NCEA Level 3 Calculus study plan

Differentiate polynomials, powers, and simple rational expressions from first principles and by rule, stating the limit definition each time

Apply the chain, product, and quotient rules to composite expressions and justify which rule applies before differentiating

Compute gradients, tangents, and normals at a given point and interpret the derivative as an instantaneous rate of change

Set up and solve basic optimisation and kinematics problems where the derivative is zero or where rate is given

Integrate power, polynomial, and simple trigonometric and exponential expressions, including negative and fractional indices

Evaluate definite integrals and apply the fundamental theorem correctly, showing antiderivative substitution limits

Use integration to find areas between curves and to solve displacement-from-velocity problems

Recognise when substitution or recognition of the derivative of a function is required and carry it out

Combine differentiation and integration in multi-step problems involving rates of change, areas, and volumes of revolution

Solve differential-equation-style problems of the form dy/dx = f(x) by direct integration with initial conditions

Interpret and communicate the meaning of a definite integral and derivative in a real context in written sentences

Identify and correct the three most frequent algebraic slips in your own working (sign errors, index errors, missing constants)

Derive and state the standard equations of circles, parabolas, ellipses, and hyperbolas in both standard and general forms

Complete the square to convert general conic equations into standard form and classify the conic from its coefficients

Find foci, vertices, directrices, eccentricity, and asymptotes for each conic type

Sketch conics accurately with labelled key features from an equation alone

Solve intersection problems between conics and lines algebraically and interpret the number of solutions geometrically

Apply the discriminant to determine tangency and to find tangent lines to conics

Use parametric forms of the parabola and ellipse to solve locus and chord problems

Translate between Cartesian and parametric representations fluently

Recall and apply exact values, reciprocal and Pythagorean identities, and compound-angle and double-angle formulae without reference

Prove identities by transforming one side into the other and state the identity used at each step

Solve trigonometric equations over a specified interval, including general solutions and those requiring rearrangement into quadratic form

Differentiate and integrate trigonometric functions and apply them to gradient and area problems

Sketch trigonometric graphs showing amplitude, period, phase shift, and asymptotes

Convert between rectangular, polar, and exponential forms of complex numbers and justify the modulus-argument choice

Apply De Moivre's theorem to powers and roots and locate roots on an Argand diagram

Perform arithmetic on complex numbers and solve quadratic equations with complex roots

Use trigonometric identities together with complex-number techniques in combined problems

Solve polynomial equations and apply the factor and remainder theorems, including complex conjugate-root pairs

Manipulate algebraic fractions, logarithms, and exponentials accurately in multi-step problems

Complete two full mixed-domain practice papers under exam timing and mark them against the schedule

Consolidate a personal error log into a one-page revision summary of the highest-risk mistakes