• Draw a left endpoint rectangle approximation of the area with n = 8. • How do Ax and the change? • If you start with only one of your eight rectangles, on the left side of your interval, and add them in one at a time, until all of the approximating rectangles are present, how much area do you add at each step? (Don't actually calculate this estimate for eight rectangles, just describe it in terms of the information you have, and explain how you would calculate it.) How is this related to Part 1 of the Fundamental Theorem of Calculus? •
• Draw a left endpoint rectangle approximation of the area with n = 8. • How do Ax and the change? • If you start with only one of your eight rectangles, on the left side of your interval, and add them in one at a time, until all of the approximating rectangles are present, how much area do you add at each step? (Don't actually calculate this estimate for eight rectangles, just describe it in terms of the information you have, and explain how you would calculate it.) How is this related to Part 1 of the Fundamental Theorem of Calculus? •
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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I need help with the bulletpoint starting with 'Draw a left endpoint rectangle ...' and its three subpoints underneath. Thank you!
![### Using the Function \( f(x) = x^2 - 2 \) for Calculus Applications
#### 1. Tangent and Secant Lines
**For \( a = 2 \) and the interval \([2, 3]\)**:
- **Find each of the following without actually calculating the equation of the tangent line**:
- **\( Δx \)**
- **\( dx \)**
- **\( Δy \)**
- **\( dy \)**
Illustrate these on your graph, ensuring each quantity is shown clearly.
#### 2. Equation of the Tangent Line
- **Find the equation of the tangent line at \( a = 2 \)** and the equation of the secant line for the interval \([2, 3]\).**
- Add these two lines to your graph.
- Illustrate the quantities \( Δx \), \( dx \), \( Δy \), and \( dy \) again.
- Explain the relationships between these quantities and the tangent and secant lines.
#### 3. Newton’s Method
- **For the same function, without calculating anything**:
- Sketch an illustration of the Newton’s Method process for finding the \( x_1 \) and \( x_2 \) approximations of the zero of the function, using the seed value \( x_0 = 2 \).
- Draw in the lines that would produce the next two approximations (approximate zero values).
- Explain how each line produces the next approximation.
- **Use Newton’s Method formula to find \( x_1 \) and \( x_2 \).**
- Compare your calculated \( x_2 \) value with the value obtained through illustration.
- **Find the zero using technology**. Discuss the accuracy and usefulness of this approximation.
#### 4. Graph Sketch and Integration
- **Sketch the graph of \( f(x) = x^2 - 2 \)**, with the area shaded under the curve and above the \( x \)-axis over the interval \([2, 3]\).
- Write a definite integral representing this shaded region’s area.
#### 5. Rectangular Approximations and Numerical Integration
- **Draw a right endpoint rectangle approximation** of the area with \( n = 4 \) divisions.
- Calculate the following quantities for your illustration:
- **\(](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fad0c56f5-1126-465b-9e5e-d8b5ed557527%2F8d12b417-702b-4f75-b36b-e4d007b3f9f4%2F688tsgx5j_processed.png&w=3840&q=75)
Transcribed Image Text:### Using the Function \( f(x) = x^2 - 2 \) for Calculus Applications
#### 1. Tangent and Secant Lines
**For \( a = 2 \) and the interval \([2, 3]\)**:
- **Find each of the following without actually calculating the equation of the tangent line**:
- **\( Δx \)**
- **\( dx \)**
- **\( Δy \)**
- **\( dy \)**
Illustrate these on your graph, ensuring each quantity is shown clearly.
#### 2. Equation of the Tangent Line
- **Find the equation of the tangent line at \( a = 2 \)** and the equation of the secant line for the interval \([2, 3]\).**
- Add these two lines to your graph.
- Illustrate the quantities \( Δx \), \( dx \), \( Δy \), and \( dy \) again.
- Explain the relationships between these quantities and the tangent and secant lines.
#### 3. Newton’s Method
- **For the same function, without calculating anything**:
- Sketch an illustration of the Newton’s Method process for finding the \( x_1 \) and \( x_2 \) approximations of the zero of the function, using the seed value \( x_0 = 2 \).
- Draw in the lines that would produce the next two approximations (approximate zero values).
- Explain how each line produces the next approximation.
- **Use Newton’s Method formula to find \( x_1 \) and \( x_2 \).**
- Compare your calculated \( x_2 \) value with the value obtained through illustration.
- **Find the zero using technology**. Discuss the accuracy and usefulness of this approximation.
#### 4. Graph Sketch and Integration
- **Sketch the graph of \( f(x) = x^2 - 2 \)**, with the area shaded under the curve and above the \( x \)-axis over the interval \([2, 3]\).
- Write a definite integral representing this shaded region’s area.
#### 5. Rectangular Approximations and Numerical Integration
- **Draw a right endpoint rectangle approximation** of the area with \( n = 4 \) divisions.
- Calculate the following quantities for your illustration:
- **\(
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