# How to find this rate of change?

1. Mar 1, 2013

### supermiedos

1. The problem statement, all variables and given/known data

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

2. Relevant equations

3. 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?

2. Mar 1, 2013

### Staff: Mentor

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).

3. Mar 1, 2013

### supermiedos

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!!

4. Mar 2, 2013

### Staff: Mentor

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

5. Mar 2, 2013

### HallsofIvy

No, this is not "bad written" (or even "badly written":tongue:). 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.)

6. Mar 3, 2013

### supermiedos

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 lenght 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)

7. Mar 4, 2013

### Staff: Mentor

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