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.
(a) Find the direction for which the directional derivative of the function f(x, y) = 9xy + 2y² is a maximum at P = (2, 1).
(Give your answer using component form or standard basis vectors. Express numbers in exact form. Use symbolic notation and
fractions where needed.)
direction: 9i+22j
Incorrect
Format:
Find the directional derivative of f(x, y, z) = -4x²z - 11 xy + 2z² at P = (-3, -3, -2) in the direction of (6,-2,0).
D(6,-2,0)| (-3,-3,-2)
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• If your answer below is a vector, input it as a column vector using the vector/matrix palette tool.
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brackets (e.g. x*y rather than xy; 2*(x+y) rather than 2(x+y)).
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. Give your answers in exact form.
ab
sin (a)
Əə
dx
What is the maximum rate of increase of f at P?
|Vƒl(-3,-3,-2)
sin (a)
Ә
2
f
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∞
f
8
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a
Ω
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Question M
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Let A(5, –1, –3) and B(1, 3, 2) be two points in R. The directional derivative of f(x, y, z) = xy + 11yz + 7zx at the point A in the direction of
the vector AB is
Use calculator and write your answer in three decimal digits, like 34.123
Calculus, Single Variable: Early Transcendentals (3rd Edition)
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