Notes, summaries, assignments, exams, and exercises for Mathematics for Baccalaureate

Statistical Goals

• Describe: Explain what's happening in the data (e.g., mean, mode, average, minimum, variation).
• Explore: Understand how different variables relate to each other.
• Draw Inference: Test hypotheses or theories to make generalizations. Important: Correlation doesn't equal causation.
• Predict: Forecast future outcomes (e.g., weather networks).
• Draw Causal Inference: Determine cause-and-effect relationships, which requires experiments.

Variables in Statistics

Variable Types

1. Categorical (e.g., color, name, religion) vs. Numerical (Discrete: whole numbers OR Continuous: decimals, e.g., movie ranking).
2. • Nominal: Categories with no inherent order.
   • Ordinal: Categories with a meaningful order.
   • Interval: Ordered, equal intervals, but zero is arbitrary

1. What does boot mean?
2. How to determine the basis of taxable and non-taxable stock dividends?
3. How to determine if a gain is long-term or short-term?
4. How does capital gain/loss netting work?
5. What is the tax rate of regular long-term capital gain? [0%, 15%, 20%]
6. Individual taxpayers can deduct net capital loss (short-term or long-term) up to $3,000 per year.
7. The maximum tax rate for collectibles is 28%; the maximum tax rate for unrecaptured Section 1250 gain is 25%.
8. How to determine amount realized? How to determine amount realized when there is debt assumption?
9. What is unrecaptured Section 1250 gain? How to calculate it?
10. What is the basis of gifts? How to determine the holding period of a gift?
11. How to determine the adjusted basis?
12. How to determine realized

Product Information and Constraints

Fran Farnsworth sells three health care products: Flex, Feelgood, and Firmup. Her costs, sales time, delivery costs, and commission for each product are as follows:

• Flex: $3 cost, 10 minutes to sell, $0.50 delivery cost, $2 commission
• Feelgood: $5 cost, 20 minutes to sell, no delivery cost, $4 commission
• Firmup: $4 cost, 12 minutes to sell, $1 delivery cost, $3 commission

Fran's weekly constraints are:

• Maximum $1500 worth of products
• Maximum $85 delivery expense
• Maximum 40 hours (2400 minutes) for sales

Linear Programming Model

Let x, y, and z represent the number of boxes of Flex, Feelgood, and Firmup sold per week, respectively. Let C be Fran's weekly commission. The LP model is:

Maximize:

C = 2x + 4y + 3z

Subject

Special Corporate Tax Regimes in Spain

This document outlines key special tax regimes under the Spanish Corporate Tax Law (TRLIS), offering significant benefits for eligible businesses.

Tax Benefits for Small Companies (TRLIS Articles 108-114)

This regime applies to companies with a turnover below €10 million in the previous year. Key consequences include:

• Reduced Tax Rates:
  • 25% for the first €300,000 of taxable base.
  • 30% for the remainder of the taxable base.
• Accelerated Depreciation:
  • For fixed assets, the maximum depreciation rate is multiplied by 2.
  • For intangible assets, the maximum repayment rate is generally multiplied by 2.
  • In case of asset reinvestment (Article 113 TRLIS), the maximum repayment rate is multiplied by 3.
• Provision for Doubtful

Geometry Fundamentals

Understanding Angles

An angle is the portion of a plane formed by two rays (semi-straight lines) sharing a common endpoint. The rays are called the sides of the angle, and the common endpoint is called the vertex.

Angle Designation

• By three letters, with the vertex letter always in the middle (e.g., ∠ABC).
• By the letter of the vertex (e.g., ∠B).
• By a number or a Greek letter, often placed near the vertex (e.g., ∠α).

Types of Angles

• Adjacent angles: Angles that share a common vertex and a common side.
• Linear Pair: Two adjacent angles whose non-common sides form a straight line (sum is 180°).
• Right angle: An angle that measures exactly 90°.
• Straight angle: An angle that measures exactly 180°.
• Complementary angles: Two angles

Morphology: Defining Linguistic Units

Morphology defines and describes its primary units: the word, the root (or stem), and the morpheme.

The Word and the Root

• The Word: The minimal free form.
• The Root (or Lexeme): The constant segment of a word that remains after removing all accompanying morphemes. It serves as the starting point for morphological analysis.

Words are categorized as variable (nouns, adjectives, verbs) or invariant (prepositions, conjunctions, adverbs).

Understanding Morphemes

The morpheme is the smallest significant morphological constituent of a word.

• Free Morphemes: Words that stand alone (e.g., "but," "y").
• Bound Morphemes: Units that cannot stand alone but function when combined (e.g., "sun-es").

Grammatical Affixes

Grammatical affixes...

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...

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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

Basic Probability Notations and Examples

Probability is the branch of mathematics that deals with the likelihood of occurrence of an event. The probability of an event ranges from 0 to 1, where:

• 0 indicates an impossible event.
• 1 indicates a certain event.

Probability is widely used in Artificial Intelligence, Machine Learning, Statistics, Data Science, and Decision Making.

1. Sample Space (S)

The sample space is the set of all possible outcomes of an experiment.

Example: When a die is rolled, S = {1, 2, 3, 4, 5, 6}.

2. Event (A)

An event is a subset of the sample space.

Example: Event A = Getting an even number, so A = {2, 4, 6}.

3. Probability of an Event

The probability of an event is calculated as:

P(A) = (Number of favorable outcomes) / (Total number