PAU Applied Mathematics for Social Sciences II - Canary Islands / ULPGC & ULL cheat sheet

A system Ax = b is consistent iff rank(A) = rank(A|b); if that common rank r equals the number of unknowns n the solution is unique, if r < n there are infinitely many solutions with n − r free parameters.

For a square matrix A: det A ≠ 0 ⇔ A is invertible ⇔ rank(A) = n ⇔ Ax = 0 has only the trivial solution ⇔ Ax = b has a unique solution for every b.

Cramer's rule: if det A ≠ 0, then x_i = det(A_i)/det A, where A_i is A with column i replaced by b; the rule requires as many equations as unknowns.

Determinant properties: det(AB) = det A · det B, det(Aᵀ) = det A, det(A⁻¹) = 1/det A, det(kA) = kⁿ det A for n×n; a zero row or two equal rows give det = 0.

Gauss–Jordan elimination reduces the augmented matrix to row-echelon form; the number of nonzero rows of the reduced coefficient part equals rank(A), and pivot columns identify basic variables.

Matrix equation AX = B with A invertible is solved by X = A⁻¹B; for XA = B use X = BA⁻¹, since matrix multiplication is not commutative.

(A + B)ᵀ = Aᵀ + Bᵀ and (AB)ᵀ = BᵀAᵀ; the inverse of a product reverses order: (AB)⁻¹ = B⁻¹A⁻¹.

For 2×2 matrices, A⁻¹ = (1/det A)·[[d, −b], [−c, a]] when A = [[a, b], [c, d]] and det A ≠ 0.

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

Standard two-variable form: optimise z = ax + by subject to linear inequalities plus x ≥ 0, y ≥ 0; the feasible region is the intersection of half-planes, a convex polygon (possibly empty or unbounded).

Fundamental theorem: if the feasible region is a nonempty bounded polygon and the objective is linear, the optimum is attained at a vertex; evaluate z at every vertex and compare.

A vertex is found by solving the 2×2 system formed by the two boundary lines that meet there; discard intersections that violate any constraint.

Corner-point method procedure: graph each constraint as an equality, test a point to fix the half-plane, list vertices, evaluate z, and select max or min.

Level curves of z = ax + by are parallel lines ax + by = k; the optimum direction is the gradient (a, b), and if a level line is parallel to an edge, every point of that edge is optimal (multiple optima).

An unbounded feasible region may still have a finite optimum; check whether the objective improves without bound along an open edge before declaring the problem unbounded.

Slack variables convert a ≤ constraint into an equality, and each slack measures unused resource; a binding constraint has slack zero at the optimum.

Duality: the dual of max z = cᵀx with Ax ≤ b, x ≥ 0 is min w = bᵀy with Aᵀy ≥ c, y ≥ 0; at the optimum z* = w* (strong duality).

Sensitivity: the shadow price of a resource is the dual variable, i.e. the rate of change of the optimal value per unit increase in that right-hand side, valid while the basis stays optimal.

Indeterminacies 0/0 and ∞/∞: factor and cancel, multiply by the conjugate for radicals, or apply L'Hôpital's rule when the quotient is differentiable and still indeterminate.

For rational functions at infinity, compare degrees: numerator degree larger gives ±∞, equal gives the ratio of leading coefficients, denominator larger gives 0.

(1 + 1/n)ⁿ → e and (1 + f(x))^{1/f(x)} → e when f(x) → 0; take logarithms for indeterminacies of type 1^∞, 0⁰, ∞⁰.

f is continuous at a iff lim_{x→a} f(x) = f(a); continuity requires the limit to exist, be finite, and equal the function value.

Derivative as a limit: f'(a) = lim_{h→0} [f(a+h) − f(a)]/h; it is the slope of the tangent line y − f(a) = f'(a)(x − a).

Rules: (uv)' = u'v + uv', (u/v)' = (u'v − uv')/v², and the chain rule [f(g(x))]' = f'(g(x))·g'(x).

Standard derivatives: (xⁿ)' = nxⁿ⁻¹, (eˣ)' = eˣ, (ln x)' = 1/x, (aˣ)' = aˣ ln a, (sin x)' = cos x, (cos x)' = −sin x.

A function is differentiable at a only if it is continuous there; continuity does not imply differentiability (e.g. |x| at 0).

Average rate of change on [a, b] is [f(b) − f(a)]/(b − a); the instantaneous rate at a point is f'(a), the marginal value in economic models.

f is increasing where f' > 0 and decreasing where f' < 0; critical points are where f' = 0 or f' does not exist.

First-derivative test: if f' changes from + to − at c, f has a local maximum at c; from − to +, a local minimum; no sign change means no extremum.

Second-derivative test: at a critical point with f'(c) = 0, f''(c) < 0 gives a local maximum and f''(c) > 0 gives a local minimum; f''(c) = 0 is inconclusive.

f is concave up (convex) where f'' > 0 and concave down where f'' < 0; inflection points are where concavity changes, so f'' = 0 or fails to exist and changes sign.

Global extrema on a closed interval [a, b]: evaluate f at all critical points in (a, b) and at both endpoints; the largest and smallest values are the global extrema.

Vertical asymptotes occur where the denominator vanishes without cancellation; horizontal asymptotes come from limits at ±∞; oblique asymptotes arise when numerator degree exceeds denominator degree by one.

Curve sketching order: domain and symmetries, intercepts, asymptotes, sign of f', sign of f'', then plot critical and inflection points.

Economic optimisation: profit P(x) = I(x) − C(x) is maximised where marginal revenue equals marginal cost, I'(x) = C'(x), provided P'' < 0.

Marginal cost C'(x) approximates the cost of the next unit; average cost C(x)/x is minimised where marginal cost equals average cost.

An antiderivative F of f satisfies F' = f; all antiderivatives are F(x) + C, and the indefinite integral is written ∫f(x)dx = F(x) + C.

Barrow's rule (fundamental theorem): if F is an antiderivative of f and f is continuous on [a, b], then ∫_a^b f(x)dx = F(b) − F(a).

Basic integrals: ∫xⁿdx = xⁿ⁺¹/(n+1) + C for n ≠ −1, ∫1/x dx = ln|x| + C, ∫eˣdx = eˣ + C, ∫aˣdx = aˣ/ln a + C.

Linearity: ∫[k f(x) ± g(x)]dx = k∫f(x)dx ± ∫g(x)dx; constants factor out of definite integrals.

Area between curves: if f(x) ≥ g(x) on [a, b], the area is ∫_a^b [f(x) − g(x)]dx; find intersection points to set the limits and split the interval where the upper curve changes.

Additivity of the definite integral: ∫_a^b f = ∫_a^c f + ∫_c^b f; also ∫_a^a f = 0 and ∫_a^b f = −∫_b^a f.

Accumulation function A(x) = ∫_a^x f(t)dt satisfies A'(x) = f(x); if f is a rate, the definite integral gives the total change over the interval.

In economic applications, the definite integral of marginal cost between quantities gives total variable cost, and the integral of marginal revenue gives total revenue change.

For an area below the x-axis the definite integral is negative; geometric area uses the absolute value, so split at the zeros of f before integrating.

Axioms: 0 ≤ P(A) ≤ 1, P(Ω) = 1, and P(A ∪ B) = P(A) + P(B) − P(A ∩ B); for disjoint events the intersection term vanishes.

Complement rule: P(Aᶜ) = 1 − P(A); also P(A ∩ Bᶜ) = P(A) − P(A ∩ B).

Conditional probability: P(A|B) = P(A ∩ B)/P(B) with P(B) > 0; rearranging gives the product rule P(A ∩ B) = P(B)·P(A|B).

Independence: A and B are independent iff P(A ∩ B) = P(A)P(B), equivalently P(A|B) = P(A); disjoint events with positive probability are never independent.

Total probability: for a partition {B₁, …, Bₙ} of Ω, P(A) = Σ P(Bᵢ)·P(A|Bᵢ).

Bayes' theorem: P(Bᵢ|A) = P(Bᵢ)P(A|Bᵢ) / Σ_j P(Bⱼ)P(A|Bⱼ); the denominator is the total probability of A.

Combinatorics: permutations P(n, r) = n!/(n−r)! count ordered selections, combinations C(n, r) = n!/[r!(n−r)!] count unordered ones; with repetition, nʳ ordered sequences.

Laplace's rule applies only when outcomes are equally likely: P(A) = number of favourable outcomes / total number of outcomes.

Sampling: with replacement the draws are independent; without replacement, use the product rule with changing conditional probabilities.

A discrete variable has a probability function p(xᵢ) with p ≥ 0 and Σp(xᵢ) = 1; a continuous variable has a density f with f ≥ 0, ∫f = 1, and P(a ≤ X ≤ b) = ∫_a^b f(x)dx.

Expectation: E(X) = Σ xᵢp(xᵢ) for discrete, E(X) = ∫x f(x)dx for continuous; E(aX + b) = aE(X) + b.

Variance: Var(X) = E(X²) − [E(X)]², and Var(aX + b) = a²Var(X); the standard deviation is σ = √Var(X).

Binomial B(n, p): P(X = k) = C(n, k)pᵏ(1−p)^{n−k}, with E(X) = np and Var(X) = np(1−p); it models the number of successes in n independent trials.

Normal N(μ, σ): standardise with Z = (X − μ)/σ, then P(X ≤ x) = P(Z ≤ (x − μ)/σ) and read values from the standard normal table Φ.

Symmetry of the normal curve: Φ(−z) = 1 − Φ(z), and P(a ≤ X ≤ b) = Φ((b−μ)/σ) − Φ((a−μ)/σ).

Normal approximation to the binomial is valid when np > 5 and n(1−p) > 5; apply the continuity correction by extending the interval by 0.5 before standardising.

For the continuous case P(X = c) = 0, so P(a ≤ X ≤ b) = P(a < X < b); the density f(x) is not itself a probability and may exceed 1.

The distribution function F(x) = P(X ≤ x) is nondecreasing with limits 0 at −∞ and 1 at +∞; for continuous X, F'(x) = f(x).