• Term taught 
  •             Michaelmas term 2017-2018
  • Lecturer
  •             Dr. Victoria Lebed
                E-mail: lebed (at) maths.tcd.ie
                Office: 1.5, School of Mathematics
                Any questions or remarks or calls for help are welcome!
  • Schedule
  •             Monday 12:00 (MacNeil Theatre, Hamilton Building)
                Tuesday 12:00 (MacNeil Theatre, Hamilton Building)
                Friday 9:00 (MacNeil Theatre, Hamilton Building)
  • Tutorials and homework
  •              Every Friday your homework assignment appears here and on the tcd.blackboard page of the module. It is to be handed in at the end of the tutorial of the following week. During the tutorial you can discuss with the tutors the homework and/or the problems from a supplementary problem sheet.
                 Homework problems: HW1 (due: week 2), HW2 (due: week 3), HW3 (due: week 4), HW4 (due: week 5), HW5 (due: week 6), HW6 (due: week 8, after the Reading Week), HW7 (due: week 9), HW8 (due: week 10), HW9 (due: week 11), HW10 (due: week 12).
                 Homework solutions: HW2, HW3, HW4, HW5, HW6, HW7, HW8, HW9, HW10.
                 Final exam problems will be very close to your homework assignments, so spending some time on your homework and trying to solve problems independently of your classmates is a key to success at the exam!
                 Here is a very rough sample of the final exam, with a partial list of the topics the real exam is likely to cover.
  • Plan
  •             1. Introduction. The notion of a function. Slides
                2. Domain and range of a function. Digression on intervals, rays, and operations on them. Slides
                3. Building new functions from old ones: arithmetic operations and composition. Slides
                4. Functions with symmetries. Parametric families of functions. Slides
                5. Classical functions: polynomial, rational, algebraic, trigonometric. Slides
                6. Inverse functions: definition, examples, uniqueness, x-y symmetry. Slides
                7. Inverse functions: graphs, existence, inverse trigonometric functions. Cf. slides from Lecture 6.
                8. The notion of limit: informal and formal definitions, two- and one-sided limits, infinite limits and vertical asymptotes. Slides
                9. Computing limits: limits and operations on functions, limits of polynomial and rational functions, indeterminate forms of type 0/0. Slides
                10. Limits at infinity and horizontal asymptotes. Slides
                11. Continuity: definitions, examples and applications (Intermediate Value Theorem, bisection method for solving equations numerically, Pancake and Ham Sandwich Theorems). Slides
                12. An overview of differential and integrable calculi. Definitions of tangent lines, differentiability and the derivative of a function. Positive and negative examples. Slides
                13. Differentiability and continuity. One-sided derivatives. Different notations for the derivatives. Derivatives of powers, linear combinations, and products. Slides
                14. Derivatives of quotients, trigonometric functions, and compositions.
                15. Derivatives of inverse functions. Implicit differentiation. Slides for Lectures 14-15.
                16. Higher derivatives. Polynomials as functions with vanishing higher derivatives. Monotony of f and the sign of f'. Slides
                17. Local and global extrema, and critical points. Extremal Value Theorem. Slides
                18. Concavity of f and the sign of f''. Analysis of polynomial and rational functions. Curvilinear asymptotes. Slides
                19. Applications of differential calculus. Newton's method for solving equations numerically. L'Hopital's rule for computing limits. Slides
                20. Exponential and logarithmic functions. Slides
                21. The notion of antiderivative and indefinite integral. Integration of linear combinations. u-substitution. Slides
                22. Integration methods for indefinite integrals: u-substitution and integration by parts. Slides
                23. Areas of plane figures via approximation by rectangles. Sigma notation as a tool for working with sums. Slides
                24. Areas of plane figures via definite and indefinite integrals. The Fundamental Theorem of Calculus relating integral and differential calculi. Slides
                25. Integration methods for definite integrals: u-substitution and integration by parts. Slides
                26. A stronger form of the Fundamental Theorem of Calculus. The Mean Value Theorem for Integrals. A summary of integral calculus. Slides
                27. Applications of integrals in mathematics: the area of a plane figure, the arc length of a plane curve, the average value of a function, estimation of an infinite sum. Slides
                28. Applications of integrals in sciences: displacement and distance traveled, work, center of gravity. Slides
  • Additional materials
  •             Exercise sheets: 1, 2, 3.
                Slides: A remarkable limit; History of calculus.
  • Grading
  •              Homework  20% +  final exam 80%.
                 There will be a single exam and a single mark for the module MA1S11, which means that half of the exam paper will be about calculus and the other half about linear algebra/discrete mathematics (the part taught by Prof. McLoughlin).

  • Suggested reading
  • You can borrow a copy from the Hamilton Library, or buy one from second-year students (for this you might have a look at the TCD noticeboard). If you are going to buy a new one, avoid the editions subtitled "Single Variable". They contain only the first 8 or 9 chapters, and you will need further chapters of the more complete editions for the second semester (MA1S12).

  • Topics covered
  •     Functions
        Limits and Derivatives
        Trigonometrical, Logarithmic, and Exponential Functions

  • If you need help
  •             Ask me directly after class, e-mail me your questions or make an appointment, talk to your tutors, attend the School of Mathematics help room (the New Seminar Room in the School of Mathematics, 1-2pm Monday to Friday). Remember that the whole system is there to help you get the best out of the classes!