PAU Applied Mathematics for Social Sciences II (Cantabria) flashcards

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

Matrix product condition

The product AB exists only when the number of columns of A equals the number of rows of B. The result has as many rows as A and as many columns as B.

Determinant of 2×2 matrix

For A = [[a,b],[c,d]], det A = ad − bc. The matrix is invertible exactly when this value is nonzero; otherwise it is singular.

Inverse of 2×2 matrix

A⁻¹ = (1/(ad−bc))·[[d,−b],[−c,a]] when ad − bc ≠ 0. Multiplying A by this matrix on either side gives the identity.

Rouché–Fröbenius theorem

A linear system is consistent exactly when the rank of the coefficient matrix equals the rank of the augmented matrix. If that common rank is below the number of unknowns, solutions are infinite.

Cramer's rule

For n equations in n unknowns with nonzero determinant, each unknown equals the determinant of the matrix with its column replaced by the constants, divided by det A.

Homogeneous system solutions

A system AX = 0 always has the trivial solution X = 0. It has nontrivial solutions exactly when rank(A) is less than the number of unknowns.

Feasible region

The set of points satisfying all constraints of a linear program, including non-negativity. In two variables it is a convex polygon, possibly unbounded or empty.

Vertex optimality principle

For a linear objective over a nonempty bounded feasible polygon, the maximum and minimum occur at vertices. Evaluating the objective at every vertex therefore solves the program.

Graphical method steps

Draw each constraint line, shade the half-plane satisfying it, intersect to find vertices, then evaluate the objective at each vertex and select the best value.

Slack variable

A non-negative variable added to a ≤ constraint to convert it into an equation: ax + by ≤ c becomes ax + by + s = c with s ≥ 0.

Unbounded linear program

When the feasible region is unbounded and the objective improves indefinitely along an open direction, no finite optimum exists. Otherwise the optimum still occurs at a vertex.

Continuity at a point

f is continuous at a when f(a) is defined, the limit of f as x approaches a exists, and that limit equals f(a). All three conditions are required.

Existence of a limit

The limit of f at a exists exactly when the left-hand and right-hand limits both exist and are equal. Differing one-sided limits produce a jump.

Removable discontinuity

A point where the limit of f exists but does not equal f(a), or f(a) is undefined. Redefining the single value at that point makes the function continuous.

Limits at infinity for rationals

Compare degrees of numerator and denominator: equal degrees give the ratio of leading coefficients, larger denominator degree gives zero, larger numerator degree gives an infinite limit.

Intermediate Value Theorem

If f is continuous on a closed interval and k lies between the endpoint values, some interior point c satisfies f(c) = k. Used to prove roots exist.

Derivative as a limit

f'(a) = lim(h→0)[f(a+h) − f(a)]/h. It gives the slope of the tangent line to the graph at x = a.

Chain rule

The derivative of a composite f(g(x)) is f'(g(x))·g'(x). The outer derivative is evaluated at the inner function and multiplied by the inner derivative.

Second derivative test

At a critical point where f' = 0, f'' > 0 indicates a local minimum and f'' < 0 a local maximum. If f'' = 0 the test is inconclusive.

Inflection point

A point where the concavity of f changes, which requires f'' to change sign there. It is not necessarily a critical point of the function.

Tangent line equation

The tangent to y = f(x) at x = a is y = f(a) + f'(a)(x − a). The normal line there has slope −1/f'(a) when f'(a) is nonzero.

Marginal cost and revenue

Marginal cost is the derivative of total cost and marginal revenue the derivative of total revenue. Profit is maximized where marginal revenue equals marginal cost.

Fundamental Theorem of Calculus

If F is a primitive of continuous f on [a,b], the definite integral of f from a to b equals F(b) − F(a). It links areas with antiderivatives.

Area between two curves

On an interval where f is above g, the enclosed area is the integral of f − g. Where curves cross, split the interval and integrate the absolute difference.

Integration by parts

The rule ∫u dv = uv − ∫v du. Choose u as the factor that becomes simpler when differentiated, such as a polynomial or a logarithm.

Primitive of 1/x

The antiderivative of 1/x is ln|x| + C. Absolute value is needed because the logarithm is defined only for positive arguments.

Mean value of a function

The average value of f on [a,b] equals (1/(b−a)) times the definite integral of f over that interval. It lies between the minimum and maximum values.

Conditional probability

P(A|B) = P(A ∩ B)/P(B) when P(B) > 0. It restricts the sample space to outcomes where B occurs.

Independence of two events

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.

Bayes' theorem

P(Aᵢ|B) = P(Aᵢ)P(B|Aᵢ) divided by the total probability of B. It reverses a conditional probability using prior probabilities.

Binomial distribution

For n independent trials with success probability p, P(X = k) = C(n,k)pᵏ(1−p)ⁿ⁻ᵏ. Mean is np and variance is np(1−p).

Standardizing a normal variable

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

Variance of a linear transform

Var(aX + b) = a²Var(X). Adding a constant shifts the distribution but does not change dispersion; scaling multiplies variance by the square.