#
Calculus and Statistics

A.Y. 2020/2021

Learning objectives

The study of the environment and impacts on health is a complex and challenging job requiring solid scientific and technical competences.

The aim of this course is to give students the basic mathematical and statistical knowledge that are necessary to cope with quantitative activities related to science of life. To reach this aim, it is important that students understand which are the internal structures and the essential procedures of Mathematics and Statistics in order to be able to apply them in their future technical and professional activities.

The aim of this course is to give students the basic mathematical and statistical knowledge that are necessary to cope with quantitative activities related to science of life. To reach this aim, it is important that students understand which are the internal structures and the essential procedures of Mathematics and Statistics in order to be able to apply them in their future technical and professional activities.

Expected learning outcomes

At the end of the course students are expected to be able to:

Develop a logical and mathematical reasoning

Solve problems with differential and integration calculus

Develop basic mathematical models

Select the most appropriate statistical procedures for scientific and laboratory applications

Students will achieve knowledge of:

Fundamental aspects of differential and integral calculus as a base for further courses in their degree program

Fundamental statistical and probabilistic methods as a base for software instruments used in biological and pharmacological laboratories.

Develop a logical and mathematical reasoning

Solve problems with differential and integration calculus

Develop basic mathematical models

Select the most appropriate statistical procedures for scientific and laboratory applications

Students will achieve knowledge of:

Fundamental aspects of differential and integral calculus as a base for further courses in their degree program

Fundamental statistical and probabilistic methods as a base for software instruments used in biological and pharmacological laboratories.

**Lesson period:** First semester
(In case of multiple editions, please check the period, as it may vary)

**Assessment methods:** Esame

**Assessment result:** voto verbalizzato in trentesimi

Course syllabus and organization

### Single session

Responsible

Lesson period

First semester

Whenever possible, some lectures will be held in presence while the remaining lectures will be held remotely, during the class timetable, via Microsoft Teams. The lectures will be recorded.

Details about the class timetable will be available in Ariel platform.

In case of emergency, lectures will only be held remotely via Microsoft Teams and will be recorded.

Exams

In case of emergency the first partial, currently planned in precence, will be taken remotely via Microsoft Teams according to the University guidelines. The second partial will only be taken remotely.

Moreover, in case of necessity, the general exam will be taken remotely as well and the duration will be one and half hours instead of two hours.

Operative details will be available in the Ariel platform.

Details about the class timetable will be available in Ariel platform.

In case of emergency, lectures will only be held remotely via Microsoft Teams and will be recorded.

Exams

In case of emergency the first partial, currently planned in precence, will be taken remotely via Microsoft Teams according to the University guidelines. The second partial will only be taken remotely.

Moreover, in case of necessity, the general exam will be taken remotely as well and the duration will be one and half hours instead of two hours.

Operative details will be available in the Ariel platform.

**Course syllabus**

Sets, operations with sets. Number sets. Intervals. Upper bound, lower bound, maximum, minimum, supremum, infimum of real sets. Absolute value, distance. Neighbourhoods. Left/right neighbourhoods.

Functions

The concept of function. Injective, surjective, bijective function. Composition of functions. Inverse function. Review of elementary functions: linear, quadratic, power, exponential, logarithmic, trigonometric and hyperbolic functions. Bounded functions. Monotonic functions, strictly monotonic functions. Maxima/minima, maximizers/minimizers: global and local. Concave/convex functions

Sequences and series

Sequences of real numbers. Recursively defined sequences. Limits of sequences. Convergent, divergent, irregular sequences. Calculation of limits. Indeterminate forms.Comparisons among infinities and among infinitesimals. The symbols ~ and o, their properties. The number e.

The concept of series. The sequence of partial sums. Behaviour of a series; convergent, divergent, irregular series. Behaviour of the geometric series. Behaviour of the harmonic series. The exponential series.

Limits of functions and continuity

Limits of functions of one real. Asymptotes. Theorem on the uniqueness of the limit. Limits of elementary functions. Indeterminate forms. The symbols ~ and o. Calculation of limits. Change of variable. Fundamental limits. Theorem on the permanence of sign. Comparison criterion. Continuity for functions of one real variable. Points of discontinuity. Weierstrass's theorem, zero-value theorem (Bolzano's theorem)

One-variable differential calculus

Difference quotient, derivative at a point; geometric meaning, equation of the tangent line. Rules on derivatives. Derivative of the inverse function. Higher derivatives. Differentiability, differential. Relationship between derivability and differentiability.

Stationary points. Necessary condition for local maximizers/minimizers (Fermat's theorem). Rolle's theorem. Lagrange's mean value theorem. Monotonicity/strict monotonicity test on an interval.

Taylor's polynomial and Maclaurin's polynomial. Taylor's theorem; Taylor's formula and Maclaurin's formula of order n, with Peano's remainder. Second sufficient condition for local maximizers/minimizers. Determination of global and local maximizers/minimizers. Study of the graph of a function.

Antiderivatives of a function. Indefinite integral: definition. Elementary integrals. Integration methods: by decomposition, by parts, by substitution. Definite integral (according to Riemann): definition, geometric meaning. Fundamental theorems of calculus. Computation of definite integrals. Mean value. Generalized integral on bounded and unbounded intervals. Comparison criterion, asymptotic comparison criterion. Absolute convergence. The Gaussian curve.

Statistics:

Descriptive statistics. Sample space, data analysis, dispersion measures. Quantitative and qualitative variables. Frequencies, box-plot.

Probability: sample space. Random events, probability of an event, probability space. Probability axioms. Stochastically independent events, positively/negatively correlated events. Bayes's theorem. Discrete and continuous random numbers Distribution function. Expected value and variance .Random vectors: covariance and correlation. Binomial, Poisson distributions. Uniform, exponential, normal, t-student distributions. Central limit theorem. Hypothesis testing.

Functions

The concept of function. Injective, surjective, bijective function. Composition of functions. Inverse function. Review of elementary functions: linear, quadratic, power, exponential, logarithmic, trigonometric and hyperbolic functions. Bounded functions. Monotonic functions, strictly monotonic functions. Maxima/minima, maximizers/minimizers: global and local. Concave/convex functions

Sequences and series

Sequences of real numbers. Recursively defined sequences. Limits of sequences. Convergent, divergent, irregular sequences. Calculation of limits. Indeterminate forms.Comparisons among infinities and among infinitesimals. The symbols ~ and o, their properties. The number e.

The concept of series. The sequence of partial sums. Behaviour of a series; convergent, divergent, irregular series. Behaviour of the geometric series. Behaviour of the harmonic series. The exponential series.

Limits of functions and continuity

Limits of functions of one real. Asymptotes. Theorem on the uniqueness of the limit. Limits of elementary functions. Indeterminate forms. The symbols ~ and o. Calculation of limits. Change of variable. Fundamental limits. Theorem on the permanence of sign. Comparison criterion. Continuity for functions of one real variable. Points of discontinuity. Weierstrass's theorem, zero-value theorem (Bolzano's theorem)

One-variable differential calculus

Difference quotient, derivative at a point; geometric meaning, equation of the tangent line. Rules on derivatives. Derivative of the inverse function. Higher derivatives. Differentiability, differential. Relationship between derivability and differentiability.

Stationary points. Necessary condition for local maximizers/minimizers (Fermat's theorem). Rolle's theorem. Lagrange's mean value theorem. Monotonicity/strict monotonicity test on an interval.

Taylor's polynomial and Maclaurin's polynomial. Taylor's theorem; Taylor's formula and Maclaurin's formula of order n, with Peano's remainder. Second sufficient condition for local maximizers/minimizers. Determination of global and local maximizers/minimizers. Study of the graph of a function.

Antiderivatives of a function. Indefinite integral: definition. Elementary integrals. Integration methods: by decomposition, by parts, by substitution. Definite integral (according to Riemann): definition, geometric meaning. Fundamental theorems of calculus. Computation of definite integrals. Mean value. Generalized integral on bounded and unbounded intervals. Comparison criterion, asymptotic comparison criterion. Absolute convergence. The Gaussian curve.

Statistics:

Descriptive statistics. Sample space, data analysis, dispersion measures. Quantitative and qualitative variables. Frequencies, box-plot.

Probability: sample space. Random events, probability of an event, probability space. Probability axioms. Stochastically independent events, positively/negatively correlated events. Bayes's theorem. Discrete and continuous random numbers Distribution function. Expected value and variance .Random vectors: covariance and correlation. Binomial, Poisson distributions. Uniform, exponential, normal, t-student distributions. Central limit theorem. Hypothesis testing.

**Prerequisites for admission**

Algeabric equations and inequalities, analytic geometry

**Teaching methods**

Face to face lectures

**Teaching Resources**

Textbook: Marco Abate - "Matematica e Statistica - Le basi per le scienze della vita". - McGraw Hill Education

Exercises posted on line

Exercises posted on line

**Assessment methods and Criteria**

Written modality, score out of 30, either with two partials or general exam.

Modality with two partials: the first test on mathematics , the second one on statistics. The two partials will be taken during the course.

Each test lasts 1 hour. The first test weights 60% of the final score, the second one weights 40% of the final score.

The general exam lasts 2 hours and consists in 3 open answer math questions and two open answer questions on statistics plus a question on theory (either on mathematics or on statistics).

Modality with two partials: the first test on mathematics , the second one on statistics. The two partials will be taken during the course.

Each test lasts 1 hour. The first test weights 60% of the final score, the second one weights 40% of the final score.

The general exam lasts 2 hours and consists in 3 open answer math questions and two open answer questions on statistics plus a question on theory (either on mathematics or on statistics).

MAT/07 - MATHEMATICAL PHYSICS - University credits: 6

Practicals: 32 hours

Lessons: 32 hours

Lessons: 32 hours

Professor:
Mariano Laura

Educational website(s)

Professor(s)