Matemática I (1 º Sem 2011/2012)

ECO , FIN , GES

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    Part I - Linear algebra and linear systems of equations

    1. Vectors
      1.  Definition, geometrical representation, elementary operations, inner product, norm, distance, Cauchy-Schwarz inequality.
      2. Linear combination and linear independence of vectors
    2. Matrices
      1. Definition, properties, sum and scalar product, transpose
      2. Product of matrices
      3. Gauss elimination method, rank
      4. Gauss method as product of matrices
    3. Systems of k linear equations with n variables
      1. No solutions
      2. Unique solution
      3. Infinite solutions
    4. Systems of  n linear equations with  n variables
      1. Determinant of a matrix
      2. Properties of determinants
      3. Inverse matrix
      4. Cramer's rule
    Part II-Mathematical analysis: real-valued functions

    1. Functions and series
      1. Definition of a function, sequences, odd and even functions
      2. Series, arithmetica and geometrical series, convergence
      3. Real-valued functions, limits, continuity, linear and non-linear functions
      4. Composition of functions
      5. Polynomial and exponential functions
    2. Derivatives
      1. Definition, geometrical interpretation
      2. Rate of change, monotonicity, elasticity
      3. Chain rule, derivative of the inverse function
      4. Implicit differentiation
      5. L'Hôpital/Cauchy's rule
    3. Polynomial approximation of functions
      1. Linear and quadratic approximations
      2. Taylor series
      3. Intermediate value theorem
      4. Mean value theorem
    4. Optimization
      1. Stationary and extreme points
      2. Second derivative classification
      3. Convexity and concavity
    5. Integral
      1. Definition
      2. Anti-derivatives: definition, properties
      3. Computation of areas
      4. Fundamental theorem of calculus
    6. Further integration methods
      1. By parts
      2. By substitution

     

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