PAU Applied Mathematics for Social Sciences II - Canary Islands / ULPGC & ULL flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Rouché–Capelli consistency criterion

A linear system Ax = b is consistent exactly when rank(A) equals rank(A|b). If that common rank equals the number of unknowns the solution is unique; if it is smaller, there are infinitely many solutions.

Cramer's rule

For a square system Ax = b with det A ≠ 0, each unknown equals x_i = det(A_i)/det A, where A_i is A with its i-th column replaced by the vector b.

Inverse of a product

If A and B are invertible matrices of the same order, then (AB)⁻¹ = B⁻¹A⁻¹. The order of the factors is reversed, because matrix multiplication is not commutative.

Rank via Gaussian elimination

Row-reduce the matrix to echelon form; the rank is the number of nonzero rows, equivalently the number of pivot positions. Row operations do not change the rank.

Solving AX = B

If A is invertible, multiply both sides on the left by A⁻¹ to get X = A⁻¹B. For XA = B multiply on the right instead, giving X = BA⁻¹.

Vertex of feasible region

A corner point of the feasible polygon, obtained by solving the two linear equations given by the boundary lines of two active constraints and checking that the point satisfies all inequalities.

Corner-point theorem

If a linear objective is optimised over a nonempty bounded feasible region, the optimal value is attained at a vertex. Evaluate the objective at every vertex and compare.

Slack variable

A nonnegative variable added to a ≤ constraint to turn it into an equation. Its value at the optimum measures the unused amount of the corresponding resource.

Shadow price

The rate of change of the optimal objective value per unit increase in a constraint's right-hand side, equal to the associated dual variable and valid while the current basis remains optimal.

Multiple optimal solutions

When the objective's level lines are parallel to one edge of the feasible polygon, every point of that edge gives the same optimal value, so the optimum is not unique.

L'Hôpital's rule

For a 0/0 or ∞/∞ indeterminacy, the limit of f/g equals the limit of f'/g', provided the latter exists. Differentiate numerator and denominator separately, not as a quotient.

Limit of rational function at infinity

Compare polynomial degrees: numerator higher gives infinite limit, denominator higher gives zero, and equal degrees give the ratio of the leading coefficients.

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 derivative, i.e. the slope of the curve at that point.

Chain rule

The derivative of a composition is [f(g(x))]' = f'(g(x))·g'(x). Differentiate the outer function at the inner value, then multiply by the inner derivative.

Number e as a limit

The sequence (1 + 1/n)ⁿ tends to e ≈ 2.71828 as n tends to infinity. This limit underlies indeterminacies of type 1 raised to infinity.

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, and if f''(c) = 0 the test is inconclusive.

Inflection point

A point where the concavity of f changes. It requires f'' to be zero or undefined there, and f'' must actually change sign across that point.

Profit maximisation 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 profit is negative there.

Global extrema on closed interval

Evaluate f at every critical point inside the open interval and at both endpoints; the largest of these values is the global maximum and the smallest the global minimum.

Horizontal asymptote

The line y = L where L is the limit of f(x) as x tends to plus or minus infinity. Rational functions may have different limits at each end.

Barrow's rule

If F is any antiderivative of a continuous f on [a, b], then the definite integral of f from a to b equals F(b) − F(a).

Area between two curves

If f(x) ≥ g(x) on [a, b], the enclosed area is the integral of f − g over that interval. Find intersection points for the limits and split where the order changes.

Accumulation function derivative

If A(x) is the integral of f from a to x, then A'(x) = f(x) whenever f is continuous. The integral of a rate gives the accumulated total change.

Total variable cost by integration

The definite integral of marginal cost between two production levels gives the change in total variable cost over that interval, since marginal cost is the derivative of cost.

Antiderivative of 1/x

The integral of 1/x with respect to x equals the natural logarithm of the absolute value of x plus a constant, valid for x different from zero.

Conditional probability formula

For events A and B with P(B) > 0, the probability of A given B is P(A ∩ B)/P(B). Rearranged, it gives the product rule P(A ∩ B) = P(B)P(A|B).

Independence criterion

Events A and B are independent exactly when P(A ∩ B) = P(A)P(B), equivalently P(A|B) = P(A) when P(B) > 0. Disjoint events with positive probability are dependent.

Law of total probability

If B₁ to Bₙ form a partition of the sample space, then P(A) equals the sum over i of P(Bᵢ) multiplied by P(A|Bᵢ).

Bayes' theorem

P(Bᵢ|A) = P(Bᵢ)P(A|Bᵢ) divided by the sum over j of P(Bⱼ)P(A|Bⱼ). It reverses the conditioning using prior and likelihood information.

Binomial distribution

For n independent trials with success probability p, P(X = k) = C(n, k)pᵏ(1−p)^{n−k}. Its mean is np and its variance is np(1−p).

Standardisation to normal Z

If X is normal with mean μ and standard deviation σ, then Z = (X − μ)/σ is standard normal. Probabilities are read from the table of Φ.

Variance shortcut formula

The variance of X equals E(X²) minus the square of E(X). For a linear change, Var(aX + b) = a²Var(X), so adding a constant does not affect spread.