7.7: Coping with Challenging Emotions
Ways to Find Support
Throughout this chapter we have explored the variety of ways our emotions can manifest themselves. We have also discussed ways to understand where our emotions stem from both internally as well as within our social environments. One thing has been made clear throughout each section of this chapter: emotions are challenging at times, and being able to control our emotions is an essential part of our communication and interpersonal relationships. When we are in tune with ourselves and have emotional awareness it can help us in many aspects of life including decision-making, conflict management, relational satisfaction, and self-esteem.
There are some emotions that are more difficult than others to manage and can not only impact our short-term emotional state but can also physically alter our personality and our sense of self. Emotions such as grief, trauma, depression, anxiety, stress, and anger are emotional experiences that can often feel invisible but are valid. During these challenging emotions, it can feel extremely difficult to express what the emotions feel like, not only to others but even ourselves. It can feel impossible to reach out for support or know where we can turn to to find empathy, but it is an extremely important step in being able to cope with these emotions.
Among our basic needs as human beings are social needs: we need to be able to find others who are like us and that we can express ourselves to. During times where we feel threatened by our emotions, finding support and being able to talk through our emotions is a vital step in creating a positive outlook. This step has the ability to create a positive change in mood and allows us to continue to function in our daily life. Finding support comes in many forms, and the kind of support we need in one instance will change to the next. Having the ability to pull from our support resources will allow us to find what we need in order to confront these feelings. It is important to remember that we are not a burden to other people, that we are not weak, and that every single person needs to be supported by others throughout our lives. We will be able to be that support for someone else when they need it one day, and provide empathy for their situation.
Some ways in which we may find support in order to help cope with challenging emotions include:
- Find someone we feel safe with. Find friends, family, or loved ones who can be there to listen to us. While it can feel really difficult to reach out, remember that the loved ones we choose to reach out to are honored to be able to help us in such important ways.
- Take care of a pet. Being able to care for an animal and feel love from an animal has significant benefits for our mood regulation both mentally and physically.
- Showcase support. Sometimes being able to be of help to others allows us to feel supported ourselves. Volunteering or providing service can help to strengthen our goals and regulate our mental outlook.
- Join a support group. Talking with and listening to other individuals who are going through similar emotions as us is a really humbling way to make sure we know we aren’t alone.
- Seek medical help. Therapy and counseling can be beneficial for many people in order to not only talk out their thoughts and emotions, but to gain feedback and support in a constructive manner. While many counselors do cost money, there are free and affordable options through colleges and universities, as well as city programs that we can use in order to get the appropriate care we deserve.


2.2: Simple Interest
In this section, you will learn to:
- Use the simple interest formula to calculate the future value of a lump sum
So that they can analyze financial scenarios to make informed decisions.
Discussing interest starts with the principal , or the amount your account starts with. This could be a starting investment or the starting amount of a loan. Interest, in its most simple form, is calculated as a percent of the principal. For example, if you borrowed $100 from a friend and agree to repay it with 5% interest, then the amount of interest you would pay would just be 5% of 100: \(\$ 100(0.05)=\$ 5\). The total amount you would repay would be $105, the original principal plus the interest.
\[I=P_{0} r\]
\[A=P_{0}+I=P_{0}+P_{0} r=P_{0}(1+r)\]
where
- \(I\) is the interest
- \(A\) is the end amount: principal plus interest
- \(P_0\) is the principal (starting amount)
- \(r\) is the interest rate (in decimal form. Example: \(5\% = 0.05\))
A friend asks to borrow $300 and agrees to repay it in 30 days with 3% interest. How much interest will you earn?
Solution
\(\begin{array}{ll} P_{0}=\$ 300 & \text{the principal } \\ r=0.03 & 3 \%\text{ rate} \\
I=\$ 300(0.03)=\$ 9. & \text{You will earn }\$ 9 \text{ interest.}\end{array}\)
One-time simple interest is only common for extremely short-term loans. For longer-term loans, it is common for interest to be paid on a daily, monthly, quarterly, or annual basis. In that case, interest would be earned regularly. For example, bonds are essentially a loan made to the bond issuer (a company or government) by you, the bondholder. In return for the loan, the issuer agrees to pay interest, often annually. Bonds have a maturity date, at which time the issuer pays back the original bond value.
Suppose your city is building a new park, and issues bonds to raise the money to build it. You obtain a $1,000 bond that pays 5% interest annually that matures in 5 years. How much interest will you earn?
Solution
Each year, you would earn 5% interest: \(\$ 1000(0.05)=\$ 50\) in interest. So over the course of five years, you would earn a total of $250 in interest. When the bond matures, you would receive back the $1,000 you originally paid, leaving you with a total of $1,250.
We can generalize this idea of simple interest over time.
\(I=P_{0} r t\)
\(A=P_{0}+I=P_{0}+P_{0} r t=P_{0}(1+r t)\)
where
- \(I\) is the interest
- \(A\) is the end amount: principal plus interest
- \(P_0\) is the principal (starting amount)
- \(r\) is the interest rate in decimal form
- \(t\) is time
The units of measurement (years, months, etc.) for the time should match the time period for the interest rate.
Interest rates are usually given as an annual percentage rate (APR) – the total interest that will be paid in the year. If the interest is paid in smaller time increments, the APR will be divided up.
For example, a \(6 \%\) APR paid monthly would be divided into twelve \(0.5 \%\) payments.
A \(4 \%\) annual rate paid quarterly would be divided into four \(1 \%\) payments.
Treasury Notes (T-notes) are bonds issued by the federal government to cover its expenses. Suppose you obtain a $1,000 T-note with a 4% annual rate, paid semi-annually, with a maturity in 4 years. How much interest will you earn?
Solution
Since interest is being paid semi-annually (twice a year), the 4% interest will be divided into two 2% payments.
\(\begin{array}{ll} P_{0}=\$ 1000 & \text{the principal } \\ r=0.02 & 2 \%\text{ rate} \\ t = 8 & \text{4 years = 8 half-years} \\
I=\$ 1000(0.02)(8)=\$ 160. & \text{You will earn }\$ 160 \text{ interest total over the four years.}\end{array}\)


6.5: Potential, Kinetic, Free, and Activation Energy - Free Energy
- Discuss the concept of free energy.
Free Energy
Since chemical reactions release energy when energy-storing bonds are broken, how is the energy associated with chemical reactions quantified and expressed? How can the energy released from one reaction be compared to that of another reaction?
A measurement of free energy is used to quantitate these energy transfers. Free energy is called Gibbs free energy (G) after Josiah Willard Gibbs, the scientist who developed the measurement. Recall that according to the second law of thermodynamics, all energy transfers involve the loss of some amount of energy in an unusable form such as heat, resulting in entropy. Gibbs free energy specifically refers to the energy associated with a chemical reaction that is available after accounting for entropy. In other words, Gibbs free energy is usable energy or energy that is available to do work.
Calculating ∆G
Every chemical reaction involves a change in free energy, called delta G (∆G). The change in free energy can be calculated for any system that undergoes a change, such as a chemical reaction. To calculate ∆G, subtract the amount of energy lost to entropy (denoted as ∆S) from the total energy change of the system. This total energy change in the system is called enthalpy and is denoted as ∆H. The formula for calculating ∆G is as follows, where the symbol T refers to absolute temperature in Kelvin (degrees Celsius + 273): G=ΔH−TΔS.
The standard free energy change of a chemical reaction is expressed as an amount of energy per mole of the reaction product (either in kilojoules or kilocalories, kJ/mol or kcal/mol; 1 kJ = 0.239 kcal) under standard pH, temperature, and pressure conditions. Standard pH, temperature, and pressure conditions are generally calculated at pH 7.0 in biological systems, 25 degrees Celsius, and 100 kilopascals (1 atm pressure), respectively. It is important to note that cellular conditions vary considerably from these standard conditions; therefore, standard calculated ∆G values for biological reactions will be different inside the cell.
Endergonic and Exergonic Reactions
If energy is released during a chemical reaction, then the resulting value from the above equation will be a negative number. In other words, reactions that release energy have a ∆G < 0. A negative ∆G also means that the products of the reaction have less free energy than the reactants because they gave off some free energy during the reaction. Reactions that have a negative ∆G and, consequently, release free energy, are called exergonic reactions. Exergonic means energy is exiting the system. These reactions are also referred to as spontaneous reactions because they can occur without the addition of energy into the system. Understanding which chemical reactions are spontaneous and release free energy is extremely useful for biologists because these reactions can be harnessed to perform work inside the cell. An important distinction must be drawn between the term spontaneous and the idea of a chemical reaction that occurs immediately. Contrary to the everyday use of the term, a spontaneous reaction is not one that suddenly or quickly occurs. The rusting of iron is an example of a spontaneous reaction that occurs slowly, little by little, over time.
If a chemical reaction requires an input of energy rather than releasing energy, then the ∆G for that reaction will be a positive value. In this case, the products have more free energy than the reactants. Thus, the products of these reactions can be thought of as energy-storing molecules. These chemical reactions are called endergonic reactions; they are non-spontaneous. An endergonic reaction will not take place on its own without the addition of free energy.
Free Energy and Biological Processes
In a living cell, chemical reactions are constantly moving towards equilibrium, but never reach it. A living cell is an open system: materials pass in and out, the cell recycles the products of certain chemical reactions into other reactions, and chemical equilibrium is never reached. In this way, living organisms are in a constant energy-requiring, uphill battle against equilibrium and entropy.
When complex molecules, such as starches, are built from simpler molecules, such as sugars, the anabolic process requires energy. Therefore, the chemical reactions involved in anabolic processes are endergonic reactions. On the other hand, the catabolic process of breaking sugar down into simpler molecules releases energy in a series of exergonic reactions. As in the example of rust above, the breakdown of sugar involves spontaneous reactions, but these reactions don’t occur instantaneously. An important concept in the study of metabolism and energy is that of chemical equilibrium. Most chemical reactions are reversible. They can proceed in both directions, releasing energy into their environment in one direction, and absorbing it from the environment in the other direction.
Key Points
- Every chemical reaction involves a change in free energy, called delta G (∆G).
- To calculate ∆G, subtract the amount of energy lost to entropy (∆S) from the total energy change of the system; this total energy change in the system is called enthalpy (∆H ): ΔG=ΔH−TΔS.
- Endergonic reactions require an input of energy; the ∆G for that reaction will be a positive value.
- Exergonic reactions release free energy; the ∆G for that reaction will be a negative value.
Key Terms
- exergonic reaction : A chemical reaction where the change in the Gibbs free energy is negative, indicating a spontaneous reaction
- endergonic reaction : A chemical reaction in which the standard change in free energy is positive, and energy is absorbed
- Gibbs free energy : The difference between the enthalpy of a system and the product of its entropy and absolute temperature


5.11: Test Yourself
Now use this Skin Test Yourself to see how much you have learned and remember.
1. The two layers that form the skin are the a) and b)
2. The special protein deposited in epidermal cells to make them waterproof is:
3. Many important skin structures are made of keratin. These include:
4. Sweat, sebaceous and mammary glands all have ducts to the outside. These kind of glands are known as:
5. What is the pigment deposited in skin cells that protects underlying skin layers from the harmful effects of the sun?
6. How does the skin help cool an animal down when it is active or in a hot environment?
7. Name two mechanisms by means of which the skin helps prevent heat loss when an animal is in a cold environment.


2.S: Geometric Optics and Image Formation (Summary)
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Key Terms
| aberration | distortion in an image caused by departures from the small-angle approximation |
| accommodation | use of the ciliary muscles to adjust the shape of the eye lens for focusing on near or far objects |
| angular magnification | ratio of the angle subtended by an object observed with a magnifier to that observed by the naked eye |
| apparent depth | depth at which an object is perceived to be located with respect to an interface between two media |
| Cassegrain design | arrangement of an objective and eyepiece such that the light-gathering concave mirror has a hole in the middle, and light then is incident on an eyepiece lens |
| charge-coupled device (CCD) | semiconductor chip that converts a light image into tiny pixels that can be converted into electronic signals of color and intensity |
| coma | similar to spherical aberration, but arises when the incoming rays are not parallel to the optical axis |
| compound microscope | microscope constructed from two convex lenses, the first serving as the eyepiece and the second serving as the objective lens |
| concave mirror | spherical mirror with its reflecting surface on the inner side of the sphere; the mirror forms a “cave” |
| converging (or convex) lens | lens in which light rays that enter it parallel converge into a single point on the opposite side |
| convex mirror | spherical mirror with its reflecting surface on the outer side of the sphere |
| curved mirror | mirror formed by a curved surface, such as spherical, elliptical, or parabolic |
| diverging (or concave) lens | lens that causes light rays to bend away from its optical axis |
| eyepiece | lens or combination of lenses in an optical instrument nearest to the eye of the observer |
| far point | furthest point an eye can see in focus |
| farsightedness (or hyperopia) | visual defect in which near objects appear blurred because their images are focused behind the retina rather than on the retina; a farsighted person can see far objects clearly but near objects appear blurred |
| first focus or object focus | object located at this point will result in an image created at infinity on the opposite side of a spherical interface between two media |
| focal length | distance along the optical axis from the focal point to the optical element that focuses the light rays |
| focal plane | plane that contains the focal point and is perpendicular to the optical axis |
| focal point | for a converging lens or mirror, the point at which converging light rays cross; for a diverging lens or mirror, the point from which diverging light rays appear to originate |
| image distance | distance of the image from the central axis of the optical element that produces the image |
| linear magnification | ratio of image height to object height |
| magnification | ratio of image size to object size |
| near point | closest point an eye can see in focus |
| nearsightedness (or myopia) | visual defect in which far objects appear blurred because their images are focused in front of the retina rather than on the retina; a nearsighted person can see near objects clearly but far objects appear blurred |
| net magnification | (MnetMnet) of the compound microscope is the product of the linear magnification of the objective and the angular magnification of the eyepiece |
| Newtonian design | arrangement of an objective and eyepiece such that the focused light from the concave mirror was reflected to one side of the tube into an eyepiece |
| object distance |
distance of the object from the central axis of the optical element that produces its image |
| objective | lens nearest to the object being examined. |
| optical axis | axis about which the mirror is rotationally symmetric; you can rotate the mirror about this axis without changing anything |
| optical power | ( P ) inverse of the focal length of a lens, with the focal length expressed in meters. The optical power P of a lens is expressed in units of diopters D; that is, \(\displaystyle 1D=1/m=1m^{−1}\) |
| plane mirror | plane (flat) reflecting surface |
| ray tracing | technique that uses geometric constructions to find and characterize the image formed by an optical system |
| real image | image that can be projected onto a screen because the rays physically go through the image |
| second focus or image focus | for a converging interface, the point where a bundle of parallel rays refracting at a spherical interface; for a diverging interface, the point at which the backward continuation of the refracted rays will converge between two media will focus |
| simple magnifier (or magnifying glass) | converging lens that produces a virtual image of an object that is within the focal length of the lens |
| small-angle approximation | approximation that is valid when the size of a spherical mirror is significantly smaller than the mirror’s radius; in this approximation, spherical aberration is negligible and the mirror has a well-defined focal point |
| spherical aberration | distortion in the image formed by a spherical mirror when rays are not all focused at the same point |
| thin-lens approximation | assumption that the lens is very thin compared to the first image distance |
| vertex | point where the mirror’s surface intersects with the optical axis |
| virtual image | image that cannot be projected on a screen because the rays do not physically go through the image, they only appear to originate from the image |
Key Equations
| Image distance in a plane mirror | \(\displaystyle d_o=−d_i\) |
| Focal length for a spherical mirror | \(\displaystyle f=\frac{R}{2}\) |
| Mirror equation | \(\displaystyle \frac{1}{d_o}+\frac{1}{d_i}=\frac{1}{f}\) |
| Magnification of a spherical mirror | \(\displaystyle m=\frac{h_i}{h_o}=−\frac{d_i}{d_o}\) |
| Sign convention for mirrors | |
| Focal length \(\displaystyle f\) |
+for concave mirror
−for conve x mirror |
| Object distance \(\displaystyle d_o\) |
+for real object
−for virtual object |
| Image distance \(\displaystyle d_i\) |
+for real image
−for virtual image |
| Magnification \(\displaystyle m\) |
+for upright image
−for inverted image |
| Apparent depth equation | \(\displaystyle h_i=(\frac{n_2}{n_1})h_o\) |
| Spherical interface equation | \(\displaystyle \frac{n_1}{d_o}+\frac{n_2}{d_i}=\frac{n_2−n_1}{R}\) |
| The thin-lens equation | \(\displaystyle \frac{1}{d_o}+\frac{1}{d_i}=\frac{1}{f}\) |
| The lens maker’s equation | \(\displaystyle \frac{1}{f}=(\frac{n_2}{n_1}−1)(\frac{1}{R_1}−\frac{1}{R_2})\) |
| The magnification m of an object | \(\displaystyle m≡\frac{h_i}{h_o}=−\frac{d_i}{d_o}\) |
| Optical power | \(\displaystyle P=\frac{1}{f}\) |
| Optical power of thin, closely spaced lenses | \(\displaystyle P_{total}=P_{lens1}+P_{lens2}+P_{lens3}+⋯\) |
| Angular magnification M of a simple magnifier | \(\displaystyle M=\frac{θ_{image}}{θ_{object}}\) |
|
Angular magnification of an object a distance L from the eye for a convex lens of focal length f held a distance ℓ from the eye |
\(\displaystyle M=(\frac{25cm}{L})(1+\frac{L−ℓ}{f})\) |
| Range of angular magnification for a given lens for a person with a near point of 25 cm | \(\displaystyle \frac{25cm}{f}≤M≤1+\frac{25cm}{f}\) |
| Net magnification of compound microscope | \(\displaystyle M_{net}=m^{obj}M^{eye}=−\frac{d^{obj}_i(f^{eye}+25cm)}{f^{obj}f^{eye}}\) |
Summary
2.1 Images Formed by Plane Mirrors
- A plane mirror always forms a virtual image (behind the mirror).
- The image and object are the same distance from a flat mirror, the image size is the same as the object size, and the image is upright.
2.2 Spherical Mirrors
- Spherical mirrors may be concave (converging) or convex (diverging).
- The focal length of a spherical mirror is one-half of its radius of curvature: \(\displaystyle f=R/2\).

- Spherical aberration occurs for spherical mirrors but not parabolic mirrors; comatic aberration occurs for both types of mirrors.
2.3 Images Formed by Refraction
This section explains how a single refracting interface forms images.
- When an object is observed through a plane interface between two media, then it appears at an apparent distance \(\displaystyle h_i\) that differs from the actual distance \(\displaystyle h_o:h_i=(n_2/n_1)h_o\).
- An image is formed by the refraction of light at a spherical interface between two media of indices of refraction \(\displaystyle n_1\) and \(\displaystyle n_2\).
- Image distance depends on the radius of curvature of the interface, location of the object, and the indices of refraction of the media.
2.4 Thin Lenses
- Two types of lenses are possible: converging and diverging. A lens that causes light rays to bend toward (away from) its optical axis is a converging (diverging) lens.
- For a converging lens, the focal point is where the converging light rays cross; for a diverging lens, the focal point is the point from which the diverging light rays appear to originate.
- The distance from the center of a thin lens to its focal point is called the focal length f.
- Ray tracing is a geometric technique to determine the paths taken by light rays through thin lenses.
- A real image can be projected onto a screen.
- A virtual image cannot be projected onto a screen.
- A converging lens forms either real or virtual images, depending on the object location; a diverging lens forms only virtual images.
2.5 The Eye
- Image formation by the eye is adequately described by the thin-lens equation.
- The eye produces a real image on the retina by adjusting its focal length in a process called accommodation.
- Nearsightedness, or myopia, is the inability to see far objects and is corrected with a diverging lens to reduce the optical power of the eye.
- Farsightedness, or hyperopia, is the inability to see near objects and is corrected with a converging lens to increase the optical power of the eye.
- In myopia and hyperopia, the corrective lenses produce images at distances that fall between the person’s near and far points so that images can be seen clearly.
2.6 The Camera
- Cameras use combinations of lenses to create an image for recording.
- Digital photography is based on charge-coupled devices (CCDs) that break an image into tiny “pixels” that can be converted into electronic signals.
2.7 The Simple Magnifier
- A simple magnifier is a converging lens and produces a magnified virtual image of an object located within the focal length of the lens.
- Angular magnification accounts for magnification of an image created by a magnifier. It is equal to the ratio of the angle subtended by the image to that subtended by the object when the object is observed by the unaided eye.
- Angular magnification is greater for magnifying lenses with smaller focal lengths.
- Simple magnifiers can produce as great as tenfold ( 10× ) magnification.
2.8 Microscopes and Telescopes
- Many optical devices contain more than a single lens or mirror. These are analyzed by considering each element sequentially. The image formed by the first is the object for the second, and so on. The same ray-tracing and thin-lens techniques developed in the previous sections apply to each lens element.
- The overall magnification of a multiple-element system is the product of the linear magnifications of its individual elements times the angular magnification of the eyepiece. For a two-element system with an objective and an eyepiece, this is
\(\displaystyle M=m^{obj}M^{eye}.\) (2.41)
where \(\displaystyle m^{obj}\) is the linear magnification of the objective and \(\displaystyle M^{eye}\) is the angular magnification of the eyepiece.
- The microscope is a multiple-element system that contains more than a single lens or mirror. It allows us to see detail that we could not to see with the unaided eye. Both the eyepiece and objective contribute to the magnification. The magnification of a compound microscope with the image at infinity is
\(\displaystyle M_{net}=−\frac{(16cm)(25cm)}{f^{obj}f^{eye}}\). (2.42)
In this equation, 16 cm is the standardized distance between the image-side focal point of the objective lens and the object-side focal point of the eyepiece, 25 cm is the normal near point distance, \(\displaystyle f^{obj}\) and \(\displaystyle f^{eye}\) are the focal distances for the objective lens and the eyepiece, respectively.
- Simple telescopes can be made with two lenses. They are used for viewing objects at large distances.
- The angular magnification M for a telescope is given by
\(\displaystyle M=−\frac{f^{obj}}{f^{eye}}\), (2.43)
where \(\displaystyle f^{obj}\) and \(\displaystyle f^{eye}\) are the focal lengths of the objective lens and the eyepiece, respectively.


28.1: Clinical Judgment Measurement Model
By the end of this section, you will be able to:
- Identify the purpose for development of the Clinical Judgment Measurement Model (CJMM)
- Explain how to apply the Clinical Judgment Measurement Model (CJMM) to the nursing practice
Historically, nursing was often viewed as a “task-oriented” career, meaning that nurses were to take orders from physicians without question and carry out nursing tasks. These tasks were usually simple, such as bed linen changes, helping patients use the toilet, and providing hygiene care. However, the profession of nursing has evolved into a more complex, autonomous career over the past several decades. Nurses now use clinical judgment , which is the thought process that allows nurses to arrive at a conclusion, based on objective and subjective information about a patient, to achieve positive patient outcomes. Many times, nurses are the care providers tasked with making clinical decisions that will significantly affect the lives of their patients. In recent years, nursing education has also evolved to better train nurses to critically think and use clinical judgment in practice. To achieve this, the clinical judgment measurement model (CJMM) was developed. This model allows nurse educators to teach, assess, and measure the development of clinical judgment skills of nursing students. It assists nursing students to connect knowledge learned in the classroom to provide exceptional clinical care in practice. This model is the new foundation for nursing critical thinking and skill development and is discussed in more detail throughout this chapter.
Purpose for Development of the CJMM
Before development of the CJMM, there was not a good way to measure the clinical judgment and decision-making skills of nursing students. Developing these skills was a priority focus of most nursing programs, but there was no tangible way to measure it to ensure it was being taught effectively. This was the basis for the development of the CJMM. Researchers at the National Council of State Boards of Nursing (NCSBN) used nursing literature and research studies, in combination with data analysis and input from nursing students, to develop the CJMM. This model is used not only as a teaching tool within nursing school curricula, it is also used as a guiding framework for the development of new types of questions on the National Council Licensure Examination (NCLEX) to assess nursing student’s clinical judgment and critical thinking skills. By using the CJMM, nurse educators can be more confident that they are preparing nursing students to enter practice with a high level of critical thinking and the ability to make sound clinical judgments when caring for their patients. It also provides students with a structured approach for decision-making, improves students’ decision-making, and supports quality and safety.
You can learn more about the new NCLEX examination that incorporates concepts from the CJMM and get prepared for test day.
Comparing Different Nursing Process Models
Before development of the CJMM, the nursing process was “ADPIE,” which stands for Assessment, Diagnosis, Planning, Implementation, and Evaluation. This nursing framework was developed in the 1950s and continues to be used in some capacity today, though most nursing schools have replaced it with the CJMM, which further expands the nursing process into more measurable components. A description of each of the steps of the original nursing framework is given in Table 28.1.
| Steps | Description |
|---|---|
| Assessment |
|
| Diagnosis |
|
| Planning |
|
| Implementation |
|
| Evaluation |
|
Using the nursing process has many benefits for nurses, patients, and other members of the healthcare team. The benefits of using the nursing process include:
- decreases omissions and duplications
- encourages collaborative management of a patient’s healthcare problems
- identifies a patient’s goals and strategies to attain them
- improves patient safety
- improves patient satisfaction
- increases the likelihood of achieving positive patient outcomes
- promotes quality patient care
- provides a guide for all staff involved to provide consistent and responsive care saves time, energy, and frustration by creating a care plan or path to follow
Learn more about the CJMM to explore new ways of testing clinical judgment in nursing as part of the NCLEX.
Application of the CJMM to Nursing Practice
There are several layers to the CJMM framework. Layer 0, at the top, is the broadest layer and the layers get more specific as the reach the bottom at layer 4. As nurses move through layers 2 through 4, they are working through a cognitive process that helps them make clinical decisions for patients, using clinical judgment to do so (as represented by the broad layers 0 and 1 at the top of the model). The more specific layers (2–4) are discussed in more detail in the following sections.
Form, Refine, and Evaluate Hypotheses
Layer 2 of the CJMM is composed of three different parts: form hypotheses, refine hypotheses, and evaluation. The nurse uses specific patient assessment findings (“cues”) to develop hypotheses, or educated guesses, about the patient’s condition. The nurse uses the cues in combination with their foundational clinical knowledge to determine potential explanations for the patient’s situation. As more cues are assessed and more information about the patient is obtained, the nurse can refine their initial hypothesis and determine its accuracy. As the nurse moves through the steps in layers 3 and 4 to make clinical decisions, they are also checking in with the parts of this layer to continuously re-evaluate the plan of care. As you will read in the next section, each of the cognitive skills in layer 3 corresponds to a component of layer 2. As the nurse assesses and cares for a patient, they can determine whether their actions have satisfied the goals of care. If the goals are not satisfied, the nurse can move again through the steps in layers 3 and 4 to revise the care plan and make different clinical decisions that, hopefully, will assist in better meeting the patient's needs.
Application of Cognitive Skills
Layer 3 of the CJMM is composed of six steps that involve a repetitious process that improves with time and nursing experience. Eventually, these steps become second nature to a more experienced nurse, but they serve as a framework for nursing students and early-career nurses to use as a more deliberate guide for making clinical decisions. These six cognitive skill steps (also known as clinical judgment functions) are described in more detail in Table 28.2.
| Cognitive Skill | Description |
|---|---|
| Recognize cues |
|
| Analyze cues |
|
| Prioritize hypotheses |
|
| Generate solutions |
|
| Take action |
|
| Evaluate outcomes |
|
Expected Responses and Behaviors
For each cognitive skill, there are specific responses and behaviors the nursing student is expected to exhibit. When expected responses are performed by the student, it indicates that the cognitive skill has been adequately demonstrated, thus the student is using clinical judgment effectively. Expected responses and behaviors by the student that would indicate they can successfully recognize cues include recognizing assessment data pertinent to the patient’s condition, identifying subtle changes in the patient’s condition, and using knowledge and experience to thoroughly assess the patient.
To analyze cues, it is expected that the student can recognize abnormal assessment findings, anticipate patient needs, identify potential complications that may arise, and begin to prioritize patient problems. When prioritizing hypotheses, the student nurse should be able to organize data and findings based on patterns and trends and prioritize the patient’s goals and needs. During the generate solutions (planning) step, the student nurse is expected to collaborate with the interprofessional care team to establish goals of care, prioritize nursing interventions based on the patient’s needs, and continuously modify the plan of care based on patient condition changes. When the student begins to the take action step, it is expected that they will accurately perform nursing interventions based on previously established patient priorities and needs, document care appropriately, and provide education to the patient and family. When evaluating outcomes, expected behaviors include reassessing the patient’s condition to determine whether goals and outcomes have been met, evaluating how effective nursing interventions were, and modifying patient goals and priorities as needed.
Consideration of Factors
While using clinical judgment to make practice decisions, it is important to do so within the context of environmental and individual factors. These factors are specific to each patient and help the nurse make informed, personalized decisions.
Environmental Factors
Environmental factors are things within the external environment that may affect the clinical decision-making process. These factors include:
- culture (e.g., diet, religion, language, literacy)
- individual medical information, including medical history, laboratory and diagnostic test results, intake and output, medications, and current treatments
- patient demographics
- resources (e.g., supplies, staffing, open beds)
- setting (e.g., hospital, long-term care, community health center)
- situational factors (e.g., safety considerations, available equipment, surroundings)
- time pressure related to emergent orders and changes in patient condition
Cultural factors must be considered as part of the clinical judgment process. Without this context, it may be difficult to ascertain the cause of the patient’s condition. For example, in some cultures, a vegetarian diet is common. In the context of this factor, the nurse must consider that a lack of protein and vitamin B12 may be the cause of certain medical issues within this population. Although considering cultural factors is important, it is even more important not to generalize information about cultures, because this can border on perpetuating stereotypes. There is a difference between generalizing information about all members of a population versus asking questions about the patient’s culture to get specific information that can help the nurse make informed clinical decisions.
Individual Factors
Individual factors that must be considered during the clinical judgment process are more related to the nurse than the patient. These factors include:
- cognitive load of the nurse (e.g., demands, job stress, problem solving skills, memory)
- nurse characteristics, including attitudes, prior experiences, amount of nursing experience
- nurse specialty, knowledge, and skills
Nurse:
Gabby, BSN
Clinical setting:
Medical-surgical unit
Years in practice:
3
Facility location:
Small community hospital in rural Georgia
After clocking in for my shift, I was notified by the charge nurse that I would be floating to a different unit for the night. She told me I would need to go to the labor and delivery unit to help out because there were several patients being admitted who would be delivering babies that night. I had only ever worked on a medical-surgical unit, so I expressed that I was concerned and uncomfortable taking care of this patient population. The charge nurse shrugged her shoulders and informed me I was floating to the unit anyway.
Once I got to the unit, the labor and delivery charge nurse told me I would be taking care of a patient who had just delivered a baby 2 hours ago. I again expressed my discomfort, because I had never taken care of postpartum mothers or babies. The charge nurse seemed surprised by this and stated, “I was told they were sending down a nurse with obstetrics experience. Let me make a call real quick.”
When the charge nurse came back from making her phone call, she informed me that there had been a mistake and a different nurse with obstetrics experience was supposed to float to the unit, not me. I breathed a huge sigh of relief and headed back to my home unit, thankful that I advocated for myself and the patients.


14.8: Physical Properties of Alkanes
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Learning Objectives
- To identify the physical properties of alkanes and describe trends in these properties.
Because alkanes have relatively predictable physical properties and undergo relatively few chemical reactions other than combustion, they serve as a basis of comparison for the properties of many other organic compound families. Let’s consider their physical properties first.
Table \(\PageIndex{1}\) describes some of the properties of some of the first 10 straight-chain alkanes. Because alkane molecules are nonpolar, they are insoluble in water, which is a polar solvent, but are soluble in nonpolar and slightly polar solvents. Consequently, alkanes themselves are commonly used as solvents for organic substances of low polarity, such as fats, oils, and waxes. Nearly all alkanes have densities less than 1.0 g/mL and are therefore less dense than water (the density of H 2 O is 1.00 g/mL at 20°C). These properties explain why oil and grease do not mix with water but rather float on its surface.
| Molecular Name | Formula | Melting Point (°C) | Boiling Point (°C) | Density (20°C)* | Physical State (at 20°C) |
|---|---|---|---|---|---|
| methane | CH 4 | –182 | –164 | 0.668 g/L | gas |
| ethane | C 2 H 6 | –183 | –89 | 1.265 g/L | gas |
| propane | C 3 H 8 | –190 | –42 | 1.867 g/L | gas |
| butane | C 4 H 10 | –138 | –1 | 2.493 g/L | gas |
| pentane | C 5 H 12 | –130 | 36 | 0.626 g/mL | liquid |
| hexane | C 6 H 14 | –95 | 69 | 0.659 g/mL | liquid |
| octane | C 8 H 18 | –57 | 125 | 0.703 g/mL | liquid |
| decane | C 10 H 22 | –30 | 174 | 0.730 g/mL | liquid |
| *Note the change in units going from gases (grams per liter) to liquids (grams per milliliter). Gas densities are at 1 atm pressure. |
Looking Closer: Gas Densities and Fire Hazards
Table \(\PageIndex{1}\) indicates that the first four members of the alkane series are gases at ordinary temperatures. Natural gas is composed chiefly of methane, which has a density of about 0.67 g/L. The density of air is about 1.29 g/L. Because natural gas is less dense than air, it rises. When a natural-gas leak is detected and shut off in a room, the gas can be removed by opening an upper window. On the other hand, bottled gas can be either propane (density 1.88 g/L) or butanes (a mixture of butane and isobutane; density about 2.5 g/L). Both are much heavier than air (density 1.2 g/L). If bottled gas escapes into a building, it collects near the floor. This presents a much more serious fire hazard than a natural-gas leak because it is more difficult to rid the room of the heavier gas.
Also shown in Table \(\PageIndex{1}\) are the boiling points of the straight-chain alkanes increase with increasing molar mass. This general rule holds true for the straight-chain homologs of all organic compound families. Larger molecules have greater surface areas and consequently interact more strongly; more energy is therefore required to separate them. For a given molar mass, the boiling points of alkanes are relatively low because these nonpolar molecules have only weak dispersion forces to hold them together in the liquid state.
Looking Closer: An Alkane Basis for Properties of Other Compounds
An understanding of the physical properties of the alkanes is important in that petroleum and natural gas and the many products derived from them—gasoline, bottled gas, solvents, plastics, and more—are composed primarily of alkanes. This understanding is also vital because it is the basis for describing the properties of other organic and biological compound families. For example, large portions of the structures of lipids consist of nonpolar alkyl groups. Lipids include the dietary fats and fatlike compounds called phospholipids and sphingolipids that serve as structural components of living tissues. These compounds have both polar and nonpolar groups, enabling them to bridge the gap between water-soluble and water-insoluble phases. This characteristic is essential for the selective permeability of cell membranes.
Key Takeaway
-
Alkanes are nonpolar compounds that are low boiling and insoluble in water.


Health Concerns During Adolescence
Nutrition
Adequate adolescent nutrition is necessary for optimal growth and development. Dietary choices and habits established during adolescence greatly influence future health, yet many studies report that teens consume few fruits and vegetables and are not receiving the calcium, iron, vitamins, or minerals necessary for healthy development.
One of the reasons for poor nutrition is anxiety about
body image,
which is a person’s idea of how his or her body looks. The way adolescents feel about their bodies can affect the way they feel about themselves as a whole. Few adolescents welcome their sudden weight increase, so they may adjust their eating habits to lose weight. Adding to the rapid physical changes, they are simultaneously bombarded by messages, and sometimes teasing, related to body image, appearance, attractiveness, weight, and eating that they encounter in the media, at home, and from their friends/peers (both in person and via social media).
Much research has been conducted on the psychological ramifications of body image on adolescents. Modern day teenagers are exposed to more media on a daily basis than any generation before them. Recent studies have indicated that the average teenager watches roughly 1500 hours of television per year, and 70% use social media multiple times a day.
As such, modern day adolescents are exposed to many representations of ideal, societal beauty. The concept of a person being unhappy with their own image or appearance has been defined as “
body dissatisfaction
.” In teenagers, body dissatisfaction is often associated with body mass, low self-esteem, and atypical eating patterns. Scholars continue to debate the effects of media on body dissatisfaction in teens. What we do know is that two-thirds of U.S. high school girls are trying to lose weight and one-third think they are overweight, while only one-sixth are actually overweight (MMWR, June 10, 2016).
Eating Disorders
Dissatisfaction with body image can explain why many teens, mostly girls, eat erratically or ingest diet pills to lose weight and why boys may take steroids to increase their muscle mass. Although eating disorders can occur in children and adults, they frequently appear during the teen years or young adulthood (National Institute of Mental Health (NIMH), 2019).
https://www.nimh.nih.gov/health/topi...rs/index.shtml
Eating disorders affect both genders, although rates among women are 2½ times greater than among men. Similar to women who have eating disorders, some men also have a distorted sense of body image, including
muscle dysmorphia
or an extreme concern with becoming more muscular.
Because of the high mortality rate, researchers are looking into the etiology of the disorder and associated risk factors. Researchers are finding that eating disorders are caused by a complex interaction of genetic, biological, behavioral, psychological, and social factors (NIMH, 2019). Eating disorders appear to run in families, and researchers are working to identify DNA variations that are linked to the increased risk of developing eating disorders. Researchers have also found differences in patterns of brain activity in women with eating disorders in comparison with healthy women. The main criteria for the most common eating disorders:
anorexia nervosa
,
bulimia nervosa
, and
binge-eating disorder
are described in the Diagnostic and Statistical Manual of Mental Disorders-Fifth Edition, DSM-5 (American Psychiatric Association, 2013).
Health Consequences of Eating Disorders
For those suffering from anorexia, health consequences include an abnormally slow heart rate and low blood pressure, which increase
s
the risk for heart failure. Additionally, there is a reduction in bone density (osteoporosis), muscle loss and weakness, severe dehydration, fainting, fatigue, and overall weakness. Anorexia nervosa has the highest mortality rate of any psychiatric disorder.  In women, suicide is much more common in those with anorexia than with most other mental disorders.
The binging and purging cycle of bulimia can affect the digestive system and lead to electrolyte and chemical imbalances that can affect the heart and other major organs. Frequent vomiting can cause inflammation and possible rupture of the esophagus, as well as tooth decay and staining from stomach acids. Lastly, binge eating disorder results in similar health risks to obesity, including high blood pressure, high cholesterol level
s
, heart disease, Type II diabetes, and gall bladder disease (National Eating Disorders Association, 2016).
Eating Disorders Treatment
To treat eating disorders, getting adequate nutrition and stopping inappropriate behaviors, such as purging, are the foundations of treatment. Treatment plans are tailored to individual needs and include medical care, nutritional counseling, medications (such as antidepressants), and individual, group, and/or family psychotherapy (NIMH, 2019). For example, the Maudsley Approach has parents of adolescents with anorexia nervosa be actively involved their child’s treatment, such as assuming responsibility for feeding their child. To eliminate binge eating and purging behaviors, cognitive behavioral therapy (CBT) assists sufferers by identifying distorted thinking patterns and changing inaccurate beliefs.
Sexual Development
Developing sexually is an expected and natural part of growing into adulthood.   During adolescence, teens strive to become comfortable with their changing bodies and to make healthy, safe decisions about which sexual activities, if any, they wish to engage in.
Earlier in the physical development section, we discussed
primary and secondary sex characteristics
. During puberty, every primary sex organ (the ovaries, uterus, penis, and testes) increases dramatically in size and matures in function. During puberty, reproduction becomes possible. Simultaneously, secondary sex characteristics develop. These characteristics are not required for reproduction, but they do signify masculinity and femininity. At birth, boys and girls have similar body shapes, but during puberty, males widen at the shoulders and females widen at the hips and develop breasts (examples of secondary sex characteristics). Sexual development is impacted by a dynamic mixture of physical and cognitive change coupled with social expectations. With physical maturation, adolescents may become alternately fascinated with and chagrined by their changing bodies, and often compare themselves to the development they notice in their peers or see in the media. For example, many adolescent girls focus on their breast development, hoping their breasts will conform to an ideal body image.
 Culture, however, shapes actual sexual behaviors.  Simply put, the most important influence on adolescents’ sexual activity is not their bodies, but their close friends, who have more influence than do sex or ethnic group norms (van de Bongardt et al., 2015).
Sexual interest and interaction are a natural part of adolescence. Sexual fantasy and
masturbation
episodes increase between the ages of 10 and 13. Masturbation is very ordinary—even young children have been known to engage in this behavior.  For adolescents, masturbation is a common way to explore their erotic potential, and this behavior can continue throughout adult life.
Sexual Interactions
Many early social interactions tend to be nonsexual—text messaging, phone calls, email—but by the age of 12 or 13, some young people may pair off and begin dating and experimenting with kissing, touching, and other physical contact, such as oral sex. The vast majority of young adolescents are not prepared emotionally or physically for oral sex and sexual intercourse. If adolescents this young do have sex, they are highly vulnerable for sexual and emotional abuse,
sexually transmitted infections (STIs)
, HIV, and early pregnancy (
https://pedsinreview.aappublications...ontent/34/1/29
). For STI’s in particular, adolescents are slower to recognize symptoms, tell partners, and get medical treatment, which puts them at risk of infertility and even death.
Adolescents ages 14 to 16 understand the consequences of unprotected sex and teen parenthood, if properly taught, but cognitively they may lack the skills to integrate this knowledge into everyday situations or consistently to act responsibly in the heat of the moment. By the age of 17, many adolescents have willingly experienced sexual intercourse. Teens who have early sexual intercourse report strong peer pressure as a reason behind their decision. Some adolescents are just curious about sex and want to experience it.
Becoming a sexually healthy adult is a developmental task of adolescence that requires integrating psychological, physical, cultural, spiritual, societal, and educational factors. It is particularly important to understand the adolescent in terms of his or her physical, emotional, and cognitive stage. Additionally, healthy adult relationships are more likely to develop when adolescent impulses are not shamed or feared. Guidance is certainly needed, but acknowledging that adolescent sexuality development is both normal and positive would allow for more open communication so adolescents can be more receptive to education concerning the risks (Tolman & McClelland, 2011).
Adolescents are receptive to their culture, to the models they see at home, in school, and in the mass media. These observations influence moral reasoning and moral behavior, which we discuss in more detail later in this module.
Freud and Sexual Development
According to Sigmund Freud, adolescents are in the genital stage of psychosexual development. This stage begins around the time that puberty starts, and ends at death. According to Freud, the genital stage is similar to the phallic stage, in that its main concern is the genitalia; however, this concern is now conscious. The genital stage comes about when the sexual and aggressive drives have returned, but the source of sexual pleasure expands outside of the mother and father (as in the Oedipus or Electra complex).
During the genital stage the ego and superego have become more developed. This allows the individual to have a more realistic way of thinking and to establish an assortment of social relations apart from the family. The genital stage is the last stage and is considered the highest level of maturity. In this stage a person’s concern shifts from primary-drive gratification (instinct) to applying secondary process-thinking to gratify desire symbolically and intellectually by means of friendships, intimate relationships, and family and adult responsibilities.
Glossary
-
anorexia nervosa
-
an eating disorder characterized by self-starvation. Affected individuals voluntarily undereat and often overexercise, depriving their vital organs of nutrition. Anorexia can be fatal
-
binge-eating disorder
-
an eating disorder characterized by recurrent episodes of eating large quantities of food (often very quickly and to the point of discomfort); a feeling of a loss of control during the binge; experiencing shame, distress or guilt afterwards; and not regularly using unhealthy compensatory measures (e.g., purging) to counter the binge eating. It is the most common eating disorder in the United States
-
body dissatisfaction
-
negative subjective evaluation of the weight and shape of one’s own body, which may predict the onset, severity, and treatment outcomes of eating disorders
-
body image
-
a person’s idea of how his or her body looks
-
bulimia nervosa
-
an eating disorder characterized by binge eating and subsequent purging, usually by induced vomiting and/or use of laxatives
-
masturbation
-
sexual self-stimulation, usually achieved by touching, stroking, or massaging the male or female genitals until this triggers an orgasm
-
muscle dysmorphia
-
sometimes called “reverse anorexia” this is an obsession with being small and underdeveloped; extreme concern with becoming more muscular
-
primary sex characteristics
-
the parts of the body that are directly involved in reproduction, including the vagina, uterus, ovaries, testicles, and penis
-
secondary sex characteristics
-
physical traits that are not directly involved in reproduction but that indicate sexual maturity, such as a man’s beard or a woman’s breasts
-
sexually transmitted infections (STIs)
-
diseases that are spread by sexual contact, including syphilis, gonorrhea, genital herpes, chlamydia, and HIV/AIDS


3.3: Using Calculus to Describe Motion
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Objects do not necessarily have a constant velocity or acceleration. We thus need to extend our description of the position and velocity of an object to a more general case. This can be done in much the same way as we introduced accelerated motion; namely by pretending that during a very small interval in time, \(∆t\), the velocity and acceleration are constant, and then considering the motion as the sum over many small intervals in time. In the limit that \(∆t\) tends to zero, this will be an accurate description.
Instantaneous and average velocity
Suppose that an object is moving with a non constant velocity, and covers a distance \(\Delta x\) in an amount of time \(\Delta t\) . We can define an average velocity , \(v^{avg}\) :
\[v^{avg}= \frac{\Delta x}{\Delta t}\]
That is, regardless of our choice of time interval, \(\Delta t\) , we can always calculate the average velocity, \(v^{avg}\) , of an object over a particular distance. That average velocity will be an average over the interval, between some time \(t\) and \(t+\Delta t\) . If we shrink the length of the time interval used to measure the velocity, and take the limit \(\Delta t\rightarrow 0\) , we can define the instantaneous velocity :
\[v = \lim_{\Delta t\to 0} \frac{\Delta x}{\Delta t}\]
The instantaneous velocity is the velocity only in that small instant in time where we choose \(\Delta x\) and \(\Delta t\) . Another way to read this equation is that the velocity, \(v\) , is the slope of the graph of \(x(t)\) . Recall that the slope is the “rise over run”, in other words the change in \(x\) divided by the corresponding change in \(t\) . Indeed, when we had no acceleration, the position as a function of time, Equation 3.1.1 , explicitly had the velocity as the slope of a linear function:
\[x(t) = v_{0x}+v_{x}t\]
If we go back to Figure \(\PageIndex{1}\), where velocity was no longer constant, we can indeed see that the graph of the velocity versus time ( \(v(t)\) ) corresponds to the instantaneous slope of the graph of position versus time ( \(x(t)\) ). For \(t<1.02\text{s}\) , the slope of the \(x(t)\) graph is positive but decreasing (as is \(v(t)\) ). At \(t=1.02\text{s}\) , the slope of \(x(t)\) is instantaneously \(0\text{m/s}\) (as is the velocity). Finally, for \(t>1.02\text{s}\) , the slope of \(x(t)\) is negative and increasing in magnitude, as is \(v(t)\) .
Leibniz and Newton were the first to develop mathematical tools to deal with calculations that involve quantities that tend to zero, as we have here for our time interval \(\Delta t\) . Nowadays, we call that field of mathematics “calculus”, and we will make use of it here. Using the vocabulary of calculus, rather than saying that “instantaneous velocity is the slope of the graph of position versus time at some point in time”, we say that “instantaneous velocity is the time derivative of position as a function of time”. We also use a slightly different notation so that we do not have to write the limit \(\lim_{\Delta t\to 0}\) :
\[ v(t)=\lim_{\Delta t\to 0} \frac{\Delta x}{\Delta t}=\frac{dx}{dt}=\frac{d}{dt} x(t)\]
where we can really think of \(dt\) as \(\lim_{\Delta t\to 0}\Delta t\) , and \(dx\) as the corresponding change in position over an infinitesimally small time interval \(dt\) .
Similarly, we introduce the instantaneous acceleration , as the time derivative of \(v(t)\) :
\[a_x(t)=\frac{dv}{dt}=\frac{d}{dt}v(t)\]
Olivia's Thoughts
When looking at a graph of position versus time, it is sometimes hard to tell at first glance whether the speed of the object is increasing or decreasing. This section gives us an easy way to figure it out. The velocity is the instantaneous slope of the graph \(x(t)\), so the speed is the “steepness” of that graph. Simply draw a few lines that are tangent to (meaning just touching) the curve, and see what happens as time increases. If the lines get steeper, the object is speeding up. If they are getting flatter, the object is slowing down.
From here, you can also figure out what the direction of the acceleration is. If an object is speeding up, the acceleration and velocity must be in the same direction (i.e. both positive or both negative). If the object is slowing down, they must be in opposite directions. Imagine the graphs in Figure \(\PageIndex{1}\) are describing the motion of a person running in heavy wind. In the graph on the left, the person is running with the wind and accelerating (\(v(t)\) and \(a(t)\) positive), and in the second graph the person is running against the wind and decelerating (\(v(t)\) positive and \(a(t)\) negative).
Using calculus to obtain acceleration from position
Suppose that we know the function for position as a function of time, and that it is given by our previous result (for the case when the acceleration \(a_{x}\) is constant):
\[x(t)=x_0+v_{0x}t+\frac{1}{2}a_{x}t^{2}\]
With our calculus-based definitions above, we should be able to recover that:
\[\begin{aligned} v(t) = v_{0x}t+at\\[4pt] a_x(t) = a_x\end{aligned}\]
Let us start by determining \(v(t)\) :
\[\begin{aligned} v(t) = \frac{dx}{dt}=\lim_{\Delta t\to 0} \frac{\Delta x}{\Delta t}\end{aligned}\]
Knowing our function \(x(t)\) , we can can rewrite this as:
\[\begin{aligned} \frac{dx}{dt}=\lim_{\Delta t\to 0} \frac{x(t+\Delta t)-x(t)}{\Delta t} \end{aligned}\]
which is the formal definition of the derivative \(\frac{dx}{dt}\) and gives us a prescription to evaluate it. We will proceed as illustrated in Figure \(\PageIndex{1}\):
- Use \(x(t)\) to determine \(\Delta x\) for a small interval \(\Delta t\)
- Divide \(\Delta x\) by \(\Delta t\)
- Take the limit, \(\lim_{\Delta t\to 0}\)
To obtain \(\Delta x\) , we introduce a start time, \(t_1\) , and an end time \(t_2\) for our time interval, such that \(t_2-t_1=\Delta t\) , centerd about a time \(t\) . The change in position is then given by:
\[\begin{aligned} \Delta x &= x(t_2) - x(t_1)\\[4pt] &=\left(x_0+v_{0x}t_2+\frac{1}{2}a_xt_2^2\right )- \left(x_0+v_{0x}t_1+\frac{1}{2}a_xt_1^2\right )\\[4pt] &=v_{0x}(t_2-t_1)+\frac{1}{2}a_x(t_2^2-t_1^2)\\[4pt] &=v_{0x}\Delta t+\frac{1}{2}a_x(t_2-t_1)(t_2+t_1)\\[4pt] &=v_{0x}\Delta t+\frac{1}{2}a_x\Delta t (t_2+t_1)\\[4pt]\end{aligned}\]
which we divide by \(\Delta t\) to get \(v(t)\) :
\[\begin{aligned} v(t) &= \frac{\Delta x}{\Delta t}\\[4pt] &=\frac{v_{0x}\Delta t+\frac{1}{2}a_x\Delta t (t_2+t_1)}{\Delta t}\\[4pt] &=v_{0x}+\frac{1}{2}a_x(t_2+t_1)\end{aligned}\]
We now take the limit where \(\Delta t\to 0\) , namely when \(t_2-t_1\) is very small. As we make the interval small, \(t_2\) and \(t_1\) will both approach the same value of time, say \(t\) , corresponding to the time at the center of the interval. In particular, the average of \(t_1\) and \(t_2\) , given by \(\frac{1}{2}(t_1+t_2)\) , will approach the time at the center of the interval, \(t\) . We thus recover the equation for instantaneous velocity:
\[\begin{aligned} v(t) &= v_{0x}+a_xt\end{aligned}\]
Of course, once you become more familiar with calculus, you will be able to directly use the formulas for derivatives to recover the answer:
\[\begin{aligned} v(t) &= \frac{d}{dt}\left( x_0+v_{0x}t+\frac{1}{2}a_xt^2 \right) \\[4pt] &= v_{0x}+at \end{aligned}\]
Similarly, we can now confirm that the acceleration is a constant, independent of time:
\[\begin{aligned} a_x(t) &= \frac{dv}{dt} = \frac{d}{dt}\left(v_{0x}+a_xt \right)\\[4pt] &=a_x\end{aligned}\]
You can easily verify that you obtain this result by first calculating \(v(t)\) at two different times, \(t_1\) and \(t_2\) , taking the difference, \(\Delta v = v(t_2)-v(t_1)\) , and then taking the limit of \(\lim_{\Delta t\to 0}\frac{\Delta v}{\Delta t}\) to get \(a_x(t)\) .
Exercise \(\PageIndex{1}\)
Chloë has been working on a detailed study of how vicuñas run, and found that their position as a function of time when they start running is well modelled by the function \(x(t)=(40\text{m/s}^{2})t^{2}+(20\text{m/s}^{3})t^{3}\) . What is the acceleration of the vicuñas?
- \(a_x(t)=40\text{m/s}^{2}\)
- \(a_x(t)=80\text{m/s}^{2}\)
- \(a_x(t)=40\text{m/s}^{2}+(20\text{m/s}^{3})t\)
- \(a_x(t)=80\text{m/s}^{2}+(120\text{m/s}^{3})t\)
"Never heard of vicuñas? Internet!
- Answer
- The acceleration is time dependent. It begins as twice the first term because position depends on \(0.5 a\). Over time, the acceleration changes by \(2\times 3 \times 20 t\), where the three is similar to the factor of 2 in constant acceleration. Therefore, the correct answer is \(a_x(t)=80\text{m/s}^{2}+(120\text{m/s}^{3})t\).
Using calculus to obtain position from acceleration
Now that we saw that we can use derivatives to determine acceleration from position, we will see how to do the reverse and use acceleration to determine position. Let us suppose that we have a constant acceleration, \(a_{x}(t)=a_{x}\) , and that we know that at time \(t=0\text{s}\) , the object had a speed of \(v_{0x}\) and was located at a position \(x_{0}\) .
Since we only know the acceleration as a function of time, we first need to find the velocity as a function of time. We start with:
\[a_{x}(t)=a_{x}=\frac{d}{dt} v(t)\]
which tells us that we know the slope (derivative) of the function \(v(t)\) , but not the actual function. In this case, we must do the opposite of taking the derivative, which in calculus is called taking the “anti-derivative” with respect to \(t\) and has the symbol \(\int dt\) . In other words, if:
\[\frac{d}{dt} v(t) =a_{x}(t)\]
then:
\[v(t) =\int a_{x}(t) dt +C\]
where as we will see later, the constant \(C\) is required in order for the function \(v(t)\) to go through the point \(v(t=0)=v_{0x}\) . All we need now is to determine how to calculate the anti-derivative, \(\int a_{x}(t) dt\) . Since in this case, \(a_{x}(t)\) is a constant, \(a_{x}\) , we can determine the anti-derivative quite easily.
For a small interval in time, \(\Delta t\) , the velocity will change by a small amount, \(\Delta v\) , such that:
\[\begin{aligned} a_x &= \frac{\Delta v}{\Delta t}\\[4pt] \Delta v &= a_x \Delta t\end{aligned}\]
If we label the start time of the interval as \(t_1\) and the end of the interval as \(t_2\) , we have:
\[\begin{aligned} \Delta t &= t_2 - t_1 \\[4pt] \Delta v &= v(t_2) - v(t_1) = a_x \Delta t\\[4pt] \therefore v(t_2) &= v(t_1)+a_x\Delta t\end{aligned}\]
If we set \(t_1=0\text{s}\) to correspond to the point where \(v(t)=v_{0x}\) , then we can write the velocity at \(t_2\) as:
\[\begin{aligned} v(t_2) = v(t=\Delta t) = v_{0x}+a_x\Delta t\end{aligned}\]
and we see that the velocity changed by an amount \(a \Delta t\) over a period of time \(\Delta t\) . Since \(a_x\) is the same at all times, this is always true, and after a period of time \(t = N\Delta t\) , the velocity will have changed by \(N a_x \Delta t\) , and we recover the original equation for velocity as a function of time when acceleration is constant:
\[\begin{aligned} v(t=N \Delta t) &= v_{0x}+Na_x\Delta t\\[4pt] \therefore v(t) &= v_{0x}+a_xt\\[4pt]\end{aligned}\]
We can identify the anti-derivative for the case where \(a(t)\) is constant:
\[\begin{aligned} \frac{d}{dt} v(t) &=a_x\\[4pt] \int a_x dt &=a_xt +C\end{aligned}\]
where the constant, \(C\) , is given by \(v_{0x}\) .
\[v(t)=C+a_{x}t=v_{0x}+a_{x}t\]
and we recover the formula for velocity when the acceleration is constant. Now that we know the velocity as a function of time, we can take one more anti-derivative with respect to time to obtain the position:
\(\begin{aligned}&v(t)=\frac{dx}{dt} \\[4pt] &\therefore x(t)=\int v(t)dt\end{aligned}\)
In the case where acceleration is constant, this gives:
\(\begin{aligned} x(t)&=\int v(t)dt \\[4pt] &=\int (v_{0x}+a_{x}t)dt \\[4pt] &=v_{0x}t+\frac{1}{2}a_{x}t^{2}+C' \end{aligned}\)
where \(C'\) is a different constant than the one we had when determining velocity. The constant is given by our initial conditions. If the object was located at position \(x = x_{0}\) at time \(t = 0\), then \(C'= x_{0}\) and we recover the equation for position as a function of time for constant acceleration:
\[x(t)=x_{0}+v_{0x}t+\frac{1}{2}a_{x}t^{2}\]
To now obtain position as a function of time, we proceed in the same manner, namely:
- Define a small interval in time \(\Delta t\)
- Calculate the corresponding change in position \(\Delta x\)
- Add \(\Delta x\) to our original position, and repeat.
Again, we have the time-derivative of the position equal to a function of time:
\[\begin{aligned} \frac{d}{dt}x(t)=v(t)=v_{0x}+a_xt\end{aligned}\]
and we need to find the anti-derivative:
\[\begin{aligned} x(t) = \int \left( v_{0x}+av_xt \right) dt \end{aligned}\]
given that, at time \(t=0\) , the position was \(x=x_0\) . After a small interval in time, \(\Delta t\) , the position will have changed by an amount \(\Delta x\) :
\[\begin{aligned} \Delta x &= v(t) \Delta t\\[4pt]\end{aligned}\]
so that the position at time \(t=\Delta t\) will be given by:
\[\begin{aligned} x(t=\Delta t) &= x_0+ \Delta x\\[4pt] & = x_0+v(t) \Delta t\end{aligned}\]
The problem here is to evaluate \(v(t)\) since the velocity changes throughout the interval. One possible choice is to evaluate the velocity at \(t = \frac{1}{2}\Delta t\) , midway in the interval, as we did before. We could also choose to use the velocity at the beginning or at the end of the time interval, as all three choices will converge to the same value when \(\Delta t \to 0\) . For now, we will leave the choice open and simply call the velocity that we use \(v_1\) to indicate that it is the velocity in the first interval. We thus write the position, \(x(t=\Delta t)\) , after a time interval \(\Delta t\) as:
\[\begin{aligned} x(t=\Delta t) = x_0+v_1\Delta t\end{aligned}\]
The position, \(x(t=2\Delta t)\) , after another interval in time \(\Delta t\) will then be given by:
\[\begin{aligned} x(t=2\Delta t) &= x(t=\Delta t)+v_2\Delta t\\[4pt] &=x_0+v_1\Delta t+v_2\Delta t\end{aligned}\]
where \(v_2\) is the velocity over the second interval in time (different than \(v_1\) , since velocity changes with time). For the Nth interval, we label the position \(x_N=x(t=N\Delta t)\) :
\[\begin{aligned} x_N=x(t=N\Delta t)&=x_0+v_1\Delta t+v_2\Delta t+\dots+v_N\Delta t\\[4pt] &=x_0+\sum_{i=1}^Nv_i\Delta t \end{aligned}\]
where we made use of the summation notation ( \(\sum\) ) to avoid writing out every term. The above equation is only correct in the limit of \(\Delta t\to 0\) , in which case it must be the anti-derivative of \(v(t)\) :
\[\begin{aligned} x(t) &= \int v(t) dt\\[4pt] &= x_0+\lim_{\Delta t\to 0}\sum_{i=1}^Nv_i\Delta t \end{aligned}\]
so that we can identify:
\[ \int v(t) dt=\lim_{\Delta t\to 0}\sum_{i=1}^Nv_i\Delta t + C \]
where it is understood that \(v_i\) is the “average” velocity in the ith interval, and the constant \(C=x_0\) ensures that \(x(t=0)=x_0\) . This is now a general definition for the anti-derivative, as we have made no specific assumption about the function \(v(t)\) . Equation 3.3.10 tells us that the anti-derivative of a function (in this case, \(v(t)\) ) can be obtained from a sum.
The sum shows the function \(v(t)\) and several intervals \(\Delta t\) . Over each interval, \(i\) , we labeled the average velocity, \(v_i\) . As the intervals shrink, \(\Delta t\to 0\) , the average velocity \(v_i\) approaches the instantaneous velocity, \(v(t)\) , at the center for the interval. Since the function \(v(t)\) is linear (for constant acceleration), the speed at the middle of an interval is exactly equal to the average speed in the interval. Taking \(v_i\) as the speed in the middle of the interval, we then see that each term in the sum, \(v_i\Delta t\) , is equal to the area between the curve \(v(t)\) and the t-axis. This is illustrated for the second term in the sum, \(v_2\Delta t\).
The anti-derivative of a function is thus related to the area between the function and the horizontal axis. If we specify limits on the horizontal axis between which we calculate the area, then the anti-derivative is called an integral . For example, if we wish to calculate the sum of \(v_i \Delta t\) for values between \(t_a\) and \(t_b\) , we would write the integral as:
\[ \int_{t_a}^{t_b}v(t) dt = \lim_{\Delta t\to 0}\sum_{i=1}^Nv_i\Delta t \]
where the sum is such that for \(i=1\) , \(v_i\) is close to \(v(t=t_a)\) and for \(i=N\) , \(v_N\) is close to \(v(t=t_b)\) . An illustration of taking the integral of \(v(t)\) is shown where the sum is shown for two different values of \(\Delta t\) . It is clear that as \(\Delta t \to 0\) , the sum becomes equal to the area between the curve and the horizontal axis.
Since \(v(t)\) is a linear function when acceleration is constant, we can easily calculate the area between the curve and the horizontal axis. In the case of a linear function, \(v(t)=v_{0x}+at\) , the area is a trapezoid, and we have:
\[\begin{aligned} \int_{t_a}^{t_b}v(t) dt &= \text{base}\times\text{average height}\\[4pt] &=(t_b-t_a)\times\frac{1}{2}\left(v(t_a)+v(t_b)\right)\\[4pt] &=(t_b-t_a)\frac{1}{2}(v_{0x}+a_xt_a+v_{0x}+a_xt_b)\\[4pt] &=\left( \frac{1}{2}(v_{0x}t_b+a_xt_at_b+v_{0x}t_b+a_xt_b^2) \right)- \left( \frac{1}{2}(v_{0x}t_a+a_xt_a^2+v_{0x}t_a+a_xt_bt_a) \right)\\[4pt] &=\left( v_{0x}t_b+\frac{1}{2}a_xt_b^2 \right)-\left( v_{0x}t_a+\frac{1}{2}a_xt_a^2 \right)\end{aligned}\]
If we define a new function, \(V(t)=v_{0x}t+\frac{1}{2}a_xt^{2}\) , then we have:
\[ \int_{t_{a}}^{t_{b}}v(t) dt = V(t_{b}) -V(t_{a})\]
In other words, given a function, \(v(t)\) , the integral of that function between two values \(t_a\) and \(t_b\) can be found by evaluating a different function, \(V(t)\) , at the end points \(t_a\) and \(t_b\) . As you will see in your calculus course, the function \(V(t)\) is precisely what we call the anti-derivative:
\[ \int v(t) dt= V(t) + C\]
which has derivative:
\[\begin{aligned} \frac{dV}{dt}=v(t)\end{aligned}\]
Note that when taking the integral, the constant \(C\) always cancels.
Exercise \(\PageIndex{2}\)
Choose the graph of \(x(t)\) for the case when acceleration is given by \(a(t) = Aω^{2}cos(ωt)\), where \(ω\) and \(A\) are positive constants. The velocity and position are zero at \(t = 0\).
- Figure A
- Figure B
- Figure C
- Answer
- Figure A. The acceleration is positive near \(t=0 \text{s}\) as described by the second derivative, which is concave up.
Exercise \(\PageIndex{3}\)
The acceleration of a cricket jumping sideways is observed to increase linearly with time, that is, \(a_{x}(t)=a_{0}+jt\) , where \(a_{0}\) and \(j\) are constants. What can you say about the velocity of the cricket as a function of time?
- it is constant
- it increases linearly with time ( \(v(t)\propto t\) )
- it increases quadratically with time ( \(v(t)\propto t^{2}\) )
- it increases with the cube of time ( \(v(t)\propto t^{3}\) )
- Answer
- C. We know velocity is linear in time ( \(v(t)\propto t\) ) when \(a\) is constant. If \(a\) becomes linearly time-dependent ( \(a(t)\propto t\) ), the velocity will change its dependence by the same change in time-dependence ( \(v(t)\propto t^{2}\) ).


1.3: The Law of Reflection
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Learning Objectives
By the end of this section, you will be able to:
- Explain the reflection of light from polished and rough surfaces
- Describe the principle and applications of corner reflectors
Whenever we look into a mirror, or squint at sunlight glinting from a lake, we are seeing a reflection. When you look at a piece of white paper, you are seeing light scattered from it. Large telescopes use reflection to form an image of stars and other astronomical objects.
The law of reflection states that the angle of reflection equals the angle of incidence:
\[θ_r=θ_i \label{law of reflection} \]
The law of reflection is illustrated in Figure \(\PageIndex{1}\), which also shows how the angle of incidence and angle of reflection are measured relative to the perpendicular to the surface at the point where the light ray strikes.
We expect to see reflections from smooth surfaces, but Figure \(\PageIndex{2}\) illustrates how a rough surface reflects light. Since the light strikes different parts of the surface at different angles, it is reflected in many different directions, or diffused. Diffused light is what allows us to see a sheet of paper from any angle, as shown in Figure \(\PageIndex{1a}\).
People, clothing, leaves, and walls all have rough surfaces and can be seen from all sides. A mirror, on the other hand, has a smooth surface (compared with the wavelength of light) and reflects light at specific angles, as illustrated in Figure \(\PageIndex{3b}\). When the Moon reflects from a lake, as shown in Figure \(\PageIndex{1c}\), a combination of these effects takes place.
When you see yourself in a mirror, it appears that the image is actually behind the mirror (Figure \(\PageIndex{4}\)). We see the light coming from a direction determined by the law of reflection. The angles are such that the image is exactly the same distance behind the mirror as you stand in front of the mirror. If the mirror is on the wall of a room, the images in it are all behind the mirror, which can make the room seem bigger. Although these mirror images make objects appear to be where they cannot be (like behind a solid wall), the images are not figments of your imagination. Mirror images can be photographed and videotaped by instruments and look just as they do with our eyes (which are optical instruments themselves). The precise manner in which images are formed by mirrors and lenses is discussed in an upcoming chapter on Geometric Optics and Image Formation .
Corner Reflectors (Retroreflectors)
A light ray that strikes an object consisting of two mutually perpendicular reflecting surfaces is reflected back exactly parallel to the direction from which it came (Figure \(\PageIndex{5}\)). This is true whenever the reflecting surfaces are perpendicular, and it is independent of the angle of incidence. Such an object is called a corner reflector , since the light bounces from its inside corner. Corner reflectors are a subclass of retroreflectors, which all reflect rays back in the directions from which they came. Although the geometry of the proof is much more complex, corner reflectors can also be built with three mutually perpendicular reflecting surfaces and are useful in three-dimensional applications.
Many inexpensive reflector buttons on bicycles, cars, and warning signs have corner reflectors designed to return light in the direction from which it originated. Rather than simply reflecting light over a wide angle, retroreflection ensures high visibility if the observer and the light source are located together, such as a car’s driver and headlights. The Apollo astronauts placed a true corner reflector on the Moon (Figure \(\PageIndex{6}\)). Laser signals from Earth can be bounced from that corner reflector to measure the gradually increasing distance to the Moon of a few centimeters per year.
Working on the same principle as these optical reflectors, corner reflectors are routinely used as radar reflectors (Figure \(\PageIndex{7}\)) for radio-frequency applications. Under most circumstances, small boats made of fiberglass or wood do not strongly reflect radio waves emitted by radar systems. To make these boats visible to radar (to avoid collisions, for example), radar reflectors are attached to boats, usually in high places.
As a counterexample, if you are interested in building a stealth airplane, radar reflections should be minimized to evade detection. One of the design considerations would then be to avoid building 90°90° corners into the airframe.


30.7: Patterns in Spectra Reveal More Quantization
Learning Objectives
By the end of this section, you will be able to:
- State and discuss the Zeeman effect.
- Define orbital magnetic field.
- Define orbital angular momentum.
- Define space quantization.
High-resolution measurements of atomic and molecular spectra show that the spectral lines are even more complex than they first appear. In this section, we will see that this complexity has yielded important new information about electrons and their orbits in atoms.
In order to explore the substructure of atoms (and knowing that magnetic fields affect moving charges), the Dutch physicist Hendrik Lorentz (1853–1930) suggested that his student Pieter Zeeman (1865–1943) study how spectra might be affected by magnetic fields. What they found became known as the Zeeman effect , which involved spectral lines being split into two or more separate emission lines by an external magnetic field, as shown in Figure \(\PageIndex{1}\). For their discoveries, Zeeman and Lorentz shared the 1902 Nobel Prize in Physics.
Zeeman splitting is complex. Some lines split into three lines, some into five, and so on. But one general feature is that the amount the split lines are separated is proportional to the applied field strength, indicating an interaction with a moving charge. The splitting means that the quantized energy of an orbit is affected by an external magnetic field, causing the orbit to have several discrete energies instead of one. Even without an external magnetic field, very precise measurements showed that spectral lines are doublets (split into two), apparently by magnetic fields within the atom itself.
Bohr’s theory of circular orbits is useful for visualizing how an electron’s orbit is affected by a magnetic field. The circular orbit forms a current loop, which creates a magnetic field of its own, \(B_{orb}\) as seen in Figure \(\PageIndex{2}\). Note that the orbital magnetic field \(B_{orb}\) and the orbital angular momentum \(L_{orb}\) are along the same line. The external magnetic field and the orbital magnetic field interact; a torque is exerted to align them. A torque rotating a system through some angle does work so that there is energy associated with this interaction. Thus, orbits at different angles to the external magnetic field have different energies. What is remarkable is that the energies are quantized—the magnetic field splits the spectral lines into several discrete lines that have different energies. This means that only certain angles are allowed between the orbital angular momentum and the external field, as seen in Figure \(\PageIndex{3}\).
We already know that the magnitude of angular momentum is quantized for electron orbits in atoms. The new insight is that the direction of the orbital angular momentum is also quantized . The fact that the orbital angular momentum can have only certain directions is called space quantization . Like many aspects of quantum mechanics, this quantization of direction is totally unexpected. On the macroscopic scale, orbital angular momentum, such as that of the moon around the earth, can have any magnitude and be in any direction.
Detailed treatment of space quantization began to explain some complexities of atomic spectra, but certain patterns seemed to be caused by something else. As mentioned, spectral lines are actually closely spaced doublets, a characteristic called fine structure , as shown in Figure \(\PageIndex{4}\). The doublet changes when a magnetic field is applied, implying that whatever causes the doublet interacts with a magnetic field. In 1925, Sem Goudsmit and George Uhlenbeck, two Dutch physicists, successfully argued that electrons have properties analogous to a macroscopic charge spinning on its axis. Electrons, in fact, have an internal or intrinsic angular momentum called intrinsic spin \(S\). Since electrons are charged, their intrinsic spin creates an intrinsic magnetic field \(B_{orb}\), which interacts with their orbital magnetic field \(B_{orb}\). Furthermore, electron intrinsic spin is quantized in magnitude and direction , analogous to the situation for orbital angular momentum. The spin of the electron can have only one magnitude, and its direction can be at only one of two angles relative to a magnetic field, as seen in Figure \(\PageIndex{5}\). We refer to this as spin up or spin down for the electron. Each spin direction has a different energy; hence, spectroscopic lines are split into two. Spectral doublets are now understood as being due to electron spin.
These two new insights—that the direction of angular momentum, whether orbital or spin, is quantized, and that electrons have intrinsic spin—help to explain many of the complexities of atomic and molecular spectra. In magnetic resonance imaging, it is the way that the intrinsic magnetic field of hydrogen and biological atoms interact with an external field that underlies the diagnostic fundamentals.
Summary
- The Zeeman effect—the splitting of lines when a magnetic field is applied—is caused by other quantized entities in atoms.
- Both the magnitude and direction of orbital angular momentum are quantized.
- The same is true for the magnitude and direction of the intrinsic spin of electrons.
Glossary
- Zeeman effect
- the effect of external magnetic fields on spectral lines
- intrinsic spin
- the internal or intrinsic angular momentum of electrons
- orbital angular momentum
- an angular momentum that corresponds to the quantum analog of classical angular momentum
- fine structure
- the splitting of spectral lines of the hydrogen spectrum when the spectral lines are examined at very high resolution
- space quantization
- the fact that the orbital angular momentum can have only certain directions
- intrinsic magnetic field
- the magnetic field generated due to the intrinsic spin of electrons
- orbital magnetic field
- the magnetic field generated due to the orbital motion of electrons


3: Quadratic and Polynomial Functions
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- 3.2: Quadratic Functions
- In this section, we will explore the family of 2nd degree polynomials, the quadratic functions. While they share many characteristics of polynomials in general, the calculations involved in working with quadratics is typically a little simpler, which makes them a good place to start our exploration of short run behavior.
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- 3.5: Real Zeros of Polynomials
- n this section, we will learn how to find good candidates to test using synthetic division. In the days before graphing technology was commonplace, mathematicians discovered a lot of clever tricks for determining the likely locations of zeros. Technology has provided a much simpler approach to narrow down potential candidates, but it is not always sufficient by itself.
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- 3.6: Complex Zeros
- When finding the zeros of polynomials, at some point you’re faced with the problem x²=−1 . While there are clearly no real numbers that are solutions to this equation, leaving things there has a certain feel of incompleteness. To address that, we will need utilize the imaginary unit, i .
Thumbnail: (CC BY; Openstax)


11: Rotational dynamics
- Understand how to use vector quantities for describing the kinematics of rotations.
- Understand how to use torque to determine the angular acceleration of an object.
- Understand conditions for static and dynamic equilibrium.
- Understand how to determine the moment of inertia of an object.
In this Chapter, we use Newton’s Second Law to develop a formalism to describe how objects rotate. In particular, we will introduce the concept of torque which plays a similar role to that of force in non-rotational dynamics. We will also introduce the concept of moment of inertia to describe how objects resist rotational motion.
A construction worker would like to lift one end of a heavy block from the ground using a bar propped against a rock on the ground as a lever. Should he place the rock close or far from the block to make it easier to lift the block?
- It will be easier to lift the block if the rock is close to the block.
- It will be easier to lift the block if the rock is far from the block.
- It does not matter where he places the rock, as long as the bar is short.
- It does not matter where he places the rock, as long as the bar is long.


1.3: New Page
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- Stephen Skripak et al.
- Virginia Tech via Virginia Tech Libraries' Open Education Initiative
A business is any activity that provides goods or services to consumers for the purpose of making a profit. Be careful not to confuse the terms revenue and profit. Revenue represents the funds an enterprise receives in exchange for its goods or services. Profit is what’s left (hopefully) after all the bills are paid. When Steve Jobs and Steve Wozniak launched the Apple I, they created Apple Computer in Jobs’ family garage in the hope of making a profit. Before we go on, let’s make a couple of important distinctions concerning the terms in our definitions. First, whereas Apple produces and sells goods (Mac, iPhone, iPod, iPad, Apple Watch), many businesses provide services . Your bank is a service company, as is your Internet provider. Hotels, airlines, law firms, movie theaters, and hospitals are also service companies. Many companies provide both goods and services. For example, your local car dealership sells goods (cars) and also provides services (automobile repairs). Second, some organizations are not set up to make profits. Many are established to provide social or educational services. Such not-for profit (or nonprofit), organizations include the United Way of America, Habitat for Humanity, the Boys and Girls Clubs, the Sierra Club, the American Red Cross, and many colleges and universities. Most of these organizations, however, function in much the same way as a business. They establish goals and work to meet them in an effective, efficient manner. Thus, most of the business principles introduced in this text also apply to nonprofits.
Query \(\PageIndex{1}\)
Business Participants and Activities
Let’s begin our discussion of business by identifying the main participants of business and the functions that most businesses perform. Then we’ll finish this section by discussing the external factors that influence a business’ activities.
Participants
Every business must have one or more owners whose primary role is to invest money in the business. When a business is being started, it’s generally the owners who polish the business idea and bring together the resources (money and people) needed to turn the idea into a business. The owners also hire employees to work for the company and help it reach its goals. Owners and employees depend on a third group of participants— customers . Ultimately, the goal of any business is to satisfy the needs of its customers in order to generate a profit for the owners.
Stakeholders
Consider your favorite restaurant. It may be an outlet or franchise of a national chain (more on franchises in a later chapter) or a local “mom and pop” without affiliation to a larger entity. Whether national or local, every business has stakeholders – those with a legitimate interest in the success or failure of the business and the policies it adopts. Stakeholders include customers, vendors, employees, landlords, bankers, and others (Figure \(\PageIndex{1}\)). All have a keen interest in how the business operates, in most cases for obvious reasons. If the business fails, employees will need new jobs, vendors will need new customers, and banks may have to write off loans they made to the business. Stakeholders do not always see things the same way – their interests sometimes conflict with each other. For example, lenders are more likely to appreciate high profit margins that ensure the loans they made will be repaid, while customers would probably appreciate the lowest possible prices. Pleasing stakeholders can be a real balancing act for any company.
Query \(\PageIndex{1}\)


7.1: Introduction
Learning Objectives
- Apply the nursing process to clients with depressive disorders
- Describe nursing assessments related to depressive disorders
- Identify common nursing problems/diagnoses related to depressive disorders
- Establish a safe environment
- Apply evidence-based practice when planning and implementing nursing care
- Describe common treatments for depressive disorders
- Identify appropriate referrals to community resources
- Evaluate the effectiveness of interventions
- Provide patient education to clients and their family members
Depression is a common illness affecting an estimated 8% of adults in the United States annually. [1] Approximately 280 million people in the world have depression. [2] This chapter will discuss causes and types of depression, treatments for depression, and steps for applying the nursing process when caring for clients with depressive disorders.
- National Alliance on Mental Illness. (2017, August). Depression . https://nami.org/About-Mental-Illness/Mental-Health-Conditions/Depression ↵
- Depression by the World Health Organization is licensed under CC BY-NC-SA 3.0 IGO ↵


7.12: Frequent-Shopper and Rewards Programs
- Analyze the various tactics of frequent-shopper or rewards programs
Customer retention is crucial to the success of any business given the high cost of acquiring a customer in the first place. It is estimated that it costs a business 5-25X more to acquire a new customer than to sell to an existing one. Further, established customers are thought to spend 67% more than new customers. [1]
It’s no wonder that savvy businesses have brought a variety of customer retention techniques together under the umbrella of formal “Frequent Shopper” or “Customer Loyalty” programs. As consumers, we all experience these techniques every day.
Some programs are based on simple point system–a purchase amount is equivalent to a number of points. Those points can then be accumulated and used as currency to make additional purchases.
A variant of this tactic, a so-called tiered reward program, is designed to foster longer-term loyalty. Airlines and car rental companies offer such programs and consumers move up the food chain by becoming Bronze, Silver and Gold members over time. With each designation, the customer receives more benefits and perks.
Another tactic is charging customers an annual fee in exchange for VIP treatment. Amazon Prime would be a good example of this type of customer loyalty program. “Prime” customers pay an annual $99 fee to participate in the program. As Amazon Prime members, customers receive free two-day shipping on millions of products with no minimum purchase required. According to the Consumer Intelligence Research, Prime members spend an average of $1,500 per year on Amazon.com, compared with $625 per year spent by Amazon customers who aren’t Prime members. [2]
Contributors and Attributions
- Frequent-Shopper and Rewards Programs. Authored by : Bob Danielson. Provided by : Lumen Learning. License : CC BY: Attribution
- So many coffee loyalty cards. Authored by : Nick Webb. Provided by : Flickr. Located at : https://flic.kr/p/5xBEEV . License : CC BY: Attribution