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

Understanding Key Statistical Measures

Statistical measures help us analyze and interpret data. They can be broadly categorized into measures of central tendency, position, and dispersion.

Measures of Central Tendency

Measures of central tendency describe the central value around which data are distributed. They provide a single value that represents the typical or central point of a dataset.

The Mode

Definition of Mode

The mode (represented by Mo) is the value with the highest absolute frequency in a dataset. It can be found for both qualitative and quantitative variables.

Mode Examples

To find the mode of a distribution:

• Example 1: 2, 3, 3, 4, 4, 4, 5, 5
  Mo = 4
• Example 2 (Bimodal/Multimodal): If a dataset has two or more values with the same highest

Understanding Financial Audit: Role and Regulations

The audit function is primarily regulated by Law 19/1988 of July 22 on the Audit of Accounts and its corresponding audit regulation, which is updated almost annually. These rules govern the external, financial, or economic-financial audit, carried out by independent professionals.

An audit involves the verification and review of financial statements (whether economic, financial, or economic-financial). This verification must be performed by individuals who, among other qualifications, are graduates in relevant fields with extensive accounting knowledge.

Auditors are ultimately independent professionals; they are not public officials and have no affiliation with the company being audited. In conducting... Continue reading "Understanding Financial Audit: Role, Regulations, and Objectives" »

Understanding Linguistic Vices

Linguistic vices refer to any defect or impairment that may present in words or sentences, hindering clear and correct communication.

Common Linguistic Vices

Here are some common linguistic errors:

• Cacophony: This is a very obvious error, consisting of the repetition of syllables or sounds that are contiguous or in close proximity, creating an unpleasant effect.
• Monotony: A linguistic vice produced by the frequent use of the same words or expressions to refer to different situations, leading to a lack of variety in language.
• Ambiguity or Amphibology: This vice involves expressing ideas so obscurely that they are not clearly understood, or can be interpreted in two or more ways, leading to confusion.
• Solecisms: These

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" »

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" »

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)

Finding the Vertex Coordinates of a Parallelogram

The points A (-2, 3, 1), B (2, -1, 3), and C (0, 1, -2) are consecutive vertices of the parallelogram ABCD.

(a) Find the Vertex Coordinates of D

If ABCD are the vertices of a parallelogram, free vectors AB and DC are equal:

• AB = (4, -4, 2)
• DC = (-x, 1 - y, -2 - z)

Equating coordinates, we have x = -4, y = 5, and z = -4. The missing point is D (-4, 5, -4).

(b) Equation of the Line Through B and Parallel to Diagonal AC

The line passes through point B (2, -1, 3) and has a direction vector AC = (2, -2, -3). Its continuous equation is:

(x - 2) / 2 = (y + 1) / -2 = (z - 3) / -3

(c) Equation of the Plane Containing the Parallelogram

We can use point B (2, -1, 3) and the vectors BA = (-4, 4, -2) and BC = (-2,... Continue reading "Solving Problems with Parallelograms, Lines, and Planes in 3D Space" »