Numbers; sequences and series; functions of one real variable; limits and continuity; differential calculus in one variable and approximation; Riemann integral.
1) M. Bramanti, C.D. Pagani, S. Salsa,"Analisi Matematica 1", Zanichelli (2008)
2) S. Salsa, A. Squellati, "Esercizi di Analisi Matematica" Vol. 1, Zanichelli (2011)
3) G. Anichini, G. Conti, "Analisi Matematica 1", Pearson (2010).
4) Carlo Sbordone, Paolo Marcellini, "Esercitazioni di matematica Volume I", Parte prima e parte seconda
Learning Objectives
Acquisition of mathematical tools needed for the description and understanding of physical phenomena. Strengthening and development of the attitude to both the analytical and logical deductive reasoning to identify the essential data in the analysis and synthesis of the presentation of possible problems.
Prerequisites
Elementary logic and algebra; literal calculus. Euclidean geometry in 2 and 3 dimenions. Lenght, area, volume of elementary hapes. Analytic geometry: polar coordinates and graphcs of elementary functions. Trigonometry.
Teaching Methods
The course includes the carrying out of lectures and classroom exercises.
The exam consists of a written practice test and a written or oral theory test. These tests are aimed at verifying the development of the attitude to the analytical and logical deductive reasoning and the learning of mathematical tools, necessary for the description and understanding of physical phenomena, both from a theoretical and from a practical point of view. During the course there will be two learning assessment tests; the passing of such tests is equivalent to passing the written practice test.
Course program
THE NUMBERS
Sets. Set operations: union, intersection, difference, complement, cartesian product, inclusion. sum. Summation formulas and properties. Sum of numbers in a geometric progression. Natural numbers. Principle of induction. Factorial of n and binomial coefficients. Newton binomial. Rational numbers. Real numbers. Absolute value. Upper and lower bounds. Supremum and infimum, maximum and minimum of a set.
REAL FUNCTIONS
Domain, image, graph. Bounded, injective, surjective, monotone functions. Composition of functions. Inverse function. Exponential, logarithmic functions. Trigonometric functions. Hyperbolic functions. Inverse trigonometric and inverse hyperbolic functions.
SEQUENCES
Formal definition. Bounded sequences. Limit of a sequence. Uniqueness of the limit. Convergent, divergent, oscillating sequences. Operation with limits. Indeterminate forms. Squeeze theorem. Permanence of sign. Monotone sequences. The number e. Use of the asymptotic. Fundamental limits.
LIMIT OF FUNCTIONS. CONTINUOUS FUNCTIONS.
Intervals in R. Definition of limit. Relations with limits of sequences. Right-handed and left-handed limit. Operations. Uniqueness, permanence of sign, squeeze theorems. Horizontal, oblique, vertical asymptotes. Fundamental limits. Continuous functions. Continuity of the sum, product, ratio and composition of functions. Weierstrass theorem. Bolzano and intermediate value theorem. Continuity of monoton functions. Inverse of continuous functions.
DIFFERENTIAL CALCULUS OF ONE VARIABLE FUNCTIONS
Derivative and geometrical interpretation. Derivatives of elementary functions. Operation with derivatives. Differentiability and continuity. Derivatives of inverse functions. Local maxima and minima. Fermat's theorem. Rolle's theorem. Lagrange's theorem. First derivative sign and monotonicity. Higher order derivatives. Convex and concave functions. Inflection points. Graphic representation of a function. L'Hospital’s rule. A sufficient condition of differentiability.
Linear approximation. Little o notation. Taylor's theorem. Peano and Lagrange forms of the remainder.
INTEGRAL CALCULUS OF ONE VARIABLE FUNCTIONS
Integral of bounded functions. Geometric interpretation. Class of integrable functions. Non-integrable functions. General properties: linearity, additivity, comparison. The integral mean value theorem. Primitives. The fundamental theorem of calculus. Methods of integration: substitution, decomposition by parts, by recurrence. Generalized integrals. Comparison and asymptotic comparison tests. Absolute convergence. Integral functions.
NUMERICAL SERIES
Convergent, divergent, oscillating series. Geometrical series. Necessary condition of convergence.
Comparison and asymptotic comparison. Root and ratio tests. Integral test. Generalized harmonic series. Absolute convergence. Leibniz criterion.