Calculus | HSC Advanced

Assessment evidence

What these examples cover

These examples show assessable skills, not an official HSC topic weighting or prediction. The 2024 NSW syllabuses have their first HSC examination in 2027; students sitting the 2026 HSC continue under the applicable 2017 syllabus.

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3 deterministic examples

Calculus questions with worked solutions

Each question is generated from an existing curriq QDSL archetype at a fixed seed. Marks and catalog content mappings travel with the generated question.

Worked example 1

3 marksadv-c1-1

Derivative from first principles Let

f(x) = x^2 - 2x + 7

Part a · 3 marks

Differentiate $f(x)$ from first principles.

Worked solution

Answer: 2 x - 2

First Principles Substitution: (f(x+h)-f(x))/h
Expand Difference Quotient: h(2x+h-b)/h
Evaluate Limit: 2 x - 2

Catalog reference: adv_jr24_first_principles_quadratic_derivative · seed 41001

Worked example 2

2 marksadv-c2-1

Differentiate an exponential quotient Differentiate

(e^x)/(x^2+9)

Part a · 2 marks

Give the simplified derivative.

Worked solution

Answer: (exp(x)((x^2 - (2 x)) + 9)) / ((x^2 + 9)^2)

Quotient Rule: ((exp(x)(x^2 + 9)) - ((2 x) exp(x))) / ((x^2 + 9)^2)
Algebraic Simplification: (exp(x)((x^2 - (2 x)) + 9)) / ((x^2 + 9)^2)

Catalog reference: adv_cssa20_differentiate_exponential_quotient · seed 41001

Worked example 3

2 marksadv-c4-1

Definite polynomial integral Evaluate

integral_(-1)^0 x(3x-4) dif x

Part a · 2 marks

Find the exact value.

Worked solution

Answer: 3

Power Rule Integration: 1 x^3 - (2 x^2)
Fundamental Theorem: 3

Catalog reference: adv_cssa20_definite_polynomial_integral · seed 41001

Content mapping and common errors

Calculus content identifiers

C1.1

Introduction to Differentiation

Understand the derivative as a limit and as the gradient of a tangent; differentiate polynomials from first principles.

Common student mistakes

·Forgetting to take the limit as h→0 in first principles
·Algebraic errors when expanding (x+h)ⁿ

C1.2

Differential Calculus

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

Common student mistakes

·Forgetting the chain rule when differentiating composite functions
·Sign errors in the quotient rule
·Differentiating ln(f(x)) as 1/f(x) instead of f′(x)/f(x)

C1.3

Applications of Differentiation

Find stationary points; solve optimisation problems; apply rates of change in context.

Common student mistakes

·Not testing the nature of stationary points (first or second derivative test)
·Forgetting to check endpoints in a closed-interval optimisation
·Not defining variables before writing an equation in word problems

C2.1

Anti-differentiation and the Indefinite Integral

Understand anti-differentiation as the reverse of differentiation; find indefinite integrals.

Common student mistakes

·Forgetting the constant of integration (+C)
·Integrating trig functions with incorrect signs (e.g., ∫sinx dx = cosx instead of −cosx)

C2.2

Definite Integrals

Use the fundamental theorem of calculus; evaluate definite integrals; find areas under and between curves.

Common student mistakes

·Not taking the absolute value when calculating area below the x-axis
·Subtracting integrals in the wrong order for area between curves

C3.1

Differential Equations

Solve first-order differential equations of the form dy/dx = f(x) and model exponential growth/decay.

Common student mistakes

·Forgetting to apply the initial condition to find C
·Misidentifying growth vs decay from the sign of the rate constant

Source and methodology

Examples are deterministic curriq catalog generations from existing QDSL archetypes at the published fixed seed. The generator copies only student-facing prompts, mark allocations, topic mappings, display tables or options, checked answers and marking-path results into this resource.

Review the official NSW Curriculum source ↗

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Common mistakes

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Question design

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Marking

NESA HSC Maths Marking Guidelines Explained

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What these examples cover

Calculus questions with worked solutions

Worked example 1

Worked example 2

Worked example 3

Calculus content identifiers

Introduction to Differentiation

Differential Calculus

Applications of Differentiation

Anti-differentiation and the Indefinite Integral

Definite Integrals

Differential Equations

Source and methodology

Other Advanced topics

From the blog

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