Maths for scientists (Calculus, first semester), TCD 2013/14
Lectures:
Mon, 16:00 - 16:50 Goldsmith Hall Lecture Theatre
Tue, 12:00 - 12:50 MacNeill Lecture Theatre (Hamilton Building)
Thu, 9:00 - 9:50 Edmund Burke Lecture Theatre (Arts Building)
Covered so far:
| Lecture | Topics covered | Textbook chapter | Materials |
|---|---|---|---|
| 1 (23/09) | Functions, graphs and some examples, vertical line test, piecewise defined functions, domains | 0.1 | [PDF] |
| 2 (24/09) | Natural domain and range of a function. Arithmetic operations on functions, and the corresponding domains. | 0.1, 0.2 | [PDF] |
| 3 (26/09) | Composition of functions. Translations, reflections, scaling, and their effect on graphs. | 0.2 | [PDF] |
| 4 (30/09) | Odd and even functions. Symmetry tests for plane curves. Parametric families: straight lines and power functions. | 0.3 | [PDF] |
| 5 (01/10) | Questions from the first tutorial. Classical types of functions: polynomials and rational functions. Examples. | 0.3 | [PDF] |
| 6 (03/10) | Classical types of functions: trigonometric functions. Inverse functions. Examples, domains and ranges. | 0.3, 0.4 | [PDF] |
| 7 (07/10) | Inverse functions. Computing inverses. Horisontal line test. Summary of the first part of the module. | 0.4 | [PDF] |
| 8 (08/10) | Limits: the informal approach and the formal approach. One-sided limits. Infinite limits. | 1.1, 1.4 | [PDF] |
| 9 (10/10) | Arithmetics of limits. Indeterminate forms of type 0/0. Infinite limits and vertical asymptotes. | 1.2 | [PDF] |
| 10 (14/10) | Limits at infinity and horisontal asymptotes. Infinite limits at infinity. Limit behaviour of polynomials, rational functions, and more complicated expressions. | 1.3 | [PDF] |
| 11 (15/10) | Continuous functions. Continuity on open and closed interval. Continuity and arithmetic operations and composition. Intermediate Value Theorem. | 1.5 | [PDF] |
| 12 (17/10) | Continuity of inverse functions. Continuity of trigonometric functions. Limit of sin(x)/x at x=0. | 1.6 | [PDF] |
| 13 (21/10) | No lecture on October 21 | N/A | N/A |
| 14 (22/10) | Historical remarks on calculus. Slope of the tangent line to a graph. Instantaneous velocity and rate of change. The derivative function. Differentiability at a point or on an open interval. | 2.1, 2.2 | [PDF] |
| 15 (24/10) | Differentiability and continuity. Differentiability on a closed interval. Other notation for derivatives. Derivatives and scalar factors, sums, and differences. Formula an-bn=(a-b)(an-1+an-2b+...+abn-2+bn-1) and derivatives of power functions with positive integer exponents. Derivatives of polynomials. | 2.2, 2.3 | [PDF] |
| 16 (28/10) | October Bank Holiday, no lecture. | N/A | N/A |
| 17 (29/10) | The binomial formula, and derivatives of power functions with positive integer exponents. Product rule. Derivative of 1/g and of the quotient f/g. Derivatives of trigonometric functions. | 2.3, 2.4, 2.5 | [PDF] |
| 18 (31/10) | Derivatives of power functions with negative integer exponents. Chain rule. Implicit differentiation, derivatives of inverse functions. Derivatives of power functions with fractional exponents. | 2.6, Appendix D | [PDF] |
| Study Week (November 4-10): make sure to use it wisely! | |||
| 19 (11/11) | Implicit differentiation: examples. Higher derivatives. Derivatives of increasing/decreasing functions. | 2.6, 3.1 | [PDF] |
| 20 (12/11) | Derivatives of increasing/decreasing functions. Relative maxima and minima. First and second derivative tests; critical points, stationary points, points of non-differentiability. | 3.1, 3.2 | [PDF] |
| 21 (14/11) | Concavity up and down, inflection points. Multiplicity of roots of polynomials, and its geometric meaning. | 3.2 | [PDF] |
| 22 (18/11) | No lecture on November 18 | N/A | N/A |
| 23 (19/11) | Graphing rational functions. Differential calculus and Extreme Value Theorem / Intermediate Value Theorem. Newton's method. | 3.3, 3.4, 3.7 | [PDF] |
| 24 (21/11) | Newton's method: examples. Exponential and logarithmic function; their derivatives. Using logarithms to simplify functions for differentiation. Summary of differential calculus. Antiderivatives. | 3.7, 4.2, 6.1, 6.2, 6.3 | [PDF] |
| 25 (25/11) | Antiderivatives. Rules for antiderivatives (linear combinations, u-substitution, integration by parts). Examples. | 4.2, 4.3, 7.2 | [PDF] |
| 26 (26/11) | Mnemonics for antiderivatives using dy=f'(x)dx. More examples for the u-substitution and for the integration by parts. | 4.3, 7.1, 7.2 | [PDF] |
| 27 (28/11) | No lecture on November 28 | N/A | N/A |
| 28 (02/12) | Areas under curves. Example of y=x2. Net signed area under the curve. Riemann sums and integrability. Definite integral and its basic properties: exchange of limits of integration is compensated by a factor (-1), linearity of the definite integral, integral over [a,a] vanishes. | 4.1, 4.4, 4.5 | [PDF] |
| 29 (03/12) | Fundamental theorem of calculus. Mean-Value theorems for derivatives and integrals. | 4.5, 4.6 | [PDF] |
| 30 (05/12) | Using u-substitution in definite integrals. Examples. Summary of integral calculus. | 4.9 | [PDF] |
| 31 (09/12) | Applications of definite integrals in geometry. Area between two curves. Arc length of a curve. Volume of a solid of revolution; area of a surface of revolution. | 5.1, 5.2, 5.4, 5.5 | [PDF] |
| 32 (10/12) | Applications of definite integrals in physics and engineering. Work, energy, centre of gravity. | 5.6, 5.7 | [PDF] |
| 33 (12/12) | No lecture on December 12. Attempt sample exam problems. | [PDF] | |
| Solutions to sample exam problems are available. | [PDF] |
Tutorials:
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 on the noticeboard next to the School of Maths entrance on the ground floor of the Hamilton building.
Tutorial times and venues:
group AA1: Thu 12 - 1, M4, Museum Building (tutor: Robert Murtagh)
group AA2: Thu 4 - 5, Salmon Lecture Theatre, Hamilton Building (tutor: Gemma Foley)
group AA3: Thu 12 - 1, Synge Lecture Theatre, Hamilton Building (tutor: Mairead Grogan)
group AA4: Thu 4 - 5, Chemistry Building Large Lecture Theatre (tutor: Brendan Bulfin)
group AA5: Wed 1 - 2, Salmon Lecture Theatre, Hamilton Building (tutor: Gemma Foley)
group AA6: Fri 2 - 3, Synge Lecture Theatre, Hamilton Building (tutor: Brendan Bulfin)
group AA7: Fri 12 - 1, Salmon Lecture Theatre, Hamilton Building (tutor: Gemma Foley)
group AA8: Tue 9 - 10, Synge Lecture Theatre, Hamilton Building (tutor: Mairead Grogan)
Tutorial sheets:
By the weekend of each week, a tutorial sheet for the next week will be posted here, as well as solutions to the previous tutorial sheet.
Tutorial 1 (Week 2, Michaelmas 2013): Problem sheet [PDF] Solutions [PDF]
Tutorial 2 (Week 3, Michaelmas 2013): Problem sheet [PDF] Solutions [PDF]
Tutorial 3 (Week 4, Michaelmas 2013): Problem sheet [PDF] Solutions [PDF]
Tutorial 4 (Week 5, Michaelmas 2013): Problem sheet [PDF] Solutions [PDF]
Tutorial 5 (Week 6, Michaelmas 2013): Problem sheet [PDF] Solutions [PDF]
No tutorial in Week 7 (study week); make sure you practise with exercises from the textbook!
Tutorial 6 (Week 8, Michaelmas 2013): Problem sheet [PDF] Solutions [PDF]
Tutorial 7 (Week 9, Michaelmas 2013): Problem sheet [PDF] Solutions [PDF]
Tutorial 8 (Week 10, Michaelmas 2013): Problem sheet [PDF] Solutions [PDF]
Tutorial 9 (Week 11, Michaelmas 2013): Problem sheet [PDF] Solutions [PDF]
Tutorial 10 (Week 12, Michaelmas 2013): Problem sheet [PDF] Solutions [PDF]
Further help with maths questions:
Of course, you may ask me directly after class or call into my office (19.32A Westland Row, access through School of Maths in Hamilton Building). In addition, the School of Mathematics runs a helproom, a walk-in service to ask maths questions you are struggling with. For the moment, the helproom hours are 1-2pm Monday to Friday. (The helproom venue is the New Seminar Room in the School of Mathematics, 20 Westland Row. It can be accessed from Hamilton building: enter the Hamilton building at the basement level, beside the shop, proceed to the right, past the sign for School of Mathematics on the left, until you see a sign for the Helproom and also signs for WISER. Enter and go up two levels. The Seminar Room is in front of you.)
About this module MA1S11:
This module is divided in two parts, covering calculus and discrete mathematics (mostly linear algebra), respectively. I am teaching calculus, Prof. McLoughlin is teaching the discrete mathematics part.
You may also have a look at last year's course page from when Prof. Sint was teaching the calculus part. There will be a single exam and a single mark for MA1S11, which means that half of the exam paper will be about calculus and the other half about linear algebra/discrete mathematics. The final mark for MA1S11 will be 80 percent based on the exam and 20 percent on continuous assessments, i.e. the tutorials and labs. You can find past exam papers here .
Synopsis
Functions.
Limits and Continuity.
The Derivative.
The Derivative in Graphing and Applications.
Integration.
Textbook
Calculus, by Howard Anton, Irl Bivens, Stephen Davis, (9th edition; publisher Wiley, New York)
You can find it in the Hamilton library [515 P2*8;4,6,7, S-LEN 515.P2*8;1-3]
Unfortunately, there exist different versions of the same book or parts of it, with different subtitles. The library has a few hardbound copies which only contain the first 8 or 9 chapters and are subtitled "Single Variable". While this is sufficient for the course I am teaching, some material needed for the second semester (MA1S12) is missing. Therefore, if you intend to buy the book it is probably best to go with the paperback edition pictured below.
 |
Calculus Late Transcendentals, 9th Edition Howard Anton, Irl Bivens, Stephen Davis ISBN: 978-0-470-39874-6 Wiley 2009, (1168 pages + appendices) |
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