Expert: Paul Klarreich - 11/30/2008

Question
A point moves along the curve y=x^2 + 1 in such a way that when x=4, the x coordinate is increasing at the rate of 5 ft/sec. At what rate is the y-coordinate changing at that time?

A) 80 ft/sec
B) 45 ft/sec
C) 32 ft/sec
D) 85 ft/sec
E) 40 ft/sec

I chose (E) because the second derivative is y=2 and when x=4 it's increasing at a rate of 5 ft/sec, which is 20, then multiplied times 2 is 40.

Answer
Questioner:   Charlene
Category:  Calculus
 
Subject:  calculus; derivatives
Question:  A point moves along the curve y=x^2 + 1 in such a way that when x=4, the x coordinate is increasing at the rate of 5 ft/sec. At what rate is the y-coordinate changing at that time?


A) 80 ft/sec
B) 45 ft/sec
C) 32 ft/sec
D) 85 ft/sec
E) 40 ft/sec

I chose (E) because the second derivative is y=2 and when x=4 it's increasing at a rate of 5 ft/sec, which is 20, then multiplied times 2 is 40.
......................................

The basic (chain) rule says:

dy   dy/dt
-- = ------
dx   dx/dt

dy/dx = 2x, and at  x = 4, this is equal to 8.

Now "the x coordinate is increasing at the rate of 5 ft/sec"
says  dx/dt = 5.


And: "At what rate is the y-coordinate changing at that time?"
means "What is dy/dt?"

So if:

dy   dy/dt
-- = ------
dx   dx/dt
    dy/dt
8 = ----
      5

Looks like dy/dt = 40.

I don't like your reasoning, but if you obviously very lucky; stop wasting your time going to college -- get out to the racetrack.