Learn basic contents and techniques from Mathematical Analysis, which are needed to study functions, to solve problems relying on integral calculus and to solve simple differential equations. When possible, these themes will be related to applied problems.
Students will learn basic concepts: limit, differentiability, study of functions; integral and its applications; differential equations.
This concepts will be used to solve concrete problems and to face simple mathematical models.
SETS Operations between sets, numerical sets. discrete sets, continuous sets. NUMBERS Natural, relative, rational, Irrational numbers. Absolute value (or modulo). Real numbers, power of the continuous. One-to-one correspondence between real numbers and points of a line. FUNCTIONS Counting, continuity, operations, functions. Variable and image, domain and codomain. Equations. unknowns and parameters. Resolution method, solutions. ALGEBRA Addition, subtraction, algebraic sum. Multiplication. Rules. Operations with monomials, binomials and polynomials. Division. Rules. Operations with modulo, greater than and less than. Power. Rules. Operations with monomials, binomials and polynomials. Measurement. Units of measurement, Systems of units, change of unity, use of powers of 10. Root. Rules. Operations with roots. Inverse operations and functions. Logarithm. Rules. Natural logarithms. Operations with Logarithms. ANALYTIC GEOMETRY Cartesian plan. Cartesian axes, coordinates of a point. Graphs of functions. Plane curves. Straight line. Straight line graph (by points). Angular coefficient, intercept. Distance between two points. Explicit equation of the straight line. Variable and Parameters. Straight lines parallel to the axes, Bisector of the 1st quadrant. Bundles of straight lines. Lines through one point, line through two points, parallel lines. Condition of parallelism in explicit form. Implicit equation of a straight line. Formulas from explicit to implicit form. Condition of parallelism in implicit form. Intersection point between two straight lines. Systems of linear equations in explicit, implicit, canonical form. Matrices and determinants. Cramer's method. Parabola. Equation, parameters, concavity, axis. Bundle of parabolas. Intersection of a parabola and a line. Exponential function. Logarithm function. TRIGONOMETRY Angles. Degrees and Radians. The Radiant. Positive and negative directions of rotation. Goniometric functions. Goniometric circumference, sine, cosine and tangent. Graphs of goniometric functions. Fundamental equation. Tangent equation. Transformations among sin (α), cos (α) and tan (α) Equations for complementary, supplementary, explementary and opposite. Trigonometry. Right triangle. Orthogonal projection of oriented segments. Goniometric functions of important angles. 0, π / 6, π / 4, π / 3, π, 3/2 π, 2π. Addition, subtraction and duplication formulas. Applications. Cartesian components of a vector. Trigonometric meaning of the angular coefficient of the straight line. MATHEMATICAL ANALYSIS Functions, limits, discontinuity. Differentials and infinitesimals Continue functions. Differential of the independent variable. Differential of the function. Derivation. Derivative in a point. Incremental ratio. Different ways to indicate a derivative. Applications. Increasing and decreasing functions. Maximum and minimum of a continuous function. The trigonometric tangent of the geometric tangent to a curve at a point. Derivative function and its primitive. Main derivatives. Rules of derivation. Calculation of derivatives. Integration. From the derivative to the primitive function. Indefinite integral. Additive constant. Immediate Integrals. Integration rules. Computation of Indefinite Integrals. The definite integral, areas calculation.