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

1. Understand the Problem

The first step to solving a two-step algebraic equation is to clearly write down the problem. This helps you visualize the solution process. For our example, we will work with the equation: -4x + 7 = 15.

2. Isolate the Variable Term Using Addition or Subtraction

The next step is to isolate the variable term (e.g., "-4x") on one side of the equation and the constants (whole numbers) on the other. To achieve this, you'll use the Additive Inverse. Find the opposite of the constant term on the same side as the variable. In our example, the constant is +7, so its additive inverse is -7.

Subtract 7 from both sides of the equation to cancel out the "+7" on the variable's side. Write "-7" below the 7 on the left side and below... Continue reading "Mastering Two-Step Algebraic Equations" »

Essential Statistical & Numerical Concepts

This document covers fundamental questions and answers across various topics in statistics, numerical methods, and data analysis. Each section provides a concise explanation of key concepts.

1. What is Interpolation?

Interpolation is a mathematical technique used to estimate unknown values that fall between known data points. It involves creating a function or model based on the given data and using it to predict values within the range of the data, rather than extrapolating outside it. This method is commonly applied in fields like data analysis, engineering, and computer graphics for filling in missing data or smoothing curves.

2. Regula Falsi Method Formula for Finding Roots

The formula for finding... Continue reading "Core Concepts in Statistics and Numerical Analysis" »

Understanding the distribution of sample means is crucial in statistics. Let's analyze two distinct scenarios.

MLB Player Salaries in 2012

In 2012, there were 855 major league baseball players. The mean salary was \(\mu = 3.44\) million dollars, with a standard deviation of \(\sigma = 4.70\) million dollars. We will examine random samples of size \(n = 50\) players to understand the distribution of their mean salaries.

Distribution of Sample Means

To describe the shape, center, and spread of the distribution of sample means, we apply the Central Limit Theorem (CLT). The CLT states that for sufficiently large sample sizes (typically \(n \ge 30\)), the sampling distribution of the sample mean will be approximately normal, irrespective of the population... Continue reading "MLB Player Salaries & Dark Chocolate's Vascular Health Impact" »

Section A: R Basics and Data Types (Weeks 1-4)

Model Questions and Answers

Q: Create a vector v (3, NA, Inf, -Inf). Explain adding 5 to v.

A: The operation is element-wise. Missing values (NA) propagate, resulting in NA. Infinite values (Inf, -Inf) remain infinite.

v <- c(3, NA, Inf, -Inf)
print(v + 5)
# Output: [1]  8 NA Inf -Inf

Q: Given vector a, write code to count and replace NAs.

A: Assuming a <- c(10, 15, NA, 20).

• Count NAs: sum(is.na(a)) → 1.
• Replace NAs (e.g., with 0): a[is.na(a)] <- 0.

Q: Explain the difference between a[a > 12] and a[which(a > 12)].

A: Both select elements greater than 12 (15, 20). However:

• a[a > 12] → [1] 15 NA (Uses logical indexing; preserves the position of NA in the original vector as NA).
• a[which(

1. Simplifying Algebraic Expressions

How to Simplify Algebraic Expressions

• Multiply numbers and letters together.
• Combine like terms (terms with the same letters and powers).

Example:
−2ac × 4bd = −8abcd

Example:
5ab − 8b²a + ba = 5ab − 8ab² + ab = −8ab² + 6ab

2. Expanding Brackets (Distributive Law)

How to Expand Brackets

• Multiply everything inside the bracket by what is outside.

Example:
−4(x + 7) = −4x − 28

Example:
5(x − 3) + 2(6 − x) = 5x − 15 + 12 − 2x = 3x − 3

3. Solving Equations

How to Solve Equations

• Get all variables (e.g., x's) on one side and numbers on the other.
• Perform the same operation on both sides of the equation.

Example:
3x + 2 = 17
3x = 15
x = 5

Example:
9x − 8 = 4x + 7
5x = 15
x = 3

4. Solving Inequalities

How

Understanding Inheritance in Python

Inheritance allows a class to use and extend the properties and methods of another class. It promotes code reusability and reduces duplication. Think of it as “child learns from parent.” 👨‍👦‍💻

Code Duplication: Program Without Inheritance

When classes share common attributes or methods (such as first_name, last_name, and get_age), implementing them separately leads to redundant code, as shown below:

import datetime

class TennisPlayer:
    def __init__(self, fname, lname, birth_year):
        self.first_name = fname
        self.last_name = lname
        self.birth_year = birth_year
        self.aces = []

    def get_age(self):
        now = datetime.datetime.now()
        return now.year -

System Types
The systems of equations can be classified by the number of solutions that can arise. According to that case may have the following cases:
· Incompatible system if it has no solution.
· Compatible system if you have any solution in this case can also distinguish between:
or compatible system determined when it has a finite number of solutions.
indeterminate or compatible system when it admits an infinite set of solutions.
Fitting and classification:

Calculating the rank of a matrix for determining

1. We can rule a line if:.
· All the coefficients are zeros.
· There are two equal lines.
• A line is proportional to another.
• A line is a linear combination of others.
Delete the third column because it is a linear... Continue reading "Matrices: multiplication, rank, determinant, inverse and Rouche-Frobenius theorem" »

Part I: Complete the Following Table

Part II: Complete the Following Sentences

1. She is from the United States. She is American.
2. She is from Nigeria. She is Nigerian.
3. She is from Germany. She is German.
4. They are from Egypt. They are Egyptian.
5. We are from Canada. We are Canadian.

Part III: Select the Correct Sentence

1. Select the correct sentence.
   • a) She lives with Carlos, and she works on Saturdays.
2. Select the correct sentence.
   • c) They are running in the park.
3. Select the correct sentence.
   • b) She is 20 years old.
4. Select the correct sentence.
   • a) We always

Fundamental Accounting Principles and Practices

Core Accounting Definitions

Assets = Liabilities + Owner’s Equity: This is the fundamental accounting equation, representing the balance of a company's financial position.

Double-Entry Accounting

The recording of both debit and credit parts of every transaction, ensuring the accounting equation remains balanced.

Objective Evidence Principle

Requires that a source document (e.g., invoice, receipt) be prepared for every entry in a journal, providing verifiable proof of a transaction.

File Maintenance

The process of arranging accounts in a general ledger, assigning account numbers, and keeping records current.

Opening an Account

Writing an account title and number on the heading of an account in the ledger.... Continue reading "Essential Accounting Concepts and Financial Reporting" »