1. CIVE2602 - Engineering Mathematics 2.2 (20 credits) Series Limits Partial Differentiation Vector Algebra Numerical methods Statistics and Probability Limits, Sequences and Partial differentiation Lecturer: Dr Duncan Borman Room 3.05 Civil Eng 3 Example classes: 13 th Oct, 20 th Oct, 10 th Nov Assessment Formative Assessment Worksheet +2 Problem Activities (problem sheets + weekly online task 10% of module mark) Exam (June) Full details available on the VLE Module taught over 2 semesters by: Dr A Sleigh Dr D Stewart Dr D Borman

2. CIVE2602 - Engineering Mathematics 2.2 Lecturer: Dr Duncan Borman Overview of topic Introduction to sequences Sequences-Recurrence relationship Convergence of a sequence Assessed Online Tasks: MathXL Lecture 1 Limits, Sequences and Partial differentiation

3. CIVE2602 - Engineering Mathematics 2.2 Why do Civil Engineers need to do maths? Understanding principles in other subjects (Fluid Dynamics, Structures, Mechanics, Architecture.. etc) Underpins lots of concepts Modelling systems and structures Limits, Sequences and Partial differentiation

4. Essential for modelling Earthquakes Modelling dam performance/failures Stress analysis of railway bridge ©Alton.art 2008, sourced from http://en.wikipedia.org/wiki/File:101.portrait.altonthompson.jpg Available under creative commons license

5. Stadiums- FEA and CFD analysis Arsenal stadium (CFD) New Wembley Chinese stadiums ©www.landscapesofengland.co.uk 2010, sourced fromwww.landscapesofengland.co.uk http://www.flickr.com/photos/bbmexplorer/5055894363/ Available under the creative commons license ©Kurt Smith, 2008, sourced from http://www.flickr.com/photos/81195835@N00/3005592094 Available under creative commons license © A. Aruninta 2008, sourced from http://commons.wikimedia.org/wiki/File:Olympic2008_watercube02_night.jpg?uselang=en-gb Available under creative commons license

6. Sequences and series Why are we interested in them? Sequences turn up in many engineering applications: e.g. Sampling a wave form – computer monitoring bridge oscillations over time e.g. When finding approximate solutions to equations that model physical phenomena (needed when using a computer) e.g. describing discrete time signals – calculations on a computer are done at fixed intervals of time governed by clock Series come about when we need to sum a sequence together

7. CIVE2602 - Engineering Mathematics 2.2 Sequences {x 1, x 2, x 3, x 4,.... } or {x 0, x 1, x 2, x 3,.... } Examples 1. a) {0, 1, 2, 3,..., n,... } Example 1. b) {0, 1, 4, 9, 16... n 2,... } n = 0 1 2 3 n n = 0 1 2 3 4 n

8. Sequences Example 1. c) {1, 2, 4, 8, 16,... 2 n,... } x 0 =2 0 =1, x 1 =2 1 =2, x 2 =2 2 =4, x 3 =2 3 =8, x 4 =2 4 =16, x 5 =2 5 =32,...... n = 0 1 2 3 4 n

9. Quick questions (2 mins) Q2 {-1, 0, 3, 8, 15..... } Q3 {0, 2, 6, 12, 20...... } Find Q4 {7, 8, 15, 34, 71...... } Q1 {-3, 0, 3, 6, 9..... }

10. Recurrence Relationship -relation between successive terms in the sequence Recurrence relation - equation that defines a sequence recursively (each term found from one before) {x 0, x 1, x 2,... x n, x n+1,...} Example x 0 = 1 x n+1 = ½ x n Recurrence relation n=0 : n=1 : n=2 : Notice that as n →∞, x n →0 n=3 : x 4 =½ x 3 = x 3 =½ x 2 = ½ x ¼ =

11. Recurrence Relationship – try this one 2mins Example x 0 = 1 x n+1 = n=0 : n=1 : n=2 : Recurrence relation n=3 : Notice that as n → ∞, x n → ∞ x 1 =3 x 0 -1 = 2 x 2 =3 x 1 -1= 6-1 = 5 x 3 =3 x 2 -1=15-1 = 14 x 4 =3 x 3 -1= 42-1= 41 Etc. What does x n approach as n→ ∞ ?

12. N.B. When we calculate more terms (e.g. x 4, x 5... etc.), we note that as n increases, successive terms x n tend to 1.41421… = In this case the sequence { x n } is said to be convergent and the limit of the sequence is Recurrence Relationship Example x 0 = 1 x n+1 = n=0 : n=1 : n=2 : Recurrence relation n=3 : 1.5 1.4166 1.4142

13. Recurrence Relationship Example (continued) x n+1 = How can we find what a Recurrence Relationship Converges to ? (without doing all the sums)

14. Recurrence Relationship Example x 0 = 1 x n+1 = n=0 : n=1 : n=2 : Recurrence relation n=3 : n=xnxn 01 1-2 210 35.2 44.46 54.21 64.10 74.049 84.024 4.46 4.21. 4.00. n n=4 : 10 5.2 -2

15. Example (continued) Algebra method -find what a Recurrence Relationship Converges x 0 = 1 x n+1 = Mult by x or Rearrange Factorise Why did we only find convergence towards x=4?

16. Recurrence relation

17. Question (h:h) Use Algebra method to find if this Recurrence Relationship Converges x 0 = 1 x n+1 = Mult by x+1 or Rearrange Factorise Converges towards x=-2 or x=4 Multiply out x 2 = 3+0.769 =3.769 x 3 = 3+1.048 =4.1048 x 4 = 3+0.979 =3.979 x 5 = 3+1.005 =4.005 x 1 = 3+2.5 =5.5

19. Recurrence Relationship

21. Multiple choice Choose A,B,C or D for each of these: Differentiate the following wrt x : 1) B D AC

22. Multiple choice Choose A,B,C or D for each of these: Differentiate the following: 2) A B C D

23. Multiple choice Choose A,B,C or D for each of these: Differentiate the following: 3) A B C D

24. Multiple choice Choose A,B,C or D for each of these: Differentiate the following wrt x : 4) A B C D

25. Multiple choice Choose A,B,C or D for each of these: Differentiate the following: 5) A B C D

26. Multiple choice Choose A,B,C or D for each of these: Differentiate the following: 1min to work out 6) A B C D

27. Convergence of a Sequence Example 1 As If the sequence { x n } does not have a limit as we say the sequence is divergent x n = 0 Therefore sequence is convergent n x n

28. Convergence of a Sequence Example 2 i.e. {0, 1, 4, 9, 16... } Divergent Example 3 i.e. {1, -1, 1, -1, 1, -1... } or -1, i.e. no unique limit Divergent Example 4 i.e. Convergent

29. Why should we care if things Converge? © vattenfall 2008, sourced from http://www.flickr.com/photos/vattenfall/5019768387/ Available under creative commons license

30. What does this sequence converge to as n→ ∞ ? Example 1 Show limit is n=1: e.g. = = 1.4996 = 1.3333 = n=2: n=3: n=4: = 1.3333 = = 1.361 =

31. Alternative method – better Example 1 Show limit is = = both will approach zero as xn xn = We want to find what limit is as n approaches infinte Example 3 Example 2

32. Convergence of a sequence Example 2 Example 3 xn xn =

33. Please can these 15 people see me after the lecture

34. Assessment VERY IMPORTANT! weekly online task (mathlab) -you will get an e-mail this week with your login id Problem sheets Exam Other resourses -VLE https://vlebb.leeds.ac.uk https://vlebb.leeds.ac.uk -Mathlab (other login information on Thursday)

35. CIVE2602 - Engineering Mathematics 2.2 Lecture 1- Summary Sequences {1, 3, 5, 7, …,n} Recurrence Relationships Convergence of a Sequence Room: 1.42 SPEME

36. Please can these 15 people see me after the lecture

37. Convergence of a Sequence A fixed pendulum has 2 stable positions - Stopped at the bottom and Stopped at the top. (when θ =0 or 180, sin θ =0 so F = 0 ->no resultant force) Iterative techniques to find these solutions would nearly always find the one at the bottom as unless the top is chosen as first guess. F = - mg sinθ ©Fraser on Flickr 2008, sourced from http://www.flickr.com/photos/fraseronflickr/2565601087/ Available on creative commons license ©Duhon 2010, sourced from http://commons.wikimedia.org/wiki/File:60_ Jahre_Fr%C3%BChjahrsvolksfest_Deutz,_Eclipse_2.jpg?uselang=en-gb Available on creative commons license