# Thread: Cylinder rate of change problem.

1. ## Cylinder rate of change problem.

I'm trying to solve the following problem:

Consider a cylinder of radius 5 whose axis is along the z axis.
i) what is the rate of change of f(x,y,z) in the direction normal to the cylinder at the point (3,-4,4)?

In the previous part of the question I was asked to find the grad of f = ln(x^2+y^2) + z but I cannot tie this in. I know that the equation of the cylinder is simply x^2 + y^2 = 25.

Any help would be much appreciated!

2. The rate of change in a specific direction (directional derivative) is the gradient dotted with a unit vector in that specific direction. That's why the previous part of the question asked you to find the gradient. Now, find a unit vector in the direction of normal to the point (3,-4,4). Then, dot the gradient with that unit vector.

3. And, when you write a surface as f(x,y,z)= constant, $\displaystyle \nabla f$ is normal to the surface at every point.

Here, your surface is given by $\displaystyle f(x,y,z)= x^2+ y^2= 25$ so $\displaystyle \nabla f= 2x\vec{i}+ 2y\vec{j}$. Evaluate that at (3, -4, 4). Don't forget that you need a unit vector in that direction.