Short Answer. Show all work that leads to your answer....

Short Answer. Show all work that leads to your answer. Partial credit may be awarded depending on the accuracy of your work, 3. A particle moves along the x-axis so that at time \(t\) its position is given by \(x(t) = t^3 - 4t + 6\) for \(t\) and where \(t\) is measured in seconds and \(x\) is measured in feet. a. Find the velocity and acceleration functions b. At \(t = 3\), is the particle speeding up or slowing down? c. What is the distance traveled over the first 8 seconds?

Transcript

00:01 Okay, so here we have s equals f of t, which is equal to, okay, 0 .01 is 1 over 100.

00:14 So we are going to have t to the 4th power over 100 minus, okay, this guy here is going to be 4 over 100.

00:25 So let me just write 1 over 25 multiplied by t cubed so t cubed of let me write this okay so t cubed over 25 perfect now for the first part of our exercise well the velocity is just the derivative of f which is t cubed over 25 minus 3 over 25 t squared t squared okay, part b of our exercise, well this one is just, okay, v of 3, which is f prime of 3, which is 27 over 25 minus 27 over 25, nice, which is equal to 0.

01:18 Okay, perfect.

01:20 Now, when is the particle moving in the positive direction? well, all.

01:26 Okay, here we need to study, this is part c, we need to study the sign of f prime.

01:33 So here, in order to do this, we are going to rewrite f prime of t as, okay, t squared over 25 multiplied by t minus three.

01:47 Okay, so thanks to this thing, it will be easier to study the sign of f prime.

01:56 So here we're gonna have the sign of t squared over 25.

02:02 Well, this is pretty easy.

02:05 This is, okay, let me put three here, three here.

02:11 And well, we know that t squared over 25 is always positive.

02:16 It's always positive, but at t equals zero, where this guy is zero, but it doesn't really matter.

02:24 Okay, now, here.

02:26 We are going to have the sign of t minus 3 and well this guy is positive here negative here here we're going to have the sign of f prime and well what can we see well the sign here is the same as the sign of t minus 3 so plus here and minus here okay so at this point well we have our particle moving in the positive direction so positive direction for t greater than three negative direction negative direction for t less than three okay so part c was pretty easy okay now we need oh this one oh this one was actually part d so let me just change this so this one was d for this one was d for c is even easier the particle is at rest at t equals 3 okay so the particle is at rest at t equals 3 because the velocity is 0 okay perfect now let's find the total distance traveled during the first 8 seconds okay so this is part e well this distance is gonna be what okay so we are gonna have negative 1 multiplied by an integral from 0 to 3 of v of t okay vot in the t plus an integral from 3 to 8 of v of t in the t okay here we have negative 1 because the particle here is moving in the negative direction well we already know an antiderivative of v of v that is an antiderivative of v is just f so here what are we going to have well here we are going to have negative f of three and here we are going to have plus f of eight minus f of three so f of eight okay so f of eight of eight minus two f of three so we just need to plug eight and three into so f but this is immediate okay perfect now part g okay so okay we already okay we already do a diagram of the motion of the particle well is this one so okay this one was part f let me just write done in part d so done in d okay perfect so now we need to solve g we need to solve part g of this of the exercise well okay this is easy this one is just v prime of three that is the acceleration is the derivative of v computed at three okay the derivative here is easy to find okay we know that v is this guy here so we are going to have okay so we are going to have 3 t squared over 25 so we are going to have 27 over 25 again so 27 over 25 and the derivative of this guy here is 6 minus 25 so we are going to have next negative 18 over 25.

06:32 So negative 18 over 25, which is, okay, this one is 9 over 25.

06:40 This is the acceleration.

06:42 Okay, now let's graph, okay, so let's graph our position, velocity and acceleration functions between 0 and 8.

06:57 Okay, so this is t.

07:00 This is okay our y -axis so i'm gonna use red for the position green for the velocity and this color for the acceleration okay so let's get started with okay with the first with the first one okay so here we are gonna have our position function well for z for t equals zero we are gonna have zero okay and for eight we are gonna have what okay let's see for eight we are gonna have a positive value so more or less our function okay let's see let's just take a look at when where our function f is increasing okay so our function f is increasing between zero and three oh let's see is actually decreasing between zero and three and then increasing so okay our position function is going to look like this one here okay perfect so three is here this is here this is our position function then we have our velocity function okay our velocity function which is a cubic function okay so this function is gonna be negative between zero and three okay zero at three okay let's see and then it's gonna be positive okay so we are using green for this one so more or less is gonna be like this and finally our acceleration function well our acceleration function is okay, so, oh, v was a cubic function.

09:08 Nothing changes, but you're not just to be precise...

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