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ACMSM098

use and apply the notation \(\left|x\right|\) for the absolute value for the real number \(x\) and the graph of \(y=\left|x\right|\)

ACMSM117

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

ACMSM099

examine the relationship between the graph of \(y=f(x)\) and the graphs of \(y=\frac1{f(x)}\), \(y=\vert f\left(x\right)\vert\) and \(y=f(\left|x\right|)\)

ACMSM118

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

ACMSM067

define the imaginary number i as a root of the equation \(x^2=-1\)

convert sums \(\mathrm a\cos\mathrm x+\mathrm b\;\sin\mathrm x\) to \(\mathrm R\;\cos{(\mathrm x\pm\mathrm\alpha)}\) or \(\mathrm R\sin{(\mathrm x\pm\mathrm\alpha)}\) and apply these to sketch graphs, solve equations of the form \(\mathrm a\cos\mathrm …

ACMSM049

prove and apply other trigonometric identities such as \(\cos3\mathrm x=4\;\mathrm c\mathrm o\mathrm s^{3\;}\mathrm x-3\cos\mathrm x\)

ACMSM137

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

ACMSM121

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

ACMSM055

define and use basic linear transformations: dilations of the form \((\mathrm x,\mathrm y)\longrightarrow({\mathrm\lambda}_1\mathrm x,{\mathrm\lambda}_2\mathrm y)\) , rotations about the origin and reflection in a line which passes through the origin, …

ACMSM130

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 \(\frac{dy}{dx}=f\left(x\right)g\left(y\right)\) …

ACMSM141

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 …

ACMSM053

calculate the determinant and inverse of 2x2 matrices and solve matrix equations of the form AX=B , where A is a 2x2 matrix and X and B are column vectors.

ACMSM042

find all solutions of \(\mathrm f\left(\mathrm a\left(\mathrm x-\mathrm b\right)\right)=\mathrm c\) where \(f\) is one of \(\sin\), \(\cos\) or \(\tan\)

ACMSM043

graph functions with rules of the form \(\mathrm y=\mathrm f(\mathrm a\left(\mathrm x-\mathrm b\right))\) where \(f\) is one of \(\sin\), \(\cos\) or \(\tan\)

ACMSM143

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 \(\mu\)

ACMSM138

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 …

ACMSM139

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 …

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