A bug crawling on a number line has velocity given by the graph below, where positive velocities indicate the bug is crawling to the right:

(a) At what time(s) does the bug change direction?

(b) On what time interval(s) is the bug moving to the right?

(c) On what time interval(s) is the bug moving to the left?

(d) At what time(s) is the bug moving most rapidly to the right?

(e) At what time(s) is the bug moving most rapidly to the left?

(f) At what time(s) is the bug moving most rapidly?

(g) On what time interval(s) is the bug speeding up?

(h) On what time interval(s) is the bug slowing down?

Answer

(a) The bug changes direction whenever *v*(*t*) goes from positive to negative, which happens at *t* = 4 and *t* = 12, or from negative to positive, which happens at *t* = 8. We do NOT include *t* = 0, because we can't tell if *v*(*t*) is negative to the left of *t* = 0. To summarize, the bug changes direction at *t* = 4, 8, and 12.

(b) The bug is moving to the right whenever *v*(*t*) is positive, which occurs on the intervals (0,4) and (8,12).

(c) The bug is moving to the left whenever *v*(*t*) is negative, which occurs on the intervals (4,8) and (12,16).

(d) The bug is moving most rapidly to the right when *t* = 10, since this is where the largest positive value of *v*(*t*) occurs.

(e) The bug is moving most rapidly to the left when *t* = 16, since this is where the most negative value of *v*(*t*) occurs.

(f) The bug is moving most rapidly when *t* = 16, since this is where |*v*(*t*)| is largest (equivalently, this is where *v*(*t*) is farthest from 0).

(g) The bug is speeding up on (0,2), (4,6), (8,10), and (12,16), since these are the intervals where *v*(*t*) is getting further from 0.

(h) The bug is slowing down on (2,4), (6,8), and (10,12), since these are the intervals where *v*(*t*) is getting closer to 0.