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

Direct Methods

1a) Gauss Elimination Method

The Gauss Elimination method consists of converting a matrix into an equivalent matrix with zeros below the main diagonal of A. The first equation will have n variables, the second n-1, and so on until the final equation, which will have only one variable. Once the zeros are achieved, the values of the variables are found by starting with the last variable and substituting back up to the first. This solves the system.

1b) Gauss-Jordan Method

The Gauss-Jordan method consists of obtaining a diagonal matrix instead of just an upper triangular matrix. Variables are obtained directly from the resulting system, without substitutions. This saving in the final step comes at the cost of slightly increasing the...

Identification of Financial Statements

The financial statements shall be clearly identified and distinguished from any other information published in the same document.

Characteristics of Financial Statements

• Accuracy: Displaying information in a clear, precise, and true manner.
• Opportunity: Must be submitted on time.
• Relevance: Only present information unique to the company's operations.
• Integrity: The information submitted must be complete and not incomplete or unreliable.
• Frequency: Financial statements should be made with a fixed periodicity.

Notes to Financial Statements

Clarifications or explanations of events or situations, quantifiable or not, are an integral part of every financial statement. They must be read together for correct interpretation....

Floating Point Systems & Numerical Error

A Floating Point (FP) System represents numbers as: x = ± (d₀ + d₁/β + d₂/β² + ... + d_t-1/β^(t-1)). The Unit Roundoff (u) is defined as ε_machine/2, where fl(1 + ε) > 1.

Rounding to Nearest

When rounding to the nearest representable number, fl(x) = x(1 + ε) where |ε|.

IEEE 754 Standard for Floating Point

Normalized Numbers

If the exponent (e) is not equal to 0, it's a normalized FP number. The value is x = (-1)^sign ⋅ β^{(e - offset)} ⋅ (1.d₁ d₂...d_t-1).

Denormalized Numbers

If the exponent (e) is 0, the number is denormalized. The value is x = (-1)^sign ⋅ β^{(e - offset + 1)} ⋅ (0.d₁ d₂...d_t-1). The sticky bit 0 is free because it is always determined by the value of exponent e.

Exceptional Values

• If

Financial Ratio Analysis

Usefulness of Ratio Analysis

1. Measure of Profitability

Ratio analysis provides context for evaluating a company's profit, such as comparing it to previous periods or industry averages. Ratios like Gross Profit Margin, Net Profit Margin, and Expense Ratio help assess profitability and identify areas for improvement.

2. Evaluation of Operational Efficiency

Ratios like Turnover Ratios and Efficiency Ratios can reveal how effectively a company manages its assets and resources, helping to identify areas of inefficiency and potential cost savings.

3. Ensure Suitable Liquidity

Ratios such as the Current Ratio and Quick Ratio measure a company's short-term liquidity, indicating its ability to meet immediate cash obligations and...

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 = &...

Inventory Costing Methods: Situational Analysis

For each of the following situations, determine whether FIFO, LIFO, or Weighted Average would apply:

• FIFO would produce the highest amount of net income in an inflationary environment.
• FIFO would produce the highest amount of assets in an inflationary environment.
• FIFO would produce the lowest amount of net income in a deflationary environment.
• Weighted Average would produce the same unit cost for assets and cost of goods sold in an inflationary environment.
• LIFO would produce the lowest amount of net income in an inflationary environment.
• Weighted Average would produce an asset value that was the same regardless of whether the environment was inflationary or deflationary.
• LIFO would produce the lowest

Grounded Theory: Strategies and Methods

Grounded theory has two main strategies:

Theoretical Sampling

Involves adding to the sample of subjects or situations encountered, additional comments, or further interviews. It seeks to find different cases.

Constant Comparative Method

It is a method to analyze the data at different times in which the researcher interacts with:

1. Classified portions of data:
   • Words
   • Phrases
   • Paragraphs

Categories arise, either intuited or implicit (they are marked, but without a name).

Constant Comparative Method

As the process of selecting pieces of data advances, categories appear with more explicit rules.

The task is to identify:

1. Emerging categories = Issues
   • Categories = Properties
2. Central category = Recurring Themes

Links between categories...

Cauchy's Mean Value Theorem

Suppose f and g are two functions such that:

• f and g are continuous on the closed interval [a, b].
• f and g are differentiable on the open interval (a, b).
• For all x in the open interval (a, b), g'(x) ≠ 0.

Then there exists a number z in the open interval (a, b) such that:

[f(b) - f(a)] / [g(b) - g(a)] = f'(z) / g'(z)

Proof of the Theorem

Proving that g(b) is not equal to g(a)

First, we show that g(b) ≠ g(a). We use a proof by contradiction; assume that g(b) = g(a). Since g satisfies the conditions of the Mean Value Theorem, there is some number c in (a, b) such that:

g'(c) = [g(b) - g(a)] / (b - a)

If we assume g(b) = g(a), then g(b) - g(a) = 0, which implies g'(c) = 0. This contradicts the third condition of our hypothesis,...

Balance Sheet: The second column in this column should note the value amount of each account. The third column space in this column records the total amount for each group. The fourth column in this column should write down the total amounts of the assets, liabilities, and stockholders' equity. Comparative balance sheet reporting shows that assets less liabilities equals capital employed. The first column should write down the amount of each of the accounts involved in the final. The second column is used to note the amount of each of the accounts up to the initial balance. The third column is used to assemble the alteration or difference of each account. The fourth column is used to denote the signs of plus or minus. Output states location...

Markowitz Portfolio Selection

The Markowitz model focuses on the relationship between portfolio risk, expected returns, and the covariance and correlation between assets. Diversification reduces the variability of expected returns. The model identifies combinations that provide the maximum return for a given level of risk. A portfolio is considered efficient when it offers the highest return for a specific risk level or the minimum risk for a target return.

Core Investment Hypotheses

• Rational Investor Behavior: Investors prefer higher returns for a given risk level.
• Risk Aversion: Investors prefer lower risk for a given return level.

The Three-Step Optimization Process

1. Determine the Efficient Frontier

The efficient frontier consists of all efficient...