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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... |
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 (... |
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 a... |
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 2... |
, 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 t... |
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 functi... |
− 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 horizo... |
.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 construct... |
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... |
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... |
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(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 ... |
. 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. ... |
. (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 ... |
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 S... |
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 23,2... |
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 logarithm... |
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 U... |
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... |
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... |
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... |
. 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 l... |
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... |
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 give... |
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 matc... |
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... ... |
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 mea... |
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 wind... |
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 ... |
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 gene... |
.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 exp... |
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 ... |
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 Exercis... |
/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. log... |
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... |
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 50... |
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 ... |
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 equati... |
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 equatio... |
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. Inv... |
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 t... |
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 fac... |
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 ... |
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 you... |
) (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. 1... |
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... |
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 algebraical... |
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 th... |
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... |
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... |
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 ... |
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... |
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 s... |
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 ... |
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 sco... |
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 F... |
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 thes... |
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.... |
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 239... |
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... |
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 t... |
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 ... |
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, fin... |
: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 determi... |
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 ... |
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 Exerc... |
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). So... |
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 ... |
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... |
(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 ... |
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 f... |
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 lo... |
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, ... |
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.... |
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 ... |
. 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 doe... |
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... |
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 20... |
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 Invers... |
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 g... |
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 x... |
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,... |
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... |
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 ... |
.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 ... |
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 ro... |
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 ... |
, 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.... |
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. M... |
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 ... |
(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) 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 kilo... |
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... |
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 ... |
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 describ... |
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, an... |
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 funct... |
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... |
/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... |
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 n... |
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