§-Module

NSWMaths Advanced← Maths Advanced syllabus

Year 12: Calculus

12 dot points across 12 inquiry questions, in syllabus order. Each dot point has a focused answer, with exam-style questions and worked answers where available.

How can we estimate the area under a curve, or a definite integral, when we only have a table of values or an integral we cannot evaluate exactly?

Use the trapezoidal rule to estimate areas and definite integrals, and determine whether the estimate is an over- or under-estimate

Trapezoidal rule. The single-application and multi-strip formulas, reading values from a table or a function, estimating a definite integral, and using concavity to decide over- or under-estimate.

16 min read →

How do we use calculus to analyse the motion of a particle moving in a straight line?

Apply calculus to motion in a straight line, with displacement, velocity and acceleration as derivatives and integrals with respect to time

Rectilinear motion. Velocity as the derivative of displacement, acceleration as the derivative of velocity, and recovering displacement from velocity by integration.

14 min read →

How do we use calculus to model the rate at which a quantity changes, recover the quantity from its rate, and read a rate graph?

Solve problems involving related and general rates of change, including integrating a given rate dQ/dt to recover a quantity with an initial condition, and interpreting rate graphs

General rates of change. Rate of change as a derivative, the chain rule for related rates, integrating a given rate dQ/dt with an initial condition to recover the quantity (tanks, dams, populations), and reading and sketching a rate graph: increasing at a decreasing rate, where the quantity is greatest or least, and net change as signed area.

18 min read →

How do we differentiate functions built from products, quotients and compositions of standard functions?

Apply the product, quotient and chain rules, and differentiate exponential, logarithmic and trigonometric functions

Differentiation rules. The power, chain, product and quotient rules, plus derivatives of exponential, logarithmic and trigonometric functions, with worked examples and exam traps.

12 min read →

How do the chain and quotient rules give the derivatives of the reciprocal trigonometric functions, and how are those derivatives used in tangent, normal and stationary point problems?

Use the rules of differentiation to find the derivatives of

\operatorname{cosec} x

\sec x

and

\cot x

The Year 12 Mathematics Advanced answer on differentiating the reciprocal trigonometric functions, new in the 2024 syllabus: why sec x differentiates to sec x tan x, cosec x to minus cosec x cot x and cot x to minus cosec squared x, the chain-rule forms, and tangent, normal and stationary-point problems, with checked worked examples.

12 min read →

How do we find antiderivatives and use the Fundamental Theorem of Calculus to evaluate definite integrals?

Find antiderivatives of standard functions, apply integration by substitution and evaluate definite integrals using the Fundamental Theorem of Calculus

Integration. Antiderivatives of standard functions, integration by substitution, definite integrals and the Fundamental Theorem of Calculus, with worked examples.

16 min read →

How do we find the area between a curve and the y-axis, and how does the reflection of exponential and logarithmic graphs in y = x help with areas?

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

The Year 12 Mathematics Advanced answer on areas bounded by a curve and the y-axis, new in the 2024 syllabus: rewriting x in terms of y, integrating with respect to y, handling regions left of the y-axis or crossing it, and using the fact that y = a^x and y = log_a x reflect in y = x to find exponential and logarithmic areas.

12 min read →

How do we use definite integrals to compute areas between curves and volumes of solids of revolution?

Calculate the area under a curve, the area between two curves, and the volume of a solid of revolution about the

x

y

axis

Areas and volumes via integration. Areas under and between curves, the disk method for volumes of revolution about the $x$ and $y$ axes.

15 min read →

How do we model continuous growth, decay and cooling using differential equations?

Establish and solve differential equations of the form

\frac{dN}{dt} = k N

and

\frac{dT}{dt} = k(T - T_a)

and apply them to growth, decay and Newton's law of cooling

Exponential modelling. The equations dN/dt = kN and dT/dt = k(T - Ta), their solutions, and worked applications to population, radioactive decay and cooling.

15 min read →

How can the derivative be used to analyse curves and solve optimisation and rate problems?

Use the first and second derivatives to find stationary points, points of inflection, and to solve optimisation and related rates problems

Applications of differentiation. Stationary points, concavity and inflection, maxima and minima word problems, and related rates with worked examples.

14 min read →

How do we differentiate and integrate exponential and logarithmic functions, and how do they appear in modelling?

Find derivatives and integrals of

e^x

and

\ln x

, including composed forms, and apply them to modelling problems

Calculus of exponential and logarithmic functions. Derivatives and integrals of e^x and ln(x), composed forms via the chain rule, and worked examples.

12 min read →

How do we differentiate and integrate trigonometric functions and use them to model periodic phenomena?

Find derivatives and integrals of

\sin

\cos

and

\tan

(with linear inside arguments) and apply them to model simple harmonic and periodic motion

Trigonometric calculus. Derivatives and integrals of sin, cos and tan, plus modelling periodic motion such as tides and oscillations.

12 min read →