Essential Formulas and Theorems in Multivariable Calculus and Differential Equations
Convergence of Sequences
Convergence Point
To resolve the limit:
Uniform Convergence
- State the Supremum Criterion:
To check for uniform convergence, the characterization of the supremum criterion is used, which states that 
converges uniformly to 
in M if and only if:
- Calculate
- Calculate the maximum (supremum) that can be reached by the absolute value in the interval marked by
- Check that the supremum tends to zero as
Power Series
Series of the form:
Convergence Study
- Calculate the radius of convergence:
If
Read MoreUnderstanding Company Assets, Liabilities, and Accounting
The assets and means that a company uses to perform its activity, i.e. to operate, are the heritage of the company.
If we consider the value of each of the elements of heritage and add them, we will have a number that represents the value of heritage.
- The goods are the material means necessary for the company’s business: buildings, machines, money.
- The rights are those effects that the company can claim: sales invoices unpaid by customers, etc.
- The obligations are the debts that the company has with
Probability and Combinatorics: Key Concepts and Examples
Probability and Combinatorics: Key Concepts
Combinatorics provides procedures and formulas necessary to count the possibilities to choose a set of items with certain characteristics.
Permutations and Variations
- By taking all elements of a finite set and ordering them in all possible ways, we have a permutation.
- A variation of order k (or k-order variation) is a group of k elements chosen from a total set, where each group is different either by the elements it contains or by the order in which they
Stock Return Analysis and Statistical Concepts
Stock Comparison: Return and Risk
Stock 1: x̄ = 9.62% and s = 23.58%
Stock 2: x̄ = 12.38% and s = 15.45%
x̄ represents the average return of a stock.
Stock 2 has a higher average return because its x̄ (12.38%) is greater than Stock 1’s x̄ (9.62%).
Stock 1 is riskier. Standard deviation (s) measures return volatility. A higher standard deviation indicates wider fluctuations and greater unpredictability. Therefore, Stock 1 is riskier (s = 23.58%) than Stock 2 (s = 15.45%).
Implication: Sharpe Ratio
Stock
Read MoreNumber Sense, Measurement, and Geometry in Early Childhood
Building Number Sense in Early Childhood
Stages of Number Development
Third Stage: Cardinality Rule
- Children aged 5 to 10 can recognize that the last number counted represents the total (cardinality) of the set.
- They do not connect the order of counting to the relative size of numbers.
- Children do exercises with numbers 1 to 9.
- They identify characteristics and qualities of the set to be counted. If sorting, they understand the desired size and can represent it in words or pictures.
Fourth Stage: Relative
Read MoreQualitative and Quantitative Research Methods: A Comprehensive Overview
Quantitative vs. Qualitative Research
| Quantitative | Qualitative | |
|---|---|---|
| Purpose | Measure, test hypotheses, count, and analyze numerically | Understand experiences, meanings, and perceptions; non-numerical |
| Data | Numbers, statistics, surveys, experiments | Words, images, interviews, observations |
| External Validity | High | Low (context-specific) |
| Internal Validity | High (controlled studies) | Can be high (case studies, in-depth analysis) |
| Reliability | High (consistent results) | Lower (subjective interpretation) |
| Examples | Surveys, experiments, |