WACE ATAR Mathematics Specialist flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

First principles derivative

The derivative at a point is the limit of the difference quotient: f'(x) = lim(h→0)[f(x+h) − f(x)]/h, provided this limit exists and is finite.

Chain rule

For composite functions, d/dx f(g(x)) = f'(g(x))·g'(x). Differentiate the outer function evaluated at the inner function, then multiply by the inner derivative.

Integration by parts

∫u dv = uv − ∫v du. Choose u so its derivative simplifies, often following the LIATE order; choose dv so it is easy to integrate.

Fundamental theorem of calculus

If F is an antiderivative of f, then the definite integral from a to b of f(x)dx equals F(b) − F(a). This links areas to antiderivatives.

Second derivative test

At a stationary point where f'(c) = 0: if f''(c) > 0 the point is a local minimum; if f''(c) < 0 it is a local maximum; if zero the test is inconclusive.

Differentiability implies continuity

If a function is differentiable at a point then it is continuous there. The converse fails: for example, the absolute value function is continuous but not differentiable at zero.

Implicit differentiation

Differentiate both sides of a relation in x and y with respect to x, applying the chain rule to y terms, then solve algebraically for dy/dx.

Dot product

a·b = |a||b|cos θ = a₁b₁ + a₂b₂ + a₃b₃. It is a scalar; it equals zero exactly when the two vectors are perpendicular.

Cross product

a×b is a vector perpendicular to both a and b, with magnitude |a||b|sin θ equal to the parallelogram area. Components: (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁).

Vector projection

The projection of a onto b is [(a·b)/(b·b)]·b. Its scalar component along b is (a·b)/|b|, which may be negative if the angle is obtuse.

Vector equation of a line

A line is r = a + t·d, where a is the position vector of a point on the line and d is a nonzero direction vector parallel to it.

Plane equation

A plane is n·(r − a) = 0, equivalently n·r = n·a, where n is a normal vector and a is a point on the plane.

Point-to-plane distance

For plane n·r = k and point with position vector p, the perpendicular distance is |n·p − k|/|n|. The absolute value ensures a nonnegative distance.

Angle between vectors

cos θ = (a·b)/(|a||b|), giving θ in the interval from 0 to π. This applies in two or three dimensions and uses the dot product.

Scalar triple product

a·(b×c) is a scalar whose absolute value is the volume of the parallelepiped formed by a, b and c. It is zero exactly when the vectors are coplanar.

Modulus and argument

For z = a + bi, modulus |z| = √(a² + b²) and argument θ satisfies tan θ = b/a, chosen in the correct quadrant, usually (−π, π].

Polar form multiplication

If z₁ = r₁cis θ₁ and z₂ = r₂cis θ₂, then z₁z₂ = r₁r₂cis(θ₁ + θ₂). Moduli multiply and arguments add; division subtracts arguments.

De Moivre's theorem

(r cis θ)ⁿ = rⁿ cis(nθ) for integer n. It computes powers and, with r set to the nth root of the modulus, the n distinct nth roots.

Conjugate and division

The conjugate of a + bi is a − bi, and z·z̄ = |z|². To divide by z, multiply numerator and denominator by z̄ to obtain a real denominator.

Euler's formula

e^{iθ} = cos θ + i sin θ. It gives cos θ = (e^{iθ} + e^{−iθ})/2 and sin θ = (e^{iθ} − e^{−iθ})/(2i), useful for identities.

Conjugate root theorem

If a polynomial has real coefficients and z is a root, then its conjugate z̄ is also a root. Non-real roots therefore occur in conjugate pairs.

Circle locus in complex plane

The equation |z − z₀| = r describes a circle of radius r centred at z₀, because the modulus is the distance between points in the complex plane.

Perpendicular bisector locus

The equation |z − z₁| = |z − z₂| describes all points equidistant from z₁ and z₂, which form the perpendicular bisector of the segment joining them.

Matrix multiplication condition

The product AB exists only when the number of columns of A equals the number of rows of B; the product then has the rows of A and columns of B.

Inverse of a 2×2 matrix

For A = [[a, b], [c, d]], A⁻¹ = (1/(ad − bc))·[[d, −b], [−c, a]], provided the determinant ad − bc is nonzero; otherwise A is singular.

Determinant and invertibility

A square matrix is invertible exactly when its determinant is nonzero. A zero determinant means the matrix is singular and its columns are linearly dependent.

Solving linear systems

The system Ax = b has a unique solution exactly when det A ≠ 0. If det A = 0, the system has either no solution or infinitely many.

Transpose product rule

For conformable matrices, (AB)ᵀ = BᵀAᵀ. The order of multiplication reverses, and for invertible matrices (A⁻¹)ᵀ = (Aᵀ)⁻¹.

Eigenvalues and eigenvectors

Eigenvalues satisfy det(A − λI) = 0, a polynomial equation in λ. A nonzero vector v is an eigenvector for λ when Av = λv.

Binomial distribution

For X ~ Bin(n, p), P(X = k) = C(n, k)pᵏ(1 − p)ⁿ⁻ᵏ. The mean is np and the variance is np(1 − p); trials are independent.

Standardising a normal variable

If X is normal with mean μ and standard deviation σ, then z = (x − μ)/σ has the standard normal distribution, allowing probabilities to be read from tables.

Standard error of the mean

The sample mean has standard deviation σ/√n, called the standard error. By the central limit theorem it is approximately normal for large n.