Saxony Abitur Mathematics flashcards

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

Power rule

For f(x) = x^n with constant exponent n, the derivative is f'(x) = n·x^(n−1). It is the basis for differentiating polynomials term by term, together with the sum and constant-factor rules.

Chain rule

For a composite function f(x) = u(v(x)), the derivative is f'(x) = u'(v(x))·v'(x). The inner derivative is multiplied on the outside; forgetting it is the most common differentiation error.

Product rule

For f(x) = u(x)·v(x), f'(x) = u'(x)·v(x) + u(x)·v'(x). Each factor is differentiated once while the other is held fixed, and the two contributions are added.

Quotient rule

For f(x) = u(x)/v(x) with v(x) ≠ 0, f'(x) = (u'(x)v(x) − u(x)v'(x))/v(x)². The order in the numerator matters: the denominator times the derivative of the numerator comes first.

Second-derivative test

If f'(x₀) = 0 and f''(x₀) < 0, then x₀ is a local maximum; if f''(x₀) > 0, it is a local minimum. If f''(x₀) = 0, the test gives no decision and the sign of f' must be checked.

Inflection point

A point where the second derivative changes sign and the function is continuous. It marks a change between convex and concave curvature; the tangent there is called the Wendetangente in German usage.

Tangent line equation

The tangent to f at x₀ is t(x) = f'(x₀)(x − x₀) + f(x₀). It touches the graph at x₀ with the same slope, and its linear approximation is used for estimates near x₀.

Main theorem of calculus

If f is continuous on [a; b] and F is any antiderivative of f, then the definite integral from a to b equals F(b) − F(a). It links areas under graphs to antiderivatives.

Area between two graphs

For f(x) ≥ g(x) on [a; b], the enclosed area is the integral of f − g from a to b. Where the graphs cross, split the interval and integrate the absolute difference on each part.

Dot product

For vectors a and b, a·b = a₁b₁ + a₂b₂ + a₃b₃ = |a||b|cos φ. It is zero exactly when the nonzero vectors are perpendicular, and it yields the angle between them.

Cross product

For vectors a and b in space, a×b is the vector (a₂b₃ − a₃b₂; a₃b₁ − a₁b₃; a₁b₂ − a₂b₁). It is perpendicular to both, and its length equals the parallelogram area.

Point–direction line form

A line is written g: x = a + t·u, where a is the position vector of a point on the line, u ≠ 0 is the direction vector, and t runs over the real numbers.

Normal form of a plane

A plane is given by n·(x − a) = 0, where n is a normal vector and a is the position vector of a point in the plane. Expanding gives the coordinate form n₁x + n₂y + n₃z = d.

Distance point to plane

For the plane n·x = d and a point with position vector p, the distance is |n·p − d|/|n|. The numerator is the plane equation evaluated at the point; the denominator normalises the normal vector.

Line–plane position test

Substitute the line x = a + t·u into the plane equation. If u·n = 0 and a satisfies the plane, the line lies in it; if u·n = 0 only, it is parallel; otherwise there is one intersection point.

Gaussian elimination

Row operations (swapping rows, adding a multiple of one row to another, scaling a row) reduce a linear system to staircase form. Back substitution then gives the solution set, which may be empty or infinite.

Determinant and invertibility

A square matrix is invertible exactly when its determinant is nonzero. For 2×2 matrices, det A = ad − bc and A⁻¹ = (1/det A)·[[d, −b], [−c, a]].

Laplace probability

When all outcomes in a finite sample space are equally likely, the probability of an event A is the number of favourable outcomes divided by the total number of outcomes: P(A) = |A|/|Ω|.

Conditional probability

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

Stochastic independence

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. Knowing one event then gives no information about the other.

Binomial distribution

X ~ B(n; p) counts successes in n independent trials with constant success probability p. Its probability mass function is P(X = k) = C(n, k)·p^k·(1 − p)^(n−k) for k = 0, 1, …, n.

Binomial mean and variance

For X ~ B(n; p), the expected value is μ = n·p and the variance is σ² = n·p·(1 − p). The standard deviation is the square root σ = √(n·p·(1 − p)).

Expected value of a discrete variable

E(X) = Σ xᵢ·P(X = xᵢ), the probability-weighted average of all values. For constants a and b, linearity gives E(aX + b) = a·E(X) + b.

Variance formula

Var(X) = E(X²) − (E(X))². For a transformed variable, Var(aX + b) = a²·Var(X); adding a constant shifts the distribution but leaves its spread unchanged.

Standard normal variable

If X has mean μ and standard deviation σ > 0, then Z = (X − μ)/σ has mean 0 and standard deviation 1. Probabilities are read from the standard normal table via Φ.

Normal approximation criterion

A binomial distribution B(n; p) may be approximated by a normal distribution with μ = n·p and σ = √(n·p·(1 − p)) when σ > 3. Continuity correction shifts the boundary by 0.5.

Hypothesis test decision rule

Choose a significance level α. Reject the null hypothesis when the probability of the observed or more extreme result under H₀ is at most α; otherwise the result is not significant at that level.

Contrapositive equivalence

The statement 'A implies B' is logically equivalent to its contrapositive 'not B implies not A'. The converse 'B implies A' is not equivalent and needs a separate proof.

Proof by contradiction

To prove a claim, assume its negation and derive a logical contradiction. The contradiction shows the negation is false, so the original claim must be true; irrationality of √2 is a classic example.

Negating quantified statements

Negating 'for all x, P(x)' gives 'there exists an x with not P(x)', and negating 'there exists x with P(x)' gives 'for all x, not P(x)'. The quantifier flips and the predicate is negated.

Counterexample

A single example that satisfies the premises but violates the conclusion disproves a universal claim. Examples can never prove a 'for all' statement, though one example proves an existence claim.

Interpretation of the derivative

In an applied context, f'(x₀) is the instantaneous rate of change at x₀, measured in units of f per unit of x. Its sign indicates increase or decrease, and its magnitude the speed of change.