Castilla-La Mancha PAU Applied Mathematics for Social Sciences II (Matemáticas Aplicadas a las Ciencias Sociales II - UCLM 2026) cheat sheet

Matrix product AB is defined only when columns of A equal rows of B; the result has size (rows of A) × (columns of B).

(AB)ᵀ = BᵀAᵀ and (AB)⁻¹ = B⁻¹A⁻¹; matrix product is not commutative, so AB ≠ BA in general.

A square matrix A is invertible iff det(A) ≠ 0; then A⁻¹ = adj(A)/det(A), where adj(A) is the transpose of the cofactor matrix.

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

For a homogeneous system AX = 0 consistency is automatic; nontrivial solutions exist iff rank(A) < n.

Cramer's rule: xᵢ = det(Aᵢ)/det(A), where Aᵢ replaces column i of A by the independent-term vector; requires det(A) ≠ 0.

A 2×2 determinant: det[[a,b],[c,d]] = ad − bc; a 3×3 determinant expands by any row or column with alternating signs +,−,+.

Gauss elimination: use elementary row operations to reach echelon form, then solve by back-substitution; number of pivots equals the rank.

Feasible region = intersection of the half-planes defined by the constraints plus the non-negativity conditions x ≥ 0, y ≥ 0.

A constraint ax + by ≤ c is satisfied by the half-plane containing the origin when c ≥ 0; test a point to decide the side.

Vertices of the feasible region are found by intersecting pairs of boundary lines and keeping only points that satisfy every constraint.

Fundamental theorem: if the objective function is linear and the feasible region is non-empty and bounded, the optimum occurs at a vertex (or along a whole edge).

Corner method: evaluate z = ax + by at every vertex; the largest value is the maximum and the smallest is the minimum.

Gradient method: the level lines ax + by = k are parallel; the optimum direction is given by the vector (a, b).

Unbounded region: maximum may not exist (z grows without limit); minimum may not exist if the region is unbounded in the decreasing direction.

An integer-programming variant requires the optimal vertex coordinates to be whole numbers; if not, test nearby feasible integer points.

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

Indeterminate forms 0/0 and ∞/∞ are resolved by factoring, rationalising, or L'Hôpital's rule (valid when the quotient is 0/0 or ∞/∞ and the derivative quotient has a limit).

Elementary equivalence: lim(x→0) sen x / x = 1 and lim(x→0) (1 + x)^(1/x) = e; also lim(x→∞) (1 + 1/x)^x = e.

A function f is continuous at x = a iff lim(x→a) f(x) = f(a); equivalently, the three conditions (f(a) defined, limit exists, they coincide) all hold.

Types of discontinuity: removable (finite limit ≠ value or value undefined), jump (different one-sided limits), essential/infinite (limit is infinite or does not exist).

Bolzano's theorem: if f is continuous on [a, b] and f(a)·f(b) < 0, there exists c in (a, b) with f(c) = 0.

Weierstrass theorem: a continuous function on a closed bounded interval attains its absolute maximum and minimum there.

At infinity: rational functions behave like the quotient of leading terms; if degrees are equal the limit is the ratio of leading coefficients.

Definition: f'(a) = lim(h→0) [f(a+h) − f(a)]/h; geometrically it is the slope of the tangent line at x = a.

Tangent line at x = a: y = f(a) + f'(a)(x − a).

Basic derivatives: (xⁿ)' = n·xⁿ⁻¹, (eˣ)' = eˣ, (ln x)' = 1/x, (sen x)' = cos x, (cos x)' = −sen x.

Product rule: (u·v)' = u'v + uv'; quotient rule: (u/v)' = (u'v − uv')/v².

Chain rule: [f(g(x))]' = f'(g(x))·g'(x); for example (e^{u})' = e^{u}·u' and (ln u)' = u'/u.

Derivative of a constant is 0; derivative of k·f is k·f'; derivative of a sum is the sum of derivatives.

Differentiability implies continuity, but continuity does not imply differentiability (e.g., f(x) = |x| at x = 0).

A function is differentiable on an open interval where its derivative exists at every point; check one-sided derivatives at breakpoints of piecewise functions.

f is increasing on an interval where f'(x) > 0 and decreasing where f'(x) < 0; solve f'(x) = 0 to find critical points.

First-derivative test: if f' changes from + to − at c, f has a local maximum at c; from − to + gives a local minimum.

Second-derivative test: if f'(c) = 0 and f''(c) < 0, local maximum at c; if f''(c) > 0, local minimum; if f''(c) = 0 the test is inconclusive.

Concavity: f is concave up (convex) where f''(x) > 0 and concave down where f''(x) < 0.

Inflection points occur where concavity changes, i.e., where f'' changes sign (f''(c) = 0 is necessary, not sufficient).

Absolute extrema on [a, b]: evaluate f at critical points in (a, b) and at the endpoints a and b; the largest and smallest values are the absolute extrema.

Optimisation word problems: build the objective function, use constraints to reduce to one variable, differentiate, solve, and verify with the second derivative.

Marginal analysis: if C(x) is a cost function, marginal cost is C'(x); profit P(x) = I(x) − C(x) is maximised when I'(x) = C'(x), provided P''(x) < 0.

Indefinite integral: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C for n ≠ −1; ∫1/x dx = ln|x| + C; ∫eˣ dx = eˣ + C.

Linearity: ∫[k·f(x) ± g(x)] dx = k∫f(x) dx ± ∫g(x) dx.

Fundamental theorem of calculus: if F is an antiderivative of f, then ∫ₐᵇ f(x) dx = F(b) − F(a).

Definite integral properties: ∫ₐᵃ f = 0; ∫ₐᵇ f = −∫ᵦᵃ f; ∫ₐᵇ f = ∫ₐᶜ f + ∫꜀ᵇ f.

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

Area between two curves y = f(x) and y = g(x) on [a, b], with f ≥ g, is ∫ₐᵇ [f(x) − g(x)] dx.

Change of variables: if u = g(x), then ∫ f(g(x))·g'(x) dx = ∫ f(u) du; adjust limits when evaluating definite integrals.

A definite integral of a positive function on [a, b] gives a positive number; a negative result signals that the function lies below the axis on that interval.

Mean: x̄ = (Σxᵢ)/n; variance s² = (Σxᵢ²)/n − x̄² (population) or divided by n − 1 (sample); standard deviation s = √s².

Coefficient of variation CV = s/x̄ (×100 for percent); used to compare dispersion between variables with different units or very different means.

Median: middle value of the ordered data (average of the two central values if n is even); mode: most frequent value.

Quartiles Q₁, Q₂, Q₃ divide ordered data into four equal parts; interquartile range IQR = Q₃ − Q₁.

Pearson's linear correlation coefficient r = σₓᵧ/(σₓ·σᵧ), where σₓᵧ = (Σxᵢyᵢ)/n − x̄·ȳ; −1 ≤ r ≤ 1.

|r| close to 1 indicates strong linear association; r = 0 indicates no linear correlation (not necessarily independence).

Least-squares regression line of y on x: ŷ = ȳ + (σₓᵧ/σₓ²)(x − x̄); slope b = σₓᵧ/σₓ², intercept a = ȳ − b·x̄.

Coefficient of determination r² is the proportion of variance in y explained by the linear regression on x.

Covariance sign matches the sign of r; positive covariance means x and y tend to move together.

P(A ∪ B) = P(A) + P(B) − P(A ∩ B); if A and B are mutually exclusive, P(A ∩ B) = 0.

Conditional probability: P(A|B) = P(A ∩ B)/P(B), defined when P(B) > 0.

Independence: P(A ∩ B) = P(A)·P(B), equivalently P(A|B) = P(A) and P(B|A) = P(B).

Total probability theorem: if B₁, …, Bₙ form a partition of the sample space, then P(A) = Σ P(Bᵢ)·P(A|Bᵢ).

Bayes' theorem: P(Bⱼ|A) = P(Bⱼ)·P(A|Bⱼ) / Σ P(Bᵢ)·P(A|Bᵢ).

Complement rule: P(Ā) = 1 − P(A); P(A ∩ B̄) = P(A) − P(A ∩ B).

Tree diagram: multiply along branches to get joint probabilities, add across branches to get total probability.

Independent events do not imply mutually exclusive events; mutually exclusive events with positive probability are never independent.

Discrete random variable: mean μ = Σxᵢ·P(X = xᵢ); variance σ² = Σxᵢ²·P(X = xᵢ) − μ².

Binomial B(n, p): P(X = k) = C(n, k)·pᵏ·(1−p)ⁿ⁻ᵏ; mean μ = np; variance σ² = np(1−p).

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

Normal N(μ, σ): standardise with z = (x − μ)/σ, then use standard normal tables for P(Z ≤ z).

P(a ≤ X ≤ b) = P(Z ≤ (b−μ)/σ) − P(Z ≤ (a−μ)/σ); use symmetry P(Z ≤ −z) = 1 − P(Z ≤ z).

Standard normal: P(−1 ≤ Z ≤ 1) ≈ 0.6827; P(−2 ≤ Z ≤ 2) ≈ 0.9545; P(−3 ≤ Z ≤ 3) ≈ 0.9973.

Binomial approximation by normal: if np ≥ 5 and n(1−p) ≥ 5, use N(np, √(np(1−p))) with continuity correction ±0.5.

A normal random variable is continuous, so P(X = x) = 0 for every single value x.