9. An object moving along a horizontal axis has its...

9. An object moving along a horizontal axis has its instantaneous velocity at time t in seconds given by the function v pictured in Figure 4.1.11, where v is measured in feet/sec. Assume that the curves that make up the parts of the graph of y = v(t) are either portions of straight lines or portions of circles. Figure 4.1.11. The graph of y = v(t), the velocity function of a moving object. a. Determine the exact total distance the object traveled on 0 ? t ? 2. b. What is the value and meaning of s(5) - s(2), where y = s(t) is the position function of the moving object? c. On which time interval did the object travel the greatest distance: [0, 2], [2, 4], or [5, 7]? d. On which time interval(s) is the position function s increasing? At which point(s) does s achieve a relative maximum?

Transcript

00:01 Kind of bear with me because there's a lot of explanation here.

00:04 Now in part a, you just want the area under the curve from 0 to 2.

00:18 Just trying to explain my work here.

00:20 And so as i'm looking at this piece, you have a, let me double check, it's a 1 by 1 triangle, so this area is 1 half.

00:30 But then you have a quarter of a circle and with a radius of 1.

00:34 So that would be pi times 1 squared, which is 1, and then you have to divide by 4.

00:40 So the first answer from 0 to 2 would be, well, i would just write it as 1 half plus pi over 4.

00:50 I don't know if you want to show a little bit more work, like how'd you get this answer is 1 half times base times height.

00:56 And how'd you get this answer is pi r squared over 4.

01:01 Then you add them together.

01:03 So there's a little bit of work for you.

01:04 So in letter b, when they ask you to do s of 5 minus s of 2, basically what they're saying, i don't know if you've learned this notation, it's the integral from 2 to 5 of v of t dt.

01:22 It's the fundamental theorem of calculus, which you may not have learned yet.

01:26 But either way, if you do the area under the curve, again, it's negative because we're below the x axis.

01:32 I guess i didn't say that yet, but up here was positive.

01:36 And you can, i think you can tell that you have another 1 by 1 triangle, which would be 1 half.

01:45 And then you have a 1 by 1 rectangle.

01:48 And then you have 1 more.

01:50 And as you add that together, 1 half plus 1 half is 1 plus 1 is 2.

01:55 So that answer would be negative 2.

01:56 Now what's the meaning is that was the net change.

02:02 Let me write that down.

02:04 The net change, or some people would call that displacement, net change in position, that's displacement for this object.

02:29 And because it's negative, it just means that you move left 2.

02:33 It's left 2.

02:35 Something like that should be sufficient.

02:38 So let's look at c, where you want the greatest distance traveled.

02:45 So we already did 0 to 2.

02:47 So let me just jot that down real quick.

02:49 0 to 2, we had the answer of 1 half plus pi over 4.

02:57 And then the next one, going from 2 to 4, we almost did.

03:02 We just went from 2 to 5...

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