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/or open-ended exercises while the function study is carried out. In the second part, complete one of the questions corresponding to a theory topic from a list of questions published on virtual. 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 sum of the scores obtained in the two parts and is published on Almaesami. Students can appear for all exams. The exam dates are published on Almaesami. Registration on Almaesami for both parts of the exam is mandatory. Learning assessment methods Detailed information on the exam method Part A (duration 2 hours): Consists of multiple choice and/or open-ended or follow-up exercises. During part A, the student cannot consult his 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 26, while the theory question has a score from 0 to 5. Multiple choice exercises are worth: +4 (correct answer), 0 (answer not given) -1 (wrong answer) Exercise with progression: from 0 to 10. Part B (duration 30 minutes). The student can only bring the pen with him and explains two theoretical topics following the path assigned by the teacher. The maximum score for this part is 5. Marking and verbalisation: The final mark is given by the sum 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 marking of the written tests, a special student reception is set up to view the homework and, at the end of this reception, the Commission proceeds to record all the valid marks. To refuse the grade it is necessary to participate in the homework viewing and communicate it verbally on the same day as the homework viewing. Exam timetable: it is published on Almaesami and visible on the Course of Study web page dedicated to exam sessions. The standard texts of some part A exam tests are distributed during the lesson 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