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# Class Note for MATH 250A with Professor Bergevin at UA 3

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Date Created: 02/06/15

MATH 250a Fall Semester 2007 Section 2 J M Cushing Thursday October 25 httprnathariz0naeducushing250ahtrnl Chapter 91 Introduction to Sequences amp Discrete Dynamical Systems Example Cultures of the bacterium Staphylococcus aureus double every 30 minutes Let x number of bacteria at time 7 Measure time in hours 71 01 2 3 6 Measure bacterla 1n umts of one m11110n 10 Example Cultures of the bacterium Staphylococcus aureus double every 30 minutes Let x number of bacteria at time 7 Measure time in hours 71 01 2 3 Measure bacteria in units of one million 106 x H 4x Bacteria numbers quadruple every hour Example Cultures of the bacterium Staphylococcus aureus double every 30 minutes Let x number of bacteria at time 7 Measure time in hours 71 01 2 3 Measure bacteria in units of one million 106 x H 4x x0 61 Initial infection by a X 106 bacteria at hour n O xn 1 2 42 x0 a This is an example of an initial value problem Each initial condition a generates a unique sequence of predicted bacteria numbers called the solution of the initial value problem xn 1 2 42 x0 a This is an example of an initial value problem Each initial condition a generates a unique sequence of predicted bacteria numbers called the solution of the initial value problem x0a x1 4x0 2 4a xn 1 2 42 x0 a This is an example of an initial value problem Each initial condition a generates a unique sequence of predicted bacteria numbers called the solution of the initial value problem x0a x14x04a 2 x2 4x1 4 a xn 1 2 42 x0 a This is an example of an initial value problem Each initial condition a generates a unique sequence of predicted bacteria numbers called the solution of the initial value problem x0a x14x04a 2 x2 4x1 4 a 3 x3 4x2 4 a xn 1 2 42 x0 a This is an example of an initial value problem Each initial condition a generates a unique sequence of predicted bacteria numbers called the solution of the initial value problem x0a x14x04a 2 x2 4x1 4 a 3 x3 4x2 4 a xn 1 2 42 a This is an example of an initial value problem Each initial condition a generates a unique sequence of predicted bacteria numbers called the solution of the initial value problem l l xn 4 a This is called a solution formula These illustrate two basic ways to define numerical sequences These illustrate two sic ways to define numerical sequences recursively These illustrate two basic ways to define numerical sequences recursively M1 f xn xoza These illustrate two basic ways to define umerical sequences recursively or by formulas M1 f xn xoza These illustrate two basic ways to define numerical sequences recursively or by for ulas xnl l fxn an gn x0 a Example Cultures of the bacterium Staphylococcus aureus double every 30 minutes An antibiotic treatment kills 6 X 105 bacteria per milligram instantaneously Example Cultures of the bacterium Staphylococcus aureus double every 30 minutes An antibiotic treatment kills 6 X 105 bacteria per milligram instantaneously A dose of d milligrams is administered every hour to a patient Who had an initial infection of a X 106 bacteria Example Cultures of the bacterium Staphylococcus aureus double every 30 minutes An antibiotic treatment kills 6 X 105 bacteria per milligram instantaneously A dose of d milligrams is administered every hour to a patient Who had an initial infection of a X 106 bacteria What dosage 61 will cure the infection Example Cultures of the bacterium Staphylococcus aureus double every 30 minutes An antibiotic treatment kills 6 X 105 bacteria per milligram instantaneously A dose of d milligrams is administered every hour to a patient Who had an initial infection of a X 106 bacteria What dosage 61 will cure the infection Since we measure 6 in millions each hour the antibiotic treatment kills 6 x 105 d 106 06 d million bacteria Example xmfzkn O d a Example an 4x 06d x0 a Takea1andd1 xn1 4x 06 x021 Mlle xn14xn 06d x0261 Takea1andd1 xn14xn 06 x021 x1 4x0 06 34 Mlle an 4x 06d x0 61 Takea1andd1 xn14xn 06 x0 1 x14x0 0634 x2 4xl 06130 MILE an 4x 06d x0 61 Takea1andd1 xn1 4x 06 x021 x1 4x0 06 34 x2 4x1 06 2130 x3 2 4x2 06514 Example an 4x 06d x0 61 Takea1andd1 xn14xn 06 x0 1 x14x0 0634 x2 4xl 06130 x3 4xZ 06514 x4 2 4x3 06 2050 Example an 4x 06d x0 a Takea1andd1 xn14xn 06 x021 x14x0 06 x2 4xl 06 x3 4xZ 06 x4 24x3 06 Does not appear to cure the infection Example x H 4x 06d Increased dosage x0 a Takea1andd1 Takea1anltld51i xn14xn 06 xn14xn x0 1 x0 1 x14x0 0634 x2 4xl 06130 x3 4xZ 06514 x4 2 4x3 06 2050 Example an 4x 0661 360261 Takea1andd1 Takea1andd51 xn1 4x 06 xn1 4x 306 x021 x021 x14x0 0634 x14x0 306094 x2 4xl 06130 x3 4xZ 06514 x4 2 4x3 06 2050 Example an 4x 0661 360 61 Takea1andd1 Takea1andd51 xn1 4x 06 xn1 4x 306 x0 1 x0 1 x1 4x0 06 34 x1 4x0 306 094 x24x1 06130 x24xl 30607 x3 2 4x2 06 514 x4 2 4x3 06 2050 Example an 4x 06d x0 61 Takea1andd1 Takea1andd51 xn1 4x 06 xn1 4x 306 x0 1 x0 1 x1 4x0 06 34 x1 4x0 306 094 x24x1 06130 x24xl 30607 x3 4xZ 06514 x3 4xZ 306 026 x4 2 4x3 06 2050 Example an 4x 06d x0 61 Takea1andd1 Takea1andd51 xn1 4x 06 xn1 4x 306 x0 1 x0 1 x1 4x0 06 34 x1 4x0 306 094 x24x1 06130 x24xl 30607 x34x2 06514 x34x2 306 x4 2 4x3 06 2050 We interpret this to mean that bacteria numbers dropped to 0 between hour 2 and 3 Example an 4x 06d x0 a Takealanddl Takealandd5l xn14xn 06 xn14xn 306 x0 1 x0 1 Conjecture There is a threshold for the dosage d somewhere between 61 l and d 51 below which the treatment does not cure the infection and above which it does Example xmlen Obd a Conjecture There is a threshold for the dosage 61 below which the treatment does not cure the infection is and above Which it does The threshold depends on the initial infection number a Example an 4x 06d x0 a Mathematically the conjecture is There is a threshold value of 039 below which the solution xn tends to 00 and above which xn tends to oo Example xmlen Ood a Mathematically the conjecture is There is a threshold value of 039 below which the solution xn tends to 00 and above which xn tends to oo We could probably easily prove this conjecture true or false and if true determine the threshold dosage d if we had a solution formula xn gn Example an 4xn 06d x0 61 Turns out it39s true that either the solution 6 either tends to 00 or to 00 as n gt 00 In other words solution sequences diverge Example x lzrxn rgt0 11 Solution formula 6 rnxo Example x lzrxn rgt0 11 Solution formula 6 Zrnxo lim xn does not exist tends to I 00 if r gt 1 n gtoo Example x lzrxn rgt0 11 Solution formula 6 rnxo lim xn does not exist tends to I 00 if r gt 1 n gtoo lim xn0 if rltl n gtoo Example x lzrxn rgt0 11 Solution formula 6 rnxo lim xn does not exist tends to I 00 if r gt 1 n gtoo lim xn0 if rltl n gtoo lim xnzxo if rl n gtoo Example x lzrxn rgt0 11 Solution formula 6 Zrnxo lim xn does not exist tends to I 00 if r gt 1 n gtoo lim x 0 if r n gtoo n lim xnzxo if rl n gtoo We say the sequence diverges Example x lzrxn rgt0 11 Solution formula 6 Zrnxo lim xn does not exist tends to I 00 if r gt 1 n gtoo lim x 0 if r ltl n gtoo n lim x x noo n 0 We say the sequ iverges We say the sequence converges Formal Definition A sequence x converges if there is a number L called the limit of the sequence for which the following is true for any small positive number 6 there is a number N so that xn Llt forall ngtN Formal Definition A sequence x converges if there is a number L called the limit of the sequence for which the following is true for any small positive number 6 there is a number N so that xn Llt forall ngtN The sequence gets and remains as close to L as you want provided you go far enough out the sequence

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