**Term taught**

**Lecturer**

E-mail: lebed (at) maths.tcd.ie

Office: 1.5, School of Mathematics

Any questions or remarks or calls for help are welcome!

**Schedule**

Tuesday 12:00 (MacNeil Theatre, Hamilton Building)

Friday 9:00 (MacNeil Theatre, Hamilton Building)

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.

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

Slides: A remarkable limit; History of calculus.

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).

- Howard Anton, Irl Bivens, Stephen Davis, "Calculus: late
transcendentals", any edition.

Limits and Derivatives

Integration

Trigonometrical, Logarithmic, and Exponential Functions

Applications