Calculus and Analysis
Calculus and Analysis tests limits, continuity, differentiation, curve sketching, optimisation, and basic integration as accumulated change or area.
Core rules
- Limit laws apply when limits exist; factor or rationalise to remove 0/0 forms. A function is continuous at a if lim_{x→a} f(x)=f(a).
- Power rule d/dx(x^n)=nx^(n−1); product, quotient, and chain rules. d/dx(sin x)=cos x, d/dx(cos x)=−sin x, d/dx(e^x)=e^x, d/dx(ln x)=1/x.
- f′(x)=0 locates stationary points; f″(x)>0 indicates a local minimum, f″(x)<0 a local maximum.
- Integration reverses differentiation: ∫x^n dx=x^(n+1)/(n+1)+C for n≠−1. Definite integrals give signed area; area between curves uses upper minus lower.
- For motion, velocity is the derivative of displacement and acceleration the derivative of velocity.
Worked example
Find and classify the stationary points of f(x)=x³−3x²+2. f′(x)=3x²−6x=3x(x−2), so x=0 or x=2. f(0)=2 and f(2)=8−12+2=−2. f″(x)=6x−6: f″(0)=−6<0, so (0,2) is a local maximum; f″(2)=6>0, so (2,−2) is a local minimum.
Traps
Do not omit +C for indefinite integrals. Differentiate the outside function and multiply by the inside derivative in chain-rule problems. A stationary point is not automatically a maximum or minimum. For definite integrals, reversing limits changes the sign. Check whether an area lies below the x-axis before treating an integral as geometric area.