Expert: Paul Klarreich - 2/17/2009

Question
Hi, I'm having trouble mostly on a). Would rate of change be the derivative of F? If so, I can't seem to find the derivative for some reason.

An object with weight W is dragged along a horizontal plane by a force acting
along a rope attached to the object. If the rope makes an angle x with the plane, then the
magnitude of the force is

F = kW / ( k sin x + cos x )

where k is a constant called the coefficient of friction.

(a) Find the rate of change of F with respect to x.
(b) When is this rate of change equal to 0?

Answer
Questioner:   Sandy
Category:  Calculus
Private:  No
 
Subject:  Rate of Change/derivatives
Question:  Hi, I'm having trouble mostly on a). Would rate of change be the derivative of F? If so, I can't seem to find the derivative for some reason.

An object with weight W is dragged along a horizontal plane by a force acting along a rope attached to the object. If the rope makes an angle x with the plane, then the magnitude of the force is

F = kW / ( k sin x + cos x )

where k is a constant called the coefficient of friction.

(a) Find the rate of change of F with respect to x.
(b) When is this rate of change equal to 0?
.................................
Generally, yes -- the rate of change is the derivative.

If:

              kW
F(x) =  ----------------
        k sin x + cos x

and k,W are constants, so that is a function of x, then you just have

F(x) = kW(k sin x + cos x)^-1

and you can use the chain rule:

F' =  - kW(k sin x + cos x)^-2(k cos x - sin x)

      -kW(k cos x - sin x)
F' = ------------------------
       (k sin x + cos x)^2

Now you can set that = 0.  [Just worry about the top.]

About the 'when', we normally use that word if the variable is 't'/