Applied mathematics II

Differential and integral calculus. Elementary algebra.

At classes, expository and inquisitive methods are used. Each chapter is motivated using practical examples. The theory is then applied through discussion and resolution of exercises.
Individual and group work skills are explored through the resolution of practical exercises, the implementation of scripts using mathematical software and the use of these codes to solve and discussion of exercises.

– Develop ability to define and analyze mathematical problems, choose the most effective solving methods, interpret and analyze results.
– Provide basic knowledge of Linear Algebra.
– Identify, understand and solve problems involving matrix calculus.
– Perform basic concepts or theory of errors.
– Apply concepts related to the numerical resolution of linear systems, nonlinear equations, polynomial interpolation and definite integrals.
– Understand the limitations of analytical techniques and develop the capacity to use numerical methods.

1. Matrices.
Definitions. Operations and properties.
2. Systems of linear equations.
Definitions. Operations with matrices and properties: matrices condensation and characteristic. Classification and resolution of systems of linear equations: i) Direct methods: Gauss elimination; ii) Iterative methods: Jacobi and Gauss-Seidel. Applications.
3. Theory of errors (brief remarks).
Introduction. Definitions.
4. Roots of nonlinear equations.
Introduction. Location of roots: graphical method and Bolzano’s theorem. Bisection and Newton’s methods: iterative rules, error and stopping criteria. Computational aspects. Applications.
5. Polynomial interpolation.
Introduction. Uniqueness of the interpolating polynomial. Interpolating polynomial using Lagrange and Newton’s forms. Interpolation error. Computational aspects. Applications.
6. Numerical integration.
Introduction. Trapezoidal and Simpson’s rules. Numerical integration error. Computational implementation.

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