National Public Radio and “Americans Doubt the Rigor and Quality of Online Education” by Allie Bidwell at US News and World Report . For practical advice on preparing to teach online, see “Benefits and Challenges of Online Education” by the Massachusetts General Hospital Institute of Health Professions, which also provides additional reading.
Keywords
asynchronous methods, face-to-face instruction (f2f), MOOCs, multimodal composition, online writing instruction (OWI), synchronous methods
Author Bios
Tiffany Bourelle is an assistant professor at the University of New Mexico, where she teaches technical communication and firstyear writing courses in both face-to-face and online environments. She is currently the co-administrator of the eComp (electronic composition) first-year writing program at UNM that she developed with Dr. Andrew Bourelle. Her research focuses on training graduate instructors to teach multimodal composition online, and her scholarship can be found in scholarly journals such as The Journal of Technical Writing and Communication, Technical Communication Quarterly, WPA: Writing Program Administration, Computers and Composition, and Kairos: A Journal of Rhetoric, Technology, and Pedagogy.
Andrew Bourelle is an assistant professor of English at the University of New Mexico. His research interests include online writing instruction, multimodal literacy, and the intersection of creative writing and composition. His scholarship has been published in Composition Forum, Computers and Composition, Journal of Teaching Writing, Kairos, Technical Communication Quarterly , and other journals and anthologies.


10.1: Center Manifold Theory
This chapter is about center manifolds, dimensional reduction, and stability of fixed points of autonomous vector fields. We begin with a motivational example.
Example \(\PageIndex{29}\)
Consider the following linear, autonomous vector field on \(\mathbb{R}^{c} \times \mathbb{R}^s\):
\(\dot{x} = Ax\),
\[\dot{y} = By, (x, y) \in \mathbb{R} \times \mathbb{R}, \label{10.1}\]
where A is a \(c \times c\) matrix of real numbers having eigenvalues with zero real part and \(B\) is a \(s \times s\) matrix of real numbers having eigenvalues with negative real part. Suppose we are interested in stability of the nonhyperbolic fixed point \((x, y) = (0, 0)\). Then that question is determined by the nature of stability of \(x = 0\) in the lower dimensional vector field:
\[\dot{x} = Ax, x \in \mathbb{R}^c. \label{10.2}\]
This follows from the nature of the eigenvalues of \(B\), and the properties that \(x\) and \(y\) are decoupled in (10) and that it is linear. More precisely, the solution of (10) is given by:
\[\begin{pmatrix} {x(t, x_{0})}&{y(t, y_{0}) } \end{pmatrix} = \begin{pmatrix} {e^{At}x_{0}}&{e^{Bt}y_{0}} \end{pmatrix}. \label{10.3}\]
From the assumption of the real parts of the eigenvalues of B having negative real parts, it follows that:
\[\lim_{t \rightarrow \infty} e^{Bt}y_{0} = 0.\]
In fact, 0 is approached at an exponential rate in time. Therefore it follows that stability, or asymptotic stability, or instability of x = 0 for (10.2) implies stability, or asymptotic stability, or instability of (x, y) = (0, 0) for (10).
It is natural to ask if such a dimensional reduction procedure holds for nonlinear systems. This might seem unlikely since, in general, nonlinear systems are coupled and the superposition principle of linear systems does not hold. However, we will see that this is not the case.
Invariant manifolds lead to a form of decoupling that results in a dimensional reduction procedure that gives, essentially, the same result as is obtained for this motivational linear example.This is the topic of center manifold theory that we now develop.
We begin by describing the set-up. It is important to realize that when applying these results to a vector field, it must be in the following form.
\(\dot{x} = Ax+f(x, y)\),
\[\dot{y} = By + g(x, y), (x, y) \in \mathbb{R}^c \times \mathbb{R}^s, \label{10.4}\]
where the matrices A and B are have the following properties:
- \(c \times c\) matrix of real numbers having eigenvalues with zero real parts,
- \(s \times s\) matrix of real numbers having eigenvalues with negative real parts,
and \(f\) and \(g\) are nonlinear functions. That is, they are of order two or higher in x and y, which is expressed in the following properties:
\[f(0, 0) = 0, Df(0, 0) = 0\]
\[g(0, 0) = 0, Dg(0, 0) = 0, \label{10.5}\]
and they are \(C^r\), \(r\) as large as required (we will explain what this means when we explicitly use this property later on).
With this set-up (x, y) = (0, 0) is a fixed point for (10.4) and we are interested in its stability properties.
The linearization of (10.4) about the fixed point is given by:
\(\dot{x} = Ax\),
\[\dot{y} = By, (x, y) \in \mathbb{R}^{c} \times \mathbb{R}^{s}, \label{10.6}\]
The fixed point is nonhyperbolic . It has a c dimensional invariant center subspace and a s dimensional invariant stable subspace given by:
\[E^c = {(x, y) \in R^{c} \times R^{s} | y = 0}, \label{10.7}\]
\[E^s = {(x, y) \in R^{c} \times R^{s} | x = 0}, \label{10.8}\]
respectively.
For the nonlinear system (Equation \ref{10.4}) there is a s dimensional, \(C^r\) passing through the origin and tangent to \(E^s\) at the origin. Moreover, trajectories in the local stable manifold inherit their behavior from trajectories in \(E^s\) under the linearized dynamics in the sense that they approach the origin at an exponential rate in time.
Similarly, there is a c dimensional \(C^r\) local center manifold that passes through the origin and is tangent to \(E^c\) are the origin. Hence, the center manifold has the form:
\[W^{c}(0) = {(x, y) \in \mathbb{R}^c \times \mathbb{R}^s | y = h(x), h(0) = 0, Dh(0) = 0}, \label{10.9}\]
which is valid in a neighborhood of the origin, i.e. for |x| sufficiently small.
We illustrate the geometry in Fig. 10.1.
The application of the center manifold theory for analyzing the behavior of trajectories near the origin is based on three theorems:
- existence of the center manifold and the vector field restricted to the center manifold,
- stability of the origin restricted to the center manifold and its relation to the stability of the origin in the full dimensional phase space,
- obtaining an approximation to the center manifold.
Theorem 5: Existence and Restricted Dynamics
There exists a \(C^r\) center manifold of \((x, y) = (0, 0)\) for Equation \ref{10.4}. The dynamics of Equation \ref{10.4} restricted to the center manifold is given by:
\[\dot{u} = Au+f(u, h(u)), u \in \mathbb{R}^c, \label{10.10}\]
for |u| sufficiently small.
A natural question that arises from the statement of this theorem is "why did we use the variable'‘u' when it would seem that 'x' would be the more natural variable to use in this situation"? Understanding the answer to this question will provide some insight and understanding to the nature of solutions near the origin and the geometry of the invariant manifolds near the origin. The answer is that "x and y are already used as variables for describing the coordinate axes in Equation \ref{10.4}. We do not want to confuse a point in the center manifold with a point on the coordinate axis. A point on the center manifold is denoted by (x, h(x)). The coordinate u denotes a parametric representation of points along the center manifold. Moreover, we will want to compare trajectories of Equation \ref{10.10} with trajectories in Equation \ref{10.4}. This will be confusing if x is used to denote a point on the center manifold. However, when computing the center manifold and when considering the (i.e. Equation \ref{10.10}) it is traditional to use the coordinate 'x', i.e. the coordinate describing the points in the center subspace. This does not cause ambiguities since we can name a coordinate anything we want. However, it would cause ambiguities when comparing trajectores in Equation \ref{10.4} with trajectories in Equation \ref{10.10}, as we do in the next theorem.
Theorem 6
- Suppose that the zero solution of Equation \ref{10.10} is stable (asymptotically stable) (unstable), then the zero solution of (10.4) is also stable (asymptotically stable) (unstable).
- Suppose that the zero solution of Equation \ref{10.10} is stable. Then if (x(t), y(t)) is a solution of (10.4) with (x(0), y(0)) sufficiently small, then there is a solution \(u(t)\) of Equation \ref{10.10} such that as \(t \rightarrow \infty\)
\(x(t) = u(t) + \mathcal{O}(e^{-\gamma t})\),
\[y(t) = h(u(t)) + \mathcal{O}(e^{-\gamma t}), \label{10.11}\]
where \(\gamma > 0\) is a constant.
Part i) of this theorem says that stability properties of the origin in the center manifold imply the same stability properties of the origin in the full dimensional equations. Part ii) gives much more precise results for the case that the origin is stable. It says that trajectories starting at initial conditions sufficiently close to the origin asymptotically approach a trajectory in the center manifold.
Now we would like to compute the center manifold so that we can use these theorems in specific examples. In general, it is not possible to compute the center manifold. However, it is possible to approximate it to "sufficiently high accuracy" so that we can verify the stability results of Theorem 6 can be confirmed. We will show how this can be done. The idea is to derive an equation that the center manifold must satisfy, and then develop an approximate solution to that equation.
We develop this approach step-by-step.
The center manifold is realized as the graph of a function,
\[y=h(x), x \in \mathbb{R}^c, y \in \mathbb{R}^s, \label{10.12}\]
i.e. any point \((x_{c}, y_{c})\), sufficiently close to the origin, that is in the center manifold satisfies \(y_{c} = h(x-{c})\). In addition, the center manifold passes through the origin (h(0) = 0) and is tangent to the center sub- space at the origin (\(Dh(0) = 0\)).
Invariance of the center manifold implies that the graph of the function \(h(x)\) must also be invariant with respect to the dynamics generated by Equation \ref{10.4}. Differentiating Equation \ref{10.12} with respect to time shows that \((\dot{x}, \dot{y})\) at any point on the center manifold satisfies
\[\dot{y} = Dh(x)\dot{x}. \label{10.13}\]
This is just the analytical manifestation of the fact that invariance of a surface with respect to a vector field implies that the vector field must be tangent to the surface.
We will now use these properties to derive an equation that must be satisfied by the local center manifold.
The starting point is to recall that any point on the local center manifold obeys the dynamics generated by . Substituting y = h(x) into gives:
\[\dot{x} = Ax + f (x, h(x)), \label{10.14}\]
\[\dot{y} = Bh(x) + g(x, h(x)), (x, y) \in \mathbb{R}^c \times \mathbb{R}^s. \label{10.15}\]
Substituting and into the invariance condition \(\dot{y} = Dh(x)\dot{x}\) gives:
\[Bh(x) + g(x, h(x)) = Dh(x) (Ax + f (x, h(x))) , \label{10.16}\]
or
\[Dh(x)(Ax+f(x, h(x)))-Bh(x)-g(x, h(x)) \equiv \mathcal{N}(h(x) = 0. \label{10.17}\]
This is an equation for \(h(x)\). By construction, the solution implies invariance of the graph of \(h(x)\), and we seek a solution satisfying the additional conditions \(h(0) = 0\) and \(Dh(0) = 0\). The basic result on approximation of the center manifold is given by the following theorem.
Theorem 7: Approximation
Let \(\phi : \mathbb{R}^c \rightarrow \mathbb{R}^s\) be a \(C^1\) mapping with
\(\phi (0) = 0, D_{\phi}(0) = 0\),
such that
\(\mathcal{N}(\phi (x)) = \mathcal{O}(|x|^{q})\) as \(x \rightarrow 0\),
for some \(q > 1\). Then
\(|h(x)-\phi (x)| = \mathcal{O}(|x|^{q})\) as \(x \rightarrow 0\).
The theorem states that if we can find an approximate solution of Equation \ref{10.17} to a specified degree of accuracy, then that approximate solution is actually an approximation to the local center manifold, to the same degree of accuracy.
We now consider some examples showing how these results are applied.
Example \(\PageIndex{30}\)
We consider the following autonomous vector field on the plane:
\(\dot{x} = x^{2}y-x^5\),
\[\dot{y} = -y+x^2, (x, y) \in \mathbb{R}^2, \label{10.18}\].
or, in matrix form:
\[\begin{pmatrix} {\dot{x}}\\ {\dot{y}} \end{pmatrix} = \begin{pmatrix} {0}&{0}\\ {0}&{-1} \end{pmatrix} \begin{pmatrix} {x}\\ {y} \end{pmatrix} + \begin{pmatrix} {x^{2}y-x^5}\\ {x^2} \end{pmatrix}, \label{10.19}\]
We are interested in determining the natural of the stability of (x, y) = (0, 0). The Jacobian associated with the linearization about this fixed point is:
\(\begin{pmatrix} {0}&{0}\\ {0}&{-1} \end{pmatrix}\),
which is nonhyperbolic, and therefore the linearization does not suffice to determine stability.
The vector field is in the form of Equation \ref{10.4}
\(\dot{x} = Ax+f(x, y)\),
\[\dot{y} = By+g(x, y), (x, y) \in \mathbb{R} \times \mathbb{R}, \label{10.20}\]
where
\[A = 0, B = 1, f(x, y)=x^{2}y-x^5, g(x, y) = x^2. \label{10.21}\]
We assume a center manifold of the form:
\[y = h(x) = ax^2 + bx^3 + \mathcal{O}(x^4), \label{10.22}\]
which satisfies h(0) = 0 ("passes through the origin") and Dh(0) = 0 (tangent to \(E^{c}\) at the origin). A center manifold of this type will require the vector field to be at least \(C^3\) (hence, the meaning of the phrase \(C^r\), r as large as necessary).
Substituting this expression into the equation for the center manifold (10.17) (using (10.21)) gives:
\[(2ax+3b^{2}+\mathcal{O}(x^3))(ax^4+bx^5+\mathcal{O}(x^6)-x^5)+ax^2+bx^3+\mathcal{O}(x^4)-x^2 = 0. \label{10.23}\]
In order for this equation to be satisfied the coefficients on each power of x must be zero. Through third order this gives:
\(x^2 : a - 1 = 0 \Rightarrow a = 1\),
\[x^3 : b = 0. \label{10.24}\]
Substituting these values into (10.22) gives the following expression for the center manifold through third order:
\[y = x^2 + \mathcal{O}(x^4). \label{10.25}\]
Therefore the vector field restricted to the center manifold is given by:
\[\dot{x} = x^4 + \mathcal{O}(x^5). \label{10.26}\]
Hence, for x sufficiently small, \(\dot{x}\) is positive for \(x \ne 0\), and therefore the origin is unstable. We illustrate the flow near the origin in Fig. 10.2.
Example \(\PageIndex{31}\)
We consider the following autonomous vector field on the plane:
\(\dot{x} = xy\),
\[\dot{y} = y+x^3, (x, y) \in \mathbb{R}^2, \label{10.27}\]
Therefore the vector field restricted to the center manifold is given by:
\[\dot{x} = x^4 + \mathcal{O}(x^5). \label{10.35}\]
Since \(\dot{x}\) is positive for x sufficiently small. the origin is unstable. We illustrate the flow near the origin in Fig. 10.3.


6.1: Modeling with Linear Functions
In this section, you will:
- Identify steps for modeling and solving.
- Build linear models from verbal descriptions.
- Build systems of linear models.
Emily is a college student who plans to spend a summer in Seattle. She has saved $3,500 for her trip and anticipates spending $400 each week on rent, food, and activities. How can we write a linear model to represent her situation? What would be the x -intercept, and what can she learn from it? To answer these and related questions, we can create a model using a linear function. Models such as this one can be extremely useful for analyzing relationships and making predictions based on those relationships. In this section, we will explore examples of linear function models.
Identifying Steps to Model and Solve Problems
When modeling scenarios with linear functions and solving problems involving quantities with a constant rate of change , we typically follow the same problem strategies that we would use for any type of function. Let’s briefly review them:
- Identify changing quantities, and then define descriptive variables to represent those quantities. When appropriate, sketch a picture or define a coordinate system.
- Carefully read the problem to identify important information. Look for information that provides values for the variables or values for parts of the functional model, such as slope and initial value.
- Carefully read the problem to determine what we are trying to find, identify, solve, or interpret.
- Identify a solution pathway from the provided information to what we are trying to find. Often this will involve checking and tracking units, building a table, or even finding a formula for the function being used to model the problem.
- When needed, write a formula for the function.
- Solve or evaluate the function using the formula.
- Reflect on whether your answer is reasonable for the given situation and whether it makes sense mathematically.
- Clearly convey your result using appropriate units, and answer in full sentences when necessary.
Building Linear Models
Now let’s take a look at the student in Seattle. In her situation, there are two changing quantities: time and money. The amount of money she has remaining while on vacation depends on how long she stays. We can use this information to define our variables, including units.
- Output: money remaining, in dollars
- Input: time, in weeks
So, the amount of money remaining depends on the number of weeks:
We can also identify the initial value and the rate of change.
- Initial Value: She saved $3,500, so $3,500 is the initial value for
- Rate of Change: She anticipates spending $400 each week, so –$400 per week is the rate of change, or slope.
Notice that the unit of dollars per week matches the unit of our output variable divided by our input variable. Also, because the slope is negative, the linear function is decreasing. This should make sense because she is spending money each week.
The rate of change is constant, so we can start with the linear model Then we can substitute the intercept and slope provided.
To find the intercept, we set the output to zero, and solve for the input.
The intercept is 8.75 weeks. Because this represents the input value when the output will be zero, we could say that Emily will have no money left after 8.75 weeks.
When modeling any real-life scenario with functions, there is typically a limited domain over which that model will be valid—almost no trend continues indefinitely. Here the domain refers to the number of weeks. In this case, it doesn’t make sense to talk about input values less than zero. A negative input value could refer to a number of weeks before she saved $3,500, but the scenario discussed poses the question once she saved $3,500 because this is when her trip and subsequent spending starts. It is also likely that this model is not valid after the intercept, unless Emily will use a credit card and goes into debt. The domain represents the set of input values, so the reasonable domain for this function is
In the above example, we were given a written description of the situation. We followed the steps of modeling a problem to analyze the information. However, the information provided may not always be the same. Sometimes we might be provided with an intercept. Other times we might be provided with an output value. We must be careful to analyze the information we are given, and use it appropriately to build a linear model.
Using a Given Intercept to Build a Model
Some real-world problems provide the intercept, which is the constant or initial value. Once the intercept is known, the intercept can be calculated. Suppose, for example, that Hannah plans to pay off a no-interest loan from her parents. Her loan balance is $1,000. She plans to pay $250 per month until her balance is $0. The intercept is the initial amount of her debt, or $1,000. The rate of change, or slope, is -$250 per month. We can then use the slope-intercept form and the given information to develop a linear model.
Now we can set the function equal to 0, and solve for to find the intercept.
The intercept is the number of months it takes her to reach a balance of $0. The -intercept is 4 months, so it will take Hannah four months to pay off her loan.
Using a Given Input and Output to Build a Model
Many real-world applications are not as direct as the ones we just considered. Instead they require us to identify some aspect of a linear function. We might sometimes instead be asked to evaluate the linear model at a given input or set the equation of the linear model equal to a specified output.
How To
Given a word problem that includes two pairs of input and output values, use the linear function to solve a problem.
- Identify the input and output values.
- Convert the data to two coordinate pairs.
- Find the slope.
- Write the linear model.
- Use the model to make a prediction by evaluating the function at a given value.
- Use the model to identify an value that results in a given value.
- Answer the question posed.
Example 1
Using a Linear Model to Investigate a Town’s Population
A town’s population has been growing linearly. In 2004 the population was 6,200. By 2009 the population had grown to 8,100. Assume this trend continues.
- ⓐ Predict the population in 2013.
- ⓑ Identify the year in which the population will reach 15,000.
- Answer
-
The two changing quantities are the population size and time. While we could use the actual year value as the input quantity, doing so tends to lead to very cumbersome equations because the intercept would correspond to the year 0, more than 2000 years ago!
To make computation a little nicer, we will define our input as the number of years since 2004:
- Input: years since 2004
- Output: the town’s population
To predict the population in 2013 we would first need an equation for the population. Likewise, to find when the population would reach 15,000, we would need to solve for the input that would provide an output of 15,000. To write an equation, we need the initial value and the rate of change, or slope.
To determine the rate of change, we will use the change in output per change in input.
The problem gives us two input-output pairs. Converting them to match our defined variables, the year 2004 would correspond to giving the point Notice that through our clever choice of variable definition, we have “given” ourselves the y -intercept of the function. The year 2009 would correspond to giving the point
The two coordinate pairs are and Recall that we encountered examples in which we were provided two points earlier in the chapter. We can use these values to calculate the slope.
We already know the y -intercept of the line, so we can immediately write the equation:
To predict the population in 2013, we evaluate our function at
If the trend continues, our model predicts a population of 9,620 in 2013.
To find when the population will reach 15,000, we can set and solve for
Our model predicts the population will reach 15,000 in a little more than 23 years after 2004, or somewhere around the year 2027.
Try It #1
A company sells doughnuts. They incur a fixed cost of $25,000 for rent, insurance, and other expenses. It costs $0.25 to produce each doughnut.
- ⓐ Write a linear model to represent the cost of the company as a function of the number of doughnuts produced.
- ⓑ Find and interpret the y -intercept.
Try It #2
A city’s population has been growing linearly. In 2008, the population was 28,200. By 2012, the population was 36,800. Assume this trend continues.
- ⓐ Predict the population in 2014.
- ⓑ Identify the year in which the population will reach 54,000.
Using a Diagram to Model a Problem
It is useful for many real-world applications to draw a picture to gain a sense of how the variables representing the input and output may be used to answer a question. To draw the picture, first consider what the problem is asking for. Then, determine the input and the output. The diagram should relate the variables. Often, geometrical shapes or figures are drawn. Distances are often traced out. If a right triangle is sketched, the Pythagorean Theorem relates the sides. If a rectangle is sketched, labeling width and height is helpful.
Example 2
Using a Diagram to Model Distance Walked
Anna and Emanuel start at the same intersection. Anna walks east at 4 miles per hour while Emanuel walks south at 3 miles per hour. They are communicating with a two-way radio that has a range of 2 miles. How long after they start walking will they fall out of radio contact?
- Answer
-
In essence, we can partially answer this question by saying they will fall out of radio contact when they are 2 miles apart, which leads us to ask a new question:
“How long will it take them to be 2 miles apart?”
In this problem, our changing quantities are time and position, but ultimately we need to know how long will it take for them to be 2 miles apart. We can see that time will be our input variable, so we’ll define our input and output variables.
- Input: time in hours.
- Output: distance in miles, and distance in miles
Because it is not obvious how to define our output variable, we’ll start by drawing a picture such as Figure 2 .
Initial Value: They both start at the same intersection so when the distance traveled by each person should also be 0. Thus the initial value for each is 0.
Rate of Change: Anna is walking 4 miles per hour and Emanuel is walking 3 miles per hour, which are both rates of change. The slope for is 4 and the slope for is 3.
Using those values, we can write formulas for the distance each person has walked.
For this problem, the distances from the starting point are important. To notate these, we can define a coordinate system, identifying the “starting point” at the intersection where they both started. Then we can use the variable, which we introduced above, to represent Anna’s position, and define it to be a measurement from the starting point in the eastward direction. Likewise, can use the variable, to represent Emanuel’s position, measured from the starting point in the southward direction. Note that in defining the coordinate system, we specified both the starting point of the measurement and the direction of measure.
We can then define a third variable, to be the measurement of the distance between Anna and Emanuel. Showing the variables on the diagram is often helpful, as we can see from Figure 3 .
Recall that we need to know how long it takes for the distance between them, to equal 2 miles. Notice that for any given input the outputs and represent distances.
Figure 2 shows us that we can use the Pythagorean Theorem because we have drawn a right angle.
Using the Pythagorean Theorem, we get:
In this scenario we are considering only positive values of so our distance will always be positive. We can simplify this answer to This means that the distance between Anna and Emanuel is also a linear function. Because is a linear function, we can now answer the question of when the distance between them will reach 2 miles. We will set the output and solve for
They will fall out of radio contact in 0.4 hours, or 24 minutes.
Q&A
Should I draw diagrams when given information based on a geometric shape?
Yes. Sketch the figure and label the quantities and unknowns on the sketch.
Example 3
Using a Diagram to Model Distance between Cities
There is a straight road leading from the town of Westborough to Agritown 30 miles east and 10 miles north. Partway down this road, it junctions with a second road, perpendicular to the first, leading to the town of Eastborough. If the town of Eastborough is located 20 miles directly east of the town of Westborough, how far is the road junction from Westborough?
- Answer
-
It might help here to draw a picture of the situation. See Figure 4 . It would then be helpful to introduce a coordinate system. While we could place the origin anywhere, placing it at Westborough seems convenient. This puts Agritown at coordinates and Eastborough at
Using this point along with the origin, we can find the slope of the line from Westborough to Agritown:
The equation of the road from Westborough to Agritown would be
From this, we can determine the perpendicular road to Eastborough will have slope Because the town of Eastborough is at the point (20, 0), we can find the equation:
We can now find the coordinates of the junction of the roads by finding the intersection of these lines. Setting them equal,
The roads intersect at the point (18, 6). Using the distance formula, we can now find the distance from Westborough to the junction.
Analysis
One nice use of linear models is to take advantage of the fact that the graphs of these functions are lines. This means real-world applications discussing maps need linear functions to model the distances between reference points.
Try It #3
There is a straight road leading from the town of Timpson to Ashburn 60 miles east and 12 miles north. Partway down the road, it junctions with a second road, perpendicular to the first, leading to the town of Garrison. If the town of Garrison is located 22 miles directly east of the town of Timpson, how far is the road junction from Timpson?
Building Systems of Linear Models
Real-world situations including two or more linear functions may be modeled with a system of linear equations . Remember, when solving a system of linear equations, we are looking for points the two lines have in common. Typically, there are three types of answers possible, as shown in Figure 5 .
How To
Given a situation that represents a system of linear equations, write the system of equations and identify the solution.
- Identify the input and output of each linear model.
- Identify the slope and y -intercept of each linear model.
- Find the solution by setting the two linear functions equal to one another and solving for or find the point of intersection on a graph.
Example 4
Building a System of Linear Models to Choose a Truck Rental Company
Jamal is choosing between two truck-rental companies. The first, Keep on Trucking, Inc., charges an up-front fee of $20, then 59 cents a mile. The second, Move It Your Way, charges an up-front fee of $16, then 63 cents a mile 4 . When will Keep on Trucking, Inc. be the better choice for Jamal?
- Answer
-
The two important quantities in this problem are the cost and the number of miles driven. Because we have two companies to consider, we will define two functions.
Input distance driven in miles Outputs cost, in dollars, for renting from Keep on Trucking
cost, in dollars, for renting from Move It Your WayInitial Value Up-front fee: and Rate of Change /mile and /mile
A linear function is of the form Using the rates of change and initial charges, we can write the equations
Using these equations, we can determine when Keep on Trucking, Inc., will be the better choice. Because all we have to make that decision from is the costs, we are looking for when Move It Your Way, will cost less, or when The solution pathway will lead us to find the equations for the two functions, find the intersection, and then see where the function is smaller.
These graphs are sketched in Figure 6 , with in blue.
To find the intersection, we set the equations equal and solve:
This tells us that the cost from the two companies will be the same if 100 miles are driven. Either by looking at the graph, or noting that is growing at a slower rate, we can conclude that Keep on Trucking, Inc. will be the cheaper price when more than 100 miles are driven, that is
Media
Access this online resource for additional instruction and practice with linear function models.


9.13: Using Video in Business
- Recognize the strengths and weaknesses of video in a business context
Remember how in the section on audio communication you learned about not being able to see and read facial expressions? Well, for better or worse, videoconferencing allows you to do just that.
If you work for a large corporation with sites all over the country or the world, it’s quite likely that you’ll be using video conferencing, especially as you move up the ladder and have colleagues who are hundreds of miles away. The first thing to think about is when to do it, and then we’ll go over how to do it most effectively.
Specifically, video conferencing is beneficial in the following situations:
- A team or committee is meeting for the first time and can’t meet in person. Your new team members will be much easier to remember and differentiate if you can put faces to their names.
- The issue at hand is emotional or sensitive, for example, discussing layoffs, discussing employee reviews, conveying and discussing bad news, and conveying and discussing especially good news. Reading each other’s expressions will be helpful in getting through the most difficult parts or celebrating the happy ones.
- There are team members who tend to dominate the conversation. Seeing others’ faces can have the effect of toning that team member down without a leader or host having to say anything.
- Your team tends to talk over one another on audio calls. Because on a video call, everyone can see each other’s faces, it’s easier to read the signals when someone is not finished speaking or is about to start. This can minimize interruptions.
- Products, demonstrations, or locations have to be shown and video files or still pictures are insufficient or unavailable.
There are not a lot of specific content—or purpose—related reasons not to use video conferencing, but do remember these two things:
- The more sophisticated the technology, the more likely you are to have bumps along the road to getting it working, at least initially.
- If it’s a meeting across time zones, some people may be in their pajamas or at home. Depending on the team, this might be okay, but do take it into consideration.
Using the Right Device
There are two basic kinds of devices for a video call:
- A smartphone or tablet
- A laptop or desktop computer
The reason we’re going to look at them separately is that the way we use them tends to be pretty specific, and the problems you may face with each are different.
Smartphones and Tablets
When making a one-on-one video call, you may be tempted to use your phone or tablet because it’s what you use for video calls in your personal life. This is fine in some situations; in fact, a phone or tablet can be really convenient if you want to show something other than just your own face. For example, using these devices can make it easier for you to show someone the space you’re in or for you to show and demonstrate products. However, these devices have some drawbacks you’ll want to be aware of:
- If you move around too much or too quickly with the device in your hand, the viewer’s screen jiggles and moves, and this can actually make people motion sick.
- The screens and camera ranges are relatively small, so you will need to be aware of how you are showing things to your colleague on the call.
Phone apps for video chatting include Facetime (which is standard on the iPhone), Google Hangouts and Google Duo, Facebook Messenger, Skype, and Signal Private Messenger.
Laptops and Desktop Computers
A laptop or desktop computer lives on the desk, so it’s great for video chatting when the main point is to see each other’s faces. You can also move items you want to show in front of the camera rather than moving the camera to the objects or places. This ensures a smoother, less sick-making experience for your viewer.
Let’s talk about how to prepare for and conduct a video call.
First, you’ll want to prepare yourself and your space.
- High-definition digital cameras are brutally clear and detailed, so take a moment to make sure you don’t have spinach in your teeth or windblown hair that might distract the person you’re talking with—and make you feel foolish when you discover it later.
- If there’s a window in your space, position yourself so you do not have the window at your back.  If you can, close blinds or curtains and rely on artificial light for the call. It’s not as nice for you, but your image will be clearer onscreen.
- Be aware of what’s behind you. A blank white wall or a file cabinet is fine. A shelf with your half-eaten sandwich or those reports you’re behind on is not so great.
- If your office has a window to the hallway outside, consider whether you have colleagues who will find it hilarious to make faces behind your back. If you do, arrange yourself and your computer to avoid that window.
Troubleshooting Your Device
Now to think about the technology. Make sure that your microphone and speakers are on and turned up, just as with an audio call. In addition, make sure that the app you are using has access to your device’s camera.
- Usually, if you download a third-party app such as Skype, it will ask for access to your camera during the installation process.
- If you are on your phone or tablet, you may need to go into the Settings menu to allow the app access to the camera.
Contributors and Attributions
- Using Video in Business. Authored by : Barbara Egel. Provided by : Lumen Learning. License : CC BY: Attribution


5.3: Importance of Biodiversity
The Biodiversity Crisis
Biologists estimate that species extinctions are currently 500–1000 times the normal, or background, rate seen previously in Earth’s history. The current high rates will cause a precipitous decline in the biodiversity of the planet in the next century or two. The loss of biodiversity will include many species we know today. Although it is sometimes difficult to predict which species will become extinct, many are listed as endangered (at great risk of extinction). However, many extinctions will affect species that biologist have not yet discovered. Most of these “invisible” species that will become extinct currently live in tropical rainforests like those of the Amazon basin. These rainforests are the most diverse ecosystems on the planet and are being destroyed rapidly by deforestation. Between 1970 and 2011, almost 20 percent of the Amazon rainforest was lost.
Biodiversity is a broad term for biological variety, and it can be measured at a number of organizational levels. Traditionally, ecologists have measured biodiversity by taking into account both the number of species and the number of individuals of each species (known as relative abundance ). However, biologists are using different measures of biodiversity, including genetic diversity, to help focus efforts to preserve the biologically and technologically important elements of biodiversity.
Biodiversity loss refers to the reduction of biodiversity due to displacement or extinction of species. The loss of a particular individual species may seem unimportant to some, especially if it is not a charismatic species like the Bengal tiger or the bottlenose dolphin. However, the current accelerated extinction rate means the loss of tens of thousands of species within our lifetimes. Much of this loss is occurring in tropical rainforests like the one pictured in Figure \(\PageIndex{1}\), which are very high in biodiversity but are being cleared for timber and agriculture. This is likely to have dramatic effects on human welfare through the collapse of ecosystems.
Biologists recognize that human populations are embedded in ecosystems and are dependent on them, just as is every other species on the planet. Agriculture began after early hunter-gatherer societies first settled in one place and heavily modified their immediate environment. This cultural transition has made it difficult for humans to recognize their dependence on living things other than crops and domesticated animals on the planet. Today our technology smooths out the harshness of existence and allows many of us to live longer, more comfortable lives, but ultimately the human species cannot exist without its surrounding ecosystems. Our ecosystems provide us with food, medicine, clean air and water, recreation, and spiritual and aesthetical inspiration.
Types of Biodiversity
A common meaning of biodiversity is simply the number of species in a location or on Earth; for example, the American Ornithologists’ Union lists 2078 species of birds in North and Central America. This is one measure of the bird biodiversity on the continent. More sophisticated measures of diversity take into account the relative abundances of species. For example, a forest with 10 equally common species of trees is more diverse than a forest that has 10 species of trees wherein just one of those species makes up 95 percent of the trees. Biologists have also identified alternate measures of biodiversity, some of which are important in planning how to preserve biodiversity.
Genetic diversity is one alternate concept of biodiversity. Genetic diversity is the raw material for evolutionary adaptation in a species and is represented by the variety of genes present within a population. A species’ potential to adapt to changing environments or new diseases depends on this genetic diversity.
It is also useful to define ecosystem diversity : the number of different ecosystems on Earth or in a geographical area. The loss of an ecosystem means the loss of the interactions between species and the loss of biological productivity that an ecosystem is able to create. An example of a largely extinct ecosystem in North America is the prairie ecosystem (Figure \(\PageIndex{2}\)). Prairies once spanned central North America from the boreal forest in northern Canada down into Mexico. They are now all but gone, replaced by crop fields, pasture lands, and suburban sprawl. Many of the species survive, but the hugely productive ecosystem that was responsible for creating our most productive agricultural soils is now gone. As a consequence, their soils are now being depleted unless they are maintained artificially at great expense. The decline in soil productivity occurs because the interactions in the original ecosystem have been lost.
Current Species Diversity
Despite considerable effort, knowledge of the species that inhabit the planet is limited. A recent estimate suggests that only 13% of eukaryotic species have been named (Table 1). Estimates of numbers of prokaryotic species are largely guesses, but biologists agree that science has only just begun to catalog their diversity. . Given that Earth is losing species at an accelerating pace, science knows little about what is being lost.
| Estimated Numbers of Described and Predicted species | ||||||
|---|---|---|---|---|---|---|
| Source: Mora et al 2011 | Source: Chapman 2009 | Source: Groombridge and Jenkins 2002 | ||||
| Described | Predicted | Described | Predicted | Described | Predicted | |
| Animals | 1,124,516 | 9,920,000 | 1,424,153 | 6,836,330 | 1,225,500 | 10,820,000 |
| Photosynthetic protists | 17,892 | 34,900 | 25,044 | 200,500 | — | — |
| Fungi | 44,368 | 616,320 | 98,998 | 1,500,000 | 72,000 | 1,500,000 |
| Plants | 224,244 | 314,600 | 310,129 | 390,800 | 270,000 | 320,000 |
| Non-photosynthetic protists | 16,236 | 72,800 | 28,871 | 1,000,000 | 80,000 | 600,000 |
| Prokaryotes | — | — | 10,307 | 1,000,000 | 10,175 | — |
| Total | 1,438,769 | 10,960,000 | 1,897,502 | 10,897,630 | 1,657,675 | 13,240,000 |
There are various initiatives to catalog described species in accessible and more organized ways, and the internet is facilitating that effort. Nevertheless, at the current rate of species description, which according to the State of Observed Species 1 reports is 17,000–20,000 new species a year, it would take close to 500 years to describe all of the species currently in existence. The task, however, is becoming increasingly impossible over time as extinction removes species from Earth faster than they can be described.
Naming and counting species may seem an unimportant pursuit given the other needs of humanity, but it is not simply an accounting. Describing species is a complex process by which biologists determine an organism’s unique characteristics and whether or not that organism belongs to any other described species. It allows biologists to find and recognize the species after the initial discovery to follow up on questions about its biology. That subsequent research will produce the discoveries that make the species valuable to humans and to our ecosystems. Without a name and description, a species cannot be studied in depth and in a coordinated way by multiple scientists.
Patterns of Biodiversity
Biodiversity is not evenly distributed on the planet. Lake Victoria contained almost 500 species of cichlids (just one family of fishes that are present in the lake) before the introduction of an exotic species in the 1980s and 1990s caused a mass extinction. All of these species were found only in Lake Victoria, which is to say they were endemic. Endemic species are found in only one location. For example, the blue jay is endemic to North America, while the Barton Springs salamander is endemic to the mouth of one spring in Austin, Texas. Endemic species with highly restricted distributions, like the Barton Springs salamander, are particularly vulnerable to extinction.
Lake Huron contains about 79 species of fish, all of which are found in many other lakes in North America. What accounts for the difference in diversity between Lake Victoria and Lake Huron? Lake Victoria is a tropical lake, while Lake Huron is a temperate lake. Lake Huron in its present form is only about 7,000 years old, while Lake Victoria in its present form is about 15,000 years old. These two factors, latitude and age, are two of several hypotheses biogeographers have suggested to explain biodiversity patterns on Earth.
Biogeography is the study of the distribution of the world’s species both in the past and in the present. The work of biogeographers is critical to understanding our physical environment, how the environment affects species, and how changes in environment impact the distribution of a species.
There are three main fields of study under the heading of biogeography: ecological biogeography, historical biogeography (called paleobiogeography), and conservation biogeography. Ecological biogeography studies the current factors affecting the distribution of plants and animals. Historical biogeography, as the name implies, studies the past distribution of species. Conservation biogeography, on the other hand, is focused on the protection and restoration of species based upon the known historical and current ecological information.
One of the oldest observed patterns in ecology is that biodiversity typically increases as latitude declines. In other words, biodiversity increases closer to the equator (Figure \(\PageIndex{3}\)).
It is not yet clear why biodiversity increases closer to the equator, but hypotheses include the greater age of the ecosystems in the tropics versus temperate regions, which were largely devoid of life or drastically impoverished during the last ice age. The greater age provides more time for speciation, the evolutionary process of creating new species. Another possible explanation is the greater energy the tropics receive from the sun. But scientists have not been able to explain how greater energy input could translate into more species. The complexity of tropical ecosystems may promote speciation by increasing the habitat complexity, thus providing more ecological niches. Lastly, the tropics have been perceived as being more stable than temperate regions, which have a pronounced climate and day-length seasonality. The stability of tropical ecosystems might promote speciation. Regardless of the mechanisms, it is certainly true that biodiversity is greatest in the tropics. There are also high numbers of endemic species.
Importance of Biodiversity
Loss of biodiversity may have reverberating consequences on ecosystems because of the complex interrelations among species. For example, the extinction of one species may cause the extinction of another. Biodiversity is important to the survival and welfare of human populations because it has impacts on our health and our ability to feed ourselves through agriculture and harvesting populations of wild animals.
Human Health
Many medications are derived from natural chemicals made by a diverse group of organisms. For example, many plants produce compounds meant to protect the plant from insects and other animals that eat them. Some of these compounds also work as human medicines. Contemporary societies that live close to the land often have a broad knowledge of the medicinal uses of plants growing in their area. For centuries in Europe, older knowledge about the medical uses of plants was compiled in herbals—books that identified the plants and their uses. Humans are not the only animals to use plants for medicinal reasons. The other great apes, orangutans, chimpanzees, bonobos, and gorillas have all been observed self-medicating with plants.
Modern pharmaceutical science also recognizes the importance of these plant compounds. Examples of significant medicines derived from plant compounds include aspirin, codeine, digoxin, atropine, and vincristine (Figure \(\PageIndex{4}\)). Many medications were once derived from plant extracts but are now synthesized. It is estimated that, at one time, 25 percent of modern drugs contained at least one plant extract. That number has probably decreased to about 10 percent as natural plant ingredients are replaced by synthetic versions of the plant compounds. Antibiotics, which are responsible for extraordinary improvements in health and lifespans in developed countries, are compounds largely derived from fungi and bacteria.
In recent years, animal venoms and poisons have excited intense research for their medicinal potential. By 2007, the FDA had approved five drugs based on animal toxins to treat diseases such as hypertension, chronic pain, and diabetes. Another five drugs are undergoing clinical trials and at least six drugs are being used in other countries. Other toxins under investigation come from mammals, snakes, lizards, various amphibians, fish, snails, octopuses, and scorpions.
Aside from representing billions of dollars in profits, these medications improve people’s lives. Pharmaceutical companies are actively looking for new natural compounds that can function as medicines. It is estimated that one third of pharmaceutical research and development is spent on natural compounds and that about 35 percent of new drugs brought to market between 1981 and 2002 were from natural compounds.
Finally, it has been argued that humans benefit psychologically from living in a biodiverse world. The chief proponent of this idea is famed entomologist E. O. Wilson. He argues that human evolutionary history has adapted us to living in a natural environment and that built environments generate stresses that affect human health and well-being. There is considerable research into the psychologically regenerative benefits of natural landscapes that suggest the hypothesis may hold some truth.
Agricultural
Since the beginning of human agriculture more than 10,000 years ago, human groups have been breeding and selecting crop varieties. This crop diversity matched the cultural diversity of highly subdivided populations of humans. For example, potatoes were domesticated beginning around 7,000 years ago in the central Andes of Peru and Bolivia. The people in this region traditionally lived in relatively isolated settlements separated by mountains. The potatoes grown in that region belong to seven species and the number of varieties likely is in the thousands. Each variety has been bred to thrive at particular elevations and soil and climate conditions. The diversity is driven by the diverse demands of the dramatic elevation changes, the limited movement of people, and the demands created by crop rotation for different varieties that will do well in different fields.
Potatoes are only one example of agricultural diversity. Every plant, animal, and fungus that has been cultivated by humans has been bred from original wild ancestor species into diverse varieties arising from the demands for food value, adaptation to growing conditions, and resistance to pests. The potato demonstrates a well-known example of the risks of low crop diversity: during the tragic Irish potato famine (1845–1852 AD), the single potato variety grown in Ireland became susceptible to a potato blight—wiping out the crop. The loss of the crop led to famine, death, and mass emigration. Resistance to disease is a chief benefit to maintaining crop biodiversity and lack of diversity in contemporary crop species carries similar risks. Seed companies, which are the source of most crop varieties in developed countries, must continually breed new varieties to keep up with evolving pest organisms. These same seed companies, however, have participated in the decline of the number of varieties available as they focus on selling fewer varieties in more areas of the world replacing traditional local varieties.
The ability to create new crop varieties relies on the diversity of varieties available and the availability of wild forms related to the crop plant. These wild forms are often the source of new gene variants that can be bred with existing varieties to create varieties with new attributes. Loss of wild species related to a crop will mean the loss of potential in crop improvement. Maintaining the genetic diversity of wild species related to domesticated species ensures our continued supply of food.
Since the 1920s, government agriculture departments have maintained seed banks of crop varieties as a way to maintain crop diversity. This system has flaws because over time seed varieties are lost through accidents and there is no way to replace them. In 2008, the Svalbard Global seed Vault, located on Spitsbergen island, Norway, (Figure) began storing seeds from around the world as a backup system to the regional seed banks. If a regional seed bank stores varieties in Svalbard, losses can be replaced from Svalbard should something happen to the regional seeds. The Svalbard seed vault is deep into the rock of the arctic island. Conditions within the vault are maintained at ideal temperature and humidity for seed survival, but the deep underground location of the vault in the arctic means that failure of the vault’s systems will not compromise the climatic conditions inside the vault.
Although crops are largely under our control, our ability to grow them is dependent on the biodiversity of the ecosystems in which they are grown. That biodiversity creates the conditions under which crops are able to grow through what are known as ecosystem services—valuable conditions or processes that are carried out by an ecosystem. Crops are not grown, for the most part, in built environments. They are grown in soil. Although some agricultural soils are rendered sterile using controversial pesticide treatments, most contain a huge diversity of organisms that maintain nutrient cycles—breaking down organic matter into nutrient compounds that crops need for growth. These organisms also maintain soil texture that affects water and oxygen dynamics in the soil that are necessary for plant growth. Replacing the work of these organisms in forming arable soil is not practically possible. These kinds of processes are called ecosystem services. They occur within ecosystems, such as soil ecosystems, as a result of the diverse metabolic activities of the organisms living there, but they provide benefits to human food production, drinking water availability, and breathable air.
Other key ecosystem services related to food production are plant pollination and crop pest control. It is estimated that honeybee pollination within the United States brings in $1.6 billion per year; other pollinators contribute up to $6.7 billion. Over 150 crops in the United States require pollination to produce. Many honeybee populations are managed by beekeepers who rent out their hives’ services to farmers. Honeybee populations in North America have been suffering large losses caused by a syndrome known as colony collapse disorder, a new phenomenon with an unclear cause. Other pollinators include a diverse array of other bee species and various insects and birds. Loss of these species would make growing crops requiring pollination impossible, increasing dependence on other crops.
Finally, humans compete for their food with crop pests, most of which are insects. Pesticides control these competitors, but these are costly and lose their effectiveness over time as pest populations adapt. They also lead to collateral damage by killing non-pest species as well as beneficial insects like honeybees, and risking the health of agricultural workers and consumers. Moreover, these pesticides may migrate from the fields where they are applied and do damage to other ecosystems like streams, lakes, and even the ocean. Ecologists believe that the bulk of the work in removing pests is actually done by predators and parasites of those pests, but the impact has not been well studied. A review found that in 74 percent of studies that looked for an effect of landscape complexity (forests and fallow fields near to crop fields) on natural enemies of pests, the greater the complexity, the greater the effect of pest-suppressing organisms. Another experimental study found that introducing multiple enemies of pea aphids (an important alfalfa pest) increased the yield of alfalfa significantly. This study shows that a diversity of pests is more effective at control than one single pest. Loss of diversity in pest enemies will inevitably make it more difficult and costly to grow food. The world’s growing human population faces significant challenges in the increasing costs and other difficulties associated with producing food.
Wild Food Sources
In addition to growing crops and raising food animals, humans obtain food resources from wild populations, primarily wild fish populations. For about one billion people, aquatic resources provide the main source of animal protein. But since 1990, production from global fisheries has declined. Despite considerable effort, few fisheries on Earth are managed sustainability.
Fishery extinctions rarely lead to complete extinction of the harvested species, but rather to a radical restructuring of the marine ecosystem in which a dominant species is so over-harvested that it becomes a minor player, ecologically. In addition to humans losing the food source, these alterations affect many other species in ways that are difficult or impossible to predict. The collapse of fisheries has dramatic and long-lasting effects on local human populations that work in the fishery. In addition, the loss of an inexpensive protein source to populations that cannot afford to replace it will increase the cost of living and limit societies in other ways. In general, the fish taken from fisheries have shifted to smaller species and the larger species are overfished. The ultimate outcome could clearly be the loss of aquatic systems as food sources.
Contributors and Attributions
- Importance of Biodiversity by OpenStax is licensed under CC BY 4.0 . Modified from the original by Matthew R. Fisher.


9.8: Capital investment analysis
One of a company’s most significant financial decisions involves the purchase of property, plant, and equipment that will be used in business operations. The costs of these assets are often very high, and they will be in place for many years to come.
Before acquiring a capital asset such as equipment, machinery, or a building, which involves a large expenditure and long-term commitment, a company should evaluate how effectively it is expected to generate a return on investment for the business. Capital investment analysis is a form of differential analysis used to determine (1) whether a fixed asset should be purchased at all, or (2) which fixed asset among a number of choices is the best investment. Three commonly used methods for evaluating capital investments will be discussed.
The first two, the average rate of return method and the cash payback method, are relatively straightforward calculations that are often used to determine whether a proposed investment meets a minimum standard for it even to be considered further.
The average rate of return method is the percentage return of net income from the proposed investment. It is calculated as follows:
\(\ \frac{\text{Average annual income}}{\text{Average investment}}\)
Each of the two amounts must first be calculated independently.
\(\ \begin{array}{c}\text { Average annual income } \\ \text { (numerator) }\end{array}=\frac{\text { Total estimated income over the asset's useful life }}{\text { Number of years in the asset's useful life }}\)
\(\ \begin{array}{l}\text { Average investment } \\ \text { (denominator) }\end{array}=\frac{\text { Book value at the beginning of the first year }^{1}+\text { Book value at the end of the last year }^{2}}{2}\)
1 The book value at the beginning of the first year is the asset’s cost.
2 The book value at the end of the last year is the asset’s residual value.
As an example, a new piece of equipment that is being considered for purchase costs $90,000 and has a residual value of $10,000. It is expected to generate revenue of $75,000 over its estimated useful life of 5 years.
\(\ \text{Average annual income} =\frac{\$ 75,000}{5}=\$ 15,000\)
\(\ \text{Average investment} =\frac{\$110,000 + $10,000}{2}=\$ 60,000\)
\(\ \bf{\text{Average investment}} =\frac{\$15,000}{$60,000}=\ 25\)%
The average rate of return of 25% should then be compared to the minimum rate of return that management requires. If the average rate of return is greater than the minimum acceptable rate, the equipment should be evaluated further since it seems promising. If it does not even meet this standard, however, it should not be purchased.
The cash payback method looks at the annual net cash inflow from the use of an asset to determine how many years it will take to recover the cost of the asset. Net cash flow includes all cash revenue generated minus all cash expenditures paid from using the asset. Depreciation is not a cash expenditure, so it would not be considered in determining net cash flow.
As an example, a new piece of equipment that is being considered for purchase costs $80,000. It is expected to generate $25,000 cash revenue each year and require cash expenditures of $5,000 to maintain.
Cash payback period $=\frac{\text { cost }}{\text { Annual nest cash flow }}=\frac{\$ 80,000}{\$ 25,000-\$ 5,000}=4$ years
The cash payback period of four years should be compared to the maximum period that management desires. If the cash payback period of four years is more than an acceptable payback period of three years, for example, the purchase should no longer be considered. If a payback period of five years is acceptable, the purchase should be looked into further.
If annual net cash flows are not expected to be equal each year, the cash payback period is determined by adding the annual expected cash flows year by year until the sum equals the initial cost of the asset.
For example, a new piece of equipment that is being considered for purchase costs $80,000. Its expected annual cash flows are as follows:
|
Year |
Net Cash Flow |
Cash Flow to Date |
|
1 |
$12,000 |
$12,000 |
| 2. |
18,000 |
30,000 |
|
3 |
24,000 |
54,000 |
|
4 |
26,000 |
80,000 |
|
5 |
30,000 |
110,000 |
|
6 |
34,000 |
144,000 |
In this case, net cash flows recover the initial cost of $80,000 after four full years. The average rate of return and the cash payback methods are relatively simple to calculate, yet they yield rather general results. Since neither considers the time value of money, they are more effective for shorter-term investments. They are often used as an initial screening to see if an investment should be immediately disqualified. If not, the investment may be analyzed further using more robust
analyses.
The
net present value
(NPV) method for evaluating a potential investment
also looks at estimated future net cash flows generated by the asset. It compares the purchase price (investment amount) to the present value of all the future net cash flows from using the asset. The investment is considered viable if the present value of the future net cash flows is greater than the purchase price. Otherwise, the investment should be avoided.
Present value factors the timing of future net cash inflows and the effect of a prevailing interest rate. An amount of cash received in the future is worth less than the same amount of cash received today. This is because cash received now may be invested at a given interest rate that causes its value to grow over time compounding, where interest is earned both on principal and on interest that has already been earned. The opportunity to invest dollars received in the future rather than today is postponed, missing out on time available to earn interest.
Determining the future value of a current amount is calculated by multiplying the amount by itself plus the interest rate. For example, the future value of $1.00 in 3 years at an interest rate of 6% would be calculated as follows:
$1.00 x 1.06 = $1.06 x 1.06 = $1.12 x 1.06 = $1.19
Note that interest is calculated on interest previously earned. This process is called compounding.
Present value works in the opposite direction. An amount in the future is known or estimated (such as a net cash inflow), and the calculation backs that amount up to its current value. The process is called discounting. For example, the present value of $1.00 to be received in 3 years at an interest rate of 6% would be calculated as follows:
\(\ \frac{\$ 1.00}{1.06}=\frac{\$ 0.94}{1.06}=\frac{\$ 0.89}{1.06}=\$ 0.84 (\text{rounded to the nearest cent})\)
The following table summarizes the present value of $1 for 10 periods for three interest rates: 6%, 8%, and 10%. Amounts are rounded to five decimal places rather than to the nearest cent.
|
Period |
6% |
8% |
10% |
|
1 |
0.94340 |
0.92593 |
0.90909 |
|
2 |
0.89000 |
0.85734 |
0.82645 |
|
3 |
0.83962 |
0.79383 |
0.75132 |
|
4 |
0.79209 |
0.73503 |
0.68302 |
|
5 |
0.74725 |
0.68058 |
0.62093 |
|
6 |
0.70495 |
0.63017 |
0.56448 |
|
7 |
0.66505 |
0.58349 |
0.51316 |
|
8 |
0.62741 |
0.54027 |
0.46651 |
|
9 |
0.59190 |
0.50025 |
0.42410 |
|
10 |
0.55840 |
0.46319 |
0.38555 |
Note that all amounts in the present value table are less than $1.00 since all represent a future cash receipt rather than the $1.00 today. The further into the future the $1.00 will be received for a given interest rate, the lower its present value.
Clearly not all future cash receipts are for $1.00. To get the present value of a different value, multiply the actual number of dollars by the present value of $1 amount given in the table at the intersection of a specified interest rate and number of years.
Examples 1 and 2 illustrate the process of discounting the future net cash flows to determine their total and comparing it to the cost of the asset.
A company is considering purchasing equipment #1 for $100,000. It is expected to provide net cash flows of $24,000 per year for the next six years for a total of $144,000. The minimum desired rate of return on the investment is 6%.
|
Year |
Undiscounted Net Cash Flow |
Present Value of $1 at 6% |
Discounted Net Cash Flow |
Since the undiscounted net cash flow amount is the same each year, the total discounted net cash flow could also be calculated by using the present value of an annuity of $1, as follows: $24,000 x 4.91731 = $118,016 Rather than multiplying $24,000 six times by six different factors, $24,000 can be multiplied once by the sum of all the factors (4.91731). The result is the same. |
|
1 |
$24,000 |
0.94340 |
$22,642 |
|
|
2 |
24,000 |
0.89000 |
21,360 |
|
|
3 |
24,000 |
0.83962 |
20,151 |
|
|
4 |
24,000 |
0.79209 |
19,010 |
|
|
5 |
24,000 |
074725 |
17,934 |
|
|
6 |
24,000 |
0.70495 |
16,919 |
|
|
Total |
$144,000 |
4.91731 |
$118,016 |
|
|
Cost |
(100,000) |
|||
|
NPV |
$18,016 |
In this case, the net present value of the future cash flows of $18,016 is greater than the cost of the asset, $100,000. The investment may be accepted since it more than pays for itself over time.
If the cost of the asset had been $130,000 rather than $100,000, the net present value would have been ($11,984), which is $118,016 - $130,000. In this case the NPV is negative and the proposed purchase should be rejected.
A company is considering purchasing equipment #2 for $100,000. It is expected to provide net cash flows of different amounts each year for the next six years for a total of $144,000. The minimum desired rate of return on the investment is 6%.
|
Year |
Undiscounted Net Cash Flow |
Present Value of $1 at 6% |
Discounted Net Cash Flow |
Since the undiscounted net cash flow amounts are different each year, the total discounted net cash flow must be calculated using six individual calculations. Each year the undiscounted net cash flow amount is multiplied by the present value of $1 factor at 6%. |
|
1 |
$34,000 |
0.94340 |
$32,076 |
|
|
2 |
30,000 |
0.89000 |
26,700 |
|
|
3 |
26,000 |
0.83962 |
21,830 |
|
|
4 |
24,000 |
0.79209 |
19,010 |
|
|
5 |
18,000 |
074725 |
13,451 |
|
|
6 |
12,000 |
0.70495 |
8,459 |
|
|
Total |
$144,000 |
$121,526 |
||
|
Cost |
(100,000) |
|||
|
NPV |
$24,526 |
In this case, the net present value of the future cash flows of $21,256 is greater than the cost of the asset, $100,000. The investment may be accepted since it more than pays for itself over time.
Net present value can be used to perform differential analysis to compare results of two or more proposed investments to determine which is more financially beneficial. Examples 3 and 4a show these comparisons.
A company is considering two different proposals for purchasing equipment. Both assets will be useful for six years. The first piece of equipment costs $100,000, and the second costs $140,000. The undiscounted cash flows appear in the two tables that follow.
|
#1 Year |
Undiscounted Net Cash Flow |
Present Value of $1 at 6% |
Discounted Net Cash Flow |
#2 Year |
Undiscounted Net Cash Flow |
Present Value of $1 at 6% |
Discounted Net Cash Flow |
|
|
1 |
$34,000 |
0.94340 |
$32,076 |
1 |
$44,000 |
0.94340 |
$41,510 |
|
|
2 |
30,000 |
0.89000 |
26,700 |
2 |
39,000 |
0.89000 |
34,710 |
|
|
3 |
26,000 |
0.83962 |
21,830 |
3 |
34,000 |
0.83962 |
28,547 |
|
|
4 |
24,000 |
0.79209 |
19,010 |
4 |
31,000 |
0.79209 |
24,555 |
|
|
5 |
18,000 |
074725 |
13,451 |
5 |
23,000 |
074725 |
17,187 |
|
|
6 |
12,000 |
0.70495 |
8,459 |
6 |
16,000 |
0.70495 |
11,279 |
|
|
Total |
$144,000 |
$121,526 |
Total |
$187,000 |
$157,788 |
|||
|
Cost |
(100,000) |
Cost |
(140,000) |
|||||
|
NPV |
$24,526 |
NPV |
$17,788 |
The second piece of equipment has a higher estimated net cash flow each year, but it also costs more to purchase. Both assets yield a positive net present value, but the first piece of equipment has a higher NPV, $21,526, vs. the NPV of the second piece, $17,788. The first piece of equipment should be purchased based on this result.
It is possible that two different investments will span two different periods; that is, one may generate cash flows for more years than the other. In order to perform a differential analysis, the number of years must be the same for both. To make them comparable, the asset with the higher number of years of cash flows is adjusted to assume that it is sold for its residual value amount in the last year that the other asset provides cash flows.
A company is considering two different proposals for purchasing equipment. The first asset provides cash flows for four years, and the second one provides cash flows for six years. Both assets cost $100,000 and have a residual value of $10,000. Both have undiscounted net cash flows of $124,000 as shown in the tables that follow.
|
#1 Year |
Undiscounted Net Cash Flow |
Present Value of $1 at 6% |
Discounted Net Cash Flow |
#2 Year |
Undiscounted Net Cash Flow |
Present Value of $1 at 6% |
Discounted Net Cash Flow |
|
|
1 |
$38,000 |
0.94340 |
$35,849 |
1 |
$34,000 |
0.94340 |
$32,076 |
|
|
2 |
34,000 |
0.89000 |
30,260 |
2 |
32,000 |
0.89000 |
28,480 |
|
|
3 |
30,000 |
0.83962 |
25,189 |
3 |
26,000 |
0.83962 |
21,830 |
|
|
4 |
22,000 |
0.79209 |
17,426 |
4 |
22,000 |
0.79209 |
17,426 |
|
|
4 (residual) |
10,000 |
0.79209 |
7,921 |
|||||
|
|
|
|
|
|||||
|
|
|
|
|
|||||
|
Total |
$124,000 |
$108,724 |
Total |
$124,000 |
$107,733 |
|||
|
Cost |
(100,000) |
Cost |
(100,000) |
|||||
|
NPV |
$8,724 |
NPV |
$7,733 |
Note that the cash flow period for the second piece of equipment is adjusted to four years to match that of the first piece of equipment. There are two net cash flows for the second piece of equipment in year four: (1) the inflow from using the equipment, and (2) the proceeds from selling it at its residual value. The cash flows for the fifth and sixth years for the second asset are not considered and are therefore greyed out in the table.
Both assets yield a positive net present value, but the first piece of equipment has a higher NPV, $8,724, vs. the NPV of the second piece, $7,733. The first piece of equipment should be purchased based on this result.
As a final illustration of two companies with different cash flow periods, note that the net present value would be identical if the annual net cash flows were the same. In this case, all are equal in years 1, 2, and 3. In year 4, they also both equal $32000: for the first asset the $32,000 is all operational cash flow, and for the second piece of equipment, the $32,000 includes $22,000 of operational cash flow and $10,000 selling price.
A company is considering two different proposals for purchasing equipment. The first asset provides cash flows for four years, and the second one provides cash flows for six years. Both assets cost $100,000 and have a residual value of $10,000. Both have undiscounted net cash flows of $124,000 as shown in the tables that follow.
|
#1 Year |
Undiscounted Net Cash Flow |
Present Value of $1 at 6% |
Discounted Net Cash Flow |
#2 Year |
Undiscounted Net Cash Flow |
Present Value of $1 at 6% |
Discounted Net Cash Flow |
|
|
1 |
$34,000 |
0.94340 |
$35,076 |
1 |
$34,000 |
0.94340 |
$32,076 |
|
|
2 |
32,000 |
0.89000 |
28,480 |
2 |
32,000 |
0.89000 |
28,480 |
|
|
3 |
26,000 |
0.83962 |
21,830 |
3 |
26,000 |
0.83962 |
21,830 |
|
|
4 |
32,000 |
0.79209 |
25,347 |
4 |
22,000 |
0.79209 |
17,426 |
|
|
4 (residual) |
10,000 |
0.79209 |
7,921 |
|||||
|
|
|
|
|
|||||
|
|
|
|
|
|||||
|
Total |
$124,000 |
$107,733 |
Total |
$124,000 |
$107,733 |
|||
|
Cost |
(100,000) |
Cost |
(100,000) |
|||||
|
NPV |
$7,733 |
NPV |
$7,733 |
Differential analysis is a useful planning tool for projecting relative results among alternatives. It encourages managers to think ahead and analyze the components of alternative outcomes with the goal of more insightful decision making.