Review Sheet for MATH 124 at UA
Review Sheet for MATH 124 at UA
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WileyPLUS MATH 124129 5th Ed WileyFLlTS Home gel Contact us Loout HughesHallett Calculus Single Variable 5e 39I Chapter 1 A Library of Functions and rJTVleW v Reading content ELI Functions and Change D12 Exponential Functions 13 New Functions from Old BIA Logarithmic Functions n ULG Powers Polynomials and Rational Functions 17 Introduction to Continuity D13 Limits DChapter Summary DReview Exercises and Problems for Chapter One Check Your Understanding DProjects for Chapter One El p Student Solutions Manual p Student Study Guide p Graphing Calculator Manual Focus on Theory p Web Quizzes 15 Trigonometric Functions Trigonometry originated as part of the study of triangles The name tri gon o metry means the measurement of threecomered gures and the rst de nitions of the trigonometric functions were in terms of triangles However the trigonometric functions can also be de ned using the unit circle a de nition that makes them periodic or repeating Many naturally occurring processes are also periodic The water level in a tidal basin the blood pressure in a heart an alternating current and the position of the air molecules transmitting a musical note all uctuate regularly Such phenomena can be represented by trigonometric functions We use the three trigonometric functions found on a calculator the sine the cosine and the tangent Radians There are two commonly used ways to represent the input of the trigonometric functions radians and degrees The formulas of calculus as you will see are neater in radians than in degrees An angle of l radian is de ned to be the angle at the center of a unit circle which cuts off an arc of length 1 measured counterclockwise See Figure l44a A unit circle has radius 1 lecmocumentsamoandmosmingsmathDesktopindex uni him 1 of238262009 63218 AM WileyPLUS lal 39 lb quot quot quot 7 39 All length 2 fl An length 1 Figure 144 Radians de ned using unit circle An angle of 2 radians cuts off an arc of length 2 on a unit circle A negative angle such as 12 radians cuts off an arc of length 12 but measured clockwise See Figure 144b It is useful to think of angles as rotations since then we can make sense of angles larger than 360 for example an angle of 720 represents two complete rotations counterclockwise Since one full rotation of 360 cuts off an arc of length 21 the circumference of the unit circle it follows that 350 2K radians 130 radians In other words 1 radian 180 rc so one radian is about 60 The word radians is often dropped so if an angle or rotation is referred to without units it is understood to be in radians Radians are useful for computing the length of an arc in any circle If the circle has radius r and the arc cuts off an angle 0 as in Figure 145 then we have the following relation Arc length s r0 Figure 145 Arc length ofa sector ofa circle The Sine and Cosine Functions leClDocuments A120andZOSettingsmathDesktopindex uni mm 2 of238262009 63218 AM WileyPLUS The two basic trigonometric functionsithe sine and cosineiare de ned using a unit circle In Figure 146 an angle of t radians is measured counterclockwise around the circle from the point 1 0 If P has coordinates x y we de ne 05 f arr21 sin y We assume that the angles are always in radians unless speci ed otherwise u n t i 205 y 5m Figure 146 The de nitions of sin tand cos I Since the equation of the unit circle is x2 y2 l we have the following fundamental identity quot31 As I increases and P moves around the circle the values of sin tand cos toscillate between 1 and l and eventually repeat as P moves through points where it has been before If t is negative the angle is measured clockwise around the circle Amplitude Period and Phase The graphs of sine and cosine are shown in Figure Notice that sine is an odd function and cosine is even The maximum and minimum values of sine and cosine are 1 and 1 because those are the maximum and minimum values of y and x on the unit circle After the pointP has moved around the complete circle once the values of cos I and sin tstart to repeat we say the functions are periodic leClDocuments MwZOand MwZOSettingsmathDesktopindex uni mm 3 of238262009 63218 AM WileyPLUS For any periodic function of time the Amplitude is half the distance between the maximum and minimum values if it exists 0 Period is the smallest time needed for the function to execute one complete cycle Amplilude 1 Figure 147 Graphs of cos tand sin I The amplitude of cos tand sin tis l and the period is 21 Why 21 Because that39s the value of I when the pointP has gone exactly once around the circle Remember that 3600 21 radians In Figure we see that the sine and cosine graphs are exactly the same shape only shifted horizontally Since the cosine graph is the sine graph shifted 152 to the left 05 339 sinyf If m L Equivalently the sine graph is the cosine graph shifted 152 to the right so 39 SmL I Functions whose graphs are the shape of a sine or cosine curve are called sinusoidal functions leClDocuments A120andZOSettingsmathDesktopindex uni mm 4 of238262009 63218 AM WileyPLUS To describe arbitrary amplitudes and periods of sinusoidal functions we use functions of the form f If A sinnCBtZII and A cosleti where lAl is the amplitude and 2 rclBl is the period The graph of a sinusoidal function is shifted horizontally by a distance lhl when t is replaced by t h or H h Functions of the form f t A sin Br C and gt A cos Br C have graphs which are shifted vertically and oscillate about the value C Example 1 Find and show on a graph the amplitude and period of the functions a y 5 sin2t b 39 quot1 1 3 cm L q L s c y12sint Solution a From Figure you can see that the amplitude of y 5 sin2t is 5 because the factor of 5 stretches the oscillations up to 5 and down to 5 The period of y sin2t is TE because when t changes from 0 to TE the quantity 2t changes from 0 to 21 so the sine function goes through one complete oscillation b Figure amp shows that the amplitude of y 5 sint 2 is again 5 because the negative sign re ects the oscillations in the t aXis but does not change how far up or down they go The period of y 5 sint2 is 41 because when tchanges from 0 to 41 the quantity t2 changes from 0 to 21 so the sine function goes through one complete oscillation c The l shifts the graphy 2 sin I up by l Sincey 2 sin I has an amplitude of2 and a period of 21 the graph ofy 1 2 sin tgoes up to 3 and down to l and has a period of 21 See Figure 150 Thusy 1 2 sin I also has amplitude 2 leClDocuments A120andZOSettingsmathDesktopindex uni mm 5 of238262009 63218 AM l WileyPLUS 1 J y 55m 2 d 39 r x 39v l Amplilude Err I l t T 139 l J Pe iod Figure 148 Amplitude 5 period TE yy 5Sintf392 a 5 equot Amplitude m f 0t Kl Ifquot 2 w 4n f 3 Period Figure 149 Amplitude 5 period 41 H I y12r sm Ix 1 Amplitude 1 v 5 I 1 t 7 Err 7 al 1 Period Figure 150 Amplitude 2 period 21 leClDocuments MwZOand MwZOSettingsmathDesktopindex uni mm 6 of238262009 63218 AM WileyPLUS Example 2 a b 0 Solution leClDocuments A120andZOSettingsmathDesktopindex uni mm 7 of238262009 63218 AM Find possible formulas for the following sinusoidal functions 3 fly I i I f h39 n 312T 3 J 39 u NJ 13 391 2 33 4 2quot quot39i hm 4r run 13w V g l 939 WileyPLUS a This function looks like a sine function of amplitude 3 so gt 3 sinBt Since the function executes one full oscillation between t 0 and t 121 when t changes by 121 the quantity Bt changes by 21 This means B 121 21 so B 16 Therefore gt 3 sint 6 has the graph shown b This function looks like an upside down cosine function with amplitude 2 so f t 2 cos Bt The function completes one oscillation between t 0 and t 4 Thus when t changes by 4 the quantity Bt changes by 21 so B 4 21c orB 152 Therefore f t 2cos1ct2 has the graph shown C This function looks like the function gt in part a but shifted a distance of 1 to the right Since gt 3 sint6 we replace tby t 1 to obtain ht 3 sint 1c6 Example 3 On July 1 2007 high tide in Boston was at midnight The water level at high tide was 99 feet later at low tide it was 01 feet Assuming the next high tide is at exactly 12 noon and that the height of the water is given by a sine or cosine curve find a formula for the water level in Boston as a function of time Solution Let y be the water level in feet and let tbe the time measured in hours from midnight The oscillations have amplitude 49 feet 99 0 l2 and period 12 so l2B 21c and B 156 Since the water is highest at midnight when t 0 the oscillations are best represented by a cosine function See Figure Q We can say leClDocuments A120andZOSettingsmathDesktopindex uni mm 8 of238262009 63218 AM WileyPLUS 12mm jam 12mm 6pm 12mm Figure 151 Function approximating the tide in Boston on July 1 2007 Example 4 Of course there s something wrong with the assumption in Example 3 that the next high tide is at noon If so the high tide would always be at noon or midnight instead of progressing slowly through the day as in fact it does The interval between successive high tides actually averages about 12 hours 24 minutes Using this give a more accurate formula for the height of the water as a function of time Solution The period is 12 hours 24 minutes 124 hours so B 2T124 giving COLTi LI 2 124 leClDocuments A120andZOSettingsmathDesktopindex uni mm 9 of238262009 63218 AM WileyPLUS Example 5 Use the information from Example g to write a formula for the water level in Boston on a day when the high tide is at 2 pm Solution When the high tide is at midnight y 7 49 BCGlIlelTJ Since 2 pm is 14 hours after midnight we replace tby t 14 Therefore on a day when the high tide is at 2 pm Ln 3 7 czrys i j 39 14 The Tangent Function If t is any number with cos t7 0 we define the tangent function as follows Figure 146 shows the geometrical meaning of the tangent function tan t is the slope of the line through the origin 0 0 and the pointP cos t sin I on the unit circle The tangent function is undefined wherever cos I 0 namely at t i rc2 i3 rc2 and it has a vertical asymptote at each of these points The function tan t is positive where sin I and cos thave the same sign The graph of the tangent is shown in Figure 152 leClDocuments A120andZOSettingsmathDesktopindex uni mm 10 of238262009 63218 AM WileyPLUS I I I l I I I tam I I I I I I I m I I I I I I I I I I I I quotI I 22 I l T39 I T I I I l I I I III I I I I 1 Period quot Figure 152 The tangent function The tangent function has period TE because it repeats every TE units Does it make sense to talk about the amplitude of the tangent function Not if we re thinking of the amplitude as a measure of the size of the oscillation because the tangent becomes in nitely large near each vertical asymptote We can still multiply the tangent by a constant but that constant no longer represents an amplitude See Figure Figure 153 Multiple oftangent leCIDocuments MIZOand MIZOSettingsmathDesktopindex uni mm 11 of238262009 63218 AM WileyPLUS The Inverse Trigonometric Functions On occasion you may need to nd a number with a given sine For example you might want to nd x such that 5in x U or such that sin 2 13 The rst of these equations has solutions x 0 in i215 The second equation also has in nitely many solutions Using a calculator and a graph we get 2 1305 234 0305 J 2H 234 J For each equation we pick out the solution between rc2 and 152 as the preferred solution For example the preferred solution to sin x 0 is x 0 and the preferred solution to sin x 03 is x 0305 We de ne the inverse sine written arcsin or sin39l as the function which gives the preferred solution For 1 S y S l arcsiny means tan y with 39 Thus the arcsine is the inverse function to the piece of the sine function having domain rt2 152 See Table 114 and Figure 154 On a calculator the arcsine function2 is usually denoted by gm l leClDocuments MwZOand MwZOSettingsmathDesktopindex uni mm 12 of238262009 63218 AM WileyPLUS Values of sin x and sin391 x x 1000 0841 0479 0000 0479 0841 1000 sin391 x K E 10 05 00 05 10 A quot3 L4 Table114 x sinx 1000 10 0841 05 0479 00 0000 05 0479 10 0841 1000 graph of the arctangent is shown in Figure 156 leClDocuments MwZOand MwZOSettingsmathDesktopindex uni mm 13 of238262009 63218 AM l I y sin 1 y sin 3 Figure 154 The arcsine function The inverse tangent written arctan or tan39l is the inverse function for the piece of the tangent function having the domain rc2 lt x lt 152 On a calculator the inverse tangent is usually denoted by tan 1 The WileyPLUS frytan 139 Figure 156 The arctangent function For any y arctan y x means tan 2 2 width m 39I The inverse cosine function written arccos or cOS39l is discussed in Problem 2 The range of the arccosine function is 0 S x S TE Exercises and Problems for Section 13 lecmocumentsamoandmosmingsmathDeskcopindex uni mm 14 of238262009 63218 AM WileyPLUS Exercises For Exercises 1 2 3 4 5 Q 7 and 2 draw the angle using a ray through the origin and determine whether aaa the sine cosine and tangent of that angle are positive negative zero or unde ned J bit 1 J l J2 TE 9959 LA J T G u l Lx 3 IJJ 8 4 9 1 Given that sin 15 12 0259 and cos TE5 0809 compute without using the trigonometric functions on your calculator the quantities in Exercises Q Q and Q You may want to draw a picture showing the angles involved and then check your answers on a calculator 13 Consider the function y 5 cos3x a What is its amplitude b What is its period c Sketch its graph Find the period and amplitude in Exercises g Q E and H 14 y 7 sin3t leClDocuments MwZOand MwZOSettingsmathDesktopindex uni mm 15 of238262009 63218 AM WileyPLUS 15 z 3 cosu4 5 16 w84sin2x rc 17 r 01 sin rct 2 For Exercises E Q E A Q Q a g E and 2 nd a possible formula for each graph 18 y 392 37139 1 aquot a y a k JI 19 y x fquot 39 J r 20 g 4 1 x x a f T J 21 h39 8 r39l 1 H QUTI39 1 39 139 22 I J 5 lecmocumentsamoandmosmingsmathDeskcopindex uni mm 16 of238262009 63218 AM WileyPLUS 23 24 25 y 26 y 27 y 3 In Exercises E 2 1 and 2 nd a solution to the equation if possible Give the answer in exact form and in decimal form leClDocuments A120andZOSettingsmathDesktopindex uni mm 17 of238262009 63218 AM WileyPLUS 28 2 5sin3x 2918cos2x13 30 8 4tan5x 31 1 8tan2x 1 3 32 8 4 sin5x Problems 33 What is the difference between sin x2 sin2 x and sinsin x Express each of the three as a composition Note sin2 x is another way of writing sin x2 34 Without a calculator or computer match the formulas with the graphs in Figure 157 a y 2 cost 152 b y 2 cost c y 2 cos I 152 i f min r x l V I If 4 M 70 Figure 157 35 a Match the functions 0 f t 0 gt 0 ht c0 kt whose values are in the table with the functions with formulas i co 15 sint ii co 05 sin I iii 0 05 sin I iV co 15 sin I b Based on the table what is the relationship between the values of gt and kt Explain this relationship using the formulas you chose for g and k leClDocuments A120andZOSettingsmathDesktopindex uni mm 18 of238262009 63218 AM WileyPLUS 36 37 38 39 40 41 c Using the formulas you chose for g and h explain why all the values of g are positive whereas all the values of h are negative t 60 65 70 75 80 f0 078 028 016 044 049 t 30 35 40 45 50 g0 164 115 074 052 054 t 50 51 52 53 54 ht 246 243 238 233 227 t 30 35 40 45 50 kt 064 015 026 048 046 A compact disc spins at a rate of 200 to 500 revolutions per minute What are the equivalent rates measured in radians per second When a car39s engine makes less than about 200 revolutions per minute it stalls What is the period of the rotation of the engine when it is about to stall What is the period of the earth39s revolution around the sun What is the period of the motion of the minute hand of a clock What is the approximate period of the moon39s revolution around the earth The Bay of Fundy in Canada has the largest tides in the world The difference between low and high water levels is 15 meters nearly 50 feet At a particular point the depth of the water y meters is given as a function of time t in hours since midnight by a What is the physical meaning of D b What is the value of A c What is the value of B Assume the time between successive high tides is 124 hours d What is the physical meaning of C 1 D 7 A 051 a What does V0 represent in terms of voltage b What is the period of this function c How many oscillations are completed in 1 second leClDocuments MwZOand MwZOSettingsmathDesktopindex uni mm 19 of238262009 63218 AM 42 In an electrical outlet the voltage V in volts is given as a function of time t in seconds by the formula V 0 1 sin 120 It 21 WileyPLUS 43 44 45 46 47 In a US household the voltage in volts in an electric outlet is given by V 156 siruiilZUm L where tis in seconds However in a European house the voltage is given in the same units by Fquot 1 Compare the voltages in the two regions considering the maximum voltage and number of cycles oscillations per second six 1 3 El A baseball hit at an angle of 0to the horizontal with initial velocity v0 has horizontal range R given by Here g is the acceleration due to gravity Sketch R as a function of 0 for 0 S 0 S 152 What angle gives the maximum range What is the maximum range A population of animals oscillates sinusoidally between a low of 700 on January 1 and a high of 900 on July 1 a Graph the population against time b Find a formula for the population as a function of time t in months since the start of the year The desert temperature H oscillates daily between 40 F at 5 am and 80 F at 5 pm Write a possible formula forH in terms of I measured in hours from 5 am The visitors39 guide to St Petersburg Florida contains the chart shown in Figure 158 to advertise their good weather Fit a trigonometric function approximately to the data where H is temperature in degrees Fahrenheit and the independent variable is time in months In order to do this you will need to estimate the amplitude and period of the data and when the maximum occurs There are many possible answers to this problem depending on how you read the graph H r i ll Jun l uh Maui Apr rluyilunc July Aug Sept M Nu Hut 100 PM b39O 7039 l r 10quot 5039 Reprinted with permission leClDocuments A120andZOSettingsmathDesktopindex uni mm 20 of238262009 63218 AM WileyPLUS Figure 158 St Petersburg where we re famous for our wonderful weather and year round sunshine 48 The pointP is rotating around a circle of radius 5 shown in Figure 159 The angle 0 in radians is given as a function of time t by the graph in Figure 160 a Estimate the coordinates of P when t 15 b Describe in words the motion of the pointP on the circle Figure 160 leClDocuments A120andZOSettingsmathDesktopindex uni mm 21 of238262009 63218 AM WileyPLUS 49 Find the area of the trapezoidal crosssection of the irrigation canal shown in Figure 161 h I 394 M v rui m Figure 161 50 For a boat to oat in a tidal bay the water must be at least 25 meters deep The depth of water around the boat dt in meters where tis measured in hours since midnight is a I 5 74 46 mil a What is the period of the tides in hours b If the boat leaves the bay at midday what is the latest time it can return before the water becomes too shallow 51 Graphy sin x y 04 andy 04 a From the graph estimate to one decimal place all the solutions of sin x 04 with rc S x S TE b Use a calculator to nd arcsin04 What is the relation between arcsin04 and each of the solutions you found in part a 0 Estimate all the solutions to sin x 04 with rc S x S TE again to one decimal place d What is the relation between arcsin04 and each of the solutions you found in part c 5239 This problem introduces the arccosine function or inverse cosine denoted by 3 1 on most calculators a Using a calculator set in radians make a table of values to two decimal places of gx arccos x forx l 08 06 0 06 081 b Sketch the graph of gx arccos x c Why is the domain of the arccosine the same as the domain of the arcsine d Why is the range of the arccosine not the same as the range of the arcsine To answer this look at how the domain of the original sine function was restricted to construct the arcsine Why can t the domain of the cosine be restricted in exactly the same way to construct the arccosine leClDocuments A120andZOSettingsmathDesktopindex uni mm 22 of238262009 63218 AM WileyPLUS Copyright vi 2009 John Wiley 3 Sons Tm All right reserve lecmocumemsamoandmosmingsmathDeskcopindex uni mm 23 of238262009 63218 AM
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