ANALISI MATEMATICA I

Module MODULO A

The lessons are complemented by exercises related to the topic of the course and both the lessons and the exercises will be carried out in frontal mode. Should teaching be carried out in mixed mode or remotely, it may be necessary to introduce changes with respect to previous statements, in line with the program planned and outlined in the Syllabus.

Lecture attendance is not compulsory but it is strongly recommended.

1. Sets of numbers. 
• Natural numbers, integer numbers, rational numbers, real numbers. Basic notions on the set N of natural numbers, on the set Z of integer numbers and on the set Q of rational numbers. The set R of real numbers. Some consequences of the axioms on the real numbers. Intervals. Absolute value of a real number. Bounds for numerical sets. The set N. Archimedean property. Density of Q and R-Q in R. Powers with real exponent.
• Complex numbers. Basic definitions. Total order. Polar coordinates in the plane. Trigonometric form of a complex number. Product and power of complex numbers in trigonometric form. Exponential form of a comples number. Product and power of complex numbers in exponential form. nth roots of a complex number. Algebraic equations.
• Applications with MATLAB. Floating-point numbers, exact arithmetic and floating-point arithmetic, complex numbers.
2. Functions and limits. 
• Functions. Basic definitions. Composite function. Inverse function. Real functions of one real variable: monotone functions, Affine functions and linear functions, even functions and odd functions, periodic functions, bounded functions and unbounded functions, global minimum and global maximum points. Operations with the functions. 
  
• Limits. Topology in R. Local minimum and loca maximum points. Limits. Theorems on limits. Algebra of limits. Indeterminate forms. Comparison theorems. One-sided limits and theorem on limits of monotone functions. Theorem on the limit of the composite function. Sequences: basic definitions, limits, sequential characterization of the limit of a function, subsequences.
  
• Applications with MATLAB. Definitions of functions, anonymous function, function handle, user-defined function, plots of functions.
3. Continuous functions and local comparison.
   
• Continuous functions. Definition of continuous function and basic results. Continuity of elementary functions and operations with continuous functions. Singularity points: removable singularity, singularity of the first kind and singularity of the second kind. Properties of continuous functions: local properties and global properties. Theorem on the existence of zeroes and its generalization, Intermediate value theorem, Weierstrass Theorem. Injectivity and strict monotonicity for continuous functions. Theorem on the continuity of the inverse function. Napier's number. Fundamental limits.
• Local comparison of functions. Bachmann-Landau symbols, comparison between infinitesimal and infinite functions. Asymptotes. Uniformly continuous functions
• Applications with MATLAB. Nonlinear equations: investment fund, state equation of a gas, population dynamics, bisection method, Newton method, fixed point iterations.
4. Differential Calculus.
• Definition of differentiable function and definition of derivative. Geometric and kinematic meaning of the first derivative. Relationship between continuity and differentiability. First finite increment formula. Derivatives of the elementary functions. One-sided derivatives. Point of non-differentiability. Definition of differential. Rules of differentiation. Theorem on the differentiability of the composite function. Theorem on the differentiability of the inverse function.
  
• Fundamental theorems of Differential Calculus and their consequences. Fermat's Theorem, Rolle's Theorem, Lagrange's Theorem and its consequences (second finite increment formula, Characterisation of the functions with identically null derivative on an interval, monotonicity test and local extrema, Test for the determination of the extrema, Characterisation of stricly monotone functions). De L'Hôpital's Theorem. Theorem on the limit of the derivative. Limit of the derivative and points of non-differentiability. Higher-order derivatives. Taylor formula with Peano's remainder and Taylor formula with Lagrange's remainder. Concave functions and convex functions: concave functions and convex functions under the differentiability assumption, inflection points, Characterisation of the concavity and the convexity with the monotonicity of the first derivative, Necessary condition for the inflection points, relationship between the concavity/convexity and the sign of the second derivative, general definition of concave function and convex function, test for the inflection points. Higher-order derivative test for the study of stationary points. Qualitative study of a function.
  
• Applications with MATLAB. Approximation of functions and data: climatology, finance, biomechanics, robotics. Approximation by Taylor's polynomials. Polynomial interpolation. Numerical differentiation: problem and examples (hydraulics, optics, electromagnetism, demography). Approximation of derivatives. Minimization of real functions of one real variable with the golden section and quadratic interpolation method.

Recommended books for the Prerequisites

[P1] C.Y. Young, Algebra and Trigonometry. Fourth Edition, Wiley (2017).

[P2] C. Y. Young, Precalculus. Third Edition, Wiley (2018).

Recommended books for the course of Mathematical Analysis I

• Recommended books for the Theory:

[T2] R.A. Adams, C. Essex, Calculus. A Complete Course, Pearson (2021).

• Recommended books for the Exercices:

[E2] R.A. Adams, C. Essex, Calculus. A Complete Course, Pearson (2021).

Self-assessment tests

During the period of delivery of the lessons, some self-assessment tests will be administered. These self-assessment tests have the task of guiding the student in the gradual learning of the contents displayed during the lessons. In addition, the self-assessment tests allow the teacher to quickly implement any additional activities aimed at supporting students in view of the exams.

Structure of the exam

The Mathematical Analysis I exam can be passed in three ways.

Mode A: mid-term tests and oral tests

There are two mid-term tests: the first at the end of the first teaching period and the second at the end of the second teaching period. The first mid-term test focuses on the contents of Module A, while the second focuses on the contents of Module B. Each mid-term test consists of a written test and an oral test. The oral exam is compulsory and can only be accessed after passing the written mid-term test. Each mid-term test is considered passed if and only if the interview relating to it has been passed, that is, if the student has obtained a score of at least 18/30. It is possible to take the second mid-term test only if the first has been previously passed. The duration of each written mid-term test is 120 minutes.

Structure of the written mid-term tests

Each written mid-term test has the same structure. Five exercises will be proposed in each written mid-term exam.

Evaluation of the mid-term tests and final grade

The maximum grade obtainable in each written mid-term test is 30/30. Each written mid-term test is passed if the student has achieved a score of at least 18/30. A pass is obtained (18/30) if and only if the student correctly solves three of the five proposed exercises. The student who, despite having passed the first mid-term test, has not passed or sustained the second mid-term test, can complete the exam following Mode B, thus supporting the partial written test relating to Module B as well as the oral test after having passed the the written test as a whole. Alternatively, the student can take the complete exam following Mode C. Each oral exam focuses on all the topics of the Module to which it refers. In the formulation of the final mark of each mid-term test, the score obtained in the written mid-term test and the evaluation of the oral test are taken into account. The final grade is the rounding up of the arithmetic average of the marks obtained in the two mid-term tests.

Mode B: written test in modules and oral test

The written test is divided into two partial tests. The first partial test focuses on the contents of Module A, while the second focuses on the contents of Module B. It is possible to take the second partial test only if the first has been previously passed. The duration of each partial test is 120 minutes. It should be noted that, following this procedure, the two partial tests cannot be sustained in the same appeal. After passing the first partial test, the student can take the second partial test in one of the following sessions and in any case no later than the third exam session. The oral exam is compulsory and can only be accessed after passing both partial tests.

Structure of partial tests

The two partial tests have the same structure. In each partial test five exercises will be proposed.

Evaluation of each part into which the written test is divided

The maximum grade obtainable in each partial test is 30/30. Each of the two partial tests is considered passed if the student has achieved a score of at least 18/30. In each partial test, passing (18/30) is obtained if and only if the student correctly solves three of the five proposed exercises. The final mark of the written test is the rounding up of the arithmetic average of the marks obtained in the two partial tests.

Oral exam and final grade

The grade with which the student is admitted to the oral test is the rounding up of the arithmetic average of the marks obtained in the two mid-term tests. The oral exam covers all the topics of the course. In formulating the final grade, the grade with which the student is admitted to the oral exam and the grade obtained in the oral exam are taken into account.

Mode C: complete written exam and compulsory oral exam

In this way, only one written test is proposed which focuses on the contents of Module A and on the contents of Module B and, after passing it, the student will have to take the oral test. The written test lasts 120 minutes.

Structure of the written test

Five exercises will be proposed in the written test. Evaluation of the written test. The maximum mark obtainable in the written test is 30/30. The written test is passed if the student has achieved a score of at least 18/30. A pass is obtained (18/30) if and only if the student correctly solves three of the five proposed exercises.

Oral exam and final grade

The oral exam covers all the topics of the course. In formulating the final grade, the grade obtained in the written test and the evaluation of the oral test are taken into account.

Note. Verification of learning can also be carried out electronically, should the conditions require it. In this case, the duration of the written test may be subject to change.

All the topics mentioned in the program can be requested during the exam.

The attendance of the lessons, the study of the recommended texts and the study of the material provided by the teacher (handouts and collections of exercises carried out and proposed) allow the student to have a clear and detailed idea of ​​the questions that may be proposed during the exam.

An adequate exposition of the theory involves the use of the rigorous language characteristic of the discipline, the exposition of simple examples and counterexamples that clarify the concepts exposed (definitions, propositions, theorems, corollaries).

The main types of exercises related to the Module A of the course of Mathematical Analysis I are:

• Finding the extrema of a numerical set.
• Exercises on complex numbers (algebraic manipulations, writing complex numbers in algebraic, trigonometric and exponential form, determination of the nth roots of a complex number, representations of geometric places in the Argand-Gauss plane, equations in the complex field).
• Calculation of limits.
  
• Study of the continuity and differentiability - and computation of derivatives - of real functions of a real variable.
• Classification of points of singularity and points of non-differentiability.
• Issues concerning the invertibility of functions and calculation of the derivative of inverse functions.
• Qualitative study of a function and applications (e.g., study of a function and its qualitative graph, proof of the existence of solutions of nonlinear equations).