16.4: Pain Assessment
By the end of this section, you will be able to:
- Describe how to collect subjective data for a pain assessment
- Identify how to collect objective data for a pain assessment
- Recognize how to document a pain assessment
Pain is a subjective experience, which means it cannot always be observed externally. Because patients do not always outwardly display signs of pain, the pain assessment must be multidimensional (Wideman et al., 2019). Both subjective and objective data need to be included in a comprehensive pain assessment to get a complete picture of the patient’s pain.
Using a variety of assessment tools allows the patient to express the quality, type, and intensity of the pain. The nurse may use a pain rating scale and different types of questions to collect subjective data. Objective data can be collected by performing a physical assessment, measuring vital signs, and observing patient behaviors such as posture and facial expressions.
A comprehensive pain assessment is the key to adequately treating pain and is a crucial nursing skill. Without comprehensive baseline data, it can be difficult to provide effective pain management for patients.
Collecting Subjective Data
Because pain is a subjective experience, most of the collected data will be subjective (Wideman et al., 2019). Information obtained from the patient and/or family members and offers important cues from their perspective is called subjective data. A comprehensive pain assessment includes questions that assess the quality, region, severity, potential cause, timing, and aggravating and relieving factors of the patient’s pain. Asking the patient open-ended questions allows them to elaborate on their pain and helps the nurse fully understand the patient’s concerns.
The PQRSTU mnemonic is often used to remember these subjective assessment questions:
- Provocation/Palliation: What makes your pain feel worse or better?
- Quality: What does the pain feel like?
- Region: Where exactly do you feel the pain? Does it radiate?
- Severity: How would you rate your pain on scale of 0 to 10?
- Timing/Treatment: When did the pain start? How long does the pain last? Have you taken anything to relieve the pain?
- Understanding: What do you think is causing the pain?
These pain assessment questions help the nurse establish a baseline assessment of the patient’s pain.
Scenario: The nurse is caring for a patient who has come into the emergency department. The nurse begins asking the patient questions about their pain including follow-up questions using PQRSTU assessment.
Nurse: Hi, my name is Sulee, and I am going to be your nurse today. What brings you in today?
Patient: My lower back is killing me. I am in so much pain.
Nurse: I am sorry to hear that. I am going to ask a few more questions about your pain so we can get an idea of what is causing it and how we can help alleviate your pain. What makes your pain worse?
Patient: Sometimes it gets worse if I am sitting for a long time.
Nurse: Do you have to sit for a long period of time often?
Patient: Yes, I sit at a desk all day at work. My back always hurts when I get home.
Nurse: What does the pain feel like?
Patient: It starts feeling sore and aching, but by the end of the day it feels like a knife is stabbing me in the back.
Nurse: Where do you feel the pain in your back?
Patient: In my lower back.
Nurse: Does the pain radiate to somewhere else in your back or in your body?
Patient: No, the pain is just across my lower back.

Patient: Probably a seven.
Nurse: Would you rate your pain differently after sitting for a long period of time?
Patient: Yes, it’s a nine when I get home from work.
Nurse: Is the pain constant or does it come and go?
Patient: The aching is constant, but the stabbing feeling comes and goes.
Nurse: Have you taken any medication to relieve the pain?
Patient: I take ibuprofen daily and sometimes I try to lie down after work.
Nurse: Does the ibuprofen and lying down relieve the pain?
Patient: Sometimes it makes it better, but the pain never goes away.
Nurse: What do you think is causing the pain?

Nurse: Thank you for sharing that information. I am going to do a few other assessments so I can better understand your pain and how we can help you.
Collecting subjective data allows the nurse to begin to understand the type and location of the patient’s pain. In addition to the PQRSTU mnemonic, there are many different pain rating scales the nurse can use depending on the patient’s cognitive and developmental level .
Numeric Rating Scale
Asking a patient to use a numeric rating scale to rate the severity of their pain from 0 to 10, with “0” being no pain and “10” being the worst pain imaginable, is a common way to assess patients for pain. Most patients find the numeric rating scale easy to use and healthcare providers find the results easy to interpret (Zambon, 2020). However, the numeric rating scale may not be appropriate for all patients. Factors such as age, native language, literacy level, and cognitive ability may prohibit patients from understanding the numeric rating scale (Zambon, 2020). Some patients may be too young or cognitively delayed, or even sedated to understand how the numbers relate to the severity of pain. Patients from various cultures and native languages may not understand the numbers on the scale and may prefer a more visual pain rating scale. The nurse must ensure the patient can understand the numeric rating scale before using it. If the patient does not understand, a different assessment tool must be used.
When using a numeric rating scale, the nurse should always accept the patient’s pain for what they say it is (Wideman et al., 2019). For example, one patient with a sprained ankle may rate their pain as a 4 out of 10, while another patient with a sprained ankle may rate their pain as an 8 out of 10. Both patients are right as their interpretation of pain is subjective (Dydyk & Grandhe, 2023).
The numeric rating scale provides the nurse with the severity of a patient’s pain but does not provide any other information (Wideman et al., 2019). Additional questions must be asked to obtain a thorough pain assessment and to assess the patient’s comfort-function goal . The comfort-function goal is an individualized patient goal identifying their acceptable pain tolerance while maintaining their daily functions. This goal provides the basis for the patient’s individualized pain treatment plan and is used to evaluate the effectiveness of interventions. Each patient’s comfort-function goal will be different. For example, one patient may have a comfort-function goal of 4 out of 10 pain severity while another patient may have a goal of 0 out of 10 pain severity.
The numeric rating scale is the most common pain rating scale used in nursing due to its simplicity and ease of understanding. However, it is important for nurses to know other pain rating options to best suit individual patient needs.
Wong-Baker FACES Pain Rating Scale
The Wong-Baker FACES Pain Rating Scale is a visual tool used to evaluate pain severity. The scale uses drawings of different faces exhibiting increasing levels of pain and was created in 1983 by two pediatric healthcare workers, Donna Wong and Connie Baker, to help children express their pain. Today, the scale can be used for anyone age three and older to visually represent their pain level (Wong-Baker FACES Foundation, 2023).
The Wong-Baker FACES Pain Rating Scale can be used for a variety of patients including children, patients of diverse cultures and native languages, and patients who are nonverbal due to its simplistic illustrations.
To use this scale, use the following evidence-based instructions. Explain to the patient that each face represents a person who has no pain (hurt), some pain, or a lot of pain. “Face 0 doesn’t hurt at all. Face 2 hurts just a little. Face 4 hurts a little more. Face 6 hurts even more. Face 8 hurts a whole lot. Face 10 hurts as much as you can imagine, although you don’t have to be crying to have this worst pain.” Ask the patient to choose the face that best represents the pain they are feeling.
Even though this pain rating scale was intended for children, research shows that many adult patients may prefer a visual scale. Patients with different native languages or patients who are cognitively delayed may have an easier time understanding a visual scale compared to a numeric rating scale .
McCaffrey Initial Pain Assessment Tool
The McCaffrey Initial Pain Assessment Tool is another assessment tool that helps patients express their pain. The McCaffrey Initial Pain Assessment Tool uses visual aids and questions including elements of the PQRSTU mnemonic to provide a comprehensive pain assessment. The McCaffrey tool provides a more comprehensive picture of a patient’s pain than the numeric rating scale or the Wong-Baker FACES scale.
The McCaffrey pain assessment tool includes questions pertaining to the severity, causes, contributing factors, and effects of pain. Images of the human body are included to help patients identify the exact location of their pain. There are detailed questions to help patients express the effects of pain such as accompanying emotions or symptoms, decreased quality of life, decreased physical activity, and poor sleep.
Collecting Objective Data
Collecting objective data is the second half of a comprehensive pain assessment. Objective data include things that the nurse can measure, such as vital signs and patient behavior. Objective data can help the nurse understand the patient’s pain rating and corroborate the subjective assessment data (Xu & Huang, 2020).
An objective pain assessment includes measuring vital signs, physical assessment, and observing for nonverbal indicators of pain, such as grimacing or moaning. It is especially important to observe for nonverbal indicators of pain in patients unable to self-report their pain, such as infants, children, patients who have a cognitive disorder, patients at end of life, patients who are non-English speaking, or patients who tend to be stoic due to cultural beliefs.
Nurses should be aware that pain can be exhibited through physical symptoms and patient behaviors. A pain assessment should be performed by the nurse prior to any patient interventions to determine if physical assessment findings are related to pain or something else. For example, a patient appears hunched over and guarding their stomach with reports of nausea and lack of appetite. The nurse should recognize that these symptoms could be physical signs of pain and perform a comprehensive pain assessment as the initial intervention.
Collecting objective data allows the nurse to corroborate the subjective assessment of a patient’s pain. An objective pain assessment is especially important if the patient is unable to answer questions about their pain.
Physical Assessment
A physical assessment is crucial to develop a pain treatment plan. Assessment of the location of the patient’s pain can give the nurse context to the potential cause of pain. The nurse should inspect the site of pain for any abnormalities such as swelling, lacerations, or discoloration. Areas of pain can be identified by palpating for any tenderness, swelling, or change in temperature. The nurse should note if the patient is guarding any body part as this could indicate the location of pain. Symptoms such as diaphoresis, nausea, vomiting, or lack of appetite can often be the result of pain. For example, a patient reports moderate pain on their upper back. Upon examining the patient’s back, the nurse finds bruising and tenderness upon palpation on the patient’s right shoulder. The physical assessment helps the nurse pinpoint a specific location of pain, which can help guide pain management.
Decreased daily activity can be another result of pain, especially if the patient is experiencing chronic unmanaged pain. Assessing the effect pain has on a patient’s ability to bathe, dress, prepare food, eat, walk, and complete other daily activities is a new standard of care that assists the interdisciplinary team in tailoring treatment goals and interventions that are customized to the patient’s situation. For example, for some patients, chronic pain affects their ability to be employed, so effective pain management is vital so they can return to work. For other patients receiving palliative care , the ability to sit up and eat a meal with loved ones without pain is an important goal.
The nurse is performing an initial assessment on a school-aged patient who has arrived to the hospital after being hit with a baseball. The nurse observes that the patient is guarding their right leg and is grimacing while sitting in the bed. The nurse inspects the patient’s right leg and identifies discoloration and swelling. The nurse observes the right leg is positioned at an odd angle and the patient cries out upon palpation. The patient is unable to move their leg and states “It hurts even more when I try to move.” The nurse recognizes that the odd angle and decreased mobility of the patient’s leg is most concerning. The nurse recognizes that further evaluation is needed to determine appropriate interventions for the patient’s pain.
Observe Expression and Posture
Nonverbal pain cues are an important part of an objective pain assessment. Expression and posture can indicate that a patient is in pain. Facial expressions such as grimacing, moaning, clenched teeth, or crying are ways for patients to express pain. Abnormal posture such as hunching over, contracting, rigidity, limited movement, or abnormal gait could also indicate that the patient is in pain (Figure 16.9).
Abnormal posture does not always indicate pain. A patient may have a stooped posture due to certain conditions, such as scoliosis or arthritis, and report no pain. The nurse should ask the patient if the abnormal posture is normal for them and always initiate a pain assessment.
Patients often exhibit abnormal posture and abnormal facial expressions together when experiencing pain. For example, a patient may exhibit a hunched posture but appear calm and relaxed and report no pain. Another patient may exhibit a hunched posture accompanied by moaning and crying. This could indicate that this is not a normal posture for the patient and may be related to pain.
When assessing patients with abnormal posture or facial expressions, the nurse should recognize that they could be potential nonverbal pain cues. Sometimes patients are in too much pain to speak or may be unable to verbally communicate. Nonverbal indicators of pain such as abnormal posture and facial expressions can be helpful to the nurse when collecting objective pain assessment data.
Inspect Skin, Muscles, and Joints
Performing an assessment of the skin, muscles, and joints can help the nurse identify specific areas of pain. Changes in skin, decreased range of motion, and an abnormal gait can all be indicative of pain. Through subjective data collection, the nurse can identify the general location of the patient’s pain. Then the nurse can assess the skin, muscles, and joints in that area to provide more data to the pain assessment.
The nurse should inspect the skin for any abnormalities such as swelling, lacerations, skin breakdown, drainage, and discoloration. Areas of pain can be identified by palpating for any tenderness, swelling, or change in temperature.
Abnormal gait, decreased range of motion, and decreased muscle strength can all be signs of pain. Just like when assessing posture and facial expressions, it is important for the nurse to remember that abnormalities in the skin, muscles, and joints do not always indicate pain. The nurse should pay particular attention to what the patient is reporting about current symptoms, as well as history of any issues. Information should be compared to expectations for the patient’s age group or that patient’s baseline. For example, an older patient may have chronic limited range of motion in the knee due to osteoarthritis, whereas a child may have new, limited range of motion due to a knee sprain that occurred during a sports activity.
Changes in skin, muscles, and joints are not always present with pain. However, these assessments can help the nurse identify a specific location of pain or potential aggravating factors.
Measure Vital Signs
Abnormal vital signs can be another nonverbal indicator of pain, especially when patients cannot show behavioral signs of pain (Ford, 2019). For example, a patient who is sedated and intubated cannot answer questions about their pain or exhibit any behaviors indicating pain. The nurse can measure vital signs along with other physical symptoms to assess pain.
Pain can cause hypertension, tachycardia, or tachypnea. Respiratory distress such as loud breathing, nasal flaring, or the use of accessory muscles can also indicate pain. Just like other nonverbal pain indicators, abnormal vital signs can also be present in the absence of pain. The nurse must use other assessment tools in addition to measuring vital signs to determine if the vital signs are related to pain (Ford, 2019).
In 1995, the American Pain Society introduced the concept of pain as the fifth vital sign to encourage providers to assess pain more frequently and prioritize pain assessments in patient care (Scher et al., 2017). There is ongoing debate as to whether treating pain as the fifth vital sign has contributed to the ongoing opioid crisis in America. Some providers believe that putting an intense focus on pain management can lead to the overprescribing of opioids, but other providers believe that pain management should be the priority in patient care. Current research shows that pain is an ongoing patient concern, and that consistent, multidimensional pain assessments are most effective in managing pain (Scher et al., 2017).
Nurses must always make pain a top priority of patient care. Nurses must include multiple assessment tools, such as measuring vital signs, in their pain assessments and individualize their assessments to meet each patient’s needs.
Assess Behavioral Cues
Behavioral cues can be important indicators of pain if the patient is unable to answer subjective assessment questions. The nurse may notice a flat affect in a patient in pain. Affect refers to the outward display of one’s emotional state. For example, a patient with a “flat affect” refers to very few facial expressions being displayed to indicate emotion (Figure 16.10).
The nurse must recognize that not every patient in pain will have the same behavioral cues. For example, one patient in pain may appear quiet and withdrawn while another patient may appear very angry and aggressive. A patient’s behavioral cues may not always match their subjective pain assessment. This is often the case with patients in chronic pain . For example, a patient who has cancer may rate their pain an 8 out of 10 while calmly eating lunch with their family. It is important for the nurse to remember that pain is whatever the patient says it is and that behavioral cues are only one aspect of the comprehensive pain assessment.
Documenting Pain Assessment
Documentation of the pain assessment, interventions, and reevaluation are key to effective, individualized pain management. Pain assessments should occur at regular intervals and a reassessment of pain should occur after any interventions (The Joint Commission, 2020).
When documenting subjective data , it should be in quotation marks and start with wording such as, “The patient reports…” or “The patient’s wife states …”. The nurse should document any subjective data stated by the patient including their pain score, what pain rating scale was used, and any follow-up information. The nurse should also document any behaviors, facial expressions, or physical attributes that could be related to pain. For example, the nurse would document “Patient states pain is a 7 out of 10 using the numeric rating scale . Patient is crying and grimacing. Patient states pain is in their right shoulder and has been ongoing for two days. Patient states heat relieves the pain for short periods of time. Patient’s right shoulder does not appear to have any skin breakdown or discoloration. Patient has decreased range of motion in their right shoulder when compared to the left shoulder.”
The nurse should also document the patient’s comfort-function goal and any communication with other healthcare providers. Documenting a pain assessment notifies the patient’s entire healthcare team of the assessment data. This allows the healthcare team to begin to plan interventions and further evaluation if needed. The team cannot initiate pain interventions without good assessment documentation, so this should be a top priority for the nurse.
Documenting Treatment
All pain management interventions need to be documented according to the policy of each healthcare facility. Pain management interventions can be pharmacological or nonpharmacological. Nonpharmacological treatments can include exercise, mind-body practices, psychological therapy, heat or cold, braces, and rehabilitation (Agency for Healthcare Research and Quality [AHRQ], 2019).
Any administered pain medication should be documented in the patient’s electronic medication administration record (eMAR) . It is important to immediately document the intervention to avoid potential errors such as an unintended repeat dose of medication. Nonpharmacological treatments should also be documented in the patient’s medical record. Documentation of all pain interventions ensures all providers are aware of pain management strategies and that the patient is getting timely and effective treatment.
Documenting Evaluation of Treatment
All pain management interventions need to be evaluated and documented to ensure they are effective. It is important to perform a follow-up assessment in the appropriate time frame. The nurse should be aware of the different mechanisms of action for different forms of pain medications to best evaluate the effectiveness. For example, administration of intravenous pain medication should be evaluated within a shorter time frame than oral pain medication due to the different durations of action. The Joint Commission guidelines state that healthcare facilities should have policies in place regarding timeliness of pain reassessments and evaluation (The Joint Commission, 2020).
The nurse should document a new pain level and any symptoms of pain following any interventions. For example, if a patient reported a pain level of 8 out of 10 before PRN pain medication was administered, the nurse evaluates the patient’s pain level after administration to ensure the pain level is decreasing and the pain medication was effective. Effective pain management involves mutual pain goals between the healthcare team and the patient and ongoing reassessment with the patient to assess the effectiveness of pain interventions (ANA Center for Ethics and Human Rights, 2018). The nurse should compare the current pain rating to the patient’s stated comfort-function goal. If the patient states that their pain does not meet their comfort-function goal, the nurse should continue to explore pain management strategies and continue to reevaluate.


1.14.75: Viscosities: Salt Solutions
The (shear) viscosities of salt solutions have been extensively studied [1,2]. A frequently cited paper (always worth reading in terms of the care taken in the experimental study) was published by Jones and Dole [3]. The dependence of the viscosity of an aqueous salt solution \(\eta(\mathrm{aq})\) (at fixed temperature and pressure) on concentration of salt \(j\) is described by the Jones-Dole equation ; equation (a) where \(\eta_{1}^{*}(\ell)\) is the viscosity of water(\(\ell\)) at the same \(\mathrm{T}\) and \(\mathrm{p}\) [3,4].
\[\eta(\mathrm{aq}) / \eta_{1}^{*}(\ell)=1+\mathrm{A} \, \mathrm{c}_{\mathrm{j}}^{1 / 2}+\mathrm{B} \, \mathrm{c}_{\mathrm{j}} \nonumber \]
Equation (a) is re-expressed as follows. By definition;
\[\psi=\left[\eta(\mathrm{aq})-\eta_{1}^{*}(\ell)\right] /\left[\eta_{1}^{*}(\ell) \, \mathrm{c}_{\mathrm{j}}^{1 / 2}\right] \nonumber \]
Hence
\[\psi=\mathrm{A}+\mathrm{B} \, \mathrm{c}_{\mathrm{j}}^{1 / 2} \nonumber \]
A plot of \(\psi\) against \(\left(\mathrm{c}_{\mathrm{j}}\right)^{1 / 2}\) has intercept \(\mathrm{A}\) and slope \(\mathrm{B}\), the Jones-Dole \(\mathrm{B}\)-viscosity coefficient [5]. The \(\mathrm{A}\) coefficient describes the impact of charge-charge interactions on the viscosity of a solution, being generally positive and estimated using the Falkenhagen equation [6-8]. The \(\mathrm{B}\) coefficient characterizes ion-solvent interactions at defined \(\mathrm{T}\) and \(\mathrm{p}\). For a 1:1 salt \(j\), the \(\mathrm{B}_{j}\) coefficient for salt \(j\) is expressed as the sum of ionic \(\mathrm{B}\) coefficients.
\[\mathrm{B}_{\mathrm{j}}=\mathrm{B}_{+}+\mathrm{B} \nonumber \]
For example, \(\mathrm{B}_{j}\) for a series of salts with a common anion, the changes in \(\mathrm{B}_{j}\) reflect changes in \(\mathrm{B}_{+}\) for the cations. The pattern in \(\mathrm{B}\left(\mathrm{R}_{4}\mathrm{N}^{+} \mathrm{~I}^{-};\mathrm{aq}; 298.2 \mathrm{~K}\right)\) reflects the changes in \(\mathrm{B}\left(\mathrm{R}_{4}\mathrm{N}^{+} \mathrm{~I}^{-};\mathrm{aq}; 298.2 \mathrm{~K}\right)\) through the series from \(\mathrm{R}\) = methyl to \(\mathrm{R}\) = n-butyl. In fact the change in this case is indicative of the change in character from ‘structure breaking’ \(\mathrm{Me}_{4}\mathrm{N}^{+}\) to hydrophobic ‘structure forming’ \(\mathrm{Bu}_{4}\mathrm{N}^{+}\) ions [9]. In broad terms a positive \(\mathrm{B}\) coefficient indicates a tendency for the solute to enhance water-water interactions and thus raise the viscosity whereas a negative coefficient indicates a tendency to induce disorder [10]. Ionic B-viscosity coefficients are linked to the hydration properties of ions. An important link was suggested by Gurney [11]. On the grounds that \(\mathrm{K}^{+}\) and \(\mathrm{Cl}^{-}\) ions are roughly the same size, Gurney argued that for aqueous solutions, \(\mathrm{B}\left(\mathrm{K}^{+}\right)=\mathrm{B}\left(\mathrm{Cl}^{-}\right)\). Hence one can estimate single-ion \(\mathrm{B}\)-viscosity coefficients. \(\mathrm{A}\) negative \(\mathrm{B}\)-ionic coefficient indicates that the ion is a ‘structure breaker’ and a positive \(\mathrm{B}\)-ionic viscosity coefficient indicates that the ion is a ‘structure former’.
The terms ‘structure breaker’ and ‘structure former’ were extensively used in the decades from 1950 to 1990. However their popularity waned towards the end of the century as more precise descriptions of ionic hydration were sought.
Footnotes
[1] We confine attention to shear viscosities (i.e. resistance to shear). The related bulk viscosities (i.e. resistance to compression) are rarely discussed in the present context.
[2] Units; dynamic viscosity; traditional unit = poise, symbol \(\mathrm{P}\)
\(\mathrm{P}=10^{-1} \text { Pas}\). But \(\mathrm{Pa} = \mathrm{~kg m}^{-1} \mathrm{~s}^{-2}\) Then \(\mathrm{P} = 10^{-1} \mathrm{~kg m}^{-1} \mathrm{~s}^{-1} SI unit; η = [kg m-1 s-1 ]
[3] G. Jones and M. Dole, J. Am. Chem. Soc., 1929, 51 ,2950.
[4] Other equations have been suggested as alternatives to the Jones-Dole equation.
- The concentration cj is replaced by the ionic strength of the salt solution; C. Wu, J. Phys Chem.,1968, 72 ,2663.
-
For non-electrolytes, the coefficient A is zero but a new term is usually added linear in the concentration of solute.
- R. Robinson and R. Mills, Viscosity of Electrolytes and Related Properties, Pergamon, London, 1965.
- W. Devine and B. M. Lowe, J.Chem.Soc.,A,1971,2113.
-
The ratio \(\eta(\mathrm{aq}) / \eta_{1}^{*}(\ell)\) is expressed as a function of the volume fraction of the solute.
- D. G. Thomas, J. Colloid. Interface Sci.,1965, 20 ,267.
- B. R. Breslau and I. F. Mikker, J. Phys. Chem.,1970, 74 ,1056.
- J. Vand, J.Phys.Chem.,1948, 52 ,277.
- D. Eagland and G. Pilling, J. Phys. Chem., 1 972, 76 ,1902.
- Add a term linear in \(\left(c_{j}\right)^{2}\) to equation (a). N. Martinus, C. D. Sinclair and C. A. Vincent, Electrochim. Acta, 1977, 22 ,1183.
-
Somewhat outside the terms of reference of the account developed here is a treatment of viscosity in terms of kinetic phenomena; e.g. in terms of Transition State Theory. S. Glasstone, K. J. Laidler and H. Eyring, Theory of Rate Processes, McGraw-Hill, New York,1941. For application of this model see for example,
- W. Good, Electrochim. Acta, 1964, 9 , 203; 1965, 10 ,1; 1966, 11 , 759,767; 1967, 12 ,103 1.
- J. C. MacDonald, Electrochim. Acta, 1972, 17 ,1965.
[5] See for example,
- R. L. Kay, K. T. Vituccio, C. Zawoyski and D. F. Evans. J. Phys. Chem.,1966, 70 , 2336.
- Y. Tamaki, Y. Ohara, M. Inabe, T. Mori and F. Numata, Bull. Chem. Soc. Jpn.,1983, 56 ,1930.
[6] H. S. Harned and B. B. Owen, The Physical Chemistry of Electrolyte Solutions, Reinhold, New York, 3rd. edition, 1958,p.240.
[7] See for example;
- A. Kacperska, S. Taniewska-Osinska, A. Bald and S. Szejgis, J. Chem. Soc. Faraday Trans.1,1989, 85 ,4147.
- A. Sacco, G. Petrella, A. Dell’Atti and A. de Giglio, J. Chem. Soc. Faraday Trans.1, 1982, 78 ,1507.
[8] An alternative approach involves calculating the A coefficient using the Falkenhagen theory and hence writing the Jones-Dole equation as follows.
\[\eta(\mathrm{aq}) / \eta_{\mathrm{I}}^{*}(\ell)-1-\mathrm{A} \,\left(\mathrm{c}_{\mathrm{j}}\right)^{1 / 2}=\mathrm{B} \, \mathrm{c}_{\mathrm{j}} \nonumber \]
In a plot of the LHS of this equation against \(\mathrm{c}_{j}\), the slope equals B. See for example, K. Tamaki, K. Suga and E. Tanihara, Bull. Chem. Soc. Jpn.,1987. 60 ,1225.
[9] B. M. Lowe and G. A. Rubienski, Electrochem. Acta, 1974, 19 ,393.
[10] E. R. Nightingale, J. Phys. Chem.,1959, 63 ,1381;1962, 66 ,894. See also, D. T. Burns, Electrochim Acta, 1965, 10 ,985.
[11] R. W. Gurney, Ionic Processes in Solution, McGraw-Hill, New York, 1953.


2.3: Limits and Continuity
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Learning Objectives
- Calculate the limit of a function of two variables.
- Learn how a function of two variables can approach different values at a boundary point, depending on the path of approach.
- State the conditions for continuity of a function of two variables.
- Verify the continuity of a function of two variables at a point.
- Calculate the limit of a function of three or more variables and verify the continuity of the function at a point.
We have now examined functions of more than one variable and seen how to graph them. In this section, we see how to take the limit of a function of more than one variable, and what it means for a function of more than one variable to be continuous at a point in its domain. It turns out these concepts have aspects that just don’t occur with functions of one variable.
Limit of a Function of Two Variables
Recall from Section 2.5 that the definition of a limit of a function of one variable:
Let \(f(x)\) be defined for all \(x≠a\) in an open interval containing \(a\). Let \(L\) be a real number. Then
\[\lim_{x→a}f(x)=L \nonumber \]
if for every \(ε>0,\) there exists a \(δ>0\), such that if \(0<|x−a|<δ\) for all \(x\) in the domain of \(f\), then
\[|f(x)−L|<ε. \nonumber \]
Before we can adapt this definition to define a limit of a function of two variables, we first need to see how to extend the idea of an open interval in one variable to an open interval in two variables.
Definition: \(\delta\) Disks
Consider a point \((a,b)∈\mathbb{R}^2.\) A \(δ\) disk centered at point \((a,b)\) is defined to be an open disk of radius \(δ\) centered at point \((a,b)\) —that is,
\[\{(x,y)∈\mathbb{R}^2∣(x−a)^2+(y−b)^2<δ^2\} \nonumber \]
as shown in Figure \(\PageIndex{1}\).
The idea of a \(δ\) disk appears in the definition of the limit of a function of two variables. If \(δ\) is small, then all the points \((x,y)\) in the \(δ\) disk are close to \((a,b)\). This is completely analogous to x being close to a in the definition of a limit of a function of one variable. In one dimension, we express this restriction as
\[a−δ<x<a+δ. \nonumber \]
In more than one dimension, we use a \(δ\) disk.
Definition: limit of a function of two variables
Let \(f\) be a function of two variables, \(x\) and \(y\). The limit of \(f(x,y)\) as \((x,y)\) approaches \((a,b)\) is \(L\), written
\[\lim_{(x,y)→(a,b)}f(x,y)=L \nonumber \]
if for each \(ε>0\) there exists a small enough \(δ>0\) such that for all points \((x,y)\) in a \(δ\) disk around \((a,b)\), except possibly for \((a,b)\) itself, the value of \(f(x,y)\) is no more than \(ε\) away from \(L\) (Figure \(\PageIndex{2}\)).
Using symbols, we write the following: For any \(ε>0\), there exists a number \(δ>0\) such that
\[|f(x,y)−L|<ε \nonumber \]
whenever
\[0<\sqrt{(x−a)^2+(y−b)^2}<δ. \nonumber \]
Proving that a limit exists using the definition of a limit of a function of two variables can be challenging. Instead, we use the following theorem, which gives us shortcuts to finding limits. The formulas in this theorem are an extension of the formulas in the limit laws theorem in the section titled The Limit Laws in single-variable calculus.
Theorem \(\PageIndex{1}\): Limit Laws for Functions of Two Variables
Let \(f(x,y)\) and \(g(x,y)\) be defined for all \((x,y)≠(a,b)\) in a neighborhood around \((a,b)\), and assume the neighborhood is contained completely inside the domain of \(f\). Assume that \(L\) and \(M\) are real numbers such that
\[\lim_{(x,y)→(a,b)}f(x,y)=L \nonumber \]
and
\[\lim_{(x,y)→(a,b)}g(x,y)=M, \nonumber \]
and let \(c\) be a constant. Then each of the following statements holds:
Constant Law:
\[\lim_{(x,y)→(a,b)}c=c \nonumber \]
Identity Laws:
\[\lim_{(x,y)→(a,b)}x=a \nonumber \]
\[\lim_{(x,y)→(a,b)}y=b \nonumber \]
Sum Law:
\[\lim_{(x,y)→(a,b)}(f(x,y)+g(x,y))=L+M \nonumber \]
Difference Law:
\[\lim_{(x,y)→(a,b)}(f(x,y)−g(x,y))=L−M \nonumber \]
Constant Multiple Law:
\[\lim_{(x,y)→(a,b)}(cf(x,y))=cL \nonumber \]
Product Law:
\[\lim_{(x,y)→(a,b)}(f(x,y)g(x,y))=LM \nonumber \]
Quotient Law:
\[\lim_{(x,y)→(a,b)}\dfrac{f(x,y)}{g(x,y)}=\dfrac{L}{M} \text{ for } M≠0 \nonumber \]
Power Law:
\[\lim_{(x,y)→(a,b)}(f(x,y))^n=L^n \nonumber \]
for any positive integer \(n\).
Root Law:
\[\lim_{(x,y)→(a,b)}\sqrt[n]{f(x,y)}=\sqrt[n]{L} \nonumber \]
for all \(L\) if \(n\) is odd and positive, and for \(L≥0\) if n is even and positive.
The proofs of these properties are similar to those for the limits of functions of one variable. We can apply these laws to finding limits of various functions.
Example \(\PageIndex{1}\): Finding the Limit of a Function of Two Variables
Find each of the following limits:
- \(\displaystyle \lim_{(x,y)→(2,−1)}(x^2−2xy+3y^2−4x+3y−6)\)
- \(\displaystyle \lim_{(x,y)→(2,−1)}\dfrac{2x+3y}{4x−3y}\)
Solution
a. First use the sum and difference laws to separate the terms:
\[\begin{align*} \lim_{(x,y)→(2,−1)}(x^2−2xy+3y^2−4x+3y−6)\\ = \left(\lim_{(x,y)→(2,−1)}x^2 \right)− \left(\lim_{(x,y)→(2,−1)}2xy \right)+ \left(\lim_{(x,y)→(2,−1)}3y^2 \right)−\left(\lim_{(x,y)→(2,−1)}4x\right) \\ + \left(\lim_{(x,y)→(2,−1)}3y \right)−\left(\lim_{(x,y)→(2,−1)}6\right). \end{align*}\]
Next, use the constant multiple law on the second, third, fourth, and fifth limits:
\[\begin{align*} =(\lim_{(x,y)→(2,−1)}x^2)−2(\lim_{(x,y)→(2,−1)}xy)+3(\lim_{(x,y)→(2,−1)}y^2)−4(\lim_{(x,y)→(2,−1)}x) \\[4pt] +3(\lim_{(x,y)→(2,−1)}y)−\lim_{(x,y)→(2,−1)}6.\end{align*}\]
Now, use the power law on the first and third limits, and the product law on the second limit:
\[\begin{align*} \left(\lim_{(x,y)→(2,−1)}x\right)^2−2\left(\lim_{(x,y)→(2,−1)}x\right) \left(\lim_{(x,y)→(2,−1)}y\right)+3\left(\lim_{(x,y)→(2,−1)}y\right)^2 \\ −4\left(\lim_{(x,y)→(2,−1)}x\right)+3\left(\lim_{(x,y)→(2,−1)}y\right)−\lim_{(x,y)→(2,−1)}6. \end{align*}\]
Last, use the identity laws on the first six limits and the constant law on the last limit:
\[\begin{align*} \lim_{(x,y)→(2,−1)}(x^2−2xy+3y^2−4x+3y−6) = (2)^2−2(2)(−1)+3(−1)^2−4(2)+3(−1)−6 \\[4pt] =−6. \end{align*}\]
b. Before applying the quotient law, we need to verify that the limit of the denominator is nonzero. Using the difference law, constant multiple law, and identity law,
\[\begin{align*} \lim_{(x,y)→(2,−1)}(4x−3y) =\lim_{(x,y)→(2,−1)}4x−\lim_{(x,y)→(2,−1)}3y \\[4pt] =4(\lim_{(x,y)→(2,−1)}x)−3(\lim_{(x,y)→(2,−1)}y) \\[4pt] =4(2)−3(−1)=11. \end{align*}\]
Since the limit of the denominator is nonzero, the quotient law applies. We now calculate the limit of the numerator using the difference law, constant multiple law, and identity law:
\[\begin{align*} \lim_{(x,y)→(2,−1)}(2x+3y) =\lim_{(x,y)→(2,−1)}2x+\lim_{(x,y)→(2,−1)}3y \\[4pt] =2(\lim_{(x,y)→(2,−1)}x)+3(\lim_{(x,y)→(2,−1)}y) \\[4pt] =2(2)+3(−1)=1. \end{align*}\]
Therefore, according to the quotient law we have
\[\begin{align*} \lim_{(x,y)→(2,−1)}\dfrac{2x+3y}{4x−3y} =\dfrac{\displaystyle \lim_{(x,y)→(2,−1)}(2x+3y)}{\displaystyle \lim_{(x,y)→(2,−1)}(4x−3y)} \\[4pt] =\dfrac{1}{11}. \end{align*}\]
Exercise \(\PageIndex{1}\):
Evaluate the following limit:
\[\lim_{(x,y)→(5,−2)}\sqrt[3]{\dfrac{x^2−y}{y^2+x−1}}. \nonumber \]
- Hint
-
Use the limit laws.
- Answer
-
\[\displaystyle \lim_{(x,y)→(5,−2)}\sqrt[3]{\dfrac{x^2−y}{y^2+x−1}}=\dfrac{3}{2} \nonumber \]
Since we are taking the limit of a function of two variables, the point \((a,b)\) is in \(\mathbb{R}^2\), and it is possible to approach this point from an infinite number of directions. Sometimes when calculating a limit, the answer varies depending on the path taken toward \((a,b)\). If this is the case, then the limit fails to exist. In other words, the limit must be unique, regardless of path taken.
Example \(\PageIndex{2}\): Limits That Fail to Exist
Show that neither of the following limits exist:
- \(\displaystyle \lim_{(x,y)→(0,0)}\dfrac{2xy}{3x^2+y^2}\)
- \(\displaystyle \lim_{(x,y)→(0,0)}\dfrac{4xy^2}{x^2+3y^4}\)
Solution
a. The domain of the function \(f(x,y)=\dfrac{2xy}{3x^2+y^2}\) consists of all points in the \(xy\)-plane except for the point \((0,0)\) (Figure \(\PageIndex{3}\)). To show that the limit does not exist as \((x,y)\) approaches \((0,0)\), we note that it is impossible to satisfy the definition of a limit of a function of two variables because of the fact that the function takes different values along different lines passing through point \((0,0)\). First, consider the line \(y=0\) in the \(xy\)-plane. Substituting \(y=0\) into \(f(x,y)\) gives
\[f(x,0)=\dfrac{2x(0)}{3x^2+0^2}=0 \nonumber \]
for any value of \(x\). Therefore the value of \(f\) remains constant for any point on the \(x\)-axis, and as \(y\) approaches zero, the function remains fixed at zero.
Next, consider the line \(y=x\). Substituting \(y=x\) into \(f(x,y)\) gives
\[f(x,x)=\dfrac{2x(x)}{3x^2+x^2}=\dfrac{2x^2}{4x^2}=\tfrac{1}{2}. \nonumber \]
This is true for any point on the line \(y=x\). If we let \(x\) approach zero while staying on this line, the value of the function remains fixed at \(\tfrac{1}{2}\), regardless of how small \(x\) is.
Choose a value for ε that is less than \(1/2\)—say, \(1/4\). Then, no matter how small a \(δ\) disk we draw around \((0,0)\), the values of \(f(x,y)\) for points inside that \(δ\) disk will include both \(0\) and \(\tfrac{1}{2}\). Therefore, the definition of limit at a point is never satisfied and the limit fails to exist.
b. In a similar fashion to a., we can approach the origin along any straight line passing through the origin. If we try the \(x\)-axis (i.e., \(y=0\)), then the function remains fixed at zero. The same is true for the \(y\)-axis. Suppose we approach the origin along a straight line of slope \(k\). The equation of this line is \(y=kx\). Then the limit becomes
\[\begin{align*} \lim_{(x,y)→(0,0)}\dfrac{4xy^2}{x^2+3y^4} = \lim_{(x,y)→(0,0)}\dfrac{4x(kx)^2}{x^2+3(kx)^4} \\ = \lim_{(x,y)→(0,0)}\dfrac{4k^2x^3}{x^2+3k^4x^4} \\ =\lim_{(x,y)→(0,0)}\dfrac{4k^2x}{1+3k^4x^2} \\ = \dfrac{\displaystyle \lim_{(x,y)→(0,0)}(4k^2x)}{\displaystyle \lim_{(x,y)→(0,0)}(1+3k^4x^2)} \\ = 0. \end{align*}\]
regardless of the value of \(k\). It would seem that the limit is equal to zero. What if we chose a curve passing through the origin instead? For example, we can consider the parabola given by the equation \(x=y^2\). Substituting \(y^2\) in place of \(x\) in \(f(x,y)\) gives
\[\begin{align*}\lim_{(x,y)→(0,0)}\dfrac{4xy^2}{x^2+3y^4} = \lim_{(x,y)→(0,0)}\dfrac{4(y^2)y^2}{(y^2)^2+3y^4} \\ = \lim_{(x,y)→(0,0)}\dfrac{4y^4}{y^4+3y^4} \\ = \lim_{(x,y)→(0,0)}1 \\ = 1. \end{align*}\]
By the same logic in part a, it is impossible to find a δ disk around the origin that satisfies the definition of the limit for any value of \(ε<1.\) Therefore,
\[\displaystyle \lim_{(x,y)→(0,0)}\dfrac{4xy^2}{x^2+3y^4} \nonumber \]
does not exist.
Exercise \(\PageIndex{2}\):
Show that
\[\lim_{(x,y)→(2,1)}\dfrac{(x−2)(y−1)}{(x−2)^2+(y−1)^2} \nonumber \]
does not exist.
- Hint
-
Pick a line with slope \(k\) passing through point \((2,1).\)
- Answer
-
If \(y=k(x−2)+1,\) then \(\lim_{(x,y)→(2,1)}\dfrac{(x−2)(y−1)}{(x−2)^2+(y−1)^2}=\dfrac{k}{1+k^2}\). Since the answer depends on \(k,\) the limit fails to exist.
Interior Points and Boundary Points
To study continuity and differentiability of a function of two or more variables, we first need to learn some new terminology.
Definition: interior and boundary points
Let \(S\) be a subset of \(\mathbb{R}^2\) (Figure \(\PageIndex{4}\)).
- A point \(P_0\) is called an interior point of \(S\) if there is a \(δ\) disk centered around \(P_0\) contained completely in \(S\).
- A point \(P_0\) is called a boundary point of \(S\) if every \(δ\) disk centered around \(P_0\) contains points both inside and outside \(S\).
Definition: Open and closed sets
Let \(S\) be a subset of \(\mathbb{R}^2\) (Figure \(\PageIndex{4}\)).
- \(S\) is called an open set if every point of \(S\) is an interior point.
- \(S\) is called a closed set if it contains all its boundary points.
An example of an open set is a \(δ\) disk. If we include the boundary of the disk, then it becomes a closed set. A set that contains some, but not all, of its boundary points is neither open nor closed. For example if we include half the boundary of a \(δ\) disk but not the other half, then the set is neither open nor closed.
Definition: connected sets and Regions
Let \(S\) be a subset of \(\mathbb{R}^2\) (Figure \(\PageIndex{4}\)).
- An open set \(S\) is a connected set if it cannot be represented as the union of two or more disjoint, nonempty open subsets.
- A set \(S\) is a region if it is open, connected, and nonempty.
The definition of a limit of a function of two variables requires the \(δ\) disk to be contained inside the domain of the function. However, if we wish to find the limit of a function at a boundary point of the domain, the \(δ\) disk is not contained inside the domain. By definition, some of the points of the \(δ\) disk are inside the domain and some are outside. Therefore, we need only consider points that are inside both the \(δ\) disk and the domain of the function. This leads to the definition of the limit of a function at a boundary point.
Definition
Let \(f\) be a function of two variables, \(x\) and \(y\), and suppose \((a,b)\) is on the boundary of the domain of \(f\). Then, the limit of \(f(x,y)\) as \((x,y)\) approaches \((a,b)\) is \(L\), written
\[\lim_{(x,y)→(a,b)}f(x,y)=L, \nonumber \]
if for any \(ε>0,\) there exists a number \(δ>0\) such that for any point \((x,y)\) inside the domain of \(f\) and within a suitably small distance positive \(δ\) of \((a,b),\) the value of \(f(x,y)\) is no more than \(ε\) away from \(L\) (Figure \(\PageIndex{2}\)). Using symbols, we can write: For any \(ε>0\), there exists a number \(δ>0\) such that
\[|f(x,y)−L|<ε\, \text{whenever}\, 0<\sqrt{(x−a)^2+(y−b)^2}<δ. \nonumber \]
Example \(\PageIndex{3}\): Limit of a Function at a Boundary Point
Prove
\[\lim_{(x,y)→(4,3)}\sqrt{25−x^2−y^2}=0. \nonumber \]
Solution
The domain of the function \(f(x,y)=\sqrt{25−x^2−y^2}\) is \(\big\{(x,y)∈\mathbb{R}^2∣x^2+y^2≤25\big\}\), which is a circle of radius \(5\) centered at the origin, along with its interior as shown in Figure \(\PageIndex{5}\).
We can use the limit laws, which apply to limits at the boundary of domains as well as interior points:
\[\begin{align*} \lim_{(x,y)→(4,3)}\sqrt{25−x^2−y^2} =\sqrt{\lim_{(x,y)→(4,3)}(25−x^2−y^2)} \\ = \sqrt{\lim_{(x,y)→(4,3)}25−\lim_{(x,y)→(4,3)}x^2−\lim_{(x,y)→(4,3)}y^2} \\ =\sqrt{25−4^2−3^2} \\ = 0 \end{align*}\]
See the following graph.
Exercise \(\PageIndex{3}\)
Evaluate the following limit:
\[\lim_{(x,y)→(5,−2)}\sqrt{29−x^2−y^2}. \nonumber \]
- Hint
-
Determine the domain of \(f(x,y)=\sqrt{29−x^2−y^2}\).
- Answer
-
\[\lim_{(x,y)→(5,−2)}\sqrt{29−x^2−y^2} = \sqrt{29−5^2−(-2)^2} = \sqrt{29−25−4} = \sqrt{0} = 0\nonumber \]
Continuity of Functions of Two Variables
In Continuity, we defined the continuity of a function of one variable and saw how it relied on the limit of a function of one variable. In particular, three conditions are necessary for \(f(x)\) to be continuous at point \(x=a\)
- \(f(a)\) exists.
- \(\displaystyle \lim_{x→a}f(x)\) exists.
- \(\displaystyle \lim_{x→a}f(x)=f(a).\)
These three conditions are necessary for continuity of a function of two variables as well.
Definition: continuous Functions
A function \(f(x,y)\) is continuous at a point \((a,b)\) in its domain if the following conditions are satisfied:
- \(f(a,b)\) exists.
- \(\displaystyle \lim_{(x,y)→(a,b)}f(x,y)\) exists.
- \(\displaystyle \lim_{(x,y)→(a,b)}f(x,y)=f(a,b).\)
Example \(\PageIndex{4}\): Demonstrating Continuity for a Function of Two Variables
Show that the function
\[f(x,y)=\dfrac{3x+2y}{x+y+1} \nonumber \]
is continuous at point \((5,−3).\)
Solution
There are three conditions to be satisfied, per the definition of continuity. In this example, \(a=5\) and \(b=−3.\)
1. \(f(a,b)\) exists. This is true because the domain of the function f consists of those ordered pairs for which the denominator is nonzero (i.e., \(x+y+1≠0\)). Point \((5,−3)\) satisfies this condition. Furthermore,
\[f(a,b)=f(5,−3)=\dfrac{3(5)+2(−3)}{5+(−3)+1}=\dfrac{15−6}{2+1}=3. \nonumber \]
2. \(\displaystyle \lim_{(x,y)→(a,b)}f(x,y)\) exists. This is also true:
\[\begin{align*} \lim_{(x,y)→(a,b)}f(x,y) =\lim_{(x,y)→(5,−3)}\dfrac{3x+2y}{x+y+1} \\ =\dfrac{\displaystyle \lim_{(x,y)→(5,−3)}(3x+2y)}{\displaystyle \lim_{(x,y)→(5,−3)}(x+y+1)} \\ = \dfrac{15−6}{5−3+1} \\ = 3. \end{align*}\]
3. \(\displaystyle \lim_{(x,y)→(a,b)}f(x,y)=f(a,b).\) This is true because we have just shown that both sides of this equation equal three.
Exercise \(\PageIndex{4}\)
Show that the function
\[f(x,y)=\sqrt{26−2x^2−y^2}\nonumber \]
is continuous at point \((2,−3)\).
- Hint
-
Use the three-part definition of continuity.
- Answer
-
- The domain of \(f\) contains the ordered pair \((2,−3)\) because \(f(a,b)=f(2,−3)=\sqrt{16−2(2)^2−(−3)^2}=3\)
- \(\displaystyle \lim_{(x,y)→(a,b)}f(x,y)=3\)
- \(\displaystyle \lim_{(x,y)→(a,b)}f(x,y)=f(a,b)=3\)
Continuity of a function of any number of variables can also be defined in terms of delta and epsilon. A function of two variables is continuous at a point \((x_0,y_0)\) in its domain if for every \(ε>0\) there exists a \(δ>0\) such that, whenever \(\sqrt{(x−x_0)^2+(y−y_0)^2}<δ\) it is true, \(|f(x,y)−f(a,b)|<ε.\) This definition can be combined with the formal definition (that is, the epsilon–delta definition ) of continuity of a function of one variable to prove the following theorems:
Theorem \(\PageIndex{2}\): The Sum of Continuous Functions is Continuous
If \(f(x,y)\) is continuous at \((x_0,y_0)\), and \(g(x,y)\) is continuous at \((x_0,y_0)\), then \(f(x,y)+g(x,y)\) is continuous at \((x_0,y_0)\).
Theorem \(\PageIndex{3}\): The Product of Continuous Functions is Continuous
If \(g(x)\) is continuous at \(x_0\) and \(h(y)\) is continuous at \(y_0\), then \(f(x,y)=g(x)h(y)\) is continuous at \((x_0,y_0).\)
Theorem \(\PageIndex{4}\): The Composition of Continuous Functions is Continuous
Let \(g\) be a function of two variables from a domain \(D⊆\mathbb{R}^2\) to a range \(R⊆R.\) Suppose \(g\) is continuous at some point \((x_0,y_0)∈D\) and define \(z_0=g(x_0,y_0)\). Let f be a function that maps \(R\) to \(R\) such that \(z_0\) is in the domain of \(f\). Last, assume \(f\) is continuous at \(z_0\). Then \(f∘g\) is continuous at \((x_0,y_0)\) as shown in Figure \(\PageIndex{7}\).
Let’s now use the previous theorems to show continuity of functions in the following examples.
Example \(\PageIndex{5}\): More Examples of Continuity of a Function of Two Variables
Show that the functions \(f(x,y)=4x^3y^2\) and \(g(x,y)=\cos(4x^3y^2)\) are continuous everywhere.
Solution
The polynomials \(g(x)=4x^3\) and \(h(y)=y^2\) are continuous at every real number, and therefore by the product of continuous functions theorem, \(f(x,y)=4x^3y^2\) is continuous at every point \((x,y)\) in the \(xy\)-plane. Since \(f(x,y)=4x^3y^2\) is continuous at every point \((x,y)\) in the \(xy\)-plane and \(g(x)=\cos x\) is continuous at every real number \(x\), the continuity of the composition of functions tells us that \(g(x,y)=\cos(4x^3y^2)\) is continuous at every point \((x,y)\) in the \(xy\)-plane.
Exercise \(\PageIndex{5}\)
Show that the functions \(f(x,y)=2x^2y^3+3\) and \(g(x,y)=(2x^2y^3+3)^4\) are continuous everywhere.
- Hint
-
Use the continuity of the sum, product, and composition of two functions.
- Answer
-
The polynomials \(g(x)=2x^2\) and \(h(y)=y^3\) are continuous at every real number; therefore, by the product of continuous functions theorem, \(f(x,y)=2x^2y^3\) is continuous at every point \((x,y)\) in the \(xy\)-plane. Furthermore, any constant function is continuous everywhere, so \(g(x,y)=3\) is continuous at every point \((x,y)\) in the \(xy\)-plane. Therefore, \(f(x,y)=2x^2y^3+3\) is continuous at every point \((x,y)\) in the \(xy\)-plane. Last, \(h(x)=x^4\) is continuous at every real number \(x\), so by the continuity of composite functions theorem \(g(x,y)=(2x^2y^3+3)^4\) is continuous at every point \((x,y)\) in the \(xy\)-plane.
Functions of Three or More Variables
The limit of a function of three or more variables occurs readily in applications. For example, suppose we have a function \(f(x,y,z)\) that gives the temperature at a physical location \((x,y,z)\) in three dimensions. Or perhaps a function \(g(x,y,z,t)\) can indicate air pressure at a location \((x,y,z)\) at time \(t\). How can we take a limit at a point in \(\mathbb{R}^3\)? What does it mean to be continuous at a point in four dimensions?
The answers to these questions rely on extending the concept of a \(δ\) disk into more than two dimensions. Then, the ideas of the limit of a function of three or more variables and the continuity of a function of three or more variables are very similar to the definitions given earlier for a function of two variables.
Definition: \(δ\)-balls
Let \((x_0,y_0,z_0)\) be a point in \(\mathbb{R}^3\). Then, a \(δ\)-ball in three dimensions consists of all points in \(\mathbb{R}^3\) lying at a distance of less than \(δ\) from \((x_0,y_0,z_0)\) —that is,
\[\big\{(x,y,z)∈\mathbb{R}^3∣\sqrt{(x−x_0)^2+(y−y_0)^2+(z−z_0)^2}<δ\big\}. \nonumber \]
To define a \(δ\)-ball in higher dimensions, add additional terms under the radical to correspond to each additional dimension. For example, given a point \(P=(w_0,x_0,y_0,z_0)\) in \(\mathbb{R}^4\), a \(δ\) ball around \(P\) can be described by
\[\big\{(w,x,y,z)∈\mathbb{R}^4∣\sqrt{(w−w_0)^2+(x−x_0)^2+(y−y_0)^2+(z−z_0)^2}<δ\big\}. \nonumber \]
To show that a limit of a function of three variables exists at a point \((x_0,y_0,z_0)\), it suffices to show that for any point in a \(δ\) ball centered at \((x_0,y_0,z_0)\), the value of the function at that point is arbitrarily close to a fixed value (the limit value). All the limit laws for functions of two variables hold for functions of more than two variables as well.
Example \(\PageIndex{6}\): Finding the Limit of a Function of Three Variables
Find
\[\lim_{(x,y,z)→(4,1,−3)}\dfrac{x^2y−3z}{2x+5y−z}. \nonumber \]
Solution
Before we can apply the quotient law, we need to verify that the limit of the denominator is nonzero. Using the difference law, the identity law, and the constant law,
\[\begin{align*}\lim_{(x,y,z)→(4,1,−3)}(2x+5y−z) =2(\lim_{(x,y,z)→(4,1,−3)}x)+5(\lim_{(x,y,z)→(4,1,−3)}y)−(\lim_{(x,y,z)→(4,1,−3)}z) \\ = 2(4)+5(1)−(−3) \\ = 16. \end{align*}\]
Since this is nonzero, we next find the limit of the numerator. Using the product law, power law, difference law, constant multiple law, and identity law,
\[\begin{align*} \lim_{(x,y,z)→(4,1,−3)}(x^2y−3z) =(\lim_{(x,y,z)→(4,1,−3)}x)^2(\lim_{(x,y,z)→(4,1,−3)}y)−3\lim_{(x,y,z)→(4,1,−3)}z \\ =(4^2)(1)−3(−3) \\ = 16+9 \\ = 25 \end{align*}\]
Last, applying the quotient law:
\[\lim_{(x,y,z)→(4,1,−3)}\dfrac{x^2y−3z}{2x+5y−z}=\dfrac{\displaystyle \lim_{(x,y,z)→(4,1,−3)}(x^2y−3z)}{\displaystyle \lim_{(x,y,z)→(4,1,−3)}(2x+5y−z)}=\dfrac{25}{16} \nonumber \]
Exercise \(\PageIndex{6}\)
Find
\[\lim_{(x,y,z)→(4,−1,3)}\sqrt{13−x^2−2y^2+z^2} \nonumber \]
- Hint
-
Use the limit laws and the continuity of the composition of functions.
- Answer
-
\[\lim_{(x,y,z)→(4,−1,3)}\sqrt{13−x^2−2y^2+z^2}=2 \nonumber \]
Key Concepts
- To study limits and continuity for functions of two variables, we use a \(δ\) disk centered around a given point.
- A function of several variables has a limit if for any point in a \(δ\) ball centered at a point \(P\), the value of the function at that point is arbitrarily close to a fixed value (the limit value).
- The limit laws established for a function of one variable have natural extensions to functions of more than one variable.
- A function of two variables is continuous at a point if the limit exists at that point, the function exists at that point, and the limit and function are equal at that point.
Glossary
- boundary point
- a point \(P_0\) of \(R\) is a boundary point if every \(δ\) disk centered around \(P_0\) contains points both inside and outside \(R\)
- closed set
- a set \(S\) that contains all its boundary points
- connected set
- an open set \(S\) that cannot be represented as the union of two or more disjoint, nonempty open subsets
- \(δ\) disk
- an open disk of radius \(δ\) centered at point \((a,b)\)
- \(δ\) ball
- all points in \(\mathbb{R}^3\) lying at a distance of less than \(δ\) from \((x_0,y_0,z_0)\)
- interior point
- a point \(P_0\) of \(\mathbb{R}\) is a boundary point if there is a \(δ\) disk centered around \(P_0\) contained completely in \(\mathbb{R}\)
- open set
- a set \(S\) that contains none of its boundary points
- region
- an open, connected, nonempty subset of \(\mathbb{R}^2\)


2.5: Native American Stories
Native American Stories
Native Americans have wonderful stories about the night skies and constellations. The Yakima Tribe, from what is today central Washington state, saw Cassiopeia as a stretched, drying elk skin. Cherokees saw two bright stars in today’s Canis Major — the great dog — as two dogs guarding the path to the land of souls. That path indicated by the Cherokee? The Milky Way. And, according to the East Coast Mesquakie Tribe and the Great Lakes Iroquois Tribe, the bowl stars of the Big Dipper in the constellation Ursa Major (the Big Bear) form a bear, with the stars of the Big Dipper’s handles as hunters. (1)
- Authored by : Florida State College at Jacksonville. License : CC BY: Attribution


1.47: Medicine for the soul- spirituality in pharmacy
At the end of this case, students will be able to:
- Recognize the role of spirituality in providing whole person patient care
- Identify spiritual concerns that may impact a patient’s medical care
- Incorporate plans to address spiritual concerns in patient care
Introduction
In delivering patient-centered care, patients should be treated holistically as people and not simply as having a disease. Acknowledging spirituality is an important aspect of caring for a patient and is considered part of whole person patient care; however, it is not often integrated into the traditional patient care plan. 1 According to recent Gallup polls, approximately 90% of people in the United States believe in God or a universal spirit; 50% of all Americans define themselves as religious and spiritual, while 25% identify as spiritual but not religious, 5% as religious but not spiritual, and only 20% as neither spiritual nor religious. 2,3
Americans who are more religious tend to have higher wellbeing. 4 During recent global and domestic crises, including the 9/11 terrorist attacks and the COVID-19 pandemic, Americans reported an increase in church attendance, improved spirituality, and praying for the pandemic to end. 5 Research in mental health has demonstrated that religious beliefs and practices have been associated with lower rates of anxiety, depression, substance abuse, and suicide, as well as faster recovery, greater meaning in life, and social support. 1 In addition to mental health, religion and spirituality have been associated with improvements related to chronic diseases including diabetes, cancer, renal and cardiovascular diseases. 6-9
Assessing a patient’s religious and/or spiritual needs can be completed through a variety of available spiritual assessment tools. Spiritual assessment tools may be very short or extremely detailed, general or disease-specific, and questions should be open-ended. Some commonly recognized spiritual assessment tools are the HOPE, 10 FICA, 11 SPIRITual Assessment, 12 CSI-Memo, 13 ACP Spiritual History, 14 and Single Question 15 tools. The Joint Commission Accreditation Standards mandate completion of a spiritual assessment, 16 and the American Psychiatric Association, in the Diagnostic and Statistical Manual of Mental Disorders (DSM-5), has a diagnostic classification of “religious or spiritual problem.” 17 Generally, spiritual assessments are recommended in the following settings: new patient visits in ambulatory care, new hospital admissions, when a patient is in crisis, when a patient receives “bad news,” when a patient is struggling with lifestyle changes, and in patients with difficult to treat diseases. 15 Notably, healthcare professionals do not need to be of the same religious background to conduct a spiritual assessment.
It is important to recognize that spiritual beliefs of patients will vary widely, even if they are part of an organized religion. 18 As healthcare professionals, there is need to provide patient-centered, whole person care, and be careful not to stereotype or pass judgment. In the realm of spirituality, this includes taking a spiritual assessment when appropriate, supporting a patient’s spiritual or religious practices (e.g., attending religious services, reading religious texts), being sensitive to a patient’s spiritual concerns, praying with a patient (if the patient requests it and the healthcare professional is comfortable doing so), or making referrals to a chaplain or other recognized religious leader. 19
In pharmacy practice, pharmacists must recognize that the spiritual beliefs of patients can influence their general health beliefs and behaviors all the way through end-of-life care. 18 For example, some individuals may favor integrative healing methods or treatments which are thought to support the human spirit. 20 Examples may include meditation, massage, prayer, tai chi, yoga, acupuncture, and natural products. In addition, spiritual beliefs may impact patient decisions regarding use of therapies such as blood products or pharmacologic agents including pain medications, contraceptives, immunizations, or medications containing animal products or gelatin. 18 A final consideration is that medications requiring administration with food may need to be adjusted when patients are fasting due to religious observances.
Case
You are the pharmacist in an ambulatory care clinic. Your physician colleague asks you to provide a medication therapy management (MTM) consultation. The physician has concerns because upon completion of the FICA spiritual assessment, she determined the patient will be fasting during the month for Ramadan.
CC: “The doctor wanted me to see you about my medications during Ramadan.”
 He is a refugee from Bangladesh and has been living in the United States for the past 15 years. He is in clinic for a routine follow-up visit.
HPI: MH has episodes of hypoglycemia at least once a month. He monitors his blood glucose once or twice a week and when he feels like his blood glucose is low. He has had no episodes of DKA or HHS and has fasted before for Ramadan; it appears his experience was positive, and he would like to fast again for 16 hours a day.
PMH: T2DM x 11 years; HTN; GERD
FH: Unknown
SH:
- History of smoking (1/2 ppd; quit seven years ago)
- Denies alcohol or illicit drug use
Medications:
- Metformin 1000 mg BID
- Liraglutide 1.2 mg subcutaneously daily
- Amlodipine 10 mg daily
- Lisinopril 10 mg daily
- Famotidine 20 mg daily PRN
Allergies: NKDA
Vitals:
- BP 135/95 mmHg
- HR 85 bpm
- RR 16/min
- Temp 97.9°F
- SpO 2 100% RA
Labs:
- SCr: 1.1 mg/dL
- POC fasting glucose: 138 mg/dL
- HgbA1c: 9.2%
SDH: MH speaks Bengali as his dominant language and completed English as second language classes upon arrival in the United States. His income is approximately $39,500/year. His wife does not work and is responsible for most of the food preparation; they follow a halal diet. They have three children at home who currently attend the local public school. MH’s neighborhood community is diverse and consists of other Bengali refugees, Polish immigrants, and Black Americans. He is a devout Muslim and lives within walking distance of the mosque.
Additional context: Your physician colleague’s FICA assessment notes the following:
- F (Faith or Beliefs): MH is a practicing Muslim who considers himself to be religious.
- I (Importance and Influence): MH’s faith is extremely important to him and greatly impacts his daily life. Specific to our meeting today, he believes that it is his duty to fast during Ramadan, but he will still take care of his diabetes and believes his current medications are allowed when he fasts. It is very important for him to be able to fast during this time.
- C (Community): MH is part of a religious community, and they are supportive of each other. He does attend religious services at a mosque. It is close to his house, and he walks there to pray five times a day when not working.
- A (Address): He would like our team to recognize that he would like to continue to fast during this time and would like to do so safely.
Case Questions
- What role would acknowledging MH’s spirituality have when caring for him?
- What challenges might you, as a healthcare professional, face when addressing MH’s spirituality?
- Why is it important to you, as the pharmacist, to conduct a spiritual assessment? What important religious or spiritual beliefs impact patient care?
- What specific spirituality-related concerns might you have regarding MH’s pharmaceutical care?
- What specific non-pharmacologic interventions could you recommend to integrate MH’s spirituality into his care plan, both in general, and during Ramadan specifically?
- What specific pharmacologic interventions could you recommend to integrate MH’s spirituality into his care plan?
Author Commentary
Spirituality and its role in patient care is often overlooked and under-valued. As the demographics of the United States change, it is expected that pharmacists and other clinicians will continue to care for patients with a wide range of different beliefs. From a global perspective, faith-based organizations (FBOs) have played a role in assisting with social determinants of health and providing healthcare services. 21,22 These FBOs have assisted in providing educational services, healthcare access, and psycho-social support/counseling. FBOs have also been integral in policy formation to achieve the United Nations Sustainable Development Goals. 21 During the COVID-19 pandemic, the Centers for Disease Control and Prevention (CDC) offered guidance for providing spiritual care to patients; however, it is important to note that this guidance may have differed from what FBOs would have recommended. 23 In addition, religious leaders were instrumental in educating their communities, specifically the African American community, on the realities of COVID-19. 24
Pharmacists should be aware of the potential impact of religion/spirituality on patient outcomes. For example, religion/spirituality has been identified as an important aspect of care for patients with HIV, including a positive association with medication adherence and clinical health outcomes, 25,26 as well as HIV prevention strategies. 27 There have also been positive associations in addiction medicine, with implications for helping to combat the current opioid epidemic and other substance use disorders, specifically alcohol use disorder. 28-30 Pharmacists should consider providing spiritual assessments to these patients and determining the role spirituality may have in assisting patients in improving their health. In addition, pharmacists are aptly situated towards recognizing and addressing spiritual or religious medication related concerns.
Patient Approaches and Opportunities
Recognizing the role of spirituality in patients’ lives will allow pharmacists to provide holistic patient-centered care. It is important to realize that pharmacists do not need to be experts on every religion or have comprehensive knowledge about integrative medicine. What is important is determining the role of spirituality in the life of each patient through a standardized spiritual assessment tool. Oftentimes, particularly in a busy practice setting, utilizing the single-question tool will provide enough information to begin a conversation. Allow the patient to lead you in a discussion about what is important for them; if they are of a particular faith/religious background listen to them about what restrictions they may have based on their religion, keeping in mind that everyone may approach religion differently.
From a pharmacologic perspective, most spiritual-based medication restrictions are in utilizing animal-derived products, gelatin or stearic acid-containing tablets/capsules, pain medications, contraceptives, immunizations, and blood products. 18,31 If you are caring for a patient who has a terminal condition or has a recent major life change, consider the role that spirituality or religion might have in their life. They may have or choose restrictions about end-of-life or palliative care, try integrative medicine, benefit from prayer, reading religious texts or simply need the support of a religious/spiritual community. 18,20 Do not hesitate to refer a patient to a chaplain or other member of the religious community for assistance. 15
Related chapters of interest:
- Saying what you mean doesn’t always mean what you say: cross-cultural communication
- More than just diet and exercise: social determinants of health and well-being
- Getting to the point: importance of immunizations for public health
- When disaster strikes: managing chaos and instilling lessons for future events
- You say medication, I say meditation: effectively caring for diverse populations
- Sweetening the deal: improving health outcomes for patients with diabetes mellitus
External resources:
-
Websites:
- Duke University. Center for Spirituality, Theology and Health. https://spiritualityandhealth.duke.edu/
- American Psychiatric Association. Mental health and faith community partnership. https://www.psychiatry.org/psychiatrists/cultural-competency/engagement-opportunities/mental-health-and-faith-community-partnership
- International Diabetes Federation and Diabetes and Ramadan International Alliance. Diabetes and Ramadan: practical guidelines 2021. https://www.idf.org/our-activities/education/diabetes-and-ramadan/healthcare-professionals.html
-
Journal articles:
- Smith KM, Hoesli TM. Effects of religious and personal beliefs on medication dosing regimens. Orthopedics 2011;34(4):292-295.
References
- Koenig HG. Religion, spirituality, and medicine: research findings and implications for clinical practice. South Med J 2004;97(12):1194-200.
- Gallup. Religion. https://news.gallup.com/poll/1690/religion.aspx . Accessed March 16, 2021.
- Lipka M, Gecewicz C. More Americans now say they’re spiritual but not religious. Pew Research Center. https://www.pewresearch.org/fact-tank/2017/09/06/more-americans-now-say-theyre-spiritual-but-not-religious/ . Accessed March 16, 2021.
- Newport F, Witters D, Agrawal S. Religious Americans enjoy higher wellbeing. https://news.gallup.com/poll/152723/religious-americans-enjoy-higher-wellbeing.aspx . Accessed March 25, 2021.
- Newport F. Religion and the COVID-19 virus in the U.S. https://news.gallup.com/opinion/polling-matters/307619/religion-covid-virus.aspx . Accessed March 25, 2021.
- Al-Ghabeesh SH, Alshraifeen AA, Saifan AR, Bashayreh IH, Alnuaimi KM, Masalha HA. Spirituality in the lives of patients with end-stage renal disease: a systematic review. J Relig Health 2018;57(6):2461-77.
- Abu HO, Ulbricht C, Ding E, et al. Association of religiosity and spirituality with quality of life in patients with cardiovascular disease: a systematic review. Qual Life Res 2018;27(11):2777-97.
- Bai M, Lazenby M. A systematic review of associations between spiritual well-being and quality of life at the scale and factor levels in studies among patients with cancer. J Palliat Med 2015;18(3):286-98.
- Lynch CP, Hernandez-Tejada MA, Strom JL, Egede LE. Association between spirituality and depression in adults with type 2 diabetes. Diabetes Educ 2012;38:427-35.
- Anandarajah G, Hight E. Spirituality and medical practice: using the HOPE questions as a practical tool for spiritual assessment. Am Fam Physician 2001;63(1):81-9.
- Borneman T, Ferrell B, Puchalski CM. Evaluation of the FICA tool for spiritual assessment. J Pain Symptom Manage 2010;40(2):163-73.
- Maugans TA. The SPIRITual history. Arch Fam Med 1996;5(1):11-6.
- Koenig HG. An 83-year-old woman with chronic illness and strong religious beliefs. JAMA 2002;288(4):487-93.
- Lo B, Quill T, Tulsky J. Discussing palliative care with patients. ACP-ASIM End-of-Life Care Consensus Panel. American College of Physicians-American Society of Internal Medicine. Ann Int Med 1999;130(9):744-9.
- Koenig HG. Spirituality in patient care: why, how, when, and what . 3 rd ed. Templeton Press; 2013.
- The Joint Commission on Accreditation of Healthcare Organizations. Evaluating your spiritual assessment process. Joint Commission: The Source 2005;3(2):6-7. https://www.professionalchaplains.org/files/resources/reading_room/evaluating_your_spiritual_assessment_process.pdf . Accessed March 17, 2021.
- American Psychiatric Association Foundation. Mental health: a guide for faith leaders. 2018. https://www.psychiatry.org/File%20Library/Psychiatrists/Cultural-Competency/Mental_Health_Guide_Tool_Kit_2018.pdf . Accessed March 17, 2021.
- Galanti G. The Joint Commission: cultural and religious sensitivity: a pocket guide for health care professionals. Joint Commission Resources. 3rd ed, 2018.
- Koenig HG. Religion, spirituality, and medicine: application to clinical practice. JAMA 2000;284(13):1708.
- Steinhorn DM, Din J, Johnson A. Healing, spirituality and integrative medicine. Ann Palliat Med 2017;6(3):237-47.
- United Nations Population Fund. Realizing the faith dividend: religion, gender, peace and security in Agenda 2030. 2016. https://www.sdgfund.org/realizing-faith-dividend-religion-gender-peace-and-security-agenda-2030 . Accessed March 18, 2021.
- Duff JF, Buckingham WW. Strengthening of partnerships between the public sector and faith-based groups. Lancet 2015;386(10005):1786-94.
- Centers for Disease Control and Prevention. COVID-19: providing spiritual and psychosocial support to people with COVID-19 at home (non-US settings). https://www.cdc.gov/coronavirus/2019-ncov/global-covid-19/providing-spiritual-support.html . Accessed March 18, 2021.
- Thompkins F, Goldblum P, Lai T, Hansell T, Barclay A, Brown LM. A culturally specific mental health and spirituality approach for African Americans facing the COVID-19 pandemic. Psychol Trauma 2020;12(5):455-6.
- Doolittle BR, Justice AC, Fiellin DA. Religion, spirituality, and HIV clinical outcomes: a systematic review of the literature. AIDS Behav 2018;22(6):1792-1801.
- Medved Kendrick H. Are religion and spirituality barriers or facilitators to treatment for HIV: a systematic review of the literature. AIDS Care 2017;29(1):1-13.
- Vigliotti V, Taggart T, Walker M, Kusmastuti S, Ransome Y. Religion, faith, and spirituality influences on HIV prevention activities: a scoping review. PLoS One 2020;15(6):e0234720. Erratum in: PLoS One 2020;15(10):e0241737.
- Beraldo L, Gil F, Ventriglio A, et al. Spirituality, religiosity and addiction recovery: current perspectives. Curr Drug Res Rev 2019;11(1):26-32.
- Dermatis H, Galanter M. The role of twelve-step-related spirituality in addiction recovery. J Relig Health 2016;55(2):510-21.
- Sliedrecht W, de Waart R, Witkiewitz K, Roozen HG. Alcohol use disorder relapse factors: a systematic review. Psychiatry Res 2019;278:97-115.
- Smith KM, Hoesli TM. Effects of religious and personal beliefs on medication dosing regimens. Orthopedics 2011;34(4):292-5.


12.3: Energy Flow Through Ecosystems
Energy capture and use
Photosynthesis
At the base of an ecosystem, primary producers are actively converting solar energy into stored chemical energy. Photosynthesis is the process of converting solar energy, water, and carbon dioxide into carbohydrates and oxygen. The process occurs in two steps: first light energy is absorbed by chlorophyll to split a molecule of water releasing hydrogen and oxygen. The second step uses the energy to convert carbon dioxide to carbohydrates.
Solar Energy
6CO 2 + 12H 2 O --------------> C 6 H 12 O 6 + 6O 2 + 6H 2 O
(chlorophyll)
The carbohydrate (C 6 H 12 O 6 ) can be converted into starch and stored by the plant. Carbohydrate can be combined with other sugar molecules to make cellulose, the basic structural material of a plant.
Video : "Photosynthesis". (Courtesy of Britannica)
Oddly enough, of all the solar radiation striking a plant, only about 1% is used in photosynthesis. The rate of photosynthesis is dependent on several things, especially the amount of light received ... up to a point. As solar radiation increases the rate of photosynthesis increases. For many plants there is an upper limit to the rate of photosynthesis. In some plants as incident solar radiation increases the rate of photosynthesis levels off, or may decrease. The increasing solar energy load causes the plant to be too hot and the need to cool the plant increases. As a result, transpiration takes over as the dominate plant process. Transpiration , the loss of water from plants, acts to cool the plant by releasing latent energy. Adequate supplies of water, carbon dioxide and the availability of nutrients in the soil affect photosynthesis.
Respiration
While photosynthesis builds stored chemical energy in a plant, respiration is the process of "burning" stored chemical energy, basically through oxidation, for maintaining plant metabolism. During plant respiration, carbohydrates combine with oxygen and is reduced to carbon dioxide, water, and heat.
C 6 H 12 O 6 + 6O 2 ---> 6CO 2 + 6H 2 O + 2830 kj
While photosynthesis operates only during day when sunshine is available, respiration goes on both night and day. Plant growth occurs so long as photosynthesis exceeds respiration.
Transpiration
Transpiration is the loss of water from plant leaves. Water exits the leaf through stomata , which are tiny pore spaces in the leaf. The rate of transpiration depends on air temperature and solar radiation. As pointed out earlier, transpiration is a cooing process for plants when temperatures or incident light rise too high and cause heating of the plant. Low humidity, often aided by windy conditions, creates a vapor gradient between the plant and the air. This too induces transpiration. Soil factors are important control over transpiration. If the pore space between soil particles are too large the soil will have poor or low soil capillary. That is, the rate of water rise is too low for plants to extract water from the soil and maintain proper moisture supply. Low soil capillary results from soil drying too. During the moist season, ample soil water is available to line soil particles to aid the movement of soil water upwards to the root zone.
However, during the dry season, a dry layer of soil develops beneath the root zone inhibiting the upward movement of capillary water causing a capillary lag . The plant ultimately wilts as it cannot extract enough water to meet the increasing demand for water during warm seasons. A small soil moisture reserve will inhibit transpiration too.
Biomass Productivity
Net biomass productivity is the difference between gross productivity (production of plant material by photosynthesis) and respiration. So long as the rate of production exceeds that of respiration, the plant will grow. Net productivity represents the amount of organic material produced by a plant. Net productivity is closely related to a number of environmental factors like climate, soils, and available nutrients. Net biomass production will be highest where there is an ample supply of moisture to meet the needs of plants. Biomass productivity is also high where soils are rich in nutrients and have a positive soil moisture balance. The figure below illustrates this well. With ample rainfall and sunlight, the tropical rain forest ranks the highest in terms of organic matter production.
Human pressure on land and vegetation resources for food, wood-based fuel, fiber, and building materials is calling into question whether the earth's net production can sustain such use. Scientists are employing satellite technology to determine the trends in human consumption of net primary production as described in the video below. Human use of earth's annual production rose from 20 to 25 percent from 1995 to 2005. As society looks to land conversion for biofuels, climate changes, and biodiversity loss occurs, a tipping point at which net production fails to meet demand may be reached in the not so distant future.
Video : Human Consumption of NPP. (Courtesy NASA GSFC)
Trophic Levels and Food Chains
The biotic elements that comprise an ecosystem fall into one of several trophic levels . The trophic level of an organism is its position in a food chain , the sequence of consumption and energy transfer through the environment. For example, a simple grazing food chain is comprised of
Plant -> herbivore -> carnivore
At the base of the food chain lies the primary producers . Primary producers are principally green plants and certain bacteria. They convert solar energy into organic energy. Above the primary producers are the consumers who ingest live plants or the prey of others. Decomposers , such as, bacteria, molds, and fungi make use of energy stored in already dead plant and animal tissues. Fungi, like mushrooms, absorb nutrients from the organisms by secreting enzymes to break up the chemical compounds that make up dead plants and animals.
Two laws of physics are important in the study of energy flow through ecosystems. The first law of thermodynamics states that energy cannot be created or destroyed; it can only be changed from one form to another. Energy for the functioning of an ecosystem comes from the Sun. Solar energy is absorbed by plants where in it is converted to stored chemical energy.
The second law of thermodynamics states that whenever energy is transformed, there is a loss energy through the release of heat. This occurs when energy is transferred between trophic levels as illustrated in a food web . When one animal feeds off another, there is a loss of heat (energy) in the process. Additional loss of energy occurs during respiration and movement. Hence, more and more energy is lost as one moves up through trophic levels. This fact lends more credence to the advantages of a vegetarian diet. For example, 1350 kilograms of corn and soybeans is capable of supporting one person if converted to beef. However, 1350 kilograms of soybeans and corn utilized directly without converting to beef will support 22 people!


13.37: Outcome- Determining IMC Objectives and Approach
What you’ll learn to do: explain factors to consider when selecting marketing communication methods to execute the strategy
It’s clear that with the growing proliferation of communication tools and methods, integrated marketing communications are the way of the future—and now. Seemingly every day brings a new social media tool or digital marketing technique to engage people in new ways. Traditional marketing communications methods and media are also stepping up their games, offering new ways to create value for companies trying to connect with their target audiences. For example, old-school conferences and trade shows now feature active mobile and social media elements that have been incorporated into their design. TV shows can sell ad space and sponsorship on air, online, and on social media feeds. Radio programs publish their podcast counterparts, complete with ads and sponsors.
For marketers, all this is great news: plenty of choices and ample opportunity to connect with customers in new ways.
But is it great news? The variety of marketing communication methods and tools can be overwhelming. How do you even get started designing an IMC program? And once you have picked an approach, how do you know you’re on the right track?
These are big questions marketers ask themselves regularly. Because marketing is a constantly evolving field, the right answer on one day might be different six months later. However, there are time-tested models that can help you apply a systematic approach to defining what you want to accomplish with IMC and how to select an approach that is best suited to your objectives.
The specific things you’ll learn in this section include:
- Discuss the AIDA model and the role of marketing communications to help move contacts toward a purchasing decision.
- Describe push vs. pull marketing strategies
- Explain the S.M.A.R.T. model for developing IMC goals and objectives
- Discuss the process of selecting marketing communication methods and tactics to fit the target audience and marketing objectives
Learning Activities
- Reading: Determining IMC Objectives and Approach
- Video: Prioritizing Marketing Communications
Contributors and Attributions
- Provided by : Lumen Learning. License : CC BY: Attribution


1.6: Primary Functions of Management
- Explain the primary functions of management.
- Differentiate between the planning, organizing, leading, and controlling functions of management.
Effective management involves four primary functions and related skill sets: planning , organizing , leading , and controlling . Although there’s a logical sequence to the functions, in practice the four functions are often performed in a dynamic manner.
For example, a manager would need to develop or reference a departmental or organizational plan prior to executing on it just as you would reference a map prior to embarking on a road trip. The proverb “if you fail to prepare you are preparing to fail” underscores the importance of this function. However, just as when road or airport closures or other factors might cause you to change your original route, unanticipated internal or external factors might cause a manager to revisit and revise the original plan, requiring a change in the other functions and associated tasks. Thus, achieving organizational goals—arriving at your intended destination—requires ongoing management of the process and an understanding of the interrelationship of the four functions.
As Figure 1 illustrates, a factor that impacts leading, for example, will have implications for controlling, planning and organizing. In summary, it is a management responsibility to ensure that unanticipated changes are factored in to the process and the integrity of the process is maintained.
Planning
Planning means defining performance goals for the organization and determining what actions and resources are needed to achieve the goals. Through planning, management defines what the future of the organization should be and how to get there. Strategic plans are long-term and affect the entire organization. A strategic plan bridges the gap between what an organization is and what it will become. Tactical plans translate strategic plans into specific actions that need to be implemented by departments throughout the organization. The tactical plan defines what has to be done, who will do it, and the resources needed to do it.
For instance, consider Company X who decided to become an elevator manufacturing and servicing company because of increased competition from Chinese steel. The management of the company set a goal of deriving the majority of its revenue from elevator-related activities. To do this, the management team made plans to create partnerships or take over existing elevator companies. The team devised plans to develop new human resources and to acquire other material resources. The company also had to divest existing steel-related resources to raise capital for the new initiative. This example is a long-term strategic plan that will take years to complete and require many changes along the way. But it starts by defining a goal and a preliminary path to achieve it.
Organizing
Once plans are made, decisions must be made about how to best implement the plans. The organizing function involves deciding how the organization will be structured (by departments, matrix teams, job responsibilities, etc.). Organizing involves assigning authority and responsibility to various departments, allocating resources across the organization, and defining how the activities of groups and individuals will be coordinated.
In the case of Company X, the management had to determine how to support two very different sets of activities in order to achieve its long-term goal. Management needed to continue steel production activities to provide continuity of funds as the emphasis gradually shifted to elevator production. It also had to develop new skills and resources to build the company’s elevator capabilities. A new organizational structure was needed that could support both business activities as one was downsized and the other built up.
Leading
Nearly everything that is accomplished in an organization is done by people. The best planning and organizing will not be effective if the people in the organization are not willing to support the plan. Leaders use knowledge, character, and charisma to generate enthusiasm and inspire effort to achieve goals. Managers must also lead by communicating goals throughout the organization, by building commitment to a common vision, by creating shared values and culture, and by encouraging high performance. Managers can use the power of reward and punishment to make people support plans and goals. Leaders inspire people to support plans, creating belief and commitment. Leadership and management skills are not the same, but they can and do appear in the most effective people.
It is very difficult to motivate people when plans involve radical change, particularly if they include downsizing and layoffs. Many people are naturally resistant to change. When the change means loss of jobs or status, people will be very resistant. At Company X, the labor unions vehemently opposed the shift from steel production to elevator manufacturing. Although the people involved in the new business functions were excited by the plans, people involved with steel production felt abandoned and demotivated. Management would have been wise to get union support for its vision of the company’s new future.
Controlling
There is a well-known military saying that says no battle plan survives contact with the enemy. This implies that planning is necessary for making preparations, but when it’s time to implement the plan, everything will not go as planned. Unexpected things will happen. Observing and responding to what actually happens is called controlling. Controlling is the process of monitoring activities, measuring performance, comparing results to objectives, and making modifications and corrections when needed. This is often described as a feedback loop , as shown in the illustration of a product design feedback loop.
Controlling may be the most important of the four management functions. It provides the information that keeps the corporate goal on track. By controlling their organizations, managers keep informed of what is happening; what is working and what isn’t; and what needs to be continued, improved, or changed. Company X had little experience in elevator manufacturing when it was making plans. It was developing new products and processes and entering new markets. The management knew it could not anticipate all the difficulties it would encounter. Close monitoring as the plan progressed allowed the company to make changes and state-of-the-art innovations that have resulted in a very successful transition.
Who Directs Each Function?
Although these functions have been introduced in a particular order, it should be apparent that the different activities happen at the same time in any one organization. The control function ensures that new plans must be created. Leaders often step up as needed when a crisis or unexpected bump demands immediate action. All managers perform all of these functions at different times, although a manager’s position or level in the organization will affect how much of his or her time is spent planning as opposed to leading or to controlling.
Contributors and Attributions
- Primary Functions of Management. Authored by : John and Lynn Bruton. Published by Lumen Learning. License : CC BY: Attribution
- Image: Key Functions of the Management Process. Authored by : John and Lynn Bruton. Published by Lumen Learning. License : CC BY: Attribution
- Image: The key functions of management . Provided by : Lumen Learning. License : CC BY: Attribution
- Business Feedback Loop . Authored by : Tomwsulcer. Located at : https://commons.wikimedia.org/wiki/F...NG_version.png
- The Management Process . Authored by : Jeff Short. Located at : https://www.youtube.com/watch?v=9Ir70kcHf-w . License : All Rights Reserved . License Terms : Standard YouTube License