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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=]xandy=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 crosssectional 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 crosssectional area
of his capillaries.12 60. Suppose, in an adult female, blood leaves the aorta
at 28 cm/ sec, and the crosssectional 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 crosssectional 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 yintercepts 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+x2—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 xyplane.
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) 25ft shadow.
b) 10ft 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 ydirection.) 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 straightline teeto—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 midline. 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 southeastfacing 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”iiilfwiéiwitiiwimm 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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