WACE ATAR Mathematics Specialist cheat sheet

Chain rule: d/dx f(g(x)) = f'(g(x))·g'(x); product rule: (uv)' = u'v + uv'; quotient rule: (u/v)' = (u'v − uv')/v².

Derivative from first principles: f'(x) = lim(h→0) [f(x+h) − f(x)]/h; this limit defines differentiability at x.

Standard derivatives: d/dx sin x = cos x, cos x = −sin x, tan x = sec²x, eˣ = eˣ, ln x = 1/x for x > 0.

Integration by substitution reverses the chain rule: ∫f(g(x))g'(x)dx = ∫f(u)du with u = g(x); change limits when definite.

Integration by parts: ∫u dv = uv − ∫v du; choose u by the LIATE order (log, inverse trig, algebraic, trig, exponential).

Fundamental theorem: if F' = f then ∫ₐᵇ f(x)dx = F(b) − F(a); the definite integral gives signed area.

A function is continuous at a if lim(x→a) f(x) = f(a); it is differentiable there only if the left and right derivatives agree.

For inverse functions, (f⁻¹)'(y) = 1/f'(x) where y = f(x); graph of f⁻¹ is the reflection of f in y = x.

Second derivative test: f'(c) = 0 and f''(c) > 0 gives a local minimum; f''(c) < 0 gives a local maximum.

Dot product: a·b = |a||b|cos θ = a₁b₁ + a₂b₂ + a₃b₃; a·b = 0 exactly when a and b are perpendicular.

Cross product magnitude |a×b| = |a||b|sin θ equals the area of the parallelogram spanned by a and b; a×b is perpendicular to both.

Component form of the cross product: a×b = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁).

Projection of a onto b: scalar component (a·b)/|b|; vector projection = [(a·b)/(b·b)]·b.

Line in vector form: r = a + t·d, where a is a point on the line and d is a direction vector parallel to the line.

Plane equation: n·(r − a) = 0, equivalently n·r = n·a; the normal n is perpendicular to every direction in the plane.

Distance from point P to a plane n·r = k is |n·p − k|/|n|, where p is the position vector of P.

Two vectors are parallel iff a = λb for some scalar λ; three vectors are coplanar iff their scalar triple product a·(b×c) = 0.

Angle between vectors: cos θ = (a·b)/(|a||b|), with θ measured in [0, π]; this formula applies in two and three dimensions.

Cartesian form z = a + bi with modulus |z| = √(a² + b²) and argument arg z = atan2(b, a), taken in (−π, π].

Polar form z = r(cos θ + i sin θ) = r·cis θ = re^{iθ}; multiplication multiplies moduli and adds arguments.

De Moivre's theorem: (r·cis θ)ⁿ = rⁿ·cis(nθ); this also yields the n distinct nth roots spaced 2π/n apart in argument.

Conjugate z̄ = a − bi satisfies z·z̄ = |z|²; the quotient w/z is rationalised by multiplying numerator and denominator by z̄.

Euler's formula e^{iθ} = cos θ + i sin θ gives cos θ = (e^{iθ} + e^{−iθ})/2 and sin θ = (e^{iθ} − e^{−iθ})/(2i).

A polynomial with real coefficients has non-real roots occurring in conjugate pairs; the conjugate of a root is also a root.

Fundamental theorem of algebra: a polynomial of degree n has exactly n roots over the complex numbers, counting multiplicity.

Loci: |z − z₀| = r is a circle centred at z₀; |z − z₁| = |z − z₂| is the perpendicular bisector of the segment joining z₁ and z₂.

Matrix product AB is defined when the column count of A equals the row count of B; the result has size (rows of A) × (columns of B).

Matrix multiplication is not commutative: in general AB ≠ BA; however (AB)C = A(BC) and A(B + C) = AB + AC hold.

The inverse of a 2×2 matrix A = [[a, b], [c, d]] is (1/(ad − bc))·[[d, −b], [−c, a]], existing only when det A ≠ 0.

Determinant of a 3×3 matrix is expanded by cofactors along any row or column; a zero determinant means the matrix is singular.

A system Ax = b has a unique solution iff det A ≠ 0; otherwise there are either no solutions or infinitely many.

Transpose rules: (AB)ᵀ = BᵀAᵀ and (A⁻¹)ᵀ = (Aᵀ)⁻¹; a symmetric matrix satisfies Aᵀ = A.

Eigenvalues satisfy det(A − λI) = 0; eigenvectors satisfy (A − λI)v = 0 for each eigenvalue λ.

Rotation matrix through angle θ about the origin: [[cos θ, −sin θ], [sin θ, cos θ]]; reflections have determinant −1, rotations determinant +1.

Binomial distribution: X ~ Bin(n, p) has P(X = k) = C(n, k)pᵏ(1 − p)ⁿ⁻ᵏ, mean np and variance np(1 − p).

Normal distribution: standardise with z = (x − μ)/σ, then use the standard normal table or calculator for probabilities.

For a sample mean, standard error is σ/√n; for large n the distribution of the sample mean is approximately normal by the central limit theorem.

Confidence interval for a mean with known σ: x̄ ± z*·(σ/√n), where z* is the standard normal critical value for the chosen confidence level.

Sample variance uses divisor (n − 1) to give an unbiased estimator; population variance uses divisor n.

Type I error rejects a true null hypothesis; Type II error fails to reject a false null hypothesis; power = 1 − P(Type II error).

Correlation coefficient r lies in [−1, 1]; r measures only the strength of linear association, not causation.

Least-squares regression line of y on x has slope b = r·(s_y/s_x) and passes through the point (x̄, ȳ).

Poisson distribution: P(X = k) = e^{−λ}λᵏ/k!, with mean and variance both equal to λ.