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MathematicsIntegration by Parts Examples: LIATE, the Tabular Method and the Present Value of a Growing Revenue StreamUse integration by parts to value a growing revenue stream, then master LIATE, repeated parts and the tabular method with exam-style examples.MathematicsOptimisation Problems in Calculus: A Step-by-Step Method with Real CasesHow to solve optimisation problems with calculus: build the objective, use the constraint, test the stationary point and check the endpoints.MathematicsQuadratic Formula Word Problems: Projectiles, Break-Even Prices and the DiscriminantHow to turn a word problem into ax² + bx + c = 0, solve it with the quadratic formula, and use the discriminant to see…MathematicsThe Chain Rule Explained: How to Differentiate a Function of a Function (with Examples)Learn the chain rule through a bakery's cost per labour hour: why rates multiply, how to set it up, and the slip that costs most exam…StatisticsChi-Square Test Explained: Goodness of Fit and Independence, Step by StepLearn the chi-square test step by step: the chi square formula, expected counts, degrees of freedom, worked goodness of fit and independence examples.InflationThe Phillips Curve Explained: Short Run, Long Run, NAIRU and the EvidenceThe Phillips curve explained: short run vs long run, the expectations-augmented equation, NAIRU, a worked example, US data 2015–2025 and exam evaluation.Development EconomicsComparative Advantage vs Absolute Advantage: Opportunity Cost, Terms of Trade and Worked ExamplesComparative advantage explained: opportunity cost (output and input methods), terms of trade range, a worked two-country example, PPCs and real data.Development EconomicsThe Lewis Model (Dual-Sector Model): Surplus Labour, the Turning Point and Structural ChangeThe Lewis model explained: surplus labour, the subsistence wage, the Lewis turning point, a worked example, China's turning point and Harris-Todaro.Development EconomicsThe Harrod–Domar Model: Formula, Worked Examples and Why It Fails for Developing CountriesThe Harrod-Domar model explained: g = s/v, the capital-output ratio, savings gap examples, knife-edge growth, real ICOR data and why aid plans fell short.Development EconomicsThe Solow Growth Model Explained: Steady State, Golden Rule and ConvergenceSolow growth model explained: steady state derivation, worked example, golden rule level of capital, convergence evidence and practice questions.Development EconomicsHuman Development Index (HDI): Formula, Worked Calculation and Latest RankingsHow the Human Development Index is calculated: goalposts, HDI formula, a worked example for Pakistan and the latest HDR 2025 rankings, with exam practice.Development EconomicsLorenz Curve and Gini Coefficient: How to Calculate and Interpret InequalityLearn how to calculate the Gini coefficient from a Lorenz curve with a worked quintile example, real World Bank data for 10 countries and exam questions.
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