How to find this rate of change?

[supermiedos]

Homework Statement

Given f(x, y, z) = 3x + 2y + 4z, find its rate of change at (1, 2, 3) on the plane y = 2.

Homework Equations

The Attempt at a Solution

Since y is constant, I don't know if I have to find ∂f/∂x or ∂f/∂z. It's a homework problem, but I don't know if it was bad written by the teacher. Could you help me please?

 

[mfb]

Rate of change with respect to what?
x and z? I think I would calculate those derivatives and evaluate them at the given point.
Maximal change for any direction? This can be evaluated, too (it has to be in the xz-plane here).

 

[supermiedos]

Rate of change with respect to what?
x and z? I think I would calculate those derivatives and evaluate them at the given point.
Maximal change for any direction? This can be evaluated, too (it has to be in the xz-plane here).

That's exactly what I tought. There are two answers here, right? If i remember, the maximal change can be calculated using gradients, right?

Btw, my teacher says slope and gradients are the same thing... How is that possible? A slope is a number, and the gradient is a vector!

 

[mfb]

Well, you could define "slope" as the gradient... looks odd, but it is just a definition issue.

If i remember, the maximal change can be calculated using gradients, right?

This is possible, right.

 

[HallsofIvy]

Homework Statement

Given f(x, y, z) = 3x + 2y + 4z, find its rate of change at (1, 2, 3) on the plane y = 2.

Homework Equations

The Attempt at a Solution

Since y is constant, I don't know if I have to find ∂f/∂x or ∂f/∂z. It's a homework problem, but I don't know if it was bad written by the teacher. Could you help me please?

No, this is not "bad written" (or even "badly written"). Since you are asked about its rate on change on y= 2, replace y by 2: f(x, z)= 3x+ 4+ 4z. The rate of change of that, at (1, 2, 3) is the length of the gradient vector evaluated at (1, 2, 3). (If the function really is what you wrote without any powers, its gradient vector is a constant and the point (1, 2, 3) is irrelevant.)

 

[supermiedos]

No, this is not "bad written" (or even "badly written"). Since you are asked about its rate on change on y= 2, replace y by 2: f(x, z)= 3x+ 4+ 4z. The rate of change of that, at (1, 2, 3) is the length of the gradient vector evaluated at (1, 2, 3). (If the function really is what you wrote without any powers, its gradient vector is a constant and the point (1, 2, 3) is irrelevant.)

I see. So, if I have a function f(x, y, z) and I'm asked to find its rate of change at (a, b, c), when, let's say, y = b. I have to find the gradient of f(x, b, y) at that point? But the length of the gradient represents the highest rate of change, right? So the problem must be more specific (since I could also find the lowest rate of change)

 

[mfb]

(since I could also find the lowest rate of change)

This is always zero with 2 or more variables, if the function is differentiable.

But the length of the gradient represents the highest rate of change, right?

Right