Construct detailed responses to the following prompts. Your answers should synthesize concepts and provide examples drawn from the source material.

1. Discuss the application of linear functions as functional models in business and economics. Use specific examples from the text, such as cost functions, profit models based on pricing, and the concept of market equilibrium, to illustrate your points.
2. Explain the process of finding a function’s relative maxima and minima. Describe the role of the first and second derivatives, defining critical points, concavity, and the Second-Derivative Test in your explanation.
3. Compare and contrast the concepts of marginal analysis and approximation by differentials. Explain how the derivative is used in both contexts to estimate changes in a function, providing examples such as estimating the cost of producing an additional unit versus estimating the change in cost for a fractional change in production.
4. Describe the process of solving a constrained optimization problem for a function of two variables using the method of Lagrange multipliers. Explain the setup of the Lagrange equations and the economic interpretation of the Lagrange multiplier, λ.
5. Outline the geometric method for solving a linear programming problem. Define the key components—objective function, decision variables, constraints, feasibility region, and corner points—and explain how they are used to find an optimal solution.