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.
Let U ⊂ Rn be open and f : U → Rm be differentiable at a ∈ U. Prove that for any vector ~v ∈ Rn,
where [Df(a)]~v denotes the total derivative (as an m × n matrix), Df(a), acting on the vector ~v.
Let 0<θ<2π, θ ≠ π. Consider the linear transformation T: C^2→C^2 given by matrix
[ cosθ -sinθ] (w.r.t standard basis)
[ sinθ cosθ]. Find the vector v1, v2 such that T v1= (e^iθ)v1, T v2= (e^-iθ)v2. Is {v1,v2} a basis for C^2? Give reason for your answer
1)Given r(t) be the vector function that describes the curve of intersection between the cylinder x^2+y^2=9and the plane x + y + z = 1.
-Find first component function is x(t) = 3 cos(πt), determine the y(t) and z(t) functions.
-Find determine the value of b so that the curve gets traversed exactly once on 0≤ t ≤ b.
Calculus, Single Variable: Early Transcendentals (3rd Edition)
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.
Linear Equation | Solving Linear Equations | What is Linear Equation in one variable ?; Author: Najam Academy;https://www.youtube.com/watch?v=tHm3X_Ta_iE;License: Standard YouTube License, CC-BY