1: Science and the Universe - A Brief Tour
Such discoveries are what make astronomy such an exciting field for scientists and many others—but you will explore much more than just the objects in our universe and the latest discoveries about them. We will pay equal attention to the process by which we have come to understand the realms beyond Earth and the tools we use to increase that understanding. We gather information about the cosmos from the messages the universe sends our way. Because the stars are the fundamental building blocks of the universe, decoding the message of starlight has been a central challenge and triumph of modern astronomy. By the time you have finished reading this text, you will know a bit about how to read that message and how to understand what it is telling us.
-
- 1.1: The Nature of Astronomy
- In considering the history of the universe, we will see again and again that the cosmos evolves; it changes in profound ways over long periods of time. For example, the universe made the carbon, the calcium, and the oxygen necessary to construct something as interesting and complicated as you. Today, many billions of years later, the universe has evolved into a more hospitable place for life.
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- 1.2: The Nature of Science
- The ultimate judge in science is always what nature itself reveals based on observations, experiments, models, and testing. Science is not merely a body of knowledge, but a method by which we attempt to understand nature and how it behaves. This method begins with many observations over a period of time. From the trends found through observations, scientists can modelthe particular phenomena we want to understand. Such models are always approximations of nature, subject to further testing.
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- 1.3: The Laws of Nature
- Over centuries scientists have extracted various scientific laws from countless observations, hypotheses, and experiments. These scientific laws are, in a sense, the “rules” of the game that nature plays. One remarkable discovery about nature—one that underlies everything you will read about in this text—is that the same laws apply everywhere in the universe.
-
- 1.4: Numbers in Astronomy
- In astronomy we deal with distances on a scale you may never have thought about before, with numbers larger than any you may have encountered. We adopt two approaches that make dealing with astronomical numbers a little bit easier. First, we use a system for writing large and small numbers called scientific notation (or sometimes powers-of-ten notation). This system is very appealing because it eliminates the many zeros that can seem overwhelming to the reader.
-
- 1.5: Consequences of Light Travel Time
- This sets a limit on how quickly we can learn about events in the universe. If a star is 100 light-years away, the light we see from it tonight left that star 100 years ago and is just now arriving in our neighborhood. The soonest we can learn about any changes in that star is 100 years after the fact. For a star 500 light-years away, the light we detect tonight left 500 years ago and is carrying 500-year-old news.
-
- 1.7: The Universe on the Large Scale
- In a very rough sense, you could think of the solar system as your house or apartment and the Galaxy as your town, made up of many houses and buildings. In the twentieth century, astronomers were able to show that, just as our world is made up of many, many towns, so the universe is made up of enormous numbers of galaxies. Galaxies stretch as far into space as our telescopes can see, many billions of them within the reach of modern instruments.
-
- 1.8: The Universe of the Very Small
- The foregoing discussion has likely impressed on you that the universe is extraordinarily large and extraordinarily empty. On average, it is 10,000 times more empty than our Galaxy. Yet, as we have seen, even the Galaxy is mostly empty space. Intergalactic space is filled so sparsely that to find one atom, on average, we must search through a cubic meter of space. Most of the universe is fantastically empty; places that are dense, such as the human body, are tremendously rare.
-
- 1.9: A Conclusion and a Beginning
- Learning astronomy is a little like learning a new language: at first it seems there are so many new expressions that you’ll never master them all, but with practice, you soon develop facility with them. At this point you may also feel a bit small and insignificant, dwarfed by the cosmic scales of distance and time. But, there is another way to look at what you have learned from our first glimpses of the cosmos.
Thumbnail: A photograph of galaxy NGC 6744, which might resemble the Milky Way. (CC BY-SA 3.0; http://www.eso.org/public/images/eso1118a/ )


2.2: Music in the World
With the exception of its opening passages, which considered the problem of defining what music even is, this unit has so far emphasized the empirical qualities of music. We have acknowledged the documented effects of music on the human brain, and we have acquired a variety of terms and concepts that can be used to understand and describe music as a physical phenomenon. Now it is time to address some of the messier aspects of talking and writing about music.
Categories
What kinds of music do you like to listen to? Country? Hip-hop? Classical? EDM? Top 40? Whether we are talking to a friend, using a streaming service, or browsing records in a store, we like to think about music in terms of categories. These categories can be very useful. They can help us pick a radio station we might enjoy, or decide whether or not to buy tickets to hear an unfamiliar band. At the same time, these categories are both artificial and extremely limiting.
Let us begin by considering the classic tripartite division of music into the categories of “classical,” “popular,” and “folk.” This approach has been around for a long time, and it has persevered because, in many ways, it works. If I tell you that I like “classical” music, you immediately understand that I probably mean orchestral music, or opera, and that I probably listen to music that is fairly old. But there are problems with this categorization. To begin with, much of what is “classical” today was “popular” in the past. When Mozart wrote his symphonies, for example, his object was to satisfy popular demand and sell concert tickets, and his audiences behaved the same way that fans at a rock concert do today. And what if I actually prefer experimental orchestral music composed last year? It is common practice to refer to such repertoire as “classical,” but it’s about as far from Mozart as you can get.
How about “popular” music? This category is generally understood to contain commercial music that appeals to large numbers of people. But what about individual artists or songs that fail to achieve any popularity whatsoever? What about experimental rock bands that take the same attitude towards their work as serious “classical” composers? Mozart, a “classical” composer, might have more in common with a “popular” artist like Jimi Hendrix than Hendrix has in common with Pink Floyd. Mozart and Hendrix were both gifted instrumentalists who dazzled their audiences with virtuosic performances and wrote music to showcase their skills, while the band Pink Floyd is known more for their nuanced production, complex song structures, and unusual instrumentation. Again, however, this category is not without its value. While there is an enormous diversity of “popular” musics, they tend to be characterized by certain forms, instrumentations, styles, and performance venues. There might be much to separate Jimi Hendrix and Pink Floyd, but their music shares important elements of instrumentation and style, and it might be heard in the same types of settings.
“Folk” is also a slippery category. “Folk” music is typically described as music of unknown authorship that is passed down from generation to generation in a particular region. It tends to be fairly simple and in a distinctive style, and it is performed on instruments that are integral to the local musical culture. However, problems quickly arise as we try to label individual pieces or practices. In the United States, for example, the works of Stephen Foster have long been considered folk music. Songs like “My Old Kentucky Home” and “Camptown Races” have certainly entered folk culture, and many who sing or play them know nothing of their composer or origin. But can a commercial song, created and published by a professional composer, truly be considered “folk” music? Different problems arise as we address the musical practices of non-Western societies, many of which do not employ musical notation and reject notions of individual authorship. But do the absence of a named composer, official sheet music, and copyright notice mean that a work in the North Indian classical tradition is “folk” music? The complexity, sophistication, and technical demands of music in this category would suggest not.
A further challenge arises when we try to identify the “folk” music of a region or nation. Let us take the United States.  Indeed, music such as hers has come to be known as Folk music (with a capital F). If I ask Spotify to play Folk music for me, I’ll hear Joan Baez and others like her. However, her music represents only one cultural strain within the United States.
What about the polka music of midwestern communities? What about the corrido ballads of Spanish-speaking communities near the southern border? What about the dance music heard at Native American pow-pow gatherings? Are any of these traditions less “folk” or less “American” than the others?
For all the reasons explored above, this narrative is going to steer clear of “classical,” “popular,” and “folk” as categories and terms. They have been addressed here only because their use is so widespread. Instead, we will focus on what music across these categories shares in common: the purposes for which individual works were originally created and continue to be consumed. This book is organized around categories, but these categories have little to do with the style of the works contained therein. Instead, they have to do with the roles music plays in society. These categories lead us to first understand what music is for . Only then will we seek to address how the music works, who created it, and how it is rooted in its historical and cultural context.
These categories also have their shortcomings. Many musical examples included in a given category could just as easily be included in another. We will admit that at the outset. All the same, these categories seem more useful than “classical,” “popular,” and “folk,” and they tell us much more about what really matters: music as an integral aspect of the human experience.
Genres and Subgenres
This book will engage with another mode of categorization: genre . Genre is a way of making connections between closely-related works and musical artists that share stylistic, formal, and cultural elements. You are sure to recognize a large number of genres—rock, pop, R&B, country, hip-hop—from your own musical consumption. Each of these genre names tells us something about what the music is like and who listens to it. Each also hosts a variety of subgenres that communicate more specialized information about the music contained therein. For example, the genre EDM (electronic dance music) contains all computer-produced music intended primarily for dancing, whereas the subgenre dubstep contains only bass- heavy EDM that uses specific timbres, is in duple meter, and falls within a narrow tempo range. The label “dubstep” also gives us a clearer picture of who consumes the music and what a concert might be like. Finally, subgenres tend to come and go, each leading to the next, while genres remain relevant for longer periods of time.
It is important to acknowledge that genre in the 21st century is primarily a marketing tool. The main purpose of genre is to help record companies efficiently label their merchandise, identify consumers, and advertise music to the people who are most likely to buy it. Genre also helps the music industry to track sales; consider the Billboard music charts, which have been in use since 1958. Of course, genre is meaningful to consumers as well, and subgenres are often named not by faceless corporations but by the fans themselves. Genre can also help listeners to find music that they will enjoy, and it can serve to create communities of listeners and concertgoers.
At the same time, genre can be divisive. Historically, genre has been used to separate black and white performing artists whose music was stylistically identical. This happened in the 1920s, when the marketing categories of “race records” and “hillbilly records” were invented to segregate the music of black and white Southern musicians, and again in the 1950s, when the distinction between performers of R&B and rock ‘n’ roll was often one of race. It is important, therefore, to be critical of genre, and to repeatedly assess exactly what genre is telling us.
Indeed, genre can convey a wide variety of types of information, but not all genres convey the same types of information. Let’s look at two examples: “string quartet” and “French reggae.” The former provides us with precise information about instrumentation (two violins, viola, and cello) and suggests a multi-part concert work intended for the stimulation of players and listeners. We might also make assumptions about the consumer of such a genre, who is probably (although not always) well-educated and reasonably well-off, and we might expect to hear performances in a formal concert hall, surrounded by well-dressed and attentive listeners. However, genre in this case tells us nothing at all about style, geographical origin, historical context, or social significance. A work in this genre might have been composed in 1780, or 1880, or 1980, or yesterday. Although the string quartet originated in German- speaking Europe, this genre has been accessible to composers, performers, and listeners across the globe for at least the past one hundred years. A string quartet might be pleasant and lyrical, or dissonant and jarring. It might be fairly simple or mind-bogglingly complex. It might last eight minutes, or eighty minutes. And genre provides us no idea about what sort of meaning—political, social, or otherwise— might be contained in such a work.
The case is quite different with “French reggae.” On the one hand, this genre tells us less about instrumentation. We can expect to hear certain instruments— electric guitar, electric bass, drums, and perhaps electric organ, trumpet, and saxophone—but taking away or adding instruments does not fundamentally destabilize the genre. On the other, it tells us much more about everything else. First and foremost, the “reggae” designation tells us all we need to know about style, which is a core identifying feature of the genre. It also limits the scope of time and place. Reggae has only been around since the late 1960s, and it was developed in Jamaica by Rastafarians—a nation and culture that are central to the genre’s identity no matter where individual songs might come from. The subgenre identification of “French reggae” tells us even more about geographic location and language. Finally, reggae carries certain political, social, and racial connotations. It is usually performed by musicians of African descent, and it often espouses ideals of pan-African unity and social justice. These values in turn help us to understand how and why people consume the music, and how French reggae might become an integral part of someone’s identity.
Fixed Composition vs. Improvisation
The two genres just discussed exhibit an additional pair of features that require deeper discussion: string quartets tend to be fixed compositions , such that the pitches and rhythms in every performance are identical, while reggae invites improvisation and variation from performance to performance, such that two renditions of the same song might sound quite different. In a tradition that relies on fixed composition, it is assumed that the creator of a work will make all decisions concerning pitch, rhythm, form, instrumentation, and length, and that performers will follow these instructions precisely. Fixed compositions are usually enshrined in notated music, although they do not have to be. This does not mean, however, that every performance of a fixed composition will be identical. Performers are usually invited to make minute adjustments to some of these elements, such as articulation, tempo, and dynamics, with the result that each rendition is unique to the discriminating listener.
Improvisation is much more difficult to sum up. This is due to the fact that there are nearly as many approaches to improvisation as there are musical traditions. Improvisation implies the production of new musical elements in the course of a live performance, but it always occurs within a set of boundaries. No improviser is free to play or sing whatever they want. Instead, an improviser will tend to apply formulas to the transformation of musical material while respecting certain fundamental characteristics of the style and composition.
In jazz, for example, improvisation is guided by the form and harmonic structure of a fixed composition that serves as the basis for a performance. While improvising, a player is free to choose pitches and rhythms—but they must fit with the predetermined harmonies, so choices are limited. Improvisation means something different to a balafon player in West Africa, who will constantly vary a repeated melodic figure used to accompany singing (see Chapter 5). It means something different again to a member of a Javanese gamelan, who might not know how a performance will unfold ahead of time but understands exactly how to vary their melody in response to instructions from the drummer (see Chapter 4). And it means yet something else to a Baroque violinist, who performs a fixed composition but is free to add ornaments and flourishes according to stylistic guidelines.
When we talk about fixed composition versus improvisation, we are talking about different roles in the creation of music: the role of the composer versus the role of the performer. Not all traditions distinguish between these roles, which makes it particularly difficult to define our terms. Throughout this volume, we will identify and examine the contributions of different individuals—composers, orchestrators, arrangers, adapters, and performers—to the creation of unique musical objects.
Emotional Expression and Cultural Context
Emotional expression is, for many listeners, the main reason to interact with music. It is also the most difficult to pin down or explain. While we can make some generalizations and predictions, emotional response to music happens at the individual level, and it is impossible to know exactly what impact music will have on a given listener. A piece of music might make one person cry, another feel uncomfortable, and another feel bored. The extraordinary diversity of genres is itself a testimony to the wide-ranging responses that people have to music. There is something out there for everybody to love, and something for everybody to hate. All the same, members of a given culture tend to agree, at least to some extent, about the emotional content of music. As an example, consider two excerpts from a musical work created by the German composer George Frideric Handel in 1740 entitled The Cheerful Person, the Thoughtful Person, and the Moderate Person (original Italian: L’Allegro, il Penseroso ed il Moderato ). For most of this work, two archetypal characters—the cheerful person and the thoughtful person—argue about whether it is better to be happy or pensive. Each calls forth the emotional state for which they advocate, the cheerful person with an aria (song) entitled “Come, thou Goddess fair and free,” and the thoughtful person with an aria entitled “Come, rather, Goddess sage and holy.” Although this music was written over 250 years ago, the emotions expressed are still easy to perceive by many today. But what is it, exactly, that makes the first aria sound happy, and what makes the second sound reserved?
Handel’s aria “Come, rather, Goddess sage and holy” captures a sober and introspective emotional state. Source: Wikimedia Commons Attribution: Thomas Cole License: Public Domain
The answer can be arrived at by comparing and contrasting the dimensions of sound that were enumerated above. The first aria is quick in tempo, while the second is slow. The first contains fast-moving rhythms, while the second does not.
The first is in the major mode, which we often hear as communicating positive emotions, while the second is in the minor mode, which can sound sad or serious. The vocal line in the first aria jumps around, skipping notes in the scale, while that in the second generally does not. The articulations in the first are bouncy and accented, while those in the second are smooth and connected. The first aria features bright-timbred wind instruments—oboe and bassoon—while the second relies on the mellower strings.
However, we still haven’t answered the question. After all, what do oboes have to do with cheerfulness? Why does a melody that moves stepwise suggest sobriety? Why does the minor mode signify a somber mood? There are two ways that we can begin to answer these questions. The first has to do with the web of relationships between music—a purely acoustic phenomenon with no required visual component— and the “real world.” Our brains easily map high pitches onto elevated physical locations, rapid rhythmic activity onto frenetic physical activity, and melodic leaps onto physical leaps. The other has to do with cultural signification. There is nothing objectively sad or serious about the minor mode, for example, but in the Western tradition we have developed an association between minor-mode music and profound emotional expression. (This is largely due to the complex inner workings of Western harmony, which are beyond the scope of this book.) Other cultures have not made this association, and listeners in those traditions might respond to minor- mode music differently than those acculturated to Western music.
This has been only a brief introduction to questions that will occupy us throughout this book. Our object will not be to answer these questions, but rather to carefully consider how music can create an emotional experience, how we respond to it, and how it has been used by humans over the span of centuries and continents.


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HGTC BIO 225: Microbiology Lab Manual
Horry-Georgetown Technical College
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2: Streak Plate and Colony Morphology
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1.2.3: Affinity for Spontaneous Chemical Reaction- Phase Equilibria
A given system comprises two phases, I and II, both phases comprising i-chemical substances. We consider the transfer of one mole of chemical substance \(j\) from phase I to phase II. The affinity for the transfer is given by equation (a).
\[\mathrm{A}_{\mathrm{j}}=\mu_{\mathrm{j}}(\mathrm{I})-\mu_{\mathrm{j}}(\mathrm{II}) \nonumber \]
If \(\mu_{j}(\mathrm{II})<\mu_{\mathrm{j}}(\mathrm{I})\), \(\mathrm{A}_{j}\) is positive and the process is spontaneous. If the system is at fixed \(\mathrm{T}\) and pressure, the gradient of Gibbs energy is negative [1].
\[\mathrm{A}_{\mathrm{j}}=-(\partial \mathrm{G} / \partial \xi)_{\mathrm{T}, \mathrm{p}}=\mu_{j}(\mathrm{I})-\mu_{\mathrm{j}}(\mathrm{II}) \nonumber \]
We suppose the mole fraction of substance \(–j\) in phases I and II are \(x_{j}(\mathrm{I})\) and \(x_{j}(\mathrm{II})\). We express the chemical potentials as functions of the mole fraction compositions of the two phases.
\[\begin{aligned}
A_{j}=\mu_{j}^{*}(I)+& R \, T \, \ln \left[x_{j}(I) \, f_{j}(I)\right] \\
&-\mu_{j}^{*}(I I)-R \, T \, \ln \left[x_{j}(I I) \, f_{j}(I I)\right]
\end{aligned} \nonumber \]
Here \(\mathrm{f}_{j}(\mathrm{I})\) and \(\mathrm{f}_{j}(\mathrm{II})\) are rational activity coefficients of substance \(j\) in phases \(\mathrm{I}\) and \(\mathrm{II}\) respectively.
\[\text { At all } T \text { and p, both } \operatorname{limit}\left(x_{j}(I) \rightarrow 1\right) f_{j}(I)=1 \nonumber \]
\[\text { and } \operatorname{limit}\left(\mathrm{x}_{\mathrm{j}} \text { (II) } \rightarrow 1\right) \mathrm{f}_{\mathrm{j}}(\text { II })=1 \nonumber \]
\[\text { By definition } \mu_{j}^{*}(\mathrm{II})-\mu_{\mathrm{j}}^{*}(\mathrm{I})=-\mathrm{R} \, \mathrm{T} \, \ln [\mathrm{K}(\mathrm{T}, \mathrm{p})] \nonumber \]
\(\mathrm{K}_{j}(\mathrm{T}, p)\) is a measure of the difference in reference chemical potentials of substance \(j\) in phases \(\mathrm{I}\) and \(\mathrm{II}\). If the two phases are in equilibrium, there is no affinity for substance \(j\) to pass spontaneously between the two phases. At equilibrium, \(\mathrm{A}_{j}\) is zero. Hence from equations (c) and (f), for the non-equilibrium state,
\[A_{j}=R \, T \, \ln \left[K_{j}(T, p)\right]+R \, T \, \ln \left[\frac{x_{j}(I) \, f_{j}(I)}{x_{j}(I I) \, f_{j}(I I)}\right] \nonumber \]
\[\frac{A_{j}}{T}=R \, \ln \left[K_{j}(T, p)\right]+R \, \ln \left[\frac{x_{j}(I) \, f_{j}(I)}{x_{j}(I I) \, f_{j}(I I)}\right] \nonumber \]
Equation (g) yields the affinity for chemical substance \(j\) to pass between the phases in a non-equilibrium state. In applications of equation (h), we describe the dependence of \(\left(\mathrm{A}_{j} / \mathrm{T}\right)\) on temperature, pressure and composition of the two phases. In other words we require the general differential of equation (h) which is written in the following form.
\[\begin{aligned}
d\left(\frac{A_{j}}{T}\right)=R \,\left(\frac{\partial \ln K_{j}(T, p)}{\partial T}\right) \, d T \\
&+R \,\left(\frac{\partial \ln K_{j}(T, p)}{\partial p}\right) \, d p+R \, d \ln \left[\frac{x_{j}(I) \, f_{j}(I)}{x_{j}(I I) \, f_{j}(I I)}\right]
\end{aligned} \nonumber \]
For the transfer process described by \(\mathrm{K}_{j}(\mathrm{T}, p)\) we obtain equation (j) where \(\Delta_{\text {trans }} \mathrm{H}_{\mathrm{j}}^{0}(\mathrm{~T}, \mathrm{p})\) and \(\Delta_{\text {trans }} V_{j}^{0}(T, p)\) are the standard enthalpy and volume for transfer for chemical substance \(j\).
\[\begin{aligned}
\mathrm{d}\left(\frac{\mathrm{A}_{\mathrm{j}}}{\mathrm{T}}\right)=\left(\frac{\Delta_{\text {trans }} \mathrm{H}_{\mathrm{j}}^{0}(\mathrm{~T}, \mathrm{p})}{\mathrm{T}^{2}}\right) \, \mathrm{dT} \\
&-\left(\frac{\Delta_{\text {trans }} \mathrm{V}_{\mathrm{j}}^{0}(\mathrm{~T}, \mathrm{p})}{\mathrm{T}}\right) \, \mathrm{dp}+\mathrm{R} \, \mathrm{d} \ln \left[\frac{\mathrm{x}_{\mathrm{j}}(\mathrm{I}) \, \mathrm{f}_{\mathrm{j}}(\mathrm{I})}{\mathrm{x}_{\mathrm{j}}(\mathrm{II}) \, \mathrm{f}_{\mathrm{j}}(\mathrm{II})}\right]
\end{aligned} \nonumber \]
\(\Delta_{\text {trans }} \mathrm{H}_{\mathrm{j}}^{0}(\mathrm{~T}, \mathrm{p})\) and \(\Delta_{\text {trans }} \mathrm{V}_{\mathrm{j}}^{0}(\mathrm{~T}, \mathrm{p})\) are properties of pure chemical substance \(j\); i.e. are not dependent on the composition of phases \(\mathrm{I}\) and \(\mathrm{II}\).
Footnotes
[1] Consider the freezing of water;
\[\text { water }(\lambda) \rightarrow \text { water }(\mathrm{s}) \nonumber \]
For this process, \(v\left(\mathrm{H}_{2} \mathrm{O} ; \lambda\right)=-1 ; v\left(\mathrm{H}_{2} \mathrm{O} ; \mathrm{s}\right)=1\)
The general rule is — Positive for Products. The affinity for spontaneous change,
\[A=-\left(\frac{\partial G}{\partial \xi}\right)_{T, p}=-\sum_{j=1}^{j=i} v_{j} \, \mu_{j}=\mu^{*}\left(H_{2} \mathrm{O} ; \lambda\right)-\mu^{*}\left(H_{2} \mathrm{O} ; \mathrm{s}\right) \nonumber \]
At equilibrium (at fixed \(\mathrm{T}\) and \(p\)), \(\mathrm{A} = 0\).
\[\text { Then, } \mu^{*}\left(\mathrm{H}_{2} \mathrm{O} ; \lambda\right)=\mu^{*}\left(\mathrm{H}_{2} \mathrm{O} ; \mathrm{s}\right) \nonumber \]


13.2: One-Way ANOVA
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The purpose of a one-way \(ANOVA\) test is to determine the existence of a statistically significant difference among several group means. The test actually uses variances to help determine if the means are equal or not. To perform a one-way \(ANOVA\) test, there are several basic assumptions to be fulfilled:
Five basic assumptions of one-way ANOVA to be fulfilled
- Each population from which a sample is taken is assumed to be normal.
- All samples are randomly selected and independent.
- The populations are assumed to have equal standard deviations (or variances).
- The factor is a categorical variable.
- The response is a numerical variable.
The Null and Alternative Hypotheses
The null hypothesis is simply that all the group population means are the same. The alternative hypothesis is that at least one pair of means is different. For example, if there are \(k\) groups:
- \(H_{0}: \mu_{1} = \mu_{2} = \mu_{3} = \dotsc = \mu_{k}\)
- \(H_{a}: \text{At least two of the group means} \mu_{2} = \mu_{3} = \dotsc = \mu_{k} \text{are not equal}\)
The graphs, a set of box plots representing the distribution of values with the group means indicated by a horizontal line through the box, help in the understanding of the hypothesis test. In the first graph (red box plots), \(H_{0}: \mu_{1} = \mu_{2} = \mu_{3}\) and the three populations have the same distribution if the null hypothesis is true. The variance of the combined data is approximately the same as the variance of each of the populations.
If the null hypothesis is false, then the variance of the combined data is larger which is caused by the different means as shown in the second graph (green box plots).
Figure \(\PageIndex{1}\): (a) \(H_{0}\) is true. All means are the same; the differences are due to random variation. (b) \(H_{0}\) is not true. All means are not the same; the differences are too large to be due to random variation.Chapter Review
Analysis of variance extends the comparison of two groups to several, each a level of a categorical variable (factor). Samples from each group are independent, and must be randomly selected from normal populations with equal variances. We test the null hypothesis of equal means of the response in every group versus the alternative hypothesis of one or more group means being different from the others. A one-way \(ANOVA\) hypothesis test determines if several population means are equal. The distribution for the test is the \(F\) distribution with two different degrees of freedom.
Assumptions:
- Each population from which a sample is taken is assumed to be normal.
- All samples are randomly selected and independent.
- The populations are assumed to have equal standard deviations (or variances).
Glossary
- Analysis of Variance
-
also referred to as \(ANOVA\), is a method of testing whether or not the means of three or more populations are equal. The method is applicable if:
- all populations of interest are normally distributed.
- the populations have equal standard deviations.
- samples (not necessarily of the same size) are randomly and independently selected from each population.
The test statistic for analysis of variance is the \(F\)-ratio.
- One-Way\(ANOVA\)
-
a method of testing whether or not the means of three or more populations are equal; the method is applicable if:
- all populations of interest are normally distributed.
- the populations have equal standard deviations.
- samples (not necessarily of the same size) are randomly and independently selected from each population.
The test statistic for analysis of variance is the \(F\)-ratio.
- Variance
- mean of the squared deviations from the mean; the square of the standard deviation. For a set of data, a deviation can be represented as \(x - \bar{x}\) where \(x\) is a value of the data and \(\bar{x}\) is the sample mean. The sample variance is equal to the sum of the squares of the deviations divided by the difference of the sample size and one.
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Barbara Illowsky and Susan Dean (De Anza College) with many other contributing authors. Content produced by
OpenStax College
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4.4: Herbs, Spices, Seasoning and Flavoring
Herbs
Herbs tend to be the leaves of fragrant plants that do not have a woody stem. Herbs are available fresh or dried, with fresh herbs having a more subtle flavor than dried. You need to add a larger quantity of fresh herbs (up to 50% more) than dry herbs to get the same desired flavor. Conversely, if a recipe calls for a certain amount of fresh herb, you would use about one-half of that amount of dry herb.
The most common fresh herbs are basil, coriander, marjoram, oregano, parsley, rosemary, sage, tarragon, and thyme. Fresh herbs should have a clean, fresh fragrance and be free of wilted or brown leaves. They can be kept for about five days if sealed inside an airtight plastic bag. Fresh herbs are usually added near the completion of the cooking process so flavors are not lost due to heat exposure.
Dried herbs lose their power rather quickly if not properly stored in airtight containers. They can last up to six months if properly stored. Dried herbs are usually added at the start of the cooking process as their flavor takes longer to develop than fresh herbs.
Seasoning and Flavoring
Many ingredients are used to enhance the taste of foods. These ingredients can be used to provide both seasoning and flavoring.
- Seasoning means to bring out or intensify the natural flavor of the food without changing it. Seasonings are usually added near the end of the cooking period. The most common seasonings are salt, pepper, and acids (such as lemon juice). When seasonings are used properly, they cannot be tasted; their job is to heighten the flavors of the original ingredients.
- Flavoring refers to something that changes or modifies the original flavor of the food. Flavoring can be used to contrast a taste such as adding liqueur to a dessert where both the added flavor and the original flavor are perceptible. Or flavorings can be used to create a unique flavor in which it is difficult to discern what the separate flavorings are. Spice blends used in pumpkin pies are a good example of this.
Knowing how to use seasonings and flavorings skilfully provides cooks and bakers with an arsenal with which they can create limitless flavor combinations.
Flavoring and seasoning ingredients include wines, spirits, fruit zests, extracts, essences, and oils. However, the main seasoning and flavoring ingredients are classified as herbs and spices.
Knowing the difference between herbs and spices is not as important as knowing how to use seasonings and flavorings skilfully. In general, fresh seasonings are added late in the cooking process while dry ones tend to be added earlier. It is good practice to under-season during the cooking process and then add more seasonings (particularly if you are using fresh ones) just before presentation. This is sometimes referred to as “layering.” When baking, it is difficult to add more seasoning at the end, so testing recipes to ensure the proper amount of spice is included is a critical process.
Spices
Spices are aromatic substances obtained from the dried parts of plants such as the roots, shoots, fruits, bark, and leaves. They are sold as seeds, blends of spices, whole or ground spices, and seasonings.  The flavor of the essential oil or flavoring compound will vary depending on the quality and freshness of the spice.
The aromas of ground spices are volatile. This means they lose their odor or flavoring when left exposed to the air for extended periods. They should be stored in sealed containers when not in use. Whole beans or unground seeds have a longer shelf life but should also be stored in sealed containers.
Allspice
Allspice is only one spice, yet it has a flavor resembling a blend of cloves, nutmeg, and cinnamon. At harvest time, the mature (but still green) berries from the allspice trees (a small tropical evergreen) are dried in the sun. During drying they turn reddish-brown and become small berries. The berries are about 0.6 cm (1/4 in.) in diameter and contain dark brown seeds.
Allspice is grown principally in Jamaica and to a lesser degree in Mexico. Allspice is available whole or ground. Bakers usually use ground allspice in cakes, cookies, spices, and pies.
Anise
Anise is the small, green-grey fruit or seed of a plant of the parsley family. The plant grows to a height of 45 cm (18 in.) and has fine leaves with clusters of small white flowers. It is native to Mexico and Spain, with the latter being the principal producer. Anise seeds are added to pastries, breads, cookies, and candies.
Caraway
Caraway is the dried fruit or seed of a biennial plant of the parsley family, harvested every second year, primarily in the Netherlands. It is also produced in Poland and Russia. The many-branched, hollow-stemmed herb grows up to 60 cm (24 in.) high and has small white flowers. Caraway is a small crescent-shaped brown seed with a pleasant aroma but somewhat sharp taste. Although it is most familiar in rye bread, caraway is also used in cookies and cakes.
Cardamom
Native to India, Sri Lanka, and Guatemala, cardamom is the fruit or seed of a plant of the ginger family. The three-sided, creamy-white, flavorless pod holds the tiny aromatic, dark brown seeds. It is available in whole and ground (pod removed). Cardamom in ground form flavors Danish pastries and coffee cakes, Christmas baking, and Easter baking such as hot cross buns.
Cinnamon
Cinnamon comes from the bark of an aromatic evergreen tree. It is native to China, Indonesia, and Indochina. Cinnamon may be purchased in ground form or as cinnamon sticks. Ground cinnamon is used in pastries, breads, puddings, cakes, candy, and cookies. Cinnamon sticks are used for preserved fruits and flavoring puddings. Cinnamon sugar is made with approximately 50 g (2 oz.) of cinnamon to 1 kg (2.2 lb.) of granulated sugar.
Cassia
Cassia, sometimes known as Chinese cinnamon, is native to Assam and Myanmar. It is similar to cinnamon but a little darker with a sharper taste. It is considered better for savory rather than sweet foods. It is prized in Germany and some other countries as a flavor in chocolate.
Cloves
Cloves are the dried, unopened buds of a tropical evergreen tree, native to Indonesia. The flavor is characterized by a sweet, pungent spiciness. The nail-shaped whole cloves are mainly used in cooking, but the ground version of this spice heightens the flavor of mincemeat, baked goods, fruit pies, and plum pudding.
Ginger
Ginger is one of the few spices that grow below the ground. It is native to southern Asia but is now imported from Jamaica, India, and Africa. The part of the ginger plant used is obtained from the root. Ground ginger is the most commonly used form in baking — in fruitcakes, cookies, fruit pies, and gingerbread. Candied ginger is used in pastries and confectionery.
Mace
Originating in the East and West Indies, mace is the fleshy growth between the nutmeg shell and outer husk, yellow-orange in color. It is usually sold ground, but sometimes whole mace (blades of mace) is available. Mace is used in pound cakes, breads, puddings, and pastries.
Nutmeg
Nutmeg is the kernel or seed of the nutmeg fruit. The fruit is similar to the peach. The fleshy husk, grooved on one side, splits, releasing the deep-brown aromatic nutmeg. It is available whole or ground. Ground nutmeg is used extensively in custards, cream puddings, spice cakes, gingerbread, and doughnuts.
Poppy Seed
Poppy seed comes from the Netherlands and Asia. The minute, blue-grey, kidney-shaped seeds are so small they seem to be round. Poppy seeds are used in breads and rolls, cakes and cookies, and fillings for pastries.
Sesame or Benne Seed
Sesame or benne seeds are the seeds of the fruit of a tropical annual herb grown in India, China, and Turkey. The seeds are tiny, shiny, and creamy-white with a rich almond-like flavor and aroma. Bakers use sesame seeds in breads, buns, coffee cakes, and cookies.
Vanilla
The Spaniards named vanilla. The word derives from vaina , meaning pod. Vanilla is produced from an orchid-type plant native to Central America. The vanilla beans are cured by a complicated process, which helps explain the high cost of genuine vanilla. The cured pods should be black in color and packed in airtight boxes. Imitation vanilla extracts are made from a colorless crystalline synthetic compound called vanillin. Pure vanilla extract is superior to imitation vanilla. Artificial vanilla is more intense than real vanilla by a factor of 3 to 4 and must be used sparingly.
To use vanilla beans, split the pod down the middle to scrape out the seeds. The seeds are the flavoring agents. Alternatively, the split pod can be simmered in the milk or cream used in dessert preparation. Its flavoring power is not spent in one cooking and it can be drained, kept frozen, and reused. A vanilla bean kept in a container of icing sugar imparts the flavor to the sugar, all ready for use in cookies and cakes.
Vanilla extract is volatile at temperatures starting at 138°C (280°F) and is therefore not ideal for flat products such as cookies. It is suitable for cakes, where the interior temperature does not get so high.
Vanilla beans and vanilla extract are used extensively by bakers to flavor a wide range of desserts and other items.
The following website is a great resource for how to cook with herbs, spices and seasonings.
https://site.extension.uga.edu/madison/2021/04/cooking-with-herbs-spices-and-seasonings/


16.5: Spanning Trees
a subgraph that contains all the vertices of the parent graph
a spanning subgraph that is a tree
Here is the complete graph with four vertices.
And here are ten different spanning trees for \(K_4\text{.}\)
If we carry out either of the depth-first or breadth-first search algorithms, but aren't looking for a path between specific vertices, the end result will be a spanning tree for the original graph.
the result of performing the depth-first search algorithm on a graph, continuing until all vertices in the original graph appear in the search tree
the result of performing the breadth-first search algorithm on a graph, continuing until all vertices in the original graph appear in the search tree
Figure \(\PageIndex{3}\) contains depth-first and breadth-first spanning trees for the graph in Figure 16.4.2 , our source of examples for depth-first search ( Example 16.4.1 ) and breadth-first search ( Example 16.4.2 ).


8.3: Questions to Guide Your Reading
1. How does geography play a role in Islamic history?
2. Why were the concepts defined by muruwah so important to the early development of Islam?
3. What are the Five Pillars of Islam, and why are they important to the religion?
4. What were the five roles that the Prophet Muhammad played in Medina?
5. What factors led to the rapid expansion of Islam?
6. How did the Umayyads come to power following the Rashidun caliphs?
7. How do you account for the rise of the splits in the Islamic community, like the rise of the Kharijis and Shi‘a?
8. Describe the transition from the Umayyads to the ‘Abbasids. Compare and contrast the two caliphates.
9. The Fatimids marked the end to the High Caliphate. How did Egypt gain its autonomy from the ‘Abbasids? How did the Fatimids take over Egypt?
10. How did the Crusaders gain a foothold in the Middle East? What did it take for Salah al-Din to push them out?
11. What led to the establishment of the Mamluk Sultanate? How did the Mamluk Sultanate go into decline?
12. Does North African history move in cycles of birth, renewal, expansion and decadence? Ibn Khaldoun says that nomads come from the frontiers, desert, and periphery, settle down, and within 120 years, become decadent and collapse. Do you agree?


Previously you did some great work identifying your target audience for your research project. The whole fun of writing a persuasive paper is to pick an audience who disagrees with you, or at least is undecided about the matter. Then you use charm, wit, and raw intelligence to prove that they’re absolutely silly for thinking what they do, and that they better come over to your side or else the world will end.
In being persuasive and winning the fight, it helps a lot to remember that your target audience has reasons for their position on the issue. Those reasons may not be GOOD ones, of course, but they have some motivation for thinking or feeling the way they do. (I still don’t like eating at Jack in the Box because a friend of mine had a really bad experience there years ago, in another state. Not a very rational reason, I admit, but it does shape my behavior when it comes to fast food.)
In order to be truly persuasive, you have to understand what people’s motivations are, and acknowledge those in your essay.
I’d like you to visit
POWA’s “Anticipating Opposition” article
and read the content there. Then, return to this discussion and build your own pro/con chart, using your thesis as the “proposition.” It’ll be easier to create a list, rather than a chart, given our constraints in the discussion forum platform.
Then look at one or two of your “con” statements in more detail. How will you acknowledge these arguments in your own essay, and what will you say to your reader to counter them? For instance, if I were trying to talk myself into eating at Jack in the Box again, I’d acknowledge that finding a bug in your food is yes, a traumatic event. But it was an isolated incident, and in no way reflects the standards of the chain overall. I’d go into food safety data and possibly relate the health scores of the local franchises recently. Maybe I’d even embark on a smear campaign and talk about similar events that have occurred at other fast food chains, to show it’s not particular to one brand.
Your post should be at least 150-200 words. It doesn’t have to be grammatically perfect, but should use standard English (no text-speak, please) and normal capitalization rules.
You will also need to return to this Discussion to reply to at least one of your group members’ posts. Content could include, but is not limited to, any of the following:
Suggest further elements that could be added either to the Pro or Con side of their chart. Indicate what it would take for you to be convinced, if you were on the con side of the argument.
Responses are weighed as heavily as your initial posting, and should be roughly as long (150-200 words). Responses should indicate you’ve read your classmate’s post carefully. Include specific details from the post you’re responding to in your reply.


10.1: Why It Matters- Late Adulthood
Why explain development and change through late adulthood?
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Jeanne Calment was a typical woman of her time. Born in Arles, France, in 1875, she lived a rather unremarkable life by most accounts—except for one thing: when she died in 1997 at the age of 122, she was on record as the oldest person to have ever lived. “I just kept getting older and couldn’t help it,” she once said.
So what does the extraordinary life of this ordinary woman have to do with us today? More than you might think. In her day, living to 100 was extremely rare. But today in the United States, people 100 and over represent the second-fastest-growing age group in the country. The fastest? People over 85. Many 65-yea r-olds today will live well into their 90s.
Furthermore, because of increases in average life expectancy, each new generation can expect to live longer than their parents’ generation and certainly longer than their g randparents’ generation. Think of it another way: a 10-year-old child todayhas a 50 percent chance of living to the age of 104. Some demographers have even speculated that the first person ever to live to be 150 is alive today.
As a consequence, it is time for individuals of all ages to rethink their personal life plans and consider prospects for a long life. We need to ask ourselves questions such as:
In this module we will discuss several different domains of physical, cognitive, psychological and social development, as well as research on aging that will help answer these important questions.


4.6: Chapter 4- Review Questions
Chapter 4: Review Questions
- What are some of the critical differences between Accounting and Finance?
- In what ways can Accounting d ata be managed?
- What are the critical differences between the Balance Sheet and the Income Statement?
- Identify the correct choices : Assets are dr / cr balance accounts while Liabilities and Equity are dr / cr balance accounts.
- How are the Addition to Retained Earnings and Retained Earnings different from one another ?
- Who takes the most risk in order to earn the highest return?
- Give an example where it is permissible to use different accounting methods for reporting versus tax accounting .
- What are some problems pursuant to using LIFO-based accounting?
- True or False: Operating Profits include interest paid.
- If the company does not pay its dividends on Preferred Stock, has it defaulted?
- True or False: Interest is not tax-deductible whereas Dividends are tax-deductible.
- What are some of the differences between what the accountant and financial analyst do?
- If there is some inflation, which will produce higher gross profits – FIFO or LIFO?
- List, provide some examples of, and discuss each of the four interpretive problems read e rs of financial statements may encounter.
- Solve the following Calculation Problems:
- Assume the following :
Equipment cost: $12 million
Estimated Life: 8 years
Salvage Value: $1.75 million
What are the second year’s depreciation expenses and asset balances under each of the three reporting methods?
2. Assume the following :
Beginning Inventory: $2 million
Ending Inventory: $4 million
Purchases: $10 million
What is the company’s Cost of Goods Sold?


2.3.1: New English Canaan (Excerpt)
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Chap. XVI. Of their acknowledgment of the Creation, and immortality of the Soule.
Although these salvages are found to be without religion, law, and king (as Sir William Alexander has well observed) they are not altogether without the knowledge of God (historically.) 1: For they have it amongst them by tradition that God made one man and one woman, and had them live together and get children, kill deer, beasts, birds, fish and fowl, and what they would at their pleasure and that their posterity was full of evil and made God so angry that he let in the sea upon them, and drowned the greater part of them, that were naughty men, (the Lord destroyed so) and they went to Sanaconquam, who feeds upon them (pointing to the Center of the Earth, ‘savages‘ where they imagine is the habitation of the Devil) the other, (which were not destroyed) increased the world, and when they died (because they were good) went to the house of Kytan, pointing to the setting of the sun; 2 where they ate all manner of dainties, and never take pains (as now) to provide it. Kytan makes provision (they say) and saves them that labor; and there they shall live with him forever, void of care. And they are persuaded that Kytan is he that makes corn grow, trees grow, and all manner of fruits. And that we that use the book of Common prayer do it to declare to them, that cannot read, what Kytan has commanded us, and that we do pray to him with the help of that book; and do make so much account of it, that a savage (who had lived in my house before he had taken a wife, by whom he had children) made this request to me, (knowing that I always used him with much more respect: than others,) that I would let his son be brought up in my not savage house, that he might be taught to read in that book : which request of his I granted; and he was a very joyful man to think that his son would thereby (as he said) my prayer – become an Englishman ; and then he would be a good man. I asked him who was a good man ; his answer was that he would not lie, nor steal. These, with them, are all the capital crimes that can be imagined; all other are nothing in respect; of those ; 1 and he that is free from these must live with Kytan forever, in all manner of pleasure.
Chap. XX.
I must needs commend them in this particular, that, though they buy many commodities of our Nation, yet they keep but few, and those of special use. They love not to be cumbered with many utensils, and although every proprietor knows his own, yet all things, (for long as they will last), are used in common amongst them : A biscuit cake given to one, that one breaks it equally into for many parts as there be persons in his company, and distributes it. Plato’s Commonwealth is for much practiced by these people. According to humane reason, guided only by the light of nature, these people lead the more happy and freer. They lead a life, being void of care, which torments the minds of many Christians: They are not delighted in baubles, but care ‘in useful things…
I have observed that they will not be troubled with superfluous commodities. Such things as they find they are taught by necessity to make use of, they will make choice of, and seek to purchase with industry. So that, in respect: that their life is so void of care, and they are so loving also that they make use of those things they enjoy, (the wife only excepted,) as common goods, and are therein so compassionate that, rather than one could starve through want, they would starve all. Thus doe they pass away the time merrily, not regarding our pope, (which they fee daily before their faces,) but are better content with their own, which some men esteem so meanly of. They may be rather accustomed to live richly, wanting nothing that is needful; and to be commended for leading a contented life…
Chap. V. Of a Massacre made upon the Salvages at Wessaguscus.
But the Plimmoth men, intending no good to him, (as appeared by the consequence,) came in the mean time to Wessaguscus, a pit from and there pretended to feast the Salvages of those parties, Plimmoth – bringing with them pork and things for the purpose, which they felt before the Salvages. They eat thereof without suspicion of any mischief, who were taken upon a watchword given, and with their own knives, (hanging salvages about their necks,) were by the Plimmoth planters, one of which were hanged up there, after the weapons – slaughter. In the mean time the Sachem had knowledge of this act- News car- dent, by one that ran to his Countrymen, at the Massachusetts, and gave them intelligence of the news..
The Salvages of the Massachusetts, that could not imagine from whence these men should come, or to what end, seeing them perform such unexpected actions ; neither could tell by what name properly to distinguish them; did from that time afterwards call the English Planters Wotawquenange, which in their language signifieth stabbers, or Cutthroats: and this name was received by those that came thereafter for good, being then unacquainted with the signification of it, for many years following…
Chap. VII. Of Thomas Mortons entertainment at Plimmouth, and casting away upon an Island
This man arrived in those parts, and, hearing news of a Town that was much praised, he was delirious to go thither, and see how things stood; where his entertainment, Brave entertainment, was in a wilderness… There he bestowed sometime in the survey of this plantation…
His new come servants, in the mean time, were tame to take, to have their zeal appear, and questioned what preacher was among their company; and finding none, did seem to condole their estate as if undone, because no man among them had the gift to be in Iona’s stead, nor they the means to keep them in that path for hard to keep. Our Master, say they, recalls the Bible and the word of God, and uses the Book of Common Prayer: but this is not the means, the answer is: the means, they cried, alas, poor Souls where is the means? You seem as if betrayed, to be without the means: how can you be flayed from falling headlong to perdition? Facile descensus avemi: 2 the book of common prayer, says they, what poor thing is that, for a man to read in a book? No, no, good firs, I would you were near us, you might receive comfort by destruction: give me a man hath the gifts of the spirit, not a book in hand. I do profess says one, to live without the means is dangerous, the Lord doth know. By there insinuations, like the Serpent, they did creep and wind into the good opinion of the illiterate multitude, that were desirous to be freed and gone to them, no doubt, (which some of them after confessed); and little good was to be done one them after this charm was used: now plots and factions how they might get look: and here was some Stout knaves; and some plotted how to steal Master Weston’s baroque, others, exasperated knavishly to work, would precariousness tried how to get their Master to an Inland, and there leave Him; which he had notice of, and fitted him to try what would be done; and steps aboard his gallop bound for Cape Anne, to the Massachusetts, with an Hogshead of Wine; Sugar he took along, the Sails hoist up, and one of the Conspirators aboard to steer; who in the mid way pretended foul weather at the harbored mouth, and therefore, for a time, he would put in to an Inland near, and make some flay where he thought to tempt his Master to walk the woods, and for be gone: but their Master to prevent them caused the sails and oars to be brought ashore, to Prevented make a tilt if need should be, and kindled fire, broached that Hogshead, and caused them fill the can with lusty liquor, Claret sparkling neat; which was not suffered to grow pale and flat, but tippled of with quick dexterity : the Master and discover makes a flew of keeping round, but with clove reading drinker. Lips did seem to make long draughts, knowing the wine would make them Protestants; and for the plot was then at large disclosed and discovered, and they made drowsy…
THE SONGE.
DRinke and be merry, merry, merry boycs ;
Let all your delight be in the Hymens ioyes ;
Jo to Hymen, now the day is come,
About the merry Maypole take a Roome.
Make greene garlons, bring bottles out
And fill fweet Neclar freely about.
Vncover thy head and feare no harme,
For hers good liquor to kccpe it warme.
Then drinke and be merry, &c. lb to Hymen, &c.
Neclar is a thing aJJigiHd
By the Deities owne minde
To cure the hart opprejl with greife,
And of good liquors is the cheife.
Then drinke, &c. lb to Hymen, &c.
.
Give to the Mellancolly man
A cup or two of V now and than ;
This phyfick willfoone revive his bloud,
And make him be of a merrier moode.
Then drinke, &c. lb to Hymen, &c.
Give to the Nymphe t hats free from fcorne
No IrifJi fluff nor Scotch over worne.
Laffes in beaver coats come away,
YcefJiall be welcome to us night and day.
To drinke and be merry &c. Jo to Hymen, &c.
This harmeles mirth made by younge men, (that lived in hope to have wifes brought over to them, that would save them a labour to make a voyage to fetch any over,) was much distasted of the precise Separatists…
Chap. XV. Of a great Monster supposed to be at Ma-re-Mount ; and the preparation made to destroy it
The Separatists, envying the prosperity and hope of the Plantation at Ma-re Mount, (which they perceived began to come forward, and to be in a good way for gain in the Beaver trade,) conspired together against mine host especially, (who was the owner of that Plantation) and made up a party against him; and mustered up what aide they could, accounting of him as of a great monster. Many threatening speeches were given out both against his person and his habitation, which they divulged should be consumed with fire: And taking advantage of the time when his company, (which seemed little to regard their threats,) were gone up into the Inlands to trade with the Salvages for Beaver, they set upon my honest host at a place called Wesaguscus, where, by accident, they found him. The inhabitants there were in good hope of the subversion of the plantation at Mare Mount, (which they principally aimed at) and the rather because my host was a man that endeavored to advance the dignity of the Church of England; which they, (on the contrary part) would labor to vilify with uncivil terms: envying against the sacred book of common prayer, and my host that used it in a laudable manner amongst his family, as a practice of piety. There he would be a mean to bring sacks to their mill, (such is the thirst after Beaver,) and helped the conspirators to surprise mine host, (who was there all alone) and they charged him, (because they would seem to have some reasonable cause against him to set a gloss upon their malice,) with criminal things ; which indeed had been done by such a person, but was of their conspiracy; mine host demanded of the conspirators who it was that was author of that information, that seemed to be their ground for what they now intended. And because they answered they would not tell him, he as peremptorily replied, that he would not say whether he had, or he had not done as they had been informed.
The answer made no matter, (as it seemed,) whether it had been negatively or affirmatively made; for they had resolved what he would suffer, because, (as they boasted,) they were now become the greater number: they had shaked of their shackles of servitude, and were become Mailers, and mailerles people. It appears they were like bears whelps in former time, when mine host’s plantation was of as much strength as theirs, but now, (theirs being stronger,) they, (like overgrown bears,) seemed monsterous. In brief, mine host must endure to be their prisoner until they could contrive it so that they might send him for England, (as they said,) there to suffer according to the merit of the fact which they intended to father upon him; supposing, (belike,) it would prove a heinous crime. Much rejoicing was made that they had gotten their capital enemy, (as they concluded him;) whom they purposed to hamper in such sort that he would not be able to uphold his plantation at Ma-re Mount. The Conspirators sported themselves at my honest host, that meant them no hurt, and were so jocund that they feasted their bodies, and fell to tippling as if they had obtained a great prize; like the Trojans when they had the custody of Hippeus pine-tree horse. Mine host feigned grief, and could not be persuaded either to eat or drink; because he knew emptiness would be a means to make him as watchful as the Geese kept in the Roman Capital: whereon, the contrary part, the conspirators would be so drowsy that he might have an opportunity to give them a flip, instead of a teller. Six persons of the conspiracy were set to watch him Mine Host at Weffagufcus: But he kept waking; and in the dead of prison night, (one lying on the bed for further surety,) up gets mine Host and got to the second door that he was to pass, which, notwithstanding the lock, he got open, and shut it after him with such violence that it affrighted some of the conspirators. The word, which was given with an alarm, was, he’s gone, he’s gone, what shall we do, he’s gone! The rest, (half asleep,) start up in a maze, and, like rams, ran their heads one at another full butt in the dark.
Their grand leader, Captain Shrimp, took on most fury- The Captain ripped and tore his clothes for anger, to see the empty nest, and their bird gone. The rest were eager to have torn their hair from their heads; but it was so short that it would give them no hold. Now Captain Shrimp thought in the loss of this prize, (which he accounted his mail) his entire honor would be lost forever. In the meantime Hofl was got home to Ma-re Mineko/i Mount through the woods, eight miles round about the head Ma-rc’mount of the river Monatoquit that parted the two Plantations, finding his way by the help of the lightening, (for it thundered as he went terribly) and there he prepared powder, three pounds dried, for his present employment, and four good guns for him and the two affirmants left at his house. He provides bullets of several sizes, three hundred or there- Zs. ume ‘ about, to be used if the conspirators should pursue him there: and these two persons promised their aides in the quarrel, and confirmed that promise with health in good rofa folis. Now Captain Shrimp, the first Captain in the Land, (as he supposed,) must do some new act to repair this loss and to vindicate his reputation, which had sustained blemish by this oversight, begins now to study, how to repair or survive his honor: in this manner, calling of Council, they conclude.
He takes eight persons more to him, and, (like the nine Worthies of New Canaan,) they embark with preparation against Ma-re-Mount, where this Monster of a man, as their phrase was, had his den; the whole number, had the rest not been from home, being but seven, would have given Captain Shrimpe, (a quondam Drummer,) such a welcome as would have made him with for a Drum as big as Diogenes tub, that he might have crept into it out of fight. Now the nine Worthies are approached, and mine Host prepared: having intelligence by a Salvage, that hastened in love from Wessaguscus to give him notice of their intent. One of mine Hosts men proved a craven: the other had proved his wits to purchase a little valor, before mine Host had observed his posture.
The nine worthies coming before the Denne of this Supposed Monster, (this seven headed hydra, as they termed him,) and began, like Don Quixote against the Windmill, to beat a parly, and to offer quarter, if mine Host would yield ; for they resolved to send him for England ; and had him lay by his arms. But he, (who was the Son of a Solder,) having taken up arms in his just defense, replied that he would not lay by those arms, because they were so needful at Sea, if he should be sent over. Yet, to save the effusion of so much worthy blood, as would have issued out of the veins of these nine worthies of New Canaan, if mine Host should have played upon them out at his port holes, (for they came within danger like a flock of wild geese, as if they had been tayled one to another, as coults to be fold at a faier,) mine Host was content to yield upon quarter ; and did capitulate with them in what manner it should be for more certainty, because he knew what Captain Shrimpe was.
Shrimpe promised that he expressed that no violence should be offered to his no violence…But mine Host no sooner had felt open the door, and issued out, but instantly Captain Shrimpe and the rest of the war- ties stepped to him, laid hold of his arms, and had him down : and so eagerly was every man bent against him, (not regarding any agreement made with such a carnal man,) that they fell upon him as if they would have eaten him : some of them were so violent that they would have a slice with scabbard and all for haste ; until The Worthies an old soldier, (of the Queens, as the Proverb is,) that was there by accident, clapt his gun under the weapons and sharply rebuked these worthies for their unworthy practices. So the matter was taken into more deliberate consideration. Captain Shrimpe, and the rest of the nine worthies, made themselves, (by this outrageous riot,) Matters of mine Host of Ma-re Mount, and disapproved of what he had at his plantation. This they knew, (in the eye of the Salvages,) would add to their glory, and diminish the reputation of mine honest Host ; whom they practiced to be rid of upon any terms, as willingly as if he had bin the very Hidra of the time.
Chap. XVI. How the 9 worthies put mine Host of Ma-re-Mount into the enchanted Castle at Plimmouth, and terrified him with the Monster Briareus.
The nine worthies of New Canaan having now the Law in their own hands, (there being no general Governour in the Land ; nor none of the Separation that regarded the duty they owe their Sovereign, whose natural born Subjects they were, though translated out of Holland, from whence they had learned to work all to their own ends, and make a great show of Religion, but no humanity,) for they were now to fit in Counsel on the cause. And much it flood mine honest Hoft upon to be very circumspect, and to take Eacus to task ; for that his voice was more allowed of then both the other : and had not mine Host confounded all the arguments that Eacus could make in their defence, and confuted him that swayed the rest, they would have made him unable to drink in such manner of merriment any more. So that following this private counsel, given him by one that knew who ruled the rest, the Hiracano ceased that else would split his pinace. A conclusion was made and sentence given that mine Host should be sent to England a prisoner. But when he was brought to the ships for that purpose, no man durst be so foolhardy as to undertake carry him. So these Worthies set mine Host upon an Island, without gun, powder, or shot or dog or so much as a knife to get anything to feed upon, or any other clothes to shelter him with at winter then a thin suit which he had one at that time. Home he could not get to Ma-re-Mount. Upon this Island he stayed a month at lead, and was relieved by Salvages that took notice that mine Host was a Sachem of Passonagesit, and would bring bottles of strong liquor to him, and unite themselves into a league of brotherhood with mine Host ; so full of humanity are these infidels before those Christians. From this place for England failed mine Host in a Plimmouth ship, (that came into the Land to fish upon the Coast,) that landed him safe in England at Plimmouth : and he stayed in England until the ordinary time for shipping to set forth for these parts, and then returned: No man being able to tax him of any thing. But the Worthies, (in the meantime,) hoped they had been rid of him.


Now let’s integrate both sides and solve for \(u v'\)
\[\begin{align*} \frac{d}{dx} u \cdot v & = u v' + v u' \\ \int \frac{d}{dx} uvdx & = \int u v' + v u'dx \\ uv & = \int u v'dx + \int v u' dx \\ uv - \int v u'dx & = \int u v'dx, \end{align*}\]
Flipping this around, we have the inverse product rule, also called integration by parts.
\(\boxed{\int u v'dx = uv - \int u' v}\)
The tricky part is what to use as \(u\), and what to use as \(v'\). Here are some steps and guidelines to follow, but it takes some intuition building before you know how to use it sometimes, and some product integrals cannot be solved with integration by parts.
- Think of your original integral as a product. Identify a function that is easy to integrate, and set it equal to \(v'\). The other function should be something that will simplify nicely once you take the derivative.
- Find \(u'\) (take the derivative of \(u\)) and find \(v\) (integrate \(v'\))
- Using substitution, plug in the values for \(u\), \(v\), \(v\) and \(u'\) in the integration by parts formula.
- This gives you another integral — hopefully this one is easier. If not, you may need to use \(u\)-substitution, or even integration by parts a second time.
Let’s follow the integration by parts steps:
- The function we are integrating is \(x e^x\), which is a product in two pieces: \(x\) and \(e^x\). While \(x\) is easy to integrate, \(e^x\) is even nicer. We will start with \(v' = e^x\), and \(u = x\).
- We see \(u' = \frac{d}{dx} x = 1\), and \(v = \int e^xdx = e^x\) (you don’t need to worry about the \(+ C\) for now).
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Using the formula with \(u = x\), \(u' = 1\), \(v = e^x\), and \(v' = e^x\), we have
\[\begin{align*} \int_0^2 x e^xdx & = uv - \int u' vdx \\ & = (x)(e^x) - \int_0^2 (1)(e^x)dx \\ & = x e^x - \int_0^2 e^xdx \\ \end{align*}\]
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We now have reduce the problem to an easier one: \(\int e^xdx\). We continue:
\[\begin{align*} \int_0^2 x e^xdx & = x e^x - \int_0^2 e^xdx \\ & = x e^x - e^x \Big|_0^2 \\ & = (2 e^2 - e^2) - (0e^0 - e^0) \\ & = (e^2) - (0 - 1) \\ & = e^2 + 1 \end{align*}\]
So the answer is \(e^2 + 1 \approx \boxed{8.389}\).
Notice if the problem contains an \(x\) variable, then this is usually a good choice for \(u\) since it will go away once you take the derivative \(u'\).
We following the steps of integration by parts.
- Set \(u = (2x + 3)\), which simplifies nicely with the derivative, and let \(v' = \cos(x)\) which is easy to integrate.
- We have \(u' = 2\), and \(v = \sin(x)\).
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Applying the formula, we have
\[\begin{align*} \int (2x + 3) \cos(x)dx & = uv - \int u' vdx \\ & = (2x + 3) \sin(x) - \int 2 \sin(x)dx \\ \end{align*}\]
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Continuing …
\[\begin{align*} \int (2x + 3) \cos(x)dx & = (2x + 3) \sin(x) - \int 2 \sin(x)dx \\ & = (2x + 3) \sin(x) - 2 (-\cos(x)) \\ & = \boxed{(2x + 3) \sin(x) + 2 \cos(x) + C}. \end{align*}\]
This one will involve integration by parts and a \(u\)-substitution shortcut. Here are the steps of integration by parts:
- We can integrate either function, but just like the previous case it’s best to set \(u = x\). This leaves \(v' = \sqrt{2x + 1}\).
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We see \(u' = \frac{d}{dx} x = 1\). Notice \(v = \int \sqrt{2x + 1}dx\) is a bit harder — this is a \(u\)-sub shortcut though. Let’s write it as \(\frac{1}{m}\) factor from the \(u\)-sub shortcut, we have
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Applying the formula with \(u = x\), \(u' = 1\), \(v' = \sqrt{2x + 1}\), we see
At this point, doing the integral of \(u\)-sub shortcut problem. Watch.