Unit 7: Foundations of Calculus
Section C: Applications of Derivatives
Welcome
Welcome to Section C: Applications of Derivatives! In this section, you’ll explore how derivatives are used to solve a wide range of real-world problems, from finding maximum and minimum values to analyzing motion and optimizing business operations.
Imagine
Imagine you’re a business analyst tasked with maximizing profits for a company. The concept of derivatives allows you to determine the optimal price and production level to achieve the highest profit, guiding the company’s strategic decisions.
Context
You’ve previously studied the basics of differential calculus, including how to calculate derivatives. Now, we’ll apply these concepts to solve real-world problems, helping you understand the practical significance of derivatives in various contexts.
Overview
This section covers the rate of change in various contexts, using derivatives to find maxima and minima, derivatives in economics (cost and revenue functions), sketching curves using first and second derivatives, and optimizing business operations using derivatives. You’ll learn to apply these concepts to analyze and solve problems involving optimization and change.
Objectives
- Understand and calculate rates of change in various contexts, applying derivatives to analyze how functions behave over time.
- Use derivatives to find maximum and minimum values of functions, solving optimization problems in various scenarios.
- Apply derivatives in economics, analyzing cost and revenue functions to optimize business operations.
- Sketch curves using first and second derivatives, understanding how these tools help analyze and visualize the behavior of functions.
- Optimize business operations using derivatives, applying these concepts to real-world problems in economics and management.
Preparatory Guidance
Definitions and Pronunciations
- Maxima and Minima: The highest and lowest points on a function’s graph, representing the function’s maximum and minimum values.
- Cost Function: A mathematical function representing the cost of producing a certain number of goods or services, often analyzed using derivatives to minimize costs.
Verbal Reading of Equations
- For finding maxima and minima, say “set the derivative equal to zero and solve for critical points to determine maximum and minimum values.”
- For cost and revenue functions, describe the process as “using derivatives to analyze and optimize business operations, finding the most efficient production levels.”
Problem-Solving Strategies
- Understand and calculate rates of change in various contexts, applying derivatives to analyze how functions behave over time.
- Use derivatives to find maximum and minimum values of functions, solving optimization problems in various scenarios.
- Apply derivatives in economics, analyzing cost and revenue functions to optimize business operations.
- Sketch curves using first and second derivatives, understanding how these tools help analyze and visualize the behavior of functions.
- Optimize business operations using derivatives, applying these concepts to real-world problems in economics and management.
Considerations
How can understanding the applications of derivatives help you analyze and optimize functions in real-world scenarios? Why are derivatives important for solving problems involving change and optimization? In what ways can you apply derivatives to improve business operations and make strategic decisions?