Figure 3.1 Differential calculus is used in Section 3.1 to model the rate at which HIV virions (as depicted) is cleared from the body during drug therapy.

Preview

In the previous chapters we have only been able to compute derivatives by directly appealing to the definition of the derivative. As you may have noticed, computing derivatives in this manner quickly becomes tedious. In this chapter we consider the rules and tools that allow us to quickly compute the derivative of any imaginable function. Learning these rules is critical, as expressed in the following admonition (from Colin Adams, Joel Hass, and Abigail Thompson, How to Ace Calculus: The Streetwise Guide, New York: W. H. Freeman, 1998):

“Know these backwards and forwards. They are to calculus what ‘Don't go through a red light’ and ‘Don't run over a pedestrian’ are to driving.”

The first three sections of this chapter provide essential basic rules for calculating derivatives, and the fourth section focuses on the important trigonometric functions. The last three sections expand these tools so that we can apply them to a variety of applications in biology and the life sciences. Applications in this chapter include predicting the growth of a fetal heart and of the Yellowstone bison population. Also, we use differential calculus to investigate the clearance rate of HIV viral particles (see Figure 3.1) from infected humans, how Northwestern crows break whelk shells, dose-response curves in the context of administering drugs and into rates of mortality due to airborne diseases, and Usain Bolt's record-breaking, 100-meter run in the Olympic Games in Beijing.

3.1 Derivatives of Polynomials and Exponentials

As we have seen in Chapters 1 and 2, we can use polynomial and exponential functions to model natural phenomena ranging from the melting of Arctic sea ice to the decay of a drug in the body. To facilitate finding the instantaneous rates of change of these processes, we now derive general rules for computing the derivatives of polynomials and exponentials.

Derivatives of y = xⁿ

In Example 1 of Section 2.7, we proved that

and we guessed that = nxⁿ⁻¹. This powerful result is known as the power rule.

At this point, we are only equipped to prove the power rule for any natural number n. Later, we shall prove the general power rule. The proof when n is a natural number involves the binomial expansion of (a + b)ⁿ (if you don't remember this expansion, look it up on the World Wide Web):

Now to the proof of the power rule for n a natural number:

Proof. If f(x) = xⁿ, then from the binomial theorem

From the definition of derivative we have

Note that if n = 0, then f(x) = xⁿ = x⁰ = 1, so f′(x) = 0 as expected, since 1 is a constant.

Derivatives of sums, differences, and scalar multiples

The limit laws from Chapter 2 allow us to quickly compute the derivatives of a sum, difference, or scalar multiple whenever we know the derivatives for f and g. In stating these laws, it is more succinct to use the “prime” rather than full Leibnitz notation.

In other words, the derivative of a sum is the sum of the derivatives, the derivative of a difference is the difference of the derivatives, and the derivative of a scalar multiple is the scalar multiple of the derivative.

Combining these elementary differentiation rules with the power rule allows us to differentiate any polynomial. Note that throughout this and subsequent chapters, we use the verb differentiate in a technical sense. To differentiate a function is to “take its derivative” using the methods of differential calculus presented in this book. It does not mean that we are trying to distinguish the function from some other function, unless we specifically say so.

In addition to being used to finding derivatives of all polynomials, the power rule and the scalar multiplication rule can be used to find derivatives of all scaling laws.

Derivatives of exponentials

Consider the function f(x) = a^x for some positive constant a > 0. To find the derivative, we use the definition of the derivative. Let x be a fixed number.

Although it is beyond the scope of this book, it can be shown that k = exists whenever a > 0. In the following example, we estimate the value of k for the case a = 2.

Since f′(x) = kf(x) for an appropriate choice of k whenever f(x) = a^x, we can ask this question: Is there a value of a such that k= 1? It turns out that the number e, which we defined in Section 1.4 as e = (1 + 1/n)ⁿ, is the appropriate choice of a. Namely,

Except for multiplying by a constant, e^x is the only function that remains unchanged under the operation of differentiation; that is, if f(x) = f′(x), then f(x) = ae^x for some real number a. This fact inspired one mathematician to write: “Who has not been amazed to learn that the function y = e^x, like a phoenix rising again from its own ashes, is its own derivative?” (Francois l'Lionnais, Great Currents of Mathematical Thought, vol. 1, New York: Dover Publications, 1962). Armed with the derivative of e^x, we can use the rules of differentiation to find the derivative of more general exponential functions f(x) = e^ax.

Proof. If a = 0, then

so the statement is true. If a is any nonzero real number, then e^ax = (e^a)^x and

To find this limit, define Δx = ah. Since h = Δx/a and h → 0 whenever Δx → 0,

Thus, we have shown that

For the last part, find the value of a such that b = e^a and finish the details on your own in Problem 27 in Problem Set 3.1.

Level 1 DRILL PROBLEMS

Differentiate the functions given in Problems 1 to 14. Assume that C is a constant.

6. f(x) = 5x³ − 5x² + 3x − 5

7. f(x) = x⁵ − 3x² − 1

8. f(x) = 2x² − 5x⁸ + 1

9. s(t) = 4e^t − 5t + 1

10. f(t) = 5 − e^2t

11. f(t) = 5.9(2.25)^t

12. f(t) = 82.1(1.85)^t

13. g(x) = Cx² + 5x + e^−2x

14. F(x) = 5e^Cx − 4x²

Determine on what intervals each function given in Problems 15 to 19 is increasing and on what intervals it is decreasing.

15. f(x) = x³ − x² + 1

16. g(x) =

17. f(x) = x⁵ + 5x⁴ − 550x³ − 2,000x² + 60,000x (round to the nearest tenth)

18. g(x) = x³ + 35x² − 125x − 9,375

19. H(w) = 2w − e^w

20. Let f(x) = x^1/2.

1. Find the derivative using the definition of the derivative.
2. Apply the power rule with n = 1/2.

21. Let f(x) = x^3/2.

1. Find the derivative using the definition of derivative. Hint: Write x^3/2 as x and rationalize the numerator.
2. Apply the power rule with n = 3/2.

Simplify the functions in Problems 22 to 26 and find their derivative whenever it is well defined.

22. g(x) = x²(x³ − 3x)

23. f(x) =

24. f(x) = (e^x − 1)(e^x + 1)

25. h(t) =

26. q(x) =

Level 2 APPLIED AND THEORY PROBLEMS

27. Prove that for any real number b > 0

28. Use the limit laws to prove the sum rule for differentiation:

29. Use the limit laws to prove the scalar multiple rule for differentiation:

for a constant c.

30. After pouring a mug full of the German beer Erdinger Weissbier (see Section 1.4), German physicist Leike (“Demonstration of Exponential Decay,” pp. 21–26) measured the height of the beer froth at regular time intervals. He estimated the height (in centimeters) of the beer froth as

where t is measured in seconds. Find

and interpret this quantity.

31. A drug that influences weight gain was tested on eight animals of the same size, age, and sex. Each animal was randomly assigned to a dose level. After two weeks, the difference W in the end and start weight (measured in decagrams) was calculated. The best-fitting quadratic equation to the data was found to be

where D is the dose level that ranges from 1 to 8.

1. Find and identify its units.
2. When does weight gain increase with dose level D?

32. Using data from 158 marine species, John Hoenig of the Virginia Institute of Marine Sciences studied how the natural mortality rate M of a species depends on the maximum observed age T (“Empirical Use of Longevity Data to Estimate Mortality Rates,” Fisheries Bulletin 82 (1983): 898–902). Using linear regression, he found

where T is measured in years. Find and interpret

33. During an outbreak of influenza at a school the number of students who became ill after t days is given by

where C is a constant.

1. If ten people were ill at the beginning of the epidemic (when t = 0), what is C?
2. At what rate is N(t) increasing when t = 5?

34. A glucose solution is administered intravenously into the blood stream of a patient at a constant rate of r milligrams/hour. As the glucose is being administered, it is converted into other substances and removed from the blood stream. Suppose the concentration of the glucose solution after t hours is given by

where k is a constant.

1. If C₀ is the initial concentration of glucose (when t = 0), what is C₀ in terms of r and k?
2. What is the rate at which the concentration of glucose is changing at time t?

35. In Problem 39 in Problem Set 1.3, we found that the length L(cm) of a pumpkin is related to the width W(cm) of a pumpkin by the allometric equation

How rapidly is length changing with regard to width for pumpkins of size 5 cm compared with those of 50 cm?

36. At the beginning of Section 1.4, we saw that the population model N(t) = 8.3(1.33)^t, where N is in millions and t is in decades starting at t = 0 representing the year 1815, can be used to describe the size of the U.S. population during most of the nineteenth century. Use this relationship to compare the total rate of growth of the U.S. population in 1815 versus the growth rate in 1865 (i.e., five decades later).

37. In Section 1.3, we found that the amount lifted (in kilograms) by an Olympic weightlifter can be predicted by the scaling law

where M is the mass of the lifter in kilograms. Find and interpret .

38. In Example 12 of Section 1.6 (changing the names of the variables from x and y to M and R), we found that the metabolic rate (in kilocalories/day) for animals ranging in size from mice to elephants is given by the function ln R = 0.75 ln M + 4.2 which yields the equation

where M is the body mass of the animal in kilograms.

1. The average California Condor weighs about 10 kg. Find and interpret .
2. The average football player weighs about 100 kg. Find and interpret .
3. Compare and discuss the quantities that you found in parts a and b.

39. The number of children newly infected with a particular pathogen that is transmitted through contact with their mothers has been modeled by the function

where N(t) is measured in thousands of individuals per year, and t is the number of years since 2000. In epidemiology, N(t) is known as an incidence function.

1. At what rate is the incidence function N changing with respect to time in the year 2010?
2. When will the incidence start to decline?

3.2 Product and Quotient Rules

Previously, we saw that the derivative of a sum equals the sum of the derivatives, and the derivative of a difference equals the difference of the derivatives. Armed with these elementary differentiation rules, we might guess that the derivative of a product is the product of the derivatives. The following simple example, however, shows this not the case. Let f(x) = x and g(x) = x², and consider their product

Because f′(x) = 1 and g′(x) = 2x, the product of the derivatives is

whereas the actual derivative of p(x) = x³ is p′(x) = 3x². Hence, our naïve guess is wrong! It is also easy to show that the derivative of a quotient is not the quotient of the derivatives. The goal of this section is to uncover the correct rules for differentiation for products and quotients of functions.

Product rule

To derive a rule for products, we appeal to our geometric intuition by considering areas where Δx > 0 and f(x) and g(x) are assumed to be increasing, positive differentiable functions of x. Note that the algebraic steps stand alone—without considering area or making the assumptions we made in the previous sentence.

Let

This product of p can be represented as the area of a rectangle:

Next, we find

The next step gives us the area of the “inverted L-shaped” region:

The key to the proof of the product rule is to rewrite this difference. We can see how to do this by looking at the area of the inverted L-shaped region in another way:

Divide both sides by Δx (where Δx ≠ 0):

The last step in deriving the product rule is to take the limit as Δx → 0:

We have just proved the product rule.

A simple way to remember the product rule is with this mnemonic: “The derivative of the product is the derivative of the first times the second plus the derivative of the second times the first.” Or sing the words of the following ditty aloud or in your mind.

Sing the product rule in time,
One prime two plus one two prime.
Isn't mathematics fun,
One prime two plus two prime one.

In the next problem, we encounter the notion that many events in biology are not certain but occur with a particular probability. Examples of this is an individual dying over a specified interval of time, a female giving birth to a particular number of individuals (e.g., the litter size of a mouse), or a molecule binding to a receptor site on a cell. You may have already learned, or will learn if you take a course in basic probability theory, that probabilities associated with independent events combine through multiplication. For example, if one flips a coin and the probability of getting heads is p = 1/2, then the probability of getting heads twice in a row is p × p = 1/2 × 1/2 = 1/4. Similarly, if one rolls a fair (honest or unbiased) six-sided die, then the probability of rolling a 3 is p = 1/6 and the probability of rolling an even number is p = 3/6 = 1/2. Thus the probability of getting a 3 on the first roll and an even number on the second roll is 1/6 × 3/6 = 3/36 = 1/12.

Although calculus and probability theory are generally taught as separate courses in college, probability is such an integral part of biological systems that it is useful for us to state and use the following fact for the examples we develop in this text.

Probability of two independent events. If p and q are the probabilities of the events 1 and 2 respectively occurring independent of one another, then pq is the probability that event 1 and event 2 both occur.

In Chapter 7, we revisit probability theory and explore how the integral calculus has become a central tool in the development of this theory.

Quotient rule

Before we derive a quotient rule, we begin with an example for finding the derivative of a reciprocal, which is a special case of a quotient that has 1 in the numerator.

We restate the result of this example for easy reference.

Combining the reciprocal and product rule, we can find the derivative of a quotient of functions. Let f and g be differentiable functions, and assume that g(x) ≠ 0.

Thus, we have derived what is known as the quotient rule.

There exist a variety of playful mnemonics that can be used to remember the quotient rule. For example, if we replace f by hi and g by lo, then we get the limmrick “lo-dee-hi less hi-dee-lo, draw the line and square below.”

Level 1 DRILL PROBLEMS

Find the derivatives in Problems 1 to 18.

1. p(x) = (3x² − 1)(7 + 2x³)

2. p(x) = (x² + 4)(1 − 3x³)

3. q(x) =

4. q(x) =

5. f(x) = x2^x

6. f(x) = x³3^x

7. f(x) = (1 + x + x²)e^x

8. f(x) = (e³ + e² + e)x²

9. F(L) = (1 + L + L³ + L⁴)(L − L²)

10. G(M) = (M − M³)(1 − 4M)

11. f(x) = (4x + 3)² Hint: Think (4x + 3)(4x + 3)

12. g(x) = (5 − 2x)²

13. f(x) =

14. g(t) =

15. f(p) = where a is a constant

16. g(m) = where b is a constant

17. F(x) =

18. G(x) =

Find the equation for the tangent line at the prescribed point for each function in Problems 19 to 24.

19. f(x) = (x³ − 2x²)(x + 2) where x = 1

20. G(x) = (x − 5)(x³ − x) where x = − 1

21. F(x) = where x = 0

22. f(x) = e^x + e^−x where x = 0

23. F(x) =

24. g(x) = x ln x where x = 1.

25. a. Differentiate the function

b. Factor the function in part a and differentiate by using the product rule. Show that the two answers are the same.

26. a. Use the quotient rule to differentiate

b. Rewrite the function in part a as f(x) = x⁻³(2x − 3) and differentiate by using the product rule.

c. Rewrite the function in part a as f(x) = 2x⁻² − 3x⁻³ and differentiate using the power rule.

d. Show that the answers to parts a, b, and c are all the same.

Level 2 APPLIED AND THEORY PROBLEMS

27. The body mass index (BMI) for an individual who weighs w pounds and is h inches tall is given by

A person with a body mass index greater than 30 is considered obese; see the BMI chart.

1. Consider all adults who are h = 63 inches tall. After setting h = 63 inches so that B is now a function of w only, find at w = 130 and interpret your results.
2. Consider all children who weigh 60 lb. After setting w = 60 lb so that B is now a function of h only, find at h = 54 and interpret your results.

28. In addition to zebra, Professor Getz's group studied springbok (a type of antelope) in Etosha National Park. In a study paralleling Example 2 of this section, they found that the probability of a springbok surviving predation and disease to the end of the eighth week of a typical year is given by, respectively, 94.9% and 98.8%. Suppose they also determined that at the end of this eight weeks the probability of dying from predation and disease is decreasing respectively by 0.6% and 0.25% per week. Assume the events of dying from predation or disease are independent.

1. Find the probability of surviving to the end of week 8.
2. Find the rate of change of probability of surviving at the end of week 8.
3. Estimate the probability of surviving to the end of week 9.

29. Suppose that Professor Getz's group found that the probability of zebra surviving both predation and disease to the end of week 4 was 98.4% and 99.6%, respectively. Suppose they also determined that at the end of week 4 the probability of a zebra dying from predation and disease is decreasing respectively by 0.4% and 0.1% per week (remember when converting percentages to numbers to shift the value by two decimal places). Assume the events of dying from predation or disease are independent.

1. Find the probability of surviving up to the end of week 4.
2. Find the rate of change of probability of surviving at the end of week 4.
3. Estimate the probability of surviving to end of week 5.

30. Consider the generalized Beverton-Holt model of population growth given by

where

and a > 0, b > 0.

1. Find g′(N).
2. Determine what values of b > 0 cause g to be increasing for all N > 0.
3. When b is outside the range of values found in part b, determine on what interval g is increasing and on what interval g is decreasing.

31. A ligand is a molecule that binds to another molecule or other chemically active structure (e.g., a receptor on a membrane) to form a larger complex. In a study of two ligands (I and II) competing for the same sites on a substrate, ligand II is added to a substrate solution that already contains ligand I. As the concentration of ligand II is increased, the concentration of ligand I bound to the substrate decreases. This one-site ligand competition process is characterized by this equation:

where T is the concentration of bound ligand I per milligram of tissue and x is the exponent of the concentration of ligand II in the solution. The constants a and b arise from the relative binding rates of the two ligands and satisfy a > 0 and b > a.

1. Compute
2. Interpret the quantities a, b, and c and sketch the graph.

32. In the 1960s, scientists at Woods Hole Oceanographic Institution measured the uptake rate of glucose by bacterial populations from the coast of Peru (R. R. Vaccaro and H. W. Jannasch, “Variations in Uptake Kinetics for Glucose by Natural Populations in Sea Water,” Limnol. Oceanogr. (1967) 12, 540–542.). In one field experiment, they found that the uptake

micrograms per hour where x is micrograms of glucose per liter. Compute and interpret f′(20) and f′(100).

33. In Example 5 of Section 2.4, we found that the predation rate of wolves on moose in North America could be modeled by

where x is measured in number of moose per square kilometer. Compute and interpret f′(0.5) and f′(2.0).

34. Cells often use receptors to transport nutrients from outside of the cell membrane to the inner cell. In Example 6 of Section 1.6, we determined that the rate, R, at which nutrients enter the cell depends on the concentration, N, of nutrients outside the cell. The function

models the amount of nutrients absorbed in one hour where a and b are positive constants.

1. Find R when N = b. What does this tell you about b?
2. Compute and interpret . When is R increasing? When is R decreasing?

35. In Problem 42 in Problem Set 2.4, we modeled how wolf densities in North America depend on moose densities with the following function

where x is number of moose per square kilometer. Determine for what x values f(x) is increasing.

36. In Problem 45 in Problem Set 2.3, two fisheries scientists (T. J. Quinn and R. B. Deriso, “Stock and Recruitment,” Chapter 3 (pp. 86–127) in Quantitative Fish Dynamics, 1999. Oxford University Press, New York, New York) found that the following stock-recruitment function provides a good fit to data pertaining to the southeast Alaska pink salmon fishery:

where y is the number of young fish recruited, and x is the number of adult fish involved in recruitment.

1. Compute .
2. Determine for what x values y is increasing and decreasing. Interpret your results.

37. This problem uses the hyperbolic secant function, denoted sech x, which is an important function in mathematics and is defined by the formula

We will discuss further in Chapter 4 how two mathematicians, W. O. Kermack and A. G. McKendrick, showed that the weekly mortality rate during the outbreak of the plague in Bombay in 1905–1906 can be reasonably well described by the function

where t is measured in weeks. Determine when the mortality rate is increasing and when the mortality rate is decreasing.

38. In a classic paper, V. A. Tucker and K. Schmidt-Koenig modeled the energy expended by a species of Australian parakeet during flight (“Flight of Birds in Relation to Energetics and Wind Directions,” The Auk 88 (1971): 97–107). They used the following function:

where v is the bird's velocity (km/h).

1. Find a formula for the rate of change of energy with respect to v.
2. At what velocity, v, is the energy expenditure neither increasing nor decreasing? Discuss the importance of this velocity.

   

   Australian budgerigar (Melopsittacus undulatus)

3.3 Chain Rule and Implicit Differentiation

We now move to the next level in terms of developing tools to differentiate functions that can be regarded as the composite of more elementary functions (discussed in Section 1.5). These tools will give us the power to differentiate functions such as the bell-shaped curve y = e^−x2, the polynomial y = (1 + 2x + x³)¹⁰¹, and the logarithm function y = ln x.

Chain rule

Suppose we were asked to find the derivative of the function y = (1 + 2x + x³)¹⁰¹. It is not practical to expand this product in order to take the derivative of a polynomial. Instead, we can use a result known as the chain rule. In order to motivate this important rule, let us consider an application.

It is known that the carbon monoxide pollution in the air is changing at the rate of 0.02 ppm (parts per million) for each person in a town whose population is growing at the rate of 1000 people per year. To find the rate at which the level of pollution is increasing with respect to time, we must compute the product

We can generalize this commonsense calculation by noting that the pollution level, L, is a function of the population size, P, which itself is a function of time, t. Thus, L as a function of time is (L P)(t) or, equivalently, L[P(t)]. With this notation, the commonsense calculation becomes:

Expressing each of these rates in terms of an appropriate derivative of L[P(t)] in Leibniz form, we obtain the following equation:

These observations anticipate the following important result known as the chain rule.

Proof. To prove the chain rule, define

It should be intuitive that G(h) is continuous at h = 0. (You will be asked to verify this statement in Problem Set 3.3). With this observation in hand, the proof of the chain rule becomes straightforward. By the definition of the derivative.

In Example 2, we found the derivative of f(x) = e^−0.01x by using the derivative of an exponential (Section 3.1), but we could also use the chain rule. It is worthwhile to restate an extended derivative rule for a natural exponential function:

We illustrate this idea with the following example.

Implicit differentiation

The equation y = explicitly defines f(x) = as a function of x for −5 ≤ x ≤ 5. The same function can also be defined implicitly by the equation x² + y² = 25, as long as we restrict y by 0 ≤ y ≤ 5 so the vertical line test is satisfied. To find the derivative of the explicit form, we use the chain rule:

To obtain the derivative of the same function in its implicit form, we simply differentiate across the equation x² + y² = 25, remembering that y is a function of x and using the chain rule:

The procedure we have just illustrated is called implicit differentiation.

More generally, given any equation involving x and y, we can differentiate both sides of the equation, use the chain rule, and solve for . This becomes particularly important when one cannot (or not easily) express y in terms of x explicitly. The next example is of this type and is included because of its historical importance and aesthetic appeal. You may think that such curves have no application to biology; but, for example, the logarithmic spiral (see Figure 1.56) encountered in Project 1C: Golden Ratio (at the end of Chapter 1) describes the growth of the nautilus mollusk.

Derivatives of logarithms

Implicit differentiation allows us to easily find the derivative of logarithms and power functions.

In the problem set of this section you are asked to prove the more general statements of this result; namely, if y = ln |x| then, for x ≠ 0, = .

The next example deals with the clearance rate—which, as we saw in Example 7 of Section 3.1 in the context of viral particles, is the constant rate of decay—of a drug from the blood stream of an individual.

If the base on the logarithm has a base other than the natural base, e, then we use the following result, which you will be asked to verify in the Problem 41 of Problem Set 3.3:

In Section 3.1, we stated the power rule and promised to prove it later in this chapter for all real numbers. We fulfill this promise in the following example.

You will encounter the proof for x < 0 in the problem set at the end of this section.

Level 1 DRILL PROBLEMS

Use the chain rule to compute the derivative dy/dx for the functions given in Problems 1 to 4.

1. y = u² + 1; u = 3x − 2

2. y = 2u² − u + 5; u = 1 − x²

3. y = ; u = x² − 9

4. y = u²; u = ln x

Differentiate each function in Problems 5 to 8 with respect to the given variable of the function.

Differentiate each function in Problems 9 to 18.

Find dy/dx by implicit differentiation in Problems 19 to 25.

19. x² + y = x³ + y³

20. xy = 25

21. xy(2x + 3y) = 2

23. (2x + 3y)² = 10

24. ln(xy) = e^2x

25. e^xy + ln y² = x

26. Consider the functions y = f(x) and y = g(x) whose graphs in red and blue, respectively, are shown in Figure 3.12.

Figure 3.12 Functions y = f(x) (red) and y = g(x) (blue).

27. The graphs of u = g(x) and y = f(u) are shown in Figure 3.13.

Figure 3.13 Chain rule with graphs.

1. Find the approximate value of u at x = 2. What is the slope of the tangent line at that point?
2. Find the approximate value of y at x = 5. What is the slope of the tangent line at that point?
3. Find the slope of y = f[g(x)] at x = 2.

28. Let g(x) = f[u(x)], where u(−3) = 5, u(−3) = 2, f(5) = 3, and f (5) = −3. Find an equation for the tangent to the graph of g at the point where x = −3.

29. Let f be a function for which

1. If g(x) = f(3x − 1), what is g′(x)?
2. If h(x) = f, what is h′(x)?

30. The cissoid of Diocles is a curve of the general form represented by the following particular equation

as illustrated in Figure 3.14.

Figure 3.14 Cissoid of Diocles.

Find the equation of the tangent line to this graph at (3, 3).

31. The folium of Descartes is a curve of the general form represented by the following particular equation

as illustrated in Figure 3.15.

Figure 3.15 Folium of Descartes.

Find the equation of the tangent line to this graph at (2, 1).

32. Another version of the folium of Descartes is given by the equation

as illustrated in Figure 3.16

Figure 3.16 Folium of Descartes.

Find at what points the tangent line is horizontal.

Level 2 APPLIED AND THEORY PROBLEMS

33. Arteriosclerosis develops when plaque forms in the arterial walls, blocking the flow of blood; this, in turn, often leads to heart attack or stroke (Figure 3.17). Model the cross-section of an artery as a circle with radius R centimeters, and assume that plaque is deposited in such a way that when the patient is t years old, the plaque is p(t) cm thick, where

Find the rate at which the cross-sectional area covered by plaque is changing with respect to time in a sixty-year-old patient.

Figure 3.17 Blood flow in a healthy artery (top) and in an artery narrowed by arteriosclerosis (bottom).

34. In Problem 38 in Problem Set 3.2, the energy expended by a species of Australian parakeet during flight was modeled by the function

where v is the bird's velocity (km/h). Assume at one point in time (say t = 0) in its flight, a parakeet's velocity is 25 km/hour and its velocity is increasing at a rate of 2 km/hour². Find the instantaneous rate of change of this parakeet's energy use at t = 0. Be sure to identify the correct units for your answer.

35. In Example 5 of Section 2.4, we found that the predation rate of wolves in North America could be modeled by

where x is measured in number of moose per square kilometer. If the current moose density is x = 0.5 and is increasing at a rate of 0.1 per year, determine the rate at which the predation rate is increasing.

36. In Problem 42 in Problem Set 2.4, we modeled how wolf densities in North America depend on moose densities with the following function

where x is the number of moose per square kilometer. If the current moose density is x = 2.0 and decreasing at a rate of 0.2 per year, determine the current rate of change of the wolf densities.

37. The proportion of a species of aphid that escapes parasitism is

where x is the density of parasitoids. If the density of parasitoids is currently 10 wasps per acre and decreasing at a rate 20 wasps per acre per day, find at what rate the likelihood of escaping parasitism is changing.

38. In Example 10 we saw that an environmental study of a certain suburban community suggested the following relationship between the population level p (thousand people) and the amount of carbon monoxide C (ppm) in the air:

Now suppose that the population p (in thousands) at time t is modeled by the function

Use the chain rule to find the rate at which the pollution level is changing after three years and compare this with the estimate obtained in Example 10.

39. The bicorn (also called the cocked hat) is a quartic curve studied by mathematician James Sylvester (1814–1897) in 1864. As illustrated in Figure 3.18, it is given by the set of points that satisfy the equation

Find the formulas for the two tangent lines at x = 1/2.

Figure 3.18 Bicorn curve.

40. Prove that if f is differentiable at u(a) and u is continuous at a, then

is continuous at h = 0.

41. Prove that

42. Prove that

for y = xⁿ where x < 0 and n is any non-zero number for which y is defined.

43. The average height of boys in the United States as a function of age is shown in Figure 3.19. This relationship is well described by the linear function

Figure 3.19 Heights in inches of boys and girls in the United States as a function of time in months.

where t is measured in months. The relationship between height and weight can be modeled by the power function

Find the instantaneous rates at which height and weight are changing at age ten years.

44. The average height of girls in the United States as a function of age is shown in Figure 3.19. This relationship is well-described by the linear function

where t is measured in months. The relationship between height and weight can be modeled by the power function

Find the instantaneous rates at which height and weight are changing at age ten years.

3.4 Derivatives of Trigonometric Functions

Many physical and biological processes change periodically over time and consequently are represented by a periodic function. A powerful result in mathematical analysis proves that periodic functions can be represented as a sum of sines and cosines; this is called the Fourier series representation. In this section, we find the derivative of these fundamental functions and their functional relatives—tangent, secant, cotangent, and cosecant. Recall from Chapter 1 that we assume the trigonometric functions are functions of real numbers or of angles measured in radians. We make this assumption because the trigonometric differentiation formulas rely on limit formulas that become more complicated if degrees, rather than radians, are used to measure angles.

Derivatives of sine and cosine

Before stating the theorem regarding the derivatives of the sine and cosine functions, we look at the graph of for small values of h. In particular, for h = 0.01, the expression has the following graph:

From this graph, it appears that the derivative of f(x) = sin x is f′(x) = cosx. This relationship appears to match up, as illustrated in Figure 3.20: sine is increasing on intervals where cosine is positive and has turning points where cosine is zero.

Figure 3.20 Graphs of cosine (blue) and sine (red).

Before verifying this assertion, we need two important limits. The first limit is presented numerically in Example 1. You will be asked to give analytical derivations of the limits in Problems 29 and 30 in Problem Set 3.4.

We can now state the derivative rule for the sine and cosine functions.

Proof. We will prove the first derivative formula using the trigonometric identity

and the definition of derivative. For a fixed x

To find the derivative of cosine, we use the trigonometric identities

and the chain rule

Periodic fluctuations in biological systems can be internally or externally driven, as the next two examples illustrate.

Derivatives of other trigonometric functions

If you know the derivatives of sine and cosine and the basic rules of differentiation, then all the other trigonometric derivatives follow.

All the additional derivative rules are proved by using the quotient rule along with formulas for the derivative of sine and cosine. Here we will obtain the derivative of the tangent function and leave the rest to Problem Set 3.4.

Notice that the derivatives of all “co” trig functions except for cosine have the “co-trig” derivative form of their corresponding trigonometric partners, but with a sign change. Thus, for example, because the derivative of tangent is secant squared, this rule implies that the derivative of cotangent is the negative of cosecant squared.

Level 1 DRILL PROBLEMS

Differentiate the functions given in Problems 1 to 20.

1. f(x) = sin x + cos x

2. g(x) = 2 sin x + tan x

3. y = sin 2x

4. y = cos 2x

5. f(t) = t² + cos t + cos

6. g(t) = 2 sec t + 3 tan t − tan

7. y = e^−x sin x

8. y = tan x²

9. f(θ) = sin² θ

10. g(θ) = cos² θ

11. y = cos x¹⁰¹

12. y = (cos x)¹⁰¹

13. p(t) = (t² + 2) sin t

14. y = x sec x

18. y = sin(2t³ + 1)

19. y = ln(sin x + cos x)

20. y = ln(sec x + tan x)

Use the given trigonometric identity in parentheses and the basic rules of differentiation to find the derivatives of the functions given in Problems 21 to 24.

Level 2 APPLIED AND THEORY PROBLEMS

29. Consider three areas as shown in Figure 3.22.

Figure 3.22 Triangles and a unit circle with subtended angle h.

1. What is the area g(h) of the blue-shaded triangle?
2. What is the area f(h) of the pink-shaded sector? Hint: The area of a sector of a circle of radius r and central angle θ measured in radians is .
3. What is the area k(h) of the green-shaded triangle?
4. Use the fact that g(h) ≤ f(h) ≤ k(h) for small h > 0 to prove that

   

   by beginning with the inequality

   

30. Prove

Hint: Multiply by 1 written as and use a fundamental trigonometric identity.

31. A researcher studying a certain species of fish in a northern lake models the population after t months of the study by the function

At what rate is the population changing after two months? Is the population growing or declining at this time?

32. In an experiment with algae, a research scientist manipulated the light in growth chambers so that

described the population density (in cells per liter) as a function of t (in hours).

1. Find a function r(t) such that

   

2. Under the assumption that the growth rate of the population is proportional to the intensity of light, what is the period of the light fluctuations in the chamber?
3. Determine at what times (if any) the population is decreasing in abundance.

33. In a more ambitious experiment with algae, a research scientist manipulated the light in algal tanks so that

describes the population density (in cells per liter) as a function of t (in hours).

1. Find a function r(t) such that

   

2. Under the assumption that the growth rate of the population is proportional to the intensity of light, what is the period of the light fluctuations in the tank?
3. Determine at what times (if any) the population is decreasing in abundance.

34. In Example 4 of Section 1.5, we modeled the tides for Toms Cove in Assateague Beach, Virginia, on August 19, 2004 with the function

where H is the height of the tide (in feet) and t is the time (in hours after midnight). Find and interpret .

35. The temperature in a swimming pool is given by

where t is measured in hours since midnight.

1. Find T′(12) and interpret.
2. During what hours is the temperature of the pool increasing?

36. The human heart goes through cycles of contraction and relaxation (called systoles). During these cycles, blood pressure goes up and down repeatedly; as the heart contracts, pressure rises, and as the heart relaxes (for a split second), the pressure drops. Consider the following approximate function for the blood pressure of a patient:

where t is measured in minutes. Find and interpret P′(t).

3.5 Linear Approximation

We have seen that the tangent line is the line that just touches a curve at a single point. In this section, we will discover that the tangent line can be used to provide a reasonable approximation to a curve. Using these linear approximations we will be able to make projections about the size of a bison population (Figure 3.23) estimate , and estimate the effects of measurement error.

Figure 3.23 The North American plain's bison (Bison bison) is one of two subspecies of bison that roamed the Great Plains. Their numbers have dropped from tens of millions to thousands over the last 200 years.

Approximating with the tangent line

We begin with an example that illustrates how well a tangent line can approximate a curve.

The difference between a tangent line and the associated curve becomes more and more negligible as you zoom into the point of contact. Thus, it seems quite reasonable to approximate the function with the tangent line in a neighborhood of the point at which this tangent is constructed. This is called the linearization of a function.

With linear approximations, we can make predictions about the future, as illustrated in the next example. We use data on the abundance of the North American bison in Yellowstone National Park going back as early as 1902. (Estimates of bison population levels in Yellowstone from 1902–1931 can be found at http://www.seattlecentral.edu/qelp/Data.html.) Annual abundances for bison in Yellowstone for the twenty-nine-year period 1902 to 1931 are shown in Figure 3.25. These data suggest that the bison population was recovering in the first part of the twentieth century after years of intense hunting in the nineteenth century.

Figure 3.25 Bison abundance in Yellowstone National Park.

In the next example, we use the data only from 1908 to 1915. However, a project involving all the data is outlined at the end of the chapter.

Using linear approximation, we can estimate the value of a function at points near known values of the function. As we will see, however, linear approximations of nonlinear functions generally get increasingly worse as we move away from the point of approximation.

The next example shows that the linear approximation of sin x in the neighborhood of 0 is very simple, but it fails badly as the approximation is pushed too far beyond 0.

Development of organisms from immature forms, like eggs or seeds, to adult forms depends on a series of biochemical reactions. These biochemical reactions are often temperature sensitive, breaking down at too low or too high temperatures. For plants and insects whose internal temperatures are largely determined by the temperature of their environment, it follows that developmental rates depend on the temperature of the environment. This temperature dependence can be measured experimentally, as the following example illustrates.

Error analysis

When a scientist makes a measurement, it is always subject to some measurement error. Hence, we have

measurement = actual value + measurement error

For instance, when the clearance rate of acetaminophen is given as 0.28 per hour, this estimate is the average of a series of measurements, each of which may vary by 0.05 or so. Consequently, when we estimate the half-life of acetaminophen, or the amount in the blood stream several hours after taking the drug, it is important to understand how small variations in the estimate 0.28 influence half-life estimates.

Example 6 illustrates how an error in the measurement of the independent variable x propagates to an error in the dependent variable y—a process called error propagation.

Consider another example.

Elasticity

Often scientists are more interested in the percent error and not the absolute error. For example, a scientist may ask this question: How does a 10% error in the measurement of the clearance rate result in a percentage error in the estimate of the half-life? If x = a is the true value of the independent variable and there is a measurement error of Δx, then the percent error in x is

With an error of Δx in the independent variable, we get an error of Δy = f(a + Δx) − f(a) in y. Hence, the percent error in y is

The ratio of the percentage error in y over the percentage error in x is given by

and can be approximated by

This quantity is used quite commonly in the analysis of biological models and, consequently, has a special name: elasticity.

Using elasticity, we can estimate with what accuracy we need to measure an independent variable to ensure a certain accuracy in the estimate of a dependent variable.

Level 1 DRILL PROBLEMS

In Problems 1 to 6 find the linear approximation of y = f(x) at the specified point, and use technology to graph the function and its linear approximation to determine whether the linear approximation tends to overestimate or underestimate y = f(x) near the specified point.

In Problems 7 to 12, estimate the indicated quantity using a linear approximation around the given point x and compare to the true value obtained using technology.

Find the sensitivity of y = f(x) at the point specified in Problems 13 to 18, and use it to estimate Δy for the given measurement error Δx.

Find the elasticity of y = f(x) at the point specified in Problems 19 to 24 and use it to estimate the percent error in y for the given percent error in x.

Level 2 APPLIED AND THEORY PROBLEMS

25. In Example 5, we saw that Roy and colleagues estimated how developmental rates of a beetle (Stethorus punctillum) varied with temperature. They modeled the rate at which egg development is completed with the following function

where T is measured in degrees Celsius. This model and the data used to parameterize the model are shown in Figure 3.28. This figure suggests that the developmental rate is approximately linear in T for temperatures near 20°C.

Figure 3.28 Developmental rate (percent development completed per day) for egg development of the beetle Stethorus punctillum(spider mite destroyer).

1. Find a linear approximation to D(T) near the value T = 20.
2. Sketch the linear approximation on top of the graph in Figure 3.28. How does this compare to the data?

26. In Example 5, we saw that Roy and colleagues estimated how developmental rates of a beetle (Stethorus punctillum) varied with temperature. They modeled the rate at which the first stage of larval development is completed with the following function

where T is measured in degrees Celsius. This model and the data used to parameterize the model are shown in Figure 3.29. This figure suggests that the developmental rate is approximately linear in T for temperatures near 20°C.

Figure 3.29 Developmental rate (percent completed per day) for the first stage of larval development for the beetle Stethorus punctillum (spider mite destroyer).

1. Find a linear approximation to D(T) near the value T = 20.
2. Sketch the linear approximation on top of the graph in Figure 3.29. How does this compare to the data?

27. If your measurement of the radius of a circle is accurate to within 3%, approximately how accurate (to the nearest percentage) is your calculation to the area A when the radius is r = 12 cm? (Recall the formula A = πr²).

28. Suppose a 12-ounce can of Coke® (i.e., Coca-Cola) has a height of 4.75 inches. If your measurement of the radius has an accuracy to within 1%, how accurate is your measurement for the volume? Check your answer by examining a Coke can.

29. An environmental study suggests that t years from now, the average level of carbon monoxide in the air will be

parts per million (ppm). By approximately how much will the carbon monoxide level change during the next six months?

30. A certain cell is modeled as a sphere. If the formulas S = 4πr² and V = πr³ are used to compute the surface area and volume of the sphere, respectively, estimate the effect of S and V produced by a 1% increase in the radius r.

31. In a model developed by John Helms (“Environmental Control of Net Photosynthesis in Naturally Grown Pinus Ponderosa Nets,” Ecology (Winter 1972, p. 92), the water evaporation E(T) for a ponderosa pine is modeled by

where T (degrees Celsius) is the surrounding air temperature.

1. Compute the elasticity of E(T) at T = 30.
2. If the temperature is increased by 5% from 30°C, estimate the corresponding percentage change in E(T).

32. In a healthy person of height x inches, the average pulse rate in beats per minute is modeled by the formula

1. Compute the sensitivity of P at x = 60.
2. Use your answer to part a to estimate the change in pulse rate that corresponds to a height change from 59 to 60 inches.
3. Compute the elasticity of P. Does it depend on x?
4. Determine how accurate the measurement of x needs to be to ensure the estimate for P has an error of less than 10%.

33. In Example 6, we showed that the half-life, T, of a drug with clearance rate x is given by

Suppose that the true value of the clearance rate of some drug is given by x = a.

1. Find the elasticity of T with respect to x.
2. If you want to estimate the half-life of this drug within an error of 2%, how accurately do you have to measure the clearance rate of the drug?

34. A drug is injected into a patient's blood stream. The concentration of the drug in the blood stream t hours after the drug is injected is modeled by the formula

where C is measured in milligrams per cubic centimeter.

1. Compute the sensitivity of C at t = 30.
2. Use your answer to part a to estimate the change in concentration over the time period from 30 to 35 minutes after injection.

35. According to Poiseuille's law, the speed of blood flowing along the central axis of an artery of radius R is modeled by the formula

where c is a constant. What percentage error (rounded to the nearest percent) will you make in the calculation of S(R) from this formula if you make a 1% error in the measurement of R?

36. The gross U.S. federal debt (in trillions of dollars) from 2000 to 2004 is given in the following table

Year	Gross federal debt
2000	5.629
2001	5.770
2002	6.198
2003	6.760
2004	7.355

Data source: From the historical tables of the Office of Management and Budget, 2006, as downloaded from www.whitehouse.gov/omb/budget/Historicals.

1. Plot the data and the linear approximation of the data at t = 0 (2000). Discuss the quality of this approximation.
2. Use a linear approximation to estimate the federal debt in 2010. Look up the actual gross federal debt to see how well the approximation worked.

37. In Example 7 of Section 2.6, the function

was used to fit data on the extent of the Arctic sea ice S(million square kilometers) as a function of years t since 1980 (corresponding to t = 0).

1. What is the sensitivity of S in 1980?
2. How has the sensitivity changed from 1980 to 2000? Does this make sense? If not, why not?

38. Consider a power function f(x) = ax^b with a > 0 and b ≠ 0. Show that the elasticity of f(x) is independent of the value x. What does it depend on? Use this answer to quickly solve the following problems.

1. If there is a 5% error in estimating the mass M of a weightlifter, what is approximately the percent error in estimating the lift L = 20.15M^2/3kg of the weightlifter?
2. If there is a 10% error in estimating the mass M of an organism, then what is approximately the percent error in estimating the metabolic rate R ∝ M^3/4kcal/hour of the organism?
3. If there is a 2% error in measuring the weight W of a person, what is approximately the percent error in estimating the body mass index B ∝ W of the person?

39. In Problem 31 in Problem Set 3.2, we used the following function (here we replace c with the constant k = e^−c−10 and x with ln y)

to model T—the concentration of bound ligand I per milligram of tissue—in terms of y representing the concentration of a second ligand II in the solution.

1. What is the elasticity of T with respect to changes in y?
2. If there is a 10% error in estimating the concentration y of ligand II, then what is the error in calculating the level of ligand I as a function of y for the case a = 1, b = 2, and k = 1.

40. In Example 2 of Section 2.4, we used the function

to represent the percent of patients exhibiting an above normal temperature response to dose x millimoles (mM) of a particular histamine.

1. What is the elasticity of R with respect to changes in x?
2. If there is a 5% error in estimating the dose x when x = −5 (mM), then what is the percent error in calculating the response R?

3.6 Higher Derivatives and Approximations

The derivative of a function can be interpreted as the instantaneous rate of change, and it yields linear approximations to the function. Since the derivative of a function is also a function, this latter function also has a derivative. What does this derivative of a derivative represent? How useful is it? The goal of this section is to answer these questions—and considerably more.

Second derivatives

The second derivative of a function f is the derivative of f′ and is denoted f″. In other words,

Equivalently, we write

or if y = f(x),

Note that is regarded as the symbol for the “operation of taking the second derivative of a function with respect to its argument x.”

What do these second derivatives represent? Consider the following definition.

Since f″ is the derivative of f′, the mean value theorem implies that if f″ > 0 on an interval, then f′ is increasing on this interval. What does this mean? In terms of tangent lines, this means that the slope of the tangent line is increasing in the interval. Hence, f is “bending upward” or, equivalently, is concave up on this interval. Alternatively, if f″ < 0, then the slope of the tangent line is decreasing and f is “bending downward” or, equivalently, is concave down.

A point on a continuous graph that separates a concave downward portion of a curve from a concave upward portion is called an inflection point. The following example illustrates this idea using data from mortality due to airborne diseases.

The curve in Example 3 is an example of a sigmoidal curve, a curve that is monotonic, with horizontal asymptotes at positive and negative infinity and a single point of inflection. Another example of a sigmoidal curve is Figure 1.28 in Section 1.4; this is a model of population growth in the United States where growth is initially exponential and then levels off asymptotically. The following graphs show both forms of sigmoidal curves.

Graphs of increasing (in blue) and decreasing (in red) sigmoidal functions.

Changing rates of change

Recall from Section 2.1 the interpretation of the derivative of a function as the rate of change of that function with respect to increasing values of the function's argument. If we go back to the origins of the calculus, we see that velocity was interpreted by Newton as the instantaneous rate of change over time of the position of a particle in space, and acceleration was interpreted as the instantaneous rate of change over time of velocity. These interpretations led Newton to formulate his famous second law on the relationship between the force acting on an object, the mass of the object, and its resulting acceleration.

The field of kinematics—that is, the motion of points or bodies through space—and the application of calculus to physics are developed in physics courses. But no calculus book is complete without at least one example dealing with acceleration, and here we focus on a biological one.

The reason for choosing a fifth-order polynomial rather than a fourth or sixth in Example 4 is that the runners accelerate strongly at the beginning and tend to decelerate as they tire at the end. Thus, acceleration is initially positive and eventually negative. Since this type of behavior is best modeled by odd-ordered rather than even-ordered polynomials, we choose an odd-ordered polynomial for distance, as it becomes odd again after taking two derivatives. We choose a fifth-order instead of third-order polynomial as it does a better job modeling the intermediate, close to zero acceleration segment that we see occur approximately around seconds 4 to 7 during the course of the race (see Problem 53 in Problem Set 3.6).

Using our interpretations of the first and second derivative, we should be able to identify the graph of one from the other.

Second-order approximations

In Section 3.5, we approximated functions with their tangent lines. While a good start, these approximations can be improved upon by using first and second derivatives.

The preceding example suggests that function obtained by taking the difference between the original function and its derivative is approximately parabolic. If we want to approximate this function by a quadratic function, how do we find the best quadratic approximation? To answer this question, we develop the following procedure for finding the best quadratic approximation to a function f around the point x = 0. To find

near x = 0, we require

To have the first derivatives of f(x) and the approximation a + bx +cx² agree at x = 0, we can take derivatives of both sides

At x = 0, we want f′(0) = b, so we define b = f′(x). Finally, to have their second derivatives agree at x = 0, we differentiate one more time:

This leads us to define c = . This gives a quadratic (second-order or parabolic) approximation at x = 0:

Let us see how well this approximation works.

In cases where the linear approximation is a horizontal line, the quadratic approximation is the first approximation to give real information about the concavity of the curve in question.

More generally, we may wish to approximate a function near a point x = a with a parabola. As an exercise, you can verify that by forcing the quadratic approximation and the function to agree up to the second derivative at x = a, you obtain the following second-order approximation of f at x = a.

Even higher derivatives

Once we have taken second derivatives, there is no reason to stop. We can attempt to take the derivative of the second derivative, and the derivative of the resulting derivatives until the function obtained is 0; or we can continue indefinitely (e.g., see parts b and c of the next example). These higher derivatives and the associated notations are defined as follows:

We conclude with an application of higher derivatives from politics.

Level 1 DRILL PROBLEMS

Find the higher derivatives indicated in Problems 1 to 12.

In Problems 13 to 18 you are given the function s (t) that specifies the position of an object on a line as a function of time t. Find expressions for the velocity and acceleration as a function of time for t ≥0. Solve for t when the acceleration is 0; use technology where needed to find the values and when multiple values exist, use the closest to t = 0.

In Problems 19 to 24 find the linear approximations of the given functions f(x) around x = a. Using second-order derivatives, determine whether the linear approximation tends to overestimate or underestimate f(x) near x = a.

In Problems 25 to 34 determine on what intervals f is increasing, decreasing, concave up, concave down, and find the points of inflection.

Find the first- and second-order approximations of y = f(x) around x = a in Problems 35 to 40. Use technology to plot the function and its approximations near x = a.

In Problems 41 to 44, identify y = f(x), y = f′(x), and y = f″(x).

45. Sketch the graph of a function with all of the following properties:

46. Sketch the graph of a function with all of the following properties:

Level 2 APPLIED AND THEORY PROBLEMS

47. The slogan of a particular home improvement company is “Improving Home Improvement.” Explain the role of derivatives in this slogan.

48. A politician claims that “Under a new law, prices would rise slower than if the law were not passed.” Explain the role of higher derivatives in this statement.

49. At the website http://www.nlreg.com/aids.htm, you can find the following figure that graphs the number of new cases of AIDS since 1980.

1. Estimate where the function is concave up and concave down.
2. Describe in words what these changes in the concavity mean for the AIDS epidemic.

   

50. In Example 2 of Section 2.4, a dose-response curve for patients responding to a dose of histamine is given by

where x is the natural logarithm of the dose in millimoles (mmol).

1. Compute
2. Determine for what dose ranges R is concave up and concave down. Interpret your results.

51.

One of the most famous women in the history of mathematics is Maria Gaetana Agnesi.

She was born in Milan, the first of twenty-one children. Her first publication was at age nine, when she wrote a Latin discourse defending higher education for women. Her most important work was a now classic calculus textbook published in 1748. Agnesi is primarily remembered for a curve defined by the equation

for a positive constant a. The curve was named versiera (from the Italian verb to turn) by Agnesi, but John Colson, an Englishman who translated her work, confused the word versiera with the word avversiera, which means “wife of the devil” in Italian; the curve has ever since been called the “witch of Agnesi.” This was particularly unfortunate because Colson wanted Agnesi's work to serve as a model for budding young mathematicians, especially young women. Graph this curve, find the points of inflection (if any), and discuss its concavity.

52. The spruce budworm is a moth whose larvae eat the leaves of coniferous trees. These insects suffer predation by birds. Ludwig and colleagues* suggested as model for the per capita predation rate, p(x):

where b is the maximum predation rate and a is the number of budworms at which the predation rate is half its maximum rate. What is the concavity of this curve, and is there a point of inflection?

53. In Example 4 of this section, we fitted a fifth-order polynomial to the data on the distance covered as a function of time during Usain Bolt's record-breaking 100-meter win in the 2008 Beijing Olympic Games. Use technology to fit a third-order polynomial to these data; then differentiate this polynomial to obtain the corresponding velocity and acceleration functions of time. Plot these three functions (displacement, velocity, acceleration) and estimate the time t_s at which Bolt switches from acceleration to deceleration during the course of the race and his velocity v_max, which is a maximum, at this switching point. What is the average deceleration from the switching point to the end of the race? Compare your values of t_s, v_max, and average deceleration with those obtained in Example 4.

54. In Example 4 of this section, we fitted a fifth-order polynomial to the data on the distance covered as a function of time during Usain Bolt's record-breaking 100-meter win in the 2008 Beijing Olympic Games. Use technology to fit a seventh-order polynomial to these data; then differentiate this polynomial to obtain the corresponding velocity and acceleration functions of time. Plot these three functions (displacement, velocity, acceleration) and estimate the time t_s at which Bolt switches from acceleration to deceleration during the course of the race and his velocityv_max, which is a maximum, at this switching point. What is the average deceleration from the switching point to the end of the race? Compare your values of t_s, v_max, and average deceleration with those obtained in Example 4.

55. In Example 4 of this section, we fitted a fifth-order polynomial to the data on the distance covered as a function of time during Usain Bolt's record-breaking 100-meter win in the 2008 Beijing Olympic Games. Use Ben Johnson's split times given in Table 2.1 in Example 3 of Section 2.1 to obtain data relating distance and time of Johnson's performance during the 1988 Seoul Olympics 100-meter final. Fit a fifth-order polynomial to the data; then differentiate this polynomial to obtain the corresponding velocity and acceleration functions of time. Plot these three functions (displacement, velocity, acceleration) and estimate the time t_s at which Johnson switches from acceleration to deceleration during the course of the race. What is Johnson's maximum velocity, which occurs at t_s, and what is his average deceleration from time t_s until the end of the race? Compare your results to Bolt's performance analyzed in Example 4.

56. Let f be a function that is twice differentiable on an interval I containing the point x = a. If there exists a K > 0 such that |f″(x)|≤K for all x in I, then show that

for all x in I. This result gives the error of the first-order approximation. Hint: Pick any point b ≠ a in I. Define

G(x) = f(x) − f(a) − f′(a) (x −a) − C(x − a)²

where C is chosen such that G(b) = 0. Differentiate G and apply the mean value theorem to G and f′.

57. Let f be a function with first- and second-order derivatives at x = a. Consider a quadratic of the form q(x) = b + c(x −a) + d(x − a)². Show that f(a) = q(a), f′(a) = q′(a), and f″(a) = q″(a) if and only if b = f(a), c = f′(a) and d = f″(a)/2.

3.7 l'Hôpital's Rule

In models of tumor or population growth, spread of rumors, and risk of being infected, one may encounter limits of the form

where lim_x→a f(x) and lim_x→a g(x) are both zero or both infinite. Such limits are called 0/0 indeterminate form and ∞/∞ indeterminate form, respectively, because their value cannot be determined without further analysis. In this section, we study an approach to handling these limits and explore some applications.

The 0/0 and ∞/∞ indeterminate forms

In 1694, French mathematician Guillaume de l'Hôpital (see Historical Quest in Problem Set 3.7) found a useful method for evaluating limits involving indeterminate forms. He considered the special case where f(a) = g(a) = 0, f and g are differentiable at x = a, and g′(a) ≠ 0. Under these assumptions, we have

The following example illustrates how replacing the original limit with a limit involving derivatives can be helpful.

This approach to computing limits is known as l'Hôpital's rule. The general statement of this rule is given in the following theorem.

We consider two applications of l'Hôpital's rule to models of population growth.

What do tumor growth, sales of mobile phones, spread of rumor or infection, and population growth have in common? They all can be modeled mathematically by specifying how the growth rate of the tumor, rumor, or population depends on its current size, frequency, or abundance. The next example looks an important family of growth functions introduced by F. J. Richards in a 1959 article that appeared in the Journal of Experimental Botany[2: 290–301].

Sometimes repeated applications of l'Hôpital's rule are necessary to get anywhere.

Note that L'Hôpital's rule is not the only way to solve the above example. We could have divided both the numerator and denominator by x² to obtain

Most examples in this section, however, do not yield to this simple procedure, so l'Hôpital's rule must be used. Before applying l'Hôpital's rule, however, we must check that the conditions of Theorem 3.1 apply. If they do not hold, then the analysis is not valid, as illustrated by the next two examples.

If you apply l'Hôpital's rule in Example 6, you obtain the WRONG answer:

Other indeterminate forms

Remember that l'Hôpital's rule itself applies only to the indeterminate forms 0/0 and ∞/∞. Other indeterminate forms, such as 1^∞, 0⁰, ∞⁰, ∞ − ∞, and 0 · ∞, can often be manipulated algebraically, or by taking logarithms, into one of the standard forms 0/0 or ∞/∞, and then evaluated using l'Hôpital's rule. In a case where we have taken the logarithm to obtain one of the standard forms, we need to remember to transform back by applying exponentiation to our solution.

We saw in Figure 3.38 that the graph of f(x) = x^1/x approaches the line y = 1 asymptotically as x → ∞, but how does f(x) behave as x → 0⁺? That is, what is

It may seem that to answer this question, we need to apply l'Hôpital's rule again, but this limit has the form 0^∞, which is simply 0 and is not indeterminate at all. Other forms that may appear to be indeterminate, but really are not, are 0/∞, ∞ · ∞, ∞ + ∞ and − ∞ − ∞.

Level 1 DRILL PROBLEMS

1. An incorrect use of l'Hôpital's rule is illustrated in the following limit computations. In each case, explain what is wrong and find the correct value of the limit.

2. Sometimes l'Hôpital's rule leads nowhere. For example, observe what happens when the rule is applied to

Use any method you wish to evaluate this limit.

Find the limits, if possible, in Problems 3 to 18.

In Problems 19 to 22, use l'Hôpital's rule to determine all horizontal asymptotes to the graph of the given function. You are NOT required to sketch the graph.

Verify the statements in Problems 23 to 25.

Level 2 APPLIED AND THEORY PROBLEMS

26. Fisheries scientists have found that a Ricker stock-recruitment relationship, which has the form

where y is a measure (also called an index) of the number of individuals recruited to the fishery each year (typically one-year-olds), and x is an index of the spawning stock biomass (sometimes measured in terms of eggs produced), provides a reasonable fit to various species. Consider the case where the parameter values are a = 5.9 and b = 0.0018.

1. What is the value of the recruitment index as x → ∞?
2. What is the maximum value of the recruitment index and at what spawning stock index value does it occur?
3. Over what range of spawning stock index values is the recruitment function concave up and over what values is it concave down?
4. Use the information obtained in parts a, b, and c to sketch this function.

27. An agronomist experimenting with a new breed of giant potato has found that individual tubers x months after planting have a biomass in kilograms given by the equation y(x) = 2e^−1/(5x) for x > 0.

1. Calculate the rate of growth of the tuber over time and determine what happens to this rate in the limit as x → 0 and x → ∞.
2. Find the time after planting when the growth rate of the tuber is maximized.
3. Show that the growth rate is positive for all x > 0 and determine the regions over which the growth is accelerating and decelerating.
4. Sketch the biomass of the potato, as well as its growth rate, indicating the important points and regions calculated in parts a, b, and c.

28. Determine which function, f(x) = xⁿ with n > 0 or g(x) = e^ax with a > 0, grows faster at ∞ by computing .

29. Determine which function, f(x) = xⁿ with n > 0 or g(x) = ln x, grows faster at ∞ by computing .

30. Consider a drug in the body whose current concentration is 1 mg/liter. In this problem, you investigate the meaning of exponential decay of the drug.

1. If one-half of the drug particles cleared the body after one hour, determine the concentration of the drug that remains after one hour.
2. If one-quarter of the drug particles cleared the body every half an hour, determine the concentration of the drug that remains after one hour.
3. If one-twentieth of the drug particles cleared the body every six minutes, determine the concentration of the drug that remains after one hour.
4. If of the drug particles cleared the body every 1/nth of an hour, determine the concentration c_n of the drug that remains after one hour.
5. Find c_n.

31. The French mathematician Guillaume de l'Hôpital (1661–1704) is best known today for the rule that bears his name, but that rule was discovered by l'Hôpital's teacher, Johann Bernoulli. Not only did l'Hôpital neglect to cite his sources in his book, but there is also evidence that he paid Bernoulli for his results and for keeping their arrangements for payment confidential. In a letter dated March 17, 1694, he asked Bernoulli “to communicate to me your discoveries”—with the request not to mention them to others: “it would not please me if they were made public.” (See D. J. Stuik, A Source Book in Mathematics, 1200–1800, Cambridge, MA: Harvard University Press, 1969, 313–316.) L'Hôpital's argument, which was originally given without using functional notation, can easily be reproduced:

First, place some conditions on the functions f and g that will make this argument true. Next, supply reasons for this argument, and give necessary conditions for the functions f and g.

32. Consider the general logistic growth function

from Example 4.

1. For v > 0, find the density N > 0 that maximizes G(N); that is, solve G′(N) = 0. The answer will depend on the parameters r, v, and K.
2. Take your answer from part a and compute its limit as v → 0.
3. Find density N > 0 that maximizes−rN log(N/K) and compare your answer to what you found in part b.

1. Find for the following expressions.

   

2. Approximate 65^1/3 and determine whether this approximation overestimates or underestimates the true answer. Justify your answers by using derivatives.
3. Find where y = x²(2x − 3)³.
4. Use the definition of derivative to calculate, showing all details, (x − 3x²).
5. Find the first- and second-order approximations to y = e^x2 at x = 0. Graph the function and its approximations.
6. In Figure 3.39, which graph represents the function and which graph the derivative?

   

   Figure 3.39 A function and its derivative.

7. Sketch the graph of a function with the following properties:

   

   What can you say about the derivative of f when x = 1?

8. The developmental rate of insects and plants as a function of temperature T can be modeled by the Briere model

   

   where T_L is the lower developmental threshold below which an individual does not develop, T_U is the upper developmental threshold above which an individual does not develop, and a is a proportionality constant. Find a linear approximation to D(T) at T = T_L, discuss what it means, and discuss where it breaks down.

9. Let f be a function defined by

   

   Determine where the function is increasing, where it is decreasing, and where the graph is concave up and where it is concave down.

10. Suppose the proportion of insect hosts escaping parasitism depends on the parasitoid density, d, and is modeled by the function f(d) = e^−.05d. Does the proportion escaping parasitism increase or decrease with parasitoid density? What is the concavity of this curve, and is there a point of inflection?
11. Determine the concavity and inflection points and use l'Hôpital's rule to find the horizontal asymptotes of the graph of

    

12. Suppose the concentration in the blood at time t of a drug injected into the body is modeled by

    

    Use l'Hôpital's rule to find the horizontal asymptote. Find the time when C′(t) = 0. Graph this curve and verify that the largest concentration occurs at the solution to C′(t) = 0.

13. Say you want to estimate the height of a tall tree. To do so, you cannot simply drop a tape measure from the top of the tree. However, you can determine the height by using a sextant to determine the angle θ between the ground and the tip of the tree at a distance of 100 feet from the base of the tree.
    1. Find the height of the tree, H, as a function of θ.
    2. If you measure an angle θ = 1.1 radians, determine the height of the tree.
    3. Determine the elasticity of the height in part b to θ. Discuss how a 10% error in measuring θ influences the estimate for the height of the tree when θ = 1.1.
14. A bacterial colony is estimated to have a population of P thousand individuals, where

    

    and t is the number of hours after a toxin is introduced.

    1. At what rate is the population changing when t = 0 and t = 1?
    2. Is the rate increasing or decreasing at t = 0 and t = 1?
    3. At what time does the population begin to decrease?
15. As we saw in Example 6 of Section 1.5, scientists at Woods Hole Oceanographic Institution measured the uptake rate of glucose by bacterial populations from the coast of Peru. In one field experiment, they found that the uptake rate can be modeled by f(x) = micrograms per hour, where x is micrograms of glucose per liter. If the current uptake rate is twelve, determine the current level of glucose and, hence, determine the rate at which this uptake rate is itself changing per unit increase in glucose.
16. The gross U.S. federal debt (in trillions of dollars) is plotted below.

    

    Regarding this debt, President Ronald Reagan stated in 1979 that the United States is “going deeper into debt at a faster rate than we ever have before.” Discuss the role of higher-order derivatives in the graph of federal debt from 1950 to the end of the graph in the context of President Reagan's statement.

17. The figure eight curve shown below is defined implicitly by the equation x⁴ = x² − y².

    

    Find all the points on this curve that have horizontal tangents.

18. Consider the functions y = f(x) (in blue) and y = g(x) (in red) whose graphs are shown below.

    

    Find f(g(x)) at x = −1 and x = 1.

19. Find the tangent line of y = (1 + sin 2x)¹⁰⁰ at x = π/2.
20. We modeled the concentration of carbon dioxide (in ppm) at the Mauna Loa Observatory of Hawaii with the function

    

    where t is months since May 1974. Find for what months h is increasing and decreasing over the interval [0, 12].