Using the conclusion of Proposition 11.3.6, prove Proposition 11.3.8 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)
=
(wx, zxsin (xy),z3x)
T.
Find
∂∂fx
,
∂∂fy
, and
∂∂fz
for f =
x2y2z + w
e2 + xy + z2
sin
( 2)
z
+ cos
(xy)
ln
( )
x2 + y2 + 1
+ ez
sin
(xyz)
+ cos
(xy)
Find
∂∂fx
,
∂∂fy
, and
∂∂fz
for f =
x2y + cos
(xy)
+ z3y
ex2+y2z sin
(x+ y)
z2 sin3
( x2+y3)
e
x2 cos
( ( ( )))
sin tan z2 + y2
xy2+z
Suppose
{ 3 2 2 3 3 3
2xy+6x-+12xy+18y3xx+2+366yy2+sin(x-)+tan(3y-) if (x,y) ⁄= (0,0)
f (x,y) = 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?