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

Def.Función: Sun, Pts cut with axes, asymptotes (vertical, if the denominator vanishes, horizontal-limits cndo tends to infinity, if radical oblique-function and the divisor is greater dividend is divided), monotony ( first derivative-points is studied before and after you leave), curvature (inflection pts) (second derivative)-is studied around-if <0-decreasing-convex,> 0-grows-concave), table values . Rouche theorem-schema-sist-ed homogeneous det-range (A) = n,, comp indet-range (A) <n. Not homogeneous-incompatible-rank (A) different rank (A / B), are comp.Que before. To solve system: 1) its rank is calculated by determinantes.2) resolves the sist x metodo gauss and is studied through the scheme. Rouche theorem-Fro benius. Discussion... Continue reading "Essential Formulas for Calculus, Linear Algebra, and Geometry" »

Understanding Accounting and Financial Statements

Accounting, within economics, is the study of business assets, the rules, and the scientific basis for the registration of economic information within an enterprise.

Key Components of Annual Accounts

Annual accounts, comprising the balance sheet, profit and loss account, the statement of changes in equity, and the memory, form a unified set. These accounts are financial statements that report on the company's results and its financial position. They include the following:

• The Balance Sheet

  Reports on the financial situation of the company at the close of a given fiscal year.

• The Profit and Loss Account

  Reports on the performance of the company for a year from its business operations.

• The Statement of

Relative Positions in 3D Space

Position of Three Planes

The relative position of three planes is determined by analyzing the rank of the system of equations.

• Intersect at one point: The rank of the coefficient matrix and the augmented matrix is 3.
• Intersect in a line: The rank of both matrices is 2.
• Parallel or Coincident: If the ranks are 1 or 2, the planes can be parallel, coincident, or form a prismatic surface.

Position of Two Lines

Given two lines with direction vectors v₁, v₂ and points P₁, P₂:

• Intersecting Lines: The lines are coplanar. The determinant of the matrix formed by vectors v₁, v₂, and P₁P₂ is zero. To find the intersection point, solve the system of their parametric equations.
• Parallel or Coincident Lines: The direction

Measures of Central Tendency:

Arithmetic Mean

Used for:

Intervals and pooled data (xi = class mark)

Data are not grouped (tables).

Data are grouped (tables) but no intervals.

Median

The middle value in a sorted dataset.

For an odd number of observations, it's the middle number.

Sort data from lowest to highest before finding the median.

Mode

The value that appears most frequently in a dataset.

Mid-Range

RM = (Maximum Value + Minimum Value) / 2

Geometric Mean

G = ⁿ√(x₁ * x₂ * ... * xn)

Harmonic Mean

H = n / ( (1/x₁) + (1/x₂) + ... + (1/xn) )

Quadratic Mean (Root Mean Square)

Q = √[ (x₁² + x₂² + ... + xn²) / n ]

Percentile

A measure indicating the value below which a given percentage of observations in a group falls.

Kth Percentile

Measures of... Continue reading "Statistical Measures: Central Tendency, Dispersion, and Form" »

Practice 4: Salaries, Overtime, SSO, PHL

Basic Salary

• = (Additional Table at $[$D$24] * Time Worked)

Total Overtime

• Fx = IF(logical test = E10 [Overtime] > 40)
• True-Value: E10 - 40
• False-value: 0

Triple Overtime

• Fx = IF Function
• Logic Test: Total Overtime [G10 > 8]
• True-Value: G10 - 8
• False-value: 0

Payment of Extra Time Triple

• Triple Overtime (H10) * Pay Per Hour (D24) * 3

Double Overtime

• = G10 (Total Overtime) - H10 (Triple Overtime)

Payment of Extra Time Double

• = J10 (Double Extra Time) * D24 (Pay per hour) * 2

Compulsory Social Security (SSO)

• = (F10 [Basic Salary] + I10 [Payment of Triple Overtime] + K10 [Payment of Double Overtime]) * 5%

Housing Policy Act (HPL)

• = (F10 [Basic Salary] + I10 [Payment of Triple Overtime] + K10 [Payment of Double Overtime]

Saw-Tooth Diagram for Inventory Management

The Saw-Tooth Diagram is a design used for input control and monitoring of stock. This graph serves to manage inventories, ordering, and the volume of flow itself.

The optimal quantity is determined according to the purchase price, storage costs, and handling volume. The optimal lot (often related to the Economic Order Quantity or EOQ, here referred to as PDID) includes the purchase price, carrying costs, PDID costs, overhead costs, and costs incurred until the item reaches the point of sale or placement.

Henry Gantt and the Gantt Chart

Henry Gantt was an American engineer belonging to the classical school of organization who planned American military transport during World War I, using bar-line graphs.... Continue reading "Project Scheduling Diagrams: Gantt, CPM, PERT, and Inventory Control" »

Least Squares Adjustment: Fundamental Models

1. Conditional Equations Model

This section outlines the formulation and solution for conditional equations in least squares adjustment.

• B: Jacobian matrix of the conditional equations with respect to observations.
• A: Vector representing the sum or constant terms of the conditional equations.
• P: Weight matrix of observations, calculated as 1 / σ₀².
• Q: Covariance matrix of observations, Q = P&supminus;¹.
• Qe: Covariance matrix of the conditional equations, Qe = B * Q * B&supT;.
• K: Vector of Lagrange multipliers, K = -Qe&supminus;¹ * A.
• V: Vector of residuals, V = Q * B&supT; * K.
• L̂: Adjusted observations, L̂ = V + Lobserved.

Linearized Conditional Observation Equation

A common form of... Continue reading "Least Squares Adjustment Models and Error Propagation" »

Fundamentals of 2D Analytical Geometry

This document outlines key concepts in the analytical geometry of straight lines within a two-dimensional plane.

Coordinate Systems and Vector Representation

An affine coordinate system (SdR) is defined by a triple (O, u, U), where O is a fixed point in the Euclidean plane (E2) serving as the origin, and (u, U) forms a basis for the vector space R² (V2R). A point P in E2 is uniquely determined by its coordinates (x, y) relative to the chosen basis, which represent its position vector OP.

Points and Segments

The coordinates of the midpoint M of a segment AB are obtained by averaging the coordinates of its endpoints. If A = (x_A, y_A) and B = (x_B, y_B), then the midpoint M = ((x_A + x_B) / 2, (y_A + y_B) / 2).

Equations

Data Visualization: Types and Applications

Graphics are a visual representation of data, prioritizing simplicity, easy interpretation, and adherence to standards.

Classification of Graphics

• Structural graphics: Representing a single set of data.
• Relational graphics: Connecting two sets of data.
• Special graphics.

Types of Graphics

• Organizational Charts: Represent the structure of an organization.
• Classification Plans: Represent elements of a total set, subdivided into smaller subsets. These can be square or pyramidal.
• Schedules: Use current statistics, ordering data over time to identify peaks and trends.
• Histograms: Represent class intervals or monthly values, highlighting extreme values.
• Function Tables: Establish mutual relationships between two sets

The definition of a weighted average:

What is a Weighted Average?

A weighted average is the result of multiplying each number in a set by a value assigned to it (its weight), and then calculating the arithmetic mean of the resulting products. It's used when the components contributing to the average are not equally important.

For example, if a teacher states that an examination is worth 40% of the final mark, another is worth 35%, and a third is worth 25%, the weighted average would be calculated as follows:

mediaPond = (ex1 * 40 + ex2 * 35 + ex3 * 25) / 100

Basically, it's an average of a dataset that allows you to define the degree of importance for each data point's contribution to the average.

If the data are 2, 3, 5, 7, 9, 6, 8, the average... Continue reading "Understanding Weighted Averages: Definition and Applications" »