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VCE Units 1–4 · Specialist Maths

VCE Specialist Maths — Advanced Calculus

Specialist extends Methods calculus significantly — implicit differentiation, inverse trigonometric functions, integration by substitution and by parts, and differential equations all appear in Units 3 and 4. These techniques are typically the highest-difficulty questions in the exam.

Key Concepts & Formulas

  • Inverse trig derivatives: d/dx(sin⁻¹x) = 1/√(1−x²), d/dx(cos⁻¹x) = −1/√(1−x²), d/dx(tan⁻¹x) = 1/(1+x²)

  • Implicit differentiation: differentiate both sides with respect to x, applying the chain rule to y terms (giving dy/dx as a factor)

  • Related rates: differentiate an equation involving multiple variables with respect to time t

  • Integration by substitution: let u = g(x), then du = g'(x)dx — transforms the integral in terms of u

  • Integration by parts: ∫u dv = uv − ∫v du — choose u to be the factor that simplifies when differentiated

  • Separable differential equations: dy/dx = f(x)g(y) — separate variables then integrate both sides

  • Newton's law of cooling: dT/dt = −k(T − T_env) — separable DE with exponential solution

  • Logistic equation: dP/dt = kP(1 − P/N) — separable; solution approaches N as t → ∞

Practice Questions

5 questions

Attempt each question before reading the hint. These are styled to match VCE exam format.

Q1.Find dy/dx by implicit differentiation: x² + y² = 5xy.

3 marks

Q2.A spherical balloon is inflated so its radius increases at 2 cm/s. Find the rate of change of volume when r = 5 cm.

3 marks
Show hint

V = (4/3)πr³; differentiate both sides with respect to t.

Q3.Evaluate ∫ x·eˣ dx using integration by parts.

3 marks

Q4.Solve the differential equation dy/dx = xy, given y = 2 when x = 0.

3 marks

Q5.Find ∫ 1/√(4 − x²) dx.

2 marks
Show hint

Recognise the inverse sine integral form.

Common Mistakes to Avoid

These are the errors that VCE students most frequently make in Advanced Calculus — and that examiners are specifically watching for.

  • In implicit differentiation, forgetting the dy/dx factor when differentiating y terms: d/dx(y²) = 2y·(dy/dx)

  • In related rates, not identifying which quantity is changing with time and which is constant

  • Choosing u and dv incorrectly in integration by parts — use LIATE order: Logarithm, Inverse trig, Algebraic, Trig, Exponential

  • In separable DEs, forgetting to include the +c constant before applying initial conditions

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