27991 - Mathematical Analysis T-1

Course contents

Number Sets:

• Natural numbers (general concepts, elements of combinatorics, Newton's binomial theorem, principle of induction),
• Integers, rational numbers, real numbers (completeness axiom and its consequences).

Function Theory Basics:

• Domain, range, injective, surjective, bijective functions; function composition; inverse functions.
• Elementary real-variable functions: nth root, power, exponential, logarithm, trigonometric functions.

Sequences in R:

• Sequence limits; fundamental theorems on limits; monotonic sequences.
• The number e; some basic sequence limits.

Numerical Series:

• Convergence issues (model series; non-negative term series, absolutely convergent series; convergence criteria).

Limits of Real-Valued Functions:

• Extension of results for sequences, some important limits.
• Function continuity: Intermediate Value Theorem, Zero Theorem, Weierstrass Theorem.

Derivation of Real-Valued Functions:

• Derivative as the limit of incremental ratios, calculation rules (derivatives of sums, differences, products, or quotients of differentiable functions).
• Derivatives of elementary functions, the relationship between differentiability and continuity.
• Theorems on differentiable functions: Rolle's Theorem, Lagrange's Theorem, Cauchy's Theorem.
• Consequences of Lagrange's Theorem and function analysis.

Indeterminate Forms and Limits:

• L'Hôpital's Rule.

Taylor's Formula with remainders according to Peano and Lagrange, applications to numerical calculations.

Indefinite Integration:

• Definition of antiderivatives, integration methods (integration by parts, variable substitution in integrals, partial fraction method).
• Necessary conditions for a function to admit an antiderivative (Darboux's Theorem).

Definite Integration:

• Definition of Riemann integrals, Riemann integral properties, Riemann integrability of continuous functions.
• Mean Value Theorem and the Fundamental Theorem of Calculus.

Ordinary Differential Equations:

• Separable variable equations, first-order linear equations (homogeneous and non-homogeneous), second-order linear equations with constant coefficients (homogeneous and with source terms).
• Applications of differential calculus.

 

Readings/Bibliography

Bertsch, Dal Passo, Giacomelli - Analisi matematica (McGraw-Hill)

Salsa, Squellati - Esercizi di Matematica I (Zanichelli)

Teaching methods

 

Lectures and classroom exercises.

Assessment methods

 

The exam is a 90-minute written test focusing on the types of exercises proposed during lectures and some theoretical questions aimed at assessing understanding of the key results presented in the course.

To take the exam, students must register on Almaesami at least four days before the test. If unable to attend, students are kindly requested to withdraw their registration.

Teaching tools

 

Exercises to be solved at home and in class with the instructor's guidance will be provided during lessons. Educational materials will be uploaded progressively on the virtual.unibo.it platform.

Office hours

See the website of Paolo Albano

See the website of Marco Mughetti