Applications of Derivatives and Integration
The exam links derivatives to tangents, rates and optimisation, then tests standard integrals, substitution, integration by parts and area under a curve.
Applications of derivatives
- Tangent slope at (x₁,y₁) is dy/dx there; normal slope is −1/(dy/dx).
- f is increasing where f′(x) > 0, decreasing where f′(x) < 0, on the whole interval in question.
- Stationary points solve f′(x) = 0. Second derivative test: f″ < 0 gives a local maximum, f″ > 0 a local minimum; if f″ = 0 fall back on the first derivative test.
Integration
∫xⁿ dx = x^(n+1)/(n+1) + C (n ≠ −1), ∫(1/x) dx = log|x| + C, ∫eˣ dx = eˣ + C, ∫sin x dx = −cos x + C. By parts: ∫uv′ dx = uv − ∫u′v dx, choosing u by the usual order (inverse, log, algebraic, trig, exponential). Definite integrals use the limits after finding the antiderivative.
Worked example
f(x) = x³ − 3x² + 4. Then f′(x) = 3x² − 6x = 3x(x − 2), so stationary points are x = 0 and x = 2. f″(x) = 6x − 6, giving f″(0) = −6 < 0 (local maximum, value 4) and f″(2) = 6 > 0 (local minimum, value 0). For integration, ∫x eˣ dx with u = x, v′ = eˣ gives x eˣ − ∫eˣ dx = x eˣ − eˣ + C.
Traps that cost marks
- Omitting the constant of integration in indefinite integrals.
- Sign slips on ∫cos x dx = sin x + C versus ∫sin x dx = −cos x + C.
- Taking the area under a curve without splitting where it crosses the axis (areas below are negative).
- Testing only stationary points when the question asks for an absolute maximum on a closed interval - check endpoints too.