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# ALGEBRA WITH APPL MATH 111

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This 243 page Class Notes was uploaded by Addison Beer on Wednesday September 9, 2015. The Class Notes belongs to MATH 111 at University of Washington taught by Staff in Fall. Since its upload, it has received 11 views. For similar materials see /class/192092/math-111-university-of-washington in Mathematics (M) at University of Washington.

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Lecture 21 Types of Interest Mong Kon Mo August 8 2007 Course Website httpwwwmathwashingtoneduwmkmomath111 Math 111 Simple Interest So far we have been talking about compounded interest Before discussing compound interest in more detail let39s look at a different type of interest simple interest With simple interest you only earn interest on the principal not on the entire balance Simple Interest An Example 77 You deposit 500 into an account paying simple interest of 3 per year 3 of 500 is 15 So after one year you will have A1 500 15 515 After two years you will have A2 500 215 530 Notice that after each year you gain 15 regardless of the current balance This means that At is an additive sequence where t is in years Simple Interest An Example So At 50015t 500003 500t 5001003t principal annual interest rate Just like the CAF this formula works even if t is not a whole number For example in 21 months the balance will be A lt21 A175 5001 003175 52625 E Compounded Interest 7 In most cases you will earn interest not just on the principal but on the entire balance In these cases we say that interest is compounded This is what we have been working with since Worksheet 18 Compounded Interest An Examl You deposit 500 into an account paying 3 per year compounded semiannualy twice a year CAUTION We know that 3 per year is NOT the same as 15 every six months However when a bank says 3 per year compounded semiannuallyquot they actually do mean 15 every six months Compounded Interest An Examle 77 Let At be the balance after t years Then A0 500 A1 50010152 A2 50010154 At 50010152t As mentioned last time this formula works even it t is not a whole number For example in 21 months the balance will be A A175 50010152175 52675 Generalizing the CAP 7 Suppose that a bank otters r 100 interest per year compounded n times a year Then they are actually offering 100 interest n times a year So the balance after t years will be At P 1 quott Some Examples You deposit 500 into an account paying 3 per year compounded daily so n 365 Q What is the balance after 21 months 0 03 355 The generalized CAF gives us At 500 lt1 A So 365075 A A175 500 lt1 03903 52695 W Some Examples You deposit 500 into an account paying 3 per year compounded every minute so n 525600 Q What is the balance after 21 months 003 525500 The generallzed CAF glves us At 500 lt1 5257600 A So 525600175 A A175 500 1 03903 gt 5257600 52695 An Observation I Notice that in each of these examples the principal and annual interest rate were the same P 500 and r 003 The only thing that changed was the number of times interest was compounded each year n 2 gave us A175 z 5267458858 n 365 gave us A175 z 5269501444 n 525600 gave us A175 z 5269512802 As n increases the balance also increases But there is a limit to how big the balance will get In this case the limit is 52695 Compounding Continuously W 7 If a bank compounds so that at any given time you have the maximum possible balance then the interest is said to be compounded continuously A crude way to do this is to just pick a big enough value of n to use If we wanted to derive the formula for the balance in an account that compounds continuously we would need some calculus Since most of us don39t know any calculus we will just use the formula without deriving it At Pe l principal annual interest rate balance after t years Compounding Continuously An E9 7 i You deposit 1400 into an account that pays 7 per year compounded continuously Q What is the balance after 55 months The continuous compounding interest formula gives us At 1400e007t A So the balance after 55 months will be A 1740080976 3 192959 1L warlt 1J7 wqmgmmg Mong Kon Mo July 27 2007 Course Website httpwwwma lhwashimg loneduNmkmoma39lh i 1 u Se Background Definition A list of numbers taken in some specified order is called a sequence Bali eymd A list of numbers taken in some specified order is called a sequence Some Examples A 1015202530 B 510204080160 C 1123581321 D 1000100110021003100 E 20 30 45 675 10125 151 875 F 50 54 58 62 66 70 Se Background Definition The numbers that make up the sequence are called terms Se Background Definition The numbers that make up the sequence are called terms When we count terms we start counting from zero Se Background Definition The numbers that make up the sequence are called terms When we count terms we start counting from zero Recall that A 1015202530 Se Background Definition The numbers that make up the sequence are called terms When we count terms we start counting from zero Recall that A 1015202530 The zeroth term of the sequence A is 10 We write A0 10 Se Background Definition The numbers that make up the sequence are called terms When we count terms we start counting from zero Recall that A 1015202530 The zeroth term of the sequence A is 10 We write A0 10 Q What are the values of A4 and A6 Se Background Definition The numbers that make up the sequence are called terms When we count terms we start counting from zero Recall that A 1015202530 The zeroth term of the sequence A is 10 We write A0 10 Q What are the values of A4 and A6 A A4 30 Se Background Definition The numbers that make up the sequence are called terms When we count terms we start counting from zero Recall that A 1015202530 The zeroth term of the sequence A is 10 We write A0 10 Q What are the values of A4 and A6 A A4 30 and A6 40 Paterns Notice that we were able to name the value of A6 even though it wasn39t given to us This is because the terms in sequence A follow as easily discernable pattern Paterns Notice that we were able to name the value of A6 even though it wasn39t given to us This is because the terms in sequence A follow as easily discernable pattern Paterns Notice that we were able to name the value of A6 even though it wasn39t given to us This is because the terms in sequence A follow as easily discernable pattern Paterns Notice that we were able to name the value of A6 even though it wasn39t given to us This is because the terms in sequence A follow as easily discernable pattern Paterns Notice that we were able to name the value of A6 even though it wasn39t given to us This is because the terms in sequence A follow as easily discernable pattern Paterns Notice that we were able to name the value of A6 even though it wasn39t given to us This is because the terms in sequence A follow as easily discernable pattern Paterns Notice that we were able to name the value of A6 even though it wasn39t given to us This is because the terms in sequence A follow as easily discernable pattern Paterns Notice that we were able to name the value of A6 even though it wasn39t given to us This is because the terms in sequence A follow as easily discernable pattern Paterns Notice that we were able to name the value of A6 even though it wasn39t given to us This is because the terms in sequence A follow as easily discernable pattern Ak1Ak5 This is called a recursive formula for sequence A Patterns Notice that we were able to name the value of A6 even though it wasn39t given to us This is because the terms in sequence A follow as easily discernable pattern Ak1Ak5 This is called a recursive formula for sequence A Translation To get from one term to the next add 5 Aditive Sequences Definition This type of sequence is called additive you add the same number to get from one term to the next The number that needs to be added to get from one term to the next is called the increment of the sequence Aditive Sequences Definition This type of sequence is called additive you add the same number to get from one term to the next The number that needs to be added to get from one term to the next is called the increment of the sequence If we rewrite the recursive formula as Ak 1 7Ak 5 we see that the difference between any two consecutive terms is always 5 which is the increment of sequence A 99417 Identifying Additive Sequence This gives us a handy test to see if a sequence is additive Mam 111 99417 Idetifying Additive Sequence This gives us a handy test to see if a sequence is additive In sequence B we compute Mam i ii 99417 Idetifying Additive Sequence This gives us a handy test to see if a sequence is additive In sequence B we compute B17B010755 Mam i ii 99417 Idetifying Additive Sequence This gives us a handy test to see if a sequence is additive In sequence B we compute Mam i ii 99417 Idetifying Additive Sequence This gives us a handy test to see if a sequence is additive In sequence B we compute B17B010755 B27B12071010 Mam i ii Idetifying Additive Sequence This gives us a handy test to see it a sequence is additive In sequence B we compute B17B010755 B27B12071010 The difference between two consecutive terms is not always the same So sequence B is not additive Identifying Additive Sequences C 1123581321 D 1000100110021003100 E 20 30 45 675 10125 151 875 F 50 54 58 62 66 70 Identifying Additive Sequences C 1123581321 D 1000100110021003100 E 20 30 45 675 10125 151 875 F 50 54 58 62 66 70 Your Turn Which of the other sequences are additive It the sequence is additive find a recursive formula for it Identifying Additive Sequences C 1123581321 D 1000100110021003100 E 20 30 45 675 10125 151 875 F 50 54 58 62 66 70 Your Turn Which of the other sequences are additive It the sequence is additive find a recursive formula for it A The only sequence that is additive is sequence F Identifying Additive Sequences C 1123581321 D 1000100110021003100 E 20 30 45 675 10125 151 875 F 50 54 58 62 66 70 Your Turn Which of the other sequences are additive If the sequence is additive find a recursive formula for it A The only sequence that is additive is sequence F The difference between two consecutive terms is always 4 Identifying Additive Sequences C 1123581321 D 1000100110021003100 E 20 30 45 675 10125 151 875 F 50 54 58 62 66 70 Your Turn Which of the other sequences are additive If the sequence is additive find a recursive formula for it A The only sequence that is additive is sequence F The difference between two consecutive terms is always 4 So the recursive formula is Fk1 Identifying Additive Sequences C 1123581321 D 1000100110021005100 E 20304567510125 151875 F 50 54 58 62 66 70 Your Turn Which of the other sequences are additive If the sequence is additive find a recursive formula for it A The only sequence that is additive is sequence F The difference between two consecutive terms is always 4 So the recursive formula is Fk1Fk4 Exicit Formulas Q What is the 100th term of sequence A Exicit Formulas Q What is the 100th term of sequence A A0 10 Exicit Formulas Q What is the 100th term of sequence A A0101005 Explicit Formulas Q What is the 100th term of sequence A A0101005 A 1 1 05 Explicit Formulas Q What is the 100th term of sequence A A0101005 A 1 1 051015 Exicit Formulas Q What is the 100th term of sequence A A0101005 A11051015 A21025 Explicit Formulas Explicit Formulas Explicit Formulas Q What is the 100th term of sequence A A0101005 A11051015 A21025 A3103 5 Explicit Formulas Q What is the 100th term of sequence A A0101005 A11051015 A21025 A3103 5 A100101005 510 In general Ak 10 5k This is called an explicit formula for sequence A Explicit Formulas Q What is the 100th term of sequence A A0101005 A11051015 A21025 A3103 5 A100101005 510 In general Ak 10 5k This is called an explicit formula for sequence A It allows us to compute the kth term without knowing all the terms that came before it Exicit Formulas F 50 54 58 62 66 70 Explicit Formulas F 50 54 58 62 66 70 Your Turn Find an explicit formula for Fk and use it to find F100 Explicit Formulas F 50 54 58 62 66 70 Your Turn Find an explicit formula for Fk and use it to find F100 A Fk 50 4k and F100 450 Explicit Formulas F 50 54 58 62 66 70 Your Turn Find an explicit formula for Fk and use it to find F100 A Fk 50 4k and F100 450 In general if S is an additive sequence with increment r then the explicit formula for the kth term is Sk s0 rk Explicit Formulas F 50 54 58 62 66 70 Your Turn Find an explicit formula for Fk and use it to find F100 A Fk 50 4k and F100 450 In general if S is an additive sequence with increment r then the explicit formula for the kth term is Sk s0 rk increment Explicit Formulas F 50 54 58 62 66 70 Your Turn Find an explicit formula for Fk and use it to find F100 A Fk 50 4k and F100 450 In general if S is an additive sequence with increment r then the explicit formula for the kth term is Sk s0 rk l starting increment term Mutiplicative Sequences What if the sequence is not additive Can we still find a recursive formula How about an explicit formula We know that B is not an additive sequence but it still follows a pattern Mutiplicative Sequences What if the sequence is not additive Can we still find a recursive formula How about an explicit formula We know that B is not an additive sequence but it still follows a pattern 31 2B0 Multiplicative Sequences What if the sequence is not additive Can we still find a recursive formula How about an explicit formula We know that B is not an additive sequence but it still follows a pattern Multiplicative Sequences What if the sequence is not additive Can we still find a recursive formula How about an explicit formula We know that B is not an additive sequence but it still follows a pattern Multiplicative Sequences What if the sequence is not additive Can we still find a recursive formula How about an explicit formula We know that B is not an additive sequence but it still follows a pattern 31 2 0 32 23 1 33 2 2 Multiplicative Sequences What if the sequence is not additive Can we still find a recursive formula How about an explicit formula We know that B is not an additive sequence but it still follows a pattern 31 2 0 32 23 1 33 2 2 Multiplicative Sequences What if the sequence is not additive Can we still find a recursive formula How about an explicit formula We know that B is not an additive sequence but it still follows a pattern 31 2 0 32 23 1 33 2 2 Bk12Bk The recursive formula means to get from one term to the next multiply by 2quot Mltiplicative Sequences Definition This type of sequence is called multiplicative you multiply by the same number to get from one term to the next The number that you multiply by to get from one term to the next is called the multiplier B k If we reerte the recurslve formula as B k l 2 we see that the ratio of any two consecutive terms is always 2 which is the multiplier of sequence B 99417 Identifying Multiplicative Sequences This gives us a handy test to see if a sequence is multiplicative Man 111 99417 Idetifying Multiplicative Sequences This gives us a handy test to see if a sequence is multiplicative In sequence C we compute C1 0 lilalli l ll 99417 Idetifying Multiplicative Sequences This gives us a handy test to see if a sequence is multiplicative In sequence C we compute lilalli l ll 99417 Idetifying Multiplicative Sequences This gives us a handy test to see if a sequence is multiplicative In sequence C we compute 99417 Idetifying Multiplicative Sequences This gives us a handy test to see if a sequence is multiplicative In sequence C we compute Identifying Multiplicative Sequences This gives us a handy test to see it a sequence is multiplicative In sequence C we compute The ratio between two consecutive terms is not always the same So sequence C is not multiplicative Identifying Multiplicative Sequences D 1000100110021005100 E 20304567510125151875 Identifying Multiplicative Sequences D 1000100110021005100 E 20304567510125151875 Your Turn Which of these sequences is multiplicative If the sequence is multiplicative find a recursive formula for it Identifying Multiplicative Sequences D 1000100110021005100 E 20304567510125151875 Your Turn Which of these sequences is multiplicative If the sequence is multiplicative find a recursive formula for it A Only sequence E is multiplicative Identifying Multiplicative Sequences D 1000100110021003400 E 20304567510125151875 Your Turn Which of these sequences is multiplicative If the sequence is multiplicative find a recursive formula for it A Only sequence E is multiplicative The ratio between two consecutive terms is always 15 Identifying Multiplicative Sequences D 1000100110021003400 E 20304567510125151875 Your Turn Which of these sequences is multiplicative If the sequence is multiplicative find a recursive formula for it A Only sequence E is multiplicative The ratio between two consecutive terms is always 15 So the recursive formula is Ek1 Identifying Multiplicative Sequences D 1000100110021003400 E 20304567510125 151875 Your Turn Which of these sequences is multiplicative If the sequence is multiplicative find a recursive formula for it A Only sequence E is multiplicative The ratio between two consecutive terms is always 15 So the recursive formula is Ek1 15Ek Exicit Formulas Q What is the 100th term of sequence B Exicit Formulas Q What is the 100th term of sequence B 30 5 Exicit Formulas Q What is the 100th term of sequence B B0515 Exicit Formulas Q What is the 100th term of sequence B Exicit Formulas Q What is the 100th term of sequence B B0515 B11025 Exicit Formulas Q What is the 100th term of sequence B Exicit Formulas Q What is the 100th term of sequence B B0515 B11025 B220225 Exicit Formulas Q What is the 100th term of sequence B B0515 B11025 B220225 B3235 Explicit Formulas B0515 B11025 B220225 B3235 Explicit Formulas In general Bk 2k 5 B0515 B11025 B220225 B3235 3100 210 5 Exicit Formulas E 20304567510125151875 Explicit Formulas E 20304567510125151875 Your Turn Find an explicit formula for Ek and use it to find E100 Explicit Formulas E 20304567510125151875 Your Turn Find an explicit formula for Ek and use it to find E100 A Ek 15k20 and E100 1510020 Explicit Formulas E 20304567510125151875 Your Turn Find an explicit formula for Ek and use it to find E100 A Ek 151620 and E100 1510020 In general if M is a multiplicative sequence with multiplier m then the explicit formula forM is Mk mk M0 Explicit Formulas E 20304567510125151875 Your Turn Find an explicit formula for Ek and use it to find E100 A Ek 151620 and E100 1510020 In general if M is a multiplicative sequence with multiplier m then the explicit formula forM is Mk mk M0 multiplier Explicit Formulas E 20304567510125151875 Your Turn Find an explicit formula for Ek and use it to find E100 A Ek 151620 and E100 1510020 In general if M is a multiplicative sequence with multiplier m then the explicit formula forM is multiplier starting term Oter Sequences We will only work with additive and multiplicative sequences in this class However most sequences are neither Oter Sequences We will only work with additive and multiplicative sequences in this class However most sequences are neither C 1123581321 Oter Sequences We will only work with additive and multiplicative sequences in this class However most sequences are neither C 1123581321 isthe Fibonacci sequence Oter Sequences We will only work with additive and multiplicative sequences in this class However most sequences are neither C 1123581321 isthe Fibonacci sequence Recursive Formula Ck 2 Ck Ck 1 Oter Sequences We will only work with additive and multiplicative sequences in this class However most sequences are neither C 1123581321 isthe Fibonacci sequence Recursive Formula Ck 2 Ck Ck 1 lt1 gtkelt17 gtk 2amp5 Explicit Formula Ck Where Is This Going It you deposit money into an account bearing r interest per year then your account balances at the end of each year form a multiplicative sequence whose multiplier depends on r We39ll develop a formula for the balance after t years Where Is This Going It you deposit money into an account bearing r interest per year then your account balances at the end of each year form a multiplicative sequence whose multiplier depends on r We39ll develop a formula for the balance after t years next time Reminders 0 Next Monday we will review for the midterm Try all the problems on the old midterms and come prepared with questions Reminders 0 Next Monday we will review for the midterm Try all the problems on the old midterms and come prepared with questions 0 Midterm is next Tuesday and will cover Worksheets 916 Reminders 0 Next Monday we will review for the midterm Try all the problems on the old midterms and come prepared with questions 0 Midterm is next Tuesday and will cover Worksheets 916 0 Office hours for next week Monday 13pm and Wednesday 12pm ngitwm Ummnqu39g turg rgm w R jtgg Smugth 3 m39n fmwgo y Mong Kon Mo June 20 2007 Course Website h lfipjwwwmathwashimg loneduNmkmomaim 1 u L hj lh 3 Announcements 0 MSC is moved to CMU B014 not B006 for the summer a My office hours are MW 12 and by appointment PkgWill tquot Key Question You are traveling along the freeway on a road trip After passing some construction work you begin to speed up Does this necessarily mean that the average speed for your trip is also going up Or is it possible for your speed to go up while your average trip speed goes down We are being asked to compare the ATS with the actual instantatneous speed We studied the ATS in detail and found that it was the slope of the diagonal line We won t study actual speed until Math 112 So instead we will 39 it with 39 quot g speed r39r39 l l ll jl lf 3quot Graphical Interpretatienwof Averae Trip Speed 600 Distance in miles 500 400 300 200 100 1O 20 30 40 5O 60 70 Time in minutes 11113111111 Average Speed Over 10 Minute Intervals Recall that AD Average Speed E Use this to complete the table below llllallr 111 Graphical Interpretatienwof Averae Speed 600 Distance in miles 500 Time in minutes 11113111111 Change In AS Q Over what time period is the AS decreasing Increasing A AS is decreasing for 0 lt t lt 40 and increasing for 40 lt t lt 75 You should now be able to answer the key question Remember ATS is decreasing for 0 lt t lt 50 and increasing for 50 lt tlt 75 a Reservoir The graph to the left shows the amounts of water in thousands of gallons drawn from a city water reservoir by customers over halthour intervals from noon to 6 pm Water flows into the reservoir at the constant rate of three thousand gallons per half 1 2 3 4 5 6 hour Time in hours since noon NCOgt010 Amount of Water omammmwmsmmmmmu in thousands of gallons 0 O Key Question Key Question If the reservoir were empty at noon could the customers be supplied with their water for the whole day It not what would be a reasonable amount of water to have in the reservoir to be sure that the customers can be supplied with water velillll Reading the Graph Correctly Q What does the point 14 represent A Between 1230 and 100 pm 4 thousand gallons of water were drawn from the reservoir This means that during the time interval 1230 to 100 pm 4 thousand gallons of water were used by consumers Water Consumption Q How much water was used over the time interval 1200 to 100 pm Time Interval Water Consumed 1200 to 1230 pm a 1230 to 100 pm a 05 to 1 544 94 A 94 thousand gallons Let 0 total amount of water that flows out of the reservoir since noon Total Flow Out Since Noon Amount of Water in thousands of gallons O NWAU ICDV 0 1 2 3 4 5 6 Time in hours since noon Total Flow Out Sice Noon Amount of Water in thousands of gallons AANNW J IOU IOU IO 1 2 3 4 5 Time in hours since noon Total Flow In Sice Noon Amount of Water in thousands of gallons AANNW J IOU IOU IO 1 2 3 4 5 Time in hours since noon vt llllll Comparing Total Flow In and Total Flow Out Amount of Water in thousands of gallons AANNW OU IOU IOU IO O 6 1 Time in hours since noon Is There Enoug Water 0 gt 1 for 0 lt t lt 27 ie between noon and 242 pm Q What does 0 gt 1 mean A More water has flowed out of the reservoir than flowed in So there is a shortage of water between noon and 242 pm You should be able to answer the first part of the key question vehilli How Much Water Is Needed Q What does 0 g 1 mean A At least as much water has flowed into the reservoir as has flowed out So we need 0 g 1 for all times t We can achieve this by adding a certain amount of water to the reservoir at noon How much should be add What affect does this have on the graph of 1 Amount of Water in thousands of gallons AANNW OU IOU IOU IO 1 2 Time in hours since noon So we need 34 thousand gallons in the reservoir at noon 6 WM Reminders Things to do before Friday 0 Printout the handout for Worksheet 3 and bring it with you on Friday 0 Keep working on Homework 1 0 Come get your extra credit 1L MW 1J5 4Ml 1 W1 Mame Mong Kon Mo July 23 2007 Course Website httpwwwma lhwashimgioneduNmkmoma39m i 1 u Situation You sell Things and you know the following o VCq 001q3 7 0135q2 06075q o MCq 003q2 7 027q 06075 0 FC 9000 where q is in hundreds of Things and VC and MC are in hundreds of dollars Then FC 90 hundreds of dollars Wrmup Questions Q Compute VC at 325 Things A VC325 0013253 7 01353252 06075325 08917 VC for 325 Things is 8917 Wrmup Questions Q Compute MC at 325 Things A MC325 0033252 7 027325 06075 0046875 MC for 325 Things is 005 per Thing Mam i ii Wrmup Questions Q VC is in hundreds of dollars while MC is in dollars per Thing Why TCq 1 7 TCq 4 1 7 q hundreds of dollars 7 dollars hundreds of Things 7 Thing 39 A Since MCq its units are Maili i ii Some Preliminary Computations Given formulas for VCq MCq and FC we can compute the following TCq VCq FC 001q3 7 0135172 06075q 90 TC Acltqgt M q 90 001q2 70135q06075 Avcltqgt Lil 001q2 7 0135q 06075 TC is in hundreds of dollars while AC and AVC are in dollars per Thing Further Questions For each of the following questions set up the equation you would solve to answer the question Solve the equation and answer the question if you can Q1 Things sell for 075 each What quantity allows you to break even TRq 075q q in hundreds of Things and TR in hundreds of dollars Solve TRq TCq for q 075q 001q3 7 0135q2 060751 90 0 001q3 7 0885q2 06075q 90 This is a cubic and we can39t solve it by hand If we could the nonnegative solutions would give us the BBQ Further Questions 02 Things sell for 075 each What quantity allows you to make backjust your variable cost Solve TRq VCq for q 075q 001q3 7 0135q2 06075q 0 001q3 7 0135172 7 01425q q001q2 7 0135q 7 01425 So q 0 or using the quadratic formula q 144839 or q 709838 A2 So to make back your variable cost you must sell 0 or 1449 Things Further Questions 03 What is your breakeven price Recall that BEP is the common yvalue of the point of intersection of MC and AC Solve MCq ACq for q 003q2 7 027q 06075 001q2 7 0135q 06075 q 003q3 7 027q2 001q3 7 0135q2 90 This is another cubic that we can39t solve by hand If we could we would plug the resulting q into either MCq or ACq to get BEP Further Questions Q4 What is your shutdown price Recall that SDP is the common yvalue of the point of intersection of MC and AVC Solve MCq AVCq for q 003072 7 027q 06075 001q2 7 0135q 06075 002072 7 0135q 0 q 0 or q 675 A4 Disregard q 0 Why SDP MC675 0151875 z 015 per ing LEMMF 1119 Cmmgwumd Mm m mema Mong Kon Mo August 3 2007 Course Website hitpjwwwmathwashingioneduwmkmoma39im 1 n Wig From last time 5000 was deposited into a bank account in your name 0 Option B 14 of the current balance was credited to the account at the end of every year 0 Option C 65 of the current balance was credited to the account at the end of every six months Wham m Mimi From last time 5000 was deposited into a bank account in your name 0 Option B 14 of the current balance was credited to the account at the end of every year 0 Option C 65 of the current balance was credited to the account at the end of every six months We found out that at the end of 25 years the first case was the better option Mimi From last time 5000 was deposited into a bank account in your name 0 Option B 14 of the current balance was credited to the account at the end of every year 0 Option C 65 of the current balance was credited to the account at the end of every six months We found out that at the end of 25 years the first case was the better option To do this we found formulas to tell us how much money would be in the account after k years Compounded Amount Formula Option B Bk 5000114k 11113111111 Compounded Amount Formula Option B Bk 5000114k 50001014k 11113111111 Compounded Amount Formula Option B Bk 5000114k 50001014k Option C Ck 500010652k 11113111111 Compounded Amount Formula Option B Bk 5000114k 50001014k Option c Ck 500010652k 50001 0065 11113111111 Compounded Amount Formula Option B Bk 5000114k 50001014k Option c Ck 500010652k 50001 0065 Suppose that we had another option 11113111111 Compounded Am unt Formula Option B Bk 5000114k 50001014k Option c Ck 500010652k 50001 0065 Suppose that we had another option Option D 325 of the current balance was credited to the account at the end of every four months Compounded Am unt Formula Option B Bk 5000114k 50001014k Option c Ck 500010652k 50001 0065 Suppose that we had another option Option D 325 of the current balance was credited to the account at the end of every four months Find a formula for Dk the amount in the account after k years by observing a pattern in the formulas for Bk and Ck Compounded Am unt Formula Option B Bk 5000114k 50001014k Option c Ck 500010652k 50001 0065 Suppose that we had another option Option D 325 of the current balance was credited to the account at the end of every four months Find a formula for Dk the amount in the account after k years by observing a pattern in the formulas for Bk and Ck Compounded Am unt Formula Option B Bk 5000114k 50001014k Option c Ck 500010652k 50001 0065 Suppose that we had another option Option D 325 of the current balance was credited to the account at the end of every four months Find a formula for Dk the amount in the account after k years by observing a pattern in the formulas for Bk and Ck Dk 5000 Compounded Am unt Formula Option B Bk 5000114k 50001014k Option c Ck 500010652k 50001 0065 Suppose that we had another option Option D 325 of the current balance was credited to the account at the end of every four months Find a formula for Dk the amount in the account after k years by observing a pattern in the formulas for Bk and Ck Dk 5000100325 Compounded Am unt Formula Option B Bk 5000114k 50001014k Option c Ck 500010652k 50001 0065 Suppose that we had another option Option D 325 of the current balance was credited to the account at the end of every four months Find a formula for Dk the amount in the account after k years by observing a pattern in the formulas for Bk and Ck Dk 50001003253k Compound Amount Formula In general if 11113111111 Compound Amount Formula In general if P principal llllallr 111 Compound Amount Formula In general it P principal amount initially deposited Compound Amount Formula In general it P principal amount initially deposited i interest rate expressed as a decimal Compound Amount Formula In general it P principal amount initially deposited i interest rate expressed as a decimal N number of times interest is compounded Compound Amount Formula In general it P principal amount initially deposited i interest rate expressed as a decimal N number of times interest is compounded A amount after N compoundings Compound Amount Formula In general it P principal amount initially deposited i interest rate expressed as a decimal N number of times interest is compounded A amount after N compoundings then AP1iN Compound Amount Formula In general it P principal amount initially deposited i interest rate expressed as a decimal N number of times interest is compounded A amount after N compoundings then A P1iN This is called the Compound Amount Formula CAF Another Option Option E 15 of the current balance was credited to the account at the end of every month Another Option Option E 15 of the current balance was credited to the account at the end of every month Q Find a formula for Ek the amount in the account after k years Use it to determine how much money will be in the account after 25 years Another Option Option E 15 of the current balance was credited to the account at the end of every month Q Find a formula for Ek the amount in the account after k years Use it to determine how much money will be in the account after 25 years A Since interest is compounded every month in k years there will have been 12k compoundings So Another Option Option E 15 of the current balance was credited to the account at the end of every month Q Find a formula for Ek the amount in the account after k years Use it to determine how much money will be in the account after 25 years A Since interest is compounded every month in k years there will have been 12k compoundings So Ek 5000 Another Option Option E 15 of the current balance was credited to the account at the end of every month Q Find a formula for Ek the amount in the account after k years Use it to determine how much money will be in the account after 25 years A Since interest is compounded every month in k years there will have been 12k compoundings So Ek 500010015 Another Option Option E 15 of the current balance was credited to the account at the end of every month Q Find a formula for Ek the amount in the account after k years Use it to determine how much money will be in the account after 25 years A Since interest is compounded every month in k years there will have been 12k compoundings So Ek 50001 0015 Another Option Option E 15 of the current balance was credited to the account at the end of every month Q Find a formula for Ek the amount in the account after k years Use it to determine how much money will be in the account after 25 years A Since interest is compounded every month in k years there will have been 12k compoundings So Ek 50001 0015 Then E25 50000015 25 x 435294 Present and Future Values Mo wants to deposit money into an interestbearing account so that he can purchase a new camera lens that costs 1500 He finds an account that pays 9 interest per year How much must he deposit so that he can afford the lens in 2 years Mo wants to deposit money into an interestbearing account so that he can purchase a new camera lens that costs 1500 He finds an account that pays 9 interest per year How much must he deposit so that he can afford the lens in 2 years Let At be the amount in Mo39s account after t years Then AU Mo wants to deposit money into an interestbearing account so that he can purchase a new camera lens that costs 1500 He finds an account that pays 9 interest per year How much must he deposit so that he can afford the lens in 2 years Let At be the amount in Mo39s account after t years Then At P1 009 Mo wants to deposit money into an interestbearing account so that he can purchase a new camera lens that costs 1500 He finds an account that pays 9 interest per year How much must he deposit so that he can afford the lens in 2 years Let At be the amount in Mo39s account after t years Then At P1 009 We want Mo wants to deposit money into an interestbearing account so that he can purchase a new camera lens that costs 1500 He finds an account that pays 9 interest per year How much must he deposit so that he can afford the lens in 2 years Let At be the amount in Mo39s account after t years Then At P1 009 We wantA2 1500 So Mo wants to deposit money into an interestbearing account so that he can purchase a new camera lens that costs 1500 He finds an account that pays 9 interest per year How much must he deposit so that he can afford the lens in 2 years Let At be the amount in Mo39s account after t years Then At P1 009 We wantA2 1500 So 1 500 100092 Mo wants to deposit money into an interestbearing account so that he can purchase a new camera lens that costs 1500 He finds an account that pays 9 interest per year How much must he deposit so that he can afford the lens in 2 years Let At be the amount in Mo39s account after t years Then At P1 009 We wantA2 1500 So 1500 101092 1500 P11881 Mo wants to deposit money into an interestbearing account so that he can purchase a new camera lens that costs 1500 He finds an account that pays 9 interest per year How much must he deposit so that he can afford the lens in 2 years Let At be the amount in Mo39s account after t years Then At P1 009 We wantA2 1500 So 1500 101092 1500 P11881 P 126252 Some Terminology We say that 126252 is the present value PV of 1500 two years from now Some Terminology We say that 126252 is the present value PV of 1500 two years from now OR Some Terminology We say that 126252 is the present value PV of 1500 two years from now OR We say that 1 500 is the future value FV of 1 26252 two years from now Your Turn Given the same interest rate of 9 per year Your Turn Given the same interest rate of 9 per year Q what is the PV of 5000 six years from now Your Turn Given the same interest rate of 9 per year Q what is the PV of 5000 six years from now 5000 101096 Your Turn Given the same interest rate of 9 per year Q what is the PV of 5000 six years from now 5000 101096 P z 298134 Your Turn Given the same interest rate of 9 per year Q what is the PV of 5000 six years from now 5000 101096 P z 298134 A So PV 298134 Your Turn Given the same interest rate of 9 per year Q what is the PV of 5000 six years from now 5000 101096 P z 298134 A So PV 298134 Q what is the FV of 1 two hundred years from now Your Turn Given the same interest rate of 9 per year Q what is the PV of 5000 six years from now 5000 101096 P z 298134 A So PV 298134 Q what is the FV of 1 two hundred years from now A200 1109200 Your Turn Given the same interest rate of 9 per year Q what is the PV of 5000 six years from now 5000 101096 P z 298134 A So PV 298134 Q what is the FV of 1 two hundred years from now A200 1109200 z 30 570 29208 Your Turn Given the same interest rate of 9 per year Q what is the PV of 5000 six years from now 5000 101096 P R 298134 A So PV 298134 Q what is the FV of 1 two hundred years from now A200 1109200 z 30 570 29208 A So FV 30 57029208 Present Population and Future Population A bacteria colony triples its popuation every 15 minutes You want to have 72900 bacteria 90 minutes from now How many bacteria do you need right now Present Population and Future Population A bacteria colony triples its popuation every 15 minutes You want to have 72900 bacteria 90 minutes from now How many bacteria do you need right now Let Bk be the number of bacteria present after k 15minute intervals have elapsed 19ft ona d Future Population A bacteria colony triples its popuation every 15 minutes You want to have 72900 bacteria 90 minutes from now How many bacteria do you need right now Let Bk be the number of bacteria present after k 15minute intervals have elapsed This means that B0 amount present right now mitian m 19ft ona d Future Population A bacteria colony triples its popuation every 15 minutes You want to have 72900 bacteria 90 minutes from now How many bacteria do you need right now Let Bk be the number of bacteria present after k 15minute intervals have elapsed This means that B0 amount present right now 36 mitian m Ptiquot d Future Population A bacteria colony triples its popuation every 15 minutes You want to have 72900 bacteria 90 minutes from now How many bacteria do you need right now Let Bk be the number of bacteria present after k 15minute intervals have elapsed This means that B0 amount present right now B6 amount present 90 minutes from now Ptiquot d Future Population A bacteria colony triples its popuation every 15 minutes You want to have 72900 bacteria 90 minutes from now How many bacteria do you need right now Let Bk be the number of bacteria present after k 15minute intervals have elapsed This means that B0 amount present right now B6 amount present 90 minutes from now So we want to know what B0 needs to be in order to have B6 72 900 A Graphical Interpretation 11301111 A Graphical Interpretation 11301111 A Graphical Interpretatlon T3 191gtX3 32 11113111111 L A Graphical Interpretatlon 73 32 T3 33 A Graphical Interpretation gtBz gtB3 gtgtB6 x3 x3 x3 x3 11301111 A Graphical Interpretatlon A Graphical Interpretation 11301111 A Graphical Interpretatlon A Graphical Interpretatlon 30 X3 31 733739 7 736 x32 B13B0 B232B0 A Graphical Interpretatlon 30 X3 31 733739 7 736 x32 x33 B13B0 B232B0 A Graphical Interpretation gt 32 gt 33 X3 36 gtnnu x3 x3 x32 A Graphical Interpretation gt 32 gt 33 X3 36 gtnnu x3 x3 x32 A Graphical Interpretation gt 32 gt 33 X3 36 gtnnu x3 x3 x32 Solving the Problem Remember that we need to find the value of B0 that makes B6 72 900 Solving the Problem Remember that we need to find the value of B0 that makes B6 72 900 72900 3630 Solving the Problem Remember that we need to find the value of B0 that makes B6 72 900 72900 36190 30 100 Solving the Problem Remember that we need to find the value of B0 that makes B6 72 900 72900 36190 30 100 So we need 100 bacteria right now Some Terminology We say that 100 bacteria is the present population PP of 72900 bacteria 90 minutes from now Some Terminology We say that 100 bacteria is the present population PP of 72900 bacteria 90 minutes from now OR Some Terminology We say that 100 bacteria is the present population PP of 72900 bacteria 90 minutes from now OR We say that 72900 bacteria is the future population FP of 100 bacteria 90 minutes from now Your Turn Q What is the PP of 59409 bacteria 150 minutes from now Your Turn Q What is the PP of 59409 bacteria 150 minutes from now 59 409 31030 Your Turn Q What is the PP of 59409 bacteria 150 minutes from now 59409 31030 30 1 Your Turn Q What is the PP of 59409 bacteria 150 minutes from now 59409 31030 30 1 Your Turn Q What is the PP of 59409 bacteria 150 minutes from now 59409 31030 30 1 A So PP 1 Your Turn Q Suppose instead that we want 81 million bacteria 7 hours from now How many bacteria should we have 6 hours from now Your Turn Q Suppose instead that we want 81 million bacteria 7 hours from now How many bacteria should we have 6 hours from now amount present in 6 hours Your Turn Q Suppose instead that we want 81 million bacteria 7 hours from now How many bacteria should we have 6 hours from now B24 amount present in 6 hours Your Turn Q Suppose instead that we want 81 million bacteria 7 hours from now How many bacteria should we have 6 hours from now B24 amount present in 6 hours amount present in 7 hours Your Turn Q Suppose instead that we want 81 million bacteria 7 hours from now How many bacteria should we have 6 hours from now B24 amount present in 6 hours B28 amount present in 7 hours Your Turn Q Suppose instead that we want 81 million bacteria 7 hours from now How many bacteria should we have 6 hours from now B24 amount present in 6 hours B28 amount present in 7 hours So B28 324 Your Turn Q Suppose instead that we want 81 million bacteria 7 hours from now How many bacteria should we have 6 hours from now B24 amount present in 6 hours B28 amount present in 7 hours So B28 34324 Your Turn Q Suppose instead that we want 81 million bacteria 7 hours from now How many bacteria should we have 6 hours from now B24 amount present in 6 hours B28 amount present in 7 hours So B28 34324 81 000000 34324 Your Turn Q Suppose instead that we want 81 million bacteria 7 hours from now How many bacteria should we have 6 hours from now B24 amount present in 6 hours B28 amount present in 7 hours So B28 34324 81 000000 34324 81000000 324 3 4 1 000 000 Your Turn Q Suppose instead that we want 81 million bacteria 7 hours from now How many bacteria should we have 6 hours from now B24 amount present in 6 hours B28 amount present in 7 hours So B28 34324 81 000000 34324 324 w 17000000 A We need 1 million bacteria 6 hours from now in order to have 81 million bacteria 7 hours from now A Generalization In general Bk 7190mm A Generalization In general Bk 4 Bk17 Bk2 x3 A Generalization In general Bk 4 Bk17 Bk2 x3 A Generalization In general Bk4gtBk17Bk27 mk 39 x3 x3 A Generalization In general Bk4gtBk17Bk27 mk 39 x3 x3 A Generalization In general Bk4gtBk17Bk27 mk 39 x3 x3 A Generalization In general Bk4gtBkl73k2 gt BUg Fj x3 x3 x3 A Generalization In general Bk4gtBk1gtBk2 gt EBUHJ39 x3 x3 x3 x3 x32 x3 A Generalization In general Bk4gtBk1gtBk2 gt EBUHJ39 x3 x3 x3 x3 x3 So Bk j 313k Lcwmmrcg 1E8 wwmggg gum Egmlmg Mong Kon Mo August 1 2007 Course Website hitpjwwwmathwashingioneduwmkmoma39im 1 n WW B A sequence is called additive if you can add the same number to get from one term to the next The number that needs to be added to get from one term to the next is called the increment of the sequence Give me an example of an additive sequence What is the recursive formula for this sequence What is the explicit formula for this sequence mm number to get from one term to the next The number that you multiply by to get from one term to the next is called the multiplier Give me an example of a multiplicative sequence What is the recursive formula for this sequence What is the explicit formula for this sequence mm

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