Grade 12 Calculus Study Guide — Limits, Derivatives & Curve Sketching

Everything you need for Paper 1 calculus: first principles, the rules, curve sketching and optimisation — with a worked exam example.

Calculus Study Guide — Paper 1

The derivative

First principles: f′(x) = lim(h→0) [f(x+h) − f(x)]/h — memorise it; it's asked almost every year (5 marks).
Power rule: d/dx(axⁿ) = anxⁿ⁻¹.
Products: (uv)′ = u′v + uv′.

Curve sketching checklist

y-intercept (x = 0) and x-intercepts (y = 0)
Stationary points: solve f′(x) = 0
Second derivative: f″(x) > 0 → local minimum, < 0 → local maximum
Concavity changes → point of inflection

Optimisation in 4 steps

Draw & label. 2. Write the formula for what you optimise. 3. Substitute to one variable. 4. Differentiate, set = 0, answer the question (with units!).

Worked example: A farmer has 60 m of fence for a rectangular camp against a river (no fence along the river). Area A = x(60 − 2x) = 60x − 2x². A′ = 60 − 4x = 0 → x = 15 m, depth 30 m, A = 450 m².

Exam tips

"Determine from first principles" ≠ "differentiate" — using the power rule there scores 0.
Tangent problems: point → gradient → y − y₁ = m(x − x₁).

Follow along (device voice)

Listen and follow along as each word is spoken

Calculus Study Guide — Paper 1 The derivative - First principles: f′ x = lim h→0 f x+h − f x /h — memorise it; it's asked almost every year 5 marks . - Power rule: d/dx axⁿ = anxⁿ⁻¹. - Products: uv ′ = u′v + uv′. Curve sketching checklist 1. y-intercept x = 0 and x-intercepts y = 0 2. Stationary points: solve f′ x = 0 3. Second derivative: f″ x 0 → local minimum, < 0 → local maximum 4. Concavity changes → point of inflection Optimisation in 4 steps 1. Draw & label. 2. Write the formula for what you optimise. 3. Substitute to one variable. 4. Differentiate, set = 0, answer the question with units . Worked example: A farmer has 60 m of fence for a rectangular camp against a river no fence along the river . Area A = x 60 − 2x = 60x − 2x². A′ = 60 − 4x = 0 → x = 15 m, depth 30 m, A = 450 m². Exam tips - "Determine from first principles" ≠ "differentiate" — using the power rule there scores 0. - Tangent problems: point → gradient → y − y₁ = m x − x₁ .