Calculus
Calculus
This domain tests limits, continuity, differentiation and its applications, and basic integration. ASSEB questions commonly ask for a derivative using chain/product/quotient rules, a tangent or normal, a maximum/minimum, or a definite integral.
Core rules
- lim_{x→a} f(x) exists only if left and right limits agree. Differentiability implies continuity, but continuity does not imply differentiability (for example |x| at 0).
- Standard derivatives: d/dx(x^n)=nx^{n−1}, sin x→cos x, cos x→−sin x, e^x→e^x, ln x→1/x.
- 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).
- For maxima/minima, solve f'(x)=0, then use f''(x) or a sign change. Check endpoints if the interval is closed.
- Indefinite integrals need +C; definite integrals use F(b)−F(a).
Worked example
Let f(x)=x^3−3x^2+2. Then f'(x)=3x^2−6x=3x(x−2), so critical points are x=0,2. Also f''(x)=6x−6. At x=0, f''(0)=−6<0, so f has a local maximum f(0)=2. At x=2, f''(2)=6>0, so f has a local minimum f(2)=8−12+2=−2. The sign chart of f' confirms: positive outside [0,2] and negative inside, so the classification is consistent.
Traps
Do not drop the chain rule's inner derivative. Do not write +C for a definite integral. For a quotient, the numerator order is u'v−uv', not uv'−u'v. A stationary point is not automatically an extremum; verify with the second derivative or sign change. When finding a tangent at x=a, evaluate both f(a) and f'(a).