Unit 4

WACE Specialist Unit 4 areas between curves: the top-minus-bottom integral, finding intersection points as limits, splitting the region where curves cross, and integrating with respect to y when convenient, with a worked example.

WACE Specialist Unit 4 central limit theorem: why the sample mean is approximately normal for large n regardless of population shape, the standardising z-score for the sample mean, and computing probabilities, with a worked example.

WACE Specialist Unit 4 confidence intervals: the interval sample mean plus or minus z times the standard error, the critical z-values, the correct repeated-sampling interpretation, the margin of error, and solving for the required sample size, with a worked example.

WACE Specialist Unit 4 growth models: the exponential equation dy/dt equal to ky and its solution, the logistic equation with a carrying capacity, the S-shaped solution curve, equilibria, and interpreting parameters, with a worked example.

WACE Specialist Unit 4 partial fractions: decomposing a proper rational function over distinct linear factors, solving for the constants, integrating each term to a logarithm, and handling improper fractions by division first, with a worked example.

WACE Specialist Unit 4 integration by substitution: choosing u, replacing dx with du over the derivative, changing limits for definite integrals, and the reverse chain rule, with a worked example.

WACE Specialist Unit 4 integration: substitution, partial fractions, trigonometric identities and the double-angle method, definite integrals, volumes of revolution, and solving separable differential equations including exponential and logistic models, with worked examples.

WACE Specialist Unit 4 trigonometric integration: using double-angle identities to integrate sine and cosine squared, the Pythagorean identity for odd powers, and product-to-sum identities, with a worked example.

WACE Specialist Unit 4 matrices: 2x2 matrix arithmetic, determinants and inverses, the matrices for rotation, reflection, dilation and shear, composition of transformations by matrix multiplication, and the geometric meaning of the determinant.

WACE Specialist proof by mathematical induction: the base case and inductive step, the structure of a rigorous induction, and applying it to summation formulas, divisibility statements and inequalities, with a worked example.

WACE Specialist Unit 4 sampling distribution: the sample mean as a random variable, its expected value equal to the population mean, the standard error sigma over root n, and why larger samples cluster more tightly, with a worked example.

WACE Specialist Unit 4 separation of variables: recognising a separable equation, moving all y terms to one side and x terms to the other, integrating both sides, and using an initial condition to fix the constant, with a worked example.

WACE Specialist Unit 4 slope fields: reading the gradient at each point from a first-order differential equation, drawing short line segments, sketching solution curves that follow the field, and recognising equilibrium solutions, with a worked example.

WACE Specialist Unit 4 statistical inference: the sampling distribution of the sample mean, its mean and standard deviation (standard error), the central limit theorem, and constructing and interpreting confidence intervals for a population mean, with a full worked example.

WACE Specialist Unit 4 vector geometry: vector and parametric equations of lines and planes, the cartesian and scalar (normal) forms, intersections, the angle between lines and planes, parallel and skew lines, and distance calculations in three dimensions.

WACE Specialist Unit 4 volumes of revolution: the disc method about the x-axis and y-axis, integrating pi times radius squared, rotating between two curves with the washer idea, and setting up the correct limits, with a worked example.