Teacher: Andrea Seppi

Exercise sessions: Alan McLeay

Tutorials: Nhat Minh Doan

Week 1 (17-21/9): Natural, integer, rational and real numbers. Functions of real numbers and examples: polynomials, trigonometric functions (sine and cosine).

Week 2 (24-28/9): Definition of limit of a sequence. Definition that a sequence has limit +∞ or -∞. Examples of sequences that have and do not have limits. Uniqueness of the limit. Monotone sequences and bounded sequences. Monotone convergence theorem: i.e. every monotone increasing sequence has a limit.

Week 3 (1-5/10): Definition of supremum of subsets of the real numbers. Properties of the supremum. Proof of monotone convergence theorem. Definition of the exponential function and some properties. Limits at +∞ of real valued functions: exponential function, power functions. Change of variables. The exponential goes to +∞ faster than any power function. Definition of hyperbolic trigonometric functions.

Week 4 (8-12/10): Definiton of limit of a function at a point in R. Example of functions which do and do not admit limits. Right and left limits. A function admits a limit at a point x if and only if it admits left and right limits and they coincide. Limit of the sum and product of functions (with proof for the sum).

Week 5 (15-19/10): Limits at infinity of the ratio of a polynomial and an exponential, and of the ratio of two polynomials. Limits of a ratio of two polynomials and a zero of the denominator. Sandwich theorem. Limits of sinx/x at 0 and at infinity.

Week 6 (22-26/10): Continuous functions. Definitions and examples. Sum, products and ratios (where defined) of continuous functions are continuous. Polynomials, exponential, trigonometric functions and hyperbolic trigonometric functions are continuous. Intermediate value theorem. The image of a continuous function defined on an interval is an interval.

Week 7 (29/10-2/11): Definition of differentiable functions and derivatives. Tangent lines. Differentiability implies continuity. Examples of derivatives: linear functions, sinus, cosinus, exponential, power functions.

Week 8 (5-9/11): Derivatives of f+g, 1/f, f^2, fg (Leibniz rule), f/g, and of the composition of two functions. Examples with trigonometric functions.

Week 9 (12-16/11): Inverse functions and their derivatives. Examples: square root, logarithm, arcsin, arccos, arctan. Properties of logarithm. Definition of power function and derivative.

Week 10 (19-23/11): De l’Hôpital rule.

Week 11 (26-30/11): Local minima and maxima. Theorem of Fermat. Right and left derivatives. Rolle and Lagrange theorems, with proof. Criteria to detect local maxima and minima.

Week 12 (3-7/12): Convexity. Several equivalent definitions. A differentiable function is convex if and only if its derivative is increasing. The second derivative.

Week 13 (10-14/12): Taylor series. Example of limits using Taylor’s expansion.

Week 14 (17-21/12): Big O notation. Taylor’s theorem. Newton’s method. Examples with Mathematica on the screen.