A hummingbird is flying back and forth with velocity given by the graph below, where positive velocities mean the hummingbird is flying North and negative velocities mean the hummingbird is flying South:

(a) At what time(s) is the hummingbird moving neither North nor South?

(b) On what time interval(s) is the hummingbird moving North?

(c) On what time interval(s) is the hummingbird moving South?

(d) At what time(s) is the hummingbird moving most rapidly North?

(e) At what time(s) is the hummingbird moving most rapidly South?

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

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

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

Answer

(a) The hummingbird is moving neither North nor South whenever *v*(*t*) = 0, which occurs at *t* = 2, 5, and 7.

(b) The hummingbird is moving North when *v*(*t*) is positive, which occurs on the interval (2,5).

(c) The hummingbird is moving South when *v*(*t*) is negative, which occurs on the intervals [0,2) and (5,7). We do include *t* = 0 in the first interval, since v(0) is negative.

(d) The hummingbird is moving most rapidly North when *v*(*t*) is positive and as large as possible. This is when *t* = 4.

(e) The hummingbird is moving most rapidly South when *v*(*t*) is at its most negative. This occurs at both *t* = 0 and *t* = 6.

(f) The hummingbird is moving most rapidly when |*v*(*t*)| is largest. This occurs when *t* = 4.

(g) The hummingbird is speeding up on all intervals where *v*(*t*) is getting farther from zero (this means its speed |*v*(*t*)| is getting bigger). For this graph, *v*(*t*) is getting farther from zero on the intervals (2,4) and (5,6).

(h) The hummingbird is slowing down on all intervals where its speed |*v*(*t*)| is getting smaller, meaning *v*(*t*) is getting closer to zero. In this case, *v*(*t*) is getting closer to zero on the intervals (0,2), (4,5), and (6,7).