## ANALISI 1

The idea of a set and the idea of a mapping between sets

Calculations with integers, with decimals, and with fractions

Solution of equations of first and second degree

Powers and the algebraic laws which they satisfy

Polynomial algebra

Graphs, axes, and coordinates

The graph of the linear function: y=mx+q, which is a line

Logarithms

Trigonometry for triangles

Written exam, at the end of the course, during the regular exam periods. The exam will be made up mostly of calculations and exercises, not theory. However, for a course of this level, it's more or less impossible to learn to do the exercises without a solid understanding of the underlying ideas.

At the exam, you can bring up to two pages, A4, both sides, with notes and formulae, but it all has to be written by hand, by you yourself. You can bring a calculator or other electronic assistance, but nothing that you could use to put yourself in contact with the internet or the outside world. Unfortunately, this restriction, which is clearly a necessity, eliminates many tools which any real engineer would use while working.

Copying will be treated as a serious offence.

Students will learn some calculational methods and some important mathematical concepts which are fundamental for science and technology in general and for computer engineering in particular.

Bounds on error terms: the big-O and little-o notations

Limits:

sequences

series

functions

Completeness of R:

Cauchy sequences

Absolutely convergent series

Nested intervals

inf and sup

Derivatives:

f(x+h)=f(x)+f'(x)+o(h)

rules for calculating derivatives:

the composition of two functions --- the chain rule

the sum of two functions

the product of two functions

the ratio of two functions

The exponential function:

you look for a function satisfying f'(x)=f(x)

e^x = 1+x+x^2/2+x^3/6+...

e^z = 1+z+z^2/2+z^3/6+... for z complex

geometric interpretation of e^z

Trigonometric functions:

sin(x) and cos(x)

Euler's formula

tan(x), sec(x), cotan(x), csc(x)

arctan(x), arcsin(x), arccos(x)

graphs of these functions

derivatives of these functions

Antiderivatives:

easy cases

integration by parts

function of ... times derivative of ...

substitution

trigonometric substitutions

decomposition of rational functions into simpler fractions

Integrals:

definition

calculating them with antiderivatives

improper integrals

Power series

Introduction to differential equations

The Laplace transform

Robert A. Adams, C. Essex: Calculus: A Complete Course

Publisher: Prentice-Hall Canada

This textbook should be good for this course, and also for the follow-up course, Analisi Matematica 2. A closer analog to the suggested Italian textbook would be Calculus: Single Variable, same authors as above, but that would cover only the present course, Analisi Matematica 1.

lectures and guided exercise sessions

The professors use linux, specifically Ubuntu. Lots of useful programs for numerical and symbolic calculation are available free with linux, and those are the programs that the professors use. So, if you have access to a computer with linux installed, it will be to your advantage.