MATHEMATICS AND ELEMENTS OF STATISTICS

Knowledge and understanding.

Students will be able to use mathematical and statistical language with rigor and understand the process of building mathematical models in a wide range of application areas of geological sciences. They will be able to acquire a wide range of techniques for calculating and manipulating the most common objects of Mathematics and Statistics, such as functions of one variable , differential calculus and the normal distribution of data.

Apply knowledge and understanding.

Students will be able to acquire skills in designing and solving a model based on a real-world problem.

Expressing judgments.
Students will be challenged to express informed judgments by evaluating the appropriateness and accuracy of mathematical techniques when applied to engineering problems. They will develop the ability to critically assess and select the most suitable mathematical methods, enhancing their problem-solving skills and engineering decision-making processes.

Communication skills.
The course places a strong emphasis on developing effective communication skills, equipping students with the ability to articulate mathematical concepts and problem-solving approaches clearly and concisely. Through exercises and collaborative discussions, students will learn to convey complex mathematical ideas to both technical and non-technical audiences, a crucial skill for success in their engineering careers.

Learning skills.

Students will actively cultivate essential learning skills, including self-directed study, problem-solving strategies, and adaptability when approaching mathematical challenges. Through a variety of exercises and assessments, students will develop the ability to independently explore and apply mathematical concepts, fostering a lifelong capacity for continued learning in engineering and related disciplines.

Use and understand mathematical language with rigour.

Understand the process of mathematical model-building in a range of application areas of geological sciences.

Demonstrate skills in designing and solving a model based on a real-world problem.

Acquire a wide range of techniques of computation and manipulation of mathematical objects, such as sequences, series, limits, differential calculus in one variable.

Face-to-face lectures and individual and group works.

Information for students with disabilities and / or SLD

To guarantee equal opportunities and in compliance with the laws in force, interested students can ask for a personal interview in order to plan any compensatory and / or dispensatory measures, based on the didactic objectives and specific needs. It is also possible to contact the referent teacher CInAP (Center for Active and Participated Integration - Services for Disabilities and / or SLD) of our Department.

No prerequisites are expected.

Mandatory

Review of Arithmetic and Algebra. Scientific notation of real numbers. Rounding for decimal and significant figures. Fractions, powers, logarithms, absolute value, and properties. Simplification of algebraic expressions. Factoring of polynomials.

Trigonometric functions. Angles: radians and degrees. Definition of sine, cosine, tangent. Properties of trigonometric functions. Trigonometric equations. Inverse functions: arcsine, arccosine, arctangent. Trigonometric identities. Trigonometric functions and triangles.

Geometry. Analytical representation of a line: explicit and implicit forms. Slope and intercept. Parallel and perpendicular lines. Intersection point between two lines. Distance between two points. Line passing through two points. Distance between a point and a line. Equation of a parabola. Parabola through three points.

Interpolation. Interpolation nodes. Interpolating polynomial. Lagrange polynomials. Linear, quadratic, and cubic interpolation. Accuracy. Applications to geological problems.

Sequences. Definition of numerical sequence. Arithmetic, geometric, harmonic sequences. Convergent, positively divergent, negatively divergent, oscillating sequences. Limits of sequences. Proof of convergence and divergence of elementary sequences using the definition. Algebra of limits. Introduction to numerical series.

Real functions. Definition of a function and its graph. Domain. Absolute minimum and maximum points. Local minimum and maximum points. Composite functions. Examples of functions: linear functions, exponential functions, logarithmic functions, absolute value function. Inverse functions. Operations between functions. Limits. Asymptotes for the graph of a function: horizontal, vertical, and oblique.

Differential calculus. Definition of the derivative of a function and its geometric interpretation. Derivatives of elementary functions. Rules of differentiation. Theorems on derivatives. Derivatives and increasing or decreasing functions. Determination of local and absolute minima and maxima. Higher-order derivatives. Concavity, convexity, and inflection points. Applications of derivatives: tangent line to a curve; position, velocity, and acceleration; constrained optimization.

Integration. Definition of antiderivative and indefinite integral. Integrals of elementary functions. Linearity of the integral operator. Integration by parts formula. Substitution formula for integration. Definite integral and fundamental theorem of integral calculus. Area under a curve. Area between two curves.

Introduction to Statistics. Brief overview on probability. Descriptive statistics. Simple data, data with frequencies. Centrality indices, dispersion indices. Mean, median, variance, quartiles. Normal distribution, Gaussian tables. Central Limit Theorem.

Contribution of teaching to the objectives of the 2030 Agenda for Sustainable Development. The course aims to provide basic knowledge of mathematical and statistical tools for different applications. The knowledge acquired will be usable in the numerous topics that make use of mathematical methods, such as models for the exploitation of sustainable energy, infrastructure design, atmospheric models to reduce the effects of climate change, in accordance with points 7, 9, 11, 12 and 13 of the 2030 Agenda.

1. Lecture notes provided during classes.

2. Matematica per le scienze, Marco Bramanti, Fulvia Confortola, Sandro Salsa. Zanichelli. 

3. Istituzioni di Matematica, Michiel Bertsch, Bollati Boringhieri.

4. Istituzioni di Matematica e Applicazioni. Paolo Marcellini, Carlo Sbordone. Liguori Editore.

Written Exam (exercises and theory), marked out of 30

Criteria for assigning marks: the following will be taken into account: clarity of presentation, completeness of knowledge, ability to connect different topics. The student must demonstrate that they have acquired sufficient knowledge of the main topics covered during the course and that they are able to carry out at least the simplest of the assigned exercises.

The following criteria will normally be followed to assign the grade:

Not approved: the student has not acquired the basic concepts and is not able to carry out the exercises.

18-23: the student demonstrates minimal mastery of the basic concepts, their skills in exposition and connection of contents are modest, they are able to solve simple exercises.

24-27: the student demonstrates good mastery of the course contents, their presentation and content connection skills are good, they solve the exercises with few errors.

28-30 cum laude: the student has acquired all the contents of the course and is able to explain them fully and connect them with a critical spirit; they solve the exercises completely and without mistakes.

Geometric meaning of the derivative. Function analysis. Convergence criterion for a geometric series. Classification of stationary points: relative minima, relative maxima, inflection points.