ACMMM122
determine \(f\left(x\right),\) given \(f^{'\;}(x)\;\) and an initial condition \(f\left(a\right)=b\)
ACMMM122 | Content Descriptions | Unit 3 | Mathematical Methods | Mathematics | Senior secondary curriculum
ACMMM151
define logarithms as indices: \(a^x=b\) is equivalent to \(x=\log_ab\) i.e. \(a^{\log_ab}=b\)
ACMMM151 | Content Descriptions | Unit 4 | Mathematical Methods | Mathematics | Senior secondary curriculum
ACMMM153
recognise the inverse relationship between logarithms and exponentials: \(y=a^x\) is equivalent to \(x=\log_ay\)
ACMMM153 | Content Descriptions | Unit 4 | Mathematical Methods | Mathematics | Senior secondary curriculum
ACMMM160
recognise and use the inverse relationship of the functions \(y=e^x\) and \(y=\ln x\)
ACMMM160 | Content Descriptions | Unit 4 | Mathematical Methods | Mathematics | Senior secondary curriculum
ACMSM118
establish and use the formula \(\int\frac1xdx=\ln{\;\vert x\;\vert}+c\) for x ≠ 0
ACMSM118 | Content Descriptions | Unit 4 | Specialist Mathematics | Mathematics | Senior secondary curriculum
ACMMM014
recognise features of the graphs of \(y=x^n\) for \(n\in\boldsymbol N,\) \(n=-1\) and \(n=½\), including shape, and behaviour as \(x\rightarrow\infty\) and \(x\rightarrow-\infty\)
ACMMM014 | Content Descriptions | Unit 1 | Mathematical Methods | Mathematics | Senior secondary curriculum
ACMMM083
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 | Content Descriptions | Unit 2 | Mathematical Methods | Mathematics | Senior secondary curriculum
ACMMM124
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 | Content Descriptions | Unit 3 | Mathematical Methods | Mathematics | Senior secondary curriculum
ACMMM125
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 | Content Descriptions | Unit 3 | Mathematical Methods | Mathematics | Senior secondary curriculum
ACMMM126
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 | Content Descriptions | Unit 3 | Mathematical Methods | Mathematics | Senior secondary curriculum
ACMEM122
generate tables of values for linear functions, including for negative values of \(x\)
ACMEM122 | Content Descriptions | Unit 3 | Essential Mathematics | Mathematics | Senior secondary curriculum
ACMEM123
graph linear functions for all values of \(x\) with pencil and paper and with graphing software.
ACMEM123 | Content Descriptions | Unit 3 | Essential Mathematics | Mathematics | Senior secondary curriculum
ACMMM046
expand \(\left(x+y\right)^n\) for small positive integers \(n\)
ACMMM046 | Content Descriptions | Unit 1 | Mathematical Methods | Mathematics | Senior secondary curriculum
ACMMM085
interpret the derivative as the slope or gradient of a tangent line of the graph of \(y=f(x)\)
ACMMM085 | Content Descriptions | Unit 2 | Mathematical Methods | Mathematics | Senior secondary curriculum
ACMMM102
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 | Content Descriptions | Unit 3 | Mathematical Methods | Mathematics | Senior secondary curriculum
ACMMM117
establish and use the formula \(\int e^xdx=e^x+c\)
ACMMM117 | Content Descriptions | Unit 3 | Mathematical Methods | Mathematics | Senior secondary curriculum
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