College Algebra, 2013a

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486 Exponential and Logarithmic Functions 29. Carbon-14 cannot be used to date inorganic material such as rocks, but there are many other methods of radiometric dating which estimate the age of rocks. One of them, Rubidium- Strontium dating, uses Rubidium-87 which decays to Strontium-87 with a half-life of 50 billion years. Use Equation 6.5 to express the amount of Rubidium-87 left from an initial 2.3 micrograms as a function of time t in billions of years. Research this and other radiometric techniques and discuss the margins of error for various methods with your classmates. 30. Use Equation 6.5 to show that k = − ln(2) h where h is the half-life of the radioactive isotope. 31. A pork roast 22 was taken out of a hardwood smoker when its internal temperature had reached 180 ◦ F and it was allowed to rest in a 75 ◦ F house for 20 minutes after which its internal temperature had dropped to 170 ◦ F. Assuming that the temperature of the roast follows Newton’s Law of Cooling (Equation 6.6), (a) Express the temperature T (in ◦ F) as a function of time t (in minutes). (b) Find the time at which the roast would have dropped to 140 ◦ F had it not been carved and eaten. 32. In reference to Exercise 44 in Section 5.3, if Fritzy the Fox’s speed is the same as Chewbacca the Bunny’s speed, Fritzy’s pursuit curve is given by y(x) = 1 4 x2 − 1 4 ln(x) − 1 4 Use your calculator to graph this path for x>0. Describe the behavior of y as x → 0 + and interpret this physically. 33. The current i measured in amps in a certain electronic circuit with a constant impressed voltage of 120 volts is given by i(t) =2− 2e −10t where t ≥ 0 is the number of seconds after the circuit is switched on. Determine the value of i as t →∞. (This is called the steady state current.) 34. If the voltage in the circuit in Exercise 33 above is switched off after 30 seconds, the current is given by the piecewise-defined function i(t) = { 2 − 2e −10t if 0 ≤ t
6.5 Applications of Exponential and Logarithmic Functions 487 35. In Exercise 26 in Section 2.3, we stated that the cable of a suspension bridge formed a parabola but that a free hanging cable did not. A free hanging cable forms a catenary and its basic shapeisgivenbyy = 1 2 (ex + e −x ). Use your calculator to graph this function. What are its domain and range? What is its end behavior? Is it invertible? How do you think it is related to the function given in Exercise 47 in Section 6.3 and the one given in the answer to Exercise 38 in Section 6.4? When flipped upside down, the catenary makes an arch. The Gateway Arch in St. Louis, Missouri has the shape y = 757.7 − 127.7 2 ) (e x 127.7 + e − x 127.7 where x and y are measured in feet and −315 ≤ x ≤ 315. Find the highest point on the arch. 36. In Exercise 6a in Section 2.5, we examined the data set given below which showed how two cats and their surviving offspring can produce over 80 million cats in just ten years. It is virtually impossible to see this data plotted on your calculator, so plot x versus ln(x) aswas done on page 480. Find a linear model for this new data and comment on its goodness of fit. Find an exponential model for the original data and comment on its goodness of fit. Year x 1 2 3 4 5 6 7 8 9 10 Number of Cats N(x) 12 66 382 2201 12680 73041 420715 2423316 13968290 80399780 37. This exercise is a follow-up to Exercise 26 which more thoroughly explores the population growth of Painesville, Ohio. According to Wikipedia, the population of Painesville, Ohio is given by Year t 1860 1870 1880 1890 1900 1910 1920 1930 1940 1950 Population 2649 3728 3841 4755 5024 5501 7272 10944 12235 14432 Year t 1960 1970 1980 1990 2000 Population 16116 16536 16351 15699 17503 (a) Use a graphing utility to perform an exponential regression on the data from 1860 through 1920 only, letting t = 0 represent the year 1860 as before. How does this calculator model compare with the model you found in Exercise 26? Use the calculator’s exponential model to predict the population in 2010. (The 2010 census gave the population as 19,563) (b) The logistic model fit to all of the given data points for the population of Painesville t years after 1860 (again, using t = 0 as 1860) is P (t) = 18691 1+9.8505e −0.03617t According to this model, what should the population of Painesville have been in 2010? (The 2010 census gave the population as 19,563.) What is the population limit of Painesville?
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College Algebra Version ⌊π⌋ Co
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Table of Contents Preface ix 1 Rela
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Table of Contents v 4.3.3 Answers .
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Preface Thank you for your interest
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xi text and we have gone to great l
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Chapter 1 Relations and Functions 1
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1.1 Sets of Real Numbers and the Ca
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1.2 Relations 21 Solution. 1. To gr
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1.2 Relations 23 lines y = 1 and y
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1.2 Relations 25 ‘connecting the
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1.2 Relations 27 (x − 2) 2 + y 2
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1.2 Relations 29 1.2.2 Exercises In
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1.2 Relations 31 29. 5 4 3 y 30. 2
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1.2 Relations 33 1.2.3 Answers 1. y
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1.2 Relations 35 13. y 14. y 3 2
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1.2 Relations 37 41. y = x 2 +1 The
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1.2 Relations 39 45. y = √ x −
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1.2 Relations 41 49. (x +2) 2 + y 2
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1.3 Introduction to Functions 43 1.
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1.3 Introduction to Functions 45 So
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1.3 Introduction to Functions 47 Al
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1.3 Introduction to Functions 51 23
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1.4 Function Notation 55 1.4 Functi
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1.4 Function Notation 57 (c) To fin
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1.4 Function Notation 59 1 − 4x x
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1.4 Function Notation 61 these limi
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1.4 Function Notation 63 1.4.2 Exer
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1.4 Function Notation 65 ⎧ ⎪⎨
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1.4 Function Notation 67 (b) How mu
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1.4 Function Notation 69 1.4.3 Answ
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1.4 Function Notation 71 18. For f(
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1.4 Function Notation 73 26. For f(
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1.4 Function Notation 75 69. C(0) =
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1.5 Function Arithmetic 77 ( ) 4. F
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1.5 Function Arithmetic 79 Next, we
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1.5 Function Arithmetic 81 r(x + h)
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1.5 Function Arithmetic 83 Solution
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1.5 Function Arithmetic 85 27. f(x)
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1.5 Function Arithmetic 87 1.5.2 An
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1.5 Function Arithmetic 89 13. For
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1.5 Function Arithmetic 91 37. 39.
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1.6 Graphs of Functions 93 1.6 Grap
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1.6 Graphs of Functions 95 In the p
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1.6 Graphs of Functions 97 The calc
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1.6 Graphs of Functions 99 The calc
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1.6 Graphs of Functions 101 on [−
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1.6 Graphs of Functions 103 1. Find
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1.6 Graphs of Functions 105 Example
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1.6 Graphs of Functions 107 1.6.2 E
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1.6 Graphs of Functions 109 In Exer
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1.6 Graphs of Functions 111 For Exe
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1.6 Graphs of Functions 113 y y 4 5
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1.6 Graphs of Functions 115 6. f(x)
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1.6 Graphs of Functions 117 17. y 1
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1.6 Graphs of Functions 119 90. h i
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1.7 Transformations 121 7 y 7 y (5,
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1.7 Transformations 123 2 to all of
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1.7 Transformations 125 2 1 (0, 0)
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1.7 Transformations 127 Solution. 1
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1.7 Transformations 129 y y 3 2 1 (
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1.7 Transformations 131 A few remar
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1.7 Transformations 133 6 y 6 y (4,
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1.7 Transformations 135 Finally, to
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1.7 Transformations 137 (a, f(a)) a
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1.7 Transformations 139 shift right
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1.7 Transformations 141 Thecomplete
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1.7 Transformations 143 64. The gra
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1.7 Transformations 145 25. y =2−
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1.7 Transformations 147 38. g(x) =f
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1.7 Transformations 149 50. y = S 1
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Chapter 2 Linear and Quadratic Func
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2.1 Linear Functions 153 P 2 5. m =
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2.1 Linear Functions 155 Equation 2
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2.1 Linear Functions 157 y 4 3 2 y
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2.1 Linear Functions 159 4. We find
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2.1 Linear Functions 161 average sp
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2.1 Linear Functions 163 2.1.1 Exer
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2.1 Linear Functions 165 40. A rest
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2.1 Linear Functions 167 61. y = 2
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2.1 Linear Functions 169 2.1.2 Answ
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2.1 Linear Functions 171 35. (a) F
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2.2 Absolute Value Functions 173 2.
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2.2 Absolute Value Functions 177 By
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2.2 Absolute Value Functions 179 Ex
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2.2 Absolute Value Functions 181 wh
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2.2 Absolute Value Functions 185 24
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2.2 Absolute Value Functions 187 31
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2.3 Quadratic Functions 189 y y (
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2.3 Quadratic Functions 191 shows t
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2.3 Quadratic Functions 193 From g(
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2.3 Quadratic Functions 195 ( a x +
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2.3 Quadratic Functions 197 To find
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2.3 Quadratic Functions 199 Neverth
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2.3 Quadratic Functions 201 15. The
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2.3 Quadratic Functions 203 2.3.2 A
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2.3 Quadratic Functions 205 8. f(x)
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2.3 Quadratic Functions 207 25. (a)
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2.4 Inequalities with Absolute Valu
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2.5 Regression 225 2.5 Regression W
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2.5 Regression 227 2. Find the leas
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2.5 Regression 229 The coefficient
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2.5 Regression 231 4. The chart bel
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2.5 Regression 233 2.5.2 Answers 1.
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Chapter 3 Polynomial Functions 3.1
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3.1 Graphs of Polynomials 237 of th
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3.1 Graphs of Polynomials 239 In or
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3.1 Graphs of Polynomials 241 Despi
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3.1 Graphs of Polynomials 243 at th
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3.1 Graphs of Polynomials 245 Theor
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3.1 Graphs of Polynomials 247 In Ex
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3.1 Graphs of Polynomials 249 37. T
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3.1 Graphs of Polynomials 251 9. f(
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3.1 Graphs of Polynomials 253 21. g
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3.1 Graphs of Polynomials 255 33. (
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3.2 The Factor Theorem and the Rema
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3.3 Real Zeros of Polynomials 269 3
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3.3 Real Zeros of Polynomials 275 f
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3.3 Real Zeros of Polynomials 277 T
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3.3 Real Zeros of Polynomials 279 (
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3.3 Real Zeros of Polynomials 281 I
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3.4 Complex Zeros and the Fundament
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Chapter 4 Rational Functions 4.1 In
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4.1 Introduction to Rational Functi
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4.2 Graphs of Rational Functions 32
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4.3 Rational Inequalities and Appli
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Chapter 5 Further Topics in Functio
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5.1 Function Composition 361 2. As
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5.1 Function Composition 363 (+)
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5.1 Function Composition 365 9. The
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5.1 Function Composition 367 Theore
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5.1 Function Composition 369 5.1.1
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5.1 Function Composition 371 50. (g
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5.1 Function Composition 373 7. For
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5.1 Function Composition 375 20. Fo
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5.1 Function Composition 377 56. (g
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5.2 Inverse Functions 379 The main
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5.2 Inverse Functions 381 y = f −
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5.2 Inverse Functions 383 (b) We ca
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5.2 Inverse Functions 385 y = f(x)
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5.2 Inverse Functions 387 = = 2x 2x
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5.2 Inverse Functions 389 We get g
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5.2 Inverse Functions 391 Using wha
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5.2 Inverse Functions 393 1. Explai
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5.2 Inverse Functions 395 26. Show
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5.3 Other Algebraic Functions 397 5
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5.3 Other Algebraic Functions 399 I
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5.3 Other Algebraic Functions 401 b
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5.3 Other Algebraic Functions 405
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5.3 Other Algebraic Functions 415 (
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Chapter 6 Exponential and Logarithm
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6.1 Introduction to Exponential and
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Page 445 and 446: 6.1 Introduction to Exponential and
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Page 505 and 506: Chapter 7 Hooked on Conics 7.1 Intr
Page 507 and 508: 7.1 Introduction to Conics 497 If t
Page 509 and 510: 7.2 Circles 499 4 y 3 2 1 −4 −3
Page 511 and 512: 7.2 Circles 501 We close this secti
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Page 527 and 528: 7.4 Ellipses 517 Minor Axis Major A
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Page 533 and 534: 7.4 Ellipses 523 From this sketch,
Page 535 and 536: 7.4 Ellipses 525 7.4.1 Exercises In
Page 537 and 538: 7.4 Ellipses 527 7.4.2 Answers x 2
Page 539 and 540: 7.4 Ellipses 529 (x − 4) 2 (y −
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7.5 Hyperbolas 537 As with the othe
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7.5 Hyperbolas 539 y 6 5 4 3 2 Jeff
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7.5 Hyperbolas 541 7.5.1 Exercises
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7.5 Hyperbolas 543 is a hyperbola.
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7.5 Hyperbolas 545 (y − 3) 2 (x
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7.5 Hyperbolas 547 19. (x − 1) 2
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Chapter 8 Systems of Equations and
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8.1 Systems of Linear Equations: Ga
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8.2 Systems of Linear Equations: Au
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8.3 Matrix Arithmetic 579 ⎡ ⎤
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8.3 Matrix Arithmetic 581 As did ma
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8.3 Matrix Arithmetic 583 While the
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8.3 Matrix Arithmetic 585 Using Ri
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8.3 Matrix Arithmetic 587 C 2 − 5
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8.3 Matrix Arithmetic 589 ( √2 2
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8.3 Matrix Arithmetic 591 8.3.1 Exe
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8.3 Matrix Arithmetic 593 28. Now l
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8.3 Matrix Arithmetic 595 8.3.2 Ans
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8.3 Matrix Arithmetic 597 [ ] −9
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8.4 Systems of Linear Equations: Ma
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8.5 Determinants and Cramer’s Rul
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8.6 Partial Fraction Decomposition
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8.7 Systems of Non-Linear Equations
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20. Solve the following system ⎧
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Chapter 9 Sequences and the Binomia
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9.1 Sequences 653 3. From {2n − 1
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9.1 Sequences 655 Solution. A good
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9.1 Sequences 657 Looking at the de
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9.1 Sequences 659 28. 0.9, 0.99, 0.
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9.2 Summation Notation 661 9.2 Summ
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9.2 Summation Notation 663 5∑ (
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9.2 Summation Notation 665 S = n (2
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9.2 Summation Notation 667 payment
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9.2 Summation Notation 669 n∑ k=1
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9.2 Summation Notation 671 In Exerc
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9.3 Mathematical Induction 673 9.3
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9.3 Mathematical Induction 675 This
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9.3 Mathematical Induction 677 it i
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9.3 Mathematical Induction 679 9.3.
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9.4 The Binomial Theorem 681 9.4 Th
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9.4 The Binomial Theorem 683 songs
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9.4 The Binomial Theorem 685 (a + b
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9.4 The Binomial Theorem 687 ∑k
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9.4 The Binomial Theorem 689 ( 0 0)
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9.4 The Binomial Theorem 691 9.4.1
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Index n th root of a complex number
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Index 1071 central angle, 701 chang
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Index 1073 reflective property, 523
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Index 1075 matrix, multiplicative,
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Index 1077 midpoint definition of,
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Index 1079 instantaneous, 161, 472
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Index 1081 inconsistent, 553 indepe
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College Algebra, 2013a

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