Chapter 3 Sections 6 & 7: Power Rule and Real-World Applications

Welcome to this week's exploration of calculus! We'll be focusing on sections 3.6 and 3.7, covering the Power Rule and its application to real-world problems. While section 3.6 builds upon previous differentiation techniques, section 3.7 is where we'll see these rules in action. Let's get started!

Section 3.6: The Power Rule

In Section 3.6, we'll be working with the Power Rule. As I mentioned in class, we won't be dwelling on this section for too long, as it closely resembles the concepts we covered in sections 3.1 through 3.4. The main idea is to understand and apply the rule effectively.

The Power Rule states that if we have a function of the form $f(x) = x^n$, where $n$ is a constant, then its derivative is:

$$f'(x) = nx^{n-1}$$

In essence, you multiply by the exponent and then reduce the exponent by 1. For example, if $f(x) = x^5$, then $f'(x) = 5x^4$.

Key points to remember:

  • The power rule applies when differentiating terms with a variable raised to a constant power.
  • Make sure you can rewrite the function to identify the power, such as converting $\sqrt{x}$ to $x^{\frac{1}{2}}$.
  • Don't forget to apply other differentiation rules in conjunction with the power rule, such as the constant multiple rule.

Here is the PowerPoint presentation for this section: Section 3-6 PowerPoint

Section 3.7: Real-World Applications of Differentiation

Section 3.7 is where things get really interesting! We'll be exploring how differentiation rules can be applied to solve real-life problems. This section will be the primary focus of tonight's class, where we'll be working through various examples together.

Here are some common types of application problems you might encounter:

  • Related Rates: These problems involve finding the rate of change of one quantity in terms of the rate of change of another. For example, determining how fast the radius of a circle is increasing if we know how fast the area is increasing.
  • Optimization: These problems involve finding the maximum or minimum value of a function. For example, maximizing profit or minimizing cost. We will use the first and second derivative tests to find critical points and determine whether they are maximums, minimums, or neither.
  • Marginal Analysis: In economics, marginal cost, marginal revenue, and marginal profit are all derivatives. They help us understand how cost, revenue, and profit change as we produce and sell more units.

For example, consider the problem of finding the dimensions of a rectangular garden with the largest area that can be enclosed by a fixed amount of fencing. This is an optimization problem where we need to maximize the area function subject to the constraint of the fixed perimeter.

Remember these steps when approaching application problems:

  1. Read the problem carefully: Identify what you are given and what you need to find.
  2. Draw a diagram: A visual representation can help you understand the relationships between the variables.
  3. Write down the relevant equations: Express the quantity you want to optimize or relate in terms of other variables.
  4. Differentiate: Use the differentiation rules we've learned to find the derivative.
  5. Solve: Find the critical points and determine the maximum or minimum value.
  6. Answer the question: Make sure you provide the answer in the context of the problem.

Here is the PowerPoint presentation for this section: Section 3-7 PowerPoint

I encourage you to review the examples in the textbook and the PowerPoint presentations. Practice is key to mastering these concepts! Good luck, and I look forward to seeing you in class.