Calculus: rates, optimisation and area
Calculus in AI SL is applied: the derivative as a gradient and a rate of change, differentiation of powers and simple functions, tangents and normals, stationary points and optimisation, kinematics, and integration as the reverse process leading to areas under curves.
Core rules
- If f(x) = axⁿ then f′(x) = anx^(n−1); constants differentiate to zero.
- Derivatives of eˣ, ln x, sin x and cos x follow the standard results used in your course; sums are differentiated term by term.
- Tangent gradient at x = a is f′(a); the normal gradient is −1/f′(a).
- Stationary points solve f′(x) = 0; test with the sign change of f′ or with f″(x).
- Kinematics: v = ds/dt and a = dv/dt.
- ∫ axⁿ dx = ax^(n+1)/(n+1) + C for n ≠ −1; a definite integral gives signed area.
Worked example
Find and classify the stationary points of f(x) = x³ − 6x² + 9x + 1. Differentiate: f′(x) = 3x² − 12x + 9 = 3(x − 1)(x − 3), so stationary points occur at x = 1 and x = 3. For the classification, f″(x) = 6x − 12, giving f″(1) = −6 < 0, a maximum, and f″(3) = 6 > 0, a minimum. Then f(1) = 5 and f(3) = 1, so the maximum point is (1, 5) and the minimum point is (3, 1).
Traps that cost marks
Omitting + C in indefinite integrals; finding f′(x) = 0 but never classifying or giving coordinates; treating a negative definite integral as an error rather than area below the axis; differentiating with the GDC in the wrong angle mode for sine and cosine; giving units of a rate incorrectly (for example, litres instead of litres per minute).