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.
{Use of Tech} v = 4 √(t +1) (mi/hr) . for 0 ≤ t ≤ 15 ; n = 5

{Use of Tech} v = 4 √(t +1) (mi/hr) . for 0 ≤ t ≤ 15 ; n = 5

Accepted Answer

Hello. In this video, we are told that a particle's velocity is given by V is equal to 5 multiplied by the square root of T + 2 for the interval from 1 to 16. We want to approximate the displacement of the particle over this interval by dividing it into 5 subintervals and using the left end points for each subinterval for the rectangular heights. So, let's go ahead and list out our given information. First, we are told that we have a velocity function that is defined as 5, multiplied by the square root of T + 2. Now, T is on the interval from 1 to 16, and we want to split this interval into 5 sub-intervals. So, if we were to create a number line for overall interval, From 1 to 16. We want to go ahead and split this into 5 subintervals, but what is going to be the width of each of these intervals? Well, the width is going to be defined as delta T, and delta T is equal to B minus A divided by N. That is the value of our overall interval divided by the amount of sub-intervals we're working with. So, B minus A is going to be defined as 16 minus 1, and this will be divided by 5. This will give us 15 divided by 5, which simplifies to be 3. So, the width of each of our intervals is going to be 3, and we are going to use this to define all of the endpoints on the number line. So, starting at 1, we are going to add 3 until we get to 16. 1 + 3 gives us 44 + 3 gives us 77 + 3 gives us 1010 + 3 gives us 13, and 13 + 3 gives us 16. Now, what's important here is that we want to work with left end points. So, we want to take the 1st 5 left endpoints on the number line, and that is going to be the values of 147, 10, and 13. What we want to do now is we want to find the value of the velocity function at these endpoints. Starting at 1 If we plug in one into the function, we get 5 multiplied by the square root of 1 + 2. And this gives us 5 multiplied by the square root of 3. Plugging in 4 gives us 5 multiplied by the square root of 4 + 2, which gives us 5 multiplied by the square root of 6. Plugging in 7 gives us 5 multiplied by the square root, a 7 + 2, which is going to be 15. Plugging in 10 gives us 5 multiplied. By the square root of 10 + 2, which simplifies to 10 square root of 3. And finally, plugging in 13 gives us 5 multiplied by 13 + 2, which simplifies to 5 multiplied by the square root of 15. Now that we have the velocity values at each of the endpoints, we are going to make the summation that defines the displacement. That is going to be the summation. The summation starting at 1 and ending at 5, a V of X sub I multiplied by delta T. This will simplify to be delta T multiplied by V1 plus V of 4. Plus V of 7 + V of 10. Plus V of 13. And by plugging in all the respective values for delta T in all the end points, that is going to leave us with 3 multiplied by 5 square root of 3, plus 5 square root of 6 + 15, plus 10 square root of 3. Plus 5 square root of 15. And with the help of a calculator, this is going to approximate to be 217.78 ft. This is going to be the total displacement for the velocity function on the interval from 1 to 16. And with that being said, the overall solution to this problem 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.

Key Concepts

Velocity and Displacement

Riemann Sums

Subintervals and Partitioning

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