1.1 Real Numbers and Functions
1.2 Data Fitting with Linear and Periodic Functions
1.3 Power Functions and Scaling Laws
1.6 Inverse Functions and Logarithms
“Mathematicians do not study objects, but relations between objects.”
Henri Poincare, 1854–1912.
Although all readers taking a first course in calculus have a background in algebra, geometry, and trigonometry, the depth of exposure and choice of material covered can be quite variable. The material in this chapter is designed to provide a common framework upon which to build an introductory course in calculus for students who have a strong interest in the life sciences. In reviewing real numbers and functions, our intention is also to develop the notation we will use throughout the book. As students, you must become familiar with this notation if you want to be fluent in reading the mathematical text in this book. We also introduce data—and concepts around working with data—early on, because this component of the mathematical modeling process is critical to testing model predictions in the context of real-world problems (as discussed in the introduction to this book). We pay particular attention to power, exponential, and logarithmic functions since these all play a critical role in the development of differential and integral calculus. Trigonometric functions are important but less fundamental, and they have been dealt with extensively in precalculus mathematics courses. Thus, we provide only a brief review; we expect students who are rusty on this topic to go back and review trigonometry functions themselves. The topics dealt with in the function building section and inverse function subsection provide the kinds of skills that are needed in model building. Finally, we introduce the notion of sequences. Sequences are important both for introducing the concept of limits and in the context of dynamics, where consecutive terms in sequences can be used to represent the changes in the state of some object over time—an idea that is central to the application of calculus to all branches of science.
You may have had a medical test in which an electrocardiograph, as shown in Figure 1.2, was used to check whether your heart was beating normally. In order to analyze graphs such as this, we need to seek unifying ideas relating graphs, data, tables, and equations. The mathematical concept that unifies these elements is the notion of a real-valued function, which is at the core of the development of both differential and integral calculus.
In this section, we discuss real numbers, functions, and basic properties of functions. To do this, we use the set notation {x: statement}, which means the set of all values of x or points x on the number line that satisfy or are defined by the statement following the colon.
Historically, the concept of numbers arose to address a need to count and keep exact records of land and property and to facilitate commerce. This process began with the counting numbers, now referred as the natural numbers, as depicted in Table 1.1.
It took a surprisingly long time for human civilization to add zero to this group to obtain the whole numbers. Negative numbers, which by some historical accounts first appeared in India and China around the seventh century, were then added to obtain the integers. The integers, however, are not closed under the operation of division: for example, −4/2 = −2 is an integer, but 4/3 is not. Ancient Egyptian surveyors were well aware of fractional numbers, but only after negative numbers were widely accepted could the set of all positive and negative fractions, called the rational numbers, be defined.
Rational numbers are extremely useful for the measurements of “continuous” traits such as weight, height, humidity, and temperature, which are often measured by counting. For instance, we measure lengths by counting the number of marked intervals (e.g., inches, centimeters) on a tape measure. By subdividing these intervals into smaller and smaller fractions, we obtain more and more accurate measurements. We might expect that if we allow for all possible fractional divisions, then we can measure the precise length of anything. It came as a shock to the Greeks that this expectation is wrong! For instance, the Greeks proved that the length of the diagonal of a unit square (i.e., sides of length one) cannot be expressed as a rational number (see the in Problem Set 1.1). Because this length corresponds to a number that cannot be found in the set of rational numbers, it is called irrational (not rational). It is denoted by the symbol and its value can be approximated as precisely as we want by bounding it above and below by sequences of rational numbers that approach it in the limit! Intuitively, if we have a ruler with all fractional divisions, we can measure arbitrarily close approximations of this length.
To deal with irrational numbers, mathematicians extended the rational numbers to a larger set of numbers that we call the real numbers, . One can think of the real numbers as living on the edge of an infinitely long ruler with demarcations at all powers of ten. A real number is a point on this line and can be represented in a decimal form with its integer part before the decimal and tenths, hundredths, thousandths, ten thousandths, and so on, after the decimal. Rational numbers on this line have decimal representations that terminate or repeat, while the irrational numbers have decimal representations that do not terminate (see Table 1.1). For example, π = 3.141592... has a decimal representation that does not terminate or repeat and, consequently, is an irrational number—as is the Euler number e that we will encounter later in this chapter. Since = 1.4142135... is irrational, its decimal representation also does not terminate or repeat itself.
Intervals of real numbers arise so frequently in calculus that it is worthwhile giving them special names and notations. An open interval from a to b is denoted
Notice that this interval includes all the real numbers between a and b but does not include a and b themselves. A closed interval from a to b is denoted
Unlike an open interval, a closed interval includes the end points. In addition to these finite intervals, we are often interested in infinite intervals. These are intervals where either the right side of the interval extends infinitely far in the positive direction or the left side extends infinitely far in the negative direction, or both. In the first case, to denote this situation, we use the symbol ∞ on the right side of the interval, and in the second case we use the symbol −∞ on the left side of the interval, as follows:
The typical graphical depictions of these intervals on the real line is shown in Figure 1.3. For infinite intervals, it is important to realize there is no number “∞” or “−∞.” These symbols are only used to indicate numbers in the interval whose magnitudes are arbitrarily large and positive or large and negative, respectively.
Often domains of functions are not a single interval but consist of the union of two or more intervals. We denote the union of two (or more) intervals with the notation ∪. For instance,
Biologists, mathematicians, and other researchers often study relationships between two quantities. The mathematical study of such relationships involves the concept of a function.
Function
A function f : X → Y is a rule that assigns to each element x of a set X (called the domain D) a unique element y of a set Y. The element y is called the image of x under f and is denoted by f(x), read as “f of x.” The set of all images f(x) for x in X is called the range R of f.
A function can also be regarded as follows:
Example 1 Identifying functions
Determine whether the following rules are functions. If one is a function, identify its domain and (if possible) its range.
Data Source: Komhyr, W. D., Harris, T. B., Waterman, L. S., Chin, J. F. S. and Thoning, K. W. (1989), Atmospheric carbon dioxide at Mauna Loa Observatory 1. NOAA Global monitoring for climatic change measurements with a nondispersive infrared analyzer, 1974–1985; J. Geop. Res., v. 94, no. D6, pp. 8533–8547. Data can be downloaded at http://www.seattlecentral.org/qelp/sets/016/016.html
Solution
Since a radius of a circle can only be nonnegative, the domain of this function is the nonnegative reals, [0, ∞). The range of this function is also [0, ∞).
Alternatively, if we identify any natural number n with n months after April 1974 until December 1985, then the domain of this function is
as there are eleven years and eight months of monthly data recordings. To determine the range, we would have to find the values of the collected data. These data are illustrated in Figure 1.5 and suggest the range is contained in the interval [327, 350]. While these data, in themselves, cannot be precisely described by a simple algebraic formula, we shall see in Section 1.3 that this function is well approximated by a simple algebraic formula.
As the preceding example and figure illustrate, functions can be represented in a variety of ways: verbally, algebraically, numerically, or graphically. Being able to move freely between these representations of a function is a skill that this book tries to cultivate.
Example 2 From words to algebraic representations
For regular strength Tylenol, each tablet contains 325 mg of acetaminophen. According to the Handbook of Basic Pharmacokinetics,* approximately 67% of the drug is removed from the body every four hours. Suppose Professor Schreiber had x mg of acetaminophen in his body four hours ago and just swallowed two more tablets of regular strength Tylenol. Write a formula in terms of x for the amount A of acetaminophen in Schreiber's body.
Solution Since 100% − 67% = 33% of the acetaminophen from four hours ago remains in Schreiber's body now, 0.33x is the amount of acetaminophen that is still in his body from four hours ago. Since Schreiber just swallowed two tablets of 325 mg per tablet, the total amount of acetaminophen in his body is
We will use this function in Section 1.7 to examine how the amount of acetaminophen in Schreiber's body varies in time whenever he takes two tablets every four hours.
In this book, unless otherwise specified, the domain of a function is the set of real numbers for which the function is a well-defined real number determined by the context of the problem. We call this the implied domain convention. For example, if f(x) = and g(y) = , we need x ≠ 2 and y ≥ 0, respectively. Alternatively, if n is the number of people in an elevator, the context requires that n is a whole number.
Example 3 From algebraic expressions to graphs
Find the domain D of the following functions.
Solution
Plants use light energy, in the form of photons, to synthesize glucose from carbon dioxide and water while excreting oxygen as a byproduct of this process called photosynthesis. Plants then use the sugars to fuel other processes associated with their maintenance and growth while the oxygen is used by animals and other creatures for respiration. Thus, photosynthesis is a key process not only for plants but also for animal life on Earth!
Example 4 From verbal descriptions to graphs
Let P(t) denote the photosynthetic activity of a leaf as function of t, where t is the number of hours after midnight. Sketch a rough graph of this function. Assume the sunrise is at 6 A.M. and the sunset is at 8 P.M.
Solution Noting that there is no photosynthetic activity prior to the sunrise, we have P(t) = 0 for 0 ≤ t ≤ 6. At sunrise, the photosynthetic activity slowly increases with the availability of light and reaches some maximum during midday. As the sun begins to set, the photosynthetic activity of the plant declines to zero and remains zero for the rest of the day. The graph of this function is shown in Figure 1.6.
In Example 1, you were asked to identify functions. We extend this question to deciding if a given graph is the graph of a function. By looking at the definition of a function, we see that its graph has one point for a given element of the domain. Graphically, this idea can be stated in terms of the following vertical line test.
Vertical Line Test
A set of points in the xy-plane is the graph of a real valued function if and only if every vertical line intersects the graph at, at most, one point.
Example 5 Vertical line test in action
Determine which of the given graphs is the graph of a function.
Solution In panel a (next page), a vertical line intersects the curve at two points for x = −0.5. Hence, this curve fails the vertical line test and is not the graph of a function. In fact, this curve is an ellipse given by the set of points that satisfy
The upper and lower halves of this ellipse can be described by the pair of functions
In panel b (next page), this curve does satisfy the vertical line test for all points x, as shown below for x = 1. In fact, recalling your trigonometric functions (see the next section), it is the graph of the function y = |sin x|.
In panel c (next page), this set of points is not the graph of a function as the vertical line at x = 1 intersects three points.
In panel d (next page), this set of points is the graph of a function as it passes the vertical line test for all x, as shown below for x = 1970. In fact, these points are the graph of the average annual temperature in New York as a function of time (in years).
In the real world sometimes functions must be defined with more than one formula; therefore, these are called piecewise-defined functions or just piecewise functions for short.
Example 6 Income tax rates
The federal income tax rates for married filing jointly in 2009 can be described as 10% for (adjusted) incomes up to $16,700, 15% for incomes between $16,701(rounding up) and $67,900, 25% for incomes between $67,901 and $137,050, 28% for incomes between $137,051 and $208,850, 33% for incomes between $208,851 and $372,950, and 35% for incomes greater than $372,950. Express the income tax rate f(x) for an individual in 2009 with adjusted income x as a piecewise function. Graph the income tax rates over the interval (0, 500000] (note the point 0 is not included).
Solution An algebraic representation of this piecewise function is given by
The graph of this piecewise function over the interval (0, 500,000] is shown in Figure 1.7. This graph consists of linear pieces with jumps between income brackets.
A particularly important piecewise-defined function is the absolute value function.
Absolute Value Function
The absolute value function y = |x| is defined by
When x is nonnegative, the absolute value of x is itself. When x is negative, the absolute value of x is the negative of itself. Hence, the graph of the absolute value function is shown in Figure 1.8.
Functions may exhibit different properties on intervals within their domain, as we now describe.
Increasing and Decreasing Functions
Let I be an interval in the domain of a function. Then
f is increasing on I if f(x) < f(y) for all x < y in I; that is, its graph rises from left to right on I
f is decreasing on I if f(x) > f(y) for all x < y in I; that is, its graph falls from left to right on I
f is constant on I if f(x) = f(y) for every x and y in I; that is, the graph is flat on I
These classifications are shown graphically in Figure 1.9.
Example 7 Classifying a function
Consider the function f defined by the following graph on the interval I = [−2, 3].
Find the intervals on which f is increasing and the intervals on which f is decreasing.
Solution The function f is decreasing on [−2, −1), increasing on (−1, 0), decreasing on (0, 2), and increasing on (2, 3]. Note that the interval is open at points where the function switches from increasing to decreasing or vice versa.
PROBLEM SET 1.1
Level 1 DRILL PROBLEMS
Determine whether the descriptions in Problems 1 to 6 represent functions. If a description is a function, find the domain and (if possible) the range.
1. a. {(4, 7), (3, 4), (5, 4), (6, 9)}
b. {6, 9, 12, 15}
2. a. {(5, 2), (7, 3), (1, 6), (7, 4)}
b. {(x, y) : y = 4x + 3}
3. a. {(x, y) : y ≤ 4x + 3}
b. {(x, y) : y = 1 if x is positive and y = −1 if x is negative}
4. a. {(x, y) : y is the closing price of IBM stock on July 1 of year x}
b. {(x, y) : x is the closing price of Apple stock on July 1 of year y}
5. a. {(x, y) : (x, y) is a point on a circle of radius 4 passing through (2, 3)}
b. {(x, y) : (x, y) is a point on an upward-opening parabola with vertex (−3, −4)}
6. a. {(x, y) : (x, y) is a point on a line passing through (2, 3) and (4, 5)}
b. {(x, y) : (x, y) is a point on a line passing through (4, 5) and (−3, 5)}
Use the vertical line test in Problems 7 to 9 to determine whether the curve is a function. Also state the probable domain and range.
Problems 10 to 12 show the output you might find for a calculator graph. The Xmin and Xmax values show the input x values and the Ymin and Ymax values show the input y values. The scale for each tick mark on the x and y axis is also shown. These values determine the extent of the box that is shown. Using these calculator images use the vertical line test to determine whether the curve is a function. Also state the probable domain and range if you assume that a curve reaching a boundary of the frame continues in the same fashion as it continues beyond the shown screen.
In Problems 13 to 18 find the domain of f and compute the indicated values or state that the corresponding x-value is not in the domain.
13. f(x) = −x2 + 2x + 3; f(0), f(1), f(−2)
14. f(x) = 3x2 + 5x − 2; f(1), f(0), f(−2)
15. f(x) = f(2), f(0), f(−3)
16. f(x) = (2x – 1)−3/2; f(1), f(), f(0)
17. f(x) =
18. f(x) =
19. Consider a function machine
that yields the table of values
Input values | Output values |
1 | 1 |
2 | 4 |
3 | 9 |
−5 | 25 |
Algebraically, define the simplest function F you can think of for input values x from the domain D = .
20. Consider a function machine
that yields the table of values
Input values | Output values |
0 | 3 |
1 | 5 |
2 | 7 |
3 | 9 |
4 | 11 |
Algebraically, define the simplest function S you can think of for input values t from the domain D = [0, ∞).
21. Suppose you are given a machine that multiplies the input value by 3 and then subtracts 7. Complete the table of values given below
Input values | Output values |
3 | 2 |
5 | |
0 | |
−3 |
and algebraically, define a function M for input values x from the domain D = .
22. Suppose there is super-secret machine the produces the table
Input values | Output values |
0 | 5 |
1 | 6 |
2 | 9 |
3 | 14 |
4 | 21 |
Algebraically define the simplest function, T, for input values t from the domain D = .
Find the domain and range for the graphs indicated in Problems 23 to 28. Also tell where the function is increasing, decreasing, and constant.
For each verbal description in Problems 29 to 36, write a rule in the form of an equation, state the domain, and then graph the function.
29. For each number x in the domain, the corresponding range value, y, is found by multiplying by 3 and then subtracting 5.
30. For each number x in the domain, the corresponding range value, y, is found by squaring and then subtracting 5 times the domain value.
31. For each number x in the domain, the corresponding range value, y, is found by subtracting the domain value from 5 and then taking the square root.
32. For each number x in the domain, the corresponding range value, y, is found by taking 5 added to 5 times the domain value and then dividing this by the result of adding 1 to the domain value.
33. From a square whose side has length x (in inches), create a new square whose side is 5 inches longer. Find an expression for the difference between the area of the two squares (in square inches) as a function of x. Graph this expression for 0 ≤ x ≤ 10.
34. From a square whose side has length x (in meters), create a new square whose side is 10 meters longer. Find an expression for the sum of the areas of the two squares (in square meters) as a function of x. Graph this expression for 0 ≤ x ≤ 10.
35. Find the area of a square as a function of its perimeter P.
36. Find the area of a circle as a function of its circumference C.
Level 2 APPLIED AND THEORY PROBLEMS
37. Recall from Example 2, a tablet of regular strength Tylenol contains 325 mg of acetaminophen and approximately 67% of the drug is removed from the body every four hours. Suppose Professor Schreiber had x mg of acetaminophen in his body four hours ago and just swallowed two more tablets of regular strength Tylenol. Write a formula in terms of x for the amount A of acetaminophen in Schreiber's body four hours from now.
38. Diazepam is a medication used for the management of anxiety disorders. Approximately 68% of the drug is removed from the body every 24 hours. Suppose a patient had x mg of diazepam in his body 24 hours ago and just took an oral dose of 30 mg. Write a formula in terms of x for the amount A of diazepam in the patient's body 24 hours from now.
39. Professor Getz mows his backyard lawn every Saturday. Draw a graph of the height of the grass in the lawn over a two-week period, beginning just after Getz mowed his lawn one Saturday.
40. Continuous morphine infusion (CMI) is a means of providing a continuous dosage of medication (morphine) for acute pain. Morphine is administered continuously by a computerized pump connected to the patient by an intravenous tube (IV). Graph the concentration of morphine in a patient's blood (in milligrams/liter) over a three-day period. Assume the patient initially had no morphine in the blood stream and the concentration of morphine was relatively constant at 1 mg/liter on the last day.
41. Biologists have found that the speed of blood in an artery is a function of the distance of the blood from the artery's central axis (Figure 1.10). According to Poiseuille's law, the speed (centimeters/second) of blood that is r centimeters from the central axis of an artery is given by the function
where R is the radius of the artery and C is a constant that depends on the viscosity of the blood and the pressure between the two ends of the blood vessel. (The law and the unit poise, a unit of viscosity, are named for the French physician Jean Louis Poiseuille, 1799–1869.) Suppose that for a certain artery
and
42. The reaction rate of an autocatalytic reaction is given by the formula
for 0 ≤ x ≤ a, where a is the initial concentration of substance A and x is the concentration of X.
43. Consider the function defined to study the rate at which animals learn when a psychology student performed an experiment in which a rat was sent repeatedly through a laboratory maze. Suppose that the time (in minutes) required for the rat to traverse the maze on the nth trial is modeled by the function
44. Consider the function defined by
45. Friend's rule is a method for calculating pediatric drug dosages in terms of a child's age (up to 12½ years). If A is the adult dose (in milligrams) and n is the age of the child (in years), then the child's dose is given by
46. Young's rule is another method for calculating pediatric drug dosages in terms of a child's age. If A is the adult dose (in milligrams) and n is the age of the child (in years), then the child's dose is given by
47. Clark's rule is a method for calculating pediatric drug dosages based on a child's weight (w) in pounds (lb). If A denotes the adult dose (in milligrams) then the corresponding child's dose is given by
48. Table 1.2 tabulates the estimated number of HIV/AIDS cases diagnosed each year in the United States from 1999 to 2002.
49.
Throughout the text, you will find problems called Historical Quest. These problems are not just historical notes to help you see mathematics and biology as living disciplines; rather, these problems are designed to involve you in the quest of pursuing great ideas in the history of science. Yes, they relate some interesting history, but they will also lead you on a quest that you may find interesting.
Even though we know little about the man himself, we do know that Pythagoras was a Greek philosopher who is sometimes described as the first true mathematician in the history of mathematics. He founded a philosophical and religious school in Croton and attracted many followers, known today as the Pythagoreans. The Pythagoreans were a secret society who had their own philosophy, religion, and way of life. This group investigated music, astronomy, geometry, and number properties. Because of their strict secrecy, much of what we know about them is legend, and it is difficult to tell what work can be attributed to Pythagoras himself. We also know that it was considered impious for a member of the Pythagoreans to claim any discovery for himself. Instead, each new idea was attributed to their founder Pythagoras. No doubt you know the Pythagorean theorem, but do you know that the Pythagoreans believed that all things are numbers and that by a number they meant the ratio of two whole numbers? For this you are to use these two ideas to prove that is an irrational number.
There is a legend (not historical fact) that one day a group of Pythagoreans were out in a boat seeking truth. One person on board came up with the following argument: Construct a right triangle with legs of length 1 unit. By the Pythagorean theorem, the length of the hypotenuse is (using modern notation) exactly units long. Is the length of this side a rational number or an irrational number? Let = . (Remember, they believed that all numbers could be expressed as the ratio of two whole numbers; thus, assume that is a rational number.) Assume that is a reduced fraction (if it is not reduced, simply reduce it and work with the reduced form). See if you can reproduce the work done in the boat; that is, show the details outlined here. Square both sides of the equation and prove that p is an even number. If p is even, then it can be written as p = 2k. Use this fact to show that q is even. Thus, the fraction is not reduced. Now, if you understand logic as did the Pythagoreans, you can see the contradiction. What is it? How can you use this information to prove that is irrational? Legend has it that this contradiction bothered those on the boat so much that they tossed the person who came up with this argument overboard—and pledged themselves to secrecy!
In the previous section we presented data about carbon dioxide (CO2) collected at the top of the Mauna Loa volcano since 1958 by the U.S. government's Climate Monitoring Diagnostics Laboratory. These data are plotted in Figure 1.5. Scientists routinely collect data involving two variables x and y and refer to such data as bivariate. In many cases, a list of bivariate points, such as the Mauna Loa CO2 data, can be modeled by a relatively simple functional relationship of the form y = f(x) that passes, if not through all points, then close by all points. The advantages of the model are that it describes the data more concisely than a list, it can make predictions for uncollected data values, and it can generate hypotheses. For instance, if we had a function that did a good job of describing how carbon dioxide concentrations fluctuate in time, then we could make predictions about future levels of carbon dioxide concentrations. The importance of these predictions stems from the fact that carbon dioxide is a greenhouse gas. It prevents the escape of heat radiating from the Earth. Consequently, carbon dioxide in the atmosphere influences the Earth's temperature, and many people would like to know what the temperature might be twenty or fifty years from now so that they can plan accordingly.
The most commonly fitted function is a linear function: a function that depicts a constant rate of change with respect to unit changes in the argument of the function. In this section, we review basic facts about linear functions and briefly discuss how to fit linear functions to data sets, a process referred to in statistics as linear regression. For instance, the data in Figure 1.5 suggest that carbon dioxide concentrations in the atmosphere are tending to increase across years. Using linear regression, we can determine at what rate this increase across years is occurring. In addition to exhibiting a linear trend, the carbon dioxide data clearly exhibit seasonal fluctuations. These seasonal fluctuations can be modeled by periodic functions. Consequently, the section continues by reviewing basic properties of periodic and trigonometric functions and fitting trigonometric functions to data sets. Using a combination of linear and trigonometric functions, we arrive at surprisingly good model of CO2 fluctuations.
Linear functions play a fundamental role in differential calculus as they can be used to approximate functions locally (i.e., over a relatively small interval of the domain of the variable x). A linear function is a function of the form
where m is the slope and b is the vertical or y-intercept of the linear function. The vertical intercept b is the value of y when x equals zero. Equivalently, it is the y-value at which the graph of y = f(x) intercepts the y-axis; that is, b = f(0). In contrast, the slope m of the line tells us that if we increase the x-value by an increment, say 0.2, then the corresponding y-value increases by m times that increment, 0.2m. Equivalently, the change in y divided by the corresponding change in x is always the constant m. This leads us to a slope formula.
Slope of a Line
A nonvertical line that contains the points P1 = (x1, y1) and P2 = (x2, y2) has slope
When the function y = mx + b is regarded as a relationship between the paired variables (x, y), x is called the independent variable and y the dependent variable, because the relationship is designed to answer this question: What value of y corresponds to a given value for x?
Example 1 From graphs to equations
Let y = f(x) be the linear function whose graph is shown in Figure 1.11. Find the equation for f(x).
Solution Looking at the graph, we see that the y intercept is given by b = 1. Since y = 1 when x = 0 and y = 0 when x = 0.5, we see that y decreases by 1 when x increases by 0.5. Thus,
and the equation of this line is
Example 2 From equations to graphs
Let y = f(x) be a linear function such that f(2) = 3 and f(−2) = −1. Write a formula for f(x) and sketch the graph.
Solution Since f(x) is linear, we write f(x) = m x + b where we need to determine the constants m and b. The slope is given by
Therefore, f(x) = x + b. To find b, we solve
Hence, y = f(x) = x + 1. To graph this function, it suffices to draw a line that passes through the points (−2, −1) and (2, 3) as shown in Figure 1.12.
Many data sets exhibit trends that can be reasonably described by linear functions. We can fit linear functions to data using either formal or informal approaches. Informal approaches include eyeballing how well a selected line passes through a given set of data or fitting a line to two suitably chosen points in the data set. Formal statistical methods provide ways for finding the best-fitting line in some well-defined mathematical sense, which we describe after the next example.
Example 3 Carbon dioxide output from electric power plants
In Figure 1.13 the carbon dioxide emissions of most of the electricity generation plants in California are plotted as a function of the heat input for the year 1997. The heat input units are a million British thermal units (i.e., 106 BTU or 1 MMBTU) and CO2 emissions are measured in metric tons.
In Table 1.3, six points that appear in Figure 1.13 are listed.
Heat input (MMBTU) | CO2 output (tons) |
45.179 × 106 | 2.685 × 106 |
1.00 × 106 | 0.058 × 106 |
1.902 × 106 | 0.113 × 106 |
3.334 × 106 | 0.197 × 106 |
0.086 × 106 | 0.005 × 106 |
13.897 × 106 | 0.826 × 106 |
Solution
Using the point-slope formula (see Problem 17 in Problem Set 1.2) for a line yields
Sketching this line over the data graph shown in Figure 1.13 yields the following graph.
This is a very good fit considering we just used the first two data points. Such a good fit does not always happen.
This is significantly smaller than the value of 2.3 × 106 tons of CO2 given in the data. Thus, the power plant represented by this point on the graph pollutes almost ten times as much as it should compared with other power plants of similar energy output.
Sometimes we can get a good fit to data by appropriately choosing two data points and finding the line that passes through these points. However, this method is quite ad hoc, because it depends on the two points selected and thus yields many different possible lines. Statisticians have solved this problem by inventing a method called linear regression. It is used to find a line that best fits that data in the following sense: The slope parameter m and y-intercept parameter b are chosen to minimize the sum of the squared vertical distances ei of the data from the line (see Figure 1.14). The values ei are called the residuals because they represent “what is left over once the linear fit has been taken into account.”
Why squared distances? To find the answer to this question and to learn the statistical underpinnings of linear regression, you need to take an introductory statistics course! However, we note without further details (see any elementary statistics text for details) that a sum-of-squares measure of the fit leads to relatively simple formulas for the slope and y-intercept of the best-fitting line (which can be easily computed with calculators, computer software, and on line web applications). Part of this simple formula is derived in Chapter 4 as an application of differentiation.
Example 4 CO2 concentrations in Hawaii
Table 1.4 describes how CO2 concentrations (in ppm) have varied from May 1974 to December 1985 at the Mauna Loa Observatory in Hawaii. A plot of these data (where time is measured in months) was given by Example 1 of Section 1.1.
STOP: Do not just read this—do it! Plotting this line against the data results in Figure 1.15.
The estimated CO2 concentration for December 2004 is 374.4 ppm. This is 374.4 − 338 = 36.4 ppm higher than the average level from May 1974 to December 1985.
Many biological and physical time series exhibit oscillatory behavior, as just shown by Example 4. These types of data sets can be described by periodic functions that repeat their values at evenly spaced intervals. More formally, we make the following definition.
A real-valued function f is periodic if there is a real number T > 0 such that
for all x. The smallest possible value of T is called the period of f. The amplitude (if it exists) of a periodic function is half of the difference between its largest and smallest values.
Example 5 Estimating periods and amplitudes
Estimate the period and amplitude for the CO2 data in Figure 1.16.
Solution A quick examination of the CO2 data reveals that the time between peaks is approximately twelve months, so the period is a year. From the plot of the residuals in Figure 1.16, we see that the largest values of the data seem to be around 3 ppm, while the smallest values are typically around −3 ppm. Hence, the amplitude is approximately (3 − (−3))/2 = 3 ppm.
Two important periodic functions that you have encountered previously in precalculus mathematics studies are the cosine and sine functions. The cosine function, y = cos x, is defined for all reals and has a range of [−1, 1]. Hence, the amplitude of cosine is 1. As with all trigonometric functions used here, we assume x is measured in radians. You may recall the an angle of 90° is a right angle, which is equal to π/2 radians. Consequently, the full period of the sine and cosine function is 2π. In other words, sin (x) = sin (x + 2π) and cos(x) = cos(x + 2π) for any value of x measured in radians. The graphs of both functions are shown in Figure 1.17. Since the graph of sine is the graph of cosine shifted to the right by π/ 2, it follows that
Curves with the shape of sine or cosine functions are called sinusoidal. An important two-parameter family of sinusoidal functions are functions of the form
where a is a real number and b is a nonzero real number. Since the range of f(x) is [−|a|,|a|], the amplitude of f(x) is |a|. To find the period T > 0 of f(x), we need to find the smallest T > 0 such that
This occurs when bT = 2π. Therefore, the period of f is 2π/b. In the following example, we put this information to use.
Example 6 Fitting the CO2 data
Consider
where a and b are positive constants.
Solution
The graph of this equation against the data is shown below.
A truly remarkable fit! Next, calculate h(12 · 31 + 11) = h(383) = f(383)+ g(383) = 2.60 + 376.2 = 378.8. According to one website, the March measurement was 381 ppm. Hence, CO2 may well be increasing slightly faster than predicted by the model, possibly due to an accelerating rate of CO2 emissions.
Other important trigonometric functions that you will encounter in this book are given by taking either reciprocals or ratios of the sine and cosine functions. For example, the tangent function is
Because cos x = 0 for odd integer multiples of π/ 2, the domain of tangent is all real numbers except these odd integer multiples of π/ 2. Furthermore, as we discuss further in Chapter 2, tan x approaches + ∞ as x approaches π/ 2 from the left and approaches −∞ as x approaches π/ 2 from the right. Consequently, the range of tangent is the entire reals. Also it does not have a well-defined amplitude but is periodic, with period π, as shown in Figure 1.18. Like tangent, the other trigonometric functions—namely, cotangent cot x = , secant sec x = , and cosecant csc x = −are not defined for all reals, have no well-defined amplitude, but also have a well-defined period as shown in Figure 1.18. Since these functions are all expressed in terms of the sine and cosine functions, and cosine is the sine function with a π/ 2 shift in its argument, the properties of all the trigonometric functions can be directly deduced from the properties of the sine function.
PROBLEM SET 1.2
Level 1 DRILL PROBLEMS
Solve for y as a function of x and graph the resulting function for Problems 1 to 10.
1. 5x − 4y − 8 = 0
2. x − 3y + 2 = 0
3. 100x − 250y + 500 = 0
4. 2x − 5y − 200 = 0
5. 3x + y − 2 = 0, −7 ≤ x ≤ 1
6. 2x − 2y + 6 = 0, 1 ≤ x ≤ 5
7. y = 4 cos x
8. y = cos (4x)
9. y=(sin x)/ 2, −8 ≤ x ≤ 8
10. y = sin(x/ 2), −8 ≤ x ≤ 8
Using the information in Problems 11 to 16, find the formula for the line y = mx + b.
11. Slope 3, passing through (1, 3)
12. Slope ; passing through (5, −2)
13. Passing through (−1, 2) and (0, 1)
14. Passing through (5, 6) and (7, 6)
15. y-intercept 4 passing through (3, 4)
16. horizontal line through (−2, 5)
17. Show that
is the equation of the line passing through the point (h, k) with slope m.
18. Derive the equation of vertical line passing through (h, k). Does this set of points represent a function?
Classify each graph in Problems 19 to 24 as a linear function or a periodic function. If it is linear, estimate the slope and write an equation of the form y = mx + b. If it is periodic, estimate the period and the amplitude and write an equation of the form y = a cos(bx), a > 0.
Match the equations in Problems 25 to 30 along with the scatter diagrams and best-fitting lines in figure panels A–F below.
25. y = 0.6x + 2
26. y = 0.5x + 2
27. y = 0.4x + 2
28. y = −0.4x + 2
29. y = −0.5x + 2
30. y = −0.7x + 2
Consider some standard trigonometric curves shown in Figure 1.19. Specify the period and amplitude for each graph (if exists) in Problems 31 to 36, and graph each curve.
31. y = cos(x + π/ 6)
32. y = 2 sin(x − π/ 4)
33. y = 2 sin 2πx
34. y = 3 cos 3πx
35. y = tan (2x − π/ 2)
36. y = tan (x/ 2 + π/ 3)
Level 2 APPLIED AND THEORY PROBLEMS
37. A life insurance table indicates that a woman who is now A years old can expect to live E years longer. Suppose that A and E are linearly related and that E = 50 when A = 24 and E = 20 when A = 60.
38. In certain parts of the world, the number of deaths N per week has been observed to be linearly related to the average concentration x of sulfur dioxide in the air. Suppose there are 97 deaths when x = 100 mg/m3 and 110 deaths when x = 500 mg/m3.
39. The chart in Figure 1.20, based on data reported in the November 1987 issue of Scientific American, shows the fat intake compared with death rates in various regions around the world.
Source: Data are selected from the following article: Leonard Cohen, “Diet and Cancer,” Scientific American, November 1987, p 44.
Which of the following lines best fits the data?
Use your choice to estimate the number of deaths per 100,000 population to be expected from an average fat intake of 150 g/day (roughly the fat intake in the United States during the period the data was collected).
40. The chart in Figure 1.21 shows a comparison of Foude number with stride length for humans, kangaroos, and others.
Data Source: Graph by Patricia J. Wynne, from “How Dinosaurs Ran,” by R. McNeill Alexander, Scientific American, April 1991, p. 132 ©1991 by Scientific American, Inc. All rights reserved.
It can be shown that the best-fitting line is one of the following:
Which do you think is the correct one? Use your choice to estimate the relative stride length that corresponds to a Froude number x = 4.
41. In a classic study by Julian Huxley, the weight X, in milligrams, of the small fiddler crab (Uca pugnax) is compared with the weight of the large claw (Y, in milligrams). The data are shown in Table 1.5.
42. The data in Table 1.6 compare the mandibles of the male stag-beetle (Cyclommatus tarandus) where X is the total length (body and mandibles) in millimeters and Y is the length of the mandibles in millimeters.
43. Table 1.7 shows the census figures (in millions) for the U.S. population since the first census.
44. Ethyl alcohol is metabolized by the human body at a constant rate (independent of concentration). Suppose the rate is 10 milliliters per hour.
45. In a 1971 study by Savini and Bodhaine, data for velocity of water versus depth were collected for the Columbia River below Grand Coulee Dam. The data are reported in Table 1.8 and were measured 13 feet from the shoreline.
Depth (feet) | Velocity (feet/second) |
0.7 | 1.55 |
2.0 | 1.11 |
2.6 | 1.42 |
3.3 | 1.39 |
4.6 | 1.39 |
5.9 | 1.14 |
7.3 | 0.91 |
8.6 | 0.59 |
9.9 | 0.59 |
10.6 | 0.41 |
11.2 | 0.22 |
Data source: Savini, J. and Bodhaine, G. L. (1971), Analysis of current meter data at Columbia River gaging stations, Washington and Oregon; USGS Water Supply Paper 1869-F.
46. Eighty-eight samples of shells of the native butter clam (Saxidomus giganteus) were collected. These clams grow to lengths of 12–13 cm and live for more than 20 years. A scatter plot of their data is given in Figure 1.22.
Source: Quantitative Environmental Learning Project (QELP) Web Site at http://seattlecentral.edu/qelp/index.html.
47. Temperature fluctuations in many parts of the world exhibit sinusoidal patterns. Consider, for example, the average monthly temperature in Chappaqua, New York, reported in Table 1.9.
48. Problem 47 illustrates temperature oscillations on a yearly time scale corresponding to the seasons. Temperatures also vary daily: cooler at night and warmer at noon. Consider, for example, the average July hourly temperature in Abeerdeen, New York, shown in the following table.
Why can an ant lift a hundred times its weight whereas a typical man can only lift about six-tenths of his weight? Why is getting wet life-threatening for a fly but not for a human? Why can a mouse fall from a skyscraper and still scurry home, while a human who falls is likely to be killed? Why are elephants' legs so much thicker relative to their length than are gazelles' legs? A class of functions called power functions provides a means to answer these questions.
Power Functions
A function f(x) is a power function if it is of the form
where a and b are real numbers. The variable x is called the base, the parameter b is called the exponent, and the parameter a is called the constant of proportionality.
Note that and x3 are power functions, while 3x is not because, in this latter case, the exponent rather than the base is the variable.
Example 1 Graphing power functions
Graph each of the following sets of functions and discuss how they differ from one another and what properties they have in common.
a. y = x2, y = x4, and y = x6
b. y = x3, y = x5, and y = x7
c. y = x1/2, y = x, and y = x3/2
d. y =
All of these graphs tend to “bend” upward and are “U-shaped.” All three of these graphs intersect at the points (0, 0), (−1, 1), and (1, 1). On the interval [−1, 1] the function with the smallest exponent grows most rapidly as you move away from x = 0, and on the intervals (−∞, 1) and (1, ∞) the function with the largest exponent increases most rapidly.
All of these graphs are “seat shaped,” bending downward for negative x and bending upward for positive x. All three of these graphs intersect at the points (0, 0), (−1, −1) and (1, 1). On the interval [−1, 1] the function with the smallest exponent grows most rapidly as you move away from x = 0, and on the intervals (−∞, 1) and (1, ∞) the function with the largest exponent grows most rapidly.
We graphed over the domain [0, ∞) of y = x1/2 and y = x3/2. All of these graphs increase as x increases and pass through the points (0, 0) and (1, 1). The graph of x1/2 becomes steeper and steeper at 0, while the graph of x3/2 becomes flatter and flatter. Moreover, the graph of x1/2 bends downward, while the graph of x3/2 bends upward.
Both of these functions pass through the point (1, 1) and approach positive or negative infinity (i.e., have a vertical asymptotes) as x approaches 0, although the function does so by approaching −∞ in the third quadrant while the “other branch” of approaches +∞ in the second quadrant. Branches of the graphs lying above the x axis, bend upward, while parts lying below bend downward.
To algebraically manipulate power functions, we review some properties of exponents.
Laws of Exponents
Let x, y, a, and b be any real numbers. Then provided that both sides of the equality are well defined, the following five rules govern the use of exponents:
Example 2 Using Laws of Exponents
Simplify the following expressions using the laws of exponents.
Solution
In his essay, “On Being the Right Size,”* John B. S. Haldane (1892–1964), noted biologist and one of the founders of the field of population genetics, wrote:
“A man coming out of a bath carries with him a film of water of about one-fiftieth of an inch in thickness. This weighs roughly a pound. A wet mouse has to carry about its own weight in water. A wet fly has to lift many times its own weight and, as everybody knows, a fly once wetted by water or any other liquid is in a very serious position indeed.”
You might wonder how Haldane came up with these conclusions. To see why, consider power laws in the context of proportionality, which is defined as follows.
Proportionality
We say that y is proportional to x if there exists some constant a > 0 such that y = a x for all x > 0. When y is proportional to x, we write
Example 3 Geometric similarity
Imagine a world in which all individuals are cubical critters of different types: one such critter is drawn in Figure 1.23. The size of each critter can be characterized using one measurement, L meters, which denotes the length of the critter in any of its three dimensions.
mass, M, is also proportional to Lb for an appropriate choice of b. In your argument, you may use the fact that 1 cubic meter of water has a mass of 1000 kilograms.
Solution
In other words, surface area is proportional to length squared and volume is proportional to length cubed.
Notice that this proportionality would not change even if we used a different density constant.
Geometrical similarity is not confined to cubical critters. So long as all dimensions of an organism scale in the same way, the organisms are geometrically similar. Moreover, for any measurement of length L (e.g., height, arm length, chest circumference), surface area S (e.g., palm surface area, cross-sectional area of a muscle), and mass M (e.g., mass of a hair or the entire body), the relationships S ∝ L2 and M ∝ L3 continue to hold.
To work with proportionality relationships, we need to remember a few basic rules. Essentially these rules have this effect: we can treat a proportionality symbol for manipulative purposes like an equality sign.
Rules of Proportionality.
Example 4 Rules of proportionality
Demonstrate that proportionality satisfies the properties listed in the box above.
Solution
Transitive property: Since x ∝ y, then there exists a constant a > 0 such that x = ay.
Since y ∝ z, then there exists a constant b > 0 such that y = bz. Therefore,
This equality implies that x ∝ z with proportionality constant ab.
Power-to-root property: If y ∝ xb, then there exists a constant a > 0 such that
y = axb. Solving for x in terms of y yields
Hence, x ∝ y1/b with proportionality constant a−1/b > 0.
General transitive property: This property is really just a simple extension of the transitive property, but it is easily demonstrated directly. If x ∝ yb and y ∝ zc, then there exist a1 > 0 and a2 > 0 such that x = a1yb and y = a2zc. Therefore, x = a1(a2zc)b = a1ab2zbc. Hence, x ∝ zbc with proportionality constant a1ab2.
Example 5 Dangers of getting wet
To understand the dangers of getting wet, it is reasonable to assume that the mass, W, of the water on your body of mass M after getting wet is proportional to the surface area, S, of your body.
Solution
In other words, W ∝ Mb for b = 2/3.
The man-sized cubical critter has mass M = 60 with W = 0.6. Substituting these values into W = a M2/3 allows us to solve for a:
The ratio of water mass to body mass for the man is
The mouse-sized critter has mass M = 0.01 kg with W = 0.04(0.01)2/3 ≈ 0.00186 kg. The ratio of water mass to body mass for the mouse is
We see that the wet cubical man has to lift only 1% of his body mass while the wet cubical mouse has to lift approximately 19% of its body mass.
Note that this calculation does not take into account that a mouse is hairier than a man and therefore likely to retain more water per surface area of body than a man. This calculation assumes that the retention properties for each unit of surface area are the same.
The graph of the ratio of water mass to body mass, y = 0.04M−1/3, is shown in Figure 1.24.
This graph illustrates that the bigger creature (i.e., M becomes larger), the amount of water one has to carry relative to one's body mass decreases. Hence, getting wet is much worse for a fly than a human.
The previous example shows how we can use geometric similarity to understand the implications of getting wet for critters of vastly different sizes—from humans to flies. In Problem 43 of Problem Set 1.3, we pose a counterpoint analysis of how smaller animals are favored when it comes to the dangers of falling from high places. Although it is true that organisms are often geometrically quite dissimilar, it turns out that in many cases analyses using the approximation of geometric similarity are quite good.
Example 6 Olympic weightlifting
The heaviest weight classes are excluded, as individuals in this class have no weight restriction and therefore are often not geometrically similar to their lighter counterparts.
Table 1.10 reports the body mass and the winning lifts (in kilograms) for the male gold medalists in the 1988, 1992, and 1996 Olympic Games. In this example, we develop a simple model relating body mass to mass lifted.
with each weightlifter and declare the individual with the largest score to be the overall winner. Use this approach to find the overall winner in the 1988 Olympics.
Solution
Since we have assumed that l ∝ S, we can conclude that
The plot l = 20.15 M2/3 against the data as shown in Figure 1.26 illustrates a remarkable fit of the model to the data.
Although geometric similarity works wonders, it is not universal. When the shape of an animal (or organ or bone) deviates from geometrical similarity with size, then we say that it scales allometrically (allo = different, metric = measure) if we can find a power law that relates one particular measure (e.g., length of the mammalian femur) denoted by, say, x to another (e.g., cross-sectional area of the mammalian femur) denoted by, say, y as the size of an individual increases. In this case, the fundamental allometric formula posits the relationship
for some constants a > 0 and b.
Haldane made the key observation in “On Being the Right Size” that a structure breaks when a load that is proportional to the volume of an organism (cubic dimension) acts on the cross-sectional area (square dimension) of the structure supporting this organism. The essence of this issue is presented in the next example.
Example 7 Breaking bones
Calculate the dimensions of a sugar cube that would crush under its own weight at the surface of the Earth where the gravitational acceleration is 9.81 m/s2, given that the sugar cube's density is 1040 kg/m3 and its crushing strength (the maximum value of K that it can resist) is 5.17 × 106 newtons/m2.
Data Source: T. A. McMahon (1975). “Allometry and Biomechanics: Limb Bones in Adult Ungulates.” The American Naturalist, Vol. 109, No. 969, pp. 547–563.
In this data set, the humerus bone of an African impala has a length of 173 mm and a diameter of 22.5 mm. Use this information to estimate the length of a wildebeest humerus whose diameter is 42.6 mm.
Solution
Using the relationship L = 21.7D2/3 with D = 42.6 yields L ≈ 264.7 mm for the length of the wildebeest humerus. The actual value from the data set is 256. Hence, our estimate from the scaling law is not too bad.
PROBLEM SET 1.3
Level 1 DRILL PROBLEMS
Simplify the functions in Problems 1 to 9, and determine whether the functions are power functions. If a function is a power function, write it in the form y = axb.
11. If y ∝ x2 and y increases from 103 to 1015, what happens to x?
12. If y ∝ 6x and x ∝ t, how does t change when y increases from 2 × 102 to 6 × 104?
13. If y ∝ 10x3 how is y proportionally related to x?
14. If x ∝ 100y and y ∝ 45z, then how does z change as x decreases from 95 to 12?
15. If x ∝ y2 and y ∝ z3, then how is x proportionally related to z?
16. If x ∝ and y ∝ z2, then how is x proportionally related to z?
Graph the functions in Problems 17 to 22. By inspection, state the intervals where the function is increasing and the intervals where it is decreasing.
23. The linear function
y = 3x + b
represents a family of functions whose graphs all look the same except for the relative placement with respect to the y-axis. On the same coordinate axis, graph the members of this family for the values b = 0, 4, −3, and and state which of these are power functions.
24. The quadratic function y = ax2 represents a family of functions whose graphs all look the same except for the relative placement with respect to the y-axis. On the same coordinate axis, graph the members of this family for the given parameter, a = 0, 4 −3, , and state which of these are power functions.
25. A spherical cell of radius r has volume V = πr3 and surface area S = 4πr2. Express V as a function of S. If S is quadrupled, what happens to r?
26. Consider a cylinder of radius r and height 5r. Express the volume and surface area of this cylinder as a function r. If r is doubled, what happens to the volume? If S is quadrupled, what happens to r?
27. Consider a cone of height h and radius h/2 at the top. Express the volume and surface area of this cone as a function of h. If h is doubled, what happens to S?
Drug doses for dogs and cats are known to scale with their surface area S. When body mass W is measured in kilograms, then surface area S in square meters is given by
where for dogs K = 10.1 and for cats K = 10.4. Further, when converting human drug doses of an average adult to pet drug doses, this formula is used:
In Problems 28 to 33, the human adult dose of a drug is given. Calculate the drug dose (rounded to the nearest milligram) that you would give your dog or cat of the indicated weight.
28. 100 mg of aspirin and your dog weighs 7 kg
29. 200 mg of aspirin and your cat weighs 4.6 kg
30. 250 mg of an antibiotic and your dog weighs 16 kg
31. 500 mg of a renal drug and your cat weighs 5.3 kg
32. 50 mg of an anticoagulant and your dog weighs 31 kg
33. 50 mg of an anticoagulant and your cat weighs 4.8 kg
Level 2 APPLIED AND THEORY PROBLEMS
34. An ant weighs approximately 1/500 ounce and can lift 1/5 ounce, which is approximately 100 times its weight. Assume that strength is proportional to the cross-section of a muscle and that all organisms on Earth (ants and humans) are geometrically similar. Using these assumptions, determine how much a 150-pound person on Earth can lift.
35. A comic book explained Superman's strength by stating that on Krypton an organism's strength is directly proportional to its body mass. Based on this assumption and assuming that Krypton ants are like Earth ants (see Problem 34), how much can a 150-pound person on Krypton lift?
36. In a sample of twenty-six trees of a particular species, wood density, D(kg/m3), is related to breaking strength, S(MPa), according to the relationship D ∝ S0.91. If one of the points that this relationship passed through was (D, S) = (300, 10), find the equation and sketch its graph.
37. A sample based on nineteen mountain ash trees of different sizes yielded a relationship between the leaf area, A(m2), of the tree and the stem diameter at breast height (DBH), d(cm). The relationship obtained was A ∝ d2.99. If one of the points that this relationship passed through was (d, A) = (30, 78), find the equation and sketch its graph.
38. In Julian Huxley's classic book Problems of Relative Growth, there are data showing an allometric relationship between the mass (C milligrams) of the large claw (chela) and that of the rest of the body (B milligrams) in the male fiddler crab (Uca pugnax). The exponent of this relationship is approximately 1.6, that is C = aB1.6, and it passes through the point (B, C) = (1000, 500). Calculate the parameter a and then graph the relationship for the growth of a large claw mass as a function of an individual's body mass (excluding claw) over the range 50 ≤ x ≤ 2200 mg.
39. In 1936, Sinnott showed that there is an allometric relationship between the length (L) and width (W) of gourds, when observed from ovary to maturity. (See Roger V. Jean, Differential Growth, Huxley's Allometric Formula and Sigmoid Growth (COMAP, Incorporated, Lexington MA, 1984) UMAP Module 635, p. 421.) He obtained the exponents of m = 0.95 for pumpkins (Cucurbita pepo) to m = 2.2 for the snake gourd (Trichosanthes). Plot these relationships on the same graph for both types of gourds over the interval [1, 50] centimeters, if they both pass through the point (L, W) = (10, 10).
40. Professor Smith's house (10 m wide, 20 m long, 4 m high—just a hovel, really) has a 30,000 watt furnace that just barely keeps him warm on cold winter nights. He's thinking of building a larger house to accommodate his growing insect collection and needs advice on the output of the new furnace. The new house will be three times as high, three times as wide, and three times as long.
41. Consider the following quote from Jonathan Swift's Gulliver's Travels:
“The reader may be pleased to observe, that, in the last article of the recovery of my liberty, the emperor stipulates to allow me a quantity of meat and drink sufficient for the support of 1724 Lilliputians. Some time after, asking a friend at court how they came to fix on that determinate number, he told me that his majesty's mathematicians, having taken the height of my body by the help of a quadrant, and finding it to exceed theirs in the proportion of twelve to one, they concluded from the similarity of their bodies, that mine must contain at least 1724 of theirs, and consequently would require as much food as was necessary to support that number of Lilliputians. By which the reader may conceive an idea of the ingenuity of that people, as well as the prudent and exact economy of so great a prince.”
Let F denote the amount of food an individual eats and L the height of an individual. This quotation implicitly assumes that F ∝ Lb for an appropriate choice of b. Find this b value and provide a biological explanation for this choice of b.
42. Suppose the main loss of energy is heat loss through the surface. For the quotation in Problem 41, determine the appropriate choice of b so that F ∝ Lb. Under the assumption, how much should the Lilliputians feed Gulliver?
43. The following quote from Haldane (On Being the Right Size, p. 424) illustrates the dangers of being large:
To the mouse and any smaller animal, [gravity] presents practically no dangers. You can drop a mouse in a thousand-yard mine shaft; and, on arriving at the bottom, it gets a slight shock and walks away. A rat would be probably killed, though it can fall safely from the eleventh story of a building; a man is killed, a horse splashes. For the resistance presented to movement by air is proportional to the surface of a moving object. Divide an animal's length, breadth, and height each by ten; its weight is reduced to a thousandth, but its surface only to a hundredth. So the resistance to falling in the case of the small animal is relatively ten times greater than the driving force.
Consider a cubical critter being dropped down a mine shaft. Let A denote the force due to air resistance that the cubical critter experiences and let M denote the critter's weight. Assume that A is proportional to surface area and M is proportional to volume.
Without doubt, the linear function y = ax + b is the most important elementary function in mathematics. In the context of calculus, its importance is equaled only by the function we introduce in this section, the exponential function. Just why this function is so critical in calculus will become apparent once we introduce the concept of a derivative. In this section, we show that the exponential function is suitable for describing how populations, income, beer froth, and the radioactivity of unstable isotopes change over time.
The following table provides data on the growth of the United States from 1815 until 1895.
Year | Population (in millions) |
1815 | 8.3 |
1825 | 11.0 |
1835 | 14.7 |
1845 | 19.7 |
1855 | 26.7 |
1865 | 35.2 |
1875 | 44.4 |
1885 | 55.9 |
1895 | 68.9 |
These data, which are plotted in Figure 1.28, indicate around a tenfold (also referred to as an order of magnitude) increase in the U.S. population size during the nineteenth century. To get a finer understanding of the actual rate of growth, we can divide the size of the population in any given year by its size one decade earlier. For example,
and
These calculations tell us that population increased by a factor of approximately 33% over both decades. Let us assume that the population increases by 33% every decade. If t corresponds to the number of decades that have elapsed since 1815 and t is a positive integer, then we might estimate the population size by
More generally, for any real t, we can model the population size N(t) at t decades after 1815 by the exponential function
The graph of N(t) is plotted in Figure 1.28 against the data, and reasonably approximates the data until 1880, after which it begins to overestimate the population size.
In the previous section, we introduced power functions y = xa for which the independent variable x is raised to some fixed power. In contrast, the function N(t) has its exponent as the independent variable t and its base is a fixed constant. Such functions are termed exponential functions.
Exponential Function
An exponential function is a function of the form
where the parameter a ≠ 1 (the base) is a positive real number and the variable x (the exponent) is a real number.
The graphs of exponential functions have three different shapes, depending on the value of the base, as shown in the following example.
Example 1 Sketching exponential functions
Sketch the exponential function
where a > 1, a = 1, and 0 < a < 1 on the same coordinate axes, and comment on each of the graphs.
Solution The graphs are shown in Figure 1.29.
The graph of y = ax passes through (0, 1) for all values of a. We also notice:
Example 1 illustrates that if the base of the exponential function is greater than one, then the exponential function is an increasing function. In the context of population change, this increase corresponds to population growth. A fundamental quantity associated with population growth is the doubling time; how long before the population size doubles. The following example illustrates this concept.
Example 2 Malthus's estimate for doubling time
In An Essay on the Principal of Population (http://www.gutenberg.org/files/4239/4239-h/4239-h.htm), Thomas Malthus wrote:
In the United States of America, where the means of subsistence have been more ample, the manners of the people more pure, and consequently the checks to early marriages fewer, than in any of the modern states of Europe, the population has been found to double itself in twenty-five years.
Let N(t) = 8.3(1.33)t be our model of population growth in the United States from 1815 onward.
Solution
Hence the population increases by just over a factor of 2.
We see that Malthus's prediction conforms reasonably well with our model. Notice that we could not test the prediction directly with data, as data are reported only in ten-year intervals.
In Example 2, we used a law of exponents (discussed earlier in Section 1.3). These laws are extremely useful for manipulating exponential functions, so you may review them. For example, using these laws, we uncover a key property of exponential functions. Namely, if f(x) = ax and h is a real number, then by the law of exponents
In other words, over any interval of length h, the exponential function changes by a fixed factor ah. In the case of Example 2, this observation implies that the population approximately doubles over any twenty-five-year period.
Exponential growth is much faster than polynomial growth, as we explore in the next example.
Example 3 Exponential growth versus polynomial growth
Use technology to graph the functions y = 2x and y = x4. Which function takes on larger values when x is large?
Solution Figure 1.30 shows both functions plotted on a the intervals [0, 1.8], [0, 10], and [0, 18]. Over the [0, 1.8] interval y = 2x initially takes on larger values than y = x4, but at the end of this interval y = x4 takes on the larger values. The graphs on the intervals [0, 10] suggest that y = x4 continues to be larger. On the interval [0, 18], we see the curves again cross at x = 16, and for x > 16, we see that y = 2x is larger.
The fact that exponential functions with base greater than one grow faster than any polynomial led the economist Thomas Malthus to make a dire prediction about the future of humankind.
Example 4 Malthus's law of misery
In An Essay on the Principle of Population,* Thomas Malthus wrote:
“Let us then take this for our rule, though certainly far beyond the truth, and allow that, by great exertion, the whole produce of the Island might be increased every twenty-five years, by a quantity of subsistence equal to what it at present produces. The most enthusiastic speculator cannot suppose a greater increase than this. In a few centuries it would make every acre of land in the Island like a garden.”
To illustrate the meaning of this quote, consider the United States to be the “island” and farms to be the “garden.” Let N(t) = 8.3(1.33)t (in millions) be the population size t decades after 1815. Assume that in 1815, the amount of food produced in this year is equivalent to 10 million yearly rations. Further, assume, as suggested by Malthus, that the production of food in the United States will increase every twenty-five years by 10 million yearly rations.
Solution
Time t (seconds) | Froth height H (centimeters) |
0 | 17.0 |
15 | 16.1 |
30 | 14.9 |
45 | 14.0 |
60 | 13.2 |
75 | 12.5 |
90 | 11.9 |
105 | 11.2 |
120 | 10.7 |
When the base of an exponential function is less than one, the exponential function is decreasing and exhibits so-called exponential decay. An amusing example of exponential decay resulted in Arnd Leike, a professor of physics at University of Munich, winning the 2002 Ig Nobel Prize. The Ig Nobel Prize is annually awarded to scientists who firstly make people laugh, and secondly make them think. Leike received his award for his paper, “Demonstration of the Exponential Decay Law Using Beer Froth” (European Journal Physics 23 (2002): 21–26.) This paper reports an experiment that Leike performed with a mug of the German beer Erdinger Weissbier. After pouring the beer, Leike measured the height of the beer froth at regular time intervals. The measured values are shown in Table 1.11.
If we consider the ratios of heights at subsequent time intervals, we find
and
Note that 0.94 represents 6% decay. If we assume, as the data suggest, every 15 seconds the height of the froth decays by a factor of 6%, then we can write an expression (formula) for the froth height and see how well it fits the data.
Example 5 Modeling the decay of beer froth
Find values for the parameters a and b of the function
that ensure the function passes through the first data point in Table 1.11 and that the height of the froth declines 6% every 15 seconds. Use technology to graph H(t) alongside the data. How well does the function fit the data? Assume that t is measured in seconds.
Solution Since the initial height of the froth is 17 cm and H(0) = ab0 = a, we set a = 17. On the other hand, assuming that the froth decays by a factor of 6% every 15 seconds means that
Hence, b = 0.941/15 ≈ 0.99588. Therefore, we have (in centimeters)
The graph is shown in Figure 1.31 and appears to fit the data very well.
One way of understanding this exponential decay is to think of the froth as a large collection of bubbles. According to our calculations, approximately every 15 seconds, 6% of the bubbles will pop, leaving only 94% of the original head of froth. As the bubbles continue to pop, there are fewer and fewer that can pop. Consequently, as shown in Figure 1.31, the number of bubbles left to pop declines to zero over time in a way that seems to be modeled rather well by a function that has a variable appearing as the exponent of some base value. For this reason, the decline is called exponential decay. In the problem set and in the next section, you will see that exponential decay arises in many biological contexts: exponential decay of a tumor following radiation therapy, exponential decay of a drug in the body, and exponential decay of endangered populations.
There is one choice of a base a for exponential functions ax that plays a particularly important role in calculus. It is Euler's number e named after the Swiss mathematician Leonhard Euler (1707–1783). Its importance will become apparent only when we get into the machinery of calculus. The following example, however, introduces you to this constant, and in providing an economic interpretation of its value.
Example 6 Continuously compounded interest
Jacob Bernoulli (1655–1705), another Swiss mathematician, discovered the irrational number e while exploring the compound interest on loans. Consider a bank account starting with one dollar. If the bank (never to be seen in the real world!) gives you 100% interest on this dollar after a year, you will have $2.00 after one year. What happens to your initial dollar if the bank compounds the interest more frequently?
Solution
Number of times compounded | Dollar amount at end of year |
1 (annually) | $2.00 |
2 (half-yearly) | $1.00(1 + 1/2)2 = $2.25 |
4 (quarterly) | $1.00(1 + 1/4)4 ≈ $2.4414 |
12 (monthly) | $1.00(1 + 1/12)12 ≈ $2.61304 |
365 (daily) | $1.00(1 + 1/365)365 ≈ $2.71457 |
8760 (hourly) | $1.00(1 + 1/8760)8760 ≈ $2.71813 |
525600 (minutely) | $1.00(1 + 1/525600)525600 ≈ $2.71828 |
The table suggests that the amount in the account is approaching some value near $2.71828 as you compound more and more frequently.
This example suggests that the quantity (1 + 1/n)n approaches a limiting value near 2.71828 as n gets very large. The actual limiting value is Euler's number e. We state this definition formally using the concept of a limit discussed in the introduction to this book. Limits are discussed extensively in Chapter 2.
Euler's Number e
Based on Example 6, we can interpret e as the value of one dollar a year from now in a continuously compounded bank account at a rate of 100%. The exponential function y = ex has many properties that are particularly convenient for calculus. We will see some of these properties in Chapter 2 when we discuss derivatives.
With continuous compounding, we compound interest not quarterly, or monthly, or daily, or even every second, but instantaneously, so that the future amount of money A in the account grows continuously. If the account initially has P dollars and an annual interest rate of r, then we define the future money A in the account t years from now as the limiting value of
as the number of compounding periods n grows without bound. We denote by A(t) the future value after t years. As with the definition of e, we denote this by the limit notation n → ∞:
These observations are now summarized.
Future Value
If P dollars are compounded n times per year at an annual rate r, then the future value after t years is given by
and if the compounding is continuous, the future value is
Since e > 1, the exponential function y = ex increases rapidly. To get a sense of how quickly this exponential function increases, you will use technology in the next example to determine how long it would take the continuously compounded bank account with one dollar to yield $10,000. An analytical approach to solving this problem is given in Section 1.6, after we introduce logarithm functions.
Example 7 Naïve approach to solving an exponential equation
Graph y = ex and y = 10,000 to solve the equation ex = 10,000.
Solution Graphing y = ex and y = 10,000 yields the following graph (with units on the y-axis in units of 10,000).
We estimate the x value at which the intersection occurs to be x ≈ 9. Hence, it would take only 9 years for a dollar to become $10,000 dollars in an account compounding continuously at a rate of 100% per year.
PROBLEM SET 1.4
Level 1 DRILL PROBLEMS
Graph the exponential functions in Problems 1.4 to 6.
1. y = 2x
2. y =
3. y = 3−x
4. y = e2x
5. y = 0.1−x
6. y = πex
Graph the exponential function y = f(x) and the polynomial function y = g(x) using the same domain (x-values) and range (y-values) in Problems 7 to 10. Estimate for what value of x that f(x) ≥ g(x).
7. f(x) = 2x and g(x) = 2x.
8. f(x) = ex and g(x) = 2x.
9. f(x) = πx and g(x) = x4 −4.
10. f(x) = 1.1x and g(x) = 5x5 + x + 1.
Use a graphical approach to estimate the solution to the equations specified in Problems 11 to 18. Note that the solution may be negative.
11. e2x = 10
12. ex/2 = 1/2
13. e−x = x
14. e−x = −3x
15. ex/3 = x
16. e−x = −x2
17. ex = −x2 + 2
18. ex = x2 + 2x
Assume that $1,000 is invested for t years at r percent interest compounded in the frequencies given in Problems 19 to 24. Calculate the future value.
19. t = 25, r = 0.07; annual compounding
20. t = 10, r = 0.12; semiannual compounding
21. t = 1, r = 0.01; continuous compounding
22. t = 5, r = 0.02; continuous compounding
23. t = (4 months), r = 0.16; monthly compounding
24. t = (6 months), r = 0.08; monthly compounding
Find the equation for the exponential function f(x) = bax that passes through the two indicated points in Problems 25 to 28.
25. (0, 2) and (1, 5)
26. (−2, 32) and (2, 8)
27. (1/2, 3) and (1, 1)
28. (−2, 1/2) and (2, 2)
Level 2 APPLIED AND THEORY PROBLEMS
29. In Example 5, we modeled the height of the beer froth (in centimeters) as H(t) = 17(0.99588)t where t is measured in seconds. Plot this function over a four-minute time interval and estimate at what time the froth is one half of its original height.
30. In Example 2, we modeled the population size (in millions) of the United States as N(t) = 8.3(1.33)t where t is measured in decades after 1815. Plot this function over a fifty-year period and estimate the time at which the population would triple in size. How does this compare to the actual time that the population tripled in size?
31. Consider $100 in a bank account that has annual interest rate of 20%.
32. Consider $1,000 in a bank account that has annual interest rate of 5%.
33. Consider a bacterial species that produces ten off-spring per day. Assume that you start with one bacterial cell, no cells die, and all cells reproduce at the same rate.
34. Consider a bacterial species that produces five off-spring per day. Assume that you start with twenty bacterial cells, no cells die, and all cells reproduce at the same rate.
35. The following functions give the population size P(t) in millions for four fictional countries where t is the number of decades since 1900.
Country 1: P1(t) = 3(1.5)t
Country 2: P2(t) = 10(1.1)t
Country 3: P3(t) = 20(0.95)t
Country 4: P4(t) = 2(1.4)t
36. The following functions give the froth height (in centimeters) of three fictional beers where t represents time (in seconds).
Beer 1: H1(t) = 20(0.99)t
Beer 2: H2(t) = 40(0.9)t
Beer 3: H3(t) = 15(0.98)t
37. Hyperthyroidism is a condition in which the thyroid gland makes too much thyroid hormone. The condition can lead to difficulty concentrating, fatigue, and weight loss. One treatment for hypothyroidism is the administration of replacement thyroid hormone such as thyroxine (T4). The concentration of this hormone in a patient's body exhibits exponential decay with a half-life of about seven days. Consider an individual that has taken 100 mcg of T4.
38. In Problem 37, we discussed hyperthyroidism. For individuals with this conditions, the body converts thyroxine (T4) into triiodothyronine (T3), which is the hormone that the body uses at the cellular level. The bodies of some individuals are unable to do this conversion; consequently these people are given injections of T3. The half-life of T3 is about ten hours. Consider an individual who has taken 100 mcg of T3.
39. Carbon-14 has a half-life of 5,730 years. How much is left of 500 g of C-14 after t years?
40. If a bacterial population initially has twenty individuals and doubles every 9.3 hours, then how many individuals will it have after three days?
41. “Whale Numbers up 12% a Year” was a headline in a 1993 Australian newspaper. In a thirteen-year study, beginning in 1981, scientists estimated that the humpback whale (Megaptera novaeangliae) population off the coast of Australia was increasing by 12% a year from a level of 350 individuals.
42. The population size (in millions) of Mexico in the early 1980s is reported in Table 1.12.
Year | Population (in millions) |
1980 | 67.38 |
1981 | 69.13 |
1982 | 70.93 |
1983 | 72.77 |
1984 | 74.66 |
1985 | 76.60 |
43. Consider an instant lottery game in which you buy a scratch-off card and there is a certain chance p of winning a prize. If you buy N scratch-off cards, the chance of not winning a prize is (1 − p)N, that is, the chance you didn't win the prize on the first card times the chance you didn't win the prize on the second card, and so on. Therefore, the chance of winning a prize with N cards is 1 − (1 − p)N.
44. In an experimental study performed at Dartmouth College, two groups of mice with tumors were treated with the chemotherapeutic drug cisplatin. Prior to the therapy, the tumor consisted of proliferating cells (also known as clonogenic cells) that grew exponentially with a doubling time of approximately 2.9 days. Assume the initial tumor size was 0.1cm3.
45. In the experimental study described in Problem 44, each of the mice was given a dosage of 10 mg/kg of cisplatin. At the time of the therapy, the average tumor size was approximately 0.5 cm3. Assume all the cells became quiescent (i.e., no longer dividing) and decay with a half-life of approximately 5.7 days.
46. By comparing the graphs of and determine which is larger:
is obviously true (since ππ = ππ). Using a graphical method, find another value (approximately) for which the given statement is true.
In calculus the number e is sometimes introduced using slopes. In Problems 47 to 49 explore this idea.
47. a. Draw the graph of y = 2x and plot the points (0, 1) and (2, 4), which are on the graph.
b. Consider the secant line passing through these points. Now, consider the slope of the secant line as the point (2, 4) slides along the curve toward the point (0, 1). Draw the line that you think will result when the point (2, 4) reaches the point (0, 1). This is the tangent line to the curve y = 2x at (0, 1).
c. Using the tangent line, and the fact that the slope of a line is RISE/RUN, estimate (to the nearest tenth) the slope of the tangent line.
48. a. Draw the graph of y = 3x and plot the points (0, 1) and (2, 9), which are on the graph.
b. Consider the secant line passing through these points. Now, consider the slope of the secant line as the point (2, 9) slides along the curve toward the point (0, 1). Draw the line that you think will result when the point (2, 9) reaches the point (0, 1). This is the tangent line to the curve y = 2x at (0, 1).
c. Using the tangent line, and the fact that the slope of a line is RISE/RUN, estimate (to the nearest tenth) the slope of the tangent line.
49. a. Draw a line passing through (0, 1) with slope 1.
b. Compare the graphs of y = 2x (Problem 47) and y = 3x (Problem 48). Now, it seems reasonable that there exists a number between 2 and 3 with the property that the slope of the tangent through (0, 1) is 1. Draw such a curve.
c. On the same coordinate axes, draw the graph of y = ex. How does this curve compare with the curve you drew in part b? In calculus, the number between 2 and 3 with the property that the slope of the tangent through (0, 1) is 1 is used as the number e.
We have reviewed basic properties of linear, periodic, exponential, and power functions. By combining these functions, we can greatly enlarge our “toolbox” of functions. With this larger toolbox, we can describe more data sets and model more biological processes. For instance, in this section we develop models of the waxing and waning of tides and the rates at which microbes consume nutrients.
The simplest way to create the graph of a new function from the graph of another function is to shift the graph vertically or horizontally.
Horizontal and Vertical Shifts
Let y = f(x) be a given function with a > 0.
Horizontal shifts:
y = f(x − a) shifts the graph of y = f(x) to the right a units
y = f(x + a) shifts the graph of y = f(x) to the left a units
Vertical shifts:
y = f(x) + a shifts the graph of y = f(x) upward a units
y = f(x) − a shifts the graph of y = f(x) downward a units
To understand why these shifts occur, consider y = f(x − a). Substituting x + a for x yields y = f(x + a − a) = f(x). Hence, the function y = f(x − a) has the same value as the function y = f(x) when you “shift x” to the right by a.
Consider the function y = f(x) whose graph is given by Figure 1.32.
Sketch the graphs of y = f(x − 0.5), y = f(x) − 0.5, and y = f(x + 1) + 1.
Solution y = f(x − 0.5) shifts the graph right 0.5 units. y = f(x) − 0.5 shifts the graph down 0.5 units. y = f(x + 1) + 1 shifts the graph left 1 unit and up 1 unit. These graphs are shown in Figure 1.33
In addition to shifting graphs, we can reflect graphs across axes.
Reflections
Let y = f(x) be a given function.
The graph of y = −f(x) is the reflection across the x axis. It is found by replacing each point (x, y) on the graph with (x, −y).
The graph of y = f(−x) is the reflection across the y axis. It is found by replacing each point (x, y) on the graph with (−x, y).
Example 2 Reflecting a function
Consider the function y = f(x) whose graph is given by
Solution
A curve can be stretched or compressed in either the x-direction, the y-direction, or both, as shown in Figure 1.34.
Stretching and Compressing
Let y = f(x) be a given function.
To sketch the graph of y = f(bx), replace each point (x, y) with (x, y). If 0 < b < 1, then we call the transformation an x-dilation (or x stretch). If b > 1, then we call the transformation an x-compression.
To sketch the graph of y = c f(x), replace each point (x, y) with (x, cy). If c > 1, then we call the transformation a y-dilation (or y stretch). If 0 < c < 1, then we call the transformation a y-compression.
Example 3 Stretching and compressing
Consider the function y = f(x) defined by
Find and sketch the functions y = f(2x) and y = 3f(2x).
Solution We begin by sketching the function y = f(x) which is 0 for negative x, linear with slope 1 from x = 0 to x = 1, and equal to 1 for x ≥ 1 to obtain the left panel below.
Since 2x < 0 if and only if x < 0, we get the function y = f(2x) equals 0 for x < 0. Since 0 ≤ 2x < 1 if and only if 0 ≤ x < 1/2, y = f(2x) = 2x for 0 ≤ x < 1/2.
Finally, y = f(2x) = 1 for x ≥ . Therefore, we have shown
Plotting this function, we get the center panel below. Hence, y = f(2x) compresses the function y = f(x) by a factor of 2 in the horizontal direction.
The function y = 3f(2x) stretches the function y = f(2x) by a factor of 3 in the vertical direction. Hence, we get
Plotting this function, we get the right panel below.
By compressing and stretching sinusoidal functions, we can model periodic phenomena like tidal movements.
Example 4 Modeling tidal movements
The tides for Toms Cove in Assateague Beach, Virginia, on August 19, 2004, are listed in the table on the left.
Assume that this can be modeled by
where T denotes the height (in feet) of the tide t hours after midnight. Find values of A, B, C and D such that the function fits the Assateague tide data.
Solution The data suggest that the period of T is approximately twelve hours, in which case B = . The amplitude of the tide is given by A = = 1.8. Since the graph of cosine is always centered around the horizontal axis, we need to vertically shift the graph up by the midtide height, D = (4.0 + 0.4)/2 = 2.2. Finally, since the high tide occurs approximately at t = 11, we can choose C = −11 to shift the graph left by 11. Putting this all together yields
To graph this function, we note that T(t) has a vertical shift of 2.2 and a horizontal shift of 11 of the cosine function. Additionally, the amplitude of the cosine function is 1.8 and the period is 12, as shown in Figure 1.35.
The easiest way to create new functions is to perform arithmetic operations on old functions. The first three of these operations result in a function whose domain is the intersection of the domains of the original functions, where the symbol ∩ is used to denote set intersection. Since division by zero is not permitted, division can further reduce the domain of the new function.
Functional Arithmetic
Let f and g be functions with domains A and B, respectively. Then,
Addition: f + g is defined by (f + g)(x) = f(x) + g(x) with domain A ∩ B
Subtraction: f − g is defined by (f − g)(x) = f(x) − g(x) with domain A ∩ B
Multiplication: fg is defined by (fg)(x) = f(x)g(x) with domain A ∩ B
Division: f/g is defined by (f/g)(x) = f(x)/g(x) with domain consisting of points x in A ∩ B such that g(x) ≠ 0
Example 5 Combining functions
Consider the functions f(x) = and g(x) = sin x. Find the domains for f + g, fg, and f/g.
Solution The domain of f(x) is [−10, 10] and the domain of g(x) is (−∞, ∞). Thus it follows that the domains of f + g and fg are [−10, 10]. Multiplying the graphs pointwise yields a graph that does not alter the domain, but taking the quotient f/g requires that we remove points on [−10, 10] where g(x) = 0; that is the values 0, ±π, ±2π, ±3π.
Two important classes of functions that we get by adding, multiplying, and dividing power functions are polynomials and rational functions.
Polynomials and Rational Functions
Let n be a whole number. A polynomial function of degree n is a function of the form
where a0, a1,..., an are constants.
A rational function is a function of the form
where division by zero is excluded, n, m, are natural numbers, a0, a1, a2,..., an, b0, b1, b2,..., bm are constants.
Rational functions arise in biology in many ways, particularly in the context of the rate at which organisms extract resources from their environments. Bacteria, for example, have special molecular receptors embedded in their cell membrane to ingest nutrients, such as glucose, into their cell bodies. These receptors “capture” nutrient molecules outside of the cell and transport them into the cell body. This process is illustrated in Figure 1.36.
The rate at which nutrients can be brought into the cell body is called the uptake rate. The uptake rate is limited by the number of receptors and the time it takes a receptor to bring a nutrient particle into the cell body. The next example derives the Michaelis-Menton uptake function named after biochemists Leonor Michaelis (1875–1947) and Maud Menten (1879–1960). In addition to describing nutrient uptake, this function is used to describe enzyme kinetics and the consumption rates of organisms such as foxes eating rabbits, or ladybugs eating aphids.
Example 6 Michaelis-Menten glucose uptake rate
This problem is divided into two parts: the derivation of the Michaelis-Menten uptake function in part a and application of this function in part b.
where a > 0 is the proportionality constant. If a bound receptor requires b hours (b is much less than 1) to bring a glucose molecule into the cell, then T satisfies
Solve for T in terms of x and derive an expression for the total uptake rate f(x) in terms of x.
Glucose concentration (micrograms per liter) | Uptake rate for 1 liter of bacteria (micrograms per hour) |
0 | 0 |
20 | 12 |
40 | 16 |
60 | 18 |
80 | 19 |
100 | 20 |
By an appropriate change of variables (see Problem set 1.5), one can use linear regression to estimate the parameters for the uptake function. Doing so yields f(x) = . How does this relate to the expression for f(x) derived in part a? Use technology to plot this function against the data. How good is the fit?
Solution
Substituting this expression into the expression N = aTx, we get
Since N is the number of molecules handled by each receptor and the cell has R receptors, the total uptake rate of the cell is
The fact that the function fits the data so well gives us confidence that the arguments used to construct the function are sound! One interesting question to ask is what happens to the uptake rate f(x) as x gets very large (i.e., approaches +∞)? In the next chapter, we will develop ideas to tackle this question.
Situations often arise in biology where the relationship between two variables x and z is mediated by a third variable y. For example, the rate z at which a population of mice or shrews grows is related to the number y of insects the animals consume per unit time, and this rate y is related to the density x of insects in the area where these animals feed. Let f be the function that relates consumption rate y to resource density x; that is y = f(x). Let g be the function that relates the per capita population growth rate z of the population to the consumption rate y; that is, z = g(y). Then by substitution, we obtain z = g(f(x)). We have expressed the growth rate as a function of resource density through the process of taking a function of a function. This process is known as composition and is shown in Figure 1.37.
Composite Functions
Let f and g be functions with domains A and B, respectively. The composite function g f is defined by
The domain of g f is the subset of A for which g f is defined.
To visualize how functional composition works, think of f g in terms of an assembly line in which f and g are arranged in series, with output f becoming the input of g.
Example 7 Composing functions
Let f(x) = 2x + 1 and g(x) = . Find the composite functions g f and f g and their domains.
Solution The function g f is defined by
Notice that g f means that f is applied first, then g is applied. Since g f is defined only for 2x + 1 ≥ 0 or x ≥ −, the domain of g f is .
The function f g is defined by
In this part, first apply g then apply f. Since f g is defined only for x ≥ 0, we see the domain of f g is [0, ∞).
Example 7 illustrates that functional composition is not, in general, commutative. That is, in general,
Sometimes it can be useful to express a function as the composite of two simpler functions.
Example 8 Decomposing functions
Express each of the following functions as the composite f g of two functions f and g.
Solution
Let g(x) = cos x + 2 and f(x) = x3. Then,
Alternatively, let g(x) = cos x and f(x) = (2 + x)3. Then,
The next example involves the composition of two well-known functions in ecology: (1) the consumption function y = f(x) that relates the rate at which an organism is able to consume a resource of density x in the environment (also known as the functional response) and (2) the per capita growth rate of an organism g(y) that is a function of the consumption rate y.
The example is based on data pertaining to the daily rates at which individual short-tailed shrews (Blarina brevicauda), as shown in Figure 1.38, gather cocoons of the European pine sawfly (Neodiprion sertifer) buried in forest-floor litter. These data, as a function of cocoon density x per thousandth acre (i.e., acres × 10−3), can be fitted reasonably well by the function
This function is equivalent to the Michaelis-Menten uptake function derived in Example 6.
Example 9 Short-tailed shrews exploiting cocoons
Consider the shrew population studied by the Canadian ecologist C. S. Holling, “Some Characteristics of Simple Types of Predation and Parasitism,” The Canadian Entomologist 91(1959): 385–398. Suppose we are given the information that under ideal conditions (i.e., when the number of sawfly cocoons per shrew is essentially unlimited) each pair of shrews produces an average of around twenty female and twenty male progeny per year.
Solution
Setting b = 100, we get the per capita growth rate as a function of cocoons consumed per day y:
PROBLEM SET 1.5
Level 1 DRILL PROBLEMS
Let y = f(x)be the function whose graph is given by Figure 1.39. Sketch the graph of the functions in Problems 1 to 6.
1. y = f(x) + 2
2. y = f(x + 1)
3. y = f(x − 2) + 1
4. y = 2 f(x + 2)
5. y = −f(x)
6. y = f(−x + 2)
Sketch the graph of the functions in Problems 7 to 10 by appropriately shifting, stretching, or translating the graph of y = cos x.
In Problems 11 to 19 sketch the graph of each function without the aid of technology.
11. y = (x − 2)2
12. y = (x − 2)2 + 1
13. y = −1.25(x − 2)2
14. y = (x + 1)3 + 5
15. y = (2x + 1)3 + 5
16. y = (2x + 1)3/2
18. y = −|x|/2
19. y = 1 − 2|x|
20. Find the indicated values given the functions
f = {(0, 1), (1, 4), (2, 7), (3, 10)}and
g = {(0, 3), (1, −1), (2, 1), (3, 3)}
21. Find the indicated values given the functions
and
22. Let f(t) be a periodic function with period p = 2π and amplitude a = 1. Show that the given functions are periodic and find their period and amplitude.
23. Let f(t) be a periodic function with period p = T and amplitude a = A. Show that the following functions are periodic and find their period and amplitude.
Express each of the functions in Problems 24 to 29 as the composition f g of two functions f and g. (Answers are not unique.)
24. y = (2x2 − 1)4
25. y =
26. y = e−x2
27. y = e1−x2
28. y = |x + 1|2 + 6
29. y =
For each of the functions in Problems 30 to 33, find f + g, fg, and, f/g. Also give the domain of each of these functions.
Level 2 APPLIED AND THEORY PROBLEMS
34. The tides for Hell Gate, Wards Island, New York, on September 6, 2004, are given by the following table:
Let
denote the height of the tide t hours after midnight. Find values of A, B, C, and D such that the function fits the Hell Gate tide data.
35. The tides for Bodega Bay, California, on March 10, 2005, are given by the following table:
Let
denote the height of the tide t hours after midnight. Find values of A, B, C, and D such that the function fits the Bodega Bay tide data.
36. Enzymes are nature's catalysts because they are compounds that enhance the rate (speed) of biochemical reactions. Enzymes are used according to the body's need for them. Some enzymes aid in blood clotting and some aid in digestion. Even enzymes within the cell are needed for specific reactions. In this problem, you will derive a model of a biochemical reaction where there is a substance (e.g., glucose) that is converted to a new substance (e.g., fructose) by an enzyme (e.g., isomerase). Let f(x) be the amount of substance produced per minute as a function of the substrate concentration x. To model this reaction rate, assume that enzymes are either “occupied” (i.e., processing a substrate particle) or are “unoccupied” (i.e., waiting to bind to another substrate particle).
Using linear regression on the transformed data, the uptake rate can be approximated by f(x) = . Graph this function against the data.
37. In several applications it has been useful to fit a function of the form y = f(x) = to a data set. Consider the change of variables given by t = 1/x and z= 1/y.
Take the reciprocals of the (x, y) data values to get the corresponding (t, z) values. Use technology to fit a line to the (t, z) data. If this line is given by z = c + dt, use your work in a to find the parameters a and b in y = .
38. Environmental studies are often concerned with the relationship between the population of an urban area and the level of pollution. Suppose it is estimated that when p hundred thousand people live in a certain city, the average daily level of carbon monoxide in the air is
ppm. Further, assume that in t years there will be
hundred thousand people in the city. Based on these assumptions, what level of air pollution should be expected in four years?
39. The volume, V, of a certain cone is given by
Suppose the height is expressed as a function of time, t, by h(t) = 2t.
40. The surface area, S, of a spherical balloon with radius r is given by
Suppose the radius is expressed as a function of time t by r(t) = 3t.
41. The ecologist C. S. Holling mentioned in Example 9 also collected data on the daily rates at which individual masked shrews (Sorex cinereus), gathered European pine sawfly cocoons in forest-floor litter. His data for this species are fitted by the functional response
where x is the density of cocoons on the forest floor. If breeding pairs for this species produce approximately four female and four male progeny per year under favorable conditions and the growth breakeven point is b = 40 cocoons per day, then find the specific form of the per capita hyperbolic growth rate r per day:g(y) = r(1 − b/y) for this species. Use this to derive the composite per capita growth rate function G = (g f)(x). Plot a graph of this composite function plotted over the interval [200, 400].
42. Suppose the number of hours between sunrise and sunset in Los Angeles, is modeled by
where n is the number of the day in the year (n = 1 on January 1 and n = 365 on December 31, except in leap years when n = 366). On what days of the year in 2011 were there approximately twelve hours of daylight in Los Angeles?
43. According to the model in Problem 42, when will the length of the day in Los Angeles be about thirteen hours?
44. In an experimental study performed at Dartmouth College, two groups of mice with tumors were treated with the chemotherapeutic drug cisplatin. In Problems 44 and 45 in Section 1.4, we modeled the growth of the volume of these tumors before and after chemotherapy. A key feature missing in the model was that tumors often consist of a mixture of quiescent (i.e., nondividing) cells and proliferating cells. To add this important component to the model, assume (as was observed in the experiments), that 99% of cells are quiescent and 1% are proliferating following chemotherapy. Furthermore, assume that the volume of tumors at the time of therapy is 0.5 cm3, the quiescent cells have a half-life of 5.7 days, and the proliferating cells have a doubling time of 2.9 days.
45. The following graph posted at the Global Warming Art website indicates regular oscillations to observed sunspot activity over the past 170 years.
Find a trigonometric function of the form y = a + b cos(cx + d), where x is in years, that provides a good fit to the data from the minimum that occured around 1844 to the minimum that occurred in 1997.
46. The following is a graph of a famous data set on the number of lynx pelts handled by the Hudson Bay Company from 1821 to 1910.
Write a trigonometric function of the form y = a + b cos(cx + d), where x is in years, that provides a good fit to these data.
Year t | N = f(t) Population size (in millions) |
1815 | 8.3 |
1825 | 11.0 |
1835 | 14.7 |
1845 | 19.7 |
1855 | 26.7 |
1865 | 35.2 |
1875 | 44.4 |
1885 | 55.9 |
1895 | 68.9 |
N Population size (in millions) | Year t = f−1(N) |
8.3 | 1815 |
11.0 | 1825 |
14.7 | 1835 |
19.7 | 1845 |
26.7 | 1855 |
35.2 | 1865 |
44.4 | 1875 |
55.9 | 1885 |
68.9 | 1895 |
Sometimes when we are given the output of a function, we want to know what inputs could generate the observed output. For instance, consider the function that assigns to each gene the protein that it encodes. If in an experimental study we observe certain proteins at high abundance, we might want to know what genes might have been expressed. At another time we might be interested in how long before a population doubles in size or reaches a specific size. Here, we introduce the concepts of one-to-one functions and inverse functions to tackle such questions. A particularly important family of inverse functions consists of logarithm functions. As you will discover, logarithmic functions are extremely convenient for examining questions about scaling laws, population growth, and radioactive decay.
Let's reexamine the U.S. population growth data introduced at the beginning of Section 1.4 and shown again in Table 1.13.
We can view this table as defining a function f(t) that associates each year t with the U.S. population size N = f(t) in that year. The domain of this function is {1815, 1825, 1835, 1845, 1855, 1865, 1875, 1885, 1895} and its range is {8.3, 11, 14.7, 19.7, 26.7, 35.2, 44.4, 55.9, 68.9}. An important feature of this function is for each value N in its range, there is only a single year t in the domain such that f(t) = N. For example, there are 19.7 million people in the United States only in 1845 and 55.9 million people in the United States only in 1885. Functions that have this feature are called one-to-one. This one-to-one feature of f(t)—that is, two different inputs lead to two different outputs (some people refer to this as two-to-two)—allows us to define a new function t = f−1(N) that sends each population size N (in millions) to the year t corresponding to that population size N. This function is the inverse function of N and is shown in Table 1.14.
While the U.S. population data function N = f(t) is one-to-one and that allows us to define an inverse function, not all functions are one-to-one. For example, consider the quadratic function y = f(x) = x2. Since there are two values of x = 2, −2 such that f(x) = 4, this function is not one-to-one.
One-to-One (Two-to-Two)
A function f : X → Y is one-to-one if f(a) = f(b) for some a and b in X implies that a = b (this can also be thought of as two different x in X to two different y in Y).
Example 1 Checking for one-to-one
Determine whether each of the following functions is one-to-one.
Solution
In Section 1.1, we used the vertical line test to determine if a given relation is a function. We have a similar test, called the horizontal line test, to determine if a given function is one-to-one.
Horizontal Line Test
A function f is one-to-one if and only if every horizontal line over the function's domain intersects the graph of y = f(x) in at most one point.
Example 2 Using the horizontal line test
Determine which of the following functions are one-to-one.
Solution
When a function y = f(x) is one-to-one, we can associate a single value in its domain with a single value in its range. This association defines the inverse function of f(x).
Inverse Function
Let f be a one-to-one function with domain D and range R. The inverse f−1 of f is the function with domain R and range D such that
Equivalently,
The second characterization of an inverse function states that if we take the output from the function f as the input to the function f−1, then we get back the input into the function f. Hence, the inverse function f−1 can be viewed as a “machine that undoes” the work of the function f.
Example 3 Using the definitions of the inverse function
Solution
Thus, g is f−1.
In part b of the previous example, we expressed the inverse function g as a function of x instead of as a function of y. This choice corresponds to the convention that x is considered the independent variable and y is the dependent variable. For the U.S. population growth example at the beginning of the section, we did not interchange the names of N and t, as N and t had different units, that is, population size (in millions) and years. In general, it is not necessary to interchange the names of x and y if we are comfortable expressing the inverse function as x = f−1(y).
To find the inverse of a function algebraically, there is a simple procedure to follow.
Finding the Inverse
Let y = f(x) be a one-to-one function with domain D and range R. To find the inverse function, follow these steps:
Step 1. Solve the equation y = f(x) for x in terms of y (provided this is possible).
Step 2. (Optional) To express the inverse function as a function of x, interchange the xs and ys in the solution from Step 1. This results in the equation y = f−1(x).
Example 4 Finding inverses
Find the inverses for the following functions.
Interchanging the roles of x and y, we get y = f−1(x) = − 1 for all x ≠ 0. Notice that the range of f(x) is all the real numbers but zero, and this range corresponds to the domain of the function f−1(x) that we found.
In words, the radius of a circle is the square root of its area divided by π. Since there are no conventions associated with the variable names A and r, we do not bother to interchange the names, especially because A stands for area and r stands for radius and we do not want to mix these up.
The idea of interchanging the roles of x and y to find the equation of an inverse function also can be used to graph an inverse function. Indeed, if (x, y) = (a, b) is a point on the graph of y = f(x), then f(a) = b. Therefore, a = f−1(b) and (x, y) = (b, a) is a point on the graph of y = f−1(x).
Graphing Inverses
If f is one-to-one, then the graph of its inverse y = f−1(x) is given by reflecting the graph of y = f(x) about the line y = x.
Example 5 Graphing inverses
Consider the function y = f(x) whose graph is shown below,
Sketch the graph of y = f−1(x) by hand.
Solution If we sketch the line y = x in black, then reflecting the graph about the line y = x yields the graph of y = f−1(x), shown in red.
Consider the exponential function y = ax where a > 0. Suppose that x1 ≠ x2 are such that ax1 = ax2. Then by the laws of exponents, 1 = ax1a−x2 = ax1−x2. Since x1 − x2 ≠ 0, a must equal 1. Hence, we may conclude that y = ax is a one-to-one function provided that a ≠ 1. Therefore, y = ax has an inverse function provided that a ≠ 1. This inverse is the logarithm function with base a.
Logarithm
Let a > 0 and a ≠ 1. Then
y = logax is the logarithm of x with base a.
The statement “y =logax” should be read as “y is the exponent on a base a that gives the value x.” Do not forget that a logarithm is an exponent.
Example 6 Using the definition of logarithm
Find x such that
Solution
In elementary work, the most commonly used base is 10, so we call a logarithm to the base 10 a common logarithm and agree to write it without using a subscript 10. Thus, part c of the previous example is usually written log x = 3. In biological applications dealing with natural growth or decay, the base e is more common. A logarithm to the base e is called a natural logarithm and is denoted by ln x. The expression ln x is often pronounced “ell en ex” or “lawn ex.” In some texts, especially those pertaining to information theory in computer science, the function log2x is of theoretical importance and it is written simply as lg x. Its use, however, is not common in differential or integral calculus.
Logarithmic Notations
To evaluate a logarithm means to find an exact answer if possible (e.g., log9 81 = 2), or to calculate a decimal approximation to a required number of decimal places. Find the keys labeled and on your calculator. Verify the following calculator evaluations using your own calculator:
Since logarithms are exponents, the following properties of logarithms follow immediately from the properties of exponents and the definition of logarithms.
Laws of Logarithms
Additive law: logax + logay = loga xy
Subtractive law: logax − logay = loga
Multiplicative law: y loga x = loga(xy)
Change of base: logb x =
Cancellation properties:
Example 7 Graphing logarithmic functions
Use technology to graph the logarithmic functions y = log x, y = ln x, and y = log2 x on the same coordinate axes. Discuss the common properties of these graphs.
Solution The graphs (using technology) are shown in Figure 1.40.
In all cases, the function has a domain of (0, ∞) and range of (−∞, ∞)—that is, the range is the real number line R. The x-intercept is (1, 0). Additionally, the graph appears to be increasing from −∞ as x increases from 0, and the steepness of the graph is decreasing with increasing x.
Example 8 Solving exponential equations
Approximate the solutions to two decimal places.
Solution Be sure to duplicate the results below using your calculator.
Logarithmic functions are key to finding half-lives of exponentially decaying quantities and to finding doubling times for exponentially growing quantities:
Exponential half-life. In a process x(t) decaying exponentially, its half-life is the time t = T that it takes to change from its current size x(0) to x(T) = x(0)/2.
Exponential doubling time. In a process x(t) growing exponentially, its doubling time is the time t = T that it takes to change from its current size x(0) to x(T) = 2x(0).
In the next example, we solve for these quantities for processes introduced in Section 1.4.
Example 9 Half-life and doubling time
Solution
Thus, it takes the froth around 168 seconds, which is almost three minutes, to decay to half its height!
Since T is in decades, the doubling of the population occurs approximately in twenty-four years and four months.
Organisms vary greatly in their body mass. On the one hand, mycoplasma, which are bacteria without cell walls, weigh less than a tenth of a picogram (i.e., less than 10−13 grams). On the other hand, an ancient sequoia weighs as much as 5,000 tons (i.e., 1010 grams). Body sizes of other species are shown in Table 1.15. Marking all of these body sizes on an axis would be exceedingly difficult (try it!). However, if we take the log of these body sizes in grams, we can mark these sizes easily on a single axis as shown in Figure 1.41. An equivalent means of representing the data is to mark the body sizes on a logarithmic scale where powers of 10 are equally spaced. Doing so yields the lower panel in Figure 1.41, which simply corresponds to replacing the log body sizes in the upper panel of Figure 1.41 with the actual body sizes. Each unit increase on the logarithmic scale thus represents a tenfold increase in the underlying quantity.
Example 10 Using the log scale
Mark the numbers 0.00005, 0.1, 20, and 60,000 on a logarithmic scale.
Solution Applying log to the numbers 0.00005, 0.1, 20, and 60,000, we get −4.3, −1, 1.3, and 4.8. Marking these log values on an x-axis yields
To turn this figure into a logarithmic scale, we replace the exponents x with 10x.
When plotting functions in the xy plane, it can be useful to plot one or both axes on the logarithmic scale.
Semi-Log and Log-Log Plots
Semi-log plots correspond to using the logarithmic scale on one axis and arith metic scale on the other axis.
Log-log plots correspond to using the logarithmic scale on both axes.
The next example illustrates how plotting on semi-log or log-log plots can transform the graphs of nonlinear functions into the graphs of linear functions.
Example 11 Exponentials and power functions on logarithmic scales
Solution
Plotting log y against x yields the left panel below, while replacing the values on the log y axis with ten raised to the power of these values produces the semi-log plot in the right panel below.
Plotting log y as a function of log x yields the left panel below, while replacing the values on the log y axis and log x axis with ten raised to the power of these values produces the log-log plot in the right panel below.
The previous example illustrates how the graphs of exponential functions and power functions are linear when using logarithmic scales appropriately.
Exponentials, Power Functions, and the Logarithmic Scale
If y = bax is an exponential function with b > 0, then log y =log b + x log a is a linear function of x. Therefore, exponential functions appear linear on a semi-log plot.
If y = axb is a power function with a > 0, then log y = log a + b log x is a linear function of log x. Therefore, power functions appear linear on log-log plots.
These observations allow us to apply data fitting techniques discuss in Section 1.2 to exponential and power functions.
Example 12 Linear regression on a logarithmic scale
The metabolic rate of an organism is the rate at which it builds up (anabolism) and breaks down (catabolism) the organic materials that constitute its body. A famous data set exhibiting an allometric scaling law for relating metabolic rate y to body mass x was first published by Max Kleiber and is reproduced here in Table 1.16.
Since the data should exhibit allometry, we would expect that there exist real numbers a > 0 and b such that
Solution
As before, we use technology to find the best-fitting line:
The ln y-intercept is (0, 4.20577) and the slope is 0.755917 ≈ . There have been many theoretical attempts to explain this scaling exponent.
The elephant will burn off approximately 53,000 kilocalories per day.
In the problem set that follows you will see that a similar approach can be used for fitting exponential functions to data.
PROBLEM SET 1.6
Level 1 DRILL PROBLEMS
Determine whether the functions defined by the tables in Problems 1 to 4 are one-to-one. For the functions that are one-to-one, write the inverse function.
Use the horizontal line test to determine which of the functions in Problems 5 to 8 is one-to-one. For the functions that are one-to-one, sketch the inverse.
Find the inverse of the functions in Problems 9 to 14. State the domain and range of the inverse.
10. y = e2x+1
11. y =(x + 1)3 − 2
12. y = exp(x2) on [0, ∞)
13. y = exp(x2) on (−∞, 0]
14. y =
Find x in Problems 15 to 19 using the definition of logarithm (do not use a calculator).
Simplify the expressions given in Problems 20 to 22.
In Problems 23 to 25 write the expressions in terms of base and simplify where possible.
Simplify the expressions in Problems 26 to 30 using the definition of logarithm (do not use a calculator).
26. log 100 + log
27. ln e + ln 1 + ln e542
28. log8 4 + log8 16 + log882.3
29. 10log 0.5
30. ln elog 1,000
Sketch the indicated points on a logarithmic scale in Problems 31 to 34.
31. 0.002, 0.5, 10, 25000
32. 0.0003, 0.01, 0.1, 1, 200
33. 0.00004, 0.2, 10, 200000
34. 7, 10, 2000, 10000000
Level 2 APPLIED AND THEORY PROBLEMS
35. In Example 6 of Section 1.5, you modeled the uptake rate of glucose by bacterial populations with the Michaelis-Menten function U = micrograms/hour where C is the concentration of glucose (mg/l) (micrograms/liter).
36. In Example 9 of Section 1.5, you modeled the per capita growth rate for a shrew population with the function F = pairs/day where C is the cocoon density (cocoons per thousandth acre).
37. In Problem 41 of Section 1.4, you modeled the population size of humpback whales off the coast of Australia with the exponential function N(t) = 350(1.12)t where t is measured in years since 1981. Estimate the doubling time for this population of whales.
38. In Problem 42 of Section 1.4, you modeled the population size (in millions) of Mexico with the function P(t) = 67.38(1.026)t, where t is years after 1980. Find the doubling time for the population.
39. In an experimental study performed at Dartmouth College, two groups of mice with tumors were treated with the chemotherapeutic drug cisplatin. Prior to the therapy, the tumor consisted of proliferating cells (also known as clonogenic cells) that grew exponentially with a doubling time of approximately 2.9 days. In Problem 44 from Section 1.4, you modeled the volume of the tumor with the function V(t) = 0.1(2)t/2.9 cm3, where t is measured in days. Find an exact expression for the time at which the tumor size is 0.5cm3.
40. In the experimental study described in Problem 39, each of these mice was given a dose of 10 mg/kg of cisplatin. At the time of the therapy, the average tumor size was approximately 0.5 cm3. Assume all the cells became quiescent (i.e., no longer dividing) and assume decay with a half-life of approximately 5.7 days. In Problem 45 from Section 1.4, you modeled the volume of this tumor with the function V(t) = 0.5(1/2)t/5.7. Find an exact expression for when the tumor size is 0.1cm3.
41. Figure 1.43 shows a plot of the weight W (in grams) versus length L (in meters) for a sample of 158 male and 167 female western hognose snakes (Heterodon nasicus) from Harvey County, Kansas. The females are represented by open circles, and the males by closed circles. The scale is log-log.
Data Source: D. R. Platt, Natural History of the Hognose Snakes Heterodon platyrhinos and Heterodon nasicus (Natural History Museum of the University of Kansas. Reprinted with permission.)
It appears that when L = 0.4 cm, the corresponding weight on the best-fitting line is W = 28 g; likewise, L = 0.6 m appears to correspond to W = 100 g. Assuming an allometric relationship W = cLm, we have
Find the allometric relationship between weight and length (round c to the nearest integer).
42. It is known that fluorocarbons have the effect of depleting ozone in the upper atmosphere. Suppose the amount Q of ozone in the atmosphere is depleted by 15% per year, so that after t years, the amount of original ozone Q0 that remains may be modeled by
43. Allison and Cicchetti reported data on body weight (in kilograms) and corresponding brain weight (in grams) for sixty-two different terrestrial mammals (no whales). A partial list of the data is given below.
44. Rivers and streams carry small solid particles of rock and mineral downhill, either suspended in the water column (“suspended load”) or bounced, rolled, or slid along the river bed (“bed load”). Solid particles are classified according to their mean diameter from smallest to largest as clay, silt, sand, pebble, cobble, and boulder. During low velocity flow, only very small particles (clay and silt) can be transported by the river, whereas during high velocity flow, much larger particles may be transported, as documented in the table below.
45. Rainbow trout taken from four different localities along the Spokane River (eastern Washington) during July, August, and October 1999 were analyzed for heavy metals for the Washington State Department of Ecology. As part of this study, the length (in millimeters) and weight (in grams) of each trout were measured, as documented in the Table below; age determinations using scales are currently underway.
Length (mm) | Weight (g) |
457 | 855 |
405 | 715 |
455 | 975 |
460 | 895 |
335 | 472 |
365 | 540 |
390 | 660 |
368 | 581 |
385 | 609 |
360 | 557 |
346 | 433 |
438 | 840 |
392 | 623 |
324 | 387 |
360 | 479 |
413 | 754 |
276 | 235 |
387 | 538 |
345 | 438 |
395 | 584 |
Data source: Johnson, A. Results from Analyzing Metals in 1999 Spokane River Fish and Crayfish Samples (Washington State Dept. of Ecology report, 2000).
46. Consider the first four entries presented in the table below, which represent one estimate of the world's population levels over the second millennium AD:
Year AD (t) | Population size x (in billions) |
1000 | 0.31 |
1250 | 0.40 |
1500 | 0.50 |
1750 | 0.79 |
Provide a semi-log plot of the points (t, ln x) and find and graph the best-fitting line through these points on the same plot. From this line, provide an estimate of the average growth rate exponent r for the population size function x(t) = cert over this period of 750 years.
47. Consider the entries presented in the table below, which represent one estimate of the world's population levels over the period 1750 to 1920:
Year AD (t) | Population size x (in billions) |
1750 | 0.79 |
1800 | 0.98 |
1850 | 1.26 |
1900 | 1.65 |
1910 | 1.75 |
1920 | 1.86 |
Provide a semi-log plot of the points ( t, ln x) and find and graph the best-fitting line through these points on the same plot. From this line, provide an estimate of the average growth rate exponent r for the population size function x(t) = cert over this 170-year period.
48. Consider the entries presented in the table below, which represent one estimate of the world's population levels over the period 1920 to 2010:
Year AD (t) | Population size x (in billions) |
1920 | 1.86 |
1930 | 2.07 |
1940 | 2.30 |
1950 | 2.52 |
1960 | 3.02 |
1970 | 3.70 |
1980 | 4.44 |
1990 | 5.27 |
1999 | 5.98 |
2010 | 6.86 |
Provide a semi-log plot of the points (t, ln x) and find and graph the best-fitting line through these points on the same plot. From this line, provide an estimate of the average growth rate exponent r for the population size function x(t) = cert over this 90-year period.
Often, experimental measurements are collected at discrete intervals of time. For example, the number of elephants in wildlife park in Africa may be counted every year to ensure that poachers are not exterminating the population. Blood may be drawn on a weekly basis from a patient infected with HIV and the number of CD4+ cells produced by the patient's immune system counted to monitor patient response to treatment. Data obtained in this regular fashion can be represented by a sequence of numbers over time. In this section, we describe the basic properties of such sequences and demonstrate that some sequences can be generated recursively using a relationship called a difference equation. These equations are formulated using a function from the natural numbers to the real numbers.
We begin with the idea of a sequence, which is simply a succession of numbers that are listed according to a given prescription or rule. Specifically, if n is a natural number, the sequence whose nth term is the number an can be written as
The number a1 is called the first term, a2 the second term,..., and an the nth term.
Sequence
A sequence is a real-valued function whose domain is the set of natural numbers.
When working with sequences, we alter the usual functional notation. For a function a from the natural to the real numbers we should write a(1), a(2), a(3),..., but for convenience we write a1, a2, a3,.... The function a(n) is written an and is called the general term.
Example 1 Finding the sequence, given the general term
Find the first five terms of the sequences whose general term is given.
d. an is the digit in the nth decimal place of the number π
Solution
To visualize a sequence, one can graph the sequence of points
in the coordinate plane. The first several terms of the first four sequences from Example 1 are graphed in Figure 1.44. Since the domain consists of the natural numbers, the graph consists of discrete points.
Beyond specifying a sequence by its general term, sequences can also be generated term by term using a rule called a difference equation, which specifies how to calculate each term in the sequence from the values of preceding terms. For example, the difference equation
generates the geometric sequence
Similarly, the difference equation
generates the arithmetic sequence
Example 2 Geometric decay of acetaminophen
Example 2 of Section 1.1 stated that a tablet of regular strength Tylenol contains 325 mg of acetaminophen and that approximately 67% of the drug in the body is removed from the body every four hours. Assume Professor Schreiber just swallowed two tablets of Tylenol. Let an be the amount in milligrams of acetaminophen in his body 4n hours after taking the two tablets.
Solution
Plotting these values produces the figure on the next page.
More generally, for any real-valued function f, we can make the following definition.
Difference Equation
Let f be a real-valued function. Then
is a difference equation.
A sequence a1, a2, a3,... satisfying this equation for all n is a solution to the difference equation. By specifying a particular value of a1 and applying the an+1 = f(an) inductively, one can generate solutions to the difference equation, provided f is well-defined at every step.
Difference equations allow us to describe how quantities evolve over discrete intervals of time. For example, the difference equation an+1 = 0.33an in Example 2 described the decay of a drug in the body. A similar equation could describe the weekly growth of a bacterial culture in a laboratory, or even a population of California condors that were reintroduced to a wild area where they had previously become extinct because of use of the pesticide DDT.
From a modeling perspective, discrete intervals of time implied by the iteration of the difference equation (e.g., daily, weekly, or annual growth rules) correspond to one of two factors: synchronized events of the system (e.g., daily injections of a drug, annual reproductive cycles in a population) or intervals separating experimental measurements of the system (e.g., daily blood cell counts, annual population counts).
Example 3 The difference equation implicit in taking repeated square roots
Enter any nonzero number into your calculator. Press the square root key and record your answer. Press again and record repeatedly. Let an denote the nth number displayed on the screen.
Solution
Thus, the difference equation in this case is an+1 = f(an) with f(x) = .
Difference equations can be used to model a variety of biological phenomena. The next two examples illustrate their usage in modeling repeated drug dosages and the purging of a lethal recessive gene from a population.
Example 4 Drug delivery
In Example 2, Professor Schreiber took only two tablets of Tylenol. However, the directions recommend taking two tablets every four to six hours and not taking more than ten tablets in twenty-four hours. Suppose Schreiber takes two tablets every four hours. To model how the amount of drug in Schreiber's body changes in time, let an be the amount of drug in his body right before taking the nth dose.
Solution
This table suggests that an is approaching a value that rounded to two decimal places is 320.15 mg.
The difference equation an+1 = 0.33an + 214.5 in Example 4 is an example of a linear difference equation: the right-hand side of the difference equation depends linearly on an. In Problem Set 1.7, you are asked to write explicit solutions for linear difference equations.
Difference equations arise in biology whenever we consider how certain quantities change over regular, discrete intervals of time:
Now we consider a model of how a particular quantity changes from one generation to the next.
The quantity to model is the proportion of a particular allele (i.e., a variant of a particular gene) responsible for a genetic disease, such as Tay-Sachs or cystic fibrosis, that has a lethal effect when untreated. The model we present is the simplest example of a class of models that traces the proportion of a particular allele a in a diploid organism that has two possible alleles a and A associated with the gene in question and thus has genotypes aa, aA= Aa and AA. These models are only valid for large populations, where the assumption that one can replace the concept of probabilities with proportions holds.
Specifically, if one flips a coin four times and represents the proportion of heads using the variable x, then it is unreasonable to assume that half of the flips were heads and half were tails (i.e., x = 0.5), since quite often one might land up with three heads and five tails (x = 3/8), five heads and three tails (x = 5/8), or values of x even closer to 0 or 1. On the other hand, if one flipped the coin a million times, then one can safely assume, to a very good approximation, that half of the flips were heads and half were tails, that is, x = 0.50.
In such models, we apply the following principle of Gregor Mendel (1822–1884), derived from his work on plant hybridization
Random mating and Mendelian inheritance principle: Under the assumptions of individuals choosing mates at random, and alleles segregating randomly and independently, it follows that if x and (1 − x) are the proportion of alleles A and a in a population, then the proportion of genotypes among the progeny, before evaluating their ability to survive, are as follows:
This accounts for all possible genotypes, which we check by adding these three genotype frequencies to obtain the value 1:
Example 5 Lethal recessive genes
Suppose a disease in humans is primarily due to the existence of a lethal recessive allele a. By lethal recessive, we mean that individuals of type aa die from the disease, whereas individuals of type AA and Aa are not affected by the disease.
Solution
In Figure 1.47, experiments on the fruit fly show that the difference equation xn+1 = does a reasonable job of describing observed frequencies of the lethal allele, Glued, in fruit flies. The observed trajectories illustrate that even if you start with the same initial conditions (i.e., 50% with Glued), random birth and death events can result in different experimental trajectories. Hence, the model can only be expected to describe what happens for the “average” experiment.
Source: Genetics 83: 793–810 August, 19iG Dynamics of Correlated Genetic Systems. I. Selection in the Region of the Glued Locus of Drosophila Melanogaster, M. T. Clegg, J. F. Kidwell, M. G. Kidwell and N. J. Daniel.
In Examples 3 and 5 we saw that for certain initial values the difference equations generating the sequences in question produced a string of constant values. Specifically, in Example 3 the difference equation an+1 = produced the sequence 1, 1, 1,... for a1 = 1 (i.e., the square root of 1 is 1) and in Example 5 the difference equation xn+1 = produced the sequence 0, 0, 0,..., when x1 = 0 (i.e., if the lethal allele is not present initially, it never appears). Such starting values are called equilibria for the equations in question.
Equilibrium
An equilibrium of the difference equation
is an initial value a1 such that f(a1) = a1. From this it easily follows that a1 = a2 = a3= ···.
Example 6 Finding equilibria
Find the equilibria for the following three difference equations. Discuss how the answers you find relate to what was observed in Examples 3, 4, and 5.
Solution
In Example 4, we observed that for the initial condition a1 = 0, the sequence an would approach this equilibrium value.
In Chapter 2, we explore more carefully the sequence approach that identified equilibria in Example 6. The next example illustrates that an equilibrium is not always approached. It is based on a model for population biology.
In 1981, Thomas Bellows investigated how the survivorship of different species of stored grain beetles depended on the population abundance x. Some of the data from this experiment are illustrated in Figure 1.48. Bellows showed that the function s(x) = with x corresponding to population density, a > 0 and b > 0, could describe all of these data sets. The function s(x) describes the fraction of grain beetles surviving as a function of population abundance. If r > 0 is the average number of progeny produced by an individual, then the population model arising from this is
with the specific form considered by Bellows given in the next example.
Source: After T. S. Bellows, “The Descriptive Properties of Some Models for Density Dependence,” Journal of Animal Ecology 50(1)(1981): 139–156. Reprinted with permission.
Example 7 Generalized Beverton-Holt dynamics
If xn is the population density in generation n, then the population model
(sometimes referred to as the generalized Beverton-Holt model) produces solutions with behavior that depends on the three parameters r, a, and b.
Solution
for x. Clearly, x = 0 is a solution. For x ≠ 0, we obtain
Thus, x = 100 is an equilibrium value regardless of the value of b > 0.
and x1 = 99, yields Figure 1.49a. It appears that the sequence is approaching the equilibrium value of x = 100.
Using technology for b = 6:
and x1 = 99, yields Figure 1.49b. Despite starting near the equilibrium abundance of x = 100, this sequence exhibits oscillatory bursts of population growth and decline without any other characterizable pattern of behavior. In Chapter 4, we will discuss methods to distinguish among these different outcomes.
Another way to visualize sequences determined by a difference equation
is via a graphical technique known as cobwebbing.
Before describing this technique, we provide a graphical characterization of equilibria.
Finding Equilibria Graphically
To find equilibria of an+1 = f(an), it suffices to look for intersection points of the graphs of y = x and y = f(x)
Cobwebbing
To create a cobweb for the difference equation an+1 = f(an) with initial condition a1, follow these steps:
Step 1. Graph the functions y = f(x) and y = x in the xy plane.
Step 2. Draw a vertical line segment from (a1, a1) to (a1, f(a1)) and draw a horizontal line segment from (a1, f(a1)) to (f(a1), f(a1)). Since a2 = f(a1), you will have ended at the point (a2, a2).
Step 3. Repeat this procedure as desired. More specifically, if you are the point (an, an), then draw a vertical line segment from (an, an) to (an, f(an)) and draw a horizontal line segment from (an, f(an)) to (f(an), f(an)) = (an+1, an+1).
Example 8 Cobwebbing square roots
Consider the difference equation an+1 = f(an) where f(x) = . Use cobwebbing to visualize the first ten terms of the sequence determined by
Solution
Cobwebbing an increasing function, such as the square root function, is relatively simple. The cobweb diagram gets more complicated when the function is both increasing and decreasing over its relevant domain (see next example).
Example 9 Cobwebbing a hump-shaped function
Use cobwebbing to visualize the first forty terms of the sequence determined by the equation
from starting value a1 = 50. Discuss the primary difference between this example and Example 8.
Solution We begin by drawing the graphs of y = f(x) = and y = x (Figure 1.52a). To visualize the first two terms of the sequence, start at (50, 50) and draw a vertical line up to the graph of y = f(x) followed by a horizontal line to the graph of y = x (Figure 1.52b). To visualize the next term, draw a vertical down from (150, 150) to the graph of y = f(x) followed by a horizontal line to the graph of y = x (Figure 1.52c). Unlike our previous cobwebbing, we see that the sequence is already exhibiting some oscillation. In fact, continuing for the remaining thirty-seven terms yields the wild web depicted in Figure 1.52d.
Level 1 DRILL PROBLEMS
Find and graph the first five terms for the sequences in Problems 1 to 10.
5. an is the nth digit of the decimal representation of the number
6. an is the nth digit of e
7. a1 = 256, an+1 =
8. a1 = 2, an+1 = a2n, n ≥2
9. a1 = −4, a2 = 6, an = an−1 + an−2, n ≥ 3
10. a1 = 1 and a2 = 2, an+1 = anan−1, n ≥ 3
Find a5 for each difference equation in Problems 11 to 20.
11. an+1 = an + 8; a1 = 0
12. an+1 = 3an; a1 = 1
13. an =
14. an =
15. an+1 = 5an + 2; a1 = 0
16. an+1 = 1 − 2an; a1 = 0
17. an+1 = 2an + 1; a1 = 8
18. an+1 =
19. an+1 =
20. an+1 = 2an(1 − an); a1 = 1
Find the equilibria of an+1 = f(an) and sketch cobwebbing diagrams for the values of a1 given in Problems 21 to 26.
21. f(x) = 2x(1 − x) with a1 = 0.1
22. f(x) = x(2 − x) with a1 = 0.4
23. f(x) = with a1 = 0.1
24. f(x) = with a1 = 3
25. f(x) = 1 + x/2 with a1 = 0
26. f(x) = with a1 = 3
Find the equilibria of an+1 = f(an) where the graph of y = f(x) is shown in Problems 27 to 30, and sketch the cobwebbing diagrams starting with the given a1 value.
Level 2 APPLIED AND THEORY PROBLEMS
31. A drug is administered into the body. At the end of each hour, the amount of drug present is half what it was at the end of the previous hour. What percentage of the drug is present at the end of four hours? At the end of n hours?
32. A friend has a really bad headache. He decides to take 500 mg of aspirin every four hours. At the end of each four-hour period, the body clears out 80% of the aspirin in his body. Let an denote the amount of aspirin in your friend's body at the time he takes the nth aspirin.
33. Consider the general case of a patient who is taking a drug for a health issue. Let A be the amount the patient takes each time and c be the fraction of the drug cleared by the body between doses. Define an to be the amount of drug in the body immediately prior to taking the nth dose.
34. A doctor has prescribed a drug for a patient. Let A be the amount the patient takes each time and c be the fraction of drug cleared by the body between doses. Define an to be the amount of drug in the body immediately after taking the nth dose.
35. The wildebeest (or gnu) is a ubiquitous species in the Serengeti of Africa. The following data about wildebeest abundance were collected by the Serengeti Research Institute.
36. The Ricker model of a dynamic salmon population is given by
where b is the total number of progeny produced per individual per generation and e−can represents the fraction of progeny that survives after accounting for the effects of adult cannibalism of very young fish. Find all the equilibria for this model and determine under what conditions they are positive. Sketch cobwebbing diagrams for b = 0.9, b = 2.0, b = 8.0, and b = 20.0. In these diagrams, let c = 1.0 and a1 = 2.
37. A simple continued fraction is an expression of the form
where b0, b1,... are real numbers. The simplest continued fraction occurs when 1 = b0 = b1 = b2 = .... This continued fraction is generated by the sequence
38.
Leonardo of Pisa, also known as Fibonacci, was one of the best mathematicians of the Middle Ages. He played an important role in reviving ancient mathematics and introduced the Hindu-Arabic place-value decimal system to Europe. His book, Liber abaci, published in 1202, introduced Arabic numerals, as well as the famous rabbit problem, for which he is best remembered today. To describe Fibonacci's rabbit problem, we consider a sequence whose nth term is defined by a difference equation. Suppose rabbits breed in such a way that each pair of adult rabbits produces a pair of baby rabbits each month.
The first month after birth, the rabbits are adolescents and produce no offspring. However, beginning the second month, the rabbits are adults, and each pair produces a pair of offspring every month. The sequence of numbers describing the number of rabbits is called the Fibonacci sequence, and it has applications in many areas, including biology and botany.
In this you are to examine some properties of the Fibonacci sequence. Let an denote the number of pairs of rabbits in the “colony” at the end of n months.
for n = 2, 3, 4,...
Compute rn for n = 1, 2, 3,..., 10.
39. Consider the difference equation xn+1 = introduced in Example 5. Let x1 be given.
40. A biologist discovers that a gene has a lethal allele a that is not purely recessive: genotypes of the form aa all die before reproducing and half the genotypes of the form Aa also die before reproducing.
41. A biologist discovers that a gene has a lethal allele a that is not purely recessive: genotypes of the form aa all die before reproducing and two thirds of the genotypes of the form Aa also die before reproducing.
42. A biologist discovers that a gene has a lethal allele a that is not purely recessive: genotypes of the form aa all die before reproducing and one third genotypes of the form Aa also die before reproducing.
43. Compare the first ten terms of the sequences obtained from the difference equations derived in Example 5 and in Problems 40, 41, and 42. What do you conclude about the effect of a lethal allele in the population when it has a partial effect on the genotypes Aa? What happens when the lethal allele kills all Aa genotypes before they have a chance to reproduce?
CHAPTER 1 REVIEW QUESTIONS
Mix and match the following functions with their corresponding graphs.
where t is the number of days from the beginning of the year. A treatment program initiated by the Department of Wildlife is 50% effective against this pollution. When does the model predict, for the first time, that the pollution will be at a level of 40?
The average weight, w, in grams, of an apple at time of harvest is the following decreasing function of the number apples:
Are either of these linear functions?
where f(t) is measured in tons per day, and t is measured in years, from 1992.
Seeing a project through on your own, or working in a small group to complete a project, teaches important skills. The following projects provide opportunities to develop such skills.
Smaller mammals and birds have faster heart rates than larger ones. If we assume that evolution has determined the best rate for each, why isn't there a single best rate? Is there a model that leads to a correct rule relating heart rates? A warm-blooded animal uses large quantities of energy to maintain body temperature because of heat loss through its body surfaces. Cold-blooded animals require very little energy when they are resting. The major energy drain on a resting warm-blooded animal seems to be maintenance of body temperature.
The amount of energy available is roughly proportional to blood flow through the lungs—the source of oxygen. Assuming the least amount of blood needed is circulated, the amount of available energy will equal the amount used. In this project, you are to develop a model of blood flow and heart pulse rates as a function of body size and validate the model using the data in Tables 1.17 and 1.18. Be sure to address the following points:
The most universal feature of living organisms is their turnover of energy. Animals, with few exceptions, obtain energy by the oxidation of organic compounds, and the rate of energy turnover (the metabolic rate) is often measured by the rate of oxygen consumption. The fact that there is a regular relationship between the metabolic rate, or rate of oxygen consumption, and the body size of animals is thoroughly familiar to biologists. In the early part of the twentieth century, French scientists realized that the heat dissipation from warm-blooded animals must be roughly proportionate to their free surface. Since smaller animals have a larger relative surface, they must also have a higher relative rate of heat production than larger animals. In this project, you are to develop a model to explore this relation. Use the data in Table 1.19 to assess the accuracy of your model and its assumptions. The project needs to address the following point:
Around 300 BC, the greatest of the ancient Greek geometers, Euclid of Alexandria, defined what he called the “extreme and mean ratio,” now better known as the golden ratio. (See Mario Livio, The Golden Ratio: the Story of Phi, the Worlds Most Astonishing Number, New York: Broadway Books, 2002, p. 3.) Euclid's ratio states:
A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater segment to the lesser segment.
Specifically, if we look at the line illustrated in Figure 1.54, this statement can be expressed mathematically as
The golden rectangle is the rectangle whose sides conform to the golden ratio. The beauty, and astonishing perfection, of the golden rectangle arises from this fact: If you add one additional edge parallel to a short side of the rectangle to form a square within the rectangle, the smaller rectangle so formed (now oriented at 90 degrees to the original rectangle) is also a golden rectangle, as illustrated in Figure 1.55.
A spiral can be constructed passing through the corners of all embedded squares of the preceding construction in a such way that the spiral is equiangular—also known as the logarithmic spiral, which has the form shown in Figure 1.56.
Leonardo da Vinci (1452–1519), one of the greatest painters of all time, so valued the aesthetic proportions of the golden rectangle that aspects of figures and forms in many of his paintings conform to golden rectangle proportions. In the twentieth century, the noted American architect Frank Lloyd Wright (1867–1959) used the logarithmic spiral in designing the Guggenheim Museum in New York City.
1. List five paintings that art historians regard as compositions containing golden rectangles.
2. The shell of the nautilus mollusk, Nautilus pompilius, has the shape of a logarithmic spiral. Find a list of at least five other natural objects that contain shapes conforming to logarithmic spirals.
3. Define, using the concept of a tangent to a curve, what is meant by an equiangular spiral.
4. From Euclid's statement regarding the extreme and mean ratio, commonly denoted by the Greek letter phi (ϕ), show that ϕ =
5. If
then use technology to calculate ϕi, i = 5,..., 10. Use the definition of ϕn to generate a relationship of the form ϕn+1 = f(ϕn) and demonstrate that an equilibrium solution ϕ = f(ϕ) is the golden ratio. To how many decimal places do the numerical values of ϕ and ϕ10 coincide?
6. If
can you find a relationship of the form ϕ′n+1 = f(ϕ′n)? Demonstrate that an equilibrium solution ϕ′ = f(ϕ′) is the golden ratio. Notice that the denominators and numerators of the consecutive fractions are the Fibonacci sequence discussed in the , Problem 38 of Section 1.7. Use this fact to write the expression for ϕ′10. Draw a conclusion.
7. Compare the value of ϕ′10 obtained in the preceding question with the golden ratio ϕ and ϕ10 obtained from the question before that. Which of ϕ10 and ϕ′10 provides the better approximation to ϕ? Can you generalize this statement to ϕnand ϕ′n as an approximation to ϕ for any n?
* W. A. Ritschel, Handbook of Basic Pharmacokinetics, 2nd ed. (Hamilton, IL: Drug Intelligence Publications, 1980): 413–426.
* John B. S. Haldane, “On Being the Right Size,” The Harper's Monthly, March 1926, pp. 424–427.
* http://www.gutenberg.org/files/4239/4239-h/4239-h.htm.
* R. F. Vaccaro and H. W. Jannasch (1967). “Variations in uptake kinetics for glucose by natural populations in seawater.” Limnology and Oceanography. 12:540–542.
* W. M. Getz, “Metaphysiological and Evolutionary Dynamics of Populations Exploiting Constant and Interactive Resources: r-K Selection Revisited,” Evolutionary Ecology 7(1993): 287–305.
* John Damuth, “Interspecific Allometry of Population Density in Mammals and Other Animals: The Independence of Body Mass and Population Energy-use,” Biological Journal of the Linnean Society, 31(2008): 193–246.