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

Derivatives of Algebraic and Transcendental Functions

• f(x) = (x³ - 3x)³: f'(x) = 3(x³ - 3x)² · (3x² - 3) = 9x⁸ - 63x⁶ + 135x⁴ - 81x²
• f(x) = 2e^x · (3x - 1): f'(x) = 2e^x · (3x - 1) + 2e^x · 3 = 6xe^x - 2e^x + 6e^x = 6xe^x + 4e^x
• f(x) = ln(x² + 6x): f'(x) = (1 / (x² + 6x)) · (2x + 6)
• f(x) = 1 / ln x: f'(x) = -1(ln x)^-2 · (1 / x) = -1 / (x ln² x)
• f(x) = 2 cos x · sin x: f'(x) = 2(-sin x) · sin x + 2 cos x · cos x = 2(cos² x - sin² x)
• f(x) = 2^{x2 - 3}: f'(x) = 2^{x2 - 3} · 2x · ln 2
• f(x) = cos³(2x): f'(x) = 3 · (cos 2x)² · (-sin 2x) · 2 = -6 cos²(2x) · sin(2x)
• f(x) = (2x³ - 3x) / (x² - 1): f'(x) = [(6x² - 3)(x² - 1) - (2x³ - 3x)(2x)] / (x² - 1)²
• f(x) = (x² - 1) / (x + 1): f'(x) = [(2x)(x + 1) - (x² - 1) · 1] / (x + 1)²
• f(x) = e^x2 / (1 - e^x): f'(x) =

Pragmatic Analysis of Academic Discourse

When we analyze a text from a pragmatic point of view, we should consider several key elements: the purpose, the channel, the genre, the sender, and the receiver.

Characteristics of Academic Discourse

The primary purpose of academic discourse is to inform. In this context, the issuer acts as a transmitter, while the receiver is positioned at either a specialized level (scientific, such as a nurse) or a declarative level for general dissemination (divulgation).

Linguistic Markers and Procedures

Key markers of academic discourse include:

• Absence of deictics and an emphasis on impersonality.
• Use of the third-person axis.
• Predominance of the indicative mode and declarative statements.
• Absence of modalization procedures.

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]

Limit Theorems and Convergence

• Root Test: lim ⁿ√a_n = L
• Ratio Test: lim a_n/a_n-1 = L
• Stolz Theorem: lim a_n/b_n = lim (a_n-a_n-1)/(b_n-b_n-1)
• Arithmetic Mean: lim (a₁ +...+ a_n)/n = lim a_n
• Geometric Mean: lim ⁿ√(a₁...a_n) = lim a_n

Convergence Criteria

• Raabe: lim n(a_n/a_n+1 - 1) > 1 (Convergent)
• Pringsheim: lim n^αa_n (if α > 1, Convergent)
• Root Test (n-th): lim ⁿ√a_n < 1 (Convergent)
• Logarithmic: lim log(1/a_n) / log(n) > 1 (Convergent)
• Absolute Convergence: If ∑|(-1)ⁿa_n| converges, then ∑(-1)ⁿa_n converges.
• Leibniz Criterion: 1) lim a_n = 0; 2) a_n+1 - a_n < 0

Series Types

• Hypergeometric Series: a_n+1/a_n = (an + b)/(n + c). Convergent if (c-b)/a > 1.
• Telescoping Series: ∑a_n = x_n - x_n+1. Sum = x₁ - lim x_n+1.

Asymptotic Equivalences

sin(a_n) ≈ a_n;... Continue reading "Essential Calculus Formulas and Convergence Theorems" »

Genetic Algorithm Implementation Functions

The following Python functions demonstrate key components of Genetic Algorithms, including crossover, selection, and population initialization.

Four-Point Crossover Method

This function performs a four-point crossover using four parents to generate offspring.

def four_point_cross(parent1, parent2, parent3, parent4):
    pos_alea = sorted(list(np.random.choice(range(1, parent1), 4, replace=False)))
    hijo1 = parent1[:pos_alea[0]] + parent2[pos_alea[0]:pos_alea[1]], parent1[:pos_alea[1:2]], parent2[pos_alea[2]:pos_alea[3]], parent1[:pos_alea[3:]]
    hijo2 = parent2[:pos_alea[0]] + parent3[pos_alea[0]:pos_alea[1]], parent2[:pos_alea[1:2]], parent3[pos_alea[2]:pos_alea[3]], parent2[:pos_alea[3:]]
    hijo3

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" »

Inventory: Key Products Waiting for Use

1. The STOCKS: key products in a situation of waiting for later use.

2. Types of Stock

Depending on the degree of transformation of materials

Depending on the degree of transformation of materials:

• Raw materials: Goods, parts or components found in the warehouse expected to be transformed into products suitable for final use or consumption.
• Work in progress (WIP): Materials in intermediate manufacturing stages that are undergoing transformation. These are ongoing products that will progress from raw material to finished product and usually require inspection or quality control before reaching the finished product stage.
• Semi-finished products: Manufactured materials used by the business that are not yet for

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" »

Quality Management Tools and Techniques

Brainstorming

Brainstorming is based on the ideas and contributions of individuals and is used to define the causes of problems, take interim measures, and achieve innovative ideas. This mode can be:

• Structured: Each person in the group gives their idea in turn (e.g., in a circle).
• Unstructured: Free flow, where each person gives their idea when they think of it.

Cause-Effect Diagram (Ishikawa Fishbone)

This diagram is used to identify and sort all the causes of a problem to specify the source and fix it. Its use is widespread and helps to:

• Identify the true causes.
• Summarize existing relationships.
• Promote process improvement and foster creativity.
• Gain a more global perspective.
• Equalize members and help analyze

Accounting: It uses information systems to register technical summaries, classify, and analyze transactions based on operations and along the timeline of the company. Accounting Steps: Identification and economic analysis of differences, quantifying the value of the operation, and booking entries to decide (journals, ledgers, and auxiliary) for the preparation of accounts, balance sheets, and analysis of balances. Characteristics of Accounting: Order, adequacy, relevance, objectivity, rigor, and standardization. Normalization: Accounting is understandable for all persons who need the information it contains. Standard Accounting Plan Types: General Types of Accounting: Microaccounting: Economic domestic, profit entities without profit motive,... Continue reading "Understanding Accounting Principles and Practices" »