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

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Matrix product condition

Product AB exists only if the number of columns of A equals the number of rows of B; the resulting matrix has as many rows as A and as many columns as B.

Inverse of a 2×2 matrix

For A = [[a,b],[c,d]] with det(A) = ad − bc ≠ 0, the inverse is (1/(ad−bc))·[[d,−b],[−c,a]]. If the determinant is zero, A has no inverse.

Rouché–Fröbenius theorem

A linear system AX = B is consistent exactly when the rank of the coefficient matrix equals the rank of the augmented matrix. If this rank equals the number of unknowns, the solution is unique.

Cramer's rule

For a square system with det(A) ≠ 0, each unknown equals the determinant of the matrix obtained by replacing its column with the independent terms, divided by det(A).

Feasible region

The set of points satisfying all constraints and non-negativity conditions of a linear program. Its vertices are the candidate points where a linear objective attains its optimum.

Corner method

Evaluate the objective function z = ax + by at every vertex of the feasible region; the largest value is the maximum and the smallest is the minimum.

Bounded versus unbounded region

A bounded feasible region always yields finite maximum and minimum values at vertices. An unbounded region may lack one or both optima, so the optimum must be verified.

Level line of objective

The set of points where ax + by = k. All level lines are parallel, and moving them in the direction of the gradient vector (a, b) increases the objective value.

L'Hôpital's rule

For indeterminate forms 0/0 or ∞/∞, the limit of f/g equals the limit of f'/g', provided the latter exists. Verify the indeterminate form before applying it.

Removable discontinuity

At x = a the limit exists and is finite but does not equal f(a), or f(a) is undefined. The discontinuity disappears if the function value is redefined as the limit.

Bolzano's theorem

If f is continuous on [a, b] and f(a) and f(b) have opposite signs, then some c in (a, b) satisfies f(c) = 0, guaranteeing a root.

Chain rule

The derivative of a composition f(g(x)) equals f'(g(x))·g'(x). For example, the derivative of e^{3x} is 3e^{3x}.

Tangent line equation

The tangent to y = f(x) at x = a is y = f(a) + f'(a)(x − a), where f'(a) is the slope of the curve at that point.

Differentiability implies continuity

If f is differentiable at a point, it is continuous there. The converse fails: f(x) = |x| is continuous but not differentiable at x = 0.

Quotient rule

The derivative of u/v is (u'v − uv')/v², valid where v ≠ 0. The numerator order matters: reversing it changes the sign of the result.

First-derivative test

If f' changes from positive to negative at c, f has a local maximum there; if it changes from negative to positive, f has a local minimum.

Second-derivative test

At a critical point c with f'(c) = 0: if f''(c) < 0 there is a local maximum; if f''(c) > 0 a local minimum; if f''(c) = 0 the test is inconclusive.

Inflection point

A point where the concavity of f changes sign. A necessary condition is f''(c) = 0, but the sign of f'' must actually change across c.

Absolute extrema on interval

To find absolute extrema of f on [a, b], evaluate f at all critical points inside the interval and at both endpoints; compare the resulting values.

Marginal profit condition

Profit P(x) = I(x) − C(x) is maximised where marginal revenue equals marginal cost, I'(x) = C'(x), provided the second derivative of P is negative.

Fundamental theorem of calculus

If F is an antiderivative of f on [a, b], then the definite integral of f from a to b equals F(b) − F(a). This links areas with antiderivatives.

Area between two curves

If f(x) ≥ g(x) on [a, b], the area enclosed between the curves is the definite integral of f(x) − g(x) from a to b.

Area with sign changes

The integral of f over [a, b] equals the net signed area. The geometric area is the integral of |f|, so split the interval at each zero of f.

Coefficient of variation

CV = s/x̄, often expressed as a percentage. It measures relative dispersion and allows comparison of variables with different units or very different means.

Pearson correlation coefficient

r = σₓᵧ/(σₓσᵧ) measures the strength and direction of linear association; it ranges from −1 to 1 and is unaffected by changes of scale.

Regression line of y on x

ŷ = ȳ + (σₓᵧ/σₓ²)(x − x̄). Its slope equals covariance divided by the variance of x, and the line passes through the point (x̄, ȳ).

Coefficient of determination

r², the square of the correlation coefficient, is the proportion of the variance of y explained by the linear regression on x; it lies between 0 and 1.

Conditional probability formula

P(A|B) = P(A ∩ B)/P(B) for P(B) > 0. It gives the probability of A given that B has occurred, restricting the sample space to B.

Independence criterion

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

Law of total probability

If B₁, …, Bₙ partition the sample space, then P(A) equals the sum over i of P(Bᵢ)·P(A|Bᵢ), combining conditional probabilities across cases.

Bayes' theorem

P(Bⱼ|A) = P(Bⱼ)P(A|Bⱼ) / Σ P(Bᵢ)P(A|Bᵢ). It reverses conditional probabilities, updating prior beliefs after observing A.

Binomial distribution parameters

For X ~ B(n, p): P(X = k) = C(n,k)pᵏ(1−p)ⁿ⁻ᵏ, with mean np and variance np(1−p). It models the number of successes in n independent trials.

Standardisation to N(0,1)

If X ~ N(μ, σ), then Z = (x − μ)/σ follows a standard normal distribution, allowing probabilities to be read from standard normal tables.

Normal approximation to binomial

When np ≥ 5 and n(1−p) ≥ 5, B(n, p) is approximated by N(np, √(np(1−p))). Apply a continuity correction of ±0.5 to the boundaries.