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 . 43. φ ( x, y ) = y − 2 x ; A (−1, −2), B (0, 0), C (1,2), and D (2,4)
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 . 43. φ ( x, y ) = y − 2 x ; A (−1, −2), B (0, 0), C (1,2), and D (2,4)
Solution Summary: The gradient field is defined as the vector field which is obtained as a gradient of the scalar-valued function, phi .
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.
43. φ(x, y) = y − 2x; A(−1, −2), B(0, 0), C(1,2), and D(2,4)
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.
Describe the two main geometric properties of the gradient V f.
Chapter 17 Solutions
Calculus: Early Transcendentals and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition) (Briggs, Cochran, Gillett & Schulz, Calculus Series)
Precalculus Enhanced with Graphing Utilities (7th Edition)
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