Figure 1.1 Mathematical models are used in Section 1.3 to identify Pocket Hercules as one of the all-time greatest weightlifters.

Preview

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.

1.1 Real Numbers and Functions

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.

Figure 1.2 Portion of an electrocardiograph.

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.

Real numbers

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.

Table 1.1 Sets of numbers

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.

Figure 1.3 Graphical representations of intervals.

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,

Functions

Biologists, mathematicians, and other researchers often study relationships between two quantities. The mathematical study of such relationships involves the concept of a function.

A function can also be regarded as follows:

• A rule that assigns a unique “output” in the set Y to each “input” from the set X (Figure 1.4a)
• A graph that corresponds to the set of ordered pairs {(x, f(x)):x D} in the xy plane (Figure 1.4b)
• A machine that into which values of x are inserted and, after some internal operations are performed, a unique value f(x) is produced (Figure 1.4c)
• An algebraic equation (Figure 1.4d)

Figure 1.4 Different representations of a function.

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.

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.

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!

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.

Piecewise-defined functions

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.

A particularly important piecewise-defined function is the absolute value function.

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.

Figure 1.8 Graph of y = |x|.

Increasing and decreasing functions

Functions may exhibit different properties on intervals within their domain, as we now describe.

These classifications are shown graphically in Figure 1.9.

Figure 1.9 Classifications of functions.

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

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

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

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

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

Figure 1.10 Cutaway view of an artery.

1. Compute the speed of the blood at the central axis of this artery.
2. Compute the speed of the blood midway between the artery's wall and central axis.
3. What is the domain for the function defined by the ordered pairs (r, S)?
4. Graph this function for S ≥ 0.

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.

1. What is the domain?
2. Graph this function for k = 3 and a = 8.

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

1. What is the domain of the function f if n is a continuous variable?
2. What is the domain of the function f in the context of the psychology experiment?
3. Graph the function f defined in part b for n on interval [1, 20].
4. What will happen to the time required for the rat to traverse the maze as the number of trials increases? Will the rat ever be able to traverse the maze in less than three minutes?

44. Consider the function defined by

1. What is the domain of the function f?
2. Suppose that during a nationwide program to immunize the population against a certain form of influenza, public health officials found the cost (in millions of dollars) of inoculating x% of the population is modeled by f. For what values of x does f(x) have a practical interpretation in this context?
3. Graph the function for its interpretable range for the problem at hand.
4. Compare the cost of inoculating the first 50% of the population with the cost for the second 50%.

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

1. What is the domain for the function defined by (n, D)?
2. Graph this function for continuous 0 ≤ n ≤ 12.5 and A = 100.
3. If a 3-year-old child receives 100 mg of a certain drug, what is the corresponding dose for a 5-year-old child?

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

1. What is the domain for the function defined by (n, D) if the formula is applied to individuals up to age 16?
2. Graph this function for continuous 0 ≤ n ≤ 16 and A = 100.
3. If a 6-year-old child receives 120 mg of a certain drug, what is the corresponding dose for an 8-year-old child?

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

1. What is the domain for the function defined by (w, D)?
2. Graph this function for A = 200 mg.
3. If a 70-lb child receives 90 mg of a certain drug, what is the corresponding dose for an adult?

48. Table 1.2 tabulates the estimated number of HIV/AIDS cases diagnosed each year in the United States from 1999 to 2002.

Table 1.2 Number of diagnosed cases of HIV/AIDS by year

1. Use these data to draw a graph of the number of cases being diagnosed each day during the period starting at the beginning of 1999 and ending at the end of 2002 for the age group 25–34. This should be done by assuming that the average daily rate each year holds at the beginning of the year and then joining these points by a “continuous” curve (i.e., a curve with no jumps or breaks). The concept of continuity will be made more precise in the Chapter 2.
2. Use these data to draw a graph of the number of new cases diagnosed each day for all age groups.

49.

1.2 Data Fitting with Linear and Periodic Functions

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

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.

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?

Fitting linear functions to data

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.

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 e_i of the data from the line (see Figure 1.14). The values e_i are called the residuals because they represent “what is left over once the linear fit has been taken into account.”

Figure 1.14 Vertical distance of data from a line.

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.

Periodic and trigonometric functions

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.

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

Figure 1.17 Graphs of cosine and sine.

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.

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.

Figure 1.18 Graphs of the trigonometric functions y = tan(x), y = cot(x), y = sec(x), and y = csc(x).

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

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.

Figure 1.19 Standard cosine, sine, and tangent curves.

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.

1. At what age may a woman expect to live 30 years longer?
2. What is the life expectancy of a newborn female child?
3. At what age is the life expectancy zero?

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.

1. What is the functional relationship between N and x?
2. Use the function in part a to find the number of deaths per week when x = 300 mg/m³. What concentration of sulfur dioxide corresponds to 100 deaths per week?
3. Research data on how air pollution affects the death rate in a population. You may find the following articles helpful: D. W. Dockery, J. Schwartz, and J. D. Spengler, “Air Pollution and Daily Mortality: Associations with Particulates and Acid Aerosols,” Environmental Research 59 (1992): 362–373; Y. S. Kim, “Air Pollution, Climate, Socioeconomics Status and Total Mortality in the United States,” Science of the Total Environment (1985): 245–256. Summarize your results in a one-paragraph essay.

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.

Figure 1.20 Fat intake and age adjusted death rates due to cancer for countries in various regions of 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?

1. y = 0.139x
2. y = 0.231x − 3
3. y = 0.981x + 1

40. The chart in Figure 1.21 shows a comparison of Foude number with stride length for humans, kangaroos, and others.

Figure 1.21 Comparison of Froude number with stride length.

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.

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.

Table 1.5 Comparison of the weight of the fiddler crab with the weight of its large claw

1. Plot the points in the table. Does this look like a linear model to you?
2. Plot the line y = 0.47x − 49 on the axis for the points you plotted in part a. Does this look like a best-fitting line? Do you think you can find a better fitting line?

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.

Table 1.6 Comparison of body weight with the length of the mandibles of the male stag-beetle

1. Plot these points. Does this look like a linear model to you?
2. Plot the line y = 0.62x − 9.7 on the axis for the points you plotted in part a. Does this look like a best-fitting line? Do you think you can find a better-fitting line?

43. Table 1.7 shows the census figures (in millions) for the U.S. population since the first census.

Table 1.7 U.S. population

1. Plot these points where 1780 represents x = 0. Does this look like a linear model can provide a good fit?
2. Plot the line y = 1.3x − 50 on the axis for the points you plotted in part a. Does this look like a best-fitting line? Do you think you can find a better-fitting line?

44. Ethyl alcohol is metabolized by the human body at a constant rate (independent of concentration). Suppose the rate is 10 milliliters per hour.

1. Express the time t (in hours) required to metabolize the effects of drinking ethyl alcohol in terms of the amount A of ethyl alcohol consumed (in milliliters).
2. How much time is required to eliminate the effects of a liter of beer containing 3% ethyl alcohol?
3. Discuss how the function in part a. can be used to determine a reasonable “cutoff” value for the amount of ethyl alcohol A that each individual may be served at a party.

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.

Table 1.8 Depth and flow of Grand Coulee Dam

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.

1. Plot these points.
2. Find the lines defined by the first two data points and the first and last data points. Plot these lines against the data and decide which fits the data better.
3. Use technology to find the best-fitting data line.
4. Estimate the velocity of the river at a depth of 12 feet and 20 feet. Discuss the answers you obtain.

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.

Figure 1.22 Plot of length and width of clam samples.

Source: Quantitative Environmental Learning Project (QELP) Web Site at http://seattlecentral.edu/qelp/index.html.

1. A pair of points on this data set are given by (1.3, 1.7) and (7.3, 8.9). These two points are drawn in black in the Figure 1.22. Sketch the line passing through these points and find the formula for this line.
2. Use your line to estimate the width of a butter clam whose length is 12 cm.

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.

1. Plot these data.
2. Approximate the amplitude of the data and the period.
3. Find a and b such that 56.25 + a cos(bx) has the same amplitude and period as your data, and plot this function against your data. Let x represent months after July.

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.

1. Plot these data.
2. Approximate the amplitude of the data and the period.
3. Find a and b such that 70.8 + a cos(bx) has the same amplitude and period as your data, and plot this function against your data. Let x represent months after July.

1.3 Power Functions and Scaling Laws

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 and their properties

Note that and x³ are power functions, while 3^x is not because, in this latter case, the exponent rather than the base is the variable.

To algebraically manipulate power functions, we review some properties of exponents.

Proportionality and geometric similarity

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.

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.

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.

Allometric scaling

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.

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 = ax^b.

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 ∝ S^0.91. 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 ∝ d^2.99. 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 = aB^1.6, 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.

1. If he assumes that the furnace size should be proportional to the volume of the house, then what size furnace should he install?
2. If heat loss depends on the surface area of exterior walls, roof, and floor exposed to the winter cold rather than on the volume of the house, then what size furnace would you recommend?

41. Consider the following quote from Jonathan Swift's Gulliver's Travels:

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 ∝ L^b. 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:

1. Determine the value of b for which ∝ M^b.
2. Graph y = M^b and discuss the implications for a falling cubical critter.

1.4 Exponential Growth

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.

Exponential growth and exponential functions

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

Figure 1.28 Population of the United States.

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 = x^a 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.

The graphs of exponential functions have three different shapes, depending on the value of the base, as shown in the following example.

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.

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) = a^x 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 a^h. 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.

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.

Table 1.11 Froth height decay

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

Exponential decay

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.

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.

The number e

There is one choice of a base a for exponential functions a^x 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.

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.

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 = e^x 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.

Since e > 1, the exponential function y = e^x 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.

Level 1 DRILL PROBLEMS

Graph the exponential functions in Problems 1.4 to 6.

1. y = 2^x

2. y =

3. y = 3^−x

4. y = e^2x

5. y = 0.1^−x

6. y = π^ex

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) = 2^x and g(x) = 2x.

8. f(x) = e^x and g(x) = 2x.

9. f(x) = π^x and g(x) = x⁴ −4.

10. f(x) = 1.1^x 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. e^2x = 10

12. e^x/2 = 1/2

13. e^−x = x

14. e^−x = −3x

15. e^x/3 = x

16. e^−x = −x²

17. e^x = −x² + 2

18. e^x = 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) = ba^x 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.

Table 1.12 Population in Mexico

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:

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 = 2^x 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 = 2^x 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 = 3^x 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 = 2^x 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 = 2^x (Problem 47) and y = 3^x (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 = e^x. 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.

1.5 Function Building

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.

Shifting, reflecting, and stretching

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.

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.

In addition to shifting graphs, we can reflect graphs across axes.

A curve can be stretched or compressed in either the x-direction, the y-direction, or both, as shown in Figure 1.34.

Figure 1.34 Stretching and compressing a graph.

By compressing and stretching sinusoidal functions, we can model periodic phenomena like tidal movements.

Figure 1.35 Graph fitting the Assateague tide data.

Adding, subtracting, multiplying, and dividing

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.

Two important classes of functions that we get by adding, multiplying, and dividing power functions are polynomials and rational functions.

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.

Figure 1.36 Cell body and receptors.

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.

Composing

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.

Figure 1.37 Composition of two functions.

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 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.

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.

Figure 1.38 Short-tailed shrew (Blarina brevicauda).

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.

Figure 1.39 Graph of f.

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

17. y = 1 − e^−x

18. y = −|x|/2

19. y = 1 − 2|x|

20. Find the indicated values given the functions

1. (f + g)(1)
2. (f − g)(2)
3. (fg)(2)
4. (f/g)(0)
5. (f g)(2)

21. Find the indicated values given the functions

and

1. (f + g)(−1)
2. (f − g)(2)
3. (fg)(9)
4. (f/g)(99)
5. (f g)(0)

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.

1. f(t + 1)−2
2. 4f(t)
3. −2f(3t)
4. 2f(t − 4) + 1

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 = e^1−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).

1. Let t be the fraction of time that an enzyme is unoccupied. Assuming that 1 − t is proportional to x, find t as a function of x.
2. Assuming that f(x) is proportional to 1 − t, find an expression for f(x).
3. Below are data for glucose-6-phosphate converted to fructose-6-phospate by the enzyme phosphoglucose isomerase.

   

   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.

1. Write an expression for z in terms of t.
2. Consider the following data set

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.

1. Find the volume when t = 2.
2. Express the volume as a function of elapsed time by finding V h.
3. If the domain of V is [0, 6], find the domain of h; that is, what are the permissible values for t?

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.

1. Find the surface area when t = 2.
2. Express the surface area as a function of elapsed time by finding S r.
3. If the domain of S is (0, 8), find the domain of r; that is, what are the permissible values for t?

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.

1. Write a function of the tumor growth as a function of t in days.
2. Graph this function over a twenty-day period and describe what you see.

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.

1.6 Inverse Functions and Logarithms

Table 1.13 U.S. population size N as a function of time t

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

Table 1.14 Year t as a function of U.S. population size N

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.

Inverse functions

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.

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.

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).

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.

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.

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).

Logarithms

Consider the exponential function y = a^x where a > 0. Suppose that x₁ ≠ x₂ are such that a^x1 = a^x2. Then by the laws of exponents, 1 = a^x1a^−x2 = a^x₁−x₂. Since x₁ − x₂ ≠ 0, a must equal 1. Hence, we may conclude that y = a^x is a one-to-one function provided that a ≠ 1. Therefore, y = a^x has an inverse function provided that a ≠ 1. This inverse is the logarithm function with base a.

The statement “y =log_ax” 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.

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.

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.

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.

Table 1.15 Body sizes of different organisms

The logarithmic scale

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.

Figure 1.41 Log body sizes (upper panel) and body sizes on a logarithmic scale (lower panel) of the organisms listed in Table 1.15.

When plotting functions in the xy plane, it can be useful to plot one or both axes on the logarithmic scale.

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.

The previous example illustrates how the graphs of exponential functions and power functions are linear when using logarithmic scales appropriately.

These observations allow us to apply data fitting techniques discuss in Section 1.2 to exponential and power functions.

In the problem set that follows you will see that a similar approach can be used for fitting exponential functions to data.

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 = e^2x+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 + log₈8^2.3

29. 10^{log 0.5}

30. ln e^{log 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).

1. If the observed uptake rate of glucose was 1 mcg/hour, find the concentration of glucose per liter.
2. Find the inverse function of the uptake rate function and be sure to identify the appropriate units for this function.

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).

1. If the observed shrew per capita growth rate was 0.02 pairs/day, find the density of cocoons.
2. Find the inverse function of the per capita growth rate function and be sure to identify the appropriate units for this function.

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.

Figure 1.43 Regression line of weight versus length.

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 = cL^m, 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

1. How long (to the nearest year) will it take before half the original ozone is depleted?
2. Suppose through the efforts of careful environmental management, the ozone depletion rate is decreased so that it takes 100 years for half the original ozone to be depleted, what is the new rate (to the nearest hundredth of a percent)?

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.

1. Plot the log of brain size against the log body weight.
2. Find the best-fitting line using least squares regression on the log-transformed data.

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.

1. Plot the log of diameter against the log of the speed.
2. Find the best-fitting line using least squares regression on the log-transformed data.

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).

1. Plot the log of weight against length.
2. Find the best-fitting line using least squares regression on the transformed data.

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) = ce^rt 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) = ce^rt 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) = ce^rt over this 90-year period.

1.7 Sequences and Difference Equations

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.

Sequences

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 a_n can be written as

The number a₁ is called the first term, a₂ the second term,..., and a_n the nth term.

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 a_n and is called the general term.

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.

Figure 1.44 Graphs of sequences.

Difference equations

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

More generally, for any real-valued function f, we can make the following definition.

Difference equations allow us to describe how quantities evolve over discrete intervals of time. For example, the difference equation a_n+1 = 0.33a_n 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).

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.

The difference equation a_n+1 = 0.33a_n + 214.5 in Example 4 is an example of a linear difference equation: the right-hand side of the difference equation depends linearly on a_n. 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:

• From one fifteen-second period to the next, as in the beer froth problem considered in Section 1.4
• From one four-hour period to the next, as in this example
• From one day to the next, as in the tumor growth problems presented in Problem Sets 1.4–1.6
• From one month to the next, as in the carbon dioxide concentration problem considered in Section 1.2
• From one year to the next, or even one decade to the next, as discussed in the U.S. population growth problem in Section 1.4

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:

• x² for type aa
• 2x(1 − x) for type Aa(the 2 arises because aA = Aa)
• (1 − x)² for type AA

This accounts for all possible genotypes, which we check by adding these three genotype frequencies to obtain the value 1:

In Figure 1.47, experiments on the fruit fly show that the difference equation x_n+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.

Figure 1.47 Data on the decline of the Glued gene in fruit flies compared with the expected rate predicted by the model in Example 5.

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 a_n+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 x_n+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.

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.

Figure 1.48 Relationship between number of survivors and initial egg density for four species of stored product beetles.

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.

Cobwebbing

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.

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).

Level 1 DRILL PROBLEMS

Find and graph the first five terms for the sequences in Problems 1 to 10.

5. a_n is the nth digit of the decimal representation of the number

6. a_n is the nth digit of e

7. a₁ = 256, a_n+1 =

8. a₁ = 2, a_n+1 = a²_n, n ≥2

9. a₁ = −4, a₂ = 6, a_n = a_n−1 + a_n−2, n ≥ 3

10. a₁ = 1 and a₂ = 2, a_n+1 = a_na_n−1, n ≥ 3

Find a₅ for each difference equation in Problems 11 to 20.

11. a_n+1 = a_n + 8; a₁ = 0

12. a_n+1 = 3a_n; a₁ = 1

13. a_n =

14. a_n =

15. a_n+1 = 5a_n + 2; a₁ = 0

16. a_n+1 = 1 − 2a_n; a₁ = 0

17. a_n+1 = 2a_n + 1; a₁ = 8

18. a_n+1 =

19. a_n+1 =

20. a_n+1 = 2a_n(1 − a_n); a₁ = 1

Find the equilibria of a_n+1 = f(a_n) 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 a_n+1 = f(a_n) 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 a_n 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 a_n 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 a_n 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^−ca_n 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 a_n denote the number of pairs of rabbits in the “colony” at the end of n months.

39. Consider the difference equation x_n+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?

1. Let y = .
   1. State the domain and range of this function.
   2. Find the inverse of this function. State its domain and range as an inverse function.
2. The maximum temperature on a fall day in Woodland, California, was 92°F at 5 PM. The minimum temperature was 52°F at 5 AM. Let T(t) be the temperature in degrees at the tth hour after midnight. Write a sinusoidal function of the form T(t) = A + B cos(C(t − D)) that could be used to describe the temperatures on this warm fall day.
3. Find the equilibrium of a_n+1 = f(a_n) where the graph of y = f(x) is shown. Sketch the cobwebbing diagram for a₁ = −0.5.

   

4. Sketch the x values 0.0001, 0.1, 0.3, and 3,000,000 on a logarithmic axis.
5. Calculate the residuals of the graphs y = x and y = + 1 with respect to the data set {(0, 0.7), (1, 1.1), (2, 2.2)}. Use the sum-of-squares criterion to decide which of these two curves fits the data best.
6. The time for cooking a roast is approximately proportional to its surface area.
   1. How does the cooking time scale with the weight of the roast?
   2. How much longer should it take to cook a 20-lb roast compared to a 10-lb roast?
7. John Damuth hypothesized a power function relationship y = cx^m between the body mass x of individuals and the density y of populations belonging to related species.* The data he presented for five species of apes, where the units of x are kilograms and y are numbers/square kilometer: the western lowland gorilla, (127, 1.8); the common chimpanzee, (45.0, 2.5); the bonobo chimpanzee, (22.7, 4.0); the Bornean orangutan, (53.0, 2.0); and the agile gibbon, (5.9, 5.1). By plotting these data on a semi-log scale, what would you estimate to the nearest integer that the power of the allometric relationship is between abundance and body mass?
8. Consider a bank account with initially $1,000. The annual interest rate for this account is 10%.
   1. Find the amount in the account one year from now if the interest is compounded once a year, twice a year, three times a year.
   2. What value is the amount in the account approaching as the number n of times the money is compounded in the year approaches ∞?
9. An exponential function y = ab^x is plotted below on a semi-log plot. Find a and b.

   

10. Hypothyroidism is a condition in which the thyroid gland makes too little thyroid hormone. One treatment for hypothyroidism is 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 who takes 100 mcg of T4 once a week. Let a_n denote the amount of T4 in the individual's body right before the nth dose.
    1. Write a difference equation for a_n.
    2. Find the equilibrium for the difference equation in part a and describe in words what this equilibrium means.
    3. In the long term, what is the greatest amount of T4 in the individual's body?
11. Consider a population with two alleles, A and a, at a single locus such that all individuals born with the genotype aa die immediately, 10% of individuals with genotype Aa die immediately, and individuals with the genotype AA always survive to reproduce. Let x_n denote the fraction a alleles in the population in generation n.
    1. Write a difference equation for x_n.
    2. Find x₂, x₂, x₄ given that the initial proportion of a is x₁ = 0.9.
12. Data points with a curve fitted to those points are shown. Decide whether the data in parts a–d are better modeled by an exponential or a logarithmic function.

    

13. Let y = f(x) be the function whose graph is given in Figure 1.53.

    

    Figure 1.53 Graph of f.

    Mix and match the following functions with their corresponding graphs.

    

    

14. As a result of recovery efforts, the size of the whooping crane population in Wood Buffalo Park grew from 20 individuals in 1941 to 518 individuals in 2007. Assume that you can model the population growth of the whooping cranes as N(t) = ae^rt individuals where t is years after 1941.
    1. Find a and r.
    2. Use the model to estimate the doubling time for the whooping crane population.
15. Find the first five terms for the given sequences.

    

16. The pollution level, P, in Lake Bowegon varies during a typical year according to the formula

    

    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?

17. The number of apples, n, in a tree is a function of the population density, d, of bees pollinating the apple orchard. This function can be modeled by the formula

    

    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?

    1. Graph the weight of the apple as a function of the density of bees. What is the domain of w?
    2. As the number of bees increases, what can you say about the average weight of an apple?
18. The amount of solids discharged from the MWRA (Massachusetts Water Resources Authority) sewage treatment plant on Deer Island (near Boston Harbor) is given by the function

    

    where f(t) is measured in tons per day, and t is measured in years, from 1992.

    1. How many more tons per day were discharged in 2002 than in 1996?
    2. Sketch the graph of f.
19. A female moth (Tinea pellionella) lays nearly 150 eggs and then dies. In one year, up to five generations may be born, and each female larva eats about 20 mg of wool. Assume that two thirds of the eggs die, and 50% of the remaining moths are females. Use an exponential population growth model to estimate the largest amount of wool that may be destroyed by the female descendants of one female over a period of one year.
20. The level of a certain pollutant in the Los Angeles area has been decreasing linearly since 1990 when a new pollution control program began. The level of pollution was 0.17 ppm in 2000 and had fallen to 0.11 ppm in 2010.
    1. Let P be the level of pollutant (in parts per million) at time t (years after 2000). Express P as a function of t.
    2. Air with a pollutant level of 0.05 ppm is considered clean. If the present trend continues, when will this clean level be achieved?