21.8A: Dalton’s Law of Partial Pressure
Dalton’s law of partial pressures states that the pressure of a mixture of gases is the sum of the pressures of the individual components.
- Infer from Dalton’s law of partial pressure the sum of partial pressures in alveoli
Key Points
- This empirical law was observed by John Dalton in 1801 and is related to the ideal gas laws.
- Atmospheric air is a mixture of nitrogen, water, oxygen, carbon dioxide, and other minor gasses. The relative concentrations of a gasses don’t change even as the pressure and volume of the total gasses change.
- Gasses flow from areas of high to low pressure, so the partial pressures of inhaled and alveolar air determine why oxygen goes into the alveoli, and why carbon dioxide leaves the alveoli.
- Dalton’s law is only completely accurate for ideal gasses.
Key Terms
- Dalton’s law : The total pressure of a mixture of gases is the sum of the partial pressures of each gas in the mixture; it is only true for ideal gases.
Dalton’s law states that the total pressure exerted by the mixture of inert (non-reactive) gases is equal to the sum of the partial pressures of individual gases in a volume of air. This empirical law was observed by John Dalton in 1801 and is related to the ideal gas laws.
Dalton’s Law in Respiration
The air in the atmosphere is a mixture of many different gases, that vary in concentration. Dalton’s law states that at any given time, the percentage of each of these gasses in the air we breathe makes its contribution to total atmospheric pressure, and this contribution will depend on how much of each gas is in the air we breathe.
Dalton’s law also implies that the relative concentration of gasses (their partial pressures) does not change as the pressure and volume of the gas mixture changes, so that air inhaled into the lungs will have the same relative concentration of gasses as atmospheric air. In the lungs, the relative concentration of gasses determines the rate at which each gas will diffuse across the alveolar membranes.
Mathematically, the pressure of a mixture of gases can be defined as the sum of the partial pressures of each of the gasses in air.
Ptotal=P1+P2+P3+⋯+Pn=n∑i=1PiPtotal=P1+P2+P3+⋯+Pn=∑i=1nPi
DALTON’S LAW
In regards to atmospheric air, Dalton’s law becomes:
Atm=PN2+PO2+PCO2+PH2O+P(other gasses)Atm=PN2+PO2+PCO2+PH2O+P(other gasses)
For the purposes of gas exchange, O 2 and CO 2 are mainly considered due to their metabolic importance in gas exchange. Because gasses flow from areas of high pressure to areas of low pressure, atmospheric air has higher partial pressure of oxygen than alveolar air (P O 2 = 159mm Hg compared to PA O 2 = 100mm Hg).
Similarly, atmospheric air has a much lower partial pressure for carbon dioxide compared to alveolar air (P CO 2 = .3mm Hg compared to PA CO 2 = 40mm Hg). These pressure differences explain why oxygen flows into the alveoli and why carbon dioxide flows out of the alveoli through passive diffusion (just as a similar process explains alveolar and arterial gas exchange).
While inhaled air is similar to atmospheric air due to Dalton’s law, exhaled air will have relative concentrations that are in between atmospheric and alveolar air due to the passive diffusion of gasses during gas exchange.
Dalton’s law is only truly applicable in every situation to ideal gasses. Therefore most gasses will not follow it exactly, especially in conditions of extremely high pressure, or in situations where intermolecular forces act to keep the gasses together.


4.10: Brand Community
- Define a brand community
A brand community is a community formed on the basis of attachment to a product or brand, stressing the connection between brand, individual identity, and culture. In the advertising and marketing world, “brand community” has become a term used to encompass a brand’s customers, fans, and advocates. Having a strong and loyal brand community can turn a small brand into a success if it is nurtured and appreciated properly.
The advertising agency Blade Creative Branding writes, “The people out in the marketplace who embrace the values of the brand, as customers and/or purchase influencers, are the brand’s true ‘owners.'” This philosophy has become especially popular with marketers creating and working on social media campaigns with easier interaction and more opportunities to tap into the brand community to leverage the brand.
Contributors and Attributions
- Brand Community. Authored by : Patrick Williams. Provided by : Lumen Learning. License : CC BY: Attribution


2.4: The Laws of Planetary Motion
Learning Objectives
By the end of the section, you will be able to:
- Describe how Tycho Brahe and Johannes Kepler contributed to our understanding of how planets move around the Sun
- Explain Kepler’s three laws of planetary motion
At about the time that Galileo was beginning his experiments with falling bodies, the efforts of two other scientists dramatically advanced our understanding of the motions of the planets. These two astronomers were the observer Tycho Brahe and the mathematician Johannes Kepler. Together, they placed the speculations of Copernicus on a sound mathematical basis and paved the way for the work of Isaac Newton in the next century.
Tycho Brahe’s Observatory
Three years after the publication of Copernicus’ De Revolutionibus , Tycho Brahe was born to a family of Danish nobility. He developed an early interest in astronomy and, as a young man, made significant astronomical observations. Among these was a careful study of what we now know was an exploding star that flared up to great brilliance in the night sky. His growing reputation gained him the patronage of the Danish King Frederick II, and at the age of 30, Brahe was able to establish a fine astronomical observatory on the North Sea island of Hven (Figure \(\PageIndex{1}\)). Brahe was the last and greatest of the pre-telescopic observers in Europe.
At Hven, Brahe made a continuous record of the positions of the Sun, Moon, and planets for almost 20 years. His extensive and precise observations enabled him to note that the positions of the planets varied from those given in published tables, which were based on the work of Ptolemy. These data were extremely valuable, but Brahe didn’t have the ability to analyze them and develop a better model than what Ptolemy had published. He was further inhibited because he was an extravagant and cantankerous fellow, and he accumulated enemies among government officials. When his patron, Frederick II, died in 1597, Brahe lost his political base and decided to leave Denmark. He took up residence in Prague, where he became court astronomer to Emperor Rudolf of Bohemia. There, in the year before his death, Brahe found a most able young mathematician, Johannes Kepler, to assist him in analyzing his extensive planetary data.
Johannes Kepler
Johannes Kepler was born into a poor family in the German province of Württemberg and lived much of his life amid the turmoil of the Thirty Years’ War (see Figure \(\PageIndex{1}\)). He attended university at Tubingen and studied for a theological career. There, he learned the principles of the Copernican system and became converted to the heliocentric hypothesis. Eventually, Kepler went to Prague to serve as an assistant to Brahe, who set him to work trying to find a satisfactory theory of planetary motion—one that was compatible with the long series of observations made at Hven. Brahe was reluctant to provide Kepler with much material at any one time for fear that Kepler would discover the secrets of the universal motion by himself, thereby robbing Brahe of some of the glory. Only after Brahe’s death in 1601 did Kepler get full possession of the priceless records. Their study occupied most of Kepler’s time for more than 20 years.
Through his analysis of the motions of the planets, Kepler developed a series of principles, now known as Kepler’s three laws, which described the behavior of planets based on their paths through space. The first two laws of planetary motion were published in 1609 in The New Astronomy . Their discovery was a profound step in the development of modern science.
The First Two Laws of Planetary Motion
The path of an object through space is called its orbit. Kepler initially assumed that the orbits of planets were circles, but doing so did not allow him to find orbits that were consistent with Brahe’s observations. Working with the data for Mars, he eventually discovered that the orbit of that planet had the shape of a somewhat flattened circle, or ellipse . Next to the circle, the ellipse is the simplest kind of closed curve, belonging to a family of curves known as conic sections (Figure \(\PageIndex{2}\)).
You might recall from math classes that in a circle, the center is a special point. The distance from the center to anywhere on the circle is exactly the same. In an ellipse, the sum of the distance from two special points inside the ellipse to any point on the ellipse is always the same. These two points inside the ellipse are called its foci (singular: focus), a word invented for this purpose by Kepler.
This property suggests a simple way to draw an ellipse (Figure \(\PageIndex{3}\)). We wrap the ends of a loop of string around two tacks pushed through a sheet of paper into a drawing board, so that the string is slack. If we push a pencil against the string, making the string taut, and then slide the pencil against the string all around the tacks, the curve that results is an ellipse. At any point where the pencil may be, the sum of the distances from the pencil to the two tacks is a constant length—the length of the string. The tacks are at the two foci of the ellipse.
The widest diameter of the ellipse is called its major axis . Half this distance—that is, the distance from the center of the ellipse to one end—is the semimajor axis , which is usually used to specify the size of the ellipse. For example, the semimajor axis of the orbit of Mars, which is also the planet’s average distance from the Sun, is 228 million kilometers.
The shape (roundness) of an ellipse depends on how close together the two foci are, compared with the major axis. The ratio of the distance between the foci to the length of the semimajor axis is called the eccentricity of the ellipse.
If the foci (or tacks) are moved to the same location, then the distance between the foci would be zero. This means that the eccentricity is zero and the ellipse is just a circle; thus, a circle can be called an ellipse of zero eccentricity. In a circle, the semimajor axis would be the radius.
Next, we can make ellipses of various elongations (or extended lengths) by varying the spacing of the tacks (as long as they are not farther apart than the length of the string). The greater the eccentricity, the more elongated is the ellipse, up to a maximum eccentricity of 1.0, when the ellipse becomes “flat,” the other extreme from a circle.
The size and shape of an ellipse are completely specified by its semimajor axis and its eccentricity. Using Brahe’s data, Kepler found that Mars has an elliptical orbit, with the Sun at one focus (the other focus is empty). The eccentricity of the orbit of Mars is only about 0.1; its orbit, drawn to scale, would be practically indistinguishable from a circle, but the difference turned out to be critical for understanding planetary motions.
Kepler generalized this result in his first law and said that the orbits of all the planets are ellipses . Here was a decisive moment in the history of human thought: it was not necessary to have only circles in order to have an acceptable cosmos. The universe could be a bit more complex than the Greek philosophers had wanted it to be.
Kepler’s second law deals with the speed with which each planet moves along its ellipse, also known as its orbital speed . Working with Brahe’s observations of Mars, Kepler discovered that the planet speeds up as it comes closer to the Sun and slows down as it pulls away from the Sun. He expressed the precise form of this relationship by imagining that the Sun and Mars are connected by a straight, elastic line. When Mars is closer to the Sun (positions 1 and 2 in Figure \(\PageIndex{4}\)), the elastic line is not stretched as much, and the planet moves rapidly. Farther from the Sun, as in positions 3 and 4, the line is stretched a lot, and the planet does not move so fast. As Mars travels in its elliptical orbit around the Sun, the elastic line sweeps out areas of the ellipse as it moves (the colored regions in our figure). Kepler found that in equal intervals of time (t), the areas swept out in space by this imaginary line are always equal; that is, the area of the region B from 1 to 2 is the same as that of region A from 3 to 4.
If a planet moves in a circular orbit, the elastic line is always stretched the same amount and the planet moves at a constant speed around its orbit. But, as Kepler discovered, in most orbits that speed of a planet orbiting its star (or moon orbiting its planet) tends to vary because the orbit is elliptical.
Kepler’s Third Law
Kepler’s first two laws of planetary motion describe the shape of a planet’s orbit and allow us to calculate the speed of its motion at any point in the orbit. Kepler was pleased to have discovered such fundamental rules, but they did not satisfy his quest to fully understand planetary motions. He wanted to know why the orbits of the planets were spaced as they are and to find a mathematical pattern in their movements—a “harmony of the spheres” as he called it. For many years he worked to discover mathematical relationships governing planetary spacing and the time each planet took to go around the Sun.
In 1619, Kepler discovered a basic relationship to relate the planets’ orbits to their relative distances from the Sun. We define a planet’s orbital period , (\(P\)), as the time it takes a planet to travel once around the Sun. Also, recall that a planet’s semimajor axis, a, is equal to its average distance from the Sun. The relationship, now known as Kepler’s third law , says that a planet’s orbital period squared is proportional to the semimajor axis of its orbit cubed, or
\[P^2 \propto a^3 \nonumber\]
When \(P\) (the orbital period) is measured in years, and a is expressed in a quantity known as an astronomical unit ( AU ), the two sides of the formula are not only proportional but equal. One AU is the average distance between Earth and the Sun and is approximately equal to \(1.55 × 10^8\) kilometers. In these units,
\[P^2=a^3 \nonumber\]
Kepler’s third law applies to all objects orbiting the Sun, including Earth, and provides a means for calculating their relative distances from the Sun from the time they take to orbit. Let’s look at a specific example to illustrate how useful Kepler’s third law is.
For instance, suppose you time how long Mars takes to go around the Sun (in Earth years). Kepler’s third law can then be used to calculate Mars’ average distance from the Sun. Mars’ orbital period (1.88 Earth years) squared, or \(P^2\), is 1.882 = 3.53, and according to the equation for Kepler’s third law, this equals the cube of its semimajor axis, or \(a^3\). So what number must be cubed to give 3.53? The answer is 1.52 (since \(1.52 \times 1.52 \times 1.52 = 3.53\)). Thus, Mars’ semimajor axis in astronomical units must be 1.52 AU. In other words, to go around the Sun in a little less than two years, Mars must be about 50% (half again) as far from the Sun as Earth is.
Example \(\PageIndex{1}\): Calculating Periods
Imagine an object is traveling around the Sun. What would be the orbital period of the object if its orbit has a semimajor axis of 50 AU?
- Answer
-
From Kepler’s third law, we know that (when we use units of years and AU)
\[P^2=a^3 \nonumber\]
If the object’s orbit has a semimajor axis of 4 AU ( a = 50), we can cube 50 and then take the square root of the result to get P:
\[ \begin{array}{l} P = \sqrt{a^3} \\ P = \sqrt{50 \times 50 \times 50} = \sqrt{125,000} = 353.6 \text{ years} \end{array}\]
Therefore, the orbital period of the object is about 350 years. This would place our hypothetical object beyond the orbit of Pluto.
Exercise \(\PageIndex{1}\)
What would be the orbital period of an asteroid (an rocky and metal object between Mars and Jupiter) with a semimajor axis of 3 AU?
- Answer
-
\[P = \sqrt{3 \times 3 \times 3} = \sqrt{27} = 5.2 \text{ years}\]
Kepler’s three laws of planetary motion can be summarized as follows:
- Kepler’s first law : Each planet moves around the Sun in an orbit that is an ellipse, with the Sun at one focus of the ellipse.
- Kepler’s second law : The straight line joining a planet and the Sun sweeps out equal areas in space in equal intervals of time.
- Kepler’s third law : The square of a planet’s orbital period is directly proportional to the cube of the semimajor axis of its orbit.
Kepler’s three laws provide a precise geometric description of planetary motion within the framework of the Copernican system. With these tools, it was possible to calculate planetary positions with greatly improved precision. Still, Kepler’s laws are purely descriptive: they do not help us understand what forces of nature constrain the planets to follow this particular set of rules. That step was left to Isaac Newton.
Example \(\PageIndex{2}\): Applying Kepler’s Third Law
Using the orbital periods and semimajor axes for Venus and Earth that are provided here, calculate \(P^2\) and \(a^3\), and verify that they obey Kepler’s third law. Venus’ orbital period is 0.62 year, and its semimajor axis is 0.72 AU. Earth’s orbital period is 1.00 year, and its semimajor axis is 1.00 AU.
- Answer
-
We can use the equation for Kepler’s third law, \(P^2 \propto a^3\). For Venus, \(P^2 = 0.62 \times 0.62 = 0.38 \text{ years}\) and \(a^3 = 0.72 \times 0.72 \times 0.72 = 0.37 \text{ AU}\) (rounding numbers sometimes causes minor discrepancies like this). The orbital period (0.38 year) approximates the semimajor axis (0.37 AU).
Therefore, Venus obeys Kepler’s third law. For Earth, \(P^2 = 1.00 \times 1.00 = 1.00 \text{ year}\) and \(a^3 = 1.00 \times 1.00 \times 1.00 = 1.00 \text{ AU}\). The orbital period (1.00 year) approximates (in this case, equals) the semimajor axis (1.00 AU). Therefore, Earth obeys Kepler’s third law.
Exercise \(\PageIndex{1}\)
Using the orbital periods and semimajor axes for Saturn and Jupiter that are provided here, calculate \(P^2\) and \(a^3\), and verify that they obey Kepler’s third law. Saturn’s orbital period is 29.46 years, and its semimajor axis is 9.54 AU. Jupiter’s orbital period is 11.86 years, and its semimajor axis is 5.20 AU.
- Answer
-
For Saturn, \(P^2 = 29.46 \times 29.46 = 867.9 \text{ years}\) and \(a^3 = 9.54 \times 9.54 \times 9.54 = 868.3 \text{ AU}\). The orbital period (867.9 years) approximates the semimajor axis (868.3 AU). Therefore, Saturn obeys Kepler’s third law.
In honor of the scientist who first devised the laws that govern the motions of planets, the team that built the first spacecraft to search for planets orbiting other stars decided to name the probe “Kepler.” To learn more about Johannes Kepler’s life and his laws of planetary motion, as well as lots of information on the Kepler Mission, visit NASA’s Kepler website and follow the links that interest you.
Key Concepts and Summary
Tycho Brahe’s accurate observations of planetary positions provided the data used by Johannes Kepler to derive his three fundamental laws of planetary motion. Kepler’s laws describe the behavior of planets in their orbits as follows: (1) planetary orbits are ellipses with the Sun at one focus; (2) in equal intervals, a planet’s orbit sweeps out equal areas; and (3) the relationship between the orbital period (\(P\)) and the semimajor axis (\(a\)) of an orbit is given by \(P^2 = a^3\) (when \(a\) is in units of AU and \(P\) is in units of Earth years).
Glossary
- astronomical unit (AU)
- the unit of length defined as the average distance between Earth and the Sun; this distance is about 1.5 × 108 kilometers
- eccentricity
- in an ellipse, the ratio of the distance between the foci to the major axis
- ellipse
- a closed curve for which the sum of the distances from any point on the ellipse to two points inside (called the foci) is always the same
- focus
- (plural: foci) one of two fixed points inside an ellipse from which the sum of the distances to any point on the ellipse is constant
- Kepler’s first law
- each planet moves around the Sun in an orbit that is an ellipse, with the Sun at one focus of the ellipse
- Kepler’s second law
- the straight line joining a planet and the Sun sweeps out equal areas in space in equal intervals of time
- Kepler’s third law
- the square of a planet’s orbital period is directly proportional to the cube of the semimajor axis of its orbit
- major axis
- the maximum diameter of an ellipse
- orbit
- the path of an object that is in revolution about another object or point
- orbital period ( P )
- the time it takes an object to travel once around the Sun
- orbital speed
- the speed at which an object (usually a planet) orbits around the mass of another object; in the case of a planet, the speed at which each planet moves along its ellipse
- semimajor axis
- half of the major axis of a conic section, such as an ellipse
Contributors and Attributions
-
Andrew Fraknoi (Foothill College), David Morrison (NASA Ames Research Center), Sidney C. Wolff (National Optical Astronomy Observatory) with many contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at https://openstax.org/details/books/astronomy ).


11.5: Axial Muscles of the Abdominal Wall and Thorax
By the end of this section, you will be able to:
- Identify the intrinsic skeletal muscles of the back and neck, and the skeletal muscles of the abdominal wall and thorax
- Identify the movement and function of the intrinsic skeletal muscles of the back and neck, and the skeletal muscles of the abdominal wall and thorax
It is a complex job to balance the body on two feet and walk upright. The muscles of the vertebral column, thorax, and abdominal wall extend, flex, and stabilize different parts of the body’s trunk. The deep muscles of the core of the body help maintain posture as well as carry out other functions. The brain sends out electrical impulses to these various muscle groups to control posture by alternate contraction and relaxation. This is necessary so that no single muscle group becomes fatigued too quickly. If any one group fails to function, body posture will be compromised.
Muscles of the Abdomen
There are four pairs of abdominal muscles that cover the anterior and lateral abdominal region and meet at the anterior midline. These muscles of the anterolateral abdominal wall can be divided into four groups: the external obliques, the internal obliques, the transversus abdominis, and the rectus abdominis (Figure 11.16 and Table 11.6).
| Movement | Target | Target motion direction | Prime mover | Origin | Insertion |
|---|---|---|---|---|---|
| Twisting at waist; also bending to the side | Vertebral column | Supination; lateral flexion | External obliques; internal obliques | Ribs 5–12; ilium | Ribs 7–10; linea alba; ilium |
| Squeezing abdomen during forceful exhalations, defecation, urination, and childbirth | Abdominal cavity | Compression | Transversus abdominis | Ilium; ribs 5–10 | Sternum; linea alba; pubis |
| Sitting up | Vertebral column | Flexion | Rectus abdominis | Pubis | Sternum; ribs 5 and 7 |
| Bending to the side | Vertebral column | Lateral flexion | Quadratus lumborum | Ilium; ribs 5–10 | Rib 12; vertebrae L1–L4 |
There are three flat skeletal muscles in the antero-lateral wall of the abdomen. The external oblique , closest to the surface, extend inferiorly and medially, in the direction of sliding one’s four fingers into pants pockets. Perpendicular to it is the intermediate internal oblique , extending superiorly and medially, the direction the thumbs usually go when the other fingers are in the pants pocket. The deep muscle, the transversus abdominis , is arranged transversely around the abdomen, similar to the front of a belt on a pair of pants. This arrangement of three bands of muscles in different orientations allows various movements and rotations of the trunk. The three layers of muscle also help to protect the internal abdominal organs in an area where there is no bone.
The linea alba is a white, fibrous band that is made of the bilateral rectus sheaths that join at the anterior midline of the body. These enclose the rectus abdominis muscles (a pair of long, linear muscles, commonly called the “sit-up” muscles) that originate at the pubic crest and symphysis, and extend the length of the body’s trunk. Each muscle is segmented by three transverse bands of collagen fibers called the tendinous intersections . This results in the look of “six-pack abs,” as each segment hypertrophies on individuals at the gym who do many sit-ups.
The posterior abdominal wall is formed by the lumbar vertebrae, parts of the ilia of the hip bones, psoas major and iliacus muscles, and quadratus lumborum muscle. This part of the core plays a key role in stabilizing the rest of the body and maintaining posture.
Career Connection
Physical Therapists
Those who have a muscle or joint injury will most likely be sent to a physical therapist (PT) after seeing their regular doctor. PTs have a master’s degree or doctorate, and are highly trained experts in the mechanics of body movements. Many PTs also specialize in sports injuries.
If you injured your shoulder while you were kayaking, the first thing a physical therapist would do during your first visit is to assess the functionality of the joint. The range of motion of a particular joint refers to the normal movements the joint performs. The PT will ask you to abduct and adduct, circumduct, and flex and extend the arm. The PT will note the shoulder’s degree of function, and based on the assessment of the injury, will create an appropriate physical therapy plan.
The first step in physical therapy will probably be applying a heat pack to the injured site, which acts much like a warm-up to draw blood to the area, to enhance healing. You will be instructed to do a series of exercises to continue the therapy at home, followed by icing, to decrease inflammation and swelling, which will continue for several weeks. When physical therapy is complete, the PT will do an exit exam and send a detailed report on the improved range of motion and return of normal limb function to your doctor. Gradually, as the injury heals, the shoulder will begin to function correctly. A PT works closely with patients to help them get back to their normal level of physical activity.
Muscles of the Thorax
The muscles of the chest serve to facilitate breathing by changing the size of the thoracic cavity (Table 11.7). When you inhale, your chest rises because the cavity expands. Alternately, when you exhale, your chest falls because the thoracic cavity decreases in size.
| Movement | Target | Target motion direction | Prime mover | Origin | Insertion |
|---|---|---|---|---|---|
| Inhalation; exhalation | Thoracic cavity | Compression; expansion | Diaphragm | Sternum; ribs 6–12; lumbar vertebrae | Central tendon |
| Inhalation;exhalation | Ribs | Elevation (expands thoracic cavity) | External intercostals | Rib superior to each intercostal muscle | Rib inferior to each intercostal muscle |
| Forced exhalation | Ribs | Movement along superior/inferior axis to bring ribs closer together | Internal intercostals | Rib inferior to each intercostal muscle | Rib superior to each intercostal muscle |
The Diaphragm
The change in volume of the thoracic cavity during breathing is due to the alternate contraction and relaxation of the diaphragm (Figure 11.17). It separates the thoracic and abdominal cavities, and is dome-shaped at rest. The superior surface of the diaphragm is convex, creating the elevated floor of the thoracic cavity. The inferior surface is concave, creating the curved roof of the abdominal cavity.
Defecating, urination, and even childbirth involve cooperation between the diaphragm and abdominal muscles (this cooperation is referred to as the “Valsalva maneuver”). You hold your breath by a steady contraction of the diaphragm; this stabilizes the volume and pressure of the peritoneal cavity. When the abdominal muscles contract, the pressure cannot push the diaphragm up, so it increases pressure on the intestinal tract (defecation), urinary tract (urination), or reproductive tract (childbirth).
The inferior surface of the pericardial sac and the inferior surfaces of the pleural membranes (parietal pleura) fuse onto the central tendon of the diaphragm. To the sides of the tendon are the skeletal muscle portions of the diaphragm, which insert into the tendon while having a number of origins including the xiphoid process of the sternum anteriorly, the inferior six ribs and their cartilages laterally, and the lumbar vertebrae and 12th ribs posteriorly.
The diaphragm also includes three openings for the passage of structures between the thorax and the abdomen. The inferior vena cava passes through the caval opening , and the esophagus and attached nerves pass through the esophageal hiatus. The aorta, thoracic duct, and azygous vein pass through the aortic hiatus of the posterior diaphragm.
The Intercostal Muscles
There are three sets of muscles, called intercostal muscles , which span each of the intercostal spaces. The principal role of the intercostal muscles is to assist in breathing by changing the dimensions of the rib cage (Figure 11.18).
The 11 pairs of superficial external intercostal muscles aid in inspiration of air during breathing because when they contract, they raise the rib cage, which expands it. The 11 pairs of internal intercostal muscles, just under the externals, are used for expiration because they draw the ribs together to constrict the rib cage. The innermost intercostal muscles are the deepest, and they act as synergists for the action of the internal intercostals.
Muscles of the Pelvic Floor and Perineum
The pelvic floor is a muscular sheet that defines the inferior portion of the pelvic cavity. The pelvic diaphragm , spanning anteriorly to posteriorly from the pubis to the coccyx, comprises the levator ani and the ischiococcygeus.
The large levator ani consists of two skeletal muscles, the pubococcygeus and the iliococcygeus (Figure 11.19). The levator ani is considered the most important muscle of the pelvic floor because it supports the pelvic viscera. It resists the pressure produced by contraction of the abdominal muscles so that the pressure is applied to the colon to aid in defecation and to the uterus to aid in childbirth (assisted by the ischiococcygeus , which pulls the coccyx anteriorly). This muscle also creates skeletal muscle sphincters at the urethra and anus.
The perineum is the diamond-shaped space between the pubic symphysis (anteriorly), the coccyx (posteriorly), and the ischial tuberosities (laterally), lying just inferior to the pelvic diaphragm (levator ani and coccygeus). Divided transversely into triangles, the anterior is the urogenital triangle , which includes the external genitals.  The perineum is also divided into superficial and deep layers with some of the muscles common to people of any sex (Figure 11.21). Females also have the compressor urethrae and the sphincter urethrovaginalis , which function to close the vagina. In males, there is the deep transverse perineal muscle that plays a role in ejaculation.


4.6: Introduction to Rational Functions
- Find asymptotes of rational functions.
- Find holes of rational functions.
- Describe behavior of graph near asymptotes.
- Describe end behavior of rational functions.
If we add, subtract, or multiply polynomial functions, we will produce another polynomial function. If, on the other hand, we divide two polynomial functions, the result may not be a polynomial. Division of polynomials gives us another important family of functions, called rational functions .
A rational function is a function which is the ratio of polynomial functions of the form
\[ r(x) = \dfrac{p(x)}{q(x)} \nonumber \]
where \(p\) and \(q\) are polynomial function and \(q(x) \neq 0\).
Recall that we have domain issues any time the denominator of a fraction is zero. In the example below, we review this concept as well as some of the algebra of rational expressions.
Find the domain of the following rational functions. Write them in the form \(\dfrac{p(x)}{q(x)}\) for polynomial functions \(p\) and \(q\) and simplify.
- \(f(x) = \dfrac{2x-1}{x+1}\)
- \(g(x) = 2 - \dfrac{3}{x+1}\)
- \(h(x) = \dfrac{2x^2-1}{x^2-1} - \dfrac{3x-2}{x^2-1}\)
- \(r(x) = \dfrac{2x^2-1}{x^2-1} \div \dfrac{3x-2}{x^2-1}\)
Solution
1. We find the zeros of the denominator and exclude them from the domain. Setting \(x+1=0\) results in \(x=-1\). The domain is \((-\infty, -1) \cup (-1,\infty)\). The function \(f(x)\) is already in simplest form since there are no common factors among the numerator and denominator.
2. We determine the domain of \(g\) by solving \(x+1=0\). The domain of \(g\) is \((-\infty, -1) \cup (-1,\infty)\). To simplify \(g(x)\), we need to get a common denominator
\[ \begin{align*} g(x) &= 2 - \dfrac{3}{x+1} \\[5pt] & = \dfrac{2}{1} - \dfrac{3}{x+1} \\[5pt]& = \dfrac{(2)(x+1)}{(1)(x+1)} - \dfrac{3}{x+1} \\[5pt]& = \dfrac{(2x+2) - 3}{x+1}\\[5pt] & = \dfrac{2x-1}{x+1} \end{align*} \nonumber\]
3. The denominators for \(h(x)\) are both \(x^2-1\) whose zeros are \(x = \pm 1\). The domain of \(h\) is \((-\infty, -1) \cup (-1,1) \cup (1, \infty)\). Since we have the same denominator in both terms, we subtract the numerators. We then factor the resulting numerator and denominator, and cancel out the common factor.
\[ \begin{align*} h(x) & = \dfrac{2x^2-1}{x^2-1} - \dfrac{3x-2}{x^2-1} \\[5pt] & =\dfrac{\left(2x^2-1\right) - \left(3x-2\right)}{x^2-1} \\[5pt] & =\dfrac{2x^2-1 - 3x+2}{x^2-1}\\[5pt] & = \dfrac{2x^2 - 3x+1}{x^2-1} \\[5pt] & = \dfrac{(2x-1)(x-1)}{(x+1)(x-1)}\\[5pt] & = \dfrac{2x-1}{x+1} & & \\ \end{align*} \nonumber\]
4. To find the domain of \(r\), it may help to temporarily rewrite \(r(x)\) as
\[ r(x) = \dfrac{\dfrac{2x^2-1}{x^2-1} }{\dfrac{3x-2}{x^2-1}\vphantom{\left(\dfrac{X}{X}\right)}}\nonumber \]
We need to set all of the denominators equal to zero which means we need to solve not only \(x^2-1= 0\), but also \(\dfrac{3x-2}{x^2-1}=0\). We find \(x = \pm 1\) for the former and \(x= \dfrac{2}{3}\) for the latter. The domain is \((-\infty, -1) \cup \left(-1,\dfrac{2}{3}\right) \cup \left(\dfrac{2}{3},1\right) \cup (1, \infty)\). We simplify \(r(x)\) by rewriting the division as multiplication by the reciprocal and then by canceling the common factor
\[ \begin{align*}r(x) & = \dfrac{2x^2-1}{x^2-1} \div \dfrac{3x-2}{x^2-1}\\[5pt] & = \dfrac{2x^2-1}{x^2-1} \cdot \dfrac{x^2-1}{3x-2}\\[5pt] & = \dfrac{\left(2x^2-1\right)\left(x^2-1\right)}{\left(x^2-1\right)(3x-2)} \\[5pt] & = \dfrac{2x^2-1}{3x-2} & & \\ \end{align*}\nonumber\]
A few remarks about the previous example are in order. Note that the the functions \(f(x)\), \(g(x)\) and \(h(x)\) all simplify to the same thing. However, only two of these functions are actually equal. For two functions to be equal, they need, among other things, to have the same domain. Since \(f(x) = g(x)\) when simplified and \(f\) and \(g\) have the same domain, they are equal functions. Even though \(h(x)\) simplifies to be the same as \(f\) and \(g\), the domain of \(h\) is different than the domain of \(f\) and \(g\), and thus \(h\) is a different function.
Vertical Asymptotes
We now turn our attention to the graphs of rational functions. Let's first focus on how these functions behave near values that are not in the domain. Consider the function \(f(x) = \dfrac{2x-1}{x+1}\) from the previous example. Using a graphing calculator, we obtain the following graph:
( note that older graphing calculator technologies will show a slightly different graph due to graphing limitations)
Note that the graph appears to 'break' at \(x=-1\). We know from our last example that \(x=-1\) is not in the domain of \(f\) which means \(f(-1)\) is undefined. When we make a table of values to study the behavior of \(f\) near \(x=-1\) we see that we can get 'near' \(x=-1\) from two directions. We can choose values a little less than \(-1\), for example \(x=-1.1\), \(x=-1.01\), \(x=-1.001\), and so on. These values are said to 'approach \(-1\) from the left .' Similarly, the values \(x=-0.9\), \(x=-0.99\), \(x=-0.999\), etc., are said to 'approach \(-1\) from the right .' If we make two tables, we find that the numerical results confirm what we see graphically.
|
|
As the \(x\) values approach \(-1\) from the left, the function values become larger and larger. We express this symbolically as \(x \rightarrow -1^{-}\), \(f(x) \rightarrow \infty\). Similarly, we conclude from the table that as \(x \rightarrow -1^{+}\), \(f(x) \rightarrow -\infty\). For this type of unbounded behavior, we say the graph of \(y=f(x)\) has a vertical asymptote of \(x = -1\). Roughly speaking, this means that near \(x=-1\), the graph looks very much like the vertical line \(x=-1\).
The line \(x=c\), where \(c\) is a real number, is called a vertical asymptote of the graph of a function \(y=f(x)\) if as \(x \rightarrow c^{-}\) or as \(x \rightarrow c^{+}\), either \(f(x) \rightarrow \infty\) or \(f(x) \rightarrow -\infty\).
Let's revisit \(h(x) = \dfrac{2x^2-1}{x^2-1} - \dfrac{3x-2}{x^2-1}\). We discussed that although \(h(x)\) simplified to \(\dfrac{2x-1}{x+1}\), the functions \(f\) and \(h\) are different since \(x=1\) is in the domain of \(f\) but \(x=1\) is not in the domain of \(h\). If we graph \(h(x)\) using a graphing calculator, we will get a graph that looks identical to \(y=f(x)\). There is a vertical asymptote at \(x=-1\), but near \(x=1\), everything seem fine. Tables of values provide numerical evidence which supports the graphical observation.
|
|
We see that as \(x \rightarrow 1^{-}\), \(h(x) \rightarrow 0.5^{-}\) and as \(x \rightarrow 1^{+}\), \(h(x) \rightarrow 0.5^{+}\). In other words, the points on the graph of \(y=h(x)\) are approaching \((1,0.5)\), which we can summarize as \(x \rightarrow 1\), \(h(x) \rightarrow 0.5\). Since \(x=1\) is not in the domain of \(h\), we put an open circle, also called a hole or removable discontinuity , at \((1,0.5)\). Below is a graph of \(h(x)\) with the hole drawn in. Note that most graphing technologies do not sketch holes.
We have seen that values excluded from the domain will yield a vertical asymptote or a hole for the graph. How do we know which graphical behavior will occur at a value not in the domain? Notice that when we simplified \(h\), we had \(h(x) = \dfrac{(2x-1)(x-1)}{(x+1)(x-1)}\). The factor \((x-1)\) cancels whereas \((x+1)\) does not. Loosely speaking, the trouble caused by \((x-1)\) in the denominator is canceled away while the factor \((x+1)\) remains to cause mischief. This is why the graph of \(y=h(x)\) has a vertical asymptote at \(x=-1\) but only a hole at \(x=1\). These observations are generalized and summarized in the theorem below, whose proof is found in Calculus.
Suppose \(r\) is a rational function which can be written as \(r(x) = \dfrac{p(x)}{q(x)}\) where \(p\) and \(q\) have no common zeros {in other words, \(r(x)\) is in lowest terms.}. Let \(c\) be a real number which is not in the domain of \(r\).
- If \(q(c) \neq 0\), then the graph of \(y=r(x)\) has a hole at \(\left(c, \dfrac{p(c)}{q(c)}\right)\).
- If \(q(c) = 0\), then the line \(x=c\) is a vertical asymptote of the graph of \(y=r(x)\).
This theorem says that if \(x=c\) is not in the domain of \(r\) but, when we simplify \(r(x)\), it no longer makes the denominator \(0\), then we have a hole at \(x=c\). Otherwise, the line \(x=c\) is a vertical asymptote of the graph of \(y=r(x)\).
Find the vertical asymptotes and/or holes in the graphs of the following rational functions. Describe the behavior of the graph near them using proper notation.
- \(f(x) = \dfrac{2x}{x^2-3}\)
- \(g(x) = \dfrac{x^2-x-6}{x^2-9}\)
- \(h(x) = \dfrac{x^2-x-6}{x^2+9}\)
- \(r(x) = \dfrac{x^2-x-6}{x^2+4x+4}\)
Solution
- To find the domain, we solve \(x^2 - 3 = 0\) and get \(x = \pm \sqrt{3}\). Since the expression \(f(x)\) is in lowest terms, there is no cancellation possible, and we conclude that the lines \(x = -\sqrt{3}\) and \(x=\sqrt{3}\) are vertical asymptotes to the graph of \(y=f(x)\). As \(x \rightarrow -\sqrt{3}^{\, -}\), \(f(x) \rightarrow -\infty\), as \(x\rightarrow -\sqrt{3}^{\, +}\), \(f(x) \rightarrow \infty\), as \(x \rightarrow \sqrt{3}^{\, -}\), \(f(x) \rightarrow -\infty\), and finally as \(x\rightarrow \sqrt{3}^{\, +}\), \(f(x) \rightarrow \infty\).
- Solving \(x^2 - 9 = 0\) gives \(x = \pm 3\). In lowest terms \(g(x) = \dfrac{x^2-x-6}{x^2-9} = \dfrac{(x-3)(x+2)}{(x-3)(x+3)} = \dfrac{x+2}{x+3}\). Since \(x+3\) continues to make trouble in the denominator, the line \(x=-3\) is a vertical asymptote of the graph of \(y=g(x)\). Since \(x-3\) cancelled out, we have a hole at \(x=3\). To find the \(y\)-coordinate of the hole, we substitute \(x=3\) into the simplified expression, \(\dfrac{x+2}{x+3}\), and find the hole is at \(\left(3, \dfrac{5}{6}\right)\). As \(x \rightarrow -3^{-}\), \(g(x) \rightarrow \infty\), as \(x \rightarrow -3^{+}\), \(g(x) \rightarrow -\infty\), as \(x \rightarrow 3^{-}\), \(g(x) \rightarrow \dfrac{5}{6}^{-}\), and as \(x \rightarrow 3^{+}\), \(g(x) \rightarrow \dfrac{5}{6}^{+}\).
3. The domain of \(h\) is all real numbers, since \(x^2+9 = 0\) has no real solutions. Hence, the graph of \(y=h(x)\) has no vertical asymptotes or holes.
4. Setting \(x^2+4x+4 = 0\) gives us \(x=-2\). Simplifying, we see \[r(x) = \dfrac{x^2-x-6}{x^2+4x+4} = \dfrac{(x-3)(x+2)}{(x+2)^2} = \dfrac{x-3}{x+2}\nonumber\] Since \(x=-2\) continues to produce a \(0\) in the denominator of the reduced function, we know \(x=-2\) is a vertical asymptote to the graph. As \(x\rightarrow -2^{-}\), \(r(x) \rightarrow \infty\) and as \(x \rightarrow -2^{+}\), \(r(x) \rightarrow -\infty\).
Our next example gives us a physical interpretation of a vertical asymptote. This type of model arises from a family of equations cheerily named 'doomsday' equations . These functions arise in Differential Equations. The unfortunate name will make sense shortly.
A mathematical model for the population \(P\), in thousands, of a certain species of bacteria, \(t\) days after it is introduced to an environment is given by \(P(t) = \dfrac{100}{(5-t)^{2}}\), \(0 \leq t < 5\).
- Find and interpret \(P(0)\).
- When will the population reach \(100,\!000\)?
- Determine the behavior of \(P\) as \(t \rightarrow 5^{-}\). Interpret this result graphically and within the context of the problem.
Solution
- Substituting \(t=0\) gives \(P(0) = \dfrac{100}{(5-0)^2} = 4\), which means \(4000\) bacteria are initially introduced into the environment.
- To find when the population reaches \(100,\! 000\), we first need to remember that \(P(t)\) is measured in thousands . In other words, \(100,\! 000\) bacteria corresponds to \(P(t) = 100\). Substituting for \(P(t)\) gives the equation \(\dfrac{100}{(5-t)^2} = 100\). Clearing denominators and dividing by \(100\) gives \((5-t)^2=1\), which, after extracting square roots, produces \(t = 4\) or \(t=6\). Of these two solutions, only \(t=4\) in in our domain, so this is the solution we keep. Hence, it takes \(4\) days for the population of bacteria to reach \(100,\! 000\).
-
To determine the behavior of \(P\) as \(t \rightarrow 5^{-}\), we can make a table
| t | P(t) |
|---|---|
| 4.9 | 10000 |
| 4.99 | 1000000 |
| 4.999 | 100000000 |
| 4.9999 |<PHONE_NUMBER>0 |
In other words, as \(t \rightarrow 5^{-}\), \(P(t) \rightarrow \infty\). Graphically, the line \(t=5\) is a vertical asymptote of the graph of \(y=P(t)\). Physically, this means that the population of bacteria is increasing without bound as we near 5 days, which cannot actually happen. For this reason, \(t=5\) is called the 'doomsday' for this population. There is no way any environment can support infinitely many bacteria, so shortly before \(t = 5\) the environment would collapse.
Horizontal Asymptotes
Now that we have thoroughly investigated behavior or rational functions around values not in the domain, let's turn our attention to end behavior of these functions. Recall the graph of the function \(y=f(x)=\dfrac{2x-1}{x+1}\) that we were discussing earlier. Notice that the graph of \(f\) seems to 'level off' on the left and right hand sides of the screen. This is a statement about the end behavior of the function. Looking at a tables of values to investigate what happens as \(x\) gets larger and larger and also more and more negative without bound, we find
|
|
|
From the tables, we see that as \(x \rightarrow -\infty\), \(f(x) \rightarrow 2\) and as \(x \rightarrow \infty\), \(f(x) \rightarrow 2\). In this case, we say the graph of \(y=f(x)\) has a horizontal asymptote of \(y=2\). This means that the end behavior of \(f\) resembles the horizontal line \(y=2\), which explains the 'leveling off' behavior we see in the calculator's graph.
The line \(y=c\), where \(c\) is a real number, is called a horizontal asymptote of the graph of a function \(y=f(x)\) if as \(x \rightarrow -\infty\) or as \(x \rightarrow \infty\), \(f(x) \rightarrow c\).
Note that in in this definition, we write \(f(x) \rightarrow c\) (not \(f(x) \rightarrow c^{+}\) or \(f(x) \rightarrow c^{-}\)) because we are unconcerned from which direction the values \(f(x)\) approach the value \(c\), just as long as they do so.
Suppose \(r\) is a rational function and \(r(x) = \dfrac{p(x)}{q(x)}\), where \(p\) and \(q\) are polynomial functions with leading coefficients \(a\) and \(b\), respectively.
- If the degree of \(p(x)\) is the same as the degree of \(q(x)\), then \(y=\dfrac{a}{b}\) is the horizontal asymptote of the graph of \(r(x)\).
- If the degree of \(p(x)\) is less than the degree of \(q(x)\), then \(y=0\) is the horizontal asymptote of the graph of \(r(x)\).
- If the degree of \(p(x)\) is greater than the degree of \(q(x)\), then the graph of \(r(x)\) has no horizontal asymptotes.
You may be wondering why the existence of horizontal asymptotes for rational functions is dependent upon the degree of the numerator and denominator. Remember that rational functions are division of polynomials. Let's revisit division for a moment to discover why. Using our example \(f(x) = \dfrac{2x-1}{x+1}\) and division, we can write \(f\) as \(\dfrac{2x-1}{x+1} = 2 - \dfrac{3}{x+1}\). As \(x\) becomes unbounded in either direction, the quantity \(\dfrac{3}{x+1}\) gets closer and closer to \(0\) so that the values of \(f(x)\) become closer and closer to \(2\) (you can verify this using tables). In symbols, as \(x \rightarrow \pm \infty\), \(f(x) \rightarrow 2\). By definition, this indicates \(f\) has a horizontal asymptote at \(y=2\).
Alternatively, we can use what we know about end behavior of polynomials to help us understand this theorem. We know the end behavior of a polynomial is determined by its leading term. Applying this to the numerator and denominator of \(f(x)\), we get that as \(x \rightarrow \pm \infty\), \(f(x) = \dfrac{2x-1}{x+1} \approx \dfrac{2x}{x} = 2\).
Applying this reasoning to the general case, suppose \(r(x) = \dfrac{p(x)}{q(x)}\) where \(a\) is the leading coefficient of \(p(x)\) and \(b\) is the leading coefficient of \(q(x)\). As \(x \rightarrow \pm \infty\), \(r(x) \approx \dfrac{ax^n}{bx^m}\), where \(n\) and \(m\) are the degrees of \(p(x)\) and \(q(x)\), respectively. If the degree of \(p(x)\) and the degree of \(q(x)\) are the same, then \(n=m\) so that \(r(x) \approx \dfrac{a}{b}\), which means \(y=\dfrac{a}{b}\) is the horizontal asymptote in this case. If the degree of \(p(x)\) is less than the degree of \(q(x)\), then \(n < m\), so \(m-n\) is a positive number, and hence, \(r(x) \approx \dfrac{a}{bx^{m-n}} \rightarrow 0\) as \(x \rightarrow \pm \infty\). If the degree of \(p(x)\) is greater than the degree of \(q(x)\), then \(n > m\), and hence \(n-m\) is a positive number and \(r(x) \approx \dfrac{ax^{n-m}}{ b}\), which becomes unbounded as \(x \rightarrow \pm \infty\).
List the horizontal asymptotes, if any, of the graphs of the following functions. Describe the behavior of the graph near them using proper notation.
- \(f(x) = \dfrac{5x}{x^2+1}\)
- \(g(x) = \dfrac{x^2-4}{x+1}\)
- \(h(x) = \dfrac{6x^3-3x+1}{5-2x^3}\)
Solution
- The numerator of \(f(x)\) is \(5x\), which has degree \(1\). The denominator of \(f(x)\) is \(x^2+1\), which has degree \(2\). Since the degree of the numerator is less than the denominator, \(y=0\) is the horizontal asymptote. As \(x \rightarrow - \infty\), \(f(x) \rightarrow 0^{-}\) and as \(x \rightarrow \infty\), \(f(x) \rightarrow 0^{+}\).
- The numerator of \(g(x)\), \(x^2-4\), has degree \(2\), but the degree of the denominator, \(x+1\), has degree \(1\). Since the degree of the numerator is larger thant he denominator, here is no horizontal asymptote. {Sit tight! We'll revisit this function and its end behavior shortly.}
- The degrees of the numerator and denominator of \(h(x)\) are both three, so \(y = \dfrac{6}{-2} = -3\) is the horizontal asymptote. As \(x \rightarrow -\infty\), \(h(x) \rightarrow -3^{+}\), and as \(x \rightarrow \infty\), \(h(x) \rightarrow -3^{-}\).
Slant Asymptotes
In the previous example, \(g(x) = \dfrac{x^2-4}{x+1}\), did not have a horizontal asymptote. What is the end behavior of this function then? Performing long division, we get \(g(x) = \dfrac{x^2-4}{x+1} = x-1 - \dfrac{3}{x+1}\). Since the term \(\dfrac{3}{x+1} \rightarrow 0\) as \(x \rightarrow \pm \infty\), it stands to reason that as \(x\) becomes unbounded, the function values \(g(x) = x-1 - \dfrac{3}{x+1} \approx x-1\). Geometrically, this means that the graph of \(y=g(x)\) should resemble the line \(y = x-1\) as \(x \rightarrow \pm \infty\). We see this play out both numerically and graphically below.
|
|
||||||||||||||||||||||||||||||
|
\(y=g(x)\) and \(y=x-1\)
|
\(y=g(x)\) and \(y=x-1\) as \(x \rightarrow \infty\) |
The way we symbolize the relationship between the end behavior of \(y=g(x)\) with that of the line \(y=x-1\) is to write 'as \(x \rightarrow \pm \infty\), \(g(x) \rightarrow x-1\).' In this case, we say the line \(y=x-1\) is a slant asymptote of \(y=g(x)\). Informally, the graph of a rational function has a slant asymptote if, as \(x \rightarrow \infty\) or as \(x \rightarrow -\infty\), the graph resembles a non-horizontal, or 'slanted' line. Formally, we define a slant asymptote as follows.
The line \(y = mx+b\) where \(m \neq 0\) is called a slant asymptote of the graph of a function \(y=f(x)\) if as \(x \rightarrow -\infty\) or as \(x \rightarrow \infty\), \(f(x) \rightarrow mx+b\).
Note that the stipulation \(m \neq 0\) in this definition is what makes the 'slant' asymptote 'slanted' as opposed to the case when \(m=0\) in which case we'd have a horizontal asymptote.
Also, since \(y=x-1\) has the end behavior that \(y \rightarrow -\infty\) as \(x \rightarrow -\infty\) and \(y \rightarrow \infty\) as \(x \rightarrow \infty\) we can describe the end of behavior of g(x) as follows:
\(g(x) \rightarrow -\infty\) as \(x \rightarrow -\infty\) and \(g(x) \rightarrow \infty\) as \(x \rightarrow \infty\)
Some mathematicians prefer to describe the end behavior with the actual asymptote and some prefer to describe the end behavior directly like was just done above.
Our next task is to determine the conditions under which the graph of a rational function has a slant asymptote, and if it does, how to find it. In the case of \(g(x) = \dfrac{x^2-4}{x+1}\), the degree of the numerator \(x^2-4\) is \(2\), which is exactly one more than the degree if its denominator \(x+1\) which is \(1\). This results in a linear quotient polynomial, and it is this quotient polynomial which is the slant asymptote.
Suppose \(r\) is a rational function and \(r(x) = \dfrac{p(x)}{q(x)}\), where the degree of \(p\) is exactly one more than the degree of \(q\). Then the graph of \(y=r(x)\) has the slant asymptote \(y=L(x)\) where \(L(x)\) is the quotient obtained by dividing \(p(x)\) by \(q(x)\).
Find the slant asymptotes of the graphs of the following functions if they exist. Describe the end behavior of the graph near them using proper notation.
- \(f(x) = \dfrac{x^2-4x+2}{1-x}\)
- \(g(x) = \dfrac{x^2-4}{x-2}\)
- \(h(x) = \dfrac{x^3+1}{x^2-4}\)
Solution
- The degree of the numerator is \(2\) and the degree of the denominator is \(1\), so \(f\) has a slant asymptote. To find it, we divide \(1-x = -x+1\) into \(x^2-4x+2\) and get a quotient of \(-x+3\), so our slant asymptote is \(y=-x+3\). As \(x\rightarrow -\infty\) or as as \(x\rightarrow \infty\), \(f(x) \rightarrow (-x+3)\).
- The degree of the numerator \(g(x) = \dfrac{x^2-4}{x-2}\) is \(2\) and the degree of the denominator is \(1\). However, \[ g(x) = \dfrac{x^2-4}{x-2} = \dfrac{(x+2)(x-2)}{(x-2)} = x+2, x \neq 2\nonumber \] so we have that the slant asymptote \(y=x+2\) is identical to the graph of \(y=g(x)\) except at \(x=2\) (where the latter has a 'hole' at \((2,4)\).) As \(x\rightarrow -\infty\) and as \(x\rightarrow \infty\), \(g(x) \rightarrow (x+2)\).
- For \(h(x) = \dfrac{x^3+1}{x^2-4}\), the degree of the numerator is \(3\) and the degree of the denominator is \(2\) so again, we are guaranteed the existence of a slant asymptote. The long division \(\left(x^3+1 \right) \div \left(x^2-4\right)\) gives a quotient of just \(x\), so our slant asymptote is the line \(y=x\). As \(x \rightarrow -\infty\) and as \(x \rightarrow \infty\), \(g(x) \rightarrow x\)
Oblique Asymptotes
We have discussed the end behavior and asymptotes of rational functions where the degree of the numerator is less than, equal, or one degree higher than the degree of the denominator. You may be wondering about the end behavior of rational functions where the degree of the numerator is a degree of at least two or larger than the degree of the denominator. We know from polynomial division that the quotient will be a polynomial. Since rational functions are division of polynomials that means that the quotient will describe the end behavior of the rational function in a similar way as slant asymptotes do.
The function \(f(x)=\dfrac{x^4+x^3-2x^2+3}{x^2+x-2}\) can be rewritten as \(f(x)=x^2+\dfrac{3}{x^2+x-2}\). As \(x \rightarrow \pm \infty\), \(f(x)\rightarrow x^2\). So, the end behavior for \(f\) is as \(x \rightarrow \pm \infty\), \(f(x) \rightarrow \infty\). We call \(y=x^2\) an oblique asymptote . When graphing \(f\) and the oblique asymptote together, visually can see that, in fact, \(f(x)\rightarrow x^2\) as \(x \rightarrow \pm \infty\).
Some mathematicians will say that horizontal and/or slant asymptotes are types of oblique asymptotes. Others will argue that oblique asymptotes are strictly polynomials of degree two or higher. We are not in a position to argue that one description is better than another. This main thing is to be aware that the end behavior of rational functions is polynomial in nature. For this text, we will summarize the end behavior into three categories of asymptotes as follows:
For a rational function, \(r(x)\), as \(x \rightarrow \pm \infty\)
- If \(r(x)\rightarrow c\) where \(c\) is a real number, then \(y=c\) is a horizontal asymptote for \(r(x)\).
- If \(r(x)\rightarrow mx+b\) where \(m\) and \(b\) are real numbers and \(m \neq0\), then \(y=mx+b\) is a slant asymptote for \(r(x)\).
- If \(r(x)\rightarrow P(x)\) where \(P(x)\) is a polynomial of degree 2 or higher, then \(y=P(x)\) is a slant asymptote for \(r(x)\).


1.2: Sets
Intuitively, a mathematical set is a collection of mathematical objects. Unfortunately this simple characterization of sets, carelessly handled, gives rise to contradictions. Some collections will turn out not to have the properties that we demand of mathematical sets. An example of how this can occur is presented in Section 1.7. We shall not develop formal set theory from scratch here. Instead, we shall assume that certain building block sets are known, and describe ways to build new sets out of these building blocks.
Our initial building blocks will be the sets of natural numbers, integers, rational numbers and real numbers. In Chapter 8 , we shall show how to build all these from the natural numbers. One can’t go much further than this, though: in order to do mathematics, one has to start with axioms that assert that the set of natural numbers exist.
If \(X\) is a set and \(x\) is an object in \(X\) , we say that \(x\) is an element, or member, of \(X\) . This is written \[x \in X .\] We write \(x \notin X\) if \(x\) is not a member of \(X\) .
There are numerous ways to define sets. If a set has few elements, it may be defined by listing. For instance, \[X=\{2,3,5,7\}\] is the set of the first four prime numbers. In the absence of any other indication, a set defined by a list is assumed to have as elements only the objects in the list. For sets with too many elements to list, we must provide the reader with a means to determine membership in the set. The author can inform the reader that not all elements of the set have been listed, but that enough information has been provided for the reader to identify a pattern for determining membership in the set. For example, let \[X=\{2,4,6,8, \ldots, 96,98\} .\] Then \(X\) is the set of positive even integers less than 100. However, using an ellipsis to define a set may not always work: it assumes that the reader will identify the pattern you wish to characterize. Although this usually works, it carries the risk that the reader is unable to correctly identify the pattern intended by the author.
Some sets are so important that they have standard names and notations that you will need to know.
Notation. Natural numbers, \(\mathbb{N}\) The natural numbers are the elements of the set \[\{0,1,2,3, \ldots\} .\] This set is denoted by \(\mathbb{N}\) .
Beware: Many authors call \(\{1,2,3, \ldots\}\) the set of natural numbers. This is a matter of definition, and there is no universal convention; logicians tend to favor our convention, and algebraists the other. In this book, we shall use \(\mathbb{N}^{+}\) to denote \(\{1,2,3, \ldots\}\) .
NOTATION. \(\mathbb{N}^{+} \mathbb{N}^{+}\) is the set of positive integers, \[\{1,2,3, \ldots\} \text {. }\] NOTATION. Integers, \(\mathbb{Z} \mathbb{Z}\) is the set of integers, \[\{\ldots,-3,-2,-1,0,1,2,3, \ldots\} \text {. }\] NOTATION. Rational numbers, \(\mathbb{Q} \mathbb{Q}\) is the set of rational numbers, \[\left\{\frac{p}{q} \text { where } p, q \in \mathbb{Z} \text { and } q \neq 0\right\} \text {. }\] NOTATION. Real numbers, \(\mathbb{R} \mathbb{R}\) is the set of real numbers. A good understanding of the real numbers requires a bit of mathematical develoment. In fact, it was only in the nineteenth century that we really came to a modern understanding of \(\mathbb{R}\) . We shall have a good deal to say about the real numbers in Chapter 8 .
A number \(x\) is positive if \(x>0\) . A number \(x\) is nonnegative if \(x \geq 0\) .
NOTATION. \(X^{+}\) If \(X\) is a set of real numbers, we use \(X^{+}\) for the positive numbers in the set \(X\) .
The notation we have presented for these sets is widely used. We introduce a final convention for set names which is not as widely recognized, but is useful for set theory.
NOTATION. \(\ulcorner n\urcorner\) is the set of all natural numbers less than \(n\) : \[\ulcorner n\urcorner=\{0,1,2, \ldots, n-1\} .\] One purpose of this notation is to canonically associate any natural number \(n\) with a set having exactly \(n\) elements.
The reader should note that we have not defined the above sets. We are assuming that you are familiar with them, and some of their properties, by virtue of your previous experience in mathematics. We shall eventually define the sets systematically in Chapter \(8 .\)
A more precise method of defining a set is to use unambiguous conditions that characterize membership in the set.
Notation. \(\{x \in X \mid P(x)\}\) Let \(X\) be a (previously defined) set, and let \(P(x)\) be a condition or property. Then the set \[Y=\{x \in X \mid P(x)\}\] is the set of elements in \(X\) which satisfy condition \(P\) . The set \(X\) is called the domain of the variable.
In words, (1.1) is read: " \(Y\) equals the set of all (little) \(x\) in (capital) \(X\) such that \(P\) is true of \(x\) ". The symbol " \(\mid\) " in (1.1) is often written instead with a colon, viz. \(\{x \in X: P(x)\}\) . In mathematics, \(P(x)\) is a often a mathematical formula. For instance, suppose \(P(x)\) is the formula " \(x^{2}=4\) ". By \(P(2)\) we mean the formula with 2 substituted for \(x\) , that is \[" 2^{2}=4 "\] If the substitution results in a true statement, we say that \(P(x)\) holds at 2 , or \(P(2)\) is true. If the statement that results from the substitution is false, for instance \(P(1)\) , we say that \(P(x)\) does not hold at 1 , or that \(P(1)\) is false.
Consider the set \[X=\{0,1,4,9, \ldots\} .\] A precise definition of the same set is the following: \[X=\left\{x \in \mathbb{N} \mid \text { for some } y \in \mathbb{N}, x=y^{2}\right\} .\] EXAMPLE 1.3. Let \(Y\) be the set of positive even integers less than 100. Then \(Y\) can be written: \[\left\{x \in \mathbb{N} \mid x<100 \text { and there is } n \in \mathbb{N}^{+} \text {such that } x=2 \cdot n\right\}\] EXAMPLE 1.4. An interval \(I\) is a non-empty subset of \(\mathbb{R}\) with the property that whenever \(a, b \in I\) and \(a<c<b\) , then \(c\) is in \(I\) . A bounded interval must have one of the four forms \[\begin{aligned} (a, b) &=\{x \in \mathbb{R} \mid a<x<b\} \\ {[a, b) } &=\{x \in \mathbb{R} \mid a \leq x<b\} \\ (a, b] &=\{x \in \mathbb{R} \mid a<x \leq b\} \\ {[a, b] } &=\{x \in \mathbb{R} \mid a \leq x \leq b\} \end{aligned}\] where in the first three cases \(a\) and \(b\) are real numbers with \(a<b\) and in the fourth case we just require \(a \leq b\) . Unbounded intervals have five forms: \[\begin{aligned} (-\infty, b) &=\{x \in \mathbb{R} \mid x<b\} \\ (-\infty, b] &=\{x \in \mathbb{R} \mid x \leq b\} \\ (b, \infty) &=\{x \in \mathbb{R} \mid x>b\} \\ {[b, \infty) } &=\{x \in \mathbb{R} \mid x \geq b\} \\ \mathbb{R} & \end{aligned}\] where \(b\) is some real number. An interval is called closed if it contains all its endpoints (both \(a\) and \(b\) in the first group of examples, just \(b\) in the first four examples of the second group), and open if it contains none of them. Notice that this makes \(\mathbb{R}\) the only interval that is both closed and open.
For the sake of brevity, an author may not explicitly identify the domain of the variable. Be careful of this, as the author is relying on the reader to make the necessary assumptions. For instance, consider the set \[X=\left\{x \mid\left(x^{2}-2\right)(x-1)\left(x^{2}+1\right)=0\right\} .\] If the domain of the variable is assumed to be \(\mathbb{N}\) , then \[X=\{1\} .\] If the domain of the variable is assumed to be \(\mathbb{R}\) , then \[X=\{1, \sqrt{2},-\sqrt{2}\} .\] If the domain of the variable is assumed to be the complex numbers, then, \[X=\{1, \sqrt{2},-\sqrt{2}, i,-i\}\] where \(i\) is the complex number \(\sqrt{-1}\) . Remember, the burden of clear communication is on the author, not the reader.
Another alternative is to include the domain of the variable in the condition defining membership in the set. So, if \(X\) is the intended domain of the set and \(P(x)\) is the condition for membership in the set, \[\{x \in X \mid P(x)\}=\{x \mid x \in X \text { and } P(x)\} .\] As long as the definition is clear, the author has some flexibility with regard to notation.
1.2.1. Set Identity.
When are two sets equal? You might be inclined to say that two sets are equal provided they are the same collection of objects. Of course this is true, but equality as a relation between objects is not very interesting. However, you have probably spent a lot of time investigating equations (which are just statements of equality), and we doubt that equality seemed trivial. This is because in general equality should be understood as a relationship between descriptions or names of objects, rather than between the objects themselves. The statement \[a=b\] is a claim that the object represented by \(a\) is the same object as that represented by \(b\) . For example, the statement \[5-3=2\] is the claim that the number represented by the arithmetic expression \(5-3\) is the same number as that represented by the numeral 2 .
In the case of sets, this notion of equality is called extensionality.
Let \(X\) and \(Y\) be sets. Then \(X=Y\) provided that every element of \(X\) is also an element of \(Y\) and every element of \(Y\) is also an element of \(X\) .
There is flexibility in how a set is characterized as long as we are clear on which objects constitute the set. For instance, consider the set equation \[\{\text { Mark Twain, Samuel Clemens }\}=\{\text { Mark Twain }\} .\] If by "Mark Twain" and "Samuel Clemens", we mean the deceased American author, these sets are equal, by extensionality, and the statement is true. The set on the left hand side of the equation has only one element since both names refer to the same person. If, however, we consider "Mark Twain" and "Samuel Clemens" as names, the statement is false, since "Samuel Clemens" is a member of the set on the left hand side of the equation, but not the right hand side. You can see that set definitions can depend on the implicit domain of the variable even if the sets are defined by listing.
Consider the following six sets: \[\begin{aligned} &X_{1}=\{1,2\} \\ &X_{2}=\{2,1\} \\ &X_{3}=\{1,2,1\} \\ &X_{4}=\{n \in \mathbb{N} \mid 0<n<3\} \\ &X_{5}=\left\{n \in \mathbb{N} \mid \text { there exist } x, y, z \in \mathbb{N}^{+} \text {such that } x^{n}+y^{n}=z^{n}\right\} \\ &X_{6}=\{0,1,2\} . \end{aligned}\] The first five sets are all equal, and the sixth is different. However, while it is obvious that \(X_{1}=X_{2}=X_{3}=X_{4}\) , the fact that \(X_{5}=X_{1}\) is the celebrated theorem of Andrew Wiles (his proof of Fermat’s last theorem).
1.2.2. Relating Sets.
In order to say anything interesting about sets, we need ways to relate them, and we shall want ways to create new sets from existing sets.
Let \(X\) and \(Y\) be sets. \(X\) is a subset of \(Y\) if every element of \(X\) is also an element of \(Y\) . This is written \[X \subseteq Y .\] Superset, \(\supseteq\) If \(X \subseteq Y\) , then \(Y\) is called a superset of \(X\) , written \[Y \supseteq X .\] In order to show two sets are equal (or that two descriptions of sets refer to the same set), you must show that they have precisely the same elements. It is often easier if the argument is broken into two simpler arguments in which you show mutual containment of the sets. In other words, saying \(X=Y\) is the same as saying \[X \subseteq Y \text { and } Y \subseteq X,\] and verifying the two separate claims in (1.6) is often easier (or at least clearer) than showing that \(X=Y\) all at once.
Let’s add a few more elementary notions to our discussion of sets.
Let \(X\) and \(Y\) be sets. \(X\) is a proper subset of \(Y\) if \[X \subseteq Y \text { and } X \neq Y \text {. }\] We write this as \[X \subsetneq Y\] or \[Y \supsetneq X .\] DEFINITION. Empty set, \(\emptyset\) The empty set is the set with no elements. It is denoted by \(\emptyset\) .
So for any set, \(X\) , \[\emptyset \subseteq X \text {. }\] (Think about why this is true). Just because \(\emptyset\) is empty does not mean it is unimportant. Indeed, many mathematical questions reduce to asking whether a particular set is empty or not. Furthermore, as you will see in Chapter 8, we can build the entire real line from the empty set using set operations.
Show that \[\{n \in \mathbb{N} \mid n \text { is odd and } n=k(k+1) \text { for some } k \in \mathbb{N}\}\] is empty.
Let’s discuss some ways to define new sets from existing sets. DEFINITION. Union, \(\cup\) Let \(X\) and \(Y\) be sets. The union of \(X\) and \(Y\) , written \(X \cup Y\) , is the set \[X \cup Y=\{x \mid x \in X \text { or } x \in Y\} .\] (Recall our discussion in Section 1.1 about the mathematical meaning of the word "or".)
Let \(X\) and \(Y\) be sets. The intersection of \(X\) and \(Y\) , written \(X \cap Y\) , is the set \[X \cap Y=\{x \mid x \in X \text { and } x \in Y\} .\] DEFINITION. Set difference, \(\backslash\) Let \(X\) and \(Y\) be sets. The set difference of \(X\) and \(Y\) , written \(X \backslash Y\) , is the set \[X \backslash Y=\{x \in X \mid x \notin Y\} .\] DEFInITION. Disjoint Let \(X\) and \(Y\) be sets. \(X\) and \(Y\) are disjoint if \[X \cap Y=\emptyset .\] Oftentimes one deals with sets that are subsets of some fixed given set \(U\) . For example, when dealing with sets of natural numbers, the set \(U\) would be \(\mathbb{N}\) .
Let \(X \subseteq U\) . The complement of \(X\) in \(U\) is the set \(U \backslash X\) . When \(U\) is understood from the context, the complement of \(X\) is written \(X^{c}\) .
What about set operations involving more than two sets? Unlike arithmetic, in which there is a default order of operations (powers, products, sums), there is not a universal convention for the order in which set operations are performed. If intersections and unions appear in the same expression, then the order in which the operations are performed can matter. For instance, suppose \(X\) and \(Y\) are disjoint, nonempty sets, and consider the expression \[X \cap X \cup Y \text {. }\] If we mean for the intersection to be executed before the union, then \[(X \cap X) \cup Y=X \cup Y .\] If, however we intend the union to be computed before the intersection, then \[X \cap(X \cup Y)=X .\] Since \(Y\) is nonempty and disjoint from \(X\) , \[(X \cap X) \cup Y \neq X \cap(X \cup Y) .\] Consequently, the order in which set operations are executed needs to be explicitly prescribed with parentheses.
Let \(X=\mathbb{N}\) and \(Y=\mathbb{Z} \backslash \mathbb{N}\) . Then \[(X \cap X) \cup Y=\mathbb{N} \cup Y=\mathbb{Z} .\] However \[X \cap(X \cup Y)=\mathbb{N} \cap \mathbb{Z}=\mathbb{N} .\] DEFINITION. Cartesian product, Direct product, \(X \times Y\) Let \(X\) and \(Y\) be sets. The Cartesian product of \(X\) and \(Y\) , written \(X \times Y\) , is the set of ordered pairs \[\{(x, y) \mid x \in X \text { and } y \in Y\} .\] The Cartesian product is also called the direct product.
Let \[X=\{1,2,3\}\] and \[Y=\{1,2\} .\] Then \[X \times Y=\{(1,1),(1,2),(2,1),(2,2),(3,1),(3,2)\} .\] Note that the order matters - that is \[(1,2) \neq(2,1) .\] So \(X \times Y\) is a set with six elements. Since direct products are themselves sets, we can easily define the direct product of more than two factors. For example, let \(X, Y\) and \(Z\) be sets, then \[(X \times Y) \times Z=\{((x, y), z) \mid x \in X, y \in Y, z \in Z\} .\] Formally, \[(X \times Y) \times Z \neq X \times(Y \times Z),\] because \(((x, y), z)\) and \((x,(y, z))\) are not the same. However in nearly every application, this distinction is not important, and mathematicians generally consider the direct product of more than two sets without regard to this detail. Therefore you will generally see the Cartesian product of three sets written without parentheses, \[X \times Y \times Z \text {. }\] In this event you may interpret the direct product as either side of statement 1.8.
With some thought, you can conclude that we have essentially described the Cartesian product of an arbitrary finite collection of sets. The elements of the Cartesian product \(X \times Y\) are ordered pairs. Our characterization of the Cartesian product of three sets, \(X, Y\) and \(Z\) , indicates that its elements could be thought of as ordered pair of elements of \(X \times Y\) and \(Z\) , respectively. From a practical point of view, it is simpler to think of elements of \(X \times Y \times Z\) as ordered triples. We generalize this as follows.
Cartesian product, Direct product, \(\prod_{i=1}^{n} X_{i}\) Let \(n \in\) \(\mathbb{N}^{+}\) , and \(X_{1}, X_{2}, \ldots, X_{n}\) be sets. The Cartesian product of \(X_{1}, \ldots, X_{n}\) , written \(X_{1} \times X_{2} \times \ldots \times X_{n}\) , is the set \[\left\{\left(x_{1}, x_{2}, \ldots, x_{n}\right) \mid x_{i} \in X_{i}, 1 \leq i \leq n\right\} .\] This may also be written \[\prod_{i=1}^{n} X_{i} .\] When we take the Cartesian product of a set \(X\) with itself \(n\) times, we write it as \(X^{n}\) : \[X^{n}:=\overbrace{X \times X \times \cdots \times X}^{n \text { times }} .\]


8.5: Checklist for Cardiovascular Assessment
Use the checklist below to review the steps for completion of a “Cardiovascular Assessment.” [1]
Steps
Disclaimer: Always review and follow agency policy regarding this specific skill.
- Gather supplies: stethoscope and watch with a second hand.
-
Perform safety steps:
- Perform hand hygiene.
- Check the room for transmission-based precautions.
- Introduce yourself, your role, the purpose of your visit, and an estimate of the time it will take.
- Confirm patient ID using two patient identifiers (e.g., name and date of birth).
- Explain the process to the patient and ask if they have any questions.
- Be organized and systematic.
- Use appropriate listening and questioning skills.
- Listen and attend to patient cues.
- Ensure the patient’s privacy and dignity.
- Assess ABCs.
-
Conduct a focused interview related to cardiovascular and peripheral vascular disease.
- Ask relevant focused questions based on patient status. See Tables 8.3a and 8.3b for example questions.
-
Inspect:
- Face, lips, and extremities for pallor or cyanosis
- Neck for jugular vein distension (JVD) in upright position or with head of bed at 30-45 degree angle
- Chest for deformities and wounds/scars on chest
- Bilateral arms/hands, noting color, warmth, movement, sensation (CWMS), edema, and color of nail beds
- Bilateral legs, noting CWMS, edema to lower legs and feet, presence of superficial distended veins, and color of nail beds
- Auscultate with both the bell and the diaphragm of the stethoscope over five auscultation areas of the heart. Auscultate the apical pulse at the fifth intercostal space, midclavicular line for one minute. Note the rate and rhythm. Identify the S1 and S2 sounds and follow up on any unexpected findings (e.g., extra sounds or irregular rhythm).
- Palpate the radial, brachial, dorsalis pedis, and posterior tibialis pulses bilaterally. Palpate the carotid pulse one side at a time. Note presence/amplitude of pulse and any unexpected findings requiring follow-up.
- Palpate the nail beds for capillary refill. Document the capillary refill time as less than or greater than 2 seconds.
- Assist the patient to a comfortable position, ask if they have any questions, and thank them for their time.
-
Ensure safety measures when leaving the room:
- CALL LIGHT: Within reach
- BED: Low and locked (in lowest position and brakes on)
- SIDE RAILS: Secured
- TABLE: Within reach
- ROOM: Risk-free for falls (scan room and clear any obstacles)
- Perform hand hygiene.
- Document the assessment findings. Report any concerns according to agency policy.
- This work is a derivative of Clinical Procedures for Safer Patient Care by British Columbia Institute of Technology licensed under CC BY 4.0 ↵


4.3: Thorstein Veblen, on Labor(1898)
-
- Last updated
- Save as PDF
“Early humans were members of a group which depended for its survival on the industrial efficiency of its members and on their singleness of purpose in making use of the material means at hand.”
NOTE ON SOURCE: This passage is from an article entitled “The Instinct of Workmanship and the Irksomeness of Labor,” published in the American Journal of Sociology in September 1898, pages 187-201. It has been abbreviated for publication here.
Introduction – Why this is important and what to look for
Veblen tries to understand how it is that we could have a collective aversion to work when, as a species, we have always had a very strong “instinct for workmanship.” He then contrasts sportsmanship (fighting) with workmanship, locating the denigration of work in the development of a “predatory culture.” As you read this passage and follow Veblen’s argument, think about how it compares with Marx’s description of the rise of capitalism and Weber’s explanation of the difference between class and status societies.
The Instinct of Workmanship [1] and the Irksomeness of Labor
It is one of the commonplaces of the revived economic theory that work is irksome. Many a discussion proceeds on this axiom that, so far as regards economic matters, people desire above all things to get the goods produced by labor and to avoid the labor by which the goods are produced. In a general way the common-sense opinion is well in accord with current theory on this head. According to the common-sense ideal, the economic beatitude lies in an unrestrained consumption of goods, without work; whereas the perfect economic affliction is unremunerated labor. We instinctively revolt at effort that goes to supply the means of life.
No one will accept the proposition when stated in this bald fashion, but even as it stands it is scarcely an overstatement of what is implied in the writings of eminent economists.
Yet, for all the apparent absurdity of the thing, there is the fact. With more or less sincerity, people currently avow an aversion to useful effort. The avowal does not cover all effort, but only such as is of some use; it is, more particularly, such effort as is vulgarly recognized to be useful labor. Less repugnance is expressed as regards effort which brings gain without giving a product that is of human use, as, for example, the effort that goes into war, politics, or other employments of a similar nature. And there is commonly no avowed aversion to sports or other similar employments that yield neither a pecuniary gain nor a useful product.
Like other animals, humans are agents that act in response to stimuli afforded by the environment in which they live. Like other species, we are creatures of habit and propensity. But in a higher degree than other species, we mentally digest the content of the habits under whose guidance we act and appreciate the trend of these habits and propensities. We are in an eminent sense an intelligent agent. By selective necessity we are endowed with a proclivity for purposeful action. We are possessed of a discriminating sense of purpose, by force of which all futility of life or of action is distasteful to us. This is a generic feature of human nature. It is not a trait that occurs sporadically in a few individuals.
Our great advantage over other species in the struggle for survival has been our superior facility in turning the forces of the environment to account. It is to our proclivity for turning the material means of life to account that we owe our positions as masters of creation. It is not a proclivity of effort, but to achievement – to the compassing of an end. Our primacy is in the last resort an industrial or economic primacy. In our economic life we are agents, not sponges; we are agents seeking in every act the accomplishment of some concrete, objective, impersonal end.
All this seems to contradict what has just been said of the conventional aversion to labor. But the contradiction is not so sheer in fact as it appears to be at first sight. Its solution lies in the fact that the aversion to labor is in great part a conventional aversion only. In the intervals of sober reflection, when not harassed by the strain of overwork, our common sense speaks unequivocally under the guidance of the instinct of workmanship . We like to see others spend their lives to some purpose, and we like to reflect that our own life is of some use. All of us have this quasi-aesthetic sense of economic or industrial merit, and to this sense of economic merit futility and inefficiency are distasteful. In its positive expression it is an impulse or instinct of workmanship; negatively it expresses itself in a deprecation of waste.
This instinct of workmanship apparently stands in sheer conflict with conventional antipathy to useful effort. The two are found together in full discord in the common run of people; but whenever a deliberate judgment is passed on conduct or one vents, the former asserts its primacy in a pervasive way which suggests that it is altogether the more generic, more abiding trait of human nature. There can scarcely be a serious question of precedence between the two. The former is a human trait necessary to the survival of the species; the latter is a habit of thought possible only in a species which has distanced all competitors, and then it prevails only by sufferance and within limits set by the former. The question between them is, Is the aversion to labor a derivative of the instinct of workmanship? And, how has it arisen and gained consistence in spite of its being at variance with that instinct?
Until recently there has been something of a consensus among those who have written on early culture, to the effect that, human beings, as they first emerged upon the properly human plane, were of a contentious disposition, inclined to isolate their own interests and purposes from those of their fellows, and with a penchant for feuds and brawls. Even today it is held that men are inclined to fight, not to work. This view would make the proclivity to purposeful action an impulse to sportsmanship rather than to workmanship.
There is much to be said for this view. If humans were by derivation a race not of workers but of sportsmen, then there is no need of explaining the conventional aversion to work. Work is unsportsmanlike and therefore distasteful, and perplexity then arises in explaining how we have in any degree become reconciled to any but a predatory life. The history of mankind, as conventionally written, has been a narrative of predatory exploits, and this history is not commonly felt to be one-sided or misinformed. And a sportsmanlike inclination to warfare is also to be found in nearly all modern communities. Similarly, the sense of honor, so-called, whether it is individual or national honor, is also an expression of sportsmanship. The prevalence of notions of honor may, therefore, be taken as evidence going in the same direction.
Yet there is a considerable body of evidence, both from cultural history and from the present-day phenomena of human life, which traverses this conventionally accepted view that makes man generically a sportsman. Obscurely but persistently, throughout the history of human culture, the great body of people have almost everywhere, in their everyday life, been at work to turn things to human use. The proximate aim of all industrial improvement has been the better performance of some workmanlike task. It will not do to say that the work accomplished is entirely due to compulsion under a predatory regime, for the most striking advances in this respect have been wrought where the coercive force of a sportsmanlike exploitation has been least.
The same view is borne out by the expressions of common sense. As has already been remarked, whenever they dispassionately take thought and pass a judgment on the value of human conduct, the common run of people approve workmanship rather than sportsmanship. At the best, they take an apologetic attitude toward the latter. Predatory exploit, simply as such, is not felt to carry its own legitimation. What meets unreserved approval is such conduct as furthers human life on the whole, rather than such as furthers the invidious or predatory interest of one as against another.
The archaic turn of mind that inclines people to commend workmanlike serviceability is the outcome of long and consistent habituation to a course of life of such a character as is reflected by this inclination.
Our life is activity; and as we act, so we think and feel. This is necessarily so, since it is we, the agent, that does the thinking and feeling. Like other species, we are a creature of habits and propensities. We are social animals and the selective process whereby we have acquired the spiritual makeup of a social animal has at the same time made us substantially a peaceful animal. We may have wandered far from the ancient position of peacefulness, but even now the traces of a peaceful trend in our everyday habits of thought and feeling are plain enough. The sight of blood and the presence of death, even of the blood or death of the lower animals, commonly strike inexperienced persons with a sickening revulsion. In the common run of cases, the habit of complacency with slaughter comes only as the result of discipline. In this respect we differ from the beasts of prey. In our unarmed frame and in the slight degree to which our muscular force is specialized for fighting, as well as in our instinctive aversion to hostile contact with ferocious beasts, we are to be classed with those animals that owe their survival to an aptitude for avoiding direct conflict with their competitors, rather than those which survive by virtue of overcoming and eating their rivals.
We are the weakest and most defenseless of all living things, and, according to the Law of the Jungle, it is our part to take advice and contrive to turn diverse things to account in ways that are incomprehensible to the rest. Without tools we are not a dangerous animal, as animals go. And we did not become a formidable animal until we had made some considerable advance in the contrivance of implements for combat. In the days before tools had been brought into effective use – that is to say, during by far the greater part of the period of human evolution – we could not be primarily agents of destruction or disturbers of the peace. We were of a peaceable and retiring disposition by force of circumstances. Tools and implements, in the early days, must have served chiefly to shape facts and objects for human use, rather than for inflicting damage and discomfort. Industry would have to develop far before it became possible for one group of men to live at the cost of another.
By selection and by training, our life, before a predatory life became possible, would act to develop and to conserve in us an instinct for workmanship. The adaptation to the environment which the situation enforced was of an industrial kind: it required us to acquire facility in shaping things and situations for human use. This does not mean the shaping of things by the individual to her own individual use simply; for archaic humans were necessarily members of a group, and during this early stage, when industrial efficiency was still inconsiderable, no group could have survived except on the basis of a sense of solidarity strong enough to throw self-interest into the background. Self-interest, as an accepted guide of action, is possible only as the concomitant of a predatory life, and a predatory life is possible only after the use of tools has developed so far as to leave a large surplus of product over what is required for the sustenance of the producers. Subsistence by predation implies something substantial to prey upon.
Early humans were members of a group which depended for its survival on the industrial efficiency of its members and on their singleness of purpose in making use of the material means at hand. Some competition between groups for the possession of the fruits of the earth and for advantageous locations there would be even at a relatively early stage, but much hostile contact between groups there could not be; not enough to shape the dominant habits of thought.
What we can do easily is what we do habitually, and this decides what we can think and know easily. We feel at home in the range of ideas which is familiar through our everyday line of action. A habitual line of action constitutes a habitual line of thought and gives the point of view from which facts and events are apprehended and reduced to a body of knowledge. A process or method of life, once understood, assimilated in thought, works into the scheme of life and becomes a norm of conduct, simply because the thinking, knowing agent is also the acting agent. What is apprehended with facility and is consistent with the process of life and knowledge is thereby apprehended as right and good.
Under the canon of conduct imposed by the instinct of workmanship, efficiency, serviceability, commends itself, and inefficiency or futility is odious. We contemplate our own conduct and that of our neighbors, and pass judgment of complacency or of dispraise. The degree of effectiveness with which we live up to the accepted standard of efficiency in great measure determines our contentment with ourselves and our situation.
Sensitiveness to rebuke or approval is a matter of selective necessity under the circumstances of associated life. Without it no group of persons could carry on a collective life in a material environment that requires shaping to our ends.
Under the guidance of this taste for good work, we are compared with one another and with the accepted ideas of efficiency and are rated and graded by the common sense of our fellows according to conventional schemes of merit and demerit. The visible achievement of one of us is, therefore, compared with that of another, and the award of esteem comes habitually to rest on an invidious comparison of persons instead of on the immediate bearing of the given line of conduct upon the approved end of action. The ground of esteem in this way shifts from a direct appreciation of the expediency of the conduct to a comparison of the abilities of different agents. Instead of a valuation of serviceability, there is a gauging of capability on the ground of visible success. It becomes the proximate end of effort to put forth evidence of power, rather than to achieve an impersonal end for its own sake, simply as an item of human use.
Over time, aggression and predatory behavior creep in, as a way of marking success…
With the increasing density of population that follows from a heightened industrial efficiency, the group passes, by force of circumstances, from the archaic condition of poverty-stricken peace to a stage of predatory life. When a group emerges into this predatory phase of its development, the employments which most occupy men’s attention are employments that involve exploit. The most serious concern of the group, and at the same time the direction in which the most spectacular effect may be achieved by the individual, is conflict with men and beasts. The assertion of a strong hand, successful aggression, usually of a destructive character, becomes the accepted basis of repute. The dominant life interest of the group throws its strong light upon this creditable employment of force and sagacity, and the other, obscurer ways of serving the group’s life fall into the background. The guiding animus of the group becomes a militant one, and men’s actions are judged from the standpoint of the fighting man.
As the predatory culture reaches a fuller development, there comes a distinction between employments. Prowess comes near being recognized as the sole virtue. Other employments, in which people are occupied with tamely shaping inert materials to human use, become unworthy and end with becoming debasing. The tame employments, those that involve no obvious destruction of life and no spectacular coercion of refractory antagonists, fall into disrepute and are relegated to those members of the community who are defective in predatory capacity; those who are lacking in massiveness, agility, or ferocity. Therefore, the able-bodied barbarian of the predatory culture, who is at all mindful of his good name, severely leaves all uneventful drudgery to the women and children of the group. He puts in his time in the manly arts of war and devotes his talents to devising ways and means of disturbing the peace. That way lies honor.
In the barbarian scheme of life, the peaceable, industrial employments are women’s work. They imply defective force, incapacity for aggression or devastation, and are therefore not of good report. In this way industrial occupations fall under a polite odium and are apprehended to be substantially ignoble. They are unsportsmanlike. Labor carries a taint, and all contamination from vulgar employments must be shunned by self-respecting men.
Where the predatory culture had developed in full consistence, the common-sense apprehension that labor is ignoble has developed into the further refinement that labor is wrong. In the further cultural development, when some wealth has been accumulated and the members of the community fall into a servile class on the one hand and a leisure class on the other, the tradition that labor is ignoble gains an added significance. It is not only a mark of inferior force, but it is also a perquisite of the poor. This is the situation today .
There is no remedy for this kind of irksomeness, short of a subversion of that cultural structure on which our canons of decency rest. Appeal may of course be made to taste and conscience to set aside the conventional aversion to labor; such an appeal is made from time to time by well-meaning persons, and some fitful results have been achieved in that way. But the commonplace, common-sense person is bound by the deliverances of common-sense decorum on this head – the heritage of an unbroken cultural line of descent that runs back to the beginning.
Questions for Contemplation and Discussion
- Is work irksome to us? Compare and contrast your answer with that provided by Veblen.
- Many of the first sociological theorists made comparisons between human animals and non-human animals. What sets us apart as a species, according to Veblen? How does this compare to what Marx thought set humans apart? What other comparisons can be made here between Veblen and Marx? Do you think Veblen had read Marx? Explain.
- What is the “instinct of workmanship”? Why has Veblen chosen the word “instinct” to define this phenomenon?
- Veblen was known primarily as an economist, and economics is the only social science he mentions in this article. What makes him a sociologist? Find passages and assertions in support.
- In this article, Veblen asserts a sociology of knowledge, of how we come to think as we do, about the things we do. According to him, what is the source of our knowledge? What is the relationship between our thoughts, beliefs, values and the context (social and environmental) in which we live? Further, what is the source of our ethical values and social norms?
- Veblen also puts forth a theory about how the gendered division of labor emerged, and with what consequences for relations between the sexes. What does he say? How does this compare with Durkheim?
- What are the social and cultural consequences of seeing manual labor as inferior to exploitation? Does Veblen offer any solutions here?
-
A note on gender-neutral language here. In most cases, where Veblen uses “man” to denote all humans, I have changed the text to reflect this (most of the time, adopting “we” and “us” instead of the generic “men”). However, in cases where it is obvious that women were not encompassed by the word, such as “sportsmanship”, the original term has been retained. In the case of workmanship, the original has sometimes been retained (as here, in the title), and at times changed to a more generic work, labor, or craft, as more appropriate to the intended meaning. ↵


1.1: Introduction
selected template will load here
This action is not available.
Buildings are constructed to support and protect activity and artifacts. Buildings may incorporate natural structures, such as the document storage and server farms housed in rooms within a large, underground limestone mine owned by Iron Mountain in Pennsylvania. While human buildings are quite prominent, many animals engage in building activities for nests, hives, etc.
While buildings may be relatively simple structures, most include other systems providing quite extensive functionality, including:
Charging stations for battery electric and plug-in vehicles.
Buildings also have systems for emergencies and security. Fire alarms are often required by regulation. Signage for evacuation and emergency lighting is common. First aid supplies are common. Video cameras for security purposes are often installed.