Analisi Matematica 1 Exam, Feb 2005

Analisi Matematica 1 - Ing. Edile-Architettura, 14 Febbraio 2005

• The document presents 5 exercises for the Analisi Matematica 1 exam.
• The exam is for students of the Edile-Architettura engineering program.
• The date of the exam is February 14, 2005.

Esercizio 1

• The exercise involves studying the graph of the function f(x) = x^2 / (x + 3).
• Key aspects to analyze include the function's domain, sign, discontinuities, points of non-derivability, asymptotes, intervals of increasing/decreasing, and concavity/convexity.

Esercizio 2

• The exercise asks to compute the definite integral of e^x / (1 + e^(2x)) from 0 to 1.

Esercizio 3

• The exercise involves determining the solutions of the differential equation y'' + 4y = sin(2x).

Esercizio 4

• The exercise discusses the convergence of the series ∑ (n^2 + n) / (n^α + n^3) * arctan(1/n) from n=1 to infinity.
• The analysis should cover all values of α ∈ ℝ.

Esercizio 5

• If f and g are derivable in x_0 ∈ ℝ, then the exercise involves proving that the product function f⋅g is derivable in x_0.
• Also the formula (f⋅g)'(x_0) = f'(x_0)g(x_0) + f(x_0)g'(x_0) must be shown.

Statics - Chapter 2: Force Vectors - 2.1 Scalars and Vectors

Scalars

• Scalars are defined by having a positive or negative numerical value.
• Length, area, volume, and time are all scalars.

Vectors

• Vectors are defined by a magnitude and direction.
• The magnitude of a vector is always positive
• Position, force, and moment are vectors.

Vector Operations

Multiplication or Division of a Vector by a Scalar

• Multiplying a vector by a scalar changes the vector's magnitude.
• A negative scalar will also reverse the direction (sense) of the vector.

Vector Addition

• Vector addition follows the parallelogram law.
• Vector V can be added to vector P using either the parallelogram law or the triangle rule.
• Parallelogram law: The resultant R is the diagonal of the parallelogram formed by P and V.
• Triangle rule: Resultant R connects the head of P to the tail of V.

Vector Subtraction

• Vector subtraction is the addition of a negative vector.
• R = A - B is equivalent to R = A + (-B).

Important points about Scalars and Vectors

• Vector magnitude is always positive, but it's direction can be reversed by multiplying with a negative scalar.
• Vector subtraction involves adding the negative.
• Vectors can be added or subtracted graphically using the parallelogram law or triangle construction.

Guía de inicio rápido de Python

¿Qué es Python?

• Python is a popular programming language created by Guido van Rossum and released in 1991.
• Python is used in web development (server-side), software development, mathematics, and system scripting.

¿Qué puede hacer Python?

• Python can be used on a server to create web applications.
• Python can be used alongside software to create workflows.
• Python can connect to database systems, and read and modify files.
• Python handles big data, and perform complex mathematics.
• Python allows for rapid prototyping and production-ready software.

¿Por qué Python?

• Python runs on multiple platforms (Windows, Mac, Linux, Raspberry Pi, etc.).
• Python's syntax is similar to the English language.
• Python's syntax allows developers to write programs with fewer lines of code than in other languages.
• Python is interpreted, meaning code executes immediately, which allows very rapid prototyping.
• Python supports procedural, object-oriented, or functional programming.

Bueno saber

• Python 3 is the latest version, although Python 2 (security updates only) remains popular.
• Python can be written in a text editor or an integrated development environment (IDE) like Thonny, Pycharm, Netbeans, or Eclipse.

Sintaxis de Python en comparación con otros lenguajes de programación

• Python emphasizes readability, similar to English with influence from mathematics.
• Python uses new lines to complete commands, unlike other languages using semicolons or parentheses.
• Python defines scope through indentation (whitespace) instead of curly braces, which are commonly used in other languages.