O How do you find the average value of a function g(x) for xe [c, d]? Make a sketch of some function g(x) for x = [c, d] on the axes to the right. What does g(x)dx represent geometrically on your graph? What does d-c represent geometrically on your graph? Draw in the height 9ag on your graph as a horizontal line. What is true about the area under g(x) and the area under 9 for xe [c, d]? Write an equation that reflects this fact.
O How do you find the average value of a function g(x) for xe [c, d]? Make a sketch of some function g(x) for x = [c, d] on the axes to the right. What does g(x)dx represent geometrically on your graph? What does d-c represent geometrically on your graph? Draw in the height 9ag on your graph as a horizontal line. What is true about the area under g(x) and the area under 9 for xe [c, d]? Write an equation that reflects this fact.
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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Question
How do you find the average value of g(x)?
![**Finding the Average Value of a Function**
**Question: How do you find the average value of a function \( g(x) \) for \( x \in [c, d] \)?**
1. **Graph Sketching:**
- Make a sketch of some function \( g(x) \) for \( x \in [c, d] \) on the axes provided.
2. **Geometric Interpretation:**
- What does \( \int_{c}^{d} g(x) \, dx \) represent geometrically on your graph?
3. **Interval Representation:**
- What does \( d - c \) represent geometrically on your graph?
4. **Average Height:**
- Draw in the height \( g_{\text{avg}} \) on your graph as a horizontal line.
- Consider what is true about the area under \( g(x) \) and the area under \( g_{\text{avg}} \) for \( x \in [c, d] \).
5. **Equation Formation:**
- Write an equation that reflects this fact.
**Axes Details:**
- A set of axes is provided for sketching. The x-axis represents the interval \( [c, d] \), and the y-axis represents the function values.
The goal is to understand the relationship between the function, its integral over the interval, and its average value, visually and analytically.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc05bac74-9392-4bb0-9ff6-f7e027936b2b%2F6a6e3b37-a04f-4ffe-83db-b4af19aa469b%2Fcovstd4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Finding the Average Value of a Function**
**Question: How do you find the average value of a function \( g(x) \) for \( x \in [c, d] \)?**
1. **Graph Sketching:**
- Make a sketch of some function \( g(x) \) for \( x \in [c, d] \) on the axes provided.
2. **Geometric Interpretation:**
- What does \( \int_{c}^{d} g(x) \, dx \) represent geometrically on your graph?
3. **Interval Representation:**
- What does \( d - c \) represent geometrically on your graph?
4. **Average Height:**
- Draw in the height \( g_{\text{avg}} \) on your graph as a horizontal line.
- Consider what is true about the area under \( g(x) \) and the area under \( g_{\text{avg}} \) for \( x \in [c, d] \).
5. **Equation Formation:**
- Write an equation that reflects this fact.
**Axes Details:**
- A set of axes is provided for sketching. The x-axis represents the interval \( [c, d] \), and the y-axis represents the function values.
The goal is to understand the relationship between the function, its integral over the interval, and its average value, visually and analytically.
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