1. Find the directional derivative of the function

at a point (-2,1) along the direction

.

The unit vector along

is

. Gradient of the given function is

. Thus the directional derivative at (x,y) is

. At (-2,1) its value is -12/5.

2. Find the directional derivative of the function

along the direction

at a point (1,1,1)

The unit vector along

is

. Gradient of the given function is

.. At (1,1,1) the gradient is

Thus the directional derivative at (1,1,1) is

3. Consider a function

. What is the direction in which maximum change of the function takes place at the point (2,2,1)?

Maximum change takes place in the direction of the gradient. In this case the unit vector along the gradient is

The level surface is

. Normal to the surface is in the direction of gradient which is

. The equation to the tangent plane is given by

where

and

are the partial derivative of the function f(x,y,z)= constant at the point

. In this case

. Thus the equation is