


Politecnico di Torino  
Academic Year 2015/16  
16ACFLZ, 16ACFLN, 16ACFLP, 16ACFLS, 16ACFLX, 16ACFMA, 16ACFMB, 16ACFMC, 16ACFMH, 16ACFMK, 16ACFMN, 16ACFMO, 16ACFMQ, 16ACFNL, 16ACFNM, 16ACFNX, 16ACFOA, 16ACFOD, 16ACFPC, 16ACFPI, 16ACFPL, 16ACFQR Mathematical analysis I 

1st degree and Bachelorlevel of the Bologna process in Aerospace Engineering  Torino 1st degree and Bachelorlevel of the Bologna process in Automotive Engineering  Torino 1st degree and Bachelorlevel of the Bologna process in Electronic And Communications Engineering  Torino Espandi... 





Subject fundamentals
This course represents a bridge between high school and college ways of teaching and learning.
The main goal of the course is to provide the students with tools to understand and elaborate logical arguments, from simple to more elaborate ones, and to introduce the elements of differential and integral calculus for functions of one variable with applications to ordinary differential equations of the first order and of the second order linear case. 
Expected learning outcomes
Ability to follow a chain of logical arguments. Essentials of differential and integral calculus for functions of one variable. Computational ability.

Prerequisites / Assumed knowledge
Numbers, equations and inequalities, analytical geometry and trigonometry. First properties of elementary functions.

Contents
Preliminaries: sets and operations with sets. Numerical sets, maxima, minima and extrema. The completeness propertiy of real numbers and its consequences.
Functions: surjectivity and injectivity; composition of functions, inverse functions. Functions of a real variable: elementary functions, monotone functions and inverse functions. (About 15 hours) Limits and continuity: theory of limits of functions and sequences. Continuous functions and related theorems. Order and comparison theorems. Algebra of limits. Indeterminate forms. Local comparison of functions. Landau symbols. Infinite and infinitesimal functions. Order of an infinity and of an infinitesimal, principal part with respect to a test function. Asymptotes. The number e. Fundamental trigonometric and exponential limits. Properties continuous functions on an interval: existence of zeros and of maxima and minima. (About 24 hours) Derivatives: geometrical and physical meaning. Computation of derivatives. List of fundamental derivatives. Derivatives and continuity. Lack of differentiability, extremal and critical points. Fermat Theorem. Properties of differentiable functions on an interval and fundamental theorems of differential calculus (Rolle and Lagrange theorems) and their consequences. De L'Hôpital rule. Taylor formula and fundamental MacLaurin formulas. Use of Taylor expansions in the local analysis of functions: comparison, extrema and convexity. (About 23 hours) Primitives: rules of computation; primitives of rational functions. Indefinite integral. Riemann integral and its properties. Classes of integrable functions. Integral mean value. Fundamental theorem of integral calculus: relations between definite integrals and primitives. Improper integrals: definitions and convergence tests. (About 21 hours) Complex numbers and differential equations: algebraic and trigonometric forms of complex numbers. Real and imaginary parts, modulus and argument. Roots of complex numbers; Fundamental theorem of Algebra. Exponentials of complex numbers and Euler formulas. Ordinary differential equations: Cauchy problem. First order ordinary differential equations, linear or with separable variables. Second order linear differential equations with constant coefficients. (About 17 hours) 
Delivery modes
60 hours of theoretical classes and 40 hours of exercise classes.

Texts, readings, handouts and other learning resources
The following course materials are to be considered as indicative. Further details will be provided by professors in class.
C. Canuto, A. Tabacco. Analisi Matematica I. SpringerVerlag Italia, 2014. S. Lancelotti. Lezioni di Analisi Matematica 1. Celid, 2013. F. Nicola. Analisi Matematica I. Appunti delle lezioni. CLUT, 2013. 
Assessment and grading criteria
The exam consists of a test, of a written part and, when required, of an oral part. The test (on a computer, time allowed one hour) consists of 20 multiplechoice questions, including theoretical ones. Each question is worth one mark and the maximum score is 20. If the score is less than 12, then the exam is failed, otherwise the student proceeds with the written part of the exam (exercises and theory, time allowed one hour). The written part has a maximum score of
13 marks. The final score of the exam is the sum of the scores of the test and of the written part, unless a further oral exam is required by the teacher (or by the student, provided the final score is at least 18/30). In this case, the final score will be based also on the oral part. 
