1.6: Legal Considerations and Ethics
Legal Considerations
As discussed earlier in this chapter, nurses can be reprimanded or have their licenses revoked for not appropriately following the Nurse Practice Act in the state they are practicing. Nurses can also be held legally liable for negligence, malpractice, or breach of patient confidentiality when providing patient care.
Negligence and Malpractice
Negligence is a “general term that denotes conduct lacking in due care, carelessness, and a deviation from the standard of care that a reasonable person would use in a particular set of circumstances.” [1] Malpractice is a more specific term that looks at a standard of care, as well as the professional status of the caregiver.” [2]
To prove negligence or malpractice, the following elements must be established in a court of law:
- Duty owed the patient
- Breach of duty owed the patient
- Foreseeability
- Causation
- Injury
- Damages [3]
To avoid being sued for negligence or malpractice, it is essential for nurses and nursing students to follow the scope and standards of practice care set forth by their state’s Nurse Practice Act; the American Nurses Association; and employer policies, procedures, and protocols to avoid the risk of losing their nursing license. Examples of nurses breach of duty that can be viewed as negligence include: [4]
- Failure to Assess: Nurses should assess for all potential nursing problems/diagnoses, not just those directly affected by the medical disease. For example, all patients should be assessed for fall risk and appropriate fall precautions implemented.
- Insufficient monitoring: Some conditions require frequent monitoring by the nurse, such as risk for falls, suicide risk, confusion, and self-injury.
-
Failure to Communicate:
- Lack of documentation: A basic rule of thumb in a court of law is that if an assessment or action was not documented, it is considered not done. Nurses must document all assessments and interventions, in addition to the specific type of patient documentation called a nursing care plan.
- Lack of provider notification: Changes in patient condition should be urgently communicated to the health care provider based on patient status. Documentation of provider notification should include the date, time, and person notified and follow-up actions taken by the nurse.
- Failure to Follow Protocols: Agencies and states have rules for reporting certain behaviors or concerns. For example, a nurse is required to report suspicion of patient, child, or elder abuse based on data gathered during an assessment.
Patient Confidentiality
In addition to negligence and malpractice, patient confidentiality is a major legal consideration for nurses and nursing students. Patient confidentiality is the right of an individual to have personal, identifiable medical information, referred to as protected health information (PHI), kept private. This right is protected by federal regulations called the Health Insurance Portability and Accountability Act (HIPAA). HIPAA was enacted in 1996 and was prompted by the need to ensure privacy and protection of personal health records and data in an environment of electronic medical records and third-party insurance payers. There are two main sections of HIPAA law, the Privacy Rule and the Security Rule. The Privacy Rule addresses the use and disclosure of individuals’ health information. The Security Rule sets national standards for protecting the confidentiality, integrity, and availability of electronically protected health information. HIPAA regulations extend beyond medical records and apply to patient information shared with others. Therefore, all types of patient information should only be shared with health care team members who are actively providing care to them.
How do HIPAA regulations affect you as a student nurse? You are required to adhere to HIPAA guidelines from the moment you begin to provide patient care. Nursing students may be disciplined or expelled by their nursing program for violating HIPAA. Nurses who violate HIPAA rules may be fired from their jobs or face lawsuits. See the following box for common types of HIPAA violations and ways to avoid them.
- Gossiping in the hallways or otherwise talking about patients where other people can hear you. It is understandable that you will be excited about what is happening when you begin working with patients and your desire to discuss interesting things that occur. As a student, you will be able to discuss patient care in a confidential manner behind closed doors with your instructor. However, as a health care professional, do not talk about patients in the hallways, elevator, breakroom, or with others who are not directly involved with that patient’s care because it is too easy for others to overhear what you are saying.
- Mishandling medical records or leaving medical records unsecured. You can breach HIPAA rules by leaving your computer unlocked for anyone to access or by leaving written patient charts in unsecured locations. You should never share your password with anyone else. Make sure that computers are always locked with a password when you step away from them and paper charts are closed and secured in an area where unauthorized people don’t have easy access to them. NEVER take records from a facility or include a patient’s name on paperwork that leaves the facility.
- Illegally or unauthorized accessing of patient files. If someone you know, like a neighbor, coworker, or family member is admitted to the unit you are working on, do not access their medical record unless you are directly caring for them. Facilities have the capability of tracing everything you access within the electronic medical record and holding you accountable. This rule holds true for employees who previously cared for a patient as a student; once your shift is over as a student, you should no longer access that patient’s medical records.
- Sharing information with unauthorized people. Anytime you share medical information with anyone but the patient themselves, you must have written permission to do so. For instance, if a husband comes to you and wants to know his spouse’s lab results, you must have permission from his spouse before you can share that information with him. Just confirming or denying that a patient has been admitted to a unit or agency can be considered a breach of confidentiality.
- Information can generally be shared with the parents of children until they turn 18, although there are exceptions to this rule if the minor child seeks birth control, an abortion, or becomes pregnant. After a child turns 18, information can no longer be shared with the parent unless written permission is provided, even if the minor is living at home and/or the parents are paying for their insurance or health care. As a general rule, any time you are asked for patient information, check first to see if the patient has granted permission.
- Texting or e-mailing patient information on an unencrypted device. Only use properly encrypted devices that have been approved by your health care facility for e-mailing or faxing protected patient information. Also, ensure that the information is being sent to the correct person, address, or phone number.
- Sharing information on social media. Never post anything on social media that has anything to do with your patients, the facility where you are working or have clinical, or even how your day went at the agency. Nurses and other professionals have been fired for violating HIPAA rules on social media. [6] , [7] , [8]
Social Media Guidelines
Nursing students, nurses, and other health care team members must use extreme caution when posting to Facebook, Instagram, Twitter, Snapchat, and other social media sites. Information related to patients, patient care, and/or health care agencies should never be posted on social media; health care team members who violate this guideline can lose their jobs and may face legal action and students can be disciplined or expelled from their nursing program. Be aware that even if you think you are posting in a private group, the information can become public.
The American Nurses Association (ANA) has established the following principles for nurses using social media: [9]
- Nurses must not transmit or place online individually identifiable patient information.
- Nurses must observe ethically prescribed professional patient-nurse boundaries.
- Nurses should understand that patients, colleagues, organizations, and employers may view postings.
- Nurses should take advantage of privacy settings and seek to separate personal and professional information online.
- Nurses should bring content that could harm a patient’s privacy, rights, or welfare to the attention of appropriate authorities.
- Nurses should participate in developing organizational policies governing online conduct.
In addition to these principles, the ANA has also provided these tips for nurses and nursing students using social media: [10]
- Remember that standards of professionalism are the same online as in any other circumstance.
- Do not share or post information or photos gained through the nurse-patient relationship.
- Maintain professional boundaries in the use of electronic media. Online contact with patients blurs this boundary.
- Do not make disparaging remarks about patients, employers, or coworkers, even if they are not identified.
- Do not take photos or videos of patients on personal devices, including cell phones.
- Promptly report a breach of confidentiality or privacy.
Read more about the ANA’s Social Media Principles .
View the Social Media Guidelines for Nurses video from the National Council of State Boards of Nursing (NCSBN) on using social media responsibly.
Code of Ethics
In addition to legal considerations, there are also several ethical guidelines for nursing care.
There is a difference between morality, ethical principles, and a code of ethics . Morality refers to “personal values, character, or conduct of individuals within communities and societies.” [11] An ethical principle is a general guide, basic truth, or assumption that can be used with clinical judgment to determine a course of action. Four common ethical principles are beneficence (do good), nonmaleficence (do no harm), autonomy (control by the individual), and justice (fairness). A code of ethics is set for a profession and makes their primary obligations, values, and ideals explicit.
The American Nursing Association (ANA) guides nursing practice with the Code of Ethics for Nurses. [12] This code provides a framework for ethical nursing care and a guide for decision-making. The Code of Ethics for Nurses serves the following purposes:
- It is a succinct statement of the ethical values, obligations, duties, and professional ideals of nurses individually and collectively.
- It is the profession’s nonnegotiable ethical standard.
- It is an expression of nursing’s own understanding of its commitment to society. [13]
The ANA Code of Ethics contains nine provisions. See a brief description of each provision in the following box.
The nine provisions of the ANA Code of Ethics are briefly described below. The full code is available to read for free at Nursingworld.org .
Provision 1: The nurse practices with compassion and respect for the inherent dignity, worth, and unique attributes of every person.
Provision 2: The nurse’s primary commitment is to the patient, whether an individual, family, group, community, or population.
Provision 3: The nurse promotes, advocates for, and protects the rights, health, and safety of the patient.
Provision 4: The nurse has authority, accountability, and responsibility for nursing practice; makes decisions; and takes action consistent with the obligation to promote health and to provide optimal care.
Provision 5: The nurse owes the same duties to self as to others, including the responsibility to promote health and safety, preserve wholeness of character and integrity, maintain competence, and continue personal and professional growth.
Provision 6: The nurse, through individual and collective effort, establishes, maintains, and improves the ethical environment of the work setting and conditions of employment that are conducive to safe, quality health care.
Provision 7: The nurse, in all roles and settings, advances the profession through research and scholarly inquiry, professional standards development, and the generation of both nursing and health policy.
Provision 8: The nurse collaborates with other health professionals and the public to protect human rights, promote health diplomacy, and reduce health disparities.
Provision 9: The profession of nursing, collectively through its professional organizations, must articulate nursing values, maintain the integrity of the profession, and integrate principles of social justice into nursing and health policy.
The ANA Center for Ethics and Human Rights
In addition to publishing the Code of Ethics, the ANA Center for Ethics and Human Rights was established to help nurses navigate ethical and value conflicts and life-and-death decisions, many of which are common to everyday practice.
Check your knowledge with the following questions:
Query \(\PageIndex{1}\)
Query \(\PageIndex{2}\)
- Missouri Department of Health & Senior Services. (n.d.). Negligence and malpractice. https://health.mo.gov/living/lpha/phnursing/negligence.php#:~:text=Negligence%20is%3A,a%20particular%20set%20of%20circumstances. ↵
- Missouri Department of Health & Senior Services. (n.d.). Negligence and malpractice . https://health.mo.gov/living/lpha/phnursing/negligence.php#:~:text=Negligence%20is%3A,a%20particular%20set%20of%20circumstances . ↵
- Missouri Department of Health & Senior Services. (n.d.). Negligence and malpractice. https://health.mo.gov/living/lpha/phnursing/negligence.php#:~:text=Negligence%20is%3A,a%20particular%20set%20of%20circumstances . ↵
- Vera, M. (2020). Nursing care plan (NCP): Ultimate guide and database . https://nurseslabs.com/nursing-care-plans/#:~:text=Collaborative%20interventions%20are%20actions%20that,to%20gain%20their%20professional%20viewpoint. ↵
- Patterson, A. (2018, July 3). Most common HIPAA violations with examples . Inspired eLearning. https://inspiredelearning.com/blog/hipaa-violation-examples/ ↵
- Karimi, H., & Masoudi Alavi, N. (2015). Florence Nightingale: The mother of nursing. Nursing and Midwifery Studies, 4 (2), e29475. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4557413/ ↵
- American Nurses Association. (n.d.). About ANA . https://www.nursingworld.org/ana/about-ana/ ↵
- American Nurses Association. (n.d.). Scope of practice . https://www.nursingworld.org/practice-policy/scope-of-practice / ↵
- American Nurses Association. (n.d.). Social media . https://www.nursingworld.org/social/ ↵
- American Nurses Association. (n.d.). Social media . https://www.nursingworld.org/social/ ↵
- American Nurses Association. (2015). Code of ethics for nurses with interpretive statements . American Nurses Association. https://www.nursingworld.org/practice-policy/nursing-excellence/ethics/code-of-ethics-for-nurses/coe-view-only/ ↵
- American Nurses Association. (2015). Code of ethics for nurses with interpretive statements . American Nurses Association. https://www.nursingworld.org/practice-policy/nursing-excellence/ethics/code-of-ethics-for-nurses/coe-view-only / ↵
- American Nurses Association. (2015). Code of ethics for nurses with interpretive statements . American Nurses Association. https://www.nursingworld.org/practice-policy/nursing-excellence/ethics/code-of-ethics-for-nurses/coe-view-only/ ↵
- American Nurses Association. (2015). Code of ethics for nurses with interpretive statements . American Nurses Association. https://www.nursingworld.org/practice-policy/nursing-excellence/ethics/code-of-ethics-for-nurses/coe-view-only/ ↵


20.5: Sculpture of the Northern Renaissance
Wood Sculpture in the Northern Renaissance
The sculpture of Veit Stoss and Tilman Riemenschneider show the evolution from the Gothic style to the styles of the Renaissance.
Describe how the sculptures of Veit Stoss and Tilman Riemenschneider mark a transition between the Gothic and Renaissance styles
Key Points
- Veit Stoss and Tilman Riemenschneider are the most famous among the wood carvers of the German-speaking states of the Northern Renaissance . Their long careers covered the transition between the Gothic and Renaissance periods, although their ornament often remained Gothic even after their compositions began to reflect Renaissance principles.
- Veit Stoss was a leading German sculptor, mostly in wood, whose style emphasized pathos and emotion, helped by his virtuoso carving of billowing drapery.
- Tilman Riemenschneider, also known for his wood sculptures, produced work in the Gothic, classical , and Mannerist styles. His tomb of Lorenz von Bibra marks a transition between the Gothic and the classical.
Key Terms
- polychromatic : Multi-colored.
- virtuoso : A person (especially a musician) with masterly ability, technique, or personal style.
The late 15th and early 16th centuries mark the transition in wood sculpture from the International Gothic to a more classical style of the Renaissance. Often seen as a revival of the classical style, the International Gothic is defined by a dignified elegance, which replaces monumentality, along with rich decorative coloring, elongated figures, and flowing lines . It also makes a more practiced use of perspective , modeling, and setting.
Through commerce and individual travel, the classical and Mannerist styles from the Italian peninsula began to make their way north. The classical style recaptures the naturalism of Graeco-Roman art, with attention to features of the body and expressive restraint. Artists working in this style often employed live models to ensure a realistic appearance of their productions. On the other hand, the Mannerist, or Late Renaissance, style jettisoned the live model and aimed for more expressive figures, resulting in heightened emotion and drama and sometime artificially elongated and ill-proportioned bodies. The wood sculptures of the German-speaking states of this time display both Italian styles in addition to more traditional International Gothic elements.
Veit Stoss (1450–1533), Tilman Riemenschneider (1460–1531), and Peter Vischer the Elder (1455–1529) are the most famous among the wood carvers of the German-speaking states of the Northern Renaissance. Their long careers covered the transition between the Gothic and Renaissance periods, although their ornamentation often remained Gothic even after their compositions began to reflect Renaissance principles.
Veit Stoss
Veit Stoss was a leading German sculptor, mostly working in wood, whose career covered the transition between the late Gothic and the Northern Renaissance. His style emphasizes pathos, often conveyed by tilted heads, downcast eyes, expressive or introspective countenances, and dynamic poses. His virtuoso carving of billowing drapery helps to underscore this sense of emotion. He had a large workshop, and in addition to his own works there are a number by pupils. He is best known for the colossal polychromatic altarpiece in St. Mary’s Basilica in Kraków, Poland, produced in collaboration with his son Stanislaw. Completed in 1489, the altarpiece was the largest triptych of its time. Its use of drama through gilding, diagonal lines, and expressive body language convey Stoss’s loyalty to the International Gothic style, while its naturalistic bodies and drapery point to an adoption of Renaissance attributes.
Better known for his polychromatic sculptures, Stoss also left some sculptures unpainted, as became the preferred appearance for a brief time in the early 16th century. This can be seen in Tobias and the Angel (c. 1504–06), a commission for Bamberg Cathedral . Like his painted sculptures, Tobias and the Angel retains Stoss’s signature expressive countenances and body language.
Tilman Riemenschneider
Tilman Riemenschneider was active in Würzburg from 1483. He was one of the most prolific and versatile sculptors of the transition period between late Gothic and Renaissance, a master in stone and limewood. His work is characterized by the expressiveness of the figurines ‘ faces (often shown with an inward look) and by their detailed and richly folded clothing. The emphasis on expression of inner emotions sets Riemenschneider’s work apart from that of his immediate predecessors.
Riemenschneider’s early success as a sculptor was due to the plasticity of his works, with great care in the modeling of garments. This way of sculpting the garments as well as the typical oval faces and almond-shaped eyes were modeled on art from the Upper Rhine region of the 1470s, implying that Riemenschneider might have learned his trade either there or at Ulm. Later works lost some of the volume of the early sculptures, allowing a more efficient production. While his earlier styles convey a strong Gothic influence, his mature style shows Mannerist characteristics.
Reimenschneider’s tomb of Lorenz von Bibra, the Prince-Bishop of Würzburg, is considered a transition work from the Gothic to the Renaissance style. Here, the figure of the interred is depicted with veristic “warts-and-all” realism , shown in his gaunt cheeks and double chin. Such attributes are reminiscent of the Roman Republic. While the top half of his body is masked in his heavy drapery (a hallmark of many medieval styles), the lower half depicts the contours of his legs, acknowledging the body beneath the garments (a hallmark of classicism). Likewise, his hands, drapery folds, sword, and scepter are rendered in a realistic manner.
- Curation and Revision. Provided by : Boundless.com. License : CC BY-SA: Attribution-ShareAlike
- 167px-LorenzvBibrawholegrave.jpg. Provided by : Wikimedia Commons. Located at : commons.wikimedia.org/w/index.php?curid=3073618. License : Public Domain: No Known Copyright
- Gothic altar veit stoss bordercropped. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/File:Gothic_altar_veit_stoss_bordercropped.jpg. License : CC BY-SA: Attribution-ShareAlike
- 423px-GNM-Veit-Stoss-Raphael-Tobias.jpg. Provided by : Wikimedia Commons. Located at : commons.wikimedia.org/w/index.php?curid=15140669%20. License : CC BY-SA: Attribution-ShareAlike
- Veit Stoss altarpiece in Kraku00f3w. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Veit_Stoss_altarpiece_in_Krak%C3%B3w. License : CC BY-SA: Attribution-ShareAlike
- Northern Renaissance. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Northern_Renaissance#Art. License : CC BY-SA: Attribution-ShareAlike
- Tilman Riemenschneider. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Tilman_Riemenschneider. License : CC BY-SA: Attribution-ShareAlike
- Lorenz von Bibra. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Lorenz_von_Bibra. License : CC BY-SA: Attribution-ShareAlike
- International Gothic. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/International_Gothic#Painting_and_sculpture. License : CC BY-SA: Attribution-ShareAlike
- German Art. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/German_art#Sculpture. License : CC BY-SA: Attribution-ShareAlike
- Veit Stoss. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Veit_Stoss. License : CC BY-SA: Attribution-ShareAlike
- Virtuoso. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/virtuoso. License : CC BY-SA: Attribution-ShareAlike


3.1.9: Katherine Mansfield (1888-1923)
The Garden Party New Zealander Modernism Best known for her modernist short stories, Katherine Mansfield was born into a prominent New Zealand family in Wellington in 1888. At 19, she moved to London, where she eventually became part of the Bloomsbury group that included Virginia Woolf and Leonard Woolf; the two later published Mansfield's short stories through their Hogarth Press. Mansfield struggled to balance her ambitions as a writer with a tumultuous love life; she had numerous love affairs with both men and women, and two brief marriages; at the time of her death, she was married to the prominent editor and critic John Middleton Murry, whom she met in 1911 and married in 1918. The last five years of Mansfield's life were dominated by her efforts to find a successful treatment for the tuberculosis that would end her life at the age of 34. Mansfield began writing short stories as a teenager in New Zealand. Her early efforts were marked by a sympathetic presentation of the Maori minority, who were often oppressed by the white colonialists. While she travelled back to New Zealand once as a young adult, most of her adult life was spent in London or travelling on the continent, where she pursued her ambition to write professionally. An accomplished cellist, she acknowledged the influence of music on her writing process. Like other modernist writers, Mansfield is less interested in plot than in the psychology of her characters, who are often frustrated, alienated, and isolated. Depicting the rich inner lives of her characters through interior monologues, she also makes use of free indirect discourse. Also a poet, Mansfield's style is characterized by her use of imagery. In the tightly constructed form of the short story, she is also notable for her frequent use, like Joyce and Woolf, of the epiphany, what Woolf refers to as "a moment of being." Along with "The Daughters of the Late Colonel" and "Miss Brill," "The Garden Party" (1922) is one of Mansfield's best-known short stories. The story is set in Mansfield's home town, Wellington; Laura Sheridan, the protagonist, is preoccupied with all of the details of planning a garden party, including her pleasure in wearing a new hat, when tragedy intervenes in the death of a local tradesman. Even as she considers the poverty in which the carter's wife and family will be left, Sheridan cannot bring herself to cancel the party. Her epiphany at the story's end suggests that she will someday grow more critical of the middle class colonial values that she and her family embody. Consider while reading:
- How does Mansfield characterize Laura?
- How would you describe the relationship between Laura and her mother?
- How is death portrayed in the story?
- What does Mansfield seem to be suggesting about class distinctions during this period?
Written by Anita Turlington


4.3: Rewarding Yourself When You Achieve Your Goals
selected template will load here
This action is not available.
The best part of goal-setting is the reward for achieving your goals. Remember the words “intrinsic” and “extrinsic” from the Motivation chapter and from earlier in this chapter? If you plan both intrinsic and extrinsic rewards for a goal, your motivation will be higher because you’ve thought in advance about the positive outcomes of your hard work. Examples of intrinsic rewards are a sense of accomplishment, pride, satisfaction - all leading toward increased self-efficacy. Examples of extrinsic rewards are going on a trip, dinner at your favorite restaurant and buying something nice for yourself. Intrinsic rewards are more powerful because no one can take those emotions and thoughts away from you, but extrinsic rewards can be fun. Why not plan for both and enjoy your accomplishments!
I made it a goal in college to have a 3.2 minimum GPA and to be active in 3-6 clubs each semester. And addition to academic learning, I also wanted to expand my knowledge on anything and everything, and to become a part of my college community. I took 18-19 credits each semester and had a 90 minute road trip commute to campus, which significantly limited my time for extracurricular activities. I ended up not taking out the necessary time to study for my classes and this hit me hard after I completed my first year and noted that my overall GPA was less than what I had hoped for. I knew I had to control of my situation and made time for both my classes and my extracurricular activities period of physical calendar work best for me. Each night, I spent just a few minutes making a to-do list of items that needed to get done the following day (studying, research, homework, etc.). From that to-do list, I'd create a detailed block time schedule for myself for the following day. I learned that unexpected things happen and that I had to prioritize each item by importance. Sometimes that meant not attending a campus event I had looked forward to because I had to finish a class assignment. So if my car suddenly broke down or there was an accident on my way to the campus library that cut into my study time, that meant I needed to make up that time by not attending a campus event or not going out for lunch with my friends before class. Prioritizing what was important really helped me make the most of my busy schedule and in the end I was glad for it because it helped me stay on track of my goals of getting good grades and being able to partake in college life at the same time.
- Rabyia Ahmed, Academic Advisor


Skip to main content
Minority Studies: A Brief Text: Study Guide for Part IV—Aging
-
Identify and differentiate among various groups of the elderly
-
Young-old
-
Old
-
Old-old
-
Oldest-old
-
Discuss the aging process
-
Biological aging
-
Social aging
-
Identify and discuss social attitudes toward the elderly
-
Age-appropriate roles
-
Age-appropriate behavior
-
Clothing
-
Activity levels
-
Sexuality
-
STDs and the elderly
-
Health and illness
-
Ability of the elderly to care for themselves and others
-
Drug abuse and alcoholism
-
Alzheimer’s disease and dementia
-
Nursing homes
-
The portrayal of the elderly in popular media
-
Use the Internet to find and display statistical information concerning aging and the elderly
-
Use the Internet to find and display statistical information concerning the current legal status of the elderly in the United States
-
Use the Internet to find and display statistical information concerning the historical legal status of the elderly in the United States
-
Discuss ageism and the forms that it takes in the US
-
Use the Internet to find and display information about anti-ageist political and social groups
-
The AARP
-
The Grey Panthers
-
Others
-
Identify and discuss theories of aging
-
Conflict theories of aging
-
Functionalist theories of aging
-
Symbolic interactionist theories of aging
-
Discuss Social Security and Medicare
-
Discuss theories of death and dying
-
Discuss euthanasia from social, political, and legal perspectives
-
Use the Internet to find and display information about euthanasia


<IP_ADDRESS>: Dulce et Decorum est
Dulce et Decorum est
Dulce et Decorum est License: Public DomainBent double, like old beggars under sacks,
Knock-kneed, coughing like hags, we cursed through sludge,
Till on the haunting flares we turned our backs,
And towards our distant rest began to trudge.
Men marched asleep. Many had lost their boots,
But limped on, blood-shod. All went lame, all blind;
Drunk with fatigue; deaf even to the hoots
Of gas-shells dropping softly behind.
Gas! GAS!
Dim through the misty panes and thick green light,
As under a green sea, I saw him drowning.
In all my dreams before my helpless sight
He plunges at me, guttering, choking, drowning.
If in some smothering dreams, you too could pace
Behind the wagon that we flung him in,
And watch the white eyes writhing in his face,
His hanging face, like a devil's sick of sin,
If you could hear, at every jolt, the blood
Come gargling from the froth-corrupted lungs
Bitter as the cud
Of vile, incurable sores on innocent tongues,—
My friend, you would not tell with such high zest
To children ardent for some desperate glory,
The old Lie: Dulce et decorum est
Pro patria mori .


2.3: Evolution of Atomic Theory
-
- Last updated
- Save as PDF
Learning Objectives
By the end of this section, you will be able to:
- Outline milestones in the development of modern atomic theory
- Summarize and interpret the results of the experiments of Thomson, Millikan, and Rutherford
- Describe the three subatomic particles that compose atoms
- Define isotopes and give examples for several elements
If matter is composed of atoms, what are atoms composed of? Are they the smallest particles, or is there something smaller? In the late 1800s, a number of scientists interested in questions like these investigated the electrical discharges that could be produced in low-pressure gases, with the most significant discovery made by English physicist J. J. Thomson using a cathode ray tube. This apparatus consisted of a sealed glass tube from which almost all the air had been removed; the tube contained two metal electrodes. When high voltage was applied across the electrodes, a visible beam called a cathode ray appeared between them. This beam was deflected toward the positive charge and away from the negative charge, and was produced in the same way with identical properties when different metals were used for the electrodes. In similar experiments, the ray was simultaneously deflected by an applied magnetic field, and measurements of the extent of deflection and the magnetic field strength allowed Thomson to calculate the charge-to-mass ratio of the cathode ray particles. The results of these measurements indicated that these particles were much lighter than atoms (Figure \(\PageIndex{1}\)).
Based on his observations, here is what Thomson proposed and why: The particles are attracted by positive (+) charges and repelled by negative (−) charges, so they must be negatively charged (like charges repel and unlike charges attract); they are less massive than atoms and indistinguishable, regardless of the source material, so they must be fundamental, subatomic constituents of all atoms. Although controversial at the time, Thomson’s idea was gradually accepted, and his cathode ray particle is what we now call an electron , a negatively charged, subatomic particle with a mass more than one thousand-times less that of an atom. The term “electron” was coined in 1891 by Irish physicist George Stoney, from “ electr ic i on .”
In 1909, more information about the electron was uncovered by American physicist Robert A. Millikan via his “oil drop” experiments. Millikan created microscopic oil droplets, which could be electrically charged by friction as they formed or by using X-rays. These droplets initially fell due to gravity, but their downward progress could be slowed or even reversed by an electric field lower in the apparatus. By adjusting the electric field strength and making careful measurements and appropriate calculations, Millikan was able to determine the charge on individual drops (Figure \(\PageIndex{2}\)).
Looking at the charge data that Millikan gathered, you may have recognized that the charge of an oil droplet is always a multiple of a specific charge, 1.6 10 −19 C. Millikan concluded that this value must therefore be a fundamental charge—the charge of a single electron—with his measured charges due to an excess of one electron (1 times 1.6 10 −19 C), two electrons (2 times 1.6 10 −19 C), three electrons (3 times 1.6 10 −19 C), and so on, on a given oil droplet. Since the charge of an electron was now known due to Millikan’s research, and the charge-to-mass ratio was already known due to Thomson’s research (1.759 10 11 C/kg), it only required a simple calculation to determine the mass of the electron as well.
Scientists had now established that the atom was not indivisible as Dalton had believed, and due to the work of Thomson, Millikan, and others, the charge and mass of the negative, subatomic particles—the electrons—were known. However, the positively charged part of an atom was not yet well understood. In 1904, Thomson proposed the “plum pudding” model of atoms, which described a positively charged mass with an equal amount of negative charge in the form of electrons embedded in it, since all atoms are electrically neutral. A competing model had been proposed in 1903 by Hantaro Nagaoka , who postulated a Saturn-like atom, consisting of a positively charged sphere surrounded by a halo of electrons (Figure \(\PageIndex{3}\)).
The next major development in understanding the atom came from Ernest Rutherford , a physicist from New Zealand who largely spent his scientific career in Canada and England. He performed a series of experiments using a beam of high-speed, positively charged alpha particles (α particles) that were produced by the radioactive decay of radium; α particles consist of two protons and two neutrons (you will learn more about radioactive decay in the chapter on nuclear chemistry). Rutherford and his colleagues Hans Geiger (later famous for the Geiger counter) and Ernest Marsden aimed a beam of α particles, the source of which was embedded in a lead block to absorb most of the radiation, at a very thin piece of gold foil and examined the resultant scattering of the α particles using a luminescent screen that glowed briefly where hit by an α particle.
What did they discover? Most particles passed right through the foil without being deflected at all. However, some were diverted slightly, and a very small number were deflected almost straight back toward the source (Figure \(\PageIndex{4}\)). Rutherford described finding these results: “It was quite the most incredible event that has ever happened to me in my life. It was almost as incredible as if you fired a 15-inch shell at a piece of tissue paper and it came back and hit you.” 1
Here is what Rutherford deduced: Because most of the fast-moving α particles passed through the gold atoms undeflected, they must have traveled through essentially empty space inside the atom. Alpha particles are positively charged, so deflections arose when they encountered another positive charge (like charges repel each other). Since like charges repel one another, the few positively charged α particles that changed paths abruptly must have hit, or closely approached, another body that also had a highly concentrated, positive charge. Since the deflections occurred a small fraction of the time, this charge only occupied a small amount of the space in the gold foil. Analyzing a series of such experiments in detail, Rutherford drew two conclusions:
- The volume occupied by an atom must consist of a large amount of empty space.
- A small, relatively heavy, positively charged body, the nucleus , must be at the center of each atom.
Link to Learning
View this simulation of the Rutherford gold foil experiment. Adjust the slit width to produce a narrower or broader beam of α particles to see how that affects the scattering pattern.
This analysis led Rutherford to propose a model in which an atom consists of a very small, positively charged nucleus, in which most of the mass of the atom is concentrated, surrounded by the negatively charged electrons, so that the atom is electrically neutral (Figure \(\PageIndex{5}\)). After many more experiments, Rutherford also discovered that the nuclei of other elements contain the hydrogen nucleus as a “building block,” and he named this more fundamental particle the proton , the positively charged, subatomic particle found in the nucleus. With one addition, which you will learn next, this nuclear model of the atom, proposed over a century ago, is still used today.
Link to Learning
The Rutherford Scattering simulation allows you to investigate the differences between a “plum pudding” atom and a Rutherford atom by firing α particles at each type of atom.
Another important finding was the discovery of isotopes. During the early 1900s, scientists identified several substances that appeared to be new elements, isolating them from radioactive ores. For example, a “new element” produced by the radioactive decay of thorium was initially given the name mesothorium. However, a more detailed analysis showed that mesothorium was chemically identical to radium (another decay product), despite having a different atomic mass. This result, along with similar findings for other elements, led the English chemist Frederick Soddy to realize that an element could have types of atoms with different masses that were chemically indistinguishable. These different types are called isotopes —atoms of the same element that differ in mass. Soddy was awarded the Nobel Prize in Chemistry in 1921 for this discovery.
One puzzle remained: The nucleus was known to contain almost all of the mass of an atom, with the number of protons only providing half, or less, of that mass. Different proposals were made to explain what constituted the remaining mass, including the existence of neutral particles in the nucleus. As you might expect, detecting uncharged particles is very challenging, and it was not until 1932 that James Chadwick found evidence of neutrons , uncharged, subatomic particles with a mass approximately the same as that of protons. The existence of the neutron also explained isotopes: They differ in mass because they have different numbers of neutrons, but they are chemically identical because they have the same number of protons. This will be explained in more detail later in this chapter.
Footnotes
- 1 Ernest Rutherford, “The Development of the Theory of Atomic Structure,” ed. J. A. Ratcliffe, in Background to Modern Science , eds. Joseph Needham and Walter Pagel, (Cambridge, UK: Cambridge University Press, 1938), 61–74. Accessed September 22, 2014, https://ia600508.us.archive.org/3/it...e032734mbp.pdf .


11.8E: T Cell Receptors
- Discuss the role of the T cell receptor (TCR)
T lymphocytes have a dual specificity: they recognize polymorphic residues of self major histocompatibility complex (MHC) molecules, which accounts for their MHC restriction; they also recognize residues of peptide antigens displayed by these MHC molecules, which is responsible for their specificity. MHC molecules and peptides form complexes on the surface of antigen presenting cells (APCs). The receptor that recognizes these peptide-MHC complexes is called the T Cell Receptor (TCR). Clones of T cells with different specificities express different TCRs.
The biochemical signals that are triggered in T cells following antigen recognition are transduced not by the TCR itself, but by invariant proteins (CD3, and zeta), which are non-covalently linked to the antigen receptor to form the TCR complex. T cells also express other membrane receptors that do not recognize antigens but participate in responses to antigens; these are collectively called ‘accessory molecules’. The physiologic role of some accessory molecules is to deliver signals to the T cells that function in concert with signals from the TCR complex to fully activate the cell.
The antigen receptor of MHC-restricted CD4 helper T cells and CD8 cytolytic T cell is a heterodimer consisting of two transmembrane polypeptide chains, designated alpha and beta, covalently linked to each other by disulfide bonds. Each alpha and beta chain consists of one variable domain (V), one constant domain (C), a hydrophobic transmembrane region, and a short cytoplasmic region. The V regions of the TCR contain short stretches of amino acids where the variability between different TCRs is concentrated, and these form the hypervariable or complementarity-determining regions (CDRs). The recognition of peptide-MHC complexes is mediated by CDRs formed by both the alpha and beta chains of the TCR.
Key Points
- Many TCRs recognize the same antigen and many antigens are recognized by the same TCR.
- The TCR is composed of two different protein chains (that is, it is a heterodimer). In 95% of T cells, this consists of an alpha (α) and beta (β) chain, whereas in 5% of T cells this consists of gamma and delta (γ/δ) chains.
- When the TCR engages with antigen and MHC, the T lymphocyte is activated through a series of biochemical events mediated by associated enzymes, co- receptors, specialized accessory molecules, and activated or released transcription factors.
Key Terms
- polymorphic: relating to polymorphism (any sense), able to have several shapes or forms
- major histocompatibility complex: MHC is a cell surface molecule that mediate interactions of immune cells with other leukocytes or body cells. MHC determines compatibility of donors for organ transplants as well as one’s susceptibility to an autoimmune disease. In humans, MHC is also called human leukocyte antigen (HLA).


4.4: Motion in Space
-
- Last updated
- Save as PDF
We have now seen how to describe curves in the plane and in space, and how to determine their properties, including their derivatives, where they are smooth, and their unit tangent vectors. All of this leads to the main goal of this chapter, which is the description of motion along plane curves and space curves. We now have all the tools we need; in this section, we put these ideas together and look at how to use them.
Motion Vectors in the Plane and in Space
Our starting point is using vector-valued functions to represent the position of an object as a function of time. All of the following material can be applied either to curves in the plane or to space curves. For example, when we look at the orbit of the planets, the curves defining these orbits all lie in a plane because they are elliptical. However, a particle traveling along a helix moves on a curve in three dimensions.
Definition: Speed, Velocity, and Acceleration
Let \(\vecs r(t)\) be a twice-differentiable vector-valued function of the parameter \(t\) that represents the position of an object as a function of time.
The velocity vector \(\vecs v(t)\) of the object is given by
\[\text{Velocity}\,=\vecs v(t)=\vecs r^{\,\prime}(t). \label{Eq1}\]
The acceleration vector \(\vecs a(t)\) is defined to be
\[\text{Acceleration}\,=\vecs a(t)=\vecs v^{\,\prime}(t)=\vecs r^{\,\prime\prime}(t). \label{Eq2}\]
The speed is defined to be
\[\mathrm{Speed}\,=‖\vecs v(t)‖=‖\vecs r^{\,\prime}(t)‖=\dfrac{ds}{dt}. \label{Eq3}\]
Since \(\vecs{r}(t)\) can be in either two or three dimensions, these vector-valued functions can have either two or three components. In two dimensions, we define \(\vecs{r}(t)=x(t) \hat{\mathbf i}+y(t) \hat{\mathbf j}\) and in three dimensions \(\vecs r(t)=x(t) \hat{\mathbf i}+y(t) \hat{\mathbf j}+z(t) \hat{\mathbf k}\). Then the velocity, acceleration, and speed can be written as shown in the following table.
| Quantity | Two Dimensions | Three Dimensions |
|---|---|---|
| Position | \(\vecs{r}(t)=x(t) \hat{\mathbf i}+y(t) \hat{\mathbf j}\) | \(\vecs{r}(t)=x(t) \hat{\mathbf i}+y(t) \hat{\mathbf j}+z(t) \hat{\mathbf k}\) |
| Velocity | \(\vecs{v}(t)=x^{\,\prime}(t) \hat{\mathbf i}+y^{\,\prime}(t) \hat{\mathbf j}\) | \(\vecs{v}(t)=x^{\,\prime}(t) \hat{\mathbf i}+y^{\,\prime}(t) \hat{\mathbf j}+z^{\,\prime}(t) \hat{\mathbf k}\) |
| Acceleration | \(\vecs{a}(t)=x^{\,\prime\prime}(t) \hat{\mathbf i}+y^{\,\prime\prime}(t) \hat{\mathbf j}\) | \(\vecs{a}(t)=x^{\,\prime\prime}(t) \hat{\mathbf i}+y^{\,\prime\prime}(t) \hat{\mathbf j}+z^{\,\prime\prime}(t) \hat{\mathbf k}\) |
| Speed | \(\|\vecs{v}(t)\|= \sqrt{(x^{\,\prime}(t))^2+(y^{\,\prime}(t))^2}\) | \(\|\vecs{v}(t)\|=\sqrt{(x^{\,\prime}(t))^2+(y^{\,\prime}(t))^2+(z^{\,\prime}(t))^2}\) |
Example \(\PageIndex{1}\): Studying Motion Along a Parabola
A particle moves in a parabolic path defined by the vector-valued function \(\vecs{r}(t)=t^2 \hat{\mathbf i}+ \sqrt{5−t^2} \hat{\mathbf j}\), where \(t\) measures time in seconds.
- Find the velocity, acceleration, and speed as functions of time.
- Sketch the curve along with the velocity vector at time \(t=1\).
- Write the parametric equations of the tangent line to this curve at the point specified by \(t=1\).
Solution
-
We use Equations \ref{Eq1}, \ref{Eq2}, and \ref{Eq3}:
\[ \begin{align*} \vecs v(t) & = \vecs r^{\,\prime}(t)=2t\hat{\mathbf i}−\dfrac{t}{\sqrt{5-t^2}}\hat{\mathbf j} \\ \vecs a(t) & =\vecs v^{\,\prime}(t)=2\hat{\mathbf i}−5(5−t^2)^{-\frac{3}{2}}\hat{\mathbf j} \\ ||\vecs v(t)||&=||\vecs r^{\,\prime}(t)|| \\ &=\sqrt{(2t)^2+\left(-\dfrac{t}{\sqrt{5-t^2}}\right)^2} \\ &=\sqrt{4t^2+\dfrac{t^2}{5-t^2}} \\ &=\sqrt{\dfrac{21t^2-4t^4}{5-t^2}}. \end{align*}\]
-
The graph of \(\vecs r(t)=t^2 \hat{\mathbf i}+ \sqrt{5−t^2} \hat{\mathbf j}\) is a portion of a parabola (Figure \(\PageIndex{1}\)).
When \(t=1\), \(\vecs r(1) = (1)^2 \mathbf{\hat i} + \sqrt{5-(1)^2} \mathbf{\hat j} \quad = \quad \mathbf{\hat i} + \sqrt{4} \mathbf{\hat j} \quad = \quad \mathbf{\hat i} + 2 \mathbf{\hat j}\).
Thus the particle would be located at the point \((1, 2)\) when \(t =1\).
The velocity vector at \(t=1\) is\[\vecs v(1)=\vecs r^{\,\prime}(1)=2(1)\hat{\mathbf i}−\frac{1}{\sqrt{5-1^2}} \hat{\mathbf j} \quad = \quad 2\hat{\mathbf i}−\frac{1}{2}\hat{\mathbf j}\]
and the acceleration vector at \(t=1\) is\[\vecs a(1)=\vecs v^{\,\prime}(1)=2\hat{\mathbf i}−5(5 - 1^2)^{-3/2}\hat{\mathbf j} \quad = \quad 2\hat{\mathbf i}−\frac{5}{8}\hat{\mathbf j}.\]
Notice that the velocity vector is tangent to the path, as is always the case. -
To find the parametric equations of the tangent line, we need the point at which it is tangent to the curve and a direction vector for the line. The point is just what we found above, \((1,2)\). And it turns out that we can use the velocity vector as a direction vector for the tangent line. This is always true, unless the velocity is the zero vector at this point.
So as we found above, \(\vecs v(1) = 2\hat{\mathbf i}−\frac{1}{2}\hat{\mathbf j}.\) This is the direction vector for this tangent line so we can write the parametric equations of the tangent line as: \[ \begin{align*} x &= 1+2t \\ y &= 2 - \frac{1}{2}t \\z &= 0 \end{align*}\]
A corresponding vector parameterization of this tangent line would be: \[ \vecs L(t) = \left(1+2t\right) \mathbf{\hat i} + \left(2 - \frac{1}{2}t\right) \mathbf{\hat j}. \]
Exercise \(\PageIndex{1}\)
A particle moves in a path defined by the vector-valued function \(\vecs r(t)=(t^2−3t)\,\hat{\mathbf i}+(2t−4)\,\hat{\mathbf j}+(t+2)\,\hat{\mathbf k}\), where \(t\) measures time in seconds and where distance is measured in feet. Find the velocity, acceleration, and speed as functions of time.
- Hint
-
Use Equations \ref{Eq1}, \ref{Eq2}, and \ref{Eq3}.
- Answer
-
\[\begin{align*}\vecs v(t)=\vecs{r}^{\,\prime}(t)&=(2t-3)\,\hat{\mathbf i}+2\,\hat{\mathbf j}+\,\hat{\mathbf k}\\ \vecs a(t)&=\vecs v^{\,\prime}(t) =2\,\hat{\mathbf i} \end{align*}\]
\[ ||\vecs{r}^{\,\prime}(t)||=\sqrt{(2t-3)^2+2^2+1^2} =\sqrt{4t^2-12t+14}\]
The units for velocity and speed are feet per second, and the units for acceleration are feet per second squared.
To gain a better understanding of the velocity and acceleration vectors, imagine you are driving along a curvy road. If you do not turn the steering wheel, you would continue in a straight line and run off the road.
Figure \(\PageIndex{2}\): At each point along a road traveled by a car, the velocity vector of the car is tangent to the path traveled by the car.
However, the fact that you must turn the steering wheel to stay on the road indicates that your velocity is always changing (even if your speed is not) because your direction is constantly changing to keep you on the road. As you turn to the right, your acceleration vector also points to the right. As you turn to the left, your acceleration vector points to the left. This indicates that your velocity and acceleration vectors are constantly changing, regardless of whether your actual speed varies (Figure \(\PageIndex{3}\)).
Projectile Motion
Now let’s look at an application of vector functions. In particular, let’s consider the effect of gravity on the motion of an object as it travels through the air, and how it determines the resulting trajectory of that object. In the following, we ignore the effect of air resistance. This situation, with an object moving with an initial velocity but with no forces acting on it other than gravity, is known as projectile motion. It describes the motion of objects from golf balls to baseballs, and from arrows to cannonballs.
First we need to choose a coordinate system. If we are standing at the origin of this coordinate system, then we choose the positive \(y\)-axis to be up, the negative \(y\) - axis to be down, and the positive \(x\) - axis to be forward (i.e., away from the thrower of the object). The effect of gravity is in a downward direction, so Newton’s second law tells us that the force on the object resulting from gravity is equal to the mass of the object times the acceleration resulting from gravity, or \(\vecs F_g=m\vecs a\), where \(\vecs F_g\) represents the force from gravity and \(\vecs a = -g\,\hat{\mathbf j}\) represents the acceleration resulting from gravity at Earth’s surface. The value of \(g\) in the English system of measurement is approximately 32 ft/sec 2 and it is approximately 9.8 m/sec 2 in the metric system. This is the only force acting on the object. Since gravity acts in a downward direction, we can write the force resulting from gravity in the form \(\vecs F_g=−mg\,\hat{\mathbf j}\), as shown in Figure \(\PageIndex{4}\).
Newton’s second law also tells us that \(F=m\vecs{a}\), where \(\vecs a\) represents the acceleration vector of the object. This force must be equal to the force of gravity at all times, so we therefore know that
\[\begin{align*} \vecs F&=\vecs F_g \\ m\vecs{a} &= -mg \,\hat{\mathbf j} \\ \vecs{a} &= -g\,\hat{\mathbf j}. \end{align*}\]
Now we use the fact that the acceleration vector is the first derivative of the velocity vector. Therefore, we can rewrite the last equation in the form
\[\vecs v^{\,\prime}(t) = -g\,\hat{\mathbf j} \]
By taking the antiderivative of each side of this equation we obtain
\[ \vecs v(t) = \int -g \,\hat{\mathbf j}\; dt = -gt\,\hat{\mathbf j} + \vecs C_1 \]
for some constant vector \(\vecs C_1\). To determine the value of this vector, we can use the velocity of the object at a fixed time, say at time \(t=0\). We call this velocity the initial velocity : \(\vecs v(0)=\vecs v_0\). Therefore, \(\vecs v(0)=−g(0)\,\hat{\mathbf j}+\vecs C_1=\vecs v_0\) and \(\vecs C_1= \vecs v_0\). This gives the velocity vector as \(\vecs v(t)=−gt\,\hat{\mathbf j}+\vecs v_0\).
Next we use the fact that velocity \(\vecs{v}(t)\) is the derivative of position \(\vecs{s}(t)\). This gives the equation
\[\vecs s^{\,\prime}(t)=-gt\,\hat{\mathbf j}+\vecs{v}_0. \]
Taking the antiderivative of both sides of this equation leads to
\[\begin{align*} \vecs s(t) &= \int -gt\,\hat{\mathbf j} + \vecs{v}_0 \;dt \\ &= -\dfrac{1}{2}gt^2 \,\hat{\mathbf j} + \vecs{v}_0 t + \vecs{C}_2 \end{align*}\]
with another unknown constant vector \(\vecs{C}_2\). To determine the value of \(\vecs{C}_2\), we can use the position of the object at a given time, say at time \(t=0\). We call this position the initial position : \(\vecs{s}(0)=\vecs{s}_0\). Therefore, \(\vecs{s}(0)=−(1/2)g(0)^2\,\hat{\mathbf j}+\vecs{v}_0(0)+\vecs{C}_2=\vecs{s}_0\). This gives the position of the object at any time as
\[ \vecs{s}(t)=−\dfrac{1}{2}gt^2 \,\hat{\mathbf j}+\vecs{v}_0 t+\vecs{s}_0. \]
Let’s take a closer look at the initial velocity and initial position. In particular, suppose the object is thrown upward from the origin at an angle \(\theta\) to the horizontal, with initial speed \(\vecs{v}_0\). How can we modify the previous result to reflect this scenario? First, we can assume it is thrown from the origin. If not, then we can move the origin to the point from where it is thrown. Therefore, \(\vecs{s}_0=\vecs{0}\), as shown in Figure \(\PageIndex{5}\).
We can rewrite the initial velocity vector in the form \(\vecs{v}_0= v_0 \cos \theta \,\hat{\mathbf i} + v_0 \sin \theta \,\hat{\mathbf j}\). Then the equation for the position function \(\vecs{s}(t)\) becomes
\[\begin{align*} \vecs{s}(t)&=-\dfrac{1}{2} gt^2\,\hat{\mathbf j} + v_0 t \cos\theta \,\hat{\mathbf i} + v_0 t \sin\theta \,\hat{\mathbf j} \\ &= v_0 t \cos\theta\,\hat{\mathbf i} + v_0 t \sin\theta \,\hat{\mathbf j} - \dfrac{1}{2} gt^2\,\hat{\mathbf j} \\ &= v_0 t \cos\theta \,\hat{\mathbf i} + \left(v_0 t \sin\theta - \dfrac{1}{2} gt^2\right)\,\hat{\mathbf j}. \end{align*}\]
The coefficient of \(\hat{\mathbf i}\) represents the horizontal component of \(\vecs{s}(t)\) and is the horizontal distance of the object from the origin at time \(t\) . The maximum value of the horizontal distance (measured at the same initial and final altitude) is called the range \(R\). The coefficient of \(\hat{\mathbf j}\) represents the vertical component of \(\vecs{s}(t)\) and is the altitude of the object at time \(t\) . The maximum value of the vertical distance is the height \(H\).
Example \(\PageIndex{2}\): Motion of a Cannonball
During an Independence Day celebration, a cannonball is fired from a cannon on a cliff toward the water. The cannon is aimed at an angle of 30° above horizontal and the initial speed of the cannonball is 600 ft/sec. The cliff is 100 ft above the water (Figure \(\PageIndex{6}\)).
- Find the maximum height of the cannonball.
- How long will it take for the cannonball to splash into the sea?
- How far out to sea will the cannonball hit the water?
Solution
We use the equation
\[\vecs{s}(t) = v_0 t \cos\theta \,\hat{\mathbf i} + \left(v_0 t \sin\theta - \dfrac{1}{2}gt^2 \right)\,\hat{\mathbf j} \]
with \(\theta=30^\circ \), \(g=32 \dfrac{\text{ft}}{\text{sec}^2}\), and \(v_0=600 \dfrac{\text{ft}}{\text{sec}}\). Then the position equation becomes
\[\begin{align*} \vecs{s}(t) &= 600 t ( \cos 30^\circ)\,\hat{\mathbf i} + \left(600t \sin30^\circ - \dfrac{1}{2}(32)t^2 \right)\,\hat{\mathbf j} \\ &= 300t\sqrt{3} \,\hat{\mathbf i} + \left( 300t - 16t^2 \right)\,\hat{\mathbf j} \end{align*}\]
-
The cannonball reaches its maximum height when the vertical component of its velocity is zero, because the cannonball is neither rising nor falling at that point. The velocity vector is
\[\begin{align*} \vecs{v}(t)&=\vecs s^{\,\prime}(t)\\ &= 300 \sqrt{3} \,\hat{\mathbf i} + (300-32t)\,\hat{\mathbf j} \end{align*} \]
Therefore, the vertical component of velocity is given by the expression \(300−32t\). Setting this expression equal to zero and solving for \(t\) gives \(t=9.375\) sec. The height of the cannonball at this time is given by the vertical component of the position vector, evaluated at \(t=9.375\).\[\begin{align*} \vecs{s}(9.375)=300(9.375)\sqrt{3}\,\hat{\mathbf i}+(300(9.375)−16(9.375)^2)\,\hat{\mathbf j}=4871.39 \,\hat{\mathbf i}+1406.25\,\hat{\mathbf j} \end{align*}\]
Therefore, the maximum height of the cannonball is 1406.39 ft above the cannon, or 1506.39 ft above sea level. -
When the cannonball lands in the water, it is 100 ft below the cannon. Therefore, the vertical component of the position vector is equal to \(−100.\) Setting the vertical component of \(\vecs s(t)\) equal to \(−100\) and solving, we obtain
\[\begin{align*} 300t-16t^2 &= -100 \\ 16t^2-300t-100&=0 \\4t^2-75-25&=0 \\ t&= \dfrac{75\pm \sqrt{(-75)^2}-4(4)(-25) }{2(4)} \\ &= \dfrac{75 \pm \sqrt{6025}}{8} \\ &= \dfrac{75 \pm 5\sqrt{241}}{8} \end{align*}\]
The positive value of \(t\) that solves this equation is approximately 19.08. Therefore, the cannonball hits the water after approximately 19.08 sec. -
To find the distance out to sea, we simply substitute the answer from part (b) into \(\vecs{s}(t)\):
\[\begin{align*} \vecs s(19.08)&=300(19.08)\sqrt{3} \,\hat{\mathbf i}+\left(300(19.08)−16(19.08)^2\right)\,\hat{\mathbf j}\\ &=9914.26\,\hat{\mathbf i}−100.7424\,\hat{\mathbf j} \end{align*}\]
Therefore, the ball hits the water about 9914.26 ft away from the base of the cliff. Notice that the vertical component of the position vector is very close to −100, which tells us that the ball just hit the water. Note that 9914.26 feet is not the true range of the cannon since the cannonball lands in the ocean at a location below the cannon. The range of the cannon would be determined by finding how far out the cannonball is when its height is 100 ft above the water (the same as the altitude of the cannon).
Exercise \(\PageIndex{2}\)
An archer fires an arrow at an angle of 40° above the horizontal with an initial speed of 98 m/sec. The height of the archer is 171.5 cm. Find the horizontal distance the arrow travels before it hits the ground.
- Hint
-
The equation for the position vector needs to account for the height of the archer in meters.
- Answer
-
967.15 m
One final question remains: In general, what is the maximum distance a projectile can travel, given its initial speed? To determine this distance, we assume the projectile is fired from ground level and we wish it to return to ground level. In other words, we want to determine an equation for the range. In this case, the equation of projectile motion is
\[\vecs{s}=v_0 t \cos\theta \,\hat{\mathbf i} + \left(v_0t\sin\theta - \dfrac{1}{2}gt^2 \right)\,\hat{\mathbf j}.\]
Setting the second component equal to zero and solving for \(t\) yields
\[\begin{align*} v_0 t \sin\theta - \dfrac{1}{2}gt^2& =0\\ t\left(v_0 \sin\theta - \dfrac{1}{2}gt\right) &=0 \end{align*}\]
Therefore, either \(t=0\) or \(t=\dfrac{2v_0\sin\theta}{g}\). We are interested in the second value of \(t\), so we substitute this into \(\vecs{s}(t)\), which gives
\[\begin{align*} \vecs{s}\left(\dfrac{2v_0\sin\theta}{g} \right) &= v_0 \left(\dfrac{2v_0\sin\theta}{g} \right) \cos\theta \,\hat{\mathbf i} + \left( v_0\left(\dfrac{2v_0\sin\theta}{g} \right)\sin\theta - \dfrac{1}{2}g\left(\dfrac{2v_0\sin\theta}{g} \right)^2 \right)\,\hat{\mathbf j} \\ &= \left(\dfrac{2v_0^2\sin\theta\cos\theta}{g} \right)\,\hat{\mathbf i} \\ &= \dfrac{v_0^2 \sin2\theta}{g}\,\hat{\mathbf i}. \end{align*}\]
Thus, the expression for the range of a projectile fired at an angle \(\theta\) is
\[R=\dfrac{v_0^2 \sin2\theta}{g}\,\hat{\mathbf i} . \]
The only variable in this expression is \( \theta\). To maximize the distance traveled, take the derivative of the coefficient of \(\hat{\mathbf i}\) with respect to \(\theta\) and set it equal to zero:
\[\begin{align*} \dfrac{d}{d\theta} \left( \dfrac{v_0^2 \sin2\theta}{g} \right)&=0\\ \dfrac{2v_0^2\cos2\theta}{g}&=0\\ \theta=45^\circ \end{align*}\]
This value of \(\theta)\) is the smallest positive value that makes the derivative equal to zero. Therefore, in the absence of air resistance, the best angle to fire a projectile (to maximize the range) is at a 45° angle. The distance it travels is given by
\[\vecs{s}\left(\dfrac{2v_0 \sin 45^\circ}{g} \right)= \dfrac{v_0^2 \sin 90^\circ}{g} \,\hat{\mathbf i} = \dfrac{v_0^2}{g}\,\hat{\mathbf i} \]
Therefore, the range for an angle of 45° is \(\dfrac{v_0^2}{g}\) units.
Key Concepts
- If \(\vecs{r}(t)\) represents the position of an object at time \(t,\) then \(\vecs{r}^{\,\prime}(t)\) represents the velocity and \(\vecs{r}′^{\,\prime}(t)\) represents the acceleration of the object at time \(t.\) The magnitude of the velocity vector is speed.
- Projectile motion can be represented by the vector-valued function \[\vecs{r}(t) = v_0 t \cos\theta \,\hat{\mathbf i} + \left(v_0 t \sin\theta - \dfrac{1}{2}gt^2 \right)\,\hat{\mathbf j}, \] where the initial velocity is given by \(\vecs{v}_0= v_0 \cos \theta \,\hat{\mathbf i} + v_0 \sin \theta \,\hat{\mathbf j}\).
Key Equations
- Velocity \[\vecs{v}(t)=\vecs{r}^{\,\prime}(t) \nonumber\]
- Acceleration \[\vecs{a}(t)=\vecs{v}^{\,\prime}(t)=\vecs{r}′^{\,\prime}(t) \nonumber\]
- Speed \[v(t)=||\vecs{v}(t)||=||\vecs{r}^{\,\prime}(t)||=\dfrac{ds}{dt} \nonumber\]
- Projectile Motion \[\vecs{r}(t) = v_0 t \cos\theta \,\hat{\mathbf i} + \left(v_0 t \sin\theta - \dfrac{1}{2}gt^2 \right)\,\hat{\mathbf j} \]
Glossary
- acceleration vector
- the second derivative of the position vector
- projectile motion
- motion of an object with an initial velocity but no force acting on it other than gravity
- velocity vector
- the derivative of the position vector
Contributors
-
Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org .
-
Edited by Paul Seeburger
Paul Seeburger added finding point \((1, 2)\) when \(t=1\) and added part 3 (finding the equations of the tangent line) in Example \(\PageIndex{1}\).
He also created Figure \(\PageIndex{1}\).


15.7: Handling Missing Values
-
- Last updated
- Save as PDF
There’s one last topic that I want to discuss briefly in this chapter, and that’s the issue of missing data . Real data sets very frequently turn out to have missing values: perhaps someone forgot to fill in a particular survey question, for instance. Missing data can be the source of a lot of tricky issues, most of which I’m going to gloss over. However, at a minimum, you need to understand the basics of handling missing data in R.
single variable case
Let’s start with the simplest case, in which you’re trying to calculate descriptive statistics for a single variable which has missing data. In R, this means that there will be
NA
values in your data vector. Let’s create a variable like that:
> partial <- c(10, 20, NA, 30)
Let’s assume that you want to calculate the mean of this variable. By default, R assumes that you want to calculate the mean using all four elements of this vector, which is probably the safest thing for a dumb automaton to do, but it’s rarely what you actually want. Why not? Well, remember that the basic interpretation of
NA
is “I don’t know what this number is”. This means that
1 + NA = NA
: if I add 1 to some number that I don’t know (i.e., the
NA
) then the answer is
also
a number that I don’t know. As a consequence, if you don’t explicitly tell R to ignore the
NA
values, and the data set does have missing values, then the output will itself be a missing value. If I try to calculate the mean of the
partial
vector, without doing anything about the missing value, here’s what happens:
> mean( x = partial )
[1] NA
Technically correct, but deeply unhelpful.
To fix this, all of the descriptive statistics functions that I’ve discussed in this chapter (with the exception of
cor()
which is a special case I’ll discuss below) have an optional argument called
na.rm
, which is shorthand for “remove NA values”. By default,
na.rm = FALSE
, so R does nothing about the missing data problem. Let’s try setting
na.rm = TRUE
and see what happens:
When calculating sums and means when missing data are present (i.e., when there are
NA
values) there’s actually an additional argument to the function that you should be aware of. This argument is called
na.rm
, and is a logical value indicating whether R should ignore (or “remove”) the missing data for the purposes of doing the calculations. By default, R assumes that you want to keep the missing values, so unless you say otherwise it will set
na.rm = FALSE
. However, R assumes that
1 + NA = NA
: if I add 1 to some number that I don’t know (i.e., the
NA
) then the answer is
also
a number that I don’t know. As a consequence, if you don’t explicitly tell R to ignore the
NA
values, and the data set does have missing values, then the output will itself be a missing value. This is illustrated in the following extract:
> mean( x = partial, na.rm = TRUE )
[1] 20
Notice that the mean is
20
(i.e.,
60 / 3
) and
not
15
. When R ignores a
NA
value, it genuinely ignores it. In effect, the calculation above is identical to what you’d get if you asked for the mean of the three-element vector
c(10, 20, 30)
.
As indicated above, this isn’t unique to the
mean()
function. Pretty much all of the other functions that I’ve talked about in this chapter have an
na.rm
argument that indicates whether it should ignore missing values. However, its behaviour is the same for all these functions, so I won’t waste everyone’s time by demonstrating it separately for each one.
Missing values in pairwise calculations
I mentioned earlier that the
cor()
function is a special case. It doesn’t have an
na.rm
argument, because the story becomes a lot more complicated when more than one variable is involved. What it does have is an argument called
use
which does roughly the same thing, but you need to think little more carefully about what you want this time. To illustrate the issues, let’s open up a data set that has missing values,
parenthood2.Rdata
. This file contains the same data as the original parenthood data, but with some values deleted. It contains a single data frame,
parenthood2
:
> load( "parenthood2.Rdata" )
> print( parenthood2 )
dan.sleep baby.sleep dan.grump day
1 7.59 NA 56 1
2 7.91 11.66 60 2
3 5.14 7.92 82 3
4 7.71 9.61 55 4
5 6.68 9.75 NA 5
6 5.99 5.04 72 6
BLAH BLAH BLAH
If I calculate my descriptive statistics using the
describe()
function
> describe( parenthood2 )
var n mean sd median trimmed mad min max BLAH
dan.sleep 1 91 6.98 1.02 7.03 7.02 1.13 4.84 9.00 BLAH
baby.sleep 2 89 8.11 2.05 8.20 8.13 2.28 3.25 12.07 BLAH
dan.grump 3 92 63.15 9.85 61.00 62.66 10.38 41.00 89.00 BLAH
day 4 100 50.50 29.01 50.50 50.50 37.06 1.00 100.00 BLAH
we can see from the
n
column that there are 9 missing values for
dan.sleep
, 11 missing values for
baby.sleep
and 8 missing values for
dan.grump
.
84
Suppose what I would like is a correlation matrix. And let’s also suppose that I don’t bother to tell R how to handle those missing values. Here’s what happens:
> cor( parenthood2 )
dan.sleep baby.sleep dan.grump day
dan.sleep 1 NA NA NA
baby.sleep NA 1 NA NA
dan.grump NA NA 1 NA
day NA NA NA 1
Annoying, but it kind of makes sense. If I don’t
know
what some of the values of
dan.sleep
and
baby.sleep
actually are, then I can’t possibly
know
what the correlation between these two variables is either, since the formula for the correlation coefficient makes use of every single observation in the data set. Once again, it makes sense: it’s just not particularly
helpful
.
To make R behave more sensibly in this situation, you need to specify the
use
argument to the
cor()
function. There are several different values that you can specify for this, but the two that we care most about in practice tend to be
"complete.obs"
and
"pairwise.complete.obs"
. If we specify
use = "complete.obs"
, R will completely ignore all cases (i.e., all rows in our
parenthood2
data frame) that have any missing values at all. So, for instance, if you look back at the extract earlier when I used the
head()
function, notice that observation 1 (i.e., day 1) of the
parenthood2
data set is missing the value for
baby.sleep
, but is otherwise complete? Well, if you choose
use = "complete.obs"
R will ignore that row completely: that is, even when it’s trying to calculate the correlation between
dan.sleep
and
dan.grump
, observation 1 will be ignored, because the value of
baby.sleep
is missing for that observation. Here’s what we get:
> cor(parenthood2, use = "complete.obs")
dan.sleep baby.sleep dan.grump day
dan.sleep 1.00000000 0.6394985 -0.89951468 0.06132891
baby.sleep 0.63949845 1.0000000 -0.58656066 0.14555814
dan.grump -0.89951468 -0.5865607 1.00000000 -0.06816586
day 0.06132891 0.1455581 -0.06816586 1.00000000
The other possibility that we care about, and the one that tends to get used more often in practice, is to set
use = "pairwise.complete.obs"
. When we do that, R only looks at the variables that it’s trying to correlate when determining what to drop. So, for instance, since the only missing value for observation 1 of
parenthood2
is for
baby.sleep
R will only drop observation 1 when
baby.sleep
is one of the variables involved: and so R keeps observation 1 when trying to correlate
dan.sleep
and
dan.grump
. When we do it this way, here’s what we get:
> cor(parenthood2, use = "pairwise.complete.obs")
dan.sleep baby.sleep dan.grump day
dan.sleep 1.00000000 0.61472303 -0.903442442 -0.076796665
baby.sleep 0.61472303 1.00000000 -0.567802669 0.058309485
dan.grump -0.90344244 -0.56780267 1.000000000 0.005833399
day -0.07679667 0.05830949 0.005833399 1.000000000
Similar, but not quite the same. It’s also worth noting that the
correlate()
function (in the
lsr
package) automatically uses the “pairwise complete” method:
> correlate(parenthood2)
CORRELATIONS
============
- correlation type: pearson
- correlations shown only when both variables are numeric
dan.sleep baby.sleep dan.grump day
dan.sleep . 0.615 -0.903 -0.077
baby.sleep 0.615 . -0.568 0.058
dan.grump -0.903 -0.568 . 0.006
day -0.077 0.058 0.006 .
The two approaches have different strengths and weaknesses. The “pairwise complete” approach has the advantage that it keeps more observations, so you’re making use of more of your data and (as we’ll discuss in tedious detail in Chapter 10 and it improves the reliability of your estimated correlation. On the other hand, it means that every correlation in your correlation matrix is being computed from a slightly different set of observations, which can be awkward when you want to compare the different correlations that you’ve got.
So which method should you use? It depends a lot on why you think your values are missing, and probably depends a little on how paranoid you are. For instance, if you think that the missing values were “chosen” completely randomly 85 then you’ll probably want to use the pairwise method. If you think that missing data are a cue to thinking that the whole observation might be rubbish (e.g., someone just selecting arbitrary responses in your questionnaire), but that there’s no pattern to which observations are “rubbish” then it’s probably safer to keep only those observations that are complete. If you think there’s something systematic going on, in that some observations are more likely to be missing than others, then you have a much trickier problem to solve, and one that is beyond the scope of this book.


8.4: Dominance
We’ve said a lot about the equivalence relation determined by Cantor’s definition of set equivalence. We’ve also, occasionally, written things like \(|A| < |B|\), without being particularly clear about what that means. It’s now time to come clean. There is actually a (perhaps) more fundamental notion used for comparing set sizes than equivalence — dominance. Dominance is an ordering relation on the class of all sets. One should probably really define dominance first and then define set equivalence in terms of it. We haven’t followed that plan for (at least) two reasons. First, many people may want to skip this section — the results of this section depend on the difficult Cantor-Bernstein-Schröder theorem 1 . Second, we will later take the view that dominance should really be considered to be an ordering relation on the set of all cardinal numbers – i.e. the equivalence classes of the set equivalence relation – not on the collection of all sets. From that perspective, set equivalence really needs to be defined before dominance.
One set is said to dominate another if there is a function from the latter into the former. More formally, we have the following
If A and B are sets, we say “\(A\) dominates \(B\)” and write \(|A| > |B|\) iff there is an injective function \(f\) with domain \(B\) and codomain \(A\).
It is easy to see that this relation is reflexive and transitive. The Cantor-Bernstein-Schröder theorem proves that it is also anti-symmetric — which means dominance is an ordering relation. Be advised that there is an abuse of terminology here that one must be careful about — what are the domain and range of the “dominance” relation? The definition would lead us to think that sets are the things that go on either side of the “dominance” relation, but the notation is a bit more honest, “\(|A| > |B|\)” indicates that the things really being compared are the cardinal numbers of sets (not the sets themselves). Thus anti-symmetry for this relation is
\[(|A| > |B|) ∧ (|B| > |A|) \implies (|A| = |B|).\]
In other words, if \(A\) dominates \(B\) and vice versa, then \(A\) and \(B\) are equivalent sets — a strict interpretation of anti-symmetry for this relation might lead to the conclusion that \(A\) and \(B\) are actually the same set, which is clearly an absurdity.
Naturally, we want to prove the Cantor-Bernstein-Schröder theorem (which we’re going to start calling the C-B-S theorem for brevity), but first it’ll be instructive to look at some of its consequences. Once we have the C-B-S theorem we get a very useful shortcut for proving set equivalences. Given sets \(A\) and \(B\), if we can find injective functions going between them in both directions, we’ll know that they’re equivalent. So, for example, we can use C-B-S to prove that the set of all infinite binary strings and the set of reals in \((0, 1)\) really are equinumerous. (In case you had some remaining doubt. . . )
It is easy to dream up an injective function from \((0, 1)\) to \(\mathbb{F}^∞_2\): just send a real number to its binary expansion, and if there are two, make a consistent choice — let’s say we’ll take the non-terminating expansion.
There is a cute thought-experiment called Hilbert’s Hotel that will lead us to a technique for developing an injective function in the other direction. Hilbert’s Hotel has \(ℵ_0\) rooms. If any countable collection of guests show up there will be enough rooms for everyone. Suppose you arrive at Hilbert’s hotel one dark and stormy evening and the “No Vacancy” light is on — there are already a denumerable number of guests there — every room is full. The clerk sees you dejectedly considering your options, trying to think of another hotel that might still have rooms when, clearly, a very large convention is in town. He rushes out and says “My friend, have no fear! Even though we have no vacancies, there is always room for one more at our establishment.” He goes into the office and makes the following announcement on the PA system. “Ladies and Gentlemen, in order to accommodate an incoming guest, please vacate your room and move to the room numbered one higher. Thank you.” There is an infinite amount of grumbling, but shortly you find yourself occupying room number \(1\).
To develop an injection from \(\mathbb{F}^∞_2\) to \((0, 1)\) we’ll use “room number \(1\)” to separate the binary expansions that represent the same real number. Move all the digits of a binary expansion down by one, and make the first digit \(0\) for (say) the terminating expansions and \(1\) for the non-terminating ones. Now consider these expansions as real numbers — all the expansions that previously coincided are now separated into the intervals \(\left(0, \dfrac{1}{2} \right)\) and \(\left(\dfrac{1}{2}, 1\right)\). Notice how funny this map is, there are now many, many, (infinitely-many) real numbers with no preimages. For instance, only a subset of the rational numbers in \(\left(0, \dfrac{1}{2} \right)\) have preimages. Nevertheless, the map is injective, so C-B-S tells us that \(\mathbb{F}^∞_2\) and \((0, 1)\) are equivalent. There are quite a few different proofs of the C-B-S theorem. The one Cantor himself wrote relies on the axiom of choice. The axiom of choice was somewhat controversial when it was introduced, but these days most mathematicians will use it without qualms. What it says (essentially) is that it is possible to make an infinite number of choices. More precisely, it says that if we have an infinite set consisting of non-empty sets, it is possible to select an element out of each set. If there is a definable rule for picking such an element (as is the case, for example, when we selected the nonterminating decimal expansion whenever there was a choice in defining the injection from \((0, 1)\) to \(\mathbb{F}^∞_2\)) the axiom of choice isn’t needed. The usual axioms for set theory were developed by Zermelo and Frankel, so you may hear people speak of the ZF axioms. If, in addition, we want to specifically allow the axiom of choice, we are in the ZFC axiom system. If it’s possible to construct a proof for a given theorem without using the axiom of choice, almost everyone would agree that that is preferable. On the other hand, a proof of the C-B-S theorem, which necessarily must be able to deal with uncountably infinite sets, will have to depend on some sort of notion that will allow us to deal with huge infinities.
The proof we will present here 2 is attributed to Julius König. König was a contemporary of Cantor’s who was (initially) very much respected by him. Cantor came to dislike König after the latter presented a well-publicized (and ultimately wrong) lecture claiming the continuum hypothesis was false. Apparently, the continuum hypothesis was one of Cantor’s favorite ideas, because he seems to have construed König’s lecture as a personal attack. Anyway. . .
König’s proof of C-B-S doesn’t use the axiom of choice, but it does have its own strangeness: a function that is not necessarily computable — that is, a function for which (for certain inputs) it may not be possible to compute an output in a finite amount of time! Except for this oddity, König’s proof is probably the easiest to understand of all the proofs of C-B-S. Before we get too far into the proof it is essential that we understand the basic setup. The Cantor-Bernstein-Schröder theorem states that whenever \(A\) and \(B\) are sets and there are injective functions \(f : A \implies B\) and \(g : B \implies A\), then it follows that \(A\) and \(B\) are equivalent. Saying \(A\) and \(B\) are equivalent means that we can find a bijective function between them. So, to prove C-B-S, we hypothesize the two injections and somehow we must construct the bijection.
Figure \(8.4.1.\) has a presumption in it — that \(A\) and \(B\) are countable — which need not be the case. Nevertheless, it gives us a good picture to work from. The basic hypotheses, that \(A\) and \(B\) are sets and we have two functions, one from \(A\) into \(B\) and another from \(B\) into \(A\), are shown. We will have to build our bijective function in a piecewise manner. If there is a non-empty intersection between \(A\) and \(B\), we can use the identity function for that part of the domain of our bijection. So, without loss of generality, we can presume that \(A\) and \(B\) are disjoint. We can use the functions \(f\) and \(g\) to create infinite sequences, which alternate back and forth between \(A\) and \(B\), containing any particular element. Suppose \(a ∈ A\) is an arbitrary element. Since \(f\) is defined on all of \(A\), we can compute \(f(a)\). Now since \(f(a)\) is an element of \(B\), and \(g\) is defined on all of \(B\), we can compute \(g(f(a))\), and so on. Thus, we get the infinite sequence
\(a, f(a), g(f(a)), f(g(f(a))), . . .\)
If the element \(a\) also happens to be the image of something under \(g\) (this may or may not be so — since \(g\) isn’t necessarily onto) then we can also extend this sequence to the left. Indeed, it may be possible to extend the sequence infinitely far to the left, or, this process may stop when one of \(f^{−1}\) or \(g^{−1}\) fails to be defined.
\(. . . g^{−1} (f^{−1} (g^{−1} (a))), f^{−1} (g^{−1} (a)), g^{−1} (a), a, f(a), g(f(a)), f(g(f(a))), . . .\)
Now, every element of the disjoint union of \(A\) and \(B\) is in one of these sequences. Also, it is easy to see that these sequences are either disjoint or identical. Taking these two facts together it follows that these sequences form a partition of \(A ∪ B\). We’ll define a bijection \(\phi : A \implies B\) by deciding what it must do on these sequences. There are four possibilities for how the sequences we’ve just defined can play out. In extending them to the left, we may run into a place where one of the inverse functions needed isn’t defined — or not. We say a sequence is an \(A\)-stopper, if, in extending to the left, we end up on an element of \(A\) that has no preimage under \(g\) (see Figure \(8.4.2\)). Similarly, we can define a \(B\)-stopper. If the inverse functions are always defined within a given sequence there are also two possibilities; the sequence may be finite (and so it must be cyclic in nature) or the sequence may be truly infinite.
Finally, here is a definition for \(\phi\).
\( \phi(x)=\left\{ \begin{array}{ll} g^{−1}(x) \;\;\;\;\text{ if } x \text{ is in a } B\text{-stopper}\\ f(x) \;\;\;\;\;\;\;\text{ otherwise} \end{array} \right.\)
Notice that if a sequence is either cyclic or infinite it doesn’t matter whether we use \(f\) or \(g^{−1}\) since both will be defined for all elements of such sequences. Also, certainly \(f\) will work if we are in an \(A\)-stopper. The function we’ve just created is perfectly well-defined, but it may take arbitrarily long to determine whether we have an element of a \(B\)-stopper, as opposed to an element of an infinite sequence. We cannot determine whether we’re in an infinite versus a finite sequence in a prescribed finite number of steps.
Exercises:
How could the clerk at the Hilbert Hotel accommodate a countable number of new guests?
Let \(F\) be the collection of all real-valued functions defined on the real line. Find an injection from \(\mathbb{R}\) to \(F\). Do you think it is possible to find an injection going the other way? In other words, do you think that \(F\) and \(\mathbb{R}\) are equivalent? Explain.
Fill in the details of the proof that dominance is an ordering relation. (You may simply cite the C-B-S theorem in proving anti-symmetry.)
We can inject \(\mathbb{Q}\) into \(\mathbb{Z}\) by sending \(± \dfrac{a}{b}\) to \(±2^a 3^b\). Use this and another obvious injection to (in light of the C-B-S theorem) reaffirm the equivalence of these sets.


12.8.5: Procedure- Turning and Positioning the Patient in Bed
Turning a patient in bed is an important task of the Home Health Aide/Personal Care Aide. It is very important to protect the patient’s skin any time they are turned or moved. Friction (rubbing of two surfaces together, such as the skin rubbing against a sheet) and shearing (when skin sticks to a surface, such as a sheet, and the muscles underneath slide in the direction the body moves) can cause skin breakdown and injury. Friction and shearing can also occur when the patient slides down in bed. To reduce friction and shearing, use a draw sheet or bed pad (these are special linens placed underneath patients to help lift them in bed) when moving a patient. When lifting or moving patients in bed use at least two people to assist as often as possible. Never pull on a patient’s body part to lift or turn them, as this can cause a serious injury.
Patients should be turned every two hours to prevent development of pressure ulcers. There are a number of positions in which patient can be placed. Alternate the positions used to help prevent pressure ulcers and to provide patient comfort. Position pillows under bony areas to prevent pressure ulcers. If patients prefer to stay in one particular position, a pillow should be placed under one buttock or hip to prevent pressure ulcers. This pillow can be rotated to the other side with the next position change. For example, the patient has a pillow placed under their right buttock at 10 am. With the 12 noon turn and position, the pillow can be moved to under the left buttock.
Positions:
-
Fowler’
s
Position:
In this position the patient is placed in a sitting up or upright position.
- This is the position patients should be placed for eating, taking medications, and if they have difficulty with swallowing or breathing.
- Pillows may be placed behind the head, under each arm, under a hip to lift the buttock from the bed, and one under each leg with heels off the surface of the bed.
-
Supine
Position
:
In this position the patient is placed in a lying or reclining position with their back against the bed and their face positioned up.
- Pillows may be placed behind the head, under a buttock, and one under each leg with heels off the surface of the bed.
-
Prone
Position
:
In this position the patient is placed face down with their belly against the bed. Arms should be positioned so they are not underneath the patient. Arms may be gently bent at an angle at the patient’s side.
- Pillows may be placed under calves, feet, arms, and head for comfort and to prevent pressure ulcers.
-
Lateral Position:
In this position the patient is placed on one side.
- Pillows should be placed between their knees, under their head, under the arm that is draped over their top leg, and under the leg that is closest to the bed to keep heels off the surface of the bed.
Pillows should be placed under bony prominences to prevent pressure ulcers.
Examples of Bony Prominences That Are at Risk for Pressure Ulcers:
- Back of the head
- Heels
- Elbows
- Ankles
- Hips
- Sacrum (lower back)
- Coccyx (tailbone)
Before repositioning the patient, Home Health Aides/Personal Care Aides should also offer the use of the restroom, bedpan, or urinal. Patients should also be offered a glass of water at this time to ensure proper hydration. Ensure the patient’s body is in proper alignment once you are done positioning them, and that pillows are used to support body parts.
Procedure: Turning the Patient in Bed
- Explain the procedure to the patient.
- Wash and dry hands. Put on gloves as needed.
- Position yourself on the side of the bed that the patient will be turned to. Adjust the bed to waist height. Lock bed brakes. Lower the head of the bed. This allows gravity to work with, instead of against you. Lower the side rail closest to you. Use proper body mechanics.
- Provide privacy and warmth for the patient. Keep them covered with a blanket.
- Ensure safety of the patient while turning. Watch position of the patient’s head and extremities (arms and legs).
-
Lifting
the
patient
toward
the
head
of
the
bed
:
Before turning or positioning a patient, and when patients slide down in bed, Home Health Aides/Personal Care Aides may have to lift them further up toward the head of the bed.
- For a patient who can assist : Put the head of the bed flat and move any pillows that will be in the path you are moving toward. Lower the side rail on the side you are standing.  Face the patient. The foot closest to where you are moving should be slightly forward and your rear leg slightly backward. Place one arm under the patient’s shoulders and the other under the patient’s thighs. Instruct the patient to bend their knees and push down against the mattress with their feet and hands. You should rock back and forth to begin your motion and start to count out loud. On the count of three, shift your weight to your front leg, and assist the patient to move up in bed, while they push down on the mattress with their hands and feet.
- Using a draw sheet/bed pad to lift the patient with two people : Put the head of the bed flat and move any pillows that will be in the path you are moving toward. Lower the side rails.  Point the foot closest to the head of the bed in that direction. Your other leg should be slightly behind you. Face the patient. Each person should roll the bed pad on their side and grasp it in their hands. Each person should shift their weight from front to back legs by rocking back and forth and then shift their weight toward their front foot (the direction to which they are going) and on the count of three, slide the patient and draw sheet toward the head of the bed.
- Using a draw sheet/ bed pad to lift the patient with a one person assist : Put the head of the bed flat and move any pillows that will be in the path you are moving toward. Stand behind the head of the bed.   Roll and grasp the top of the draw sheet/bed pad on either side of the patient’s shoulders. While pulling the draw sheet up toward you, rock your weight from your front to your back foot. On the count of three, pull the bed pad up toward the head of the bed. If possible, ask the patient to assist by pushing down on the bed with their feet while you lift up.
-
Moving
the
patient
to
the
side
of
the
bed
:
Before turning a patient, and when performing procedures such as a bed bath or
mouth
care, you should position the patient closest to the side of the bed you are on. This prevents you from having to extend your reach, which reduces your personal injury risk. Moving the patient to the side of the bed before turning them also ensures that they will not end up too far on the edge of the other side of the bed, which increases their chance of falling off the bed.
- Moving the patient in segments : To move the patient in segments, you will first move the upper part of the body, and then the trunk area, and finally the legs and feet. To move the upper part of the body, place the hand closest to the patient underneath their neck and grasp their far shoulder (the shoulder furthest from you) from underneath their body. With your other arm, place your hand under the middle of their back, near their waist area. Gently move their upper body toward you. While moving the patient, watch for proper body mechanics. Rock backward and shift your weight to your back leg. To move their torso, place one arm under their waist and one under their thigh. Rock backward, shift your weight to your rear leg, and move them toward you. To move their legs and feet, place your arms under their thighs and calves. Rock backward and shift your weight to your rear leg. Take care not to let their feet drag or get caught underneath them.
- Moving the patient with a draw sheet/ bed pad with a 2 person assist : To move the patient using a draw sheet/bed pad, roll up the draw sheet/bed pad on the side closest to you. This provides you with a grip with which to move the patient. Another worker should do the same on the other side. On the count of three, rock backward, placing your weight on your rear leg, and pull the patient toward you, lifting slightly so the patient does not drag across the bed.
- Moving the patient with a draw sheet/ bed pad with one person : To move the patient using a draw sheet/bed pad, roll up the draw sheet/bed pad on the side closest to you. This provides you with a grip with which to move the patient. One hand should have the draw sheet/bed pad near the patient’s shoulders, with the other hand grasping the draw sheet near the patient’s hips. On the count of three, rock backward, placing your weight on your rear leg, and pull the patient toward you, using the draw sheet/bed pad to move the patient.
- To turn the patient toward you : Stand on the side of the bed to which you want to move the patient. Ensure the side rail is up on the side to which the patient is turning from. Lower the side rail on the side closest to you.Cross the patient’s arms over their chest and bend/cross the patient’s leg nearest you over the farthest leg. Stand with your feet about 12 inches apart, bend at the knees, and place one hand on the patient’s shoulder and one on the patient’s hip. Rock backward and shift your weight to your rear leg. Roll the patient toward you.
- To turn the patient away from you : Stand on the side of the bed opposite of the side you want to move the patient. Ensure the side rail is up on the side to which the patient will be turning. Cross the patient’s arms over their chest. Cross and slightly bend the patient’s leg furthest from you over their lower leg. Stand with your feet about 12 inches apart, bend at the knees, and place one hand on the patient’s shoulder with the other on their hip. Roll the patient in one smooth motion so that their entire body rolls as one. As you roll them, shift your weight from your back to your front leg. You can also use the bed pad to turn the patient toward you by gripping the bed pad near the patient’s shoulder and hip and rolling the patient. Keep the patient in good alignment at all times.
- Place the patient’s head, arms, and legs correctly. The top knee should be flexed with a pillow between the legs. The ankle and foot should be supported. Support arms and legs as necessary with pillows or rolled towels or blankets. Ensure the patient is not lying on their arm by adjusting their shoulder. Position pillows underneath all bony prominences to prevent pressure ulcer development. A pillow should be placed at the small of their back to prevent them from rolling onto their back.
- Adjust pillows and head height for comfort. Ensure the patient’s body is properly aligned.
- Ensure that the patient’s mouth, face, and nose are not obstructed by pillows so they can properly breathe.
- Return bed to proper safe height (lowest position). Ensure side rails are up for patient safety.
- Remove your gloves. Wash and dry your hands.
- Document completion of task and record any changes in condition or behavior. Report and record any observations made about skin condition.


10.7: Inserting Videos in PowerPoint
Learning Outcomes
- Insert an online video into a PowerPoint slide
Karen is creating a work presentation to introduce her team to a few different concepts. She begins by adding in her video slides first. Later she will fill in the rest of the presentation around them. The first video is about using creative commons licensed material. She has already opened a new presentation in PowerPoint and has a video she wishes to insert into her slide.
Placing the cursor on the small box with the six different choices, she selects the Insert Video icon.
The insert video window will appear and since Karen knows the video she wants is on YouTube so she searches in the YouTube box. She searches for Creative Commons and presses the Enter key. Her search results appear and she scrolls through to find the video she wants.
Karen selects the video and clicks the Insert button (she can also double-click on the video to insert). Once the video is inserted, she has a variety of options. Let’s take a look at what she can choose.
- Video Tools. This is a special tab that automatically appears when inserting a video.
- Video Styles. Karen can select a variety of different styles to display the video in this slide.
- Resizing Tool. Karen wants to make the video appear larger on the slide. She can click the Height and Width arrows to increase the size.
- Resizing by Dragging. Alternatively Karen can resize the video by clicking on a corner or side and dragging the cursor to the desired size.
Karen decides she doesn’t like the hard edges on the video. She moves up to the video style ribbon and selects the Visual Effects button.
She selects the Soft Edge > 2.5 Point option. To play the video, she double-clicks on it and waits until she sees the YouTube red icon.
Now when Karen clicks the red button, the video plays.
Contributors and Attributions
- Inserting Videos in PowerPoint. Authored by : Sherri Pendleton. Provided by : Lumen Learning. License : CC BY: Attribution


15.3: Introduction to Staying Connected
selected template will load here
This action is not available.
Now that we understand the tools and the processes by which we connect with other human beings—that is, the social communication model and the methods of communication—we can look at them in terms of this crazy, technological world we live in. Our coworkers, colleagues, and customers are out there waiting to communicate and be communicated with, and they’re doing it in a variety of ways that are vastly different than they were twenty years ago.
More times than not, you don’t even get to choose the tool you’re using to communicate your message, so in this section, we’re going to look at the ways you can digitally communicate with colleagues and customers, the strengths and weaknesses of those tools, and how to best leverage them to be understood.