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7) If the acceleration of an object is given by a(t) = rvir" + 4, find the average rate of change of its velocity between =0 and t = 1....

Question

7) If the acceleration of an object is given by a(t) = rvir" + 4, find the average rate of change of its velocity between =0 and t = 1.

7) If the acceleration of an object is given by a(t) = rvir" + 4, find the average rate of change of its velocity between =0 and t = 1.



Answers

Average velocity The function $s(t)$ represents the position of an object at time $t$ moving along a line. Suppose $s(1)=84$ and $s(4)=144 .$ Find the average velocity of the object over the interval of time [1,4].

Problem. Number six X is equal to one plus three. Teen. Why is equal to two minus 40 as it is equal to seven plus plus team where T is equal to two fabulosity is equal to the X over 18 i plus D y over 80 j plus these it over DT K which is equal to three. I minus four j lost one. Yeah, plus one K. The acceleration is equal to at the revision for the being with respect to time which is equal to see the speed is equal to the norm. Off a huge is equal to three square the past negative pores Where the bus one square to sit With Jews, we're route off 26 at velocity and at times equal to tow. The velocity is the same because is not functioning tea and the acceleration is the same because no function also, Inti

Hello, everyone. A particle moving along the line. It's a displacement according to the function, uh, ex ft. Uh, except he is equal to a T squared minus tootie plus four, where X is measured in meters and tease, measured in seconds. But in the average velocity or the time period, A T is equal to 02 I find it from 0 to 2. So the average velocity, um, is just a fancy way of saying to slope. So if you want to find the average velocity from 0 to 2, we're just going to plug. It was going to find this slope. Essentially. So when we plug in, um, well, this little was just why? To minus lie one over x two minutes, x one. In this case, no, uh, that X two is two. And why x one is zero. And I would have to find the corresponding my values. Several good to plug in to into our equation to get why, too. So we get, uh, t squared. So two squared minus two times two. Bless for which is equal to four. Um, so we have four of the top eyes. Why? To um And then for? Why one? We're going to plug in zero, so you get zero squared plus for so is also equal to four. So get four minus four, divided by two minus zero, which is equal to zero. So our average velocity over this time period is going to be equal to zero meters per second. So thank you for watching, and I hope this helps.

So there's two different things going on here. Whenever we hear average rate of change. This is similar to the average rate of change. This is similar to finding the slope between two points. So this would be similar to just f. of two -F of one. All over 2 -1. Okay, so average rate of change effort to is just two times two plus seven minus two times one plus seven. All over two minus one is one. So this will be for plus 7, 11 minus 9/1 or just to two will be the average rate of change. But when I hear the instantaneous rate of change, this is similar to the derivative at a point or slope at a point. Mhm. So I'm gonna take the first derivative of each term, first derivative of two t constant role. Just bring down the concept is to first derivative at seven through constant zero. So two plus zero is just too, so at each of the endpoints just plug in at one but there's nothing to plug in. So it's just too, so same as the average rate of change and also add to when I plug into, there's nothing to plug in. Also the same. So the average rate of change of the instantaneous rate of change are all the same in this case. So this probably means this is a straight line

Were given a function of time except T equals B times to you the fourth, where B is a constant and we're going to find the instantaneous velocity as well as the average velocity and find a relationship between them. So the instantaneous velocity is just the time derivative of this function, which is the position. So the instantaneous velocity is Deed 80 of B T to the fourth. We can pull be out because it's just a constant and we have the derivative of t of the fourth. Now do you have The fourth is a power function and from calculus. What this means is we'll pull down a factor of four, which multiplies be and we'll subtract one from the exponents, which gives us three. And we're left with just t huge. So we have four being t cubed and the average velocity we're going to write with a bar over the bar. So this is going to equal Delta X over Delta T. And what this means is it's the value of the position at time T minus the value of position at time zero over the time, minus the time a time zero, which is just zero. And so this is equal to B T to the fourth minus this same quantity a t zero, which is just zero over T minus zero, which gives us me do you the fourth over tea which is equal to B t cubed. And we see that the instantaneous velocity and the average velocity are not the same. In fact, we see that the average velocity is equal toe 1/4 the instantaneous velocity. So the average velocity is equal to 1/4 the instantaneous velocity, and we have obtained a relationship between them.


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