I'm currently working on the question:
Find the rate of change of the funtion $f(x,y,z) = x^2y-xz$ along the curve $y=x^2$, $z=x$ in the direction of decreasing x at the point (2,4,2).
Now, I know that to find the rate of change, we'd need to determine $D_uT$ of f(x,y,z), but what I'm unsure of now is how to go about the extra addition of the curve. I was thinking that a solution would be to replace y and x in f(x,y,z) from the information of the curve so that is would instead be $f(x,y,z) = x^4-x^2$. Then, we'd determine $D_uT$, plug in our points (2,4,2) and that'd be our solution? However, I have a feeling this method is missing something. Also, I'm unsure how to go about the point that the rate of change needs to be in the direction of decreasing x. How would we go about that particular clause?
Any help is greatly appreciated,
thank you.
Best Answer
The first step is to find the direction that we want the derivative in. The extra curve step is not as menacing as it sounds.
We want to find the tangent to the curve described by $y=x^2$, $z=x$ at the given point. This curve could be written as $r=\langle x, x^2, x\rangle$. Then $r'=\langle 1, 2x, 1\rangle$. When $x=2$ the tangent vector will simply be $\langle 1, 4, 1\rangle$.
Since the problem specifies that we want the "decreasing $x$" direction, we assume we should use the negative of this tangent, $\langle -1, -4, -1\rangle$. Remember this is still the tangent of the curve, just antiparallel to the first we found.
Now that you have the direction, it is simply a matter of calculating the derivative in that direction.