Madrid PAU Applied Mathematics for Social Sciences II cheat sheet

Order of a matrix is rows × columns; A(m×n)·B(n×p) is defined only when inner dimensions match and yields order m×p.

(AB)ᵀ = BᵀAᵀ and (AB)⁻¹ = B⁻¹A⁻¹ for invertible square matrices; in general AB ≠ BA.

A is invertible iff det(A) ≠ 0; for 2×2, A⁻¹ = (1/det A)·[[d, −b], [−c, a]] with A = [[a, b], [c, d]].

Rouché–Frobenius: 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 infinitely many.

If rank(A) ≠ rank(A|B) the system is inconsistent (no solution); if rank(A) = rank(A|B) = n, Cramer's rule gives xᵢ = det(Aᵢ)/det(A).

A homogeneous system AX = 0 is always consistent (trivial solution); it has nontrivial solutions iff det(A) = 0.

Gauss elimination reduces the augmented matrix to row echelon form using elementary row operations, which preserve the solution set.

det of a triangular or diagonal matrix equals the product of its diagonal entries; swapping two rows changes the sign of the determinant.

Matrix equation AX = B with A invertible has unique solution X = A⁻¹B.

Standard form: maximize/minimize z = c₁x₁ + … + cₙxₙ subject to linear constraints and non-negativity xᵢ ≥ 0.

Feasible region is the intersection of half-planes defined by the constraints; it is a convex polygon (or unbounded polyhedron).

Fundamental theorem: if an optimum exists on a bounded feasible region, it is attained at a vertex (corner point) of the region.

Corner-point method: evaluate the objective function at every vertex; the largest value is the maximum and the smallest is the minimum.

A constraint line ax + by = c divides the plane; test a point (e.g. the origin) to decide which side satisfies the inequality.

Redundant constraints do not change the feasible region; the vertices are found by solving pairs of binding constraint lines simultaneously.

Unbounded feasible region: the optimum may still exist if the objective improves toward a vertex, otherwise the problem is unbounded.

Integer programming adds xᵢ ∈ ℤ; the optimum is the best integer point inside the feasible region, not necessarily a real-valued vertex.

lim(x→a) f(x) exists iff left and right limits are equal; f is continuous at a iff lim(x→a) f(x) = f(a).

Indeterminate forms 0/0 and ∞/∞ can be resolved by factoring, rationalizing, or L'Hôpital's rule: lim f/g = lim f'/g'.

Derivative rules: (xⁿ)' = n·xⁿ⁻¹, (eˣ)' = eˣ, (ln x)' = 1/x, (sin x)' = cos x, (cos x)' = −sin x.

Product rule (uv)' = u'v + uv'; quotient rule (u/v)' = (u'v − uv')/v²; chain rule (f(g(x)))' = f'(g(x))·g'(x).

f is increasing where f' > 0 and decreasing where f' < 0; critical points satisfy f'(x) = 0 or f' undefined.

Second derivative test: f''(a) > 0 gives a local minimum, f''(a) < 0 gives a local maximum at critical point a.

f is concave up where f'' > 0 and concave down where f'' < 0; inflection points occur where f'' changes sign.

Asymptotes: vertical where denominator → 0; horizontal lim(x→±∞) f(x) = L; oblique when degree of numerator exceeds denominator by one.

Fundamental theorem: ∫ₐᵇ f(x) dx = F(b) − F(a) where F is any antiderivative of f.

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

∫ eˣ dx = eˣ + C, ∫ sin x dx = −cos x + C, ∫ cos x dx = sin x + C.

Area between y = f(x) and y = g(x) on [a, b] is ∫ₐᵇ |f(x) − g(x)| dx; identify intersection points to set the limits.

Area between a curve and the x-axis: integrate f where f ≥ 0 and subtract the integral where f < 0, or integrate |f|.

Integration by parts: ∫ u dv = uv − ∫ v du; useful for x·eˣ and x·ln x type integrands.

Improper integrals ∫ₐ^∞ f dx = lim(b→∞) ∫ₐᵇ f dx; convergence depends on whether the limit is finite.

Definite integral properties: ∫ₐᵃ f = 0, ∫ₐᵇ f = −∫ᵦᵃ f, and ∫ₐᵇ f = ∫ₐᶜ f + ∫ᶜᵇ f.

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

Complement rule P(Aᶜ) = 1 − P(A); for independent events P(A ∩ B) = P(A)·P(B).

Conditional probability P(A|B) = P(A ∩ B)/P(B) for P(B) > 0.

Total probability: P(B) = Σ P(Aᵢ)·P(B|Aᵢ) over a partition {Aᵢ} of the sample space.

Bayes' theorem: P(Aᵢ|B) = P(Aᵢ)·P(B|Aᵢ) / Σ P(Aⱼ)·P(B|Aⱼ).

Events A and B are independent iff P(A ∩ B) = P(A)·P(B), equivalently P(A|B) = P(A).

Mutually exclusive events satisfy P(A ∩ B) = 0, so P(A ∪ B) = P(A) + P(B).

For equiprobable outcomes, P(A) = number of favourable outcomes / total number of outcomes.

For a discrete variable, Σ p(xᵢ) = 1; mean μ = Σ xᵢ·p(xᵢ) and variance σ² = Σ xᵢ²·p(xᵢ) − μ².

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

Normal N(μ, σ): standardize with z = (x − μ)/σ, then use the standard normal table Φ(z).

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

Normal approximation to binomial: valid when np ≥ 5 and n(1−p) ≥ 5; use μ = np, σ = √(np(1−p)).

Continuity correction: P(X ≤ k) ≈ Φ((k + 0.5 − μ)/σ), P(X ≥ k) ≈ 1 − Φ((k − 0.5 − μ)/σ).

A continuous variable has P(X = a) = 0; probabilities are areas under the density curve.

Linear transformation: if X has mean μ and sd σ, then aX + b has mean aμ + b and sd |a|σ.

Central limit theorem: for large n the sample mean X̄ is approximately N(μ, σ/√n), where σ/√n is the standard error.

Confidence interval for a mean (known σ or large n): X̄ ± z_(α/2)·σ/√n; common values z = 1.645 (90%), 1.96 (95%), 2.576 (99%).

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; required sample size n = (z_(α/2)·σ / E)² rounded up.

Increasing confidence level widens the interval; increasing sample size narrows it (width ∝ 1/√n).

Sample standard deviation uses divisor n − 1: s² = Σ(xᵢ − x̄)²/(n − 1).

A 95% confidence interval does not mean 95% probability that μ lies in this specific interval; it refers to the procedure's long-run coverage.

For small n with unknown σ, use the t-distribution with n − 1 degrees of freedom instead of z.

Order of a matrix is rows × columns; A(m×n)·B(n×p) is defined only when inner dimensions match and yields order m×p.

(AB)ᵀ = BᵀAᵀ and (AB)⁻¹ = B⁻¹A⁻¹ for invertible square matrices; in general AB ≠ BA.

A is invertible iff det(A) ≠ 0; for 2×2, A⁻¹ = (1/det A)·[[d, −b], [−c, a]] with A = [[a, b], [c, d]].

Rouché–Frobenius: 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 infinitely many.

If rank(A) ≠ rank(A|B) the system is inconsistent (no solution); if rank(A) = rank(A|B) = n, Cramer's rule gives xᵢ = det(Aᵢ)/det(A).

A homogeneous system AX = 0 is always consistent (trivial solution); it has nontrivial solutions iff det(A) = 0.

Gauss elimination reduces the augmented matrix to row echelon form using elementary row operations, which preserve the solution set.

det of a triangular or diagonal matrix equals the product of its diagonal entries; swapping two rows changes the sign of the determinant.

Matrix equation AX = B with A invertible has unique solution X = A⁻¹B.

Standard form: maximize/minimize z = c₁x₁ + … + cₙxₙ subject to linear constraints and non-negativity xᵢ ≥ 0.

Feasible region is the intersection of half-planes defined by the constraints; it is a convex polygon (or unbounded polyhedron).

Fundamental theorem: if an optimum exists on a bounded feasible region, it is attained at a vertex (corner point) of the region.

Corner-point method: evaluate the objective function at every vertex; the largest value is the maximum and the smallest is the minimum.

A constraint line ax + by = c divides the plane; test a point (e.g. the origin) to decide which side satisfies the inequality.

Redundant constraints do not change the feasible region; the vertices are found by solving pairs of binding constraint lines simultaneously.

Unbounded feasible region: the optimum may still exist if the objective improves toward a vertex, otherwise the problem is unbounded.

Integer programming adds xᵢ ∈ ℤ; the optimum is the best integer point inside the feasible region, not necessarily a real-valued vertex.

lim(x→a) f(x) exists iff left and right limits are equal; f is continuous at a iff lim(x→a) f(x) = f(a).

Indeterminate forms 0/0 and ∞/∞ can be resolved by factoring, rationalizing, or L'Hôpital's rule: lim f/g = lim f'/g'.

Derivative rules: (xⁿ)' = n·xⁿ⁻¹, (eˣ)' = eˣ, (ln x)' = 1/x, (sin x)' = cos x, (cos x)' = −sin x.

Product rule (uv)' = u'v + uv'; quotient rule (u/v)' = (u'v − uv')/v²; chain rule (f(g(x)))' = f'(g(x))·g'(x).

f is increasing where f' > 0 and decreasing where f' < 0; critical points satisfy f'(x) = 0 or f' undefined.

Second derivative test: f''(a) > 0 gives a local minimum, f''(a) < 0 gives a local maximum at critical point a.

f is concave up where f'' > 0 and concave down where f'' < 0; inflection points occur where f'' changes sign.

Asymptotes: vertical where denominator → 0; horizontal lim(x→±∞) f(x) = L; oblique when degree of numerator exceeds denominator by one.

Fundamental theorem: ∫ₐᵇ f(x) dx = F(b) − F(a) where F is any antiderivative of f.

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

∫ eˣ dx = eˣ + C, ∫ sin x dx = −cos x + C, ∫ cos x dx = sin x + C.

Area between y = f(x) and y = g(x) on [a, b] is ∫ₐᵇ |f(x) − g(x)| dx; identify intersection points to set the limits.

Area between a curve and the x-axis: integrate f where f ≥ 0 and subtract the integral where f < 0, or integrate |f|.

Integration by parts: ∫ u dv = uv − ∫ v du; useful for x·eˣ and x·ln x type integrands.

Improper integrals ∫ₐ^∞ f dx = lim(b→∞) ∫ₐᵇ f dx; convergence depends on whether the limit is finite.

Definite integral properties: ∫ₐᵃ f = 0, ∫ₐᵇ f = −∫ᵦᵃ f, and ∫ₐᵇ f = ∫ₐᶜ f + ∫ᶜᵇ f.

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

Complement rule P(Aᶜ) = 1 − P(A); for independent events P(A ∩ B) = P(A)·P(B).

Conditional probability P(A|B) = P(A ∩ B)/P(B) for P(B) > 0.

Total probability: P(B) = Σ P(Aᵢ)·P(B|Aᵢ) over a partition {Aᵢ} of the sample space.

Bayes' theorem: P(Aᵢ|B) = P(Aᵢ)·P(B|Aᵢ) / Σ P(Aⱼ)·P(B|Aⱼ).

Events A and B are independent iff P(A ∩ B) = P(A)·P(B), equivalently P(A|B) = P(A).

Mutually exclusive events satisfy P(A ∩ B) = 0, so P(A ∪ B) = P(A) + P(B).

For equiprobable outcomes, P(A) = number of favourable outcomes / total number of outcomes.

For a discrete variable, Σ p(xᵢ) = 1; mean μ = Σ xᵢ·p(xᵢ) and variance σ² = Σ xᵢ²·p(xᵢ) − μ².

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

Normal N(μ, σ): standardize with z = (x − μ)/σ, then use the standard normal table Φ(z).

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

Normal approximation to binomial: valid when np ≥ 5 and n(1−p) ≥ 5; use μ = np, σ = √(np(1−p)).

Continuity correction: P(X ≤ k) ≈ Φ((k + 0.5 − μ)/σ), P(X ≥ k) ≈ 1 − Φ((k − 0.5 − μ)/σ).

A continuous variable has P(X = a) = 0; probabilities are areas under the density curve.

Linear transformation: if X has mean μ and sd σ, then aX + b has mean aμ + b and sd |a|σ.

Central limit theorem: for large n the sample mean X̄ is approximately N(μ, σ/√n), where σ/√n is the standard error.

Confidence interval for a mean (known σ or large n): X̄ ± z_(α/2)·σ/√n; common values z = 1.645 (90%), 1.96 (95%), 2.576 (99%).

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; required sample size n = (z_(α/2)·σ / E)² rounded up.

Increasing confidence level widens the interval; increasing sample size narrows it (width ∝ 1/√n).

Sample standard deviation uses divisor n − 1: s² = Σ(xᵢ − x̄)²/(n − 1).

A 95% confidence interval does not mean 95% probability that μ lies in this specific interval; it refers to the procedure's long-run coverage.

For small n with unknown σ, use the t-distribution with n − 1 degrees of freedom instead of z.

Order of a matrix is rows × columns; A(m×n)·B(n×p) is defined only when inner dimensions match and yields order m×p.

(AB)ᵀ = BᵀAᵀ and (AB)⁻¹ = B⁻¹A⁻¹ for invertible square matrices; in general AB ≠ BA.

A is invertible iff det(A) ≠ 0; for 2×2, A⁻¹ = (1/det A)·[[d, −b], [−c, a]] with A = [[a, b], [c, d]].

Rouché–Frobenius: 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 infinitely many.

If rank(A) ≠ rank(A|B) the system is inconsistent (no solution); if rank(A) = rank(A|B) = n, Cramer's rule gives xᵢ = det(Aᵢ)/det(A).

A homogeneous system AX = 0 is always consistent (trivial solution); it has nontrivial solutions iff det(A) = 0.

Gauss elimination reduces the augmented matrix to row echelon form using elementary row operations, which preserve the solution set.

det of a triangular or diagonal matrix equals the product of its diagonal entries; swapping two rows changes the sign of the determinant.

Matrix equation AX = B with A invertible has unique solution X = A⁻¹B.

Standard form: maximize/minimize z = c₁x₁ + … + cₙxₙ subject to linear constraints and non-negativity xᵢ ≥ 0.

Feasible region is the intersection of half-planes defined by the constraints; it is a convex polygon (or unbounded polyhedron).

Fundamental theorem: if an optimum exists on a bounded feasible region, it is attained at a vertex (corner point) of the region.

Corner-point method: evaluate the objective function at every vertex; the largest value is the maximum and the smallest is the minimum.

A constraint line ax + by = c divides the plane; test a point (e.g. the origin) to decide which side satisfies the inequality.

Redundant constraints do not change the feasible region; the vertices are found by solving pairs of binding constraint lines simultaneously.

Unbounded feasible region: the optimum may still exist if the objective improves toward a vertex, otherwise the problem is unbounded.

Integer programming adds xᵢ ∈ ℤ; the optimum is the best integer point inside the feasible region, not necessarily a real-valued vertex.

lim(x→a) f(x) exists iff left and right limits are equal; f is continuous at a iff lim(x→a) f(x) = f(a).

Indeterminate forms 0/0 and ∞/∞ can be resolved by factoring, rationalizing, or L'Hôpital's rule: lim f/g = lim f'/g'.

Derivative rules: (xⁿ)' = n·xⁿ⁻¹, (eˣ)' = eˣ, (ln x)' = 1/x, (sin x)' = cos x, (cos x)' = −sin x.

Product rule (uv)' = u'v + uv'; quotient rule (u/v)' = (u'v − uv')/v²; chain rule (f(g(x)))' = f'(g(x))·g'(x).

f is increasing where f' > 0 and decreasing where f' < 0; critical points satisfy f'(x) = 0 or f' undefined.

Second derivative test: f''(a) > 0 gives a local minimum, f''(a) < 0 gives a local maximum at critical point a.

f is concave up where f'' > 0 and concave down where f'' < 0; inflection points occur where f'' changes sign.

Asymptotes: vertical where denominator → 0; horizontal lim(x→±∞) f(x) = L; oblique when degree of numerator exceeds denominator by one.

Fundamental theorem: ∫ₐᵇ f(x) dx = F(b) − F(a) where F is any antiderivative of f.

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

∫ eˣ dx = eˣ + C, ∫ sin x dx = −cos x + C, ∫ cos x dx = sin x + C.

Area between y = f(x) and y = g(x) on [a, b] is ∫ₐᵇ |f(x) − g(x)| dx; identify intersection points to set the limits.

Area between a curve and the x-axis: integrate f where f ≥ 0 and subtract the integral where f < 0, or integrate |f|.

Integration by parts: ∫ u dv = uv − ∫ v du; useful for x·eˣ and x·ln x type integrands.

Improper integrals ∫ₐ^∞ f dx = lim(b→∞) ∫ₐᵇ f dx; convergence depends on whether the limit is finite.

Definite integral properties: ∫ₐᵃ f = 0, ∫ₐᵇ f = −∫ᵦᵃ f, and ∫ₐᵇ f = ∫ₐᶜ f + ∫ᶜᵇ f.

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

Complement rule P(Aᶜ) = 1 − P(A); for independent events P(A ∩ B) = P(A)·P(B).

Conditional probability P(A|B) = P(A ∩ B)/P(B) for P(B) > 0.

Total probability: P(B) = Σ P(Aᵢ)·P(B|Aᵢ) over a partition {Aᵢ} of the sample space.

Bayes' theorem: P(Aᵢ|B) = P(Aᵢ)·P(B|Aᵢ) / Σ P(Aⱼ)·P(B|Aⱼ).

Events A and B are independent iff P(A ∩ B) = P(A)·P(B), equivalently P(A|B) = P(A).

Mutually exclusive events satisfy P(A ∩ B) = 0, so P(A ∪ B) = P(A) + P(B).

For equiprobable outcomes, P(A) = number of favourable outcomes / total number of outcomes.

For a discrete variable, Σ p(xᵢ) = 1; mean μ = Σ xᵢ·p(xᵢ) and variance σ² = Σ xᵢ²·p(xᵢ) − μ².

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

Normal N(μ, σ): standardize with z = (x − μ)/σ, then use the standard normal table Φ(z).

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

Normal approximation to binomial: valid when np ≥ 5 and n(1−p) ≥ 5; use μ = np, σ = √(np(1−p)).

Continuity correction: P(X ≤ k) ≈ Φ((k + 0.5 − μ)/σ), P(X ≥ k) ≈ 1 − Φ((k − 0.5 − μ)/σ).

A continuous variable has P(X = a) = 0; probabilities are areas under the density curve.

Linear transformation: if X has mean μ and sd σ, then aX + b has mean aμ + b and sd |a|σ.

Central limit theorem: for large n the sample mean X̄ is approximately N(μ, σ/√n), where σ/√n is the standard error.

Confidence interval for a mean (known σ or large n): X̄ ± z_(α/2)·σ/√n; common values z = 1.645 (90%), 1.96 (95%), 2.576 (99%).

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; required sample size n = (z_(α/2)·σ / E)² rounded up.

Increasing confidence level widens the interval; increasing sample size narrows it (width ∝ 1/√n).

Sample standard deviation uses divisor n − 1: s² = Σ(xᵢ − x̄)²/(n − 1).

A 95% confidence interval does not mean 95% probability that μ lies in this specific interval; it refers to the procedure's long-run coverage.

For small n with unknown σ, use the t-distribution with n − 1 degrees of freedom instead of z.