Figure 2.1 Biologists count spawning adult sockeye salmon to obtain a stock-recruitment relationship (see Figure 2.43).

Preview

“Mathematics, in one view, is the science of infinity.”

Philip Davis and Reuben Hersh,
The Mathematical Experience, Birkhäuser
Boston, 1981, p. 152.

Calculus, one of the great intellectual achievements of humankind, came to fruition through the work of Sir Isaac Newton (1642–1727) and Gottfried Wilhelm Leibniz (1646–1716). It consists of two parts, differential calculus and integral calculus, both of which hinge on the concept of a limit in which the behavior of a function is described as its argument approaches a selected limiting value. In this chapter, we first discuss some of the basic properties of a limit and the associated concept of continuity. Using limits, we then introduce the main concept in differential calculus: the derivative. This concept is one of the fundamental ideas in mathematics and a cornerstone of modern scientific thought. It allows us to come to grips with the phenomenon of change in the value of variables over increasingly small intervals of time or space (or any other independent variable—variables representing such things as the velocity of a stooping falcon or the density of a population of fish). By finding the derivative of functions, we calculate the slopes of tangent lines to these functions and examine rates of change defined by these functions. Applications considered in this chapter include enzyme kinetics, biodiversity, foraging of hummingbirds, wolf predation, and population dynamics of the sockeye salmon depicted in Figure 2.1.

2.1 Rates of Change and Tangent Lines

One of the fundamental concepts in calculus is that of the limit. Intuitively, “taking a limit” corresponds to investigating the value of a function as you get closer and closer to a specified point without actually reaching that point. To “motivate limits,” we begin by showing how they arise when we consider rates of change and the tangent lines to graphs of functions.

Rates of change

Rates of change describe how quantities change with respect to a variable such as time or body mass. For instance, in countries where overpopulation is an issue, projections are constantly made about the population growth rate. For a patient receiving drug treatment, physicians perform experiments to estimate the clearance rate of the drug (i.e., the rate at which the drug leaves the blood stream). To calculate a rate of change, we first select an interval of interest and then compute the average rate of change over that interval using the equation in the box below. To calculate the rate of change at a point of interest requires taking limits, as discussed after the next example.

The instantaneous rate of change of f(x) at x = a (if it exists) is defined to be the limiting value of the average rate of change of f on smaller and smaller intervals starting at x = a. For instance, in Example 1, we would estimate the instantaneous rate of change of the population in Mexico in 1980 to be 1.75 million per year. In other words, the population is growing at a rate of 1.75 million per year in 1980. More precisely, we have the following definition.

How to calculate such limits is one of the challenges of differential calculus, a problem that we tackle further in Sections 2.2 and 2.3.

This example indicates that a linear fit to an oscillating function may provide a reasonable estimate of rates of change averaged over several oscillations; however, it is a poor estimate of the instantaneous rate of change because that depends on the particular stage of each oscillation.

Velocity

Now we consider concepts central to the history of calculus, concepts that motivated much of the work of Newton: the velocity (considered here) and the acceleration (considered in Sections 3.6 and 5.1) of an object moving along a line. For example, the object may be an athlete running along a racing track, or a coffee mug falling to the ground.

If you have ever watched a track meet, you may have wondered how fast the winners ran during their races. In the next example, we find out exactly how fast the world's best 100-meter sprinters are and how their velocities change during the race.

Figure 2.2 Usian Bolt is widely regarded as the fastest person ever, earning him the nickname “Lightning Bolt.”

We have all seen objects fall for different periods of time, whether it be a coffee mug falling off a table or a peregrine falcon stooping from high altitudes. The next example uses a basic physical principle (developed in greater detail in Section 5.1) to determine how the distance traveled by a falling object changes in time.

Tangent lines

A linear function is a function with constant slope, which raises this question: What is the slope of a nonlinear function at a point? To answer this question, we need to solve the tangent problem, whose solution resides in the following words of Leibniz quoted by David Berlinski (Infinite Ascent: A Short History of Mathematics (2008) Random House, NY):

We have only to keep in mind that to find a tangent means to draw the line that connects two points of the curve at an infinitely small distance

As illustrated in Figure 2.4 and stated by Leibniz, the tangent line of y = f(x) at a point (a, f(a)) (in blue) can be approximated by secant lines (in red) passing through the points (a, f(a)) and (a + h, f(a + h)) as h ≠ 0 gets closer and closer to 0. The slope of this secant line is given by

Figure 2.4 Approximating the tangent line with a secant line.

By letting h get arbitrarily close to 0, we get the

where the symbol can be interpreted as taking h > 0 arbitrarily close to, but not equal to 0. This limit may or may not exist, so not every curve will have a tangent line at every point.

Sometimes it is possible to algebraically determine the slope of the tangent line.

Level 1 DRILL PROBLEMS

Find the average rate of change for the functions in Problems 1 to 6 on the specified intervals.

1. f(x) = 4 − 3x on [−3, 2]

2. f(x) = 5 on [−3, 3]

3. f(x) = 3x² on [1, 3]

4. f(x) = −2x² + x + 4 on [1, 4]

5. f(x) = on [4, 9]

6. f(x) = on [1, 5]

Approximate the instantaneous rate of change for the functions in Problems 7 to 12 at the indicated point. These are the same functions as those given in Problems 1 to 6.

7. f(x) = 4 − 3x at x = −3

8. f(x) = 5 at x = 3

9. f(x) = 3x² at x = 1

10. f(x) = −2x² + x + 4 at x = 4

11. f(x) = at x = 4

12. f(x) = at x = 1

Find the instantaneous velocity of a cup falling from the specified height h feet and time t second in Problems 13 to 16. Use the fact that the height of the cup as a function of time t is given by f(t) = h − 16t² feet.

13. h = 64 feet; t = 1 seconds

14. h = 64 feet; t = 2 seconds

15. h = 4 feet; t = 1/2 seconds

16. h = 1,600 feet; t = 10 seconds

Trace the curves in Problems 17 to 22 onto your own paper and draw the secant line passing through P and Q. Next, imagine h → 0 and draw the tangent line at P, assuming that Q moves along the curve to the point P. Finally, estimate the slope of the curve at P using the slope of the tangent line you have drawn.

Estimate the tangent line for y = f(x) in Problems 23 to 36 by approximating secant slopes to estimate the limiting value. Graph both the function and the tangent line on the same plot.

23. f(x) = x² − x + 1 at x = −1

24. f(x) = 4 − x² at x = 0

25. f(x) = sin at x = 0.5

26. f(x) = cos at x = 0.5

27. f(x) = tan at x = 0.5

28. f(x) = sin at x = 1

29. f(x) = at x = 3

30. f(x) = e^x at x = 0

31. f(x) = ln x at x = 9

32. f(x) = tan x at x = 0

33. f(x) = tan at x = 0.95

34. f(x) = ln at x = 1

35. f(x) = ln at x = 0.1

36. f(x) = e^−x at x = 0

Algebraically determine the tangent line for y = f(x) at the point specified in Problems 37 to 42. Graph both y = f(x) and the tangent line on the same plot.

37. f(x) = 3x − 7 at x = 3

38. f(x) = x² at x = −1

39. f(x) = 3x² at x = −2

40. f(x) = x³ at x = 1

41. at x = 9. Hint: Multiply by

42. at x = 5

Level 2 APPLIED AND THEORY PROBLEMS

Find the average rate of change of the given functions over the specified intervals in Problems 43 to 46. Be sure to specify units and briefly state the meaning of the average rate of change.

43. P(t) = 8.3 × 1.33^t is the number (in millions) of people living in the United States t decades after 1815 over intervals [0, 2] and [2, 4].

44. L(x) = 20.15 x^2/3 is the number of kilograms lifted by Olympic Gold Medal weightlifters weighing x kilograms over intervals [56, 75] and [100, 110].

45. The height H(t) of beer froth after t seconds over intervals [0, 30] and [60, 90]. Use the data in Table 2.2.

Table 2.2 Height of beer froth as a function of time after pouring

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

Data Source: Arnd Leike, “Demonstration of the Exponential Decay Law Using Beer Froth,” European Journal Physics 23 (2002): 21–26.

46. The height f(x), in feet, of the tide at time x hours, where the graph of y = f(x) is provided in Figure 2.9, over intervals [0.25, 0.50] and [1, 2]

Figure 2.9 Tidal height.

Approximate the instantaneous rates of change of the given functions at the points specified in Problems 47 to 50. These are the same functions as those in Problems 43 to 46.

47. P(t) = 8.3(1.33)^t is the number (in millions) of people living in the United States t decades after 1815 at the points t = 0 and t = 2.

48. L(x) = 20.15x^2/3 is the number of kilograms lifted by Olympic Gold Medal weightlifters weighing x kilograms at the points x = 56 and x = 100.

49. Use the data in Table 2.2 to estimate the instantaneous rate of change of height H(t) of beer froth after 0 and 60 seconds respectively.

50. The height f(x), in feet, of the tide at time x hours where the graph of y = f(x) is in given in Figure 2.9 at the points x = 0.25 and x = 1.

51. From the data presented in Table 2.1 in Example 3, calculate Ben Johnson's velocities at the end of each of the 10-meter splits during the course of his race and discuss how his velocity changes over the course of the race.

52. The population of a particular bacterial colony was determined to be given by the function

thousand individuals t hours after observation began. Find the rate at which the colony was growing after exactly five hours.

53. The biomass of a particular bush growing in a field is given by the function B(t) = 1 + 10t^1/2 kg, where t is years. Find the rate at which the bush is increasing in biomass after exactly sixteen years.

54. An environmental study of a certain suburban community suggests that t years from now, the average level of CO₂ in the air can be modeled by the formula

parts per million.

1. By how much will the CO₂ level change in the first year?
2. By how much will the CO₂ level change over the next (second) year?
3. At what rate will the CO₂ level be changing with respect to time exactly one year from now?

2.2 Limits

In defining the instantaneous rate of change and the slope of a tangent line, we use the notation for a limit in Section 2.1. The concept of a limit is one of the foundations of both differential and integral calculus. Thus, we devote this section to exploring the concept of limits in the context of functions before proceeding to the calculus itself.

Introduction to Limits

We begin our study of limits with the following mathematically “informal” definition.

Note that sometimes f(x) may not be defined at x = a but may be defined for all x as close as you like to the value of a.

This definition is made mathematically precise at the end of this section. At several other places in this book we favor informal over formal definitions, using the following statement of the historian E. T. Bell as our justification (see Men of Mathematics, New York: Simon & Schuster, 1937, p. 98):

To the early developers of the calculus the notions of variables and limits were intuitive; to us they are extremely subtle concepts hedged about with thickets of semi-metaphysical mysteries concerning the nature of numbers....

Our goal is to provide definitions that make the concepts usable without getting caught up in the technicalities of formal definitions.

Figure 2.12 Graph of the function e^−1/x2.

The existence of a limit f(x) = L can be interpreted in terms of choosing the appropriate window for viewing a function.

The statement f(x) = L can fail in two ways. First, f(x) exists but does not equal L. Second, the f(x) does not exist. In other words, no matter what L we choose, the statement f(x) = L is false. You typically encounter the first failure when someone is testing you on this material or when someone has defined a function such that f(a) ≠ f(x). The second failure is more interesting.

We occasionally rely on technology to compute limits. Although technology almost always steers us in the right direction, cases exist where it drives us to incorrect conclusions.

One-sided limits

The definition of the limit of f(x) as x approaches a requires that f(x) approach the same value independent of whether x approaches a from the right or the left. In this sense, f(x) is a “two-sided” limit. One-sided limits, on the other hand, only require that f(x) approach a value as x approaches a from the left or the right.

A two-sided limit cannot exist if the corresponding one-sided limits are different. Conversely, it can be shown that if the two one-sided limits of a given function f as x → a⁻ and x → a⁺ both exist and are equal, then the two-sided limit, f(x), exists. These observations are so important that we restate them as follows:

One-sided limits are particularly useful when considering piece-wise defined functions (see Section 1.1 for a definition).

In the next example, the piece-wise defined function describes the feeding behavior of the copepod Calanus pacificus, illustrated in Figure 2.23. Planktonic copepods are small crustaceans found in the sea. These organisms play an important role in global ecology as they are a major food source for small fish, whales, and sea birds. It is believed that they form the largest animal biomass on Earth. Given their importance, scientists are interested in understanding how their feeding rate depends on availability of resources. In a classic ecology paper, C. S. Holling classified feeding rates into three types. (See his article, “The Functional Response of Invertebrate Predators to Prey Density,” Memoirs of the Entomological Society of Canada 48 (1966): 1–86.) The first type, so-called type I, assumes that organisms consume at a rate proportional to the amount of food available until they achieve a maximal feeding rate. The type II feeding rate was studied in Example 6 of Section 1.6.

Figure 2.23 The planktonic copepod Calanus pacificus as seen under an electron microscope. Copepods such as this are believed by some scientists to form the largest animal biomass on Earth.

Limits: A formal perspective (optional)

This section can be omitted by those not going on to major in mathematics at the undergraduate level. Our informal definition of the limit provides valuable intuition that allows you to develop a working knowledge of this fundamental concept. For theoretical work, however, the intuitive definition will not suffice, because it gives no precise, quantifiable meaning to the terms “arbitrarily close to L” and “sufficiently close to a.” In the nineteenth century, leading mathematicians, including Augustin-Louis Cauchy (1789–1857) and Karl Weierstrass (1815–1897), sought to put calculus on a sound logical foundation by giving precise definitions for the foundational ideas of calculus. The following definition, derived from the work of Cauchy and Weierstrass, gives precision to the definition of a limit.

Figure 2.25 The epsilon-delta definition of limit.

Behind the formal language is a fairly straightforward idea. Given any > 0 specifying a desired degree of proximity to L, a number δ > 0 is found that determines how close x must be to a to ensure that f(x) is within of L. This is shown in Figure 2.25.

Because the Greek letters (epsilon) and δ (delta) are traditionally used in this context, the formal definition of limit is sometimes called the epsilon-delta definition of the limit. The goal of this subsection is to show how it can be used rigorously to establish a variety of results.

One can view this definition as setting up an adversarial relationship between two individuals. One person shouts out a value of > 0. The opponent has to come up with a δ > 0 such that f(x) is within of L whenever x is within δ of a. This relationship is illustrated in Figure 2.26.

Figure 2.26 Formal definition of limit: f(x) = L.

Note that f(a) does not need to equal the value of L in order for f(x) = f(a). If f(x) = f(a), then the epsilon-delta argument confirms that f(x) is continuous at x = a. On the other hand, if f(x) ≠ f(a), then the epsilon-delta argument confirms that f(x) is not continuous at x = a.

Notice that whenever x is within δ units of a (but not equal to a), the point (x, f(x)) on the graph of f must lie in the rectangle (shaded region) formed by the intersection of the horizontal band of width 2 centered at L and the vertical band of width 2δ centered at a. The smaller the interval around the proposed limit L, generally the smaller the δ interval will need to be for f(x) to lie in the interval. If such a δ can be found no matter how small is, then L must be the limit. The following examples illustrate epsilon-delta proofs, one in which the limit exists and one in which it does not.

Level 1 DRILL PROBLEMS

Given the functions f and g defined by the graphs in Figure 2.29, find the limits in Problems 1 to 4.

Figure 2.29 Graphs of the functions f and g.

Given the functions defined by the graphs in Figure 2.30, find the limits, if they exist, in Problems 5 to 8. If the limits do not exist, discuss why.

Figure 2.30 Graphs of the functions F and G.

Describe each figure in Problems 9 to 12 with a one-sided limit statement. For example, for Problem 9, the answer is f(x) = 2.

Approximate the limits by filling in the appropriate values in the tables in Problems 13 to 15 using a one-sided statement.

Determine the limits in Problems 16 to 24. If the limit exists, explain how you found the limit. If the limit does not exist, explain why.

25. Consider the function

whose graph is given Figure 2.31. Does f(x) exist? If so, what is it? If not, why not?

Figure 2.31 Graph of the function f(x) = .

26. Consider the function

whose graph is given Figure 2.32. Does f(x) exist? If so, what is it? If not, why not?

Figure 2.32 Graph of the function f(x) = .

27. Consider the statement (4 + x) = 5. How close does x need to be to 1 to ensure that 4 + x is within the given distance of 5?

28. Consider the statement x² = 4. How close does x need to be to 2 to ensure that x² is within the given distance of 4?

29. Consider the statement = 0. How close does x need to be to 0 to ensure that is within the given distance of 0?

30. Consider the statement ln x = 1. How close does x need to be to e to ensure that ln x = 1 is within the given distance of 1?

31. Consider the function

1. Use technology to graph y = f(x) over the interval [−2, 2].
2. Use technology to graph y = f(x) over the interval [−0.1, 0.1]. Based on your graph, guess the value of f(x).
3. Use technology to graph y = f(x) over the interval [−10⁻⁷, 10⁻⁷]. Based on your graph, guess the value of f(x).
4. Most technologies can keep track of sixteen or fewer digits. In light of this observation, discuss what might be happening in parts b and c.

32. Consider the function f(x)) = .

1. Use technology to graph y = f(x) over the interval [−2, 2].
2. Use technology to graph y = f(x) over the interval [−0.1, 0.1]. Based on your graph, guess the value of f(x).
3. Use technology to graph y = f(x) over the interval [−10⁻⁷, 10⁻⁷]. Based on your graph, guess the value of f(x).
4. Most technologies can keep track of sixteen or fewer digits. In light of this observation, discuss what might be happening in parts b and c.

In Problems 33 to 38, prove the limit exists using the formal definition of the limit.

Level 2 APPLIED AND THEORY PROBLEMS

39. The federal income tax rates for singles in 2010 is shown in Table 2.3.

Table 2.3 Schedule X—Single

Express the income tax f(x) for an individual in 2010 with adjusted income x dollars as a piecewise defined function.

1. Graph y = f(x) over the interval [0, 500,000].
2. Determine at what values of a, f(x) does not exist.

40. In 2011, the U. S. postal rates were 44 cents for the first ounce or fraction of an ounce, and 17 cents for each additional ounce or fraction of an ounce up to 3.5 ounces. Let p represent the total amount of postage (in cents) for a letter weighing x ounces.

1. Graph y = p(x) over the interval [0, 3.5] ounces.
2. Determine at what values of a, f(x) does not exist.

41. A wildlife ecologist who studied the rate at which wolves kill moose in Yellowstone National Park found that when moose were plentiful, wolves killed moose at the rate of one moose per wolf every twenty-five days. (Note this doesn't mean that wolves only eat every twenty-five days, because they hunt in packs and share kills.) However, when the density of moose drops below x = 3 per km², then the rate at which wolves kill moose is proportional to the density. Construct a type I functional response f(x) (see Example 7) such that f(x) has a limit at x = 3.

42. A student looking at Figure 2.24 decided that the following function might provide a better fit to the data:

Find values for the parameters a and b that ensure f(x) has limits at x = 150 and x = 300.

2.3 Limit Laws and Continuity

Having defined limits, we are ready to develop some tools to verify their existence and to compute them more readily. In some cases, taking the limit of a function reduces to evaluating the function at the limit point, and in other cases we cannot find the limit by evaluation. In this section, we find when evaluation is acceptable and when it is not.

Properties of limits

With a definition of a limit in hand, it is important to understand how the definition acts under functional arithmetic. For instance, if f(x) = L and g(x) = M, then f(x) and g(x) can be made arbitrarily close to L and M, respectively, for all x sufficiently close but not equal to a. Hence, f(x)g(x) must be arbitrarily close to LM for all x sufficiently close but not equal to a. Therefore, it is reasonable to conjecture that the limit of the product f · g is the product LM of the limits. Indeed, this is true and can be proved using the formal definition of the limit. In fact, limits satisfy all the arithmetic properties that you would think they should, as summarized in the following limit laws.

The preceding example illustrates that applying the product and sum limit laws repeatedly allows us to quickly compute limits of polynomials and rational functions as x approaches a by evaluating them at the value a, provided a is in the domain.

Proof. We have previously shown that c = c (the limit of a constant is a constant) and x = a. By applying the limit law for products repeatedly, we have xⁿ = aⁿ for n = 1, 2, 3,.... Let p(x) = b₀ + b₁x + b₂x² + ... b_nxⁿ be a polyno mial. Then

Thus, we have shown p(x) = p(a) for any polynomial. You will be asked to prove the result for a rational function in problem 34 in Problem Set 2.3.

In Example 3b of Section 2.2, we computed a limit by “squeezing” a function for an unknown limit between functions whose limits we understand. The squeeze theorem provides a general approach for computing limits in this manner. This theorem, stated below, is sometimes called the sandwich theorem or pinching theorem.

Continuity at a point

Example 2 illustrated how easy it is to find limits f(x) when f is a polynomial or rational function and x = a is in the domain of f. One simply evaluates f at x = a. When one is able to evaluate a limit so easily, a function is continuous at x = a. The idea of continuity corresponds to the intuitive notion of a curve “without breaks or jumps”.

In general, continuity of a function f(x) at x = a can fail in three ways. First, the limit f(x) = L may be well defined, but either f(a) is not defined or f(a) ≠ L. When this occurs, f has a point discontinuity at x = a. These discontinuities can be repaired by redefining f at x = a by f(a) = L. For example, in Example 4a, we repaired a point discontinuity at x = 0. The second type of discontinuity is a jump discontinuity where f(x) = L and f(x) = M with L ≠ M. The left and right limits don't agree, there is no way to repair the discontinuity. Example 4a illustrates this type of discontinuity at x = 1. The final type of discontinuity is an essential discontinuity, which occurs when one or both of the one-sided limits, f(x) or f(x), do not exist or are infinite. In Example 3a and Figure 2.17 in Section 2.2, we showed that f(x) = x cos exhibits an essential discontinuity at x = 0.

With the concept of continuity, we are able to get a limit law for a composition of functions.

The next examples illustrate why g(x) existing does not suffice for g[f(x)] to exist.

Using the limit laws, we can derive some laws of continuity.

Proof. We will illustrate the proof of the continuity property for products. All other parts follow in a similar manner. Assume that f and g are continuous at a. Then f(x) = f(a) and g(x) = g(a). Hence

Therefore, fg is continuous at x = a.

Since we have shown that f(x) = f(a) for polynomial and rational functions at points in their domain, these functions are continuous at all points on their domain. As it turns out, this statement holds for all elementary functions.

Combining continuity theorems with the limit laws, we can compute limits that we could not otherwise find.

Intermediate Value Theorem

The function f is said to be continuous on the open interval (a, b) if it is continuous for each number (i.e., at each point) in this interval. Note that the end points are not part of open intervals. If f is also continuous from the right at a, we say it is continuous on the half-open interval[a, b). Similarly, f is continuous on the half-open interval (a, b] if it is continuous at each number between a and b and is continuous from the left at the end point b. Finally, f is continuous on the closed interval [a, b] if it is continuous at each number between a and b and is both continuous from the right at a and continuous from the left at b.

The graphs of functions that are continuous on an interval cannot have any breaks or gaps as shown by the following theorem.

This theorem says that if f is a continuous function on some closed interval [a, b], then f(x) must take on all values between f(a) and f(b). The intermediate value theorem is extremely useful in ensuring that we can solve certain nonlinear equations. Consider the following example.

While the intermediate value theorem allows us to prove when a function has a root, sometimes we need to solve for these roots numerically. One numerical approach is the bisection method.

Equations that take the form of finding the roots of polynomials, such as the problem in the previous example of finding the largest root of a fifth-order polynomial, are known as algebraic equations. Equations that involve exponential or trigonometric functions, such as x sin x − 1/2 = 0 or − π = 0, are known as transcendental equations and generally have no analytical solutions but need to be solved numerically.

Level 1 DRILL PROBLEMS

Determine the limits f(x), f(x) and f(x) in Problems 1 to 6. If they do not exist, discuss why.

3. f(x) = x/|x| with a = 0.

4. f(x) = x²/|x| with a = 0.

5. f(x) defined by the graph with a = 1.

6. f(x) defined by the graph with a = 0.

Find the limits in Problems 7 to 14. Justify each step with limit laws and the appropriate results from this section.

The graph of a function f is shown in Problems 15 to 18. Determine at what points f is not continuous and whether f can be redefined at these points to make it continuous. Explain briefly.

Each function in Problems 19 to 22 is defined for all x > 0, except at x = 2. In each case, find the value that should be assigned to f(2), if any, to guarantee that f will be continuous at 2. Explain briefly.

In Problems 23 to 28, use the intermediate value theorem to prove the following equations have at least one solution.

29. Use the bisection method explicated in Example 10 to find the following roots of the polynomial

to an accuracy of 3 decimal places.

1. smallest root
2. second smallest root
3. largest negative root
4. smallest positive root

In Problems 30 to 33, use the squeeze theorem, limit laws, and continuity of elementary functions to find the limit.

Level 2 APPLIED AND THEORY PROBLEMS

34. Prove that if p and q are polynomial functions with q(a) ≠ 0, then

35. Use limit laws to prove if f and g are continuous at x = a, then f + g is continuous at x = a.

36. Use limit laws to prove if f and g are continuous at x = a, then f − g is continuous at x = a.

37. Use limit laws to prove if f and g are continuous at x = a and g(a) ≠ 0, then is continuous at x = a.

38. Use limit laws to prove if f is continuous at x = a and g is continuous at x = f(a), then g f is continuous at x = a.

39. Why does the cubic equation x³ + ax² + bx + c = 0 have at least one root for any values of a, b, and c?

40. For any constants a, b, why does the equation xⁿ = a + b cos(x) always have at least one real root when n is an odd integer but not necessarily when n is an even integer?

41. Consider an organism that can move freely between two spatial locations. In one location, call it patch 1, the number of progeny produced per individual is

where N is the population size. In the other location, call it patch 2, the number of progeny produced per individual is always 5. Assume that all individuals in the population move to the patch that allows them to produce the greatest number of progeny. Let g(N) represent the number of progeny produced per individual for such a population.

1. Write an explicit expression for g(N) such that g(N) = f(N) whenever N < c for some constant c > 0 and g(N) = 5 whenever N ≥ c.
2. Determine the value of c that ensures g(N) is continuous for N ≥ 0.

42. As discussed in Example 11, recent research suggests that stabilizing carbon dioxide concentrations in the atmosphere at 450 parts per million (ppm) could limit global warming to 2°C. Use the bisection method and the carbon dioxide concentration model

where x is months after December 1973, to find the second time that this model predicts carbon dioxide levels of 450 ppm. Get a prediction that is accurate to 2 decimal places.

43. Use the bisection method to find the last time that the model in Problem 42 predicts carbon dioxide levels of 450 ppm. Get a prediction that is accurate to 2 decimal places.

44. Scientists believe that it will be extremely difficult to rein in carbon emissions enough to stabilize the atmospheric CO₂ concentration at 450 parts per million, as discussed in Example 11, and think that even 550 ppm will be a challenge. Use the bisection method to find the first time that the model in Problem 42 predicts carbon dioxide levels of 550 ppm. Get a prediction that is accurate to 2 decimal places.

45. Fisheries scientists often use data to establish a stock-recruitment relationship of the general form y = f(x), where x is the number of adult fish participating in the spawning process (i.e., the laying and fertilizing of eggs) that occurs on a seasonal basis each year and y is the number of young fish recruited to the fishery as a result of hatching from the eggs and surviving through to the life stage at which they become part of the fishery (i.e., available for harvesting). Two fisheries scientists found that the following stock-recruitment function provides a good fit to data pertaining to the southeast Alaska pink salmon fishery:

Use the bisection method to find the spawning stock level x that is expected to recruit 10,000 individuals to the fishery. (Hint: For the value of y in question, find the root of the equation y − f(x) = 0.) For more information on the research study, see T. J. Quinn and R. B. Deriso, Quantitative Fish Dynamics (New York: Oxford University Press, 1977).

46. Use the bisection method to find the spawning stock level x that is expected to recruit 20,000 individuals to the fishery modeled by the stock-recruitment function given in Problem 45.

47. Use the bisection method to find the spawning stock level x that is expected to recruit 5,000 individuals to the fishery modeled by the stock-recruitment function given in Problem 45.

2.4 Asymptotes and Infinity

In Chapter 1, we introduced the notion of infinity and represented it with the symbols ∞ and −∞. This symbol was used by the Romans to represent the number 1,000 (a big number to them). It was not until 1650, however, that it was first used by John Wallis (1616–1703) to represent an uncountably large number. From childhood many of us come to think of infinity as endlessness, which in a sense it is since infinity is a not a number. To mathematicians, however, infinity is a much more complicated idea than simply endlessness. As the famous mathematician David Hilbert (1862–1943) said, “The infinite! No other question has ever moved so profoundly the spirit of man” (in J. R. Newman, ed., The World of Mathematics, New York: Simon & Schuster, 1956, 1593).

In this section, we tackle limits involving the infinite in two ways. First, we determine under what conditions functions approach a limiting value as their argument becomes arbitrarily positive or arbitrarily negative. Second, we study functions whose value becomes arbitrarily large as their arguments approach a finite value where the function is not well defined.

Horizontal asymptotes

To understand the behavior of functions as their argument becomes very positive or negative (i.e., further from the origin in either direction), we introduce horizontal asymptotes.

Understanding the asymptotic behavior of a function can help us graph and interpret it. The next example involves dose–response curves that arise in many kinds of biological experiments. For example, the x axis of a dose–response curve may represent concentration of a drug or hormone delivered at various “doses.” The y axis represents the response, which could be many things, depending on the experiment. For example, the response might be the activity of an enzyme, accumulation of an intracellular messenger, voltage drop across a cell membrane, secretion of a hormone, increase in heart rate, or contraction of a muscle. In the next example, dose is the amount of a histamine (measured in millimoles) administered to a patient, and response is the percentage of patients exhibiting above-normal temperatures.

Limits at infinity can be very nonintuitive, as the following example shows.

Vertical asymptotes

Many functions, such as rational functions, logarithms, and certain power functions, are not defined at isolated values. As the argument of the function gets close to these isolated values, the function may become arbitrarily large or arbitrarily negative and exhibit a vertical asymptote.

Combining the information about horizontal and vertical asymptotes can provide a relatively complete sense of the graph of a function, as illustrated in the next example. This example, described by Francois Messier (“Ungulate Population Models with Predation: A Case Study with the North American Moose.” Ecology 75 (1994): 478–488), examines wolf-moose interactions over a broad spectrum of moose densities throughout North America. One of his primary objectives was to determine how the predation rate of moose by wolves depends on moose density. Messier found that a Michaelis-Menton function provides a good fit to moose-killing rates by wolves as a function of moose density. Recall that we encountered the Michaelis-Menton function in Example 6 of Section 1.5 for modeling bacterial nutrient-uptake rates.

Infinite limits at infinity

As x gets larger and larger without bound, the value of f might also get larger and larger without bound. In such a case, it is natural to say that f(x) approaches infinity as x approaches infinity.

Part b of the previous example and Example 3 illustrate the subtlety of taking the limits of the form f(x) − g(x) when f(x) = ∞ and g(x) = ∞. In the previous example with f(x) = x and g(x) = x², the limit of the difference was −∞. In Example 3 with f(x) = and g(x) = x, the limit of the difference was 0. The following example illustrates that any limiting value is possible.

Level 1 DRILL PROBLEMS

In Problems 1 to 24, find the specified limits.

For in Problems 25 to 30, determine how close x > a needs to be to a to ensure that f(x) ≥ 1,000,000.

For f(x) = L in Problems 31 to 34, determine how negative x needs to be to ensure that |f(x) − L| ≤ 0.05.

For the limit f(x) = ∞ in Problems 35 to 38, determine how large x needs to be to ensure that f(x) > 1,000,000.

Level 2 APPLIED AND THEORY PROBLEMS

39. In Example 8, we showed that N(t)= ∞ whereN(t) = 8.3(1.33)^t represents U.S. population size in millions t decades after 1815. To see that N(t) can get arbitrarily large for x sufficiently large, do the following:

1. Determine how large t needs to be to ensure that N (t) ≥ 500,000,000.
2. Determine how large t needs to be to ensure that N(t) ≥ 1,000,000,000.

40. In Example 5 of Section 1.4, we modeled the height of beer froth with the function H(t) = 17(0.99588)^tcm where t is measured in seconds.

1. Determine L such that H(t) = L.
2. Determine how large t needs to be to ensure that H(t) is within 0.1 of L.
3. Determine how large t needs to be to ensure that H(t) is within 0.01 of L.

41. In Example 6 of Section 1.5, we modeled the uptake rate of glucose by bacterial populations with the function f(x) = mg per hour where x is measured in milligrams per liter.

1. Find the horizontal and vertical asymptotes of f(x). Interpret the horizontal asymptote(s).
2. Graph f(x) for all values of x.

42. In Example 5, we examined how the predation rate of wolves depended on moose density. Messier also studied how wolf densities in North America depend on moose densities. He found that the following function provides a good fit to the data:

where x is number of moose per square kilometer.

1. Find the horizontal and vertical asymptotes of f(x). Interpret the horizontal asymptotes.
2. Graph f(x) for all values of x > 0.

43. In Problem 42, you were asked to find L such that f(x) = L.

1. Determine how large x needs to be to ensure that f(x) is within 0.1 of L.
2. Determine how large x needs to be to ensure that f(x) is within 0.01 of L.

44. The von Bertalanffy growth curve is used to describe how the size L(usually in terms of length) of an animal changes with time. The curve is given by

where t measures time after birth and a, b, and t₀ are positive parameters. We will derive this curve in Chapter 6. To better understand the meaning of the parameters t₀ and b, carry out these steps.

1. Evaluate L(t₀). What does this imply about the meaning of t₀?
2. Find L(t). What does this limit say about the biological meaning of a?
3. Graph L(t) and discuss how an organism grows according to this curve.

45. At the beginning of the twentieth century, several notable biologists, including G. F. Gause and T. Carlson, studied the population dynamics of yeast. For example, Carlson grew yeast under constant environmental conditions in a flask; he regularly monitored their population densities.* In Chapter 6, we will show that the following function describes the growth of the population:

where N(t) is the population density and t is time in hours. Find N(t) and discuss the meaning of this limit.

46. The following equation is used to calculate the average firing rate f of a neuron (in spikes per second) as a function of the concentration x of neurotransmitters perfusing its synapses.

Find the horizontal asymptote; then find the values of x such that f(x) is within 0.5% of its asymptotic values.

47. The following equation is used to calculate the average firing rate f of a neuron (in spikes per second) as a function of the concentration x of neurotransmitters perfusing its synapses.

Find the values of x such that f(x) is within 0.5% of its asymptotic values.

48. Compare the solutions obtained to Problems 46 and 47 and decide which of these represents a tighter on-off switch of the neuron, that is, from firing at its maximum rate to being inactive. What do you conclude in terms of which of the parameters a, b, and c in the function

controls the narrowness of the range of x over which on-off switching occurs? Note that this function is called the logistic function and will be encountered in many different examples in upcoming chapters.

2.5 Sequential Limits

In Section 1.7, we considered sequences a₁, a₂,... of real numbers, which can be used to model drug concentrations, population dynamics, and population genetics. In some cases, these sequences converged to a limiting value as n got very large. In this section, we study the limits of sequences, their relationship to continuity, and a convergence theorem, as well as how these concepts can be used to understand the asymptotic behavior of difference equations. While limits of functions form the basis of differentiation as we shall soon see, limits of sequences form the basis of integration (as we will discuss in Chapter 5).

Sequential limits and continuity

For sequences, there is only one type of limit to consider: the sequential limit, defined as the limiting value of a_n as n → ∞.

As in our previous limit definitions, the existence of a sequential limit implies that we can make a_n as close to L as we like, provided that n is sufficiently large. But how do we verify this statement? What is meant by sufficiently large? The following example illustrates the answer to this question.

There is an important relationship between limits of sequences and limits of functions. This relationship is most useful for proving discontinuity of a function.

One direction of this theorem is clear. If a_n = a, then a_n can be made arbitrarily close to a for n sufficiently large. Therefore, if f(x) = L, then f(a_n) is arbitrarily close to L for n sufficiently large. Hence, if f(x) = L, then f(a_n) = L. If you are feeling sufficiently adventuresome, try proving this direction using formal definitions of limits. To do so, you will have to come up with a formal definition of sequential limits. The other direction of the sequential continuity theorem is more subtle, and the ideas of the proof are beyond the scope of this text.

Asymptotic behavior of difference equations

When we introduced sequences in Section 1.7, we considered a special class of sequences that arises through a difference equation

where a₁ is specified and f is a function. In some instances, we can actually find explicit expressions for the sequence defined by the difference equation and take the limit.

In part b of Example 4, we saw that sometimes it is useful to find a_n by finding f(a_n) for an appropriate choice of a continuous one-to-one function f.

Returning to our a_n notation, recall from Section 1.7, that a point a is an equilibrium of a difference equation a_n+1 = f(a_n) if f(a) = a. In Example 4a and Example 5, the only solution to the equation f(a) = a is a = 0, and the sequences generated by these difference equations converge to this equilibrium. In Example 4b, the equation = a has two solutions: a = 0 and a = 1, and the sequence we examined converged to the latter rather than former equilibrium.

To see why this convergence to equilibria occurs, consider a sequence a_n that satisfies the difference equation

where f is a continuous function. Assume that a_n = a. By the sequential continuity theorem, we have

Hence, the limiting value a is an equilibrium for this difference equation.

One of the most important models in population biology is the discrete logistic model:

where the parameter r > 0 is called the intrinsic rate of growth and K > 0 is called the environmental carrying capacity.

Examples 6 and 7 illustrate that the existence of equilibria for a difference equation does not ensure the convergence of the sequences generated by it. This raises the question, when do the sequences generated by a difference equation converge to an equilibrium? In general, this is a hard question. The following theorem, stated without proof, provides a criterion that ensures convergence of solutions of a difference equation. Later, when we have covered the basics of derivatives of functions, we will present another criterion that ensures convergence of a sequence to an equilibrium. We make two definitions before stating this theorem. We say that a sequence is increasing (respectively, decreasing) if a₁ ≤ a₂ ≤ a₃ ≤ ... (respectively, a₁ ≥ a₂ ≥ a₃...).

The next example illustrates the application of the monotone convergence theorem to the Beverton-Holt stock-recruitment model. This model has been used extensively by fisheries scientists to formulate models of fish, such as Pacific salmon, that breed once and then die.Stock-recruitment curves describe how a spawning stock of N individuals in one generation contributes recruits R (i.e., new individuals) to the next generation. An example of a Beverton-Holt function fitted to data obtained for sockeye salmon spawning in Karluk Lake, Alaska, is shown in Figure 2.43.

Figure 2.43 Sockeye salmon (Oncorhynchus nerka) on the left; and, on the right, the relationship between recruits and stock for sockeye salmon in Karluk Lake, Alaska.

Data Source: John A. Gulland, Fish Stock Assessment: A Manual of Basic Methods (New York: Wiley, 1983).

In Example 5 of Section 1.7, we developed a population genetics model under the assumption that individuals could carry and pass on a recessive lethal allele. In formulating this model, we assumed that two alleles, A and a, exist and determine three possible genotypes: AA, Aa, and aa. If the allele a is the recessive lethal, then this implies that genotype aa is the least viable since, by definition, aa individuals all die before they are able to reproduce. In the next example, we assume that all three genotypes are viable, but that the so-called heterozygous genotype Aa is the least viable. Extreme instances of heterozygous inviability arise when different genotypes can mate but produce infertile offspring, such as mules, which are produced when horses mate with donkeys.

Fibonacci famously posed the following problem (reworded here) over 800 years ago. Suppose a newly born pair of rabbits, one male and one female, are put in an enclosed, but very large, field. Further, suppose all rabbits are able to mate at the age of one month and that the impregnated female gives birth to a male-female pair one month later. Ignoring questions relating to inbreeding, assuming that rabbits never die and there is always enough food for them to eat, what is the asymptotic annual rate of increase of the rabbits in the enclosed field? This is the problem we address in the next example.

Level 1 DRILL PROBLEMS

Determine whether the sequential limits in Problems 1 to 8 exist. If they exist, find the limit. If they do not exist, explain briefly why.

Consider the sequences defined in Problems 9 to 14.

1. Find a_n.
2. Determine how large n needs to be to ensure that |a_n − L| < 0.001.

All the sequences in Problems 15 to 18 satisfy a_n = ∞. Determine how large n has to be to ensure that a_n ≥ 1,000,000.

15. a_n = 2n

16. a_n = n²

17. a_n = 2ⁿ − 10,000

18. a_n =

Find the sequences determined by the difference equation x_n+1 = f(x_n) with the initial condition x₁ specified in Problems 19 to 24. Determine a_n. Justify your answer.

19. f(x) = x + 2 with x₁ = 0

20. f(x) = with x₁ = 27

21. f(x) = with x₁ = 100

22. f(x) = x² with x₁ = 1.00001

23. f(x) = x² with x₁ = 0.99999

24. f(x) = 4x² with x₁ = 1

Find the equilibrium of the difference equations in Problems 25 to 28 and use technology to determine which of the specified sequences converge to one of the equilibria.

Use the monotone convergence theorem in Problems 29 to 34 to determine the limits of the following specified sequences.

Level 2 APPLIED AND THEORY PROBLEMS

35. In Example 5 of Section 2.5 we introduced a model for the frequency of lethal recessive alleles in a population. The model is given by

where x_n is the frequency of the recessive allele in the population.

1. If x₁ = 0.25, then verify that x_n = satisfies the difference equation.
2. Find x_n.
3. Determine how large n needs to be to ensure that x_n ≤ 0.1.
4. Determine how large n needs to be to ensure that x_n ≤ 0.001.

36. Let us consider the lethal recessive allele model in greater generality.

1. Verify that x_n = satisfies the difference equation x_n+1 = for any choice of x₁.
2. For any x₁, find x_n.
3. Assuming that x₁ lies in (0, 1), determine how large n needs to be to ensure that x_n ≤ 0.01.

37. In Example 5 of Section 1.7, we discussed a population genetics model under the assumption that there were two alleles, A and a, that determined three possible genetic types, AA, Aa, and aa. We assume that genetic type Aa(so called heterozygote) is the least viable. If the genotype aa produces nine times more progeny than genotype AA progeny, then the frequency x_n of allele a at time n can be modeled by x_n+1 = f(x_n) where the graph of f is given by

1. Determine what happens to x_n in the long term if x₁ = 0.91.
2. Determine what happens to x_n in the long term if x₁ = 0.89.
3. As reported in a 1972 Science article, Foster and others experimentally examined changes in two chromosomal frequencies in Drosophila melanogaster. Data from a set of experiments are graphed next:

   

   These experimentally determined graphs show how the frequency of an allele changes over generations for different initial conditions. Discuss whether these experiments are consistent with the model predictions.

38. The Beverton-Holt model has been used extensively by fisheries. This model assumes that populations are competing for a single limiting resource and reproduce at discrete moments in time. If we let N_n denote the population abundance in the nth year (or generation), r > 0 the maximal per capita growth rate, and a > 0 as a competition coefficient, then the model is given by

with r > 0 and a > 0.

1. Assume N₁ = 1 and find the first four terms of the sequence.
2. Guess the explicit expression for the sequence and verify that your guess is correct.
3. Find the equilibria of this difference equation.
4. By considering the N_n, determine under what conditions the population goes extinct (i.e., converges to 0) versus under what conditions it is able to persist (i.e., converge to a positive equilibrium N > 0) if it is initially at N₁ = 1.

In the Fibonacci rabbit problem laid out in Example 10, suppose only a proportion p of the females that could become pregnant actually do become pregnant each month. (We assume the population starts with a large number of pairs so that when we refer to proportions, we are actually thinking of whole numbers of pairs rather than, say one-third of two pairs, which makes no biological sense.) What is the annual rate of increase in Problems 39 to 43 if p equals the specified value?

39. 3/4

40. 2/3

41. 1/2

42. 1/3

43. 1/4

44. Consider the Fibonacci sequence in Example 10. Show that

and hence apply the monotone convergence theorem to find out what happens to the sequences of even elements of a_n.

45. Use technology to calculate the first twenty points of the sequence a_n+1 = a_n + ra_n(1 − a_n) with a₁ = 0.5 for the cases r = 0.9, r = 1.5 and r = 2.1. How does this fit in with the discussion in the solution to part e of Example 7.

46. Revisiting the sockeye salmon stock-recruitment relationship considered in Example 8, we see from Figure 2.43 that we could just as credibly fit the Ricker function

If x is the stock in one generation and y is the stock recruited in the next generation, then this relationship is actually a population model of the form

If the population is now at the level x₁ = 100 individuals, use technology to generate the number of individuals that you expect in the next ten generations. Deduce the equilibrium value and check this value by using technology or the bisection method to solve the equation x = 3.7xe^−0.01x. In Figure 2.47, the solid line is fit to the data using a Ricker functional form that is similar to the Beverton-Holt form considered in Example 8.

Figure 2.47 Sockeye salmon stock-recruitment.

Data Source: John A. Gulland, 1983.Fish Stock Assessment: A Manual of Basic Methods, Wiley.

2.6 Derivative at a Point

In this section, we introduce one of the major concepts in calculus, the idea of a derivative. Although functions are fundamental and limits are essential, they simply laid the foundation for the excitement to come. To motivate the idea of the derivative at a point, let us recast Example 1 of Section 2.1 using different notation. If we let f(x) represent the population of Mexico in year x, then the average population between the years 1980 and 1985 can be found by

where a = 1980 (the base year) and h = 5 (the duration of the time interval). We also found the average rate of change over smaller and smaller intervals to guess that the instantaneous rate of change of the Mexico population in 1980 would be 1.75 million per year. This idea can be written as

In Chapter 1 and Section 2.1, we previewed the notion of the derivative at a point by defining the tangent line for f(x) at a point x = a to be the limit of the slope of the secant lines

Slopes of tangent lines and instantaneous rates of changes have the same formula, and it is this limiting process that is the basis for the concept of the derivative.

Example 1 illustrates two facts. First, the derivative of a constant function is zero. Intuitively this makes sense, as a constant function by definition does not change and, consequently, its rate of change should be zero. Second, the derivative of a linear function is the slope of the linear function. Intuitively, this makes sense as the rate at which the function is increasing is given by the slope of the function. More interestingly, Example 1 illustrates that we can explicitly compute the slope (equivalently, the instantaneous rate of change) of a quadratic function.

The derivatives in Example 1 were pretty straightforward to compute, but other derivatives require certain algebraic procedures to compute, as illustrated by the next example.

Slopes of tangent lines

The definition of the derivative was inspired directly by the slope of the tangent line. Using derivatives, we can find the tangent line.

In Example 3, the tangent line intersects the graph of the function in exactly one point. This unique intersection is not typical, as illustrated in the next example.

Instantaneous rates of change

In Section 2.1, we defined

to be the average rate of change of f over the interval [a, b]. Taking the limit as b approaches a yields the instantaneous rate of change

In the next example, we relate this definition of the instantaneous rate of change to the derivative.

The solution to Example 5 allows us to equate the derivative with an instantaneous rate of change.

Differentiability and continuity

If a function f(x) is differentiable at a point a, then it is also continuous at a point a, as stated in the following theorem.

The reverse of this theorem, however, is not true. To fully appreciate differentiability, it is useful to understand examples of where continuity holds but the function is not differentiable.

A biological model where we have continuity but nondifferentiability at a point is illustrated in the following example.

Example 9 illustrates that continuity does not ensure differentiability and that differentiability can fail in at least two ways. The limit of the slopes of the secant lines might not converge or this limit may become infinitely large. While continuity does not imply differentiability, Theorem 2.6 ensures that the opposite is true; that is, differentiability ensures continuity. Hence, differentiability can be viewed as an improvement over continuity.

Level 1 DRILL PROBLEMS

Using the definition of a derivative, find the derivatives specified in Problems 1 to 10.

1. f′(−2) where f(x) = 3x − 2

2. f′(3) where f(x) = 5 − 2x

3. f′(1) where f(x) = −x²

4. f′ (0) where f (x) = x + x²

Find the tangent line at the specified point and graph the tangent line and the corresponding function in Problems 11 to 20. Notice these functions are the same as those given in Problems 1 to 10.

11. f(x) = 3 x − 2

12. f(x) = 5 − 2x at x = 3

13. f(x) = −x² at x = 1

14. f(x) = x + x² at x = 0

17. f(x) = x³ at x = −1

18. f(x) = x³ + 1 at x = 2

19. f(x) = at x = 9

20. f(x) = at x = 5

Determine at which values of x in Problems 21 to 26 that f is not differentiable. Explain briefly.

25. f(x) = |x − 2|

26. f(x) = 2|x + 1|

27. Let f(x) =

1. Sketch the graph of f.
2. Show that f is continuous, but not differentiable, at x = 1.

28. Give an example of a function that is continuous on (−∞, ∞) but is not differentiable at x = 5.

29. Consider the function defined by

Sketch this graph and find all points where the graph is continuous but not differentiable.

30. Consider the function defined by

Sketch this graph and find all points where the graph is continuous but not differentiable.

Level 2 APPLIED AND THEORY PROBLEMS

31. A baseball is thrown upward and its height at time t in seconds is given by

1. Find the velocity of the baseball after two seconds.
2. Find the time at which the baseball hits the ground.
3. Find the velocity of the baseball when it hits the ground.

32. A ball is thrown directly upward from the edge of a cliff and travels in such a way that t seconds later, its height above the ground at the base of the cliff is

feet.

1. Find the velocity of the ball after two seconds.
2. When does the ball hit the ground, and what is its impact velocity?
3. When does the ball have a velocity of zero? What physical interpretation should be given to this time?

33. If the data in Figure 2.53 represents a set of measurements relating enzyme activity to temperature in degrees Celsius, and the quadratic equation

provides a good fit to this data, then find A′(50) and discuss its meaning.

Figure 2.53 Enzyme activity as a function of temperature.

34. In Example 7 we discussed how the extent of Arctic sea ice is modeled by the function

where t is years after 1980. Use the definition of the derivative to compute S′(20). How does this value compare to S′(30) found in Example 7? What does this comparison suggest about the rate at which arctic ice is being lost?

35. In 2010, W. B. Grant published an article about the prevalence of multiple sclerosis in three U.S. communities and the role of vitamin D. The best-fitting quadratic relationship to the data published in this article, relating prevalence of multiple sclerosis (MS) to latitude (exposure to sunlight, hence vitamin D synthesis decreases with latitude) is shown in Figure 2.54. This quadratic function is given by

where x is latitude.

Figure 2.54 Prevalence of MS (in cases per 100,000) as a function of latitude in the United States.

Data Source: W. B. Grant, “The Prevalence of Multiple Sclerosis in 3 US Communities: The Role of Vitamin D” [letter].Prevention of Chronic Diseases 7 (2010): A89.

1. Find P′(40).
2. Find P′(45).
3. Interpret and compare the numbers that you found in a and b.

36. An environmental study of a certain suburban community suggests that t years from now, the average level of carbon monoxide in the air can be modeled by the formula

parts per million.

1. At what rate will the carbon monoxide level be changing with respect to time one year from now?
2. By how much will the carbon monoxide level change during the first year?

37. Perelson and colleagues studied the viral load of HIV patients during antiviral drug treatment.* They estimated the viral load of the typical patient to be

particles per milliliter on day t after the drug treatment.

1. Estimate V′(2).
2. Describe the units of V′(2) and interpret this quantity.

38. Stock-recruitment data and a fitted Beverton-Holt function for sockeye salmon in Karluk Lake, Alaska, were was shown in Figure 2.43. The fitted function was

where x is the current stock size and y is the number of recruits for the next year. To determine the number of recruits produced per individual, consider the function

1. Algebraically find g′(10).
2. Describe the units of g′(10), and discuss the meaning of this quantity.

39. In Example 6 of Section 1.5, we developed the Michaelis-Menton model for the rate at which an organism consumes its resource. For bacterial populations in the ocean, this model was given by

where x is the concentration of glucose (micrograms per liter) in the environment. To determine the rate of glucose consumption per microgram of glucose in the environment, consider the function

1. Algebraically compute g′(0) and g′(20).
2. Describe the meaning of the derivatives that you computed.

40. In Example 5 of Section 2.4, we found that the rate at which wolves kill moose can be modeled by

where x is measured in number of moose per square kilometer. To determine the per capita killing rate of moose, consider the function

1. Algebraically compute g′(1) and g′(2).
2. Describe the meaning of the derivatives that you computed.

2.7 Derivatives as Functions

Our notion f′(a) for the derivative at the point x = a suggests that we can think of f′ as a function. Indeed this is true.

When given numerical data, we can estimate the derivative of the data using the definition of the derivative with the smallest possible h value.

The solution to part b of Example 2 suggests that whenever P(t) has the general exponential form P(t) = ab^t, then the derivative has the same form but differing by some constant; that is, P′(t) = cP(t), where for Example 3 we obtained c = 0.026. This equation is an example of what is known as a differential equation, as it relates a function to its derivative. You will learn more about differential equations in Chapters 6 and 8. In Chapter 3, we will verify that the derivative of an exponential function is a constant multiple of the exponential function.

Notational alternatives

The primed-function notation f′ that we have been using to denote derivatives is but one of several used in various texts. Newton used a “dot” notation, which we will not consider here. The notation that mathematicians prefer was developed by Leibniz. This notation is inspired by the following presentation of the derivative.

Let Δx represent a small change in x. The change of y = f(x) over the interval [x, x + Δx] is given by

The average rate of change of y = f(x) over the interval [x, x + Δx] is given by

Hence, the derivative of f at x is

Leibniz represented this limit as

where in some sense dy corresponds to an “infinitesimal” change in y and dx represents an “infinitesimal” change in x. Commonly used variations in notation that you will find in this and other calculus texts include

To indicate the derivative at the point x = a using Leibniz notation, we use the cumbersome expression

The next example draws upon research undertaken by ecologist Nathan Sanders and colleagues, who assessed the number of local ant species along an elevational gradient, Kyle Canyon, in the Spring Mountains of Nevada to obtain a measure called species richness. These data, illustrated in Figure 2.58, are plotted in terms of number of species of ants as a function of elevation (in kilometers).

Mean Value Theorem

To understand what the derivative tells us about the shape of a function, we need the mean value theorem (MVT). The proof of this theorem is given as a series of challenging exercises in Problem set 2.7.

It is worth noting from Figure 2.61 that E′(t) is above the average rate of energy intake for t < 20 and below the average for t > 20. Hence, as we explore in more detail in Chapter 4, the hummingbird may consider leaving the patch after 20 seconds.

Derivatives and graphs

Using the mean value theorem, we can prove the following two facts about the relationship of the sign of the derivative f′ to the graph of y = f(x).

To prove these properties, assume that f′ > 0 on (a, b). Take any two points x₂ > x₁ in the interval (a, b). By the mean value theorem, there exists a point c in the interval [x₁, x₂] such that

Since f′(c) > 0, we have

Since x₂ − x₁ > 0, we have f(x₂) − f(x₁) > 0. Equivalently, f(x₂) > f(x₁). Therefore, f is increasing on the interval [a, b]. The case of f′ < 0 on [a, b] can be proved similarly, and it appears as an exercise in Problem Set 2.7.

For a function y = f(x), a turning point is an x value where the function switches from increasing to decreasing, or vice versa. More precisely, if f is continuous on (a, b), then c in (a, b) is a turning point provided that either (i) f is increasing on (a, c) and decreasing on (c, b), or (ii) f is decreasing on (a, c) and increasing on (c, b). When f is differentiable on (a, b), turning points correspond to where the derivative f′ changes sign. For example, if f′(x) > 0 on (a, c) and f′ (x) < 0 on (c, b), then c is a turning point as the function switches from increasing to decreasing at x = c.

Figure 2.62 Graph of y = f(x).

Level 1 DRILL PROBLEMS

Use the definition of a derivative to find f′(x) for the functions in Problems 1 to 8.

Use the derivatives found in Problems 1 to 8 to find the values requested in Problems 9 to 16.

Find at what point the slope of the instantaneous rate of change equals the average rate of change over the specified intervals in Problems 17 to 22. Also, provide a sketch that illustrates this relationship.

17. f(x) = 8 over the interval [−5, 5]

18. f(x) = 3x − 2 over the interval [3, 4]

19. f(x) = −x² over the interval [−1, 1]

20. f(x) = x + x² over the interval [0, 1]

21. f(x) = over the interval [1, 2]

22. f(x) = over the interval [1, 4].

Mix and match the graphs in Problems 23 to 28 with the graphs labeled (A) to (F), which are the derivative graphs.

For each of the functions given in Problems 29 to 34, find intervals for which f is increasing and intervals for which f is decreasing.

29. f(x) = x² − x + 1

30. f(x) = 5 − x²

31. f(x) = x³ + x

32. f(x) = 8 − x³

33. Let f be the function for which the graph of the derivative y = f′(x) = g(x) is given by

34. Let f be the function for which the graph of the derivative y = f′(x) = g(x) is given by

35. For the graph y = g(x) given in Problem 33, estimate all values of c in [−3, 0] such that

36. For the graph y = g(x) given in Problem 34, estimate all values of c in [−2, 2] such that

In each of Problems 37 to 40, the graph of a function f′ is given. Draw a possible graph of f.

Level 2 APPLIED AND THEORY PROBLEMS

41. Let f be differentiable on the interval (a, b). Use the mean value theorem to prove if f′ < 0 on [a, b], then f is decreasing on (a, b).

42. Rolle's theorem: Let f be differentiable on (a, b) and continuous on [a, b]. Assume f(a) = f(b) = 0. Without using the mean value theorem, argue that there exists a c in (a, b) such that f′(c) = 0.

43. Use Rolle's theorem to prove the mean value theorem.

44. A baseball is hit upward and its height at time t in seconds is given by

1. Find the velocity of the baseball after t seconds.
2. Find the time at which the velocity of the ball is 0.
3. Find the height of the ball at which the velocity is 0.

45. To study the response of nerve fibers to a stimulus, a biologist models the sensitivity, S, of a particular group of fibers by the function

where t is the number of days since the excitation began.

1. Over what time period is sensitivity increasing? When is it decreasing?
2. Graph S′(t).

46. During the time period 1905–1940, hunters virtually wiped out all large predators on the Kaibab Plateau near the Grand Canyon in northern Arizona. The data for the deer population, P, over this period of time are as follows:

1. Estimate P′(t) for 1905 ≤ t ≤ 1939.
2. Graph and interpret P′(t).

47. In 1913, Carlson studied a growing culture of yeast (see Problem 45 in Problem Set 2.4 and Section 6.1). The table of population densities N(t) at one-hour intervals is shown here:

1. Estimate N′(t) for 0 ≤ t ≤ 17.
2. Graph N′(t) and briefly interpret this graph.

48. Our ruby-throated hummingbird has entered another patch of flowers, and the energy she is getting as a function of time in the patch is plotted below.

1. Find the average energy intake rate.
2. Estimate at what time the instantaneous energy intake rate equals the average intake rate.

49. Two radar patrol cars are located at fixed positions 6 miles apart on a long, straight road where the speed limit is 65 miles per hour. A sports car passes the first patrol car traveling at 60 miles per hour; then 5 minutes later, it passes the second patrol car going 65 miles per hour. Analyze this situation to show that at some time between the two clockings, the sports car exceeded the speed limit. Hint: Use the MVT.

50. Suppose two race cars begin at the same time and finish at the same time. Analyze this situation to show that at some point in the race they had the same speed.

1. Find at what point the instantaneous rate of change of the function f(x) = x³ − x equals the average rate of change over the interval [−1, 2]. Provide a sketch that illustrates this relationship.
2. In the 2012 Summer Olympics in London, swimmer Ye Shiwen swam the last 50 meters of her 400-meter individual medley final quicker than the winner of the men's race, Ryan Lochte. The 50-meter split times for both swimmers are shown in the following table:

   

   1. Find the average velocity of both swimmers in the last split of the race.
   2. Find the average velocity of both swimmers over the entire race.
3. Use the definition of derivative to find f′(x) for f(x) = .
4. Find the average rate of change of f(x) = x² − 2x + 1 on [1, 3] and the instantaneous rate of change at x = 1.
5. Consider f(x) = 9 − x² and g(x) = ln x.
   1. Graph y = f(x) and y = g(x) on the same coordinate axes.
   2. Plot the point P(2, ln 2) on the graph of g. Graphically, estimate the position of the line tangent to g at the point P.
   3. Plot the point Q(2, 5) on the graph of f. Algebraically find the line tangent to f at the point Q. Use the equation of this tangent line to show that it “kisses” the graph of f at the point Q.
6. Find with the suggested methods.
   1. Graphically
   2. Using a table of values
   3. By algebraic simplification
   4. By using the informal definition of limit
   5. Using technology
7. Let f(x) =

   Find the requested limits.

   

8. Evaluate the sequential limits, if they exist.

   

9. The graph of a function f′ is given. Draw a possible graph of f.

   

10. An environmental study of a certain suburban community suggests that t years from now, the average level of carbon dioxide in the air can be modeled by the formula

    

    parts per million.

    1. At what rate will the CO₂ level be changing with respect to time one year from now?
    2. By how much will the CO₂ level change in the first year?
    3. By how much will the CO₂ level change over the next (second) year?
11. Consider the function y =
    1. Find the horizontal asymptote(s) of this function.
    2. Find the right-hand side and left-hand side limits of this function at all of its vertical asymptotes.
12. The canopy height (in meters) of a tropical elephant grass (Pennisetum purpureum) is modeled by

    

    where t is the number of days after mowing.

    1. Sketch the graph of h(t).
    2. Sketch the graph of h′(t).
    3. Approximately when was the canopy height growing most rapidly? Least rapidly?
13. The concentration C(t) of a drug in a patient's blood stream is given by

    

    1. Estimate C′(t) for t = 0, 0.1,... 0.9.
    2. Sketch C′(t) and interpret it.
14. Suppose that systolic blood pressure of a patient t years old is modeled by

    

    for 0 ≤ t ≤ 65, where P(t) is measured in millimeters of mercury.

    1. Sketch the graph of y = P(t).
    2. Using the graph in part a, sketch the graph of y = P′(t).
15. Find the limit and determine how positive x needs to be to ensure that is within one millionth of the limiting value L.
16. Whales have difficulty finding mates in the vast oceans of the world when their population numbers drop below a critical value. Thus, a model of the growth of whale populations from one whale generation to the next is going to be relatively more robust at intermediate whale densities than at low densities when finding mates is a problem, or at high densities when competition for food is a problem. The form of a hypothetical function f in the difference equation

    

    that reflects the above properties is illustrated in Figure 2.65, where a_n is the density of the whales in generation n(units are whales per 1000 sq km).

    

    Figure 2.65 A function, f, modeling the growth of a hypothetical whale population.

    Determine a_n when a₁ = 55. Justify your answer.

17. Give a proof that the function f(x) = xe^−x − 0.1 has at least one positive root.
18. Sketch a graph of a function y = f(x) on the interval [−2, 2] such that f(−2) = f(0) = f(2) = 0, > 0 on [−2, −1) and (1, 2], and < 0 on (−1, 1).
19. The average population growth rate of Mexico from 1981 to 1983 was 1.82 million per year and from 1983 to 1985 was 1.915 million per year. Assume the population size N(t) as a function of time and N′(t) are continuous. Prove that at some point in time between 1981 and 1985, the instantaneous rate of population growth was 1.85 million per year.
20. Consider the graph of f shown in Figure 2.66. On the interval [−3, 4] find the points of discontinuity as well as places where the derivative does not exist. Explain your reasoning.

    

    Figure 2.66 A function, f, on the interval [−3, 4].