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−4 −6 6 8 (1, 0) (0, − 4) (6, − 10) 6. One of the fundamental themes of calculus is to find the slope of the tangent line to a curve at a point. To see how this on the graph of the can be done, consider the point quadratic function f x x2. 2, 4 (2, 4) y 5 4 3 2 1 −3 −2 −1 1 2 3 x (a) Find the slope of the line joining 2, 4 slope of the tangent line at than the slope of the line through (b) Find the slope of the line joining 2, 4 slope of the tangent line at than the slope of the line through and and and 2, 4 3, 9? 3, 9. Is the greater than or less 2, 4 2, 4 1, 1. Is the greater than or less 2, 4 2, 4 2, 4 1, 1? 2.1, 4.41. and greater than or 2, 4 and and (c) Find the slope of the line joining Is the slope of the tangent line at less than the slope of the line through 2.1, 4.41? (d) Find the slope of the line joining 2 h, f 2 h in terms of the nonzero number 2, 4 and h. (e) Evaluate the slope formula from part (d) for h 1, 1, and 0.1. Compare these values with those in parts (a)–(c). (f) What can you conclude the slope of the tangent line at 2, 4 to be? Explain your answer. 215 333202_020R.qxd 12/7/05 9:43 AM Page 216 7. Use the form f x x kqx r function that (a) passes through the point the right and (b) passes through the point to the right. (There are many correct answers.) to create a cubic 2, 5 and rises to 3, 1 and falls 8. The multiplicative inverse of z zm 1. such that each complex number. z zm Find the multiplicative inverse of is a complex number (a) z 1 i (b) z 3 i (c) z 2 8i 9. Prove that the product of a complex number a bi and its complex conjugate is a real number. 10. Match the graph of the rational function given by (b) Deter |
mine the effect on the graph of f if a 0 and b is varied. 12. The endpoints of the interval over which distinct vision is possible is called the near point and far point of the eye (see figure). With increasing age, these points normally change. The table shows the approximate near points (in inches) for various ages (in years). y x Object blurry Object clear Object blurry Near point Far point f x ax b cx d with the given conditions. (a) y (b) y FIGURE FOR 12 (c) y x x (d) y x x (i) a > 0 (ii) a > 0 (iii) a < 0 (iv 11. Consider the function given by f x ax x b2. (a) Determine the effect on the graph of a is varied. Consider cases in which negative. f if b 0 and a is positive and a is 216 Age, x Near point, y 16 32 44 50 60 3.0 4.7 9.8 19.7 39.4 (a) Use the regression feature of a graphing utility to find a quadratic model for the data. Use a graphing utility to plot the data and graph the model in the same viewing window. (b) Find a rational model for the data. Take the reciprocals of the near points to generate the points Use the regression feature of a graphing utility to find a linear model for the data. The resulting line has the form x, 1y. 1 y ax b. Solve for Use a graphing utility to plot the data and graph the model in the same viewing window. y. (c) Use the table feature of a graphing utility to create a table showing the predicted near point based on each model for each of the ages in the original table. How well do the models fit the original data? (d) Use both models to estimate the near point for a person who is 25 years old. Which model is a better fit? (e) Do you think either model can be used to predict the near point for a person who is 70 years old? Explain. 333202_0300.qxd 12/7/05 10:24 AM Page 217 Exponential and Logarithmic Functions 33 Logarithmic Functions and Their Graphs Exponential Functions and Their Graphs Properties of Logarithms 3.3 3.1 3.2 3.4 3.5 Exponential and Logarithmic Equations Exponential |
and Logarithmic Models Carbon dating is a method used to determine the ages of archeological artifacts up to 50,000 years old. For example, archeologists are using carbon dating to determine the ages of the great pyramids of Egypt AT I O N S Exponential and logarithmic functions have many real-life applications. The applications listed below represent a small sample of the applications in this chapter. • Computer Virus, Exercise 65, page 227 • Galloping Speeds of Animals, • IQ Scores, Exercise 85, page 244 Exercise 47, page 266 • Data Analysis: Meteorology, • Average Heights, • Forensics, Exercise 70, page 228 Exercise 115, page 255 Exercise 63, page 268 • Sound Intensity, Exercise 90, page 238 • Carbon Dating, Exercise 41, page 266 • Compound Interest, Exercise 135, page 273 217 333202_0301.qxd 12/7/05 10:25 AM Page 218 218 Chapter 3 Exponential and Logarithmic Functions 3.1 Exponential Functions and Their Graphs What you should learn • Recognize and evaluate exponential functions with base a. • Graph exponential functions and use the One-to-One Property. • Recognize, evaluate, and graph exponential functions with base e. • Use exponential functions to model and solve real-life problems. Why you should learn it Exponential functions can be used to model and solve real-life problems. For instance, in Exercise 70 on page 228, an exponential function is used to model the atmospheric pressure at different altitudes. © Comstock Images/Alamy Exponential Functions So far, this text has dealt mainly with algebraic functions, which include polynomial functions and rational functions. In this chapter, you will study two types of nonalgebraic functions—exponential functions and logarithmic functions. These functions are examples of transcendental functions. Definition of Exponential Function The exponential function with base a f is denoted by f x ax where a > 0, a 1, and x is any real number. a 1 The base function, not an exponential function. is excluded because it yields f x 1x 1. This is a constant You have evaluated 43 64 and for integer and rational values of For example, you x, know that you need to interpret forms with irrational exponents. For the purposes of this text, it is sufficient to think of However, to evaluate for any real number ax 412 2. 4x x. a2 (where 2 1.4142 |
1356 ) as the number that has the successively closer approximations a1.4, a1.41, a1.414, a1.4142, a1.41421,.... Example 1 Evaluating Exponential Functions Use a calculator to evaluate each function at the indicated value of x. Function a. b. c. f x 2x f x 2x f x 0.6x Solution Value x 3.1 x x 3 2 Function Value f 3.1 23.1 f 2 f 3 0.632 2 a. b. c. Graphing Calculator Keystrokes Display 2 2.6 > > 3.1 ENTER ENTER > 3 2 ENTER 0.1166291 0.1133147 0.4647580 The HM mathSpace® CD-ROM and Eduspace® for this text contain additional resources related to the concepts discussed in this chapter. When evaluating exponential functions with a calculator, remember to enclose fractional exponents in parentheses. Because the calculator follows the order of operations, parentheses are crucial in order to obtain the correct result. Now try Exercise 1. 333202_0301.qxd 12/7/05 10:25 AM Page 219 Section 3.1 Exponential Functions and Their Graphs 219 Graphs of Exponential Functions The graphs of all exponential functions have similar characteristics, as shown in Examples 2, 3, and 5. Example 2 Graphs of y ax In the same coordinate plane, sketch the graph of each function. a. f x 2x b. gx 4x Solution The table below lists some values for each function, and Figure 3.1 shows the graphs of the two functions. Note that both graphs are increasing. Moreover, the graph of is increasing more rapidly than the graph of f x 2x. gx 4x x 2x 4x 3 1 8 1 64 2 1 4 1 16 16 Now try Exercise 11. Exploration Note that an exponential function f x ax is a constant raised to a variable power, whereas a power function is a variable raised to a constant power. Use a graphing utility to graph each pair of functions in the same viewing window. Describe any similarities and differences in the graphs. gx xn a. y1 b. y1 2x, y2 3x, y2 x2 x3 y g(x) = 4x 16 14 12 10 8 6 4 2 The table in Example 2 was evaluated by hand. You could, of course |
, use a graphing utility to construct tables with even more values. f(x) = 2x Example 3 Graphs of y a –x − 4 −3 −2 −1 −2 1 2 3 4 FIGURE 3.1 G(x) = 4−x y 16 14 12 10 8 6 4 F(x) = 2−x − 4 −3 −2 −1 −2 1 2 3 4 FIGURE 3.2 x x In the same coordinate plane, sketch the graph of each function. G x 4x F x 2x b. a. Solution The table below lists some values for each function, and Figure 3.2 shows the graphs of the two functions. Note that both graphs are decreasing. Moreover, the graph of is decreasing more rapidly than the graph of F x 2x. Gx 4x x 2x 4x 2 1 4 16 16 3 1 8 1 64 Now try Exercise 13. tions In Example 3, note that by using one of the properties of exponents, the funcFx 2x and Fx 2x 1 2x can be rewritten with positive exponents. and Gx 4x 1 4x Gx 4x x 1 2 1 4 x 333202_0301.qxd 12/7/05 3:30 PM Page 220 220 Chapter 3 Exponential and Logarithmic Functions Comparing the functions in Examples 2 and 3, observe that Fx 2x f x Gx 4x gx. and F g G is a reflection (in the -axis) of the graph of The Consequently, the graph of and have the same relationship. The graphs in Figures 3.1 and 3.2 graphs of y ax are typical of the exponential functions They have one y x -intercept and one horizontal asymptote (the -axis), and they are continuous. The basic characteristics of these exponential functions are summarized in Figures 3.3 and 3.4. y ax. and f. y y Graph of y ax, a > 1, • Domain: Notice that the range of an exponential function is which means that values of x. ax > 0 0,, for all y = ax (0, 1) FIGURE 3.3 y y = a −x (0, 1) FIGURE 3.4 x x • Range: • Intercept: 0, 0, 1 • Increasing • x -axis is a horizontal asymptote ax → 0 x→ • Continuous as Graph of • Domain: |
y ax, a > 1, • Range: • Intercept: 0, 0, 1 • Decreasing • x -axis is a horizontal asymptote ax→ 0 x→ • Continuous as From Figures 3.3 and 3.4, you can see that the graph of an exponential function is always increasing or always decreasing. As a result, the graphs pass the Horizontal Line Test, and therefore the functions are one-to-one functions. You can use the following One-to-One Property to solve simple exponential equations. For a > 0 and a 1, ax ay if and only if x y. One-to-One Property Example 4 Using the One-to-One Property a. b. 9 3x1 32 3x1 2 x 1 1 x 1 2 x 8 ⇒ 2x 23 ⇒ x 3 Now try Exercise 45. Original equation 9 32 One-to-One Property Solve for x. 333202_0301.qxd 12/7/05 10:25 AM Page 221 Section 3.1 Exponential Functions and Their Graphs 221 In the following example, notice how the graph of y a x can be used to sketch the graphs of functions of the form f x b ± axc. Example 5 Transformations of Graphs of Exponential Functions Each of the following graphs is a transformation of the graph of f x 3x. a. Because gx 3x1 f x 1, the graph of can be obtained by shifting g the graph of f one unit to the left, as shown in Figure 3.5. b. Because shifting the graph of hx 3x 2 f x 2, f the graph of downward two units, as shown in Figure 3.6. k the graph of kx 3x f x, h can be obtained by reflecting can be obtained by c. Because the graph of f in the -axis, as shown in Figure 3.7. x d. Because graph of jx 3x f x, f y in the -axis, as shown in Figure 3.8. the graph of can be obtained by reflecting the j g(x) = 3x + 1 y 3 2 1 f(x) = 3 x x x −2 −1 1 FIGURE 3.5 Horizontal shift y 2 1 −1 −2 f(x) = 3x 1 2 k(x) = −3x −2 y 2 1 f(x) = 3x − (x) = 3 x |
− 2 FIGURE 3.6 Vertical shift y 4 3 2 1 j(x) = 3−x f(x) = 3x −2 −1 1 2 x x FIGURE 3.7 Reflection in x-axis FIGURE 3.8 Reflection in y-axis Now try Exercise 17. Notice that the transformations in Figures 3.5, 3.7, and 3.8 keep the -axis as a horizontal asymptote, but the transformation in Figure 3.6 yields a new horizontal asymptote of Also, be sure to note how the -intercept is affected by each transformation. y 2. y x 333202_0301.qxd 12/7/05 10:25 AM Page 222 222 Chapter 3 Exponential and Logarithmic Functions y 3 2 The Natural Base e (1, e) In many applications, the most convenient choice for a base is the irrational number e 2.718281828.... f(x) = ex (−1, e−1) (0, 1) (−2, e−2) −2 −1 FIGURE 3.9 x 1 f(x) = 2e0.24x 3 − 2 −1 1 2 3 4 FIGURE 3.10 (x) = e−0.58x 1 2 − 4 −3 − 2 −1 1 2 3 4 FIGURE 3.11 This number is called the natural base. The function given by is called the natural exponential function. Its graph is shown in Figure 3.9. Be sure is the constant you see that for the exponential function 2.718281828..., whereas is the variable. f x ex, e x f x e x Exploration Use a graphing utility to graph viewing window. Using the trace feature, explain what happens to the graph of 1 1xx in the same increases. e and as y1 y2 x y1 Example 6 Evaluating the Natural Exponential Function Use a calculator to evaluate the function given by value of a. x. x 2 x 0.25 x 1 b. c. f x ex at each indicated d. x 0.3 Solution Function Value f 2 e2 f 1 e1 f 0.25 e0.25 f 0.3 e0.3 a. b. c. d. Graphing Calculator Keystrokes 2 ENTER ex ex ex ex 1 ENTER 0.25 ENTER 0.3 ENTER Display 0.1353353 0 |
.3678794 1.2840254 0.7408182 x x Now try Exercise 27. Example 7 Graphing Natural Exponential Functions Sketch the graph of each natural exponential function. a. f x 2e0.24x b. gx 1 2e0.58x Solution To sketch these two graphs, you can use a graphing utility to construct a table of values, as shown below. After constructing the table, plot the points and connect them with smooth curves, as shown in Figures 3.10 and 3.11. Note that the graph in Figure 3.10 is increasing, whereas the graph in Figure 3.11 is decreasing. x f x gx 3 2 1 0 1 2 3 0.974 1.238 1.573 2.000 2.542 3.232 4.109 2.849 1.595 0.893 0.500 0.280 0.157 0.088 Now try Exercise 35. 333202_0301.qxd 12/7/05 10:25 AM Page 223 Exploration Use the formula A P1 r n nt P $3000, years, and to calculate the amount in an account when r 6%, t 10 compounding is done (a) by the day, (b) by the hour, (c) by the minute, and (d) by the second. Does increasing the number of compoundings per year result in unlimited growth of the amount in the account? Explain. m 1 10 100 1,000 10,000 100,000 1,000,000 10,000,000 1 1 m m 2 2.59374246 2.704813829 2.716923932 2.718145927 2.718268237 2.718280469 2.718281693 e Section 3.1 Exponential Functions and Their Graphs 223 Applications One of the most familiar examples of exponential growth is that of an investment earning continuously compounded interest. Using exponential functions, you can develop a formula for interest compounded times per year and show how it leads to continuous compounding. n Suppose a principal compounded once a year. If the interest is added to the principal at the end of the year, the new balance is invested at an annual interest rate r, P P1 P1 is P Pr P1 r. This pattern of multiplying the previous principal by successive year, as shown below. 1 r is then repeated each Year Balance After Each Compounding P1 P2 P3... |
Pt P1 r P1 P2 P1 rt 1 r P1 r1 r P1 r2 1 r P1 r21 r P1 r3 To accommodate more frequent (quarterly, monthly, or daily) compounding of interest, let be the number of compoundings per year and let be the number of years. Then the rate per compounding is and the account balance after t years is rn n t A P1 r n nt. n Amount (balance) with compoundings per year If you let the number of compoundings approaches what is called continuous compounding. In the formula for compoundings per year, let increase without bound, the process n This produces m nr. n mrt nt A P1 r n P1 r mr mrt P1 1 m P1 1 m mrt. Amount with compoundings per year n Substitute mr for n. Simplify. Property of exponents increases without bound, the table at the left shows that 1 1mm → e From this, you can conclude that the formula for continuous m As m →. as compounding is A Pert. Substitute e for 1 1mm. 333202_0301.qxd 12/7/05 10:25 AM Page 224 224 Chapter 3 Exponential and Logarithmic Functions Be sure you see that the annual interest rate must be written in decimal form. For instance, 6% should be written as 0.06. Formulas for Compound Interest t After years, the balance r interest rate A 1. For compoundings per year: n (in decimal form) is given by the following formulas. A P1 r n nt in an account with principal P and annual 2. For continuous compounding: A Pe rt Example 8 Compound Interest A total of $12,000 is invested at an annual interest rate of 9%. Find the balance after 5 years if it is compounded a. quarterly. b. monthly. c. continuously. Solution a. For quarterly compounding, you have balance is n 4. So, in 5 years at 9%, the nt A P1 r n 12,0001 0.09 4 4(5) Formula for compound interest Substitute for r,P, n, and t. $18,726.11. Use a calculator. b. For monthly compounding, you have n 12. So, in 5 years at 9%, the balance is nt A P1 r n 12,0001 |
0.09 12 12(5) Formula for compound interest Substitute for r,P, n, and t. $18,788.17. Use a calculator. c. For continuous compounding, the balance is A Pert 12,000e0.09(5) $18,819.75. Formula for continuous compounding Substitute for r,P, and t. Use a calculator. Now try Exercise 53. In Example 8, note that continuous compounding yields more than quarterly or monthly compounding. This is typical of the two types of compounding. That is, for a given principal, interest rate, and time, continuous compounding will always yield a larger balance than compounding times a year. n 333202_0301.qxd 12/7/05 10:25 AM Page 225 ) 10 9 8 7 6 5 4 3 2 1 FIGURE 3.12 Section 3.1 Exponential Functions and Their Graphs 225 Example 9 Radioactive Decay Radioactive Decay P ( ( P = 10 t/24,100 1 2 In 1986, a nuclear reactor accident occurred in Chernobyl in what was then the Soviet Union. The explosion spread highly toxic radioactive chemicals, such as plutonium, over hundreds of square miles, and the government evacuated the city and the surrounding area. To see why the city is now uninhabited, consider the model (24,100, 5) (100,000, 0.564) t 50,000 100,000 Years of decay P 101 2 t24,100 which represents the amount of plutonium that remains (from an initial amount t of 10 pounds) after years. Sketch the graph of this function over the interval t 100,000, represents 1986. How much of the 10 from pounds will remain in the year 2010? How much of the 10 pounds will remain after 100,000 years? where t 0 t 0 to P Solution The graph of this function is shown in Figure 3.12. Note from this graph that plutonium has a half-life of about 24,100 years. That is, after 24,100 years, half of the original amount will remain. After another 24,100 years, one-quarter of the original amount will remain, and so on. In the year 2010 there will still be t 24, P 101 22424,100 101 20.0009959 9.993 pounds of plutonium remaining. After 100,000 years, there will still be P 101 2100,00024,100 101 24.14 |
94 0.564 pound of plutonium remaining. Now try Exercise 67. W RITING ABOUT MATHEMATICS Identifying Exponential Functions Which of the following functions generated the two tables below? Discuss how you were able to decide. What do these functions have in common? Are any of them the same? If so, explain why. a. d. x 2(x3) f1 x 1 x 7 f4 2 x x 8 1 x 7 2x 2 b. f2 e. f5 (x3) x 1 x 82x 2 c. f3 f. f6 x gx 1 7.5 0 8 1 9 2 3 11 15 x hx 2 1 32 16 0 8 1 4 2 2 Create two different exponential functions of the forms with y-intercepts of 0, 3. y abx and y cx d 333202_0301.qxd 12/7/05 10:25 AM Page 226 226 Chapter 3 Exponential and Logarithmic Functions 3.1 Exercises The HM mathSpace® CD-ROM and Eduspace® for this text contain step-by-step solutions to all odd-numbered exercises. They also provide Tutorial Exercises for additional help. VOCABULARY CHECK: Fill in the blanks. 1. Polynomials and rational functions are examples of ________ functions. 2. Exponential and logarithmic functions are examples of nonalgebraic functions, also called ________ functions. 3. The exponential function given by f x ex is called the ________ ________ function, and the base e is called the ________ base. 4. To find the amount A t in an account after years with principal P and an annual interest rate compounded r n times per year, you can use the formula ________. 5. To find the amount A t in an account after years with principal P and an annual interest rate compounded r continuously, you can use the formula ________. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1– 6, evaluate the function at the indicated value of Round your result to three decimal places. x. Function 1. 2. 3. 4. 5. 6. f x 3.4x f x 2.3x f x 5x f x 2 3 gx 50002x f x 2001.212x 5x Value x |
5.6 x 3 2 x x 3 10 x 1.5 x 24 In Exercises 7–10, match the exponential function with its graph. [The graphs are labeled (a), (b), (c), and (d).] (a) (c) y 6 4 −4 − 2 −2 x 2 4 y 6 4 (b) y 6 4 2 −2 −2 (d) 4 6 2 y 6 4 2 −4 −2 −2 7. 9. f x 2x f x 2x x 2 4 −4 −2 −2 2 4 8. 10. f x 2x 1 f x 2x2 x x In Exercises 11–16, use a graphing utility to construct a table of values for the function. Then sketch the graph of the function. x f x 1 f x 6x f x 2 x1 f x 1 2 f x 6x f x 4x3 3 x 16. 14. 15. 11. 13. 12. 2 In Exercises 17–22, use the graph of transformation that yields the graph of f g. to describe the 17. 18. 19. 20. 21. 22. f x 3 x, f x 4x, f x 2x, f x 10 x, f x 7 x, f x 0.3x, 2 gx 3x4 gx 4x 1 gx 5 2 x gx 10 x3 gx 7 2 gx 0.3x 5 x6 In Exercises 23–26, use a graphing utility to graph the exponential function. 23. 25. y 2x 2 y 3x2 1 24. 26. y 3x y 4x1 2 In Exercises 27–32, evaluate the function at the indicated value of Round your result to three decimal places. x. Function 27. 28. 29. 30. 31. 32. hx ex f x ex f x 2e5x f x 1.5ex2 f x 5000e0.06x f x 250e0.05x Value x 3 4 x 3.2 x 10 x 240 x 6 x 20 333202_0301.qxd 12/7/05 10:25 AM Page 227 Section 3.1 Exponential Functions and Their Graphs 227 In Exercises 33–38, use a graphing utility to construct a table of values for the function. Then sketch the graph of the function. 33. 35 |
. 37. f x ex f x 3ex4 f x 2ex2 4 34. 36. 38. f x ex f x 2e0.5x f x 2 e x5 In Exercises 39– 44, use a graphing utility to graph the exponential function. 39. 41. 43. y 1.085x st 2e0.12t gx 1 ex 40. 42. 44. y 1.085x st 3e0.2t hx e x2 In Exercise 45–52, use the One-to-One Property to solve the equation for x. 45. 47. 49. 51. 3x1 27 2x2 1 32 e3x2 e3 ex23 e2x 46. 48. 50. 52. 125 2x3 16 x1 1 5 e2x1 e4 ex 26 e5x Compound Interest A table to determine the balance rate for years and compounded In Exercises 53–56, complete the for dollars invested at n times per year. P r t 1 2 4 12 365 Continuous n A 53. 54. 55. 56. years P $2500, r 2.5%, t 10 P $1000, r 4%, t 10 P $2500, r 3%, t 20 P $1000, r 6%, t 40 years years years Compound Interest table to determine the balance rate for years, compounded continuously. In Exercises 57– 60, complete the for $12,000 invested at A r t 62. Trust Fund A deposit of $5000 is made in a trust fund that pays 7.5% interest, compounded continuously. It is specified that the balance will be given to the college from which the donor graduated after the money has earned interest for 50 years. How much will the college receive? 63. Inflation If the annual rate of inflation averages 4% over the next 10 years, the approximate costs of goods or services during any year in that decade will be modeled by Ct P1.04t, is the present cost. The price of an oil change for your car is presently $23.95. Estimate the price 10 years from now. is the time in years and where C P t 64. Demand The demand equation for a product is given by p 50001 4 4 e0.002x where p is the price and x is the number of units. (a) Use a graphing utility to graph the demand function for x > 0 and p > 0 |
. (b) Find the price p for a demand of x 500 units. (c) Use the graph in part (a) to approximate the greatest price that will still yield a demand of at least 600 units. 65. Computer Virus The number of computers infected by a computer virus increases according to the model Vt 100e4.6052t, V1, and (c) (b) where is the time in hours. Find (a) V2. V1.5, V t P 66. Population The population (in millions) of Russia from 1996 to 2004 can be approximated by the model P 152.26e0.0039t, t 6 corresponding (Source: Census Bureau, International Data Base) where represents the year, with t to 1996. (a) According to the model, is the population of Russia increasing or decreasing? Explain. (b) Find the population of Russia in 1998 and 2000. (c) Use the model to predict the population of Russia in 2010. Q 67. Radioactive Decay Let 226Ra represent a mass of radioactive (in grams), whose half-life is 1599 years. years is radium The quantity of radium present after Q 251 (a) Determine the initial quantity (when t1599. t 0 ). t 2 10 20 30 40 50 t A (b) Determine the quantity present after 1000 years. (c) Use a graphing utility to graph the function over the interval t 0 to t 5000. 57. 59. r 4% r 6.5% 58. 60. r 6% r 3.5% 61. Trust Fund On the day of a child’s birth, a deposit of $25,000 is made in a trust fund that pays 8.75% interest, compounded continuously. Determine the balance in this account on the child’s 25th birthday. 68. Radioactive Decay Let represent a mass of carbon (in grams), whose half-life is 5715 years. The quan- 14 14C tity of carbon 14 present after years is t5715. Q t (a) Determine the initial quantity (when (b) Determine the quantity present after 2000 years. (c) Sketch the graph of this function over the interval t 0 to t 10,000. Q 101 t 0 ). 2 333202_0301.qxd 12/7/05 10:25 AM Page 228 228 Chapter 3 Exponential and Logarithmic Functions Model |
It Synthesis 69. Data Analysis: Biology To estimate the amount of defoliation caused by the gypsy moth during a given year, a forester counts the number of egg masses on 1 of an acre (circle of radius 18.6 feet) in the fall. The 40 percent of defoliation the next spring is shown in the table. y (Source: USDA, Forest Service) x Egg masses, x Percent of defoliation, y 0 25 50 75 100 12 44 81 96 99 A model for the data is given by y 100 1 7e0.069x. (a) Use a graphing utility to create a scatter plot of the data and graph the model in the same viewing window. (b) Create a table that compares the model with the sample data. (c) Estimate the percent of defoliation if 36 egg masses are counted on 1 40 acre. 2 3 (d) You observe that of a forest is defoliated the following spring. Use the graph in part (a) to estimate the number of egg masses per acre. 1 40 70. Data Analysis: Meteorology A meteorologist measures P (in (in pascals) at altitude the atmospheric pressure kilometers). The data are shown in the table. h True or False? the statement is true or false. Justify your answer. In Exercises 71 and 72, determine whether 71. The line y 2 is an asymptote for the graph of f x 10 x 2. e 271,801. 99,990 72. Think About It nents to determine which functions (if any) are the same. In Exercises 73–76, use properties of expo- 73. 75. 3x f x 3x2 gx 3x 9 hx 1 9 f x 164x gx 1 x2 4 hx 1622x 74. 76. f x 4x 12 gx 22x6 hx 644x f x ex 3 gx e3x hx ex3 77. Graph the functions given by y 3x and y 4x and use the graphs to solve each inequality. (a) 4x < 3x (b) 4x > 3x 78. Use a graphing utility to graph each function. Use the graph to find where the function is increasing and decreasing, and approximate any relative maximum or minimum values. f x x 2ex gx x23x (b) (a) 79. Graphical Analysis Use |
a graphing utility to graph f x 1 0.5 x x and gx e0.5 in the same viewing window. What is the relationship between increases and decreases without bound? and as g x f 80. Think About It Which functions are exponential? (a) 3x (b) 3x 2 (c) 3x (d) 2x Altitude, h Pressure, P Skills Review 0 5 10 15 20 101,293 54,735 23,294 12,157 5,069 A model for the data is given by P 107,428e 0.150h. (a) Sketch a scatter plot of the data and graph the model on the same set of axes. (b) Estimate the atmospheric pressure at a height of 8 kilometers. In Exercises 81 and 82, solve for y. 81. x 2 y 2 25 82. x y 2 In Exercises 83 and 84, sketch the graph of the function. 83. f x 2 9 x 84. f x 7 x 85. Make a Decision To work an extended application analyzing the population per square mile of the United (Data States, visit this text’s website at college.hmco.com. Source: U.S. Census Bureau) 333202_0302.qxd 12/7/05 10:28 AM Page 229 Section 3.2 Logarithmic Functions and Their Graphs 229 3.2 Logarithmic Functions and Their Graphs What you should learn • Recognize and evaluate logarithmic functions with base a. • Graph logarithmic functions. • Recognize, evaluate, and graph natural logarithmic functions. • Use logarithmic functions to model and solve real-life problems. Why you should learn it Logarithmic functions are often used to model scientific observations. For instance, in Exercise 89 on page 238, a logarithmic function is used to model human memory. © Ariel Skelley/Corbis Remember that a logarithm is an exponent. So, to evaluate the loga x, logarithmic expression you need to ask the question, “To what power must be raised to obtain ”x? a Logarithmic Functions In Section 1.9, you studied the concept of an inverse function. There, you learned that if a function is one-to-one—that is, if the function has the property that no horizontal line intersects the graph of the function |
more than once—the function must have an inverse function. By looking back at the graphs of the exponential functions introduced in Section 3.1, you will see that every function of the form f x ax passes the Horizontal Line Test and therefore must have an inverse function. This inverse function is called the logarithmic function with base a. Definition of Logarithmic Function with Base a For a 1, and if and only if x ay. a > 0, x > 0, y loga x The function given by f x loga x x. Read as “log base of ” a is called the logarithmic function with base a. The equations y loga x and x a y are equivalent. The first equation is in logarithmic form and the second is in can be exponential form. For example, the logarithmic equation can rewritten in exponential form as be rewritten in logarithmic form as The exponential equation log5 125 3. When evaluating logarithms, remember that a logarithm is an exponent. is the exponent to which must be raised to obtain For 2 log3 9 53 125 9 32. x. a because 2 must be raised to the third power to get 8. This means that instance, log2 8 3 loga x Example 1 Evaluating Logarithms Use the definition of logarithmic function to evaluate each logarithm at the indicated value of x. f x log2 x, f x log4 x, x 32 x 2 b. d. f x log3 x, f x log10 x, x 1 x 1 100 Solution a. f 32 log2 32 5 f 1 log3 1 0 f 2 log4 2 1 f 1 log10 1 100 100 2 2 because because because because 25 32. 30 1. 412 4 2. 102 1 10 2 1 100. Now try Exercise 17. a. c. b. c. d. 333202_0302.qxd 12/7/05 10:28 AM Page 230 230 Chapter 3 Exponential and Logarithmic Functions Exploration Complete the table for f x 10 x. 2 1 0 1 2 x f x Complete the table for f x log x. 1 100 1 10 1 10 100 x f x Compare the two tables. What is the relationship between f x 10 x and f x log x? The logarithmic function with base 10 is called the common logarithmic function. It is den |
oted by or simply by log. On most calculators, this log10. Example 2 shows how to use a calculator to evaluate function is denoted by common logarithmic functions. You will learn how to use a calculator to calculate logarithms to any base in the next section. LOG Example 2 Evaluating Common Logarithms on a Calculator Use a calculator to evaluate the function given by x 1 x 2.5 3 x 10 b. a. c. f x log x d. at each value of x 2 x. Solution Function Value f 10 log 10 log f 1 f 2.5 log 2.5 f 2 log2 1 3 3 a. b. c. d. Graphing Calculator Keystrokes Display LOG 10 ENTER LOG LOG LOG 1 3 ENTER 2.5 ENTER 2 ENTER 1 0.4771213 0.3979400 ERROR Note that the calculator displays an error message (or a complex number) when The reason for this is that there is no real number you try to evaluate power to which 10 can be raised to obtain log2. 2. Now try Exercise 23. The following properties follow directly from the definition of the logarith- mic function with base a. Properties of Logarithms 1. because loga 1 0 loga a 1 loga a x x 2. 3. and loga x loga y, 4. If because a0 1. a1 a. a log a x x Inverse Properties then x y. One-to-One Property Example 3 Using Properties of Logarithms a. Simplify: log4 1 b. Simplify: log7 7 c. Simplify: 6log 620 Solution a. Using Property 1, it follows that log7 7 1. b. Using Property 2, you can conclude that c. Using the Inverse Property (Property 3), it follows that log4 1 0. 6log 620 20. Now try Exercise 27. You can use the One-to-One Property (Property 4) to solve simple logarithmic equations, as shown in Example 4. 333202_0302.qxd 12/7/05 10:28 AM Page 231 Section 3.2 Logarithmic Functions and Their Graphs 231 Example 4 Using the One-to-One Property a. b. c. log3 x log3 12 x 12 Original equation One-to-One Property log2x 1 log x ⇒ 2x 1 x ⇒ x 1 log4 |
x2 6 log4 10 ⇒ x2 6 10 ⇒ x2 16 ⇒ x ±4 Now try Exercise 79. Graphs of Logarithmic Functions f(x) = 2x y = x g(x) = log 2 x y 10 10 −2 −2 FIGURE 3.13 Vertical asymptote: x = 0 f(x) = log 10 FIGURE 3.14 To sketch the graph of functions are reflections of each other in the line y loga x, y x. you can use the fact that the graphs of inverse Example 5 Graphs of Exponential and Logarithmic Functions In the same coordinate plane, sketch the graph of each function. a. f x 2x b. gx log2 x Solution a. For f x 2x, construct a table of values. By plotting these points and connecting them with a smooth curve, you obtain the graph shown in Figure 3.13. x f x 2x. Because gx log2 x obtained by plotting the points g curve. The graph of shown in Figure 3.13. is the inverse function of f x, x f x 2x, is and connecting them with a smooth as the graph of in the line y x, g f is a reflection of the graph of Now try Exercise 31. Example 6 Sketching the Graph of a Logarithmic Function Sketch the graph of the common logarithmic function vertical asymptote. f x log x. Identify the Solution Begin by constructing a table of values. Note that some of the values can be obtained without a calculator by using the Inverse Property of Logarithms. Others require a calculator. Next, plot the points and connect them with a smooth curve, as shown in Figure 3.14. The vertical asymptote is y ( -axis). x 0 Without calculator With calculator x fx log x 1 100 2 1 10 1 1 0 10 1 2 5 8 0.301 0.699 0.903 Now try Exercise 37. 333202_0302.qxd 12/7/05 10:28 AM Page 232 232 Chapter 3 Exponential and Logarithmic Functions The nature of the graph in Figure 3.14 is typical of functions of the form f x loga x, a > 1. x -intercept and one vertical asymptote. They have one x > 1. Notice how slowly the graph rises for The basic characteristics of logarithmic graphs are |
summarized in Figure 3.15. y = loga x (1, 0) 1 2 y 1 −1 FIGURE 3.15 Graph of • Domain: • Range: y loga x, a > 1 0,, 1, 0 • x -intercept: • Increasing x • One-to-one, therefore has an • inverse function y -axis is a vertical asymptote loga x → 0. x → as • Continuous • Reflection of graph of about the line y x y a x f x ax and gx loga x. are shown below to illus- The basic characteristics of the graph of f x ax 0, trate the inverse relation between, 0,1 • Domain: -intercept: • Range: y x • • In the next example, the graph of f x b ± loga functions of the form the graph results in a horizontal shift of the vertical asymptote. is used to sketch the graphs of Notice how a horizontal shift of -axis is a horizontal asymptote y loga x x c. ax → 0 as x →. You can use your understanding of transformations to identify vertical asymptotes of logarithmic functions. For instance, in Example 7(a) the graph of gx f x 1 of the vertical asymptote of x 1, the vertical asymptote of the graph of f x. shifts the graph one unit to the right. So, is gx one unit to the right of f x Example 7 Shifting Graphs of Logarithmic Functions The graph of each of the functions is similar to the graph of f x log x. a. Because gx logx 1 f x 1, the graph of g can be obtained by shifting the graph of f one unit to the right, as shown in Figure 3.16. b. Because hx 2 log x 2 f x, the graph of h can be obtained by shifting the graph of f two units upward, as shown in Figure 3.17. y 1 f(x) = log x (1, 0) x 1 (2, 0) −1 FIGURE 3.16 g(x) = log(x − 1) Now try Exercise 39. y 2 1 (1, 2) h(x) = 2 + log x f(x) = log x (1, 0) 2 x FIGURE 3.17 333202_0302.qxd 12/7/ |
05 10:28 AM Page 233 Section 3.2 Logarithmic Functions and Their Graphs 233 The Natural Logarithmic Function By looking back at the graph of the natural exponential function introduced in Section 3.1 on page 388, you will see that is one-to-one and so has an inverse function. This inverse function is called the natural logarithmic x function and is denoted by the special symbol ln read as “the natural log of ” or “el en of ” Note that the natural logarithm is written without a base. The base is understood to be f x ex e. x, x. The Natural Logarithmic Function The function defined by f x loge x ln x, x > 0 y = x is called the natural logarithmic function. y f(x) = ex (1, e) (e, 1) 3 2 (0, 1) ( ) 1 −1, e −2 −1 −1 −2 x 2 3 (1, 0) ( ) 1, −1 e g(x) = f −1(x) = ln x Reflection of graph of y x the line FIGURE 3.18 f x ex about Notice that as with every other logarithmic function, the domain of the natural logarithmic function is the set of positive real numbers—be sure you see that ln negative numbers. is not defined for zero or for x The definition above implies that the natural logarithmic function and the natural exponential function are inverse functions of each other. So, every logarithmic equation can be written in an equivalent exponential form and every exponential equation can be written in logarithmic form. That is, and x e y y ln x are equivalent equations. Because the functions given by f x e x and gx ln x tions of each other, their graphs are reflections of each other in the line This reflective property is illustrated in Figure 3.18. are inverse funcy x. On most calculators, the natural logarithm is denoted by LN, as illustrated in Example 8. Example 8 Evaluating the Natural Logarithmic Function Use a calculator to evaluate the function given by x 0.3 x 1 x 2 b. a. c. f x ln x for each value of x. d. x 1 2 Solution Function Value Graphing Calculator Keystrokes Display a |
. b. c. d. f 2 ln 2 f 0.3 ln 0.3 f 1 ln1 f 1 2 ln1 2 LN LN LN LN Now try Exercise 61. 2 ENTER.3 ENTER 1 1 ENTER 0.6931472 –1.2039728 ERROR 2 ENTER 0.8813736 In Example 8, be sure you see that gives an error message on most calculators. (Some calculators may display a complex number.) This occurs because the domain of ln is the set of positive real numbers (see Figure 3.18). So, ln1 The four properties of logarithms listed on page 230 are also valid for is undefined. x ln1 natural logarithms. 333202_0302.qxd 12/7/05 10:28 AM Page 234 234 Chapter 3 Exponential and Logarithmic Functions Properties of Natural Logarithms 1. because e0 1. e1 e. because ln 1 0 ln e 1 ln e x x 2. 3. and ln x ln y, eln x x then x y. Inverse Properties One-to-One Property 4. If Example 9 Using Properties of Natural Logarithms Use the properties of natural logarithms to simplify each expression. a. ln 1 e b. eln 5 c. ln 1 3 d. 2 ln e Solution 1 ln e a. ln e1 1 Inverse Property b. eln 5 5 Inverse Property c. ln 1 3 0 3 0 Property 1 d. 2 ln e 21) 2 Property 2 Now try Exercise 65. Example 10 Finding the Domains of Logarithmic Functions Find the domain of each function. a. f x lnx 2 b. gx ln2 x c. hx ln x 2 f Solution a. Because is b. Because is c. Because g, 2. ln x 2 real numbers except lnx 2 is defined only if x 2 > 0, it follows that the domain of 2,. The graph of f is shown in Figure 3.19. ln2 x is defined only if g 2 x > 0, is shown in Figure 3.20. The graph of it follows that the domain of is defined only if x 0. x 2 > 0, The graph of it follows that the domain of h is shown in Figure 3.21. |
h is all f(x) = ln(x − 21 −2 −3 −4 y 2 g(x) = ln(2 − x) −1 x 1 2 −1 −1 FIGURE 3.19 FIGURE 3.20 Now try Exercise 69. h(x) = ln x2 y 4 2 −2 2 4 x −4 FIGURE 3.21 333202_0302.qxd 12/7/05 10:28 AM Page 235 Section 3.2 Logarithmic Functions and Their Graphs 235 Memory Model f t( ) Application Example 11 Human Memory Model f(t) = 75 − 6ln(t + 1) Students participating in a psychology experiment attended several lectures on a subject and were given an exam. Every month for a year after the exam, the students were retested to see how much of the material they remembered. The average scores for the group are given by the human memory model 80 70 60 50 40 30 20 10 FIGURE 3.22 4 10 6 Time (in months) 8 t 12 f t 75 6 lnt 1, 0 ≤ t ≤ 12 where t is the time in months. The graph of f a. What was the average score on the original b. What was the average score at the end of c. What was the average score at the end of t 2 t 6 months? months? is shown in Figure 3.22. t 0 exam? Solution a. The original average score was f 0 75 6 ln0 1 75 6 ln 1 75 60 75. Substitute 0 for t. Simplify. Property of natural logarithms Solution b. After 2 months, the average score was f 2 75 6 ln2 1 75 6 ln 3 75 61.0986 68.4. Substitute 2 for t. Simplify. Use a calculator. Solution c. After 6 months, the average score was f 6 75 6 ln6 1 75 6 ln 7 75 61.9459 63.3. Substitute 6 for t. Simplify. Use a calculator. Solution Now try Exercise 89. W RITING ABOUT MATHEMATICS Analyzing a Human Memory Model Use a graphing utility to determine the time in months when the average score in Example 11 was 60. Explain your method of solving the problem. Describe another way that you can use a graphing utility to determine the answer. 333202_0302.qxd 12/7/05 |
3:32 PM Page 236 236 Chapter 3 Exponential and Logarithmic Functions 3.2 Exercises VOCABULARY CHECK: Fill in the blanks. 1. The inverse function of the exponential function given by 2. The common logarithmic function has base ________. fx ax is called the ________ function with base a. 3. The logarithmic function given by fx ln x is called the ________ logarithmic function and has base ________. 4. The Inverse Property of logarithms and exponentials states that loga ax x and ________. 5. The One-to-One Property of natural logarithms states that if ln x ln y, then ________. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1– 8, write the logarithmic equation in exponential form. For example, the exponential form of log5 25 2 52 25. is log4 64 3 log7 1 2 log32 4 2 log36 6 1 49 5 2 1. 3. 5. 7. 2. 4. 6. 8. 3 log3 81 4 log 1 1000 log16 8 3 log8 4 2 4 3 In Exercises 9 –16, write the exponential equation in logarithmic form. For example, the logarithmic form of 23 8 log2 8 3. is 9. 11. 13. 15. 53 125 8114 3 62 1 36 70 1 10. 12. 14. 16. 82 64 932 27 43 1 64 103 0.001 In Exercises 17–22, evaluate the function at the indicated value of without using a calculator. x Function f x log2 x f x log16 x f x log7 x f x log x gx loga x gx logb x 17. 18. 19. 20. 21. 22. Value x 16 x 4 x 1 x 10 x a2 x b3 In Exercises 23–26, use a calculator to evaluate at the indicated value of decimal places. f x log x Round your result to three x. 23. 25. x 4 5 x 12.5 24. 26. x 1 500 x 75.25 In Exercises 27–30, use the properties of logarithms to simplify the expression. 27. 29. log3 |
34 log 28. 30. log1.5 1 9log915 In Exercises 31–38, find the domain, -intercept, and vertical asymptote of the logarithmic function and sketch its graph. x 31. 33. 35. 37. f x log4 x y log3 x 2 f x log6 y logx 5 x 2 32. 34. 36. gx log6 x hx log4 y log5 x 3 x 1 4 38. y logx to In Exercises 39– 44, use the graph of match the given function with its graph. Then describe the f relationship between the graphs of and [The graphs are labeled (a), (b), (c), (d), (e), and (f).] gx log3 x g. (a) –3 y 3 2 –1 –2 (b) y 3 2 1 x 1 –4 –3 –2 –1 –1 1 –2 (c) y (d) y 4 3 2 1 –1 –2 –1 –1 –2 1 2 3 x x 333202_0302.qxd 12/7/05 3:33 PM Page 237 Section 3.2 Logarithmic Functions and Their Graphs 237 (e) y (f1 –1 –2 1 3 4 x In Exercises 73–78, use a graphing utility to graph the function. Be sure to use an appropriate viewing window. 73. 75. 77. fx logx 1 fx lnx 1 fx ln x 2 74. 76. 78. fx logx 1 fx lnx 2 fx 3 ln x 1 In Exercises 79–86, use the One-to-One Property to solve the equation for x. 39. 41. 43. f x log3 x 2 f x log3 f x log3 1 x x 2 40. 42. 44. f x log3 x f x log3 f x log3 x 1 x 79. 81. 83. 85. x 1 log2 4 log2 log2x 1 log 15 lnx 2 ln 6 lnx2 2 ln 23 82. 80. x 3 log2 9 log2 log5x 3 log 12 lnx 4 ln 2 84. 86. lnx2 x ln 6 In Exercises 45–52, write the logarithmic equation in exponential form. |
45. 47. 49. 51. 0.693... ln 1 2 ln 4 1.386... ln 250 5.521... ln 1 0 46. 48. 50. 52. ln 2 0.916... 5 ln 10 2.302... ln 679 6.520... ln e 1 In Exercises 53– 60, write the exponential equation in logarithmic form. 53. 55. 57. 59. e3 20.0855... e12 1.6487... e0.5 0.6065... ex 4 54. 56. 58. 60. e2 7.3890... e13 1.3956... e4.1 0.0165... e2x 3 In Exercises 61–64, use a calculator to evaluate the function x. at the indicated value of Round your result to three decimal places. Function f x ln x f x 3 ln x gx 2 ln x gx ln x 61. 62. 63. 64. Value x 18.42 x 0.32 x 0.75 x 1 2 In Exercises 65– 68, evaluate value of without using a calculator. x gx ln x at the indicated 65. 67. x e3 x e23 66. 68. x e2 x e52 In Exercises 69–72, find the domain, -intercept, and vertical asymptote of the logarithmic function and sketch its graph. x 69. 71. f x lnx 1 gx lnx 70. 72. hx lnx 1 f x ln3 x Model It 87. Monthly Payment The model t 12.542 ln x x 1000, x > 1000 approximates the length of a home mortgage of $150,000 at 8% in terms of the monthly payment. In the model, is the monthly payment in dollars (see figure). is the length of the mortgage in years and x t t 30 25 20 15 10,000 4,000 6,000 8,000 10,000 Monthly payment (in dollars) x (a) Use the model to approximate the lengths of a $150,000 mortgage at 8% when the monthly payment is $1100.65 and when the monthly payment is $1254.68. (b) App |
roximate the total amounts paid over the term of the mortgage with a monthly payment of $1100.65 and with a monthly payment of $1254.68. (c) Approximate the total interest charges for a monthly payment of $1100.65 and for a monthly payment of $1254.68. (d) What is the vertical asymptote for the model? Interpret its meaning in the context of the problem. 333202_0302.qxd 12/7/05 10:28 AM Page 238 238 Chapter 3 Exponential and Logarithmic Functions 88. Compound Interest A principal 91 2% and K compounded continuously, increases to an amount times is given by the original principal after years, where t ln K0.095. (a) Complete the table and interpret your results. invested at P, t t 1 2 4 6 8 10 12 K t (b) Sketch a graph of the function. 89. Human Memory Model Students in a mathematics class were given an exam and then retested monthly with an equivalent exam. The average scores for the class are given f t 80 17 logt 1, by the human memory model 0 ≤ t ≤ 12 where (a) Use a graphing utility to graph the model over the is the time in months. t specified domain. (b) What was the average score on the original exam t 0? (c) What was the average score after 4 months? (d) What was the average score after 10 months? 90. Sound Intensity The relationship between the number of in watts per square I decibels and the intensity of a sound meter is 10 log I 1012. (a) Determine the number of decibels of a sound with an intensity of 1 watt per square meter. (b) Determine the number of decibels of a sound with an intensity of 102 watt per square meter. (c) The intensity of the sound in part (a) is 100 times as great as that in part (b). Is the number of decibels 100 times as great? Explain. Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 91 and 92, determine whether 91. You can determine the graph of by graphing and reflecting it about the -axis. f x log6 x x gx 6x 92. The graph of f x log3 x contains the point 27, 3. f f 93. g g. In |
Exercises 93–96, sketch the graph of and and describe the relationship between the graphs of and What is the relationship between the functions and gx log3 x gx log5 x gx ln x gx log x f x 3x, f x 5x, f x ex, f x 10 x, 96. 94. 95. g? f g 97. Graphical Analysis Use a graphing utility to graph f in the same viewing window and determine which and. is increasing at the greater rate as What can you conclude about the rate of growth of the natural logarithmic function? gx x gx 4x f x ln x, f x ln x, approaches (b) (a) x 98. (a) Complete the table for the function given by fx ln x x. 1 5 10 102 104 106 x f x (b) Use the table in part (a) to determine what value f x approaches as x increases without bound. (c) Use a graphing utility to confirm the result of part (b). 99. Think About It The table of values was obtained by evaluating a function. Determine which of the statements may be true and which must be falsea) y is an exponential function of x. (b) y is a logarithmic function of x. (c) x is an exponential function of y. (d) y is a linear function of x. 100. Writing Explain why loga x is defined only for 0 < a < 1 and a > 1. In Exercises 101 and 102, (a) use a graphing utility to graph the function, (b) use the graph to determine the intervals in which the function is increasing and decreasing, and (c) approximate any relative maximum or minimum values of the function. f x ln x hx lnx 2 1 101. 102. Skills Review In Exercises 103–108, evaluate f x 3x 2 and f g2 gx x3 1. 103. the function for 105. fg6 107. f g7 104. f g1 0 f g 108. g f 3 106. 333202_0303.qxd 12/7/05 10:29 AM Page 239 3.3 Properties of Logarithms Section 3.3 Properties of Logarithms 239 What you should learn • Use the change-of-base formula to rewrite and evaluate logarithmic expressions. • Use |
properties of logarithms to evaluate or rewrite logarithmic expressions. • Use properties of logarithms to expand or condense logarithmic expressions. • Use logarithmic functions to model and solve real-life problems. Why you should learn it Logarithmic functions can be used to model and solve real-life problems. For instance, in Exercises 81–83 on page 244, a logarithmic function is used to model the relationship between the number of decibels and the intensity of a sound. Change of Base Most calculators have only two types of log keys, one for common logarithms (base 10) and one for natural logarithms (base ). Although common logs and natural logs are the most frequently used, you may occasionally need to evaluate logarithms to other bases. To do this, you can use the following change-of-base formula. e Change-of-Base Formula a, Let loga x b, and be positive real numbers such that can be converted to a different base as follows. x a 1 and b 1. Then Base b loga x logb x logb a Base 10 loga x log x log a Base e loga x ln x ln a One way to look at the change-of-base formula is that logarithms to base a are simply constant multiples of logarithms to base The constant multiplier is 1logba. b. Example 1 Changing Bases Using Common Logarithms a. b. AP Photo/Stephen Chernin log4 25 log 25 log 4 1.39794 0.60206 2.3219 log2 12 log 12 log 2 loga x log x log a Use a calculator. Simplify. 1.07918 0.30103 3.5850 Now try Exercise 1(a). Example 2 Changing Bases Using Natural Logarithms a. b. log4 25 ln 25 ln 4 3.21888 1.38629 2.3219 log2 12 ln 12 ln 2 loga x ln x ln a Use a calculator. Simplify. 2.48491 0.69315 3.5850 Now try Exercise 1(b). 333202_0303.qxd 12/7/05 3:35 PM Page 240 240 Chapter 3 Exponential and Logarithmic Functions Properties of Logarithms a You know from the preceding section that the |
logarithmic function with base is the inverse function of the exponential function with base So, it makes sense that the properties of exponents should have corresponding properties involving logarithms. For instance, the exponential property has the corresponding logarithmic property a0 1 loga1 0. a. There is no general property that can be used to rewrite u ± v. loga u v loga loga u loga v. Specifically, is not equal to Properties of Logarithms Let be a positive number such that and are positive real numbers, the following properties are true. a 1, a v n and let be a real number. If u 1. Product Property: loga 2. Quotient Property: loga Logarithm with Base a uv loga u loga v u v loga u loga v 3. Power Property: loga un n loga u Natural Logarithm lnuv ln u ln v u ln v ln u ln v ln un n ln u For proofs of the properties listed above, see Proofs in Mathematics on page 278. Example 3 Using Properties of Logarithms Write each logarithm in terms of ln 2 and ln 3. 2 27 a. ln 6 ln b. Solution a. ln 6 ln2 3 b. ln ln 2 ln 3 2 27 ln 2 ln 27 ln 2 ln 33 ln 2 3 ln 3 Rewrite 6 as 2 3. Product Property Quotient Property Rewrite 27 as 33. Power Property Now try Exercise 17. Example 4 Using Properties of Logarithms Find the exact value of each expression without using a calculator. a. log5 35 b. ln e6 ln e2 Solution a. log5 35 log5 513 1 3 log5 5 1 3 1 1 3 b. ln e6 ln e2 ln e6 e2 ln e4 4 ln e 41 4 Now try Exercise 23 Historical Note John Napier, a Scottish mathematician, developed logarithms as a way to simplify some of the tedious calculations of his day. Beginning in 1594, Napier worked about 20 years on the invention of logarithms. Napier was only partially successful in his quest to simplify tedious calculations. Nonetheless, the development of logarithms was a step forward and received immediate recognition. 333202_0303 |
.qxd 12/7/05 10:29 AM Page 241 Section 3.3 Properties of Logarithms 241 Rewriting Logarithmic Expressions The properties of logarithms are useful for rewriting logarithmic expressions in forms that simplify the operations of algebra. This is true because these properties convert complicated products, quotients, and exponential forms into simpler sums, differences, and products, respectively. Example 5 Expanding Logarithmic Expressions Expand each logarithmic expression. a. log4 5x3y b. ln 3x 5 7 Exploration Use a graphing utility to graph the functions given by Solution a. ln x lnx 3 y1 b. and ln y2 x x 3 in the same viewing window. Does the graphing utility show the functions with the same domain? If so, should it? Explain your reasoning. ln 3x 5 7 log4 5x3y log4 5 log4 x3 log4 y log4 5 3 log4 x log4 y 3x 512 7 ln3x 512 ln 7 1 2 ln3x 5 ln 7 ln Product Property Power Property Rewrite using rational exponent. Quotient Property Power Property Now try Exercise 47. In Example 5, the properties of logarithms were used to expand logarithmic expressions. In Example 6, this procedure is reversed and the properties of logarithms are used to condense logarithmic expressions. Example 6 Condensing Logarithmic Expressions Condense each logarithmic expression. a. c. 1 2 log x 3 logx 1 log2 x log2 x 1 1 3 b. 2 lnx 2 ln x Solution 2 log x 3 logx 1 log x12 logx 13 1 a. logxx 13 b. 2 lnx 2 ln x lnx 22 ln x c. 1 3 log2 x log2 ln x 22 x log2 xx 1 x 1 1 3 xx 113 log2 log2 3xx 1 Now try Exercise 69. Power Property Product Property Power Property Quotient Property Product Property Power Property Rewrite with a radical. 333202_0303.qxd 12/7/05 10:29 AM Page 242 242 Chapter 3 Exponential and Logarithmic Functions Application x One method of determining how the - and -values for a set of nonlinear data are related is to take the natural logarithm |
of each of the - and -values. If the points are graphed and fall on a line, then you can determine that the - and -values are related by the equation ln y m ln x y y x x y where m is the slope of the line. Example 7 Finding a Mathematical Model x y The table shows the mean distance and the period (the time it takes a planet to orbit the sun) for each of the six planets that are closest to the sun. In the table, the mean distance is given in terms of astronomical units (where Earth’s mean distance is defined as 1.0), and the period is given in years. Find an equation that y relates and x. Planet Mean distance, x Period, y Mercury Venus Earth Mars Jupiter Saturn 0.387 0.723 1.000 1.524 5.203 9.537 0.241 0.615 1.000 1.881 11.863 29.447 Solution The points in the table above are plotted in Figure 3.23. From this figure it is not y clear how to find an equation that relates and To solve this problem, take the y natural logarithm of each of the - and -values in the table. This produces the following results. x. x ln x Planet Mercury 0.949 1.423 ln y Venus 0.324 0.486 Earth Mars Jupiter Saturn 0.000 0.421 1.649 2.255 0.000 0.632 2.473 3.383 Now, by plotting the points in the second table, you can see that all six of the points appear to lie in a line (see Figure 3.24). Choose any two points to 0, 0, determine the slope of the line. Using the two points you can determine that the slope of the line is 0.421, 0.632 and m 0.632 0 0.421 0 1.5 3 2. By the point-slope form, the equation of the line is ln y 3 X ln x. You can therefore conclude that 2 ln x. Y 3 2 X, where Y ln y and Now try Exercise 85. Planets Near the Sun y Saturn Mercury Venus Earth Jupiter Mars x 30 25 20 15 10 Mean distance (in astronomical units) 10 FIGURE 3.23 ln y 3 2 1 Earth Venus Mercury FIGURE 3.24 Saturn Jupiter 3 ln y = ln x 2 Mars 1 |
2 3 ln x 333202_0303.qxd 12/7/05 3:36 PM Page 243 Section 3.3 Properties of Logarithms 243 3.3 Exercises VOCABULARY CHECK: In Exercises 1 and 2, fill in the blanks. 1. To evaluate a logarithm to any base, you can use the ________ formula. 2. The change-of-base formula for base e is given by loga x ________. In Exercises 3–5, match the property of logarithms with its name. uv loga u loga v 3. 4. 5. loga ln un n ln u loga u v loga u loga v (a) Power Property (b) Quotient Property (c) Product Property PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1–8, rewrite the logarithm as a ratio of (a) common logarithms and (b) natural logarithms. 1. 3. 5. 7. log5 x log15 x logx 3 10 log2.6 x 2. 4. 6. 8. log3 x log13 x logx 3 4 log7.1 x In Exercises 9–16, evaluate the logarithm using the change-of-base formula. Round your result to three decimal places. 9. 11. 13. 15. log3 7 log12 4 log9 0.4 log15 1250 10. 12. 14. 16. log7 4 log14 5 log20 0.125 log3 0.015 17. In Exercises 17–22, use the properties of logarithms to rewrite and simplify the logarithmic expression. 42 34 log2 9 log 300 6 e2 log4 8 log5 1 ln5e6 21. 19. 20. 22. 18. ln 250 32. 3 ln e4 33. 34. 35. 36. 37. 38. ln 1 e ln 4e3 ln e2 ln e5 2 ln e6 ln e5 log5 75 log5 3 log4 2 log4 32 In Exercises 39–60, use the properties of logarithms to expand the expression as a sum, difference, and |
/or constant multiple of logarithms. (Assume all variables are positive.) 39. log4 5x 41. log8 x 4 log5 5 x ln z ln xyz2 43. 45. 47. 49. ln zz 12, z > 1 In Exercises 23–38, find the exact value of the logarithmic expression without using a calculator. (If this is not possible, state the reason.) 23. 25. 27. 29. 31. log3 9 log2 48 log4 161.2 9 log3 ln e4.5 24. 26. 28. 30. log5 1 125 log6 36 log3 810.2 16 log2 51. 53. 55. 57. 59., a > 1 a 1 9 log2 ln 3x y x 4y z 5 ln log5 x 2 y 2z3 ln 4x3x2 3 40. 42. log3 10z y 2 log10 44. log6 1 z3 46. 48. 50. 52. 54. 56. 58. ln 3t log 4x2 y lnx 2 1 x3, x > 1 ln 6 x 2 1 lnx 2 y3 x y4 z4 xy4 z5 log10 log2 60. ln x 2x 2 333202_0303.qxd 12/7/05 3:36 PM Page 244 244 Chapter 3 Exponential and Logarithmic Functions In Exercises 61–78, condense the expression to the logarithm of a single quantity. Model It 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. ln x ln 3 ln y ln t log4 z log4 y log5 8 log5 t x 4 2 log2 z 2 2 3 log7 1 4 log3 5x 4 log6 2x ln x 3 lnx 1 2 ln 8 5 lnz 4 log x 2 log y 3 log z 3 log3 x 4 log3 y 4 log3 z ln x 4lnx 2 lnx 2 4ln z lnz 5 2 lnz 5 2 lnx 3 ln x lnx2 1 1 3 23 ln x lnx 1 lnx 1 log8 y 2 log8 y 4 |
log8 1 3 log4 1 2 x 1 2 log4 y 1 x 1 6 log4 x In Exercises 79 and 80, compare the logarithmic quantities. If two are equal, explain why. 79. 80. log2 32 log2 4 log7, log2 32 4, log2 32 log2 4 70, log7 35, 1 2 log7 10 Sound Intensity In Exercises 81–83, use the following information. The relationship between the number of decibels in watts per square meter is given by and the intensity of a sound I 84. Human Memory Model Students participating in a psychology experiment attended several lectures and were given an exam. Every month for a year after the exam, the students were retested to see how much of the material they remembered. The average scores for the group can be modeled by the human memory model f t 90 15 logt 1, 0 ≤ t ≤ 12 where t is the time in months. (a) Use the properties of logarithms to write the func- tion in another form. (b) What was the average score on the original exam t 0? (c) What was the average score after 4 months? (d) What was the average score after 12 months? (e) Use a graphing utility to graph the function over the specified domain. (f) Use the graph in part (e) to determine when the average score will decrease to 75. (g) Verify your answer to part (f) numerically. 85. Galloping Speeds of Animals Four-legged animals run with two different types of motion: trotting and galloping. An animal that is trotting has at least one foot on the ground at all times, whereas an animal that is galloping has all four feet off the ground at some point in its stride. The number of strides per minute at which an animal breaks from a trot to a gallop depends on the weight of the animal. Use the table to find a logarithmic equation that relates an animal’s weight (in pounds) and its lowest galloping speed (in strides per minute). x y 10 log I 1012. 81. Use the properties of logarithms to write the formula in simpler form, and determine the number of decibels of a 106 sound with an intensity of watt per square meter. 82. Find the difference in loudness between an average office 1.26 107 watt per square meter and watt with |
an intensity of a broadcast studio with an intensity of per square meter. 3.16 105 83. You and your roommate are playing your stereos at the same time and at the same intensity. How much louder is the music when both stereos are playing compared with just one stereo playing? Weight, x Galloping Speed, y 25 35 50 75 500 1000 191.5 182.7 173.8 164.2 125.9 114.2 333202_0303.qxd 12/7/05 10:29 AM Page 245 78 C 21 C. 86. Comparing Models A cup of water at an initial temperais placed in a room at a constant temperature ture of of The temperature of the water is measured every 5 minutes during a half-hour period. The results are recordwhere is the time (in ed as ordered pairs of the form T minutes) and 0, 78.0, 25, 42.4, is the temperature (in degrees Celsius). 20, 46.3, 5, 66.0, 30, 39.6 15, 51.2, 10, 57.5, t, T, t is given 93. Proof Prove that logb u v logb u logb v. Section 3.3 Properties of Logarithms 245 Synthesis True or False? statement is true or false given that answer. In Exercises 87–92, determine whether the Justify your f x ln x. 87. 88. 89. 90. f 0 0 f ax f a f x, f x 2 f x f 2, f x < 0, then then v u2. 0 < x < 1. 91. If 92. If a > 0, x > 0 x > 2 94. Proof Prove that logb un n logb u. In Exercises 95–100, use the change -of-base formula to rewrite the logarithm as a ratio of logarithms. Then use a graphing utility to graph both functions in the same viewing window to verify that the functions are equivalent. 95. 97. 99. f x log2 x f x log12 x f x log11.8 x 96. 98. 100. f x log4 x f x log14 x f x log12.4 x 101. Think About It Consider the functions below. f x ln x 2, gx ln x ln 2, hx ln x ln 2 Which two functions should have identical |
graphs? Verify your answer by sketching the graphs of all three functions on the same set of coordinate axes. 102. Exploration For how many integers between 1 and 20 can the natural logarithms be approximated given that ln 2 0.6931, ln 3 1.0986, 1.6094? Approximate these logarithms (do not use a calculator). ln 5 and Skills Review In Exercises 103–106, simplify the expression. 3 2x 2 3y xyx1 y11 24xy2 16x3y 18x3y4318x3y43 106. 105. 103. 104. (a) The graph of the model for the data should be asymptotic with the graph of the temperature of the room. Subtract the room temperature from each of the temperatures in the ordered pairs. Use a graphing utility to plot the data points t, T (b) An exponential model for the data and t, T 21 t, T 21. by T 21 54.40.964t. Solve for with the plot of the original data. T and graph the model. Compare the result (c) Take the natural logarithms of the revised temperatures. t, lnT 21 Use a graphing utility to plot the points and observe that the points appear to be linear. Use the regression feature of the graphing utility to fit a line to these data. This resulting line has the form lnT 21 at b. T. Use the properties of the logarithms to solve for Verify that the result is equivalent to the model in part (b). (d) Fit a rational model to the data. Take the reciprocals of y the -coordinates of the revised data points to generate the points t,. 1 T 21 Use a graphing utility to graph these points and observe that they appear to be linear. Use the regression feature of a graphing utility to fit a line to these data. The resulting line has the form 1 T 21 at b. Solve for rational function and the original data points. and use a graphing utility to graph the T, (e) Write a short paragraph explaining why the transformations of the data were necessary to obtain each model. Why did taking the logarithms of the temperatures lead to a linear scatter plot? Why did taking the reciprocals of the temperature lead to a linear scatter plot? In Exercises 107–110, solve the equation. |
107. 109. 3x2 2x 1 0 x 4 2 3x 1 108. 110. 4x2 5x 1 0 2x 3 5 x 1 333202_0304.qxd 12/7/05 10:31 AM Page 246 246 Chapter 3 Exponential and Logarithmic Functions 3.4 Exponential and Logarithmic Equations What you should learn • Solve simple exponential and logarithmic equations. • Solve more complicated exponential equations. • Solve more complicated logarithmic equations. • Use exponential and logarithmic equations to model and solve real-life problems. Why you should learn it Exponential and logarithmic equations are used to model and solve life science applications. For instance, in Exercise 112, on page 255, a logarithmic function is used to model the number of trees per acre given the average diameter of the trees. © James Marshall/Corbis Introduction So far in this chapter, you have studied the definitions, graphs, and properties of exponential and logarithmic functions. In this section, you will study procedures for solving equations involving these exponential and logarithmic functions. There are two basic strategies for solving exponential or logarithmic equations. The first is based on the One-to-One Properties and was used to solve simple exponential and logarithmic equations in Sections 3.1 and 3.2. The the following second is based on the Inverse Properties. For properties are true for all and a > 0 and are defined. for which a 1, x y and loga y loga x One-to-One Properties if and only if x y. if and only if a x a y loga x loga y Inverse Properties alog a x x loga a x x x y. Example 1 Solving Simple Equations Original Equation a. b. c. d. e. f. 2x 32 ln x ln 3 0 1 x 9 3 e x 7 ln x 3 log x 1 Rewritten Equation 2x 25 ln x ln 3 3x 32 ln e x ln 7 e ln x e3 10 log x 101 Now try Exercise 13. Solution Property x 5 x 3 x 2 x ln 7 x e3 x 101 1 10 One-to-One One-to-One One-to-One Inverse Inverse Inverse The strategies used in Example 1 are summarized as follows. Strategies for Sol |
ving Exponential and Logarithmic Equations 1. Rewrite the original equation in a form that allows the use of the One-to-One Properties of exponential or logarithmic functions. 2. Rewrite an exponential equation in logarithmic form and apply the Inverse Property of logarithmic functions. 3. Rewrite a logarithmic equation in exponential form and apply the Inverse Property of exponential functions. 333202_0304.qxd 12/7/05 10:31 AM Page 247 Section 3.4 Exponential and Logarithmic Equations 247 Solving Exponential Equations Example 2 Solving Exponential Equations Solve each equation and approximate the result to three decimal places if necessary. a. ex2 e3x4 32 x 42 b. Solution a. ex2 e3x4 x2 3x 4 x2 3x 4 0 x 1x Write original equation. One-to-One Property Write in general form. Factor. Set 1st factor equal to 0. Set 2nd factor equal to 0. x 1 and x 4. Check these in the original equation. b. The solutions are 32 x 42 2 x 14 log2 2 x log2 14 x log2 14 x ln 14 ln 2 3.807 Write original equation. Divide each side by 3. Take log (base 2) of each side. Inverse Property Change-of-base formula The solution is x log2 14 3.807. Now try Exercise 25. Check this in the original equation. In Example 2(b), the exact solution is x log2 14 and the approximate An exact answer is preferred when the solution is an solution is intermediate step in a larger problem. For a final answer, an approximate solution is easier to comprehend. x 3.807. Example 3 Solving an Exponential Equation Solve e x 5 60 and approximate the result to three decimal places. Remember that the natural logarithmic function has a base of e. Solution e x 5 60 e x 55 ln ex ln 55 Write original equation. Subtract 5 from each side. Take natural log of each side. x ln 55 4.007 Inverse Property The solution is x ln 55 4.007. Check this in the original equation. Now try Exercise 51. 333202_0304.qxd 12/7/05 10:31 AM Page 248 248 Chapter 3 Exponential and Logarithmic Functions Example 4 Solving |
an Exponential Equation Solve 232t5 4 11 and approximate the result to three decimal places. Solution 232t5 4 11 232t5 15 32t5 15 2 log3 32t5 log3 2t 5 log3 15 2 15 2 log3 7.5 2t 5 log3 7.5 t 5 1 2 2 t 3.417 t 5 1 2 Write original equation. Add 4 to each side. Divide each side by 2. Take log (base 3) of each side. Inverse Property Add 5 to each side. Divide each side by 2. Use a calculator. Remember that to evaluate a logarithm such as you need to use the change-of-base formula. log3 7.5, log3 7.5 ln 7.5 ln 3 1.834 The solution is 2 log3 7.5 3.417. Check this in the original equation. Now try Exercise 53. When an equation involves two or more exponential expressions, you can still use a procedure similar to that demonstrated in Examples 2, 3, and 4. However, the algebra is a bit more complicated. Example 5 Solving an Exponential Equation of Quadratic Type Solve e 2x 3e x 2 0. Algebraic Solution e 2x 3e x 2 0 e x2 3e x 2 0 e x 2e x 1 0 e x 2 0 Write original equation. Write in quadratic form. Factor. Set 1st factor equal to 0. x ln 2 Solution Set 2nd factor equal to 0. e x 1 0 x 0 x ln 2 0.693 Solution The solutions are these in the original equation. y e2x 3ex 2. Graphical Solution Use a graphing utility to graph Use the zero or root feature or the zoom and trace features of the graphing utility to approximate the values of for which y 0. In Figure 3.25, you can see that the zeros occur at x 0 and and at x 0.693. So, the solutions are x 0.693. x 0 x y = e2x − 3e x + 2 3 and x 0. Check 3 3 Now try Exercise 67. −1 FIGURE 3.25 333202_0304.qxd 12/7/05 10:31 AM Page 249 Section 3.4 Exponential and Logarithmic Equations 249 Solving Logarithmic Equations To solve a logarithmic equation, you |
can write it in exponential form. ln x 3 eln x e 3 x e 3 Logarithmic form Exponentiate each side. Exponential form This procedure is called exponentiating each side of an equation. Example 6 Solving Logarithmic Equations a. b. Remember to check your solutions in the original equation when solving equations to verify that the answer is correct and to make sure that the answer lies in the domain of the original equation. ln x 2 e ln x e 2 x e 2 5x 1 log3 5x 1 x 7 4x 8 x 2 log3 x 7 Original equation Exponentiate each side. Inverse Property Original equation One-to-One Property Add x and 1 to each side. Divide each side by 4. Original equation Quotient Property of Logarithms c. log6 3x 14 log6 5 log6 2x log6 2x log63x 14 5 3x 14 5 2x One-to-One Property 3x 14 10x 7x 14 x 2 Now try Exercise 77. Cross multiply. Isolate x. Divide each side by 7. Example 7 Solving a Logarithmic Equation Solve 5 2 ln x 4 and approximate the result to three decimal places. Solution 5 2 ln x 4 2 ln x 1 ln x 1 2 eln x e12 x e12 x 0.607 Now try Exercise 85. Write original equation. Subtract 5 from each side. Divide each side by 2. Exponentiate each side. Inverse Property Use a calculator. 333202_0304.qxd 12/7/05 10:31 AM Page 250 250 Chapter 3 Exponential and Logarithmic Functions Example 8 Solving a Logarithmic Equation Solve 2 log5 3x 4. Solution 2 log5 3x 4 log5 3x 2 5 log5 3x 52 3x 25 x 25 3 x 25 3. The solution is Write original equation. Divide each side by 2. Exponentiate each side (base 5). Inverse Property Divide each side by 3. Check this in the original equation. Now try Exercise 87. Because the domain of a logarithmic function generally does not include all real numbers, you should be sure to check for extraneous solutions of logarithmic equations. Notice in Example 9 that the logarithmic part of the equation is condensed into a single logarithm before exponentiating each side of |
the equation. Example 9 Checking for Extraneous Solutions Solve log 5x logx 1 2. Algebraic Solution log 5x logx 1 2 log5xx 1 2 10 log5x 25x 102 5x2 5x 100 x2 x 20 0 x 5x Write original equation. Product Property of Logarithms Exponentiate each side (base 10). Inverse Property Write in general form. Factor. Set 1st factor equal to 0. Solution Set 2nd factor equal to 0. x 4 Solution The solutions appear to be you check these in the original equation, you can see that is the only solution. However, when x 5 and x 4. x 5 Now try Exercise 99. and 2 graphing utility y2 log 5x logx 1 Graphical Solution Use to graph a y1 in the same viewing window. From the graph shown in Figure 3.26, it appears that the graphs intersect at one point. Use the intersect feature or the zoom and trace features to determine that the graphs intersect at x 5. approximately So, Verify that 5 is an exact solution algebraically. the solution is 5, 2. y1 = log 5x + log(x − 1) y2 = 2 9 5 0 −1 FIGURE 3.26 In Example 9, the domain of x > 0 so the domain of the original equation is x > 1, real numbers greater than 1, the solution Figure 3.26 verifies this concept. log 5x is x 4 logx 1 and the domain of x > 1. is Because the domain is all is extraneous. The graph in 333202_0304.qxd 12/7/05 10:31 AM Page 251 Section 3.4 Exponential and Logarithmic Equations 251 Applications Example 10 Doubling an Investment You have deposited $500 in an account that pays 6.75% interest, compounded continuously. How long will it take your money to double? Solution Using the formula for continuous compounding, you can find that the balance in the account is A Pert A 500e0.0675t. A 1000 and solve the Let A 1000. To find the time required for the balance to double, let t. resulting equation for 500e0.0675t 1000 e0.0675t 2 ln e0.0675t ln 2 0.0675t ln 2 t ln 2 0.0675 t 10.27 Divide each side by 0.0675. |
Divide each side by 500. Take natural log of each side. Use a calculator. Inverse Property The balance in the account will double after approximately 10.27 years. This result is demonstrated graphically in Figure 3.27. Doubling an Investment (10.27, 1000 GTON A = 500e 0.0675t (0, 500) A 1100 ) 900 700 500 300 100 2 4 6 8 Time (in years) t 10 FIGURE 3.27 Now try Exercise 107. In Example 10, an approximate answer of 10.27 years is given. Within the years, does not make ln 20.0675 context of the problem, the exact solution, sense as an answer. Exploration The effective yield of a savings plan is the percent increase in the balance after 1 year. Find the effective yield for each savings plan when $1000 is deposited in a savings account. a. 7% annual interest rate, compounded annually b. 7% annual interest rate, compounded continuously c. 7% annual interest rate, compounded quarterly d. 7.25% annual interest rate, compounded quarterly Which savings plan has the greatest effective yield? Which savings plan will have the highest balance after 5 years? 333202_0304.qxd 12/7/05 10:31 AM Page 252 252 Chapter 3 Exponential and Logarithmic Functions Endangered Animal Species y Example 11 Endangered Animals 450 400 350 300 250 200 10 FIGURE 3.28 The number of endangered animal species in the United States from 1990 to 2002 can be modeled by y y 119 164 ln t, 10 ≤ t ≤ 22 t 10 t where represents the year, with corresponding to 1990 (see Figure 3.28). During which year did the number of endangered animal species reach 357? (Source: U.S. Fish and Wildlife Service) 14 12 20 16 Year (10 ↔ 1990) 18 t 22 Solution 119 164 ln t y 119 164 ln t 357 164 ln t 476 ln t 476 164 eln t e476164 t e476164 t 18 Write original equation. Substitute 357 for y. Add 119 to each side. Divide each side by 164. Exponentiate each side. Inverse Property Use a calculator. The solution is number of endangered animals reached 357 in 1998. Because t 18. t 10 represents 1990, it follows that the W RITING ABOUT MATHEMATICS Comparing Mathematical Models The table shows the U.S. Postal Service rates y for sending |
an express mail package x 5 for selected years from 1985 through 2002, where (Source: U.S. Postal Service) represents 1985. Year, x Rate, y 5 8 11 15 19 21 22 10.75 12.00 13.95 15.00 15.75 16.00 17.85 Now try Exercise 113. a. Create a scatter plot of the data. Find a linear model for the data, and add its graph to your scatter plot. According to this model, when will the rate for sending an express mail package reach $19.00? b. Create a new table showing values for ln x and ln y and create a scatter plot of these transformed data. Use the method illustrated in Example 7 in Section 3.3 to find a model for the transformed data, and add its graph to your scatter plot. According to this model, when will the rate for sending an express mail package reach $19.00? c. Solve the model in part (b) for y, and add its graph to your scatter plot in part (a). Which model better fits the original data? Which model will better predict future rates? Explain. 333202_0304.qxd 12/7/05 10:31 AM Page 253 Section 3.4 Exponential and Logarithmic Equations 253 3.4 Exercises VOCABULARY CHECK: Fill in the blanks. 1. To ________ an equation in means to find all values of x x for which the equation is true. 2. To solve exponential and logarithmic equations, you can use the following One-to-One and Inverse Properties. if and only if ________. if and only if ________. (a) (b) (c) (d) ax ay loga x loga y aloga x loga ax ________ ________ 3. An ________ solution does not satisfy the original equation. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1–8, determine whether each solution (or an approximate solution) of the equation. -value is a x 2. 23x1 32 x 1 (a) x 2 (b) 1. 3. 4. 5. 6. 7. 8. 42x7 64 x 5 (a) x 2 (b) 3e x2 75 (a) 2 ln 6 (b |
) (b) (b) (c) log4 (a) x 2 e25 x 2 ln 25 x 1.219 (c) 2e5x2 12 x 1 (a) 5 x ln 6 5 ln 2 x 0.0416 3x 3 x 21.333 x 4 x 64 3 x 3 10 x 1021 x 17 x 102 3 (c) ln2x 3 5.8 (a) (c) log2 (a) x 1 2 x 1 2 x 163.650 (c) lnx 1 3.8 x 1 e3.8 (a) x 45.701 x 1 ln 3.8 (b) (b) (b) (c) 3 ln 5.8 3 e5.8 In Exercises 9–20, solve for x. 9. 11. 13. 15. 17. 19. 4x 16 1 x 32 2 ln x ln 2 0 e x 2 ln x 1 log4 x 3 10. 12. 14. 16. 18. 20. 3x 243 1 x 64 4 ln x ln 5 0 e x 4 ln x 7 log5 x 3 In Exercises 21–24, approximate the point of intersection g. of the graphs of Then solve the equation fx gx algebraically to verify your approximation. and f 21. f x 2x gx 8 22. f x 27x gx 9 y 12 4 −4 −8 −4 g f x 4 8 −8 −4 y g f x 4 8 12 8 4 −4 23. f x log3 x gx 2 y 24. f x lnx 4 gx 0 y 4 g f 4 8 12 x 12 8 4 −4 g f 8 x 12 333202_0304.qxd 12/7/05 10:31 AM Page 254 254 Chapter 3 Exponential and Logarithmic Functions In Exercises 25–66, solve the exponential equation algebraically. Approximate the result to three decimal places. 25. 27. 29. 31. 33. 35. 37. 39. 41. 43. 45. 47. 49. 51. 53. 55. 57. 59. 61. 63. 65. ex ex22 ex23 ex2 43x 20 2ex 10 ex 9 19 32x 80 5t2 0.20 3x1 27 23x 565 8103x 12 35x1 21 |
e3x 12 500ex 300 7 2ex 5 623x1 7 9 e2x 4ex 5 0 e2x 3ex 4 0 20 2 500 100 e x2 3000 2 e2x 1 0.065 365 1 0.10 12 365t 4 12t 2 26. 28. 30. 32. 34. 36. 38. 40. 42. 44. 46. 48. 50. 52. 54. 56. 58. 60. 62. 64. 66. e2x ex28 ex2 ex22x 25x 32 4ex 91 6x 10 47 65x 3000 43t 0.10 2x3 32 82x 431 510 x6 7 836x 40 e2x 50 1000e4x 75 14 3ex 11 8462x 13 41 e2x 5ex 6 0 e2x 9ex 36 0 350 7 400 1 ex 119 e6x 14 4 2.471 40 16 0.878 26 9t 3t 30 In Exercises 67–74, use a graphing utility to graph and solve the equation. Approximate the result to three decimal places. Verify your result algebraically. 67. 69. 71. 73. 6e1x 25 3e3x2 962 e0.09t 3 e 0.125t 8 0 68. 70. 72. 74. 4ex1 15 0 8e2x3 11 e1.8x 7 0 e2.724x 29 In Exercises 75–102, solve the logarithmic equation algebraically. Approximate the result to three decimal places. 75. 77. 79. 81. 83. 85. ln x 3 ln 2x 2.4 log x 6 3 ln 5x 10 lnx 2 1 7 3 ln x 5 76. 78. 80. 82. 84. 86. ln x 2 ln 4x 1 log 3z 2 2 ln x 7 lnx 8 5 2 6 ln x 10 87. 89. 91. 92. 93. 94. 95. 96. 97. 98. 99. 100. 101. 102. x 2 11 5 log10 ln x lnx 1 1 90. 88. 0.5x 11 6 log3 ln x lnx 1 2 ln x lnx 2 1 ln x lnx 3 1 lnx 5 lnx 1 lnx 1 lnx |
1 lnx 2 ln x 2x 3 log2 log2 logx 6 log2x 1 logx 4 log x logx 2 x 2 log2 log2 x log2 x 1 1 log4 x log4 2 x 8 2 log3 x log3 log 8x log1 x 2 log 4x log12 x 2 x 4 x 6 In Exercises 103–106, use a graphing utility to graph and solve the equation. Approximate the result to three decimal places. Verify your result algebraically. 103. 105. 7 2x 3 ln x 0 104. 106. 500 1500ex2 10 4 lnx 2 0 Compound Interest In Exercises 107 and 108, $2500 is compounded invested in an account at interest rate continuously. Find the time required for the amount to (a) double and (b) triple. r, 109. Demand The demand equation for a microwave oven is given by p 500 0.5e0.004x. Find the demand p $300. (b) x for a price of (a) p $350 and 110. Demand The demand equation for a hand-held elec- tronic organizer is p 50001 4 4 e0.002x. for a price of (a) p $600 and x Find the demand p $400. (b) 111. Forest Yield The yield t acre) for a forest at age years is given by V (in millions of cubic feet per V 6.7e48.1t. (a) Use a graphing utility to graph the function. (b) Determine the horizontal asymptote of the function. Interpret its meaning in the context of the problem. (c) Find the time necessary to obtain a yield of 1.3 million cubic feet. 21 107. r 0.085 108. r 0.12 333202_0304.qxd 12/7/05 10:31 AM Page 255 Section 3.4 Exponential and Logarithmic Equations 255 112. Trees per Acre The number N of trees of a given species per acre is approximated by the model N 68100.04x, is the average diameter of the trees (in inches) 3 feet above the ground. Use the model to approximate the average diameter of the N 21. trees in a test plot when 5 ≤ x ≤ 40 where x 113. Medicine The number y of hospitals in the United States from 1995 to 2002 can be modeled |
by y 7312 630.0 ln t, 5 ≤ t ≤ 12 t represents the year, with where corresponding to 1995. During which year did the number of hospitals reach 5800? (Source: Health Forum) t 5 y 114. Sports The number of daily fee golf facilities in the United States from 1995 to 2003 can be modeled by y 4381 1883.6 ln t, represents the year, with corresponding to 1995. During which year did the number of daily fee golf facilities reach 9000? (Source: National Golf Foundation) 5 ≤ t ≤ 13 where t 5 t 115. Average Heights The percent of American males between the ages of 18 and 24 who are no more than x inches tall is modeled by m mx 100 1 e0.6114x69.71 f and the percent of American females between the ages of 18 and 24 who are no more than inches tall is modeled by x f x 100 1 e0.66607x64.51. (Source: U.S. National Center for Health Statistics) (a) Use the graph to determine any horizontal asymptotes of the graphs of the functions. Interpret the meaning in the context of the problem 100 80 60 40 20 f(x) m(x) x 75 55 70 65 60 Height (in inches) (b) What is the average height of each sex? 116. Learning Curve mathematical model for the proportion responses after trials was found to be n In a group project in learning theory, a of correct P P 0.83 1 e0.2n. (a) Use a graphing utility to graph the function. (b) Use the graph to determine any horizontal asymptotes of the graph of the function. Interpret the meaning of the upper asymptote in the context of this problem. (c) After how many trials will 60% of the responses be correct? Model It 117. Automobiles Automobiles are designed with crumple zones that help protect their occupants in crashes. The crumple zones allow the occupants to move short distances when the automobiles come to abrupt stops. The greater the distance moved, the fewer g’s the crash victims experience. (One g is equal to the acceleration due to gravity. For very short periods of time, humans have withstood as much as 40 g’s.) In crash tests with vehicles moving at 90 kilometers per hour, analysts measured the numbers of g’s experienced during deceleration by crash dummies that were permitted |
to move meters during impact. The data are shown in the table. x x 0.2 0.4 0.6 0.8 1.0 g’s 158 80 53 40 32 A model for the data is given by y 3.00 11.88 ln x 36.94 x where y is the number of g’s. (a) Complete the table using the model. 0.2 0.4 0.6 0.8 1.0 x y (b) Use a graphing utility to graph the data points and the model in the same viewing window. How do they compare? (c) Use the model to estimate the distance traveled during impact if the passenger deceleration must not exceed 30 g’s. (d) Do you think it is practical to lower the number of g’s experienced during impact to fewer than 23? Explain your reasoning. 333202_0304.qxd 12/7/05 10:31 AM Page 256 256 Chapter 3 Exponential and Logarithmic Functions 118. Data Analysis An object at a temperature of 160 C was removed from a furnace and placed in a room at 20 C. The temperature of the object was measured each hour h and recorded in the table. A model for the data is given The graph of this model is by shown in the figure. T 20 1 72h. T Hour, h Temperature, T 0 1 2 3 4 5 160 90 56 38 29 24 (a) Use the graph to identify the horizontal asymptote of the model and interpret the asymptote in the context of the problem. (b) Use the model to approximate the time when the temperature of the object was 100 C. 122. The logarithm of the quotient of two numbers is equal to the difference of the logarithms of the numbers. 123. Think About It Is it possible for a logarithmic equation to have more than one extraneous solution? Explain. 124. Finance You are investing dollars at an annual interest rate of compounded continuously, for years. Which of the following would result in the highest value of the investment? Explain your reasoning. r, P t (a) Double the amount you invest. (b) Double your interest rate. (c) Double the number of years. 125. Think About It Are the times required for the investments in Exercises 107 and 108 to quadruple twice as long as the times for them to double? Give a |
reason for your answer and verify your answer algebraically. 126. Writing Write two or three sentences stating the general guidelines that you follow when solving (a) exponential equations and (b) logarithmic equations. Skills Review In Exercises 127–130, simplify the expression. 127. 128. 129. 130. 48x2y 5 32 225 325 315 3 10 2 In Exercises 131–134, sketch a graph of the function. T 160 140 120 100 80 60 40 20 ( Synthesis 1 2 3 5 4 Hour 6 7 8 h 131. 132. 133. 134 gx 2x, gx x 3, x2 1, x2 4 True or False? In Exercises 119–122, rewrite each verbal statement as an equation. Then decide whether the statement is true or false. Justify your answer. 119. The logarithm of the product of two numbers is equal to the sum of the logarithms of the numbers. 120. The logarithm of the sum of two numbers is equal to the product of the logarithms of the numbers. 121. The logarithm of the difference of two numbers is equal to the difference of the logarithms of the numbers. In Exercises 135–138, evaluate the logarithm using the change-of-base formula. Approximate your result to three decimal places. 135. 136. log6 9 log3 4 137. log34 5 138. log8 22 333202_0305.qxd 12/7/05 10:33 AM Page 257 Section 3.5 Exponential and Logarithmic Models 257 3.5 Exponential and Logarithmic Models What you should learn • Recognize the five most com- mon types of models involving exponential and logarithmic functions. • Use exponential growth and decay functions to model and solve real-life problems. • Use Gaussian functions to model and solve real-life problems. • Use logistic growth functions to model and solve real-life problems. • Use logarithmic functions to model and solve real-life problems. Why you should learn it Exponential growth and decay models are often used to model the population of a country. For instance, in Exercise 36 on page 265, you will use exponential growth and decay models to compare the populations of several countries. Introduction The five most common types of mathematical models involving exponential functions and logarithmic functions are as follows. 1. |
Exponential growth model: 2. Exponential decay model: 3. Gaussian model: 4. Logistic growth model: y ae bx, b > 0 y aebx, b > 0 y ae(xb) 2c y a 1 berx 5. Logarithmic models: y a b ln x, y a b log x The basic shapes of the graphs of these functions are shown in Figure 3.29. y 4 3 2 1 −1 −1 −2 y = ex y = e−x 1 2 3 x −3 −2 −1 y 4 3 2 1 −1 −2 y 2 y = e−x 2 x 1 −1 x 1 −1 EXPONENTIAL GROWTH MODEL EXPONENTIAL DECAY MODEL GAUSSIAN MODEL y 3 2 1 −1 −1 y = 3 1 + e−5x x 1 y 2 1 y = 1 + ln x y 2 1 y = 1 + log x −1 x 1 x 1 2 −1 −2 −1 −2 LOGISTIC GROWTH MODEL FIGURE 3.29 NATURAL LOGARITHMIC MODEL COMMON LOGARITHMIC MODEL Alan Becker/Getty Images You can often gain quite a bit of insight into a situation modeled by an exponential or logarithmic function by identifying and interpreting the function’s asymptotes. Use the graphs in Figure 3.29 to identify the asymptotes of the graph of each function. 333202_0305.qxd 12/7/05 10:33 AM Page 258 258 Chapter 3 Exponential and Logarithmic Functions Exponential Growth and Decay Example 1 Digital Television D 100 80 60 40 20 ) FIGURE 3.30 D 100 80 60 40 20 ) FIGURE 3.31 Digital Television Estimates of the numbers (in millions) of U.S. households with digital television from 2003 through 2007 are shown in the table. The scatter plot of the data is shown in Figure 3.30. (Source: eMarketer) t 7 Year 2003 2004 2005 2006 2007 Households 44.2 49.0 55.5 62.5 70.3 4 5 3 Year (3 ↔ 2003) 6 Digital Television D 30.92e0.1171t, 3 ≤ t ≤ 7 An exponential growth model that approximates these data is given by D is the number of households (in millions) and where represents 2003. Compare the |
values given by the model with the estimates shown in the table. According to this model, when will the number of U.S. households with digital television reach 100 million? t 3 Solution The following table compares the two sets of figures. The graph of the model and the original data are shown in Figure 3.31. t 7 Year 2003 2004 2005 2006 2007 Households Model 44.2 43.9 49.0 49.4 55.5 55.5 62.5 62.4 70.3 70.2 4 5 3 Year (3 ↔ 2003) 6 To find when the number of U.S. households with digital television will reach in the model and solve for 100 million, let D 100 t. Te c h n o l o g y Some graphing utilities have an exponential regression feature that can be used to find exponential models that represent data. If you have such a graphing utility, try using it to find an exponential model for the data given in Example 1. How does your model compare with the model given in Example 1? 30.92e0.1171t D 30.92e0.1171t 100 e0.1171t 3.2342 ln e0.1171t ln 3.2342 0.1171t 1.1738 Write original model. Let D 100. Divide each side by 30.92. Take natural log of each side. Inverse Property t 10.0 Divide each side by 0.1171. According to the model, the number of U.S. households with digital television will reach 100 million in 2010. Now try Exercise 35. 333202_0305.qxd 12/7/05 10:33 AM Page 259 Section 3.5 Exponential and Logarithmic Models 259 In Example 1, you were given the exponential growth model. But suppose this model were not given; how could you find such a model? One technique for doing this is demonstrated in Example 2. Example 2 Modeling Population Growth In a research experiment, a population of fruit flies is increasing according to the law of exponential growth. After 2 days there are 100 flies, and after 4 days there are 300 flies. How many flies will there be after 5 days? Solution y Let be the number of flies at time From the given information, you know that y 100. Substituting this information into the model t 2 y aebt and produces t. when y 300 t 4 when 100 ae2b and 300 |
ae4b. To solve for b, solve for a 100 ae2b in the first equation. a 100 e2b Solve for a in the first equation. Then substitute the result into the second equation. 300 ae4b 300 100 e2be4b e2b 300 100 ln 3 2b ln 3 b 1 2 Write second equation. Substitute 100 e2b for a. Divide each side by 100. Take natural log of each side. Solve for b. Using b 1 and the equation you found for a, you can determine that 2 ln 3 a 100 e212 ln 3 100 e ln 3 100 3 33.33. a 33.33 y 33.33e 0.5493t So, with and Substitute 1 2 ln 3 for b. Simplify. Inverse Property Simplify. b 1 2 ln 3 0.5493, the exponential growth model is as shown in Figure 3.32. This implies that, after 5 days, the population will be y 33.33e 0.54935 520 flies. 600 500 400 300 200 100 Fruit Flies (5, 520) y = 33.33e 0.5493t (4, 300) (2, 100) 1 2 4 3 Time (in days) 5 t FIGURE 3.32 Now try Exercise 37. 333202_0305.qxd 12/7/05 10:33 AM Page 260 260 Chapter 3 Exponential and Logarithmic Functions Carbon Dating R 10−12 t = 0 R = e−t/8223 1 1012 o i 1 t a 2R ( 10−12 ) t = 5,700 t = 19,000 10−13 FIGURE 3.33 t 5,000 15,000 Time (in years) The carbon dating model in Example 3 assumed that the carbon 14 to carbon 12 ratio was one part in 10,000,000,000,000. Suppose an error in measurement occurred and the actual ratio was one part in 8,000,000,000,000. The fossil age corresponding to the actual ratio would then be approximately 17,000 years. Try checking this result. 1012. In living organic material, the ratio of the number of radioactive carbon isotopes (carbon 14) to the number of nonradioactive carbon isotopes (carbon 12) is about 1 to When organic material dies, its carbon 12 content remains fixed, whereas its radioactive carbon 14 begins to decay |
with a half-life of about 5700 years. To estimate the age of dead organic material, scientists use the following formula, which denotes the ratio of carbon 14 to carbon 12 present at t any time (in years). R 1 Carbon dating model The graph of is shown in Figure 3.33. Note that decreases as R t increases. 1012 et 8223 R Example 3 Carbon Dating Estimate the age of a newly discovered fossil in which the ratio of carbon 14 to carbon 12 is R 1 1013. Solution In the carbon dating model, substitute the given value of following. R to obtain the 1 1012et 8223 R et 8223 1 1012 1013 et 8223 1 10 ln et 8223 ln 1 10 Write original model. Let R 1 1013. Multiply each side by 1012. Take natural log of each side. t 8223 2.3026 Inverse Property t 18,934 Multiply each side by 8223. So, to the nearest thousand years, the age of the fossil is about 19,000 years. Now try Exercise 41. b The value of in the exponential decay model determines the decay of radioactive isotopes. For instance, to find how much of an initial 10 grams of isotope with a half-life of 1599 years is left after 500 years, substitute this information into the model y aebt. 226Ra y aebt 10 10eb1599 ln 1 2 Using the value of b found above and 1 2 a 1599b 10, the amount left is b ln 1 2 1599 y 10eln121599500 8.05 grams. 333202_0305.qxd 12/7/05 10:33 AM Page 261 Section 3.5 Exponential and Logarithmic Models 261 Gaussian Models As mentioned at the beginning of this section, Gaussian models are of the form y aexb 2c. This type of model is commonly used in probability and statistics to represent populations that are normally distributed. The graph of a Gaussian model is called a bell-shaped curve. Try graphing the normal distribution with a graphing utility. Can you see why it is called a bell-shaped curve? For standard normal distributions, the model takes the form y 1 ex22. 2 The average value for a population can be found from the bell-shaped curve by observing where the maximum value of the function occurs. The -value corresponding to the maximum value of the function represents the average |
value of the independent variable—in this case, y- y- x. x Example 4 SAT Scores In 2004, the Scholastic Aptitude Test (SAT) math scores for college-bound seniors roughly followed the normal distribution given by y 0.0035ex518 225,992, 200 ≤ x ≤ 800 x where From the graph, estimate the average SAT score. is the SAT score for mathematics. Sketch the graph of this function. (Source: College Board) Solution The graph of the function is shown in Figure 3.34. On this bell-shaped curve, the maximum value of the curve represents the average score. From the graph, you can estimate that the average mathematics score for college-bound seniors in 2004 was 518. 0.003 0.002 0.001 SAT Scores 50% of population x = 518 200 400 600 800 Score x FIGURE 3.34 Now try Exercise 47. 333202_0305.qxd 12/7/05 10:33 AM Page 262 Chapter 3 Exponential and Logarithmic Functions 262 y Decreasing rate of growth Increasing rate of growth x Logistic Growth Models Some populations initially have rapid growth, followed by a declining rate of growth, as indicated by the graph in Figure 3.35. One model for describing this type of growth pattern is the logistic curve given by the function y a 1 ber x y is the population size and is the time. An example is a bacteria culture where that is initially allowed to grow under ideal conditions, and then under less favorable conditions that inhibit growth. A logistic growth curve is also called a sigmoidal curve. x FIGURE 3.35 Example 5 Spread of a Virus On a college campus of 5000 students, one student returns from vacation with a contagious and long-lasting flu virus. The spread of the virus is modeled by 5000 y 1 4999e0.8t, t ≥ 0 is the total number of students infected after days. The college will where cancel classes when 40% or more of the students are infected. y t a. How many students are infected after 5 days? b. After how many days will the college cancel classes? Solution a. After 5 days, the number of students infected is y 5000 1 4999e0.85 5000 1 4999e4 54. b. Classes are canceled when the number infected is 0.405000 2000. 2000 5000 1 4999e0.8t 1 4999e0.8t 2.5 e |
0.8t 1.5 4999 ln e0.8t ln 0.8t ln 1.5 4999 1.5 4999 ln 1.5 4999 t 1 0.8 t 10.1 So, after about 10 days, at least 40% of the students will be infected, and the college will cancel classes. The graph of the function is shown in Figure 3.36. Now try Exercise 49. 2500 2000 1500 1000 500 Flu Virus (10.1, 2000) (5, 54) t 2 FIGURE 3.36 6 8 10 4 Time (in days) 12 14 333202_0305.qxd 12/7/05 10:33 AM Page 263 On December 26, 2004, an earthquake of magnitude 9.0 struck northern Sumatra and many other Asian countries. This earthquake caused a deadly tsunami and was the fourth largest earthquake in the world since 1900. Section 3.5 Exponential and Logarithmic Models 263 Logarithmic Models Example 6 Magnitudes of Earthquakes On the Richter scale, the magnitude of an earthquake of intensity R I is given by R log I I0 1 I0 where is the minimum intensity used for comparison. Find the intensities per unit of area for each earthquake. (Intensity is a measure of the wave energy of an earthquake.) a. Northern Sumatra in 2004: b. Southeastern Alaska in 2004: R 9.0 R 6.8 Solution a. Because 1 I0 and I 9.0 log 1 109.0 10log I R 9.0, you have Substitute 1 for I0 and 9.0 for R. Exponentiate each side. I 109.0 100,000,000. Inverse Property b. For R 6.8, you have 6.8 log I 1 106.8 10log I Substitute 1 for I0 and 6.8 for R. Exponentiate each side. I 106.8 6,310,000. Inverse Property Note that an increase of 2.2 units on the Richter scale (from 6.8 to 9.0) represents an increase in intensity by a factor of 1,000,000,000 6,310,000 158. Year Population, P In other words, the intensity of the earthquake in Sumatra was about 158 times greater than that of the earthquake in Alaska. 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 92.23 106.02 123.20 132.16 |
151.33 179.32 203.30 226.54 248.72 281.42 Now try Exercise 51. W RITING ABOUT MATHEMATICS Comparing Population Models The populations P (in millions) of the United States for the census years from 1910 to 2000 are shown in the table at the left. Least squares regression analysis gives the best quadratic model for these data as P 1.0328t 2 9.607t 81.82, as your conclusion. Which model better fits the data? Describe how you reached and the best exponential model for these data (Source: U.S. Census Bureau) P 82.677e0.124t 10 333202_0305.qxd 12/7/05 10:33 AM Page 264 264 Chapter 3 Exponential and Logarithmic Functions 3.5 Exercises VOCABULARY CHECK: Fill in the blanks. 1. An exponential growth model has the form ________ and an exponential decay model has the form ________. 2. A logarithmic model has the form ________ or ________. 3. Gaussian models are commonly used in probability and statistics to represent populations that are ________ ________. 4. The graph of a Gaussian model is ________ shaped, where the ________ ________ is the maximum -value of the graph. y 5. A logistic curve is also called a ________ curve. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1–6, match the function with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f ).] (a) y (b) y Compound Interest In Exercises 7–14, complete the table for a savings account in which interest is compounded continuously. 6 4 2 −c) y (d) y 12 8 4 4 2 −2 2 4 6 −8 −4 x 4 8 (e) y (f) y x x Initial Investment 7. $1000 8. $750 9. $750 10. $10,000 11. $500 12. $600 13. 14. Amount After 10 Years Annual % Rate Time to Double 3.5% 101 2% 73 4 yr 12 yr $1505.00 4.5% $2000.00 2% $19,205.00 $ |
10,000.00 that must be invested at rate Compound Interest In Exercises 15 and 16, determine the P principal compounded monthly, so that $500,000 will be available for retirement in years. r 71 r 12%, t 40 2%, t 20 15. 16. r, t 6 4 2 6 −12 −6 x 6 12 −2 −2 2 4 1. 3. y 2e x4 y 6 logx 2 5. y lnx 1 2. 4. y 6ex4 y 3ex2 25 6. y 4 1 e2x Compound Interest In Exercises 17 and 18, determine the time necessary for $1000 to double if it is invested at r interest rate compounded (a) annually, (b) monthly, (c) daily, and (d) continuously. x 17. r 11% 18. r 101 2% 19. Compound Interest Complete the table for the time t necessary for dollars to triple if interest is compounded continuously at rate r. P 2% 4% 6% 8% 10% 12% r t 20. Modeling Data Draw a scatter plot of the data in Exercise 19. Use the regression feature of a graphing utility to find a model for the data. 333202_0305.qxd 12/7/05 10:33 AM Page 265 21. Compound Interest Complete the table for the time t P necessary for dollars to triple if interest is compounded annually at rate r. 2% 4% 6% 8% 10% 12% r t 22. Modeling Data Draw a scatter plot of the data in Exercise 21. Use the regression feature of a graphing utility to find a model for the data. 23. Comparing Models If $1 is invested in an account over a repre10-year period, the amount in the account, where or sents the time in years, is given by A e0.07t depending on whether the account pays simple 71 interest at or continuous compound interest at 7%. 2% Graph each function on the same set of axes. Which grows at a higher rate? (Remember that is the greatest integer function discussed in Section 1.6.) A 1 0.075 t t t 24. Comparing Models 10-year period, the amount in the account, where sents the time in years, is given by If $1 is invested in an account over a repre- t A 1 0.06 t or A 1 0.055 365 365t depending |
on whether the account pays simple interest at 51 compounded daily. Use a 6% or compound interest at 2% graphing utility to graph each function in the same viewing window. Which grows at a higher rate? Radioactive Decay for the radioactive isotope. In Exercises 25–30, complete the table Isotope 226Ra 226Ra 14C 14C 239Pu 239Pu 25. 26. 27. 28. 29. 30. Half-life (years) Initial Quantity 1599 1599 5715 5715 24,100 24,100 10 g 3 g Amount After 1000 Years 1.5 g 2 g 2.1 g 0.4 g In Exercises 31–34, find the exponential model that fits the points shown in the graph or table. y aebx 31. y 32. y 10 8 6 4 2 (3, 10) (0, 1) x 8 6 4 2 (4, 5) ( )1 20 Section 3.5 Exponential and Logarithmic Models 265 33. x y 0 5 4 1 34. x y 0 1 3 1 4 35. Population The population thousands) of Pittsburgh, Pennsylvania from 2000 through 2003 can be where represents the year, modeled by with (Source: U.S. Census Bureau) P 2430e0.0029t, corresponding to 2000. t 0 (in P t (a) According to the model, was the population of Pittsburgh increasing or decreasing from 2000 to 2003? Explain your reasoning. (b) What were the populations of Pittsburgh in 2000 and 2003? (c) According to the model, when will the population be approximately 2.3 million? Model It 36. Population The table shows the populations (in millions) of five countries in 2000 and the projected populations (in millions) for the year 2010. (Source: U.S. Census Bureau) Country 2000 2010 Bulgaria Canada China United Kingdom United States 7.8 31.3 1268.9 59.5 282.3 7.1 34.3 1347.6 61.2 309.2 or y aebt (a) Find the exponential growth or decay model y aebt for the population of each country by letting correspond to 2000. Use the model to predict the population of each country in 2030. t 0 (b) You can see that the populations of the United States and the United Kingdom are growing at different rates. What constant in the equation y aebt is determined by these different growth rates? Discuss |
the relationship between the different growth rates and the magnitude of the constant. (c) You can see that the population of China is increasing while the population of Bulgaria is y aebt decreasing. What constant in the equation reflects this difference? Explain. 333202_0305.qxd 12/7/05 10:33 AM Page 266 266 Chapter 3 Exponential and Logarithmic Functions 37. Website Growth The number y of hits a new search- engine website receives each month can be modeled by y 4080ekt t represents the number of months the website has where been operating. In the website’s third month, there were 10,000 hits. Find the value of and use this result to predict the number of hits the website will receive after 24 months. k, 38. Value of a Painting The value V (in millions of dollars) of a famous painting can be modeled by V 10ekt t represents the year, with corresponding to where 1990. In 2004, the same painting was sold for $65 million. Find the value of and use this result to predict the value of the painting in 2010. k, t 0 39. Bacteria Growth The number N of bacteria in a culture is modeled by N 100ekt t is the time in hours. If where estimate the time required for the population to double in size. when N 300 t 5, 40. Bacteria Growth The number N of bacteria in a culture is modeled by N 250ekt t is the time in hours. If t 10, where estimate the time required for the population to double in size. N 280 when 41. Carbon Dating (a) The ratio of carbon 14 to carbon 12 in a piece of wood Estimate the age of R 1814. discovered in a cave is the piece of wood. (b) The ratio of carbon 14 to carbon 12 in a piece of paper R 11311. Estimate the age of the buried in a tomb is piece of paper. 42. Radioactive Decay Carbon 14 dating assumes that the carbon dioxide on Earth today has the same radioactive content as it did centuries ago. If this is true, the amount of 14C absorbed by a tree that grew several centuries ago should be the same as the amount of absorbed by a tree growing today. A piece of ancient charcoal contains only 15% as much radioactive carbon as a piece of modern charcoal. How long ago was the tree burned to make the ancient charcoal if the half-life of is 5715 years? 14C 14C |
43. Depreciation A 2005 Jeep Wrangler that costs $30,788 new has a book value of $18,000 after 2 years. (a) Find the linear model V mt b. (b) Find the exponential model V aekt. (c) Use a graphing utility to graph the two models in the same viewing window. Which model depreciates faster in the first 2 years? (d) Find the book values of the vehicle after 1 year and after 3 years using each model. (e) Explain the advantages and disadvantages of using each model to a buyer and a seller. 44. Depreciation A Dell Inspiron 8600 laptop computer that costs $1150 new has a book value of $550 after 2 years. (a) Find the linear model V mt b. (b) Find the exponential model V aekt. (c) Use a graphing utility to graph the two models in the same viewing window. Which model depreciates faster in the first 2 years? (d) Find the book values of the computer after 1 year and after 3 years using each model. (e) Explain the advantages and disadvantages to a buyer and a seller of using each model. 45. Sales The sales S (in thousands of units) of a new CD years are t burner after it has been on the market for modeled by St 1001 ekt. Fifteen thousand units of the new product were sold the first year. (a) Complete the model by solving for k. (b) Sketch the graph of the model. (c) Use the model to estimate the number of units sold after 5 years. 46. Learning Curve The management at a plastics factory has found that the maximum number of units a worker can produce in a day is 30. The learning curve for the number N of units produced per day after a new employee has t worked days is modeled by N 301 ekt. After 20 days on the job, a new employee produces 19 units. (a) Find the learning curve for this employee (first, find the value of ). k (b) How many days should pass before this employee is producing 25 units per day? 47. IQ Scores The IQ scores from a sample of a class of returning adult students at a small northeastern college roughly follow the normal distribution y 0.0266ex1002450, 70 ≤ x ≤ 115 where x is the IQ score. (a) Use a graphing utility to graph the function. (b) From the graph in |
part (a), estimate the average IQ score of an adult student. 333202_0305.qxd 12/7/05 10:33 AM Page 267 48. Education The time (in hours per week) a student utilizes a math-tutoring center roughly follows the normal distribution y 0.7979ex5.420.5, 4 ≤ x ≤ 7 where x is the number of hours. (a) Use a graphing utility to graph the function. (b) From the graph in part (a), estimate the average num- ber of hours per week a student uses the tutor center. 49. Population Growth A conservation organization releases 100 animals of an endangered species into a game preserve. The organization believes that the preserve has a carrying capacity of 1000 animals and that the growth of the pack will be modeled by the logistic curve pt 1000 1 9e0.1656t where t is measured in months (see figure). 1200 1000 800 600 400 200 2 4 6 8 10 12 14 16 18 Time (in years) t (a) Estimate the population after 5 months. (b) After how many months will the population be 500? (c) Use a graphing utility to graph the function. Use the graph to determine the horizontal asymptotes, and interpret the meaning of the larger -value in the context of the problem. p 50. Sales After discontinuing all advertising for a tool kit in 2000, the manufacturer noted that sales began to drop according to the model S 500,000 1 0.6ekt S t 0 where represents 2000. In 2004, the company sold 300,000 units. represents the number of units sold and (a) Complete the model by solving for k. (b) Estimate sales in 2008. Section 3.5 Exponential and Logarithmic Models 267 Geology In Exercises 51 and 52, use the Richter scale R log I I0 for measuring the magnitudes of earthquakes. 51. Find the intensity of an earthquake measuring 1 Richter scale (let ). I I0 R on the (a) Centeral Alaska in 2002, (b) Hokkaido, Japan in 2003, R 4.2 R (c) Illinois in 2004, R 7.9 R 8.3 52. Find the magnitude of each earthquake of intensity (let I I0 (a) (c) ). 1 I 80,500,000 I 251,200 (b) I 48,275,000 Intensity of Sound In |
Exercises 53–56, use the following information for determining sound intensity. The level of sound in decibels, with an intensity of, is given by, I 10 log I I0 I0 is an intensity of where watt per square meter, corresponding roughly to the faintest sound that can be heard by the human ear. In Exercises 53 and 54, find the level of sound. 1012 53. (a) (b) (c) (d) 54. (a) (b) (c) (d) I 1010 I 105 I 108 I 100 I 1011 I 102 I 104 I 102 m2 m2 m2 watt per (quiet room) watt per (busy street corner) watt per (quiet radio) watt per m2 watt per m2 watt per (threshold of pain) m2 (rustle of leaves) (jet at 30 meters) watt per watt per m2 m2 (door slamming) (siren at 30 meters) 55. Due to the installation of noise suppression materials, the noise level in an auditorium was reduced from 93 to 80 decibels. Find the percent decrease in the intensity level of the noise as a result of the installation of these materials. 56. Due to the installation of a muffler, the noise level of an engine was reduced from 88 to 72 decibels. Find the percent decrease in the intensity level of the noise as a result of the installation of the muffler. pH log [H], pH Levels by hydrogen ion concentration hydrogen per liter) of a solution. In Exercises 57– 62, use the acidity model given where acidity (pH) is a measure of the (measured in moles of [H] H 2.3 105. 57. Find the pH if 58. Find the pH if H 11.3 106. 333202_0305.qxd 12/7/05 10:33 AM Page 268 268 Chapter 3 Exponential and Logarithmic Functions 59. Compute 60. Compute H H for a solution in which pH 5.8. for a solution in which pH 3.2. 61. Apple juice has a pH of 2.9 and drinking water has a pH of 8.0. The hydrogen ion concentration of the apple juice is how many times the concentration of drinking water? 62. The pH of a solution is decreased by one unit. The hydrogen ion concentration is increased by what factor? 63. Forensics At 8 |
:30 A.M., a coroner was called to the home of a person who had died during the night. In order to estimate the time of death, the coroner took the person’s temperature twice. At 9:00 A.M. the temperature was 85.7F, 82.8F. and at 11:00 a.m. the temperature was From these two temperatures, the coroner was able to determine that the time elapsed since death and the body temperature were related by the formula t 10 ln T 70 98.6 70 t where is the time in hours elapsed since the person died T is the temperature (in degrees Fahrenheit) of the and person’s body. Assume that the person had a normal body at death, and that the room temperature of 70F. temperature was a constant (This formula is derived from a general cooling principle called Newton’s Law of Cooling.) Use the formula to estimate the time of death of the person. 98.6F 71 2% 64. Home Mortgage A $120,000 home mortgage for 35 years at has a monthly payment of $809.39. Part of the monthly payment is paid toward the interest charge on the unpaid balance, and the remainder of the payment is used to reduce the principal. The amount that is paid toward the interest is u M M Pr 12 1 r 12 12t and the amount that is paid toward the reduction of the principal is v M Pr 12 1 r 12 12t. In these formulas, interest rate, years. M P is the is the size of the mortgage, is the monthly payment, and is the time in r t (a) Use a graphing utility to graph each function in the same viewing window. (The viewing window should show all 35 years of mortgage payments.) (b) In the early years of the mortgage, is the larger part of the monthly payment paid toward the interest or the principal? Approximate the time when the monthly payment is evenly divided between interest and principal reduction. (c) Repeat parts (a) and (b) for a repayment period of 20 years M $966.71. What can you conclude? 65. Home Mortgage The total interest r mortgage of dollars at interest rate P u for years is paid on a home t u P 1 rt 1 1 r12 12t 1. Consider a $120,000 home mortgage at 71 2%. (a) Use a graphing utility to graph the total interest function. (b) Approx |
imate the length of the mortgage for which the total interest paid is the same as the size of the mortgage. Is it possible that some people are paying twice as much in interest charges as the size of the mortgage? 66. Data Analysis The table shows the time (in seconds) required to attain a speed of miles per hour from a standing start for a car. s t Speed, s Time, t 30 40 50 60 70 80 90 3.4 5.0 7.0 9.3 12.0 15.8 20.0 Two models for these data are as follows. 40.757 0.556s 15.817 ln s 1.2259 0.0023s 2 t1 t2 (a) Use the regression feature of a graphing utility to find a for the data. linear model and an exponential model t4 t3 (b) Use a graphing utility to graph the data and each model in the same viewing window. (c) Create a table comparing the data with estimates obtained from each model. (d) Use the results of part (c) to find the sum of the absolute values of the differences between the data and the estimated values given by each model. Based on the four sums, which model do you think better fits the data? Explain. 333202_0305.qxd 12/7/05 10:33 AM Page 269 Synthesis True or False? statement is true or false. Justify your answer. In Exercises 67–70, determine whether the 67. The domain of a logistic growth function cannot be the set of real numbers. 68. A logistic growth function will always have an -intercept. x 69. The graph of f x 4 1 6e2 x 5 is the graph of gx 4 1 6e2x shifted to the right five units. 70. The graph of a Gaussian model will never have an x -intercept. 71. Identify each model as linear, logarithmic, exponential, logistic, or none of the above. Explain your reasoning. (a) y (bc) y 8 6 4 2 −2 −2 (e) y 12 10 d) y 6 5 4 3 2 1 (f Section 3.5 Exponential and Logarithmic Models 269 72. Writing Use your school’s library, the Internet, or some other reference source to write a paper describing John Napier’s work with logarithms. Skills Review In |
Exercises 73–78, (a) plot the points, (b) find the distance between the points, (c) find the midpoint of the line segment joining the points, and (d) find the slope of the line passing through the points. 73. 74. 75. 76. 77. 78. 1, 2, 0, 5 4, 3, 6, 1 3, 3, 14, 2 7, 0, 10, 4 4, 0, 3 1 2, 1, 2 7 3, 1 3, 1 6 4 3 In Exercises 79–88, sketch the graph of the equation. 79. 80. 81. 82. 83. 84. 85. 86. 87. 88. y 10 3x y 4x 1 y 2x2 3 y 2x2 7x 30 3x2 4y 0 x2 8y 0 y 4 1 3x y x2 x 2 x2 y 82 25 x 42 y 7 4 In Exercises 89–92, graph the exponential function. 89. 90. 91. 92. f x 2 x1 5 f x 2x1 1 f x 3x 4 f x 3 x 4 93. Make a Decision To work an extended application analyzing the net sales for Kohl’s Corporation from 1992 to 2004, visit this text’s website at college.hmco.com. (Data Source: Kohl’s Illinois, Inc.) 333202_030R.qxd 12/7/05 10:34 AM Page 270 270 Chapter 3 Exponential and Logarithmic Functions 3 Chapter Summary What did you learn? Section 3.1 Recognize and evaluate exponential functions with base a (p. 218). Graph exponential functions and use the One-to-One Property (p. 219). Recognize, evaluate, and graph exponential functions with base e (p. 222). Use exponential functions to model and solve real-life problems (p. 223). Section 3.2 Recognize and evaluate logarithmic functions with base a (p. 229). Graph logarithmic functions (p. 231). Recognize, evaluate, and graph natural logarithmic functions (p. 233). Use logarithmic functions to model and solve real-life problems (p. 235). Section 3.3 Use the change-of-base formula to rewrite and evaluate logarithmic expressions (p. 239). Use properties of logarithms to evaluate or rewrite log |
arithmic expressions (p. 240). Use properties of logarithms to expand or condense logarithmic expressions (p. 241). Use logarithmic functions to model and solve real-life problems (p. 242). Section 3.4 Solve simple exponential and logarithmic equations (p. 246). Solve more complicated exponential equations (p. 247). Solve more complicated logarithmic equations (p. 249). Use exponential and logarithmic equations to model and solve real-life problems (p. 251). Section 3.5 Recognize the five most common types of models involving exponential and logarithmic functions (p. 257). Use exponential growth and decay functions to model and solve real-life problems (p. 258). Use Gaussian functions to model and solve real-life problems (p. 261). Use logistic growth functions to model and solve real-life problems (p. 262). Use logarithmic functions to model and solve real-life problems (p. 263). Review Exercises 1–6 7–26 27–34 35–40 41–52 53–58 59–68 69, 70 71–74 75–78 79–94 95, 96 97–104 105–118 119–134 135, 136 137–142 143–148 149 150 151, 152 333202_030R.qxd 12/7/05 10:35 AM Page 271 3 Review Exercises In Exercises 1–6, evaluate the function at the indicated 3.1 value of Round your result to three decimal places. x. Function f x 6.1x f x 30x f x 20.5x f x 1278x5 f x 70.2x f x 145x 1. 2. 3. 4. 5. 6. Value x 2.4 x 3 x x 1 x 11 x 0.8 In Exercises 7–10, match the function with its graph. [The graphs are labeled (a), (b), (c), and (d).] (a) y 1 −3 −2 −1 1 2 3 −2 −3 −4 −5 (c) y 5 4 3 2 1 −3 −2 −1 1 2 3 x x (b) −3 −2 −1 (d3 −2 −1 21 3 x x 7. 9. f x 4x f x 4x 8. 10. f x 4x f x 4x |
1 In Exercises 11–14, use the graph of transformation that yields the graph of f g. to describe the 11. 12. 13. 14. f x 5x, f x 4x gx 5x1 gx 4x 3 gx 1 x2 x gx 8 2 2 3 In Exercises 15–22, use a graphing utility to construct a table of values for the function. Then sketch the graph of the function. 15. 17. f x 4x 4 f x 2.65x1 16. 18. f x 4x 3 f x 2.65x1 Review Exercises 271 19. 21. f x 5 x2 4 f x 1 x 3 2 20. 22. f x 2 x6 5 f x 1 x2 5 8 In Exercises 23–26, use the One-to-One Property to solve the equation for x. 23. 3x2 1 9 25. e5x7 e15 81 x2 1 3 e82x e3 24. 26. In Exercises 27–30, evaluate the function given by at the indicated value of decimal places. fx ex Round your result to three x. 27. 29. x 8 x 1.7 28. 30. x 5 8 x 0.278 In Exercises 31–34, use a graphing utility to construct a table of values for the function. Then sketch the graph of the function. 31. 33. hx ex2 f x e x2 32. 34. hx 2 ex2 st 4e2t, t > 0 Compound Interest table to determine the balance rate In Exercises 35 and 36, complete the for dollars invested at n P times per year. for years and compounded A r t 1 2 4 12 365 Continuous n A 35. 36. P $3500, r 6.5%, t 10 years P $2000, r 5%, t 30 years 37. Waiting Times The average time between incoming calls at a switchboard is 3 minutes. The probability of waiting t less than minutes until the next incoming call is approxiFt 1 et 3. mated by the model A call has just come in. Find the probability that the next call will be within F (a) minute. 1 2 (b) 2 minutes. (c) 5 minutes. 38. Depreciation After t years, the value originally cost $14,000 is given by V Vt 14,0003 of |
a car that t. 4 (a) Use a graphing utility to graph the function. (b) Find the value of the car 2 years after it was purchased. (c) According to the model, when does the car depreciate most rapidly? Is this realistic? Explain. 333202_030R.qxd 12/7/05 3:39 PM Page 272 272 Chapter 3 Exponential and Logarithmic Functions 39. Trust Fund On the day a person is born, a deposit of $50,000 is made in a trust fund that pays 8.75% interest, compounded continuously. (a) Find the balance on the person’s 35th birthday. (b) How much longer would the person have to wait for the balance in the trust fund to double? Q 40. Radioactive Decay Let 241Pu represent a mass of plutonium 241 (in grams), whose half-life is 14.4 years. The quantity of plutonium 241 present after years is given by Q 1001 (a) Determine the initial quantity (when t14.4. t 0 ). t 2 (b) Determine the quantity present after 10 years. (c) Sketch the graph of this function over the interval t 0 to t 100. In Exercises 41– 44, write the exponential equation in 3.2 logarithmic form. 41. 43. 43 64 e0.8 2.2255... 42. 44. 2532 125 e0 1 In Exercises 45– 48, evaluate the function at the indicated value of without using a calculator. x Function f x log x gx log9 x gx log2 x f x log4 x 45. 46. 47. 48. Value x 1000 x 3 x 1 8 x 1 4 In Exercises 49–52, use the One-to-One Property to solve the equation for x. x 7 log4 14 log4 lnx 9 ln 4 49. 51. 50. 52. 3x 10 log8 5 log8 ln2x 1 ln 11 -intercept, and vertical In Exercises 65–68, find the domain, asymptote of the logarithmic function and sketch its graph. x 65. 67. f x ln x 3 hx lnx2 66. 68. f x lnx 3 f x 1 4 ln x 69. Antler Spread The antler spread (in inches) and h |
(in inches) of an adult male American elk shoulder height h 116 loga 40 176. are related by the model Approximate the shoulder height of a male American elk with an antler spread of 55 inches. a 70. Snow Removal The number of miles of roads cleared s of snow is approximated by the model s 25 13 lnh12 2 ≤ h ≤ 15, ln 3 is the depth of the snow in inches. Use this model h where s to find when h 10 inches. 3.3 In Exercises 71–74, evaluate the logarithm using the change-of-base formula. Do each exercise twice, once with common logarithms and once with natural logarithms. Round your the results to three decimal places. 71. 73. log4 9 log12 5 72. 74. log12 200 log3 0.28 In Exercises 75–78, use the properties of logarithms to rewrite and simplify the logarithmic expression. 75. log 18 77. ln 20 76. 78. 1 log2 12 ln3e4 In Exercises 79–86, use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.) In Exercises 53–58, find the domain, -intercept, and vertical asymptote of the logarithmic function and sketch its graph. x 53. 55. 57. gx log7 x f x logx 3 f x 4 logx 5 54. gx log5 x 56. f x 6 log x 58. f x logx 3 1 79. 81. 83. 85. log3 log5 5x 2 6 3x ln x2y2z lnx 3 xy 80. 82. 84. 86. log7 log 7x4 x 4 ln 3xy2 lny 1 4 2, y > 1 In Exercises 59–64, use a calculator to evaluate the function given by at the indicated value of Round your result to three decimal places if necessary. f x ln x x. 59. 61. x 22.6 x e12 63. x 7 5 60. 62. x 0.98 x e7 64. x 3 8 In Exercises 87–94, condense the expression to the logarithm of a single quantity. |
88. 87. 89. log2 5 log2 x ln x 1 4 ln y x 4 7 log8 y 1 3 log8 2 ln2x 1 2 ln x 1 1 93. 94. 5 ln x 2 ln x 2 3 ln x 90. 92. 91. log6 y 2 log6 z 3 ln x 2 lnx 1 2 log x 5 logx 6 333202_030R.qxd 12/7/05 10:35 AM Page 273 Review Exercises 273 95. Climb Rate The time (in minutes) for a small plane to climb to an altitude of feet is modeled by t h t 50 log 18,000 18,000 h where 18,000 feet is the plane’s absolute ceiling. (a) Determine the domain of the function in the context of the problem. (b) Use a graphing utility to graph the function and identify any asymptotes. (c) As the plane approaches its absolute ceiling, what can be said about the time required to increase its altitude? (d) Find the time for the plane to climb to an altitude of 4000 feet. 96. Human Memory Model Students in a learning theory study were given an exam and then retested monthly for 6 months with an equivalent exam. The data obtained in t is the study are given as the ordered pairs is the the time in months after the initial exam and average score for the class. Use these data to find a logarithmic equation that relates and 1, 84.2, 2, 78.4, 3, 72.1, 4, 68.5, 5, 67.1, 6, 65.3 where s t, s, s. t 3.4 In Exercises 97–104, solve for x. 97. 99. 101. 103. 8x 512 ex 3 log4 x 2 ln x 4 98. 100. 102. 104. 6x 1 216 ex 6 log6 x 1 ln x 3 In Exercises 105–114, solve the exponential equation algebraically. Approximate your result to three decimal places. 105. 107. 109. 111. 113. ex 12 e4x ex 23 2 x 13 35 45 x 68 e2x 7ex 10 0 106. 108. 110. 112. 114. e3x 25 14e3x2 560 6 x 28 8 212x 190 e2x 6ex 8 0 In Exercises |
115–118, use a graphing utility to graph and solve the equation. Approximate the result to three decimal places. 115. 117. 20.6x 3x 0 25e0.3x 12 116. 118. 40.2x x 0 4e 1.2 x 9 124. 126. lnx 8 3 ln x ln 5 4 x 2 123. 125. 127. 128. 129. 130. ln x ln 3 2 lnx 1 2 log8 log6 log1 x 1 logx 4 2 x 1 log8 x 2 log6 x log6 x 2 log8 x 5 In Exercises 131–134, use a graphing utility to graph and solve the equation. Approximate the result to three decimal places. 131. 133. 2 lnx 3 3x 8 4 lnx 5 x 10 132. 134. 6 logx2 1 x 0 x 2 logx 4 0 135. Compound Interest You deposit $7550 in an account that pays 7.25% interest, compounded continuously. How long will it take for the money to triple? 136. Meteorology The speed of the wind S (in miles per hour) near the center of a tornado and the distance (in miles) the tornado travels are related by the model S 93 log d 65. On March 18, 1925, a large tornado struck portions of Missouri, Illinois, and Indiana with a wind speed at the center of about 283 miles per hour. Approximate the distance traveled by this tornado. d In Exercises 137–142, match the function with its 3.5 graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).] (a) y (b8 −6 −4 −2 −2 x 2 −8 −6 −4 −2 x 2 (c) y (d) y 8 6 4 2 −4 −2 −2 2 4 6 (e) y 10 8 6 4 2 x x −4 −2 2 4 6 y 3 2 −1 −2 −3 1 2 3 (f) x x In Exercises 119–130, solve the logarithmic equation algebraically. Approximate the result to three decimal places. 119. 121. ln 3x 8.2 2 ln 4x 15 120. 122. ln 5x 7.2 4 ln 3x 15 3 2 1 − |
1 −2 1 2 3 4 5 6 333202_030R.qxd 12/7/05 10:35 AM Page 274 274 Chapter 3 Exponential and Logarithmic Functions 137. 139. y 3e2x3 y lnx 3 141. y 2ex4 23 138. 140. 142. y 4e2x3 y 7 logx 3 6 1 2e2x y In Exercises 143 and 144, find the exponential model y aebx that passes through the points., 0, 1 4, 3 144. 5, 5 2 143. 0, 2, 145. Population The population P of South Carolina (in thousands) from 1990 through 2003 can be modeled by P 3499e0.0135t, t 0 corresponding to 1990. According to this model, when will the population reach 4.5 million? (Source: U.S. Census Bureau) represents the year, with where t 234U 146. Radioactive Decay The half-life of radioactive uranium II is about 250,000 years. What percent of a present amount of radioactive uranium II will remain after 5000 years? 147. Compound Interest A deposit of $10,000 is made in a savings account for which the interest is compounded continuously. The balance will double in 5 years. (a) What is the annual interest rate for this account? (b) Find the balance after 1 year. 148. Wildlife Population A species of bat is in danger of becoming extinct. Five years ago, the total population of the species was 2000. Two years ago, the total population of the species was 1400. What was the total population of the species one year ago? 149. Test Scores The test scores for a biology test follow a normal distribution modeled by y 0.0499ex71 2128, 40 ≤ x ≤ 100 where x is the test score. (a) Use a graphing utility to graph the equation. (b) From the graph in part (a), estimate the average test score. 150. Typing Speed In a typing class, the average number N of words per minute typed after weeks of lessons was found to be t N 157 1 5.4e0.12t. Find the time necessary to type (a) 50 words per minute and (b) 75 words per minute. 151. Sound Intensity The relationship between the number in watts per and the intensity of a sound I of decibels square centimeter is 10 log 1016. I Determine the |
intensity of a sound in watts per square centimeter if the decibel level is 125. 152. Geology On the Richter scale, the magnitude R of an earthquake of intensity I is given by R log I I0 1 R. R 8.4 (a) I0 is the minimum intensity used for where comparison. Find the intensity per unit of area for each value of (b) R 6.85 (c) R 9.1 Synthesis True or False? whether the equation is true or false. Justify your answer. In Exercises 153 and 154, determine 153. 154. logb b2x 2x lnx y ln x ln y 155. The graphs of and Which of the four values are negative? Which are are shown where b, c, d. positive? Explain your reasoning. y e kt k a, (a) (c) y 3 2 (0, 1) (b) y 3 y = eat (0, 1) y = ebt −2 −1 −1 1 2 y 3 2 (0, 1) y = ect −2 −1 −1 1 2 x x −2 −1 −1 1 2 (d) y 3 2 (0, 1) −2 −1 −1 y = edt 1 2 x x 333202_030R.qxd 12/7/05 10:35 AM Page 275 3 Chapter Test Chapter Test 275 Take this test as you would take a test in class. When you are finished, check your work against the answers given in the back of the book. In Exercises 1– 4, evaluate the expression. Approximate your result to three decimal places. 1. 12.42.79 2. 432 3. e710 4. e3.1 In Exercises 5–7, construct a table of values. Then sketch the graph of the function. f x 6 x2 f x 1 e2x f x 10x 7. 5. 6. 8. Evaluate (a) log7 70.89 and (b) 4.6 ln e2. In Exercises 9–11, construct a table of values. Then sketch the graph of the function. Identify any asymptotes. f x log x 6 f x 1 lnx 6 f x lnx 4 10. 11. 9. In Exercises 12–14, evaluate the logarithm using the change-of-base |
formula. Round your result to three decimal places. 12. log7 44 13. log25 0.9 14. log24 68 In Exercises 15–17, use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. 15. log2 3a4 16. ln 5x 6 17. log 7x2 yz3 In Exercises 18–20, condense the expression to the logarithm of a single quantity. 18. 20. log3 13 log3 y 2 ln x lnx 5 3 ln y 19. 4 ln x 4 ln y In Exercises 21– 26, solve the equation algebraically. Approximate your result to three decimal places. Exponential Growth y (9, 11,277) (0, 2745) 2 4 6 8 10 t 5x 1 25 5 1025 8 e4x 18 4 ln x 7 21. 23. 25. 22. 3e5x 132 24. ln x 1 2 26. log x log8 5x 2 12,000 10,000 8,000 6,000 4,000 2,000 FIGURE FOR 27 27. Find an exponential growth model for the graph shown in the figure. 28. The half-life of radioactive actinium 227Ac is 21.77 years. What percent of a present amount of radioactive actinium will remain after 19 years? 29. A model that can be used for predicting the height H on his or her age is age of the child in years. H 70.228 5.104x 9.222 ln x, x (Source: Snapshots of Applications in Mathematics) where 1 4 (in centimeters) of a child based is the ≤ x ≤ 6, (a) Construct a table of values. Then sketch the graph of the model. (b) Use the graph from part (a) to estimate the height of a four-year-old child. Then calculate the actual height using the model. 333202_030R.qxd 12/7/05 10:35 AM Page 276 276 Chapter 3 Exponential and Logarithmic Functions 3 Cumulative Test for Chapters 1–3 Take this test to review the material from earlier chapters. When you are finished, check your work against the answers given in the back of the book. 1, 1. segment joining the points and the distance between the points |
. Find the coordinates of the midpoint of the line 1. Plot the points 3, 4 and x 2 4 In Exercises 2– 4, graph the equation without using a graphing utility. 2. x 3y 12 0 3. y x 2 9 4. y 4 x 5. Find an equation of the line passing through 1 2, 1 and 3, 8. y 4 2 −2 −4 FIGURE FOR 6 6. Explain why the graph at the left does not represent as a function of x. 7. Evaluate (if possible) the function given by for each value. (a) f 6 (b) f 2 (c. Compare the graph of each function with the graph of y 3x. (Note: It is not necessary to sketch the graphs.) (a) r x 1 3x 2 (b) hx 3x 2 (c) gx 3x 2 In Exercises 9 and 10, find (a) is the domain of f x x 3, f/g? 9. gx 4x 1 f gx, (b) f gx, (c) fgx, and (d) f/gx. What 10. f x x 1, gx x2 1 In Exercises 11 and 12, find (a) function. f g and (b) g f. Find the domain of each composite 11. f x 2x2, gx x 6 12. f x x 2, gx x 13. Determine whether hx 5x 2 has an inverse function. If so, find the inverse function. 14. The power produced by a wind turbine is proportional to the cube of the wind speed A wind speed of 27 miles per hour produces a power output of 750 kilowatts. Find P S. the output for a wind speed of 40 miles per hour. 15. Find the quadratic function whose graph has a vertex at the point 4, 7. 8, 5 and passes through In Exercises 16–18, sketch the graph of the function without the aid of a graphing utility. 16. hx x 2 4x 17. f t 1 4tt 22 18. gs s2 4s 10 In Exercises 19–21, find all the zeros of the function and write the function as a product of linear factors. 19. 20. 21. f x x3 2x2 4x 8 f x x4 4x3 21x |
2 f x 2x4 11x3 30x2 62x 40 333202_030R.qxd 12/7/05 10:35 AM Page 277 Cumulative Test for Chapters 1–3 277 22. Use long division to divide 6x3 4x2 by 2x2 1. 23. Use synthetic division to divide 2x 4 3x3 6x 5 by x 2. 24. Use the Intermediate Value Theorem and a graphing utility to find intervals one unit is guaranteed to have a zero. gx x3 3x2 6 in length in which the function Approximate the real zeros of the function. In Exercises 25–27, sketch the graph of the rational function by hand. Be sure to identify all intercepts and asymptotes. 25. 27. f x 2x x2 9 f x x3 3x2 4x 12 x2 x 2 26. f x x2 4x 3 x2 2x 3 In Exercises 28 and 29, solve the inequality. Sketch the solution set on the real number line. 28. 3x3 12x ≤ 0 29. 1 x 1 ≥ 1 x 5 In Exercises 30 and 31, use the graph of to describe the transformation that yields the graph of f x 2 gx 2.2x 4 gx 2 f x 2.2x, x3 g. x, 30. 31. f 5 5 In Exercises 32–35, use a calculator to evaluate the expression. Round your result to three decimal places. 32. log 98 33. log6 7 34. ln31 35. ln40 5 36. Use the properties of logarithms to expand lnx2 16 x 4, where x > 4. 37. Write 2 ln x 1 2 lnx 5 as a logarithm of a single quantity. In Exercises 38–40, solve the equation algebraicially. Approximate the result to three decimal places. 38. 6e2x 72 39. e2x 11ex 24 0 40. lnx 2 3 41. The sales S (in billions of dollars) of lottery tickets in the United States from 1997 through 2003 are shown in the table. (Source: TLF Publications, Inc.) (a) Use a graphing utility to create a scatter plot of the data. Let represent the year, t with t 7 corresponding to 1997. (b) Use the regression feature |
of the graphing utility to find a quadratic model for the data. (c) Use the graphing utility to graph the model in the same viewing window used for the scatter plot. How well does the model fit the data? (d) Use the model to predict the sales of lottery tickets in 2008. Does your answer Year Sales, S 1997 1998 1999 2000 2001 2002 2003 35.5 35.6 36.0 37.2 38.4 42.0 43.5 TABLE FOR 41 seem reasonable? Explain. 42. The number N of bacteria in a culture is given by the model when t where estimate the time required for the N 420 t 8, N 175ekt, is the time in hours. If population to double in size. 333202_030R.qxd 12/7/05 3:39 PM Page 278 Proofs in Mathematics Each of the following three properties of logarithms can be proved by using properties of exponential functions. Slide Rules The slide rule was invented by William Oughtred (1574–1660) in 1625. The slide rule is a computational device with a sliding portion and a fixed portion. A slide rule enables you to perform multiplication by using the Product Property of Logarithms. There are other slide rules that allow for the calculation of roots and trigonometric functions. Slide rules were used by mathematicians and engineers until the invention of the hand-held calculator in 1972. 278 Properties of Logarithms (p. 240) a 1, Let be a positive number such that and are positive real numbers, the following properties are true. a v n and let be a real number. If u 1. Product Property: 2. Quotient Property: Logarithm with Base a loga uv loga u loga v u v loga u loga v loga 3. Power Property: loga un n loga u Proof Let Natural Logarithm lnuv ln u ln v u ln v ln u ln v ln un n ln u x loga u and y loga v. The corresponding exponential forms of these two equations are ax u and ay v. To prove the Product Property, multiply u and v to obtain uv axay axy. The corresponding logarithmic form of uv axy is loga uv x y. So, loga uv loga u loga v. To prove the Quotient Property, divide |
u by v to obtain u v ax ay a xy. The corresponding logarithmic form of uv axy is loga uv x y. So, loga u v loga u loga v. To prove the Power Property, substitute follows. ax for u in the expression loga un, as loga un loga axn loga anx nx n loga u So, loga un n loga u. Substitute ax for u. Property of exponents Inverse Property of Logarithms Substitute loga u for x. 333202_030R.qxd 12/7/05 10:35 AM Page 279 P.S. Problem Solving This collection of thought-provoking and challenging exercises further explores and expands upon concepts learned in this chapter. y x. y1 x, 1. Graph the exponential function given by a 0.5, 1.2, and 2.0. Which of these curves intersects the line y x? for which the curve Determine all positive numbers intersects the line y ax y ax for a y4 y2 y3 x3, ex and x 2. Use a graphing utility to graph x2, x. functions function increases at the greatest rate as approaches and each of the y5 Which? 3. Use the result of Exercise 2 to make a conjecture about the and is a natural. 4. Use the results of Exercises 2 and 3 to describe what is implied when it is stated that a quantity is growing exponentially. ex rate of growth of x number and approaches y xn, where y1 n 5. Given the exponential function f x ax show that (a) (b) f u v f u f v. f 2x f x2. 6. Given that f x ex ex 2 and gx ex ex 2 show that f x2 gx2 1. 7. Use a graphing utility the function given by function. n. y ex n (read “ n! to compare the graph of with the graph of each given is defined as factorial” (a) y1 (b) y2 (c) y3 1 x 1! 1 x 1! 1 x 1! x2 2! x2 2! x3 3! 8. Identify the pattern of successive polynomials given in Exercise 7. Extend the pattern one more term and compare the graph of the resulting polynomial function with the graph of What do you think |
this pattern implies? y ex. 9. Graph the function given by f x e x ex. From the graph, the function appears to be one-to-one. Assuming that the function has an inverse function, find f 1x. f 1x if 10. Find a pattern for f x ax 1 ax 1 where a > 0, a 1. 11. By observation, identify the equation that corresponds to the graph. Explain your reasoning. y 8 6 4 −4 −2 −2 x 2 4 (a) (b) (c) y 6ex22 6 1 ex2 y 61 ex 22 y 12. You have two options for investing $500. The first earns 7% compounded annually and the second earns 7% simple interest. The figure shows the growth of each investment over a 30-year period. (a) Identify which graph represents each type of invest- ment. Explain your reasoning ( 4000 3000 2000 1000 20 25 30 t 5 10 15 Year (b) Verify your answer in part (a) by finding the equations that model the investment growth and graphing the models. (c) Which option would you choose? Explain your reasoning. 13. Two different samples of radioactive isotopes are decaying. as well as respectively. Find the time required The isotopes have initial amounts of k2, half-lives of for the samples to decay to equal amounts. and and c2, k1 c1 279 333202_030R.qxd 12/7/05 10:35 AM Page 280 14. A lab culture initially contains 500 bacteria. Two hours later, the number of bacteria has decreased to 200. Find the exponential decay model of the form B B0akt that can be used to approximate the number of bacteria t after hours. 15. The table shows the colonial population estimates of the (Source: U.S. American colonies from 1700 to 1780. Census Bureau) Year Population 1700 1710 1720 1730 1740 1750 1760 1770 1780 250,900 331,700 466,200 629,400 905,600 1,170,800 1,593,600 2,148,100 2,780,400 In each of the following, let the year with t 0 t, corresponding to 1700. y represent the population in (a) Use the regression feature of a graphing utility to find an exponential model for the data. (b) Use the regression feature of the graphing utility to find a quadratic model |
for the data. (c) Use the graphing utility to plot the data and the models from parts (a) and (b) in the same viewing window. (d) Which model is a better fit for the data? Would you use this model to predict the population of the United States in 2010? Explain your reasoning. 16. Show that 17. Solve logax logab x ln x2 ln x2. 1 loga 1 b. 18. Use a graphing utility to compare the graph of y ln x with the graph of each given (a) the function given by function. y1 y2 y3 b) (c) x 12 x 12 1 3 x 13 280 19. Identify the pattern of successive polynomials given in Exercise 18. Extend the pattern one more term and compare the graph of the resulting polynomial function What do you think the pattern with the graph of implies? y ln x. 20. Using y ab x and y axb take the natural logarithm of each side of each equation. What are the slope and -intercept of the line relating and ln y What are the slope and -intercept of the for and line relating y ab x? ln x y axb? ln y for y y x In Exercises 21 and 22, use the model y 80.4 11 ln x, 100 ≤ x ≤ 1500 which approximates the minimum required ventilation rate in terms of the air space per child in a public school classroom. In the model, is the air space per child in cubic feet and is the ventilation rate per child in cubic feet y per minute. x 21. Use a graphing utility to graph the model and approximate the required ventilation rate if there is 300 cubic feet of air space per child. 22. A classroom is designed for 30 students. The air conditioning system in the room has the capacity of moving 450 cubic feet of air per minute. (a) Determine the ventilation rate per child, assuming that the room is filled to capacity. (b) Estimate the air space required per child. (c) Determine the minimum number of square feet of floor space required for the room if the ceiling height is 30 feet. In Exercises 23–26, (a) use a graphing utility to create a scatter plot of the data, (b) decide whether the data could best be modeled by a linear model, an exponential model, or a logarith |
mic model, (c) explain why you chose the model you did in part (b), (d) use the regression feature of a graphing utility to find the model you chose in part (b) for the data and graph the model with the scatter plot, and (e) determine how well the model you chose fits the data. 1, 2.0, 1.5, 3.5, 2, 4.0, 4, 5.8, 6, 7.0, 8, 7.8 1, 4.4, 1.5, 4.7, 2, 5.5, 4, 9.9, 6, 18.1, 8, 33.0 1, 7.5, 1.5, 7.0, 2, 6.8, 4, 5.0, 6, 3.5, 8, 2.0 25. 26. 1, 5.0, 1.5, 6.0, 2, 6.4, 4, 7.8, 6, 8.6, 8, 9.0 23. 24. 44 333202_0400.qxd 12/7/05 10:59 AM Page 281 Trigonometry 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 Radian and Degree Measure Trigonometric Functions: The Unit Circle Right Triangle Trigonometry Trigonometric Functions of Any Angle Graphs of Sine and Cosine Functions Graphs of Other Trigonometric Functions Inverse Trigonometric Functions Applications and Models Airport runways are named on the basis of the angles they form with due north, measured in a clockwise direction. These angles are called bearings and can be determined using trigonometry AT I O N S Trigonometric functions have many real-life applications. The applications listed below represent a small sample of the applications in this chapter. • Speed of a Bicycle, Exercise 108, page 293 • Respiratory Cycle, Exercise 73, page 330 • Security Patrol, Exercise 97, page 351 • Machine Shop Calculations, • Data Analysis: Meteorology, • Navigation, Exercise 69, page 310 Exercise 75, page 330 Exercise 29, page 360 • Sales, Exercise 88, page 320 • Predator-Prey Model, Exercise 77, page 341 • Wave Motion, Exercise 60, page 362 281 333202_0401.qxd 12/7/05 11:01 AM Page 282 282 Chapter 4 Trigonometry |
4.1 Radian and Degree Measure What you should learn • Describe angles. • Use radian measure. • Use degree measure. • Use angles to model and solve real-life problems. Why you should learn it You can use angles to model and solve real-life problems. For instance, in Exercise 108 on page 293, you are asked to use angles to find the speed of a bicycle. Angles As derived from the Greek language, the word trigonometry means “measurement of triangles.” Initially, trigonometry dealt with relationships among the sides and angles of triangles and was used in the development of astronomy, navigation, and surveying. With the development of calculus and the physical sciences in the 17th century, a different perspective arose—one that viewed the classic trigonometric relationships as functions with the set of real numbers as their domains. Consequently, the applications of trigonometry expanded to include a vast number of physical phenomena involving rotations and vibrations. These phenomena include sound waves, light rays, planetary orbits, vibrating strings, pendulums, and orbits of atomic particles. The approach in this text incorporates both perspectives, starting with angles and their measure. e a l si d e r m i n T Initial side Vertex Angle FIGURE 4.1 y Terminal side Initial side x Angle in Standard Position FIGURE 4.2 © Wolfgang Rattay/Reuters/Corbis An angle is determined by rotating a ray (half-line) about its endpoint. The starting position of the ray is the initial side of the angle, and the position after rotation is the terminal side, as shown in Figure 4.1. The endpoint of the ray is the vertex of the angle. This perception of an angle fits a coordinate system in which the origin is the vertex and the initial side coincides with the positive x -axis. Such an angle is in standard position, as shown in Figure 4.2. Positive angles are generated by counterclockwise rotation, and negative angles by clockwise rotation, as shown in Figure 4.3. Angles are labeled with Greek letters In Figure 4.4, note that angles have the same initial and terminal sides. Such angles are coterminal. (theta), as well as uppercase letters (beta), and (alpha), A, B, and and C. y y y Positive angle (counterclockwise) x Negative angle (clockwise) α β x α x β FIGURE 4.3 FIGURE 4.4 C |
oterminal Angles The HM mathSpace® CD-ROM and Eduspace® for this text contain additional resources related to the concepts discussed in this chapter. 333202_0401.qxd 12/7/05 11:01 AM Page 283 Section 4.1 Radian and Degree Measure 283 y Radian Measure r θ r s = r x The measure of an angle is determined by the amount of rotation from the initial side to the terminal side. One way to measure angles is in radians. This type of measure is especially useful in calculus. To define a radian, you can use a central angle of a circle, one whose vertex is the center of the circle, as shown in Figure 4.5. radius when 1 radian Arc length FIGURE 4.5 Definition of Radian One radian is the measure of a central angle r in length to the radius of the circle. See Figure 4.5. Algebraically, this means that that intercepts an arc equal s s r where is measured in radians. y Because the circumference of a circle is 2r units, it follows that a central angle of one full revolution (counterclockwise) corresponds to an arc length of 2 radians 1 radian r s 2r. 3 radians r r r r x 6 radians 4 radians r 5 radians FIGURE 4.6 r 2 One revolution around a circle of radius corresponds to an radians because angle of 2r r s r 2 radians. Moreover, because there are just over six radius lengths in a full circle, as shown in Figure 4.6. Because the units of measure for and are the has no units—it is simply a real number. same, the ratio sr s r 2 6.28, Because the radian measure of an angle of one full revolution is 2, you can obtain the following. 1 2 1 4 1 6 revolution revolution revolution 2 2 2 4 2 6 radians 2 3 radians radians These and other common angles are shown in Figure 4.7. π 6 π 2 FIGURE 4.7 π 4 π π 3 π2 Recall that the four quadrants in a coordinate system are numbered I, II, III, lie in each are acute angles and and IV. Figure 4.8 on page 284 shows which angles between 0 and of the four quadrants. Note that angles between 0 and angles between are obtuse angles. 2 2 and 2 333202_0401 |
.qxd 12/7/05 11:01 AM Page 284 284 Chapter 4 Trigonometry θ = π 2 Quadrant II π < < π θ 2 Quadrant Quadrant III < < θ π Quadrant IV The phrase “the terminal side of lies in a quadrant” is often abbreviated by simply saying that “ lies in a quadrant.” The terminal sides of the “quadrant 2, angles” do and not lie within quadrants. 32, 0, θ = π 3 2 FIGURE 4.8 Two angles are coterminal if they have the same initial and terminal sides. and You can find an angle that is coterminal to a given angle by adding or is coterminal with For instance, the angles 0 and 136. subtracting has infinitely many coterminal angles. For instance, (one revolution), as demonstrated in Example 1. A given angle are coterminal, as are the angles 6 6 2 2 6 where 2n n is an integer. Example 1 Sketching and Finding Coterminal Angles a. For the positive angle 136, subtract 2 to obtain a coterminal angle 13 6 2. 6 See Figure 4.9. 3 4 b. For the positive angle 2 5. 4 c. For the negative angle 2 4. 3 2 3 π 2 θ= π13 6 π π 3 2 π 6 0 π 34, subtract 2 to obtain a coterminal angle See Figure 4.10. 23, add 2 to obtain a coterminal angle See Figure 4.11 FIGURE 4.9 FIGURE 4.10 FIGURE 4.11 Now try Exercise 17. 333202_0401.qxd 12/7/05 11:01 AM Page 285 Section 4.1 Radian and Degree Measure 285 Two positive angles 2. other) if their sum is of each other) if their sum is and are complementary (complements of each Two positive angles are supplementary (supplements. See Figure 4.12. β α β α Complementary Angles FIGURE 4.12 Supplementary Angles Example 2 Complementary and Supplementary Angles If possible, find the complement and the supplement of (a) 25 and (b) 45. Solution a. The complement of 5 2 10 5 2 The supplement of 5 2 5 5 45 b. Because 10. 25 is 4 |
10 25 is 2 3. 5 5 2, is greater than it has no complement. (Remember that complements are positive angles.) The supplement is 4 5 5 5 4 5. 5 Now try Exercise 21. Degree Measure. A second way to measure angles is in terms of degrees, denoted by the symbol of a complete A measure of one degree (1 ) is equivalent to a rotation of revolution about the vertex. To measure angles, it is convenient to mark degrees on the circumference of a circle, as shown in Figure 4.13. So, a full revolution 360, (counterclockwise) corresponds to a quarter revolution to Because and so on. radians corresponds to one complete revolution, degrees and a half revolution to 90, 2 180, 1 360 x radians are related by the equations 360 2 rad and 180 rad. From the latter equation, you obtain 1 180 rad and 1 rad 180 which lead to the conversion rules at the top of the next page. y 1 ° ° 90 = (360 ) 4 1 (360 )° 60° = 6 1 45° = 8 30° = 0° 360° 330° θ 315° (360 )° 1 (360 )° 12 240° 270° 300° 120° 135° 150° 180° 210° 225° FIGURE 4.13 333202_0401.qxd 12/7/05 11:01 AM Page 286 286 Chapter 4 Trigonometry Conversions Between Degrees and Radians 1. To convert degrees to radians, multiply degrees by 2. To convert radians to degrees, multiply radians by rad 180. 180 rad. To apply these two conversion rules, use the basic relationship (See Figure 4.14.) rad 180. π 6 30° π 2 90° FIGURE 4.14 π 4 45° π 180° π 3 60° π2 360° When no units of angle measure are specified, radian measure is implied. For instance, if you write 2, you imply that 2 radians. Te c h n o l o g y With calculators it is convenient to use decimal degrees to denote fractional parts of degrees. Historically, however, fractional parts of degrees were expressed in minutes and seconds, using the prime ( ) and double prime ( ) notations, respectively. That is, Example 3 Converting from Degrees to Radians a. b. c. 135 135 deg rad 180 deg 540 540 deg rad 180 deg 270 270 deg rad 180 deg |
3 radians 4 3 radians 3 2 radians Now try Exercise 47. Example 4 Converting from Radians to Degrees 1 one minute 1 60 1 one second 1 3600 1 1 64 32 47. Consequently, an angle of 64 degrees, 32 minutes, and 47 seconds is represented by Many calculators have special keys for converting an angle in degrees, minutes, and seconds to decimal degree form, and vice versa. D M S a. b. c. rad180 deg rad rad 2 2 rad180 deg rad 9 9 rad 2 2 360 2 rad 2 rad180 deg rad 90 810 114.59 Multiply by 180. Now try Exercise 51. If you have a calculator with a “radian-to-degree” conversion key, try using it to verify the result shown in part (c) of Example 4. Multiply by 180. Multiply by 180. Multiply by 180. Multiply by 180. Multiply by 180. 333202_0401.qxd 12/7/05 11:01 AM Page 287 Section 4.1 Radian and Degree Measure 287 Applications The radian measure formula, a circle. sr, can be used to measure arc length along Arc Length For a circle of radius by s r r, a central angle intercepts an arc of length given s Length of circular arc is measured in radians. Note that if where measure of equals the arc length. r 1, then s, and the radian Example 5 Finding Arc Length A circle has a radius of 4 inches. Find the length of the arc intercepted by a central angle of as shown in Figure 4.15. 240, Solution To use the formula s r, 240 240 deg rad 180 deg first convert 4 3 radians 240 to radian measure. Then, using a radius of 44 3 s r r 4 16 3 inches, you can find the arc length to be 16.76 inches. Note that the units for radian measure, which has no units. r are determined by the units for because r is given in Now try Exercise 87. The formula for the length of a circular arc can be used to analyze the motion of a particle moving at a constant speed along a circular path. Linear and Angular Speeds Consider a particle moving at a constant speed along a circular arc of radius r. the particle is is the length of the arc traveled in time then the linear speed of If t, v s Linear speed v arc length time s t. Moreover |
, if s, length the particle is is the angle (in radian measure) corresponding to the arc then the angular speed (the lowercase Greek letter omega) of Angular speed central angle time. t s θ = 240° r = 4 FIGURE 4.15 Linear speed measures how fast the particle moves, and angular speed measures how fast the angle changes. By dividing the t, formula for arc length by you can establish a relationship v between linear speed and as shown. angular speed s r r t, s t v r 333202_0401.qxd 12/7/05 11:01 AM Page 288 288 Chapter 4 Trigonometry Example 6 Finding Linear Speed The second hand of a clock is 10.2 centimeters long, as shown in Figure 4.16. Find the linear speed of the tip of this second hand as it passes around the clock face. 10.2 cm Solution In one revolution, the arc length traveled is s 2r 210.2 20.4 centimeters. Substitute for r. FIGURE 4.16 The time required for the second hand to travel this distance is t 1 minute 60 seconds. So, the linear speed of the tip of the second hand is Linear speed s t 20.4 centimeters 60 seconds 1.068 centimeters per second. Now try Exercise 103. 50 ft FIGURE 4.17 Example 7 Finding Angular and Linear Speeds A Ferris wheel with a 50-foot radius (see Figure 4.17) makes 1.5 revolutions per minute. a. Find the angular speed of the Ferris wheel in radians per minute. b. Find the linear speed of the Ferris wheel. Solution a. Because each revolution generates 1.52 3 radians per minute. In other words, the angular speed is 2 radians, it follows that the wheel turns Angular speed t 3 radians 1 minute 3 radians per minute. b. The linear speed is Linear speed s t r t 503 feet 1 minute 471.2 feet per minute. Now try Exercise 105. 333202_0401.qxd 12/7/05 11:01 AM Page 289 Section 4.1 Radian and Degree Measure 289 A sector of a circle is the region bounded by two radii of the circle and their intercepted arc (see Figure 4.18). θ r FIGURE 4.18 Area of a Sector of a Circle A For a circle of radius r, is given by A 1 2 r2 the area of a sector of the circle with central angle |
where is measured in radians. Example 8 Area of a Sector of a Circle A sprinkler on a golf course fairway is set to spray water over a distance of 70 feet and rotates through an angle of (see Figure 4.19). Find the area of the fairway watered by the sprinkler. 120 120° 70 ft FIGURE 4.19 to radian measure as follows. Multiply by 180. 120 Solution First convert 120 120 deg rad 180 deg 2 3 radians 23 r2 Then, using A 1 2 1 2 4900 3 7022 3 and r 70, the area is Formula for the area of a sector of a circle Substitute for and r. Simplify. Simplify. 5131 square feet. Now try Exercise 107. 333202_0401.qxd 12/7/05 11:01 AM Page 290 290 Chapter 4 Trigonometry 4.1 Exercises The HM mathSpace® CD-ROM and Eduspace® for this text contain step-by-step solutions to all odd-numbered exercises. They also provide Tutorial Exercises for additional help. VOCABULARY CHECK: Fill in the blanks. 1. ________ means “measurement of triangles.” 2. An ________ is determined by rotating a ray about its endpoint. 3. Two angles that have the same initial and terminal sides are ________. 4. One ________ is the measure of a central angle that intercepts an arc equal to the radius of the circle. 2 are ________ angles, and angles that measure between 5. Angles that measure between 0 and 2 and are ________ angles. 6. Two positive angles that have a sum of are ________ angles. have a sum of 2 are ________ angles, whereas two positive angles that 7. The angle measure that is equivalent to 1 360 of a complete revolution about an angle’s vertex is one ________. 8. The ________ speed of a particle is the ratio of the arc length traveled to the time traveled. 9. The ________ speed of a particle is the ratio of the change in the central angle to time. 10. The area of a sector of a circle with radius and central angle where r, is measured in radians, is given by the formula ________. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1– 6, estimate |
the angle to the nearest one-half radian. 1. 3. 5. 2. 4. 6. In Exercises 7–12, determine the quadrant in which each angle lies. (The angle measure is given in radians.) 18. (a) 8. (a) 11 8 (b) 9 8 7. (a) 9. (a) 5 12 (b) 7 5 (b) 2 10. (a) 1 (b) 11 9 11. (a) 3.5 (b) 2.25 12. (a) 6.02 (b) 4.25 In Exercises 13–16, sketch each angle in standard position. 13. (a) 15. (a) 5 4 11 6 (b) 2 3 14. (a) 7 4 (b) 3 16. (a) 4 (b) 5 2 (b) 7 In Exercises 17–20, determine two coterminal angles (one positive and one negative) for each angle. Give your answers in radians. 17. (a 19. (a) 20. (a) (b) (b π11 6 θ = − (b) 12 (b) 2 15 333202_0401.qxd 12/7/05 11:01 AM Page 291 In Exercises 21–24, find (if possible) the complement and supplement of each angle. 41. (a) 42. (a) 240 420 (b) (b) 180 230 Section 4.1 Radian and Degree Measure 291 21. (a) 3 23. (a) 1 (b) 3 4 (b) 2 22. (a) 12 (b) 11 12 24. (a) 3 (b) 1.5 In Exercises 25–30, estimate the number of degrees in the angle. 25. 27. 29. 26. 28. 30. In Exercises 31–34, determine the quadrant in which each angle lies. 31. (a) 32. (a) 33. (a) 34. (a) 130 8.3 132 50 260 (b) (b) (b) (b) 285 257 30 336 3.4 In Exercises 35–38, sketch each angle in standard position. 35. (a) 37. (a) 30 405 (b) (b) 150 480 36. (a) 38. (a) 270 750 |
(b) (b) 120 600 In Exercises 39– 42, determine two coterminal angles (one positive and one negative) for each angle. Give your answers in degrees. 39. (a) (b) θ = −36° θ = 45° 40. (a) (b) θ = 420− ° θ = 120° In Exercises 43– 46, find (if possible) the complement and supplement of each angle. 43. (a) 45. (a) 18 79 (b) (b) 115 150 44. (a) 46. (a) 3 130 (b) (b) 64 170 In Exercises 47–50, rewrite each angle in radian measure as (Do not use a calculator.) a multiple of. 47. (a) 30 20 49. (a) (b) 150 240 (b) 48. (a) 315 270 50. (a) (b) 120 144 (b) In Exercises 51–54, rewrite each angle in degree measure. (Do not use a calculator.) 51. (a) 53. (a) 3 2 7 3 (b) (b) 7 6 11 30 52. (a) 54. (a) 7 12 11 6 (b) (b) 9 34 15 In Exercises 55–62, convert the angle measure from degrees to radians. Round to three decimal places. 55. 57. 59. 61. 115 216.35 532 0.83 56. 58. 60. 62. 87.4 48.27 345 0.54 In Exercises 63–70, convert the angle measure from radians to degrees. Round to three decimal places. 7 15 8 4.2 2 63. 65. 67. 69. 5 11 13 2 4.8 0.57 64. 66. 68. 70. In Exercises 71–74, convert each angle measure to decimal degree form. 71. (a) 72. (a) 73. (a) 74. (a) 54 45 245 10 85 18 30 135 36 (b) (b) (b) (b) 128 30 2 12 330 25 408 16 20 In Exercises 75–78, convert each angle measure to form. D M S 75. (a) 76. (a) 77. (a) 78. (a) 240.6 345.12 2.5 0.355 (b) |
(b) 145.8 0.45 3.58 (b) (b) 0.7865 333202_0401.qxd 12/7/05 11:01 AM Page 292 292 Chapter 4 Trigonometry In Exercises 79–82, find the angle in radians. City 29 96. San Francisco, California Seattle, Washington Latitude 37 47 36 N 47 37 18 N 97. Difference in Latitudes Assuming that Earth is a sphere of radius 6378 kilometers, what is the difference in the latitudes of Syracuse, New York and Annapolis, Maryland, where Syracuse is 450 kilometers due north of Annapolis? 98. Difference in Latitudes Assuming that Earth is a sphere of radius 6378 kilometers, what is the difference in the latitudes of Lynchburg, Virginia and Myrtle Beach, South Carolina, where Lynchburg is 400 kilometers due north of Myrtle Beach? 99. Instrumentation The pointer on a voltmeter is 6 centimeters in length (see figure). Find the angle through which the pointer rotates when it moves 2.5 centimeters on the scale. 10 in. 79. 6 81. 32 5 7 80. 82. 10 75 60 In Exercises 83–86, find the radian measure of the central angle of a circle of radius that intercepts an arc of length s. r Radius r Arc Length s 83. 27 inches 84. 14 feet 85. 14.5 centimeters 86. 80 kilometers 6 inches 8 feet 25 centimeters 160 kilometers In Exercises 87–90, find the length of the arc on a circle of radius intercepted by a central angle. r Radius r Central Angle 6 cm 2 ft 87. 15 inches 88. 9 feet 89. 3 meters 90. 20 centimeters 180 60 1 radian 4 radian In Exercises 91–94, find the area of the sector of the circle with radius and central angle. r Radius r 91. 4 inches 92. 12 millimeters 93. 2.5 feet 94. 1.4 miles Central Angle 3 4 225 330 Distance Between Cities In Exercises 95 and 96, find the distance between the cities. Assume that Earth is a sphere of radius 4000 miles and that the cities are on the same longitude (one city is due north of the other). City 95. Dallas, Texas Omaha, Nebraska Latitude 32 47 39 N 41 15 50 N Not drawn to scale FIGURE FOR 99 FIGURE FOR 100 100. Electric Hoist An electric hoist is being used to lift a beam ( |
see figure). The diameter of the drum on the hoist is 10 inches, and the beam must be raised 2 feet. Find the number of degrees through which the drum must rotate. 101. Angular Speed A car is moving at a rate of 65 miles per hour, and the diameter of its wheels is 2.5 feet. (a) Find the number of revolutions per minute the wheels are rotating. (b) Find the angular speed of the wheels in radians per minute. 102. Angular Speed A two-inch-diameter pulley on an electric motor that runs at 1700 revolutions per minute is connected by a belt to a four-inch-diameter pulley on a saw arbor. (a) Find the angular speed (in radians per minute) of each pulley. (b) Find the revolutions per minute of the saw. 333202_0401.qxd 12/7/05 11:01 AM Page 293 103. Linear and Angular Speeds A 71 4 -inch circular power saw rotates at 5200 revolutions per minute. (a) Find the angular speed of the saw blade in radians per minute. (b) Find the linear speed (in feet per minute) of one of the 24 cutting teeth as they contact the wood being cut. 104. Linear and Angular Speeds A carousel with a 50-foot diameter makes 4 revolutions per minute. (a) Find the angular speed of the carousel in radians per minute. (b) Find the linear speed of the platform rim of the carousel. 105. Linear and Angular Speeds The diameter of a DVD is approximately 12 centimeters. The drive motor of the DVD player is controlled to rotate precisely between 200 and 500 revolutions per minute, depending on what track is being read. (a) Find an interval for the angular speed of a DVD as it rotates. (b) Find an interval for the linear speed of a point on the outermost track as the DVD rotates. 106. Area A car’s rear windshield wiper rotates The total length of the wiper mechanism is 25 inches and wipes the windshield over a distance of 14 inches. Find the area covered by the wiper. 125. 107. Area A sprinkler system on a farm is set to spray water over a distance of 35 meters and to rotate through an angle of Draw a diagram that shows the region that can be irrigated with the sprinkler. Find the area of the region. 140. Model It 108. Speed of |
a Bicycle The radii of the pedal sprocket, the wheel sprocket, and the wheel of the bicycle in the figure are 4 inches, 2 inches, and 14 inches, respectively. A cyclist is pedaling at a rate of 1 revolution per second. 14 in. 2 in. 4 in. (a) Find the speed of the bicycle in feet per second and miles per hour. (b) Use your result from part (a) to write a function for the distance (in miles) a cyclist travels in terms of the number of revolutions of the pedal sprocket. n d Section 4.1 Radian and Degree Measure 293 Model It (co n t i n u e d ) (c) Write a function for the distance (in miles) a cyclist travels in terms of the time (in seconds). Compare this function with the function from part (b). d t (d) Classify the types of functions you found in parts (b) and (c). Explain your reasoning. Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 109–111, determine whether 109. A measurement of 4 radians corresponds to two complete revolutions from the initial side to the terminal side of an angle. 110. The difference between the measures of two coterminal if expressed in degrees radians if expressed in angles is always a multiple of and is always a multiple of radians. 360 2 111. An angle that measures 1260 lies in Quadrant III. 112. Writing In your own words, explain the meanings of (a) an angle in standard position, (b) a negative angle, (c) coterminal angles, and (d) an obtuse angle. 113. Think About It A fan motor turns at a given angular speed. How does the speed of the tips of the blades change if a fan of greater diameter is installed on the motor? Explain. 114. Think About It Is a degree or a radian the larger unit of measure? Explain. 115. Writing If the radius of a circle is increasing and the magnitude of a central angle is held constant, how is the length of the intercepted arc changing? Explain your reasoning. 116. Proof Prove that the area of a circular sector of radius r A 1 is measured in 2 where r2, is with central angle radians. Skills Review In Exercises 117–120, simplify the radical expression. 117. 119. 4 42 22 62 55 210 172 92 |
118. 120. In Exercises 121–124, sketch the graphs of specified transformation. f x x 25 f x 2 x5 123. 121. 122. 124. f x x5 4 f x x 35 y x5 and the 333202_0402.qxd 12/8/05 8:18 AM Page 294 294 Chapter 4 Trigonometry 4.2 Trigonometric Functions: The Unit Circle What you should learn • Identify a unit circle and describe its relationship to real numbers. • Evaluate trigonometric functions using the unit circle. • Use the domain and period to evaluate sine and cosine functions. • Use a calculator to evaluate trigonometric functions. Why you should learn it Trigonometric functions are used to model the movement of an oscillating weight. For instance, in Exercise 57 on page 300, the displacement from equilibrium of an oscillating weight suspended by a spring is modeled as a function of time. The Unit Circle The two historical perspectives of trigonometry incorporate different methods for introducing the trigonometric functions. Our first introduction to these functions is based on the unit circle. Consider the unit circle given by x2 y 2 1 Unit circle as shown in Figure 4.20. y (0, 1) − ( 1, 0) (1, 0) x (0, 1)− FIGURE 4.20 Imagine that the real number line is wrapped around this circle, with positive numbers corresponding to a counterclockwise wrapping and negative numbers corresponding to a clockwise wrapping, as shown in Figure 4.21 Richard Megna/Fundamental Photographs FIGURE 4.21 θ x (1, 0) (1, 0, y 1, 0. the real number t corresponds to a point corresponds to the point 2, ference of As the real number line is wrapped around the unit circle, each real number on the circle. For example, the real number 0 Moreover, because the unit circle has a circum2 In general, each real number also corresponds to the point t standard position) whose radian measure is With this interpretation of r 1 length formula of the arc intercepted by the angle ) indicates that the real number, (in the arc is the length also corresponds to a central angle given in radians. s r 1, 0. (with t, t. t 333202_0402.qxd 12/7/05 11:02 AM Page 295 Section 4.2 Trigonometric Functions: The Unit Circle 295 The Trigonometric |
Functions From the preceding discussion, it follows that the coordinates are two t. functions of the real variable You can use these coordinates to define the six trigonometric functions of and t. y x sine cosecant cosine secant tangent cotangent These six functions are normally abbreviated sin, csc, cos, sec, tan, and cot, respectively. Definitions of Trigonometric Functions t Let be a real number and let ding to x, y t. be the point on the unit circle correspon- sin t y cos t x csc t 1 y, y 0 sec t 1 x, x 0 tan t y x cot t x y,, x 0 y 0 In the definitions of the trigonometric functions, note that the tangent and corresponds are undefined. Similarly, For instance, because secant are not defined when to it follows that the cotangent and cosecant are not defined when t 0 y 0. and csc 0 are undefined. x, y 1, 0, cot 0 For instance, because x, y 0, 1, corresponds to sec2 tan2 t 2 x 0. and In Figure 4.22, the unit circle has been divided into eight equal arcs, corre- t sponding to -values of,, 5 4 3 4,, 0 and 2. Similarly, in Figure 4.23, the unit circle has been divided into 12 equal arcs, corresponding to -values of t 0, 11, 6 and 2. To verify the points on the unit circle in Figure 4.22, note that also lies on the line circle produces the following. y x. So, substituting x for y x2 x2 1 2x2 1 Because the point is in the first quadrant, x in the equation of the unit 2 2 2 2, x2 1 2 2 2 and because x ± 2 2 y x, you also have y 2 2. You can use similar reasoning to verify the rest of the points in Figure 4.22 and the points in Figure 4.23. x, y Using the coordinates in Figures 4.22 and 4.23, you can easily evaluate t the trigonometric functions for common -values. This procedure is demonstrated in Examples 1 and 2. You should study and learn these exact function values for common -values because they will help you in later sections to perform calculations quickly and easily. t Note in the definition at the right that the functions in the |
second row are the reciprocals of the corresponding functions in the first row0, 1, 0) x (1, 00, 1)− (, − 2 2 ) 2 2 FIGURE 4.22 ( − 0, 1, 0) x (1, 00, 1)− FIGURE 4.23 ( 333202_0402.qxd 12/7/05 11:02 AM Page 296 296 Chapter 4 Trigonometry Example 1 Evaluating Trigonometric Functions Evaluate the six trigonometric functions at each real number. a. t 6 b. t 5 4 c. t 0 d. t Solution For each -value, begin by finding the corresponding point circle. Then use the definitions of trigonometric functions listed on page 295. x, y on the unit t a. t 6 corresponds to the point x, y 3 2,. 1 2 sin 6 y 1 2 cos 6 x tan 6 y x 3 2 12 32 1 3 corresponds to the point 1 y 1 12 2 csc sec 6 6 1 x 2 3 32 12 23 3 3 3 3 x, y cot cos x tan y x 2 2 22 22 5 4 5 4 5 4 csc sec 5 4 5 4 5 4 1 cot 22 22 1 b. t 5 4 sin c. t 0 corresponds to the point x, y 1, 0. sin 0 y 0 cos 0 x 1 tan 0 y x 0 1 0 csc 0 1 y sec 0 1 x cot 0 x y is undefined. 1 1 1 is undefined. d. t corresponds to the point x, y 1, 0. sin y 0 cos x 1 tan y x 0 1 0 Now try Exercise 23. csc 1 y sec 1 x cot x y is undefined. 1 1 1 is undefined. 333202_0402.qxd 12/7/05 11:02 AM Page 297 Exploration With your graphing utility in radian and parametric modes, enter the equations X1T = cos T and Y1T = sin T and use the following settings. Tmin = 0, Tmax = 6.3, Tstep = 0.1 Xmin = -1.5, Xmax = 1.5, Xscl = 1 Ymin = -1, Ymax = 1, Yscl = 1 1. Graph the entered equations and describe the graph. 2. Use the trace feature to move the cursor around the graph. What do the -values represent? What do |
the x- and values represent? y- t 3. What are the least and greatest values of and y? x y (0, 1) (−1, 0) (1, 0) x −1 ≤ y ≤ 1 (0, −1) −1 ≤ x ≤ 1 FIGURE 4.24,... Section 4.2 Trigonometric Functions: The Unit Circle 297 Example 2 Evaluating Trigonometric Functions Evaluate the six trigonometric functions at t. 3 Solution Moving clockwise around the unit circle, it follows that to the point sin x, y 12, 32. csc 3 2 3 2 3 3 23 3 t 3 corresponds cos tan 3 1 2 32 12 3 sec 2 3 3 cot 3 12 32 1 3 3 3 Now try Exercise 25. Domain and Period of Sine and Cosine The domain of the sine and cosine functions is the set of all real numbers. To determine the range of these two functions, consider the unit circle shown in Figure 4.24. Because Moreover, x, y 1 ≤ x ≤ 1. because So, the values of sine and cosine also range between and 1 ≤ y ≤ 1 1 is on the unit circle, you know that cos t x. and it follows that sin t y r 1, and 1. 1 1 ≤ ≤ y sin t ≤ ≤ 1 1 and 1 1 ≤ ≤ x cos t ≤ ≤ 1 1 2 t in the interval to each value of completes a second Adding revolution around the unit circle, as shown in Figure 4.25. The values of sint 2 Similar results can be obtained for repeated revolutions (positive or negative) on the unit circle. This leads to the general result correspond to those of cost 2 cos t. sin t and and 0, 2 sint 2n sin t and cost 2n cos,...,... t. for any integer and real number Functions that behave in such a repetitive (or cyclic) manner are called periodic. n π π t =, 3,... x t = 0, 2,...,... 7 t = 4 ππ,... ππ 7, 4 π + 4,... FIGURE 4.25 Definition of Periodic Function A function is periodic if there exists a positive real number such that c f ft c f t in the domain of The smallest number f. t for all called the period of f. c for which f is periodic is 333202_0402.qxd 12 |
/7/05 11:02 AM Page 298 298 Chapter 4 Trigonometry Recall from Section 1.5 that a function f t f t. if f is even if f t f t, and is odd Even and Odd Trigonometric Functions The cosine and secant functions are even. sect sec t cost cos t The sine, cosecant, tangent, and cotangent functions are odd. sint sin t tant tan t csct csc t cott cot t Example 3 Using the Period to Evaluate the Sine and Cosine From the definition of periodic function, it follows that the sine and cosine functions are peri2. odic and have a period of The other four trigonometric functions are also periodic, and will be discussed further in Section 4.6. Te c h n o l o g y When evaluating trigonometric functions with a calculator, remember to enclose all fractional angle measures in parentheses. For instance, if you want to 6, sin evaluate should enter you for 13 6 7 2 cos 7 2 sin t 4 5, a. Because 2 b. Because 4, 6 13 6 sin2 sin 6 6 1 2. you have sin, 2 you have cos4 cos 2 2 0. c. For because the sine function is odd. sint 4 5 Now try Exercise 31. Evaluating Trigonometric Functions with a Calculator When evaluating a trigonometric function with a calculator, you need to set the calculator to the desired mode of measurement (degree or radian). Most calculators do not have keys for the cosecant, secant, and cotangent key with their respecfunctions. To evaluate these functions, you can use the tive reciprocal functions sine, cosine, and tangent. For example, to evaluate csc8, use the fact that x 1 csc 8 1 sin8 and enter the following keystroke sequence in radian mode. SIN 6 ENTER. SIN 8 x 1 ENTER Display 2.6131259 These keystrokes yield the correct value of 0.5. Note that some calculators automatically place a left parenthesis after trigonometric functions. Check the user’s guide for your calculator for specific keystrokes on how to evaluate trigonometric functions. Example 4 Using a Calculator Function 2 3 sin a. Mode Calculator Keystrokes Display Radian SIN 2 3 ENTER 0.8660254 b. cot 1. |
5 Radian TAN 1.5 x 1 ENTER 0.0709148 Now try Exercise 45. 333202_0402.qxd 12/7/05 11:02 AM Page 299 Section 4.2 Trigonometric Functions: The Unit Circle 299 4.2 Exercises VOCABULARY CHECK: Fill in the blanks. t 1. Each real number corresponds to a point 2. A function f x, y is ________ if there exists a positive real number such that on the ________ ________. c f t c f t for all t in the domain of f. 3. The smallest number c f is periodic is called the ________ of f. 4. A function f is ________ if and ________ if f t f t. for which a function f t f t PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1– 4, determine the exact values of the six trigonometric functions of the angle. 1. ( 8 15 −, 17 17 ( y 2. y θ x 3. y 4. y θ x ( 12 13, − 5 13 ( ( − 4 5, − ( 3 5 ( 12 5, 13 13 ( θ x θ x In Exercises 5–12, find the point corresponds to the real number x, y t. on the unit circle that 5. 7. 9. 11. 8. 10. t 3 t 5 4 t 5 3 12. t 19. 21. t 11 6 t 3 2 20. t 5 3 22. t 2 In Exercises 23–28, evaluate (if possible) the six trigonometric functions of the real number. 23. 25. 27. t 3 4 t 2 t 4 3 24. 26. 28 In Exercises 29–36, evaluate the trigonometric function using its period as an aid. 29. 31. 33. 35. cos sin 5 8 3 cos15 2 sin9 4 30. 32. 34. 36. sin cos 5 9 4 19 6 cos8 3 sin In Exercises 37– 42, use the value of the trigonometric function to evaluate the indicated functions. sint 3 8 sin t (a) 37. 38. In Exercises 13–22, evaluate (if possible) the sine, cosine, and tangent of the real number. 13 |
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