00:01
A function of time that describes the position of particle.
00:05
So f of t is equal t squared minus 7t squared plus 21 t now we know that the velocity of something is the rate of change of its position or so we can represent that by making a function b of t and making that the derivative of f of t.
00:29
So f prime of t, which is the rate of change of f.
00:34
Now let's compute that.
00:35
We can compute that using each term.
00:38
You can use the power rule.
00:41
3g squared minus 14t plus 21.
00:54
And this is your velocity at any time to your velocity function.
01:01
So given this velocity function, we can find the velocity at any time t, given that we know t.
01:08
So fee of one, which is what the second part is asking, is 3 times 1 squared, 3 times 1 squared minus 14 times 1 plus 21.
01:27
And once you plug this into a calculator, 21 minus 14 is 7, 7 plus 3 is 10.
01:34
So the velocity at t equals 1 is 10 meters meters per second.
01:47
And this is your answer for part 2.
01:53
Now, part 3 asks when the particle is at rest.
01:56
So when the particle is at rest, the velocity is zero.
02:01
So all we have to do is set the velocity function equal to zero and find our values for t.
02:06
This corresponds to finding the value of time for when velocity equals 0.
02:15
T plus 21 is equal to 0.
02:28
Now, we have a quadratic, and we can solve this using a variety of ways.
02:36
However, in solving this equation, we get that t is actually imaginary.
02:42
We have a real component, 7 over 3, and an radical 14 over 3.
02:55
So since we have an imaginary solution and no real solutions, we know that the velocity never equals 0.
03:05
So the particle is never at rest.
03:11
So your answer here is going to be never.
03:13
Part d asks when the particle is moving in the positive direction.
03:18
So you can find this by solving for v of t is greater than 0.
03:25
When the particle is moving in a positive direction, that means the velocity is positive.
03:33
So velocity has to be greater than zero since it can't be here.
03:41
So if we take a look at the function right here in its quali.
03:46
It's standard quadratic form, we can see that it never equals zero...