Calculus
What you must understand
Calculus in Maths Methods covers differentiation and integration of polynomial, exponential, logarithmic and trigonometric functions, and their applications. Candidates must know derivative rules (power, product, quotient, chain), the relationship between the derivative and gradient of a tangent, stationary points and their classification via the second derivative or a sign table, and the interpretation of the derivative as a rate of change. For integration, you need antiderivatives, the definite integral as a signed area, and applications to areas between curves and average values. You should also handle kinematics: displacement, velocity and acceleration linked by differentiation.
Traps that appear
- Forgetting the chain rule factor when differentiating composite functions such as e^(3x) or sin(2x).
- Confusing the derivative of ln x (which is 1/x) with the derivative of aˣ.
- Not checking endpoints when finding global maxima or minima on a closed interval.
- Treating a definite integral as unsigned area; regions below the axis contribute negatively.
- Omitting the constant of integration for indefinite integrals.
- Mixing up velocity and speed, or setting acceleration to zero when asked for maximum velocity.
Worked example
Find the stationary points of f(x) = x³ − 3x² + 1 and classify them. f′(x) = 3x² − 6x = 3x(x − 2), so stationary points at x = 0 and x = 2. f″(x) = 6x − 6: f″(0) = −6 (local maximum), f″(2) = 6 (local minimum). Rule of thumb: differentiate, solve f′(x) = 0, then test the second derivative - but fall back to a sign table if f″ = 0.