Calculating the Cost Function

This page focuses on determining the coefficients of the cost function K(x) = ax³ + bx² + cx + d using given data points.

The process involves:

1. Setting up a system of equations using the given data points.
2. Solving the system of equations to find the values of a, b, and c.
3. Using the known fixed cost $d = 24$ to complete the function.

Example: Using data points (4, 68), (6, 122), and (7, 166), we create a system of equations: 64a + 16b + 4c = 44 216a + 36b + 6c = 98 343a + 49b + 7c = 142

Through algebraic manipulation and substitution, the coefficients are determined:

a = 1 b = -10 c = 35 d = 24 (given)

Highlight: The final cost function is derived as K(x) = x³ - 10x² + 35x + 24

This method demonstrates how to construct a cost function from empirical data, a crucial skill in business economics and financial modeling.

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.

Function Analysis

This section introduces a new problem focusing on analyzing a quadratic function: f(x) = x² - 10x² + 9

The analysis includes:

a) Symmetry b) Behavior at infinity c) Y-intercept d) Roots (zeros)

Example: Symmetry analysis f$-x$ = $-x$² - 10$-x$² + 9 = x² - 10x² + 9 = f(x) This shows that the function is symmetric about the y-axis.

The behavior at infinity is determined by examining the leading term (x⁴), indicating that f(x) approaches positive infinity as x approaches either positive or negative infinity.

The y-intercept is found by calculating f(0) = 9, so the y-intercept is at (0, 9).

Highlight: This comprehensive analysis provides a complete picture of the function's behavior and key characteristics.

Root Finding and Function Construction

This final section covers two main topics:

1. Finding the roots of the quadratic function f(x) = x² - 10x² + 9 using the p-q formula.

Formula: For a quadratic equation ax² + bx + c = 0, x = -p/2 ± √$(p/2)² - q$, where p = b/a and q = c/a

The roots are calculated as x₁ = 5 + √16 and x₂ = 5 - √16.

2. Constructing a function with given roots:

Given roots: x₁ = 1, x₂ = -2, x₃ = 4

The function is constructed as: f(x) = $x - 1$$x + 2$$x - 4$

Example: Expanding this gives f(x) = x³ - 3x² - 6x + 8

Highlight: This demonstrates how to both analyze existing functions and construct new functions with specific properties, essential skills in advanced mathematics and its applications.

Page 8-9: Function Analysis and Additional Problems

These final pages cover supplementary mathematical analysis and additional practice problems.

Highlight: Includes detailed analysis of function symmetry and behavior at infinity

Example: Finding zeros of higher-degree polynomials using factoring and algebraic methods

Page 7: Maximum Profit and Unit Costs

This page determines the maximum profit point and variable unit costs.

Highlight: Maximum profit occurs at x = 6,000 units with a profit of 42,000 EUR.

Formula: Variable unit cost function: k(x) = x² - 10x + 35

Page 8: Function Analysis Techniques

This page covers advanced function analysis techniques.

Example: Analysis of symmetry, behavior at infinity, and finding zeros using the p-q formula.

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.