23.5: Faraday’s Law of Induction- Lenz’s Law
Learning Objectives
By the end of this section, you will be able to:
- Calculate emf, current, and magnetic fields using Faraday’s Law.
- Explain the physical results of Lenz’s Law
Faraday’s and Lenz’s Law
Faraday’s experiments showed that the emf induced by a change in magnetic flux depends on only a few factors. First, emf is directly proportional to the change in flux \(\Delta \Phi\). Second, emf is greatest when the change in time \(\Delta t\) is smallest—that is, emf is inversely proportional to \(\Delta t\). Finally, if a coil has \(N\) turns, an emf will be produced that is \(N\) times greater than for a single coil, so that emf is directly proportional to \(N\). The equation for the emf induced by a change in magnetic flux is \[emf = -N \frac{\Delta \Phi}{\Delta t}.\label{23.3.1}\] This relationship is known as Faraday's law of induction . The units for emf are volts, as is usual.
The minus sign in Faraday’s law of induction is very important. The minus means that the emf creates a current I and magnetic field B that oppose the change in flux \(\Delta \Phi\) -- this is known as Lenz's law . The direction (given by the minus sign) of the emfis so important that it is called Lenz's law after the Russian Heinrich Lenz (1804–1865), who, like Faraday and Henry,independently investigated aspects of induction. Faraday was aware of the direction, but Lenz stated it so clearly that he is credited for its discovery. (See Figure 1.)
PROBLEM-SOLVING STRATEGY FOR LENZ'S LAW:
To use Lenz’s law to determine the directions of the induced magnetic fields, currents, and emfs:
- Make a sketch of the situation for use in visualizing and recording directions.
- Determine the direction of the magnetic field B.
- Determine whether the flux is increasing or decreasing.
- Now determine the direction of the induced magnetic field B. It opposes the change in flux by adding or subtracting from the original field.
- Use RHR-2 to determine the direction of the induced current I that is responsible for the induced magnetic field B.
- The direction (or polarity) of the induced emf will now drive a current in this direction and can be represented as current emerging from the positive terminal of the emf and returning to its negative terminal.
For practice, apply these steps to the situations shown in Figure 1 and to others that are part of the following text material.
Applications of Electromagnetic Induction
There are many applications of Faraday’s Law of induction, as we will explore in this chapter and others. At this juncture, let us mention several that have to do with data storage and magnetic fields. A very important application has to do with audio and video recording tapes . A plastic tape, coated with iron oxide, moves past a recording head. This recording head is basically a round iron ring about which is wrapped a coil of wire—an electromagnet (Figure 2). A signal in the form of a varying input current from a microphone or camera goes to the recording head. These signals (which are a function of the signal amplitude and frequency) produce varying magnetic fields at the recording head. As the tape moves past the recording head, the magnetic field orientations of the iron oxide molecules on the tape are changed thus recording the signal. In the playback mode, the magnetized tape is run past another head, similar in structure to the recording head. The different magnetic field orientations of the iron oxide molecules on the tape induces an emf in the coil of wire in the playback head. This signal then is sent to a loudspeaker or video player.
Similar principles apply to computer hard drives, except at a much faster rate. Here recordings are on a coated, spinning disk. Read heads historically were made to work on the principle of induction. However, the input information is carried in digital rather than analog form – a series of 0’s or 1’s are written upon the spinning hard drive. Today, most hard drive readout devices do not work on the principle of induction, but use a technique known as giant magnetoresistance. (The discovery that weak changes in a magnetic field in a thin film of iron and chromium could bring about much larger changes in electrical resistance was one of the first large successes of nanotechnology.) Another application of induction is found on the magnetic stripe on the back of your personal credit card as used at the grocery store or the ATM machine. This works on the same principle as the audio or video tape mentioned in the last paragraph in which a head reads personal information from your card.
Another application of electromagnetic induction is when electrical signals need to be transmitted across a barrier. Consider the cochlear implant shown below. Sound is picked up by a microphone on the outside of the skull and is used to set up a varying magnetic field. A current is induced in a receiver secured in the bone beneath the skin and transmitted to electrodes in the inner ear. Electromagnetic induction can be used in other instances where electric signals need to be conveyed across various media.
Another contemporary area of research in which electromagnetic induction is being successfully implemented (and with substantial potential) is transcranial magnetic simulation. A host of disorders, including depression and hallucinations can be traced to irregular localized electrical activity in the brain. In transcranial magnetic stimulation , a rapidly varying and very localized magnetic field is placed close to certain sites identified in the brain. Weak electric currents are induced in the identified sites and can result in recovery of electrical functioning in the brain tissue.
Sleep apnea (“the cessation of breath”) affects both adults and infants (especially premature babies and it may be a cause of sudden infant deaths [SID]).  A cessation of more than 20 seconds can be very dangerous. Stroke, heart failure, and tiredness are just some of the possible consequences for a person having sleep apnea. The concern in infants is the stopping of breath for these longer times. One type of monitor to alert parents when a child is not breathing uses electromagnetic induction. A wire wrapped around the infant’s chest has an alternating current running through it. The expansion and contraction of the infant’s chest as the infant breathes changes the area through the coil. A pickup coil located nearby has an alternating current induced in it due to the changing magnetic field of the initial wire. If the child stops breathing, there will be a change in the induced current, and so a parent can be alerted.
MAKING CONNECTIONS: CONSERVATION OF ENERGY:
Lenz’s law is a manifestation of the conservation of energy. The induced emf produces a current that opposes the change in flux, because a change in flux means a change in energy. Energy can enter or leave, but not instantaneously. Lenz’s law is a consequence. As the change begins, the law says induction opposes and, thus, slows the change. In fact, if the induced emf were in the same direction as the change in flux, there would be a positive feedback that would give us free energy from no apparent source—conservation of energy would be violated.
Example \(\PageIndex{1}\): Calculating Emf: How Great is the Induced Emf?
Calculate the magnitude of the induced emf when the magnet in Figure 1a is thrust into the coil, given the following information: the single loop coil has a radius of 6.00 cm and the average value of \(B\cos{\theta}\) (this is given, since the bar magnet’s field is complex) increases from 0.0500 T to 0.250 T in 0.100 s.
Strategy:
To find the magnitute of emf, we use Faraday’s law of induction as stated by \(emf = -N\frac{\Delta \Phi}{\Delta t}\), but without the minus sign that indicates direction: \[emf = N\frac{\Delta \Phi}{\Delta t}.\]
Solution:
We are given that \(N = 1\) and \(\Delta t = 0.100 s\) but we must determine the change in flux \(\Delta \Phi\) before we can find emf. Since the area of the loop is fixed, we see that \[\Delta \Phi \left(BA\cos{\theta}\right) = A\Delta \left(B\cos{\theta}\right).\label{23.3.2}\] Now \(\Delta \left(B\cos{\theta}\right) = 0.200 T\), since it was given that \(B\cos{\theta}\) changes from 0.0500 to 0.250 T. The area of the loop is \(A = \pi r^{2} = \left(3.14...\right)\left(0.060 m\right)^{2} = 1.13 \times 10^{-2} m^{2}\). Thus, \[\Delta \Phi = \left(1.13 \times 10^{-2} m^{2}\right)\left(0.200 T\right).\] Entering the determined values into the expression for emf gives \[Emf = N\frac{\Delta \Phi}{\Delta t} = \frac{\left(1.13 \times 10^{-2}m^{2}\right)\left(0.200 T\right)}{0.100 s} = 22.6mV.\]
Discussion:
While this is an easily measured voltage, it is certainly not large enough for most practical applications. More loops in the coil, a stronger magnet, and faster movement make induction the practical source of voltages that it is.
Summary
- Faraday’s law of induction states that the emfinduced by a change in magnetic flux is \[emf = N\frac{\Delta \Phi}{\Delta t}\] when flux changes by \(\Delta \Phi\) in a time \(Delta t\).
- If emf is induced in a coil, \(N\) is its number of turns.
- The minus sign means that the emf creates a current \(I\) and magnetic field \(B\) that oppose the change in flux \(\Delta \Phi\) -- this opposition is known as Lenz’s law.
Glossary
- Faraday’s law of induction
- the means of calculating the emf in a coil due to changing magnetic flux, given by \(emf = -N\frac{\Delta \Phi}{\Delta t}\)
- Lenz’s law
- the minus sign in Faraday’s law, signifying that the emf induced in a coil opposes the change in magnetic flux


9: Fossils
Fossils are defined as any evidence of past life; for our purposes we will consider something to be a fossil if it is unusual in that it is either ancient or has been altered from its original form in a way that makes it more likely to be preserved. Fossils are very important because they tell us about how organisms changed and evolved through time, how they interacted with each other and their environment, and how Earth’s climate and environments have changed through time.
- List the major types of fossils and modes of preservation
- Describe the morphology of fossils using the appropriate terminology
- Identify the major types of invertebrate body fossils in hand sample and thin section
- Identify the most common trace fossils and apply the ichnofacies concept to interpret depositional environments
Chapter thumbnail shows a fossil gastropod ( Masha Milshina via Wikimedia Commons ; CC BY-SA 4.0 ).
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- 9.2: Types of Preservation
- Robust hard parts (teeth, bone, shell) and even soft parts (muscle, skin, other tissue) can be preserved. Replacement occurs when the original organic material is replaced with another material. Molds and casts are void and infills formed after organisms. Recrystallization occurs when a mineral changes to a more stable form. Carbonization happens when all the volatiles are driven off leaving only a carbon film. Permineralization occurs when mineral precipitate into void spaces.
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- 9.4: Major Fossil-Forming Groups (Invertebrates)
- The overwhelming majority of fossils come from invertebrates that have hard skeletal material. The most common and easily recognized are sponges, corals, brachiopods, bryozoans, mollusks (bivalves, gastropods, and cephalopods), certain arthropods, and echinoderms.
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- 9.6: Trace Fossils
- Trace fossils include tracks, trails, burrows, and any other fossilized behaviors. Trace fossils are named on the morphology of the feature, not on the interpreted trace maker. The ichnofacies concept groups associations of trace fossils together that likely record behaviors that worked in a particular environment.


What you’ll learn to do: explain emotional, social, and moral development during middle childhood
Children in middle childhood are starting to make friends in more sophisticated ways. They are choosing friends for specific characteristics, including shared interests, sense of humor, and being a good person. That is quite a departure from the earlier days of playing with the people in your group just because they are there. Children in middle childhood are starting to realize that there are benefits to friendships, and there are sometimes difficulties as well. In this section, we’ll examine some aspects of these relationships.
Contributors and Attributions
CC licensed content, Shared previously


1.1: Whole Numbers
- know the difference between numbers and numerals
- know why our number system is called the Hindu-Arabic numeration system
- understand the base ten positional number system
- be able to identify and graph whole number
Numbers and Numerals
We begin our study of introductory mathematics by examining its most basic building block, the number .
A number is a concept. It exists only in the mind.
The earliest concept of a number was a thought that allowed people to mentally picture the size of some collection of objects. To write down the number being conceptualized, a numeral is used.
A numeral is a symbol that represents a number.
In common usage today we do not distinguish between a number and a numeral. In our study of introductory mathematics, we will follow this common usage.
The following are numerals. In each case, the first represents the number four, the second represents the number one hundred twenty-three, and the third, the number one thousand five. These numbers are represented in different ways.
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Hindu-Arabic numerals
4, 123, 1005 -
Roman numerals
IV, CXXIII, MV - Egyptian numerals:
Practice Set A
Do the phrases "four," "one hundred twenty-three," and "one thousand five" qualify as numerals? Yes or no?
- Answer
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Yes. Letters are symbols. Taken as a collection (a written word), they represent a number.
The Hindu-Arabic Numeration System
Our society uses the Hindu-Arabic numeration system . This system of numeration began shortly before the third century when the Hindus invented the numerals
0 1 2 3 4 5 6 7 8 9
About a thousand years later, in the thirteenth century, a mathematician named Leonardo Fibonacci of Pisa introduced the system into Europe. It was then popularized by the Arabs. Thus, the name, Hindu-Arabic numeration system.
The Base Ten Positional Number System
The Hindu-Arabic numerals 0 1 2 3 4 5 6 7 8 9 are called digits . We can form any number in the number system by selecting one or more digits and placing them in certain positions. Each position has a particular value. The Hindu mathematician who devised the system about A.D. 500 stated that "from place to place each is ten times the preceding."
It is for this reason that our number system is called a positional number system with base ten .
When numbers are composed of more than three digits, commas are sometimes used to separate the digits into groups of three.
These groups of three are called periods and they greatly simplify reading numbers.
In the Hindu-Arabic numeration system, a period has a value assigned to each or its three positions, and the values are the same for each period. The position values are
Thus, each period contains a position for the values of one, ten, and hundred. Notice that, in looking from right to left, the value of each position is ten times the preceding. Each period has a particular name.
As we continue from right to left, there are more periods. The five periods listed above are the most common, and in our study of introductory mathematics, they are sufficient.
The following diagram illustrates our positional number system to trillions. (There are, to be sure, other periods.)
In our positional number system, the value of a digit is determined by its position in the number.
Find the value of 6 in the number 7,261.
Solution
Since 6 is in the tens position of the units period, its value is 6 tens.
6 tens \(=60\)
Find the value of 9 in the number 86,932,106,005.
Solution
Since 9 is in the hundreds position of the millions period, its value is 9 hundred millions.
9 hundred millions = 9 hundred million
Find the value of 2 in the number 102,001.
Solution
Since 2 is in the ones position of the thousands period, its value is 2 one thousands.
2 one thousands = 2 thousand
Practice Set B
Find the value of 5 in the number 65,000.
- Answer
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five thousand
Practice Set B
Find the value of 4 in the number 439,997,007,010.
- Answer
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four hundred billion
Practice Set B
Find the value of 0 in the number 108.
- Answer
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zero tens, or zero
Whole Numbers
Numbers that are formed using only the digits
0 1 2 3 4 5 6 7 8 9
are called whole numbers . They are
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, …
The three dots at the end mean "and so on in this same pattern."
Graphing Whole Numbers
Whole numbers may be visualized by constructing a number line . To construct a number line, we simply draw a straight line and choose any point on the line and label it 0.
This point is called the origin . We then choose some convenient length, and moving to the right, mark off consecutive intervals (parts) along the line starting at 0. We label each new interval endpoint with the next whole number.
We can visually display a whole number by drawing a closed circle at the point labeled with that whole number. Another phrase for visually displaying a whole number is graphing the whole number. The word graph means to "visually display."
Graph the following whole numbers: 3, 5, 9.
Specify the whole numbers that are graphed on the following number line. The break in the number line indicates that we are aware of the whole numbers between 0 and 106, and 107 and 872, but we are not listing them due to space limitations.
Solution
The numbers that have been graphed are
0, 106, 873, 874
Practice Set C
Graph the following whole numbers: 46, 47, 48, 325, 327.
- Answer
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Practice Set C
Specify the whole numbers that are graphed on the following number line.
- Answer
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4, 5, 6, 113, 978
A line is composed of an endless number of points. Notice that we have labeled only some of them. As we proceed, we will discover new types of numbers and determine their location on the number line.
Exercises
Exercise \(\PageIndex{1}\)
What is a number?
- Answer
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concept
Exercise \(\PageIndex{2}\)
What is a numeral?
Does the word "eleven" qualify as a numeral?
- Answer
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Yes, since it is a symbol that represents a number.
Exercise \(\PageIndex{3}\)
How many different digits are there?
Our number system, the Hindu-Arabic number system, is a number system with base ?
- Answer
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positional; 10
Exercise \(\PageIndex{4}\)
Numbers composed of more than three digits are sometimes separated into groups of three by commas. These groups of three are called.
In our number system, each period has three values assigned to it. These values are the same for each period. From right to left, what are they?
- Answer
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ones, tens, hundreds
Exercise \(\PageIndex{5}\)
Each period has its own particular name. From right to left, what are the names of the first four?
In the number 841, how many tens are there?
- Answer
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4
Exercise \(\PageIndex{6}\)
In the number 3,392, how many ones are there?
In the number 10,046, how many thousands are there?
- Answer
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0
Exercise \(\PageIndex{7}\)
In the number 779,844,205, how many ten millions are there?
In the number 65,021, how many hundred thousands are there?
- Answer
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0
For following problems, give the value of the indicated digit in the given number.
Exercise \(\PageIndex{8}\)
5 in 599
1 in 310,406
- Answer
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ten thousand
Exercise \(\PageIndex{9}\)
9 in 29,827
6 in 52,561,001,100
- Answer
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6 ten millions = 60 million
Exercise \(\PageIndex{10}\)
Write a two-digit number that has an eight in the tens position.
Write a four-digit number that has a one in the thousands position and a zero in the ones position.
- Answer
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1,340 (answers may vary)
Exercise \(\PageIndex{11}\)
How many two-digit whole numbers are there?
How many three-digit whole numbers are there?
- Answer
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900
Exercise \(\PageIndex{12}\)
How many four-digit whole numbers are there?
Is there a smallest whole number? If so, what is it?
- Answer
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yes; zero
Exercise \(\PageIndex{13}\)
Is there a largest whole number? If so, what is it?
Another term for "visually displaying" is ?
- Answer
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graphing
Exercise \(\PageIndex{14}\)
The whole numbers can be visually displayed on a .
Graph (visually display) the following whole numbers on the number line below: 0, 1, 31, 34.
- Answer
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Exercise \(\PageIndex{15}\)
Construct a number line in the space provided below and graph (visually display) the following whole numbers: 84, 85, 901, 1006, 1007.
Specify, if any, the whole numbers that are graphed on the following number line.
- Answer
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61, 99, 100, 102
Exercise \(\PageIndex{16}\)
Specify, if any, the whole numbers that are graphed on the following number line.


<IP_ADDRESS>: Pattern-Recognition Receptors (PRRs)
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State the function of the following as they relate to innate immunity.
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pattern recognition receptors (PRRs)
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endocytic pattern recognition receptors
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signaling pattern recognition receptors
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danger-associated molecular patterns
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danger recognition receptors
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inflammasome
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pyroptosis
Name 2 endocytic PRRs.
Name 2 signaling PRRs found on host cell surfaces.
Name 2 signaling PRRs found in the endosomes of phagocytic cells.
Name 2 signaling PRRs found on the host cell cytoplasm.
Briefly describe the major difference between the effect of the cytokines produced in response to PAMPs that bind to cell surface signaling PRRs and endosomal PRRs.
In order to recognize PAMPs, various body cells have a variety of corresponding receptors called pattern-recognition receptors or PRRs (see Figure \(\PageIndex{5}\)) capable of binding specifically to conserved portions of these molecules. Cells that typically have pattern recognition receptors include macrophages, dendritic cells, endothelial cells, mucosal epithelial cells, and lymphocytes.
Many pattern-recognition receptors are located on the surface of these cells where they can interact with PAMPs on the surface of microbes. Others PRRs are found within the phagolysosomes of phagocytes where they can interact with PAMPs located within microbes that have been phagocytosed. Some PRRs are found in the cytosol of the cell.
There are two functionally different major classes of pattern-recognition receptors: endocytic pattern-recognition receptors and signaling pattern-recognition receptors.
Endocytic (Phagocytic) Pattern-Recognition Receptors
Endocytic pattern-recognition receptors, also called phagocytic pattern-recognition receptors, are found on the surface of phagocytes and promote the attachment of microorganisms to phagocytes leading to their subsequent engulfment and destruction. They include:
1. Mannose receptors
Mannose receptors on the surface of phagocytes bind to various microbial carbohydrates such as those rich in mannose or fucose, and to N-acetylglucosamine (NAG). Human glycoproteins and glycolipids typically have terminal N-acetylglucosamine and sialic acid groups. C-type lectins found on the surface of phagocytes are mannose receptors (see Figure \(\PageIndex{6}\)) .
It is now thought that mannose receptors may be quite important in removing potentially harmful mannose-containing glycoproteins such as lysosomal hydrolases that are produced in increased amounts during inflammation.
2. Dectin-1
Dectin-1 recognizes beta-glucans (polymers of glucose) commonly found in fungal cell walls.
3. Scavenger receptors
Scavenger receptors found on the surface of phagocytic cells bind to bacterial cell wall components such as LPS, peptidoglycan and teichoic acids (see Figure \(\PageIndex{7}\)) . There are also scavenger receptors for certain components of other types of microorganisms, as well as for stressed, infected, or injured cells. Scavenger receptors include CD-36, CD-68, and SRB-1.
4. Opsonin receptors
Opsonins are soluble molecules produced as a part of the body's immune defenses that bind microbes to phagocytes. One portion of the opsonin binds to a PAMP on the microbial surface and another portion binds to a specific receptor on the phagocytic cell.
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Acute phase proteins circulating in the plasma, such as:
- mannose-binding lectin (also called mannose-binding protein) that binds to various microbial carbohydrates such as those rich in mannose or fucose, and to N-acetylglucosamine (NAG); and
- C-reactive protein (CRP) that binds to phosphorylcholine portion of teichoic acids and lipopolysaccharides of bacterial and fungal cell walls. It also binds to the phosphocholine found on the surface of damaged or dead human cells.
- Complement pathway proteins, such as C3b (see Figure \(\PageIndex{8}\)) and C4b recognize a variety of PAMPS.
- Surfactant proteins in the alveoli of the lungs, such as SP-A and SP-D are opsonins.
- During adaptive immunity, the antibody molecule IgG can function as an opsonin (see Figure \(\PageIndex{16}\)) .
5. N -formyl Met receptors
N-formyl methionine is the first amino acid produced in bacterial proteins since the f-met-tRNA in bacteria has an anticodon complementary to the AUG start codon (see Figure \(\PageIndex{17}\)) . This form of the amino acid is not typically seen in mammalian proteins. FPR and FPRL1 are N -formyl receptors on neutrophils and macrophages. Binding of N -formyl Met to its receptor promotes the motility and the chemotaxis of these phagocytes. It also promotes phagocytosis.
Signaling Pattern-Recognition Receptors
Signaling pattern-recognition receptors bind a number of microbial molecules: LPS, peptidoglycan, teichoic acids, flagellin, pilin, unmethylated cytosine-guanine dinucleotide or CpG sequences from bacterial and viral genomes; lipoteichoic acid, glycolipids, and zymosan from fungi; double-stranded viral RNA, and certain single-stranded viral RNAs. Binding of microbial PAMPs to signaling PRRs promotes the production of:
- inflammatory cytokines, such as such as interleukin-1 (IL-1), tumor necrosis factor-alpha (TNF-alpha), and interleukin-12 (IL-12);
- antiviral cytokines called type-1 interferons (IFN), such as IFN-alpha and IFN-beta;
- chemotactic factors, such as the chemokines interleukin-8 (IL-8), MCP-1, and RANTES; and
- antimicrobial peptides, such as human defensins ) and cathelicidins .
These molecules are crucial to initiating innate immunity and adaptive immunity.
1. Signaling PRRs found on cell surfaces (see Figure \(\PageIndex{5}\)) :
A series of signaling pattern-recognition receptors known as toll-like receptors (TLRs) are found on the surface of a variety of defense cells and other cells. These TLRs play a major role in the induction of innate immunity and contribute to the induction of adaptive immunity.
Different combinations of TLRs appear in different cell types and may occur in pairs. Different TLRs directly or indirectly bind different microbial molecules. For example:
a. TLR-2 - recognizes peptidoglycan, bacterial lipoproteins, lipoteichoic acid (Gram-positive bacteria), and porins (gram-negative bacteria).
b. TLR-4 - recognizes lipopolysaccharide (Gram-negative bacteria), fungal mannans, viral envelope proteins, parasitic phospholipids, heat-shock proteins.
c. TLR-5 - recognizes bacterial flagellin;
d. TLR-1/TLR-2 pairs - binds to bacterial lipopeptides, lipomannans (mycobacteria) lipoteichoic acids (Gram-positive bacteria), cell wall beta gucans (bacteria and fungi), zymosan (fungi) and glycosylphosphatidylinositol (GPI)-anchored proteins (protozoa).
e. TLR-2/TL6 pairs - also binds to bacterial lipopeptides, lipomannans (mycobacteria) lipoteichoic acids (Gram-positive bacteria), cell wall beta gucans (bacteria and fungi), zymosan (fungi) and glycosylphosphatidylinositol (GPI)-anchored proteins (protozoa).
Many of the TLRs, especially those that bind to bacterial and fungal cell wall components, stimulate the transcription and translation of inflammatory cytokines such as interleukin-1 (IL-1), tumor necrosis factor-alpha (TNF-alpha), and interleukin-12 (IL-12), as well as chemokines such as interleukin-8 (IL-8), MCP-1, and RANTES. These cytokines trigger innate immune defenses such as inflammation, fever, and phagocytosis in order to provide an immediate response against the invading microorganism (see Figure \(\PageIndex{9}\)) . Because cytokines such as IL-I, TNF-alpha, and IL-12 that trigger an inflammatory response, they are often referred to as inflammatory cytokines. Chemokines are a group of cytokines that enable the migration of leukocytes from the blood to the tissues at the site of inflammation. To counter inflammation, anti-inflammatory cytokines such as IL-1 receptor antagonist, IL-4, and IL-10 are produced.
Another cell surface PRR is CD14. CD14 is found on monocytes, macrophages, and neutrophils and promotes the ability of TLR-4 to respond to LPS. LPS typically binds to LPS-binding protein in the plasma and tissue fluid. The LPS-binding protein promotes the binding of LPS to the CD14 receptors. At that point the LPS-binding protein comes off and the LPS-CD14 bind to TLR-4. Interaction of LPS and CD14 with TLR-4 leads to an elevated synthesis and secretion of inflammatory cytokines such as IL-1, IL-6, IL-8, TNF-alpha, and platelet-activating factor (PAF). These cytokines then bind to cytokine receptors on target cells and initiate inflammation and activate both the complement pathways and the coagulation pathway (see Figure \(\PageIndex{9}\)) .
The signaling process for the CD14 and TLR-4 response to LPS is shown in Figure \(\PageIndex{15}\) .
TLRs also participate in adaptive immunity by triggering various secondary signals needed for humoral immunity (the production of antibodies ) and cell-mediated immunity (the production of cytotoxic T-lymphocytes, activated macrophages, and additional cytokines ). Without innate immune responses there could be no adaptive immunity.
a. T-independent (TI) antigens allow B-lymphocytes to mount an antibody response without the requirement of interaction with effector T4-lymphocytes. The resulting antibody molecules are generally of the IgM isotype and do not give rise to a memory response. There are two basic types of T-independent antigens: TI-1 and TI-2. TI-1 antigens are pathogen-associated molecular patterns (PAMPs) such as lipopolysaccharide (LPS) from the outer membrane of the gram-negative cell wall and lipoteichoic acids from the gram-positive cell wall. These antigens activate B-lymphocytes by binding to their specific toll-like receptors rather than to B-cell receptors (see Figure \(\PageIndex{11}\)) . Antibody molecules generated against TI-1 antigens are often called "natural antibodies" because they are always being made against bacteria present in the body.
b. The activation of naive T-lymphocytes requires co-stimulatory signals involving the interaction of accessory molecules on antigen-presenting cells or APCs with their corresponding ligands on T-lymphocytes. These co-stimulatory molecules are only synthesized when toll-like receptors on APCs bind to pathogen-associated molecular patterns of microbes (see Figure \(\PageIndex{12}\)) .
2. Signaling PRRs found in the membranes of the endosomes (phagolysosomes ) used to degrade pathogens (see Figure \(\PageIndex{5}\)) :
a. TLR-3 - binds double-stranded viral RNA;
b. TLR-7 - binds single-stranded viral RNA, such as in HIV, rich in guanine/uracil nucleotide pairs;
c. TLR-8 - binds single-stranded viral RNA;
d. TLR-9 - binds unmethylated cytosine-guanine dinucleotide sequences (CpG DNA) found in bacterial and viral genomes but uncommon or masked in human DNA and RNA.
Most of the TLRs that bind to viral components trigger the synthesis of cytokines called interferons that block viral replication within infected host cells as well as inflammatory cytokines.
3. Signaling PRRs and DRRs found in the cytoplasm (see Figure \(\PageIndex{5}\))
Pattern-recognition receptors or PRRs found in the cytoplasm include:
a. NODs (nucleotide-binding oligomerization domain)
NOD proteins, including NOD-1 and NOD-2, are cytostolic proteins that allow intracellular recognition of peptidoglycan components.
1. NOD-1 recognizes peptidoglycan containing the muramyl dipeptide NAG-NAM-gamma-D-glutamyl-meso diaminopimelic acid, part of the peptidoglycan monomer in common gram-negative bacteria and just a few gram-positive bacteria.
2. NOD-2 recognizes peptidoglycan containing the muramyl dipeptide NAG-NAM-L-alanyl-isoglutamine found in practically all bacteria (see Figure \(\PageIndex{5}\)) .
As macrophages phagocytose either whole bacteria or peptidoglycan fragments released during bacterial growth, the peptidoglycan is broken down into muramyl dipeptides. Binding of the muramyl dipetides to NOD-1 or NOD-2 leads to the activation of genes coding for inflammatory cytokines such as IL-1, TNF-alpha, IL-8, and IL-12 in a manner similar to the cell surface TLRs. Activation of NOD-2 also induces the production of antimicrobial peptides such as defensins as well as microbicidal reactive oxygen species (ROS).
b. CARD-containing proteins
CARD (caspase activating and recruitment domain)-containing proteins, such as RIG-1 (retinoic acid-inducible gene-1) and MDA-5 (melanoma differentiation-associated gene-5), are cytoplasmic sensors of viral RNA molecules that trigger the synthesis of type-1 interferons, antiviral cytokines that block viral replication within infected host cells in a manner similar to the endosomal TLRs. RIG-1 recognizes 5'-PPPs on viral RNAs. The 5'-PPPs on host cell RNAs are either capped or removed and are not recognized by RIG-1. Rig-1 and MDA-5 can also, through another regulatory pathway, stimulate the production of inflammatory cytokines.
Detection of PAMPs by PRRs in the cytosol trigger the formation of multi-protein complexes called inflammasomes which, in turn, leads to the activation of caspase-1. Caspase-1 triggers the formation of inflammatory cytokines and can also result in an inflammatory response-induced cell suicide called pyroptosis. Pyroptosis, unlike apoptosis, leads to the release of PAMPS as well as inflammatory cytokines from the lysed cell.
Pyroptosis is initiated by PAMPs binding to pattern-recognition receptors (PRRs) on various defense cells which then triggers the production of inflammatory cytokines and type-1 interferons. Other PRRs, called nod-like receptors (NLRs) located in the cytosol of these defense cells recognize PAMPs and DAMPs that have entered the host cell’s cytosol. Some NLRs trigger the production of inflammatory cytokines while others activate caspase 1-dependent pyroptosis of the cell causing the release of its intracellular inflammatory cytokines (see Figure \(\PageIndex{1}\)) . The binding of PAMPs or DAMPs to their respective NLRs triggers the assembly of multiprotein complexes called inflammasomes in the cytosol of the host cell. It is these inflammasomes that activate caspase 1 and induce inflammation and pyroptosis. Pyroptosis results in production of proinflammatory cytokines, rupture of the cell’s plasma membrane, and subsequent release of proinflammatory intracellular contents. It plays an essential role in innate immunity by promoting inflammation to control microbial infections. At highly elevated levels, however, it can cause considerable harm to the body and even death.
c. Danger recognition receptors or DRRs
Danger recognition receptors or DRRs found in the cytoplasm recognize danger-associated molecular patterns (DAMPS) in the cytosol such as altered membrane phospholipids , and materials released from damaged phagosomes and damaged lysosomes, including antibodies bound to microbes from opsonization . DAMPs are also produced as a result of tissue injury during cancer, heart attack, and stroke. Detection of DAMPs by DRRs in the cytosol also triggers the activation of inflammasomes , release of inflammatory cytokines, and pyroptosis.
4. Secreted signaling PRRs found in plasma and tissue fluid
In addition to the PRRs found on or within cells, there are also secreted pattern-recognition receptors. These PRRs bind to microbial cell walls and enable them to activate the complement pathways, as well as by phagocytes. For example, mannan-binding lectin -also known as mannan-binding protein - is synthesized by the liver and released into the bloodstream as part of the acute phase response discussed later in Unit 4. Here it can bind to the carbohydrates on bacteria, yeast, some viruses, and some parasites (see Figure \(\PageIndex{6}\)) . This, in turn, activates the lectin complement pathway (discussed later in Unit 4) and results in the production of a variety of activated complement proteins that are able to trigger inflammation, chemotactically attract phagocytes to the infection site, promote the attachment of antigens to phagocytes via enhanced attachment or opsonization, and cause lysis of gram-negative bacteria and infected or transformed human cells.
Other secreted PRRs include C-reactive protein (CRP), surfactant protein A (SP-A), surfactant protein D (SP-D), collectin liver 1 (CL-L1), and ficolins.
Compare and contrast the functions of endocytic pattern-recognition receptors and signaling pattern-recognition receptors.
Compare and contrast signaling pattern-recognition receptors found on cell surfaces with those found in the membranes of endosomes (phagolysosomes).
|
Concept Map for PRRs and DRRs
|
Summary
- Early induced innate immunity begins 4 - 96 hours after exposure to an infectious agent and involves the recruitment of defense cells as a result of pathogen-associated molecular patterns or PAMPS binding to pattern-recognition receptors or PRRs and danger-associated molecular patterns or DAMPs binding to danger-recognition receptors or DRRs.
- Endocytic pattern-recognition receptors are found on the surface of phagocytes and promote the attachment of microorganisms to phagocytes leading to their subsequent engulfment and destruction. They include mannose receptors, scavenger receptors, and opsonin receptors.
- Binding of microbial PAMPs to signaling PRRs promotes the production of inflammatory cytokines, antiviral cytokines called type-1 interferons (IFN), chemotactic factors, and antimicrobial peptides. They include toll-like receptors (TLRs) and NODs.
- PRRs found on the surface of the body’s cells typically bind to surface PAMPs on microbes and stimulate the production of inflammatory cytokines.
- PRRs found within cellular phagolysosomes (endosomes) typically detect nucleic acid PAMPs released during the phagocytic destruction of viruses and stimulate the production of antiviral cytokines called type-1 interferons.
- PRRs and DRRs found within the cytoplasm of host cells typically trigger the formation of multi-protein complexes called inflammasomes which, in turn, triggers the formation of inflammatory cytokines and can also leads to an inflammatory response-induced cell suicide called pyroptosis.
- PRRs circulating in the blood and tissue fluid activate the complement pathways and may function as opsonins.


7: Systems
In this chapter, we will investigate matrices and their inverses, and various ways to use matrices to solve systems of equations. First, however, we will study systems of equations on their own: linear and nonlinear, and then partial fractions.
-
- 7.1: Prelude to Systems of Equations and Inequalities
- In this chapter, we will investigate matrices and their inverses, and various ways to use matrices to solve systems of equations. First, however, we will study systems of equations on their own: linear and nonlinear, and then partial fractions. We will not be breaking any secret codes here, but we will lay the foundation for future courses.
-
- 7.2: Systems of Linear Equations - Two Variables
- A system of linear equations consists of two or more equations made up of two or more variables such that all equations in the system are considered simultaneously. The solution to a system of linear equations in two variables is any ordered pair that satisfies each equation independently. Systems of equations are classified as independent with one solution, dependent with an infinite number of solutions, or inconsistent with no solution.
-
- 7.5: Partial Fractions
- Decompose a ratio of polynomials by writing the partial fractions. Solve by clearing the fractions, expanding the right side, collecting like terms, and setting corresponding coefficients equal to each other, then setting up and solving a system of equations. The decomposition with repeated linear factors must account for the factors of the denominator in increasing powers. The decomposition with a nonrepeated irreducible quadratic factor needs a linear numerator over the quadratic factor.
-
- 7.6: Matrices and Matrix Operations
- To solve a systems of equations, we can use a matrix, which is a rectangular array of numbers. A row in a matrix is a set of numbers that are aligned horizontally. A column in a matrix is a set of numbers that are aligned vertically. Each number is an entry, sometimes called an element, of the matrix. Matrices (plural) are enclosed in [ ] or ( ), and are usually named with capital letters.
-
- 7.7: Solving Systems with Gaussian Elimination
- A matrix can serve as a device for representing and solving a system of equations. To express a system in matrix form, we extract the coefficients of the variables and the constants, and these become the entries of the matrix. We use a vertical line to separate the coefficient entries from the constants, essentially replacing the equal signs. When a system is written in this form, we call it an augmented matrix.
-
- 7.8: Solving Systems with Inverses
- A matrix that has a multiplicative inverse is called an invertible matrix. Only a square matrix may have a multiplicative inverse, as reversibility is a requirement. Not all square matrices have an inverse. We will look at two methods for finding the inverse of a 2×2 matrix and a third method that can be used on both 2×2 and 3×3 matrices.
-
- 7.9: Solving Systems with Cramer's Rule
- In this section, we will study two more strategies for solving systems of equations. A determinant is a real number that can be very useful in mathematics because it has multiple applications, such as calculating area, volume, and other quantities. Here, we will use determinants to reveal whether a matrix is invertible by using the entries of a square matrix to determine whether there is a solution to the system of equations. Cramer’s Rule to solve a system of equations in two & three variables.
-
- 7.E: Systems of Equations and Inequalities (Exercises)
- In this chapter, we will investigate matrices and their inverses, and various ways to use matrices to solve systems of equations. First, however, we will study systems of equations on their own: linear and nonlinear, and then partial fractions. We will not be breaking any secret codes here, but we will lay the foundation for future courses.
-
- 7.R: Systems of Equations and Inequalities (Review)
- In this chapter, we will investigate matrices and their inverses, and various ways to use matrices to solve systems of equations. First, however, we will study systems of equations on their own: linear and nonlinear, and then partial fractions. We will not be breaking any secret codes here, but we will lay the foundation for future courses.


13.3: Polarity in a parallel circuit
Just as in series circuits, electrical current flows “from negative to positive” through each of the load components in a parallel circuit. As illustrated in Figure \(\PageIndex{1}\), electrons leave the negative terminal of the source and flow from negative to positive through each of the load resistors. Note that the polarity of each of the resistors is the same as the polarity of the source.
Polarity is always expressed from one point of a circuit relative to another point with a different electrical potential. Note that in Figure 6 the top side of each resistor, which is marked negative, is in effect the same point. No difference in potential exists between any of these like terminals.
Also notice that the individual currents through each resistor (I 1 , I 2 , I 3 ) together constitute the total current (I T ) drawn from the source. When the total current required to operate each of these parallel loads exceeds the current output rating of the one source, you will need to increase the source output.
Polarity test for parallel voltage sources
Voltage sources are connected in parallel whenever it is necessary to deliver a current output greater than the current output a single source of supply can provide, without increasing voltage across a load.
- Power sources are connected in series to increase the voltage output .
- Conversely, power sources are connected in parallel to increase the current capacity .
An advantage of parallel-connected power sources is that one source can be removed for maintenance or repairs while reduced power to the load is maintained. If a 6 V battery has a maximum current output of 1 A, and if it is necessary to supply a load requiring 2 A, then you can connect a second 6 V battery in parallel with the first.
If there is any doubt about the polarity of the two batteries, then you can do a simple voltmeter test for correct polarity.
- Tie one side of the power sources together.
- Before connecting the paralleling jumper between the remaining two terminals, insert a voltmeter between these two points. See Figure \(\PageIndex{2}\).
- If the polarity is incorrect (Figure \(\PageIndex{2}\)b), the voltmeter indicates two times the source voltage, because the equal EMFs aid each other. Do NOT connect across these terminals.
Caution!
Since there is a difference in potential between these two points, connecting a paralleling jumper between them would result in a short circuit!
If the polarity is correct (Figure \(\PageIndex{2}\)a), then the voltmeter indicates 0 V because the EMFs oppose each other. You may connect a paralleling jumper between these two points.
Now complete the Learning Task Self-Test.


1.8: Histograms, Frequency Polygons, and Time Series Graphs
For most of the work you do in this book, you will use a histogram to display the data. One advantage of a histogram is that it can readily display large data sets. A rule of thumb is to use a histogram when the data set consists of 100 values or more.
A histogram consists of contiguous (adjoining) boxes. It has both a horizontal axis and a vertical axis. The horizontal axis is labeled with what the data represents (for instance, distance from your home to school). The vertical axis is labeled either frequency or relative frequency (or percent frequency or probability). The graph will have the same shape with either label. The histogram (like the stemplot) can give you the shape of the data, the center, and the spread of the data.
The relative frequency is equal to the frequency for an observed value of the data divided by the total number of data values in the sample. (Remember, frequency is defined as the number of times an answer occurs.) If:
- \(f\) is frequency
- \(n\) is total number of data values (or the sum of the individual frequencies), and
- \(RF\) is relative frequency,
then:
\[n = \Sigma f \label{2.3.1}\]
\[RF=\dfrac{f}{n} \label{2.3.2}\]
For example, if three students in Mr. Ahab's English class of 40 students received from 90% to 100%, then, f = 3, n = 40, and RF = fn = 340 = 0.075. 7.5% of the students received 90–100%. 90–100% are quantitative measures.
To construct a histogram, first decide how many bars or intervals, also called classes or bins, represent the data. Many histograms consist of five to 15 bars or classes for clarity. The number of bars needs to be chosen. Choose a starting point for the first interval to be less than the smallest data value. A convenient starting point is a lower value carried out to one more decimal place than the value with the most decimal places. For example, if the value with the most decimal places is 6.1 and this is the smallest value, a convenient starting point is \(6.05 (6.1 – 0.05 = 6.05)\). We say that 6.05 has more precision. If the value with the most decimal places is 2.23 and the lowest value is 1.5, a convenient starting point is \(1.495 (1.5 – 0.005 = 1.495)\). If the value with the most decimal places is 3.234 and the lowest value is 1.0, a convenient starting point is \(0.9995 (1.0 – 0.0005 = 0.9995)\). If all the data happen to be integers and the smallest value is two, then a convenient starting point is \(1.5 (2 - 0.5 = 1.5)\). Also, when the starting point and other boundaries are carried to one additional decimal place, no data value will fall on a boundary. The next two examples go into detail about how to construct a histogram using continuous data and how to create a histogram using discrete data.
If:
- \(max\) is maximum (largest) value of the data
- \(min\) is minimum (smallest) value of the data,
- \(starting \ point\) is a lower value carried out to one more decimal place than the value with the most decimal places, the adjustment
- \(ending \ point\) is an upper value carried out to one more decimal place than the value with the most decimal places, the adjustment
- \(bars\) is the number of bars in the histogram,
To calculate width of each bar or class interval:
\[starting \ point = min - adjustment \label{2.3.3}\]
\[ending\ point = max + adjustment \label{2.3.4}\]
\[class \ interval = \dfrac{ending \ point - starting \ point}{bars} \label{2.3.5}\]
The following data are the heights (in inches to the nearest half inch) of 100 male semiprofessional soccer players.  Since the data with the most decimal places has one decimal (for instance, 61.5), we want our starting point to have two decimal places. Since the numbers 0.5, 0.05, 0.005, etc. are convenient numbers, use 0.05 and subtract it from 60, the smallest value, for the convenient starting point.
60 – 0.05 = 59.95 which is more precise than, say, 61.5 by one decimal place. The starting point is, then, 59.95.
The largest value is 74, so 74 + 0.05 = 74.05 is the ending value.
Next, calculate the width of each bar or class interval. To calculate this width, subtract the starting point from the ending value and divide by the number of bars (you must choose the number of bars you desire). Suppose you choose eight bars.
\[\dfrac{74.05−59.95}{8}=1.76\]
We will round up to two and make each bar or class interval two units wide. Rounding up to two is one way to prevent a value from falling on a boundary. Rounding to the next number is often necessary even if it goes against the standard rules of rounding. For this example, using 1.76 as the width would also work. A guideline that is followed by some for the width of a bar or class interval is to take the square root of the number of data values and then round to the nearest whole number, if necessary. For example, if there are 150 values of data, take the square root of 150 and round to 12 bars or intervals.
The boundaries are:
- 59.95
- 59.95 + 2 = 61.95
- 61.95 + 2 = 63.95
- 63.95 + 2 = 65.95
- 65.95 + 2 = 67.95
- 67.95 + 2 = 69.95
- 69.95 + 2 = 71.95
- 71.95 + 2 = 73.95
- 73.95 + 2 = 75.95
The heights 60 through 61.5 inches are in the interval 59.95–61.95. The heights that are 63.5 are in the interval 61.95–63.95. The heights that are 64 through 64.5 are in the interval 63.95–65.95. The heights 66 through 67.5 are in the interval 65.95–67.95. The heights 68 through 69.5 are in the interval 67.95–69.95. The heights 70 through 71 are in the interval 69.95–71.95. The heights 72 through 73.5 are in the interval 71.95–73.95. The height 74 is in the interval 73.95–75.95.
The following histogram displays the heights on the x -axis and relative frequency on the y -axis.
The following data are the shoe sizes of 50 male students. The sizes are discrete data since shoe size is measured in whole and half units only. Construct a histogram and calculate the width of each bar or class interval. Suppose you choose six bars.
9; 9; 9.5; 9.5; 10; 10; 10; 10; 10; 10; 10.5; 10.5; 10.5; 10.5; 10.5; 10.5; 10.5; 10.5
11; 11; 11; 11; 11; 11; 11; 11; 11; 11; 11; 11; 11; 11.5; 11.5; 11.5; 11.5; 11.5; 11.5; 11.5
12; 12; 12; 12; 12; 12; 12; 12.5; 12.5; 12.5; 12.5; 14
Answer
Smallest value: 9
Largest value: 14
Convenient starting value: 9 – 0.05 = 8.95
Convenient ending value: 14 + 0.05 = 14.05
\(\frac{14.05-8.95}{6}\) = 0.85
The calculations suggests using 0.85 as the width of each bar or class interval. You can also use an interval with a width equal to one.
The following data are the number of books bought by 50 part-time college students at ABC College. The number of books is discrete data , since books are counted.
1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1
2; 2; 2; 2; 2; 2; 2; 2; 2; 2
3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3
4; 4; 4; 4; 4; 4
5; 5; 5; 5; 5
6; 6
Eleven students buy one book. Ten students buy two books. Sixteen students buy three books. Six students buy four books. Five students buy five books. Two students buy six books.
Because the data are integers, subtract 0.5 from 1, the smallest data value and add 0.5 to 6, the largest data value. Then the starting point is 0.5 and the ending value is 6.5.
Next, calculate the width of each bar or class interval. If the data are discrete and there are not too many different values, a width that places the data values in the middle of the bar or class interval is the most convenient. Since the data consist of the numbers 1, 2, 3, 4, 5, 6, and the starting point is 0.5, a width of one places the 1 in the middle of the interval from 0.5 to 1.5, the 2 in the middle of the interval from 1.5 to 2.5, the 3 in the middle of the interval from 2.5 to 3.5, the 4 in the middle of the interval from _______ to _______, the 5 in the middle of the interval from _______ to _______, and the _______ in the middle of the interval from _______ to _______ .
Answer
Calculate the number of bars as follows:
\(\dfrac{6.5 - 0.5}{\text{number of bars}}\) = 1
where 1 is the width of a bar. Therefore, bars = 6.
The following histogram displays the number of books on the x -axis and the frequency on the y -axis.
Go to [link] . There are calculator instructions for entering data and for creating a customized histogram. Create the histogram for Example .
-
Press
Y=
. PressCLEAR
to delete any equations. -
Press
STAT
then1:EDIT
. If L1 has data in it, arrow up into the name L1, pressCLEAR
and then arrow down. If necessary, do the same for L2. - Into L1, enter 1, 2, 3, 4, 5, 6.
- Into L2, enter 11, 10, 16, 6, 5, 2.
-
Press
WINDOW
. Set Xmin = .5, Xscl = (6.5 – .5)/6, Ymin = –1, Ymax = 20, Yscl = 1, Xres = 1. -
Press
2 nd
thenY=
. Start by pressing4:Plotsoff
andENTER
. -
Press
2 nd
thenY=
. Press1:Plot1
. PressENTER
. Arrow down toTYPE
. Arrow to the 3 rd picture (histogram). PressENTER
. - Arrow down to Xlist: Enter L1 (2 nd 1). Arrow down to Freq. Enter L2 (2 nd 2).
-
Press
GRAPH
. -
Use the
TRACE
key and the arrow keys to examine the histogram.
The following data are the number of sports played by 50 student athletes. The number of sports is discrete data since sports are counted.
1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1
2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2
3; 3; 3; 3; 3; 3; 3; 3
20 student athletes play one sport. 22 student athletes play two sports. Eight student athletes play three sports.
Fill in the blanks for the following sentence. Since the data consist of the numbers 1, 2, 3, and the starting point is 0.5, a width of one places the 1 in the middle of the interval 0.5 to _____, the 2 in the middle of the interval from _____ to _____, and the 3 in the middle of the interval from _____ to _____.
Answer
1.5
1.5 to 2.5
2.5 to 3.5
Using this data set, construct a histogram.
| 9.95 | 10 | 2.25 | 16.75 | 0 |
| 19.5 | 22.5 | 7.5 | 15 | 12.75 |
| 5.5 | 11 | 10 | 20.75 | 17.5 |
| 23 | 21.9 | 24 | 23.75 | 18 |
| 20 | 15 | 22.9 | 18.8 | 20.5 |
Answer
Some values in this data set fall on boundaries for the class intervals. A value is counted in a class interval if it falls on the left boundary, but not if it falls on the right boundary. Different researchers may set up histograms for the same data in different ways. There is more than one correct way to set up a histogram.
The following data represent the number of employees at various restaurants in New York City. Using this data, create a histogram.
22; 35; 15; 26; 40; 28; 18; 20; 25; 34; 39; 42; 24; 22; 19; 27; 22; 34; 40; 20; 38 and 28
Use 10–19 as the first interval.
Count the money (bills and change) in your pocket or purse. Your instructor will record the amounts. As a class, construct a histogram displaying the data. Discuss how many intervals you think is appropriate. You may want to experiment with the number of intervals.
Frequency Polygons
Frequency polygons are analogous to line graphs, and just as line graphs make continuous data visually easy to interpret, so too do frequency polygons. To construct a frequency polygon, first examine the data and decide on the number of intervals, or class intervals, to use on the x -axis and y -axis. After choosing the appropriate ranges, begin plotting the data points. After all the points are plotted, draw line segments to connect them.
A frequency polygon was constructed from the frequency table below.
| Lower Bound | Upper Bound | Frequency | Cumulative Frequency |
|---|---|---|---|
| 49.5 | 59.5 | 5 | 5 |
| 59.5 | 69.5 | 10 | 15 |
| 69.5 | 79.5 | 30 | 45 |
| 79.5 | 89.5 | 40 | 85 |
| 89.5 | 99.5 | 15 | 100 |
The first label on the x -axis is 44.5. This represents an interval extending from 39.5 to 49.5. Since the lowest test score is 54.5, this interval is used only to allow the graph to touch the x -axis. The point labeled 54.5 represents the next interval, or the first “real” interval from the table, and contains five scores. This reasoning is followed for each of the remaining intervals with the point 104.5 representing the interval from 99.5 to 109.5. Again, this interval contains no data and is only used so that the graph will touch the x -axis. Looking at the graph, we say that this distribution is skewed because one side of the graph does not mirror the other side.
Construct a frequency polygon of U.S. Presidents’ ages at inauguration shown in the Table.
| Age at Inauguration | Frequency |
|---|---|
| 41.5–46.5 | 4 |
| 46.5–51.5 | 11 |
| 51.5–56.5 | 14 |
| 56.5–61.5 | 9 |
| 61.5–66.5 | 4 |
| 66.5–71.5 | 2 |
Answer
The first label on the x -axis is 39. This represents an interval extending from 36.5 to 41.5. Since there are no ages less than 41.5, this interval is used only to allow the graph to touch the x -axis. The point labeled 44 represents the next interval, or the first “real” interval from the table, and contains four scores. This reasoning is followed for each of the remaining intervals with the point 74 representing the interval from 71.5 to 76.5. Again, this interval contains no data and is only used so that the graph will touch the x -axis. Looking at the graph, we say that this distribution is skewed because one side of the graph does not mirror the other side.
Frequency polygons are useful for comparing distributions. This is achieved by overlaying the frequency polygons drawn for different data sets.
We will construct an overlay frequency polygon comparing the scores from Example with the students’ final numeric grade.
| Lower Bound | Upper Bound | Frequency | Cumulative Frequency |
|---|---|---|---|
| 49.5 | 59.5 | 5 | 5 |
| 59.5 | 69.5 | 10 | 15 |
| 69.5 | 79.5 | 30 | 45 |
| 79.5 | 89.5 | 40 | 85 |
| 89.5 | 99.5 | 15 | 100 |
| Lower Bound | Upper Bound | Frequency | Cumulative Frequency |
|---|---|---|---|
| 49.5 | 59.5 | 10 | 10 |
| 59.5 | 69.5 | 10 | 20 |
| 69.5 | 79.5 | 30 | 50 |
| 79.5 | 89.5 | 45 | 95 |
| 89.5 | 99.5 | 5 | 100 |
Suppose that we want to study the temperature range of a region for an entire month. Every day at noon we note the temperature and write this down in a log. A variety of statistical studies could be done with this data. We could find the mean or the median temperature for the month. We could construct a histogram displaying the number of days that temperatures reach a certain range of values. However, all of these methods ignore a portion of the data that we have collected.
One feature of the data that we may want to consider is that of time. Since each date is paired with the temperature reading for the day, we don‘t have to think of the data as being random. We can instead use the times given to impose a chronological order on the data. A graph that recognizes this ordering and displays the changing temperature as the month progresses is called a time series graph.
Constructing a Time Series Graph
To construct a time series graph, we must look at both pieces of our paired data set . We start with a standard Cartesian coordinate system. The horizontal axis is used to plot the date or time increments, and the vertical axis is used to plot the values of the variable that we are measuring. By doing this, we make each point on the graph correspond to a date and a measured quantity. The points on the graph are typically connected by straight lines in the order in which they occur.
The following data shows the Annual Consumer Price Index, each month, for ten years. Construct a time series graph for the Annual Consumer Price Index data only.
| Year | Jan | Feb | Mar | Apr | May | Jun | Jul |
|---|---|---|---|---|---|---|---|
| 2003 | 181.7 | 183.1 | 184.2 | 183.8 | 183.5 | 183.7 | 183.9 |
| 2004 | 185.2 | 186.2 | 187.4 | 188.0 | 189.1 | 189.7 | 189.4 |
| 2005 | 190.7 | 191.8 | 193.3 | 194.6 | 194.4 | 194.5 | 195.4 |
| 2006 | 198.3 | 198.7 | 199.8 | 201.5 | 202.5 | 202.9 | 203.5 |
| 2007 | 202.416 | 203.499 | 205.352 | 206.686 | 207.949 | 208.352 | 208.299 |
| 2008 | 211.080 | 211.693 | 213.528 | 214.823 | 216.632 | 218.815 | 219.964 |
| 2009 | 211.143 | 212.193 | 212.709 | 213.240 | 213.856 | 215.693 | 215.351 |
| 2010 | 216.687 | 216.741 | 217.631 | 218.009 | 218.178 | 217.965 | 218.011 |
| 2011 | 220.223 | 221.309 | 223.467 | 224.906 | 225.964 | 225.722 | 225.922 |
| 2012 | 226.665 | 227.663 | 229.392 | 230.085 | 229.815 | 229.478 | 229.104 |
| Year | Aug | Sep | Oct | Nov | Dec | Annual |
|---|---|---|---|---|---|---|
| 2003 | 184.6 | 185.2 | 185.0 | 184.5 | 184.3 | 184.0 |
| 2004 | 189.5 | 189.9 | 190.9 | 191.0 | 190.3 | 188.9 |
| 2005 | 196.4 | 198.8 | 199.2 | 197.6 | 196.8 | 195.3 |
| 2006 | 203.9 | 202.9 | 201.8 | 201.5 | 201.8 | 201.6 |
| 2007 | 207.917 | 208.490 | 208.936 | 210.177 | 210.036 | 207.342 |
| 2008 | 219.086 | 218.783 | 216.573 | 212.425 | 210.228 | 215.303 |
| 2009 | 215.834 | 215.969 | 216.177 | 216.330 | 215.949 | 214.537 |
| 2010 | 218.312 | 218.439 | 218.711 | 218.803 | 219.179 | 218.056 |
| 2011 | 226.545 | 226.889 | 226.421 | 226.230 | 225.672 | 224.939 |
| 2012 | 230.379 | 231.407 | 231.317 | 230.221 | 229.601 | 229.594 |
Answer
The following table is a portion of a data set from www.worldbank.org. Use the table to construct a time series graph for CO 2 emissions for the United States.
| Ukraine | United Kingdom | United States | |
|---|---|---|---|
| 2003 | 352,259 | 540,640 | 5,681,664 |
| 2004 | 343,121 | 540,409 | 5,790,761 |
| 2005 | 339,029 | 541,990 | 5,826,394 |
| 2006 | 327,797 | 542,045 | 5,737,615 |
| 2007 | 328,357 | 528,631 | 5,828,697 |
| 2008 | 323,657 | 522,247 | 5,656,839 |
| 2009 | 272,176 | 474,579 | 5,299,563 |
Uses of a Time Series Graph
Time series graphs are important tools in various applications of statistics. When recording values of the same variable over an extended period of time, sometimes it is difficult to discern any trend or pattern. However, once the same data points are displayed graphically, some features jump out. Time series graphs make trends easy to spot.
Review
A histogram is a graphic version of a frequency distribution. The graph consists of bars of equal width drawn adjacent to each other. The horizontal scale represents classes of quantitative data values and the vertical scale represents frequencies. The heights of the bars correspond to frequency values. Histograms are typically used for large, continuous, quantitative data sets. A frequency polygon can also be used when graphing large data sets with data points that repeat. The data usually goes on y -axis with the frequency being graphed on the x -axis. Time series graphs can be helpful when looking at large amounts of data for one variable over a period of time.Glossary
References
- Data on annual homicides in Detroit, 1961–73, from Gunst & Mason’s book ‘Regression Analysis and its Application’, Marcel Dekker
- “Timeline: Guide to the U.S. Presidents: Information on every president’s birthplace, political party, term of office, and more.” Scholastic, 2013. Available online at www.scholastic.com/teachers/a...-us-presidents (accessed April 3, 2013).
- “Presidents.” Fact Monster. Pearson Education, 2007. Available online at http://www.factmonster.com/ipka/A0194030.html (accessed April 3, 2013).
- “Food Security Statistics.” Food and Agriculture Organization of the United Nations. Available online at http://www.fao.org/economic/ess/ess-fs/en/ (accessed April 3, 2013).
- “Consumer Price Index.” United States Department of Labor: Bureau of Labor Statistics. Available online at http://data.bls.gov/pdq/SurveyOutputServlet (accessed April 3, 2013).
- “CO2 emissions (kt).” The World Bank, 2013. Available online at http://databank.worldbank.org/data/home.aspx (accessed April 3, 2013).
- “Births Time Series Data.” General Register Office For Scotland, 2013. Available online at www.gro-scotland.gov.uk/stati...me-series.html (accessed April 3, 2013).
- “Demographics: Children under the age of 5 years underweight.” Indexmundi. Available online at http://www.indexmundi.com/g/r.aspx?t=50&v=2224&aml=en (accessed April 3, 2013).
- Gunst, Richard, Robert Mason. Regression Analysis and Its Application: A Data-Oriented Approach . CRC Press: 1980.
- “Overweight and Obesity: Adult Obesity Facts.” Centers for Disease Control and Prevention. Available online at http://www.cdc.gov/obesity/data/adult.html (accessed September 13, 2013).
- Frequency
- the number of times a value of the data occurs
- Histogram
- a graphical representation in \(x-y\) form of the distribution of data in a data set; \(x\) represents the data and \(y\) represents the frequency, or relative frequency. The graph consists of contiguous rectangles.
- Relative Frequency
- the ratio of the number of times a value of the data occurs in the set of all outcomes to the number of all outcomes


8.7: Nuclear Fusion
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As we saw in the preceding section, when the nuclei of heavy atoms split, energy is released. For light atoms, the opposite is true; when these nuclei combine (fuse together), energy is released. This is the process of nuclear fusion. Fusion of light elements, mostly hydrogen, is the force that powers energy release in the sun and in sun-like stars. Imagine the sun as a huge sphere of hydrogen. Because a star is so massive, the gravitation pull on the hydrogen atoms is sufficient to overcome the repulsion between the two nuclei to force them together to form an unstable nucleus. This immediately ejects a positron, leaving deuterium, , and releasing a significant amount of energy. In the cascade of reactions deuterium fuses with another hydrogen to give , and two of these combine to form helium, ejecting two high-energy protons in the process.
In stars that are larger and heavier than our sun, the “triple alpha process” is the dominant nuclear reaction. In this, helium nuclei fuse to eventually form carbon, releasing significant energy in the process.
One of the great challenges in physics and engineering today is to replicate fusion of this sort under controlled conditions, harvesting the energy released and converting it, indirectly, into electrical power. The extremely high temperatures and pressures that are required to initiate and sustain fusion reactions thwarted, thus far, attempts to build a fusion reactor that is “break even” in terms of the energy released relative to the energy required to produce the fusion events. Uncontrolled fusion is certainly possible, and fusion bombs exist, but these typically use an advanced fission bomb to create the temperatures and pressures necessary to promote the fusion of the lighter elements. Clearly, this approach does not work in the laboratory! Work on fusion reactors continues at a fast pace and includes novel approaches such as aneutronic fusion reactions that utilize proton-boron fusion to produce charged particles rather than a barrage of neutrons. The advantage here is that few neutrons are produced, reducing the need for shielding, and the charged particles formed can potentially be captured directly as electricity.


7.11: VII Glossary
Admission: Refers to an initial visit or contact with a client.
Brief: A short session to share a plan, discuss team formation, assign roles and responsibilities, establish expectations and climate, and anticipate outcomes and contingencies.
Closed-loop communication: A communication strategy used to ensure that information conveyed by the sender is heard by the receiver and completed.
Continuity of care: The use of information on past events and personal circumstances to make current care appropriate for each individual. [1]
Cultural diversity: A term used to describe cultural differences among clients, family members, and health care team members.
Cultural humility: A humble and respectful attitude toward individuals of other cultures that pushes one to challenge their own cultural biases, realize they cannot possibly know everything about other cultures, and approach learning about other cultures as a lifelong goal and process. [2]
CUS statements: Assertive statements that are well-recognized by all staff across a health care agency as implementation of the two-challenge rule. These assertive statements are “I am Concerned – I am Uncomfortable – This is a Safety issue!” [3]
Debrief: An informal information exchange session designed to improve team performance and effectiveness through reinforcement of positive behaviors and reflecting on lessons learned after a significant event occurs.
DESC: A tool used to help resolve conflict. DESC is a mnemonic that stands for Describe the specific situation or behavior and provide concrete data, Express how the situation makes you feel/what your concerns are using “I” messages, Suggest other alternatives and seek agreement, and Consequences are stated in terms of impact on established team goals while striving for consensus.
Discharge: The completion of care and services in a health care facility and the client is sent home (or to another health care facility).
Feedback: Information is provided to a team member for the purpose of improving team performance. Feedback should be timely, respectful, specific, directed towards improvement, and considerate. [4]
Handoff reports: A transfer and acceptance of patient care responsibility achieved through effective communication. It is a real-time process of passing patient specific information from one caregiver to another, or from one team of caregivers to another, for the purpose of ensuring the continuity and safety of the patient’s care. [5]
Huddle: A brief meeting during a shift to reestablish situational awareness, reinforce plans already in place, and adjust the teamwork plan as needed.
I’M SAFE: A tool used to assess one’s own safety status, as well as that of other team members in their ability to provide safe patient care. It is a mnemonic standing for personal safety risks as a result of Illness, Medication, Stress, Alcohol and Drugs, Fatigue, and Eating and Elimination.
Interdisciplinary care conferences: Meetings where interprofessional team members professionally collaborate, share their expertise, and plan collaborative interventions to meet client needs.
Interprofessional collaborative practice: Multiple health workers from different professional backgrounds working together with patients, families, caregivers, and communities to deliver the highest quality of care.
I-PASS: A mnemonic used as a structured communication tool among interprofessional team members. I-PASS stands for Illness severity, Patient summary, Action list, Situation awareness, and Synthesis by the receiver.
ISBARR: A mnemonic for the components of Introduction, Situation, Background, Assessment, Request/Recommendations, and Repeat back. [6] , [7]
Mutual support: The ability to anticipate and support team members’ needs through accurate knowledge about their responsibilities and workload.
Shared mental model: The actions of a team leader that ensure all team members have situation awareness and are “on the same page” as situations evolve on the unit. [8]
Situation awareness: The awareness of a team member knowing what is going on around them. [9]
Situation monitoring: The process of continually scanning and assessing the situation to gain and maintain an understanding of what is going on around you. [10]
STEP tool: A situation monitoring tool used to know what is going on with you, your patients, your team, and your environment. STEP stands for Status of the patients, Team members, Environment, and Progress Toward Goal. [11]
TeamSTEPPS®: An evidence-based framework used to optimize team performance across the health care system. It is a mnemonic standing for Team Strategies and Tools to Enhance Performance and Patient Safety. [12]
Two-challenge rule: A strategy for advocating for patient safety that includes a team member assertively voicing their concern at least two times to ensure that it has been heard by the decision-maker.
- Kim, S. Y. (2017). Continuity of care. Korean Journal of Family Medicine, 38 (5), 241. https://doi.org/10.4082/kjfm.2<IP_ADDRESS> ↵
- American Nurses Association. (2021). Nursing: Scope and standards of practice (4th ed.). American Nurses Association. ↵
- AHRQ. (2020, January). Pocket guide: TeamSTEPPS. https://www.ahrq.gov/teamstepps/instructor/essentials/pocketguide.html ↵
- AHRQ. (2020, January). Pocket guide: TeamSTEPPS. https://www.ahrq.gov/teamstepps/instructor/essentials/pocketguide.html ↵
- Starmer, A. J., Spector, N. D., Srivastava, R., Allen, A. D., Landrigan, C. P., Sectish, T. C., & I-Pass Study Group. (2012). Transforming pediatric GME. Pediatrics, 129 (2), 201-204. https://www.ipassinstitute.com/hubfs/I-PASS-mnemonic.pdf ↵
- Institute for Healthcare Improvement (n.d.). ISBAR trip tick . http://www.ihi.org/resources/Pages/Tools/ISBARTripTick.aspx ↵
- Grbach, W., Vincent, L., & Struth, D. (2008). Curriculum developer for simulation education. QSEN Institute. https://qsen.org/reformulating-sbar-to-i-sbar-r/ ↵
- AHRQ. (2020, January). Pocket guide: TeamSTEPPS. https://www.ahrq.gov/teamstepps/instructor/essentials/pocketguide.html ↵
- AHRQ. (2020, January). Pocket guide: TeamSTEPPS. https://www.ahrq.gov/teamstepps/instructor/essentials/pocketguide.html ↵
- AHRQ. (2020, January). Pocket guide: TeamSTEPPS. https://www.ahrq.gov/teamstepps/instructor/essentials/pocketguide.html ↵
- AHRQ. (2020, January). Pocket guide: TeamSTEPPS. https://www.ahrq.gov/teamstepps/instructor/essentials/pocketguide.htm l ↵
- AHRQ. (2020, January). Pocket guide: TeamSTEPPS. https://www.ahrq.gov/teamstepps/instructor/essentials/pocketguide.html ↵


8.6: Key Words and Terms
|
10 nm fiber |
double helix |
mutations |
|
30 nm fiber |
euchromatin |
non-histone proteins |
|
5' -to-3' replication |
F and Hfr plasmids |
nuclear proteins |
|
antiparallel DNA strands |
F- strain |
nucleosomes |
|
bacterial conjugation |
F+ strain |
recipient cell |
|
base ratios |
fertility plasmid |
replication |
|
beads-on-a-string |
heterochromatin |
S. pneumonia type III-S |
|
chromatin |
Hfr strain |
S. pneumonia type II-R |
|
chromosomes |
histone octamer |
semi-conservative replication |
|
conjugation tube |
histone proteins |
sex pili |
|
conservative replication |
influenza |
solenoid fiber |
|
deoxyribonuclease |
karyotype |
spindle fibers |
|
discontinuous replication |
levels of chromatin packing |
transforming principle |
|
dispersive replication |
mating bridge |
X & Y chromosomes |
|
DNA |
metaphase chromatin |
X-ray crystallography |
|
donor cell |
mitosis & meiosis |
X-ray diffraction |


4.7: Application and Presentation Roles
The application and presentation roles in WAI-ARIA change the way assistive technologies interact with web content. Both have “use with caution” warnings. Their use and where and when to use them are described here.
Application Role
The application role is used when there is not a corresponding widget interaction pattern available to provide semantics for a custom widget.
Imagine, for instance, a file manager application embedded in a web page, which does not have widget roles specifically defined. It may have many of the functions a typical file manager might have on a Windows, Mac, or Linux system. It might have the typical File, Edit, and View menus that most applications have, including browsers. Those menus in the file manager should function like these same menus in other applications. When the application role is used in a container containing the embedded file manager, keystrokes are intercepted and repurposed to operate the file manager, instead of the browser and the assistive technology.
When in the file manager application, this behaviour may be desirable. But, defined with the application role, all of the standard screen reader shortcut keys are also disabled, so the user is no longer able to navigate the pages by headings, or landmarks, for instance, while inside the application. This may be fine in such a case because the screen reader user will likely temporarily want shortcut keys to file manager functions, and not those of the browser or screen reader.
If, however, the application role is used to contain a carousel widget, for example, then browser and assistive technology functionality may be unnecessarily disabled, potentially creating barriers. A carousel widget typically has limited functionality. For example, carousels may contain scripted Arrow keys to move back and forth between slides, between headings within each slide for added structure, or link to another section of the site presented in a slide. In such cases, screen reader users would be unable to navigate through the slides by listing headings or links, using their screen reader’s default heading and link list functionality. By removing the application role, the scripted next/previous link, as well as the heading and the links could be used to navigate the carousel.
The bottom line is to use the application role carefully. Be sure it is not creating more barriers than it is intended to prevent.
Presentation Role
Much like the application role disables default keyboard functionality, the presentation role (and its synonym
role="none"
), theoretically, removes the default semantics from children of the element it applies to.
So, for instance, if you have a list with
role="presentation"
, it should not announce as a list, and its list items should not announce as list items. However, nested lists within those suppressed list items will announce as usual.
There are a couple of intended exceptions where the presentation role will not remove default semantics:
-
When
role="presentation"
is not applied to elements that have tab focus, such as links, form elements, and elements that have tabindex defined, or - Where an element has been modified with any of the 21 global states or properties
Where
role="presentation"
is applied to a parent element, all of its child elements should inherit that role, but not all of its grandchildren. For example, if

is used then the semantics for each of its

elements will be ignored. But, if an

contains a sublist, that list would be announced as usual.
It should be noted that current support for the presentation role is spotty across browsers and assistive technologies, and you are likely to find it not all that useful if you’re trying to develop with cross browser compatibility. Typically, tables, images, and headings are affected by the presentation role, while other elements like lists, forms, and links are not, or only partially affected. If you are trying to hide elements completely from screen readers, you might consider using either
aria-hidden
or CSS
display:none
.
Three common uses for
role="presentation"
include:
- Hiding a decorative image. It is equivalent to giving the image null alt text.
- Suppressing table semantics for tables used for layout in circumstances where the table semantics do not convey meaningful relationships.
- Eliminating semantics of intervening orphan elements in the structure of a composite widget, such as a tablist, menu, or tree as demonstrated in the example above.
Source: WAI-ARIA Authoring Best Practices
There are also a number of WAI-ARIA
roles that act like the presentation role
, and these suppress the default semantics for the elements to which they are applied. For instance, if a
tablist
is created from a

, and
role="tab"
is applied to each of the list items within that

, their default
listitem
role will be replaced with the
tab
role, without the need to set them as presentational.
The following JSFiddle examples have been created for cross browser testing of the presentation role. Navigate through each example with ChromeVox + Chrome. If you have them available, also navigate them with JAWS + IE and NVDA + FF to understand the varied support for the presentation role. Below, the fiddle is a listing of support for current versions of these screen readers.
Skip JSFiddle
Screen Reader Output from the Above Demo
NVDA (2018.1.1) + Edge (41.16299.248.0)
By keyboard, only the link is announced. Mouseover, though, and all elements are announced.
- Link: not announced
- List: not announced or keyboard focusable
- Headings: not announced or keyboard focusable
- Table: not announced or keyboard focusable
- Image: not announced or keyboard focusable
- Form: “Combo box opt three collapsed”
NVDA (2018.1.1) + FireFox (59.0.2)
- Link: announces as usual
- List: values announced but not bullets or the list itself
- Headings: announce as usual
- Table: not announced.
- Image: not announced
- Form: announces opts but not the combobox
JAWS (18) + Edge (41.16299.248.0)
- Link: focusable, but not announced
- List: not focusable, not announced
- Headings: does not announce first heading, but does announce second heading
- Table: not announced
- Image: not announced
- Form: not focusable, not announced
ChromeVox (53.0.2784.5)
- Link: focusable, but not read (using Tab key) Announces as usual when using CVOX + Arrows
- List: Skips over list (using Arrow key) except when link receives focus first, then Arrow key announces the numbers in the list. Announces numbers but not as a list when using CVOX + Arrows
- Headings: Does not announce the first heading, but does announce the second (Arrows and CVOX+Arrows)
- Table: not announced.
- Image: reads alt text “Ryerson Chang School” both Arrow and CVOX + Arrow
- Form: focusable, not announced (using Tab key), Arrow keys announces “Combobox. Opt 3, 3 of 3.”
Suggested Reading:


31.6: Review Questions
A client in the clinic is prescribed aluminum hydroxide by the provider. The nurse should include which of the following in the client’s education?
- Answer
-
Take the medication 2 hours before or after other medications.
The nurse is administering IV cimetidine to a client admitted to the ICU with a GI bleed. Which of the following adverse effects is the client at increased risk for when receiving cimetidine parenterally?
- Answer
-
Hypotension
The nurse is educating a client on a new prescription of omeprazole. What does the nurse include in teaching?
- Answer
-
The medication decreases the production of gastric acids and pepsin.
The nurse is educating the client on their newly prescribed medication misoprostol. Which of the following statements by the client indicates a priority need for additional follow-up?
- Answer
-
“I recently stopped taking my birth-control pills.”
A client being treated for GERD arrives at the clinic with complaints of constipation and flatulence. Which of the following medications is likely causing these symptoms?
- Answer
-
Calcium carbonate
The nurse is discharging a client with instructions from the provider to take calcium carbonate to treat their indigestion. Which of the following statements by the client indicates the need for additional instruction?
- Answer
-
“I need to take this every day for it to work.”
The nurse is preparing to administer a dose of lansoprazole to a client with an ulcer. Which of the following statements by the client indicates the client needs additional instruction on this medication?
- Answer
-
“I know my pain will improve right after taking this.”
The nurse is providing discharge instructions to a client with a gastric ulcer who is taking sucralfate. What information regarding administration is important to stress to the client?
- Answer
-
Take this medication 4 times a day; do not miss a dose.
The nurse is reviewing lab values for a client who has been taking over-the-counter antacid medications. Which of the following findings could be attributed to antacid use?
- Answer
-
Hypermagnesemia
The nurse is educating a client who is taking pantoprazole. What information should be included in the education?
- Answer
-
Swallow the medication whole, do not chew.


4.8: Chapter 4 Excercises
P4.1 (4 pts)
Many classrooms are equipped with
video projectors that can be connected to portable, laptop
computers for use during class meetings.
a. Based upon your experience with such systems, what is the
probability of projection system failure over the course of a year
of regular classes in such a room?
b. Develop a fault tree of potential causes for a classroom video
projection system.
P4.2 (8 pts)
Appearing below is a series of roof inspection condition summaries, where 1 is excellent and 5 is poor. Note that an inspection 1997.5 occurred in the second six months of 1997, whereas 1997 occurred in the first six months of 1997. The roof was replaced in 1985. Answer the questions below. You might use software aids, such as EXCEL or MATLAB, for this problem.
| DATE | CONDITION |
| 1985 | 1 |
| 1985.5 | 1 |
| 186.5 | 2 |
| 1997 | 2 |
| 1997.5 | 2 |
| 1998 | 2 |
| 1998.5 | 2 |
| 1999 | 3 |
| 1999.5 | 3 |
| 2000 | 4 |
| 2000.5 | 4 |
| 2001 | 4 |
| 2002 | 4 |
| 2002.5 | 4 |
| 2003 | 5 |
a. Estimate an ordinary least squares regression deterioration model of the form: Condition = \(a + b(age)\) where age is the age of the roof in years. Report your parameter estimates, standard errors, t-statistics, and \(R^2\) values. Note that there is a gap in the data from 1985 to 1996!
b. Suppose I have a comparable roof that is 12 years old. What would your regression model in (a) predict for its condition? What would it predict for age 18? At what age is the condition expected to become 5?
c. Plot the data and your regression line.
d. Do you think a non-linear regression model would fit the data better? Try a quadratic model (Condition = \(a + b(age) + c(age^2)\) and an exponential model (Condition = \(a*age^b\)) and discuss your results. Which model has a better-adjusted \(R^2\)? Which model would you use in practice for deterioration prediction?
P4.3 (16 pts)
Formulate a simple Markov process model of roof condition. Assume that transitions occur every six months and can either be a return to current condition or a transition to the next worse condition (except for state 5 which is an absorbing state without an exit in this deterioration model…)
a. Draw your process model as a series of states (in circles) and transition possibilities (as arrows) for five states corresponding to roof conditions 1 to 5.
b. Assume the probability of remaining in state 1 is 0.93 in any one transition. Estimate (from the data above) or calculate (when appropriate) the remaining transition probabilities and mark them on your process diagram.
c. Develop a Markov transition matrix for your process.
d. Suppose you start at time 0 in state 1. Calculate the probability of being in each state for the next twenty years (or 40 transitions) based on your model.
e. Suppose you believe that a roof must be replaced when the roof condition reaches state 5. Starting with a new roof (state 1 in time 0), plot the probability of being in state 5 as a function of time.
f. Calculate the expected service time of the roof based on your data in part e. You can assume that the expected service time is when the probability of entering state 5 reaches 50%.
g. If you ran your model to the limit (infinite time), what is the probability of being in each state?
h. How does your Markov process model compare with your linear regression model in Question 1? In particular, is the expected service time different? Is the Markov model non-linear? Why or why not? Which is preferable and why?
P4.4 (6 pts)
Suppose I have the simple piping
system shown below:
For this simple system, I develop a fault tree for failure analysis
as:
Suppose further that the estimated failure probabilities of the four sub-events are independent and are as follows (left to right in the figure):
| Event | D empty | A broken | B blocked | C blocked |
| Event Probability | 0.15 | 0.05 | 0.1 | 0.1 |
P4.1 (4 pts)
Many classrooms are equipped with
video projectors that can be connected to portable, laptop
computers for use during class meetings.
a. Based upon your experience with such systems, what is the
probability of projection system failure over the course of a year
of regular classes in such a room?
b. Develop a fault tree of potential causes for a classroom video
projection system.
P4.6 (4 pts)
With the growth of internet service providers, a researcher decides to examine whether there is a correlation between cost of internet service per month (rounded to the nearest dollar) and degree of customer satisfaction (on a scale of 1 - 10 with a 1 being not at all satisfied and a 10 being extremely satisfied). The researcher only includes programs with comparable types of services. A sample of the data is provided below.
| Dollars | Satisfaction |
| 11 | 6 |
| 19 | 8 |
| 17 | 10 |
| 15 | 4 |
| 9 | 9 |
| 5 | 6 |
| 12 | 3 |
| 19 | 5 |
| 22 | 2 |
| 25 | 10 |
a. Plot the data. Do you think dollars and satisfaction are related (or correlated)?
b. Estimate a linear regression with Satisfaction = a + b*dollars. Discuss your results.
P4.7 (3 pts)
Which of the following are transition matrices for Markov processes? Explain.
a. \begin{bmatrix}
.4 & .3 & .3 \\
.2 & .4 & .4 \\
.6 & .1 & .3 \\
\end{bmatrix}
b. \begin{bmatrix}
.2 & .3 & .5 \\
.6 & .1 & .2 \\
.7 & .1 & .3 \\
\end{bmatrix}
c. \begin{bmatrix}
.25 & .15 & .3 & .4\\
.5 & 0 & .15 &
.3 \\
.15 & .35 & .4 & .2 \\
.1 & .5 & .2 & .2 \\
\end{bmatrix}


7.2: Siliceous Sedimentary Rocks
As the name implies, siliceous sedimentary rocks are enriched in silica . We will avoid the numerous mineral and gem names applied to siliceous material (jasper, flint, agate, chalcedony, opal, etc.) and instead use the terms microcrystalline quartz or chert to refer to any/all of them.
Formation of siliceous sediments and sedimentary rocks
Ultimately, all dissolved silica can trace its origin back to the weathering and/or dissolution of existing silicate rocks or input from hydrothermal sources. Once in solution, there are several possibilities for the formation of siliceous sediments:
1. Direct precipitation from water has been reported in modern nonmarine settings and, although not active today, the abundance of chert in Precambrian banded iron formations suggests that this process was active in the ocean in the geologic past.
2. Biogenic silica forms when organisms such as sponges, radiolarians, and/or diatoms extract silica from the water to build their skeletons.
Common siliceous sedimentary rocks
Bedded cherts form in the deep ocean as siliceous skeletons rain down from shallow waters. Although siliceous organisms occur across the entirety of the ocean, their contribution is generally overwhelmed by calcareous skeletal material. However, in the deep ocean (below the carbonate compensation depth) or in areas where upwelling occurs, the water chemistry can result in the accumulation of siliceous oozes and environmental conditions that are not favorable for carbonate deposition.
Nodular cherts form in carbonates and other sedimentary rocks when primary components of the rock are replaced by microcrystalline quartz during diagenesis.
Banded Iron Formations (BIFs) are largely Precambrian deposits that are composed of alternating layers of chert and iron oxides (typically hematite or magnetite). Although the exact mechanism for BIF formation is not well understood, most workers agree that they reflect profoundly different ocean chemistry during the Precambrian (less oxygen) that allowed for upwelling of silica and iron-rich waters and regular changes in oxygen levels that resulted in the precipitation of iron oxides. Beyond that, there is ongoing discussion of the nature of diagenetic/metamorphic overprinting of iron-bearing phases and the direct/indirect involvement of microorganisms beyond the production of oxygen.
Silcretes are resistant duricrusts formed at or near the sediment surface by the accumulation of silica through pedogenic processes, groundwater precipitation, or evaporation in ephemeral lakes.