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

Vector Determination by Length and Angle

V = <||V|| Cosθ, ||V|| Sinθ> ---> ||V||Cosθi + ||V||Sinθj

Example:

a) Find the vector of length 2 that makes an angle of π/4 with the positive x-axis.

b) Find the angle that the vector V = -₃ i + j makes with the positive x-axis.

a) <||V||Cosθ , ||V||Sinθ> = <2cos45, 2sin45> ---> <₂, ₂>

b) Normalize... ||V|| = _{(-3)² + 1²} = ₄ = 2 -----> V/||V|| = <-_3/2 , 1/2> = <cosθ, sinθ> ----> cosθ = -_3/2, sinθ = 1/2 ---> θ = 5π/6

Dot Product

If U = <U₁, U₂> and V = <V₁, V₂>, then the dot product is U•V = U₁V₁ + U₂V₂.

Example:

a) U = <3, 5>, V = <-1, 2> -----> U•V = (3)(-1) + (5)(2) ---> U•V = -3 + 10 --> U•V = 7

b) U = &... Continue reading "Vector Operations, Dot and Cross Products, and Lines" »

Multiples consist of using the multiples of a company similar to yours to apply to your numbers and get an approximation of what yours is worth. It is not the most accurate in my opinion.

An example would be the EBITDA multiple, you multiply EBITDA by that figure obtained by dividing the value of the company you are comparing yourself to by its EBITDA.

This method is useful because although the estimated value of your company is not precise, with this method you can see the evolution over time.

250. The increase in provisions is not considered an outflow of cash, so it goes to the income statement.

– 150. The following are considered cash outflows

Add the cash paid instantly by the customer recorded in the cash flow statement as cash received... Continue reading "Valuation Methods and Cash Flow Analysis" »

What is a One-Way Function?

A one-way function is a function that is easy to compute in one direction but difficult to compute in the reverse. For example, given an input, a hash is easy to compute. However, given the output, it is extremely difficult (time-consuming) to determine the input.

What is a Trapdoor Function?

A trapdoor function is like a one-way function; however, a trapdoor function can reverse the one-way function if the trapdoor is known. For example, finding the two prime divisors of 6,895,601 is difficult. However, if you know that 1,931 is one of the numbers, it will be easy to divide 6,895,601 by 1,931 to determine the answer. In this case, 1,931 is the trapdoor.

What Makes RSA Difficult to Break?

The large number factorization... Continue reading "Understanding Cryptographic Functions and RSA Security" »

Essential Business and Financial Vocabulary

This list provides key English financial and business terms translated into Catalan and Spanish for professional reference.

General Business Terms

• 1. Track = Seguir
• 2. Reliability = Fiabilitat, és fiable
• 3. Shareholders = Accionistes
• 4. Invoice = Factures per cobrar
• 5. Receipt = Factures per pagar
• 6. Stakeholder = Inversor
• 7. Lender = Prestamista
• 8. Suppliers = Proveïdor
• 9. Reliability = Que no hi ha risc
• 10. Meeting = Complir amb
• 11. Showcasing = Mostrar
• 12. Often issued = Treure a la llum
• 13. Quarterly intervals = 4t trimestre

Credit and Debt Terminology

• 14. Creditworthy = Creditibilitat
• 15. Creditor = Persona que inverteix
• 16. Resources = Recursos
• 17. Balance sheet = Fulla de càlcul
• 18. Running out = Quedar-se sense
• 19.

1. Exponential Distribution: Tax Office Service Time

The service time (in minutes) at the local government tax office is exponentially distributed with λ = 0.025. (15 points)

• a) What are the mean and standard deviation?
  μ = σ = 1/λ = 40 minutes
• b) Find the probability that the service time will be greater than 40 minutes.
  e^{-0.025 × 40} = 0.3678
• c) Find the probability that the service time will be between 20 and 35 minutes.
  (1 - e^{-0.025 × 35}) - (1 - e^{-0.025 × 20}) = 0.583 – 0.393 = 0.19

2. Uniform Distribution: Barcelona Subway Waiting Time

The time a person waits for the Barcelona subway is uniformly distributed from 0 to 6 minutes. (15 points)

• a) What is the probability of each outcome?
  p = 1/(b - a) = 1/6
• b) What are the mean and standard deviation

Sample Characteristics: Measures of Location and Variation

Measures of Location

• Average

  The average is a central value, often referred to as a measure of central tendency.

• Mean

  • Also known as the arithmetic mean.
  • Found by dividing the sum of all values by the total number of values.
  • Formula: Mean = (Sum of Quantities) / (Number of Quantities)
  • Example: Find the mean of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
    • Sum of values: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55
    • Number of values = 10
    • Mean of values = 55 / 10 = 5.5

• Median

  The median of a set is the middle number when the values are arranged in order from lowest to highest.

• Mode

  • The mode is the value that occurs most frequently in a set of values, also known as the modal value.
  • Example: Find the mode of 1, 2, 2, 3,

10 reasons to use games in language teaching

1.Games create a context for meaningful communication. 2. This meaningful communication serves as a basis for comprehensible input. 3. Games add interest to what learners find boring. 4. Games can be used with all language skills. 5. Games offer a fun experience. 6. Games encourage participation from all learners. 7. Games are learner-centered activities. 8. Games work outside of class. 9. Games promote cooperative learning. 10. Games fit into multiple intelligence theory.

Using Games in Teaching English to Young Learners: How to Choose a Game

Students may wish to play games purely for fun. Teachers need more convincing reasons. The key to a successful language game is that the rules are clear. The... Continue reading "Benefits of Using Games, Songs, and Stories in Language Teaching" »

Problem 4: Bank Savings Distribution

The average savings in the clients' accounts of a certain bank follow a normal distribution N(50,000, 15,000) (mean = 50,000 €, standard deviation = 15,000 €). The bank distributes its clients into three categories:

• Category A: Those clients that have less than 20,000 € in their savings account
• Category B: Those clients that have between 20,000 € and 80,000 € in their savings account
• Category C: Those clients that have more than 80,000 € in their savings account

Calculate:

4a) What percentage of clients belong to Category A?

Use the standard normal variable Z = (X - μ) / σ.

For X = 20,000 €:

Z = (20,000 - 50,000) / 15,000 = -30,000 / 15,000 = -2.000

The cumulative probability Φ(-2.00) ≈ 0.0228,... Continue reading "Normal Distribution Problems: Z‑Scores, Percentages & Parameters" »