§-Module

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 →