Graph the function f(x) over the given interval. Partition the interval into 4 subintervals of equal length. Then add to your sketch the rectangles associated with the Riemann sum £ fck) Axk , k-1 using the indicated point in the kth subinterval for c. f(x) = 2x +2, [0, 2], left-hand endpoint 8. 7+ 4+ 3+ 0.5 1 1.5 7+ 6- 5- 2 0.5 1 15 7+ 6+ 5+ 4- 3- 1+ 0.5 1 1.5 7- 6+ 5- 1+ 0.5 7+ 6- 1.5
Graph the function f(x) over the given interval. Partition the interval into 4 subintervals of equal length. Then add to your sketch the rectangles associated with the Riemann sum £ fck) Axk , k-1 using the indicated point in the kth subinterval for c. f(x) = 2x +2, [0, 2], left-hand endpoint 8. 7+ 4+ 3+ 0.5 1 1.5 7+ 6- 5- 2 0.5 1 15 7+ 6+ 5+ 4- 3- 1+ 0.5 1 1.5 7- 6+ 5- 1+ 0.5 7+ 6- 1.5
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Title: Understanding Riemann Sums through Graphical Representation**
**Content:**
This educational module focuses on understanding the concept of Riemann sums using graphical methods. The function f(x) defined as f(x) = 2x + 2 is considered over the interval [0, 2]. The task is to partition this interval into 4 subintervals of equal length and visualize the Riemann sum using the left-hand endpoints for the calculation of each subinterval.
**Graph Descriptions:**
1. **Initial Graph:**
- The first graph displays the linear function f(x) = 2x + 2 over the interval [0, 2]. The x-axis ranges from 0 to 2, and the y-axis ranges from 0 to 8.
2. **Riemann Sum Visualization with Rectangles:**
- The subsequent graphs demonstrate the partitioning of the interval into four subintervals: [0, 0.5], [0.5, 1], [1, 1.5], and [1.5, 2].
- For each subinterval, rectangles are drawn using the function values at the left-hand endpoints: f(0), f(0.5), f(1), and f(1.5). These rectangles approximate the area under the function curve, representing the Riemann sum. Each rectangle’s height corresponds to the function value at the left endpoint of the subinterval, and its width is 0.5.
**Detailed Explanation:**
1. **Subinterval [0, 0.5]:**
- The left-hand endpoint is 0. The function value f(0) = 2. A rectangle is drawn with a width of 0.5 and height of 2.
2. **Subinterval [0.5, 1]:**
- The left-hand endpoint is 0.5. The function value f(0.5) = 3. A rectangle is drawn with a width of 0.5 and height of 3.
3. **Subinterval [1, 1.5]:**
- The left-hand endpoint is 1. The function value f(1) = 4. A rectangle is drawn with a width of 0.5 and height of 4.
4. **Subinterval [1.5, 2]:**
- The left-hand endpoint](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F34715e37-b64b-42a1-b0ce-fdbcb06d67b0%2Fa5551baf-11fd-4c09-a171-296f66aa5c11%2Folc91dk_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Understanding Riemann Sums through Graphical Representation**
**Content:**
This educational module focuses on understanding the concept of Riemann sums using graphical methods. The function f(x) defined as f(x) = 2x + 2 is considered over the interval [0, 2]. The task is to partition this interval into 4 subintervals of equal length and visualize the Riemann sum using the left-hand endpoints for the calculation of each subinterval.
**Graph Descriptions:**
1. **Initial Graph:**
- The first graph displays the linear function f(x) = 2x + 2 over the interval [0, 2]. The x-axis ranges from 0 to 2, and the y-axis ranges from 0 to 8.
2. **Riemann Sum Visualization with Rectangles:**
- The subsequent graphs demonstrate the partitioning of the interval into four subintervals: [0, 0.5], [0.5, 1], [1, 1.5], and [1.5, 2].
- For each subinterval, rectangles are drawn using the function values at the left-hand endpoints: f(0), f(0.5), f(1), and f(1.5). These rectangles approximate the area under the function curve, representing the Riemann sum. Each rectangle’s height corresponds to the function value at the left endpoint of the subinterval, and its width is 0.5.
**Detailed Explanation:**
1. **Subinterval [0, 0.5]:**
- The left-hand endpoint is 0. The function value f(0) = 2. A rectangle is drawn with a width of 0.5 and height of 2.
2. **Subinterval [0.5, 1]:**
- The left-hand endpoint is 0.5. The function value f(0.5) = 3. A rectangle is drawn with a width of 0.5 and height of 3.
3. **Subinterval [1, 1.5]:**
- The left-hand endpoint is 1. The function value f(1) = 4. A rectangle is drawn with a width of 0.5 and height of 4.
4. **Subinterval [1.5, 2]:**
- The left-hand endpoint
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