Introduction to Measurement and Observation
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Notes, abstracts, papers, exams and problems of Mathematics
Understanding Functions: Definitions, Properties, and Types
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Function
A function defines the relationship between an initial set and a final set, so that each element of the initial set (independent variable) corresponds to a single element of the final set (dependent variable).
Domain of the Function
The domain of a function is the set of possible values that the independent variable (e.g., coins) can take.
Range of the Function
The range of a function is the set of possible values that the dependent variable (e.g., drinks) can represent.
A function can be represented by tables, graphs, and algebraic formulas.
Increasing and Decreasing Functions
- A function is increasing on an interval if for any pair of values a and b in this interval, where a < b, the rate of change is positive.
- A function is decreasing
Essential Trigonometric Identities and Formulas
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Pythagorean Identities:
sin (a + b) = sin(a) · cos(b) + cos(a) · sin(b) cos (a + b) = cos(a) · cos(b) - sin(a) · sin(b) tan (a + b) = (tan(a) + tan(b)) / (1 - tan(a)tan(b)) sin(2a) = 2 · sin(a) · cos(a) cos(2a) = cos2(a) - sin2(a) tan(2a) = 2tan(a) / (1 - tan2(a)) sin(a / 2) = ±√((1 - cos(a)) / 2) cos(a / 2) = ±√((1 + cos(a)) / 2) tan(a / 2) = ±√((1 - cos(a)) / (1 + cos(a))) sin(a)sin(b) = 2sin((a + b) / 2) · cos((a - b) / 2) sin(a) - sin(b) = 2cos((a + b) / 2) · sin((a - b) / 2) cos(a) + cos(b) = 2cos((a + b) / 2) · cos((a - b) / 2) cos(a) - cos(b) = -2sin((a + b) / 2) · sin((a - b) / 2) |
Basic Trigonometric Identities:
sin2(x) + cos2(x) = 1 1 + tan2(x) = sec2(x) 1 + cot2(x) = csc2(x) tan(x) = sin(x) / cos( |
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Key Commercial Documents Explained
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Differences Between Delivery Notes and Invoices
Delivery Note: A provisional document justifying the dispatch of goods. It does not include VAT.
Invoice: A definitive document providing legal accreditation. It is valid for any claim and includes VAT.
Another important difference is that invoices are legally required to be kept for 6 years, while retaining delivery notes is not mandatory for the same period.
Sales Transaction Documentation
Common documents involved in sales transactions include:
- The order sheet
- The delivery note
- The invoice
- The expenses sheet
- Remittance advice
- Receipt
- Voucher or promissory note
- Check
- Bill of exchange
What Are Quantity Discounts (Rappels)?
These are discounts granted by the seller to the buyer for purchasing goods exceeding... Continue reading "Key Commercial Documents Explained" »
Understanding Sequences, Progressions, and Functions in Math
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Understanding Sequences, Progressions, and Functions
Sequences
Sequences are unlimited strings of real numbers. Each of the numbers that form a sequence is a term and is designated with a letter and an index that indicates its position in the sequence. The general term is the algebraic expression used to calculate any term, depending on the index.
Recurrent Sequences
Recurrent sequences are those in which terms are defined based on one given earlier, according to a known algebraic expression.
Arithmetic Progressions
A sequence of rational numbers is an arithmetic progression if each term is obtained from the previous one by adding a fixed number, or difference, usually represented by *d*. The general term is: W = A1 + (n-1) * d.
Geometric Progressions
A... Continue reading "Understanding Sequences, Progressions, and Functions in Math" »
Solving Problems with Parallelograms, Lines, and Planes in 3D Space
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Finding the Vertex Coordinates of a Parallelogram
The points A (-2, 3, 1), B (2, -1, 3), and C (0, 1, -2) are consecutive vertices of the parallelogram ABCD.
(a) Find the Vertex Coordinates of D
If ABCD are the vertices of a parallelogram, free vectors AB and DC are equal:
- AB = (4, -4, 2)
- DC = (-x, 1 - y, -2 - z)
Equating coordinates, we have x = -4, y = 5, and z = -4. The missing point is D (-4, 5, -4).
(b) Equation of the Line Through B and Parallel to Diagonal AC
The line passes through point B (2, -1, 3) and has a direction vector AC = (2, -2, -3). Its continuous equation is:
(x - 2) / 2 = (y + 1) / -2 = (z - 3) / -3
(c) Equation of the Plane Containing the Parallelogram
We can use point B (2, -1, 3) and the vectors BA = (-4, 4, -2) and BC = (-2,... Continue reading "Solving Problems with Parallelograms, Lines, and Planes in 3D Space" »
Understanding Key Financial Ratios for Businesses
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Working Capital
Working Capital measures the capacity for payment in the ordinary course of business activity. It's calculated as: Current Assets (CA) - Current Liabilities (CL)
- CA > CL: Positive Working Capital. The business has the potential for investment. Working Capital should never exceed 10% of CA, as these are idle funds.
- CA < CL: Negative Working Capital. This may indicate a suspension of payments or insolvency. It usually signifies mismanagement in the negotiation of ordinary business activity, but it doesn't always mean a bad situation.
Acid Test
The Acid Test measures a company's capacity to meet all of its short-term debts. It's calculated as: (Current Liabilities - Treasury) / Available. This indicates immediate liquidity;... Continue reading "Understanding Key Financial Ratios for Businesses" »
Understanding Asset and Liability Valuation Concepts
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Valuation Criteria
Historical Cost
For Assets: The purchase price or production cost. This includes the amount of cash paid or payable, plus the fair value of any other consideration given for the acquisition. All costs directly related to the acquisition and necessary to bring the asset to operating condition are included.
For Liabilities: The value corresponding to the consideration received in exchange for incurring the debt. In some cases, it is the amount of cash expected to be paid to settle the liability in the ordinary course of business.
Fair Value
The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between knowledgeable, willing market participants at the measurement date. It is determined... Continue reading "Understanding Asset and Liability Valuation Concepts" »
Effective Questionnaire Design for Market Research
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Sampling Stage
Sample Size:
- Depends on population size
- The type of sample
- Feature (parameter) of the population that is analyzed
- The maximum permissible error in the estimation of the parameters
The Form: The information collected can be classified as:
- Acts or behaviors that can be compared
- Information: analyzes the degree of respondents' knowledge on specific topics
- Opinions or Judgments: such as perceived service quality
- Attitudes or predispositions of mind: We are looking for what is thought in relation to something
- Motives or explanations for specific behaviors: the question is, why certain views or acts?
- Possible future behavior: it may ask whether or not to consume a prepared product.
Concept and Structure of Questionnaires
The questionnaire is a way... Continue reading "Effective Questionnaire Design for Market Research" »
Key Concepts in Health, Nutrition, and Statistics
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Food and Nutrition
Food is the act of providing sustenance to the human body, while nutrition encompasses the physiological processes by which the body receives, transforms, and utilizes the chemical components in food.
Inherited Diseases
- Hemophilia
- Huntington's disease
- Cystic fibrosis
- Color blindness
- Phenylketonuria
Measures of Dispersion
Measures of dispersion indicate how close the data are to the average.
Vaccines
Vaccines are preparations containing killed or attenuated microorganisms. They are introduced into our bodies to produce antibodies that kill the organism, providing immunity.
Common Pathogens
Pathogens such as bacteria and fungi are the most common causes of infectious diseases.
Health and Disease
Health is a state of complete physical, mental,... Continue reading "Key Concepts in Health, Nutrition, and Statistics" »