§-Module

Unit 3: Further calculus and statistics

8 dot points across 3 inquiry questions. Click any dot point for a focused answer with worked past exam questions where available.

Topic 2: Integrals

Find antiderivatives of standard functions including polynomial, exponential and trigonometric forms, evaluate definite integrals using the Fundamental Theorem of Calculus, and recognise the definite integral as the limit of a Riemann sum

A focused answer to the QCE Mathematical Methods Unit 3 dot point on integration. Covers the standard antiderivatives, the linear-inside-argument shortcut, the Fundamental Theorem of Calculus as the bridge between differentiation and integration, and the Riemann-sum definition of the definite integral, with worked Paper 1 and Paper 2 examples QCAA examiners reward.

9 min answer →

Apply the definite integral to find the area under a curve, the area between two curves, the average value of a function, and to solve kinematics problems involving displacement, velocity and acceleration

A focused answer to the QCE Mathematical Methods Unit 3 dot point on the applications of integration. Covers area under a curve, area between two curves (including curves that cross), the average value of a function, and the kinematics chain (integrate acceleration for velocity, integrate velocity for displacement), with worked Paper 2 and PSMT-style examples.

9 min answer →

Topic 1: Further differentiation and applications

Differentiate exponential and logarithmic functions, including compositions of the form

e^{f(x)}

and

\ln(f(x))

, and apply the derivatives to model and analyse rates of change

A focused answer to the QCE Mathematical Methods Unit 3 dot point on differentiating exponential and logarithmic functions. Covers the derivatives of $e^x$ , $a^x$ , $\ln x$ and $\log_a x$ , the chain rule generalisations $e^{f(x)}$ and $\ln(f(x))$ , and the application to rates of change, with worked Paper 1 and Paper 2 examples.

8 min answer →

Differentiate trigonometric functions, including compositions of the form

\sin(f(x))

\cos(f(x))

and

\tan(f(x))

, working in radians

A focused answer to the QCE Mathematical Methods Unit 3 dot point on differentiating trigonometric functions. Sets out the standard derivatives of $\sin x$ , $\cos x$ and $\tan x$ in radians, the chain rule generalisations, why radian measure is required for calculus, and the exact-value and Paper 1 fluency QCAA examiners reward in IA2 and the EA.

7 min answer →

Use the first and second derivative to analyse the behaviour of a function (intervals of increase and decrease, stationary points and their nature, concavity and inflection), and apply the derivative to solve optimisation and rates of change problems in context

A focused answer to the QCE Mathematical Methods Unit 3 dot point on the applications of differentiation. Sets out how to use the first and second derivative to classify stationary points, walks through the optimisation method (model, constrain, differentiate, classify, check), and the related rates approach that QCAA examiners reward in PSMTs and EA extended response.

9 min answer →

Apply the product, quotient and chain rules, including in combination, to differentiate functions built from polynomial, exponential, logarithmic and trigonometric components

A focused answer to the QCE Mathematical Methods Unit 3 dot point on the product, quotient and chain rules. Sets out each rule, walks through worked combinations of polynomial, exponential, logarithmic and trigonometric functions, and identifies the order-of-operations and simplification traps that QCAA examiners reward in Paper 1 short response.

8 min answer →

Topic 3: Discrete random variables

Define a discrete random variable and its probability distribution, calculate the expected value

E(X)

and the variance

\mathrm{Var}(X)

and standard deviation, and recognise the Bernoulli distribution as the single-trial case

A focused answer to the QCE Mathematical Methods Unit 3 dot point on discrete random variables. Covers the probability distribution and its conditions ( $p_i \geq 0$ and $\sum p_i = 1$ ), the calculation of $E(X)$ and $\mathrm{Var}(X)$ from a distribution table, and the Bernoulli distribution as the single-trial case, with QCAA IA2-style worked examples.

8 min answer →

Recognise the binomial distribution

X \sim \mathrm{Bin}(n, p)

as the count of successes in

n

independent Bernoulli trials, apply the binomial probability formula and use CAS, and use the formulas

E(X) = np

and

\mathrm{Var}(X) = np(1 - p)

A focused answer to the QCE Mathematical Methods Unit 3 dot point on the binomial distribution. Defines the binomial conditions (BINS), states the probability formula, gives the mean $np$ and variance $np(1 - p)$ , and walks through both by-hand Paper 1 calculations and CAS-supported Paper 2 calculations including $P(X \leq k)$ , $P(X \geq k)$ and modelling applications.

9 min answer →