The accompanying figure shows the velocity v = f(t) of a particle moving on a coordinate line. a). when does the particle move forward? move backward? speed up? slow down?

We are given graph of velocity v=f(t).

a) In the interval [0,1) and (5,7) velocity is positive.

So, it would move forward between 0 to 1 second and between 5 to 7 seconds.

In the interval (1,2) and (3,5) velocity is negative.

So, it would move backward between 1 to 2 second and between 3 to 5 seconds.

Slope in [0,2) and (6,7) is negative so, speed down between 0 to 2 and 6 to 7 seconds.

Slope between 3 and 6 seconds is positive so speed up between 3 to 6 seconds.

b) Slope in [0,2) and (6,7) is negative. So, acceleration 0 to 2 and 6 to 7 seconds are negative and

Slope between 3 and 6 seconds is positive. So acceleration between 3 to 6 seconds is positive.

Between 2 to 3  and 7 to 9 seconds velocity is constant. So, acceleration between 2 to 3  and 7 to 9 seconds is 0.

c) Slope between 3 and 6 seconds is positive so speed up between 3 to 6 seconds. So particle would move between 3 to 6 seconds.

d) Between 7 to 9 seconds velocity is 0. So, particle stand still between 7 to 9 seconds (7,9]

Hello there!

First let's define what we know:

if the velocity is positive, you are moving forward.

If the velocity is negatieve, you are moving backwards.

If the slope is negative in the case of positive velocity or positive in the case of negative velocity, you are slowing down.

If the slope is positive in the case of positive velocity, or negative in the case of negative velocity, you are speeding up.

if the slope is 0, the velocity is constant

if the velocity is 0, the position is constant.

Let's analyze the graph with this info:

in the segment 0 seconds to 1 seconds, you can see that the velocity is positive and the slope is negative, so the particle is moving forward and slowing down,

from  1 seconds to 2 seconds, the velocity is negative and also the slope, so the particle is going backwards and speeding up

from 2 seconds to 3 seconds the velocity is negative and the slope is 0, so the particle is  moving backwards at constant velocity.

from 3 to 5 seconds, the velocity is negative and the slope is positive, so the particle is  moving backwards and the velocity is decreacing.

from 5 to 6 seconds the velocity is positive and also the slope, so the particle is  moving forward and the speed is increacing.

from 6 to 7 seconds, the velocity is positive and the slope is negative, so the particle is moving forward and the velociti is slowing down.

from 7 to 9 seconds the velocity is 0, so the particle is not moving.

the formula of a slope:

from a table we have

substitute:

the slope of f(x) is -1.

the graph of funcfion g(x) is a horizontal line. the slope of a horizontal line is equal 0. therefore the slope of g(x) is 0.

-1 < 0

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