3.1: Introduction to Light Microscopy
- Understand the importance of the microscope in microbiology.
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Learn and be able to identify the parts of the microscope and their purpose.
- Learn how to calculate total magnification.
- Learn the difference between dry and oil objectives.
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Define magnification, resolution, and contrast.
- Understand the difference between the virtual and real image generated by the microscope.
- Gain experience using a compound light microscope: successfully turning the microscope on, focusing a specimen using oil immersion, and cleaning/caring for the microscope.
Introduction
As early as 460BC, Greek-age philosophers and physicians, like Hippocrates, hypothesized that infection and disease had natural causes found within people and their environments. Their suspicions wouldn't be proven true, however, until the 1600s with the development of the microscope. Antone van Leeuwenhoek, a Dutch merchant, is credited for being the first person to develop a microscope powerful enough to view microscopic organisms in 1675. His observation of pond water and subsequent discovery of microscopic single-celled organisms, which he called animalcules, helped launch the field of microbiology. With the development of more sophisticated and powerful microscopes, microscopy is now a key tool in the microbiology field. Microscopes are used for clinical diagnoses, observation and determination of pathogenic effects on human cells, identification of microbes in environmental samples, university research, and more. (Hartline, 2023) Thus, learning how to use and apply the microscope is a significant portion of this course.
There are three main principles of microscopy: Magnification, Resolution, and Contrast. Magnification is the ability of a lens to enlarge specimen images in comparison to the real, physical object. Objective lenses with different magnification powers are used to observe specimens. For example, an objective lens with a power of 40x will make the specimen appear 40 times larger than its original size. Magnification is a powerful tool for observing microbes, but the use of objective lenses alone is not sufficient enough to produce a clear, usable image. Resolution is the ability to identify two distinct points or regions in a microscopy image, ie the ability to identify separate bacterial cells clearly. High-resolution images will appear clear and sharp, while low-resolution images will be fuzzy. Resolution can be increased by altering the pathway of light through the microscope. The more light centered on a specimen, the clearer the image will be. We can increase the ability of the objective lens to gather light through the use of immersion oil.
Immersion Oil is a unique oil used solely on high power, wet or oil objective lenses (for this course it will be the 100x objective lens). The oil physically touches the microscope slide and objective lens, altering light refraction, and decreasing light scattering. Thus, the oil increases light exposure of the specimen and increases the resolution of the final image. With magnification and resolution perfected, microscopy still encounters the problem of contrast. Contrast is the ability to see the specimen in relation to the color of the background. Most microbes are transparent and difficult to impossible to see on a microscope. To increase the contrast of the microbial cell body versus slide background, special dyes (called stains) are used. These stains will target the external cell body, internal cell structures, or even slide background, adding color to the sample and increasing contrast.
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| Magnification: Through the use of lenses and light, microscopic species, like yeast (pictured above), can be easily observed. | Resolution: Sections of eye tissue were observed by microscopy. The image on the left has poor resolution and is fuzzy. The image on the right has high resolution and individual regions of eye tissue can be observed and labeled. | Contrast: By using stains, like the Gram Stain above, transparent bacterial cells can be easily observed against their background. |
Table 1. These images help demonstrate the three main principles of microbiology. Magnification: increasing the size of microbes, like yeast, using lenses and light. Resolution: the ability to see cells or specimen regions as distinct, separate images. Contrast: the ability to see a specimen clearly against the color of the background.
Figure 1. A) Immersion oil will physically touch the slide/specimen and oil-safe objective. B) Resolution can be increased through the use of oil immersion. Oil helps decrease light scattering/refraction and increase illumination of the specimen, which will make cells and specimen regions clearer.
Parts of a Microscope
In this course, you will use a Compound Light Microscope . This type of microscope uses visible light to observe specimens and is also called a brightfield microscope. Compound light microscopes are made with 2 lenses that magnify or compound their strength. Advantages of this microscope include: the ability to view living and heat prepped specimens, straightforward operation, and the ability to easily update equipment with technological advancements. Understanding the parts of the microscope and how to use them are key to success in this course.
Table 2: Compound Light Microscope Parts and Functions
Table 2: Compound Light Microscope Parts and Functions
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1. Eyepiece and Ocular Lens: The eye piece is a structural component that contains the ocular lens and is the part of the microscope an individual looks into. The ocular lens is a magnifying lens with a power of 10x. |
| 2. Nose Piece: A structural component that contains and rotates the objective lenses. |
| 3. Objective Lenses: Magnifying lenses with different magnification power that observe the specimen. |
| 4. Coarse Adjustment Knob: Adjusts the focus of the microscope by moving the stage in large increments, creating a focused "big picture" where cell details are not yet observable but the overall cell body is clearly focused. Typically used with lower power objectives like the 4x objective lense. |
| 5. Fine Adjustment Knob: Adjusts the focus of the microscope by moving the stage in small increments, creating a sharp image of individual cells and cell details. Typically used with higher power objectives like the 10x, 40x, and 100x objective lenses. |
| 6./9. Mechanical Stage and Stage Clips: Movable metal clips are used to hold a microscope slide in place on a platform called the stage. |
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7. Illuminator: Light source, typically a high-intensity light bulb located below the stage in the base of the microscope. |
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8a. Diaphragm: Adjusts the amount of light that passes between the condenser and specimen, altering specimen brightness. 8b. Condenser: Light passes from the illuminator to the condenser which will focus the light on the specimen. |
Compound Light Microscope Handling and Care Instructions:
Improper handling and care of a microscope can damage the equipment. Make sure you read and understand the following steps to use a microscope safely and correctly.
To take the microscope out of the microscope cabinet:
- Use your dominant hand to hold the arm of the microscope, and your non-dominant hand to hold the base of the microscope. (See Figure 2)
- Carry the microscope upright, flush with your body. That way the ocular lenses located in the eyepiece do not fall out and prevent other microscope structures from shifting.
- Make sure the electrical cord is wrapped or removed from the microscope, avoid tripping on the cord.
To return the microscope to the microscope cabinet:
- Turn off the microscope.
- Remove your slide. Pre-made slides go back in the Bio 225 slide-holder box. Student prepared slides are disposed of in the hard plastic sharps bin.
- Turn the revolving nosepiece to the lowest power objective, the 4x.
- Lower the stage all the way down.
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Use lens cleaner and lens paper to wipe down the 100x oil objective.
- The 4x, 10x, and 40x objectives can be wiped down with dry lens paper. Lens cleaner should not be used on these "dry" objectives.
- Disconnect from power source.
- Wrap cord around the microscope and cover with the protective plastic sheet.
- Place the microscope back in the cabinet and close the cabinet door.
Figure 2. All movable microscopes should be properly supported during transport to prevent equipment damage or personal injury. When moving the microscope, one hand should always support the base and one hand should hold/support the arm.
Total Magnification
The compound light microscope is made with a pair of ocular lenses and multiple objective lenses that have different magnification powers. The Ocular Lenses are located in the eye piece of a microscope, the part of the microscope an individual looks into. The Objective Lenses are connected to a rotating nose piece and magnify the specimen at variable strengths. Since the objective lenses are the magnifying lens that physically observes the specimen, these lenses create the real image . The ocular lens observes the magnified, mirrored image created by the objectives, meaning the ocular lens looks at a virtual image. The virtual image is produced by combining the magnification power of the ocular lens and objective lens, a value called the total magnification power. Total magnification is important as it tells us how many times the original size of the specimen has been magnified. Total magnification should always be included in microscopy photos and even in the observation illustrations students will draw in this course. Total magnification is calculated with the following formula:
Ocular Lens Magnification x Objective Lens Magnification = Total Magnification
The ocular lens of the compound light microscope always has a magnification power of 10x. In this course we will use objective lenses with magnification powers of: 4x, 10x, 40x, and 100x. Thus, total magnification for this course will be:
| Ocular Lens Magnification | Objective Lens Magnification | Total Magnification |
| 10x | 4x | 40x |
| 10x | 10x | 100x |
| 10x | 40x | 400x |
| 10x | 100x | 1,000x |
Table 3. Total magnification is determined by multiplying the power of the ocular lens by the objective lens. In this course, we will use the 4x, 10x, 40x, and 100x objective lens. The power of the ocular lens is always 10x for the compound light microscope.
Focusing A Compound Light Microscope
Read the following steps on how to focus a compound light microscope before completing the microscope activity.
- Follow instructions from "Compound Light Microscope Handling and Care Instructions:" to collect a microscope from the cabinet.
- Place the microscope on a level surface, the lab bench, and plug it in such that the cord does not create a tripping hazard.
- Turn the light on using the light switch located in the microscope base.
- Adjust the eye piece and ocular lenses to best fit your face and viewing comfort.
- Place a slide on the stage of the microscope and properly position the stage clips such that the clips "hug" the slide on either side and are not on top of the slide.
- Move the stage control knobs to position the specimen directly over the path of the light.
- Using the 4x objective, turn the course focus knob to raise the stage.
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Look through the microscope and use the course focus knob to focus on the sample.
- You should be able to focus on the "big picture" at this step. Cell structures and details will not be visible, but groups of cell bodies should be in focus.
- Use the stage control knobs to slowly move the slide left, right, up, or down to find your specimen.
- Switch to the 10x objective lens without changing any other settings on the microscope.
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Using the 10x objective,
turn the fine focus knob only to sharpen the image and focus on cell shape and structure.
- Try not to use the coarse knob, you risk losing your specimen and focus.
- When the 10x image is sharp and focused, move to the 40x objective without changing any other settings on the microscope.
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Using the 40x objective,
turn the fine focus knob only to modify the focus.
- The specimen should mostly be in focus and only minor adjustments should be necessary.
- When the 40x image is sharp and focused, move the nose piece such that the 40x dry and 100x oil objective straddle each end of the slide, meaning no objective lens should be touching the slide. (See Figure 3)
- Add a small amount of immersion oil directly to your specimen/slide then slowly move the 100x oil immersion objective lense into place.
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Using the 100x oil immersion lens,
turn the fine focus knob only to modify the focus.
- The specimen should mostly be in focus and only minor adjustments should be necessary.
- Record observations then follow instructions from "Compound Light Microscope Handling and Care Instructions" to clean the microscope and return it the the cabinet.
NOTE: If your sample goes out of focus and is "lost", clean your objective lenses, wipe the oil off, go back to the 4x objective (Step 7), and re-start the focusing process.
** You CAN NOT switch between the 40x dry and 100x oil lens after oil has been added to the slide. The 40x objective is a "dry" lens meaning no liquids should touch it or it will be damaged. The 100x objective is a "wet" oil lens meaning it is specially crafted to handle liquids. Switching between the 40x and 100x lens after applying oil is going to damage the 40x objective. If you lose your sample when using the 100x objective, you must clean the lens, slide, and re-start the focusing process.
** You can change the light exposure (brightness) by using the the rheostat (a dial) located on the microscope base. This may help you focus the image when switching between objectives.
Figure 3. Oil should never touch the dry 40x objective. To place oil on a slide, move the 40x objective away from the slide, but do not bring the 100x oil objective into place yet. Instead, the 40x and 100x objective should straddle each side of the slide, leaving the specimen uncovered. Add oil to the uncovered specimen then slowly move the 100x oil objective into place.
Prokaryotic Microorganisms
Prokaryotes are ubiquitous, found everywhere, and play a key role in human welfare and disease. By using the microscope, we can visualize these potentially beneficial or pathogenic microbes to better understand their cell structure and function. In this course, we will specifically be using the microscope to observe different bacterial species. All bacterial cells have significant differences in their cell shape, arrangement, and size. These differences make up the bacteria's cell morphology. Bacterial cell morphology contributes to their pathogenicity and can be used to assist with clinical diagnosis and research. Common cell shapes we can see with a microscope are shown in Figure 4.
Figure 4. Magnification of bacteria reveals their cell shape and arrangement, their cell morphology. In this course, students will observe coccus (round cells) and bacillus (rod-shaped cells).
Malaria Diagnostic Tests
Malaria is a blood parasite caused by protozoans in the Plasmodium genus. The parasite is transmitted from mosquitoes to humans and most commonly found in regions of Asia and Africa. The disease is characterized by red blood cell lysis leading to extreme fever, chills, and organ failure or death. The “gold standard” for diagnosing malaria infections is using a microscope to identify the Plasmodium parasite.
Patients suspected to have malaria will need to submit a blood sample for microscopic evaluation. The blood samples will be spread on a microscope in a thin or thick blood smear and stained with dyes like Giemsa. Giemsa stains use multiple dyes, like eosin and methylene blue, to stain human red blood cells and the protozoan parasite. This stain allows us to see if the parasite is present inside the red blood cells.
Microscopic evaluation of malaria has many advantages. The equipment and stains needed to observe malaria are widely available. After staining, determining if the parasite is present in a patient’s blood can be completed within a few hours as the staining procedure is simple and quick to complete. False negatives and false positives are very unlikely to occur in this method as the stain is highly specific for the parasite.
Attributions
"Microbiology Laboratory Manual: Labs, 1.4 Microscopy" by Dr. Rosanna Hartline , LibreTexts: Biology , West Hills College Lemoore is licensed under CC BY-NC-SA 4.0
"Microbiology Laboratory Manual: Labs, 1.5 Get to Know the Microscope and Microbes" by Dr. Rosanna Hartline , LibreTexts: Biology , West Hills College Lemoore is licensed under CC BY-NC-SA 4.0
"Microbiology Textbook: Chapter 1, An Invisible World " by OpenStax , Digital ISBN 13: 978-1-947172-2 is licensed under CC BY 4.0
"Microbiology Textbook: Chapter 2, How We See the Invisible World" by Openstax , Digital ISBN 13: 978-1-947172-23-4 is licensed under CC BY 4
"Microbiology Textbook: Chapter 3, The Cell" by OpenStax , Digital ISBN 13: 978-1-947172-23-4 is licensed under CC BY 4.0
"Microbiology Textbook: Chapter 4, Prokaryotic Diversity" by OpenStax , Digital ISBN 13: 978-1-947172-23-4 is licensed under CC BY 4.0
"Microbiology Textbook: Chapter 25, Circulatory and Lymphatic System Infections" by Openstax , Use of CDC Materials Statement is licensed under CC BY 4.0
"Malaria: Malaria Diagnostic Tests" by Centers for Disease Control and Prevention , Use of CDC Materials Statement is in the Public Domain
"DPDx-Laboratory Identification of Parasite of Public Health Concern" by Centers for Disease Control and Prevention , Use of CDC Materials Statement is in the Public Domain
Image Citations
Figure 1, "Microbiology Textbook: Chapter 2, How We See the Invisible World" by Openstax , Digital ISBN 13: 978-1-947172-23-4 is licensed under CC BY 4
Figure 2, Modified From: "Microscope - school laboratory (1).jpg" by Kvr.lohith, WikiMedia Commons is licensed under CC BY-SA 4.0
Figure 3, Modified From: "µBrews Episode 3 – Focusing & Changing Lenses on a Microscope" by Sui Generis Brewing , Creative Commons Attribution license (reuse allowed) , YouTube is licensed under CC BY
Figure 4, "Microbiology Textbook: Chapter 1, An Invisible World " by OpenStax , Digital ISBN 13: 978-1-947172-2 is licensed under CC BY 4.0
Table 1, Modified From:
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"Microbiology Textbook: Chapter 1, An Invisible World " by OpenStax , Digital ISBN 13: 978-1-947172-2 is licensed under CC BY 4.0
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"Opthalmology AMD Super Resolution Cremer.png" by Andy Nestl, Wikimedia Commons is licensed under CC BY-SA 3.0
- "Gram stain 01.jpg" by Y tambe, WikiMedia Commons is licensed under CC BY-SA 3.0
Table 2, "Microbiology Textbook: Chapter 2, How We See the Invisible World" by Openstax , Digital ISBN 13: 978-1-947172-23-4 is licensed under CC BY 4


2.2: Patient-Centered Care
By the end of this section, you will be able to:
- Define the fundamentals of patient-centered care
- Explain benefits of patient-centered care
- Describe ways in which patient-centered care is a more transparent form of healthcare delivery
Picture this: a patient walks into a hospital or clinic, feeling vulnerable and uncertain about what lies ahead. Suddenly, the patient is greeted by a team of healthcare providers who prioritize their unique needs, preferences, and values above all else. They listen attentively to the patient’s concerns, involve the patient in decision-making, and tailor their care to align with the patient’s goals. That is the power of patient-centered care. It goes beyond treating the physical symptoms: it recognizes that the patient is a whole person with individual emotions, needs, and desires.
Patient-centered care puts the patient at the heart of the healthcare experience, empowering them to actively participate in their own care and leading to better outcomes, increased satisfaction, and a truly collaborative partnership with their healthcare team.
Fundamentals of Patient-Centered Care
A fundamental concept in nursing that places the patient at the forefront of healthcare delivery is patient-centered care (AACN, 2023). It recognizes the importance of understanding and respecting the unique needs, preferences, values, and beliefs of each individual patient. Patient-centered care takes a holistic view of the patient, considering their physical, psychological, emotional, and social well-being. Nurses assess and address not only the patient’s medical condition but also their personal and psychosocial needs. This approach recognizes that patients are unique individuals with their own life experiences, values, and beliefs that impact their health and healthcare decisions.
Patient as a Team Member
Patient-centered care involves a collaborative approach in which nurses work together with patients, their families, and other interdisciplinary team members . Shared decision-making empowers patients to actively participate in their care by providing information, discussing treatment options, and jointly deciding on the best course of action as a member of their own healthcare team. Nurses act as advocates, facilitating communication between patients and other healthcare professionals.
Although the patient is a member of their own healthcare team, patient-centered care ultimately recognizes patient autonomy, or the concept that patients have the right to make decisions about their own health care. Each patient is unique, with distinct physical, emotional, and social needs. Nurses should respect patients’ choices, values, and preferences and involve them in the decision-making process. This requires effective communication and active listening to understand and address the patient’s concerns. Nurses should strive to provide personalized care that considers the patient’s specific circumstances. This involves tailoring interventions, treatment plans, and education to meet the patient’s individual needs, cultural background, and beliefs.
Communication is crucial in patient-centered care. Nurses should use clear and concise language, avoiding medical jargon, and ensure that patients understand the information provided. Active listening skills are essential to comprehend patients’ concerns, fears, and expectations. Open and honest communication builds trust and fosters a therapeutic relationship between the nurse and patient. Respecting privacy and maintaining confidentiality are also integral aspects of patient-centered care.
Scenario: Registered nurse (RN) walks into the patient’s room. The patient just received news from the APRN that he has a diagnosis of chronic obstructive pulmonary disease (COPD) . The RN is coming to talk to the patient about his new diagnosis, provide education, and help him establish an effective treatment plan.
Nurse: Hello, I’m going to be your nurse today. I understand you have COPD. Have you heard of COPD before?
Patient: Hello. I just got the news about having COPD, and honestly, it’s hitting me hard. I used to love hiking and gardening, but now it feels like even simple tasks are a struggle.
Nurse: COPD can be tough, and it’s normal to feel overwhelmed. One thing that’s important is to stay active. Let’s talk about your hobbies and activities. What is the one thing that matters most to you right now?
Patient: Well, I miss hiking more than anything.
Nurse: I can tell how much hiking means to you. Let’s work on a plan to get you back on the trails, one step at a time. First, we’ll focus on managing your symptoms. We’ll go over the medications and how to use them effectively so you can breathe better.
Patient: Thank you. That’s helpful, but I also know I need to quit smoking. I’ve tried a few times, but it has been really difficult.
Nurse: You’re absolutely right. Quitting smoking is vital for managing COPD. I’ll connect you with some written and online resources to make it easier. We can talk to your provider to see if medication is an appropriate option. We’ll also talk about creating a smoke-free environment at home.
Patient: I appreciate that. But, let’s get back to hiking.
Nurse: I hear you.  We’ll start with an exercise plan tailored to your abilities, gradually building your strength and lung capacity. I’ll also provide techniques that you can use if you get short of breath during your hikes.
Patient: All of this sounds really helpful. But honestly, I’m worried I’ll forget some of it. Is there a way we can put this plan down in writing?
Nurse: Of course! We’ll create a personalized COPD action plan that outlines our goals and the steps we’re taking. I’ll make sure you have it in writing, along with some reliable resources for reference.
Patient: That sounds like a great plan. Thanks for being here for me and working on this together.
Nurse: You’re welcome! Remember, I’m here to support you and your goals. Let’s focus on getting you back on those hiking trails and enjoying the outdoors again.
Coordinated Health Care
Patient-centered care extends beyond individual encounters. Nurses should strive to provide continuity of care by promoting coordinated transitions between different healthcare settings and providers in order to improve patient outcomes. This includes clearly communicating patient information, practicing proper handoff procedures, and ensuring that patients feel supported throughout their healthcare journey. By promoting coordination and continuity, nurses help prevent fragmented and disjointed care.
Benefits of Patient-Centered Care
There are multiple benefits of patient-centered care, both for the patients and for other involved parties.
- It enhances the overall patient experience, leading to higher levels of satisfaction. Effective communication promotes trust and rapport between patients and healthcare providers. When patients feel respected, heard, and involved in their care, they are more likely to feel satisfied with the healthcare services they receive.
- It contributes to the job satisfaction of healthcare providers, including nurses. When nurses are able to establish meaningful connections with patients, collaborate in decision-making, and witness positive health outcomes, it can increase their professional fulfillment and motivation.
- It can lead to cost savings in the long run, although patient-centered care requires an investment of time and resources. By actively involving patients in their care and tailoring interventions to their needs, unnecessary tests, procedures, and hospital readmissions may be reduced. Additionally, improved adherence to treatment plans can prevent complications and the need for more expensive interventions.
Another look at the details and benefits of patient-centered care is presented in this brief but informative video.
Improved Patient Outcomes
When patients are actively involved in their care and their preferences are considered, it can lead to improved health outcomes. Patients who feel empowered and engaged are more likely to adhere to treatment plans, manage chronic conditions effectively, and achieve better overall health. Table 2.3 lists some ways in which patient-centered care leads to improved patient outcomes .
| Outcome | Description | Example |
|---|---|---|
| Enhanced treatment adherence | Patients who are actively involved in their care and have a good understanding of their conditions and treatment options are more likely to adhere to prescribed treatments and medication regimens. This improved treatment adherence leads to better health outcomes and improved individual patient well-being. | A 67-year-old patient with type 2 diabetes actively participates in setting her treatment goals, collaborates with her healthcare team to make treatment decisions, receives comprehensive education, and follows a personalized care plan that aligns with her lifestyle. This patient-centered approach enhances her treatment adherence, resulting in better blood sugar control and improved overall health. |
| Increased patient safety | Patient-centered care encourages patients to ask questions, voice concerns, and provide important information about their health history and current conditions. This collaborative approach helps identify and address potential safety risks, reducing medical errors and improving patient safety. | A 43-year-old male patient is scheduled for a surgical procedure, and informed consent is a priority. The medical team thoroughly discusses the surgery, explaining risks and benefits in plain language. The patient’s preferences, like anesthesia and pain management, are considered, and he actively participates in medication reconciliation. This approach enhances patient safety by ensuring clear communication, tailored care, and a comprehensive understanding of the procedure. |
| Enhanced patient communication and trust | Patient-centered care prioritizes clear and effective communication between healthcare providers and patients in order to build trust. Trusting relationships foster a safe environment for patients to ask questions, seek clarification, and actively participate in their care. | A 43-year-old female patient with a new breast cancer diagnosis visits her healthcare provider. During the routine checkup, the provider takes the time to actively listen to the patient’s concerns regarding her cancer treatment options and the lifestyle challenges they encounter. They engage in a collaborative conversation about treatment options. Complex medical information is conveyed in a clear, understandable manner, with visual aids to assist with comprehension. Importantly, the patient’s preferences, including dietary choices and exercise routines, are respected and integrated into the treatment plan. The provider listens patiently and with a nonjudgmental attitude. By respecting the patient’s preferences, a sense of ownership over their health is cultivated, strengthening the patient-provider relationship. The ideal outcome is that the patient experiences enhanced communication and trust with their healthcare provider. |
| Patient empowerment and self-management | By providing education, information, and support, patients are equipped with the knowledge and tools to self-manage their conditions effectively. Empowered patients are more likely to engage in healthy behaviors, make informed decisions, and take responsibility for their own well-being. |
A 73-year-old male patient is diagnosed with high blood pressure. A nurse provides education on practical tools and strategies for self-management. This includes guidance on meal planning, exercise routines, blood pressure monitoring, and medication adherence. The patient is encouraged to actively monitor his blood pressure, record his progress, and attend regular follow-up appointments. Regular follow-up appointments allow him to discuss his achievements, challenges, and any adjustments needed in his care plan.
As a result of this patient-centered approach, the patient becomes increasingly empowered. He possesses the knowledge, skills, and confidence to make informed choices about his daily activities, diet, and medication. With the ongoing support and partnership of the nurse, the patient successfully manages his diabetes, achieving improved blood pressure control and an enhanced overall quality of life. |
| Improved patient satisfaction | Patient-centered care focuses on meeting the unique needs and preferences of each patient. By actively involving patients in their care, listening to their concerns, and respecting their values and choices, nurses help patients feel heard, valued, and respected. They are more likely to have positive experiences with their healthcare providers and increased overall satisfaction. |
A 24-year-old female seeks care for a chronic health condition at a healthcare facility committed to patient-centered care. Upon arrival, the patient is greeted by a warm and empathetic receptionist who takes the time to ensure the patient is checked-in and ready for her appointment on time. During her consultation with the healthcare providers, the patient feels actively engaged in her care. Her treatment plan is developed collaboratively between the patient and her treatment team, incorporating her unique preferences and values. Her providers explain her condition and treatment options clearly, ensuring that she fully comprehends her choices. The patient feels respected and heard throughout this process.
As the patient progresses through her treatment plan, she experiences consistent communication with her healthcare team, including follow-up calls and emails. This proactive approach assures the patient that her well-being is a top priority for the healthcare facility. The patient's overall experience reflects the principles of patient-centered care, where her values, needs, and preferences are honored. As a result, she not only experiences improved health outcomes but also expresses high satisfaction with her healthcare journey. Her confidence in the healthcare system grows, contributing to her commitment to the treatment plan. |
Improved Population Outcomes
Patient-centered care is a healthcare approach that prioritizes the individual patient’s needs, preferences, and values. While its primary focus is on the individual patient, patient-centered care has a broader impact by improving the overall quality of care provided and promoting better health outcomes for the whole community. Table 2.4 lists some ways in which patient-centered care improves population outcomes.
| Outcome | Description | Example |
|---|---|---|
| Enhanced preventive care and health promotion | Nurses work collaboratively with patients to develop personalized preventive strategies and educate them about disease prevention and health maintenance. | A nurse establishes regular community health fairs that offer free or low-cost health screenings, such as blood pressure checks, cholesterol tests, and glucose monitoring. |
| Improved access to care | Nurses advocate for equitable access to healthcare services, facilitate referrals, and help patients navigate the healthcare system. | A nurse helps set up a telemedicine center in a rural community, equipped with technology such as videoconferencing equipment and diagnostic tools like digital stethoscopes and remote monitoring devices. |
| Reduction of health disparity | Nurses recognize that different populations may face varying healthcare challenges and provide individualized care to address these disparities. Nurses understand and respect diverse patient needs and cultural backgrounds. | A nurse creates a community program that collaborates with local farmers’ markets to provide fresh, affordable produce to low-income residents. Cooking classes are also offered to teach healthy meal preparation on a budget. |
| Empowerment of communities | Nurses collaborate with community leaders and organizations, educate communities about health promotion and disease prevention, and facilitate access to resources. | A neighborhood has limited access to fresh, healthy food options and a lack of green spaces. The residents in this community have expressed a desire for better nutrition and a stronger sense of community. A nurse helps establish a community-led urban garden initiative to address these concerns and empower the residents. |
| Feedback and quality improvement | Nurses actively seek patient input, listen to their concerns, and use their feedback to make necessary improvements in care delivery. | A nurse works at a community health clinic. The nurse institutes a program for the clinic where multiple ways are given for patients to provide feedback, including suggestion boxes, online surveys, and in-person feedback sessions. The goal is to make it easy for patients to share their experiences and suggestions. |
Transparency of Healthcare Information
Patient-centered care emphasizes the importance of providing patients with accurate, understandable, and relevant information about their health conditions, treatment options, and care plans. This includes discussing risks, benefits, and alternatives to treatment in a transparent manner. Using transparent communication ensures that patients have the necessary information to make autonomous decisions about their care. This approach ensures that patients are fully informed and actively involved in all relevant decisions. By empowering patients with knowledge, nurses can ensure that they actively participate in decision-making and have a clear understanding of their healthcare journey.
Use of Technology
Modern technology facilitates patient-centered care by making it easier for patients to access their medical records, test results, and other relevant health information. This transparency allows patients to review their healthcare information, verify its accuracy, and actively participate in discussions about their care. Access to medical records enables patients to stay informed, engage in meaningful conversations with healthcare providers, and make informed decisions.
Better Resource Allocation
One broader benefit of transparent communication is that it can contribute to improved resource allocation in a number of ways. Resource allocation refers to the equitable and efficient distribution of healthcare resources to meet the diverse healthcare needs of patients and communities. When patients can easily access information about healthcare services, costs, and outcomes, they can make more informed decisions about their treatment options. This allows the patient to consider the value and cost-effectiveness of different interventions and choose the most appropriate and cost-efficient course of action. In addition, transparent healthcare information enables patients to understand their medical history, test results, and treatment plans. With access to this information, patients can effectively communicate with healthcare providers, reducing the risk of duplicate or otherwise unnecessary tests or procedures. Informed decision-making thus reduces unnecessary utilization of resources on treatments that may not provide significant benefits, thereby optimizing resource allocation.
Transparent communication holds healthcare systems, providers, and payers accountable for their resource allocation decisions. When patients have access to information about costs, quality, and outcomes, the monitoring and evaluation of healthcare services are facilitated. This transparency can drive providers and healthcare systems to optimize their resource allocation, improve efficiency, and deliver value-based care.
Increased transparency of healthcare information also enables patients to compare prices, quality, and outcomes across different providers or healthcare organizations. It allows them to compare such factors as accreditation standards and hospital rankings. This promotes competition and market forces within the healthcare system, as providers strive to deliver high-quality care at competitive prices. This competitive environment can incentivize providers to allocate resources efficiently and improve the value of services they offer.


3: Biopsychology
Biopsychology is the application of the principles of biology to the study of physiological, genetic, and developmental mechanisms of behavior in humans and other animals. It typically investigates at the level of neurons, neurotransmitters, brain circuitry, hormones and the evolutionary and developmental processes that underlie normal and abnormal behavior.
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- 3.1: Prelude to Biopsychology
- A device’s internal workings are often distinct from its user interface on the outside. For example, we don’t think about microchips and circuits when we turn up the volume on a mobile phone; instead, we think about getting the volume just right. Similarly, the inner workings of the human body are often distinct from the external expression of those workings. It is the job of psychologists to find the connection between these—for example, to figure out how the firings of millions of neurons beco
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- 3.2: Human Genetics
- Psychological researchers study genetics in order to better understand the biological basis that contributes to certain behaviors. While all humans share certain biological mechanisms, we are each unique. And while our bodies have many of the same parts—brains and hormones and cells with genetic codes—these are expressed in a wide variety of behaviors, thoughts, and reactions.
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- 3.3: Cells of the Nervous System
- Learning how the cells and organs (like the brain) function, help us understand the biological basis behind human psychology. The nervous system is composed of two basic cell types: glial cells (also known as glia) and neurons. Glial cells, which outnumber neurons ten to one, are traditionally thought to play a supportive role to neurons, both physically and metabolically.
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- 3.4: Parts of the Nervous System
- The nervous system can be divided into two major subdivisions: the central nervous system (CNS) and the peripheral nervous system (PNS), shown in the figure below. The CNS is comprised of the brain and spinal cord; the PNS connects the CNS to the rest of the body. In this section, we focus on the peripheral nervous system; later, we look at the brain and spinal cord.
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- 3.5: The Brain and Spinal Cord
- The brain is a remarkably complex organ comprised of billions of interconnected neurons and glia. It is a bilateral, or two-sided, structure that can be separated into distinct lobes. Each lobe is associated with certain types of functions, but, ultimately, all of the areas of the brain interact with one another to provide the foundation for our thoughts and behaviors. In this section, we discuss the overall organization of the brain and the functions associated with different brain areas.
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- 3.6: The Endocrine System
- The endocrine system consists of a series of glands that produce chemical substances known as hormones. Like neurotransmitters, hormones are chemical messengers that must bind to a receptor in order to send their signal. However, unlike neurotransmitters, which are released in close proximity to cells with their receptors, hormones are secreted into the bloodstream and travel throughout the body, affecting any cells that contain receptors for them.
Contributors and Attributions
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Rose M. Spielman with many significant contributors. The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the creative commons license and may not be reproduced without the prior and express written consent of Rice University. For questions regarding this license, please contact<EMAIL_ADDRESS>.Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at<EMAIL_ADDRESS>.


15.7B: Control of the Pituitary Gland by the Hypothalamus
The pituitary gland consists of the anterior pituitary and the posterior pituitary.
- Explain how the hypothalamus controls the pituitary gland
Key Points
- While the pituitary gland is known as the master endocrine gland, both of its lobes are under the control of the hypothalamus: the anterior pituitary receives its signals from the parvocellular neurons, and the posterior pituitary receives its signals from the magnocellular neurons.
- The pituitary gland is connected by a system of blood vessels to the hypothalamus. This system of blood vessels is known as the hypophyseal portal system, and it allows endocrine communication between the two structures.
- The mechanism for hormone transport via hypothalamoportal vessels involves cells that are regulated by different nuclei in the hypothalamus; for instance, neurons that release neurotransmitters as hormones in the connective link between the pituitary and the brain.
Key Terms
- pituitary gland : An endocrine gland, about the size of a pea, that sits in a small, bony cavity at the base of the brain whose secretions control the other endocrine glands and influence growth, metabolism, and maturation.
- hypothalamus : A region of the forebrain located below the thalamus, forming the basal portion of the diencephalon, that regulates body temperature, some metabolic processes, and governs the autonomic nervous system.
- hypophyseal portal system : The system of blood vessels that link the hypothalamus and the anterior pituitary in the brain.
The pituitary gland consists of two components: the anterior pituitary and the posterior pituitary, and is functionally linked to the hypothalamus by the pituitary stalk (also named the infundibular stem, or simply the infundibulum).
Whilst the pituitary gland is known as the master endocrine gland, both of the lobes are under the control of the hypothalamus: the anterior pituitary receives its signals from the parvocellular neurons, and the posterior pituitary receives its signals from magnocellular neurons.
The anterior lobe of the pituitary receives
hypothalamic-releasing hormones from the hypothalamus that bind with receptors on endocrine cells in the anterior pituitary that regulate the release of adrenal hormones into the circulatory system. Hormones from the hypothalamus are rapidly degraded in the anterior pituitary, which prevents them from entering the circulatory system.
The posterior lobe of the pituitary gland develops as an extension of the hypothalamus. As such, it is not capable of producing its own hormones; instead, it stores hypothalamic hormones for later release into the systemic circulation.
Pituitary gland : The anterior and posterior lobes of the pituitary (hypophysis) gland are shown.


1.3: Structural Organization of the Human Body
By the end of the section, you will be able to:
- Describe the structure of the human body in terms of six levels of organization
- List the eleven organ systems of the human body and identify at least one organ and one major function of each
Before you begin to study the different structures and functions of the human body, it is helpful to consider its basic architecture; that is, how its smallest parts are assembled into larger structures. It is convenient to consider the structures of the body in terms of fundamental levels of organization that increase in complexity: atoms, molecules, organelles, cells, tissues, organs, organ systems, and organisms (Figure \(\PageIndex{1}\)).
The Levels of Organization
To study the smallest level of organization, scientists consider the simplest building blocks of matter: atoms and molecules. The chemical level of organization considers these two building block as atoms bond to form molecules with three dimensional structures. All matter in the universe is composed of one or more unique pure substances called elements, familiar examples of which are hydrogen, oxygen, carbon, nitrogen, calcium, and iron. The smallest unit of any of these pure substances (elements) is an atom. Atoms are made up of subatomic particles such as the proton, electron and neutron. Two or more atoms combine to form a molecule, such as the water molecules, proteins, and sugars found in living things. Molecules are the chemical building blocks of all body structures.
The cellular level is considered when a variety of molecules combine to form the fluid and organelles of a body cell. A cell is the smallest independently functioning unit of a living organism. Even bacteria, which are extremely small, independently-living organisms, have a cellular structure. Each bacterium is a single cell. All living structures of human anatomy contain cells, and almost all functions of human physiology are performed in cells or are initiated by cells. A human cell, such as a smooth muscle cell, typically consists of flexible membranes that enclose cytoplasm , a water-based cellular fluid together with a variety of tiny functioning units called organelles .
The tissue level can be studied when a community of similar cells form a body tissue. A tissue is a group of many similar cells (though sometimes composed of a few related types) that work together to perform a specific function. For example, when many smooth muscle cells come together both structurally and functionally, these cell collectively form a layer of smooth muscle tissue.
An organ is an anatomically distinct structure of the body composed of two or more tissue types, which forms the organ level of organization. Each organ performs one or more specific physiological functions. The human bladder, which is composed of smooth muscle tissue, transitional epithelial tissue, and several types of connective tissue serves the function of storing urine produced by the kidneys.
An organ system level is a group of organs that work together to perform major functions or meet physiological needs of the body. In the organ example above, both the kidneys and the bladder are organs of the urinary system. The kidneys produce urine, which is moved to the bladder by the ureters. Urine can then leave the bladder, and the body, through the urethra. These four organs work together to rid the body of liquid waste.
This book covers eleven distinct organ systems in the human body (Figure \(\PageIndex{2}\) and Figure \(\PageIndex{3}\)). Assigning organs to organ systems can be imprecise since organs that “belong” to one system can also have functions integral to another system. In fact, most organs contribute to more than one system.
The eleven distinct organ systems in the human body covered in this book seen in Figure \(\PageIndex{2}\) and Figure \(\PageIndex{3}\) include:
- The integumentary system functions to enclose internal body structures and is the site of many sensory receptors. Some organs of this system include skin, hair, and nails.
- The skeletal system supports the body and enables movement (with the help of the muscular system). Bones are the major organs of this system.
- The muscular system enables movement (with the help of the skeletal system) and also helps to maintain body temperature. Skeletal muscles are the major organs of this system, which are connected to bones by tendons.
- The nervous system detects and processes sensory information and activates bodily responses. The main organs that perform these functions are the brain, spinal cord, and peripheral nerves.
- The endocrine system is responsible for secreting hormones to regulate bodily processes. Some major organs of the endocrine system include the pituitary gland, thyroid gland, pancreas, adrenal glands, testes, and ovaries.
- The cardiovascular system delivers oxygen and nutrients to tissues while also removing wastes, and helps to equalize temperature in the body. This is accomplished by the heart and blood vessels.
- The lymphatic system functions to return fluids back to the blood and defends the body against pathogens. Major structures and organs of this system include the thymus, lymph nodes, the spleen, and lymphatic vessels.
- The respiratory system delivers oxygen to blood and removes carbon dioxide from the body. The nasal cavity, trachea, and lungs all work together to carry out these functions.
- The digestive system processes food for use by the body and removes wastes from undigested foods. Some organs of the digestive system include the stomach, liver, gallbladder, large intestine, and small intestine.
- The urinary system controls water balance in the body and removes wastes from the blood and excretes them. The kidneys and bladder are major organs of this system.
- The reproductive system produces sex hormones and gametes. The male reproductive system also functions to deliver gametes to the female, while the female reproductive system supports an embryo/fetus until birth and produces milk for an infant. Organs of the male reproductive system include the epididymis and testes. Organs of the female reproductive system include the mammary glands, ovaries, and the uterus.
While only some functions and major organs of each system have been listed above, each of these organ systems will be covered in much greater detail in the following chapters of this book.
The organism level , when many organ systems work harmoniously together to perform the functions of an independent organism, is the highest level of organization in the study of human anatomy. An organism is a living being that has a cellular structure and that can independently perform all physiologic functions necessary for life. In multicellular organisms, including humans, all cells, tissues, organs, and organ systems of the body work together to maintain the life and health of the organism.
Concept Review
Life processes of the human body are maintained at several levels of structural organization. These include the chemical, cellular, tissue, organ, organ system, and the organism level. Higher levels of organization are built from lower levels. Therefore, molecules combine to form cells, cells combine to form tissues, tissues combine to form organs, organs combine to form organ systems, and organ systems combine to form organisms.
Review Questions
Q. The smallest independently functioning unit of an organism is a(n) ________.
A. cell
B. molecule
C. organ
D. tissue
- Answer
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Answer: A
Q. A collection of tissues that performs a specific function is an ________.
A. organ
B. organelle
C. organism
D. organ system
- Answer
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Answer: A
Q. The body system responsible for structural support and movement is the ________.
A. cardiovascular system
B. endocrine system
C. muscular system
D. skeletal system
- Answer
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Answer: D
Critical Thinking Questions
Q. Name the six levels of organization of the human body.
- Answer
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A. Chemical, cellular, tissue, organ, organ system, organism.
Q. The female ovaries and the male testes are a part of which body system? Can these organs be members of more than one organ system? Why or why not?
- Answer
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A. The female ovaries and the male testes are parts of the reproductive system. But they also secrete hormones, as does the endocrine system, therefore ovaries and testes function within both the endocrine and reproductive systems.
Glossary
- cardiovascular system
- organ system that delivers oxygen and nutrients to tissues while also removing wastes, and helps to equalize temperature in the body
- cell
- smallest independently functioning unit of all organisms; in animals, a cell contains cytoplasm, composed of fluid and organelles
- cellular level
- a variety of molecules combine to form the fluid and organelles of a body cell
- chemical level
- atoms bond to form molecules with three dimensional structures
- cytoplasm
- a water-based cell fluid
- digestive system
- organ system that processes food for use by the body and removes wastes from undigested foods
- endocrine system
- organ system that is responsible for secreting hormones to regulate bodily processes
- integumentary system
- organ system that functions to enclose internal body structures and is the site of many sensory receptors
- lymphatic system
- organ system that returns fluids back to the blood and defends the body against pathogens
- muscular system
- organ system that enables movement and maintains body temperature
- nervous system
- organ system that detects and processes sensory information and activates bodily responses
- organ
- functionally distinct structure composed of two or more types of tissues
- organ level
- two or more different tissues combine to form an organ
- organ system level
- group of organs that work together to carry out a particular function
- organelles
- a variety of tiny functioning units within a cell
- organism
- living being that has a cellular structure and that can independently perform all physiologic functions necessary for life
- reproductive system
- organ system that produces sex hormones and gametes
- respiratory system
- organ system that delivers oxygen to blood and removes carbon dioxide from the body
- skeletal system
- organ system that supports the body and enables movement
- tissue
- group of similar or closely related cells that act together to perform a specific function
- tissue level
- a community of similar cells form a body tissue
- urinary system
- organ system that controls water balance in the body and removes wastes from the blood and excretes them
Contributors and Attributions
-
OpenStax Anatomy & Physiology (CC BY 4.0). Access for free at https://openstax.org/books/anatomy-and-physiology


7.3: Polar Lattice
The polar lattice is a lattice dual of the direct lattice, which is the ancestor of the reciprocal lattice. It was introduced by Auguste Bravais in a " mémoire" presented to the Académie de Sciences de Paris on 11 December 1848.
The construction of the polar lattice is essentially the same as that of the reciprocal lattice, but the parameter along a row of the polar lattice is V 2 /3 / d ( hkl ) instead of 1/ d ( hkl ). The polar lattice has thus the same dimensions as the direct lattice, namely Ångstroms, instead of Ångstroms -1 , like the reciprocal lattice.
- The unit cell of the polar lattice has the same volume as that of the direct lattice.
- The scalar product of the basis vectors of the direct and polar lattice is V 2 / 3 δ ij , where δ is Kroneker's tensor and the indices i and j point to the basis vectors.
The polar lattice was introduced to facilitate the morphological study of crystals.


43.0: Prelude to Animal Reproduction and Development
Animal reproduction is necessary for the survival of a species. In the animal kingdom, there are innumerable ways that species reproduce. Asexual reproduction produces genetically identical organisms (clones), whereas in sexual reproduction, the genetic material of two individuals combines to produce offspring that are genetically different from their parents. During sexual reproduction the male gamete (sperm) may be placed inside the female’s body for internal fertilization, or the sperm and eggs may be released into the environment for external fertilization. Seahorses provide an example of the latter. Following a mating dance, the female lays eggs in the male seahorse’s abdominal brood pouch where they are fertilized. The eggs hatch and the offspring develop in the pouch for several weeks.


11.4: Civil Rights in an Affluent Society
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FIgure \(\PageIndex{1}\): This segregated drinking fountain was located on the grounds of the Halifax County courthouse in North Carolina. Photograph, April 1938. Wikimedia .
Education was but one aspect of the nation’s Jim Crow machinery. African Americans had been fighting against a variety of racist policies, cultures, and beliefs in all aspects of American life. And while the struggle for black inclusion had few victories before World War II, the war and the Double V campaign for victory against fascism abroad and racism at home, as well as the postwar economic boom led, to rising expectations for many African Americans. When persistent racism and racial segregation undercut the promise of economic and social mobility, African Americans began mobilizing on an unprecedented scale against the various discriminatory social and legal structures.
While many of the civil rights movement’s most memorable and important moments, such as the sit-ins, the Freedom Rides, and especially the March on Washington, occurred in the 1960s, the 1950s were a significant decade in the sometimes tragic, sometimes triumphant march of civil rights in the United States. In 1953, years before Rosa Parks’s iconic confrontation on a Montgomery city bus, an African American woman named Sarah Keys publicly challenged segregated public transportation. Keys, then serving in the Women’s Army Corps, traveled from her army base in New Jersey back to North Carolina to visit her family. When the bus stopped in North Carolina, the driver asked her to give up her seat for a white customer. Her refusal to do so landed her in jail in 1953 and led to a landmark 1955 decision, Sarah Keys v. Carolina Coach Company , in which the Interstate Commerce Commission ruled that “separate but equal” violated the Interstate Commerce Clause of the U.S. Constitution. Poorly enforced, it nevertheless gave legal coverage for the Freedom Riders years later and motivated further assaults against Jim Crow.
But if some events encouraged civil rights workers with the promise of progress, others were so savage they convinced activists that they could do nothing but resist. In the summer of 1955, two white men in Mississippi kidnapped and brutally murdered fourteen-year-old Emmett Till. Till, visiting from Chicago and perhaps unfamiliar with the “etiquette” of Jim Crow, allegedly whistled at a white woman named Carolyn Bryant. Her husband, Roy Bryant, and another man, J. W. Milam, abducted Till from his relatives’ home, beat him, mutilated him, shot him, and threw his body in the Tallahatchie River. Emmett’s mother held an open-casket funeral so that Till’s disfigured body could make national news. The men were brought to trial. The evidence was damning, but an all-white jury found the two not guilty. Mere months after the decision, the two boasted of their crime, in all of its brutal detail, in Look magazine. “They ain’t gonna go to school with my kids,” Milam said. They wanted “to make an example of [Till]—just so everybody can know how me and my folks stand.” 16 The Till case became an indelible memory for the young black men and women soon to propel the civil rights movement forward.
On December 1, 1955, four months after Till’s death and six days after the Keys v. Carolina Coach Company decision, Rosa Parks refused to surrender her seat on a Montgomery city bus and was arrested. Montgomery’s public transportation system had longstanding rules requiring African American passengers to sit in the back of the bus and to give up their seats to white passengers if the buses filled. Parks was not the first to protest the policy by staying seated, but she was the first around whom Montgomery activists rallied.
Montgomery’s black population, under the leadership of local ministers and civil rights workers, formed the Montgomery Improvement Association (MIA) and coordinated an organized boycott of the city’s buses. The Montgomery Bus Boycott lasted from December 1955 until December 20, 1956, when the Supreme Court ordered their integration. The boycott not only crushed segregation in Montgomery’s public transportation, it energized the entire civil rights movement and established the leadership of the MIA’s president, a recently arrived, twenty-six-year-old Baptist minister named Martin Luther King Jr.
Motivated by the success of the Montgomery boycott, King and other African American leaders looked to continue the fight. In 1957, King helped create the Southern Christian Leadership Conference (SCLC) to coordinate civil rights groups across the South and buoy their efforts organizing and sustaining boycotts, protests, and other assaults against southern Jim Crow laws.
As pressure built, Congress passed the Civil Rights Act of 1957, the first such measure passed since Reconstruction. The act was compromised away nearly to nothing, although it did achieve some gains, such as creating the Department of Justice’s Civil Rights Commission, which was charged with investigating claims of racial discrimination. And yet, despite its weakness, the act signaled that pressure was finally mounting on Americans to confront the legacy of discrimination.
Despite successes at both the local and national level, the civil rights movement faced bitter opposition. Those opposed to the movement often used violent tactics to scare and intimidate African Americans and subvert legal rulings and court orders. For example, a year into the Montgomery bus boycott, angry white southerners bombed four African American churches as well as the homes of King and fellow civil rights leader E. D. Nixon. Though King, Nixon, and the MIA persevered in the face of such violence, it was only a taste of things to come. Such unremitting hostility and violence left the outcome of the burgeoning civil rights movement in doubt. Despite its successes, civil rights activists looked back on the 1950s as a decade of mixed results and incomplete accomplishments. While the bus boycott, Supreme Court rulings, and other civil rights activities signaled progress, church bombings, death threats, and stubborn legislators demonstrated the distance that still needed to be traveled.


7.1: Assess the place of reading in your learning journey
Reading and the online learning journey
Online learning typically requires you to interact with a larger amount of written material than traditional in-class courses. This can benefit your growth as a lifelong learner by developing your skills in selecting relevant reading material, approaching it purposefully, and managing the information you read. Consider the following principles as a guide as you approach reading:
- Not all reading material requires equal time or attention. Unlike a novel, where you give most pages equal time in order to understand the story, much of your professional reading is focused on finding and using relevant information. This means that you may not read every word in available readings. Some information may require a close and careful reading, while other information may be skimmed to find key points.
- Before you begin reading, identify your purpose for reading. What do you need to learn from this reading? This will determine how you approach the reading material.
- Use questions to guide your reading. In the next sections of this module, you will learn a strategy called SQ3R that can guide you through the process of using questions to guide your reading.
- Develop a system for identifying important information and taking notes. You have already explored systems for online information management. Consider how you will mark key learnings in the texts that you read, and organize this information in a form where you can easily access it again.
Go to the next chapter to explore reading strategies in more detail.


6.5.E: Problems on Repeated Differentiation and Taylor Expansions
Complete all details in the proof of Theorem 1. What is the motivation for introducing the auxiliary functions \(h_{t}\) and \(g_{s}\) in this particular way?
Is symbolic "multiplication" in Note 2 always commutative? (See Example (A).) Why was it possible to collect "similar" terms
\[\frac{\partial^{2} f}{\partial x \partial y} d x d y \text { and } \frac{\partial^{2} f}{\partial y \partial x} d y d x\]
in Example (B)? Using (5), find the general formula for \(d^{3} f.\) Expand it!
Carry out the induction in Theorem 2 and Corollary 2. (Use a suitable notation for subscripts: \(k_{1} k_{2} \ldots\) instead of \(j k \ldots\).)
Do Example(C) with \(m=3\) (instead of \(m=2\)) and with \(\vec{p}=(0,0)\). Show that \(R_{m} \rightarrow 0,\) i.e., \(f\) admits a Taylor series about \(\vec{p}.\)
Do it in the following two ways.
(i) Use Theorem 2.
(ii) Expand \(\sin y\) as in Problem 6(a) in Chapter 5, §6, and then multiply termwise by \(x.\)
Give an estimate for \(R_{3}\).
Use Theorem 2 to expand the following functions in powers of \(x-3\) and \(y+2\) exactly (choosing \(m\) so that \(R_{m}=0\)).
(i) \(f(x, y)=2 x y^{2}-3 y^{3}+y x^{2}-x^{3}\);
(ii) \(f(x, y)=x^{4}-x^{3} y^{2}+2 x y-1\);
(iii) \(f(x, y)=x^{5} y-a x y^{5}-x^{3}\).
For the functions of Problem 15 in §4, give their Taylor expansions up to \(R_{2},\) with
\[\vec{p}=\left(1, \frac{\pi}{4}, 1\right)\]
in case (i) and
\[\vec{p}=\left(e, \frac{\pi}{4} e\right)\]
in (ii). Bound \(R_{2}\).
(Generalized Taylor theorem.) Let \(\vec{u}=\vec{x}-\vec{p} \neq \overrightarrow{0}\) in \(E^{\prime}\)(\(E^{\prime}\) need not be \(E^{n}\) or \(C^{n}\)); let \(I=L[\vec{p}, \vec{x}].\) Prove the following statement:
If \(f : E^{\prime} \rightarrow E\) and the derived functions \(D_{\vec{u}}^{i} f(i \leq m)\) are relatively continuous on \(I\) and have \(\vec{u}\)-directed derivatives on \(I-Q\) (\(Q\) countable), then formula (6) and Note 3 hold, with \(d^{i} f(\vec{p} ; \vec{u})\) replaced by \(D_{\vec{u}}^{i} f(\vec{p})\).
[Hint: Proceed as in Theorem 2 without using the chain rule or any partials or components. Instead of (8), prove that \(h^{(i)}(t)=D_{\vec{u}}^{i} f(\vec{p}+t \vec{u})\) on \(J-Q^{\prime}, Q^{\prime}= g^{-1}[Q]\).]
(i) Modify Problem 7 by setting
\[\vec{u}=\frac{\vec{x}-\vec{p}}{|\vec{x}-\vec{p}|}.\]
Thus expand \(f(\vec{x})\) in powers of \(|\vec{x}-\vec{p}|.\)
(ii) Deduce Theorem 2 from Problem 7, using Corollary 2.
Given \(f : E^{2}\left(C^{2}\right) \rightarrow E, f \in C D^{m}\) on an open set \(A,\) and \(\vec{s} \in A,\) prove that \(\left(\forall \vec{u} \in E^{2}\left(C^{2}\right)\right)\)
\[d^{i} f(\vec{s} ; \vec{u})=\sum_{j=0}^{i}\left(\begin{array}{c}{i} \\ {j}\end{array}\right) u_{1}^{j} u_{2}^{i-j} D_{k_{1} \dots k_{i}} f(\vec{s}), \quad 1 \leq i \leq m,\]
where the \(\left(\begin{array}{l}{i} \\ {j}\end{array}\right)\) are binomial coefficients, and in the \(j\)th term,
\[k_{1}=k_{2}=\cdots=k_{j}=2\]
and
\[k_{j+1}=\cdots=k_{i}=1.\]
Then restate formula (6) for \(n=2\).
[Hint: Use induction, as in the binomial theorem.]
\(\Rightarrow\) Given \(\vec{p} \in E^{\prime}=E^{n}\left(C^{n}\right)\) and \(f : E^{\prime} \rightarrow E,\) prove that \(f \in C D^{1}\) at \(\vec{p}\) iff \(f\) is differentiable at \(\vec{p}\) and
\[(\forall \varepsilon>0)(\exists \delta>0)\left(\forall \vec{x} \in G_{\vec{p}}(\delta)\right) \quad\left\|d^{1} f(\vec{p} ; \cdot)-d^{1} f(\vec{x} ; \cdot)\right\|<\varepsilon,\]
with norm \(\|\) as in Definition 2 in §2. (Does it apply?)
[Hint: If \(f \in C D^{1},\) use Theorem 2 in §3. For the converse, verify that
\[\varepsilon \geq\left|d^{1} f(\vec{p} ; \vec{t})-d^{1} f(\vec{x} ; \vec{t})\right|=\left|\sum_{k=1}^{n}\left[D_{k} f(\vec{p})-D_{k} f(\vec{x})\right] t_{k}\right|\]
if \(\vec{x} \in G_{\vec{p}}(\delta)\) and \(|\vec{t}| \leq 1.\) Take \(\vec{t}=\vec{e}_{k},\) to prove continuity of \(D_{k} f\) at \(\vec{p}\).]
Prove the following.
(i) If \(\phi : E^{n} \rightarrow E^{m}\) is linear and \([\phi]=\left(v_{i k}\right),\) then
\[\|\phi\|^{2} \leq \sum_{i, k}\left|v_{i k}\right|^{2}.\]
(ii) If \(f : E^{n} \rightarrow E^{m}\) is differentiable at \(\vec{p},\) then
\[\|d f(\vec{p} ; \cdot)\|^{2} \leq \sum_{i, k}\left|D_{k} f_{i}(\vec{p})\right|^{2}.\]
(iii) Hence find a new converse proof in Problem 10 for \(f : E^{n} \rightarrow E^{m}\).
Consider \(f : C^{n} \rightarrow C^{m},\) too.
[Hints: (i) By the Cauchy-Schwarz inequality, \(|\phi(\vec{x})|^{2} \leq|\vec{x}|^{2} \sum_{i, k}\left|v_{i k}\right|^{2}.\) (Why?) (ii) Use part (i) and Theorem 4 in §3.]
(i) Find \(d^{2} u\) for the functions of Problem 10 in §4, in the "variable" and "mapping" notations.
(ii) Do it also for
\[u=f(x, y, z)=\left(x^{2}+y^{2}+z^{2}\right)^{-\frac{1}{2}}\]
and show that \(D_{11} f+D_{22} f+D_{33} f=0\).
(iii) Does the latter hold for \(u=\arctan \frac{y}{x}?\)
Let \(u=g(x, y), x=r \cos \theta, y=r \sin \theta\) (passage to polars).
Using "variables" and then the "mappings" notation, prove that if \(g\) is differentiable, then
(i) \(\frac{\partial u}{\partial r}=\cos \theta \frac{\partial u}{\partial x}+\sin \theta \frac{\partial u}{\partial y}\) and
(ii) \(|\nabla g(x, y)|^{2}=\left(\frac{\partial u}{\partial r}\right)^{2}+\left(\frac{1}{r} \frac{\partial u}{\partial \theta}\right)^{2}\).
(iii) Assuming \(g \in C D^{2},\) express \(\frac{\partial^{2} u}{\partial r \partial \theta}, \frac{\partial^{2} u}{\partial r^{2}},\) and \(\frac{\partial^{2} u}{\partial \theta^{2}}\) as in (i).
Let \(f, g : E^{1} \rightarrow E^{1}\) be of class \(C D^{2}\) on \(E^{1}.\) Verify (in "variable" notation, too) the following statements.
(i) \(D_{11} h=a^{2} D_{22} h\) if \(a \in E^{1}\) (fixed) and
\[h(x, y)=f(a x+y)+g(y-a x).\]
(ii) \(x^{2} D_{11} h(x, y)+2 x y D_{12} h(x, y)+y^{2} D_{22} h(x, y)=0\) if
\[h(x, y)=x f\left(\frac{y}{x}\right)+g\left(\frac{y}{x}\right).\]
(iii) \(D_{1} h \cdot D_{21} h=D_{2} h \cdot D_{11} h\) if
\[h(x, y)=g(f(x)+y)\]
Find \(D_{12} h,\) too.
Assume \(E^{\prime}=E^{n}\left(C^{n}\right)\) and \(E^{\prime \prime}=E^{m}\left(C^{m}\right).\) Let \(f : E^{\prime} \rightarrow E^{\prime \prime}\) and \(g : E^{\prime \prime} \rightarrow E\) be twice differentiable at \(\vec{p} \in E^{\prime}\) and \(\vec{q}=f(\vec{p}) \in E^{\prime \prime}\), respectively, and set \(h=g \circ f\).
Show that \(h\) is twice differentiable at \(\vec{p},\) and
\[d^{2} h(\vec{p} ; \vec{t})=d^{2} g(\vec{q} ; \vec{s})+d g(\vec{q} ; \vec{v}),\]
where \(\vec{t} \in E^{\prime}, \vec{s}=d f(\vec{p} ; \vec{t}),\) and \(\vec{v}=\left(v_{1}, \ldots, v_{m}\right) \in E^{\prime \prime}\) satisfies
\[v_{i}=d^{2} f_{i}(\vec{p} ; \vec{t}), \quad i=1, \ldots, m.\]
Thus the second differential is not invariant in the sense of Note 4 in §4.
[Hint: Show that
\[D_{k l} h(\vec{p})=\sum_{j=1}^{m} \sum_{i=1}^{m} D_{i j} g(\vec{q}) D_{k} f_{i}(\vec{p}) D_{l} f_{j}(\vec{p})+\sum_{i=1}^{m} D_{i} g(\vec{q}) D_{k l} f_{i}(\vec{p}).\]
Proceed.]
Continuing Problem 15, prove the invariant rule:
\[d^{r} h(\vec{p} ; \vec{t})=d^{r} g(\vec{q} ; \vec{s}),\]
if \(f\) is a first-degree polynomial and \(g\) is \(r\) times differentiable at \(\vec{q}\).
[Hint: Here all higher-order partials of \(f\) vanish. Use induction.]


17.4: Interpretation of Gauss’ Law and vector calculus
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In this section, we provide a little more theoretical background and intuition on Gauss’ Law, as well as its connection to vector calculus (which is beyond the scope of this textbook, but interesting to have a feeling for). Very generally, Gauss’ Law is a statement that connects a property of a vector field to the “source” of that field. We think of mass as the source for the gravitational field, and we think of charge as the source for the electric field. The property of the field that we considered in this case was its “flux out of a closed surface”.
Recall that determining the flux of a field out of a closed surface is equivalent to counting the net number of field lines that exit that closed surface. Field lines must start on a positive charge and must end on a negative charge. Thus, if there is a net number of field lines exiting the surface, there must be a positive charge in the volume defined by the surface (a “source” of field lines). If there is a net number of field lines entering the surface, then the volume defined by the surface must enclose a negative charge (a “sink” of field lines). Gauss’ Law is simply a statement that the number of field lines entering/exiting a closed surface is proportional to the amount of charge enclosed in that volume.
The flux out of a closed surface is tightly connected to the vector calculus concept of “divergence”, which describes whether field lines are diverging (spreading out or getting closer together). When a point charge is present, field lines will emanate radially from that point charge; in other words, they will diverge. We say that the electric field has non-zero divergence if there is a source of the electric field in that position of space. The key difference between the concept of divergence and that of “flux out of a closed surface”, is that divergence is a local property of the field (it is true at a point), whereas the flux out of a surface must be calculated using a finite volume and makes it challenging to define the field at a specific position. Gauss’s Law defined using flux is thus not as useful for describing how the field changes at specific positions, and is usually limited to situations with a high degree of symmetry.
The divergence, \(\nabla \cdot \vec E\) , of a vector field, \(\vec E\) , at some position is defined as:
\[\begin{aligned}\nabla\cdot\vec E =\frac{\partial E}{\partial x}+\frac{\partial E}{\partial y}+\frac{\partial E}{\partial z}\end{aligned}\]
and corresponds to the sum of three partial derivatives evaluated at that position in space. Gauss’ Theorem (also called the Divergence Theorem) states that:
\[\begin{aligned} \int_V \nabla \cdot \vec E = \oint_S \vec E \cdot d\vec A\end{aligned}\]
where the \(V\) ( \(S\) ) on the integral indicates whether the sum (integral) should be carried out over a volume, \(V\) , or over a closed surface, \(S\) , as we have practised in this chapter. While it is not important at this level to understand the theorem in detail, the point is that one can convert a “flux over a closed surface” into an integral of the divergence of the field. In other words, we can convert a global property (flux) to a local property (divergence). Gauss’ Law in terms of divergence can be written as:
\[\nabla\cdot\vec E =\frac{\rho}{\epsilon_0}\quad\text{(Local version of Gauss' Law)}\]
where \(\rho\) is the charge per unit volume at a specific position in space. This is the version of Gauss’ Law that is usually seen in advanced textbooks and in Maxwell’s unified theory of electromagnetism. This version of Gauss’s Law relates a local property of the field (its divergence) to a local property of charge at that position in space (the charge per unit volume at that position in space). If we integrate both sides of the equation over volume, we recover the original formulation of Gauss’ Law: the left hand side, by the Divergence Theorem, leads to flux when integrated over volume, whereas on the right hand side, the integral over volume of charge per unit volume, \(\rho\) , will give the total charge enclosed in that volume, \(Q^{enc}\) :
\[\begin{aligned} \int_V \left(\nabla \cdot \vec E \right)dV&= \int_V \left(\frac{\rho}{\epsilon_0}\right) dV\\[4pt] \oint_S \vec E \cdot d\vec A &=\frac{Q^{enc}}{\epsilon_0}\end{aligned}\]


6.13: Agenda Setting
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The media can and does have a strong effect on what people think about. By highlighting certain events repeatedly, they create a sense of urgency about those issues even though this is not always an accurate reflection of reality .
Reality is what is actually happening in the world pertaining to the economy, society, politics and science. The media selectively highlights certain events and gives them prominence. The criteria for the selection depends very much on the ideology of the media editors and their vested interests. By highlighting certain events an uncritical audience will perceive and construct the mediated reality as reality. One example is the royal wedding of Prince William to Kate Middleton, which mesmerised the world, including parts of the world that have no connection to the British monarchy. Discerning audiences may well ask why the wedding of two people, who are in no way related to them, is important enough to justify worldwide media attention, or indeed, how and why it might be relevant to them?
The agenda setting theory was formally developed by McCombs and Shaw (1972) when they studied the US Presidential Election of 1968. Their analysis of the news and media coverage found a strong correlation to the opinions held by the voters in Chapel Hill, North Carolina. Similar findings were found in the media study surrounding the murder trial of OJ Simpson (Salwen & Driscoll, 1997) and the attempted impeachment of US President Clinton (Yioutas & Segvic, 2003).
These videos explain the agenda setting theory in detail:
A YouTube element has been excluded from this version of the text. You can view it online here: https://opentextbc.ca/mediastudies101/?p=38
From calleno16 under Standard YouTube License
A YouTube element has been excluded from this version of the text. You can view it online here: https://opentextbc.ca/mediastudies101/?p=38
From CSIxxTwee under Standard YouTube License
A YouTube element has been excluded from this version of the text. You can view it online here: https://opentextbc.ca/mediastudies101/?p=38
From MEF via emjay23 under CC License
Discussion
- Obviously journalists cannot include everything that happens in the world on any given day in their Newspaper TV or internet news program.
- Investigate a cross section of media sources on a given day – compare how and why the different sources choose, and then mediate, the events for their expected audience.
- Then consider the question, are they accurately representing the importance of the issues of the day?
References
McCombs, M.E and Shaw, D.L. (1972) The agenda-setting function of mass media. Public Opinion Quarterly,36 (2).
Salwen, M. B., & Driscoll, P. D. (1997). Consequences of third‐person perception in support of press restrictions in the OJ Simpson trial. Journal of communication, 47(2), 60-78.
Yioutas, J., & Segvic, I. (2003). Revisiting the Clinton/Lewinsky scandal: The convergence of agenda setting and framing. Journalism & Mass Communication Quarterly, 80(3), 567-582.


3.6: Costume (Bunad)
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Costume (
Bunad
)
Norwegian-American Irene Gilbertson, 1969 Wool, cotton, brass Gift of Irene Gilbertson MHAHS 2015.011.0001 |
This decorated dress is a costume known as a bunad , a Norwegian rural folk garment. Irene Gilbertson created this particular outfit to wear when she took tickets and was a host at Mount Horeb’s annual performance of The Song of Norway , a play based on the life of Edvard Grieg. Begun in 1969, the popular production became an outlet for local talent and a way for the community to attract visitors to the area. Irene Gilbertson’s paternal grandparents emigrated from Norway. She was a pillar of the community and involved in many different groups, including being a long-time participant of The Song of Norway .
This beautifully decorated dress is a costume called a Bunad. This particular bunad—Norwegian rural folk garment—is decorated in the Gudbrandsdahlen style. It was handcrafted and sewn by Irene Gilbertson and consists of seven pieces. This particular outfit includes: a bodice, a blouse, a small hat, a purse, 2 pins of silver metal filigree (necklaces), and cufflinks. The dress, purse, and hat are blue with intricately ornate multi-color flowers.
Bunad’s are a specific movement of style of dress in Norwegian communities. They come in many forms as they are related to their specific regions. This particular garment is styled in the Gudbrandsahlen style (Gudbrandalsbunad) that originates from the Lillehammer region. This specific style could mean that it is double woven, which is usually the easiest way to distinguish it from other bunad styles. These costumes hold rich traditions within Norwegian communities, both literally and metaphorically. They were sometimes used to display wealth as they could be richly decorated with gold and silvers, but are now more commonly used as religious confirmation presents or as costumes in performances.
Irene Gilbertson, and the memory of all her accomplishments, remains a prominent member of Mt. Horeb, Wisconsin. She was born and raised in Madison where she met her husband Otto. They then owned and operated Gilbertson Hardware in Mt. Horeb for 43 years. She was a pillar of the community and was heavily involved in many different groups. Her impressive resume includes: long-term membership with the Evangelical Lutheran Church, Matron of the Mt. Horeb Eastern Star, organized the Rainbow for Girls, a charter member for Mound View Garden Club, but most importantly she was a charter member and curator for Mt. Horeb’s Historical Society and participated in the Song of Norway for 14 years. She clearly loved being active and engaging with her community, and participated and created groups that continue to last in Mt. Horeb today.
In 1969 Irene Gilbertson began her participation in the Song of Norway, a small stage production. This play was originated in 1944 as an operetta on Broadway and portrays the life of Edvard Grieg—a composer who struggled to create an authentic Norwegian national music. The play was popular among these Norwegian communities and was performed throughout the state. It was often used in festivities and celebrations that focused on the towns’ Norwegian heritage. The play was so popular that Hollywood attempted to ‘perfect it’ in a film adaptation, that ultimately flopped. This bunad is what Irene Gilbertson created and wore for Song of Norway, and follows the Gudbrandsahled style—however, this dress was typically black, but Irene Gilbertson preferred blue. It is through the generous donation of her family that Irene Gilbertson’s bunad has a home at the Mt. Horeb Historical Society.


2.13: Exercises
Use an index card or a piece of paper folded lengthwise, and cover up the right-hand column of the exercises below. Read each exercise in the left-hand column, answer it in your mind, then slide the index card down to reveal the answer and see if you’re right! For every exercise you missed, figure out why you missed it before moving on.
| Q | A |
|---|---|
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1. Is the set { Will, Smith } the same as the set { Smith, Will }? |
Yes indeed. |
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2. Is the ordered pair (Will, Smith) the same as (Smith, Will)? |
No. Order matters with ordered pairs (hence the name), and with any size tuple for that matter. |
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3. Is the set { { Luke, Leia }, Han } the same as the set { Luke, { Leia, Han } }? |
No. For instance, the first set has Han as a member but the second set does not. (Instead, it has another set as a member, and that inner set happens to include Han.) |
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4. What’s the first element of the set { Cowboys, Redskins, Steelers }? |
The question doesn’t make sense. There is no “first element" of a set. All three teams are equally members of the set, and could be listed in any order. |
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5.
Let
\(G\)
be { Matthew, Mark, Luke, John },
\(J\)
be { Luke, Obi-wan, Yoda },
\(S\)
be the set of all Star Wars characters, and
\(F\)
be the four gospels from the New Testament.
|
No. |
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6. Is \(J \subseteq S\) ? |
Yes. |
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7. Is Yoda \(\in J\) ? |
Yes. |
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8. Is Yoda \(\subseteq J\) ? |
No. Yoda isn’t even a set, so it can’t be a subset of anything. |
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9. Is { Yoda } \(\subseteq J\) ? |
Yes. The (unnamed) set that contains only Yoda is in fact a subset of \(J\) . |
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10. Is { Yoda } \(\in J\) ? |
No. Yoda is one of the elements of \(J\) , but { Yoda } is not. In other words, \(J\) contains Yoda, but \(J\) does not contain a set which contains Yoda (nor does it contain any sets at all, in fact). |
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11. Is \(S \subseteq J\) ? |
No. |
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12. Is \(G \subseteq F\) ? |
Yes, since the two sets are equal. |
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13. Is \(G \subset F\) ? |
No, since the two sets are equal, so neither is a proper subset of the other. |
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14. Is \(\varnothing \subseteq S\) ? |
Yes, since the empty set is a subset of every set. |
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15. Is \(\varnothing \subseteq \varnothing\) ? |
Yes, since the empty set is a subset of every set. |
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16. Is \(F \subseteq \Omega\) ? |
Yes, since every set is a subset of \(\Omega\) . |
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17. Is \(F \subset \Omega\) ? |
Yes, since every set is a subset of \(\Omega\) , and \(F\) is certainly not equal to \(\Omega\) . |
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18. Suppose \(X\) = { Q, \(\varnothing\) , { Z } }. Is \(\varnothing \in X\) ? Is \(\varnothing \subseteq X\) ? |
Yes and yes. The empty set is an element of \(X\) because it’s one of the elements, and it’s also a subset of \(X\) because it’s a subset of every set. Hmmm. |
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19. Let \(A\) be { Macbeth, Hamlet, Othello }, \(B\) be { Scrabble, Monopoly, Othello }, and \(T\) be { Hamlet, Village, Town }.What’s \(A \cup B\) ? |
{ Macbeth, Hamlet, Othello, Scrabble, Monopoly }. (The elements can be listed in any order.) |
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20. What’s \(A \cap B\) ? |
{ Othello }. |
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21. What’s \(A \cap \overline{B}\) ? |
{ Macbeth, Hamlet }. |
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22. What’s \(B \cap T\) ? |
\(\varnothing\) . |
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23. What’s \(B \cap \overline{T}\) ? |
\(B\) . (which is { Scrabble, Monopoly, Othello }.) |
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24. What’s \(A \cup (B \cap T)\) ? |
{ Hamlet, Othello, Macbeth }. |
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25. What’s \((A \cup B) \cap T\) ? |
{ Hamlet }. (Note: not the same answer as in item 24 now that the parens are placed differently.) |
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26. What’s \(A - B\) ? |
{ Macbeth, Hamlet }. |
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27. What’s \(T - B\) ? |
Simply \(T\) , since the two sets have nothing in common. |
| 28. What’s \(T \times A\) ? | { (Hamlet, Macbeth), (Hamlet, Hamlet), (Hamlet, Othello), (Village, Macbeth), (Village, Hamlet), (Village, Othello), (Town, Macbeth), (Town, Hamlet), (Town, Othello) }. The order of the ordered pairs within the set is not important; the order of the elements within each ordered pair is important. |
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29. What’s \((B \cap B) \times (A \cap T)\) ? |
{ (Scrabble, Hamlet), (Monopoly, Hamlet), (Othello, Hamlet) }. |
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30. What’s \(|A \cup B \cup T|\) ? |
7. |
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31. What’s \(|A \cap B \cap T|\) ? |
0. |
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32. What’s \(|(A \cup B \cup T) \times (B \cup B \cup B)|\) ? |
21. (The first parenthesized expression gives rise to a set with 7 elements, and the second to a set with three elements ( \(B\) itself). Each element from the first set gets paired with an element from the second, so there are 21 such pairings.) |
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33. Is \(A\) an extensional set, or an intensional set? |
The question doesn’t make sense. Sets aren’t “extensional" or “intensional"; rather, a given set can be described extensionally or intensionally. The description given in item 19 is an extensional one; an intensional description of the same set would be “The Shakespeare tragedies Stephen studied in high school." |
|
34. Recall that \(G\) was defined as { Matthew, Mark, Luke, John }. Is this a partition of \(G\) ?
|
No, because the sets are not collectively exhaustive (Mark is missing). |
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35. Is this a partition of \(G\) ?
|
No, because the sets are neither collectively exhaustive (John is missing) nor mutually exclusive (Luke appears in two of them). |
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36. Is this a partition of \(G\) ?
|
Yes. (Trivia: this partitions the elements into the synoptic gospels and the non-synoptic gospels). |
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37. Is this a partition of \(G\) ?
|
Yes. (This partitions the elements into the gospels which feature a Christmas story and those that don’t). |
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38. Is this a partition of \(G\) ?
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Yes. (This partitions the elements into the gospels that were written by Jews, those that were written by Greeks, those that were written by Romans, and those that were written by Americans). |
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39. What’s the power set of { Rihanna }? |
{ { Rihanna }, \(\varnothing\) }. |
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40. Is { peanut, jelly } \(\in \mathbb{P}\) ({ peanut, butter, jelly }? |
Yes, since { peanut, jelly } is one of the eight subsets of { peanut, butter, jelly }. (Can you name the other seven?)
|
| 41. Is it true for every set \(S\) that \(S \in \mathbb{P}(S)\) ? | Yep. |


6: Geometric Shapes
In the waterworks industry geometric shapes are prevalent. For example, a pipe is a cylinder with the opening of a circle, a sedimentation basin can be a rectangle, and an aqueduct might be a trapezoid. These are a few of the shapes that we will cover in this text. It is important to understand how to work with these shapes to be able to calculate areas, volumes, perimeters, and circumferences.
We will first look at geometric shapes and how to calculate the corresponding area of these shapes. The answer when determining the area of a geometric shape should always be given in square feet (sqft or ft\(^2\)). Other units can be used such as, square meters or square inches, but these are not typically used in the waterworks industry. Remember though that the units you are given in a question (or in “real life” for that matter, may not be the units you need to calculate square feet. Therefore, you should always convert the “given units” to feet before calculating the area. The first shape we will look at is the CIRCLE. Pipes are the backbone to water distribution systems and can also be found in treatment plants. Pipe lines carry raw water to the treatment plant, transmission lines can carry water from treatment plants to storage tanks, and distribution pipe lines delivery water to customers.