📈

Calculus Fundamentals for IGCSE

Sep 8, 2025

Overview

This lecture covers the fundamentals of rates of change, including equations, the chain rule, examples, and how to calculate percentage changes, with a focus on calculus applications for IGCSE Math.

Basics of Rates of Change

  • The rate of change is expressed as d(something)/dt, e.g., dV/dt is the rate of change of volume with respect to time.
  • Positive rate of change means a quantity is increasing; negative means decreasing.
  • If dt/dx = u, then dx/dt = 1/u by reciprocation.

Chain Rule for Rates of Change

  • The chain rule states: dy/dt = dy/dx × dx/dt.
  • For three variables: dy/dt = dy/ds × ds/du × du/dt (telescoping product).

Small Changes and Delta Notation

  • Delta (Δ) represents a small change; e.g., Δx is a small change in x.
  • Δy/Δx ≈ dy/dx when changes are small.
  • From Δy/Δx ≈ dy/dx, two forms: Δy = (dy/dx) × Δx or Δx = Δy × (dx/dy).

Worked Example: Changing Radius

  • Given r = t² + 2, to find rate of change between t=2 and t=2.1:
    • Calculate Δr/Δt = [r(2.1) - r(2)] / (2.1 - 2) = 4.1 cm/s.
  • For interval t to t+Δt: Δr/Δt = 2t + Δt cm/s.
  • At t=2, instantaneous rate: dr/dt = 2t = 4 cm/s.

Percentage Change

  • Percentage change in x: (Δx/x) × 100%.
  • Percentage change in y: (Δy/y) × 100%.

Example: Area of a Circle

  • Area, A = Ï€r²; given dr/dt = 3 cm/s and r = 5 cm.
  • Use chain rule: dA/dt = (dA/dr) × (dr/dt) = (2Ï€r) × 3 = 30Ï€ cm²/s.

Example: Approximate Percentage Change

  • Given y = 3x² - 2x - 3 and x increases by p% when x=2.
  • Find dy/dx at x=2: dy/dx = 6x - 2 = 10.
  • Δx = (p/100) × 2; Δy ≈ dy/dx × Δx = 10 × (2p/100) = 0.2p.
  • y at x=2 is 5; percentage change in y: (0.2p/5) × 100 = 4p%.

Key Terms & Definitions

  • Rate of Change — How fast a quantity changes with respect to another, denoted as d(something)/dt.
  • Chain Rule — A rule for finding the derivative of a composition of functions: dy/dt = dy/dx × dx/dt.
  • Delta (Δ) — Symbol representing a small finite change in a quantity.
  • Percentage Change — The change in a variable as a percent of its original value: (Δx/x) × 100%.

Action Items / Next Steps

  • Review and practice rate of change problems using the chain rule and delta notation.
  • Prepare for upcoming questions on rates of change in the next lecture.

Full transcript