PAU Applied Mathematics for Social Sciences II (Cantabria) cheat sheet

Matrix product AB is defined only when columns of A equal rows of B; entry (i,j) of AB is the dot product of row i of A with column j of B, so generally AB ≠ BA.

Inverse of a 2×2 matrix A = [[a,b],[c,d]] is (1/(ad−bc))·[[d,−b],[−c,a]], valid only when det A = ad − bc ≠ 0.

A square matrix is invertible if and only if det A ≠ 0; det A = 0 means the columns are linearly dependent and the matrix is singular.

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

Rouché–Fröbenius: a system AX = B is inconsistent if rank(A) < rank(A|B); it is consistent with a unique solution if rank(A) = rank(A|B) = n (number of unknowns).

If rank(A) = rank(A|B) = r < n, the system has infinitely many solutions depending on n − r free parameters.

For a homogeneous system AX = 0 consistency is automatic; nontrivial solutions exist exactly when rank(A) < n, i.e. det A = 0 for square A.

Cramer's rule for n equations in n unknowns: xᵢ = det(Aᵢ)/det(A), where Aᵢ replaces column i of A by the constant vector B; requires det A ≠ 0.

Matrix rank equals the number of nonzero rows in any row-echelon form; elementary row operations (swapping, scaling, adding a multiple) preserve rank and the solution set.

A linear program in two variables maximizes or minimizes z = ax + by subject to linear constraints plus non-negativity x ≥ 0, y ≥ 0; the feasible region is a convex polygon.

Fundamental theorem: if the feasible region is nonempty and bounded and the objective is linear, the optimum is attained at a vertex (corner point) of the feasible region.

Graphical method: draw each constraint as a line, shade the half-plane satisfying it, find the vertices of the intersection polygon, then evaluate z at each vertex and compare.

A constraint written ax + by ≤ c is satisfied by the half-plane containing the origin when c ≥ 0; test the point (0,0) to decide which side to shade.

If the feasible region is unbounded, the objective may have no finite maximum; check whether z improves without bound along an open direction.

A vertex is the intersection of two constraint lines; solve the 2×2 system and keep the point only if it satisfies all remaining constraints.

The optimal value is unique in value, but if the objective is parallel to a binding constraint edge, every point of that edge is optimal (multiple optimal solutions).

Slack variables convert ax + by ≤ c into ax + by + s = c with s ≥ 0; the simplex method moves between basic feasible solutions improving z at each step.

Integer programming adds x, y ∈ ℤ; the integer optimum is not simply the rounded continuous optimum, so vertices of the integer hull must be checked.

lim(x→a) f(x) exists and equals L if and only if both one-sided limits exist and are equal: lim(x→a⁻) f(x) = lim(x→a⁺) f(x) = L.

f is continuous at x = a when three conditions hold: f(a) is defined, lim(x→a) f(x) exists, and the limit equals f(a).

Indeterminate forms 0/0 and ∞/∞ are resolved with L'Hôpital's rule (for differentiable f, g near a) or by algebraic factorization and cancellation.

lim(x→±∞) of a rational function is determined by the highest-degree terms: same degree gives ratio of leading coefficients, denominator degree higher gives 0, numerator higher gives ±∞.

A removable discontinuity occurs when the limit exists but differs from f(a) or f(a) is undefined; a jump discontinuity occurs when one-sided limits differ.

Exponential limits: lim(x→∞)(1 + 1/x)^x = e and lim(x→0)(1 + x)^(1/x) = e; lim(x→0) sin x / x = 1.

Intermediate Value Theorem: if f is continuous on [a,b] and k lies between f(a) and f(b), there exists c ∈ (a,b) with f(c) = k.

Domain of a function excludes points where a denominator vanishes and requires the radicand of an even root to be non-negative; the range is the set of attained values.

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

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

Derivatives of elementary functions: (xⁿ)' = n xⁿ⁻¹, (eˣ)' = eˣ, (ln x)' = 1/x, (sin x)' = cos x, (cos x)' = −sin x.

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

Second derivative test: at a critical point, f'' > 0 gives a local minimum, f'' < 0 gives a local maximum; f'' = 0 is inconclusive.

f is concave up where f'' > 0 and concave down where f'' < 0; inflection points are where concavity changes (f'' changes sign).

Tangent line at x = a: y = f(a) + f'(a)(x − a); normal line slope is −1/f'(a) when f'(a) ≠ 0.

Optimization: build the objective as a function of one variable using the constraint, differentiate, solve f' = 0, and verify with the second derivative or by comparing values.

L'Hôpital's rule applies to 0/0 or ∞/∞ only; for 0·∞, ∞ − ∞ or 1^∞ rewrite as a quotient or take logarithms before applying it.

Marginal analysis: if C(x) is total cost, marginal cost is C'(x); profit is maximized where marginal revenue equals marginal cost, R'(x) = C'(x).

A primitive of f is any F with F' = f; the indefinite integral ∫f(x)dx = F(x) + C, and all primitives differ by a constant.

Barro's rule (Fundamental Theorem of Calculus): if F is a primitive of continuous f on [a,b], then ∫ₐᵇ f(x)dx = F(b) − F(a).

Standard primitives: ∫xⁿdx = xⁿ⁺¹/(n+1) + C for n ≠ −1, ∫(1/x)dx = ln|x| + C, ∫eˣdx = eˣ + C, ∫sin x dx = −cos x + C.

Area between two curves: ∫ₐᵇ [f(x) − g(x)]dx where f is the upper curve on [a,b]; if the curves cross, split the interval at the intersection points and integrate |f − g|.

Area between a curve and the x-axis: integrate |f(x)| on each subinterval where f keeps a constant sign, so regions below the axis contribute positive area.

Area between a curve x = g(y) and the y-axis over [c,d] is ∫꜀ᵈ |g(y)|dy; choose the variable that avoids splitting the region.

Integration by parts: ∫u dv = uv − ∫v du; choose u as the factor that simplifies when differentiated (polynomials, ln x).

Change of variables: ∫f(g(x))g'(x)dx = ∫f(u)du with u = g(x); for definite integrals change the limits to u-values.

Mean value of f on [a,b] is (1/(b−a))∫ₐᵇ f(x)dx; the average value lies between the minimum and maximum of f on the interval.

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

Complement rule: P(Aᶜ) = 1 − P(A); conditional probability P(A|B) = P(A ∩ B)/P(B), defined when P(B) > 0.

Independence: 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 {Aᵢ}; Bayes' theorem gives P(Aᵢ|B) = P(Aᵢ)P(B|Aᵢ)/P(B).

A discrete random variable has P(X = xᵢ) = pᵢ with pᵢ ≥ 0 and Σpᵢ = 1; its mean is μ = Σxᵢpᵢ and variance σ² = Σxᵢ²pᵢ − μ².

Binomial distribution B(n,p): P(X = k) = C(n,k)pᵏ(1−p)ⁿ⁻ᵏ, with mean np and variance np(1−p).

Normal distribution N(μ,σ): standardize with z = (x − μ)/σ and use the standard normal table; P(μ−σ < X < μ+σ) ≈ 0.6826.

For a continuous variable, probability equals area under the density: P(a ≤ X ≤ b) = ∫ₐᵇ f(x)dx, and P(X = a) = 0 for any single point.

Counting: permutations of n distinct objects are n!; combinations of n taken k are C(n,k) = n!/(k!(n−k)!), used when order does not matter.

A density function f satisfies f(x) ≥ 0 for all x and ∫f(x)dx over the whole real line equals 1; the distribution function is F(x) = P(X ≤ x) = ∫₋∞ˣ f(t)dt.

For a continuous variable, E[X] = ∫x·f(x)dx and Var(X) = E[X²] − (E[X])², with E[X²] = ∫x²f(x)dx.

Linearity of expectation always holds: E[aX + b] = aE[X] + b and E[X + Y] = E[X] + E[Y], even without independence.

Variance rules: Var(aX + b) = a²Var(X); if X and Y are independent, Var(X + Y) = Var(X) + Var(Y).

The standard normal table gives Φ(z) = P(Z ≤ z) for Z ~ N(0,1); use symmetry Φ(−z) = 1 − Φ(z) to handle negative z.

For a normal variable, P(X ≤ x) = Φ((x − μ)/σ); the 68–95–99.7 rule approximates probabilities within one, two and three standard deviations of the mean.

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

Standard deviation is σ = √Var(X); it measures dispersion in the same units as X, while variance is in squared units.