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.
Represent the line segment from P to Q by a vector-valued function. (P corresponds to t = 0. Q corresponds to t = 1.)
P(-9, -2, -1), Q(−4, −8, −9)
r(t) =
Represent the line segment from P to Q by a set of parametric equations. (Enter your answers as a comma-separated list of equations.)
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Consider the function f(x,y) = 3 -
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.
(Type exact answers, using radicals as needed.)
(a,b) = (2,0)
(a,b)= (0,2)
-3√2
-8√/2
u=
V =
W =
√2 √2
2 2
√2 √2
2
2
22
3x²
2
- 4y², whose graph is a paraboloid (see figure). Complete parts (a) and (b).
√2 √2
2
3√2
3√2
-8√2
8√√2
(a,b) = (1,1)
- 11√2
2
√√2
2
11√2
2
The position vector r describes the path of an object moving in the xy-plane.
Position Vector
Point
r(t) = 2 cos ti + 2 sin tj
(VZ, V2)
(a) Find the velocity vector, speed, and acceleration vector of the object.
v(t)
=
s(t)
a(t) =
(b) Evaluate the velocity vector and acceleration vector of the object at the given point.
a(#) =
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