Find the directional derivative of at the point
(-3, 2) in the direction .
The gradient of f is:
&nabla f = &lang
, &rang
&nabla f (-3, 2) = &lang, &rang
The directional derivative is:

&nabla f = &lang,&rang
&nabla f (-3, 2) = &lang -480 , 270 &rang
directional derivative = -6.173

The work

2.

Consider the function .
Find the gradient of f:
&lang, , &rang
Find the gradient of f at
the point (-3, -1, -3).
&lang, , &rang
Find the rate of change of the function f at the point (-3, -1,-3) in the direction
.

&nabla f = &lang&rang
&nabla f (-3, -1, -3) = &lang -28, 6, -75&rang
rate of change = 19.24

The work

3.

Find the directional derivative of at the point
(-3, -3, -5) in the direction of the vector v = &lang -1, -5, -1 &rang .

-2.31

The work

4.

Find the directional derivative of at the point (2, 5, 3) in the direction of the origin.

24.33

The work

5.

Find the maximum rate of change of f(x,y,z) = x + y/z at the point
(2, 2, 2) and the direction in which it occurs.
Maximum rate of change:
Direction (unit vector) in which it occurs:
&lang, ,&rang

Max rate change: 1.225
Direction Vector:
&lang 0.8165, 0.4082, -0.40825&rang

The work

6.

Find the maximum rate of change of at the point (3, 5) and the direction in which it occurs.
Maximum rate of change:
Direction (unit vector) in which it occurs:
&lang, &rang

Maximum rate change: 0.343
Direction Vector: &lang 0.5145, 0.8575&rang

The work

7.

Consider the function .
Find the the directional derivative of f at the point
( 3, -2 ) in the direction given by the angle
.

Find the unit vector which describes the direction in which f is
increasing most rapidly at ( 3, -2 ).
(, ).

Directional derivative: -19.856

Unit vector: ( 0.6, -0.8 )

The work

8.

Suppose f (x,y) = x/y, P = (-1, 2) and v = -4i - 4j.

A. Find the gradient of f.
&nabla f = i + j

B. Find the gradient of f at the point P.
(&nabla f) (P) = i + j

C. Find the directional derivative of f at P in the direction of v.

D. Find the maximum rate of change of f at P.

E. Find the (unit) direction vector in which the maximum rate of change occurs
at P. u = i + j

A) (1/y)i + (-x/y^2)j

B) .5i + .25j

C) -0.5303

D) 0.559

E) 0.894i + 0.447j

The work

9.

Suppose that you are climbing a hill whose shape is given by
,
and that you are at the point
(20, 60, 300).
In which direction (unit vector) should you proceed initially in order
to reach the top of the hill fastest?
&lang,&rang
If you climb in that direction, at what
angle above the horizontal will you be climbing initially (radian
measure)?

Proceed: &lang -0.2577, -0.9662&rang

1.4905 rad

The work

10.

Consider a function f (x,y) at the point (6, 3).
At that point the function has directional derivatives:
in the direction (parallel to) &lang 7, 3 &rang, and
in the direction (parallel to) &lang 6, 4 &rang .
The gradient of f at the point (6, 3 ) is
(, ).

( 0.6, 0.6)

The work

11.

Find equations of the tangent plane and normal line to the surface
at the point (7, -10, 10).
Tangent Plane: (make the coefficient of x equal to 1).
= 0.
Normal line: &lang 7, , &rang
+ t&lang 1, , &rang .

Find equations of the tangent plane and normal line to the surface
at the point (-9, 0, 0).
Tangent Plane: (make the coefficient of x equal to 1).
= 0.
Normal line: &lang -9, , &rang
+ t&lang , , 1 &rang .