Review Sheet for MATH 124 with Professor Wood at UA
Review Sheet for MATH 124 with Professor Wood at UA
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Date Created: 02/06/15
WileyPLUS MATH 124129 5th Ed l I Chapter 1 A Library of Functions v Reading content 11 Functions and Change a D 13 New Functions from Old D 14 Logarithmic Functions D 15 Trigonometric Functions D 16 Powers Polynomials and Rational Functions D 17 Introduction to Continuity D 18 Limits Chapter Summary Review Exercises and Problems for Chapter One DCheck Your Understanding D Projects for Chapter One El Student Solutions Manual p Student Study Guide Graphing Calculator Manual gt Focus on Theow p Web Quizzes Emanuel 12 Exponential Functions Population Growth The population of Nevada from 2000 to 2006 is given in Table E To see how the population is growing we look at the increase in population in the third column If the population had been growing linearly all the numbers in the third column would be the same Table 16 Population of Nevada Estimated 200k2006 Year Population Change in millions population millions 2000 2020 2001 2093 0073 2002 2168 0075 2003 2246 0078 2004 2327 0081 2005 2411 0084 fileClDocumentsZUandZUSettingsmathDesktopindexunihtm 1 onZ8262009 62803 AM WileyPLUS 2006 2498 0087 Suppose we divide each year s population by the previous year s population For example Population in 39 U1 i 1 3927 Population in LLJLIU 20 million 39 d U quot c ii quot 711172 393quot FF 39 39r u 39 P 111qu I 111 Ln In hn 11 5 P 1111atin in 211131 2 093 1niin The fact that both calculations give 1036 shows the population grew by about 36 between 2000 and 2001 and between 2001 and 2002 Similar calculations for other years show that the population grew by a factor of about 1036 or 36 every year Whenever we have a constant growth factor here 1036 we have eXponential growth The population tyears after 2000 is given by the exponential function P 203 11139 If we assume that the formula holds for 50 years the population graph has the shape shown in Figure 116 Since the population is growing faster and faster as time goes on the graph is bending upward we say it is concave up Even eXponential functions which climb slowly at first such as this one eventually climb extremely quickly To recognize that a table oft and P values comes from an eXponential function P Poat look for ratios of P values that are constant for equally spaced tvalues fileClDocumentsZUandZUSettlngsmathDesktoplndexunlhtm Z onZ8262009 62803 AM WileyPLUS P popula on in millions P 20201036 S L if i 39 39 7 39 years since 2000 IU 1U 2U T50 4L BU Figure 116 Population of Nevada estimated Exponential growth Concavity We have used the term concave up5 to describe the graph in Figure 116 In words The graph of a function is concave up if it bends upward as we move left to right it is concave down if it bends downward See Figure 117 for four possible shapes A line is neither concave up 1101 concave dOWIl f Concave 39 down mitigate 1 Figure 117 Concavity ofa graph Elimination of a Drug from the Body Now we look at a quantity which is decreasing exponentially instead of increasing When a patient is fileClDocumentsZUandZUSettingsmathDesktopindexunihtm 3 onZ8262009 62803 AM WileyPLUS given medication the drug enters the bloodstream As the drug passes through the liver and kidneys it is metabolized and eliminated at a rate that depends on the particular drug For the antibiotic ampicillin approximately 40 of the drug is eliminated every hour A typical dose of ampicillin is 250 mg Suppose Q f t where Q is the quantity of ampicillin in mg in the bloodstream at time t hours since the drug was given At t 0 we have Q 250 Since every hour the amount remaining is 60 of the previous amount we have 10 250 391 2 5031 w and after thours Q 5quot er 2 230 1111 5 r This is an exponential decay function Some values of the function are in Table u its graph is in Figure Table 17 I Q hours mg 0 250 l 150 2 90 3 54 4 324 5 194 fileClDocumentsZUandZUSettlngsmathDesktoplndexunlhtm 4 onZ8262009 62803 AM WileyPLUS Figure 118 Drug elimination Exponential decay Notice the way the function in Figure is decreasing Each hour a smaller quantity of the drug is removed than in the previous hour This is because as time passes there is less of the drug in the body to be removed Compare this to the exponential growth in Figure where each step upward is larger than the previous one Notice however that both graphs are concave up The General Exponential Function We say P is an exponential function of t with base a if P PDQ r where P0 is the initial quantity when t 0 and a is the factor by which P changes when t increases by l Ifa gt 1 we have exponential growth if0 lt a lt l we have exponential decay Provided a gt 0 the largest possible domain for the exponential function is all real numbers The reason we do not want a S 0 is that for example we cannot define a 2 if a lt 0 Also we do not usually have a 1 since P P011 P0 is then a constant function fileClDocumentsZUandZUSettlngsmathDesktoplndexunlhtm 5 onZ8262009 62803 AM WileyPLUS The value of a is closely related to the percent growth or decay rate For example if a 103 then P is growing at 3 ifa 094 thenP is decaying at 6 Exanu e1 Suppose that Q ft is an eXponential function oft Iff20 882 andf23 914 a Find the base b Find the growth rate c Evaluatef25 SoM on a Let 2 Quit Substituting t 20 Q 882 and t 23 Q 914 gives two equations for Q0 and a r j n EU n n 23 35 Ema and 313 1 gnu Dividing the two equations enables us to eliminate Q0 b Since a 1012 the growth rate is 0012 12 fileClDocumentsZUandZUSettlngsmathDesktoplndexunlhtm 6 onZ8262009 62803 AM WileyPLUS C We want to evaluatef25 Q0a25 Q0l 01225 First we nd Q0 from the equa on Q cjil1123quot Solving gives Q0 695 Thus 5 233 5951012 35 HalfLife and Doubling Time Radioactive substances such as uranium decay exponentially A certain percentage of the mass disintegrates in a given unit of time the time it takes for half the mass to decay is called the halflife of the substance A wellknown radioactive substance is carbonl4 which is used to date organic objects When a piece of wood or bone was part of a living organism it accumulated small amounts of radioactive carbonl4 Once the organism dies it no longer picks up carbonl4 Using the halflife of carbonl4 about 5730 years we can estimate the age of the object We use the following de nitions The halflife of an exponentially decaying quantity is the time required for the quantity to be reduced by a factor of one half The doubling time of an eXponentially increasing quantity is the time required for the quantity to double The Family of Exponential Functions The formulaP Poat gives a family of eXponential functions with positive parameters Po the initial quantity and a the base or growthdecay factor The base tells us whether the function is increasing a gt 1 or decreasing 0 lt a lt 1 Since a is the factor by whichP changes when tis increased by l fileClDocumentsZUandZUSettingsmathDesktopindexunihtm 7 onZ8262009 62803 AM WileyPLUS large values of a mean fast growth values of a near 0 mean fast decay See Figures 119 and 120 All members of the family P Poat are concave up P 5 3 2 4ch m ll 139 1 39m quoti ll r J I 1 3 I39 xquot 20 j my 111 Figure 13919 Exponential growth P at for a gt 1 P l in quot H W 051 l k x num39 06 M i n939 quotWax 02 my my 0 r quot 7 1 24681012 Figure 13920 Exponential decay P at for 0 lt a lt l fileClDocumentsZUandZUSettlngsmathDesktoplndexunlhtm 8 onZ8262009 62803 AM WileyPLUS ExmnMeZ Figure 121 is the graph of three exponential functions What can you say about the values of the siX constants a b c d p q H l r 111111 lI y i t d quot y 2 a b J Figure 121 SoM on All the constants are positive Since a c 17 represent yintercepts we see that a 0 because these graphs intersect on the y aXis In addition a c lt 17 since y p qx crosses the yaXis above the other two Since y a bx is decreasing we have 0 lt b lt l The other functions are increasing so 1 lt d and l lt q Exponential Functions with Base e The most frequently used base for an eXponential function is the famous number 6 271828 This base is used so often that you will find an ex button on most scientific calculators At first glance this is all somewhat mysterious Why is it convenient to use the base 271828 The full answer to that question must wait until Chapter i where we show that many calculus formulas come out neatly when 6 is used as the base We often use the following result fileClDocumentsZUandZUSettingsmathDesktopindexunihtm 9 onZ8262009 62803 AM WileyPLUS Any exponential growth function can be written for some a gt 1 and k gt 0 in the form P Fgcxf 01 P 2 P39 and any exponential decay function can be written for some 0 lt a lt l and k gt 0 as raga or gas 5 where P0 and Q0 are the initial quantities We say thatP and Q are growing or decaying at a con nuousi rate of k For example k 002 corresponds to a continuous rate of 2 Example 3 Convert the functions P 8051 and Q Se39OZI into the form y your Use the results to explain the shape of the graphs in Figures 122 and 123 l 2 3 4 5 7 Figure 122 An eXponential growth function fileCDocumentsZUandZUSettingsmathDesktopindexunihtm 10 of ZZ8262009 62803 AM WileyPLUS Q 5 4 3 2 Egan 1 ERR 2 41 539 10 Figure 123 An exponential decay function Solution We have r at P GUquot 93913 2 Jig5 Thus P is an exponential growth function with P0 1 and a 165 The function is increasing and its graph is concave up similar to those in Figure 119 Also 139 4 511 if 5m 3193 so Q is an eXponential decay function with Q0 5 and a 0819 The function is decreasing and its graph is concave up similar to those in Figure 120 fileClDocumentsZUandZUSettlngsmathDesktoplndexunlhtm 11 of ZZ8262009 62803 AM WileyPLUS ExmnMe4 The quantity Q of a drug in a patient s body at time tis represented for positive constants S and k by the function Q Sl am For t 2 0 describe how Q changes with time What does S represent SoM on The graph of Q is shown in Figure Initially none of the drug is present but the quantity increases with time Since the graph is concave down the quantity increases at a decreasing rate This is realistic because as the quantity of the drug in the body increases so does the rate at which the body excretes the drug Thus we eXpect the quantity to level off Figure shows that S is the saturation level The line Q S is called a horizontal asymptote D quantiiy of drug Saturation level 539 r Iquot l I l l 2 3 4 5 Figure 124 Buildup ofthe quantity ofa drug in body 1 time in hours Exercises and Problems for Section E fileClDocumentsZUandZUSettlngsmathDesktoplndexunlhtm 12 of ZZ8262009 62803 AM WileyPLUS Exercises 5 P 5107 fileCDocumentsZUandZUSettingsmathDesktopindexunihtm 13 of ZZ8262009 62803 AM The functions in Exercises 5 6 P 77092 a In Exercises 1 g 1 and A decide whether the graph is concave up concave down or neither l h Q 1 and represent exponential growth or decay What is the initial quantity What is the growth rate State if the growth rate is continuous WileyPLUS 7 P 32e003t 8 P 15e006 Write the functions in Problems 2 m u and Q in the form P Poat Which represent exponential growth and which represent exponential decay 9 P 15e025 10 p ZeOSI 11 p POeOQI 12 P 7en In Problems Q and let f t Qoat Q0l rt a Find the base a b Find the percentage growth rate r 13 f5 7594 and f7 17086 14 f002 2502 and f005 2506 15 A town has a population of 1000 people at time t 0 In each of the following cases write a formula for the population P of the town as a function of year t a The population increases by 50 people a year b The population increases by 5 a year fileClDocumentsZUandZUSettlngsmathDesktoplndexunlhtm 14 of ZZ8262009 62803 AM WileyPLUS 16 Identify the xintervals on which the function graphed in Figure 125 is a Increasing and concave up b Increasing and concave down c Decreasing and concave up 1 Decreasing and concave down Figure 125 Problems 17 An airfreshener starts with 30 grams and evaporates In each of the following cases write a formula for the quantity Q grams of airfreshener remaining tdays after the start and sketch a graph of the function The decrease is a 2 grams a day b 12 a day fileClDocumentsZUandZUSettingsmathDesktopindexunihtm 15 of ZZ8262009 62803 AM WileyPLUS 18 19 20 21 In 2007 the world s population reached 67 billion and was increasing at a rate of 12 per year Assume that this growth rate remains constant In fact the growth rate has decreased since 1987 a Write a formula for the world population in billions as a function of the number of years since 2007 b Use your formula to estimate the population of the world in the year 2020 c Sketch a graph of world population as a function of years since 2007 Use the graph to estimate the doubling time of the population of the world A photocopy machine can reduce copies to 80 of their original size By copying an already reduced copy further reductions can be made a If a page is reduced to 80 what percent enlargement is needed to return it to its original size b Estimate the number of times in succession that a page must be copied to make the final copy less than 15 of the size of the original When a new product is advertised more and more people try it However the rate at which new people try it slows as time goes on a Graph the total number of people who have tried such a product against time b What do you know about the concavity of the graph Sketch reasonable graphs for the following Pay particular attention to the concavity of the graphs a The total revenue generated by a car rental business plotted against the amount spent on advertising b The temperature of a cup of hot coffee standing in a room plotted as a function of time Give a possible formula for the functions in Problems 2 Q a and g fileClDocumentsZUandZUSettlngsmathDesktoplndexunlhtm 16 of ZZ8262009 62803 AM WileyPLUS 22 23 24 25 26 27 H l a A population P grows at a continuous rate of 2 a year and starts at 1 million Write P in the form P Poem with P0 k constants b Plot the population in part a against time When the Olympic Games were held outside Mexico City in 1968 there was much discussion about the effect the high altitude 7340 feet would have on the athletes Assuming air pressure decays eXponentially by 04 every 100 feet by what percentage is air pressure reduced by moving from sea level to Mexico City fileClDocumentsZUandZUSettlngsmathDesktoplndexunlhtm 17 of ZZ8262009 62803 AM WileyPLUS 28 29 30 31 32 During April 2006 Zimbabwe s in ation rate averaged 067 a day This means that on average prices went up by 067 from one day to the next a By what percentage did prices in Zimbabwe increase in April of 2006 b Assuming the same rate all year what was Zimbabwe s annual in ation rate during 2006 a The halflife of radium226 is 1620 years Write a formula for the quantity Q of radium left after tyears if the initial quantity is Q0 b What percentage of the original amount of radium is left after 500 years In the early 1960s radioactive strontium90 was released during atmospheric testing of nuclear weapons and got into the bones of people alive at the time If the halflife of strontium90 is 29 years what fraction of the strontium90 absorbed in 1960 remained in people s bones in 1990 A certain region has a population of 10000000 and an annual growth rate of 2 Estimate the doubling time by guessing and checking Aircraft require longer takeoff distances called takeoff rolls at high altitude airports because of diminished air density The table shows how the takeoff roll for a certain light airplane depends on the airport elevation Takeoff rolls are also strongly in uenced by air temperature the data shown assume a temperature of 00C Determine a formula for this particular aircraft that gives the takeoff roll as an eXponential function of airport elevation Elevation Sea 1000 2000 3000 4000 ft level Takeoff 670 734 805 882 967 roll ft fileClDocumentsZUandZUSettlngsmathDesktoplndexunlhtm 18 of ZZ8262009 62803 AM WileyPLUS 33 Each of the functions g h k in Table Q is increasing but each increases in a different way Which of the graphs in Figure 126 best fits each function Tabe18 t g0 ht W 1 23 10 22 2 24 20 25 3 26 29 28 4 29 37 31 5 33 44 34 6 38 50 37 ta H U Figure 126 34 Each of the functions in Table decreases but each decreases in a different way Which of the graphs in Figure 127 best fits each function Table 19 fileC1DocumentsZUandZUSettingsmathDesktopindexunihtm 19 of ZZ8262009 62803 AM WileyPLUS x f x 800 hx 1 100 220 93 2 90 214 91 3 81 208 88 4 73 202 84 5 66 196 79 6 60 190 73 13 lb I W Figure 127 35 a Which if any of the functions in the following table could be linear Find formulas for those functions b Which if any of these functions could be exponential Find formulas for those functions x f x 800 W 12 16 37 2 17 24 34 1 0 20 36 31 1 21 54 28 2 18 81 25 fileClDocumentsZUandZUSettlngsmathDesktoplndexunlhtm 20 of ZZ8262009 62803 AM WileyPLUS 36 The median price P ofa home rose from 60000 in 1980 to 180000 in 2000 Let tbe the number of years since 1980 a Assume the increase in housing prices has been linear Give an equation for the line representing price P in terms of I Use this equation to complete column a of Table 110 Use units of1000 b If instead the housing prices have been rising exponentially nd an equation of the form P Poat to represent housing prices Complete column b of Table 110 c On the same set of axes sketch the functions represented in column a and column b of Table 110 d Which model for the price growth do you think is more realistic Table 110 t a 13 Linear Exponential growth growth price price in in 1000 5 1000 units units 0 60 60 10 20 180 180 30 40 fileClDocumentsZUandZUSettlngsmathDesktoplndexunlhtm 21 of ZZ8262009 62803 AM WileyPLUS 37 Estimate graphically the doubling time of the exponentially growing population shown in Figure 128 Check that the doubling time is independent of where you start on the graph Show algebraically that if P Poat doubles between time tand time t d then dis the same number for any I population 51 000 6f 000 JD 000 20000 llmE yearst l2313915h 7i 3399 Figure 128 fileClDocumentsZUandZUSettlngsmathDesktoplndexunlhtm 22 of ZZ8262009 62803 AM
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