FUNCTIONS WITH ONE VARIABLE A
cod. 13462

Academic year 2007/08
1° year of course - First semester
Professor
Academic discipline
Analisi matematica (MAT/05)
Field
Formazione matematica
Type of training activity
Basic
56 hours
of face-to-face activities
7 credits
hub: -
course unit
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Learning objectives

Provide the basic tools of Mathematical Analysis

Prerequisites

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Course unit content

<br />REAL NUMBERS. AXIOMATIC DEFINITION OF REAL NUMBERS, MAXIMUM, MINIMUM, LEAST UPPER AND GREATEST LOWER BOUND; WHOLE PART AND MODULUS OF REAL NUMBERS; POWERS, ROOTS, NTH ROOTS OF NON-NEGATIVE NUMBERS; RATIONAL AND IRRATIONAL NUMBERS; INTERVALS, DISTANCE; NEIGHBOURHOODS, CLUSTER POINTS, ISOLATED POINTS, INTERIOR POINTS; CLOSED SETS, OPEN SETS, FRONTIER. INJECTIVE, SURJECTIVE, AND BIJECTIVE FUNCTIONS, INVERSE FUNCTION; GRAPHS; REAL FUNCTIONS OF A REAL VARIABLE, MONOTONE FUNCTIONS, EXPONENTIAL AND LOGARITHMIC FUNCTIONS; TRIGONOMETRIC FUNCTIONS. LIMITS. LIMITS OF FUNCTIONS WITH REAL VALUES, UNIQUENESS OF THE LIMIT, LIMITS OF THE RESTRICTIONS; LIMIT OF THE SUM, PRODUCT, QUOTIENT OF TWO FUNCTIONS; PERMANENCE OF THE SIGN, COMPARISON THEOREMS; RIGHT AND LEFT LIMIT; LIMITS OF MONOTONE FUNCTIONS; ORDERS OF INFINITESIMALS AND INFINITIES, ASYMPTOTES. CONTINUOUS FUNCTIONS. CONTINUITY OF REAL FUNCTIONS OF A REAL VARIABLE, RESTRICTIONS OF CONTINUOUS FUNCTIONS, COMPOSITION OF CONTINUOUS FUNCTIONS; SUM, PRODUCT, QUOTIENT OF CONTINUOUS FUNCTIONS; EXAMPLES OF CONTINUOUS FUNCTIONS; DISCONTINUITY, EXAMPLES OF DISCONTINUOUS FUNCTIONS; ZERO THEOREM; CONTINUITY AND INTERVALS; CONTINUITY AND MONOTONY; CONTINUITY OF INVERSE FUNCTIONS; WEIERSTRASS THEOREM. DIFFERENTIAL CALCULUS. DIFFERENCE QUOTIENTS, DERIVATIVES, RIGHT AND LEFT DERIVATIVES; GEOMETRICAL SIGNIFICANCE OF THE DERIVATIVE; DERIVATION RULES: DERIVATIVES OF THE SUM, PRODUCT, QUOTIENT OF TWO FUNCTIONS; DERIVATIVES OF COMPOUND FUNCTIONS AND INVERSE FUNCTIONS; DERIVATIVES OF ELEMENTARY FUNCTIONS; RELATIVE MAXIMUMS AND MINIMUMS; STATIONARY POINTS; RELATIONSHIP BETWEEN MONOTONY AND SIGN OF THE DERIVATIVE; ROLLE'S THEOREM AND LAGRANGE'S THEOREM AND THEIR GEOMETRICAL INTERPRETATION, CAUCHY'S THEOREM AND L`HOPITAL'S THEOREM; CONVEX FUNCTIONS, DERIVATIVES OF CONVEX FUNCTIONS, RELATIONSHIP BETWEEN CONVEXITY AND SIGN OF THE SECOND DERIVATIVE; TAYLOR'S FORMULA WITH PEANO, LAGRANGE AND INTEGRAL REMAINDER; STUDY OF LOCAL MAXIMUMS AND MINIMUMS WITH CALCULATION OF SUCCESSIVE DERIVATIVES. INTEGRALS. PARTITIONS OF AN INTERVAL; UPPER AND LOWER INTEGRAL, INTEGRABLE FUNCTIONS IN AN INTERVAL, INTEGRABILITY OF CONTINUOUS FUNCTIONS AND MONOTONE FUNCTIONS; GEOMETRICAL INTERPRETATION OF THE INTEGRAL; PROPERTIES OF INTEGRALS; MEAN OF AN INTEGRABLE FUNCTION; INTEGRALS ON DIRECTED INTERVALS; FUNDAMENTAL THEOREM OF INTEGRAL CALCULUS; PRIMITIVES, INDEFINITE INTEGRALS; INTEGRATION BY PARTS AND BY SUBSTITUTION; INTEGRALS OF RATIONAL FUNCTIONS. "

Full programme

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Bibliography

 J. Cecconi, G. Stampacchia,  Analisi Matematica 1, Ed. Liguori, 1974;<br />M. Giaquinta, G. Modica,  Analisi Matematica 1: Funzioni di una variabile, Ed. Pitagora, 1998;<br />E. Giusti, Analisi Matematica 1, Ed. Boringhieri, 1983.

Teaching methods

Teaching method: classroom lectures and classroom exercises<br />Assessment method: written and oral examination

Assessment methods and criteria

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Other information

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