Notes, summaries, assignments, exams, and problems for Mathematics

Function

A function defines the relationship between an initial set and a final set, so that each element of the initial set (independent variable) corresponds to a single element of the final set (dependent variable).

Domain of the Function

The domain of a function is the set of possible values that the independent variable (e.g., coins) can take.

Range of the Function

The range of a function is the set of possible values that the dependent variable (e.g., drinks) can represent.

A function can be represented by tables, graphs, and algebraic formulas.

Increasing and Decreasing Functions

• A function is increasing on an interval if for any pair of values a and b in this interval, where a < b, the rate of change is positive.
• A function is decreasing

Fundamentals of Ratios and Proportions

Defining Ratios and Proportions

• Ratio: A ratio of two numbers is their quotient.
• Proportion: A proportion exists when we have two ratios whose quotients are equal.

Key Properties of Proportions

The terms in a proportion are often referred to as means and extremes (or ends).

Example: $2/4 = 3/6$

The Fundamental Property of Proportions states that the product of the extremes is equal to the product of the means. This property is essential for finding an unknown term.

Constant of Proportionality

The constant of proportionality is the ratio (quotient) of any of the corresponding terms in the proportion.

Understanding Proportionality

Magnitudes and Measurement

A magnitude is a measurable characteristic, such as:

• Length
• Volume
• Weight
• Mass
• Temperature

Direct

Differences Between Delivery Notes and Invoices

Delivery Note: A provisional document justifying the dispatch of goods. It does not include VAT.

Invoice: A definitive document providing legal accreditation. It is valid for any claim and includes VAT.

Another important difference is that invoices are legally required to be kept for 6 years, while retaining delivery notes is not mandatory for the same period.

Sales Transaction Documentation

Common documents involved in sales transactions include:

• The order sheet
• The delivery note
• The invoice
• The expenses sheet
• Remittance advice
• Receipt
• Voucher or promissory note
• Check
• Bill of exchange

What Are Quantity Discounts (Rappels)?

These are discounts granted by the seller to the buyer for purchasing goods exceeding... Continue reading "Key Commercial Documents Explained" »

1.1 Theory of Sets

Theory of sets. Why is it said to be a mathematical system and language? It is said to be a mathematical system because it contains a set of operations, theorems, functions and relations, and it underpins areas such as algebra, geometry, calculus and more.

Set theory is an appropriate tool for structured thinking and for developing the capacity to analyze and design solutions for particular problems. It allows focusing on a problem as a whole by removing what is irrelevant and highlighting the essentials.

Set theory facilitates the visualization of relationships between all component parts of a problem as well as each part individually. It lets us combine elements within its own methodology and use deductive reasoning together... Continue reading "Set Theory Fundamentals: Definitions, Notation and Examples" »

Project Management Fundamentals: PERT and CPM

HL (Total Slack) is the time that can delay the completion of an activity without affecting the start of another and the final term. HM (Marginal Slack) refers to the most unfavorable slack case in the book.

The Critical Path

The Critical Path is the longest and most winding route. It defines the sequence that marks the current and critical events in a double-arrow network. An activity is considered critical when it has zero slack time. An event is critical when its TE (Earliest Time) and TL (Latest Time) are equal. It is essential to pay attention to critical paths, as any delay can affect the entire project. Both initial and final events are critical.

PERT: Probabilistic System

PERT is a probabilistic... Continue reading "Project Management Techniques: PERT and CPM Analysis" »

Understanding Sequences, Progressions, and Functions

Sequences

Sequences are unlimited strings of real numbers. Each of the numbers that form a sequence is a term and is designated with a letter and an index that indicates its position in the sequence. The general term is the algebraic expression used to calculate any term, depending on the index.

Recurrent Sequences

Recurrent sequences are those in which terms are defined based on one given earlier, according to a known algebraic expression.

Arithmetic Progressions

A sequence of rational numbers is an arithmetic progression if each term is obtained from the previous one by adding a fixed number, or difference, usually represented by *d*. The general term is: W = A1 + (n-1) * d.

Geometric Progressions

A... Continue reading "Understanding Sequences, Progressions, and Functions in Math" »

Regulations for Checks and Payment Methods

1. The Drawer: The policyholder or drawer is the individual who issues the check for charge.
2. Signature and Seal: The firm extends the check, which must be signed and stamped by a seal.
3. Bearer Checks: A check payable to the bearer is collected by the person who holds this collection.
4. Endorsement Clauses: A check can be endorsed unless it has been completed with a "not to order" clause.
5. Bank Checks: A bank check is one where the bank always guarantees that it will be paid.
6. Guarantor Responsibilities: The endorsement is characterized by the guarantor responding in the same way as the person guaranteed.
7. Transmission of Rights: Endorsement is characterized by the transmission of all rights of the check.
8. Insufficient

The Architecture of Basilica San Lorenzo

The Basilica San Lorenzo was built during the Quattrocento by the architect Filippo Brunelleschi, with construction conducted between 1422 and 1470. It is inspired by the Church of the Holy Cross (Gothic). The structure is divided into three naves and lateral chapels. In the middle of the transept is a cupola, which is surrounded by chapels. Opposite the middle of the transept is the main chapel.

Geometric and Mathematical Design

Its plan is a Latin cross with three naves and side chapels, forming a basilica that is almost extremely longitudinal. In the transept, which has little development, he set a dome with a lantern. The space was mathematically and geometrically modulated by the circle inscribed in

Study Focus: Techniques for systematizing repair processes and improving duration.

Study Methods: Analyzes processes by studying the sequence of movements and operations executed by employees. The study considers all factors impacting the result: job, equipment and tools, facilities, and workers.

Process to Follow for Improvement

• Selecting work to perform
• Record facts
• Examine actions
• Develop a new improved method
• Implement the new method

Record of Activities

This involves recording, through direct observation, all operations and tasks that are part of the job being studied.

Dingbats Actions

• Operation (Round): Performed when changing the properties of a piece during assembly or disassembly.
• Shipping (Arrow): When a piece is displaced.
• Inspection (Square)