Introduction: Number sets. Rational and real numbers. Density property. Intervals. Absolute value and its properties. Supremum and infimum. Powers and roots. Powers with real exponent (4 hours)
Functions: Functions, domain, codomain, image and graph. Bounded, injective, surjective and bijective functions. Composed function. Inverse function. Monotonic functions. Even, odd and periodic functions. Exponential and logarithmic functions. (6 hours)
Limits: Euclidean distance. Neighbourhoods. Accumulation and isolated points. Open and closed sets. Interior, exterior and boundary points. Definition of limit (various cases). Uniqueness of limit (proof). Limit from the right and from the left. Theorem of permanence of sign for limits (proof).Comparison theorem (dim.). Convergent functions are eventually bounded (proof). Local maxima and minima. Algebra of limits. Indeterminate forms. Limits of composed functions. Particular limits (proof). (6 hours)
Sequences: Convergent, divergent and irregular sequences. Permanence of sign. Convergent sequences are regular. Comparison theorem. Limits of monotonic sequences. Nepier’s number. Limits of subsequences. Infinities and infinitesimals Comparison of infinities and infinitesimals. Order of infinity and of infinitesimal. Asymptotic behaviour (6 hours)
More about limits of functions: Particular limits (proof). “Bridge” theorem and its consequences. Indeterminate forms. Asymptotes. (2 hours)
Continuous functions of one real variable: Definition of continuous function. Continuity from the right and from the left. Theorem of permanence of the sign for contnuous functions (proof).Composition of continuous functions. Points of discontinuity. Continuity of monotonic functions. Theorem of zeros (proof).Corollary (proof).Intermediate values theorem (proof).Relationship between invertibility and monotonicity for continuous functions. Continuity of inverse function. Weierstrass’ theorem. (12 hours)
Differential calculus: Secant and tangent line. Difference quotient and derivative. Geometric meaning of the derivative. Derivative of elementary functions. Differentiable functions are continuous (proof).Functions of class C1 . Points with vertical tangent line. Right and left derivatives and their geometric meaning. Edges and cusps. Algebra of derivatives. Derivative of a product (proof).of a ratio (proof).and of composed function (proof).Derivative of inverse function (proof).Derivative of log(x), arcsin(x), arccos(x),arctan(x) . Derivative of log|x|, loga|x|,ax . Fermat’s theorem (proof).Stationary points. Local extreme point of a function. Rolle’s theorem (proof).Lagrange’s mean value theorem (proof).Monotonic behaviour of a function and sign of its derivative (proof).Theorem of de l'Hopital. Corollary to the theorem of de l'Hopital (proof).Derivatives of higher order. Convex and concave functions. Convex functions and continuity . Geometric meaning of convexity. Convexity and sign of the second derivative. Inflection points. Inflection points with vertical tangent line. Inflection points and second derivative. Study of the graph of a function. Taylor’s polynomial. MacLaurin ‘s polynomial of exponential, logarithmus, sine and cosine. Peano’s theorem. Consequences of Peano’s theorem (proof).Taylor’s polynomial in computation of limits of functions. Errors and Lagrange’s formula of remainder (16 hours)
Integration: Definition of Riemann integral. Superior and inferior sums. Integrability of continuous and bounded functions. Relationship between integral and area under the graph of a function. Properties of integrals. Mean value theorem (proof).Definition of integral function. Fundamental theorem of integral calculus (proof).Definition of primitive function. Primitive functions of a given function differ by an additive constant (proof).Computation of definite integral trough a primitive (proof).Indefinite integrals. Primitive functions of elementary functions. Integration by part (proof).Integration by substi