11–12Mathematics Advanced 11–12 Syllabus

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Year 11

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Introduction to differentiation

MAO-WM-01
develops understanding and fluency in mathematics through exploring and connecting mathematical concepts, choosing and applying mathematical techniques to solve problems, and communicating their thinking and reasoning coherently and clearly
MAV-11-06
interprets the meaning of the derivative and determines the derivative of functions to solve problems

Introduction to differentiation

Estimating change

Define the average rate of change of $y$ with respect to $x$ for a function $y = f (x)$ over the domain $[a, b]$ as $\frac{Δ y}{Δ x} = \frac{change in y}{change in x}$ , that is $\frac{Δ y}{Δ x} = \frac{f (b) - f (a)}{b - a}$ , and recognise $\frac{f (b) - f (a)}{b - a}$ as the gradient of the secant through $(a, f (a))$ and $(b, f (b))$ on the graph of $y = f (x)$
Recognise speed as a rate of change of distance with respect to time
Use the definition for average rate of change to determine the average speed of an object from a given distance–time graph
Describe the difference between the average speed of an object and its instantaneous speed
Determine that the instantaneous speed of an object at time $t$ can be approximated by the average speed between its position at time $t$ and its position some time later, and explain how this approximation can be improved
Relate the instantaneous speed of an object to the gradient of the tangent at that point on its distance–time graph
Estimate the instantaneous speed of an object from its distance–time graph
Recognise when modelling with a linear function that its gradient is the rate of change and determine the rate of change for linear functions in practical situations
Recognise when modelling with a non-linear function that the rate of change is not constant and is represented by the gradient of the tangent to the curve at each point on the curve
Estimate the instantaneous rate of change of a non-linear function at a given point from a given graph of a practical situation

The derivative

Examine the gradient of a curve at a point on the curve using graphing applications
Approximate the gradient of a curve $f (x) = x^{n}$ at a point $P (c, f (c))$ by considering the gradient of the secant through $P$ and $Q (c + h, f (c + h))$ as the magnitude of $h$ approaches zero, using graphing applications or a spreadsheet
Infer that $n x^{n - 1}$ is the gradient of $f (x) = x^{n}$ and verify the result using graphing applications
Define $f' (x)$ , for any function $f (x)$ and any value $x$ , to be the gradient of the tangent to the curve $y = f (x)$ at the point $P (x, f (x))$ if the tangent exists and is not vertical
Refer to $f' (x)$ as the derivative of $f (x)$ or the gradient, or derived, function of $f (x)$
Define differentiation as the process of finding the derivative of a function
Find derivatives of constant and linear functions
Define the derivative of the function $f (x)$ from first principles, as the limiting value of the gradient of the secant $\frac{f (x + h) - f (x)}{h}$ as $h$ approaches zero, when this limiting value exists, and use the notation $f' (x) = \underset{h \to 0}{l i m} \frac{f (x + h) - f (x)}{h}$
Use first principles to find the derivative of quadratic functions

Calculations with the derivative

Use the notation $\frac{dy}{dx}$ and $y'$ for the derivative of $y$ when $y$ is a function of $x$
Use the notation $\frac{d}{dx} (f (x))$ and $f' (x)$ for the derivative of a function $f (x)$
Use the formula $\frac{d}{dx} (x^{n}) = n x^{n - 1}$ for all real values of $n$
Apply the fact that the derivative of a sum is the sum of the derivatives: $\frac{d}{dx} (f (x) + g (x)) = f' (x) + g' (x)$ , and the derivative of a multiple of a function is the multiple of its derivative: $\frac{d}{dx} (k \cdot f (x)) = k \cdot f' (x)$
Use the rules for differentiation to find equations of tangents and normals to a curve at points on the curve
Find points on a curve where the tangent or normal has a given gradient
Examine and use the relationship between the angle of inclination of a line or tangent to a curve, $θ,$ with the positive $x$ -axis, and the gradient, $m$ , of that line or tangent, and establish that $\tan θ = m$
Apply the product rule: if $y = uv$ , where $u$ and $v$ are both differentiable functions of $x$ , then $\frac{dy}{dx} = u \frac{dv}{dx} + v \frac{du}{dx}$ or if $h (x) = f (x) g (x)$ for differentiable functions $f (x)$ and $g (x)$ then $h' (x) = f (x) g' (x) + f' (x) g (x)$
Apply the quotient rule: if $y = \frac{u}{v}$ , where $u$ and $v$ are both functions of $x,$ then $\frac{dy}{dx} = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^{2}}$ or if $h (x) = \frac{f (x)}{g (x)}$ then $h' (x) = \frac{g (x) f^{'} (x) - f (x) g^{'} (x)}{[{g (x)]}^{2}}$
Apply the chain rule: if $y$ is a differentiable function of $u$ , and $u$ is a differentiable function of $x$ , then $\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$ or if $h (x) = f (g (x))$ for differentiable functions $f (x)$ and $g (x)$ then $h' (x) = f' (g (x)) g' (x)$
Identify and apply the product, quotient or chain rule, or a combination of the rules, as appropriate to differentiate a given function

Graphical applications of the derivative

Interpret $f (x)$ as increasing at $x = c$ when $f' (c) > 0$ and decreasing at $x = c$ when $f' (c) < 0$
Describe the behaviour of a function at a point as stationary when the tangent at the point is parallel to the $x$ -axis, and recognise that $f (x)$ is stationary at $x = c$ when $f' (c) = 0$
Graph $y = f^{'} (x)$ for a given graph of a function $y = f (x)$
Numerically estimate the value of the derivative at a point on the graph of a power of $x$ , with and without the use of digital tools
Identify stationary points on the graph of a cubic function, and the values of $x$ for which the function is increasing and/or decreasing, by first calculating the derivative, and justify conclusions

The derivative as a rate of change

Interpret $f' (c)$ as the instantaneous rate of change of the function $f (x)$ at $x = c$
Define and distinguish between displacement and distance and between velocity and speed
Use graphs of functions and their derivatives, without the use of algebraic techniques, to describe and interpret physical phenomena
Use the notation $\frac{dx}{dt}$ or $\dot{x}$ to represent the velocity of a particle with displacement $x$ from a point as a function of time $t$
Solve problems by determining the velocity of a particle moving in a straight line, given its displacement from a point as a function of time

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Year 12

Welcome to the NSW Curriculum website

11–12Mathematics Advanced 11–12 Syllabus

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