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ACMMM068

recognise and use the recursive definition of an arithmetic sequence: \(t_{n+1}=t_n+d\)

ACMMM069

use the formula \(t_n=t_1+\left(n-1\right)d\) for the general term of an arithmetic sequence and recognise its linear nature

ACMMM070

use arithmetic sequences in contexts involving discrete linear growth or decay, such as simple interest

ACMMM071

establish and use the formula for the sum of the first \(n\) terms of an arithmetic sequence.

ACMMM072

recognise and use the recursive definition of a geometric sequence:\(t_{n+1}=rt_n\)

ACMMM073

use the formula \(t_n=r^{n-1}t_1\) for the general term of a geometric sequence and recognise its exponential nature

ACMMM074

understand the limiting behaviour as \(n\rightarrow\infty\) of the terms \(t_n\) in a geometric sequence and its dependence on the value of the common ratio \(r\)

ACMMM075

establish and use the formula \(S_n=t_1\frac{r^n-1}{r-1}\) for the sum of the first \(n\) terms of a geometric sequence

ACMMM076

use geometric sequences in contexts involving geometric growth or decay, such as compound interest.

ACMMM077

interpret the difference quotient \(\frac{f\left(x+h\right)-f(x)}h\) as the average rate of change of a function \(f\)

ACMMM078

use the Leibniz notation \(\delta x\) and \(\delta y\) for changes or increments in the variables \(x\) and \(y\)

ACMMM079

use the notation \(\frac{\delta y}{\delta x}\) for the difference quotient \(\frac{f\left(x+h\right)-f(x)}h\) where \(y=f(x)\)

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 \(y=f(x)\)

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

ACMMM082

define the derivative \(f'\left(x\right)\) as \(\lim_{h\rightarrow0}\frac{f\left(x+h\right)-f(x)}h\)

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 \(y=f(x)\)

ACMMM084

interpret the derivative as the instantaneous rate of change

ACMMM085

interpret the derivative as the slope or gradient of a tangent line of the graph of \(y=f(x)\)

ACMMM086

estimate numerically the value of a derivative, for simple power functions

ACMMM087

examine examples of variable rates of change of non-linear functions

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