Approximating displacement The velocity of an object is given ...

Question

Approximating displacement The velocity of an object is given by the following functions on a specified interval. Approximate the displacement of the object on this interval by subdividing the interval into n subintervals. Use the left endpoint of each subinterval to compute the height of the rectangles.
v = 2t + 1(m/s), for 0 ≤ t ≤ 8 ; n = 2

v = 2t + 1(m/s), for 0 ≤ t ≤ 8 ; n = 2

Accepted Answer

Hello. In this video, we are told that a particle's velocity is given by V is equal to 14 minus 1 for the interval from 2 to 10. We want to approximate the displacement of the particle over this interval by dividing it into two subintervals and using left end points for each subinterval for the rectangular heights. So, let's go ahead and list out our given information. First, we are given a velocity function, which is defined as 4T minus 1. We are going to work on the interval for time from 2 to 10, and we are told to split this interval into two sub-intervals. So, if we were to make a number line for our interval, this is going to be a number line going from 2 to 10. And we want to split this interval in half because we want two sub-intervals. Now, the first thing we want to do is you want to go ahead and define the width of each of our intervals. That width is going to be defined as delta T. Delta T will have the definition of B minus A divided by N, which is going to be the value of our interval divided by the amount of sub-intervals we are working with. So, the width of our intervals is going to be 10 minus 2 divided by 2, which is going to simplify to 8 divided by 2, which is equal to 4. And so what this means is that our intervals are going to have a width of 4. So, our endpoints are going to be 2 + 4, which gives us 6, plus 4, which gives us 10. Now, the important thing to note here is that we are going to work with left end points. And so what this means is that the end points that we are going to work with for the summation of the displacement is going to be 2 and 6. So, let's go ahead and define the summation that will give us the value of the displacement. That summation is going to be the sum starting from 1 and ended at 2 of the velocity of each endpoint multiplied by delta T. This is going to be rewritten as delta T multiplied by V of 2. Plus 5 of 6. And so in order to give V2 and V of 6, we have to plug in our endpoints into the given velocity function. If we plug in 2 into the velocity function, we get 4 multiplied by 2 minus 1, and this is going to be simplified to give us the value of 7. Plugging in 6 into the velocity function gives us VF 6, which is going to be 4 multiplied by 6 minus 1, and that gives us a value of 23. And recall that delta T is equal to 4. So we can rewrite the summation to be 4, multiplied by the quantity, 7 + 23. This is going to further simplify to give us the value of 4 multiplied by 30, which is equal to 120 m. So the total displacement for the velocity function from 2 to 10 is going to be 120 m, and with that being said, the overall solution is going to be a. So I hope this video helps you in understanding how to approach this problem, and we will go ahead and see you all in the next video.

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