Calculus: limits, differentiation and integration
What this domain tests
The calculus questions carry substantial weight: limits and continuity, derivative rules and their interpretation, curve analysis, and definite integration as signed area. You should move confidently between algebraic, graphical and applied interpretations.
Core rules
- Limit: substitute first; if you obtain 0/0, factor, rationalise or simplify before taking the limit. A function is continuous at a if the limit exists and equals f(a).
- Derivative rules: power rule; product rule (uv)' = u'v + uv'; quotient rule (u/v)' = (u'v - uv')/v^2; chain rule d/dx f(g(x)) = f'(g(x))g'(x).
- Standard derivatives: d/dx(e^x) = e^x, d/dx(ln x) = 1/x, d/dx(sin x) = cos x, d/dx(cos x) = -sin x.
- Stationary points: solve f'(x) = 0; classify with the sign change of f' or with f''. The tangent at x = a is y - f(a) = f'(a)(x - a).
- Integration: integration reverses differentiation; for definite integrals use F(b) - F(a). Area between a curve and the x-axis is the integral of |f(x)|, so split at zeros.
Worked example
Let f(x) = x ln x - x for x > 0. Then f'(x) = ln x + x(1/x) - 1 = ln x. Setting f'(x) = 0 gives x = 1. Since f''(x) = 1/x > 0, this is a local minimum, with f(1) = 0 - 1 = -1. For the integral, note that x ln x - x is an antiderivative of ln x, so the integral from 1 to e of ln x dx equals [x ln x - x] from 1 to e = (e - e) - (0 - 1) = 1.
Traps that cost marks
- Omitting the chain-rule factor, especially with e^(kx), ln(kx) and trigonometric composites.
- Forgetting + C in indefinite integrals, or dropping the evaluation limits.
- Treating signed area as geometric area when the curve crosses the x-axis.
- Using L'Hopital's rule without checking the 0/0 or infinity/infinity form.