11–12Mathematics Advanced 11–12 Syllabus (2024)

Record of changes

Implementation from 2026

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Overview Rationale Aim Outcomes Content Assessment Glossary Teaching and learning support

Content

Year 11

Year 12

Teaching advice

Examples

Integral calculus

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-12-05
solves problems involving indefinite and definite integrals

Integral calculus

Primitive functions

Define a primitive of a function $f (x)$ as a function $F (x)$ whose derivative $F' (x) = f (x)$ and recognise the process of finding the primitive as the reverse of differentiation
Recognise that a function whose derivative is everywhere Loading is a Loading
Prove by differentiation that a primitive of $f (x) = x^{n}$ is $F (x) = \frac{x^{n + 1}}{n + 1}$ , for all real $n \neq - 1$
Prove by differentiation that if $F (x)$ and $G (x)$ are primitives of $f (x)$ and $g (x)$ , and $k$ is a constant, then $F (x) + G (x)$ is a primitive of $f (x) + g (x)$ , and $kF (x)$ is a primitive of $kf (x)$
Recognise that primitives of a function $f (x)$ are not unique, and that any two primitives of $f (x)$ differ by a constant, so that if $F (x)$ is a primitive of $f (x)$ , the general primitive of $f (x)$ is $F (x) + C$ , for some constant $C$
Determine the primitive of a given function $f (x)$ , where $f (x)$ is a sum of functions of the form $k x^{n}$ for all real $n \neq - 1$
Determine the primitive function for functions of the form $f (x) = {(ax + b)}^{n}$ , for all real $n \neq - 1$ , where $a$ and $b$ are constants
Use algebraic manipulation to express given functions in forms suitable for determining primitive functions
Determine $f (x)$ , given $f' (x)$ and an initial condition $f (a) = b$ where $a$ and $b$ are constants

The definite integral

Examine for a function $f (x)$ , which indicates the rate of change of a quantity, the meaning of $\sum_{a}^{b} f (x) Δ x$ , where the interval $a \leq x \leq b$ is divided into subintervals of length $Δ x$ , and describe $\sum_{a}^{b} f (x) Δ x$ as an estimate of the total change in that quantity over the interval $a \leq x \leq b$
Consider the definite integral as $\int_{a}^{b} f (x) d x = \lim_{Δ x \to 0} \sum_{a}^{b} f (x) Δ x$ , noting that this implies that the result of a definite integral will be negative when $f (x) \leq 0$ throughout the interval $a \leq x \leq b$
Define informally that a function is continuous on the interval $a \leq x \leq b$ if it can be drawn between the two endpoints of the interval without taking the pen off the paper
Graph the region between the continuous function $y = f (x)$ and the $x$ -axis, where $f (x) \geq 0$ on the interval $a \leq x \leq b$
Use a graphing application to compare different methods of approximating the area, $A$ , of the region between the continuous function $y = f (x)$ and the $x$ -axis, where $f (x) \geq 0$ on the interval $a \leq x \leq b$ , by summing the areas of trapezia or rectangles each of width $Δ x = \frac{b - a}{n}$ and approximate height $f (x)$ for any $x$ lying in its base, and observe the effect on the precision of the approximation of $A$ as the number $n$ of subintervals of $a \leq x \leq b$ increases, that is as $Δ x \to 0$
Evaluate the definite integral $\int_{a}^{b} f (x) dx$ by calculating areas using geometrical formulas, where the shape of $f (x)$ allows such calculations, in cases where $f (x) \geq 0$ throughout $a \leq x \leq b$ , $f (x) \leq 0$ throughout $a \leq x \leq b$ or where $f (x)$ changes sign in the interval $a \leq x \leq b$

The Fundamental Theorem of Calculus

Consider the function defined by $A (x) = \int_{a}^{x} f (t) d t$ and use a graphing application to recognise that $A (x)$ is a primitive of $f (x)$
Recognise the Fundamental Theorem of Calculus as $\int_{a}^{b} f (x) dx = (F (x)] \binom{b}{a} = F (b) - F (a)$ for a continuous function $f$ on the interval $a \leq x \leq b$ where $F (x)$ is any primitive of $f (x)$

Indefinite integrals

Use the notation $\int f (x) d x$ for the general primitive of $f (x),$ called the indefinite integral of $f (x)$ , so that $\int f (x) d x = F (x) + C$ , for some constant $C$ , where $F (x)$ is any primitive of $f (x)$
Recognise Loading as the process of finding the indefinite integral of a function
Use the formula $\int x^{n} dx = \frac{1}{n + 1} x^{n + 1} + C$ for real $n \neq - 1$
Use the identities $\int (f (x) + g (x)) d x = \int f (x) d x + \int g (x) d x$ and $\int k f (x) d x = k \int f (x) d x$ for primitives
Prove by differentiation, and apply $\int u^{n} \frac{du}{dx} dx = \frac{1}{n + 1} u^{n + 1} + C$ , where $u$ is a function of $x$ , or $\int f' (x) {(f (x)]}^{n} dx = \frac{1}{n + 1} {(f (x)]}^{n + 1} + C$ , for real $n \neq - 1$

Integration with exponential functions

Establish and use the formula $\int e^{x} dx = e^{x} + C$
Establish and use the formula $\int e^{ax + b} dx = \frac{1}{a} e^{ax + b} + C$ , where $a$ and $b$ are constants and $a \neq 0$
Establish and use the formula $\int a^{x} dx = \frac{a^{x}}{\ln a} + C$ , where $a$ is a constant and $a > 0$
Establish and use $\int e^{u} \frac{du}{dx} dx = e^{u} + C$ , where $u$ is a function of $x$ , or $\int f^{'} (x) e^{f (x)} dx = e^{f (x)} + C$
Find primitives of functions involving Loading

Integration with logarithmic functions

Derive and use the formula $\int \frac{1}{x} d x = \ln | x | + C$ where $x \neq 0$
Establish and use the formula $\int \frac{1}{ax + b} dx = \frac{1}{a} \ln (ax + b| + C$ , where $a$ and $b$ are constants and $a \neq 0$
Establish and use $\int \frac{u^{'}}{u} dx = \ln (u| + C$ , where $u$ is a function of $x$ , or $\int \frac{f^{'} (x)}{f (x)} d x = \ln | f (x) | + C$ , on a domain where $f (x) \neq 0$

Integration with trigonometric functions

Establish and use the formulas $\int \sin x d x = - \cos x + C$ , $\int \cos x d x = \sin x + C$ and $\int \sec^{2} x d x = \tan x + C$
Establish and use indefinite integrals of the form $\int f (ax + b) dx$ , where $a$ and $b$ are constants and $a \neq 0$ , and $f (x) = \sin x$ , $f (x) = \cos x$ and $f (x) = \sec^{2} x$
Determine indefinite integrals of the form $\int f' (x) \sin f (x) dx$ , $\int f' (x) \cos f (x) dx$ and $\int f' (x) \sec^{2} f (x) d x$

Areas and the definite integral

Apply $\int_{a}^{b} f (x) d x = F (b) - F (a)$ , where $F (x)$ is a primitive of $f (x)$ , to calculate definite integrals and solve related theoretical problems involving functions within the scope of the Mathematics Advanced course
Describe, in the case where $f (x) \geq 0$ for all values of $x$ in the interval $a \leq x \leq b$ , the area bounded by the graph of the continuous function $y = f (x)$ , the $x$ -axis and the lines $x = a$ and $x = b$ , as $\int_{a}^{b} f (x) dx$
Recognise, in the case where $f (x) \leq 0$ for all values of $x$ in the interval $a \leq x \leq b$ , the area bounded by the graph of the continuous function $y = f (x)$ , the $x$ -axis and the lines $x = a$ and $x = b$ , as $(\int_{a}^{b} f (x) dx|$ or $- \int_{a}^{b} f (x) d x$
Conclude, for a continuous function $y = f (x)$ on the interval $a \leq x \leq b$ , that $\int_{a}^{b} f (x) dx =$ (area of regions between curve and $x$ -axis lying above the $x$ -axis) $-$ (area of regions between curve and the $x$ -axis lying below the $x$ -axis)
Use definite integrals to solve problems involving the areas of regions bounded by the graph of the continuous function $y = f (x)$ , the $x$ -axis and the lines $x = a$ and $x = b$ , in cases where $f (x) \geq 0$ throughout $a \leq x \leq b$ , $f (x) \leq 0$ throughout $a \leq x \leq b$ or where $f (x)$ changes sign in the interval $a \leq x \leq b$ , with or without the graph provided
Use definite integrals to solve problems involving the areas of regions bounded by the graph of the continuous function $y = f (x)$ , the $y$ -axis and the lines $y = a$ and $y = b$ , in cases where $x \geq 0$ throughout $a \leq y \leq b$ , $x \leq 0$ throughout $a \leq y \leq b$ or where $x$ changes sign in the interval $a \leq y \leq b$ with or without the graph provided
Use the fact that the graphs of $y = a^{x}$ and $y = \log_{a} x$ are reflections of each other in the line $y = x$ to solve problems involving areas between the $x$ -axis or $y$ -axis and a curve involving either an exponential or logarithmic function
Recognise and use the result, where $f (x)$ is continuous on the interval $a \leq x \leq c$ , $\int_{a}^{b} f (x) dx + \int_{b}^{c} f (x) dx = \int_{a}^{c} f (x) dx$ for all $c$ such that $a \leq b \leq c$
Define and use the result $\int_{a}^{b} f (x) dx = - \int_{b}^{a} f (x) dx$ , where $f (x)$ is continuous on the interval $a \leq x \leq b$
Recognise and use symmetry, particularly Loading and Loading , to simplify and solve integration problems
Use the Loading to approximate integrals
Use an online computational application to evaluate definite and indefinite integrals involving functions within and beyond the scope of the Mathematics Advanced course
Model and solve practical problems involving integrals and areas of regions bounded by a curve and the $x$ -axis, or by a curve and the $y$ -axis, involving functions within the scope of the Mathematics Advanced course

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