Page 1: Performance Assessment in Analysis 2

This page introduces a series of calculus problems focusing on function analysis and optimization.

1. Function Analysis:
   • Students are asked to analyze a given function, including its domain, symmetry, and derivatives.
   • The task involves finding zeros, poles, and Extrempunkte berechnen.

Vocabulary: Extrempunkte (extreme points) are the maximum and minimum points of a function.

2. Inflection Point and Tangent:
   • A cubic function with a given inflection point is presented.
   • The goal is to determine the equation of the tangent at the inflection point.

Definition: An inflection point is where a function's curvature changes from concave to convex or vice versa.

3. Cubic Function Reconstruction:
   • Given specific properties of a cubic function, students must derive its equation.

Example: The function has a local maximum at (1,4), a zero at x=-1, and an inflection point at x=3.

4. Optimization Problem:
   • A rectangular area optimization problem is presented using a quadratic function.

Highlight: This is a classic Extremwertaufgaben maximaler Flächeninhalt Rechteck problem.

5. Cost Optimization:
   • A real-world problem involving yacht production costs is introduced.
   • The task is to find the production quantity that minimizes unit costs.

Vocabulary: Stückkosten (unit costs) are the total costs divided by the number of units produced.