Saarland Abitur Technik-Mathematik flashcards

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

Vector director

Non-zero vector parallel to a line; together with one point of the line it determines the line completely in vector, parametric and continuous forms.

Vector normal

Non-zero vector perpendicular to a plane; its components are exactly the coefficients of x, y and z in the general equation of the plane.

Producto escalar

Operation on two vectors giving the number |u||v|cos θ; zero result means the vectors are perpendicular, and it yields the angle between them.

Producto vectorial

Operation on two vectors in space giving a third vector perpendicular to both, with modulus |u||v|sin θ; used to build normals and to compute areas of parallelograms.

Producto mixto

Scalar u·(v×w), equal to the determinant of the three vectors as rows; it gives the signed volume of the parallelepiped and vanishes when they are coplanar.

Vectores coplanarios

Three or more vectors lying parallel to a single plane; three of them are coplanar exactly when their mixed product is zero.

Ecuaciones paramétricas de la recta

Form x = a₁ + t·v₁, y = a₂ + t·v₂, z = a₃ + t·v₃, where (a₁,a₂,a₃) is a point, v the direction vector and t the real parameter.

Ecuación general del plano

Equation Ax + By + Cz + D = 0, where (A,B,C) is a normal vector; substituting a point of the plane fixes the constant D.

Posición relativa de rectas

Classification of two lines as coincident, parallel, intersecting or skew, decided by comparing their direction vectors and checking whether a vector joining points lies in their span.

Rectas que se cruzan

Two lines in space that are neither parallel nor intersecting, so they lie in different parallel planes and share no point.

Ángulo entre recta y plano

For a nonzero line direction v and plane normal n, the non-obtuse angle alpha satisfies sin(alpha)=abs(v·n)/(|v||n|). It is complementary to the acute or right angle between the line direction and the normal.

Distancia de un punto a un plano

Length of the perpendicular segment from the point to the plane, computed as |Ax₀+By₀+Cz₀+D| divided by the modulus of the normal vector (A,B,C).

Límite de una función

Value L that f(x) approaches as x approaches a point; it may exist even when the function is undefined or takes another value at that point.

Continuidad

Property of a function at a point where the limit exists, the function is defined there, and both values coincide; differentiability implies it but not conversely.

Indeterminación

Expression such as 0/0, ∞/∞, ∞−∞ or 0·∞ whose limit cannot be read off directly and requires algebraic manipulation, L'Hôpital's rule or equivalent techniques.

Regla de L'Hôpital

Method for indeterminate quotients: replace the limit of f/g by the limit of the quotient of derivatives f'/g', provided the original quotient has form 0/0 or ∞/∞ and the theorem’s differentiability, nonzero denominator-derivative and derivative-limit hypotheses hold.

Derivada

Limit of the difference quotient, giving the slope of the tangent line and the instantaneous rate of change of a function at a point.

Derivabilidad

Existence of the derivative at a point; it requires continuity there, and at a breakpoint the two one-sided derivatives must agree.

Recta tangente

Line touching a curve at a point with slope equal to the derivative there; its equation is y − f(a) = f'(a)(x − a).

Optimización

Search for maxima or minima of a function, usually by solving f'(x) = 0 and classifying the critical points with the sign of the second derivative or a sign table.

Teorema de Bolzano

If a function is continuous on a closed interval and takes values of opposite sign at its ends, then it vanishes at some interior point; used to locate roots.

Teorema de Rolle

For a function continuous on [a,b] and differentiable on (a,b) with equal endpoint values, there is an interior point where the derivative is zero.

Primitiva

Function F whose derivative is the given function f; all primitives differ from F by an additive constant, forming the indefinite integral.

Integral definida

Limit of Riemann sums over an interval, equal to F(b) − F(a) for any primitive F; it gives the signed area between the curve and the x-axis.

Regla de Barrow

Fundamental theorem statement used in practice: the definite integral from a to b equals a primitive evaluated at b minus the same primitive evaluated at a.

Integración por partes

Technique based on ∫u dv = uv − ∫v du, useful when the integrand is a product such as a polynomial times an exponential or logarithm.

Área entre dos curvas

Region bounded by two graphs between two intersection abscissas, computed as the integral of the absolute difference of the functions.

Volumen de un sólido de revolución

Space generated by rotating a region about an axis; for rotation about OX the volume is π times the integral of f(x)² over the interval.

Vector Space

A set closed under addition and scalar multiplication, satisfying axioms such as associativity, commutativity, and existence of a zero vector and additive inverses.

Linear Independence

A property of a set of vectors where no nontrivial linear combination equals the zero vector. Equivalently, no vector in the set is a combination of the others.