17.2: Mapping Genomes
Skills to Develop
- Define genomics
- Describe genetic and physical maps
- Describe genomic mapping methods
Genomics is the study of entire genomes, including the complete set of genes, their nucleotide sequence and organization, and their interactions within a species and with other species. Genome mapping is the process of finding the locations of genes on each chromosome. The maps created by genome mapping are comparable to the maps that we use to navigate streets. A genetic map is an illustration that lists genes and their location on a chromosome. Genetic maps provide the big picture (similar to a map of interstate highways) and use genetic markers (similar to landmarks). A genetic marker is a gene or sequence on a chromosome that co-segregates (shows genetic linkage) with a specific trait. Early geneticists called this linkage analysis. Physical maps present the intimate details of smaller regions of the chromosomes (similar to a detailed road map). A physical map is a representation of the physical distance, in nucleotides, between genes or genetic markers. Both genetic linkage maps and physical maps are required to build a complete picture of the genome. Having a complete map of the genome makes it easier for researchers to study individual genes. Human genome maps help researchers in their efforts to identify human disease-causing genes related to illnesses like cancer, heart disease, and cystic fibrosis. Genome mapping can be used in a variety of other applications, such as using live microbes to clean up pollutants or even prevent pollution. Research involving plant genome mapping may lead to producing higher crop yields or developing plants that better adapt to climate change.
Genetic Maps
The study of genetic maps begins with linkage analysis , a procedure that analyzes the recombination frequency between genes to determine if they are linked or show independent assortment. The term linkage was used before the discovery of DNA. Early geneticists relied on the observation of phenotypic changes to understand the genotype of an organism. Shortly after Gregor Mendel (the father of modern genetics) proposed that traits were determined by what are now known as genes, other researchers observed that different traits were often inherited together, and thereby deduced that the genes were physically linked by being located on the same chromosome. The mapping of genes relative to each other based on linkage analysis led to the development of the first genetic maps.
Observations that certain traits were always linked and certain others were not linked came from studying the offspring of crosses between parents with different traits. For example, in experiments performed on the garden pea, it was discovered that the color of the flower and shape of the plant’s pollen were linked traits, and therefore the genes encoding these traits were in close proximity on the same chromosome. The exchange of DNA between homologous pairs of chromosomes is called genetic recombination , which occurs by the crossing over of DNA between homologous strands of DNA, such as nonsister chromatids. Linkage analysis involves studying the recombination frequency between any two genes. The greater the distance between two genes, the higher the chance that a recombination event will occur between them, and the higher the recombination frequency between them. Two possibilities for recombination between two nonsister chromatids during meiosis are shown in Figure \(\PageIndex{1}\). If the recombination frequency between two genes is less than 50 percent, they are said to be linked.
The generation of genetic maps requires markers, just as a road map requires landmarks (such as rivers and mountains). Early genetic maps were based on the use of known genes as markers. More sophisticated markers, including those based on non-coding DNA, are now used to compare the genomes of individuals in a population. Although individuals of a given species are genetically similar, they are not identical; every individual has a unique set of traits. These minor differences in the genome between individuals in a population are useful for the purposes of genetic mapping. In general, a good genetic marker is a region on the chromosome that shows variability or polymorphism (multiple forms) in the population.
Some genetic markers used in generating genetic maps are restriction fragment length polymorphisms (RFLP), variable number of tandem repeats (VNTRs), microsatellite polymorphisms , and the single nucleotide polymorphisms (SNPs). RFLPs (sometimes pronounced “rif-lips”) are detected when the DNA of an individual is cut with a restriction endonuclease that recognizes specific sequences in the DNA to generate a series of DNA fragments, which are then analyzed by gel electrophoresis. The DNA of every individual will give rise to a unique pattern of bands when cut with a particular set of restriction endonucleases; this is sometimes referred to as an individual’s DNA “fingerprint.” Certain regions of the chromosome that are subject to polymorphism will lead to the generation of the unique banding pattern. VNTRs are repeated sets of nucleotides present in the non-coding regions of DNA. Non-coding, or “junk,” DNA has no known biological function; however, research shows that much of this DNA is actually transcribed. While its function is uncertain, it is certainly active, and it may be involved in the regulation of coding genes. The number of repeats may vary in individual organisms of a population. Microsatellite polymorphisms are similar to VNTRs, but the repeat unit is very small. SNPs are variations in a single nucleotide.
Because genetic maps rely completely on the natural process of recombination, mapping is affected by natural increases or decreases in the level of recombination in any given area of the genome. Some parts of the genome are recombination hotspots, whereas others do not show a propensity for recombination. For this reason, it is important to look at mapping information developed by multiple methods.
Physical Maps
A physical map provides detail of the actual physical distance between genetic markers, as well as the number of nucleotides. There are three methods used to create a physical map: cytogenetic mapping, radiation hybrid mapping, and sequence mapping. Cytogenetic mapping uses information obtained by microscopic analysis of stained sections of the chromosome. It is possible to determine the approximate distance between genetic markers using cytogenetic mapping, but not the exact distance (number of base pairs). Radiation hybrid mapping uses radiation, such as x-rays, to break the DNA into fragments. The amount of radiation can be adjusted to create smaller or larger fragments. This technique overcomes the limitation of genetic mapping and is not affected by increased or decreased recombination frequency. Sequence mapping resulted from DNA sequencing technology that allowed for the creation of detailed physical maps with distances measured in terms of the number of base pairs. The creation of genomic libraries and complementary DNA (cDNA) libraries (collections of cloned sequences or all DNA from a genome) has sped up the process of physical mapping. A genetic site used to generate a physical map with sequencing technology (a sequence-tagged site, or STS) is a unique sequence in the genome with a known exact chromosomal location. An expressed sequence tag (EST) and a single sequence length polymorphism (SSLP) are common STSs. An EST is a short STS that is identified with cDNA libraries, while SSLPs are obtained from known genetic markers and provide a link between genetic maps and physical maps.
Integration of Genetic and Physical Maps
Genetic maps provide the outline and physical maps provide the details. It is easy to understand why both types of genome mapping techniques are important to show the big picture. Information obtained from each technique is used in combination to study the genome. Genomic mapping is being used with different model organisms that are used for research. Genome mapping is still an ongoing process, and as more advanced techniques are developed, more advances are expected. Genome mapping is similar to completing a complicated puzzle using every piece of available data. Mapping information generated in laboratories all over the world is entered into central databases, such as GenBank at the National Center for Biotechnology Information (NCBI). Efforts are being made to make the information more easily accessible to researchers and the general public. Just as we use global positioning systems instead of paper maps to navigate through roadways, NCBI has created a genome viewer tool to simplify the data-mining process.
Scientific Method Connection: How to Use a Genome Map Viewer
Problem statement : Do the human, macaque, and mouse genomes contain common DNA sequences?
Develop a hypothesis.
To test the hypothesis, click here .
In Search box on the left panel, type any gene name or phenotypic characteristic, such as iris pigmentation (eye color). Select the species you want to study, and then press Enter. The genome map viewer will indicate which chromosome encodes the gene in your search. Click each hit in the genome viewer for more detailed information. This type of search is the most basic use of the genome viewer; it can also be used to compare sequences between species, as well as many other complicated tasks.
Is the hypothesis correct? Why or why not?
Link to Learning
Online Mendelian Inheritance in Man (OMIM) is a searchable online catalog of human genes and genetic disorders. This website shows genome mapping information, and also details the history and research of each trait and disorder. Click this link to search for traits (such as handedness) and genetic disorders (such as diabetes).
Summary
Genome mapping is similar to solving a big, complicated puzzle with pieces of information coming from laboratories all over the world. Genetic maps provide an outline for the location of genes within a genome, and they estimate the distance between genes and genetic markers on the basis of recombination frequencies during meiosis. Physical maps provide detailed information about the physical distance between the genes. The most detailed information is available through sequence mapping. Information from all mapping and sequencing sources is combined to study an entire genome.
Glossary
- cytogenetic mapping
- technique that uses a microscope to create a map from stained chromosomes
- expressed sequence tag (EST)
- short STS that is identified with cDNA
- genetic map
- outline of genes and their location on a chromosome
- genetic marker
- gene or sequence on a chromosome with a known location that is associated with a specific trait
- genetic recombination
- exchange of DNA between homologous pairs of chromosomes
- genome mapping
- process of finding the location of genes on each chromosome
- cDNA library
- collection of cloned cDNA sequences
- genomic library
- collection of cloned DNA which represents all of the sequences and fragments from a genome
- genomics
- study of entire genomes including the complete set of genes, their nucleotide sequence and organization, and their interactions within a species and with other species
- linkage analysis
- procedure that analyzes the recombination of genes to determine if they are linked
- microsatellite polymorphism
- variation between individuals in the sequence and number of repeats of microsatellite DNA
- physical map
- representation of the physical distance between genes or genetic markers
- radiation hybrid mapping
- information obtained by fragmenting the chromosome with x-rays
- restriction fragment length polymorphism (RFLP)
- variation between individuals in the length of DNA fragments generated by restriction endonucleases
- sequence mapping
- mapping information obtained after DNA sequencing
- single nucleotide polymorphism (SNP)
- variation between individuals in a single nucleotide
- variable number of tandem repeats (VNTRs)
- variation in the number of tandem repeats between individuals in the population


2.4: A Close Look at Water Chemistry
1. Hydrogen Bonds, the Polarity and Properties of Water
Hydrogen bonds are a subcategory of electrostatic interaction (i.e., formed by the attraction of oppositely charges). As noted above, water molecules attract one another ( cohere ) because of strong electrostatic interactions that form H-bonds. Because of water’s polar covalent nature, it is able to attract positively and negatively charged groups of solutes, making it a good solvent. Solutes (water-soluble molecules) or molecular surfaces attracted to water are hydrophilic . Lipids like fats and oils are not polar molecules and therefore do not dissolve in water; they are hydrophobic (from hydro, water; phobic, fearing) .
Soluble salts like NaCl dissolve because the Cl- and Na+ ions more strongly attract the partial positive and negative charges (respectively) of water molecules. The result is that the ions separate. We call this separation of salt ionization. The ionization of NaCl dissolving in water is shown below.
Water is also a good solvent for macromolecules (proteins, nucleic acids) with exposed polar chemical groups on their surfaces that attract water molecules, as shown below.
125 Water, Hydrogen & ionic Bonds
In addition to being a good solvent, we recognize the following properties of water, all of which result from its polar nature and H-bonding abilities:
- Cohesion : the ability of water molecules to stick together via H-bonds.
- High Surface tension : water’s high cohesion means that it can be hard to break the surface; think the water strider, an insect that ‘walks’ on water.
- Adhesion : this results from water’s electrostatic interactions with ions and the partial charges on polar covalent molecules or functional groups. Adhesion explains water’s solvent properties and (at least in part) capillary action where water molecules ‘crawl’ along hydrophilic surfaces, often against the force of gravity.
- High specific heat : The cohesion of water molecules is so strong that it takes a lot of energy to separate the molecules and make them move faster, i.e., to heat water; specifically it takes 1 Kcal , (1 Calorie , with a capital C) to heat a gram of water 1oC. Incidentally, high specific heat also explains why water “holds its heat” (i.e., stays hotter longer that the pot that it’s in!).
- High heat of vaporization : It takes even more energy per gram of water to turn it into water vapor!
2. Water Ionization and pH
One last property of water: it ionizes weakly to form H+ and OH- ions, - or more correctly, pairs of water molecules form H3O+ and OH- ions. You can think of this as happening in the following two reactions:
Acid molecules added to water dissociate and release protons. This drives reaction #2, forming more H3O+ ions in the solution, in turn driving reaction #1 forward. A pH meter measures the relative acidity or concentration of protons in a solution. Acidic solutions have a pH below 7.0 (neutrality).
Bases ionizing in water release OH- (hydroxyl) ions. The increase in OH- ions removes protons from the solution, driving both reaction in reverse and raising the pH of the solution.
To summarize acid-base chemistry:
When dissolved in water
- Acids release H+
- Bases accept H+
Since the pH of a solution is the negative logarithm of the hydrogen ion concentration,
PH and solution
- at pH 7.0, a solution is neutral
- below a pH of 7.0, a solution is acidic
- above a pH of 7.0, a solution is basic
Check a basic chemistry book to remind yourself of the relationship between pH and the [H+] in a solution!


6.3: Mid term Exam 2
Mid term Exam 2 (30 %)
Instructions
Answer all the questions in this paper
1. (a). Draw the block diagram of a DMA controller (10 marks)
(b). explain its functioning (5 marks)
2.Describe the architecture of a shared memory multiprocessor (5 marks)
Grading Scheme
1.(a). Correct diagram 10 marks
(b). Correct function 2 marks each (max 10)
2. Any correct answer 2 marks each (max 10)
Feedback
1. (a). Draw the block diagram of a DMA controller
(b). Explain its functioning
During any given bus cycle, one of the system components connected to the system bus is given control of the bus. This component is said to be the master during that cycle and the component it is communicating with is said to be the slave. The CPU with its bus control logic is normally the master, but other specially designed components can gain control of the bus by sending a bus request to the CPU. After the current bus cycle is completed the CPU will return a bus grant signal and the component sending the request will become the master.
2. Describe the architecture of a shared memory multiprocessor
- Processors have their own connection to memory
- Processors are capable of independent execution and control
- Have a single OS for the whole system, support both processes and threads, and appear as a common multiprogrammed system
- Can be used to run multiple sequential programs concurrently or parallel programs
- Suitable for parallel programs where threads can follow different code (task-level-parallelism)


16.9: Corrosion Gallery
Corrosion gallery
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Corrosion of a nail The nails are immersed in agar which forms a moist solid gel. The agar also contains phenolphthalein and hexacyanoiron(III) Fe(CN 6 ) – which forms a deep blue color ("prussian blue") in the presence of Fe 2 + . The blue colors are clearly associated with those parts of the nail that have been stressed, thus faciliting the anodic release of Fe 2 + from the metal. The the pink color shows the cathodic regions that have been made alkaline by the reaction O 2 + 2 H 2 O + 4 e – → 4 OH – This clearly shows the separation between the anodic and cathodic processes in corrosion. [Illustration from U of West Indies: link] |
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Water distribution main If you live in the older part of a city where the mains are 50-100 years old, the water you drink may well have passed through a pipe in this condition! Severe corrosion like this is more common in areas where the water is acidic. Such water comes from mountain snowmelt and runoff, and usually acquires its acidity from dissolved atmospheric carbon dioxide. Waters from rivers, lakes, and especially groundwaters from wells have usually been in sufficiently long contact with carbonate-containing sediments to have been neutralized. Water-works administrators like to make the water slightly alkaline and slightly supersaturated in calcium carbonate in order to maintain a thin coating of solid carbonate on the interior of the pipe which acts to protect it from corrosion. |
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Corrosion of reinforcing bars in concrete All large concrete structures contain steel reinforcing bars ("rebars") that help ensure structural integrity under varying load conditions and especially during earthquakes. Intrusion of water, even in the form of fog or mists, can lead to serious corrosion damage, as seen in this picture of this column which supports a highway overpass. |
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Corrosion at metallic joints The picture shows two steel structural members joined by cast iron flanges which have been bolted together. For some reason, one of the pieces has become more anodic than the other, leading to extensive corrosion of the upper part. |
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Bacterial-assisted corrosion This gas pipe was buried in a red soil that contained iron pyrites (FeS.) The bacterium thiobacillus ferrooxidans derives its energy by oxidizing Fe 2 + to the more soluble Fe 3 + , transferring the electrons to O 2 . It also oxidizes the sulfur, producing sulfuric acid. The resulting chemical cocktail has eaten a hole into the pipe. |
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| These galvanized bolts were used to join wooden beams together. Subsequent movement of the beams due to varying load conditions abraded the zinc coating. A lack of oxygen near the centers of the bolts also likely contributed to the corrosion by preventing the formation of a protective oxide film. |
Pitting corrosion
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12.8: Review Questions
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1.
What are the cultural expectations about normative behavior and achievements associated with age, such as starting a career or having a first child, influenced by?
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rank-order stability
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mean-level change
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the maturity principle
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the social clock
2 .
What theory predicts that when young people commit to certain life roles such as spouse or store assistant manager, corresponding changes in personality traits to support the demands of these roles will occur?
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the maturity principle
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social investment theory
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the quarter-life crisis
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rank-order stability
3.
What age range was theorized to serve as a "second adulthood" for women by write Gail Sheehy?
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ages 20 to 30
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ages 30 to 40
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ages 40 to 50
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ages 50 to 60
4 .
Studies using Marcia’s identity status have consistently found that identity commitment was most strongly encouraged by which life event?
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graduating from college
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becoming a parent
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getting married
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starting a career
5.
The relevance of a person’s identity in a particular situation is called identity ________.
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salience
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regard
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diffusion
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moratorium
6 .
What term describes the overlapping experience of different aspects of our identity, such as ethnicity, gender, and sexuality?
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intersectionality
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identity confusion
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identity centrality
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moratorium-achievement-moratorium-achievement (MAMA) cycle
7.
For British families, what is the most common expected reason for young adults to leave the parental home?
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employment or education opportunities
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marriage and plans for parenthood
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financial independence
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conflict in the home of origin
8 .
In early adulthood, friendship networks usually
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expand
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shrink
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do not change in size
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become irrelevant
9.
What is the most common source of disagreement between young adults and their parents?
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parents' retirement plans, including moving homes
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parents' religion and their insistence on the child maintaining those beliefs
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emerging adult's sexual orientation or romantic relationships
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emerging adult’s financial decisions
10 .
Developmental trajectories are determined by the interaction of
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family and friends
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friends and neighbors
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behavior and context
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attitude and personality
11.
According to Super’s lifespan theory of career development, what occurs in early adulthood?
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Career exploration is followed by career establishment.
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Career establishment is followed by career exploration.
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College students should choose a major aligned with their career goals.
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Intimacy should be developed before vocational identity.
12 .
Life balance is the extent to which a person feels able to ________.
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maintain their identity within intimate relationships
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live independently
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find a job aligned with their education
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spend the desired amount of time on activities that matter
13.
Research on the relationship between identity and intimacy in modern young adults finds that which of the following is true?
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Intimacy predicts the development of identity for people of any sex.
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Identity predicts the development of intimacy for people of any sex.
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Intimacy predicts the development of identity, but only for men.
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Identity predicts the development of intimacy, but only for women.
14 .
What contextual factor can make it more likely that a person will get married earlier in young adulthood?
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high level of education
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being career-focused
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living in a rural area
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being non-heterosexual
15.
Which trend with regard to cohabiting with a romantic partner to whom one is not married is true?
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More young adults now live with a cohabiting romantic partner than live with a spouse.
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Cohabitation is far more common among the LGBTQ+ population than among those who identify as heterosexual.
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Cohabitation becomes less common as people increase in their financial income and socioeconomic status.
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People who are very religious are more likely to cohabit as a “test for marriage and hedge against divorce” than those who are not.


11.1.3A: Minority Groups
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The term “minority” is applied to various groups who hold few or no positions of power in a given society.
Learning Objectives
- Define a minority group
Key Points
- The term minority doesn’t necessarily refer to a numeric minority. Women, for example, make up roughly half the population but are often considered a minority group.
- Affirmative action refers to policies that take factors, such as race, religion, sexual orientation, or national origin into account in order to benefit an underrepresented group.
Key Terms
- affirmative action : A policy or program providing advantages for people of a minority group who are seen to have traditionally been discriminated against, with the aim of creating a more egalitarian society through preferential access to education, employment, health care, social welfare, etc.
- minority group : a sociological category that is differentiated, defined, and often discriminated against by those who hold the majority of positions of social power
- minority : Categories of persons who hold few or no positions of social power in a given society.
In the social sciences, “minority” does not just refer to a statistical measure and can instead refer to categories of persons who hold few or no positions of social power in a given society.
Sociologist Louis Wirth defined a minority group as “a group of people who, because of their physical or cultural characteristics, are singled out from the others in the society in which they live for differential and unequal treatment, and who therefore regard themselves as objects of collective discrimination. ” This definition includes both objective and subjective criteria: membership of a minority group is objectively ascribed by society, based on an individual’s physical or behavioral characteristics, such as ethnicity and race or gender and sexuality. It is also subjectively applied by its members, who may use their status as the basis of group identity or solidarity.
Minority group status is also categorical in nature: an individual who exhibits the physical or behavioral characteristics of a given minority group will be accorded the status of that group and be subject to the same treatment as other members of that minority group.
Racial or Ethnic Minorities
Every large society contains ethnic minorities: subgroups that share a common heritage, which often consists of a shared language, culture (often including a religion), or ideology that stresses common ancestry or endogamy. In this case, while minority status can be conditioned by a clear numerical difference, more significantly it refers to issues of political power. In some places, subordinate ethnic groups may constitute a numerical majority, such as blacks in South Africa under apartheid.
In addition to long-established ethnic minority populations in various nation-states, ethnic minorities may consist of more recent migrant, indigenous, or landless nomadic communities residing within, or between, a particular national territory.
Gender and Sexuality Minorities
Recognition of lesbian, gay, bisexual, and transgender people as a minority group or groups has gained prominence in the Western world since the nineteenth century. The abbreviation “LGBT” is currently used to group these identities together. The term queer is sometimes understood as an umbrella term for all non-normative sexualities and gender expressions but does not always signify a minority; rather, as with many gay rights activists of the 1960s and 1970s, it sometimes represents an attempt to highlight sexual diversity in everyone.
While in most societies the numbers of men and women are roughly equal, the status of women as a oppressed group has led some, such as feminists and other participants in women’s rights movements, to identify them as a minority group.
Religious Minorities
Persons belonging to religious minorities have a faith which is different from that held by the majority population or the population group that is in power. It is now accepted in many multicultural societies around the world that people should have the freedom to choose their own religion, including not having any religion (atheism or agnosticism), and including the right to convert from one religion to another. However in some countries this freedom is still either formally restricted or subject to cultural bias from the majority population.
People with Disabilities
The disability rights movement has contributed to an understanding of people with disabilities as a minority or a coalition of minorities who are disadvantaged by society, not just as people who are disadvantaged by their impairments.
Advocates of disability rights emphasize differences in physical or psychological functioning, rather than inferiority: for example, some people with autism argue for acceptance of neuro-diversity, in the same way in which opponents of racism argue for acceptance of ethnic diversity. The deaf community is often regarded as a linguistic and cultural minority rather than a group with disabilities, and some deaf people do not see themselves as having a disability at all. Rather, they are disadvantaged by technologies and social institutions that are designed to cater to the dominant, hearing-unimpaired group.
Affirmative Action
Affirmative action is a controversial issue, which refers to policies that take factors including race, religion, gender, sexual orientation, or national origin into consideration in order to benefit an underrepresented group. This is usually justified as countering the effects of a history of discrimination. Affirmative action can, for example, take the form of a government program to provide immigrant or minority groups who primarily speak a marginalized language with extra teaching in the majority language, so that they are better able to compete for places at university or for jobs.
Ethnic Minority Disparities in Cancer Treatment : Illinois Congressman Jesse Jackson discusses minority health research at the National Institute of Health.
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2.7: Hardanger Fiddle Performance, 1940
An audio element has been excluded from this version of the text. You can listen to it online here: https://wisc.pb.unizin.org/driftless/?p=1101
This recording is of fiddler Dagny Andrea Veum Quisling (1869-1952) of Madison playing a Halling, an athletic rural Norwegian dance that involves leaps and kicks performed by young men.
Quisling was born in Fyredal, Telemark, Norway, a stronghold for Hardanger fiddlers. Her husband was Sverre Quisling, a physician in Madison. Sometimes called Andrea, Quisling was the only woman active in the Kappleik (Competitions) of the Hardanger Fiddle Association of America. She competed in a 1921 Kappleik in Mount Horeb. Please visit the HFAA for more information: www.hfaa.org/Home/articles-on...-united-states
1921, Mt. Horeb, WI
Over 8,000 attend this Kappleik!
Winners: Eilef Smedal , Harald Smedal
Participants: Tolleif Strand (Chippewa Falls, WI), Eilef Smedal (La Crosse, WI), Harald Smedal (McFarland, WI), Andrea Quisling (Madison, WI), Hans Fykerud (Stoughton, WI), Bjørgulv Bjørnaraa (Wanke, MN), Sjur Bjotveit (Chicago, IL), Helge Thyresen (Dentybow, MN), John Gudvangen (Chicago, IL), I.G. Nordgaard (Setesdal), Olav Smedal (Albert Lea, MN), S. Indeleggen (Chicago, IL), Gunnar Helland (Chippewa Falls, WI)
This recording was produced by Helene Stratman-Thomas for the Library of Congress on May 16, 1940.
Many thanks to Professor James Leary for this information and recording.


7: Trigonometric Identities and Equations
In this chapter, we discuss how to manipulate trigonometric equations algebraically by applying various formulas and trigonometric identities. We will also investigate some of the ways that trigonometric equations are used to model real-life phenomena.
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- 7.0: Prelude to Trigonometric Identities and Equations
- Math is everywhere, even in places we might not immediately recognize. For example, mathematical relationships describe the transmission of images, light, and sound. Such phenomena are described using trigonometric equations and functions. In this chapter, we discuss how to manipulate trigonometric equations algebraically by applying various formulas and trigonometric identities.
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- 7.2: Sum and Difference Identities
- In this section, we will learn techniques that will enable us to solve useful problems. The formulas that follow will simplify many trigonometric expressions and equations. Keep in mind that, throughout this section, the termformula is used synonymously with the word identity.
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- 7.3: Double-Angle, Half-Angle, and Reduction Formulas
- In this section, we will investigate three additional categories of identities. Double-angle identities are derived from the sum formulas of the fundamental trigonometric functions: sine, cosine, and tangent. Reduction formulas are especially useful in calculus, as they allow us to reduce the power of the trigonometric term. Half-angle formulas allow us to find the value of trigonometric functions involving half-angles, whether the original angle is known or not.
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- 7.4: Sum-to-Product and Product-to-Sum Formulas
- From the sum and difference identities, we can derive the product-to-sum formulas and the sum-to-product formulas for sine and cosine. The product-to-sum formulas can rewrite products of sines, products of cosines, and products of sine and cosine as sums or differences of sines and cosines. We can also derive the sum-to-product identities from the product-to-sum identities using substitution. The sum-to-product formulas are used to rewrite sum or difference as products of sines and cosines.
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- 7.5: Solving Trigonometric Equations
- In earlier sections of this chapter, we looked at trigonometric identities. Identities are true for all values in the domain of the variable. In this section, we begin our study of trigonometric equations to study real-world scenarios such as the finding the dimensions of the pyramids.
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- 7.6: Modeling with Trigonometric Equations
- Many natural phenomena are also periodic. For example, the phases of the moon have a period of approximately 28 days, and birds know to fly south at about the same time each year. So how can we model an equation to reflect periodic behavior? First, we must collect and record data. We then find a function that resembles an observed pattern and alter the function to get adependable model. Here. we will take a deeper look at specific types of periodic behavior and model equations to fit data.


5.10: Rondo Form - Five-Part Rondo Form
A five-part rondo contains three refrains separated by two episodes: ABACA. (We will occasionally use these short-hand letter designations (A, B, and so on) to label rondos, but get in the habit of labeling the sections as “Refrain 1,” “Refrain 2,” “Episode 1,” etc., to avoid ambiguity with lower-level, embedded forms.)
The episodes of a five-part rondo are interior themes or second-theme complexes. It is more common to find a second-theme complex in Episode 1 than it is to find one in Episode 2. Often, the returns of the refrain are ornamented or abridged.
The older rondeau , from which this formal type derives, could feature many more refrains and episodes. Occasionally, therefore, you will find a seven-, or even nine-part rondo in the classical style.


7.5: Profit Maximization in an Oligopoly
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To introduce oligopoly, consider an example where there are only two firms that supply the market, Firm A and Firm B. This is the simplest form of oligopoly (a duopoly). To simplify the example further, assume that both firms have identical variable cost functions \(VC= 20Q_{i}\), where \(i \in [A, B]\). This means that the marginal cost facing each firm is \(MC = 20\). Finally, suppose that the inverse demand curve for this industry is \(P = 200 - 2Q_{A}-2Q_{B}\)
Just to be clear, \(Q_{A}\) is the amount that Firm A produces and \(Q_{B}\) is the amount that Firm B produces.
Because the inverse demand curve is linear, it is easy to find marginal revenue. Note that Firm A can only choose \(Q_{A}\). The amount of \(Q_{B}\) is outside of Firm A’s control, so the \(Q_{B}\) component of the industry inverse demand curve becomes part of the intercept of the inverse demand curve that Firm A actually faces. With this in mind, the MR for Firm A is
\(MR_{A}=200-4Q_{A}-2Q_{B}\).
Set \(MR=MC\) for Firm A to find profit maximizing quantity for Firm A conditional on Firm B’s output choice
\(200-4Q_{A}-2Q_{B} = 200 \Rightarrow Q_{A} = 45 - \dfrac{1}{2} Q_{B}\).
This is known as the reaction function for Firm A. It indicates Firm A’s optimal quantity choice as a function of Firm B’s quantity. In other words, it says how Firm A’s quantity will react to a change in Firm B’s quantity.
Next, repeat the above steps to find profit maximizing quantity for Firm B conditional on Firm A’s output choice. MR for Firm B is
\(MR_{B} = 200 - 2Q_{A} -4Q_{B}\).
Set MR = MC for Firm B and solve for \(Q_{B}\) to get Firm B’s reaction function.
\(200-2Q_{A} - 4Q_{B} = 20 \Rightarrow Q_{B} = 45 \dfrac{1}{2} Q_{A}\)
Cournot Nash Equilibrium
A Cournot Nash equilibrium occurs where the reaction functions for these two firms intersect (see Figure \(\PageIndex{1}\)). To find the equilibrium, we can substitute Firm B’s reaction function into the reaction function for Firm A:
\(Q_{A} = 45 - \dfrac{1}{2}(45 - \dfrac{1}{2}Q_{A})
Now there is one equation and one unknown. Solving for \(Q_{A}\) provides \(Q_{A} = 30\). Putting this back into Firm B’s reaction function, we find that \(Q_{B} = 30\) as well.
Finally, we can find the price at the Cournot Nash Equilibrium by putting these quantities into the industry inverse demand curve to get
\(P = 200 - 2(30) - 2(30)= 80.\)
A Nash equilibrium refers to a situation wherein each agent chooses a strategy that maximizes his or her payoffs conditional on the strategies of all other agents. In the example above, the strategy variable is the quantity to be placed on the market. The Cournot equilibrium is a Nash equilibrium because 30 units is the optimal quantity to be placed on the market by Firm A, given that Firm B places 30 units on the market and vice versa . This type of equilibrium, is named after John Forbes Nash, Jr. , a mathematician who was awarded the Nobel Prize in Economics for this idea. The model in this example is one of Cournot oligopoly, wherein competition is on quantity. This model is named after another mathematician, Antoine Augustin Cournot .
The Prisoners’ Dilemma
The Cournot Nash equilibrium outcome is not optimal from the standpoint of the two oligopolists because there exists another feasible outcome that provides each firm with a higher level profits. To demonstrate, suppose that the two firms merged. Now there would be just one combined quantity to put on the market \(Q_{M} = Q_{A} + Q_{B}\). The inverse demand equation for the combined firm would be
\(P = 200 - 2Q_{M}.\)
Assuming that the merger does not impact marginal cost, the marginal cost would still be \(MC=20\), but marginal revenue for the merged firm would now be \(P-200 -4Q_{M}\). Setting marginal revenue equal to marginal cost and solving for \(Q_{M}\) results in the merged firm (now a monopolist) putting a quantity of 45 units on the market and charging a price of \(P=200-2(45) = $110\). The point here is that technically it would be feasible for the oligopolists to each put half of this amount, 22.5 units, on the market. The industry price would be $110, and they would each be more profitable.
- At the Cournot Nash equilibrium, each firm makes profits above fixed costs of \((80-20) \times 30 = $1800\).
- By each putting half of the monopoly quantity on the market, each firm would make profits above fixed costs of \((110-20) \times 22.5 = $2025\)
The prisoners’ dilemma is a metaphor commonly used in the social sciences, including economics, to help understand problems such as that facing the oligopolists in Cournot competition. In the prisoners’ dilemma, two criminals are apprehended and placed in separate cells. The prosecutor has enough evidence to put each away for two years. However, these criminals are guilty of an offence that carries an eight year sentence if evidence is sufficient to convict. The prosecutor needs one or both of the prisoners to turn state’s evidence. The payoffs facing these prisoners is summarized in the following table
| Prisoner 1 (rows), Prisoner 2 (cols.) | Prisoner 2 Tells | Prisoner 2 Doesn’t Tell |
|---|---|---|
| Prisoner 1 Tells | -6, -6 | 0, -8 |
| Prisoner 1 Doesn’t Tell | -8, 0 | -2, -2 |
Note: Prisoner 1’s payoff is the first number in each cell (no pun intended), Prisoner 2’s payoff is the second number.
The prosecutor meets the prisoners separately and offers to reduce the sentence by two years in return for implicating the other in the more serious crime. Thus, if one prisoner implicates his friend while the other remains silent, the one that turns state’s evidence will get a suspended sentence (zero years), while the other will get eight years. If both prisoners implicate each other, both get six-year sentences. If both remain silent, both get two-year sentences.
The Nash equilibrium in the prisoners’ dilemma is for each prisoner to tell on the other. As a result, each prisoner gets six years. Had the prisoners kept quiet, they would have gotten only two years each, something better from their perspective. The reason the prisoners have a dilemma is that if I think you will tell on me, it is certainly in my best interest to tell on you. Similarly, if I think you will not tell on me, it is still in my interest to tell on you and get the suspended sentence. Moreover, I know that if I do not tell on you, your best course of action is to tell on me. Thus, each prisoner’s best action given his expectation of the other’s action is to tell.
The Cournot model of oligopoly is like the prisoners’ dilemma. In our example of the duopolists above, placing half of the monopoly quantity on the market is analogous to “not telling” in the prisoners’ dilemma. Now, suppose that Firm B decides to do this and sets its quantity at \(Q_{B} = 22.5\). Firm A uses its reaction function to set its quantity as
\(Q_{A} = 45 \dfrac{1}{2}(22.5) = 33.75.\)
In this case, Firm B’s profits above fixed costs are $1,518.75. Firm A’s profits above fixed costs are $2,278.13. This is much better than what Firm A could get by matching Firm 2’s quantity at 22.5 units and much worse for Firm B in comparison to what it could have achieved had it simply put its Cournot equilibrium quantity on the market. Thus, in a Cournot oligopoly, firms have an incentive to put more on the market than that which optimizes profits for the industry as a whole. This problem is compounded as more and more firms to the Cournot oligopoly. In fact, the market price approaches the competitive price (\(P=MC\)) and profits go to zero as the number of firms in a Cournot oligopoly becomes large. Demonstration \(\PageIndex{2}\) shows what happens to total industry profits at the Cournot Nash equilibrium as the number of firms in the industry increases. Notice that the prisoners’ dilemma becomes even more pronounced as the number of firms increases.
Demonstration \(\PageIndex{2}\). Profitability of a Cournot Oligopoly Relative to a Monopoly as Number of Firms Change*
*In Demonstration \(\PageIndex{2}\), the market inverse demand curve is \(P = 200 - 2 \Sigma Q_{i}\) and \(MC = 20\) for all firms.
Alternative Bertrand Version of the Oligopoly
In the Cournot model, firms compete by setting quantities. The Bertrand model is an alternative formulation of the oligopolists’ problem and differs in that the firms compete by setting prices instead. This model is named after Joseph Bertrand, a mathematician who is credited with formalizing this model. Again, to keep things simple, assume that there are two firms. Furthermore, assume that the products produced by each firm are identical. This permits us to take the example above and adapt it to the Bertrand framework.
If Firm A sets its price lower than Firm B’s price (\(P_{A} < P_{B}\)), then Firm A faces a direct demand function of
\(Q_{A} = 100 - \dfrac{1}{2} P_{A}.\)
If Firm A sets its price above Firm B’s price, then Firm A’s demand is
\(Q_{A} = 0\)
Finally, if Firm A’s price matches Firm B’s price (\(P_{A} = P_{B}\)), then consumers randomly choose to patronize Firm A or Firm B with equal probability. In this case, Firm A stands to get about half the market, so its demand is
\(Q_{A} = 0.5(100 - \dfrac{1}{2}P_{A}).\)
The demand facing Firm B is similarly structured. Firm B gets the entire market if \(P_{B} < P_{A}\), sells nothing if \(P_{B} > P_{A}\), and splits the market 50/50 with A if \(P_{B} = P_{A}\). As before, assume that each firm faces a constant marginal cost of MC = $20.
The Bertrand Nash equilibrium outcome occurs where \(P_{A} = P_{B} = MC\). In this case, profits to each firm are zero, and the oligopoly outcome is the same as that which would have occurred under perfect competition. Demonstration \(\PageIndex{3}\) reflects the scenario just described and shows why.
Suppose that Firm A and Firm B have each chosen the monopoly price of $110. Each makes $2,025. This cannot be a Nash equilibrium because $110 is not the best price for Firm A given that Firm B charges $110. Firm A could undercut Firm B by charging a price of $80, in which case Firm A’s profits would go up to $3,600. Of course, Firm B would undercut Firm A and the process would continue until each firm’s price fell to its marginal cost of $20. This is the Bertrand Nash Equilibrium. No firm would cut below its marginal cost.
Demonstration \(\PageIndex{3}\). Bertrand Duopoly
| Price | Quantity | Profits | |
| Firm A | 80.00 | 30.00 | 1800.00 |
| Firm B | 80.00 | 30.00 | 1800.00 |
The Folk Theorem
Both Cournot and Bertrand outcomes typify the prisoners’ dilemma because equilibrium outcomes do not maximize industry profits. In each case, there is a feasible outcome (sharing the market at the monopoly price and quantity) that makes firms better off than the Nash equilibrium profits.
Might firms be able to coordinate tacitly on a quantity less than the Cournot quantity or on a price higher than the Bertrand price? Suppose, for sake of argument, that Firm A and B are in Bertrand competition and each has matched the others price at $80. This price is above the Bertrand Nash Equilibrium price of $20 per unit, but it corresponds to the market price the firms would get at the Cournot Nash Equilibrium. If, in the current period, Firm A raises its price from $80 to $110, the monopoly price, how should Firm B respond? Pretend that you are Firm B. You might reason as follows:
- If I keep my price at $80 and do not raise my price to match Firm A, I will have the entire market during this period. My profits will be $3,600. This is certainly better than $2,025 I would receive if I did match Firm A’s higher price.
- However, if I do not match, I cannot expect that Firm A will not continue to keep its price high in the next period. When it sees that I have not matched its price increase and it loses market share, it will certainly cut is price back to $80 during the next period or possibly even lower. I will be back to getting $1,800 in profits (or possibly something worse).
- If I do match Firm A’s price increase, Firm A will probably continue to keep its price high at $110. Thus I will be getting $2,025 each period as opposed to $1,800. Over the long-term, matching the price increase would pay off.
In this highly stylized example, the decision to match the price increase boils down to taking a one-time windfall of $3,600 (if Firm B does not match) or a $2,025 in profits in every period going forward (if Firm B does match).
The folk theorem states that for an indefinitely repeated prisoners’ dilemma (such as Bertrand) and provided that discount rates are not too high, an outcome preferable to the Nash equilibrium (e.g., a price above the Bertrand price) can be sustained over time. One way firms reinforce the folk theorem is to match their competitors’ pricing decisions tit-for-tat . Tit-for-tat means that one agent responds in kind to another. In the Bertrand example, Firm B plays tit-for-tat if it cuts (raises) its price whenever Firm A cuts (raises) its price. Firm B would want Firm A to know that it intends to play tit-for-tat in setting its prices. Thus, it might advertise to make sure Firm A is aware that it will face tit-for-tat retaliation in the event it decides to cut its price.
Many advertisements that you see send a subtle message to competitors that they can expect tit-for-tat retaliation in the event of a price cut. For example, consider the following Walmart advertisement: go to youtube.com . The advertisement is designed to make you feel like you will be getting the best price if you shop at Walmart and that the whole supply chain is organized to keep prices low. However, this add could also sends a subtle message to competing retailers. The message is as follows: If you cut your price, Walmart will match it. In other words, think carefully before making a price cut because Walmart will respond tit-for-tat!
Oligopolists may appeal to the folk theorem and avoid the prisoners’ dilemma inherent in the Cournot and Bertrand models provided certain conditions are met. The theorem itself mentions two conditions.
- The first is that all firms expect the competitive interaction to continue indefinitely. The folk theorem works because future profits serve as an incentive not to undercut competitors today. If firms know that competition will be for this period and this period only, then there is no incentive not to undercut. Moreover, if firms know they will compete for a known and finite number of periods, the incentive not to undercut unravels. This is because all firms expect the others to undercut them in the last period, which means it would be best to undercut in the second-to-last period, which means it would best to undercut in the third-to-last period, and so forth. The folk theorem only works if firms expect to be in competition for the foreseeable future.
- Another requirement is that discount rates not be too high. For the folk theorem to apply, future returns that result from not undercutting your competitor’s price must be large enough to offset the immediate gains you will receive if you do undercut a competitor. If discount rates are high, then returns in the future have less value today and the folk theorem is less likely to apply.
The rapidity with which firms can respond to price changes of competitors also affect whether the folk theorem can hold. If firms play tit-for-tat, then when one firm undercuts, it can expect retaliation by other firms. If it takes a long time for other firms to retaliate, then undercutting becomes more attractive regardless of the discount rate.
It also helps if firms are similar. This is because it is illegal for firms to collude to fix prices. There must be some price point that all firms can agree upon tacitly, without formally colluding to fix a price. The existence of such a point will be more likely if firms are reasonably similar in terms of costs and product characteristics.


1.3: Meter Usage
Safe Meter Usage
Using an electrical meter safely and efficiently is perhaps the most valuable skill an electronics technician can master, both for the sake of their own personal safety and for proficiency at their trade. It can be daunting at first to use a meter, knowing that you are connecting it to live circuits which may harbor life-threatening levels of voltage and current. This concern is not unfounded, and it is always best to proceed cautiously when using meters. Carelessness more than any other factor is what causes experienced technicians to have electrical accidents.
The most common piece of electrical test equipment is a meter called the multimeter . Multimeters are so named because they have the ability to measure a multiple of variables: voltage, current, resistance , and often many others, some of which cannot be explained here due to their complexity. In the hands of a trained technician, the multimeter is both an efficient work tool and a safety device. In the hands of someone ignorant and/or careless, however, the multimeter may become a source of danger when connected to a “live” circuit.
There are many different brands of multimeters, with multiple models made by each manufacturer sporting different sets of features. Most meters used in industry today have digital displays, rather than a needle style. These Digital Multimeters (DMMs) allow us to measure voltage, current and resistance. The multimeter shown here in the following illustrations is a “generic” design, not specific to any manufacturer, but general enough to teach the basic principles of use:
The rotary selector switch (now set in the Off position) has five different measurement positions it can be set in: two “V” settings, two “A” settings, and one setting in the middle with a funny-looking “horseshoe” symbol on it representing “resistance.” The “horseshoe” symbol is the Greek letter “Omega” (Ω), which is the common symbol for the electrical unit of ohms.
Of the two “V” settings and two “A” settings, you will notice that each pair is divided into unique markers with either a pair of horizontal lines (one solid, one dashed), or a dashed line with a squiggly curve over it. The parallel lines represent “DC” while the squiggly curve represents “AC.” The “V” of course stands for “voltage” while the “A” stands for “amperage” (current). The meter uses different techniques, internally, to measure DC than it uses to measure AC, and so it requires the user to select which type of voltage (V) or current (A) is to be measured. Although we haven’t discussed alternating current (AC) in any technical detail, this distinction in meter settings is an important one to bear in mind.
There are three different sockets on the multimeter face into which we can plug our test leads . Test leads are nothing more than specially-prepared wires used to connect the meter to the circuit under test. The wires are coated in a color-coded (either black or red) flexible insulation to prevent the user’s hands from contacting the bare conductors , and the tips of the probes are sharp, stiff pieces of wire:
The black test lead always plugs into the black socket on the multimeter: the one marked “COM” for “common.” The red test lead plugs into either the red socket marked for voltage and resistance, or the red socket marked for current, depending on which quantity you intend to measure with the multimeter.
To see how this works, let’s look at a couple of examples showing the meter in use. First, we’ll set up the meter to measure DC voltage from a battery:
Note that the two test leads are plugged into the appropriate sockets on the meter for voltage, and the selector switch has been set for DC “V”. Now, we’ll take a look at an example of using the multimeter to measure AC voltage from a household electrical power receptacle (wall socket):
The only difference in the setup of the meter is the placement of the selector switch: it is now turned to AC “V”. Since we’re still measuring voltage, the test leads will remain plugged in the same sockets. In both of these examples, it is imperative that you not let the probe tips come in contact with one another while they are both in contact with their respective points on the circuit. If this happens, a short-circuit will be formed, creating a spark and perhaps even a ball of flame if the voltage source is capable of supplying enough current! The following image illustrates the potential for hazard:
This is just one of the ways that a meter can become a source of hazard if used improperly.
Voltage measurement is perhaps the most common function a multimeter is used for. It is certainly the primary measurement taken for safety purposes (part of the lock-out/tag-out procedure), and it should be well understood by the operator of the meter. Being that voltage is always relative between two points, the meter must be firmly connected to two points in a circuit before it will provide a reliable measurement. That usually means both probes must be grasped by the user’s hands and held against the proper contact points of a voltage source or circuit while measuring.
Because a hand-to-hand shock current path is the most dangerous, holding the meter probes on two points in a high-voltage circuit in this manner is always a potential hazard. If the protective insulation on the probes is worn or cracked, it is possible for the user’s fingers to come into contact with the probe conductors during the time of test, causing a bad shock to occur. If it is possible to use only one hand to grasp the probes, that is a safer option. Sometimes it is possible to “latch” one probe tip onto the circuit test point so that it can be let go of and the other probe set in place, using only one hand. Special probe tip accessories such as spring clips can be attached to help facilitate this.
Remember that meter test leads are part of the whole equipment package, and that they should be treated with the same care and respect that the meter itself is. If you need a special accessory for your test leads, such as a spring clip or other special probe tip, consult the product catalog of the meter manufacturer or other test equipment manufacturer. Do not try to be creative and make your own test probes, as you may end up placing yourself in danger the next time you use them on a live circuit.
Also, it must be remembered that digital multimeters usually do a good job of discriminating between AC and DC measurements, as they are set for one or the other when checking for voltage or current. As we have seen earlier, both AC and DC voltages and currents can be deadly, so when using a multimeter as a safety check device you should always check for the presence of both AC and DC, even if you’re not expecting to find both! Also, when checking for the presence of hazardous voltage, you should be sure to check all pairs of points in question.
For example, suppose that you opened up an electrical wiring cabinet to find three large conductors supplying AC power to a load. The circuit breaker feeding these wires (supposedly) has been shut off, locked, and tagged. You double-checked the absence of power by pressing the Start button for the load. Nothing happened, so now you move on to the third phase of your safety check: the meter test for voltage.
First, you check your meter on a known source of voltage to see that its working properly. Any nearby power receptacle should provide a convenient source of AC voltage for a test. You do so and find that the meter indicates as it should. Next, you need to check for voltage among these three wires in the cabinet. But voltage is measured between two points, so where do you check?
The answer is to check between all combinations of those three points. As you can see, the points are labeled “A”, “B”, and “C” in the illustration, so you would need to take your multimeter (set in the voltmeter mode) and check between points A & B, B & C, and A & C. If you find voltage between any of those pairs, the circuit is not in a Zero Energy State. But wait! Remember that a multimeter will not register DC voltage when its in the AC voltage mode and vice versa, so you need to check those three pairs of points in each mode for a total of six voltage checks in order to be complete!
However, even with all that checking, we still haven’t covered all possibilities yet. Remember that hazardous voltage can appear between a single wire and ground (in this case, the metal frame of the cabinet would be a good ground reference point) in a power system. So, to be perfectly safe, we not only have to check between A & B, B & C, and A & C (in both AC and DC modes), but we also have to check between A & ground, B & ground, and C & ground (in both AC and DC modes)! This makes for a grand total of twelve voltage checks for this seemingly simple scenario of only three wires. Then, of course, after we’ve completed all these checks, we need to take our multimeter and re-test it against a known source of voltage such as a power receptacle to ensure that its still in good working order.
Using a multimeter to check for resistance is a much simpler task. The test leads will be kept plugged in the same sockets as for the voltage checks, but the selector switch will need to be turned until it points to the “horseshoe” resistance symbol. Touching the probes across the device whose resistance is to be measured, the meter should properly display the resistance in ohms:
One very important thing to remember about measuring resistance is that it must only be done on de-energized components! When the meter is in “resistance” mode, it uses a small internal battery to generate a tiny current through the component to be measured. By sensing how difficult it is to move this current through the component, the resistance of that component can be determined and displayed. If there is any additional source of voltage in the meter-lead-component-lead-meter loop to either aid or oppose the resistance-measuring current produced by the meter, faulty readings will result. In a worse-case situation, the meter may even be damaged by the external voltage.
The “resistance” mode of a multimeter is very useful in determining wire continuity as well as making precise measurements of resistance. When there is a good, solid connection between the probe tips (simulated by touching them together), the meter shows almost zero Ω. If the test leads had no resistance in them, it would read exactly zero:
If the leads are not in contact with each other, or touching opposite ends of a broken wire, the meter will indicate infinite resistance (usually by displaying dashed lines or the abbreviation “O.L.” which stands for “open loop”):
By far the most hazardous and complex application of the multimeter is in the measurement of current. The reason for this is quite simple: in order for the meter to measure current, the current to be measured must be forced to go through the meter. This means that the meter must be made part of the current path of the circuit rather than just be connected off to the side somewhere as is the case when measuring voltage. In order to make the meter part of the current path of the circuit, the original circuit must be “broken” and the meter connected across the two points of the open break. To set the meter up for this, the selector switch must point to either AC or DC “A” and the red test lead must be plugged in the red socket marked “A”. The following illustration shows a meter all ready to measure current and a circuit to be tested:
Now, the circuit is broken in preparation for the meter to be connected:
The next step is to insert the meter in-line with the circuit by connecting the two probe tips to the broken ends of the circuit, the black probe to the negative (-) terminal of the 9-volt battery and the red probe to the loose wire end leading to the lamp:
This example shows a very safe circuit to work with. 9 volts hardly constitutes a shock hazard, and so there is little to fear in breaking this circuit open (bare handed, no less!) and connecting the meter in-line with the flow of electrons. However, with higher power circuits, this could be a hazardous endeavor indeed. Even if the circuit voltage was low, the normal current could be high enough that an injurious spark would result the moment the last meter probe connection was established.
Another potential hazard of using a multimeter in its current-measuring (“ammeter”) mode is failure to properly put it back into a voltage-measuring configuration before measuring voltage with it. The reasons for this are specific to ammeter design and operation. When measuring circuit current by placing the meter directly in the path of current, it is best to have the meter offer little or no resistance against the flow of electrons. Otherwise, any additional resistance offered by the meter would impede the electron flow and alter the circuits operation. Thus, the multimeter is designed to have practically zero ohms of resistance between the test probe tips when the red probe has been plugged into the red “A” (current-measuring) socket. In the voltage-measuring mode (red lead plugged into the red “V” socket), there are many mega-ohms of resistance between the test probe tips, because voltmeters are designed to have close to infinite resistance (so that they don’t draw any appreciable current from the circuit under test).
When switching a multimeter from current- to voltage-measuring mode, its easy to spin the selector switch from the “A” to the “V” position and forget to correspondingly switch the position of the red test lead plug from “A” to “V”. The result—if the meter is then connected across a source of substantial voltage—will be a short-circuit through the meter!
To help prevent this, most multimeters have a warning feature by which they beep if ever there’s a lead plugged in the “A” socket and the selector switch is set to “V”. As convenient as features like these are, though, they are still no substitute for clear thinking and caution when using a multimeter.
All good-quality multimeters contain fuses inside that are engineered to “blow” in the event of excessive current through them, such as in the case illustrated in the last image. Like all overcurrent protection devices, these fuses are primarily designed to protect the equipment (in this case, the meter itself) from excessive damage, and only secondarily to protect the user from harm. A multimeter can be used to check its own current fuse by setting the selector switch to the resistance position and creating a connection between the two red sockets like this:
A good fuse will indicate very little resistance while a blown fuse will always show “O.L.” (or whatever indication that model of multimeter uses to indicate no continuity). The actual number of ohms displayed for a good fuse is of little consequence, so long as its an arbitrarily low figure.
So now that we’ve seen how to use a multimeter to measure voltage, resistance, and current, what more is there to know? Plenty! The value and capabilities of this versatile test instrument will become more evident as you gain skill and familiarity using it. There is no substitute for regular practice with complex instruments such as these, so feel free to experiment on safe, battery-powered circuits.


8.5: Exercises
Exercise \(\PageIndex{1}\)
Find the values of \((i)^i\) .
- Answer
-
We can write \(i\) in polar coordinates as \(\exp(i\pi/2).\) Hence, \[\begin{align} (i)^i &= \exp\Big\{i \ln\big[\exp(i\pi/2)\big]\Big\} \\ &= \exp\left\{i \left[\frac{i\pi}{2} + 2 \pi i n\right]\right\}, \quad n \in \mathbb{Z} \\ &= \exp\left[- 2\pi\left(n+\frac{1}{4}\right) \right], \quad n \in \mathbb{Z}.\end{align}\]
Exercise \(\PageIndex{2}\)
Prove that \(\ln(z)\) has branch points at \(z = 0\) and \(z = \infty\) .
- Answer
-
Let \(z = r\exp(i\theta)\), where \(r > 0\). The values of the logarithm are \[\ln(z) = \ln(r) + i (\theta + 2\pi n), \;\;\;n \in \mathrm{Z}.\] For each \(n\), note that the first term is the real part and the second term is the imaginary part of a complex number \(w_n\). The logarithm in the first term can be taken to be the real logarithm.
For \(z \rightarrow 0\), we have \(r \rightarrow 0\) and hence \(\ln(r)\rightarrow -\infty\). This implies that \(w_n\) lies infinitely far to the left of the origin on the complex plane. Therefore, \(w_n \rightarrow \infty\) (referring to the complex infinity) regardless of the value of \(n\). Likewise, for \(z \rightarrow \infty\), we have \(r \rightarrow \infty\) and hence \(\ln(r)\rightarrow +\infty\). This implies that \(w_n\) lies infinitely far to the right of the origin on the complex plane, so \(w_n \rightarrow \infty\) regardless of the value of \(n\). Therefore, \(0\) and \(\infty\) are both branch points of the complex logarithm.
Exercise \(\PageIndex{3}\)
For each of the following multi-valued functions, find all the possible function values, at the specified \(z\) :
-
\(z^{1/3}\)
at
\(z = 1\)
.
-
\(z^{3/5}\)
at
\(z = i\)
.
-
\(\ln(z+i)\)
at
\(z = 1\)
.
-
\(\cos^{-1}(z)\)
at
\(z = i\)
Exercise \(\PageIndex{4}\)
For the square root operation \(z^{1/2}\) , choose a branch cut. Then show that both the branch functions \(f_\pm(z)\) are analytic over all of \(\mathbb{C}\) excluding the branch cut.
Exercise \(\PageIndex{5}\)
Consider \(f(z) = \ln(z+a) - \ln(z-a)\) . For simplicity, let \(a\) be a positive real number. As discussed in Section 8.4, we can write this as \[f(z) = \ln\left|\frac{z+a}{z-a}\right| + i(\theta_+ - \theta_-), \qquad \theta_\pm \equiv \mathrm{arg}(z\pm a).\] Suppose we represent the arguments as \(\theta_+ \in (-\pi,\pi)\) and \(\theta_- \in (-\pi,\pi)\) . Explain why this implies a branch cut consisting of a straight line joining \(a\) with \(-a\) . Using this representation, calculate the change in \(f(z)\) over an infinitesimal loop encircling \(z = a\) or \(z = -a\) . Calculate also the change in \(f(z)\) over a loop of radius \(R \gg a\) encircling the origin (and thus enclosing both branch points).


20.5: Embryonic Development of the Respiratory System
By the end of this section, you will be able to:
- Create a timeline of the phases of respiratory development in the fetus
- Propose reasons for fetal breathing movements
- Explain how the lungs become inflated after birth
Development of the respiratory system begins early in the fetus. It is a complex process that includes many structures, most of which arise from the endoderm. Towards the end of development, the fetus can be observed making breathing movements. Until birth, however, the mother provides all of the oxygen to the fetus as well as removes all of the fetal carbon dioxide via the placenta.
Time Line
The development of the respiratory system begins at about week 4 of gestation. By week 28, enough alveoli have matured that a baby born prematurely at this time can usually breathe on its own. The respiratory system, however, is not fully developed until early childhood, when a full complement of mature alveoli is present.
Weeks 4–7
Respiratory development in the embryo begins around week 4. Ectodermal tissue from the anterior head region invaginates posteriorly to form olfactory pits, which fuse with endodermal tissue of the developing pharynx. An olfactory pit is one of a pair of structures that will enlarge to become the nasal cavity. At about this same time, the lung bud forms. The lung bud is a dome-shaped structure composed of tissue that bulges from the foregut. The foregut is endoderm just inferior to the pharynx, surrounded by splanchnic mesoderm. Eventually, the epithelial tissues of the trachea, bronchi, and lungs form from the endoderm while the smooth muscle and connective tissues of the trachea, bronchi, and lungs, as well as the visceral pleura, develop from the splanchnic mesoderm. The laryngotracheal tube is a structure that forms from the longitudinal extension of the lung bud as development progresses. The portion of this structure nearest the pharynx becomes the trachea, whereas the distal end becomes more bulbous and bifurcates, forming bronchial buds. A bronchial bud is one of a pair of structures that will eventually become the bronchi and all other lower respiratory structures (Figure \(\PageIndex{1}\)).
Weeks 7–16
Bronchial buds continue to branch as development progresses until all of the segmental bronchi have been formed. Beginning around week 13, the lumens of the bronchi begin to expand in diameter. By week 16, respiratory bronchioles form. The fetus now has the beginnings of all major lung structures involved in the airway.
Weeks 16–24
Once the respiratory bronchioles form, further development includes extensive vascularization, or the development of the blood vessels, as well as the formation of alveolar ducts and alveolar precursors. At about week 19, the respiratory bronchioles have formed. In addition, cells lining the respiratory structures begin to differentiate to form type I and type II alveolar cells. Once type II alveolar cells have differentiated, they begin to secrete small amounts of pulmonary surfactant. Around week 20, fetal breathing movements may begin.
Weeks 24–Term
Major growth and maturation of the respiratory system occurs from week 24 until term. More alveolar precursors develop, and larger amounts of pulmonary surfactant are produced. Surfactant levels are not generally adequate to create effective lung compliance, or a change in lung volume as the result of a change in pressure, until about the eighth month of pregnancy. The respiratory system continues to expand, and the surfaces that will form the respiratory membrane develop further. At this point, pulmonary capillaries have formed and continue to expand, creating a large surface area for gas exchange. The major milestone of respiratory development occurs at around week 28, when sufficient alveolar precursors have matured so that a baby born prematurely at this time can usually breathe on its own. However, alveoli continue to develop and mature into childhood. A full complement of functional alveoli does not form until around 8 years of age.
Fetal “Breathing”
Although the function of fetal breathing movements is not entirely clear, they can be observed starting at 20–21 weeks of development. Fetal breathing movements involve muscle contractions that cause the inhalation of amniotic fluid and exhalation of the same fluid, with pulmonary surfactant and mucus. Fetal breathing movements are not continuous and may include periods of frequent movements and periods of no movements. Maternal factors can influence the frequency of breathing movements. For example, high blood glucose levels, called hyperglycemia, can boost the number of breathing movements. Conversely, low blood glucose levels, called hypoglycemia, can reduce the number of fetal breathing movements. Tobacco use is also known to lower fetal breathing rates. Fetal breathing may help tone the muscles in preparation for breathing movements once the fetus is born. It may also help the alveoli to form and mature. Fetal breathing movements are considered a sign of robust health.
Birth
Prior to birth, the lungs are filled with amniotic fluid, mucus, and surfactant. As the fetus is squeezed through the birth canal, the fetal thoracic cavity is compressed, expelling much of this fluid. Some fluid remains, however, but is rapidly absorbed by the body shortly after birth. The first inhalation occurs within 10 seconds after birth and not only serves as the first inspiration, but also acts to inflate the lungs. Pulmonary surfactant is critical for inflation to occur, as it reduces the surface tension of the alveoli. Preterm birth around 26 weeks frequently results in severe respiratory distress, although with current medical advancements, some babies may survive. Prior to 26 weeks, sufficient pulmonary surfactant is not produced, and the surfaces for gas exchange have not formed adequately; therefore, survival is low.
DISORDERS OF THE...
Respiratory System: Respiratory Distress Syndrome
Respiratory distress syndrome (RDS) primarily occurs in infants born prematurely. Up to 50 percent of infants born between 26 and 28 weeks and fewer than 30 percent of infants born between 30 and 31 weeks develop RDS. RDS results from insufficient production of pulmonary surfactant, thereby preventing the lungs from properly inflating at birth. A small amount of pulmonary surfactant is produced beginning at around 20 weeks; however, this is not sufficient for inflation of the lungs. As a result, dyspnea occurs and gas exchange cannot be performed properly.
The primary cause of RDS is premature birth, which may be due to a variety of known or unknown causes. Other risk factors include gestational diabetes, cesarean delivery, second-born twins, and family history of RDS. The presence of RDS can lead to other serious disorders, such as septicemia (infection of the blood) or pulmonary hemorrhage. Therefore, it is important that RDS is immediately recognized and treated to prevent death and reduce the risk of developing other disorders.
Medical advances have resulted in an improved ability to treat RDS and support the infant until proper lung development can occur. At the time of delivery, treatment may include resuscitation and intubation if the infant does not breathe on his or her own. These infants would need to be placed on a ventilator to mechanically assist with the breathing process. If spontaneous breathing occurs, application of nasal continuous positive airway pressure (CPAP) may be required. In addition, a laboratory synthesized form of pulmonary surfactant is typically administered. Death due to RDS has been reduced by 50 percent due to the introduction of pulmonary surfactant therapy. Other therapies may include corticosteroids, supplemental oxygen, and assisted ventilation. Supportive therapies, such as temperature regulation, nutritional support, and antibiotics, may be administered to the premature infant as well.
Concept Review
The development of the respiratory system in the fetus begins at about 4 weeks and continues into childhood. Ectodermal tissue in the anterior portion of the head region invaginates posteriorly, forming olfactory pits, which ultimately fuse with endodermal tissue of the early pharynx. At about this same time, a protrusion of endodermal tissue extends anteriorly from the foregut, producing a lung bud, which continues to elongate until it forms the laryngotracheal bud. The proximal portion of this structure will mature into the trachea, whereas the bulbous end will branch to form two bronchial buds. These buds then branch repeatedly, so that at about week 16, all major airway structures are present. Development progresses after week 16 as respiratory bronchioles and alveolar ducts form, and extensive vascularization occurs. Type I alveolar cells also begin to take shape. Type II alveolar cells develop and begin to produce small amounts of surfactant. As the fetus grows, the respiratory system continues to expand as more alveoli develop and more surfactant is produced. Beginning at about week 36 and lasting into childhood, alveolar precursors mature to become fully functional alveoli. At birth, compression of the thoracic cavity forces much of the fluid in the lungs to be expelled. The first inhalation inflates the lungs. Fetal breathing movements begin around week 20 or 21, and occur when contractions of the respiratory muscles cause the fetus to inhale and exhale amniotic fluid. These movements continue until birth, which may help to tone the muscles in preparation for breathing after birth, and are a sign of good health.
Review Questions
Q. The olfactory pits form from which of the following?
A. mesoderm
B. cartilage
C. ectoderm
D. endoderm
- Answer
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C
Q. A full complement of mature alveoli are present by ________.
A. early childhood, around 8 years of age
B. birth
C. 37 weeks
D. 16 weeks
- Answer
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A
Q. If a baby is born prematurely before type II alveolar cells produce sufficient pulmonary surfactant, which of the following might you expect?
A. difficulty expressing fluid
B. difficulty inflating the lungs
C. difficulty with pulmonary capillary flow
D. no difficulty as type I alveolar cells can provide enough surfactant for normal breathing
- Answer
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B
Q. When do fetal breathing movements begin?
A. around week 20
B. around week 37
C. around week 16
D. after birth
- Answer
-
A
Critical Thinking Questions
Q. During what timeframe does a fetus have enough mature structures to breathe on its own if born prematurely? Describe the other structures that develop during this phase.
- Answer
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At about week 28, enough alveolar precursors have matured so that a baby born prematurely at this time can usually breathe on its own. Other structures that develop about this time are pulmonary capillaries, expanding to create a large surface area for gas exchange. Alveolar ducts and alveolar precursors have also developed.
Q. Describe fetal breathing movements and their purpose.
- Answer
-
Fetal breathing movements occur due to the contraction of respiratory muscles, causing the fetus to inhale and exhale amniotic fluid. It is thought that these movements are a way to “practice” breathing, which results in toning the muscles in preparation for breathing after birth. In addition, fetal breathing movements may help alveoli to form and mature.
Glossary
- bronchial bud
- structure in the developing embryo that forms when the laryngotracheal tube extends and branches to form two bulbous structures
- foregut
- endoderm of the embryo towards the head region
- laryngotracheal tube
- tube that forms from the lung bud, has a tracheal end and bulbous bronchial buds at the distal end
- lung bud
- median dome that forms from the endoderm of the foregut
- olfactory pit
- invaginated ectodermal tissue in the anterior portion of the head region of an embryo that will form the nasal cavity
Contributors and Attributions
-
OpenStax Anatomy & Physiology (CC BY 4.0). Access for free at https://openstax.org/books/anatomy-and-physiology


7.6: Quantitative Chemical Analysis
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Learning Objectives
By the end of this section, you will be able to:
- Describe the fundamental aspects of titrations and gravimetric analysis.
- Perform stoichiometric calculations using typical titration and gravimetric data.
In the 18th century, the strength (actually the concentration) of vinegar samples was determined by noting the amount of potassium carbonate, K 2 CO 3 , which had to be added, a little at a time, before bubbling ceased. The greater the weight of potassium carbonate added to reach the point where the bubbling ended, the more concentrated the vinegar.
We now know that the effervescence that occurred during this process was due to reaction with acetic acid, CH 3 CO 2 H, the compound primarily responsible for the odor and taste of vinegar. Acetic acid reacts with potassium carbonate according to the following equation:
The bubbling was due to the production of CO 2 .
The test of vinegar with potassium carbonate is one type of quantitative analysis —the determination of the amount or concentration of a substance in a sample. In the analysis of vinegar, the concentration of the solute (acetic acid) was determined from the amount of reactant that combined with the solute present in a known volume of the solution. In other types of chemical analyses, the amount of a substance present in a sample is determined by measuring the amount of product that results.
Titration
The described approach to measuring vinegar strength was an early version of the analytical technique known as titration analysis . A typical titration analysis involves the use of a buret (Figure \(\PageIndex{1}\)) to make incremental additions of a solution containing a known concentration of some substance (the titrant ) to a sample solution containing the substance whose concentration is to be measured (the analyte ). The titrant and analyte undergo a chemical reaction of known stoichiometry, and so measuring the volume of titrant solution required for complete reaction with the analyte (the equivalence point of the titration) allows calculation of the analyte concentration. The equivalence point of a titration may be detected visually if a distinct change in the appearance of the sample solution accompanies the completion of the reaction. The halt of bubble formation in the classic vinegar analysis is one such example, though, more commonly, special dyes called indicators are added to the sample solutions to impart a change in color at or very near the equivalence point of the titration. Equivalence points may also be detected by measuring some solution property that changes in a predictable way during the course of the titration. Regardless of the approach taken to detect a titration’s equivalence point, the volume of titrant actually measured is called the end point . Properly designed titration methods typically ensure that the difference between the equivalence and end points is negligible. Though any type of chemical reaction may serve as the basis for a titration analysis, the three described in this chapter (precipitation, acid-base, and redox) are most common. Additional details regarding titration analysis are provided in the chapter on acid-base equilibria.
Example 4.14: Titration Analysis
The end point in a titration of a 50.00-mL sample of aqueous HCl was reached by addition of 35.23 mL of 0.250 M NaOH titrant. The titration reaction is:
What is the molarity of the HCl?
Solution
As for all reaction stoichiometry calculations, the key issue is the relation between the molar amounts of the chemical species of interest as depicted in the balanced chemical equation. The approach outlined in previous modules of this chapter is followed, with additional considerations required, since the amounts of reactants provided and requested are expressed as solution concentrations.
For this exercise, the calculation will follow the following outlined steps:
The molar amount of HCl is calculated to be:
Using the provided volume of HCl solution and the definition of molarity, the HCl concentration is:
Note: For these types of titration calculations, it is convenient to recognize that solution molarity is also equal to the number of milli moles of solute per milli liter of solution:
Using this version of the molarity unit will shorten the calculation by eliminating two conversion factors:
Exercise \(\PageIndex{1}\)
A 20.00-mL sample of aqueous oxalic acid, H 2 C 2 O 4 , was titrated with a 0.09113- M solution of potassium permanganate, KMnO 4 (see net ionic equation below).
A volume of 23.24 mL was required to reach the end point. What is the oxalic acid molarity?
- Answer
-
0.2648 M
Gravimetric Analysis
A gravimetric analysis is one in which a sample is subjected to some treatment that causes a change in the physical state of the analyte that permits its separation from the other components of the sample. Mass measurements of the sample, the isolated analyte, or some other component of the analysis system, used along with the known stoichiometry of the compounds involved, permit calculation of the analyte concentration. Gravimetric methods were the first techniques used for quantitative chemical analysis, and they remain important tools in the modern chemistry laboratory.
The required change of state in a gravimetric analysis may be achieved by various physical and chemical processes. For example, the moisture (water) content of a sample is routinely determined by measuring the mass of a sample before and after it is subjected to a controlled heating process that evaporates the water. Also common are gravimetric techniques in which the analyte is subjected to a precipitation reaction of the sort described earlier in this chapter. The precipitate is typically isolated from the reaction mixture by filtration, carefully dried, and then weighed (Figure \(\PageIndex{2}\)). The mass of the precipitate may then be used, along with relevant stoichiometric relationships, to calculate analyte concentration.
Example 4.15: Gravimetric Analysis
A 0.4550-g solid mixture containing MgSO 4 is dissolved in water and treated with an excess of Ba(NO 3 ) 2 , resulting in the precipitation of 0.6168 g of BaSO 4 .
What is the concentration (mass percent) of MgSO 4 in the mixture?
Solution
The plan for this calculation is similar to others used in stoichiometric calculations, the central step being the connection between the moles of BaSO 4 and MgSO 4 through their stoichiometric factor. Once the mass of MgSO 4 is computed, it may be used along with the mass of the sample mixture to calculate the requested percentage concentration.
The mass of MgSO 4 that would yield the provided precipitate mass is
The concentration of MgSO 4 in the sample mixture is then calculated to be
Exercise \(\PageIndex{1}\)
What is the percent of chloride ion in a sample if 1.1324 g of the sample produces 1.0881 g of AgCl when treated with excess Ag + ?
- Answer
-
23.76%
The elemental composition of hydrocarbons and related compounds may be determined via a gravimetric method known as combustion analysis . In a combustion analysis, a weighed sample of the compound is heated to a high temperature under a stream of oxygen gas, resulting in its complete combustion to yield gaseous products of known identities. The complete combustion of hydrocarbons, for example, will yield carbon dioxide and water as the only products. The gaseous combustion products are swept through separate, preweighed collection devices containing compounds that selectively absorb each product (Figure \(\PageIndex{3}\)). The mass increase of each device corresponds to the mass of the absorbed product and may be used in an appropriate stoichiometric calculation to derive the mass of the relevant element.
Example 4.16: Combustion Analysis
Polyethylene is a hydrocarbon polymer used to produce food-storage bags and many other flexible plastic items. A combustion analysis of a 0.00126-g sample of polyethylene yields 0.00394 g of CO 2 and 0.00161 g of H 2 O. What is the empirical formula of polyethylene?
Solution
The primary assumption in this exercise is that all the carbon in the sample combusted is converted to carbon dioxide, and all the hydrogen in the sample is converted to water:
Note that a balanced equation is not necessary for the task at hand. To derive the empirical formula of the compound, only the subscripts x and y are needed.
First, calculate the molar amounts of carbon and hydrogen in the sample, using the provided masses of the carbon dioxide and water, respectively. With these molar amounts, the empirical formula for the compound may be written as described in the previous chapter of this text. An outline of this approach is given in the following flow chart:
The empirical formula for the compound is then derived by identifying the smallest whole-number multiples for these molar amounts. The H-to-C molar ratio is
and the empirical formula for polyethylene is CH 2 .
Exercise \(\PageIndex{1}\)
A 0.00215-g sample of polystyrene, a polymer composed of carbon and hydrogen, produced 0.00726 g of CO 2 and 0.00148 g of H 2 O in a combustion analysis. What is the empirical formula for polystyrene?
- Answer
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CH