Revenue and Profit Functions

This section covers the calculation and graphing of the revenue and profit functions, as well as the break-even analysis.

1. Revenue Function: E(x) = 22x This linear function is derived directly from the given revenue per unit.

2. Profit Function: G(x) = E(x) - K(x) = 22x - (x³ - 10x² + 35x + 24) = -x³ + 10x² - 13x - 24

Definition: The profit function represents the difference between revenue and total costs.

3. Break-even Analysis:
   • Nutzenschwelle (Break-even point): (3, 0)
   • Nutzengrenze (Shutdown point): (8, 0)

Vocabulary:

• Nutzenschwelle: The point where revenue equals total costs
• Nutzengrenze: The point beyond which the business should cease operations

4. Graphing: The page includes instructions for graphing the cost, revenue, and profit functions on the same coordinate system, providing a visual representation of their relationships.

Highlight: Graphing these functions together allows for a clear visual analysis of the business's financial performance across different production quantities.

Profit Maximization and Variable Costs

This page focuses on analyzing the profit function to determine the maximum profit and calculating the variable cost function.

1. Profit Maximization: The maximum profit is estimated from the graph to be at the point (6, 42), meaning a profit of 42,000 € is achieved at a production quantity of 6,000 units.

Highlight: This graphical method provides a quick estimation of the optimal production quantity for maximum profit.

2. Variable Cost Function: The variable cost function is derived from the total cost function by removing the fixed costs:

   K(x) = x³ - 10x² + 35x + 24 Kv(x) = x³ - 10x² + 35x

   The variable unit cost function is then: kv(x) = x² - 10x + 35

Definition: Variable costs are those that change with the level of production, as opposed to fixed costs which remain constant.

This analysis provides crucial insights for managerial decision-making, helping to determine optimal production levels and understand cost structures.

Cost, Revenue and Profit Functions

This section introduces the fundamental concepts of cost, revenue, and profit functions in business economics. It provides a practical example of calculating these functions for a manufacturing company.

The problem presents a third-degree polynomial cost function K(x) = ax³ + bx² + cx + d, where x represents the production quantity. Fixed costs are given as 24 GE (monetary units), and revenue per unit is 22 GE. The task involves calculating various aspects of this business scenario.

Vocabulary: GE (Geldeinheit) - Monetary unit, ME (Mengeneinheit) - Quantity unit

Example: 1 GE = 1,000 EUR, 1 ME = 1,000 pieces

The problem is broken down into several steps, including:

a) Calculating the total cost function b) Determining the revenue function c) Graphing the cost and revenue functions d) Deriving the profit function e) Calculating break-even points f) Graphing the profit function g) Estimating the maximum profit h) Determining the variable unit cost function

Highlight: This comprehensive problem covers all major aspects of cost, revenue, and profit analysis, providing a practical application of these concepts.