← Maths Advanced syllabus

NSWMaths Advanced

Year 12: Calculus

8 dot points across 8 inquiry questions. Click any dot point for a focused answer with worked past exam questions where available.

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
A focused answer to the HSC Maths Advanced dot point on applications of differentiation. Stationary points, concavity and inflection, maxima and minima word problems, and related rates with worked examples.
10 min answer →

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 $x$ or $y$ axis
A focused answer to the HSC Maths Advanced dot point on areas and volumes via integration. Areas under and between curves, the disk method for volumes of revolution about the $x$ and $y$ axes.
9 min answer →

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
A focused answer to the HSC Maths Advanced dot point on differentiation rules. The power, chain, product and quotient rules, plus derivatives of exponential, logarithmic and trigonometric functions, with worked examples and exam traps.
8 min answer →

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
A focused answer to the HSC Maths Advanced dot point on exponential modelling. The equations dN/dt = kN and dT/dt = k(T - Ta), their solutions, and worked applications to population, radioactive decay and cooling.
9 min answer →

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
A focused answer to the HSC Maths Advanced dot point on integration. Antiderivatives of standard functions, integration by substitution, definite integrals and the Fundamental Theorem of Calculus, with worked examples.
10 min answer →

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
A focused answer to the HSC Maths Advanced dot point on the calculus of exponential and logarithmic functions. Derivatives and integrals of e^x and ln(x), composed forms via the chain rule, and worked examples.
8 min answer →

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
A focused answer to the HSC Maths Advanced dot point on rectilinear motion. Velocity as the derivative of displacement, acceleration as the derivative of velocity, and recovering displacement from velocity by integration.
8 min answer →

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
A focused answer to the HSC Maths Advanced dot point on trigonometric calculus. Derivatives and integrals of sin, cos and tan, plus modelling periodic motion such as tides and oscillations.
8 min answer →