Engineering Mathematics I (MA1E01) **NOT CURRENT ACADEMIC YEAR**
Index
Jump to:
- Organization
- Assesment
- Help with math questions
- Textbook
- List of lectures
- Tutorials
- Projects
- Model Exam
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 Mondays, Tuesdays and Thursdays at 11am in Hamilton building, theather 3 (McN3).
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%.
Tutorials (exercise classes contributing to the final mark) will start in the second week of teaching. Group lists for tutorials will be available in due course.
Please make sure to hand your exercises before the tutorial starts.
Help with math questions
Of course, every student is encouraged to ask any question about the material covered in my lectures either after the lectures or directly in my office in the Lloyd building 2.20.
The school of maths also run a maths helproom. Everyday between 1pm and 2pm you can find people in the new seminar room of the school of maths (off the Hamilton Building) that will help you with your questions about math (not to do your homework).
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.
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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.
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Week 1
- 25.09.2017 (chapter 0.1): Practicalities. Functions (graphs, tables of values, algebraic expression). (Natural) domain of a function.
- 26.09.2017 (chapter 0.1, 0.2): Intervals, natural domain and range of a function. Operations with functions: sum, difference, product, division. How domains change under operations.
- 28.09.2017 (chapter 0.2, 0.3): Composition of functions. Translation, reflection and strectches/compressions as compositions. Polynomials.
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Week 2
- 02.10.2017 (chapter 0.3, 0.4): Measuring angles and trigonometric functions. Inverse functions.
- 03.10.2017 (chapter 1.1): Intuitive idea of limit.
- 04.10.2017 (chapter 1.2, 1.4): Rigorous approach to limits. Computing limits.
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Week 3
- 09.10.2017 (chapters 1.3, 1.5): Infinite limits and limits at infinity. Continuity and Bolzano’s theorem.
- 10.10.2017 (chapters 2.1, 2.2): Rate of change. Definition of derivative.
- 12.10.2017 (chapters 2.5, 2.6): Techniques of diferenciation (I).
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Week 4
- 16.10.2017: Ophelia!
- 17.10.2017 (chapter 2.6): Techniques of differentiation (II).
- 19.10.2017 (chapter 2.7): Mostly implicit diferentiation.
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Week 5
- 23.10.2017: (chapter 2.7, 2.8): Implicit differentiation, linear approximation of functions.
- 24.10.2017: (chapter 3.1): Finding maxima or minima of functions.
- 26.10.2017: (chapter 3.2,3.4): More maxima and minima of functions.
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Week 6
- 30.10.2017: Bank holiday.
- 31.10.2017: (chapter 3.3): Rational functions and asymptotes.
- 02.11.2017: (chapter 3.5): Maximum/minimum problems.
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Week 7
- Study week
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Week 8
- 13.11.2017: (chapter 3.8): Rolle’s theorem, Mean value theorem.
- 14.11.2017: (chapter 6.1): Exponential functions and the number \( e \).
- 16.11.2017: (chapter 6.1, 6.1): Exponentials and logarithms.
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Week 9
- 20.11.2017: (chapter 4.1): The area problem.
- 21.11.2017: (This is better explained in M. Spivak, chapter 13): Upper and lower sums. Definition of definite integral.
- 23.11.2017: (chapter 4.6): The fundamental theorem of calculus.
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Week 10
- 27.11.2017: (chapter 4.2, 4.5): Properties of definite integrals
- 28.11.2017: (chapter 4.2): Techniques of integration I.
- 30.11.2017: (chapter 4.3): Techniques of integration II.
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Week 11
- 04.12.2017: (chapter 5.1): Applications of integrals I: Area between curves.
- 05.12.2017: (chapter 5.2): Applications of integrals II: Solids of revolutions, volumes by slicing.
Recap
- Limits: Download PDF.
- Minimum principles: Download PDF.
Tutorials
- Week 1: There are no tutorials the first week. Tutorials starts week 2, so either 4th or 6th of October, depending on your group.
- Week 2: Functions, domain and range of a function. Operations with functions. Polynomials. Problem set.
- Week 3: Inverse functions. Computing limits. Problem set.
- Week 4: Derivatives. Problem set.
- Week 5: Derivatives. Implicit functions. Problem set.
- Week 6: Linear approximations of functions. Maxima and minima. Problem set.
- Week 7: Study week.
- Week 8: Asymptotes. Maximization/minimization problems. Problem set.
- Week 9: Mean value theorem. Problem set.
- Week 10: Integrals and areas Problem set.
- Week 11: Techniques of integration Problem set.
- Week 12: Applications of integrals Problem set.
Projects
Rules
- 14.12.2017, at 15:00 (very sharp), in Goldsmith Hall.
- Material: Bring your poster/slides/whatever printed, and tape to put it on the wall.
- e-mail me the material (<alberto.ramos_at_maths.tcd.ie>) BEFORE 14.12. Add in the subject your poster number and in the body of the mail the Names/IDs of all members of the group.
- Take a slook at this
Group list
- Group 10 (The Kepler laws): Eimear, Seán, Sadhbh, Aliya, Megan, Emily.
- Group 15 (The Kepler laws): Karen Morrissey, Ruth Mullen, Aimee Goodwin, Ellen Rose Tighe and Caoimhe O’Hare.
- Group 20 (The Kepler laws): Holly Leech, Orla Keenan, Laura Murphy and Anna Eniko Illesi.
- Group 25 (The Kepler laws): Sinead quinn, Eimhear Hughes and Una O’Neill.
- Group 30 (The fundamental theorem of algebra): Elizabeth Bolger, Laura Stack, Evan Magee, Cian Rellis, Garry Tiscovschi and Emily Broderick.
- Group 35 (The fundamental theorem of algebra): Dylan Hunt, Úna Raeside, Ellie O’Sullivan and Finn Voorhees.
- Group 40 (Galois theory): Leo Devlin, Matthew Conway, and Joseph O’Connell.
- Group 45 (Galois theory): Johnny Scalon, Conor O’Donnell, Conor Maher and Jack Byrne
- Group 50 (Galois theory): David Carroll, Michael Diamond and Sam Butler.
- Group 55 (Linear regression): Jack O’Donoghue, Kevin Digan, Cian Kellet, Rebecca Sutherland, Kate O’Connell, Aoife Hurd and Lisa Callagh
- Group 60 (Linear regression): Leo Cooke and Cormac Doyle.
- Group 65 (Linear regression): James Sutton, Julie Gallagher, Orla Kearney and Leontina Schelling.
- Group 70 (Fourier analysis): Gearoid Farrell, Conor Bryce, Keshav Sapkota, Charlie Maguire, David Dunne, Conrad Vandlik.
- Group 75 (Fourier analysis): Ronan McAlister, Nicholas Cody, Nollaig Butler and James Toland.
- Group 80 (Fourier analysis): Bella Salajeva, Jakub Kluza, Cian Skelly, Bríd O’Donnell, Áine Deasley and Muireann Farrell.
- Group 85 (The hyperreal line): Rebecca Katherine Armstrong, Galina Kennedy, Ella Keough, Carmen Galante and Abieyuwa Enehikhare Obasuyi.
- Group 90 (The hyperreal line): James Nangle, Conor O’Farrelly, Conor Corry, Diarmuid Coffey, Rory Clarke, Oscar Crowley.
- Group 95 (The hyperreal line): Alistair Tidey, Grant Arnott, Ruairi o Connor, Patrick junghenn, Oran keeling, Mathew lynch.
- Group 100 (Fixed point theorem’s): Anneliese Walsh, Ciara Coyne, Claudia Alonso Vicedo, Elvira Guiomard, Katie Henn, Méabh Childs.
- Group 105 (Fixed point theorem’s): Kate Fagan, Louise Fagan, Emma Grimes and Wajeeha Shahbaz.
- Group 110 (Fixed point theorem’s): Seán Byrne, Jack Hartnett, Conor Kirwan, Conor Dunne and Islam Alagtal.
- Group 115 (Propagation of errors): Charlotte Brehm, Aaron Heery, Fernando Valpuesta, Jean Luc Aho, Eoin Burke and Mohanad M. Jawadah.
- Group 120 (Propagation of errors): Aislinn Dunleavy, Ben Broderick, Isobel Duffy and Denis McCambridge.
- Group 125 (Calculus of variations): Alexis Flaherty, Miriam Kelly, Gabriela Panek, and Jessica Bagnall.
- Group 130 (Calculus of variations): Wei Ying Chang, Kiera Cullen, Victoria Watterson, Emma Gallagher, Aisling Cashell, Rachel O’Grady.
- Group 135 (Hamilton’s least action principle): Aaron Hannon, Conor O’hEocha.
- Group 140 (Hamilton’s least action principle): Laoise Kinsella, Nasir Said, Rayyan Mehunood, Cormac Dunne.
- Group 145 (Hamilton’s least action principle): Erik Slattery, Rahul Seth, Kyle Mendez, Oisin Callan and Erika Galligan.
- Group 150 (Complex numbers, quaternions and rotations): Alex Pocock, Alex mcGinn, John Bennet and Oisin Dougan.
- Group 155 (Complex numbers, quaternions and rotations): Naoise Barry, Jakub Pyszka, Mellisa Dumitrache, Carolín Laoide-Kemp and Euan Reid.
- Group 160 (Complex numbers, quaternions and rotations): Calum McGrath, Luke Corcoran, David Berry, Paddy Bryne, Niall Lee, Daniel DiMasio.
- Group 163 (Complex numbers, quaternions and rotations): Tom Sheehy, Alison McDonnell, Eoin Murray and me Mark Hanrahan.
- Group 165 (Trascendental numbers): Jessica Ojomor and Jeffrey Dargan.
- Group 170 (Trascendental numbers): Jack Goodwin and David Mckenna.
- Group 175 (Trascendental numbers): Mabroor Ahmed, Andrew Downey, Ahad Khalil and Jack Lowe
- Group 180 (Newton and other Fractals): Laura Owens, Orlaith Murphy, Kelly Augustt, Anuk Mornoey and Erica Markey.
- Group 185 (Newton and other Fractals): Mark Kelly, Uchenna Udemezue, Ankit Vijay, Francesco Digeonimo.
- Group 190 (The Golden Ratio): Tom Mulligan, Cian Murphy, Jack O’Leary, Ian Black, Conor Dalton, Aodhán Buggy.
- Group 195 (The Golden Ratio): Rory Byrne, Conall Percy
- Group 200 (The Golden Ratio): Robert lynch, Darragh Toal, Alfie duggan, Aideen Boardman.
- Group 203 (The Golden ratio): Billy McKenna, Mark Cunningham, Mark Theunissen, Edward O’Sullivan, Conor McKeon and David Hughes.
- Group 205 (Mersenne Primes): Alexander O’Boyle, Cathal Healy, Mikolaj Piotrowski and Colm O’Brien.
- Group 208 (Mersenne Primes): Kai cussen, Pheilim kenny, Ben quirey, Eoin lynch.
- Group 210 (Supertasks, Infinity and Fractals): Morgan Hurley O’ Dwyer, Adam Walsh and Xin Shu.
- Group 215 (Banach-tarski paradox): Arthur Odlum, Sean Ojejinmi, Liam lysaght, Yann Blake and Ali El-kmati.
- Group 220 (Banach-tarski paradox): Seán Gallagher and Stephen Callaghan.
- Group 225 (Euler’s identity): Taron Wright, Conor Duffy, Ronan Friel, Ciarán Kelleher.
- Group 230 (Hexagons in nature): Adam Delaney, Daniel Lavin, Harry Thompson, Catherine O Connell, Jack Rudden Kelly.
- Group 233 (Hexagons in nature): Dylan McCaul, Alex Davies, Ciaran Finnegan, Philip McDowell, Dylan McCaul and Matthew Burke.
- Group 235 (The 4th dimension): Eusang cho, Franklin Lukason, Minaal Billavara, Radu Stefan Cristea, Van Aizel Bergado and Tom He.
- Group 240 (The Mathematical Origins of Common transcendentals): Anbu Sundar and Adam Clarke.
- Group 245 (The Napkin Ring Problem): Daniel Flood, Karl Weldon, Conchubhair Guiney, Sean Farrell and Stephen Byrne.
- Group 250 (The Napkin Ring Problem): Erik Slattery, Rahul Seth, Kyle Mendez, Oisin Callan, Erika Galligan.
- Group 255 (Göedel incompletness theorem): Conall Daly, Eoghan O’Doherty, Bradley Richardson.
- Group 260 (The pareto distribution): Sean Daly and Brian Walsh.
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?. Assigned. Group A: Eimear, Seán, Sadhbh, Aliya, Megan, Emily. Group B: Karen Morrissey, Ruth Mullen, Aimee Goodwin, Ellen Rose Tighe and Caoimhe O’Hare. Group C: Holly Leech, Orla Keenan, Laura Murphy and Anna Eniko Illesi. Group D: Sinead quinn, Eimhear Hughes and Una O’Neill. Guidelines.
- The fundamental theorem of algebra: A polynomial of degree n has exactly n complex roots (if one counts the multiplicity). Assigned Group A: Elizabeth Bolger, Laura Stack, Evan Magee, Cian Rellis and Emily Broderick. Group B: Dylan Hunt, Úna Raeside, Ellie O’Sullivan and Finn Voorhees. Guidelines.
- Galois theory: There is no explicit formula for the roots of a polynomial of degree 5 or higher!. Assigned. Group A: Leo Devlin, Matthew Conway, and Joseph O’Connell. Group B: Johnny Scalon, Conor O’Donnell, Conor Maher and Jack Byrne. Group C: David Carroll, Michael Diamond and Sam Butler. Guidelines
- Linear regression: Using polynomials to estimate/model the relationship between variables. Assigned. Group A: Jack O’Donoghue, Kevin Digan, Cian Kellet, Rebecca Sutherland, Kate O’Connell, Aoife Hurd and Lisa Callagh. Group B: Leo Cooke and Cormac Doyle. Group C: James Sutton, Julie Gallagher, Orla Kearney and Leontina Schelling. Guidelines
- Fourier analysis: Representing a function as a sum of trigonometric functions. Assigned: Group A: Gearoid Farrell, Conor Bryce, Keshav Sapkota, Charlie Maguire, David Dunne, Conrad Vandlik. Group B: Ronan McAlister, Nicholas Cody, Nollaig Butler and James Toland. Group C: Bella Salajeva, Jakub Kluza, Cian Skelly, Bríd O’Donnell, Áine Deasley and Muireann Farrell. Guidelines
- The hyperreal line: Extending the real line to fit infinite(simals). Group A: Rebecca Katherine Armstrong, Galina Kennedy, Ella Keough, Carmen Galante and Abieyuwa Enehikhare Obasuyi. Group B: James Nangle, Conor O’Farrelly, Conor Corry, Diarmuid Coffey, Rory Clarke, Oscar Crowley. Group C: Alistair Tidey, Grant Arnott, Ruairi o Connor, Patrick junghenn, Oran keeling, Mathew lynch. Guidelines
- Fixed point theorem’s: Brouwer fixed-point theorem, Banach fixed-point theorem, and its amazing consequences. Assigned. Group A: Anneliese Walsh, Ciara Coyne, Claudia Alonso Vicedo, Elvira Guiomard, Katie Henn, Méabh Childs. Group B: Kate Fagan, Louise Fagan, Emma Grimes and Wajeeha Shahbaz. Group C: Seán Byrne, Jack Hartnett, Conor Kirwan, Conor Dunne and Islam Alagtal. Guidelines
- Propagation of errors: Using linear approximations of functions to “propagate” uncertainties. Assigned. Group A: Charlotte Brehm, Aaron Heery, Fernando Valpuesta, Jean Luc Aho, Eoin Burke and Mohanad M. Jawadah. Group B: Aislinn Dunleavy, Ben Broderick, Isobel Duffy and Denis McCambridge Guidelines
- Calculus of variations: Instead of finding the point at with a function is maximum/minimum, calculus of variations determines the function that makes a functional maximum/minimum. Assigned. Group A: Alexis Flaherty, Miriam Kelly, Gabi Pansnek, and Jessica Bagnall. Group B: Wei Ying Chang, Kiera Cullen, Victoria Watterson, Emma Gallagher, Aisling Cashell, Rachel O’Grady.
- Hamilton’s least action principle: Newtonian laws are in fact equivalent to the principle of least action. This is the basis of Lagrangian and Hamiltonian mechanics. Assigned. Group A: Aaron Hannon, Conor O’hEocha. Group B: Laoise Kinsella, Nasir Said, Rayyan Mehunood, Cormac Dunne. Group C: Erik Slattery, Rahul Seth, Kyle Mendez, Oisin Callan and Erika Galligan.
- Complex numbers, quaternions and rotations: How complex numbers can be used to describe rotations in a plane, and quaternions to describe rotations in 3D. Assigned. Group A: Alex Pocock, Alex mcGinn, John Bennet and Oisin Dougan. Group B: Naoise Barry, Jakub Pyszka, Mellisa Dumitrache, Carolín Laoide-Kemp and Euan Reid. Group C: Calum McGrath, Luke Corcoran, David Berry, Paddy Bryne, Niall Lee, Daniel DiMasio.
- Trascendental numbers: \( e \) is irrational… Even worse… It is not an algebraic number (like almost every real…). Group A: Jessica Ojomor and Jeffrey Dargan. Group B: Jack Goodwin and David Mckenna. Group C: Mabroor Ahmed, Andrew Downey, Ahad Khalil and Jack Lowe.
- Newton (and other Fractals): What are fractals? Why they are useful to describe natural phenomena? Group A: Laura Owens, Orlaith Murphy, Kelly Augustt, Anuk Mornoey and Erica Markey. Group B: Mark Kelly, Uchenna Udemezue, Ankit Vijay, Francesco Digeonimo.
- The Golden Ratio: Group A: Tom Mulligan, Cian Murphy, Jack O’Leary, Ian Black, Conor Dalton, Aodhán Buggy. Group B: Rory Byrne, Conall Percy. Group C: Robert lynch, Darragh Toal, Alfie duggan, Aideen Boardman.
- Mersenne Primes: Alexander O’Boyle, Cathal Healy, Mikolaj Piotrowski and Colm O’Brien.
- Supertasks, Infinity and Fractals: Morgan Hurley O’ Dwyer, Adam Walsh and Xin Shu.
- Banach-tarski paradox: Group A: Arthur Odlum, Sean Ojejinmi, Liam lysaght, Yann Blake and Ali El-kmati. Group B: Seán Gallagher and Stephen Callaghan.
- Euler’s identity: Taron Wright, Conor Duffy, Ronan Friel, Ciarán Kelleher.
- Hexagons in nature: Adam Delaney, Daniel Lavin, Harry Thompson, Catherine O Connell, Jack Rudden Kelly.
- The 4th dimension: Eusang cho, Franklin Lukason, Minaal Billavara, Radu Stefan Cristea, Van Aizel Bergado and Tom He.
- The Mathematical Origins of Common transcendentals: Anbu Sundar and Adam Clarke.
- The Napkin Ring Problem: Group A: Daniel Flood, Karl Weldon, Conchubhair Guiney, Sean Farrell and Stephen Byrne. Group B: Erik Slattery, Rahul Seth, Kyle Mendez, Oisin Callan, Erika Galligan.
Model exam
- An example of an exam. Solution
- May exam (solved and commented).