Worked examples: VCE Mathematical Methods Formula Sheets

Formula index with worked examples

1. Mensuration · trapezium and triangle

   Formula or reference: Trapezium area = (a + b)h/2. Triangle area = bc sin A/2.

   Units and conditions: Use square units. h is perpendicular to the parallel sides, and A is the angle between b and c.

   Worked example: A trapezium with bases 5 m and 9 m and height 3 m has area 21 m². A triangle with sides 6 m and 4 m enclosing 30° has area 6 m².

2. Mensuration · cylinder and cone

   Formula or reference: Cylinder curved area = 2πrh. Cylinder volume = πr²h. Cone volume = πr²h/3.

   Units and conditions: Use square or cubic units as appropriate. Curved area excludes the end discs.

   Worked example: With r = 2 cm and h = 6 cm, the cylinder’s curved area is 24π cm² and its volume is 24π cm³. The cone of the same base and height has volume 8π cm³.

3. Mensuration · pyramid and sphere

   Formula or reference: Pyramid volume = Ah/3. Sphere volume = 4πr³/3.

   Units and conditions: A is base area and h is perpendicular height. Volume uses cubic units.

   Worked example: A pyramid of base area 12 cm² and height 6 cm has volume 24 cm³. A sphere of radius 3 cm has volume 36π cm³.

4. Calculus · powers

   Formula or reference: d(xⁿ)/dx = nx^(n−1). ∫ xⁿ dx = x^(n+1)/(n + 1) + C for n ≠ −1.

   Units and conditions: Apply on the real domain of the expression and include C for an indefinite integral.

   Worked example: d(x³)/dx = 3x². Conversely, ∫ 3x² dx = x³ + C.

5. Calculus · powers of linear expressions

   Formula or reference: d[(ax + b)ⁿ]/dx = an(ax + b)^(n−1). ∫ (ax + b)ⁿ dx = (ax + b)^(n+1)/(a(n + 1)) + C for a ≠ 0 and n ≠ −1.

   Units and conditions: Work on an interval where the function is defined.

   Worked example: The derivative of (2x + 1)³ is 6(2x + 1)². An antiderivative of (2x + 1)³ is (2x + 1)⁴/8 + C.

6. Calculus · exponential and logarithm

   Formula or reference: d(e^(ax))/dx = ae^(ax). ∫ e^(ax) dx = e^(ax)/a + C for a ≠ 0. d(ln x)/dx = 1/x. ∫ (1/x) dx = ln x + C for x > 0.

   Units and conditions: For the stated logarithm form use x > 0.

   Worked example: The derivative of e^(3x) is 3e^(3x), and an antiderivative of e^(3x) is e^(3x)/3. At x = 2, the derivative of ln x is 1/2.

7. Calculus · trigonometric derivatives

   Formula or reference: d[sin(ax)]/dx = a cos(ax). d[cos(ax)]/dx = −a sin(ax). d[tan(ax)]/dx = a/cos²(ax) = a sec²(ax).

   Units and conditions: Use radians. The tangent derivative is defined where cos(ax) ≠ 0.

   Worked example: For a = 2, the derivatives at x = 0 of sin(2x), cos(2x) and tan(2x) are 2, 0 and 2, respectively.

8. Calculus · trigonometric antiderivatives

   Formula or reference: ∫ sin(ax) dx = −cos(ax)/a + C. ∫ cos(ax) dx = sin(ax)/a + C, for a ≠ 0.

   Units and conditions: Use radians and include the integration constant.

   Worked example: ∫ sin(3x) dx = −cos(3x)/3 + C and ∫ cos(3x) dx = sin(3x)/3 + C.

9. Calculus · product rule

   Formula or reference: d(uv)/dx = u′v + uv′

   Units and conditions: The derivative has units of output per input where applicable.

   Worked example: For y = x²eˣ, y′ = 2xeˣ + x²eˣ = eˣ(2x + x²).

10. Calculus · quotient rule

    Formula or reference: d(u/v)/dx = (vu′ − uv′)/v² for v ≠ 0.

    Units and conditions: State any excluded denominator values.

    Worked example: For y = x/(x + 1), y′ = [(x + 1) − x]/(x + 1)² = 1/(x + 1)² for x ≠ −1.

11. Calculus · chain rule

    Formula or reference: dy/dx = (dy/du)(du/dx)

    Units and conditions: Differentiate the outer function at u, then multiply by the derivative of u.

    Worked example: For y = sin(x²), take u = x², so dy/dx = 2x cos(x²).

12. Calculus · Newton’s method

    Formula or reference: x(next) = x(current) − f(x(current))/f′(x(current))

    Units and conditions: The derivative at the current iterate must be non-zero. Convergence depends on the starting value.

    Worked example: For f(x) = x² − 2, starting at x = 1 gives the next iterate 1 − (−1)/2 = 1.5.

13. Calculus · trapezium rule

    Formula or reference: Approximate the integral by (h/2) times the sum of both endpoint heights and twice each interior height, with h = (b − a)/n.

    Units and conditions: Subintervals must have equal width. The result uses input units times output units.

    Worked example: For f(x) = x² from 0 to 2 with n = 2, h = 1 and the estimate is (0 + 2 × 1 + 4)/2 = 3.

14. Probability · complement and union

    Formula or reference: P(A) = 1 − P(A′). P(A ∪ B) = P(A) + P(B) − P(A ∩ B).

    Units and conditions: Probabilities are unitless between zero and one.

    Worked example: If P(A) = 0.6, P(B) = 0.4 and P(A ∩ B) = 0.2, the union has probability 0.8 and A’s complement has probability 0.4.

15. Probability · conditional probability

    Formula or reference: P(A | B) = P(A ∩ B)/P(B) when P(B) > 0.

    Units and conditions: Divide by the probability of the conditioning event.

    Worked example: If P(A ∩ B) = 0.12 and P(B) = 0.30, P(A | B) = 0.40.

16. Probability · mean and variance

    Formula or reference: μ = E(X). Var(X) = σ² = E[(X − μ)²] = E(X²) − μ².

    Units and conditions: Mean has the units of X. Variance has squared units.

    Worked example: For equally likely values 0 and 2, μ = 1 and E(X²) = 2, so variance is 1.

17. Probability · binomial coefficient

    Formula or reference: C(n,k) = n!/[k!(n − k)!]

    Units and conditions: Use integer n ≥ 0 and k between zero and n inclusive.

    Worked example: C(5,2) = 5!/(2!3!) = 10.

18. Probability · discrete distribution

    Formula or reference: P(X = x) = p(x). μ = Σ x p(x). σ² = Σ (x − μ)² p(x).

    Units and conditions: The probabilities sum to one.

    Worked example: If X is 0 with probability 0.75 and 4 with probability 0.25, μ = 1 and σ² = 0.75 × 1² + 0.25 × 3² = 3.

19. Probability · binomial distribution

    Formula or reference: P(X = k) = C(n,k)pᵏ(1 − p)^(n−k). μ = np. σ² = np(1 − p).

    Units and conditions: Use n independent trials with the same success probability p.

    Worked example: For four fair trials, P(X = 2) = 6/16 = 0.375, μ = 2 and σ² = 1.

20. Probability · continuous distribution

    Formula or reference: P(a < X < b) is the integral of f(x) from a to b. μ is the integral of xf(x), and σ² of (x − μ)²f(x), over the full support.

    Units and conditions: A probability density integrates to one. Variance has squared units.

    Worked example: For a uniform density of 1 on [0,1], the probability of (0, 1/2) is 1/2. Integrating x gives μ = 1/2, and integrating (x − 1/2)² gives σ² = 1/12.

21. Sampling · sample proportions

    Formula or reference: P̂ = X/n. E(P̂) = p. sd(P̂) = √[p(1 − p)/n].

    Units and conditions: Use a binomial model for X. Proportions are unitless.

    Worked example: If n = 100 and p = 0.40, the mean is 0.40 and standard deviation is about 0.049. In a sample with 43 successes the observed proportion is 0.43.

22. Sampling · approximate confidence interval

    Formula or reference: p̂ ± z√[p̂(1 − p̂)/n]

    Units and conditions: Use the critical z for the requested confidence level. Check the conditions for the normal approximation.

    Worked example: With 40 successes in 100 trials and z = 1.96, the approximate 95% interval is 0.40 ± 0.096, or (0.304, 0.496).

More worked examples

VCAA links separate formula sheets alongside its Mathematical Methods examination 1 and 2 sample materials. A supplied formula sheet is different from a student's bound reference: the authorised-materials rules permit a bound reference for Methods examination 2, not examination 1.

1. Chain rule with an exponential

   Relationship: d/dx[e^(g(x))] = g′(x)e^(g(x)).

   Differentiate f(x) = e^(2x−1) and find f′(½).

   Show worked response and error check

   f′(x) = 2e^(2x−1). At x = ½ the exponent is zero, so f′(½) = 2. This is the tangent gradient at that x-value.

   Check: The derivative of the inner function contributes the factor 2.

2. A binomial probability

   Relationship: For X ~ Bin(n,p), P(X=k) = C(n,k)p^k(1−p)^(n−k).

   Four independent trials each have success probability 0.5. Find the probability of exactly two successes.

   Show worked response and error check

   P(X=2) = C(4,2) × 0.5² × 0.5² = 6/16 = 0.375. Independence and constant probability justify the model.

   Check: Exactly two differs from at least two; the latter also includes three and four successes.

3. Inverse functions and restrictions

   Relationship: To invert a one-to-one function, solve y = f(x) for x and swap the variables; the range becomes the inverse domain.

   Find the inverse of f(x) = x² + 1 with domain x ≥ 0.

   Show worked response and error check

   y − 1 = x². The domain restriction selects x = √(y − 1), so f⁻¹(x) = √(x − 1) with domain x ≥ 1 and range y ≥ 0.

   Check: Using both signs does not define a single inverse function for this restricted original domain.