Expert: Alon Mandes - 5/15/2009

Question
Consider these statements written in ordinary language:

A The speed of the car is proportional to the distance it has traveled.
B The car is speeding up.
C The car is slowing down.
D The car always travels the same distance in the same time interval.
E We are driving backwards.
F Our acceleration is decreasing.

Denoting by s(t) the distance covered by the car at time t, and letting k denote a constant, match these statements with the following mathematical statements by entering the letters A through E on the appropriate boxes:


s''<0
s' is a constant
s'<0
s'''<0
s''>0
s'=ks


Can you help me distribute the appropriate wording with the correct higher derivative that is listed. I don't know how to choose by reading a the statement and matching it with its derivative.

THANKS!!

Answer
A -> s'=ks
Explanation :
If s(t) is the position then the velocity is the derivative.

B -> s''>0
Explanation :
Acceleration is s''(t). When speeding up S''>0 , when slowing down
s''<0

C -> s''<0

D -> s' is a constant
Explanation :
The car always travels the same distance in the same time interval
means that the velocity is constant, & velocity is S' . No acceleration

E -> s'<0
Explanation :
Driving backward means that the velocity is negative . We also could
accelerate by driving backward .

F -> s'''<0
The acceleration is the 2nd derivative . The 3rd derivative will
determine if the acceleration is increasing or decreasing. Note that
whether s'''>0 or s'''<0 still s''>0 !!!

Alon.