Basque PAU Mathematics II cheat sheet

A matrix product AB exists only when columns of A equal rows of B; the result has order (rows of A)×(columns of B).

(AB)^T = B^T A^T and (AB)^{-1} = B^{-1}A^{-1}; order reverses, so never write A^{-1}B^{-1}.

A is invertible iff det(A) ≠ 0; then A^{-1} = adj(A)^T / det(A), where adj(A) is the cofactor matrix.

det(AB) = det(A)·det(B); det(kA) = k^n·det(A) for an n×n matrix A; det(A^T) = det(A).

For 3×3 determinants use Sarrus or cofactor expansion along the row or column with most zeros.

A square system Ax = b is compatible determined iff det(A) ≠ 0; solution x = A^{-1}b.

Rouché–Frobenius: the system is compatible iff rank(A) = rank(A|b); if that common rank equals the number of unknowns it is determined, otherwise indeterminate with n − rank parameters.

Elementary row operations preserve rank but change the determinant; swapping two rows multiplies det by −1, adding a multiple of a row leaves det unchanged.

For a homogeneous system Ax = 0, the trivial solution always exists; nontrivial solutions exist iff det(A) = 0.

u·v = |u||v|cos θ = u1v1 + u2v2 + u3v3; u ⊥ v iff u·v = 0.

|u×v| = |u||v|sin θ equals the area of the parallelogram spanned by u and v; u×v is perpendicular to both.

The scalar triple product [u,v,w] = u·(v×w) gives the signed volume of the parallelepiped; the vectors are coplanar iff it is zero.

Line forms: vector r = a + t·d; parametric x = a1 + t·d1, etc.; continuous (x − a1)/d1 = (y − a2)/d2 = (z − a3)/d3.

Plane: n·(r − p) = 0, i.e. A(x − x0) + B(y − y0) + C(z − z0) = 0 with normal n = (A,B,C).

Distance from point P to plane Ax + By + Cz + D = 0 is |Ax0 + By0 + Cz0 + D| / √(A² + B² + C²).

Distance from point P to line through A with direction d is |AP × d| / |d|.

Angle between two lines uses their direction vectors; angle between a line and a plane uses sin α = |d·n|/(|d||n|).

Line–plane position: substitute the parametric line into the plane equation; one t value means intersecting, no solution means parallel, identity means the line lies in the plane.

lim_{x→a} f(x) = L exists iff both one-sided limits exist and are equal to L.

L'Hôpital applies only to 0/0 or ∞/∞; differentiate numerator and denominator separately, not by the quotient rule.

(1 + 1/n)^n → e and (1 + f(x))^{1/f(x)} → e when f(x) → 0; take logarithms for indeterminate powers 1^∞, 0^0, ∞^0.

f is continuous at a iff lim_{x→a} f(x) = f(a): the limit exists, the value exists, and they coincide.

f is differentiable at a iff the derivative limit exists; differentiability implies continuity, but continuity does not imply differentiability.

Rolle: if f is continuous on [a,b], differentiable on (a,b) and f(a) = f(b), then f′(c) = 0 for some c in (a,b).

Mean value theorem: under continuity on [a,b] and differentiability on (a,b), f′(c) = (f(b) − f(a))/(b − a) for some c.

Asymptotes: vertical where the denominator vanishes without cancellation; horizontal from lim_{x→±∞} f(x); oblique y = mx + n with m = lim f(x)/x and n = lim (f(x) − mx).

For one-sided differentiability at a breakpoint, compare the lateral derivatives; a corner or cusp means non-differentiable.

Fundamental theorem: if F′ = f and f is continuous on [a,b], then ∫_a^b f(x) dx = F(b) − F(a).

Barro's rule (integral mean value): ∫_a^b f(x) dx = (b − a)·f(c) for some c in [a,b] when f is continuous.

Area between curves: integrate (upper − lower) between consecutive intersection points; split the interval where the curves cross.

Volume of revolution about the x-axis: V = π∫_a^b [f(x)]² dx; about the y-axis use shells V = 2π∫_a^b x·f(x) dx.

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

Substitution: ∫f(g(x))g′(x) dx = ∫f(u) du with u = g(x); change the limits when the integral is definite.

Improper integrals: ∫a^∞ f = lim{b→∞} ∫_a^b f; the integral converges iff the limit is finite.

Partial fractions: decompose a proper rational function into simple fractions before integrating; if the degree of the numerator is ≥ the denominator, divide first.

Average value of f on [a,b] is (1/(b − a))∫_a^b f(x) dx.

P(A ∪ B) = P(A) + P(B) − P(A ∩ B); for mutually exclusive events the intersection term is zero.

Conditional probability P(A|B) = P(A ∩ B)/P(B), valid when P(B) > 0; A and B are independent iff P(A ∩ B) = P(A)P(B).

Total probability: P(B) = Σ P(Ai)P(B|Ai) over a partition {Ai}; Bayes then gives P(Ai|B) = P(Ai)P(B|Ai)/P(B).

Binomial B(n,p): P(X = k) = C(n,k)p^k(1−p)^{n−k}, mean np, variance np(1−p).

Normal N(μ,σ): standardize with z = (x − μ)/σ, then use the standard normal table; P(−1 ≤ z ≤ 1) ≈ 0.6827, P(−2 ≤ z ≤ 2) ≈ 0.9545.

Approximating a binomial by a normal is valid for large n; apply the continuity correction of ±0.5.

Sample mean X̄ of size n from N(μ,σ) is N(μ, σ/√n); the standard error is σ/√n.

Confidence interval for the mean with known σ: x̄ ± z_{α/2}·σ/√n; for 95% use z = 1.96.

Standard deviation is the square root of variance; variance is E(X²) − [E(X)]².

The rank of a matrix is the order of the largest non-zero minor; it equals the number of non-zero rows in row-echelon form.

Cramer's rule: x_i = det(A_i)/det(A), where A_i replaces column i of A by the vector b; requires det(A) ≠ 0.

A matrix is symmetric if A = A^T and antisymmetric if A = −A^T; an antisymmetric matrix of odd order has determinant zero.

(A + B)^T = A^T + B^T but (A + B)² ≠ A² + 2AB + B² unless A and B commute.

A matrix is idempotent if A² = A and involutive if A² = I; such conditions let you solve for unknown parameters.

Eigenvalues satisfy det(A − λI) = 0; the trace equals the sum of eigenvalues and the determinant equals their product.

Matrix equations: isolate the unknown matrix using inverse operations, remembering that multiplication does not commute.

Parallel vectors have proportional components; perpendicular vectors have zero dot product.

Equation of a plane through three non-collinear points: normal n = (B − A) × (C − A), then n·(r − A) = 0.

Line–line position in space: parallel if direction vectors are proportional; intersecting if the system has a solution; skew if not parallel and no solution.

Plane–plane intersection is a line whose direction is the cross product of the two normals.

Distance between parallel planes Ax + By + Cz + D1 = 0 and Ax + By + Cz + D2 = 0 is |D1 − D2|/√(A² + B² + C²).

Projection of u onto v is (u·v/|v|²)·v; the symmetric point of P about a plane is found by projecting P along the normal.

Area of a triangle with vertices A, B, C is ½|(B − A) × (C − A)|.

Standard limits: lim_{x→0} sin x/x = 1, lim_{x→0}(1 − cos x)/x² = ½, lim_{x→0}(e^x − 1)/x = 1, lim_{x→0} ln(1 + x)/x = 1.

Derivatives: (sin x)′ = cos x, (cos x)′ = −sin x, (tan x)′ = 1/cos²x, (e^x)′ = e^x, (ln x)′ = 1/x, (arctan x)′ = 1/(1 + x²).

Chain rule: (f∘g)′ = f′(g(x))·g′(x); logarithmic differentiation simplifies products and powers.

Tangent line at x = a: y − f(a) = f′(a)(x − a); the normal has slope −1/f′(a).

Critical points where f′(x) = 0 or undefined; classify with the sign change of f′ or with f″ (f″ > 0 minimum, f″ < 0 maximum).

Inflection points require f″(x) = 0 with a sign change of f″.

A function is increasing where f′ > 0 and decreasing where f′ < 0 on an interval.

Probability density f satisfies f ≥ 0 and ∫f = 1 over its support; P(a ≤ X ≤ b) = ∫_a^b f(x) dx.

For a density, E(X) = ∫x·f(x) dx and Var(X) = ∫x²f(x) dx − [E(X)]².

The uniform distribution on [a,b] has density 1/(b − a), mean (a + b)/2 and variance (b − a)²/12.

The exponential distribution with rate λ has density λe^{−λx} for x ≥ 0, mean 1/λ and variance 1/λ².

The cumulative distribution F(x) = ∫_{−∞}^x f(t) dt is non-decreasing and satisfies F′(x) = f(x) at continuity points.

For a normal approximation, use z-scores rounded to two decimals and read the table for the area to the left of z.

Regression line of y on x: y = ȳ + (cov(x,y)/var(x))(x − x̄); the correlation coefficient r lies in [−1, 1].