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

Fundamental Geometric Concepts

Basic Geometric Elements

• Line: A straight path without beginning or end, extending infinitely in both directions.
• Ray: A straight path that has a beginning point but no end, extending infinitely in one direction.
• Segment: A straight path delimited by two distinct endpoints.
• Angle: The opening formed by two rays that share a common starting point (vertex).

Relationships Between Lines

• Intersecting Lines (Secant): Lines that cross each other at a single point.
• Parallel Lines: Lines that lie in the same plane and never intersect, having nothing in common.
• Coincident Lines: Lines that occupy the exact same position, sharing all points in common.
• Perpendicular Lines: Lines that intersect to form four equal (90-degree) angles,

Understanding Mathematical Functions

A relationship based on two numerical variables, usually called X and Y. X is the independent variable, and Y is the dependent variable. The function is usually denoted by y = f(x), associating each value of x with a single value of y: x → y = f(x).

Domain and Image of a Function

Domain: The domain of a function f(x) consists of all values of X for which the function is defined.

Image (Range): Im(f) is the set of values that the function takes. That is, the set of values y for which there exists an x such that f(x) = y.

Methods of Representing Functions

Ways to represent functions:

• Through its graphical expression
• Through a statement
• By a table of values
• By an analytical expression or formula

Calculating Different

Exercise 1: Calculating Monthly Interest Rate

A person invested $12,000 in an institution and received $13,008.00 after seven months. What is the equivalent monthly interest rate that the investor earned?

Solution:

• Initial Capital (C) = $12,000
• Time (n) = 7 months
• Future Value (FV) = $13,008.00

Using the formula: FV / C = (1 + i * n)

13,008.00 / 12,000.00 = (1 + i * 7)

1. 0840 = 1 + 7i

2. 0840 = 7i

i = 0.0840 / 7

i = 0.012 or 1.2% per month

Exercise 2: Equivalent Financial Return

An investment of $15,000 is made for a period of three months at a simple interest rate of 26% per annum. What amount must be invested for two months at a linear rate of 18% per year to achieve the same financial return?

Solution:

Investment 1:

• C = $15,000
• n = 3 months
• i (annual)

Direct Methods

1a) Gauss Elimination Method

The Gauss Elimination method consists of converting a matrix into an equivalent matrix with zeros below the main diagonal of A. The first equation will have n variables, the second n-1, and so on until the final equation, which will have only one variable. Once the zeros are achieved, the values of the variables are found by starting with the last variable and substituting back up to the first. This solves the system.

1b) Gauss-Jordan Method

The Gauss-Jordan method consists of obtaining a diagonal matrix instead of just an upper triangular matrix. Variables are obtained directly from the resulting system, without substitutions. This saving in the final step comes at the cost of slightly increasing the... Continue reading "Direct and Indirect Methods: Gauss and Jacobi" »

Essential Language and Literary Concepts

Vocabulary Definitions

• Ratification: To confirm something is true or valid.
• Sullen: Bad-tempered and sulky; gloomy.
• Persuade: To induce someone to do something through reasoning or argument.
• Obey: To comply with a command, instruction, or law.
• Prescribed: To lay down as a rule or course of action to be followed; to determine.
• Literal: Taking words in their most basic sense without metaphor or allegory.
• Prolific: Producing much fruit, foliage, or many offspring; producing many works.
• Clause: A distinct article or provision in a document, such as a contract or law.
• Maize: A cereal grain, especially corn, that is already mature.
• Diligence: Careful and persistent work or effort.
• Ninety: The cardinal number equivalent

Troubleshooting Bibliographic Reference

Daniel Robert Sippe & Bulfin: Planning and Production Control, Chapter 3

Troubleshooting Problem Definition

Problem: When I expect something to happen and it does not.

Approaches to Resolution

How to Solve:

1. Acquittal: Ignore the issue.
2. Resolution: Use common sense.
3. Solution: Use the USA method (the best answer).
4. Dissolution: Redesign to eliminate the cause.

Participation in the Problem

Who is involved:

• Owners of the Problem: Those who interact with and live with the problem.
• Analysts: Those who study the solutions.

Focus: The Six-Step Problem Solving Process

Step 1: Identification of the Problem

1. Owners of the Problem
2. Analyst
3. Need / Opportunity
4. Overall Purpose:
   • Levels
   • Goals
   • Objectives
5. Assumptions

Step 2: Understanding the

Parties Involved in a Bill of Exchange

The issuance and circulation of a bill of exchange involves the following parties:

• The Drawer: The creditor who issues the bill of exchange, instructing the debtor to pay.
• The Drawee: The debtor who is obligated to pay the bill of exchange at maturity. The drawee may accept or reject the payment order. If accepted, the drawee becomes the acceptor.
• The Payee/Holder/Beneficiary: The person who holds the bill of exchange and is entitled to receive payment.

Form of Bills of Exchange

A bill of exchange must be issued on official forms or stamped paper issued by the State. The amount should be proportional to the value stated on the bill. Improper formatting may cause difficulties in pursuing action against the debtor... Continue reading "Understanding Bills of Exchange: Parties, Form, and Accounting" »

Diraya System Objectives and Elements

Diraya Objectives: Integrated into a single Health History (HC); manage appointment schedules, and monitor treatments (ttos).

Diraya Elements: Unique health history, unifying access to services, structured information, accessibility to services, benefits, and information.

User Database (BDU) Functions

BDU: Administrative data, medical service, health service, and management of the health card.

BDU Functions: Identify citizens, know the insurance status of persons, and facilitate the free choice of doctor.

Minimum Data Set (MDS) and CMBDA Rules

MDS: A set of unprocessed basic data, collected in the shortest possible time, with the greatest range of applications; a centralized information system.

MDS Requirements:

Item 12

What Does the Acronym ICT Stand For?

ICT stands for Installation of Telephone and Common Infrastructure.

ICT Networks in a Building

• Food Network
• Distribution Network
• Dispersion Network
• Interior User Network

Elements Inside a RITI or RITS

The interior of a RITI or RITS contains the following elements:

• Point of Interconnection: Consists mainly of an input terminal strip where multi-pair cables from each operator arrive, and output strips where they connect the cables that go toward secondary multi-pair registers.
• End-point Distribution: A set of TLCA distributors, and if required, an amplifier. It is responsible for carrying the TLCA signal to the building's vertical risers and amplifying the signal to reach adequate quality levels.
• Elements of electrical

Time Study in Work Measurement

The study of time as one of the major work measurement techniques was formalized by Taylor.

Technical Insolvency

A firm is in a state of technical insolvency when the economic profit generated by its assets is not sufficient to meet the interest on its debts.

Quota Sampling

Quota sampling falls under the category of non-probability sampling.

Commercial Experimentation Technique

Analysis of Variance

The technique used in most commercial experimentation is Analysis of Variance.

Strategic Behavior and Social Groups

The set of shared values of a social group determines your preference for a particular type of strategic behavior.

Investment Risk

If a certain level of risk exists between two alternative investments, preference... Continue reading "Time Study, Sampling, and Business Management Concepts" »