# Mathematical Methods (Version 8.4)

### Rationale

Mathematics is the study of order, relation and pattern. From its origins in counting and measuring it has evolved in highly sophisticated and elegant ways to become the language now used to describe much of the modern world.

### Links to Foundation to Year 10

In Mathematical Methods, there is a strong emphasis on mutually reinforcing proficiencies in Understanding, Fluency, Problem solving and Reasoning.

### Representation of General capabilities

The seven general capabilities of Literacy, Numeracy, Information and Communication technology (ICT) capability, Critical and creative thinking, Personal and social capability, Ethical understanding, and Intercultural understanding are identified where they offer opportunities to add depth and richness to student learning.

### Structure of Mathematical Methods

Mathematical Methods is organised into four units. The topics broaden students’ mathematical experience and provide different scenarios for incorporating mathematical arguments and problem solving. The units provide a blending of algebraic and geometric thinking.

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## Unit 1

### Unit 1 Description

This unit begins with a review of the basic algebraic concepts and techniques required for a successful introduction to the study of calculus. The basic trigonometric functions are then introduced. Simple relationships between variable quantities are reviewed, and these are used to introduce the key concepts of a function and its graph. The study of inferential statistics begins in this unit with a review of the fundamentals of probability and the introduction of the concepts of conditional probability and independence. Access to technology to support the computational aspects of these topics is assumed.

### Unit 1 Learning Outcomes

By the end of this unit, students:

• understand the concepts and techniques in algebra, functions, graphs, trigonometric functions and probability
• solve problems using algebra, functions, graphs, trigonometric functions and probability
• apply reasoning skills in the context of algebra, functions, graphs, trigonometric functions and probability
• interpret and evaluate mathematical information and ascertain the reasonableness of solutions to problems
• communicate their arguments and strategies when solving problems.

### Unit 1 Content Descriptions

#### Lines and linear relationships:

determine the coordinates of the midpoint of two points (ACMMM001)

examine examples of direct proportion and linearly related variables (ACMMM002)

recognise features of the graph of $$y=mx+c$$, including its linear nature, its intercepts and its slope or gradient (ACMMM003)

find the equation of a straight line given sufficient information; parallel and perpendicular lines (ACMMM004)

solve linear equations. (ACMMM005)

examine examples of quadratically related variables (ACMMM006)

recognise features of the graphs of $$y=x^2$$, $$y=a{(x-b)}^2+c$$, and $$y=a\left(x-b\right)\left(x-c\right)$$ including their parabolic nature, turning points, axes of symmetry and intercepts (ACMMM007)

solve quadratic equations using the quadratic formula and by completing the square (ACMMM008)

find the equation of a quadratic given sufficient information (ACMMM009)

find turning points and zeros of quadratics and understand the role of the discriminant (ACMMM010)

recognise features of the graph of the general quadratic $$y=ax^2+bx+c$$ (ACMMM011)

#### Inverse proportion:

examine examples of inverse proportion (ACMMM012)

recognise features of the graphs of $$y=\frac1x$$ and $$y=\frac a{x-b}$$, including their hyperbolic shapes, and their asymptotes. (ACMMM013)

#### Powers and polynomials:

recognise features of the graphs of $$y=x^n$$ for $$n\in\boldsymbol N,$$ $$n=-1$$ and $$n=½$$, including shape, and behaviour as $$x\rightarrow\infty$$ and $$x\rightarrow-\infty$$ (ACMMM014)

identify the coefficients and the degree of a polynomial (ACMMM015)

expand quadratic and cubic polynomials from factors (ACMMM016)

recognise features of the graphs of $$y=x^3$$, $$y=a{(x-b)}^3+c$$ and $$y=k(x-a)(x-b)(x-c)$$, including shape, intercepts and behaviour as $$x\rightarrow\infty$$ and $$x\rightarrow-\infty$$ (ACMMM017)

factorise cubic polynomials in cases where a linear factor is easily obtained (ACMMM018)

solve cubic equations using technology, and algebraically in cases where a linear factor is easily obtained. (ACMMM019)

#### Graphs of relations:

recognise features of the graphs of $$x^2+y^2=r^2$$ and $$\left(x-a\right)^2+\left(y-b\right)^2=r^2$$, including their circular shapes, their centres and their radii (ACMMM020)

recognise features of the graph of $$y^2=x$$ including its parabolic shape and its axis of symmetry. (ACMMM021)

#### Functions:

understand the concept of a function as a mapping between sets, and as a rule or a formula that defines one variable quantity in terms of another (ACMMM022)

use function notation, domain and range, independent and dependent variables (ACMMM023)

understand the concept of the graph of a function (ACMMM024)

examine translations and the graphs of $$y=f\left(x\right)+a$$ and $$y=f(x+b)$$ (ACMMM025)

examine dilations and the graphs of $$y=cf\left(x\right)$$ and $$y=f\left(kx\right)$$ (ACMMM026)

recognise the distinction between functions and relations, and the vertical line test. (ACMMM027)

#### Cosine and sine rules:

review sine, cosine and tangent as ratios of side lengths in right-angled triangles (ACMMM028)

understand the unit circle definition of $$\cos\theta,\;\sin\theta$$ and $$\tan\theta$$ and periodicity using degrees (ACMMM029)

examine the relationship between the angle of inclination of a line and the gradient of that line (ACMMM030)

establish and use the sine and cosine rules and the formula $$Area=\frac12bc\sin A$$ for the area of a triangle. (ACMMM031)

#### Circular measure and radian measure:

define and use radian measure and understand its relationship with degree measure (ACMMM032)

calculate lengths of arcs and areas of sectors in circles. (ACMMM033)

#### Trigonometric functions:

understand the unit circle definition of $$\cos\theta,\;\sin\theta$$ and $$\tan\theta$$ and periodicity using radians (ACMMM034)

recognise the exact values of $$\cos\theta,\;\sin\theta$$ and $$\tan\theta$$ at integer multiples of $$\frac\pi6$$ and $$\frac\pi4$$ (ACMMM035)

recognise the graphs of $$y=\sin x,\;y=\cos x,$$ and $$y=\tan x$$ on extended domains (ACMMM036)

examine amplitude changes and the graphs of $$y=a\sin x$$ and $$y=a\cos x$$ (ACMMM037)

examine period changes and the graphs of $$y=\sin bx,\;$$, $$y=\cos bx$$, and $$y=\tan bx$$ (ACMMM038)

examine phase changes and the graphs of $$y=\sin{(x+c)}$$, $$y=\cos{(x+c)}$$ and $$y=\tan{(x+c)}$$ and the relationships $$\sin\left(x+\frac\pi2\right)=\cos x$$ and $$\cos\left(x-\frac\pi2\right)=\sin x$$ (ACMMM039)

prove and apply the angle sum and difference identities (ACMMM041)

identify contexts suitable for modelling by trigonometric functions and use them to solve practical problems (ACMMM042)

solve equations involving trigonometric functions using technology, and algebraically in simple cases. (ACMMM043)

#### Combinations:

understand the notion of a combination as an unordered set of $$r$$ objects taken from a set of $$n$$ distinct objects (ACMMM044)

use the notation $$\begin{pmatrix}n\\r\end{pmatrix}$$ and the formula $$\begin{pmatrix}n\\r\end{pmatrix}=\frac{n!}{r!\left(n-r\right)!}$$ for the number of combinations of $$r$$ objects taken from a set of $$n$$ distinct objects (ACMMM045)

expand $$\left(x+y\right)^n$$ for small positive integers $$n$$ (ACMMM046)

recognise the numbers $$\begin{pmatrix}n\\r\end{pmatrix}$$ as binomial coefficients, (as coefficients in the expansion of $$\left(x+y\right)^n)$$ (ACMMM047)

use Pascal’s triangle and its properties. (ACMMM048)

#### Language of events and sets:

review the concepts and language of outcomes, sample spaces and events as sets of outcomes (ACMMM049)

use set language and notation for events, including $$\overline A$$ (or $$A'$$) for the complement of an event $$A,$$ $$A?B$$ for the intersection of events $$A$$ and $$B$$, and $$A?B$$ for the union, and recognise mutually exclusive events (ACMMM050)

use everyday occurrences to illustrate set descriptions and representations of events, and set operations. (ACMMM051)

#### Review of the fundamentals of probability:

review probability as a measure of ‘the likelihood of occurrence’ of an event (ACMMM052)

review the probability scale: $$0\leq P(A)\leq1$$ for each event $$A,$$ with $$P\left(A\right)=0$$ if $$A$$ is an impossibility and $$P\left(A\right)=1$$ if $$A$$ is a certaint (ACMMM053)

review the rules: $$P\left(\overline A\right)=1-P\left(A\right)$$ and $$P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)-P\left(A\cap B\right)$$ (ACMMM054)

use relative frequencies obtained from data as point estimates of probabilities. (ACMMM055)

#### Conditional probability and independence:

understand the notion of a conditional probability and recognise and use language that indicates conditionality (ACMMM056)

use the notation $$P(A\vert B)$$ and the formula $$P(A\vert B)=P(A\cap B) / P(B)$$ (ACMMM057)

understand the notion of independence of an event $$A$$ from an event $$B$$, as defined by $$P(A\vert B)=P(A)$$ (ACMMM058)

establish and use the formula $$P(A\cap B)=P(A)P(B)$$ for independent events $$A$$ and $$B$$, and recognise the symmetry of independence (ACMMM059)

use relative frequencies obtained from data as point estimates of conditional probabilities and as indications of possible independence of events. (ACMMM060)

## Unit 2

### Unit 2 Description

The algebra section of this unit focuses on exponentials and logarithms. Their graphs are examined and their applications in a wide range of settings are explored. Arithmetic and geometric sequences are introduced and their applications are studied. Rates and average rates of change are introduced, and this is followed by the key concept of the derivative as an ‘instantaneous rate of change’. These concepts are reinforced numerically, by calculating difference quotients both geometrically, as slopes of chords and tangents, and algebraically. Calculus is developed to study the derivatives of polynomial functions, with simple applications of the derivative to curve sketching, calculating slopes and equations of tangents, determining instantaneous velocities and solving optimisation problems.

Access to technology to support the computational aspects of these topics is assumed.

### Unit 2 Learning Outcomes

By the end of this unit, students:

• understand the concepts and techniques used in algebra, sequences and series, functions, graphs and calculus
• solve problems in algebra, sequences and series, functions, graphs and calculus
• apply reasoning skills in algebra, sequences and series, functions, graphs and calculus
• interpret and evaluate mathematical and statistical information and ascertain the reasonableness of solutions to problems
• communicate arguments and strategies when solving problems.

### Unit 2 Content Descriptions

#### Indices and the index laws:

review indices (including fractional indices) and the index laws (ACMMM061)

use radicals and convert to and from fractional indices (ACMMM062)

understand and use scientific notation and significant figures. (ACMMM063)

#### Exponential functions:

establish and use the algebraic properties of exponential functions (ACMMM064)

recognise the qualitative features of the graph of $$y=a^x(a>0)$$ including asymptotes, and of its translations $$y=a^x+b$$ and $$y=a^{x+c}$$ (ACMMM065)

identify contexts suitable for modelling by exponential functions and use them to solve practical problems (ACMMM066)

solve equations involving exponential functions using technology, and algebraically in simple cases. (ACMMM067)

#### Arithmetic sequences:

recognise and use the recursive definition of an arithmetic sequence: $$t_{n+1}=t_n+d$$ (ACMMM068)

use the formula $$t_n=t_1+\left(n-1\right)d$$ for the general term of an arithmetic sequence and recognise its linear nature (ACMMM069)

use arithmetic sequences in contexts involving discrete linear growth or decay, such as simple interest (ACMMM070)

establish and use the formula for the sum of the first $$n$$ terms of an arithmetic sequence. (ACMMM071)

#### Geometric sequences:

recognise and use the recursive definition of a geometric sequence:$$t_{n+1}=rt_n$$ (ACMMM072)

use the formula $$t_n=r^{n-1}t_1$$ for the general term of a geometric sequence and recognise its exponential nature (ACMMM073)

understand the limiting behaviour as $$n\rightarrow\infty$$ of the terms $$t_n$$ in a geometric sequence and its dependence on the value of the common ratio $$r$$ (ACMMM074)

establish and use the formula $$S_n=t_1\frac{r^n-1}{r-1}$$ for the sum of the first $$n$$ terms of a geometric sequence (ACMMM075)

use geometric sequences in contexts involving geometric growth or decay, such as compound interest. (ACMMM076)

#### Rates of change:

interpret the difference quotient $$\frac{f\left(x+h\right)-f(x)}h$$ as the average rate of change of a function $$f$$ (ACMMM077)

use the Leibniz notation $$\delta x$$ and $$\delta y$$ for changes or increments in the variables $$x$$ and $$y$$ (ACMMM078)

use the notation $$\frac{\delta y}{\delta x}$$ for the difference quotient $$\frac{f\left(x+h\right)-f(x)}h$$ where $$y=f(x)$$ (ACMMM079)

interpret the ratios $$\frac{f\left(x+h\right)-f(x)}h$$ and $$\frac{\delta y}{\delta x}$$ as the slope or gradient of a chord or secant of the graph of $$y=f(x)$$ (ACMMM080)

#### The concept of the derivative:

examine the behaviour of the difference quotient $$\frac{f\left(x+h\right)-f(x)}h$$ as $$h\rightarrow0$$ as an informal introduction to the concept of a limit (ACMMM081)

define the derivative $$f'\left(x\right)$$ as $$\lim_{h\rightarrow0}\frac{f\left(x+h\right)-f(x)}h$$ (ACMMM082)

use the Leibniz notation for the derivative: $$\frac{dy}{dx}=\lim_{\mathit{δx}\rightarrow0}\frac{\delta y}{\delta x}$$ and the correspondence $$\frac{dy}{dx}=f'\left(x\right)$$ where $$y=f(x)$$ (ACMMM083)

interpret the derivative as the instantaneous rate of change (ACMMM084)

interpret the derivative as the slope or gradient of a tangent line of the graph of $$y=f(x)$$ (ACMMM085)

#### Computation of derivatives:

estimate numerically the value of a derivative, for simple power functions (ACMMM086)

examine examples of variable rates of change of non-linear functions (ACMMM087)

establish the formula $$\frac d{dx}\left(x^n\right)=nx^{n-1}$$ for positive integers $$n$$ by expanding $${(x+h)}^n$$ or by factorising $${(x+h)}^n-x^n$$ (ACMMM088)

#### Properties of derivatives:

understand the concept of the derivative as a function (ACMMM089)

recognise and use linearity properties of the derivative (ACMMM090)

calculate derivatives of polynomials and other linear combinations of power functions. (ACMMM091)

#### Applications of derivatives:

find instantaneous rates of change (ACMMM092)

find the slope of a tangent and the equation of the tangent (ACMMM093)

construct and interpret position-time graphs, with velocity as the slope of the tangent (ACMMM094)

sketch curves associated with simple polynomials; find stationary points, and local and global maxima and minima; and examine behaviour as $$x\rightarrow\infty$$ and $$x\rightarrow-\infty$$ (ACMMM095)

solve optimisation problems arising in a variety of contexts involving simple polynomials on finite interval domains. (ACMMM096)

#### Anti-derivatives:

calculate anti-derivatives of polynomial functions and apply to solving simple problems involving motion in a straight line. (ACMMM097)

## Unit 3

### Unit 3 Description

In this unit the study of calculus continues with the derivatives of exponential and trigonometric functions and their applications, together with some differentiation techniques and applications to optimisation problems and graph sketching. It concludes with integration, both as a process that reverses differentiation and as a way of calculating areas. The fundamental theorem of calculus as a link between differentiation and integration is emphasised. In statistics, discrete random variables are introduced, together with their uses in modelling random processes involving chance and variation. This supports the development of a framework for statistical inference.

Access to technology to support the computational aspects of these topics is assumed.

### Unit 3 Learning Outcomes

By the end of this unit, students:

• understand the concepts and techniques in calculus, probability and statistics
• solve problems in calculus, probability and statistics
• apply reasoning skills in calculus, probability and statistics
• interpret and evaluate mathematical and statistical information and ascertain the reasonableness of solutions to problems.
• communicate their arguments and strategies when solving problems.

### Unit 3 Content Descriptions

#### Exponential functions:

estimate the limit of $$\frac{a^h-1}h$$ as $$h\rightarrow0$$ using technology, for various values of $$a\;>0$$ (ACMMM098)

recognise that $$e$$ is the unique number $$a$$ for which the above limit is 1 (ACMMM099)

establish and use the formula $$\frac d{dx}\left(e^x\right)=e^x$$ (ACMMM100)

use exponential functions and their derivatives to solve practical problems.  (ACMMM101)

#### Trigonometric functions:

establish the formulas $$\frac d{dx}\left(\sin x\right)=\cos x,\;\text{ and }\frac d{dx}\left(\cos x\right)=-\sin x$$ by numerical estimations of the limits and informal proofs based on geometric constructions (ACMMM102)

use trigonometric functions and their derivatives to solve practical problems.  (ACMMM103)

#### Differentiation rules:

understand and use the product and quotient rules  (ACMMM104)

understand the notion of composition of functions and use the chain rule for determining the derivatives of composite functions (ACMMM105)

apply the product, quotient and chain rule to differentiate functions such as $$xe^x$$, $$\tan x,$$, $$\frac1{x^n}$$, $$x\sin x,\text{ }e^{-x}\sin x$$ and $$f(ax+b)$$ (ACMMM106)

#### The second derivative and applications of differentiation:

use the increments formula: $$\delta y\cong\frac{dy}{dx}\times\delta x$$ to estimate the change in the dependent variable $$y$$ resulting from changes in the independent variable $$x$$ (ACMMM107)

understand the concept of the second derivative as the rate of change of the first derivative function (ACMMM108)

recognise acceleration as the second derivative of position with respect to time (ACMMM109)

understand the concepts of concavity and points of inflection and their relationship with the second derivative (ACMMM110)

understand and use the second derivative test for finding local maxima and minima (ACMMM111)

sketch the graph of a function using first and second derivatives to locate stationary points and points of inflection (ACMMM112)

solve optimisation problems from a wide variety of fields using first and second derivatives. (ACMMM113)

#### Anti-differentiation:

recognise anti-differentiation as the reverse of differentiation (ACMMM114)

use the notation $$\int f\left(x\right)dx$$ for anti-derivatives or indefinite integrals (ACMMM115)

establish and use the formula $$\int x^ndx=\frac1{n+1}x^{n+1}+c$$ for $$n\neq-1$$ (ACMMM116)

establish and use the formula  $$\int e^xdx=e^x+c$$ (ACMMM117)

establish and use the formulas, $$\int\sin xdx=-\cos x+c$$ and $$\int\cos xdx=\sin x+c$$ (ACMMM118)

recognise and use linearity of anti-differentiation (ACMMM119)

determine indefinite integrals of the form $$\int f\left(ax+b\right)dx$$ (ACMMM120)

identify families of curves with the same derivative function (ACMMM121)

determine $$f\left(x\right),$$ given $$f^{'\;}(x)\;$$ and an initial condition $$f\left(a\right)=b$$ (ACMMM122)

determine displacement given velocity in linear motion problems.  (ACMMM123)

#### Definite integrals:

examine the area problem, and use sums of the form $$\sum\nolimits_if\left(x_i\right)\;\delta x_i$$ as area under the curve $$y=f(x)$$ (ACMMM124)

interpret the definite integral $$\int_a^bf\left(x\right)dx\;$$ as area under the curve $$y=f\left(x\right)$$ if $$f\left(x\right)>0\;$$ (ACMMM125)

recognise the definite integral $$\int_a^bf\left(x\right)dx\;\;$$ as a limit of sums of the form $$\sum\nolimits_if\left(x_i\right)\;\delta x_i$$ (ACMMM126)

interpret $$\int_a^bf\left(x\right)dx\;$$ as a sum of signed areas (ACMMM127)

recognise and use the additivity and linearity of definite integrals.  (ACMMM128)

#### Fundamental theorem:

understand the concept of the signed area function $$F\left(x\right)=\int_a^xf\left(t\right)dt$$ (ACMMM129)

understand and use the theorem $$F'\left(x\right)=\frac d{dx}\left(\int_a^xf\left(t\right)dt\right)=f\left(x\right)$$, and illustrate its proof geometrically (ACMMM130)

understand the formula $$\int_a^b{f\left(x\right)dx=F\left(b\right)-F(a)}$$ and use it to calculate definite integrals. (ACMMM131)

#### Applications of integration:

calculate the area under a curve (ACMMM132)

calculate total change by integrating instantaneous or marginal rate of change (ACMMM133)

calculate the area between curves in simple cases (ACMMM134)

determine positions given acceleration and initial values of position and velocity (ACMMM135)

#### General discrete random variables:

understand the concepts of a discrete random variable and its associated probability function, and their use in modelling data  (ACMMM136)

use relative frequencies obtained from data to obtain point estimates of probabilities associated with a discrete random variable (ACMMM137)

recognise uniform discrete random variables and use them to model random phenomena with equally likely outcomes (ACMMM138)

examine simple examples of non-uniform discrete random variables (ACMMM139)

recognise the mean or expected value of a discrete random variable as a measurement of centre, and evaluate it in simple cases (ACMMM140)

recognise the variance and standard deviation of a discrete random variable as a measures of spread, and evaluate them in simple cases (ACMMM141)

use discrete random variables and associated probabilities to solve practical problems.  (ACMMM142)

#### Bernoulli distributions:

use a Bernoulli random variable as a model for two-outcome situations (ACMMM143)

identify contexts suitable for modelling by Bernoulli random variables (ACMMM144)

recognise the mean $$p$$ and variance $$p(1-p)$$ of the Bernoulli distribution with parameter $$p$$ (ACMMM145)

use Bernoulli random variables and associated probabilities to model data and solve practical problems.  (ACMMM146)

#### Binomial distributions:

understand the concepts 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 (ACMMM147)

identify contexts suitable for modelling by binomial random variables (ACMMM148)

determine and use the probabilities $$\mathrm P\left(\mathrm X=\mathrm r\right)=\begin{pmatrix}\mathrm n\\\mathrm r\end{pmatrix}\mathrm p^\mathrm r{(1-\mathrm p)}^{\mathrm n-\mathrm r}$$ associated with the binomial distribution with parameters $$n$$ and $$p$$ ; note the mean $$np$$ and variance $$np(1-p)$$ of a binomial distribution (ACMMM149)

use binomial distributions and associated probabilities to solve practical problems. (ACMMM150)