A cat climbs up and down a tree with velocity given by the graph below. When the velocity is positive it means the cat is climbing up the tree.

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

(b) At what time(s) is the cat climbing most rapidly upwards?

(c) At what time(s) is the cat climbing most rapidly downwards?

(d) What is the cat's fastest speed on the interval [0,7]?

(e) What is the cat's slowest speed on the interval [0,7]?

(f) At what time(s) is the cat highest in the tree?

Answer

(a) The cat changes direction whenever its velocity changes from positive to negative or from negative to positive. The only time this happens on [0,7] is at *t* = 5, where the cat changes from climbing up the tree (positive velocity) to climbing down the tree (negative velocity).

(b) The cat is climbing most rapidly upwards when *t* = 0 and when *t* = 4, since these are the times when the cat's velocity is largest.

(c) The cat is climbing most rapidly downwards when *t* = 6, since this is when the cat's velocity is most negative.

(d) The cat's fastest speed on the interval [0,7] is 2 feet per second, which occurs at *t* = 0 and *t* = 4.

(e) The slowest the cat ever goes on [0,7] is 0 feet per second.

(f) From *t* = 0 to *t* = 5 the cat is climbing up, and from *t* = 5 to *t* = 7 the cat is climbing down. The cat is highest in the tree when *t* = 5, since this is when it stops climbing up and starts climbing down.