89332 - Applied Mathematics and Statistics

Learning outcomes

Upon completion of the course, students will possess basic knowledge of differential and integral calculus for functions of one real variable, the basics of calculus for functions of several variables, and methods for solving differential equations. Specifically, students will be able to: - represent data or functions graphically; - perform applications of differential and integral calculus for functions of one or more real variables. Students will also have a basic understanding of computer programming and will therefore be able to create simple programs for analyzing numerical data. Upon completion of the course, students will be familiar with elementary statistics and will be able to calculate the mean, median, and mode of statistical variables, as well as understand some fundamental distributions (e.g., Gaussian).

Course contents

Differential and integral calculus for functions of one real variable and the first elements of calculus for functions of several variables as well as methods for solving differential equations.

The course introduces the fundamental tools of calculus and mathematical analysis, providing the theoretical and practical foundations required to address quantitative problems arising in biotechnology and the life sciences.

The first part of the course reviews the essential concepts of algebra and analytic geometry needed for the study of functions. It then introduces real-valued functions of a real variable, with particular emphasis on domains, ranges, elementary functions, function composition and inversion, and graphical representation.

The concepts of limits and continuity are subsequently developed, together with the fundamental properties of continuous functions and the main theoretical results. Differential calculus includes the definition of the derivative, its geometric and practical interpretation, differentiation rules, and the derivatives of elementary and composite functions. These tools are applied to the qualitative analysis of functions, including monotonicity, concavity, local and global extrema, inflection points, and graph sketching. Applications to optimization and mathematical models relevant to biology and biotechnology are discussed throughout the course.

The section on integral calculus covers both indefinite and definite integrals, the principal techniques of integration, and the Fundamental Theorem of Calculus. Integration is presented both as the inverse operation of differentiation and as a tool for computing areas, average values, cumulative quantities, and other measures of scientific interest.

The course also introduces the basic concepts of multivariable calculus. Topics include functions of several variables, graphical representation of surfaces and level curves, partial derivatives, the gradient, the total differential, and local analysis of multivariable functions, together with an introduction to unconstrained optimization.

A dedicated module covers ordinary differential equations, with particular emphasis on first-order separable and linear equations, initial value problems, and elementary analytical solution methods. Simple mathematical models describing biological phenomena, such as population growth, exponential decay, transport processes, and basic kinetic models, are presented as applications.

The course also introduces the fundamentals of scientific programming and numerical computation. Students learn to develop simple computer programs for numerical data analysis, graphical visualization of functions and datasets, and the numerical solution of elementary mathematical problems.

Finally, the course provides an introduction to descriptive statistics and elementary probability. Students learn how to summarize datasets using measures of central tendency and variability, including the mean, median, mode, variance, standard deviation, and percentiles, as well as appropriate graphical representations. Basic concepts of probability are introduced together with the probability distributions most commonly used in the life sciences, with particular emphasis on the normal (Gaussian) distribution, the binomial distribution, and other fundamental discrete and continuous distributions.

Course Topics

• Review of elementary algebra and analytic geometry.

• Real-valued functions of one real variable: domain, range, graphs, and elementary functions.

• Limits and continuity.

• Differential calculus: derivatives, differentiation rules, fundamental theorems, and applications.

• Qualitative analysis of functions and optimization problems.

• Integral calculus: indefinite and definite integrals, techniques of integration, and the Fundamental Theorem of Calculus.

• Applications of integration to areas, average values, and cumulative quantities.

• Functions of several variables: domains, surfaces, level curves, partial derivatives, gradients, and differentials.

• Introduction to optimization of multivariable functions.

• Ordinary differential equations: first-order equations and an introduction to second-order linear equations.

• Elementary mathematical models in biotechnology and the life sciences.

• Introduction to scientific programming for numerical computation and data visualization.

• Descriptive statistics: measures of central tendency and variability, and graphical representation of data.

• Elementary probability and the main probability distributions, with particular emphasis on the normal (Gaussian) distribution.

Readings/Bibliography

Metodi Matematici per le Scienze applicate, C. Bisi, R. Fioresi

Teaching methods

Lectures

Assessment methods

Written and oral test

Students with Specific Learning Disorders (SLD) or temporary or permanent disabilities are strongly encouraged to contact the University's dedicated support office as early as possible (https://site.unibo.it/studenti-con-disabilita-e-dsa/it). The office will work with the students concerned to identify any appropriate accommodations. Any proposed accommodations must be submitted to the course instructor for approval at least 15 days in advance. The instructor will assess their suitability in relation to the intended learning outcomes and academic requirements of the course.

The use of artificial intelligence (AI) tools is not permitted in any form during assessment. Any use of such tools will be considered a breach of academic integrity.

Teaching tools

virtuale.it

Office hours

See the website of Rita Fioresi