OVERVIEW

This intensive course provides the theoretical foundations and analytical tools essential for engineering. Structured sequentially, the curriculum moves from core functions to advanced calculus:

• Precalculus & Functions: Analysis of elementary functions, domains, image, injectivity, monotonicity, graph manipulations, and non-linear inequalities.

• Limits & Continuity: Theory of neighborhoods, algebraic and exponential indeterminate forms, fundamental limits, continuity, and asymptotes.

• Differential Calculus: Derivatives computation, points of non-differentiability, core theorems (Rolle, Cauchy, Lagrange, de l'Hôpital), and advanced curve sketching (monotonicity, extrema, convexity).

• Integral Calculus: Indefinite integrals (by parts, substitution, rational maps), definite integrals, and improper integrals.

• Differential Equations: First-order ODEs, second-order linear ODEs with constant coefficients, and Cauchy problems for engineering modeling.

AIMS AND CONTENT

LEARNING OUTCOMES

The aim of this teaching unit is to provide a practical working tool for students where rigorous Calculus is needed. The main focus is on the study of functions of one real variable (continuity, derivative, maxima/minima, integration). The last part of the teaching unit is oriented towards basic ordinary differential equations (for example separation of variables, linear first-order, and constant coefficients ODE).

AIMS AND LEARNING OUTCOMES

The course aims to provide engineering students with a solid mathematical foundation in single-variable calculus and ordinary differential equations. It focuses on developing rigorous analytical thinking, the ability to manipulate and interpret mathematical graphs, and the skills needed to set up and solve quantitative models for engineering applications.

By the end of this course, students will be able to:

• Analyze and Manipulate Functions: Identify domain, image, monotonicity, and invertibility of elementary functions, and deduce the behavior of composite or transformed functions from their graphs.

• Evaluate Limits and Continuity: Resolve algebraic and exponential indeterminate forms using fundamental limits and asymptotic analysis, and classify points of discontinuity.

• Apply Differential Calculus: Compute derivatives accurately, apply core theorems (Rolle, Lagrange, Cauchy, de l'Hôpital), and utilize first and second derivatives to map and sketch complex functions.

• Solve Integrals: Compute definite, indefinite, and improper integrals using standard methodologies such as substitution, integration by parts, and rational fraction decomposition.

• Model with Differential Equations: Solve first-order ordinary differential equations and second-order linear ODEs with constant coefficients, applying initial conditions to resolve Cauchy problems.

PREREQUISITES

To successfully follow this course, students should have a solid command of high school mathematics, specifically in the following areas:

• Algebra: Mastery of algebraic manipulations, factoring polynomials, solving first and second-degree equations, and handling systems of equations.

• Inequalities: Confidence in solving linear, quadratic, fractional, and literal inequalities, as well as systems of inequalities.

• Analytic Geometry: Knowledge of the Cartesian coordinate system, equations and properties of lines, circles, parabolas, ellipses, and hyperbolas.

• Trigonometry: Understanding of trigonometric functions (sinx, cosx, tanx), fundamental trigonometric identities, and solving trigonometric equations and inequalities.

• Basic Functions: Familiarity with the concepts of powers, roots, absolute values, exponentials, and logarithms.

TEACHING METHODS

The course is structured to balance theoretical rigor with practical problem-solving skills, ensuring a comprehensive understanding of mathematical analysis. The teaching unit is divided into two main components:

• Theoretical Lectures: These sessions focus on the core conceptual framework of the course. Students will be introduced to formal definitions, fundamental theorems, and their main proofs, establishing the logical rigor required for engineering disciplines.

• Exercise Sessions: Designed to bridge theory and practice, these interactive lessons focus on discussing and solving practical examples alongside the students. The primary objective is to enhance calculus skills, master computational techniques, and prepare students for written examinations through guided problem-solving and test simulations.

The total instructional hours are distributed as follows to optimize student learning and engagement:

• In-Presence Lectures: 48 hours of face-to-face classroom teaching, covering the core theoretical syllabus and standard exercises.

• Online Lectures: 12 hours of digital teaching, designed to complement the in-presence material.

Working students and students with certified SLD (Specific Learning Disorders), disability or other special educational needs are advised to contact the teacher at the beginning of the teaching unit to agree on teaching and examination arrangements so to take into account individual learning patterns, while respecting the teaching objectives.

Contacts of the teacher and the School's disability contact person can be found at the following link Comitato di Ateneo per l’inclusione delle studentesse e degli studenti con disabilità o con DSA | UniGe | Università di Genova

SYLLABUS/CONTENT

Module 1: Foundations and Theory of Functions

• Introduction to Functions: Definitions, domain and image, injectivity, surjectivity, invertibility, and monotone functions. Finding the inverse of a function.

• Elementary Functions: Properties and algebraic behavior of powers, roots, exponentials, logarithms, and absolute values.

• Graph Manipulation: Techniques for shifting, stretching, and reflecting graphs. Sketching y=x​, y=ln(f(x)), and y=1/f(x) starting from the graph of y=f(x).

• Equations and Inequalities: Advanced review and remarks on radical, exponential, and logarithmic inequalities.

Module 2: Limits and Continuity

• Introduction to Limits: Notion of neighborhoods, formal definitions, and verification of limits.

• Continuity: Definition of continuous functions, classification of discontinuity points, and the algebra of limits.

• Asymptotic Behavior & Indeterminate Forms: Evaluation of algebraic indeterminate forms and determination of vertical, horizontal, and oblique asymptotes.

• Advanced Limits: Fundamental (remarkable) limits and resolution of exponential indeterminate forms (1∞, 00, ∞0).

Module 3: Differential Calculus and Applications

• The Derivative: Definition, geometric meaning, and computation of derivatives for elementary and composite functions. Derivative of the inverse function.

• Differential Theorems: De l'Hôpital’s Theorem for indeterminate forms. The fundamental theorems of Rolle, Cauchy, and Lagrange.

• Applications of Differentiability: Local and absolute extrema, intervals of monotonicity, convexity (concavity), and inflection points. Study of points where differentiability fails (cusps, corners, vertical tangents). Advanced curve sketching.

Module 4: Integral Calculus

• Indefinite Integrals: Antiderivatives, fundamental integration formulas, integration by parts, and integration by substitution.

• Rational Maps: Integration of rational fractions via partial fraction decomposition.

• Definite and Improper Integrals: The Riemann integral, the Fundamental Theorem of Calculus, evaluation of areas, and the study of convergence for generalized (improper) integrals.

Module 5: Ordinary Differential Equations (ODEs)

• First-Order ODEs: Introduction to differential equations, separable variables, first-order linear equations, and the formulation and solution of the Cauchy (initial value) problem.

• Second-Order ODEs: Linear homogeneous and non-homogeneous differential equations of order two with constant coefficients (method of undetermined coefficients).

RECOMMENDED READING/BIBLIOGRAPHY

Claudio Canuto, Anita Tabacco: "Mathematical Analysis 1", Pearson

LESSONS

Class schedule

The timetable for this course is available here: Portale EasyAcademy

EXAMS

EXAM DESCRIPTION

The final examination consists of an oral exam. 

ASSESSMENT METHODS

During the exam, students will be asked to solve calculus exercises on the blackboard and will be tested on their knowledge of the main definitions, theorems, and proofs covered during the lectures.