t 0 3 6 9 12
v(t) 32 35 37 38 41
Using the chart above, find an upper estimate for the distance traveled using n=2 and n=4
To find an upper estimate for the distance traveled using n=2 and n=4, we'll first calculate the average rate of change between each pair of consecutive data points.
For n=2:
Average rate of change = (v(3) - v(0)) / (3 - 0) = (35 - 32) / (3 - 0) = 3 / 3 = 1
We'll multiply this rate by the interval length: 1 * (3 - 0) = 3.
So, for n=2, the upper estimate for the distance traveled is 3.
For n=4:
Average rate of change = (v(6) - v(0)) / (6 - 0) = (37 - 32) / (6 - 0) = 5 / 6
We'll multiply this rate by the interval length: (5 / 6) * (6 - 0) = 5.
So, for n=4, the upper estimate for the distance traveled is 5.
To find an upper estimate for the distance traveled using the given chart, we can use the concept of finite differences. Finite differences allow us to estimate the rate of change between consecutive data points.
The given chart shows the time "t" in the first column and the corresponding velocity "v(t)" in the second column. To estimate the distance traveled, we need to find the average velocity between pairs of consecutive time points.
For n=2:
To estimate the upper distance traveled between time intervals t=0 and t=3, we calculate the average velocity over this time period. The average velocity can be found by taking the change in velocity (finite difference) and dividing it by the change in time. Here's how to calculate it:
Average velocity between t=0 and t=3:
= (v(t=3) - v(t=0)) / (t=3 - t=0)
= (35 - 32) / (3 - 0)
= 3 / 3
= 1 unit/time
Since the average velocity is 1 unit/time, we can multiply this by the time interval to estimate the distance traveled. In this case, the time interval is 3:
Upper estimate for distance traveled with n=2:
= average velocity * time interval
= 1 * 3
= 3 units
Therefore, the upper estimate for the distance traveled with n=2 is 3 units.
For n=4:
Similarly, to estimate the upper distance traveled between time intervals t=0 and t=6, we calculate the average velocity over this time period:
Average velocity between t=0 and t=6:
= (v(t=6) - v(t=0)) / (t=6 - t=0)
= (37 - 32) / (6 - 0)
= 5 / 6
= 0.83... unit/time
We multiply the average velocity by the time interval (6) to estimate the distance traveled:
Upper estimate for distance traveled with n=4:
= average velocity * time interval
= 0.83... * 6
= 5 units (rounded to the nearest unit)
So, the upper estimate for the distance traveled with n=4 is 5 units.