Saarland Abitur Technik-Mathematik key terms

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.

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.

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.

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.

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.

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

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.

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

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.

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

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.

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).

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.

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.

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.

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.

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

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

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

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.

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.

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.

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

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.

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.

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

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

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.

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

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.