Worked examples: SACE Stage 2 Mathematical Methods Formula Sheet

Selected worked examples

1. Calculus · quotient rule

   Formula or reference: (u/v)′ = (u′v − uv′)/v²

   Units and conditions: v must be nonzero at the input.

   Worked example: For y = x/(x + 2), y′ = 2/(x + 2)². At x = 2 the gradient is 1/8.

2. Probability · binomial complement

   Formula or reference: P(X ≥ 1) = 1 − P(X = 0) = 1 − (1 − p)ⁿ.

   Units and conditions: Use independent trials with one success probability p.

   Worked example: In three trials at p = 0.20, at least one success has probability 1 − 0.8³ = 0.488.