The function
s
(
t
)
describes the motion of a particle along a line:
s
(
t
)
=
18
t
−
t
2
.

a) find the velocity function
v
(
t
)
of the particle at any time
t
≥
0.

b) identify the time interval in which the particle is moving in a positive direction.

c) identify the time interval in which the particle is moving in a negative direction.

d) identify the time at which the particle changes direction.

a)

b)

c)

d)

Explanation:

We have the following parts:

(a) find the velocity function of the particle at any time t ≥0 )

For this case w ejust need to take the derivate of s(t) and we got:

(b) identify the time interval(s) in which the particle is moving in a positive direction

The partcile is moving in the positive direction when the velocity is higher than 0. We have a quadratic equation for the velocity so we can solve for the t intercepts  like this:

For this case

And if we replace we got:

And now we can see on which intervals we have the velocity positive or negative:

So the is positive between

(c) identify the time interval(s) in which the particle is moving in a negative direction

From the before part we see that the velocity is negative just on this interval:

(d) identify the time(s) at which the particle changes direction. s(t) = t³ - 5t²+4t

The points where the particle changes direction are given by the critical point because are the points where the derivate is 0 and we have the change of direction and on this case are:

weather forecasts are done by collecting data about the current state of the atmosphere mainly the temperature, humidity and wind and using meteorology, to determine the atmosphere that evolves over time. the h and l mean the high and low pressure. the red strip is a warm front. the blue strip is a cold front. the blue and red strip is a stationary front. the purple strip is a occluded front.

explanation: