0### **Lesson Plan: Applications of Derivatives**
**Grade Level**: 12th Grade
**Subject**: Mathematics
**Topic**: Applications of Derivatives
**Duration**: 60 minutes
---
### **Learning Objectives:**
By the end of the lesson, students will be able to:
- Understand the concept of derivatives and their real-life applications.
- Use derivatives to solve problems related to rate of change, increasing and decreasing functions, and maxima/minima.
- Apply derivatives to curve sketching and optimization problems.
---
### **Lesson Structure:**
#### **1. Introduction to Derivatives (10 minutes):**
- **Definition**: Introduce the derivative as a rate of change of a function with respect to a variable.
> **Derivative**: It measures how a function changes as its input changes.
- **Formula**:
\[
f'(x) = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}
\]
- **Basic Idea**: Link derivatives to real-life scenarios, such as the speed of a moving car (change of position over time) or the rate of change in temperature.
#### **2. Applications of Derivatives (20 minutes):**
##### **2.1 Rate of Change**:
- **Concept**: The derivative is used to calculate the rate of change of one quantity with respect to another.
- **Example**: The rate of change of the area of a circle as its radius changes.
\[
\text{Given}: A = \pi r^2 \quad \text{Find:} \frac{dA}{dr}
\]
- Solution:
\[
\frac{dA}{dr} = 2\pi r
\]
This shows that the area of the circle changes at a rate proportional to its radius.
##### **2.2 Increasing and Decreasing Functions**:
- **Concept**: The derivative tells us if a function is increasing or decreasing.
- **Rule**:
- If \( f'(x) > 0 \), the function is increasing.
- If \( f'(x) < 0 \), the function is decreasing.
- **Example**: Find the intervals in which \( f(x) = x^3 - 3x^2 + 1 \) is increasing or decreasing.
- Solution:
\[
f'(x) = 3x^2 - 6x
\]
Solve \( f'(x) = 0 \) to find critical points and analyze the sign of \( f'(x) \) on intervals.
##### **2.3 Maxima and Minima**:
- **Concept**: Derivatives are used to find the maximum or minimum values of functions.
- **Example**: Find the maximum or minimum value of \( f(x) = -2x^2 + 4x + 6 \).
- Solution:
\[
f'(x) = -4x + 4
\]
Set \( f'(x) = 0 \) to find critical points, then use the second derivative test to determine whether it is a maximum or minimum.
#### **3. Optimization Problems (15 minutes):**
- **Concept**: Optimization involves finding the maximum or minimum values of a function under given conditions.
- **Example Problem**: A farmer wants to fence a rectangular field with a fixed perimeter of 100 meters. What dimensions will maximize the area of the field?
- Solution:
- Let the length of the rectangle be \( l \) and the width be \( w \). The perimeter is given by:
\[
2l + 2w = 100 \quad \Rightarrow \quad l + w = 50
\]
- Express the area \( A = l \times w \) as a function of \( l \):
\[
A = l(50 - l) = 50l - l^2
\]
- Find \( l \) that maximizes the area by differentiating \( A(l) \) and solving \( A'(l) = 0 \).
\[
A'(l) = 50 - 2l = 0 \quad \Rightarrow \quad l = 25
\]
- Thus, the dimensions that maximize the area are 25 meters by 25 meters.
#### **4. Curve Sketching (10 minutes):**
- **Concept**: Derivatives can be used to analyze the behavior of curves (increasing, decreasing, concavity) and sketch their graphs.
- **Example**: Sketch the graph of \( f(x) = x^3 - 3x^2 \) using its first and second derivatives.
- First derivative for increasing/decreasing behavior.
- Second derivative for concavity and inflection points.
#### **5. Conclusion and Homework (5 minutes):**
- **Conclusion**: Recap the main applications of derivatives: rates of change, maxima/minima, optimization, and curve sketching.
- **Homework**: Solve problems related to maxima and minima from the textbook (e.g., find the maximum profit, minimum cost, etc.).
---
### **Materials Needed:**
- Whiteboard and markers
- Graphing calculator (optional)
- Textbook exercises on derivatives
---
This lesson introduces students to practical applications of derivatives, helping them connect abstract mathematical concepts with real-world scenarios.
