Unit 4: Further calculus, and statistical inference

10 dot points, in syllabus order. Each dot point has a focused answer, with exam-style questions and worked answers where available.

Topic 1: Integration and applications of integration

Apply integration techniques including substitution, integration by partial fractions, and trigonometric integrals, and use them to evaluate definite integrals and compute areas and volumes of solids of revolution

Integration techniques. Covers integration by substitution and the change-of-limits method, partial-fraction decomposition for rational integrands, trigonometric integrals, and volumes of revolution, with a fully verified worked example and the substitution-limits mistake QCAA markers watch for.

8 min read →

Determine areas between curves and volumes of solids of revolution generated by rotating a region about the x-axis or the y-axis, setting up the correct definite integral and evaluating it

Areas and volumes. Covers the area between two curves, the disc formula for rotation about each axis, and rotating a region bounded by two curves, with a verified worked example and the squaring trap.

8 min read →

Use Simpson's rule to approximate the value of a definite integral or an area, applying the rule with an even number of subintervals and recognising when a numerical method is required

Simpson's rule. Covers the rule and its weighting pattern, the requirement for an even number of subintervals, choosing the strip width, and when numerical integration is needed, with a verified worked example and the coefficient-pattern trap.

8 min read →

Evaluate integrals using integration by parts and integrate trigonometric expressions using identities such as double-angle and Pythagorean identities and products of sines and cosines

Integration by parts and trigonometric integrals. Covers the parts formula and the LIATE choice, repeated parts, and reducing powers of sine and cosine with double-angle and Pythagorean identities, with a verified worked example and the wrong-choice trap.

8 min read →

Topic 2: Rates of change and differential equations

Differentiate implicitly defined relations and solve related rates problems using the chain rule, including contexts involving volumes and surface areas of cones, spheres and cylinders

Implicit differentiation and related rates. Covers differentiating relations not solved for y, the chain rule link between rates, the four-step related-rates method, and geometric volume contexts, with a verified worked example and the differentiate-then-substitute trap.

8 min read →

Construct and interpret slope (direction) fields of a first-order differential equation, sketch solution curves through given points, and relate the qualitative behaviour of solutions to the field

Slope fields. Covers building a direction field from dy/dx, reading slopes at grid points, sketching solution curves through given initial points, and identifying equilibrium solutions, with a verified worked example and the curve-crossing trap.

8 min read →

Model and solve practical situations with first-order differential equations, including exponential growth and decay, Newton's law of cooling and the logistic equation, and interpret the long-term behaviour of solutions

Differential equation models. Covers exponential growth and decay, Newton's law of cooling, the logistic equation and its carrying capacity, and the long-term behaviour of each model, with a verified worked example and the equilibrium-interpretation trap.

7 min read →

Formulate and solve first-order differential equations using separation of variables, including growth and decay and the logistic model, and interpret solutions in applied rates-of-change contexts

Differential equations. Covers separation of variables, the general and particular solution, exponential growth and decay, Newton's law of cooling and the logistic model, with a verified worked example and the constant-of-integration mistake QCAA markers penalise.

8 min read →

Topic 3: Statistical inference

Construct and interpret confidence intervals for a population mean using the sample mean and standard error, choosing the appropriate confidence level, and understand the meaning of the confidence level in repeated sampling

Confidence intervals. Covers the structure of an interval estimate, the critical value and margin of error, how confidence level and sample size affect width, the correct repeated-sampling interpretation, and a fully verified worked example with the common interpretation mistake QCAA penalises.

8 min read →

Understand the distribution of the sample mean, apply the central limit theorem to describe its shape, mean and standard deviation, and use these to compute probabilities for sample means drawn from a population

Sampling distribution of the mean. Covers the mean and standard error of the sample mean, the central limit theorem, standardising to compute probabilities, and how sample size affects spread, with a verified worked example and the standard-error mistake QCAA markers watch for.

8 min read →