# 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 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)