PAU Applied Math ACS II (Galicia) flashcards

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

Matrix product condition

The product A·B is defined only when the number of columns of A equals the number of rows of B. The result has rows of A and columns of B; entry (i,j) is the dot product of row i of A and column j of B.

Determinant of a 2×2 matrix

For A = [[a,b],[c,d]], det A = ad − bc. The matrix is invertible exactly when this value is nonzero; if det A = 0 the matrix is singular and has no inverse.

Rouché–Frobenius theorem

A linear system AX = B is consistent iff rank(A) equals rank(A|B). If both equal n (number of unknowns) the solution is unique; if both are less than n there are infinitely many solutions.

Cramer's rule

For a square system AX = B with det A ≠ 0, each unknown is xᵢ = det Aᵢ / det A, where Aᵢ is A with column i replaced by the vector of independent terms B.

Limit existence criterion

The limit of f at a exists only when the left-hand limit and the right-hand limit both exist and are equal. The value f(a) itself need not exist for the limit to exist.

Derivative as a limit

The derivative at a is f'(a) = lim(h→0)[f(a+h) − f(a)]/h. It gives the slope of the tangent line and the instantaneous rate of change of the function at that point.

Second derivative test

At a critical point where f'(x) = 0: if f''(x) > 0 the point is a local minimum, if f''(x) < 0 it is a local maximum, and if f''(x) = 0 the test is inconclusive.

Tangent line equation

The tangent to y = f(x) at x = a is y = f(a) + f'(a)(x − a). The point of tangency is (a, f(a)) and the slope is the derivative evaluated at a.

Fundamental Theorem of Calculus

If F is any antiderivative of a continuous f, then the definite integral from a to b equals F(b) − F(a). This links areas under curves with antiderivatives and enables exact computation.

Area between two curves

If f(x) ≥ g(x) on [a,b], the area between them is ∫ₐᵇ [f(x) − g(x)] dx. When the curves cross inside the interval, split the integral at each intersection point.

Integration by parts

The rule ∫u dv = uv − ∫v du reverses the product rule. Choose u by the LIATE order (logarithmic, inverse trigonometric, algebraic, trigonometric, exponential) to make the remaining integral simpler.

Conditional probability formula

For events A and B with P(B) > 0, the probability of A given B is P(A|B) = P(A∩B)/P(B). It restricts the sample space to outcomes where B has occurred.

Independence of two events

Events A and B are independent exactly when P(A∩B) = P(A)·P(B), which is equivalent to P(A|B) = P(A) whenever P(B) > 0. Independence is not the same as mutual exclusivity.

Bayes' theorem

For a partition A₁,…,Aₙ, P(Aᵢ|B) = P(Aᵢ)·P(B|Aᵢ) / Σⱼ P(Aⱼ)·P(B|Aⱼ). It updates prior probabilities using observed evidence and the total probability in the denominator.

Binomial distribution formula

If X counts successes in n independent trials with success probability p, then P(X=k) = C(n,k)·pᵏ·(1−p)ⁿ⁻ᵏ. The mean is np and the variance is np(1−p).

Standardising a normal variable

If X ~ N(μ,σ), the transformation z = (x − μ)/σ gives a standard normal Z ~ N(0,1). Then P(X ≤ x) = Φ(z), read from the standard normal table.

Mean and variance of univariate data

The mean is x̄ = Σxᵢ/n. The variance is σ² = Σxᵢ²/n − x̄², and the standard deviation is its positive square root σ. The coefficient of variation is σ/x̄.

Pearson correlation coefficient

The correlation r = s_xy/(s_x·s_y) measures the strength and direction of linear association. It always lies between −1 and 1; values near 0 indicate weak linear relationship only.

Regression line of y on x

The least-squares line is y = ȳ + (s_xy/s_x²)(x − x̄). Its slope is b = s_xy/s_x² and its intercept is a = ȳ − b·x̄; the line always passes through (x̄, ȳ).

Interquartile range and outliers

The IQR = Q₃ − Q₁ measures the spread of the middle 50% of the data. Values below Q₁ − 1.5·IQR or above Q₃ + 1.5·IQR are commonly classified as outliers.

Linear model interpretation

In y = mx + c, the slope m is the constant rate of change per unit increase in x, and c is the value of y when x = 0. Use it when change per unit is constant.

Exponential growth and decay

The model y = y₀·e^{kt} describes growth when k > 0 and decay when k < 0. The doubling time for growth is ln2/k, and the half-life for decay is also ln2/|k|.

Break-even point

The break-even quantity is the value of x satisfying Revenue(x) = Cost(x). Profit is P(x) = R(x) − C(x), so break-even corresponds to P(x) = 0; the marginal profit is P'(x).

Quadratic vertex in models

For y = ax² + bx + c, the vertex occurs at x = −b/(2a). It gives the maximum when a < 0 and the minimum when a > 0, which is useful for optimisation in context.

Feasible region in linear programming

The feasible region is the set of points satisfying every constraint simultaneously, including non-negativity. It is a convex polygon whose vertices come from solving pairs of constraint equations.

Vertex theorem of linear programming

If the feasible region is nonempty and bounded, the objective function z = ax + by attains its maximum and minimum at vertices. Evaluate z at every vertex and compare.

Unbounded feasible region

When the feasible region is unbounded, the optimum may fail to exist or may be infinite. Check the objective value along unbounded edges before concluding that a vertex gives the optimum.

Redundant constraint

A constraint is redundant if removing it does not change the feasible region. At the optimum it never binds; identifying redundant constraints simplifies the graphical solution.

L'Hôpital's rule

For indeterminate forms 0/0 or ∞/∞, lim f/g = lim f'/g', provided the latter limit exists. Differentiate numerator and denominator separately; the rule is not the quotient rule.

Global extrema on closed interval

A continuous function on [a,b] attains its global maximum and minimum either at critical points inside (a,b) or at the endpoints a and b. Evaluate f at all these candidates.

Binomial approximated by normal

If X ~ B(n,p) with np > 5 and n(1−p) > 5, then X is approximately N(np, √(np(1−p))). Apply a continuity correction by using boundaries such as k ± 0.5.

Total probability theorem

If A₁,…,Aₙ partition the sample space, then P(B) = Σ P(Aᵢ)·P(B|Aᵢ). It computes the probability of B by weighting each conditional probability by its prior.