In Example 1 , evaluate D ‒ u f (3, 2) and D − v f (3, 2). Example 1 Computing directional derivatives Consider the paraboloid z = f ( x, y ) = 1 4 ( x 2 + 2 y 2 ) + 2 . Let P 0 be the point (3, 2) and consider the unit vectors u = 〈 1 2 , 1 2 〉 and v = 〈 1 2 , − 3 2 〉 a. Find the directional derivative of f at P 0 in the directions of u and v.
In Example 1 , evaluate D ‒ u f (3, 2) and D − v f (3, 2). Example 1 Computing directional derivatives Consider the paraboloid z = f ( x, y ) = 1 4 ( x 2 + 2 y 2 ) + 2 . Let P 0 be the point (3, 2) and consider the unit vectors u = 〈 1 2 , 1 2 〉 and v = 〈 1 2 , − 3 2 〉 a. Find the directional derivative of f at P 0 in the directions of u and v.
Solution Summary: The author evaluates the values of D_-uf(3,2) and
In Example 1, evaluate D‒u f(3, 2) and D−vf(3, 2).
Example 1 Computing directional derivatives
Consider the paraboloid z = f(x, y) =
1
4
(
x
2
+
2
y
2
)
+
2
. Let P0 be the point (3, 2) and consider the unit vectors
u =
〈
1
2
,
1
2
〉
and v =
〈
1
2
,
−
3
2
〉
a. Find the directional derivative of f at P0 in the directions of u and v.
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(0, 0, 0), Q(4, 5, 4)
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.)
Represent the line segment from P to Q by a vector-valued function. (P corresponds to t = 0. Q corresponds tot = 1.)
P(0, 0, 0), Q(4, 6, 6)
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.)
Represent the line segment from P to Q by a vector-valued function. (P corresponds to t = 0. Q corresponds to t = 1.)
P(−7, −5, −1), Q(−1, −9, −6)
(a) r(t) =
(b) Represent the line segment from P to Q by a set of parametric equations. (Enter your answers as a comma-separated list of equations.)
Calculus, Single Variable: Early Transcendentals (3rd Edition)
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