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

Slope, Inclination, and Lines

PENDING

Slope

Inclination and Tilt

This refers to the straight tilt or angle of a line.

Parallel Lines Definition

Two lines are parallel if their slopes are equal: L₁ || L₂ if m₁ = m₂.

To find the angle (theta), the slope is used, often in conjunction with trigonometric tables or functions.

Perpendicular Lines Definition

Two lines are perpendicular if they form an angle of 90⁰. This occurs when the product of their slopes is -1:

(m₁)(m₂) = -1

General Equation of a Line

To formulate the general equation of a line, two components are typically needed:

• The slope (m)
• A specific point on the line (x₁, y₁)

The variables (x, y) in the equation represent any point on that line.

Midpoint of a Line Segment

The coordinates of the midpoint... Continue reading "Core Concepts in Analytical Geometry: Lines, Conics, and Algebra" »

Critical Path Method (CPM) vs. PERT

HL (time to completion may be delayed without affecting the beginning of another and final term), HM (the worst case of free float.) Critical path: the path of longer duration and is defined by the sequence of activities that mark critical events in the network. These activities are styled with a double arrow. An activity is critical when it has no slack time, and an event is critical when its TE (earliest time) and TL (latest time) are equal. We must pay critical attention to these routes, as any delay may affect the entire project. Beginning and ending events are critical.

PERT (Program Evaluation Review Technique): It is a probabilistic system that assigns each possible activity three durations: pessimistic,... Continue reading "Critical Path Method (CPM) vs. Program Evaluation Review Technique (PERT)" »

Levels of Number Understanding

Level 1

At this level, number figures are symbols linked to real-world objects.

Level 2

Children try to find correspondence between written figures and quantitative aspects of objects.

Level 3

Numbers, especially single-digit numbers, represent counts of objects. Two-digit numbers are not yet seen as composed of constituent figures.

Level 4

Two-digit numbers systematically represent the total quantity, but each digit is treated as a separate number without understanding place value. For example, in 16, '6' might represent six objects and '1' might represent one object. Alternatively, '6' might represent sets of 6, and '1' might relate to a single unit or group.

Level 5

Each digit in a two-digit number represents an amount... Continue reading "Key Concepts in Math Education" »

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" »

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" »

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" »

Understanding Invoices

An invoice is a document derived from a legally established commercial operation. To create invoices, information from the order note and delivery note will be taken into account.

Invoice Issuance Essentials

Key considerations for issuing invoices include:

• Issuance Requirement: Issue invoices for all operations undertaken in your business activity. It's crucial to keep a copy of each invoice, even in cases of self-consumption, with certain exceptions.
• Mandatory Issuance Cases: Invoices must be issued in all circumstances for the following operations:
  • When the recipient requests it.
  • For goods delivered to another Member State of the European Union.
  • When recipients are Public Administrations.
  • For work executions.
  • For deliveries of