28622 - Mathematical Analysis T-A

Learning outcomes

Know the methodological-operational aspects of mathematical analysis, with particular regard to the functions of a real variable, in order to be able to use this knowledge to interpret and describe engineering problems.

Course contents

 

 

Premises:

N, Z, Q, R, order relations: minimum and maximum, upper and lower extremes of a subset of R. Density of Q in R.

Domain and condominium of a function, functions, graph, injectivity, surjectivity, image, counter image, inverse function, compound function.

Elementary functions (integer exponent function, n-ma root, exponential, logarithm, circular and inverse functions, hyperbolic functions, absolute value function).

Complex numbers The field of complex numbers, algebraic form, module and argument, trigonometric form, roots (*), algebraic equations in the complex field.

 

Limits

Surroundings, points of accumulation.

Finite and infinite limits of functions of real variable with real values, right and left limit.

Properties of the limit: uniqueness, locality, local limitation; algebraic properties of the limit and comparison theorem. Limits of monotone functions.

Indeterminate forms: infinite and infinitesimal. Symbols of Landau.

Notable limits (*).

Continuity

Continuous functions of real variable with values in R. Continuity of the compound function. Permanence of the sign.

Properties of continuous functions defined on intervals: Weierstrass theorem, Bolzano theorem, theorem of zeros (*), theorem on invertibility and monotony, continuity theorem of the inverse function.

 

Derivation and applications

Geometric and mechanical interpretation of the derivative, derivatives of higher order, derivatives of elementary functions.

Rules of derivation: derivative of the sum of functions (*), Leibniz's rule (*), derivative of the reciprocal function (*), derivative of the inverse function (*), derivative of the compound function.

Properties of differentiable functions on intervals: Rolle's theorem (*), Lagrange's theorem, functions with zero derivative and constant functions (*), primitive, monotony theorem and sign of derivative (*). De l'Hopital's theorem for indeterminate forms.

Convex functions: definition and geometric interpretation, theorem on convexity and monotony of the first derivative, theorem on convexity and sign of the second derivative.

Approximation of regular functions with Taylor's formula. Taylor's polynomial, uniqueness of the polynomial of degree less than or equal to n which approximates a function of order n (*), Taylor's formula with the remainder of Peano (proof in cases n = 1 and n = 2), properties of the derivatives of the polynomial by Taylor; Taylor's formula with Lagrange's remainder, Taylor's formula of elementary functions: exp (x) (*), cos (x) (*), sin (x) (*), cosh (x) (*), senh ( x) (*), (1 + x) ^ a (*), 1 / (1-x) (*), 1 / (1 + x) (*), 1 / (1-x ^ 2) (* ), 1 / (1 + x ^ 2) *, log (1 + x) (*), application to the limits of indeterminate forms.

Qualitative analysis of functions. Asymptotes: vertical, horizontal, oblique; singular points of the first and second species, angular points, cusps, local extremes, stationary points, internal extremes are stationary (*), sufficient conditions (by means of the derivatives) for a point to be extremely local (*), inflection points : geometric definition and interpretation, necessary conditions and sufficient conditions (by means of derivatives) for a point to be inflected.

 

Integration and applications

Definition of Riemann integral for bounded functions defined on bounded and closed intervals. Properties of the integral: linearity, monotony, additivity.

Classification of integrable functions according to Riemann on limited and closed intervals (continuous functions except a finite number of points; monotone functions). The Dirichlet function. Theorem of the integral mean (*).

Integral function and primitive function. The fundamental theorem of integral calculus for continuous functions (*). Torricelli's rule (*). Integration theorem by parts (*) and integration theorem by substitution (*).

Integration of rational functions.

Generalized Riemann integral. Comparison criterion for the convergence of the generalized integral of a positive function. Addability of 1 / x ^ a (*).

Readings/Bibliography

Texts / Bibliography Texts / Bibliography M.Bramanti, C.D. Pagani, S. Salsa: Mathematical Analysis 1 (Zanichelli)

Teaching methods

Lectures and frontal exercises

Assessment methods

The exam takes place in written form and consists of two parts to be taken in the same session. In the first part the student solves multiple choice and guided exercises. In the second part he carries out in full one of the multiple choice exercises of his own task and answers two theory questions. The use of any electronic device connected to the Internet during the exam is prohibited, under penalty of cancellation of the exam itself. The final score is the arithmetic mean of the scores obtained in the two parts and is published on Almaesami. Students can present themselves to all exam sessions. The dates of the exams are published on Almaesami. Registration on Almaesami is mandatory for both parts of the exam. Learning assessment procedures Detailed information on the examination procedure Part A (duration 2 hours): It consists of multiple choice and guided answer exercises. During part A, the student can consult their own textbooks and notes and cannot use any type of calculator. The use of any other electronic device is prohibited. The maximum score for this test is 30. The student who reaches the admission threshold 18/30 is admitted to part B. The multiple choice exercises are worth: +5 (correct answer), 0 (answer not given) -1 (wrong answer) Guided answer exercise: from 0 to 10. Part B (duration 1 hour). The student can carry only the pen, he exposes two theory topics following the outline assigned by the teacher. The maximum score for this part is 30. Each question is worth from 0 (answer not given or off topic) to 15. Marks and minutes: The final mark is given by the arithmetic mean of the scores obtained in the two tests. Scores above 30/30 will be recorded as 30/30 cum laude on Almaesami. At the end of the correction of the written tests, a specific student reception is set for the examination of the assignments and, at the end of this reception, the Commission proceeds to record all valid marks. To refuse the grade, it is necessary to participate in the homework view and communicate it verbally on the day of the homework view. Exam dates: it is published on Almaesami and visible on the web page of the Degree Program dedicated to exam sessions. The standard texts of some part A exams are distributed and lecture and published in the virtual spaces of the course.

Teaching tools

 

 

The texts of some exams of part A are available on the digital platform of the course Reception hours Consult the Cataldo Grammatico website [https://www.unibo.it/sitoweb/cataldo.grammatico]

Office hours

See the website of Cataldo Grammatico