PAU Applied Math ACS II (Aragón) cheat sheet

Order of a matrix is rows × columns; A + B requires identical orders, while AB requires columns of A equal rows of B, giving order (rows A) × (columns B).

det(AB) = det(A)·det(B); det(Aᵀ) = det(A); det(kA) = kⁿ·det(A) for an n×n matrix; det(A⁻¹) = 1/det(A) when det(A) ≠ 0.

A is invertible iff det(A) ≠ 0 iff rank(A) = n. For 2×2, A⁻¹ = (1/(ad−bc))·[[d, −b], [−c, a]].

A square system AX = B is compatible determined iff rank(A) = rank(A|B) = n; compatible indeterminate iff rank(A) = rank(A|B) < n; incompatible iff rank(A) < rank(A|B).

Rouché–Capelli: compare rank(A) with rank of the augmented matrix; the number of free parameters is n − rank(A).

Cramer's rule: if det(A) ≠ 0, then xᵢ = det(Aᵢ)/det(A), where Aᵢ replaces column i of A by the vector B.

Gauss elimination: reduce the augmented matrix to echelon form using row operations; zero rows in the coefficient part with nonzero constant term signal incompatibility.

Matrix equation AX = B solves as X = A⁻¹B when A is invertible; for AXB = C use X = A⁻¹CB⁻¹, respecting non-commutativity.

Powers of a matrix: compute A², A³ to detect a pattern, then prove the general form Aⁿ by induction.

Standard form: optimize z = c₁x₁ + … + cₙxₙ subject to linear constraints and xᵢ ≥ 0; the feasible region is the intersection of half-planes, a convex polygon.

Fundamental theorem: if a linear program has an optimal solution, at least one optimal solution occurs at a vertex of the feasible region.

Graphical method: draw each constraint line, shade the feasible side, find vertices by solving pairs of binding constraint equations, then evaluate z at each vertex.

Bounded feasible region guarantees both maximum and minimum; an unbounded region may give an unbounded objective or an optimum at a vertex.

A constraint is binding at a vertex when it holds with equality there; the optimum lies at the intersection of at least two binding constraints in two-variable problems.

Slope test: the objective line c₁x₁ + c₂x₂ = k is shifted parallel; the optimum vertex is the last contact point with the feasible region in the improving direction.

Integer programming: solve the relaxed linear program, then test integer points near the optimal vertex; round the vertex coordinates only after checking feasibility.

Sensitivity: the reduced cost of a non-basic variable indicates how much its coefficient must improve before it can enter an optimal basis.

lim(x→a) f(x) exists iff both one-sided limits exist and are equal; continuity at a requires f(a) defined, the limit existing, and lim(x→a) f(x) = f(a).

Indeterminations 0/0 and ∞/∞: apply L'Hôpital's rule lim f/g = lim f'/g' when the quotient is differentiable near a and the derivative quotient limit exists.

Notable limits: lim(x→0) sin x/x = 1; lim(x→0) (1 − cos x)/x² = 1/2; lim(x→∞) (1 + 1/x)ˣ = e; lim(x→0) (eˣ − 1)/x = 1; lim(x→0) ln(1+x)/x = 1.

lim(x→0) (1 + x)^(1/x) = e; rewrite indeterminate powers f^g as e^(g·ln f) before applying L'Hôpital.

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

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

Derivatives: (sin x)' = cos x, (cos x)' = −sin x, (tan x)' = 1/cos²x, (ln x)' = 1/x, (eˣ)' = eˣ, (aˣ)' = aˣ ln a, (arctan x)' = 1/(1+x²).

Implicit differentiation: differentiate both sides with respect to x treating y as y(x), then solve for y'.

For piecewise functions, continuity and differentiability at the join point require matching one-sided limits of f and of f'.

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

Barrow's rule with substitution: ∫ₐᵇ f(g(x))g'(x) dx = ∫_{g(a)}^{g(b)} f(u) du; change the limits when substituting.

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

Area between curves: A = ∫ₐᵇ |f(x) − g(x)| dx, splitting the interval at intersection points where the upper function changes.

Area with respect to y: A = ∫_c^d |x_right(y) − x_left(y)| dy, useful when the region is bounded by x = g(y).

Improper integrals: ∫₁^∞ 1/xᵖ dx converges iff p > 1; ∫₀¹ 1/xᵖ dx converges iff p < 1.

Volume of revolution about the x-axis: V = π∫ₐᵇ [f(x)]² dx (disks); about the y-axis use shells V = 2π∫ₐᵇ x·f(x) dx.

Mean value of f on [a, b] is (1/(b−a))∫ₐᵇ f(x) dx; the function F(x) = ∫ₐˣ f(t) dt satisfies F'(x) = f(x).

Axioms: P(Ω) = 1, P(∅) = 0, and P(A ∪ B) = P(A) + P(B) − P(A ∩ B); if A and B are mutually exclusive, P(A ∪ B) = P(A) + P(B).

Conditional probability: P(A|B) = P(A ∩ B)/P(B) with 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) for a partition {Aᵢ}; Bayes: P(Aᵢ|B) = P(Aᵢ)P(B|Aᵢ) / Σ P(Aⱼ)P(B|Aⱼ).

Complement rule: P(Aᶜ) = 1 − P(A); use it for 'at least one' events, P(at least one) = 1 − P(none).

Binomial B(n, p): P(X = k) = C(n, k)pᵏ(1−p)^(n−k), mean np, variance np(1−p); models counts of successes in n independent trials.

Poisson P(λ): P(X = k) = e^(−λ)λᵏ/k!, mean and variance both λ; used for rare events in a fixed interval.

Continuous variable: P(a ≤ X ≤ b) = ∫ₐᵇ f(x) dx, total area 1, and P(X = a) = 0 for any single point.

Normal N(μ, σ): standardize z = (x − μ)/σ, then use the standard normal table; P(μ − σ < X < μ + σ) ≈ 0.6827, P(μ − 2σ < X < μ + 2σ) ≈ 0.9545.

Sampling distribution of the mean: if X ~ N(μ, σ), then X̄ ~ N(μ, σ/√n); the standard error of the mean is σ/√n.

Confidence interval for μ with known σ: x̄ ± z_(1−α/2)·σ/√n; for 95% use z = 1.96, for 99% use z = 2.576.

Confidence interval with unknown σ and small n: x̄ ± t_(n−1, 1−α/2)·s/√n, using the t distribution with n − 1 degrees of freedom.

Margin of error E = z·σ/√n; the sample size needed for a target E is n = (zσ/E)², rounded up.

Hypothesis test for μ: statistic z = (x̄ − μ₀)/(σ/√n) or t = (x̄ − μ₀)/(s/√n); reject H₀ when the statistic falls in the critical region.

Two-tailed test at level α: reject H₀ if |z| > z_(1−α/2); one-tailed: reject if z > z_(1−α) (right) or z < −z_(1−α) (left).

The p-value is the probability, under H₀, of a statistic at least as extreme as observed; reject H₀ when p-value < α.

Type I error is rejecting a true H₀ (probability α); type II error is failing to reject a false H₀ (probability β); power is 1 − β.

Order of a matrix is rows × columns; A + B requires identical orders, while AB requires columns of A equal rows of B, giving order (rows A) × (columns B).

det(AB) = det(A)·det(B); det(Aᵀ) = det(A); det(kA) = kⁿ·det(A) for an n×n matrix; det(A⁻¹) = 1/det(A) when det(A) ≠ 0.

A is invertible iff det(A) ≠ 0 iff rank(A) = n. For 2×2, A⁻¹ = (1/(ad−bc))·[[d, −b], [−c, a]].

A square system AX = B is compatible determined iff rank(A) = rank(A|B) = n; compatible indeterminate iff rank(A) = rank(A|B) < n; incompatible iff rank(A) < rank(A|B).

Rouché–Capelli: compare rank(A) with rank of the augmented matrix; the number of free parameters is n − rank(A).

Cramer's rule: if det(A) ≠ 0, then xᵢ = det(Aᵢ)/det(A), where Aᵢ replaces column i of A by the vector B.

Gauss elimination: reduce the augmented matrix to echelon form using row operations; zero rows in the coefficient part with nonzero constant term signal incompatibility.

Matrix equation AX = B solves as X = A⁻¹B when A is invertible; for AXB = C use X = A⁻¹CB⁻¹, respecting non-commutativity.

Powers of a matrix: compute A², A³ to detect a pattern, then prove the general form Aⁿ by induction.

Standard form: optimize z = c₁x₁ + … + cₙxₙ subject to linear constraints and xᵢ ≥ 0; the feasible region is the intersection of half-planes, a convex polygon.

Fundamental theorem: if a linear program has an optimal solution, at least one optimal solution occurs at a vertex of the feasible region.

Graphical method: draw each constraint line, shade the feasible side, find vertices by solving pairs of binding constraint equations, then evaluate z at each vertex.

Bounded feasible region guarantees both maximum and minimum; an unbounded region may give an unbounded objective or an optimum at a vertex.

A constraint is binding at a vertex when it holds with equality there; the optimum lies at the intersection of at least two binding constraints in two-variable problems.

Slope test: the objective line c₁x₁ + c₂x₂ = k is shifted parallel; the optimum vertex is the last contact point with the feasible region in the improving direction.

Integer programming: solve the relaxed linear program, then test integer points near the optimal vertex; round the vertex coordinates only after checking feasibility.

Sensitivity: the reduced cost of a non-basic variable indicates how much its coefficient must improve before it can enter an optimal basis.

lim(x→a) f(x) exists iff both one-sided limits exist and are equal; continuity at a requires f(a) defined, the limit existing, and lim(x→a) f(x) = f(a).

Indeterminations 0/0 and ∞/∞: apply L'Hôpital's rule lim f/g = lim f'/g' when the quotient is differentiable near a and the derivative quotient limit exists.

Notable limits: lim(x→0) sin x/x = 1; lim(x→0) (1 − cos x)/x² = 1/2; lim(x→∞) (1 + 1/x)ˣ = e; lim(x→0) (eˣ − 1)/x = 1; lim(x→0) ln(1+x)/x = 1.

lim(x→0) (1 + x)^(1/x) = e; rewrite indeterminate powers f^g as e^(g·ln f) before applying L'Hôpital.

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

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

Derivatives: (sin x)' = cos x, (cos x)' = −sin x, (tan x)' = 1/cos²x, (ln x)' = 1/x, (eˣ)' = eˣ, (aˣ)' = aˣ ln a, (arctan x)' = 1/(1+x²).

Implicit differentiation: differentiate both sides with respect to x treating y as y(x), then solve for y'.

For piecewise functions, continuity and differentiability at the join point require matching one-sided limits of f and of f'.

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

Barrow's rule with substitution: ∫ₐᵇ f(g(x))g'(x) dx = ∫_{g(a)}^{g(b)} f(u) du; change the limits when substituting.

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

Area between curves: A = ∫ₐᵇ |f(x) − g(x)| dx, splitting the interval at intersection points where the upper function changes.

Area with respect to y: A = ∫_c^d |x_right(y) − x_left(y)| dy, useful when the region is bounded by x = g(y).

Improper integrals: ∫₁^∞ 1/xᵖ dx converges iff p > 1; ∫₀¹ 1/xᵖ dx converges iff p < 1.

Volume of revolution about the x-axis: V = π∫ₐᵇ [f(x)]² dx (disks); about the y-axis use shells V = 2π∫ₐᵇ x·f(x) dx.

Mean value of f on [a, b] is (1/(b−a))∫ₐᵇ f(x) dx; the function F(x) = ∫ₐˣ f(t) dt satisfies F'(x) = f(x).

Axioms: P(Ω) = 1, P(∅) = 0, and P(A ∪ B) = P(A) + P(B) − P(A ∩ B); if A and B are mutually exclusive, P(A ∪ B) = P(A) + P(B).

Conditional probability: P(A|B) = P(A ∩ B)/P(B) with 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) for a partition {Aᵢ}; Bayes: P(Aᵢ|B) = P(Aᵢ)P(B|Aᵢ) / Σ P(Aⱼ)P(B|Aⱼ).

Complement rule: P(Aᶜ) = 1 − P(A); use it for 'at least one' events, P(at least one) = 1 − P(none).

Binomial B(n, p): P(X = k) = C(n, k)pᵏ(1−p)^(n−k), mean np, variance np(1−p); models counts of successes in n independent trials.

Poisson P(λ): P(X = k) = e^(−λ)λᵏ/k!, mean and variance both λ; used for rare events in a fixed interval.

Continuous variable: P(a ≤ X ≤ b) = ∫ₐᵇ f(x) dx, total area 1, and P(X = a) = 0 for any single point.

Normal N(μ, σ): standardize z = (x − μ)/σ, then use the standard normal table; P(μ − σ < X < μ + σ) ≈ 0.6827, P(μ − 2σ < X < μ + 2σ) ≈ 0.9545.

Sampling distribution of the mean: if X ~ N(μ, σ), then X̄ ~ N(μ, σ/√n); the standard error of the mean is σ/√n.

Confidence interval for μ with known σ: x̄ ± z_(1−α/2)·σ/√n; for 95% use z = 1.96, for 99% use z = 2.576.

Confidence interval with unknown σ and small n: x̄ ± t_(n−1, 1−α/2)·s/√n, using the t distribution with n − 1 degrees of freedom.

Margin of error E = z·σ/√n; the sample size needed for a target E is n = (zσ/E)², rounded up.

Hypothesis test for μ: statistic z = (x̄ − μ₀)/(σ/√n) or t = (x̄ − μ₀)/(s/√n); reject H₀ when the statistic falls in the critical region.

Two-tailed test at level α: reject H₀ if |z| > z_(1−α/2); one-tailed: reject if z > z_(1−α) (right) or z < −z_(1−α) (left).

The p-value is the probability, under H₀, of a statistic at least as extreme as observed; reject H₀ when p-value < α.

Type I error is rejecting a true H₀ (probability α); type II error is failing to reject a false H₀ (probability β); power is 1 − β.