Mathematics A 1000-1MATA1
Definitions of elementary functions: sine, cosine, tangent, exponential function, logarithms, root functions. Inverse trigonometric functions. Limits of functions. Continuity: intermediate value theorem, Weierstrass Maximum Principle. Definition of the derivative. Tangent to the graph of a function. Differentiation rules. Sketching graphs of functions: monotonicity, convexity, inflection points, locally extreme values. Taylor formula with Lagrange remainder and Peano remainder, approximation, error estimates. Taylor series. Primitive function. Simple integrals. Basic techniques of integration: substitution and integration by parts. Vectors in the plane and in the 3-dimensional space. Scalar product and vector product. Matrices: addition, multiplication by a number and by a matrix. Determinants. Linearity with regard to columns or rows, skew symmetry, row reduction, Cramer's formulae, eigenvalues and eigenvectors. Orthogonal matrices, symmetric matrices, transposed matrix.
Prerequisities: basic high school algebra, solving of linear systems with 2 or 3 unknowns, canonical form of a quadratic trinomial, second degree equations and inequalities, polynomial factorization by finding rational roots or by grouping terms in simple cases.
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