Find the max rate of change of f at the given point and the direction in which it occurs.
Attempt:
Gradient vector at
How would you find the direction in which f occurs?
If at , then among all possible directional derivatives of at , the derivative in the direction of at has the largest value. The value of this largest directional derivative is at .
To see why this is so, you should know the directional derivative of in the direction of at is denoted by and is defined (for our purposes) by
You should also be familiar with the following definition of the dot product of two vectors and [/tex]\mathbf b[/tex]
where is the angle between and .
Now, we can re-write the definition of the directional derivative as
Because is a unit vector, and
From the above equation, is maximum when , which means and are parallel.
Returning to your problem, first find at the given point .
To find the maximum rate of change of , find .
To find the direction in which the maximum rate of change of occurs, find the unit vector in the direction of .
In your case, is already a unit vector because . In other words,
and your problem is solved.