CHAPTER SUMMARY AND REVIEW PROBLEMS CHAPTER SUMMARY AND REVIEW PROBLEMS. Chapter 4 Review Chapter Summary and Review Problems 357

Transcription

1 Chapter 4 Review Chapter Summary and Review Problems 357 THE SPREAD OF AN EPIDEMIC MARKET RESEARCH OPTIMAL HOLDING TIME 48. An epidemic spreads throughout a community so that t weeks after its outbreak, the number of residents who have been infected is given by a function of the form A f(t), where A is the total number of susceptible residents. Show that 1 Ce kt the epidemic is spreading most rapidly when half of the susceptible residents have been infected. 49. Use the graphing utility of your calculator to sketch the graph of f(x) x(e x e 2x ). Use zoom and trace to find the largest value of f(x). What happens to f(x) as x 50. A company is trying to use television advertising to expose as many people as possible to a new product in a large metropolitan area with 2 million possible viewers. A model for the number of people N (in millions) who are aware of the product after t days is found to be N 2(1 e 0.037t ). Use a graphing utility to graph this function. What happens as t? (Suggestion: Set the range on your viewing screen to [0, 200]10 by [0, 3]1.) 51. Suppose you win a parcel of land whose market value t years from now is estimated to be V(t) 20,000te 0.4t dollars. If the prevailing interest rate remains constant at 7% compounded continuously, when will it be most advantageous to sell the land? (Use a graphing utility and zoom and trace to make the required determination.) CHAPTER SUMMARY AND REVIEW PROBLEMS CHAPTER SUMMARY AND REVIEW PROBLEMS IMPORTANT TERMS, SYMBOLS, AND FORMULAS Exponential function: f(x) b x Rules of exponents: b x b y b x y b x y (b x ) y b xy b x b y The number e lim nfi 1 n 1 n 2.718

2 358 Chapter 4 Exponential and Logarithmic Functions Q(t) Q(t) Q 0 Q 0 t t Exponential growth: Q(t) Q 0 e kt Q (t) kq(t) Percentage rate of change 100k The natural logarithm: y ln x if and only if x e y Inverse relationship: e ln x x and ln e x x Properties of logarithms: ln u ln v if and only if u v u ln uv ln u ln v ln ln u ln v ln u r r ln u v Conversion formula for logarithms: log b x ln x ln b Doubling time and half-life Differentiation formulas: d and dx (ln x) 1 x d and dx (ex ) e x Logarithmic differentiation Simple interest formula B(t) P(1 rt) Exponential decay: Q(t) Q 0 e kt Q (t) kq(t) Percentage rate of change 100k b 0, b 1 d 1 du [ln u(x)] dx u(x) dx d dx [eu(x) ] e u(x)du dx (r expressed as a decimal)

3 Chapter 4 Review Chapter Summary and Review Problems 359 Interest compounded k times per year: Balance B(t) P Effective rate Present value B 1 r k kt 1 r k k 1 1 r k kt Continuously compounded interest: Balance B(t) Pe rt Effective rate e r 1 Present value Be rt Optimal holding time Q(t) Q(t) B B A 0 t B B 1 + A 0 ln A Bk t Learning curve: Q(t) B Ae kt Logistic curve: Q(t) B 1 Ae Bkt REVIEW PROBLEMS 1. Sketch the graph of each of the following functions: (a) f(x) 5e x (b) f(x) 5 2e x (c) f(x) e 3x (d) f(x) 3 2e 2x 1 e 2x 2. (a) Find f(4) if f(x) Ae kx and f(0) 10, f(1) 25. (b) Find f(3) if f(x) Ae kx and f(1) 3, f(2) 10. (c) Find f(9) if f(x) 30 Ae kx and f(0) 50, f(3) (d) Find f(10) if f(t) and f(0) 3, f(5) 2. 1 Ae kt

4 360 Chapter 4 Exponential and Logarithmic Functions 3. The value of a certain industrial machine is decreasing exponentially. If the machine was originally worth $50,000 and was worth $20,000 when it was 5 years old, how much will it be worth when it is 10 years old? 4. It is estimated that if x thousand dollars are spent on advertising, approximately Q(x) 50 40e 0.1x thousand units of a certain commodity will be sold. (a) Sketch the relevant portion of this sales function. (b) How many units will be sold if no money is spent on advertising? (c) How many units will be sold if $8,000 is spent on advertising? (d) How much should be spent on advertising to generate sales of 35,000 units? (e) According to this model, what is the most optimistic sales projection? 5. The daily output of a worker on the job for t weeks is Q(t) 120 Ae kt units. Initially the worker could produce 30 units per day, and after 8 weeks the worker could produce 80 units per day. How many units could the worker produce per day after 4 weeks? 6. It is estimated that t years from now the population of a certain country will be 30 P(t) million. 1 2e 0.05t (a) Sketch the graph of P(t). (b) What is the current population? (c) What will be the population 20 years from now? (d) What will happen to the population in the long run? 7. Evaluate the following expressions without using tables or a calculator. (a) ln e 5 (b) e ln 2 3 ln 4 ln 2 (c) e (d) ln 9e 2 ln 3e 2 8. Solve for x. (a) 8 2e 0.04x (b) 5 1 4e 6x (c) 4 ln x 8 (d) 5 x e 3 9. The number of bacteria in a certain culture grows exponentially. If 5,000 bacteria were initially present and 8,000 were present 10 minutes later, how long will it take for the number of bacteria to double? 10. Differentiate the following functions: (a) f(x) 2e 3x 5 (b) f(x) x 2 e x (c) g(x) ln x 2 4x 1 (d) h(x) x ln x 2

5 Chapter 4 Review Chapter Summary and Review Problems 361 t (e) f(t) (f) g(t) t log 3 (t 2 ) ln 2t 11. How quickly will $2,000 grow to $5,000 when invested at an annual interest rate of 8% if interest is compounded: (a) Quarterly (b) Continuously 12. Which investment has the greater effective interest rate: 8.25% per year compounded quarterly or 8.20% per year compounded continuously? 13. How much should you invest now at an annual interest rate of 6.25% so that your balance 10 years from now will be $2,000 if interest is compounded: (a) Monthly (b) Continuously 14. Find the present value of $8,000 payable 10 years from now if the annual interest rate is 6.25% and interest is compounded: (a) Semiannually (b) Continuously 15. A bank compounds interest continuously. What (nominal) interest rate does it offer if $1,000 grows to $2, in 12 years? 16. A certain bank offers an interest rate of 6% per year compounded annually. A competing bank compounds its interest continuously. What (nominal) interest rate should the competing bank offer so that the effective interest rates of the two banks will be equal? 17. An environmental study of a certain suburban community suggests that t years from now, the average level of carbon monoxide in the air will be Q(t) 4e 0.03t parts per million. (a) At what rate will the carbon monoxide level be changing with respect to time 2 years from now? (b) At what percentage rate will the carbon monoxide level be changing with respect to time t years from now? Does this percentage rate of change depend on t or is it constant? 18. A manufacturer can produce cameras at a cost of $40 apiece and estimates that if they are sold for p dollars apiece, consumers will buy approximately D(p) 800e 0.01p cameras per week. At what price should the manufacturer sell the cameras to maximize profit? 19. Determine where each function is increasing and decreasing, and where its graph is concave up and concave down. Find the relative extrema and inflection points and draw the graph.

6 362 Chapter 4 Exponential and Logarithmic Functions (a) f(x) xe 2x (b) f(x) e x e x 4 (c) f(x) (d) f(x) ln (x 2 1) 1 e x 20. Suppose that you own a coin collection whose value t years from now will be V(t) 2,000e t dollars. If the prevailing interest rate remains constant at 7% per year compounded continuously, when will it be most advantageous for you to sell the collection and invest the proceeds? 21. A Cro-Magnon cave painting at Lascaux, France, is approximately 15,000 years old. Approximately what ratio of 14 C to 12 C would you expect to find in a fossil found in the same cave as the painting? 22. In 1389, Pierre d Arcis, the bishop of Troyes, wrote a memo to the pope, accusing a colleague of passing off a certain cloth, cunningly painted as the burial shroud of Jesus Christ. Despite this early testimony of forgery, the image on the cloth was so compelling that many people regarded it as a sacred relic. Known as the shroud of Turin, the cloth was subjected to carbon dating in If authentic, the cloth would have to be approximately 1,960 years old. (a) If the shroud were actually 1,960 years old, what percentage of the 14 C would remain? (b) Scientists determined that 92.3% of the shroud s original 14 C remained. How old is the shroud? 23. Instant coffee is made by adding boiling water (212 F) to coffee mix. If the air temperature is 70 F, Newton s law of cooling says that after t minutes, the temperature of the coffee will be given by a function of the form f(t) 70 Ae kt. After cooling for 5 minutes, the coffee is still 15 F too hot to drink, but 2 minutes later it is just right. What is this ideal temperature for drinking? 24. It is sometimes useful for actuaries to be able to project mortality rates within a given population. A formula once used for computing the mortality rate D(t) for women in the age group is D(t) (D )e 0.162t where t is the number of years after a fixed base year and D 0 is the mortality rate when t 0. (a) Suppose the initial mortality rate of a particular group is (8 deaths per 1,000 women). What is the mortality rate of this group 10 years later? What is the rate 25 years later? (b) Sketch the graph of the mortality function D(t) for the group in part (a) for 0 t 25.

7 Chapter 4 Review Chapter Summary and Review Problems Investors are often interested in knowing how long it takes for a particular investment to double. A simple means for making this determination is the rule of 70, which says: The doubling time of an investment with an annual interest rate r 70 (expressed as a decimal) compounded continuously is given by d. r (a) For interest rate r, use the formula B Pe rt to find the doubling time for r 4, 6, 9, 10, and 12. In each case, compare the value with the value obtained from the rule of 70. (b) Some people prefer a rule of 72 and others use a rule of 69. Test these alternative rules as in part (a) and write a paragraph on which rule you would prefer to use. 26. The aerobic rating of a person x years old is (ln x 2) A(x) 110 for x 10 x At what age is aerobic capacity maximized? 27. A naturalist at an animal sanctuary has determined that the function f(x) (ln x)2 4e x provides a good measure of the number of animals in the sanctuary that are x years old. Sketch the graph of f(x) for x 0 and find the most likely age among the animals (that is, the age for which f(x) is largest). 28. The Ice Man is the name given a Neolithic corpse found frozen in an Alpine glacier. He was originally thought to be from the Bronze Age because of the hatchet he was carrying. However, the hatchet proved to be made of copper rather than bronze. Read an article on the Bronze Age and determine the least age of the Ice Man assuming that he dates before the Bronze Age. What is the largest percentage of 14 C that can remain in a sample taken from his body? 29. Fick s law* says that f(t) C(1 e kt ), where f(t) is the concentration of solute inside a cell at time t, C is the (constant) concentration of solute surrounding the cell, and k is a positive constant. Suppose that for a particular cell, the concentration on the inside of the cell after 2 hours is 0.8% of the concentration outside the cell. (a) What is k? * Fick s law plays an important role in ecology. For instance, see M. D. LaGrega, P. L. Buckingham, and J. C. Evans, Hazardous Waste Management, McGraw-Hill, Inc., New York, 1994, pages 95, 464, and especially page 813, where the authors discuss contaminant transport through landfill.

8 364 Chapter 4 Exponential and Logarithmic Functions (b) What is the percentage rate of change of f(t) at time t? (c) Write a paragraph on the role played by Fick s law in ecology. 30. A child falls into a lake where the water temperature is 3 C. Her body temperature after t minutes in the water is T(t) 35e 0.32t. She will lose consciousness when her body temperature reaches 27 C. How long do rescuers have to save her? How fast is her body temperature dropping at the time it reaches 27 C? 31. When a chain, a telephone line, or a TV cable is strung between supports, the curve it forms is called a catenary. A typical catenary curve is y 0.125(e 4x e 4x ) (a) Sketch the catenary curve. (b) Catenary curves are important in architecture. Read an article on the Gateway Arch to the West in St. Louis, Missouri, and write a paragraph on the use of the catenary shape in its design.* 32. The effect of temperature on the reaction rate of a chemical reaction is given by the Arrhenius equation k A e E 0/RT where k is the rate constant, T (in degrees Kelvin) is the temperature, and R is the gas constant. The quantities A and E 0 are fixed once the reaction is specified. Let k 1 and k 2 be the reaction rate constants associated with temperatures T 1 and T 2. Find an expression for ln k 1 in terms of E 0, R, T 1, and T 2. k The double declining balance formula in accounting is V(t) V L t where V(t) is the value after t years of an article that originally cost V 0 dollars and L is a constant, called the useful life of the article. (a) A refrigerator costs $875 and has a useful life of 8 years. What is its value after 5 years? What is its annual rate of depreciation? (b) In general, what is the percentage rate of change of V(t)? 34. A cool drink is removed from a refrigerator on a hot summer day and placed in a room whose temperature is 30 Celsius. According to a law of physics, the tem- * A good place to start is the article by William V. Thayer, The St. Louis Arch Problem, UMAP Modules 1983: Tools for Teaching, Consortium for Mathematics and Its Applications, Inc., Lexington, MA, 1984.

9 Chapter 4 Review Chapter Summary and Review Problems 365 perature of the drink t minutes later is given by a function of the form f(t) 30 Ae kt. Show that the rate of change of the temperature of the drink with respect to time is proportional to the difference between the temperature of the room and that of the drink. 35. In chemistry, the acidity of a solution is measured by its ph value, which is defined by ph log 10 [H 3 O ], where [H 3 O ] is the hydronium ion concentration (moles/liter) of the solution. On average, milk has a ph value that is three times the ph value of a lime, which in turn has half the ph value of an orange. If the average ph of an orange is 3.2, what is the average hydronium ion concentration of a lime? 36. A population model employed at one time by the U.S. Census Bureau uses the formula P(t) e t to estimate the population of the United States (in millions) for every tenth year from the base year Thus, for instance, t 0 corresponds to 1790, t 1 to 1800, t 10 to 1890, and so on. The figures exclude Alaska and Hawaii. (a) Use this formula to compute the population of the United States for the years 1790, 1800, 1830, 1860, 1880, 1900, 1920, 1940, 1960, 1980, 1990, and (b) Sketch the graph of P(t). When does this model predict that the population of the United States will be increasing most rapidly? (c) Use an almanac or some other source to find the actual population figures for the years listed in part (a). Does the given population model seem to be accurate? Write a paragraph describing some possible reasons for any major differences between the predicted population figures and the actual census figures. 37. Use a graphing utility to draw the graphs of y 2 x, y 3 x, y 5 x, and y (0.5) x on the same set of axes. How does a change in base affect the graph of the exponential function? (Suggestion: Use [ 3, 3]1 by [ 3, 3]1.) 38. Use a graphing utility to draw the graphs of y 3 x, y 3 x, and y 3 x on the same set of axes. How do these graphs differ? (Suggestion: Use [ 3, 3]1 by [ 3, 3]1.) 39. Use a graphing utility to draw the graphs of y 3 x and y 4 ln x on the same axes. Then use trace and zoom to find all points of intersection of the two graphs. 40. Solve the equation log 5 (x 5) log 2 x 2 log 10 (x 2 2x) with three decimal place accuracy.

10 366 Chapter 4 Exponential and Logarithmic Functions 41. Use a graphing utility to draw the graphs of y ln (1 x 2 1 ) and y on the same axes. Do these graphs ever intersect? x 42. Make a table for the quantities ( n) n 1 and ( n 1) n, with n 8, 9, 12, 20, 25, 31, 37, 38, 43, 50, 100, and 1,000. Which of the two quantities seems to be larger? Do you think this inequality holds for all n 8?