Core Numerical Methods: Algorithms for Computation
Implementing Euler’s Method
- Start
- Define function f(x,y)
- Read values of initial condition (x0 and y0), number of steps (n), and calculation point (xn)
- Calculate step size (h) = (xn – x0) / n
- Set i = 0
- Loop:
- yn = y0 + h * f(x0 + i*h, y0)
- y0 = yn
- i = i + 1
- Display yn as result
- Stop
Implementing Runge-Kutta 4th Order Method
- Start
- Define function f(x,y)
- Read values of initial condition (x0 and y0), number of steps (n), and calculation point (xn)
- Calculate step size (h) = (xn – x0) / n
- Set i = 0
- Loop:
- k1
Intangible Capital’s Impact on Investment-Q Relation
Intangible Capital and the Investment-Q Relation
Based on the research by Peters and Taylor (2016).
Introduction: Shifting Investment Landscape
The neoclassical theory of investment was developed when firms primarily owned physical assets. Consequently, empirical tests of this theory have historically focused on physical capital. However, the U.S. economy has since shifted toward service and technology-based industries, where intangible assets have become increasingly important.
The Core Research Question
Is
Read MoreEssential Business Math Formulas & Concepts
Applied Business Mathematics Formulas
1. Polynomials, Exponents, and Roots
Exponent Rules:
- am ⋅ an = am+n
- (am)n = am⋅n
- a-n = 1/an
- am/an = am-n
Factoring Techniques:
- Common Factor: ab + ac = a(b + c)
- Difference of Squares: a2 – b2 = (a – b)(a + b)
- Quadratic Factorization: ax2 + bx + c = (px + q)(rx + s)
Quadratic Formula:
x = (-b ± √(b2 – 4ac)) / 2a
Example: Solving Quadratic Equations
- Solve x2 – 5x + 6 = 0
- Using the quadratic formula:
- x = (-(-5) ± √((-5)2 – 4(1)(6))) / 2(1)
- x = (5 ± 1) / 2
- Solutions: x
Foundational Statistical Concepts for Data Analysis
Types of Research Questions
Making an Estimate About the Population
- What is the average number of hours that students study each week?
- What proportion of all Singaporean students is enrolled in a university?
Testing a Claim About the Population
- Does the majority of students qualify for student loans?
- Is the average course load for a university student greater than 20 units?
Comparing Two Sub-Populations or Investigating a Relationship Between Two Variables in the Population
- In University X, do female students
Industrial Process Engineering: Distillation, Roles, and Asset Depreciation
Distillation Column Design Principles
The FUG Method for Distillation Column Design
The FUG method (Fenske–Underwood–Gilliland) is a shortcut technique used to estimate the number of stages and reflux ratio in distillation column design. It’s a crucial tool for preliminary design.
Fenske Equation: Minimum Stages
The Fenske equation calculates the minimum number of theoretical stages at total reflux. This provides a theoretical lower bound for the number of stages required for a given separation.
Read MoreEssential Math Foundations: Shapes, Space, Numbers
Quadrilaterals: Shapes & Properties
What is a Quadrilateral?
A quadrilateral is a four-sided polygon.
The interior angles of a quadrilateral add up to 360°.
Types of Quadrilaterals
| Type | Properties |
|---|---|
| Square | All sides equal, all angles 90°, opposite sides parallel |
| Rectangle | Opposite sides equal and parallel, all angles 90° |
| Rhombus | All sides equal, opposite angles equal, diagonals bisect at right angles |
| Parallelogram | Opposite sides equal and parallel, opposite angles equal |
| Trapezium | One pair of opposite sides |