ACMMM127
interpret \(\int_a^bf\left(x\right)dx\;\) as a sum of signed areas
ACMMM127 | Content Descriptions | Unit 3 | Mathematical Methods | Mathematics | Senior secondary curriculum
ACMMM162
establish and use the formula \(\int\frac1xdx=\ln\;x\;+c\) for \(x>0\)
ACMMM162 | Content Descriptions | Unit 4 | Mathematical Methods | Mathematics | Senior secondary curriculum
ACMSM067
define the imaginary number i as a root of the equation \(x^2=-1\)
ACMSM067 | Content Descriptions | Unit 2 | Specialist Mathematics | Mathematics | Senior secondary curriculum
ACMMM077
interpret the difference quotient \(\frac{f\left(x+h\right)-f(x)}h\) as the average rate of change of a function \(f\)
ACMMM077 | Content Descriptions | Unit 2 | Mathematical Methods | Mathematics | Senior secondary curriculum
ACMMM081
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 | Content Descriptions | Unit 2 | Mathematical Methods | Mathematics | Senior secondary curriculum
ACMMM095
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 | Content Descriptions | Unit 2 | Mathematical Methods | Mathematics | Senior secondary curriculum
ACMSM048
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 …
ACMSM048 | Content Descriptions | Unit 2 | Specialist Mathematics | Mathematics | Senior secondary curriculum
ACMMM020
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 | Content Descriptions | Unit 1 | Mathematical Methods | Mathematics | Senior secondary curriculum
ACMMM107
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 | Content Descriptions | Unit 3 | Mathematical Methods | Mathematics | Senior secondary curriculum
ACMMM130
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 | Content Descriptions | Unit 3 | Mathematical Methods | Mathematics | Senior secondary curriculum
ACMSM049
prove and apply other trigonometric identities such as \(\cos3\mathrm x=4\;\mathrm c\mathrm o\mathrm s^{3\;}\mathrm x-3\cos\mathrm x\)
ACMSM049 | Content Descriptions | Unit 2 | Specialist Mathematics | Mathematics | Senior secondary curriculum
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
ACMSM137 | Content Descriptions | Unit 4 | Specialist Mathematics | Mathematics | Senior secondary curriculum
ACSPH025
Energy transfers and transformations in mechanical systems (for example, internal and external combustion engines, electric motors) always result in some heat loss to the environment, so that the usable energy is reduced and the system cannot be 100 percent …
ACSPH025 | Content Descriptions | Unit 1 | Physics | Science | Senior secondary curriculum
ACMMM013
recognise features of the graphs of \(y=\frac1x\) and \(y=\frac a{x-b}\), including their hyperbolic shapes, and their asymptotes.
ACMMM013 | Content Descriptions | Unit 1 | Mathematical Methods | Mathematics | Senior secondary curriculum
ACMMM021
recognise features of the graph of \(y^2=x\) including its parabolic shape and its axis of symmetry.
ACMMM021 | Content Descriptions | Unit 1 | Mathematical Methods | Mathematics | Senior secondary curriculum
ACMMM026
examine dilations and the graphs of \(y=cf\left(x\right)\) and \(y=f\left(kx\right)\)
ACMMM026 | Content Descriptions | Unit 1 | Mathematical Methods | Mathematics | Senior secondary curriculum
ACMMM115
use the notation \(\int f\left(x\right)dx\) for anti-derivatives or indefinite integrals
ACMMM115 | Content Descriptions | Unit 3 | Mathematical Methods | Mathematics | Senior secondary curriculum
ACMMM129
understand the concept of the signed area function \(F\left(x\right)=\int_a^xf\left(t\right)dt\)
ACMMM129 | Content Descriptions | Unit 3 | Mathematical Methods | Mathematics | Senior secondary curriculum
ACMMM161
establish and use the formula \(\frac d{dx}\left(\ln x\right)=\frac1x\)
ACMMM161 | Content Descriptions | Unit 4 | Mathematical Methods | Mathematics | Senior secondary curriculum
ACMSM121
integrate expressions of the form \(\frac{\pm1}{\sqrt[{}]{a^2-x^2}}\) and \(\frac a{a^2+x^2}\)
ACMSM121 | Content Descriptions | Unit 4 | Specialist Mathematics | Mathematics | Senior secondary curriculum