PAU Applied Math ACS II (Galicia) cheat sheet

Matrix product A·B exists only if columns of A equal rows of B; entry (i,j) is the dot product of row i of A with column j of B. Generally AB ≠ BA.

Determinant of 2×2 A=[[a,b],[c,d]] is ad − bc; A is invertible iff det A ≠ 0.

Inverse of 2×2: A⁻¹ = (1/det A)·[[d,−b],[−c,a]]; swap the main diagonal, change signs on the anti-diagonal.

Cramer's rule: for AX=B with det A ≠ 0, xᵢ = det Aᵢ / det A, where Aᵢ replaces column i by the independent-term vector B.

Rouché–Frobenius: let r = rank(A), r* = rank(A|B). If r ≠ r*, inconsistent; if r = r* = n, unique solution; if r = r* < n, infinitely many solutions with n − r free parameters.

Gaussian elimination: use elementary row operations to reach row-echelon form, then back-substitute; a row [0 0 … 0 | k] with k ≠ 0 signals inconsistency.

Matrix rank equals the number of nonzero rows in row-echelon form, and also the highest order of a nonvanishing minor.

Homogeneous system AX=0 always has the trivial solution; it has nontrivial solutions iff det A = 0 (square case).

Determinant properties: det(AB) = det A·det B; det(Aᵀ) = det A; det(A⁻¹) = 1/det A; swapping two rows changes the sign.

Limit at a point exists iff left and right limits agree; lim(x→a) f(x) may differ from f(a) without affecting the limit.

Indeterminate forms 0/0 and ∞/∞: factor, rationalise, or apply L'Hôpital (differentiate numerator and denominator separately) when the hypotheses hold.

Continuity at a: f(a) defined, lim(x→a) f(x) exists, and the limit equals f(a); all three conditions are required.

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

Differentiation rules: (u·v)' = u'v + uv'; (u/v)' = (u'v − uv')/v²; chain rule (f∘g)' = f'(g(x))·g'(x).

Critical points: solve f'(x) = 0 (and points where f' fails to exist); classify with f'' - f''>0 local minimum, f''<0 local maximum, f''=0 inconclusive.

On a closed interval [a,b], the global extrema of a continuous function occur at critical points or at the endpoints a and b.

For optimisation problems: build the objective function, express it in one variable using the constraint, differentiate, solve, and check the feasible domain.

Marginal cost/revenue = derivative of total cost/revenue; average cost C(x)/x is minimised when marginal cost equals average cost.

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

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

Area between y = f(x) (upper) and y = g(x) (lower) from a to b is ∫ₐᵇ [f(x) − g(x)]dx; when curves cross, split the interval at intersection points.

Area between a curve and the x-axis: integrate |f(x)|, splitting where f changes sign.

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

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

Area bounded by two curves: find intersection points first by solving f(x) = g(x); these give the integration limits.

The definite integral of a rate of change gives net change: ∫ₐᵇ f'(x)dx = f(b) − f(a).

Conditional probability: P(A|B) = P(A∩B)/P(B), defined for 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} of the sample space.

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

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

Normal: X ~ N(μ,σ); standardise z = (x − μ)/σ and use the standard normal table; P(X ≤ x) = Φ(z).

Symmetry of the normal: Φ(−z) = 1 − Φ(z); P(a ≤ X ≤ b) = Φ(z_b) − Φ(z_a).

Binomial approximation by normal is valid when np > 5 and n(1−p) > 5, using μ = np, σ = √(np(1−p)) with continuity correction.

Complement rule: P(at least one) = 1 − P(none); useful for binomial questions phrased as 'at least'.

Univariate: mean x̄ = Σxᵢ/n; variance σ² = Σxᵢ²/n − x̄²; standard deviation σ = √σ²; coefficient of variation CV = σ/x̄.

For grouped data, use class midpoints xᵢ weighted by frequency fᵢ; N = Σfᵢ.

Median: middle value of ordered data; for grouped data use the cumulative frequency to locate the median class.

Quartiles Q₁, Q₂, Q₃; interquartile range IQR = Q₃ − Q₁; outliers often flagged beyond Q₁ − 1.5·IQR or Q₃ + 1.5·IQR.

Bivariate: covariance s_xy = (Σxᵢyᵢ)/n − x̄·ȳ; Pearson r = s_xy/(s_x·s_y), with −1 ≤ r ≤ 1.

Regression line of y on x: y = ȳ + (s_xy/s_x²)(x − x̄); slope b = s_xy/s_x², intercept a = ȳ − b·x̄.

The regression line passes through (x̄, ȳ); the coefficient of determination r² is the proportion of variance explained.

Correlation measures linear association only; r near 0 does not rule out a nonlinear relationship; correlation does not imply causation.

Model cycle: identify variables, state assumptions, build the function or system, solve, interpret, and validate against the original context.

Linear model y = mx + c: m is the rate of change per unit; c is the initial value; use when the change is constant.

Exponential model y = y₀·e^{kt} or y = y₀·aᵗ: growth when k > 0, decay when k < 0; doubling time = ln2/k.

Quadratic model y = ax² + bx + c: vertex at x = −b/(2a) gives the maximum (a<0) or minimum (a>0).

Break-even: solve Revenue(x) = Cost(x); profit P(x) = R(x) − C(x); marginal profit = P'(x).

Logistic-type or constrained models: state the domain of the variable (e.g. x ≥ 0, x integer) and check that the solution lies in the feasible domain.

Always translate the mathematical answer back into the context with units; reject solutions that violate the assumptions (negative quantities, non-integer people).

Sensitivity check: vary a parameter slightly and observe how the result changes to judge the robustness of the model.

Standard form: maximise or minimise z = ax + by subject to a system of linear inequalities and non-negativity constraints x ≥ 0, y ≥ 0.

Graph each constraint as a line; test a point (e.g. the origin) to shade the half-plane that satisfies the inequality; the feasible region is the intersection.

Vertices of the feasible region are found by solving pairs of constraint lines simultaneously.

Fundamental theorem of linear programming: if the feasible region is nonempty and bounded, the optimal value of the objective function occurs at a vertex.

Method: compute z at each vertex; the largest value is the maximum, the smallest is the minimum.

If the feasible region is unbounded, the optimum may not exist or may be infinite; check the behaviour along the unbounded edges.

If the objective line is parallel to a constraint edge, every point on that edge gives the same optimal value (multiple optimal solutions).

A constraint that never binds at the optimum is redundant; removing it does not change the solution.

Matrix product A·B exists only if columns of A equal rows of B; entry (i,j) is the dot product of row i of A with column j of B. Generally AB ≠ BA.

Determinant of 2×2 A=[[a,b],[c,d]] is ad − bc; A is invertible iff det A ≠ 0.

Inverse of 2×2: A⁻¹ = (1/det A)·[[d,−b],[−c,a]]; swap the main diagonal, change signs on the anti-diagonal.

Cramer's rule: for AX=B with det A ≠ 0, xᵢ = det Aᵢ / det A, where Aᵢ replaces column i by the independent-term vector B.

Rouché–Frobenius: let r = rank(A), r* = rank(A|B). If r ≠ r*, inconsistent; if r = r* = n, unique solution; if r = r* < n, infinitely many solutions with n − r free parameters.

Gaussian elimination: use elementary row operations to reach row-echelon form, then back-substitute; a row [0 0 … 0 | k] with k ≠ 0 signals inconsistency.

Matrix rank equals the number of nonzero rows in row-echelon form, and also the highest order of a nonvanishing minor.

Homogeneous system AX=0 always has the trivial solution; it has nontrivial solutions iff det A = 0 (square case).

Determinant properties: det(AB) = det A·det B; det(Aᵀ) = det A; det(A⁻¹) = 1/det A; swapping two rows changes the sign.

Limit at a point exists iff left and right limits agree; lim(x→a) f(x) may differ from f(a) without affecting the limit.

Indeterminate forms 0/0 and ∞/∞: factor, rationalise, or apply L'Hôpital (differentiate numerator and denominator separately) when the hypotheses hold.

Continuity at a: f(a) defined, lim(x→a) f(x) exists, and the limit equals f(a); all three conditions are required.

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

Differentiation rules: (u·v)' = u'v + uv'; (u/v)' = (u'v − uv')/v²; chain rule (f∘g)' = f'(g(x))·g'(x).

Critical points: solve f'(x) = 0 (and points where f' fails to exist); classify with f'' - f''>0 local minimum, f''<0 local maximum, f''=0 inconclusive.

On a closed interval [a,b], the global extrema of a continuous function occur at critical points or at the endpoints a and b.

For optimisation problems: build the objective function, express it in one variable using the constraint, differentiate, solve, and check the feasible domain.

Marginal cost/revenue = derivative of total cost/revenue; average cost C(x)/x is minimised when marginal cost equals average cost.

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

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

Area between y = f(x) (upper) and y = g(x) (lower) from a to b is ∫ₐᵇ [f(x) − g(x)]dx; when curves cross, split the interval at intersection points.

Area between a curve and the x-axis: integrate |f(x)|, splitting where f changes sign.

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

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

Area bounded by two curves: find intersection points first by solving f(x) = g(x); these give the integration limits.

The definite integral of a rate of change gives net change: ∫ₐᵇ f'(x)dx = f(b) − f(a).

Conditional probability: P(A|B) = P(A∩B)/P(B), defined for 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} of the sample space.

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

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

Normal: X ~ N(μ,σ); standardise z = (x − μ)/σ and use the standard normal table; P(X ≤ x) = Φ(z).

Symmetry of the normal: Φ(−z) = 1 − Φ(z); P(a ≤ X ≤ b) = Φ(z_b) − Φ(z_a).

Binomial approximation by normal is valid when np > 5 and n(1−p) > 5, using μ = np, σ = √(np(1−p)) with continuity correction.

Complement rule: P(at least one) = 1 − P(none); useful for binomial questions phrased as 'at least'.

Univariate: mean x̄ = Σxᵢ/n; variance σ² = Σxᵢ²/n − x̄²; standard deviation σ = √σ²; coefficient of variation CV = σ/x̄.

For grouped data, use class midpoints xᵢ weighted by frequency fᵢ; N = Σfᵢ.

Median: middle value of ordered data; for grouped data use the cumulative frequency to locate the median class.

Quartiles Q₁, Q₂, Q₃; interquartile range IQR = Q₃ − Q₁; outliers often flagged beyond Q₁ − 1.5·IQR or Q₃ + 1.5·IQR.

Bivariate: covariance s_xy = (Σxᵢyᵢ)/n − x̄·ȳ; Pearson r = s_xy/(s_x·s_y), with −1 ≤ r ≤ 1.

Regression line of y on x: y = ȳ + (s_xy/s_x²)(x − x̄); slope b = s_xy/s_x², intercept a = ȳ − b·x̄.

The regression line passes through (x̄, ȳ); the coefficient of determination r² is the proportion of variance explained.

Correlation measures linear association only; r near 0 does not rule out a nonlinear relationship; correlation does not imply causation.

Model cycle: identify variables, state assumptions, build the function or system, solve, interpret, and validate against the original context.

Linear model y = mx + c: m is the rate of change per unit; c is the initial value; use when the change is constant.

Exponential model y = y₀·e^{kt} or y = y₀·aᵗ: growth when k > 0, decay when k < 0; doubling time = ln2/k.

Quadratic model y = ax² + bx + c: vertex at x = −b/(2a) gives the maximum (a<0) or minimum (a>0).

Break-even: solve Revenue(x) = Cost(x); profit P(x) = R(x) − C(x); marginal profit = P'(x).

Logistic-type or constrained models: state the domain of the variable (e.g. x ≥ 0, x integer) and check that the solution lies in the feasible domain.

Always translate the mathematical answer back into the context with units; reject solutions that violate the assumptions (negative quantities, non-integer people).

Sensitivity check: vary a parameter slightly and observe how the result changes to judge the robustness of the model.

Standard form: maximise or minimise z = ax + by subject to a system of linear inequalities and non-negativity constraints x ≥ 0, y ≥ 0.

Graph each constraint as a line; test a point (e.g. the origin) to shade the half-plane that satisfies the inequality; the feasible region is the intersection.

Vertices of the feasible region are found by solving pairs of constraint lines simultaneously.

Fundamental theorem of linear programming: if the feasible region is nonempty and bounded, the optimal value of the objective function occurs at a vertex.

Method: compute z at each vertex; the largest value is the maximum, the smallest is the minimum.

If the feasible region is unbounded, the optimum may not exist or may be infinite; check the behaviour along the unbounded edges.

If the objective line is parallel to a constraint edge, every point on that edge gives the same optimal value (multiple optimal solutions).

A constraint that never binds at the optimum is redundant; removing it does not change the solution.