Biotechnologies

Activity type

Discipline matematiche, fisiche, informatiche e statistiche

SSD

MATH-04/A

Year

1st year

Semester

1st semester

CFU

3

Hours distribution

24 classroom hours

Lecturers

GIADA BASILE

Activity type

Discipline matematiche, fisiche, informatiche e statistiche

SSD

MATH-03/A

Year

1st year

Semester

1st semester

CFU

3

Hours distribution

24 classroom hours

Lecturers

GIADA BASILE

General skills
This course concerns the applications of the fundamentals tools of calculus and probability and to the solution of problems emerging within Applied Sciences, with a specific reference to Biotecnology.
The main goal is learning the basic concepts of differential and integral calculus, discrete and continuous probability, as well as their application to biological sciences.
It is assumed that students entering the course know the basics of elementary mathematics from the high school. The fact that students actually have the mentioned pre-knowledge is certified by the entrance test.

The course includes both lectures and exercise sessions, aiming to test the ability of the students to apply the theoretical knowledge to the solution of concrete problems.

Specific skills

A) Knowledge and understanding
• Knowledge and understanding of the concept of limit and of the fundamentals of differential and integral calculus.
• Knowledge and understanding of the fundamentals of probability theory.
• Knowledge and understanding of diagnostic tests for the analysis of medical data.

B) Applying knowledge and understanding
• Ability to properly use the specific terminology of mathematics and probability.
• Ability to translate a concrete problem, appearing e.g. in the context of Biological Sciences, to a corresponding mathematical problem, by a suitable procedure involving approximation, abstraction, and modeling.
• Ability to use deductive reasoning in an abstract setting.
• Ability to recognize the mathematical tools and concepts appearing within other courses (specifically: Physics, Chemistry, Biology) and to properly use them.
• Ability to find the most convenient procedure to solve simple mathematical problems.

C) Making judgements
• Ability to autonomously formulate examples to mathematical statements.
• Ability to self-questioning.
• Ability to autonomously evaluate the validity of a theoretical model, through suitable probabilistic tests on the empirical data collected in a laboratory.


D) Communication skills
• Ability to communicate what has been learned through written themes.
• Ability to formulate a logically structured speech.

E) Learning skills
• Learning the specific terminology.
• Ability to make the logical connections between the topics covered.
• Ability to identify the most relevant topics in a subject.

The main goal of the course is to provide basic mathematical notions (particularly concerning calculus and probability theory) for mathematical modeling, quantitative treatment, and resolution of typical problems in applied sciences. Specifically, students will need to acquire the fundamental tools that will subsequently be applied in the analysis of experimental data.
The following points constitute essential requirements for a positive evaluation:
i) Knowledge and understanding of the fundamental concepts of differential and integral calculus, probability theory, and diagnostic tests for the analysis of medical data;
ii) Ability to apply knowledge and understanding, particularly in properly using the specific terminology of calculus and probability, translating a concrete problem into mathematical terms, applying inductive reasoning to abstract problems, using mathematical methods proficiently, and choosing the most suitable procedures for resolution;
iii) Autonomy of judgment regarding the validity of mathematical statements and quantitative models;
iv) Communicative and logical skills in the presentation and discussion of problems and resolution procedures.

The course requires basic knowledge of elementary mathematics acquired in high school.

Module: Mathematics 1
Probability theory. Events (definition, intersection and union of events, mutually exclusive events). Probability of an event. Properties of the probability function.
Conditional probability and independent events. Bayes' theorem.
Random variables. Cumulative distribution function.
Discrete random variables. Probability mass function. Expectation and variance of a discrete distribution. Uniform, Bernoulli, binomial, and Poisson distributions. Elements of combinatorics, Stirling’s formula. Poisson distribution as a limit of the binomial distribution.
Continuous random variables. Probability density function. Expectation and variance of a continuous distribution. Uniform, exponential, and normal distributions.
Introduction to descriptive statistics.




Module: Mathematics 2
Set theory and operations. Numerical sets. Real line and geometric representation. Intervals. Neighborhoods. Open and closed sets.
Functions: domain of definition, domain, image. Injective, surjective, and bijective functions. Inverse function. Composite function.
Real functions of a real variable. Graph of a function. Transformations of the graph of a function.
Even, odd, and periodic functions. Monotonic functions. Elementary functions and their graphs.
Limits of functions. Limits at finite points and at infinity. Algebra of limits. Notable limits.
Continuous functions. Intermediate value theorem.
Derivative of a function and geometric interpretation. Right and left derivatives. Differentiable functions. Derivatives of elementary functions. Derivative of the sum, product, and quotient of functions. Derivative of the composite function and the inverse function.
Local and global maxima and minima. Bounded functions. Concave and convex functions. Concave and convex sets. Higher-order derivatives. Taylor’s formula and its application to limit computation. Use of derivatives in function analysis.
The definite integral. Primitive of a function. The fundamental theorem of calculus. Integral of elementary functions. Integration techniques: by parts and by substitution.

Module: Mathematics 1
Sheldon M. Ross "Introduction to Probability and Statistics for Engineers and Scientists", Elsevier 2014


Module: Mathematics 2
C. Neuhauser , M. Roper, "Calculus For Biology And Medicine" Pearson Education, 2018

Module: Mathematics 1
C. Neuhauser , M. Roper, "Calculus For Biology And Medicine" Pearson Education, 2018 (solo in inglese)


Module: Mathematics 2
"Mathematical Analysis I": Vol. 1, Springer, Claudio Canuto, Anita Tabacco

Lectures (60%), examples and exercises (40%).

Attendance at lessons is strongly recommended for a good understanding of the course content.

The final exam includes a written test, aiming in verifying the acquired knowledge, and an optional oral exam. The written test consists of open-ended questions and requires a minimum time of 120 minutes. The oral exam consists of a variable-length interview. The final grade is the average of the grades from the written and oral exams (or the written test only).

Computation of limits of functions with basic techniques. Evaluation of minima and/or maxima (both local and global) of functions. Analysis of graphs of functions. Integral calculus with the usage of the integration techniques provided during the course.

Module: Mathematics 1
N/D
Module: Mathematics 2
N/D