22.1: Stern-Gerlach Experiment
In 1920, Otto Stern and Walter Gerlach designed an experiment that unintentionally led to the discovery that electrons have their own individual, continuous spin even as they move along their orbital of an atom. The experiment was done by putting a silver foil in an oven to vaporize its atoms. The silver atoms were collected into a beam that passed through an inhomogeneous magnetic field. The result was that the magnetic beam split the beam into two (and only two) separate ones. The Stern–Gerlach experiment demonstrated that the spatial orientation of angular momentum is quantized into two components (up and down). Thus an atomic-scale system was shown to have intrinsically quantum properties. The experiment is normally conducted using electrically neutral particles such as silver atoms. This avoids the large deflection in the path of a charged particle moving through a magnetic field and allows spin-dependent effects to dominate.
If the particle is treated as a classical spinning magnetic dipole, it will precess in a magnetic field because of the torque that the magnetic field exerts on the dipole. If it moves through a homogeneous magnetic field, the forces exerted on opposite ends of the dipole cancel each other out and the trajectory of the particle is unaffected. However, if the magnetic field is inhomogeneous then the force on one end of the dipole will be slightly greater than the opposing force on the other end, so that there is a net force which deflects the particle’s trajectory. If the particles were classical spinning objects, one would expect the distribution of their spin angular momentum vectors to be random and continuous. Each particle would be deflected by an amount proportional to its magnetic moment, producing some density distribution on the detector screen. Instead, the particles passing through the Stern–Gerlach apparatus are equally distributed among two possible values, with half of them ending up at an upper spot (“spin up”), and the other half at the lower spot (“spin down”). Since the particles are deflected by a magnetic field, spin is a magnetic property that is associated to some intrinsic form of angular momentum. As we saw in chapter 6, the quantization of the angular momentum gives energy levels that are \((2\ell+1)\)-fold degenerate. Since along the direction of the magnet we observe only two possible eigenvalues for the spin, we conclude the following value for \(s\):
\[ 2s+1=2 \quad\Rightarrow\quad s=\dfrac{1}{2}. \label{23.1.1} \]
The Stern-Gerlach experiment proves that electrons are spin-\(\dfrac{1}{2}\) particles. These have only two possible spin angular momentum values measured along any axis, \(+\dfrac {\hbar }{2}\) or \(-\dfrac {\hbar }{2}\), a purely quantum mechanical phenomenon. Because its value is always the same, it is regarded as an intrinsic property of electrons, and is sometimes known as “intrinsic angular momentum” (to distinguish it from orbital angular momentum, which can vary and depends on the presence of other particles).
The act of observing (measuring) the momentum along the \(z\) direction corresponds to the operator \(\hat{S}_z\), which project the value of the total spin operator \(\hat{S}^2\) along the \(z\) axis. The eigenvalues of the projector operator are:
\[ \hat{S}_z \phi = \hbar m_s \phi, \label{23.1.2} \]
where \(m_s=\left\{-s,+s\right\}=\left\{-\dfrac{1}{2},+\dfrac{1}{2}\right\}\) is the spin quantum number along the \(z\) component. The eigenvalues for the total spin operator \(\hat{S}^2\)—similarly to the angular momentum operator \(\hat{L}^2\) seen in Equation 22.3.6 —are:
\[ \hat{S}^2 \phi = \hbar^2 s(s+1) \phi, \label{23.1.3} \]
The initial state of the particles in the Stern-Gerlach experiment is given by the following wave function:
\[ \phi = c_1\, \phi_{\uparrow} + c_2 \,\phi_{\downarrow}, \label{23.1.4} \]
where \(\uparrow=+\dfrac{\hbar}{2}\), \(\downarrow=-\dfrac{\hbar}{2}\), and the coefficients \(c_1\) and \(c_2\) are complex numbers. In this initial state, spin can point in any direction. The expectation value of the operator \(\hat{S}_z\) (the quantity that the Stern-Gerlach experiment measures), can be obtained using Equation 22.2.6 :
\[ \begin{aligned} <S_z> &= \int \phi^{*} \hat{S}_z \phi \, d\mathbf{s} \\ &= +\dfrac{\hbar}{2} \vert c_1\vert^2 -\dfrac{\hbar}{2} \vert c_2\vert^2, \end{aligned} \label{23.1.5} \]
where the integration is performed along a special coordinate \(\mathbf{s}\) composed of only two values, and the coefficient \(c_1\) and \(c_2\) are complex numbers. Applying the normalization condition, Equation 6.2.4 we can obtain:
\[ |c_{1}|^{2}+|c_{2}|^{2}=1 \quad\longrightarrow\quad |c_{1}|^{2}=|c_{2}|^{2}=\dfrac{1}{2}. \label{23.1.6} \]
This equation is not sufficient to determine the values of the coefficients since they are complex numbers. Equation \ref{23.1.6}, however, tells us that the squared magnitudes of the coefficients can be interpreted as probabilities of outcome from the experiment. This is true because their values are obtained from the normalization condition, and the normalization condition guarantees that the system is observed with probability equal to one. Summarizing, since we started with random initial directions, each of the two states, \(\phi_{\uparrow}\) and \(\phi_{\downarrow}\), will be observed with equal probability of \(\dfrac{1}{2}\).


1.1: Welcome to Computer Applications and Concepts
Welcome to ITE 115 – Introduction to Computer Applications and Concepts
Computers are important tools in nearly every profession, so almost everyone can benefit by knowing how they work and how to use them. In this course, you will learn the basics of computer hardware, software, and networking. You will also develop basic skills in using Windows and Microsoft Office, and for creating your own web pages. Finally, you will learn how to use computers safely, and you will consider ethical issues related to computer usage.
Course Requisite Technical Skills
In order to succeed in this online course, you must be comfortable working with technology. At a minimum, you must possess the following technical skills:
- Basic knowledge and moderate proficiency about the operation of a computer and software installation.
- Ability to use File Manager/Windows Explorer.
- Ability to use the Internet in an effective and efficient manner, including installation and management of browser plug-ins.
- Proficient in searching the web for information
- Ability to download and print information from websites and download, view, and/or print PDF files
- Working knowledge of the Blackboard learning management system.
Course Objectives
- Identify and discuss the four main functions of computer hardware: input, processing, output, and storage.
- Identify and describe major hardware components.
- Identify, describe and use communications and networking terminology and technology to include Internet operations and its uses.
- Describe the major operating system functions and demonstrate usage of operating system services to include: disk management, file management, and memory management.
- Identify and discuss computer ethics and security issues.
- Demonstrate searching, validation, and evaluation of information found online and in library databases.
- Identify, discuss and use important categories of application software: word processing, spreadsheet, database management, and presentation graphics.
- Identify and discuss software development processes and programming languages.
- Create a Web site with xhtml tags and post to a Web server
CC licensed content, Original
- Welcome. Authored by : Jim Shannon. Provided by : Extended Learning Institute of Northern Virginia Community College. License : CC BY: Attribution
CC licensed content, Shared previously
- 5. Authored by : Rosa Menkman. Located at : https://www.flickr.com/photos/r00s/896695552/ . License : CC BY-NC: Attribution-NonCommercial


2.5: Art and Identity- Introduction
-
- Last updated
- Save as PDF
One of the more important themes emerging from the last century has been the individual’s search for identity. For example, genealogical websites have proliferated and special television programs are devoted to the subject. Since it first aired on PBS in 2012, Henry Louis Gates Jr.’s Finding Your Roots has been a popular program. The British version, The Guardian , has been successful since 2006.
Some anthropologists suggest that the deep-rooted interest in identity or ancestry is partly shaped by evolutionary forces dating back to early humans supporting each other in extended family groups. Anthropologist Dwight Read theorizes that the Neolithic people were the first to understand the concept of the family tree and the perception of self in a family unit and in society. 1 If connected through blood, people have the tendency to be more willing to care for each other; a common interest and support system is readily realized within a clan or a group.
Early humans created two- and three-dimensional likenesses of themselves in their environment to help understand who they were in relation to the other members of their group. Con- temporary humans do the same; they make records of themselves with family members, most commonly in photographs and Selfies, and on Instagram. It is the same fundamental concept and placement in an environment that collectively identifies who we are in society, for example, in social gatherings, organizations, and religious settings. This means, above all, that we must place ourselves within the world in order to obtain identity. Children search for their identity at a very young age by observing and recognizing their parents and family members. Their markings with in a simple drawing of self and family similar to those of early humans help them to vindicateand confirm who they are and how they are perceived by their family group.
Like children, artists sometimes explore their identity through self-portraits and symbolically in works of art that relate to ancestry or culture. Doing so allows them to take a look inside their core and see how they fit within their contemporary culture; this investigation of self plays an important role in how artists understand their environment and the world.
Vincent van Gogh is known as a person who spent much of his time in solitude. He painted more than thirty self-portraits between the years 1886 and 1889, placing him among the most prolific self-portraitists of all time.  (Figures 8.1, 8.2, and 8.3) While Van Gogh used the study of his own image to help develop his skills as an artist, these self portraits also give us insights into the artist’s life and well being, how he fit in society, and his place among the groups with whom he associated.
Like Van Gogh, Pablo Picasso painted a number of self-portraits. Throughout his career, Picasso painted various likenesses that reflected changes in himself, his style, his artistic development, as well as in his life style and beliefs—all of which may be viewed closely from the content of his paintings. (Figures 8.4 and 8.5) The first self-portrait, painted in 1901 while he was establishing himself as an artist in Paris, France, and still spending time in Barcelona, Spain, reflects the somber mode and tones of his Blue Period (1901-1904). The second, dated to 1906, at the very end of his Rose Period (1904-1906), Picasso depicts himself as the artist who by that time was moving in artistic circles, gaining respect, and acquiring patrons.
Frida Kahlo (1907-1954, Mexico) used the iconography of her Mexican heritage to paint herself and the pain that had become an integral part of her life following a bus accident at the age of 18 in which she suffered numerous injuries. She identified as a group member of her country, with Mexican culture and ancestry, and as belonging to the female gender. Kahlo’s self-portraits are dramatic, bloody, brutal, and at times overtly political. ( Self-Portrait , Frida Kahlo: upload.wikimedia.org/Wikipedia/en/1/1e/ Frida_Kahlo_%28self_portrait%29.jpg) In seeking her roots, she voiced concern for her country as it struggled for an independent cultural identity. She spoke to her country and people through her art. Kahlo’s art was inspired by her public beliefs and personal sufferings; she wanted her art to speak from her consciousness.
Although self-portraits of today may be slightly different from those of earlier decades, they still depict self-exploration and identity through society and groups that communicate who we are. Cai Guo-Qiang (b. 1958, China, lives USA) exploded small charges of gunpowder to create an image of himself. ( Self- Portrait: A Subjugated Soul , Cai Guo-Qiang: http:// www.caiguoqiang.com/sites/default/files/styles/medium/public/1989_SelfPortrait_0389_001ltr-web. jpg) Different from those by Van Gogh, Picasso, and Kahlo, Cai’s self-portrait does not have any likeness or resemblance to his personal features, but it too sends a message about our society and how Cai relates to it. For example, the artist associates the lack of identifying information, rendering him anonymous, with contemporary society, and the fired gunpowder with both chaos and transformation.
Despite the distance in time that separates early and modern humans, the search for their place in society and who they are remains of fascination and a mystery to all humans regardless of their time in history.
- Ghose, Tia (Oct. 26, 2012). Why we care about our ancestry, Live Science. http://www.livescience.com/24313-why-ancestry.html


1.6: Estimates and Fermi Calculations
- Estimate the values of physical quantities.
On many occasions, physicists, other scientists, and engineers need to make estimates for a particular quantity. Other terms sometimes used are guesstimates , order-of-magnitude approximations , back-of-the-envelope calculations , or Fermi calculations . (The physicist Enrico Fermi mentioned earlier was famous for his ability to estimate various kinds of data with surprising precision.) Will that piece of equipment fit in the back of the car or do we need to rent a truck? How long will this download take? About how large a current will there be in this circuit when it is turned on? How many houses could a proposed power plant actually power if it is built? Note that estimating does not mean guessing a number or a formula at random. Rather, estimation means using prior experience and sound physical reasoning to arrive at a rough idea of a quantity’s value. Because the process of determining a reliable approximation usually involves the identification of correct physical principles and a good guess about the relevant variables, estimating is very useful in developing physical intuition. Estimates also allow us perform “sanity checks” on calculations or policy proposals by helping us rule out certain scenarios or unrealistic numbers. They allow us to challenge others (as well as ourselves) in our efforts to learn truths about the world.
Many estimates are based on formulas in which the input quantities are known only to a limited precision. As you develop physics problem-solving skills (which are applicable to a wide variety of fields), you also will develop skills at estimating. You develop these skills by thinking more quantitatively and by being willing to take risks. As with any skill, experience helps. Familiarity with dimensions (see Table 1.5.1 ) and units (see Table 1.3.1 and Table 1.3.2 ), and the scales of base quantities (see Figure 1.2.3 ) also helps.
To make some progress in estimating, you need to have some definite ideas about how variables may be related. The following strategies may help you in practicing the art of estimation:
- Get big lengths from smaller lengths . When estimating lengths, remember that anything can be a ruler. Thus, imagine breaking a big thing into smaller things, estimate the length of one of the smaller things, and multiply to get the length of the big thing. For example, to estimate the height of a building, first count how many floors it has. Then, estimate how big a single floor is by imagining how many people would have to stand on each other’s shoulders to reach the ceiling. Last, estimate the height of a person. The product of these three estimates is your estimate of the height of the building. It helps to have memorized a few length scales relevant to the sorts of problems you find yourself solving. For example, knowing some of the length scales in Figure 1.2.3 might come in handy. Sometimes it also helps to do this in reverse—that is, to estimate the length of a small thing, imagine a bunch of them making up a bigger thing. For example, to estimate the thickness of a sheet of paper, estimate the thickness of a stack of paper and then divide by the number of pages in the stack. These same strategies of breaking big things into smaller things or aggregating smaller things into a bigger thing can sometimes be used to estimate other physical quantities, such as masses and times.
- Get areas and volumes from lengths . When dealing with an area or a volume of a complex object, introduce a simple model of the object such as a sphere or a box. Then, estimate the linear dimensions (such as the radius of the sphere or the length, width, and height of the box) first, and use your estimates to obtain the volume or area from standard geometric formulas. If you happen to have an estimate of an object’s area or volume, you can also do the reverse; that is, use standard geometric formulas to get an estimate of its linear dimensions.
- Get masses from volumes and densities . When estimating masses of objects, it can help first to estimate its volume and then to estimate its mass from a rough estimate of its average density (recall, density has dimension mass over length cubed, so mass is density times volume). For this, it helps to remember that the density of air is around 1 kg/m 3 , the density of water is 10 3 kg/m 3 , and the densest everyday solids max out at around 10 4 kg/m 3 . Asking yourself whether an object floats or sinks in either air or water gets you a ballpark estimate of its density. You can also do this the other way around; if you have an estimate of an object’s mass and its density, you can use them to get an estimate of its volume.
- If all else fails, bound it . For physical quantities for which you do not have a lot of intuition, sometimes the best you can do is think something like: Well, it must be bigger than this and smaller than that. For example, suppose you need to estimate the mass of a moose. Maybe you have a lot of experience with moose and know their average mass offhand. If so, great. But for most people, the best they can do is to think something like: It must be bigger than a person (of order 10 2 kg) and less than a car (of order 10 3 kg). If you need a single number for a subsequent calculation, you can take the geometric mean of the upper and lower bound—that is, you multiply them together and then take the square root. For the moose mass example, this would be $$\left(10^{2} \times 10^{3}\right)^{0.5} = 10^{2.5} = 10^{0.5} \times 10^{2} \approx 3 \times 10^{2}\; kg \ldotp$$The tighter the bounds, the better. Also, no rules are unbreakable when it comes to estimation. If you think the value of the quantity is likely to be closer to the upper bound than the lower bound, then you may want to bump up your estimate from the geometric mean by an order or two of magnitude.
- One “sig. fig.” is fine . There is no need to go beyond one significant figure when doing calculations to obtain an estimate. In most cases, the order of magnitude is good enough. The goal is just to get in the ballpark figure, so keep the arithmetic as simple as possible.
- Ask yourself: Does this make any sense? Last, check to see whether your answer is reasonable. How does it compare with the values of other quantities with the same dimensions that you already know or can look up easily? If you get some wacky answer (for example, if you estimate the mass of the Atlantic Ocean to be bigger than the mass of Earth, or some time span to be longer than the age of the universe), first check to see whether your units are correct. Then, check for arithmetic errors. Then, rethink the logic you used to arrive at your answer. If everything checks out, you may have just proved that some slick new idea is actually bogus.
Estimate the total mass of the oceans on Earth.
Strategy
We know the density of water is about 10 3 kg/m 3 , so we start with the advice to “get masses from densities and volumes.” Thus, we need to estimate the volume of the planet’s oceans. Using the advice to “get areas and volumes from lengths,” we can estimate the volume of the oceans as surface area times average depth, or V = AD. We know the diameter of Earth from Figure 1.4 and we know that most of Earth’s surface is covered in water, so we can estimate the surface area of the oceans as being roughly equal to the surface area of the planet. By following the advice to “get areas and volumes from lengths” again, we can approximate Earth as a sphere and use the formula for the surface area of a sphere of diameter d—that is, A = \(\pi d^{2}\), to estimate the surface area of the oceans. Now we just need to estimate the average depth of the oceans. For this, we use the advice: “If all else fails, bound it.” We happen to know the deepest points in the ocean are around 10 km and that it is not uncommon for the ocean to be deeper than 1 km, so we take the average depth to be around (10 3 x 10 4 ) 0.5 ≈ 3 x 10 3 m. Now we just need to put it all together, heeding the advice that “one ‘sig. fig.’ is fine.”
Solution
We estimate the surface area of Earth (and hence the surface area of Earth’s oceans) to be roughly
\[A = \pi d^{2} = \pi \left(10^{7}\; m\right)^{2} \approx 3 \times 10^{14}\; m^{2} \ldotp\]
Next, using our average depth estimate of D = 3 x 10 3 m, which was obtained by bounding, we estimate the volume of Earth’s oceans to be
\[V = AD = \left(3 \times 10^{14}\; m^{2}\right)\left(3 \times 10^{3}\; m\right) = 9 \times 10^{17}m^{3} \ldotp\]
Last, we estimate the mass of the world's oceans to be
\[M = \rho V = \left(10^{3}\; kg/m^{3}\right) \left(9 \times 10^{17}\; m^{3}\right) = 9 \times 10^{20}\; kg \ldotp\]
Thus, we estimate that the order of magnitude of the mass of the planet’s oceans is 10 21 kg.
Significance
To verify our answer to the best of our ability, we first need to answer the question: Does this make any sense? From Figure 1.4, we see the mass of Earth’s atmosphere is on the order of 10 19 kg and the mass of Earth is on the order of 10 25 kg. It is reassuring that our estimate of 10 21 kg for the mass of Earth’s oceans falls somewhere between these two. So, yes, it does seem to make sense. It just so happens that we did a search on the Web for “mass of oceans” and the top search results all said 1.4 x 10 21 kg, which is the same order of magnitude as our estimate. Now, rather than having to trust blindly whoever first put that number up on a website (most of the other sites probably just copied it from them, after all), we can have a little more confidence in it.
Figure 1.4 says the mass of the atmosphere is 10 19 kg. Assuming the density of the atmosphere is 1 kg/m 3 , estimate the height of Earth’s atmosphere. Do you think your answer is an underestimate or an overestimate? Explain why.
How many piano tuners are there in New York City? How many leaves are on that tree? If you are studying photosynthesis or thinking of writing a smartphone app for piano tuners, then the answers to these questions might be of great interest to you.  However, these are exactly the sorts of estimation problems that people in various tech industries have been asking potential employees to evaluate their quantitative reasoning skills. If building physical intuition and evaluating quantitative claims do not seem like sufficient reasons for you to practice estimation problems, how about the fact that being good at them just might land you a high-paying job?
For practice estimating relative lengths, areas, and volumes, check out this PhET simulation, titled “ Estimation .”


11.11: Sunk Costs
Learning Outcomes
- Recognize sunk costs
There is a piece of machinery sitting in your warehouse. No matter what product your company decides to make, that piece of machinery is worthless. It is old, broken and outdated. But your supervisor comes to you and says “but it cost $50,000.” So what?  We call that a sunk cost.
From a personal financial standpoint, here are some examples of sunk cost to help make the concept make more sense:
Sunk costs are those costs that happened and there is not one thing we can do about it. These costs are never relevant in our decision making process because they already happened!
 Let’s look at an example:
Hupana Running Company bought a machine for $50,000 that was used to make water bottles. The water bottle line never really took off, so Hupana decided to discontinue the line, and focus their efforts on the running shoes that were the profitable portion of the business. Even though, in hindsight, they should have never added a line of water bottles to their product line, the $50,000 they spent on the machine is a sunk cost.
It would be senseless for Hupana to keep making the water bottles to try to recoup the already spent $50,000, but they could try to sell the machine to another company who already makes water bottles!! No matter what they do with the machine, the 50K is a sunk cost, not to be recovered, and irrelevant to any decisions the company makes now. So we need to just let it go.
Practice Questions


12: Connectivity, Communications and Technology
selected template will load here
This action is not available.
In the words of Rucker (1983, 108) ‘the human race is a single vast tapestry, linked by our shared food and air’. In this sense, it is correct that the entire human race is connected through the material world. It is wrong, however, to assume that such connections create any kind of unity. In international relations, when we think of humanity, we do not think of a single, homogenous, peaceful body, but of a number of distinct factions competing, coercing and cooperating to achieve their own end goals. These factions may be groupings such as ethnic, racial or religious divisions or they may be nation-states. They can also be anywhere on a scale from very large to very small. Importantly, however, none of these groupings exist independently of the individual humans within them. The individual is the basic unit at which humanity exists. In this way, individuals are symbiotic with the wider system, with each playing a role in shaping and influencing the other. Humanity consists not only of human bodies, but also of the ideas, the convictions, and the wills contained within human minds. Given this definition, what does it mean for humanity to be connected? In a physical sense, a disconnection has always been present. Each human mind is contained within a human body that exists separately from all others. It is, however, on the metaphysical plain – that of ideas, convictions, and wills – that humanity can be connected. The uniting of many individuals for a common cause, for example, represents a connection of minds leading to action. Such unity can of course arise by complete chance or through non-conscious actions. However, more powerful connections arise when the unity stems from conscious interaction. Central to the concept of connectivity, therefore, is the ability to communicate with others, which we do more and more today via digital means.


2.40: Doctrine of the Affections
Here you’ll find a brief explanation of the Doctrine of the Affections. Like the use of basso continuo, the practice of composing music that expressed a single emotion (affect) is unique to the Baroque era. Later composers wanted the freedom to express contrasting emotions in a single piece of music. One of the most noticeable results of Baroque composers’ adherence to the Doctrine of Affections was the practice of breaking a longer text up into shorter phrases and setting each as a separate movement with music designed to express a single emotion or affect.
For example, the Utrecht Te Deum by George Frideric Handel (a late Baroque composer) consists of ten separate movements, while the Te Deum No. 2 in C by Franz Joseph Haydn (a composer of the late Classical era) is a single piece of music with no separate movements. Note that those two texts are not exactly the same. Handel’s Te Deum is in English while Haydn’s is in the original Latin. But they serve to illustrate the impact of the Doctrine of Affections on music composition in the Baroque.
Introduction
The doctrine of the affections , also known as the doctrine of affects , doctrine of the passions , theory of the affects , or by the German term Affektenlehre (after the German Affekt ; plural Affekte ) was a theory in the aesthetics of painting, music, and theatre, widely used in the Baroque era (1600–1750) (Harnoncourt 1983; Harnoncourt 1988). Literary theorists of that age, by contrast, rarely discussed the details of what was called “pathetic composition,” taking it for granted that a poet should be required to “wake the soul by tender strokes of art” (Rogerson 1953, p. 68). The doctrine was derived from ancient theories of rhetoric and oratory (Buelow 2001). Some pieces or movements of music express one Affekt throughout; however, a skillful composer like Johann Sebastian Bach could express different affects within a movement (Boetticher 2010).
History and Definition
The doctrine of the affections was an elaborate theory based on the idea that the passions could be represented by their outward visible or audible signs. It drew largely on elements with a long previous history, but first came to general prominence in the mid-seventeenth century amongst the French scholar-critics associated with the Court of Versailles, helping to place it at the centre of artistic activity for all of Europe (Rogerson 1953, p. 70). The term itself, however, was only first devised in the twentieth century by German musicologists Hermann Kretzschmar, Harry Goldschmidt, and Arnold Schering, to describe this aesthetic theory (Buelow 2001; Nagley and Bujić 2002).
René Descartes held that there were six basic affects, which can be combined together into numerous intermediate forms (Descartes 1649, p. 94):
- Admiration (admiration)
- Amour (love)
- Haine (hatred)
- Désir (desire)
- Joie (joy)
- Tristesse (sorrow)
Another authority also mentions sadness, anger, and jealousy (Buelow 2001).
Lorenzo Giacomini (1552–1598) in his Orationi e discorsi defined an affection as “a spiritual movement or operation of the mind in which it is attracted or repelled by an object it has come to know as a result of an imbalance in the animal spirits and vapours that flow continually throughout the body” (Giacomini Tebalducci Malespini 1597).
“Affections are not the same as emotions; however, they are a spiritual movement of the mind” (Palisca 1991, p. 3).
A prominent Baroque proponent of the Doctrine of the Affections was Johann Mattheson (Poultney 1996, p. 107).
Contributors and Attributions
- Authored by : Elliott Jones. Provided by : Santa Ana College. Located at : http://www.sac.edu . License : CC BY: Attribution
- Doctrine of the affections. Provided by : Wikipedia. Located at : http://en.Wikipedia.org/wiki/Doctrine_of_the_affections . License : CC BY-SA: Attribution-ShareAlike


13.2: Convergence and the Central Limit Theorem
The Central Limit Theorem
The central limit theorem (CLT) asserts that if random variable \(X\) is the sum of a large class of independent random variables, each with reasonable distributions, then \(X\) is approximately normally distributed. This celebrated theorem has been the object of extensive theoretical research directed toward the discovery of the most general conditions under which it is valid. On the other hand, this theorem serves as the basis of an extraordinary amount of applied work. In the statistics of large samples, the sample average is a constant times the sum of the random variables in the sampling process . Thus, for large samples, the sample average is approximately normal—whether or not the population distribution is normal. In much of the theory of errors of measurement, the observed error is the sum of a large number of independent random quantities which contribute additively to the result. Similarly, in the theory of noise, the noise signal is the sum of a large number of random components, independently produced. In such situations, the assumption of a normal population distribution is frequently quite appropriate.
We consider a form of the CLT under hypotheses which are reasonable assumptions in many practical situations. We sketch a proof of this version of the CLT, known as the Lindeberg-Lévy theorem, which utilizes the limit theorem on characteristic functions, above, along with certain elementary facts from analysis. It illustrates the kind of argument used in more sophisticated proofs required for more general cases.
Consider an independent sequence \(\{X_n: 1 \le n\}\) of random variables. Form the sequence of partial sums
\(S_n = \sum_{i = 1}^{n} X_i\) \(\forall n \ge 1\) with \(E[S_n] = \sum_{i = 1}^{n} E[X_i]\) and \(\text{Var} [S_n] = \sum_{i = 1}^{n} \text{Var} [X_i]\)
Let \(S_n^*\) be the standardized sum and let \(F_n\) be the distribution function for \(S_n^*\). The CLT asserts that under appropriate conditions, \(F_n (t) \to \phi(t)\) as \(n \to \infty\) for all \(t\). We sketch a proof of the theorem under the condition the \(X_i\) form an iid class.
Central Limit Theorem (Lindeberg-Lévy form)
If \(\{X_n: 1 \le n\}\) is iid, with
\(E[X_i] = \mu\), \(\text{Var} [X_i] = \sigma^2\), and \(S_n^* = \dfrac{S_n - n\mu}{\sigma \sqrt{n}}\)
then
\(F_n (t) \to \phi (t)\) as \(n \to \infty\), for all \(t\)
IDEAS OF A PROOF
There is no loss of generality in assuming \(\mu = 0\). Let \(\phi\) be the common characteristic function for the \(X_i\), and for each \(n\) let \(\phi_n\) be the characteristic function for \(S_n^*\). We have
\(\varphi (t) = E[e^{itX}]\) and \(\varphi_n (t) = E[e^{itS_n^*}] = \varphi^n (t/\sigma \sqrt{n})\)
Using the power series expansion of \(\varphi\) about the origin noted above, we have
\(\varphi (t) = 1 - \dfrac{\sigma^2 t^2}{2} + \beta (t)\) where \(\beta (t) = o (t^2)\) as \(t \to 0\)
This implies
\([\varphi (t/\sigma \sqrt{n}) - (1 - t^2/2n)] = [\beta (t /\sigma \sqrt{n})] = o(t^2/\sigma^2 n)\)
so that
\(n[\varphi (t/\sigma \sqrt{n}) - (1 - t^2/2n)] \to 0\) as \(n \to \infty\)
A standard lemma of analysis ensures
\((1 - \dfrac{t^2}{2n})^n \to e^{-t^2/2}\) as \(n \to \infty\)
so that
\(\varphi (t/\sigma \sqrt{n}) \to e^{-t^2/2}\) as \(n \to \infty\) for all \(t\)
By the convergence theorem on characteristic functions, above, \(F_n(t) \to \phi (t)\).
— □
The theorem says that the distribution functions for sums of increasing numbers of the X i converge to the normal distribution function, but it does not tell how fast. It is instructive to consider some examples, which are easily worked out with the aid of our m-functions.
Demonstration of the central limit theorem
Discrete examples
We first examine the gaussian approximation in two cases. We take the sum of five iid simple random variables in each case. The first variable has six distinct values; the second has only three. The discrete character of the sum is more evident in the second case. Here we use not only the gaussian approximation, but the gaussian approximation shifted one half unit (the so called continuity correction for integer-values random variables). The fit is remarkably good in either case with only five terms.
A principal tool is the m-function diidsum (sum of discrete iid random variables). It uses a designated number of iterations of mgsum.
Example \(\PageIndex{1}\) First random variable
X = [-3.2 -1.05 2.1 4.6 5.3 7.2]; PX = 0.1*[2 2 1 3 1 1]; EX = X*PX' EX = 1.9900 VX = dot(X.^2,PX) - EX^2 VX = 13.0904 [x,px] = diidsum(X,PX,5); % Distribution for the sum of 5 iid rv F = cumsum(px); % Distribution function for the sum stairs(x,F) % Stair step plot hold on plot(x,gaussian(5*EX,5*VX,x),'-.') % Plot of gaussian distribution function % Plotting details (see Figure 13.2.1)
Figure 13.2.1.
Distribution for the sum of five iid random variables.
Example \(\PageIndex{2}\) Second random variable
X = 1:3; PX = [0.3 0.5 0.2]; EX = X*PX' EX = 1.9000 EX2 = X.^2*PX' EX2 = 4.1000 VX = EX2 - EX^2 VX = 0.4900 [x,px] = diidsum(X,PX,5); % Distribution for the sum of 5 iid rv F = cumsum(px); % Distribution function for the sum stairs(x,F) % Stair step plot hold on plot(x,gaussian(5*EX,5*VX,x),'-.') % Plot of gaussian distribution function plot(x,gaussian(5*EX,5*VX,x+0.5),'o') % Plot with continuity correction % Plotting details (see Figure 13.2.2)
Figure 13.2.2.
Distribution for the sum of five iid random variables.
As another example, we take the sum of twenty one iid simple random variables with integer values. We examine only part of the distribution function where most of the probability is concentrated. This effectively enlarges the x-scale, so that the nature of the approximation is more readily apparent.
Example \(\PageIndex{3}\) Sum of twenty-one iid random variables
X = [0 1 3 5 6]; PX = 0.1*[1 2 3 2 2]; EX = dot(X,PX) EX = 3.3000 VX = dot(X.^2,PX) - EX^2 VX = 4.2100 [x,px] = diidsum(X,PX,21); F = cumsum(px); FG = gaussian(21*EX,21*VX,x); stairs(40:90,F(40:90)) hold on plot(40:90,FG(40:90)) % Plotting details (see Figure 13.2.3)
Figure 13.2.3.
Distribution for the sum of twenty one iid random variables.
Absolutely continuous examples
By use of the discrete approximation, we may get approximations to the sums of absolutely continuous random variables. The results on discrete variables indicate that the more values the more quickly the conversion seems to occur. In our next example, we start with a random variable uniform on (0, 1).
Example \(\PageIndex{4}\) Sum of three iid, uniform random variables.
Suppose \(X\) ~ uniform (0, 1). Then \(E[X] = 0.5\) and \(\text{Var} [X] = 1/12\).
tappr Enter matrix [a b] of x-range endpoints [0 1] Enter number of x approximation points 100 Enter density as a function of t t<=1 Use row matrices X and PX as in the simple case EX = 0.5; VX = 1/12; [z,pz] = diidsum(X,PX,3); F = cumsum(pz); FG = gaussian(3*EX,3*VX,z); length(z) ans = 298 a = 1:5:296; % Plot every fifth point plot(z(a),F(a),z(a),FG(a),'o') % Plotting details (see Figure 13.2.4)
Figure 13.2.4.
Distribution for the sum of three iid uniform random variables.
For the sum of only three random variables, the fit is remarkably good. This is not entirely surprising, since the sum of two gives a symmetric triangular distribution on (0, 2). Other distributions may take many more terms to get a good fit. Consider the following example.
Example \(\PageIndex{5}\) Sum of eight iid random variables
Suppose the density is one on the intervals (-1, -0.5) and (0.5, 1). Although the density is symmetric, it has two separate regions of probability. From symmetry. \(E[X] = 0\). Calculations show \(\text{Var}[X] = E[X^2] = 7/12\). The MATLAB computations are:
tappr Enter matrix [a b] of x-range endpoints [-1 1] Enter number of x approximation points 200 Enter density as a function of t (t<=-0.5)|(t>=0.5) Use row matrices X and PX as in the simple case [z,pz] = diidsum(X,PX,8); VX = 7/12; F = cumsum(pz); FG = gaussian(0,8*VX,z); plot(z,F,z,FG) % Plottting details (see Figure 13.2.5)
Figure 13.2.5
. Distribution for the sum of eight iid uniform random variables.
Although the sum of eight random variables is used, the fit to the gaussian is not as good as that for the sum of three in Example 13.2.4 . In either case, the convergence is remarkable fast—only a few terms are needed for good approximation.
Convergence phenomena in probability theory
The central limit theorem exhibits one of several kinds of convergence important in probability theory, namely convergence in distribution (sometimes called weak convergence). The increasing concentration of values of the sample average random variable A n with increasing \(n\) illustrates convergence in probability . The convergence of the sample average is a form of the so-called weak law of large numbers . For large enough n the probability that \(A_n\) lies within a given distance of the population mean can be made as near one as desired. The fact that the variance of \(A_n\) becomes small for large n illustrates convergence in the mean (of order 2).
\(E[|A_n - \mu|^2] \to 0\) as \(n \to \infty\)
In the calculus, we deal with sequences of numbers. If \(\{a_n: 1 \le n\}\) s a sequence of real numbers, we say the sequence converges iff for \(N\) sufficiently large \(a_n\) approximates arbitrarily closely some number \(L\) for all \(n \ge N\). This unique number \(L\) is called the limit of the sequence. Convergent sequences are characterized by the fact that for large enough \(N\), the distance \(|a_n - a_m|\) between any two terms is arbitrarily small for all \(n\), \(m \ge N\). Such a sequence is said to be fundamental (or Cauchy ). To be precise, if we let \(\epsilon > 0\) be the error of approximation, then the sequence is
- Convergent iff there exists a number \(L\) such that for any \(\epsilon > 0\) there is an \(N\) such that
\(|L - a_n| \le \epsilon\) for all \(n \ge N\)
- Fundamental iff for any \(\epsilon > 0\) there is an \(N\) such that
\(|a_n - a_m| \le \epsilon\) for all \(n, m \ge N\)
As a result of the completeness of the real numbers, it is true that any fundamental sequence converges (i.e., has a limit). And such convergence has certain desirable properties. For example the limit of a linear combination of sequences is that linear combination of the separate limits; and limits of products are the products of the limits.
The notion of convergent and fundamental sequences applies to sequences of real-valued functions with a common domain. For each \(x\) in the domain, we have a sequence
\(\{f_n (x): 1 \le n\}\) of real numbers. The sequence may converge for some \(x\) and fail to converge for others.
A somewhat more restrictive condition (and often a more desirable one) for sequences of functions is uniform convergence. Here the uniformity is over values of the argument \(x\). In this case, for any \(\epsilon > 0\) there exists an \(N\) which works for all \(x\) (or for some suitable prescribed set of \(x\)).
These concepts may be applied to a sequence of random variables , which are real-valued functions with domain \(\Omega\) and argument \(\omega\). Suppose \(\{X_n: 1 \le n\}\) is is a sequence of real random variables. For each argument \(\omega\) we have a sequence \(\{X_n (\omega): 1 \le n\}\) of real numbers. It is quite possible that such a sequence converges for some ω and diverges (fails to converge) for others. As a matter of fact, in many important cases the sequence converges for all \(\omega\) except possibly a set (event) of probability zero. In this case, we say the seqeunce converges almost surely (abbreviated a.s.). The notion of uniform convergence also applies. In probability theory we have the notion of almost uniform convergence. This is the case that the sequence converges uniformly for all \(\omega\) except for a set of arbitrarily small probability.
The notion of convergence in probability noted above is a quite different kind of convergence. Rather than deal with the sequence on a pointwise basis, it deals with the random variables as such. In the case of sample average, the “closeness” to a limit is expressed in terms of the probability that the observed value \(X_n (\omega)\) should lie close the the value \(X(\omega)\) of the limiting random variable. We may state this precisely as follows:
A sequence \(\{X_n: 1 \le n\}\) converges to X in probability , designated \(X_n \stackrel{P}\longrightarrow X\) iff for any \(\epsilon > 0\).
\(\text{lim}_n P(|X - X_n| > \epsilon) = 0\)
There is a corresponding notion of a sequence fundamental in probability.
The following schematic representation may help to visualize the difference between almost-sure convergence and convergence in probability. In setting up the basic probability model, we think in terms of “balls” drawn from a jar or box. Instead of balls, consider for each possible outcome \(\omega\) a “tape” on which there is the sequence of values \(X_1 (\omega)\), \(X_2 (\omega)\), \(X_3 (\omega)\), \(\cdot\cdot\cdot\).
- If the sequence of random variable converges a.s. to a random variable \(X\), then there is an set of “exceptional tapes” which has zero probability. For all other tapes, \(X_n (\omega) \to X(\omega)\). This means that by going far enough out on any such tape, the values \(X_n (\omega)\) beyond that point all lie within a prescribed distance of the value \(X(\omega)\) of the limit random variable.
- If the sequence converges in probability, the situation may be quite different. A tape is selected. For \(n\) sufficiently large, the probability is arbitrarily near one that the observed value \(X_n (\omega)\) lies within a prescribed distance of \(X(\omega)\). This says nothing about the values \(X_m (\omega)\) on the selected tape for any larger \(m\). In fact, the sequence on the selected tape may very well diverge.
It is not difficult to construct examples for which there is convergence in probability but pointwise convergence for no \(\omega\). It is easy to confuse these two types of convergence. The kind of convergence noted for the sample average is convergence in probability (a “weak” law of large numbers). What is really desired in most cases is a.s. convergence (a “strong” law of large numbers). It turns out that for a sampling process of the kind used in simple statistics, the convergence of the sample average is almost sure (i.e., the strong law holds). To establish this requires much more detailed and sophisticated analysis than we are prepared to make in this treatment.
The notion of mean convergence illustrated by the reduction of \(\text{Var} [A_n]\) with increasing \(n\) may be expressed more generally and more precisely as follows. A sequence \(\{X_n: 1 \le n\}\) converges in the mean of order \(p\) to \(X\) iff
\(E[|X - X_n|^p] \to 0\) as \(n \to \infty\) designated \(X_n \stackrel{L^p}\longrightarrow X\); as \(n \to \infty\)
If the order \(p\) is one, we simply say the sequence converges in the mean. For \(p = 2\), we speak of mean-square convergence .
The introduction of a new type of convergence raises a number of questions.
- There is the question of fundamental (or Cauchy) sequences and convergent sequences.
- Do the various types of limits have the usual properties of limits? Is the limit of a linear combination of sequences the linear combination of the limits? Is the limit of products the product of the limits?
- What conditions imply the various kinds of convergence?
- What is the relation between the various kinds of convergence?
Before sketching briefly some of the relationships between convergence types, we consider one important condition known as uniform integrability . According to the property (E9b) for integrals
\(X\) is integrable iff \(E[I_{\{|X_i|>a\}} |X_t|] \to 0\) as \(a \to \infty\)
Roughly speaking, to be integrable a random variable cannot be too large on too large a set. We use this characterization of the integrability of a single random variable to define the notion of the uniform integrability of a class.
Definition
An arbitray class \(\{X_t: t \in T\}\) is uniformly integrable (abbreviated u.i.) with respect to probability measure \(P\) iff
\(\text{sup}_{t \in T} E[I_{\{|X_i| > a\}} | X_t|] \to 0\) as \(a \to \infty\)
This condition plays a key role in many aspects of theoretical probability.
The relationships between types of convergence are important. Sometimes only one kind can be established. Also, it may be easier to establish one type which implies another of more immediate interest. We simply state informally some of the important relationships. A somewhat more detailed summary is given in PA, Chapter 17. But for a complete treatment it is necessary to consult more advanced treatments of probability and measure.
Relationships between types of convergence for probability measures
Consider a sequence \(\{X_n: 1 \le n\}\) of random variables.
It converges almost surely iff it converges almost uniformly.
If it converges almost surely, then it converges in probability.
It converges in mean, order \(p\), iff it is uniformly integrable and converges in probability.
If it converges in probability, then it converges in distribution (i.e. weakly).
Various chains of implication can be traced. For example
- Almost sure convergence implies convergence in probability implies convergence in distribution.
- Almost sure convergence and uniform integrability implies convergence in mean \(p\).
We do not develop the underlying theory. While much of it could be treated with elementary ideas, a complete treatment requires considerable development of the underlying measure theory. However, it is important to be aware of these various types of convergence, since they are frequently utilized in advanced treatments of applied probability and of statistics.


4.1: Introduction
After completing this section, you should be able to use the terms “ether,” “diethyl ether ” and “ethyl ether ” appropriately in context.
Ethers and Epoxides
Ethers (R–O–R′) , like the alcohols, are organic derivatives of water, but they have two organic groups bonded to the same oxygen atom rather than one. The organic groups might be alkyl, aryl, or vinylic, and the oxygen atom might be in an open chain or a ring.
Perhaps the most well-known ether is diethyl ether, which has a long history of medicinal use as an anesthetic and industrial use as a solvent. Other useful ethers include anisole, a pleasant-smelling aromatic ether used in perfumery, and tetrahydrofuran (THF), a cyclic ether often used as a solvent.
Thiols and Sulfides
Thiols (R–S–H, otherwise called thio alcohols or mercaptans) and sulfides (R–S–R′) are sulfur analogs of alcohols and ethers, respectively. Both functional groups are found in various biomolecules, although not as commonly as their oxygen-containing relatives.
Make certain that you can define, and use in context, the key terms below.
- ether (\(\ce{R-O-R′}\))
- sulfide (\(\ce{R-S-R′}\))
- thiol (\(\ce{R-S-H}\))
As defined in the textbook, an “ether” is a substance with the general formula (\(\ce{R-O-R′}\)) where \(\ce{R}\) and \(\ce{R′}\) are alkyl, aryl, vinyl or allyl groups. However, the word “ether” is also commonly used to refer to the specific compound, \(\ce{CH3-CH2-O-CH2-CH3}\), which is correctly called “diethyl ether.” Further confusion can arise because some chemists refer to “diethyl ether ” as “ethyl ether.” In this course, “ether ” will be used to refer to the class of compounds with the structure (\(\ce{R-O-R′}\)) ; diethyl ether will refer to the compound, \(\ce{CH3-CH2-O-CH2-CH3}\); and “ethyl ether ” will not be used.


13.19: Reading- Customer Relationship Management (CRM) Systems and IMC
Using CRM to Support Marketing Communication
Earlier in this course we introduced customer relationship management (CRM) systems, which serve key functions for marketing, sales, and account management. These systems capture data about customers as well as an organization’s interactions with these customers. They also provide tools to help marketers and salespeople better manage customer relationships and meet their customers’ needs. CRM systems generally capture and maintain information about prospective as well as current customers, making them very useful to both marketing and sales processes.
The overall business goals of CRM systems are to help organizations 1) capture new leads and move them through the sales process; 2) support and manage relationships with current customers to maximize their lifetime value to the company; and 3) boost productivity and lower the overall costs of marketing, sales, and account management.
CRM systems can be complicated to implement because they are intended to support a complex set of processes and business functions. At times, the systems themselves are so sophisticated that organizations never fully use all their capabilities. However, a wide selection of CRM systems are now available at different levels of pricing and complexity. As managers refine their understanding of how CRM systems can help them achieve their business and marketing objectives, they can identify suitable systems and implementation approaches to fit their needs.
CRM Uses in Sales and Marketing
CRM systems are transformational for marketing communication because they allow marketers to use customer data to personalize their interactions to fit the unique needs of individuals. When marketers or salespeople know more about the customer–thanks to information the CRM telling them who the customer is, how she found the company, what information she has requested, and so forth–they can anticipate that person’s needs and tailor the next set of interactions to help her progress through the decision-making process.
Information gained through CRM initiatives can support the development of sales and marketing strategy by developing the organization’s knowledge in key areas: identifying customer segments, improving customer retention, improving product offerings (by better understanding customer needs), improving the customer experience, and identifying the organization’s most profitable customers.
In the following video, Amazon CEO Jeff Bezos explains how his company captures data about what products customers buy to tailor the shopping experience every time someone visits the Amazon Web site. The customer data, captured in Amazon’s CRM system, feeds a “recommendation engine” to suggest products Amazon thinks customers will want, based on their prior purchases and the purchase histories of other customers who buy the same types of things they do.
You can view the transcript for “Personalize Recommendations Online” here (opens in new window) .
Marketing and IMC
CRM systems for marketing help an organization identify and target potential clients and generate leads. A key marketing capability is tracking and measuring the effectiveness of multichannel campaigns, including email, search, social media, telephone, direct mail, and other channels. The CRM system can monitor which individuals click, respond, and participate in any call to action. It also reports overall campaign metrics such as clicks, responses, leads generated, deals closed, and revenue. Many CRM systems are capable of tracking customer interactions and nurturing relationships from first contact through the closed sale and beyond, providing a 360-degree view of the customer relationship.
Marketing automation uses data from a CRM system to help marketers coordinate and manage marketing interaction across multiple digital-marketing channels—email, Web sites, social media, etc. Marketers use marketing automation systems to design and execute marketing campaigns targeted to specific segments based on various criteria. Campaigns might target individuals in the CRM system by job title, industry, or geography, for example, or some combination of individual traits. Campaigns may also be designed around the stage of the sales process, so that everyone at a certain stage receives a weekly email or other touch point to provide further information and help nudge the contact toward the buying decision.
Marketing automation is particularly good for automating repetitive tasks, so that marketers can maintain interaction and build relationships with a large number of contacts simultaneously. But because CRM data tracks what each prospective customer is looking for and where they are in the decision process, marketers can orchestrate an appropriate set of automated interactions to keep the relationship “warm” and eventually move them toward a sale.
Based on the characteristics and behavior of the prospective customers captured in the CRM system, marketers can also direct the system to conduct “lead scoring.” Lead scoring involves assigning points to marketing or sales leads to help marketers prioritize who is most ready to buy and move them toward a buying decision. For instance, B2B marketers could define lead scoring as follows:
- 8 points to every contact that resides within a targeted geography
- 7 points for sharing an email address
- 8 points for sharing job title
- 15 points to any contact who holds a position as vice president or higher in their organization
- 3 points each time a contact visits the company’s Web site
- 5 points each time a contact downloads a document or views a video on the Web site
The system calculates the total score for each lead, indicating which contacts are the best-prepared targets for making a sale. Marketing and sales team members then target appropriate marketing campaigns and outreach opportunities designed to help leads continue progressing through the sales process.
Sales Force Automation
Sales force automation (SFA) is another function of many CRM systems. It involves using software to streamline all phases of the sales process, minimizing the time that sales representatives need to spend on each phase. This enables a business to use fewer sales representatives to manage their clients. At the core of SFA is a contact management system for tracking and recording every stage in the sales process for each prospective client, from initial contact to final disposition. Many SFA applications also provide reporting and analytical tools to give salespeople and managers insights into specific sales opportunities, territories, and forecasting future sales. Some also offer workflow automation tools to help streamline sales-related processes and improve both the quality and efficiency of sales teams’ interactions with prospective customers.
As with marketing automation, sales force automation can help salespeople streamline the task of communicating with prospective customers during the sales process. A salesperson can can set up automated triggers to send specific communications to sales leads at various points in the process. For example, when an individual requests a product demonstration, a sales automation system might automatically send a set of communications to set up a virtual meeting for the demo. The system could then check whether that lead has downloaded a marketing piece comparing product features versus competitors’. If not, the system would automatically send that informational piece in a follow-up email message.
CRM and Marketing Automation in Action
The following video explains how a Precor, a B2B gym-equipment company, uses its CRM system with marketing automation to better understand and anticipate customer needs. It translates the information from the CRM system to more efficient marketing, better customer service, and increased sales. This video was produced by Salesforce.com, the CRM company that provides the two systems Precor uses: Sales Cloud (CRM) and Pardot (marketing automation system). Notice that the video not only trumpets the value of these systems, but the video itself is a polished piece of marketing communication (promoting Salesforce.com).
Contributors and Attributions
- Revision and adaptation. Provided by : Lumen Learning. License : CC BY-SA: Attribution-ShareAlike
- CRM and Personal Selling, from Boundless Marketing. Provided by : Boundless. Located at : https://courses.lumenlearning.com/boundless-marketing/ . License : CC BY-SA: Attribution-ShareAlike
- Personalized Recommendations Online. Provided by : BBC. Located at : https://youtu.be/I952kdcpE3Y . License : CC BY-NC-ND: Attribution-NonCommercial-NoDerivatives
- Pardot Customer Story: Precor. Provided by : Pardot. Located at : https://youtu.be/GVH4THFmjrU?list=PLNmYf_HyHDDo5uLP-Fowrm5LIKNM8zj0d . License : All Rights Reserved . License Terms : Standard YouTub license


6.A: Quantum Mechanics (Answers)
-
- Last updated
- Save as PDF
Check Your Understanding
7.1. \(\displaystyle (3+4i)(3−4i)=9−16i^2=25\)
7.2. \(\displaystyle A=\sqrt{2/L}\)
7.3. \(\displaystyle (1/2−1/π)/2=9%\)
7.4. \(\displaystyle 4.1×10^{−8}eV; 1.1×10^{−5}nm\)
7.5. \(\displaystyle 0.5mω^2x^2ψ(x)∗ψ(x)\)
7.6. None. The first function has a discontinuity; the second function is double-valued; and the third function diverges so is not normalizable.
7.7. a. 9.1%;
b. 25%
7.8. a. 295 N/m;
b. 0.277 eV
7.9. \(\displaystyle ⟨x⟩=0\)
7.10. \(\displaystyle L_{proton}/L_{electron}=\sqrt{m_e/m_p}=2.3%\)
Conceptual Questions
1. \(\displaystyle 1/\sqrt{L}\), where \(\displaystyle L=length\); 1/L , where \(\displaystyle L=length\)
3. The wave function does not correspond directly to any measured quantity. It is a tool for predicting the values of physical quantities.
5. The average value of the physical quantity for a large number of particles with the same wave function.
7. Yes, if its position is completely unknown. Yes, if its momentum is completely unknown.
9. No. According to the uncertainty principle, if the uncertainty on the particle’s position is small, the uncertainty on its momentum is large. Similarly, if the uncertainty on the particle’s position is large, the uncertainty on its momentum is small.
11. No, it means that predictions about the particle (expressed in terms of probabilities) are time-independent.
13. No, because the probability of the particle existing in a narrow (infinitesimally small) interval at the discontinuity is undefined.
15. No. For an infinite square well, the spacing between energy levels increases with the quantum number n . The smallest energy measured corresponds to the transition from n = 2 to 1, which is three times the ground state energy. The largest energy measured corresponds to a transition from \(\displaystyle n=∞\) to 1, which is infinity. (Note: Even particles with extremely large energies remain bound to an infinite square well—they can never “escape”)
17. No. This energy corresponds to \(\displaystyle n=0.25\), but n must be an integer.
19. Because the smallest allowed value of the quantum number n for a simple harmonic oscillator is 0. No, because quantum mechanics and classical mechanics agree only in the limit of large nn.
21. Yes, within the constraints of the uncertainty principle. If the oscillating particle is localized, the momentum and therefore energy of the oscillator are distributed.
23. doubling the barrier width
25. No, the restoring force on the particle at the walls of an infinite square well is infinity.
Problems
27. \(\displaystyle ∣ψ(x)∣^2sin^2ωt\)
29. (a) and (e), can be normalized
31. a. \(\displaystyle A=\sqrt{2α/π}\);
b. \(\displaystyle probability=29.3%\);
c. \(\displaystyle ⟨x⟩=0⟨x⟩=0\);
d. \(\displaystyle ⟨p⟩=0\);
e. \(\displaystyle ⟨K⟩=α^2ℏ^2/2m\)
33. a. \(\displaystyle Δp≥2.11×10^{−34}N⋅s\);
b. \(\displaystyle Δv≥6.31×10^{−8}m\);
c. \(\displaystyle Δv/\sqrt{k_BT/m_α}=5.94×10^{−11}\)
35. \(\displaystyle Δτ≥1.6×10^{−25}s\)
37. a. \(\displaystyle Δf≥1.59MHz\);
b. \(\displaystyle Δω/ω_0=3.135×10^{−9}\)
39. Carrying out the derivatives yields \(\displaystyle k^2=\frac{ω^2}{c^2}\).
41. Carrying out the derivatives (as above) for the sine function gives a cosine on the right side the equation, so the equality fails. The same occurs for the cosine solution.
43. \(\displaystyle E=ℏ^2k^2/2m\)
45. \(\displaystyle ℏ^2k^2ℏ\); The particle has definite momentum and therefore definite momentum squared.
47. 9.4 eV, 64%
49. 0.38 nm
51. 1.82 MeV
53. 24.7 nm
55. \(\displaystyle 6.03Å\)
57. a.
The wave functions for the n=1 through n=5 states of the electron in an infinite square well are shown. Each function is displaced vertically by its energy, measured in meV. The n=1 state is the first half wave of the sine function. The n=2 function is the first full wave of the sine function. The n=3 function is the first one and a half waves of the sine function. The n=4 function is the first two waves of the sine function. The n=5 function is the first two and a half waves of the sine function. ;
b. \(\displaystyle λ_{5→3}=12.9nm,λ_{3→1}=25.8nm,λ_{4→3}=29.4nm\)
59. proof
61. \(\displaystyle 6.662×10^{14}Hz\)
63. \(\displaystyle n≈2.037×10^{30}\)
65. \(\displaystyle ⟨x⟩=0.5mω^2⟨x^2⟩=ℏω/4\); \(\displaystyle ⟨K⟩=⟨E⟩−⟨U⟩=ℏω/4\)
67. proof
69. A complex function of the form, \(\displaystyle Ae^{iϕ}\), satisfies Schrӧdinger’s time-independent equation. The operators for kinetic and total energy are linear, so any linear combination of such wave functions is also a valid solution to Schrӧdinger’s equation. Therefore, we conclude that Equation 7.68 satisfies Equation 7.61, and Equation 7.69 satisfies Equation 7.63.
71. a. 4.21%;
b. 0.84%;
c. 0.06%
73. a. 0.13%;
b. close to 0%
75. 0.38 nm
Additional Problems
77. proof
79. a. 4.0 %;
b. 1.4 %;
c. 4.0%;
d. 1.4%
81. a. \(\displaystyle t=mL^2/h=2.15×10^{26}years\);
b. \(\displaystyle E_1=1.46×10^{−66}J,K=0.4J\)
83. proof
85. 1.2 N/m
87. 0
Challenge Problems
89. 19.2µm;11.5µm19.2µm;11.5µm
91. 3.92%
93. proof


7.1.4A: The Functionalist Perspective on Deviance
-
- Last updated
- Save as PDF
Functionalism claims that deviance help to create social stability by presenting explanations of non-normative and normative behaviors.
Learning Objectives
- Describe the functionalist view of deviance in society
Key Points
- A structural functionalist approach emphasizes social solidarity, divided into organic and mechanical typologies, and stability in social structures.
- Deviance provides the key to understanding the disruption and recalibration of society that occurs over time. Some traits that could cause social disruption will be stigmatized.
- Systems of deviance create norms and tell members of a given society on how to behave by laying out patterns of acceptable and unacceptable behavior.
- Deviance allows for group majorities to unite around their worldview, often at the expense of those marked as deviant. Social parameters create boundaries between populations and enable an “us-versus-them” mentality within the two groups.
- Being marked as deviant can actually bolster solidarity within the marked community as members take pride and ownership in their stigmatized identity.
- Some traits will be stigmatized and can potentially cause social disruption. However, as traits become more mainstream, society will gradually adjust to incorporate the formerly stigmatized traits.
Key Terms
- structural functionalism : A sociological approach that looks at society through a macro-level orientation, which is a broad focus on the social structures that shape society as a whole.
- population : A count of the number of residents within a political or geographical boundary, such as a town, a nation, or the world or of the number of individuals belonging to a particular group.
- Social Parameters : The given rules and norms in a given social situation.
What function does deviance play in society? This is a question asked by sociologists subscribing to the school of structural functionalism. Structural functionalism has its roots in the very origins of sociological thought and the development of sociology as a discipline. Though precursors of structural functionalism have been in existence since the mid-1800’s, structural functionalism was solidified by Émile Durkheim in the late nineteenth century. A structural functionalist approach emphasizes social solidarity, divided into organic and mechanical typologies, and stability in social structures. Structural functionalists ask “How does any given social phenomenon contribute to social stability?” This question cannot be answered without investigating deviance.
Functions of Crime : This is a short clip from the “Functions of Crime” segment of their new seven-part DVD “Short Cuts to Sociology: Crime and Deviance. ”
For the structural functionalist, deviance serves two primary roles in creating social stability. First, systems of recognizing and punishing deviance create norms and tell members of a given society how to behave by laying out patterns of acceptable and unacceptable behavior. In order to avoid unsettling society, one must be aware of what behaviors are marked as deviant. Second, these social parameters create boundaries between populations and enable an “us-versus-them” mentality within different groups. Deviance allows for the majorities to unite around their normativity, at the expense of those marked as deviant. Conversely, being marked as deviant can actual bolster solidarity within the marked community, as members take pride and ownership in their stigmatized identity and create cohesive units of their own (for example, members of the LGBT community unifying around Pride).
From a structural functionalist perspective, then, how does society change, particularly in regards to establishing norms and deviant behaviors? Deviance provides the key to understanding the disruption and re-calibration of society that occurs over time. Some traits will be stigmatized and can potentially cause social disruption. However, as traits become more mainstream, society will gradually adjust to incorporate the formerly stigmatized traits.
Take, again, the example of homosexuality. In urban America 50 years ago, homosexual behavior was considered deviant. On the one hand, this fractured society into those marked as homosexuals and those unmarked as normative heterosexuals. While this us-versus-them mentality solidified social identities and solidarities within the two categories, there was an overarching social schism. As time went on, homosexuality has come to be accepted as somewhat more mainstream. Accordingly, what originally appears as a fracturing of society actually reinforces social stability by enabling mechanisms for social adjustment and development.


16.12: Regulating Gene Expression in Cell Development - Gene Expression in Stem Cells
- Discuss the types of cell division that can occur to add cells during development
Adding cells through cellular division
Stem cells are undifferentiated biological cells found in multicellular organisms, that can differentiate into specialized cells (asymmetric division) or can divide to produce more stem cells (symmetric division). In mammals, there are two broad types of stem cells: embryonic stem cells, which are isolated from the inner cell mass of blastocysts, and adult stem cells, which are found in various tissues. In adult organisms, stem cells and progenitor cells act as a repair system for the body by replenishing adult tissues. In a developing embryo, stem cells can differentiate into all of the specialized cells (including ectoderm, endoderm and mesoderm cells) but also maintain the normal turnover of regenerative organs, such as blood, skin, or intestinal tissues. The pathway that is taken to produced specialized cells included: the embryonic cells develop from totipotent cells, to pluripotent cells which undergo differentiation and become more specialized. The key component however, in the ability to maintain tissues is the ability to maintain a key of stem cells.
There are three accessible sources of autologous adult stem cells in humans: (1) bone marrow, which requires extraction by harvesting (i.e., drilling into bone); (2) adipose tissue (lipid cells), which requires extraction by liposuction; and (3) blood, which requires extraction through apheresis (wherein blood is drawn from the donor, passed through a machine that extracts the stem cells, and returned to the donor). Stem cells can also be taken from umbilical cord blood just after birth. Of all the stem cell types, autologous harvesting involves the least risk. By definition, autologous cells are obtained from one’s own body, just as one may bank his or her own blood for elective surgical procedures. Highly plastic adult stem cells are routinely used in medical therapies, for example in bone marrow transplantation. Stem cells can now be artificially grown and differentiated into specialized cell types with characteristics consistent with muscle or nerve cells through cell culture. Embryonic cell lines and autologous embryonic stem cells generated through therapeutic cloning have also been proposed as promising candidates for future therapies.
Symmetric and asymmetric cell division
To ensure self-renewal, stem cells undergo two types of cell division: symmetric and asymmetric. Symmetric division gives rise to two identical daughter cells both endowed with stem cell properties. Asymmetric division, on the other hand, produces only one stem cell and a progenitor cell with limited self-renewal potential. Progenitors can go through several rounds of cell division themselves before terminally differentiating into a mature cell.. It is possible that the molecular distinction between symmetric and asymmetric division lies in differential segregation of cell membrane proteins between the daughter cells. An alternative theory is that stem cells remain undifferentiated due to environmental cues in their particular niche. Stem cells differentiate when they leave that niche or no longer receive those signals.
An asymmetric cell division produces two daughter cells with different cellular fates. This is in contrast to normal symmetric cell divisions, which give rise to daughter cells of equivalent fates. Notably, stem cells divide asymmetrically to give rise to two distinct daughter cells: one copy of the original stem cell as well as a second daughter programmed to differentiate into a non-stem cell fate.
In principle, there are two mechanisms by which distinct properties may be conferred on the daughters of a dividing cell. In one, the daughter cells are initially equivalent but a difference is induced by signaling between the cells, from surrounding cells, or from the precursor cell. This mechanism is known as extrinsic asymmetric cell division. Extrinsic factors involve interactions with neighboring cells and the micro and macro environment of the precursor cell.
In the second mechanism, the prospective daughter cells are inherently different at the time of division of the mother cell. Because this latter mechanism does not depend on interactions of cells with each other or with their environment, it must rely on intrinsic asymmetry. The term asymmetric cell division usually refers to such intrinsic asymmetric divisions. Intrinsic factors generally involve differing amounts of cell-fate determinants being distributed into each daughter cell.
Animals are made up of a vast number of distinct cell types. During development, the zygote undergoes many cell divisions that give rise to various cell types, including embryonic stem cells. Asymmetric divisions of these embryonic cells gives rise to one cell of the same potency (self-renewal), and another that may be of the same potency or stimulated to further differentiate into specialized cell types such as neurons. Asymmetric division of stem cells plays a key role in development by allowing for the differentiation of a subset of daughter cells while maintaining stem cell pluripotency. Since it can be controlled by both intrinsic and extrinsic factors, upon delineating these particular factors it may be possible to use this knowledge in applications of tissue and whole organ generation.
Key Points
- Symmetric cell division of stem cells ensures that a constant pool of stem cells is available by giving rise to two identical daughter cells both endowed with stem cell properties.
- Asymmetric division of stem cells results in the production of only one stem cell and a progenitor cell with limited self-renewal potential.
- Progenitor cells that are produced via asymmetric cell division will go through additional rounds of cell division until they are terminally differentiated into a mature, specialized cell.
- Asymmetric division can be controlled by both intrinsic and extrinsic factors.
- Intrinsic factors involve differing amounts of cell-fate determinants being distributed into each daughter cell, while extrinsic factors involve interactions with neighboring cells and the micro and macro environment of the precursor cell.
Key Terms
- totipotency : the ability of a cell to produce differentiated cells upon division
- progenitor cell : a biological cell that, like a stem cell, has a tendency to differentiate into a specific type of cell, but is already more specific than a stem cell and is pushed to differentiate into its “target” cell.
- autologous : derived from part of the same individual (i.e. from the recipient rather than the donor)
- morula : a spherical mass of blastomeres that forms following the splitting of a zygote; it becomes the blastula
- pluripotent : able to develop into more than one mature cell or tissue type, but not all


6.4: The Greatest Common Factor
Factoring Method
In the last two types of problems, we knew one of the factors and were able to determine the other factor through division. Suppose, now, we’re given the product without any factors. Our problem is to find the factors, if possible. This procedure and the previous two procedures are based on the distributive property.
We will use the distributive property in reverse.
\(\underbrace{a b+a c}_{\text {product }}=\underbrace{a(b+c)}_{\text {factors }}\)
We notice that in the product, \(a\) is common to both terms. (In fact, \(a\) is a common factor of both terms.) Since \(a\) is common to both terms, we will factor it out and write
\(a ()\)
Now we need to determine what to place inside the parentheses. This is the procedure of the previous section. Divide each term of the product by the known factor \(a\).
\(\dfrac{ab}{a} = b\) and \(\dfrac{ac}{a} = c\)
Thus, \(b\) and \(c\) are the required terms of the other factor. Hence,
\(ab + ac = a(b+c)\)
When factoring a monomial from a polynomial, we seek out factors that are not only common to each term of the polynomial but factors that have these properties:
- The numerical coefficients are the largest common numerical coefficients.
- The variables possess the largest exponents common to all the variables.
Greatest Common Factor
A monomial factor that meets the above two requirements is called the greatest common factor of the polynomial.
Sample Set A
Factor \(3x - 18\)
The greatest common factor is \(3\).
\(\begin{array}{flushleft}
3x-18&=&3 \cdot x - 3 \cdot 6&\text{Factor out } 3\\
3x-18&=&3()&\text{Divide each term of the product by } 3\\
&&&\dfrac{3x}{3} = x \text{ and } \dfrac{-18}{3} = -6\\
&&&\text{(Try to perform this division mentally.}\\
3x-18&=&3(x-6)
\end{array}\)
Factor \(9x^3+18x^2+27x\)
Notice that \(9x\) is the greatest common factor.
\(9x^3 + 18x^2 + 27x = 9x \cdot x^2 + 9x \cdot 2x + 9x \cdot 3\). Factor out \(9x\)
\(9x^3 + 18x^2 + 27x = 9x()\) Mentally divide \(9x\) into each term of the product\)
\(9x^3 + 18x^2 + 27x = 9x(x^2+2x+3)\)
Factor \(10x^2y^3 - 20xy^4 - 35y^5\).
Notice that \(5y^3\) is the greatest common factor. Factor out \(5y^3\).
\(10x^2y^3-20xy^4-35y^5 = 5y^3()\)
Mentally divide \(5y^3\) into each term of the product and place the resulting quotients inside the ( ).
\(10x^2y^3-20xy^4-35y^5=5y^3(2x^2-4xy-7y^2)\)
Factor \(-12x^5 + 8x^3 - 4x^2\).
We see that the greatest common factor is \(-4x^2\).
\(-12x^5 + 8x^3 - 4x^2 = -4x^2()\)
Mentally dividing \(-4x^2\) into each term of the product, we get
\(-12x^5 + 8x^3 - 4x^2 = -4x^2(3x^3 - 2x + 1\)
Practice Set A
Factor \(4x−48\).
- Answer
-
\(4(x−12)\)
Factor \(6y^3 + 24y^2 + 36y\)
- Answer
-
\(6y(y^2 + 4y + 6)\)
Factor \(10a^5b^4 - 14a^4b^5-8b^6\)
- Answer
-
\(2b^4(5a^5 - 7a^4b - 4b^2\)
Factor \(-14m^4 + 28m^2 - 7m\)
- Answer
-
\(-7m(2m^2 - 4m + 1\)
Consider this problem: factor \(Ax+Ay\). Surely, \(Ax+Ay=A(x+y)\). We know from the very beginning of our study of algebra that letters represent single quantities. We also know that a quantity occurring within a set of parentheses is to be considered as a single quantity. Suppose that the letter \(A\) is representing the quantity \((a+b)\). Then we have
\(Ax + Ay = A(x+y)\)
\((a+b)x + (a+b)y = (a+b)(x+y)\)
When we observe the expression
\((a+b)x + (a+b)y\)
we notice that \((a+b)\) is common to both terms. Since it is common, we factor it out.
\((a+b)( )\)
As usual, we determine what to place inside the parentheses by dividing each term of the product by \((a+b)\).
\(\dfrac{(a+b)x}{(a+b)} = x\) and \(\dfrac{(a+b)y}{(a+b)} = y\)
This is a forerunner of the factoring that will be done in Section 5.4.
Sample Set B
Factor \((x−7)a+(x−7)b\).
Notice that \((x-7)\) is the greatest common factor. Factor out \((x-7)\).
\((x-7)a + (x-7)b = (x-7)( )\)
Then, \(\dfrac{(x-7)a}{(x-y)} = a \text{ and } \dfrac{(x-7)b}{(x-7)} = b\)
\((x-7)a+(x-7)b = (x-7)(a+b)\)
Factor \(3x^2(x+1)-5x(x+1)\).
Notice that \(x\) and \((x+1)\) are common to both terms. Factor them out. We'll perform this factorization by letting \(A = x(x+1)\). Then we have
\(3xA-5A = A(3x-5)\)
But \(A = x(x+1)\), so
\(3x^2(x+1)-5x(x+1) = x(x+1)(3x-5)\)
Practice Set B
Factor \((y+4)a+(y+4)b\).
- Answer
-
\((y+4)(a+b)\)
Factor \(8m^3(n-4) - 6m^2(n-4)\)
- Answer
-
\(2m^2(n-4)(4m-3)\)
Exercises
For the following problems, factor the polynomials.
\(9a+18\)
- Answer
-
\(9(a+2)\)
\(6a+24\)
\(8b+12\)
- Answer
-
\(4(2b+3)\)
\(16x+12\)
\(4x−6\)
- Answer
-
\(2(2x−3)\)
\(8x−14\)
\(21y−28\)
- Answer
-
\(7(3y−4)\)
\(16f−36\)
\(12x^2 + 18x\)
- Answer
-
\(6x(2x+3)\)
\(10y^2 + 15y\)
\(8y^2 + 18\)
- Answer
-
\(2(4y^2 + 9)\)
\(7x^2 - 21\)
\(3y^2 - 6\)
- Answer
-
\(3(y^2 - 2)\)
\(2x^2-2\)
\(6y^2-6y\)
- Answer
-
\(6y(y−1)\)
\(ax^2-a\)
\(by^2 + b\)
- Answer
-
\(b(y^2 + 1)\)
\(7by^2 + 14b\)
\(5a^2x^2 + 10x\)
- Answer
-
\(5x(a^2x + 2)\)
\(24ax^2 + 28a\)
\(10x^2 + 5x - 15\)
- Answer
-
\(5(2x^2 + x - 3)\)
\(12x^2 - 8x - 16\)
\(15y^3 - 24y + 9\)
- Answer
-
\(3(5y^3 - 8y + 3)\)
\(ax^2 + ax + a\)
\(by^3 + by^2 + by + b\)
- Answer
-
\(b(y^3 + y^2 + y + 1)\)
\(2y^2 + 6y + 4xy\)
\(9x^2 + 6xy + 4x\)
- Answer
-
\(x(9x+6y+4)\)
\(30a^2b^2 + 40a^2b^2 + 50a^2b^2\)
\(13x^2y^5c - 26x^2y^5c - 39x^2y^5\)
- Answer
-
\(13x^2y^5(-c-3)\)
\(-4x^2-12x-8\)
\(-6y^3 - 8y^2 - 14y + 10\)
- Answer
-
\(-2(3y^3 + 4y^2 + 7y - 5)\)
\(Ab+Ac\)
\(Nx+Ny\)
- Answer
-
\(N(x+y)\)
\(Qx+Qy\)
\(Ax−Ay\)
- Answer
-
\(A(x−y)\)
\((x+4)b+(x+4)c\)
\((x−9)a+(x−9)b\)
- Answer
-
\((x−9)(a+b)\)
\((2x+7)a+(2x+7)b\)
\((9a−b)w−(9a−b)x\)
- Answer
-
\((9a−b)(w−x)\)
\((5−v)X+(5−v)Y\)
\(3x^5y^4 - 12x^3y^4 + 27x^5y^3 - 6x^2y^6\)
- Answer
-
\(3x^2y^3(x^3y - 4xy + 9x^3-2y^3)\)
\(8a^3b^{15} + 24a^2b^{14} + 48a^3b^6 - 20a^3b^7 + 80a^4b^6 - 4a^3b^6 - 4a^3b^7 + 4a^2b\)
\(-8x^3y^2 - 3x^3y^2 + 16x^4y^3 + 2x^2y\)
- Answer
-
\(-x^2y(11xy-16x^2y^2-2)\)
Exercises for Review
A quantity plus \(21\)% more of that quantity is \(26.25\). What is the original quantity?
Solve the equation \(6(t−1)=4(5−s)\) if \(s=2\).
- Answer
-
\(t=3\)
Given that \(4a^3\) is a factor of \(8a^3 - 12a^2\), find the other factor.


3.11: Geometric Sequences
- Identify geometric sequences.
- Find a given term in a geometric sequence.
- Find the th term of a geometric sequence.
- Find the sum of a finite geometric sequence.
- Use geometric sequences to solve real-world applications.
One of the concerns when investing is the doubling time, which is length of time it takes for the value of the investment to be twice, or double, that of its starting value. A shorter doubling times means the investment gets bigger, sooner. For example, if you invest \(\$ 200\) in an account with an 8 -year doubling time, then in 8 years the value of the account will be double the starting amount, or \(2 \times \$ 200=\$ 400\). After another 8 years (for a total of 16 years) the investment would be twice its value after the first 8 years, or \(2 \times(2 \times \$ 400)=2 \times(\$ 400)=\$ 800\). Every 8 years, the investment would double again, so after the third 8 -year period, the investment would be worth \(2 \times 2 \times(2 \times \$ 400)=\$ 1,600\). This process exhibits exponential growth, an application of geometric sequences, which is explored in this section.
Identifying Geometric Sequences
We know what a sequence is, but what makes a sequence a geometric sequence? In an arithmetic sequence, each term is the previous term plus the constant difference. So, you add a (possibly negative) number at each step. In a geometric sequence , though, each term is the previous term multiplied by the same specified value, called the common ratio . In the sequence
Each term in this arithmetic sequence is the previous term plus 5.
Each term in this geometric sequence is the previous term times 2.
In the sequence \(\{3,6,12,24,48,96,192,384,728,1456\}\), the numbers get big fairly quickly, and stay positive. However, that's not always the case with geometric sequences. Depending on the value of the common ratio, the terms could increase each time (like in the one shown in Figure 3.51), or the terms can get smaller each time, or the terms can alternate between positive and negative values. It all depends on the value of the common ratio, \(r\).
Consider this geometric sequence:
Each term is the previous term times 5, which means the common ratio is 5. This common ratio is larger than 1, and so the terms increase each time. Now, look at this geometric sequence:
Each term is the previous term times −3, and the sign of the terms alternate from positive to negative. Then, there’s this geometric sequence:
Each term is the previous term times , and the terms decrease each time. What we should take away from these three examples is if the common ratio is a positive number larger than 1, then the sequence increases. If the common ratio is a negative number, then the sign of the terms alternates between positive and negative. If the common ratio is between 0 and 1, then the terms decrease.
Two special cases of geometric sequences are when the constant ratio is 1 and when the common ratio is 0. When the constant ratio is 1, every term of the sequence is the same, as in . This is referred to as a constant sequence . When the constant ratio is 0, the first term can be any number, but every term after the first term is 0, as in .
For each sequence, determine if the sequence is a geometric sequence. If so, identify the common ratio.
- Answer
-
- In the sequence , the jump from 5 to 20 is a multiplication by 4, as is the next jump to 80, and the next to 320. Each term is 4 times the previous term. Since each term is 4 times the previous, this is a geometric sequence. The common ratio is 4.
- In the sequence , notice that 6 is −3 times −2. The jump from 6 to −12 is another multiplication by negative. So, if this is a geometric sequence, each term should be the previous term times −2. But the change from 24 to 11 is not a multiplication by −2, This means the sequence is not a geometric sequence.
- In the sequence , the change from 4 to 2 is a multiplication by , as is the next jump, from 2 to 1, as is the next from 1 to . Each term is times the previous term. Since each term is times the previous, this is a geometric sequence. The common ratio is .
For each sequence, determine if the sequence is a geometric sequence. If so, identify the common ratio.
- \(\left\{ { - 1, - 5, - 25, - 125, - 625, - 3125,,...} \right\}\)
- \(\left\{ { - 3,6, - 12,24,11,33} \right\}\)
- \(\left\{ { - 500,50, - 5,\frac{1}{2}, - \frac{1}20,...} \right\}\)
As with arithmetic sequences, the first term of a geometric sequence is labeled
The
th term of the geometric sequence,
Return to the sequence
. We observe that the first term is 3, so
| , Place in Sequence |
|
Value of Term |
Term Written as
|
|---|---|---|---|
| 1 |
|
3 | |
| 2 |
|
6 | |
| 3 |
|
12 | |
| 4 |
|
24 | |
| 5 |
|
48 | |
|
|
|
In the following geometric sequences, determine the indicated term of the geometric sequence with a given first term and common ratio.
-
Determine the
term of the geometric sequence with9 th a 1 × 6 and .a 1 × 6 -
Determine the
term of the geometric sequence with11 th 2 a 1 and .= 2
- Answer
-
-
Using
a 1 r n − 1 a n with= a 1 r n − 1 6 a 1 , , and , we calculate= 6
.a 1 r 9 − 1 = 6 × ( 3 ) 9 − 1 = 6 × ( 3 ) 8 = 6 × 6561 = 39366 a 9 = a 1 r 9 − 1 = 6 × ( 3 ) 9 − 1 = 6 × ( 3 ) 8 = 6 × 6561 = 39366 The
9 th term of the geometric sequence with9 th 6 a 1 and is= 6 39366 a 9 .= 39366 -
Using
a 1 r n − 1 a n with= a 1 r n − 1 2 a 1 , , and , we calculate= 2 a 11 = a 1 r 11 − 1 = 2 × ( − 5 ) 11 − 1 = 2 × ( − 5 ) 10 = 2 × 9,765,625 = 19,531, 250 a 11 = a 1 r 11 − 1 = 2 × ( − 5 ) 11 − 1 = 2 × ( − 5 ) 10 = 2 × 9,765,625 = 19,531,250
-
Using
In the following geometric sequences, determine the indicated term of the geometric sequence with a given first term and common ratio.
- Determine the \(12\text{th}\)
- Determine the \(5\text{th}\)
Geometric Sequences
Finding the Sum of a Finite Geometric Sequence
As with arithmetic sequences, it is possible to add the terms of the geometric sequence. Like arithmetic sequences, the formula for the finite sum of the terms of a geometric sequence has a straightforward formula.
The sum of the first
terms of a finite geometric sequence, written
-
What is the sum of the first 13 terms of the geometric sequence with first term
and common ratio ?5 -
What is the sum of the first 7 terms of the geometric sequence with first term
and common ratio16 ?1 8
- Answer
-
- The sum of the first 13 terms of this geometric sequence is 1,328,600.
-
Using
,16 , and , we find that the sum is: \[\begin{gathered}1 8
s_7=a_1\left(\frac{1-r^{n-1}}{1-r}\right)=16 \times\left(\frac{1-\left(\frac{1}{8}\right)^{7-1}}{1-\left(\frac{1}{8}\right)}\right)=16 \times\left(\frac{1-\left(\frac{1}{8}\right)^6}{\frac{7}{8}}\right)=16 \times\left(\frac{1-\frac{1}{262,144}}{\frac{7}{8}}\right)=16 \times\left(\frac{\frac{262,143}{262,144}}{\frac{7}{8}}\right) \\=16 \times\left(\frac{262,143}{229,376}\right)=\frac{262,143}{14,336}=18.2856\end{gathered}\]The sum of the first 7 terms of this geometric sequence is 18.2856.
- What is the sum of the first 10 terms of the geometric sequence with first term \({a_1} = 7" style="font-size:1.1rem;\)
- What is the sum of the first 6 terms of the geometric sequence with first term \({a_1} = 27\)
Using Geometric Sequences to Solve Real-World Applications
Geometric sequences have a multitude of applications, one of which is compound interest. Compound interest is something that happens to money deposited into an account, be it savings or an individual retirement account, or IRA. The interest on the account is calculated and added to the account at regular intervals. This means the interest that was earned later gains its own interest. This allows the money to grow faster. If that interest is added every month, we say it is compounded monthly. If the interest is added daily, then we say it is compounded daily. The amount of money that is deposited into the account is called the principal and is denoted . The account earns money on that principal. The amount it earns is a percentage of the money in the account. The interest rate, expressed as a decimal, is denoted .
If you deposit
dollars in an account that earns interest compounded yearly, then the amount in the account,
, after
years is calculated with the formula:
. This is a geometric sequence, with constant ratio
and first term
Daryl deposits $1,000 in an account earning interest compounded yearly. How much money is in the account after 25 years?
- Answer
-
Using with
, , and1000 , we find that . After 25 years, there is in the account.25
Sophia deposited $4,000 in an account that earns 5.5% interest compounded yearly. After 20 years, Sophia withdrew all the money in the account to pay for her child’s college. How much money was in the account when Sophia withdrew the money?
Another application of geometric sequences is exponential growth. This arises in biology quite frequently, especially in relation to bacterial cultures, but also with other organism population models. In bacterial cultures, the time it takes the population to double is often recorded. This time to double is the same, regardless of how big the population gets. So, if the population doubles after 3 hours, it doubles again after another 3 hours, and again after another 3 hours, and so on. Put into geometric sequence language, it has a common ratio of 2.
When Escherichia coli ( E. coli ) is in a broth culture at 37°C, the population of E. coli doubles in number with 30 organisms, how many E. coli bacteria are present in the culture after 16 hours?
- Answer
-
Since the population is doubling every 20 minutes, this is a geometric sequence situation with common ratio . The culture begins with 30 organisms, so
30 a 1 . The time,16 hours, is 48 twenty-minute periods, so we’re looking for the 48th term in the sequence. Using these values in the geometric sequence formula gives= 30 a 48 = a 1 r n − 1 = 30 × 2 48 − 1 = 30 × 2 47 = 30 × ( 1.40737 × 10 14 ) = 4.22212 × 10 15 .a 48 = a 1 r n − 1 = 30 × 2 48 − 1 = 30 × 2 47 = 30 × ( 1.40737 × 10 14 ) = 4.22212 × 10 15 So, after 16 hours, the culture contains E. coli organisms. That’s more than 4,000 trillion bacteria.
When Streptococcus lactis ( S. lactis ) is in a milk culture at 37°C, the population of S. lactis doubles in number every 30 minutes. If the culture began with 15 organisms, how many S. lactis bacteria are present in the culture after 20 hours?
A player places one grain of rice on the first square of a chess board. On the second square, the player places 2 grains of rice. On the third square, the player places 4 grains of rice. On each successive square of the board, the player doubles the number of grains of rice placed on the chess board. When the player places the last rice on the 64th square, how many total grains of rice have been placed on the board?
- Answer
-
Since the number of grains of rice is doubled at each step, this is a geometric sequence with first term
1 a 1 and common ratio . Rice is placed on 64 total squares, so we want the sum of the first 64 terms. Using this information and the formula, the total number of grains of rice on the board will be:= 1 s 64 = a 1 ( 1 − r n − 1 1 − r ) = 1 × ( 1 − 2 64 − 1 1 − 2 ) = ( 1 − 2 63 − 1 ) = − ( 1 − 2 63 ) = − ( − 9.2233720369 × 10 18 ) = 9.2233720369 × 10 18 s 64 = a 1 ( 1 − r n − 1 1 − r ) = 1 × ( 1 − 2 64 − 1 1 − 2 ) = ( 1 − 2 63 − 1 ) = − ( 1 − 2 63 ) = − ( − 9.2233720369 × 10 18 ) = 9.2233720369 × 10 18 That’s a 20-digit number!
You have a square 1 meter on each side. You begin by coloring one half of the square blue. Then you color half the remaining area blue. Then you color half the remaining area blue once more. At each step, you color half the remaining area. What is the total area you have colored blue after performing this process 15 times?
Sum of a Finite Geometric Sequence
Check Your Understanding
- Is the following a geometric sequence? Explain. {3, 6, 12, 24, 48, 96, 192}
- Find the common ratio of the geometric sequence {3, −30, 300, −3,000, …}.
- In a geometric sequence, the first term is 10 and the common ratio is 1.5. What is the 15th term?
- What is the sum of the first 100 terms of the geometric sequence with first term 4 and common ratio 0.3?
- $15,000 is deposited in an account the yields 4.2% interest compounded annually. How much is in the account after 17 years?