MATHEMATICAL ANALYSIS I
Academic year and teacher
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- Versione italiana
- Academic year
- 2017/2018
- Teacher
- CHIARA BOITI
- Credits
- 12
- Didactic period
- Annualità Singola
- SSD
- MAT/05
Training objectives
- The course of Mathematical Analysis I is to provide basic knowledge of differential and integral calculus for functions of one variable. Also the basics of differential equations necessary to face courses of the second year will be given.
The main knowledge provided by the course are:
sequences, continuous and differentiable functions, Riemann integrals and generalized integrals, numerical series, ordinary differential equations linear or with separable variables.
The main skills that students will acquire are: knowing how to calculate limits of sequences; knowing how to calculate limits and derivatives of functions in order to be able to plot the graph of a function; knowing how to calculate integrals of functions via, for example, the method of integration by parts or the change of variables; knowing how to study the convergence of a series via, for example, the comparison criterium, or the root or ratio criteria; knowing how to solve linear differential equations of the first order, or of any order, but with constant coefficients; knowing how to solve differential equations with separable variables. Prerequisites
- Notion of mathematics usually teached in the secondary school: equations and inequalities (of first and second degree, with absolute values, roots, rational), logarithms, exponential, trigonometry. Such contents
are handled during the pre-course in Mathematics, that will be held one week before the start of classes, and will be quickly retrieved during the first lessons. Course programme
- The total duration of the course is 96 hours.
Brief mentions of set theory and trigonometry (1 hour). Introduction of N, Z, Q and R (1 hour and 1/2)
Mathematical induction (2 hous and 1/2). Basic notions on complex numbers (2 hours and 1/2). Notion of function, limit and continuity; properties of continuous functions (26 hours). Numerical sequences and series (14 hours). Differential calculus for functions of one variable (13 hours). Theorems of Rolle, Cauchy and Lagrange (3 hours). Graph of functions (3 hours and 1/2). Riemann integrals for functions of one variable (18 hours). First resolving methods for ordinary differenzial equations linear or with separable variables (13 hours). Didactic methods
- Lessons will be held at the blackboard.
There will be held lectures and exercises.
In performing the exercises we will also try to engage students. They will be assigned exercises to do at home. Learning assessment procedures
- Each exam session consists of a written test (total) and an oral examination. These tests are aimed at verifying the learning themes of the course.
More specifically, the written test is designed to examine the ability of resolution of exercises, while the oral test aims to capture even the theoretical part of the course. For the written exam 4 exercises are given (with topics that can range throughout the course program). Students are admitted to the oral with an evaluation of the written test of at least 18/30.
Alternatively, instead of the total written test, students can do partial written tests: one about the first part of the course; the second one about the second part of the course; and the third partial written test at the end of the course (about the third part of the course). Each partial test consists of 3-4 exercises.
If the average of the three partial written tests is at least 18, then the student can avoid the total written exam.
With partial tests students can take the oral exam in any session of exams, while with the total written test they can do the oral examination only during the same session.The partial tests are valid for one year.
If the student does not pass the oral exam, then he must redo both the total written test and the oral exam. Reference texts
- M. Bertsch, R. Dal Passo, L. Giacomelli
``Analisi Matematica''
McGraw-Hill (last version)
