Mathematik — Analysis (Calculus) for the Abitur Exam

Common failure modes

These are the patterns that cause most candidates to lose marks on this topic. Recognising them in advance is half the work.

• !Forgetting to apply the chain rule when differentiating composite functions
• !Integration errors: missing the constant of integration, or incorrect limits for definite integrals
• !Not showing the complete Kurvendiskussion (function analysis) steps: domain, zeros, extrema, inflection points, asymptotes

Study tips

• 1Master all six differentiation rules: Ableitungsregel for power, product (Produktregel), quotient (Quotientenregel), chain (Kettenregel), sum, and constant-factor rule.
• 2Practise the complete Kurvendiskussion (function analysis) template: Definitionsbereich → Nullstellen → Extremstellen → Wendepunkte → Verhalten für x → ±∞.
• 3For Extremwertaufgaben (optimisation): define the objective function, find the domain, differentiate, set to zero, verify it is a maximum/minimum.

Sample Abitur Mathematik — Analysis (Calculus) questions

These sample items mirror the format and difficulty of real Abitur questions. Practice with thousands more on the free Koydo question bank.

1. 1

   Gegeben: f(x) = x³ − 3x. Bestimmen Sie die Extremstellen. (Given: f(x) = x³ − 3x. Find the extrema.)

   • AKeine Extremstellen (No extrema)
   • BMinimum bei x = 0
   • CMaximum bei x = −1, Minimum bei x = 1Correct
   • DMaximum bei x = 1, Minimum bei x = −1
   Why this answer?

   f'(x) = 3x² − 3 = 0 → x² = 1 → x = ±1. f''(x) = 6x. At x = −1: f''(−1) = −6 < 0 → Hochpunkt (maximum). At x = 1: f''(1) = 6 > 0 → Tiefpunkt (minimum). Maxima at x = −1, Minima at x = 1.