2.9: Exercises
In the following drawing, the green spheres represent atoms of a certain element. The purple spheres represent atoms of another element. If the spheres of different elements touch, they are part of a single unit of a compound. The following chemical change represented by these spheres may violate one of the ideas of Dalton’s atomic theory. Which one?
Which postulate of Dalton’s theory is consistent with the following observation concerning the weights of reactants and products? When 100 grams of solid calcium carbonate is heated, 44 grams of carbon dioxide and 56 grams of calcium oxide are produced.
Identify the postulate of Dalton’s theory that is violated by the following observations: 59.95% of one sample of titanium dioxide is titanium; 60.10% of a different sample of titanium dioxide is titanium.
Samples of compound X, Y, and Z are analyzed, with results shown here.
| Compound | Description | Mass of Carbon | Mass of Hydrogen |
|---|---|---|---|
| X | clear, colorless, liquid with strong odor | 1.776 g | 0.148 g |
| Y | clear, colorless, liquid with strong odor | 1.974 g | 0.329 g |
| Z | clear, colorless, liquid with strong odor | 7.812 g | 0.651 g |
Do these data provide example(s) of the law of definite proportions, the law of multiple proportions, neither, or both? What do these data tell you about compounds X, Y, and Z?
The existence of isotopes violates one of the original ideas of Dalton’s atomic theory. Which one?
How are electrons and protons similar? How are they different?
How are protons and neutrons similar? How are they different?
Predict and test the behavior of α particles fired at a “plum pudding” model atom.
(a) Predict the paths taken by α particles that are fired at atoms with a Thomson’s plum pudding model structure. Explain why you expect the α particles to take these paths.
(b) If α particles of higher energy than those in (a) are fired at plum pudding atoms, predict how their paths will differ from the lower-energy α particle paths. Explain your reasoning.
(c) Now test your predictions from (a) and (b). Open the Rutherford Scattering simulation and select the “Plum Pudding Atom” tab. Set “Alpha Particles Energy” to “min,” and select “show traces.” Click on the gun to start firing α particles. Does this match your prediction from (a)? If not, explain why the actual path would be that shown in the simulation. Hit the pause button, or “Reset All.” Set “Alpha Particles Energy” to “max,” and start firing α particles. Does this match your prediction from (b)? If not, explain the effect of increased energy on the actual paths as shown in the simulation.
Predict and test the behavior of α particles fired at a Rutherford atom model.
(a) Predict the paths taken by α particles that are fired at atoms with a Rutherford atom model structure. Explain why you expect the α particles to take these paths.
(b) If α particles of higher energy than those in (a) are fired at Rutherford atoms, predict how their paths will differ from the lower-energy α particle paths. Explain your reasoning.
(c) Predict how the paths taken by the α particles will differ if they are fired at Rutherford atoms of elements other than gold. What factor do you expect to cause this difference in paths, and why?
(d) Now test your predictions from (a), (b), and (c). Open the Rutherford Scattering simulation and select the “Rutherford Atom” tab. Due to the scale of the simulation, it is best to start with a small nucleus, so select “20” for both protons and neutrons, “min” for energy, show traces, and then start firing α particles. Does this match your prediction from (a)? If not, explain why the actual path would be that shown in the simulation. Pause or reset, set energy to “max,” and start firing α particles. Does this match your prediction from (b)? If not, explain the effect of increased energy on the actual path as shown in the simulation. Pause or reset, select “40” for both protons and neutrons, “min” for energy, show traces, and fire away. Does this match your prediction from (c)? If not, explain why the actual path would be that shown in the simulation. Repeat this with larger numbers of protons and neutrons. What generalization can you make regarding the type of atom and effect on the path of α particles? Be clear and specific.
In what way are isotopes of a given element always different? In what way(s) are they always the same?
Write the symbol for each of the following ions:
(a) the ion with a 1+ charge, atomic number 55, and mass number 133
(b) the ion with 54 electrons, 53 protons, and 74 neutrons
(c) the ion with atomic number 15, mass number 31, and a 3− charge
(d) the ion with 24 electrons, 30 neutrons, and a 3+ charge
Write the symbol for each of the following ions:
(a) the ion with a 3+ charge, 28 electrons, and a mass number of 71
(b) the ion with 36 electrons, 35 protons, and 45 neutrons
(c) the ion with 86 electrons, 142 neutrons, and a 4+ charge
(d) the ion with a 2+ charge, atomic number 38, and mass number 87
Open the Build an Atom simulation and click on the Atom icon.
(a) Pick any one of the first 10 elements that you would like to build and state its symbol.
(b) Drag protons, neutrons, and electrons onto the atom template to make an atom of your element.
State the numbers of protons, neutrons, and electrons in your atom, as well as the net charge and mass number.
(c) Click on “Net Charge” and “Mass Number,” check your answers to (b), and correct, if needed.
(d) Predict whether your atom will be stable or unstable. State your reasoning.
(e) Check the “Stable/Unstable” box. Was your answer to (d) correct? If not, first predict what you can do to make a stable atom of your element, and then do it and see if it works. Explain your reasoning.
Open the Build an Atom simulation
(a) Drag protons, neutrons, and electrons onto the atom template to make a neutral atom of Oxygen-16 and give the isotope symbol for this atom.
(b) Now add two more electrons to make an ion and give the symbol for the ion you have created.
Open the Build an Atom simulation
(a) Drag protons, neutrons, and electrons onto the atom template to make a neutral atom of Lithium-6 and give the isotope symbol for this atom.
(b) Now remove one electron to make an ion and give the symbol for the ion you have created.
Determine the number of protons, neutrons, and electrons in the following isotopes that are used in medical diagnoses:
(a) atomic number 9, mass number 18, charge of 1−
(b) atomic number 43, mass number 99, charge of 7+
(c) atomic number 53, atomic mass number 131, charge of 1−
(d) atomic number 81, atomic mass number 201, charge of 1+
(e) Name the elements in parts (a), (b), (c), and (d).
The following are properties of isotopes of two elements that are essential in our diet. Determine the number of protons, neutrons and electrons in each and name them.
(a) atomic number 26, mass number 58, charge of 2+
(b) atomic number 53, mass number 127, charge of 1−
Give the number of protons, electrons, and neutrons in neutral atoms of each of the following isotopes:
(a)
(b)
(c)
(d)
(e)
Give the number of protons, electrons, and neutrons in neutral atoms of each of the following isotopes:
(a)
(b)
(c)
(d)
(e)
Click on the site and select the “Mix Isotopes” tab, hide the “Percent Composition” and “Average Atomic Mass” boxes, and then select the element boron.
(a) Write the symbols of the isotopes of boron that are shown as naturally occurring in significant amounts.
(b) Predict the relative amounts (percentages) of these boron isotopes found in nature. Explain the reasoning behind your choice.
(c) Add isotopes to the black box to make a mixture that matches your prediction in (b). You may drag isotopes from their bins or click on “More” and then move the sliders to the appropriate amounts.
(d) Reveal the “Percent Composition” and “Average Atomic Mass” boxes. How well does your mixture match with your prediction? If necessary, adjust the isotope amounts to match your prediction.
(e) Select “Nature’s” mix of isotopes and compare it to your prediction. How well does your prediction compare with the naturally occurring mixture? Explain. If necessary, adjust your amounts to make them match “Nature’s” amounts as closely as possible.
Repeat Exercise 2.20 using an element that has three naturally occurring isotopes.
An element has the following natural abundances and isotopic masses: 90.92% abundance with 19.99 amu, 0.26% abundance with 20.99 amu, and 8.82% abundance with 21.99 amu. Calculate the average atomic mass of this element.
Average atomic masses listed by IUPAC are based on a study of experimental results. Bromine has two isotopes, 79 Br and 81 Br, whose masses (78.9183 and 80.9163 amu, respectively) and abundances (50.69% and 49.31%, respectively) were determined in earlier experiments. Calculate the average atomic mass of bromine based on these experiments.
Variations in average atomic mass may be observed for elements obtained from different sources. Lithium provides an example of this. The isotopic composition of lithium from naturally occurring minerals is 7.5% 6 Li and 92.5% 7 Li, which have masses of 6.01512 amu and 7.01600 amu, respectively. A commercial source of lithium, recycled from a military source, was 3.75% 6 Li (and the rest 7 Li). Calculate the average atomic mass values for each of these two sources.
The average atomic masses of some elements may vary, depending upon the sources of their ores. Naturally occurring boron consists of two isotopes with accurately known masses ( 10 B, 10.0129 amu and 11 B, 11.00931 amu). The actual atomic mass of boron can vary from 10.807 to 10.819, depending on whether the mineral source is from Turkey or the United States. Calculate the percent abundances leading to the two values of the average atomic masses of boron from these two countries.
The 18 O: 16 O abundance ratio in some meteorites is greater than that used to calculate the average atomic mass of oxygen on earth. Is the average mass of an oxygen atom in these meteorites greater than, less than, or equal to that of a terrestrial oxygen atom?
Explain why the symbol for an atom of the element oxygen and the formula for a molecule of oxygen differ.
Explain why the symbol for the element sulfur and the formula for a molecule of sulfur differ.
Write the molecular and empirical formulas of the following compounds:
(a)
(b)
(c)
(d)
Write the molecular and empirical formulas of the following compounds:
(a)
(b)
(c)
(d)
Determine the empirical formulas for the following compounds:
(a) caffeine, C 8 H 10 N 4 O 2
(b) sucrose, C 12 H 22 O 11
(c) hydrogen peroxide, H 2 O 2
(d) glucose, C 6 H 12 O 6
(e) ascorbic acid (vitamin C), C 6 H 8 O 6
Determine the empirical formulas for the following compounds:
(a) acetic acid, C 2 H 4 O 2
(b) citric acid, C 6 H 8 O 7
(c) hydrazine, N 2 H 4
(d) nicotine, C 10 H 14 N 2
(e) butane, C 4 H 10
Write the empirical formulas for the following compounds:
(a)
(b)
Open the Build a Molecule simulation and select the “Larger Molecules” tab. Select an appropriate atom's “Kit” to build a molecule with two carbon and six hydrogen atoms. Drag atoms into the space above the “Kit” to make a molecule. A name will appear when you have made an actual molecule that exists (even if it is not the one you want). You can use the scissors tool to separate atoms if you would like to change the connections. Click on “3D” to see the molecule, and look at both the space-filling and ball-and-stick possibilities.
(a) Draw the structural formula of this molecule and state its name.
(b) Can you arrange these atoms in any way to make a different compound?
Use the Build a Molecule simulation to repeat Exercise 2.34, but build a molecule with two carbons, six hydrogens, and one oxygen.
(a) Draw the structural formula of this molecule and state its name.
(b) Can you arrange these atoms to make a different molecule? If so, draw its structural formula and state its name.
(c) How are the molecules drawn in (a) and (b) the same? How do they differ? What are they called (the type of relationship between these molecules, not their names)?
Use the Build a Molecule simulation to repeat Exercise 2.34, but build a molecule with three carbons, seven hydrogens, and one chlorine.
(a) Draw the structural formula of this molecule and state its name.
(b) Can you arrange these atoms to make a different molecule? If so, draw its structural formula and state its name.
(c) How are the molecules drawn in (a) and (b) the same? How do they differ? What are they called (the type of relationship between these molecules, not their names)?
Write a sentence that describes how to determine the number of moles of a compound in a known mass of the compound if we know its molecular formula.

(a) Which has the largest number of molecules? Explain why.
(b) Which has the greatest mass? Explain why.
Which contains the greatest mass of oxygen: 0.75 mol of ethanol (C 2 H 5 OH), 0.60 mol of formic acid (HCO 2 H), or 1.0 mol of water (H 2 O)? Explain why.
Which contains the greatest number of moles of oxygen atoms: 1 mol of ethanol (C 2 H 5 OH), 1 mol of formic acid (HCO 2 H), or 1 mol of water (H 2 O)? Explain why.
How are the molecular mass and the molar mass of a compound similar and how are they different?
Calculate the molar mass of each of the following compounds:
(a) hydrogen fluoride, HF
(b) ammonia, NH 3
(c) nitric acid, HNO 3
(d) silver sulfate, Ag 2 SO 4
(e) boric acid, B(OH) 3
Calculate the molar mass of each of the following:
(a) S 8
(b) C 5 H 12
(c) Sc 2 (SO 4 ) 3
(d) CH 3 COCH 3 (acetone)
(e) C 6 H 12 O 6 (glucose)
Calculate the empirical or molecular formula mass and the molar mass of each of the following minerals:
(a) limestone, CaCO 3
(b) halite, NaCl
(c) beryl, Be 3 Al 2 Si 6 O 18
(d) malachite, Cu 2 (OH) 2 CO 3
(e) turquoise, CuAl 6 (PO 4 ) 4 (OH) 8 (H 2 O) 4
Calculate the molar mass of each of the following:
(a) the anesthetic halothane, C 2 HBrClF 3
(b) the herbicide paraquat, C 12 H 14 N 2 Cl 2
(c) caffeine, C 8 H 10 N 4 O 2
(d) urea, CO(NH 2 ) 2
(e) a typical soap, C 17 H 35 CO 2 Na
Determine the number of moles of compound and the number of moles of each type of atom in each of the following:
(a) 25.0 g of propylene, C 3 H 6
(b) 3.06 10 −3 g of the amino acid glycine, C 2 H 5 NO 2
(c) 25 lb of the herbicide Treflan, C 13 H 16 N 2 O 4 F (1 lb = 454 g)
(d) 0.125 kg of the insecticide Paris Green, Cu 4 (AsO 3 ) 2 (CH 3 CO 2 ) 2
(e) 325 mg of aspirin, C 6 H 4 (CO 2 H)(CO 2 CH 3 )
Determine the mass of each of the following:
(a) 0.0146 mol KOH
(b) 10.2 mol ethane, C 2 H 6
(c) 1.6 10 −3 mol Na 2 SO 4
(d) 6.854 10 3 mol glucose, C 6 H 12 O 6
(e) 2.86 mol Co(NH 3 ) 6 Cl 3
Determine the number of moles of the compound and determine the number of moles of each type of atom in each of the following:
(a) 2.12 g of potassium bromide, KBr
(b) 0.1488 g of phosphoric acid, H 3 PO 4
(c) 23 kg of calcium carbonate, CaCO 3
(d) 78.452 g of aluminum sulfate, Al 2 (SO 4 ) 3
(e) 0.1250 mg of caffeine, C 8 H 10 N 4 O 2
Determine the mass of each of the following:
(a) 2.345 mol LiCl
(b) 0.0872 mol acetylene, C 2 H 2
(c) 3.3 10 −2 mol Na 2 CO 3
(d) 1.23 10 3 mol fructose, C 6 H 12 O 6
(e) 0.5758 mol FeSO 4 (H 2 O) 7
The approximate minimum daily dietary requirement of the amino acid leucine, C 6 H 13 NO 2 , is 1.1 g. What is this requirement in moles?
Determine the mass in grams of each of the following:
(a) 0.600 mol of oxygen atoms
(b) 0.600 mol of oxygen molecules, O 2
(c) 0.600 mol of ozone molecules, O 3
A 55-kg woman has 7.5 10 −3 mol of hemoglobin (molar mass = 64,456 g/mol) in her blood. How many hemoglobin molecules is this? What is this quantity in grams?
Determine the number of atoms and the mass of zirconium, silicon, and oxygen found in 0.3384 mol of zircon, ZrSiO 4 , a semiprecious stone.
Determine which of the following contains the greatest mass of hydrogen: 1 mol of CH 4 , 0.6 mol of C 6 H 6 , or 0.4 mol of C 3 H 8 .
Determine which of the following contains the greatest mass of aluminum: 122 g of AlPO 4 , 266 g of Al 2 Cl 6 , or 225 g of Al 2 S 3 .
Diamond is one form of elemental carbon. An engagement ring contains a diamond weighing 1.25 carats (1 carat = 200 mg). How many atoms are present in the diamond?
The Cullinan diamond was the largest natural diamond ever found (January 25, 1905). It weighed 3104 carats (1 carat = 200 mg). How many carbon atoms were present in the stone?
One 55-gram serving of a particular cereal supplies 270 mg of sodium, 11% of the recommended daily allowance. How many moles and atoms of sodium are in the recommended daily allowance?
A certain nut crunch cereal contains 11.0 grams of sugar (sucrose, C 12 H 22 O 11 ) per serving size of 60.0 grams. How many servings of this cereal must be eaten to consume 0.0278 moles of sugar?
A tube of toothpaste contains 0.76 g of sodium monofluorophosphate (Na 2 PO 3 F) in 100 mL.
(a) What mass of fluorine atoms in mg was present?
(b) How many fluorine atoms were present?
Which of the following represents the least number of molecules?
(a) 20.0 g of H 2 O (18.02 g/mol)
(b) 77.0 g of CH 4 (16.06 g/mol)
(c) 68.0 g of C 3 H 6 (42.08 g/mol)
(d) 100.0 g of N 2 O (44.02 g/mol)
(e) 84.0 g of HF (20.01 g/mol)


8.1.5F: Religion
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Social class is associated with individuals’ religious affiliations and practices but not with religiosity itself.
Learning Objectives
- Explain how social class relates to religious affiliation, denomination and religiosity
Key Points
- Social class is an indicator of religious affiliation, with upper class members concentrated in formal denominations and lower class members concentrated in informal denominations.
- Social class is not an indicator of religiosity; members of each social class practice their faiths with a range of intensities.
- Income, and therefore social class, is related to an individual’s denomination. Religion is also strongly linked to level of education.
Key Terms
- Religious Affiliation : The measure of which religious denomination a person identifies with or practices
- religiosity : An index of how strongly religious a person is
Social class, measured by socioeconomic status, is associated with individuals’ religious affiliations and practices. This affiliation has more to do with how religion is practiced rather than degree of religiosity. Members of lower classes tend to be affiliated with more fundamentalist religions and sect-like groups. Members of the middle class tend to belong to more formal churches. For example, American Presbyterians and Episcopalians (two highly formal Protestant denominations), tend to have above average socioeconomic statuses. Methodists and Lutherans (two moderately formal Protestant denominations) tend to have about average SES. Baptists and members of Protestant fundamentalist sects (which tend to be decentralized and informal) have below average SES. Variations in SES across denomination reveal a correlation between religious affiliation and social class.
Social class is not significantly correlated to religiosity, an index of how strongly religious a person is. Religiosity is measured by tracking frequency of church attendance, church group involvement, frequency of prayer, and other such markers of strength of religious practice. Members of each social class show a range of religiosity.
On the other hand, income, and therefore social class, is related to an individual’s denomination. When one looks at average income by religion, there are clear differences. The highest-earning religion on average is Judaism, with an average income of $72,000 in 2000. This is dramatically higher than average; the next highest-earning denomination is Unitarianism at $56,000. Jehovah’s Witness, Church of God, and Seventh Day Adventists are at the bottom of the income distribution, with $24,000, $26,000, and $31,000 respectively.
Religion is also linked with education. 72% of Unitarian and 67% of Hindu adherents are college graduates, while only 12% of Jehovah’s Witness and 15% of Church of God members graduated from college.


23.10: For Further Exploration
Articles
Death of Stars
Hillebrandt, W., et al. “How To Blow Up a Star.” Scientific American (October 2006): 42. On supernova mechanisms.
Irion, R. “Pursuing the Most Extreme Stars.” Astronomy (January 1999): 48. On pulsars.
Kalirai, J. “New Light on Our Sun’s Fate.” Astronomy (February 2014): 44. What will happen to stars like our Sun between the main sequence and the white dwarf stages.
Kirshner, R. “Supernova 1987A: The First Ten Years.” Sky & Telescope (February 1997): 35.
Maurer, S. “Taking the Pulse of Neutron Stars.” Sky & Telescope (August 2001): 32. Review of recent ideas and observations of pulsars.
Zimmerman, R. “Into the Maelstrom.” Astronomy (November 1998): 44. About the Crab Nebula.
Gamma-Ray Bursts
Fox, D. & Racusin, J. “The Brightest Burst.” Sky & Telescope (January 2009): 34. Nice summary of the brightest burst observed so far, and what we have learned from it.
Nadis, S. “Do Cosmic Flashes Reveal Secrets of the Infant Universe?” Astronomy (June 2008): 34. On different types of gamma-ray bursts and what we can learn from them.
Naeye, R. “Dissecting the Bursts of Doom.” Sky & Telescope (August 2006): 30. Excellent review of gamma-ray bursts—how we discovered them, what they might be, and what they can be used for in probing the universe.
Zimmerman, R. “Speed Matters.” Astronomy (May 2000): 36. On the quick-alert networks for finding afterglows.
Zimmerman, R. “Witness to Cosmic Collisions.” Astronomy (July 2006): 44. On the Swift mission and what it is teaching astronomers about gamma-ray bursts.
Websites
Death of Stars
Crab Nebula: http://chandra.harvard.edu/xray_sources/crab/crab.html . A short, colorfully written introduction to the history and science involving the best-known supernova remant.
Introduction to Neutron Stars: https://www.astro.umd.edu/~miller/nstar.html . Coleman Miller of the University of Maryland maintains this site, which goes from easy to hard as you get into it, but it has lots of good information about corpses of massive stars.
Introduction to Pulsars (by Maryam Hobbs at the Australia National Telescope Facility): http://www.atnf.csiro.au/outreach/education/everyone/pulsars/index.html .
Magnetars, Soft Gamma Repeaters, and Very Strong Magnetic Fields: http://solomon.as.utexas.edu/magnetar.html . Robert Duncan, one of the originators of the idea of magnetars, assembled this site some years ago.
Gamma-Ray Bursts
Brief Intro to Gamma-Ray Bursts (from PBS’ Seeing in the Dark ): http://www.pbs.org/seeinginthedark/astronomy-topics/gamma-ray-bursts.html .
Discovery of Gamma-Ray Bursts: http://science.nasa.gov/science-news/science-at-nasa/1997/ast19sep97_2/ .
Missions to Detect and Learn More about Gamma-Ray Bursts:
- Fermi Space Telescope: https://www.nasa.gov/content/fermi/overview .
- INTEGRAL Spacecraft: http://www.esa.int/Science_Exploration/Space_Science/Integral_overview .
- SWIFT Spacecraft: https://swift.gsfc.nasa.gov/about_swift/ .
Videos
Death of Stars
BBC interview with Antony Hewish: http://www.bbc.co.uk/archive/scientists/10608.shtml . (40:54).
Black Widow Pulsars: The Vengeful Corpses of Stars: https://www.youtube.com/watch?v=Fn-3G_N0hy4 . A public talk in the Silicon Valley Astronomy Lecture Series by Dr. Roger Romani (Stanford University) (1:01:47).
Hubblecast 64: It all ends with a bang!: http://www.spacetelescope.org/videos/hubblecast64a/ . HubbleCast Program introducing Supernovae with Dr. Joe Liske (9:48).
Space Movie Reveals Shocking Secrets of the Crab Pulsar: hubblesite.org/newscenter/archive/releases/2002/24/video/c/. A sequence of Hubble and Chandra Space Telescope images of the central regions of the Crab Nebula have been assembled into a very brief movie accompanied by animation showing how the pulsar affects its environment; it comes with some useful background material (40:06).
Gamma-Ray Bursts
Gamma-Ray Bursts: The Biggest Explosions Since the Big Bang!: https://www.youtube.com/watch?v=ePo_EdgV764 . Edo Berge in a popular-level lecture at Harvard (58:50).
Gamma-Ray Bursts: Flashes in the Sky: https://www.youtube.com/watch?v=23EhcAP3O8Q . American Museum of Natural History Science Bulletin on the Swift satellite (5:59).
Overview Animation of Gamma-Ray Burst: http://news.psu.edu/video/296729/2013/11/27/overview-animation-gamma-ray-burst . Brief Animation of what causes a long-duration gamma-ray burst (0:55).


3.2.1: Double Integrals Part 1 (Exercises)
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In exercises 1 and 2, use the midpoint rule with \(m = 4\) and \(n = 2\) to estimate the volume of the solid bounded by the surface \(z = f(x,y)\) , the vertical planes \(x = 1\) , \(x = 2\) , \(y = 1\) , and \(y = 2\) , and the horizontal plane \(z = 0\) .
1) \(f(x,y) = 4x + 2y + 8xy\)
- Answer
- \(27\)
2) \(f(x,y) = 16x^2 + \frac{y}{2}\)
In exercises 3 and 4, estimate the volume of the solid under the surface \(z = f(x,y)\) and above the rectangular region \(R\) by using a Riemann sum with \(m = n = 2\) and the sample points to be the lower left corners of the sub-rectangles of the partition.
3) \(f(x,y) = \sin x - \cos y\), \(R = [0, \pi] \times [0, \pi]\)
- Answer
- \(0\)
4) \(f(x,y) = \cos x + \cos y\), \(R = [0, \pi] \times [0, \frac{\pi}{2}]\)
5) Use the midpoint rule with \(m = n = 2\) to estimate \(\iint_R f(x,y) \,dA\), where the values of the function f on \(R = [8,10] \times [9,11]\) are given in the following table.
| \(y\) | |||||
|---|---|---|---|---|---|
| \(x\) | 9 | 9.5 | 10 | 10.5 | 11 |
| 8 | 9.8 | 5 | 6.7 | 5 | 5.6 |
| 8.5 | 9.4 | 4.5 | 8 | 5.4 | 3.4 |
| 9 | 8.7 | 4.6 | 6 | 5.5 | 3.4 |
| 9.5 | 6.7 | 6 | 4.5 | 5.4 | 6.7 |
| 10 | 6.8 | 6.4 | 5.5 | 5.7 | 6.8 |
- Answer
- \(21.3\)
6) The values of the function \(f\) on the rectangle \(R = [0,2] \times [7,9]\) are given in the following table. Estimate the double integral \(\iint_R f(x,y)\,dA\) by using a Riemann sum with \(m = n = 2\). Select the sample points to be the upper right corners of the sub-squares of R .
| \(y_0 = 7\) | \(y_1 = 8\) | \(y_2 = 9\) | |
|---|---|---|---|
| \(x_0 = 0\) | 10.22 | 10.21 | 9.85 |
| \(x_1 = 1\) | 6.73 | 9.75 | 9.63 |
| \(x_2 = 2\) | 5.62 | 7.83 | 8.21 |
7) The depth of a children’s 4-ft by 4-ft swimming pool, measured at 1-ft intervals, is given in the following table.
- Estimate the volume of water in the swimming pool by using a Riemann sum with \(m = n = 2\). Select the sample points using the midpoint rule on \(R = [0,4] \times [0,4]\).
-
Find the average depth of the swimming pool.
\(y\) \(x\) 0 1 2 3 4 0 1 1.5 2 2.5 3 1 1 1.5 2 2.5 3 2 1 1.5 1.5 2.5 3 3 1 1 1.5 2 2.5 4 1 1 1 1.5 2
- Answer
-
a. 28 \(\text{ft}^3\)
b. 1.75 ft.
8) The depth of a 3-ft by 3-ft hole in the ground, measured at 1-ft intervals, is given in the following table.
- Estimate the volume of the hole by using a Riemann sum with \(m = n = 3\) and the sample points to be the upper left corners of the sub-squares of \(R\).
-
Find the average depth of the hole.
\(y\) \(x\) 0 1 2 3 0 6 6.5 6.4 6 1 6.5 7 7.5 6.5 2 6.5 6.7 6.5 6 3 6 6.5 5 5.6
9) The level curves \(f(x,y) = k\) of the function \(f\) are given in the following graph, where \(k\) is a constant.
- Apply the midpoint rule with \(m = n = 2\) to estimate the double integral \(\iint_R f(x,y)\,dA\), where \(R = [0.2,1] \times [0,0.8]\).
-
Estimate the average value of the function \(f\) on \(R\).
- Answer
-
a. Answers will vary somewhat, but should be close to these: \(f(0.4,0.2) \approx 0.125\), \(f(0.4,0.6) \approx− 0.2\), \(f(0.8,0.2) \approx 0.6\), and \(f(0.8,0.6) \approx 0.3\).
Since the area of each rectangle is \(\Delta A_i = (0.4)(0.4) = 0.16 \text{ units}^2\), \(\displaystyle \iint_R f(x,y)\,dA = \int_{0.2}^{1}\int_{0}^{0.8} f(x,y) \,dy\,dx \approx f(0.4,0.2)(0.16)+f(0.4,0.6)(0.16) +f(0.8,0.2) (0.16) + f(0.8,0.6)(0.16) = 0.132\)
b. \(f_{ave} = \dfrac{\int_{0.2}^{1}\int_{0}^{0.8} f(x,y) \,dy\,dx}{\text{area of }R}\approx \dfrac{\int_{0.2}^{1}\int_{0}^{0.8} f(x,y) \,dy\,dx}{0.64} \approx 0.206\)
10) The level curves \(f(x,y) = k\) of the function \(f\) are given in the following graph, where \(k\) is a constant.
- Apply the midpoint rule with \(m = n = 2\) to estimate the double integral \(\iint_R f(x,y)\,dA\), where \(R = [0.1,0.5] \times [0.1,0.5]\).
-
Estimate the average value of the function
f
on \(R\).
11) The solid lying under the surface \(z = \sqrt{4 - y^2}\) and above the rectangular region\( R = [0,2] \times [0,2]\) is illustrated in the following graph. Evaluate the double integral \(\iint_Rf(x,y)\), where \(f(x,y) = \sqrt{4 - y^2}\) by finding the volume of the corresponding solid.
- Answer
- \(2\pi\)
12) The solid lying under the plane \(z = y + 4\) and above the rectangular region \(R = [0,2] \times [0,4]\) is illustrated in the following graph. Evaluate the double integral \(\iint_R f(x,y)\,dA\), where \(f(x,y) = y + 4\), by finding the volume of the corresponding solid.
In the exercises 13 - 20, calculate the integrals by reversing the order of integration.
13) \(\displaystyle \int_{-1}^1\left(\int_{-2}^2 (2x + 3y + 5)\,dx \right) \space dy\)
- Answer
- \(40\)
14) \(\displaystyle \int_0^2\left(\int_0^1 (x + 2e^y + 3)\,dx \right) \space dy\)
15) \(\displaystyle \int_1^{27}\left(\int_1^2 (\sqrt[3]{x} + \sqrt[3]{y})\,dy \right) \space dx\)
- Answer
- \(\frac{81}{2} + 39\sqrt[3]{2}\)
16) \(\displaystyle \int_1^{16}\left(\int_1^8 (\sqrt[4]{x} + 2\sqrt[3]{y})\,dy \right) \space dx\)
17) \(\displaystyle \int_{\ln 2}^{\ln 3}\left(\int_0^1 e^{x+y}\,dy \right) \space dx\)
- Answer
- \(e - 1\)
18) \(\displaystyle \int_0^2\left(\int_0^1 3^{x+y}\,dy \right) \space dx\)
19) \(\displaystyle \int_1^6\left(\int_2^9 \frac{\sqrt{y}}{x^2}\,dy \right) \space dx\)
- Answer
- \(15 - \frac{10\sqrt{2}}{9}\)
20) \(\displaystyle \int_1^9 \left(\int_4^2 \frac{\sqrt{x}}{y^2}\,dy \right)\,dx\)
In exercises 21 - 34, evaluate the iterated integrals by choosing the order of integration.
21) \(\displaystyle \int_0^{\pi} \int_0^{\pi/2} \sin(2x)\cos(3y)\,dx \space dy\)
- Answer
- \(0\)
22) \(\displaystyle \int_{\pi/12}^{\pi/8}\int_{\pi/4}^{\pi/3} [\cot x + \tan(2y)]\,dx \space dy\)
23) \(\displaystyle \int_1^e \int_1^e \left[\frac{1}{x}\sin(\ln x) + \frac{1}{y}\cos (\ln y)\right] \,dx \space dy\)
- Answer
- \((e − 1)(1 + \sin 1 − \cos 1)\)
24) \(\displaystyle \int_1^e \int_1^e \frac{\sin(\ln x)\cos (\ln y)}{xy} \,dx \space dy\)
25) \(\displaystyle \int_1^2 \int_1^2 \left(\frac{\ln y}{x} + \frac{x}{2y + 1}\right)\,dy \space dx\)
- Answer
- \(\frac{3}{4}\ln \left(\frac{5}{3}\right) + 2 (\ln 2)^2 - \ln 2\)
26) \(\displaystyle \int_1^e \int_1^2 x^2 \ln(x)\,dy \space dx\)
27) \(\displaystyle \int_1^{\sqrt{3}} \int_1^2 y \space \arctan \left(\frac{1}{x}\right) \,dy \space dx\)
- Answer
- \(\frac{1}{8}[(2\sqrt{3} - 3) \pi + 6 \space \ln 2]\)
28) \(\displaystyle \int_0^1 \int_0^{1/2} (\arcsin x + \arcsin y)\,dy \space dx\)
29) \(\displaystyle \int_0^1 \int_1^2 xe^{x+4y}\,dy \space dx\)
- Answer
- \(\frac{1}{4}e^4 (e^4 - 1)\)
30) \(\displaystyle \int_1^2 \int_0^1 xe^{x-y}\,dy \space dx\)
31) \(\displaystyle \int_1^e \int_1^e \left(\frac{\ln y}{\sqrt{y}} + \frac{\ln x}{\sqrt{x}}\right)\,dy \space dx\)
- Answer
- \(4(e - 1)(2 - \sqrt{e})\)
32) \(\displaystyle \int_1^e \int_1^e \left(\frac{x \space \ln y}{\sqrt{y}} + \frac{y \space \ln x}{\sqrt{x}}\right)\,dy \space dx\)
33) \(\displaystyle \int_0^1 \int_1^2 \left(\frac{x}{x^2 + y^2} \right)\,dy \space dx\)
- Answer
- \(-\frac{\pi}{4} + \ln \left(\frac{5}{4}\right) - \frac{1}{2} \ln 2 + \arctan 2\)
34) \(\displaystyle \int_0^1 \int_1^2 \frac{y}{x + y^2}\,dy \space dx\)
In exercises 35 - 38, find the average value of the function over the given rectangles.
35)\(f(x,y) = −x +2y\), \(R = [0,1] \times [0,1]\)
- Answer
- \(\frac{1}{2}\)
36) \(f(x,y) = x^4 + 2y^3\), \(R = [1,2] \times [2,3]\)
37) \(f(x,y) = \sinh x + \sinh y\), \(R = [0,1] \times [0,2]\)
- Answer
- \(\frac{1}{2}(2 \space \cosh 1 + \cosh 2 - 3)\).
38) \(f(x,y) = \arctan(xy)\), \(R = [0,1] \times [0,1]\)
39) Let \(f\) and \(g\) be two continuous functions such that \(0 \leq m_1 \leq f(x) \leq M_1\) for any \(x ∈ [a,b]\) and \(0 \leq m_2 \leq g(y) \leq M_2\) for any\( y ∈ [c,d]\). Show that the following inequality is true:
\[m_1m_2(b-a)(c-d) \leq \int_a^b \int_c^d f(x) g(y)\,dy dx \leq M_1M_2 (b-a)(c-d). \nonumber \]
In exercises 40 - 43, use property v. of double integrals and the answer from the preceding exercise to show that the following inequalities are true.
40) \(\frac{1}{e^2} \leq \iint_R e^{-x^2 - y^2} \space dA \leq 1\), where \(R = [0,1] \times [0,1]\)
41) \(\frac{\pi^2}{144} \leq \iint_R \sin x \cos y \space dA \leq \frac{\pi^2}{48}\), where \(R = \left[ \frac{\pi}{6}, \frac{\pi}{3}\right] \times \left[ \frac{\pi}{6}, \frac{\pi}{3}\right]\)
42) \(0 \leq \iint_R e^{-y}\space \cos x \space dA \leq \frac{\pi}{2}\), where \(R = \left[0, \frac{\pi}{2}\right] \times \left[0, \frac{\pi}{2}\right]\)
43) \(0 \leq \iint_R (\ln x)(\ln y) \,dA \leq (e - 1)^2\), where \(R = [1, e] \times [1, e] \)
44) Let \(f\) and \(g\) be two continuous functions such that \(0 \leq m_1 \leq f(x) \leq M_1\) for any \(x ∈ [a,b]\) and \(0 \leq m_2 \leq g(y) \leq M_2\) for any \(y ∈ [c,d]\). Show that the following inequality is true:
\[(m_1 + m_2) (b - a)(c - d) \leq \int_a^b \int_c^d |f(x) + g(y)| \space dy \space dx \leq (M_1 + M_2)(b - a)(c - d) \nonumber \]
In exercises 45 - 48, use property v. of double integrals and the answer from the preceding exercise to show that the following inequalities are true.
45) \(\frac{2}{e} \leq \iint_R (e^{-x^2} + e^{-y^2}) \,dA \leq 2\), where \(R = [0,1] \times [0,1]\)
46) \(\frac{\pi^2}{36}\iint_R (\sin x + \cos y)\,dA \leq \frac{\pi^2 \sqrt{3}}{36}\), where \(R = [\frac{\pi}{6}, \frac{\pi}{3}] \times [\frac{\pi}{6}, \frac{\pi}{3}]\)
47) \(\frac{\pi}{2}e^{-\pi/2} \leq \iint_R (\cos x + e^{-y})\,dA \leq \pi\), where \(R = [0, \frac{\pi}{2}] \times [0, \frac{\pi}{2}]\)
48) \(\frac{1}{e} \leq \iint_R (e^{-y} - \ln x) \,dA \leq 2\), where \(R = [0, 1] \times [0, 1]\)
In exercises 49 - 50, the function \(f\) is given in terms of double integrals.
- Determine the explicit form of the function \(f\).
- Find the volume of the solid under the surface \(z = f(x,y)\) and above the region \(R\).
- Find the average value of the function \(f\) on \(R\).
- Use a computer algebra system (CAS) to plot \(z = f(x,y)\) and \(z = f_{ave}\) in the same system of coordinates.
49) [T] \(f(x,y) = \int_0^y \int_0^x (xs + yt) ds \space dt\), where \((x,y) \in R = [0,1] \times [0,1]\)
- Answer
-
a. \(f(x,y) = \frac{1}{2} xy (x^2 + y^2)\);
b. \(V = \int_0^1 \int_0^1 f(x,y)\,dx \space dy = \frac{1}{8}\);
c. \(f_{ave} = \frac{1}{8}\);d.
50) [T] \(f(x,y) = \int_0^x \int_0^y [\cos(s) + \cos(t)] \, dt \space ds\), where \((x,y) \in R = [0,3] \times [0,3]\)
51) Show that if \(f\) and \(g\) are continuous on \([a,b]\) and \([c,d]\), respectively, then
\(\displaystyle \int_a^b \int_c^d |f(x) + g(y)| dy \space dx = (d - c) \int_a^b f(x)\,dx\)
\(\displaystyle + \int_a^b \int_c^d g(y)\,dy \space dx = (b - a) \int_c^d g(y)\,dy + \int_c^d \int_a^b f(x)\,dx \space dy\).
52) Show that \(\displaystyle \int_a^b \int_c^d yf(x) + xg(y)\,dy \space dx = \frac{1}{2} (d^2 - c^2) \left(\int_a^b f(x)\,dx\right) + \frac{1}{2} (b^2 - a^2) \left(\int_c^d g(y)\,dy\right)\).
53) [T] Consider the function \(f(x,y) = e^{-x^2-y^2}\), where \((x,y) \in R = [−1,1] \times [−1,1]\).
- Use the midpoint rule with \(m = n = 2,4,..., 10\) to estimate the double integral \(I = \iint_R e^{-x^2 - y^2} dA\). Round your answers to the nearest hundredths.
- For \(m = n = 2\), find the average value of f over the region R . Round your answer to the nearest hundredths.
- Use a CAS to graph in the same coordinate system the solid whose volume is given by \(\iint_R e^{-x^2-y^2} dA\) and the plane \(z = f_{ave}\).
- Answer
-
a. For \(m = n = 2\), \(I = 4e^{-0.5} \approx 2.43\)
b. \(f_{ave} = e^{-0.5} \simeq 0.61\);c.
54) [T] Consider the function \(f(x,y) = \sin (x^2) \space \cos (y^2)\), where \((x,y \in R = [−1,1] \times [−1,1]\).
- Use the midpoint rule with \(m = n = 2,4,..., 10\) to estimate the double integral \(I = \iint_R \sin (x^2) \cos (y^2) \space dA\). Round your answers to the nearest hundredths.
- For \(m = n = 2\), find the average value of \(f\)over the region R. Round your answer to the nearest hundredths.
- Use a CAS to graph in the same coordinate system the solid whose volume is given by \(\iint_R \sin(x^2) \cos(y^2) \space dA\) and the plane \(z = f_{ave}\).
In exercises 55 - 56, the functions \(f_n\) are given, where \(n \geq 1\) is a natural number.
- Find the volume of the solids \(S_n\) under the surfaces \(z = f_n(x,y)\) and above the region \(R\).
- Determine the limit of the volumes of the solids \(S_n\) as \(n\) increases without bound.
55) \(f(x,y) = x^n + y^n + xy, \space (x,y) \in R = [0,1] \times [0,1]\)
- Answer
-
a. \(\frac{2}{n + 1} + \frac{1}{4}\)
b. \(\frac{1}{4}\)
56) \(f(x,y) = \frac{1}{x^n} + \frac{1}{y^n}, \space (x,y) \in R = [1,2] \times [1,2]\)
57) Show that the average value of a function \(f\) on a rectangular region \(R = [a,b] \times [c,d]\) is \(f_{ave} \approx \frac{1}{mn} \sum_{i=1}^m \sum_{j=1}^n f(x_{ij}^*,y_{ij}^*)\),where \((x_{ij}^*,y_{ij}^*)\) are the sample points of the partition of \(R\), where \(1 \leq i \leq m\) and \(1 \leq j \leq n\).
58) Use the midpoint rule with \(m = n\) to show that the average value of a function \(f\) on a rectangular region \(R = [a,b] \times [c,d]\) is approximated by
\[f_{ave} \approx \frac{1}{n^2} \sum_{i,j =1}^n f \left(\frac{1}{2} (x_{i=1} + x_i), \space \frac{1}{2} (y_{j=1} + y_j)\right). \nonumber \]
59) An isotherm map is a chart connecting points having the same temperature at a given time for a given period of time. Use the preceding exercise and apply the midpoint rule with \(m = n = 2\) to find the average temperature over the region given in the following figure.
- Answer
- \(56.5^{\circ}\) F; here \(f(x_1^*,y_1^*) = 71, \space f(x_2^*, y_1^*) = 72, \space f(x_2^*,y_1^*) = 40, \space f(x_2^*,y_2^*) = 43\), where \(x_i^*\) and \(y_j^*\) are the midpoints of the subintervals of the partitions of \([a,b]\) and \([c,d]\), respectively.


47 .
How would you describe the “positivity effect” regarding changes in perception in middle adulthood? What are the benefits of this perceptual shift? Use examples in your answer.
48 .
Menopause is a significant transition that women experience during middle adulthood, with both physical and emotional effects. How does culture influence the way women experience and cope with menopause? Discuss the role of societal expectations, cultural beliefs, and support systems in shaping the experiences of women going through this stage.
49 .
Evaluate how creativity changes during middle adulthood. What factors influence whether creativity remains stable, grows, or declines during this stage? Include considerations such as work, hobbies, and life experiences.  What strategies could you suggest to help reignite their creativity?
50 .
Examine the role of mental and emotional well-being in physical health during middle adulthood. How do stress, relationships, and work-life balance contribute to overall health outcomes? Imagine you are a health coach working with a middle-aged client who is struggling with these challenges. What specific recommendations would you give them to maintain positive mental health and overall well-being? Provide examples from the text that show how individuals can maintain positive mental health while navigating the challenges of middle adulthood.
51 .
In this chapter, you learned that job satisfaction may be impaired by remaining within the same organization for a long period of time. According to a 2023 report by insurance brokerage Gallagher (2023), more than half of employers surveyed reported an increase in employee turnover. Based on what you have learned in this chapter, what can businesses do to increase employee satisfaction and retention? Explain why you make these recommendations.


14.4: The speed of a wave
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In the previous section we found that the speed of a transverse wave in a rope is related to the ratio of the tension in the rope to the linear mass density of the rope:
\[\begin{aligned} v&= \sqrt{\frac{F_T}{\mu}}\end{aligned}\]
The speed of a wave in any medium is usually given by a ratio, where the numerator is a measure of how easy it is to deform the medium, and the denominator is measure of the inertia of the medium. For a rope, the tension is a measure of how stiff the rope is. A higher tension makes it more difficult to disturb the rope from equilibrium and it will “snap back” faster when disturbed, so the pulse will propagate faster. The heavier the rope, the harder it will be for the disturbance to propagate as the rope has more inertia, which will slow down the pulse.
The only way that a wave can propagate through a medium is if that medium can be deformed and the particles in the medium can be displaced from their equilibrium position, much like simple harmonic oscillators. The wave will propagate faster if those oscillators have a stiff spring constant and there is a strong force trying to restore them to equilibrium. However, if those oscillators have a large inertia, even with a large restoring force, they will accelerate back to their equilibrium with a smaller acceleration.
In general, the speed of a wave is given by:
\[\begin{aligned} v=\sqrt{\frac{\text{Stiffness of medium}}{\text{Inertia of medium}}}\end{aligned}\]
For example, the speed of longitudinal pressure waves in a solid is given by:
\[\begin{aligned} v=\sqrt{\frac{E}{\rho}}\end{aligned}\]
where \(E\) is the “elastic (or Young’s) modulus” for the material, and \(\rho\) is the density of the material. The elastic modulus of a solid is a measure of the material’s resistance to being deformed when a force (or pressure) is exerted on it. The more easily it is deformed, the lower its elastic modulus will be.
For the propagation of longitudinal pressure waves through a fluid, the speed is given by:
\[\begin{aligned} v=\sqrt{\frac{B}{\rho}}\end{aligned}\]
where \(B\) is the bulk modulus of the liquid, and \(\rho\) its density.
Exercise \(\PageIndex{1}\)
A wave will propagate faster through...
- ice.
- water.
- Answer
- A.


16.2A: Soil Composition
- Explain soil composition
Plants obtain inorganic elements from the soil, which serves as a natural medium for land plants. Soil is the outer, loose layer that covers the surface of Earth. Soil quality, a major determinant, along with climate, of plant distribution and growth, depends not only on the chemical composition of the soil, but also the topography (regional surface features) and the presence of living organisms.
Soil consists of these major components:
- inorganic mineral matter, about 40 to 45 percent of the soil volume
- organic matter, about 5 percent of the soil volume
- water, about 25 percent of the soil volume
- air, about 25 percent of the soil volume
The amount of each of the four major components of soil depends on the quantity of vegetation, soil compaction, and water present in the soil. A good, healthy soil has sufficient air, water, minerals, and organic material to promote and sustain plant life.
The organic material of soil, called humus, is made up of microorganisms (dead and alive), and dead animals and plants in varying stages of decay. Humus improves soil structure, providing plants with water and minerals. The inorganic material of soil is composed of rock, slowly broken down into smaller particles that vary in size. Soil particles that are 0.1 to 2 mm in diameter are sand. Soil particles between 0.002 and 0.1 mm are called silt, and even smaller particles, less than 0.002 mm in diameter, are called clay. Some soils have no dominant particle size, containing a mixture of sand, silt, and humus; these soils are called loams .
Key Points
- The chemical composition of the soil, the topography, and the presence of living organisms determines the quality of soil.
- In general, soil contains 40-45% inorganic matter, 5% organic matter, 25% water, and 25% air.
- In order to sustain plant life, the proper mix of air, water, minerals, and organic material is required.
- Humus, the organic material in soil, is composed of microorganisms (dead and alive) and decaying plants.
- The inorganic material of soil is composed of rock, which is broken down into small particles of sand (0.1 to 2 mm), silt (0.002 to 0.1 mm), and clay (less than 0.002 mm).
- Loam is a soil that is a mix sand, silt, and humus.
Key Terms
- loam : soil with no dominant particle size that contains a mixture of sand, silt, and humus
- humus : a large group of natural organic compounds found in the soil composed of decaying plants and dead and living microorganisms


1.4: Definitions of Elementary Number Theory
1.4.1: Even and Odd
If you divide a number by \(2\) and it comes out even (i.e. with no remainder) the number is said to be even. So the word even is related to division. It turns out that the concept even is better understood through thinking about multiplication.
An integer \(n\) is even exactly when there is an integer \(m\) such that \(n = 2m\).
You should note that there is a “two-way street” sort of quality to this definition – indeed with most, if not all, definitions. If a number is even, then we are guaranteed the existence of another integer half as big. On the other hand, if we can show that another integer half as big exists, then we know the original number is even. This two-wayness means that the definition is what is known as a biconditional , a concept which we’ll revisit in Section 2.2 .
A lot of people don’t believe that \(0\) should be counted as an even number. Now that we are armed with a precise definition, we can answer this question easily. Is there an integer \(x\) such that \(0 = 2x\)? Certainly! let \(x\) also be \(0\). (Notice that in the definition, nothing was said about \(m\) and \(n\) being distinct from one another.)
An integer is odd if it isn’t even. That is, amongst integers, there are only two possibilities: even or odd. We can also define oddness without reference to “even.”
An integer \(n\) is odd exactly when there is an integer \(m\) such that \(n = 2m + 1\).
1.4.2: Decimal and Base-\(n\) Notation
You can also identify even numbers by considering their decimal representation. Recall that each digit in the decimal representation of a number has a value that depends on its position. For example, the number \(3482\) really means \(3 · 10^3 + 4 · 10^2 + 8 · 10^1 + 2 · 10^0\). This is also known as place notation. The fact that we use the powers of \(10\) in our place notation is probably due to the fact that most humans have \(10\) fingers. It is possible to use any number in place of \(10\). In Computer Science there are \(3\) other bases in common use: \(2\), \(8\) and \(16\) – these are known (respectively) as binary, octal and hexadecimal notation. When denoting a number using some base other than \(10\), it is customary to append a subscript indicating the base. So, for example, \(1011_2\) is binary notation meaning \(1 · 2^3 + 0 · 2^2 + 1 · 2^1 + 1 · 2^0\) or \(8 + 2 + 1 = 11\). No matter what base we are using, the rightmost digit of the number multiplies the base raised to the \(0\)-th power. Any number raised to the \(0\)-th power is \(1\), and the rightmost digit is consequently known as the units digit. We are now prepared to give some statements that are equivalent to our definition of even. These statements truly don’t deserve the designation “theorem,” they are immediate consequences of the definition.
An integer is even if the units digit in its decimal representation is one of \(0\), \(2\), \(4\), \(6\) or \(8\).
An integer is even if the units digit in its binary representation is \(0\).
For certain problems it is natural to use some particular notational system. For example, the last theorem would tend to indicate that binary numbers are useful in problems dealing with even and odd. Given that there are many different notations that are available to us, it is obviously desirable to have means at our disposal for converting between them. It is possible to develop general rules for converting a base-\(a\) number to a base-\(b\) number (where a and b are arbitrary) but it is actually more convenient to pick a “standard” base (and since we’re human we’ll use base-\(10\)) and develop methods for converting between an arbitrary base and the “standard” one. Imagine that in the not-too-distant future we need to convert some numbers from the base-\(7\) system used by the Seven-lobed Amoebazoids from Epsilon Eridani III to the base-\(12\) scheme favored by the Dodecatons of Alpha-Centauri IV. We will need a procedure for converting base-\(7\) to base-\(10\) and another procedure for converting from base-\(10\) to base-\(12\). In the School House Rock episode “Little Twelve Toes” they describe base-\(12\) numeration in a way that is understandable for elementary school children – the digits they use are \(\{1, 2, 3, 4, 5, 6, 7, 8, 9, δ, \epsilon \}\), the last two digits (which are pronounced “dec” and “el”) are necessary since we need single symbols for the things we ordinarily denote using \(10\) and \(11\).
Converting from some other base to decimal is easy. You just use the definition of place notation. For example, to find what \(451663_7\) represents in decimal, just write
\(4·7^5+5·7^4+1·7^3+6·7^2+6·7+3 = 4·16807+5·2401+1·343+6·49+6·7+3 = 79915.\)
(Everything in the line above can be interpreted as a base-\(10\) number, and no subscripts are necessary for base-\(10\).)
Converting from decimal to some other base is harder. There is an algorithm called “repeated division” that we’ll explore a bit in the exercises for this section. For the moment, just verify that \(3δ2 \epsilon 7_{12}\) is also a representation of the number more conventionally written as \(79915\).
1.4.3: Divisibility
The notion of being even has an obvious generalization. Suppose we asked whether \(3\) divided evenly into a given number. Presumably, we could make a definition of what it meant to be threeven , but rather than doing so (or engaging in any further punnery) we shall instead move to a general definition. We need a notation for the situation when one number divides evenly into another. There are many ways to describe this situation in English, but essentially just one in “math,” we use a vertical bar – not a fraction bar. Indeed the difference between this vertical bar and the fraction symbol (\(|\) versus \(/\)) needs to be strongly stressed. The vertical bar when placed between two numbers is a symbol which asks the question “Does the first number divide evenly (i.e. with no remainder) into the second?” On the other hand the fraction bar asks you to actually carry out some division. The value of \(2 | 5\) is false, whereas the value of \(2/5\) is \(.4\).
As was the case in defining even, it turns out that it is best to think of multiplication, not division, when making a formal definition of this concept. Given any two integers \(n\) and \(d\) we define the symbol \(d|n\) by
\(d|n\) exactly when \(∃k ∈ Z\) such that \(n = kd\)
In spoken language the symbol \(d|n\) can be translated in a variety of ways:
- \(d\) is a divisor of \(n\).
- \(d\) divides \(n\) evenly.
- \(d\) is a factor of \(n\).
- \(n\) is an integer multiple of \(d\).
Although, by far the most popular way of expressing this concept is to just say “\(d\) divides \(n\).”
1.4.4: Floor and Ceiling
Suppose there is an elevator with a capacity of \(1300\) pounds. A large group of men who all weigh about \(200\) pounds want to ascend in it. How many should ride at a time? This is just a division problem, \(1300/200\) gives \(6.5\) men should ride together. Well, obviously putting half a person on an elevator is a bad idea – should we just round-up and let \(7\) ride together? Not if the \(1300\) pound capacity rating doesn’t have a safety margin! This is an example of the kind of problem in which the floor function is used. The floor function takes a real number as input and returns the next lower integer.
Suppose after a party we have \(43\) unopened bottles of beer. We’d like to store them in containers that hold \(12\) bottles each. How many containers will we need? Again, this is simply a division problem \(– 43/12 = 3.58333\). So we need \(3\) boxes and another \(7\) twelfths of a box. Obviously, we really need \(4\) boxes – at least one will have some unused space in it. In this sort of situation, we’re dealing with the ceiling function. Given a real number, the ceiling function rounds it up to the next integer.
Both of these functions are denoted using symbols that look very much like absolute value bars. The difference lies in some small horizontal strokes.
If \(x\) is a real number, its floor is denoted \(\lfloor x \rfloor\), and its ceiling is denoted \(\lceil x \rceil\). Here are the formal definitions:
\(y = \lfloor x \rfloor\) exactly when \(y ∈ Z\) and \(y ≤ x < y + 1\).
\(y = \lceil x \rceil\) exactly when \(y ∈ Z\) and \(y − 1 < x ≤ y\).
Basically, the definition of floor says that \(y\) is an integer that is less than or equal to \(x\), but \(y + 1\) definitely exceeds \(x\). The definition of ceiling can be paraphrased similarly.
1.4.5: Div and mod
In the next section, we’ll discuss the so-called division algorithm – this may be over-kill since you certainly already know how to do division! Indeed, in the U.S., long division is usually first studied in the latter half of elementary school, and division problems that don’t involve a remainder may be found as early as the first grade. Nevertheless, we’re going to discuss this process in sordid detail because it gives us a good setting in which to prove relatively easy statements. Suppose you are setting-up a long division problem in which the integer \(n\) is being divided by a positive divisor \(d\). (If you want to divide by a negative number, just divide by the corresponding positive number and then throw an extra minus sign on at the end.)
\( \;\;\;\underline{\;\;q\;\;\;} \\ d| \;\;n \\ \;\;\;\;... \\ \;\;\underline{\;\;\;\;\;\;\;\;} \\ \;\;\;\;\;r\)
Recall that the answer consists of two parts, a quotient \(q\), and a remainder \(r\). Of course, \(r\) may be zero, but also, the largest \(r\) can be is \(d − 1\). The assertion that this answer uniquely exists is known as the quotient-remainder theorem :
Given integers \(n\) and \(d > 0\), there are unique integers \(q\) and \(r\) such that \(n = qd + r\) and \(0 ≤ r < d\).
The words “div” and “mod” that appear in the title of this subsection provide mathematical shorthand for \(q\) and \(r\). Namely, “\(n \text{ mod } d\)” is a way of expressing the remainder \(r\), and “\(n \text{ div } d\)” is a way of expressing the quotient \(q\).
If two integers, \(m\) and \(n\), leave the same remainder when you divide them by \(d\), we say that they are congruent modulo \(d\). One could express this by writing \(n \text{ mod } d = m \text{ mod } d \), but usually we adopt a shorthand notation
\[n ≡ m (\text{mod } d).\]
If one is in a context in which it is completely clear what \(d\) is, it’s acceptable to just write \(n ≡ m\).
The “\(\text{mod}\)” operation is used quite a lot in mathematics. When we do computations modulo some number\(d\), (this is known as “modular arithmetic” or, sometimes, “clock arithmetic”) some very nice properties of “\(\text{mod}\)” come in handy
\[x + y \text{ mod } d = (x \text{ mod } d + y \text{ mod } d) \text{ mod } d\]
and
\[x · y \text{ mod } d = (x \text{ mod } d · y \text{ mod } d) \text{ mod } d.\]
These rules mean that we can either do the operations first, then reduce the answer \(\text{mod } d\) or we can do the reduction \(\text{mod } d\) first and then do the operations (although we may have to do one more round of reduction \(\text{mod } d\)).
For example, if we are working \(\text{mod } 10\), and want to compute \(87·96 \text{ mod } 10\), we can instead just compute \(7 · 6 \text{ mod } 10\), which is \(2\).
1.4.6: Binomial Coefficients
A “binomial” is a polynomial with \(2\) terms, for example \(x + 1\) or \(a + b\). The numbers that appear as the coefficients when one raises a binomial to some power are – rather surprisingly – known as binomial coefficients
Let’s have a look at the first several powers of \(a + b\).
\[\begin{array} (a + b)^0 &= 1 \\ (a + b)^1 &= a + b \\ (a + b)^2 &= a^2 + 2ab + b^2 \end{array}\]
To go much further than the second power requires a bit of work, but try the following
Multiply \((a+b)\) and \((a^2 + 2ab+b^2)\) in order to determine \((a+b)^3\). If you feel up to it, multiply \((a^2+2ab+b^2)\) times itself in order to find \((a+b)^4\).
Since we’re interested in the coefficients of these polynomials, it’s important to point out that if no coefficient appears in front of a term that means the coefficient is \(1\).
These binomial coefficients can be placed in an arrangement known as Pascal’s triangle 1 , which provides a convenient way to calculate small binomial coefficients
Notice that in the triangle there is a border on both sides containing 1’s and that the numbers on the inside of the triangle are the sum of the two numbers above them. You can use these facts to extend the triangle.
Add the next two rows to the Pascal triangle in Figure \(1.4.1\).
Binomial coefficients are denoted using a somewhat strange-looking symbol. The number in the \(k\)-th position in row number \(n\) of the triangle is denoted \(\binom{n}{k}\). This looks a little like a fraction, but the fraction bar is missing. Don’t put one in! It’s supposed to be missing. In spoken English, you say “\(n\) choose \(k\)” when you encounter the symbol \(\binom{n}{k}\).
There is a formula for the binomial coefficients – which is nice. Otherwise we’d need to complete a pretty huge Pascal triangle in order to compute something like \(\binom{52}{5}\). The formula involves factorial notation. Just to be sure we are all on the same page, we’ll define factorials before proceeding.
The symbol for factorials is an exclamation point following a number. This is just a short-hand for expressing the product of all the numbers up to a given one. For example \(7!\) means \(1 · 2 · 3 · 4 · 5 · 6 · 7\). Of course, there’s really no need to write the initial \(1\) — also, for some reason people usually write the product in decreasing order \((7! = 7 · 6 · 5 · 4 · 3 · 2)\).
The formula for a binomial coefficient is
\[\binom{n}{k} = \dfrac{n!}{k! · (n − k)!} \]
For example
\[\binom{5}{3} = \dfrac{5!}{3! · (5 − 3)!} = \dfrac{1 · 2 · 3 · 4 · 5}{ (1 · 2 · 3) · (1 · 2)} = 10 \]
A slightly more complicated example (and one that gamblers are fond of) is
\[\begin{array} \binom{52}{5} &= \dfrac{52!}{5! · (52 − 5)!} \\ &= \dfrac{1 · 2 · 3 · · · · 52}{(1 · 2 · 3 · 4 · 5) · (1 · 2 · 3 · · · · 47)} \\ &= \dfrac{48 · 49 · 50 · 51 · 52}{1 · 2 · 3 · 4 · 5} \\ &= 2598960. \end{array}\]
The reason that a gambler might be interested in the number we just calculated is that binomial coefficients do more than just give us the coefficients in the expansion of a binomial. They also can be used to compute how many ways one can choose a subset of a given size from a set. Thus \(\binom{52}{5}\) is the number of ways that one can get a \(5\) card hand out of a deck of \(52\) cards.
There are seven days in a week. In how many ways can one choose a set of three days (per week)
Exercises:
An integer \(n\\) is doubly-even if it is even, and the integer \(m\) guaranteed to exist because \(n\) is even is itself even. Is \(0\) doubly-even? What are the first \(3\) positive, doubly-even integers?
Dividing an integer by two has an interesting interpretation when using binary notation: simply shift the digits to the right. Thus, \(22 = 10110_2\) when divided by two gives \(1011_2\) which is \(8 + 2 + 1 = 11\). How can you recognize a doubly-even integer from its binary representation?
The octal representation of an integer uses powers of \(8\) in place notation. The digits of an octal number run from \(0\) to \(7\), one never sees \(8\)’s or \(9\)’s. How would you represent \(8\) and \(9\) as octal numbers? What octal number comes immediately after \(777_8\)? What (decimal) number is \(777_8\)?
One method of converting from decimal to some other base is called repeated division . One divides the number by the base and records the remainder – one then divides the quotient obtained by the base and records the remainder. Continue dividing the successive quotients by the base until the quotient is smaller than the base. Convert \(3267\) to base-\(7\) using repeated division. Check your answer by using the meaning of base-\(7\) place notation. (For example \(54321_7\) means \(5 · 7^4 + 4 · 7^3 + 3 · 7^2 + 2 · 7^1 + 1 · 7^0\).)
State a theorem about the octal representation of even numbers.
In hexadecimal (base-\(16\)) notation one needs \(16\) “digits,” the ordinary digits are used for \(0\) through \(9\), and the letters \(\text{A}\) through \(\text{F}\) are used to give single symbols for \(10\) through \(15\). The first \(32\) natural number in hexadecimal are:
\(1,2,3,4,5,6,7,8,9,\text{A},\text{B},\text{C},\text{D},\text{E},\text{F},10,11,12,13,14,15,16,17,18,19,1\text{A},1\text{B},1\text{C},1\text{D},1\text{E},1\text{F},20\).
Write the next \(10\) hexadecimal numbers after \(\text{AB}\).
For conversion between the three bases used most often in Computer Science we can take binary as the “standard” base and convert using a table look-up. Each octal digit will correspond to a binary triple, and each hexadecimal digit will correspond to a \(4\)-tuple of binary numbers. Complete the following tables. (As a check, the \(4\)-tuple next to \(A\) in the table for hexadecimal should be \(1010\) – which is nice since \(A\) is really \(10\) so if you read that as “ten-ten” it is a good aid to memory.)
| octal | binary |
|---|---|
| 0 | 000 |
| 1 | 001 |
| 2 | |
| 3 | |
| 4 | |
| 5 | |
| 6 | |
| 7 |
| hexadecimal | binary |
|---|---|
| 0 | 0000 |
| 1 | 0001 |
| 2 | 0010 |
| 3 | |
| 4 | |
| 5 | |
| 6 | |
| 7 | |
| 8 | |
| 9 | |
| \(\text{A}\) | |
| \(\text{B}\) | |
| \(\text{C}\) | |
| \(\text{D}\) | |
| \(\text{E}\) | |
| \(\text{F}\) |
Use the tables from the previous problem to make the following conversions.
- Convert \(757_8\) to binary.
- Convert \(1007_8\) to hexadecimal.
- Convert \(100101010110_2\) to octal.
- Convert \(1111101000110101_2\) to hexadecimal.
- Convert \(FEED_{16}\) to binary.
- Convert \(FFFFFF_{16}\) to octal.
Try the following conversions between various number systems:
- Convert \(30\) (base \(10\)) to binary.
- Convert \(69\) (base \(10\)) to base \(5\).
- Convert \(12223\) to binary.
- Convert \(12347\) to base \(10\).
- Convert \(EEED_{15}\) to base \(12\). (Use \(\{1, 2, 3 . . . 9, d, e\}\) as the digits in base \(12\).)
- Convert \(6789\) to hexadecimal.
It is a well known fact that if a number is divisible by \(3\), then \(3\) divides the sum of the (decimal) digits of that number. Is this result true in base \(7\)? Do you think this result is true in any base?
Suppose that \(340\) pounds of sand must be placed into bags having a \(50\) pound capacity. Write an expression using either floor or ceiling notation for the number of bags required.
True or false?
\(\lfloor \dfrac{n}{d} \rfloor < \lceil \dfrac{n}{d} \rceil \)
for all integers \(n\) and \(d > 0\). Support your claim.
What is the value of \(\lceil π \rceil^2 − \lceil π^2 \rceil\)?
Assuming the symbols \(n\), \(d\), \(q\) and \(r\) have meanings as in the quotient-remainder theorem (Theorem \(1.4.3\)). Write expressions for \(q\) and \(r\), in terms of \(n\) and \(d\) using floor and/or ceiling notation.
Calculate the following quantities:
- \(3 \text{ mod } 5\)
- \(37 \text{ mod } 7\)
- \(1000001 \text{ mod } 100000\)
- \(6 \text{ div } 6\)
- \(7 \text{ div } 6\)
- \(1000001 \text{ div } 2\)
Calculate the following binomial coefficients:
- \(\binom{3}{0}\)
- \(\binom{7}{7}\)
- \(\binom{13}{5}\)
- \(\binom{13}{8}\)
- \(\binom{52}{7}\)
An ice cream shop sells the following flavors: chocolate, vanilla, strawberry, coffee, butter pecan, mint chocolate chip and raspberry. How many different bowls of ice cream – with three scoops – can they make?


The majority of western musicals performed today derive from Greek roots in theater and performance. However, many other forms of musical theater existed to east in Asia such as Chinese, Japanese, and Taiwanese Operas. The first recorded Chinese opera was known as the Canjun Opera and was supposedly performed during the Zhoa Dynasty sometime between 319 AD – 351 AD. Another eastern form of musical theater is Noh. Noh Is the Japanese term for “talent” or “skill” and is used to describe a Japanese musical. It has been performed since the mid-14 th century and is still practiced today in specific Noh theaters. Taiwanese Opera or Koa-á-hì is the only known form of drama to emerge from Taiwan as early as the 18 th century. Most of the songs are stories and folktales with occasional supernatural elements.


10.1: Introduction
In the previous chapter, we examined various ways in which nations can be represented by musical means. Our examples included national anthems, pieces of art music that draw from folk sources, and a complete musical tradition with national significance. In this chapter, we will turn our attention to the role of music in political action, and we will see how music can not only represent a nation but also directly influence its character and future.
As we already discussed in the context of national anthems, music can bring people together and inspire them to feel sympathetic toward one another. Music—especially participatory music—can create community. Music can also inspire an emotional response in the listener. These capacities are combined to great effect when music is used to urge a group of people to think or act along specific political lines.
Sometimes, music is employed in support of a political candidate or ideology. The most prominent examples of this include campaign songs, music played at political rallies, and the soundtracks to political advertisements. In all of these cases, music is used to inspire sympathy with a cause or candidate and win the support of the listener. However, the creators of the music in question do not always have a say in its political use. More often than not, music is coopted for political purposes and used in ways that the original artist had never imagined (and may not approve of). In such cases, we see the capacity of music to express meaning beyond its creator’s intent.
Other times, music is used to combat a dominant political force. The most prominent example of this is the protest song , which is used to inspire resistance to an ongoing political action. When music is used to protest ruling powers, however, there is always an element of danger. Those who position themselves in opposition to governments or bosses risk serious reprisal. For this reason, protest music is not always easy to decipher. Sometimes it is overt in its meaning, clearly indicating the objectionable situation and the desired action. Sometimes, however, it is covert , instead disguising its meaning so that only certain listeners will receive the message. In such cases, it can be difficult to determine exactly what a piece of music “means” or even to prove that it is subversive at all. We will examine both overt and covert examples in this chapter.


1.1: An Introduction to the Human Body
Introduction
Though you may approach a course in biology of aging strictly as a requirement for your field of study, the knowledge you gain in this course will serve you well in many aspects of your life. An understanding of aging is not only fundamental to any career in human services, but it can also benefit your own health. Familiarity with the human body can help you make healthful choices and prompt you to take appropriate action when signs of illness arise. Your knowledge in this field will help you understand changes to the human body through out your own and others life times. Additionally you will gain a knowledge base so that you can better understand future news about nutrition, medications, medical devices, and procedures. At some point, everyone will have a problem with some aspect of his or her body and your knowledge can help you to be a better parent, spouse, partner, friend, colleague, or caregiver.
This chapter begins with an overview of the human body and a preview of the body functions. It then covers the characteristics of life and how the body works to maintain stable conditions. Next the chapter will cover the standard methods used to study aging, rates of aging, and general changes to the body as it ages. It ends with standard terms related to aging that will serve as a foundation for more comprehensive information covered later in the text.
Contributors and Attributions
- Chapter 1: An Introduction to the Human Body. Authored by : OpenStax College. Provided by : Rice University. Located at : openstaxcollege.org/files/textbook_version/low_res_pdf/13/col11496-lr.pdf. Project : Anatomy & Physiology. License : CC BY: Attribution . License Terms : Download for free at http://cnx.org/content/col11496/latest/ .