PAU Applied Math ACS II (La Rioja) cheat sheet

Order of a matrix is rows × columns; A(m×n)·B(n×p) is defined and gives order m×p. Matrix product is not commutative: AB ≠ BA in general.

Inverse exists only for square matrices with det(A) ≠ 0; then A⁻¹ = adj(A)/det(A) and A·A⁻¹ = I. Singular means det(A) = 0.

2×2 inverse formula: for A = [[a,b],[c,d]], A⁻¹ = (1/(ad−bc))·[[d,−b],[−c,a]].

Rouché–Fröbenius: system AX = B is consistent iff rank(A) = rank(A|B). If rank = number of unknowns, unique solution; if rank < unknowns, infinitely many.

For homogeneous systems AX = 0 consistency is automatic; nontrivial solutions exist iff rank(A) < number of unknowns, i.e. det(A) = 0 for square A.

Cramer's rule: xᵢ = det(Aᵢ)/det(A), where Aᵢ replaces column i of A by B; valid only when det(A) ≠ 0.

Rank = number of nonzero rows in row-echelon form = order of largest nonvanishing minor; row operations preserve rank.

Matrix equation AX + B = C solves as X = A⁻¹(C − B) when A is invertible; check dimensions before multiplying.

Standard form: optimize z = c₁x₁ + … + cₙxₙ subject to linear inequalities (constraints) and xᵢ ≥ 0 (non-negativity).

Feasible region of a two-variable problem is the intersection of half-planes; it is a convex polygon (possibly unbounded or empty).

Fundamental theorem: if a finite optimum exists, at least one optimal solution is a vertex of the feasible region. Evaluate z at every vertex and compare.

Vertices are found by intersecting pairs of constraint boundary lines and keeping points that satisfy all constraints simultaneously.

Bounded feasible region ⇒ both maximum and minimum exist. Unbounded region ⇒ objective may grow without limit in some direction; test vertices and rays.

Slope rule: for z = ax + by, the level lines have slope −a/b; the optimum vertex is where this slope fits between the slopes of the two edges meeting there.

A constraint is redundant if removing it does not change the feasible region; it never determines an optimal vertex.

Integer programming variant: after solving the continuous problem, round and test feasible integer points near the optimum; the LP bound is not always attainable.

lim(x→a) f(x) exists iff both one-sided limits exist and are equal; value of f(a) itself is irrelevant to the limit.

Indeterminations 0/0 and ∞/∞: apply L'Hôpital (lim f/g = lim f′/g′) or factor/divide by highest power. For x→∞ rational functions, compare degrees.

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

f is continuous at a iff lim(x→a) f(x) = f(a): the limit exists, f(a) is defined, and they coincide. Otherwise classify as removable, jump, or essential.

Derivative definition: f′(a) = lim(h→0) [f(a+h) − f(a)]/h; differentiability at a implies continuity at a, but not conversely.

Rolle's theorem: f continuous on [a,b], differentiable on (a,b), f(a) = f(b) ⇒ exists c with f′(c) = 0. Mean value theorem drops the f(a)=f(b) condition.

Monotonicity: f′ > 0 on an interval ⇒ increasing; f′ < 0 ⇒ decreasing. Critical points are where f′ = 0 or f′ does not exist.

Second derivative test: f′(c)=0 and f″(c)>0 ⇒ local minimum; f″(c)<0 ⇒ local maximum; f″(c)=0 ⇒ inconclusive, use sign change of f′.

Concavity and inflection: f″ > 0 ⇒ convex (concave up); f″ < 0 ⇒ concave down; inflection points require a change of sign of f″.

Global extrema on [a,b]: compare f at critical points inside (a,b) and at the endpoints a and b.

Barro's rule (fundamental theorem): if F′ = f, then ∫ₐᵇ f(x) dx = F(b) − F(a); the definite integral is independent of the chosen primitive.

Area between y = f(x) and the x-axis on [a,b] is ∫ₐᵇ |f(x)| dx; split at the zeros of f so each piece is positive.

Area between two curves: ∫ₐᵇ [f(x) − g(x)] dx where f is the upper curve on that interval; split where the curves cross.

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

Substitution: ∫ f(g(x))g′(x) dx = ∫ f(u) du with u = g(x); for definite integrals change the limits to u-values.

Improper integrals: ∫ₐ^∞ f = lim(b→∞) ∫ₐᵇ f; convergent iff the limit is finite. ∫₁^∞ 1/xᵖ converges iff p > 1.

Volumes of revolution about OX: V = π∫ₐᵇ [f(x)]² dx (disc method); about OY with inverse function use V = π∫ [x(y)]² dy.

Even/odd symmetry: if f is odd, ∫₋ₐᵃ f = 0; if f is even, ∫₋ₐᵃ f = 2∫₀ᵃ f.

Axioms: P(Ω)=1, P(∅)=0, 0 ≤ P(A) ≤ 1; P(Aᶜ) = 1 − P(A); P(A∪B) = P(A) + P(B) − P(A∩B).

Conditional probability: P(A|B) = P(A∩B)/P(B), P(B) > 0. 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(Aᵢ)P(B|Aᵢ) over a partition. Bayes: P(Aᵢ|B) = P(Aᵢ)P(B|Aᵢ) / Σ P(Aⱼ)P(B|Aⱼ).

Discrete random variable: probability function p(xᵢ) ≥ 0 with Σp(xᵢ)=1; mean μ = Σxᵢp(xᵢ); variance σ² = Σxᵢ²p(xᵢ) − μ².

Binomial B(n,p): P(X=k) = C(n,k)pᵏ(1−p)ⁿ⁻ᵏ, μ = np, σ² = np(1−p); models number of successes in n independent trials.

Continuous variable: density f(x) ≥ 0 with ∫f = 1; P(a ≤ X ≤ b) = ∫ₐᵇ f(x)dx; P(X = c) = 0 always.

Normal N(μ,σ): standardize with Z = (X−μ)/σ; use tables for P(Z ≤ z). For sample means, σ is replaced by σ/√n.

Mean and variance properties: E(aX+b) = aE(X)+b; Var(aX+b) = a²Var(X); Var(X) = E(X²) − [E(X)]².

Sampling distribution of the mean: for samples of size n from N(μ,σ), X̄ ~ N(μ, σ/√n); σ/√n is the standard error.

Central limit theorem: for large n (n ≥ 30) X̄ is approximately normal regardless of the population distribution.

Confidence interval for μ with known σ: x̄ ± z_(α/2)·σ/√n, where z_(α/2) is the standard normal critical value (1.96 at 95%, 2.576 at 99%).

Unknown σ with small n: use t distribution with n−1 degrees of freedom: x̄ ± t_(n−1,α/2)·s/√n, where s is the sample standard deviation.

Maximum error of estimate E = z_(α/2)·σ/√n; required sample size n = [z_(α/2)·σ/E]², always rounded up.

Confidence level 1−α means the procedure captures μ in 100(1−α)% of repeated samples; it is not the probability that this particular interval contains μ.

Raising confidence level widens the interval; increasing n or reducing σ narrows it. Width = 2E.

Confidence interval for a proportion p: p̂ ± z_(α/2)·√(p̂(1−p̂)/n), valid when np̂ ≥ 5 and n(1−p̂) ≥ 5.