Analyse I (anglais)

Course: MATH-101(en) Analysis 1 (English),  6 credits (ECTS)

Teacher : T Mountford

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Lectures

• Mondays, 8h15-10h00, room PO 01 (click on the room's name for the position in the EPFL campus).
• Wednesdays, 8h15-10h00, room CO2 (click on the room's name for the position in the EPFL campus).

• Wednesdays, 10h15-12h00, rooms CM010, CO121, CO122, CO123, CO124.

• CM 010: Section MT MX     Assistants: Mathurin Froment Adrian Cardaba
  
• CO 121: Sections CGL  EL GL SC   Assistants: C Camus-Emschwiller  Allessandro D'Urso
  
• CO 122:GM Assistants:Erik Algarp Juliette Sikking
  
• CO 123 SV SIE Assistants Ainur Zhaikan  Matthieu Charbonnier
  
• CO 124 Sections IN Assistants H Balc  K. Dinev
  

Below is a list of useful links, documents, tools.

Lecture 1 (click on the lecture name to be sent to the video for it):

• Motivations;
• Proofs;
• Sets;
• Number sets;
• Properties of the real numbers.
  

Lecture 2 (click on the lecture name to be sent to the video for it):

• More notation on number sets and intervals ;
  
• Upper and lower bounds: definitions, properties, examples;
• Maximum and minimum: definitions, properties, examples;
• Supremum and infimum: definitions, properties, examples;.
• Axiom 2.22 and its consequences
  
• Pages 10-18 approximately

Lecture 3 (click on the lecture name to be sent to the video for it):

• More results on inf/sup;
• Subset of the natural numbers always have minima;
• Integral part of a real number;
• Density of the rational numbers in the real numbers.
• Pages 17-end of chapter 2
  

Lecture 4 (click on the lecture name to be sent to the video for it):

• Triangular inequality over the reals;
• Extended real line;
• Complex numbers and operations among them;
• Absolute value of complex numbers and triangular inequality;
• Polar form of a complex number.

Lecture 5 (click on the lecture name to be sent to the video for it):

• Solving equations over the complex numbers;
• Sequences: definitions and examples;
• Induction;
  
• Bernoulli's inequality.

Lecture 6

• Definition of limit and examples
  
• Algebra of Limits
  
• Squeeze Theorem
  
• 
  

• Recursive sequences and their limits
  
• Infinite limits

Lecture 8 (click on the lecture name to be sent to the video for it):

• Sequences that approach infinity
• Quotient test;
• limsup/liminf
• Subsequences: definitions and properties;
• Cauchy criterion
  
  

• Series
• Convergence;
• 2 gendarme4, comnparison
  

• convergence of series 1/n^k for k > 1
  
• Proposition 5.20
  
• Alternating series proposition
  
• Cauchy d'Alemebert criterion
  

• Examples
• Review
• Functions (start of chapter 6)
  

• Basic definitions about functions;
  
• Examples.
• Pointed neighborhood;
  
• Definition of Limit
  

• Algebra of limits
  
• Limits of functions including infinity
  

Lecture 14:

• continuity
• Composition of functions.
• Composition and continuity.
  
• Examples.
• infinite limits
  

• Infinite limits.
• Composition of functions.
• Composition and continuity.
  
• Examples.
• One sided limits

In lecture one we complete the IVT and then discuss the range of a continuous function on intervals.We then give necessary and sufficient conditions for a function on an interval to be injective (trhat the function be strictly monotone. We then show that the inverse is continuous.

Finally we define e^x and its inverse log(x).

  Date: Wednesday November 13
Starting Time: 10h20.
Duration: 60 minutes.
Place: the exercise session rooms (see below for further info).

The mock exam will be held on campus starting at 10:20, on November 15 and it will last 60 minutes.
The exam consists of 7 QCM (multiple choice questions) and 4 T/F questions and 1 open questions. 

No book or calculator allowed. You should just plan to use pen, pencil, and your brain.
This is the only simulation of the exam you will have before the final exam in January.

Lecture 18

• Differentiablility basic properties
  
• Derivatives of compositions

Lecture 19:

• differnetiability of e^x
  
• finding extrema;

• Mean Value Theorem.
• Local extrema and stationary points;
• How to find the global max/min of a continuous function over a bounded interval;.

• Applications of the Mean Value Theorem;
• l'Hopital rule;
• Taylor's approximation formula;
• Examples.

• s.

• Applications of the Mean Value Theorem;
• Taylor's approximation formula;;
• Higher orders and stationary points;
• Chapter 7.

• Properties of Integral;
• Fundamental Theorem of calculus
  
• Substitution

Lecture 24:

• Substitution
• Integration by Parts
  
• Integration of rational functions
  

• Integration of rational functions
• Improper integrals
  

• Properties of improper integrals;
• Comparison theorem for improper integrals;
• Power series: definition;
• Domain of convergence, derivative, integrals;
• Taylor series.
  

• More examples of integration by substitution;
• Integrating rational functions;
• Improper integrals: basic definitions.

• Properties of improper integrals;
• Comparison theorem for improper integrals;
• Power series: definition;
• Domain of convergence, derivative, integrals;
• Taylor series.