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Teacher
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PALUMBO BIAGIO
(syllabus)
INTRODUCTORY TOPICS
Outline of hypothetical-deductive systems. Outline of a constructive definition of the set of reals. Axiomatic definition of ℝ. Field axioms and their consequences. Order axioms. Definition of inequality symbols. Transitive property of inequality. Tricotomy law and other consequences of the order axioms. Intervals. Representation of real numbers on a line. Upper bound, maximum and supremum of a set; lower bound, minimum and infimum. Characteristic properties of inf and sup. Completeness axiom (existence of sup) and consequences. Module symbol. Module of a product and of a fraction. Triangular inequality. Inductive sets. Unlimiteness of inductive sets. Intersection of a family of inductive sets. Definition of the set ℕ. Definition of sets ℤ and ℚ. Induction method. Examples of proofs by induction: Bernoulli's inequality and its generalizations, formulas concerning summations and products, etc. Recursive definitions: natural and integer exponent powers, factorial, summation and product. Power properties. Existence of irrational numbers. Decimal representation of real numbers. k-permutations, permutations, combinations. Binomial coefficients and their properties (law of complementary terms, Stifel's law). Index translation in a summation. Newton's binomial formula. Integer and decimal part. Density of rationals in reals.
ELEMENTARY FUNCTIONS
Informal and formal definition of function. Domain, image, range. Limited and unlimited functions. Symmetries. Functions obtainable from equations in implicit form. Some examples of elementary functions: linear function, second-degree polynomial function, power functions with positive and negative integer, n-th root function and its reciprocal. Module function and module combinations. Sign function. Floor and mantissa functions. Periodic functions. Monotony in intervals. Inverse functions. Graphic meaning of the inversion. Exponential and logarithm. Euler's number "e". Trigonometric functions and their inverses. Hyperbolic functions and their inverses. Composite functions; decomposition of a composite function and determination of the domain.
LIMITS AND CONTINUITY
Neighborhood of a point, complete and incomplete. Circular neighborhoods. Informal illustration of the limit. Definition of finite limit for x → a (finite). Examples of verification and shortcuts. Limit uniqueness. Operations on limits. Examples of non-existence of the limit. Product limit of an infinitesimal function by a limited one. Definition of a continuous function at a point and in an interval. Geometric meaning of continuity in intervals. Continuity of the sum, of the product, etc. of continuous functions. Continuity of constant functions and linear functions. Continuity of polynomial and rational functions. Sign-preserving theorem. Continuity of the inverse function. Resolution of indeterminate forms 0/0 through the simplification of a common factor. Squeeze principle. Limit for x → 0 of sen x / x, and other limits related to it. Limits regarding logarithms and exponentials. Right and left limits, both finite and infinite. Calculation of right and left limits. Right and left continuity. Continuity in semi-open and closed intervals. Infinite limits for x → a finite. Infinite limits for x → a from right or left. Theorem on the infinite limit of the reciprocal function. Theorems on infinite limits (sums and products of infinite limits). Finite limit to infinity and its graphic meaning. Horizontal asymptotes. Infinite limits at infinity. Calculation of the infinity limit of a ratio of polynomials and other examples with irrational functions. Extension of the squeeze principle for x → +∞. Limits for x → ± ∞ of irrational functions. Infinity behavior of logarithm and exponential. Theorem of the existence of zeros. Intermediate-value theorem and its extensions. Existence and uniqueness of the n-th root of a positive number. Absolute maxima and minima of a function in an interval. Weierstrass theorem. Constant-sign theorem and application to the resolution of inequalities of any type. Demonstration of the existence of roots for not elementary solvable equations. Infinity order of infinity for x → +∞. Classification of discontinuities. Continuity of composite functions.
DERIVATIVES AND APPLICATIONS
Tangent line problem and instantaneous speed problem. Incremental ratio. Derivative of a function at a point. Examples of calculating the derivative with the definition. Continuity of derivable functions. Derivative intended as a function (derivative in a generic point of the domain). Direct calculation of the generic derivative for some elementary functions. Derivation rules: derivative of a function multiplied by a constant, derivative of a sum and of a product. Derivative of the reciprocal function and of a relationship. Derivatives of other elementary functions obtainable from the rules of derivations. Derivative of a composite function. Derivative of x^a for every real exponent a. Derivation of the inverse function. Calculation of the inverse derivative of given functions (including inverse trigonometric and hyperbolic functions). Incremental ratio on the right and on the left. Right and left derivative. Relative maximum and minimum. Higher order derivatives. Fermat's theorem (vanishing of the derivative at an interior extremum). Rolle's theorem: graphic meaning and examples of applicability and non-applicability. Lagrange theorem: graphic and kinematic meaning. Shortcut theorem (calculation of the right or left derivative as the limit of the derivative), and application to the study of angular points. Monotonicity theorem of derivable functions and the null-derivative theorem. Search for relative maxima and minima using the derivative. Inflection points with horizontal tangent line. Examples of curve sketching. Proof of existence (and possible uniqueness) of root of non solvable equations. Oblique asymptotes. Proof of inequalities from monotonicity properties. Proof of identities from the null-derivative theorem. Non-derivability points: angular points, cusps, inflection points with vertical tangent. Examples of functions with terms in modulus. Functions with cusps and inflection points with vertical tangent. Trigonometric functions. Functions with discontinuities.
INTEGRALS
Computing the area of a plane figure. Partition of a closed and limited interval. Lower and upper integral sums. Separation of lower and upper integral sum. Definition of integral as a separating element between lower and upper integral sums. Integrability of continuous functions. Examples of calculating integrals only with definition, and geometric interpretations. Norm of a partition. Definition of integral through an unique integral sum (as a limit), and equivalence of the two definitions. Monotonicity properties. Linearity properties. Generalized monotonicity properties. Additiveness with respect to the integration interval. Extensions of the integral symbol to cases a = b and a b. Generalized additivity to n intervals. Use of the differential symbol within an integral; critical remarks on the use of Leibniz symbols. Mean value theorem. Definition of integral function. Remarks on the domain of an integral function. First fundamental theorem of integral calculus. Primitives. Primitive uniqueness theorem. Second fundamental theorem of integral calculus and application examples with immediately integrable functions. Indefinite integral symbol. Some immediate integrals and other similar cases (integration of f(ax)). Computing areas of figures delimited by two or more curves. Other examples of integrals that can be computed with simple transformations of the integrand function. Integration by substitution. Examples of definite and indefinite integrals solvable by substitution. Examples of integration of irrational functions. Other particular integrals to use as immediate. Decomposition in partial fractions. Weighted mean-value theorem and applications.Improper integrals (integral of a continue constant-sign function in [a, + ∞)). Integrals of rational functions in sine and cosine. Integration by parts, with application examples. Explicit expression of an integral function containing modules. Study of the properties of a non elementary computable integral function. Various techniques for indefinite integration.
TAYLOR'S FORMULA
Definition of Taylor polynomial of order n. Explicit expression of the Taylor polynomial. Derivation, integration and substitution in Taylor polynomials. Examples of Taylor polynomials. Taylor's formula, with error expressed in integral form and in the Lagrange form. Examples of approximate calculation of function values, with error valutation. The error in Taylor's formula is an infinitesimal of order n. De L’Hôpital rule: examples of applications and cases of non-applicability. Symbol "little-o": definition and main algebraic properties. The Taylor polynomial is the only one for which the error is o((x - a)^n). Theorems on the use of "little-o". Examples of manipulation of expressions containing "little-o": Taylor polynomials of products and composite functions. Application to indeterminate forms 0/0. Linearization formulas and applications. Other cases of indeterminate forms solvable with the De L’Hôpital rule: 0^0, 1^∞, ∞^0.
COMPLEX NUMBERS
Abstract definition of the set ℂ as a set of ordered pairs of real numbers. Verification of field properties. Subfield ℂ_0 and isomorphism with the field ℝ of real numbers. Imaginary unit. Representation of a complex number (x, y) in the algebraic form x + iy. Conjugate of a complex number. Module of a complex number. Graphical representation of complex numbers (Gauss plane). Polar coordinates. Trigonometric representation of a complex number. Argument of a complex number. Principal argument. Graphic meaning of module and argument. Main properties of the module in the complex field. Product and ratio of complex numbers written in trigonometric form. De Moivre's formula. Roots in the complex field.
SEQUENCES AND SERIES
Definition of sequence. Limit of sequences, finite and infinite. Verification examples. Theorems on sequence limits (uniqueness, limit of a sum, squeeze principle, etc.). Boundness of convergent sequences. Monotone sequences. Definiively valid properties of sequences. Regularity of monotone sequences (in the two cases of bounded and unbounded successions). Subsequences. Limit of a subsequence, and application to the proof of non-existence of a limit. Bolzano-Weierstrass theorem. The number "e" as a sequence limit. Definition of numerical infinite series. Convergent series and its sum. Divergent and indeterminate series. Examples of determination of the character (and possibly sum) of a series only with the definition. Necessary condition for the convergence of a series. Linearity theorem, also if one of the series is divergent. Associative properties for the series, and related counterexamples. Replacement or cancellation of a finite number of terms in a series. Regularity of series with nonnegative terms. Telescoping series. Geometric series. Divergence of the harmonic series. Convergence tests for series with positive or nonnegative terms. Comparison test. Report test. Root test. Integral test. Generalized harmonic series. Limit comparison test. Special case of limit comparison test with the generalized harmonic series (infinitesimal order test). Weak version of the limit comparison test. Convergence tests for general series: absolute convergence test, Leibniz test for alternating series.
SOME TOPICS IN NUMBER THEORY
Divisibility in ℕ and in ℤ. Prime and composite numbers numbers. The unique factorization theorem. Elementary proofs of irrationality from the unique factorization theorem (integer roots, logarithms of integers in integer base). Irrationality of e.
(reference books)
B. Palumbo, M.C. Signorino: FUNZIONI ALGEBRICHE E TRASCENDENTI, ed. Accademica (Roma, 2018)
M. Amar, A.M. Bersani: ANALISI MATEMATICA I, esercizi e richiami di teoria. Ed. La Dotta (Bologna, 2013)
Exercises and former tests distributed by the teacher
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Università Roma Tre