# Mathematical Methods

### 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.

## 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)