37643 - Mathematical Analysis T (F-Z)

Learning outcomes

At the end of the course, after a positive result at the final exam, the student should be able to know and to use the fundamental concepts and techniques of calculus concerning the functions of one real variable: limits, continuous functions, differential calculus, integral calculus.

Course contents

• Introduction. Properties of the real numbers (cardinality, total order) and of the relevant subsets (N, Z, Q), intervals. Definition of function in one variable, injectivity, surjectivity, invertibility,  inverse function, composition of function, monotone functions. Elementary functions (powers, roots, exponentials, logarithms, hyperbolic functions, trigonometric functions and inverse trigonometric functions, absolute value functions).
• Complex numbers   Definiton of the field of the complex numbers.  Algebraic form. Modulus and argument of a complex number. Exponential form of a complex number. De Moivre's formula. Complex roots of a complex number. Algebraic equations in C.
• Limits Accumulation point, definitions of limit for real functions in one variable, one-sided limits. Elementary properties of limits: unicity, locality. Algebraic properties of the limit, Sandwich theorem, limits of monotone functions. Indeterminacy. Landau symbols
• Continuity Definition of a continuous function of one real variable. The Weierstrass theorem, the Bolzano theorem and the intermediate value theorem. Continuity of the composition of two continuous functions. Continuity of the inverse function.
• Differential calculus and applications Definition of a differentiable function and of derivative of a function. The algebra of derivatives. The mean value theorems and their application to study the monotonicity of a function. Higher order derivatives. Hospital theorem for indeterminate limits. Taylor's formula. Asymptotes. Relative maxima and minima of a function: definitions, necessary conditions, sufficient conditions. Convex functions.
• Integration Definition of the Riemann integral. Properties of the integral: linearity, additivity, monotonicity, the mean value theorem. The fundamental theorems of the integral calculus. The theorems of integration by substitution and of integration by parts. Improper integrals. 
• Differential equations. Linear differential equations. The general solutions of homogeneous and nonhomogeneous linear differential equations. The Cauchy problem. Solution of  linear differential equations (of order one , of order n with constant coefficients).  The sympatghy method and the method of Lagrange for the search of a particular solution to a inhomogeneous linear differential equation Differential equations solvable by separation.

Readings/Bibliography

Simonetta Abenda: Analisi Matematica -Progetto Leonardo (Soc. Ed. Esculapio)

Simonetta Abenda: Esercizi di Analisi Matematica -Progetto Leonardo (Soc. Ed. Esculapio)

Marco Bramanti, Carlo Domenico Pagani, Sandro Salsa, Analisi Matematica 1. Ed. Zanichelli.

Tom M. Apostol, Mathematical Analysis, Addison-Wesley Publishing Company

Teaching methods

Part of the lessons will be dedicated to the theoretical aspects of the program. The remaining part of the lessons will be dedicated to the solution of exercises and applications.

Assessment methods

 The examination is written and consists of two parts. It is obligatory to enrol in the Amaesami list of exams for both parts of the exam.  

 The first part lasts for 2 and 1/2 hours and it consists of multiple choice and open exercises. Students may use their own books and notes. It is forbidden to use any electronic device. The highest rank of this part is 16. If the student achieves 6.1/16 or more in the A part, he/she may partecipate in the B part.

The B part lasts for 1 h and the student may take only the pen with him/herself. In this part, the student must solve one of the multiple choice exercises of his/her A part and must answer to two theoretical questions following the proposed draft. The highest rank for this part of the exam is 21.

The final mark is obtained adding the part A and part B marks. Final marks greater than 30 will correspond to 30/30 cum laude on Almaesami.

The student may check his work during a special office hours before the verbalization of all valid marks.

Further piece of information on exams is available in the web pages  http://www.unibo.it/docenti/simonetta.abenda .

The dates of the exams are published on Almaesami.

Facsimiles of the part A written examination are avalaible in the Alma Campus collection.

Teaching tools

 

 In the Alma Campus collection students may find either  a fac-simile of the exam and exercises useful for the OFA exam.

Links to further information

http://www.unibo.it/docenti/simonetta.abenda

Office hours

See the website of Simonetta Abenda