Basics of Math Modeling
Basics of Math Modeling MMAT 3320
University of Texas-Pan American (UTPA)
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Date Created: 10/29/15
1 RECURRENCE RELATIONS A sequence is a function from the set n0 n0 1 n0 2 where no is an integer into the set of real numbers Frequently n0 is O or 1 If s is a sequence and n is an element ofthe domain of s sn is frequently denoted by sn and is called the nth term ofthe sequence s EXAMPLE 1 For the sequence a defined by an n2 4n 3 find the first five terms 31 M32 M33 M34 M35 A recurrence relation or difference equation for a sequence s is a formula that expresses sn in terms of one or more of the previous terms of the sequence for all terms from some index on A sequence is called a solution of a recurrence relation if its terms satisfy the recurrence relation The initial conditions specify the terms that precede the first term where the recurrence relation takes effect EXAMPLE 2 Find the first four terms of the recurrence relation with initial condition given below sn 2sn11s13 S1 S2 53 54 EXAMPLE 3 Forthe sequence a in Example 1 a complete the following a1 a2a1 asa2 a4as a5a4 b find a recurrence relation with initial condition for a an an71 a1 MMAT 3320 NOTES SECTION 1 PAGE 1 EXAMPLE 4 Given the sequence b defined by bn 062 3 b0 b1 b2 and b3 b Find a recurrence relation with initial condition for sequence b c Sketch the graph of sequence b MMAT 3320 NOTES SECTION 1 PAGE 2 EXAMPLE 5 Suppose c is a sequence having c1 40 c2 55 c3 70 and c4 85 a find a recurrence relation with initial condition for c b find a nonrecursive formula for on cn c Sketch the graph of sequence c MMAT 3320 NOTES SECTION 1 PAGE 3 EXAMPLE 6 Let F be the sequence such that Fn is the product of the first n positive integers a Find F5 and F6 b Find a recurrence relation with initial condition for F Note In the sequence F defined above Fn is more commonly denoted by n and is called n factorial Also we define O to be 1 EXAMPLE 7 Consider the following recurrence relation with initial condition n1 i on gn4 1 911 Find the value of 91533 Defining 0 to be 1 makes certain formulas nicer in probability and combinatorics MMAT 3320 NOTES SECTION 1 PAGE 4 EXAMPLE 8 Consider the recurrence relation with initial condition given by hnhn71n 1h10 a Use Excel to find the value of h78 A B 1 n hn 2 1 O 3 A21 BZA3 1 4 Copy formula above down Copy formula above down MMAT 3320 NOTES SECTION 1 PAGE 5 b Use the builtin sequence feature of the Tl83 Plus calculator to nd the value of has Press the Mode key and change to Seq sequence graphing in the four line of the Mode menu see lelt below Exit the m h m z u mlt e 5 Iquot e m a o z a o m i 239 o o line andthe valueW ofthe initial condition for unMin on thethird line see right below Sci Eng Phu H ZPhu 8123456789 nMin1 Degree HunEun1n1 uCnMinEB VCn vltnM1n an To verify our original table of results press TBLSET 2nd V ndow enter 1 for the TblStart value enter 1 for the ATbl value and select Auto for both the lndpnt and Depend values see le be Pressing TABLE 2nd Graph we see atable ofvalues with n starting 1 and incrementing by 1 see middle below By scrolling down with own arrow key you can see additional values of the table see right below 539 m a c 5 ml 5 i3 5 Tu enter u pvessan 7 Tu enter n pvesslhe men key quotquot Enter unlylhe value the braceswill be insened m yuu MMAT 3320 NOTES SECTION 1 PAGE 6 To answer the question press TBLSET 2nd Wndow again and change TblStart to 93 see lelt below Press TABLE 2nd Graph to J I H v y THBLE SETUP TblStart93 4Tb11 n As a bonus we can easily obtain sequential graphs Press the V ndow key and enter the values below NINDDU nM1n1 nMax18 PlotStart1 PlotSteP1 Xm1n8 Xmax18 X5511 Press the Graph key to View the graph in the selected window see lelt below Press the Trace key and use the le arrow and right I L L Lnkl anuw 1 Changing the window see below NINDDU nM1n1 nMa BB PlotStart1 PlotSteP1 Xm1n Xmax188 Xscl18 Vs X 888 1IBBB Wmmemmwmwwmmmmmmmmw MMAT 3320 NOTES SECTION 1 PAGE 7 and regraphing see lelt below press the Trace key and enter 93 for n to answer the original question see right below EXAMPLE 9 The Fibonacci sequence f is de ned by the following recurrence relation with initial conditions 7 fm21f oyfl1 a Find the value of f7 MMAT 3320 NOTES SECTION 1 PAGE 8 b Use the Tl83 Plus sequence feature to nd the value of f mu Plotz Plot nMinB HuCnEun1un uCnMinE1B VCn vltnMin an ml c Complete the following Excel spreadsheet for determining the Fibonacci sequence A B n fn MMAT 3320 NOTES SECTION 1 PAGE 9 HOMEWORK 1 2 0399 0399 MMAT 3320 NOTES Find the first four terms of the sequence a defined by an n2 2 For the sequence c defined by cn nl find the value of c5 c0 Consider the sequence b defined by the following recurrence relation with initial condition bnbn14 b13 Find the value of b5 Find a nonrecursive formula for bn Consider the sequence d defined by the following recurrence relation with initial condition dn 3dn4 5 d0 7 Find the value of d5 Use Excel to find the value of d20 The Lucas sequence L is defined by the following recurrence relation with initial conditions Ln Ln71 Ln72 L1 1 L2 3 Find the tenth term of the Lucas sequence Consider the sequence e defined by the following recurrence relation with initial condition en Zena 3en72 e1 4 e2 5 Use the sequence feature ofthe Tl83 calculator to find the value of 820 Use Excel to find the value of e15 Consider the sequence g defined by the following recurrence relation with initial condition Find the value of g235 SECTION 1 PAGE 10 8 TRIGONOMETRIC RATIOS Given a right triangle and angle 6 as one of its acute angles quotoppquot denotes the leg opposite 6 quotadjquot denotes the other leg ie the leg quotadjacentquot to 6 and quothypquot denotes the hypotenuse Define the following cos 6 a dJ hyp sin e opp hy tan 6 0 pP ad adj Because corresponding sides of similar triangles are in proportion the quotsizequot of the right triangle does not affect the definition EXAMPLE 1 When the angle of elevation of the sun is 70 a flagpole casts a shadow that is 15 feet long Find the height of the flagpole Round answer to the nearest foot We use the following abbreviations cos for cosine sin for sine and tan for tangent MMAT 3320 NOTES SECTION 8 PAGE 1 EXAMPLE 2 San Bernardino California is 100 miles due north of San Diego California Yuma Arizona is S 56 E from San Bernardino and due east of San Diego How far is Yuma from San Bernardino Round to the nearest mile MMAT 3320 NOTES SECTION 8 PAGE 2 Inverse trigonometric functions 0 For 0 lt X lt 1 cos 1X is the acute angle 6 satisfying cos 6 X o For 0 lt X lt 1 sin 1X is the acute angle 6 satisfying sin 6 X o For X gt 0 tan 1X is the acute angle 6 satisfying tan 6 X Inverse trigonometric functions are useful in right triangle problems in which one is trying to find the measure of one of the acute angles in the right triangle EXAMPLE 3 An isosceles triangle has sides of length 151 inches 151 inches and 124 inches Find the measure of its verteX angle Round answer to the nearest degree cos391X is also denoted by arccos X sin391X by arcsin X and tan391X by arctan X MMAT 3320 NOTES SECTION 8 PAGE 3 EXAMPLE 4 Find the measure of the acute angle formed by a diagonal of a cube and a diagonal of its face Note Since all cubes are similar you can assign any length to the edge of the cube MMAT 3320 NOTES SECTION 8 PAGE 4 Some trigonometric applications involve vectors EXAMPLE 5 The wind is blowing from the north at 545 kilometers per hour The pilot of an airplane finds that if the plane is headed in the direction of 74 from due north then the plane travels due east Find the plane39s air speed and ground speed Round to the nearest tenth MMAT 3320 NOTES SECTION 8 PAGE 5 EXAMPLE 6 A wind blowing from the east causes a small plane flying with an air speed of 70 miles per hour to travel due south with a ground speed of 60 miles per hour Find the plane39s heading clockwise from due north Round to the nearest integer MMAT 3320 NOTES SECTION 8 PAGE 6 HOMEWORK 1 MMAT 3320 NOTES From a point on the ground 115 meters away from the foot of a tower the angle of elevation of the top of the tower is 682 How high is the tower Round to the nearest meters The base of an extension ladder is 391 feet from the base of a building and makes an angle of 58 with the ground How long is the ladder Round to the nearest tenth of a foot A flagpole stands atop a 65foot building Form the position of an observer whose eyes are 5 feet above the ground the angles of elevation of the top and bottom of the flagpole are 47 and 41 respectively Find the length of the flagpole Round to the nearest foot McAllen is located 3 miles west and 7 miles south of Edinburg Find the bearing of McAllen with respect to Edinburg Round to the nearest integer A regular pentagon has a perimeter of 3850 inches How far is it from the center of the pentagon to each vertex Round to the nearest hundredth An airplane with an air speed of 160 kilometers per hour is headed due west Because of a wind from the south the plane39s ground speed is 170 kilometers per hour Find the plane39s course measured clockwise from due north Round to the nearest integer Find the speed ofthe wind Round to the nearest integer SECTION 8 PAGE 7 16 FINDING TANGENT LINES ALGEBRAICALLY Recall The equation of the circle with center hk and radius r is given by x h2 y k2 r2 EXAMPLE 1 Use properties of algebra and geometry to find the equation ofthe line tangent to the graph of the circle x2y2 14X4y280 at the point 31 a Use completing the square to write the equation of the circle in standard form b Find the center of the circle c Find the slope ofthe radius from the center to the point of tangency d Find the slope ofthe tangent line e Find the equation of the tangent line in the form y mx b MMAT 3320 NOTES SECTION 16 PAGE 1 EXAMPLE 2 Find the equation of the line tangent to the ellipse 7x2 3y2 103 at the point 25 a Let m denote the slope of the tangent line at 25 Find the equation ofthis line b For the nonlinear system consisting of the equations of the ellipse and the line eliminate y and obtain a quadratic equation in x MMAT 3320 NOTES SECTION 16 PAGE 2 c By inspection of the original graph the ellipse and the line intersect in one point only Thus the quadratic equation from part b must have a double root Find the value of m that produces this double root d Find the equation of the desired tangent line MMAT 3320 NOTES SECTION 16 PAGE 3 Descartes39 method for finding a tangent line at a point on a curve involves using a circle whose center is on the Xaxis and which passes through the given point in such a way that the tangent line to the circle at the given point coincides with the tangent line to the curve at the given point EXAMPLE 3 For the parabola y2 4X find the slope of the tangent line at 96 using Descartes39 method a Let h0 denote the center of the circle passing through 96 Find the equation ofthis circle in standard form b For the nonlinear system consisting of the equations of the circle and the original parabola eliminate y and obtain a quadratic equation in x MMAT 3320 NOTES SECTION 16 PAGE 4 c By inspection of the original graph the circle and the parabola intersect in one point only Thus the quadratic equation from part b must have a double root Find the value of h that produces this double root d Use the result from part c to find the equation of the tangent line to the parabola at the point 96 MMAT 3320 NOTES SECTION 16 PAGE 5 HOMEWORK 1 MMAT 3320 NOTES Use properties of algebra and geometry to find the equation of the line tangent to the circle x2 y2 8X 12y 37 O at the point 1211 Write the final answer in slopeintercept form y mx b Find the equation of the line tangent to the parabola y x2 at 39 using the following method Let m denote the slope of the tangent line at 39 Find the equation ofthis line For the nonlinear system consisting of the equations of the parabola and the line eliminate y and obtain a quadratic equation in x By inspection of the original graph the parabola and the line intersect in one point only Thus the quadratic equation from part b must have a double root Find the value of m that produces this double root Find the equation of the desired tangent line For the hyperbola 5X2 2y2 13 do the following steps to find the slope ofthe tangent line at 3 4 using Descartes39 method Let h0 denote the center of the circle passing through 3 4 Find the equation ofthis circle in standard form For the nonlinear system consisting of the equations of the circle and the original hyperbola eliminate y and obtain a quadratic equation in x By inspection of the original graph the circle and the hyperbola intersect in one point only Thus the quadratic equation from part b must have a double root Find the value of h that produces this double root Use the result from part c to find the equation of the tangent line to the hyperbola at the point 3 4 SECTION 16 PAGE 6 5 EXPONENTIAL MODELS A geometric sequence is a sequence with a recurrence relation given by an ram where r is a constant called the common ratio Assuming the starting index of this geometric sequence is 0 we have a1 ra0 a2 ra1 rra0 a3 ra2 rrra0 an rr r r a0rn A model based on a geometric sequence is called a discrete exponential model since the isolated points of the graph of an a0rn lie on an exponential graph A continuous exponential model is one with a continuous independent variable and has equation y yorx Sometimes the last equation is written in the form y yoekx EXAMPLE 1 Your savings account currently has 2300 in it If money grows at 5 compounded annually how much will be in your account after 9 years a Set up a recurrence relation with initial condition which models this problem Let an denote the amount in dollars in your saving account n years from now b Solve the recurrence relation with initial condition in part a MMAT 3320 NOTES SECTION 5 PAGE 1 MMAT 3320 NOTES Answer the question using the result of part b u a each year E 39 39 L 39 mean wer39 quot answer in the spreadsheet Number of years Amount in account 2415 00 2535 75 2662 54 coooummewmuo 3568 06 EXAMPLE 2 According to an internet site the 2000 Census recorded 569463 residents in the Hidalgo Countymetro area a 485 percent rise over the 383545 residents in 19 0 Let pquot denote the population in Hidalgo County amp Mc AllenEM MSA Pnrulzmnn isnnznnn the Hidalgo Countymetro area n years alter 990 Letting r denote the annual growth rate we have the recurrence relation n quotgt1 Find the two initial conditions SECTION 5 PAGE 2 b Find the annual growth rate Express as a percentage rounded to 2 decimal places c Assuming that the same annual growth rate were continue after the year 2000 predict what the population in the Hidalgo Countymetro area will be in the year 2023 Round to the nearest thousand MMAT 3320 NOTES SECTION 5 PAGE 3 Radioactive substances decay exponentially and so 0 lt r lt 1 The half life of a radioactive substance is the amount of time it takes for the radioactive substance to decay to half of its original amount EXAMPLE 3 The amount of carbon 14 in a living creature begins to decay exponentially once the creature dies The halflife of carbon 14 is about 5730 years a Set up an exponential model for this problem Be sure to declare variables MMAT 3320 NOTES SECTION 5 PAGE 4 b A fossil is found in which 89 of the original carbon 14 has decayed Estimate how long ago that the creature died Round to the nearest hundred years MMAT 3320 NOTES SECTION 5 PAGE 5 EXAMPLE 4 The take for a movie at the box office during each of the first six weeks of release is given in the table below Number of weeks Revenue in since release millions of dollars 902 513 296 196 143 82 a Which is better a linear model or an exponential model CDU ILOONA b Use the better model to answer the following If the movie studio plans to withdraw the film from theaters when the box office revenues drop to 250000 per week how many weeks will the movie run in theatrical release MMAT 3320 NOTES SECTION 5 PAGE 6 HOMEWORK 90594 MMAT 3320 NOTES If you deposit 750 at 4 compounded annually how much money is in your account after 3 years how long does it take until your account has 1200 how much interest does your money earn during its first 5 years how much interest does your money earn during its first 10 years You borrow 5000 and must pay back 7800 in 6 years What annual compound rate of interest are you being charged Express as a percentage rounded to 2 decimal places The number of bacteria in a colony triples every day Find the hourly growth rate Express as a percentage rounded to 2 decimal places The population of a town was 13900 in 1970 15000 in 1980 16200 in 1990 and 18900 in 2000 Use exponential regression to find an exponential equation expressing the population P in terms of t the number of years since 1970 Round values in equation to 5 decimal places To the nearest hundred predict what the population of the town will be in the year 2010 To the nearest year find the doubling time of the population in other words find how long it takes for the population to double A certain radioactive substance has decayed to 90 of its original amount after 47 years What percentage of the original substance remains after 67 years Round to the nearest percentage point Find the halflife of this substance Round to the nearest year A study found that athletes who rested after exhausting exercise had a halflife of lactic acid removal of about 23 minutes How much lactic acid has been removed after 10 minutes of rest Round to the nearest percentage point How long would it take to remove 85 of the lactic acid Round to the nearest minute SECTION 5 PAGE 7 9 TRIGONOMETRIC VALUES OF SPECIAL ACUTE ANGLES EXAMPLE 1 Consider the following isosceles right triangle a Find the length ofthe remaining leg 45 b Find the length ofthe hypotenuse 45 I c Find the value of cos 45 EXAMPLE 2 Consider the following 30 60 90 triangle a Find the length of the hypotenuse b Find the length of the remaining leg 0 Find the value of sin 60 d Find the value of tan 30 MMAT 3320 NOTES SECTION 9 PAGE 1 EXAMPLE 3 Considerthe following 15 75 90 triangle 1 5 J54 Find the value of cos 15 MMAT 3320 NOTES SECTION 9 PAGE 2 EXAMPLE 4 Does 2sin 30 equal sin 60 Notation When n is a positive integer sinquote is an abbreviationquot for sin equot For example 2 o oz 121 sIn 30 sIn 30 2 4 EXAMPLE 5 Find the exact value of cos330 Notice how powers of trigonometric values are entered differently on the calculator than how they are written roximation ofcos33C kin exact result 39om Ex 4 cos36A 3 8 6495198528 6495198528 EXAMPLE 6 Approximate the value of tan849 to 4 places alter the decimal point quot Stmtiav statements huld tut cusquote and lanquote MMAT 3320 NOTES SECTION 9 PAGE 3 HOMEWORK 0094 FOP00quot DP05900 F DP05901 FD MMAT 3320 NOTES Find the exact value of each ofthe following sin 45 b cos 30 tan 60 d sin 30 tan 45 f cos 60 Consider the following 15 75 90 triangle 1 5 F 13 Find the exact value of sin 15 Find the exact value of tan 15 Find the exact value of cos 75 Find the exact value of sin 75 Find the exact value of tan 75 Find the exact value of each ofthe following cos230 sin230 cos245 sin245 cos260 sin260 cos215 sin215 cos275 sin275 Based on the results of the previous problem make a conjecture Based on earlier results complete the following cos 30 sin cos 45 sin cos 60 sin cos 15 sin cos 75 sin Based on the results of the previous problem make a conjecture SECTION 9 PAGE 4 Based on your conjecture complete the following cos 18 sin c Use your calculator to verify your answer in part b 7 Based on earlier results complete the following a sin 300 cos b sin 45 cos c sin 60 cos d sin 15 cos e sin 75 cos 8 Make a conjecture based on the results of the previous problem ANSWERS 1a Q 2 1b 2 1c J3 1d 1 2 1e 1 1f 1 2 2a J5 4 2b 2 J5 2c E 4 21 M 4 2e 2 J3 MMAT 3320 NOTES SECTION 9 PAGE 5 14 THE SINE CURVE We have seen how the trigonometric functions help model situations based on measurements of triangles There is however much more to the trigonometry story Here we will look at the sine function and see how it can be used to model periodic phenomena EXAMPLE 1 Complete the following table for each value of x note x is in radians and then use the results to sketch the graph ofy sin x X X MMAT 3320 NOTES SECTION 14 PAGE 1 The sine curve has the equation y sin x where X is measured in radians Below is one period of its graph 2n EXAMPLE 2 Sketch the graph of one period of y 2 sin x MMAT 3320 NOTES SECTION 14 PAGE 2 EXAMPLE 3 Sketch the graph of one period of y sin X 1 EXAMPLE 4 Sketch the graph of one period of y sin 2x MMAT 3320 NOTES SECTION 14 PAGE 3 EXAMPLE 5 Sketch the graph of one period of y sin x g EXAMPLE 6 Our goal is to sketch one period ofthe graph of 2 n y 3srn2x 5 1 a Sketch the graph of one period ofy sin x 2n MMAT 3320 NOTES SECTION 14 PAGE 4 b Sketch the graph of one period ofy gsin x 1 7 77 2n n t a i 1 7 2 c Sketch the graph of one perIod ofy 5srn X 1 1 7 77 7 2n n Z 1 7 MMAT 3320 NOTES SECTION 14 PAGE 5 d Sketch the graph of one period ofy gsin 2X 1 n 2n 2n MMAT 3320 NOTES SECTION 14 PAGE 6 EXAMPLE 7 Below is one period of a sine curve Find an equation for this sine curve MMAT 3320 NOTES SECTION 14 PAGE 7 EXAMPLE 8 Below is one period of a sine curve Find an equation for this sine curve MMAT 3320 NOTES SECTION 14 PAGE 8 HOMEWORK 1 MMAT 3320 NOTES Sketch one period of the graph of each ofthe following sine curves 1 X sinx 1 b sin 3 y 4 y 2 y3sinx d ysin3X rc Each of the following shows the graph of one period of a sine curve Find an equation for this sine curve 1 SECTION 14 PAGE 9 ANSWERS 1a MMAT 3320 NOTES SECTION 14 PAGE 10 3 LINEAR MODELS An arithmetic sequence is a sequence with a recurrence relation given by an am d where d is a constant called the common difference Assuming the starting index of this arithmetic sequence is 0 we have a1 a0 d a2a1da0dd a3a2da0ddd ana0ddddna0 A model based on an arithmetic sequence is called a discrete linear model since the isolated points of the graph of an dn a0 lie on a line with slope d and yintercept a0 A continuous linear model is one with a continuous independent variable and has equation y mx b EXAMPLE 1 A certain longdistance telephone company charges 245 for a collect call lasting up to one minute and 065 for each additional minute or fraction thereof a Let n denote the length of the call in minutes and let pn denote the price ofthe collect call in dollars Find a linear model b How much does a collect call lasting 6 minutes and 12 seconds cost MMAT 3320 NOTES SECTION 3 PAGE 1 c How long can a collect call last if you do not want its cost to exceed ten dollars EXAMPLE 2 Over a 15year period a piece of equipment depreciates linearly from its original cost of 20 000 to its scrap value of 2000 a Find a linear model Be sure to declare all variables b By how much does the equipment depreciate each year c How much is the equipment worth 11 years alter being bought MMAT 3320 NOTES SECTION 3 PAGE 2 EXAMPLE 3 The following chart was given to new nurses at a hospital concerning the ideal weight of an quotaveragequot man at various heights Hei ht ininches 62 64 66 68 70 72 Weightin pounds 124 133 142 151 160 169 a Construct a scatterplot on your calculator w HDDN Note By pressing the TRACE key and using the arrow keys you are able to see the coordinates of each data point puma n xsa 39l39151 5quot Let h represent the height in inches and let w denote the weight in pounds of an quotaveragequot male Find a linear model MMAT 3320 NOTES SECTION 3 PAGE 3 Assuming that the same linear trend applies to an quotaveragequot ma whose height is more than 6 feet what weight should the quotaveragequot male who is 6 feet 5 inches tall have 6 EXAMPLE 4 The following table gives some measurements for the rate of chirping in chirps per minute ofthe striped ground cricket at various Fahrenheit temperatures Temp F l 89 72 93 84 81 75 7o 82 Chirps l 78 60 79 73 63 62 59 68 Tem 69 83 80 83 81 84 76 C rps 61 65 60 69 64 68 57 a Create a scatter plot Plotz not Df f Store the temperatures in L1 Store the Spa n number of chIrps in L2 In the Stat Plot menu select Plot1 and set up a scatter plot ofthe data In the Zoom menu select ZoomStat quot5 quot First make sure that all fundiuns are cleaved in the v menu MMAT 3320 NOTES SECTION 3 PAGE 4 5quot Use linear regression to nd a linear model expressing the number of chirps per minute c in terms of the Fahrenheit temperature F ound values to 2 places alter the decimal point Also recreate the scatter plot with the regression line included LinReg In the Stat menu CALC submenu enter X363 the command LinRegaxb L1L2Yi The regression equation is stored under Y1 in theY menu Press the GRAPH key c If you hear a striped ground cricket make 85 chirps per minute what temperature do you think it is Round to the nearest degree quotquot vi can be mm m the VAR S menu VVVAR S submenu Funcliun subrsubmenu MMAT 3320 NOTES SECTION 3 PAGE 5 HOMEWORK 1 MMAT 3320 NOTES A computer technician charges a company a 125 charge for each office visit and 60 for each hour or fraction thereof of labor Let dn denote the number of dollars charged for an office visit with n hours of labor Set up a recurrence relation with initial condition Solve the recurrence relation in part a How much does the technician charge for an office visit with 5 hours and 15 minutes of labor At a price of 39 cents a can 237 cans of soda can be sold per day at a convenience store If the price is raised to 57 cents per can then only 192 cans can be sold per day Let cn denote the number of cans of soda that can be sold per day at a price of n cents per can Assuming a linear model find a nonrecursive formula for on If the price is raised to 69 cents per can how many cans of soda can be sold per day What must the price be raised to in order for the demand to drop to 107 cans per day According to the 2002 IRS Form 1040 a single person whose taxable income is between 27950 and 67700 inclusive in 2002 is taxed 10 of the first 6000 of taxable income plus 15 of the next 21950 of taxable income plus 27 of the taxable income over 27950 Let t denote the number of dollars taxed on a single person whose taxable income was n dollars in 2002 Find the taxed amount for a person whose taxable income in 2002 was 27950 Find an equation expressing t in terms of n where n is between 27950 and 67700 inclusive How much to the nearest dollar is a single person whose taxable income was 57125 in 2002 taxed A friend tells you that he was taxed 10000 on his 2002 IRS Form 1040 What was his taxable income for 2002 Round to the nearest dollar SECTION 3 PAGE 6 4 QUADRATIC MODELS A quadratic model is one in which the rst differences form an arithmetic sequence and so the second differences are constant A recurrence relation for a discrete quadratic model is ofthe form an am d en where e 7 0 Using the iterative method one can show that the solution of this recurrence relation with initial condition aEl is given by the quadratic sequence an 1enZJrGeerjnJran A continuous discrete model has the form fx ax2 bx c where a 7 0 EXAMPLE 1 Below are pictures of the rst four hexagonal numbers 0 Let hn denote the nm hexagonal number ie the number of dots in the picture of the nm hexagonal number a Find a recurrence relation with initial condition for sequence h hnhn4 MMAT 3320 NOTES SECTION 4 PAGE 1 b Find a nonrecursive formula for hn 0 Which hexagonal number has 8911 dots MMAT 3320 NOTES SECTION 4 PAGE 2 EXAMPLE 2 We wish to find an equation relating braking distance to speed a Complete the following data table Speed Braking distance First Second mph in car lengths Differences Differences 20 4 30 6 4O 9 50 60 18 7O 24 For instance if you drive your car at 40 mph it will take you 9 car lengths to stop The rst 3 car lengths ofthese 9 car lengths result from your reaction time ie you will have traveled 3 car lengths before you even press the pedal b Find an equation expressing d the breaking distance in car lengths in terms of s the speed ofthe car in miles per hour c If you are traveling at 120 miles per hour and you suddenly have to brake how many car lengths would it take you to stop MMAT 3320 NOTES SECTION 4 PAGE 3 EXAMPLE 3 Galileo studied falling objects at the start of the 17th century In his studies he determined the number of feet that the object fell during each second and discovered that these measurements were in proportion to the odd numbers a Explain how a quadratic model is appropriate b Consider an experiment in which a rock was dropped from the top of a 400foot building It was observed that the rock dropped 16 feet during the first second Complete the following table Elapsed time Height above First Second in seconds ground in feet Differences Differences O 1 4 5 c Find a quadratic equation expressing h the height of the rock above the ground in feet in terms oft the number of seconds elapsed MMAT 3320 NOTES SECTION 4 PAGE 4 d How far did the rock drop during the first 35 seconds e Find the average speed of the rock during the first 15 seconds of its flight Height of rock above ground initially Height of rock above ground after 15 seconds Distance rock dropped during first 15 seconds Average speed of rock during first 15 seconds f Find the average speed of the rock during the last quarter second of its flight MMAT 3320 NOTES SECTION 4 PAGE 5 HOMEWORK 1 MMAT 3320 NOTES Derive the formula at the beginning of the section using the iterative process On your calculator plot the original data from Example 2 along the equation that we found to verify that the equation does fit the data On your calculator plot the original data from the first two columns of Example 3b along the equation that we found in part c to verify that the equation does fit the data A lumber company owns 50000 acres of forest land In the first year of operation they plan to log 1200 acres The next year they plan to log 1400 acres Each subsequent year they plan to log 200 more acres than in the previous year Let an denote the total number of acres logged after n years Find a recurrence relation with initial condition that models this problem Solve the recurrence relation with initial condition in part a How many years will it take to log the entire 50000 acres In a roundrobin tournament each team must play every other team once Let tn denote the total number of matches to schedule in a round robin tournament with n teams Find a recurrence relation with initial condition that models this problem Solve the recurrence relation with initial condition in part a A bowling league has 22 fourperson teams and wants to schedule a roundrobin tournament When two teams compete each player is required to pay 900 for lane fees What total amount of money will the players in the league pay forthis roundrobin tournament Referring to part c if every team plays one match a week how many weeks will it take to complete the roundrobin tournament SECTION 4 PAGE 6 6 When a small business sells its widgets at p per item the amount of profit P in dollars is given by P 23 15p2 25905p 3759 a If the business sells each widget for 20 cents how much profit will it make b If the business sells each widget for 70 cents how much profit will it make c lfthe business sells each widget for 1 dollar how much profit will it make d Graph the profit function in the window O1 500500 and use the trace feature to find the unit price that the business should charge in order to maximize its profit e Explain in business terms why a unit price that is too low produces negative profit f Explain in business terms why a unit price that is too high produces a negative profit MMAT 3320 NOTES SECTION 4 PAGE 7 ANSWERS 1 a1ande1 a 1 e2ande1de2 aga1de3ande1de2de3 nde1de2de3den anandd de1e2e3en numes nannde162e3eI I nannde123n anannde7quotquot2 1 1 anannd en2 en aquot lenz rlen rnd r aEl 2 2 a len2 1ed naEl 2 2 2 w 3 w MMAT 3320 NOTES SECTION 4 PAGE 8 10 TRIGONOMETRIC FUNCTIONS OF TRIGONOMETRIC ANGLES We now want to extend our concept of angle to include measures of any number of degrees positive negative orzero Definition Given two rays having a common endpoint a trigonometric angle is the amount of rotation needed to move the first ray called the initial side to the second ray called the terminal side A positive angle is generated by a counterclockwise rotation a negative angle is generated by a clockwise rotation a zero angle is generated by no rotation Note To specify a trigonometric angle in addition to its sides we need a curved arrow extending from its initial side to its terminal side 540 Definition In the Cartesian plane an angle is said to be in standard position if its vertex is at the origin and its initial side coincides with the positive xaxis Angles in standard position whose terminal sides coincide are called coterminal angles Angles in standard position whose terminal sides lie on one of the axes are called quadrantal angles Nonquadrantal angles are said to be in a certain quadrant if their terminal sides lie in that quadrant EXAMPLE 1 Draw two other angles in standard position that are coterminal with the given angle Are these angles quadrantal Explain MMAT 3320 NOTES SECTION 10 PAGE 1 We now extend our definitions ofthe cosine sine and tangent functions Definition Given an angle 0 in standard position choose any point xy different from 00 that lies on the terminal side of 0 Let r be defined by r X2 y2 Note that r is always positive Then x cos0 sin0l and tan0 r r x Note The triangle associated with this definition is given by In the case of quadrantal angles the reference triangle quotcollapses39 into a degenerate case EXAMPLE 2 Approximate cos 165 to the nearest hundredth MMAT 3320 NOTES SECTION 10 PAGE 2 EXAMPLE 3 Find the exact values of cos 270 sin 270 and tan 270 MMAT 3320 NOTES SECTION 10 PAGE 3 EXAMPLE 4 Find the exact value of sin 240 EXAMPLE 5 Find the exact value of cos 315 MMAT 3320 NOTES SECTION 10 PAGE 4 EXAMPLE 6 Suppose cos 6 g and 6 doesn39t lie in Quadrant Find the exact value of tan 6 MMAT 3320 NOTES SECTION 10 PAGE 5 EXAMPLE 7 Suppose tan 6 g and 6 doesn39t lie in Quadrant IV Find the exact value of sin 6 MMAT 3320 NOTES SECTION 10 PAGE 6 HOMEWORK 1 MMAT 3320 NOTES nances Find the smallest positive angle that is coterminal with 2000 Find the largest angle less than 2000 that is a quadrantal angle Find the exact value of each ofthe following tan 180 cos 120 sin 300 tan 225 sin 90 tan 330 Suppose sin 6 and 6 doesn t lie in Quadrant lll Find the exact value of cos 6 Suppose tan 6 5 and 6 doesn t lie in Quadrant Find the exact value of sin 6 SECTION 10 PAGE 7 19 OPTIMIZATION PROBLEMS EXAMPLE 1 A parcel delivery service will deliver a package if and only if the length plus girth distance around does not exceed 108 inches Find the dimensions of a rectangular box with square ends that satisfies the delivery services restriction and has maximum volume a Draw a picture b To maximize the volume of the box we will want for the length plus girth to be exactly 108 inches if it was less than 108 inches we could increase the length for example to increase the volume Complete the following table to explore different possible box sizes 12 c Tabular Approach Create a spreadsheet to provide additional possible box dimensions estimate the size of box with the maximum volume 1 3 4 MMAT 3320 NOTES SECTION 19 PAGE 1 9 S MMAT 3320 NOTES Suppose x denotes the width of a box Determine the length girth and volume oft e box Which valuess of x make sense in the context ofthis application Graphing Approach In the Graph menu store the volume function under Y1 and set up the minimum and maximum xvalues of the domain in the V ndow menu choosing an appropriate scale for the x tick marks Don39t wor about the vaues in the V ndom Moog 39 menu yet mu Plotz not Ifyou try graphing using this window the graph does not 39ii window since t e maxi value ofy isn39t large enough enu select ZoomFit This command will adjust Into the From the s in the V ndow menl so that the graph quot tsquot n the screen the y value Go back to the Wndow menu and adjust the maximum value ofy and theyscae 0 our iki The1 re graph the function 11 NDDU 1nB SECTION 19 PAGE 2 Under the CALC graphing menu select quotmaximumquot Follow the Lnnu poIntto the right of the highest point The calculator responds with an approximation for the hi hest oint We see that the function achieves its maximum value when x 18 Thus to achieve the maximum volume the width of the box should be 18 inches and it follows that the length of the box would then be 36 inches Note that the maximum volume of the box is 11664 cubic inches In I I 44 r the nnr a in box that maximizes its volume MMAT 3320 NOTES SECTION 19 PAGE 3 EXAMPLE 2 Solve the following problem using the calculus approach A manufacturer wants to produce a can that will hold 20 cubic inches in the form of a right circular cylinder Find the dimensions of the can that will use the smallest amount of material Round each dimension to 2 places after the decimal point MMAT 3320 NOTES SECTION 19 PAGE 4 EXAMPLE 3 Solve the following problem using the calculator graphing approach A river flows west to east Alphaville is 7 miles north of the river Betaburg is located 9 miles east and 3 miles south of Alphaville The two towns wish to build a bridge on the river so that the sum of the distance from Alphaville to the bridge and the distance from Betaburg to the bridge is as small as possible Determine where to place the bridge MMAT 3320 NOTES SECTION 19 PAGE 5 HOMEWORK Do problems 13 using a the tabular approach b the graphing calculator approach and c the calculus approach Use the method of your choice for problems 4 and 5 1 Redo Example 1 using a cylindrical tube Round to 2 places after the decimal point 2 If 1200 square centimeters of material is available to make a box with a square base and an open top find the largest possible volume of the box 3 A rectangular storage container with an open top is to have a volume of 10 cubic meters The length of its base is twice the width Material for the base costs 10 per square meter Material for the sides costs 6 per square meter Find the cost of materials for the cheapest such container 4 A Norman window consists of a rectangular pane topped with a semicircular pane Of all Norman windows having a perimeter of 16 feet which one allows in the most light Round to 2 places after the decimal point 5 An island is located 5 kilometers offshore Thirteen kilometers down shore from the point on the beach nearest the island is a bird39s nest Birds use 14 times as much energy to fly over water as over land If the bird is released from the island and wishes to get to its nest at what point on the beach should the bird fly in order to minimize the total amount of energy it uses Assume the beach is straight Round to 2 places after the decimal point MMAT 3320 NOTES SECTION 19 PAGE 6 2 SOLVING RECURRENCE RELATIONS EXAMPLE 1 Consider the recurrence relation with initial conditions given a DWNOEUIhWNA O MMAT 3320 NOTES by an 2am aniz a0 0 a1 3 Use Excel to try to find a nonrecursive solution Formulas Values A B 1 n an 1 2 recursively 2 3 O O 3 4 A31 3 4 5 l 2B4 B3 5 6 l l 6 7 l l 7 8 l l 8 9 l l 9 10 l l 10 From the table of values it appears that a solution of this recurrence relation with initial conditions is an 3n We can check to see if our answer is reasonable by inserting a third column containing the values ofthe nonrecursive sequence Formulas Values A B C A B C n an an 1 n an an recursively nonrecursively 2 recursively nonrecursively 0 0 3A3 3 0 0 0 A31 3 l 4 1 3 3 l 2B4 B3 l 5 2 6 6 l l l 6 3 9 9 l l l 7 4 12 12 l l l 8 5 15 15 l l l 9 6 18 18 l l l 10 7 21 21 SECTION 2 PAGE 1 Verify that an 3n is indeed a solution of this recurrence relation with initial conditions Note that an 3n is a solution ofthis recurrence relation since 2am aH 23n 1 3n 2 6n 1 3n 2 6n 6 3n 6 3n an Also we have that a0 30 O and a1 31 3 Thus the initial conditions are also satisfied EXAMPLE 2 Find the solution of the following recurrence relation with initial condition an 2am ao 4 List the first few terms and make a guess at the formula Use an iterative approach to develop the formula MMAT 3320 NOTES SECTION 2 PAGE 2 EXAMPLE 3 Find the solution of the following recurrence relation with initial condition an n2an4 a0 1 MMAT 3320 NOTES SECTION 2 PAGE 3 EXAMPLE 4 Find the solution of the following recurrence relation with initial condition an an4 5 a1 13 MMAT 3320 NOTES SECTION 2 PAGE 4 EXAMPLE 5 Forthe quadratic sequence fn an2 bn c complete the following table n fn an2 bn c First differences Second differences ICDU ILOON a What is true about the first differences b What is true about the second differences MMAT 3320 NOTES SECTION 2 PAGE 5 EXAMPLE 6 We wish to find a closedform formula for the sum of the first n positive integers a Complete the following table 1 2 3 n First differences Second differences ICDU ILOONAD b Notice that the values in this table represent those of a quadratic sequence By equating corresponding positions in the table above and the table in the previous example for quadratic sequences find the quadratic expression that equals the sum of the first n positive integers MMAT 3320 NOTES SECTION 2 PAGE 6 c Regression commands on Tl83 Plus calculator can be used to nd linear quadratic cubic and quartic equations that lit given data St the data L L3 mum IE Enter command at QuadReg L1 home screen Lz Result of command QuadReg uaxzbxc 5 5 R 2 1 UIHDDU Set uE window for plot Thus we have that Xmln B equation u 2 w R R vaiue An R vaiue um indicateslhallhe equaliun m the data pe eclly MMAT 3320 NOTES SECTION 2 PAGE 7 The following formulas will be useful in solving certain recurrence relations 123 n 21 122232 n2nn12n1 6 132333 n3n2n12 4 rn11 1rr2r 1forr 1 r EXAMPLE 7 Find the solution of the following recurrence relation with initial condition an an16na05 MMAT 3320 NOTES SECTION 2 PAGE 8 HOMEWORK 1 906 0399 9 a 0 MMAT 3320 NOTES Create a spreadsheet like that in Example 1 which quotcheckquot the solution found in Example 2 Example 3 Example 4 Example 7 Verify the second and third formulas at the top of page 8 of the notes using the technique given in Examples 5 and 6ab using the regression commands on the Tl83 Plus calculator Verify the fourth formula at the top of page 8 ofthe notes as follows Let S 1 r r2 r Multiply both sides of this equation by r and simplify the righthand side Subtract corresponding sides of the original equation from this quotneV39 equation simplify the righthand side and then solve for S assuming that rdoesn39t equal 1 Solve each of the following recurrence relations with initial conditions an am 7 a0 52 bn 5bn1b0 125 cn cm 24n2 c0 60 en nen1e0 1 fn n1f01 on gm 12n3go 0 SECTION 2 PAGE 9
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