8️⃣ Revision and Final-Exam Preparation

Mathematical Fundamentals · MS01-001-G

Learning Objectives

By the end of this revision session, you should be able to:

  • Recall the essential results from Modules 1–7.
  • Connect concepts across topics — sequences ↔︎ interest, functions ↔︎ optimisation.
  • Identify the appropriate mathematical method without a chapter label.
  • Practise on mixed problems under exam-like conditions.

Module 1 Recap — Numerical Calculations

Key Formula — Percentage Change

\[\Delta\% = \frac{V_{\text{final}} - V_{\text{initial}}}{V_{\text{initial}}} \times 100\]

Successive rates multiply: \((1+t_1)(1+t_2) \neq 1+t_1+t_2\)

Takeaway: Always divide by the initial value. Index \(= \dfrac{\text{current}}{\text{reference}} \times 100\).

Module 2 Recap — Algebra and Equations

Key Identities

  • \((a+b)^2 = a^2 + 2ab + b^2\)
  • \((a-b)^2 = a^2 - 2ab + b^2\)
  • \((a+b)(a-b) = a^2 - b^2\)

Takeaway: Factoring reveals structure; expanding removes parentheses. To solve \(ax+b=0\): isolate \(x\) by inverse operations.

Module 3 Recap — Graphs, Systems, Inequalities

Key Formula — Equation of a Line

\[y = ax + b\]

Slope \(a = \dfrac{y_2 - y_1}{x_2 - x_1}\) · Intercept \(b\) = value at \(x = 0\)

Takeaway: Two lines intersect where \(a_1 x + b_1 = a_2 x + b_2\). Multiplying an inequality by a negative reverses the sign.

Module 4 Recap — Numerical Sequences

Key Formulas

Type General term Common element
Arithmetic \(u_n = u_0 + nd\) Constant difference \(d\)
Geometric \(u_n = u_0 \times q^n\) Constant ratio \(q\)

Takeaway: If increments are equal → arithmetic; if ratios are equal → geometric.

Module 5 Recap — Financial Mathematics

Simple vs. Compound Interest

Regime Formula Growth
Simple \(C_n = C_0(1 + nt)\) Linear
Compound \(C_n = C_0(1 + t)^n\) Exponential

Takeaway: Compound interest reinvests — over time it significantly outperforms simple interest.

Module 6 Recap — Functions

Discriminant for \(ax^2+bx+c = 0\)

\[\Delta = b^2 - 4ac\]

\(\Delta > 0\): two roots · \(\Delta = 0\): one root · \(\Delta < 0\): no real roots

Takeaway: Check domain (forbidden values) before evaluating any function. Affine = linear; polynomial = curved; exponential = rapid growth.

Module 7 Recap — Optimisation

Finding an Extremum

  1. Differentiate: \(f'(x) = \ldots\)
  2. Solve \(f'(x) = 0\) → critical point \(x^*\)
  3. Sign table of \(f'\) → nature: max (\(+\) to \(-\)) or min (\(-\) to \(+\))

Takeaway: For \(f(x) = ax^2+bx+c\), the vertex is at \(x^* = -\dfrac{b}{2a}\); maximum if \(a < 0\), minimum if \(a > 0\).

Method Identification Guide

Given a scenario, pick the right tool:

Scenario Module Tool
Price change, index, discount 1 Percentage change, factor
Break-even, cost = revenue 2 First-degree equation
Two competing models, intersection 3 System of equations
Fixed-amount periodic growth 4 Arithmetic sequence
Fixed-rate periodic growth 4 Geometric sequence
Investment, loan interest 5 Simple or compound interest
Model type for a phenomenon 6 Function identification
Maximise profit / minimise cost 7 Derivative, sign table

Strategy: Read the problem → identify the structure → pick the module.

Exam Strategy

  • Read the entire question before writing anything — identify which module the problem belongs to.
  • Show every step — partial credit is awarded for correct method even if arithmetic fails.
  • Check your answer: Does the sign make sense? Is the unit correct? Is the percentage relative to the initial value?
  • Formula sheet: Know where every formula is — do not derive from scratch under time pressure.
  • Time allocation: Spend no more than 2–3 minutes on any single sub-question. Move on and return.

Summary

  1. No chapter label in the exam — practise identifying the method from the context alone, not from the section heading.

  2. Connect the modules: percentages (M1) reappear in financial maths (M5); sequences (M4) underpin compound interest (M5); functions (M6) are the objects optimised in M7.

  3. The formula sheet is your safety net — use it, but understand what each formula computes so you apply it correctly.

📖 Full Reference

These slides provide a cross-module revision overview for the final exam.

For complete theory, worked examples, and collapsible exercises for each topic, return to the individual modules:

→ Module 8: Revision and Final-Exam Preparation