Rates of Change in Applications

Rates of Change in Applications
[ Home ] [ Up ]

3.8

1. A variable that is defined only at isolated points is called a discrete variable. For example, the size of a calculus class is discrete, since it must be an integer. A variable that can assume an interval of values is called continuous. In the six applications discussed in the text, in which cases is time a discrete variable and in which cases is it continuous? Which functions are discrete and which are continuous? Among the six examples given in the text, in which cases were the rates of change truly derivatives? Explain this answer in terms of discrete and continuous variables.

2. An important model of population growth is the so-called logistic equation x' (t) = x(t)[1 - x(t)]. Here, x(t) represents not the actual population size but the proportion of sustainable capacity: for instance, x(t) = 0.5 means that the population is half of the total number of organisms that the environment can support and x(t) = 1.1 means that there are 10% more organisms than the available resources can support. Note that the differential equation here is the same as was used to describe an autocatalytic chemical reaction. The equation has two competing contributions to the rate of change x' (t). The term x(t) by itself would mean that the larger x(t) is, the faster the population (or concentration of chemical) grows. This is balanced by the term 1 - x(t), which indicates that the closer x(t) gets to 1, the slower the population growth is. With these two terms together, the model has the property that for small x(t), slightly larger x(t) means greater growth, but as x(t) approaches 1, the growth tails off. Explain in terms of population growth and the concentration of chemical why the model is reasonable.

3. Corporate deficits and debt are frequently in the news, but the terms are often confused with each other. To take an example, suppose a company finishes a fiscal year owing $5,000. That is their debt. Suppose that in the following year the company has revenues of $106,000 and expenses of $109,000. The company's deficit for the year is $3,000 and the company's debt has increased to $8,000. Briefly explain why deficit is the derivative of debt.

4. Many people find the healthy heart example (Example 8.4) surprising. To make it more believable, explain how your level of activity affects your heart rate. Also, explain how breathing rate and emotional status can affect heart rate. Given these and many other factors, discuss whether you should expect your heart rate to be exactly constant.

5. Suppose that the charge in an electrical circuit is Q(t) = e^{- 2t}(cos 3t - 2sin 3t) coulombs. Find the current.

6. Suppose that the charge in an electrical circuit is Q(t) = e^t(3cos 2t + sin 2t) coulombs. Find the current.

7. Suppose that the charge at a particular location in an electrical circuit is Q(t) = e^{- 3t}cos 2t + 4sin 3t coulombs. What happens to this function as t ? Explain why the term e^{- 3t}cos 2t is called a transient term and 4sin 3t is known as the steady-state or asymptotic value of the charge function. Find the transient and steady-state values of the current function.

8. As in exercise 7, find the steady-state and transient values of the current function if the charge function is given by Q(t) = e^{- 2t}(cos t - 2sin t) + te^{- 3t} + 2cos 4t.

9. If the concentration of a chemical changes according to the equation x' (t) = 2x(t)[4 - x(t)], find the concentration x(t) for which the reaction rate is a maximum.

10. If the concentration of a chemical changes according to the equation x' (t) = 0.5x(t)[5 - x(t)], find the concentration x(t) for which the reaction rate is a maximum.

11. Show that in exercise 9, the maximum concentration is 4 if 0 < x(0) < 4. Find the maximum concentration in exercise 10.

12. Find the equation for an autocatalytic reaction in which the maximum concentration is x(t) = 16 and the reaction rate equals 12 when x(t) = 8 .

13. Mathematicians often study equations of the form x' (t) = rx(t)[1 - x(t)] instead of the more complicated x'(t) = cx(t)[K - x(t)], justifying the simplification with the statement that the second equation “reduces to” the first equation. Starting with y' (t) = cy(t)[K - y(t)], substitute y(t) = Kx(t) and show that the equation reduces to the form x' (t) = rx(t) [1 - x(t)]. How does the constant r relate to the constants c andK?

14. Suppose a chemical reaction follows the equation x'(t) = cx(t)[K - x(t)]. Suppose that at time t = 4 the concentration is x(4) = 2 and the reaction rate is x' (4) = 3. At time t = 6, suppose that the concentration is x(6) = 4 and the reaction rate is x' (6) = 4. Find the values of c and K for this chemical reaction.

15. In a general second-order chemical reaction, chemicals A and B (the reactants) combine to form chemical C (the product). If the initial concentrations of the reactants A and B are a and b, respectively, then the concentration x(t) of the product satisfies the equation x' (t) = [a - x(t)][b - x(t)]. What is the rate of change of the product when x(t) = a ? At this value, is the concentration of product increasing, decreasing or staying the same? Assuming that a < b and there is no product present when the reaction starts, explain why the maximum concentration of product is x(t) = a .

16. For the second-order reaction defined in exercise 15, find the (mathematical) value of x(t) that minimizes the reaction rate. Show that the reaction rate for this value of x(t) is negative. Explain why the concentration x(t) would never get this large, so that this mathematical solution is not physically relevant. Explain why x(t) must be between 0 and a, and find the maximum and minimum reaction rates on this closed interval.

17. It can be shown that a solution of the equation x' (t) =

[a - x(t)][b - x(t)] is given by
.
Find x(0), the initial concentration of chemical, and , the limiting concentration of chemical (assume a < b ). Graph x(t) on the interval [0, ) and describe in words how the concentration of chemical changes over time.

18. For the solution in exercise 17, find and graph x' (t). Compute and describe in words how the reaction rate changes

over time.

In exercises 19-22, the mass of the first x meters of a thin rod is given by the function m(x) on the indicated interval. Find the linear density function for the rod. Based on what you find, briefly describe the composition of the rod.

19. m(x) = 4x - sin x grams for 0 x 6

20. m(x) = (x -1)³ + 6x grams for 0 x 2

21. m(x) = 4x grams for 0 x 2

22. m(x) = 4x² grams for 0 x 2

In exercises 23-26, you are given times (in seconds) at which peaks in a heart patient's EKG occurred. Analyze the heart rate and diagnose the patient as resting comfortably or in danger of a heart attack.

23. 0.0, 0.99, 2.12, 3.19, 4.12, 5.17, 6.08, 6.95, 8.16, 9.26, 10.24, 11.28, 12.22, 13.14, 14.21, 15.18

24. 0.0, 0.99, 2.02, 3.01, 4.02, 5.03, 6.01, 6.99, 8.01, 9.02, 10.02, 11.02, 12.02, 13.01, 14.01, 15.02

25. 0.0, 0.98, 1.96, 2.90, 3.80, 4.74, 5.72, 6.70, 7.70, 8.72, 9.74, 10.80, 11.90, 13.06, 14.08, 15.10

26. 0.0, 0.98, 1.96, 3.02, 3.98, 5.04, 6.22, 7.17, 8.15, 9.17, 10.13, 11.08, 12.11, 13.06, 14.08, 15.10

27. If the cost of manufacturing x items is C(x) = x³ + 20x² + 90x + 15, find the marginal cost function and compare the marginal cost at x = 50 and the actual cost of manufacturing the 50th item.

28. If the cost of manufacturing x items is C(x) = x⁴ + 14x² + 60x + 35, find the marginal cost function and compare the marginal cost at x = 50 and the actual cost of manufacturing the 50th item.

29. If the cost of manufacturing x items is C(x) = x³ + 21x² + 110x + 20, find the marginal cost function and compare the marginal cost at x = 100 and the actual cost of manufacturing the 100th item.

30. If the cost of manufacturing x items is C(x) = x³ + 11x² + 40x + 10, find the marginal cost function and compare the marginal cost at x = 100 and the actual cost of manufacturing the 100th item.

31. Suppose the cost of manufacturing x items is C(x) = x³ - 30x² + 300x + 100 dollars. Find the inflection point and discuss the significance of this value in terms of the cost of manufacturing.

32. A baseball team owner has determined that if tickets are priced at $10, the average attendance at a game will be 27,000 and if tickets are priced at $8, the average attendance will be 33,000. Using a linear model, we would then estimate that tickets priced $9 would produce an average attendance of 30,000. Discuss whether or not you think the use of a linear model here is reasonable. Then, using the linear model, determine the price at which the revenue is maximized.

33. The learning curve f (t) = 80/(1 + 3e^-0.4t) was discussed in Example 8.6. Show that and show that f (t) is an increasing function for t > 0. Discuss whether you think both results are reasonable properties for a learning curve.

34. Suppose the maximum score that a person will ever score onatest is 90. Further suppose that the person scores 60 after 2hours of studying. Find a learning curve of the form f (t) = a/(1 + 3e^{- bt}) for this person on this test.

35. Suppose that a person scores f (t) = 180/(2 + 4e^-0.2t) after t hours of studying. What is the person's score after 3 hours of studying? Find f ' (3) and estimate how many additional points the person would earn by studying a fourth hour. Find f ' (10) and estimate how many additional points the person would earn by studying an eleventh hour.

36. Suppose that a person scores f (t) = 10/(1 + 4e^-0.5t) after t hours of studying. What is the person's score after 4 hours of studying? Find f ' (4) and estimate how many additional points the person would earn by studying a fifth hour. Find f ' (10) and estimate how many additional points the person would earn by studying an eleventh hour.

37. The function f (t) = a/(1 + 3e^{- bt}) has also been used to model the spread of a rumor. Suppose that a = 70 and b = 0.2. Compute f (2), the percentage of the population that has heard the rumor after 2 hours. Compute f ' (2) and describe what it represents. Compute and describe what it represents.

38. After an injection, the concentration of drug in a muscle varies according to a function of time, f (t). Suppose that t is measured in hours and f (t) = e^-0.02t - e^-0.42t. Determine the time when the maximum concentration of drug occurs.

39. Suppose that the size of the pupil of an animal is given by f (x) (mm), where x is the intensity of the light on the pupil. If

show that f (x) is a decreasing function. Interpret this result in terms of the response of the pupil to light.

40. Suppose that the body temperature 1 hour after receiving x mg of a drug is given by for 0 x 6. The absolute value of the derivative, | T' (x)|, is defined as the sensitivity of the body to the drug dosage. Find the dosage which maximizes sensitivity.

41. Let C(x) be the cost of manufacturing x items and define (x) = C(x)/x as the average cost function. Suppose that C(x) = 0.01x² + 40x + 3600. Show that C' (100) < (100) and show that increasing the production (x) by 1 will decrease the average cost.

42. For the cost function in exercise 41, show that C' (1000) > (1000) and show that increasing the production (x) by 1 will increase the average cost.

43. For the cost function in exercise 41, prove that average cost is minimized at the x - value, where C' (x) = (x) .

44. If the cost function is linear, C(x) = a + bx with a and b positive, show that there is no minimum average cost and that C'(x) (x) for all x.

45. Let R(x) be the revenue and C(x) be the cost from manufacturing x items. Profit is defined as P(x) = R(x) - C(x). Show that at the value of x that maximizes profit, marginal revenue equals marginal cost.

46. Find the maximum profit if R(x) = 10x -0.001x² dollars and C(x) = 2x + 5000 dollars.

47. In the titration of a weak acid and strong base, the pH is given by c + ln where c is a constant (closely related to the acid dissociation constant) and f is the fraction ( 0 < f < 1 ) of converted acid (see Harris' Quantitative Chemical Analysis for more details). Find the value of f at which the rate of change of pH is the smallest. What happens as f approaches 1?

48. In exercise 47, you found the significance of one inflection point of a titration curve. A second inflection point, called the equivalence point, corresponds to f = 1. In the generalized titration curve below, identify on the graph both inflection points and briefly explain why chemists prefer to measure the equivalence point and not the inflection point of exercise 47. (Note: the horizontal axis of a titration curve indicates the amount of base added to the mixture. This is directly proportional to the amount of converted acid in the region where 0 < f < 1 .)

49. Epidemiology is the study of the spread of infectious diseases. A simple model for the spread of fatal diseases such as AIDS divides people into the categories of susceptible (but not exposed), exposed (but not infected) and infected. The proportions of people in each category at time t are denoted S(t), E(t) and I(t), respectively. The general equations for this model are
S' (t) = mI(t) - bS(t)I(t),
E' (t) = bS(t)I(t) - aE(t),
I' (t) = aE(t) - mI(t),
where m, b and a are positive constants. Notice that each equation gives the rate of change of one of the categories. Each rate of change has both a positive and negative term. Explain why the positive term represents people who are entering the category and the negative term represents people who are leaving the category. In the first equation, the term mI(t) represents people who have died from the disease (the constant m is the reciprocal of the life expectancy of someone with the disease). This term is slightly artificial: the assumption is that the population is constant, so that when one person dies, a baby is born who is not exposed or infected. The dynamics of the disease is that susceptible (healthy) people get infected by contact with infected people. Explain why the number of contacts between susceptible people and infected people is proportional to S(t) and I(t). The term bS(t)I(t), then, represents susceptible people who have been exposed by contact with infected people. Explain why this same term shows up as a positive in the second equation. Explain the rest of the remaining two equations in this fashion. (Hint: The constant a represents the reciprocal of the average latency period. In the case of AIDS, this would be how long it takes an HIV-positive person to actually develop AIDS.)

50. Without knowing how to solve differential equations (we hope you will go far enough in your study of mathematics to learn to do so!), we can nonetheless deduce some important properties of the solutions of differential equations. For example, consider the equation for an autocatalytic reaction x' (t) = x(t)[1 - x(t)]. Suppose x(0) lies between 0 and 1. Show that x' (0) is positive by determining the possible values of x(0)[1 - x(0)]. Explain why this indicates that the value of x(t) will increase from x(0), and will continue to increase as long as 0 < x(t) < 1. Explain why if x(0) < 1 and x(t) > 1 for some t > 0, then it must be true that x(t) = 1 for some t > 0. However, if x(t) = 1, then x' (t) = 0 and the solution x(t) stays constant (equal to 1). Therefore, we can conjecture that . Similarly, show that if x(0) > 1, then x(t) decreases and we could again conjecture that . Changing equations, suppose that x' (t) = -0.05x(t) + 2. This is a model of an experiment in which a radioactive substance is decaying at the rate of 5% but the substance is being replenished at the constant rate of 2. Find the value of x(t) for which x' (t) = 0. Pick various starting values of x(0) less than and greater than the constant solution and determine if the solution x(t) will increase or decrease. Based on these conclusions, conjecture the value of , the limiting amount of radioactive substance in the experiment.