2.2: How an FTIR Spectrometer Operates
FTIR spectrometers (Fourier Transform Infrared Spectrometer) are widely used in organic synthesis, polymer science, petrochemical engineering, pharmaceutical industry and food analysis. In addition, since FTIR spectrometers can be hyphenated to chromatography, the mechanism of chemical reactions and the detection of unstable substances can be investigated with such instruments.
Introduction
The range of Infrared region is 12800 ~ 10 cm -1 and can be divided into near-infrared region (12800 ~ 4000 cm -1 ), mid-infrared region (4000 ~ 200 cm -1 ) and far-infrared region (50 ~ 1000 cm -1 ). The discovery of infrared light can be dated back to the 19th century. Since then, scientists have established various ways to utilize infrared light. Infrared absorption spectroscopy is the method which scientists use to determine the structures of molecules with the molecules’ characteristic absorption of infrared radiation. Infrared spectrum is molecular vibrational spectrum. When exposed to infrared radiation, sample molecules selectively absorb radiation of specific wavelengths which causes the change of dipole moment of sample molecules. Consequently, the vibrational energy levels of sample molecules transfer from ground state to excited state. The frequency of the absorption peak is determined by the vibrational energy gap. The number of absorption peaks is related to the number of vibrational freedom of the molecule. The intensity of absorption peaks is related to the change of dipole moment and the possibility of the transition of energy levels. Therefore, by analyzing the infrared spectrum, one can readily obtain abundant structure information of a molecule. Most molecules are infrared active except for several homonuclear diatomic molecules such as O 2 , N 2 and Cl 2 due to the zero dipole change in the vibration and rotation of these molecules. What makes infrared absorption spectroscopy even more useful is the fact that it is capable to analyze all gas, liquid and solid samples. The common used region for infrared absorption spectroscopy is 4000 ~ 400 cm -1 because the absorption radiation of most organic compounds and inorganic ions is within this region.
FTIR spectrometers are the third generation infrared spectrometer. FTIR spectrometers have several prominent advantages: (1) The signal-to-noise ratio of spectrum is significantly higher than the previous generation infrared spectrometers. (2) The accuracy of wavenumber is high. The error is within the range of ± 0.01 cm -1 . (3) The scan time of all frequencies is short (approximately 1 s). (4) The resolution is extremely high (0.1 ~ 0.005 cm -1 ). (5) The scan range is wide (1000 ~ 10 cm -1 ). (6) The interference from stray light is reduced. Due to these advantages, FTIR Spectrometers have replaced dispersive IR spectrometers.
Development of IR Spectrometers
Up till FTIR spectrometers, there have been three generations of IR spectrometers.
- The first generation IR spectrometer was invented in late 1950s. It utilizes prism optical splitting system. The prisms are made of NaCl. The requirement of the sample’s water content and particle size is extremely strict. Further more, the scan range is narrow. Additionally, the repeatability is fairly poor. As a result, the first generation IR spectrometer is no longer in use.
- The second generation IR spectrometer was introduced to the world in 1960s. It utilizes gratings as the monochrometer. The performance of the second generation IR spectrometer is much better compared with IR spectrometers with prism monochrometer, But there are still several prominent weaknesses such as low sensitivity, low scan speed and poor wavelength accuracy which rendered it out of date after the invention of the third generation IR spectrometer.
- The invention of the third generation IR spectrometer, Fourier transform infrared spectrometer, marked the abdication of monochrometer and the prosperity of interferometer. With this replacement, IR spectrometers became exceptionally powerful. Consequently, various applications of IR spectrometer have been realized.
Dispersive IR Spectrometers
To understand the powerfulness and usefulness of FTIR spectrometer, it is essential to have some background information of dispersive IR Spectrometer. The basic components of a dispersive IR spectrometer include a radiation source, monochromator, and detector. The common IR radiation sources are inert solids that are heated electrically to promote thermal emission of radiation in the infrared region of the electromagnetic spectrum. The monochromator is a device used to disperse or separate a broad spectrum of IR radiation into individual narrow IR frequencies.
Generally, dispersive spectrometers have a double-beam design with two equivalent beams from the same source passing through the sample and reference chambers as independent beams. These reference and sample beams are alternately focused on the detector by making use of an optical chopper, such as, a sector mirror. One beam will proceed, traveling through the sample, while the other beam will pass through a reference species for analytical comparison of transmitted photon wavefront information.
After the incident radiation travels through the sample species, the emitted wavefront of radiation is dispersed by a monochromator (gratings and slits) into its component frequencies. A combination of prisms or gratings with variable-slit mechanisms, mirrors, and filters comprise the dispersive system. Narrower slits gives better resolution by distinguishing more closely spaced frequencies of radiation and wider slits allow more light to reach the detector and provide better system sensitivity. The emitted wavefront beam (analog spectral output) hits the detector and generates an electrical signal as a response.
Detectors are devices that convert the analog spectral output into an electrical signal. These electrical signals are further processed by the computer using mathematical algorithm to arrive at the final spectrum. The detectors used in IR spectrometers can be classified as either photon/quantum detectors or thermal detectors.
It is the absorption of IR radiation by the sample, producing a change of IR radiation intensity, which gets detected as an off-null signal (e.g. different from reference signal). This change is translated into the recorder response through the actions of synchronous motors. Each frequency that passes through the sample is measured individually by the detector which consequently slows the process of scanning the entire IR region. A block diagram of a classic dispersive IR spectrometer is shown in Figure \(\PageIndex{1}\).
FTIR Spectrometers
The Components of FTIR Spectrometers
A common FTIR spectrometer consists of a source, interferometer, sample compartment, detector, amplifier, A/D convertor, and a computer. The source generates radiation which passes the sample through the interferometer and reaches the detector. Then the signal is amplified and converted to digital signal by the amplifier and analog-to-digital converter, respectively. Eventually, the signal is transferred to a computer in which Fourier transform is carried out. Figure \(\PageIndex{2}\) is a block diagram of an FTIR spectrometer.
The major difference between an FTIR spectrometer and a dispersive IR spectrometer is the Michelson interferometer.
Michelson Interferometer
The Michelson interferometer, which is the core of FTIR spectrometers, is used to split one beam of light into two so that the paths of the two beams are different. Then the Michelson interferometer recombines the two beams and conducts them into the detector where the difference of the intensity of these two beams are measured as a function of the difference of the paths. Figure \(\PageIndex{3}\) is a schematic of the Michelson Interferometer.
A typical Michelson interferometer consists of two perpendicular mirrors and a beamsplitter. One of the mirror is a stationary mirror and another one is a movable mirror. The beamsplitter is designed to transmit half of the light and reflect half of the light. Subsequently, the transmitted light and the reflected light strike the stationary mirror and the movable mirror, respectively. When reflected back by the mirrors, two beams of light recombine with each other at the beamsplitter.
If the distances travelled by two beams are the same which means the distances between two mirrors and the beamsplitter are the same, the situation is defined as zero path difference (ZPD). But imagine if the movable mirror moves away from the beamsplitter, the light beam which strikes the movable mirror will travel a longer distance than the light beam which strikes the stationary mirror. The distance which the movable mirror is away from the ZPD is defined as the mirror displacement and is represented by ∆. It is obvious that the extra distance travelled by the light which strikes the movable mirror is 2∆. The extra distance is defined as the optical path difference (OPD) and is represented by delta. Therefore,
\[\delta =2\Delta \label{1}\]
It is well established that when OPD is the multiples of the wavelength , constructive interference occurs because crests overlap with crests, troughs with troughs. As a result, a maximum intensity signal is observed by the detector. This situation can be described by the following equation:
\[\delta =n\lambda \label{2}\]
with n = 0,1,2,3...
In contrast, when OPD is the half wavelength or half wavelength add multiples of wavelength, destructive interference occurs because crests overlap with troughs. Consequently, a minimum intensity signal is observed by the detector. This situation can be described by the following equation:
\[\delta =(n+\dfrac{1}{2})\lambda \label{3} \]
with n = 0,1,2,3...
These two situations are two extreme situations. If the OPD is neither n-fold wavelengths nor (n+1/2)-fold wavelengths, the interference should be between constructive and destructive. So the intensity of the signal should be between maximum and minimum. Since the mirror moves back and forth, the intensity of the signal increases and decreases which gives rise to a cosine wave. The plot is defined as an interferogram. When detecting the radiation of a broad band source rather than a single-wavelength source, a peak at ZPD is found in the interferogram. At the other distance scanned, the signal decays quickly since the mirror moves back and forth. Figure \(\PageIndex{4}\)(a) shows an interferogram of a broad band source.
Fourier Transform of Interferogram to Spectrum
The interferogram is a function of time and the values outputted by this function of time are said to make up the time domain. The time domain is Fourier transformed to get a frequency domain, which is deconvolved to product a spectrum. Figure \(\PageIndex{4}\) shows the Fast Fourier transform from an interferogram of polychromatic light to its spectrum.
The Fourier Transform
The first one who found that a spectrum and its interferogram are related via a Fourier transform was Lord Rayleigh. He made the discover in 1892. But the first one who successfully converted an interferogram to its spectrum was Fellgett who made the accomplishment after more than half a century. Fast Fourier transform method on which the modern FTIR spectrometer based was introduced to the world by Cooley and Turkey in 1965 . It has been applied widely to analytical methods such as infrared spectrometry, nuclear magnetic resonance and mass spectrometry due to several prominent advantages which are listed in Table \(\PageIndex{1}\) .
Table \(\PageIndex{1}\).
Advantages of Fourier Transform over Continuous-Wave Spectrometry
Fourier transform, named after the French mathematician and physicist Jean Baptiste Joseph Fourier, is a mathematical method to transform a function into a new function. The following equation is a common form of the Fourier transform with unitary normalization constants:
in which t is time, i is the square root of -1.
The following equation is another form of the Fourier transform(cosine transform) which applies to real, even functions:
The following equation shows how f(t) is related to F(v) via a Fourier transform:
An Alternative Explanation of the Fourier Transform in FTIR Spectrometers
The math description of the Fourier transform can be tedious and confusing. An alternative explanation of the Fourier transform in FTIR spectrometers is provided here before we jump into the math description to give you a rough impression which may help you understand the math description.
The interferogram obtained is a plot of the intensity of signal versus OPD. A Fourier transform can be viewed as the inversion of the independent variable of a function. Thus, Fourier transform of the interferogram can be viewed as the inversion of OPD. The unit of OPD is centimeter, so the inversion of OPD has a unit of inverse centimeters, cm -1 . Inverse centimeters are also known as wavenumbers. After the Fourier transform, a plot of intensity of signal versus wavenumber is produced. Such a plot is an IR spectrum. Although this explanation is easy to understand, it is not perfectly rigorous.
Simplified Math Description of the Fourier Transform in FTIR
The wave functions of the reflected and transmitted beams may be represented by the general form of:
\[E_{1}=rtc E_{m}\times \cos(\nu t-2\pi kx) \label{7}\]
and
\[E_{1}=rtc E_{m}\times \cos[\nu t-2\pi k(\nu x+\Delta d)] \label{8}\]
where
- \(\Delta{d}\) is the path difference,
- \(r\) is the reflectance (amplitude) of t he beam splitter,
- \(t\) is the transmittance, and
- \(c\) is the polarization constant.
The resultant wave function of their superposition at the detector is represented as:
\[E=E_{1}+E_{2}=2(r\times t\times c\times E_{m})\times \cos(\nu t-2\pi kx)\cos(\pi k\Delta d) \label{9}\]
where E m , , ν, and k are the amplitude, frequency and wave number of the IR radiation source.
The intensity (\(I\)) detected is the time average of \(E_2\) and is written as
\[I=4r^{2}t^{2}c^{2}E_{m}^{2}cos^{2}(\nu t-2\pi kx)\cos^{2}(\pi k\Delta d) \label{10}\]
Since the time average of the first cosine term is just ½, then
\[I=2I(k)\cos^{2}(\pi k\Delta d) \label{11}\]
and
\[I(\Delta d)=I(k)[1+\cos(2\pi k\Delta d)]\label{12}\]
wh ere \(I(k)\) is a constant that depends only upon \(k\) and \(I(∆d)\) is the interferogram .
From \(I(∆d)\) we can get \(I(k)\) using Fourier transform as follows:
\[I(\Delta d)-I(\infty)=\int_{0}^{k_{m}}I(k)\cos(2\Pi k\Delta d)dk \label{13}\]
Letting K m → ∞, we can write
\[I(k)=\int_{0}^{\infty}[I(\Delta d)-I(\infty)]\cos(2\Pi k\Delta d)d\Delta d \label{14}\]
The physically measured information recorded at the detector produces an interferogram, which provides information about a response change over time within the mirror scan distance. Therefore, the interferogram obtained at the detector is a time domain spectrum. This procedure involves sampling each position, which can take a long time if the signal is small and the number of frequencies being sampled is large.
In terms of ordinary frequency, \(\nu\), the Fourier transform of this is given by (angular frequency \(\omega= s\pi \nu\)):
\[f(\nu )=\int_{-\infty}^{\infty}f(t)e^{-i2\Pi \nu t}dt \label{15}\]
The inverse Fourier transform is given by:
\[f(\nu )=\int_{-\infty}^{\infty}f(t)e^{+i2\pi \nu t}dt \label{16}\]
The interferogram is transformed into IR absorption spectrum (Figure \(\PageIndex{5}\)) that is commonly recognizable with absorption intensity or % transmittance plotted against the wavelength or wavenumber. The ratio of radiant power transmitted by the sample (I) relative to the radiant power of incident light on the sample (I 0 ) results in quantity of Transmittance, (T). Absorbance (A) is the logarithm to the base 10 of the reciprocal of the transmittance (T):
\[A=log_{10}\dfrac{1}{T}=-log_{10}T=-log_{10}\dfrac{I}{I_{0}} \label{17}\]
Hands-on Operation of an FTIR Spectrometer
Step 1 : The first step is sample preparation . The standard method to prepare solid sample for FTIR spectrometer is to use KBr. About 2 mg of sample and 200 mg KBr are dried and ground. The particle size should be unified and less than two micrometers. Then, the mixture is squeezed to form transparent pellets which can be measured directly. For liquids with high boiling point or viscous solution, it can be added in between two NaCl pellets. Then the sample is fixed in the cell by skews and measured. For volatile liquid sample, it is dissolved in CS 2 or CCl 4 to form 10% solution. Then the solution is injected into a liquid cell for measurement. Gas sample needs to be measured in a gas cell with two KBr windows on each side. The gas cell should first be vacuumed. Then the sample can be introduced to the gas cell for measurement.
Step 2: The second step is getting a background spectrum by collecting an interferogram and its subsequent conversion to frequency data by inverse Fourier transform. We obtain the background spectrum because the solvent in which we place our sample will have traces of dissolved gases as well as solvent molecules that contribute information that are not our sample. The background spectrum will contain information about the species of gases and solvent molecules, which may then be subtracted away from our sample spectrum in order to gain information about just the sample. Figure \(\PageIndex{6}\) shows an example of an FTIR background spectrum.
The background spectrum also takes into account several other factors related to the instrument performance, which includes information about the source, interferometer, detector, and the contribution of ambient water (note the two irregular groups of lines at about 3600 cm –1 and about 1600 cm –1 in Figure \(\PageIndex{6}\)) and carbon dioxide (note the doublet at 2360 cm –1 and sharp spike at 667 cm –1 in Figure \(\PageIndex{6}\)) present in the optical bench.
Step 3: Next, we collect a single-beam spectrum of the sample, which will contain absorption bands from the sample as well as the background (gaseous or solvent).
Step 4: The ratio between the single-beam sample spectrum and the single beam background spectrum gives the spectrum of the sample (Figure \(\PageIndex{7}\)).
Step 5: Data analysis is done by assigning the observed absorption frequency bands in the sample spectrum to appropriate normal modes of vibrations in the molecules.
Portable FTIR Spectrometers
Despite of the powerfulness of traditional FTIR spectrometers, they are not suitable for real-time monitoring or field use. So various portable FTIR spectrometers have been developed. Below are two examples.
Ahonen et al developed a portable, real-time FTIR spectrometer as a gas analyzer for industrial hygiene use. The instrument consists of an operational keyboard, a control panel, signal and control processing electronics, an interferometer, a heatable sample cell and a detector. All the components were packed into a cart. To minimize the size of the instrument, the resolution of FTIR spectrometer was sacraficed. But it is good enough for the use of industrial hygiene. The correlation coefficient of hygienic effect between the analyzer and adsorption tubes is about 1 mg/m 3 .
Korb et al developed a portable FTIR spectrometer which only weighs about 12.5 kg so that it can be held by hand. Moreover, the energy source of the instrument is battery so that the mobility is significantly enhanced. Besides, the instrument can function well within the temperature range of 0 to 45 o C and the humidity range of 0 to 100%. Additionally, this instrument resists vibration. It works well in an operating helicopter. Consequently, this instrument is excellent for the analysis of radiation from the surface and atmosphere of the Earth. The instrument is also very stable. After a three-year operation, it did not lose optical alignment. The reduction of size was implemented by a creative design of optical system and accessory components. Two KBr prisms were used to constitute the interferometer cavity. Optical coatings replaced the mirrors and beam splitter in the interferometer. The optical path is shortened with a much more compact packaging of components. A small, low energy consuming interferometer drive was designed. It is also mass balanced to resist vibration. The common He-Ne tube was replaced by a smaller laser diode.
References
- Science , 2 1, 1983 , 297
- W.D. Perkins, "Fourier Transform-Infrared Spectroscopy”. Part 1. Instrumentation. Topics in Chemical Instrumentation. Ed. Frank A. Settle, Jr. Journal of Chemical Education , 63 :1, January 1986: A5-A10.
- D.A. Skoog and J.J. Leary. “Principles of Instrumental Analysis, 4 th Ed.”, Harcourt Brace Jovanovich. Philadelphia, PA, 1992. Chapter 12.
- F. Daniels, J.W. Williams, P. Bender, R.A. Alberty, C.D. Cornwell, J. E. Harriman. "Experimental Physical Chemistry, 7th Ed.”, McGraw-Hill, New York, NY, 1970.
- J.W. Cooley and J.W. Turkey, Math. Comp. , 1965 , 19, 9
- A.G. Marshall, Acc. Chem. Res. , 1985 , 18, 316
- A. R. Korb, P. Dybwad, W. Wadsworth, J. W. Salisbury, Applied Optics , 1996 , 35, 1679
- I. Ahonen, H. Riipinen, A. Roos, Analyst , 1996 , 121, 1253
- D. W. Ball, Field Guide to Spectroscopy, SPIE Publication, Bellingham, 2006
- V. Saptari, Fourier-Transform Spectroscopy Instrumentation Engineering , SPIE Publication, Bellingham, 2003
- P. R. Griffiths, J. A. de Haseth, Fourier Transform Infrared Spectrometry , Wiley, New York, 1986
- B. C. Smith, Fundamentals of Fourier Transform Infrared Spectroscopy , CRC press, 1996
- B. Stuart, Modern Infrared Spectroscopy , Wiley, New York, 1996
- A. L. Smith, Applied Infrared Spectroscopy : Fundamentals, Techniques, and Analytical Problem-solving , Wiley, New York, 1979
- P. R. Griffiths, Chemical Infrared Fourier Transform Spectroscopy , Wiley, New York, 1975
Contributors and Attributions
- Nancy Birkner (UCD), Qian Wang (UCD)


3.1: Review Concepts from Chapter 2
Vocabulary
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Multiple Choice Questions on Gender, Number and Verb “to be”
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बातचीत “Conversation”
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धन्यवाद !


26.1: Maternal Health
By the end of this section, you should be able to:
- Assess the global and national health status of childbearing clients.
- Examine major risk factors influencing the health of childbearing clients.
- Create evidence-based educational interventions to promote self-care for health promotion, illness prevention, and illness management of childbearing clients.
- Identify Healthy People 2030 goals established for childbearing clients.
- Describe health promotion and disease prevention actions applicable to maternal health.
- Discuss evidence-based strategies for integrating sociocultural and linguistically responsive health promotion and disease prevention interventions in maternal health clinical practice.
Maternal health begins at preconception and continues throughout the pregnancy. Childbearing is a critical period that requires special attention, as it impacts the health of the pregnant person and the child. This section describes the current health status of childbearing clients in the United States and globally, identifies risk factors that influence their health, and describes educational interventions and Healthy People 2030 goals specific to this population. It also reviews health promotion and disease prevention actions applicable to maternal health and health literacy efforts related to this population.
National and Global Health Status of Childbearing Clients
The infant mortality rate is an important measure of the well-being of infants, children, and pregnant clients. This is the number of children dying under 1 year of age divided by the number of live births that year. The infant mortality rate is associated with factors such as maternal health, quality of and access to medical care, socioeconomic conditions, and public health practices.
In the United States, the 2020 infant mortality rate was 5.4 deaths per 1,000 live births (Centers for Disease Control and Prevention [CDC], 2022g). The top five causes of infant death in 2020 were congenital anomaly , preterm birth and low birth weight (LBW), sudden infant death syndrome (SIDS), injuries such as suffocation, and maternal pregnancy complications. Many of these deaths may have been prevented by improving the childbearing client’s health. According to the CDC (2022g), about 1 in every 33 babies has a congenital disorder. Some examples include cleft lip or palate, heart defects, and hearing loss. Managing health conditions and adopting healthy behaviors before pregnancy can increase the chances of a healthy baby. Preterm birth occurs when a baby is born before 37 weeks of pregnancy. In 2021, preterm birth affected about 1 of every 10 infants born in the United States, increasing from 10.1 percent in 2020 to 10.5 percent in 2021. In Black women, the rate of preterm births is about 50 percent higher than that among White or Hispanic women (CDC, 2022q).
The CDC (2023g) also monitors trends in pregnancy-related mortality ratios in the United States through the Pregnancy Mortality Surveillance System. Since 1987, the number of reported pregnancy-related deaths has increased from 7.2 deaths per 100,000 live births to 17.6 deaths per 100,000 live births in 2019 (CDC, 2023g). The most common causes of these deaths from 2017 to 2019 included infections, cardiomyopathy, hemorrhage, and other cardiovascular and non-cardiovascular medical conditions (CDC, 2023g). Non-Hispanic Native Hawaiian or other Pacific Islanders, non-Hispanic Blacks, and non-Hispanic American Indian or Alaska Native (AIAN) clients had two or more times more pregnancy-related deaths than non-Hispanic White, non-Hispanic Asian, or Hispanic clients (CDC, 2023g).
Globally, the infant mortality rate in 2021 was 28 deaths per 1,000 live births (CDC, 2023g). An estimated 287,000 maternal deaths occurred in 2020, with an overall global maternal mortality rate of 223 maternal deaths per 100,000 live births (WHO, 2023c). This equates to about 800 maternal deaths daily and one every two minutes globally. The WHO (2023c) indicates that “almost all of these deaths occurred in low-resource settings, and most could have been prevented” (para. 1). The WHO (2022a) emphasized that maternal mortality is a health indicator that “shows very wide gaps between rich and poor and between countries” (para. 1). This has led to the World Health Organization et al. (2023) developing a Sustainable Development Goal , Target 3.1: Reduce the global maternal mortality ratio to less than 70 per 100,000 live births with a target date of 2030.
The U.S. Department of Health and Human Services Office of Disease Prevention and Health Promotion (ODPHP, n.d.-g) indicated that in 2016, women in the United States were more likely to die from childbirth than those in other developed countries. Subsequently, a study by Tikkanen et al. (2020) found that the United States has the highest maternal mortality rate among developed countries despite the fact that most maternal deaths are preventable. The study also found that about one-third of the pregnancy-related deaths happen after the client has given birth, which is considered the period up to a year post-birth.
(See Tikkanen et al., 2020.)
Risk Factors Influencing the Health of Childbearing Clients
Existing health conditions, age, and the client’s lifestyle can make a pregnancy high risk. Some risk factors related to health issues can occur before and during pregnancy, which the Eunice Kennedy Shriver National Institute of Child Health and Human Development (2018) refers to as “conditions of pregnancy” (para. 7).
Existing Health Conditions
Table 26.1 presents some of the most common existing health conditions that can lead to a high-risk pregnancy. Because each pregnancy is individual, an existing condition may not lead to a risky pregnancy. It is necessary for the nurse, however, to be aware of risk factors.
| Existing Condition | Management | Complications |
|---|---|---|
| Hypertension |
|
|
| Polycystic ovary syndrome (PCOS) |
|
|
| Diabetes |
|
|
| Kidney disease |
|
|
| Autoimmune disease, such as lupus, multiple sclerosis (MS), and myasthenia gravis (MG) |
|
|
| Thyroid disease (hypothyroidism or hyperthyroidism) |
|
|
| Obesity (adult body mass index [BMI] of 30 or higher) |
|
|
| Human immunodeficiency virus (HIV), acquired immunodeficiency syndrome (AIDS) |
|
|
| Sexually transmitted infections (STIs) |
|
|
| Zika infection |
|
|
Age
The U.S. birth rate per 1,000 females ages 15 to 19 decreased steadily from its peak of 61.8 births in 1991 to 15.4 in 2020 (Osterman et al., 2022). The birth rate for women ages 40 to 44 was 11.8, for women ages 45 to 49 was 0.9, and for women ages 50 and over was 1.0 (Osterman et al., 2022). Age at pregnancy can correlate with pregnancy risks and childbirth and fetal issues.
Pregnant adolescents are more likely to develop pregnancy-related high blood pressure and anemia and to have preterm labor and delivery. Adolescents may have unknown STIs, and they are less likely to seek prenatal care , which can lead to other risk factors. For example, a pregnant adolescent may not know to take certain medications or may not realize the importance of good nutrition (Chacko, 2023).
First-time pregnancies after age 35 are considered “older first-time mothers” and consist of nine percent of births in the United States (Fretts, 2022). These childbearing clients are at higher risk for spontaneous abortion or pregnancy loss, ectopic pregnancy, chromosomal abnormalities (particularly cardiac anomalies), placental problems, LBW newborns, and preterm delivery. Fretts (2022) also indicated that childbearing clients “40 years or older are at a sixfold increased risk of maternal death when compared with women less than 20 years of age” (para. 48); Black childbearing clients aged 40 or older are over three times more likely to die during pregnancy than White childbearing clients in this age group.
Lifestyle Factors
Drinking alcohol during pregnancy can increase the baby’s risk for problems such as fetal alcohol spectrum disorders (FASDs) and SIDS. FASDs are completely preventable; if a client does not drink alcohol during pregnancy, their child will not have an FASD (CDC, 2022s). Alcohol use can also lead to miscarriage or stillbirth. There is no safe amount of alcohol to drink during pregnancy (CDC, 2022s). Different FASD diagnoses are based on particular symptoms, as shown in Table 26.2.
| Diagnosis | Symptoms |
|---|---|
| Fetal alcohol syndrome (FAS) | Central nervous system (CNS) problems, minor facial features, growth problems; problems with learning, memory, attention span, communication, vision, and/or hearing. May have difficulties in school and trouble getting along with others. |
| Alcohol-related neurodevelopmental disorder (ARND) | Intellectual disabilities; problems with behavior and learning. May have difficulties in school, particularly with math, memory, attention, judgment, and poor impulse control. |
| Alcohol-related birth defects (ARBD) | May have problems with the heart, kidneys, bones, and/or hearing. |
| Neurobehavioral disorder associated with prenatal alcohol exposure (ND-PAE) |
The delivering client must have consumed more than 13 alcoholic drinks per month of pregnancy (that is, any 30-day period of pregnancy) or more than two alcoholic drinks in one setting. For diagnosis with ND-PAE, the child must have problems in:
|
Smoking can have adverse health effects before, during, and after pregnancy. The CDC (2020a) indicates that childbearing clients who smoke tobacco have more difficulty becoming pregnant. Smoking tobacco during pregnancy can lead to preterm birth, LBW, miscarriage, congenital disorders, and SIDS. Pregnant clients exposed to secondhand smoke are more likely to have LBW babies. Babies exposed to secondhand tobacco smoke are more likely to die from SIDS and have weaker lungs, which can lead to additional health problems (CDC, 2020a). Childbearing clients who smoke marijuana may double their risk of stillbirth; using marijuana during pregnancy can interfere with fetal brain development (Chang, 2023).
Childbearing clients who use other drugs during pregnancy can experience problems with fetal and infant health. Use of opioids such as heroin, diverted or misused prescription opioids, or other morphine-like drugs is associated with substantial maternal, fetal, and neonatal risks (Seligman et al., 2023). Cocaine use by the pregnant client also leads to preterm birth and LBW. Methamphetamine exposure during pregnancy has been associated with maternal and neonatal morbidity and mortality (Chang, 2023). Self-reported data from 2019 indicated that 6.6 percent of pregnant clients reported prescription opioid use during pregnancy, and 21.2 percent of these reported drug misuse (Ko et al., 2020). The drug overdose mortality rate per 100,000 for pregnant or postpartum persons almost doubled from 2017 (6.56) to 2020 (11.85) in the United States (Bruzelius & Martins, 2022).
Neonatal abstinence syndrome (NAS) is a treatable condition in newborns exposed to certain drugs, particularly opioids, while in utero. The term neonatal opioid withdrawal syndrome (NOWS) specifically describes the symptoms experienced by infants exposed to opioids. Both NAS and NOWS include intense irritability, difficulty with feeding, respiratory issues, and seizures. To mitigate or prevent negative outcomes associated with NAS and NOWS, nurses and health care providers must engage the pregnant client with opioid and substance use disorders in substance use treatment and other supportive services as part of prenatal care (National Center on Substance Abuse and Child Welfare, n.d.).
Challenges and Complications of Childbearing
Table 26.3 lists other challenges and risks that can arise during pregnancy, childbirth, and the postpartum period for the childbearing client. Understanding the challenges and complications of childbearing is crucial for health care professionals. The goal is to support informed decision-making, promote proactive prenatal care, and ultimately contribute to the well-being of childbearing clients and their babies. Due to the risks and concerns with a rising rate of cesarean section deliveries, the WHO (2018) and ACOG et al. (2014) have suggested interventions to reduce the rate of unnecessary cesarean deliveries.
| Challenge | Risk Factors | Complications |
|---|---|---|
| Multiple gestation |
|
|
| Preeclampsia |
|
|
| Gestational diabetes |
|
|
| Preterm birth |
|
|
| Delivery via cesarean section |
|
|
| Maternal infections |
|
|
| Postpartum hemorrhage |
|
|
| Postpartum depression |
|
|
In some cases, fetal anomalies or genetic conditions may be detected and treated in utero or immediately following birth. For example, spina bifida may be repaired before birth (Bowman, 2022). Certain heart problems common among infants with Down syndrome will require immediate surgical correction following birth (Altman, 2022).
Educational Interventions to Promote Self-Care in Childbearing Clients
Self-care for the childbearing client begins preconception and involves many of the same health promotion activities clients should engage in throughout their lifespan, including during adolescence. Education regarding risk assessment and screening, health promotion and counseling, and any appropriate topic based on identified risk factors will promote optimal health promotion and self-care during pregnancy. Once a pregnancy is identified, education moves toward anticipatory guidance, or what the childbearing client may expect to occur during pregnancy. This may include the normal discomforts during pregnancy, body changes, weight gain expectations, signs and symptoms of problems and when/how to report them, available community resources, nutritional needs, newborn care, and other topics (Lockwood & Magriples, 2023).
Healthy People 2030 Goals for Childbearing Clients
The overarching goal of Healthy People 2030 related to pregnancy and childbirth is to “prevent pregnancy complications and maternal deaths and improve women’s health before, during, and after pregnancy” (ODPHP, n.d.-g, para. 1). Healthy People 2023 includes several pregnancy and childbirth objectives related to this goal.
Healthy People 2030 Pregnancy and Childbirth Objectives focus on the broad goals of preventing pregnancy complications and maternal deaths while also helping women stay healthy before, during, and after pregnancy. They specifically address prenatal care; drug, alcohol, and tobacco abstinence during pregnancy; family planning; reduction of sexually transmitted infections; and vaccination of pregnant clients.
Health Promotion and Disease Prevention Activities to Improve the Health of Childbearing Clients
Health promotion activities can help prevent complications during pregnancy and childbirth, promote healthy behaviors throughout the lifespan, and improve the client’s and child’s overall health outcomes. The goal is to empower this population to take an active role in their health and well-being while promoting positive health outcomes for the pregnant person and child. A range of health promotion activities is available, including:
- Promoting healthy lifestyles (e.g., healthy diet, exercise, abstaining from alcohol and tobacco)
- Encouraging regular prenatal care
- Providing education on childbirth and parenting
- Supporting breastfeeding
- Providing education to decrease postpartum depression
Health promotion activities can also address social determinants of health (SDOH), such as access to health care, housing, and nutrition, ensuring these clients have the resources necessary to maintain good health. Collaboration among health care providers, community organizations, and other partners will provide childbearing clients with the support and resources they need for a healthy and successful pregnancy and birth.
Childbearing clients can take prevention efforts before and during pregnancy. The Client Teaching feature below lists some of the most important preventive measures for childbearing clients.
The community health nurse may teach childbearing clients the following:
- Get regular medical checkups.
- Eat healthy foods and maintain a healthy weight.
- Make sure any medical conditions are under control.
- Take a vitamin with folic acid every day.
- Do not smoke, drink alcohol, or use drugs.
- Talk with your doctor about any medications you are taking or thinking about taking, including prescription and over-the-counter medications and dietary and herbal products.
- Avoid exposure to toxic secondhand smoke, chemicals, and fumes.
-
Get tested for infectious diseases and get these recommended
vaccination
s (Lockwood & Magriples, 2023):
- COVID-19
- Influenza , if pregnant during flu season, regardless of stage of pregnancy
- Tetanus and diphtheria ( Tdap ) immunizations and boosters should be up-to-date; receive Tdap in the third trimester of each pregnancy to protect the infant from pertussis regardless of prior maternal vaccination (Yawetz, 2023)
- Wash your hands often, especially after contact with diapers or secretions associated with a child who is in daycare.
- Avoid kissing children under age 6 on the mouth or cheek and sharing food or drinks with young children; clean all surfaces and toys that may come into contact with children’s secretions.
- Avoid contact with those who have fevers that could be infectious.
- Avoid changing or cleaning cat litter boxes.
- Avoid x-rays and other radiation.
- Use condoms during sex to prevent exposure to STDs.
- Avoid ingesting contaminated, undercooked, or cured meat or meat products; soil-contaminated fruits or vegetables; and contaminated unfiltered water.
- Wash utensils and surfaces that are used to prepare raw fish with hot, soapy water.
- If not immune to varicella infection, Varizig (VZIG) is recommended during pregnancy to prevent maternal complications.
- Avoid Zika virus by postponing travel to areas with ongoing mosquito transmission of the virus.
(See Office of the Surgeon General, 2020.)
Even if a childbearing client takes prevention measures, there is still a risk of having a child with a congenital disorder . Following healthy habits early and consistently throughout the pregnancy gives the best chance of having a healthy baby (New York State Department of Health, 2017).
Primary Prevention
Primary prevention for childbearing clients refers to activities aimed at preventing health problems before they occur. Some examples of primary prevention strategies to improve the health of childbearing clients include:
- Healthy lifestyle promotion
- Immunizations
- Preconception care
- Education on safe sex and family planning
- Screening for infectious diseases
Secondary Prevention
Secondary prevention for childbearing clients refers to activities aimed at detecting and treating health problems at an early stage, before they are even problematic, to prevent subclinical disease progression. Some examples of secondary prevention strategies to improve the health of childbearing clients include:
- Prenatal care
- Screening for genetic disorders
- Screening for gestational diabetes
- Early detection and treatment of infections
- Screening for postpartum depression
Tertiary Prevention
Tertiary prevention for childbearing clients refers to activities aimed at managing and treating health problems that have already occurred and preventing their recurrence or further complications. Some examples of tertiary prevention strategies to improve the health of childbearing clients include:
- Management of chronic health conditions, such as diabetes, hypertension, and asthma
- Treatment of postpartum complications, such as postpartum hemorrhage, infection, and thromboembolism
- Support for breastfeeding
- Mental health treatment
- Family planning to prevent unintended pregnancy in the postpartum period
Table 26.4 summarizes these three levels of prevention and incudes primordial and quaternary prevention as well.
| Levels of Prevention | Specific Example of Level of Prevention |
|---|---|
| Primordial | A state’s governmental policy increases taxes on cigarettes to discourage smoking. |
| Primary | Local advertisements are provided via different forms of media depicting the dangers of smoking to childbearing clients and their babies. |
| Secondary | A local women’s health center offers free tobacco cessation programs to childbearing clients. |
| Tertiary | The nurse provides care to a LBW newborn whose mother smoked during pregnancy and educates the mother to prevent further complications. |
| Quaternary | Over-screening for gestational diabetes during the first half of pregnancy is prevented by targeting early pregnancy screening to those at increased risk of undiagnosed type 2 diabetes. |
Integration of Sociocultural and Linguistically Responsive Interventions in Maternal Health
Providing high-quality maternal health care requires the nurse to integrate sociocultural and linguistically responsive interventions. Pregnancy and childbirth are influenced by a complex set of social, cultural, and linguistic factors that can affect health outcomes for the client and the baby. Nurses must consider their clients’ sociocultural and linguistic backgrounds and provide care sensitive to their needs and preferences. This section explores the reasons for integrating sociocultural and linguistically responsive interventions in maternal health care and provides examples of interventions that can improve health outcomes for childbearing clients.
Sociocultural Interventions
Addressing the sociocultural preferences of maternal clients is necessary to provide client-centered care . The WHO (2017) Standards for Improving Quality of Maternal and Newborn Care in Health Facilities recommend that health care professionals participate in regular education to improve their interpersonal communication, counseling skills, and cultural competence to encourage respectful maternity care. The Joint Commission, the National Institutes of Health, the International Confederation of Midwives, and the International Federation of Gynecology and Obstetrics have issued similar initiatives (Hodin, 2018).
Interventions to provide culturally appropriate maternal health services, specifically related to clients’ ethnicity , language, and religion , have positively affected pregnant people's continual use of maternity care, particularly prenatal care (Jones et al., 2017). In a review of 15 studies to examine how such interventions affected the use of pregnancy care, 10 studies found this to be true, and four themes evolved to describe the barriers and facilitators to ensuring socioculturally appropriate maternity care. These included (Jones et al., 2017, pp. 5–7):
- Access—Interventions should consider broader economic, geographic, and social factors that affect ethnic minority groups’ access to service.
- Community participation—Interventions should include community participation to understand problems with existing services and potential solutions from a community perspective as well as in developing and implementing interventions.
- Person-centered care —Respectful, person-centered care should be at the core of all interventions.
- Continuum of care —Interventions should be spread all along the continuum of care. For example, rather than focusing interventions on prenatal care, effective partnerships should extend interventions between the socioculturally appropriate service and all other health care professionals clients and their families encounter along the continuum of care from detection of pregnancy until after birth.
Several sociocultural interventions can be integrated into maternal health to positively impact maternal health outcomes. Providing cultural competency training for all health care workers will help them better understand the cultural beliefs, practices, and values of the populations they serve and promote the provision of care that is sensitive to that population’s needs and preferences. Community-based interventions, such as peer groups and community health workers, can help connect maternal clients with culturally appropriate resources and support. Addressing stigma and discrimination related to pregnancy and childbirth, such as discrimination against teen or single parents or those with HIV , can help promote equitable and respectful care for all clients. By providing each of these interventions, the nurse can work to improve the health outcomes for childbearing clients and their children.
Aligning Early Childhood and Medicaid is a national initiative funded by the Robert Wood Johnson Foundation that works to better align Medicaid and state agencies responsible for early childhood programs. Its goal is to improve the health and social outcomes of low-income infants, young children, and families. The initiative included eight states: Colorado, Minnesota, New Jersey, New York, Oregon, Rhode Island, Vermont, and Washington. A specific intervention taken within this initiative involved addressing racial and ethnic disparities in maternal and child health through home visiting programs.
Watch the video, and then respond to the following questions.
- Describe three sociocultural nursing interventions described in the video that could be used to positively impact the health of maternal clients in any sociocultural population.
- Now, connect these three nursing interventions to one of the five social determinants of health (SDOH).
SDOH).
Linguistically Responsive Interventions
A variety of interventions are available to provide linguistically responsive care, some of which are included in Culturally and Linguistically Responsive Nursing Care. Nurses can use the services of professional medical interpreters to ensure that the client can communicate with others effectively. Communication is essential when educating the client about prenatal care, labor and delivery , and postpartum care . Nurses should provide educational materials written in the client’s preferred language that consider the client’s sociocultural beliefs and practices. They can provide visual aids to explain more complex concepts to clients with limited English proficiency and use plain language to ensure clients understand the information being provided. Nurses should encourage clients to ask questions and clarify doubts regarding the pregnancy or other issues.
Nurses should also assess their clients’ health literacy levels and communicate accordingly to allow clients to best understand their health conditions and improve their health outcomes. Health literacy skills such as “reading, listening, analyzing, decision-making, and using these skills in health situations” (Solhi et al., 2019, p. 3), regardless of the client’s educational level or general reading ability, will have a definite impact on the health outcomes of a client and their child. The maternal client must be involved in health promotion and preventive care, but if they do not understand health concepts, they will have difficulty making informed decisions (Solhi et al., 2019). Even for those with health literacy skills, the first pregnancy may be full of surprises if their education is not specifically in this realm. Also, the maternal client’s understanding of health information directly impacts their child’s health. Solhi et al. (2019) determined that adopting interventions to increase health literacy in maternal clients, such as those previously described, promoted physical and mental self-care during pregnancy, particularly over time.


7.3: Hyperbolic Geometry with Curvature k < 0
We may do the same gentle scaling of the Poincaré model of hyperbolic geometry as we did in the previous section to the disk model of elliptic geometry. In particular, for each negative number \(k \lt 0\) we construct a model for hyperbolic geometry with curvature \(k\text{.}\)
We define the space \(\mathbb{D}_k\) to be the open disk of radius \(\dfrac{1}{\sqrt{|k|}}\) centered at the origin in \(\mathbb{C}\text{.}\) That is, \(\mathbb{D}_k\) consists of all \(z\) in \(\mathbb{C}\) such that \(|z| \lt \dfrac{1}{\sqrt{|k|}}\text{.}\) In this setting, the circle at infinity is the boundary circle \(|z| = \dfrac{1}{\sqrt{|k|}}\text{.}\)
The group \({\cal H}_k\) consists of all Möbius transformations that send \(\mathbb{D}_k\) to itself. The geometry \((\mathbb{D}_k,{\cal H}_k)\) with \(k \lt 0\) is called hyperbolic geometry with curvature \(k\text{.}\) Pushing analogy with the elliptic case, we may define the group of transformations to consist of all Möbius transformations with this property: if \(z\) and \(z^*\) are symmetric with respect to the circle at infinity then \(T(z)\) and \(T(z^*)\) are also symmetric with respect to the circle at infinity. Noting that the point symmetric to \(z\) with respect to this circle is \(\displaystyle z^* = \dfrac{1}{|k|\overline{z}} = -\dfrac{1}{k\overline{z}}\text{,}\) we draw the satisfying conclusion that \(T \in {\cal H}_k\) if and only if the following holds:
\[ \text{if}~z^* = -\dfrac{1}{k\overline{z}}~~ \text{then}~~ T(z^*) =-\dfrac{1}{k\overline{T(z)}}\text{.} \]
Thus, the group \({\cal H}_k\) in the hyperbolic case has been defined precisely as the group \({\cal S}_k\) in the elliptic case. Furthermore, one can show that transformations in \({\cal H}_k\) have the form
\[ T(z) = e^{i\theta}\dfrac{z-z_0}{1+k\overline{z_0}z}\text{,} \]
where \(z_0\) is a point in \(\mathbb{D}_k\text{.}\)
Straight lines in this geometry are the clines in \(\mathbb{C}^+\) orthogonal to the circle at infinity. By Theorem \(3.2.3\), a straight line in \((\mathbb{D}_k,{\cal H}_k)\) is precisely a cline with the property that if it goes through \(z\) then it goes through its symmetric point \(\dfrac{-1}{k\overline{z}}\text{.}\)
The arc-length and area formulas also get tweaked by the scale factor, and now look identical to the formulas for elliptic geometry with curvature \(k\text{.}\)
The arc-length of a smooth curve \(\boldsymbol{r}\) in \(\mathbb{D}_k\) is
\[ {\cal L}(\boldsymbol{r}) = \int_a^b \dfrac{2|\boldsymbol{r}^\prime(t)|}{1 + k|\boldsymbol{r}(t)|^2}~dt\text{.} \]
The area of a region \(R\) given in polar coordinates is computed by the formula
\[ A(R) = \iint_R \dfrac{4r}{(1+kr^2)^2}dr d\theta\text{.} \]
As in Chapter 5 when \(k\) was fixed at \(-1\), the area formula is a bear to use, and one may convert to an upper half-plane model to determine the area of a \(\dfrac{2}{3}\)-ideal triangle in \(\mathbb{D}_k\text{.}\) The ambitious reader might follow the methods of Section 5.5 to show that the area of a \(\dfrac{2}{3}\)-ideal triangle in \((\mathbb{D}_k,{\cal H}_k)\) (\(k \lt 0\)) with interior angle \(\alpha\) is \(-\dfrac{1}{k}(\pi - \alpha)\text{.}\)
With this formula in hand, we can derive the area of any triangle in \(\mathbb{D}_k\) in terms of its angles, exactly as we did in Chapter 5.
In hyperbolic geometry with curvature \(k\text{,}\) the area of a triangle with angles \(\alpha, \beta\text{,}\) and \(\gamma\) is
\[ A =\dfrac{1}{k}(\alpha+\beta+\gamma -\pi)\text{.} \]
Observing Negative Curvature
Suppose we are located at a point \(z\) in a hyperbolic universe with curvature \(k\text{.}\) We see in the distance a hyperbolic line \(L\) that seems to extend indefinitely. We might intuitively see the point \(w\) on the line that is closest to us, as suggested in Figure \(7.3.1\). Now suppose we look down the road a bit to a point \(v\text{.}\) If \(v\) is close to \(w\) the angle \(\angle wzv\) will be close to \(0\). As \(v\) gets further and further away from \(w\text{,}\) the angle will grow, getting closer and closer to the angle \(\theta = \angle wzu\text{,}\) where \(u\) is an ideal point of the line \(L\text{.}\)
A curious fact about hyperbolic geometry is that this angle \(\theta\text{,}\) which is called the angle of parallelism of \(\boldsymbol{z}\) to \(\boldsymbol{L}\), is a function of \(z\)'s distance \(d\) to \(L\text{.}\) In Section 5.4 we saw that \(\cosh(d) = \dfrac{1}{\sin(\theta)}\) in \((\mathbb{D},{\cal H})\text{.}\) In particular, one may deduce the distance \(d\) to the line \(L\) by computing \(\theta\text{.}\) No such analogy exists in Euclidean geometry. In a Euclidean world, if one looks farther and farther down the line \(L\text{,}\) the angle will approach \(90^{\circ}\text{,}\) no matter one's distance \(d\) from the line. The following theorem provides another formula relating the angle of parallelism to a point's distance to a line.
In hyperbolic geometry with curvature \(k\text{,}\) the hyperbolic distance \(d\) of a point \(z\) to a hyperbolic line \(L\) is related to the angle of parallelism \(\theta\) by the formula
\[ \tan(\dfrac{\theta}{2}) = e^{-\sqrt{|k|}d}\text{.} \]
- Proof
-
For this proof, let \(s = \dfrac{1}{\sqrt{|k|}}\text{.}\) Note that \(s\) is the Euclidean radius of the circle at infinity in the disk model for hyperbolic geometry with curvature \(k\text{.}\) Since angles and lines and distances are preserved, assume \(z\) is the origin and \(L\) is orthogonal to the positive real axis, intersecting it at the point \(x\) (with \(0 \lt x \lt s\)).
Recall the half-angle formula
\[ \tan(\dfrac{\theta}{2}) =\dfrac{\tan(\theta)}{\sec(\theta)+1}\text{.} \]
According to Figure \(7.3.2\), \(\tan(\theta) = \dfrac{r}{s} \text{,}\) where \(r\) is the Euclidean radius of the circle containing the hyperbolic line \(L\text{.}\) Furthermore, \(\sec(\theta) = \dfrac{(x+r)}{s}\text{,}\) so
\[ \tan(\dfrac{\theta}{2}) = \dfrac{r}{x+r+s}\text{.} \label{7.3.8} \]
We may express \(x\) and \(r\) in terms of the hyperbolic distance \(d\) from \(0\) to \(x\text{.}\) In Exercise \(7.3.2\) we prove the hyperbolic distance from \(0\) to \(x\) in \((\mathbb{D}_k,{\cal H}_k)\) is
\[ d = s \ln\left(\dfrac{s + x}{s-x}\right) \]
so that
\[ x = s \cdot \dfrac{e^{d/s} - 1}{e^{d/s}+1}\text{.} \]
Express \(r\) in terms of \(d\) by first expressing it in terms of \(x\text{.}\) Note that segment \(x x^*\) is a diameter of the circle containing \(L\text{,}\) where \(x^* = \dfrac{-1}{kx}\) is the point symmetric to \(x\) with respect to the circle at infinity. Thus, \(r\) is half the distance from \(x\) to \(x^*\text{:}\)
\[ r = -\dfrac{1 + kx^2}{2kx}\text{.} \]
Replacing \(k\) with \(-\dfrac{1}{s^2}\text{,}\) we have
\[ r=\dfrac{s^2-x^2}{2x}\text{.} \]
One checks that after writing \(x\) in terms of \(d\text{,}\) \(r\) is given by
\[ r = \dfrac{2se^{d/s}}{e^{2d/s}-1}\text{.} \]
Substitute this expression for \(r\) into the equation labeled (\(\ref{7.3.8}\)) in this proof, and after a dose of satisfactory simplifying one obtains the desired result:
\[ \tan(\dfrac{\theta}{2}) = e^{-d/s}\text{.} \]
Since \(s = \dfrac{1}{\sqrt{|k|}}\) this completes the proof.
Parallax
If a star is relatively close to the Earth, then as the Earth moves in its annual orbit around the Sun, the star will appear to move relative to the backdrop of the more distant stars. In the idealized picture that follows, \(e_1\) and \(e_2\) denote the Earth's position at opposite points of its orbit, and the star \(s\) is orthogonal to the plane of the Earth's orbit. The angle \(p\) is called the parallax , and in a Euclidean universe, \(p\) determines the star's distance from the Sun, \(D\text{,}\) by the equation \(D = \dfrac{d}{\tan(p)}\text{,}\) where \(d\) is the Earth's distance from the Sun.
We may determine \(p\) by observation, and \(d\) is the radius of Earth's orbit around the Sun (\(d\) is about \(8.3\) light-minutes.) In practice, \(p\) is quite small, so a working formula is \(D = \dfrac{d}{p}\text{.}\) The first accurate measurement of parallax was recorded in \(1837\) by Friedrich Bessel (\(1784\)-\(1846\)). He found the stellar parallax of \(0.3\) arc seconds (\(1\) arc second \(= \dfrac{1}{3600}^{\circ}\)) for star \(61\) Cygni, which put the star at about \(10.5\) light-years away.
If we live in a hyperbolic universe with curvature \(k\text{,}\) a detected parallax puts a bound on how curved the universe can be. Consider Figure \(7.3.3\). As before, \(e_1\) and \(e_2\) represent the position of the Earth at opposite points of its orbit, so that the distance between them is \(2d\text{,}\) or about \(16.6\) light-minutes. Assume star \(s\) is on the positive real axis and we have detected a parallax \(p\text{,}\) so that angle \(\angle e_2se_1 = 2p\text{.}\)
The angle \(\alpha = \angle e_1e_2s\) in Figure \(7.3.3\) is less than the angle of parallelism \(\theta = \angle e_1e_2u\text{.}\) Noting that \(\tan(x)\) is an increasing function and applying Lobatchevsky's formula it follows that
\[ \tan(\dfrac{\alpha}{2}) \lt \tan(\dfrac{\theta}{2}) = e^{-\sqrt{|k|}2d}\text{.} \]
We may solve this inequality for \(|k|\text{:}\)
\begin{align*} \tan(\dfrac{\alpha}{2}) & \lt e^{-\sqrt{|k|}2d} & \\ \ln(\tan(\dfrac{\alpha}{2})) & \lt -\sqrt{|k|}2d & \ln(x) ~\text{is increasing}\\ \bigg[ \dfrac{\ln(\tan(\dfrac{\alpha}{2}))}{2d}\bigg]^2 & > |k|.& x^2~ \text{is decreasing for}~x\lt 0 \end{align*}
To get a bound for \(|k|\) in terms of \(p\text{,}\) note that \(\alpha \approx \dfrac{\pi}{2} - 2p\) (the triangles used in stellar parallax have no detectable angular deviation from \(180^{\circ}\)), so
\[ |k| \lt \left[\dfrac{\ln(\tan(\dfrac{\pi}{4}-p))}{2d}\right]^2\text{.} \]
We remark that for values of \(p\) near \(0\), the expression \(\ln(\tan(\dfrac{\pi}{4}-p))\) has linear approximation equal to \(-2p\text{,}\) so a working bound for \(k\text{,}\) which appeared in Schwarzschild's \(1900\) paper [27] , is \(|k| \lt (\dfrac{p}{d})^2\text{.}\)
Exercises
Prove that for \(k \lt 0\text{,}\) any transformation in \({\cal H}_k\) has the form
\[ T(z) = e^{i\theta}\dfrac{z-z_0}{1+k\overline{z_0}z}\text{,} \]
where \(\theta\) is any real number and \(z_0\) is a point in \(\mathbb{D}_k\text{.}\)
- Hint
-
Follow the derivation of the transformations in \({\cal H}\) found in Chapter 5.
Assume \(k \lt 0\) and let \(s = \dfrac{1}{\sqrt{|k|}}\text{.}\) Derive the following measurement formulas in \((\mathbb{D}_k,{\cal H}_k)\text{.}\)
-
The length of a line segment from \(0\) to \(x\text{,}\) where \(0 \lt x \lt s\) is
\[ d_{k}(0,x) = s\ln\left(\dfrac{s+x}{s-x}\right)\text{.} \]
Hint: Evaluate the integral by partial fractions. - The circumference of the circle centered at the origin with hyperbolic radius \(r\) is \(c=2\pi s\sinh(\dfrac{r}{s})\text{.}\)
- The area of the circle centered at the origin with hyperbolic radius \(r\) is \(A = 4\pi s^2 \sinh^2(\dfrac{r}{2s})\text{.}\)
Let's investigate the idea that the hyperbolic formulas in Exercise \(7.3.2\) for distance, circumference, and area approach Euclidean formulas when \(k \to 0^-\text{.}\)
- Show that the hyperbolic distance \(d_{k}(0,x)\) from \(0\) to \(x\text{,}\) where \(0 \lt x \lt s,\) approaches \(2x\) as \(k \to 0^-\text{.}\)
- Show that the hyperbolic circumference of a circle with hyperbolic radius \(r\) approaches \(2\pi r\) as \(k \to 0^-\text{.}\)
- Show that the hyperbolic area of this circle approaches \(\pi r^2\) as \(k \to 0^-\text{.}\)
Triangle trigonometry in \((\mathbb{D}_k,{\cal H}_k)\text{.}\)
Suppose we have a triangle in \((\mathbb{D}_k,{\cal H}_k)\) with side lengths \(a,b,c\) and angles \(\alpha, \beta,\) and \(\gamma\) as pictured in Figure \(5.4.4\). Suppose further that \(s = \dfrac{1}{\sqrt{|k|}}\text{.}\)
-
Prove the hyperbolic law of cosines in \((\mathbb{D}_k,{\cal H}_k)\text{:}\)
\[ \cosh(\dfrac{c}{s})=\cosh(\dfrac{a}{s})\cosh(\dfrac{b}{s})-\sinh(\dfrac{a}{s})\sinh(\dfrac{b}{s})\cos(\gamma)\text{.} \]
-
Prove the hyperbolic law of sines in \((\mathbb{D}_k,{\cal H}_k)\text{:}\)
\[ \dfrac{\sinh(\dfrac{a}{s})}{\sin(\alpha)}=\dfrac{\sinh(\dfrac{b}{s})}{\sin(\beta)}=\dfrac{\sinh(\dfrac{c}{s})}{\sin(\gamma)}\text{.} \]
As \(k \lt 0\) approaches \(0\), the formulas of hyperbolic geometry \((\mathbb{D}_k,{\cal H}_k)\) approach those of Euclidean geometry. What happens to Lobatchevsky's formula as \(k\) approaches \(0\)? What must be the angle of parallelism \(\theta\) in the limiting case? Is this value of \(\theta\) independent of \(d\text{?}\)
Bessel determined a parallax of \(p = .3\) arcseconds for the star \(61\) Cygni. Convert this angle to radians and use it to estimate a bound for the curvature constant \(k\) if the universe is hyperbolic. The units for this bound should be light years\(^{-2}\) (convert the units for the Earth-Sun distance to light-years).
The smallest detectable parallax is determined by the resolving power of our best telescopes. Search the web to find the smallest detected parallax to date, and use it to estimate a bound on \(k\) if the universe is hyperbolic.


12.3: The Function of Nervous Tissue
Learning Objectives
- Distinguish the major functions of the nervous system: sensation, integration, and response
- List the sequence of events in a simple sensory receptor–motor response pathway
Having looked at the components of nervous tissue, and the basic anatomy of the nervous system, next comes an understanding of how nervous tissue is capable of communicating within the nervous system. Before getting to the nuts and bolts of how this works, an illustration of how the components come together will be helpful. An example is summarized in Figure \(\PageIndex{1}\).
Fi gure 12.3.1: Testing the Water. (1) The sensory neuron has endings in the skin that sense a stimulus such as water temperature. The strength of the signal that starts here is dependent on the strength of the stimulus. (2) The graded potential from the sensory endings, if strong enough, will initiate an action potential at the initial segment of the axon (which is immediately adjacent to the sensory endings in the skin). (3) The axon of the peripheral sensory neuron enters the spinal cord and contacts another neuron in the gray matter. The contact is a synapse where another graded potential is caused by the release of a chemical signal from the axon terminals. (4) An action potential is initiated at the initial segment of this neuron and travels up the sensory pathway to a region of the brain called the thalamus. Another synapse passes the information along to the next neuron. (5) The sensory pathway ends when the signal reaches the cerebral cortex. (6) After integration with neurons in other parts of the cerebral cortex, a motor command is sent from the precentral gyrus of the frontal cortex. (7) The upper motor neuron sends an action potential down to the spinal cord. The target of the upper motor neuron is the dendrites of the lower motor neuron in the gray matter of the spinal cord. (8) The axon of the lower motor neuron emerges from the spinal cord in a nerve and connects to a muscle through a neuromuscular junction to cause contraction of the target muscle.
Imagine you are about to take a shower in the morning before going to school. You have turned on the faucet to start the water as you prepare to get in the shower. After a few minutes, you expect the water to be a temperature that will be comfortable to enter. So you put your hand out into the spray of water. What happens next depends on how your nervous system interacts with the stimulus of the water temperature and what you do in response to that stimulus.
Found in the skin of your fingers or toes is a type of sensory receptor that is sensitive to temperature, called a thermoreceptor . When you place your hand under the shower (Figure \(\PageIndex{2}\)), the cell membrane of the thermoreceptors changes its electrical state (voltage). The amount of change is dependent on the strength of the stimulus (how hot the water is). This is called a graded potential . If the stimulus is strong, the voltage of the cell membrane will change enough to generate an electrical signal that will travel down the axon. You have learned about this type of signaling before, with respect to the interaction of nerves and muscles at the neuromuscular junction. The voltage at which such a signal is generated is called the threshold , and the resulting electrical signal is called an action potential . In this example, the action potential travels—a process known as propagation —along the axon from the axon hillock to the axon terminals and into the synaptic end bulbs. When this signal reaches the end bulbs, it causes the release of a signaling molecule called a neurotransmitter .
The neurotransmitter diffuses across the short distance of the synapse and binds to a receptor protein of the target neuron. When the molecular signal binds to the receptor, the cell membrane of the target neuron changes its electrical state and a new graded potential begins. If that graded potential is strong enough to reach threshold, the second neuron generates an action potential at its axon hillock. The target of this neuron is another neuron in the thalamus of the brain, the part of the CNS that acts as a relay for sensory information. At another synapse, neurotransmitter is released and binds to its receptor. The thalamus then sends the sensory information to the cerebral cortex , the outermost layer of gray matter in the brain, where conscious perception of that water temperature begins.
Within the cerebral cortex, information is processed among many neurons, integrating the stimulus of the water temperature with other sensory stimuli, with your emotional state (you just aren't ready to wake up; the bed is calling to you), memories (perhaps of the lab notes you have to study before a quiz). Finally, a plan is developed about what to do, whether that is to turn the temperature up, turn the whole shower off and go back to bed, or step into the shower. To do any of these things, the cerebral cortex has to send a command out to your body to move muscles (Figure \(\PageIndex{3}\)).
Figure \(\PageIndex{3}\): The Motor Response. On the basis of the sensory input and the integration in the CNS, a motor response is formulated and executed.
A region of the cortex is specialized for sending signals down to the spinal cord for movement. The upper motor neuron is in this region, called the precentral gyrus of the frontal cortex , which has an axon that extends all the way down the spinal cord. At the level of the spinal cord at which this axon makes a synapse, a graded potential occurs in the cell membrane of a lower motor neuron . This second motor neuron is responsible for causing muscle fibers to contract. In the manner described in the chapter on muscle tissue, an action potential travels along the motor neuron axon into the periphery. The axon terminates on muscle fibers at the neuromuscular junction. Acetylcholine is released at this specialized synapse, which causes the muscle action potential to begin, following a large potential known as an end plate potential. When the lower motor neuron excites the muscle fiber, it contracts. All of this occurs in a fraction of a second, but this story is the basis of how the nervous system functions.
CAREER CONNECTIONS: Neurophysiologist
Understanding how the nervous system works could be a driving force in your career. Studying neurophysiology is a very rewarding path to follow. It means that there is a lot of work to do, but the rewards are worth the effort.
The career path of a research scientist can be straightforward: college, graduate school, postdoctoral research, academic research position at a university. A Bachelor’s degree in science will get you started, and for neurophysiology that might be in biology, psychology, computer science, engineering, or neuroscience. But the real specialization comes in graduate school. There are many different programs out there to study the nervous system, not just neuroscience itself. Most graduate programs are doctoral, meaning that a Master’s degree is not part of the work. These are usually considered five-year programs, with the first two years dedicated to course work and finding a research mentor, and the last three years dedicated to finding a research topic and pursuing that with a near single-mindedness. The research will usually result in a few publications in scientific journals, which will make up the bulk of a doctoral dissertation. After graduating with a Ph.D., researchers will go on to find specialized work called a postdoctoral fellowship within established labs. In this position, a researcher starts to establish their own research career with the hopes of finding an academic position at a research university.
Other options are available if you are interested in how the nervous system works. Especially for neurophysiology, a medical degree might be more suitable so you can learn about the clinical applications of neurophysiology and possibly work with human subjects. An academic career is not a necessity. Biotechnology firms are eager to find motivated scientists ready to tackle the tough questions about how the nervous system works so that therapeutic chemicals can be tested on some of the most challenging disorders such as Alzheimer’s disease or Parkinson’s disease, or spinal cord injury.
Others with a medical degree and a specialization in neuroscience go on to work directly with patients, diagnosing and treating mental disorders. You can do this as a psychiatrist, a neuropsychologist, a neuroscience nurse, or a neurodiagnostic technician, among other possible career paths.
Chapter Review
Sensation starts with the activation of a sensory ending, such as the thermoreceptor in the skin sensing the temperature of the water. The sensory endings in the skin initiate an electrical signal that travels along the sensory axon within a nerve into the spinal cord, where it synapses with a neuron in the gray matter of the spinal cord. The temperature information represented in that electrical signal is passed to the next neuron by a chemical signal that diffuses across the small gap of the synapse and initiates a new electrical signal in the target cell. That signal travels through the sensory pathway to the brain, passing through the thalamus, where conscious perception of the water temperature is made possible by the cerebral cortex. Following integration of that information with other cognitive processes and sensory information, the brain sends a command back down to the spinal cord to initiate a motor response by controlling a skeletal muscle. The motor pathway is composed of two cells, the upper motor neuron and the lower motor neuron. The upper motor neuron has its cell body in the cerebral cortex and synapses on a cell in the gray matter of the spinal cord. The lower motor neuron is that cell in the gray matter of the spinal cord and its axon extends into the periphery where it synapses with a skeletal muscle in a neuromuscular junction.
Review Questions
Q. If a thermoreceptor is sensitive to temperature sensations, what would a chemoreceptor be sensitive to?
A. light
B. sound
C. molecules
D. vibration
Answer: C
Q. Which of these locations is where the greatest level of integration is taking place in the example of testing the temperature of the shower?
A. skeletal muscle
B. spinal cord
C. thalamus
D. cerebral cortex
Answer: D
Q. How long does all the signaling through the sensory pathway, within the central nervous system, and through the motor command pathway take?
A. 1 to 2 minutes
B. 1 to 2 seconds
C. fraction of a second
D. varies with graded potential
Answer: C
Q. What is the target of an upper motor neuron?
A. cerebral cortex
B. lower motor neuron
C. skeletal muscle
D. thalamus
Answer: B
Critical Thinking Questions
Q. Sensory fibers, or pathways, are referred to as “afferent.” Motor fibers, or pathways, are referred to as “efferent.” What can you infer about the meaning of these two terms (afferent and efferent) in a structural or anatomical context?
A. Afferent means “toward,” as in sensory information traveling from the periphery into the CNS. Efferent means “away from,” as in motor commands that travel from the brain down the spinal cord and out into the periphery.
Q. If a person has a motor disorder and cannot move their arm voluntarily, but their muscles have tone, which motor neuron—upper or lower—is probably affected? Explain why.
A. The upper motor neuron would be affected because it is carrying the command from the brain down.
Glossary
- action potential
- change in voltage of a cell membrane in response to a stimulus that results in transmission of an electrical signal; unique to neurons and muscle fibers
- cerebral cortex
- outermost layer of gray matter in the brain, where conscious perception takes place
- graded potential
- change in the membrane potential that varies in size, depending on the size of the stimulus that elicits it
- lower motor neuron
- second neuron in the motor command pathway that is directly connected to the skeletal muscle
- neurotransmitter
- chemical signal that is released from the synaptic end bulb of a neuron to cause a change in the target cell
- precentral gyrus of the frontal cortex
- region of the cerebral cortex responsible for generating motor commands, where the upper motor neuron cell body is located
- propagation
- movement of an action potential along the length of an axon
- thalamus
- region of the central nervous system that acts as a relay for sensory pathways
- thermoreceptor
- type of sensory receptor capable of transducing temperature stimuli into neural action potentials
- threshold
- membrane voltage at which an action potential is initiated
- upper motor neuron
- first neuron in the motor command pathway with its cell body in the cerebral cortex that synapses on the lower motor neuron in the spinal cord


4.4: Partially Ordered Sets
Terminology
Poset.
A relation is an partially ordered set (poset) if and only if the relation is reflexive, anti-symmetric, and transitive.
Consider the following poset defined by listing the relations explicitly.
\(R=\{(a,a), \) \((a,d), \) \((a,f), \) \((a,g), \) \((a,h),\) \((b,b), \) \((b,d), \) \((b,e), \) \((b,f), \) \((b,g), \) \((b,h),\) \((c,c), \) \((c,d), \) \((c,f), \) \((c,g), \) \((c,h),\) \((d,d), \) \((d,f), \) \((d,g), \) \((d,h),\) \((e,e), \) \((e,g), \) \((e,h),\) \((f,f), \) \((g,g), \) \((h,h) \}.\)
Another way to define the elements of the relation is a Hasse diagram. Figure \(\PageIndex{2}\) shows the Hasse diagram for \(R.\) The rules for constructing these diagrams are in List \(\PageIndex{3}\) .
List \(\PageIndex{3}\) Constructing a Hasse Diagram
- If \((a,b) \in R\) then \(a\) is below \(b\) on the diagram.
-
If \((a,b) \in R\) then there is a line from \(a\) to \(b\)
- unless \(a=b\) (reflexive), or
- unless \((a,c), (c,b) \in R\) (transitive set).
Note that constructing a Hasse diagram requires sorting the elements of the set. Each level of the diagram in some sense is “before” the levels above it. Because of this ordering in posets we write \(a \preceq b\) to mean \(aRb\text{.}\)
Practice
Checkpoint \(\PageIndex{4}\)
Determine whether the following relation is a poset. Let the relation \(R\) on the set of numbers \(\{1,2,3,6\}\) be defined by \(aRb\) if and only if \(a|b.\)
Checkpoint \(\PageIndex{5}\)
Determine whether the following relation is a poset. Let the relation \(R\) on the set of numbers \(\{1,2,3,5,6,10,15,30\}\) be defined by \(aRb\) if and only if \(a|b.\)
Checkpoint \(\PageIndex{6}\)
Determine whether the following relation is a poset. Let the relation \(R\) on the set of positive integers be defined by \(aRb\) if and only if \(a|b.\)
Checkpoint \(\PageIndex{7}\)
Determine whether the following relation is a poset. Let the relation \(R\) on the set of real numbers be defined by \(aRb\) if and only if \(a \le b.\)
Checkpoint \(\PageIndex{8}\)
Determine whether the following relation is a poset. Consider the set \(P(X)\) for the set \(X=\{a,b,c\}.\) Let the relation \(R\) be defined by \(A\) is related to \(B\) if and only if \(A \subseteq B\text{.}\)
Checkpoint \(\PageIndex{9}\)
Draw the Hasse diagrams for the following posets.
- \(\subseteq\) on \(P(\{0,1\}).\) Note it may help to list all the elements first.
- \(|\) (i.e., \(aRb\) iff \(a|b\)) on \(\{1,2,3,6\}.\)
- \(|\) on \(\{1,2,3,5,6,10,15,30\}.\)
- \(X=\{ \)(0,0,0), (1,0,0), (0,1,0), (0,0,1), (0,1,1), (1,0,1), (1,1,0), (1,1,1) \(\}\) with the relation \((a_1,a_2,a_3) \preceq (b_1,b_2,b_3)\) iff \(a_i \le b_i\) for all \(i.\)
Poset Properties
Terminology
Two elements \(a,b\) of a poset are comparable if and only if either \(a \preceq b\) or \(b \preceq a.\)
For the poset represented by the Hasse diagram in Figure \(\PageIndex{2}\) the elements d and g are comparable . The elements b and f are incomparable.
A partially ordered set is a totally ordered set if and only if all pairs of elements are comparable.
An element \(c\) is a least upper bound of two elements \(a\) and \(b\text{,}\) denoted \(a \wedge b\text{,}\) if and only if \(a \preceq c,\) \(b \preceq c\) and if \(a \preceq d,\) \(b \preceq d\) for any other element then \(c \preceq d.\)
For the poset represented by the Hasse diagram in Figure \(\PageIndex{2}\) \(lub(a,b)=d\text{.}\) Note h is another upper bound of a and b, but \(d \preceq h\text{.}\)
An element \(c\) is a greatest lower bound of two elements \(a\) and \(b\text{,}\) denoted \(a \vee b\text{,}\) if and only if \(c \preceq a,\) \(c \preceq b\) and if \(d \preceq a,\) \(d \preceq b\) for any other element then \(d \preceq c.\)
For the poset represented by the Hasse diagram in Figure \(\PageIndex{2}\) \(glb(d,e)=b\text{.}\) There are no other lower bounds for this pair.
An element \(a\) in a poset is a minimal element if and only if there does not exist an element \(b\) such that \(b \preceq a.\)
For the poset represented by the Hasse diagram in Figure \(\PageIndex{2}\) the minimal elements are a,b, and c. a is a minimal element because there is no element x such that \(x \preceq a\text{.}\) Notice where the minimal elements are in the Hasse diagram.
An element \(a\) in a poset is a maximal element if and only if there does not exist an element \(b\) such that \(a \preceq b.\)
For the poset represented by the Hasse diagram in Figure \(\PageIndex{2}\) the maximal elements are f,g, and h. f is a maximal element because there is no element x such that \(f \preceq x\text{.}\)
Practice
Checkpoint \(\PageIndex{21}\)
Find two comparable elements in each of the relations in Checkpoint \(\PageIndex{9 }\)
Checkpoint \(\PageIndex{22}\)
Find two incomparable elements in each of the relations in Checkpoint \(\PageIndex{9 }\)
Checkpoint \(\PageIndex{23}\)
Find all minimal and maximal elements of the posets in Problem Checkpoint \(\PageIndex{9 }\)
Checkpoint \(\PageIndex{24}\)
Consider the poset given in Figure \(\PageIndex{2 }\) Find the following upper and lower bounds.
(a) Find \(f \vee h\)
(b) Find \(a \wedge e\)
(c) Find \(a \wedge d\)
(d) Does \(d \wedge e\) exist?
(e) Does \(g \vee h\) exist?
A poset in which every pair of elements has both a least upper bound and a greatest lower bound is a lattice
Checkpoint \(\PageIndex{26 }\)
Which of the posets in Checkpoint \(\PageIndex{9 }\) are lattices?
Checkpoint \(\PageIndex{27}\)
Why is the name “lattice” appropriate?


11: Personality
-
- 11.1: Prelude to Personality
- While Bill Clinton was making his political ascendance, his half-brother, Roger Clinton, was arrested numerous times for drug charges, including possession, conspiracy to distribute cocaine, and driving under the influence, serving time in jail. Two brothers, raised by the same people, took radically different paths in their lives. Why did they make the choices they did? What internal forces shaped their decisions? Personality psychology can help us answer these questions and more.
-
- 11.2: What Is Personality?
- The word personality comes from the Latin word persona. Personality refers to the long-standing traits and patterns that propel individuals to consistently think, feel, and behave in specific ways. Our personality is what makes us unique individuals. Each person has an idiosyncratic pattern of enduring, long-term characteristics and a manner in which he or she interacts with other individuals and the world around them. Our personalities are thought to be long term, stable, and not easily changed
-
- 11.3: Freud and the Psychodynamic Perspective
- Sigmund Freud is probably the most controversial and misunderstood psychological theorist. When reading Freud’s theories, it is important to remember that he was a medical doctor, not a psychologist. There was no such thing as a degree in psychology at the time that he received his education, which can help us understand some of the controversy over his theories today.
-
- 11.4: Neo-Freudians- Adler, Erikson, Jung, and Horney
- Freud attracted many followers who modified his ideas to create new theories about personality. These theorists, referred to as neo-Freudians, generally agreed with Freud that childhood experiences matter, but deemphasized sex, focusing more on the social environment and effects of culture on personality. Four notable neo-Freudians include Alfred Adler, Erik Erikson, Carl Jung, and Karen Horney.
-
- 11.5: Learning Approaches
- In contrast to the psychodynamic approaches of Freud and the neo-Freudians, which relate personality to inner (and hidden) processes, the learning approaches focus only on observable behavior. This illustrates one significant advantage of the learning approaches over psychodynamics: Because learning approaches involve observable, measurable phenomena, they can be scientifically tested.
-
- 11.6: Humanistic Approaches
- As the “third force” in psychology, humanism is touted as a reaction both to the pessimistic determinism of psychoanalysis, with its emphasis on psychological disturbance, and to the behaviorists’ view of humans passively reacting to the environment, which has been criticized as making people out to be personality-less robots. It does not suggest that psychoanalytic, behaviorist, and other points of view are incorrect but argues that these perspectives do not recognize the depth and meaning of h
-
- 11.7: Biological Approaches
- How much of our personality is in-born and biological, and how much is influenced by the environment and culture we are raised in? Psychologists who favor the biological approach believe that inherited predispositions as well as physiological processes can be used to explain differences in our personalities. Some aspects of our personalities are largely controlled by genetics; however, environmental factors and maturation can affect the ways in which children’s personalities are expressed.
-
- 11.8: Trait Theorists
- Trait theorists believe personality can be understood via the approach that all people have certain traits, or characteristic ways of behaving. Trait theorists attempt to explain our personality by identifying our stable characteristics and ways of behaving. They have identified important dimensions of personality. The Five Factor Model is the most widely accepted trait theory today. The five factors are openness, conscientiousness, extroversion, agreeableness, and neuroticism.
-
- 11.9: Cultural Understandings of Personality
- The culture in which you live is one of the most important environmental factors that shapes your personality. Western ideas about personality may not be applicable to other cultures. In fact, there is evidence that the strength of personality traits varies across cultures. Individualist cultures and collectivist cultures place emphasis on different basic values. People who live in individualist cultures tend to believe that independence, competition, and personal achievement are important.
-
- 11.10: Personality Assessment
- Personality tests are techniques designed to measure one’s personality. They are used to diagnose psychological problems as well as to screen candidates for college and employment. There are two types of personality tests: self-report inventories and projective tests. The MMPI is one of the most common self-report inventories. It asks a series of true/false questions that are designed to provide a clinical profile of an individual.
Thumbnail: Sigmund Freud was an Austrian neurologist and the founder of psychoanalysis, a clinical method for treating psychopathology through dialog between a patient and a psychoanalyst.


4.2: Lesson 4.2: Note-Taking Part 2, Key Information and Formats
Perhaps the most useful learning tools of all are notes taken from both lectures and course materials. By annotating for key information, then condensing it, students not only create personalized summaries but also aid their memory by using both visual and kinesthetic learning styles. Add auditory, too, if the notes are taken from a lecture or video.
Often, students are unsure about what constitutes “key information.” Here is a list of items to highlight or annotate for in textbooks and a list of items to listen for in lectures.
Key Information in Textbooks
In addition to paying attention to all the items listed in Lesson 3.2, Getting the Most Out of Your Textbooks, the following elements of a textbook chapter are especially important in helping you discern key information:
- Introductions
- Summaries
- Study questions
- Topic sentences (as the speaker in the video on “Skim Reading,” exercise 3.2, also in Lesson 3.2 notes: sometimes the reader has to read the first two or even three sentences of a paragraph or section to get the entire main topic).
- Anything that is bolded, or in some other way set off from the default print size and style. Sub-topic titles are good examples of this.
- “ Side bars ,” which are boxes of related information. These might include statistics, brief biographies of authors or persons of note related to the chapter content, price points on brochures for businesses, charts, graphs, photographs, and/or illustrations. They are typically a different color or in some other way set off for attention but not as the focal point of the text. Pay attention to the captions or legends that might accompany graphics. In this e-text, the exercises are set apart in side bars.
- Glossary terms that may be incorporated in the margins or otherwise set apart.
- Some textbooks include outlines of each chapter’s main points in the introductory section.
Key Information in Lectures
As the lecturer, live or video, presents the material, there are two types of key information cues to be aware of.
NONVERBAL CUES
As covered in the previous lesson on listening skills, a speaker will often have unique facial and body nonverbal cues that alert you to several things, as you learn to “read” your professor:
- Stances or movements that alert you to when he/she will shift to a different topic or subtopic.
- Other cues that alert you to when the information is of special significance (including verbal clues, below).
VERBAL CLUES
- Pay attention to when the speaker uses any of the transition clues used in reading comprehension, as listed in Lesson 3.3, context clues. Click on common clue words for a separate list.
- Many speakers also announce when they are adding information or changing topics in various other ways.
For tips on how to deal with fast talkers, when you should NOT take notes, what note-taking formats work well, and other advice, complete the exercise below.
UNIT 4, EXERCISE 2.1
- Download and read Study Tip #9-Taking Useful Class Notes, by Dan Hodges, found on the PDF file with all of his articles. Scroll down to #9.
- Number each of the 7 sections for easy reference.
- As you read, annotate (highlight, circle, underline, write marginal notes, etc.) for items that you especially want to remember from each of the 7 topics.
- Transfer the notes to a Cornell Note-Taking form . Hodges also includes instructions for creating your own form. Hodges features the Cornell format in section 4, “What Style of Note-Taking Works Well?”
Example:
KEY POINTS DETAILS
section 1: Why take notes? Good notes=better grades
(add the rest of the details)
UNIT 4, EXERCISE 2.2
There are, of course, as many ways to take notes as there are note-takers with creative minds! For this exercise, do an online search using the key words, “fun and creative ways to take notes YouTube.” Preview as many as you like. Pick one (these are all fairly short), then take notes on it (!) in the creative way suggested. Be sure to cite the YouTube source (title and URL) in your notes.


6.6E: Exercises
Practice Makes Perfect
Use the Zero Product Property
In the following exercises, solve.
1. \((3a−10)(2a−7)=0\)
- Answer
-
\(a=\frac{10}{3},\; a=\frac{7}{2}\)
2. \((5b+1)(6b+1)=0\)
3. \(6m(12m−5)=0\)
- Answer
-
\(m=0,\; m=\frac{5}{12}\)
4. \(2x(6x−3)=0\)
5. \((2x−1)^2=0\)
- Answer
-
\(x=\frac{1}{2}\)
6. \((3y+5)^2=0\)
Solve Quadratic Equations by Factoring
In the following exercises, solve.
7. \(5a^2−26a=24\)
- Answer
-
\(a=−\frac{4}{5},\; a=6\)
8. \(4b^2+7b=−3\)
9. \(4m^2=17m−15\)
- Answer
-
\(m=\frac{5}{4},\; m=3\)
10. \(n^2=5−6n\)
11. \(7a^2+14a=7a\)
- Answer
-
\(a=−1,\; a=0\)
12. \(12b^2−15b=−9b\)
13. \(49m^2=144\)
- Answer
-
\(m=\frac{12}{7},\; m=−\frac{12}{7}\)
14. \(625=x^2\)
15. \(16y^2=81\)
- Answer
-
\(y=−\frac{9}{4},\; y=\frac{9}{4}\)
16. \(64p^2=225\)
17. \(121n^2=36\)
- Answer
-
\(n=−\frac{6}{11},\; n=\frac{6}{11}\)
18. \(100y^2=9\)
19. \((x+6)(x−3)=−8\)
- Answer
-
\(x=2,\; x=−5\)
20. \((p−5)(p+3)=−7\)
21. \((2x+1)(x−3)=−4x\)
- Answer
-
\(x=\frac{3}{2},\; x=−1\)
22. \((y−3)(y+2)=4y\)
23. \((3x−2)(x+4)=12x\)
- Answer
-
\(x=\frac{3}{2},\; x=−1\)
24. \((2y−3)(3y−1)=8y\)
25. \(20x^2−60x=−45\)
- Answer
-
\(x=−\frac{2}{3}\)
26. \(3y^2−18y=−27\)
27. \(15x^2−10x=40\)
- Answer
-
\(x=2,\; x=−\frac{4}{3}\)
28. \(14y^2−77y=−35\)
29. \(18x^2−9=−21x\)
- Answer
-
\(x=−\frac{3}{2},\; x=\frac{1}{3}\)
30. \(16y^2+12=−32y\)
31. \(16p^3=24p^2-9p\)
- Answer
-
\(p=0,\; p=\frac{3}{4}\)
32. \(m^3−2m^2=−m\)
33. \(2x^3+72x=24x^2\)
- Answer
-
\(x=0,\space x=6\)
34. \(3y^3+48y=24y^2\)
35. \(36x^3+24x^2=−4x\)
- Answer
-
\(x=0,\space x=\frac{1}{3}\)
36. \(2y^3+2y^2=12y\)
Solve Equations with Polynomial Functions
In the following exercises, solve.
37. For the function, \(f(x)=x^2−8x+8\), ⓐ find when \(f(x)=−4\) ⓑ Use this information to find two points that lie on the graph of the function.
- Answer
-
ⓐ \(x=2\) or \(x=6\) ⓑ \((2,−4)\) \((6,−4)\)
38. For the function, \(f(x)=x^2+11x+20\), ⓐ find when \(f(x)=−8\) ⓑ Use this information to find two points that lie on the graph of the function.
39. For the function, \(f(x)=8x^2−18x+5\), ⓐ find when \(f(x)=−4\) ⓑ Use this information to find two points that lie on the graph of the function.
- Answer
-
ⓐ \(x=\frac{3}{2}\) or \(x=\frac{3}{4}\)
ⓑ \((\frac{3}{2},−4)\) \((\frac{3}{4},−4)\)
40. For the function, \(f(x)=18x^2+15x−10\), ⓐ find when \(f(x)=15\) ⓑ Use this information to find two points that lie on the graph of the function.
In the following exercises, for each function, find: ⓐ the zeros of the function ⓑ the \(x\)-intercepts of the graph of the function ⓒ the \(y\)-intercept of the graph of the function.
41. \(f(x)=9x^2−4\)
- Answer
-
ⓐ \(x=\frac{2}{3}\) or \(x=−\frac{2}{3}\)
ⓑ \((\frac{2}{3},0)\), \((−\frac{2}{3},0)\)
ⓒ \((0,−4)\)
42. \(f(x)=25x^2−49\)
43. \(f(x)=6x^2−7x−5\)
- Answer
-
ⓐ \(x=\frac{5}{3}\) or \(x=−\frac{1}{2}\)
ⓑ \((\frac{5}{3},0)\), \((−\frac{1}{2},0)\)
ⓒ \((0,−5)\)
44. \(f(x)=12x^2−11x+2\)
Solve Applications Modeled by Quadratic Equations
In the following exercises, solve.
45. The product of two consecutive odd integers is \(143\). Find the integers.
- Answer
-
\(−13,\space −11\) and \(11,\space 13\)
46. The product of two consecutive odd integers is \(195\). Find the integers.
47. The product of two consecutive even integers is \(168\). Find the integers.
- Answer
-
\(−14,\space −12\) and \(12,\space 14\)
48. The product of two consecutive even integers is \(288\). Find the integers.
49. The area of a rectangular carpet is \(28\) square feet. The length is three feet more than the width. Find the length and the width of the carpet.
- Answer
-
\(−4\) and \(7\)
50. A rectangular retaining wall has area \(15\) square feet. The height of the wall is two feet less than its length. Find the height and the length of the wall.
51. The area of a bulletin board is \(55\) square feet. The length is four feet less than three times the width. Find the length and the width of the a bulletin board.
- Answer
-
\(5,\space 11\)
52. A rectangular carport has area \(150\) square feet. The height of the carport is five feet less than twice its length. Find the height and the length of the carport.
53.  The length of one side of the pennant is two feet longer than the length of the other side. Find the length of the two sides of the pennant.
- Answer
-
\(6,\space 8\)
54.  The hypotenuse is \(15\) feet. One leg is three more than the other. Find the lengths of the legs.
55. A reflecting pool is shaped like a right triangle, with one leg along the wall of a building. The hypotenuse is \(9\) feet longer than the side along the building. The third side is \(7\) feet longer than the side along the building. Find the lengths of all three sides of the reflecting pool.
- Answer
-
\(8,\space 15,\space 17\)
56. A goat enclosure is in the shape of a right triangle. One leg of the enclosure is built against the side of the barn. The other leg is \(4\) feet more than the leg against the barn. The hypotenuse is \(8\) feet more than the leg along the barn. Find the three sides of the goat enclosure.
57. Juli is going to launch a model rocket in her back yard. When she launches the rocket, the function \(h(t)=−16t^2+32t\) models the height, \(h\), of the rocket above the ground as a function of time, \(t\). Find:
ⓐ the zeros of this function which tells us when the rocket will hit the ground. ⓑ the time the rocket will be \(16\) feet above the ground.
- Answer
-
ⓐ 0, 2 ⓑ 1
58. Gianna is going to throw a ball from the top floor of her middle school. When she throws the ball from \(48\) feet above the ground, the function \(h(t)=−16t^2+32t+48\) models the height, \(h\), of the ball above the ground as a function of time, \(t\). Find:
ⓐ the zeros of this function which tells us when the ball will hit the ground. ⓑ the time(s) the ball will be \(48\) feet above the ground. ⓒ the height the ball will be at \(t=1\) seconds which is when the ball will be at its highest point.
Writing Exercises
59. Explain how you solve a quadratic equation. How many answers do you expect to get for a quadratic equation?
- Answer
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Answers will vary.
60. Give an example of a quadratic equation that has a GCF and none of the solutions to the equation is zero.
Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ Overall, after looking at the checklist, do you think you are well-prepared for the next section? Why or why not?