Gradient fields on curves For the potential function φ and points A , B , C , and D on the level curve φ ( x, y ) = 0, complete the following steps. a. Find the gradient field F =∇ φ . b. Evaluate F at the points A , B , C , and D . c. Plot the level curve φ ( x , y ) = 0 and the vectors F at the points A , B , C , and D . 45. φ ( x, y ) = − y + sin x ; A (π/2, 1), B ( π , 0), C (3π/2, −1), and D (2π, 0)
Gradient fields on curves For the potential function φ and points A , B , C , and D on the level curve φ ( x, y ) = 0, complete the following steps. a. Find the gradient field F =∇ φ . b. Evaluate F at the points A , B , C , and D . c. Plot the level curve φ ( x , y ) = 0 and the vectors F at the points A , B , C , and D . 45. φ ( x, y ) = − y + sin x ; A (π/2, 1), B ( π , 0), C (3π/2, −1), and D (2π, 0)
Gradient fields on curves For the potential function φ and points A, B, C, and D on the level curve φ(x, y) = 0, complete the following steps.
a. Find the gradient field F =∇φ.
b. Evaluate F at the points A, B, C, and D.
c. Plot the level curve φ(x, y) = 0 and the vectors F at the points A, B, C, and D.
45. φ(x, y) = −y + sin x; A(π/2, 1), B(π, 0), C(3π/2, −1), and D(2π, 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.
I would need help with a, b, and c as mention below.
(a) Find the gradient of f.(b) Evaluate the gradient at the point P.(c) Find the rate of change of f at P in the direction of the vector u.
A. Find the gradient of f.
Vf
Note: Your answers should be expressions of x and y; e.g. "3x - 4y"
B. Find the gradient of f at the point P.
(Vƒ) (P) =
Note: Your answers should be numbers
Suppose f (x, y) = , P = (1, −1) and v = 2i – 2j.
=
C. Find the directional derivative of f at P in the direction of V.
Duf =
Note: Your answer should be a number
D. Find the maximum rate of change of f at P.
Note: Your answer should be a number
u=
E. Find the (unit) direction vector in which the maximum rate of change occurs at P.
Explain how a directional derivative is formed from the two partial derivatives f, and f,.
Choose the correct answer below.
O A. Form the dot product between the unit direction vector u and the gradient of the function.
O B. Form the dot product between the unit direction vectors.
O C. Form the sum between the unit direction vector u and the gradient of the function.
O D. Form the dot product between the unit coordinate vectors.
Click to select your answer.
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