Directional derivatives Consider the function f ( x , y ) = 2 x 2 + y 2 , whose graph is a paraboloid (see figure). a. Fill in the table with the values of the directional derivative at the points (a, b ) in the directions given by the unit vectors u , v , and w . ( a , b ) = (1,0) ( a , b ) = (1,1) ( a,b ) = (1,2) u = 〈 1 , 0 〉 v = 〈 2 2 , 2 2 〉 w = 〈 0 , 1 〉 b. Interpret each of the directional derivatives computed in part (a) at the point (1, 0).
Directional derivatives Consider the function f ( x , y ) = 2 x 2 + y 2 , whose graph is a paraboloid (see figure). a. Fill in the table with the values of the directional derivative at the points (a, b ) in the directions given by the unit vectors u , v , and w . ( a , b ) = (1,0) ( a , b ) = (1,1) ( a,b ) = (1,2) u = 〈 1 , 0 〉 v = 〈 2 2 , 2 2 〉 w = 〈 0 , 1 〉 b. Interpret each of the directional derivatives computed in part (a) at the point (1, 0).
Directional derivatives Consider the function
f
(
x
,
y
)
=
2
x
2
+
y
2
, whose graph is a paraboloid (see figure).
a. Fill in the table with the values of the directional derivative at the points (a, b) in the directions given by the unit vectorsu, v, and w.
(a,b) = (1,0)
(a,b) = (1,1)
(a,b) = (1,2)
u
=
〈
1
,
0
〉
v
=
〈
2
2
,
2
2
〉
w
=
〈
0
,
1
〉
b. Interpret each of the directional derivatives computed in part (a) at the point (1, 0).
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
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