Year 12 maths: Mathematics Methods ATAR Year 12 · Units 3 and 4

Topic 3.1: Further differentiation and applications

3.1.1estimate the limit of (𝑎^ℎ − 1)/ℎ as ℎ → 0, using technology, for various values of 𝑎 > 0
- Limit of the exponential difference quotient
3.1.2identify that 𝑒 is the unique number 𝑎 for which the above limit is 1
- The number e as the base giving a limit of 1
3.1.3establish and use the formula 𝑑/𝑑𝑥 (𝑒^𝑥) = 𝑒^𝑥
- Level 1: Differentiation of mixed functions
3.1.4use exponential functions of the form 𝐴𝑒^(𝑘𝑥) and their derivatives to solve practical problems
- Level 1: Differentiation word problems
- Level 2: Differentiation word problems
3.1.5establish the formulas 𝑑/𝑑𝑥 (sin 𝑥) = cos 𝑥 and 𝑑/𝑑𝑥 (cos 𝑥) = −sin 𝑥 by graphical treatment, numerical estimations of the limits, and informal proofs based on geometric constructions
3.1.6use trigonometric functions and their derivatives to solve practical problems
- Level 2: Differentiation word problems
- Level 3: Differentiation word problems
3.1.7examine and use the product and quotient rules
3.1.8examine the notion of composition of functions and use the chain rule for determining the derivatives of composite functions
3.1.9apply the product, quotient and chain rule to differentiate functions such as 𝑥𝑒^𝑥, tan 𝑥, 𝑥^𝑛, 𝑥 sin 𝑥, 𝑒^(−𝑥) sin 𝑥 and 𝑓(𝑎𝑥 − 𝑏)
3.1.10use the increments formula: 𝛿𝑦 ≈ (𝑑𝑦/𝑑𝑥) × 𝛿𝑥 to estimate the change in the dependent variable 𝑦 resulting from changes in the independent variable 𝑥
- Increments formula: linear approximation
3.1.11apply the concept of the second derivative as the rate of change of the first derivative function
- Level 1: Stationary points, inflection points and intervals of increase or decrease
3.1.12identify acceleration as the second derivative of position with respect to time
- Level 3: Differentiation word problems
3.1.13examine the concepts of concavity and points of inflection and their relationship with the second derivative
- Level 1: Stationary points, inflection points and intervals of increase or decrease
3.1.14apply the second derivative test for determining local maxima and minima
- Level 1: Local minimum and local maximum values
- Level 2: Local minimum and local maximum values
3.1.15sketch the graph of a function using first and second derivatives to locate stationary points and points of inflection
3.1.16solve optimisation problems from a wide variety of fields using first and second derivatives

Topic 3.2: Integrals

3.2.1identify anti-differentiation as the reverse of differentiation
- Level 1: Indefinite integration and antiderivatives
3.2.2use the notation ∫ 𝑓(𝑥)𝑑𝑥 for anti-derivatives or indefinite integrals
- Level 1: Indefinite integration and antiderivatives
3.2.3establish and use the formula ∫ 𝑥^𝑛 𝑑𝑥 = 𝑥^(𝑛+1)/(𝑛+1) + 𝑐 for 𝑛 ≠ −1
- Indefinite integration of polynomials
- Indefinite integration of surds
3.2.4establish and use the formula ∫ 𝑒^𝑥 𝑑𝑥 = 𝑒^𝑥 + 𝑐
- Indefinite integration of polynomials
3.2.5establish and use the formulas ∫ sin 𝑥 𝑑𝑥 = −cos 𝑥 + 𝑐 and ∫ cos 𝑥 𝑑𝑥 = sin 𝑥 + 𝑐
- Indefinite integration of trigonometric functions
3.2.6identify and use linearity of anti-differentiation
- Indefinite integration of polynomials
3.2.7determine indefinite integrals of the form ∫ 𝑓(𝑎𝑥 − 𝑏)𝑑𝑥
- Level 1: Indefinite integration and antiderivatives
3.2.8identify families of curves with the same derivative function
- Level 2: Finding a curve from a stationary point
3.2.9determine 𝑓(𝑥), given 𝑓′(𝑥) and an initial condition 𝑓(𝑎) = 𝑏
- Level 2: Finding a curve from a stationary point
- Level 3: Finding a curve from a stationary point
3.2.10examine the area problem and use sums of the form ∑_𝑖 𝑓(𝑥_𝑖) 𝛿𝑥_𝑖 to estimate the area under the curve 𝑦 = 𝑓(𝑥)
- Riemann sums: estimating area under a curve
3.2.11identify the definite integral ∫_𝑎^𝑏 𝑓(𝑥)𝑑𝑥 as a limit of sums of the form ∑_𝑖 𝑓(𝑥_𝑖) 𝛿𝑥_𝑖
- Definite integration
3.2.12interpret the definite integral ∫_𝑎^𝑏 𝑓(𝑥)𝑑𝑥 as area under the curve 𝑦 = 𝑓(𝑥) if 𝑓(𝑥) > 0
- Level 1: Definite integration
3.2.13interpret ∫_𝑎^𝑏 𝑓(𝑥)𝑑𝑥 as a sum of signed areas
- Level 2: Definite integration
3.2.14apply the additivity and linearity of definite integrals
- Definite integration
3.2.15examine the concept of the signed area function 𝐹(𝑥) = ∫_𝑎^𝑥 𝑓(𝑡)𝑑𝑡
- Level 2 - Definite integration
3.2.16apply the theorem: 𝐹′(𝑥) = 𝑑/𝑑𝑥 (∫_𝑎^𝑥 𝑓(𝑡)𝑑𝑡) = 𝑓(𝑥), and illustrate its proof geometrically
- Level 2: Definite integration
3.2.17develop the formula ∫_𝑎^𝑏 𝑓′(𝑥)𝑑𝑥 = 𝑓(𝑏) − 𝑓(𝑎) and use it to calculate definite integrals
- Definite integration
3.2.18calculate total change by integrating instantaneous or marginal rate of change
- Level 1: Average and instantaneous rates of change
3.2.19calculate the area under a curve
- Level 1: Definite integration
3.2.20calculate the area between curves determined by functions of the form 𝑦 = 𝑓(𝑥)
- Level 2: Definite integration
3.2.21determine displacement given velocity in linear motion problems
- Displacement given velocity
3.2.22determine positions given linear acceleration and initial values of position and velocity.
- Position given acceleration and initial conditions

Topic 3.3: Discrete random variables

3.3.1develop the concepts of a discrete random variable and its associated probability function, and their use in modelling data
- Level 1: Discrete random variables
- Identify discrete probability distributions
3.3.2use relative frequencies obtained from data to obtain point estimates of probabilities associated with a discrete random variable
- Calculate probabilities given a discrete probability distribution
3.3.3identify uniform discrete random variables and use them to model random phenomena with equally likely outcomes
- Identify discrete probability distributions
3.3.4examine simple examples of non-uniform discrete random variables
- Identify discrete probability distributions
3.3.5define the mean or expected value of a discrete random variable and interpret it as a measure of location, and evaluate it in simple cases
- Level 1: Discrete random variables
3.3.6define the variance and standard deviation of a discrete random variable and interpret them as measures of spread, and evaluate them using technology
- Level 1: Discrete random variables
3.3.7examine the effects of linear changes of scale and origin on the mean and the standard deviation
- Linear transformation of mean and standard deviation
3.3.8use discrete random variables and associated probabilities to solve practical problems
- Calculate probabilities given a discrete probability distribution
3.3.9use a Bernoulli random variable as a model for two-outcome situations
- Bernoulli distribution: mean and variance
- Bernoulli distribution: modelling practical problems
3.3.10identify contexts suitable for modelling by Bernoulli random variables
- Bernoulli distribution - modelling practical problems
3.3.11determine the mean 𝑝 and variance 𝑝(1 − 𝑝)of the Bernoulli distribution with parameter 𝑝
- Bernoulli distribution: mean and variance
3.3.12use Bernoulli random variables and associated probabilities to model data and solve practical problems
- Bernoulli distribution: modelling practical problems
3.3.13examine the concept of Bernoulli trials and the concept of a binomial random variable as the number of ‘successes’ in 𝑛 independent Bernoulli trials, with the same probability of success 𝑝 in each trial
- Level 1: Multi-stage probability experiments
3.3.14identify contexts suitable for modelling by binomial random variables
- Level 1: Multi-stage probability experiments
3.3.15determine and use the probabilities 𝑃(𝑋 = 𝑥) = 𝑛C𝑥 · 𝑝^𝑥(1 − 𝑝)^(𝑛−𝑥) associated with the binomial distribution with parameters 𝑛 and 𝑝; note the mean 𝑛𝑝 and variance 𝑛𝑝(1 − 𝑝) of a binomial distribution
- Level 2: Multi-stage probability experiments
3.3.16use binomial distributions and associated probabilities to solve practical problems
- Level 1: Multi-stage probability experiments
- Level 2: Multi-stage probability experiments

Topic 4.1: The logarithmic function

4.1.1define logarithms as indices: 𝑎^𝑥 = 𝑏 is equivalent to 𝑥 = log_𝑎 𝑏 i.e. 𝑎^(log_𝑎 𝑏) = 𝑏
- Evaluation of logarithms
4.1.2establish and use the algebraic properties of logarithms
- Level 1: Logarithm laws
- Simplification of logarithmic expressions
4.1.3examine the inverse relationship between logarithms and exponentials: 𝑦 = 𝑎^𝑥 is equivalent to 𝑥 = log_𝑎 𝑦
- Equations involving exponents
4.1.4interpret and use logarithmic scales
- Interpreting and using logarithmic scales
4.1.5solve equations involving indices using logarithms
- Equations involving exponents
4.1.6identify the qualitative features of the graph of 𝑦 = log_𝑎 𝑥 (𝑎 > 1), including asymptotes, and of its translations 𝑦 = log_𝑎 𝑥 + 𝑏 and 𝑦 = log_𝑎(𝑥 − 𝑐)
- Graph features of translated logarithmic functions
4.1.7solve simple equations involving logarithmic functions algebraically and graphically
4.1.8identify contexts suitable for modelling by logarithmic functions and use them to solve practical problems
- Level 2: Logarithmic equations and domains
4.1.9define the natural logarithm ln 𝑥 = log_𝑒 𝑥
- Evaluation of logarithms
4.1.10examine and use the inverse relationship of the functions 𝑦 = 𝑒^𝑥 and 𝑦 = ln 𝑥
4.1.11establish and use the formula 𝑑/𝑑𝑥 (ln 𝑥) = 1/𝑥
- Level 1: Differentiation of logarithmic functions
4.1.12establish and use the formula ∫ (1/𝑥) 𝑑𝑥 = ln 𝑥 + 𝑐, for 𝑥 > 0
- Level 1: Indefinite integration and antiderivatives
4.1.13determine derivatives of the form 𝑑/𝑑𝑥 (ln 𝑓(𝑥)) and integrals of the form ∫ (𝑓′(𝑥)/𝑓(𝑥)) 𝑑𝑥 for 𝑓(𝑥) > 0
- Level 2: Differentiation of logarithmic functions
4.1.14use logarithmic functions and their derivatives to solve practical problems
- Level 3: Differentiation word problems

Topic 4.2: Continuous random variables and the normal distribution

4.2.1use relative frequencies and histograms obtained from data to estimate probabilities associated with a continuous random variable
- Understanding histograms
4.2.2examine the concepts of a probability density function, cumulative distribution function, and probabilities associated with a continuous random variable given by integrals; examine simple types of continuous random variables and use them in appropriate contexts
- Level 1: Probabilities from polynomial probability density functions
- Level 1: Probabilities from exponential probability density functions
4.2.3define the expected value, variance and standard deviation of a continuous random variable and evaluate them in simple cases, using technology where required
- Level 1: Statistics from polynomial probability density functions
- Level 1: Statistics from exponential probability density functions
4.2.4examine the effects of linear changes of scale and origin on the mean and the standard deviation
- Linear transformation of mean and standard deviation
4.2.5identify contexts, such as naturally occurring variation, that are suitable for modelling by normal random variables
- Contexts suitable for a normal model
4.2.6identify features of the graph of the probability density function of the normal distribution with mean μ and standard deviation σ and the use of the standard normal distribution
- Normal distribution: qualitative graph features
4.2.7calculate probabilities and quantiles associated with a given normal distribution using technology, and use these to solve practical problems
- Normal distribution: probabilities and quantiles

Topic 4.3: Interval estimates for proportions

4.3.1examine the concept of a random sample
- The concept of a random sample
4.3.2discuss sources of bias in samples, and procedures to ensure randomness
- Sources of bias and ensuring randomness
4.3.3use graphical displays of simulated data to investigate the variability of random samples from various types of distributions, including uniform, normal and Bernoulli
- Understanding histograms
- Summary and spread of data
4.3.4examine the concept of the sample proportion 𝑝̂ as a random variable whose value varies between samples, and the formulas for the mean 𝑝 and standard deviation √(𝑝(1−𝑝)/𝑛) of the sample proportion 𝑝̂
- Sample proportion: mean and standard deviation
4.3.5examine the approximate normality of the distribution of 𝑝̂ for large samples
- Approximate normality of the sample proportion
4.3.6simulate repeated random sampling, for a variety of values of 𝑝 and a range of sample sizes, to illustrate the distribution of 𝑝̂ and the approximate standard normality of (𝑝̂ − 𝑝)/√(𝑝(1−𝑝)/𝑛), where the closeness of the approximation depends on both 𝑛 and 𝑝
- Simulated sampling distribution of the sample proportion
4.3.7examine the concept of an interval estimate for a parameter associated with a random variable
- Confidence interval and margin of error for a proportion
4.3.8use the approximate confidence interval (𝑝̂ − 𝑧√(𝑝̂(1−𝑝̂)/𝑛), 𝑝̂ + 𝑧√(𝑝̂(1−𝑝̂)/𝑛)) as an interval estimate for 𝑝, where 𝑧 is the appropriate quantile for the standard normal distribution
- Confidence interval and margin of error for a proportion
4.3.9define the approximate margin of error 𝐸 = 𝑧√(𝑝̂(1−𝑝̂)/𝑛) and understand the trade-off between margin of error and level of confidence
- Confidence interval and margin of error for a proportion
4.3.10use simulation to illustrate variations in confidence intervals between samples and to show that most, but not all, confidence intervals contain 𝑝
- Coverage of repeated confidence intervals

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Year 12: Mathematics Methods ATAR