Using the conclusion of Proposition 20.3.6, prove Proposition 20.3.8 from the
geometric description of the dot product, the one which says the dot product
is the product of the lengths of the vectors and the cosine of the included
angle which is no larger than π.
Find the largest value of the directional derivative of f
(x,y,z)
= ln
( )
x + y2
+
z2 at the point
(1,1,1)
.
Find the smallest value of the directional derivative of f
(x,y,z)
=
xsin
( )
4xy2
+ z2 at the point
(1,1,1)
.
An ant falls to the top of a stove having temperature T
(x,y)
= x2 sin
(x+ y)
at the point
(2,3)
. In what direction should the ant go to minimize the
temperature? In what direction should he go to maximize the temperature?
Find the partial derivative with respect to y of the function f
(x,y,z,w)
=
( 2 2 3 )
y ,z sin (xy ),z x
T.
Find the partial derivative with respect to x of the function f
(x,y,z,w)
=
( 3 )
wx, zxsin (xy),z x
T.
Find
∂f
∂x
,
∂f
∂y
, and
∂f
∂z
for f =
(a) x2y2z + w
(b) e2 + xy + z2
(c) sin
( 2)
z
+ cos
(xy)
(d) ln
( )
x2 + y2 + 1
+ ez
(e) sin
(xyz)
+ cos
(xy)
Find
∂∂fx
,
∂∂fy
, and
∂∂fz
for f =
(a) x2y + cos
(xy)
+ z3y
(b) ex2+y2z sin
(x+ y)
(c) z2 sin3
( x2+y3)
e
(d) x2 cos
( ( ( )))
sin tan z2 + y2
(e) xy2+z
Suppose
{
2xy+6x3+12xy2+18yx2+36y3+sin(x3)+tan(3y3)
f (x,y) = 3x2+6y2 if (x,y) ⁄= (0,0)
0 if (x,y) = (0,0).
Find
∂f
∂x
(0,0)
and
∂f
∂y
(0,0)
.
Why must the vector in the definition of the directional derivative be a unit vector?
Hint: Suppose not. Would the directional derivative be a correct manifestation of
steepness?