Calculus and Analysis
Calculus and Analysis in the Pancyprian Core Mathematics paper covers limits, differentiation with its applications to graphs and optimisation, and integration as accumulated area. The question types are standard, so most lost marks come from a missing constant of integration, a dropped chain-rule factor, or a stationary point classified without evidence.
Limits and continuity
- For a polynomial, the limit at a point equals the value of the function there.
- For a 0/0 form, factorise and cancel before substituting.
- For limits at infinity, divide numerator and denominator by the highest power of x present.
Differentiation
- (xⁿ)′ = nxⁿ⁻¹; (sin x)′ = cos x; (cos x)′ = −sin x; (eˣ)′ = eˣ; (ln x)′ = 1/x.
- Apply the product, quotient and chain rules - the inner derivative from the chain rule is the most commonly forgotten factor.
- The tangent at x = a has gradient f′(a); the normal there has gradient −1/f′(a).
- Stationary points solve f′(x) = 0 and are classified by the sign change of f′ or by the sign of f″.
Integration
- ∫xⁿ dx = xⁿ⁺¹/(n + 1) + C for n ≠ −1, and the constant is mandatory for indefinite integrals.
- Definite integral: ∫ from a to b of f(x) dx = F(b) − F(a), where no constant is needed.
- Area between a curve and the x-axis is the integral of |f|; split the interval wherever the curve crosses the axis.
- Area between two curves: integrate (upper − lower) between the intersection points.
Worked example
Find and classify the stationary points of f(x) = x³ − 3x² + 1. Differentiate: f′(x) = 3x² − 6x = 3x(x − 2), so x = 0 or x = 2. Then f″(x) = 6x − 6, giving f″(0) = −6 < 0, so (0, 1) is a local maximum, and f″(2) = 6 > 0, so (2, −3) is a local minimum.
Traps that cost marks
- Omitting + C in an indefinite integral.
- Differentiating a constant to 1 instead of 0, or ignoring the inner derivative in sin 3x or e^(x²).
- Evaluating F(b) − F(a) as F(b − a).
- Declaring a stationary point a maximum without checking, and forgetting to compare endpoint values on a closed interval.