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adverse drug event
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when an individual is harmed by a drug
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adverse effect
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an effect of a drug that is undesired
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biologic
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a drug isolated from natural resources and developed through biomolecular science, immunology, and genetic engineering and produced through biotechnological processes
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biosimilar
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a drug that is synthetically produced and has similar properties to a specific biologic; although it does go through testing, it does not go through the same rigorous testing that a biologic does
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brand name
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the unique identifier of a drug assigned by the drug company and marketed to consumers; also known as a trade name
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chemical name
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a method for identifying drugs built around the drug’s specific chemical structure or composition; often of most use to the chemist or pharmacist
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complementary and alternative medicine (CAM)
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alternative medicine refers to using a treatment instead of mainstream conventional medicine. Complementary therapy refers to using both alternative and conventional medicine together. Treatments may include massage, acupuncture, acupressure, mind–body interventions, herbs, vitamins, or dietary supplements.
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counterfeit drug
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a product that is illegally manufactured or mislabeled regarding the identity or source and appears to be a genuine product; may be harmful to the client
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drug
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a chemical that exerts an effect on the living body; a compound used in the prevention, treatment, diagnosis, or cure of a condition or disease
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drug formulary
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the list of prescription drugs that are covered by an insurance plan or carried by a medical institution
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drug prototype
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use of one drug within a class to represent all other drugs within the class—a “class representative”
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enteral administration
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the administration of medications into the gastrointestinal (GI) tract
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generic name
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the name of a drug, usually derived from the chemical name, that uniquely identifies the drug
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intramuscular injection
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the administration of medication into the muscle of a client (usually deltoid or ventrogluteal in an adult and the vastus lateralis in an infant under age 2)
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intravenous
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the administration of a medication directly into the bloodstream through a vein
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over-the-counter (OTC) drug
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a drug or medication available without prescription
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parenteral administration
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the administration of medication elsewhere besides the enteral (GI) route
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percutaneous administration
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the application of drugs to the skin or mucous membranes
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pharmacogenetics
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the study of the body’s hereditary response to drugs
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pharmacokinetics
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the movement of a drug through the body, or “what the body does to the drug”
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pharmacologic classification
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refers to how a group of drugs can be organized by how they work in the body—the mechanism of action
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pharmacology
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the study of medicines (or drugs) or the study of the biological effects of chemicals on the body
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side effect
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an effect of a drug that is undesired
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subcutaneous injection
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the administration of a medication into adipose tissue
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synthetic drug
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a chemical compound produced in a laboratory by a drug manufacturer or illegally by individuals for illicit purposes
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teratogenic
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harmful to a fetus by causing severe malformations or death
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therapeutic classification
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refers to how a group of drugs can be organized by the diagnosis or disease being treated
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therapeutic effects
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the effects that are expected and desired from a particular medication


6.2: Music in the Nineteenth Century
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Classical Music |
Nineteenth-Century Music |
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Musical Style, Performing Forces, and Forms
The nineteenth century is marked by a great diversity in musical styles, from the conservative to the progressive. As identified by the style comparison chart above, nineteenth century melodies continue to be tuneful and are perhaps even more songlike than classical style melodies, although they may contain wider leaps. They still use sequences, which are often as a part of modulation from one key to another. Melodies use more chromatic (or “colorful”) pitches from outside the home key and scale of a composition. Along with the continuing emphasis on tuneful melodies comes predominantly homophonic textures, although as compositions use more instruments, there are also increasing numbers of accompanying, but relatively interesting, musical lines.
Harmonies in nineteenth-century music are more dissonant than ever. More chords add a fourth note to the triad, making them more dissonant and chromatic. These dissonances may be sustained for some time before resolving to a chord that is consonant. One composition may modulate between several keys, and these keys often have very different pitch contents. Such modulations tend to disorient the listening and add to the chaos of the musical selection. Composers were in effect “pushing the harmonic envelope.”
The lengths of nineteenth-century musical compositions ran from the min- ute to the monumental. Songs and short piano pieces might be only a couple of minutes long, although they were sometimes grouped together in cycles or collections. On the other hand, symphonies and operas grow in size. By the end of the century, a typical symphony might be an hour long, with the operas of Verdi, Wagner, and Puccini clocking in at several hours each. Performing forces reflected similar extremes. There is much nineteenth-century music for solo piano or solo voice with piano accompaniment. The piano achieved a modern form, with the full eighty-eight-note keyboard that is still used today and an iron frame that allowed for greater string tension and a wider range of dynamics. Crescendos and decrescendos became more common, alongside more tempo fluctuations, even within compositions. As we will see, Fryderyk Chopin was the first composer to make prevalent use of rubato as a performance instruction in his musical scores.
During the nineteenth century, the industrial revolution facilitated and enabled marked improvements to many musical instruments besides the piano with its improved and updated iron frame and tempered metal strings. Efficient valves were added to the trumpet and a general improvement in metal works tightened tolerances and metal fittings of all brass instruments. Along with the many improvements to instruments, new instruments were researched and created, including the piccolo, English horn, tuba, contrabassoon, and saxophone.
Orchestras also increased in size and became more diverse in makeup, thereby allowing composers to exploit even more divergent dynamics and timbres. With or- chestral compositions requiring over fifty (and sometimes over 100) musicians, a conductor was important, and the first famous conductors date from this period. In fact, generally-speaking, the nineteenth-century orchestra looked not unlike what you might see today at most concerts by most professional orchestras (see Figure 6.9). Nineteenth-century composers knew well the forms and genres used by their predecessors, most prominently the music of Beethoven, but also the music of com- posers such as Mozart, Handel, Haydn, and Bach. They continued to compose in these forms and genres, while sometimes transforming them into something quite different, especially among those composers who identified themselves as progressives, as opposed to conservatives. The wider nineteenth-century interest in emotion and in exploring connections between all of the arts led to musical scores with more poetic or prose instructions from the composer. It also led to more program music, which as you will recall, is instrumental music that represents something “extra musical,” that is, something outside of music itself, such as nature, a literary text, or a painting. Nineteenth-century critics and philosophers sustained expansive debates about ways in which listeners might hear music as related to the extra musical. Extra musical influences, from the characteristic title to a narrative attached to a musical score, guided composers and listeners as they composed and heard musical forms.
Genres of Instrumental Music
Some nineteenth-century compositions use titles similar to those found in clas- sical style music, such as “Symphony No. 3,” “Concerto, Op. 3,” or “String Quartet in C Minor.” These compositions are sometimes referred to as examples of abso- lute music (that is, music for the sake of music). Program music with titles came in several forms. Short piano compositions were described as “character pieces” and took on names reflecting their emotional mood, state, or reference. Orchestral program music included the program symphony and the symphonic poem (also known as the tone poem). The program symphony was a multi-movement com- position for orchestra that represented something extra musical, a composition such as Hector Berlioz’s Symphonie fantastique (discussed below). A symphonic or tone poem was a one-movement composition for orchestra, again with an extra musical referent, such as Bedřich Smetana’s Moldau .
Genres of Vocal Music
Opera continued to be popular in the nineteenth century and was dominated by Italian styles and form, much like it had been since the seventeenth century. Ital- ian opera composer Giacomo Rossini even rivaled Beethoven in popularity. By the 1820s, however, other national schools were becoming more influential. Carl Maria von Weber’s German operas enhanced the role of the orchestra, whereas French grand opera by Meyerbeer and others was marked by the use of large choruses and elaborate sets. Later in the century, composers such as Giuseppe Verdi and Richard Wagner would synthesize and transform opera into an even more dramatic genre.
Other large-scale choral works in the tradition of the Baroque cantata and ora- torio were written for civic choirs which would sometimes band together into larg- er choral ensembles in annual choral festivals. The song for voice and piano saw revived interest, and art songs were chief among the music performed in the home for private and group entertainment. The art song is a composition for solo voice and piano that merges poetic and musical concerns. It became one of the most popular genres of nineteenth-century Romanticism, a movement that was always looking for connections between the arts. Sometimes these art songs were grouped into larger collections called song cycles or, in German, Liederkreis . Among the important composers of early nineteenth-century German Lieder were Robert and Clara Schumann, Johannes Brahms, and Franz Schubert.


17.12: Spectroscopy of Alcohols and Phenols
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Objectives
After completing this section, you should be able to
- identify the two most prominent absorptions seen in the infrared spectra of alcohols and phenols.
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describe the characteristic feature of the proton NMR spectra of alcohols and phenols.
- explain how deuterium oxide (D 2 O) can be used to assist in the identification of the signal caused by the presence of the O-H proton in the 1 H NMR spectrum of an alcohol.
- predict the general form (i.e., number of peaks, approximate chemical shifts, and splitting pattern) of the proton NMR of a given alcohol or phenol.
- describe the two most common initial fragmentations in the mass spectra of alcohols.
- use spectral data (infrared, NMR, mass spectroscopy) to assist in the identification of an unknown alcohol or phenols. You may use tables of characteristic absorptions as an aid to accomplishing this objective.
Key Terms
Make certain that you can define, and use in context, the key terms below.
- alpha cleavage
- dehydration
Study Notes
The 1 H NMR chemical shifts for phenols are not particularly distinctive. However, one expects the \(\ce{-OH}\) signal to be in the 4–7 ppm range, while the aromatic protons (see Section 15.7) are expected to be found at 7–8 ppm.
In a mass spectrometer, alcohols fragment in two characteristic ways: alpha cleavage and dehydration. From the equation showing H-Y elimination, you can see that the dehydration of an alcohol in a mass spectrometer is essentially the same as the dehydration of an alcohol in a normal chemical reaction. “Alpha cleavage” refers to the breaking of the bond between the oxygen-bearing carbon atom and one of the neighboring carbons.
Infrared Spectroscopy
The IR spectrum of aliphatic alcohols have a distinctive O-H stretch in the range of 3300 to 3400 cm -1 . This peak tends to be very strong and very broad. This exact position of the peak is dependent on the amount of hydrogen bonding in the alcohol. In addition alcohol have a strong C-O stretch near 1000 cm -1 . In the IR spectra of 1-butanol, show below, the O-H stretch appears at 3300 cm -1 and the C-O stretch appears at 1073 cm -1 .
Figure IR8. IR spectrum of 1-butanol. Source: SDBSWeb : http://riodb01.ibase.aist.go.jp/sdbs/ (National Institute of Advanced Industrial Science and Technology of Japan, 14 July 2008)
Peak shapes are sometimes very useful in recognizing what kind of bond is present. The rounded shape of most O-H stretching modes occurs because of hydrogen bonding between different hydroxy groups. Because protons are shared to varying extent with neighboring oxygens, the covalent O-H bonds in a sample of alcohol all vibrate at slightly different frequencies and show up at slightly different positions in the IR spectrum. Instead of seeing one sharp peak, you see a broad set of multiple overlapping peaks. The broadness of the O-H peak makes it very easy to distinguish in an IR spectrum.
The IR spectrum of phenols the O-H stretch appears at roughly 3500 cm -1 . In addition, the IR spectra will show the bands typical for aromatic compounds in the region of 1500-1600 cm -1 .
The IR Spectrum of Phenol
Exercise \(\PageIndex{1}\)
Assume that you have just converted cyclohexanol into cyclohexanone. How would you use IR spectroscopy to tell if the conversion was successful? What changes would you expect to see between the starting material and the product?
- Answer
-
The cyclohexanol starting material would have a distinctive O-H stretch between 3300 and 3400 cm -1 . If the conversion were successful this peak would not be present in the product. Also, the IR spectrum of the product would have a strong C=O stretch between 1700 and 1800 cm -1 .
1 H Nuclear Magnetic Resonance Spectroscopy
Alcohols
- Protons on carbon adjacent to the alcohol oxygen show up in the region of 3.4-4.5 ppm. The electronegativity of the alcohol oxygen de-shields these protons causing them to appear downfield when compared to alkane protons.
- Protons directly attached to the alcohol oxygen often appear in the region of 2.0 to 2.5 ppm. These peaks tend to appear as short, broad singlets. The position of the -OH peak can vary depending on the conditions, such as the NMR solvent used, the concentration of the alcohol, the purity of the alcohol, temperature, and if any water is present in the sample.
Notice that the alcohol proton is not involved in spin-spin splitting. This is usually true unless the alcohol is exceptionally pure. Most samples of alcohols contain minute amounts of acidic impurities which catalyze the exchange of protons causing the splitting effects to be removed. This is why alcohol protons typically appears as a singlet in NMR spectrum. The 1 H NMR spectrum of propanol shows the -CH 2 - attached to the alcohol as a triplet at 3.58 ppm. This shows that the signal is being split by the adjacent -CH 2 - group and not the alcohol -OH. The signal for the proton attached to the oxygen appears as a singlet at 2.26 ppm. If this signal was involved in spin-spin splitting with the adjacent -CH 2 - group it would appear as a triplet.
Source: SDBSWeb : http://sdbs.db.aist.go.jp (National Institute of Advanced Industrial Science and Technology, 28 June 2017
Phenol
- Protons attached to the aromatic ring in phenols show up near the aromatic region of an NMR spectrum (7-8 ppm). These peaks will have splitting typical for aromatic protons.
- The protons directly attached to the alcohol oxygen of phenols appear in the region of 3 to 8 ppm. These peaks tend to appear as short, broad singlets similarly to other alcohols.
Determining the Position of an -OH Peak in 1 H NMR
The position of the -OH absorption for alcohols and phenols can be easily determined through the addition a few drops of deuterium oxide, D 2 O to the NMR sample tube.
After addition, the OH proton will be rapidly exchanged for a deuterium. Because deuterium atoms do not produce peaks in a typical NMR spectrum the original -OH peak will disappear. This technique is sometimes called a "D 2 O shake" due to the mixing required after D 2 O has been added to the NMR sample tube.
13 C Nuclear Magnetic Resonance Spectroscopy
Alcohols
- Carbons adjacent to the alcohol oxygen show up in the distinctive region of 50-65 ppm in 13 C NMR spectrum.
Phenols
- Due to the electronegative oxygen, the aromatic carbon attached to the -OH group is shifted downfield to 155 ppm.
- The other carbons in the phenol ring appear in the region typical for aromatic carbons of 125-150 ppm.
Mass Spectra
Alcohols
An alcohol's molecular ion is small or non-existent. Alpha-cleavage of the C-C bond next to the oxygen usually occurs. A loss of H 2 O may occur as in the spectra below.3-Pentanol (C 5 H 12 O) with MW = 88.15
3-Pentanol shows three significant fragment ions. Alpha-fragmentation (loss of an ethyl radical) forms the m/z=59 base peak. Loss of water from this gives a m/z=41 fragment, and loss of ethene from m/z=59 gives a m/z=31 fragment.
Phenol
Phenol (C 6 H 6 O) with MW = 94.11
Phenol exhibit a strong molecular ion peak. The presence of an aromatic ring and an OH gives phenol a unique fragmentation pattern. In particular loss of CO (M – 28) and loss of a formyl radical (HCO·, M – 29) are common observed.
From an outside source
Exercise \(\PageIndex{1}\)
1) From mass spectroscopy analysis it was determined that a compound has the general formula C 5 H 12 O. Given the following 1 H NMR spectrum, draw the structure. The integration values of each group of signals is given on the spectrum.
2) How would you expect the NMR spectrum show in question one to change if D 2 O was added to the NMR sample tube.
3) From mass spectroscopy analysis it was determined that a compound has the general formula C3H 8 O. Given the following 1 H NMR spectrum, draw the structure. The integration values of each group of signals is given on the spectrum.
- Answer
-
1) 3-Pentanol
Exercise \(\PageIndex{2}\)
How would you expect the NMR spectrum shown in Exercise 1 to change if D 2 O was added to the NMR sample tube.
- Answer
-
The -OH peak at 4.5 ppm would dissappear due to dueterium exchange.
Exercise \(\PageIndex{3}\)
From mass spectroscopy analysis it was determined that a compound has the general formula C3H 8 O. Given the following 1 H NMR spectrum, draw the structure. The integration values of each group of signals is given on the spectrum.
- Answer
-
1-Propanol
Contributors and Attributions
-
William Reusch, Professor Emeritus ( Michigan State U. ), Virtual Textbook of Organic Chemistry
- James Kabrhel (University of Wisconsin - Green Bay, Sheboygan Campus)


15.6: Chapter 7
Your Turn
7.1
1. \(4 \times 15 = 60\)
7.2
1. \(4 \times 15 \times 3 \times 2 = 360\)
7.3
1. \(3 \times 10 \times 10 \times 10 \times 10 \times 10 \times 26 \times 26 \times 26 = {\text{5,272,800,000}}\)
7.4
1. 120
7.5
1. 720
2. 132
3. 70
7.6
1. 30
2. 24,024
3. 5,814
7.7
1. \(_{15}{P_3} = 2{,}730\)
7.8
1. combination
2. combination
7.9
1. 15
2. 45
3. 2,002
7.10
1. 30,856
2. 111,930
7.11
1. 290,004
7.12
1. \(\{\heartsuit ,\spadesuit ,\clubsuit ,\diamondsuit\}\)
2. {A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K}
3. {1, 2, 3, 4}
4. {3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
7.13
1. Independent
2. Dependent
7.14
1. {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}
7.15
1. {J\(\heartsuit\) Q\(\heartsuit\), J\(\heartsuit\) K\(\heartsuit\), Q\(\heartsuit\) J\(\heartsuit\), Q\(\heartsuit\) K\(\heartsuit\), K\(\heartsuit\) J\(\heartsuit\), K\(\heartsuit\) Q\(\heartsuit\)}
7.16
1. \(P(\text{H} < 5) = 1\)
2. \(0 < P(\text{H} < 4) < 1\)
3. \(P(\text{H} ≥ 5) = 0\)
7.17
1. \(\frac{1}{6}\)
2. \(\frac{2}{3}\)
3. \(\frac{1}{2}\)
7.18
1. \(\frac{5}16\)
7.19
1. \(\frac{1}{3}\)
2. \(\frac{1}{2}\)
3. \(\frac{2}{3}\)
7.20
1. \(\frac13
(click for details)
= 1.3\% \)
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7.21
1. theoretical
2. subjective
3. empirical
7.22
1. \(\frac1316\)
7.23
1. \(\frac14443680 = \frac{3}910\)
7.24
1. \(1 - \frac
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\approx 10.7\%\)
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2. \(\frac
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\approx 0.86\%\)
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7.25
1. \(13{:}3\)
2. \(12{:}4 = 3{:}1\)
7.26
1. The odds for \(E\) are \(4{:}1\) and the odds against \(E\) are \(1{:}4\).
7.27
1. \(P(E) = \frac{1}16\)
2. \(P(E) = \frac2.53.5 \approx 0.714\)
7.28
1. Mutually exclusive
2. Not mutually exclusive
3. Mutually exclusive
7.29
1. \(\frac{4}10 + \frac{2}10 = \frac{3}{5}\)
2. Not appropriate; the events are not mutually exclusive.
3. \(\frac{3}10 + \frac{2}10 = \frac{1}{2}\)
7.30
1. \(\frac{1}{2}\)
2. \(\frac{1}{2}\)
3. \(\frac{2}{3}\)
7.31
1. \(\frac{1}{3}\)
2. \(1\)
3. \(\frac{2}{3}\)
7.32
1. \(\frac{3}28\)
2. \(\frac{3}56\)
3. \(\frac{3}28\)
7.33
1.
| Roll | Probability |
|---|---|
| 1 | \(\frac{1}{3}\) |
| 2 | \(\frac{1}{4}\) |
| 3 | \(\frac{1}{6}\) |
| 4 | \(\frac{1}12\) |
| 5 | \(\frac{1}12\) |
| 6 | \(\frac{1}12\) |
7.34
1. Not binomial (more than two outcomes)
2. Not binomial (not independent)
3. Binomial
4. Not binomial (number of trials isn’t fixed)
7.35
1. 0.146
2. 0.190
3. 0.060
7.36
1. 0.2972
2. 0.6615
3. 0.0919
4. 0.5207
5. 0.3643
7.37
1. \(\frac10{3}\)
2. \(\frac{3}{2}\)
3. \(\frac25{4}\) = $6.25
7.38
1. If you roll the special die many times, the mean of the numbers showing will be around 3.33.
2. If you repeat the coin-flipping experiment many times, the mean of the number of heads you get will be around 1.5.
3. If you play this game many times, the mean of your winnings will be around $10.
7.39
1. If the player bets on 7, the expected value is \(- \$ 0.17\).
2. If the player bets on 12, the expected value is \(- \$ 0.14\).
3. If the player bets on any craps, the expected value is \(- \$ 0.11\).
The best bet for the player is any craps; the best bet for the casino is the bet on 7.
The best bet for the player is any craps; the best bet for the casino is the bet on 7.
Check Your Understanding
1. 540
2. 14
3. 1,024
4. 800
5. 25,920
6. 120
7. 120
8. 1,320
9. 1,680
10. \(_{15}{P_4} = 32{,}760\)
11. permutations
12. combinations
13. 66
14. 560
15. 20
16. 560
17. {0, 1, 2, 3, 4, 5, 6}
18.
| Crinkle Fries | Curly Fries | Onion Rings | |
|---|---|---|---|
| 8 Nuggets | 8 nuggets with crinkle fries | 8 nuggets with curly fries | 8 nuggets with onion rings |
| 12 Nuggets | 12 nuggets with crinkle fries | 12 nuggets with curly fries | 12 nuggets with onion rings |
19.
20. {8 nuggets with crinkle fries, 8 nuggets with curly fries, 8 nuggets with onion rings, 12 nuggets with crinkle fries, 12 nuggets with curly fries, 12 nuggets with onion rings}
21. {history with Anderson, history with Burr, English with Carter, sociology with Johnson, sociology with Kirk, sociology with Lambert}
22. \(\frac{1}{4}\)
23. \(\frac{1}{2}\)
24. \(\frac{1}{2}\)
25. 0
26. theoretically
27. subjectively
28. empirically
29. \({\text{number of heads}} > 20\)
30. 89.9%
31. \(\frac
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\approx 0.278\%\)
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32. \(\frac
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\approx 0.079\%\)
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33. \(\frac
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\approx 0.16\%\)
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34. \(\frac
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\approx 1.67\%\)
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35. \(\frac
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\approx 1.9\%\)
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36. \(\frac
(click for details)
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\approx 1.9\%\)
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37. \(1{:}2\)
38. \(5{:}1\)
39. \(1{:}1\)
40. \(3{:}5\)
41. \(11{:}2\)
42. \(\frac{4}13\)
43. \(\frac{5}12\)
44. \(\frac{3}{5}\)
45. \(\frac{3}{5}\)
46. \(\frac{3}{5}\)
47. \(\frac{2}{5}\)
48. \(\frac{2}{5}\)
49. \(\frac{3}10\)
50. \(\frac{1}{4}\)
51. \(\frac{1}{9}\)
52. \(\frac{1}{6}\)
53. \(\frac{2}15\)
54. \(\frac{4}45\)
55. \(\frac{8}45\)
56. No (more than two outcomes)
57. Yes
58. No (number of trials is not fixed)
59. 0.9453
60. 0.1366
61. 0.8230
62. 4.1
63. If you roll this die many times, the mean of the numbers rolled will be around 4.1.
64. $0.417
65. If you play this game many times, the mean of the amount won/lost each time will be about 42 cents.
66. Yes; the expected value is positive.


1.4: More Complex Argument Structures
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So far we have seen that an argument consists of a premise (typically more than one) and a conclusion. However, very often arguments and explanations have a more complex structure than just a few premises that directly support the conclusion. For example, consider the following argument:
No one living in Pompeii could have survived the eruption of Mt. Vesuvius. The reason is simple: the lava was flowing too fast and there was nowhere to go to escape it in time. Therefore, this account of the eruption, which claims to have been written by an eyewitness living in Pompeii, was not actually written by an eyewitness.
The main conclusion of this argument—the statement that depends on other statements as evidence but doesn’t itself provide any evidence for any other statement—is:
A. This account of the eruption of Mt. Vesuvius was not actually written by an eyewitness.
However, the argument’s structure is more complex than simply having a couple of premises that provide evidence directly for the conclusion. Rather, some statement provides evidence directly for the main conclusion, but that statement itself is supported by another statement. To determine the structure of an argument, we must determine which statements support which. We can use our premise and conclusion indicators to help with this. For example, the passage contains the phrase, “the reason is...” which is a premise indicator, and it also contains the conclusion indicator, “therefore.” That conclusion indicator helps us to identify the main conclusion, but the more important thing to see is that statement A does not itself provide evidence or support for any of the other statements in the argument, which is the clearest reason why statement A is the main conclusion of the argument. The next question we must answer is: which statement most directly supports A? What most directly supports A is:
B. No one living in Pompeii could have survived the eruption of Mt. Vesuvius.
However, there is also a reason offered in support of B. That reason is that:
C. The lava from Mt. Vesuvius was flowing too fast and there was nowhere for someone living in Pompeii to go in order to escape it in time.
So the main conclusion (A) is directly supported by B, and B is supported by C. Since B acts as a premise for the main conclusion but is also itself the conclusion of further premises, we refer to B as an intermediate conclusion. The important thing to recognize here is that one and the same statement can act as both a premise and a conclusion. Statement B is a premise that supports the main conclusion (A), but it is also itself a conclusion that follows from C. Here is how we would put this complex argument into standard form (using numbers this time, as we always do when putting an argument into standard form):
1. The lava from Mt. Vesuvius was flowing too fast and there was nowhere for someone living in Pompeii to go in order to escape it in time.
2. Therefore, no one living in Pompeii could have survived the eruption of Mt. Vesuvius. (from 1)
3. Therefore, this account of the eruption of Mt. Vesuvius was not actually written by an eyewitness. (from 2)
Notice that at the end of statement 2 I have written in parentheses “from 1” (and likewise at the end of statement 3 I have written “from 2”). This is a shorthand way of saying: “this statement follows from statement 1.” We will use this convention as a way of keeping track of the structure of the argument. It may also help to think about the structure of an argument spatially, as figure 1 shows:
The main argument here (from 2 to 3) contains a subargument, in this case the argument from 1 to 2. A subargument , as the term suggests, is a part of an argument that provides indirect support for the main argument. The main argument is simply the argument whose conclusion is the main conclusion.
Another type of structure that arguments can have is when two or more premises provide direct but independent support for the conclusion. Here is an example of an argument with that structure: I know that Wanda rode her bike to work today because when she arrived at work she had her right pant leg rolled up (which cyclists do in order to keep their pants legs from getting caught in the chain). Moreover, our coworker, Bob, who works in accounting, saw her riding towards work at 7:45 am.
The conclusion of this argument is “Wanda rode her bike to work today” and there are two premises that provide independent support for it: the fact that Wanda had her pant leg cuffed and the fact that Bob saw her riding her bike. Here is the argument in standard form:
- Wanda arrived at work with her right pant leg rolled up.
- Cyclists often roll up their right pant leg.
- Bob saw Wanda riding her bike towards work at 7:45.
- Therefore, Wanda rode her bike to work today. (from 1-2, 3 independently)
Again, notice that next to statement 4 of the argument I have written the premises from which that conclusion follows. In this case, in order to avoid any ambiguity, I have noted that the support for the conclusion comes independently from statements 1 and 2, on the one hand, and from statement 3, on the other hand. It is important to point out that an argument or subargument can be supported by one or more premises. We see this in the present argument since the conclusion (4) is supported jointly by 1 and 2, and singly by 3. As before, we can represent the structure of this argument spatially, as figure 2 shows:
There are endless different argument structures that can be generated from these few simple patterns. At this point, it is important to understand that arguments can have these different structures and that some arguments will be longer and more complex than others. Determining the structure of very complex arguments is a skill that takes some time to master. Even so, it may help to remember that any argument structure ultimately traces back to some combination of these.
Exercise
Write the following arguments in standard form and show how the argument is structured using a diagram like the ones I have used in this section.
- There is nothing wrong with prostitution because there is nothing wrong with consensual sexual and economic interactions between adults.

- There was someone in this cabin recently because there was warm water in the tea kettle and because there was wood still smoldering in the fireplace. But the person couldn’t have been Tim because Tim has been with me the whole time. Therefore, there must be someone else in these woods.
- It is possible to be blind and yet run in the Olympic Games since Marla Runyan did it at the 2000 Sydney Olympics.
- The train was late because it had to take a longer, alternate route since the bridge was out.
- Israel is not safe if Iran gets nuclear missiles since Iran has threatened multiple times to destroy Israel and if Iran had nuclear missiles it would be able to carry out this threat. Moreover, since Iran has been developing enriched uranium, they have the key component needed for nuclear weapons—every other part of the process of building a nuclear weapon is simple compared to that. Therefore, Israel is not safe.
- Since all professional hockey players are missing front teeth and Martin is a professional hockey player, it follows that Martin is missing front teeth. And since almost all professional athletes who are missing their front teeth have false teeth, it follows that Martin probably has false teeth.
- Anyone who eats the crab rangoon at China Food restaurant will probably have stomach troubles afterward. It has happened to me every time, which is why it will probably happen to other people as well. Since Bob ate the crab rangoon at China Food restaurant, he will probably have stomach troubles afterward.
- Albert and Caroline like to go for runs in the afternoon in Hyde Park. Since Albert never runs alone, we know that any time Albert is running, Caroline is running too. But since Albert looks like he has just run (since he is panting hard), it follows that Caroline must have ran too.
- Just because Jeremy’s prints were on the gun that killed Tim and the gun was registered to Jeremy, it doesn’t follow that Jeremy killed Tim since Jeremy’s prints would certainly be on his own gun and someone else could have stolen Jeremy’s gun and used it to kill Tim.


4.2E: Graphs of Exponential Functions (Exercises)
Section 4.2 Exercise
Match each function with one of the graphs below.
1. \(f\left(x\right)=2\left(0.69\right)^{x}\)
2. \(f\left(x\right)=2\left(1.28\right)^{x}\)
3. \(f\left(x\right)=2\left(0.81\right)^{x}\)
4. \(f\left(x\right)=4\left(1.28\right)^{x}\)
5. \(f\left(x\right)=2\left(1.59\right)^{x}\)
6. \(f\left(x\right)=4\left(0.69\right)^{x}\)
If all the graphs to the right have equations with form \(f\left(x\right)=ab^{x}\),
7. Which graph has the largest value for \(b\)?
8. Which graph has the smallest value for \(b\)?
9. Which graph has the largest value for \(a\)?
10. Which graph has the smallest value for \(a\)?
Sketch a graph of each of the following transformations of \(f\left(x\right)=2^{x}\)
11. \(f\left(x\right)=2^{-x}\)
12. \(g\left(x\right)=-2^{x}\)
13. \(h\left(x\right)=2^{x} +3\)
14. \(f\left(x\right)=2^{x} -4\)
15. \(f\left(x\right)=2^{x-2}\)
16. \(k\left(x\right)=2^{x-3}\)
Starting with the graph of \(f\left(x\right)=4^{x}\), find a formula for the function that results from
17. Shifting \(f(x)\) 4 units upwards
18. Shifting \(f(x)\) 3 units downwards
19. Shifting \(f(x)\) 2 units left
20. Shifting \(f(x)\) 5 units right
21. Reflecting \(f(x)\) about the x-axis
22. Reflecting \(f(x)\) about the y-axis
Describe the long run behavior, as \(x \to \infty\) and \(x \to -\infty\) of each function
23. \(f\left(x\right)=-5\left(4^{x} \right)-1\)
24. \(f\left(x\right)=-2\left(3^{x} \right)+2\)
25. \(f\left(x\right)=3\left(\dfrac{1}{2} \right)^{x} -2\)
26. \(f\left(x\right)=4\left(\dfrac{1}{4} \right)^{x} +1\)
27. \(f\left(x\right)=3\left(4\right)^{-x} +2\)
28. \(f\left(x\right)=-2\left(3\right)^{-x} -1\)
Find a formula for each function graphed as a transformation of \(f\left(x\right)=2^{x}\).
29. 30.
31. 32.
Find an equation for the exponential function graphed.
33. 34.
35. 36.
- Answer
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1. B
3. A
5. E
7. D
9. C
11.
13.
15.
17. \(y = 4^x + 4\)
19. \(y = 4^{x + 2}\)
21. \(y = -4^x\)
23. As \(x \to \infty\), \(f(x) \to -\infty\). As \(x \to -\infty\) \(f(x) \to -1\)
25. As \(x \to \infty\), \(f(x) \to -2\). As \(x \to -\infty\) \(f(x) \to \infty\)
27. As \(x \to \infty\), \(f(x) \to 2\). As \(x \to -\infty\) \(f(x) \to \infty\)
29. \(y = -2^{x + 2} + 1 = -4(2)^x + 1\)
31. \(y = -2(2)^{-x} + 3\)
33. \(y = -2(3)^x + 7\)
35. \(y = 2(\dfrac{1}{2})^x - 4\)


6: Measurement of Constructs
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Theoretical propositions consist of relationships between abstract constructs. Testing theories (i.e., theoretical propositions) require measuring these constructs accurately, correctly, and in a scientific manner, before the strength of their relationships can be tested. Measurement refers to careful, deliberate observations of the real world and is the essence of empirical research. While some constructs in social science research, such as a person’s age, weight, or a firm’s size, may be easy to measure, other constructs, such as creativity, prejudice, or alienation, may be considerably harder to measure. In this chapter, we will examine the related processes of conceptualization and operationalization for creating measures of such constructs.


7: Probability Last updated Save as PDF Page ID 129483 OpenStax OpenStax 7.0: Introduction 7.1: The Multiplication Rule for Counting 7.1.0: Exercises 7.2: Permutations 7.2.0: Exercises 7.3: Combinations 7.3.0: Exercises 7.4: Tree Diagrams, Tables, and Outcomes 7.4.0: Exercises 7.5: Basic Concepts of Probability 7.5.0: Exercises 7.6: Probability with Permutations and Combinations 7.6.0: Exercises 7.7: What Are the Odds? 7.7.0: Exercises 7.8: The Addition Rule for Probability 7.8.0: Exercises 7.9: Conditional Probability and the Multiplication Rule 7.9.0: Exercises 7.10: The Binomial Distribution 7.10.0: Exercises 7.11: Expected Value 7.11.0: Exercises 7.12: Chapter Summary 7.12.0: Key Terms 7.12.1: Key Concepts 7.12.2: Formula Review 7.12.3: Projects 7.12.4: Chapter Review 7.12.5: Chapter Test Thumbnail: Two standard gaming dice. (CC BY-SA 3.0 Unported; Steaphan Greene via Wikipedia )