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

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" »

Chapter 1: Core Accounting Fundamentals

Key Financial Statements Equations

• Balance Sheet (BS) Equation: Assets = Liabilities + Stockholders' Equity (SHE)
• Income Statement (IS) Equation: Revenues - Expenses = Net Income (NI)
• Retained Earnings (RE) Equation: Beginning RE + Net Income - Dividends = Ending RE

Debit and Credit Rules

• A Debit increases: Expenses, Assets, Dividends (DEAD)
• A Credit increases: Liabilities, Revenues, Equity (CLEAR)

Objective of Financial Reporting

1. Provide information useful to equity investors and creditors.
2. Maintain the Entity Perspective (separating the company from its owners/people).
3. Ensure Decision Usefulness.

Adjusting Entries: Deferrals and Accruals

Deferrals (Cash Paid/Received Before Recognition)

Prepaid Expenses

Meaning: Paid... Continue reading "Core Financial Accounting Concepts and Principles" »

Math Reference Cheat Sheet

Rules for Multiplying and Dividing Integers

Special Rules for Integers

• Even amount of negative integers: Product/quotient is positive.
• Odd amount of negative integers: Product/quotient is negative.

Classifying Decimals

• Terminating decimals: Have a remainder of zero; the decimal stops.
• Non-terminating decimals: Continue infinitely.
  • Repeating: Continues infinitely and repeats a pattern (denoted by bar notation). Examples: 0.333..., 0.49090..., 0.166...
  • Non-repeating: Continues infinitely but does not repeat a pattern. Examples: Pi (π) and Euler's number (e).

Calculating Percent Error

% Error = [(Estimated Value... Continue reading "Essential Math Cheat Sheet: Integers, Decimals, and Properties" »

Digital Signal Processing

Load the data file: load('exam.mat');

Parameters

• x = xBPFI; % Input signal
• Fs = 10e3; % Sampling frequency
• Ts = 1/Fs;
• t = (0:length(x)-1) * Ts; % Time vector
• n_f = 10000;
• f = (0:n_f-1)*Fs/n_f; % Frequency vector

Input Signal Analysis

Plotting the input signal in the time and frequency domains:

plot(t,x); xlabel('t (s)'); ylabel('x(t) (V)'); title('Input signal in TIME');
X = fft(x,n_f)/length(x);
figure(); subplot(2,1,1); plot(f,abs(X)); xlabel('f (Hz)'); ylabel('|X(f)| (V)'); title('Input signal in FREQUENCY');
subplot(2,1,2); plot(f,angle(X)); xlabel('f (Hz)'); ylabel('\angleX(f) (rad)');

Bandpass Filter Design

Configuring the filter parameters and applying a Hamming window:

• Tbw = 150; fpl = 2400; fcl = (fpl-Tbw/2)/Fs; wcl = 2*

Metallurgical sample preparation for microscopy is a crucial step in analyzing the microstructure and properties of metallic materials. Proper sample preparation is essential to obtain accurate and meaningful results. The process involves several key steps:

1. Sample Selection for Microscopy

Choose a representative portion of the material to be analyzed. Ensure that the sample is free from external contaminants and has a flat surface for preparation.

2. Precision Cutting Techniques

Use a precision cutting method to obtain a small section of the material for analysis. Common cutting techniques include abrasive cutting (using a saw with abrasive blades), wire cutting, or electrical discharge machining (EDM) for hard materials.

3. Sample Mounting for

Introduction to Statistics

Statistics is the science of collecting, organizing, analyzing, and interpreting data to make informed decisions.

Types of Statistics

• Descriptive Statistics: Methods of organizing, visualizing, and summarizing information from samples or populations.
• Inferential Statistics: Methods of using information from a sample to draw conclusions regarding the population.

Example: A survey of 2,000 students (3rd to 12th grade) found that they devoted an average of 7 hours and 38 minutes each day to using electronic media.

Key Definitions

• Data: Information coming from observations, counts, measurements, or responses.
• Population: The collection of all outcomes, measurements, or responses (sometimes called a census). A numerical description

Bond Characteristics

• Coupon: The interest payment made by the bond issuer, usually expressed as an annual percentage of the bond's face value.
• Par (Face Value): The amount the bondholder receives when the bond matures, typically $1,000.
• Term to Maturity: The time remaining until the bond's maturity date when the issuer must repay the bond's par value.
• Denomination: The face value of the bond, usually in increments of $1,000.
• Quotation: Bonds are quoted as a percentage of their face value (e.g., a bond quoted at 95 is selling for 95% of $1,000, or $950).

Bond Prices, Yield to Maturity (YTM), Current Yield, and Rate of Return (HPR)

• Bond Prices: The market price of a bond depends on interest rates. Prices and interest rates have an inverse relationship.

Essential Financial Statement Formulas

• Stockholder Equity: Total Assets – Total Liabilities
• Retained Earnings: Net Income – Dividends Declared
• Net Income: Sales Revenue – Expenses
• Gross Profit: Sales Revenue – Cost of Goods Sold (COGS)
• Cost of Goods Sold (COGS) based on Rate: (1 – Gross Profit Rate) × Net Sales

Inventory Accounting Adjustments (LIFO and FIFO)

• Calculating LIFO Reserve: FIFO Ending Inventory Cost – LIFO Ending Inventory Cost
• Adjusting Balance Sheet from LIFO to FIFO Inventory: LIFO Reserve + LIFO Inventory

Accounting Estimates and Depreciation

Changing Accounting Estimates

• Book Value at Date of Change: Costs – Accumulated Depreciation
• New Remaining Useful Life: Original Life – Years Depreciated + Additional Years
• Depreciation