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The velocity graph of an accelerating car is shown.

(a) Use the Midpoint Rule to estimate the average velocity of the car during the first 12 seconds.

(b) At what time was the instantaneous velocity equal to the average velocity?

(A). Average velocity $=45.33 \mathrm{km} / \mathrm{h}$

(B). Click to see

Applications of Integration

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Oregon State University

Harvey Mudd College

University of Nottingham

Idaho State University

in this problem, it is given that we lost photograph of an accelerating car is shown. E use the midpoint rule to estimate the average we lost two of the car during the 1st 12 seconds. And secondly, it is that asked that at what time was the instantaneous we lost is equal to the average velocity. So let me make point rule is used for N equals 23 here. Right? Because we have three sectors for 30 seconds, searches 48. And uh so we have three parts. Right? All right, so now average velocity will be what others have lost is given by burn upon the that there is nothing. But I'm not telling gentle, I'm telling me to my wrist fever. Right? So because DLT I'm using format pointing here. Right, So now here one day uh minus zero integration of three equals to 0 to 12 V D D. T. And we will find using uh mid control that is the minus A by end. That is given by four certainly comes out to be four. So there will be four V values interval. Food. Right? For the time. So 0-4 than we have second interval for to it. Then we have it was moved right. So now here to even Star Stephen studies what Cuban studies? Midpoint of 0 to 4 interval. Right? So here it is too, I'm writing for 4 to 8, it is mid pointers six and what P 3- That is 8 to 12 we have gen in between. Right, So we will use here again, this midpoint rule is completed when we will right here, the situation that is I equals to integration of 0-12 VDDT equals to tell TFT even start plus FP two stuff yes ft 33 strap right? So now here at F two, that is F D one style that is, we can right here at two, Right at at T equals to the Stephen Starr equals to two. We we can see here in the graph that is 15 km/h and F six at 6, 13 seconds, you can see the speed is 37 kilometers product and F 10 is 49 km, whereas in the subject in the graph, you can say, So now here, I'm talking about don't see this red part and uh made you understand that what I'm writing here, I'll tell you. So Here I equals two integration of 0 to 12 V D D T equals to four and 2 15 plus 37 plus 49. And this, this will give us answer. S +404. So now why I'm writing this, Okay, so here it is seconds, right, 40 seconds and these are the values with, with kilometer better. So you need to take care of units, but why I'm not taking this care, let me, Let me tell you, let us say this is let me let me right here, 404 is kilometers what? Uh and this is let me right here like this kilometre for all, and this is in seconds. Right? So, so first of all we needed to find the average velocity right? So everyone has to be need to find So here they unit is this right? So let me not convert it into this, We can convert it into this, but I I'm not converting because after that you need to multiply again with the 3600, right, so that's why I'm not doing so, so let me see how it works. Let us see. So here I have this right, 404 kilometer parter in two seconds is given to us and again, if we talk about this 12, right, so well, yeah, the trail is the seconds. Alright, so 2nd, 2nd car talk, so there's me no need. Right? So that's why I'm not converting it into this. You can also convert it into this, but you need to multiply it again with by 3600, so I'm taking average velocity good, you can see Yeah, definitely. Yeah, yeah, so we can take this is kilometer, so this is the average velocity. So this we needed to find out. Yes, so this is a velocity ridge velocity. So this is for the part, this is the answer, right? And now, for the second part time, at which beat equals to have is velocity is nothing, but this is 5.3 seconds we can see in the graph that is 5.3 seconds. Where is 500 seconds? Somewhere here? 5.3 seconds. And the average velocity 34 to somewhere, right? I'm not able to read like it should be like this. This should be edited in this way, Right? So this one show you if I .3 seconds. Great. No. Uh, this is how we solve this problem. I hope you understood the concept. Thanks for watching.