Abitur Mathematics flashcards

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

Derivative of e^(g(x))

By the chain rule, d/dx e^(g(x)) = g'(x)·e^(g(x)). The exponential reproduces itself and the inner derivative multiplies in front.

Product rule

For differentiable u and v, (u·v)' = u'·v + u·v'. Differentiate one factor at a time and add the two resulting products.

Quotient rule

For v(x) ≠ 0, (u/v)' = (u'·v − u·v') / v². The order in the numerator matters: derivative of the top times bottom minus top times derivative of the bottom.

Chain rule

d/dx f(g(x)) = f'(g(x))·g'(x). Differentiate the outer function at the inner function and multiply by the inner derivative.

Second derivative test

If f'(x₀) = 0 and f''(x₀) < 0, then f has a local maximum at x₀; if f''(x₀) > 0, it is a local minimum.

Inflection point condition

A point where f''(x₀) = 0 and f'' changes sign is an inflection point. The sign change is essential; f'' = 0 alone is not sufficient.

Definite integral as area

For f ≥ 0 on [a,b], ∫ₐᵇ f(x) dx is the area between the graph and the x-axis. Regions below the axis contribute negative values.

Substitution rule

∫ f(g(x))·g'(x) dx = F(g(x)) + C where F' = f. For definite integrals, also change the limits from x-values to g(x)-values.

Tangent line equation

The tangent to f at x₀ is y = f(x₀) + f'(x₀)·(x − x₀). It passes through the point of tangency with the local slope.

Dot product

a·b = a₁b₁ + a₂b₂ + a₃b₃ = |a||b|cos φ. It equals zero exactly when the two vectors are perpendicular.

Cross product

a×b is a vector perpendicular to both a and b; its magnitude |a×b| = |a||b|sin φ equals the area of the spanned parallelogram.

Plane equation in normal form

A plane is given by n·(x − p) = 0, where n is a normal vector and p is a point in the plane. Expanded, this reads n·x = d.

Distance point to plane

d = |n·(q − p)| / |n|, where p lies in the plane and n is its normal. In coordinate form it is |aq₁+bq₂+cq₃−d| / √(a²+b²+c²).

Distance between skew lines

For lines p₁ + s·u₁ and p₂ + t·u₂ with direction vectors u₁, u₂, the distance is |(p₂ − p₁)·(u₁×u₂)| / |u₁×u₂|.

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.

Gaussian elimination

Row operations (swapping, scaling, adding multiples) transform a linear system into row-echelon form without changing its solution set. Back substitution then yields the solutions.

Angle between line and plane

The angle φ satisfies sin φ = |u·n| / (|u||n|), where u is the line direction and n the plane normal. The line-plane angle is the complement of the line-normal angle.

Binomial probability formula

P(X = k) = C(n,k)·p^k·(1−p)^(n−k) for k = 0,…,n. It counts k successes in n independent trials with constant success probability p.

Binomial mean and variance

For X ~ B(n,p): E(X) = n·p and Var(X) = n·p·(1−p), so the standard deviation is σ = √(np(1−p)).

Conditional probability

P(A|B) = P(A∩B)/P(B) for P(B) > 0. It is the probability of A given that B has occurred, restricted to the reduced sample space B.

Independence of events

Events A and B are independent exactly when P(A∩B) = P(A)·P(B), equivalently P(A|B) = P(A) whenever P(B) > 0.

Standardisation

z = (x − μ)/σ converts a normal variable into the standard normal Z with mean 0 and variance 1. Then P(X ≤ x) = Φ(z).

Standard normal symmetry

Because the standard normal density is symmetric, Φ(−z) = 1 − Φ(z). This lets you read only positive z-values from the table.

One-sided upper-tail test

For H₀: p ≤ p₀ against H₁: p > p₀, reject H₀ at level α if the observed count k satisfies P(X ≥ k | p₀) ≤ α.

Type-I and type-II errors

A type-I error rejects a true null hypothesis and has probability at most α. A type-II error fails to reject a false null hypothesis.

Law of total probability

If A₁,…,Aₙ partition the sample space, then P(B) = Σ P(Aᵢ)·P(B|Aᵢ). Multiply along tree branches and add across branches.

Exponential growth model

f(t) = a·e^(kt) with k > 0 for growth and k < 0 for decay. The doubling time is T = ln 2 / k; the half-life uses |k|.

Logistic growth

f(t) = S/(1 + c·e^(−kt)) grows towards the saturation limit S. The growth rate peaks at the inflection point where f = S/2.

Rate as derivative

If f(t) describes a quantity, f'(t) is its instantaneous rate of change with units (unit of f) per (unit of t). The integral of the rate gives the total change.

Average value of a function

The mean value of f over [a,b] is (1/(b−a))·∫ₐᵇ f(x) dx. It is the constant height giving the same area over the interval.

Optimisation with a constraint

Use the constraint to express the objective in one variable, differentiate, solve f' = 0, and verify the extremum with f'' or by checking the interval endpoints.

Correlation coefficient r

r lies in [−1,1] and measures the strength of a linear relationship between two data sets. Values near ±1 indicate a strong linear fit; near 0, a weak one.