1.5E: Exercises for Equations of Lines and Planes in Space
-
- Last updated
- Save as PDF
In exercises 1 - 4, points \( P\) and \( Q\) are given. Let \( L\) be the line passing through points \( P\) and \( Q\) .
a. Find the vector equation of line \( L\) .
b. Find parametric equations of line \( L\) .
c. Find symmetric equations of line \( L\) .
d. Find parametric equations of the line segment determined by \( P\) and \( Q\) .
1) \( P(−3,5,9), \quad Q(4,−7,2)\)
- Answer:
-
a. \(\vecs r=⟨−3,5,9⟩+t⟨7,−12,−7⟩, \quad t∈R;\)
b. \( x=−3+7t, \quad y=5−12t, \quad z=9−7t, \quad t∈R;\)
c. \(\dfrac{x+3}{7}=\dfrac{y−5}{−12}=\dfrac{z−9}{−7};\)
d. \(x=−3+7t, \quad y=5−12t, \quad z=9−7t, \quad 0 \le t \le 1\)
2) \( P(4,0,5), \quad Q(2,3,1)\)
3) \( P(−1,0,5), \quad Q(4,0,3)\)
- Answer:
-
a. \(\vecs r=⟨−1,0,5⟩+t⟨5,0,−2⟩, \quad t∈R;\)
b. \( x=−1+5t,y=0,z=5−2t, \quad t∈R;\)
c. \(\dfrac{x+1}{5}=\dfrac{z−5}{−2}, \quad y=0;\)
d. \(x=−1+5t, \quad y=0, \quad z=5−2t, \quad t∈[0,1]\)
4) \( P(7,−2,6), \quad Q(−3,0,6)\)
For exercises 5 - 8, point \( P\) and vector \(\vecs v\) are given. Let \( L\) be the line passing through point \( P\) with direction \(\vecs v\) .
a. Find parametric equations of line \( L\) .
b. Find symmetric equations of line \( L\) .
c. Find the intersection of the line with the \(xy\) -plane.
5) \( P(1,−2,3),\,\vecs v=⟨1,2,3⟩\)
- Answer:
-
a. \(x=1+t, \quad y=−2+2t, \quad z=3+3t, \quad t∈R;\)
b. \( \dfrac{x−1}{1}=\dfrac{y+2}{2}=\dfrac{z−3}{3};\)
c. \((0,−4,0)\)
6) \( P(3,1,5), \,\vecs v=⟨1,1,1⟩\)
7) \( P(3,1,5), \,\vecs v=\vecd{QR},\) where \( Q(2,2,3)\) and \( R(3,2,3)\)
- Answer:
-
a. \(x=3+t, \quad y=1, \quad z=5, \quad t∈R;\)
b. \( y=1, \quad z=5;\)
c. The line does not intersect the \(xy\)-plane.
8) \( P(2,3,0), \,\vecs v=\vecd{QR},\) where \( Q(0,4,5)\) and \( R(0,4,6)\)
For exercises 9 and 10, line \( L\) is given.
a. Find a point \( P\) that belongs to the line and a direction vector \(\vecs v\) of the line. Express \(\vecs v\) in component form.
b. Find the distance from the origin to line \( L\) .
9) \( x=1+t, \quad y=3+t, \quad z=5+4t, \quad t∈R\)
- Answer:
-
a. A possible point and direction vector are \(P(1,3,5)\) and \(\vecs v=⟨1,1,4⟩\), but these answers are not unique.
b. \( \sqrt{3} \) units
10) \( −x=y+1, \quad z=2\)
11) Find the distance between point \( A(−3,1,1)\) and the line of symmetric equations
\( x=−y=−z.\)
- Answer:
- \( \frac{2\sqrt{2}}{\sqrt{3}} = \frac{2\sqrt{6}}{3} \) units
12) Find the distance between point \( A(4,2,5)\) and the line of parametric equations
\( x=−1−t, \; y=−t, \; z=2, \; t∈R.\)
For exercises 13 - 14, lines \( L_1\) and \( L_2\) are given.
a. Verify whether lines \( L_1\) and \( L_2\) are parallel.
b. If the lines \( L_1\) and \( L_2\) are parallel, then find the distance between them.
13) \( L_1:x=1+t, \quad y=t, \quad z=2+t, \quad t∈R\) and \(L_2:x−3=y−1=z−3\)
- Answer:
-
a. Parallel;
b. \( \frac{\sqrt{2}}{\sqrt{3}} = \frac{\sqrt{6}}{3} \) units
14) \( L_1:x=2, \quad y=1, \quad z=t, \quad t∈R\) and \( L_2:x=1, \quad y=1, \quad z=2−3t, \quad t∈R\)
15) Show that the line passing through points \( P(3,1,0)\) and \( Q(1,4,−3)\) is perpendicular to the line with equation \( x=3t, \quad y=-32+8t, \quad z=−9+6t, \quad t∈R.\)
- Answer:
-
\( \vecd{PQ} = \langle -2, 3, -3 \rangle\) is the direction vector of the line through points \(P\) and \(Q\), and the direction vector of the line defined by the parametric equations above is \(\vecs v = \langle 3, 8, 6 \rangle.\)
Since \(\vecs v \cdot \vecd{PQ} = -6 + 24 - 18 = 0\), the two direction vectors are orthogonal.
Now all we need to show is that the two lines intersect.
The line through points \( P(3,1,0)\) and \( Q(1,4,−3)\) has parametric equations: \(x = 3 - 2u\), \(y = 1 + 3u\), and \(z = -3u\).
Setting the \(x\)- and \(z\)-coordinates of the two lines equal, we obtain the system of equations:
\[3t = 3 - 2u \quad\text{and}\quad -9 + 6t = -3u \nonumber\]
Solving this system using substitution gives us, \(u = -3\) and \(t = 3\). Plugging these values of \(t\) and \(u\) back into the parametric equations of these two lines gives us the intersection point with coordinates \(\left(9, -8, 9\right)\) on both lines.
Therefore the lines intersect and the line through points \(P\) and \(Q\) with direction vector \(\vecd{PQ} \) is perpendicular to the other line.
16) Are the lines of equations \( x=−2+2t, \quad y=−6, \quad z=2+6t, \quad t∈R\) and \( x=−1+t, \quad y=1+t, \quad z=t, \quad t∈R,\) perpendicular to each other?
17) Find the point of intersection of the lines of equations \( x=−2y=3z\) and \( x=−5−t, \quad y=−1+t, \quad z=t−11, \quad t∈R.\)
- Answer:
- \( (−12,6,−4)\)
18) Find the intersection point of the \(x\)-axis with the line of parametric equations \( x=10+t, \quad y=2−2t, \quad z=−3+3t, \quad t∈R.\)
For exercises 19 - 22, lines \( L_1\) and \( L_2\) are given. Determine whether the lines are equal , parallel but not equal , skew , or intersecting .
19) \( L_1:x=y−1=−z\) and \( L_2:x−2=−y=\dfrac{z}{2}\)
- Answer:
- The lines are skew.
20) \( L_1:x=2t, \quad y=0, \quad z=3, \quad t∈R\) and \( L_2:x=0, \quad y=8+s, \quad z=7+s, \quad s∈R\)
21) \( L_1:x=−1+2t, \quad y=1+3t, \quad z=7t, \quad t∈R\) and \( L_2:x−1=\frac{2}{3}(y−4)=\frac{2}{7}z−2\)
- Answer:
- The lines are equal.
22) \( L_1:3x=y+1=2z\) and \( L_2:x=6+2t, \quad y=17+6t, \quad z=9+3t, \quad t∈R\)
23) Consider line \( L\) of symmetric equations \( x−2=−y=\dfrac{z}{2}\) and point \( A(1,1,1).\)
a. Find parametric equations for a line parallel to \( L\) that passes through point \( A\).
b. Find symmetric equations of a line skew to \( L\) and that passes through point \( A\).
c. Find symmetric equations of a line that intersects \( L\) and passes through point \( A\).
- Answer:
-
a. \(x=1+t, \quad y=1−t, \quad z=1+2t, \quad t∈R\)
b. For instance, the line passing through \( A\) with direction vector \( j:x=1,z=1\)
c. For instance, the line passing through \( A\) and point \( (2,0,0)\) that belongs to \( L\) is a line that intersects; \( L:\frac{x−1}{−1}=y−1=z−1\)
24) Consider line \( L\) of parametric equations \( x=t, \quad y=2t, \quad z=3, \quad t∈R.\)
a. Find parametric equations for a line parallel to \( L\) that passes through the origin.
b. Find parametric equations of a line skew to \( L\) that passes through the origin.
c. Find symmetric equations of a line that intersects \( L\) and passes through the origin.
For exercises 25 - 28, point \( P\) and vector \(\vecs n\) are given.
a. Find the standard form of the equation of the plane that passes through \( P\) and has normal vector \(\vecs n\) .
b. Find the general form of the equation of the plane that passes through \( P\) and has normal vector \(\vecs n\) .
25) \( P(0,0,0), \quad \vecs n=3\mathbf{\hat i}−2\mathbf{\hat j}+4\mathbf{\hat k}\)
- Answer:
-
a. \(3x−2y+4z=0\)
b. \(3x−2y+4z=0\)
26) \( P(3,2,2), \quad \vecs n=2\mathbf{\hat i}+3\mathbf{\hat j}−\mathbf{\hat k}\)
27) \( P(1,2,3), \quad \vecs n=⟨1,2,3⟩\)
- Answer:
-
a. \((x−1)+2(y−2)+3(z−3)=0\)
b. \(x+2y+3z−14=0\)
28) \( P(0,0,0), \quad \vecs n=⟨−3,2,−1⟩\)
For exercises 29 - 32, the equation of a plane is given.
a. Find normal vector \(\vecs n\) to the plane. Express \(\vecs n\) using standard unit vectors.
b. Find the intersections of the plane with each of the coordinate axes (its intercepts).
c. Sketch the plane.
29) [T] \( 4x+5y+10z−20=0\)
- Answer:
-
a. \(\vecs n=4\mathbf{\hat i}+5\mathbf{\hat j}+10\mathbf{\hat k}\)
b. \((5,0,0), \,(0,4,0),\) and \( (0,0,2)\)c.
30) \( 3x+4y−12=0\)
31) \( 3x−2y+4z=0\)
- Answer:
-
a. \(\vecs n=3\mathbf{\hat i}−2\mathbf{\hat j}+4\mathbf{\hat k}\)
b. \((0,0,0)\)c.
32) \( x+z=0\)
33) Given point \( P(1,2,3)\) and vector \(\vecs n=\mathbf{\hat i}+\mathbf{\hat j}\), find point \( Q\) on the \(x\)-axis such that \( \vecd{PQ}\) and \(\vecs n\) are orthogonal.
- Answer:
- \( (3,0,0)\)
34) Show there is no plane perpendicular to \(\vecs n=\mathbf{\hat i}+\mathbf{\hat j}\) that passes through points \( P(1,2,3)\) and \( Q(2,3,4)\).
35) Find parametric equations of the line passing through point \( P(−2,1,3)\) that is perpendicular to the plane of equation \( 2x−3y+z=7.\)
- Answer:
- \( x=−2+2t, \quad y=1−3t, \quad z=3+t, \quad t∈R\)
36) Find symmetric equations of the line passing through point \( P(2,5,4)\) that is perpendicular to the plane of equation \( 2x+3y−5z=0.\)
37) Show that line \( \dfrac{x−1}{2}=\dfrac{y+1}{3}=\dfrac{z−2}{4}\) is parallel to plane \( x−2y+z=6\).
38) Find the real number \( α\) such that the line of parametric equations \( x=t, \quad y=2−t, \quad z=3+t, \quad t∈R\) is parallel to the plane of equation \( αx+5y+z−10=0.\)
For exercises 39 - 42, the equations of two planes are given.
a. Determine whether the planes are parallel , orthogonal , or neither .
b. If the planes are neither parallel nor orthogonal, then find the measure of the angle between the planes. Express the answer in degrees rounded to the nearest integer.
c. If the planes intersect, find the line of intersection of the planes, providing the parametric equations of this line.
39) [T] \( x+y+z=0, \quad 2x−y+z−7=0\)
- Answer:
-
a. The planes are neither parallel nor orthogonal.
b. \(62°\)
c. \(x = -1 + 2t\)
\(y = -4 + t\)
\(z = 5 - 3t\)
40) \( 5x−3y+z=4, \quad x+4y+7z=1\)
41) \( x−5y−z=1, \quad 5x−25y−5z=−3\)
- Answer:
- a. The planes are parallel.
42) [T] \( x−3y+6z=4, \quad 5x+y−z=4\)
For exercises 43 - 46, determine whether the given line intersects with the given plane. If they do intersect, state the point of intersection.
43) Plane: \(2x + y - z = 11\) Line: \(x = 1 + t, \quad y = 3 - 2t, \quad z = 2 +4t\)
- Answer:
- They intersect at point \( (-1, 7, -6) \).
44) Plane: \(-x + 2y + z = 2\) Line: \(x = 1 + 2t, \quad y = -2 + t, \quad z = 5 - 3t\)
- Answer:
- They intersect at point \( \left(-\frac{1}{3}, \, -\frac{8}{3}, \, 7\right) \).
45) Plane: \(x - 3y + 2z = 4\) Line: \(x = 2 - t, \quad y = t, \quad z = 4 +2t\)
- Answer:
- The line does not intersect with this plane.
46) Plane: \(x - 3y + 2z = 10\) Line: \(x = 2 - t, \quad y = t, \quad z = 4 +2t\)
- Answer:
- The line is actually fully contained in this plane, so every point on the line is on the plane. For example, when \(t = 0\) we have the point, \((2, 0, 4)\).
47) Show that the lines of equations \( x=t, \quad y=1+t, \quad z=2+t, \quad t∈R,\) and \( \dfrac{x}{2}=\dfrac{y−1}{3}=z−3\) are skew, and find the distance between them.
- Answer:
- \( \frac{1}{\sqrt{6}} = \frac{\sqrt{6}}{6}\) units
48) Show that the lines of equations \( x=−1+t, \quad y=−2+t, \quad z=3t, \quad t∈R,\) and \( x=5+s, \quad y=−8+2s, \quad z=7s, \quad s∈R\) are skew, and find the distance between them.
49) Consider point \( C(−3,2,4)\) and the plane of equation \( 2x+4y−3z=8\).
a. Find the radius of the sphere with center \(C\) tangent to the given plane.
b. Find point \(P\) of tangency.
- Answer:
-
a. \(r = \frac{18}{\sqrt{29}} = \frac{18\sqrt{29}}{29}\)
b. \(P\left(−\frac{51}{29},\frac{130}{29},\frac{62}{29}\right)\)
50) Consider the plane of equation \( x−y−z−8=0.\)
a. Find the equation of the sphere with center \(C\) at the origin that is tangent to the given plane.
b. Find parametric equations of the line passing through the origin and the point of tangency.
51) Two children are playing with a ball. The girl throws the ball to the boy. The ball travels in the air, curves \( 3\) ft to the right, and falls \( 5\) ft away from the girl (see the following figure). If the plane that contains the trajectory of the ball is perpendicular to the ground, find its equation.
- Answer:
- \( 4x−3y=0\)
52) [T] John allocates \( d\) dollars to consume monthly three goods of prices \( a,b\), and \( c\). In this context, the budget equation is defined as \( ax+by+cz=d,\) where \( x≥0,\, y≥0\), and \( z≥0\) represent the number of items bought from each of the goods. The budget set is given by \( \big\{(x,y,z)\,|\,ax+by+cz≤d,\;x≥0,\;y≥0,\;z≥0\big\},\) and the budget plane is the part of the plane of equation \( ax+by+cz=d\) for which \( x≥0,\,y≥0\), and \( z≥0\). Consider \( a=$8, \,b=$5, \,c=$10,\) and \( d=$500.\)
a. Use a CAS to graph the budget set and budget plane.
b. For \( z=25,\) find the new budget equation and graph the budget set in the same system of coordinates.
53) [T] Consider \(\vecs r(t)=⟨\sin t,\cos t,2t⟩\) the position vector of a particle at time \( t∈[0,3]\), where the components of \(\vecs r\) are expressed in centimeters and time is measured in seconds. Let \( \vecd{OP}\) be the position vector of the particle after \( 1\) sec.
a. Determine the velocity vector \(\vecs v(1)\) of the particle after \( 1\) sec.
b. Find the scalar equation of the plane that is perpendicular to \( v(1)\) and passes through point \( P\). This plane is called the normal plane to the path of the particle at point \( P\).
c. Use a CAS to visualize the path of the particle along with the velocity vector and normal plane at point \( P\).
- Answer:
-
a. \(\vecs v(1)=⟨\cos 1,−\sin 1, 2⟩\)
b. \( (\cos 1)(x−\sin 1)−(\sin 1)(y−\cos 1)+2(z−2)=0\)
c.
54) [T] A solar panel is mounted on the roof of a house. The panel may be regarded as positioned at the points of coordinates (in meters) \( A(8,0,0), \, B(8,18,0), \, C(0,18,8),\) and \( D(0,0,8)\) (see the following figure).
a. Find the general form of the equation of the plane that contains the solar panel by using points \( A, \, B,\) and \( C\), and show that its normal vector is equivalent to \( \vecd{AB}×\vecd{AD}.\)
b. Find parametric equations of line \( L_1\) that passes through the center of the solar panel and has direction vector \(\vecs s=\frac{1}{\sqrt{3}}\mathbf{\hat i}+\frac{1}{\sqrt{3}}\mathbf{\hat j}+\frac{1}{\sqrt{3}}\mathbf{\hat k},\) which points toward the position of the Sun at a particular time of day.
c. Find symmetric equations of line \( L_2\) that passes through the center of the solar panel and is perpendicular to it.
d. Determine the angle of elevation of the Sun above the solar panel by using the angle between lines \( L_1\) and \( L_2\).
Contributors
Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org .
Problems 15 and 43 - 46 created by Paul Seeburger


3.5: Chapter Summary
The Renaissance period was truly a time of great discovery in science, music, society, and the visual arts. The reemergence and renewed interests of Greek and Roman history/culture is still current in today’s modern society. Performing music outside of the church in courts and the public really began to thrive in the Renais- sance and continues today in the music industry. Many of the master works, both sacred and secular, from the Renaissance are still appreciated and continue to be the standard for today’s music industry. Songs of love, similar to Renaissance chan- sons, are still composed and performed today. The beauty of Renaissance music, as well as the other arts, is reintroduced and appreciated in modern-day theater performances and visually in museums. The results of the Protestant Reformation are still felt today, and the struggles between contemporary and traditional church worship continues very much as it did during the Renaissance. As we continue our reading and study of music through the Baroque period, try to recall the changes and trends of the Medieval and Renaissance eras and how they thread their way through history to today. Music and the Arts do not just occur; they evolve and also remain the same.


11.1: Module Introduction
Children’s lullabies were quite popular in decades past. Many of us learned a number of these lullabies as children, though they do not seem to be as popular today. In Rhymes for the Nursery, published by sisters Jane and Ann Taylor in 1806 in London was a simple rhyme about a star. Many of us have heard and even sung the first verse of what we call Twinkle, twinkle little star. And some of these questions in this children’s lullaby are still the same astronomers and astrophysicists are asking today.
The Star
Twinkle, twinkle little star,
How I wonder what you are,
Up above the world so high,
Like a diamond in the sky.
When the blazing sun is gone,
When he nothing shines upon,
Then you show your little light,
Twinkle, twinkle, all the night.
Then the traveller in the dark,
Thanks you for your tiny spark,
He could not see which way to go,
If you did not twinkle so.

As your bright and tiny spark,
Lights the traveller in the dark.
Though I know not what you are,
Twinkle, twinkle, little star.
Twinkle, twinkle, little star.
How I wonder what you are.
Up above the world so high,
Like a diamond in the sky.
Twinkle, twinkle, little star.
How I wonder what you are.
How I wonder what you are.
Jane Taylor
The Star,
Rhymes for the Nursery
, 1806
This module looks at the characteristics of stars, the terminology astronomers use to describe the stars, and the types and classifications of stars.
Objectives
Upon completion of this module, the student will be able to:Upon completion of this module, the student will be able to:
- Identify the basic stellar characteristics
- Describe how astronomers use parallax
- Identify terms, such as light-year, parsec, stellar luminosity, and magnitude system
- Detail Stellar Types and those responsible for the development of the stellar types
- Describe how the H-R Diagram works and what it shows
- Identify the different types of binary stars and variable stars
- Differentiate between globular and open clusters


43.2: Finding and Interpreting the Mean as the Balance Point
Lesson
Let's look at another way to understand the mean of a data set.
Exercise \(\PageIndex{1}\): Which One Doesn't Belong: Division
Which expression does not belong? Be prepared to explain your reasoning.
\(\frac{8+8+4+4}{4}\qquad\frac{10+10+4}{4}\qquad\frac{9+9+5+5}{4}\qquad\frac{6+6+6+6+6}{5}\)
Exercise \(\PageIndex{2}\): Travel Times (Part 1)
Here is the data set from an earlier lesson showing how long it takes for Diego to walk to school, in minutes, over 5 days. The mean number of minutes is 11.
\(12\qquad 7\qquad 13\qquad 9\qquad 14\)
- Represent Diego’s data on a dot plot. Mark the location of the mean with a triangle.
-
The mean can also be seen as a
measure of center
that balances the points in a data set. If we find the distance between every point and the mean, add the distances on each side of the mean, and compare the two sums, we can see this balancing.
-
Record the distance between each point and 11 and its location relative to 11.
time in minutes distance from 11 left of 11 or right of 11? \(12\) \(7\) \(13\) \(9\) \(14\) Table \(\PageIndex{1}\) -
Sum of distances left of 11:___________ Sum of distances right of 11:___________
What do you notice about the two sums?
-
Record the distance between each point and 11 and its location relative to 11.
-
Can another point that is
not
the mean produce similar sums of distances?
Let’s investigate whether 10 can produce similar sums as those of 11.-
Complete the table with the distance of each data point from 10.
time in minutes distance from 10 left of 10 or right of 10? \(12\) \(7\) \(13\) \(9\) \(14\) Table \(\PageIndex{2}\) -
Sum of distances left of 10:___________ Sum of distances right of 10:___________
What do you notice about the two sums?
-
Complete the table with the distance of each data point from 10.
- Based on your work so far, explain why the mean can be considered a balance point for the data set.
Exercise \(\PageIndex{3}\): Travel Times (Part 2)
- Here are dot plots showing how long Diego’s trips to school took in minutes—which you studied earlier—and how long Andre’s trips to school took in minutes. The dot plots include the means for each data set, marked by triangles.
- Which of the two data sets has a larger mean? In this context, what does a larger mean tell us?
- Which of the two data sets has larger sums of distances to the left and right of the mean? What do these sums tell us about the variation in Diego’s and Andre’s travel times?
- Here is a dot plot showing lengths of Lin's trips to school.
- Calculate the mean of Lin’s travel times.
-
Complete the table with the distance between each point and the mean as well whether the point is to the left or right of the mean.
time in minutes distance from the mean left or right of the mean? \(22\) \(18\) \(11\) \(8\) \(11\) Table \(\PageIndex{3}\) - Find the sum of distances to the left of the mean and the sum of distances to the right of the mean.
- Use your work to compare Lin’s travel times to Andre’s. What can you say about their average travel times? What about the variability in their travel times?
Summary
The mean is often used as a measure of center of a distribution. This is because the mean of a distribution can be seen as the “balance point” for the distribution. Why is this a good way to think about the mean? Let’s look at a very simple set of data on the number of cookies that each of eight friends baked:
\(19\qquad 20\qquad 20\qquad 21\qquad 21\qquad 22\qquad 22\qquad 23\)
Here is a dot plot showing the data set.
The distribution shown is completely symmetrical. The mean number of cookies is 21, because \((19+20+20+21+21+22+22+23)\div 8=21\). If we mark the location of the mean on the dot plot, we can see that the data points could balance at 21.
In this plot, each point on either side of the mean has a mirror image. For example, the two points at 20 and the two at 22 are the same distance from 21, but each pair is located on either side of 21. We can think of them as balancing each other around 21.
Similarly, the points at 19 and 23 are the same distance from 21 but are on either side of it. They, too, can be seen as balancing each other around 21.
We can say that the distribution of the cookies has a center at 21 because that is its balance point, and that the eight friends, on average, baked 21 cookies.
Even when a distribution is not completely symmetrical, the distances of values below the mean, on the whole, balance the distances of values above the mean.
Glossary Entries
Definition: Average
The average is another name for the mean of a data set.
For the data set 3, 5, 6, 8, 11, 12, the average is 7.5.
\(3+5+6+8+11+12=45\)
\(45\div 6=7.5\)
Definition: Mean
The mean is one way to measure the center of a data set. We can think of it as a balance point. For example, for the data set 7, 9, 12, 13, 14, the mean is 11.
To find the mean, add up all the numbers in the data set. Then, divide by how many numbers there are. \(7+9+12+13+14=55\) and \(55\div 5=11\).
Definition: Measure of Center
A measure of center is a value that seems typical for a data distribution.
Mean and median are both measures of center.
Practice
Exercise \(\PageIndex{4}\)
On school days, Kiran walks to school. Here are the lengths of time, in minutes, for Kiran’s walks on 5 school days:
\(16\qquad 11\qquad 18\qquad 12\qquad 13\)
- Create a dot plot for Kiran’s data.
- Without calculating, decide if 15 minutes would be a good estimate of the mean. If you think it is a good estimate, explain your reasoning. If not, give a better estimate and explain your reasoning.
- Calculate the mean for Kiran’s data.
-
In the table, record the distance of each data point from the mean and its location relative to the mean.
time in minutes distance from the mean left or right of the mean? \(16\) \(11\) \(18\) \(12\) \(13\) Table \(\PageIndex{4}\) - Calculate the sum of all distances to the left of the mean, then calculate the sum of distances to the right of the mean. Explain how these sums show that the mean is a balance point for the values in the data set.
Exercise \(\PageIndex{5}\)
Noah scored 20 points in a game. Mai's score was 30 points. The mean score for Noah, Mai, and Clare was 40 points. What was Clare's score? Explain or show your reasoning.
Exercise \(\PageIndex{6}\)
Compare the numbers using >, <, or =.
- \(-2\underline{\qquad} 3\)
- \(|-12|\underline{\qquad} |15|\)
- \(3\underline{\qquad} -4\)
- \(|15|\underline{\qquad} |-12|\)
- \(7\underline{\qquad} -11\)
- \(-4\underline{\qquad} |5|\)
(From Unit 7.1.7)
Exercise \(\PageIndex{7}\)
- Plot \(\frac{2}{3}\) and \(\frac{3}{4}\) on a number line.
- Is \(\frac{2}{3}<\frac{3}{4}\), or is \(\frac{3}{4}<\frac{2}{3}\)? Explain how you know.
(From Unit 7.1.3)
Exercise \(\PageIndex{8}\)
Select all the expressions that represent the total area of the large rectangle.
- \(5(x+y)\)
- \(5+xy\)
- \(5x+5y\)
- \(2(5+x+y)\)
- \(5xy\)
(From Unit 6.2.5)


2.6: The Most Probable Population Set at Constant N, V, and T
-
- Last updated
- Save as PDF
We are imagining that we can examine a collection of \(N\) distinguishable molecules and determine the energy of each molecule in the collection at any particular instant. If we do so, we find the population set, \(\{N_1,\ N_2,\dots ,N_i,\dots .\}\), that characterizes the system at that instant. In Section 3.9 , we introduce the idea that the most probable population set, \(\{N^{\textrm{⦁}}_1,N^{\textrm{⦁}}_2,\dots N^{\textrm{⦁}}_i,.,,,\}\), or its proxy, \(\{NP\left({\epsilon }_1\right),NP\left({\epsilon }_2\right),\dots ,NP\left({\epsilon }_i\right),\dots .\}\), is the best prediction we can make about the outcome of a future replication of this measurement. In Section 20.2 , we hypothesize that the properties of the system when it is characterized by the most probable population set are indistinguishable from the properties of the system at equilibrium.
Now let us show that this hypothesis is implied by the central limit theorem. We suppose that the population set that characterizes the system varies from instant to instant and that we can find this population set at any given instant. The population set that we find at a particular instant comprises a random sample of \(N\) molecular energies. For this sample, we can find the average energy from
\[\overline{\epsilon }=\sum^{\infty }_{i=1}{\left(\frac{N_i}{N}\right)}{\epsilon }_i \nonumber \]
The expected value of the molecular energy is \[\left\langle \epsilon \right\rangle =\sum^{\infty }_{i=1}{P_i{\epsilon }_i} \nonumber \]
It is important that we remember that \(\overline{\epsilon }\) and \(\left\langle \epsilon \right\rangle\) are not the same thing. There is a distribution of \(\overline{\epsilon }\) values, one \(\overline{\epsilon }\) value for each of the possible population sets \(\{N_1,\ N_2,\dots ,N_i,\dots .\}\). In contrast, when \(N\), \(V\), and \(T\) are fixed, the expected value, \(\left\langle \epsilon \right\rangle\), is a constant; the value of \(\left\langle \epsilon \right\rangle\) is completely determined by the values of the variables that determine the state of the system and fix the probabilities \(P_i\). If our theory is to be useful, the value of \(\left\langle \epsilon \right\rangle\) must be the per-molecule energy that we observe for the macroscopic system we are modeling.
According to the central limit theorem, the average energy of a randomly selected sample, \(\overline{\epsilon }\), approaches the expected value for the distribution, \(\left\langle \epsilon \right\rangle\), as the number of molecules in the sample becomes arbitrarily large. In the present instance, we hypothesize that the most probable population set, or its proxy, characterizes the equilibrium system. When \(N\) is sufficiently large, this hypothesis implies that the probability of the \(i^{th}\) energy level is given by \(P_i={N^{\textrm{⦁}}_i}/{N}\). Then the expected value of a molecular energy is
\[\left\langle \epsilon \right\rangle =\sum^{\infty }_{i=1}{P_i{\epsilon }_i}=\sum^{\infty }_{i=1}{\left(\frac{N^{\textrm{⦁}}_i}{N}\right){\epsilon }_i} \nonumber \]
Since the central limit theorem asserts that \(\overline{\epsilon }\) approaches \(\left\langle \epsilon \right\rangle\) as \(N\) becomes arbitrarily large:
\[0={\mathop{\mathrm{lim}}_{N\to \infty } \left(\overline{\epsilon }-\left\langle \epsilon \right\rangle \ \right)\ }={\mathop{\mathrm{lim}}_{N\to \infty } \sum^{\infty }_{i=1}{\left(\frac{N_i}{N}-P_i\right)}\ }{\varepsilon }_i={\mathop{\mathrm{lim}}_{N\to \infty } \sum^{\infty }_{i=1}{\left(\frac{N_i}{N}-\frac{N^{\textrm{⦁}}_i}{N}\right)}{\epsilon }_i\ } \nonumber \]
One way for the limit of this sum to be zero is for the limit of every individual term to be zero. If the \({\epsilon }_i\) were arbitrary, this would be the only way that the sum could always be zero. However, the \({\epsilon }_i\) and the \(P_i\) are related, so we might think that the sum is zero because of these relationships.
To see that the limit of every individual term must in fact be zero, we devise a new distribution. We assign a completely arbitrary number, \(X_i\), to each energy level. Now the \(i^{th}\) energy level is associated with an \(X_i\) as well as an \({\epsilon }_i\). We have an \(X\) distribution as well as an energy distribution. We can immediately calculate the expected value of \(X\). It is
\[\left\langle X\right\rangle =\sum^{\infty }_{i=1}{P_iX_i} \nonumber \]
When we find the population set \(\{N_1,\ N_2,\dots ,N_i,\dots .\}\), we can calculate the corresponding average value of \(X\). It is \[\overline{X}=\sum^{\infty }_{i=1}{\left(\frac{N_i}{N}\right)}X_i \nonumber \]
The central limit theorem applies to any distribution. So, it certainly applies to the \(X\) distribution; the average value of \(X\) approaches the expected value of \(X\) as \(N\) becomes arbitrarily large:
\[0={\mathop{\mathrm{lim}}_{N\to \infty } \left(\overline{X}-\left\langle X\right\rangle \ \right)\ }={\mathop{\mathrm{lim}}_{N\to \infty } \sum^{\infty }_{i=1}{\left(\frac{N_i}{N}-P_i\right)}\ }X_i={\mathop{\mathrm{lim}}_{N\to \infty } \sum^{\infty }_{i=1}{\left(\frac{N_i}{N}-\frac{N^{\textrm{⦁}}_i}{N}\right)}X_i\ } \nonumber \]
Now, because the \(X_i\) can be chosen completely arbitrarily, the only way that the limit of this sum can always be zero is that every individual term becomes zero.
In the limit as \(N\to \infty\), we find that
\[{N_i}/{N}\to {N^{\textrm{⦁}}_i}/{N} \nonumber \]
As the number of molecules in the equilibrium system becomes arbitrarily large, the fraction of the molecules in each energy level at an arbitrarily selected instant approaches the fraction in that energy level in the equilibrium-characterizing most-probable population set, \(\{N^{\textrm{⦁}}_1,N^{\textrm{⦁}}_2,\dots N^{\textrm{⦁}}_i\dots \}\). In other words, the only population sets that we have any significant chance of observing in a large equilibrium system are population sets whose occupation fractions, \({N_i}/{N}\), are all very close to those, \({N^{\textrm{⦁}}_i}/{N}\), in the equilibrium-characterizing population set. Estimating \(P_i\) as the ratio \({N_i}/{N}\) gives essentially the same result whichever of these population sets we use. Below, we see that the \({\epsilon }_i\) and the \(P_i\) determine the thermodynamic properties of the system. Consequently, when we calculate any observable property of the macroscopic system, each of these population sets gives the same result.
Since the only population sets that we have a significant chance of observing are those for which
\[{N_i}/{N}\approx {N^{\textrm{⦁}}_i}/{N} \nonumber \]
we frequently say that we can ignore all but the most probable population set. What we have in mind is that the most probable population set is the only one we need in order to calculate the macroscopic properties of the equilibrium system. We are incorrect, however, if we allow ourselves to think that the most probable population set is necessarily much more probable than any of the others. Nor does the fact that the \({N_i}/{N}\) are all very close to the \({N^{\textrm{⦁}}_i}/{N}\) mean that the \(N_i\) are all very close to the \(N^{\textrm{⦁}}_i\). Suppose that the difference between the two ratios is \({10}^{-10}\). If \(N={10}^{20}\), the difference between \(N_i\) and \(N^{\textrm{⦁}}_i\) is \({10}^{10}\), which probably falls outside the range of values that we usually understand by the words “very close.”
We develop a theory that includes a mathematical model for the probability that a molecule has any one of its quantum-mechanically possible energies. It turns out that we are frequently interested in macroscopic systems in which the number of energy levels greatly exceeds the number of molecules. For such systems, we find \(NP_i\ll 1\), and it is no longer possible to say that a single most-probable population set, \(\{N^{\textrm{⦁}}_1,N^{\textrm{⦁}}_2,\dots N^{\textrm{⦁}}_i,\dots \}\), describes the equilibrium state of the system. When it is very unlikely that any energy level is occupied by more than one molecule, the probability of any population set in which any \(N_i\) is greater than one becomes negligibly small. We can approximate the total probability sum as
\[1={\left(P_1+P_2+\dots +P_i+\dots \right)}^N\approx \sum_{\{N_i\}}{N!}P^{N_1}_1P^{N_2}_2\dots P^{N_i}_i\dots \nonumber \]
However, the idea that the proxy, \(\{NP\left({\epsilon }_1\right),NP\left({\epsilon }_2\right),\dots ,NP\left({\epsilon }_i\right),\dots .\}\), describes the equilibrium state of the system remains valid. In these circumstances, a great many population sets can have essentially identical properties; the properties calculated from any of these are indistinguishable from each other and indistinguishable from the properties calculated from the proxy. Since the equilibrium properties are fixed, the value of these extended products is fixed. For any of the population sets available to such a system at equilibrium, we have
\[P^{N_1}_1P^{N_2}_2\dots P^{N_i}_i\dots =P^{{NP}_1}_1P^{{NP}_2}_2\dots P^{{NP}_i}_i\dots =\mathrm{constant} \nonumber \]
It follows that, for some constant, \(c\), we have
\[c=\sum^{\infty }_{i=1}{NP_i{ \ln P_i\ }}=N\sum^{\infty }_{i=1}{P_i{ \ln P_i\ }} \nonumber \]
As it evolves, we see that the probability of finding a molecule in an energy level is the central feature of our theory.


3.1.2: Exploring Circles
Lesson
Let's explore circles.
Exercise \(\PageIndex{1}\): How Do You Figure?
Here are two figures.
Figure C looks more like Figure A than Figure B. Sketch what Figure C might look like. Explain your reasoning.
Exercise \(\PageIndex{2}\): Sorting Round Objects
Your teacher will give you some pictures of different objects.
- How could you sort these pictures into two groups? Be prepared to share your reasoning.
- Work with your partner to sort the pictures into the categories that your class has agreed on. Pause here so your teacher can review your work.
- What are some characteristics that all circles have in common?
- Put the circular objects in order from smallest to largest.
- Select one of the pictures of a circular object. What are some ways you could measure the actual size of your circle?
Are you ready for more?
On January 3rd, Earth is 147,500,000 kilometers away from the Sun. On July 4th, Earth is 152,500,000 kilometers away from the Sun. The Sun has a radius of about 865,000 kilometers.
Could Earth’s orbit be a circle with some point in the Sun as its center? Explain your reasoning.
Exercise \(\PageIndex{3}\): Measuring Circles
Priya, Han, and Mai each measured one of the circular objects from earlier.
- Priya says that the bike wheel is 24 inches.
- Han says that the yo-yo trick is 24 inches.
- Mai says that the glow necklace is 24 inches.
- Do you think that all these circles are the same size?
- What part of the circle did each person measure? Explain your reasoning.
Exercise \(\PageIndex{4}\): Drawing Circles
Spend some time familiarizing yourself with the tools that are available in this applet.
Draw and label each circle.
- Circle A, with a diameter of 6 cm.
- Circle B, with a radius of 5 cm. Pause here so your teacher can review your work.
- Circle C, with a radius that is equal to Circle A’s diameter.
- Circle D, with a diameter that is equal to Circle B’s radius.
- Use a compass to recreate one of these designs.
Summary
A circle consists of all of the points that are the same distance away from a particular point called the center of the circle.
A segment that connects the center with any point on the circle is called a radius . For example, segments \(QG\), \(QH\), \(QI\), and \(QJ\) are all radii of circle 2. (We say one radius and two radii.) The length of any radius is always the same for a given circle. For this reason, people also refer to this distance as the radius of the circle.
A segment that connects two opposite points on a circle (passing through the circle’s center) is called a diameter . For example, segments \(AB\), \(CD\), and \(EF\) are all diameters of circle 1. All diameters in a given circle have the same length because they are composed of two radii. For this reason, people also refer to the length of such a segment as the diameter of the circle.
The circumference of a circle is the distance around it. If a circle was made of a piece of string and we cut it and straightened it out, the circumference would be the length of that string. A circle always encloses a circular region. The region enclosed by circle 2 is shaded, but the region enclosed by circle 1 is not. When we refer to the area of a circle, we mean the area of the enclosed circular region.
Glossary Entries
Definition: Circle
A circle is made out of all the points that are the same distance from a given point.
For example, every point on this circle is 5 cm away from point \(A\), which is the center of the circle.
Definition: Circumference
The circumference of a circle is the distance around the circle. If you imagine the circle as a piece of string, it is the length of the string. If the circle has radius \(r\) then the circumference is \(2\pi r\).
The circumference of a circle of radius 3 is \(2\cdot\pi\cdot 3\), which is \(6\pi\), or about 18.85.
Definition: Diameter
A diameter is a line segment that goes from one edge of a circle to the other and passes through the center. A diameter can go in any direction. Every diameter of the circle is the same length. We also use the word diameter to mean the length of this segment.
Definition: Radius
A radius is a line segment that goes from the center to the edge of a circle. A radius can go in any direction. Every radius of the circle is the same length. We also use the word radius to mean the length of this segment.
For example, \(r\) is the radius of this circle with center \(O\).
Practice
Exercise \(\PageIndex{5}\)
Use a geometric tool to draw a circle. Draw and measure a radius and a diameter of the circle.
Exercise \(\PageIndex{6}\)
Here is a circle with center \(H\) and some line segments and curves joining points on the circle.
Identify examples of the following. Explain your reasoning.
- Diameter
- Radius
Exercise \(\PageIndex{7}\)
Lin measured the diameter of a circle in two different directions. Measuring vertically, she got 3.5 cm, and measuring horizontally, she got 3.6 cm. Explain some possible reasons why these measurements differ.
Exercise \(\PageIndex{8}\)
A small, test batch of lemonade used \(\frac{1}{4}\) cup of sugar added to 1 cup of water and \(\frac{1}{4}\) cup of lemon juice. After confirming it tasted good, a larger batch is going to be made with the same ratios using 10 cups of water. How much sugar should be added so that the large batch tastes the same as the test batch?
(From Unit 2.1.1)
Exercise \(\PageIndex{9}\)
The graph of a proportional relationship contains the point with coordinates \((3,12)\). What is the constant of proportionality of the relationship?
(From Unit 2.4.4)


2.10: Gas Heat – Basic Principle
Click play on the following audio player to listen along as you read this section.
One of the most common ways to heat a home or other type of building, is with a gas-fired furnace . In contrast to an electric furnace, a gas fired furnace converts the stored chemical energy in methane rich natural gas to produce heat. While natural gas is cleaner than other fossil fuels, it still emits carbon when consumed, and so it is a greenhouse gas emitter. Its chief benefit is that it is often a cheaper alternative to expensive electric heating bills.
Natural gas is used to provide space heating by heating either air, in a forced-air system , or water, in a hydronic heating system . Both systems rely on either pumps or fans to push either the heated air or water throughout the building.
Basic Principle
A gas fired forced-air furnace operates on a similar principle as a central electric furnace. Both require a control circuit to sense the temperature of the space to be heated, and a fan to circulate air throughout that space. The furnace is usually located in the basement or crawl space of a house.
If the room temperature falls too low, the main thermostat clicks on and opens the main gas valve which allows the combustible gas to fill the heat exchanger . The inflowing gas is ignited via a pilot light, direct spark ignition, or an electrically heated hot-surface igniter similar to a diesel engine glow plug. Once the temperature in the heat exchanger has risen to a sufficient level, a local thermostat or timer switch engages the blower fan .
An upper-limit safety thermostat is installed in the plenum chamber to disengage the main gas valve should the blower fan fail to engage, and the temperature rise too high.
The blower fan pushes the heated air through the ducting system and the building, usually via floor registers. Since hot air rises, it is always more efficient to install heat registers as low as possible and near places of high ambient temperature loss, such as windows and exterior walls or doors. Return-air registers draw cooled air back down to the furnace and the heat exchanger to be warmed again.
As the blower fan draws the return air, it pushes it through a filter before heating it again. This helps maintain healthy air quality levels in the building, and filters should be replaced regularly.
If a hydronic heating system is used instead of a forced-air system, a pump will push the heated water to radiators throughout the building to dissipate their thermal energy.
One of the by-products of combusting gasses to produce heat, is smoke, which needs to be vented safely away from the house. The heat exchanger keeps the toxic byproducts of combustion separate from the air that is circulated throughout the building. Most of the heat is transferred to the circulating air, but some still lost through exhaust. Older heat exchangers had an efficiency of roughly 50%, meaning half the heat produced was wasted up the chimney. Modern designs have improved efficiency to approximately 70% to 80%, and recent high-efficiency furnaces can achieve heat distribution efficiencies above 90%.


3.4: What a logarithm is
- How is the base-\(10\) logarithm defined?
- What is the “natural logarithm” and how is it different from the base-\(10\) logarithm?
- How can we solve an equation that involves \(e\) to some unknown quantity?
In Section 1.7 , we introduced the idea of an inverse function. The fundamental idea is that \(f\) has an inverse function if and only if there exists another function \(g\) such that \(f\) and \(g\) “undo” one another's respective processes. In other words, the process of the function \(f\) is reversible, and reversing \(f\) generates a related function \(g\text{.}\)
More formally, recall that a function \(y = f(x)\) (where \(f : A \to B\)) has an inverse function if and only if there exists another function \(g : B \to A\) such that \(g(f(x)) = x\) for every \(x\) in \(A\text{,}\) and \(f(g(y)) = y\) for every \(y\) in \(B\text{.}\) We know that given a function \(f\text{,}\) we can use the Horizontal Line Test to determine whether or not \(f\) has an inverse function. Finally, whenever a function \(f\) has an inverse function, we call its inverse function \(f^{-1}\) and know that the two equations \(y = f(x)\) and \(x = f^{-1}(y)\) say the same thing from different perspectives.
Let \(P(t)\) be the “powers of 10” function, which is given by \(P(t) = 10^t\text{.}\)
- Complete Table \(\PageIndex{1}\) to generate certain values of \(P\text{.}\)
| \(t\) | -3 | -2 | -1 | 0 | 1 | 2 | 3 |
| \(y = P(t) = 10^t\) |
2. Why does \(P\) have an inverse function?
3. Since \(P\) has an inverse function, we know there exists some other function, say \(L\text{,}\) such that writing “\(y = P(t)\)” says the exact same thing as writing “\(t = L(y)\)”. In words, where \(P\) produces the result of raising \(10\) to a given power, the function \(L\) reverses this process and instead tells us the power to which we need to raise \(10\text{,}\) given a desired result. Complete Table \(\PageIndex{2}\) to generate a collection of values of \(L\text{.}\)
| \(y\) | \(10^{-3}\) | \(10^{-2}\) | \(10^{-1}\) | \(10^{0}\) | \(10^{1}\) | \(10^{2}\) | \(10^{3}\) |
| \(L(y)\) |
4. What are the domain and range of the function \(P\text{?}\) What are the domain and range of the function \(L\text{?}\)
The base-\(10\) logarithm
The powers-of-\(10\) function \(P(t) = 10^t\) is an exponential function with base \(b \gt 1\text{.}\) As such, \(P\) is always increasing, and thus its graph passes the Horizontal Line Test, so \(P\) has an inverse function. We therefore know there exists some other function, \(L\text{,}\) such that writing \(y = P(t)\) is equivalent to writing \(t = L(y)\text{.}\) For instance, we know that \(P(2)=100\) and \(P(-3)=\frac{1}{1000}\text{,}\) so it's equivalent to say that \(L(100) = 2\) and \(L(\frac{1}{1000}) = -3\text{.}\) This new function \(L\) we call the base \(10\) logarithm , which is formally defined as follows.
Given a positive real number \(y\text{,}\) the base-\(10\) logarithm of \(y\) is the power to which we raise \(10\) to get \(y\text{.}\) We use the notation “\(\log_{10}(y)\)” to denote the base-\(10\) logarithm of \(y\text{.}\)
The base-\(10\) logarithm is therefore the inverse of the powers of \(10\) function. Whereas \(P(t) = 10^t\) takes an input whose value is an exponent and produces the result of taking \(10\) to that power, the base-\(10\) logarithm takes an input number we view as a power of \(10\) and produces the corresponding exponent such that \(10\) to that exponent is the input number.
In the notation of logarithms, we can now update our earlier observations with the functions \(P\) and \(L\) and see how exponential equations can be written in two equivalent ways. For instance,
each say the same thing from two different perspectives. The first says “\(100\) is \(10\) to the power \(2\)”, while the second says “\(2\) is the power to which we raise \(10\) to get \(100\)”. Similarly,
If we rearrange the statements of the facts in Equation (3.4.1), we can see yet another important relationship between the powers of \(10\) and base-\(10\) logarithm function. Noting that \(\log_{10}(100) = 2\) and \(100 = 10^2\) are equivalent statements, and substituting the latter equation into the former, we see that
In words, Equation (3.4.3) says that “the power to which we raise \(10\) to get \(10^2\text{,}\) is \(2\)”. That is, the base-\(10\) logarithm function undoes the work of the powers of \(10\) function.
In a similar way, if we rearrange the statements in Equation (3.4.2), we can observe that by replacing \(-3\) with \(\log_{10}(\frac{1}{1000})\) we have
In words, Equation (3.4.4) says that “when \(10\) is raised to the power to which we raise \(10\) in order to get \(\frac{1}{1000}\text{,}\) we get \(\frac{1}{1000}\)”.
We summarize the key relationships between the powers-of-\(10\) function and its inverse, the base-\(10\) logarithm function, more generally as follows.
- The domain of \(P\) is the set of all real numbers and the range of \(P\) is the set of all positive real numbers.
- The domain of \(L\) is the set of all positive real numbers and the range of \(L\) is the set of all real numbers.
- For any real number \(t\text{,}\) \(\log_{10}(10^t) = t\text{.}\) That is, \(L(P(t)) = t\text{.}\)
- For any positive real number \(y\text{,}\) \(10^{\log_{10}(y)} = y\text{.}\) That is, \(P(L(y)) = y\text{.}\)
- \(10^0 = 1\) and \(\log_{10}(1) = 0\text{.}\)
The base-\(10\) logarithm function is like the sine or cosine function in this way: for certain special values, it's easy to know the value of the logarithm function. While for sine and cosine the familiar points come from specially placed points on the unit circle, for the base-\(10\) logarithm function, the familiar points come from powers of \(10\text{.}\) In addition, like sine and cosine, for all other input values, (a) calculus ultimately determines the value of the base-\(10\) logarithm function at other values, and (b) we use computational technology in order to compute these values. For most computational devices, the command
log(y)
produces the result of the base-\(10\) logarithm of \(y\text{.}\)
It's important to note that the logarithm function produces exact values. For instance, if we want to solve the equation \(10^t = 5\text{,}\) then it follows that \(t = \log_{10}(5)\) is the exact solution to the equation. Like \(\sqrt{2}\) or \(\cos(1)\text{,}\) \(\log_{10}(5)\) is a number that is an exact value. A computational device can give us a decimal approximation, and we normally want to distinguish between the exact value and the approximate one. For the three different numbers here, \(\sqrt{2} \approx 1.414\text{,}\) \(\cos(1) \approx 0.540\text{,}\) and \(\log_{10}(5) \approx 0.699\text{.}\)
For each of the following equations, determine the exact value of the unknown variable. If the exact value involves a logarithm, use a computational device to also report an approximate value. For instance, if the exact value is \(y = \log_{10}(2)\text{,}\) you can also note that \(y \approx 0.301\text{.}\)
- \(\displaystyle 10^t = 0.00001\)
- \(\displaystyle \log_{10}(1000000) = t\)
- \(\displaystyle 10^t = 37\)
- \(\displaystyle \log_{10}(y) = 1.375\)
- \(\displaystyle 10^t = 0.04\)
- \(\displaystyle 3 \cdot 10^t + 11 = 147\)
- \(\displaystyle 2\log_{10}(y) + 5 = 1\)
The natural logarithm
The base-\(10\) logarithm is a good starting point for understanding how logarithmic functions work because powers of \(10\) are easy to mentally compute. We could similarly consider the powers of \(2\) or powers of \(3\) function and develop a corresponding logarithm of base \(2\) or \(3\text{.}\) But rather than have a whole collection of different logarithm functions, in the same way that we now use the function \(e^t\) and appropriate scaling to represent any exponential function, we develop a single logarithm function that we can use to represent any other logarithmic function through scaling. In correspondence with the natural exponential function, \(e^t\text{,}\) we now develop its inverse function, and call this inverse function the natural logarithm .
Given a positive real number \(y\text{,}\) the natural logarithm of \(y\) is the power to which we raise \(e\) to get \(y\text{.}\) We use the notation “\(\ln(y)\)” to denote the natural logarithm of \(y\text{.}\)
We can think of the natural logarithm, \(\ln(y)\text{,}\) as the “base-\(e\) logarithm”. For instance,
and
The former equation is true since “the power to which we raise \(e\) to get \(e^{-1}\) is \(-1\)”; the latter equation is true since “when we raise \(e\) to the power to which we raise \(e\) to get \(2\text{,}\) we get \(2\)”. The key relationships between the natural exponential and the natural logarithm function are investigated in Activity \(\PageIndex{3}\)
Let \(E(t) = e^t\) and \(N(y) = \ln(y)\) be the natural exponential function and the natural logarithm function, respectively.
- What are the domain and range of \(E\text{?}\)
- What are the domain and range of \(N\text{?}\)
- What can you say about \(\ln(e^t)\) for every real number \(t\text{?}\)
- What can you say about \(e^{\ln(y)}\) for every positive real number \(y\text{?}\)
- Complete Table \(\PageIndex{5}\) and Table \(\PageIndex{6}\) with both exact and approximate values of \(E\) and \(N\text{.}\) Then, plot the corresponding ordered pairs from each table on the axes provided in Figure \(\PageIndex{7}\) and connect the points in an intuitive way. When you plot the ordered pairs on the axes, in both cases view the first line of the table as generating values on the horizontal axis and the second line of the table as producing values on the vertical axis 1 ; label each ordered pair you plot appropriately.
| \(t\) | \(-2\) | \(-1\) | \(0\) | \(1\) | \(2\) |
| \(E(t)=e^t\) | \(e^{-2} \approx 0.135\) |
| \(y\) | \(e^{-2}\) | \(e^{-1}\) | \(1\) | \(e^1\) | \(e^2\) |
| \(N(y)=\ln(y)\) | \(-2\) |
\(f(t) = b^t\) revisited
In Section 3.1 and Section 3.2 , we saw that that function \(f(t) = b^t\) plays a key role in modeling exponential growth and decay, and that the value of \(b\) not only determines whether the function models growth (\(b \gt 1\)) or decay (\(0 \lt b \lt 1\)), but also how fast the growth or decay occurs. Furthermore, once we introduced the natural base \(e\) in Section 3.3 , we realized that we could write every exponential function of form \(f(t) = b^t\) as a horizontal scaling of the function \(E(t) = e^t\) by writing
for some value \(k\text{.}\) Our development of the natural logarithm function in the current section enables us to now determine \(k\) exactly.
Determine the exact value of \(k\) for which \(f(t) = 3^t = e^{kt}\text{.}\)
Solution . Since we want \(3^t = e^{kt}\) to hold for every value of \(t\) and \(e^{kt} = (e^k)^t\text{,}\) we need to have \(3^t = (e^k)^t\text{,}\) and thus \(3 = e^k\text{.}\) Therefore, \(k\) is the power to which we raise \(e\) to get \(3\text{,}\) which by definition means that \(k = \ln(3)\text{.}\)
In modeling important phenomena using exponential functions, we will frequently encounter equations where the variable is in the exponent, like in Example 3.4.8 where we had to solve \(e^k = 3\text{.}\) It is in this context where logarithms find one of their most powerful applications. Activity 3.4.4 provides some opportunities to practice solving equations involving the natural base, \(e\text{,}\) and the natural logarithm.
Solve each of the following equations for the exact value of the unknown variable. If there is no solution to the equation, explain why not.
- \(\displaystyle e^t = \frac{1}{10}\)
- \(\displaystyle 5e^{t}=7\)
- \(\displaystyle \ln(t) = -\frac{1}{3}\)
- \(\displaystyle e^{1-3t} = 4\)
- \(\displaystyle 2\ln(t) + 1 = 4\)
- \(\displaystyle 4 - 3e^{2t} = 2\)
- \(\displaystyle 4 + 3e^{2t} = 2\)
- \(\displaystyle \ln(5 - 6t) = -2\)
Summary
- The base-\(10\) logarithm of \(y\text{,}\) denoted \(\log_{10}(y)\text{,}\) is defined to be the power to which we raise \(10\) to get \(y\text{.}\) For instance, \(\log_{10}(1000) = 3\text{,}\) since \(10^3 = 1000\text{.}\) The function \(L(y) = \log_{10}(y)\) is thus the inverse of the powers-of-\(10\) function, \(P(t) = 10^t\text{.}\)
- The natural logarithm \(N(y) = \ln(y)\) differs from the base-\(10\) logarithm in that it is the logarithm with base \(e\) instead of \(10\text{,}\) and thus \(\ln(y)\) is the power to which we raise \(e\) to get \(y\text{.}\) The function \(N(y) = \ln(y)\) is the inverse of the natural exponential function \(E(t) = e^t\text{.}\)
-
The natural logarithm often enables us solve an equation that involves \(e\) to some unknown quantity. For instance, to solve \(2e^{3t-4} + 5 = 13\text{,}\) we can first solve for \(e^{3t-4}\) by subtracting \(5\) from each side and dividing by \(2\) to get
\[ e^{3t-4} = 4\text{.} \nonumber \]
This last equation says “\(e\) to some power is \(4\)”. We know that it is equivalent to say
\[ \ln(4) = 3t-4\text{.} \nonumber \]Since \(\ln(4)\) is a number, we can solve this most recent linear equation for \(t\text{.}\) In particular, \(3t = 4 + \ln(4)\text{,}\) so
\[ t = \frac{1}{3}(4 + \ln(4))\text{.} \nonumber \]
Exercises
Recall that when a function \(y = f(x)\) has an inverse function, the two equations \(y = f(x)\) and \(x = f^{-1}(y)\) say the same thing from different perspectives: the first equation expresses \(y\) in terms of \(x\text{,}\) while the second expresses \(x\) in terms of \(y\text{.}\) When \(y = f(x) = e^x\text{,}\) we know its inverse is \(x = f^{-1}(y) = \ln(y)\text{.}\) Through logarithms, we now have the ability to find the inverse of many different exponential functions. In particular, because exponential functions and their transformations are either always increasing or always decreasing, any function of the form \(y = f(x) = ae^{-kx} + c\) will have an inverse function.
Find the inverse function for each given function by solving algebraically for \(x\) as a function of \(y\text{.}\) In addition, state the domain and range of the given function and the domain and range of of the inverse function.
- \(\displaystyle y = g(x) = e^{-0.25x}\)
- \(\displaystyle y = h(x) = 2e^{x} + 1\)
- \(\displaystyle y = r(x) = 21 + 15e^{-0.1x}\)
- \(\displaystyle y = s(x) = 72 - 40e^{-0.05x}\)
- \(\displaystyle y = u(x) = -5e^{3x-4} + 8\)
- \(\displaystyle y = w(x) = 3\ln(x) + 4\)
- \(\displaystyle y = z(x) = -0.2 \ln(2x - 5) + 1\)
We've seen that any exponential function \(f(t) = b^t\) (\(b \gt 0\text{,}\) \(b \ne 1\)) can be written in the form \(f(t) = e^{kt}\) for some real number \(k\text{,}\) and this is because \(f(t) = b^t\) is a horizontal scaling of the function \(E(t) = e^{t}\text{.}\) In this exercise, we explore how the natural logarithm can be scaled to achieve a logarithm of any base.
Let \(b \gt 1\text{.}\) Because the function \(y = f(t) = b^t\) has an inverse function, it makes sense to define its inverse like we did when \(b = 10\) or \(b = e\text{.}\) The base-\(b\) logarithm , denoted \(\log_b(y)\) is defined to be the power to which we raise \(b\) to get \(y\text{.}\) Thus, writing \(y = f(t) = b^t\) is the same as writing \(t = f^{-1}(y) = \log_b(y)\text{.}\)
In
Desmos
, the natural logarithm function is given by
ln(t)
, while the base-\(10\) logarithm by
log(t)
. To get a logarithm of a different base, such as a base-\(2\) logarithm, type
log_2(t)
(the underscore will generate a subscript; then use the right arrow to get out of subscript mode).
In a new
Desmos
worksheet, enter
V(t) = k * ln(t)
and accept the slider for \(k\text{.}\) Set the lower and upper bounds for the slider to \(0.01\) and \(15\text{,}\) respectively.
- Define \(f(t) = \log_2(t)\) in Desmos . Can you find a value of \(k\) for which \(\log_2(t) = k\ln(t)\text{?}\) If yes, what is the value? If not, why not?
- Repeat (a) for the functions \(g(t) = \log_3(t)\text{,}\) \(h(t) = \log_5(t)\text{,}\) and \(p(t) = \log_{1.25}(t)\text{.}\) What pattern(s) do you observe?
- True or false: for any value of \(b \gt 1\text{,}\) the function \(\log_b(t)\) can be viewed as a vertical scaling of \(\ln(t)\text{.}\)
- Compute the following values: \(\frac{1}{\ln(2)}\text{,}\) \(\frac{1}{\ln(3)}\text{,}\) \(\frac{1}{\ln(5)}\text{,}\) and \(\frac{1}{\ln(1.25)}\text{.}\) What do you notice about these values compared to those of \(k\) you found in (a) and (b)?
A can of soda is removed from a refrigerator at time \(t = 0\) (in minutes) and its temperature, \(F(t)\text{,}\) in degrees Fahrenheit, is computed at regular intervals. Based on the data, a model is formulated for the object's temperature, given by
- Determine the exact time when the soda's temperature is \(50^\circ\text{.}\)
- Is there ever a time when the soda's temperature is \(36^\circ\text{?}\) Why or why not?
- For the model, its domain is the set of all positive real numbers, \(t \gt 0\text{.}\) What is its range?
- Find a formula for the inverse of the function \(y = F(t)\text{.}\) What is the meaning of this function?
<1> Note that when we take this perspective for plotting the data in Table 3.4.6, we are viewing \(N\) as a function of \(t\text{,}\) writing \(N(t) = \ln(t)\) in order to plot the function on the \(t\)-\(y\) axes


6.2: Divisions of the Skeletal System
By the end of this section, you will be able to:
- Distinguish between the axial skeleton and appendicular skeleton
- Discuss the functions of the axial skeleton and appendicular skeleton
- Define the axial skeleton and its components
- Define the appendicular skeleton and its components
The skeletal system includes all of the bones, cartilages, and ligaments of the body that support and give shape to the body and body structures. The skeleton consists of the bones of the body. For adults, there are 206 bones in the skeleton. Younger individuals have higher numbers of bones because some bones fuse together during childhood and adolescence to form an adult bone. The primary functions of the skeleton are to provide a rigid, internal structure that can support the weight of the body against the force of gravity, and to provide a structure upon which muscles can act to produce movements of the body. The lower portion of the skeleton is specialized for stability during walking or running. In contrast, the upper skeleton has greater mobility and ranges of motion, features that allow you to lift and carry objects or turn your head and trunk.
In addition to providing for support and movements of the body, the skeleton has protective and storage functions. It protects the internal organs, including the brain, spinal cord, heart, lungs, and pelvic organs. The bones of the skeleton serve as the primary storage site for important minerals such as calcium and phosphate. The bone marrow found within bones stores fat and houses the blood-cell producing tissue of the body.
The skeleton is subdivided into two major divisions—the axial and appendicular.
The Axial Skeleton
The axial skeleton forms the vertical, central axis of the body and includes all bones of the head, neck, chest, and back (Figure \(\PageIndex{1}\)). It serves to protect the brain, spinal cord, heart, and lungs. It also serves as the attachment site for muscles that move the head, neck, and back, and for muscles that act across the shoulder and hip joints to move their corresponding limbs.
The axial skeleton of the adult consists of 80 bones, including the skull , the vertebral column , and the thoracic cage . The skull has two main divisions, the cranial portion and the facial portion, and is formed by 22 bones. Also associated with the head are an additional seven bones, including the hyoid bone and the ear ossicles (three small bones found in each middle ear). The vertebral column consists of 24 bones, each called a vertebra , plus the sacrum and coccyx . The thoracic cage includes the 12 pairs of ribs , and the sternum , the flattened bone of the anterior chest.
The Appendicular Skeleton
The appendicular skeleton includes all bones of the upper and lower limbs, plus the bones that attach each limb to the axial skeleton. There are 126 bones in the appendicular skeleton of an adult. The bones that attach the upper limbs to the axial skeleton, include the clavicle and scapula, and are collectively known as the pectoral (shoulder) girdle. Bones of the upper limb include the humerus, ulna, radius, carpals, metacarpals and phalanges. The pelvic girdle, which attaches the lower limbs to the axial skeleton, is made up of the hip bones (coxae; singular coxa). Bones of the lower limb include the femur, patella, tibia, fibula, tarsals, metatarsals, and phalanges. The bones of the appendicular skeleton serve as attachment sites for muscles that move the appendages. The bones of the lower limb are structured to bear the weight of the body as well. The bones of the appendicular skeleton are covered in more detail in a separate chapter.
Concept Review
The skeletal system includes all of the bones, cartilages, and ligaments of the body. It serves to support the body, protect the brain and other internal organs, and provides a rigid structure upon which muscles can pull to generate body movements. It also stores fat and the tissue responsible for the production of blood cells. The skeleton is subdivided into two parts. The axial skeleton forms a vertical axis that includes the head, neck, back, and chest. It has 80 bones and consists of the skull, vertebral column, and thoracic cage. The adult vertebral column consists of 24 vertebrae plus the sacrum and coccyx. The thoracic cage is formed by 12 pairs of ribs and the sternum. The appendicular skeleton consists of 126 bones in the adult and includes all of the bones of the upper and lower limbs plus the bones that anchor each limb to the axial skeleton.
Review Questions
Q. Which of the following is part of the axial skeleton?
A. shoulder bones
B. thigh bone
C. foot bones
D. vertebral column
- Answer
-
Answer: D
Q. Which of the following is a function of the axial skeleton?
A. allows for movement of the wrist and hand
B. protects nerves and blood vessels at the elbow
C. supports trunk of body
D. allows for movements of the ankle and foot
- Answer
-
Answer: C
Q. The axial skeleton ________.
A. consists of 126 bones
B. forms the vertical axis of the body
C. includes all bones of the body trunk and limbs
D. includes only the bones of the lower limbs
- Answer
-
Answer: B
Critical Thinking Question
Q. Define the two divisions of the skeleton.
- Answer
-
A. The axial skeleton forms the vertical axis of the body and includes the bones of the head, neck, back, and chest of the body. It consists of 80 bones that include the skull, vertebral column, and thoracic cage. The appendicular skeleton consists of 126 bones and includes all bones of the upper and lower limbs.
Q. Discuss the functions of the axial skeleton.
- Answer
-
A. The axial skeleton supports the head, neck, back, and chest of the body and allows for movements of these body regions. It also gives bony protections for the brain, spinal cord, heart, and lungs; stores fat and minerals; and houses the blood-cell producing tissue.
Glossary
- appendicular skeleton
- all bones of the upper and lower limbs, plus the girdle bones that attach each limb to the axial skeleton
- axial skeleton
- central, vertical axis of the body, including the skull, vertebral column, and thoracic cage
- coccyx
- small bone located at inferior end of the adult vertebral column that is formed by the fusion of four coccygeal vertebrae; also referred to as the “tailbone”
- ear ossicles
- three small bones located in the middle ear cavity that serve to transmit sound vibrations to the inner ear
- hyoid bone
- small, U-shaped bone located in upper neck that does not contact any other bone
- ribs
- thin, curved bones of the chest wall
- sacrum
- single bone located near the inferior end of the adult vertebral column that is formed by the fusion of five sacral vertebrae; forms the posterior portion of the pelvis
- skeleton
- bones of the body
- skull
- bony structure that forms the head, face, and jaws, and protects the brain; consists of 22 bones
- sternum
- flattened bone located at the center of the anterior chest
- thoracic cage
- consists of 12 pairs of ribs and sternum
- vertebra
- individual bone in the neck and back regions of the vertebral column
- vertebral column
- entire sequence of bones that extend from the skull to the tailbone
Contributors and Attributions
-
OpenStax Anatomy & Physiology (CC BY 4.0). Access for free at https://openstax.org/books/anatomy-and-physiology


7.2: Expressionism and Serialism
While the Impressionist composers attempted to move further away from romantic forms and romantic harmony, some Expressionist composers succeeded in completely eliminating harmony and tonal melody (melody based on a particular key) from their music. The resultant sounds were often not very melodically and harmonically pleasant to hear and, as a result, the Expressionist style of music did not (and still does not) appeal to the majority of audiences. The name of this style period can be confusing for some. The Expressionist period was not a time when composers sought to express themselves emotionally in a romantic, beautiful, or programmatic way. Due to the nature of the sounds produced by the system of composition described below, Expressionism seems more appropriate for evoking more extreme, and sometimes even harsh, emotions. Using this experimental style of writing, composers such as Arnold Schoenberg (1874-1951) attempted to intentionally eliminate what we call tonality; music that is based on scales and the progression (movement) of chords from one to another.
In Edward Munch’s famous painting, The Scream (Figure 7.2.1), we see an excellent example of the parallel movement of expressionism taking place in the visual arts. Expressionists looked inward, specifically to the anxiety they felt towards the outside world. This was in stark contrast to the impressionists, who looked to the beauty of nature for inspiration. Expressionist paintings relied instead on stark colors and harsh swirling brushstrokes to convey the artist’s reaction to the ugliness of the modern world. Abstract Expressionism took this concept to a greater extreme, by abandoning shape altogether for pure abstraction. This style is typified by the works of the American painter Jackson Pollock (see Image 7.2.2).
Many of the early works of Austrian-born Arnold Schoenberg (1874-1951) exemplified an expressionistic musical style. He is most famous for his experiments with atonality, that is, music without a tonal center. His music was highly dissonant and sounded quite radical when compared to earlier music, which utilized dissonance only as a means to eventually return to the stasis of consonance. However, Schoenberg saw dissonance not as a means to an end, but as the end itself. His music invited the listener to revel in various levels of dissonance, and many listeners were never able to adjust.
Born in Austria, and of Jewish descent, Schoenberg was already composing by the age of nine. While in his teens, he studied composition with the Austrian composer and conductor Alexander Zemlinsky. In 1901 he moved to Berlin where he was befriended and mentored by the German composer Richard Strauss. Three years later in 1903, Schoenberg returned to Austria and began a long association with the renowned composer Gustav Mahler who became one of his strongest supporters.
License | Fair Use
In 1909, Schoenberg composed the first complete work that completely did away with tonality. This piano composition was one of three that together are listed as his Opus 11 and was the first piece we now refer to as being completely atonal (without tonality). Schoenberg’s most-important atonal compositions include: Five Orchestral Pieces (1909), Pierrot Lunaire (1912), Die Jakobsleiter (Jacob’s Ladder - begun in 1917 but never finished), Die glückliche Hand (The Lucky Hand - 1924), and Erwartung (Expectation - 1924) for soprano and orchestra.
Schoenberg famously developed a system whereby the twelve notes of the chromatic scale were randomly organized into scale units that he called the twelve-tone row. These rows could then be further “serialized” (organized in random fashion) by a number of different techniques. This idea of assigning values to musical information is called serialism . In 1921 Schoenberg composed his Piano Suite opus 25, the first composition written using the 12-tone method. Each 12-tone composition is built from a series of 12 different pitches that may be arranged in a number of different ways. The original row may be played forward, backwards (retrograde), upside down (inverted), and backwards and inverted (retrograde inversion). All of the melodies and harmonies in a 12-tone piece must be derived in some way from the original row or from fragments of the original row.
In 1925 Schoenberg was hired by the Prussian Academy of Arts in Berlin to teach composition, and he would most likely have continued his career as teacher and composer in Europe were it not for the rise of the Nazi party and their subsequent persecution of European Jews. In 1933 he was released from the Academy and moved first to Paris and then to Boston. In 1934 he settled in California and held teaching positions first at the University of Southern California (1935-36) and then the University of Central Los Angeles (1936-44).
After immigrating to the United States, Schoenberg reconnected with the Jewish faith he had abandoned as a young man. The sadness he felt because of the personal accounts of the horrible treatment experienced by so many Jews during World War II led to his composition of A Survivor from Warsaw, which was composed for orchestra, male chorus, and narrator. The piece was completed in September 1947 and the entire piece is built on a twelve-tone row. This important work is Schoenberg’s dramatization of a tragic story he heard from surviving Polish Jews who were victims of Nazi atrocities during World War II. Schoenberg created a story about a number of Jews who survived the war by living in the sewers of Warsaw. Interestingly, among Schoenberg’s many and very specific performance instructions is the request that the narrator not attempt to sing his part throughout the performance.
|
LISTENING GUIDE For audio, go to: |
| Composer : Arnold Schoenberg |
| Composition : A Survivor from Warsaw |
| Date : 1947 |
| Genre : 12-tone composition for small orchestra, male chorus, and narrator |
| Form : through-composed |
| Nature of Text : Narration of Germans’ treatment of Jews in Warsaw during WWII |
| Performing Forces : orchestra, male chorus, and narrator |
| Timing | Performing Forces, Melody and Texture |
| 0:35 |
Trumpet introductory fanfare built from 12-tone row. Trumpets, snare drum, clarinets. Irregular rhythmic figures built from 12-tone row. 12-tone chordal structures built from 12-tone row. |
| 0:46 |
Celli (cellos) enter with rhythmic motif. Brief medodic motifs move between celli, woodwinds, trumpets, and strings. Rhythms are derived from the 12-tone row and are irregular. 12-tone based chordal structures continue throughout piece. |
| 1:06 | Xylophone added. |
| 1:16 |
Clarinet added. Clarinet completes instrumental introduction. |
| 1:21 |
Narrator enters. Instrumentation and dynamics are altered to match rise and fall of phrases in narration. |
| 1:57 | French Horn enters. |
| 2:19 |
Narration. Narration much more intense and trumpet fanfare underscores this change. |
| 3:09 |
Narration. Bass drum begins a steady pulse with snare drum and xylophone irregular rhythms as drama in narration increases. |
| 3:19 |
Narration switches to German. Narrator begins to shout in German. |
| 3:38 | Narration switches back to English. Strings play tremolo in background. |
| 4:05 |
Narration becomes more introspective. Strings become more lyrical to underscore change in story. |
| 4:20 |
Orchestra. Orchestra interlude decreases the intensity of the moment. |
| 4:38 | Narrator returns. 7:41 Brass join chorus. Intensity in Chorus and Orchestra build. 7:52 Brass continue as chorus ends. Brass and strings build to big climactic moment and conclude piece at 8:01. |
| 6:01 |
Narration. As narrator says “faster and faster” the music begins to accelerate as well. |
| 6:19 |
Male chorus. Men begin to sing the Jewish prayer Shema Yisroel accompanied by strings. Brass and woodwinds are used as interjections throughout this section. |
| 7:41 |
Brass join chorus. Intensity in Chorus and Orchestra build. |
| 7:52 |
Brass continue as chorus ends. Brass and strings build to big climactic moment and conclude piece at 8:01. |
Schoenberg’s ideas were further developed by his two famous students, Alban Berg and Anton Webern. Together, the three came to be known as the Second Viennese School, in reference to the first Viennese School, which consisted of Hadyn, Mozart, and Beethoven. Born in Vienna, Alban Berg began studying with Schoenberg at the age of 19 and soon became known for his unique compositional style, which fused post-romantic concepts with Schoenberg’s cutting edge twelve-tone techniques. Heavily influenced by Richard Wagner, Berg held on to techniques such as the leitmotif and sought to couch his harmonic ideas in tried-and-true forms such as the sonata and fugue. Although he composed many famous pieces, such as his Violin Concerto and his unfinished opera Lulu, he initially made his fame with Wozzeck, an opera based on the drama Woyzeck by German playright Georg Buchner. Berg served during World War I, and much of Wozzeck was composed in 1917, during a period of leave from the Austro-Hungarian Army. The opera consists of three acts, each with five scenes organized around the variations of a musical idea, such as the variations of a theme, a chord, or a rhythmic pattern. Berg himself adapted the libretto from Buchner’s original play.
The story of the opera centers on the title character Wozzeck. Like the main character in many romantic operas, he is a tragic figure. However, whereas the operas of the nineteenth century often depicted gods and mythical figures, the story of Wozzeck is couched in a sense of realism and addresses the type of societal problems that Berg may himself have encountered during World War I, problems such as apathy and human cruelty. The character of Wozzeck is that of a pitiful and unremarkable soldier who is tormented by his captain and used for and subjected to medical experiments by a sadistic doctor. Wozzeck, who is often given to hallucinations, eventually goes mad and kills his love interest, Marie, who has been unfaithful. The opera ends after Wozzeck drowns trying to clean the murder weapon in a pond and wading out too far.
Listen to the recording below of act 3, scene 2, the scene in which Wozzeck kills Marie. The scene features a variation on a single note, namely B.
|
LISTENING GUIDE For audio, go to: |
| Composer : Alban Berg |
| Composition : Wozzeck |
| Date : 1924 |
| Genre : Opera |
| Form : variation on a single note |
| Nature of Text : Wozzeck and Marie walk by a pond. Wozzeck stabs Marie in throat with a knife. |
| Performing Forces : orchestra, singers |
| Timing | Performing Forces, Melody, and Texture | Text and Form |
| 0:00 | Instrumental introduction evoking a low, and ominous feeling. | Orchestra. |
| 0:24 |
Wozzeck and Marie enter. Marie wishes to leave. A syrupy melody in the strings reflects Wozzeck’s pleas to Marie to sit down. |
Marie: Dort links geht’s in die Stadt. ‘s ist noch weit. Komm schneller! Wozzeck: Du Sollst dableiben, Marie. Kom, setz’ Dich. Marie: Abe rich muss fort. |
| 0:45 | Marie leaps up, saying, “I must go!” and low ominous notes play underneath as Wozzeck lures her back. |
Wozzeck: Komm. Bist weit gegangen, Marie.  Und so dunkel. – Weisst noch, Marie, wi lang’ es jetzt ist, dass wir uns kennen? Marie: Zu Pfingsten drei Jahre. Wozzeck: Und was meinst, wie lang’ es noch dauern wird? Marie: Ich muss fort. Wozzeck: Furchst Dich, Marie? Und bist doch fromm! Und gut! Und true! |
| 2:06 | A sweet melody in the strings evokes the line by Wozzeck “What sweet lips you have, Marie.” | Wozzeck: Was Du fur susse Lippen hast, Marie! Den Himmel gab’ ich drum und die Seligkeit, wenn ich Dich noch of so kussen durft! Abe rich darf nicht! Was zitterst? |
| 2:57 | Wozzeck says, “Those who are cold shiver no more. You will not shiver in the morning dew,” fortelling Marie’s death. She asks what he means and the music ceases creating a tense silence. |
Marie: Der Nachttau fallt. Wozzeck: Wer kalt ist, den friert nicht meher! Dich wird beim Morgentau nicht frieren. Marie: Was sagst Du da? Wozzeck: nix. |
| 3:40 | The music begins to build as Wozzeck prepares to kill Marie. |
Marie: Wie der Mond rot aufgeht! Wozzeck: Wie ein blutig Eisen! Marie: Was zitterst? Was Willst? |
| 4:07 | The music echoes Wozzeck word by word as he says, “No one, Marie! If not me, then no one!” After the act is done, the orchestra dies down to a single note and Wozzeck exclaims, “Dead!” |
Wozzeck: Ich nicht, Marie! Und kein Andrer auch nicht! Marie: Hilfe! Wozzeck: Tot! |
| 5:00 |
Orchestra Orchestral interlude |


15.12: Janus v. AFSCME
- Discuss the impact of the Supreme Court decision in Janus v. AFSCME
On June 27, 2018 the Supreme Court decided the case of Janus v. AFSCME (American Federation of State, County, and Municipal Employees), ruling that agency fees—the fees that unions charge non-members that are covered by collective bargaining agreements—violate those worker’s First Amendment right to free speech. The case was settled in a 5–4 vote with Neil Gorsuch casting the deciding vote. The Trump Administration also filed a brief in support of Janus.
The plaintiff in the case was Mark Janus, a child support specialist at the Illinois Department of Healthcare and Family Services. He joined a lawsuit originally filed by then Illinois governor Bruce Rauner (Republican) in 2015. A federal judge ruled that Rauner had no standing; since he wasn’t paying agency fees, he was insufficiently connected to or harmed by the law and couldn’t proceed. The National Right to Work Legal Defense Foundation and the Liberty Justice Center (funded by Rauner) recruited Janus to carry the case forward.
For perspective on the organizations and interests behind the litigation, read The New York Times article Behind a Key Anti-Labor Case, a Web of Conservative Donors.
A legal precedent for agency fees was established in the unanimous 1977 Supreme Court case Abood v. Detroit Board of Education, which distinguished between union membership dues, which must be voluntary and may contribute to political activity and agency or “fair share” fees—generally a percentage of union dues—that are required to cover the cost of union representation, including collective bargaining and arbitration. The Supreme Court heard related arguments in the 2016 Friedrichs v. California Teachers Association case that ended in a tie. Janus’ position was that the activities “agency fees” pay for are inherently political and support union policies he doesn’t agree with. Specifically, Janus claimed that the fees violate his First Amendment right of freedom of speech. The union countered that if employees didn’t pay agency fees, they would gain an unfair benefit—essentially, a free ride.
The majority opinion rejected the stare decisis principles (precedent) set by Abood, stating the decision was “poorly reasoned” and that “the practice of states and public-sector unions collecting agency fees from nonconsenting employees was a violation of the First Amendment, and that no further agency fees or other forms of payment to a public-sector union could be collected, nor could attempts be made to collect such payments from employees without their consent.” [1] In a dissenting opinion, Justice Kagan, joined by Justices Sotomayor, Ginsburg, and Breyer, “faulted the Majority for upsetting the balance that Abood brought to public-sector labor relations, and for disregarding stare decisis principles.” [2] Justice Kagan argued that “The balance Abood struck between public employers’ interests and public employees’ expression is right at home in First Amendment doctrine.” [3] Differences of opinion aside, the result is that public sector unions can no longer charge agency fees and workers must affirmatively opt into union membership, rather than the default being membership.
So what are the implications for the private sector? As a Washington Examiner headline summarizes, “Extending [the] Supreme Court’s Janus decision to private-sector unions [would be an] uphill battle.” [4] In the article, Harvard University labor professor Benjamin Sachs notes that Supreme Court precedent [interpretation of the Constitution] “is generally held to apply to things the government does, not things that private-sector employers do.” However, the article also mentioned that the National Right to Work Legal Defense Foundation followed up the Janus ruling with a request for the Supreme Court to consider a case involving private-sector union fees and quoted Right to Work Foundation president Mark Mix stating the “While this [the Janus] victory represents a massive step forward in the fight to protect American workers from forced unionism, that fight is far from over. There remains much work to do to both enforce and expand upon this historic victory over coercive unionism.” That said, with the percentage of unionized workers in the private sector declining steadily, the primary battleground will likely remain the public sector. For perspective, Bureau of Labor Statistics data shows that the union membership rate of public-sector workers is approximately 34%—over five times the percent of private-sector workers. [5] Education news site The74 described the situation prior to the court’s decision: “Janus is a threat to labor because government employees are most of what’s left in the movement.” [6]
The Janus decision not only affected unions, it will have a ripple effect on the politicians, institutions and other interests that unions traditionally support. As The New York Times noted, the Janus decision will also impact “a vast network of groups dedicated to advancing liberal policies and candidates.” [7] Indeed, liberal activists claim that eliminating the financial support to groups working for civil and immigrant rights and supporting voter registration and Democratic campaigns was the primary goal of conservative groups behind the Janus case. And those groups don’t dispute that fact.
Unions were engaged in contingency planning—internally and with their stakeholder groups—well in advance of the Supreme Court decision. For example, the Service Employees International Union (SEIU) reduced its 2018 budget by 30% and has been working with leaders of liberal groups for two years to develop plans to leverage people power (approximately 2 million members) and the union’s fund-raising capacity to partially offset the loss of financial support. Although organizations will have to be more selective about the projects they pursue and some programs will have to be scaled back or eliminated, the consensus is that the loss of union contributions will be offset by wealthy liberal donors, individual contributions and the “people power” of union members and other voters. [8]
-
Janus v. American Federation of State, County, and Municipal Employees, Council 31
. Oyez. Accessed July 22, 2019. ↵
-
Ibid. ↵
-
Janus v. AFSCME
. Ballotpedia. Accessed July 22, 2019. ↵
-
Langford, James. "
Extending Supreme Court's Janus Decision to Private-Sector Unions an Uphill Battle
." Washington Examiner. Juy 17, 2018. Accessed July 22, 2019. ↵
-
Bureau of Labor Statistics. "
Union Members Summary
." January 18, 2019. Accessed July 22, 2019. ↵
-
Cantor, David. "
Labor in the Age of Janus: 6 Things to Keep in Mind About American Unions on the Eve of a Pivotal Supreme Court Case
." The74. June 12, 2018. Accessed July 22, 2019. ↵
-
Scheiber, Noam. "Supreme Court Labor Decision Wasn't Just a Loss for Unions." The New York Times. July 1, 2018. Accessed July 22, 2019. ↵
-
Ibid. ↵


Summary
In this chapter we defined verbal communication as an agreed-upon and rule-governed system of symbols used to share meaning. These symbols are arbitrary, ambiguous, and abstract. The rules that dictate our use and understanding of symbols include phonology, semantics, syntactics, and pragmatics. As you recall there are distinct differences between written and spoken forms of verbal communication in terms of levels of formality, synchronicity, recording, and privacy. Yet, new technologies are beginning to blur some of these differences. Finally, verbal communication is central to our identity as humans and it allows us to define reality, organize ideas and experiences into categories, help us think, and shape out attitudes about the world.
Discussion Questions
-
In what ways do you define yourself as a person? What kinds of definitions do you have for yourself? What do you think would happen if you changed some of your self-definitions?
-
How do advances in technology impact verbal communication? What are some examples?
-
How does popular culture impact our verbal communication? What are some examples?
-
When you use text messages or email, are you formal or informal?
-
In what situations/contexts would it be appropriate to speak formally rather than informally? Why?
-
To what extent do you believe that verbal communication drives thought, or vice versa?
Key Terms
-
abstract
-
ambiguous
-
arbitrary
-
archived
-
asynchronous
-
connotative meaning
-
context
-
denotative meaning
-
formal
-
informal
-
phonology
-
pragmatics
-
reclaim
-
rule-governed
-
semantics
-
symbols
-
synchronous
-
syntactics
-
verbal communication
Contributions and Affiliations
-
Survey of Communication Study.
Authored by
: Scott T Paynton and Linda K Hahn.
Provided by
: Humboldt State University.
Located at
: en.wikibooks.org/wiki/Survey_of_Communication_Study.
License
:
CC BY-SA: Attribution-ShareAlike


Every English verb has five different forms, but only two of the forms have a tense feature. The tensed forms are indicated with a morphosyntactic feature, either [+past] or [-past]. The form that a verb in the V-head position takes depends on what tense feature is in the T-head position, among other things.
|
bare
|
(non-tensed)
|
eat
|
walk
|
sing
|
take
|
|
[-past]
|
(tensed)
|
eats
|
walks
|
sings
|
takes
|
|
[+past]
|
(tensed)
|
ate
|
walked
|
sang
|
took
|
|
past participle
|
(non-tensed)
|
eaten
|
walked
|
sung
|
taken
|
|
present participle
|
(non-tensed)
|
eating
|
walking
|
singing
|
taking
|
But there are some quirks of English that can make things confusing:
Is it bare or [-past]?
For just about every verb, the [-past] form is recognizable in the 3rd-person-singular form (
she eats/walks/sings/takes
). The 1st & 2nd-person forms
(
I eat/walk/sing/take
and
You eat/walk/sing/take
) look just like the bare form (
eat/walk/sing/take
).
If you’re looking at a verb and can’t tell if it’s in the bare form or the
[-past] form, give it a 3rd-person subject and then look for the
–s
morpheme:
I
want
to
visit
Saskatoon.
(bare or [-past]? Can’t tell: they’re ambiguous)
She
wants
to
visit
Saskatoon.
(
wants
is [-past],
visit
is bare)
Is it [+past] or past participle?
For many English verbs, the [+past] form (
She
bought
a donut
) and the past participle form (
She has
bought
a donut
) are the same.
If you’re looking at a verb and can’t tell if it’s in the [+past] form or the past participle form, try replacing it with the verb
eat
:
She
bought
a donut after she had
walked
the dog.
( [+past] or past participle? Can’t tell: they’re ambiguous)
She
ate
a donut after she had
eaten
the dog.
(Silly, but grammatical)
The form
ate
is [+past], while
eaten
is past participle,
so we can conclude that, in that sentence,
bought
is [+past] while
walked
is the past participle)
What about auxiliaries?
The modal auxiliaries never change their form: they occupy the T-head position in their own right.
The non-modal auxiliaries, like main verbs, change their form depending on what tense feature is in the T-head position, among other things.
|
bare
|
(non-tensed)
|
be
|
have
|
do
|
|
[-past]
|
(tensed)
|
am/are/is
|
has
|
does
|
|
[+past]
|
(tensed)
|
was/were
|
had
|
did
|
|
past participle
|
(non-tensed)
|
been
|
had
|
done
|
|
present participle
|
(non-tensed)
|
being
|
having
|
doing
|