Expert: Paul Klarreich - 4/11/2006

Question
Hi,
A point (x,y) is moving along a curve y=f(x). At the instant when the slope of the curve is -1/3, the x-coordinate of the point is increasing at the rate of 5 units per second. The rate of cange, in units per second, of the y-coordinate of the point is -5/3.Why is this true?Please show me how to solve this problem.
Thank you very much.

Answer
Hi, Jeff,

Time is getting short, I know.  Already I was reminded of this when my brother-in-law told us he will be grading AP exam papers again this June.  (Alas, I can't help you -- he is a professor of music, not math.)

Question:  Hi,
A point (x,y) is moving along a curve y=f(x). At the instant when the slope of the curve is -1/3, the x-coordinate of the point is increasing at the rate of 5 units per second. The rate of change, in units per second, of the y-coordinate of the point is -5/3.Why is this true?Please show me how to solve this problem.
Thank you very much.
------------------------------
Here is one form of the chain rule:

dy     dy/dt
--  =  -----
dx     dx/dt

So your second and third sentences say, when translated into the notation of derivatives:

At the instant when dy/dx = -1/3, dx/dt = 5. Then dy/dt = -5/3.

Need I say more?