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

Taylor Series

f(x+h) = f(x) + f'(x)h + f''(x)h² /2! + f'''(x)h³ /3! + ...

Secant Method (Find Root)

1. Rewrite as f(x)=0
2. Set initial points
3. f'(x₂) = (f(x₂) - f(x₁)) / (x₂-x₁)
4. New point becomes x₂
5. Iterate

LU Decomposition

A = LU

Crout's

Identity top

Doolittle

Identity bottom

1. Set up LU
2. Find components through matrix multiplication and solve for variables

Solving for b matrix

LUX=B

1. Set UX = Y
2. Solve LY = B using algebra
3. Solve UX = Y using algebra

Norm

Absolute value of the largest row sum

Condition

||A||_inf||A^-1||_inf

Gauss-Seidel

1. Set each equation to a variable
2. Use any initial values for the first equation
3. Use the x₁ from the first equation to solve x₂
4. Iterate (Fast version of Jacobi Method)

Discrete Least Squares Approximation

AX = B

1. Make matrix A by [ n ∑(x) ∑(x²) ...]
2. Make

E-Business and Its Impact

The Changing Landscape of Business

The rise of the internet has dramatically transformed the business landscape, giving rise to the concept of e-business. E-business encompasses a wide range of activities, including B2B (business-to-business) and B2C (business-to-consumer) transactions, supply chain management, and customer relationship management.

Key Characteristics of E-Business

• Integration: E-business systems integrate various business functions and channels, such as brick-and-mortar stores, online platforms, and mobile apps.
• Flexibility: E-business frameworks are designed to be flexible, open, and adaptable to changing market conditions.
• Ubiquity: E-business infrastructure enables ubiquitous commerce, allowing businesses

Single-Factor ANOVA is used to determine whether three or more populations have equal means.

ANOVA Assumptions:

• The population values are normally distributed.
• The variances for each population are equal.
• The samples are independent.
• The data measurement is interval or ratio level.

ANOVA is an analysis of variance. It needs to have equal variance in order to test the means.

Single-Factor ANOVA: analysis of variance design in which independent samples are obtained from two or more levels of a single factor for the purpose of testing whether the levels have equal means.

Type of data used in ANOVA: ratio or interval data (numbers).

H₀: all means are equal.

H₁: there is a difference in means.

Theory: Practicality.

In Multifactor ANOVA:

• Groups: rows (router

Fixed Costs

Fixed costs are not a function of the output. They do not vary with the output. They cannot be avoided until the operation of the firm is closed. They are contractual (prime).

Recurring and Non-Recurring Costs

• Recurring: Predetermined expenses for running the business, e.g., salaries, repairs, maintenance.
• Non-Recurring: Not predetermined, regular budgeting, e.g., repair of a machine. Sometimes it is also planned.

Cost Estimation

Cost estimation is the process of finding an estimate, an approximation of a value, which will be used for some purpose, though it is completely uncertain and unstable. Estimation is typically a value from statistics used to estimate the value of a corresponding population parameter.

Learning Curve

The learning... Continue reading "Cost Estimation and Management Decision Areas in Business" »

Qualitative and Quantitative Variables

Qualitative: variables that are not numerical. They represent categories or groups that the data can fall into. Nominal: the categories do not have a natural order or ranking. The key characteristic of nominal variables is that the different categories are mutually exclusive and there is no inherent order to the categories. Ordinal: the categories have a logical or natural order. However, the distances between the categories are not necessarily meaningful.

Quantitative: variables that represent quantities and can be measured numerically. Discrete: a type of quantitative variable that can take on a countable number of distinct values. These are typically values that you can list or count. They often represent... Continue reading "Understanding Data Types and Sampling Methods" »

The purse is one of our most popular products for women. Its made of leather and comes in a variety of colours: black, green , red. It has got three separate sections—one holds banknotes, another holds credit cards and the third which has got a zip, holds coins. IT’S A large purse, but it fits easily into a normal-sized handbag….The purse comes with a three-year guarantee..

The purse is one of our most popular products for women. Its made of leather and comes in a variety of colours: black, green , red. It has got three separate sections—one holds banknotes, another holds credit cards and the third which has got a zip, holds coins. IT’S A large purse, but it fits easily into a normal-sized handbag….The purse comes with a three-year... Continue reading "holea" »

Practicing material for Quiz #5 – Discrete Probability Distribution SOLUTIONS

1. The manager of a baseball team has determined that the number of walks, x, issued in a game by one of the pitchers is described by the probability distribution given below.

1. This pitcher issues as few as 0 walks and as many as 5 walks in a game.

1. Determine the following probabilities

i. P(x = 2) = 0.15

ii. P(x > 4) = 1 – 0.10 = 0.90

iii. P(x > 5) = 0

iv. P(2 < x < 4) = 0.15+0.45+0.15 = 0.75

1. Calculate the mean for this discrete probability distribution, µ = 2.85 walks

µ = Sxp(x) = 0(0.05) + 1(0.10) + 2(0.15) + 3(0.45) + 4(0.15) + 5(0.10) = 2.85

1. The average, over time, of walks issued in a game by one of the pitchers is 2.

Nanotechnology

is the study, design and creation of materials and devices by means of a control of matter on a nanometric scale.

Nanotechnology in Computing

The manufacture of nanochips, capable of storing a huge number of transistors, will enable the design of devices which are much smaller and more powerful.

Nanomedicine

There exists the possibility of building tiny devices, which in sufficient amounts will be able to circulate around the human body detecting, at an early stage, diseases such as cancer.

Nanoindustry

Machines can be designed that make use of waste to generate themselves or generate devices which make use of energy in a more efficient way.

Location, Production and Using Up of Materials

Globalization is the process by which the world... Continue reading "Nanotechnology and Globalization: Impacts and Opportunities" »

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... Continue reading "Financial Ratio Analysis: Uses, Benefits, and Limitations" »

Vocabulary