Learning outcomes of the course unit
The purpose of the course is to provide students with a clear understanding
of the basic ideas of calculus.
Course contents summary
The real numbers system. The absolute value, the triangle inequality. Upper and lower bounds, suprema and infima. Completeness of the real numbers. Mathematical induction. The binomial theorem.
Functions. Domain and range of a function, restriction, composition, inverse functions. Monotone functions. Bounded/unbounded functions. Elementary functions: polinomials, power functions, exponentials, logarithms, trigonometric functions. Odd and even functions, traslations, dilations, reflections.
Elemetary topology of the real line: intervals, neighborhoods, open sets, closed sets, accumulation points.
Limits of sequences. Limits of monotone sequences. The number "e", compound interest.
Limits and continuity of functions. One-sided limits. The composition theorem for limits. The algebraic rules for limits.
The behaviour of continuous functions on intervals: intermediate value theorem, inverse function theorem,
.Definition of the derivative, examples of derivatives. Interpretations of the derivative, differentiability and linear approximation. Properties of derivatives. The Rolle's theorem, the Lagrange mean value theorem, the Cauchy mean value theorem. Local maxima and minima. Higher order derivatives. Concavity and inflection points. De L'Hopital rule. The Landau symbols, order of magnitude. Stirling's formula. Taylor polinomials.
Indefinite integration. Integration by substitution, integration by parts. Rational functions.
Definite integration: the Riemann integral. The interpretation of definite integrals. Integrability of monotone functions and of continuous functions. The fundamental theorem of calculus.
Improper integrals. Convergence of integrals of nonnegative functions. Absolute convergence. Comparison tests.
Infinite series. Geometric and harmonic series. Convergence of series of nonnnegative terms, absolute convergence, ratio test. Comparison tests. Alternating sign series. Taylor series.
Differential equations. Slope fields. Separation of variables. Linear first order differential equations. Linear second order constant coefficients differential equations: oscillations.
C. Canuto - A. Tabacco, Analisi matematica I, Springer Italia
M. Bramanti - C.D. Pagani - S. Salsa, Analisi Matematica 1, Zanichelli