Calculus
Calculus at Higher Level covers differentiation, rates of change, tangents and normals, maxima and minima, integration, and area. Questions often require you to connect algebra, graphs, and context.
Core rules
- Differentiation: power rule; product rule; quotient rule; chain rule. Know (\frac{d}{dx}\sin x=\cos x), (\frac{d}{dx}\cos x=-\sin x), (\frac{d}{dx}e^x=e^x), and (\frac{d}{dx}\ln x=\frac1x).
- At (x=a), the tangent slope is (f'(a)). The normal slope is (-\frac1{f'(a)}).
- Turning points: solve (f'(x)=0). Use (f''(x)) or a sign table to classify. Check endpoints for global maxima and minima.
- Integration: (\int x^n,dx=\frac{x^{n+1}}{n+1}+C) for (n\ne-1). For a definite integral, evaluate (F(b)-F(a)).
- Area under a curve is found by integration; regions below the (x)-axis give a negative integral, so split the interval if necessary.
- Rates of change: velocity is (\frac{ds}{dt}); acceleration is (\frac{dv}{dt}).
Worked example
For (f(x)=x^3-3x), (f'(x)=3x^2-3=3(x-1)(x+1)). Turning points at (x=1) and (x=-1). Since (f''(x)=6x), (f''(1)=6>0), so (x=1) is a local minimum; (f''(-1)=-6<0), so (x=-1) is a local maximum.
Traps
Omitting (+C) in indefinite integration; misapplying the chain rule; confusing maxima with minima; forgetting the normal slope is the negative reciprocal; treating area below the axis as positive without splitting; not including units in a rate-of-change answer.