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hw2-4 - Chapteri Functions and Graphs EXAMPLE ‘3 Home...

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Unformatted text preview: Chapteri Functions and Graphs EXAMPLE ‘3 Home Range. The home range of an animal is defined as the region to which the animal confines its movements. It has been hypothesized in statistical studies* that the area H of that region can be approximated by the function H 2 W141 where W is the weight of the animal. Graph the function. Solution We can approximate function values using a power key YX on a calculator. We see that H 2 WM 2 W141/100 = 1«GO/‘Wmi The graph is shown below. Note that the function values increase from left to right. As body weight increases, the area over which the animal moves increases. *J. M. Emlen, Ecology: An Evolutionary Approach, p. 200 (Addison—Wesley, Reading, MA, 1973). fiXeruse Set 1.3 Graph. Using the same set of axes, graph the pair of equations. [\J U) 1.y=]x|andy=ix+3| 5y:— 2.y=|xlandy=lx+1| X x 3.y=\/)—candy=\/x+1 —2 —3 4.y=\/)_candy=\/x—2 x region tistical ona = 10 = 9 y x2 y x —- 1 3 1 11 y = V} 12 y = m 2 2 x — 9 x — 4 = 14. = 13. f(x) x + 3 g(x) _ 2 2 2 x — 1 x — 25 = 16. = 15' f(x) x — 1 £06) x + 5 Convert to expressions with rational exponents. 17. Vx3 18. Vx5 19. 5 a3 20. V4 b2 21. \7/l 22. \“/E 23 1 24- 1 ' 3 t4 .W 25 1 26 —1— ‘ t ' Vfi 1 27. ———— 28. Vx3 + 4 sz + 7 Convert to radical notation. 29. x15 30. t” 31. y2/3 32. t2/5 33. tHZ/S 34' yH2/3 35.1?” 36. If“ 37. (2‘17/6 38. m—19/6 39 (x2 __ 3)_l/2 40. (3/2 + 7)‘“1/4 l ' l 41' 2273 42- W—4/5 Simplify. 43. 93/2 44. 165/2 45. 642/3 46- 82’3 47. 163’4 48. 255/2 Determine the domain of the function. 2 _ 2 _ 49- f(x) = "7:? 50. f(x) = : + 24 51- f(X) = x3 52. f(x) = x4 + 7 x2—5x+6 53. f(x) = m ,4 54.f(x) = V2x — 6 55. "iétn . Ema. Refer to Example 9. The territorial area of an animal is defined as its defended region, or exclusive region. For example, a lion has a cer— w 10‘ 20 3o 40 50 100- x2+6x+5 1.3 Rational and Radical Functions 39 tain region over which it is ruler. The area T of that region can be approximated by the power function I 2 W131 where W is the weight of the animal. Complete the table of approximate function values and graph the function. 56. 3.; {7,3153 l mu According to Zipf’s Law, the number of cities with a population greater than S is inversely proportional to 5. In 2000, there were 48 U.S. Cities with a population greater than 350,000. Estimate the number of U.S. cities with a population greater than 200,000.8 57. 4157047ka Stature flaw. A person whose mass is 75 kg has surface area approximated by f(h) = 0.144h1/2, where f(h) is measured in square meters and h is the person’s height in centimeters.9 a) Find the approximate surface area of a person Whose mass is 75 kg and whose height is 180 cm. b) Find the approximate surface area of a person whose mass is 75 kg and whose height is 170 cm. c) Graph the function f(l1) for 0 S h s 200. 58. Mwsmm. The body mass y (in kilograms) of a theropod dinosaur may be approximated by the function y = 0.73x3‘63, where x is the total length of the dinosaur (in meters).10 a) Find the body mass of Coelophysis baari, which has a total length of 2.7 m. b) Find the body mass of Sinraptor dongi, which has a total length of 7 In. BUS. Bureau of the Census. 9U.S. Oncology 10F. Seebacher, “A New Method to Calculate Allometric Length- Mass Relationships of Dinosaurs,” Journal of Vertebrate Pale— ontology, Vol. 21, pp. 51—60 (2001). 40 Chapter1 ‘ Functions and Graphs c) Suppose a therapod has a body mass of 5000 kg. Find its total length. _ 5 The velocity of blood in a blood vessel is inversely proportional to the cross-sectional area of the blood vessel. This relationship is called the law of conti— nuity.11 Area Velocity 59. Suppose, in an adult male, blood leaves the aorta at 30 cm/ sec, and the cross—sectional area of the aorta is 3 cm2. Given that blood travels in the capillaries at 0.026 cm/sec, find the total cross-sectional area of his capillaries.12 60. Suppose, in an adult female, blood leaves the aorta at 28 cm/ sec, and the cross-sectional area of the aorta is 2.8 cm2. Given that blood travels in the capillaries at 0.025 cm/ sec, find the total cross— sectional area of her capillaries. Solve. 9 61. x + 7 + — = 0 (Hint: Multiply both sides by x.) x l 1 62.1——=——2 W W “NA. Campbell and J.B. Reece, Biology (Benjamin Cum— mings, New York, 2002). 12Notice this is the total cross-sectional area of all capillaries, not the cross—sectional area of a single capillary. . 3 Pollution control has become a very important concern in all countries. If controls are not put in place, it has been predicted that the function 63. P = 1000?” + 14,000 will describe the average pollution, in particles of pollution per cubic centimeter, in most cities at time t, in years, where t = 0 corresponds to 1970 and t = 37 corresponds to 2007. a) Predict the pollution in 2007, 2010, and 2020. b) Graph the function over the interval [0, 50]. UN 64. At most, how many y-intercepts can a function have? Explain. WV 65. Explain the difference between a rational function and a polynomial function. Is every polynomial function a rational function? Use the ZERO feature or the INTERSECT feature to ap- proximate the zeros of the function to three decimal places. 66. f(x) = %(lx — 4| + Ix — 7|) — 4 67.f(x)= \/7—x2 68.f(x)=lx+1l+|x-2|—5 69. f(x) — ix 1| Ix 2| 70. f(x) = Ix -_ ll »— lx— 2[ — 3 ‘ 46 Chapteri ~ Functions and Graphs 0 Technology Connection uni. , Let’s determine the acute angle whose sine is 0.2. On the 11—83 and many other graphers, this is done using the SIN'1 key. If the grapher is in degree mode, then the answer is approximately 11.537°. If the calculator is in radian mode, then the answer is approximately 0.20136 radians. The C057‘ or TAN" key can be used to find an angle if we know its cosine or tan- gent, respectively. EXERCISES Use a grapher to approximate the acute angle in degrees. 1. Findtifsint= 0.12. 2. Findtif cos t = 0.73. 3. Find t iftan t = 1.24. Use a grapher to approximate the acute angle in radians. 4. Find t ifsin t = 0.85. 5. Find tif cos t = 0.62.. 6. Find t if tan t = 0.45. {Exercise Set 1.4 In Exercises 1—6, convert from degrees into radians. Draw a picture of each angle on the xy~plane. 1. 120° 2. 150° 3. 240° 4. 300° 5. 540° 6. —450° Draw a picture of each angle on the xy-plane. 7. 377/4 8. 77T/6 9. 377/2 10. 377' 11. —’7T/3 12. —117T/15 In Exercises 7—12, convert from radians into degrees. At times we wish to know an angle in a right triangle when we are given the lengths of two sides. We can find the angle by using the definition of a trigonometric function. EXAMPhE @ Find the angle of elevation of the sun if a building 25 ft high casts a a) 25-ft shadow. b) 10-ft shadow. So/ufion 25 a) We can see from the figure that tan t = 55 = l. The angle must be 45° since tan 45° = 1. 25 b) In this case, tant=—1—O= 2.5. We have not seen any special angle whose tangent is 2.5. To find the angle we use the TAN" key on a calculator to find t % 68.20°. 25 ft Determine if the following pairs of angles are coterminal. 13. 15° and 395° 14. 225° and —135° 15. 107° and —107° 16. 1400 and 440° 17. 7r/2 and37T/2 18. 77/2 and —377/2 19. 77/6 and —577/6 20. 377/4 and —77/4 Use a calculator to find the values of the following trigonometric functions. 21. sin 34° 22. sin 82° 23. cos 12° 24. cos 41° iangle 1d the on. m ifa : angle :n any gle we .20°. aa 25. tan 5° 26. tan 68° 27. cot 340 28. cot 56° 29. sec 23° . 30. csc 72° 31. sin(7r/5) 32. cos(27T/5) 33. tan(7'r/7) 34. cot(37r/ll) 35. sec(37T/8) 36. csc(477/13) 37. sin(2.3) 38. cos(0.81) Use a calculator to find the degree measure of an acute angle whose trigonometric function is given. 39. sin t = 0.45 40. sin t = 0.87 41. cost = 0.34 42. cos t = 0.72 43. tan t = 2.34 44. tan t = 0.84 Use a calculator to find the radian measure of an acute angle whose trigonometric function is given. \ 45. sin t = 0.59 46. sin t = 0.26 47. cos t = 0.60 48. cos t = 0.78 49. tan t = 0.11 50. tan t = 1.26 Solve for the missing side x. 52. 15 20° 40 x X I O 15 X I x 1.4 51. 53. 54. 1.4 Trigonometric Functions 47 Solve for the missing angle t. Express your answer in degrees. 55. 56. 20 25 so 40 - 57. 58. 50 9.3 30 18 59. Use a Sum Identity to find cos 75°. k 60. f i(7.‘l€?}’§9é‘ft’f$. Honeybees communicate the location of food sources to other bees in a hive through an elaborate dance.15 Through such a dance, the hive learns that a food source is 200 m away at an angle 20° north of the sun, which is rising due east. Find the x~ and y—coordinates of the food source. (Think of east as the positive x—direction and north as the positive y-direction.) y (X.y) fFood 200m; I ‘ ll)’ IL 0 l 20 . > X X 61. {Elude oia 130ml. On a 5—mi stretch of highway, the road decreases in elevation at an angle of 4°. How much lower is a car after traveling on this part of the highway? (Remember that there are 5280 feet in a mile.) 15Karl von Frisch, The Dance Language and Orientation of Bees, Harvard University Press, 1971. :; ‘ ‘V .1 48 Chapteri - Functions and Graphs 62. {trade nit; Hand. The tangent of a road’s angle of 65. Suppose f = 4,000,000 Hz, (1 = 100 Hz, and t, elevation t is called the grade of the road; the grade t = 65°. Determine the blood velocity through the is often expressed as a percentage. Suppose a high- vessel. way through a mountain pass has a grade of 5% i * and is 6 mi long from the base to the top of the j pass. How much higher is the pass than the base? 63. mtg". A certain hole on a golf course is 330 yd long with a 40° dogleg, as illustrated in the figure. The distance from the tee to the center of the dogleg is 180 yd, while the distance from the center of the dogleg to the green is 150 yd. 66. Washington Mommn’nt. While standing in the Mall in Washington, DC, a tourist observes the angle of elevation to the top of the Washington Monument to be 67°. After moving 1012 ft farther away from the Washington Monument, the angle of elevation l i! . a) Find x. changes to 24°. ll 5 b) Find y. ‘ c) Find 2, the straight-line tee-to—green distance. a) Use the small triangle to find x in terms of h. b) Use the large triangle to find the height of the Washington Monument. 67. iiropoétéons {sffétfl {:1wa ’ii'émgies. Consider the adjacent 30—60—90 right triangles, each with hypotenuse of length 2, shown in the figure. a) Explain why the two triangles form one equilat~ eral triangle. b) Explain why the short leg of each triangle has 1? 2M 3.: Ultrasound measures the velocity of length 1- f 1: blood (m cm /5) through a blood vessel using c) Use the Pythagorean theorem to find the length ‘ . of the long leg. v = 77:000d 56C t (1) Explain how this figure gives the trigonometric f ' functions of 77/ 6 and 7r/ 3. In this formula, f is the emitted ultrasound beam frequency, d is the Doppler shift (or the difference between the emitted and received beam frequencies), and t is the angle between the ultrasound beam and the blood vessel.16 64. Suppose f = 5,000,000 Hz, 01 = 200 Hz, and t = 60°. Determine the blood velocity through the vessel. 16Triton Technology, Inc. [all le of :nt on gift“ [9 lat— 68. Proportions ofé‘fi~4¢3—-<}ljl 'li'irmglcs. Consider the 45—45—90 right triangle shown in the figure, with a leg of length 1. a) Explain Why the other leg also has length 1. b) Use the Pythagorean theorem to find the length of the hypotenuse. c) Explain how this figure gives the trigonometric functions of 77/4, 69. Refer to the figure below. a) Use the small right triangle to show that tan t = 5/7. b) Use the large right triangle to show that tan t = 10/ 14. UN c) Why don’t the trigonometric functions depend on the size of the triangle? 10 ’fl‘igtmomemt‘: identities. Many interrelationships between the trigonometric functions can be found, as shown in the following exercises. 70. fiiéittipr‘ot‘trl. Use the definitions of the trigonometric functions to derive the Reciprocal Identities. 71. raw. Use the definitions of the trigonometric functions to derive the Ratio Identities. 72. ‘wizmi ltlrirztityjor Sign; Refer to the figure below to answer these questions.17 a) Show that u = sin t and v = cos t. b) Use geometry to show that r = s. c) Show that w = sin 5 cos t. 17R.B. Nelson, Proofs Without Words II (Mathematical Associa- tion of America. Washington, DC, 2000). 1.4 Trigonometric Functions 49 (1) Show that x = cos 5 sin 1:. e) Conclude that sin(s + t) = (w + x)/1 = sin 5 cos t + cos 5 sin t. 73. 7e?! itirtrztityjos' {Lo/4,3524%. Use the figure from the previous exercise. a) Show that u = sin t and v = cos t. b) Use geometry to show that r = s. c) Show that y = cos 5 cos t. CD Show that z = sin 5 sin t. e) Conclude that cos(s + t) = (y — z)/l = cos 5 cos t — sin 5 sin t. 74-. (,Jrgflmrtimr a) Use the figure to show that W t ' t cos —— =sm. 2 b) Show that sin<g — t) = cos t. Opposite to t Adjacent to 7r/2 — t 75. £631,527 mu. Show that 1 + tanzt = seczt. (Hint: Begin With the Pythagorean Identity of Theorem 6 and divide both sides by coszt.) 76. li).l1$’u1;,;o2mm. Show that 1 + cotzt = csczt. 77. Moravia-«Angle for £25323, Show that sin 2t = 2 sin t cos t. (Hint: Let s = t and use a Sum Identity.) 58 Chapteri Functions and Graphs Sketch the following angles. 1. 577/4 2. —57r/6 3. —7T 4. 27r 5.137r/6 6. ~77r/4 Use a unit circle to compute the following trigonomet- ric functions. 7. COS(9’7T/2) 8. sin(57T/4) 9. sin(~57r/6) . cos(—57T/4) 11. cos 577 12. sin 677 13. tan(—4iT/3) 14. tan(—77r/3) Use a calculator to evaluate the following trigonometric functions. 15. COS 125° 16. sin 164° 17. tan(—220°) 18. COS(”253D) 19. sec 286° 20. csc 312° 21. sin(1.27'r) 22. tan(—2.37r) 23. COS("1.91) 24. sin(—2.04) Find all solutions of the given equation. . 1 25. smt= -— 2 26. sin t = —1 27. sin 2t = O 28. 2 sin<t + 1) 3 7T 29. 3t + — COS< 4) 30. cos(Zt) = O 31. cos(3t) = 1 t 32.2 cos<3> = —\/3 33. 25in2t — 55m: — 3 = 0 34. coszx + 5 cosx = 6 35. coszx + 5 cosx = -6 36. sinzt—Zsint—3=O For the following functions, find the amplitude, period, and mid-line. Also, find the maximum and minimum. 37.y=25in2t+4 38.y=3cosZt—3 39. y = 5 cos(t/Z) + 1 40. y = 3 sin(t/3) + 2 1 . 1 41. y = 3 51n(3t) — 3 42. y = 3 cos(4t) + 2 43- 3’ = 4 5111(7”) + 2 44. y = 3 cos(377t) — 2 For each of the following graphs, determine if the func- tion should be modeled by either y = a sin bt + k or y = a cos bt + h. Then find a, b, and k. 45. 2 1C- ift'lilfliilfi Swim Radiation. The annual radiation (in megajoules per square centimeter) for certain land areas of the northern hemisphere may be modeled with the equation19 R = 0.339 + 0.808 cos l cos 5 — 0.196 sin 1 sin 5 — 0.482 cos a sin 5. In this equation, I is the latitude (between 30° and 60°) and s is the slope of the ground (between 0° and 60°). Also, a is the aspect, or the direction that the slope faces. For a slope facing due north, a = 0°, and for a slope facing south, a = 180°. For a slope facing either east or west, a = 90°. 49. Find the annual radiation of north—facing land at 40° north latitude with a 30° slope. 50. Find the annual radiation of south~facing land at 30° north latitude with a 20° slope. 51. Find the annual radiation of southeast-facing land at 50° north latitude with a 55° slope. 52. Find the annual radiation of flat land at 50° north latitude. 1mg Capacity. As we breathe, our lungs increase and decrease in volume. The volume of air that we inhale and exhale with each breath is called tidal volume. The maximum possible tidal volume is called the vital ca- pacity, normally approached during strenuous physical activity. Even at vital capacity, the lungs are never completely drained of air; the minimum volume of the lungs is called the residual volume.20 53. Suppose a man watching television breathes once every 5 sec. His average lung capacity is 2500 mL, and his tidal volume is 500 mL. Express the 19B. McCune and D. Keon, “Equations for potential annual di— rect incident radiation and heat load,” Joumal of Vegetation Science, Vol. 13, pp. 603—606 (2002). 206.]. Borden, K. S. Harris, and L]. Raphael, Speech Science Primer; 4th ed. (Lippincott Williams & Wilkins, Philadel— ‘ phia, 2003). 1.5 tW Trigonometric Functions and the Unit Circle 59 volume of his lungs using the model V(t) = a cos lot + k, where time 0 corresponds to the lungs at their largest capacity. 54. A woman undergoes her ordinary strenuous work- out, breathing once every 2 sec. Her tidal volume is 3400 mL, and her residual volume is 1100 mL. Ex- press the volume of her lungs using the model V(t) = a cos bt + h, Where time 0 corresponds to the lungs at their largest capacity. 55. Explain why a periodic model like the cosine function may be reasonable for describing lung capacity. 56. Body limipm'uttirtt. In a laboratory experiment, the body temperature T of rats was measured.21 lThe body temperatures of the rats varied between 35.33°C and 36.87°C during the course of the day. Assuming that the peak body temperature occurred at t = 0, model the body temperature with a func- tion of the form T(t) = a cos bt + k. Sound Waves. The pitch of a sound wave is measured by its frequency Humans can hear sounds in the range from 20 to 20,000 Hz, while dogs can hear sounds as high as 40,000 Hz. The loudness of the sound is deter~ mined by the amplitude.22 57. The note A above middle C on a piano generates a sound modeled by the function g(t) = 4 sin(88077t), where t is in seconds. Find the frequency of A above middle C. 58. The note A below middle C on a piano generates a sound modeled by the function g(t) = 4 sin(4407Tt), where tis in seconds. Find the frequency of A below middle C. 59. Blood Pressure. During a period of controlled breathing, the systolic blood pressure p of a volunteer averaged 143 mmHg with an amplitude of 5.3 mmHg and a frequency of 0.172 Hz. Assum— ing that the blood pressure was highest when t = 0, find a model p(t) = a cos bt + k for blood pressure as a function of time. 60. Blood Pressure. During an episode of sleep apnea, the systolic blood pressure averaged 137 mmHg with an amplitude of 6.7 mmHg and a frequency of 0.079 Hz. Assuming that the blood pressure was highest when t = 0, find a model 21H, Takeuchi, A. Enzo, and H. Minamitani, “Circadian rhythm changes in heart rate variability during chronic sound stress,” Medical and Biological Engineering and Computing, Vol. 39, pp. 113—117 (2001). 22NA. Campbell and J.B. Reece, Biology (Benjamin Cum- mings, New York, 2002). Chapteri Functions and Graphs p(t) = (1 cos bi + k for blood pressure as a function of time.23 Using a calculator, find the x— and y—coordinates of the following points on the unit circle. 23M. Javorka, 1. Zila, K. Javorka, and A. Calkovska, “Do the os- cillations of cardiovascular parameters persist during volun- tary apnea in humans?” Physiological Research, Vol. 51, pp. 227—238 (2002). 65. Compute sin 105°. (Hint: Use a Sum Identity and the fact that 105° = 45° + 60°.) 66. Compute cos 165°. (cos(t + 71'), sin(t + 7r)) 67. a”iiilf-wiéiwitiiwimm a) Use the figure to explain why sin(t + 71') = —sin t and cos(t + 77') = —cos t. b) Rederive the results of part (21) using Sum Identities. c) Show that tan(t + 71') = tan t. . win ' .Jma Consider the function g(t) = a sin bi + k, where a and b are positive. a) Show that the maximum and minimum of g(t) are k + a and la — a, respectively. b) Show that the mid—line is the line y = la. c) Show that the amplitude of g(t) is a. . 5%,:th a) Use a unit circle to explain why sin t =2 sin(t + 277) for all numbers 1;. b) Let g(t) = a sin bt + k. Show that g(t + Zw/b) = g(t). (W c) Why does the result of part (b) imply that the period of g(t) is 27T/b? in :im li‘m. Basilar fibers in the ear detect sound, and they vary in length, tension, and density throughout the basilar membrane. A fiber is affected most by sound frequencies near the fundamental frequency f of the fiber, which is approximately 1 T sz d" In this formula, L is the length of the fiber, T is the ten— sion of the fiber, and d is the density of the fiber.24 UN 70. At the...
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