Pre-Calculus and Calculus
Core content
Candidates should understand functions and their behaviour, including polynomial, rational, exponential, logarithmic, and trigonometric models. Know limits, one-sided limits, continuity, and asymptotic behaviour. For differentiation, know the limit definition, power, product, quotient, and chain rules, plus tangent lines, rates of change, extrema, and optimisation. For integration, understand antiderivatives, the fundamental theorem of calculus, definite integrals as signed area, and basic techniques. Sequences, series, vectors, parametrics, and polar coordinates may also appear.
Trap watch
A limit can exist while the function value is undefined or different. Check one-sided limits at piecewise boundaries. The derivative does not exist at a cusp, corner, vertical tangent, or discontinuity. Apply the chain rule to every composite factor, and do not forget the constant of integration. For a definite integral, reversing the limits changes the sign. Distinguish average rate of change from instantaneous rate of change. When finding extrema on a closed interval, test the endpoints as well as critical points.
Worked example
Find lim(x->2) (x^2 - 4)/(x - 2). Factor the numerator as (x - 2)(x + 2), cancel the common factor for x not equal to 2, and evaluate x + 2 at x = 2 to obtain 4. The original expression is undefined at x = 2, but the limit exists.
Rule of thumb: before differentiating or integrating, check continuity, domain, and the hypotheses that justify the rule.