§-Unit

Unit 3: Further calculus and introduction to statistics

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

Topic 1: Differentiation of exponential and logarithmic functions

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

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 read →

Topic 2: Differentiation of trigonometric functions and differentiation rules

Differentiate trigonometric functions, including compositions of the form

\sin(f(x))

\cos(f(x))

and

\tan(f(x))

, working in radians

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 read →

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

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 read →

Topic 3: Further applications of differentiation

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

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 read →

Topic 4: Introduction to integration

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

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 read →

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

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 read →

Topic 5: 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

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 read →

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 a graphics calculator, and use the formulas

E(X) = np

and

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

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 graphics-calculator-supported Paper 2 calculations including $P(X \leq k)$ , $P(X \geq k)$ and modelling applications.

9 min read →