Engineering Mathematics I (Michalemas term 19/20)

Index

Jump to:

• Organization
• Assesment
  • Projects
  • Tutorials assessment
  • Written Exam
• Help with math questions
• Textbook
• List of lectures
  • Recap
• Tutorials
• Ideas for projects
• Project Evaluation
• Project Groups
• Model Exams

Organization

The module runs for the first half (12 weeks) of the academic year and comprises of three lectures and one tutorial per week (total of 44 hours contact time).

Lectures take place:

• Mon. 10:00 @McNeil
• Tue. 9:00 @McNeil
• Thu. 11:00 @McNeil

Assessment

Weekly continuous assessment together with a team project contributes 20% towards the final grade with the end-of-year final written two-hour examination contributing 80%.

Projects

The final year group project is mandatory in order to have any grade in the continuous assessment (i.e without a project, one gets a 0 in CA no matter the tutorial grades).

The project should be about something that goes beyond what has been covered in class (just explaining something that has already been covered is not enough). Here you have some ideas. When you decide the title of your project make sure to let me know (to see if this is appropriate) by email. You should also attach a list of the members of the group.

Obviously you are encouraged to ask me any doubts or help with your project. I will try to provide some guidelines if you are interested in something in particular.

Tutorials assessment

Each week exercises will be available in this web page. You have to solve the exercises and hand them to your tutor before the tutorial starts. During the tutorials you are encouraged to ask any questions about the difficulties you found while doing the exercises, or about any other material covered in class.

The names of the tutors are:

• Dovydas Mickus MICKUSD@tcd.ie: Wed. 9am and 3pm @M21
• Marlon Brenes Navarro brenesnm@tcd.ie: Wed. and Fri. at 9am @M20
• Michael McDermott mcdermm6@tcd.ie: Fri. at 10am @TBSI (B2.21)

Written exam

The final year written exam consists on 4-5 exercises that have to be solved without any calculator/books/tables.

Help with math questions

Of course, every student is encouraged to ask any question during or after the or directly in my office in the Lloyd building 2.20.

Textbook

• Main text for the course: Calculus, by Howard Anton, Irl Bivens, Stephen Davis.

There are several copies in the Hamilton library. There exist different versions and editions of the same book or parts of it, with different subtitles. Some of them only contain the first 8 or 9 chapters and are subtitled “Single Variable”. This is sufficient for the course I am teaching, but some material needed for the second semester (1E2) is missing. If you intend to buy the book probably is better that you make sure to buy an edition that covers both 1E1 and 1E2. There is also the possibility to buy an on-line version.

• Additional interesting reference books:
  • Calculus - An Intuitive and Physical Approach, by Morris Kline.
  • Calculus, by M. Spivak.
  • Mathematics Its Content, Methods, and Meaning, by M. A. Lavrent’ev, A. D. Aleksandrov, A. N. Kolmogorov.
  • The cartoon guide to Calculus, by Larry Gonick.

Lectures

A brief overview of what has been covered in each class. In some cases I will provide some notes. This is just some handouts that I use to prepare the class. They are never a substitute to study a book. When no notes are provided I will write the chapters of the book that I have used.

• Week 1 Some notes

  • 9.09.2019 (chapter 0.1): Practicalities. (Natural) domain of a function.
  • 10.09.2019 (chapter 0.1/0.2): More on domains. Operations with functions.
  • 12.09.2019 Transformations of functions as compositions.

• Week 2 Some notes

  • 16.09.2019 (chapter 0.3): Elementary functions I: polynomials.
  • 17.09.2019 (chapter 0.3): Elementary functions II: trigonometric. functions.
  • 19.09.2017: (chapter 6.1/0.4): Elementary functions III: Exponential and log. Summary of basic properties of functions.

• Week 3 Some notes

  • 23.09.2019: Intuitive notion of limit.
  • 24.09.2019: Rigorous definition of limit.
  • 26.09.2019: Computation of Limits.

• Week 4 Some notes

  • 30.09.2019: One sided limits and continuity.
  • 01.10.2019: Continuity and limits at infinity.
  • 03.10.2019: Sequences.

• Week 5 notes

  • 7.10.2019: Limit definition of derivative.
  • 8.10.2019: Computation of derivatives.
  • 10.10.2019: More derivatives and implicit diferentiations.

• Week 6

  • 14.10.2019: (chapter 2.7, 2.8): Implicit differentiation, linear approximation of functions.
  • 15.10.2019: (chapter 3.1): Finding maxima or minima of functions.
  • 17.10.2019: (chapter 3.5): Maximum/minimum problems.

• Week 7

  • Study week

• Week 8

  • 28.10.2019: Bank Holiday.
  • 29.10.2019: Newton method, L’hopital.
  • 31.10.2019: Rational functions and asymptotes.

• Week 9

  • 4.11.2019: (chapter 3.8): Rolle’s theorem, Mean value theorem.
  • 5.11.2019: (chapter 4.1): The area problem.
  • 7.11.2019: (This is better explained in M. Spivak, chapter 13): Upper and lower sums. Definition of definite integral.

• Week 10

  • 12.11.2019: (chapter 4.4): Integrals and areas.
  • 13.11.2019: (chapter 4.2): Techniques of integration I.
  • 15.11.2019: (chapter 4.3): Techniques of integration II.

Recap

Small sumaries or nice reads related to the most important topics of the course.

• Who gave you the Epsilons?, by Judith V. Grabiner in The American Mathematical Monthly: Nice interesting read about the rigorous definition of limit and continuity.
• Limits: Download PDF.
• Integrals and areas: Download PDF

Tutorials

• Week 1: There are no tutorials the first week. Tutorials starts week 2, so the week of 16th September.
• Week 2: Functions, domain and range of a function. Operations with functions. Problem set. Solutions.
• Week 3: Elementary functions. Inverse functions. Problem set. Solutions.
• Week 4: Limits. Problem set. Solutions.
• Week 5: More limits, continuity, sequences. Problem set. Solutions.
• Week 6: Derivatives. Problem set. Solutions.
• Week 8: Minimum principles. Problem set. Solutions.
• Week 9: Applications of derivatives Problem set. Solutions.
• Week 10: Mean value theorem. Areas and partitions. Problem set. Solutions.
• Week 11: Techniques of integration. Problem set.

Ideas for projects

Here are some ideas for the projects

Some ideas for projects

• The Kepler laws: What do they say? How do they derive from the laws of motion and the law of gravity?. Guidelines.
• The fundamental theorem of algebra: A polynomial of degree n has exactly n complex roots (if one counts the multiplicity). Guidelines.
• Galois theory: There is no explicit formula for the roots of a polynomial of degree 5 or higher!. Guidelines
• Linear regression: Using polynomials to estimate/model the relationship between variables. Guidelines
• Fourier analysis: Representing a function as a sum of trigonometric functions. Guidelines
• The hyperreal line: Extending the real line to fit infinite(simals). Guidelines
• Fixed point theorem’s: Brouwer fixed-point theorem, Banach fixed-point theorem, and its amazing consequences. Guidelines
• Propagation of errors: Using linear approximations of functions to “propagate” uncertainties. Guidelines

Project Evaluation

Information will follow

Project Groups

• Group 100 (Josephus Problem): Ian O’Flynn, James Meaney and Emer Muldoon.

• Group 105 (Squaring the circle): Fergal Riordan, Matthew Pyper.

• Group 110 (Category theory): Adriano Oliveri Orioles, Alì Saim, Siobhán Bannister, Mia Brzakovic, Chi Chun Ko, Rogelio Ramel.

• Group 115 (Complex Numbers): Erin Croniger, Martin Gutierrez, Aaron Fay, Alex Denby.

• Group 120 (Complex numbers): Jennifer Banayo, Nicoleta Marcu, Isabella Irving.

• Group 125 (A Hierarchy of Infinite Cardinals): Seán Higginbotham.

• Group 130 (Nash Equilibrium): Berndett Finna, Molly Hand, Caitriona McNamara.

• Group 135 (Fourier Analysis and its Applications): Leon Keating, Luca Paolozzi, Patrick Goodwin.

• Group 140 (Hexagons in Nature): Jayne Holton, Amanda Norris, Emma McLoghlin, Oscar Whelan.

• Group 243 (4 colour theorem): Matthew Corcoran, Eamonn O’ Broin.

• Group 150 (MNaxwell equations): Pranav Beeputh, Ritchie Sun, Shang Xia, Caelan Kelleher.

• Group 155 (Narcissistic Numbers): Aleksandra Zyrun, Weiyun Yuan, Aimée Ward, Stacey Torres.

• Group 160 (The squared square puzzle): Ben O’Sullivan, Fergus O’Brien, Danny Nolan, Johnathan Landy, Conor Neill.

• Group 165 (The riemann hypothesis): Conor Powderly, Finn Reid, Peter Power, Reuben Strunz, Aidan Marrinan.

• Group 170 (The Mandelbrot set): William Dempsey, Fionn Moran, Michael Cummins, Leon Campbell.

• Group 175 (The mobius strip): Emma McCabe, Jessica Okoye, Nurul Adawiyah Mohamed Ismail, Pik Shun (Jessica) Wong.

• Group 185 (Transcendental Numbers): Eimear Kennedy, Laoise Brady, Rían Spillane, Manasvi Ghanta, Jack Loscher and Sarah McCarthy.

• Group 195 (Complex Numbers, Quaternions and Rotations): Anna Schwer, Ciara Morgan, Maya O’Riordan, Alex Reid, Niamh Shanahan.

• Group 200 (Golden ratio): Alfie Hales, Sean Duffy, Tom Hawkins and Denny Moriarty.

• Group 210 (Fibonacci sequence and the golden ratio): Salman Alsaffar, Labib Alwasi, Erik Ryan, Jyothis Jenu, Brian Burke.

• Group 220 (The Borsuk-Ulam Theorem): Jamie Columb, Mark O’Connor, Angus Cheng, Sean Cooney.

• Group 230 (Methods of finding exoplanets): Tarek Eid, …

• Group 240 (Galois Theory): Alia Binismail, Maël Bonnec, Alp Gurpinar.

• Group 250 (Fundamental theorem of arithmetic): Stephen Briggs, Sean Banks, Robert Murphy, Daire o neill, Adam Kielty.

• Group 260 (Dzhanibekov Effect): Rhys Farrington, Rory Staunton, Ronan Slattery, Andy Rohu.

• Group 270 (OpenPGP and TLS encryption): Peter O’Flynn, Liam Hart, Johnny Kennedy, Pranav Manish Vora, John Grimes.

• Group 280 (Lorentz transformations): Donnchadh Griffin Carroll, Jack Twomey, Christopher Dehaene.

• Group 290 (Navier – Stokes flow equations): Jan Szkaradek, Toni Mockler, James Kelly, Mihai Mesteru.

• Group 300 (Fractals): Matthew Kaye Mellor

• Group 310 (Kepler laws): Louise Cafferty, Heather Mitchell.

• Group 320 (Kepler laws): Caroline Kelly, Sean Dowling, Isabelle Bryans, Katherine Bolger and Olwyn Hughes.

• Group 330 (Georg Cantor’s Theorems/Paradox): Sean Higginbotham, …

• Group 340 (Eulers Identity): Isabelle McGrath, Leah Paul, Aoife Roche, Emma Patterson, Sophie Weldon.

• Group 350 (Aeronautical mathematics): Daragh Scanlan, Simon Delany Calleja.

• Group 360 (Euler’s identity): David O’Leary, Cillian Smith, Olivia Strong, Faye Prendergast and Hedda Tokerud Jerve.

• Group 370 (Quaternions): Ruairi Grant, Caoimhe McManus, Stephen O’Sullivan, Nora Muller and Tom Aherne.

• Group 375 (The Mandelbrot set): Ross O Neill, Daniel Drummond.

• Group 385 (Boolean Algebra): Sean Thomas Sheridan, Louis Mulville and Cian Ó Feinneadha.

• Group 395 (Transformations of Inverse Functions): Stanislav Miladinov.

• Group 405 (Mercury: Earth’s closest neighbour?): Alex Kennedy, Anna Woodcock, Louise Caslin, Charlie Molony.

• *Group 415 (Lorentz transformations): Andrew Coyle, Adam Khalaf, Alex Doyle and Stephen Lowe.

• Group 425 (Fractals): Finn O’sullivan, Hannah Kerr, Miles Dixon, Matthew Kaye Mellor.

• Group 435 (Kepler laws and their Applications): Colm Deeny, Hugh Byrne

• Group 445 (Kepler laws): Andrew Daly, Martin Hutz.

• Group 455 (Kepler laws): Alan Keogan, Claire Dempsey, Eimear Conway, Michael McEvoy, Emily Curran.

• Group 465 (The Toeplitz conjecture): Joe Brown, Una Cotter, Anna Nolan, Scott Atkins and Caelan McEvoy.

• Group 475 (Complex Numbers): Isobel Forde, Jacqueline McCarrick, Emily woods, Ava cudmore and Moya Whelan

• Group 485 (Chinese Postman Problem): Alice Debert and Sarah Mateu.

• Group 495 (The Number Theory and the History of Numbers): Fionn Donnelly, Jane McHugh, Hannah Burgess, Matthew Campbell, David Porter.

• Group 505 (Distortion of time by gravity): Aaron Lawlor, Brian Lee, Cian Lawlor.

• Group 515 (Hyper real numbers): Ella Campbell, Louise Kennedy, Niamh Crilly, Kate Davitt, Moya Whelan, Ciaran Healy.

• Group 520 (The Mathematics of Cryptography): Euan Selkirk, Dylan O Maoldomhnaigh, Eoin Redmond, Luke O Hagan, Cian O Criochain.

• Group 525 (Fourier Analysis): Tadhg-Lorcan Oude Essink, Shane Murphy, Scott Yamada-Brennan, Conn Murphy, Levente Paksy.

• Group 530 (Prisoner’s Dilemma): Fiyin Adeniranye, Zachary Barte, Oisín de Barra, Moyo Mobolaji, Emmanuel Ojelabi.

• Group 535 (): Thomas Hartigan, Alex Kennedy, Alastair Philip.

• Group 540 (Kepler laws): Ryan O’Neill, Oscar Langan, Conor Ryan, Jack Delaney, Tim Farrelley.

• Group 545 (Kepler laws): Katherine Hughes, Paraic O’Reilly, Tadhg McNevin.

• Group 560 (The Taylor Series): Nikita gaydenko, Mikhail Vaganov, Levon Sandukhchyan, Mark Shteingardt.

• Group 565 (Kepler laws): Ben Horkan, David Gallagher.

• Group 570 (Rolling of Polygons): Eddy Motrea, Fionn O’Connor, and Mihailo Manojlovic.

• Group 575 (Kepler laws): Jim Hickey, Alex Nash.

• Group 580 (Gini Index): David Murphy, Max Lynch, Tom Fitzgerald, Toby Hudson-Fowler.

• Group 585 (Kepler Laws): Will O’Callaghan, Sean McKenna, David Yau, Ciaran McKay, Luke Gilmartin, Hugo Crighton.

• Group 590 (Fractals and the coastline rule): Ross Brady, Andrew Butler, Rían Cathcart.

• Group 595 (P vs. NP): Maeve Sheehan Galvão, Jack O’Sullivan, Lucy O’Sullivan, Siddarth Sethuraman.

• Group 600 (Julia sets): Sebastien Dunne Fulmer, Pedro Quaresma, Oscar Toomey.

• Group 605 (The fundamental Theorem of Algebra): Garbhan McCormack, Peter Coyle, John Coreless, Hugo Moran and Jack Cleary.

• Group 610 (Propagation of Uncertainty): Fiona Brogan, Ciaran Lynch and Thomas Watson

Model exams

An example of an exam. Solution