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

1. Statistics: Classes and Basic Concepts

Statistics is the science that deals with data collection, organization, and analysis, as well as the predictions that can be made from it.

Descriptive Statistics

Descriptive statistics is concerned with collecting data from a set, organizing it into tables, and calculating numerical summaries that comprehensively describe the studied whole.

Inferential Statistics

Inferential statistics aims to draw conclusions about a population based on the results of a sample and the reliability of these findings.

2. Statistical Variables and Characteristics

Qualitative variables or characteristics are those that cannot be measured and are described in words.

Quantitative variables or characteristics are those that can be... Continue reading "Statistical Foundations: Concepts, Variables, and Data Visualization" »

Triangles

• Median: The segment connecting a vertex to the midpoint of the opposite side.
• Centroid (G): The intersection point of the three medians. It divides each median into two segments.
• Height (Altitude): The perpendicular segment from a vertex to the opposite side or its extension.
• Orthocenter: The point where the three altitudes intersect.
• Perpendicular Bisector: The line perpendicular to a segment at its midpoint.
• Angle Bisector: The ray that divides an angle into two equal angles.
• Circumcenter: The intersection point of the three perpendicular bisectors. It is equidistant from the three vertices.
• Incenter: The intersection point of the three angle bisectors. It is equidistant from the sides of the triangle.

Polygons

• Polygon: A plane region bounded

Rolle's Theorem

If the function y = f(x) is continuous on [a, b], differentiable on (a, b), and f(a) = f(b), there is at least one x₀ ∈ (a, b) such that f'(x₀) = 0.

Proof of Rolle's Theorem

The continuity of y = f(x) on the closed interval [a, b] implies the existence of an absolute maximum M and an absolute minimum m, according to the Weierstrass theorem. Two cases may occur:

• Case 1: The maximum M is in (a, b), the minimum m is in (a, b), or both are in (a, b).
• Case 2: M and m are at the endpoints a and b.

Suppose Case 1, where M is the value of the function at a point in the open interval (a, b). Since the function is differentiable, the derivative must vanish: if f(x₀) = M, then f'(x₀) = 0. The same occurs if m is reached at a point... Continue reading "Rolle and Lagrange Theorems: Calculus Principles Explained" »

Longitudinal Profile: Analyzing Cutting Contours

Example: Consider a horizontal area in a waterway. It is clear that the profile should show a sharp break rather than a continuous stretch. To correct this, we define a break in the ground using breaklines.

System Representation of Height

To represent height, we utilize geometric tools and random points known as "stitches," each assigned a specific height. The density of these points determines the accuracy of the model.

Accuracy and Human Perception

The human eye can perceive a difference of 0.25 mm with an error margin of 0.2 mm. Consequently, the visible error depends on the map scale. For example, what is the visible error at a scale of 1:50,000?

Distance Definitions

• Dn (Natural Distance): The stretch

Phases of a File Report

Depending on the activity or inactivity of files, a file in healthcare (HC) can be categorized into three stages:

• Active Archive
• Passive File
• Historical Archive

• Active file: This is one that meets the active healthcare compliance (HHCC), i.e., records subject to continued use and consultation.
• Archive candidate: This is one that meets the inactive HHCC, i.e., one that you have to transfer from active files when a query in a time interval of 3 to 5 years is nil. These documents remain here until disposal or transfer to the historical archive.
• Historical Archive: This is where documentation is transferred from the passive file after 10 years without being consulted. These documents are kept permanently because they were not

Linguistic Sign Features

Root: The root is the carpenter-joiner and wood SIGIF.

Morpheme: It is the -er meaning 'person who works with' and the -o meaning 'male', while -s refers to the plural.

Characteristics of the Linguistic Sign

• Biplane: It consists of two levels. The signifier (expression level) is the string of sounds or spellings (e.g., the signifier of 'niño' is /n/ /i/ /ñ/ /o/). The signified (content level) is the concept in our minds associated with the signifier (e.g., 'niño' means 'young person').
• Arbitrary: The meaning of 'young person' varies by language: 'niño' in Spanish, 'warm' in Quechua, and 'boy' in English.
• Articulate: It is composed of smaller parts.

Monemes: Units of Meaning

The first articulation consists of units that... Continue reading "Linguistic Signs, Monemes, and Literary Analysis Essentials" »

Fundamental Concepts of Probability

• Random Experience: An experiment whose outcome depends on chance.
• Random Event: An event that may or may not occur depending on chance.
• Sample Space (E): The set of all possible outcomes of a random experiment.
• Event: Any subset of E, including individual elementary events, the empty set, and the certain event.

Set Operations in Probability

• Union (A∪B): An event comprising elements of A or B; verified when at least one occurs.
• Intersection (A∩B): An event consisting of elements common to both A and B.
• Difference (A\B): An event consisting of elements in A that are not in B.
• Complementary Event (A'): The opposite event (E \ A).
• Mutually Exclusive Events: Two events are inconsistent if they have no common elements

Simple Capitalization

That type of capitalization is characterized by the fact that the interests that occur in each period are not added to the capital to produce new interests in the following period. That is, simple interest capitalizations are not productive.

Properties of Simple Capitalization

• The capital that remains invested in each time period is always the same.
• Therefore, the interests produced in each period are also equal.

Financial Laws

To verify two or more capitals invested at different points in time, one must find their value at a single moment. To keep the funds at the same time, we need to use a financial law. They can be of two types:

• Financial Law of Update: Used to calculate the value of capital at a later time and move it to

Payment Process

Payments can be made in cash or deferred. Cash payments can be made in different ways:

• Cash: A receipt will be issued to prove delivery.
• Bank Transfer: Cash deposited directly into a current account.
• Checks
• Credit and Debit Cards
• Letters of Credit

Other common payment instruments include:

• Bills of Exchange
• Promissory Notes

Checks

A check is a document regulated by law, instructing a bank to pay a specified amount from the drawer's funds to the payee. The drawer is entitled to dispose of these funds by check.

Key Parties in a Check Transaction

• Drawer (Maker): The person or entity who issues the check.
• Drawee (Bank): The bank ordered to pay the check.
• Payee (Holder): The person or entity to whom the check is payable.
• Endorser: The party who transmits

Checks: A Comprehensive Overview

What is a Check?

A check is a document instructing a bank (the drawee) to pay a specific amount of money to the check holder (the payee) from the account of the person who wrote the check (the drawer).

Parties Involved

• Drawer: The person who writes and signs the check, authorizing the payment.
• Drawee: The bank or financial institution where the drawer has an account.
• Payee/Holder: The person or entity to whom the check is made payable.
• Guarantor: A person who guarantees payment if the drawer's account has insufficient funds.
• Endorser: The payee who signs the back of the check to transfer ownership.
• Endorsee: The person to whom the check is endorsed.

Types of Checks

Forms of Writing

• Bearer: Payable to the person holding