PAU Mathematics II (Galicia) cheat sheet

Matrix product AB exists only when columns of A equal rows of B; (AB)^T = B^T A^T and (AB)^{-1} = B^{-1}A^{-1} when both are invertible.

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

For 3×3 matrices, det(A) = a11(a22a33 − a23a32) − a12(a21a33 − a23a31) + a13(a21a32 − a22a31); Sarrus' rule gives the same value.

Determinant properties: det(AB) = det(A)det(B), det(A^T) = det(A), det(kA) = k^n det(A) for n×n, det(A^{-1}) = 1/det(A).

Rouché–Capelli: a system AX = B is consistent iff rank(A) = rank(A|B). If this common rank equals the number of unknowns, the solution is unique; if less, there are infinitely many solutions with n − r free parameters.

Cramer's rule for a square consistent system with det(A) ≠ 0: x_i = det(A_i)/det(A), where A_i replaces column i by the constant vector B.

Elementary row operations (swap rows, multiply a row by a nonzero scalar, add a multiple of one row to another) preserve the solution set; use Gauss elimination to reach row echelon form.

A homogeneous system AX = 0 is always consistent; it has nontrivial solutions iff rank(A) < n, i.e. iff det(A) = 0 for square A.

rank(A) equals the order of the largest nonvanishing minor, and also the number of nonzero rows in row echelon form.

Given A(x1,y1,z1) and B(x2,y2,z2), vector AB = (x2−x1, y2−y1, z2−z1); its modulus is √(dx²+dy²+dz²).

Dot product: u·v = u1v1+u2v2+u3v3 = |u||v|cos θ; u ⊥ v iff u·v = 0. Orthogonal projection of u onto v is (u·v/|v|²)v.

Cross product u×v is perpendicular to both; |u×v| = |u||v|sin θ equals the area of the parallelogram spanned by u and v. Its components are (u2v3−u3v2, u3v1−u1v3, u1v2−u2v1).

u, v, w are coplanar iff the scalar triple product [u,v,w] = u·(v×w) = 0; its absolute value gives the volume of the parallelepiped.

Line forms: vector r = a + t·d; parametric x = a1+td1, y = a2+td2, z = a3+td3; continuous (x−a1)/d1 = (y−a2)/d2 = (z−a3)/d3.

Plane through point P0 with normal n = (A,B,C): A(x−x0)+B(y−y0)+C(z−z0) = 0, i.e. Ax+By+Cz+D = 0.

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

Distance from point P to plane Ax+By+Cz+D = 0 is |Ax0+By0+Cz0+D|/√(A²+B²+C²); distance from P to line through A with direction d is |AP×d|/|d|.

Relative positions: two lines may be coincident, parallel, intersecting, or skew (check whether d1×d2 = 0, then whether the triple product of d1, d2 and a connecting vector vanishes).

L'Hôpital's rule applies to 0/0 and ∞/∞ indeterminate forms: lim f/g = lim f'/g', provided the latter limit exists.

Notable limits: lim(x→0) sin x/x = 1; lim(x→0)(1−cos x)/x² = 1/2; lim(x→0)(e^x−1)/x = 1; lim(x→0) ln(1+x)/x = 1; lim(x→∞)(1+1/x)^x = e.

A function is continuous at a iff lim(x→a⁻) f = lim(x→a⁺) f = f(a); differentiability at a implies continuity there, but not conversely.

Derivative rules: (fg)' = f'g+fg'; (f/g)' = (f'g−fg')/g²; chain rule (f∘g)' = f'(g(x))·g'(x).

Standard derivatives: (x^n)' = nx^{n−1}; (e^x)' = e^x; (ln x)' = 1/x; (sin x)' = cos x; (cos x)' = −sin x; (tan x)' = 1/cos²x; (arctan x)' = 1/(1+x²).

Fermat's theorem: if f has a local extremum at an interior point and is differentiable there, then f'(a) = 0. Critical points plus endpoints are the candidates for extrema.

Rolle's theorem needs continuity on [a,b], differentiability on (a,b) and f(a) = f(b); the mean value theorem then gives f'(c) = (f(b)−f(a))/(b−a) for some c in (a,b).

Concavity: f'' > 0 means convex (concave up), f'' < 0 means concave down; inflection points require a sign change of f''.

Asymptotes: vertical where the denominator vanishes with a nonfinite limit; horizontal if lim(x→±∞) f = L; oblique y = mx+n with m = lim f(x)/x and n = lim (f(x)−mx).

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

Barrett's rule (reduction): ∫ f^n g' dx = f^n g − n∫ f^{n−1} f' g dx; integration by parts is ∫ u dv = uv − ∫ v du.

Area between curves: A = ∫_a^b |f(x) − g(x)| dx, where a and b are the intersection abscissas of the two curves.

Volume of revolution about the x-axis: V = π∫_a^b [f(x)]² dx (disc method). About the y-axis with f invertible: V = π∫_c^d [g(y)]² dy, where g is the inverse.

Volume by parallel cross-sections: V = ∫_a^b A(x) dx, where A(x) is the cross-sectional area perpendicular to the x-axis.

Arc length of y = f(x) on [a,b]: L = ∫_a^b √(1+[f'(x)]²) dx.

Improper integrals: ∫_1^∞ dx/x^p converges iff p > 1; ∫_0^1 dx/x^p converges iff p < 1.

Even/odd symmetry: ∫_{−a}^{a} f = 2∫_0^a f if f is even, and 0 if f is odd.

Substitution: ∫ f(g(x))g'(x)dx = ∫ f(u)du with u = g(x); change the limits accordingly for definite integrals.

Permutations without repetition: P(n,r) = n!/(n−r)!; combinations: C(n,r) = n!/(r!(n−r)!). Permutations with repetition of a multiset: n!/(n1!n2!…nk!).

P(A∪B) = P(A)+P(B)−P(A∩B); for mutually exclusive events the last term vanishes.

Conditional probability: P(A|B) = P(A∩B)/P(B); A and B are independent iff P(A∩B) = P(A)P(B), equivalently P(A|B) = P(A).

Total probability: P(B) = Σ P(A_i)P(B|A_i) for a partition {A_i}; Bayes' theorem gives P(A_i|B) = P(A_i)P(B|A_i)/P(B).

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

Binomial conditions: fixed number n of independent trials, two outcomes per trial, constant success probability p.

Geometric distribution: P(X = k) = (1−p)^{k−1}p, the probability that the first success occurs on trial k; mean 1/p.

Hypergeometric sampling without replacement from N items with K successes: P(X = k) = C(K,k)C(N−K,n−k)/C(N,n).

Complement rule: P(A) = 1 − P(A^c); use it when 'at least one' appears in the event.

Sampling distribution of the mean: E(X̄) = μ and Var(X̄) = σ²/n; by the central limit theorem X̄ is approximately normal for large n regardless of the population shape.

Confidence interval for μ with known σ: x̄ ± z_{α/2}·σ/√n; with unknown σ and small n, use x̄ ± t_{n−1,α/2}·s/√n.

Confidence interval for a proportion: p̂ ± z_{α/2}√(p̂(1−p̂)/n), valid for large n.

Margin of error E = z_{α/2}σ/√n; the sample size needed for a target E is n = (z_{α/2}σ/E)², rounded up.

Common critical values: z_{0.025} = 1.96 for 95% confidence, z_{0.05} = 1.645 for one-tailed α = 0.05, z_{0.005} = 2.576 for 99% confidence.

Hypothesis test for μ with known σ: statistic Z = (x̄ − μ0)/(σ/√n), compared with the normal distribution; for unknown σ use T = (x̄ − μ0)/(s/√n) with n−1 degrees of freedom.

The p-value is the probability, under H0, of a result at least as extreme as the observed one; reject H0 when p < α.

Errors: a type I error rejects a true H0 (probability α); a type II error fails to reject a false H0 (probability β); power is 1 − β.

One-tailed tests place α entirely in one tail (z_α); two-tailed tests split α/2 into each tail (z_{α/2}).