NCEA Level 3 Calculus study guide

NCEA Level 3 Calculus · Work through your exam, one topic at a time.

NCEA Level 3 Calculus is the highest of the three NCEA senior mathematics subjects in New Zealand's national qualification framework, taken in Year 13 and typically used as a prerequisite pathway into engineering, physical sciences, mathematics, and economics at tertiary level. It is designed for students who are comfortable with algebraic manipulation and who intend to study a mathematically demanding degree. The subject builds on Level 2 Mathematics with Calculus and extends into formal differentiation and integration techniques and their applications, conic sections and coordinate geometry, trigonometric functions and identities, and complex numbers alongside algebra. Assessment combines internally assessed standards, where investigation and modelling work is graded by your school, and externally assessed standards examined at the end of the year. Topics on this exam: Differentiation and integration (calculus methods), Conic sections and coordinate geometry, Trigonometric functions and identities, and Complex numbers and algebra. The NCEA Level 3 Calculus study guide has 4 sections and about 858 words.

Differentiation and Integration Methods

This domain covers AS91578 (differentiation) and AS91579 (integration) and tests calculus as a problem-solving tool, not just routine differentiation. You need chain, product and quotient rules, derivatives of sin, cos, tan, e^x and ln x, implicit and parametric differentiation, rates of change, stationary points and optimisation. For integration, know standard results (including ∫1/x dx = ln x + c and ∫e^{kx} dx), substitution, definite integrals, areas between curves, volumes of revolution and simple separable differential equations. Differentiate y=x^2 ln x and find the tangent at x=1.

269 words, about 1 min 02

Conic Sections and Coordinate Geometry

Parabola y^2=4ax: focus (a,0), directrix x=-a. Ellipse x^2/a^2+y^2/b^2=1 (a b): c^2=a^2-b^2, foci (±c,0), eccentricity e=c/a<1. Hyperbola x^2/a^2-y^2/b^2=1: c^2=a^2+b^2, foci (±c,0), e=c/a 1. Rectangular hyperbola xy=c^2 has parametric form (ct,c/t).

206 words, about 1 min 03

Trigonometric Functions and Identities

Know sin^2θ+cos^2θ=1, 1+tan^2θ=sec^2θ, 1+cot^2θ=csc^2θ; compound angles sin(A±B), cos(A±B), tan(A±B); double angles sin2θ=2sinθcosθ, cos2θ=cos^2θ-sin^2θ=2cos^2θ-1=1-2sin^2θ. General solutions: if sinθ=k then θ=nπ+(-1)^n α; if cosθ=k then θ=2nπ±α; if tanθ=k then θ=nπ+α. Solve 2sin^2x+sinx-1=0, 0≤x≤2π. Factor: (2sinx-1)(sinx+1)=0.

177 words, under 1 min 04

Complex Numbers and Algebra

z=a+bi, i^2=-1, conjugate z̄=a-bi. Modulus z =√(a^2+b^2); argument arg z measured from positive real axis. Polar form z=r(cosθ+isinθ)=r cisθ. Then z1z2 = z1 z2 , arg(z1z2)=arg z1+arg z2; division subtracts arguments.

206 words, about 1 min