# MATHEMATICAL ANALYSIS cod. 00013

1° year of course - First semester
Professor
Analisi matematica (MAT/05)
Field
Formazione matematico-fisica
Type of training activity
Basic
84 hours
of face-to-face activities
9 credits
hub: PARMA
course unit
in - - -

## Learning objectives

Basic notions of calculus (limits, derivatives, integrals). The student should be able to plot the graph of a function, and to handle standard integration methods

None

## Course unit content

Limits for sequences. Differential and integral calculus for real functions of one real variable

## Full programme

Sets and numbers. Basic set theory, operations between sets. Number systems: N, Z, Q, R, C. Representation of real numbers on a line; maximum, minimum, supremum, infimum of a set of real numbers; integer part and absolute value of real numbers; powers and roots. Complex numbers in various forms.

Functions: injective, surjective and bijective functions. Composition of functions; inverse function. Graphs. Real functions of one real variable. Monotonic functions. Powers with real exponent. Exponential and logarithmic functions. Angles; trigonometrical functions. Cardinality.

Sequences and series. Limits of sequences. Numeric series and convergence.

Limits and continuity. Limits of real functions of one real variable; properties. Continuity of real functions of one real variable; properties of continuous functions.

Differential calculus. Derivative and its geometric interpretation. Derivation rules (sum, product, ratio, inverse); chain rule; derivatives of the elementary functions. Relative and absolute maxima and minima; stationary points; monotony and the sign of the derivative. Main theorems (Fermat, Rolle, Lagrange aka mean-value, De l'Hopital); higher order derivatives; Taylor series development. Graphs.

Integral calculus. Primitive of a function defined in an interval; indefinite integrals. Geometric interpretation. Main properties. Fundamental theorem of the integral calculus. Integration techniques: by parts, by substitution; integration of rational functions.

## Bibliography

M. Bertsch, R. Dal Passo, L. Giacomelli, Analisi Matematica, Mc Graw-Hill