Università Roma Tre

REAL NUMBERS AND OTHER INTRODUCTORY SUBJECTS
Axiomatic Definition of R. Field Axioms and Consequences. Elemental operations and their properties. Order axioms and consequences. Inequality Symbols. Intervals. Infinity of real numbers. Representing real numbers on one straight line. Increasing and minor, maximum and minimum, upper and lower extremes of a set of real numbers. Property features of the lower extremity and the upper extremity.
Axiom of completeness (as an axiom of the existence of the upper extremity). Existence of the lower end. Extremes of separate sets (theory 1.18).
Extreme upper sum of sums (theorem 1.19 without dim.). Module. Triangular inequality for the sum and for the difference (theorem 1.21 without dim.). Inductive sets. Intersection of a family of inductive sets. Unlimited inductive sets. Definition of the set N. Definition of Z and Q sets. Real numbers archimedea property.
Induction Principle. Closing N versus adding and multiplying. Possibility of subtraction in N (theory 1.27 without dim.). Z properties (theorems 1.28, 1.29 and 1.30 without dim.). Definition of finite set (theory 1.31 without dim.).
The principle of good ordering (theory 1.32 without dim.). Definitions for induction (power to natural exponent and its principal factorial, factorial and factor factor, summation and production). Power to Full Exponent Negative and Ad rational exponent. Combinative calculation: simple layouts, simple permutations, simple combinations.
Binomial coefficients and their properties. Newton binomial formula (exclude binomial coefficient with  not integer). Full part and decimal place. Division with rest in N and Z. Density of rational in real. Existence of numbers irrational (without dim.). Irrationality of 2.
Density of the irrational in the real. Understand algebraic and transcendent numbers.
ELEMENTAL FUNCTIONS
Intuitive function concept and strict definition. Domain and coding. Image. Graph of a function. Functions injecting, persuasive, biotic. Limited and unlimited functions. Equal and odd functions. Decomposition of a function like sum of a even and odd function. Summaries, differences. Products and relationships of even and odd functions. Functions rational algebras, whole and fractured. Irrational algebraic functions. Function module and other functions with terms in module.
Sign Function. Full Part Function and Decimal Part. Periodic functions. Strongly increasing functions, not decreasing, closely decreasing, not increasing. Reverse function. Relationship between inverse function graphs. Monotony and inequality of inverse function (theory 2.3 and 2.4 without dim.). Elemental definitions of exponential and logarithm and theirs graphics. Properties of exponential and logarithm. Using different bases. Number e. Natural logarithms. Measure angles and radii of radiant circumference. Elemental definitions of the main goniometric functions (breast, cosine, tangent, cotangent) and their charts. Principal goniometry formulas. Reverse Goniometric Functions and Their Graphics. Functions hyperbolic and hyperbolic inverse, and their graphs. Composition of functions. Domain of a composite function.
LIMITS AND CONTINUITY
Around a point. Interspersed. Circular circles. Finite limit intuitive concept of a function for x  finite.
Definition of finite limit for x  finite and its graphical interpretation. Verify a limit with the definition.
Limit uniqueness theorem. Sign permanence theorem (first formulation). Operation Theorem Limits (only dim. Part (s)). Product limit between an infinitesima and a limited function (theory 3.4). Function continues at one point. Sign permanence theorem (second formulation). Continuity of sum, difference, product and relationship of continuous functions. Continuity in a set. Continuity of elementary functions. Continuity of the reverse function and composite function (theory 3.7 and 3.8 without dim.). Resolution of unspecified shapes by 0/0 eliminating a common factor. Comparison Theorem. Limit for x  0 of (sen x) / x. Other remarkable limits containing goniometric functions. Significant limits containing logarithms and exponentials. Partial Limits. Continuity to the right and to the left at one point. Continuity in a generic range, even semi-open or closed. Limit  for x  finite and its interpretation graphics, with examples of verification. Endless limits only from right or left. Limit of the reciprocal of an infinitesima function (Theory 3.11). Calculation of vertical asymptotics. Somes and products of infinite limits, or of a finite limit and an infinite. Form indefinite  /  and  ∙ 0. Finished limit for x  ±  and its graphical interpretation, with verification examples. Calculation of asymptotes horizontally. Limit ±  for x  ±  and its graphical interpretation, with verification examples. Infinite Behavior of Elementary Functions. Comparison of infinite logarithms, exponentials and powers. Infinite order of a function, perx  +  and for x  a. An infinitesimal order of a function, for x  a and for x  + .
Classification of discontinuities. Bolzano - Cauchy (the existence of zeros). Intermediate theorem. Interval values theorem extensions (at unlimited intervals, infinite limits, etc.). Existence and uniqueness of the n-th root. Signal theorem constantant application to resolution of inequalities. Continuous Function Restriction Theorem. Weierstrass theorem (existence of extremes). Dirichlet.
DERIVATE
Function Tribe problem and instantaneous velocity problem. Incremental ratio. Derivative of a function at a point. Continuous function of derivable functions. Deriving rules. Derivatives of some elementary functions (polynomials, powers to rational exponent, goniometric, exponential functions, etc.). Derivative of the composite function. Derivative of the reverse function and its graphic interpretation. Derivatives of inverse inverse and inverse hyperbolic functions. Right and left incremental ratio. Right and left derivatives. Logarithmic derivative. Top Derivatives. Points of minimum relative maximum. Fermat theorem (canceling the derivative at a relative extreme point). Rolle theorem. Lagrange theorem. Direct calculation of the right or left derivative as the derivative limit (theorem 4.7). Cauchy theorem. The monotonic theorem of the derivative functions and the theorem of the derivative nothing. Applications for the demonstration of mortality and eventual uniqueness of roots of equations and identity demonstration. Determination of extremorelative points by studying the sign of the derivative. Horizontal tangent flush points. Non-derivative points: point-angle, cuspid, vertical tangent flanks. Determination of oblique asymptotes. Applications for the study of functions and the plotting of related graphs. Differentiation of a function of a variable and the equivalence between differentiation ederaability. Noun on Leibniz.
INTEGRAL
Symbols of the areas. Geometric definition of the sub-area of a non-negative function. Subdivisions (opations) of a closed and limited range. Lower and higher integer sums of a continuous function on a limited range. Refining a partition. Separation of integral sums. Integral lower and upper. Integral of a continuous function on a closed and restricted range. Examples of calculating an integral with only definition.Number of a partition. Definition of integral of a continuous function on a closed interval and limited as limit. Equivalence of the two integral definitions (theorem 5.3 without dim.). Integral property: monotony, narrow monotony, linearity, additivity versus integration range. Integral symbol extensions (first extreme extremes equal to the second). Theorem of the media and its graphic interpretation. Triangular inequality for integers. Definition of integral function. Domain of an integral function. First fundamental theorem of integral calculus. Primitive of a function. Indefinite integer symbol. Primitive uniqueness on a range of less than constant. According to the fundamental theorem of integral calculus. Weighted average theorem. Immediate Integral. Consideration of the use of the "dx" symbol within an integral. Replacement integration (theory 5.16 without dim.). Partial integration. Other major integrals. Noun on improper integrals. Integration techniques for particular function classes: simple decomposition and application to integration of rational functions, integral rationalizations through appropriate substitutions. Study of integral function properties. Taylor's of Order n (Definition and Expression). Taylor polynomial uniqueness. Polynomial of Taylordella derivative, integer, and function f (cx) (part (iii) of theor 7.2 only in case a = 0). Full form of the rest in Taylor's form. Lagrange shape of the rest. Applies to approximate calculation of function values. Disaggregates the number and its approximate computation. De L'Hôpital rule in the case of indefinite form 0 / 0. De L'Hôpital's rule of extension for the indefinite form  /  (theory 7.8 without dim.). Unspecified forms of the type exponential. "Small" symbol and algebraic rules for manipulating it. Linearization Formulas. Taylor formula with the rest expressed as "or small". Polynomial uniqueness, the rest of which is either (xn) (theorem 7.11 without dim.). Taylor polynomials of function products. Application to the resolution of indefinite shapes.
COMPLEX NUMBERS
Definition of C as the set of ordered pairs of real numbers. Sum and product of complex numbers. Verify the field properties. Identification between R and subchapter C0. Imaginary unit. Algebraic form of a complex number. Imaginary numbers. Conjugated complex numbers. Amount and product of conjugated numbers (dim. 8.3 without dim.). Cennosull unavailability of the order in C. Representation of complex numbers in the Gauss plan. Polar coordinates. Formulas from polar coordinates to Cartesian coordinates and vice versa. Argument of a complex number. Main reason. Trigonometric representation of a complex number. Module properties in the campus complex (theory 8.5 without dim.). Product and ratio of complex numbers written in trigonometric form. DeMoivre Formula. Roots in complex field.
SUCCESSIONS AND SERIES
Definition. Limit of a succession for n  . Convergent, divergent and indefinite sequences. Regular adventures. Theorems about succession limits. Monotone successions. Regularity of monotone sequences (both cases of limited and unlimited sequences). Subsequences. Regularity of underdevelopment. Existence of underdevelopment. Successions defined by recurrence (exclude the Babylonian method). The number is expressed as a limit. Definitions series. Partial sums. Convergent, divergent, indeterminate series. Regularity of the series in terms of a constant sign. Condition required for convergence (exclude the part of the paragraph relating to the application of the Cauchy criterion). Operations with the series. Associativity. Invariance of the character of a series for replacing or deleting a number of terms. Telescopic series. Geometric Series and Applications. Harmonic Series. Number and expressed as a series. Criterion Convergence for Non-Negative Series: Comparison, Relationship, Root, Unintelligent Comparison, Asymptotic Comparison (also a weak version) of the infinitesimal order. Convergence criteria for any sign atermine series: absolute convergence (dim dim. II of theory 9.28), Leibniz criterion. Criterion of the Infinite Order for Improper Integral. Decimal representation of real numbers. Note on the use of other bases. The rules of the pearl fraction generating a periodic number (dim only of the 2nd case of the theory 9.47).
NUMBERS OF THEORY QUESTIONS
Divisibility in N and Z. Transitive property of the divisibility. First and Compound Numbers. Single factor faktorization (without dim.). Infinity of the prime numbers. Notes on function  (x) and on the prime number theorem. Simple case studies that can be demonstrated through unique factorization (roots and logarithms). Irrationality of e.

B. Palumbo - M.C. Signorino, Funzioni algebriche e trascendenti, ed. Accademica, 2015.
Theorems are meant for demonstration, unless otherwise indicated.