Calculus
Calculus practice develops the ability to move fluently between a function, its rate of change, and its accumulated total. Expect differentiation of composite, implicit and parametric forms, evaluation of definite and improper integrals, and questions that require you to justify a limit before differentiating.
Differentiation
- Chain rule: d/dx f(g(x)) = f'(g(x))·g'(x). The inner derivative is the most commonly dropped factor.
- Implicit differentiation: differentiate both sides with respect to x, treating y as y(x), then collect the dy/dx terms.
- Logarithmic differentiation pays off for products of powers such as x^x or (sin x)^x.
- L'Hôpital requires 0/0 or ∞/∞ plus the theorem's differentiability, nonzero denominator-derivative and derivative-limit hypotheses; the indeterminate form alone is not sufficient.
Integration
- Substitution reverses the chain rule; for a definite integral, change the limits at the same time.
- Integration by parts: ∫u dv = uv − ∫v du. Choose u so that du simplifies.
- The Fundamental Theorem converts a definite integral into an antiderivative difference; improper integrals need an explicit limit.
- Partial fractions for rational functions, after comparing the degrees of numerator and denominator.
Worked example
Evaluate ∫₀¹ x·e^{2x} dx. Take u = x, dv = e^{2x} dx, so du = dx and v = e^{2x}/2. ∫ x·e^{2x} dx = x·e^{2x}/2 − ∫ (e^{2x}/2) dx = x·e^{2x}/2 − e^{2x}/4. At x = 1: e²/2 − e²/4 = e²/4. At x = 0: 0 − 1/4. Subtract: e²/4 − (−1/4) = (e² + 1)/4 ≈ 2.097.
Traps that cost marks
- Omitting +C on an indefinite integral.
- Forgetting to change the limits after substitution, then substituting back into x-limits.
- Splitting an improper integral across a singularity without taking one-sided limits.
- Sign slips when subtracting the lower limit; write the bracket out explicitly.
- Losing a factor when the constant multiplies the whole term rather than just the outer function.