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The velocity v(t) in the table below is increasing for 0_t_12.


t 0 3 6 9 12

v(t) 26 27 30 35 40

AFind an upper estimate for the total distance traveled using

n=4 subdivisions: distance traveled =

n=2 subdivisions: distance traveled =


Find a lower estimate for the total distance traveled using n=4.

distance traveled =

Verified Answer :

When n=4 subdivisions distance traveled = 40 X 12 = 480


When n=2 subdivisions distance traveled = 30 X 6 = 180 

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Answer:

To find an upper estimate for the total distance traveled, we estimate the distance of each interval using the largest velocity on the interval. In this case, the velocity is increasing this means that we are gonna use the velocity at the right end of each interval. Also, from the table, we can see that each interval is \(\Delta t = 3\) because \(n=4\)

So, \((distance)=3(27+30+35+40) = 396m\); this is the distance traveled.

If we consider \(n=2\), then each interval is \(\Delta t = 6\):

So, \((distance)=6(30+40) = 420m\); distance traveled.

Similarly,  to find a lower estimate, we can use the same process, but the difference is that we start from right to left, using the velocity at the beginning of each interval:

\((distance)=3(26+27+30+35) = 354m\)

Therefore, the answers are:

Upper estimates:

  • For n = 4, we have 396m traveled.
  • For n = 2, we have 420m traveled.

Lower estimate:

  • For n = 4, we hace 354m traveled.

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