Page 1.1.1: Advice For Success
We first start off our journey by noting that it is important to be aware of how to succeed in mathematics. Although the following is not a comprehensive list, I hope that it serves as a guideline on how to excel in your math classes. We will try and practice many of these skills here in our boot camp! Whenever we do so, we will highlight it and make a note.
1) In mathematics, we should try to see many examples and non-examples. In doing so, we obtain a clearer idea on the topic that is being discussed.
2) Whenever we have a definition or a theorem, try to translate the definition or theorem in words. Often, students tend to memorize a formula by repeating the symbols, but instead we should be able to explain the formula in words! For instance, instead of saying "The Pythagorean Theorem states that \( a^2 + b^2 = c^2 \) ", a more detailed answer would be "The Pythagorean Theorem states that for any right triangle, the square length of one leg plus the square length of the other leg equals to the square length of the hypotenuse". At first, the second suggestion may seem like it is a lot of work for a little value. However, in your journey in calculus, you will be required to demonstrate a strong grasp on interpreting what a formula may mean.
3) Try to reflect on every rule/theorem. Can you understand why the rule/theorem makes intuitive sense?
4) Form a study group since a good habit to get into is to try discussing the material with someone. Mathematics is a language and if we only write math, then we have never practiced speaking it! There is a rich vocabulary associated with mathematics and the act of forcing yourself to find the right words and explain an idea will give you a better command of the material.
5) Math is not a spectator sport! To learn math, you must practice, practice and practice. It is easy to attend lectures and watch your professor work out all the examples. But just because you watched someone work out, it does not mean that you, yourself, can also work out! Remember, your instructors have an excellent command of the material. Just because they may make a problem look easy, it does not mean you can replicate their process on the day of the exam without practicing first.
6) Although this may sound excessive, perhaps for your math courses we have four separate notebooks. In one notebook, we should copy down the lecture notes. When we go home, we should actively copy these lecture notes into our second notebook. Doing so forces us to go over the lecture and grants us an opportunity to solve the problems done in class. This is important because many students do not realize that it is one thing to read someone's solution and understand it. However, it is a completely different thing to be able to come up with that solution, on your own, from scratch. Students (myself included) often read their notes and think they can imitate the same type of thinking on the test because our instructor makes it look so easy. However, this is often not the case. We really don't know how much we actually know until we are forced to come up with an answer on our own. For some students, the first time they are actually trying a problem on their own is during the exam. If the first time we are actually trying a problem from scratch is on the exam, then it is not going to go well! The third notebook should contain all the homework problems worked out in as much details as possible. And finally, the forth notebook should contain summaries/outlines of the textbook. The textbook can offer a different viewpoint from the instructor and serves as another place where you can actively study.
7) Make use of your professor's office hours, the Learning Commons ( https://www.qc.cuny.edu/academics/qclc/ ) and the Math Lab ( https://www.qc.cuny.edu/academics/math/getting-help/ ).
8) Here is a more comprehensive list of tips: https://mathstat.slu.edu/resources/success-in-mathematics .
Again, all of the above is merely advice. It is important to find which combination of strategies works best for you. The one thing to keep in mind is that simply attending the lecture might not guarantee good grades! There is a lot of work to be done outside of the classroom and hopefully the above list can point you in the right direction.


26.1: Evolution of Seed Plants
Skills to Develop
- Explain when seed plants first appeared and when gymnosperms became the dominant plant group
- Describe the two major innovations that allowed seed plants to reproduce in the absence of water
- Discuss the purpose of pollen grains and seeds
- Describe the significance of angiosperms bearing both flowers and fruit
The first plants to colonize land were most likely closely related to modern day mosses (bryophytes) and are thought to have appeared about 500 million years ago. They were followed by liverworts (also bryophytes) and primitive vascular plants—the pterophytes—from which modern ferns are derived. The lifecycle of bryophytes and pterophytes is characterized by the alternation of generations, like gymnosperms and angiosperms; what sets bryophytes and pterophytes apart from gymnosperms and angiosperms is their reproductive requirement for water. The completion of the bryophyte and pterophyte life cycle requires water because the male gametophyte releases sperm, which must swim—propelled by their flagella—to reach and fertilize the female gamete or egg. After fertilization, the zygote matures and grows into a sporophyte, which in turn will form sporangia or "spore vessels." In the sporangia, mother cells undergo meiosis and produce the haploid spores. Release of spores in a suitable environment will lead to germination and a new generation of gametophytes.
In seed plants, the evolutionary trend led to a dominant sporophyte generation, and at the same time, a systematic reduction in the size of the gametophyte: from a conspicuous structure to a microscopic cluster of cells enclosed in the tissues of the sporophyte. Whereas lower vascular plants, such as club mosses and ferns, are mostly homosporous (produce only one type of spore), all seed plants, or spermatophytes , are heterosporous. They form two types of spores: megaspores (female) and microspores (male). Megaspores develop into female gametophytes that produce eggs, and microspores mature into male gametophytes that generate sperm. Because the gametophytes mature within the spores, they are not free-living, as are the gametophytes of other seedless vascular plants. Heterosporous seedless plants are seen as the evolutionary forerunners of seed plants.
Seeds and pollen—two critical adaptations to drought, and to reproduction that doesn’t require water—distinguish seed plants from other (seedless) vascular plants. Both adaptations were required for the colonization of land begun by the bryophytes and their ancestors. Fossils place the earliest distinct seed plants at about 350 million years ago. The first reliable record of gymnosperms dates their appearance to the Pennsylvanian period, about 319 million years ago (Figure \(\PageIndex{1}\)). Gymnosperms were preceded by progymnosperms , the first naked seed plants, which arose about 380 million years ago. Progymnosperms were a transitional group of plants that superficially resembled conifers (cone bearers) because they produced wood from the secondary growth of the vascular tissues; however, they still reproduced like ferns, releasing spores into the environment. Gymnosperms dominated the landscape in the early (Triassic) and middle (Jurassic) Mesozoic era. Angiosperms surpassed gymnosperms by the middle of the Cretaceous (about 100 million years ago) in the late Mesozoic era, and today are the most abundant plant group in most terrestrial biomes.
Pollen and seed were innovative structures that allowed seed plants to break their dependence on water for reproduction and development of the embryo, and to conquer dry land. The pollen grains are the male gametophytes, which contain the sperm (gametes) of the plant. The small haploid (1 n ) cells are encased in a protective coat that prevents desiccation (drying out) and mechanical damage. Pollen grains can travel far from their original sporophyte, spreading the plant’s genes. The seed offers the embryo protection, nourishment, and a mechanism to maintain dormancy for tens or even thousands of years, ensuring germination can occur when growth conditions are optimal. Seeds therefore allow plants to disperse the next generation through both space and time. With such evolutionary advantages, seed plants have become the most successful and familiar group of plants, in part because of their size and striking appearance.
Evolution of Gymnosperms
The fossil plant Elkinsia polymorpha , a "seed fern" from the Devonian period—about 400 million years ago—is considered the earliest seed plant known to date. Seed ferns (Figure \(\PageIndex{2}\)) produced their seeds along their branches without specialized structures. What makes them the first true seed plants is that they developed structures called cupules to enclose and protect the ovule —the female gametophyte and associated tissues—which develops into a seed upon fertilization. Seed plants resembling modern tree ferns became more numerous and diverse in the coal swamps of the Carboniferous period.
Fossil records indicate the first gymnosperms (progymnosperms) most likely originated in the Paleozoic era, during the middle Devonian period: about 390 million years ago. Following the wet Mississippian and Pennsylvanian periods, which were dominated by giant fern trees, the Permian period was dry. This gave a reproductive edge to seed plants, which are better adapted to survive dry spells. The Ginkgoales, a group of gymnosperms with only one surviving species—the Gingko biloba— were the first gymnosperms to appear during the lower Jurassic. Gymnosperms expanded in the Mesozoic era (about 240 million years ago), supplanting ferns in the landscape, and reaching their greatest diversity during this time. The Jurassic period was as much the age of the cycads (palm-tree-like gymnosperms) as the age of the dinosaurs. Gingkoales and the more familiar conifers also dotted the landscape. Although angiosperms (flowering plants) are the major form of plant life in most biomes, gymnosperms still dominate some ecosystems, such as the taiga (boreal forests) and the alpine forests at higher mountain elevations (Figure \(\PageIndex{3}\)) because of their adaptation to cold and dry growth conditions.
Seeds and Pollen as an Evolutionary Adaptation to Dry Land
Unlike bryophyte and fern spores (which are haploid cells dependent on moisture for rapid development of gametophytes), seeds contain a diploid embryo that will germinate into a sporophyte. Storage tissue to sustain growth and a protective coat give seeds their superior evolutionary advantage. Several layers of hardened tissue prevent desiccation, and free reproduction from the need for a constant supply of water. Furthermore, seeds remain in a state of dormancy—induced by desiccation and the hormone abscisic acid—until conditions for growth become favorable. Whether blown by the wind, floating on water, or carried away by animals, seeds are scattered in an expanding geographic range, thus avoiding competition with the parent plant.
Pollen grains (Figure \(\PageIndex{4}\)) are male gametophytes and are carried by wind, water, or a pollinator. The whole structure is protected from desiccation and can reach the female organs without dependence on water. Male gametes reach female gametophyte and the egg cell gamete though a pollen tube: an extension of a cell within the pollen grain. The sperm of modern gymnosperms lack flagella, but in cycads and the Gingko , the sperm still possess flagella that allow them to swim down the pollen tube to the female gamete; however, they are enclosed in a pollen grain.
Evolution of Angiosperms
Undisputed fossil records place the massive appearance and diversification of angiosperms in the middle to late Mesozoic era. Angiosperms (“seed in a vessel”) produce a flower containing male and/or female reproductive structures. Fossil evidence (Figure \(\PageIndex{5}\)) indicates that flowering plants first appeared in the Lower Cretaceous, about 125 million years ago, and were rapidly diversifying by the Middle Cretaceous, about 100 million years ago. Earlier traces of angiosperms are scarce. Fossilized pollen recovered from Jurassic geological material has been attributed to angiosperms. A few early Cretaceous rocks show clear imprints of leaves resembling angiosperm leaves. By the mid-Cretaceous, a staggering number of diverse flowering plants crowd the fossil record. The same geological period is also marked by the appearance of many modern groups of insects, including pollinating insects that played a key role in ecology and the evolution of flowering plants.
Although several hypotheses have been offered to explain this sudden profusion and variety of flowering plants, none have garnered the consensus of paleobotanists (scientists who study ancient plants). New data in comparative genomics and paleobotany have, however, shed some light on the evolution of angiosperms. Rather than being derived from gymnosperms, angiosperms form a sister clade (a species and its descendents) that developed in parallel with the gymnosperms. The two innovative structures of flowers and fruit represent an improved reproductive strategy that served to protect the embryo, while increasing genetic variability and range. Paleobotanists debate whether angiosperms evolved from small woody bushes, or were basal angiosperms related to tropical grasses. Both views draw support from cladistics studies, and the so-called woody magnoliid hypothesis—which proposes that the early ancestors of angiosperms were shrubs—also offers molecular biological evidence.
The most primitive living angiosperm is considered to be Amborella trichopoda , a small plant native to the rainforest of New Caledonia, an island in the South Pacific. Analysis of the genome of A. trichopoda has shown that it is related to all existing flowering plants and belongs to the oldest confirmed branch of the angiosperm family tree. A few other angiosperm groups called basal angiosperms, are viewed as primitive because they branched off early from the phylogenetic tree. Most modern angiosperms are classified as either monocots or eudicots, based on the structure of their leaves and embryos. Basal angiosperms, such as water lilies, are considered more primitive because they share morphological traits with both monocots and eudicots.
Flowers and Fruits as an Evolutionary Adaptation
Angiosperms produce their gametes in separate organs, which are usually housed in a flower . Both fertilization and embryo development take place inside an anatomical structure that provides a stable system of sexual reproduction largely sheltered from environmental fluctuations. Flowering plants are the most diverse phylum on Earth after insects; flowers come in a bewildering array of sizes, shapes, colors, smells, and arrangements. Most flowers have a mutualistic pollinator, with the distinctive features of flowers reflecting the nature of the pollination agent. The relationship between pollinator and flower characteristics is one of the great examples of coevolution.
Following fertilization of the egg, the ovule grows into a seed. The surrounding tissues of the ovary thicken, developing into a fruit that will protect the seed and often ensure its dispersal over a wide geographic range. Not all fruits develop from an ovary; such structures are “false fruits.” Like flowers, fruit can vary tremendously in appearance, size, smell, and taste. Tomatoes, walnut shells and avocados are all examples of fruit. As with pollen and seeds, fruits also act as agents of dispersal. Some may be carried away by the wind. Many attract animals that will eat the fruit and pass the seeds through their digestive systems, then deposit the seeds in another location. Cockleburs are covered with stiff, hooked spines that can hook into fur (or clothing) and hitch a ride on an animal for long distances. The cockleburs that clung to the velvet trousers of an enterprising Swiss hiker, George de Mestral, inspired his invention of the loop and hook fastener he named Velcro.
Evolution Connection: Building Phylogenetic Trees with Analysis of DNA Sequence Alignments
All living organisms display patterns of relationships derived from their evolutionary history. Phylogeny is the science that describes the relative connections between organisms, in terms of ancestral and descendant species. Phylogenetic trees, such as the plant evolutionary history shown in Figure \(\PageIndex{6}\), are tree-like branching diagrams that depict these relationships. Species are found at the tips of the branches. Each branching point, called a node, is the point at which a single taxonomic group (taxon), such as a species, separates into two or more species.
Phylogenetic trees have been built to describe the relationships between species since Darwin’s time. Traditional methods involve comparison of homologous anatomical structures and embryonic development, assuming that closely related organisms share anatomical features during embryo development. Some traits that disappear in the adult are present in the embryo; for example, a human fetus, at one point, has a tail. The study of fossil records shows the intermediate stages that link an ancestral form to its descendants. Most of these approaches are imprecise and lend themselves to multiple interpretations. As the tools of molecular biology and computational analysis have been developed and perfected in recent years, a new generation of tree-building methods has taken shape. The key assumption is that genes for essential proteins or RNA structures, such as the ribosomal RNA, are inherently conserved because mutations (changes in the DNA sequence) could compromise the survival of the organism. DNA from minute amounts of living organisms or fossils can be amplified by polymerase chain reaction (PCR) and sequenced, targeting the regions of the genome that are most likely to be conserved between species. The genes encoding the ribosomal RNA from the small 18S subunit and plastid genes are frequently chosen for DNA alignment analysis.
Once the sequences of interest are obtained, they are compared with existing sequences in databases such as GenBank, which is maintained by The National Center for Biotechnology Information. A number of computational tools are available to align and analyze sequences. Sophisticated computer analysis programs determine the percentage of sequence identity or homology. Sequence homology can be used to estimate the evolutionary distance between two DNA sequences and reflect the time elapsed since the genes separated from a common ancestor. Molecular analysis has revolutionized phylogenetic trees. In some cases, prior results from morphological studies have been confirmed: for example, confirming Amborella trichopoda as the most primitive angiosperm known. However, some groups and relationships have been rearranged as a result of DNA analysis.
Summary
Seed plants appeared about one million years ago, during the Carboniferous period. Two major innovations—seed and pollen—allowed seed plants to reproduce in the absence of water. The gametophytes of seed plants shrank, while the sporophytes became prominent structures and the diploid stage became the longest phase of the lifecycle. Gymnosperms became the dominant group during the Triassic. In these, pollen grains and seeds protect against desiccation. The seed, unlike a spore, is a diploid embryo surrounded by storage tissue and protective layers. It is equipped to delay germination until growth conditions are optimal. Angiosperms bear both flowers and fruit. The structures protect the gametes and the embryo during its development. Angiosperms appeared during the Mesozoic era and have become the dominant plant life in terrestrial habitats.
Glossary
- flower
- branches specialized for reproduction found in some seed-bearing plants, containing either specialized male or female organs or both male and female organs
- fruit
- thickened tissue derived from ovary wall that protects the embryo after fertilization and facilitates seed dispersal
- ovule
- female gametophyte
- pollen grain
- structure containing the male gametophyte of the plant
- pollen tube
- extension from the pollen grain that delivers sperm to the egg cell
- progymnosperm
- transitional group of plants that resembled conifers because they produced wood, yet still reproduced like ferns
- seed
- structure containing the embryo, storage tissue and protective coat
- spermatophyte
- seed plant; from the Greek sperm (seed) and phyte (plant)


1.3: Relations
Relations play a fundamental role in probability theory, as in most other areas of mathematics.
Definitions and Constructions
Suppose that \(S\) and \(T\) are sets. A relation from \(S\) to \(T\) is a subset of the product set \(S \times T\).
- The domain of \(R\) is the set of first coordinates: \(\text{domain}(R) = \left\{x \in S: (x, y) \in R \text{ for some } y \in T\right\}\).
- The range of \(R\) is the set of second coordinates: \(\text{range}(R) = \left\{y \in T: (x, y) \in R \text{ for some } x \in S\right\}\).
A relation from a set \(S\) to itself is a relation on \(S\).
As the name suggests, a relation \(R\) from \(S\) into \(T\) is supposed to define a relationship between the elements of \(S\) and the elements of \(T\), and so we usually use the more suggestive notation \(x\,R\,y\) when \((x, y) \in R\). Note that the domain of \(R\) is the projections of \( R \) onto \( S \) and the range of \( R \) is the projection of \( R \) onto \( T \).
Basic Examples
Suppose that \(S\) is a set and recall that \(\mathscr{P}(S)\) denotes the power set of \(S\), the collection of all subsets of \(S\). The membership relation \(\in\) from \(S\) to \(\mathscr{P}(S)\) is perhaps the most important and basic relationship in mathematics. Indeed, for us, it's a primitive (undefined) relationship—given \(x\) and \(A\), we assume that we understand the meaning of the statement \(x \in A\).
Another basic primitive relation is the equality relation \(=\) on a given set of objects \(S\). That is, given two objects \(x\) and \(y\), we assume that we understand the meaning of the statement \(x = y\).
Other basic relations that you have seen are
- The subset relation \(\subseteq\) on \(\mathscr{P}(S)\).
- The order relation \(\le\) on \(\R\)
These two belong to a special class of relations known as partial orders that we will study in the next section. Note that a function \(f\) from \(S\) into \(T\) is a special type of relation. To compare the two types of notation (relation and function), note that \(x\,f\,y\) means that \(y = f(x)\).
Constructions
Since a relation is just a set of ordered pairs, the set operations can be used to build new relations from existing ones.
if \(Q\) and \(R\) are relations from \(S\) to \(T\), then so are \(Q \cup R\), \(Q \cap R\), \(Q \setminus R\).
- \(x(Q \cup R)y\) if and only if \(x\,Q\,y\) or \(x\,R\,y\).
- \(x(Q \cap R)y\) if and only if \(x\,Q\,y\) and \(x\,R\,y\).
- \(x(Q \setminus R)y\) if and only if \(x\,Q\,y\) but not \(x\,R\,y\).
- If \(Q \subseteq R\) then \(x\,Q\,y\) implies \(x\,R\,y\).
If \(R\) is a relation from \(S\) to \(T\) and \(Q \subseteq R\), then \(Q\) is a relation from \(S\) to \(T\).
The restriction of a relation defines a new relation.
If \(R\) is a relation on \(S\) and \(A \subseteq S\) then \(R_A = R \cap (A \times A)\) is a relation on \(A\), called the restriction of \(R\) to \(A\).
The inverse of a relation also defines a new relation.
If \(R\) is a relation from \(S\) to \(T\), the inverse of \(R\) is the relation from \(T\) to \(S\) defined by \[ y\,R^{-1}\,x \text{ if and only if } x\,R\,y \]
Equivalently, \(R^{-1} = \{(y, x): (x, y) \in R\}\). Note that any function \(f\) from \(S\) into \(T\) has an inverse relation , but only when the \(f\) is one-to-one is the inverse relation also a function (the inverse function). Composition is another natural way to create new relations from existing ones.
Suppose that \(Q\) is a relation from \(S\) to \(T\) and that \(R\) is a relation from \(T\) to \(U\). The composition \(Q \circ R\) is the relation from \(S\) to \(U\) defined as follows: for \(x \in S\) and \(z \in U\), \(x(Q \circ R)z\) if and only if there exists \(y \in T\) such that \(x\,Q\,y\) and \(y\,R\,z\).
Note that the notation is inconsistent with the notation used for composition of functions, essentially because relations are read from left to right, while functions are read from right to left. Hopefully, the inconsistency will not cause confusion, since we will always use function notation for functions.
Basic Properties
The important classes of relations that we will study in the next couple of sections are characterized by certain basic properties. Here are the definitions:
Suppose that \(R\) is a relation on \(S\).
- \(R\) is reflexive if \(x\,R\,x\) for all \(x \in S\).
- \(R\) is irreflexive if no \(x \in S\) satisfies \(x\,R\,x\).
- \(R\) is symmetric if \(x\,R\,y\) implies \(y\,R\,x\) for all \(x, \, y \in S\).
- \(R\) is anti-symmetric if \(x\,R\,y\) and \(y\,R\,x\) implies \(x = y\) for all \(x, \, y \in S\).
- \(R\) is transitive if \(x\,R\,y\) and \(y\,R\,z\) implies \(x\,R\,z\) for all \(x, \, y, \, z \in S\).
The proofs of the following results are straightforward, so be sure to try them yourself before reading the ones in the text.
A relation \(R\) on \(S\) is reflexive if and only if the equality relation \(=\) on \(S\) is a subset of \(R\).
Proof
This follows from the definitions. \( R \) is reflexive if and only if \( (x, x) \in R \) for all \( x \in S \).
A relation \(R\) on \(S\) is symmetric if and only if \(R^{-1} = R\).
Proof
Suppose that \( R \) is symmetric. If \( (x, y) \in R \) then \( (y, x) \in R \) and hence \( (x, y) \in R^{-1} \). If \( (x, y) \in R^{-1} \) then \( (y, x) \in R \) and hence \( (x, y) \in R \). Thus \( R = R^{-1} \). Conversely, suppose \( R = R^{-1} \). If \( (x, y) \in R \) then \( (x, y) \in R^{-1} \) and hence \( (y, x) \in R \).
A relation \(R\) on \(S\) is transitive if and only if \(R \circ R \subseteq R\).
Proof
Suppose that \( R \) is transitive. If \( (x, z) \in R \circ R \) then there exists \( y \in S \) such that \( (x, y) \in R \) and \( (y, z) \in R \). But then \( (x, z) \in R \) by transitivity. Hence \( R \circ R \subseteq R \). Conversely, suppose that \( R \circ R \subseteq R \).  Hence \( R \) is transitive.
A relation \(R\) on \(S\) is antisymmetric if and only if \(R \cap R^{-1}\) is a subset of the equality relation \(=\) on \(S\).
Proof
Restated, this result is that \( R \) is antisymmetric if and only if \( (x, y) \in R \cap R^{-1} \) implies \( x = y \). Thus suppose that \( R \) is antisymmetric. If \( (x, y) \in R \cap R^{-1} \) then \( (x, y) \in R \) and \( (x, y) \in R^{-1} \). But then \( (y, x) \in R \) so by antisymmetry, \( x = y \). Conversely suppose that \( (x, y) \in R \cap R^{-1} \) implies \( x = y \). If \( (x, y) \in R \) and \( (y, x) \in R \) then \( (x, y) \in R^{-1} \) and hence \( (x, y) \in R \cap R^{-1} \). Thus \( x = y \) so \( R \) is antisymmetric.
Suppose that \(Q\) and \(R\) are relations on \(S\). For each property below, if both \(Q\) and \(R\) have the property, then so does \(Q \cap R\).
- reflexive
- symmetric
- transitive
Proof
- Suppose that \( Q \) and \( R \) are reflexive. Then \( (x, x) \in Q \) and \( (x, x) \in R \) for each \( x \in S \) and hence \( (x, x) \in Q \cap R \) for each \( x \in S \). Thus \( Q \cap R \) is reflexive.
- Suppose that \( Q \) and \( R \) are symmetric. If \( (x, y) \in Q \cap R \) then \( (x, y) \in Q \) and \( (x, y) \in R \). Hence \( (y, x) \in Q \) and \( (y, x) \in R \) so \( (y, x) \in Q \cap R \). Hence \( Q \cap R \) is symmetric.
- Suppose that \( Q \) and \( R \) are transitive.  Hence \( (x, z) \in Q \) and \( (x, z) \in R \) so \( (x, z) \in Q \cap R \). Hence \( Q \cap R \) is transitive.
Suppose that \(R\) is a relation on a set \(S\).
- Give an explicit definition for the property \(R\) is not reflexive .
- Give an explicit definition for the property \(R\) is not irreflexive .
- Are any of the properties \(R\) is reflexive , \(R\) is not reflexive , \(R\) is irreflexive , \(R\) is not irreflexive equivalent?
Answer
- \( R \) is not reflexive if and only if there exists \( x \in S \) such that \( (x, x) \notin R \).
- \( R \) is not irreflexive if and only if there exists \( x \in S \) such that \( (x, x) \in R \).
- Nope.
Suppose that \(R\) is a relation on a set \(S\).
- Give an explicit definition for the property \(R\) is not symmetric .
- Give an explicit definition for the property \(R\) is not antisymmetric .
- Are any of the properties \(R\) is symmetric , \(R\) is not symmetric , \(R\) is antisymmetric , \(R\) is not antisymmetric equivalent?
Answer
- \( R \) is not symmetric if and only if there exist \( x, \, y \in S \) such that \( (x, y) \in R \) and \( (y, x) \notin R \).
- \( R \) is not antisymmetric if and only if there exist distinct \( x, \, y \in S \) such that \( (x, y) \in R \) and \( (y, x) \in R \).
- Nope.
Computational Exercises
Let \(R\) be the relation defined on \(\R\) by \(x\,R\,y\) if and only if \(\sin(x) = \sin(y)\). Determine if \(R\) has each of the following properties:
- reflexive
- symmetric
- transitive
- irreflexive
- antisymmetric
Answer
- yes
- yes
- yes
- no
- no
The relation \(R\) in the previous exercise is a member of an important class of equivalence relations.
Let \(R\) be the relation defined on \(\R\) by \(x\,R\,y\) if and only if \(x^2 + y^2 \le 1\). Determine if \(R\) has each of the following properties:
- reflexive
- symmetric
- transitive
- irreflexive
- antisymmetric
Answer
- no
- yes
- no
- no
- no


5.6: Telescope Mounts
Telescope Mount is the telescope’s support structure. The two main types of telescope mounts are alt-azimuth and equatorial mounts. The Alt-azimuth mounts move in two directions: left to right, or up and down. The word “alt-azimuth” is a combination of altitude and azimuth. Equatorial mounts track the apparent motion of the stars by aligning one of its axes parallel to Earth’s axis.
Go-To Telescope Mounts
Go-To’s telescope mounts enable the telescope to find objects in the sky using a microprocessor, encoders and motors. A Global Positioning System (GPS) as a part of the Go-To system tells the mount and microprocessor where the telescope is located, what objects are visible, and where they are in the sky.


24.3B: Glomerular Filtration
Glomerular filtration is the renal process whereby fluid in the blood is filtered across the capillaries of the glomerulus.
- Explain the process of glomerular filtration in the kidneys
Key Points
- The formation of urine begins with the process of filtration. Fluid and small solutes are forced under pressure to flow from the glomerulus into the capsular space of the glomerular capsule.
- The Bowman’s capsule is the filtration unit of the glomerulus and has tiny slits in which filtrate may pass through into the nephron. Blood entering the glomerulus has filterable and non-filterable components.
- Filterable blood components include water, nitrogenous waste, and nutrients that will be transferred into the glomerulus to form the glomerular filtrate.
- Non-filterable blood components include blood cells, albumins, and platelets, that will leave the glomerulus through the efferent arteriole.
- Glomerular filtration is caused by the force of the difference between hydrostatic and osmotic pressure (though the glomerular filtration rate includes other variables as well).
Key Terms
- glomerulus : A small, intertwined group of capillaries within nephrons of the kidney that filter the blood to make urine.
- hydrostatic pressure : The pushing force exerted by the pressure in a blood vessel. It is the primary force that drives glomerular filtration.
Glomerular filtration is the first step in urine formation and constitutes the basic physiologic function of the kidneys. It describes the process of blood filtration in the kidney, in which fluid, ions, glucose, and waste products are removed from the glomerular capillaries.
Many of these materials are reabsorbed by the body as the fluid travels through the various parts of the nephron, but those that are not reabsorbed leave the body in the form of urine.
Glomerulus Structure
Glomerulus structure : A diagram showing the afferent and efferent arterioles bringing blood in and out of the Bowman’s capsule, a cup-like sac at the beginning of the tubular component of a nephron.
Blood plasma enters the afferent arteriole and flows into the glomerulus, a cluster of intertwined capillaries. The Bowman’s capsule (also called the glomerular capsule) surrounds the glomerulus and is composed of visceral (simple squamous epithelial cells—inner) and parietal (simple squamous epithelial cells—outer) layers.
The visceral layer lies just beneath the thickened glomerular basement membrane and is made of podocytes that form small slits in which the fluid passes through into the nephron. The size of the filtration slits restricts the passage of large molecules (such as albumin) and cells (such as red blood cells and platelets) that are the non-filterable components of blood.
These then leave the glomerulus through the efferent arteriole, which becomes capillaries meant for kidney–oxygen exchange and reabsorption before becoming venous circulation. The positively charged podocytes will impede the filtration of negatively charged particles as well (such as albumins).
The Mechanisms of Filtration
The process by which glomerular filtration occurs is called renal ultrafiltration. The force of hydrostatic pressure in the glomerulus (the force of pressure exerted from the pressure of the blood vessel itself) is the driving force that pushes filtrate out of the capillaries and into the slits in the nephron.
Osmotic pressure (the pulling force exerted by the albumins) works against the greater force of hydrostatic pressure, and the difference between the two determines the effective pressure of the glomerulus that determines the force by which molecules are filtered. These factors will influence the glomeruluar filtration rate, along with a few other factors.


7.4: Tobacco Use
Tobacco use is the leading preventable cause of disease, disability, and death in the United States. According to the Centers for Disease Control and Prevention (CDC), cigarette smoking results in more than 480,000 premature deaths in the United States each year—about 1 in every 5 U.S. deaths—and an additional 16 million people suffer with a serious illness caused by smoking. In fact, for every one person who dies from smoking, about 30 more suffer from at least one serious tobacco-related illness.
The harmful effects of smoking extend far beyond the smoker. Exposure to secondhand smoke can cause serious diseases and death. Each year, an estimated 88 million nonsmoking Americans are regularly exposed to secondhand smoke and almost 41,000 nonsmokers die from diseases caused by secondhand smoke exposure.
How Does Tobacco Affect the Brain?
Cigarettes and other forms of tobacco—including cigars, pipe tobacco, snuff, and chewing tobacco—contain the addictive drug nicotine. Nicotine is readily absorbed into the bloodstream when a tobacco product is chewed, inhaled, or smoked. A typical smoker will take 10 puffs on a cigarette over the period of about 5 minutes that the cigarette is lit. Thus, a person who smokes about 1 pack (25 cigarettes) daily gets 250 “hits” of nicotine each day.
Upon entering the bloodstream, nicotine immediately stimulates the adrenal glands to release the hormone epinephrine (adrenaline). Epinephrine stimulates the central nervous system and increases blood pressure, respiration, and heart rate.
Similar to other addictive drugs like cocaine and heroin, nicotine increases levels of the neurotransmitter dopamine, which affects the brain pathways that control reward and pleasure. For many tobacco users, long-term brain changes induced by continued nicotine exposure result in addiction—a condition of compulsive drug seeking and use, even in the face of negative consequences. Studies suggest that additional compounds in tobacco smoke, such as acetaldehyde, may enhance nicotine’s effects on the brain.
When an addicted user tries to quit, he or she experiences withdrawal symptoms including irritability, attention difficulties, sleep disturbances, increased appetite, and powerful cravings for tobacco. Treatments can help smokers manage these symptoms and improve the likelihood of successfully quitting.
Electronic Cigarettes
What are they?
E-cigarettes are battery-operated devices that typically produce a flavored nicotine vapor that looks like tobacco smoke.
Are they safe?
Although e-cigarette vapor does not contain the tar currently responsible for most lung cancer and other lung diseases, it has been shown to contain known carcinogens and toxic chemicals (such as formaldehyde and acetaldehyde), as well as potentially toxic metal nanoparticles from the vaporizing mechanism. There are currently no accepted measures to confirm their purity or safety, and the long-term health consequence of e-cigarette use remain unknown. NIDA is developing research programs to help answer these questions.
In addition, the U.S. Food and Drug Administration (FDA) has established a new rule for e-cigarettes and their liquid solutions in an effort to help protect the public from the dangers of tobacco use. Because e-cigarettes contain nicotine derived from tobacco, they are now subject to government regulation as tobacco products, including the requirement that both in-store and online purchasers be at least 18 years of age (see “ Government Regulation of Tobacco Extended to All Tobacco Products “).
Can they help people quit smoking traditional cigarettes?
Because they deliver nicotine without burning tobacco, e-cigarettes are thought by many to be a safer alternative to conventional cigarettes, and some people even think they may help smokers lower nicotine cravings while they are trying to quit smoking. However, studies of the effectiveness of e-cigarettes have not shown they help with smoking cessation. It has also been suggested that they could perpetuate the nicotine addiction and actually interfere with quitting.
In fact, early evidence suggests that e-cigarette use may not only put users at risk for nicotine addiction but also serve as an introduction to nicotine that could lead to use of regular cigarettes and other tobacco products. A recent study showed that students who have used e-cigarettes by the time they start 9th grade are more likely than others to start smoking traditional cigarettes and other smokable tobacco products within the next year.
What Other Adverse Effects Does Tobacco Have on Health?
Cigarette smoking accounts for about one-third of all cancers, including 85-90 percent of lung cancer cases.  Smokeless tobacco (such as chewing tobacco and snuff) also increases the risk of cancer, especially oral cancers. In addition to cancer, smoking causes lung diseases such as chronic bronchitis and emphysema, and increases the risk of heart disease, including stroke, heart attack, vascular disease, and aneurysm. Smoking has also been linked to leukemia, cataracts, and pneumonia.
Although nicotine is addictive and can be toxic if ingested in high doses, it does not cause cancer—other chemicals are responsible for most of the severe health consequences of tobacco use. Tobacco smoke is a complex mixture of chemicals such as carbon monoxide, tar, formaldehyde, cyanide, and ammonia—many of which are known carcinogens. Carbon monoxide increases the chance of cardiovascular diseases. Tar exposes the user to an increased risk of lung cancer, emphysema, and bronchial disorders.
Pregnant women who smoke cigarettes run an increased risk of miscarriage, stillborn or premature infants, or infants with low birthweight. Maternal smoking may also be associated with learning and behavioral problems in children. Smoking more than one pack of cigarettes per day during pregnancy nearly doubles the risk that the affected child will become addicted to tobacco if that child starts smoking.
While we often think of medical consequences that result from direct use of tobacco products, passive or secondary smoke also increases the risk for many diseases. Secondhand smoke, also known as environmental tobacco smoke, consists of exhaled smoke and smoke given off by the burning end of tobacco products.
Nonsmokers exposed to secondhand smoke at home or work increase their risk of developing heart disease by 25–30 percent and lung cancer by 20–30 percent.
In addition; secondhand smoke causes health problems in both adults and children, such as coughing, overproduction of phlegm, reduced lung function and respiratory infections, including pneumonia and bronchitis. Each year about 150,000 – 300,000 children younger than 18 months old experience respiratory tract infections caused by secondhand smoke. Children exposed to secondhand smoke are at an increased risk of ear infections, severe asthma, respiratory infections and death. In fact, more than 100,000 babies have died in the past 50 years from sudden infant death syndrome (SIDS), and other health complications as a result of parental smoking. Children who grow up with parents who smoke are more likely to become smokers, thus placing themselves (and their future families) at risk for the same health problems as their parents when they become adults.
There is no safe level of exposure to tobacco smoke.
Contributors and Attributions
Public Domain Content
- Drug Facts - Cigarettes and Other Tobacco Products. Authored by: National Institute on Drug Abuse. Provided by: National Institutes of Health. Located at: https://www.drugabuse.gov/publications/drugfacts/cigarettes-other-tobacco-products#reg . License: Public Domain: No Known Copyright


8.2: Thermodynamics of Solutions
Make sure you thoroughly understand the following essential ideas:
- Describe the two fundamental processes that must occur whenever a solute dissolves in a solvent, and discuss the effects of the absorption or release of energy on the extent of these processes.
- Another factor entering into the process of solution formation is the increase (or occasionally, the decrease) in the entropy — that is, the degree to which thermal energy is dispersed or "diluted". Explain this in your own terms.
- Explain how the adage "like dissolves like" reflects the effects mentioned above. What is the principal physical property of a molecule that defines this "likeness"?
- What do we mean when we describe a liquid such as water as "associated"? Explain how this relates to the the solubility of solutes in such liquids.
You may recall that in the earlier unit on phase equilibria, we pointed out that aggregations of molecules that are more disordered tend to be the ones that are favored at higher temperature, whereas those that possess the lowest potential energy are favored at lower temperatures. This is a general principle that applies throughout the world of matter; the stable form at any given temperature will always be that which leads to the best balance between low potential energy and high molecular disorder. To see how these considerations are applied to solutions, think about the individual steps that must be carried out when a solute is dissolved in a solvent:
- If the solute is a solid or liquid, it must first be dispersed — that is, its molecular units must be pulled apart. This requires energy, and so this step always works against solution formation.
- The solute must then be introduced into the solvent. Whether this is energetically favorable or unfavorable depends on the nature of the solute and solvent. If the solute is A and the solvent is B, then what is important is the strength of the attractive forces between A-A and B-B molecules, compared to those between A-B pairs; if the latter are greater, then the potential energy will be lower when the substances are mixed and solution formation will be favored.
If step 2 releases more energy than is consumed in step 1 , this will favor solution formation, and we can generally expect the solute to be soluble in the solvent. Even if the dissolution process is slightly endothermic, there is a third important factor, the entropy increase, that will very often favor the dissolved state.
Entropy of Solution
As anyone who has shuffled a deck of cards knows, disordered arrangements of objects are statistically more favored simply because there are more ways in which they can be realized. And as the number of objects increases, the more does statistics govern their most likely arrangements. The numbers of objects (molecules) we deal with in Chemistry is so huge that their tendency to become as spread out as possible becomes overwhelming. However, in doing so, the thermal energy they carry with them is also spread and dispersed, so the availability of this energy, as measured by the temperature, is also of importance. Chemists use the term "entropy" to denote this aspect of molecular randomness.
Readers of this section who have had some exposure to thermodynamics will know that solubility, like all equilibria, is governed by the Gibbs free energy change for the process, which incorporates the entropy change at a fundamental level. A proper understanding of these considerations requires some familiarity with thermodynamics, which most students do not encounter until well into their second semester of Chemistry. If you are not there yet, do not despair; you are hereby granted temporary permission to think of molecular "disorder" and entropy simply in terms of "spread-outedness".
Thus in the very common case in which a small quantity of solid or liquid dissolves in a much larger volume of solvent, the solute becomes more spread out in space, and the number of equivalent ways in which the solute can be distributed within this volume is greatly increased. This is the same as saying that the entropy of the solute increases.
If the energetics of dissolution are favorable, this increase in entropy means that the conditions for solubility will always be met. Even if the energetics are slightly endothermic, the entropy effect can still allow the solution to form, although perhaps limiting the maximum concentration that can be achieved. In such a case, we may describe the solute as being slightly soluble in a certain solvent. What this means is that a greater volume of solvent will be required to completely dissolve a given mass of solute.
Enthalpy of Solution
Polar molecules are those in which electric charge is distributed asymmetrically. The most familiar example is ordinary water, in which the highly electronegative oxygen atom pulls part of the electric charge cloud associated with each O–H bond closer to itself. Although the H 2 O molecule is electrically neutral overall, this charge imbalance gives rise to a permanent electric dipole moment .
Chemists use the term "Associated" liquids to refer to liquids in which the effects of hydrogen bonding dominate the local structure. Water is the most important of these, but ammonia NH 3 and hydrogen cyanide HCN are other common examples.
Thus liquid water consists of an extended network of H 2 O molecules linked together by dipole-dipole attractions that we call hydrogen bonds . Because these are much weaker than ordinary chemical bonds, they are continually being disrupted by thermal forces. As a result, the extended structure is highly disordered (in contrast to that of solid ice) and continually changing.
When a solute molecule is introduced into an associated liquid, a certain amount of energy must be expended in order to break the local hydrogen-bond structure and make space for the new molecule. If the solute is itself an ion or a polar molecule, new ion-dipole or dipole-dipole attractions come into play. In favorable cases these may release sufficient potential energy to largely compensate for the energy required to incorporate the solute into the structure.
An extreme example of this occurs when ammonia dissolves in water. Each NH 3 molecule can form three hydrogen bonds, so the resulting solution is even more hydrogen-bonded than is pure water — accounting for the considerable amount of heat released in the process and the extraordinarily large solubility of ammonia in water.
Nonpolar solutes are Sparingly Soluble in Water: The Hydrophobic effect
When a nonpolar solute such as oxygen or hexane is introduced into an associated liquid, we might expect that the energy required to break the hydrogen bonds to make space for the new molecule is not compensated by the formation of new attractive interactions, suggesting that the process will be energetically unfavorable. We can therefore predict that solutes of these kinds will be only sparingly soluble in water, and this is indeed the case.
It turns out, however, that this is not an entirely correct explanation for the small solubility of non polar solutes in water. It is now known that the H 2 O molecules that surround a non-polar intruder and find themselves unable to form energy-lowering polar or hydrogen-bonded interactions with it will rearrange themselves into a configuration that maximizes the hydrogen bonding between the water molecules themselves. In doing so, this creates a cage-like shell around the solute molecule. In terms of the energetics of the process, these new H 2 O-H 2 O interactions largely compensate for the lack of solute-H 2 O interactions.
However, this shell of highly organized water molecules exacts its own toll on the solubility by reducing the entropy of the system. Dissolution of a solute normally increases the entropy by spreading the solute molecules (and the thermal energy they contain) through the larger volume of the solvent. But in this case, the H 2 O molecules within the highly structured shell surrounding the solute molecule are themselves constrained to this location, and their number is sufficiently great to reduce the entropy by far more than the dissolved solute increases it.
The implications of the hydrophobic effect extend far beyond the topic of solubility. It governs the way that proteins fold, the formation of soap bubbles, and the formation of cell membranes. The small solubility of a non polar solute in an associated liquid such as water results more from the negative entropy change rather than from energetic considerations. This phenomenon is known as the hydrophobic effect . In the next section, we will explore the ways in which these energy-and-entropy considerations come together in various kinds of solutions.


1.4: Goals for Infrastructure Management
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- Donald Coffelt and Chris Hendrickson
- Carnegie Mellon University via Donald Coffelt and Chris Hendrickson
Just as the number and extent of infrastructure systems are complex, so are the goals that are pursued for any particular infrastructure system. One common goal suggested is to ensure ‘sustainable’ infrastructure. One interpretation of sustainability is simply to have facilities with great longevity. However, this is often not a realistic goal. First, managers must be sensitive to the amount of resources required to construct and maintain any particular facility. Longevity requires greater capital investment for initial construction. Second, the requirements for facilities are likely to change over time. For example, the legal size and weight of trucks can change over time (usually with an increase), which may make existing bridges functionally obsolete since they cannot support larger trucks. Third, the usage of facilities may decline to such an extent that maintaining an existing facility is not beneficial.
For most infrastructure systems, managers adopt a planning horizon for longevity decision making. Such planning horizons can vary from a short period (such as a year or two) to decades (for infrastructures such as ports or buildings). Each organization involved in infrastructure management may have its own planning horizon for such decision making.
In practice, the goals for infrastructure management are complex and multiple. Most critically, facilities are expected to provide acceptable performance to a variety of users. For example, a local roadway might accommodate a variety of motorized vehicles (such as buses, cars, and trucks - moving or parked), bicycles and pedestrians (particularly at intersections). Deterioration of the facility can affect acceptable performance, as with the development of potholes, uneven surfaces and cracking for pavement. Extreme events such as earthquakes, hurricanes, flooding or terrorist activities can require immediate attention and response.
Typically, goals for infrastructure management can be categorized as economic, environmental and social. Economic impacts include the direct and indirect costs of managing and operating the infrastructure system, the economic development potential for the system (including employment) and any user or non-user benefits stemming from the system. Environmental impacts are associated with ecological system uses, emissions to the environment (especially toxic chemicals and greenhouse gas emissions), and non-renewable resource use. Social impacts pertain to equity of benefits, social justice and individual development (including employment). This ‘Triple Bottom Line’ of goals is common for many social investments.


2.R: Linear Functions(Review)
2.1 Linear Functions
1) Determine whether the algebraic equation is linear: \(2 x+3 y=7\)
- Answer
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Yes
2) Determine whether the algebraic equation is linear: \(6 x^{2}-y=5\)
3) Determine whether the function is increasing or decreasing: \(f(x)=7 x-2\)
- Answer
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Increasing
4) Determine whether the function is increasing or decreasing: \(g(x)=-x+2\)
5) Given each set of information, find a linear equation that satisfies the given conditions, if possible. Passes through \((7,5)\) and \((3,17)\)
- Answer
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\(y=-3 x+26\)
6) Given each set of information, find a linear equation that satisfies the given conditions, if possible. \(x\) -intercept at \((6,0)\) and \(y\) -intercept at \((0,10)\)
7) Find the slope of the line shown in the line graph.
- Answer
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\(3\)
8) Find the slope of the line graphed.
9) Write an equation in slope-intercept form for the line shown.
- Answer
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\(y=2 x-2\)
10) Does the following table represent a linear function? If so, find the linear equation that models the data.
| \(x\) | –4 | 0 | 2 | 10 |
|---|---|---|---|---|
| \(g(x)\) | 18 | –2 | –12 | –52 |
11) Does the following table represent a linear function? If so, find the linear equation that models the data.
| \(x\) | 6 | 8 | 12 | 26 |
|---|---|---|---|---|
| \(g(x)\) | –8 | –12 | –18 | –46 |
- Answer
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Not linear
12) On June 1st , a company has \(\$4,000,000\) profit. If the company then loses \(150,000\) dollars per day thereafter in the month of June, what is the company’s profit n th day after June 1st ?
2.2 Graphs of Linear Functions
For the exercises 1-2, determine whether the lines given by the equations below are parallel, perpendicular, or neither parallel nor perpendicular:
1) \(\begin{align*} 2x-6y &=12 \\ -x+3y &=1 \end{align*} \nonumber\)
- Answer
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parallel
2) \(\begin{align*} y &=\dfrac{1}{3} x-2 \\ 3x+y &=-9 \end{align*} \nonumber \)
For the exercises 3-4, find the \(x\)- and \(y\)- intercepts of the given equation
3) \(7 x+9 y=-63\)
- Answer
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\((-9,0); (0,-7)\)
4) \(f(x)=2 x-1\)
For the exercises 5-11, use the descriptions of the pairs of lines to find the slopes of Line 1 and Line 2. Is each pair of lines parallel, perpendicular, or neither?
5)
Line 1: Passes through \((5,11)\) and \((10,1)\)
Line \(2 :\) Passes through \((-1,3)\) and \((-5,11)\)
- Answer
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Line 1 : \(m=-2\); Line 2 : \(m=-2\); Parallel
6)
Line 1: Passes through \((8,-10)\) and \((0,-26)\)
Line 2: Passes through \((2,5)\) and \((4,4)\)
7) Write an equation for a line perpendicular to \(f(x)=5 x-1\) and passing through the point \((5,20)\).
- Answer
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\(y=-0.2 x+21\)
8) Find the equation of a line with a \(y\) -intercept of \((0,2)\) and slope \(-\dfrac{1}{2}\).
9) Sketch a graph of the linear function \(f(t)=2t-5\).
- Answer
-
10) Find the point of intersection for the \(2\) linear functions: \(\begin{align*} x &=y+6 \\ 2x-y &=13 \end{align*}\)
11) A car rental company offers two plans for renting a car.
Plan A: \(25\) dollars per day and \(10\) cents per mile
Plan B: \(50\) dollars per day with free unlimited mileage
How many miles would you need to drive for plan B to save you money?
- Answer
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\(250\)
2.3 Modeling with Linear Functions
1) Find the area of a triangle bounded by the \(y\) axis, the line \(f(x)=10-2x\), and the line perpendicular to \(f\) that passes through the origin.
2) A town’s population increases at a constant rate. In 2010 the population was \(55,000\). By 2012 the population had increased to \(76,000\). If this trend continues, predict the population in 2016.
- Answer
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\(118,000\)
3) The number of people afflicted with the common cold in the winter months dropped steadily by \(50\) each year since 2004 until 2010. In 2004, \(875\) people were inflicted.
4) Find the linear function that models the number of people afflicted with the common cold \(C\) as a function of the year, \(t\). When will no one be afflicted?
For the exercises 5-6, use the graph in the Figure below showing the profit, \(y\), in thousands of dollars, of a company in a given year, \(x\), where \(x\) represents years since 1980.
5) Find the linear function \(y\), where \(y\) depends on \(x\), the number of years since 1980.
- Answer
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\(y=-300 x+11,500\)
6) Find and interpret the \(y\)-intercept.
7) For the following exercise, consider this scenario: In 2004, a school population was \(1,700\). By 2012 the population had grown to \(2,500\). Assume the population is changing linearly.
- How much did the population grow between the year 2004 and 2012?
- What is the average population growth per year?
- Find an equation for the population, \(P\), of the school \(t\) years after 2004.
- Answer
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- \(800\)
- \(100\) students per year
- \(P(t)=100 t+1700\)
For the exercises 8-9, consider this scenario: In 2000, the moose population in a park was measured to be \(6,500\). By 2010, the population was measured to be \(12,500\). Assume the population continues to change linearly.
8) Find a formula for the moose population, \(P\).
9) What does your model predict the moose population to be in 2020?
- Answer
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\(18,500\)
For the exercises 10-, consider this scenario: The median home values in subdivisions Pima Central and East Valley (adjusted for inflation) are shown in the Table below. Assume that the house values are changing linearly.
| Year | Pima Central | East Valley |
|---|---|---|
| 1970 | 32,000 | 120,250 |
| 2010 | 85,000 | 150,000 |
10) In which subdivision have home values increased at a higher rate?
11) If these trends were to continue, what would be the median home value in Pima Central in 2015?
- Answer
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\(\$91,625\)
2.4 Fitting Linear Models to Data
1) Draw a scatter plot for the data in the Table below. Then determine whether the data appears to be linearly related.
| 0 | 2 | 4 | 6 | 8 | 10 |
| –105 | –50 | 1 | 55 | 105 | 160 |
2) Draw a scatter plot for the data in the Table. If we wanted to know when the population would reach \(15,000\), would the answer involve interpolation or extrapolation?
| Year | Population |
|---|---|
| 1990 | 5,600 |
| 1995 | 5,950 |
| 2000 | 6,300 |
| 2005 | 6,600 |
| 2010 | 6,900 |
- Answer
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Extrapolation
3) Eight students were asked to estimate their score on a \(10\)-point quiz. Their estimated and actual scores are given in the Table. Plot the points, then sketch a line that fits the data.
| Predicted | Actual |
|---|---|
| 6 | 6 |
| 7 | 7 |
| 7 | 8 |
| 8 | 8 |
| 7 | 9 |
| 9 | 10 |
| 10 | 10 |
| 10 | 9 |
4) Draw a best-fit line for the plotted data.
- Answer
-
For the exercises 5-6, consider the data in the Table below, which shows the percent of unemployed in a city of people \(25\) years or older who are college graduates is given below, by year.
| Year | Percent Graduates |
|---|---|
| 2000 | 6.5 |
| 2002 | 7.0 |
| 2005 | 7.4 |
| 2007 | 8.2 |
| 2010 | 9.0 |
5) Determine whether the trend appears to be linear. If so, and assuming the trend continues, find a linear regression model to predict the percent of unemployed in a given year to three decimal places.
6) In what year will the percentage exceed \(12\%\)?
- Answer
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Midway through 2024
7) Based on the set of data given in the Table, calculate the regression line using a calculator or other technology tool, and determine the correlation coefficient to three decimal places.
| \(x\) | 17 | 20 | 23 | 26 | 29 |
|---|---|---|---|---|---|
| \(y\) | 15 | 25 | 31 | 37 | 40 |
8) Based on the set of data given in the Table, calculate the regression line using a calculator or other technology tool, and determine the correlation coefficient to three decimal places.
| \(x\) | 10 | 12 | 15 | 18 | 20 |
|---|---|---|---|---|---|
| \(y\) | 36 | 34 | 30 | 28 | 22 |
- Answer
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\(y=-1.294 x+49.412; r=-0.974\)
For the exercises 9-12, consider this scenario: The population of a city increased steadily over a ten-year span. The following ordered pairs show the population and the year over the ten-year span (population, year) for specific recorded years: \((3,600,2000) ;(4,000,2001) ;(4,700,2003) ;(6,000,2006)\)
9) Use linear regression to determine a function \(y\), where the year depends on the population, to three decimal places of accuracy.
10) Predict when the population will hit \(12,000\).
- Answer
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Early in 2022
11) What is the correlation coefficient for this model to three decimal places of accuracy?
12) According to the model, what is the population in 2014?
- Answer
-
\(7,660\)
Practice Test
1) Determine whether the following algebraic equation can be written as a linear function: \(2 x+3 y=7\)
- Answer
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Yes
2) Determine whether the following function is increasing or decreasing: \(f(x)=-2 x+5\)
3) Determine whether the following function is increasing or decreasing: \(f(x)=7 x+9\)
- Answer
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Increasing
4) Given the following set of information, find a linear equation satisfying the conditions, if possible. Passes through \((5,1)\) and \((3,-9)\)
5) Given the following set of information, find a linear equation satisfying the conditions, if possible. \(x\) intercept at \((-4,0)\) and \(y\) -intercept at \((0,-6)\)
- Answer
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\(y=-1.5 x-6\)
6) Find the slope of the line in the Figure below.
- Answer
-
\(y=-2 x-1\)
8) Does the Table represent a linear function? If so, find a linear equation that models the data.
| \(x\) | –6 | 0 | 2 | 4 |
| \(g(x)\) | 14 | 32 | 38 | 44 |
9) Does the Table represent a linear function? If so, find a linear equation that models the data.
| \(x\) | 1 | 3 | 7 | 11 |
| \(g(x)\) | 4 | 9 | 19 | 12 |
- Answer
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No
10) At 6 AM, an online company has sold \(120\) items that day. If the company sells an average of \(30\) items per hour for the remainder of the day, write an expression to represent the number of items that were sold \(n\) after 6 AM.
For the exercises 11-22, determine whether the lines given by the equations below are parallel, perpendicular, or neither parallel nor perpendicular:
11) \(\begin{align*}{y=\dfrac{3}{4} x-9} \\ {-4 x-3 y=8}\end{align*}\)
- Answer
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Perpendicular
12) \(\begin{align*}{-2 x+y=3} \\ {3 x+\dfrac{3}{2} y=5}\end{align*}\)
13) Find the \(x\) - and \(y\) -intercepts of the equation \(2 x+7 y=-14\).
- Answer
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\((-7,0) ;(0,-2)\)
14) Given below are descriptions of two lines. Find the slopes of Line 1 and Line 2. Is the pair of lines parallel, perpendicular, or neither?
Line \(1 :\) Passes through \((-2,-6)\) and \((3,14)\)
Line \(2 :\) Passes through \((2,6)\) and \((4,14)\)
15) Write an equation for a line perpendicular to \(f(x)=4 x+3\) and passing through the point \((8,10)\).
- Answer
-
\(y=-0.25 x+12\)
16) Sketch a line with a \(y\)-intercept of \((0,5)\) and slope \(-\dfrac{5}{2}\).
17) Graph of the linear function \(f(x)=-x+6\).
- Answer
-
18) For the two linear functions, find the point of intersection: \(\begin{align*} x &=y+2 \\ 2x-3y &=-1 \end{align*}\)
19) A car rental company offers two plans for renting a car.
Plan A: \(\$ 25\) per day and \(\$ 0.10\) per mile
Plan B: \(\$ 40\) per day with free unlimited mileage
How many miles would you need to drive for plan B to save you money?
- Answer
-
\(150\)
20) Find the area of a triangle bounded by the \(y\) axis, the line \(f(x)=12-4 x,\) and the line perpendicular to \(f\) that passes through the origin.
21) A town’s population increases at a constant rate. In 2010 the population was \(65,000\). By 2012 the population had increased to \(90,000\). Assuming this trend continues, predict the population in 2018.
- Answer
-
\(165,000\)
22) The number of people afflicted with the common cold in the winter months dropped steadily by \(25\) each year since 2002 until 2012. In 2002, \(8,040\) people were inflicted. Find the linear function that models the number of people afflicted with the common cold \(C\) as a function of the year, \(t\). When will less than \(6,000\) people be afflicted?
For the exercises 23-24, use the graph in Figure, showing the profit, \(y\), in thousands of dollars, of a company in a given year, \(x\), where \(x\) represents years since 1980.
23) Find the linear function \(y\), where \(y\) depends on \(x\), the number of years since 1980.
- Answer
-
\(y=875 x+10,675\)
24) Find and interpret the \(y\)-intercept.
25) In 2004, a school population was \(1250\). By 2012 the population had dropped to \(875\). Assume the population is changing linearly.
- How much did the population drop between the year 2004 and 2012?
- What is the average population decline per year?
- Find an equation for the population, \(P\), of the school \(t\) years after 2004.
- Answer
-
- \(375\)
- dropped an average of \(46.875,\) or about \(47\) people per year
- \(y=-46.875 t+1250\)
26) Draw a scatter plot for the data provided in the Table. Then determine whether the data appears to be linearly related.
| 0 | 2 | 4 | 6 | 8 | 10 |
| –450 | –200 | 10 | 265 | 500 | 755 |
27) Draw a best-fit line for the plotted data.
- Answer
-
For the exercises 28-29, use the Table below, which shows the percent of unemployed persons \(25\) years or older who are college graduates in a particular city, by year.
| Year | Percent Graduates |
|---|---|
| 2000 | 8.5 |
| 2002 | 8.0 |
| 2005 | 7.2 |
| 2007 | 6.7 |
| 2010 | 6.4 |
28) Determine whether the trend appears linear. If so, and assuming the trend continues, find a linear regression model to predict the percent of unemployed in a given year to three decimal places.
29) In what year will the percentage drop below \(4\%\)?
- Answer
-
Early in 2018
30) Based on the set of data given in the Table below, calculate the regression line using a calculator or other technology tool, and determine the correlation coefficient. Round to three decimal places of accuracy.
| \(x\) | 16 | 18 | 20 | 24 | 26 |
|---|---|---|---|---|---|
| \(y\) | 106 | 110 | 115 | 120 | 125 |
For the exercises 31-33, consider this scenario: The population of a city increased steadily over a ten-year span. The following ordered pairs shows the population (in hundreds) and the year over the ten-year span, (population, year) for specific recorded years: \((4,500,2000) ;(4,700,2001) ;(5,200,2003) ;(5,800,2006)\)
31) Use linear regression to determine a function y, where the year depends on the population. Round to three decimal places of accuracy.
- Answer
-
\(y=0.00455 x+1979.5\)
32) Predict when the population will hit \(20,000\).
33) What is the correlation coefficient for this model?
- Answer
-
\(r=0.999\)


13.1: Electric Fields and Capacitance
Admittedly, the concept of a “field” is somewhat abstract. At least with electric current it isn’t too difficult to envision tiny particles called electrons moving their way between the nuclei of atoms within a conductor, but a “field” doesn’t even have mass, and need not exist within matter at all.
Despite its abstract nature, almost every one of us has direct experience with fields, at least in the form of magnets. Have you ever played with a pair of magnets, noticing how they attract or repel each other depending on their relative orientation? There is an undeniable force between a pair of magnets, and this force is without “substance.” It has no mass, no color, no odor, and if not for the physical force exerted on the magnets themselves, it would be utterly insensible to our bodies. Physicists describe the interaction of magnets in terms of magnetic fields in the space between them. If iron filings are placed near a magnet, they orient themselves along the lines of the field, visually indicating its presence.
The Electric Fields
The subject of this chapter is electric fields (and devices called capacitors that exploit them), not magnetic fields, but there are many similarities. Most likely you have experienced electric fields as well. Chapter 1 of this book began with an explanation of static electricity, and how materials such as wax and wool—when rubbed against each other—produced a physical attraction. Again, physicists would describe this interaction in terms of electric fields generated by the two objects as a result of their electron imbalances. Suffice it to say that whenever a voltage exists between two points, there will be an electric field manifested in the space between those points.
The Field Force and the Field Flux
Fields have two measures: a field force and a field flux . The field force is the amount of “push” that a field exerts over a certain distance. The field flux is the total quantity, or effect, of the field through space. Field force and flux are roughly analogous to voltage (“push”) and current (flow) through a conductor, respectively, although field flux can exist in totally empty space (without the motion of particles such as electrons) whereas current can only take place where there are free electrons to move. Field flux can be opposed in space, just as the flow of electrons can be opposed by resistance. The amount of field flux that will develop in space is proportional to the amount of field force applied, divided by the amount of opposition to flux. Just as the type of conducting material dictates that conductor’s specific resistance to electric current, the type of insulating material separating two conductors dictates the specific opposition to field flux.
Normally, electrons cannot enter a conductor unless there is a path for an equal amount of electrons to exit (remember the marble-in-tube analogy?). This is why conductors must be connected together in a circular path (a circuit) for continuous current to occur. Oddly enough, however, extra electrons can be “squeezed” into a conductor without a path to exit if an electric field is allowed to develop in space relative to another conductor. The number of extra free electrons added to the conductor (or free electrons taken away) is directly proportional to the amount of field flux between the two conductors.
The Capacitors Electric Field
Capacitors are components designed to take advantage of this phenomenon by placing two conductive plates (usually metal) in close proximity with each other. There are many different styles of capacitor construction, each one suited for particular ratings and purposes. For very small capacitors, two circular plates sandwiching an insulating material will suffice. For larger capacitor values, the “plates” may be strips of metal foil, sandwiched around a flexible insulating medium and rolled up for compactness. The highest capacitance values are obtained by using a microscopic-thickness layer of insulating oxide separating two conductive surfaces. In any case, though, the general idea is the same: two conductors, separated by an insulator.
The schematic symbol for a capacitor is quite simple, being little more than two short, parallel lines (representing the plates) separated by a gap. Wires attach to the respective plates for connection to other components. An older, obsolete schematic symbol for capacitors showed interleaved plates, which is actually a more accurate way of representing the real construction of most capacitors:
When a voltage is applied across the two plates of a capacitor, a concentrated field flux is created between them, allowing a significant difference of free electrons (a charge) to develop between the two plates:
As the electric field is established by the applied voltage, extra free electrons are forced to collect on the negative conductor, while free electrons are “robbed” from the positive conductor. This differential charge equates to a storage of energy in the capacitor, representing the potential charge of the electrons between the two plates. The greater the difference of electrons on opposing plates of a capacitor, the greater the field flux, and the greater “charge” of energy the capacitor will store.
Because capacitors store the potential energy of accumulated electrons in the form of an electric field, they behave quite differently than resistors (which simply dissipate energy in the form of heat) in a circuit. Energy storage in a capacitor is a function of the voltage between the plates, as well as other factors which we will discuss later in this chapter. A capacitor’s ability to store energy as a function of voltage (potential difference between the two leads) results in a tendency to try to maintain voltage at a constant level. In other words, capacitors tend to resist changes in voltage drop. When voltage across a capacitor is increased or decreased, the capacitor “resists” the change by drawing current from or supplying current to the source of the voltage change, in opposition to the change .
To store more energy in a capacitor, the voltage across it must be increased. This means that more electrons must be added to the (-) plate and more taken away from the (+) plate, necessitating a current in that direction. Conversely, to release energy from a capacitor, the voltage across it must be decreased. This means some of the excess electrons on the (-) plate must be returned to the (+) plate, necessitating a current in the other direction.
Just as Isaac Newton’s first Law of Motion (“an object in motion tends to stay in motion; an object at rest tends to stay at rest”) describes the tendency of a mass to oppose changes in velocity, we can state a capacitor’s tendency to oppose changes in voltage as such: “A charged capacitor tends to stay charged; a discharged capacitor tends to stay discharged.” Hypothetically, a capacitor left untouched will indefinitely maintain whatever state of voltage charge that its been left it. Only an outside source (or drain) of current can alter the voltage charge stored by a perfect capacitor:
Practically speaking, however, capacitors will eventually lose their stored voltage charges due to internal leakage paths for electrons to flow from one plate to the other. Depending on the specific type of capacitor, the time it takes for a stored voltage charge to self-dissipate can be a long time (several years with the capacitor sitting on a shelf!).
When the voltage across a capacitor is increased, it draws current from the rest of the circuit, acting as a power load. In this condition the capacitor is said to be charging , because there is an increasing amount of energy being stored in its electric field. Note the direction of electron current with regard to the voltage polarity:
Conversely, when the voltage across a capacitor is decreased, the capacitor supplies current to the rest of the circuit, acting as a power source. In this condition the capacitor is said to be discharging . Its store of energy—held in the electric field—is decreasing now as energy is released to the rest of the circuit. Note the direction of electron current with regard to the voltage polarity:
If a source of voltage is suddenly applied to an uncharged capacitor (a sudden increase of voltage), the capacitor will draw current from that source, absorbing energy from it, until the capacitor’s voltage equals that of the source. Once the capacitor voltage reached this final (charged) state, its current decays to zero. Conversely, if a load resistance is connected to a charged capacitor, the capacitor will supply current to the load, until it has released all its stored energy and its voltage decays to zero. Once the capacitor voltage reaches this final (discharged) state, its current decays to zero. In their ability to be charged and discharged, capacitors can be thought of as acting somewhat like secondary-cell batteries.
The choice of insulating material between the plates, as was mentioned before, has a great impact upon how much field flux (and therefore how much charge) will develop with any given amount of voltage applied across the plates. Because of the role of this insulating material in affecting field flux, it has a special name: dielectric . Not all dielectric materials are equal: the extent to which materials inhibit or encourage the formation of electric field flux is called the permittivity of the dielectric.
The measure of a capacitor’s ability to store energy for a given amount of voltage drop is called capacitance . Not surprisingly, capacitance is also a measure of the intensity of opposition to changes in voltage (exactly how much current it will produce for a given rate of change in voltage). Capacitance is symbolically denoted with a capital “C,” and is measured in the unit of the Farad, abbreviated as “F.”
Convention, for some odd reason, has favored the metric prefix “micro” in the measurement of large capacitances, and so many capacitors are rated in terms of confusingly large microFarad values: for example, one large capacitor I have seen was rated 330,000 microFarads!! Why not state it as 330 milliFarads? I don’t know.
The Capacitor’s Obsolete Name
An obsolete name for a capacitor is condenser or condensor . These terms are not used in any new books or schematic diagrams (to my knowledge), but they might be encountered in older electronics literature. Perhaps the most well-known usage for the term “condenser” is in automotive engineering, where a small capacitor called by that name was used to mitigate excessive sparking across the switch contacts (called “points”) in electromechanical ignition systems.
Review
- Capacitors react against changes in voltage by supplying or drawing current in the direction necessary to oppose the change.
- When a capacitor is faced with an increasing voltage, it acts as a load : drawing current as it absorbs energy (current going in the negative side and out the positive side, like a resistor).
- When a capacitor is faced with a decreasing voltage, it acts as a source : supplying current as it releases stored energy (current going out the negative side and in the positive side, like a battery).
- The ability of a capacitor to store energy in the form of an electric field (and consequently to oppose changes in voltage) is called capacitance . It is measured in the unit of the Farad (F).
- Capacitors used to be commonly known by another term: condenser (alternatively spelled “condensor”).


9.6: Suite for Piano
Here is some additional information on Schoenberg’s Suite for Piano. Our playlist features the Trio, which is a portion of movement 5. Please note that this suite was the first piece composed entirely in Schoenberg’s twelve-tone technique. It’s also worth noting the connection via genre to the suites we studied in the Baroque era.
Arnold Schoenberg’s Suite for Piano (German: Suite für Klavier ), Op. 25, is a twelve tone piece for piano composed between 1921 and 1923.
The work is the earliest in which Schoenberg employs a row of “12 tones related only to one another” in every movement: the earlier 5 Stücke , Op. 23 (1920–23) employs a 12-tone row only in the final Waltz movement, and the Serenade , Op. 24 uses a single row in its central Sonnet .
The Basic Set of the Suite for Piano consists of the following succession: E–F–G–D ♭ –G ♭ –E ♭ –A ♭ –D–B–C–A–B ♭ .
In form and style the work echoes many features of the Baroque suite.
Schoenberg’s Suite has six movements:
- Präludium (1921)
- Gavotte (1923)
- Musette (1923)
- Intermezzo (1921–1923)
- Menuett. Trio (1923)
- Gigue (1923)
A performance of the entire Suite für Klavier takes around 16 minutes.
In this work Schoenberg employs transpositions and inversions of the row for the first time: the sets employed are P-0, I-0, P-6, I-6 and their retrogrades. Arnold Whittall has suggested that “The choice of transpositions at the sixth semitone—the tritone—may seem the consequence of a desire to hint at ‘tonic-dominant’ relationships, and the occurrence of the tritone G-D ♭ in all four sets is a hierarchical feature which Schoenberg exploits in several places.”
The Suite for Piano was first performed by Schoenberg’s pupil Eduard Steuermann in Vienna on 25 February 1924. Steuermann made a commercial recording of the work in 1957. The first recording of the Suite for Piano to be released was made by Niels Viggo Bentzon some time before 1950.
The Gavotte movement contains, “a parody of a baroque keyboard suite that involves the cryptogram of Bach’s name as an important harmonic andmelodic device (Stuckenschmidt 1977, 108; Lewin 1982–83, n.9)” and a related quotation of Schoenberg’s op. 19/vi.
Edward T. Cone (1972) has catalogued what he believes to be a number of mistakes in Reinhold Brinkmann’s 1968 revised edition of Schoenberg’s piano music, one of which is in measure number five of the Suite’s “Gavotte”, G ♭ instead of G ♮ . Henry Klumpenhouwer invokes Sigmund Freud’s concept of parapraxes (i.e., mental slips) to suggest a psychological context explaining the deviation from the note predicted from the tone row.
Contributors and Attributions
- Authored by : Elliott Jones. Provided by : Santa Ana College. Located at : http://www.sac.edu . License : CC BY: Attribution
- Suite for Piano (Schoenberg). Provided by : Wikipedia. Located at : http://en.Wikipedia.org/wiki/Suite_for_Piano_(Schoenberg) . License : CC BY-SA: Attribution-ShareAlike


10.1C: The Vocabulary Epidemiology
- Compare and contrast the following concepts: epidemic, endemic, pandemic; incidence vs prevalence; morbidity vs mortality; incubation, latency, acute, decline and convalescent periods
Epidemiology, literally meaning “the study of what is upon the people”, is derived from Greek: epi , meaning “upon, among”, demos , meaning “people, district”, and logos , meaning “study, word, discourse”, suggesting that it applies only to human populations. However, the term is widely used in studies of zoological populations (veterinary epidemiology) and of plant populations (botanical or plant disease epidemiology).
Outbreak is a term used in epidemiology to describe an occurrence of disease greater than would otherwise be expected at a particular time and place. It may affect a small and localized group or impact upon thousands of people across an entire continent. An asymptomatic carrier (healthy carrier or just carrier) is a person or other organism that has contracted an infectious disease, but who displays no symptoms. Although unaffected by the disease themselves, carriers can transmit it to others. A number of animal species act as vectors of human diseases.
EPIDEMIC, ENDEMIC OR PANDEMIC?
The distinction between “epidemic” and “endemic” was first drawn by Hippocrates, to distinguish between diseases that are “visited upon” a population ( epidemic ) from those that “reside within” a population ( endemic ). A pandemic is an epidemic of infectious disease that has spread through human populations across a large region; for instance multiple continents, or even worldwide. The term epidemiology is now widely applied to cover the description and causation of not only epidemic disease, but of disease in general, and even many non-disease health-related conditions, such as high blood pressure and obesity.
INCIDENCE VS. PREVALENCE
Incidence is a measure of the risk of developing some new condition within a specified period of time. Although sometimes loosely expressed simply as the number of new cases during a time period, it is better expressed as the incidence rate which is the number of new cases per population in a given time period. Incidence should not be confused with prevalence , which is a measure of the total number of cases of disease in a population rather than the rate of occurrence of new cases. Thus, incidence conveys information about the risk of contracting the disease, whereas prevalence indicates how widespread the disease is. Prevalence is the proportion of the total number of cases to the total population and is more a measure of the burden of the disease on society.
MORBIDITY VS. MORTALITY
Morbidity is a diseased state, disability, or poor health due to any cause. The term may be used to refer to the existence of any form of disease, or to the degree that a health condition affects the patient. In epidemiology, the term morbidity rate can refer to either the incidence rate, or the prevalence of a disease, or medical condition. This measure of sickness is contrasted with the mortality rate of a condition, which is the proportion of people dying during a given time interval.
PHASES OF DISEASES
Epidemiologists are interested in determining the progression of a disease. In an infectious disease, the incubation period is the time between infection and the appearance of symptoms (acute period). The latency period is the time between infection and the ability of the disease to spread to another person, which may precede, follow, or be simultaneous with the appearance of symptoms. In most illnesses, the acute period is followed by the decline period (symptoms get better) and convalescent (or recovery) period.
Some viruses also exhibit a dormant phase, called viral latency , in which the virus hides in the body in an inactive state. For example, varicella zoster virus causes chickenpox in the acute phase; after recovery from chickenpox, the virus may remain dormant in nerve cells for many years, and later cause herpes zoster (shingles).
Key Points
- Outbreak is a term used in epidemiology to describe an occurrence of disease greater than would otherwise be expected at a particular time and place.
- An asymptomatic carrier is a person or other organism that has contracted an infectious disease, but who displays no symptoms.
- Diseases that are “visited upon” a population are epidemic, whereas those that “reside within” a population are endemic. A pandemic is an epidemic of infectious disease that has spread through human populations across a large region.
- Incidence is a measure of the risk of developing some new condition within a specified period of time. Prevalence is a measure of the total number of cases of disease in a population.
- Morbidity is a diseased state, disability, or poor health due to any cause. The mortality rate of a condition is the proportion of people dying from it during a given time interval.
- The progression of an infection usually follows these phases: infection, incubation period, acute period, decline period, and convalescent period.
Key Terms
- epidemiology: The branch of a science dealing with the spread and control of diseases, computer viruses, concepts, etc., throughout populations or systems.


15.25D: Cytomegalovirus Infections
- Describe the route of transmission and risks associated with the human cytomegalovirus (HCMV)
Cytomegalovirus (CMV) is a viral genus of the viral family known as Herpesviridae or herpesviruses. The species that infects humans is commonly known as human CMV (HCMV) or human herpesvirus-5 (HHV-5), and is the most studied of all cytomegaloviruses. All herpesviruses can stay latent for long durations. Although they may be found throughout the body, CMV infections are frequently associated with the salivary glands in humans and other mammals. Other CMV viruses are found in several mammal species, but species isolated from animals differ from HCMV in terms of genomic structure, and have not been reported to cause human disease.
HCMV is found throughout all geographic locations and socioeconomic groups, and infects between 50% and 80% of adults in the United States (40% worldwide) as indicated by the presence of antibodies in much of the general population. Seroprevalence is age-dependent: 58.9% of individuals aged 6 and older are infected with CMV while 90.8% of individuals aged 80 and older are positive for HCMV. HCMV is also the virus most frequently transmitted to a developing fetus. HCMV infection is more widespread in developing countries and in communities with lower socioeconomic status; it represents the most significant viral cause of birth defects in industrialized countries. Major areas of risk of infection include prenatal or postnatal infants and immunocompromised individuals, such as organ transplant recipients, persons with leukemia, or those infected with human immunodeficiency virus (HIV).
CMV is generally transmitted from infected people to others through direct contact with body fluids, such as urine, saliva, vaginal secretions, and semen. Although they may be found throughout the body, HCMV infections are frequently associated with the salivary glands. HCMV infection is typically unnoticed in healthy people, but can be life-threatening for the immunocompromised, such as HIV-infected persons, organ transplant recipients, or newborn infants.
CMV, like all herpesviruses, can stay latent for long durations of time. The initial introduction of CMV generally gives way to an extended period of infection during which there is no detectable clinical illness. Severe impairment of the body’s immune system reactivates the virus from this dormant state. CMV persists in the host because the viral genome encodes multiple proteins that interfere with major histocompatibility complex (MHC) class I presentation of viral antigens. One viral protein blocks translocation of peptides into the lumen of the endoplasmic reticulum, while two other viral proteins cause degradation of MHC class I proteins before they reach the cell surface. In AIDS patients, CMV can cause loss of vision (cotton wool spots), pneumonia, and hepatitis. Transplant patients with CMV are also susceptible to pneumonia and hepatitis.
A vaccine against (CMV) is currently under investigation. Because CMV can cause congenital infection, considerable effort has been made towards the development of a vaccine, with particular emphasis on protection for pregnant women. There are currently three licensed anti-HCMV drugs target the viral DNA polymerase, pUL54. Ganciclovir (GCV) acts as nucleoside analogue. Its antiviral activity requires phosphorylation by the HCMV protein kinase, pUL97. The second drug, Cidofovir (CDV), is a nucleotide analogue, which is already phosphorylated and thus active. Finally, Foscarnet (FOS) has a different mode of action. It directly inhibits polymerase function by blocking the pyrophosphate binding site of pUL54.
Key Points
- Cytomegalovirus (CMV) is a viral genus of the viral family known as Herpesviridae or herpesviruses.
- CMV infections are frequently associated with the salivary glands
- HCMV is found throughout all geographic locations and socioeconomic groups, and infects between 50% and 80% of adults in the United States
- CMV is generally transmitted from infected people to others through direct contact with body fluids, such as urine, saliva, vaginal secretions, and semen.
Key Terms
- immunocompromised : Having an immune system that has been impaired by disease or treatment.
- Seroprevalence : Seroprevalence is the number of persons in a population who test positive for a specific disease based on serology (blood serum) specimens; often presented as a percent of the total specimens tested or as a proportion per 100,000 persons tested.
- major histocompatibility complex : MHC is a cell surface molecule that mediate interactions of immune cells with other leukocytes or body cells. MHC determines compatibility of donors for organ transplants as well as one’s susceptibility to an autoimmune disease. In humans, MHC is also called human leukocyte antigen (HLA).


8.E: Exercises
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Exercise \(\PageIndex{1}\)
In the following, polar coordinates \((r,θ)\) for a point in the plane are given. Find the corresponding Cartesian coordinates.
- \((2,\pi /4)\)
- \((-2, \pi/4)\)
- \((3, \pi/3)\)
- \((-3, \pi/3)\)
- \((2,5\pi /6)\)
- \((-2, 11\pi /6)\)
- \((2,\pi /2)\)
- \((1,3\pi /2)\)
- \((-3, 3\pi /4)\)
- \((3, 5\pi /4)\)
- \((-2, \pi /6)\)
Exercise \(\PageIndex{2}\)
Consider the following Cartesian coordinates \((x, y)\). Find polar coordinates corresponding to these points.
- \((-1,1)\)
- \((\sqrt{3},-1)\)
- \((0,2)\)
- \((-5,0)\)
- \((-2\sqrt{3},2)\)
- \((2,-2)\)
- \((-1,\sqrt{3})\)
- \((-1,-\sqrt{3})\)
Exercise \(\PageIndex{3}\)
The following relations are written in terms of Cartesian coordinates \((x, y)\). Rewrite them in terms of polar coordinates, \((r,\theta )\).
- \(y=x^2\)
- \(y=2x+6\)
- \(x^2+y^2=4\)
- \(x^2-y^2=1\)
Exercise \(\PageIndex{4}\)
Use a calculator or computer algebra system to graph the following polar relations.
- \(r=1-\sin (2\theta ),\:\theta\in [0,2\pi ]\)
- \(r=\sin (4\theta ),\:\theta\in [0,2\pi ]\)
- \(r=\cos (3\theta )+\sin (2\theta ),\: \theta\in [0,2\pi]\)
- \(r=\theta,\:\theta\in [0,15]\)
Exercise \(\PageIndex{5}\)
Graph the polar equation \(r = 1+\sinθ\) for \(θ ∈ [0, 2π]\).
Exercise \(\PageIndex{6}\)
Graph the polar equation \(r = 2+\sinθ\) for \(θ ∈ [0, 2π]\).
Exercise \(\PageIndex{7}\)
Graph the polar equation \(r = 1+2 \sinθ\) for \(θ ∈ [0, 2π]\).
Exercise \(\PageIndex{8}\)
Graph the polar equation \(r = 2+\sin(2θ)\) for \(θ ∈ [0, 2π]\).
Exercise \(\PageIndex{9}\)
Graph the polar equation \(r = 1+\sin(2θ)\) for \(θ ∈ [0, 2π]\).
Exercise \(\PageIndex{10}\)
Graph the polar equation \(r = 1+\sin(3θ)\) for \(θ ∈ [0, 2π]\).
Exercise \(\PageIndex{11}\)
Describe how to solve for \(r\) and \(θ\) in terms of \(x\) and \(y\) in polar coordinates.
Exercise \(\PageIndex{12}\)
This problem deals with parabolas, ellipses, and hyperbolas and their equations. Let \(l\), \(e > 0\) and consider \[r=\frac{l}{1\pm e\cos\theta}\nonumber\] Show that if \(e = 0\), the graph of this equation gives a circle. Show that if \(0 < e < 1\), the graph is an ellipse, if \(e = 1\) it is a parabola and if \(e > 1\), it is a hyperbola.
Exercise \(\PageIndex{13}\)
The following are the cylindrical coordinates of points, \((r,θ,z)\). Find the Cartesian and spherical coordinates of each point.
- \((5,\frac{5\pi}{6},-3)\)
- \((3,\frac{\pi}{3},4)\)
- \((4,\frac{2\pi}{3},1)\)
- \((2,\frac{3\pi}{4},-2)\)
- \((3,\frac{3\pi}{2},-1)\)
- \((8,\frac{11\pi}{6},-11)\)
Exercise \(\PageIndex{14}\)
The following are the Cartesian coordinates of points, \((x, y,z)\). Find the cylindrical and spherical coordinates of these points.
- \((\frac{5}{2}\sqrt{2},\frac{5}{2}\sqrt{2},-3)\)
- \((\frac{3}{2},\frac{3}{2}\sqrt{3},2)\)
- \((-\frac{5}{2}\sqrt{2},\frac{5}{2}\sqrt{2},11)\)
- \((-\frac{5}{2},\frac{5}{2}\sqrt{3},23)\)
- \((-\sqrt{3},-1,-5)\)
- \((\frac{3}{2},-\frac{3}{2}\sqrt{3},-7)\)
- \((\sqrt{2},\sqrt{6},2\sqrt{2})\)
- \((-\frac{1}{2}\sqrt{3},\frac{3}{2},1)\)
- \((-\frac{3}{4}\sqrt{2},\frac{3}{4}\sqrt{2},-\frac{3}{2}\sqrt{3})\)
- \((-\sqrt{3}1,2\sqrt{3})\)
- \((-\frac{1}{4}\sqrt{2},\frac{1}{4}\sqrt{6},-\frac{1}{2}\sqrt{2})\)
Exercise \(\PageIndex{15}\)
The following are spherical coordinates of points in the form \((ρ,φ,θ)\). Find the Cartesian and cylindrical coordinates of each point.
- \((4,\frac{\pi}{4},\frac{5\pi}{6})\)
- \((2,\frac{\pi}{3},\frac{2\pi}{3})\)
- \((3,\frac{5\pi}{6},\frac{3\pi}{2})\)
- \((4,\frac{\pi}{2},\frac{7\pi}{4})\)
- \((4,\frac{2\pi}{3},\frac{\pi}{6})\)
- \((4,\frac{3\pi}{4},\frac{5\pi}{3})\)
Exercise \(\PageIndex{16}\)
Describe the surface \(φ = π/4\) in Cartesian coordinates, where \(φ\) is the polar angle in spherical coordinates.
Exercise \(\PageIndex{17}\)
Describe the surface \(θ = π/4\) in spherical coordinates, where \(θ\) is the angle measured from the positive \(x\) axis.
Exercise \(\PageIndex{18}\)
Describe the surface \(r=5\) in Cartesian coordinates, where \(r\) is one of the cylindrical coordinates.
Exercise \(\PageIndex{19}\)
Describe the surface \(\rho =4\) in Cartesian coordinates, where \(\rho\) is the distance to the origin.
Exercise \(\PageIndex{20}\)
Give the cone described by \(z=\sqrt{x^2+y^2}\) in cylindrical coordinates and in spherical coordinates.
Exercise \(\PageIndex{21}\)
The following are described in Cartesian coordinates. Rewrite them in terms of spherical coordinates.
- \(z=x^2+y^2\)
- \(x^2-y^2=1\)
- \(z^2+x^2+y^2=6\)
- \(z=\sqrt{x^2+y^2}\)
- \(y=x\)
- \(z=x\)
Exercise \(\PageIndex{22}\)
The following are described in Cartesian coordinates. Rewrite them in terms of cylindrical coordinates.
- \(z=x^2+y^2\)
- \(x^2-y^2=1\)
- \(z^2+x^2+y^2=6\)
- \(z=\sqrt{x^2+y^2}\)
- \(y=x\)
- \(z=x\)