### 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

For all content areas of Specialist Mathematics, the proficiency strands of the F–10 curriculum are still very much applicable and should be inherent in students’ learning of the subject. The strands of Understanding, Fluency, Problem solving and Reasoning are essential and mutually reinforcing.### 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 Specialist Mathematics

Specialist Mathematics is structured over four units. The topics in Unit 1 broaden students’ mathematical experience and provide different scenarios for incorporating mathematical arguments and problem solving.### Glossary

### Achievement standards

## Unit 4

### Unit 4 Description

Unit 4 of Specialist Mathematics contains three topics: ‘Integration and applications of integration’, ‘Rates of change and differential equations’ and ‘Statistical inference’.

In Unit 4, the study of differentiation and integration of functions continues, and the calculus techniques developed in this and previous topics are applied to simple differential equations, in particular in biology and kinematics. These topics demonstrate the real-world applications of the mathematics learned throughout Specialist Mathematics.

In this unit all of the students’ previous experience working with probability and statistics is drawn together in the study of statistical inference for the distribution of sample means and confidence intervals for sample means.

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

### Unit 4 Learning Outcomes

By the end of this unit, students:

- understand the concepts and techniques in applications of calculus and statistical inference
- apply reasoning skills and solve problems in applications of calculus and statistical inference
- communicate their arguments and strategies when solving problems
- construct proofs of results
- interpret mathematical and statistical information and ascertain the reasonableness of their solutions to problems.

### Unit 4 Content Descriptions

#### Topic 1: Integration and applications of integration

#### Integration techniques:

integrate using the trigonometric identities \(\mathrm s\mathrm i\mathrm n^2x=\frac12(1-\mathrm c\mathrm o\mathrm s\;2x)\), \(\mathrm c\mathrm o\mathrm s^2x=\frac12(1+\mathrm c\mathrm o\mathrm s\;2x)\) and \(1+\;\mathrm t\mathrm a\mathrm n^2x=\mathrm s\mathrm e\mathrm c^2x\) (ACMSM116)

use substitution \(u = g(x)\) to integrate expressions of the form \(f\left(g\left(x\right)\right)g'\left(x\right)\) (ACMSM117)

establish and use the formula \(\int\frac1xdx=\ln{\;\vert x\;\vert}+c\) for x ≠ 0 (ACMSM118)

find and use the inverse trigonometric functions: arcsine, arccosine and arctangent (ACMSM119)

find and use the derivative of the inverse trigonometric functions: arcsine, arccosine and arctangent (ACMSM120)

integrate expressions of the form \(\frac{\pm1}{\sqrt[{}]{a^2-x^2}}\) and \(\frac a{a^2+x^2}\) (ACMSM121)

use partial fractions where necessary for integration in simple cases (ACMSM122)

integrate by parts. (ACMSM123)

#### Applications of integral calculus:

calculate areas between curves determined by functions (ACMSM124)

determine volumes of solids of revolution about either axis (ACMSM125)

use numerical integration using technology \(f\left(t\right)=\lambda e^{-\lambda t}\) for \(t\geq0\) of the exponential random variable with parameter \(\lambda>0,\), and use the exponential random variables and associated probabilities and quantiles to model data and solve practical problems. (ACMSM127)

use numerical integration using technology (ACMSM126)

#### Topic 2: Rates of change and differential equations

use implicit differentiation to determine the gradient of curves whose equations are given in implicit form (ACMSM128)

Related rates as instances of the chain rule: \(\frac{dy}{dx}=\frac{dy}{du}\times\frac{du}{dx}\) (ACMSM129)

solve simple first-order differential equations of the form \(\frac{dy}{dx}=f(x)\), differential equations of the form \(\frac{dy}{dx}=g\left(y\right)\) and, in general, differential equations of the form \(\frac{dy}{dx}=f\left(x\right)g\left(y\right)\) using separation of variables (ACMSM130)

examine slope (direction or gradient) fields of a first order differential equation (ACMSM131)

formulate differential equations including the logistic equation that will arise in, for example, chemistry, biology and economics, in situations where rates are involved. (ACMSM132)

#### Modelling motion:

examine momentum, force, resultant force, action and reaction (ACMSM133)

consider constant and non-constant force (ACMSM134)

understand motion of a body under concurrent forces (ACMSM135)

consider and solve problems involving motion in a straight line with both constant and non-constant acceleration, including simple harmonic motion and the use of expressions \(\frac{dv}{dt}\), \(v\frac{dv}{dx}\) and \(\frac{d(\frac12v^2)}{dx}\) for acceleration. (ACMSM136)

#### Topic 3: Statistical inference

#### Sample means:

examine the concept of the sample mean **X** as a random variable whose value varies between samples where **X** is a random variable with mean **μ** and the standard deviation **σ**
(ACMSM137)

simulate repeated random sampling, from a variety of distributions and a range of sample sizes, to illustrate properties of the distribution of \(\overline X\;\) across samples of a fixed size \(n\), including its mean \(\mu\), its standard deviation \(\sigma/\sqrt[{}]n\) (where \(\mu\) and \(\sigma\) are the mean and standard deviation of X), and its approximate normality if \(n\) is large (ACMSM138)

simulate repeated random sampling, from a variety of distributions and a range of sample sizes, to illustrate the approximate standard normality of \(\frac{\overline X-\mu}{s/\sqrt[{}]n}\) for large samples \(\left(n\geq30\right)\), where \(s\) is the sample standard deviation. (ACMSM139)

#### Confidence intervals for means:

understand the concept of an interval estimate for a parameter associated with a random variable (ACMSM140)

examine the approximate confidence interval \(\left(\overline{\mathrm X}\;–\frac{\mathrm z\mathrm s}{\sqrt[{}]n},\;\;\overline{\mathrm X}+\frac{\mathrm z\mathrm s}{\sqrt[{}]n}\right),\), as an interval estimate for \(\mu\) ,the population mean, where \(z\) is the appropriate quantile for the standard normal distribution (ACMSM141)

use simulation to illustrate variations in confidence intervals between samples and to show that most but not all confidence intervals contain \(\mu\) (ACMSM142)

use \(\overline x\) and \(s\) to estimate \(\mu\) and \(\sigma\), to obtain approximate intervals covering desired proportions of values of a normal random variable and compare with an approximate confidence interval for \(\mu\) (ACMSM143)

collect data and construct an approximate confidence interval to estimate a mean and to report on survey procedures and data quality. (ACMSM144)