Unit 4: Further calculus and statistical inference
7 dot points across 3 inquiry questions. Click any dot point for a focused answer with worked past exam questions where available.
Topic 2: Trigonometric functions and integration applications
- Apply the definite integral to compute the area between curves (including curves that change relative order), the average value of a function, and kinematics quantities (displacement, distance, position) from velocity and acceleration
A focused answer to the QCE Maths Methods Unit 4 dot point on the applications of integration. Area between curves, average value of a function, displacement and distance from velocity, position from acceleration with initial conditions, with worked PSMT-style examples.
9 min answer → - Integrate trigonometric functions including $\sin(kx)$, $\cos(kx)$ and $\sec^2(kx)$, and apply the linear reverse-chain rule for integrals of the form $f(ax+b)$
A focused answer to the QCE Maths Methods Unit 4 dot point on integrating trigonometric functions. Antiderivatives of $\sin(kx)$, $\cos(kx)$ and $\sec^2(kx)$ with the $1/k$ reverse-chain factor, definite-integral evaluation with exact values at standard angles, and worked PSMT-style applications.
8 min answer →
Topic 3: Continuous random variables, the normal distribution, and statistical inference
- Define a continuous random variable, its probability density function (pdf), cumulative distribution function (cdf), and compute probabilities, expected value (mean), variance and standard deviation as definite integrals
A focused answer to the QCE Maths Methods Unit 4 dot point on continuous random variables. Defines the pdf, cdf, mean, variance and standard deviation as integrals, including the normalisation condition and a worked PSMT-style example.
9 min answer → - Apply the normal distribution $N(\mu, \sigma^2)$ and the standardisation $Z = (X - \mu)/\sigma$ to compute normal probabilities and inverse probabilities, including the empirical 68-95-99.7 rule
A focused answer to the QCE Maths Methods Unit 4 dot point on the normal distribution. Standardisation, the empirical rule, normal probability and inverse-normal calculations, and worked PSMT and EA examples.
9 min answer → - Apply the sampling distribution of the sample proportion $\hat{p}$ (mean $p$, standard deviation $\sqrt{p(1-p)/n}$) and construct approximate confidence intervals $\hat{p} \pm z^* \sqrt{\hat{p}(1-\hat{p})/n}$ for a population proportion
A focused answer to the QCE Maths Methods Unit 4 dot point on sample proportions and confidence intervals. The sampling distribution of $\hat{p}$, the normal approximation, the CI formula with standard $z^*$ values, and worked Paper 2 / PSMT examples.
9 min answer →
Topic 1: Further differentiation and applications
- Apply the product, quotient and chain rules to differentiate composite functions involving exponential, logarithmic, polynomial and trigonometric pieces, including logarithmic differentiation and the differentiation of inverse functions
A focused answer to the QCE Maths Methods Unit 4 dot point on further differentiation. Logarithmic differentiation for products and powers, derivatives of inverse functions via $1 / f'(x)$, and the standard PSMT and EA contexts in which the further rules appear.
9 min answer → - Apply implicit differentiation to find $\frac{dy}{dx}$ from equations relating $x$ and $y$ that cannot be expressed in the form $y = f(x)$, and apply differentiation to related rates problems
A focused answer to the QCE Maths Methods Unit 4 dot point on implicit differentiation and related rates. The four-step procedure for related rates, the chain-rule treatment of $y(x)$, and PSMT contexts where two or more time-dependent quantities are related geometrically.
9 min answer →