ACMMM065
recognise the qualitative features of the graph of \(y=a^x(a>0)\) including asymptotes, and of its translations \(y=a^x+b\) and \(y=a^{x+c}\)
ACMMM065 | Content Descriptions | Unit 2 | Mathematical Methods | Mathematics | Senior secondary curriculum
ACMMM082
define the derivative \(f'\left(x\right)\) as \(\lim_{h\rightarrow0}\frac{f\left(x+h\right)-f(x)}h\)
ACMMM082 | Content Descriptions | Unit 2 | Mathematical Methods | Mathematics | Senior secondary curriculum
ACMMM079
use the notation \(\frac{\delta y}{\delta x}\) for the difference quotient \(\frac{f\left(x+h\right)-f(x)}h\) where \(y=f(x)\)
ACMMM079 | Content Descriptions | Unit 2 | Mathematical Methods | Mathematics | Senior secondary curriculum
ACMMM080
interpret the ratios \(\frac{f\left(x+h\right)-f(x)}h\) and \(\frac{\delta y}{\delta x}\) as the slope or gradient of a chord or secant of the graph of \(y=f(x)\)
ACMMM080 | Content Descriptions | Unit 2 | Mathematical Methods | Mathematics | Senior secondary curriculum
ACMMM088
establish the formula \(\frac d{dx}\left(x^n\right)=nx^{n-1}\) for positive integers \(n\) by expanding \({(x+h)}^n\) or by factorising \({(x+h)}^n-x^n\)
ACMMM088 | Content Descriptions | Unit 2 | Mathematical Methods | Mathematics | Senior secondary curriculum
ACMMM078
use the Leibniz notation \(\delta x\) and \(\delta y\) for changes or increments in the variables \(x\) and \(y\)
ACMMM078 | Content Descriptions | Unit 2 | 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
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
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