# Mathematical analysis 1 (2008/2009)

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Course code
4S00030
Credits
6
Coordinator
Sisto Baldo
Teaching is organised as follows:
mod.1 3 MAT/05-MATHEMATICAL ANALYSIS 1° Q - solo 1° Anno, 2° Q Sisto Baldo
mod.2 3 MAT/05-MATHEMATICAL ANALYSIS 1° Q - solo 1° Anno, 2° Q Sisto Baldo

#### Learning outcomes

Module: mod.2
-------
In the second unit of the course, we present the basic results of differential
and integral calculus in one variable. We conclude with the theory of series of real numbers.

Module: mod.1
-------
This course is devoted to the presentation of the basic notions of differential and integral calculus for functions of one real variable.
In the first unit, the real numbers are introduced together with the notions and main properties of limits, continuity and derivative.

#### Syllabus

Module: mod.2
-------
Mean value theorem. Sign of the first derivative and monotonicity. De L'Hopital Theorem. Taylor's theorem (with error term in Peano and Lagrange form). Taylor
polynomials of some elementary functions, Taylor series.

The problem of computing areas: integration in the sense of Riemann. Integrability of monotonic and continuous functions. The fundamental theorem of calculus.
Integration rules. Generalized integrals.

Solution of some first order O.D.E.: linear equations, separation of variables.

Infinite series of real numbers: some convergence criteria. Basic properties
of power series.

Module: mod.1
-------
The numeric sets N, Z, Q. The real line R: completeness axiom.
Maximum and minimum of a subset of R, supremum and infimum.
Real functions of one real variable: domain, codomain, image, graph.
Some simple manipulation of graphs, basic functions and their graphs. Trigonometric functions, inverse trigonometric functions, exponential and logarithmic functions.

Limits: from the naive idea to the rigorous definition. Infinite limits, limits
at infinity.

Sequences and their limits. Sequential characterization of limits of real functions. Limits of increasing sequences. Some fundamental limits.

Continuous functions. Basic theorems on continuous functions.

Slope of a graph at a point: intuitive and rigorous definition
of the derivative of a function. Applications of derivatives.
Derivation of elementary functions and derivation rules.
Convex functions and first/second derivatives of a function.

#### Assessment methods and criteria

Module: mod.2
-------
Written test and oral exam.

Module: mod.1
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Written test and oral exam.

 Author Title Publisher Year ISBN Note Adams, R. Calcolo differenziale. [volume 1] Funzioni di una variabile reale (Edizione 3) Ambrosiana 2003 884081261X