Calculus

This is the last of the questions I didn't understand on my worksheet, please please help?

the function s(t) describes the motion of a particle moving along a line. For each function, (a) find the velocity function of the particle at any time t _ 0,
(b) identify the time interval(s) when the particle is
moving in a positive direction, (c) identify the time interval(s)
when the particle is moving in a negative direction, and (d) identify the time(s) when the particle changes its direction

s(t)=t^2-7t+10

Question

Calculus

This is the last of the questions I didn't understand on my worksheet, please please help?

the function s(t) describes the motion of a particle moving along a line. For each function, (a) find the velocity function of the particle at any time t >_ 0,
(b) identify the time interval(s) when the particle is
moving in a positive direction, (c) identify the time interval(s)
when the particle is moving in a negative direction, and (d) identify the time(s) when the particle changes its direction

s(t)=t^2-7t+10

mertsj · Answer

The velocity function is the first derivative of the distance.

loser66 · Answer

why are you stuck? and where?

anonymous · Answer

I don't understand

anonymous · Answer

@satellite73

anonymous · Answer

The position function is $s(t)$. The rate of change of position is the derivative of $s(t)$, or $s'(t)$. The "rate of change of position" is the definition of "velocity," so we call $s'(t)=v(t)$.

To find the velocity function, it's only a matter of differentiating $s(t)$ with respect to $t$.
$$s(t)=t^2-7t+10~~\Rightarrow~~s'(t)=v(t)=2t-7$$

anonymous · Answer

thank you:)

anonymous · Answer

and to find b do I have to graph it?

kainui · Answer

Nope, but if you don't immediately know it would be wise to graph it so you start to develop an intuition of what a graph might look like and where it is moving in a positive direction so that later you can do it without graphing it and just sort of vaguely visualize it in your head.

anonymous · Answer

How can I solve it without graphing?

kainui · Answer

Think about it. This is something you need to figure out on your own because it's a simple concept that you are 100% capable of seeing. Ask yourself: when is the particle moving in the positive direction?

kainui · Answer

After you graph it you will be able to figure out on your own clearly how to solve it without graphing it. Think about the slope. Derive your own understanding.

anonymous · Answer

@Loser66  please help me with this last question, please

loser66 · Answer

what I can help? for a) velocity is derivative of displacement, so , just take s'(t)

anonymous · Answer

"Moving in a positive direction" seems to refer to "when $s(t)$ is increasing." Likewise, "moving in a negative direction" sounds like "when $s(t)$ is decreasing." Apply the first derivative test.

anonymous · Answer

Also, the "times when the particle changes direction," assuming what I said earlier is right, means "find the local extrema," i.e. when $s(t)$ stops increasing/decreasing and beings to decrease/increase, respectively.

loser66 · Answer

I think so, too. That's why I ask him "why are you stuck" . He worked on it many times yesterday.

anonymous · Answer

aaah ok.

anonymous · Answer

I wasn't sure

anonymous · Answer

thank you so much