6.3: Reference Material
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This chapter was edited by Paula Fortier, with content contributions by Tizoc Chavez, Zachary W. Dresser, Blake Earle, Morgan Deane, Paula Fortier, Larry A. Grant, Mariah Hepworth, Jun Suk Hyun, and Leah Richier.
Recommended citation: Tizoc Chavez et al., “World War I and Its Aftermath,” Paula Fortier, ed., in The American Yawp , eds. Joseph Locke and Ben Wright (Stanford, CA: Stanford University Press, 2018).
Recommended Reading
- Capozzola, Christopher. Uncle Sam Wants You: World War I and the Making of the Modern American Citizen. New York: Oxford University Press, 2010.
- Coffman, Edward M. The War to End All Wars: The American Military Experience in World War I. New York: Oxford University Press, 1968.
- Cooper, John Milton. Breaking the Heart of the World: Woodrow Wilson and the Fight for the League of Nations. New York: Cambridge University Press, 2001.
- Dawley, Alan. Changing the World: American Progressives in War and Revolution. Princeton, NJ: Princeton University Press, 2003.
- Doenecke, Justus D. Nothing Less than War: A New History of America’s Entry into World War I. Lexington: University Press of Kentucky, 2011.
- Freeberg, Ernest. Democracy’s Prisoners: Eugene V. Debs, the Great War, and the Right to Dissent. Cambridge, MA: Harvard University Press, 2008.
- Fussell, Paul. The Great War and Modern Memory. New York: Oxford University Press, 1975.
- Gerwarth, Robert, and Erez Manela, eds. Empires at War: 1911–1923. New York: Oxford University Press, 2015.
- Greenwald, Maurine W. Women, War, and Work: The Impact of World War I on Women Workers in the United States. Westport, CT: Greenwood, 1980.
- Hahn, Steven. A Nation Under Our Feet: Black Political Struggles in the Rural South from Slavery to the Great Migration. Cambridge, MA: Harvard University Press, 2003.
- Hawley, Ellis. The Great War and the Search for Modern Order. New York: St. Martin’s Press, 1979.
- Jensen, Kimberly. Mobilizing Minerva: American Women in the First World War. Champaign: University of Illinois Press, 2009.
- Keene, Jennifer. Doughboys, The Great War, and the Remaking of America. Baltimore: Johns Hopkins University Press, 2001.
- Kennedy, David. Over Here: The First World War and American Society. New York: Oxford University Press, 1980.
- Knock, Thomas J. To End All Wars: Woodrow Wilson and the Quest for a New World Order. New York: Oxford University Press, 1992.
- MacMillan, Margaret. The War That Ended Peace: The Road to 1914. New York: Random House, 2014.
- Manela, Erez. The Wilsonian Movement: Self-Determination and the International Origins of Anticolonial Nationalism. New York: Oxford University Press, 2007.
- Montgomery, David. The Fall of the House of Labor: The Workplace, the State, and American Labor Activism, 1865–1925. Cambridge, UK: Cambridge University Press, 1988.
- Murphy, Paul. World War I and the Origins of Civil Liberties in the United States. New York: Norton, 1979.
- Neiberg, Michael S. The Path to War: How the First World War Created Modern America. New York: Oxford University Press, 2016.
- Rosenberg, Emily. Financial Missionaries to the World: The Politics and Culture of Dollar Diplomacy, 1900–1930. Durham, NC: Duke University Press, 2003.
- Smith, Tony. Why Wilson Matters: The Origin of American Liberal Internationalism and Its Crisis Today. Princeton, NJ: Princeton University Press, 2017.
- Tuttle, William. Race Riot: Chicago in the Red Summer of 1919. Champaign: University of Illinois Press, 1970.
- Wilkerson, Isabel. The Warmth of Other Sons: The Epic Story of America’s Great Migration. New York: Vintage Books, 2010.
- Williams, Chad L. Torchbearers of Democracy: African American Soldiers in the World War I Era. Chapel Hill: University of North Carolina Press, 2010.
Notes
- David Stevenson, The First World War and International Politics (London: Oxford University Press, 1988); David Stevenson, Cataclysm: The First World War as Political Tragedy (New York: Basic Books, 2004).
- George Washington, Farewell Address, Annals of Congress , 4th Congress, 2869–2870.
- Paul Koistinen, Mobilizing for Modern War: The Political Economy of American Warfare, 1865–1919 (Lawrence: University Press of Kansas, 1997).
- John S. D. Eisenhower, Intervention! The United States and the Mexican Revolution, 1913–1917 (New York: Norton, 1995); Friedrich Katz, The Secret War in Mexico: Europe, the United States, and the Mexican Revolution (Chicago: University of Chicago Press, 1981)
- Arthur S. Link, Wilson: The Struggle for Neutrality, 1914–1915 (Princeton, NJ: Princeton University Press, 1960).
- Michael S. Neiberg, Fighting the Great War: A Global History (Cambridge, MA: Harvard University Press, 2005).
- American Federation of Labor, Report of the Proceedings of the Annual Convention (Washington, DC: Law Reporter, 1917), 112.
- Christopher Capozzola, Uncle Sam Wants You: World War I and the Making of the Modern American Citizen (New York: Oxford University Press, 2010)
- Albert Gallitin Love, Defects Found in Drafted Men (Washington, DC: U.S. Government Printing Office, 1920), 73
- Dawley, Changing the World .
- Susan Zeiger, In Uncle Sam’s Service: Women Workers with the American Expeditionary Force, 1917–1919 (Philadelphia: University of Pennsylvania Press, 2004), 2–4.
- Lettie Gavin, American Women in World War I: They Also Served (Boulder: University Press of Colorado, 1997); Kimberly Jensen, Mobilizing Minerva: American Women in the First World War (Chicago: University of Illinois Press, 2008), 170–172.
- Gavin, American Women , 129–240.
- Nikki Brown, Private Politics and Public Voices: Black Women’s Activism from World War I to the New Deal (Bloomington: Indiana University Press, 2006), 66–107
- David Kennedy, Over Here: The First World War and American Society (New York: Oxford University Press, 1980)
- Neiberg, Fighting the Great War
- Ibid.
- Ibid.
- Ibid.
- Nancy K. Bristow, American Pandemic: The Lost Worlds of the 1918 Influenza Epidemic (New York: Oxford University Press, 2012); Alfred W. Crosby, America’s Forgotten Pandemic: The Influenza of 1918 (Cambridge, UK: Cambridge University Press, 2003).
- Bristow, American Pandemic; Crosby, America’s Forgotten Pandemic .
- Dawley, Changing the World ).
- Thomas J. Knock, To End All Wars: Woodrow Wilson and the Quest for a New World Order (New York: Oxford University Press, 1992).
- John Milton Cooper, Breaking the Heart of the World: Woodrow Wilson and the Fight for the League of Nations (Cambridge, UK: Cambridge University Press, 2001).
- Ibid.
- David Fromkin, A Peace to End All Peace: The Fall of the Ottoman Empire and the Creation of the Modern Middle East (New York: Holt, 1989).
- Moshik Temkin, The Sacco-Vanzetti Affair: America on Trial (New Haven, CT: Yale University Press, 2009).
- Isabel Wilkerson, The Warmth of Other Suns: The Epic Story of America’s Great Migration (New York: Vintage Books, 2011).
- W. E. B. Du Bois, “Returning Soldiers,” The Crisis (May 1919): 14.
- William Tuttle, Race Riot: Chicago in the Red Summer of 1919 (Champaign: University of Illinois Press, 1970); Cameron McWhirter, Red Summer: The Summer of 1919 and the Awakening of Black America (New York: Holt, 2011).


9.3: §63. Latin Verbs of the First Conjugation
Like Latin nouns, Latin verbs can be grouped by pattern or type, so as to make them much easier to learn. In Chapters 2 and 3, we learned nouns in groups that we called DECLENSIONS. For some two thousand years, Latin verb groups have been known as CONJUGATIONS (literally, forms “joined together”). Don’t let the label alarm you; once again, we are merely talking about separate types or categories.
The 1st conjugation is an ideal place to begin, since it is the easiest and most regular. It is not the category to which vidē re belongs; all verbs in this first class will have a PRESENT INFINITIVE in – ā re , like voc ā re (“to call”) or port ā re (“to carry”). Because the – ā – of that infinitive ending is a long vowel, we should pronounce the form as vocā́ re . (Don’t use those marks in writing the word; they are merely pronunciation guides.) The regular simplicity of the 1st conjugation is best seen in the PERFECT PARTICIPLE, which almost always assumes a form like voc ā tus or port ā tus . That reassuring fact makes this conjugation thoroughly predictable—a great blessing, since it is the source of at least a thousand English words. Remember it, then, as the “ vowel A ” conjugation, characterized by the two forms -ARE and -ATUS .
| cantare, cantatus | sing | putare, putatus | think, reckon |
| clamare, clamatus | shout | secare, sectus | cut |
| mutare, mutatus | change | servare, servatus | save |
| plicare, plicatus | fold | stare, status | stand |
| portare, portatus | carry | vocare, vocatus | call |
This is a mere sampling of literally hundreds of Latin verbs that follow this pattern. Just to keep you honest, we have included one that departs from type: the perfect participle of secare does not show the usual -atus ending, but instead is sectus . If you apply the memory trick given in §62, you can easily remember sectus from the English word section (“something cut”).
At first glance, you may not spot a great many English derivatives from the verbs in Table 9.1. The real linguistic dividends start to accrue when you combine these verb bases with the prefixes of the last chapter (§59). Though portare by itself may suggest only the French-derived portage (< late Latin portaticum ), the prefixes will soon bring to mind such words as import , important , export , report , deportation (a “carrying away”), disport (to “carry apart” from work)—the source, believe it or not, of our common word sport . A similar assault on putare will yield compute, computer, deputation, dispute, impute, reputable, disreputable, and reputation, as well as more learned words like putative . In earlier chapters we saw that the scope of Latin noun and adjective derivatives is rich and varied; but the extent of Latin verb derivatives in English is simply mind-boggling.
If you know any French, you will recognize cantare as the source of chanter; indeed, it is usual for 1st conjugation Latin verbs in -are to become 1st conjugation French verbs in -er . Thus English chant can be seen as an Old French loan-word, whereas recant and incantation come directly from Latin. [1] Therefore enchant and incant are perfect doublets, while enchantment and incantation are closely related. You may be startled to realize that a sea shanty and a Bach cantata are brothers under the skin.
Don’t be surprised if you find spelling changes in Latin verb derivatives. Our verbs claim (“shout”), exclaim (“shout out”), proclaim (“shout forth”) and reclaim (“shout back”) are all derivatives of clamare . But observe how the corresponding noun derivatives, all from the perfect participle, show the correct Latin spelling: exclamation, proclamation, reclamation. A parallel is seen in derivatives of vocare . From the present stem we get convoke (“call together”), evoke (“call out ”), invoke (“call in”), provoke (“call forth”), and revoke (“call back), whereas the perfect participle provides us with convocation , invocation , and provocation . (We’ll meet later such interesting forms as irrevocable .) Notice, by the way, that reclaim and revoke are very close in etymological meaning, and do have some kinship in meaning today.
Sometimes the English derivatives can be thoroughly disguised. From plicare you would soon discover complicate (“fold together”), implicate (“fold in”) and explication (“a folding out”), and you might then track down implicit and explicit ; but you would need an etymological dictionary to uncover plait , pleat , ply (all < plicatus ), imply , employ (both < implicare ), deploy , display (both < displicare ), apply , appliqu é (both < applicatus < ad-plicatus ), and the noun exploit (< explicatus ). If you then learned that the root plic- occurs also in complex , accomplice , simplicity , and multiply (= Germanic manifold ), you would realize that you had unfolded a very complicated pattern.
The first time around, you are likely to make a few mistakes. You would be wrong to think that servare is the source of English serve (< L servire ); but it is the origin of conserve, conservation, preserve, reservation, reservoir, etc. The important Latin verb stare , status is cognate with—but is not the source of—the Germanic English verb stand ; it has given us the derivatives status , station , stature , constant , instant , stance , and inconstancy . There is absolutely no way that you can grasp all these words at once; some of them will be carefully explained in later chapters. The only expectation at present is that you will try to learn the two necessary Latin forms and the basic English meanings for each of the verbs on the table, making intelligent use of any English clues that you can spot.
To review and stress that all-important key to the perfect participle, here is an “ -ion ” guide to the ten 1st conjugation verbs on Table 9.1: incantation, exclamation, mutation, complication, deportation, reputation, section, conservation, station, and vocation.
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Sometimes English will “correct” or “re-Latinize” the spelling of a word taken from French. The term
descant
, for instance, was derived from L
discantus
(“singing apart”) through F
deschant
. ↵


2.4: Conclusion
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Diplomacy in the modern era, an era sometimes called the ‘long peace’ (Gaddis 1989) due to the absence of major war since 1945, has deepened and widened in complexity. Nowadays, it would be ill advised to base a description of diplomacy on actions short of, or in response to, war between states. Diplomacy today is integral to ensuring that our period of long peace gets longer and that the world we live in is as conducive as possible to the progress of the individual, as well as the state. As today’s world is more linked and interdependent than ever before, effective and skilful diplomacy is vital to ensure that humankind can navigate an ever-growing list of shared challenges such as climate change, pandemics, transnational terrorism and nuclear proliferation that may be our undoing if left unresolved. So, while you may not know the names of many of those engaged in diplomatic endeavours, nor see much of their hard work credited in the media, their work is more important than ever to all of us.


27.4.1: Call by Value vs. Call by Reference
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Overview
In call by value , a parameter acts within the function as a new local variable initialized to the value of the argument (a local (isolated) copy of the argument). In call by reference , the argument variable supplied by the caller can be affected by actions within the called function. [1]
Discussion
Call by Value
Within most current programming languages, parameters are passed by value by default, with the argument as a copy of the calling value. Arguments are isolated, and functions are free to make changes to parameter values without any risk of impact to the calling function. Consider the following pseudocode:
Function Main Declare Real fahrenheit Assign fahrenheit = 100 Output "Main fahrenheit = " & fahrenheit Call ChangeFahrenheit(fahrenheit) Output "Main fahrenheit = " & fahrenheit End Function ChangeFahrenheit (Real fahrenheit) Output "ChangeFahrenheit fahrenheit = " & fahrenheit Assign fahrenheit = 0 Output "ChangeFahrenheit fahrenheit = " & fahrenheit End
Output
Main fahrenheit = 100 ChangeFahrenheit fahrenheit = 100 ChangeFahrenheit fahrenheit = 0 Main fahrenheit = 100
In English, the Main function assigns the value 100 to the variable fahrenheit, displays that value, and then calls ChangeFahrenheit passing a copy of that value. The called function displays the argument, changes it, and displays it again. Execution returns to the calling function, and Main displays the value of the original variable. With call by value, the variable fahrenheit in the calling function and the parameter fahrenheit in the called function refer to different memory addresses, and the called function cannot change the value of the variable in the calling function.
Call by Reference
If a programming language uses or supports call by reference, the variable in the calling function and the parameter in the called function refer to the same memory address, and the called function may change the value of the variable in the calling function. Using the same code example as above, call by reference output would change to:
Main fahrenheit = 100 ChangeFahrenheit fahrenheit = 100 ChangeFahrenheit fahrenheit = 0 Main fahrenheit = 0
Programming languages that support both call by value and call by reference use some type of key word or symbol to indicate which parameter passing method is being used.
| Language | Call By Value | Call by Reference |
|---|---|---|
| C++ | default |
use
¶meter
in called function
|
| C# | default |
use
ref parameter
in calling and called functions
|
| Java | default | applies to arrays and objects |
| JavaScript | default | applies to arrays and objects |
| Python | default | applies to arrays (lists) and mutable objects |
Arrays and objects are covered in later chapters.
Key Terms
- call by reference
- Parameters passed by calling functions may be modified by called functions.
- call by value
- Parameters passed by calling functions cannot be modified by called functions.


3.E: Orbits and Gravity (Exercises)
For Further Exploration
Articles
Brahe and Kepler
Christianson, G. “The Celestial Palace of Tycho Brahe.” Scientific American (February 1961): 118.
Gingerich, O. “Johannes Kepler and the Rudolphine Tables.” Sky & Telescope (December 1971): 328. Brief article on Kepler’s work.
Wilson, C. “How Did Kepler Discover His First Two Laws?” Scientific American (March 1972): 92.
Newton
Christianson, G. “Newton’s Principia : A Retrospective.” Sky & Telescope (July 1987): 18.
Cohen, I. “Newton’s Discovery of Gravity.” Scientific American (March 1981): 166.
Gingerich, O. “Newton, Halley, and the Comet.” Sky & Telescope (March 1986): 230.
Sullivant, R. “When the Apple Falls.” Astronomy (April 1998): 55. Brief overview.
The Discovery of Neptune
Sheehan, W., et al. “The Case of the Pilfered Planet: Did the British Steal Neptune?” Scientific American (December 2004): 92.
Websites
Brahe and Kepler
Johannes Kepler: His Life, His Laws, and Time: kepler.nasa.gov/Mission/JohannesKepler/. From NASA’s Kepler mission.
Johannes Kepler: http://www.britannica.com/biography/Johannes-Kepler . Encyclopedia Britannica article.
Johannes Kepler: www-history.mcs.st-andrews.ac...es/Kepler.html. MacTutor article with additional links.
Noble Dane: Images of Tycho Brahe: http://www.mhs.ox.ac.uk/tycho/index.htm . A virtual museum exhibit from Oxford.
Newton
Sir Isaac Newton: www-groups.dcs.st-and.ac.uk/~...es/Newton.html. MacTutor article with additional links.
Sir Isaac Newton: http://www.luminarium.org/sevenlit/n.../newtonbio.htm . Newton Biography at the Luminarium.
The Discovery of Neptune
Adams, Airy, and the Discovery of Neptune: http://www.mikeoates.org/lassell/adams-airy.htm . A defense of Airy’s role by historian Alan Chapman.
Mathematical Discovery of Planets: www-groups.dcs.st-and.ac.uk/~...and_Pluto.html. MacTutor article.
Videos
Brahe and Kepler
“Harmony of the Worlds.” This third episode of Carl Sagan’s TV series Cosmos focuses on Kepler and his life and work.
Tycho Brahe, Johannes Kepler, and Planetary Motion: https://www.youtube.com/watch?v=x3ALuycrCwI . German-produced video, in English (14:27).
Newton
Beyond the Big Bang: Sir Isaac Newton’s Law of Gravity: http://www.history.com/topics/enligh...law-of-gravity . From the History Channel (4:35).
Sir Isaac Newton versus Bill Nye: Epic Rap Battles of History: https://www.youtube.com/watch?v=8yis7GzlXNM . (2:47).
The Discovery of Neptune
Richard Feynman: On the Discovery of Neptune: https://www.youtube.com/watch?v=FgXQffVgZRs . A brief black-and-white Caltech lecture (4:33).
Collaborative Group Activities
- An eccentric, but very rich, alumnus of your college makes a bet with the dean that if you drop a baseball and a bowling ball from the tallest building on campus, the bowling ball would hit the ground first. Have your group discuss whether you would make a side bet that the alumnus is right. How would you decide who is right?
- Suppose someone in your astronomy class was unhappy about his or her weight. Where could a person go to weigh one-fourth as much as he or she does now? Would changing the unhappy person’s weight have any effect on his or her mass?
- When the Apollo astronauts landed on the Moon, some commentators commented that it ruined the mystery and “poetry” of the Moon forever (and that lovers could never gaze at the full moon in the same way again). Others felt that knowing more about the Moon could only enhance its interest to us as we see it from Earth. How do the various members of your group feel? Why?
- Figure \(3.5.2\) shows a swarm of satellites in orbit around Earth. What do you think all these satellites do? How many categories of functions for Earth satellites can your group come up with?
- The Making Connections feature box Astronomy and the Poets discusses how poets included the most recent astronomical knowledge in their poetry. Is this still happening today? Can your group members come up with any poems or songs that you know that deal with astronomy or outer space? If not, perhaps you could find some online, or by asking friends or roommates who are into poetry or music.
Review Questions
- State Kepler’s three laws in your own words.
- Why did Kepler need Tycho Brahe’s data to formulate his laws?
- Which has more mass: an armful of feathers or an armful of lead? Which has more volume: a kilogram of feathers or a kilogram of lead? Which has higher density: a kilogram of feathers or a kilogram of lead?
- Explain how Kepler was able to find a relationship (his third law) between the orbital periods and distances of the planets that did not depend on the masses of the planets or the Sun.
- Write out Newton’s three laws of motion in terms of what happens with the momentum of objects.
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Which major planet has the largest . . .
- semimajor axis?
- average orbital speed around the Sun?
- orbital period around the Sun?
- eccentricity?
- Why do we say that Neptune was the first planet to be discovered through the use of mathematics?
- Why was Brahe reluctant to provide Kepler with all his data at one time?
- According to Kepler’s second law, where in a planet’s orbit would it be moving fastest? Where would it be moving slowest?
- The gas pedal, the brakes, and the steering wheel all have the ability to accelerate a car—how?
- Explain how a rocket can propel itself using Newton’s third law.
- A certain material has a mass of 565 g while occupying 50 cm3 of space. What is this material? (Hint: Use Table \(3.2.1\).)
- To calculate the momentum of an object, which properties of an object do you need to know?
- To calculate the angular momentum of an object, which properties of an object do you need to know?
- What was the great insight Newton had regarding Earth’s gravity that allowed him to develop the universal law of gravitation?
- Which of these properties of an object best quantifies its inertia: velocity, acceleration, volume, mass, or temperature?
- Pluto’s orbit is more eccentric than any of the major planets. What does that mean?
- Why is Tycho Brahe often called “the greatest naked-eye astronomer” of all time?
Thought Questions
- Is it possible to escape the force of gravity by going into orbit around Earth? How does the force of gravity in the International Space Station (orbiting an average of 400 km above Earth’s surface) compare with that on the ground?
- What is the momentum of an object whose velocity is zero? How does Newton’s first law of motion include the case of an object at rest?
- Evil space aliens drop you and your fellow astronomy student 1 km apart out in space, very far from any star or planet. Discuss the effects of gravity on each of you.
- A body moves in a perfectly circular path at constant speed. Are there forces acting in such a system? How do you know?
- As friction with our atmosphere causes a satellite to spiral inward, closer to Earth, its orbital speed increases. Why?
- Use a history book, an encyclopedia, or the internet to find out what else was happening in England during Newton’s lifetime and discuss what trends of the time might have contributed to his accomplishments and the rapid acceptance of his work.
- Two asteroids begin to gravitationally attract one another. If one asteroid has twice the mass of the other, which one experiences the greater force? Which one experiences the greater acceleration?
- How does the mass of an astronaut change when she travels from Earth to the Moon? How does her weight change?
- If there is gravity where the International Space Station (ISS) is located above Earth, why doesn’t the space station get pulled back down to Earth?
- Compare the density, weight, mass, and volume of a pound of gold to a pound of iron on the surface of Earth.
- If identical spacecraft were orbiting Mars and Earth at identical radii (distances), which spacecraft would be moving faster? Why?
Figuring for Yourself
- By what factor would a person’s weight be increased if Earth had 10 times its present mass, but the same volume?
- Suppose astronomers find an earthlike planet that is twice the size of Earth (that is, its radius is twice that of Earth’s). What must be the mass of this planet such that the gravitational force ( F gravity) at the surface would be identical to Earth’s?
- What is the semimajor axis of a circle of diameter 24 cm? What is its eccentricity?
- If 24 g of material fills a cube 2 cm on a side, what is the density of the material?
- If 128 g of material is in the shape of a brick 2 cm wide, 4 cm high, and 8 cm long, what is the density of the material?
- If the major axis of an ellipse is 16 cm, what is the semimajor axis? If the eccentricity is 0.8, would this ellipse be best described as mostly circular or very elongated?
- What is the average distance from the Sun (in astronomical units) of an asteroid with an orbital period of 8 years?
- What is the average distance from the Sun (in astronomical units) of a planet with an orbital period of 45.66 years?
- In 1996, astronomers discovered an icy object beyond Pluto that was given the designation 1996 TL 66. It has a semimajor axis of 84 AU. What is its orbital period according to Kepler’s third law?


4.2: The SIS Model
The SI model may be extended to the SIS model, where an infective can recover and become susceptible again. We assume that the probability that an infective recovers during time \(\Delta t\) is given by \(\gamma \Delta t\) . Then the total number of infective people that recover during time \(\Delta t\) is given by \(I \times \gamma \Delta t\) , and
\[I(t+\Delta t)=I(t)+\beta \Delta t S(t) I(t)-\gamma \Delta t I(t) \nonumber \]
or as \(\Delta t \rightarrow 0\)
\[\frac{d I}{d t}=\beta S I-\gamma I \nonumber \]
which we diagram as
\[S \underset{\gamma I}{\stackrel{\beta S I}{\rightleftharpoons}} I \text {. } \nonumber \]
Using \(S+I=N\) , we eliminate \(S\) from (4.2.2) to obtain
\[\frac{d I}{d t}=(\beta N-\gamma) I\left(1-\frac{\beta}{\beta N-\gamma} I\right) \nonumber \]
which is again a logistic equation, but now with growth rate \(\beta N-\gamma\) and carrying capacity \(N-\gamma / \beta\) . In the SIS model, an epidemic will occur if \(\beta N>\gamma\) . And if an epidemic does occur, then the disease becomes endemic with the number of infectives at equilibrium given by \(I_{*}=N-\gamma / \beta\) , and the number of susceptibles given by \(S_{*}=\gamma / \beta\) .
In general, an important metric for whether or not an epidemic will occur is called the basic reproductive ratio. The basic reproductive ratio is defined as the expected number of people that a single infective will infect in an otherwise susceptible population. To compute the basic reproductive ratio, define \(l(t)\) to be the probability that an individual initially infected at \(t=0\) is still infective at time \(t\) . Since the probability of being infective at time \(t+\Delta t\) is equal to the probability of being infective at time \(t\) multiplied by the probability of not recovering during time \(\Delta t\) , we have
\[l(t+\Delta t)=l(t)(1-\gamma \Delta t) \nonumber \]
or as \(\Delta t \rightarrow 0\)
\[\frac{d l}{d t}=-\gamma l \nonumber \]
With initial condition \(l(0)=1\)
\[l(t)=e^{-\gamma t} \nonumber \]
Now, the expected number of secondary infections produced by a single primary infective over the time period \((t, t+\Delta t)\) is given by the probability that the primary infective is still infectious at time \(t\) multiplied by the expected number of secondary infections produced by a single infective during time \(\Delta t\) ; that is, \(l(t) \times S(t) \beta \Delta t\) . Here, the definition of the basic reproductive ratio assumes that the entire population is susceptible so that \(S(t)=N\) . Therefore, the expected number of secondary infectives produced by a single primary infective in a completely susceptible population is
\[\begin{aligned} \int_{0}^{\infty} \beta l(t) N d t &=\beta N \int_{0}^{\infty} e^{-\gamma t} d t \\[4pt] &=\frac{\beta N}{\gamma} \end{aligned} \nonumber \]
The basic reproductive ratio, written as \(\mathcal{R}_{0}\) , is therefore defined as
\[\mathcal{R}_{0}=\frac{\beta N}{\gamma} \nonumber \]
and from (4.2.4), we can see that in the SIS model an epidemic will occur if \(R_{0}>1\) . In other words, an epidemic can occur if an infected individual in an otherwise susceptible population will on average infect more than one other individual.
We have also seen an analogous definition of the basic reproductive ratio in our previous discussion of age-structured populations \((\S 2.5)\) . There, the basic reproductive ratio was the number of female offspring expected from a new born female over her lifetime; the population size would grow if this value was greater than unity.
In the SIS model, after an epidemic occurs the population reaches an equilibrium between susceptible and infective individuals. The effective basic reproductive ratio of this steady-state population can be defined as \(\beta S_{*} / \gamma\) , and with \(S_{*}=\gamma / \beta\) this ratio is evidently unity. Clearly, for a population to be in equilibrium, an infective individual must infect on average one other individual before he or she recovers.


13.9.2: Multiply and Divide Rational Expressions
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Learning Objectives
By the end of this section, you will be able to:
- Multiply rational expressions
- Divide rational expressions
Note
Before you get started, take this readiness quiz.
If you miss a problem, go back to the section listed and review the material.
-
Multiply: \(\frac{14}{15}·\frac{6}{35}\).
If you missed this problem, review Exercise 1.6.13 . -
Divide: \(\frac{14}{15}÷\frac{6}{35}\).
If you missed this problem, review Exercise 1.6.22 . -
Factor completely: \(2x^2−98\).
If you missed this problem, review Exercise 7.5.10 . -
Factor completely: \(10n^3+10\).
If you missed this problem, review Exercise 7.5.19 . -
Factor completely: \(10p^2−25pq−15q^2\).
If you missed this problem, review Exercise 7.5.28 .
Multiply Rational Expressions
To multiply rational expressions, we do just what we did with numerical fractions. We multiply the numerators and multiply the denominators. Then, if there are any common factors, we remove them to simplify the result.
Definition: MULTIPLICATION OF RATIONAL EXPRESSIONS
If p,q,r,s are polynomials where \(q \ne 0\) and \(s \ne 0\)
\(\frac{p}{q}·\frac{r}{s}=\frac{pr}{qs}\)To multiply rational expressions, multiply the numerators and multiply the denominators.
We’ll do the first example with numerical fractions to remind us of how we multiplied fractions without variables.
Example \(\PageIndex{1}\)
Multiply: \(\frac{10}{28}·\frac{8}{15}\).
- Answer
-
Multiply the numerators and denominators. Look for common factors, and then remove them. Simplify.
Example \(\PageIndex{2}\)
Mulitply: \(\frac{6}{10}·\frac{15}{12}\).
- Answer
-
\(\frac{3}{4}\)
Example \(\PageIndex{3}\)
Mulitply: \(\frac{20}{15}·\frac{6}{8}\).
- Answer
-
1
Remember, throughout this chapter, we will assume that all numerical values that would make the denominator be zero are excluded. We will not write the restrictions for each rational expression, but keep in mind that the denominator can never be zero. So in this next example, \(x \ne 0\) and \(y \ne 0\).
Example \(\PageIndex{4}\)
Mulitply: \(\frac{2x}{3y^2}·\frac{6xy^3}{x^{2}y}\).
- Answer
-
Multiply. Factor the numerator and denominator completely, and then remove common factors. Simplify.
Example \(\PageIndex{5}\)
Mulitply: \(\frac{3pq}{q^2}·\frac{5p^{2}q}{6pq}\).
- Answer
-
\(\frac{5p^2}{2q}\)
Example \(\PageIndex{6}\)
Mulitply: \(\frac{6x^{3}y}{7x^2}·\frac{2xy}{3x^{2}y}\).
- Answer
-
\(\frac{12y^3}{7}\)
How to Multiply Rational Expressions
Example \(\PageIndex{7}\)
Mulitply: \(\frac{2x}{x^2-7x+12}·\frac{x^2−9}{6x^2}\).
- Answer
-
Example \(\PageIndex{8}\)
Mulitply: \(\frac{5x}{x^2+5x+6}·\frac{x^2−4}{10x}\).
- Answer
-
\(\frac{x−2}{2(x+3)}\)
Example \(\PageIndex{9}\)
Mulitply: \(\frac{9x^2}{x^2+11x+30}·\frac{x^2−36}{3x^2}\).
- Answer
-
\(\frac{3(x−6)}{x+5}\)
Definition: MULTIPLY A RATIONAL EXPRESSION.
- Factor each numerator and denominator completely.
- Multiply the numerators and denominators.
- Simplify by dividing out common factors.
Example \(\PageIndex{10}\)
Multiply: \(\frac{n^2−7n}{n^2+2n+1}·\frac{n+1}{2n}\).
- Answer
-
\(\frac{n^2−7n}{n^2+2n+1}·\frac{n+1}{2n}\) Factor each numerator and denominator. \(\frac{n(n−7)}{(n+1)(n+1)}·\frac{n+1}{2n}\) Multiply the numerators and the denominators. \(\frac{n(n−7)(n+1)}{(n+1)(n+1)2n}\) Simplify. \(\frac{n−7}{2(n+1)}\)
Example \(\PageIndex{11}\)
Multiply: \(\frac{x^2−25}{x^2−3x−10}·\frac{x+2}{x}\).
- Answer
-
\(\frac{x+5}{x}\)
Example \(\PageIndex{12}\)
Multiply: \(\frac{x^2−4x}{x^2+5x+6}·\frac{x+2}{x}\).
- Answer
-
\(\frac{x−4}{x+3}\)
Example \(\PageIndex{13}\)
Multiply: \(\frac{16−4x}{2x−12}·\frac{x^2−5x−6}{x^2−16}\).
- Answer
-
\(\frac{16−4x}{2x−12}·\frac{x^2−5x−6}{x^2−16}\) Factor each numerator and denominator. \(\frac{4(4−x)}{2(x−6)}·\frac{(x−6)(x+1)}{(x−4)(x+4)}\) Multiply the numerators and the denominators. \(\frac{4(4−x)(x−6)(x+1)}{2(x−6)(x−4)(x+4)}\) Simplify. \(−\frac{2(x+1)}{(x+4)}\)
Example \(\PageIndex{14}\)
Multiply: \(\frac{12x−6x^2}{x^2+8x}·\frac{x^2+11x+24}{x^2−4}\).
- Answer
-
\(−\frac{6(x+3)}{x+2}\)
Example \(\PageIndex{15}\)
Multiply: \(\frac{9v−3v^2}{9v+36}·\frac{v^2+7v+12}{v^2−9}\).
- Answer
-
\(−\frac{v}{3}\)
Example \(\PageIndex{16}\)
Multiply: \(\frac{2x−6}{x^2−8x+15}·\frac{x^2−25}{2x+10}\).
- Answer
-
Factor each numerator and denominator. Multiply the numerators and denominators. Remove common factors. Simplify.
Example \(\PageIndex{17}\)
Multiply: \(\frac{3a−21}{a^2−9a+14}·\frac{a^2−4}{3a+6}\).
- Answer
-
1
Example \(\PageIndex{18}\)
Multiply: \(\frac{b^2−b}{b^2+9b−10}·\frac{b^2−100}{b^2−10b}\).
- Answer
-
1
Divide Rational Expressions
To divide rational expressions we multiply the first fraction by the reciprocal of the second, just like we did for numerical fractions.
Remember, the reciprocal of \(\frac{a}{b}\) is \(\frac{b}{a}\). To find the reciprocal we simply put the numerator in the denominator and the denominator in the numerator. We “flip” the fraction.
Definition: DIVISION OF RATIONAL EXPRESSIONS
If p,q,r,s are polynomials where \(q \ne 0\), \(r \ne 0\), \( s \ne 0\)
\(\frac{p}{q}÷\frac{r}{s}=\frac{p}{q}·\frac{s}{r}\)
To divide rational expressions multiply the first fraction by the reciprocal of the second.
How to Divide Rational Expressions
Example \(\PageIndex{19}\)
Divide: \(\frac{x+9}{6−x}÷\frac{x^2−81}{x−6}\).
- Answer
-
Example \(\PageIndex{20}\)
Divide: \(\frac{c+3}{5−c}÷\frac{c^2−9}{c−5}\).
- Answer
-
\(−\frac{1}{c−3}\)
Example \(\PageIndex{21}\)
Divide: \(\frac{2−d}{d−4}÷\frac{4−d^2}{4−d}\).
- Answer
-
\(−\frac{1}{2+d}\)
Definition: DIVIDE RATIONAL EXPRESSIONS.
- Rewrite the division as the product of the first rational expression and the reciprocal of the second.
- Factor the numerators and denominators completely.
- Multiply the numerators and denominators together.
- Simplify by dividing out common factors.
Example \(\PageIndex{22}\)
Divide: \(\frac{3n^2}{n^2−4n}÷\frac{9n^2−45n}{n^2−7n+10}\).
- Answer
-
Rewrite the division as the product of the first rational expression and the reciprocal of the second. Factor the numerators and denominators and then multiply. Simplify by dividing out common factors.
Example \(\PageIndex{23}\)
Divide: \(\frac{2m^2}{m^2−8m}÷\frac{8m^2+24m}{m^2+m−6}\).
- Answer
-
\(\frac{(m−2)}{4(m−8)}\)
Example \(\PageIndex{24}\)
Divide: \(\frac{15n^2}{3n^2+33n}÷\frac{5n−5}{n^2+9n−22}\).
- Answer
-
\(\frac{n(n−2)}{n−1}\)
Remember, first rewrite the division as multiplication of the first expression by the reciprocal of the second. Then factor everything and look for common factors.
Example \(\PageIndex{25}\)
Divide: \(\frac{2x^2+5x−12}{x^2−16}÷\frac{2x^2−13x+15}{x^2−8x+16}\).
- Answer
-
\(\frac{2x^2+5x−12}{x^2−16}÷\frac{2x^2−13x+15}{x^2−8x+16}\) Rewrite the division as the product of the first rational expression and the reciprocal of the second. \(\frac{2x^2+5x−12}{x^2−16}·\frac{x^2−8x+16}{2x^2−13x+15}\) Factor the numerators and denominators and then multiply. \(\frac{(2x−3)(x+4)(x−4)(x−4)}{(x−4)(x+4)(2x−3)(x−5)}\) Simplify. \(\frac{(x−4)}{(x−5)}\)
Example \(\PageIndex{26}\)
Divide: \(\frac{3a^2−8a−3}{a^2−25}÷\frac{3a^2−14a−5}{a^2+10a+25}\).
- Answer
-
\(\frac{(a−3)(a+5)}{(a−5)(a−5)}\)
Exercise \(\PageIndex{27}\)
Divide: \(\frac{4b^2+7b−2}{1−b^2}÷\frac{4b^2+15b−4}{b^2−2b+1}\).
- Answer
-
\(−\frac{(b+2)(b−1)}{(1+b)(b+4)}\)
Example \(\PageIndex{28}\)
Divide: \(\frac{p^3+q^3}{2p^2+2pq+2q^2}÷\frac{p^2−q^2}{6}\).
- Answer
-
\(\frac{p^3+q^3}{2p^2+2pq+2q^2}÷\frac{p^2−q^2}{6}\) Rewrite the division as the product of the first rational expression and the reciprocal of the second. \(\frac{p^3+q^3}{2p^2+2pq+2q^2}·\frac{6}{p^2−q^2}\) Factor the numerators and denominators and then multiply. \(\frac{(p+q)(p^2−pq+q^2)6}{2(p^2+pq+q^2)(p−q)(p+q)}\) Simplify. \(\frac{3(p^2−pq+q^2)}{(p−q)(p^2+pq+q^2)}\)
Example \(\PageIndex{29}\)
Divide: \(\frac{x^3−8}{3x^2−6x+12}÷\frac{x^2−4}{6}\).
- Answer
-
\(\frac{2(x^2+2x+4)}{(x+2)(x^2−2x+4)}\)
Example \(\PageIndex{30}\)
Divide: \(\frac{2z^2}{z^2−1}÷\frac{z^3−z^2+z}{z^3−1}\).
- Answer
-
\(\frac{2z(z^2+z+1)}{(z+1)(z^2−z+1)}\)
Before doing the next example, let’s look at how we divide a fraction by a whole number. When we divide \(\frac{3}{5}÷4\)
\[\begin{array}{c} {\frac{3}{5}÷4}\\ {\frac{3}{5}÷\frac{4}{1}}\\ {\frac{3}{5}·\frac{1}{4}}\\ \nonumber \end{array}\]
We do the same thing when we divide rational expressions.
Example \(\PageIndex{31}\)
\(\frac{a^2−b^2}{3ab}÷(a^2+2ab+b^2)\).
- Answer
-
\(\frac{a^2−b^2}{3ab}÷(a^2+2ab+b^2)\) Write the second expression as a fraction. \(\frac{a^2−b^2}{3ab}÷\frac{a^2+2ab+b^2}{1}\) Rewrite the division as the first expression times the reciprocal of the second expression. \(\frac{a^2−b^2}{3ab}·\frac{1}{a^2+2ab+b^2}\) Factor the numerators and the denominators, and then multiply. \(\frac{(a−b)(a+b)1}{3ab·(a+b)(a+b)}\) Simplify. \(\frac{a−b}{3ab(a+b)}\)
Example \(\PageIndex{32}\)
\(\frac{2x^2−14x−16}{4}÷(x2+2x+1)\).
- Answer
-
\(\frac{x−8}{2(x+1)}\)
Example \(\PageIndex{33}\)
\(\frac{y^2−6y+8}{y^2−4y}÷(3y2−12y)\).
- Answer
-
\(\frac{y−2}{3y(y−4)}\)
Example \(\PageIndex{34}\)
\(\frac{\frac{6x^2−7x+2}{4x−8}}{\frac{2x^2−7x+3}{x^2−5x+6}}\).
- Answer
-
\(\frac{\frac{6x^2−7x+2}{4x−8}}{\frac{2x^2−7x+3}{x^2−5x+6}}\) Rewrite with a division sign. \(\frac{6x^2−7x+2}{4x−8}÷\frac{2x^2−7x+3}{x^2−5x+6}\) Rewrite as product of first times reciprocal of second. \(\frac{6x^2−7x+2}{4x−8}·\frac{x^2−5x+6}{2x^2−7x+3}\) Factor the numerators and the denominators, and then multiply \(\frac{(2x−1)(3x−2)(x−2)(x−3)}{4(x−2)(2x−1)(x−3)}\) Simplify. \(\frac{3x−2}{4}\)
Example \(\PageIndex{35}\)
\(\frac{\frac{3x^2+7x+2}{4x+24}}{\frac{3x^2−14x−5}{x^2+x−30}}\).
- Answer
-
\(\frac{x+2}{4}\)
Example \(\PageIndex{36}\)
\(\frac{\frac{y^2−36}{2y^2+11y−6}}{\frac{2y^2−2y−60}{8y−4}}\).
- Answer
-
\(\frac{2}{y+5}\)
If we have more than two rational expressions to work with, we still follow the same procedure. The first step will be to rewrite any division as multiplication by the reciprocal. Then we factor and multiply.
Example \(\PageIndex{37}\)
\(\frac{3x−6}{4x−4}·\frac{x^2+2x−3}{x^2−3x−10}÷\frac{2x+12}{8x+16}\).
- Answer
-
Rewrite the division as multiplication by the reciprocal. Factor the numerators and the denominators, and then multiply. Simplify by dividing out common factors. Simplify.
Example \(\PageIndex{38}\)
\(\frac{4m+4}{3m−15}·\frac{m^2−3m−10}{m^2−4m−32}÷\frac{12m−36}{6m−48}\).
- Answer
-
\(\frac{2(m+1)(m+2)}{3(m+4)(m−3)}\)
Example \(\PageIndex{39}\)
\(\frac{2n^2+10n}{n−1}÷\frac{n^2+10n+24}{n^2+8n−9}·\frac{n+4}{8n^2+12n}\).
- Answer
-
\(\frac{(n+5)(n+9)}{2(n+6)(2n+3)}\)
Key Concepts
-
Multiplication of Rational Expressions
- If p,q,r,s are polynomials where \(q \ne 0\) and \(s \ne 0\), then \(\frac{p}{q}·\frac{r}{s}=\frac{pr}{qs}\)
- To multiply rational expressions, multiply the numerators and multiply the denominators
-
Multiply a Rational Expression
- Factor each numerator and denominator completely.
- Multiply the numerators and denominators.
- Simplify by dividing out common factors.
-
Division of Rational Expressions
- If p,q,r,s are polynomials where \(q \ne 0\), \(r \ne 0\), \( s \ne 0\), then \(\frac{p}{q}÷\frac{r}{s}=\frac{p}{q}·\frac{s}{r}\)
- To divide rational expressions multiply the first fraction by the reciprocal of the second.
-
Divide Rational Expressions
- Rewrite the division as the product of the first rational expression and the reciprocal of the second.
- Factor the numerators and denominators completely.
- Multiply the numerators and denominators together.
- Simplify by dividing out common factors.


12.6: The Competitive World of Retailing
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3. What are the different kinds of retail operations?
Some 15 million Americans are engaged in retailing. Of this number, almost half work in service businesses such as barbershops, lawyers’ offices, and amusement parks. Although most retailers are involved in small businesses, most sales are made by the giant retail organizations, such as Walmart, Target, and Macy’s. Half of all retail sales come from fewer than 10 percent of all retail businesses. This small group employs about 40 percent of all retail workers. Retailers feel the impact of changes in the economy more than many other types of businesses. Survival depends on keeping up with changing lifestyles and customer shopping patterns. In recent years, online retailing trends have significantly impacted retailing organizations, providing more opportunity for smaller retailers and more competition for larger retailers.
Types of Retail Operations
There is a great deal of variety in retail operations. The major types of retailers are described in Table 12.1 , which divides them into two main categories: in-store and nonstore retailing. Examples of in-store retailing include Walmart, Target, Macy’s, and Neiman Marcus. These retailers get most of their revenue from people who come to the store to buy what they want. Many in-store retailers also do some catalog and telephone sales.
| Retailing Takes Many Forms | ||
|---|---|---|
| Types of In-Store Retailing | Description | Examples |
| Department store | Houses many departments under one roof with each treated as a separate buying center to achieve economies of buying, promotion, and control | Macy’s, Nordstrom, Bloomingdale’s, Kohl’s |
| Specialty store | Specializes in a category of merchandise and carries a complete assortment | Toys “R” Us, Zales Jewelers |
| Convenience store | Offers convenience goods with long store hours and quick checkout | 7-Eleven, Circle K |
| Supermarket | Specializes in a wide assortment of food, with self-service | Safeway, Kroger, Winn-Dixie |
| Discount store | Competes on the basis of low prices and high turnover; offers few services | Walmart, Target |
| Off-price retailer | Sells at prices 25 percent or more below traditional department store prices in a spartan environment | TJ Maxx, HomeGoods |
| Factory outlet | Owned by manufacturer; sells closeouts, factory seconds, and canceled orders | Levi Strauss, Dansk |
| Catalog store | Sends catalogs to customers and displays merchandise in showrooms where customers can order from attached warehouse | Ikea |
| Types of Nonstore Retailing | Description | Examples |
| Vending machine | Sells merchandise by machine | Canteen |
| Direct selling | Sells face-to-face, usually in the person’s home | Avon, Amway |
| Direct-response marketing | Attempts to get immediate consumer sale through media advertising, catalogs, pop-up ads, or direct mail | K-Tel Music, Ronco |
| Home shopping networks | Selling via cable television | Home Shopping Network, QVC |
| Internet retailing (e-retailing) | Selling over the internet | Bluefly.com, landsend.com, gap.com, Amazon.com, Wayfair.com, Dell.com |
Table 12.1
Nonstore retailing includes vending, direct selling, direct-response marketing, home shopping networks, and internet retailing. Vending uses machines to sell food and other items, usually as a convenience in institutions such as schools and hospitals.
Atmosphere and Retail Image
In considering retailing as a distribution strategy (place in the 5Ps), it is important to understand that place includes more than channel members or logistics. It also includes atmospherics—the image of the actual retailing store (or, in the case of nonstore retailing, the platform from which the product is offered, such as a website or vending machine). An important task in retailing is to create this image. Marketers combine the store’s merchandise mix, service level, and atmosphere to make up a retail image. Atmosphere refers to the physical layout and décor of the store. They can create a relaxed or busy feeling, a sense of luxury, a friendly or cold attitude, and a sense of organization or clutter.
These are the most influential factors in creating a store’s atmosphere:
- Employee type and density: Employee type refers to an employee’s general characteristics—for instance, neat, friendly, knowledgeable, or service-oriented. Density is the number of employees per 1,000 square feet of selling space. A discount retailer such as Target has a low employee density that creates a “do-it-yourself” casual atmosphere.
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Merchandise type and density:
The type of merchandise carried and how it is displayed add to the atmosphere the retailer is trying to create. A prestigious retailer such as Saks or Nordstrom carries the best brand names and displays them in a neat, uncluttered arrangement. Other retailers such as Dollar Tree may display goods in a more cluttered, crowded, disheveled way because their target market (lower-income individuals) equates clutter with open markets (and with lower prices and “deals”).
Exhibit 12.6 Whether peering through department store windows, buying holiday gifts, or going on a spending spree, people love to shop. Shopping makes people feel good, and a growing body of research suggests that shopping activates key areas of the brain, boosting one’s mood—at least until the bill arrives. Feelings of pleasure and satisfaction derived from a buying binge may be linked to brain chemicals that produce a “shopping high.” How might retailers use atmosphere to stimulate consumers’ natural impulse to shop? (Montgomery County Planning Commission/ Flickr/ Attribution 2.0 Generic (CC BY 2.0))
- Fixture type and density: Fixtures can be elegant (rich woods) or trendy (chrome and smoked glass), or they can be old, beat-up tables, as in an antique store. The fixtures should be consistent with the general atmosphere the store is trying to create. By displaying its merchandise on tables and shelves rather than on traditional pipe racks, the Gap creates a relaxed and uncluttered atmosphere that enables customers to see and touch the merchandise more easily. In addition to traditional display racks, Cabela’s retail stores feature two 5,000-gallon aquariums stocked with carp, trout, and other fish and a diorama featuring elephants, lions, zebras, hyenas, and other animals. A typical Cabela’s has several million customers a year. It is not unusual for someone to drive many miles to get to a Cabela’s, where you can often see license plates from many states and Canadian provinces. 1
- Sound: Sound can be pleasant or unpleasant for a customer. Classical music at a nice Italian restaurant helps create ambiance, just as country and western music does at a truck stop. Music can also entice customers to stay in the store longer and buy more, or it can encourage them to eat quickly and leave a table for others.
- Odors: Smell can either stimulate or detract from sales. The wonderful smell of pastries and breads entices bakery customers, as does the smell of freshly brewed coffee in a shopping mall. Conversely, customers can be repulsed by bad odors, such as cigarette smoke, musty smells, antiseptic odors, and overly powerful room deodorizers.
EXPANDING AROUND THE GLOBE
Creative Retailing at Selfridges
To steer traffic to its flagship store in London, Selfridges sought divine intervention—that is, a 50-foot statue of Jesus. The small-scale replica of Rio de Janeiro’s famous monument gazed down on shoppers during a month-long Brazilian-themed promotion.
Combined with a radical redesign of the retail space that makes each of Selfridges’ four outlets feel more like a collection of quirky boutiques than one gargantuan marketplace, stunts like the Brazil 40° celebration have transformed the once-staid 95-year-old British retail chain into a premier arbiter of hip. Selfridges’ success has spurred retailers worldwide to take a closer look. “A department store chief who has not made his way to Selfridges to study its operation,” says Arnold Aronson, former CEO of Saks Fifth Avenue, “is an executive not doing his job.”
Typically, department stores develop their own merchandising strategies, resulting in a retail space crowded with Tommy Hilfiger, Ralph Lauren, and other predictable names arranged in displays that rarely vary from one chain to the next. Selfridges, however, operates on the theory that no one understands a product better than the designer or vendor that created it. So individual designers are allotted space in Selfridges and asked to create in-store displays that highlight their work. Traditional “departments” such as shoes, cosmetics, and men’s business wear have been organized by lifestyle—youth, sports, or women’s contemporary. This helps expose customers to merchandise they might not otherwise see.
Recently, Selfridges asked a tattoo and body-piercing parlor called Metal Morphosis to set up shop next to some women’s fashion vendors. Metal Morphosis was such a huge hit with shoppers en route to the clothing racks that it will soon expand to other Selfridges outlets.
Selfridges is also known for its “happenings.” They recently opened a low-cost interfaith charity shop within the confines of their luxury brand Oxford street store in London. Performance artist Miranda July was involved in the creation of this shop-within-a-shop, which partners with Islamic, Jewish, and other faith groups to promote the charity store. Ironically, shoppers can find bargain-priced donated blouses just feet away from some priced at over $3,000.
Critical Thinking Questions
- Selfridges opened a new store described as a “silver blob” or “spaceship.” The building has no straight lines and is covered with 15,000 anodized aluminum disks. The atrium is an array of high-gloss white elevators and balconies that are all slanted to avoid “the atrium look.” Do you think Selfridges is becoming too cool or hip? What impact will this have on sales?
- Would Selfridges be successful in the United States? Why or why not?
Sources: “The Secrets Behind Our House,” http://www.selfridges.com/US/en , accessed September 27, 2017; Barry Toberman, “Norwood Delight as Interfaith Shop at Selfridges Brings in the Punters,” The Jewish Chronicle, https://www.thejc.com , September 1, 2017; Hannah Ellis-Petersen, “Miranda July Curates Interfaith Charity Shop Opening up in Selfridges,” The Guardian, https://www.theguardian.com , August 30, 2017.
CONCEPT CHECK
- Describe at least five types of in-store retailing and four forms of nonstore retailing.
- What factors most influence a retail store’s atmosphere?


4.7: The Fall of Humanity (Genesis 2-21-3-24. The King James Version of the Bible)
2:21 And the LORD God caused a deep sleep to fall upon Adam, and he slept: and he took one of his ribs, and closed up the flesh instead thereof;
2:22 And the rib, which the LORD God had taken from man, made he a woman, and brought her unto the man.
2:23 And Adam said, This is now bone of my bones, and flesh of my flesh: she shall be called Woman, because she was taken out of Man.
2:24 Therefore shall a man leave his father and his mother, and shall cleave unto his wife: and they shall be one flesh.
2:25 And they were both naked, the man and his wife, and were not ashamed.
3:1 Now the serpent was more subtle than any beast of the field which the LORD God had made. And he said unto the woman, Yea, hath God said, Ye shall not eat of every tree of the garden?
3:2 And the woman said unto the serpent, We may eat of the fruit of the trees of the garden:
3:3 But of the fruit of the tree which is in the midst of the garden, God hath said, Ye shall not eat of it, neither shall ye touch it, lest ye die.
3:4 And the serpent said unto the woman, Ye shall not surely die:
3:5 For God doth know that in the day ye eat thereof, then your eyes shall be opened, and ye shall be as gods, knowing good and evil.
3:6 And when the woman saw that the tree was good for food, and that it was pleasant to the eyes, and a tree to be desired to make one wise, she took of the fruit thereof, and did eat, and gave also unto her husband with her; and he did eat.
3:7 And the eyes of them both were opened, and they knew that they were naked; and they sewed fig leaves together, and made themselves aprons.
3:8 And they heard the voice of the LORD God walking in the garden in the cool of the day: and Adam and his wife hid themselves from the presence of the LORD God amongst the trees of the garden.
3:9 And the LORD God called unto Adam, and said unto him, Where art thou?
3:10 And he said, I heard thy voice in the garden, and I was afraid, because I was naked; and I hid myself.
3:11 And he said, Who told thee that thou wast naked? Hast thou eaten of the tree, whereof I commanded thee that thou shouldest not eat?
3:12 And the man said, The woman whom thou gavest to be with me, she gave me of the tree, and I did eat.
3:13 And the LORD God said unto the woman, What is this that thou hast done? And the woman said, The serpent beguiled me, and I did eat.
3:14 And the LORD God said unto the serpent, Because thou hast done this, thou art cursed above all cattle, and above every beast of the field; upon thy belly shalt thou go, and dust shalt thou eat all the days of thy life:
3:15 And I will put enmity between thee and the woman, and between thy seed and her seed; it shall bruise thy head, and thou shalt bruise his heel.
3:16 Unto the woman he said, I will greatly multiply thy sorrow and thy conception; in sorrow thou shalt bring forth children; and thy desire shall be to thy husband, and he shall rule over thee.
3:17 And unto Adam he said, Because thou hast hearkened unto the voice of thy wife, and hast eaten of the tree, of which I commanded thee, saying, Thou shalt not eat of it: cursed is the ground for thy sake; in sorrow shalt thou eat of it all the days of thy life;
3:18 Thorns also and thistles shall it bring forth to thee; and thou shalt eat the herb of the field;
3:19 In the sweat of thy face shalt thou eat bread, till thou return unto the ground; for out of it wast thou taken: for dust thou art, and unto dust shalt thou return.
3:20 And Adam called his wife’s name Eve; because she was the mother of all living.
3:21 Unto Adam also and to his wife did the LORD God make coats of skins, and clothed them.
3:22 And the LORD God said, Behold, the man is become as one of us, to know good and evil: and now, lest he put forth his hand, and take also of the tree of life, and eat, and live for ever:
3:23 Therefore the LORD God sent him forth from the garden of Eden, to till the ground from whence he was taken.
3:24 So he drove out the man; and he placed at the east of the garden of Eden Cherubims, and a flaming sword which turned every way, to keep the way of the tree of life.


12: Nonparametric Statistics
Because distribution-free tests do not assume normality, they can be less susceptible to non-normality and extreme values. Therefore, they can be more powerful than the standard tests of means that assume normality.
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- 12.1: Benefits of Distribution Free Tests
- Tests assuming normality can have particularly low power when there are extreme values or outliers. A contributing factor is the sensitivity of the mean to extreme values. Although transformations can ameliorate this problem in some situations, they are not a universal solution. Tests assuming normality often have low power for leptokurtic distributions. Transformations are generally less effective for reducing kurtosis than for reducing skew.
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- 12.4: Randomization Association
- A significance test for Pearson's r is described in the section inferential statistics for b and r . The significance test described in that section assumes normality. This section describes a method for testing the significance of r that makes no distributional assumptions.
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- 12.5: Fisher's Exact Test
- The chapter on Chi Square showed one way to test the relationship between two nominal variables. A special case of this kind of relationship is the difference between proportions. This section shows how to compute a significance test for a difference in proportions using a randomization test.
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- 12.6: Rank Randomization Two Conditions
- The major problem with randomization tests is that they are very difficult to compute. Rank randomization tests are performed by first converting the scores to ranks and then computing a randomization test. The primary advantage of rank randomization tests is that there are tables that can be used to determine significance. The disadvantage is that some information is lost when the numbers are converted to ranks. Rank randomization tests are generally less powerful than randomization tests.
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- 12.12: Spearman Rank Correlation
- Use Spearman rank correlation when you have two ranked variables, and you want to see whether the two variables covary; whether, as one variable increases, the other variable tends to increase or decrease. You also use Spearman rank correlation if you have one measurement variable and one ranked variable; in this case, you convert the measurement variable to ranks and use Spearman rank correlation on the two sets of ranks.
Contributors and Attributions
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Online Statistics Education: A Multimedia Course of Study ( http://onlinestatbook.com/ ). Project Leader: David M. Lane, Rice University.


17: Molecular Cloning
Molecular cloning refers to the isolation and amplification of a specific DNA sequence in vivo . It uses a host organism (usually bacteria) to produce a large number of identical copies of a recombinant DNA sequence that can be used for other molecular biology applications.
Also known as
Cloning
Samples needed
Concentrated solutions of a cloning vector (plasmid) and DNA containing the sequence to be cloned. The vector needs to contain an origin of replication specific to the organism into which the vector will be transformed, along with a gene encoding a selectable marker (often an antibiotic resistance gene).
Method
There are many methods for cloning a piece of DNA, but we will focus here on a classical molecular cloning process using restriction enzymes. The principles apply to most other cloning methods. You can see a diagram depicting the steps to be described in Figure 7.1 in J.C. Lessard's chapter "Molecular Cloning." [1] First, the DNA to be cloned (called the insert) is obtained and isolated through PCR, DNA synthesis, or an alternative method. The vector and insert are then digested with restriction enzymes, producing ends that are complementary to each other. The digested DNA is purified. The digested vector and insert are incubated with DNA ligase and transformed into a host (usually bacteria). Host cells that take up the plasmid are selected for by plating on agar plates containing a chemical/compound that is toxic to cells lacking the plasmid. Cell colonies that grow on the plates can be expanded and the plasmid DNA can be purified from them. Restriction enzyme digestion (see interpretation), PCR, or DNA sequencing is often used to determine which plasmids contain the desired insert DNA.
Controls
Typically, a ligation reaction containing only the vector + DNA ligase is included, to make sure that bacterial colonies recovered after transformation don’t contain a religated vector without an insert.
Interpretation
In this cloning experiment, alkaline protease genes from two haloalkaliphilic bacteria strains isolated in India (OK-5 and OM-6) were PCR amplified using genomic DNA from the bacteria as a template. The PCR products and the pET21a+ vector were both digested with Bam HI and Sal I restriction enzymes and the products were purified. Following ligation, transformation into E. coli, and plating on ampicillin-containing solid media, colonies were isolated, grown in ampicillin-containing liquid media, and plasmids were purified from each culture.
Purified plasmids were then digested with Bam HI and Sal I restriction enzymes and the products were separated by gel electrophoresis. The results from lanes 2 and 3 show that the alkaline protease gene from the OK-5 strain was successfully cloned into pET21a+, while the results from lanes 4 and 5 show that the alkaline protease gene from the OM-6 strain was successfully cloned into the plasmid.
Image Descriptions
Figure 1 image description:
A map of the pET21a+ cloning vector. Unique cloning sites are indicated by lines labeled with the corresponding restriction enzyme recognition sequences. ↵
Figure 2 image description:
An agarose gel, stained with the intercalating agent ethidium bromide, showing the restriction enzyme digestion products of recombinant vectors created in a cloning experiment. Plasmids containing an insert produce two bands on the gel. ↵
Thumbnail
"BamHI.png" ↗ by Simon Caulton is licensed under CC BY-SA 4.0 ↗.
Image description: Restriction of DNA by BamHI enzyme. The enzyme cuts the DNA at the sequence GGATCC and leaves 'sticky ends' that can attach to complementary sticky ends.


5.13: Introduction to Rational Decision Making vs. Other Types of Decision Making
What you’ll learn to do: explain the concept of “rational decision making” and contrast it with prospect theory, bounded rationality, heuristics, and robust decisions
Though everyone makes decisions, not everyone goes about the process in the same way. In fact, not everyone even uses a “process” to make decisions. There are various decision-making styles, and we will focus on the rational decision-making model. We will also become familiar with a common process that many groups and individuals follow when making decisions. Though almost everyone will agree that decision making should be rational, there are also some important contrasting ideas that often balance out the “rational” aspects to the process.
Contributors and Attributions
CC licensed content, Original
- Introduction to Rational Decision Making. Authored by : David Thompson, PhD and Lumen Learning. License : CC BY: Attribution


5.6: Differentials. Taylor’s Theorem and Taylor’s Series
Recall (Theorem 2 of §1) that a function \(f\) is differentiable at \(p\) iff
\[\Delta f=f^{\prime}(p) \Delta x+\delta(x) \Delta x,\]
with \(\lim _{x \rightarrow p} \delta(x)=\delta(p)=0.\) It is customary to write \(df\) for \(f^{\prime}(p) \Delta x\) and \(o(\Delta x)\) for \(\delta(x) \Delta x; df\) is called the differential of \(f\) (at \(p\) and \(x\)). Thus
\[\Delta f=d f+o(\Delta x);\]
i.e., \(df\) approximates \(\Delta f\) to within \(o(\Delta x)\).
More generally, given any function \(f : E^{1} \rightarrow E\) and \(p, x \in E^{1},\) we define
\[d^{n} f=d^{n} f(p, x)=f^{(n)}(p)(x-p)^{n}, \quad n=0,1,2, \ldots,\]
where \(f^{(n)}\) is the \(n\) th derived function (Definition 2 in §1); \(d^{n} f\) is called the nth differential, or differential of order n, of \(f\) (at \(p\) and \(x\)). In particular, \(d^{1} f=f^{\prime}(p) \Delta x=d f.\) By our conventions, \(d^{n} f\) is always defined, as is \(f^{(n)}\).
As we shall see, good approximations of \(\Delta f\) (suggested by Taylor) can often be obtained by using higher differentials (1), as follows:
\[\Delta f=d f+\frac{d^{2} f}{2 !}+\frac{d^{3} f}{3 !}+\cdots+\frac{d^{n} f}{n !}+R_{n}, \quad n=1,2,3, \ldots,\]
where
\[R_{n}=\Delta f-\sum_{k=1}^{n} \frac{d^{k} f}{k !} \quad \text {(the "remainder term")}\]
is the error of the approximation. Substituting the values of \(\Delta f\) and \(d^{k} f\) and transposing \(f(p),\) we have
\[f(x)=f(p)+\frac{f^{\prime}(p)}{1 !}(x-p)+\frac{f^{\prime \prime}(p)}{2 !}(x-p)^{2}+\cdots+\frac{f^{(n)}(p)}{n !}(x-p)^{n}+R_{n}.\]
Formula (3) is known as the nth Taylor expansion of \(f\) about \(p\) (with remainder term \(R_{n}\) to be estimated). Usually we treat \(p\) as fixed and \(x\) as variable. Writing \(R_{n}(x)\) for \(R_{n}\) and setting
\[P_{n}(x)=\sum_{k=0}^{n} \frac{f^{(k)}(p)}{k !}(x-p)^{k},\]
we have
\[f(x)=P_{n}(x)+R_{n}(x).\]
The function \(P_{n} : E^{1} \rightarrow E\) so defined is called the nth Taylor polynomial for \(f\) about \(p.\) Thus (3) yields approximations of \(f\) by polynomials \(P_{n}, n=1,2,3, \ldots.\) This is one way of interpreting it. The other (easy to remember) one is (2), which gives approximations of \(\Delta f\) by the \(d^{k} f.\) It remains, however, to find a good estimate for \(R_{n}.\) We do it next.
Let the function \(f : E^{1} \rightarrow E\) and its first n derived functions be relatively continuous and finite on an interval I and differentiable on \(I-Q\) (Q countable). Let \(p, x \in I.\) Then formulas (2) and (3) hold, with
\[R_{n}=\frac{1}{n !} \int_{p}^{x} f^{(n+1)}(t) \cdot(x-t)^{n} d t \quad\left(\text {"integral form of } R_{n} \text{"}\right)\]
and
\[\left|R_{n}\right| \leq M_{n} \frac{|x-p|^{n+1}}{(n+1) !} \text { for some real } M_{n} \leq \sup _{t \in I-Q}\left|f^{(n+1)}(t)\right|.\]
- Proof
-
By definition, \(R_{n}=f-P_{n},\) or
\[R_{n}=f(x)-f(p)-\sum_{k=1}^{n} f^{(k)}(p) \frac{(x-p)^{k}}{k !}.\]
We use the right side as a "pattern" to define a function \(h : E^{1} \rightarrow E.\) This time, we keep \(x\) fixed \((\text { say }, x=a \in I)\) and replace \(p\) by a variable \(t.\) Thus we set
\[h(t)=f(a)-f(t)-\frac{f^{\prime}(t)}{1 !}(a-t)-\cdots-\frac{f^{(n)}(t)}{n !}(a-t)^{n} \text { for all } t \in E^{1}.\]
Then \(h(p)=R_{n}\) and \(h(a)=0.\) Our assumptions imply that \(h\) is relatively continuous and finite on \(I,\) and differentiable on \(I-Q.\) Differentiating (4), we see that all cancels out except for one term
\[h^{\prime}(t)=-f^{(n+1)}(t) \frac{(a-t)^{n}}{n !}, \quad t \in I-Q. \quad \text {(Verify!)}\]
Hence by Definitions 1 and 2 of §5,
\[-h(t)=\frac{1}{n !} \int_{t}^{a} f^{(n+1)}(s)(a-s)^{n} d s \quad \text { on } I\]
and
\[\frac{1}{n !} \int_{p}^{a} f^{(n+1)}(t)(a-t)^{n} d t=-h(a)+h(p)=0+R_{n}=R_{n} \quad\left(\text {for } h(p)=R_{n}\right).\]
As \(x=a,\) (3') is proved.
Next, let
\[M=\sup _{t \in I-Q}\left|f^{(n+1)}(t)\right|.\]
If \(M<+\infty,\) define
\[g(t)=M \frac{(t-a)^{n+1}}{(n+1) !} \text { for } t \geq a \text { and } g(t)=-M \frac{(a-t)^{n+1}}{(n+1) !} \text { for } t \leq a.\]
In both cases,
\[g^{\prime}(t)=M \frac{|a-t|^{n}}{n !} \geq\left|h^{\prime}(t)\right| \text { on } I-Q \text { by (5).}\]
Hence, applying Theorem 1 in §4 to the functions \(h\) and \(g\) on the interval \([a, p]\) (or \([p, a]),\) we get
\[|h(p)-h(a)| \leq|g(p)-g(a)|,\]
or
\[\left|R_{n}-0\right| \leq M \frac{|a-p|^{n+1}}{(n+1) !}.\]
Thus (3") follows, with \(M_{n}=M\).
Finally, if \(M=+\infty,\) we put
\[M_{n}=\left|R_{n}\right| \frac{(n+1) !}{|a-p|^{n+1}}<M. \quad \square\]
For real functions, we obtain some additional estimates of \(R_{n}\).
If \(f\) is real and \(n+1\) times differentiable on \(I\), then for \(p \neq x\) \((p, x \in I),\) there are \(q_{n}, q_{n}^{\prime}\) in the interval \((p, x)\) (respectively, \((x, p) )\) such that
\[R_{n}=\frac{f^{(n+1)}\left(q_{n}\right)}{(n+1) !}(x-p)^{n+1}\]
and
\[R_{n}=\frac{f^{(n+1)}\left(q_{n}^{\prime}\right)}{n !}(x-p)\left(x-q_{n}^{\prime}\right)^{n}.\]
(Formulas (5') and (5") are known as the Lagrange and Cauchy forms of \(R_{n},\) respectively.)
- Proof
-
Exactly as in the proof of Theorem 1, we obtain the function \(h\) and formula (5). By our present assumptions, \(h\) is differentiable (hence continuous) on \(I,\) so we may apply to it Cauchy's law of the mean (Theorem 2 of §2) on the interval \([a, p]\) (or \([p, a]\) if \(p<a ),\) where \(a=x \in I\).
For this purpose, we shall associate \(h\) with another suitable function \(g\) (to be specified later). Then by Theorem 2 of §2, there is a real \(q \in(a, p)\) (respectively, \(q \in(p, a))\) such that
\[g^{\prime}(q)[h(a)-h(p)]=h^{\prime}(q)[g(a)-g(p)].\]
Here by the previous proof, \(h(a)=0, h(p)=R_{n},\) and
\[h^{\prime}(q)=-\frac{f^{(n+1)}}{n !}(a-q)^{n}.\]
Thus
\[g^{\prime}(q) \cdot R_{n}=\frac{f^{(n+1)}(q)}{n !}(a-q)^{n}[g(a)-g(p)].\]
Now define \(g\) by
\[g(t)=a-t, \quad t \in E^{1}.\]
Then
\[g(a)-g(p)=-(a-p) \text { and } g^{\prime}(q)=-1,\]
so (6) yields (5") (with \(q_{n}^{\prime}=q\) and \(a=x)\).
Similarly, setting \(g(t)=(a-t)^{n+1},\) we obtain (5'). (Verify!) Thus all is proved. \(\quad \square\)
Note 1. In (5') and (5"), the numbers \(q_{n}\) and \(q_{n}^{\prime}\) depend on \(n\) and are different in general \(\left(q_{n} \neq q_{n}^{\prime}\right),\) for they depend on the choice of the function \(g\). Since they are between \(p\) and \(x,\) they may be written as
\[q_{n}=p+\theta_{n}(x-p) \text { and } q_{n}^{\prime}=p+\theta_{n}^{\prime}(x-p),\]
where \(0<\theta_{n}<1\) and \(0<\theta_{n}^{\prime}<1.\) (Explain!)
Note 2. For any function \(f : E^{1} \rightarrow E,\) the Taylor polynomials \(P_{n}\) are partial sums of a power series, called the Taylor series for \(f\) (about \(p).\) We say that \(f\) admits such a series on a set \(B\) iff the series converges to \(f\) on \(B;\) i.e.,
\[f(x)=\lim _{n \rightarrow \infty} P_{n}(x)=\sum_{n=1}^{\infty} \frac{f^{(n)}(p)}{n !}(x-p)^{n} \neq \pm \infty \text { for } x \in B.\]
This is clearly the case iff
\[\lim _{n \rightarrow \infty} R_{n}(x)=\lim _{n \rightarrow \infty}\left[f(x)-P_{n}(x)\right]=0 \text { for } x \in B;\]
briefly, \(R_{n} \rightarrow 0.\) Thus
\[f \text { admits a Taylor series (about p) iff } R_{n} \rightarrow 0.\]
Caution: The convergence of the series alone (be it pointwise or uniform) does not suffice. Sometimes the series converges to a sum other than \(f(x);\) then (7) fails. Thus all depends on the necessary and sufficient condition: \(R_{n} \rightarrow 0\).
We say that \(f\) is of class \(\mathrm{CD}^{n}\), or continuously differentiable \(n\) times, on a set \(B\) iff \(f\) is \(n\) times differentiable on \(B,\) and \(f^{(n)}\) is relatively continuous on \(B\). Notation: \(f \in \mathrm{CD}^{n}\) (on \(B).\)
If this holds for each \(n \in N,\) we say that \(f\) is infinitely differentiable on \(B\) and write \(f \in \mathrm{CD}^{\infty}\) (on \(B).\)
The notation \(f \in \mathrm{CD}^{0}\) means that \(f\) is finite and relatively continuous (all on \(B).\)
(a) Let
\[f(x)=e^{x} \text { on } E^{1}.\]
Then
\[(\forall n) \quad f^{(n)}(x)=e^{x},\]
so \(f \in \mathrm{CD}^{\infty}\) on \(E^{1}.\) At \(p=0, f^{(n)}(p)=1,\) so we obtain by Theorem 1' (using (5') and Note 1)
\[e^{x}=1+\frac{x}{1 !}+\frac{x^{2}}{2 !}+\cdots+\frac{x^{n}}{n !}+\frac{e^{\theta_{n} x}}{(n+1) !} x^{n+1}, \quad 0<\theta_{n}<1.\]
Thus on an interval \([-a, a]\),
\[e^{x} \approx 1+\frac{x}{1 !}+\frac{x^{2}}{2 !}+\cdots+\frac{x^{n}}{n !}\]
to within an error \(R_{n}(>0 \text { if } x>0)\) with
\[\left|R_{n}\right|<e^{a} \frac{a^{n+1}}{(n+1) !},\]
which tends to 0 as \(n \rightarrow+\infty.\) For \(a=1=x,\) we get
\[e=1+\frac{1}{1 !}+\frac{1}{2 !}+\cdots+\frac{1}{n !}+R_{n} \text { with } 0<R_{n}<\frac{e^{1}}{(n+1) !}.\]
Taking \(n=10,\) we have
\[e \approx 2.7182818 | 011463845 \ldots\]
with a nonnegative error of no more than
\[\frac{e}{11 !}=0.00000006809869 \ldots;\]
all digits are correct before the vertical bar.
(b) Let
\[f(x)=e^{-1 / x^{2}} \text { with } f(0)=0.\]
As \(\lim _{x \rightarrow 0} f(x)=0=f(0), f\) is continuous at \(0.\) We now show that \(f \in \mathrm{CD}^{\infty}\) on \(E^{1}.\)
For \(x \neq 0,\) this is clear; moreover, induction yields
\[f^{(n)}(x)=e^{-1 / x^{2}} x^{-3 n} S_{n}(x),\]
where \(S_{n}\) is a polynomial in \(x\) of degree 2\((n-1)\) (this is all we need know about \(S_{n}).\) A repeated application of L'Hôpital's rule then shows that
\[\lim _{x \rightarrow 0} f^{(n)}(x)=0 \text { for each } n.\]
To find \(f^{\prime}(0),\) we have to use the definition of a derivative:
\[f^{\prime}(0)=\lim _{x \rightarrow 0} \frac{f(x)-f(0)}{x-0},\]
or by L'Hôpital's rule,
\[f^{\prime}(0)=\lim _{x \rightarrow 0} \frac{f^{\prime}(x)}{1}=0.\]
Using induction again, we get
\[f^{(n)}(0)=0, \quad n=1,2, \ldots.\]
Thus, indeed, \(f\) has finite derivatives of all orders at each \(x \in E^{1},\) including \(x=0,\) so \(f \in \mathrm{CD}^{\infty}\) on \(E^{1},\) as claimed.
Nevertheless, any attempt to use formula (3) at \(p=0\) yields nothing. As all \(f^{(n)}\) vanish at \(0,\) so do all terms except \(R_{n}.\) Thus no approximation by polynomials results - we only get \(P_{n}=0\) on \(E^{1}\) and \(R_{n}(x)=e^{-1 / x^{2}}\). \(R_{n}\) does not tend to 0 except at \(x=0,\) so \(f\) admits no Taylor series about 0 (except on \(E=\{0\}).\)
Taylor's theorem also yields sufficient conditions for maxima and minima, as we see in the following theorem.
Let \(f : E^{1} \rightarrow E^{*}\) be of class \(\mathrm{CD}^{n}\) on \(G_{p}(\delta)\) for an even number \(n \geq 2,\) and let
\[f^{(k)}(p)=0 \text { for } k=1,2, \ldots,\text{ } n-1,\]
while
\[f^{(n)}(p)<0 \text { (respectively, } f^{(n)}(p)>0).\]
Then \(f(p)\) is the maximum (respectively, minimum) value of \(f\) on some \(G_{p}(\varepsilon)\) \(\varepsilon \leq \delta.\)
If, however, these conditions hold for some odd \(n \geq 1\) (i.e., the first nonvanishing derivative at \(p\) is of odd order), \(f\) has no maximum or minimum at \(p.\)
- Proof
-
As
\[f^{(k)}(p)=0, \quad k=1,2, \ldots, \text{ } n-1,\]
Theorem 1' (with \(n\) replaced by \(n-1)\) yields
\[f(x)=f(p)+f^{(n)}\left(q_{n}\right) \frac{(x-p)^{n}}{n !} \quad \text { for all } x \in G_{p}(\delta),\]
with \(q_{n}\) between \(x\) and \(p\).
Also, as \(f \in \mathrm{CD}^{n}, f^{(n)}\) is continuous at \(p.\) Thus if \(f^{(n)}(p)<0,\) then \(f^{(n)}<0\) on some \(G_{p}(\varepsilon), 0<\varepsilon \leq \delta.\) However, \(x \in G_{p}(\varepsilon)\) implies \(q_{n} \in G_{p}(\varepsilon),\) so
\[f^{(n)}\left(q_{n}\right)<0,\]
while
\[(x-p)^{n} \geq 0 \text { if } n \text { is even.}\]
It follows that
\[f^{(n)}\left(q_{n}\right) \frac{(x-p)^{n}}{n !} \leq 0,\]
and so
\[f(x)=f(p)+f^{(n)}\left(q_{n}\right) \frac{(x-p)^{n}}{n !} \leq f(p) \quad \text { for } x \in G_{p}(\varepsilon),\]
i.e., \(f(p)\) is the maximum value of \(f\) on \(G_{p}(\varepsilon),\) as claimed.
Similarly, in the case \(f^{(n)}(p)>0,\) a minimum would result.
If, however, \(n\) is odd, then \((x-p)^{n}\) is negative for \(x<p\) but positive for \(x>p .\) The same argument then shows that \(f(x)<f(p)\) on one side of \(p\) and \(f(x)>f(p)\) on the other side; thus no local maximum or minimum can exist at \(p.\) This completes the proof. \(\quad \square\)
(a') Let
\[f(x)=x^{2} \text { on } E^{1} \text { and } p=0.\]
Then
\[f^{\prime}(x)=2 x \text { and } f^{\prime \prime}(x)=2>0,\]
so
\[f^{\prime}(0)=0 \text { and } f^{\prime \prime}(0)=2>0.\]
By Theorem 2, \(f(p)=0^{2}=0\) is a minimum value.
It turns out to be a minimum on all of \(E^{1}\). Indeed, \(f^{\prime}(x)>0\) for \(x>0\), and \(f^{\prime}<0\) for \(x<0,\) so \(f\) strictly decreases on \((-\infty, 0)\) and increases on \((0,+\infty).\)
Actually, even without using Theorem 2, the last argument yields the answer.
(b') Let
\[f(x)=\ln x \text { on }(0,+\infty).\]
Then
\[f^{\prime}(x)=\frac{1}{x}>0 \text { on all of }(0,+\infty).\]
This shows that \(f\) strictly increases everywhere and hence can have no maximum or minimum anywhere. The same follows by the second part of Theorem 2, with \(n=1\).
(b") In Example (b'), consider also
\[f^{\prime \prime}(x)=-\frac{1}{x^{2}}<0.\]
In this case, \(f^{\prime \prime}\) has no bearing on the existence of a maximum or minimum because \(f^{\prime} \neq 0.\) Still, the formula \(f^{\prime \prime}<0\) does have a certain meaning. In fact, if \(f^{\prime \prime}(p)<0\) and \(f \in \mathrm{CD}^{2}\) on \(G_{p}(\delta),\) then (using the same argument as in Theorem 2) the reader will easily find that
\[f(x) \leq f(p)+f^{\prime}(p)(x-p) \quad \text { for } x \text { in some } G_{p}(\varepsilon), 0<\varepsilon \leq \delta.\]
since \(y=f(p)+f^{\prime}(p)(x-p)\) is the equation of the tangent at \(p\), it follows that \(f(x) \leq y;\) i.e., near \(p\) the curve lies below the tangent at \(p.\)
Similarly, \(f^{\prime \prime}(p)>0\) and \(f \in \mathrm{CD}^{2}\) on \(G_{p}(\delta)\) implies that the curve near \(p\) lies above the tangent.


5.2: Education around the World
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Education Values Vary Geographically
International differences in education systems are not solely a financial issue. The value placed on education, the amount of time devoted to it, and the distribution of education within a country also play a role in those differences. For example, students in South Korea spend 220 days a year in school, compared to the 180 days a year of their United States counterparts (Pellissier 2010). As of 2006, the United States ranked fifth among twenty-seven countries for college participation, but ranked sixteenth in the number of students who receive college degrees (National Center for Public Policy and Higher Education 2006). These statistics may be related to how much time is spent on education in the United States.
Then there is the issue of educational distribution within a nation. In December 2010, the results of a test called the Program for International Student Assessment (PISA), which is administered to fifteen-year-old students worldwide, were released. Those results showed that students in the United States had fallen from fifteenth to twenty-fifth in the rankings for science and math (National Public Radio 2010). Students at the top of the rankings hailed from Shanghai, Finland, Hong Kong, and Singapore.
Analysts determined that the nations and city-states at the top of the rankings had several things in common. For one, they had well-established standards for education with clear goals for all students. They also recruited teachers from the top 5 to 10 percent of university graduates each year, which is not the case for most countries (National Public Radio 2010).
Finally, there is the issue of social factors. One analyst from the Organization for Economic Cooperation and Development, the organization that created the test, attributed 20 percent of performance differences and the United States’ low rankings to differences in social background. Researchers noted that educational resources, including money and quality teachers, are not distributed equitably in the United States. In the top-ranking countries, limited access to resources did not necessarily predict low performance. Analysts also noted what they described as “resilient students,” or those students who achieve at a higher level than one might expect given their social background. In Shanghai and Singapore, the proportion of resilient students is about 70 percent. In the United States, it is below 30 percent. These insights suggest that the United States’ educational system may be on a descending path that could detrimentally affect the country’s economy and its social landscape (National Public Radio 2010).
Education in Finland
With public education in the United States under such intense criticism, why is it that Singapore, South Korea, and especially Finland (which is culturally most similar to us), have such excellent public education? Over the course of thirty years, the country has pulled itself from among the lowest rankings by the Organization of Economic Cooperation (OEDC) to first in 2012, and remains, as of 2014, in the top five. Contrary to the rigid curriculum and long hours demanded of students in South Korea and Singapore, Finnish education often seems paradoxical to outside observers because it appears to break a lot of the rules we take for granted. It is common for children to enter school at seven years old, and children will have more recess and less hours in school than U.S. children—approximately 300 less hours. Their homework load is light when compared to all other industrialized nations (nearly 300 fewer hours per year in elementary school). There are no gifted programs, almost no private schools, and no high-stakes national standardized tests (Laukkanen 2008; LynNell Hancock 2011).
Prioritization is different than in the United States. There is an emphasis on allocating resources for those who need them most, high standards, support for special needs students, qualified teachers taken from the top 10 percent of the nation's graduates and who must earn a Master's degree, evaluation of education, balancing decentralization and centralization.
"We used to have a system which was really unequal," stated the Finnish Education Chief in an interview. "My parents never had a real possibility to study and have a higher education. We decided in the 1960s that we would provide a free quality education to all. Even universities are free of charge. Equal means that we support everyone and we’re not going to waste anyone’s skills." As for teachers, "We don’t test our teachers or ask them to prove their knowledge. But it’s true that we do invest in a lot of additional teacher training even after they become teachers" (Gross-Loh, 2014).
Yet over the past decade Finland has consistently performed among the top nations on the PISA. Finland’s school children didn’t always excel. Finland built its excellent, efficient, and equitable educational system in a few decades from scratch, and the concept guiding almost every educational reform has been equity. The Finnish paradox is that by focusing on the bigger picture for all, Finland has succeeded at fostering the individual potential of most every child.
"We created a school system based on equality to make sure we can develop everyone’s potential. Now we can see how well it’s been working. Last year the OECD tested adults from twenty-four countries measuring the skill levels of adults aged sixteen to sixty-five on a survey called the Programme for International Assessment of Adult Competencies (PIAAC), which tests skills in literacy, numeracy, and problem solving in technology-rich environments. Finland scored at or near the top on all measures."
Formal and Informal Education
As already mentioned, education is not solely concerned with the basic academic concepts that a student learns in the classroom. Societies also educate their children, outside of the school system, in matters of everyday practical living. These two types of learning are referred to as formal education and informal education.
Formal education describes the learning of academic facts and concepts through a formal curriculum. Arising from the tutelage of ancient Greek thinkers, centuries of scholars have examined topics through formalized methods of learning. Education in earlier times was only available to the higher classes; they had the means for access to scholarly materials, plus the luxury of leisure time that could be used for learning. The Industrial Revolution and its accompanying social changes made education more accessible to the general population. Many families in the emerging middle class found new opportunities for schooling.
The modern U.S. educational system is the result of this progression. Today, basic education is considered a right and responsibility for all citizens. Expectations of this system focus on formal education, with curricula and testing designed to ensure that students learn the facts and concepts that society believes are basic knowledge.
In contrast, informal education describes learning about cultural values, norms, and expected behaviors by participating in a society. This type of learning occurs both through the formal education system and at home. Our earliest learning experiences generally happen via parents, relatives, and others in our community. Through informal education, we learn how to dress for different occasions, how to perform regular life routines like shopping for and preparing food, and how to keep our bodies clean.
Cultural transmission refers to the way people come to learn the values, beliefs, and social norms of their culture. Both informal and formal education include cultural transmission. For example, a student will learn about cultural aspects of modern history in a U.S. History classroom. In that same classroom, the student might learn the cultural norm for asking a classmate out on a date through passing notes and whispered conversations.
Access to Education
Another global concern in education is universal access. This term refers to people’s equal ability to participate in an education system. On a world level, access might be more difficult for certain groups based on class or gender (as was the case in the United States earlier in the nation’s history, a dynamic we still struggle to overcome). The modern idea of universal access arose in the United States as a concern for people with disabilities. In the United States, one way in which universal education is supported is through federal and state governments covering the cost of free public education. Of course, the way this plays out in terms of school budgets and taxes makes this an often-contested topic on the national, state, and community levels. In viewing the table picture below you may need to enlarge your screen to view.
 (Graph courtesy of Census of Governments: Survey of School System Finances 2012)
A precedent for universal access to education in the United States was set with the 1972 U.S. District Court for the District of Columbia’s decision in Mills v. Board of Education of the District of Columbia. This case was brought on the behalf of seven school-age children with special needs who argued that the school board was denying their access to free public education. The school board maintained that the children’s “exceptional” needs, which included mental retardation and mental illness, precluded their right to be educated for free in a public school setting. The board argued that the cost of educating these children would be too expensive and that the children would therefore have to remain at home without access to education.
This case was resolved in a hearing without any trial. The judge, Joseph Cornelius Waddy, upheld the students’ right to education, finding that they were to be given either public education services or private education paid for by the Washington, D.C., board of education. He noted that
Constitutional rights must be afforded citizens despite the greater expense involved … the District of Columbia’s interest in educating the excluded children clearly must outweigh its interest in preserving its financial resources. … The inadequacies of the District of Columbia Public School System whether occasioned by insufficient funding or administrative inefficiency, certainly cannot be permitted to bear more heavily on the “exceptional” or handicapped child than on the normal child ( Mills v. Board of Education 1972).
Today, the optimal way to include differently abled students in standard classrooms is still being researched and debated. “Inclusion” is a method that involves complete immersion in a standard classroom, whereas “mainstreaming” balances time in a special-needs classroom with standard classroom participation. There continues to be social debate surrounding how to implement the ideal of universal access to education.
Summary
Educational systems around the world have many differences, though the same factors—including resources and money—affect every educational system. Educational distribution is a major issue in many nations, including in the United States, where the amount of money spent per student varies greatly by state. Education happens through both formal and informal systems; both foster cultural transmission. Universal access to education is a worldwide concern.
Critical Reflection
Has there ever been a time when your formal and informal educations in the same setting were at odds? How did you overcome that disconnect?
Do you believe free access to schools has achieved its intended goal? Explain.
Further Research
Though it’s a struggle, education is continually being improved in the developing world. To learn how educational programs are being fostered worldwide, explore the Education section of the Center for Global Development’s website.
References
Darling-Hammond, Linda. 2010. "What We Can Learn from Finland's Successful School Reform." NEA Today Magazine. Retrieved December 12, 2014. (www.nea.org/home/40991.htm)..
Durkheim, Emile. 1898 [1956]. Education and Sociology . New York: Free Press.
Gross-Loh, Christine. 2014. "Finnish Education Chief: 'We Created a School System Based on Equality.'" The Atlantic. Retrieved December 12, 2014. ( http://www.theatlantic.com/education...ngle_page=true ).
Mills v. Board of Education , 348 DC 866 (1972).
National Center for Public Policy and Higher Education. 2006. Measuring UP: The National Report Card on Higher Education . Retrieved December 9, 2011 (www.eric.ed.gov/PDFS/ED493360.pdf).
National Public Radio. 2010. “Study Confirms U.S. Falling Behind in Education.” All Things Considered , December 10. Retrieved December 9, 2011 (www.npr.org/2010/12/07/13188...d-In-Education).
Pellissier, Hank. 2010. “High Test Scores, Higher Expectations, and Presidential Hype.” Great Schools . Retrieved January 17, 2012 ( http://www.greatschools.org/students...ean-schools.gs ).
Rampell, Catherine. 2009. “Of All States, New York’s Schools Spend Most Money Per Pupil.” Economix . Retrieved December 15, 2011 ( http://economix.blogs.nytimes.com/20...ney-per-pupil/ ).
U.S. Census Bureau. 2014. "Public Education Finances 2012." Retrieved December 12, 2014. ( http://www2.census.gov/govs/school/12f33pub.pdf ).
World Bank. 2011. “Education in Afghanistan.” Retrieved December 14, 2011 (go.worldbank.org/80UMV47QB0).


12: Introduction to Calculus
Calculus is the broad area of mathematics dealing with such topics as instantaneous rates of change, areas under curves, and sequences and series. Underlying all of these topics is the concept of a limit, which consists of analyzing the behavior of a function at points ever closer to a particular point, but without ever actually reaching that point. Calculus has two basic applications: differential calculus and integral calculus.
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- 12.3: Finding Limits - Properties of Limits
- Graphing a function or exploring a table of values to determine a limit can be cumbersome and time-consuming. When possible, it is more efficient to use the properties of limits, which is a collection of theorems for finding limits. Knowing the properties of limits allows us to compute limits directly.
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- 12.4: Continuity
- A function that remains level for an interval and then jumps instantaneously to a higher value is called a stepwise function. This function is an example. A function that has any hole or break in its graph is known as a discontinuous function. A stepwise function, such as parking-garage charges as a function of hours parked, is an example of a discontinuous function. We can check three different conditions to decide if a function is continuous at a particular number.


7.6: Antimicrobial Resistance
Antimicrobial resistance refers to the adaptation of pathogens (microbes) to the current best treatments we have available for them. Antibiotic resistance can happen in bacteria, antifungal resistance in fungi, antiviral resistance for viruses, and antiparasitic resistance for parasites (worms) - although the most concerning examples are currently in bacteria and fungi. Resistance to drugs occurs when the microorganism has adapted to the way that the drug works to inactivate or kill the pathogen. Sometimes this occurs when other, healthy bacteria have also been killed off due to the treatment, allowing harmful bacteria to proliferate. These pathogens can then become extremely difficult or impossible to treat (CDC, 2022c).


1.8: Reading- The Value of Marketing
Marketing can mobilize attitudes and behavior around a common vision. It is a powerful medium for expression, creativity, and sharing across an increasingly global society. Marketing can be an agent of change in the diffusion of ideas and innovation. It can also be self-serving and manipulative, playing on human fears and insecurities to separate people from their money and from one another.
With all this in mind, what value does marketing provide?
Marketing remains an active, dynamic field because it serves useful purposes for organizations, individuals, and society.
Marketing Can Benefit Organizations
As explained earlier in this module, organizations use marketing to identify, satisfy, and retain customers. Marketing helps businesses know which problems to solve and which products, services, and experiences to offer. Effective marketing drives product improvements and determines the terms of profitable transactions. Marketing efforts help organizations build and sustain productive relationships with the people and groups they serve. Without effective marketing, companies become islands that retain no meaningful connection to their customers. When an organization loses its audience, eventually it ceases to exist.
Marketing Can Benefit People
Individuals are not just targets of marketing; they can also be beneficiaries . Marketing helps people navigate the world around them to find the things that address their wants and needs. Marketing is responsible for the creation of products that delight people, improve their productivity, and alter their quality of life. In recent years, marketing has contributed to the pervasive information now available to help people make advantageous consumer choices. Marketing reduces the friction and hassle around transactions. Imagine, without marketing you would never know that many of the products you need exist, let alone how to find them.
Marketing and Society
Because marketing is grounded in commercial, profit-seeking behaviors, some would argue that society is a net loser rather than a net winner when it comes to marketing influence. However, effective marketing helps create the conditions for healthy competition and market efficiency, where companies and consumers communicate and exchange mutual value.
Like virtually any tool, marketing can be used for noble purposes or nefarious ones. Regardless, it has a pervasive presence in the modern world. As you learn about marketing and begin to practice its principles, you will see more clearly how it influences your daily life. You will identify opportunities where marketing skills can help you become more effective at achieving your personal and professional goals.
Contributors and Attributions
- The Value of Marketing. Authored by : Lumen Learning. License : CC BY: Attribution
- Lucky Strike Ad (LIFE 1936-12-21). Authored by : Anton Raath. Located at : https://www.flickr.com/photos/fugue/3952296569/ . License : CC BY-SA: Attribution-ShareAlike
- Nike Red Laces. Authored by : Omar 18. Provided by : Wikimedia. Located at : commons.wikimedia.org/wiki/File:Nike_(RED)_Laces.jpg. License : CC BY-SA: Attribution-ShareAlike