ATARMAxxing
← All subjects
Ma
HSC · HSC Year 12

Mathematics Advanced

Functions, calculus, financial mathematics and statistics — full HSC papers with fully worked solutions.

20full-length model exams with mark-by-mark answer guides
20detailed note sets — ~120 pages across every topic
64exam-style practice questions with worked solutions
60flashcards for every key term & formula
7official past papers

Preview it all free. Unlock when you're ready.

Read every note, sit every exam, check every answer — the moment you unlock it. From $20 once for one subject — yours for life.

See all plans

One-time payment in AUD · lifetime access · by purchasing you agree to our Terms.

What's inside

Sample revision note

Applications of Derivatives: Tangents, Normals, and Rates of Change

The Gradient of a Curve at a Point

The gradient of a tangent to a curve at a given point is found by evaluating the derivative of the function at that point. This is one of the most fundamental applications of differential calculus in the NSW HSC Mathematics Advanced course.

If y = f(x), then the derivative f'(x) (also written as dy/dx) gives a gradient function — a formula that outputs the instantaneous gradient at any value of x on the curve.

To find the gradient at a specific point, substitute the x-coordinate of that point into the derivative.

  • Step 1: Differentiate f(x) to obtain f'(x).
  • Step 2: Substitute the given x-value into f'(x).
  • Step 3: The result is the gradient m of the tangent at that point.

Worked Example: Find the gradient of the tangent to y = x3 − 4x + 1 at the point where x = 2.

  • Differentiate: dy/dx = 3x2 − 4
  • Substitute x = 2: dy/dx = 3(2)2 − 4 = 3(4) − 4 = 12 − 4 = 8
  • Verification: 3 × 4 = 12; 12 − 4 = 8. Confirmed.
  • The gradient of the tangent at x = 2 is 8.

Note: the gradient of a curve at a point refers to the gradient of the tangent at that point — the two phrases are used interchangeably in HSC contexts.

Equation of the Tangent Line

Once the gradient m of the tangent is known, the equation of the tangent line can be found using the point-gradient formula:

y − y1 = m(x − x1)

where (x1, y1) is the point of tangency (the point on the curve where the tangent touches).

Worked Example: Find the equation of the tangent to y = x3 − 4x + 1 at the point where x = 2.

  • Find the y-coordinate: Substitute x = 2 into the original curve:
    y = (2)3 − 4(2) + 1 = 8 − 8 + 1 = 1
    Verification: 8 − 8 = 0; 0 + 1 = 1. Confirmed. Point of tangency: (2, 1).
  • Gradient: From the previous section, m = 8.
  • Apply point-gradient formula:
    y − 1 = 8(x − 2)
    y − 1 = 8x − 16
    y = 8x − 15

The equation of the tangent is y = 8x − 15.

In some HSC questions the point of tangency is given directly as a coordinate pair, and you only need to verify it lies on the curve before proceeding. Always substitute back to check: at x = 2, y = 8(2) − 15 = 16 − 15 = 1. This matches the curve value, confirming the equation is correct.

Tangent lines are linear. Their equation is always in the form y = mx + b or equivalently ax + by + c = 0. The HSC typically accepts either form unless the question specifies otherwise.

Sample exam question

The graph of y = f(x), where f(x) = 2x - 6, is used to sketch the reciprocal graph y = 1/f(x). At which value does the reciprocal graph have a vertical asymptote?

  • A. x = -3
  • B. x = 0
  • C. x = 3
  • D. y = 3
Show the worked answer

Answer: C

A reciprocal graph y = 1/f(x) has a vertical asymptote wherever f(x) = 0. Solving 2x - 6 = 0 gives x = 3, so the vertical asymptote is x = 3.

All 20 practice exams

  1. Exam 1 — Functions and graphing techniques (rational, exponential, logarithmic) as the dominant theme; Differential and integral calculus including the second derivative and areas; Trigonometric functions and graph transformations
  2. Exam 2 — Trigonometric Functions and Graphs (transformations, equations, modelling, area); Differential Calculus (product rule, tangents, stationary points, concavity/inflection); Integral Calculus (area between curves, trapezoidal rule, applications)
  3. Exam 3 — Calculus (differential calculus, the second derivative, integral calculus) — ~50 of 90 marks; Financial Mathematics — reducing-balance loan modelling; Statistical Analysis — normal distribution, bivariate regression, continuous random variables
  4. Exam 4 — Financial Mathematics — Modelling Financial Situations (reducing-balance loans via recurrence, superannuation annuities, depreciation and geometric series): 26 dedicated marks; Statistical Analysis — Descriptive Statistics, Bivariate Data Analysis and Random Variables (continuous PDFs and the normal distribution): 33 marks; Calculus — Differential Calculus, the Second Derivative and Integral Calculus (curve sketching, tangents, area and the trapezoidal rule): 17 marks
  5. Exam 5 — Statistical Analysis — Descriptive Statistics & Bivariate Data (5-number summary, boxplots, outliers, least-squares regression, correlation, interpretation); Statistical Analysis — Random Variables (discrete probability distributions, E(X), Var(X), linear transformations; the normal distribution, z-scores, empirical rule); Calculus — differential (product/chain rule, stationary points, second derivative, concavity) and integral (area under/between curves, trapezoidal rule)
  6. Exam 6 — Functions (Graphing Techniques) — rational functions, transformations, curve sketching with calculus; Calculus — differential, second derivative, integral (area + trapezoidal rule); Trigonometric functions and graphs
  7. Exam 7 — Trigonometric Functions and Graphs; Calculus (Differential & Integral); Financial Mathematics
  8. Exam 8 — Calculus: Differential Calculus, The Second Derivative, Integral Calculus (heavily weighted); Trigonometric Functions and Graphs; Financial Mathematics (Modelling Financial Situations)
  9. Exam 9 — Financial Mathematics (Modelling Financial Situations) — annuities, reducing-balance loans, depreciation, present-value comparison, recurrence/geometric savings models; Calculus — differential (stationary points, second-derivative test, concavity/inflection) and integral (definite integrals, area); Statistical Analysis — least-squares regression and correlation, continuous random variables, normal distribution / empirical rule
  10. Exam 10 — Statistical Analysis weighted heaviest (~40 of 90 marks): descriptive statistics, bivariate data analysis, discrete and continuous random variables; Full Year 12 coverage: graphing techniques, trigonometric functions, differential and integral calculus, the second derivative, financial mathematics; Two 9-mark multi-part modelling/application questions (bivariate+RV market data; particle kinematics via calculus)
  11. Exam 11 — Functions and Graphing Techniques (rational, quadratic, inverse, transformations); Differential and Integral Calculus (including the second derivative and curve sketching); Trigonometric Functions and Graphs
  12. Exam 12 — Trigonometric Functions and Graphs (graphing, equations, identities, modelling); Differential and Integral Calculus including the second derivative; Financial Mathematics — modelling financial situations (annuities/superannuation)
  13. Exam 13 — Differential Calculus (product/quotient/chain rules, stationary points); The Second Derivative (concavity, points of inflection, curve sketching); Integral Calculus (areas, trapezoidal rule, kinematics)
  14. Exam 14 — Financial Mathematics (Modelling Financial Situations) — reducing-balance loans, annuities, recurrence/series; Calculus — differentiation, second derivative & curve sketching, integration and kinematics; Trigonometric Functions and Graphs — modelling periodic phenomena
  15. Exam 15 — Statistical Analysis (Descriptive Statistics) — five-number summary, outliers, standard deviation; Statistical Analysis (Bivariate Data) — Pearson's r, least-squares regression, prediction and reliability; Statistical Analysis (Random Variables) — discrete distributions and the normal distribution / empirical rule
  16. Exam 16 — Functions (Graphing Techniques) — rational functions, asymptotes, area between curves, transformations, curve sketching; Calculus — differentiation rules, second-derivative concavity, integration, optimisation; Financial Mathematics — reducing-balance loan / annuity modelling
  17. Exam 17 — Trigonometric Functions and Graphs (transformations, equations, identities, modelling); Differential and Integral Calculus (including the second derivative and trig calculus); Financial Mathematics (compound interest, annuities, reducing-balance loans)
  18. Exam 18 — Calculus emphasis: differential calculus, the second derivative, integral calculus; Realistic NESA Section II weighting across all Year 12 Mathematics Advanced modules; Multi-part modelling/application questions worth 5+ marks (tide, loan/superannuation, bivariate)
  19. Exam 19 — Financial Mathematics — Modelling Financial Situations (annuities, reducing-balance loans, compound interest, recurrence); Differential & Integral Calculus (second derivative, curve sketching, areas, volumes of revolution); Statistical Analysis (normal distribution / z-scores, bivariate data & least-squares regression)
  20. Exam 20 — Statistical Analysis — Descriptive Statistics & Bivariate Data Analysis (Q15, Q16); Statistical Analysis — Random Variables: continuous PDFs and the normal distribution (Q17, Q18); Calculus — differentiation, the second derivative and integration (Q11, Q13, Q19)

All 20 revision notes

  • Applications of Derivatives: Tangents, Normals, and Rates of Change
  • Applications of Integration: Displacement, Velocity, and Acceleration
  • Areas Under and Between Curves
  • Curve Sketching Using Calculus
  • Derivatives of Trigonometric, Exponential, and Logarithmic Functions
  • Differentiation Rules: Product, Quotient, and Chain
  • Integration Rules: Indefinite and Definite Integrals
  • Stationary Points and the Second Derivative Test
  • Combining and Squaring Functions
  • Reciprocal and Absolute Value Graphs
  • Transformations of Functions
  • Graphs of Sine, Cosine, and Tangent
  • Solving Trigonometric Equations Graphically and Algebraically
  • Trigonometric Identities and Applications
  • Annuities and Loan Repayments
  • Simple and Compound Interest
  • Applications of the Normal Distribution
  • Correlation, Regression, and the Least-Squares Line
  • Discrete Random Variables and Probability Distributions
  • The Normal Distribution and z-Scores