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## Basic Skills in Higher Mathematics

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**Basic Skills in Higher Mathematics**Mathematics 1(H)Outcome 4 Robert GlenAdviser in Mathematics**Mathematics 1(Higher)**Outcome 4 Define and interpret math. models involving recurrence relations. Recurrencerelations**Mathematics 1(Higher)**Outcome 4 Define and interpret math. models involving recurrence relations. Recurrencerelations PC index Click on the PC you want PC(a) Writing recurrence relations PC(b) Limits**Mathematics 1(Higher)**Outcome 4 Define and interpret math. models involving recurrence relations. Writing Recurrencerelations PC(a)**Mathematics 1(Higher)**Outcome 4 Define and interpret math. models involving recurrence relations. PC(a) Define and interpret a recurrence relation in a mathematical model Example 1 A manufacturer claims that Zap! kills 90% of household germs. Even if this is true, 20 000 new germs are produced in a kitchen each day. There are ungerms in the kitchen at the start of one particular day. Write a recurrence relation for un+1, the number of germs in the kitchen at the start of the following day.**Mathematics 1(Higher)**Outcome 4 Define and interpret math. models involving recurrence relations. PC(a) Define and interpret a recurrence relation in a mathematical model Example 1 1 At the beginning of Day n thereare ungerms in the kitchen. If Zap! is used at the beginning of Day n it will kill 90% of the germs. 2 3 10% of the germs will be left. We can write this as 0.1 un . 4 20 000 new germs are produced during the day so the total is now 5 un+1= 20 000 0.1 un +**Mathematics 1(Higher)**Outcome 4 Define and interpret math. models involving recurrence relations. PC(a) Define and interpret a recurrence relation in a mathematical model Example 2 In a pond 3/10 of the tadpoles are eaten by fish each day. Come on in, boys!**Mathematics 1(Higher)**Outcome 4 Define and interpret math. models involving recurrence relations. PC(a) Define and interpret a recurrence relation in a mathematical model Example 2 In a pond 3/10 of the tadpoles are eaten by fish each day. During the night 750 new tadpoles are hatched. There are untadpoles in the pond at the start of one particular day. Write a recurrence relation for un+1, the number of tadpoles in the pond at the start of the following day.**Mathematics 1(Higher)**Outcome 4 Define and interpret math. models involving recurrence relations. PC(a) Define and interpret a recurrence relation in a mathematical model Example 2 1 At the beginning of Day n thereare untadpoles in the pond. During the day 3/10 of the tadpoles are eaten. 2 3 7/10 of the tadpoles will be left. We can write this as 0.7 un . 4 750 new tadpoles are hatched during the night so the total is now 5 un+1= 750 0.7 un +**Mathematics 1(Higher)**Outcome 4 Define and interpret math. models involving recurrence relations. PC(a) Define and interpret a recurrence relation in a mathematical model 125% of the trees in a forest are cut down each month.**Mathematics 1(Higher)**Outcome 4 Define and interpret math. models involving recurrence relations. PC(a) Define and interpret a recurrence relation in a mathematical model 125% of the trees in a forest are cut down each month. Next time you click, the answer will appear here. Answerun+1= 0.75un + 50 50 new trees are planted each month. There are untrees in the forest at the start of one particular month. Write a recurrence relation for un+1, the number of trees in the forest at the start of the following month.**Mathematics 1(Higher)**Outcome 4 Define and interpret math. models involving recurrence relations. PC(a) Define and interpret a recurrence relation in a mathematical model 2A bank loses 5% of its customers each month.**Mathematics 1(Higher)**Outcome 4 Define and interpret math. models involving recurrence relations. PC(a) Define and interpret a recurrence relation in a mathematical model 2A bank loses 5% of its customers each month. Next time you click, the answer will appear here. Answerun+1= 0.95un + 350 It expects to gain 350 new customers each month. There are uncustomers on the bank’s list at the start of one particular month. Write a recurrence relation for un+1, the number of customers on the bank’s list at the start of the following month.**Mathematics 1(Higher)**Outcome 4 Define and interpret math. models involving recurrence relations. PC(a) Define and interpret a recurrence relation in a mathematical model 3During a Martian invasion20% of the aliens die each hour in Earth’s atmosphere. Next time you click, the answer will appear here. Answerun+1= 0.8un + 5000 50 spaceships each containing100 Martians land each hour. There are unMartians on Earth at the start of one particular hour. Write a recurrence relation for un+1, the number of Martians on Earth at the start of the following hour.**Mathematics 1(Higher)**Outcome 4 Define and interpret math. models involving recurrence relations. PC(a) Define and interpret a recurrence relation in a mathematical model 3During a Martian invasion20% of the aliens die each hour in Earth’s atmosphere. Answerun+1= 0.8un + 5000 50 spaceships each containing100 Martians land each hour. There are unMartians on Earth at the start of one particular hour. Write a recurrence relation for un+1, the number of Martians on Earth at the start of the following hour.**Mathematics 1(Higher)**Outcome 4 Define and interpret math. models involving recurrence relations. PC(a) Define and interpret a recurrence relation in a mathematical model 3During a Martian invasion20% of the aliens die each hour in Earth’s atmosphere. 50 spaceships each containing100 Martians land each hour. There are unMartians on Earth at the start of one particular hour. Write a recurrence relation for un+1, the number of Martians on Earth at the start of the following hour.**Mathematics 1(Higher)**Outcome 4 Define and interpret math. models involving recurrence relations. PC(a) Define and interpret a recurrence relation in a mathematical model 3During a Martian invasion15% of the aliens die each hour in Earth’s atmosphere. 50 spaceships each containing100 Martians land each hour. There are unMartians on Earth at the start of one particular hour. Write a recurrence relation for un+1, the number of Martians on Earth at the start of the following hour.**Mathematics 1(Higher)**Outcome 4 Define and interpret math. models involving recurrence relations. PC(a) Define and interpret a recurrence relation in a mathematical model 3During a Martian invasion15% of the aliens die each hour in Earth’s atmosphere. 50 spaceships each containing100 Martians land each hour. There are unMartians on Earth at the start of one particular hour. Write a recurrence relation for un+1, the number of Martians on Earth at the start of the following hour.**Mathematics 1(Higher)**Outcome 4 Define and interpret math. models involving recurrence relations. PC(a) Define and interpret a recurrence relation in a mathematical model 3During a Martian invasion15% of the aliens die each hour in Earth’s atmosphere. 50 spaceships each containing100 Martians land each hour. There are unMartians on Earth at the start of one particular hour. Write a recurrence relation for un+1, the number of Martians on Earth at the start of the following hour.**Mathematics 1(Higher)**Outcome 4 Define and interpret math. models involving recurrence relations. PC(a) Define and interpret a recurrence relation in a mathematical model 3During a Martian invasion15% of the aliens die each hour in Earth’s atmosphere. The end 50 spaceships each containing100 Martians land each hour. There are unMartians on Earth at the start of one particular hour. Write a recurrence relation for un+1, the number of Martians on Earth at the start of the following hour.**The end**50 spaceships each containing100 Martians land each hour.**Mathematics 1(Higher)**Outcome 4 Define and interpret math. models involving recurrence relations. PC(a) Define and interpret a recurrence relation in a mathematical model End of PC(a) Now do the examples on page47 of the Basic Skills booklet Click here for answersto limit questions (PC(b))**Mathematics 1(Higher)**Outcome 4 Define and interpret math. models involving recurrence relations. Limits PC(b)**Mathematics 1(Higher)**Outcome 4 Define and interpret math. models involving recurrence relations. PC(b) Find and interpret the limit of ………………. a recurrence relation Example In a pond 3/10 of the tadpoles are eaten by fish each day. Come on in, boys!**Mathematics 1(Higher)**Outcome 4 Define and interpret math. models involving recurrence relations. PC(b) Find and interpret the limit of ………………. a recurrence relation Example In a pond 3/10 of the tadpoles are eaten by fish each day. During the night 750 new tadpoles are hatched. There are untadpoles in the pond at the start of one particular day. Write a recurrence relation for un+1, the number of tadpoles in the pond at the start of the following day.**Mathematics 1(Higher)**Outcome 4 Define and interpret math. models involving recurrence relations. PC(b) Find and interpret the limit of ………………. a recurrence relation Example 1 1 At the beginning of Day n thereare untadpoles in the pond. During the day 3/10 of the tadpoles are eaten. 2 3 7/10 of the tadpoles will be left. We can write this as 0.7 un . 4 750 new tadpoles are hatched during the night so the total is now 5 un+1= 750 0.7 un +**Mathematics 1(Higher)**Outcome 4 Define and interpret math. models involving recurrence relations. PC(b) Find and interpret the limit of ………………. a recurrence relation What do you notice aboutthese answers as n increases? Example 1 un+1= 0.7 un + 750 Suppose there were 1000 tadpoles in the pond at the start of Day 1 ( ie u1 = 1000). u10 = (0.7 u9) + 750 = 2439. 1450 u2 = (0.7 1000) + 750 = u11 = (0.7 u10) + 750 = u3 = (0.7 1450) + 750 = 2457. 1765 2470. u4 = (0.7 1765) + 750 = u12 = (0.7 u11) + 750 = 1985. u5= (0.7 1985.) + 750 = 2479. u13 = (0.7 u12) + 750 = 2139. u14 = (0.7 u13) + 750 = 2485. 2247. u6= (0.7 2139.) + 750 = 2323. u15 = (0.7 u14) + 750 = u7 = (0.7 2247.) + 750 = 2489. u16 = (0.7 u15) + 750 = u8= (0.7 2323.) + 750 = 2376. 2492. u17 = (0.7 u16) + 750 = u9 = (0.7 2376.) + 750 = 2413. 2495.**Mathematics 1(Higher)**Outcome 4 Define and interpret math. models involving recurrence relations. PC(b) Find and interpret the limit of ………………. a recurrence relation What do you notice aboutthese answers as n increases? Example 1 un+1= 0.7 un + 750 Suppose there were 10000 tadpoles in the pond at the start of Day 1 ( ie u1 = 10000). u10 = (0.7 u9) + 750 = 2802. 7750 u2 = (0.7 10000) + 750 = u11 = (0.7 u10) + 750 = u3 = (0.7 7750) + 750 = 2711. 6175 2648 u4 = (0.7 6175) + 750 = u12 = (0.7 u11) + 750 = 5072. u5= (0.7 5072.) + 750 = 2603. u13 = (0.7 u12) + 750 = 4300 u14 = (0.7 u13) + 750 = 2572. 3760. u6= (0.7 4300.) + 750 = 3382. u15= (0.7 u14) + 750 = u7 = (0.7 3760.) + 750 = 2550. u16= (0.7 u15) + 750 = u8= (0.7 3382.) + 750 = 3117. 2535. u17= (0.7 u16) + 750 = u9 = (0.7 3117.) + 750 = 2932. 2524.**Mathematics 1(Higher)**Outcome 4 Define and interpret math. models involving recurrence relations. PC(b) Find and interpret the limit of ………………. a recurrence relation Example 1 un+1= 0.7 un + 750 Notice that when u1= 1000 the number of tadpoles in the pond seems to be settling at around 2500. When u1= 10000the number of tadpoles in the pond still seems to settle at around 2500. The same would happen with any other starting number. As n increases, the number of tadpoles gets closer and closer to 2500. We say that the limit of the sequence is about 2500.**Mathematics 1(Higher)**Outcome 4 Define and interpret math. models involving recurrence relations. PC(b) Find and interpret the limit of ………………. a recurrence relation Example 1 un+1= 0.7 un + 750 u1= 1000 u1= 10000 un 2500 un We say that the limit of the sequence is about 2500.**Mathematics 1(Higher)**Outcome 4 Define and interpret math. models involving recurrence relations. PC(b) Find and interpret the limit of ………………. a recurrence relation Example 1 We need a way of finding limits without doing all this calculation. un+1= 0.7 un + 750 Call the limit of this sequence L. Suppose n is a very large number un L un+1 L and Since un+1= 0.7 un + 750 This means that the number of tadpoles in the pond will settle at 2500. L = 0.7 L + 750 0.3 L = 750 L = = 2500**Mathematics 1(Higher)**Outcome 4 Define and interpret math. models involving recurrence relations. PC(b) Find and interpret the limit of ………………. a recurrence relation We need a formula for anyrecurrence relation like this Example 1 un+1= aun+ b un+1= 0.7 un + 750 Call the limit of this sequence L. Suppose n is a very large number un L un+1 L and Since un+1= aun + b NB Limits only occur when -1 a 1 L = aL + b (1 - a) L = b L =**Mathematics 1(Higher)**Outcome 4 Define and interpret math. models involving recurrence relations. PC(b) Find and interpret the limit of ………………. a recurrence relation For each of these recurrence relations say how you know a limit exists, then calculate the limit. In all cases a limit exists because a < 1 a Answers 1 L= 2500 1un+1 = 0.8 un+ 500 2un+1 = 0.2 un+ 1000 2L= 1250 3un+1 = 0.1 un+ 750 3 L= 2500 4un+1 = 0.75 un+ 3000 4 L= 12000 5un+1 = 0.85 un+ 300 5 L= 2000 6un+1 = 0.25 un+ 150 6 L= 200**Mathematics 1(Higher)**Outcome 4 Define and interpret math. models involving recurrence relations. PC(b) Find and interpret the limit of ………………. a recurrence relation Find the limit of each recurrence relation youfound in PC(a). Say what each limit means in the context of the question. Click here to find the questions.**Mathematics 1(Higher)**Outcome 4 Define and interpret math. models involving recurrence relations. PC(b) Find and interpret the limit of ………………. a recurrence relation Answers to limit questions 1L = 200 The number of trees will settle at 200. 2L = 7000 The number of customers will settle at 7000. 3 L = 200 The number of Martians will settle at 25000.**Mathematics 1(Higher)**Outcome 4 Define and interpret math. models involving recurrence relations. PC(a) Define and interpret a recurrence relation in a mathematical model End of PC(b) Now do the examples on page51 of the Basic Skills booklet