Your velocity is given by v(t)=1t2+4 in m/sec, with t in seconds. Estimate the distance, s, traveled between t=0 and t=8. Use the average of the left and right sums with 4 subdivisions.
Answer: s=
so, each interval is of width 2, making the approximation
2((f(0)+f(2))/2 + ... + (f(6)+f(8))/2)
= 2(f(0)/2 + f(2)+f(4)+f(6) + f(8)/2)
so plug and chug
To estimate the distance traveled between t=0 and t=8 using the average of left and right sums with 4 subdivisions, follow these steps:
1. Calculate the width of each subdivision:
- Divide the total time interval (8 seconds) by the number of subdivisions (4).
- In this case, the width of each subdivision is 8/4 = 2 seconds.
2. Calculate the velocity at the left point of each subdivision:
- Substitute the left point of each subdivision into the velocity function v(t) = 1t^2 + 4.
- For the first subdivision (t=0), the velocity is v(0) = 1(0)^2 + 4 = 4 m/sec.
- For the second subdivision (t=2), the velocity is v(2) = 1(2)^2 + 4 = 9 m/sec.
- For the third subdivision (t=4), the velocity is v(4) = 1(4)^2 + 4 = 20 m/sec.
- For the fourth subdivision (t=6), the velocity is v(6) = 1(6)^2 + 4 = 40 m/sec.
3. Calculate the velocity at the right point of each subdivision:
- Substitute the right point of each subdivision into the velocity function v(t) = 1t^2 + 4.
- For the first subdivision (t=2), the velocity is v(2) = 1(2)^2 + 4 = 9 m/sec.
- For the second subdivision (t=4), the velocity is v(4) = 1(4)^2 + 4 = 20 m/sec.
- For the third subdivision (t=6), the velocity is v(6) = 1(6)^2 + 4 = 40 m/sec.
- For the fourth subdivision (t=8), the velocity is v(8) = 1(8)^2 + 4 = 132 m/sec.
4. Calculate the average velocity for each subdivision:
- Add the velocity at the left point and the velocity at the right point, then divide by 2 to get the average velocity.
- For the first subdivision, the average velocity is (4 + 9) / 2 = 6.5 m/sec.
- For the second subdivision, the average velocity is (9 + 20) / 2 = 14.5 m/sec.
- For the third subdivision, the average velocity is (20 + 40) / 2 = 30 m/sec.
- For the fourth subdivision, the average velocity is (40 + 132) / 2 = 86 m/sec.
5. Calculate the distance traveled in each subdivision:
- Multiply each average velocity by the width of the subdivision to get the distance traveled.
- For the first subdivision, the distance traveled is 6.5 m/sec * 2 sec = 13 meters.
- For the second subdivision, the distance traveled is 14.5 m/sec * 2 sec = 29 meters.
- For the third subdivision, the distance traveled is 30 m/sec * 2 sec = 60 meters.
- For the fourth subdivision, the distance traveled is 86 m/sec * 2 sec = 172 meters.
6. Sum up the distances traveled in each subdivision to get the estimated total distance:
- Add up the distances traveled in each subdivision: 13 + 29 + 60 + 172 = 274 meters.
Therefore, the estimated distance traveled between t=0 and t=8, using the average of left and right sums with 4 subdivisions, is s = 274 meters.
To estimate the distance traveled using the average of the left and right sums, we need to calculate the width of each subdivision and then calculate the sum of the velocities at each subdivision point.
First, let's calculate the width of each subdivision:
Width of each subdivision = (8 - 0) / 4 = 2
Now, let's calculate the average velocity at each subdivision point:
For the left sum, we consider the velocity at the left endpoint of each subdivision.
For the right sum, we consider the velocity at the right endpoint of each subdivision.
Subdivision 1: t = 0
Velocity at left endpoint (v(0)) = 1(0)^2 + 4 = 4 m/sec
Velocity at right endpoint (v(2)) = 1(2)^2 + 4 = 9 m/sec
Subdivision 2: t = 2
Velocity at left endpoint (v(2)) = 1(2)^2 + 4 = 9 m/sec
Velocity at right endpoint (v(4)) = 1(4)^2 + 4 = 20 m/sec
Subdivision 3: t = 4
Velocity at left endpoint (v(4)) = 1(4)^2 + 4 = 20 m/sec
Velocity at right endpoint (v(6)) = 1(6)^2 + 4 = 37 m/sec
Subdivision 4: t = 6
Velocity at left endpoint (v(6)) = 1(6)^2 + 4 = 37 m/sec
Velocity at right endpoint (v(8)) = 1(8)^2 + 4 = 68 m/sec
Now, let's calculate the distance covered in each subdivision using the average of left and right velocities:
Subdivision 1: (4 + 9) / 2 * (2 - 0) = 13 * 2 = 26 m
Subdivision 2: (9 + 20) / 2 * (4 - 2) = 29 * 2 = 58 m
Subdivision 3: (20 + 37) / 2 * (6 - 4) = 57 * 2 = 114 m
Subdivision 4: (37 + 68) / 2 * (8 - 6) = 105 * 2 = 210 m
Finally, let's calculate the total distance traveled by summing the distances covered in each subdivision:
Total distance, s = 26 + 58 + 114 + 210 = 408 m
So, using the average of the left and right sums with 4 subdivisions, the estimated distance traveled between t = 0 and t = 8 is 408 meters.