on -The velocity in ft/s of an object moving along a line is given by v = = f(t) or the interval 0≤ t ≤ 6 (see figure), where t is measured in seconds. H a. Divide the Interval [0, 6] into n 3 subintervals, [0, 2], [2, 4] and [4, 6]. On each subinterval, assume the object moves at a constant velocity equal to the value of v evaluated at the right endpoint of the subinterval, and use these approximations to estimate the displacement of the object on [0, 6] (see part (a) of the figure). b. Repeat part (a) for n = = 6 subintervals (see part (b) of the figure). 80v=f(t) 60 MM 40 20+ 80 v= f(t)- 60 40 20 0 2 4 (a) 6 1 0 2 4 (b) 6 1
on -The velocity in ft/s of an object moving along a line is given by v = = f(t) or the interval 0≤ t ≤ 6 (see figure), where t is measured in seconds. H a. Divide the Interval [0, 6] into n 3 subintervals, [0, 2], [2, 4] and [4, 6]. On each subinterval, assume the object moves at a constant velocity equal to the value of v evaluated at the right endpoint of the subinterval, and use these approximations to estimate the displacement of the object on [0, 6] (see part (a) of the figure). b. Repeat part (a) for n = = 6 subintervals (see part (b) of the figure). 80v=f(t) 60 MM 40 20+ 80 v= f(t)- 60 40 20 0 2 4 (a) 6 1 0 2 4 (b) 6 1
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 5 and 9 ) show all work, thank you! question 5 answers :
(a) = 340 ft (b)= 330ft
Question 9 answer :
Left sum 34 and right
sum 24.
![**Velocity and Displacement Estimation Using Subintervals**
**Problem Statement:**
5. The velocity \( v \) in feet per second (ft/s) of an object moving along a line is given by the function \( v = f(t) \) over the interval \( 0 \leq t \leq 6 \), where \( t \) is measured in seconds.
**Tasks:**
a. Divide the interval \([0, 6]\) into \( n = 3 \) subintervals: \([0, 2]\), \([2, 4]\), and \([4, 6]\). For each subinterval, assume the object moves at a constant velocity equal to the value of \( v \) evaluated at the right endpoint of the subinterval. Use these approximations to estimate the displacement of the object over \([0, 6]\). This process is illustrated in part (a) of the figure.
b. Repeat part (a) for \( n = 6 \) subintervals. This process is shown in part (b) of the figure.
**Graphical Explanation:**
- **Graph (a):** Represents the function \( v = f(t) \) over the interval \([0, 6]\), divided into three subintervals. Each subinterval is represented by a rectangle whose height corresponds to the velocity at the right endpoint. The approximate displacement is given by the sum of the areas of these rectangles.
- **Graph (b):** Similar to Graph (a), but the interval \([0, 6]\) is divided into six smaller subintervals. Again, rectangles represent velocity at the right endpoints, providing a finer approximation of displacement. The sum of the areas of these rectangles offers a more precise estimate compared to part (a).
These tasks demonstrate how dividing intervals into subintervals and evaluating the function at specific points can help approximate the displacement of an object using methods such as Riemann sums.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3ea0963c-64ff-46a6-b97c-79944ef211ae%2Fb8e2fcfa-0dc5-4e7e-af12-a34ed5d76f82%2Fyuzm3zd_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Velocity and Displacement Estimation Using Subintervals**
**Problem Statement:**
5. The velocity \( v \) in feet per second (ft/s) of an object moving along a line is given by the function \( v = f(t) \) over the interval \( 0 \leq t \leq 6 \), where \( t \) is measured in seconds.
**Tasks:**
a. Divide the interval \([0, 6]\) into \( n = 3 \) subintervals: \([0, 2]\), \([2, 4]\), and \([4, 6]\). For each subinterval, assume the object moves at a constant velocity equal to the value of \( v \) evaluated at the right endpoint of the subinterval. Use these approximations to estimate the displacement of the object over \([0, 6]\). This process is illustrated in part (a) of the figure.
b. Repeat part (a) for \( n = 6 \) subintervals. This process is shown in part (b) of the figure.
**Graphical Explanation:**
- **Graph (a):** Represents the function \( v = f(t) \) over the interval \([0, 6]\), divided into three subintervals. Each subinterval is represented by a rectangle whose height corresponds to the velocity at the right endpoint. The approximate displacement is given by the sum of the areas of these rectangles.
- **Graph (b):** Similar to Graph (a), but the interval \([0, 6]\) is divided into six smaller subintervals. Again, rectangles represent velocity at the right endpoints, providing a finer approximation of displacement. The sum of the areas of these rectangles offers a more precise estimate compared to part (a).
These tasks demonstrate how dividing intervals into subintervals and evaluating the function at specific points can help approximate the displacement of an object using methods such as Riemann sums.
![**9. Approximating Area from a Graph**
*Objective:* Approximate the area of the region bounded by the graph (see figure) and the x-axis by dividing the interval \([1, 7]\) into \(n = 6\) subintervals. Use a left and right Riemann sum to obtain two different approximations.
*Graph Explanation:*
- The graph displays a curve within the coordinate plane from \(x = 1\) to \(x = 7\).
- The area beneath the curve and above the x-axis is shaded, indicating the region to be approximated.
- The curve begins at approximately \(y = 10\) when \(x = 1\) and decreases as \(x\) increases to 7.
*Approach:*
1. **Left Riemann Sum:**
- Use the left endpoints of each subinterval to calculate the height of the rectangles for approximation.
2. **Right Riemann Sum:**
- Use the right endpoints of each subinterval to determine the rectangle heights for another approximation.
This method helps in visualizing and calculating the approximate area under the curve using two different perspectives, aiding in understanding the concept of integration.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3ea0963c-64ff-46a6-b97c-79944ef211ae%2Fb8e2fcfa-0dc5-4e7e-af12-a34ed5d76f82%2F6lywa6w_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**9. Approximating Area from a Graph**
*Objective:* Approximate the area of the region bounded by the graph (see figure) and the x-axis by dividing the interval \([1, 7]\) into \(n = 6\) subintervals. Use a left and right Riemann sum to obtain two different approximations.
*Graph Explanation:*
- The graph displays a curve within the coordinate plane from \(x = 1\) to \(x = 7\).
- The area beneath the curve and above the x-axis is shaded, indicating the region to be approximated.
- The curve begins at approximately \(y = 10\) when \(x = 1\) and decreases as \(x\) increases to 7.
*Approach:*
1. **Left Riemann Sum:**
- Use the left endpoints of each subinterval to calculate the height of the rectangles for approximation.
2. **Right Riemann Sum:**
- Use the right endpoints of each subinterval to determine the rectangle heights for another approximation.
This method helps in visualizing and calculating the approximate area under the curve using two different perspectives, aiding in understanding the concept of integration.
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