1.4: More on Using Functions to Describe Relationships
Focus: Describing the Graphical Behavior of Functions
If a function describing a relationship between two quantities does not have a constant rate of change, then a linear function is not appropriate to model the relationship. In that case, the rate of increase or decrease is getting larger or smaller in size (or magnitude). The function may change from increasing to decreasing. We need to develop more tools to describe this aspect of the function and explore the graphical behavior of the function.
Consider the function \(y=f(x)=x^2 - 4x+5\). Exploring how the values of the input and output change with a table.
| x | 0 | 1 | 2 | 3 | 4 | 5 |
| y | 5 | 2 | 1 | 2 | 5 | 10 |
Notice how the change in the output \(y\) changes for each increase in \(x\) of one.
| Interval | [0,1] | [1,2] | [2,3] | [3,4] | [4,5] |
| \(\Delta x \) | +1 | +1 | +1 | +1 | +1 |
| \(\Delta y \) | -3 | -1 | 1 | 3 | 5 |
| \( \dfrac{\Delta y}{\Delta x} \) | \( \dfrac{-3}{1}=-3 \) | \( \dfrac{-1}{1}=-1 \) | \( \dfrac{1}{1}=1 \) | \( \dfrac{3}{1}=3 \) | \( \dfrac{5}{1}=5 \) |
On the interval \([2, \infty) \), a larger value of x gives a larger value of y. This is how we defined an increasing function in section 1.3. On the graph below, the graph moves upward (for a larger \(y\) ) as we move from left to right (for a larger \(x\) ) when the function is increasing on this interval. On the interval \( (-\infty,2) \), a larger value of x gives a smaller value of y. This is how we defined an decreasing function in section 1.3. On the graph below, the graph moves downward (for a smaller \(y\)) as we move from left to right (for a larger \(x\)) when the function is decreasing on this interval. Be aware that the change in \(x\), \( \Delta x \), does not always have to be \( \Delta x = 1\) as shown on the graph to help clarify the different rates of change.
Key Point: Increasing/Decreasing Functions on a Graph
If a function is increasing on an interval, the graph moves upward from left to right.
If a function is decreasing on an interval, the graph moves upward from left to right.
Moreover, in the introductory example above, notice how the average rates of change on each successive interval get bigger or smaller in size. The rates of increase are getting larger in size on the interval \([2, \infty)\) from \( \dfrac{\Delta y}{\Delta x} = \dfrac{1}{1} \) on [2,3] to \( \dfrac{\Delta y}{\Delta x} = \dfrac{3}{1} \) on [3,4] and \( \dfrac{\Delta y}{\Delta x} = \dfrac{5}{1} \) on [4,5]. Notice how the graph curves upward on the interval \([2, \infty)\) getting steeper and steeper as we move from left to right corresponding to the larger and larger rates of increase as shown on the graph below and the table above. In general, if a function has rates of increase that are getting larger in size over an interval, the shape of that portion of the graph will curve upward from left to right resembling the shape of the graph below.
On the interval \((-\infty, 2)\), the rates of decrease are getting smaller in size, decreasing in size from \( \dfrac{\Delta y}{\Delta x} = \dfrac{-5}{1} \) on [-1,0] to \( \dfrac{\Delta y}{\Delta x} = \dfrac{-3}{1} \) on [0,1] and \( \dfrac{\Delta y}{\Delta x} = \dfrac{-1}{1} \) on [1,2]. The graph moves downward with a smaller and smaller magnitude steepness as we move from left to right along the graph corresponding to the smaller and smaller magnitude rates of decrease as shown in the the graph below and the table above. In general, if a function has rates of decrease that are getting smaller in size over an interval, the shape of that portion of the graph will curve downward from left to right resembling the shape of the graph below.
Consider a second function \(y=g(x)=-(x-2)^2 +5\). Exploring how the values of the input and output change with a table.
| x | 0 | 1 | 2 | 3 | 4 | 5 |
| y | 1 | 4 | 5 | 4 | 1 | -4 |
Notice how the change in the output \(y\) changes for each increase in \(x\) of one.
| Interval | [0,1] | [1,2] | [2,3] | [3,4] | [4,5] |
| \(\Delta x \) | +1 | +1 | +1 | +1 | +1 |
| \(\Delta y \) | +3 | +1 | -1 | -3 | -5 |
| \( \dfrac{\Delta y}{\Delta x} \) | \( \dfrac{3}{1}=3 \) | \( \dfrac{1}{1}=1 \) | \( \dfrac{-1}{1}=-1 \) | \( \dfrac{-3}{1}=-3 \) | \( \dfrac{-5}{1}=-5 \) |
On the interval \((-\infty, 2)\), a larger value of x gives a larger value of y. So the function \(g\) is increasing on \((-\infty, 2)\). On the graph below, notice the graph moves upward (for a larger \(y\) ) as we move from left to right. On the interval \([2, \infty)\), a larger value of x gives a smaller value of y. So the function \(g\) is decreasing on \([2, \infty)\). On the graph below, notice the graph moves downward (for a smaller \(y\) ) as we move from left to right.
On the interval \((-\infty, 2)\), the rates of increase are getting smaller in size from \( \dfrac{\Delta y}{\Delta x} = \dfrac{5}{1} \) on [-1,0] to \( \dfrac{\Delta y}{\Delta x} = \dfrac{3}{1} \) on [0,1] and \( \dfrac{\Delta y}{\Delta x} = \dfrac{1}{1} \) on [1,2]. Notice how the graph curves upward on the interval \((-\infty, 2)\) getting less and less steep as we move from left to right corresponding to the smaller and smaller rates of increase as shown on the graph below and the table above. In general, if a function has rates of increase that are getting smaller over an interval, the shape of that portion of the graph will curve upward from left to right resembling the shape of the graph below.
On the interval \([2, \infty )\), the rates of decrease are getting larger in size from \( \dfrac{\Delta y}{\Delta x} = \dfrac{-1}{1} \) on [2,3] to \( \dfrac{\Delta y}{\Delta x} = \dfrac{-3}{1} \) on [3,4] and \( \dfrac{\Delta y}{\Delta x} = \dfrac{-5}{1} \) on [4,5]. The graph moves downward with a larger and larger magnitude steepness as we move from left to right along the graph corresponding to the larger and larger magnitude rates of decrease as shown in the the graph below and the table above. In general, if a function has rates of decrease that are getting larger in size over an interval, the shape of that portion of the graph will curve downward from left to right resembling the shape of the graph below.
A company's sales over 4 weeks, in thousands of dollars, is shown in the graph below.
a) Are the company's sales increasing or decreasing over the 4 weeks?
b) Describe what it happening to the rate of change in sales over time. Is the rate of increase or decrease getting larger or smaller in size?
Solution
a) The company's sales are increasing over time.
b) The rates of increase in sales in getting smaller as time passes.
The value of the stock of a company reached $600 per share at the start of the year, then began to decrease. The rate of decrease in value became larger and larger in size over the next few months. Draw a graph that illustrates the decrease in value of the stock over this time.
Solution
The graph of the function \(p(t)\) is shown below.
- On what intervals is the function increasing?
- On what intervals is the function decreasing?
- On what interval(s) is the rate of increase in \(f\) getting larger?
- On what interval(s) is the rate of increase in \(f\) getting smaller?
- On what interval(s) is the rate of decrease in \(f\) getting larger in size?
- On what interval(s) is the rate of decrease in \(f\) getting smaller in size?
Solution
- The function is increasing on the intervals \((1, 3)\) and \((4, \infty)\).
- The function is decreasing on the intervals \((-\infty, 1)\) and \((3, 4)\).
- The rate of increase in f is getting larger on \((1,2)\) and \((4, \infty)\).
- The rate of increase in f is getting smaller on \((-2,3)\).
- The rate of decrease in f is getting larger in size on \((3,3.5)\) .
- The rate of decrease in f is getting smaller in size on \((-\infty ,1)\) and \((3.5,4)\).
Note that all values of x are approximated from the graph.
The graph of the function \(y = f(x)\) is given below.
- On what interval(s) is \(f\) f increasing?
- On what interval(s) is \(f\) decreasing?
- On what interval(s) is the rate of increase in \(f\) getting larger?
- On what interval(s) is the rate of increase in \(f\) getting smaller?
- On what interval(s) is the rate of decrease in \(f\) getting larger in size?
- On what interval(s) is the rate of decrease in \(f\) getting smaller in size?
- Answer
-
- The function \(f\) increasing \((-2,0)\) and \((3.5, \infty)\).
- The function \(f\) decreasing \((- \infty ,-2)\) and \((0,3.5 )\).
- The rate of increase in f is getting larger on \((-2,-1)\) and \((3.5, \infty)\).
- The rate of increase in f is getting smaller on \((-1,0)\).
- The rate of decrease in f is getting larger in size on \((0,2)\) .
- The rate of decrease in f is getting smaller in size on \((- \infty ,-2)\) and \((2,3.5)\).
Note that all values of x are approximated from the graph.
At the start of the year, a smartphone app had 50,000 monthly users. The number of users increased at smaller and smaller rate until April 1. After that, the number of users began to decrease. From April 1 to June 1, the rate of decrease in users became larger and larger in size over time. From June 1 to September 1, the rate of decrease in users became smaller and smaller in size over time before leveling off at 40,000 users. Sketch a graph for a function \(U = g(t)\) that describes the relationship between the number of users \(U\) in \(t\) months since the start of the year.
- Answer
-
Focus: A Problem with Using a Function to Describe a Relationship
Consider the function \(y = f(x) = \dfrac{3}{x-2}\) . For the input \(x = 4\), there is no problem with the corresponding output. The output is a number. \[y = f(4) = \dfrac{3}{4-2}=\dfrac{3}{2} \nonumber\ \]However, for the input \(x = 2\), the output is undefined. \[y = f(2) = \dfrac{3}{2-2}=\dfrac{3}{0} \nonumber\ \] If we are attempting to describe a relationship, such as the value of a car or the height of a ball, an undefined value does not make sense in the context of the application. To address this problem, we must specify which inputs of a function give defined outputs. These specified inputs are the only ones we should use for the function. To help us quantify the inputs and defined outputs of a function, we introduce the concept of the domain and range of a function.
Definition: Domain and Range
The domain of a function is the set of all inputs of a function that give a defined output.
The range of a function is the set of all defined outputs for a given set of inputs.
The function \(y=f(x)\) is described by the table below. Find the domain and range of the function.
| \(x\) | 0 | 1 | 2 | 3 |
| \(y=f(x)\) | 5 | 7 | 6 | 5 |
Solution
The domain is a set or list of all the inputs. The domain of \(f\) is {0,1,2,3}. The range is the set of defined outputs for the given domain. The range of \(f\) is {5,6,7}.
Notice that we did not need to list the output of \(y=5\) twice even though it appeared as the output corresponding to two different inputs. Also, the curly brackets {} tell us that this a list of the included numbers.
The function \(y=g(x)\) is described by the graph below. Find the domain and range of the function.
Solution
On a graph, the inputs are listed on the horizontal axis where there are points with defined values of \(x\) for all values of x in between 1 and 4. Since there are an infinite number of values of \(x\) in this range, we can to use interval notation or an inequality to express this set. The domain of \(g\) is (1,4]. We could also use the inequality \(1 < x \le 4\). Notice that we have included the input \(x=4\) corresponding to the closed circle at this endpoint. However, we have excluded the input \(x=2\) corresponding to the open circle at this endpoint.
On a graph, the outputs are listed on the vertical axis where there are defined values of \(y\) for all values of y in between -1 and 3. The range of \(g\) is [-1,3].
The function \(y=h(x)\) is described by the graph below. Find the domain and range of the function.
Solution
Since the graph opens upward on both ends, the graph continues left and right for larger and smaller values of x than those shown on the graph. Therefore, there are points on the graph for all real values of \(x\) along the horizontal axis. The domain is \( (-\infty , \infty) \).
The smallest output \(y\) along the vertical axis is \(y= -4\). Since the graph opens upward on both ends, the graph continues upward for larger values of \(y\) than those shown on the graph. The range is \( (-4 , \infty) \).
If we are given a formula for a function, such as \(f(x)=x^2+3x\), either a domain will be given or one is implied by all the inputs that give a defined output. Let's explore each numerical operation, the corresponding restrictions entailed to get a defined output, and the effect on the domain with a function that involves each operation.
| Operation | Example | Restrictions with the Operation | Explanation | Domain of the Example Function |
| Addition | \(f(x) = x + 2 \) | none | We can add any number and 2 and get a real number | \(D\) is the set of all real numbers. \[D= (-\infty , \infty) \nonumber \] |
| Subtraction | \(f(x) = x - 2 \) | none | We can subtract any number and 2 and get a real number | \(D\) is the set of all real numbers. \[D= (-\infty , \infty) \nonumber \] |
| Multiplication | \(f(x) = 2x \) | none | We can multiply any number and 2 and get a real number | \(D\) is the set of all real numbers . \[D= (-\infty , \infty) \nonumber \] |
| Division | \(f(x)=\dfrac{2}{x} \) | The denominator \(\neq 0\) | We can't divide 2 by zero and get a real number | \(D\) is the set of all real numbers where \( x \neq 0 \) \[= ( -\infty ,0 ) \cup (0 , \infty) \nonumber \] |
| Power (whole number only) | \(f(x) = x^2 \) | none | We can raise any number to a whole number power | \(D\) is the set of all real numbers \[D = (-\infty , \infty ) \nonumber \] |
| Square & Even Roots | \(f(x) = \sqrt{x} \) | "The input of the radical" \(\ge 0\) (radicand \(\ge 0\) ) | We can't take the square or even root of a negative number | \[D= [0 ,\infty) \nonumber \] |
| Cube & Odd Roots | \(f(x) = \sqrt[3]{x} \) | none | We can take the square or even root of any number | \(D\) is the set of all real numbers \[= (-\infty , \infty) \nonumber \] |
At this point, the only functions with restrictions on the input are those involving division and even roots. For a function with multiple operations, we need to identify the operations in the function and corresponding restrictions on the input. For example, the function \( f(x)= 2x+1 \) involves multiplication and addition neither of which has a restiction. Therefore, the domain is \( (-\infty , \infty ) \).
To determine the domain of a function given by a formula identify the restrictions on the input \(x\) corresponding to each operation in the formula.
Find the domain of the following functions.
- \( f(x)= x^2+3x-2 \)
- \( g(x)= \dfrac{3}{x-2} \)
- \( h(x)= \sqrt{2x-1} \)
Solution
- Since the function \( f(x)= x^2+3x-2 \) involves the operations of addition, subtraction, multiplication, and a power, there are no restrictions on the input. Therefore, the domain of \(f\) is \( (-\infty , \infty ) \)
- The function \( g(x)= \dfrac{3}{x-2} \) involves a subtraction which has no corresponding restrictions. However, the division has the restriction that the denominator \( x - 2 \neq 0\) . Solving for \(x\) , \( x \neq 2\). Therefore, the domain of \(g\) is all real numbers where \( x \neq 2\) or \( (-\infty , 2) \cup (2, \infty )\) in interval notation.
- The only operation in \( h(x)= \sqrt{2x-1} \) that has a restriction is the square root where the input of the radical \( 2x-1 \ge 0\). Solving for x by adding 1 to both sides \[2x \ge 1 \nonumber \] Then dividing by 2 on both sides\[x \ge \dfrac{1}{2} \nonumber \] Therefore the domain is \( [\dfrac{1}{2} , \infty) \).
Now You Try: Exercise \(\PageIndex{3}\)
Find the domain of the following functions.
- \( f(x)= 2x^3+4 \)
- \( g(x)= \dfrac{5x+1}{4x-3} \)
- \( h(x)= \sqrt{2-7x} \)
- \( k(x)= \dfrac{5x+1}{\sqrt{2-7x}} \)
- Answer
-
- The domain of \(f\) is \( (-\infty , \infty ) \)
- The domain of \(g\) is all real numbers where \( x \neq \dfrac{3}{4}\) or \( (-\infty , \dfrac{3}{4}) \cup (\dfrac{3}{4}, \infty )\)
- The domain of \(h\)is \( (-\infty, \dfrac{2}{7}] \).
- The domain of \(k\) is \( (-\infty, \dfrac{2}{7}) \).
Identifying the domain of a function is key to addressing the original problem of undefined outputs by eliminating the inputs whose corresponding outputs are undefined. We could also identify the range if we wanted, but it is usually unnecessary. If we wanted to find the range of the function \( y = f(x)= 2x^3+4 \) exercise 1.4.3a for example, we have two options. We could use the graph to determine the range.
From the graph, the range of outputs \(y\) includes all real numbers on the y-axis. Therefore, the domain of \(f\) is \( (-\infty , \infty ) \). The other option is to use \(f\) to write a formula for the \(x\) if possible, then determine the restrictions on \(y\). Solving for \(y\) by subtracting 4 from both sides \[y -4 = 2x^3 \nonumber \] Then dividing by 2 \[\dfrac{y -4}{2} = x^3 \nonumber \] Then taking the cube root of both sides \[\sqrt[3]{\dfrac{y -4}{2}} = x \nonumber \] This is a formula for \(x\) in terms of \(y\). None of the operations have a restriction. The division is always by 2. Therefore, \(y\) can be any real number and so the range of \(f\) is \( (-\infty , \infty ) \).
Focus: Another Problem with Using Functions to Describe Relationships in Applications
Suppose we used the function \(V=f(t)=40000 - 4000t\) to describe the value \(V\) of a car \(t\) years after it was purchased. Although we could evaluate \(f(-2)\) mathematically, the input \(t=-2\) may not make sense in the context of this application. Two years before the car was purchased may not make sense if the car was brand new when purchased. Although the input \(t=12\) does make sense in the context of this application as \(t=12\) years since it was purchased, the output \(V = f(12)=40000 - 4000(12) = -8000\) does not make sense as the value of a car. The value of a car can't be negative. No one will pay some one else just to have their car (ignoring taxes). A person could argue that the minimum value must be at least a few hundred dollars since it could be sold as scrap metal. Furthermore, it is important to keep in mind the limitations of those models we create. If additional information or data revealed that the linear function that is being used to describe the value of a car is valid for only the first four years, we would need to further restrict the domain accordingly.
If a function is used to model an application, the domain must be restricted so that
- the inputs satisfy all mathematical restrictions
- the inputs make sense as the quantity the input represents in the application
- the inputs have corresponding outputs that make sense as the quantity the output represents in the application
- the inputs reflect any other limitations on the quantities the input and the corresponding output represent.
The function \(V=f(t)=40000 - 4000t\) gives the value \(V\) of a new car \(t\) years after it was purchased years after it was purchased. Find the domain of the function in this application.
Solution
Mathematically, there are no restrictions on the input since the operations in the function are subtraction and multiplication. The input \(t\) makes sense as the years since it was purchased when \(t \ge 0\) since the car was purchased when it was new. The smallest value of the output \(V\) that makes sense as a value of a car is \(V = 0\) since the value can't be negative. Since the value is decreasing over time, the largest value of the car would be \(V=f(0) = 40,000 - 4000(0) = $40,000 \) when the car is brand new. Therefore, the value of the car \(V\) must be in the interval \(0 \le V \le 40,000\). However, the domain is a restriction on the input \(t\) not the output \(V\). We can use the formula for \(V\) to write this double inequality in terms of \(t\), \[0 \le 40000 - 4000t \le 40,000 \nonumber \] Solving for t by subtracting 40,000 from all sides of the double inequality \[-40000 \le -4000t \le 0 \nonumber \] Then dividing by -4000 on all sides of the double inequality and switching the direction of the inequality signs \[10 \ge t \ge 0 \nonumber \] The domain of the application is the interval where \(0 \le t \le 10 \) that satisfies both of these restrictions.
Note that we could have also used the graph of \(V=f(t)\) to approximate the values of t where \(V \ge 0 \) if we were unable to solve the double inequality \(0 \le V \le 40,000\) analytically.
Every function the models a relationship between two quantities in an application should be accompanied by a domain which tells us the range of values for the input for which the model is appropriate and valid.
Example \(\PageIndex{9}\)
The height \(h\) of a ball thrown upward at \( 64 \dfrac{ft}{sec}\) from the ground after t seconds is modeled by the function \(h = s(t) = -16t^2+64t \). Find the domain of the application.
Solution
Mathematically, there are no restrictions on the input since the operations in the function are addition, multiplication, and powers. The input \(t\) makes sense as the seconds since it was thrown when \(t \ge 0\) since a negative value of \(t\), such as \(t = -2\), would before referring to a the time before the ball was thrown. The smallest value of the output \(h\) that makes sense as a height is \(h = 0\) since the ball can't go below the ground. Therefore, we must have \( h \ge 0\). Substituting the formula for \(h\) into this inequality, we have \(h = -16t^2+64t \ge 0\). We will learn later in this course how to solve this type of inequality analytically. In the mean time, we can solve it graphically. On the graph below \( h \ge 0\) when \(0 \le t \le 4 \). The domain of the application is the interval where \(0 \le t \le 4 \) that satisfies both of these restrictions on the input \(t\).
Suppose the cost (in thousands) of removing \(p\) percent of a pollutant from an acre foot of water at a waste water treatment facility is approximated with the model \(C(p)=\dfrac{1.25p+1250}{100-p}\). Find the domain of the application.
Solution
Mathematically, there is a restriction on the input for the division, the denominator \(100 - p \neq 0 \). Solving for p, we have \(p \neq 100 \). The input \(p\) makes sense as the percent removed when \(p \ge 0\) since it is impossible to remove a negative percentage of the pollutant. The largest percent of the pollutant removed is 100% if we remove it all. Therefore, \(0 \le p \le 100 \). The smallest value of the output \(C\) that makes sense is \(C = 0\). Without any additional information, we can't identify an upper bound on the the cost. Therefore, we must have \( C \ge 0\) for the output \(C\) to make sense as a cost. Substituting the formula for \(C\) into this inequality, we have \(\dfrac{1.25p+1250}{100-p} \ge 0\). This is another type of inequality that we will later learn how to solve analytically in this course. In the mean time, we can solve this inequality graphically. On the graph below, \( C \ge 0\) when \( p \le 100 \). The domain of the application is the interval where \(0 \le p < 100 \) that satisfies all three of these restrictions.
Suppose a shoe manufacturer uses test market data to estimate that the number of shoes, N, that they can sell at a particular price p with the function \(N=f(p) = 300 - 0.12p \). Find the domain of the application.
- Answer
-
The domain of the application is the interval where \(0 \le p \le 2,500 \).
Mathematically, there are no restrictions. The input \(p\) makes sense as a price of shoes when \(p \ge 0\) since the price charged can't be negative. The manufacturer can't sell a negative number of shoes, so the output \(N \ge 0\). The values of the input that have a corresponding output that satisfy this restriction are \(p \le = 2,500 \).
A pair of shoes that costs $2,500 is not realistic in most cases. If we had actual test market data on the prices and number of shoes sold, we should restrict the domain even further to a realistic interval of prices for the shoes.
Important Topics of this Section
- Describing varying rates of change on a graph
- Definition of domain
- Definition of range
- Domain and Range from graphs
- Domain and Range from tables
- Domain from a formula
- Domain of an application


1.5: §5. The Unique Nature of English
As we shall see in later chapters, the English language has had a most unusual history, having been subjected over the centuries (like the British Isles themselves) to waves of foreign influence. On account of this historical diversity, English is magnificently supplied with the raw material of words, often having several synonyms for a single concept. Because the grammatical structure of the language is Germanic, the functional linking words (prepositions, conjunctions, and articles) are solidly Germanic: to, from, and, but, however, when, since, the, an. So are the most common everyday nouns, verbs, and adjectives—for example, words for family relationships ( mother, father, sister, brother) , for number concepts ( one, two, ten, hundred); and for fundamental aspects of daily life ( house, home, bread, water, hay, harvest, cow, horse, sun, moon; eat, drink, talk, laugh, make love; good, bad, old, young). Not all Anglo-Saxon vocabulary is limited to words of one or two syllables; but we seldom have any problem understanding the longer Germanic forms, such as wonderful, womanhood, lighthearted, nevertheless, overwhelmingly, and unfriendliness.
Alongside this huge stock of native words, one finds at least an equal number of Latin (and/or Greek) loan-words, which entered English at various stages in its evolution. These can be very simple in form and obvious in meaning: mile, wine, cheese, city, school, church, farm, joy, grief, nice, fine, poor. (Here we see proof that not all short, blunt words are Anglo-Saxon.) Some common English words are spelled exactly as they were in the Latin of Julius Caesar: area, focus, actor, index, forum, consensus, data, item, video, referendum. Many others have been adapted to English morphology, but still clearly reveal their classical origins. As a general rule, words of this type tend to be longer than their Anglo-Saxon fellows, and they may often be learned or scholarly: desiccated, exiguous, refulgent, concatenation. Unless we have some knowledge of Latin roots, these may strike us as exotic aliens, polysyllabic and obscure. [1] When they are used skilfully and appropriately, Latin and Greek loan-words can vastly enrich our speech and writing, providing greater clarity and precision, adding subtle connotations that extend the boundaries of meaning, or creating images that could not otherwise be expressed. Unfortunately, the classical vocabulary in English is particularly subject to abuse, for these loan-words are the stock in trade of pedants, posers, and propaganda artists—indeed, of all who avoid plain language in order to hide or bend the truth.
The dual heritage of English, Anglo-Saxon and Latin, has given the language a great many LEARNED VARIANTS , synonyms that offer more scholarly, polite, or devious alternatives for ordinary concepts. Sometimes the variant may be needed in order to avoid any risk of ambiguity—in scientific and medical usage, for instance. In other contexts, the learned variant may be coy or pretentious. Here we meet the sub-class of EUPHEMISMS , expressions that stem from a desire to cloak ugly or embarrassing concepts in less offensive language. Consider the following pairs, where the blunt Anglo-Saxon word is followed by a more refined Latin counterpart:
| work | labo(u)r | smell | odo(u)r, fragrance |
| sweat | perspire | spit | expectorate |
| chew | masticate | drool | salivate |
| piss | urinate | shit | defecate |
| begin | commence | end | terminate |
| break | fracture | stitch | suture |
| womb | uterus | balls | testicles |
Quite deliberately, this list is a mixed bag. Some of the variants offer only a minor elevation in tone; others enable us to avoid a crude expression or a downright obscenity. Several of the less down-to-earth alternatives are clearly medical terms or technical jargon—does anyone ever expectorate outside the Baseball Rulebook? Still others, like commence and terminate , belong to the world of bureaucratic doublespeak. As a class, learned variants cannot be given blanket approval or condemnation; their appropriateness depends on the situation and the context.
The English language has often adapted Latin adjectives to fill an obvious need, where a simple Germanic noun has no suitable adjectival form. A case in point is suggested by the pair of synonyms listed above— work and labor. To find an English word that means “full of work” or “involving work,” we must resort to the Latin-derived laborious, because there is no workful or worksome or workly. Although there is a Germanic smelly— not to mention stinking —the connotations are negative; if we want an adjective to describe a pleasant “scent” (Latin) or “aroma” (Greek), we can use odorous or fragrant.
Compare these four Germanic nouns with their Latin adjectival companions:
| sun | solar | moon | lunar |
| earth | terrestrial | sky | celestial |
Astronomers and space scientists would be lost without the Latin forms. Of course, we do have the Germanic adjectives sunny and earthy (as well as earthly) ; but they are very different in meaning from solar and terrestrial.
What percentage of English vocabulary is derived from Latin and Greek? There is really no easy answer to that question, because it makes a big difference whether we are talking about all the words in a huge dictionary (including the highly technical jargon of science and medicine), or the words that an educated person would recognize in reading, or the words that most of us use in daily conversation. Some idea of the debt can be gained from the following crude estimate: of the 20,000 most common words in English, approximately half (10,000 words) are derived from Latin, either directly or through French. A much lower number (but at least 2,000) can be traced back to Greek.
To say that roughly half our standard vocabulary is Latin does not mean that half the words in any given sentence will be Latin. That is because the plain and simple Germanic words are used far more frequently; there may well be two or three occurrences of a word like the or and in the same sentence. Any writer whose Latin word frequency is above 30%, even in a short passage, is likely to be writing complex or technical English.
Let us try a little experiment, analyzing a few sample passages of English in order to test actual practice.
If we are communicating with children, we naturally use words that are short and easily understood. These are likely to be of Anglo-Saxon origin:
Humpty Dumpty sat on a wall,
Humpty Dumpty had a great fall;
All the King’s horses, and all the King’s men
Couldn’t put Humpty together again.
Latin derivatives are not all long and learned. The lone specimen in this little verse could hardly be less exotic: it is the plain-sounding wall , which came into the English language at a very early date.
See if you can spot the three non-Germanic words in this nursery rhyme:
Hey, diddle, diddle!
The cat and the fiddle,
The cow jumped over the moon;
The little dog laughed
To see such sport,
And the dish ran away with the spoon.
You will agree, I think, that the alien words are well disguised. One of them, believe it or not, is cat , which is probably derived from Late Latin cattus . The other non-Germanic words are sport and dish —the last an interesting example, because the original source was the Greek diskos ( δίσκο ς ), which became the Latin discus (our word discus , and the source also of English disc or disk ). For our present purposes, we can say that dish is derived from Latin. If it isn’t too ponderous to apply statistics to Mother Goose, let us note that the frequency of Latin vocabulary in “Hey, diddle, diddle” is 10% (3 words out of 30).
In the great tradition of plain English style, best represented by the Authorized or King James Version of the Bible, Latin vocabulary is about as infrequent as in Mother Goose. Here is that 1611 version of the Twenty-third Psalm, with the Latin words highlighted in bold type:
The Lord is my shepherd; I shall not want. He maketh me to lie down in green pastures : he leadeth me beside the still waters. He restoreth my soul: he leadeth me in the paths of righteousness for his name’s sake. Yea, though I walk through the valley of the shadow of death, I will fear no evil: for thou art with me: thy rod and thy staff they comfort me. Thou preparest a table before me in the presence of mine enemies : thou anointest my head with oil ; my cup runneth over. Surely goodness and mercy shall follow me all the days of my life: and I will dwell in the house of the Lord for ever.
In this clear and beautiful piece of English prose, the Latin frequency is 9.3% (11 words out of 118)—almost identical to that of our nursery rhyme. William Shakespeare, who was still alive when the King James Bible was published, had a similar preference for plain Anglo-Saxon diction, despite his enormous vocabulary. His famous Sonnet 18 (“Shall I compare thee to a summer’s day?”) has a Latin frequency of only 11.4 % (13 words out of 114); in florid theatrical passages, his Latin frequency may run to 20% or more.
John Bunyan (1628-1688) was profoundly influenced by the English Bible; one can hardly imagine simpler language than the opening paragraph of Pilgrim’s Progress :
As I walked through the wilderness of this world, I lighted on a certain place where was a den, and I laid me down in that place to sleep: and, as I slept, I dreamed a dream. I dreamed, and behold I saw a man clothed with rags, standing in a certain place , with his face from his own house, a book in his hand, and a great burden upon his back. I looked, and saw him open the book and read therein; and, as he read, he wept, and trembled ; and not being able longer to contain , he brake out with a lamentable cry , saying, “What shall I do?”
Here Bunyan’s Latin frequency is exactly 10% (11 words out of 110 ).
In contrast, let us consider a selection from a great nineteenth-century stylist, Samuel Taylor Coleridge: [3]
As the result of all my reading and meditation , I abstracted two critical aphorisms , deeming them to comprise the conditions and criteria of poetic style ; first, that not the poem which we have read, but that to which we return , with the greatest pleasure , possesses the genuine power , and claims the name of essential poetry . Second , that whatever lines can be translated into other words of the same language , without diminution of their significance , either in sense , or association , or in any worthy feeling, are so far vicious in their diction . ( Biographia Literaria , 1817)
In this passage, which is obviously worlds removed from Pilgrim’s Progress in purpose and tone, the combined frequency of Latin and Greek vocabulary is 31.9% (29 out of 91 words). The classical density of Coleridge’s prose is over three times that of Bunyan.
You may find even higher frequencies in a modern textbook or technical treatise. Here, for example, is a job advertisement that appeared in a recent national publication:
At a whopping 44.2% frequency (23 words out of 52), the passage proves that Latin and Greek derivatives are alive and well in contemporary English. But this is hardly a typical piece of English prose. In standard, non-technical writing, most good stylists today try not to sound too heavy and academic. Precise and unusual Latin words will be far more effective if they are met in plain Anglo-Saxon surroundings.
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We shall meet a delicious label that is applied facetiously to words of this type—the Latin derivative “sesquipedalian.” ↵
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Exceptions are hard to find. Bastard is from Late Latin, through Old French. The plain-sounding turd, a four-letter classic, is a surprising Latin derivative.  Luckily the cause was doomed to failure, or we might now find ourselves speechless at moments of great stress. ↵
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Latin derivatives are here shown in bold type, Greek derivatives in bold italic. ↵


8.1: What makes a school effective?
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By: Amy West
Learning Objectives
- Students should know the reasons why effectiveness in education is important.
- Students should be able to understand the characteristics which make a school effective.
- Given a list of characteristics, students should be able to identify whether they are effective or ineffective.
Introduction
People often talk about the importance of education and how schools need to improve. However, that is usually as far as the idea goes. This is because everyone has different definitions for what an effective school is and how to improve ineffective schools. Just because a student goes to school every day, it does not mean the school is doing the best job they can. There is always room for improvement, even in the best of schools.
Essentially, an effective school is one which is conducted in a safe environment by qualified teachers (Ashley, 2006). Everyone in the education field should have goals and high expectations for the school and its students (Ashley, 2006). Students should not only be taught academics, but also life skills (Ashley, 2006). There are many schools which are not at this level of effectiveness, though. Ineffective schools are most commonly found in high poverty areas (NEA, 2001). They typically are not well funded, do not have enough technology, and do not have highly qualified leaders (NEA, 2001).
Improving ineffective schools should be one of America's top priorities. A good education is the foundation of every qualified professional in this country. There have been many studies done to find out what makes high performing schools effective (Sadker, Zittleman, 2006). These studies show that there are six top factors which make a school effective, among other characteristics.
Positive Classroom Environment
The classroom environment is a very important aspect of an effective school. If a student does not feel safe or comfortable while in their classroom, they will not be able to focus on what is being taught.
It is essential that students feel safe while in school (Sadker, Zittleman, 2006). Even though 98% of teachers and 93% of students say they feel safe in their school, safety is still a topic of debate (Sadker, Zittleman, 2006). Making schools safe begins with teaching students respect, for themselves and others (Sadker, Zittleman, 2006). Another key step in safety is to identify student problems early, before the student turns to violence to solve them (Sadker, Zittleman, 2006).Aside from safety, it is also valuable for students to feel at ease while at school (Ashley, 2006). They should feel able to approach their teacher and other students (Ashley, 2006). The learning environment should also be open, clean, colorful, and inviting (Armstrong, 2002).
Qualified Educational Leaders
Having qualified leaders in the field of education is essential for success. These are the people who are around the children for approximately 35 hours each week. They are the people who will teach, inspire, and guide the students and they need to be qualified to do so.
The school leadership begins with the principal. The principal should be open and honest with teachers and staff (Ashley, 2006). They should make themselves approachable so teachers feel comfortable approaching them with new ideas (Ashley, 2006). Principals should allow their teachers to be creative and innovative in the classroom (Ashley, 2006). The other aspect of school leadership is the teachers. Schools should require that all of their teachers are highly qualified and fully certified (NEA, 2001). There also needs to be incentives for teachers who meet these requirements and do their job well (Lockheed, Levin, 1993). Incentives could include mentoring programs to help teachers adjust and loan forgiveness for those who teach in low performing schools (NEA, 2001).
Setting Goals
Setting goals for schools and students is an important process. It helps to motivate people when there is something specific to aim for. It is also a good way to show the clear path to success. The principal should work with the teachers to develop a plan and mission for the school (Sadker, Zittleman, 2006). Once they have done this, they should share it with the students (Sadker, Zittleman, 2006). Goals should include specific strategies and techniques for improvement and progress (NEA, 2001). Also, everyone should be pushed to work hard in order to achieve the goals and be successful and effective (Ashley, 2006).
High Expectations for Students
It is a known fact that people often live up to what is expected of them. If you do not expect much from a person, they often will not give you much. This is why it is crucial for teachers to expect their students to do well so there is a standard for students to uphold.
Each student should be expected to do their best on each assignment, despite previous performance or social background (Ashley, 2006). There is no reason that any student should be held to a lower standard than any other student (Ashley, 2006). When a teacher holds low expectations for a student, they tend to treat them differently (Sadker, Zittleman, 2006). They may even give those students less praise and less communication (Sadker, Zittleman, 2006). Once teachers have made their expectations clear to the students, they need to develop objectives that the students can excel in (Sadker, Zittleman, 2006).
Conducting and Implementing Research and Funding
It is necessary for schools to conduct research for many reasons. They need to use their research findings in order to find out what works in schools and what does not. Then, schools need to receive funding in order to implement plans that support the research findings.
Research should be conducted in many different ways in order to reach each student (Lockheed, Levin, 1993). Everyone in the field of education should pay close attention to the findings in order to figure out where next to go with the school curriculum (Lockheed, Levin, 1993). If the research shows that something is not working, the subject or technique should be discarded and a new plan should be implemented (Lockheed, Levin, 1993).
One of the biggest problems when it comes to funding in education is the difference of funds spent on students in high poverty areas compared to those in better financed schools (NEA, 2001). Low performing schools need the funding the most due to all of the resources they are already lacking (NEA, 2001). However, all schools need more funding in order to provide everything necessary for each student to get the best education possible (NEA, 2001).
Monitoring Student Progress
One of the best ways to see if the school curriculum is effective is by monitoring student progress. If a large majority of students are not progressing, it probably means that there needs to be massive curriculum changes. On the other hand, if only a few students are struggling, it most likely is a sign that they need extra help.
The best plan to have when monitoring progress is to take a well rounded approach since one test is not a good measure for everyone (NEA, 2001). Progress reports should be made so students can know their own strengths and weaknesses, as well as where they are now and future goals (Sadker, Zittleman, 2006). Students should be assessed in comparison to national averages as well as to see if they know the material (Sadker, Zittleman, 2006). There have also been studies done which show that doing homework is a good way to monitor progress and raise student achievement scores (Sadker, Zittleman, 2006).
Other Factors
Early Start
Typically, when a child starts attending school is based on their age. However, a child's age does not have much to do with their ability to learn. Some children may be ready to start learning earlier than the minimum required age. Educational studies have shown that children who attend early start schools receive better grades and have higher IQs as adults (Sadker, Zittleman, 2006).
Extended Learning
It is common sense that more time spent in school means more time spent learning. This exra learning time could help students who are struggling to catch up with their classmates (NEA, 2001). Other methods to extend learning include longer school years, more testing, more graded homework assignments, and more after school programs (NEA, 2001).
Better Behavior
One of the biggest complaints from well behaved students is that their teachers spend too much time disciplining students with behavioral issues (Ashley, 2006). When teachers have to stop their lesson in order to discipline a student, it causes a huge disruption for the rest of the class. Teachers need to be taught how to handle all of these minor interruptions, as well as major disciplinary issues, so that valuable learning time is not lost (Ashley, 2006).
Smaller Classes and Schools
Whenever a student is in the position to receive extra support and attention, they are likely to excel in their schoolwork. Small class size, particularly during elementary school, typically has a positive effect on students because they receive more individual attention (NEA, 2001). Studies have shown that, for older students, being in smaller schools is most beneficial (Sadker, Zittleman, 2006). It has been proven that students who attend small schools are more likely to pass their classes and attend college (Sadker, Zittleman, 2006).
Community Partnerships
The saying, "It takes a village to raise a child," is true when it comes to a child's education, as well. A child will only benefit if teachers, parents, policy makers, and any other community figures are involved in their education (Ashley, 2006). It is also beneficial if all of these people involved share their ideas with each other about how to improve education (Ashley, 2006). It is extremely important for parents to have chances to be involved with their child's education by opportunities such as skill workshops and volunteer programs (NEA, 2001).
Student Driven Curriculum
There have been studies which have shown that when students have a hand in developing the curriculum, they are more invested in it (Armstrong, 2002). This is also a good way to develop assignments in which all learning styles are addressed (Armstrong, 2002). After all, the students are the ones who know how they learn best. Also, students have said that they learn better when assignments are individualized and engaging (Armstrong, 2002).
Technology
Using technology at schools and for homework can be a great thing for students. Technology can be an excellent motivational tool for students who are otherwise disinterested (NEA, 2001). Schools can also use technology to keep track of student behavior and to monitor progress (NEA, 2001).
Planning for the Future
It is vital that schools help prepare their students for the future. Students should leave school with the knowledge that lifelong learning is the key to success (Armstrong, 2002). Also, students who leave high school to enter the "real world" should have been taught the skills necessary for surviving it (Armstrong, 2002).
Conclusion
Education is the key to success in this country. It is vital that we do everything we can in order to ensure that each student gets the best education available. This is important because these students are going to be entering professional fields and, possibly, even teaching the next generation.
Simply attending school does not guarantee a good education. There are many factors which must be used together in order to build the best education possible. It is necessary for schools to have positive, open environments with highly qualified leaders. It is also crucial for schools to research progress in order to see what is or is not working so they are able to build a balanced curriculum.
A school does not just need to offer an education, they need to offer each student the best education they possibly can. By implementing as many of these factors as they can, a school can give each student a well rounded, successful, and effective education.
References
Armstrong, Sara (February 2004). What makes a good school: Students speak up at leadership forum. Edutopia , Retrieved September 10, 2008, from http://www.edutopia.org/what-makes-good-school-students-speak-leadership-forum
Ashley, D (June 2006). Creating an effective school. Curriculum Management Update , Retrieved September 10, 2008, from http://www.teachingexpertise.com/articles/creating-an-effective-school-1034
Lockheed, Levin (1993). What do we know about school effectiveness and school improvement. World Bank , Retrieved September 10, 2008, from www1.worldbank.org/education/est/resources/topic%20papers/Types%20of%20policy.doc
NEA (February 2001). America's top education priority: Lifting up low performing schools. National Education Association , Retrieved September 10, 2008, from www.nea.org/priorityschools/priority.html
Sadker, D, & Zittleman, K (2006). What makes a school effective?. McGraw Hill , Retrieved September 10, 2008, from http://www.education.com/reference/article/Ref_What_Makes_School/#


21.5B: Inspiration
Inhalation is the flow of air into an organism that is due to a pressure difference between the atmosphere and alveolus.
LEARNING OBJECTIVE
Outline the mechanics of inspiration
Key Takeaways
Key Points
- In humans, inspiration is the flow of air into an organism from the external environment, through the airways, and into the alveoli.
- Inhalation begins with the onset of a contraction of the diaphragm, which results in expansion of the thoracic and pleural cavities and a decrease in pressure (also called an increase in negative pressure).
- There are many accessory muscles involved in inhalation—such as external intercostal muscles, scalene muscles, the sternocleidomastoid muscle, and the trapezius muscle.
- Breathing only with the accessory muscles instead of the diaphragm is considered inefficient, and provides much less air during inhalation.
- The negative pressure in the pleural cavity is enough to hold the lungs open in spite of the inherent elasticity of the tissue. The thoracic cavity increases in volume causing a drop in the pressure (a partial vacuum) within the lung itself.
- As long as pressure within the alveoli is lower than atmospheric pressure, air will continue to move inwardly, but as soon as the pressure is stabilized air movement stops.
Key Terms
- inspiration : The drawing of air into the lungs, accomplished in mammals by elevation of the chest walls and flattening of the diaphragm.
- accessory muscles : Muscles that help expand small parts of the thoracic cavity, either working in addition to the diaphragm or substituting for it if the diaphragm becomes injured.
- intrapleural pressure : The pressure inside the pleural cavity, which is negative compared to outside air and becomes even more negative during inspiration.
Inspiration refers to inhalation—it is the flow of the respiratory current into an organism. In humans it is the movement of ambient air through the airways and into the alveoli of the lungs.
The Process of Inspiration
Inspiration begins with the contraction of the diaphragm, which results in expansion of the thoracic cavity and the pleural cavity. The pleural cavity normally has a lower pressure compared to ambient air (–3 mmHg normally and typically –6 mmHg during inspiration), so when it expands, the pressure inside the lungs drops.
Pressure and volume are inversely related to each other, so the drop in pressure inside the lung increases the volume of air inside the lung by drawing outside air into the lung. As the volume of air inside the lung increases, the lung pushes back against the expanded pleural cavity as a result of the drop in intrapleural pressure (pressure inside the pleural cavity).
The force of the intrapleural pressure is even enough to hold the lungs open during inpiration despite the natural elastic recoil of the lung. The alveolar sacs also expand as a result of being filled with air during inspiration, which contributes to the expansion inside the lung.
Eventually, the pressure inside the lung becomes less negative as the volume inside the lung increases and, when pressure and volume stabilize, air movement stops, inspiration ends, and expiration (exhalation) will begin. Deeper breaths have higher tidal volumes and require a greater drop in intrapleural pressure compared to shallower breaths.
Respiratory System: Resistance in any part of the respiratory tract can cause problems.
Accessory Muscles of Inspiration
The diaphragm is the primary muscle involved in breathing, however several other muscles play a role in certain circumstances. These muscles are referred to as accessory muscles of inhalation.
- External intercostal muscles: Muscles located between the ribs that help the thoracic cavity and pleural cavity expand during quiet and forced inspiration.
- Scalene muscles: Muscles in the neck that lift the upper ribs (and thoracic cavity around the upper ribs) to help with breathing. They provide a mechanism for inspiration when the diaphragm is injured and can’t contract normally.
- Sternocleidomastoid muscle: Muscles that connect the sternum to the neck and allow for rotation and turning of the head. They can lift the upper ribs like the scalene muscles can.
- Trapezius muscle: Muscles in the shoulders that retract the scapula and expand the upper part of the thoracic cavity.
The accessory muscles assist breathing by expanding the thoracic cavity in a similar way to the diaphragm. However, they expand a much smaller part of the thoracic cavity compared to diaphragm. Therefore they should not be used as the primary mechanism of inhalation, because they take in much less air compared to the diaphragm resulting in a much lower tidal volume.
For example, singers need a lot of air to support the powerful voice production needed for singing. A common problem in novice singers is breathing with the accessory muscles of the neck, shoulder, and ribs instead of the diaphragm, which gives them a much smaller air supply than what is needed to sing properly.


10.3: Adding Summary
In the Page Settings of each LibreTexts page is a summary block that will accept approximately three lines of text (Figure \(\PageIndex{1}\)). Think of the summary as a tweet (like on Twitter) on the subject of the page. It is almost the same length as a tweet and has a similar impact.
Adding a Summary to a page is not necessary; you and your text can live without them if you like. However, they are beneficial in several ways:
- If you do a search on our platform, the summary will show up next to the page to help searchers identify the page(s) that best addresses their search
- The summaries of Sections (on pages of content of a text) will show up on the Table of Contents (in Chapter Overview) in PDFs and on chapter pages when viewed online (Figure \(\PageIndex{2}\)).
- The summaries of Chapters will show up in the Table of Contents in PDFs and if you hover your mouse over the chapter thumbnails of a textbook.
- The summaries are also used by Google to aid in search results to guide traffic to your texts (i.e., Search Engine Optimization).


9.2: Defining Emotions
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In this section, we will define emotions and explain the difference between two different types of emotions: primary and secondary.
Emotions are physiological, behavioral, and/or communicative reactions to stimuli that are cognitively processed and experienced as emotional (Planlap, Fitness, & Fehr, 2006). Emotions are often internally experienced through physiological changes such as increased heart rate, a tense stomach, or a cold chill. These physiological reactions may not be noticeable by others and are therefore intrapersonal unless we exhibit some change in behavior that clues others into our internal state or we verbally or nonverbally communicate our internal state. Sometimes our behavior is voluntary—we ignore someone, which may indicate we are angry with them—or involuntary—we fidget or avoid eye contact while talking because we are nervous.
When we communicate our emotions, we call attention to ourselves and provide information to others that may inform how they should react. For example, when someone we care about displays behaviors associated with sadness, we are likely to know that we need to provide support (Planlap, Fitness, & Fehr, 2006). We learn, through socialization, how to read and display emotions, although some people are undoubtedly better at reading emotions than others. However, as with most aspects of communication, we can all learn to become more competent with increased knowledge and by learning the difference between our primary and secondary emotions.
Since these emotions require more processing, they are more easily influenced by thoughts and can be managed, which means we can become more competent communicators by becoming more aware of how we experience and express secondary emotions. Although there is more cultural variation in the meaning and expression of secondary emotions, they are still universal in that they are experienced by all cultures. It’s hard to imagine what our lives would be like without emotions, and in fact many scientists believe we wouldn’t be here without them.


1.2A: Life Functions
All life forms require certain core elements as well as physical and chemical factors from the ecosystem
for biochemical functioning.
- Categorize living organisms and non-living things
Key Points
- Living organisms are systems made from cells and are capable of life functions such as reproduction, metabolism, response to stimuli, and homeostasis.
- Nonliving things are either living things that have died or inanimate objects that were never alive.
- The nonliving (abiotic) components of an ecosystem include energy, oxygen, water, nutrients, and temperature.
- Living organisms have ranges of ecosystem conditions where they can perform all physiological functions of life. Organisms cannot perform these functions as well or at all outside of these conditions, but some organisms have developed ways to adapt to environmental changes. Migration and hibernation are two examples of these adaptations.
- Organisms called extremophiles can assume forms enabling them to withstand freezing, dehydration, starvation, high levels of radiation, and other physical or chemical challenges. Extremophiles can survive exposure to such conditions for weeks, months, years, or even centuries.
- All life forms require certain core chemical elements for biochemical structure and function. These include carbon, hydrogen, nitrogen, oxygen, phosphorus, and sulfur—the elemental macronutrients for all organisms—often represented by the acronym CHNOPS.
Key Terms
- extremophile : A microorganism that can survive in extreme environmental conditions.
- organism : A contiguous living system that can metabolize, grow, respond to stimuli, reproduce, and adapt to its environment.
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life
: A status given to any entity with the properties of reproduction and metabolism, including
animals, plants, fungi, bacteria, and archaea (single-celled microorganisms).
Living Organisms
An organism is a living system capable of growth, reproduction,metabolism, response to stimuli, adaptation and homeostasis.
Cells are the basic unit of life that can be found in every living organism. Organisms can be unicellular or multicellular and include plants, animals, bacteria, archaea, and fungi. Multicellular organisms are typically made up of organ systems, organs, tissues, and cells. More complex organisms are capable of communication.
The world is made up of both living (biotic) and nonliving things. Nonliving things are living things that have died or inanimate (abiotic) objects that were never alive to begin with. Living and nonliving things can be distinguished by function. Nonliving things are not capable of growth, reproduction, response to stimuli, adaptation, or homeostasis.
It is unknown whether viruses are actually alive. They are capable of reproduction (within a host) and adaptation, but they are not capable of homeostasis or metabolic functions. Because viruses exhibit some but not all characteristics of organisms, they cannot be fully considered living things.
Ecosystems Are Important for Life
The abiotic components of an ecosystem are the physical and chemical factors necessary for life—energy (sunlight or chemical energy), water, temperature, oxygen, and nutrients. In most ecosystems, the environmental conditions vary throughout the day or over the course of seasons. Organisms who live in an ecosystem must be able to survive its range of conditions, called the “range of tolerance.” Outside the range of tolerance are the “zones of physiological stress,” where survival and reproduction are possible but not optimal. Beyond these zones are the “zones of intolerance,” where life for that organism is impossible. Many organisms can adapt to intolerance in their environment through seasonal migration, hibernation, or other adaptations.
Many microorganisms have evolved adaptations that allow them to survive in zones that are intolerant for most other organisms, such as extreme heat or cold, dehydration, starvation, or high levels of radiation. These microorganisms are called extremophiles, and they thrive outside the ranges where life is commonly found. They excel at exploiting uncontested sources of energy. While all organisms are composed of nearly identical types of molecules, evolution has enabled extremophile microbes to cope with this wide range of physical and chemical conditions.
Rainforest Trees : Plant growth in the Hoh Rainforest
Chemical Requirements for Life
All life forms require certain core chemical elements for biochemical functioning. These include carbon, hydrogen, nitrogen, oxygen, phosphorus, and sulfur—the elemental macronutrients for all organisms. Together these make up carbohydrates, nucleic acids, proteins, and lipids, the four categories of molecules that make up the structure and function of organic life. The most widespread and important of these elements is carbon, which holds molecules together through multiple, stable covalent bonds. Organic molecules generally contain carbon, and inorganic molecules generally do not contain carbon (with the key exception of carbon dioxide). Organic molecules and their properties form the basic structure of cells and allow physiological processes to occur.


1.5: Right Triangle Trigonometry
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Learning Objectives
- Use right triangles to evaluate trigonometric functions.
- Find function values for 30°(\(\dfrac{\pi}{6}\)),45°(\(\dfrac{\pi}{4}\)),and 60°(\(\dfrac{\pi}{3}\)).
- Use equal cofunctions of complementary angles.
- Use the definitions of trigonometric functions of any angle.
- Use right-triangle trigonometry to solve applied problems.
Mt. Everest, which straddles the border between China and Nepal, is the tallest mountain in the world. Measuring its height is no easy task and, in fact, the actual measurement has been a source of controversy for hundreds of years. The measurement process involves the use of triangles and a branch of mathematics known as trigonometry. In this section, we will define a new group of functions known as trigonometric functions, and find out how they can be used to measure heights, such as those of the tallest mountains.
We have previously defined the sine and cosine of an angle in terms of the coordinates of a point on the unit circle intersected by the terminal side of the angle:
\[ \begin{align*} \cos t &= x \\ \sin t &=y \end{align*} \]
In this section, we will see another way to define trigonometric functions using properties of right triangles .
Using Right Triangles to Evaluate Trigonometric Functions
In earlier sections, we used a unit circle to define the trigonometric functions . In this section, we will extend those definitions so that we can apply them to right triangles. The value of the sine or cosine function of \(t\) is its value at \(t\) radians. First, we need to create our right triangle. Figure \(\PageIndex{1}\) shows a point on a unit circle of radius 1. If we drop a vertical line segment from the point \((x,y)\) to the x -axis, we have a right triangle whose vertical side has length \(y\) and whose horizontal side has length \(x\). We can use this right triangle to redefine sine, cosine, and the other trigonometric functions as ratios of the sides of a right triangle.
We know
\[ \cos t= \frac{x}{1}=x \]
Likewise, we know
\[ \sin t= \frac{y}{1}=y \]
These ratios still apply to the sides of a right triangle when no unit circle is involved and when the triangle is not in standard position and is not being graphed using \((x,y)\) coordinates. To be able to use these ratios freely, we will give the sides more general names: Instead of \(x\),we will call the side between the given angle and the right angle the adjacent side to angle \(t\). (Adjacent means “next to.”) Instead of \(y\),we will call the side most distant from the given angle the opposite side from angle \(t\). And instead of \(1\),we will call the side of a right triangle opposite the right angle the hypotenuse . These sides are labeled in Figure \(\PageIndex{2}\).
Understanding Right Triangle Relationships
Given a right triangle with an acute angle of \(t\),
\[\begin{align} \sin (t) &= \dfrac{\text{opposite}}{\text{hypotenuse}} \label{sindef}\\ \cos (t) &= \dfrac{\text{adjacent}}{\text{hypotenuse}} \label{cosdef}\\ \tan (t) &= \dfrac{\text{opposite}}{\text{adjacent}} \label{tandef}\end{align}\]
A common mnemonic for remembering these relationships is SohCahToa, formed from the first letters of “ S ine is o pposite over h ypotenuse, C osine is a djacent over h ypotenuse, T angent is o pposite over a djacent.”
how to: Given the side lengths of a right triangle and one of the acute angles, find the sine, cosine, and tangent of that angle
- Find the sine as the ratio of the opposite side to the hypotenuse.
- Find the cosine as the ratio of the adjacent side to the hypotenuse.
- Find the tangent is the ratio of the opposite side to the adjacent side.
Example \(\PageIndex{1}\): Evaluating a Trigonometric Function of a Right Triangle
Given the triangle shown in Figure \(\PageIndex{3}\), find the value of \(\cos α\).
Solution
The side adjacent to the angle is 15, and the hypotenuse of the triangle is 17, so via Equation \ref{cosdef}:
\[\begin{align*} \cos (α) &= \dfrac{\text{adjacent}}{\text{hypotenuse}} \\[4pt] &= \dfrac{15}{17} \end{align*}\]
Exercise \(\PageIndex{1}\)
Given the triangle shown in Figure \(\PageIndex{4}\), find the value of \(\sin t\).
- Answer
-
\(\frac{7}{25}\)
Relating Angles and Their Functions
When working with right triangles, the same rules apply regardless of the orientation of the triangle. In fact, we can evaluate the six trigonometric functions of either of the two acute angles in the triangle in Figure \(\PageIndex{5}\). The side opposite one acute angle is the side adjacent to the other acute angle, and vice versa.
We will be asked to find all six trigonometric functions for a given angle in a triangle. Our strategy is to find the sine, cosine, and tangent of the angles first. Then, we can find the other trigonometric functions easily because we know that the reciprocal of sine is cosecant, the reciprocal of cosine is secant, and the reciprocal of tangent is cotangent.
how to: Given the side lengths of a right triangle, evaluate the six trigonometric functions of one of the acute angles
- If needed, draw the right triangle and label the angle provided.
- Identify the angle, the adjacent side, the side opposite the angle, and the hypotenuse of the right triangle.
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Find the required function:
- sine as the ratio of the opposite side to the hypotenuse
- cosine as the ratio of the adjacent side to the hypotenuse
- tangent as the ratio of the opposite side to the adjacent side
- secant as the ratio of the hypotenuse to the adjacent side
- cosecant as the ratio of the hypotenuse to the opposite side
- cotangent as the ratio of the adjacent side to the opposite side
Example \(\PageIndex{2}\): Evaluating Trigonometric Functions of Angles Not in Standard Position
Using the triangle shown in Figure \(\PageIndex{6}\), evaluate \( \sin α, \cos α, \tan α, \sec α, \csc α,\) and \( \cot α\).
Solution
\[ \begin{align*} \sin α &= \dfrac{\text{opposite } α}{\text{hypotenuse}} = \dfrac{4}{5} \\ \cos α &= \dfrac{\text{adjacent to }α}{\text{hypotenuse}}=\dfrac{3}{5} \\ \tan α &= \dfrac{\text{opposite }α}{\text{adjacent to }α}=\dfrac{4}{3} \\ \sec α &= \dfrac{\text{hypotenuse}}{\text{adjacent to }α}= \dfrac{5}{3} \\ \csc α &= \dfrac{\text{hypotenuse}}{\text{opposite }α}=\dfrac{5}{4} \\ \cot α &= \dfrac{\text{adjacent to }α}{\text{opposite }α}=\dfrac{3}{4} \end{align*}\]
Exercise \(\PageIndex{2}\)
Using the triangle shown in Figure \(\PageIndex{7}\), evaluate \( \sin t, \cos t,\tan t, \sec t, \csc t,\) and \(\cot t\).
- Answer
-
\[\begin{align*} \sin t &= \frac{33}{65}, \cos t= \frac{56}{65},\tan t= \frac{33}{56}, \\ \\ \sec t &= \frac{65}{56},\csc t= \frac{65}{33},\cot t= \frac{56}{33} \end{align*}\]
Finding Trigonometric Functions of Special Angles Using Side Lengths
We have already discussed the trigonometric functions as they relate to the special angles on the unit circle. Now, we can use those relationships to evaluate triangles that contain those special angles. We do this because when we evaluate the special angles in trigonometric functions, they have relatively friendly values, values that contain either no or just one square root in the ratio. Therefore, these are the angles often used in math and science problems. We will use multiples of \(30°, 60°,\) and \(45°\), however, remember that when dealing with right triangles, we are limited to angles between \(0° \text{ and } 90°\).
Suppose we have a \(30°,60°,90°\) triangle, which can also be described as a \(\frac{π}{6}, \frac{π}{3},\frac{π}{2}\) triangle. The sides have lengths in the relation \(s,\sqrt{3}s,2s.\) The sides of a \(45°,45°,90° \)triangle, which can also be described as a \(\frac{π}{4},\frac{π}{4},\frac{π}{2}\) triangle, have lengths in the relation \(s,s,\sqrt{2}s.\) These relations are shown in Figure \(\PageIndex{8}\).
We can then use the ratios of the side lengths to evaluate trigonometric functions of special angles.
Given trigonometric functions of a special angle, evaluate using side lengths.
- Use the side lengths shown in Figure \(\PageIndex{8}\) for the special angle you wish to evaluate.
- Use the ratio of side lengths appropriate to the function you wish to evaluate.
Example \(\PageIndex{3}\): Evaluating Trigonometric Functions of Special Angles Using Side Lengths
Find the exact value of the trigonometric functions of \(\frac{π}{3}\), using side lengths.
Solution
\[\begin{align*} \sin (\dfrac{π}{3}) &= \dfrac{\text{opp}}{\text{hyp}}=\dfrac{\sqrt{3}s}{2s}=\dfrac{\sqrt{3}}{2} \\ \cos (\dfrac{π}{3}) &= \dfrac{\text{adj}}{\text{hyp}}=\dfrac{s}{2s}=\dfrac{1}{2} \\ \tan (\dfrac{π}{3}) &= \dfrac{\text{opp}}{\text{adj}} =\dfrac{\sqrt{3}s}{s}=\sqrt{3} \\ \sec (\dfrac{π}{3}) &= \dfrac{\text{hyp}}{\text{adj}} = \dfrac{2s}{s}=2 \\ \csc (\dfrac{π}{3}) &= \dfrac{\text{hyp}}{\text{opp}} =\dfrac{2s}{\sqrt{3}s}=\dfrac{2}{\sqrt{3}}=\dfrac{2\sqrt{3}}{3} \\ \cot (\dfrac{π}{3}) &= \dfrac{\text{adj}}{\text{opp}}=\dfrac{s}{\sqrt{3}s}=\dfrac{1}{\sqrt{3}}=\dfrac{\sqrt{3}}{3} \end{align*}\]
Exercise \(\PageIndex{3}\)
Find the exact value of the trigonometric functions of \(\frac{π}{4}\) using side lengths.
- Answer
-
\( \sin (\frac{π}{4})=\frac{\sqrt{2}}{2}, \cos (\frac{π}{4})=\frac{\sqrt{2}}{2}, \tan (\frac{π}{4})=1,\)
\( \sec (\frac{π}{4})=\sqrt{2}, \csc (\frac{π}{4})=\sqrt{2}, \cot (\frac{π}{4}) =1 \)
Using Equal Cofunction of Complements
If we look more closely at the relationship between the sine and cosine of the special angles relative to the unit circle, we will notice a pattern. In a right triangle with angles of \(\frac{π}{6}\) and \(\frac{π}{3}\), we see that the sine of \(\frac{π}{3}\), namely \(\frac{\sqrt{3}}{2}\), is also the cosine of \(\frac{π}{6}\), while the sine of \(\frac{π}{6}\), namely \(\frac{1}{2},\) is also the cosine of \(\frac{π}{3}\) (Figure \(\PageIndex{9}\)).
\[\begin{align*} \sin \frac{π}{3} &= \cos \frac{π}{6}=\frac{\sqrt{3}s}{2s}=\frac{\sqrt{3}}{2} \\ \sin \frac{π}{6} &= \cos \frac{π}{3}=\frac{s}{2s}=\frac{1}{2} \end{align*}\]
This result should not be surprising because, as we see from Figure \(\PageIndex{9}\), the side opposite the angle of \(\frac{π}{3}\) is also the side adjacent to \(\frac{π}{6}\), so \(\sin (\frac{π}{3})\) and \(\cos (\frac{π}{6})\) are exactly the same ratio of the same two sides, \(\sqrt{3} s\) and \(2s.\) Similarly, \( \cos (\frac{π}{3})\) and \( \sin (\frac{π}{6})\) are also the same ratio using the same two sides, \(s\) and \(2s\).
The interrelationship between the sines and cosines of \(\frac{π}{6}\) and \(\frac{π}{3}\) also holds for the two acute angles in any right triangle, since in every case, the ratio of the same two sides would constitute the sine of one angle and the cosine of the other. Since the three angles of a triangle add to π, and the right angle is \(\frac{π}{2}\), the remaining two angles must also add up to \(\frac{π}{2}\). That means that a right triangle can be formed with any two angles that add to \(\frac{π}{2}\)—in other words, any two complementary angles. So we may state a cofunction identity : If any two angles are complementary, the sine of one is the cosine of the other, and vice versa. This identity is illustrated in Figure \(\PageIndex{10}\).
Using this identity, we can state without calculating, for instance, that the sine of \(\frac{π}{12}\) equals the cosine of \(\frac{5π}{12}\), and that the sine of \(\frac{5π}{12}\) equals the cosine of \(\frac{π}{12}\). We can also state that if, for a certain angle \(t, \cos t= \frac{5}{13},\) then \( \sin (\frac{π}{2}−t)=\frac{5}{13}\) as well.
COFUNCTION IDENTITIES
The cofunction identities in radians are listed in Table \(\PageIndex{1}\).
| \( \cos t= \sin (\frac{π}{2}−t)\) | \( \sin t= \cos (\dfrac{π}{2}−t)\) |
| \( \tan t= \cot (\dfrac{π}{2}−t) \) | \( \cot t= \tan (\dfrac{π}{2}−t)\) |
| \( \sec t= \csc (\dfrac{π}{2}−t) \) | \( \csc t= \sec (\dfrac{π}{2}−t)\) |
how to: Given the sine and cosine of an angle, find the sine or cosine of its complement.
- To find the sine of the complementary angle, find the cosine of the original angle.
- To find the cosine of the complementary angle, find the sine of the original angle.
Example \(\PageIndex{4}\): Using Cofunction Identities
If \( \sin t = \frac{5}{12},\) find \(( \cos \frac{π}{2}−t)\).
Solution
According to the cofunction identities for sine and cosine,
\[ \sin t= \cos (\dfrac{π}{2}−t). \nonumber\]
So
\[ \cos (\dfrac{π}{2}−t)= \dfrac{5}{12}. \nonumber\]
Exercise \(\PageIndex{4}\)
If \(\csc (\frac{π}{6})=2,\) find \( \sec (\frac{π}{3}).\)
Solution
2
Using Trigonometric Functions
In previous examples, we evaluated the sine and cosine in triangles where we knew all three sides. But the real power of right-triangle trigonometry emerges when we look at triangles in which we know an angle but do not know all the sides.
how to: Given a right triangle, the length of one side, and the measure of one acute angle, find the remaining sides
- For each side, select the trigonometric function that has the unknown side as either the numerator or the denominator. The known side will in turn be the denominator or the numerator.
- Write an equation setting the function value of the known angle equal to the ratio of the corresponding sides.
- Using the value of the trigonometric function and the known side length, solve for the missing side length.
Example \(\PageIndex{5}\): Finding Missing Side Lengths Using Trigonometric Ratios
Find the unknown sides of the triangle in Figure \(\PageIndex{11}\).
Solution
We know the angle and the opposite side, so we can use the tangent to find the adjacent side.
\[ \tan (30°)= \dfrac{7}{a} \nonumber\]
We rearrange to solve for \(a\).
\[\begin{align} a &=\dfrac{7}{ \tan (30°)} \\ & =12.1 \end{align} \nonumber\]
We can use the sine to find the hypotenuse.
\[ \sin (30°)= \dfrac{7}{c} \nonumber\]
Again, we rearrange to solve for \(c\).
\[\begin{align*} c &= \dfrac{7}{\sin (30°)} =14 \end{align*}\]
Exercise \(\PageIndex{5}\):
A right triangle has one angle of \(\frac{π}{3}\) and a hypotenuse of 20. Find the unknown sides and angle of the triangle.
- Answer
-
\(\mathrm{adjacent=10; opposite=10 \sqrt{3}; }\) missing angle is \(\frac{π}{6}\)
Using Right Triangle Trigonometry to Solve Applied Problems
Right-triangle trigonometry has many practical applications. For example, the ability to compute the lengths of sides of a triangle makes it possible to find the height of a tall object without climbing to the top or having to extend a tape measure along its height. We do so by measuring a distance from the base of the object to a point on the ground some distance away, where we can look up to the top of the tall object at an angle. The angle of elevation of an object above an observer relative to the observer is the angle between the horizontal and the line from the object to the observer's eye. The right triangle this position creates has sides that represent the unknown height, the measured distance from the base, and the angled line of sight from the ground to the top of the object. Knowing the measured distance to the base of the object and the angle of the line of sight, we can use trigonometric functions to calculate the unknown height. Similarly, we can form a triangle from the top of a tall object by looking downward. The angle of depression of an object below an observer relative to the observer is the angle between the horizontal and the line from the object to the observer's eye. See Figure \(\PageIndex{12}\).
how to: Given a tall object, measure its height indirectly
- Make a sketch of the problem situation to keep track of known and unknown information.
- Lay out a measured distance from the base of the object to a point where the top of the object is clearly visible.
- At the other end of the measured distance, look up to the top of the object. Measure the angle the line of sight makes with the horizontal.
- Write an equation relating the unknown height, the measured distance, and the tangent of the angle of the line of sight.
- Solve the equation for the unknown height.
Example \(\PageIndex{6}\): Measuring a Distance Indirectly
To find the height of a tree, a person walks to a point 30 feet from the base of the tree. She measures an angle of 57° 57° between a line of sight to the top of the tree and the ground, as shown in Figure \(\PageIndex{13}\). Find the height of the tree.
Solution
We know that the angle of elevation is \(57°\) and the adjacent side is 30 ft long. The opposite side is the unknown height.
The trigonometric function relating the side opposite to an angle and the side adjacent to the angle is the tangent. So we will state our information in terms of the tangent of \(57°\), letting \(h\) be the unknown height.
\[\begin{array}{cl} \tan θ = \dfrac{\text{opposite}}{\text{adjacent}} & \text{} \\ \tan (57°) = \dfrac{h}{30} & \text{Solve for }h. \\ h=30 \tan (57°) & \text{Multiply.} \\ h≈46.2 & \text{Use a calculator.} \end{array} \]
The tree is approximately 46 feet tall.
Exercise \(\PageIndex{6}\):
How long a ladder is needed to reach a windowsill 50 feet above the ground if the ladder rests against the building making an angle of \(\frac{5π}{12}\) with the ground? Round to the nearest foot.
- Answer
-
About 52 ft
media:
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Key Equations
Cofunction Identities
\[\begin{align*} \cos t &= \sin ( \frac{π}{2}−t) \\ \sin t &= \cos (\frac{π}{2}−t) \\ \tan t &= \cot (\frac{π}{2}−t) \\ \cot t &= \tan (\frac{π}{2}−t) \\ \sec t &= \csc (\frac{π}{2}−t) \\ \csc t &= \sec (\frac{π}{2}−t) \end{align*}\]
Key Concepts
- We can define trigonometric functions as ratios of the side lengths of a right triangle. See Example .
- The same side lengths can be used to evaluate the trigonometric functions of either acute angle in a right triangle. See Example .
- We can evaluate the trigonometric functions of special angles, knowing the side lengths of the triangles in which they occur. See Example .
- Any two complementary angles could be the two acute angles of a right triangle.
- If two angles are complementary, the cofunction identities state that the sine of one equals the cosine of the other and vice versa. See Example .
- We can use trigonometric functions of an angle to find unknown side lengths.
- Select the trigonometric function representing the ratio of the unknown side to the known side. See Example .
- Right-triangle trigonometry permits the measurement of inaccessible heights and distances.
- The unknown height or distance can be found by creating a right triangle in which the unknown height or distance is one of the sides, and another side and angle are known. See Example .
Glossary
- adjacent side
- in a right triangle, the side between a given angle and the right angle
- angle of depression
- the angle between the horizontal and the line from the object to the observer’s eye, assuming the object is positioned lower than the observer
- angle of elevation
- the angle between the horizontal and the line from the object to the observer’s eye, assuming the object is positioned higher than the observer
- opposite side
- in a right triangle, the side most distant from a given angle
- hypotenuse
- the side of a right triangle opposite the right angle


4.5: Promotional Sign (Painting on Plywood)
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Promotional Sign (Painting on Plywood)
Norwegian-American Oljanna Cunneen, c. 1984 Plywood, paint Gift of Mount Horeb Area Chamber of Commerce MHAHS 2003.116.0003 |
Who best to greet visitors to the “Troll Capital of the World” than a smiling, one-toothed, white haired troll? Painted on plywood by Oljanna Cunneen, this double sided, four toed troll served as a promotional piece along Main Street. With a crowned girl tucked into his hair and a bird perched on his hands, this dapper fellow reflects both Cunneen’s sense of humor and her tireless promotion of the area’s Norwegian-American identify. More than a dozen of these plywood cutouts once adorned Mount Horeb’s Main Street, dubbed the “Trollway” in the mid- to late-1980s.
The stitching together of Scandinavian and American cultures to create a unique, authentic, and personal ethnic identity frames the story of Mount Horeb. The Trollway is a unique result of the influences. While the character of the troll emerges from Scandinavian folklore, troll sculptures and cutouts, and the brand of “Trolltown U.S.A.” incorporates elements of American highway culture as well. The relationship between the trolls and oddities such as the Forevertron in Sumpter, WI, the Tin Man Mailbox in Black Earth, WI, and a sculpture of Paul Bunyan and Babe the Ox in Bemidji, MN, as well as the commercial strip of downtown Las Vegas, reveal how Mount Horeb aligns itself with the American tradition of the roadside wonder.
Attractions such as Carhenge, Cadillac Ranch, or Wall Drug have long been used to draw motorists off of the highway and towards downtowns or Main Streets to foster local economies. The Trollway joins in this tradition of branding a place as unique and welcoming, and also seeks to draw motorists into town to see these works of art such as Michael Feeney’s troll sculptures or Oljanna Cunneen’s troll dolls. Another example of this, although at a much larger scale, is the Las Vegas strip, famous for its extensive signage and neon lights. Despite their many and obvious differences, the Las Vegas strip is a distant relative of the Trollway in how it aims to capture the attention and imagination of passers-by in order to draw them into the main commercial parts of town.


1.7: Atomic Structure and Electron Configuration
As we learned in Section 1.1, modern atomic theory places protons and neutrons in the nucleus of an atom and electrons are placed in a diffuse cloud surrounding this nucleus. As chemists and physicists began examining the structure of atoms, however, it became apparent that all of the electrons in atoms were not equivalent. The electrons were not randomly placed in one massive “cloud”, rather they seemed to be arranged in distinct energy levels and energy was required to move electrons from a lower to a higher energy level. A mathematical model of atomic structure was developed in the early nineteenth century that defined these energy levels as quantum levels, and today this description is generally referred to as quantum mechanics.
According to the quantum model of the atom, electron for the known elements can reside in seven different quantum levels, denoted by the principal quantum number n , where n has a value of one to seven. As the quantum number increases, the average energy of the electrons having that quantum number also increases. Each of the seven rows in the periodic table corresponds to a different quantum number. The first row ( n = 1) can only accommodate two electrons. Thus an element in the first row of the periodic table can have no more than two electrons (hydrogen has one, and helium has two). The second row ( n = 2) can accommodate eight electrons and an element in the second row of the periodic table will have two electrons in the first level (it is full) and up to eight electrons in the second level.
Quantum theory also tells us that the electrons in a given energy level are not all equivalent. Within an energy level electrons reside within sublevels (or subshells). The sublevels for any given level are identified by the letters, s, p, d and f and the total number of sublevels is also given by the quantum number, n . The s sublevel can accommodate two electrons, the p holds six, the d holds 10 and the f can hold 14. The elements in the first row in the periodic table ( n = 1) have electrons only in the 1s sublevel ( n = 1, therefore there can only be one sublevel). The single electron in hydrogen would be identified as 1s1 and the two electrons in helium would be identified as 1s 2 . Fluorine, in the second row of the periodic table (n = 2), has an atomic number of nine and therefore has nine electrons. The electrons in fluorine are arranged, two in the first level (1s 2 ), two in the 2s suborbital (2s 2 ) and five 2p suborbital (2p 5 ). If we were to write the electron configuration for fluorine, we would write it as 1s 2 2s 2 2p 5 . Each of the sublevels in an atom is also associated with an orbital, where an orbital is simply a region of space where the electron is likely to be found.


7.4: Electrical Energy Stored in a Capacitor
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Learning Objectives
By the end of this section, you will be able to:
- Explain how energy is stored in a capacitor
- Use energy relations to determine the energy stored in a capacitor network
Most of us have seen dramatizations of medical personnel using a defibrillator to pass an electrical current through a patient’s heart to get it to beat normally. Often realistic in detail, the person applying the shock directs another person to “make it 400 joules this time.” The energy delivered by the defibrillator is stored in a capacitor and can be adjusted to fit the situation. SI units of joules are often employed. Less dramatic is the use of capacitors in microelectronics to supply energy when batteries are charged (Figure \(\PageIndex{1}\)). Capacitors are also used to supply energy for flash lamps on cameras.
The energy \(U_C\) stored in a capacitor is electrostatic potential energy and is thus related to the charge Q and voltage V between the capacitor plates. A charged capacitor stores energy in the electrical field between its plates. As the capacitor is being charged, the electrical field builds up. When a charged capacitor is disconnected from a battery, its energy remains in the field in the space between its plates.
To gain insight into how this energy may be expressed (in terms of Q and V ), consider a charged, empty, parallel-plate capacitor; that is, a capacitor without a dielectric but with a vacuum between its plates. The space between its plates has a volume Ad , and it is filled with a uniform electrostatic field E . The total energy \(U_C\) of the capacitor is contained within this space. The energy density \(u_E\) in this space is simply \(U_C\) divided by the volume Ad . If we know the energy density, the energy can be found as \(U_C = u_E(Ad)\). We will learn in Electromagnetic Waves (after completing the study of Maxwell’s equations) that the energy density \(u_E\) in a region of free space occupied by an electrical field E depends only on the magnitude of the field and is
\[u_E = \frac{1}{2} \epsilon_0E^2. \nonumber \]
If we multiply the energy density by the volume between the plates, we obtain the amount of energy stored between the plates of a parallel-plate capacitor \(U_C = u_E(Ad) = \frac{1}{2}\epsilon_0E^2Ad = \frac{1}{2}\epsilon_0\frac{V^2}{d^2}Ad = \frac{1}{2}V^2\epsilon_0 \frac{A}{d} = \frac{1}{2}V^2C\).
In this derivation, we used the fact that the electrical field between the plates is uniform so that \(E = V/d\) and \(C = \epsilon_0A/d\). Because \(C = Q/V\), we can express this result in other equivalent forms:
\[U_C = \frac{1}{2}V^2C = \frac{1}{2}\frac{Q^2}{C} = \frac{1}{2}QV. \label{8.10} \]
The expression in Equation \ref{8.10} for the energy stored in a parallel-plate capacitor is generally valid for all types of capacitors. To see this, consider any uncharged capacitor (not necessarily a parallel-plate type). At some instant, we connect it across a battery, giving it a potential difference \(V = q/C\) between its plates. Initially, the charge on the plates is \(Q = 0\). As the capacitor is being charged, the charge gradually builds up on its plates, and after some time, it reaches the value Q . To move an infinitesimal charge dq from the negative plate to the positive plate (from a lower to a higher potential), the amount of work dW that must be done on dq is \(dW = W \, dq = \frac{q}{C} dq\).
This work becomes the energy stored in the electrical field of the capacitor. In order to charge the capacitor to a charge Q , the total work required is
\[W = \int_0^{W(Q)} dW = \int_0^Q \frac{q}{C}dq = \frac{1}{2}\frac{Q^2}{C}. \nonumber \]
Since the geometry of the capacitor has not been specified, this equation holds for any type of capacitor. The total work W needed to charge a capacitor is the electrical potential energy \(U_C\) stored in it, or \(U_C = W\). When the charge is expressed in coulombs, potential is expressed in volts, and the capacitance is expressed in farads, this relation gives the energy in joules.
Knowing that the energy stored in a capacitor is \(U_C = Q^2/(2C)\), we can now find the energy density \(u_E\) stored in a vacuum between the plates of a charged parallel-plate capacitor. We just have to divide \(U_C\) by the volume Ad of space between its plates and take into account that for a parallel-plate capacitor, we have \(E = \sigma/\epsilon_0\) and \(C = \epsilon_0 A/d\). Therefore, we obtain
\[u_E = \frac{U_C}{Ad} = \frac{1}{2} \frac{Q^2}{C} \frac{1}{Ad} = \frac{1}{2} \frac{Q^2}{\epsilon_0A/d} \frac{1}{Ad} = \frac{1}{2} \frac{1}{\epsilon_0} \left(\frac{Q}{A}\right)^2 = \frac{\sigma^2}{2\epsilon_0} = \frac{(E\epsilon_0)^2}{2\epsilon_0} = \frac{\epsilon_0}{2}E^2 \nonumber \]
We see that this expression for the density of energy stored in a parallel-plate capacitor is in accordance with the general relation expressed in Equation \ref{8.9}. We could repeat this calculation for either a spherical capacitor or a cylindrical capacitor—or other capacitors—and in all cases, we would end up with the general relation given by Equation \ref{8.9}.
Energy Stored in a Capacitor
Calculate the energy stored in the capacitor network in Figure 8.3.4a when the capacitors are fully charged and when the capacitances are \(C_1 = 12.0 \, \mu F, \, C_2 = 2.0 \, \mu F\), and \(C_3 = 4.0 \, \mu F\), respectively.
Strategy
We use Equation \ref{8.10} to find the energy \(U_1, \, U_2\), and \(U_3\) stored in capacitors 1, 2, and 3, respectively. The total energy is the sum of all these energies.
Solution We identify \(C_1 = 12.0 \, \mu F\) and \(V_1 = 4.0 \, V, \, C_2 = 2.0 \, \mu F\) and \(V_2 = 8.0 \, V, \, C_3 = 4.0 \, \mu F\) and \(V_3 = 8.0 \, V\). The energies stored in these capacitors are
\[U_1 = \frac{1}{2}C_1V_1^2 = \frac{1}{2}(12.0 \, \mu F)(4.0 \, V)^2 = 96 \, \mu J, \nonumber \]
\[U_2 = \frac{1}{2}C_2V_2^2 = \frac{1}{2}(2.0 \, \mu F)(8.0 \, V)^2 = 64 \, \mu J, \nonumber \]
\[U_3 = \frac{1}{2}C_3V_3^2 = \frac{1}{2}(4.0 \, \mu F)(8.0 \, V)^2 = 130 \, \mu J, \nonumber \]
The total energy stored in this network is
\[U_C = U_1 + U_2 + U_3 = 96 \, \mu J + 64 \, \mu J + 130 \, \mu J = 0.29 \, mJ. \nonumber \]
Significance
We can verify this result by calculating the energy stored in the single \(4.0-\mu F\) capacitor, which is found to be equivalent to the entire network. The voltage across the network is 12.0 V. The total energy obtained in this way agrees with our previously obtained result, \(U_C = \frac{1}{2}CV^2 = \frac{1}{2}(4.0 \, \mu F)(12.0 \, V)^2 = 0.29 \, mJ\)
Exercise \(\PageIndex{1}\)
The potential difference across a 5.0-pF capacitor is 0.40 V. (a) What is the energy stored in this capacitor? (b) The potential difference is now increased to 1.20 V. By what factor is the stored energy increased?
- Answer
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a. \(4.0 \times 10^{-13}J\); b. 9 times
In a cardiac emergency, a portable electronic device known as an automated external defibrillator (AED) can be a lifesaver. A defibrillator (Figure \(\PageIndex{2}\)) delivers a large charge in a short burst, or a shock, to a person’s heart to correct abnormal heart rhythm (an arrhythmia). A heart attack can arise from the onset of fast, irregular beating of the heart—called cardiac or ventricular fibrillation. Applying a large shock of electrical energy can terminate the arrhythmia and allow the body’s natural pacemaker to resume its normal rhythm. Today, it is common for ambulances to carry AEDs. AEDs are also found in many public places. These are designed to be used by lay persons. The device automatically diagnoses the patient’s heart rhythm and then applies the shock with appropriate energy and waveform. CPR (cardiopulmonary resuscitation) is recommended in many cases before using a defibrillator.
Example \(\PageIndex{2}\): Capacitance of a Heart Defibrillator
A heart defibrillator delivers \(4.00 \times 10^2 J\) of energy by discharging a capacitor initially at \(1.00 \times 10^4 V\). What is its capacitance?
Strategy
We are given \(U_C\) and V , and we are asked to find the capacitance C . We solve Equation \ref{8.10} for C and substitute.
Solution
Solving this expression for C and entering the given values yields \(C = 2\frac{U_C}{V^2} = 2\frac{4.00 \times 10^2 J}{(1.00 \times 10^4V)^2} = 8.00 \, \mu F\).


1: Introduction to Biology
Biology is the study of living organisms, including their physical structure (from cells to whole individuals), chemical processes (such as metabolism), physiology, and development. This is a very broad definition because the scope of biology is vast. Biologists may study anything from the microscopic view of a cell to ecosystems and the whole living planet. Listening to the daily news, you will quickly realize how many aspects of biology are discussed every day. For example, recent news topics include Escherichia coli outbreaks in spinach and Salmonella contamination in peanut butter. Other subjects include efforts toward finding a cure for AIDS, Alzheimer’s disease, and cancer. On a global scale, many researchers are committed to finding ways to protect the planet, solve environmental issues, and reduce the effects of climate change. All of these diverse endeavors are related to different facets of the discipline of biology.
What does the study of biology share with other scientific disciplines? Science (from the Latin scientia , meaning “knowledge”) can be defined as knowledge that covers general truths or the operation of general laws, especially when acquired and tested by the scientific method. It becomes clear from this definition that the application of the scientific method plays a major role in science. The scientific method is a method of research with defined steps that include experiments and careful observation.
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- 1.1: The Scientific Method
- Biologists study the living world by posing questions about it and seeking science-based responses. This approach is common to other sciences as well and is often referred to as the scientific method. The scientific method was used even in ancient times, but it was first documented by England’s Sir Francis Bacon (1561–1626), who set up inductive methods for scientific inquiry. The scientific method is not exclusively used by biologists but can be applied to almost all fields of study as a logica
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- 1.2: What is Ecology?
- Ecology is the study of how organisms interact with each other and with the physical environment in which they live. It is a large field of study and incorporates research at many spatial and temporal scales. Examples of ecological research include impacts of climatic change on species range distributions, patterns of infectious disease outbreaks, the effect of nutrient availability on ecosystem function, etc. In his 1911 book My First Summer in the Sierra, John Muir famously wrote ‘when we try
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- 1.3: What is Evolution?
- At its simplest, evolution is any change in heritable traits in a population of organisms across generations. These changes may be the result of natural selection, genetic drift, mutation, etc – processes that we will consider in depth later in the course. All populations experience evolutionary change, as influenced by their environment, their interactions with other organisms, and random chance. Like ecology, we can consider evolution at different scales.
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- 1.4: Linking Ecology and Evolution
- In 1973, evolutionary biologist Theodosius Dobzhansky wrote a now-famous essay titled Nothing in Biology Makes Sense Except in the Light of Evolution. In this essay, Dobzhansky argued that evolution is the underlying unifying theme for all of biological study and that understanding life on earth necessarily requires considering evolutionary processes and impacts. The title of this essay became a famous quote highlighting the foundational importance of evolution to all biological concepts and pro